L \\nonumber \\\\&[{\\Lambda }_{\\rm {L}}]_{i,i}=1, \\ \\ i\\le L \\nonumber \\\\& [{\\Lambda }_{\\rm {R}}]_{i,i}=0, \\ \\ i> L.$ Therefore, ${\\bf {F}}_{\\rm {A}}$ and ${\\bf {G}}_{\\rm {A}}$ do not affect the optimal solution.","Moreover, the unitary matrices ${\\bf {V}}_{{\\bf {F}}_{\\rm {A}}}$ and ${\\bf {V}}_{\\rm {RG}}$ do not affect the optimal solution as ${\\bf {\\tilde{F}}}_{\\rm {D}}$ in the constraint is unitary invariant.","Based on the above the discussion and (REF ), the remaining task to maximize the singular values of matrix $[{\\bf {U}}_{\\rm {RG}}^{\\rm {H}}{\\bf {H}}{\\bf {U}}_{{\\bf {F}}_{\\rm {A}}}]_{1:L,1:L}$ .","Note that ${\\bf {U}}_{\\rm {RG}}$ and ${\\bf {U}}_{{\\bf {F}}_{\\rm {A}}}$ are unitary matrices, for the optimal solution, the left eigenvectors of its first $ L $ largest singular values of $ {\\bf F}_{\\rm A} $ should have the maximum inner product with $[{\\bf {V}}_{\\mathcal {H}}]_{:,1:L}$ i.e., $ [{\\bf {U}}_{{\\bf {F}}_{\\rm {A}}}]_{:,1:L}^{\\rm {opt}}={\\rm {arg\\,max}}\\lbrace \\Vert [{\\bf {V}}_{\\mathcal {H}}]_{:,1:L} [{\\bf {U}}_{{\\bf {F}}_{\\rm {A}}}]_{:,1:L}^{\\rm {H}}\\Vert _{\\rm {F}}^2\\rbrace .$ Similarly for the optimal solution, the left eigenvectors of its first $ L $ largest singular values of $ {\\bf {R}}_{\\rm {n}}^{1/2} {\\bf {G}}_{\\rm {A}}^{\\mathrm {H}} $ should have the maximum inner product with $[{\\bf {U}}_{\\mathcal {H}}]_{:,1:L}$ , i.e., $ [{\\bf {U}}_{{\\bf {R}}{\\bf {G}}}]_{:,1:L}^{\\rm {opt}}={\\rm {arg\\,max}}\\lbrace \\Vert [{\\bf {U}}_{\\mathcal {H}}]_{:,1:L} [{\\bf {U}}_{{\\bf {R}}{\\bf {G}}}]_{:,1:L}^{\\rm {H}}\\Vert _{\\rm {F}}^2\\rbrace .$"],["Analog Transceiver Design","For fixed ${\\mathbf {\\Lambda }_\\mathrm {G}}$ and ${\\bf {Q}}_{\\rm {G}}$ , the optimization problem (REF ) can be transferred into the following vector variable optimization problem $\\min _{\\mathbf {r}} \\;\\; & {\\mathbf {r}}^{W} {\\mathbf {r}} - \\mathbf {p}^ - {\\mathbf {r}}^{p} + q \\\\\\text{s.t.}","\\;\\;\\; & \\mathbf {r}^{K}_{i} \\mathbf {r} = a^2, \\quad i = 1,2,\\ldots , NL.$ The vector $ \\mathbf {r} $ is constructed via vectorizing $ \\mathbf {G}_{\\mathrm {A}} $ , i.e., $\\mathbf {r} = \\big [ \\Re \\lbrace \\mathrm {vec}{( \\mathbf {G}_{\\mathrm {A}} )} \\rbrace ^ \\, \\Im \\lbrace \\mathrm {vec}{( \\mathbf {G}_{\\mathrm {A}} )} \\rbrace ^]^$ and the matrices $ \\mathbf {W}, \\, \\mathbf {K}_{i} $ and vector $ \\mathbf {p} $ are defined as follows: $\\mathbf {W} & =\\begin{bmatrix}\\; \\Re \\lbrace \\mathbf {I} \\otimes { \\mathbf {R}_{\\mathrm {n}} } \\rbrace & - \\Im \\lbrace \\mathbf {I} \\otimes { \\mathbf {R}_{\\mathrm {n}} } \\rbrace \\;\\; \\\\\\; \\Im \\lbrace \\mathbf {I} \\otimes { \\mathbf {R}_{\\mathrm {n}} } \\rbrace & \\Re \\lbrace \\mathbf {I} \\otimes { \\mathbf {R}_{\\mathrm {n}} } \\rbrace \\;\\;\\end{bmatrix}, \\\\\\mathbf {K}_{i} & = \\mathrm {diag} \\Big \\lbrace \\bigl [ \\mathbf {0}_{(i-1) \\times 1}^ 1 ,\\mathbf {0}_{(NL - 1) \\times 1}^ 1 ,\\mathbf {0}_{(NL - i) \\times 1}^] \\Big \\rbrace ,$ and $\\mathbf {p} =\\begin{bmatrix}\\Re \\lbrace \\big ( \\mathbf {I} \\otimes { \\mathbf {R}_{\\mathrm {n}}^{ {1}/{2}} } \\big )^{\\mathrm {H}} \\mathrm {vec}{ \\big ( [{\\bf {U}}_{\\mathcal {H}}]_{:,1:L} {\\mathbf {\\Lambda }_\\mathrm {G}} {\\bf {Q}}_{\\rm {G}} \\big ) } \\rbrace \\\\\\Im \\lbrace \\big ( \\mathbf {I} \\otimes { \\mathbf {R}_{\\mathrm {n}}^{ {1}/{2}} } \\big )^{\\mathrm {H}} \\mathrm {vec}{ \\big ( [{\\bf {U}}_{\\mathcal {H}}]_{:,1:L} {\\mathbf {\\Lambda }_\\mathrm {G}} {\\bf {Q}}_{\\rm {G}} \\big ) } \\rbrace \\end{bmatrix}.$ The constant scalar, $q$ , in (REF ) equals $q = || \\mathrm {vec}{ \\big ( [{\\bf {U}}_{\\mathcal {H}}]_{:,1:L} {\\mathbf {\\Lambda }_\\mathrm {G}} {\\bf {Q}}_{\\rm {G}} \\big ) } ||_2^2$ .","Note that because of the constant modulus constraints, the term ${\\bf {r}}^{\\rm {T}}{\\bf {r}}$ is a constant.","As a result, for a constant real scalar, $\\eta $ , the objective function in (REF ) is equivalent to ${\\mathbf {r}}^\\mathbf {W}+\\eta {\\bf {I}}) {\\mathbf {r}} - \\mathbf {p}^ - {\\mathbf {r}}^{p} + q$ .","As the constant modulus constraints in (REF ) are all quadratic equalities, the optimization problem (REF ) is nonconvex.","Following the idea of [42], an iterative algorithm is proposed via iteratively updating constraints to guarantee the constant modulus constraints.","Specifically, at the $n$ th iteration each constraint $\\mathbf {r}^{K}_{i} \\mathbf {r} = a^2$ is replaced by $\\mathbf {\\tilde{r}}_{(n-1)}^{K}_{i} \\mathbf {r}_{(n)} = a^2$ where $\\mathbf {\\tilde{r}}_{(n-1)}$ is a vector computed based on ${\\bf {r}}$ computed in the $(n-1)$ th iteration.","After stacking $\\mathbf {\\tilde{r}}_{(n-1)}^{K}_{i}$ for $i = 1,2,\\ldots NL$ in ${\\bf {P}}_{(n-1)}$ , optimization problem (REF ) is transferred to $\\min _{\\mathbf {r}_{(n)}} \\;\\; & {\\mathbf {r}}_{(n)}^\\mathbf {W}+\\eta {\\bf {I}}) {\\mathbf {r}}_{(n)} - \\mathbf {p}^_{(n)} - {\\mathbf {r}}_{(n)}^{p} + q \\\\\\text{s.t.}","\\;\\;\\; & {\\bf {P}}_{(n-1)}\\mathbf {r}_{(n)} = a^2{\\bf {1}},$ where the matrix $ \\mathbf {P}_{(n-1)} $ is defined as $&[ \\mathbf {P}_{(n-1)} ]_{\\ell ,j} \\nonumber \\\\=&{\\left\\lbrace \\begin{array}{ll}\\cos \\big ( \\angle [{\\rm {vec}}({\\bf {G}}_{{\\rm {A}},(n-1)})]_{\\ell } \\big ) & \\text{if } \\ell = j, \\; \\ell \\le NL \\\\\\sin \\big ( \\angle [ {\\rm {vec}}({\\bf {G}}_{{\\rm {A}},(n-1)}) ]_{\\ell } \\big ) & \\text{if } j = \\ell +NL, \\; \\ell \\le NL \\\\0 & \\text{otherwise.}\\end{array}\\right.","}$ The vector $ \\mathbf {1} $ is a column vector with all elements equal to 1.","As proved in [42], when $\\eta \\ge \\sigma _{\\max } NL / 8 + ||\\mathbf {p}||_2^2$ , where $ \\sigma _{\\max } $ is the largest eigenvalue of $ { \\mathbf {R}_{\\mathrm {n}} }$ , the optimal solution of the iterative optimization (REF ) minimizes the objective function and satisfies the constant modulus constraints asymptotically.","As (REF ) is convex at each iteration, based on its KKT conditions, at the $n$ th iteration the optimal solution of (REF ) is ${\\mathbf {r}}_{(n)} =( {\\mathbf {W}} + \\eta \\mathbf {I} )^{-1}\\left({\\bf {q}}+\\frac{\\lambda }{2}{\\bf {P}}_{(n-1)}^{\\rm {T}}\\right)$ with $\\frac{\\lambda }{2}&=\\left( {\\bf {P}}_{(n-1)}( {\\mathbf {W}} + \\eta \\mathbf {I} )^{-1}{\\bf {P}}_{(n-1)}^{\\rm {T}}\\right)^{-1}\\nonumber \\\\& \\ \\ \\ \\ \\ \\times \\left(a^2{\\bf {1}}-{\\bf {P}}_{(n-1)}( {\\mathbf {W}} + \\eta \\mathbf {I} )^{-1}{\\bf {q}}\\right).$ In a nutshell, the iterative algorithm is given in Algorithm REF .","Using the iterative algorithm, the numerical result of analog processor can be found."]],"string":"[\n [\n \"A Framework on Hybrid MIMO Transceiver Design based on Matrix-Monotonic\\n Optimization\"\n ],\n [\n \"Abstract Hybrid transceiver can strike a balance between complexity and performance of multiple-input multiple-output (MIMO) systems.\",\n \"In this paper, we develop a unified framework on hybrid MIMO transceiver design using matrix-monotonic optimization.\",\n \"The proposed framework addresses general hybrid transceiver design, rather than just limiting to certain high frequency bands, such as millimeter wave (mmWave) or terahertz bands or relying on the sparsity of some specific wireless channels.\",\n \"In the proposed framework, analog and digital parts of a transceiver, either linear or nonlinear, are jointly optimized.\",\n \"Based on matrix-monotonic optimization, we demonstrate that the combination of the optimal analog precoders and processors are equivalent to eigenchannel selection for various optimal hybrid MIMO transceivers.\",\n \"From the optimal structure, several effective algorithms are derived to compute the analog transceivers under unit modulus constraints.\",\n \"Furthermore, in order to reduce computation complexity, a simple random algorithm is introduced for analog transceiver optimization.\",\n \"Once the analog part of a transceiver is determined, the closed-form digital part can be obtained.\",\n \"Numerical results verify the advantages of the proposed design.\"\n ],\n [\n \"Introductions\",\n \"The great success of multiple-input multiple-output (MIMO) technology makes it widely accepted for current and future high data-rate communication systems[1].\",\n \"Acting as a pillar to satisfy data hungry applications, a natural question is how to reduce the cost of MIMO technology, especially that of large scale antenna arrays.\",\n \"The traditional setting of one radio-frequency (RF) chain per antenna element is too expensive for large-scale MIMO systems, especially at high frequencies, such as millimeter wave bands or Terahertz bands.\",\n \"Hybrid analog-digital architecture is promising to alleviate the straits and strike a balance between the cost and the performance of practical MIMO systems.\",\n \"A typical hybrid analog-digital MIMO transceiver consists of four components, i.e., digital precoder, analog precoder, analog processor, and digital processor [2].\",\n \"In the early transceiver design, hybrid MIMO technology is often referred to antenna selection [3], [4] to reap spatial diversity.\",\n \"In these works, analog switches are used in the radio-frequency domain.\",\n \"Phase-shifter based soft antenna selection [5], [6], [7] has been proposed to improve performance for correlated MIMO channels.\",\n \"Nowadays, the phase-shifter based hybrid structure has been widely used.\",\n \"For a phase shifter, only signal phase, instead of both magnitude and phase, can be adjusted.\",\n \"Thus, the optimization of a MIMO transceiver with phase shifters becomes complicated due to the constant-modulus constraints on analog precoder and analog processor.\",\n \"It has been shown in [8] that the performance of a full-digital system can be achieved when the number of shifters is doubled in a phase-shifter based hybrid structure.\",\n \"However, this can hardly be practical due to the requirement on a large number of phase shifters, especially in large-scale MIMO systems.\",\n \"As a matter of fact, the phase shifters in large-scale MIMO systems have been considered to be a burden sometimes.\",\n \"Thus, sub-connected hybrid structure has emerged as an alternative option [9], [10] and it has received much attention recently [9], [11], [12], [10], [13], [14].\",\n \"Unit modulus and discrete phase make the optimization of analog transceivers nonconvex and thus difficult to address [3], [4].\",\n \"There have been some works on hybrid transceiver optimization considering different design limitations and requirements.\",\n \"Their motivation is to exploit the underlying structures of the hybrid transceiver to achieve high performance but with low complexity.\",\n \"Early hybrid transceiver design is based on approximating digital transceivers in terms of the norm difference between all-digital design and the hybrid counterpart.\",\n \"For the millimeter wave (mmWave) band channels, which are usually with sparsity, an orthogonal matching pursuit (OMP) algorithm has been used in signal recovery for the hybrid transceiver [15].\",\n \"In order to overcome the non-convexity in hybrid transceiver optimization, some distinct characteristics of mmWave channels must be exploited [15].\",\n \"This methodology is a compromise on the constant-modulus constraint, which has been validated in different environments, including multiuser and relay scenarios [16], [17].\",\n \"However, it has been found later on that the OMP algorithm cannot achieve the optimal solution sometimes.\",\n \"A singular-value-decomposition (SVD) based descent algorithm [18] has been proposed, which is nearly optimal.\",\n \"An alternative fast constant-modulus algorithm [19] has also been developed to reduce the gap between the analog and digital precoders.\",\n \"The above methods are hard for complex scenarios due to high computation complexity [20], [21], [22].\",\n \"Therefore, based on the idea of unitary matrix rotation, several algorithms [23], [24] have been proposed to improve the approximation performance while maintaining a relative low complexity at the same time.\",\n \"On the other hand, some works for hybrid precoding design are based on codebooks, which relax the problem into a convex optimization problem [25].\",\n \"However, the codebook-based algorithm suffers performance loss if channel state information (CSI) is inaccurate [26].\",\n \"In order to reduce the complexity of codebook design and the impact of partial CSI, special structures of massive MIMO channels [27], [28], can be exploited.\",\n \"Recently, an angle-domain based method has been proposed from the viewpoint of array signal processing [29], [30], which provides a useful insight on hybrid analog and digital signal processing.\",\n \"Based on the concept of the angle-domain design, some mathematical approaches, such as matrix decomposition algorithm, have been developed [31], [32].\",\n \"Energy efficient hybrid transceiver design for Rayleigh fading channels has been investigated in [33].\",\n \"Hybrid transceiver optimization with partial CSI and with discrete phases has been discussed in [34] and [35], respectively.\",\n \"Hybrid MIMO transceivers are not only limited to mmWave frequency bands or terahertz frequency bands but also potentially work in other frequency bands.\",\n \"The transceiver itself could either be linear or nonlinear.\",\n \"Moreover, the performance metrics for MIMO transceiver could be different, including capacity, mean-squared error (MSE), bit-error rate (BER), etc.\",\n \"A unified framework on hybrid MIMO transceiver optimization will be of great interest.\",\n \"In this paper, we will develop a unified framework for hybrid linear and nonlinear MIMO transceiver optimization.\",\n \"Our main contributions are summarized as follows.\",\n \"Both linear and nonlinear transceivers with Tomlinson-Harashima precoding (THP) or deci-sion-feedback detection (DFD) are taken into account in the proposed framework for hybrid MIMO transceiver optimization.\",\n \"Different from the existing works in which a single performance metric is considered for hybrid MIMO transceiver designs, more general performance metrics are considered.\",\n \"Based on matrix-monotonic optimization framework, the optimal structures of both digital and analog transceivers with respect to different performance metrics have been analytically derived.\",\n \"From the optimal structures, the optimal analog precoder and processor correspond to selecting eigenchannels, which facilitates the analog transceiver design.\",\n \"Furthermore, several effective analog design algorithms have been proposed.\",\n \"The rest of this paper is organized as follows.\",\n \"In Section II, a general hybrid system model and the MSE matrices corresponding to different transceivers are introduced.\",\n \"In Section III, a unified hybrid transceiver is discussed in detail and the related transceiver optimization is present.\",\n \"In Section IV, the optimal structure of digital transceivers is derived based on matrix-monotonic optimization.\",\n \"In Section V, basic properties of the optimal analog precoder and processor are investigated, based on which effective algorithms to compute the analog transceiver are proposed.\",\n \"Next, in Section VI, simulation results are provided to demonstrate the performance advantages of the proposed algorithms.\",\n \"Finally, Figure: General hybrid MIMO transceiver.conclusions are drawn in Section VII.\",\n \"Notations: In this paper, scalars, vectors, and matrices are denoted by non-bold, bold lower-case, and bold upper-case letters, respectively.\",\n \"The notations ${\\\\bf X}^{\\\\rm {H}}$ and ${\\\\rm {Tr}}({\\\\bf X})$ denote the Hermitian and the trace of a complex matrix ${\\\\bf X}$ , respectively.\",\n \"Matrix ${\\\\bf X}^{\\\\frac{1}{2}}$ is the Hermitian square root of a positive semi-definite matrix ${\\\\bf X}$ .\",\n \"The expression $ \\\\mathrm {diag} \\\\lbrace \\\\mathbf {X} \\\\rbrace $ denotes a square diagonal matrix with the same diagonal elements as matrix $ \\\\mathbf {X} $ .\",\n \"The $ i $ th row and the $ j $ th column of a matrix are denoted as $ [\\\\cdot ]_{i,:} $ and $ [\\\\cdot ]_{:,j} $ , respectively, and the element in the $ k $ th row and the $ \\\\ell $ th column of a matrix is denoted as $ [\\\\cdot ]_{k,\\\\ell } $ .\",\n \"In the following derivations, ${\\\\Lambda }$ always denotes a diagonal matrix (square or rectangular diagonal matrix) with diagonal elements arranged in a nonincreasing order.\",\n \"Representation $ \\\\mathbf {A} \\\\preceq \\\\mathbf {B} $ means that the matrix $ \\\\mathbf {B} - \\\\mathbf {A} $ is positive semidefinite.\",\n \"The real and imaginary parts of a complex variable are represented by $ \\\\Re \\\\lbrace \\\\cdot \\\\rbrace $ and $ \\\\Im \\\\lbrace \\\\cdot \\\\rbrace $ , respectively, and statistical expectation is denoted by $\\\\mathbb {E}\\\\lbrace \\\\cdot \\\\rbrace $ .\"\n ],\n [\n \"General Structure of Hybrid MIMO Transceiver\",\n \"In this section, we will first introduce the system model of MIMO hybrid transceiver designs.\",\n \"Then a general signal model is introduced, which includes nonlinear transceiver with THP or DFD and linear transceiver as its special cases.\",\n \"Based on the general signal model, the general linear minimum mean-squared error (LMMSE) processor and data estimation mean-squared error (MSE) matrix are derived, which are the basis for the subsequent hybrid MIMO transceiver design.\"\n ],\n [\n \"System Model\",\n \"As shown in Fig.\",\n \"REF , we consider a point-to-point hybrid MIMO system where the source and the destination are equipped with $ N $ and $ M $ antennas, respectively.\",\n \"Without loss of generality, it is assumed that both the source and the destination have $L$ RF chains.\",\n \"A transmit data vector ${\\\\bf {a}}$ $\\\\in \\\\mathbb {C}^{D\\\\times 1}$ is first processed by a unit with feedback operation and then goes through a digital precoder $ {\\\\bf F}_{\\\\rm D} \\\\in \\\\mathbb {C}^{L \\\\times D} $ and an analog precoder $ {\\\\bf F}_{\\\\rm A} \\\\in \\\\mathbb {C}^{N \\\\times L} $ .\",\n \"This is a more general model as it includes both linear precoder and nonlinear precoder as its special cases.\",\n \"For the nonlinear transceiver with THP at source, the feedback matrix $ {\\\\bf B}^{\\\\rm Tx} $ is strictly lower triangular.\",\n \"The key idea behind THP is to exploit feedback operations to pre-eliminate mutual interference between different data streams.\",\n \"In order to control transmit signals in a predefined region, a modulo operation is introduced for the feedback operation [36].\",\n \"Based on lattice theory, it can be proved that the modulo operation is equivalent to adding an auxiliary complex vector ${\\\\bf {d}}$ whose element is with integer imaginary and real parts [36], [37].\",\n \"The vector ${\\\\bf {d}}$ makes sure ${\\\\bf {x}}={\\\\bf a} + {\\\\bf d}$ in a predefined region [36], [37].\",\n \"Based on this fact, the output vector ${\\\\bf {b}}$ of the feedback unit satisfies the following equation ${\\\\bf b} = ({\\\\bf a} + {\\\\bf d})-{\\\\bf {B}}^{\\\\rm {Tx}}{\\\\bf {b}},$ that is ${\\\\bf {b}}=({\\\\bf {I}}+{\\\\bf {B}}^{\\\\rm {Tx}})^{-1}\\\\underbrace{({\\\\bf a} + {\\\\bf d})}_{\\\\triangleq {\\\\bf {x}}}.$ It is worth noting that ${\\\\bf {d}}$ can be perfectly removed by a modulo operation [36], [37] and thus recovering ${\\\\bf {x}}$ is equivalent to recovering ${\\\\bf {a}}$ .\",\n \"On the other hand, for linear precoder, there is no feedback operation, i.e., ${\\\\bf {B}}^{\\\\rm {Tx}}={\\\\bf {0}}$ and ${\\\\bf {d}}={\\\\bf {0}}$ [38].\",\n \"Moreover, based on (REF ) we have ${\\\\bf {b}}={\\\\bf {a}}$ .\",\n \"Then, the received signal ${\\\\bf {y}}$ at the destination is ${\\\\bf {y}}={\\\\bf {H}}{\\\\bf {F}}_{\\\\rm {A}}{\\\\bf {F}}_{\\\\rm {D}}({\\\\bf I}+{\\\\bf B}^{\\\\rm Tx})^{-1}{\\\\bf {x}}+{\\\\bf {n}},$ where $ {\\\\bf n} $ is an $ M \\\\times 1 $ additive Gaussian noise vector with zero mean and covariance $ {\\\\bf R}_{\\\\rm n} $ , $ {\\\\bf H} $ is an $ M \\\\times N $ channel matrix, and $ {\\\\bf B}^{\\\\rm Tx} $ is a general feedback matrix at source, which is determined by the types of precoders.\",\n \"It is worth noting that $ {\\\\bf B}^{\\\\rm Tx} = {\\\\bf 0} $ corresponds to linear precoder without feedback operation.\",\n \"As shown in Fig.\",\n \"REF , after analog and digital processing at the destination, the recovered signal is given by ${\\\\bf {\\\\hat{x}}}^{\\\\rm {General}}={\\\\bf {G}}_{\\\\rm {D}}{\\\\bf {G}}_{\\\\rm {A}}{\\\\bf {y}} - {\\\\bf B}^{\\\\rm Rx}{\\\\bf x},$ where $ {\\\\bf G}_{\\\\rm A} \\\\in \\\\mathbb {C}^{L \\\\times M} $ is an analog processor, $ {\\\\bf G}_{\\\\rm D} \\\\in \\\\mathbb {C}^{D \\\\times L} $ is a digital processor, and $ {\\\\bf B}^{\\\\rm Rx} $ is a general feedback matrix at the destination.\",\n \"Note that since the analog precoder $ {\\\\bf F}_{\\\\rm A} $ and analog processor $ {\\\\bf G}_{\\\\rm A} $ are implemented through phase shifters, they are restricted to constant-modulus matrices with constant magnitude elements.\",\n \"For DFD at the receiver, the decision feedback matrix $ {\\\\bf B}^{\\\\rm Rx} $ in (REF ) is a strictly lower-triangular matrix.\",\n \"For linear detection, the feedback matrix in (REF ) is an all-zero matrix, i.e., $ {\\\\bf B}^{\\\\rm Rx} = {\\\\bf 0} $ .\",\n \"Based on (REF ) and (REF ), the recovered signal vector can be rewritten as ${\\\\bf {\\\\hat{x}}}= {\\\\bf {G}}_{\\\\rm {D}}{\\\\bf {G}}_{\\\\rm {A}}{\\\\bf {H}}{\\\\bf {F}}_{\\\\rm {A}} {\\\\bf {F}}_{\\\\rm {D}}( {\\\\bf I} + {\\\\bf B}^{\\\\rm Tx} )^{-1}{\\\\bf {x}} -{\\\\bf B}^{\\\\rm Rx}{\\\\bf {x}} + {\\\\bf {G}}_{\\\\rm {D}}{\\\\bf {G}}_{\\\\rm {A}}{\\\\bf {n}}.$ This is a general signal model and includes nonlinear hybrid transceivers with THP or DFD and linear hybrid transceiver as its special cases.\",\n \"More specifically, for a linear hybrid transceiver, there is no feedback, either at the source or at the destination, i.e., $ {\\\\bf B}^{\\\\rm Tx} = {\\\\bf B}^{\\\\rm Rx} = {\\\\bf 0} $ .\",\n \"Therefore, the recovered signal in (REF ) becomes ${\\\\bf {\\\\hat{x}}}^{\\\\rm Linear} = {\\\\bf {G}}_{\\\\rm {D}}{\\\\bf {G}}_{\\\\rm {A}}{\\\\bf {H}}{\\\\bf {F}}_{\\\\rm {A}}{\\\\bf {F}}_{\\\\rm {D}}{\\\\bf {x}}+{\\\\bf {G}}_{\\\\rm {D}}{\\\\bf {G}}_{\\\\rm {A}}{\\\\bf {n}}.$ For the nonlinear transceiver with THP at the source and linear decision at the destination, i.e., $ {\\\\bf B}^{\\\\rm Rx} = {\\\\bf 0} $ [36], [37], the detected signal vector in (REF ) becomes ${\\\\bf {\\\\hat{x}}}^{\\\\rm THP} = {\\\\bf {G}}_{\\\\rm {D}}{\\\\bf {G}}_{\\\\rm {A}}{\\\\bf {H}}{\\\\bf {F}}_{\\\\rm {A}}{\\\\bf {F}}_{\\\\rm {D}}({\\\\bf {I}}+{\\\\bf {B}}^{\\\\rm Tx})^{-1}{\\\\bf {x}}+ {\\\\bf {G}}_{\\\\rm {D}} {\\\\bf {G}}_{\\\\rm {A}} {\\\\bf {n}}.$ For the nonlinear transceiver with DFD at the destination and a linear precoder at the source, i.e., $ {\\\\bf B}^{\\\\rm Tx} = {\\\\bf 0} $ , the detected signal vector in (REF ) becomes ${\\\\bf {\\\\hat{x}}}^{\\\\rm DFD} = ({\\\\bf {G}}_{\\\\rm {D}}{\\\\bf {G}}_{\\\\rm {A}}{\\\\bf {H}}{\\\\bf {F}}_{\\\\rm {A}}{\\\\bf {F}}_{\\\\rm {D}}-{\\\\bf {B}}^{\\\\rm Rx}){\\\\bf {x}}+{\\\\bf {G}}_{\\\\rm {D}}{\\\\bf {G}}_{\\\\rm {A}}{\\\\bf {n}}.$\"\n ],\n [\n \"Unified MSE Matrix for Different Precoders and Processors\",\n \"Based on the general signal model in (REF ), the general MSE matrix of the recovered signal at the destination equals $& {\\\\Phi }_{\\\\rm {MSE}}({\\\\bf {G}}_{\\\\rm {D}},{\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {F}}_{\\\\rm {A}},{\\\\bf {F}}_{\\\\rm {D}},{\\\\bf B}^{\\\\rm Tx},{\\\\bf B}^{\\\\rm Rx}) \\\\nonumber \\\\\\\\= \\\\, & \\\\mathbb {E}\\\\lbrace ({\\\\bf {\\\\hat{x}}}-{\\\\bf {x}})({\\\\bf {\\\\hat{x}}}-{\\\\bf {x}})^{\\\\rm {H}}\\\\rbrace \\\\nonumber \\\\\\\\=\\\\, & \\\\mathbb {E}\\\\lbrace \\\\left( {\\\\bf {G}}_{\\\\rm {D}}{\\\\bf {G}}_{\\\\rm {A}}{\\\\bf {H}}{\\\\bf {F}}_{\\\\rm {A}} {\\\\bf {F}}_{\\\\rm {D}}({\\\\bf B}^{\\\\rm Rx}+{\\\\bf {I}})^{-1}{\\\\bf {x}}-{\\\\bf {x}}-{\\\\bf {G}}_{\\\\rm {D}}{\\\\bf {G}}_{\\\\rm {A}}{\\\\bf {n}} \\\\right)\\\\nonumber \\\\\\\\&\\\\times \\\\left( {\\\\bf {G}}_{\\\\rm {D}}{\\\\bf {G}}_{\\\\rm {A}}{\\\\bf {H}}{\\\\bf {F}}_{\\\\rm {A}} {\\\\bf {F}}_{\\\\rm {D}}({\\\\bf B}^{\\\\rm Rx}+{\\\\bf {I}})^{-1}{\\\\bf {x}}-{\\\\bf {x}}-{\\\\bf {G}}_{\\\\rm {D}}{\\\\bf {G}}_{\\\\rm {A}}{\\\\bf {n}} \\\\right)^{\\\\rm {H}}\\\\rbrace \\\\nonumber \\\\\\\\= \\\\, & \\\\mathbb {E} \\\\lbrace \\\\big ( {\\\\bf {G}}_{\\\\rm {D}}{\\\\bf {G}}_{\\\\rm {A}}{\\\\bf {H}}{\\\\bf {F}}_{\\\\rm {A}} {\\\\bf {F}}_{\\\\rm {D}} - ({\\\\bf B}^{\\\\rm Rx}+{\\\\bf {I}})( {\\\\bf I} + {\\\\bf B}^{\\\\rm Tx} ) \\\\big ) {\\\\bf b} {\\\\bf b}^{\\\\rm H}\\\\nonumber \\\\\\\\& \\\\times \\\\big ( {\\\\bf {G}}_{\\\\rm {D}}{\\\\bf {G}}_{\\\\rm {A}}{\\\\bf {H}}{\\\\bf {F}}_{\\\\rm {A}} {\\\\bf {F}}_{\\\\rm {D}} - ({\\\\bf B}^{\\\\rm Rx}+{\\\\bf {I}})( {\\\\bf I} + {\\\\bf B}^{\\\\rm Tx} ) \\\\big )^{\\\\rm H} \\\\rbrace \\\\nonumber \\\\\\\\& \\\\, + {\\\\bf {G}}_{\\\\rm {D}}{\\\\bf {G}}_{\\\\rm {A}}{\\\\bf {R}}_{\\\\rm {n}}{\\\\bf {G}}_{\\\\rm {A}}^{\\\\rm {H}}{\\\\bf {G}}_{\\\\rm {D}}^{\\\\rm {H}},$ where the third equality is based on ${\\\\bf {b}}=({\\\\bf B}^{\\\\rm Rx}+{\\\\bf {I}})^{-1}{\\\\bf {x}}$ given in (REF ).\",\n \"Based on lattice theory, each element of ${\\\\bf b}$ is identical and independent distributed, i.e., $\\\\mathbb {E}\\\\lbrace {\\\\bf b} {\\\\bf b}^{\\\\rm H}\\\\rbrace \\\\propto {\\\\bf {I}}$ [37].\",\n \"Thus, for notational simplicity, we can assume $\\\\mathbb {E}\\\\lbrace {\\\\bf b} {\\\\bf b}^{\\\\rm H}\\\\rbrace ={\\\\bf {I}}$ in the following derivations.\",\n \"Denote ${\\\\bf {B}}={\\\\bf B}^{\\\\rm Rx}+{\\\\bf B}^{\\\\rm Tx}+{\\\\bf B}^{\\\\rm Rx}{\\\\bf B}^{\\\\rm Tx}$ , then $({\\\\bf B}^{\\\\rm Rx}+{\\\\bf {I}})( {\\\\bf I} + {\\\\bf B}^{\\\\rm Tx} )={\\\\bf {I}}+{\\\\bf {B}}.$ It is obvious that ${\\\\bf {B}}$ is a strictly lower-triangular matrix based on the definitions of ${\\\\bf B}^{\\\\rm Tx}$ and ${\\\\bf B}^{\\\\rm Rx}$ , which implies that using nonlinear precoding at transmitter and nonlinear detection at the receiver at the same time is equivalent to just one of two.\",\n \"Therefore, nonlinear precoding at the transmitter and nonlinear detection at the receiver are equivalent and only one is enough.\",\n \"Direct matrix derivation [38] yields that the optimal $ {\\\\bf G}_{\\\\rm D} $ will be ${\\\\bf {G}}_{\\\\rm D}^{\\\\rm opt} & = ({\\\\bf {I}}+{\\\\bf B}) ({\\\\bf {G}}_{\\\\rm {A}}{\\\\bf {H}}{\\\\bf {F}}_{\\\\rm {A}}{\\\\bf {F}}_{\\\\rm {D}})^{\\\\rm {H}}\\\\nonumber \\\\\\\\& \\\\ \\\\ \\\\ \\\\times [({\\\\bf {G}}_{\\\\rm {A}}{\\\\bf {H}}{\\\\bf {F}}_{\\\\rm {A}}{\\\\bf {F}}_{\\\\rm {D}})({\\\\bf {G}}_{\\\\rm {A}}{\\\\bf {H}}{\\\\bf {F}}_{\\\\rm {A}}{\\\\bf {F}}_{\\\\rm {D}})^{\\\\rm {H}}+{\\\\bf {G}}_{\\\\rm {A}} {\\\\bf {R}}_{\\\\rm {n}}{\\\\bf {G}}_{\\\\rm {A}}^{\\\\rm {H}}]^{-1}.$ That is, the general MSE matrix can be further simplified into $& {\\\\Phi }_{\\\\rm {MSE}}({\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {F}}_{\\\\rm {A}},{\\\\bf {F}}_{\\\\rm {D}},{\\\\bf B}) \\\\nonumber \\\\\\\\= \\\\, & {\\\\Phi }_{\\\\rm {MSE}}({\\\\bf {G}}_{\\\\rm {D}}^{\\\\rm {opt}},{\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {F}}_{\\\\rm {A}},{\\\\bf {F}}_{\\\\rm {D}},{\\\\bf B}^{\\\\rm Tx},{\\\\bf B}^{\\\\rm Rx}) \\\\nonumber \\\\\\\\= \\\\, & ( {\\\\bf I} + {\\\\bf B}) ({\\\\bf I} + {\\\\bf F}_{\\\\rm D}^{\\\\rm H} {\\\\bf F}_{\\\\rm A}^{\\\\rm H} {\\\\bf H}^{\\\\rm H} {\\\\bf G}_{\\\\rm A}^{\\\\rm H}({\\\\bf G}_{\\\\rm A} {\\\\bf R}_{\\\\rm n} {\\\\bf G}_{\\\\rm A}^{\\\\rm H})^{-1} {\\\\bf G}_{\\\\rm A} {\\\\bf H} {\\\\bf F}_{\\\\rm A}{\\\\bf F}_{\\\\rm D})^{-1}\\\\nonumber \\\\\\\\&\\\\times ({\\\\bf I} + {\\\\bf B} )^{\\\\rm {H}} \\\\nonumber \\\\\\\\\\\\preceq \\\\, & {\\\\Phi }_{\\\\rm {MSE}}({\\\\bf {G}}_{\\\\rm {D}},{\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {F}}_{\\\\rm {A}},{\\\\bf {F}}_{\\\\rm {D}},{\\\\bf B}^{\\\\rm Tx},{\\\\bf B}^{\\\\rm Rx}),$ for any $ {\\\\bf G}_{\\\\rm D} $ .\",\n \"If $ {\\\\bf B}= {\\\\bf 0} $ in (REF ) and (REF ), the results are reduced to linear transceiver.\",\n \"Specifically, the corresponding digital LMMSE processor for linear transceiver is given as follows ${\\\\bf {G}}_{\\\\rm D,L}^{\\\\rm {opt}} = & ({\\\\bf {G}}_{\\\\rm {A}}{\\\\bf {H}}{\\\\bf {F}}_{\\\\rm {A}}{\\\\bf {F}}_{\\\\rm {D}})^{\\\\rm {H}}[({\\\\bf {G}}_{\\\\rm {A}}{\\\\bf {H}}{\\\\bf {F}}_{\\\\rm {A}}{\\\\bf {F}}_{\\\\rm {D}})({\\\\bf {G}}_{\\\\rm {A}}{\\\\bf {H}}{\\\\bf {F}}_{\\\\rm {A}}{\\\\bf {F}}_{\\\\rm {D}})^{\\\\rm {H}}\\\\nonumber \\\\\\\\&+{\\\\bf {G}}_{\\\\rm {A}}{\\\\bf {R}}_{\\\\rm {n}}{\\\\bf {G}}_{\\\\rm {A}}^{\\\\rm {H}}]^{-1},$ and the MSE matrix for linear transceiver is $& {\\\\Phi }_{\\\\rm {MSE}}^{\\\\rm {L}}({\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {F}}_{\\\\rm {A}},{\\\\bf {F}}_{\\\\rm {D}})\\\\nonumber \\\\\\\\=&{\\\\Phi }_{\\\\rm {MSE}}({\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {F}}_{\\\\rm {A}},{\\\\bf {F}}_{\\\\rm {D}},{\\\\bf 0})\\\\nonumber \\\\\\\\= &\\\\left[{\\\\bf {I}} + {\\\\bf {F}}_{\\\\rm {D}}^{\\\\rm {H}} {\\\\bf {F}}_{\\\\rm {A}}^{\\\\rm {H}} {\\\\bf {H}}^{\\\\rm {H}} {\\\\bf {G}}_{\\\\rm {A}}^{\\\\rm {H}}({\\\\bf {G}}_{\\\\rm {A}}{\\\\bf {R}}_{\\\\rm {n}}{\\\\bf {G}}_{\\\\rm {A}}^{\\\\rm {H}})^{-1} {\\\\bf {G}}_{\\\\rm {A}}{\\\\bf {H}}{\\\\bf {F}}_{\\\\rm {A}}{\\\\bf {F}}_{\\\\rm {D}}\\\\right]^{-1}\\\\nonumber \\\\\\\\\\\\triangleq &\\\\left[{\\\\bf {I}}+{\\\\Gamma }({\\\\bf {G}}_{\\\\rm {A}}, {\\\\bf {F}}_{\\\\rm {D}}, {\\\\bf {F}}_{\\\\rm {A}})\\\\right]^{-1},$ where ${\\\\Gamma }({\\\\bf {G}}_{\\\\rm {A}}, {\\\\bf {F}}_{\\\\rm {D}}, {\\\\bf {F}}_{\\\\rm {A}})={\\\\bf {F}}_{\\\\rm {D}}^{\\\\rm {H}} {\\\\bf {F}}_{\\\\rm {A}}^{\\\\rm {H}} {\\\\bf {H}}^{\\\\rm {H}} {\\\\bf {G}}_{\\\\rm {A}}^{\\\\rm {H}}({\\\\bf {G}}_{\\\\rm {A}}{\\\\bf {R}}_{\\\\rm {n}}{\\\\bf {G}}_{\\\\rm {A}}^{\\\\rm {H}})^{-1} {\\\\bf {G}}_{\\\\rm {A}}{\\\\bf {H}}{\\\\bf {F}}_{\\\\rm {A}}{\\\\bf {F}}_{\\\\rm {D}}$ , which is signal-to-noise ratio for single antenna case.\",\n \"For the nolinear transceivers, ${\\\\bf {B}}={\\\\bf B}^{\\\\rm Tx}$ for THP or ${\\\\bf {B}}={\\\\bf B}^{\\\\rm Rx}$ for DFD in (REF )-(REF ).\",\n \"Based on (REF ) and (REF ), the general MSE matrix for nonlinear transceivers can also be written in the following unified formula ${\\\\Phi }_{\\\\rm {MSE}}({\\\\bf {G}}_{\\\\rm {A}}, {\\\\bf {F}}_{\\\\rm {A}}, {\\\\bf {F}}_{\\\\rm {D}}, {\\\\bf {B}})&= ({\\\\bf {I}} + {\\\\bf {B}}) {\\\\Phi }_{\\\\rm {MSE}}^{\\\\rm {L}}({\\\\bf {G}}_{\\\\rm {A}}, {\\\\bf {F}}_{\\\\rm {A}}, {\\\\bf {F}}_{\\\\rm {D}}) \\\\nonumber \\\\\\\\&\\\\times ({\\\\bf {I}} + {\\\\bf {B}})^{\\\\rm H},$ which turns into the MSE matrix in (REF ) when ${\\\\bf {B}}={\\\\bf {0}}$ .\",\n \"In the following, we will investigate unified hybrid MIMO transceiver optimization, which is applicable to various objective functions based on on the general MSE matrix (REF ).\"\n ],\n [\n \"The Unified Hybrid MIMO Transceiver Optimization\",\n \"Because of the multi-objective optimization nature for MIMO systems with multiple data streams, there are different kinds of objectives that reflect different design preferences [39].\",\n \"All can be regarded as a matrix monotonic function of the data estimation MSE matrix in (REF ) [40].\",\n \"A function $f(\\\\cdot )$ is a matrix monotone increasing function if $ f({\\\\bf X}) \\\\ge f({\\\\bf Y}) $ for $ {\\\\bf X} \\\\succeq {\\\\bf Y} \\\\succeq {\\\\bf 0}$ [40].\",\n \"To avoid case-by-case discussion, we will investigate in depth hybrid MIMO transceiver optimization with different performance metrics from a unified viewpoint, in this section.\",\n \"Based on the MSE matrix in (REF ), the unified hybrid MIMO transceiver design can be formulated in the following form $\\\\min _{{\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {F}}_{\\\\rm {A}},{\\\\bf {F}}_{\\\\rm {D}},{\\\\bf {B}}} \\\\ & f \\\\left({\\\\Phi }_{\\\\rm {MSE}}({\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {F}}_{\\\\rm {A}},{\\\\bf {F}}_{\\\\rm {D}},{\\\\bf {B}})\\\\right) \\\\nonumber \\\\\\\\{\\\\rm {s.t.}}\",\n \"\\\\qquad & {\\\\rm {Tr}}({\\\\bf {F}}_{\\\\rm {A}}{\\\\bf {F}}_{\\\\rm {D}}{\\\\bf {F}}_{\\\\rm {D}}^{\\\\rm {H}}{\\\\bf {F}}_{\\\\rm {A}}^{\\\\rm {H}})\\\\le P \\\\nonumber \\\\\\\\& {\\\\bf {F}}_{\\\\rm {A}} \\\\in \\\\mathcal {F}, \\\\ \\\\ {\\\\bf {G}}_{\\\\rm {A}} \\\\in \\\\mathcal {G},$ where $f(\\\\cdot )$ is a matrix monotone increasing function [40].\",\n \"The sets $\\\\mathcal {F}$ and $\\\\mathcal {G}$ are the feasible analog precoder set and analog processor set satisfying constant-modulus constraint, and $ P $ denotes the maximum transmit power at the source.\"\n ],\n [\n \"Specific Objective Functions\",\n \"There are many ways to choose the matrix monotone increasing function.\",\n \"In this subsection, we will investigate the properties of different objective functions in (REF ).\",\n \"One group of matrix monotone increasing functions can be expressed as $&f_1\\\\left({\\\\Phi }_{\\\\rm {MSE}}({\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {F}}_{\\\\rm {A}},{\\\\bf {F}}_{\\\\rm {D}},{\\\\bf {B}})\\\\right)\\\\nonumber \\\\\\\\&=f_{\\\\rm {Schur}} \\\\left({\\\\bf {d}}({\\\\Phi }_{\\\\rm {MSE}}({\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {F}}_{\\\\rm {A}},{\\\\bf {F}}_{\\\\rm {D}},{\\\\bf {B}}))\\\\right),$ where ${\\\\bf {d}}({\\\\Phi }_{\\\\rm {MSE}}({\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {F}}_{\\\\rm {A}},{\\\\bf {F}}_{\\\\rm {D}},{\\\\bf {B}}))$ is a vector consisting of the diagonal elements of the matrix ${\\\\Phi }_{\\\\rm {MSE}}({\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {F}}_{\\\\rm {A}},{\\\\bf {F}}_{\\\\rm {D}},{\\\\bf {B}})$ and $f_{\\\\rm {Schur}}(\\\\cdot )$ is a function of a vector satisfying one the following four properties discussed in Appendix : Multiplicatively Schur-convex Multiplicatively Schur-concave Additively Schur-convex Additively Schur-concave.\",\n \"Many widely used metrics can be regarded as a special case of this group of functions [39], [37], [36].\",\n \"Conclusion 1: For linear transceiver, the feedback matrix ${\\\\bf {B}}$ in (REF ) is an all-zero matrix, i.e., $ {\\\\bf B}^{\\\\rm {opt}} = {\\\\bf 0} $ .\",\n \"For nonlinear transceiver, from Appendix the optimal feedback matrix ${\\\\bf {B}}$ for $f_1(\\\\cdot )$ is ${\\\\bf {B}}^{\\\\rm {opt}}={\\\\rm {diag}}\\\\lbrace [[{\\\\bf {L}}]_{1,1},\\\\cdots ,[{\\\\bf {L}}]_{L,L}]^{\\\\rm {T}}\\\\rbrace {\\\\bf {L}}^{-1}-{\\\\bf {I}},$ where ${\\\\bf {L}}$ is a lower triangular matrix of the following Cholesky decomposition ${\\\\Phi }_{\\\\rm {MSE}}^{\\\\rm {L}}({\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {F}}_{\\\\rm {A}},{\\\\bf {F}}_{\\\\rm {D}})={\\\\bf {L}}{\\\\bf {L}}^{\\\\rm {H}}.$ It has been proved in [40] and [38] that for nonlinear transceiver design each data stream will have the same performance if $f_{\\\\rm {Schur}}(\\\\cdot )$ in (REF ) is multiplicatively Schur-convex.\",\n \"On the other hand, if $f_{\\\\rm {Schur}}(\\\\cdot )$ in (REF ) is multiplicatively Schur-concave, for nonlinear transceiver design the objective function includes geometrically weighted signal-to-noise-plus-interference-ratio (SINR) maximization as its special case.\",\n \"If $f_{\\\\rm {Schur}}(\\\\cdot )$ in (REF ) is additively Schur-convex, the objective function includes the the maximum MSE minimization and the minimum BER with the same constellation on each data stream as special cases.\",\n \"If $f_{\\\\rm {Schur}}(\\\\cdot )$ in (REF ) is additively Schur-concave, the objective function includes weighted MSE minimization as its special case.\",\n \"Additive Schur functions are usually used for linear transceivers (${\\\\bf {B}}={\\\\bf {0}}$ in (REF )) since closed-form solutions can be obtained in this case.\",\n \"Besides the above group of matrix monotone increasing functions, we can choose one to reflect capacity and MSE for linear transceivers.\",\n \"Capacity is one of the most popular performance metrics in MIMO transceiver optimization.\",\n \"It can be expressed as the form of MSE matrix considering the well-known relationship between the MSE matrix and capacity [40], i.e., $ C = -{\\\\rm log}| {\\\\bf \\\\Phi }_{\\\\rm MSE} | $ .\",\n \"Then, the objective can be given as $& f_{\\\\rm 2} (\\\\cdot )= {\\\\rm {log}}|{\\\\Phi }_{\\\\rm {MSE}}^{\\\\rm {L}}({\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {F}}_{\\\\rm {A}},{\\\\bf {F}}_{\\\\rm {D}})|.$ MSE is another widely used performance metric that demonstrates how accurately a signal can be recovered.\",\n \"The corresponding weighted MSE minimization objective is $& f_{\\\\rm 3} (\\\\cdot )= {\\\\rm {Tr}}\\\\left[{\\\\bf {A}}^{\\\\rm {H}}{\\\\Phi }_{\\\\rm {MSE}}^{\\\\rm {L}}({\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {F}}_{\\\\rm {A}},{\\\\bf {F}}_{\\\\rm {D}}){\\\\bf {A}}\\\\right],$ where $ {\\\\bf A} $ is a general, not necessarily diagonal, weight matrix, even if it is often diagonal in many applications.\"\n ],\n [\n \"Hybrid MIMO Transceiver Optimization\",\n \"Denote ${\\\\Pi }_{\\\\rm {L}} & =( {\\\\bf {G}}_{\\\\rm {A}} {\\\\bf {R}}_{\\\\rm {n}} {\\\\bf {G}}_{\\\\rm {A}}^{\\\\rm {H}} )^{-1/2}{\\\\bf {G}}_{\\\\rm {A}}{\\\\bf {R}}_{\\\\rm {n}}^{1/2}, \\\\nonumber \\\\\\\\{\\\\Pi }_{\\\\rm {R}} & = {\\\\bf {F}}_{\\\\rm {A}} ( {\\\\bf {F}}_{\\\\rm {A}}^{\\\\rm {H}} {\\\\bf {F}}_{\\\\rm {A}} )^{-\\\\frac{1}{2}}, \\\\nonumber $ and ${\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}} & = ( {\\\\bf {F}}_{\\\\rm {A}}^{\\\\rm {H}} {\\\\bf {F}}_{\\\\rm {A}} )^{\\\\frac{1}{2}}{\\\\bf { F}}_{\\\\rm {D}}{\\\\bf {Q}}^{\\\\rm {H}},$ where ${\\\\bf {Q}}$ is a unitary matrix to be determined by digital transceiver optimization in the next section.\",\n \"Then (REF ) can be rewritten as $ {\\\\Gamma }({\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {F}}_{\\\\rm {D}},{\\\\bf {F}}_{\\\\rm {A}})={\\\\bf {Q}}^{\\\\rm {H}}{\\\\tilde{\\\\Gamma }}({\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}},{\\\\bf {F}}_{\\\\rm {A}}){\\\\bf {Q}},$ where $& {\\\\tilde{\\\\Gamma }}({\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}},{\\\\bf {F}}_{\\\\rm {A}})= {\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}}^{\\\\rm {H}}{\\\\Pi }_{\\\\rm {R}}^{\\\\rm {H}}{\\\\bf {H}}^{\\\\rm {H}}{\\\\bf {R}}_{\\\\rm {n}}^{-1/2}{\\\\Pi }_{\\\\rm {L}}^{\\\\rm {H}}{\\\\Pi }_{\\\\rm {L}} {\\\\bf {R}}_{\\\\rm {n}}^{-1/2}{\\\\bf {H}}{\\\\Pi }_{\\\\rm {R}}{\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}}.$ The optimal ${\\\\bf {B}}$ is usually a function of ${\\\\Gamma }({\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {F}}_{\\\\rm {D}},{\\\\bf {F}}_{\\\\rm {A}})$ , for all objective functions as demonstrated by (REF ) for $f_{\\\\rm {Schur}}(\\\\cdot )$ in (REF ).\",\n \"From (REF ), we can conclude that the optimal ${\\\\bf {B}}$ is a function of ${\\\\bf {Q}}^{\\\\rm {H}}{\\\\tilde{\\\\Gamma }}({\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}},{\\\\bf {F}}_{\\\\rm {A}}){\\\\bf {Q}}$ .\",\n \"Therefore, using (REF ) and (REF ), the objective function of (REF ) can be expressed in terms of ${\\\\tilde{\\\\Gamma }}({\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}},{\\\\bf {F}}_{\\\\rm {A}})$ as $& f \\\\left( {\\\\Phi }_{\\\\rm {MSE}}({\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {F}}_{\\\\rm {A}},{\\\\bf {F}}_{\\\\rm {D}},{\\\\bf {B}}^{\\\\rm {opt}}) \\\\right) \\\\nonumber \\\\\\\\= & f \\\\big ( ({\\\\bf {I}}+{\\\\bf {B}}^{\\\\rm opt}) ({\\\\bf I} + {\\\\bf {Q}}^{\\\\rm {H}}{\\\\tilde{\\\\Gamma }} ({\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}},{\\\\bf {F}}_{\\\\rm {A}}) {\\\\bf {Q}} )^{-1} ({\\\\bf {I}}+{\\\\bf {B}}^{\\\\rm opt})^{\\\\rm {H}} \\\\big ) \\\\nonumber \\\\\\\\\\\\triangleq & f_{{\\\\rm S}} \\\\left({\\\\bf {Q}}^{\\\\rm {H}}{\\\\tilde{\\\\Gamma }}({\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}},{\\\\bf {F}}_{\\\\rm {A}}){\\\\bf {Q}}\\\\right).$ After introducing ${\\\\tilde{\\\\Gamma }}({\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}},{\\\\bf {F}}_{\\\\rm {A}})$ and a new auxiliary matrix $ {\\\\bf Q} $ , the objective function is transferred into $ f_{{\\\\rm S}}(\\\\cdot )$ rather than $ f(\\\\cdot ) $ .\",\n \"Note that this new function notation, $ f_{{\\\\rm S}}(\\\\cdot )$ , is defined only for notational simplicity and it explicitly expresses the objective as a function of matrix variables ${\\\\bf Q}$ and ${\\\\tilde{\\\\Gamma }}$ .\",\n \"Therefore, the optimization problem in (REF ) is further rewritten into the following one $\\\\min _{{\\\\bf {Q}},{\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}},{\\\\bf {F}}_{\\\\rm {A}}} \\\\ & f_{{\\\\rm S}} \\\\left({\\\\bf {Q}}^{\\\\rm {H}}{\\\\tilde{\\\\Gamma }}({\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}},{\\\\bf {F}}_{\\\\rm {A}}){\\\\bf {Q}}\\\\right) \\\\nonumber \\\\\\\\{\\\\rm {s.t.}}\",\n \"\\\\ \\\\ \\\\ \\\\ \\\\ \\\\ & {\\\\rm {Tr}}({\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}}{\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}}^{\\\\rm {H}})\\\\le P \\\\nonumber \\\\\\\\& {\\\\bf {F}}_{\\\\rm {A}} \\\\in \\\\mathcal {F}, \\\\ \\\\ {\\\\bf {G}}_{\\\\rm {A}} \\\\in \\\\mathcal {G}.$ We will discuss in detail how to solve the optimization problem (REF ) with respect to ${\\\\bf {Q}}$ , ${\\\\bf {G}}_{\\\\rm {A}}$ ,${\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}}$ , and ${\\\\bf {F}}_{\\\\rm {A}}$ subsequently.\",\n \"In (REF ), ${\\\\bf {B}}$ has been formulated as a function of ${\\\\bf {Q}}$ , ${\\\\bf {G}}_{\\\\rm {A}}$ ,${\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}}$ , and ${\\\\bf {F}}_{\\\\rm {A}}$ .\",\n \"When ${\\\\bf {Q}}$ , ${\\\\bf {G}}_{\\\\rm {A}}$ ,${\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}}$ , and ${\\\\bf {F}}_{\\\\rm {A}}$ are calculated, the optimal ${\\\\bf {B}}$ can be directly derived based on (REF ).\"\n ],\n [\n \"Digital Transceiver Optimization\",\n \"In the following, we focus on the digital transceiver optimization for the optimization problem (REF ).\",\n \"More specifically, we first derive the optimal unitary matrix ${\\\\bf {Q}}$ and then find the optimal ${\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}}$ .\"\n ],\n [\n \"Optimal $ {\\\\bf {Q}} $\",\n \"At the beginning of this section, two fundamental definitions are given based on the following eigenvalue decomposition (EVD) and SVD $&{\\\\tilde{\\\\Gamma }}({\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}},{\\\\bf {F}}_{\\\\rm {A}})={\\\\bf {U}}_{{\\\\tilde{\\\\Gamma }}}{\\\\Lambda }_{{\\\\tilde{\\\\Gamma }}}{\\\\bf {U}}_{{\\\\tilde{\\\\Gamma }}}^{\\\\rm {H}} \\\\nonumber \\\\\\\\&{\\\\bf {A}}={\\\\bf {U}}_{{\\\\bf {A}}}{\\\\Lambda }_{{\\\\bf {A}}}{\\\\bf {V}}_{{\\\\bf {A}}}^{\\\\rm {H}},$ where $ {\\\\Lambda }_{\\\\rm \\\\Phi } $ and $ {\\\\Lambda }_{\\\\rm A} $ denote a diagonal matrix with the diagonal elements in nondecreasing order.\",\n \"Denote ${\\\\bf {U}}_{\\\\rm {GMD}}$ as the unitary matrix that makes the lower triangular matrix ${\\\\bf {L}}$ in (REF ) has the same diagonal elements.\",\n \"It has been shown in [39], [40], [38] that the optimal ${\\\\bf {Q}}$ for the first group of matrix-monotonic functions can be expressed as ${\\\\bf {Q}}^{\\\\rm {opt}} = \\\\left\\\\lbrace {\\\\begin{array}{l}{{\\\\bf {U}}_{{\\\\tilde{\\\\Gamma }}}{\\\\bf {U}}_{\\\\text{GMD}}^{\\\\rm {H}}}\\\\ \\\\ \\\\text{if} \\\\ f(\\\\cdot ) \\\\ \\\\text{is multiplicatively Schur-convex} \\\\\\\\{{\\\\bf {U}}_{{\\\\tilde{\\\\Gamma }}}} \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\text{if} \\\\ f(\\\\cdot ) \\\\ \\\\text{is multiplicatively Schur-concave} \\\\\\\\{{\\\\bf {U}}_{{\\\\tilde{\\\\Gamma }}}{\\\\bf {U}}_{\\\\text{DFT}}^{\\\\rm {H}}}\\\\ \\\\ \\\\ \\\\text{if} \\\\ f(\\\\cdot ) \\\\ \\\\text{is additively Schur-convex}\\\\\\\\{{\\\\bf {U}}_{{\\\\tilde{\\\\Gamma }}}} \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\text{if} \\\\ f(\\\\cdot ) \\\\ \\\\text{is additively Schur-concave}.\\\\end{array}} \\\\right.$ The above results are obtained by directly manipulating with the objective function $f(\\\\cdot )$ in (REF ), and thus the optimal ${\\\\bf {Q}}$ varies with the matrix-monotone increasing function in (REF ).\",\n \"For the capacity maximization in (REF ), the objective function of (REF ) can be written as $& f_{\\\\rm S, 2}(\\\\cdot ) = -{\\\\rm {log}} |{\\\\bf {Q}}^{\\\\rm {H}} {\\\\tilde{\\\\Gamma }} ({\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}},{\\\\bf {F}}_{\\\\rm {A}}) {\\\\bf {Q}} + {\\\\bf {I}}|.$ Since the function in (REF ) is independent of $ {\\\\bf Q} $ as long as it is a unitary matrix, the optimal $ {\\\\bf Q} $ , namely $ {\\\\bf Q}^{\\\\rm opt} $ , can be any unitary matrix with proper dimension.\",\n \"For the weighted MSE minimization given by (REF ), the objective function of (REF ) can be rewritten as $f_{\\\\rm S, 3} (\\\\cdot ) & ={\\\\rm {Tr}} [{\\\\bf {A}}^{\\\\rm {H}} ({\\\\bf {Q}}^{\\\\rm {H}} {\\\\tilde{\\\\Gamma }} ({\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}},{\\\\bf {F}}_{\\\\rm {A}}) {\\\\bf {Q}}+ {\\\\bf {I}})^{-1} {\\\\bf {A}}].$ Based on the EVD and SVD defined in (REF ) and the matrix inequality in Appendix , the optimal ${\\\\bf {Q}}$ is ${\\\\bf {Q}}^{\\\\rm opt} = {\\\\bf {U}}_{{\\\\tilde{\\\\Gamma }}}{\\\\bf { U}}_{{\\\\bf {A}}}^{\\\\rm {H}}.$ We have to stress that it is still hard to find the closed-form expression for the optimal ${\\\\bf {Q}}$ for an arbitrary function $f(\\\\cdot )$ .\",\n \"However, most of the meaningful and popular metric functions have been shown included in one of the above function families, and are with the closed-form expression for optimal ${\\\\bf {Q}}$ .\"\n ],\n [\n \"Optimal ${\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}}$\",\n \"After substituting the optimal ${\\\\bf {Q}}$ into the objective function of (REF ), the objective function becomes a function of the eigenvalues of ${\\\\tilde{\\\\Gamma }}({\\\\bf {G}}_{\\\\rm {A}}, {\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}}, {\\\\bf {F}}_{\\\\rm {A}})$ , i.e., $& f_{\\\\rm {S}}\\\\left({\\\\bf {Q}}^{\\\\rm {H}}{\\\\tilde{\\\\Gamma }}({\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}},{\\\\bf {F}}_{\\\\rm {A}}){\\\\bf {Q}}\\\\right)\\\\triangleq f_{\\\\rm {E}} \\\\left({\\\\lambda } \\\\Big ( {\\\\tilde{\\\\Gamma }}({\\\\bf {G}}_{\\\\rm {A}}, {\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}}, {\\\\bf {F}}_{\\\\rm {A}}) \\\\Big ) \\\\right),$ where ${\\\\lambda }({X})=[\\\\lambda _1({X}), \\\\cdots , \\\\lambda _L({X})]^{\\\\rm {T}}$ and $\\\\lambda _i({X})$ is the $ i $ th largest eigenvalue of ${X}$ .\",\n \"It is worth highlighting that for $f_{{\\\\rm {S}},1}(\\\\cdot )$ and $f_{{\\\\rm {S}},3}(\\\\cdot )$ based on (REF ) and (REF ) we can directly have (REF ).\",\n \"For $f_{{\\\\rm {S}},2}(\\\\cdot )$ the optimal ${\\\\bf {Q}}$ can be an arbitrary unitary matrix, minimizing $f_{{\\\\rm {S}},2}(\\\\cdot )$ mathematically equals to minimizing $-\\\\sum _{l=1}^L{\\\\rm {log}}\\\\left(1+{\\\\lambda }_l ( {\\\\tilde{\\\\Gamma }}({\\\\bf {G}}_{\\\\rm {A}}, {\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}}, {\\\\bf {F}}_{\\\\rm {A}}))\\\\right)$ for any ${\\\\bf {Q}}$ .\",\n \"In other words, (REF ) always holds for these kinds of functions discussed above.\",\n \"Note that the definition in (REF ) follows from the facts that the unitary matrix in ${\\\\tilde{\\\\Gamma }}({\\\\bf {G}}_{\\\\rm {A}}, {\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}}, {\\\\bf {F}}_{\\\\rm {A}})$ has been removed by the optimal ${\\\\bf {Q}}$ and only its eigenvalues remain to be optimized.\",\n \"Therefore, the unified hybrid MIMO transceiver optimization in (REF ) is simplified to $\\\\min _{{\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}},{\\\\bf {F}}_{\\\\rm {A}}} \\\\ \\\\ & f_{\\\\rm {E}}\\\\left({\\\\lambda }\\\\left({\\\\tilde{\\\\Gamma }}({\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}},{\\\\bf {F}}_{\\\\rm {A}})\\\\right)\\\\right) \\\\nonumber \\\\\\\\{\\\\rm {s.t.}}\",\n \"\\\\ \\\\ \\\\ & {\\\\rm {Tr}}({\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}}{\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}}^{\\\\rm {H}})\\\\le P \\\\nonumber \\\\\\\\& {\\\\bf {F}}_{\\\\rm {A}} \\\\in \\\\mathcal {F}, \\\\ \\\\ {\\\\bf {G}}_{\\\\rm {A}} \\\\in \\\\mathcal {G}.$ By applying the obtained results of ${\\\\bf Q}^{\\\\rm opt}$ and the fact that $ f(\\\\cdot ) $ is a matrix-monotone increasing function, it can be concluded from the discussion in [39], [38] that $ f_{{\\\\rm E}}(\\\\cdot ) $ is a vector-decreasing function for $f_{{\\\\rm {S}},1}(\\\\cdot )$ .\",\n \"Moreover, substituting the optimal ${\\\\bf {Q}}$ into the objective function of (REF ), for $f_{{\\\\rm {S}},2}(\\\\cdot )$ and $f_{{\\\\rm {S}},3}(\\\\cdot )$ we have $f_{{\\\\rm {E}}}(\\\\cdot )&=-\\\\sum _{l=1}^L{\\\\rm {log}}\\\\left(1+{\\\\lambda }_l ( {\\\\tilde{\\\\Gamma }}({\\\\bf {G}}_{\\\\rm {A}}, {\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}},{\\\\bf {F}}_{\\\\rm {A}}))\\\\right), \\\\\\\\f_{{\\\\rm {E}}}(\\\\cdot )&=\\\\sum _{l=1}^L\\\\frac{\\\\lambda _l({\\\\bf {A}})}{1+{\\\\lambda }_l ( {\\\\tilde{\\\\Gamma }}({\\\\bf {G}}_{\\\\rm {A}}, {\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}}, {\\\\bf {F}}_{\\\\rm {A}}))},$ respectively, which implies that $ f_{{\\\\rm E}}(\\\\cdot ) $ is also vector-decreasing.\",\n \"In a nutshell, based on ${\\\\bf Q}^{\\\\rm opt}$ we can conclude that $ f_{{\\\\rm E}}(\\\\cdot ) $ in (REF ) is a vector-decreasing function.\",\n \"Thus, from (REF ), the optimization becomes maximizing the eigenvalues of ${\\\\tilde{\\\\Gamma }}({\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}},{\\\\bf {F}}_{\\\\rm {A}})$ .\",\n \"Each eigenvalue of ${\\\\tilde{\\\\Gamma }}({\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}},{\\\\bf {F}}_{\\\\rm {A}})$ corresponds to SNR of an eigenchannel.\",\n \"In problem (REF ), the variables are still matrix variables.\",\n \"To simplify the optimization, we will first derive the diagonalizable structure of the optimal matrix variables.\",\n \"Based on the derived optimal structure, the dimensionality of the optimization problem are reduced significantly.\",\n \"In order to derive the optimal structure and to avoid tedious case-by-case discussion, we consider a multi-objective optimization problem in the following.\",\n \"Its Pareto optimal solution set contains all the optimal solutions of different types of transceiver optimizations.\",\n \"In particular, as discussed in [40], the optimal solution of problem (REF ) with a specific objective function, i.e., $f_1(\\\\cdot )$ , $f_{2}(\\\\cdot )$ , or $f_3(\\\\cdot )$ , must be in the Pareto optimal solution set of the following vector optimization (multi-objective) problem $& \\\\ \\\\ \\\\max _{{\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}},{\\\\bf {F}}_{\\\\rm {A}}} \\\\ \\\\ {\\\\lambda }\\\\left({\\\\tilde{\\\\Gamma }}({\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}},{\\\\bf {F}}_{\\\\rm {A}})\\\\right) \\\\nonumber \\\\\\\\& \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ {\\\\rm {s.t.}}\",\n \"\\\\ \\\\ \\\\ \\\\ \\\\ \\\\ {\\\\rm {Tr}}({\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}}{\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}}^{\\\\rm {H}})\\\\le P \\\\nonumber \\\\\\\\& \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ {\\\\bf {F}}_{\\\\rm {A}} \\\\in \\\\mathcal {F}, \\\\ \\\\ {\\\\bf {G}}_{\\\\rm {A}} \\\\in \\\\mathcal {G}.$ Equivalently, the vector optimization problem in (REF ) can be rewritten as the following matrix-monotonic optimization problem $& \\\\ \\\\ \\\\max _{{\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}},{\\\\bf {F}}_{\\\\rm {A}}} \\\\ \\\\ {\\\\tilde{\\\\Gamma }}({\\\\bf {G}}_{\\\\rm {A}},{\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}},{\\\\bf {F}}_{\\\\rm {A}}) \\\\nonumber \\\\\\\\& \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ {\\\\rm {s.t.}}\",\n \"\\\\ \\\\ \\\\ \\\\ \\\\ {\\\\rm {Tr}}({\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}}{\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}}^{\\\\rm {H}})\\\\le P \\\\nonumber \\\\\\\\& \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ {\\\\bf {F}}_{\\\\rm {A}} \\\\in \\\\mathcal {F}, \\\\ \\\\ {\\\\bf {G}}_{\\\\rm {A}} \\\\in \\\\mathcal {G}.$ It is worth noting that optimization (REF ) aims at maximizing a positive semi-definite matrix.\",\n \"Generally speaking, maximizing a positive semi-definite matrix includes two tasks, i.e., maximizing its eigenvalues and choosing a proper EVD unitary matrix.\",\n \"Note that in (REF ) there is no need to optimize the EVD unitary matrix, because the constraints can remain satisfied if only EVD unitary matrix changes.\",\n \"Using the definitions in (REF ) and given analog precoder ${\\\\bf {F}}_{\\\\rm {A}}$ and analog processor ${\\\\bf {G}}_{\\\\rm {A}}$ , problem (REF ) is a standard matrix-monotonic optimization with respect to ${\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}}$ .\",\n \"It follows $& \\\\ \\\\ \\\\max _{{\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}}} \\\\ \\\\ {\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}}^{\\\\rm {H}}{\\\\Pi }_{\\\\rm {R}}^{\\\\rm {H}}{\\\\bf {H}}^{\\\\rm {H}}{\\\\bf {R}}_{\\\\rm {n}}^{-1/2}{\\\\Pi }_{\\\\rm {L}}^{\\\\rm {H}}{\\\\Pi }_{\\\\rm {L}} {\\\\bf {R}}_{\\\\rm {n}}^{-1/2}{\\\\bf {H}}{\\\\Pi }_{\\\\rm {R}}{\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}} \\\\nonumber \\\\\\\\& \\\\ \\\\ \\\\ {\\\\rm {s.t.}}\",\n \"\\\\ \\\\ \\\\ \\\\ {\\\\rm {Tr}}({\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}}{\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}}^{\\\\rm {H}})\\\\le P.$ Based on the matrix-monotonic optimization theory developed in [40], the optimal solution of (REF ) satisfies the following diagonalizable structure.\",\n \"Conclusion 2: Defining the following SVD, $&{\\\\Pi }_{\\\\rm {L}}{\\\\bf {R}}_{\\\\rm {n}}^{-1/2}{\\\\bf {H}}{\\\\Pi }_{\\\\rm {R}}={\\\\bf {U}}_{{\\\\mathbb {H}}}{\\\\Lambda }_{\\\\mathbb {H}}{\\\\bf {V}}_{\\\\mathbb {H}}^{\\\\rm {H}},$ with the diagonal elements of ${\\\\Lambda }_{\\\\mathbb {H}}$ in decreasing order, the optimal ${\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}}$ satisfies ${\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}}^{\\\\rm opt} = {\\\\bf {V}}_{\\\\mathbb {H}} {\\\\Lambda }_{\\\\bf {F}} {\\\\bf {U}}_{\\\\rm {Arb}}^{\\\\rm {H}},$ where ${\\\\Lambda }_{\\\\bf {F}}$ is a diagonal matrix determined by the specific objective functions, e.g., sum MSE, capacity maximization, etc., as discussed in the previous section.\",\n \"The unitary matrix ${\\\\bf {U}}_{\\\\rm {Arb}}$ can be an arbitrary unitary matrix.\",\n \"Thus far by using Conclusion 2, the optimal ${\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}}$ can be obtained by conducting basic manipulations as in [40] on optimizing ${\\\\Lambda }_{\\\\bf {F}}$ given a specific objective function.\",\n \"As a result, the remaining key task is to optimize the analog precoder and processor, which is the focus of the following section.\"\n ],\n [\n \"Analog Transceiver Optimization\",\n \"Based on the optimal solution of digital precoder given in the previous section, we optimize the analog precoder and processor under constant-modulus constraints.\",\n \"In the following, the optimal structure of the analog transceiver is first derived.\",\n \"Different from existing works, we show that the analog precoder and processor design can be decoupled by using the optimal transceiver structure.\",\n \"This optimal structure greatly simplifies the involved analog transceiver design.\",\n \"For the analog transceiver optimization in (REF ) and using (REF ), we have the following matrix-monotonic optimization problem $& \\\\ \\\\ \\\\max _{{\\\\bf {F}}_{\\\\rm {A}},{\\\\bf {G}}_{\\\\rm {A}}} \\\\ \\\\ {\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}}^{\\\\rm {H}}{\\\\Pi }_{\\\\rm {R}}^{\\\\rm {H}}{\\\\bf {H}}^{\\\\rm {H}}{\\\\bf {R}}_{\\\\rm {n}}^{-1/2}{\\\\Pi }_{\\\\rm {L}}^{\\\\rm {H}}{\\\\Pi }_{\\\\rm {L}} {\\\\bf {R}}_{\\\\rm {n}}^{-1/2}{\\\\bf {H}}{\\\\Pi }_{\\\\rm {R}}{\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}} \\\\nonumber \\\\\\\\& \\\\ \\\\ \\\\ {\\\\rm {s.t.}}\",\n \"\\\\ \\\\ \\\\ \\\\ \\\\ \\\\ {\\\\bf {F}}_{\\\\rm {A}} \\\\in \\\\mathcal {F}, \\\\ \\\\ {\\\\bf {G}}_{\\\\rm {A}} \\\\in \\\\mathcal {G}.$ Denote the SVDs $ {\\\\bf {R}}_{\\\\rm {n}}^{-1/2}{\\\\bf {H}}& \\\\triangleq {\\\\bf {U}}_{{\\\\mathcal {H}}}{\\\\Lambda }_{\\\\mathcal {H}}{\\\\bf {V}}_{\\\\mathcal {H}}^{\\\\rm {H}},\\\\\\\\{\\\\bf {R}}_{\\\\rm {n}}^{1/2} {\\\\bf {G}}_{\\\\rm {A}}^{\\\\rm {H}}& \\\\triangleq {\\\\bf {U}}_{{\\\\bf {R}}{\\\\bf {G}}} {\\\\Lambda }_{{\\\\bf {R}}{\\\\bf {G}}}{\\\\bf {V}}_{{\\\\bf {R}}{\\\\bf {G}}}^{\\\\rm {H}}.$ In Appendix , we prove the following conclusion on the optimal structure of ${\\\\bf {F}}_{\\\\rm {A}}$ and ${\\\\bf {G}}_{\\\\rm {A}}$ .\",\n \"Conclusion 3: Let the SVD of ${\\\\bf {F}}_{\\\\rm {A}}$ be ${\\\\bf {F}}_{\\\\rm {A}} \\\\triangleq {\\\\bf {U}}_{{\\\\bf {F}}_{\\\\rm {A}}}{\\\\Lambda }_{{\\\\bf {F}}_{\\\\rm {A}}}{\\\\bf {V}}_{{\\\\bf {F}}_{\\\\rm {A}}}^{\\\\rm {H}}.$ The singular values in ${\\\\Lambda }_{{\\\\bf {F}}_{\\\\rm {A}}}$ do not affect the objective function in (REF ), and the unitary matrix ${\\\\bf {U}}_{{\\\\bf {F}}_{\\\\rm {A}}}$ for the optimal ${\\\\bf {F}}_{\\\\rm {A}}$ satisfies $ [{\\\\bf {U}}_{{\\\\bf {F}}_{\\\\rm {A}}}]_{:,1:L}^{\\\\rm {opt}}={\\\\rm {arg\\\\,max}}\\\\lbrace \\\\Vert [{\\\\bf {V}}_{\\\\mathcal {H}}]_{:,1:L} [{\\\\bf {U}}_{{\\\\bf {F}}_{\\\\rm {A}}}]_{:,1:L}^{\\\\rm {H}}\\\\Vert _{\\\\rm {F}}^2\\\\rbrace .$ On the other hand, denote the SVD of $ {\\\\bf {R}}_{\\\\rm {n}}^{1/2} {\\\\bf {G}}_{\\\\rm {A}}^{\\\\rm {H}} $ as ${\\\\bf {R}}_{\\\\rm {n}}^{1/2} {\\\\bf {G}}_{\\\\rm {A}}^{\\\\rm {H}} \\\\triangleq {\\\\bf {U}}_{{\\\\bf {R}}{\\\\bf {G}}} {\\\\Lambda }_{{\\\\bf {R}}{\\\\bf {G}}}{\\\\bf {V}}_{{\\\\bf {R}}{\\\\bf {G}}}^{\\\\rm {H}}.$ The singular values in ${\\\\Lambda }_{{\\\\bf {R}}{\\\\bf {G}}}$ do not affect the objective in (REF ), and the unitary matrix ${\\\\bf {U}}_{{\\\\bf {R}}{\\\\bf {G}}}$ for the optimal ${\\\\bf {G}}_{\\\\rm {A}}$ satisfies $[{\\\\bf {U}}_{{\\\\bf {R}}{\\\\bf {G}}}]_{:,1:L}^{\\\\rm {opt}}={\\\\rm {arg\\\\,max}}\\\\lbrace \\\\Vert [{\\\\bf {U}}_{\\\\mathcal {H}}]_{:,1:L} [{\\\\bf {U}}_{{\\\\bf {R}}{\\\\bf {G}}}]_{:,1:L}^{\\\\rm {H}}\\\\Vert _{\\\\rm {F}}^2\\\\rbrace .$ Based on the optimal structure given in Conclusion 3, in the following two kinds of algorithms are proposed to compute the analog precoder and processor.\",\n \"The first one is based on phase projection, which provides better performance while the second one based on a heuristic random selection, is with low complexity.\"\n ],\n [\n \"Phase Projection Based Algorithm\",\n \"Analog Precoder Design From Conclusion 3, the optimal analog precoder should select the first $L$ -best eigenchannels.\",\n \"It is challenging to directly optmize ${\\\\bf {F}}_{\\\\rm {A}}$ based on (REF ) because of the SVD of a constant-modulus matrix.\",\n \"Alternatively, we resort to finding a matrix in the constant-modulus space with the minimum distance to the space spanned by $[{\\\\bf {V}}_{\\\\mathcal {H}}]_{:,1:L} $ .\",\n \"Then, the corresponding optimization problem of analog precoder design can be formulated as $ &\\\\min _{{\\\\bf {F}}_{\\\\rm {A}},{\\\\mathrm {\\\\mathbf {\\\\Lambda }_A}},{\\\\bf {Q}}_{\\\\rm {A}}} \\\\ \\\\ \\\\Vert [{\\\\bf {V}}_{\\\\mathcal {H}}]_{:,1:L} {\\\\mathrm {\\\\mathbf {\\\\Lambda }_A}}{\\\\bf {Q}}_{\\\\rm {A}}-{\\\\bf {F}}_{\\\\rm {A}}\\\\Vert _{\\\\rm {F}}^2\\\\nonumber \\\\\\\\& \\\\ \\\\ \\\\ \\\\ {\\\\rm {s.t.}}\",\n \"\\\\ \\\\ \\\\ \\\\ \\\\ \\\\ {\\\\bf {Q}}_{\\\\rm {A}}{\\\\bf {Q}}_{\\\\rm {A}}^{\\\\rm {H}}={\\\\bf {I}}\\\\nonumber \\\\\\\\& \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ {\\\\bf {F}}_{\\\\rm {A}} \\\\in \\\\mathcal {F}.$ Different from the existing work [24], the diagonal matrix $ \\\\mathrm {\\\\mathbf {\\\\Lambda }_A} $ and the unitary matrix $ \\\\mathrm {\\\\mathbf {Q}_A} $ in our work are jointly optimized to make $ {\\\\bf F}_{\\\\rm A} $ as close as possible in the space spanned by $ [{\\\\bf {V}}_{\\\\mathcal {H}}]_{:,1:L} $ in terms of Frobenius norm.\",\n \"As there is no constraint on the diagonal matrix $ \\\\mathrm {\\\\mathbf {\\\\Lambda }_A} $ , given matrices $ {\\\\bf Q}_{\\\\rm A} $ and $ {\\\\bf F}_{\\\\rm A} $ , the optimal $ \\\\mathbf {\\\\Lambda }_{\\\\rm A} $ is $\\\\mathbf {\\\\Lambda }_{\\\\rm {A}}^{\\\\rm {opt}}= \\\\mathrm {diag} \\\\Big \\\\lbrace \\\\Re \\\\big ( {\\\\mathrm {\\\\mathbf {Q}_A}} {\\\\mathrm {\\\\mathbf {F}_A^H}} [{\\\\bf {V}}_{\\\\mathcal {H}}]_{:,1:L} \\\\big ) \\\\Big \\\\rbrace .$ Then we rewrite the objective function in (REF ) as $&\\\\Vert [{\\\\bf {V}}_{\\\\mathcal {H}}]_{:,1:L} \\\\mathbf {\\\\Lambda }_{\\\\rm {A}}{\\\\bf {Q}}_{\\\\rm {A}}-{\\\\bf {F}}_{\\\\rm {A}}\\\\Vert _{\\\\rm {F}}^2 \\\\nonumber \\\\\\\\=&{\\\\rm {Tr}}([{\\\\bf {V}}_{\\\\mathcal {H}}]_{:,1:L}\\\\mathbf {\\\\Lambda }_{\\\\rm {A}}^{\\\\rm {opt}}(\\\\mathbf {\\\\Lambda }_{\\\\rm {A}}^{\\\\rm {opt}})^{\\\\rm {H}}[{\\\\bf {V}}_{\\\\mathcal {H}}]_{:,1:L}^{\\\\rm {H}})\\\\nonumber \\\\\\\\&+{\\\\rm {Tr}}({\\\\bf {F}}_{\\\\rm {A}}{\\\\bf {F}}_{\\\\rm {A}}^{\\\\rm {H}})-2\\\\Re \\\\lbrace {\\\\rm {Tr}}({\\\\bf {F}}_{\\\\rm {A}}^{\\\\rm {H}}[{\\\\bf {V}}_{\\\\mathcal {H}}]_{:,1:L} \\\\mathbf {\\\\Lambda }_{\\\\rm {A}}^{\\\\rm {opt}}{\\\\bf {Q}}_{\\\\rm {A}})\\\\rbrace .$ To minimize (REF ) given ${\\\\Lambda }_{\\\\rm {A}}$ and ${\\\\bf {F}}_{\\\\rm {A}}$ , the term $\\\\Re \\\\lbrace {\\\\rm {Tr}}({\\\\bf {F}}_{\\\\rm {A}}^{\\\\rm {H}}[{\\\\bf {V}}_{\\\\mathcal {H}}]_{:,1:L} {\\\\mathrm {\\\\mathbf {\\\\Lambda }_A}^{\\\\rm {opt}}}{\\\\bf {Q}}_{\\\\rm {A}})\\\\rbrace $ should be maximized.\",\n \"By applying the matrix inequality [41], the optimal $ {\\\\bf Q}_{\\\\rm A} $ is $\\\\mathbf {Q}_{\\\\mathrm {A}}^{\\\\mathrm {opt}} = \\\\mathbf {V}_{\\\\mathrm {Q}} \\\\mathbf {U}_{\\\\mathrm {Q}}^\\\\mathrm {H},$ where $\\\\mathbf {V}_{\\\\mathrm {Q}}$ and $\\\\mathbf {U}_{\\\\mathrm {Q}}$ are defined based on the following SVD ${\\\\bf {F}}_{\\\\rm {A}}^\\\\mathrm {H} \\\\, [{\\\\bf {V}}_{\\\\mathcal {H}}]_{:,1:L} {\\\\mathbf {\\\\Lambda }_\\\\mathrm {A}} = \\\\mathbf {U}_{\\\\mathrm {Q}} \\\\mathbf {\\\\Sigma }_{\\\\mathrm {Q}} \\\\mathbf {V}_{\\\\mathrm {Q}}^\\\\mathrm {H}.$ Now that for given $ {\\\\bf Q}_{\\\\rm A} $ and $\\\\mathbf {\\\\Lambda }_\\\\mathrm {A} $ , the optimal analog precoder $ \\\\mathrm {\\\\mathbf {F}_A} $ is [20] ${\\\\bf F}_{\\\\rm A}^{\\\\rm opt} = {\\\\bf P}_{\\\\mathcal {F}} \\\\left( [ {\\\\bf {V}}_{\\\\mathcal {H}}]_{:,1:L} {\\\\mathrm {\\\\mathbf {\\\\Lambda }_A}} {\\\\bf {Q}}_{\\\\rm {A}} \\\\right),$ where the phase projection ${\\\\bf P}_{\\\\mathcal {F}}( \\\\mathrm {\\\\mathbf {A}} )$ is defined as $\\\\left[ {\\\\bf P}_{\\\\mathcal {F}}( \\\\mathrm {\\\\mathbf {A}} ) \\\\right]_{i,j} ={\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\left[ \\\\mathrm {\\\\mathbf {A}} \\\\right]_{i,j} / | \\\\left[ \\\\mathrm {\\\\mathbf {A}} \\\\right]_{i,j} |, & \\\\text{if} \\\\ \\\\left[ \\\\mathrm {\\\\mathbf {A}} \\\\right]_{i,j} \\\\ne 0 \\\\\\\\1, & \\\\text{otherwise}.\\\\end{array}\\\\right.\",\n \"}$ Using (REF ), (REF ) and (REF ), the phased projection based analog precoder optimization is proposed in Algorithm REF .\",\n \"[t] Analog Precoder Design [1] Left singular matrix of equivalent channel $ [{\\\\bf {V}}_{\\\\mathcal {H}}]_{:,1:L} $ , algorithm threshold $ \\\\zeta $ .\",\n \"Initialize ${\\\\bf {F}}_{\\\\rm {A}}$ with $\\\\mathbf {P}_{\\\\mathcal {F}} \\\\lbrace [{\\\\bf {V}}_{\\\\mathcal {H}}]_{:,1:L} \\\\rbrace $ .\",\n \"the decrement of the objective function in (REF ) is larger than $ \\\\zeta $ Calculate ${\\\\mathbf {\\\\Lambda }_{\\\\rm A}}$ based on (REF ).\",\n \"Calculate $\\\\mathbf {Q}_{\\\\mathrm {A}} $ based on (REF ).\",\n \"Calculate $\\\\mathbf {F}_{\\\\mathrm {A}}$ based on (REF ).\",\n \"Update the decrement value of the objective function in (REF ).\",\n \"$ \\\\mathbf {F}_{\\\\mathrm {A}} $ .\",\n \"Analog Processor Design Based on Conclusion 3, the optimal structure of analog processor is similar to the analog precoder, but a bit complicated in that the noise variance is tangled in the analog processor formulation.\",\n \"In this case, the left singular matrix of $ {\\\\bf {R}}_{\\\\rm {n}}^{1/2} {\\\\bf {G}}_{\\\\rm {A}}^{\\\\rm {H}} $ is required to match the first $ L $ column of left singular matrix of effective channel, i.e., $ [{\\\\bf {U}}_{\\\\mathcal {H}}]_{:,1:L} $ .\",\n \"[ht] Iterative Analog Processor Design [1] The matrix $ [{\\\\bf {U}}_{\\\\mathcal {H}}]_{:,1:L} $ , the unitary matrix $ \\\\mathbf {Q}_{\\\\mathrm {G}} $ , the diagonal matrix $ \\\\mathbf {\\\\Lambda }_{\\\\mathrm {G}} $ , controlling factor $ \\\\eta $ , and convergent threshold $ \\\\upsilon $ .\",\n \"Compute $ \\\\mathbf {W} $ in (REF ).\",\n \"Initialize constant-modulus processor as ${\\\\mathbf {r}}_{(0)} = \\\\frac{\\\\sqrt{2}}{2}\\\\mathbf {1} $ .\",\n \"The decrement of the objective function in (REF ) is larger than $ \\\\upsilon $ Calculate $ \\\\mathbf {P} $ using (REF ) based on $ \\\\mathbf {G}_{\\\\mathrm {A}} $ computed in the previous iteration.\",\n \"Find out the optimal solution of (REF ) based on (REF ).\",\n \"Update the decrement of the objective function in (REF ).\",\n \"Construct $ \\\\mathbf {G}_{\\\\mathrm {A}} $ based on the optimal solution in Step 5.\",\n \"$ {\\\\mathbf {G}}_{\\\\mathrm {A}} $ .\",\n \"Thus, analogous to the analog precoder design in (REF ), we have the following optimization problem $\\\\min _{{\\\\bf {G}}_{\\\\rm {A}},{\\\\mathrm {\\\\mathbf {\\\\Lambda }_G}},{\\\\bf {Q}}_{\\\\rm {G}}} \\\\;\\\\; & \\\\big \\\\Vert [{\\\\bf {U}}_{\\\\mathcal {H}}]_{:,1:L}{\\\\mathbf {\\\\Lambda }_\\\\mathrm {G}} {\\\\bf {Q}}_{\\\\rm {G}} - {\\\\bf {R}}_{\\\\rm {n}}^{1/2} {\\\\bf {G}}_{\\\\rm {A}}^\\\\mathrm {H} \\\\big \\\\Vert _{\\\\rm {F}}^2\\\\nonumber \\\\\\\\{\\\\rm {s.t.}}\",\n \"\\\\quad \\\\;\\\\; & {\\\\bf {Q}}_{\\\\rm {G}}{\\\\bf {Q}}_{\\\\rm {G}}^{\\\\rm {H}} = {\\\\bf {I}}\\\\nonumber \\\\\\\\& {\\\\bf {G}}_{\\\\rm {A}} \\\\in \\\\mathcal {G}.$ The optimization of unitary matrix $ {\\\\bf {Q}}_{\\\\rm {G}} $ and diagonal matrix $ {\\\\mathbf {\\\\Lambda }_\\\\mathrm {G}} $ in (REF ) is exactly the same as that for the analog precoder optimization.\",\n \"However, the optimization of the analog processor, ${\\\\bf {G}}_{\\\\rm {A}}$ , in (REF ) is different.\",\n \"When noises from different antennas are correlated the analog processor design is more challenging than the analog precoder design.\",\n \"In order to overcome this challenge, problem (REF ) is relaxed to minimize an upper bound of the original objective function.\",\n \"Applying $& \\\\big \\\\Vert [{\\\\bf {U}}_{\\\\mathcal {H}}]_{:,1:L}{\\\\mathbf {\\\\Lambda }_\\\\mathrm {G}} {\\\\bf {Q}}_{\\\\rm {G}} - {\\\\bf {R}}_{\\\\rm {n}}^{1/2} {\\\\bf {G}}_{\\\\rm {A}}^\\\\mathrm {H} \\\\big \\\\Vert _{\\\\mathrm {F}}^2 \\\\\\\\\\\\le \\\\;\\\\; & \\\\lambda _{\\\\mathrm {max}}( \\\\mathbf {R}_{\\\\mathrm {n}} )\\\\big \\\\Vert {\\\\bf {R}}_{\\\\rm {n}}^{-1/2} [{\\\\bf {U}}_{\\\\mathcal {H}}]_{:,1:L}{\\\\mathbf {\\\\Lambda }_\\\\mathrm {G}} {\\\\bf {Q}}_{\\\\rm {G}} - {\\\\bf {G}}_{\\\\rm {A}}^\\\\mathrm {H} \\\\big \\\\Vert _{\\\\mathrm {F}}^2,$ the objective function of (REF ) is relaxed with $ \\\\big \\\\Vert {\\\\bf {R}}_{\\\\rm {n}}^{-1/2} [{\\\\bf {U}}_{\\\\mathcal {H}}]_{:,1:L} {\\\\mathbf {\\\\Lambda }_\\\\mathrm {G}} {\\\\bf {Q}}_{\\\\rm {G}} - {\\\\bf {G}}_{\\\\rm {A}}^\\\\mathrm {H} \\\\big \\\\Vert _{\\\\mathrm {F}}^2$ .\",\n \"Note that solving (REF ) is the same as that for the analog precoder design.\",\n \"It is obvious that this relaxation is tight when $ {\\\\bf R}_{\\\\rm n} = \\\\sigma _n^2 {\\\\bf I} $ .\",\n \"This relaxation may result in some performance loss.\",\n \"Inspired by the work in [42], an iterative algorithm is also proposed to compute ${\\\\bf {G}}_{\\\\rm {A}}$ .\",\n \"The constant modulus constraints is asymptotically satisfied via iteratively updating an additional constraint.\",\n \"This iterative algorithm is given in Algorithm REF , and detailed derivation is given in Appendix .\"\n ],\n [\n \"Random Algorithm\",\n \"The proposed phase projection based analog transceiver design suffers from high computation complexity.\",\n \"This may prohibit the proposed analog transceiver design from practical implementation.\",\n \"In order to reduce complexity, we can randomly generate analog precoder and processor matrices to avoid the heavy computations involved in the phase projection based algorithms.\",\n \"In this random algorithm, we randomly select multiple matrices in the column or row space of ${\\\\bf H}^{\\\\rm {H}}$ and use their phase projections as the candidates for the analog transceiver design.\",\n \"Then the best candidate matrix is chosen according to some criterion.\",\n \"Specifically, the random algorithm consists of three steps.\",\n \"First, a series of parameter matrices, denoted by $ \\\\lbrace \\\\mathbf {R}_k\\\\rbrace $ and $\\\\lbrace {\\\\mathbf {T}}_k\\\\rbrace $ , are generated, whose elements are randomly generated following a specific distribution e.g., uniform distribution or Gaussian distribution.\",\n \"Secondly, a series of candidate analog precoder and processor matrices are computed based on the parameter matrices.\",\n \"Specifically, based on the parameter matrices and after computing ${\\\\bf {H}}^{\\\\rm {H}}{\\\\bf {R}}_k$ and ${\\\\mathbf {T}}_k{\\\\bf {H}}^{\\\\rm {H}}$ , the constant-modulus candidate matrices are obtained using their phase projections.\",\n \"Finally, the analog precoder and processor are chosen from these candidates according to the determinant of a certain matrix version SNR matrix.\",\n \"The procedure is detailed in Algorithm REF .\",\n \"[t] Random Algorithm for Analog Transceiver Design [1] The number of transmitter antennas $ N $ , number of RF-Chain $ L $ , selection number $ K $ , probability density function $ f_\\\\mathrm {trans} (x) $ , and ${\\\\bf {H}}$ Generate $ K $ parameter matrices, $ {\\\\bf R}_{1},\\\\ldots ,{\\\\bf R}_{K} \\\\in \\\\mathbb {C}^{ M \\\\times L} $ , whose elements are randomly generated based on $ f_\\\\mathrm {trans} (x) $ .\",\n \"Rotate the channel as $ \\\\mathrm {\\\\mathbf {H}^H} {\\\\bf R}_{k} $ .\",\n \"Calculate ${\\\\bf F}_{k} = \\\\mathrm {\\\\mathbf {P}}_{\\\\mathcal {F}} \\\\left( \\\\mathrm {\\\\mathbf {H}^H} {\\\\bf R}_{k} \\\\right) $ .\",\n \"${\\\\bf {F}}_{\\\\rm {max}}={\\\\arg \\\\max }_{{\\\\bf {F}}_i} \\\\left|{ \\\\bf F}_{i}^{\\\\mathrm {H}} {\\\\bf H}^{\\\\mathrm {H}} \\\\mathrm { \\\\mathbf {R}_n^{-1} } {\\\\bf H}{\\\\bf F}_{i} \\\\right|$ .\",\n \"Generate $ K $ parameter matrices $ {\\\\bf T}_{1},\\\\ldots ,{\\\\bf T}_{K} \\\\in \\\\mathbb {C}^{ L \\\\times N } $ randomly based on $ f_\\\\mathrm {trans} (x) $ .\",\n \"Rotate the channel as $ {\\\\bf T}_{k} \\\\mathrm {\\\\mathbf {H}^H} $ .\",\n \"Calculate $ {\\\\bf G}_{k} = \\\\mathrm {\\\\mathbf {P}}_{\\\\mathcal {F}} \\\\left( {\\\\bf T}_{k} \\\\mathrm {\\\\mathbf {H}^H} \\\\right) $ .\",\n \"$ {\\\\bf {G}}_{\\\\max }={\\\\arg \\\\max }_{{\\\\bf {G}}_i}\\\\bigl \\\\vert {\\\\bf G}_{i} \\\\mathrm { \\\\mathbf {R}_n^{-1/2} } {\\\\bf H} {\\\\bf H}^{\\\\mathrm {H}} \\\\mathrm { \\\\mathbf {R}_n^{-1/2} } {\\\\bf G}_{i}^{\\\\mathrm {H}} \\\\bigr \\\\vert $ .\",\n \"$ {\\\\bf F}_\\\\mathrm {A} = {\\\\bf F}_\\\\mathrm {max} $ , $ {\\\\bf G}_\\\\mathrm {A} = {\\\\bf G}_\\\\mathrm {max} $\"\n ],\n [\n \"Simulation Results\",\n \"In this part, some numerical results are provided to assess the performance of the proposed hybrid transceiver design.\",\n \"As our algorithms are applicable to any frequency band, both microwave frequency band and mmWave frequency band are simulated.\",\n \"In addition, quantization of phase shifters is also taken into account.\",\n \"Figure: Spectral efficiency comparison of 5 different hybrid transceiver design methods.\",\n \"Here, 32×16 32 \\\\times 16 mmWave channel model is adopted in the simulation.\",\n \"Both the transmitter and receiver are equipped with L=4 L = 4 RF-chains and the system is conveying D=4 D = 4 data streams.More specifically, both mmWave channel model $ {\\\\mathrm {\\\\mathbf {H}_{m}}} $ and classic Rayleigh channel model $ {\\\\mathrm {\\\\mathbf {H}_{r}}} $ are tested.\",\n \"For mmWave channel, $ {\\\\mathrm {\\\\mathbf {H}_{m}}} $ , the uniformed linear arrays (ULA) is adopted.\",\n \"Unless otherwise specified, it is assumed that 1) the mmWave channel has $ N_{\\\\rm cl} = 2 $ clusters with each of them containing $ N_\\\\mathrm {path} = 5 $ paths; 2) the azimuth angle spread of transmitter is restricted to $ 7.5^{\\\\circ } $ at the mean of azimuth angle $ \\\\hat{\\\\theta } = 45^{\\\\circ } $ , and the receiver is omni-directional; 3) the path loss factors obey the standard Gaussian distribution; 4) the inter-antenna spacing $ d $ equals to half-wavelength.\",\n \"The channel is normalized to meet $ \\\\mathbb {E}{ \\\\left\\\\lbrace \\\\Vert {\\\\bf H}_{\\\\rm m} \\\\Vert _{\\\\rm {F}}^2 \\\\right\\\\rbrace } = NM $ .\",\n \"For the random phase algorithm, we set $ K = 10 $ , which means that the best analog precoder and processor are selected from 10 candidates and uniform distribution is utilized, i.e., $ f_{\\\\mathrm { trans }}(x) = 1$ for $0 \\\\le x \\\\le 1 $ .\",\n \"We average the result over 2,000 independent trials.\",\n \"The transmitting power is denoted as $ P_{\\\\mathrm {Tx}} $ .\",\n \"OMP and MaGiQ algorithms refers to the corresponding algorithms in [15] and [24], respectively.\",\n \"The analog precoder and processor for the direct phase algorithm are obtained by phase projection.\",\n \"Figure: Spectral efficiency comparison of 5 different hybrid transceiver design methods.\",\n \"The 32×16 32 \\\\times 16 mmWave channel model, which involves N cl =3 N_{\\\\rm cl} = 3 clusters with N path =5 N_{\\\\rm path} = 5 multipath at each cluster, is adopted in the simulation.\",\n \"Both the transmitter and receiver are equipped with L=6 L = 6 RF-chains and the system is conveying D=4 D = 4 data streams.Figure: Spectral efficiency comparison of 5 different hybrid transceiver design methods.\",\n \"The 32×16 32 \\\\times 16 Rayleigh channel model is adopted in the simulation.\",\n \"Both the transmitter and receiver are equipped with L=6 L = 6 RF-chains and the system is conveying D=4 D = 4 data streams.Fig.\",\n \"REF demonstrates spectral efficiency versus the transmit power for different algorithms, where the hybrid transceiver is with $ N = 32 $ transmit antennas, $ M = 16 $ receive antennas, and 4 data streams.\",\n \"Both the transmitter and receiver are equipped with $ L = 4 $ RF-chains.\",\n \"From Fig.\",\n \"REF , the proposed phased projection based hybrid transceiver design outperforms the other hybrid transceiver design algorithms.\",\n \"The performance of the proposed algorithm is very close to the full digital one.\",\n \"Fig.\",\n \"REF shows the performance of the hybrid transceiver design with 6 RF-chains for channel with $ N_{\\\\rm cl} = 3 $ clusters, each with $ N_{\\\\rm path} = 5 $ paths.\",\n \"From this figure, the proposed phase projection algorithm works well for different numbers of RF-chains and performs very close to the full-digital one and it is better than that of other hybrid transceiver designs.\",\n \"It is worth noting that the direct phase projection method performs Figure: Spectral efficiency comparison of 5 different hybrid transceiver design methods concerning 2 bits quantization of phase shifters.\",\n \"The 32×16 32 \\\\times 16 mmWave channel model is adopted in the simulation.\",\n \"Both the transmitter and receiver are equipped with L=4 L = 4 RF-chains and the system is conveying D=4 D = 4 data streams.even better than OMP and MaGiQ.\",\n \"This is because the error bound of the method decreases when the number of RF chains increases [20].\",\n \"However, as the limitation that the number of data streams should be equal to that of RF-chains [24] is not satisfied in this case, MaGiQ algorithm is the worst at high SNR.\",\n \"The following simulations focus on Rayleigh channels at micro-wave bands.\",\n \"Under this circumstance, the $ 32 \\\\times 16 $ system is adopted with $ L = 6 $ RF-chains are in use transferring $ D = 4 $ data streams.\",\n \"After performing extensive simulation compared with randomly generated codebooks or DFT codebook, we found that the codebook constructed by the phase projection, i.e., $ \\\\mathcal {C} = {\\\\bf P}_{\\\\mathcal {F}}( \\\\mathrm {\\\\mathbf {H}}) $ , has much better performance.\",\n \"This codebook is used for performance comparison in the following simulation.\",\n \"Fig.\",\n \"REF compares the performance for the different algorithms under Rayleigh channels.\",\n \"Figure: Spectral efficiency of random algorithm under mmWave channel.\",\n \"L=6 L = 6 RF-chains are assumed to be equipped both transmitter and receiver.\",\n \"Both 32×16 32 \\\\times 16 and 64×16 64 \\\\times 16 system are involved during the simulation.\",\n \"The number of data streams is set to be D=4 D = 4 .From the figure, the proposed algorithm obtains nearly the optimal performance as the full-digital one.\",\n \"The proposed algorithm performs better than MaGiQ algorithm in [24].\",\n \"Moreover, it is worth noting that even with the carefully chosen codebook, the OMP algorithm exhibits a large performance gap compared with the full-digital one, which indicates that the OMP algorithm is not suitable for micro-wave frequency bands.\",\n \"As the practical analog phase shifters are often implemented by digital controller with finite resolution, Fig.\",\n \"REF compares the performance of different hybrid transceiver designs for $ 32 \\\\times 16 $ mmWave channel when phase quantization is taken into account.\",\n \"Each hybrid transceiver design only uses the phase shifter with 2-bit resolution and $ L = 4 $ .\",\n \"From the figure the performance of the proposed hybrid transceiver design still outperforms other hybrid transceiver designs with finite resolution phase shifters.\",\n \"In Fig.\",\n \"REF , both $ 32 \\\\times 16 $ and $ 64 \\\\times 16 $ mmWave channels are used to assess the performance.\",\n \"In this case, the number of RF-chains is 6.\",\n \"From Fig.\",\n \"REF , with the same number of transmit antennas, the random algorithm is worse than that of the phase projection based algorithm.\",\n \"Although the random algorithm suffers nearly $ 5\\\\,{\\\\rm {dB}} $ performance loss comparing with the full-digital one, by involving more antennas at base station, e.g., $ N = 64 $ , the performance of random algorithm will be comparable to the performance corresponding to the full-digital transmitter with 32 antennas.\",\n \"This implies that we can obtain appropriate performance using the low complexity random algorithm by simply increasing the number of transmit antennas.\",\n \"Because of its low complexity, the random algorithm will be a friendly algorithm for hardware realization.\",\n \"Fig.\",\n \"REF shows the BER performances of different kinds of hybrid MIMO transceiver designs for $ 32 \\\\times 16 $ Raleigh channel with 4 RF chains.\",\n \"In this case, there are 4 data streams and 16-QAM is used.\",\n \"From this figure, at high SNR, the BER performance of the hybrid nonlinear transceiver design is much better than that of the hybrid linear transceiver design.\",\n \"Furthermore, the hybrid nonlinear transceivers with THP and DFD have almost the same BER performance because of the duality between precoder design and processor design.\",\n \"Figure: BERs of the linear hybrid transceiver for capacity maximization, nonlinear transceiver with DFD and nonlinear transceiver THP.\",\n \"The 32×16 32 \\\\times 16 Rayleigh channel model is involved in the simulation.\",\n \"Both transmitter and receiver are equipped with L=4 L = 4 RF-chains transferring D=4 D = 4 data streams simultaneously.\"\n ],\n [\n \"Conclusions\",\n \"In this paper, we have investigated the hybrid digital and analog transceiver design for MIMO system based on matrix-monotonic optimization theory.\",\n \"We have proposed a unified framework for both linear and nonlinear transceivers.\",\n \"Based on the matrix-monotonic optimization theory, the optimal transceiver structure for various MIMO transceivers has been derived, from which the function of analog transceiver part can be regarded as eigenchannel selection.\",\n \"Using the derived optimal structure, effective algorithms have been proposed considering the constant-modulus constraint.\",\n \"Finally, it is shown that the proposed algorithms outperform existing hybrid transceiver designs.\"\n ],\n [\n \"Preliminary Definition of Majorization Theory\",\n \"In this appendix, some fundamental functions in majorization theory are defined for the convenience of unified framework analysis.\",\n \"These definitions are also given in [38] and in order to make the paper self-contained, they are also given here.\",\n \"Definition 1 [41]: For a $K\\\\times 1$ vector $ {\\\\bf x} \\\\in \\\\mathbb {R}^{K} $ , the $ \\\\ell $ th largest element of $ {\\\\bf x} $ is denoted as $ {x}_{[\\\\ell ]} $ , and in other words, we have $ {x}_{[1]} \\\\ge {x}_{[2]} \\\\ge \\\\cdots \\\\ge {x}_{[K]} $ .\",\n \"Based on this definition, for two vectors $ {\\\\bf x}, {\\\\bf y} \\\\in \\\\mathbb {R}^{K} $ , it state that $ {\\\\bf y} $ majorizes $ {\\\\bf x} $ additively, denoted by $ {\\\\bf x} \\\\prec _{+} {\\\\bf y} $ , if and only if the following properties are satisfied $\\\\sum _{k = 1}^{p} {x}_{[k]} \\\\le \\\\sum _{k = 1}^{p} {y}_{[k]}, \\\\; p = 1,2,\\\\ldots , K-1 \\\\; \\\\text{and} \\\\; \\\\sum _{k = 1}^{K} {x}_{[k]} = \\\\sum _{k = 1}^{K} {y}_{[k]}.$ Definition 2 [41]: A function $ f(\\\\cdot ) $ is Schur-convex if and only if it satisfies the following property ${\\\\bf x} \\\\prec _{+} {\\\\bf y} \\\\, \\\\Longrightarrow \\\\, f( {\\\\bf x} ) \\\\le f( {\\\\bf y} ).$ On the other hand, a function $ f(\\\\cdot ) $ is additively Schur-concave if $ -f(\\\\cdot ) $ is additively Schur-convex.\",\n \"Definition 3 [38]: For two $K\\\\times 1$ vectors $ {\\\\bf x}, {\\\\bf y} \\\\in \\\\mathbb {R}^{K} $ with nonnegative elements, it states that the vector $ {\\\\bf y} $ majorizes vector $ {\\\\bf x} $ multiplicatively, i.e., $ {\\\\bf x} \\\\prec _{\\\\times } {\\\\bf y} $ , if and only if the following properties are satisfied $\\\\prod _{k = 1}^{p} {x}_{[k]} \\\\le \\\\prod _{k = 1}^{p} {y}_{[k]}, \\\\; p = 1,2,\\\\ldots , K-1 \\\\; \\\\text{and} \\\\; \\\\prod _{k = 1}^{K} {x}_{[k]} = \\\\prod _{k = 1}^{K} {y}_{[k]}.$ Definition 4 [38]: A function $ f(\\\\cdot ) $ is multiplicatively Schur-convex if and only if it satisfies the following property ${\\\\bf x} \\\\prec _{+} {\\\\bf y} \\\\, \\\\Longrightarrow \\\\, f( {\\\\bf x} ) \\\\le f( {\\\\bf y} ).$ On the other hand, a function $ f(\\\\cdot ) $ is multiplicatively Schur-concave if $ -f(\\\\cdot ) $ is multiplicatively Schur-convex.\"\n ],\n [\n \"The optimal ${\\\\bf {B}}$\",\n \"Note that this optimal ${\\\\bf {B}}$ for nonlinear transceiver was previously obtained in [36] when function belongs to the family of multiplicatively Schur-concave/convex functions defined in Appendix A.\",\n \"The following presents a slightly different proof of the optimal B, which generalizes the result to the case with an arbitrary monotone increasing function $f(\\\\cdot )$ .\",\n \"Here, the function f operates only on the diagonal elements of ${\\\\Phi }_{\\\\rm {MSE}}({\\\\bf {G}}_{\\\\rm {A}}, {\\\\bf {F}}_{\\\\rm {A}}, {\\\\bf {F}}_{\\\\rm {D}},{\\\\bf {B}})$ and ${\\\\bf {B}}$ is restricted as a strictly lower triangular matrix which specifies the use of nonlinear transceiver.\",\n \"Based on the Cholesky decomposition ${\\\\Phi }_{\\\\rm {MSE}}^{\\\\rm {L}}({\\\\bf {G}}_{\\\\rm {A}}, {\\\\bf {F}}_{\\\\rm {A}}, {\\\\bf {F}}_{\\\\rm {D}}) ={\\\\bf {L}}{\\\\bf {L}}^{\\\\rm {H}},$ we have $&{\\\\Phi }_{\\\\rm {MSE}}({\\\\bf {G}}_{\\\\rm {A}}, {\\\\bf {F}}_{\\\\rm {A}}, {\\\\bf {F}}_{\\\\rm {D}}, {\\\\bf {B}})\\\\nonumber \\\\\\\\&= ({\\\\bf {I}} + {\\\\bf {B}}) {\\\\Phi }_{\\\\rm {MSE}}^{\\\\rm {L}}({\\\\bf {G}}_{\\\\rm {A}}, {\\\\bf {F}}_{\\\\rm {A}}, {\\\\bf {F}}_{\\\\rm {D}}) ({\\\\bf {I}} + {\\\\bf {B}})^{\\\\rm H}\\\\nonumber \\\\\\\\&=({\\\\bf {I}} + {\\\\bf {B}}){\\\\bf {L}}{\\\\bf {L}}^{\\\\rm {H}}({\\\\bf {I}} + {\\\\bf {B}})^{\\\\rm H},$ based on which the $n$ th diagonal element of ${\\\\Phi }_{\\\\rm {MSE}}({\\\\bf {G}}_{\\\\rm {A}}, {\\\\bf {F}}_{\\\\rm {A}}, {\\\\bf {F}}_{\\\\rm {D}}, {\\\\bf {B}})$ equals $[{\\\\Phi }_{\\\\rm {MSE}}({\\\\bf {G}}_{\\\\rm {A}}, {\\\\bf {F}}_{\\\\rm {A}}, {\\\\bf {F}}_{\\\\rm {D}}, {\\\\bf {B}})]_{n,n}&=[({\\\\bf {I}} + {\\\\bf {B}}){\\\\bf {L}}]_{n,:}[({\\\\bf {I}} + {\\\\bf {B}}){\\\\bf {L}}]_{n,:}^{\\\\rm {H}}\\\\nonumber \\\\\\\\&=\\\\Vert [({\\\\bf {I}} + {\\\\bf {B}}){\\\\bf {L}}]_{n,:}\\\\Vert ^2.$ In addition, as ${\\\\bf {B}}$ is strictly lower triangular it can be calculated that the last element of the vector $[({\\\\bf {I}} + {\\\\bf {B}}){\\\\bf {L}}]_{n,:}$ equals $[{\\\\bf {L}}]_{n,n}$ , i.e., $[({\\\\bf {I}} + {\\\\bf {B}}){\\\\bf {L}}]_{n,:}=[\\\\cdots ,[{\\\\bf {L}}]_{n,n}].$ Therefore, from (REF ) to (REF ) the following relationship holds $[{\\\\Phi }_{\\\\rm {MSE}}({\\\\bf {G}}_{\\\\rm {A}}, {\\\\bf {F}}_{\\\\rm {A}}, {\\\\bf {F}}_{\\\\rm {D}}, {\\\\bf {B}})]_{n,n}&=\\\\Vert [({\\\\bf {I}} + {\\\\bf {B}}){\\\\bf {L}}]_{n,:}\\\\Vert ^2\\\\ge [{\\\\bf {L}}]_{n,n}^2.$ It is obvious that the above inequality can be achieved with equality as $[{\\\\Phi }_{\\\\rm {MSE}}({\\\\bf {G}}_{\\\\rm {A}}, {\\\\bf {F}}_{\\\\rm {A}}, {\\\\bf {F}}_{\\\\rm {D}}, {\\\\bf {B}})]_{n,n}= [{\\\\bf {L}}]_{n,n}^2$ when the following equality holds for different $n$ $({\\\\bf {I}}+{\\\\bf {B}}){\\\\bf {L}} ={\\\\rm {diag}}\\\\lbrace [[{\\\\bf {L}}]_{1,1},\\\\cdots ,[{\\\\bf {L}}]_{L,L}]^{\\\\rm {T}}\\\\rbrace ,$ based on which the optimal ${\\\\bf {B}}$ equals ${\\\\bf {B}}^{\\\\rm {opt}} ={\\\\rm {diag}}\\\\lbrace [[{\\\\bf {L}}]_{1,1},\\\\cdots ,[{\\\\bf {L}}]_{L,L}]^{\\\\rm {T}}\\\\rbrace {\\\\bf {L}}^{-1}-{\\\\bf {I}}.$\"\n ],\n [\n \"Fundamental Matrix Inequalities\",\n \"In this appendix, two fundamental matrix inequalities are given.\",\n \"For two positive semi-definite matrices ${X}$ and ${Y}$ , there are following EVDs defined ${X}&={\\\\bf {U}}_{X} {\\\\Lambda }_{X} {\\\\bf {U}}^{\\\\rm {H}}_{X} \\\\ \\\\ \\\\text{with} \\\\ \\\\ {\\\\Lambda }_{X}\\\\searrow \\\\nonumber \\\\\\\\{Y}&={\\\\bf {U}}_{Y} {\\\\Lambda }_{Y} {\\\\bf {U}}^{\\\\rm {H}}_{Y} \\\\ \\\\ \\\\text{with} \\\\ \\\\ {\\\\Lambda }_{Y}\\\\searrow \\\\nonumber \\\\\\\\{Y}&={\\\\bf {\\\\bar{U}}}_{Y}{\\\\bar{\\\\Lambda }}_{Y} {\\\\bf {\\\\bar{U}}}^{\\\\rm {H}}_{Y} \\\\ \\\\ \\\\text{with} \\\\ \\\\ {\\\\bar{\\\\Lambda }}_{Y} \\\\nearrow .$ For the trace of the two matrices, we have the following fundamental matrix inequalities [40] $&\\\\sum _{i=1}^{N}\\\\lambda _{i-1+N}({X}) \\\\lambda _i({Y})\\\\le {\\\\rm {Tr}}({X}{Y}) \\\\le \\\\sum _{i=1}^{N}\\\\lambda _i({X}) \\\\lambda _i({Y}),$ where $ \\\\lambda _i( { \\\\mathbf {X} } ) $ is the $ i $ th ordered eigenvalue of $ { \\\\mathbf {X} } $ , and the left equality holds when ${\\\\bf {U}}_{X}={\\\\bf {\\\\bar{U}}}_{Y}$ .\",\n \"On the other hand, the right equality holds when ${\\\\bf {U}}_{X}={\\\\bf { U}}_{Y}$ .\"\n ],\n [\n \"Optimal Structure of Analog Transceiver\",\n \"It is worth noting that the nonzero singular values of the matrix, $ {\\\\Pi }_{\\\\rm {R}} = {\\\\bf {F}}_{\\\\rm {A}} ({\\\\bf {F}}_{\\\\rm {A}}^{\\\\rm {H}}{\\\\bf {F}}_{\\\\rm {A}})^{-\\\\frac{1}{2}} $ , are all ones.\",\n \"Similarly for ${\\\\Pi }_{\\\\rm {L}}=({\\\\bf {G}}_{\\\\rm {A}}{\\\\bf {R}}_{\\\\rm {n}}{\\\\bf {G}}_{\\\\rm {A}}^{\\\\rm {H}})^{-1/2}{\\\\bf {G}}_{\\\\rm {A}}{\\\\bf {R}}_{\\\\rm {n}}^{1/2}$ , the nonzero singular values of ${\\\\Pi }_{\\\\rm {L}}$ are all ones.\",\n \"It implies that the singular values of ${\\\\bf {F}}_{\\\\rm {A}}$ and ${\\\\bf {G}}_{\\\\rm {A}}{\\\\bf {R}}_{\\\\rm {n}}^{1/2}$ do not affect the optimization problem.\",\n \"Based on the SVDs ${\\\\bf {R}}_{\\\\rm {n}}^{-1/2}{\\\\bf {H}}={\\\\bf {U}}_{{\\\\mathcal {H}}}{\\\\Lambda }_{\\\\mathcal {H}}{\\\\bf {V}}_{\\\\mathcal {H}}^{\\\\rm {H}}$ , ${\\\\bf {R}}_{\\\\rm {n}}^{1/2} {\\\\bf {G}}_{\\\\rm {A}}^{\\\\rm {H}} = {\\\\bf {U}}_{{\\\\bf {R}}{\\\\bf {G}}} {\\\\Lambda }_{{\\\\bf {R}}{\\\\bf {G}}}{\\\\bf {V}}_{{\\\\bf {R}}{\\\\bf {G}}}^{\\\\rm {H}}$ , and ${\\\\bf {F}}_{\\\\rm {A}}={\\\\bf {U}}_{{\\\\bf {F}}_{\\\\rm {A}}} {\\\\Lambda }_{{\\\\bf {F}}_{\\\\rm {A}}}{\\\\bf {V}}_{{\\\\bf {F}}_{\\\\rm {A}}}^{\\\\rm {H}}$ with the singular values in decreasing order, the objective function in (REF ) becomes ${\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}}^{\\\\rm {H}}{\\\\bf {V}}_{{\\\\bf {F}}_{\\\\rm {A}}}{\\\\Lambda }_{\\\\rm {R}}^{\\\\rm {T}}{\\\\bf {U}}_{{\\\\bf {F}}_{\\\\rm {A}}}^{\\\\rm {H}}{\\\\bf {H}}^{\\\\rm {H}}{\\\\bf {U}}_{\\\\rm {RG}}{\\\\Lambda }_{\\\\rm {L}}^{\\\\rm {T}}{\\\\Lambda }_{\\\\rm {L}}{\\\\bf {U}}_{\\\\rm {RG}}^{\\\\rm {H}}{\\\\bf {H}}{\\\\bf {U}}_{{\\\\bf {F}}_{\\\\rm {A}}}{\\\\Lambda }_{\\\\rm {R}}{\\\\bf {V}}_{{\\\\bf {F}}_{\\\\rm {A}}}^{\\\\rm {H}}{\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}}$ where the diagonal elements of the diagonal matrices ${\\\\Lambda }_{\\\\rm {R}}$ and ${\\\\Lambda }_{\\\\rm {L}}$ satisfies $&[{\\\\Lambda }_{\\\\rm {R}}]_{i,i}=1, \\\\ \\\\ i\\\\le L \\\\nonumber \\\\\\\\& [{\\\\Lambda }_{\\\\rm {R}}]_{i,i}=0, \\\\ \\\\ i> L \\\\nonumber \\\\\\\\&[{\\\\Lambda }_{\\\\rm {L}}]_{i,i}=1, \\\\ \\\\ i\\\\le L \\\\nonumber \\\\\\\\& [{\\\\Lambda }_{\\\\rm {R}}]_{i,i}=0, \\\\ \\\\ i> L.$ Therefore, ${\\\\bf {F}}_{\\\\rm {A}}$ and ${\\\\bf {G}}_{\\\\rm {A}}$ do not affect the optimal solution.\",\n \"Moreover, the unitary matrices ${\\\\bf {V}}_{{\\\\bf {F}}_{\\\\rm {A}}}$ and ${\\\\bf {V}}_{\\\\rm {RG}}$ do not affect the optimal solution as ${\\\\bf {\\\\tilde{F}}}_{\\\\rm {D}}$ in the constraint is unitary invariant.\",\n \"Based on the above the discussion and (REF ), the remaining task to maximize the singular values of matrix $[{\\\\bf {U}}_{\\\\rm {RG}}^{\\\\rm {H}}{\\\\bf {H}}{\\\\bf {U}}_{{\\\\bf {F}}_{\\\\rm {A}}}]_{1:L,1:L}$ .\",\n \"Note that ${\\\\bf {U}}_{\\\\rm {RG}}$ and ${\\\\bf {U}}_{{\\\\bf {F}}_{\\\\rm {A}}}$ are unitary matrices, for the optimal solution, the left eigenvectors of its first $ L $ largest singular values of $ {\\\\bf F}_{\\\\rm A} $ should have the maximum inner product with $[{\\\\bf {V}}_{\\\\mathcal {H}}]_{:,1:L}$ i.e., $ [{\\\\bf {U}}_{{\\\\bf {F}}_{\\\\rm {A}}}]_{:,1:L}^{\\\\rm {opt}}={\\\\rm {arg\\\\,max}}\\\\lbrace \\\\Vert [{\\\\bf {V}}_{\\\\mathcal {H}}]_{:,1:L} [{\\\\bf {U}}_{{\\\\bf {F}}_{\\\\rm {A}}}]_{:,1:L}^{\\\\rm {H}}\\\\Vert _{\\\\rm {F}}^2\\\\rbrace .$ Similarly for the optimal solution, the left eigenvectors of its first $ L $ largest singular values of $ {\\\\bf {R}}_{\\\\rm {n}}^{1/2} {\\\\bf {G}}_{\\\\rm {A}}^{\\\\mathrm {H}} $ should have the maximum inner product with $[{\\\\bf {U}}_{\\\\mathcal {H}}]_{:,1:L}$ , i.e., $ [{\\\\bf {U}}_{{\\\\bf {R}}{\\\\bf {G}}}]_{:,1:L}^{\\\\rm {opt}}={\\\\rm {arg\\\\,max}}\\\\lbrace \\\\Vert [{\\\\bf {U}}_{\\\\mathcal {H}}]_{:,1:L} [{\\\\bf {U}}_{{\\\\bf {R}}{\\\\bf {G}}}]_{:,1:L}^{\\\\rm {H}}\\\\Vert _{\\\\rm {F}}^2\\\\rbrace .$\"\n ],\n [\n \"Analog Transceiver Design\",\n \"For fixed ${\\\\mathbf {\\\\Lambda }_\\\\mathrm {G}}$ and ${\\\\bf {Q}}_{\\\\rm {G}}$ , the optimization problem (REF ) can be transferred into the following vector variable optimization problem $\\\\min _{\\\\mathbf {r}} \\\\;\\\\; & {\\\\mathbf {r}}^{W} {\\\\mathbf {r}} - \\\\mathbf {p}^ - {\\\\mathbf {r}}^{p} + q \\\\\\\\\\\\text{s.t.}\",\n \"\\\\;\\\\;\\\\; & \\\\mathbf {r}^{K}_{i} \\\\mathbf {r} = a^2, \\\\quad i = 1,2,\\\\ldots , NL.$ The vector $ \\\\mathbf {r} $ is constructed via vectorizing $ \\\\mathbf {G}_{\\\\mathrm {A}} $ , i.e., $\\\\mathbf {r} = \\\\big [ \\\\Re \\\\lbrace \\\\mathrm {vec}{( \\\\mathbf {G}_{\\\\mathrm {A}} )} \\\\rbrace ^ \\\\, \\\\Im \\\\lbrace \\\\mathrm {vec}{( \\\\mathbf {G}_{\\\\mathrm {A}} )} \\\\rbrace ^]^$ and the matrices $ \\\\mathbf {W}, \\\\, \\\\mathbf {K}_{i} $ and vector $ \\\\mathbf {p} $ are defined as follows: $\\\\mathbf {W} & =\\\\begin{bmatrix}\\\\; \\\\Re \\\\lbrace \\\\mathbf {I} \\\\otimes { \\\\mathbf {R}_{\\\\mathrm {n}} } \\\\rbrace & - \\\\Im \\\\lbrace \\\\mathbf {I} \\\\otimes { \\\\mathbf {R}_{\\\\mathrm {n}} } \\\\rbrace \\\\;\\\\; \\\\\\\\\\\\; \\\\Im \\\\lbrace \\\\mathbf {I} \\\\otimes { \\\\mathbf {R}_{\\\\mathrm {n}} } \\\\rbrace & \\\\Re \\\\lbrace \\\\mathbf {I} \\\\otimes { \\\\mathbf {R}_{\\\\mathrm {n}} } \\\\rbrace \\\\;\\\\;\\\\end{bmatrix}, \\\\\\\\\\\\mathbf {K}_{i} & = \\\\mathrm {diag} \\\\Big \\\\lbrace \\\\bigl [ \\\\mathbf {0}_{(i-1) \\\\times 1}^ 1 ,\\\\mathbf {0}_{(NL - 1) \\\\times 1}^ 1 ,\\\\mathbf {0}_{(NL - i) \\\\times 1}^] \\\\Big \\\\rbrace ,$ and $\\\\mathbf {p} =\\\\begin{bmatrix}\\\\Re \\\\lbrace \\\\big ( \\\\mathbf {I} \\\\otimes { \\\\mathbf {R}_{\\\\mathrm {n}}^{ {1}/{2}} } \\\\big )^{\\\\mathrm {H}} \\\\mathrm {vec}{ \\\\big ( [{\\\\bf {U}}_{\\\\mathcal {H}}]_{:,1:L} {\\\\mathbf {\\\\Lambda }_\\\\mathrm {G}} {\\\\bf {Q}}_{\\\\rm {G}} \\\\big ) } \\\\rbrace \\\\\\\\\\\\Im \\\\lbrace \\\\big ( \\\\mathbf {I} \\\\otimes { \\\\mathbf {R}_{\\\\mathrm {n}}^{ {1}/{2}} } \\\\big )^{\\\\mathrm {H}} \\\\mathrm {vec}{ \\\\big ( [{\\\\bf {U}}_{\\\\mathcal {H}}]_{:,1:L} {\\\\mathbf {\\\\Lambda }_\\\\mathrm {G}} {\\\\bf {Q}}_{\\\\rm {G}} \\\\big ) } \\\\rbrace \\\\end{bmatrix}.$ The constant scalar, $q$ , in (REF ) equals $q = || \\\\mathrm {vec}{ \\\\big ( [{\\\\bf {U}}_{\\\\mathcal {H}}]_{:,1:L} {\\\\mathbf {\\\\Lambda }_\\\\mathrm {G}} {\\\\bf {Q}}_{\\\\rm {G}} \\\\big ) } ||_2^2$ .\",\n \"Note that because of the constant modulus constraints, the term ${\\\\bf {r}}^{\\\\rm {T}}{\\\\bf {r}}$ is a constant.\",\n \"As a result, for a constant real scalar, $\\\\eta $ , the objective function in (REF ) is equivalent to ${\\\\mathbf {r}}^\\\\mathbf {W}+\\\\eta {\\\\bf {I}}) {\\\\mathbf {r}} - \\\\mathbf {p}^ - {\\\\mathbf {r}}^{p} + q$ .\",\n \"As the constant modulus constraints in (REF ) are all quadratic equalities, the optimization problem (REF ) is nonconvex.\",\n \"Following the idea of [42], an iterative algorithm is proposed via iteratively updating constraints to guarantee the constant modulus constraints.\",\n \"Specifically, at the $n$ th iteration each constraint $\\\\mathbf {r}^{K}_{i} \\\\mathbf {r} = a^2$ is replaced by $\\\\mathbf {\\\\tilde{r}}_{(n-1)}^{K}_{i} \\\\mathbf {r}_{(n)} = a^2$ where $\\\\mathbf {\\\\tilde{r}}_{(n-1)}$ is a vector computed based on ${\\\\bf {r}}$ computed in the $(n-1)$ th iteration.\",\n \"After stacking $\\\\mathbf {\\\\tilde{r}}_{(n-1)}^{K}_{i}$ for $i = 1,2,\\\\ldots NL$ in ${\\\\bf {P}}_{(n-1)}$ , optimization problem (REF ) is transferred to $\\\\min _{\\\\mathbf {r}_{(n)}} \\\\;\\\\; & {\\\\mathbf {r}}_{(n)}^\\\\mathbf {W}+\\\\eta {\\\\bf {I}}) {\\\\mathbf {r}}_{(n)} - \\\\mathbf {p}^_{(n)} - {\\\\mathbf {r}}_{(n)}^{p} + q \\\\\\\\\\\\text{s.t.}\",\n \"\\\\;\\\\;\\\\; & {\\\\bf {P}}_{(n-1)}\\\\mathbf {r}_{(n)} = a^2{\\\\bf {1}},$ where the matrix $ \\\\mathbf {P}_{(n-1)} $ is defined as $&[ \\\\mathbf {P}_{(n-1)} ]_{\\\\ell ,j} \\\\nonumber \\\\\\\\=&{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\cos \\\\big ( \\\\angle [{\\\\rm {vec}}({\\\\bf {G}}_{{\\\\rm {A}},(n-1)})]_{\\\\ell } \\\\big ) & \\\\text{if } \\\\ell = j, \\\\; \\\\ell \\\\le NL \\\\\\\\\\\\sin \\\\big ( \\\\angle [ {\\\\rm {vec}}({\\\\bf {G}}_{{\\\\rm {A}},(n-1)}) ]_{\\\\ell } \\\\big ) & \\\\text{if } j = \\\\ell +NL, \\\\; \\\\ell \\\\le NL \\\\\\\\0 & \\\\text{otherwise.}\\\\end{array}\\\\right.\",\n \"}$ The vector $ \\\\mathbf {1} $ is a column vector with all elements equal to 1.\",\n \"As proved in [42], when $\\\\eta \\\\ge \\\\sigma _{\\\\max } NL / 8 + ||\\\\mathbf {p}||_2^2$ , where $ \\\\sigma _{\\\\max } $ is the largest eigenvalue of $ { \\\\mathbf {R}_{\\\\mathrm {n}} }$ , the optimal solution of the iterative optimization (REF ) minimizes the objective function and satisfies the constant modulus constraints asymptotically.\",\n \"As (REF ) is convex at each iteration, based on its KKT conditions, at the $n$ th iteration the optimal solution of (REF ) is ${\\\\mathbf {r}}_{(n)} =( {\\\\mathbf {W}} + \\\\eta \\\\mathbf {I} )^{-1}\\\\left({\\\\bf {q}}+\\\\frac{\\\\lambda }{2}{\\\\bf {P}}_{(n-1)}^{\\\\rm {T}}\\\\right)$ with $\\\\frac{\\\\lambda }{2}&=\\\\left( {\\\\bf {P}}_{(n-1)}( {\\\\mathbf {W}} + \\\\eta \\\\mathbf {I} )^{-1}{\\\\bf {P}}_{(n-1)}^{\\\\rm {T}}\\\\right)^{-1}\\\\nonumber \\\\\\\\& \\\\ \\\\ \\\\ \\\\ \\\\ \\\\times \\\\left(a^2{\\\\bf {1}}-{\\\\bf {P}}_{(n-1)}( {\\\\mathbf {W}} + \\\\eta \\\\mathbf {I} )^{-1}{\\\\bf {q}}\\\\right).$ In a nutshell, the iterative algorithm is given in Algorithm REF .\",\n \"Using the iterative algorithm, the numerical result of analog processor can be found.\"\n ]\n]"},"id":{"kind":"string","value":"1808.08526"}}},{"rowIdx":278,"cells":{"text":{"kind":"list like","value":[["Data-dependent Learning of Symmetric/Antisymmetric Relations for\n Knowledge Base Completion"],["Abstract Embedding-based methods for knowledge base completion (KBC) learn representations of entities and relations in a vector space, along with the scoring function to estimate the likelihood of relations between entities.","The learnable class of scoring functions is designed to be expressive enough to cover a variety of real-world relations, but this expressive comes at the cost of an increased number of parameters.","In particular, parameters in these methods are superfluous for relations that are either symmetric or antisymmetric.","To mitigate this problem, we propose a new L1 regularizer for Complex Embeddings, which is one of the state-of-the-art embedding-based methods for KBC.","This regularizer promotes symmetry or antisymmetry of the scoring function on a relation-by-relation basis, in accordance with the observed data.","Our empirical evaluation shows that the proposed method outperforms the original Complex Embeddings and other baseline methods on the FB15k dataset."],["Introduction","Large-scale knowledge bases, such as YAGO [16], Freebase [1], and WordNet [5] are utilized in knowledge-oriented applications such as question answering and dialog systems.","Facts are stored in these knowledge bases as triplets of form $(\\text{\\textit {subject entity}},\\text{\\textit {relation}},\\text{\\textit {object entity}})$ .","Although a knowledge base may contain more than a million facts, many facts are still missing [12].","Knowledge base completion (KBC) aims to find such missing facts automatically.","In recent years, vector embedding of knowledge bases has been actively pursued as a promising approach to KBC.","In this approach, entities and relations are embedded into a vector space as their representations, in most cases as vectors and sometimes as matrices.","A variety of methods have been proposed, each of which computes the likeliness score of given triplets using different vector/matrix operations over the representations of entities and relations involved.","In most of the previous methods, the scoring function is designed to cover general non-symmetric relations, i.e., relations $r$ such that for some entities $e_1$ and $e_2$ , triplet $(e_1,r,e_2)$ holds but not $(e_2, r, e_1)$ .","This reflects the fact that the subject and object in a relation are not interchangeable in general (e.g., parent_of).","However, the degree of symmetry differs from relation to relation.","In particular, a non-negligible number of symmetric relations exist in knowledge bases (e.g., sibling_of).","Moreover, many non-symmetric relations in knowledge base are actually antisymmetric, in the sense that for every distinct pair $e_1, e_2$ of entities, if $(e_1,r,e_2)$ holds, then $(e_2, r, e_1)$ never holds.Note that even if a relation is antisymmetric in the above sense, its truth-value matrix may not be antisymmetric.","For relations that show certain regularities such as above, the expressiveness of models to capture general relations might be superfluous, and a model with less parameters might be preferable.","It thus seems desirable to encourage the scoring function to produce sparser models, if the observed data suggests a relation being symmetric or antisymmetric.","As we do not assume any background knowledge about individual relations, the choice between these contrasting properties must be made solely from the observed data.","Further, we do not want the model to sacrifice the expressiveness to cope with relations that are neither purely symmetric or antisymmetric.","Complex Embeddings (ComplEx) [21] are one of the state-of-the-art methods for KBC.","ComplEx represents entities and relations as complex vectors, and it can model general non-symmetric relations thanks to the scoring function defined by the Hermitian inner product of these vectors.","However, for symmetric relations, the imaginary parts in relation vectors are redundant parameters since they only contribute to non-symmetry of the scoring function.","ComplEx is thus not exempt from the issue we mentioned above: the lack of symmetry/antisymmetry consideration for individual relations.","Our experimental results show that this issue indeed impairs the performance of ComplEx.","In this paper, we propose a technique for training ComplEx relation vectors adaptively to the degree of symmetry/antisymmetry observed in the data.","Our method is based on L1 regularization, but not in the standard way; the goal here is not to make a sparse, succinct model but to adjust the degree of symmetry/antisymmetry on the relation-by-relation basis, in a data-driven fashion.","In our model, L1 regularization is imposed on the products of coupled parameters, with each parameter contributing to either the symmetry or antisymmetry of the learned scoring function.","Experiments with synthetic data show that our method works as expected: Compared with the standard L1 regularization, the learned functions is more symmetric for symmetric relations and more antisymmetric for antisymmetric relations.","Moreover, in KBC tasks on real datasets, our method outperforms the original ComplEx with standard L1 and L2 regularization, as well as other baseline methods."],["Background","Let $\\mathbb {R}$ be the set of reals, and $\\mathbb {C}$ be the set of complex numbers.","Let $i \\in \\mathbb {C}$ denote the imaginary unit.","$[\\mathbf {v}]_j$ denotes the $j$ th component of vector $\\mathbf {v}$ , and $[\\mathbf {M}]_{jk}$ denotes the $(j,k)$ -element of matrix $\\mathbf {M}$ .","A superscript T (e.g., $\\mathbf {v}^{\\mathrm {T}}$ ) represents vector/matrix transpose.","For a complex scalar, vector, or matrix $\\mathbf {Z}$ , $\\overline{\\mathbf {Z}}$ represent its complex conjugate, with $\\mathop {\\text{Re}}(\\mathbf {Z})$ and $\\mathop {\\text{Im}}(\\mathbf {Z})$ denoting its real and imaginary parts, respectively."],["Knowledge Base Completion","Let $\\mathcal {E}$ and $\\mathcal {R}$ respectively be the sets of entities and the (names of) binary relations over entities in an incomplete knowledge base.","Suppose a relational triplet $(s,r,o)$ is not in the knowledge base for some $s, o \\in \\mathcal {E}$ and $r \\in \\mathcal {R}$ .","The task of KBC is to determine the truth value of such an unknown triplet; i.e., whether relation $r$ holds between subject entity $s$ and object entity $o$ .","A typical approach to KBC is to learn a scoring function $\\phi (s, r, o)$ to estimate the likeliness of an unknown triplet $(s, r, o)$ , using as training data the existing triplets in the knowledge base and their truth values.","A higher score indicates that the triplet is more likely to hold.","The scoring function $\\phi $ is usually parameterized, and the task of learning $\\phi $ is recast as that of tuning the model parameters.","To indicate this explicitly, model parameters $\\mathbf {\\Theta }$ are sometimes included in the arguments of the scoring function, as in $\\phi (s, r, o; \\mathbf {\\Theta })$ ."],["Complex Embeddings (ComplEx)","The embedding-based approach to KBC defines the scoring function in terms of the vector representation (or, embeddings) of entities and relations.","In this approach, model parameters $\\mathbf {\\Theta }$ consist of these representation vectors.","ComplEx [21] is one of the latest embedding-based methods for KBC.","It represents entities and relations as complex vectors.","Let $\\mathbf {e}_j, \\mathbf {w}_r \\in \\mathbb {C}^d$ respectively denote the $d$ -dimensional complex vector representations of entity $j \\in \\mathcal {E}$ and relation $r \\in \\mathcal {R}$ .","The scoring function of ComplEx is defined by $\\phi (s, r, o; \\mathbf {\\Theta }) & = \\mathop {\\text{Re}}\\left( \\mathbf {e}_s^{\\mathrm {T}}\\mathop {\\text{diag}}(\\mathbf {w}_r) \\overline{\\mathbf {e}_o} \\right) \\\\& = \\mathop {\\text{Re}}\\left( \\langle \\mathbf {w}_r, \\mathbf {e}_s, \\overline{\\mathbf {e}_o} \\rangle \\right) \\nonumber \\\\& = \\langle \\mathop {\\text{Re}}(\\mathbf {w}_r), \\mathop {\\text{Re}}(\\mathbf {e}_s), \\mathop {\\text{Re}}(\\mathbf {e}_o) \\rangle \\nonumber \\\\& \\qquad + \\langle \\mathop {\\text{Re}}(\\mathbf {w}_r), \\mathop {\\text{Im}}(\\mathbf {e}_s), \\mathop {\\text{Im}}(\\mathbf {e}_o) \\rangle \\nonumber \\\\& \\qquad + \\langle \\mathop {\\text{Im}}(\\mathbf {w}_r), \\mathop {\\text{Re}}(\\mathbf {e}_s), \\mathop {\\text{Im}}(\\mathbf {e}_o) \\rangle \\nonumber \\\\& \\qquad - \\langle \\mathop {\\text{Im}}(\\mathbf {w}_r), \\mathop {\\text{Im}}(\\mathbf {e}_s), \\mathop {\\text{Re}}(\\mathbf {e}_o) \\rangle , $ where $\\mathop {\\text{diag}}(\\mathbf {v})$ denotes a diagonal matrix with the diagonal given by vector $\\mathbf {v}$ , and $\\langle \\mathbf {u}, \\mathbf {v}, \\mathbf {w}\\rangle = \\left( \\sum _{k=1}^d [\\mathbf {u}]_k [\\mathbf {v}]_k [\\mathbf {w}]_k \\right) $ , with $\\mathbf {\\Theta } = \\lbrace \\mathbf {e}_j \\in \\mathbb {C}^d \\mid j \\in \\mathcal {E} \\rbrace \\cup \\lbrace \\mathbf {w}_r \\in \\mathbb {C}^d \\mid r \\in \\mathcal {R} \\rbrace $ .","The use of complex vectors and Hermitian inner product makes ComplEx both expressive and computationally efficient."],["Roles of Real/Imaginary Parts in Relation Vectors","Many relations in knowledge bases are either symmetric or antisymmetric.","For example, all 18 relations in WordNet are either symmetric (4 relations) or antisymmetric (14 relations).","Also, relations that take different “types” of entities as the subject and object are necessarily antisymmetric; take relation born_in for example, which is defined for a person and a location.","Clearly, if $(\\textit {Barack\\_Obama}, \\textit {born\\_in}, \\textit {Hawaii})$ holds, then $( \\textit {Hawaii}, \\textit {born\\_in}, \\textit {Barack\\_Obama})$ does not.","Now, let us look closely at the scoring function of ComplEx given by Eq.","(REF ).","We observe the following: If the relation vector $\\mathbf {w}_r$ is a real vector, then $\\phi (s,r,o) = \\phi (o,r,s)$ for any $s, o\\in \\mathcal {E}$ ; i.e., the scoring function $\\phi (s,r,o)$ is symmetric with respect to $s$ and $o$ .","This can be seen by substituting $\\mathop {\\text{Im}}(\\mathbf {w}_r) = \\mathbf {0}$ in Eq.","(), in which case the last two terms vanish.","If, to the contrary, $\\mathbf {w}_r$ is purely imaginary, $\\phi (s,r,o)$ is antisymmetric in $s$ and $o$ , in the sense that $\\phi (s,r,o) = -\\phi (o,r,s)$ .","Again, this can be verified with Eq.","(), but this time by substituting $\\mathop {\\text{Re}}(\\mathbf {w}_r) = \\mathbf {0}$ .","As we see from these two cases, the real parts in the components of $\\mathbf {w}_r$ are responsible for making the scoring function $\\phi $ symmetric, whereas the imaginary parts in $\\mathbf {w}_r$ are responsible for making it antisymmetric.","Each relation has a different degree of symmetry/antisymmetry, but the original ComplEx, which is usually trained with L2 regularization, does not take this difference into account.","Specifically, L2 regularization is equivalent to making a prior assumption that all parameters, including the real and imaginary parts of relation vectors, are independent.","As we have discussed above, this independence assumption is unsuited for symmetric and antisymmetric relations.","For instance, we expect the vector for symmetric relations to have a large number of nonzero real parts and zero imaginary parts."],["Multiplicative L1 Regularization for Coupled Parameters","On the basis of the observation above, we introduce a new regularization term for training ComplEx vectors.","This term encourages individual relation vectors to be more symmetric or antisymmetric in accordance with the observed data.","The resulting objective function is $\\underset{\\mathbf {\\Theta }}{\\mathrm {min}} \\sum _{(s, r, o) \\in \\Omega } \\log (1 + \\exp (-y_{rso} \\phi (s, r, o; \\mathbf {\\Theta }))) \\nonumber \\\\+ \\lambda ( \\alpha R_1(\\mathbf {\\Theta }) + (1 - \\alpha ) R_2(\\mathbf {\\Theta }))$ where $\\Omega $ is the training samples of triplets; $y_{rso} \\in \\lbrace +1, -1\\rbrace $ gives the truth value of the triplet $(s, r, o)$ ; hyperparameter $\\lambda \\ge 0$ determines the overall weight on the regularization terms; and $\\alpha \\in [0, 1]$ (also a hyperparameter) controls the balance between two regularization terms $R_1$ and $R_2$ .","These terms are defined as follows: $R_1(\\mathbf {\\Theta }) & = \\sum _{r \\in \\mathcal {R}} \\sum _{k=1}^d \\left| \\mathop {\\text{Re}}([\\mathbf {w}_r]_k) \\cdot \\mathop {\\text{Im}}([\\mathbf {w}_r]_k) \\right|, \\\\R_2(\\mathbf {\\Theta }) & = \\left\\Vert \\mathbf {\\Theta } \\right\\Vert ^2_2.","$ In Eq.","(), $\\mathbf {\\Theta }$ is treated as a vector, with all the parameters it contains as the vector components.","Eq.","(REF ) differs from the original ComplEx objective in that it introduces the proposed regularizer $R_1$ , which is a form of L1-norm penalty (see Equation (REF )).","In general, L1-norm penalty terms promote producing sparse solutions for the model parameters and are used for feature selection or the improve the interpretability of the model.","Note, however, that our L1 penalty encourages sparsity of pairwise products.","This means that only one of the coupled parameters needs to be propelled towards zero to minimize the summarized in Eq.","(REF ).","To distinguish from the standard L1 regularization, we call the regularization term in $R_1$ multiplicative L1 regularizer, since it is based on the L1 norm of the vector of the product of the real and imaginary parts of each component.","As we explain in the next subsection, standard L1 regularization implies the independence of parameters as a prior.","By contrast, our regularization term $R_1$ dictates the interaction between the real and imaginary parts of a component in a relation vector; if, as a result of L1 regularization, one of these parts falls to zero, the other can freely move to minimize the objective (REF ).","This encourages selecting either of the coupled parameters to be zero, but not necessarily both.","Unlike the standard L1 regularization, the proposed regularization term is non-convex, and makes the optimization harderNotice that the objective function in ComplEx is already non-convex without a regularization term..","However, in our experiments reported below, multiplicative L1 regularization outperforms the standard one in KBC, and is robust against random initialization.","Since the real and imaginary parts of a relation vector govern the symmetry/antisymmetry of the scoring function for the relation, this L1 penalty term is expected to help guide learning a vector for relation $r$ in accordance with whether $r$ is symmetric, antisymmetric, or neither of them, as observed in the training data.","For example, if the data suggests $r$ is likely to be symmetric, our L1 regularizer should encourage the imaginary parts to be zero while allowing the real parts to take on arbitrary values.","Because parameters are coupled componentwise, the proposed model can also cope with non-symmetric, non-antisymmetric relations with different degree of symmetry/antisymmetry."],["MAP Interpretation","MAP estimation finds the best model parameters $\\hat{\\mathbf {\\Theta }}$ by maximizing a posterior distribution: $\\hat{\\mathbf {\\Theta }} & = \\underset{\\mathbf {\\Theta }}{\\rm argmax} \\log p(\\mathbf {\\Theta } | \\mathcal {D}) \\nonumber \\\\& = \\underset{\\mathbf {\\Theta }}{\\rm argmax} \\log p(\\mathcal {D} | \\mathbf {\\Theta }) + \\log p(\\mathbf {\\Theta }), $ where $\\mathcal {D}$ is the observed data.","The first term represents the likelihood function, and the second term represents the prior distribution of parameters.","Our objective function Eq.","(REF ) can also be viewed as MAP estimation in the form of Eq.","(REF ); the first term in our objective corresponds to the likelihood function, and the regularizer terms define the prior.","Let us discuss the prior distribution implicitly assumed by using the proposed multiplicative L1 regularizer $R_1$ .For brevity, we neglect the regularizer $R_2$ and focus on $R_1$ in this discussion.","Let $C, C^{\\prime }, C^{\\prime \\prime }, \\ldots $ denote constants.","Our multiplicative L1 regularization is equivalent to assuming the prior $p(\\mathbf {\\Theta }) & = \\prod _{r\\in \\mathcal {R}} p(\\mathbf {w}_r) \\\\ \\multicolumn{2}{l}{\\text{with}}\\\\p(\\mathbf {w}_r) & = \\prod _{k=1}^d C \\exp \\left( - \\frac{ \\left|\\mathop {\\text{Re}}([\\mathbf {w}_r]_k) \\cdot \\mathop {\\text{Im}}([\\mathbf {w}_r]_k) \\right| }{C^{\\prime }} \\right).$ In other words, a 0-mean Laplacian prior is assumed on the distribution of $\\mathop {\\text{Re}}([\\mathbf {w}_r]_k) \\cdot \\mathop {\\text{Im}}([\\mathbf {w}_r]_k)$ .","The equivalence can be seen by $\\log p(\\mathbf {\\Theta }) & = \\log \\prod _{r\\in \\mathcal {R}} p( \\mathbf {w}_r) \\nonumber \\\\& = \\log \\!","\\prod _{r\\in \\mathcal {R}} \\prod _{k=1}^d \\!","C \\exp \\!","\\left( \\!\\!","- \\frac{ \\left|\\mathop {\\text{Re}}([\\mathbf {w}_r]_k) \\cdot \\mathop {\\text{Im}}([\\mathbf {w}_r]_k) \\right| }{C^{\\prime }} \\right) \\nonumber \\\\& = - C^{\\prime \\prime } \\!","\\sum _{r\\in \\mathcal {R}} \\sum _{k=1}^d \\left|\\mathop {\\text{Re}}([\\mathbf {w}_r]_k) \\cdot \\mathop {\\text{Im}}([\\mathbf {w}_r]_k) \\right| + C^{\\prime \\prime \\prime }.$ Neglecting the scaling factor $C^{\\prime \\prime }$ and the constant term $C^{\\prime \\prime \\prime }$ , we see that this is equal to the regularizer $R_1$ in Eq.","(REF ).","Now, suppose the standard L1 regularizer $R_{\\text{std L1}}(\\mathbf {\\Theta }) = \\sum _{r\\in \\mathcal {R}} \\sum _{k=1}^d ( \\left| \\mathop {\\text{Re}}([\\mathbf {w}_r]_k) \\right| + \\left| \\mathop {\\text{Im}}([\\mathbf {w}_r]_k) \\right|)$ is instead of the proposed $R_1(\\Theta )$ .","In terms of MAP estimation, in this case, its use is equivalent to assuming a prior distribution $p(\\mathbf {w}_r) = \\prod _{k=1}^d p(\\mathop {\\text{Re}}([\\mathbf {w}_r]_k)) \\, p(\\mathop {\\text{Im}}([\\mathbf {w}_r]_k))$ where both $p(\\mathop {\\text{Re}}([\\mathbf {w}_r]_k))$ and $p(\\mathop {\\text{Im}}([\\mathbf {w}_r]_k))$ obey a 0-mean Laplacian distribution.","Notice that the distribution (REF ) assumes the independence of the real and imaginary parts of components.","This assumption can be harmful if parameters are (positively or negatively) correlated with each other, which is indeed the case with symmetric or antisymmetric relations."],["Training Procedures","Optimizing our objective function (Eq.","(REF )) is difficult with standard online optimization methods, such as stochastic gradient descent.","In this paper, we extend the Regularized Dual Averaging (RDA) algorithm [22], which can produce sparse solutions effectively and is used in learning sparse word representations [4], [17].","Let us indicate the parameter value at time $t$ by superscript $(t)$ .","RDA keeps track of the online average subgradients at time $t$ : $\\sbox {\\mathbf {g}}\\widetilde{\\usebox {}}^{(t)} = (1/t) \\sum _{\\tau =1}^t \\mathbf {g}^{(\\tau )}$ , where $\\mathbf {g}^{(t)}$ is the subgradient at time $t$ .","In this paper, we calculate the subgradients $\\mathbf {g}^{(t)}$ in terms of only the loss function and L2 norm penalty term; i.e., they are the derivatives of the objective without the L1 penalty term.","The update formulas for multiplicative L1 regularizer are given as follows: $\\mathop {\\text{Re}}([\\mathbf {w}_r]_k^{(t+1)}) & = { {\\left\\lbrace \\begin{array}{ll}0, & \\text{if } \\left| [ \\sbox {\\mathbf {g}_r}\\widetilde{\\usebox {}}]_k^{(t)} \\right| \\le \\beta \\left| \\mathop {\\text{Im}}([\\mathbf {w}_r]_k^{(t)}) \\right|, \\\\\\gamma , & \\text{otherwise,}\\end{array}\\right.}}","\\\\\\mathop {\\text{Im}}([\\mathbf {w}_r]_k^{(t+1)}) & = { {\\left\\lbrace \\begin{array}{ll}0, & \\text{if } \\left| [ \\sbox {\\mathbf {g^{\\prime }}_r}\\widetilde{\\usebox {}} ]_k^{(t)} \\right| \\le \\beta \\left| \\mathop {\\text{Re}}([\\mathbf {w}_r]_k^{(t)}) \\right|, \\\\\\gamma ^{\\prime } & \\text{otherwise,}\\end{array}\\right.","}}$ where $\\beta = \\lambda \\alpha $ is a constant, $\\mathbf {g}_r, \\mathbf {g}^{\\prime }_r\\in \\mathbb {R}^d$ are the real and imaginary parts of the subgradients with respect to relation $r$ , and $\\gamma & = - \\eta t \\left( [\\sbox {\\mathbf {g}_r }\\widetilde{\\usebox {}} ]_k^{(t)} - \\beta \\left| \\mathop {\\text{Im}}([\\mathbf {w}_r]_k^{(t)}) \\right| \\mathop {\\text{sign}}([\\sbox {\\mathbf {g} _r}\\widetilde{\\usebox {}} ]_k^{(t)} ) \\right), \\\\\\gamma ^{\\prime } & = - \\eta t \\left( [\\sbox {\\mathbf {g}^{\\prime }_r}\\widetilde{\\usebox {}}]_k^{(t)} - \\beta \\left| \\mathop {\\text{Re}}([\\mathbf {w}_r]_k^{(t)}) \\right| \\mathop {\\text{sign}}([\\sbox {\\mathbf {g}^{\\prime }_r}\\widetilde{\\usebox {}} ]_k^{(t)} )\\right).$ From these formulas, we notice the interaction of the real and imaginary parts of a component; the imaginary part appears in the update formula for the real part, and vice versa.","The term $\\beta |\\mathop {\\text{Im}}([\\mathbf {w}_r^{(t)}]_k)|$ can be regarded as the strength of L1 regularizer specialized for $\\mathop {\\text{Re}}([\\mathbf {w}_r]_k)$ at time $t$ ; if $\\mathop {\\text{Im}}([\\mathbf {w}_r]_k)=0$ , then $\\mathop {\\text{Re}}([\\mathbf {w}_r]_k)$ keeps a nonzero value $\\gamma $ .","Likewise, if $\\mathop {\\text{Re}}([\\mathbf {w}_r]_k)=0$ , $\\mathop {\\text{Im}}([\\mathbf {w}_r]_k)$ is free to take on a nonzero value.","Notice that the above update formulas are applied only to relation vectors.","Entity vectors are learned with a standard optimization method."],["Knowledge Base Embedding","RESCAL [14] is an embedding-based KBC method whose scoring function is formulated as $\\mathbf {e}_{s}^{\\rm T}\\mathbf {W}_{r}\\mathbf {e}_{o}$ , where $\\mathbf {e}_s, \\mathbf {e}_o \\in \\mathbb {R}^d$ are the vector representations of entities $s$ and $o$ , respectively, and (possibly non-symmetric) matrix $\\mathbf {W}_r \\in \\mathbb {R}^{d\\times d}$ represents a relation $r$ .","Although RESCAL is able to output non-symmetric scoring functions, each relation vector $\\mathbf {W}_r$ holds $d^2$ parameters.","This can be problematic both in terms of overfitting and computational cost.","To avoid this problem, several methods have been proposed recently.","DistMult [23] restricts the relation matrix to be diagonal, $\\mathbf {W}_r = \\mathop {\\text{diag}}(\\mathbf {w}_r)$ , and it can compute the likelihood score in time $O(d)$ by way of $\\phi (s,r,o) = \\mathbf {e}_s^{\\mathrm {T}}\\mathop {\\text{diag}}(\\mathbf {w}_r) \\mathbf {e}_o $ .","However, this form of function is necessarily symmetric in $s$ and $o$ ; i.e., $\\phi (s,r,o)=\\phi (o,r,s)$ .","To reconcile efficiency and expressiveness, [21] complex proposed ComplEx, using the complex-valued representations and Hermitian inner product to define the scoring function (Eq.","(REF )).","ComplEx is founded on the unitary diagonalization of normal matrices [20].","Unlike DistMult, the scoring function can be nonsymmetric in $s$ and $o$ .","[6] eq found that ComplEx is equivalent to another state-of-the-art KBC method, Holographic Embeddings (HolE) [13].","ANALOGY [9] also assumes that $\\mathbf {W}_r$ is real normal.","They showed that any real normal matrix can be block-diagonalized, where each diagonal block is either a real scalar or a $2\\times 2$ real matrix of form $\\begin{pmatrix} a & b \\\\ -b & a \\end{pmatrix}$ .","Notice that this $2\\times 2$ matrix is exactly the real-valued encoding of a complex value $a+i b$ .","In this sense, ANALOGY can be regarded as a hybrid of ComplEx (corresponding to the $2\\times 2$ diagonal blocks) and DistMult (real scalar diagonal elements).","It can also be viewed as an attempt to reduce of the number of parameters in ComplEx, by constraining some of the imaginary parts of the vector components to be zero.","Although the idea resembles our proposed method, in ANALOGY, the reduced parameters (the number of scalar diagonal components) is a hyperparameter.","By contrast, our approach lets the data adjust the number of parameters for individual relations, by means of the multiplicative L1 regularizer.","Figure: Visualization of the vector representations trained on synthetic data.Each column represents a complex-valued vector component,with the upper and lower cells representing the real and imaginary parts, respectively.The black cells represent non-zero values."],["Sparse Modeling","Sparse modeling is often used to improve model interpretability or to enhance performance when the size of training data is insufficient relative to the number of model parameters.","Lasso [19] is a sparse modeling technique for linear regression.","Lasso uses L1 norm penalty to effectively find which parameters can be dispensed with.","Following Lasso, several variants for inducing structural sparsity have been proposed, e.g., [18], [7].","One of them, the exclusive group Lasso [7] uses an $l_{1, 2}$ -norm penalty, to enforce sparsity at an intra-group level.","An interesting connection exists between our regularization terms in Eq.","(REF ) and the exclusive group Lasso.","Let $R_j(\\mathbf {w}_r)$ , $j=1,2$ represents the terms in $R_j(\\mathbf {\\Theta })$ that are concerned with relation $r$ .","When $\\alpha = 2/3$ , we can show that the regularizer terms $\\alpha R_1(\\mathbf {w}_r) + (1 - \\alpha ) R_2 (\\mathbf {w}_r) = (1/3) \\sum _{k=1}^d ( \\mathop {\\text{Re}}([\\mathbf {w}_r]_k) + \\mathop {\\text{Im}}([\\mathbf {w}_r]_k) )^2$ .","The right-hand side can be seen as an instance of the exclusive group Lasso.","Word representation learning [10], [15] has proven useful for a variety of natural language processing tasks.","Sparse modeling has been applied to improve the interpretability of the learned word vectors while maintaining the expressive power of the model [4], [17].","Unlike our proposed method, however, the improvement of the model performance was not the main focus."],["Demonstrations on Synthetic Data","We conducted experiments with synthetic data to verify that our proposed method can learn symmetric and antisymmetric relations, given such data.","We randomly created a knowledge base of 3 relations and 50 entities.","We generated a total of 6,712 triplets, and sampled 5369 triplets as training set, then one-half of the remaining triplet as validation set and the other as test set.","The truth values of triplets were determined such that the first relation was symmetric, the second was antisymmetric, and the last was neither symmetric or antisymmetric.","We compared the standard and multiplicative L1 regularizers on this dataset.","For the standard L1 regularization, the regularizer $R_{\\text{std}}$ (Eq.","(REF )) was used in place of $R_1$ (Eq.","(REF )) in the objective function (REF ).","The dimension of the embedding space was set to $d=50$ .","For the multiplicative L1 regularizer, hyperparameters were set as follows: $\\alpha = 1.0, \\lambda = 0.05, \\eta = 0.1$ .","Figure REF displays which real and imaginary parts of the learned relation vectors have non-zero values.","The multiplicative L1 regularizer produced the expected results: most of the imaginary parts were zero in the symmetric relation and the real parts were zero in the antisymmetric relation.","Table REF shows the triplet classification accuracy on the test set.","Triplet classification is the task of predicting the truth value of given triplets in the test set.","For a given triplet, the prediction of the systems was determined by the sign of the output score ($+$ = true, $-$ = false).","The multiplicative L1 regularizer (`ComplEx w/ mul L1') outperformed the standard L1 regularizer (`ComplEx w/ std L1') considerably for both symmetric and antisymmetric relations."],["Real Datasets: WN18 and FB15k","Following previous work, we used the WordNet (WN18) and Freebase (FB15k) datasets to verify the benefits of our proposed method.","The dataset statistics are shown in Table REF .","Because the datasets contain only positive triplets, (pseudo-)negative samples must be generated in this experiment.","In this experiment, negative samples were generated by replacing the subject $s$ and object $o$ in a positive triplet $(s,r,o)$ with a randomly sampled entity from $\\mathcal {E}$ .","Table: Dataset statistics for FB15k and WN18.Table: Results on the WN18 and FB15k datasets: (Filtered and raw) MRR and filtered Hits@{1,3,10}\\lbrace 1,3,10\\rbrace (%).","* and ** denote the results reported in and , respectively.For evaluation, we performed the entity prediction task.","In this task, an incomplete triplet is given, which is generated by hiding one of the entities, either $s$ or $o$ , from a positive triplet.","The system must output the rankings of entities in $\\mathcal {E}$ for the missing $s$ or $o$ in the triplet, with the goal of placing (unknown) true $s$ or $o$ higher in the rankings.","Systems that learn a scoring function $\\phi (s,r,o)$ can use the score for computing the rankings.","The quality of the output rankings is measured by two standard evaluation measures for the KBC task: Mean Reciprocal Rank (MRR) and Hits@1, 3 and 10.","We here report results in both the filtered and raw settings [2] for MRR, but only filtered values for Hits@$n$ ."],["Experimental Setup","We selected the hyperparameters $\\lambda $ , $\\alpha $ , and $\\eta $ via grid search such that they maximize the filtered MRR on the validation set.","The ranges for the grid search were as follows: $\\lambda \\in \\lbrace 0.01, 0.001, 0.0001, 0 \\rbrace $ , $\\alpha \\in \\lbrace 0, 0.3, 0.5, 0.7, 1.0 \\rbrace $ , $\\eta \\in \\lbrace 0.1, 0.05 \\rbrace $ .","During the training, learning rate $\\eta $ was tuned with AdaGrad [3], both for entity and relation vectors.","The maximum number of training epochs was set to 500 and the dimension of the vector space was $d=200$ .","The number of negative triplets generated per positive training triplet was 10 for FB15k and 5 for WN18.","Table: Results on WN18 when the size of training data is reduced to a halfFigure: The box plots for the variance of filtered MRR on the FB15k dataset.Figure: The scatter plots showing the degree of sparsity against the symmetry score for each relation."],["Results","We compared our proposed model (`ComplEx w/ mul L1') with ComplEx and ComplEx with standard L1 regularization (`ComplEx w/ std L1').","Note that our model reduces to ComplEx when $\\alpha =0$ .","Table REF shows the results.","The results for TransE, DistMult, HolE, and ANALOGY are transcribed from the literature [21], [9].","All the compared models, except for DistMult, can produce a non-symmetric scoring function.","For most of the evaluation metrics, the multiplicative L1 regularizer (`ComplEx w/ mul L1') outperformed or was competitive to the best baseline.","Of particular interest, its performance on FB15k was much better than that of WN18.","Indeed, the accuracy of the original ComplEx is already quite high for WN18.","This difference between two datasets comes from the percentage of infrequent relations in two datasets.","Most of the 1,345 relations in FB15k are infrequent, i.e., each of them has only several dozen triplets in training data.","By contrast, of the 18 relations in WN18, most have more than one thousand training triplets.","Because sparse modeling is, in general, much effective when training data is scarce, this difference in the proportion of infrequent relations should have contributed to the different performance improvements on the two datasets.","To support this hypothesis, we ran additional experiments with WN18, by reducing the number of training samples to one half.","The test data remained the same as in the previous experiment.","The results are shown in Table REF .","The standard and multiplicative L1 regularizers work better than the original ComplEx and the improvement in each evaluation metric is greater than that observed with all training data.","The proposed method also outperformed the standard L1 regularization consistently.","The reliability of the result in Table REF was verified by computing the variance of the filtered MRR scores over 8 trials on FB15k.","For each of the trials, different random choices were used to generate the initial values of the representation vectors and the order of samples to process.","The result, shown in Figure REF , confirms that random initial values have little effect on the result; there is no overlapping MRR range among the compared methods, and the proposed method (`ComplEx w/ mul L1') consistently outperformed the standard L1 (`ComplEx w/ std L1') and vanilla ComplEx with only L2 regularization."],["Analysis","We analyzed how the two L1 regularizers, standard and multiplicative, work for symmetric and antisymmetric relations on FB15k.","Because of the large number of relations FB15k contains, manually extracting symmetric/antisymmetric relations is difficult.","To quantify the degree of symmetry of each relation $r$ , we define its “symmetry score” by $ \\mathop {\\text{sym}}(r) = |\\mathcal {T}^{\\text{sym}}_r| / |\\mathcal {T}_r| $ where $\\mathcal {T}_r & = \\lbrace (s, r, o) \\mid (s, r, o) \\in \\Omega , y_{rso} = +1 \\rbrace , \\\\\\mathcal {T}^{\\text{sym}}_r & = \\lbrace (s, r, o) \\mid (s, r, o) \\in \\Omega , y_{rso} = y_{ros} = +1 \\rbrace ,$ and compute this score for all relations in the FB15k training set.","By definition, symmetric relation $r$ should give $\\text{sym}(r)=1.0$ and antisymmetric relation $r$ should give $\\text{sym}(r)=0.0$ .","Thus, this score should indicate the degree of symmetry/antisymmetry of relations.","In Figure REF , we plot the percentage of non-zero elements in the real and imaginary parts of its representation vector $\\mathbf {w}_r$ against $\\text{sym}(r)$ for each relation $r$ .","Panels (a) and (b) are for $\\mathop {\\text{Re}}(\\mathbf {w}_r)$ and $\\mathop {\\text{Im}}(\\mathbf {w}_r)$ , respectively.","It is expected that the higher (resp.","lower) the symmetry is, the more imaginary (resp.","real) parts become zero.","With the multiplicative L1 regularizer (denoted by `mul L1' and green crosses in the figure), the density of real parts correlates with symmetry, and the density of imaginary parts inversely correlates with symmetry.","By contrast, correlation is weak or not observable for the standard L1 regularizer (`std L1'; blue circles).","As well as demonstrated on synthetic data, we conclude that the gains in Table REF come mainly from the more desirable representations for symmetric/antisymmetric relations learned with the multiplicative L1 regularization.","However, compared to the results on synthetic data shown in Figure REF , many non-zero values exist in $\\mathop {\\text{Re}}(\\mathbf {w}_r)$ and $\\mathop {\\text{Im}}(\\mathbf {w}_r)$ even for completely symmetric/antisymmetric relations.","We suspect that this is related to the fact that an antisymmetric relations does not imply an antisymmetric truth-value matrix; even if a relation is antisymmetric, there are many entity pairs $(e_1, e_2)$ such that neither of $(e_1, r, e_2)$ or $(e_2, r, e_1)$ holds."],["Conclusion","In this paper, we have presented a new regularizer for ComplEx to encourage vector representations of relations to be more symmetric or antisymmetric in accordance with data.","In the experiments, the proposed regularizer improved over the original ComplEx (with only L2 regularization) on FB15k, as well as WN18 with limited training data.","Recently, researchers have been making attempt to leverage background knowledge to improve KBC, such as by incorporating information of entity types, hierarchy, and relation attributes into the models [8], [11].","Our method does not assume background knowledge but adapt to symmetry/asymmetry of relations in a data-driven manner.","However, learning more diverse knowledge about relations from data should be an interesting future research topic.","We would also like to consider different type of regularization for entity vectors.","As another future research direction, we would like to develop a better strategy for sampling negative triplets that are suitable for our method."],["Acknowledgments","We are grateful to the anonymous reviewers for useful comments.","MS thanks partial support by JSPS Kakenhi Grant 15H02749."]],"string":"[\n [\n \"Data-dependent Learning of Symmetric/Antisymmetric Relations for\\n Knowledge Base Completion\"\n ],\n [\n \"Abstract Embedding-based methods for knowledge base completion (KBC) learn representations of entities and relations in a vector space, along with the scoring function to estimate the likelihood of relations between entities.\",\n \"The learnable class of scoring functions is designed to be expressive enough to cover a variety of real-world relations, but this expressive comes at the cost of an increased number of parameters.\",\n \"In particular, parameters in these methods are superfluous for relations that are either symmetric or antisymmetric.\",\n \"To mitigate this problem, we propose a new L1 regularizer for Complex Embeddings, which is one of the state-of-the-art embedding-based methods for KBC.\",\n \"This regularizer promotes symmetry or antisymmetry of the scoring function on a relation-by-relation basis, in accordance with the observed data.\",\n \"Our empirical evaluation shows that the proposed method outperforms the original Complex Embeddings and other baseline methods on the FB15k dataset.\"\n ],\n [\n \"Introduction\",\n \"Large-scale knowledge bases, such as YAGO [16], Freebase [1], and WordNet [5] are utilized in knowledge-oriented applications such as question answering and dialog systems.\",\n \"Facts are stored in these knowledge bases as triplets of form $(\\\\text{\\\\textit {subject entity}},\\\\text{\\\\textit {relation}},\\\\text{\\\\textit {object entity}})$ .\",\n \"Although a knowledge base may contain more than a million facts, many facts are still missing [12].\",\n \"Knowledge base completion (KBC) aims to find such missing facts automatically.\",\n \"In recent years, vector embedding of knowledge bases has been actively pursued as a promising approach to KBC.\",\n \"In this approach, entities and relations are embedded into a vector space as their representations, in most cases as vectors and sometimes as matrices.\",\n \"A variety of methods have been proposed, each of which computes the likeliness score of given triplets using different vector/matrix operations over the representations of entities and relations involved.\",\n \"In most of the previous methods, the scoring function is designed to cover general non-symmetric relations, i.e., relations $r$ such that for some entities $e_1$ and $e_2$ , triplet $(e_1,r,e_2)$ holds but not $(e_2, r, e_1)$ .\",\n \"This reflects the fact that the subject and object in a relation are not interchangeable in general (e.g., parent_of).\",\n \"However, the degree of symmetry differs from relation to relation.\",\n \"In particular, a non-negligible number of symmetric relations exist in knowledge bases (e.g., sibling_of).\",\n \"Moreover, many non-symmetric relations in knowledge base are actually antisymmetric, in the sense that for every distinct pair $e_1, e_2$ of entities, if $(e_1,r,e_2)$ holds, then $(e_2, r, e_1)$ never holds.Note that even if a relation is antisymmetric in the above sense, its truth-value matrix may not be antisymmetric.\",\n \"For relations that show certain regularities such as above, the expressiveness of models to capture general relations might be superfluous, and a model with less parameters might be preferable.\",\n \"It thus seems desirable to encourage the scoring function to produce sparser models, if the observed data suggests a relation being symmetric or antisymmetric.\",\n \"As we do not assume any background knowledge about individual relations, the choice between these contrasting properties must be made solely from the observed data.\",\n \"Further, we do not want the model to sacrifice the expressiveness to cope with relations that are neither purely symmetric or antisymmetric.\",\n \"Complex Embeddings (ComplEx) [21] are one of the state-of-the-art methods for KBC.\",\n \"ComplEx represents entities and relations as complex vectors, and it can model general non-symmetric relations thanks to the scoring function defined by the Hermitian inner product of these vectors.\",\n \"However, for symmetric relations, the imaginary parts in relation vectors are redundant parameters since they only contribute to non-symmetry of the scoring function.\",\n \"ComplEx is thus not exempt from the issue we mentioned above: the lack of symmetry/antisymmetry consideration for individual relations.\",\n \"Our experimental results show that this issue indeed impairs the performance of ComplEx.\",\n \"In this paper, we propose a technique for training ComplEx relation vectors adaptively to the degree of symmetry/antisymmetry observed in the data.\",\n \"Our method is based on L1 regularization, but not in the standard way; the goal here is not to make a sparse, succinct model but to adjust the degree of symmetry/antisymmetry on the relation-by-relation basis, in a data-driven fashion.\",\n \"In our model, L1 regularization is imposed on the products of coupled parameters, with each parameter contributing to either the symmetry or antisymmetry of the learned scoring function.\",\n \"Experiments with synthetic data show that our method works as expected: Compared with the standard L1 regularization, the learned functions is more symmetric for symmetric relations and more antisymmetric for antisymmetric relations.\",\n \"Moreover, in KBC tasks on real datasets, our method outperforms the original ComplEx with standard L1 and L2 regularization, as well as other baseline methods.\"\n ],\n [\n \"Background\",\n \"Let $\\\\mathbb {R}$ be the set of reals, and $\\\\mathbb {C}$ be the set of complex numbers.\",\n \"Let $i \\\\in \\\\mathbb {C}$ denote the imaginary unit.\",\n \"$[\\\\mathbf {v}]_j$ denotes the $j$ th component of vector $\\\\mathbf {v}$ , and $[\\\\mathbf {M}]_{jk}$ denotes the $(j,k)$ -element of matrix $\\\\mathbf {M}$ .\",\n \"A superscript T (e.g., $\\\\mathbf {v}^{\\\\mathrm {T}}$ ) represents vector/matrix transpose.\",\n \"For a complex scalar, vector, or matrix $\\\\mathbf {Z}$ , $\\\\overline{\\\\mathbf {Z}}$ represent its complex conjugate, with $\\\\mathop {\\\\text{Re}}(\\\\mathbf {Z})$ and $\\\\mathop {\\\\text{Im}}(\\\\mathbf {Z})$ denoting its real and imaginary parts, respectively.\"\n ],\n [\n \"Knowledge Base Completion\",\n \"Let $\\\\mathcal {E}$ and $\\\\mathcal {R}$ respectively be the sets of entities and the (names of) binary relations over entities in an incomplete knowledge base.\",\n \"Suppose a relational triplet $(s,r,o)$ is not in the knowledge base for some $s, o \\\\in \\\\mathcal {E}$ and $r \\\\in \\\\mathcal {R}$ .\",\n \"The task of KBC is to determine the truth value of such an unknown triplet; i.e., whether relation $r$ holds between subject entity $s$ and object entity $o$ .\",\n \"A typical approach to KBC is to learn a scoring function $\\\\phi (s, r, o)$ to estimate the likeliness of an unknown triplet $(s, r, o)$ , using as training data the existing triplets in the knowledge base and their truth values.\",\n \"A higher score indicates that the triplet is more likely to hold.\",\n \"The scoring function $\\\\phi $ is usually parameterized, and the task of learning $\\\\phi $ is recast as that of tuning the model parameters.\",\n \"To indicate this explicitly, model parameters $\\\\mathbf {\\\\Theta }$ are sometimes included in the arguments of the scoring function, as in $\\\\phi (s, r, o; \\\\mathbf {\\\\Theta })$ .\"\n ],\n [\n \"Complex Embeddings (ComplEx)\",\n \"The embedding-based approach to KBC defines the scoring function in terms of the vector representation (or, embeddings) of entities and relations.\",\n \"In this approach, model parameters $\\\\mathbf {\\\\Theta }$ consist of these representation vectors.\",\n \"ComplEx [21] is one of the latest embedding-based methods for KBC.\",\n \"It represents entities and relations as complex vectors.\",\n \"Let $\\\\mathbf {e}_j, \\\\mathbf {w}_r \\\\in \\\\mathbb {C}^d$ respectively denote the $d$ -dimensional complex vector representations of entity $j \\\\in \\\\mathcal {E}$ and relation $r \\\\in \\\\mathcal {R}$ .\",\n \"The scoring function of ComplEx is defined by $\\\\phi (s, r, o; \\\\mathbf {\\\\Theta }) & = \\\\mathop {\\\\text{Re}}\\\\left( \\\\mathbf {e}_s^{\\\\mathrm {T}}\\\\mathop {\\\\text{diag}}(\\\\mathbf {w}_r) \\\\overline{\\\\mathbf {e}_o} \\\\right) \\\\\\\\& = \\\\mathop {\\\\text{Re}}\\\\left( \\\\langle \\\\mathbf {w}_r, \\\\mathbf {e}_s, \\\\overline{\\\\mathbf {e}_o} \\\\rangle \\\\right) \\\\nonumber \\\\\\\\& = \\\\langle \\\\mathop {\\\\text{Re}}(\\\\mathbf {w}_r), \\\\mathop {\\\\text{Re}}(\\\\mathbf {e}_s), \\\\mathop {\\\\text{Re}}(\\\\mathbf {e}_o) \\\\rangle \\\\nonumber \\\\\\\\& \\\\qquad + \\\\langle \\\\mathop {\\\\text{Re}}(\\\\mathbf {w}_r), \\\\mathop {\\\\text{Im}}(\\\\mathbf {e}_s), \\\\mathop {\\\\text{Im}}(\\\\mathbf {e}_o) \\\\rangle \\\\nonumber \\\\\\\\& \\\\qquad + \\\\langle \\\\mathop {\\\\text{Im}}(\\\\mathbf {w}_r), \\\\mathop {\\\\text{Re}}(\\\\mathbf {e}_s), \\\\mathop {\\\\text{Im}}(\\\\mathbf {e}_o) \\\\rangle \\\\nonumber \\\\\\\\& \\\\qquad - \\\\langle \\\\mathop {\\\\text{Im}}(\\\\mathbf {w}_r), \\\\mathop {\\\\text{Im}}(\\\\mathbf {e}_s), \\\\mathop {\\\\text{Re}}(\\\\mathbf {e}_o) \\\\rangle , $ where $\\\\mathop {\\\\text{diag}}(\\\\mathbf {v})$ denotes a diagonal matrix with the diagonal given by vector $\\\\mathbf {v}$ , and $\\\\langle \\\\mathbf {u}, \\\\mathbf {v}, \\\\mathbf {w}\\\\rangle = \\\\left( \\\\sum _{k=1}^d [\\\\mathbf {u}]_k [\\\\mathbf {v}]_k [\\\\mathbf {w}]_k \\\\right) $ , with $\\\\mathbf {\\\\Theta } = \\\\lbrace \\\\mathbf {e}_j \\\\in \\\\mathbb {C}^d \\\\mid j \\\\in \\\\mathcal {E} \\\\rbrace \\\\cup \\\\lbrace \\\\mathbf {w}_r \\\\in \\\\mathbb {C}^d \\\\mid r \\\\in \\\\mathcal {R} \\\\rbrace $ .\",\n \"The use of complex vectors and Hermitian inner product makes ComplEx both expressive and computationally efficient.\"\n ],\n [\n \"Roles of Real/Imaginary Parts in Relation Vectors\",\n \"Many relations in knowledge bases are either symmetric or antisymmetric.\",\n \"For example, all 18 relations in WordNet are either symmetric (4 relations) or antisymmetric (14 relations).\",\n \"Also, relations that take different “types” of entities as the subject and object are necessarily antisymmetric; take relation born_in for example, which is defined for a person and a location.\",\n \"Clearly, if $(\\\\textit {Barack\\\\_Obama}, \\\\textit {born\\\\_in}, \\\\textit {Hawaii})$ holds, then $( \\\\textit {Hawaii}, \\\\textit {born\\\\_in}, \\\\textit {Barack\\\\_Obama})$ does not.\",\n \"Now, let us look closely at the scoring function of ComplEx given by Eq.\",\n \"(REF ).\",\n \"We observe the following: If the relation vector $\\\\mathbf {w}_r$ is a real vector, then $\\\\phi (s,r,o) = \\\\phi (o,r,s)$ for any $s, o\\\\in \\\\mathcal {E}$ ; i.e., the scoring function $\\\\phi (s,r,o)$ is symmetric with respect to $s$ and $o$ .\",\n \"This can be seen by substituting $\\\\mathop {\\\\text{Im}}(\\\\mathbf {w}_r) = \\\\mathbf {0}$ in Eq.\",\n \"(), in which case the last two terms vanish.\",\n \"If, to the contrary, $\\\\mathbf {w}_r$ is purely imaginary, $\\\\phi (s,r,o)$ is antisymmetric in $s$ and $o$ , in the sense that $\\\\phi (s,r,o) = -\\\\phi (o,r,s)$ .\",\n \"Again, this can be verified with Eq.\",\n \"(), but this time by substituting $\\\\mathop {\\\\text{Re}}(\\\\mathbf {w}_r) = \\\\mathbf {0}$ .\",\n \"As we see from these two cases, the real parts in the components of $\\\\mathbf {w}_r$ are responsible for making the scoring function $\\\\phi $ symmetric, whereas the imaginary parts in $\\\\mathbf {w}_r$ are responsible for making it antisymmetric.\",\n \"Each relation has a different degree of symmetry/antisymmetry, but the original ComplEx, which is usually trained with L2 regularization, does not take this difference into account.\",\n \"Specifically, L2 regularization is equivalent to making a prior assumption that all parameters, including the real and imaginary parts of relation vectors, are independent.\",\n \"As we have discussed above, this independence assumption is unsuited for symmetric and antisymmetric relations.\",\n \"For instance, we expect the vector for symmetric relations to have a large number of nonzero real parts and zero imaginary parts.\"\n ],\n [\n \"Multiplicative L1 Regularization for Coupled Parameters\",\n \"On the basis of the observation above, we introduce a new regularization term for training ComplEx vectors.\",\n \"This term encourages individual relation vectors to be more symmetric or antisymmetric in accordance with the observed data.\",\n \"The resulting objective function is $\\\\underset{\\\\mathbf {\\\\Theta }}{\\\\mathrm {min}} \\\\sum _{(s, r, o) \\\\in \\\\Omega } \\\\log (1 + \\\\exp (-y_{rso} \\\\phi (s, r, o; \\\\mathbf {\\\\Theta }))) \\\\nonumber \\\\\\\\+ \\\\lambda ( \\\\alpha R_1(\\\\mathbf {\\\\Theta }) + (1 - \\\\alpha ) R_2(\\\\mathbf {\\\\Theta }))$ where $\\\\Omega $ is the training samples of triplets; $y_{rso} \\\\in \\\\lbrace +1, -1\\\\rbrace $ gives the truth value of the triplet $(s, r, o)$ ; hyperparameter $\\\\lambda \\\\ge 0$ determines the overall weight on the regularization terms; and $\\\\alpha \\\\in [0, 1]$ (also a hyperparameter) controls the balance between two regularization terms $R_1$ and $R_2$ .\",\n \"These terms are defined as follows: $R_1(\\\\mathbf {\\\\Theta }) & = \\\\sum _{r \\\\in \\\\mathcal {R}} \\\\sum _{k=1}^d \\\\left| \\\\mathop {\\\\text{Re}}([\\\\mathbf {w}_r]_k) \\\\cdot \\\\mathop {\\\\text{Im}}([\\\\mathbf {w}_r]_k) \\\\right|, \\\\\\\\R_2(\\\\mathbf {\\\\Theta }) & = \\\\left\\\\Vert \\\\mathbf {\\\\Theta } \\\\right\\\\Vert ^2_2.\",\n \"$ In Eq.\",\n \"(), $\\\\mathbf {\\\\Theta }$ is treated as a vector, with all the parameters it contains as the vector components.\",\n \"Eq.\",\n \"(REF ) differs from the original ComplEx objective in that it introduces the proposed regularizer $R_1$ , which is a form of L1-norm penalty (see Equation (REF )).\",\n \"In general, L1-norm penalty terms promote producing sparse solutions for the model parameters and are used for feature selection or the improve the interpretability of the model.\",\n \"Note, however, that our L1 penalty encourages sparsity of pairwise products.\",\n \"This means that only one of the coupled parameters needs to be propelled towards zero to minimize the summarized in Eq.\",\n \"(REF ).\",\n \"To distinguish from the standard L1 regularization, we call the regularization term in $R_1$ multiplicative L1 regularizer, since it is based on the L1 norm of the vector of the product of the real and imaginary parts of each component.\",\n \"As we explain in the next subsection, standard L1 regularization implies the independence of parameters as a prior.\",\n \"By contrast, our regularization term $R_1$ dictates the interaction between the real and imaginary parts of a component in a relation vector; if, as a result of L1 regularization, one of these parts falls to zero, the other can freely move to minimize the objective (REF ).\",\n \"This encourages selecting either of the coupled parameters to be zero, but not necessarily both.\",\n \"Unlike the standard L1 regularization, the proposed regularization term is non-convex, and makes the optimization harderNotice that the objective function in ComplEx is already non-convex without a regularization term..\",\n \"However, in our experiments reported below, multiplicative L1 regularization outperforms the standard one in KBC, and is robust against random initialization.\",\n \"Since the real and imaginary parts of a relation vector govern the symmetry/antisymmetry of the scoring function for the relation, this L1 penalty term is expected to help guide learning a vector for relation $r$ in accordance with whether $r$ is symmetric, antisymmetric, or neither of them, as observed in the training data.\",\n \"For example, if the data suggests $r$ is likely to be symmetric, our L1 regularizer should encourage the imaginary parts to be zero while allowing the real parts to take on arbitrary values.\",\n \"Because parameters are coupled componentwise, the proposed model can also cope with non-symmetric, non-antisymmetric relations with different degree of symmetry/antisymmetry.\"\n ],\n [\n \"MAP Interpretation\",\n \"MAP estimation finds the best model parameters $\\\\hat{\\\\mathbf {\\\\Theta }}$ by maximizing a posterior distribution: $\\\\hat{\\\\mathbf {\\\\Theta }} & = \\\\underset{\\\\mathbf {\\\\Theta }}{\\\\rm argmax} \\\\log p(\\\\mathbf {\\\\Theta } | \\\\mathcal {D}) \\\\nonumber \\\\\\\\& = \\\\underset{\\\\mathbf {\\\\Theta }}{\\\\rm argmax} \\\\log p(\\\\mathcal {D} | \\\\mathbf {\\\\Theta }) + \\\\log p(\\\\mathbf {\\\\Theta }), $ where $\\\\mathcal {D}$ is the observed data.\",\n \"The first term represents the likelihood function, and the second term represents the prior distribution of parameters.\",\n \"Our objective function Eq.\",\n \"(REF ) can also be viewed as MAP estimation in the form of Eq.\",\n \"(REF ); the first term in our objective corresponds to the likelihood function, and the regularizer terms define the prior.\",\n \"Let us discuss the prior distribution implicitly assumed by using the proposed multiplicative L1 regularizer $R_1$ .For brevity, we neglect the regularizer $R_2$ and focus on $R_1$ in this discussion.\",\n \"Let $C, C^{\\\\prime }, C^{\\\\prime \\\\prime }, \\\\ldots $ denote constants.\",\n \"Our multiplicative L1 regularization is equivalent to assuming the prior $p(\\\\mathbf {\\\\Theta }) & = \\\\prod _{r\\\\in \\\\mathcal {R}} p(\\\\mathbf {w}_r) \\\\\\\\ \\\\multicolumn{2}{l}{\\\\text{with}}\\\\\\\\p(\\\\mathbf {w}_r) & = \\\\prod _{k=1}^d C \\\\exp \\\\left( - \\\\frac{ \\\\left|\\\\mathop {\\\\text{Re}}([\\\\mathbf {w}_r]_k) \\\\cdot \\\\mathop {\\\\text{Im}}([\\\\mathbf {w}_r]_k) \\\\right| }{C^{\\\\prime }} \\\\right).$ In other words, a 0-mean Laplacian prior is assumed on the distribution of $\\\\mathop {\\\\text{Re}}([\\\\mathbf {w}_r]_k) \\\\cdot \\\\mathop {\\\\text{Im}}([\\\\mathbf {w}_r]_k)$ .\",\n \"The equivalence can be seen by $\\\\log p(\\\\mathbf {\\\\Theta }) & = \\\\log \\\\prod _{r\\\\in \\\\mathcal {R}} p( \\\\mathbf {w}_r) \\\\nonumber \\\\\\\\& = \\\\log \\\\!\",\n \"\\\\prod _{r\\\\in \\\\mathcal {R}} \\\\prod _{k=1}^d \\\\!\",\n \"C \\\\exp \\\\!\",\n \"\\\\left( \\\\!\\\\!\",\n \"- \\\\frac{ \\\\left|\\\\mathop {\\\\text{Re}}([\\\\mathbf {w}_r]_k) \\\\cdot \\\\mathop {\\\\text{Im}}([\\\\mathbf {w}_r]_k) \\\\right| }{C^{\\\\prime }} \\\\right) \\\\nonumber \\\\\\\\& = - C^{\\\\prime \\\\prime } \\\\!\",\n \"\\\\sum _{r\\\\in \\\\mathcal {R}} \\\\sum _{k=1}^d \\\\left|\\\\mathop {\\\\text{Re}}([\\\\mathbf {w}_r]_k) \\\\cdot \\\\mathop {\\\\text{Im}}([\\\\mathbf {w}_r]_k) \\\\right| + C^{\\\\prime \\\\prime \\\\prime }.$ Neglecting the scaling factor $C^{\\\\prime \\\\prime }$ and the constant term $C^{\\\\prime \\\\prime \\\\prime }$ , we see that this is equal to the regularizer $R_1$ in Eq.\",\n \"(REF ).\",\n \"Now, suppose the standard L1 regularizer $R_{\\\\text{std L1}}(\\\\mathbf {\\\\Theta }) = \\\\sum _{r\\\\in \\\\mathcal {R}} \\\\sum _{k=1}^d ( \\\\left| \\\\mathop {\\\\text{Re}}([\\\\mathbf {w}_r]_k) \\\\right| + \\\\left| \\\\mathop {\\\\text{Im}}([\\\\mathbf {w}_r]_k) \\\\right|)$ is instead of the proposed $R_1(\\\\Theta )$ .\",\n \"In terms of MAP estimation, in this case, its use is equivalent to assuming a prior distribution $p(\\\\mathbf {w}_r) = \\\\prod _{k=1}^d p(\\\\mathop {\\\\text{Re}}([\\\\mathbf {w}_r]_k)) \\\\, p(\\\\mathop {\\\\text{Im}}([\\\\mathbf {w}_r]_k))$ where both $p(\\\\mathop {\\\\text{Re}}([\\\\mathbf {w}_r]_k))$ and $p(\\\\mathop {\\\\text{Im}}([\\\\mathbf {w}_r]_k))$ obey a 0-mean Laplacian distribution.\",\n \"Notice that the distribution (REF ) assumes the independence of the real and imaginary parts of components.\",\n \"This assumption can be harmful if parameters are (positively or negatively) correlated with each other, which is indeed the case with symmetric or antisymmetric relations.\"\n ],\n [\n \"Training Procedures\",\n \"Optimizing our objective function (Eq.\",\n \"(REF )) is difficult with standard online optimization methods, such as stochastic gradient descent.\",\n \"In this paper, we extend the Regularized Dual Averaging (RDA) algorithm [22], which can produce sparse solutions effectively and is used in learning sparse word representations [4], [17].\",\n \"Let us indicate the parameter value at time $t$ by superscript $(t)$ .\",\n \"RDA keeps track of the online average subgradients at time $t$ : $\\\\sbox {\\\\mathbf {g}}\\\\widetilde{\\\\usebox {}}^{(t)} = (1/t) \\\\sum _{\\\\tau =1}^t \\\\mathbf {g}^{(\\\\tau )}$ , where $\\\\mathbf {g}^{(t)}$ is the subgradient at time $t$ .\",\n \"In this paper, we calculate the subgradients $\\\\mathbf {g}^{(t)}$ in terms of only the loss function and L2 norm penalty term; i.e., they are the derivatives of the objective without the L1 penalty term.\",\n \"The update formulas for multiplicative L1 regularizer are given as follows: $\\\\mathop {\\\\text{Re}}([\\\\mathbf {w}_r]_k^{(t+1)}) & = { {\\\\left\\\\lbrace \\\\begin{array}{ll}0, & \\\\text{if } \\\\left| [ \\\\sbox {\\\\mathbf {g}_r}\\\\widetilde{\\\\usebox {}}]_k^{(t)} \\\\right| \\\\le \\\\beta \\\\left| \\\\mathop {\\\\text{Im}}([\\\\mathbf {w}_r]_k^{(t)}) \\\\right|, \\\\\\\\\\\\gamma , & \\\\text{otherwise,}\\\\end{array}\\\\right.}}\",\n \"\\\\\\\\\\\\mathop {\\\\text{Im}}([\\\\mathbf {w}_r]_k^{(t+1)}) & = { {\\\\left\\\\lbrace \\\\begin{array}{ll}0, & \\\\text{if } \\\\left| [ \\\\sbox {\\\\mathbf {g^{\\\\prime }}_r}\\\\widetilde{\\\\usebox {}} ]_k^{(t)} \\\\right| \\\\le \\\\beta \\\\left| \\\\mathop {\\\\text{Re}}([\\\\mathbf {w}_r]_k^{(t)}) \\\\right|, \\\\\\\\\\\\gamma ^{\\\\prime } & \\\\text{otherwise,}\\\\end{array}\\\\right.\",\n \"}}$ where $\\\\beta = \\\\lambda \\\\alpha $ is a constant, $\\\\mathbf {g}_r, \\\\mathbf {g}^{\\\\prime }_r\\\\in \\\\mathbb {R}^d$ are the real and imaginary parts of the subgradients with respect to relation $r$ , and $\\\\gamma & = - \\\\eta t \\\\left( [\\\\sbox {\\\\mathbf {g}_r }\\\\widetilde{\\\\usebox {}} ]_k^{(t)} - \\\\beta \\\\left| \\\\mathop {\\\\text{Im}}([\\\\mathbf {w}_r]_k^{(t)}) \\\\right| \\\\mathop {\\\\text{sign}}([\\\\sbox {\\\\mathbf {g} _r}\\\\widetilde{\\\\usebox {}} ]_k^{(t)} ) \\\\right), \\\\\\\\\\\\gamma ^{\\\\prime } & = - \\\\eta t \\\\left( [\\\\sbox {\\\\mathbf {g}^{\\\\prime }_r}\\\\widetilde{\\\\usebox {}}]_k^{(t)} - \\\\beta \\\\left| \\\\mathop {\\\\text{Re}}([\\\\mathbf {w}_r]_k^{(t)}) \\\\right| \\\\mathop {\\\\text{sign}}([\\\\sbox {\\\\mathbf {g}^{\\\\prime }_r}\\\\widetilde{\\\\usebox {}} ]_k^{(t)} )\\\\right).$ From these formulas, we notice the interaction of the real and imaginary parts of a component; the imaginary part appears in the update formula for the real part, and vice versa.\",\n \"The term $\\\\beta |\\\\mathop {\\\\text{Im}}([\\\\mathbf {w}_r^{(t)}]_k)|$ can be regarded as the strength of L1 regularizer specialized for $\\\\mathop {\\\\text{Re}}([\\\\mathbf {w}_r]_k)$ at time $t$ ; if $\\\\mathop {\\\\text{Im}}([\\\\mathbf {w}_r]_k)=0$ , then $\\\\mathop {\\\\text{Re}}([\\\\mathbf {w}_r]_k)$ keeps a nonzero value $\\\\gamma $ .\",\n \"Likewise, if $\\\\mathop {\\\\text{Re}}([\\\\mathbf {w}_r]_k)=0$ , $\\\\mathop {\\\\text{Im}}([\\\\mathbf {w}_r]_k)$ is free to take on a nonzero value.\",\n \"Notice that the above update formulas are applied only to relation vectors.\",\n \"Entity vectors are learned with a standard optimization method.\"\n ],\n [\n \"Knowledge Base Embedding\",\n \"RESCAL [14] is an embedding-based KBC method whose scoring function is formulated as $\\\\mathbf {e}_{s}^{\\\\rm T}\\\\mathbf {W}_{r}\\\\mathbf {e}_{o}$ , where $\\\\mathbf {e}_s, \\\\mathbf {e}_o \\\\in \\\\mathbb {R}^d$ are the vector representations of entities $s$ and $o$ , respectively, and (possibly non-symmetric) matrix $\\\\mathbf {W}_r \\\\in \\\\mathbb {R}^{d\\\\times d}$ represents a relation $r$ .\",\n \"Although RESCAL is able to output non-symmetric scoring functions, each relation vector $\\\\mathbf {W}_r$ holds $d^2$ parameters.\",\n \"This can be problematic both in terms of overfitting and computational cost.\",\n \"To avoid this problem, several methods have been proposed recently.\",\n \"DistMult [23] restricts the relation matrix to be diagonal, $\\\\mathbf {W}_r = \\\\mathop {\\\\text{diag}}(\\\\mathbf {w}_r)$ , and it can compute the likelihood score in time $O(d)$ by way of $\\\\phi (s,r,o) = \\\\mathbf {e}_s^{\\\\mathrm {T}}\\\\mathop {\\\\text{diag}}(\\\\mathbf {w}_r) \\\\mathbf {e}_o $ .\",\n \"However, this form of function is necessarily symmetric in $s$ and $o$ ; i.e., $\\\\phi (s,r,o)=\\\\phi (o,r,s)$ .\",\n \"To reconcile efficiency and expressiveness, [21] complex proposed ComplEx, using the complex-valued representations and Hermitian inner product to define the scoring function (Eq.\",\n \"(REF )).\",\n \"ComplEx is founded on the unitary diagonalization of normal matrices [20].\",\n \"Unlike DistMult, the scoring function can be nonsymmetric in $s$ and $o$ .\",\n \"[6] eq found that ComplEx is equivalent to another state-of-the-art KBC method, Holographic Embeddings (HolE) [13].\",\n \"ANALOGY [9] also assumes that $\\\\mathbf {W}_r$ is real normal.\",\n \"They showed that any real normal matrix can be block-diagonalized, where each diagonal block is either a real scalar or a $2\\\\times 2$ real matrix of form $\\\\begin{pmatrix} a & b \\\\\\\\ -b & a \\\\end{pmatrix}$ .\",\n \"Notice that this $2\\\\times 2$ matrix is exactly the real-valued encoding of a complex value $a+i b$ .\",\n \"In this sense, ANALOGY can be regarded as a hybrid of ComplEx (corresponding to the $2\\\\times 2$ diagonal blocks) and DistMult (real scalar diagonal elements).\",\n \"It can also be viewed as an attempt to reduce of the number of parameters in ComplEx, by constraining some of the imaginary parts of the vector components to be zero.\",\n \"Although the idea resembles our proposed method, in ANALOGY, the reduced parameters (the number of scalar diagonal components) is a hyperparameter.\",\n \"By contrast, our approach lets the data adjust the number of parameters for individual relations, by means of the multiplicative L1 regularizer.\",\n \"Figure: Visualization of the vector representations trained on synthetic data.Each column represents a complex-valued vector component,with the upper and lower cells representing the real and imaginary parts, respectively.The black cells represent non-zero values.\"\n ],\n [\n \"Sparse Modeling\",\n \"Sparse modeling is often used to improve model interpretability or to enhance performance when the size of training data is insufficient relative to the number of model parameters.\",\n \"Lasso [19] is a sparse modeling technique for linear regression.\",\n \"Lasso uses L1 norm penalty to effectively find which parameters can be dispensed with.\",\n \"Following Lasso, several variants for inducing structural sparsity have been proposed, e.g., [18], [7].\",\n \"One of them, the exclusive group Lasso [7] uses an $l_{1, 2}$ -norm penalty, to enforce sparsity at an intra-group level.\",\n \"An interesting connection exists between our regularization terms in Eq.\",\n \"(REF ) and the exclusive group Lasso.\",\n \"Let $R_j(\\\\mathbf {w}_r)$ , $j=1,2$ represents the terms in $R_j(\\\\mathbf {\\\\Theta })$ that are concerned with relation $r$ .\",\n \"When $\\\\alpha = 2/3$ , we can show that the regularizer terms $\\\\alpha R_1(\\\\mathbf {w}_r) + (1 - \\\\alpha ) R_2 (\\\\mathbf {w}_r) = (1/3) \\\\sum _{k=1}^d ( \\\\mathop {\\\\text{Re}}([\\\\mathbf {w}_r]_k) + \\\\mathop {\\\\text{Im}}([\\\\mathbf {w}_r]_k) )^2$ .\",\n \"The right-hand side can be seen as an instance of the exclusive group Lasso.\",\n \"Word representation learning [10], [15] has proven useful for a variety of natural language processing tasks.\",\n \"Sparse modeling has been applied to improve the interpretability of the learned word vectors while maintaining the expressive power of the model [4], [17].\",\n \"Unlike our proposed method, however, the improvement of the model performance was not the main focus.\"\n ],\n [\n \"Demonstrations on Synthetic Data\",\n \"We conducted experiments with synthetic data to verify that our proposed method can learn symmetric and antisymmetric relations, given such data.\",\n \"We randomly created a knowledge base of 3 relations and 50 entities.\",\n \"We generated a total of 6,712 triplets, and sampled 5369 triplets as training set, then one-half of the remaining triplet as validation set and the other as test set.\",\n \"The truth values of triplets were determined such that the first relation was symmetric, the second was antisymmetric, and the last was neither symmetric or antisymmetric.\",\n \"We compared the standard and multiplicative L1 regularizers on this dataset.\",\n \"For the standard L1 regularization, the regularizer $R_{\\\\text{std}}$ (Eq.\",\n \"(REF )) was used in place of $R_1$ (Eq.\",\n \"(REF )) in the objective function (REF ).\",\n \"The dimension of the embedding space was set to $d=50$ .\",\n \"For the multiplicative L1 regularizer, hyperparameters were set as follows: $\\\\alpha = 1.0, \\\\lambda = 0.05, \\\\eta = 0.1$ .\",\n \"Figure REF displays which real and imaginary parts of the learned relation vectors have non-zero values.\",\n \"The multiplicative L1 regularizer produced the expected results: most of the imaginary parts were zero in the symmetric relation and the real parts were zero in the antisymmetric relation.\",\n \"Table REF shows the triplet classification accuracy on the test set.\",\n \"Triplet classification is the task of predicting the truth value of given triplets in the test set.\",\n \"For a given triplet, the prediction of the systems was determined by the sign of the output score ($+$ = true, $-$ = false).\",\n \"The multiplicative L1 regularizer (`ComplEx w/ mul L1') outperformed the standard L1 regularizer (`ComplEx w/ std L1') considerably for both symmetric and antisymmetric relations.\"\n ],\n [\n \"Real Datasets: WN18 and FB15k\",\n \"Following previous work, we used the WordNet (WN18) and Freebase (FB15k) datasets to verify the benefits of our proposed method.\",\n \"The dataset statistics are shown in Table REF .\",\n \"Because the datasets contain only positive triplets, (pseudo-)negative samples must be generated in this experiment.\",\n \"In this experiment, negative samples were generated by replacing the subject $s$ and object $o$ in a positive triplet $(s,r,o)$ with a randomly sampled entity from $\\\\mathcal {E}$ .\",\n \"Table: Dataset statistics for FB15k and WN18.Table: Results on the WN18 and FB15k datasets: (Filtered and raw) MRR and filtered Hits@{1,3,10}\\\\lbrace 1,3,10\\\\rbrace (%).\",\n \"* and ** denote the results reported in and , respectively.For evaluation, we performed the entity prediction task.\",\n \"In this task, an incomplete triplet is given, which is generated by hiding one of the entities, either $s$ or $o$ , from a positive triplet.\",\n \"The system must output the rankings of entities in $\\\\mathcal {E}$ for the missing $s$ or $o$ in the triplet, with the goal of placing (unknown) true $s$ or $o$ higher in the rankings.\",\n \"Systems that learn a scoring function $\\\\phi (s,r,o)$ can use the score for computing the rankings.\",\n \"The quality of the output rankings is measured by two standard evaluation measures for the KBC task: Mean Reciprocal Rank (MRR) and Hits@1, 3 and 10.\",\n \"We here report results in both the filtered and raw settings [2] for MRR, but only filtered values for Hits@$n$ .\"\n ],\n [\n \"Experimental Setup\",\n \"We selected the hyperparameters $\\\\lambda $ , $\\\\alpha $ , and $\\\\eta $ via grid search such that they maximize the filtered MRR on the validation set.\",\n \"The ranges for the grid search were as follows: $\\\\lambda \\\\in \\\\lbrace 0.01, 0.001, 0.0001, 0 \\\\rbrace $ , $\\\\alpha \\\\in \\\\lbrace 0, 0.3, 0.5, 0.7, 1.0 \\\\rbrace $ , $\\\\eta \\\\in \\\\lbrace 0.1, 0.05 \\\\rbrace $ .\",\n \"During the training, learning rate $\\\\eta $ was tuned with AdaGrad [3], both for entity and relation vectors.\",\n \"The maximum number of training epochs was set to 500 and the dimension of the vector space was $d=200$ .\",\n \"The number of negative triplets generated per positive training triplet was 10 for FB15k and 5 for WN18.\",\n \"Table: Results on WN18 when the size of training data is reduced to a halfFigure: The box plots for the variance of filtered MRR on the FB15k dataset.Figure: The scatter plots showing the degree of sparsity against the symmetry score for each relation.\"\n ],\n [\n \"Results\",\n \"We compared our proposed model (`ComplEx w/ mul L1') with ComplEx and ComplEx with standard L1 regularization (`ComplEx w/ std L1').\",\n \"Note that our model reduces to ComplEx when $\\\\alpha =0$ .\",\n \"Table REF shows the results.\",\n \"The results for TransE, DistMult, HolE, and ANALOGY are transcribed from the literature [21], [9].\",\n \"All the compared models, except for DistMult, can produce a non-symmetric scoring function.\",\n \"For most of the evaluation metrics, the multiplicative L1 regularizer (`ComplEx w/ mul L1') outperformed or was competitive to the best baseline.\",\n \"Of particular interest, its performance on FB15k was much better than that of WN18.\",\n \"Indeed, the accuracy of the original ComplEx is already quite high for WN18.\",\n \"This difference between two datasets comes from the percentage of infrequent relations in two datasets.\",\n \"Most of the 1,345 relations in FB15k are infrequent, i.e., each of them has only several dozen triplets in training data.\",\n \"By contrast, of the 18 relations in WN18, most have more than one thousand training triplets.\",\n \"Because sparse modeling is, in general, much effective when training data is scarce, this difference in the proportion of infrequent relations should have contributed to the different performance improvements on the two datasets.\",\n \"To support this hypothesis, we ran additional experiments with WN18, by reducing the number of training samples to one half.\",\n \"The test data remained the same as in the previous experiment.\",\n \"The results are shown in Table REF .\",\n \"The standard and multiplicative L1 regularizers work better than the original ComplEx and the improvement in each evaluation metric is greater than that observed with all training data.\",\n \"The proposed method also outperformed the standard L1 regularization consistently.\",\n \"The reliability of the result in Table REF was verified by computing the variance of the filtered MRR scores over 8 trials on FB15k.\",\n \"For each of the trials, different random choices were used to generate the initial values of the representation vectors and the order of samples to process.\",\n \"The result, shown in Figure REF , confirms that random initial values have little effect on the result; there is no overlapping MRR range among the compared methods, and the proposed method (`ComplEx w/ mul L1') consistently outperformed the standard L1 (`ComplEx w/ std L1') and vanilla ComplEx with only L2 regularization.\"\n ],\n [\n \"Analysis\",\n \"We analyzed how the two L1 regularizers, standard and multiplicative, work for symmetric and antisymmetric relations on FB15k.\",\n \"Because of the large number of relations FB15k contains, manually extracting symmetric/antisymmetric relations is difficult.\",\n \"To quantify the degree of symmetry of each relation $r$ , we define its “symmetry score” by $ \\\\mathop {\\\\text{sym}}(r) = |\\\\mathcal {T}^{\\\\text{sym}}_r| / |\\\\mathcal {T}_r| $ where $\\\\mathcal {T}_r & = \\\\lbrace (s, r, o) \\\\mid (s, r, o) \\\\in \\\\Omega , y_{rso} = +1 \\\\rbrace , \\\\\\\\\\\\mathcal {T}^{\\\\text{sym}}_r & = \\\\lbrace (s, r, o) \\\\mid (s, r, o) \\\\in \\\\Omega , y_{rso} = y_{ros} = +1 \\\\rbrace ,$ and compute this score for all relations in the FB15k training set.\",\n \"By definition, symmetric relation $r$ should give $\\\\text{sym}(r)=1.0$ and antisymmetric relation $r$ should give $\\\\text{sym}(r)=0.0$ .\",\n \"Thus, this score should indicate the degree of symmetry/antisymmetry of relations.\",\n \"In Figure REF , we plot the percentage of non-zero elements in the real and imaginary parts of its representation vector $\\\\mathbf {w}_r$ against $\\\\text{sym}(r)$ for each relation $r$ .\",\n \"Panels (a) and (b) are for $\\\\mathop {\\\\text{Re}}(\\\\mathbf {w}_r)$ and $\\\\mathop {\\\\text{Im}}(\\\\mathbf {w}_r)$ , respectively.\",\n \"It is expected that the higher (resp.\",\n \"lower) the symmetry is, the more imaginary (resp.\",\n \"real) parts become zero.\",\n \"With the multiplicative L1 regularizer (denoted by `mul L1' and green crosses in the figure), the density of real parts correlates with symmetry, and the density of imaginary parts inversely correlates with symmetry.\",\n \"By contrast, correlation is weak or not observable for the standard L1 regularizer (`std L1'; blue circles).\",\n \"As well as demonstrated on synthetic data, we conclude that the gains in Table REF come mainly from the more desirable representations for symmetric/antisymmetric relations learned with the multiplicative L1 regularization.\",\n \"However, compared to the results on synthetic data shown in Figure REF , many non-zero values exist in $\\\\mathop {\\\\text{Re}}(\\\\mathbf {w}_r)$ and $\\\\mathop {\\\\text{Im}}(\\\\mathbf {w}_r)$ even for completely symmetric/antisymmetric relations.\",\n \"We suspect that this is related to the fact that an antisymmetric relations does not imply an antisymmetric truth-value matrix; even if a relation is antisymmetric, there are many entity pairs $(e_1, e_2)$ such that neither of $(e_1, r, e_2)$ or $(e_2, r, e_1)$ holds.\"\n ],\n [\n \"Conclusion\",\n \"In this paper, we have presented a new regularizer for ComplEx to encourage vector representations of relations to be more symmetric or antisymmetric in accordance with data.\",\n \"In the experiments, the proposed regularizer improved over the original ComplEx (with only L2 regularization) on FB15k, as well as WN18 with limited training data.\",\n \"Recently, researchers have been making attempt to leverage background knowledge to improve KBC, such as by incorporating information of entity types, hierarchy, and relation attributes into the models [8], [11].\",\n \"Our method does not assume background knowledge but adapt to symmetry/asymmetry of relations in a data-driven manner.\",\n \"However, learning more diverse knowledge about relations from data should be an interesting future research topic.\",\n \"We would also like to consider different type of regularization for entity vectors.\",\n \"As another future research direction, we would like to develop a better strategy for sampling negative triplets that are suitable for our method.\"\n ],\n [\n \"Acknowledgments\",\n \"We are grateful to the anonymous reviewers for useful comments.\",\n \"MS thanks partial support by JSPS Kakenhi Grant 15H02749.\"\n ]\n]"},"id":{"kind":"string","value":"1808.08361"}}},{"rowIdx":279,"cells":{"text":{"kind":"list like","value":[["Intrinsic magnetic topological insulators in van der Waals layered\n MnBi$_2$Te$_4$-family materials"],["Abstract The interplay of magnetism and topology is a key research subject in condensed matter physics and material science, which offers great opportunities to explore emerging new physics, like the quantum anomalous Hall (QAH) effect, axion electrodynamics and Majorana fermions.","However, these exotic physical effects have rarely been realized in experiment, due to the lacking of suitable working materials.","Here we predict that van der Waals layered MnBi$_2$Te$_4$-family materials show two-dimensional (2D) ferromagnetism in the single layer and three-dimensional (3D) $A$-type antiferromagnetism in the bulk, which could serve as a next-generation material platform for the state-of-art research.","Remarkably, we predict extremely rich topological quantum effects with outstanding features in an experimentally available material MnBi$_2$Te$_4$, including a 3D antiferromagnetic topological insulator with the long-sought topological axion states, the type-II magnetic Weyl semimetal (WSM) with simply one pair of Weyl points, and the high-temperature intrinsic QAH effect.","These striking predictions, if proved experimentally, could profoundly transform future research and technology of topological quantum physics."],["Intrinsic magnetic topological insulators in van der Waals layered MnBi$_2$ Te$_4$ -family materials Jiaheng Li$^{1,2}$ Yang Li$^{1,2}$ Shiqiao Du$^{1,2}$ Zun Wang$^{1,2}$ Bing-Lin Gu$^{1,2,3}$ Shou-Cheng Zhang$^{4}$ Ke He$^{1,2}$ kehe@tsinghua.edu.cn Wenhui Duan$^{1,2,3}$ dwh@phys.tsinghua.edu.cn Yong Xu$^{1,2,5}$ yongxu@mail.tsinghua.edu.cn $^1$ State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, People's Republic of China $^2$ Collaborative Innovation Center of Quantum Matter, Beijing 100084, People's Republic of China $^3$ Institute for Advanced Study, Tsinghua University, Beijing 100084, People's Republic of China $^4$ Department of Physics, McCullough Building, Stanford University, Stanford, California 94305-4045, USA $^5$ RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan The interplay of magnetism and topology is a key research subject in condensed matter physics and material science, which offers great opportunities to explore emerging new physics, like the quantum anomalous Hall (QAH) effect, axion electrodynamics and Majorana fermions.","However, these exotic physical effects have rarely been realized in experiment, due to the lacking of suitable working materials.","Here we predict that van der Waals layered MnBi$_2$ Te$_4$ -family materials show two-dimensional (2D) ferromagnetism in the single layer and three-dimensional (3D) $A$ -type antiferromagnetism in the bulk, which could serve as a next-generation material platform for the state-of-art research.","Remarkably, we predict extremely rich topological quantum effects with outstanding features in an experimentally available material MnBi$_2$ Te$_4$ , including a 3D antiferromagnetic topological insulator with the long-sought topological axion states, the type-II magnetic Weyl semimetal (WSM) with simply one pair of Weyl points, and the high-temperature intrinsic QAH effect.","These striking predictions, if proved experimentally, could profoundly transform future research and technology of topological quantum physics.","The correlation between topology and symmetry as a central fundamental problem of modern physics has attracted intensive research interests in condensed matter physics and material science since the discovery of topological insulators (TIs) [1], [2].","One essential symmetry is the time reversal symmetry (TRS), which is crucial to many kinds of topological quantum states of matter like TIs and is broken in the presence of magnetism.","Intriguingly, the interplay between magnetism and topology in materials could generate a variety of exotic topological quantum states [1], [2], [3], including the quantum anomalous Hall (QAH) effect showing dissipationless chiral edge states [4], [5], [6], the topological axion states displaying quantized magnetoelectric effects [7], [8], [9], and Majorana fermions obeying non-abelian statistics [2], [10], [11].","In this context, great research effort has been devoted to explore the novel topological quantum physics, which is of profound importance to fundamental science and future technologies, like dissipationless topological electronics and topological quantum computation [1], [2].","One key subject that is of crucial significance to the whole research community is to develop topological quantum materials (TQMs) showing the coexistence of topology and other quantum phases (e.g.","magnetism, ferroelectricity, charge density wave, and superconductivity), coined “composite TQMs” (CTQMs), which include magnetic TQMs (MTQMs) as an important subset.","Currently, very limited number of MTQMs are experimentally available, including magnetically doped TIs and magnetic topological heterostructures [6], [12], [13], [14], whose material fabrication, experimental measurement and property optimization are quite challenging.","Since the major topological properties of these existing MTQMs depend sensitively on delicate magnetic doping or proximity effects, only little preliminary experimental progress has been achieved until now, leaving many important physical effects not ready for practical use or still unproved.","For instance, the QAH effect was only observed in magnetically doped (Bi$_{x}$ Sb$_{1-x}$ )$_2$ Te$_3$ thin films at very low working temperatures by fine tuning chemical compositions [6], [12].","While a previous theory predicted the existence of topological axion states at the surface of a three-dimensional (3D) antiferromagnetic (AFM) TI [15], no 3D AFM TI has been realized experimentally, as far as we know.","For future research and applications, people should go beyond the existing strategy for building MTQMs and try to design intrinsic MTQMs in no need of introducing alloy/doping or heterostructures.","Noticeably, van der Waals (vdW) layered materials represent a large family of materials with greatly tunable properties by quantum size effects or vdW heterojunctions [16], in which a variety of quantum phases in different spatial dimensions have been discovered by the state-of-the-art research [17], [18], [19], [20], [21], [22].","For instance, both two-dimensional (2D) and 3D TI states were previously found in the tetradymite-type Bi$_2$ Te$_3$ -class materials [17], [18], and 2D intrinsic magnetism was recently found in ultrathin films of CrI$_3$ [23] and Cr$_2$ Ge$_2$ Te$_6$ [24] .","However, these topological materials are not magnetic; those magnetic material are not topological.","It is highly desirable to incorporate the magnetic and topological states together into the same vdW material, so as to obtain layered intrinsic MTQMs, which are able to inherit advantages of vdW-family materials.","In this work, based on first-principles (FP) calculations, we find a series of layered intrinsic MTQMs from the tetradymite-type MnBi$_2$ Te$_4$ -related ternary chalcogenides (MB$_2$ T$_4$ : M = transition-metal or rare-earth element, B = Bi or Sb, T = Te, Se or S), in which the intralayer exchange coupling is ferromagnetic (FM), giving 2D ferromagnetism in their septuple layer (SL); while the interlayer exchange coupling is antiferromagnetic, forming 3D $A$ -type antiferromagnetism in their vdW layered bulk.","Strikingly, we predict that plenty of novel 2D and 3D intrinsic magnetic topological states with outstanding features could be naturally realized in an experimentally available material MnBi$_2$ Te$_4$ , including the 3D AFM TI in the bulk together with the long-sought topological axion states on the (001) surface, large-gap intrinsic QAH insulators ($E_{\\textit {g}} \\sim $ 38 meV) with dissipaltionless and patternable chiral edge states in thin films, the type-II magnetic WSM with a single pair of Weyl points in the FM bulk, and possibly Majorana fermions if interacting with superconductivity.","The tetradymite-type MnBi$_2$ Te$_4$ crystalizes in a rhombohedral layered structure with the space group $R\\overline{3}m$ , and each layer has a triangular lattice with atoms ABC-stacked along the out-of-plane direction, the same as Bi$_2$ Te$_3$ .","Slightly differently, one layer MnBi$_2$ Te$_4$ includes seven atoms in a unit cell, forming a Te-Bi-Te-Mn-Te-Bi-Te SL, which can be viewed as intercalating a Mn-Te bilayer into the center of a Bi$_2$ Te$_3$ quintuple layer (Fig.","1a).","Noticeably, the mixing between Mn and Bi in this compound would create unstable valence states Mn$^{+3}$ and Bi$^{+2}$ , which is energetically unfavorable.","Thus the formation of alloys could be avoided, leading to stoichiometric compounds.","This physical mechanism also explains the stability of compounds like SnBi$_2$ Te$_4$ and PbBi$_2$ Te$_4$ and is applicable to many other MB$_2$ T$_4$ -family materials.","Importantly, our FP calculations predicted that a series of MB$_2$ T$_4$ compounds are energetically and dynamically stable, including M = Ti, V, Mn, Ni, Eu, etc., as confirmed by FP total-energy and phonon calculations [25], which could be fabricated by experiment.","These stable compounds are characterized by an insulating band gap in their single layers, offering a collection of layered intrinsic magnetic (topological) insulators.","In fact, MnBi$_2$ Te$_4$ is now experimentally available [26], which is selected to introduce properties of the material family.","Mn has a valence charge of $+$ 2 by losing its two 4$s$ electrons.","The remaining five 3$d$ electrons fill up the spin-up Mn-$d$ levels according to the Hund's rule, introducing 5 $\\mu _{B}$ magnetic moment (mostly from Mn) per unit cell.","By comparing different magnetic structures, we found that the magnetic ground state is a 2D ferromagnetism with an out-of-plane easy axis in the single layer (depicted Fig.","1a), in agreement with the previous work [27].","Each Mn atom is bonded with six neighboring Te atoms, which form a slightly distorted edge-sharing octahedron.","According to the Goodenough-Kanamori rule, the superexchange interactions between Mn-Te-Mn with a bonding angle of $\\sim 94^{\\circ }$ are ferromagnetic, similar as in CrI$_3$ and Cr$_2$ Ge$_2$ Te$_6$ [23], [24].","Intralayer ferromagnetic exchange couplings was also found in other stable MB$_2$ T$_4$ -family members, but the easy axis can be varied to in-plane (e.g.","for M = V), displaying rich 2D magnetic features.","Intriguingly, magnetic and topological states are well incorporated together into MnBi$_2$ Te$_4$ , where Mn introduces magnetism and the Bi-Te layers could generate topological states similar as Bi$_2$ Te$_3$ [17], as schematically depicted in Fig.","1b.","The exchange splitting between spin-up and spin-down Mn $d$ bands are extremely large ($>$ 7 eV) caused by the large magnetic moment of Mn.","Thus Mn $d$ -bands are far away from the band gap, and only Bi/Te $p$ -bands are close to the Fermi level.","The single layer has a 0.73 eV direct (1.36 eV indirect) band gap when including (excluding) spin-orbit coupling (SOC) (Figs.","1c and 1d).","While the single layer is topologically trivial ferromagnetic insulator, extremely interesting topological quantum physics emerges in the bulk and thin films, as we will demonstrate.","For the layered bulk, we found that the interlayer magnetic coupling is AFM, giving a $A$ -type AFM ground state (depicted in Fig.","2a) [25], which is similar as in Cr$_2$ Ge$_2$ Te$_6$ [24] and explained by the interlayer super-superexchange coupling.","The spatial inversion symmetry $P$ (i.e.","$P_1$ centered at O$_1$ ) is preserved, but the TRS $\\Theta $ gets broken.","There exist two new symmetries: $P_2 \\Theta $ and $S=\\Theta T_{1/2}$ , where $P_2$ is an inversion operation centered at O$_2$ and $T_{1/2}$ is a lattice translation (depicted in Fig.","2a).","The band structures with and without SOC are presented in Figs.","2b and 2c.","Here every band is (at least) doubly degenerate, which is ensured by the $P_2 \\Theta $ symmetry [28].","In the presence of $S$ , a $\\mathcal {Z}_2$ classification becomes feasible [15].","However, in contrast to the time-reversal invariant case, there is a $\\mathcal {Z}_2$ invariant for the $k_z = 0$ plane but not for the $k_z = \\pi $ plane.","$\\mathcal {Z}_2 = 1$ corresponds to a novel 3D AFM TI phase [15], which has not been experimentally confirmed.","Here the parity criteria can be applied to determine $\\mathcal {Z}_2$ due to the $P_1$ symmetry [29].","Interestingly, the parities of the valence band maximum (VBM) and conduction band minimum (CBM) at $\\Gamma $ are opposite, and both change signs by the SOC effects (Figs.","2b and 2c), implying a band inversion.","By varying the SOC strength, the band gap first closes and then reopens at $\\Gamma $ (Fig.","2d), showing simply one band inversion and thus implying a topological phase transition.","The SOC-induced band reversal is happened between Bi $p_z^+$ and Te $p_z^-$ , essentially the same as for Bi$_2$ Te$_3$ [17].","Our Wannier charge center (WCC) calculations for the $k_z = 0$ plane revealed $\\mathcal {Z}_2 = 1$ ($\\mathcal {Z}_2 = 0$ ) for with (without) SOC, confirming that MnBi$_2$ Te$_4$ is a 3D AFM TI.","The global band gap is $\\sim $ 0.16 eV direct at $Z$ , and the direct gap at $\\Gamma $ is $\\sim $ 0.18 eV.","One prominent feature of the AFM TI is the existence of 2D gapless surface states protected by $S$ , which is confirmed by the surface-state calculations (Figs.","2f and 2g).","The surface states are indeed gapless on the (100) termination.","However, they become gapped on the (001) termination due to the $S$ symmetry breaking.","The intrinsically gapped (001) surfaces are promising for probing the long-sought topological axion states, which give the topological quantized magnetoelectric effect related to an axion field with $\\theta = \\pi $ [7], [8], [9], [15].","Previous work proposed to probe the novel states by adding opposite out-of-plane ferromagnetism onto the bottom and top surfaces of 3D TIs [7], [14].","Such kind of TRS-broken surface states are naturally provided by even-layer MnBi$_2$ Te$_4$ films (when with negligible hybridizations between top and bottom surface states), benefitting from their $A$ -type AFM structure.","An additional requirement is to open a band gap at side surfaces, which is realized in relatively thin films or by breaking the $S$ symmetry on the side surfaces (e.g.","controlling surface morphology).","The simplified proposal together with the suitable material candidate MnBi$_2$ Te$_4$ could greatly facilitate the research of axion electrodynamics.","The AFM ground state of MnBi$_2$ Te$_4$ could be tuned, for instance, by applying an external magnetic field, to other magnetic structures.","Then the material symmetry changes, leading to distinct topological phases.","This concept is demonstrated by studying the simple FM ordering along the out-of-plane direction (Fig.","3a).","The FM structure has $P_1$ but neither $\\Theta $ nor $P_2 \\Theta $ , leading to spin-split bands, nonzero Berry curvatures and possibly nonzero topological Chern numbers that correspond to novel topological phases, like 3D QAH insulators [30] and WSMs [31], [32], [33], [22], [3].","The band structure of FM MnBi$_2$ Te$_4$ (Fig.","3b) displays a pair of band crossings at W/W$^{\\prime }$ along the $\\overline{\\textit {Z}}$ -$\\Gamma $ -Z line.","The band crossings are induced by interlayer orbital hybridizations and protected by the $C_3$ rotational symmetry.","Our WCC calculations found that W is a momentum-space monopole with a topological charge of +1 (i.e.","an Berry phase of $2\\pi $ ) (Fig.","3e), and its time-reversal partner W$^{\\prime }$ has an opposite topological charge of -1, indicating that the system is a topological WSM.","While in most of the momentum space the electron pocket is located above the hole pocket (e.g.","along the F-W-L line), the Weyl cones get tilted along the $\\overline{\\textit {Z}}$ -$\\Gamma $ -Z line (Fig.","3d), leaving some part of the electron pocket below the hole pocket, which is the characteristic feature of the type-II WSM [22].","In contrast to the time-reversal-invariant WSMs that must have even pairs of Weyl points, this ferromagnetic WSM represents the simplest one, hosting only a pair of Weyl points.","Moreover, our surface-state calculations clearly demonstrated the existence of Fermi arcs on the (100) and ($\\bar{1}$ 00) terminations (Figs.","3f and 3g), which is the fingerprint of the WSM.","Importantly, the Weyl points are well separated in the momentum space and very close to the Fermi level, advantageous for experimental observations.","The vdW layered materials are featured by tunable quantum size effects.","For AFM MnBi$_2$ Te$_4$ films, even layers do not possess $P$ and $\\Theta $ symmetries, but have $P_2 \\Theta $ , ensuring double degeneracy in every band.","Differently, odd layers have $P$ but neither $\\Theta $ nor $P_2 \\Theta $ , leading to spin-split bands.","The different symmetries lead to distinct topological properties in even and odd layers.","Specifically, the topological Chern number $\\mathcal {C} = 0$ is required by $P_2 \\Theta $ in even layers, $\\mathcal {C} \\ne 0$ is allowed in odd layers.","In contrast to the single and 3-layer films, where a trivial insulating gap is opened by quantum confinement effects, we found that the 5-layer film is an intrinsic QAH insulator with $\\mathcal {C} = 1 $ , as confirmed by the appearance of an quantized Hall conductance and chiral edge states within the bulk gap (Figs.","4a-c).","We also calculated a 7-layer film, which is also a QAH insulator with $\\mathcal {C} = 1 $ .","The exchange splitting introduced by the magnetic Mn-layers together with the strong SOC effects in the Bi-Te bands cooperatively induce the QAH effect, similar as in the magnetically doped Bi$_2$ Te$_3$ -class TIs [5].","However, the present material has an intrinsic magnetism and does not need uncommon mechanisms (e.g.","the Van-Vleck mechanism [5]) to form ferromagnetism in insulating states.","Moreover, since the magnetic doping is not needed, disorder-induced magnetic domains and potential fluctuations, that deteriorate the QAH effect, are avoided.","Furthermore, the QAH gap of the 5-layer film is 38 meV, greater than the room-temperature thermal energy of 26 meV, enabling a high working temperature.","The thickness dependence behaviors can be understood as follows.","There are intrinsically gapped surface states on both sides of thick films, as obtained for the (001) semi-infinite surface.","These surface gaps are opened by the TRS-breaking field near the surface, which have half quantized Hall conductances $\\sigma _{xy} = e^2/2h$ or $-e^2/2h$ when the magnetism in the surface layer is up- or down-oriented, respectively.","Thus, the Hall conductances of bottom and top surfaces is cancelled in even layers, giving axion insulators with $\\mathcal {C} = 0$ or topological axion states mentioned above; while they get added in odd layers, giving QAH insulators with $\\mathcal {C} = 1$ .","This physical picture, consistent with our calculation results, suggests an oscillation of $\\mathcal {C}$ in even and odd layers.","Thus chiral edge states always appear near step edges (Fig.","4d), which can be used for dissipationless conduction.","Based on this unique feature, the chiral edge states could be selectively patterned by controlling the film thickness or step edges, advantageous for building dissipationless circuits.","Looking back to the history of the TI research, the first- and second-generation TIs are the HgTe/CdTe quantum wells [34] and Bi-Sb alloys, respectively, which are very complex and difficult to study theoretically and experimentally.","Research interests have been increased exponentially since the discovery of the third-generation TIs in the intrinsic Bi$_2$ Te$_3$ -class materials [17], [18].","A very similar situation is faced by the research of magnetic topological physics.","Currently, experimental works are majorally based on magnetically doped TIs and magnetic topological heterostructures, which are quite challenging and have led to little preliminary progress.","Looking forwards, the research progress is expected to be greatly prompted by discovering intrinsic MTQMs that are simple and easy to control.","The vdW layered MnBi$_2$ Te$_4$ -family materials satisfy all these material traits.","More importantly, this material family could host extremely rich topological quantum states in different spatial dimensions (like 3D AFM TIs, time-reversal invariant TIs and topological semimetals, 2D QAH and QSH insulators, etc.)","and are promising for investigating other exotic emerging physics (like Majorana fermions), which are thus perfect next-generation MTQMs for future research.","Figure: Monolayer MnBi 2 _2Te 4 _4 (MBT).","(a) The side view of monolayer MBT.","(b) A schematic diagram of band structure in MBT including thin films and 3D bulk.","(c,d) Band structures of monolayer MBT calculated by HSE06 without/with spin-orbital coupling (SOC).","The blue (green) curve in the (1c) represents the spin-up (spin-down) states.","(e) A schematic diagram of XBT materials family.","The red arrows represent the magnetic moment, whose length and direction represent the magnitude and easy magnetic axis of magnetic atoms.Figure: AFM MBT.","(a) The side view of bulk AFM MBT crystal structure.The black dot O 1 _1 represents inversion center located at the magnetic Mn atoms.","The black dot O 2 _2 represents symmetry operation which combines inversion operator and the lattice translation T 1/2 T_{1/2} along c axis.","The schematic diagram shows MBT is stacked in the way combining the AFM order and ABC-stacking sequence.","Brillouin zone used in the calculation are showed below.","(b), (c) Band structures of 3D bulk AFM MBT without/with SOC.","The parity of wavefunction at high-symmetry points Γ\\Gamma is showed by the minus and plus signs.","Minus (plus) sign means odd (even) parity.","(d) Band gap at Γ\\Gamma calculated by artificially changing the strength of SOC.","Band gap closing happens at artificial SOC strength around 81%.","(e) Wannier charge centers along k 2 k_2 on the k 3 k_3=0 plane.","(f), (g) The calculated surface states of the AFM TI MBT on the (001) and (100) faces, respectively.","The inset of (f) shows gapped surface states on the (001) face clearly, however,the inset of (g) shows no apparent gap on the (100) face.Figure: FM MBT.","(a) Atomic structure of FM MBT.","Dashed rhombus represents the unit cell.","(b) Band structure of FM MBT.","(c) Brillouin zone of FM MBT.","The red and blue points represent a pair of type-II Weyl points W and W ' ^{\\prime }.","(d) Zoom-in band structure around Weyl point W from different directions.","(e) Motion of the sum of WCCs on the sphere around Type-II Weyl point W. (f, g) Fermi arcs on the (100) and (1 ¯00\\bar{1}00) terminations.","Fermi energy is fixed at the energy level of Weyl point W.Figure: MBT thin films.","(a) Band structure, (b) Hall conductance σ xy \\sigma _{xy} as a function of energy and (c) edge states of the 5-layer MBT.","(d) A schematic diagram showing dissipationless edge channels on step edges of MBT."]],"string":"[\n [\n \"Intrinsic magnetic topological insulators in van der Waals layered\\n MnBi$_2$Te$_4$-family materials\"\n ],\n [\n \"Abstract The interplay of magnetism and topology is a key research subject in condensed matter physics and material science, which offers great opportunities to explore emerging new physics, like the quantum anomalous Hall (QAH) effect, axion electrodynamics and Majorana fermions.\",\n \"However, these exotic physical effects have rarely been realized in experiment, due to the lacking of suitable working materials.\",\n \"Here we predict that van der Waals layered MnBi$_2$Te$_4$-family materials show two-dimensional (2D) ferromagnetism in the single layer and three-dimensional (3D) $A$-type antiferromagnetism in the bulk, which could serve as a next-generation material platform for the state-of-art research.\",\n \"Remarkably, we predict extremely rich topological quantum effects with outstanding features in an experimentally available material MnBi$_2$Te$_4$, including a 3D antiferromagnetic topological insulator with the long-sought topological axion states, the type-II magnetic Weyl semimetal (WSM) with simply one pair of Weyl points, and the high-temperature intrinsic QAH effect.\",\n \"These striking predictions, if proved experimentally, could profoundly transform future research and technology of topological quantum physics.\"\n ],\n [\n \"Intrinsic magnetic topological insulators in van der Waals layered MnBi$_2$ Te$_4$ -family materials Jiaheng Li$^{1,2}$ Yang Li$^{1,2}$ Shiqiao Du$^{1,2}$ Zun Wang$^{1,2}$ Bing-Lin Gu$^{1,2,3}$ Shou-Cheng Zhang$^{4}$ Ke He$^{1,2}$ kehe@tsinghua.edu.cn Wenhui Duan$^{1,2,3}$ dwh@phys.tsinghua.edu.cn Yong Xu$^{1,2,5}$ yongxu@mail.tsinghua.edu.cn $^1$ State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, People's Republic of China $^2$ Collaborative Innovation Center of Quantum Matter, Beijing 100084, People's Republic of China $^3$ Institute for Advanced Study, Tsinghua University, Beijing 100084, People's Republic of China $^4$ Department of Physics, McCullough Building, Stanford University, Stanford, California 94305-4045, USA $^5$ RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan The interplay of magnetism and topology is a key research subject in condensed matter physics and material science, which offers great opportunities to explore emerging new physics, like the quantum anomalous Hall (QAH) effect, axion electrodynamics and Majorana fermions.\",\n \"However, these exotic physical effects have rarely been realized in experiment, due to the lacking of suitable working materials.\",\n \"Here we predict that van der Waals layered MnBi$_2$ Te$_4$ -family materials show two-dimensional (2D) ferromagnetism in the single layer and three-dimensional (3D) $A$ -type antiferromagnetism in the bulk, which could serve as a next-generation material platform for the state-of-art research.\",\n \"Remarkably, we predict extremely rich topological quantum effects with outstanding features in an experimentally available material MnBi$_2$ Te$_4$ , including a 3D antiferromagnetic topological insulator with the long-sought topological axion states, the type-II magnetic Weyl semimetal (WSM) with simply one pair of Weyl points, and the high-temperature intrinsic QAH effect.\",\n \"These striking predictions, if proved experimentally, could profoundly transform future research and technology of topological quantum physics.\",\n \"The correlation between topology and symmetry as a central fundamental problem of modern physics has attracted intensive research interests in condensed matter physics and material science since the discovery of topological insulators (TIs) [1], [2].\",\n \"One essential symmetry is the time reversal symmetry (TRS), which is crucial to many kinds of topological quantum states of matter like TIs and is broken in the presence of magnetism.\",\n \"Intriguingly, the interplay between magnetism and topology in materials could generate a variety of exotic topological quantum states [1], [2], [3], including the quantum anomalous Hall (QAH) effect showing dissipationless chiral edge states [4], [5], [6], the topological axion states displaying quantized magnetoelectric effects [7], [8], [9], and Majorana fermions obeying non-abelian statistics [2], [10], [11].\",\n \"In this context, great research effort has been devoted to explore the novel topological quantum physics, which is of profound importance to fundamental science and future technologies, like dissipationless topological electronics and topological quantum computation [1], [2].\",\n \"One key subject that is of crucial significance to the whole research community is to develop topological quantum materials (TQMs) showing the coexistence of topology and other quantum phases (e.g.\",\n \"magnetism, ferroelectricity, charge density wave, and superconductivity), coined “composite TQMs” (CTQMs), which include magnetic TQMs (MTQMs) as an important subset.\",\n \"Currently, very limited number of MTQMs are experimentally available, including magnetically doped TIs and magnetic topological heterostructures [6], [12], [13], [14], whose material fabrication, experimental measurement and property optimization are quite challenging.\",\n \"Since the major topological properties of these existing MTQMs depend sensitively on delicate magnetic doping or proximity effects, only little preliminary experimental progress has been achieved until now, leaving many important physical effects not ready for practical use or still unproved.\",\n \"For instance, the QAH effect was only observed in magnetically doped (Bi$_{x}$ Sb$_{1-x}$ )$_2$ Te$_3$ thin films at very low working temperatures by fine tuning chemical compositions [6], [12].\",\n \"While a previous theory predicted the existence of topological axion states at the surface of a three-dimensional (3D) antiferromagnetic (AFM) TI [15], no 3D AFM TI has been realized experimentally, as far as we know.\",\n \"For future research and applications, people should go beyond the existing strategy for building MTQMs and try to design intrinsic MTQMs in no need of introducing alloy/doping or heterostructures.\",\n \"Noticeably, van der Waals (vdW) layered materials represent a large family of materials with greatly tunable properties by quantum size effects or vdW heterojunctions [16], in which a variety of quantum phases in different spatial dimensions have been discovered by the state-of-the-art research [17], [18], [19], [20], [21], [22].\",\n \"For instance, both two-dimensional (2D) and 3D TI states were previously found in the tetradymite-type Bi$_2$ Te$_3$ -class materials [17], [18], and 2D intrinsic magnetism was recently found in ultrathin films of CrI$_3$ [23] and Cr$_2$ Ge$_2$ Te$_6$ [24] .\",\n \"However, these topological materials are not magnetic; those magnetic material are not topological.\",\n \"It is highly desirable to incorporate the magnetic and topological states together into the same vdW material, so as to obtain layered intrinsic MTQMs, which are able to inherit advantages of vdW-family materials.\",\n \"In this work, based on first-principles (FP) calculations, we find a series of layered intrinsic MTQMs from the tetradymite-type MnBi$_2$ Te$_4$ -related ternary chalcogenides (MB$_2$ T$_4$ : M = transition-metal or rare-earth element, B = Bi or Sb, T = Te, Se or S), in which the intralayer exchange coupling is ferromagnetic (FM), giving 2D ferromagnetism in their septuple layer (SL); while the interlayer exchange coupling is antiferromagnetic, forming 3D $A$ -type antiferromagnetism in their vdW layered bulk.\",\n \"Strikingly, we predict that plenty of novel 2D and 3D intrinsic magnetic topological states with outstanding features could be naturally realized in an experimentally available material MnBi$_2$ Te$_4$ , including the 3D AFM TI in the bulk together with the long-sought topological axion states on the (001) surface, large-gap intrinsic QAH insulators ($E_{\\\\textit {g}} \\\\sim $ 38 meV) with dissipaltionless and patternable chiral edge states in thin films, the type-II magnetic WSM with a single pair of Weyl points in the FM bulk, and possibly Majorana fermions if interacting with superconductivity.\",\n \"The tetradymite-type MnBi$_2$ Te$_4$ crystalizes in a rhombohedral layered structure with the space group $R\\\\overline{3}m$ , and each layer has a triangular lattice with atoms ABC-stacked along the out-of-plane direction, the same as Bi$_2$ Te$_3$ .\",\n \"Slightly differently, one layer MnBi$_2$ Te$_4$ includes seven atoms in a unit cell, forming a Te-Bi-Te-Mn-Te-Bi-Te SL, which can be viewed as intercalating a Mn-Te bilayer into the center of a Bi$_2$ Te$_3$ quintuple layer (Fig.\",\n \"1a).\",\n \"Noticeably, the mixing between Mn and Bi in this compound would create unstable valence states Mn$^{+3}$ and Bi$^{+2}$ , which is energetically unfavorable.\",\n \"Thus the formation of alloys could be avoided, leading to stoichiometric compounds.\",\n \"This physical mechanism also explains the stability of compounds like SnBi$_2$ Te$_4$ and PbBi$_2$ Te$_4$ and is applicable to many other MB$_2$ T$_4$ -family materials.\",\n \"Importantly, our FP calculations predicted that a series of MB$_2$ T$_4$ compounds are energetically and dynamically stable, including M = Ti, V, Mn, Ni, Eu, etc., as confirmed by FP total-energy and phonon calculations [25], which could be fabricated by experiment.\",\n \"These stable compounds are characterized by an insulating band gap in their single layers, offering a collection of layered intrinsic magnetic (topological) insulators.\",\n \"In fact, MnBi$_2$ Te$_4$ is now experimentally available [26], which is selected to introduce properties of the material family.\",\n \"Mn has a valence charge of $+$ 2 by losing its two 4$s$ electrons.\",\n \"The remaining five 3$d$ electrons fill up the spin-up Mn-$d$ levels according to the Hund's rule, introducing 5 $\\\\mu _{B}$ magnetic moment (mostly from Mn) per unit cell.\",\n \"By comparing different magnetic structures, we found that the magnetic ground state is a 2D ferromagnetism with an out-of-plane easy axis in the single layer (depicted Fig.\",\n \"1a), in agreement with the previous work [27].\",\n \"Each Mn atom is bonded with six neighboring Te atoms, which form a slightly distorted edge-sharing octahedron.\",\n \"According to the Goodenough-Kanamori rule, the superexchange interactions between Mn-Te-Mn with a bonding angle of $\\\\sim 94^{\\\\circ }$ are ferromagnetic, similar as in CrI$_3$ and Cr$_2$ Ge$_2$ Te$_6$ [23], [24].\",\n \"Intralayer ferromagnetic exchange couplings was also found in other stable MB$_2$ T$_4$ -family members, but the easy axis can be varied to in-plane (e.g.\",\n \"for M = V), displaying rich 2D magnetic features.\",\n \"Intriguingly, magnetic and topological states are well incorporated together into MnBi$_2$ Te$_4$ , where Mn introduces magnetism and the Bi-Te layers could generate topological states similar as Bi$_2$ Te$_3$ [17], as schematically depicted in Fig.\",\n \"1b.\",\n \"The exchange splitting between spin-up and spin-down Mn $d$ bands are extremely large ($>$ 7 eV) caused by the large magnetic moment of Mn.\",\n \"Thus Mn $d$ -bands are far away from the band gap, and only Bi/Te $p$ -bands are close to the Fermi level.\",\n \"The single layer has a 0.73 eV direct (1.36 eV indirect) band gap when including (excluding) spin-orbit coupling (SOC) (Figs.\",\n \"1c and 1d).\",\n \"While the single layer is topologically trivial ferromagnetic insulator, extremely interesting topological quantum physics emerges in the bulk and thin films, as we will demonstrate.\",\n \"For the layered bulk, we found that the interlayer magnetic coupling is AFM, giving a $A$ -type AFM ground state (depicted in Fig.\",\n \"2a) [25], which is similar as in Cr$_2$ Ge$_2$ Te$_6$ [24] and explained by the interlayer super-superexchange coupling.\",\n \"The spatial inversion symmetry $P$ (i.e.\",\n \"$P_1$ centered at O$_1$ ) is preserved, but the TRS $\\\\Theta $ gets broken.\",\n \"There exist two new symmetries: $P_2 \\\\Theta $ and $S=\\\\Theta T_{1/2}$ , where $P_2$ is an inversion operation centered at O$_2$ and $T_{1/2}$ is a lattice translation (depicted in Fig.\",\n \"2a).\",\n \"The band structures with and without SOC are presented in Figs.\",\n \"2b and 2c.\",\n \"Here every band is (at least) doubly degenerate, which is ensured by the $P_2 \\\\Theta $ symmetry [28].\",\n \"In the presence of $S$ , a $\\\\mathcal {Z}_2$ classification becomes feasible [15].\",\n \"However, in contrast to the time-reversal invariant case, there is a $\\\\mathcal {Z}_2$ invariant for the $k_z = 0$ plane but not for the $k_z = \\\\pi $ plane.\",\n \"$\\\\mathcal {Z}_2 = 1$ corresponds to a novel 3D AFM TI phase [15], which has not been experimentally confirmed.\",\n \"Here the parity criteria can be applied to determine $\\\\mathcal {Z}_2$ due to the $P_1$ symmetry [29].\",\n \"Interestingly, the parities of the valence band maximum (VBM) and conduction band minimum (CBM) at $\\\\Gamma $ are opposite, and both change signs by the SOC effects (Figs.\",\n \"2b and 2c), implying a band inversion.\",\n \"By varying the SOC strength, the band gap first closes and then reopens at $\\\\Gamma $ (Fig.\",\n \"2d), showing simply one band inversion and thus implying a topological phase transition.\",\n \"The SOC-induced band reversal is happened between Bi $p_z^+$ and Te $p_z^-$ , essentially the same as for Bi$_2$ Te$_3$ [17].\",\n \"Our Wannier charge center (WCC) calculations for the $k_z = 0$ plane revealed $\\\\mathcal {Z}_2 = 1$ ($\\\\mathcal {Z}_2 = 0$ ) for with (without) SOC, confirming that MnBi$_2$ Te$_4$ is a 3D AFM TI.\",\n \"The global band gap is $\\\\sim $ 0.16 eV direct at $Z$ , and the direct gap at $\\\\Gamma $ is $\\\\sim $ 0.18 eV.\",\n \"One prominent feature of the AFM TI is the existence of 2D gapless surface states protected by $S$ , which is confirmed by the surface-state calculations (Figs.\",\n \"2f and 2g).\",\n \"The surface states are indeed gapless on the (100) termination.\",\n \"However, they become gapped on the (001) termination due to the $S$ symmetry breaking.\",\n \"The intrinsically gapped (001) surfaces are promising for probing the long-sought topological axion states, which give the topological quantized magnetoelectric effect related to an axion field with $\\\\theta = \\\\pi $ [7], [8], [9], [15].\",\n \"Previous work proposed to probe the novel states by adding opposite out-of-plane ferromagnetism onto the bottom and top surfaces of 3D TIs [7], [14].\",\n \"Such kind of TRS-broken surface states are naturally provided by even-layer MnBi$_2$ Te$_4$ films (when with negligible hybridizations between top and bottom surface states), benefitting from their $A$ -type AFM structure.\",\n \"An additional requirement is to open a band gap at side surfaces, which is realized in relatively thin films or by breaking the $S$ symmetry on the side surfaces (e.g.\",\n \"controlling surface morphology).\",\n \"The simplified proposal together with the suitable material candidate MnBi$_2$ Te$_4$ could greatly facilitate the research of axion electrodynamics.\",\n \"The AFM ground state of MnBi$_2$ Te$_4$ could be tuned, for instance, by applying an external magnetic field, to other magnetic structures.\",\n \"Then the material symmetry changes, leading to distinct topological phases.\",\n \"This concept is demonstrated by studying the simple FM ordering along the out-of-plane direction (Fig.\",\n \"3a).\",\n \"The FM structure has $P_1$ but neither $\\\\Theta $ nor $P_2 \\\\Theta $ , leading to spin-split bands, nonzero Berry curvatures and possibly nonzero topological Chern numbers that correspond to novel topological phases, like 3D QAH insulators [30] and WSMs [31], [32], [33], [22], [3].\",\n \"The band structure of FM MnBi$_2$ Te$_4$ (Fig.\",\n \"3b) displays a pair of band crossings at W/W$^{\\\\prime }$ along the $\\\\overline{\\\\textit {Z}}$ -$\\\\Gamma $ -Z line.\",\n \"The band crossings are induced by interlayer orbital hybridizations and protected by the $C_3$ rotational symmetry.\",\n \"Our WCC calculations found that W is a momentum-space monopole with a topological charge of +1 (i.e.\",\n \"an Berry phase of $2\\\\pi $ ) (Fig.\",\n \"3e), and its time-reversal partner W$^{\\\\prime }$ has an opposite topological charge of -1, indicating that the system is a topological WSM.\",\n \"While in most of the momentum space the electron pocket is located above the hole pocket (e.g.\",\n \"along the F-W-L line), the Weyl cones get tilted along the $\\\\overline{\\\\textit {Z}}$ -$\\\\Gamma $ -Z line (Fig.\",\n \"3d), leaving some part of the electron pocket below the hole pocket, which is the characteristic feature of the type-II WSM [22].\",\n \"In contrast to the time-reversal-invariant WSMs that must have even pairs of Weyl points, this ferromagnetic WSM represents the simplest one, hosting only a pair of Weyl points.\",\n \"Moreover, our surface-state calculations clearly demonstrated the existence of Fermi arcs on the (100) and ($\\\\bar{1}$ 00) terminations (Figs.\",\n \"3f and 3g), which is the fingerprint of the WSM.\",\n \"Importantly, the Weyl points are well separated in the momentum space and very close to the Fermi level, advantageous for experimental observations.\",\n \"The vdW layered materials are featured by tunable quantum size effects.\",\n \"For AFM MnBi$_2$ Te$_4$ films, even layers do not possess $P$ and $\\\\Theta $ symmetries, but have $P_2 \\\\Theta $ , ensuring double degeneracy in every band.\",\n \"Differently, odd layers have $P$ but neither $\\\\Theta $ nor $P_2 \\\\Theta $ , leading to spin-split bands.\",\n \"The different symmetries lead to distinct topological properties in even and odd layers.\",\n \"Specifically, the topological Chern number $\\\\mathcal {C} = 0$ is required by $P_2 \\\\Theta $ in even layers, $\\\\mathcal {C} \\\\ne 0$ is allowed in odd layers.\",\n \"In contrast to the single and 3-layer films, where a trivial insulating gap is opened by quantum confinement effects, we found that the 5-layer film is an intrinsic QAH insulator with $\\\\mathcal {C} = 1 $ , as confirmed by the appearance of an quantized Hall conductance and chiral edge states within the bulk gap (Figs.\",\n \"4a-c).\",\n \"We also calculated a 7-layer film, which is also a QAH insulator with $\\\\mathcal {C} = 1 $ .\",\n \"The exchange splitting introduced by the magnetic Mn-layers together with the strong SOC effects in the Bi-Te bands cooperatively induce the QAH effect, similar as in the magnetically doped Bi$_2$ Te$_3$ -class TIs [5].\",\n \"However, the present material has an intrinsic magnetism and does not need uncommon mechanisms (e.g.\",\n \"the Van-Vleck mechanism [5]) to form ferromagnetism in insulating states.\",\n \"Moreover, since the magnetic doping is not needed, disorder-induced magnetic domains and potential fluctuations, that deteriorate the QAH effect, are avoided.\",\n \"Furthermore, the QAH gap of the 5-layer film is 38 meV, greater than the room-temperature thermal energy of 26 meV, enabling a high working temperature.\",\n \"The thickness dependence behaviors can be understood as follows.\",\n \"There are intrinsically gapped surface states on both sides of thick films, as obtained for the (001) semi-infinite surface.\",\n \"These surface gaps are opened by the TRS-breaking field near the surface, which have half quantized Hall conductances $\\\\sigma _{xy} = e^2/2h$ or $-e^2/2h$ when the magnetism in the surface layer is up- or down-oriented, respectively.\",\n \"Thus, the Hall conductances of bottom and top surfaces is cancelled in even layers, giving axion insulators with $\\\\mathcal {C} = 0$ or topological axion states mentioned above; while they get added in odd layers, giving QAH insulators with $\\\\mathcal {C} = 1$ .\",\n \"This physical picture, consistent with our calculation results, suggests an oscillation of $\\\\mathcal {C}$ in even and odd layers.\",\n \"Thus chiral edge states always appear near step edges (Fig.\",\n \"4d), which can be used for dissipationless conduction.\",\n \"Based on this unique feature, the chiral edge states could be selectively patterned by controlling the film thickness or step edges, advantageous for building dissipationless circuits.\",\n \"Looking back to the history of the TI research, the first- and second-generation TIs are the HgTe/CdTe quantum wells [34] and Bi-Sb alloys, respectively, which are very complex and difficult to study theoretically and experimentally.\",\n \"Research interests have been increased exponentially since the discovery of the third-generation TIs in the intrinsic Bi$_2$ Te$_3$ -class materials [17], [18].\",\n \"A very similar situation is faced by the research of magnetic topological physics.\",\n \"Currently, experimental works are majorally based on magnetically doped TIs and magnetic topological heterostructures, which are quite challenging and have led to little preliminary progress.\",\n \"Looking forwards, the research progress is expected to be greatly prompted by discovering intrinsic MTQMs that are simple and easy to control.\",\n \"The vdW layered MnBi$_2$ Te$_4$ -family materials satisfy all these material traits.\",\n \"More importantly, this material family could host extremely rich topological quantum states in different spatial dimensions (like 3D AFM TIs, time-reversal invariant TIs and topological semimetals, 2D QAH and QSH insulators, etc.)\",\n \"and are promising for investigating other exotic emerging physics (like Majorana fermions), which are thus perfect next-generation MTQMs for future research.\",\n \"Figure: Monolayer MnBi 2 _2Te 4 _4 (MBT).\",\n \"(a) The side view of monolayer MBT.\",\n \"(b) A schematic diagram of band structure in MBT including thin films and 3D bulk.\",\n \"(c,d) Band structures of monolayer MBT calculated by HSE06 without/with spin-orbital coupling (SOC).\",\n \"The blue (green) curve in the (1c) represents the spin-up (spin-down) states.\",\n \"(e) A schematic diagram of XBT materials family.\",\n \"The red arrows represent the magnetic moment, whose length and direction represent the magnitude and easy magnetic axis of magnetic atoms.Figure: AFM MBT.\",\n \"(a) The side view of bulk AFM MBT crystal structure.The black dot O 1 _1 represents inversion center located at the magnetic Mn atoms.\",\n \"The black dot O 2 _2 represents symmetry operation which combines inversion operator and the lattice translation T 1/2 T_{1/2} along c axis.\",\n \"The schematic diagram shows MBT is stacked in the way combining the AFM order and ABC-stacking sequence.\",\n \"Brillouin zone used in the calculation are showed below.\",\n \"(b), (c) Band structures of 3D bulk AFM MBT without/with SOC.\",\n \"The parity of wavefunction at high-symmetry points Γ\\\\Gamma is showed by the minus and plus signs.\",\n \"Minus (plus) sign means odd (even) parity.\",\n \"(d) Band gap at Γ\\\\Gamma calculated by artificially changing the strength of SOC.\",\n \"Band gap closing happens at artificial SOC strength around 81%.\",\n \"(e) Wannier charge centers along k 2 k_2 on the k 3 k_3=0 plane.\",\n \"(f), (g) The calculated surface states of the AFM TI MBT on the (001) and (100) faces, respectively.\",\n \"The inset of (f) shows gapped surface states on the (001) face clearly, however,the inset of (g) shows no apparent gap on the (100) face.Figure: FM MBT.\",\n \"(a) Atomic structure of FM MBT.\",\n \"Dashed rhombus represents the unit cell.\",\n \"(b) Band structure of FM MBT.\",\n \"(c) Brillouin zone of FM MBT.\",\n \"The red and blue points represent a pair of type-II Weyl points W and W ' ^{\\\\prime }.\",\n \"(d) Zoom-in band structure around Weyl point W from different directions.\",\n \"(e) Motion of the sum of WCCs on the sphere around Type-II Weyl point W. (f, g) Fermi arcs on the (100) and (1 ¯00\\\\bar{1}00) terminations.\",\n \"Fermi energy is fixed at the energy level of Weyl point W.Figure: MBT thin films.\",\n \"(a) Band structure, (b) Hall conductance σ xy \\\\sigma _{xy} as a function of energy and (c) edge states of the 5-layer MBT.\",\n \"(d) A schematic diagram showing dissipationless edge channels on step edges of MBT.\"\n ]\n]"},"id":{"kind":"string","value":"1808.08608"}}},{"rowIdx":280,"cells":{"text":{"kind":"list like","value":[["Spontaneous and Gravitational Baryogenesis"],["Abstract Some problems of spontaneous and gravitational baryogenesis are discussed.","Gravity modification due to the curvature dependent term in gravitational baryogensis scenario is considered.","It is shown that the interaction of baryonic fields with the curvature scalar leads to strong instability of the gravitational equations of motion and as a result to noticeable distortion of the standard cosmology."],["Introduction","Observations show that at least the region of the Universe around us is matter-dominated.","Though we understand how the matter-antimatter asymmetry may be created, the concrete mechanism is yet unknown.","The amount of antimatter is very small and it can be explained as the result of high energy collisions in space.","The existence of large regions of antimatter in our neighbourhood would produce high energy radiation as a consequence of matter-antimatter annihilation, which is not observed.","Any initial asymmetry at inflation could not solve the problem of observed excess of matter over antimatter, because the energy density associated with baryonic number would not allow for sufficiently long inflation.","On the other hand, matter and antimatter seem to have similar properties and therefore we could expect a matter-antimatter symmetric Universe.","A satisfactory model of our Universe should be able to explain the origin of the local observed matter-antimatter asymmetry.","The term baryogenesis means the generation of the asymmetry between baryons (basically protons and neutrons) and antibaryons (antiprotons and antineutrons).","In 1967 Andrey Sakharov pointed out 3 ingredients, today known as Sakharov principles, to produce a matter-antimatter asymmetry from an initially symmetric Universe.","These conditions include: 1) non-conservation of baryonic number; 2) breaking of symmetry between particles and antiparticles; 3) deviation from thermal equilibrium.","However, not all of three Sakharov principles are strictly necessary.","In what follows we briefly discuss some features of spontaneous baryogenesis (SBG) and concentrate in more detail on gravitational baryogenesis (GBG).","Both these mechanisms do not demand an explicit C and CP violation and can proceed in thermal equilibrium.","Moreover, they are usually most efficient in thermal equilibrium.","The statement that the cosmological baryon asymmetry can be created by spontaneous baryogenesis in thermal equilibrium was mentioned in the original paper by Cohen and Kaplan[1] and developed in subsequent papers[2], [3], for review see[4], [5].","The term \"spontaneous\" is related to spontaneous breaking of a global $U(1)$ -symmetry, which ensures the conservation of the total baryonic number in the unbroken phase.","This symmetry is supposed to be spontaneously broken and in the broken phase the Lagrangian density acquires the additional term ${\\cal L}_{SB} = (\\partial _{\\mu } \\theta ) J^{\\mu }_B\\, ,$ where $\\theta $ is the Goldstone field and $J^{\\mu }_B$ is the baryonic current of matter fields, which becomes non-conserved.","For a spatially homogeneous field, $\\theta = \\theta (t)$ , the Lagrangian is reduced to the simple form ${\\cal L}_{SB} = \\dot{\\theta }\\, n_B\\,, \\ \\ \\ \\ n_B\\equiv J^0_B,$ where time component of a current is the baryonic number density of matter, so it is tempting to identify ${\\dot{\\theta }}$ with the chemical potential, $ \\mu _B$ , of the corresponding system.","However, such identification is questionable and depends upon the representation chosen for the fermionic fields[6], [7].","It is heavily based on the assumption ${\\dot{\\theta }\\approx const}$ , which is relaxed in the work[8].","But still the scenario is operative and presents a beautiful possibility to create an excess of particles over antiparticles in the Universe.","Subsequently the idea of gravitational baryogenesis (GBG) was put forward [9], where the scenario of SBG was modified by the introduction of the coupling of the baryonic current to the derivative of the curvature scalar $R$ : ${\\cal L}_{GBG} = \\frac{1}{M^2} (\\partial _\\mu R ) J^\\mu _B\\, ,$ where $M$ is a constant parameter with the dimension of mass.","In the presented talk we demonstrate that the addition of the curvature dependent term (REF ) to the Hilbert-Einstein Lagrangian of General Relativity (GR) leads to higher order gravitational equations of motion, which are strongly unstable with respect to small perturbations.","The effects of this instability may drastically distort not only the usual cosmological history, but also the standard Newtonian gravitational dynamics.","We discovered such instability for scalar baryons [10] and found similar effect for the more usual spin one-half baryons (quarks) [11]."],["Gravitational baryogenesis with scalar baryons ","Let us start from the model where baryonic number is carried by scalar field $\\phi $ with potential $U(\\phi , \\phi ^* )$ .","An example with baryonic current of fermions will be considered in the next section.","The action of the scalar model has the form: $A = \\int d^4 x\\, \\sqrt{-g} \\left[ \\frac{m_{Pl}^2}{16\\pi } R + \\frac{1}{M^2} (\\partial _{\\mu } R) J^{\\mu } -g^{\\mu \\nu } \\partial _{\\mu }\\phi \\, \\partial _{\\nu }\\phi ^* + U(\\phi , \\phi ^*)\\right] - A_m\\, ,$ where $m_{Pl}=1.22\\cdot 10^{19}$ GeV is the Planck mass, $A_m$ is the matter action, $J^\\mu = g^{\\mu \\nu }J_\\nu $ , and $g^{\\mu \\nu }$ is the metric tensor of the background space-time.","We assume that initially the metric has the usual GR form and study the emergence of the corrections due to the instability described below.","In contrast to scalar electrodynamics, the baryonic current of scalars is not uniquely defined.","In electrodynamics the form of the electric current is dictated by the conditions of gauge invariance and current conservation, which demand the addition to the current of the so called sea-gull term proportional to $e^2 A_\\mu |\\phi |^2$ , where $A_\\mu $ is the electromagnetic potential.","On the other hand, a local $U(1)$ -symmetry is not imposed on the theory determined by action (REF ).","It is invariant only with respect to a $U(1)$ transformations with a constant phase.","As a result, the baryonic current of scalars is considerably less restricted.","In particular, we can add to the current an analogue of the sea-gull term, $ \\sim (\\partial _\\mu R)\\,|\\phi |^2 $ , with an arbitrary coefficient.","In our paper [10] we study the following two extreme possibilities, when the sea-gull term is absent and the current is not conserved, or the sea-gull term is included with the coefficient ensuring current conservation.","In both cases no baryon asymmetry can be generated without additional interactions.","It is trivially true in the second case, when the current is conserved, but it is also true in the first case despite the current non-conservation, simply because the non-zero divergence $D_\\mu J^\\mu $ does not change the baryonic number of $\\phi $ but only leads to redistribution of particles $\\phi $ in the phase space.","So to create any non-zero baryon asymmetry we have to introduce an interaction of $\\phi $ with other particles which breaks conservation of $B$ by making the potential $U$ non-invariant with respect to the phase rotations of $\\phi $ , as it is described below.","If the potential $U(\\phi )$ is not invariant with respect to the $U(1)$ -rotation, $\\phi \\rightarrow \\exp {(i \\beta )} \\phi $ , the baryonic current defined in the usual way $J_{ \\mu } = i q (\\phi ^* \\partial _{\\mu }\\phi - \\phi \\partial _{\\mu }\\phi ^*)$ is not conserved.","Here $q$ is the baryonic number of $\\phi $ and we omitted index $B$ in current $J_{\\mu }$ .","With this current and Lagrangian (REF ) the equation for the curvature scalar, $R$ , takes the form: $\\frac{m_{Pl}^2}{16\\pi }\\, R + \\frac{1}{M^2}\\left[ (R + 3 D^2) D_{\\alpha } J^{ \\alpha } + J^{\\alpha } \\,D_{\\alpha }R \\right] -D_{\\alpha } \\phi \\, D^{\\alpha } \\phi ^* + 2 U(\\phi ) = - \\frac{1}{2} \\, T_{\\mu }^{\\mu }\\, ,$ where $D_\\mu $ is the covariant derivative in metric $g_{\\mu \\nu }$ (of course, for scalars $D_\\mu = \\partial _\\mu $ ) and $T_{\\mu \\nu }$ is the energy-momentum tensor of matter obtained from action $A_m$ .","According to definition (REF ), the current divergence is: $D_{\\mu } J^{\\mu } = \\frac{2q^2}{M^2} \\left[ D_{\\mu } R\\, (\\phi ^* D^{\\mu }\\phi + \\phi D^{\\mu }\\phi ^*) + |\\phi |^2 D^2 R \\right]+ i q \\left(\\phi \\frac{\\partial U}{\\partial \\phi } - \\phi ^* \\frac{\\partial U}{\\partial \\phi ^*} \\right)\\,.$ If the potential of $U$ is invariant with respect to the phase rotation of $\\phi $ , i.e.","$U = U(|\\phi |)$ , the last term in this expression disappears.","Still the current remains non-conserved, but this non-conservation does not lead to any cosmological baryon asymmetry.","Indeed, the current non-conservation is proportional to the product $\\phi ^* \\phi $ , so it can produce or annihilate an equal number of baryons and antibaryons.","To create cosmological baryon asymmetry we need to introduce new types of interactions, for example, the term in the potential of the form: $ U_4 = \\lambda _4 \\phi ^4 + \\lambda _4^* \\phi ^{*4} $ .","This potential is surely non invariant w.r.t.","the phase rotation of $\\phi $ and can induce the B-non-conserving process of transition of two scalar baryons into two antibaryons, $2 \\phi \\rightarrow 2 \\bar{\\phi }$ .","Let us consider solution of the above equation of motion in cosmology.","The metric of the spatially flat cosmological FRW background can be taken as: $ds^2=dt^2 - a^2(t) d{\\bf r}^2\\, .$ In the homogeneous case the equation for the curvature scalar (REF ) takes the form: $\\frac{m_{Pl}^2}{16\\pi }\\, R + \\frac{1}{M^2}\\left[ (R + 3 \\partial _t^2 + 9 H \\partial _t) D_{\\alpha } J^{\\alpha } +\\dot{R} \\, J^0 \\right]= - \\frac{ T^{(tot)}}{2} \\,,$ where $J^0$ is the baryonic number density of the $\\phi $ -field, $H = \\dot{a}/a$ is the Hubble parameter, and $T^{(tot)}$ is the trace of the energy-momentum tensor of matter including contribution from the $\\phi $ -field.","In the homogeneous and isotropic cosmological plasma $T^{(tot)} = \\rho - 3 P\\, ,$ where $\\rho $ and $P$ are respectively the energy density and the pressure of the plasma.","For relativistic plasma $\\rho = \\pi ^2 g_* T^4/30$ with $T$ and $g_*$ being the plasma temperature and the number of particle species in the plasma.","The Hubble parameter is expressed through $\\rho $ as $H^2 = 8\\pi \\rho /(3m_{Pl}^2) \\sim T^4/m_{Pl}^2$ .","The covariant divergence of the current is given by the expression (REF ).","In the homogeneous case we are considering it takes the form: $D_{\\alpha } J^{\\alpha } = \\frac{2q^2}{M^2} \\left[ \\dot{R}\\, (\\phi ^* \\dot{\\phi }+ \\phi \\dot{\\phi }^*) +(\\ddot{R} + 3H \\dot{R})\\, \\phi ^*\\phi \\right]+ i q \\left(\\phi \\frac{\\partial U}{\\partial \\phi } - \\phi ^* \\frac{\\partial U}{\\partial \\phi ^*} \\right)\\,.$ To derive the equation of motion for the classical field $R$ in the cosmological plasma we have to take the expectation values of the products of the quantum operators $\\phi $ , $\\phi ^*$ , and their derivatives.","Performing the thermal averaging, we find $\\langle \\phi ^* \\phi \\rangle = \\frac{T^2}{12}\\, , \\ \\ \\ \\langle \\phi ^* \\dot{\\phi }+ \\dot{\\phi }^* \\phi \\rangle = 0\\, .$ Substituting these average values into Eq.","(REF ) and neglecting the last term in Eq.","(REF ) we obtain the fourth order differential equation: $\\frac{m_{Pl}^2}{16\\pi }\\, R +\\frac{q^2}{6 M^4} \\left(R + 3 \\partial _t^2 + 9 H \\partial _t \\right)\\left[\\left(\\ddot{R} + 3H \\dot{R}\\right) T^2 \\right] +\\frac{1}{M^2} \\dot{R} \\, \\langle J^0 \\rangle = - \\frac{T^{(tot)}}{2} \\, .$ Here $\\langle J^0 \\rangle $ is the thermal average value of the baryonic number density of $\\phi $ .","It is assumed to be zero initially and generated as a result of GBG.","We neglect this term, since it is surely small initially and probably subdominant later.","Anyhow it does not noticeably change the exponential rise of $R$ at the onset of the instability.","Eq.","(REF ) can be further simplified if the variation of $R(t)$ is much faster than the universe expansion rate or in other words $\\ddot{R} / \\dot{R} \\gg H$ .","Correspondingly the temperature may be considered adiabatically constant.","The validity of these assumption is justified a posteriori after we find the solution for $R(t)$ .","Keeping only the linear in $R$ terms and neglecting higher powers of $R$ , such as $R^2$ or $H R$ , we obtain the linear differential equation of the fourth order: $\\frac{d^4 R}{dt^4} + \\mu ^4 R = - \\frac{1}{2} \\, T^{(tot)}\\,,\\ {\\rm where }\\ \\ \\mu ^4 = \\frac{m_{Pl}^2 M^4}{8 \\pi q^2 T^2}\\,.$ The homogeneous part of this equation has exponential solutions $R \\sim \\exp (\\lambda t)$ with $\\lambda = | \\mu | \\exp \\left( i\\pi /4 + i \\pi n /2 \\right),$ where $n = 0,1,2,3$ .","There are two solutions with positive real parts of $\\lambda $ .","This indicates that the curvature scalar is exponentially unstable with respect to small perturbations, so $R$ should rise exponentially fast with time and quickly oscillate around this rising function.","Now we need to check if the characteristic rate of the perturbation explosion is indeed much larger than the rate of the universe expansion, that is: $(Re\\, \\lambda )^4 > H^4 = \\left( \\frac{ 8\\pi \\rho }{ 3 m^2_{Pl}}\\right)^2 = \\frac{16 \\pi ^6 g_*^2}{2025}\\,\\frac{ T^8}{m_{Pl}^4},$ where $\\rho = \\pi ^2 g_* T^4 /30$ is the energy density of the primeval plasma at temperature $T$ and $g_* \\sim 10 - 100$ is the number of relativistic degrees of freedom in the plasma.","This condition is fulfilled if $\\frac{2025}{2^9 \\pi ^7 q^2 g_*^2}\\frac{m_{Pl}^6 M^4}{T^{10}} > 1\\,,$ or, roughly speaking, if $T \\le m_{Pl}^{3/5} M^{2/5} $ .","Let us stress that at these temperatures the instability is quickly developed and the standard cosmology would be destroyed.","If we want to preserve the successful big bang nucleosynthesis (BBN) results and impose the condition that the development of the instability was longer than the Hubble time at the BBN epoch at $T \\sim 1 $ MeV, then $M$ should be extremely small, $M < 10^{-32}$ MeV.","The desire to keep the standard cosmology at smaller $T$ would demand even tinier $M$ .","A tiny $M$ leads to a huge strength of coupling (REF ).","It surely would lead to pronounced effects in stellar physics."],["Gravitational baryogenesis with fermions ","Let us now generalize results, obtained for scalar baryons, to realistic fermions.","We start from the action in the form $A= \\int d^4x \\sqrt{-g} \\left[\\frac{m_{Pl}^2}{16 \\pi } \\,R - {\\cal L}_{m}\\right]\\,$ with $ \\nonumber {\\cal L}_{m} &= &\\frac{i}{2} (\\bar{Q} \\gamma ^\\mu \\nabla _\\mu Q - \\nabla _\\mu \\bar{Q}\\, \\gamma ^\\mu Q) - m_Q\\bar{Q}\\,Q\\\\&+&\\frac{i}{2} (\\bar{L} \\gamma ^\\mu \\nabla _\\mu L - \\nabla _\\mu \\bar{L} \\gamma ^\\mu L)- m_L\\bar{L}\\,L \\\\ \\nonumber &+& \\frac{g}{m_X^2}\\left[(\\bar{Q}\\,Q^c)(\\bar{Q} L) + (\\bar{Q}^cQ)(\\bar{L} Q) \\right]+ \\frac{f}{m_0^2} (\\partial _{\\mu } R) J^{\\mu } + {\\cal L}_{other}\\,,$ where $Q$ is the quark (or quark-like) field with non-zero baryonic number, $L$ is another fermionic field (lepton), $\\nabla _\\mu $ is the covariant derivative of Dirac fermion in tetrad formalism.","$m_0$ is a constant parameter with dimension of mass and $f$ is dimensionless coupling constant which is introduced to allow for an arbitrary sign of the curvature dependent term in the above expression.","$J^{\\mu } = \\bar{Q} \\gamma ^{\\mu } Q$ is the quark current with $\\gamma ^{\\mu }$ being the curved space gamma-matrices, ${\\cal L}_{other}$ describes all other forms of matter.","The four-fermion interaction between quarks and leptons is introduced to ensure the necessary non-conservation of the baryon number with $m_X$ being a constant parameter with dimension of mass and $g$ being a dimensionless coupling constant.","In grand unified theories $m_X$ may be of the order of $10^{14}-10^{15}$ GeV.","Varying the action (REF ) over metric, $g^{\\mu \\nu }$ , and taking trace with respect to $\\mu $ and $\\nu $ , we obtain the following equation of motion for the curvature scalar: $ \\nonumber - \\frac{m_{Pl}^2}{8\\pi } R &=& m_{Q} \\bar{Q} Q + m_L \\bar{L} L+\\frac{2g}{m_X^2}\\left[(\\bar{Q}\\,Q^c)(\\bar{Q} L) + (\\bar{Q}^cQ)(\\bar{L} Q) \\right] \\\\&-&\\frac{2f}{m_0^2} (R + 3D^2) D_{\\alpha } J^{\\alpha } + T_{other}\\,,$ where $T_{other} $ is the trace of the energy momentum tensor of all other fields.","At relativistic stage, when masses are negligible, we can take $T_{matter} = 0$ .","The average expectation value of the interaction term proportional to $g$ is also small, so the contribution of all matter fields may be neglected.","As we see in what follows, kinetic equation leads to an explicit dependence on $R$ of the current divergence, $D_\\alpha J^\\alpha $ , if the current is not conserved.","As a result we obtain 4th order equation for $R$ .","As previously, we study solutions of Eq.","(REF ) in cosmology in homogeneous and isotropic FRW background with the metric $ds^2=dt^2 - a^2(t) d{\\bf r}^2 $ .","The curvature is a function only of time and the covariant derivative acting on a vector $V^\\alpha $ , which depends only on time and has only time component, has the form: $D_\\alpha V^\\alpha = (\\partial _t + 3 H) V^t ,$ where $H = \\dot{a}/a$ is the Hubble parameter.","As an example let us consider the reaction $q_1 + q_2 \\leftrightarrow \\bar{q}_3 + l_4$ , where $q_1$ and $q_2$ are quarks with momenta $q_1$ and $q_2$ , while $\\bar{q}_3$ and $l_4$ are antiquark and lepton with momenta $q_3$ and $l_4$ .","We use the same notations for the particle symbol and for the particle momentum.","The kinetic equation for the variation of the baryonic number density $n_B \\equiv J^t$ through this reaction in the FRW background has the form: $(\\partial _t +3 H) n_B = I_B^{coll},$ where the collision integral for space and time independent interaction is equal to: $&&I^{coll}_B =- 3 B_q (2\\pi )^4 \\int \\,d\\nu _{q_1,q_2} \\,d\\nu _{\\bar{q}_3, l_4}\\delta ^4 (q_1 +q_2 -q_3 - l_4)\\nonumber \\\\&& \\left[ |A( q_1+q_2\\rightarrow \\bar{q}_3 +l_4)|^2f_{q_1} f_{q_2} -|A( \\bar{q}_3 +l_4 \\rightarrow q_1+q_2 ) |^2f_{\\bar{q}_3} f_{l_4}\\right],$ where $ A( a \\rightarrow b)$ is the amplitude of the transition from state $a$ to state $b$ , $B_q$ is the baryonic number of quark, $f_a$ is the phase space distribution (the occupation number), and $d\\nu _{q_1,q_2} =\\frac{d^3 q_1}{2E_{q_1} (2\\pi )^3 }\\, \\frac{d^3 q_2}{2E_{q_2} (2\\pi )^3 } ,$ where $E_q = \\sqrt{ q^2 + m^2}$ is the energy of particle with three-momentum $q$ and mass $m$ .","The element of phase space of final particles, $d\\nu _{\\bar{q}_3, l_4} $ , is defined analogously.","We neglect the Fermi suppression factors and the effects of gravity in the collision integral.","This is generally a good approximation.","The calculations are strongly simplified if quarks and leptons are in equilibrium with respect to elastic scattering and annihilation.","In this case their distribution functions take the form $f = \\frac{1}{e^{(E/T - \\xi }) + 1} \\approx e^{-E/T + \\xi },$ with $\\xi = \\mu /T$ being dimensionless chemical potential, different for quarks, $\\xi _q$ , and leptons, $\\xi _l$ .","The assumption of kinetic equilibrium is well justified since it is usually enforced by very efficient elastic scattering.","Equilibrium with respect to annihilation, say, into two channels: $2\\gamma $ and $3\\gamma $ , implies the usual relation between chemical potentials of particles and antiparticles, $\\bar{\\mu }= -\\mu $ .","The baryonic number density is given by the expression: $ \\nonumber n_B &=& \\int \\frac{d^3 q}{2 E_q\\, (2\\pi )^3} (f_q - f_{\\bar{q}}) \\\\&=& \\frac{g_S B_q}{6} \\left(\\mu T^2 + \\frac{\\mu ^3}{ \\pi ^2}\\right) =\\frac{g_S B_q T^3}{6}\\,\\left(\\xi + \\frac{\\xi ^3}{\\pi ^2}\\right) \\,,$ where $T$ is the cosmological plasma temperature, $g_S$ and $B_q$ are respectively the number of the spin states and the baryonic number of quarks.","We can use another representation of the quark field: $Q_2 = \\exp ( i f R /m_0^2 )\\, Q$ analogously to what is done in our paper [8].","Written in terms of $Q_2$ Lagrangian (REF ) would not contain terms proportional to $f /m_0^2$ , but dependence on such terms would reappear in the interaction term as: $\\frac{2g}{m_X^2}\\left[ e^{-3ifR/m_0^2}\\, (\\bar{Q}_2\\,Q_2^c)(\\bar{Q}_2 L) +e^{3ifR/m_0^2}\\,(\\bar{Q}_2^cQ_2)(\\bar{L} Q_2) \\right] .$ Nevertheless we obtain the same fourth order equation for the evolution of curvature, as for non-rotated field $Q$ .","Since the transition amplitudes, which enter the collision integral, are obtained by integration over time of the Lagrangian operator (REF ), taken between the initial and final states, the energy conservation delta-function in Eq.","(REF ) would be modified due to time dependent factors $\\exp [\\pm 3 ifR(t)/m_0^2]$ .","In the simplest case, which is usually considered in gravitational (and spontaneous) baryogenesis, a slowly changing $\\dot{R}$ is taken, so we can approximate $R(t) \\approx \\dot{R}(t)\\,t$ .","In this case the energy is not conserved but the energy conservation condition is trivially modified, as $ \\nonumber &&\\delta [ E(q_1) + E(q_2) - E(q_3) - E(l_4) ] \\rightarrow \\\\&&\\rightarrow \\delta [ E(q_1) + E(q_2) - E(q_3) - E(l_4) - 3f\\dot{R}(t) /m_0^2\\,].$ Thus the energy is non-conserved due to the action of the external field $R(t)$ .","Delta-function (REF ) is not precise, but the result is pretty close to it, if $\\dot{R}(t)$ changes very little during the effective time of the relevant reactions.","If the dimensionless chemical potentials $\\xi _q$ and $\\xi _l$ , as well as $ f\\dot{R}(t) /m_0^2 /T $ , are small, the collision integral can be written as: $I^{coll}_B \\approx \\frac{C_I g^2 T^8}{m_X^4}\\,\\left[ \\frac{3 f\\dot{R}(t)}{ m_0^2\\,T} - 3\\xi _q + \\xi _l \\right] ,$ where $C_I$ is a positive dimensionless constant.","The factor $T^8$ appears for reactions with massless particles and the power eight is found from dimensional consideration.","Because of conservation of the sum of baryonic and leptonic numbers $\\xi _l = -\\xi _q/3 $ .","The case of an essential variation of $\\dot{R} (t)$ is analogous to fast variation of $\\dot{\\theta }(t)$ studied in our paper [8].","Clearly, it is much more complicated technically.","Here we consider only the simple situation with quasi-stationary background and postpone more realistic time dependence of $R(t)$ for the future work.","For small chemical potential the baryonic number density (REF ) is equal to $n_B \\approx \\frac{g_s B_q}{6}\\, \\xi _q T^3\\, ,$ and if the temperature adiabatically decreases in the course of the cosmological expansion, according to $\\dot{T} = - H T$ , equation (REF ) turns into $\\dot{\\xi }_q = \\Gamma \\left[ \\frac{9 f\\dot{R}(t)}{10m_0^2\\, T} - \\xi _q \\right],$ where $\\Gamma \\sim g^2 T^5/m_X^4 $ is the rate of B-nonconserving reactions.","If $\\Gamma $ is in a certain sense large, this equation can be solved in stationary point approximation as $\\xi _q=\\xi _q^{eq} - \\dot{\\xi }_q^{eq}/\\Gamma \\, , \\ \\ {\\rm where} \\ \\ \\xi _q^{eq} = \\frac{9}{10} \\frac{f \\dot{R}}{m_0^2 T}\\, .$ If we substitute $\\xi _q^{eq} $ into Eq.","(REF ) we arrive to the fourth order equation for $R$ .","According to the comment below Eq.","(REF ), the contribution of thermal matter into this equation can be neglected, and we arrive to the very simple fourth order differential equation: $\\frac{d^4 R}{dt^4} = \\lambda ^4 R,$ where $\\lambda ^4 = C_\\lambda m_{Pl}^2 m_0^4 /T^2 $ with $ C_\\lambda = 5/(36\\pi f^2 g_s B_q )$ .","Deriving this equation we neglected the Hubble parameter factor in comparison with time derivatives of $R$ .","It is justified a posteriori because the calculated $\\lambda $ is much larger than $ H $ .","Evidently equation (REF ) has extremely unstable solution with instability time by far shorter than the cosmological time.","This instability would lead to an explosive rise of $R$ , which may possibly be terminated by the nonlinear terms proportional to the product of $H$ to lower derivatives of $R$ .","Correspondingly one may expect stabilization when $HR \\sim \\dot{R}$ , i.e.","$H\\sim \\lambda $ .","Since $\\dot{H} + 2 H^2 = - R/6,$ $H$ would also exponentially rise together with $R$ , $ H \\sim \\exp (\\lambda t )$ and $\\lambda H \\sim R$ .","Thus stabilization may take place at $R \\sim \\lambda ^2 \\sim m_{Pl} m_0^2 / T$ .","This result should be compared with the normal General Relativity value $ R_{GR} \\sim T_{matter} /m_{Pl}^2 $ , where $T_{matter}$ is the trace of the energy-momentum tensor of matter."],["Discussion and conclusion ","For more accurate analysis numerical solution will be helpful, which we will perform in another work.","The problem is complicated because the assumption of slow variation of $\\dot{R}$ quickly becomes broken and the collision integral in time dependent background is not so simply tractable as the usual stationary one.","The technique for treating kinetic equation in non-stationary background is presented in Ref. [8].","For evaluation of $R(t)$ in this case numerical calculations are necessary, which will be presented elsewhere.","Here we describe only the basic features of the new effect of instability in gravitational baryogenesis.","To conclude we have shown that gravitational baryogenesis in the simplest versions discussed in the literature is not realistic because the instability of the emerging gravitational equations destroys the standard cosmology.","Some stabilization mechanism is strongly desirable.","Probably stabilization may be achieved in a version of $F(R)$ -theory.","Acknowledgement This work was supported by the RSF Grant N 16-12-10037.","The author expresses sincere gratitude to Harald Fritzsch for his invitation and for the opportunity to present the talk at the Conference on Particles and Cosmology.","She would like to thank Kok Khoo Phua for his kind hospitality at NTU, Singapore."]],"string":"[\n [\n \"Spontaneous and Gravitational Baryogenesis\"\n ],\n [\n \"Abstract Some problems of spontaneous and gravitational baryogenesis are discussed.\",\n \"Gravity modification due to the curvature dependent term in gravitational baryogensis scenario is considered.\",\n \"It is shown that the interaction of baryonic fields with the curvature scalar leads to strong instability of the gravitational equations of motion and as a result to noticeable distortion of the standard cosmology.\"\n ],\n [\n \"Introduction\",\n \"Observations show that at least the region of the Universe around us is matter-dominated.\",\n \"Though we understand how the matter-antimatter asymmetry may be created, the concrete mechanism is yet unknown.\",\n \"The amount of antimatter is very small and it can be explained as the result of high energy collisions in space.\",\n \"The existence of large regions of antimatter in our neighbourhood would produce high energy radiation as a consequence of matter-antimatter annihilation, which is not observed.\",\n \"Any initial asymmetry at inflation could not solve the problem of observed excess of matter over antimatter, because the energy density associated with baryonic number would not allow for sufficiently long inflation.\",\n \"On the other hand, matter and antimatter seem to have similar properties and therefore we could expect a matter-antimatter symmetric Universe.\",\n \"A satisfactory model of our Universe should be able to explain the origin of the local observed matter-antimatter asymmetry.\",\n \"The term baryogenesis means the generation of the asymmetry between baryons (basically protons and neutrons) and antibaryons (antiprotons and antineutrons).\",\n \"In 1967 Andrey Sakharov pointed out 3 ingredients, today known as Sakharov principles, to produce a matter-antimatter asymmetry from an initially symmetric Universe.\",\n \"These conditions include: 1) non-conservation of baryonic number; 2) breaking of symmetry between particles and antiparticles; 3) deviation from thermal equilibrium.\",\n \"However, not all of three Sakharov principles are strictly necessary.\",\n \"In what follows we briefly discuss some features of spontaneous baryogenesis (SBG) and concentrate in more detail on gravitational baryogenesis (GBG).\",\n \"Both these mechanisms do not demand an explicit C and CP violation and can proceed in thermal equilibrium.\",\n \"Moreover, they are usually most efficient in thermal equilibrium.\",\n \"The statement that the cosmological baryon asymmetry can be created by spontaneous baryogenesis in thermal equilibrium was mentioned in the original paper by Cohen and Kaplan[1] and developed in subsequent papers[2], [3], for review see[4], [5].\",\n \"The term \\\"spontaneous\\\" is related to spontaneous breaking of a global $U(1)$ -symmetry, which ensures the conservation of the total baryonic number in the unbroken phase.\",\n \"This symmetry is supposed to be spontaneously broken and in the broken phase the Lagrangian density acquires the additional term ${\\\\cal L}_{SB} = (\\\\partial _{\\\\mu } \\\\theta ) J^{\\\\mu }_B\\\\, ,$ where $\\\\theta $ is the Goldstone field and $J^{\\\\mu }_B$ is the baryonic current of matter fields, which becomes non-conserved.\",\n \"For a spatially homogeneous field, $\\\\theta = \\\\theta (t)$ , the Lagrangian is reduced to the simple form ${\\\\cal L}_{SB} = \\\\dot{\\\\theta }\\\\, n_B\\\\,, \\\\ \\\\ \\\\ \\\\ n_B\\\\equiv J^0_B,$ where time component of a current is the baryonic number density of matter, so it is tempting to identify ${\\\\dot{\\\\theta }}$ with the chemical potential, $ \\\\mu _B$ , of the corresponding system.\",\n \"However, such identification is questionable and depends upon the representation chosen for the fermionic fields[6], [7].\",\n \"It is heavily based on the assumption ${\\\\dot{\\\\theta }\\\\approx const}$ , which is relaxed in the work[8].\",\n \"But still the scenario is operative and presents a beautiful possibility to create an excess of particles over antiparticles in the Universe.\",\n \"Subsequently the idea of gravitational baryogenesis (GBG) was put forward [9], where the scenario of SBG was modified by the introduction of the coupling of the baryonic current to the derivative of the curvature scalar $R$ : ${\\\\cal L}_{GBG} = \\\\frac{1}{M^2} (\\\\partial _\\\\mu R ) J^\\\\mu _B\\\\, ,$ where $M$ is a constant parameter with the dimension of mass.\",\n \"In the presented talk we demonstrate that the addition of the curvature dependent term (REF ) to the Hilbert-Einstein Lagrangian of General Relativity (GR) leads to higher order gravitational equations of motion, which are strongly unstable with respect to small perturbations.\",\n \"The effects of this instability may drastically distort not only the usual cosmological history, but also the standard Newtonian gravitational dynamics.\",\n \"We discovered such instability for scalar baryons [10] and found similar effect for the more usual spin one-half baryons (quarks) [11].\"\n ],\n [\n \"Gravitational baryogenesis with scalar baryons \",\n \"Let us start from the model where baryonic number is carried by scalar field $\\\\phi $ with potential $U(\\\\phi , \\\\phi ^* )$ .\",\n \"An example with baryonic current of fermions will be considered in the next section.\",\n \"The action of the scalar model has the form: $A = \\\\int d^4 x\\\\, \\\\sqrt{-g} \\\\left[ \\\\frac{m_{Pl}^2}{16\\\\pi } R + \\\\frac{1}{M^2} (\\\\partial _{\\\\mu } R) J^{\\\\mu } -g^{\\\\mu \\\\nu } \\\\partial _{\\\\mu }\\\\phi \\\\, \\\\partial _{\\\\nu }\\\\phi ^* + U(\\\\phi , \\\\phi ^*)\\\\right] - A_m\\\\, ,$ where $m_{Pl}=1.22\\\\cdot 10^{19}$ GeV is the Planck mass, $A_m$ is the matter action, $J^\\\\mu = g^{\\\\mu \\\\nu }J_\\\\nu $ , and $g^{\\\\mu \\\\nu }$ is the metric tensor of the background space-time.\",\n \"We assume that initially the metric has the usual GR form and study the emergence of the corrections due to the instability described below.\",\n \"In contrast to scalar electrodynamics, the baryonic current of scalars is not uniquely defined.\",\n \"In electrodynamics the form of the electric current is dictated by the conditions of gauge invariance and current conservation, which demand the addition to the current of the so called sea-gull term proportional to $e^2 A_\\\\mu |\\\\phi |^2$ , where $A_\\\\mu $ is the electromagnetic potential.\",\n \"On the other hand, a local $U(1)$ -symmetry is not imposed on the theory determined by action (REF ).\",\n \"It is invariant only with respect to a $U(1)$ transformations with a constant phase.\",\n \"As a result, the baryonic current of scalars is considerably less restricted.\",\n \"In particular, we can add to the current an analogue of the sea-gull term, $ \\\\sim (\\\\partial _\\\\mu R)\\\\,|\\\\phi |^2 $ , with an arbitrary coefficient.\",\n \"In our paper [10] we study the following two extreme possibilities, when the sea-gull term is absent and the current is not conserved, or the sea-gull term is included with the coefficient ensuring current conservation.\",\n \"In both cases no baryon asymmetry can be generated without additional interactions.\",\n \"It is trivially true in the second case, when the current is conserved, but it is also true in the first case despite the current non-conservation, simply because the non-zero divergence $D_\\\\mu J^\\\\mu $ does not change the baryonic number of $\\\\phi $ but only leads to redistribution of particles $\\\\phi $ in the phase space.\",\n \"So to create any non-zero baryon asymmetry we have to introduce an interaction of $\\\\phi $ with other particles which breaks conservation of $B$ by making the potential $U$ non-invariant with respect to the phase rotations of $\\\\phi $ , as it is described below.\",\n \"If the potential $U(\\\\phi )$ is not invariant with respect to the $U(1)$ -rotation, $\\\\phi \\\\rightarrow \\\\exp {(i \\\\beta )} \\\\phi $ , the baryonic current defined in the usual way $J_{ \\\\mu } = i q (\\\\phi ^* \\\\partial _{\\\\mu }\\\\phi - \\\\phi \\\\partial _{\\\\mu }\\\\phi ^*)$ is not conserved.\",\n \"Here $q$ is the baryonic number of $\\\\phi $ and we omitted index $B$ in current $J_{\\\\mu }$ .\",\n \"With this current and Lagrangian (REF ) the equation for the curvature scalar, $R$ , takes the form: $\\\\frac{m_{Pl}^2}{16\\\\pi }\\\\, R + \\\\frac{1}{M^2}\\\\left[ (R + 3 D^2) D_{\\\\alpha } J^{ \\\\alpha } + J^{\\\\alpha } \\\\,D_{\\\\alpha }R \\\\right] -D_{\\\\alpha } \\\\phi \\\\, D^{\\\\alpha } \\\\phi ^* + 2 U(\\\\phi ) = - \\\\frac{1}{2} \\\\, T_{\\\\mu }^{\\\\mu }\\\\, ,$ where $D_\\\\mu $ is the covariant derivative in metric $g_{\\\\mu \\\\nu }$ (of course, for scalars $D_\\\\mu = \\\\partial _\\\\mu $ ) and $T_{\\\\mu \\\\nu }$ is the energy-momentum tensor of matter obtained from action $A_m$ .\",\n \"According to definition (REF ), the current divergence is: $D_{\\\\mu } J^{\\\\mu } = \\\\frac{2q^2}{M^2} \\\\left[ D_{\\\\mu } R\\\\, (\\\\phi ^* D^{\\\\mu }\\\\phi + \\\\phi D^{\\\\mu }\\\\phi ^*) + |\\\\phi |^2 D^2 R \\\\right]+ i q \\\\left(\\\\phi \\\\frac{\\\\partial U}{\\\\partial \\\\phi } - \\\\phi ^* \\\\frac{\\\\partial U}{\\\\partial \\\\phi ^*} \\\\right)\\\\,.$ If the potential of $U$ is invariant with respect to the phase rotation of $\\\\phi $ , i.e.\",\n \"$U = U(|\\\\phi |)$ , the last term in this expression disappears.\",\n \"Still the current remains non-conserved, but this non-conservation does not lead to any cosmological baryon asymmetry.\",\n \"Indeed, the current non-conservation is proportional to the product $\\\\phi ^* \\\\phi $ , so it can produce or annihilate an equal number of baryons and antibaryons.\",\n \"To create cosmological baryon asymmetry we need to introduce new types of interactions, for example, the term in the potential of the form: $ U_4 = \\\\lambda _4 \\\\phi ^4 + \\\\lambda _4^* \\\\phi ^{*4} $ .\",\n \"This potential is surely non invariant w.r.t.\",\n \"the phase rotation of $\\\\phi $ and can induce the B-non-conserving process of transition of two scalar baryons into two antibaryons, $2 \\\\phi \\\\rightarrow 2 \\\\bar{\\\\phi }$ .\",\n \"Let us consider solution of the above equation of motion in cosmology.\",\n \"The metric of the spatially flat cosmological FRW background can be taken as: $ds^2=dt^2 - a^2(t) d{\\\\bf r}^2\\\\, .$ In the homogeneous case the equation for the curvature scalar (REF ) takes the form: $\\\\frac{m_{Pl}^2}{16\\\\pi }\\\\, R + \\\\frac{1}{M^2}\\\\left[ (R + 3 \\\\partial _t^2 + 9 H \\\\partial _t) D_{\\\\alpha } J^{\\\\alpha } +\\\\dot{R} \\\\, J^0 \\\\right]= - \\\\frac{ T^{(tot)}}{2} \\\\,,$ where $J^0$ is the baryonic number density of the $\\\\phi $ -field, $H = \\\\dot{a}/a$ is the Hubble parameter, and $T^{(tot)}$ is the trace of the energy-momentum tensor of matter including contribution from the $\\\\phi $ -field.\",\n \"In the homogeneous and isotropic cosmological plasma $T^{(tot)} = \\\\rho - 3 P\\\\, ,$ where $\\\\rho $ and $P$ are respectively the energy density and the pressure of the plasma.\",\n \"For relativistic plasma $\\\\rho = \\\\pi ^2 g_* T^4/30$ with $T$ and $g_*$ being the plasma temperature and the number of particle species in the plasma.\",\n \"The Hubble parameter is expressed through $\\\\rho $ as $H^2 = 8\\\\pi \\\\rho /(3m_{Pl}^2) \\\\sim T^4/m_{Pl}^2$ .\",\n \"The covariant divergence of the current is given by the expression (REF ).\",\n \"In the homogeneous case we are considering it takes the form: $D_{\\\\alpha } J^{\\\\alpha } = \\\\frac{2q^2}{M^2} \\\\left[ \\\\dot{R}\\\\, (\\\\phi ^* \\\\dot{\\\\phi }+ \\\\phi \\\\dot{\\\\phi }^*) +(\\\\ddot{R} + 3H \\\\dot{R})\\\\, \\\\phi ^*\\\\phi \\\\right]+ i q \\\\left(\\\\phi \\\\frac{\\\\partial U}{\\\\partial \\\\phi } - \\\\phi ^* \\\\frac{\\\\partial U}{\\\\partial \\\\phi ^*} \\\\right)\\\\,.$ To derive the equation of motion for the classical field $R$ in the cosmological plasma we have to take the expectation values of the products of the quantum operators $\\\\phi $ , $\\\\phi ^*$ , and their derivatives.\",\n \"Performing the thermal averaging, we find $\\\\langle \\\\phi ^* \\\\phi \\\\rangle = \\\\frac{T^2}{12}\\\\, , \\\\ \\\\ \\\\ \\\\langle \\\\phi ^* \\\\dot{\\\\phi }+ \\\\dot{\\\\phi }^* \\\\phi \\\\rangle = 0\\\\, .$ Substituting these average values into Eq.\",\n \"(REF ) and neglecting the last term in Eq.\",\n \"(REF ) we obtain the fourth order differential equation: $\\\\frac{m_{Pl}^2}{16\\\\pi }\\\\, R +\\\\frac{q^2}{6 M^4} \\\\left(R + 3 \\\\partial _t^2 + 9 H \\\\partial _t \\\\right)\\\\left[\\\\left(\\\\ddot{R} + 3H \\\\dot{R}\\\\right) T^2 \\\\right] +\\\\frac{1}{M^2} \\\\dot{R} \\\\, \\\\langle J^0 \\\\rangle = - \\\\frac{T^{(tot)}}{2} \\\\, .$ Here $\\\\langle J^0 \\\\rangle $ is the thermal average value of the baryonic number density of $\\\\phi $ .\",\n \"It is assumed to be zero initially and generated as a result of GBG.\",\n \"We neglect this term, since it is surely small initially and probably subdominant later.\",\n \"Anyhow it does not noticeably change the exponential rise of $R$ at the onset of the instability.\",\n \"Eq.\",\n \"(REF ) can be further simplified if the variation of $R(t)$ is much faster than the universe expansion rate or in other words $\\\\ddot{R} / \\\\dot{R} \\\\gg H$ .\",\n \"Correspondingly the temperature may be considered adiabatically constant.\",\n \"The validity of these assumption is justified a posteriori after we find the solution for $R(t)$ .\",\n \"Keeping only the linear in $R$ terms and neglecting higher powers of $R$ , such as $R^2$ or $H R$ , we obtain the linear differential equation of the fourth order: $\\\\frac{d^4 R}{dt^4} + \\\\mu ^4 R = - \\\\frac{1}{2} \\\\, T^{(tot)}\\\\,,\\\\ {\\\\rm where }\\\\ \\\\ \\\\mu ^4 = \\\\frac{m_{Pl}^2 M^4}{8 \\\\pi q^2 T^2}\\\\,.$ The homogeneous part of this equation has exponential solutions $R \\\\sim \\\\exp (\\\\lambda t)$ with $\\\\lambda = | \\\\mu | \\\\exp \\\\left( i\\\\pi /4 + i \\\\pi n /2 \\\\right),$ where $n = 0,1,2,3$ .\",\n \"There are two solutions with positive real parts of $\\\\lambda $ .\",\n \"This indicates that the curvature scalar is exponentially unstable with respect to small perturbations, so $R$ should rise exponentially fast with time and quickly oscillate around this rising function.\",\n \"Now we need to check if the characteristic rate of the perturbation explosion is indeed much larger than the rate of the universe expansion, that is: $(Re\\\\, \\\\lambda )^4 > H^4 = \\\\left( \\\\frac{ 8\\\\pi \\\\rho }{ 3 m^2_{Pl}}\\\\right)^2 = \\\\frac{16 \\\\pi ^6 g_*^2}{2025}\\\\,\\\\frac{ T^8}{m_{Pl}^4},$ where $\\\\rho = \\\\pi ^2 g_* T^4 /30$ is the energy density of the primeval plasma at temperature $T$ and $g_* \\\\sim 10 - 100$ is the number of relativistic degrees of freedom in the plasma.\",\n \"This condition is fulfilled if $\\\\frac{2025}{2^9 \\\\pi ^7 q^2 g_*^2}\\\\frac{m_{Pl}^6 M^4}{T^{10}} > 1\\\\,,$ or, roughly speaking, if $T \\\\le m_{Pl}^{3/5} M^{2/5} $ .\",\n \"Let us stress that at these temperatures the instability is quickly developed and the standard cosmology would be destroyed.\",\n \"If we want to preserve the successful big bang nucleosynthesis (BBN) results and impose the condition that the development of the instability was longer than the Hubble time at the BBN epoch at $T \\\\sim 1 $ MeV, then $M$ should be extremely small, $M < 10^{-32}$ MeV.\",\n \"The desire to keep the standard cosmology at smaller $T$ would demand even tinier $M$ .\",\n \"A tiny $M$ leads to a huge strength of coupling (REF ).\",\n \"It surely would lead to pronounced effects in stellar physics.\"\n ],\n [\n \"Gravitational baryogenesis with fermions \",\n \"Let us now generalize results, obtained for scalar baryons, to realistic fermions.\",\n \"We start from the action in the form $A= \\\\int d^4x \\\\sqrt{-g} \\\\left[\\\\frac{m_{Pl}^2}{16 \\\\pi } \\\\,R - {\\\\cal L}_{m}\\\\right]\\\\,$ with $ \\\\nonumber {\\\\cal L}_{m} &= &\\\\frac{i}{2} (\\\\bar{Q} \\\\gamma ^\\\\mu \\\\nabla _\\\\mu Q - \\\\nabla _\\\\mu \\\\bar{Q}\\\\, \\\\gamma ^\\\\mu Q) - m_Q\\\\bar{Q}\\\\,Q\\\\\\\\&+&\\\\frac{i}{2} (\\\\bar{L} \\\\gamma ^\\\\mu \\\\nabla _\\\\mu L - \\\\nabla _\\\\mu \\\\bar{L} \\\\gamma ^\\\\mu L)- m_L\\\\bar{L}\\\\,L \\\\\\\\ \\\\nonumber &+& \\\\frac{g}{m_X^2}\\\\left[(\\\\bar{Q}\\\\,Q^c)(\\\\bar{Q} L) + (\\\\bar{Q}^cQ)(\\\\bar{L} Q) \\\\right]+ \\\\frac{f}{m_0^2} (\\\\partial _{\\\\mu } R) J^{\\\\mu } + {\\\\cal L}_{other}\\\\,,$ where $Q$ is the quark (or quark-like) field with non-zero baryonic number, $L$ is another fermionic field (lepton), $\\\\nabla _\\\\mu $ is the covariant derivative of Dirac fermion in tetrad formalism.\",\n \"$m_0$ is a constant parameter with dimension of mass and $f$ is dimensionless coupling constant which is introduced to allow for an arbitrary sign of the curvature dependent term in the above expression.\",\n \"$J^{\\\\mu } = \\\\bar{Q} \\\\gamma ^{\\\\mu } Q$ is the quark current with $\\\\gamma ^{\\\\mu }$ being the curved space gamma-matrices, ${\\\\cal L}_{other}$ describes all other forms of matter.\",\n \"The four-fermion interaction between quarks and leptons is introduced to ensure the necessary non-conservation of the baryon number with $m_X$ being a constant parameter with dimension of mass and $g$ being a dimensionless coupling constant.\",\n \"In grand unified theories $m_X$ may be of the order of $10^{14}-10^{15}$ GeV.\",\n \"Varying the action (REF ) over metric, $g^{\\\\mu \\\\nu }$ , and taking trace with respect to $\\\\mu $ and $\\\\nu $ , we obtain the following equation of motion for the curvature scalar: $ \\\\nonumber - \\\\frac{m_{Pl}^2}{8\\\\pi } R &=& m_{Q} \\\\bar{Q} Q + m_L \\\\bar{L} L+\\\\frac{2g}{m_X^2}\\\\left[(\\\\bar{Q}\\\\,Q^c)(\\\\bar{Q} L) + (\\\\bar{Q}^cQ)(\\\\bar{L} Q) \\\\right] \\\\\\\\&-&\\\\frac{2f}{m_0^2} (R + 3D^2) D_{\\\\alpha } J^{\\\\alpha } + T_{other}\\\\,,$ where $T_{other} $ is the trace of the energy momentum tensor of all other fields.\",\n \"At relativistic stage, when masses are negligible, we can take $T_{matter} = 0$ .\",\n \"The average expectation value of the interaction term proportional to $g$ is also small, so the contribution of all matter fields may be neglected.\",\n \"As we see in what follows, kinetic equation leads to an explicit dependence on $R$ of the current divergence, $D_\\\\alpha J^\\\\alpha $ , if the current is not conserved.\",\n \"As a result we obtain 4th order equation for $R$ .\",\n \"As previously, we study solutions of Eq.\",\n \"(REF ) in cosmology in homogeneous and isotropic FRW background with the metric $ds^2=dt^2 - a^2(t) d{\\\\bf r}^2 $ .\",\n \"The curvature is a function only of time and the covariant derivative acting on a vector $V^\\\\alpha $ , which depends only on time and has only time component, has the form: $D_\\\\alpha V^\\\\alpha = (\\\\partial _t + 3 H) V^t ,$ where $H = \\\\dot{a}/a$ is the Hubble parameter.\",\n \"As an example let us consider the reaction $q_1 + q_2 \\\\leftrightarrow \\\\bar{q}_3 + l_4$ , where $q_1$ and $q_2$ are quarks with momenta $q_1$ and $q_2$ , while $\\\\bar{q}_3$ and $l_4$ are antiquark and lepton with momenta $q_3$ and $l_4$ .\",\n \"We use the same notations for the particle symbol and for the particle momentum.\",\n \"The kinetic equation for the variation of the baryonic number density $n_B \\\\equiv J^t$ through this reaction in the FRW background has the form: $(\\\\partial _t +3 H) n_B = I_B^{coll},$ where the collision integral for space and time independent interaction is equal to: $&&I^{coll}_B =- 3 B_q (2\\\\pi )^4 \\\\int \\\\,d\\\\nu _{q_1,q_2} \\\\,d\\\\nu _{\\\\bar{q}_3, l_4}\\\\delta ^4 (q_1 +q_2 -q_3 - l_4)\\\\nonumber \\\\\\\\&& \\\\left[ |A( q_1+q_2\\\\rightarrow \\\\bar{q}_3 +l_4)|^2f_{q_1} f_{q_2} -|A( \\\\bar{q}_3 +l_4 \\\\rightarrow q_1+q_2 ) |^2f_{\\\\bar{q}_3} f_{l_4}\\\\right],$ where $ A( a \\\\rightarrow b)$ is the amplitude of the transition from state $a$ to state $b$ , $B_q$ is the baryonic number of quark, $f_a$ is the phase space distribution (the occupation number), and $d\\\\nu _{q_1,q_2} =\\\\frac{d^3 q_1}{2E_{q_1} (2\\\\pi )^3 }\\\\, \\\\frac{d^3 q_2}{2E_{q_2} (2\\\\pi )^3 } ,$ where $E_q = \\\\sqrt{ q^2 + m^2}$ is the energy of particle with three-momentum $q$ and mass $m$ .\",\n \"The element of phase space of final particles, $d\\\\nu _{\\\\bar{q}_3, l_4} $ , is defined analogously.\",\n \"We neglect the Fermi suppression factors and the effects of gravity in the collision integral.\",\n \"This is generally a good approximation.\",\n \"The calculations are strongly simplified if quarks and leptons are in equilibrium with respect to elastic scattering and annihilation.\",\n \"In this case their distribution functions take the form $f = \\\\frac{1}{e^{(E/T - \\\\xi }) + 1} \\\\approx e^{-E/T + \\\\xi },$ with $\\\\xi = \\\\mu /T$ being dimensionless chemical potential, different for quarks, $\\\\xi _q$ , and leptons, $\\\\xi _l$ .\",\n \"The assumption of kinetic equilibrium is well justified since it is usually enforced by very efficient elastic scattering.\",\n \"Equilibrium with respect to annihilation, say, into two channels: $2\\\\gamma $ and $3\\\\gamma $ , implies the usual relation between chemical potentials of particles and antiparticles, $\\\\bar{\\\\mu }= -\\\\mu $ .\",\n \"The baryonic number density is given by the expression: $ \\\\nonumber n_B &=& \\\\int \\\\frac{d^3 q}{2 E_q\\\\, (2\\\\pi )^3} (f_q - f_{\\\\bar{q}}) \\\\\\\\&=& \\\\frac{g_S B_q}{6} \\\\left(\\\\mu T^2 + \\\\frac{\\\\mu ^3}{ \\\\pi ^2}\\\\right) =\\\\frac{g_S B_q T^3}{6}\\\\,\\\\left(\\\\xi + \\\\frac{\\\\xi ^3}{\\\\pi ^2}\\\\right) \\\\,,$ where $T$ is the cosmological plasma temperature, $g_S$ and $B_q$ are respectively the number of the spin states and the baryonic number of quarks.\",\n \"We can use another representation of the quark field: $Q_2 = \\\\exp ( i f R /m_0^2 )\\\\, Q$ analogously to what is done in our paper [8].\",\n \"Written in terms of $Q_2$ Lagrangian (REF ) would not contain terms proportional to $f /m_0^2$ , but dependence on such terms would reappear in the interaction term as: $\\\\frac{2g}{m_X^2}\\\\left[ e^{-3ifR/m_0^2}\\\\, (\\\\bar{Q}_2\\\\,Q_2^c)(\\\\bar{Q}_2 L) +e^{3ifR/m_0^2}\\\\,(\\\\bar{Q}_2^cQ_2)(\\\\bar{L} Q_2) \\\\right] .$ Nevertheless we obtain the same fourth order equation for the evolution of curvature, as for non-rotated field $Q$ .\",\n \"Since the transition amplitudes, which enter the collision integral, are obtained by integration over time of the Lagrangian operator (REF ), taken between the initial and final states, the energy conservation delta-function in Eq.\",\n \"(REF ) would be modified due to time dependent factors $\\\\exp [\\\\pm 3 ifR(t)/m_0^2]$ .\",\n \"In the simplest case, which is usually considered in gravitational (and spontaneous) baryogenesis, a slowly changing $\\\\dot{R}$ is taken, so we can approximate $R(t) \\\\approx \\\\dot{R}(t)\\\\,t$ .\",\n \"In this case the energy is not conserved but the energy conservation condition is trivially modified, as $ \\\\nonumber &&\\\\delta [ E(q_1) + E(q_2) - E(q_3) - E(l_4) ] \\\\rightarrow \\\\\\\\&&\\\\rightarrow \\\\delta [ E(q_1) + E(q_2) - E(q_3) - E(l_4) - 3f\\\\dot{R}(t) /m_0^2\\\\,].$ Thus the energy is non-conserved due to the action of the external field $R(t)$ .\",\n \"Delta-function (REF ) is not precise, but the result is pretty close to it, if $\\\\dot{R}(t)$ changes very little during the effective time of the relevant reactions.\",\n \"If the dimensionless chemical potentials $\\\\xi _q$ and $\\\\xi _l$ , as well as $ f\\\\dot{R}(t) /m_0^2 /T $ , are small, the collision integral can be written as: $I^{coll}_B \\\\approx \\\\frac{C_I g^2 T^8}{m_X^4}\\\\,\\\\left[ \\\\frac{3 f\\\\dot{R}(t)}{ m_0^2\\\\,T} - 3\\\\xi _q + \\\\xi _l \\\\right] ,$ where $C_I$ is a positive dimensionless constant.\",\n \"The factor $T^8$ appears for reactions with massless particles and the power eight is found from dimensional consideration.\",\n \"Because of conservation of the sum of baryonic and leptonic numbers $\\\\xi _l = -\\\\xi _q/3 $ .\",\n \"The case of an essential variation of $\\\\dot{R} (t)$ is analogous to fast variation of $\\\\dot{\\\\theta }(t)$ studied in our paper [8].\",\n \"Clearly, it is much more complicated technically.\",\n \"Here we consider only the simple situation with quasi-stationary background and postpone more realistic time dependence of $R(t)$ for the future work.\",\n \"For small chemical potential the baryonic number density (REF ) is equal to $n_B \\\\approx \\\\frac{g_s B_q}{6}\\\\, \\\\xi _q T^3\\\\, ,$ and if the temperature adiabatically decreases in the course of the cosmological expansion, according to $\\\\dot{T} = - H T$ , equation (REF ) turns into $\\\\dot{\\\\xi }_q = \\\\Gamma \\\\left[ \\\\frac{9 f\\\\dot{R}(t)}{10m_0^2\\\\, T} - \\\\xi _q \\\\right],$ where $\\\\Gamma \\\\sim g^2 T^5/m_X^4 $ is the rate of B-nonconserving reactions.\",\n \"If $\\\\Gamma $ is in a certain sense large, this equation can be solved in stationary point approximation as $\\\\xi _q=\\\\xi _q^{eq} - \\\\dot{\\\\xi }_q^{eq}/\\\\Gamma \\\\, , \\\\ \\\\ {\\\\rm where} \\\\ \\\\ \\\\xi _q^{eq} = \\\\frac{9}{10} \\\\frac{f \\\\dot{R}}{m_0^2 T}\\\\, .$ If we substitute $\\\\xi _q^{eq} $ into Eq.\",\n \"(REF ) we arrive to the fourth order equation for $R$ .\",\n \"According to the comment below Eq.\",\n \"(REF ), the contribution of thermal matter into this equation can be neglected, and we arrive to the very simple fourth order differential equation: $\\\\frac{d^4 R}{dt^4} = \\\\lambda ^4 R,$ where $\\\\lambda ^4 = C_\\\\lambda m_{Pl}^2 m_0^4 /T^2 $ with $ C_\\\\lambda = 5/(36\\\\pi f^2 g_s B_q )$ .\",\n \"Deriving this equation we neglected the Hubble parameter factor in comparison with time derivatives of $R$ .\",\n \"It is justified a posteriori because the calculated $\\\\lambda $ is much larger than $ H $ .\",\n \"Evidently equation (REF ) has extremely unstable solution with instability time by far shorter than the cosmological time.\",\n \"This instability would lead to an explosive rise of $R$ , which may possibly be terminated by the nonlinear terms proportional to the product of $H$ to lower derivatives of $R$ .\",\n \"Correspondingly one may expect stabilization when $HR \\\\sim \\\\dot{R}$ , i.e.\",\n \"$H\\\\sim \\\\lambda $ .\",\n \"Since $\\\\dot{H} + 2 H^2 = - R/6,$ $H$ would also exponentially rise together with $R$ , $ H \\\\sim \\\\exp (\\\\lambda t )$ and $\\\\lambda H \\\\sim R$ .\",\n \"Thus stabilization may take place at $R \\\\sim \\\\lambda ^2 \\\\sim m_{Pl} m_0^2 / T$ .\",\n \"This result should be compared with the normal General Relativity value $ R_{GR} \\\\sim T_{matter} /m_{Pl}^2 $ , where $T_{matter}$ is the trace of the energy-momentum tensor of matter.\"\n ],\n [\n \"Discussion and conclusion \",\n \"For more accurate analysis numerical solution will be helpful, which we will perform in another work.\",\n \"The problem is complicated because the assumption of slow variation of $\\\\dot{R}$ quickly becomes broken and the collision integral in time dependent background is not so simply tractable as the usual stationary one.\",\n \"The technique for treating kinetic equation in non-stationary background is presented in Ref. [8].\",\n \"For evaluation of $R(t)$ in this case numerical calculations are necessary, which will be presented elsewhere.\",\n \"Here we describe only the basic features of the new effect of instability in gravitational baryogenesis.\",\n \"To conclude we have shown that gravitational baryogenesis in the simplest versions discussed in the literature is not realistic because the instability of the emerging gravitational equations destroys the standard cosmology.\",\n \"Some stabilization mechanism is strongly desirable.\",\n \"Probably stabilization may be achieved in a version of $F(R)$ -theory.\",\n \"Acknowledgement This work was supported by the RSF Grant N 16-12-10037.\",\n \"The author expresses sincere gratitude to Harald Fritzsch for his invitation and for the opportunity to present the talk at the Conference on Particles and Cosmology.\",\n \"She would like to thank Kok Khoo Phua for his kind hospitality at NTU, Singapore.\"\n ]\n]"},"id":{"kind":"string","value":"1808.08577"}}},{"rowIdx":281,"cells":{"text":{"kind":"list like","value":[["Densest vs. jammed packings of 2D bent-core trimers"],["Abstract We identify the maximally dense lattice packings of tangent-disk trimers with fixed bond angles ($\\theta = \\theta_0$) and contrast them to both their nonmaximally-dense-but-strictly-jammed lattice packings as well as the disordered jammed states they form for a range of compression protocols.","While only $\\theta_0 = 0,\\ 60^\\circ,\\ \\rm{and}\\ 120^\\circ$ trimers can form the triangular lattice, maximally-dense maximally-symmetric packings for all $\\theta_0$ fall into just two categories distinguished by their bond topologies: half-elongated-triangular for $0 < \\theta_0 < 60^\\circ$ and elongated-snub-square for $60^\\circ < \\theta_0 < 120^\\circ$.","The presence of degenerate, lower-symmetry versions of these densest packings combined with several families of less-dense-but-strictly-jammed lattice packings act in concert to promote jamming."],["Introduction","Jamming of anisotropic constituents has attracted great interest [1], [2], [3], [4], [5] for two reasons.","The first is that understanding how anistropy affects jamming is critical because most real granular materials are composed of anisotropic grains.","The second is that constituent-particle anistropy affects systems' jamming phenomenology and their thermal-solidification phenomenology in similar ways, and hence studying the jamming of grains of a given shape can provide insight into the thermal solidification of similarly shaped molecules and/or colloids [6], [7], [8], [9].","Such studies are maximally effective when they are complemented by identifying the particles' densest possible packings since the differences between densest and jammed packings are often analogous to the differences between crystals and glasses formed via thermal solidification [10], [11], [9].","Bent-core trimers are a simple model for multiple liquid-crystal-forming [12] and glass-forming [13], [14], [15], [16] molecules.","As illustrated in Figure REF , their shape can be characterized using three parameters: the bond angle $\\theta _0$ , the ratio $r$ of end-monomer radius to center-monomer radius, and the ratio $R$ of intermonomer bond length to center-monomer diameter.","For example, para-, meso-, and ortho-terphenyl correspond to the molecule shown in Fig.","REF (with $\\theta _0 = 0^\\circ ,\\ 60^\\circ ,\\ \\rm {and}\\ 120^\\circ $ , respectively), and the popular Lewis-Wahnstrom model [17] for OTP implements this molecular geometry with $R = 2^{-1/6},\\ r = 1,\\ \\theta _0 = 105^\\circ $ .","It is well known that the properties of systems composed of such molecules depend strongly on all three of these parameters; for example, the three terphenyl isomers form very differently structured bulk solids under the same preparation protocol [13], as do xylenes [14], diphenylcycloalkenes [15] and homologous series of cyclic stilbenes [16].","The structural isomers and near-isomers of more complicated small molecules also often exhibit very different solidification behavior, e.g.","the trisnapthylbenzenes which have attracted great interest in recent years because they have been shown to form quasi-ordered glasses when vapor-deposited [18], [19], [20], [21], [22].","Figure: Rigid bent-core disk-trimers with bond angle θ 0 \\theta _0.","Panel (a) shows the general geometry with unspecified (r,Rr,\\ R).","Here we study the r=R=1r = R = 1 case shown in panel (b).Our understanding of such phenomena and hence our ability to engineer crystallizability/glass-formability at the molecular level remains very limited.","One of the reasons why this is so is that only a few theoretical studies have isolated the role played by molecular shape using simple models.","Molecules like those studied in Refs.","[18], [19], [20], [21], [22] tend to form liquid-crystalline phases with columnar order [21].","Studying packing of 2D models for these molecules corresponds to studying the in-plane ordering of such anisotropic phases.","Optimal packing of molecules with the geometry shown in Figure REF has been investigated only minimally; Ref.","[23] reported the densest packings of 2D $R = 1/2$ trimers as a function of $r$ and $\\theta _0$ .","The tangent-disk ($r = R = 1$ ) case shown in Fig.","REF (b) is of considerable interest because it allows straightforward connection to results obtained for monomers – and hence isolation of the role played by the bond and angular constraints – while remaining a reasonable minimal model for terphenyl-shaped molecules.","In this paper, we identify and characterize the densest lattice packings of 2D bent-core tangent-disk trimers as a function of their bond angle $\\theta _0$ , and contrast them to both their nonmaximally-dense-but-strictly-jammed lattice packings and the disordered jammed packings they form under dynamic compression.","Figure: Structure of the putatively densest trimer packings.","Panel (a): lattice configuration for 0≤θ 0 ≤60 ∘ 0 \\le \\theta _0 \\le 60^\\circ .","Panel (b): lattice configuration for 60 ∘ ≤θ 0 ≤120 ∘ 60^\\circ \\le \\theta _0 \\le 120^\\circ .","Panel (c): Postulated maximal density φ max (θ 0 )\\phi _{max}(\\theta _0) [Eqs.",", , ]."],["Densest Packings","The only $\\theta _0$ allowing formation of the triangular lattice [which is the densest possible 2D disk packing, with $\\phi = \\phi _{tri} = \\pi /(2\\sqrt{3}) \\simeq .9069$ ] are $0^\\circ ,\\ 60^\\circ ,\\ \\rm {and}\\ 120^\\circ $ .","In this section, we identify the densest packings for all $\\theta _0$ .","A potential geometry of the densest packings for $0 \\le \\theta _0 \\le 60^\\circ $ is shown in Fig.","REF (a).","Black circles indicate the monomer positions for a reference trimer centered at the origin.","For this range of $\\theta _0$ , another similarly oriented trimer can be centered at $\\vec{c}_2 = [1/2,\\sqrt{3}/2]$ , with its leftmost and rightmost monomers respectively at $\\vec{l}_2 = [1/2-\\cos (\\theta _0),\\sqrt{3}/2+\\sin (\\theta _0)]$ and $\\vec{r}_2 = [3/2,\\sqrt{3}/2]$ .","Then the center monomer of a third trimer with this orientation can be placed at $\\vec{c}_3 = [2+\\cos (\\theta ),-\\sin (\\theta )]$ .","A horizontally oriented unit cell with lattice vectors $\\vec{b}_1, \\vec{b}_2$ is obtained by rotating $\\vec{c}_2,\\ \\vec{c}_3$ through the angle $\\delta (\\theta _0) = \\tan ^{-1}(\\sin (\\theta _0)/[2+\\cos (\\theta _0)])$ .","The area of this unit cell is $A_1(\\theta _0) = \\det \\left( \\left[ \\begin{array}{c} \\vec{b}_1 \\\\ \\vec{b}_2 \\end{array} \\right] \\right) = \\det \\left( \\left[ \\begin{array}{c} \\vec{c}_2 \\\\ \\vec{c}_3 \\end{array} \\right] \\right),$ which yields the packing fraction of lattices with this geometry: $\\phi _1(\\theta _0) = \\displaystyle \\frac{3\\pi }{4A_1(\\theta _0)} = \\left( \\displaystyle \\frac{3}{2 + \\cos (\\theta _0) + \\sin (\\theta _0)/\\sqrt{3}} \\right) \\phi _{tri}.$ $\\phi _1(\\theta _0)$ is maximal [$\\phi _1 = \\phi _{tri}$ ] at $\\theta _0 = 0\\ \\rm {and}\\ 60^\\circ $ , and minimal [$\\phi _1 = 3\\pi /[4(1+\\sqrt{3})] \\simeq .862 \\simeq .951\\phi _{tri}$ ] at $\\theta _0 = 30^\\circ $ .","The factors of 3 in the numerators in Eq.","REF reflect the fact that there are three monomers per trimer.","For $\\theta _0 > 60^\\circ $ , it is impossible to center a second trimer at $[1/2,\\sqrt{3}/2]$ .","We postulate that the densest packings correspond to the double lattices described by Kuperberg, Torquato and Jiao [24], [25].","Double lattice packings consist of two lattices related by a displacement plus a $180^\\circ $ rotation of all constituents about their centers of inversion symmetry.","They are often the densest possible packings for both convex [24] and concave [25] particles.","Trimers are inversion-symmetric about the centroids of their center monomers.","Fig.","REF (b) shows our postulated lattice geometry.","The leftmost monomer of a second trimer may be placed at $\\vec{l}_2 = [1-\\cos (\\theta _0),\\sin (\\theta _0)]$ .","Then its center monomer lies at $\\vec{c}_2 = [2 - \\cos (\\theta _0), \\sin (\\theta _0)]$ and its rightmost monomer at $\\vec{r}_2 = [2,0]$ .","This trimer is related to the reference trimer by a $180^\\circ $ rotation about its inversion center plus displacement by $\\vec{c}_2$ .","The lattice vectors for this geometry are $\\vec{b}_1 = [1/2 - \\cos (\\theta _0), \\sqrt{3}/2 + \\sin (\\theta _0)]$ and $\\vec{b}_2 = [3,0]$ .","Its unit-cell area $A_2(\\theta _0)$ is $A_2(\\theta _0) = \\det \\left( \\left[ \\begin{array}{c} \\vec{b}_1 \\\\ \\vec{b}_2 \\end{array} \\right] \\right)$ and its packing fraction is $\\phi _2(\\theta _0) = \\displaystyle \\frac{6\\pi }{4A_2(\\theta _0)} = \\left( \\displaystyle \\frac{1}{1/2 + \\sin (\\theta _0)/\\sqrt{3}} \\right) \\phi _{tri}.$ $\\phi _2(\\theta _0)$ is maximal [$\\phi _2 = \\phi _{tri}$ ] at $\\theta _0 = 60^\\circ \\ \\rm {and}\\ 120^\\circ $ and minimal [$\\phi _2 = \\pi /(2+\\sqrt{3}) \\simeq .842 \\simeq .928\\phi _{tri}$ ] at $\\theta _0 = 90^\\circ $ .","The factor of 6 (rather than 3) in Eq.","REF reflects the fact that these packings have 2 trimers per lattice cell.","Our postulated maximal packing density for 2D bent-core trimers is $\\phi _{max}(\\theta _0) = \\bigg { \\lbrace } \\begin{array}{ccc}\\phi _1(\\theta _0) & , & 0 \\le \\theta _0 \\le 60^\\circ \\\\\\\\\\phi _2(\\theta _0) & , & 60^\\circ \\le \\theta _0 \\le 120^\\circ \\end{array}.$ The variation of $\\phi _{max}$ with $\\theta _0$ is illustrated in Fig.","REF (c).","As discussed above, minimal $\\phi _{max}(\\theta _0)$ occur at the $\\theta _0$ that are most distant from those commensurable with the triangular lattice, i.e.","$30^\\circ $ and $90^\\circ $ .","Kuperberg [24] identified a lower bound $\\phi _K = \\sqrt{3}/2$ for the maximal packing density of identical convex particles.","For $71.4^\\circ \\lesssim \\theta _0 \\lesssim 108.6^\\circ $ , $\\phi _{max} \\le \\phi _K$ , indicating trimers' concavity plays a critical role in decreasing $\\phi _{max}$ for (at least) this range of $\\theta _0$ .","The role of interlocking phenomena specific to concave particles [2], [25] in determining $\\phi _{max}(\\theta _0)$ and the jamming density $\\phi _J(\\theta _0)$ will be discussed further below.","Figure: Putatively densest packings for 2D bent-core trimers.","The top panels show the molecular geometries for θ 0 =0,15 ∘ ,30 ∘ ,45 ∘ ,60 ∘ ,75 ∘ ,90 ∘ ,105 ∘ , and 120 ∘ \\theta _0 = 0,\\ 15^\\circ ,\\ 30^\\circ ,\\ 45^\\circ ,\\ 60^\\circ ,\\ 75^\\circ ,\\ 90^\\circ ,\\ 105^\\circ ,\\ \\rm {and}\\ 120^\\circ .","The bottom panels show the bond/contact topologies for the same systems, with noncovalent contacts indicated by blue lines.","Black parallelograms show the unit cells, which are primitive cells for θ 0 <60 ∘ \\theta _0 < 60^\\circ and contain 2 trimers for θ 0 ≥60 ∘ \\theta _0 \\ge 60^\\circ .","Green lines show covalent bonds.Figure REF shows the lattice packings associated with these motifs for several representative values of $\\theta _0$ .","For $0 < \\theta _0 < 60^\\circ $ these consist of triple layers of triangular lattice separated by lines of “gap” defects.","The gaps are necessary to accommodate the incommensurability of the 3-body fixed-angle ($\\theta = \\theta _0$ ) constraints with the triangular lattice.","The size and shape of the gaps varies with $\\theta _0$ and determines $\\phi _{max}(\\theta _0)$ , but their overall orientation does not change.","For $60^\\circ < \\theta _0 < 120^\\circ $ the impossibility of centering a second trimer at $[1/2,\\sqrt{3}/2]$ (Fig.","REF ) makes forming triple layers of triangular lattice impossible; instead, the maximally dense packings are composed of double layers of triangular lattice separated by lines of gap defects.","This larger concentration of gaps is responsible for the lower $\\phi _{max}$ for $\\theta _0 > 60^\\circ $ .","For example, $[\\phi _{tri}-\\phi _{max}(90^\\circ )]/[\\phi _{tri}-\\phi _{max}(30^\\circ )] = 2(\\sqrt{3}-1) \\simeq 1.46$ is close to the value ($3/2$ ) that might be naively expected from the $\\theta _0 > 60^\\circ $ packings' larger gap concentration, and in fact $\\displaystyle \\frac{\\phi _{tri}-\\phi _{max}(90^\\circ )}{\\phi _{tri}-\\phi _{max}(30^\\circ )} \\cdot \\displaystyle \\frac{\\phi _{max}(30^\\circ )}{\\phi _{max}(90^\\circ )} = \\displaystyle \\frac{3}{2}.$ Further insight into the structure of these lattice packings can be gained by examining the topology of their bond/contact network.","As shown in the bottom panels of Fig.","REF , the bond/contact network is composed of triangles corresponding to monomers in close-packed layers and parallelograms corresponding to monomers bordering gaps.","For $\\theta _0 = 0^\\circ ,\\ 60^\\circ ,\\ \\rm {and}\\ 120^\\circ $ , the opposite corners corners of the parallelograms form additional contacts as the gaps close.","The average monomer coordination numbers are $Z_{mon} = \\bigg { \\lbrace } \\begin{array}{lcl}6 & , & \\theta _0 = 0,\\ 60^\\circ ,\\ \\rm {or}\\ 120^\\circ \\\\16/3 & , & 0 < \\theta _0 < 60^\\circ \\\\5 & , & 60^\\circ < \\theta _0 < 120^\\circ \\end{array},$ where $Z_{mon}$ includes both covalent bonds and noncovalent contacts.","The lower coordination and less efficient packing for $60^\\circ < \\theta _0 < 120^\\circ $ are both consistent with the idea that concavity plays a more important role in these systems [11], [25]; both trends arise from the inability of a reference trimer to form a bond-triangle on its concave side with a monomer belonging to a second trimer when $\\theta _0 > 60^\\circ $ , i.e.","they arise from the difference between the arrangements depicted in Fig.","REF (a-b).","One might expect that the parallelogramic bond/contact arrangements depicted in Fig.","REF are associated with soft shear modes (as they often are in monomeric systems).","In these maximally dense lattice packings, however, the fixed-angle constraints prevent the parallelograms from being sheared without violating hard-disc overlaps [26], [27].","Moreover, since monomers are highly overconstrained in these packings, trimers are necessarily also highly overconstrained.","We will show below that the dense lattice packings identified here are in fact all strictly jammed [27].","Figure: Degeneracy of the densest lattice packings.","The top panels illustrate one of the degeneracies for θ 0 =45 ∘ \\theta _0 = 45^\\circ : (a) shows the arrangement depicted in Fig.","(a), while (b) shows a degenerate flipped-and-shifted version of this arrangement with the same φ=φ max (θ 0 )\\phi = \\phi _{max}(\\theta _0).","The bottom panels illustrate one of the degeneracies for θ 0 =90 ∘ \\theta _0 = 90^\\circ : (c) shows the arrangement depicted in Fig.","(b), while (d) shows a degenerate φ=φ max (θ 0 )\\phi = \\phi _{max}(\\theta _0) arrangement with different symmetry and gap topology.A key factor influencing both jamming and glass-formation in systems of constituents that are able to crystallize is competition of degenerate crystalline structures.","For example, monodisperse sphere packings jam far more readily than monodisperse disk packings [28] because there are two incommensurate close-packed lattices in 3D (i.e.","FCC and BCC) but only a single close-packed lattice in 2D (i.e.","the triangular lattice).","Finding the geometries shown in Figs.","REF -REF did not address the question of degeneracy.","It turns out that these lattices are highly degenerate.","Figure REF illustrates how these degeneracies arise.","For $0 < \\theta _0 < 60^\\circ $ , rotating the trimers in alternating layers (here depicted in blue and red) by $180^\\circ $ about their inversion-symmetry centers and then shifting them by [$1-\\cos (\\theta _0), \\sin (\\theta _0)$ ] produces a lattice with the same $\\phi = \\phi _{max}(\\theta _0)$ .","For $60 < \\theta _0 < 120^\\circ $ , rotating the blue trimers so that their concave sides point away from rather than towards the centroids of the red trimers they double-contact achieves the same effect.","Figure: Panel (a): Schematic depiction of our adaptive shrinking cells for n tri =4n_{tri} = 4.","Periodic images of the n tri n_{tri} trimers are present in the ASC algorithm but are not shown here.","Panel (b): Comparison of analytic φ max (θ 0 )\\phi _{max}(\\theta _0) [Eqs.",", , ; solid curve] to ASC results for n tri =2n_{tri} = 2 (red *) and n tri =4n_{tri} = 4 (green x) with X final ≥7X_{final} \\ge 7 (cf.","Sec.",").","n tri =3n_{tri} = 3 is not considered here because lattices with bases containing an odd number of trimers cannot be double lattices and hence are generally less than maximally dense for θ 0 >60 ∘ \\theta _0 > 60^\\circ .","The data for n tri =2 and 4n_{tri} = 2\\ \\rm {and}\\ 4 overlap; for clarity, results are presented for alternating values of θ 0 \\theta _0.","Panel (c): One of the degenerate maximally dense n tri =4n_{tri} = 4 lattice packings for θ 0 =90 ∘ \\theta _0 = 90^\\circ contains alternating layers of the degenerate φ=φ max (90 ∘ )\\phi = \\phi _{max}(90^\\circ ) n tri =2n_{tri} = 2 lattices shown in Fig.","(c-d).The degenerate lattices depicted in Fig, REF (a-b) and (c-d) differ in their symmetries and gap topologies.","They represent competing ordered structures that are essentially trimeric – they are not present for monomers or dimers because their existence requires the fixed-angle ($\\theta = \\theta _0$ ) constraints.","Other degenerate arrangements with $\\phi = \\phi _{max}(\\theta _0)$ also exist.","We expect that there are in fact infinitely many of them, just as there are infinitely many variants of the close-packed lattice in 3D.","The arguments of Ref.","[28] suggest that all this degeneracy should strongly promote jamming / suppress crystallization in bent-core-trimeric systems relative to their monomeric and dimeric counterparts.","Below, we will investigate the degree to which this is true.","Jennings et.","al.","[23] found that for $R = 1/2$ trimers, double-lattice packings are not optimally dense for some $\\theta _0$ and $r$ .","In these special cases, the densest packings are lattices with bases containing more than two trimers.","To see whether this is true for our $r = R = 1$ systems, we identify maximally dense lattice packings for bases of various sizes using a variant of Torquato et.","al.","'s adaptive shrinking cell (ASC) algorithm [10], [11].","Figure REF (a) shows our ASC geometry for periodic cells containing $n_{tri}$ trimers.","The algorithm we use to obtain both optimally-dense and less-dense packings is described in detail below (in Section ).","Figure REF (b) compares our analytic prediction for $\\phi _{max}(\\theta _0)$ to the maximal-density lattice packings found from ASC runs for $n_{tri} \\le 4$ .","For both $n_{tri} = 2$ and $n_{tri} = 4$ , ASC results converge (within our numerical precision) to our putatively densest configurations or their degenerate counterparts.","These results indicate a key difference between the densest packings of the $r = R = 1$ trimers considered here and those of the $R = 1/2$ trimers studied in Ref.","[23] (wherein larger bases produce denser lattice packings for some $r$ and $\\theta _0$ .)","The simpler behavior for $r = R = 1$ appears to result from a reduction in the number of ways that small numbers of trimers can fit together when the trimers are composed of monodisperse tangent disks as opposed to bidisperse overlapping disks.","Another important difference associated with the abovementioned degeneracies appears for $n_{tri} > 2$ .","As shown in Fig.","REF (c), packings with $\\phi = \\phi _{max}(\\theta _0)$ may be formed by alternating layers of the degenerate structures identified above (Fig.","REF ).","Increasingly complicated arrangements of this type become possible as $n_{tri}$ increases.","This effect is analogous to the increasing number of distinguishable ways to stack $N_l$ layers of triangular lattice to form 3D close-packed structures as $N_l$ increases, and should further promote jamming."],["Statics: nonoptimally dense strictly jammed lattice packings","Having identified the maximally dense lattice packings, we now characterize the less-dense strictly-jammed lattice packings of these systems.","Our ASC algorithm [illustrated in Fig.","REF (a)] is implemented as follows.","Since translational invariance implies that trimer 0 can be centered at the origin ($x_0 = y_0 = 0$ ) without loss of generality, systems have $N_{dof} = 4 + 3(n_{tri}-1)$ degrees of freedom: the cell shape parameters $\\alpha , \\beta , \\gamma $ and the trimer-arrangement variables [$\\phi $ , and $x_i, y_i, \\phi _i$ for $i = 1, 2, ..., n_{tri}-1$ ].","Starting values of $\\alpha $ and $\\beta $ are chosen to be sufficiently large that all $n_{tri}$ trimers are able to freely rotate within the cell while $\\gamma = 0$ .","Strictly jammed packings are obtained through four types of moves: (1) random incremental changes of ($\\alpha , \\beta , \\gamma $ ) accompanied by affine displacements of the trimer centers [($x_i, y_i$ ) for $i = 1,\\ 2,\\ ...,\\ n_{tri}-1$ ]; (2) single-particle moves consisting of random incremental changes of $\\phi $ or of ($x_i, y_i, \\phi _i$ ) for some $i = 1,\\ 2,\\ ...,\\ n_{tri}-1$ ; (3) collective moves consisting of rigid translations or rotations of ($j < n_{tri}$ )-trimer subsets of trimers $1,\\ 2,\\ ...,\\ n_{tri}-1$ ; (4) changes of ($\\alpha , \\beta , \\gamma $ ) that preserve volume.","Type (1) moves are accepted if they reduce the cell volume $A = \\alpha \\beta \\cos (\\gamma )$ without producing any particle overlaps, while moves of types (2-4) are accepted if they produce no particle overlaps.","The initial maximal increment sizes for these moves are fixed at $\\lbrace |\\delta \\alpha | = .05\\cdot 2^{-X},\\ |\\delta \\beta | = .05\\cdot 2^{-X},\\ |\\delta \\phi | = 2.5^\\circ \\cdot 2^{-X},\\ |\\delta x_i |= .25\\cdot 2^{-X},\\ |\\delta y_i| = .25\\cdot 2^{-X},\\ |\\delta \\phi _i| = 2.5^\\circ \\cdot 2^{-X} \\rbrace $ with $X = 0$ .","After the process of compressing the system using moves of types (1-3) has converged [i.e.","no more moves of these types are being accepted], the system is collectively jammed [27] for the given value of $X$ .","Then moves of type (4) are used to check for strict jamming.","Successful type-4 moves indicate the system is not strictly jammed; when they occur, moves of types (1-3) are begun again.","This process repeats itself until the system is strictly jammed with respect to moves of types (1-4) for the given value of $X$ .","Then $X$ is increased by 1 and the process begins again.","This is repeated until satisfactory convergence is achieved; the data in Figure REF (b) indicate the maximally dense packings found by our algorithm that are strictly jammed for $X_{final} \\ge 7$ .","Figure: Analytic and ASC results for strictly jammed lattice packings.","Panel (a): Colored curves indicate the analytic φ i (θ 0 )\\phi _i(\\theta _0) (Table ) while black symbols indicate n tri =2n_{tri} = 2 ASC results.","Panel (b): ASC results for n tri =4n_{tri} = 4.As discussed in Refs.","[10], [11], improved performance of the algorithm can be obtained by adjusting the $X$ -increment and/or adopting more complicated Monte Carlo schemes such as allowing occasional acceptance of moves that increase $A$ .","To obtain a wide range of packings including both optimally and nonoptimally dense geometries, it is necessary to employ a wide range of trimer-arrangement initial conditions for each $\\theta _0$ .","This requirement combined with the fact that the computational complexity of the above-discussed type-3 moves is roughly exponential in $N_{dof}$ prohibited extending our comprehensive ASC studies to $n_{tri} > 4$ .","However, studies of selected $\\theta _0$ for $n_{tri} = 6$ found no packings with $\\phi > \\phi _{max}(\\theta _0)$ , and as we will show below, much insight can be combined by combining $n_{tri} \\le 4$ ASC studies with large-$n_{tri}$ molecular dynamics simulations.","Figure: θ 0 =θ min \\theta _0 = \\theta _{min} configurations and bond topologies of the seven nonoptimally-dense n tri =2n_{tri} = 2 lattice packings (families 3-9 in Table , Fig. )","that remain stable for n tri =4n_{tri} = 4.","The maximally-dense lattices (families 1-2) were illustrated in Figs.","-.","Note that it is the bond topologies that define the tiling types listed in Table .Table: Families of strictly jammed lattice packings for n tri =2n_{tri} = 2, in order of decreasing minimal density φ i (θ min )\\phi _i(\\theta _{min}).","Families 1 and 2 are respectively the maximally-symmetric maximally-dense lattice packings identified above [Fig.","(a-b); Eqs.",", ] for 0≤θ 0 ≤60 ∘ 0 \\le \\theta _0 \\le 60^\\circ and 60 ∘ ≤θ 0 ≤120 ∘ 60^\\circ \\le \\theta _0 \\le 120^\\circ .","pp and qq are respectively equal to |30 ∘ -θ 0 ||30^\\circ - \\theta _0| and 2sin -1 [(22+2cos[θ 0 ]) -1 ]2\\sin ^{-1}[(2\\sqrt{2+2\\cos [\\theta _0]})^{-1}].","The “tiling types” describe the lattices' bond topology for all θ 0 \\theta _0 within the range, but in general only precisely describe the lattices' geometrical structure for specific θ 0 \\theta _0; for example, type 5's bond topology is always that of the square lattice but its geometry is only that of the square lattice for θ 0 =90 ∘ \\theta _0 = 90^\\circ .","All families except for 10(a-b) remain strictly jammed for n tri =4n_{tri} = 4; their n tri =4n_{tri} = 4 versions are simply two adjacent copies of the n tri =2n_{tri} = 2 lattices.As shown in Figure REF (a), at least ten distinct families of strictly jammed lattice packings exist for $n_{tri} = 2$ .","Each family represents a continuous set of lattice packings sharing a common bond topology (Figure REF ).","Since the families can be distinguished by their bond topologies, it is convenient to associate them with distinct categories of planar tilings [29].","Moreover, for each family $i$ , these ASC results allowed us to identify exact analytic expressions for the packing fraction $\\phi _i(\\theta _0)$ .","Results are summarized in Table REF .","All families except for 7b and 10b [30] share two common features: (i) their $\\phi _i(\\theta _0)$ are maximized at their “endpoint” $\\theta _0$ (e.g.","$\\theta _0 = 0\\ \\rm {and}\\ 60^\\circ $ for family 1) and minimized at their respective $\\theta _{min}$ ; (ii) they reduce to the triangular lattice at at least one of the three $\\theta _0$ allowing for it ($0,\\ 60^\\circ ,\\ \\rm {and}\\ 120^\\circ $ ).","Because the various $\\phi _i(\\theta _0)$ vary continuously, feature (ii) means that the various families' densities converge to each other as $\\theta _0$ approaches these special values.","Despite this convergence, the associated lattices remain distinct and incommensurable.","We expect that this incommensurability strongly promotes jamming in bulk systems.","Some of the nonmaximally dense families correspond to familiar 2D lattice structures, e.g.","family 5 is the square lattice for $\\theta _0 = 90^\\circ $ and family 10a (10b) is the kagome lattice for $\\theta _0 = 60^\\circ $ ($\\theta _0 = 0$ ).","Others represent less-familiar forms, such as family 8 which possesses a very intriguing bond topology consisting of five-sided polygons as well as the usual triangles and parallelograms.","The wide variety of mechanically stable arrangements with different symmetries and bond topologies suggests that these systems' equilibrium phase diagrams may be especially rich [31], potentially including entropically-driven solid-solid transitions, various liquid-crystalline phases (for example, $\\theta _0 = 90^\\circ \\ \\rm {and}\\ 120^\\circ $ systems might respectively form thermodynamically stable bent-core tetratic and hexatic liquid crystals), and KTHNY-style continuous melting transitions [32], [33].","Figure REF (b) presents our ASC results for $n_{tri} = 4$ .","The most obvious difference from the $n_{tri} = 2$ results is the elimination of the lowest-$\\phi $ strictly-jammed packings; this occurs because the larger basis allows for shear modes that destabilize the kagome-like lattices (family 10).","A second obvious difference is that there are many additional families.","Visual inspection indicates that a large fraction of these are formed by combining two of the families discussed above.","There are very many such combinations, making exhaustive cataloguing of them (as we did for $n_{tri} = 2$ ) prohibitively difficult, and continuing our ASC studies to even larger $n_{tri}$ would of course exacerbate this issue.","Instead we will test the extent to which the ideas presented here are useful by looking for local structural motifs within large-$n_{tri}$ jammed configurations that correspond to the nonoptimally-dense families."],["Dynamics: disordered and partially-ordered jammed packings","None of the above discussion addresses the dynamics of the jamming process.","Since the dynamics of systems' jamming transitions naturally relate to the dynamics of their glass transitions [34], we now examine the compression-rate dependence of our model trimers' athermal solidification behavior using molecular dynamics (MD) simulations.","Each of the $n_{tri}$ simulated trimers contains three monomers of mass $m$ .","The trimers are rigid; bond lengths and angles are held fixed by holonomic constraints.","Monomers on different trimers interact via a harmonic potential $U_{H}(r) = 10\\epsilon (1 - r/\\sigma )^2 \\Theta (\\sigma -r)$ , where $\\epsilon $ is the energy scale of the pair interactions, $\\sigma $ is monomer diameter, and $\\Theta $ is the Heaviside step function.","Initial states are generated by placing the trimers randomly within a square cell, with periodic boundary conditions applied along both directions.","Then Newton's equations of motion are integrated with a timestep $\\delta t = .005\\tau $ , where the unit of time is $\\tau =\\sqrt{m\\sigma ^2/\\epsilon }$ .","Systems are equlibrated at $k_BT/\\epsilon = 1$ and $\\phi = \\exp (-1)\\phi _{tri}$ until intertrimer structure has converged, then cooled to $T=0$ at a rate $10^{-4}(\\epsilon /k_B)/\\tau $ .","After cooling, systems are hydrostatically compressed at a true strain rate $\\dot{\\epsilon }$ , i.e.","the cell side length $L$ is varied as $L = L_0 \\exp (-\\dot{\\epsilon }t)$ .","To maintain near-zero temperature during compression, we employ overdamped dynamics with the equation of motion $m\\ddot{\\vec{r_i}} = \\vec{F} - \\Gamma \\dot{\\vec{r_i}} +h(\\lbrace \\vec{r} , \\dot{\\vec{}}{r} \\rbrace )$ where $\\vec{r}_i$ is the position of monomer $i$ , $\\vec{F}$ is the force arising from the harmonic pair interactions, the damping coefficient $\\Gamma = 10^4\\dot{\\epsilon }$ , and the $h(\\lbrace \\vec{r} , \\dot{\\vec{}}{r} \\rbrace )$ term enforces trimer rigidity [35].","Jamming is defined to occur when the nonkinetic part of the pressure $P$ exceeds $P_{thres}= 10^{-4}\\epsilon /\\sigma ^2$ ; choosing a lower (higher) value of $P_{thres}$ lowers (raises) $\\phi _J(\\theta _0)$ , but does not qualitatively change any of the results presented herein.","We choose to identify jamming with the emergence of a finite bulk modulus rather than with the vanishing of soft modes because proper handling of soft modes associated with trimeric “rattlers” [36] is highly nontrivial.","All MD simulations are performed using LAMMPS [37].","Figure: Dynamical jamming.","Panels (a) and (b) show φ J (θ 0 )\\phi _J(\\theta _0) and the ratio φ J (θ 0 )/φ max (θ 0 )\\phi _J(\\theta _0)/\\phi _{max}(\\theta _0) for several strain rates, and panel (c) shows shows φ J (θ 0 ;ϵ ˙)\\phi _J(\\theta _0; \\dot{\\epsilon }) for several characteristic values of θ 0 \\theta _0.","The dotted gray line in panel (a) shows φ max (θ 0 )\\phi _{max}(\\theta _0).","Dotted lines in panel (c) indicate fits to Eq.",".","All results are averaged over 9 independently prepared n tri =400n_{tri} = 400 systems.Figure REF shows results for the rate-dependent $\\phi _J(\\theta _0)$ .","As shown in panel (a), all systems jam at densities well below the monomeric value $\\phi _J^{mon} \\simeq .84$ [36]; the reduced $\\phi _J$ relative to monomers are caused by the frozen-in 2-body and 3-body constraints [8].","For all compression rates, values of $\\phi _J(\\theta _0)$ clearly follow trends in $\\phi _{max}(\\theta _0)$ , exhibiting maxima at $\\theta _0 = 0,\\ 60^\\circ \\ \\rm {and}\\ 120^\\circ $ and minima at $\\theta _0 \\simeq 30^\\circ \\ \\rm {and}\\ \\simeq 90^\\circ $ .","There are clearly two separate branches of $\\phi _J(\\theta _0)$ : one for $0 \\le \\theta _0 \\le 60^\\circ $ and one for $60^\\circ \\le \\theta _0 \\le 120^\\circ $ .","However, values of $\\phi _J(\\theta _0)$ do not simply track $\\phi _{max}(\\theta _0)$ .","Specifically, closed ($\\theta _0 = 120^\\circ $ ) trimers have a much higher $\\phi _J$ than their open ($\\theta _0 = 0\\ \\rm {or}\\ 60^\\circ $ ) counterparts even though their $\\phi _{max}$ are identical.","More generally, while the first branch of $\\phi _J(\\theta _0)$ is close to symmetric about $\\theta _0 = 30^\\circ $ , the second branch is clearly asymmetric; $\\phi _J(90^\\circ + \\psi ) > \\phi _J(90^\\circ - \\psi )$ , increasingly so as $\\psi $ increases from zero towards $30^\\circ $ .","While it is not surprising that $\\theta _0 = 120^\\circ $ trimers are the best crystal-formers (cf.","Fig.","REF ) – they are compact and threefold-symmetric whereas lower $\\phi _J$ are expected for small-$\\theta _0$ systems owing to their larger aspect ratios [1], [8], our results show that this effect propagates downward in $\\theta _0$ as far as $\\theta _0 \\simeq 90^\\circ $ .","A useful metric characterizing the strength of the physical processes promoting disorder in these systems' solid-state morphologies is $f(\\theta _0; \\dot{\\epsilon }) = \\phi _J(\\theta _0; \\dot{\\epsilon })/\\phi _{max}(\\theta _0)$ .","This quantity is unity when systems crystallize into their maximal-density lattices during compression, and smaller when systems jam at $\\phi < \\phi _{max}(\\theta _0)$ due to the presence of disorder.","Roughly speaking, characterizing the decrease of $f(\\theta _0; \\dot{\\epsilon })$ with increasing $\\dot{\\epsilon }$ provides insight into the kinetics of the solidification process, while characterizing its variation with $\\theta _0$ in the low-$\\dot{\\epsilon }$ limit provides insight into how the strength of frustration/degeneracy-related effects varies with molecular shape.","Panel (b) presents results for $f(\\theta _0; \\dot{\\epsilon })$ for all systems.","For all strain rates considered here, trends in $f(\\theta _0; \\dot{\\epsilon })$ are opposite those in $\\phi _J(\\theta _0; \\dot{\\epsilon })$ and $\\phi _{max}(\\theta _0)$ .","Minima in the former correspond to maxima in the latter, e.g.","maxima in $f$ occur at $\\theta _0 = 30^\\circ \\ \\rm {and}\\ 90^\\circ $ .","One potential reason for this is that grains of crystals with incompatible local ordering corresponding to the different families discussed above are more likely to form at densities slightly below $\\phi _J(\\theta _0)$ and then jam as systems are further compressed for the systems with lower $f(\\theta _0; \\dot{\\epsilon })$ .","The fact that differences between several of the $\\phi _i(\\theta _0)$ are maximal for $\\theta _0 = 30^\\circ \\ \\rm {and}\\ 90^\\circ $ and minimal for $\\theta _0 = 0,\\ 60^\\circ ,\\ \\rm {and}\\ 120^\\circ $ supports this hypothesis; competition between differently ordered grains should be greater when their densities are closer due to the abovementioned convergence of the $\\phi _i(\\theta _0)$ .","Another potential reason is that jamming dynamics are controlled by the trimer mobility $\\mu $ and that $\\mu (\\phi ,\\theta _0)$ depends far more strongly on $\\phi $ than on $\\theta _0$ for $\\phi < \\phi _J$ .","It would be interesting to test this idea in thermalized versions of these systems by constructing isomobility curves in ($\\theta _0,\\phi $ )-space.","Panel (c) illustrates the compression-rate dependence of jamming in more detail for the $\\theta _0$ corresponding to extrema of $\\phi _J$ : $0,\\ 30^\\circ ,\\ 60^\\circ ,\\ 90^\\circ ,\\ \\rm {and}\\ 120^\\circ $ .","For all systems, results are well fit by $\\phi _J(\\theta _0; \\dot{\\epsilon }) \\simeq \\phi _{J}^0(\\theta _0)\\exp \\left[ -\\left( \\displaystyle \\frac{\\dot{\\epsilon }}{\\dot{\\epsilon }_0} \\right)^\\gamma \\right],$ where $\\phi _J^0$ is the quasistatic-limiting value of $\\phi _J$ , $\\dot{\\epsilon }_0$ is a characteristic rate, and $\\gamma $ describes the strength of the rate dependence.","While rigorously determining the exact functional form of $\\phi _J(\\theta _0; \\dot{\\epsilon })$ would require more computational resources than we currently possess and is beyond our present scope, the fact that the best-fit values of $\\dot{\\epsilon }_0$ and $\\gamma $ remain within relatively narrow ranges for all $0 \\le \\theta _0 \\le 120^\\circ $ – respectively $3\\cdot 10^{-4} \\lesssim \\dot{\\epsilon }_0\\tau \\lesssim 7\\cdot 10^{-4}$ and $.65 \\lesssim \\gamma \\lesssim .85$ – suggests that Eq.","REF is a useful approximate form for all $\\theta _0$ .","Similar compression-rate dependencies of $\\phi _J$ are found for dynamical jamming of a wide variety of systems [38].","Figure: Jammed packings of n tri =400n_{tri} = 400 bent-core trimers generated by dynamic compression.","Rows from top to bottom show typical marginally-jammed states for θ 0 =0,30 ∘ ,60 ∘ ,90 ∘ , and 120 ∘ \\theta _0 = 0,\\ 30^\\circ ,\\ 60^\\circ ,\\ 90^\\circ ,\\ \\rm {and}\\ 120^\\circ .","The left and right columns respectively indicate results for ϵ ˙τ=10 -5.5 and 10 -7 \\dot{\\epsilon }\\tau = 10^{-5.5}\\ \\rm {and}\\ 10^{-7}.","Colors from red to blue indicate trimers 1,2,...,4001,\\ 2,\\ ...,\\ 400.Finally we turn to a qualitative characterization of how $\\theta _0$ and $\\dot{\\epsilon }$ affect jammed systems' microstructure.","Typical marginally-jammed packings for the five characteristic $\\theta _0$ discussed above are shown in Figure REF .","For $\\theta _0 = 0,\\ 60^\\circ ,\\ \\rm {and}\\ 120^\\circ $ , triangular-crystalline grains are clearly visible.","This is consistent with the well-known result [28] that 2D systems of monodisperse disks have a strong propensity to crystallize.","Contrasting the high-$\\dot{\\epsilon }$ and low-$\\dot{\\epsilon }$ snapshots for these $\\theta _0$ suggests that systems jam via a two-stage, two-length-scale process.","First, randomly-oriented crystalline grains form and grow to a size that depends on both $\\theta _0$ and $\\dot{\\epsilon }$ .","Since these grains cannot be compressed further, they effectively behave as single nearly-rigid particles as compression continues.","The degree to which grains grow prior to jamming is kinetically limited and is a key factor producing the rate dependence of $\\phi _J(\\theta _0)$ .","Higher compression rates lead to smaller grain sizes and greater intergrain misorientation, just as is the case in monomeric systems [28], [38].","The results for $\\theta _0 = 30^\\circ \\ \\rm {and}\\ 90^\\circ $ are less easy to interpret since only basic features such as the decrease in the typical size of interstitials with decreasing $\\dot{\\epsilon }$ [28] are immediately apparent.","The $\\theta _0 = 90^\\circ $ systems clearly possess some grains with square-lattice ordering, showing that the nonoptimally-dense families identified above (Table REF , Figs.","REF -REF ) do indeed play a role in these systems' jamming phenomenology.","However, the clear crystallization kinetics observed for $\\theta _0 = 0,\\ 60^\\circ ,\\ \\rm {and}\\ 120^\\circ $ are absent here, perhaps because these systems' lower absolute $\\phi _J$ and $\\phi _{max}$ values make them appear more disordered overall.","The effects of degeneracy may also be larger at these $\\theta _0$ .","These issues might be resolved by going to much lower $\\dot{\\epsilon }$ , but doing so is not yet computationally feasible."],["Discussion and Conclusions","In this paper, we examined the athermal solidification behavior of 2D bent-core tangent-disk trimers.","We found that trends in $\\phi _J(\\theta _0)$ closely follow those in $\\phi _{max}(\\theta _0)$ , but with additional effects related to symmetry and degeneracy superposed.","We reported two distinct regimes of packing/jamming phenomenology, identifying the source of the difference between them as the ability (inability) of a reference trimer to form a bond-triangle on its concave side with a monomer belonging to a second trimer when $\\theta _0 < 60^\\circ $ ($\\theta _0 > 60^\\circ $ ).","Well-packed systems with $\\theta _0 > 60^\\circ $ are generally less dense and less hyperstatic than their $\\theta _0 < 60^\\circ $ counterparts.","Another key insight was that deviations of $\\theta _0$ away from the values allowing formation of the triangular lattice ($0^\\circ ,\\ 60^\\circ ,\\ \\rm {and}\\ 120^\\circ $ ) do not by themselves frustrate crystalline order.","Instead, crystals belonging to several families distinguished by their differing bond topologies can form.","We believe that it is the presence of these competing families combined with the extensive degeneracy of the densest lattices that frustrates crystallization and promotes jamming in these systems, Our work complements recent studies of the thermal solidification of Lewis-Wahnstrom-like models [39], [40], which illustrated several nontrivial effects of trimeric structure (e.g.","that its enhancement of the interfacial energy between crystalline and liquid phases promotes glass-formation) but did not attempt to connect their findings to the models' optimal-packing or jamming-related phenomenology.","One of the principal goals of soft materials science is designing materials that possess tunable solid morphology.","Designing custom pair interactions that yield nontriangular 2D-crystalline or non-close-packed 3D-crystalline ground states has attracted significant interest in recent years [41], [42], [43].","Our 2D bent-core tangent-disk trimers provide a simple example of how the same goal may be achieved with hard-disk or hard-sphere pair interactions by controlling 2-body and 3-body correlations, i.e.","by imposing covalent bonding and controlling the bond angle $\\theta _0$ .","The potential relevance to real systems is that controlling the bond angle (or analogous shape parameters) of small molecules [13], [14], [16], [15], [19], [20], [21], [22] is often easier than controlling the pair interactions of their constituent atoms.","Another key tuning parameter for soft materials is their degree of (dis)order.","Recent work [16], [15] has shown that the best glass-formers (crystal-formers) in homologous series of small molecules are those with the highest (lowest) ratio $T_g/T_m$ .","One might naively guess that an athermal version of this principle applicable to bent-core trimer molecules is that the best glass-formers (crystal-formers) are those with the lowest (highest) ratio $\\phi _J(\\theta _0)/\\phi _{max}(\\theta _0)$ .","However, the dynamic-compression results we presented here suggest that this is not the case.","Moreover, the multiplicity of nonoptimally-dense-but-strictly-jammed lattice packings these systems can form suggests that they may possess multiple thermodynamically stable crystalline phases and/or nontrivial liquid-crystalline phases [12].","We conclude that further characterizing these systems' equilibrium phase behavior, e.g.","by accurately determining their $\\phi _{melt}(\\theta _0)$ using techniques like those of Refs.","[32], [33], is necessary to flesh out this issue further.","This material is based upon work supported by the National Science Foundation under Grant DMR-1555242."]],"string":"[\n [\n \"Densest vs. jammed packings of 2D bent-core trimers\"\n ],\n [\n \"Abstract We identify the maximally dense lattice packings of tangent-disk trimers with fixed bond angles ($\\\\theta = \\\\theta_0$) and contrast them to both their nonmaximally-dense-but-strictly-jammed lattice packings as well as the disordered jammed states they form for a range of compression protocols.\",\n \"While only $\\\\theta_0 = 0,\\\\ 60^\\\\circ,\\\\ \\\\rm{and}\\\\ 120^\\\\circ$ trimers can form the triangular lattice, maximally-dense maximally-symmetric packings for all $\\\\theta_0$ fall into just two categories distinguished by their bond topologies: half-elongated-triangular for $0 < \\\\theta_0 < 60^\\\\circ$ and elongated-snub-square for $60^\\\\circ < \\\\theta_0 < 120^\\\\circ$.\",\n \"The presence of degenerate, lower-symmetry versions of these densest packings combined with several families of less-dense-but-strictly-jammed lattice packings act in concert to promote jamming.\"\n ],\n [\n \"Introduction\",\n \"Jamming of anisotropic constituents has attracted great interest [1], [2], [3], [4], [5] for two reasons.\",\n \"The first is that understanding how anistropy affects jamming is critical because most real granular materials are composed of anisotropic grains.\",\n \"The second is that constituent-particle anistropy affects systems' jamming phenomenology and their thermal-solidification phenomenology in similar ways, and hence studying the jamming of grains of a given shape can provide insight into the thermal solidification of similarly shaped molecules and/or colloids [6], [7], [8], [9].\",\n \"Such studies are maximally effective when they are complemented by identifying the particles' densest possible packings since the differences between densest and jammed packings are often analogous to the differences between crystals and glasses formed via thermal solidification [10], [11], [9].\",\n \"Bent-core trimers are a simple model for multiple liquid-crystal-forming [12] and glass-forming [13], [14], [15], [16] molecules.\",\n \"As illustrated in Figure REF , their shape can be characterized using three parameters: the bond angle $\\\\theta _0$ , the ratio $r$ of end-monomer radius to center-monomer radius, and the ratio $R$ of intermonomer bond length to center-monomer diameter.\",\n \"For example, para-, meso-, and ortho-terphenyl correspond to the molecule shown in Fig.\",\n \"REF (with $\\\\theta _0 = 0^\\\\circ ,\\\\ 60^\\\\circ ,\\\\ \\\\rm {and}\\\\ 120^\\\\circ $ , respectively), and the popular Lewis-Wahnstrom model [17] for OTP implements this molecular geometry with $R = 2^{-1/6},\\\\ r = 1,\\\\ \\\\theta _0 = 105^\\\\circ $ .\",\n \"It is well known that the properties of systems composed of such molecules depend strongly on all three of these parameters; for example, the three terphenyl isomers form very differently structured bulk solids under the same preparation protocol [13], as do xylenes [14], diphenylcycloalkenes [15] and homologous series of cyclic stilbenes [16].\",\n \"The structural isomers and near-isomers of more complicated small molecules also often exhibit very different solidification behavior, e.g.\",\n \"the trisnapthylbenzenes which have attracted great interest in recent years because they have been shown to form quasi-ordered glasses when vapor-deposited [18], [19], [20], [21], [22].\",\n \"Figure: Rigid bent-core disk-trimers with bond angle θ 0 \\\\theta _0.\",\n \"Panel (a) shows the general geometry with unspecified (r,Rr,\\\\ R).\",\n \"Here we study the r=R=1r = R = 1 case shown in panel (b).Our understanding of such phenomena and hence our ability to engineer crystallizability/glass-formability at the molecular level remains very limited.\",\n \"One of the reasons why this is so is that only a few theoretical studies have isolated the role played by molecular shape using simple models.\",\n \"Molecules like those studied in Refs.\",\n \"[18], [19], [20], [21], [22] tend to form liquid-crystalline phases with columnar order [21].\",\n \"Studying packing of 2D models for these molecules corresponds to studying the in-plane ordering of such anisotropic phases.\",\n \"Optimal packing of molecules with the geometry shown in Figure REF has been investigated only minimally; Ref.\",\n \"[23] reported the densest packings of 2D $R = 1/2$ trimers as a function of $r$ and $\\\\theta _0$ .\",\n \"The tangent-disk ($r = R = 1$ ) case shown in Fig.\",\n \"REF (b) is of considerable interest because it allows straightforward connection to results obtained for monomers – and hence isolation of the role played by the bond and angular constraints – while remaining a reasonable minimal model for terphenyl-shaped molecules.\",\n \"In this paper, we identify and characterize the densest lattice packings of 2D bent-core tangent-disk trimers as a function of their bond angle $\\\\theta _0$ , and contrast them to both their nonmaximally-dense-but-strictly-jammed lattice packings and the disordered jammed packings they form under dynamic compression.\",\n \"Figure: Structure of the putatively densest trimer packings.\",\n \"Panel (a): lattice configuration for 0≤θ 0 ≤60 ∘ 0 \\\\le \\\\theta _0 \\\\le 60^\\\\circ .\",\n \"Panel (b): lattice configuration for 60 ∘ ≤θ 0 ≤120 ∘ 60^\\\\circ \\\\le \\\\theta _0 \\\\le 120^\\\\circ .\",\n \"Panel (c): Postulated maximal density φ max (θ 0 )\\\\phi _{max}(\\\\theta _0) [Eqs.\",\n \", , ].\"\n ],\n [\n \"Densest Packings\",\n \"The only $\\\\theta _0$ allowing formation of the triangular lattice [which is the densest possible 2D disk packing, with $\\\\phi = \\\\phi _{tri} = \\\\pi /(2\\\\sqrt{3}) \\\\simeq .9069$ ] are $0^\\\\circ ,\\\\ 60^\\\\circ ,\\\\ \\\\rm {and}\\\\ 120^\\\\circ $ .\",\n \"In this section, we identify the densest packings for all $\\\\theta _0$ .\",\n \"A potential geometry of the densest packings for $0 \\\\le \\\\theta _0 \\\\le 60^\\\\circ $ is shown in Fig.\",\n \"REF (a).\",\n \"Black circles indicate the monomer positions for a reference trimer centered at the origin.\",\n \"For this range of $\\\\theta _0$ , another similarly oriented trimer can be centered at $\\\\vec{c}_2 = [1/2,\\\\sqrt{3}/2]$ , with its leftmost and rightmost monomers respectively at $\\\\vec{l}_2 = [1/2-\\\\cos (\\\\theta _0),\\\\sqrt{3}/2+\\\\sin (\\\\theta _0)]$ and $\\\\vec{r}_2 = [3/2,\\\\sqrt{3}/2]$ .\",\n \"Then the center monomer of a third trimer with this orientation can be placed at $\\\\vec{c}_3 = [2+\\\\cos (\\\\theta ),-\\\\sin (\\\\theta )]$ .\",\n \"A horizontally oriented unit cell with lattice vectors $\\\\vec{b}_1, \\\\vec{b}_2$ is obtained by rotating $\\\\vec{c}_2,\\\\ \\\\vec{c}_3$ through the angle $\\\\delta (\\\\theta _0) = \\\\tan ^{-1}(\\\\sin (\\\\theta _0)/[2+\\\\cos (\\\\theta _0)])$ .\",\n \"The area of this unit cell is $A_1(\\\\theta _0) = \\\\det \\\\left( \\\\left[ \\\\begin{array}{c} \\\\vec{b}_1 \\\\\\\\ \\\\vec{b}_2 \\\\end{array} \\\\right] \\\\right) = \\\\det \\\\left( \\\\left[ \\\\begin{array}{c} \\\\vec{c}_2 \\\\\\\\ \\\\vec{c}_3 \\\\end{array} \\\\right] \\\\right),$ which yields the packing fraction of lattices with this geometry: $\\\\phi _1(\\\\theta _0) = \\\\displaystyle \\\\frac{3\\\\pi }{4A_1(\\\\theta _0)} = \\\\left( \\\\displaystyle \\\\frac{3}{2 + \\\\cos (\\\\theta _0) + \\\\sin (\\\\theta _0)/\\\\sqrt{3}} \\\\right) \\\\phi _{tri}.$ $\\\\phi _1(\\\\theta _0)$ is maximal [$\\\\phi _1 = \\\\phi _{tri}$ ] at $\\\\theta _0 = 0\\\\ \\\\rm {and}\\\\ 60^\\\\circ $ , and minimal [$\\\\phi _1 = 3\\\\pi /[4(1+\\\\sqrt{3})] \\\\simeq .862 \\\\simeq .951\\\\phi _{tri}$ ] at $\\\\theta _0 = 30^\\\\circ $ .\",\n \"The factors of 3 in the numerators in Eq.\",\n \"REF reflect the fact that there are three monomers per trimer.\",\n \"For $\\\\theta _0 > 60^\\\\circ $ , it is impossible to center a second trimer at $[1/2,\\\\sqrt{3}/2]$ .\",\n \"We postulate that the densest packings correspond to the double lattices described by Kuperberg, Torquato and Jiao [24], [25].\",\n \"Double lattice packings consist of two lattices related by a displacement plus a $180^\\\\circ $ rotation of all constituents about their centers of inversion symmetry.\",\n \"They are often the densest possible packings for both convex [24] and concave [25] particles.\",\n \"Trimers are inversion-symmetric about the centroids of their center monomers.\",\n \"Fig.\",\n \"REF (b) shows our postulated lattice geometry.\",\n \"The leftmost monomer of a second trimer may be placed at $\\\\vec{l}_2 = [1-\\\\cos (\\\\theta _0),\\\\sin (\\\\theta _0)]$ .\",\n \"Then its center monomer lies at $\\\\vec{c}_2 = [2 - \\\\cos (\\\\theta _0), \\\\sin (\\\\theta _0)]$ and its rightmost monomer at $\\\\vec{r}_2 = [2,0]$ .\",\n \"This trimer is related to the reference trimer by a $180^\\\\circ $ rotation about its inversion center plus displacement by $\\\\vec{c}_2$ .\",\n \"The lattice vectors for this geometry are $\\\\vec{b}_1 = [1/2 - \\\\cos (\\\\theta _0), \\\\sqrt{3}/2 + \\\\sin (\\\\theta _0)]$ and $\\\\vec{b}_2 = [3,0]$ .\",\n \"Its unit-cell area $A_2(\\\\theta _0)$ is $A_2(\\\\theta _0) = \\\\det \\\\left( \\\\left[ \\\\begin{array}{c} \\\\vec{b}_1 \\\\\\\\ \\\\vec{b}_2 \\\\end{array} \\\\right] \\\\right)$ and its packing fraction is $\\\\phi _2(\\\\theta _0) = \\\\displaystyle \\\\frac{6\\\\pi }{4A_2(\\\\theta _0)} = \\\\left( \\\\displaystyle \\\\frac{1}{1/2 + \\\\sin (\\\\theta _0)/\\\\sqrt{3}} \\\\right) \\\\phi _{tri}.$ $\\\\phi _2(\\\\theta _0)$ is maximal [$\\\\phi _2 = \\\\phi _{tri}$ ] at $\\\\theta _0 = 60^\\\\circ \\\\ \\\\rm {and}\\\\ 120^\\\\circ $ and minimal [$\\\\phi _2 = \\\\pi /(2+\\\\sqrt{3}) \\\\simeq .842 \\\\simeq .928\\\\phi _{tri}$ ] at $\\\\theta _0 = 90^\\\\circ $ .\",\n \"The factor of 6 (rather than 3) in Eq.\",\n \"REF reflects the fact that these packings have 2 trimers per lattice cell.\",\n \"Our postulated maximal packing density for 2D bent-core trimers is $\\\\phi _{max}(\\\\theta _0) = \\\\bigg { \\\\lbrace } \\\\begin{array}{ccc}\\\\phi _1(\\\\theta _0) & , & 0 \\\\le \\\\theta _0 \\\\le 60^\\\\circ \\\\\\\\\\\\\\\\\\\\phi _2(\\\\theta _0) & , & 60^\\\\circ \\\\le \\\\theta _0 \\\\le 120^\\\\circ \\\\end{array}.$ The variation of $\\\\phi _{max}$ with $\\\\theta _0$ is illustrated in Fig.\",\n \"REF (c).\",\n \"As discussed above, minimal $\\\\phi _{max}(\\\\theta _0)$ occur at the $\\\\theta _0$ that are most distant from those commensurable with the triangular lattice, i.e.\",\n \"$30^\\\\circ $ and $90^\\\\circ $ .\",\n \"Kuperberg [24] identified a lower bound $\\\\phi _K = \\\\sqrt{3}/2$ for the maximal packing density of identical convex particles.\",\n \"For $71.4^\\\\circ \\\\lesssim \\\\theta _0 \\\\lesssim 108.6^\\\\circ $ , $\\\\phi _{max} \\\\le \\\\phi _K$ , indicating trimers' concavity plays a critical role in decreasing $\\\\phi _{max}$ for (at least) this range of $\\\\theta _0$ .\",\n \"The role of interlocking phenomena specific to concave particles [2], [25] in determining $\\\\phi _{max}(\\\\theta _0)$ and the jamming density $\\\\phi _J(\\\\theta _0)$ will be discussed further below.\",\n \"Figure: Putatively densest packings for 2D bent-core trimers.\",\n \"The top panels show the molecular geometries for θ 0 =0,15 ∘ ,30 ∘ ,45 ∘ ,60 ∘ ,75 ∘ ,90 ∘ ,105 ∘ , and 120 ∘ \\\\theta _0 = 0,\\\\ 15^\\\\circ ,\\\\ 30^\\\\circ ,\\\\ 45^\\\\circ ,\\\\ 60^\\\\circ ,\\\\ 75^\\\\circ ,\\\\ 90^\\\\circ ,\\\\ 105^\\\\circ ,\\\\ \\\\rm {and}\\\\ 120^\\\\circ .\",\n \"The bottom panels show the bond/contact topologies for the same systems, with noncovalent contacts indicated by blue lines.\",\n \"Black parallelograms show the unit cells, which are primitive cells for θ 0 <60 ∘ \\\\theta _0 < 60^\\\\circ and contain 2 trimers for θ 0 ≥60 ∘ \\\\theta _0 \\\\ge 60^\\\\circ .\",\n \"Green lines show covalent bonds.Figure REF shows the lattice packings associated with these motifs for several representative values of $\\\\theta _0$ .\",\n \"For $0 < \\\\theta _0 < 60^\\\\circ $ these consist of triple layers of triangular lattice separated by lines of “gap” defects.\",\n \"The gaps are necessary to accommodate the incommensurability of the 3-body fixed-angle ($\\\\theta = \\\\theta _0$ ) constraints with the triangular lattice.\",\n \"The size and shape of the gaps varies with $\\\\theta _0$ and determines $\\\\phi _{max}(\\\\theta _0)$ , but their overall orientation does not change.\",\n \"For $60^\\\\circ < \\\\theta _0 < 120^\\\\circ $ the impossibility of centering a second trimer at $[1/2,\\\\sqrt{3}/2]$ (Fig.\",\n \"REF ) makes forming triple layers of triangular lattice impossible; instead, the maximally dense packings are composed of double layers of triangular lattice separated by lines of gap defects.\",\n \"This larger concentration of gaps is responsible for the lower $\\\\phi _{max}$ for $\\\\theta _0 > 60^\\\\circ $ .\",\n \"For example, $[\\\\phi _{tri}-\\\\phi _{max}(90^\\\\circ )]/[\\\\phi _{tri}-\\\\phi _{max}(30^\\\\circ )] = 2(\\\\sqrt{3}-1) \\\\simeq 1.46$ is close to the value ($3/2$ ) that might be naively expected from the $\\\\theta _0 > 60^\\\\circ $ packings' larger gap concentration, and in fact $\\\\displaystyle \\\\frac{\\\\phi _{tri}-\\\\phi _{max}(90^\\\\circ )}{\\\\phi _{tri}-\\\\phi _{max}(30^\\\\circ )} \\\\cdot \\\\displaystyle \\\\frac{\\\\phi _{max}(30^\\\\circ )}{\\\\phi _{max}(90^\\\\circ )} = \\\\displaystyle \\\\frac{3}{2}.$ Further insight into the structure of these lattice packings can be gained by examining the topology of their bond/contact network.\",\n \"As shown in the bottom panels of Fig.\",\n \"REF , the bond/contact network is composed of triangles corresponding to monomers in close-packed layers and parallelograms corresponding to monomers bordering gaps.\",\n \"For $\\\\theta _0 = 0^\\\\circ ,\\\\ 60^\\\\circ ,\\\\ \\\\rm {and}\\\\ 120^\\\\circ $ , the opposite corners corners of the parallelograms form additional contacts as the gaps close.\",\n \"The average monomer coordination numbers are $Z_{mon} = \\\\bigg { \\\\lbrace } \\\\begin{array}{lcl}6 & , & \\\\theta _0 = 0,\\\\ 60^\\\\circ ,\\\\ \\\\rm {or}\\\\ 120^\\\\circ \\\\\\\\16/3 & , & 0 < \\\\theta _0 < 60^\\\\circ \\\\\\\\5 & , & 60^\\\\circ < \\\\theta _0 < 120^\\\\circ \\\\end{array},$ where $Z_{mon}$ includes both covalent bonds and noncovalent contacts.\",\n \"The lower coordination and less efficient packing for $60^\\\\circ < \\\\theta _0 < 120^\\\\circ $ are both consistent with the idea that concavity plays a more important role in these systems [11], [25]; both trends arise from the inability of a reference trimer to form a bond-triangle on its concave side with a monomer belonging to a second trimer when $\\\\theta _0 > 60^\\\\circ $ , i.e.\",\n \"they arise from the difference between the arrangements depicted in Fig.\",\n \"REF (a-b).\",\n \"One might expect that the parallelogramic bond/contact arrangements depicted in Fig.\",\n \"REF are associated with soft shear modes (as they often are in monomeric systems).\",\n \"In these maximally dense lattice packings, however, the fixed-angle constraints prevent the parallelograms from being sheared without violating hard-disc overlaps [26], [27].\",\n \"Moreover, since monomers are highly overconstrained in these packings, trimers are necessarily also highly overconstrained.\",\n \"We will show below that the dense lattice packings identified here are in fact all strictly jammed [27].\",\n \"Figure: Degeneracy of the densest lattice packings.\",\n \"The top panels illustrate one of the degeneracies for θ 0 =45 ∘ \\\\theta _0 = 45^\\\\circ : (a) shows the arrangement depicted in Fig.\",\n \"(a), while (b) shows a degenerate flipped-and-shifted version of this arrangement with the same φ=φ max (θ 0 )\\\\phi = \\\\phi _{max}(\\\\theta _0).\",\n \"The bottom panels illustrate one of the degeneracies for θ 0 =90 ∘ \\\\theta _0 = 90^\\\\circ : (c) shows the arrangement depicted in Fig.\",\n \"(b), while (d) shows a degenerate φ=φ max (θ 0 )\\\\phi = \\\\phi _{max}(\\\\theta _0) arrangement with different symmetry and gap topology.A key factor influencing both jamming and glass-formation in systems of constituents that are able to crystallize is competition of degenerate crystalline structures.\",\n \"For example, monodisperse sphere packings jam far more readily than monodisperse disk packings [28] because there are two incommensurate close-packed lattices in 3D (i.e.\",\n \"FCC and BCC) but only a single close-packed lattice in 2D (i.e.\",\n \"the triangular lattice).\",\n \"Finding the geometries shown in Figs.\",\n \"REF -REF did not address the question of degeneracy.\",\n \"It turns out that these lattices are highly degenerate.\",\n \"Figure REF illustrates how these degeneracies arise.\",\n \"For $0 < \\\\theta _0 < 60^\\\\circ $ , rotating the trimers in alternating layers (here depicted in blue and red) by $180^\\\\circ $ about their inversion-symmetry centers and then shifting them by [$1-\\\\cos (\\\\theta _0), \\\\sin (\\\\theta _0)$ ] produces a lattice with the same $\\\\phi = \\\\phi _{max}(\\\\theta _0)$ .\",\n \"For $60 < \\\\theta _0 < 120^\\\\circ $ , rotating the blue trimers so that their concave sides point away from rather than towards the centroids of the red trimers they double-contact achieves the same effect.\",\n \"Figure: Panel (a): Schematic depiction of our adaptive shrinking cells for n tri =4n_{tri} = 4.\",\n \"Periodic images of the n tri n_{tri} trimers are present in the ASC algorithm but are not shown here.\",\n \"Panel (b): Comparison of analytic φ max (θ 0 )\\\\phi _{max}(\\\\theta _0) [Eqs.\",\n \", , ; solid curve] to ASC results for n tri =2n_{tri} = 2 (red *) and n tri =4n_{tri} = 4 (green x) with X final ≥7X_{final} \\\\ge 7 (cf.\",\n \"Sec.\",\n \").\",\n \"n tri =3n_{tri} = 3 is not considered here because lattices with bases containing an odd number of trimers cannot be double lattices and hence are generally less than maximally dense for θ 0 >60 ∘ \\\\theta _0 > 60^\\\\circ .\",\n \"The data for n tri =2 and 4n_{tri} = 2\\\\ \\\\rm {and}\\\\ 4 overlap; for clarity, results are presented for alternating values of θ 0 \\\\theta _0.\",\n \"Panel (c): One of the degenerate maximally dense n tri =4n_{tri} = 4 lattice packings for θ 0 =90 ∘ \\\\theta _0 = 90^\\\\circ contains alternating layers of the degenerate φ=φ max (90 ∘ )\\\\phi = \\\\phi _{max}(90^\\\\circ ) n tri =2n_{tri} = 2 lattices shown in Fig.\",\n \"(c-d).The degenerate lattices depicted in Fig, REF (a-b) and (c-d) differ in their symmetries and gap topologies.\",\n \"They represent competing ordered structures that are essentially trimeric – they are not present for monomers or dimers because their existence requires the fixed-angle ($\\\\theta = \\\\theta _0$ ) constraints.\",\n \"Other degenerate arrangements with $\\\\phi = \\\\phi _{max}(\\\\theta _0)$ also exist.\",\n \"We expect that there are in fact infinitely many of them, just as there are infinitely many variants of the close-packed lattice in 3D.\",\n \"The arguments of Ref.\",\n \"[28] suggest that all this degeneracy should strongly promote jamming / suppress crystallization in bent-core-trimeric systems relative to their monomeric and dimeric counterparts.\",\n \"Below, we will investigate the degree to which this is true.\",\n \"Jennings et.\",\n \"al.\",\n \"[23] found that for $R = 1/2$ trimers, double-lattice packings are not optimally dense for some $\\\\theta _0$ and $r$ .\",\n \"In these special cases, the densest packings are lattices with bases containing more than two trimers.\",\n \"To see whether this is true for our $r = R = 1$ systems, we identify maximally dense lattice packings for bases of various sizes using a variant of Torquato et.\",\n \"al.\",\n \"'s adaptive shrinking cell (ASC) algorithm [10], [11].\",\n \"Figure REF (a) shows our ASC geometry for periodic cells containing $n_{tri}$ trimers.\",\n \"The algorithm we use to obtain both optimally-dense and less-dense packings is described in detail below (in Section ).\",\n \"Figure REF (b) compares our analytic prediction for $\\\\phi _{max}(\\\\theta _0)$ to the maximal-density lattice packings found from ASC runs for $n_{tri} \\\\le 4$ .\",\n \"For both $n_{tri} = 2$ and $n_{tri} = 4$ , ASC results converge (within our numerical precision) to our putatively densest configurations or their degenerate counterparts.\",\n \"These results indicate a key difference between the densest packings of the $r = R = 1$ trimers considered here and those of the $R = 1/2$ trimers studied in Ref.\",\n \"[23] (wherein larger bases produce denser lattice packings for some $r$ and $\\\\theta _0$ .)\",\n \"The simpler behavior for $r = R = 1$ appears to result from a reduction in the number of ways that small numbers of trimers can fit together when the trimers are composed of monodisperse tangent disks as opposed to bidisperse overlapping disks.\",\n \"Another important difference associated with the abovementioned degeneracies appears for $n_{tri} > 2$ .\",\n \"As shown in Fig.\",\n \"REF (c), packings with $\\\\phi = \\\\phi _{max}(\\\\theta _0)$ may be formed by alternating layers of the degenerate structures identified above (Fig.\",\n \"REF ).\",\n \"Increasingly complicated arrangements of this type become possible as $n_{tri}$ increases.\",\n \"This effect is analogous to the increasing number of distinguishable ways to stack $N_l$ layers of triangular lattice to form 3D close-packed structures as $N_l$ increases, and should further promote jamming.\"\n ],\n [\n \"Statics: nonoptimally dense strictly jammed lattice packings\",\n \"Having identified the maximally dense lattice packings, we now characterize the less-dense strictly-jammed lattice packings of these systems.\",\n \"Our ASC algorithm [illustrated in Fig.\",\n \"REF (a)] is implemented as follows.\",\n \"Since translational invariance implies that trimer 0 can be centered at the origin ($x_0 = y_0 = 0$ ) without loss of generality, systems have $N_{dof} = 4 + 3(n_{tri}-1)$ degrees of freedom: the cell shape parameters $\\\\alpha , \\\\beta , \\\\gamma $ and the trimer-arrangement variables [$\\\\phi $ , and $x_i, y_i, \\\\phi _i$ for $i = 1, 2, ..., n_{tri}-1$ ].\",\n \"Starting values of $\\\\alpha $ and $\\\\beta $ are chosen to be sufficiently large that all $n_{tri}$ trimers are able to freely rotate within the cell while $\\\\gamma = 0$ .\",\n \"Strictly jammed packings are obtained through four types of moves: (1) random incremental changes of ($\\\\alpha , \\\\beta , \\\\gamma $ ) accompanied by affine displacements of the trimer centers [($x_i, y_i$ ) for $i = 1,\\\\ 2,\\\\ ...,\\\\ n_{tri}-1$ ]; (2) single-particle moves consisting of random incremental changes of $\\\\phi $ or of ($x_i, y_i, \\\\phi _i$ ) for some $i = 1,\\\\ 2,\\\\ ...,\\\\ n_{tri}-1$ ; (3) collective moves consisting of rigid translations or rotations of ($j < n_{tri}$ )-trimer subsets of trimers $1,\\\\ 2,\\\\ ...,\\\\ n_{tri}-1$ ; (4) changes of ($\\\\alpha , \\\\beta , \\\\gamma $ ) that preserve volume.\",\n \"Type (1) moves are accepted if they reduce the cell volume $A = \\\\alpha \\\\beta \\\\cos (\\\\gamma )$ without producing any particle overlaps, while moves of types (2-4) are accepted if they produce no particle overlaps.\",\n \"The initial maximal increment sizes for these moves are fixed at $\\\\lbrace |\\\\delta \\\\alpha | = .05\\\\cdot 2^{-X},\\\\ |\\\\delta \\\\beta | = .05\\\\cdot 2^{-X},\\\\ |\\\\delta \\\\phi | = 2.5^\\\\circ \\\\cdot 2^{-X},\\\\ |\\\\delta x_i |= .25\\\\cdot 2^{-X},\\\\ |\\\\delta y_i| = .25\\\\cdot 2^{-X},\\\\ |\\\\delta \\\\phi _i| = 2.5^\\\\circ \\\\cdot 2^{-X} \\\\rbrace $ with $X = 0$ .\",\n \"After the process of compressing the system using moves of types (1-3) has converged [i.e.\",\n \"no more moves of these types are being accepted], the system is collectively jammed [27] for the given value of $X$ .\",\n \"Then moves of type (4) are used to check for strict jamming.\",\n \"Successful type-4 moves indicate the system is not strictly jammed; when they occur, moves of types (1-3) are begun again.\",\n \"This process repeats itself until the system is strictly jammed with respect to moves of types (1-4) for the given value of $X$ .\",\n \"Then $X$ is increased by 1 and the process begins again.\",\n \"This is repeated until satisfactory convergence is achieved; the data in Figure REF (b) indicate the maximally dense packings found by our algorithm that are strictly jammed for $X_{final} \\\\ge 7$ .\",\n \"Figure: Analytic and ASC results for strictly jammed lattice packings.\",\n \"Panel (a): Colored curves indicate the analytic φ i (θ 0 )\\\\phi _i(\\\\theta _0) (Table ) while black symbols indicate n tri =2n_{tri} = 2 ASC results.\",\n \"Panel (b): ASC results for n tri =4n_{tri} = 4.As discussed in Refs.\",\n \"[10], [11], improved performance of the algorithm can be obtained by adjusting the $X$ -increment and/or adopting more complicated Monte Carlo schemes such as allowing occasional acceptance of moves that increase $A$ .\",\n \"To obtain a wide range of packings including both optimally and nonoptimally dense geometries, it is necessary to employ a wide range of trimer-arrangement initial conditions for each $\\\\theta _0$ .\",\n \"This requirement combined with the fact that the computational complexity of the above-discussed type-3 moves is roughly exponential in $N_{dof}$ prohibited extending our comprehensive ASC studies to $n_{tri} > 4$ .\",\n \"However, studies of selected $\\\\theta _0$ for $n_{tri} = 6$ found no packings with $\\\\phi > \\\\phi _{max}(\\\\theta _0)$ , and as we will show below, much insight can be combined by combining $n_{tri} \\\\le 4$ ASC studies with large-$n_{tri}$ molecular dynamics simulations.\",\n \"Figure: θ 0 =θ min \\\\theta _0 = \\\\theta _{min} configurations and bond topologies of the seven nonoptimally-dense n tri =2n_{tri} = 2 lattice packings (families 3-9 in Table , Fig. )\",\n \"that remain stable for n tri =4n_{tri} = 4.\",\n \"The maximally-dense lattices (families 1-2) were illustrated in Figs.\",\n \"-.\",\n \"Note that it is the bond topologies that define the tiling types listed in Table .Table: Families of strictly jammed lattice packings for n tri =2n_{tri} = 2, in order of decreasing minimal density φ i (θ min )\\\\phi _i(\\\\theta _{min}).\",\n \"Families 1 and 2 are respectively the maximally-symmetric maximally-dense lattice packings identified above [Fig.\",\n \"(a-b); Eqs.\",\n \", ] for 0≤θ 0 ≤60 ∘ 0 \\\\le \\\\theta _0 \\\\le 60^\\\\circ and 60 ∘ ≤θ 0 ≤120 ∘ 60^\\\\circ \\\\le \\\\theta _0 \\\\le 120^\\\\circ .\",\n \"pp and qq are respectively equal to |30 ∘ -θ 0 ||30^\\\\circ - \\\\theta _0| and 2sin -1 [(22+2cos[θ 0 ]) -1 ]2\\\\sin ^{-1}[(2\\\\sqrt{2+2\\\\cos [\\\\theta _0]})^{-1}].\",\n \"The “tiling types” describe the lattices' bond topology for all θ 0 \\\\theta _0 within the range, but in general only precisely describe the lattices' geometrical structure for specific θ 0 \\\\theta _0; for example, type 5's bond topology is always that of the square lattice but its geometry is only that of the square lattice for θ 0 =90 ∘ \\\\theta _0 = 90^\\\\circ .\",\n \"All families except for 10(a-b) remain strictly jammed for n tri =4n_{tri} = 4; their n tri =4n_{tri} = 4 versions are simply two adjacent copies of the n tri =2n_{tri} = 2 lattices.As shown in Figure REF (a), at least ten distinct families of strictly jammed lattice packings exist for $n_{tri} = 2$ .\",\n \"Each family represents a continuous set of lattice packings sharing a common bond topology (Figure REF ).\",\n \"Since the families can be distinguished by their bond topologies, it is convenient to associate them with distinct categories of planar tilings [29].\",\n \"Moreover, for each family $i$ , these ASC results allowed us to identify exact analytic expressions for the packing fraction $\\\\phi _i(\\\\theta _0)$ .\",\n \"Results are summarized in Table REF .\",\n \"All families except for 7b and 10b [30] share two common features: (i) their $\\\\phi _i(\\\\theta _0)$ are maximized at their “endpoint” $\\\\theta _0$ (e.g.\",\n \"$\\\\theta _0 = 0\\\\ \\\\rm {and}\\\\ 60^\\\\circ $ for family 1) and minimized at their respective $\\\\theta _{min}$ ; (ii) they reduce to the triangular lattice at at least one of the three $\\\\theta _0$ allowing for it ($0,\\\\ 60^\\\\circ ,\\\\ \\\\rm {and}\\\\ 120^\\\\circ $ ).\",\n \"Because the various $\\\\phi _i(\\\\theta _0)$ vary continuously, feature (ii) means that the various families' densities converge to each other as $\\\\theta _0$ approaches these special values.\",\n \"Despite this convergence, the associated lattices remain distinct and incommensurable.\",\n \"We expect that this incommensurability strongly promotes jamming in bulk systems.\",\n \"Some of the nonmaximally dense families correspond to familiar 2D lattice structures, e.g.\",\n \"family 5 is the square lattice for $\\\\theta _0 = 90^\\\\circ $ and family 10a (10b) is the kagome lattice for $\\\\theta _0 = 60^\\\\circ $ ($\\\\theta _0 = 0$ ).\",\n \"Others represent less-familiar forms, such as family 8 which possesses a very intriguing bond topology consisting of five-sided polygons as well as the usual triangles and parallelograms.\",\n \"The wide variety of mechanically stable arrangements with different symmetries and bond topologies suggests that these systems' equilibrium phase diagrams may be especially rich [31], potentially including entropically-driven solid-solid transitions, various liquid-crystalline phases (for example, $\\\\theta _0 = 90^\\\\circ \\\\ \\\\rm {and}\\\\ 120^\\\\circ $ systems might respectively form thermodynamically stable bent-core tetratic and hexatic liquid crystals), and KTHNY-style continuous melting transitions [32], [33].\",\n \"Figure REF (b) presents our ASC results for $n_{tri} = 4$ .\",\n \"The most obvious difference from the $n_{tri} = 2$ results is the elimination of the lowest-$\\\\phi $ strictly-jammed packings; this occurs because the larger basis allows for shear modes that destabilize the kagome-like lattices (family 10).\",\n \"A second obvious difference is that there are many additional families.\",\n \"Visual inspection indicates that a large fraction of these are formed by combining two of the families discussed above.\",\n \"There are very many such combinations, making exhaustive cataloguing of them (as we did for $n_{tri} = 2$ ) prohibitively difficult, and continuing our ASC studies to even larger $n_{tri}$ would of course exacerbate this issue.\",\n \"Instead we will test the extent to which the ideas presented here are useful by looking for local structural motifs within large-$n_{tri}$ jammed configurations that correspond to the nonoptimally-dense families.\"\n ],\n [\n \"Dynamics: disordered and partially-ordered jammed packings\",\n \"None of the above discussion addresses the dynamics of the jamming process.\",\n \"Since the dynamics of systems' jamming transitions naturally relate to the dynamics of their glass transitions [34], we now examine the compression-rate dependence of our model trimers' athermal solidification behavior using molecular dynamics (MD) simulations.\",\n \"Each of the $n_{tri}$ simulated trimers contains three monomers of mass $m$ .\",\n \"The trimers are rigid; bond lengths and angles are held fixed by holonomic constraints.\",\n \"Monomers on different trimers interact via a harmonic potential $U_{H}(r) = 10\\\\epsilon (1 - r/\\\\sigma )^2 \\\\Theta (\\\\sigma -r)$ , where $\\\\epsilon $ is the energy scale of the pair interactions, $\\\\sigma $ is monomer diameter, and $\\\\Theta $ is the Heaviside step function.\",\n \"Initial states are generated by placing the trimers randomly within a square cell, with periodic boundary conditions applied along both directions.\",\n \"Then Newton's equations of motion are integrated with a timestep $\\\\delta t = .005\\\\tau $ , where the unit of time is $\\\\tau =\\\\sqrt{m\\\\sigma ^2/\\\\epsilon }$ .\",\n \"Systems are equlibrated at $k_BT/\\\\epsilon = 1$ and $\\\\phi = \\\\exp (-1)\\\\phi _{tri}$ until intertrimer structure has converged, then cooled to $T=0$ at a rate $10^{-4}(\\\\epsilon /k_B)/\\\\tau $ .\",\n \"After cooling, systems are hydrostatically compressed at a true strain rate $\\\\dot{\\\\epsilon }$ , i.e.\",\n \"the cell side length $L$ is varied as $L = L_0 \\\\exp (-\\\\dot{\\\\epsilon }t)$ .\",\n \"To maintain near-zero temperature during compression, we employ overdamped dynamics with the equation of motion $m\\\\ddot{\\\\vec{r_i}} = \\\\vec{F} - \\\\Gamma \\\\dot{\\\\vec{r_i}} +h(\\\\lbrace \\\\vec{r} , \\\\dot{\\\\vec{}}{r} \\\\rbrace )$ where $\\\\vec{r}_i$ is the position of monomer $i$ , $\\\\vec{F}$ is the force arising from the harmonic pair interactions, the damping coefficient $\\\\Gamma = 10^4\\\\dot{\\\\epsilon }$ , and the $h(\\\\lbrace \\\\vec{r} , \\\\dot{\\\\vec{}}{r} \\\\rbrace )$ term enforces trimer rigidity [35].\",\n \"Jamming is defined to occur when the nonkinetic part of the pressure $P$ exceeds $P_{thres}= 10^{-4}\\\\epsilon /\\\\sigma ^2$ ; choosing a lower (higher) value of $P_{thres}$ lowers (raises) $\\\\phi _J(\\\\theta _0)$ , but does not qualitatively change any of the results presented herein.\",\n \"We choose to identify jamming with the emergence of a finite bulk modulus rather than with the vanishing of soft modes because proper handling of soft modes associated with trimeric “rattlers” [36] is highly nontrivial.\",\n \"All MD simulations are performed using LAMMPS [37].\",\n \"Figure: Dynamical jamming.\",\n \"Panels (a) and (b) show φ J (θ 0 )\\\\phi _J(\\\\theta _0) and the ratio φ J (θ 0 )/φ max (θ 0 )\\\\phi _J(\\\\theta _0)/\\\\phi _{max}(\\\\theta _0) for several strain rates, and panel (c) shows shows φ J (θ 0 ;ϵ ˙)\\\\phi _J(\\\\theta _0; \\\\dot{\\\\epsilon }) for several characteristic values of θ 0 \\\\theta _0.\",\n \"The dotted gray line in panel (a) shows φ max (θ 0 )\\\\phi _{max}(\\\\theta _0).\",\n \"Dotted lines in panel (c) indicate fits to Eq.\",\n \".\",\n \"All results are averaged over 9 independently prepared n tri =400n_{tri} = 400 systems.Figure REF shows results for the rate-dependent $\\\\phi _J(\\\\theta _0)$ .\",\n \"As shown in panel (a), all systems jam at densities well below the monomeric value $\\\\phi _J^{mon} \\\\simeq .84$ [36]; the reduced $\\\\phi _J$ relative to monomers are caused by the frozen-in 2-body and 3-body constraints [8].\",\n \"For all compression rates, values of $\\\\phi _J(\\\\theta _0)$ clearly follow trends in $\\\\phi _{max}(\\\\theta _0)$ , exhibiting maxima at $\\\\theta _0 = 0,\\\\ 60^\\\\circ \\\\ \\\\rm {and}\\\\ 120^\\\\circ $ and minima at $\\\\theta _0 \\\\simeq 30^\\\\circ \\\\ \\\\rm {and}\\\\ \\\\simeq 90^\\\\circ $ .\",\n \"There are clearly two separate branches of $\\\\phi _J(\\\\theta _0)$ : one for $0 \\\\le \\\\theta _0 \\\\le 60^\\\\circ $ and one for $60^\\\\circ \\\\le \\\\theta _0 \\\\le 120^\\\\circ $ .\",\n \"However, values of $\\\\phi _J(\\\\theta _0)$ do not simply track $\\\\phi _{max}(\\\\theta _0)$ .\",\n \"Specifically, closed ($\\\\theta _0 = 120^\\\\circ $ ) trimers have a much higher $\\\\phi _J$ than their open ($\\\\theta _0 = 0\\\\ \\\\rm {or}\\\\ 60^\\\\circ $ ) counterparts even though their $\\\\phi _{max}$ are identical.\",\n \"More generally, while the first branch of $\\\\phi _J(\\\\theta _0)$ is close to symmetric about $\\\\theta _0 = 30^\\\\circ $ , the second branch is clearly asymmetric; $\\\\phi _J(90^\\\\circ + \\\\psi ) > \\\\phi _J(90^\\\\circ - \\\\psi )$ , increasingly so as $\\\\psi $ increases from zero towards $30^\\\\circ $ .\",\n \"While it is not surprising that $\\\\theta _0 = 120^\\\\circ $ trimers are the best crystal-formers (cf.\",\n \"Fig.\",\n \"REF ) – they are compact and threefold-symmetric whereas lower $\\\\phi _J$ are expected for small-$\\\\theta _0$ systems owing to their larger aspect ratios [1], [8], our results show that this effect propagates downward in $\\\\theta _0$ as far as $\\\\theta _0 \\\\simeq 90^\\\\circ $ .\",\n \"A useful metric characterizing the strength of the physical processes promoting disorder in these systems' solid-state morphologies is $f(\\\\theta _0; \\\\dot{\\\\epsilon }) = \\\\phi _J(\\\\theta _0; \\\\dot{\\\\epsilon })/\\\\phi _{max}(\\\\theta _0)$ .\",\n \"This quantity is unity when systems crystallize into their maximal-density lattices during compression, and smaller when systems jam at $\\\\phi < \\\\phi _{max}(\\\\theta _0)$ due to the presence of disorder.\",\n \"Roughly speaking, characterizing the decrease of $f(\\\\theta _0; \\\\dot{\\\\epsilon })$ with increasing $\\\\dot{\\\\epsilon }$ provides insight into the kinetics of the solidification process, while characterizing its variation with $\\\\theta _0$ in the low-$\\\\dot{\\\\epsilon }$ limit provides insight into how the strength of frustration/degeneracy-related effects varies with molecular shape.\",\n \"Panel (b) presents results for $f(\\\\theta _0; \\\\dot{\\\\epsilon })$ for all systems.\",\n \"For all strain rates considered here, trends in $f(\\\\theta _0; \\\\dot{\\\\epsilon })$ are opposite those in $\\\\phi _J(\\\\theta _0; \\\\dot{\\\\epsilon })$ and $\\\\phi _{max}(\\\\theta _0)$ .\",\n \"Minima in the former correspond to maxima in the latter, e.g.\",\n \"maxima in $f$ occur at $\\\\theta _0 = 30^\\\\circ \\\\ \\\\rm {and}\\\\ 90^\\\\circ $ .\",\n \"One potential reason for this is that grains of crystals with incompatible local ordering corresponding to the different families discussed above are more likely to form at densities slightly below $\\\\phi _J(\\\\theta _0)$ and then jam as systems are further compressed for the systems with lower $f(\\\\theta _0; \\\\dot{\\\\epsilon })$ .\",\n \"The fact that differences between several of the $\\\\phi _i(\\\\theta _0)$ are maximal for $\\\\theta _0 = 30^\\\\circ \\\\ \\\\rm {and}\\\\ 90^\\\\circ $ and minimal for $\\\\theta _0 = 0,\\\\ 60^\\\\circ ,\\\\ \\\\rm {and}\\\\ 120^\\\\circ $ supports this hypothesis; competition between differently ordered grains should be greater when their densities are closer due to the abovementioned convergence of the $\\\\phi _i(\\\\theta _0)$ .\",\n \"Another potential reason is that jamming dynamics are controlled by the trimer mobility $\\\\mu $ and that $\\\\mu (\\\\phi ,\\\\theta _0)$ depends far more strongly on $\\\\phi $ than on $\\\\theta _0$ for $\\\\phi < \\\\phi _J$ .\",\n \"It would be interesting to test this idea in thermalized versions of these systems by constructing isomobility curves in ($\\\\theta _0,\\\\phi $ )-space.\",\n \"Panel (c) illustrates the compression-rate dependence of jamming in more detail for the $\\\\theta _0$ corresponding to extrema of $\\\\phi _J$ : $0,\\\\ 30^\\\\circ ,\\\\ 60^\\\\circ ,\\\\ 90^\\\\circ ,\\\\ \\\\rm {and}\\\\ 120^\\\\circ $ .\",\n \"For all systems, results are well fit by $\\\\phi _J(\\\\theta _0; \\\\dot{\\\\epsilon }) \\\\simeq \\\\phi _{J}^0(\\\\theta _0)\\\\exp \\\\left[ -\\\\left( \\\\displaystyle \\\\frac{\\\\dot{\\\\epsilon }}{\\\\dot{\\\\epsilon }_0} \\\\right)^\\\\gamma \\\\right],$ where $\\\\phi _J^0$ is the quasistatic-limiting value of $\\\\phi _J$ , $\\\\dot{\\\\epsilon }_0$ is a characteristic rate, and $\\\\gamma $ describes the strength of the rate dependence.\",\n \"While rigorously determining the exact functional form of $\\\\phi _J(\\\\theta _0; \\\\dot{\\\\epsilon })$ would require more computational resources than we currently possess and is beyond our present scope, the fact that the best-fit values of $\\\\dot{\\\\epsilon }_0$ and $\\\\gamma $ remain within relatively narrow ranges for all $0 \\\\le \\\\theta _0 \\\\le 120^\\\\circ $ – respectively $3\\\\cdot 10^{-4} \\\\lesssim \\\\dot{\\\\epsilon }_0\\\\tau \\\\lesssim 7\\\\cdot 10^{-4}$ and $.65 \\\\lesssim \\\\gamma \\\\lesssim .85$ – suggests that Eq.\",\n \"REF is a useful approximate form for all $\\\\theta _0$ .\",\n \"Similar compression-rate dependencies of $\\\\phi _J$ are found for dynamical jamming of a wide variety of systems [38].\",\n \"Figure: Jammed packings of n tri =400n_{tri} = 400 bent-core trimers generated by dynamic compression.\",\n \"Rows from top to bottom show typical marginally-jammed states for θ 0 =0,30 ∘ ,60 ∘ ,90 ∘ , and 120 ∘ \\\\theta _0 = 0,\\\\ 30^\\\\circ ,\\\\ 60^\\\\circ ,\\\\ 90^\\\\circ ,\\\\ \\\\rm {and}\\\\ 120^\\\\circ .\",\n \"The left and right columns respectively indicate results for ϵ ˙τ=10 -5.5 and 10 -7 \\\\dot{\\\\epsilon }\\\\tau = 10^{-5.5}\\\\ \\\\rm {and}\\\\ 10^{-7}.\",\n \"Colors from red to blue indicate trimers 1,2,...,4001,\\\\ 2,\\\\ ...,\\\\ 400.Finally we turn to a qualitative characterization of how $\\\\theta _0$ and $\\\\dot{\\\\epsilon }$ affect jammed systems' microstructure.\",\n \"Typical marginally-jammed packings for the five characteristic $\\\\theta _0$ discussed above are shown in Figure REF .\",\n \"For $\\\\theta _0 = 0,\\\\ 60^\\\\circ ,\\\\ \\\\rm {and}\\\\ 120^\\\\circ $ , triangular-crystalline grains are clearly visible.\",\n \"This is consistent with the well-known result [28] that 2D systems of monodisperse disks have a strong propensity to crystallize.\",\n \"Contrasting the high-$\\\\dot{\\\\epsilon }$ and low-$\\\\dot{\\\\epsilon }$ snapshots for these $\\\\theta _0$ suggests that systems jam via a two-stage, two-length-scale process.\",\n \"First, randomly-oriented crystalline grains form and grow to a size that depends on both $\\\\theta _0$ and $\\\\dot{\\\\epsilon }$ .\",\n \"Since these grains cannot be compressed further, they effectively behave as single nearly-rigid particles as compression continues.\",\n \"The degree to which grains grow prior to jamming is kinetically limited and is a key factor producing the rate dependence of $\\\\phi _J(\\\\theta _0)$ .\",\n \"Higher compression rates lead to smaller grain sizes and greater intergrain misorientation, just as is the case in monomeric systems [28], [38].\",\n \"The results for $\\\\theta _0 = 30^\\\\circ \\\\ \\\\rm {and}\\\\ 90^\\\\circ $ are less easy to interpret since only basic features such as the decrease in the typical size of interstitials with decreasing $\\\\dot{\\\\epsilon }$ [28] are immediately apparent.\",\n \"The $\\\\theta _0 = 90^\\\\circ $ systems clearly possess some grains with square-lattice ordering, showing that the nonoptimally-dense families identified above (Table REF , Figs.\",\n \"REF -REF ) do indeed play a role in these systems' jamming phenomenology.\",\n \"However, the clear crystallization kinetics observed for $\\\\theta _0 = 0,\\\\ 60^\\\\circ ,\\\\ \\\\rm {and}\\\\ 120^\\\\circ $ are absent here, perhaps because these systems' lower absolute $\\\\phi _J$ and $\\\\phi _{max}$ values make them appear more disordered overall.\",\n \"The effects of degeneracy may also be larger at these $\\\\theta _0$ .\",\n \"These issues might be resolved by going to much lower $\\\\dot{\\\\epsilon }$ , but doing so is not yet computationally feasible.\"\n ],\n [\n \"Discussion and Conclusions\",\n \"In this paper, we examined the athermal solidification behavior of 2D bent-core tangent-disk trimers.\",\n \"We found that trends in $\\\\phi _J(\\\\theta _0)$ closely follow those in $\\\\phi _{max}(\\\\theta _0)$ , but with additional effects related to symmetry and degeneracy superposed.\",\n \"We reported two distinct regimes of packing/jamming phenomenology, identifying the source of the difference between them as the ability (inability) of a reference trimer to form a bond-triangle on its concave side with a monomer belonging to a second trimer when $\\\\theta _0 < 60^\\\\circ $ ($\\\\theta _0 > 60^\\\\circ $ ).\",\n \"Well-packed systems with $\\\\theta _0 > 60^\\\\circ $ are generally less dense and less hyperstatic than their $\\\\theta _0 < 60^\\\\circ $ counterparts.\",\n \"Another key insight was that deviations of $\\\\theta _0$ away from the values allowing formation of the triangular lattice ($0^\\\\circ ,\\\\ 60^\\\\circ ,\\\\ \\\\rm {and}\\\\ 120^\\\\circ $ ) do not by themselves frustrate crystalline order.\",\n \"Instead, crystals belonging to several families distinguished by their differing bond topologies can form.\",\n \"We believe that it is the presence of these competing families combined with the extensive degeneracy of the densest lattices that frustrates crystallization and promotes jamming in these systems, Our work complements recent studies of the thermal solidification of Lewis-Wahnstrom-like models [39], [40], which illustrated several nontrivial effects of trimeric structure (e.g.\",\n \"that its enhancement of the interfacial energy between crystalline and liquid phases promotes glass-formation) but did not attempt to connect their findings to the models' optimal-packing or jamming-related phenomenology.\",\n \"One of the principal goals of soft materials science is designing materials that possess tunable solid morphology.\",\n \"Designing custom pair interactions that yield nontriangular 2D-crystalline or non-close-packed 3D-crystalline ground states has attracted significant interest in recent years [41], [42], [43].\",\n \"Our 2D bent-core tangent-disk trimers provide a simple example of how the same goal may be achieved with hard-disk or hard-sphere pair interactions by controlling 2-body and 3-body correlations, i.e.\",\n \"by imposing covalent bonding and controlling the bond angle $\\\\theta _0$ .\",\n \"The potential relevance to real systems is that controlling the bond angle (or analogous shape parameters) of small molecules [13], [14], [16], [15], [19], [20], [21], [22] is often easier than controlling the pair interactions of their constituent atoms.\",\n \"Another key tuning parameter for soft materials is their degree of (dis)order.\",\n \"Recent work [16], [15] has shown that the best glass-formers (crystal-formers) in homologous series of small molecules are those with the highest (lowest) ratio $T_g/T_m$ .\",\n \"One might naively guess that an athermal version of this principle applicable to bent-core trimer molecules is that the best glass-formers (crystal-formers) are those with the lowest (highest) ratio $\\\\phi _J(\\\\theta _0)/\\\\phi _{max}(\\\\theta _0)$ .\",\n \"However, the dynamic-compression results we presented here suggest that this is not the case.\",\n \"Moreover, the multiplicity of nonoptimally-dense-but-strictly-jammed lattice packings these systems can form suggests that they may possess multiple thermodynamically stable crystalline phases and/or nontrivial liquid-crystalline phases [12].\",\n \"We conclude that further characterizing these systems' equilibrium phase behavior, e.g.\",\n \"by accurately determining their $\\\\phi _{melt}(\\\\theta _0)$ using techniques like those of Refs.\",\n \"[32], [33], is necessary to flesh out this issue further.\",\n \"This material is based upon work supported by the National Science Foundation under Grant DMR-1555242.\"\n ]\n]"},"id":{"kind":"string","value":"1808.08488"}}},{"rowIdx":282,"cells":{"text":{"kind":"list like","value":[["How many labeled license plates are needed?"],["Abstract Training a good deep learning model often requires a lot of annotated data.","As a large amount of labeled data is typically difficult to collect and even more difficult to annotate, data augmentation and data generation are widely used in the process of training deep neural networks.","However, there is no clear common understanding on how much labeled data is needed to get satisfactory performance.","In this paper, we try to address such a question using vehicle license plate character recognition as an example application.","We apply computer graphic scripts and Generative Adversarial Networks to generate and augment a large number of annotated, synthesized license plate images with realistic colors, fonts, and character composition from a small number of real, manually labeled license plate images.","Generated and augmented data are mixed and used as training data for the license plate recognition network modified from DenseNet.","The experimental results show that the model trained from the generated mixed training data has good generalization ability, and the proposed approach achieves a new state-of-the-art accuracy on Dataset-1 and AOLP, even with a very limited number of original real license plates.","In addition, the accuracy improvement caused by data generation becomes more significant when the number of labeled images is reduced.","Data augmentation also plays a more significant role when the number of labeled images is increased."],["Introduction","License plate recognition is one of the most important components of modern intelligent transportation systems.","It has attracted the attention of many researchers.","However, most existing algorithms[3], [13], [14], [18] can only work normally under certain conditions.","For example, some recognition systems require sophisticated hardware to shoot high-quality images, while other systems require the vehicle to slowly pass through a fixed access opening or even stop.","Accurately detecting license plates and recognizing characters in an open environment is a challenging task.","The main difficulties are different license plate fonts and colors, character distortion caused by the image capture process and non-uniform illumination, and low-quality images caused by occlusion or motion blur.","In this paper, we propose a license plate recognition system, in which we cope with challenge such as, low light, low resolution, motion blur, and other harsh conditions.","Fig.","REF shows the license plates which can be correctly recognized by our proposed method.","From top to bottom are the license plate images affected by the shooting angle, uneven illumination, low resolution, detection error and motion blur.","Figure: The complex license plates images.In general, supervised learning requires a large amount of labeled data in order to achieve good results.","However, real data is not easy to obtain, the acquisition process is slow, and the data needs to be processed and annotated before it can be used for training.","To achieve a higher accuracy of the annotation, manual inspection is also required.","However, the acquisition of a large amount of real data and manual annotations is very expensive.","Therefore, data generation is very important for the training of license plate recognition network.","We believe that the information contained in a small number of real license plates is sufficient to recognize most of the existing license plate images.","However, there is no clear common understanding on how much labeled data is needed to get satisfactory performance.","In this paper, we try to address such a question in vehicle license plate character recognition.","The main contributions of this paper can be summarized as the following three points: 1.We propose various methods of data generation and data augmentation.","As long as we have a few labeled license plate images, a large amount of generated data can be created.","We can achieve and even exceed the recognition accuracy and results of systems trained only on real images.","2.We compare the performance of various data generation and data augmentation methods to find that both data generation methods and data augmentation methods can significantly improve license plate recognition accuracy.","Data augmentation plays a larger role in accuracy improvement when there are many labeled license plates but when the number of labelled license plates is small, data generation more significantly increases accuracy.","3.We apply a network that is modified from DenseNet to license plate recognition to reduce network parameters and inference time and improve accuracy.","The rest of paper is arranged as follows.","In Section 2 we review the related works briefly.","In Section 3 we describe the details of networks used in our approach.","Experimental results are provided in Section 4, and conclusions are drawn in Section 5."],["Related Work","The section introduces previous work on license plate recognition and GANs."],["License Plate Recognition","Existing license plate recognition systems are either text segmentation-based [3], [6], or non-segmentation-based [13].","Methods that depend on segmentation first preprocess the license plate image and then segment individual characters through image processing.","After this, each character is classified by a convolutional neural network.","This method is very dependent on the accuracy of text segmentation, and the recognition speed is slower.","A recognition method that does not require segmentation is proposed by Li et al.","[13].","It is composed of a deep convolutional network and a Long Short-Term Memory(LSTM), where the deep CNN is directly applied for feature extraction, and a bidirectional LSTM network is applied for sequence labeling.","DenseNet[9] is a highly efficient convolutional neural network.","Because of its low parameter number and fast inference time, DenseNet is widely used.","Our method is also a segmentation-free approach based on the framework proposed by [9], where DenseNet is applied for feature extraction.","Data generation is used in license plate recognition to improve the accuracy of recognition.","The labeled license plates generated by CycleGAN as a pre-training data set for the recognition network are used in [18], and the model is fine-tuned with the real license plate data set.","This data generation method can significantly improve the recognition accuracy.","License plate detection and recognition is combined in [14], and it finally improves the recognition speed and recognition accuracy of the system."],["Generative Adversarial Networks","Generative adversarial networks(GANs) [2], [15] train a generator and discriminator alternatively.","The output of the discriminator acts as a generator's loss function.","Zhu et al.","[21] propose Cycle-Consistent Adversarial Networks(CycleGAN), which learns the mapping relationship from one domain to another and is mainly used for the style conversion of pictures.","Wasserstein GANs(WGAN) [1] are proposed to improve the stability of GAN training.","Applying Wasserstein loss to CycleGAN, creating CycleWGAN, also improves its training stability in [18].","Gradient penalties in WGAN(WGAN-GP) [5] are proposed to solve the WGAN generator weight distribution problem."],["Data Generation For Training","A large number of real labeled images are often difficult to obtain, so the role of data generation is very significant [17].","The synthesized images are used to train scene text detection networks [7] and recognition networks [11].","The generated data is shown to improve the performance of person detection [20], font recognition [19], and semantic segmentation [16].","However, when the difference between the generated data and the real data is very large, the performance is poor when applied to a real scene.","Therefore, [18] applies CycleGAN to convert the style of license plate generated by the script into a real license plate, which can greatly reduce the gap between the generated image and the real image.","We apply data generation and data augmentation methods at the same time, and use the data generated by different methods directly as the training set for recognition network.","Therefore we need very little real data.","In this section, the pipeline of the proposed method is described.","We train the GAN model using synthetic images and real images simultaneously.","We then use the generated images to train a model modified from DenseNet."],["CycleGAN","CycleGAN [21] learns to translate an image from a source domain X to a target domain Y in the absence of paired examples.","Our goal is to train a mapping relationship G between the script license plate domain X and the real license plate domain Y. CycleGAN contains two mapping functions $G:X\\rightarrow Y$ and $Y\\rightarrow X$ , and associated adversarial discriminators D$_Y$ , D$_X$ .","The techniques proposed in WGAN [1] are applied in CycleGAN, and CycleWGAN is proposed in [18].","WGAN points out why the traditional GAN is difficult to converge and improve during training, which greatly reduces the training difficulty and speeds up the convergence.","There are two main improvements: the first one is to remove the log from the loss function, and the second is to perform weight clipping after each iteration to update the weight, and limit the weight to a range (eg, the limit range is [-0.1, +0.1].","Outside weights are trimmed to -0.1 or +0.1).","CycleWGAN solves the problem of training instability and collapse mode, which makes the result more diverse.","We apply the techniques in WGAN-GP [5] to CycleWGAN and propose the CycleWGAN-GP.","WGAN-GP also proposes an improvement plan based on WGAN.","WGAN reduces the training difficulty of GAN, but it is still difficult to converge in some conditions, and the generated pictures are worse than DCGAN.","WGAN-GP applies gradient penalty, and solves the above problem along with the problems of vanishing gradient and exploding gradient during training.","It also converges faster than CycleWGAN and produces higher quality pictures.","We apply a CycleGAN equipped with WGAN and WGAN-GP techniques to train the mapping relationship between the fake license plate and the real license plate.","First of all, we apply OpenCV scripts to generate synthetic license plates as a source domain X, and then choose real license plates without labels as a target domain Y.","Before the training of CycleWGAN-GP, these license plates are randomly cropped and randomly flipped horizontally or vertically."],["Recognition network design","DenseNet is a densely connected convolutional neural network.","In this network, there is a direct connection between any two layers.","The input of each layer of the network is the union of the output of all previous layers, and the feature map learned by this layer is also directly transmitted to all subsequent layers.","DenseNet allows the input of lth Layer to directly affect all subsequent layers.","Its output is: $x_l=H_l([x_0,x_1,...,x_{l-1}]) $ where $H_l(\\cdot )$ refers to a composite function of three consecutive operations: batch normalization (BN) [10], followed by a rectified linear unit (ReLU), and a $3\\times 3$ convolution (Conv).","Additionally, since each layer contains the output information of all previous layers, it only needs a few feature maps, so the number of parameter of DenseNet is greatly reduced compared to other models.","Table: Construction of recognition network.","The output size represents w×\\times h×\\times c. Note that each ¡°conv¡± layer shown in the table corresponds the sequence BN-ReLU-ConvOur network structure is shown in Table REF , which is different from the network structure of [9], because the input license plate image is smaller and is a gray scale image of 136$\\times $ 36, so the network only has 3 dense blocks.","The transition layers used in our network consist of a batch normalization layer and an 1$\\times $ 1 convolutional layer followed by a 2$\\times $ 2 average pooling layer.","A 1$\\times $ 1 convolution can be introduced as bottleneck layer before each 3$\\times $ 3 convolution to reduce the number of input feature-maps.","To improve model compactness, we reduce the number of feature-maps from 192 to 128 at transition layers 2.","The last DenseNet layer is followed by a fully-connected layer with 68 neurons for the 68 classes of label, including 31 Chinese characters, 26 letters, 10 digits and ¡°blank¡±.","We train the networks with stochastic gradient descent (SGD).","The labelling loss is derived using Connectionist Temporal Classification (CTC) [4].","The optimization algorithm Adam [12] is then applied, as it converges quickly and does not require a complicated learning rate schedule.","Another advantage of using the modified DenseNet network is that it does not require the Long Short-Term Memory(LSTM) networks.","The use of LSTM complicates the solution and increases computational cost."],["Experiment","In this section, we conduct experiments to verify the effectiveness of the proposed methods.","Our network is implemented capitalizing keras.","The experiments are trained on a NVIDIA Tesla P40 with 24GB memory and are tested on a NVIDIA GTX745 GPU with 4GB memory."],["Dataset","The image in the Dataset-1 [18] are captured from a wide variety of real traffic monitoring scenes under various viewpoints, blurring and illumination.","Dataset-1 contains a training set of 203,774 plates and a test set of 9,986 plates.","The first character of Chinese license plates is a Chinese character which represents the province.","While there are 31 abbreviations for all of the provinces, Dataset-1 contains 30 classes of them.","The second data set is the application-oriented license plate (AOLP) [8] benchmark database, which has 2049 images of Taiwan license plates.","This database is categorized into three subsets: access control (AC) with 681 samples, traffic law enforcement (LE) with 757 samples, and road patrol (RP) with 611 samples.","The recognition network is shown in Table REF .","We implement it with Keras.","The images are resized to 136$\\times $ 36 and converted to gray scale and then fed to the recognition network.","We change the last layer of fully connected layers to 68 neurons according to the 68 classes of characters-33 Chinese characters, 24 letters, 10 digits and \"blank\".","We train the networks with SGD and learning rate of 0.0001.","The labelling loss is derived using CTC.","We set the training batch size as 256 and predicting size as 1."],["Evaluation Criterion","In this work, we evaluate the model's performance in terms of recognition accuracy and character recognition accuracy, which is similar to Wang et al.[18].","Recognition accuracy is defined as : $RA = \\frac{Number\\ of\\ correctly\\ recognized\\ license\\ plates}{Number\\ of\\ all\\ license\\ plates}$ Character recognition accuracy is defined as: $CRA = \\frac{Number\\ of\\ correctly\\ recognized\\ characters}{Number\\ of\\ all\\ characters}$ Figure: Three data generation methods (a)Examples of license plates generated by OpenCV scripts.","(b)Examples of license plates generated by CycleWGAN.","(c)Examples of license plates generated by CycleWGAN-GP."],["GAN Training and Testing","Three data generation methods are shown in Fig.","REF .","To train CycleWGAN, first we use the OpenCV scripts to generate 1000 blue fake license plates as a source domain X, and then select 1000 real blue license plates from Dataset-1 as a target domain Y.","We train the CycleWGAN model with these fake license plates and real license plates.","The training real plates do not require character labels.","All the images are resized to 143$\\times $ 143, cropped to 128$\\times $ 128 and randomly flipped for data augmentation.","We use Adam with $L_{1}=0.9$ , $L_{2}=0.999$ and learning rate of 0.0001.","We stop training after 300,000 steps and save the model.","When testing, first we use the OpenCV scripts to generate 40,000 blue fake license plates, and then we apply the last checkpoint to generate 40,000 license plates.","The same goes for CycleWGAN-GP.","Finally we get 80,000 blue license plates generated by CycleWGAN and CycleWGAN-GP."],["Data Augmentation","The six data augmentation methods are proposed in order to increase the training data of the recognition network.","The data was augmented through affine transformation, motion blurring, uneven lighting, stretching, erosion and dilation, downsampling and the application of gaussian noise.","Examples of these transformations are shown in Fig.","REF .","A real license plate image randomly passes through the six data augmentation methods, allowing for the creation of much more training data.","First, we select a small number of labeled real license plates from Dataset-1, such as 300.","And then using data augmentation methods in Fig.","REF , we generate 80,000 augmented license plates with these selected real license plates."],["Mixed Training Data","Our mixed training data consists of four parts, including 40,000 license plates generated by OpenCV scripts, 40,000 license plates generated by CycleWGAN, 40,000 license plates generated by CycleWGAN-GP, and 80,000 license plates augmented from a small number of labeled real license plates.","All 200,000 training images are generated with license plate character labels.","The license plates that need manual labeling are only selected from Dataset-1.","After converting the training data to gray scale, 400,000 more training images are obtained by flipping pixels in order to simulate gray images of yellow and green license plates.","Then, these images are fed to the recognition network modified from DenseNet.","Figure: Six data augmentation methods(a)Affine transformation (b)Motion blur (c)Uneven light (d)Stretching transformation (e)Erosion and dilation (f)Down sampling and gaussian noise."],["Performance Evaluation on Dataset-1","With the above methods, our mixed training data is generated from 300, 700, 3,333, 4,750 and 6,000 real license plates selected from Dataset-1 training set respectively.","Our baseline is the [18] using the license plate images generated by the CycleWGAN pre-training recognition network, and then using 9,000, 50,000 and 200,000 real labeled license plate images in a fine-tuning model.","From the results in Table REF , it is concluded that when data generation, data augmentation and DenseNet are used, we only need 300 real labeled license plates to achieve the effect of 200,000 real license plates.","In the same way, when the number of real license plates reaches 4,750, the final recognition accuracy has reached 99.0%, an increase of 1.4%.","When the number of real license plate images exceeds 4,750, license plate recognition accuracy and character recognition accuracy are not improving.","We conjecture that 4,750 real images contain enough information to recognize most of the license plates.","Thus, by increasing the number of real license plates, the total amount of information after data augmentation will not change, and the recognition accuracy will not increase any further."],["Performance Evaluation on Data Generation","In order to evaluate the effect of the data generated by different methods, we train the models using synthetic data generated by script, CycleGAN, CycleWGAN, and CycleWGAN-GP respectively.","The results are shown in Table REF .","When we only use the data set generated by script for training, the recognition accuracy on the test set of Dataset-1 is 42.2%.","As shown in Fig.","REF , our synthetic license plates generated by script also contain noise such as low light, low resolution, motion blur.","The CycleGAN images achieve a recognition accuracy of 51.2%.","Accuracy is not much improved because of the instability and lack of diversity in CycleGAN training.","As shown in Fig.","REF and Fig.","REF , the CycleWGAN and CycleWGAN-GP images display more various styles and colors, and part of them can not really distinguish from real images.","The CycleWGAN and CycleWGAN-GP images achieve a recognition accuracy of 62.5% and 64.5% respectively.","We also compare the impact of data generation and data augmentation on accuracy.","When the number of real license plates is 3333, the recognition accuracy of the augmented data on Dataset-1 is 97.9%, far exceeding the recognition accuracy of the generated data.","Table: Single data augmentation recognition results with 3333 real images compare with .","Recognition accuracy (%), character recognition accuracy (%) are shown.","\"CRA-C\" is the recognition accuracy (%) of the Chinese characters of the first character, and \"CRA-NC\" is the recognition accuracy (%) of the letters and numbers of the last six characters.In order to understand how much number of real license plates improves recognition accuracy, we compare data augmentation results from 60 to 6000 real license plates.","The result in Table REF shows that the greater the number of real license plates, the higher the recognition accuracy obtained.","Up to 4750, the highest recognition accuracy of the Dataset-1 is 99.0%.","Even if the number of real license plates is increased from 4750, the result is no longer improved.","In order to understand the impact of GAN on recognition accuracy, we did some additional comparative experiments.","It can also be seen in the Table REF that training data composed of data augmentation and data generation can get better results than training data composed of only data augmentation.","The conclusion is that the recognition accuracy of augmented data can be improved with data generation.","In addition, the fewer real license plates, the more recognition accuracy increases contributed from generated data.","Table: The effect of the number of plates and data generation on the results.","Recognition accuracy (%), character recognition accuracy (%) are shown.","RA(A) indicates the recognition accuracy of the data augmentation.","CRA(A) indicates the character recognition accuracy of the data augmentation.","RA(A+G) indicates the recognition accuracy of the training data composed of data augmentation and data generation.","CRA(A+G) indicates the character recognition accuracy of the mixed training data composed of data augmentation and data generation."],["Performance Evaluation on AOLP","For the application-oriented license plate(AOLP) dataset, the experiments are carried out by using license plates from different sub-datasets for training and test.","This data set is divided into three sub-datasets: access control (AC), traffic law enforcement (LE), and road patrol (RP).","For example, in Table REF , we use the license plates from the LE and RP sub-datasets to train the DenseNet, and test its performance on the AC sub-dataset.","Similarly, AC and RP are used for training and LE for test, and so on.","Since there is no AOLP license plate font, only the data augmentation methods are used, without script and GAN generated license plates.","In Table REF , through data augmentation and DenseNet, our method achieves the highest recognition accuracy on the AOLP dataset.","Table: The accuracy of other methods compared with the proposed method on dataset AOLP.","Recognition accuracy (%), character recognition accuracy (%) are shown.","AOLP is categorized into three subsets: access control (AC) with 681 samples, traffic law enforcement (LE) with 757 samples, and road patrol (RP) with 611 samplesIn this paper, we have investigated how many real labeled license plates are needed to train the license plate recognition system.","We have proposed three data generation methods and six data augmentation methods in order to fully obtain all the information in a small number of images.","The experimental results show that the proposed method only requires 300 real labeled license plates to achieve the effect achieved by 200,000 real license plates.","The result shows that the greater the number of real license plates, the higher the recognition accuracy obtained.","Up to 4750, the highest recognition accuracy of the Dataset-1 is 99.0%.","Even if the number of real license plates is increased furthermore, the result is no longer improved.","Additionally, training data composed of both augmented and generated data can achieve better results than training data composed of only augmented data.","Furthermore, the fewer real license plates, the more recognition accuracy increases contributed from generated data."]],"string":"[\n [\n \"How many labeled license plates are needed?\"\n ],\n [\n \"Abstract Training a good deep learning model often requires a lot of annotated data.\",\n \"As a large amount of labeled data is typically difficult to collect and even more difficult to annotate, data augmentation and data generation are widely used in the process of training deep neural networks.\",\n \"However, there is no clear common understanding on how much labeled data is needed to get satisfactory performance.\",\n \"In this paper, we try to address such a question using vehicle license plate character recognition as an example application.\",\n \"We apply computer graphic scripts and Generative Adversarial Networks to generate and augment a large number of annotated, synthesized license plate images with realistic colors, fonts, and character composition from a small number of real, manually labeled license plate images.\",\n \"Generated and augmented data are mixed and used as training data for the license plate recognition network modified from DenseNet.\",\n \"The experimental results show that the model trained from the generated mixed training data has good generalization ability, and the proposed approach achieves a new state-of-the-art accuracy on Dataset-1 and AOLP, even with a very limited number of original real license plates.\",\n \"In addition, the accuracy improvement caused by data generation becomes more significant when the number of labeled images is reduced.\",\n \"Data augmentation also plays a more significant role when the number of labeled images is increased.\"\n ],\n [\n \"Introduction\",\n \"License plate recognition is one of the most important components of modern intelligent transportation systems.\",\n \"It has attracted the attention of many researchers.\",\n \"However, most existing algorithms[3], [13], [14], [18] can only work normally under certain conditions.\",\n \"For example, some recognition systems require sophisticated hardware to shoot high-quality images, while other systems require the vehicle to slowly pass through a fixed access opening or even stop.\",\n \"Accurately detecting license plates and recognizing characters in an open environment is a challenging task.\",\n \"The main difficulties are different license plate fonts and colors, character distortion caused by the image capture process and non-uniform illumination, and low-quality images caused by occlusion or motion blur.\",\n \"In this paper, we propose a license plate recognition system, in which we cope with challenge such as, low light, low resolution, motion blur, and other harsh conditions.\",\n \"Fig.\",\n \"REF shows the license plates which can be correctly recognized by our proposed method.\",\n \"From top to bottom are the license plate images affected by the shooting angle, uneven illumination, low resolution, detection error and motion blur.\",\n \"Figure: The complex license plates images.In general, supervised learning requires a large amount of labeled data in order to achieve good results.\",\n \"However, real data is not easy to obtain, the acquisition process is slow, and the data needs to be processed and annotated before it can be used for training.\",\n \"To achieve a higher accuracy of the annotation, manual inspection is also required.\",\n \"However, the acquisition of a large amount of real data and manual annotations is very expensive.\",\n \"Therefore, data generation is very important for the training of license plate recognition network.\",\n \"We believe that the information contained in a small number of real license plates is sufficient to recognize most of the existing license plate images.\",\n \"However, there is no clear common understanding on how much labeled data is needed to get satisfactory performance.\",\n \"In this paper, we try to address such a question in vehicle license plate character recognition.\",\n \"The main contributions of this paper can be summarized as the following three points: 1.We propose various methods of data generation and data augmentation.\",\n \"As long as we have a few labeled license plate images, a large amount of generated data can be created.\",\n \"We can achieve and even exceed the recognition accuracy and results of systems trained only on real images.\",\n \"2.We compare the performance of various data generation and data augmentation methods to find that both data generation methods and data augmentation methods can significantly improve license plate recognition accuracy.\",\n \"Data augmentation plays a larger role in accuracy improvement when there are many labeled license plates but when the number of labelled license plates is small, data generation more significantly increases accuracy.\",\n \"3.We apply a network that is modified from DenseNet to license plate recognition to reduce network parameters and inference time and improve accuracy.\",\n \"The rest of paper is arranged as follows.\",\n \"In Section 2 we review the related works briefly.\",\n \"In Section 3 we describe the details of networks used in our approach.\",\n \"Experimental results are provided in Section 4, and conclusions are drawn in Section 5.\"\n ],\n [\n \"Related Work\",\n \"The section introduces previous work on license plate recognition and GANs.\"\n ],\n [\n \"License Plate Recognition\",\n \"Existing license plate recognition systems are either text segmentation-based [3], [6], or non-segmentation-based [13].\",\n \"Methods that depend on segmentation first preprocess the license plate image and then segment individual characters through image processing.\",\n \"After this, each character is classified by a convolutional neural network.\",\n \"This method is very dependent on the accuracy of text segmentation, and the recognition speed is slower.\",\n \"A recognition method that does not require segmentation is proposed by Li et al.\",\n \"[13].\",\n \"It is composed of a deep convolutional network and a Long Short-Term Memory(LSTM), where the deep CNN is directly applied for feature extraction, and a bidirectional LSTM network is applied for sequence labeling.\",\n \"DenseNet[9] is a highly efficient convolutional neural network.\",\n \"Because of its low parameter number and fast inference time, DenseNet is widely used.\",\n \"Our method is also a segmentation-free approach based on the framework proposed by [9], where DenseNet is applied for feature extraction.\",\n \"Data generation is used in license plate recognition to improve the accuracy of recognition.\",\n \"The labeled license plates generated by CycleGAN as a pre-training data set for the recognition network are used in [18], and the model is fine-tuned with the real license plate data set.\",\n \"This data generation method can significantly improve the recognition accuracy.\",\n \"License plate detection and recognition is combined in [14], and it finally improves the recognition speed and recognition accuracy of the system.\"\n ],\n [\n \"Generative Adversarial Networks\",\n \"Generative adversarial networks(GANs) [2], [15] train a generator and discriminator alternatively.\",\n \"The output of the discriminator acts as a generator's loss function.\",\n \"Zhu et al.\",\n \"[21] propose Cycle-Consistent Adversarial Networks(CycleGAN), which learns the mapping relationship from one domain to another and is mainly used for the style conversion of pictures.\",\n \"Wasserstein GANs(WGAN) [1] are proposed to improve the stability of GAN training.\",\n \"Applying Wasserstein loss to CycleGAN, creating CycleWGAN, also improves its training stability in [18].\",\n \"Gradient penalties in WGAN(WGAN-GP) [5] are proposed to solve the WGAN generator weight distribution problem.\"\n ],\n [\n \"Data Generation For Training\",\n \"A large number of real labeled images are often difficult to obtain, so the role of data generation is very significant [17].\",\n \"The synthesized images are used to train scene text detection networks [7] and recognition networks [11].\",\n \"The generated data is shown to improve the performance of person detection [20], font recognition [19], and semantic segmentation [16].\",\n \"However, when the difference between the generated data and the real data is very large, the performance is poor when applied to a real scene.\",\n \"Therefore, [18] applies CycleGAN to convert the style of license plate generated by the script into a real license plate, which can greatly reduce the gap between the generated image and the real image.\",\n \"We apply data generation and data augmentation methods at the same time, and use the data generated by different methods directly as the training set for recognition network.\",\n \"Therefore we need very little real data.\",\n \"In this section, the pipeline of the proposed method is described.\",\n \"We train the GAN model using synthetic images and real images simultaneously.\",\n \"We then use the generated images to train a model modified from DenseNet.\"\n ],\n [\n \"CycleGAN\",\n \"CycleGAN [21] learns to translate an image from a source domain X to a target domain Y in the absence of paired examples.\",\n \"Our goal is to train a mapping relationship G between the script license plate domain X and the real license plate domain Y. CycleGAN contains two mapping functions $G:X\\\\rightarrow Y$ and $Y\\\\rightarrow X$ , and associated adversarial discriminators D$_Y$ , D$_X$ .\",\n \"The techniques proposed in WGAN [1] are applied in CycleGAN, and CycleWGAN is proposed in [18].\",\n \"WGAN points out why the traditional GAN is difficult to converge and improve during training, which greatly reduces the training difficulty and speeds up the convergence.\",\n \"There are two main improvements: the first one is to remove the log from the loss function, and the second is to perform weight clipping after each iteration to update the weight, and limit the weight to a range (eg, the limit range is [-0.1, +0.1].\",\n \"Outside weights are trimmed to -0.1 or +0.1).\",\n \"CycleWGAN solves the problem of training instability and collapse mode, which makes the result more diverse.\",\n \"We apply the techniques in WGAN-GP [5] to CycleWGAN and propose the CycleWGAN-GP.\",\n \"WGAN-GP also proposes an improvement plan based on WGAN.\",\n \"WGAN reduces the training difficulty of GAN, but it is still difficult to converge in some conditions, and the generated pictures are worse than DCGAN.\",\n \"WGAN-GP applies gradient penalty, and solves the above problem along with the problems of vanishing gradient and exploding gradient during training.\",\n \"It also converges faster than CycleWGAN and produces higher quality pictures.\",\n \"We apply a CycleGAN equipped with WGAN and WGAN-GP techniques to train the mapping relationship between the fake license plate and the real license plate.\",\n \"First of all, we apply OpenCV scripts to generate synthetic license plates as a source domain X, and then choose real license plates without labels as a target domain Y.\",\n \"Before the training of CycleWGAN-GP, these license plates are randomly cropped and randomly flipped horizontally or vertically.\"\n ],\n [\n \"Recognition network design\",\n \"DenseNet is a densely connected convolutional neural network.\",\n \"In this network, there is a direct connection between any two layers.\",\n \"The input of each layer of the network is the union of the output of all previous layers, and the feature map learned by this layer is also directly transmitted to all subsequent layers.\",\n \"DenseNet allows the input of lth Layer to directly affect all subsequent layers.\",\n \"Its output is: $x_l=H_l([x_0,x_1,...,x_{l-1}]) $ where $H_l(\\\\cdot )$ refers to a composite function of three consecutive operations: batch normalization (BN) [10], followed by a rectified linear unit (ReLU), and a $3\\\\times 3$ convolution (Conv).\",\n \"Additionally, since each layer contains the output information of all previous layers, it only needs a few feature maps, so the number of parameter of DenseNet is greatly reduced compared to other models.\",\n \"Table: Construction of recognition network.\",\n \"The output size represents w×\\\\times h×\\\\times c. Note that each ¡°conv¡± layer shown in the table corresponds the sequence BN-ReLU-ConvOur network structure is shown in Table REF , which is different from the network structure of [9], because the input license plate image is smaller and is a gray scale image of 136$\\\\times $ 36, so the network only has 3 dense blocks.\",\n \"The transition layers used in our network consist of a batch normalization layer and an 1$\\\\times $ 1 convolutional layer followed by a 2$\\\\times $ 2 average pooling layer.\",\n \"A 1$\\\\times $ 1 convolution can be introduced as bottleneck layer before each 3$\\\\times $ 3 convolution to reduce the number of input feature-maps.\",\n \"To improve model compactness, we reduce the number of feature-maps from 192 to 128 at transition layers 2.\",\n \"The last DenseNet layer is followed by a fully-connected layer with 68 neurons for the 68 classes of label, including 31 Chinese characters, 26 letters, 10 digits and ¡°blank¡±.\",\n \"We train the networks with stochastic gradient descent (SGD).\",\n \"The labelling loss is derived using Connectionist Temporal Classification (CTC) [4].\",\n \"The optimization algorithm Adam [12] is then applied, as it converges quickly and does not require a complicated learning rate schedule.\",\n \"Another advantage of using the modified DenseNet network is that it does not require the Long Short-Term Memory(LSTM) networks.\",\n \"The use of LSTM complicates the solution and increases computational cost.\"\n ],\n [\n \"Experiment\",\n \"In this section, we conduct experiments to verify the effectiveness of the proposed methods.\",\n \"Our network is implemented capitalizing keras.\",\n \"The experiments are trained on a NVIDIA Tesla P40 with 24GB memory and are tested on a NVIDIA GTX745 GPU with 4GB memory.\"\n ],\n [\n \"Dataset\",\n \"The image in the Dataset-1 [18] are captured from a wide variety of real traffic monitoring scenes under various viewpoints, blurring and illumination.\",\n \"Dataset-1 contains a training set of 203,774 plates and a test set of 9,986 plates.\",\n \"The first character of Chinese license plates is a Chinese character which represents the province.\",\n \"While there are 31 abbreviations for all of the provinces, Dataset-1 contains 30 classes of them.\",\n \"The second data set is the application-oriented license plate (AOLP) [8] benchmark database, which has 2049 images of Taiwan license plates.\",\n \"This database is categorized into three subsets: access control (AC) with 681 samples, traffic law enforcement (LE) with 757 samples, and road patrol (RP) with 611 samples.\",\n \"The recognition network is shown in Table REF .\",\n \"We implement it with Keras.\",\n \"The images are resized to 136$\\\\times $ 36 and converted to gray scale and then fed to the recognition network.\",\n \"We change the last layer of fully connected layers to 68 neurons according to the 68 classes of characters-33 Chinese characters, 24 letters, 10 digits and \\\"blank\\\".\",\n \"We train the networks with SGD and learning rate of 0.0001.\",\n \"The labelling loss is derived using CTC.\",\n \"We set the training batch size as 256 and predicting size as 1.\"\n ],\n [\n \"Evaluation Criterion\",\n \"In this work, we evaluate the model's performance in terms of recognition accuracy and character recognition accuracy, which is similar to Wang et al.[18].\",\n \"Recognition accuracy is defined as : $RA = \\\\frac{Number\\\\ of\\\\ correctly\\\\ recognized\\\\ license\\\\ plates}{Number\\\\ of\\\\ all\\\\ license\\\\ plates}$ Character recognition accuracy is defined as: $CRA = \\\\frac{Number\\\\ of\\\\ correctly\\\\ recognized\\\\ characters}{Number\\\\ of\\\\ all\\\\ characters}$ Figure: Three data generation methods (a)Examples of license plates generated by OpenCV scripts.\",\n \"(b)Examples of license plates generated by CycleWGAN.\",\n \"(c)Examples of license plates generated by CycleWGAN-GP.\"\n ],\n [\n \"GAN Training and Testing\",\n \"Three data generation methods are shown in Fig.\",\n \"REF .\",\n \"To train CycleWGAN, first we use the OpenCV scripts to generate 1000 blue fake license plates as a source domain X, and then select 1000 real blue license plates from Dataset-1 as a target domain Y.\",\n \"We train the CycleWGAN model with these fake license plates and real license plates.\",\n \"The training real plates do not require character labels.\",\n \"All the images are resized to 143$\\\\times $ 143, cropped to 128$\\\\times $ 128 and randomly flipped for data augmentation.\",\n \"We use Adam with $L_{1}=0.9$ , $L_{2}=0.999$ and learning rate of 0.0001.\",\n \"We stop training after 300,000 steps and save the model.\",\n \"When testing, first we use the OpenCV scripts to generate 40,000 blue fake license plates, and then we apply the last checkpoint to generate 40,000 license plates.\",\n \"The same goes for CycleWGAN-GP.\",\n \"Finally we get 80,000 blue license plates generated by CycleWGAN and CycleWGAN-GP.\"\n ],\n [\n \"Data Augmentation\",\n \"The six data augmentation methods are proposed in order to increase the training data of the recognition network.\",\n \"The data was augmented through affine transformation, motion blurring, uneven lighting, stretching, erosion and dilation, downsampling and the application of gaussian noise.\",\n \"Examples of these transformations are shown in Fig.\",\n \"REF .\",\n \"A real license plate image randomly passes through the six data augmentation methods, allowing for the creation of much more training data.\",\n \"First, we select a small number of labeled real license plates from Dataset-1, such as 300.\",\n \"And then using data augmentation methods in Fig.\",\n \"REF , we generate 80,000 augmented license plates with these selected real license plates.\"\n ],\n [\n \"Mixed Training Data\",\n \"Our mixed training data consists of four parts, including 40,000 license plates generated by OpenCV scripts, 40,000 license plates generated by CycleWGAN, 40,000 license plates generated by CycleWGAN-GP, and 80,000 license plates augmented from a small number of labeled real license plates.\",\n \"All 200,000 training images are generated with license plate character labels.\",\n \"The license plates that need manual labeling are only selected from Dataset-1.\",\n \"After converting the training data to gray scale, 400,000 more training images are obtained by flipping pixels in order to simulate gray images of yellow and green license plates.\",\n \"Then, these images are fed to the recognition network modified from DenseNet.\",\n \"Figure: Six data augmentation methods(a)Affine transformation (b)Motion blur (c)Uneven light (d)Stretching transformation (e)Erosion and dilation (f)Down sampling and gaussian noise.\"\n ],\n [\n \"Performance Evaluation on Dataset-1\",\n \"With the above methods, our mixed training data is generated from 300, 700, 3,333, 4,750 and 6,000 real license plates selected from Dataset-1 training set respectively.\",\n \"Our baseline is the [18] using the license plate images generated by the CycleWGAN pre-training recognition network, and then using 9,000, 50,000 and 200,000 real labeled license plate images in a fine-tuning model.\",\n \"From the results in Table REF , it is concluded that when data generation, data augmentation and DenseNet are used, we only need 300 real labeled license plates to achieve the effect of 200,000 real license plates.\",\n \"In the same way, when the number of real license plates reaches 4,750, the final recognition accuracy has reached 99.0%, an increase of 1.4%.\",\n \"When the number of real license plate images exceeds 4,750, license plate recognition accuracy and character recognition accuracy are not improving.\",\n \"We conjecture that 4,750 real images contain enough information to recognize most of the license plates.\",\n \"Thus, by increasing the number of real license plates, the total amount of information after data augmentation will not change, and the recognition accuracy will not increase any further.\"\n ],\n [\n \"Performance Evaluation on Data Generation\",\n \"In order to evaluate the effect of the data generated by different methods, we train the models using synthetic data generated by script, CycleGAN, CycleWGAN, and CycleWGAN-GP respectively.\",\n \"The results are shown in Table REF .\",\n \"When we only use the data set generated by script for training, the recognition accuracy on the test set of Dataset-1 is 42.2%.\",\n \"As shown in Fig.\",\n \"REF , our synthetic license plates generated by script also contain noise such as low light, low resolution, motion blur.\",\n \"The CycleGAN images achieve a recognition accuracy of 51.2%.\",\n \"Accuracy is not much improved because of the instability and lack of diversity in CycleGAN training.\",\n \"As shown in Fig.\",\n \"REF and Fig.\",\n \"REF , the CycleWGAN and CycleWGAN-GP images display more various styles and colors, and part of them can not really distinguish from real images.\",\n \"The CycleWGAN and CycleWGAN-GP images achieve a recognition accuracy of 62.5% and 64.5% respectively.\",\n \"We also compare the impact of data generation and data augmentation on accuracy.\",\n \"When the number of real license plates is 3333, the recognition accuracy of the augmented data on Dataset-1 is 97.9%, far exceeding the recognition accuracy of the generated data.\",\n \"Table: Single data augmentation recognition results with 3333 real images compare with .\",\n \"Recognition accuracy (%), character recognition accuracy (%) are shown.\",\n \"\\\"CRA-C\\\" is the recognition accuracy (%) of the Chinese characters of the first character, and \\\"CRA-NC\\\" is the recognition accuracy (%) of the letters and numbers of the last six characters.In order to understand how much number of real license plates improves recognition accuracy, we compare data augmentation results from 60 to 6000 real license plates.\",\n \"The result in Table REF shows that the greater the number of real license plates, the higher the recognition accuracy obtained.\",\n \"Up to 4750, the highest recognition accuracy of the Dataset-1 is 99.0%.\",\n \"Even if the number of real license plates is increased from 4750, the result is no longer improved.\",\n \"In order to understand the impact of GAN on recognition accuracy, we did some additional comparative experiments.\",\n \"It can also be seen in the Table REF that training data composed of data augmentation and data generation can get better results than training data composed of only data augmentation.\",\n \"The conclusion is that the recognition accuracy of augmented data can be improved with data generation.\",\n \"In addition, the fewer real license plates, the more recognition accuracy increases contributed from generated data.\",\n \"Table: The effect of the number of plates and data generation on the results.\",\n \"Recognition accuracy (%), character recognition accuracy (%) are shown.\",\n \"RA(A) indicates the recognition accuracy of the data augmentation.\",\n \"CRA(A) indicates the character recognition accuracy of the data augmentation.\",\n \"RA(A+G) indicates the recognition accuracy of the training data composed of data augmentation and data generation.\",\n \"CRA(A+G) indicates the character recognition accuracy of the mixed training data composed of data augmentation and data generation.\"\n ],\n [\n \"Performance Evaluation on AOLP\",\n \"For the application-oriented license plate(AOLP) dataset, the experiments are carried out by using license plates from different sub-datasets for training and test.\",\n \"This data set is divided into three sub-datasets: access control (AC), traffic law enforcement (LE), and road patrol (RP).\",\n \"For example, in Table REF , we use the license plates from the LE and RP sub-datasets to train the DenseNet, and test its performance on the AC sub-dataset.\",\n \"Similarly, AC and RP are used for training and LE for test, and so on.\",\n \"Since there is no AOLP license plate font, only the data augmentation methods are used, without script and GAN generated license plates.\",\n \"In Table REF , through data augmentation and DenseNet, our method achieves the highest recognition accuracy on the AOLP dataset.\",\n \"Table: The accuracy of other methods compared with the proposed method on dataset AOLP.\",\n \"Recognition accuracy (%), character recognition accuracy (%) are shown.\",\n \"AOLP is categorized into three subsets: access control (AC) with 681 samples, traffic law enforcement (LE) with 757 samples, and road patrol (RP) with 611 samplesIn this paper, we have investigated how many real labeled license plates are needed to train the license plate recognition system.\",\n \"We have proposed three data generation methods and six data augmentation methods in order to fully obtain all the information in a small number of images.\",\n \"The experimental results show that the proposed method only requires 300 real labeled license plates to achieve the effect achieved by 200,000 real license plates.\",\n \"The result shows that the greater the number of real license plates, the higher the recognition accuracy obtained.\",\n \"Up to 4750, the highest recognition accuracy of the Dataset-1 is 99.0%.\",\n \"Even if the number of real license plates is increased furthermore, the result is no longer improved.\",\n \"Additionally, training data composed of both augmented and generated data can achieve better results than training data composed of only augmented data.\",\n \"Furthermore, the fewer real license plates, the more recognition accuracy increases contributed from generated data.\"\n ]\n]"},"id":{"kind":"string","value":"1808.08410"}}},{"rowIdx":283,"cells":{"text":{"kind":"list like","value":[["Single Image Dehazing Based on Generic Regularity"],["Abstract This paper proposes a novel technique for single image dehazing.","Most of the state-of-the-art methods for single image dehazing relies either on Dark Channel Prior (DCP) or on Color line.","The proposed method combines the two different approaches.","We initially compute the dark channel prior and then apply a Nearest-Neighbor (NN) based regularization technique to obtain a smooth transmission map of the hazy image.","We consider the effect of airlight on the image by using the color line model to assess the commitment of airlight in each patch of the image and interpolate at the local neighborhood where the estimate is unreliable.","The NN based regularization of the DCP can remove the haze, whereas, the color line based interpolation of airlight effect makes the proposed system robust against the variation of haze within an image due to multiple light sources.","The proposed method is tested on benchmark datasets and shows promising results compared to the state-of-the-art, including the deep learning based methods, which require a huge computational setup.","Moreover, the proposed method outperforms the recent deep learning based methods when applied on images with sky regions."],["Introduction","Images captured by the camera, are often affected by haze due to several atmospheric conditions such as fog, smoke, multiple light sources, the scattering of light, etc.","Presence of haze in an image obscure the clarity and hence needs to be restored.","The flux of light per unit area received by the camera from the scene is attenuated along the line of sight.","This redirection of light lessens the immediate scene transmission and replaces with a layer of scattered light known as airlight.","Such scattering of light due to rigid particles floating in the air reduces the visibility of the scene.","Figure: Sample result of the proposed approach.","(a) Input hazy image.","(b) Dehazed image using our approach.Image methods endeavor to recover original scene radiance by expelling the impact of haze from the picture(as shown as an example in Figure 1).","Since the effect of scattering of light at any pixel depends on the depth of the pixel, the degradation of the image due to haze is spatial-variant.","However, recovering the effect of scene radiance from a pixel on an object is perplexing as the measure of fog and haze relies upon the separation between the object and the camera.","In this way, global enhancement strategies do not function admirably for a scene image, because of the presence of objects at different depth and the background.","The conventional methods exploited multiple images to overcome the problem.","For example, the method of Narasimhan et.al.","[1] require multiple pictures of a similar scene taken under various climate conditions.","The method of Shwartz et.al.","[2] requires images with the various level of polarization.","Figure: NO_CAPTIONFigure: Result of dehazing an image.","(a) input haze image.","(b) refined transmission map after NN regularization (c) final haze-free image (d) depth map of the output Haze-free Image.However, images taken in various climatic conditions may be unavailable always.","Recently researches are going on for single image dehazing, where reference images for the given hazy images are unrequired.","After the introduction of the concept of Dark Channel Prior(DCP), people started dehazing an image without the help of a reference image [3].","Tan et.al.","proposed a DCP based strategy for the single image by maximizing the local image contrast adopting a Markov Random Field (MRF) framework [3].","Fattal [4] used DCP to estimate the medium transmittance accepting the surface shading, furthermore, medium transmission capacities are locally measurably uncorrelated.","He et al.","[5] proposed the concept of DCP for estimating transmission map of the image.","The DCP based approaches produce good result, but cannot handle locales where shading segment does not shift fundamentally contrasted with noise.","Nair et al.","[6] proposed a DCP based dehazing technique using a center surround filter to reduce the execution time.","Another line of thought was introduced by Fattal in [7], where the color line was used for dehazing.","The color line based methods approximate transmission map based on the shift of the direction of atmospheric light from the origin.","However, both the approaches fail to retain the original color of the objects in the hazy image.","Currently, some deep learning based methods have been proposed, producing much better results compared to the handcrafted features like DCP and color-line [8], [9], [10].","However, in case of images where sky background is present, the deep learning based methods fail to dehaze properly, as the learned features are not capable of fixing the haze at high depth areas of the image.","In this paper, the contribution over the state-of-the-art is three-folds.","Unlike the state-of-the-art methods, the proposed method for a single image assesses the benefit of the generic regularity of the image in which, pixels with moderate intensities of small image patches exhibit concordant distributions in RGB space.","Our second contribution is to substitute the commonly used soft matting technique [11] in assessing the transmission map for haze removal, by first discovering some settled points in a maximum filter and then applying Nearest-Neighbor (NN) regularization.","One example of using the transmission map in haze removal by the proposed method is shown in Figure 2.","We utilize pixels of small image patches as a rule accompanying a one-dimensional dispersion in RGB space, to approximate the intensity at each patch and interject at places wherein the estimate is not generally unreliable.","We do not consider the atmospheric light-weight.","Finally, unlike the state-of-the-art methods, we do not consider and estimate the transmission of the medium.","Alternatively, we approximate the additional airlight present in the image patch and eliminate that to clear the haze.","We interject and regularize the fractional evaluations of pixel intensity values into an entire transmission map, and outline an Adaptive Naive Bayes Classification approach to deal with pixels with a significant depth (such as sky region).","Contrary to the conventional field models which comprise of normal coupling between adjacent pixels, we settle the transmission in isolated locales by fluctuating the number of pixels.","We take the likelihood of more number of pixels to get the suitable pixel intensity value.","The rest of the paper is organised as follows.","Section 2 describes the problem and a survey on the existing literature.","Section 3 displays the mathematical and numerical foundation for the proposed strategy for dehazing.","Section 4 weighs up the proposed algorithm.","Experimental results and concluding comments are given in Sections 5 and 6 respectively."],["Related Work","Dehazing is a task of image remaking; the corruption of a hazy image is a direct result of the suspended particles in the turbid air.","Single image dehazing is a challenging task which draws significant attention from researchers of the fields of Computer Graphics and Image Processing.","The physical model often used to characterize the haze formation that causes degradation of an image, is known as Koschmieder's atmospheric scattering model [12]: $\\ I(x) = t(x)J(x) + (1 - t(x))A,$ where $I(x)$ is the observed intensity value of pixel $x$ in image $I$ ; $J(x)$ is the scene radiance of a haze-free image at $x$ and $t(x)$ is the medium transmission describing the portion of the light which reaches the camera without scattering.","$A$ is the airlight, a global vector quantity describing the ambient light.","The goal of haze removal task is to recover $J$ , $A$ and $t$ from $I$ .","In (REF ), the transmittance $t(x)$ does not change much for a given wavelength in a hazy image.","The term $J(x)t(x)$ of (REF ) is termed as direct attenuation [3], which provides the information about the quantity of radiance received by the observer.","The second part, called airlight, provides an approximation of the measure of the atmospheric light included due to the dissipating course of the viewer.","Several attempts have been made for fog and haze expulsion in outdoor scene pictures [13].","Notwithstanding, the considerable significant advance in the research on haze removal methods was the idea of DCP.","It is observed that, in the more considerable part of the local neighborhood regions which do not contain the sky, a few pixels in many instances get extremely low-intensity value at the minimum one color (RGB) channel (called \"dark pixels\").","In case of hazy image, the intensity of these dark pixels in the corresponding channel is contributed by the airlight.","Along these lines, these dark pixels can explicitly give a precise estimation of the fog's transmission.","In any case, the progression to reclassify the transmission approximation utilizing soft matting technique often leads to transmission underestimate.","When the density of haze varies smoothly in space, the effect of the homogeneous atmosphere transmission $t$ can be shown as: $\\ t(x) = e^{\\beta d(x)},$ where $d(x)$ is the depth at pixel $x$ and $\\beta $ is the scattering coefficient (in three dimensional space) of the atmosphere.","So, (REF ) specifies the amount of radiance received by the observer, which is attenuated exponentially with the depth.","Several algorithms for image dehazing follow the image formation model (REF ) to dehaze images by recouping $J$ .","However, to regain haze-free image, both (REF ) and (REF ) are followed.","First $J$ is obtained by subtracting a constant value corresponding to the dark pixels and then estimate the value of transmission $t$ precisely by assuming $A$ is known.","Haze lessens the contrast in the picture, and different techniques depend on this perception for rebuilding.","He et al.","[14] amplifies the contrast in each patch, while keeping up a global coherent image.","This method enhanced the contrast for image dehazing and as a result, distant objects to the viewer seemed to be smooth and over saturated, causing the dehazed image look artificial.","In [15], Park et al.","evaluated the amount of haze in the image, from the contrast between the RGB channels, which diminishes as haze increments.","This postulation of estimation of amount of haze is erroneous in the greay area.","In [16], Zhu et al.","assessed the haze in view of the perception that hazy regions are portrayed by high brightness and low saturation.","Some efforts have been made by utilizing a prior on the depth of the picture, for haze expulsion.","A smoothness prior is utilized in [17], expecting the image to be smooth except the pixels with depth discontinuities.","Nishino et al.","[18] expected the albedo and depth as factually free and together approximated them using priors on both.","The albedo-prior accepted the distribution of gradients in images of characteristic scenes displaying a heavy-tail dispersion, and it is estimated as a summed up ordinary appropriation and further approximated as Gaussian distribution.","The depth-prior is scene-dependent and is picked physically, either as piece-wise consistent for urban scenes or definitely changing for non-urban scenes.","The DCP based approaches [14] accept that within a small image patche there will be no less than one pixel with a dark color channel and utilize this negligible incentive as an approximate of the haze.","This prior works extremely well, with the exception of in bright regions of the scene where intensity values of the considerable number of channels are likewise high.","Hence, DCP cannot manage the sky area of the images in light of the fact that the dull channel pixels are potentially inaccessible in those splendid picture areas.","Also, DCP based techniques are tedious because of the soft matting procedure involved in the method.","Recently, several techniques have been proposed to address the shortcoming of DCP [14].","In [19], color ellipsoids are fitted in the RGB color space.","These ellipsoids are utilized to arrange a unified approach to deal with the constraints of the DCP based techniques, and another strategy was proposed to assess the transmission in every ellipsoid.","In [5], [20], color lines were fit in RGB space, searching for little fixes with a steady transmission.","The traditional color line based methods depend on the assumption that pixels in a haze-free image constitute color lines in RGB space [12].","These lines go through the inception and come from color varieties inside items.","As appeared in [5], the color lines in foggy scenes do not go through the starting point any longer, because of the added substance fog part.","Some significant attempts have been made using some priors other than the DCP and color line [21], [22], [23], [24].","In [23], globally guided regularization technique is applied for dehazing.","In [24], contrast of the image is enhanced using boundary conditions.","Tang et al.","proposed a learning based technique for image dehazing [22].","However, the DCP and colorline based methods continue to dominate the other methods due to better performances, except images with high depth area (such as sky area).","Riaz et al.","proposed a DCP based approach [25] where the inefficacy of DCP at sky area is handled by limiting the contrast at the sky region.","However, in most of the cases limiting the contrast causes loss of minute color information of the image.","The proposed method follows the concepts of DCP [14] and color line [5].","In addition to applying the above two priors, the proposed method applies a Nearest Neighbor (NN) regularization technique which helps reduce the execution time.","The domain-adaptive nature of the patches helps in maintaining the original color and contrast of the image.","Finally, the adaptive nature of the size of the mask help in maintaining the original color in high depth areas of the image (area which is far from the camera, such as, sky area).","Recently, deep learning based techniques are being successfully applied in all areas of Computer vision.","A few efforts have been made to apply deep learning techniques for haze removal [9], [10], [8].","Unlike the traditional image dehazing techniques, AOD-Net in [10] proposes an end-to-end system for image dehazing, where a simple CNN is introduced to directly estimate the haze-free image, without obtaining the transmission map of the image.","Dehazenet is proposed in [9], where a transformation map of the input image is obtained from a CNN, by a trainable model.","The CNN based methods usually give better results for image dehazing, compared to the traditional prior based models.","However, for scene images where objects are far from the camera, the CNN based methods do not work well.","Also, the deep network-based methods need a huge computational set up, which may not be affordable in many cases, e.g., hand-held devices.","Next we discuss the mathematical background of the proposed method."],["Background of the Proposed Approach","We assess the transmission by considering $A$ not constant for the scene.","We observe that (REF ) expect the pixel intensity $I(x)$ to be radiometrically-straight.","Hence, additionally to adjust methods that rely upon this course of action illustrate, our system requires the reversal of the procurement nonlinearities.","In this section we explain the mathematical background of the proposed approach and the significance of the approach in the image dehazing task."],["Dark Channel Prior (DCP)","The DCP is based on the perception on haze-free images: the most of the cases, the patches taken from a haze-free image must have no less than one color channel having a low intensity incentive at a few pixels [14].","According to this perception, for any image $J$ , we characterize: $J^{dark}(x) = \\min _{c \\in \\lbrace r,g,b \\rbrace }(\\min _{y \\in \\Omega (x) } J^{c}(y)),$ where $\\Omega (x)$ is the neighborhood patch centered at $x$ , $J^c$ is the color channel of the picture $J$ and $J^{dark}$ the dark channel of $J$ .","The intensity of $J^{dark}$ is low and tends to zero if $J$ is a haze-free image.","The low intensities in the haze-free locale are due to the following three elements: Shadows of objects: For example, the shadows of leaves, trees and rocks in landscape images or the shadows of buildings in cityscape images; Dark objects or surfaces: For example, dark tree trunk and stone; Colorful objects or surfaces: Color of any object (for example, green grass/tree/plant, blue water surface or red flower/leaf/wall) mostly tends to be very close to one of the color planes such as, Red, Green, Blue, Cyan, Magenta, Yellow, Black and White.","However, the DCP based techniques work well for images with high contrast and low depth region.","For images consisting of regions with high depth value, DCP based methods fail to maintain the original color of the region.","For example, sky region is usually bright and hence, do not have dark intensity values.","We can deal with the sky locales by utilizing the Haze atmospheric scattering model (REF ), which is applied in the proposed model and discussed in the next subsection.","Figure: (a), (c) Input hazy images.","(b), (d) Depth map of the Input Hazed Images"],["Nearest-Neighbor (NN) Regularization","In Figure 3, we can observe that the transmission map is of an even consistency but is having unanticipated depth jumps in the hazy image.","In the image-dehazing context, we assume that comparable pixels have an almost identical transmission value, according to [5], [26].","In order to proficiently quantify the similarity between two image pixels, we have to think about the spatial variation and smoothness.","Given an image pixel $x$ , we characterize its element vector for similitude estimation in light of NN regularization as given below: $f(x) = (R,G,B,\\lambda X,\\lambda Y)^T,$ where the intensities of $x$ are depicted as $R$ , $G$ and $B$ in the RGB color space, respectively; $X$ and $Y$ are the spatial coordinates of $x$ , and $\\lambda $ is the balancing factor.","Despite the fact that we utilize indistinguishable features from [27], however, is not the same as matting, so we adjust our technique to the important and broadly utilized conditions for matting utilizing (REF )."],["Color line model","Colors in the small patches of a haze-free image generally lie on the line going through the starting point as shown by Omer et al.","[12], if we represent colors as coordinates in the the RGB space.","But the line is shifted from the origin by $A$ due to the additive airlight component in the case of hazy images.","We assume that $A$ is constant.","As we expect piecewise smooth depth $d$ which gives us a smooth scattering coefficient $\\beta $ which verifies that $t(x)$ is piecewise smooth which is smooth at pixels that correspond to the same object.","And $t(x)$ is also varying smoothly and slowing in the scene known from equation (REF ) except at depth discontinuities.","This assumption of piecewise smooth geometry was used by Carr et al.","[28] and Nishino et al.","[18].","So, the equation (1) for a small patch can be rewritten as: $I(x) = J(x)t + (1 - t)A, x \\in \\Omega ,$ where $t$ is a fixed transmission value in the patch $\\Omega $ , and the airlight component is known to us.","If we estimate the color line directly, then the measurement of color line direction will be erroneous if some dark pixels are present.","In the proposed model, as the image is first subjected to DCP, will be less prone to error."],["Adaptive Naive Bayes Classification Model","Naive Bayes is a conditional likelihood model.","Given an issue occurrence to be characterized, spoken to by a vector ${\\displaystyle \\mathbf {x} =(x_{1},\\dots ,x_{n})}$ representing $n$ features (independent factors), it relegates to this case probabilities $p(C_{k} | x_{1},\\dots ,x_{n})$ for each one of the $K$ conceivable results or classes ${\\displaystyle C_{k}}$ .","The predicament with the above expression is that, if the quantity of highlights $n$ is substantial or if an element can go up against a considerable number of values, at that point applying such a model on likelihood tables is troublesome.","We, therefore, reformulate the model to make it less demanding to deal with.","Appropriating Bayes' theorem, the conditional likelihood can be calculated as: $p(C_{k} | \\mathbf {x}) = \\frac{p(C_{k}) \\cdot p(\\mathbf {x} | C_{k})}{p(\\mathbf {x})}.$ The denominator of (6) can be considered as a constant.","The numerator is proportional to the conditional likelihood $p(C_{k} | x_{1},\\dots ,x_{n})$ , which can be revised as given below, utilizing the chain lead for numerous implementations of the meaning of conditional likelihood: $p(C_{k}, x_{1},\\dots ,x_{n}) &= p(x_{1},\\dots ,x_{n}, C_{k}) \\\\&= p(x_{1}|x_{2},\\dots ,x_{n}, C_{k}) \\cdot p(x_{2},\\dots ,x_{n}, C_{k}) \\\\&= p(x_{1}|x_{2},\\dots ,x_{n}, C_{k}) \\cdot p(x_{2}|x_{3},\\dots ,x_{n},\\\\& \\quad C_{k}) \\cdot p(x_{3},\\dots ,x_{n}, C_{k})\\\\&= p(x_{1}|x_{2},\\dots ,x_{n}, C_{k}) \\cdot p(x_{2}|x_{3},\\dots ,x_{n},\\\\& \\quad C_{k}) \\cdot p(x_{3}|x_{4},\\dots ,x_{n}, C_{k})\\\\&= \\dots \\\\&= p(x_{1}|x_{2},\\dots ,x_{n}, C_{k}) \\cdot p(x_{2}|x_{3},\\dots ,x_{n},\\\\& \\quad C_{k}) \\cdots p(x_{n-1}|x_{n},C_{k}) \\cdot p(x_{n}|C_{k}) \\cdot p(C_{k})\\\\& \\quad $ The “naive\" conditional independence presumptions are connected here for the proposed dehazing approach.","Let us expect that each component ${\\displaystyle x_{i}}$ is restrictively independent of each different feature ${\\displaystyle x_{j}}$ for ${\\displaystyle j\\ne i}$ , given the classification ${\\displaystyle C}$ .","Which implies, $p(x_{i} | x_{i+1},\\dots ,x_{n}, C_{k}) = p(x_{i} | C_{k})$ .","Along these lines, the proposed conditional likelihood model can be communicated as $p(C_{k} | x_{1},\\dots ,x_{n}) &\\propto p(C_{k})p(x_{1}|C_{k})p(x_{2}|C_{k}) \\dots \\\\&= p(C_{k}) \\prod ^{n}_{i=1} p(x_{i}|C_{k}).$ where $\\propto $ denotes proportionality.","For evaluating the colorline heading, we develop a model that assigns the value in view of the component vector for similitude estimation in condition (4), where the element vectors are directly free.","In attendance, we shift the quantity of pixels given to the Naive Bayes Classification Model.","By changing the number of pixels we take the probability of more number of pixels by finding the probabilities utilizing the Naive Bayes' condition to compute the posterior probability for every pixel based on the inherent vector in (4).","The pixel with the most astounding posterior probability obtain the result of the proposed expectation model.","In this section, we explain the steps to dehaze an image utilizing the nearby fix demonstrate in (REF ) and its related transmission estimation system in (REF ).","We started with a brief review of the mathematical background in the previous section.","Now we demonstrate the way the mathematical models can be used to check the input image and consider small patches of pixels as possible fixes that satisfy (REF ).","As articulated in the previous section, pixels that relate to an almost planar surface lie on a color line in RGB space depicted by (REF ) and (REF ).","In this way, in each fix we run a Naive Bayes Classification Model unequivocally built using (REF ) that scans for a line bolstered by a significant number of pixels.","We at that point check whether the line found is reliable with our arrangement show by testing it against a rundown of conditions postured by the model.","A line that concludes everyone of these tests effectively is then utilized for assessing the transmission as per (REF ) and (REF ).","The subsequent values at that point are allocated to every one of the pixels that help in finding the color line.","We evaluate the transmission in patches where we neglect to discover a line that meets every one of the conditions by utilizing the NN Regularization and reapplying the above strides on the relating patch.","Next we discuss the process of estimating transmission map of an image using DCP."],["Transmission Estimation Using Dark Channel Prior","We believe that the transmission $t$ within a neighborhood patch $\\Omega (x)$ is steady.","Performing the minimum operation in the neighborhood patch of the hazy picture in (REF ), we have the following: $\\min _{y \\in \\Omega (x)}(I^c (y)) = \\min _{y \\in \\Omega (x) }(J^c (y))t + (1 - t)A^c.$ The above equation can be derived further as: $\\min _{y \\in \\Omega (x) }(\\frac{I^c (y)}{A^c}) = \\min _{y \\in \\Omega (x) }(\\frac{J^c (y)}{A^c})t + (1 - t).$ Considering the min operation again on the above condition, in order to get the minimum of the three colors in RGB channel, we have: $\\min _{c}(\\min _{y \\in \\Omega (x)} (\\frac{I^c (y)}{A^c}) = \\min _{c}(\\min _{y \\in \\Omega (x) }(\\frac{J^c (y)}{A^c})t) + (1 - t).$ As indicated by (REF ), the dark channel $J^{dark}$ of the haze-free pixel intensity $J$ should tend to zero.","Also, as $A^c$ is a positive constant, thus: $\\min _{c}\\Big (\\min _{y \\in \\Omega (x) }(\\frac{J^c (y)}{A^c})\\Big ) = 0.$ From (REF ) and (REF ), we get: $\\ t = 1 - \\min _{c}\\Big (\\min _{y \\in \\Omega (x) }(\\frac{I^c (y)}{A^c})\\Big ).$ The above mathematical formulation gives us an estimation of the transmission.","As stated earlier, the DCP is not a appropriate prior for the sky locales.","Since the sky is at boundless infinite depth and practically has zero transmission, the (REF ) smoothly tackles both sky and non-sky regions in the proposed approach.","We dehaze the image from the assessed approximation of transmission obtained from (REF ) and utilize the neighborhood local patch model.","Reasonably, even in clear environment, the atmosphere around us is not completely free of any molecule.","Along these lines, the haze still exist when we have a look at inaccessible items.","In addition, the presence of haze is a major prompt for a human to comprehend depth."],["Dehazing Algorithm","In this component, we illustrate the initiatives for dehazing an image using our method.","First, we dehaze the image by removing the dark pixels from the image without applying a soft matting algorithm [11] to convalesce the transmission.","We signify the transmission outline $t(x)$ and get the accompanying cost function: $E(t) = t^TLt + \\lambda (t - \\tilde{t})^{T+1},$ where $L$ is the Matting Laplacian matrix introduced by Levin [29], and $\\lambda $ is a regularization parameter.","We analyse the transmission estimations of a few points with low depth values, and propose recuperating the transmission estimations of the rest of the points by finding their closest match from the arrangement of exact focuses in light of the built k-d tree with the 5-dimensional component vectors characterized in (REF ).","Conforming transmission maps along with the input image is shown in Figure 3, from which we can perceive that the transmission outline is significantly smooth aside from unanticipated depth jumps.","As specified earlier, the dehazing issue is extremely under-constrained.","Hence, we have to surmise a few assumptions from (REF ), (REF ) and (REF ) on the normal transmission map.","The refined transmission outline exercising the above strategy accomplishes to capture the sharp edge discontinuities and outline the shape of the items.","With the transmission outline, we can remunerate the scene radiance as per (REF ).","Yet, the direct attenuation may be near to zero when the transmission is near to zero.","In this way, we limit the transmission $t(x)$ to a lower bound $t_0$ , so that a specific measure of haze are retained.","The scene radiance $J(x)$ is eventually recouped by: $J(x) = \\frac{I(x) - \\hat{A}}{max(t(x),t_{0})} + \\hat{A},$ where $\\hat{A}$ denotes the estimated airlight of the image.","A traditional range of $t_{0}$ falls in the interval [0.09, 0.1], which is fixed experimentally.","Since the scene radiance is generally unlike the atmospheric light, the image after disturbance by haze evacuation looks diminish.","Hence, we increse the insight into $J(x)$ for illustration.","We examine the information from the input image and consider limited windows of pixels as hopeful patches following (REF ).","Pixels relating to an almost-planar mono-chromatic surface lie on a color line in RGB space as indicated by (REF ).","We evaluate the color line by applying “Naive Bayes Classifier\" on the intensity values in RGB space and get two focuses $\\vec{p_{1}}$ and $\\vec{p_{2}}$ lying on the straight line and an arrangement of inliers.","The condition of the line will be $\\vec{P} = \\vec{P_{0}} + \\rho \\vec{D} $ , where $\\rho $ represent the parameter of the line, $\\vec{P_{0}} = \\vec{p_{1}}$ and the heading proportion $\\vec{D} = \\frac{\\vec{p_{2}} - \\vec{p_{1}}}{\\mid \\mid ( \\vec{p_{2}} - \\vec{p_{1}} )\\mid \\mid }$ .","We examine the assessed line with conditions in particular, noteworthy number of inliers, positive incline of $\\vec{D}$ and unimodality analogously in [5] and eliminate the questionable ones.","Figure 4 demonstrates the effect of applying the Naive-Bayes Classifier and the proposed NN regularization technique on a sample hazy image.","The second row of Figure 4 illustrates the effect of applying different patch sizes on the same image.","Figure: NO_CAPTIONFigure: Result of the proposed method with varying patch size.","(a) input haze image.","(b) Output Image without Nearest Neighbour Regularization (c) Output image without applying Naive Bayes' Classification Model (d) pixels are given to Naive Bayes Classification Model with patch size 7 ×\\times 7 (e) 15 ×\\times 15 and (f) 30 ×\\times 30."],["Computing $\\hat{A}$","We can make our estimates for $\\hat{A}$ using the Dark Channel [3] of the patch.","Dark Channel of the specific patch $\\Omega $ is determined as follows (following [30]: $Dark(\\Omega ) = \\min _{x \\in \\Omega (x) }\\Big (\\min _{c \\in {R,G,B}} I_c(x)\\Big ),$ where $I_c$ is the $c$ th color channel, for $c\\in \\lbrace R,G,B\\rbrace $ .","We have already defined the direction ratio $\\vec{D}$ in the previous section, considering the standard conditions, significant line support, positive reflect of $\\vec{D}$ , unimodality and valid transmission according to (REF ) and (REF ) as discussed in [5].","We check whether the visible pixels efficiently found are consistent with our formation model by running it following (REF ), (REF ) and (REF ).","With respect to $\\vec{D}$ , we conventionally consider the normal to the inclined plane from the precise origin as determined as $\\vec{N} = \\frac{\\vec{p_{2}} \\times \\vec{p_{1}}}{\\mid \\mid ( \\vec{p_{2}} \\times \\vec{p_{1}} )\\mid \\mid }$ .","We compute $\\hat{A}$ by minimizing the following error from the existing corresponding normals from the above defination: $E(\\hat{A}) = \\frac{1}{\\Omega } \\cdot \\sum _{i \\in {\\Omega }} (N_i \\cdot \\hat{A}),$ where the line parameters are denoted by $N_i \\cdot \\hat{A}$ of the patch pixels.","The $\\cdot $ typically denotes the dot-product in RGB space.","This reliable measure consists of projecting the line parameters onto an error function, $E(\\hat{A})$ .","Therefore, (REF ) vanishes over uniformly distributed pixels and becomes positive.","Therefore, we have $\\frac{\\partial E}{\\partial \\hat{A}} = \\hat{A} \\cdot \\Big (\\sum _{i} N_iN_i^T \\Big ) = 0.$ From the (REF ), we need the non-trivial solution as $\\hat{A}$ is a non-null vector which can be acquired by registering a covariance matrix from the normals and afterwards obtaining the result as a covariance matrix similar to the smallest eigenvalue."],["Estimating the Magnitude of Airlight","We can retrieve the magnitude $a(x)$ of the airlight component, from the evaluated $\\hat{A}$ and acquire the color line correlating with the patch by limiting the accompanying error: $\\ E_{line} (\\rho , s) = \\min _{\\rho , s} \\mid \\mid {P_{0}} + \\rho D - s \\hat{A} \\mid \\mid ^2.$ Here, airlight component is denoted by $s$ .","The solution of (REF ) is obtained following [7].","The registered airlight portion is then approved with the accompanying conditions: vast intersection and convergence angles, close intersection, substantial range and shading changeability.","Out of them, vast intersection angle and close convergence are done likewise approaches as detailed in [7].","We estimate $J$ from the refined transmission map and the output image obtained from the proposed method and a black image $E$ .","From the intermediate outcomes, it can be observed that the underlying $J$ has noticeable artefacts in the sky locale, which is progressively exterminating to amid the improvement.","The outcome is given in Figure 5, exhibiting the gradual decline in the graph alongside the images as output.","Figure: The convergence of the proposed approach.","The objective function in Eq.","() is monotonically declining.","The transitional consequences of JJ and 10×E10 \\times E at the given iteration."],["Interpolation of Estimate","For assessing the airlight part $a(x)$ we dispose of a couple of patches and introduce the estimation of airlight at each pixel.","This is done by limiting the accompanying capacity: $\\begin{aligned}\\psi (a(x)) = \\sum _{\\Omega } \\sum _{x \\in {\\Omega }} \\frac{(a(\\textbf {x}) - \\widetilde{a} (\\textbf {x})^2}{(\\sigma _a (\\Omega ))^2} + \\beta \\cdot \\sum _{x} \\frac{a(\\textbf {x})}{\\mid \\mid I(\\textbf {x})\\mid \\mid } \\\\+ \\alpha \\cdot \\sum _{\\Omega } \\sum _{x \\in {L(x)}} \\frac{(a(\\textbf {x}) - \\vec{a} (\\textbf {x})^2}{\\mid \\mid I(\\textbf {x}) - I(\\textbf {y}) \\mid \\mid ^2}.\\end{aligned}$ Here, $\\vec{a}(x)$ is the assessed magnitude of the airlight component, $L(x)$ is the local neighborhood of $\\textbf {x}$ and $a(\\textbf {x})$ represents the adequate segment to be registered and $\\sigma _a(\\Omega )$ is the error difference of the estimate of $\\vec{a}(x)$ inside the patch $\\Omega $ .","The initial two terms constitute the capacity utilized as a part of [5].","To obtain a better result, we include the last term which guarantees that the part would be a little portion of $I(\\textbf {x})$ .","The latter term of (9) is utilized to change the segment with the intensity of $I(\\textbf {x})$ .","Presently, to limit the vitality work given by condition (20), we change over this to the following structure: $\\Psi (a) = (a - \\widetilde{a})^T \\cdot \\Sigma (a - \\widetilde{a}) + \\alpha a^T La + \\beta b^T a,$ where $a$ and $\\widetilde{a} $ denote the vector forms of $a(\\textbf {x})$ and $\\widetilde{a}(\\textbf {x})$ respectively, $\\Sigma $ denote the covariance grid of the pixels where the approximation is made and $L$ is the Laplacian framework of the diagram built by considering each pixel as a vertex and interfacing the neighboring vertices.","The influence of the edge between the pair of vertices $x$ and $y$ is $\\frac{1}{\\mid \\mid I(\\textbf {x}) - I(\\textbf {y})\\mid \\mid ^2}$ .","Here $\\alpha $ and $\\beta $ are scalars controlling the significance of each term."],["Haze Removal","Airlight at every pixel is now obtained by figuring $a(\\textbf {x})\\hat{A}$ .","Consequently, the immediate transmission can be recouped from $J(\\textbf {x}) t(\\textbf {x}) = I(\\textbf {x}) - a(\\textbf {x})\\hat{A}.$ As we do not have $t(x)$ unequivocally, we elevate the contrast utilizing the airlight and attempt to recoup $J(\\textbf {x})$ .","For instance, if the recouped image is $R_im (\\textbf {x})$ , at pixel $x$ : $R_im (\\textbf {x}) = \\frac{J(\\textbf {x}) t(\\textbf {x})}{1 - Y(a(\\textbf {x})\\hat{A})},$ where $Y$ is given by the following equation: $Y(I(\\textbf {x})) = 0.2989 I_R (\\textbf {x}) + 0.5870 I_G (\\textbf {x}) + 0.1140 I_B (\\textbf {x}),$ where $Y(I(\\textbf {x}))$ processes the luma at the pixel $x$ .","The idea is to upgrade the pixel $x$ relying upon how much intensity (brightness) is expelled out from it.","In many cases, the image remains dull even after the above operation.","So we utilize gamma revision to reestablish the natural shine of the image."],["RESULTS","We assess the performance of the proposed method on the Berkeley Segmentation Dataset (BSDS300), containing natural images [31].","This is a diverse dataset of clear open air characteristic images and consequently represents the moderate scenes that may have been affected by haze.","We have evaluated airlight part in condition (17).","We utilize similar parameters for every images: in condition (10) we set $\\lambda $ = 0.1 and we scale $\\frac{1}{\\sigma ^2 (x)}$ in the interim [0,1] to maintain a strategic distance from numeric issues.","For experimenting on the synthetic dataset, we utilize the dataset proposed in [5].","The dataset contains eleven dehazed pictures, manufactured distance maps and relating reenacted fog pictures.","An indistinguishably appropriated zero-mean Gaussian commotion with three one of a kind clamour level: n = 0:01; 0:025; 0:05 was added to these images (with picture control scaled to [0; 1]).","Table: NO_CAPTION"],["Quantitative Analysis","Table I condenses the L1 errors on non-sky pixels of the transmission maps and the fog free images of the synthetic dataset.","The results of four sample images are given in the table.","We utilize a progression of assessment criteria as distant as to separate between each coordinate of the haze-free image along with the result.","Notwithstanding the extensively used mean square error (MSE) and the structural similarity (SSIM) [32] measures, we utilized some more assessment frameworks, for example, weighted peak signal-to-noise ratio (WSNR) and peak signal-to-noise ratio (PSNR) [33].","Our method is compared with seven state-of-the-art methods (including deep learning based methods) when applied on the hazy images of the BSDS dataset, results of which is shown in Table II.","Table: The average results of MSE, SSIM, PSNR and WSNR on the hazy Images.In Table II, the last three lines demonstrate the results of applying deep learning based techniques.","As we can observe, the effects of applying the proposed method on hazy images is much better than the best in class handmade approaches.","Additionally, the proposed method is comparable to the deep-learning based procedures as well.","Table: The average results of MSE, SSIM, PSNR and WSNR on the Images with the Sky Region.Figure: Graphs showing the performance of the proposed approach compared to the state-of-the-art, with increasing depth value.","(a) Images without sky area, (b) Images with sky area.In Table III, the result of implementing the proposed method on images containing sky and other high depth regions, compared to the state-of-the-art methods, are shown.","Figure 7 shows how the accuracy of the proposed methods vary with respect to depth.","We can observe that, as the depth increases, the proposed method gives much better result compared to the state-of-the-art, including the deep learning based methods.","Figure: Comparison on natural images: [Left] Input Hazy Images[Right] Our result.","Middle columns display results by several competing methods."],["Qualitative results","In Figure 8, we compare the results of the proposed approach with the state-of-the-art single image dehazing techniques [5], [14], [34], [35], [36].","As noted by [5], the picture after fog evacuation may look diminish, since the scene radiance is typically not as splendid as the airlight.","The techniques in [5], [14] provide good results, yet do not have some miniaturized scale differentiate when contrasted with [34] and to our our method.","In the consequence of [36] there are artefacts in the boundary in between portions.","Figure: Comparison on natural images: [Left] Input Hazy Images[Right] Our result.","Middle columns display results by several competing methods.Figure 9 demonstrates a comparison between the outcomes obtained by the proposed technique and the state-of-the-art (explicitly, profound deep-learning based techniques) when applied on images containing sky region.","The proposed approach outperforms the competing techniques on this kind of images.","Our assumption in regards to having a fog-free pixel in each haze line does not enclose as clear by a few cloudy pixels that set a most extreme radius, e.g.","the red structures.","In spite of that, the transmission in those territories is evaluated accurately because of the regularization that proliferates the depth data spatially from the other fog lines.","Dehazing the sky area in a hazy image is really challenging, in light of the fact that clouds and mist are similar normal phenomenons with a similar air diffusing illustrate.","This issue is facilitated, in any case proceeds in DCP [14], Non-Local Dehazing [34], DehazeNet [9] and MSCNN [8] results.","While MSCNN makes the opposite curio of overenhancement: see the sky region of Yosemite for an example (Figure 9).","AOD-Net [10] can oust the obscurity, without displaying fake shading tones or turned dissent shapes.","In any case, AOD-Net does not explicitly consider the treatment of white scenes(can be found in sky range of Yosemite and Building in Figure 9).","Our method appears to be capable of finding the sky region to keep the shading, and ensures a decent dehazing sway in different locale."],["Conclusion","In this paper, we proposed a method for Image De-Hazing in view of the dim channel earlier and shading lines pixel normality on the regular images.","Dark Channel Prior technique depends on insights of the regular outdoor pictures.","We got a local nearby formation model that reasons this consistency in foggy/ hazy scenes and portrayed how it is utilized for assessing the scene transmission.","We utilize the image forming equation to recuperate haze free picture.","We have accepted commitment $A$ to be consistent.","We computed transmission and in addition profundity discontinuities.","The proposed dehazing scheme delivers comparable outcomes with the recent deep-learning based strategies which require gigantic computational setup.","In addition, the proposed technique outperforms the state-of-the-art deep learning based approaches when applied on images with sky regions.","Stretching out the recommended model to apply on real-time images may be a conceivable future research heading, which may be extended to apply on hand held devices having less computing power."]],"string":"[\n [\n \"Single Image Dehazing Based on Generic Regularity\"\n ],\n [\n \"Abstract This paper proposes a novel technique for single image dehazing.\",\n \"Most of the state-of-the-art methods for single image dehazing relies either on Dark Channel Prior (DCP) or on Color line.\",\n \"The proposed method combines the two different approaches.\",\n \"We initially compute the dark channel prior and then apply a Nearest-Neighbor (NN) based regularization technique to obtain a smooth transmission map of the hazy image.\",\n \"We consider the effect of airlight on the image by using the color line model to assess the commitment of airlight in each patch of the image and interpolate at the local neighborhood where the estimate is unreliable.\",\n \"The NN based regularization of the DCP can remove the haze, whereas, the color line based interpolation of airlight effect makes the proposed system robust against the variation of haze within an image due to multiple light sources.\",\n \"The proposed method is tested on benchmark datasets and shows promising results compared to the state-of-the-art, including the deep learning based methods, which require a huge computational setup.\",\n \"Moreover, the proposed method outperforms the recent deep learning based methods when applied on images with sky regions.\"\n ],\n [\n \"Introduction\",\n \"Images captured by the camera, are often affected by haze due to several atmospheric conditions such as fog, smoke, multiple light sources, the scattering of light, etc.\",\n \"Presence of haze in an image obscure the clarity and hence needs to be restored.\",\n \"The flux of light per unit area received by the camera from the scene is attenuated along the line of sight.\",\n \"This redirection of light lessens the immediate scene transmission and replaces with a layer of scattered light known as airlight.\",\n \"Such scattering of light due to rigid particles floating in the air reduces the visibility of the scene.\",\n \"Figure: Sample result of the proposed approach.\",\n \"(a) Input hazy image.\",\n \"(b) Dehazed image using our approach.Image methods endeavor to recover original scene radiance by expelling the impact of haze from the picture(as shown as an example in Figure 1).\",\n \"Since the effect of scattering of light at any pixel depends on the depth of the pixel, the degradation of the image due to haze is spatial-variant.\",\n \"However, recovering the effect of scene radiance from a pixel on an object is perplexing as the measure of fog and haze relies upon the separation between the object and the camera.\",\n \"In this way, global enhancement strategies do not function admirably for a scene image, because of the presence of objects at different depth and the background.\",\n \"The conventional methods exploited multiple images to overcome the problem.\",\n \"For example, the method of Narasimhan et.al.\",\n \"[1] require multiple pictures of a similar scene taken under various climate conditions.\",\n \"The method of Shwartz et.al.\",\n \"[2] requires images with the various level of polarization.\",\n \"Figure: NO_CAPTIONFigure: Result of dehazing an image.\",\n \"(a) input haze image.\",\n \"(b) refined transmission map after NN regularization (c) final haze-free image (d) depth map of the output Haze-free Image.However, images taken in various climatic conditions may be unavailable always.\",\n \"Recently researches are going on for single image dehazing, where reference images for the given hazy images are unrequired.\",\n \"After the introduction of the concept of Dark Channel Prior(DCP), people started dehazing an image without the help of a reference image [3].\",\n \"Tan et.al.\",\n \"proposed a DCP based strategy for the single image by maximizing the local image contrast adopting a Markov Random Field (MRF) framework [3].\",\n \"Fattal [4] used DCP to estimate the medium transmittance accepting the surface shading, furthermore, medium transmission capacities are locally measurably uncorrelated.\",\n \"He et al.\",\n \"[5] proposed the concept of DCP for estimating transmission map of the image.\",\n \"The DCP based approaches produce good result, but cannot handle locales where shading segment does not shift fundamentally contrasted with noise.\",\n \"Nair et al.\",\n \"[6] proposed a DCP based dehazing technique using a center surround filter to reduce the execution time.\",\n \"Another line of thought was introduced by Fattal in [7], where the color line was used for dehazing.\",\n \"The color line based methods approximate transmission map based on the shift of the direction of atmospheric light from the origin.\",\n \"However, both the approaches fail to retain the original color of the objects in the hazy image.\",\n \"Currently, some deep learning based methods have been proposed, producing much better results compared to the handcrafted features like DCP and color-line [8], [9], [10].\",\n \"However, in case of images where sky background is present, the deep learning based methods fail to dehaze properly, as the learned features are not capable of fixing the haze at high depth areas of the image.\",\n \"In this paper, the contribution over the state-of-the-art is three-folds.\",\n \"Unlike the state-of-the-art methods, the proposed method for a single image assesses the benefit of the generic regularity of the image in which, pixels with moderate intensities of small image patches exhibit concordant distributions in RGB space.\",\n \"Our second contribution is to substitute the commonly used soft matting technique [11] in assessing the transmission map for haze removal, by first discovering some settled points in a maximum filter and then applying Nearest-Neighbor (NN) regularization.\",\n \"One example of using the transmission map in haze removal by the proposed method is shown in Figure 2.\",\n \"We utilize pixels of small image patches as a rule accompanying a one-dimensional dispersion in RGB space, to approximate the intensity at each patch and interject at places wherein the estimate is not generally unreliable.\",\n \"We do not consider the atmospheric light-weight.\",\n \"Finally, unlike the state-of-the-art methods, we do not consider and estimate the transmission of the medium.\",\n \"Alternatively, we approximate the additional airlight present in the image patch and eliminate that to clear the haze.\",\n \"We interject and regularize the fractional evaluations of pixel intensity values into an entire transmission map, and outline an Adaptive Naive Bayes Classification approach to deal with pixels with a significant depth (such as sky region).\",\n \"Contrary to the conventional field models which comprise of normal coupling between adjacent pixels, we settle the transmission in isolated locales by fluctuating the number of pixels.\",\n \"We take the likelihood of more number of pixels to get the suitable pixel intensity value.\",\n \"The rest of the paper is organised as follows.\",\n \"Section 2 describes the problem and a survey on the existing literature.\",\n \"Section 3 displays the mathematical and numerical foundation for the proposed strategy for dehazing.\",\n \"Section 4 weighs up the proposed algorithm.\",\n \"Experimental results and concluding comments are given in Sections 5 and 6 respectively.\"\n ],\n [\n \"Related Work\",\n \"Dehazing is a task of image remaking; the corruption of a hazy image is a direct result of the suspended particles in the turbid air.\",\n \"Single image dehazing is a challenging task which draws significant attention from researchers of the fields of Computer Graphics and Image Processing.\",\n \"The physical model often used to characterize the haze formation that causes degradation of an image, is known as Koschmieder's atmospheric scattering model [12]: $\\\\ I(x) = t(x)J(x) + (1 - t(x))A,$ where $I(x)$ is the observed intensity value of pixel $x$ in image $I$ ; $J(x)$ is the scene radiance of a haze-free image at $x$ and $t(x)$ is the medium transmission describing the portion of the light which reaches the camera without scattering.\",\n \"$A$ is the airlight, a global vector quantity describing the ambient light.\",\n \"The goal of haze removal task is to recover $J$ , $A$ and $t$ from $I$ .\",\n \"In (REF ), the transmittance $t(x)$ does not change much for a given wavelength in a hazy image.\",\n \"The term $J(x)t(x)$ of (REF ) is termed as direct attenuation [3], which provides the information about the quantity of radiance received by the observer.\",\n \"The second part, called airlight, provides an approximation of the measure of the atmospheric light included due to the dissipating course of the viewer.\",\n \"Several attempts have been made for fog and haze expulsion in outdoor scene pictures [13].\",\n \"Notwithstanding, the considerable significant advance in the research on haze removal methods was the idea of DCP.\",\n \"It is observed that, in the more considerable part of the local neighborhood regions which do not contain the sky, a few pixels in many instances get extremely low-intensity value at the minimum one color (RGB) channel (called \\\"dark pixels\\\").\",\n \"In case of hazy image, the intensity of these dark pixels in the corresponding channel is contributed by the airlight.\",\n \"Along these lines, these dark pixels can explicitly give a precise estimation of the fog's transmission.\",\n \"In any case, the progression to reclassify the transmission approximation utilizing soft matting technique often leads to transmission underestimate.\",\n \"When the density of haze varies smoothly in space, the effect of the homogeneous atmosphere transmission $t$ can be shown as: $\\\\ t(x) = e^{\\\\beta d(x)},$ where $d(x)$ is the depth at pixel $x$ and $\\\\beta $ is the scattering coefficient (in three dimensional space) of the atmosphere.\",\n \"So, (REF ) specifies the amount of radiance received by the observer, which is attenuated exponentially with the depth.\",\n \"Several algorithms for image dehazing follow the image formation model (REF ) to dehaze images by recouping $J$ .\",\n \"However, to regain haze-free image, both (REF ) and (REF ) are followed.\",\n \"First $J$ is obtained by subtracting a constant value corresponding to the dark pixels and then estimate the value of transmission $t$ precisely by assuming $A$ is known.\",\n \"Haze lessens the contrast in the picture, and different techniques depend on this perception for rebuilding.\",\n \"He et al.\",\n \"[14] amplifies the contrast in each patch, while keeping up a global coherent image.\",\n \"This method enhanced the contrast for image dehazing and as a result, distant objects to the viewer seemed to be smooth and over saturated, causing the dehazed image look artificial.\",\n \"In [15], Park et al.\",\n \"evaluated the amount of haze in the image, from the contrast between the RGB channels, which diminishes as haze increments.\",\n \"This postulation of estimation of amount of haze is erroneous in the greay area.\",\n \"In [16], Zhu et al.\",\n \"assessed the haze in view of the perception that hazy regions are portrayed by high brightness and low saturation.\",\n \"Some efforts have been made by utilizing a prior on the depth of the picture, for haze expulsion.\",\n \"A smoothness prior is utilized in [17], expecting the image to be smooth except the pixels with depth discontinuities.\",\n \"Nishino et al.\",\n \"[18] expected the albedo and depth as factually free and together approximated them using priors on both.\",\n \"The albedo-prior accepted the distribution of gradients in images of characteristic scenes displaying a heavy-tail dispersion, and it is estimated as a summed up ordinary appropriation and further approximated as Gaussian distribution.\",\n \"The depth-prior is scene-dependent and is picked physically, either as piece-wise consistent for urban scenes or definitely changing for non-urban scenes.\",\n \"The DCP based approaches [14] accept that within a small image patche there will be no less than one pixel with a dark color channel and utilize this negligible incentive as an approximate of the haze.\",\n \"This prior works extremely well, with the exception of in bright regions of the scene where intensity values of the considerable number of channels are likewise high.\",\n \"Hence, DCP cannot manage the sky area of the images in light of the fact that the dull channel pixels are potentially inaccessible in those splendid picture areas.\",\n \"Also, DCP based techniques are tedious because of the soft matting procedure involved in the method.\",\n \"Recently, several techniques have been proposed to address the shortcoming of DCP [14].\",\n \"In [19], color ellipsoids are fitted in the RGB color space.\",\n \"These ellipsoids are utilized to arrange a unified approach to deal with the constraints of the DCP based techniques, and another strategy was proposed to assess the transmission in every ellipsoid.\",\n \"In [5], [20], color lines were fit in RGB space, searching for little fixes with a steady transmission.\",\n \"The traditional color line based methods depend on the assumption that pixels in a haze-free image constitute color lines in RGB space [12].\",\n \"These lines go through the inception and come from color varieties inside items.\",\n \"As appeared in [5], the color lines in foggy scenes do not go through the starting point any longer, because of the added substance fog part.\",\n \"Some significant attempts have been made using some priors other than the DCP and color line [21], [22], [23], [24].\",\n \"In [23], globally guided regularization technique is applied for dehazing.\",\n \"In [24], contrast of the image is enhanced using boundary conditions.\",\n \"Tang et al.\",\n \"proposed a learning based technique for image dehazing [22].\",\n \"However, the DCP and colorline based methods continue to dominate the other methods due to better performances, except images with high depth area (such as sky area).\",\n \"Riaz et al.\",\n \"proposed a DCP based approach [25] where the inefficacy of DCP at sky area is handled by limiting the contrast at the sky region.\",\n \"However, in most of the cases limiting the contrast causes loss of minute color information of the image.\",\n \"The proposed method follows the concepts of DCP [14] and color line [5].\",\n \"In addition to applying the above two priors, the proposed method applies a Nearest Neighbor (NN) regularization technique which helps reduce the execution time.\",\n \"The domain-adaptive nature of the patches helps in maintaining the original color and contrast of the image.\",\n \"Finally, the adaptive nature of the size of the mask help in maintaining the original color in high depth areas of the image (area which is far from the camera, such as, sky area).\",\n \"Recently, deep learning based techniques are being successfully applied in all areas of Computer vision.\",\n \"A few efforts have been made to apply deep learning techniques for haze removal [9], [10], [8].\",\n \"Unlike the traditional image dehazing techniques, AOD-Net in [10] proposes an end-to-end system for image dehazing, where a simple CNN is introduced to directly estimate the haze-free image, without obtaining the transmission map of the image.\",\n \"Dehazenet is proposed in [9], where a transformation map of the input image is obtained from a CNN, by a trainable model.\",\n \"The CNN based methods usually give better results for image dehazing, compared to the traditional prior based models.\",\n \"However, for scene images where objects are far from the camera, the CNN based methods do not work well.\",\n \"Also, the deep network-based methods need a huge computational set up, which may not be affordable in many cases, e.g., hand-held devices.\",\n \"Next we discuss the mathematical background of the proposed method.\"\n ],\n [\n \"Background of the Proposed Approach\",\n \"We assess the transmission by considering $A$ not constant for the scene.\",\n \"We observe that (REF ) expect the pixel intensity $I(x)$ to be radiometrically-straight.\",\n \"Hence, additionally to adjust methods that rely upon this course of action illustrate, our system requires the reversal of the procurement nonlinearities.\",\n \"In this section we explain the mathematical background of the proposed approach and the significance of the approach in the image dehazing task.\"\n ],\n [\n \"Dark Channel Prior (DCP)\",\n \"The DCP is based on the perception on haze-free images: the most of the cases, the patches taken from a haze-free image must have no less than one color channel having a low intensity incentive at a few pixels [14].\",\n \"According to this perception, for any image $J$ , we characterize: $J^{dark}(x) = \\\\min _{c \\\\in \\\\lbrace r,g,b \\\\rbrace }(\\\\min _{y \\\\in \\\\Omega (x) } J^{c}(y)),$ where $\\\\Omega (x)$ is the neighborhood patch centered at $x$ , $J^c$ is the color channel of the picture $J$ and $J^{dark}$ the dark channel of $J$ .\",\n \"The intensity of $J^{dark}$ is low and tends to zero if $J$ is a haze-free image.\",\n \"The low intensities in the haze-free locale are due to the following three elements: Shadows of objects: For example, the shadows of leaves, trees and rocks in landscape images or the shadows of buildings in cityscape images; Dark objects or surfaces: For example, dark tree trunk and stone; Colorful objects or surfaces: Color of any object (for example, green grass/tree/plant, blue water surface or red flower/leaf/wall) mostly tends to be very close to one of the color planes such as, Red, Green, Blue, Cyan, Magenta, Yellow, Black and White.\",\n \"However, the DCP based techniques work well for images with high contrast and low depth region.\",\n \"For images consisting of regions with high depth value, DCP based methods fail to maintain the original color of the region.\",\n \"For example, sky region is usually bright and hence, do not have dark intensity values.\",\n \"We can deal with the sky locales by utilizing the Haze atmospheric scattering model (REF ), which is applied in the proposed model and discussed in the next subsection.\",\n \"Figure: (a), (c) Input hazy images.\",\n \"(b), (d) Depth map of the Input Hazed Images\"\n ],\n [\n \"Nearest-Neighbor (NN) Regularization\",\n \"In Figure 3, we can observe that the transmission map is of an even consistency but is having unanticipated depth jumps in the hazy image.\",\n \"In the image-dehazing context, we assume that comparable pixels have an almost identical transmission value, according to [5], [26].\",\n \"In order to proficiently quantify the similarity between two image pixels, we have to think about the spatial variation and smoothness.\",\n \"Given an image pixel $x$ , we characterize its element vector for similitude estimation in light of NN regularization as given below: $f(x) = (R,G,B,\\\\lambda X,\\\\lambda Y)^T,$ where the intensities of $x$ are depicted as $R$ , $G$ and $B$ in the RGB color space, respectively; $X$ and $Y$ are the spatial coordinates of $x$ , and $\\\\lambda $ is the balancing factor.\",\n \"Despite the fact that we utilize indistinguishable features from [27], however, is not the same as matting, so we adjust our technique to the important and broadly utilized conditions for matting utilizing (REF ).\"\n ],\n [\n \"Color line model\",\n \"Colors in the small patches of a haze-free image generally lie on the line going through the starting point as shown by Omer et al.\",\n \"[12], if we represent colors as coordinates in the the RGB space.\",\n \"But the line is shifted from the origin by $A$ due to the additive airlight component in the case of hazy images.\",\n \"We assume that $A$ is constant.\",\n \"As we expect piecewise smooth depth $d$ which gives us a smooth scattering coefficient $\\\\beta $ which verifies that $t(x)$ is piecewise smooth which is smooth at pixels that correspond to the same object.\",\n \"And $t(x)$ is also varying smoothly and slowing in the scene known from equation (REF ) except at depth discontinuities.\",\n \"This assumption of piecewise smooth geometry was used by Carr et al.\",\n \"[28] and Nishino et al.\",\n \"[18].\",\n \"So, the equation (1) for a small patch can be rewritten as: $I(x) = J(x)t + (1 - t)A, x \\\\in \\\\Omega ,$ where $t$ is a fixed transmission value in the patch $\\\\Omega $ , and the airlight component is known to us.\",\n \"If we estimate the color line directly, then the measurement of color line direction will be erroneous if some dark pixels are present.\",\n \"In the proposed model, as the image is first subjected to DCP, will be less prone to error.\"\n ],\n [\n \"Adaptive Naive Bayes Classification Model\",\n \"Naive Bayes is a conditional likelihood model.\",\n \"Given an issue occurrence to be characterized, spoken to by a vector ${\\\\displaystyle \\\\mathbf {x} =(x_{1},\\\\dots ,x_{n})}$ representing $n$ features (independent factors), it relegates to this case probabilities $p(C_{k} | x_{1},\\\\dots ,x_{n})$ for each one of the $K$ conceivable results or classes ${\\\\displaystyle C_{k}}$ .\",\n \"The predicament with the above expression is that, if the quantity of highlights $n$ is substantial or if an element can go up against a considerable number of values, at that point applying such a model on likelihood tables is troublesome.\",\n \"We, therefore, reformulate the model to make it less demanding to deal with.\",\n \"Appropriating Bayes' theorem, the conditional likelihood can be calculated as: $p(C_{k} | \\\\mathbf {x}) = \\\\frac{p(C_{k}) \\\\cdot p(\\\\mathbf {x} | C_{k})}{p(\\\\mathbf {x})}.$ The denominator of (6) can be considered as a constant.\",\n \"The numerator is proportional to the conditional likelihood $p(C_{k} | x_{1},\\\\dots ,x_{n})$ , which can be revised as given below, utilizing the chain lead for numerous implementations of the meaning of conditional likelihood: $p(C_{k}, x_{1},\\\\dots ,x_{n}) &= p(x_{1},\\\\dots ,x_{n}, C_{k}) \\\\\\\\&= p(x_{1}|x_{2},\\\\dots ,x_{n}, C_{k}) \\\\cdot p(x_{2},\\\\dots ,x_{n}, C_{k}) \\\\\\\\&= p(x_{1}|x_{2},\\\\dots ,x_{n}, C_{k}) \\\\cdot p(x_{2}|x_{3},\\\\dots ,x_{n},\\\\\\\\& \\\\quad C_{k}) \\\\cdot p(x_{3},\\\\dots ,x_{n}, C_{k})\\\\\\\\&= p(x_{1}|x_{2},\\\\dots ,x_{n}, C_{k}) \\\\cdot p(x_{2}|x_{3},\\\\dots ,x_{n},\\\\\\\\& \\\\quad C_{k}) \\\\cdot p(x_{3}|x_{4},\\\\dots ,x_{n}, C_{k})\\\\\\\\&= \\\\dots \\\\\\\\&= p(x_{1}|x_{2},\\\\dots ,x_{n}, C_{k}) \\\\cdot p(x_{2}|x_{3},\\\\dots ,x_{n},\\\\\\\\& \\\\quad C_{k}) \\\\cdots p(x_{n-1}|x_{n},C_{k}) \\\\cdot p(x_{n}|C_{k}) \\\\cdot p(C_{k})\\\\\\\\& \\\\quad $ The “naive\\\" conditional independence presumptions are connected here for the proposed dehazing approach.\",\n \"Let us expect that each component ${\\\\displaystyle x_{i}}$ is restrictively independent of each different feature ${\\\\displaystyle x_{j}}$ for ${\\\\displaystyle j\\\\ne i}$ , given the classification ${\\\\displaystyle C}$ .\",\n \"Which implies, $p(x_{i} | x_{i+1},\\\\dots ,x_{n}, C_{k}) = p(x_{i} | C_{k})$ .\",\n \"Along these lines, the proposed conditional likelihood model can be communicated as $p(C_{k} | x_{1},\\\\dots ,x_{n}) &\\\\propto p(C_{k})p(x_{1}|C_{k})p(x_{2}|C_{k}) \\\\dots \\\\\\\\&= p(C_{k}) \\\\prod ^{n}_{i=1} p(x_{i}|C_{k}).$ where $\\\\propto $ denotes proportionality.\",\n \"For evaluating the colorline heading, we develop a model that assigns the value in view of the component vector for similitude estimation in condition (4), where the element vectors are directly free.\",\n \"In attendance, we shift the quantity of pixels given to the Naive Bayes Classification Model.\",\n \"By changing the number of pixels we take the probability of more number of pixels by finding the probabilities utilizing the Naive Bayes' condition to compute the posterior probability for every pixel based on the inherent vector in (4).\",\n \"The pixel with the most astounding posterior probability obtain the result of the proposed expectation model.\",\n \"In this section, we explain the steps to dehaze an image utilizing the nearby fix demonstrate in (REF ) and its related transmission estimation system in (REF ).\",\n \"We started with a brief review of the mathematical background in the previous section.\",\n \"Now we demonstrate the way the mathematical models can be used to check the input image and consider small patches of pixels as possible fixes that satisfy (REF ).\",\n \"As articulated in the previous section, pixels that relate to an almost planar surface lie on a color line in RGB space depicted by (REF ) and (REF ).\",\n \"In this way, in each fix we run a Naive Bayes Classification Model unequivocally built using (REF ) that scans for a line bolstered by a significant number of pixels.\",\n \"We at that point check whether the line found is reliable with our arrangement show by testing it against a rundown of conditions postured by the model.\",\n \"A line that concludes everyone of these tests effectively is then utilized for assessing the transmission as per (REF ) and (REF ).\",\n \"The subsequent values at that point are allocated to every one of the pixels that help in finding the color line.\",\n \"We evaluate the transmission in patches where we neglect to discover a line that meets every one of the conditions by utilizing the NN Regularization and reapplying the above strides on the relating patch.\",\n \"Next we discuss the process of estimating transmission map of an image using DCP.\"\n ],\n [\n \"Transmission Estimation Using Dark Channel Prior\",\n \"We believe that the transmission $t$ within a neighborhood patch $\\\\Omega (x)$ is steady.\",\n \"Performing the minimum operation in the neighborhood patch of the hazy picture in (REF ), we have the following: $\\\\min _{y \\\\in \\\\Omega (x)}(I^c (y)) = \\\\min _{y \\\\in \\\\Omega (x) }(J^c (y))t + (1 - t)A^c.$ The above equation can be derived further as: $\\\\min _{y \\\\in \\\\Omega (x) }(\\\\frac{I^c (y)}{A^c}) = \\\\min _{y \\\\in \\\\Omega (x) }(\\\\frac{J^c (y)}{A^c})t + (1 - t).$ Considering the min operation again on the above condition, in order to get the minimum of the three colors in RGB channel, we have: $\\\\min _{c}(\\\\min _{y \\\\in \\\\Omega (x)} (\\\\frac{I^c (y)}{A^c}) = \\\\min _{c}(\\\\min _{y \\\\in \\\\Omega (x) }(\\\\frac{J^c (y)}{A^c})t) + (1 - t).$ As indicated by (REF ), the dark channel $J^{dark}$ of the haze-free pixel intensity $J$ should tend to zero.\",\n \"Also, as $A^c$ is a positive constant, thus: $\\\\min _{c}\\\\Big (\\\\min _{y \\\\in \\\\Omega (x) }(\\\\frac{J^c (y)}{A^c})\\\\Big ) = 0.$ From (REF ) and (REF ), we get: $\\\\ t = 1 - \\\\min _{c}\\\\Big (\\\\min _{y \\\\in \\\\Omega (x) }(\\\\frac{I^c (y)}{A^c})\\\\Big ).$ The above mathematical formulation gives us an estimation of the transmission.\",\n \"As stated earlier, the DCP is not a appropriate prior for the sky locales.\",\n \"Since the sky is at boundless infinite depth and practically has zero transmission, the (REF ) smoothly tackles both sky and non-sky regions in the proposed approach.\",\n \"We dehaze the image from the assessed approximation of transmission obtained from (REF ) and utilize the neighborhood local patch model.\",\n \"Reasonably, even in clear environment, the atmosphere around us is not completely free of any molecule.\",\n \"Along these lines, the haze still exist when we have a look at inaccessible items.\",\n \"In addition, the presence of haze is a major prompt for a human to comprehend depth.\"\n ],\n [\n \"Dehazing Algorithm\",\n \"In this component, we illustrate the initiatives for dehazing an image using our method.\",\n \"First, we dehaze the image by removing the dark pixels from the image without applying a soft matting algorithm [11] to convalesce the transmission.\",\n \"We signify the transmission outline $t(x)$ and get the accompanying cost function: $E(t) = t^TLt + \\\\lambda (t - \\\\tilde{t})^{T+1},$ where $L$ is the Matting Laplacian matrix introduced by Levin [29], and $\\\\lambda $ is a regularization parameter.\",\n \"We analyse the transmission estimations of a few points with low depth values, and propose recuperating the transmission estimations of the rest of the points by finding their closest match from the arrangement of exact focuses in light of the built k-d tree with the 5-dimensional component vectors characterized in (REF ).\",\n \"Conforming transmission maps along with the input image is shown in Figure 3, from which we can perceive that the transmission outline is significantly smooth aside from unanticipated depth jumps.\",\n \"As specified earlier, the dehazing issue is extremely under-constrained.\",\n \"Hence, we have to surmise a few assumptions from (REF ), (REF ) and (REF ) on the normal transmission map.\",\n \"The refined transmission outline exercising the above strategy accomplishes to capture the sharp edge discontinuities and outline the shape of the items.\",\n \"With the transmission outline, we can remunerate the scene radiance as per (REF ).\",\n \"Yet, the direct attenuation may be near to zero when the transmission is near to zero.\",\n \"In this way, we limit the transmission $t(x)$ to a lower bound $t_0$ , so that a specific measure of haze are retained.\",\n \"The scene radiance $J(x)$ is eventually recouped by: $J(x) = \\\\frac{I(x) - \\\\hat{A}}{max(t(x),t_{0})} + \\\\hat{A},$ where $\\\\hat{A}$ denotes the estimated airlight of the image.\",\n \"A traditional range of $t_{0}$ falls in the interval [0.09, 0.1], which is fixed experimentally.\",\n \"Since the scene radiance is generally unlike the atmospheric light, the image after disturbance by haze evacuation looks diminish.\",\n \"Hence, we increse the insight into $J(x)$ for illustration.\",\n \"We examine the information from the input image and consider limited windows of pixels as hopeful patches following (REF ).\",\n \"Pixels relating to an almost-planar mono-chromatic surface lie on a color line in RGB space as indicated by (REF ).\",\n \"We evaluate the color line by applying “Naive Bayes Classifier\\\" on the intensity values in RGB space and get two focuses $\\\\vec{p_{1}}$ and $\\\\vec{p_{2}}$ lying on the straight line and an arrangement of inliers.\",\n \"The condition of the line will be $\\\\vec{P} = \\\\vec{P_{0}} + \\\\rho \\\\vec{D} $ , where $\\\\rho $ represent the parameter of the line, $\\\\vec{P_{0}} = \\\\vec{p_{1}}$ and the heading proportion $\\\\vec{D} = \\\\frac{\\\\vec{p_{2}} - \\\\vec{p_{1}}}{\\\\mid \\\\mid ( \\\\vec{p_{2}} - \\\\vec{p_{1}} )\\\\mid \\\\mid }$ .\",\n \"We examine the assessed line with conditions in particular, noteworthy number of inliers, positive incline of $\\\\vec{D}$ and unimodality analogously in [5] and eliminate the questionable ones.\",\n \"Figure 4 demonstrates the effect of applying the Naive-Bayes Classifier and the proposed NN regularization technique on a sample hazy image.\",\n \"The second row of Figure 4 illustrates the effect of applying different patch sizes on the same image.\",\n \"Figure: NO_CAPTIONFigure: Result of the proposed method with varying patch size.\",\n \"(a) input haze image.\",\n \"(b) Output Image without Nearest Neighbour Regularization (c) Output image without applying Naive Bayes' Classification Model (d) pixels are given to Naive Bayes Classification Model with patch size 7 ×\\\\times 7 (e) 15 ×\\\\times 15 and (f) 30 ×\\\\times 30.\"\n ],\n [\n \"Computing $\\\\hat{A}$\",\n \"We can make our estimates for $\\\\hat{A}$ using the Dark Channel [3] of the patch.\",\n \"Dark Channel of the specific patch $\\\\Omega $ is determined as follows (following [30]: $Dark(\\\\Omega ) = \\\\min _{x \\\\in \\\\Omega (x) }\\\\Big (\\\\min _{c \\\\in {R,G,B}} I_c(x)\\\\Big ),$ where $I_c$ is the $c$ th color channel, for $c\\\\in \\\\lbrace R,G,B\\\\rbrace $ .\",\n \"We have already defined the direction ratio $\\\\vec{D}$ in the previous section, considering the standard conditions, significant line support, positive reflect of $\\\\vec{D}$ , unimodality and valid transmission according to (REF ) and (REF ) as discussed in [5].\",\n \"We check whether the visible pixels efficiently found are consistent with our formation model by running it following (REF ), (REF ) and (REF ).\",\n \"With respect to $\\\\vec{D}$ , we conventionally consider the normal to the inclined plane from the precise origin as determined as $\\\\vec{N} = \\\\frac{\\\\vec{p_{2}} \\\\times \\\\vec{p_{1}}}{\\\\mid \\\\mid ( \\\\vec{p_{2}} \\\\times \\\\vec{p_{1}} )\\\\mid \\\\mid }$ .\",\n \"We compute $\\\\hat{A}$ by minimizing the following error from the existing corresponding normals from the above defination: $E(\\\\hat{A}) = \\\\frac{1}{\\\\Omega } \\\\cdot \\\\sum _{i \\\\in {\\\\Omega }} (N_i \\\\cdot \\\\hat{A}),$ where the line parameters are denoted by $N_i \\\\cdot \\\\hat{A}$ of the patch pixels.\",\n \"The $\\\\cdot $ typically denotes the dot-product in RGB space.\",\n \"This reliable measure consists of projecting the line parameters onto an error function, $E(\\\\hat{A})$ .\",\n \"Therefore, (REF ) vanishes over uniformly distributed pixels and becomes positive.\",\n \"Therefore, we have $\\\\frac{\\\\partial E}{\\\\partial \\\\hat{A}} = \\\\hat{A} \\\\cdot \\\\Big (\\\\sum _{i} N_iN_i^T \\\\Big ) = 0.$ From the (REF ), we need the non-trivial solution as $\\\\hat{A}$ is a non-null vector which can be acquired by registering a covariance matrix from the normals and afterwards obtaining the result as a covariance matrix similar to the smallest eigenvalue.\"\n ],\n [\n \"Estimating the Magnitude of Airlight\",\n \"We can retrieve the magnitude $a(x)$ of the airlight component, from the evaluated $\\\\hat{A}$ and acquire the color line correlating with the patch by limiting the accompanying error: $\\\\ E_{line} (\\\\rho , s) = \\\\min _{\\\\rho , s} \\\\mid \\\\mid {P_{0}} + \\\\rho D - s \\\\hat{A} \\\\mid \\\\mid ^2.$ Here, airlight component is denoted by $s$ .\",\n \"The solution of (REF ) is obtained following [7].\",\n \"The registered airlight portion is then approved with the accompanying conditions: vast intersection and convergence angles, close intersection, substantial range and shading changeability.\",\n \"Out of them, vast intersection angle and close convergence are done likewise approaches as detailed in [7].\",\n \"We estimate $J$ from the refined transmission map and the output image obtained from the proposed method and a black image $E$ .\",\n \"From the intermediate outcomes, it can be observed that the underlying $J$ has noticeable artefacts in the sky locale, which is progressively exterminating to amid the improvement.\",\n \"The outcome is given in Figure 5, exhibiting the gradual decline in the graph alongside the images as output.\",\n \"Figure: The convergence of the proposed approach.\",\n \"The objective function in Eq.\",\n \"() is monotonically declining.\",\n \"The transitional consequences of JJ and 10×E10 \\\\times E at the given iteration.\"\n ],\n [\n \"Interpolation of Estimate\",\n \"For assessing the airlight part $a(x)$ we dispose of a couple of patches and introduce the estimation of airlight at each pixel.\",\n \"This is done by limiting the accompanying capacity: $\\\\begin{aligned}\\\\psi (a(x)) = \\\\sum _{\\\\Omega } \\\\sum _{x \\\\in {\\\\Omega }} \\\\frac{(a(\\\\textbf {x}) - \\\\widetilde{a} (\\\\textbf {x})^2}{(\\\\sigma _a (\\\\Omega ))^2} + \\\\beta \\\\cdot \\\\sum _{x} \\\\frac{a(\\\\textbf {x})}{\\\\mid \\\\mid I(\\\\textbf {x})\\\\mid \\\\mid } \\\\\\\\+ \\\\alpha \\\\cdot \\\\sum _{\\\\Omega } \\\\sum _{x \\\\in {L(x)}} \\\\frac{(a(\\\\textbf {x}) - \\\\vec{a} (\\\\textbf {x})^2}{\\\\mid \\\\mid I(\\\\textbf {x}) - I(\\\\textbf {y}) \\\\mid \\\\mid ^2}.\\\\end{aligned}$ Here, $\\\\vec{a}(x)$ is the assessed magnitude of the airlight component, $L(x)$ is the local neighborhood of $\\\\textbf {x}$ and $a(\\\\textbf {x})$ represents the adequate segment to be registered and $\\\\sigma _a(\\\\Omega )$ is the error difference of the estimate of $\\\\vec{a}(x)$ inside the patch $\\\\Omega $ .\",\n \"The initial two terms constitute the capacity utilized as a part of [5].\",\n \"To obtain a better result, we include the last term which guarantees that the part would be a little portion of $I(\\\\textbf {x})$ .\",\n \"The latter term of (9) is utilized to change the segment with the intensity of $I(\\\\textbf {x})$ .\",\n \"Presently, to limit the vitality work given by condition (20), we change over this to the following structure: $\\\\Psi (a) = (a - \\\\widetilde{a})^T \\\\cdot \\\\Sigma (a - \\\\widetilde{a}) + \\\\alpha a^T La + \\\\beta b^T a,$ where $a$ and $\\\\widetilde{a} $ denote the vector forms of $a(\\\\textbf {x})$ and $\\\\widetilde{a}(\\\\textbf {x})$ respectively, $\\\\Sigma $ denote the covariance grid of the pixels where the approximation is made and $L$ is the Laplacian framework of the diagram built by considering each pixel as a vertex and interfacing the neighboring vertices.\",\n \"The influence of the edge between the pair of vertices $x$ and $y$ is $\\\\frac{1}{\\\\mid \\\\mid I(\\\\textbf {x}) - I(\\\\textbf {y})\\\\mid \\\\mid ^2}$ .\",\n \"Here $\\\\alpha $ and $\\\\beta $ are scalars controlling the significance of each term.\"\n ],\n [\n \"Haze Removal\",\n \"Airlight at every pixel is now obtained by figuring $a(\\\\textbf {x})\\\\hat{A}$ .\",\n \"Consequently, the immediate transmission can be recouped from $J(\\\\textbf {x}) t(\\\\textbf {x}) = I(\\\\textbf {x}) - a(\\\\textbf {x})\\\\hat{A}.$ As we do not have $t(x)$ unequivocally, we elevate the contrast utilizing the airlight and attempt to recoup $J(\\\\textbf {x})$ .\",\n \"For instance, if the recouped image is $R_im (\\\\textbf {x})$ , at pixel $x$ : $R_im (\\\\textbf {x}) = \\\\frac{J(\\\\textbf {x}) t(\\\\textbf {x})}{1 - Y(a(\\\\textbf {x})\\\\hat{A})},$ where $Y$ is given by the following equation: $Y(I(\\\\textbf {x})) = 0.2989 I_R (\\\\textbf {x}) + 0.5870 I_G (\\\\textbf {x}) + 0.1140 I_B (\\\\textbf {x}),$ where $Y(I(\\\\textbf {x}))$ processes the luma at the pixel $x$ .\",\n \"The idea is to upgrade the pixel $x$ relying upon how much intensity (brightness) is expelled out from it.\",\n \"In many cases, the image remains dull even after the above operation.\",\n \"So we utilize gamma revision to reestablish the natural shine of the image.\"\n ],\n [\n \"RESULTS\",\n \"We assess the performance of the proposed method on the Berkeley Segmentation Dataset (BSDS300), containing natural images [31].\",\n \"This is a diverse dataset of clear open air characteristic images and consequently represents the moderate scenes that may have been affected by haze.\",\n \"We have evaluated airlight part in condition (17).\",\n \"We utilize similar parameters for every images: in condition (10) we set $\\\\lambda $ = 0.1 and we scale $\\\\frac{1}{\\\\sigma ^2 (x)}$ in the interim [0,1] to maintain a strategic distance from numeric issues.\",\n \"For experimenting on the synthetic dataset, we utilize the dataset proposed in [5].\",\n \"The dataset contains eleven dehazed pictures, manufactured distance maps and relating reenacted fog pictures.\",\n \"An indistinguishably appropriated zero-mean Gaussian commotion with three one of a kind clamour level: n = 0:01; 0:025; 0:05 was added to these images (with picture control scaled to [0; 1]).\",\n \"Table: NO_CAPTION\"\n ],\n [\n \"Quantitative Analysis\",\n \"Table I condenses the L1 errors on non-sky pixels of the transmission maps and the fog free images of the synthetic dataset.\",\n \"The results of four sample images are given in the table.\",\n \"We utilize a progression of assessment criteria as distant as to separate between each coordinate of the haze-free image along with the result.\",\n \"Notwithstanding the extensively used mean square error (MSE) and the structural similarity (SSIM) [32] measures, we utilized some more assessment frameworks, for example, weighted peak signal-to-noise ratio (WSNR) and peak signal-to-noise ratio (PSNR) [33].\",\n \"Our method is compared with seven state-of-the-art methods (including deep learning based methods) when applied on the hazy images of the BSDS dataset, results of which is shown in Table II.\",\n \"Table: The average results of MSE, SSIM, PSNR and WSNR on the hazy Images.In Table II, the last three lines demonstrate the results of applying deep learning based techniques.\",\n \"As we can observe, the effects of applying the proposed method on hazy images is much better than the best in class handmade approaches.\",\n \"Additionally, the proposed method is comparable to the deep-learning based procedures as well.\",\n \"Table: The average results of MSE, SSIM, PSNR and WSNR on the Images with the Sky Region.Figure: Graphs showing the performance of the proposed approach compared to the state-of-the-art, with increasing depth value.\",\n \"(a) Images without sky area, (b) Images with sky area.In Table III, the result of implementing the proposed method on images containing sky and other high depth regions, compared to the state-of-the-art methods, are shown.\",\n \"Figure 7 shows how the accuracy of the proposed methods vary with respect to depth.\",\n \"We can observe that, as the depth increases, the proposed method gives much better result compared to the state-of-the-art, including the deep learning based methods.\",\n \"Figure: Comparison on natural images: [Left] Input Hazy Images[Right] Our result.\",\n \"Middle columns display results by several competing methods.\"\n ],\n [\n \"Qualitative results\",\n \"In Figure 8, we compare the results of the proposed approach with the state-of-the-art single image dehazing techniques [5], [14], [34], [35], [36].\",\n \"As noted by [5], the picture after fog evacuation may look diminish, since the scene radiance is typically not as splendid as the airlight.\",\n \"The techniques in [5], [14] provide good results, yet do not have some miniaturized scale differentiate when contrasted with [34] and to our our method.\",\n \"In the consequence of [36] there are artefacts in the boundary in between portions.\",\n \"Figure: Comparison on natural images: [Left] Input Hazy Images[Right] Our result.\",\n \"Middle columns display results by several competing methods.Figure 9 demonstrates a comparison between the outcomes obtained by the proposed technique and the state-of-the-art (explicitly, profound deep-learning based techniques) when applied on images containing sky region.\",\n \"The proposed approach outperforms the competing techniques on this kind of images.\",\n \"Our assumption in regards to having a fog-free pixel in each haze line does not enclose as clear by a few cloudy pixels that set a most extreme radius, e.g.\",\n \"the red structures.\",\n \"In spite of that, the transmission in those territories is evaluated accurately because of the regularization that proliferates the depth data spatially from the other fog lines.\",\n \"Dehazing the sky area in a hazy image is really challenging, in light of the fact that clouds and mist are similar normal phenomenons with a similar air diffusing illustrate.\",\n \"This issue is facilitated, in any case proceeds in DCP [14], Non-Local Dehazing [34], DehazeNet [9] and MSCNN [8] results.\",\n \"While MSCNN makes the opposite curio of overenhancement: see the sky region of Yosemite for an example (Figure 9).\",\n \"AOD-Net [10] can oust the obscurity, without displaying fake shading tones or turned dissent shapes.\",\n \"In any case, AOD-Net does not explicitly consider the treatment of white scenes(can be found in sky range of Yosemite and Building in Figure 9).\",\n \"Our method appears to be capable of finding the sky region to keep the shading, and ensures a decent dehazing sway in different locale.\"\n ],\n [\n \"Conclusion\",\n \"In this paper, we proposed a method for Image De-Hazing in view of the dim channel earlier and shading lines pixel normality on the regular images.\",\n \"Dark Channel Prior technique depends on insights of the regular outdoor pictures.\",\n \"We got a local nearby formation model that reasons this consistency in foggy/ hazy scenes and portrayed how it is utilized for assessing the scene transmission.\",\n \"We utilize the image forming equation to recuperate haze free picture.\",\n \"We have accepted commitment $A$ to be consistent.\",\n \"We computed transmission and in addition profundity discontinuities.\",\n \"The proposed dehazing scheme delivers comparable outcomes with the recent deep-learning based strategies which require gigantic computational setup.\",\n \"In addition, the proposed technique outperforms the state-of-the-art deep learning based approaches when applied on images with sky regions.\",\n \"Stretching out the recommended model to apply on real-time images may be a conceivable future research heading, which may be extended to apply on hand held devices having less computing power.\"\n ]\n]"},"id":{"kind":"string","value":"1808.08610"}}},{"rowIdx":284,"cells":{"text":{"kind":"list like","value":[["A Dichotomous Analysis of Unemployment Welfare"],["Abstract In an economy which could not accommodate the full employment of its labor force, it employs some labor but does not employ others.","The bipartition of the labor force is random, and we characterize it by an axiom of equal employment opportunity.","We value each employed individual by his or her marginal contribution to the production function; we also value each unemployed individual by the potential marginal contribution the person would make if the market hired the individual.","We then use the aggregate individual value to distribute the net production to the unemployment welfare and the employment benefits.","Using real-time balanced-budget rule as a constraint and policy stability as an objective, we derive a scientific formula which describes a fair, debt-free, and asymptotic risk-free tax rate for any given unemployment rate and national spending level.","The tax rate minimizes the asymptotic mean, variance, semi-variance, and mean absolute deviation of the underlying posterior unemployment rate.","The allocation rule stimulates employment and boosts productivity.","Under some symmetry assumptions, we even find that an unemployed person should enjoy equivalent employment benefits, and the tax rate goes with this welfare equality.","The tool employed is the cooperative game theory in which we assume many players.","The players are randomly bipartitioned, and the payoff varies with the partition.","One could apply the fair distribution rule and valuation approach to other profit-sharing or cost-sharing situations with these characteristics.","This framework is open to alternative identification strategies and other forms of equal opportunity axiom."],["INTRODUCTION","The problem with which we are concerned relates to the following typical situation: consider an economy which could not achieve full employment of its labor force, and therefore some people are employed, and others are not.","As the employed receives wages and employment benefits (e.g., pension, health insurance, social security, education allowances, paid vacation), should the unemployed receive some unemployment welfare?","If YES, how much is fair?","In a specific jurisdiction system, the term “unemployment welfare\" here may also mean “unemployment benefits,\" “unemployment insurance,\" or even “unemployment compensation.\"","In an advanced economy, the answer to the first question is likely YES.","This paper answers the second question by justifying a fair share of unemployment welfare for the unemployed and deriving a fair tax rate for the employed.","Fair unemployment welfare and a fair tax rate are among the most fundamental topics of our society.","The fair-division problem arises in various real-world settings.","For a simple motivating example, let us consider a $k$ -out-of-$n$ redundant system in engineering which has $n$ identical components, any $k$ of which being in good condition makes the system work properly.","When valuing the importance of each component (either working or standby), one may intuitively claim that these components should be equally valued.","A very similar situation occurs in a simple majority voting where not all the voters would support the proposal to vote; thus, the proposal could be passed or failed.","Nevertheless, voters are supposed to have the same voting power no matter what they support and for whom they vote.","For another example, in the health insurance industry, not all of the policyholders are ill and use the insurance to cover their medical expenses.","The question is how to fairly share the total medical cost among both the ill policyholders and the non-ill ones.","In a labor market, we have a similar, but more complicated situation: on the one hand, the market could not hire all of its labor force even though everyone in the market would like to be employed; on the other hand, the participants in the market have heterogeneous performance in the production.","There are four common features in these examples: a coalition of players with cooperative nature, a random bipartition of the players, a payoff associated with the partition, and an objective to share the payoff with all the players.","This paper derives a solution for situations with these features.","In the $k$ -out-of-$n$ redundant system and the simple majority voting, we expect equality of outcome.","We face a few challenges to deal with when fairly distributing the welfare and benefits, both generated by the employed.","First of all, fairness may be an abstract but vague concept.","We believe that “fairness\" is bound with the equality of employment opportunity, not with the equality of outcome, nor with the equality of productivity.","Furthermore, we also believe that everyone in the labor market could contribute in some way; but the opportunities are limited.","Thus, unemployment is not a fault of the unemployed, nor a flaw of the labor market, but a self-adjustment mechanism toward the efficiency of the market.","Secondly, we attempt to apply a taxation policy to the labor market which operates in an ever-changing economy and with ever-changing productivity.","For it to be useful, the tax and division rule should be not only fair to all people but also be able to cope with the uncertainty and sustainability in generating and distributing the net production.","Ideally, it should balance the account of value generated and that of value paid in each employment contingency.","Thirdly, in a perfect world, a fair tax rate should depend only on observed data, to avoid any excessively political bargaining and costly strategic voting.","One major issue, however, is the non-observability of the heterogeneous-agent production function in all employment scenarios at all the time.","Another data issue is the desynchronization between the unemployment rate and the tax rate; the former is high-frequency data while the latter has a lower frequency.","Often in a yearly time-frame, policymakers determine the tax rate after observing most monthly unemployment rates.","Vast literature (e.g., Kornhauser 1995; Fleurbaey and Maniquet 2006) from various aspects has studied the fairness in taxation and unemployment payments.","In particular, Shapley (1953) proposes an influential axiom of fairness to develop a fair-division method, called the Shapley value, which is widely used in distributing employment compensation and welfare (see, for example, Moulin 2004; Devicienti 2010; Giorgi and Guandalini 2018; Krawczyk and Platkowski 2018).","Beneath the pillars of the Shapley value and the Shapley axiom, however, are two underlying assumptions: players' unanimous participation in the production, and distributor's complete information about the production function.","Recently, Hu (2002, 2006, 2018) relax the unanimity assumption and generalize the Shapley value, using some non-informative probability distributions for the dichotomy or bipartition of the players.","In particular, Hu (2018) proposes using a Beta-Binomial distribution to address the equality of opportunity.","This current research also capitalizes on the Beta-Binomial distributions.","Furthermore, we do not assume the complete information about the production function.","We only need its value at one observation which does occur.","The advantage of our approach is twofold.","On the one hand, we provide a game-theoretical micro-foundation for a fair-division solution to distribute the unemployment welfare and employment benefits.","One can apply the solution concept to many similar situations without substantive alternations, and one may also extend the framework using other identification schemes, rather than the tax policy stability or unemployment rate minimization, detailed in Section .","On the other hand, the fair tax rate we provide is simple enough to be used in practice.","It relies only on the unemployment rate and a reserved portion of production, which is not for personal use.","The total unemployment welfare depends only on the tax rate and the observed production.","We attempt to immunize our solution from any unnecessary randomness, hypotheticals, ambiguity, and latency.","These include, but are not limited to, the competitive and cooperative features of the labor market, endogenous employment search behavior, non-linear schedule of tax rates, the exact sizes of the labor market and time-varying unemployment population.","With this simplicity in hand, a certain level of abstraction is necessary and any application of the fair solution should accommodate to the concrete reality.","We organize the remainder of the paper as follows.","Section applies the framework of dichotomous valuation (or simply, “D-value\") in Hu (2002, 2006, 2018) to value each person in the labor market, assuming equal employment opportunity.","The two sides of D-value depend on two unknown parameters and are aggregated separately for the unemployed and the employed labor.","Next, Section formulates a set of fair divisions of the net production, using the aggregate components of D-value.","In this section, we base the set of fair tax rates on two accounting identities for a balanced budget.","In Section , by maximizing the stability of the tax rate or minimizing the expected posterior unemployment rate, we identify a specific relation between the fair tax rate and the unemployment rate.","The particular solution is robust under a few other criteria.","Section lists three other applications (namely, voting power, health insurance, and highway toll) of the distribution rule, and Section suggests several ways to extend this framework.","The account is self-contained, and the proofs are in the Appendix."],["DICHOTOMOUS VALUATION","Before our formal discussions, let us introduce a few notations.","For a general economy, we assume that its labor force consists of the employed labor and the unemployed labor, ignoring any part-time labor.","Let $\\mathbb {N} = \\lbrace 1, 2,\\cdots ,n\\rbrace $ denote the set of people in the labor force, indexed as $1,2,..., n$ , and let $\\mathbf {S} \\subseteq \\mathbb {N}$ denote the random subset of the employed labor in $\\mathbb {N}$ .","For any subset $T$ of $\\mathbb {N}$ , let $|T|$ denote its cardinality; for notational simplicity, we often use $n$ for $|\\mathbb {N}|$ , $t$ for $|T|$ , and $s$ for $|\\mathbf {S}|$ .","Let us also write the employment rate as $\\omega =\\frac{s}{n}$ , which is one minus the unemployment rate.","Besides, we employ the vinculum (overbar) in naming a singleton set; for example, “$\\overline{i}$ \" is for the singleton set $\\lbrace i \\rbrace $ .","Also, we use “$\\setminus $ \" for set subtraction, “$\\cup $ \" for set union, and $\\beta (\\cdot , \\cdot )$ for a two-parameter Beta function.","The Appendix defines $\\Delta $ , $\\Delta _1, \\cdots , \\Delta _9$ as shorthand notations."],["Equality of Employment Opportunity","Equal employment opportunity is widely acknowledged and is the starting point or axiom for us to study fairness and welfare.","In the United States, for example, equal employment opportunity has been enacted to prohibit federal contractors from discriminating against employees by race, sex, creed, religion, color, or national origin since President Lyndon Johnson signed Executive Order 11246 in 1965.","In the literatureBesides the justification from equality of opportunity, unemployment welfare could also come from other considerations (e.g., Sandmo 1998, Tzannatos and Roddis 1998, Vodopivec 2004).","These include, but are not limited to, social protection, insurance of income flow, poverty prevention, the efficiency of the labor market, and political considerations.","In this paper, we focus solely on equal employment opportunity., there are abundant qualitative descriptions, informal or formal, about the equal opportunity (e.g., Friedman and Friedman 1990; Roemer 1998; Rawls 1999).","In this section, we introduce a quantitative and probabilistic version of equal opportunity whereby the employment opportunity is assumed equitable for all persons in the labor force.","We assume three layers of uncertainty for the random subset $\\mathbf {S}$ while maintaining the equality of employment opportunity.","In the first layer, the employment size $|\\mathbf {S}|$ follows a binomial distribution with parameters $(n, p)$ .","When independence is assumed, $p$ is the probability of any given person being employed.","In the second layer, the unknown parameter $p$ has a prior Beta distribution with hyperparameters $(\\theta , \\rho )$ , where $\\theta >0$ and $\\rho >0$ are to be determined by estimation or specification.","Thus, the joint probability density of $p\\in (0,1)$ and $|\\mathbf {S}|=s$ is $ \\frac{p^{\\theta -1}(1-p)^{\\rho -1}}{\\beta (\\theta , \\rho )}\\left(\\begin{array}{c}n \\\\s\\end{array}\\right)p^s (1-p)^{n-s}=\\frac{n!}{s!(n-s)!","}\\frac{p^{\\theta +s-1}(1-p)^{\\rho +n-s-1}}{\\beta (\\theta , \\rho )}.$ Eq.","(REF ) implies the following marginal probability density for $|\\mathbf {S}|=s$ : $ \\begin{array}{rcl}\\mathbb {P}(|\\mathbf {S}| = s)& = &\\mathop {{\\int _0^1}}\\frac{n!}{s!(n-s)!","}\\frac{p^{\\theta +s-1}(1-p)^{\\rho +n-s-1}}{\\beta (\\theta , \\rho )} \\mathrm {d} p \\\\&=&\\frac{n!}{s!(n-s)!","}\\frac{\\beta (\\theta +s, \\rho +n-s)}{\\beta (\\theta , \\rho )}, \\hspace{28.45274pt} s=0,1,\\cdots ,n.\\end{array}$ In the third layer, for any given employment size $s$ , all subsets of size $s$ have the same probability of being $\\mathbf {S}$ .","As there are $\\frac{n!}{s!(n-s)!","}$ subsets of size $s$ in $\\mathbb {N}$ , the probability for the employment scenario $\\mathbf {S}=T$ is $ \\mathbb {P}(\\mathbf {S}=T)=\\left\\lbrace \\begin{array}{ll}\\frac{\\mathbb {P}(|\\mathbf {S}| = s)}{\\frac{n!}{s!(n-s)!}}","= \\frac{\\beta (\\theta +s,\\rho +n-s)}{\\beta (\\theta ,\\rho )},\\quad &\\mathrm {if} \\ t=s;\\\\\\end{array}0, & \\mathrm {otherwise}.\\right.","\\\\$ Clearly, the equality of employment opportunity is implied in the assumed triple-layered uncertainty.","Furthermore, equal opportunity is also assumed for all coalitions of the same size.","By Eq.","(REF ) and Eq.","(REF ), the posterior density function of $p$ given $|\\mathbf {S}|=s$ is $\\frac{\\frac{n!}{s!(n-s)!","}\\frac{p^{\\theta +s-1}(1-p)^{\\rho +n-s-1}}{\\beta (\\theta , \\rho )}}{\\frac{n!}{s!(n-s)!","}\\frac{\\beta (\\theta +s, \\rho +n-s)}{\\beta (\\theta , \\rho )}}=\\frac{p^{\\theta +s-1}(1-p)^{\\rho +n-s-1}}{\\beta (\\theta +s, \\rho +n-s)}.$ Thus, the posterior employment rate follows a Beta distribution with parameters $(\\theta +s, \\rho +n-s)$ .","In the following, let us use $\\tilde{p}_{n,\\omega }$ to denote the posterior employment rate given the observance of $|\\mathbf {S}|=n \\omega $ .","In contrast, $p$ is the unobservable prior employment rate, and $\\omega $ is the observable employment rate."],["Value of the Employed and the Unemployed","For any $T\\subseteq \\mathbb {N}$ , we use a heterogeneous-agent production function $v(T)$ to measure the aggregate productivity when $\\mathbf {S}=T$ .","We assume the net-profit production function $v(T)$ excludes the labor cost which compensates the time and efforts devoted by the employed labor in producing $v(T)$ .","To isolate the added value by the labor force alone, we also assume that $v(T)$ excludes the cost of consumed physical and financial resources.","Both the labor cost and the resource cost are exempt from taxation in a firm.","Thus, without loss of generality, we may assume that $v(\\emptyset )=0$ for the empty set $\\emptyset $ .","However, $v(T)$ does not necessarily increase with $T$ nor with its size $|T|$ .","To retain its labor or to minimize its labor turnover, a firm would share part of its net profit with its employees.","To keep things simple, we use the term “employment benefits\" to denote the employees' profit-sharing part in $v(T)$ , in contrast to the term “unemployment welfare.\"","Let us formally introduce two components of the D-value.","For any $i \\in \\mathbb {N}$ , to analyze its marginal effect on the value generating process, we consider two jointly exhaustive and mutually exclusive events: Event 1: $i \\in {\\mathbf {S}}$ .","Then, $i$ 's marginal effect is $v({\\mathbf {S}}) - v( {\\mathbf {S}} \\setminus \\overline{i})$ , called marginal gain, in that he or she contributes $v({\\mathbf {S}}) - v( {\\mathbf {S}} \\setminus \\overline{i})$ to the production, due to his or her existence in ${\\mathbf {S}}$ .","The expected marginal gain is $ \\gamma _i[v] \\stackrel{\\mbox{def}}{=} \\mathbb {E} \\left[v(\\mathbf {S}) - v(\\mathbf {S} \\setminus \\overline{i}) \\right].$ In the above, “def\" is for definition and “$\\mathbb {E}$ \" for expectation under the probability distribution specified by Eq.","(REF ).","Event 2: $i \\notin {\\mathbf {S}}$ .","Then, $i$ 's marginal effect is $v({\\mathbf {S}}\\cup \\overline{i}) - v({\\mathbf {S}})$ in that ${\\mathbf {S}}$ faces a marginal loss $v({\\mathbf {S}}\\cup \\overline{i}) - v({\\mathbf {S}})$ , due to $i$ 's absence from the employed labor force ${\\mathbf {S}}$ .","In other words, the person would increase the production by $v({\\mathbf {S}}\\cup \\overline{i} ) - v({\\mathbf {S}})$ if the market included him or her in $\\mathbf {S}$ .","The expected marginal loss is then $\\lambda _i[v] \\stackrel{\\mbox{def}}{=}\\mathbb {E} \\left[v({\\mathbf {S}}\\cup \\overline{i}) - v({\\mathbf {S}}) \\right].$ We let $\\gamma _i[v]$ be the employment benefits $i$ receives when he or she is employed, and let $\\lambda _i[v]$ be the unemployment welfare $i$ receives when he or she is unemployed.","Note that, even if $i$ always remains employed, both $\\mathbf {S}$ and $\\mathbf {S}\\setminus \\overline{i}$ change daily, if not hourly; thus $i$ 's marginal gain is not a constant.","Similarly, even if $i$ remains unemployed for a while, $\\mathbf {S}$ , $\\mathbf {S}\\cup \\overline{i}$ , and $i$ 's marginal loss are not constant.","To account for this uncertainty, we have already taken expectations in Eq.","(REF ) and Eq.","(REF ) when defining $\\gamma _i[v]$ and $\\lambda _i[v]$ .","A few key points worth mentioning to help us understand the profit-sharing strategy.","First, in addition to receiving employment benefits, the employed labor also receives the reimbursement for labor cost, which compensates its human capital usage in generating $v(\\mathbf {S})$ .","Human capital also accumulates in prior employment and pre-employment education.","Labor, physical, and financial costs are not part of the generated value $v(\\mathbf {S})$ .","The unemployed labor, however, only receives unemployment welfare.","Secondly, if $i \\in \\mathbf {S}$ , then $v(\\mathbf {S} \\setminus \\overline{i})$ is not observable while we observe $v( \\mathbf {S})$ .","Similarly, when $j \\notin \\mathbf {S}$ , we cannot observe both $v(\\mathbf {S} \\cup \\overline{j})$ and $v( \\mathbf {S})$ simultaneously.","Thus, we need to transform the aggregate marginals into observable forms, such as those in Theorem REF .","Thirdly, the aggregate employment benefits $\\sum \\limits _{i\\in \\mathbf {S}} \\left[v(\\mathbf {S})-v(\\mathbf {S}\\setminus \\overline{i}) \\right]$ is not necessarily equal to $v(\\mathbf {S})$ , the value collectively generated by $\\mathbf {S}$ .","Thus, we distribute some of the surpluses $v(\\mathbf {S}) - \\sum \\limits _{i\\in \\mathbf {S}}\\left[v(\\mathbf {S})-v(\\mathbf {S}\\setminus \\overline{i}) \\right]$ to the unemployed labor $\\mathbb {N}\\setminus \\mathbf {S}$ .","The distribution is not through personal giving but government taxation and unemployment payment systems.","This distribution channel also appeals to us for the aggregate benefits and aggregate welfare at the national level, as stated in Theorem REF .","The aggregate components of the D-value are $\\begin{array}{rcl}\\end{array}\\sum \\limits _{i\\in \\mathbb {N}} \\gamma _i [v]&=&\\mathbb {E} \\left[ \\sum \\limits _{i\\in \\mathbf {S}} \\left( v(\\mathbf {S}) - v(\\mathbf {S}\\setminus \\overline{i}) \\right) \\right] \\\\$ = n(+n,)(, )v(N) + TN: T= N t(+-1)-n+n-t-1 (+t, +n-t)(,)v(T), iN i [v] = E [ i NS (v(Si) - v(S) ) ] = TN:T= t(+-1)-n(-1)+t-1 (+t, +n-t)(,) v(T) - n(,+n)(,)v()."],["ACCOUNTING IDENTITIES FOR A BALANCED BUDGET","By Eq.","(REF ), the expected production is $ \\mathbb {E} \\left[ v(\\mathbf {S}) \\right]= \\sum \\limits _{T\\subseteq \\mathbb {N}} \\frac{\\beta (\\theta +t, \\rho +n-t)}{\\beta (\\theta ,\\rho )} v(T).$ Out of the $2^n$ employment scenarios for $\\mathbf {S}$ , we observe only one.","Let us consider this particular scenario at $\\mathbf {S}=T$ , which occurs with probability $\\frac{\\beta (\\theta +t, \\rho +n-t)}{\\beta (\\theta ,\\rho )}$ and which generates the value $v(T)$ .","After the production, we face a challenge of fairly dividing $v(T)$ between $T$ and $\\mathbb {N}\\setminus T$ , whom both have an entitlement to $v(T)$ .","Our division rule should fully respect the entitlement claim: each employed person receives his or her expected marginal gain, and each unemployed person receives his or her expected marginal loss.","As a contrast, the Shapley value distributes $v(\\mathbb {N})$ to all players in $\\mathbb {N}$ ; Hu(2018) formulates solutions to divide $\\mathbb {E} \\left[ v(\\mathbf {S}) \\right]$ and $v(\\mathbb {N})-\\mathbb {E} \\left[ v(\\mathbf {S}) \\right]$ .","Besides $\\gamma _i [v]$ and $\\lambda _i [v]$ , we should reserve a portion of $v(T)$ for the common good of the economy and society."],["A Real-Time Balanced Budget Rule","As noted above, we purposely divide the net production $v(T)$ into three components.","The first one is for employment benefits.","We compare the coefficients of $v(T)$ in Eq.","(REF ) and that of $\\mathbb {E} \\left[ \\sum \\limits _{i\\in \\mathbf {S}} \\left(v(\\mathbf {S}) - v(\\mathbf {S}\\setminus \\overline{i}) \\right) \\right]$ in Eq.","(REF ), ignoring the probability density for the scenario; the employed labor $T$ should retain $\\frac{t (\\theta +\\rho -1)-n\\theta }{\\rho +n-t-1} v(T)$ as their employment benefits.","The rest, $\\left[1 - \\frac{t(\\theta +\\rho -1)-n\\theta }{\\rho +n-t-1} \\right] v(T)$ , pays to the government as “tax\"In practice, not all unemployment welfare comes from the taxation system.","In the United States, for example, it is funded by a compulsory governmental insurance system, which manages the collection and payment accounts.","However, the contribution to and distribution from the insurance system are de facto of a type of payroll tax.","In Australia, unemployment welfare, as part of social security benefits, is funded through the taxation system..","Besides, we assume both $\\lambda _i[v]$ and $\\gamma _i[v]$ are tax-exempt in order to avoid any double taxation.","Thus, we define the tax rate $\\tau $ as $\\tau \\stackrel{\\mathrm {def}}{=} 1 - \\frac{s(\\theta +\\rho -1)-n\\theta }{\\rho +n-s-1},\\qquad s=1,\\cdots , n-1.$ For the time being, the rate relies on the labor market size $n$ and the employment rate $\\omega = \\frac{|\\mathbf {S}|}{n}$ , but not on the production $v(\\mathbf {S})$ .","In this definition, we exclude two extreme but unlikely cases at $\\omega =0$ and $\\omega =1$ .In reality, the two cases have zero probability to occur, even though the probability model (REF ) gives a very slight chance.","Secondly, we compare the coefficients of $v(T)$ in Eq.","(REF ) and that of $\\mathbb {E} \\left[ \\sum \\limits _{i \\in \\mathbb {N}\\setminus \\mathbf {S}} \\left(v(\\mathbf {S}\\cup \\overline{i}) - v(\\mathbf {S}) \\right) \\right]$ in Eq.","(REF ); the unemployed labor $\\mathbb {N}\\setminus T$ should claim $\\frac{t(\\theta +\\rho -1) - n(\\theta -1)}{\\theta +t-1} v(T)$ from the tax revenue $\\tau v(T)$ as its unemployment welfare.","Thirdly, we assume that a reserved proportion, $\\delta v(T)$ , is not individually and not directly distributed to the labor force $\\mathbb {N}$ .","As a consequence, the tax rate $\\tau $ includes both the reserve ratio $\\delta $ and the proportion to the unemployment welfare, i.e., $ \\tau \\equiv \\delta + \\frac{s(\\theta +\\rho -1)-n(\\theta -1)}{\\theta +s-1}.$ The reserve $\\delta v(T)$ is meant to serve the general public's interests and to have broad appeal, rather than the individual needs.","More specifically, $\\delta v(T)$ includes, but is not limited to, the payments to the population who are not in the labor force, to the corporate equity earnings not used for employment benefits, to the public administration and national defense, to the public welfare, to the tax deficit from the past, to the future development, and so on.","By admitting the corporate equity earnings to the reserve $\\delta v(T)$ , we have deliberately ignored firm-specific re-distribution processes of the net corporate earnings, and have purposely avoided the associated corporate income taxation.","In practice, government expenditureincluding that in all levels of government but excluding all unemployment welfare payments.","and corporate earnings are two major components of the reserve $\\delta v(T)$ .","When the government has other sources of tax revenue, a pro rata share of its expenditure will come from $\\delta v(T)$ .","Besides, the government could implement a countercyclical fiscal policy by adjusting its spending level in $\\delta v(T)$ such that it counteracts the ratio of the corporate earnings to the production.","Indeed, by Eq.","(REF ), $\\tau $ automatically reacts positively to the change of the corporate earning ratio, other things remaining the same.","Put in another way, the real-time balanced budget rule implied in Eq.","(REF ) and Eq.","(REF ) forbids any borrowing between different labor market scenarios or forbids any inter-temporal borrowing.","Thus, this sustainable taxation policy meets the needs of the present market scenario without compromising the ability of future market scenarios to meet their own needs.","In practice, however, it is challenging, if not impossible, to enforce or enact the balanced budget rule at the labor market scenario level or on a real-time basis.","In the United States, for example, the employment rate $\\omega $ changes daily and is recorded monthly by the Bureau of Labor Statistics; as a policy variable, the tax rate $\\tau $ changes yearly.","Striving for the balanced budget rule to the highest degree, one could minimize the variance of the employment rate $\\omega $ within a yearly time frame.","In a perfect situation, the employment rate closely follows a degenerate probability distribution and remains almost constant within the year.","We discuss the variance minimization in Section .","In contrast to the exhaustive distribution of the net production in the national accounts, a household could still maximize its utility through inter-temporal borrowing, saving, lending, and consuming."],["Sets of Feasible Solutions","Although the terms “tax rate\" and “tax rule\" have been used loosely and interchangeably about $\\tau $ , we must distinguish between them to have an accurate discussion.","As a tax rule or tax policy, $\\tau $ is a function of $(n, \\omega , \\delta )$ , subject to the equality of employment opportunity and the balance of budget as specified in Eq.","(REF ) and Eq.","(REF ).","In contrast, as a tax rate, $\\tau $ is merely the value of the function at a specific $(n,\\omega , \\delta )$ .","For a given triple of $(n, \\omega , \\delta )$ , there are three indeterminates $(\\theta , \\rho , \\tau )$ in the system of two equations Eq.","(REF ) and Eq.","(REF ).","Let $\\Omega _{n,\\omega ,\\delta }$ denote the set of all feasible combinations of $(\\theta , \\rho , \\tau )$ which satisfy both Eq.","(REF ) and Eq.","(REF ): $\\Omega _{n,\\omega ,\\delta }\\stackrel{\\mathrm {def}}{=} \\ \\left\\lbrace (\\theta ,\\rho , \\tau ) \\left|\\begin{array}{l}\\tau = 1 - \\frac{n \\omega (\\theta +\\rho -1)-n\\theta }{\\rho +n- n \\omega -1}, \\\\\\tau = \\delta + \\frac{n \\omega (\\theta +\\rho -1)-n(\\theta -1)}{\\theta + n \\omega -1}, \\\\0 \\le \\tau \\le 1,\\ \\theta >0,\\ \\rho >0.\\end{array}\\right.\\right\\rbrace .$ From now on, a “fair tax rate\" could mean a solution of $\\tau $ in $\\Omega _{n,\\omega ,\\delta }$ ; or it could be a limit of tax rates which satisfy Eq.","(REF ) and Eq.","(REF ).","In general, for a reasonable $\\delta $ , a finite $n$ , and a $\\omega \\in (0,1)$ , there are infinitely many solutions in $\\Omega _{n,\\omega ,\\delta }$ .","In this case, we need one more restriction to solve $(\\theta ,\\rho , \\tau )$ uniquely.","To do so, one could capitalize on the statistical relation between $\\omega $ and $(\\theta , \\rho )$ : the prior mean and mode of $\\omega $ are $\\frac{\\theta }{\\theta +\\rho }$ and $\\frac{\\theta -1}{\\theta +\\rho -2}$ , respectively (e.g., Johnson et al.","1995, Chapter 21).","Alongside this direction, for example, we could set $\\frac{\\theta }{\\theta +\\rho }$ (or $\\frac{\\theta -1}{\\theta +\\rho -2}$ ) to be the historical average (or mode, respectively) of $\\omega $ in the previous year.","Alternatively, we could set $\\frac{\\theta }{\\theta +\\rho }$ or $\\frac{\\theta -1}{\\theta +\\rho -2}$ to be a target employment rate or the natural employment rate.","However, this type of identification schemes requires additional input (e.g., historical average, target employment rate, or natural employment rate).","Furthermore, to derive a unique solution from $\\Omega _{n,\\omega ,\\delta }$ or its boundary, we should base our deviation on observed data only.","One concern in the input $(n, \\omega , \\delta )$ is the size of the labor market $n$ .","Even though the size $n$ is not random in our model of equal employment opportunity, it is likely a time-varying latent variable in that there is no clear cut between entry to and exit from the labor market; and many depressed unemployed persons may not actively seek new positions.","In practice, $n$ changes daily whereas $\\tau $ most likely changes yearly.","No matter how it changes and how much latent it is, however, we are confident that $n$ is a large number in the general economy.","Thus, in contrast to Eq.","(REF ) and Eq.","(REF ), we seek a fair tax rule which is valid for all large $n$ 's but is not specific to a particular $n$ .","As a result, the tax rule we are targeting should not involve $n$ , and we may write it as $\\tau (\\omega , \\delta ): (0,1) \\times (0,1) \\rightarrow (0,1)$ .","To visualize the solution set $\\Omega _{n,\\omega ,\\delta }$ , let us write $(\\theta ,\\rho )$ in terms of $(n, \\delta , \\tau , \\omega )$ using Lemma REF in the Appendix.","Figure REF (a-b) plots the feasible solution sets for $n=10000$ , $\\delta =.1$ , and any $\\omega \\in (0,1)$ .","Note that there is a straight line which dramatically raises both $\\theta $ and $\\rho $ .","From Figure REF (c-d), we observe that both $\\theta $ and $\\rho $ drop sharply on the other side of the straight line.","As a matter of fact, any point on the other side of the straight line does not represent a fair solution, owing to the positivity requirement of $\\theta $ and $\\rho $ .","Actually, the straight line has a tax rule $\\tau (\\omega , \\delta ) = 1-\\omega + \\delta \\omega $ .","This linear relation between $\\tau $ and $\\omega $ can be seen from Lemma REF : on the one hand, when $\\Delta \\equiv \\omega +\\tau -\\delta \\omega -1 > 0$ , both $\\theta =\\frac{n^2 \\omega \\Delta _1 + n\\Delta _2 +\\Delta _3 }{n\\Delta + \\Delta _3}$ and $\\rho = \\frac{n^2 (1-\\omega ) \\Delta _1 + n\\Delta _4 + \\Delta _3}{n\\Delta + \\Delta _3}$ increase to $+\\infty $ as $n\\rightarrow \\infty $ , due to the positivity of $\\Delta _1 \\equiv \\delta \\omega -\\omega -\\tau +2> 0$ ; on the other hand, when $\\Delta <0$ , both $\\theta $ and $\\rho $ decrease to $-\\infty $ as $n\\rightarrow \\infty $ .","Thus, the singular line is expressed as $\\Delta =0$ , or $\\tau (\\omega , \\delta ) = 1-\\omega +\\delta \\omega $ .","Furthermore, for any fair tax rule on the side of $\\Delta >0$ , the posterior employment rate $\\tilde{p}_{n,\\omega }$ has a degenerated limit distribution as $n\\rightarrow \\infty $ .","This is claimed in Theorem REF .","The next section answers a related question: which tax rule makes the distribution convergence fastest.","The answer happens to be the solution on the singular line.","Figure: Solutions of Eq.","() and Eq.","() for n=10000,δ=.1,ω∈(0,1)n=10000,\\ \\delta =.1,\\ \\omega \\in (0,1).For any fair tax rule $\\tau (\\omega , \\delta ) \\in (1-\\omega +\\delta \\omega ,1)$ , as $n\\rightarrow \\infty $ , $\\tilde{p}_{n,\\omega }$ converges in distribution to the degenerate probability distribution with mass at $\\omega $ ."],["AN ASYMPTOTIC RISK-FREE TAX RATE","In this section, we derive the limit fair tax rule $\\tau (\\omega , \\delta ) =1-\\omega +\\delta \\omega $ from a few different angles.","At a minimum, a good tax rule should not discourage employment incentives and productivity, such as detailed in Theorem REF and REF .","On the other end, we expect a good rule to be robust and optimal under multiple criteria.","We study five criteria in Theorem REF , REF , REF , REF , and REF , any of which uniquely identifies the solution.","They either minimize the employment market risk or maximize the employment expectation, under a given market capacity and a budget balance constraint.","First, we should heavily exploit the observed market behavior.","While an efficient labor market stimulates productivity $v(\\mathbf {S})$ , a higher employment rate does not imply higher productivity and vice versa — the production function $v(\\mathbf {S})$ does not necessarily increase with the employment size $|\\mathbf {S}|$ .","Acemoglu and Shimer (2000) find that a moderate level of unemployment could boost productivity by improving the quality of jobs.","Indeed, it fosters peer pressure in producing $v(\\mathbf {S})$ , allows workers to move on from declining firms, and enables rising companies and the economy as a whole to optimally respond to external shocks.","Therefore, our tax rule would not merely target a higher expectation of employment rate, but grounds its assumption in the observed market behavior, and lets the market itself respond with a higher employment rate, to the maximum extent permitted by the budget rule and the market capacity.","Secondly, from a statistical viewpoint, our tax rule relies on a realization of $p$ , not on the uncertainty of the realized value.","A natural way to factor out this uncertainty is to study the posterior rate $\\tilde{p}_{n,\\omega }$ , where both $s$ and $\\omega $ are no longer random.","As Theorem REF indicates, $\\omega $ is informative and indicative in revealing the central tendency of the posterior labor market, delineated by $\\tilde{p}_{n,\\omega }$ .","Asymptotic dispersion of $\\tilde{p}_{n,\\omega }$ and the market's response to the tax rule $\\tau (\\omega , \\delta )$ are among other essential ingredients in the complete profile of $\\tilde{p}_{n,\\omega }$ .","Thus, we set optimal criteria to minimize dispersion measures or maximize the expected market response."],["Zero Asymptotic Posterior Variance","Tax rate stability creates the right environment for a balance of payments, reduces the uncertainty of the labor market, and creates confidence in technology and human capital investment.","As a function of $\\omega $ , $\\tau $ migrates the risk from the unemployment rate to the tax rate.","With the absence of other exogenous shocks, indeed, stability in the tax rate is equivalent to that in the unemployment rate.","The same argument is also valid when the exogenous shocks are orthogonal to that in $\\omega $ .","When we use the variance of $\\tilde{p}_{n,\\omega }$ to measure its instability, Theorem REF states that the tax rule $\\tau (\\omega , \\delta ) = 1 - \\omega + \\delta \\omega $ minimizes the asymptotic variance of $\\tilde{p}_{n,\\omega }$ .","Furthermore, it is also the limit of variance-minimizing tax rules for finite labor markets.","We add the restriction “$\\theta , \\rho \\ge \\frac{1}{n}$ \" in the finite labor markets to ensure the positivity of the hyperparameters $(\\theta ,\\rho )$ .","As $n\\rightarrow \\infty $ , the tax rule $\\tau (\\omega , \\delta ) = 1 - \\omega + \\delta \\omega $ minimizes the limit variance of $\\sqrt{n} \\tilde{p}_{n,\\omega }$ , i.e., $\\mathop {\\mathrm {argmin}}\\limits _{\\tau } \\lim \\limits _{n\\rightarrow \\infty } \\mathrm {VAR} \\left(\\sqrt{n} \\tilde{p}_{n,\\omega } \\right)= 1 -\\omega +\\delta \\omega .$ Besides, $\\lim \\limits _{n\\rightarrow \\infty } \\mathop {\\mathrm {argmin}}\\limits _{\\tau } \\mathrm {VAR} \\left(\\sqrt{n} \\tilde{p}_{n,\\omega } \\bigm \\vert \\theta , \\rho \\ge \\frac{1}{n} \\right)= 1 -\\omega +\\delta \\omega .$ Furthermore, both the minimum limit variance and the limit minimum variance are zero.","We provide a few comments to help clarify any potential misunderstandings about the theorem.","First, a stable $\\sqrt{n} \\tilde{p}_{n,\\omega }$ is achieved at $n=\\infty $ where the limit variance is zero.","However, $\\tilde{p}_{n,\\omega }$ is still exposed to exogenous shocks, such as those studied in Pissarides (1992) and Blanchard (2000).","Secondly, it is worth emphasizing that $1-\\omega +\\delta \\omega $ is the limit tax rule as $n\\rightarrow \\infty $ .","For a large but finite $n$ , a small positive number could be added to $1-\\omega +\\delta \\omega $ to ensure the positivitySee details in the proof of Theorem REF .","of $\\theta $ and $\\rho $ .","That small positive number, however, is negligible; thus, we can practically use the rule $\\tau (\\omega , \\delta ) = 1-\\omega +\\delta \\omega $ without any addition.","Moreover, a higher-order approximation $\\tau =1-\\omega +\\delta \\omega +\\frac{\\omega (1-\\omega )(1-\\delta )^2}{n}$ could be an excellent alternativeSee the proof of Theorem REF for details.","to $\\tau =1-\\omega +\\delta \\omega $ .","Thirdly, with a zero or near zero variance in the unemployment rate, labor mobility means one layoff and one new hire should almost coincide to ensure the total employment size $|\\mathbf {S}|$ remains nearly constant.","It also means that the sizes of employment and labor market change proportionally so that their ratio remains unchanged.","Lastly, though the posterior distribution is skewed, the tax rule minimizes both the overall risk and the one-sided risk of $\\sqrt{n} \\tilde{p}_{n,\\omega }$ , as stated in Theorem REF .","In particular, a policymaker's concern is on the downside risk only.","As $n\\rightarrow \\infty $ , the tax rule $\\tau (\\omega , \\delta ) = 1 - \\omega + \\delta \\omega $ minimizes both the limit lower semivariance and the limit upper semivariance of $\\sqrt{n} \\tilde{p}_{n,\\omega }$ ."],["Consistency and Robustness","The above fair tax rule also captures several striking features of the labor market.","In the first place, it is the policymaker's best response to the market to stimulate employment within the framework of Eq.","(REF ) and Eq.","(REF ).","In the second place, we can also derive it by minimizing statistical dispersion measures other than the posterior variance or semivariance.","Meanwhile, it helps mitigate income inequality.","The rule $\\tau (\\omega , \\delta ) = 1 - \\omega + \\delta \\omega $ is an effective taxation strategy to maximally boost the employment size without breaking the opportunity equality and the budget balance.","For an economic policymaker, one primary concern is the forward-looking employment profile $\\tilde{p}_{n,\\omega }$ .","By Theorem REF , the mean of $\\tilde{p}_{n,\\omega }$ converges to $\\omega $ as $n\\rightarrow \\infty $ for any fair rule $\\tau (\\omega , \\delta ) \\in (1-\\omega +\\delta \\omega ,1)$ .","When $n$ is finitely large, they respond adversely to an increasing $\\tau $ (cf Theorem REF ).","Consequently, to maximize the posterior mean, we should minimize the tax rate $\\tau $ while still maintaining the conditions $\\tau (\\omega , \\delta ) \\in (1-\\omega +\\delta \\omega ,1)$ , $\\theta >0$ and $\\rho >0$ .","Thus, the limit of fair tax rates which maximize the mean of $\\tilde{p}_{n,\\omega }$ would be $1-\\omega +\\delta \\omega $ .","As a remark, the condition $\\omega >.5$ in Theorem REF is satisfied in the general economy.","For any $\\omega \\in (.5,1)$ and a finitely large $n$ , the mean of $\\tilde{p}_{n,\\omega }$ reacts negatively to an increasing tax rate $\\tau \\in (1-\\omega +\\delta \\omega ,1)$ .","As a consequence, $\\lim \\limits _{n\\rightarrow \\infty } \\mathop {\\mathrm {argmax}}\\limits _{\\tau } \\mathrm {MEAN}\\left(\\tilde{p}_{n,\\omega } \\bigm \\vert \\theta , \\rho \\ge \\frac{1}{n} \\right)=1 -\\omega +\\delta \\omega .$ Furthermore, the tax rule also minimizes the mean absolute deviation (thereafter, MAD) from the mean as $n \\rightarrow \\infty $ .","For a Beta distribution, especially with large parameters, MAD is a more robust measure of statistical dispersion than the variance.","The MAD around the mean for the posterior $\\tilde{p}_{n,\\omega }$ is (e.g., Gupta and Nadarajah 2004, page 37): $\\mathbb {E} \\left[ \\left|\\tilde{p}_{n,\\omega } - \\mathbb {E} (\\tilde{p}_{n,\\omega }) \\right| \\right]=\\frac{2(\\theta +s)^{\\theta +s} (\\rho +n-s)^{\\rho +n-s}}{\\beta (\\theta +s, \\rho +n-s)(\\theta +\\rho +n)^{\\theta +\\rho +n}}.$ In the next theorem, we identify the same tax rule by minimizing the asymptotic MAD.","As $n\\rightarrow \\infty $ , the tax rule $\\tau (\\omega , \\delta ) = 1 - \\omega + \\delta \\omega $ minimizes the MAD of $n \\tilde{p}_{n,\\omega }$ around the mean, i.e.","$\\lim \\limits _{n\\rightarrow \\infty } \\mathop {\\mathrm {argmin}}\\limits _{\\tau }\\mathbb {E} \\left[n \\left|\\tilde{p}_{n,\\omega } - \\mathbb {E} (\\tilde{p}_{n,\\omega }) \\right| \\biggm \\vert \\theta , \\rho \\ge \\frac{1}{n} \\right]= 1 -\\omega +\\delta \\omega .$ Besides, $\\mathop {\\mathrm {argmin}}\\limits _{\\tau } \\lim \\limits _{n\\rightarrow \\infty }\\mathbb {E} \\left[n \\left|\\tilde{p}_{n,\\omega } - \\mathbb {E} (\\tilde{p}_{n,\\omega }) \\right| \\biggm \\vert \\theta , \\rho \\ge \\frac{1}{n} \\right]= 1 -\\omega +\\delta \\omega .$"],["Equality of Outcome with Symmetric Production","To see the relation between productivity and the tax rate, we introduce a partial ordering in the labor market.","For any $i, j\\in \\mathbb {N}$ with $i\\ne j$ , we say $i$ uniformly outperforms $j$ in $v$ if $v(T\\cup \\overline{i}) - v(T) \\ge v(T\\cup \\overline{j}) - v(T)$ for any $T\\subseteq \\mathbb {N} \\setminus \\overline{i} \\setminus \\overline{j}$ ; and $v(T)-v(T\\setminus \\overline{i}) \\ge v(T)-v(T \\setminus \\overline{j})$ for any $T\\subseteq \\mathbb {N}$ with $i,j \\in T$ .","With these two inequality conditions, $i$ has higher marginal productivity than $j$ in all comparable employment contingencies $-$ either both employed or both unemployed.","As productivity is highly valued in Eq.","(REF ) and Eq.","(REF ), $j$ should receive fewer employment benefits and less unemployment welfare than $i$ does.","This is formally claimed in Theorem REF .","Besides, the theorem does not require the Beta-Binomial distribution, as long as $i$ and $j$ alone have the same chance of being employed; other players in the labor market may have unequal employment opportunities.","The theorem is valid for all fair tax rules, including the special one $\\tau (\\omega , \\delta )=1-\\omega +\\delta \\omega $ .","If $i\\in \\mathbb {N}$ uniformly outperforms $j\\in \\mathbb {N}$ in $v$ and they have equal employment opportunity, then $\\gamma _i[v] \\ge \\gamma _j[v]$ and $\\lambda _i[v] \\ge \\lambda _j[v]$ .","We say $i, j\\in \\mathbb {N}$ are symmetric in the production function $v$ if they uniformly outperform each other.","By Theorem REF , $\\lambda _i[v] =\\lambda _j[v]$ and $\\gamma _i[v] = \\gamma _j[v]$ if $i$ and $j$ are symmetric and they have equal employment opportunity.","In other words, they should receive the same amount of unemployment welfare if both are unemployed; they should also receive the same amount of employment benefits if both are employed.","Without any further analysis of or any prior knowledge about the production function, symmetry among the unemployed (or the employed) could be a reasonable a priori assumption to distribute the unemployment welfare (or employment benefits, respectively).","For example, the Cobb-Douglas production function is symmetric among all employed persons in the labor market.","Besides, if we assume symmetry among the employed labor and also assume symmetry among the unemployed labor, then the tax rule $\\tau (\\omega , \\delta ) = 1 - \\omega + \\delta \\omega $ eliminates the income inequality, when the income is either employment benefits or unemployment welfare.","Theorem REF affirms this equality of outcome, but an employed individual and an unemployed one may not be symmetric in $v$ .","Furthermore, the theorem does not restrict the size of $n$ and the specific probability distribution for the equal employment opportunity.","In the $k$ -out-of-$n$ redundant system mentioned in Section , accordingly, the $n$ components (either working or standby) are equally important if they have equal quality.","Section offers a few alternative probability distributions for the equal employment opportunity.","Assume equal employment opportunity in $\\mathbb {N}$ .","If all employed individuals are symmetric in $v$ and all unemployed individuals are also symmetric in $v$ , then $\\tau (\\omega , \\delta ) = 1 - \\omega + \\delta \\omega $ if and only if an unemployed person's unemployment welfare equals an employee's employment benefits."],["Labor Cost","We could use the symmetry to calibrate the labor cost, which is excluded from $v({\\mathbf {S}})$ .","For example, let the labor cost for $i \\in {\\mathbf {S}}$ be the minimum wage requirement from all $j \\in \\mathbb {N}\\setminus {\\mathbf {S}}$ , who either are symmetric with $i$ or uniformly outperforms $i$ in $v$ .","That is, some $j$ from $\\mathbb {N}\\setminus {\\mathbf {S}}$ can do $i$ 's work without compromising the production $v$ .","The minimum wage is called reservation wage, below which $j$ is unwilling to work.","At this minimum market replacement cost, ${\\mathbf {S}}$ can switch $i$ with someone else from $\\mathbb {N} \\setminus {\\mathbf {S}}$ without sacrificing its net profit.","In fact, $j$ does not need to be symmetric with or outperform $i$ as long as the probability of $v({\\mathbf {S}}\\cup \\overline{j}\\setminus \\overline{i}) \\ge v({\\mathbf {S}}\\cup \\overline{i}\\setminus \\overline{j})$ is significantly large and the person has a willingness to accept (e.g., Horowitz and McConnell 2003).","The probability of such $\\mathbf {S}$ could be difficult for job interviewers to approximate and estimate.","At the other end, in order to avoid creating an undue disincentive to work, $j$ 's unemployment welfare must be less than the reservation wage plus employment benefit.","Moreover, to ensure every one of $\\mathbb {N}$ is in the labor market, the welfare is necessary to bind to the incentive-compatibility constraint.","This incentive requirement places a lower bound for the labor cost."],["OTHER APPLICATIONS","In an abstract sense, the above account is a fair-division solution for the following game-theoretic setting: there are a large number of players; the players are randomly divided into two groups; the payoff comes from one group.","A wide range of applications falls into this type of games.","In this section, we analyze three applications other than labor markets to show how to use the formula derived in the last three sections."],["Voting Power","In a voting game (e.g., Shapley 1962), $v: 2^\\mathbb {N} \\rightarrow \\lbrace 0, 1\\rbrace $ is a monotonically increasing set function.","Let $\\mathbf {S}$ denote the random subset of voters who vote for the proposal.","The proposal passes when $v(\\mathbf {S})=1$ ; otherwise, it blocks when $v(\\mathbf {S})=0$ .","However, $v$ should not mean “production\" or alike.","No matter the outcome, Hu (2006) describes $\\gamma _i [v]$ as $i$ 's probability of turning a blocked result to a passed one, and $\\lambda _i [v]$ the probability of turning a passed result to being blocked.","Thus, the sum of $\\lambda _i [v]$ and $\\gamma _i [v]$ quantifies $i$ 's power in the game.","The ratio $\\delta $ plays a role in some circumstances.","Let us consider, for example, $10\\%$ of the voters approve just a proposal before a referendum voting on the proposal, and assume the number of other voters' support ballots follows a Beta-Binomial distribution.","Many voting games are symmetric.","In these cases, the equality of outcome becomes an egalitarian allocation of power."],["Health Insurance","Health insurance has two types of policyholders: some are ill and use the insurance to cover their medical expenses; others are healthy and do not use the insurance.","Let $\\mathbf {S}$ denote the random set of ill policyholders, $v(\\mathbf {S})$ be the total medical expenses with copays deducted, and $\\tilde{\\delta }v(\\mathbf {S})$ be the surcharge paid to the insurance company.","Let $\\delta = - \\tilde{\\delta }$ .","Then the total expenses except for the copays, $(1 - \\delta ) v(\\mathbf {S})$ , are billed to all insurance policyholders.","If $\\tau =1-\\omega +\\delta \\omega $ and $v$ is symmetric among the two types of policyholders, respectively, then by the equality of outcome, the cost to buy the insurance policy would be $\\frac{(1-\\delta ) \\mathbb {E} \\left[ v(\\mathbf {S}) \\right]}{n}$ per policyholder.","We take expectation on $\\frac{(1-\\delta ) v(\\mathbf {S})}{n}$ because the policyholders pay it upfront.","On the contrary, the unemployment welfare and employment benefits payments come after the production.","In this example, patients pay the predetermined copays.","In the labor market studied before, the labor costs are exempt from the distribution of the net production; they, however, have an indirect effect on $\\tau $ through the corporate earning ratio and $\\delta $ .","In the next example, we use the equality of outcome to derive the same type of payments as copays and labor cost."],["Highway Toll","The I-66 highway inside the Capital Beltway of the Washington metropolitan area has enforced a dynamic toll rule during rush hours: a carpool driver pays no toll, but a solo driver pays a dynamic toll fee, say $\\xi (n, \\omega )$ .","Here $n$ is the number of cars in the segment of the highway, and $\\omega $ is the percentage of solo drivers in the traffic.","Let $g(n)$ be the average carpool driver's cost in the traffic when the traffic volume is $n$ cars.","It is likely a nonlinear increasing function of $n$ .","An excellent choice of $g(n)$ is the expected driving time in hours multiplied by the average hourly pay rate, plus expenses on gas and vehicle depreciation.","Also, let $\\mathbf {S}$ denote the random set of solo drivers.","Then, $v(\\mathbf {S}) = ng(n) - n(1-\\omega ) g\\left(n(1-\\omega ) \\right) - n\\omega \\xi (n, \\omega ) $ is the total cost of over-traffic generated by the solo drivers, with toll fees deducted.","The production function $v$ is symmetric among all solo drivers and also symmetric among all carpool drivers.","By the equality of outcome, each driver shares the same cost $\\frac{v(\\mathbf {S})}{n}$ .","As a carpool driver pays no toll, his or her shared cost should exactly offset the extra cost caused by the solo drivers, which is $g(n) - g \\left(n(1-\\omega ) \\right)$ .","Finally, equation $\\frac{v(\\mathbf {S})}{n}=g(n) - g\\left(n(1-\\omega ) \\right)$ implies $\\xi (n, \\omega ) = g\\left(n(1-\\omega ) \\right).$ In this example, an administration surcharge $\\delta $ may apply; the carpooled passengers are free-riders, but their costs are exempt from $v(\\mathbf {S})$ ."],["CONCLUSIONS","In this paper, a fair-division solution is proposed to allocate the unemployment welfare in an economy, where the heterogeneous-agent production function is almost unknown.","We interpret “fairness\" as equal employment opportunity and model it by a Beta-Binomial probability distribution.","Our “sustainability\" is meant to be free of debt and free of surplus in the taxation budget.","To justify the value of the unemployed labor, we capitalize on the D-value concept in Hu (2018).","The D-value specifies how much of the net production to be retained with the employed labor, and what portion to be distributed to the unemployed.","Finally, we postulate that the labor market is static and identify a sustainable tax policy by minimizing the asymptotic variance of the posterior employment rate.","The policy can also be uniquely determined by minimizing the asymptotic posterior mean of the unemployment rate, or minimizing the downside risk of the posterior employment rate, or minimizing the posterior mean absolute deviation.","Surprisingly, the tax rule is not only simple enough for practical use but also motivates the unemployed to seek employment and the employed to improve productivity.","One could extend this framework in several ways.","One way is to re-specify the probability distribution of equal employment opportunity, for example, by any of the following re-specifications.","First, we can replace the two-parameter Beta distribution with a four-parameter one or a Beta rectangular distribution.","Secondly, we can let $\\theta $ and $\\rho $ be some functions of other unknown parameters.","Thirdly, we could substitute the Beta-Binomial distribution with a Dirichlet-Multinomial distribution or a Beta-Geometric distribution.","Fourthly, we could randomize $p$ without involving $(\\theta , \\rho )$ , by generating two independent three-parameter Gamma random variables $X$ and $Y$ .","Then, the ratio $X/(X+Y)$ is a Beta random variable.","In any of the four cases, however, we need additional identification restrictions to figure out a $\\tau (\\omega , \\delta )$ precisely.","From other angles, we could apply other identification schemes or other objective functions to find a unique fair tax rule.","From a statistical viewpoint, one could try the maximum likelihood estimation using the twelve months' data prior to determining the policy tax rate, or minimize the ex-ante risk of $\\omega $ , or apply the statistical methods mentioned in Section REF .","From an economic viewpoint, one could minimize the Gini coefficient of the Beta distribution of $\\tilde{p}_{n,\\omega }$ .","From a strategic game-theoretical viewpoint, one could seek a bargaining solution from the feasible solution set $\\Omega _{n,\\omega ,\\delta }$ , which may be particularly useful when $n$ is small.","Finally, a policymaker could treat the reserve ratio $\\delta $ as endogenous, for example, letting it be an increasing function of $\\omega $ .","He or she could also place a heavier weight on the marginal gain than on the marginal loss, in order to stimulate employment.","The simple static model, however, ignores several important aspects of a real labor market.","First, it does not capture the dynamic features of the income inequality, nor its rational response to the tax rule.","Secondly, while preventing the fungibility of borrowing funds from the future reduces the risk that a government administration piles up its national debt, it impairs that administration's ability (especially, monetary policy) to intervene in the economy.","The government, however, can still moderately stimulate the economy during a recession by adjusting the reserve ratio $\\delta $ .","Thirdly, the postulation of labor market efficiency does neglect the recent development of the incomplete-market theory (e.g., Magill and Quinzii 1996).","Fourthly, a multi-criteria objective function may be a viable alternative to the minimum-variance one, especially when there is a high unemployment rate $1-\\omega $ or a large $\\delta $ .","Lastly, a single tax rule $\\tau (\\omega , \\delta )$ could have overly simplified the complexity of the taxation system, which is also affected by other determinants.","These are just a few challenges our framework introduces, which require further development.","In summary, the fair and sustainable tax policy studied here has a solid theoretical underpinning, together with simplicity in practical use, consistency with productivity and employment incentives, and robustness to similar objectives.","When applying this framework to a real fair-division problem, one should also consider the benefits of alternative probability distributions for equal opportunity, alternative objective functions, alternative restrictions, and dynamic thinking."],["APPENDIX","Our focus in this paper is to study the relationship between the fair tax rate $\\tau $ and the employment rate $\\omega $ when the labor market size $n$ is large.","To analyze the limit behavior of $\\tau $ and $\\omega $ for a large $n$ , we only need the relevant asymptotic approximations.","We say two functions $f(n)=O \\left(g(n) \\right)$ if $\\limsup \\limits _{n\\rightarrow \\infty } \\left|\\frac{f(n)}{g(n)} \\right| < \\infty $ ; and we say $f(n) \\approx g(n)$ if $\\lim \\limits _{n\\rightarrow \\infty } \\frac{f(n)}{g(n)} =1$ .","For simplicity, let us denote the following shorthands: $\\begin{array}{lcl}\\Delta &\\equiv &\\omega + \\tau - \\delta \\omega - 1, \\\\\\end{array}\\Delta _1&\\equiv &\\delta \\omega - \\omega -\\tau + 2\\equiv 1 - \\Delta , \\\\$ 2- 2 - 2 + 2 + -1, 3(1- )(-) = --+ 2, 4- + + - 2+ 2 - 2 + - 2 + 3-1, 5- + - 2 + - 2 + 4-2 2+4, 6-+ +-1 2 + 3, 7-- + 2+ - 1 4 + (1-) 3, 8-- + + 3-2 3+5, 9 -2-+2+ 4-3 + 8.","$$ The following lemma, to be used in other proofs, re-writes Eq.","(REF ) and Eq.","(REF ).","In terms of $(n,\\delta ,\\tau , \\omega )$ , we can solve $(\\theta ,\\rho )$ from Eq.","(REF ) and Eq.","(REF ) as $ \\left\\lbrace \\begin{array}{l}\\theta =\\frac{n^2 \\omega \\Delta _1 + n\\Delta _2 +\\Delta _3 }{n\\Delta + \\Delta _3}, \\\\\\end{array}\\rho =\\frac{n^2 (1-\\omega ) \\Delta _1 + n\\Delta _4 + \\Delta _3}{n\\Delta + \\Delta _3}.\\right.\\qquad \\mathrm {(A.1)}$ In this proof, we use the following relation about Beta functions: $\\left\\lbrace \\begin{array}{lcll}\\beta (x-1, y+1) &=& \\frac{y}{x-1} \\beta (x, y), &x>1,\\ y>0; \\\\\\beta (x+1, y-1) &=& \\frac{x}{y-1} \\beta (x,y), &x>0,\\ y>1.","\\\\\\end{array}\\right.$ First, the expected aggregate marginal gain and loss are: $\\begin{array}{rcl}\\mathbb {E}\\left[ \\sum \\limits _{i\\in \\mathbf {S}} \\left( v(\\mathbf {S})-v(\\mathbf {S}\\setminus \\overline{i}) \\right) \\right]\\end{array}&=&\\sum \\limits _{T\\subseteq \\mathbb {N}} \\mathbb {P}(\\mathbf {S}=T) \\sum \\limits _{i\\in T} \\left[ v(T)-v(T\\setminus \\overline{i}) \\right] \\\\$ = i N TN: i T P(S=T) [ v(T)-v(Ti) ] = i N i[v], E[ i NS ( v(Si)-v(S) ) ] = TN P(S=T) i NT [ v(Ti)-v(T) ] = i N T N: i T P(S=T) [ v(Ti)-v(T) ] = i N i[v].","$$ Next, by Eq.","(REF )$-$ (REF ), we re-write the expected marginal gain and loss as: $ \\begin{array}{rcl}\\end{array}\\gamma _i [v]&=& \\sum \\limits _{T\\subseteq \\mathbb {N}: T\\ni i} \\frac{\\beta (\\theta +t,\\rho +n-t)}{\\beta (\\theta ,\\rho )}[v(T)-v(T\\setminus \\overline{i})] \\\\$ Z=Ti= TN: Ti (+t,+n-t)(,) v(T) - ZN i (+|Z|+1, +n-|Z|-1)(,) v(Z), i [v] =ZNi (+|Z|,+n-|Z|)(,)[v(Zi) -v(Z)] T=Zi=TN: Ti (+t-1,+n-t+1)(,) v(T) - ZNi (+|Z|,+n-|Z|)(,) v(Z).","By Eq.","(REF ), the aggregate value of the employed labor $\\sum \\limits _{i \\in \\mathbb {N}} \\gamma _i [v]$ is $\\begin{array}{rcl}&&\\sum \\limits _{i\\in \\mathbb {N}} \\sum \\limits _{T\\subseteq \\mathbb {N}: T\\ni i} \\frac{\\beta (\\theta +t, \\rho +n-t)}{\\beta (\\theta ,\\rho )}v(T)- \\sum \\limits _{i\\in \\mathbb {N}} \\sum \\limits _{Z\\subseteq \\mathbb {N} \\setminus \\overline{i}} \\frac{\\beta (\\theta +|Z|+1, \\rho +n-|Z|-1)}{\\beta (\\theta ,\\rho )} v(Z) \\\\\\end{array}&\\stackrel{T=Z}{=}& \\sum \\limits _{T\\subseteq \\mathbb {N}: T\\ne \\emptyset }\\sum \\limits _{i\\in T} \\frac{\\beta (\\theta +t, \\rho +n-t)}{\\beta (\\theta ,\\rho )}v(T)- \\sum \\limits _{T\\subseteq \\mathbb {N}: T\\ne \\mathbb {N}} \\sum \\limits _{i \\in \\mathbb {N} \\setminus T} \\frac{\\beta (\\theta +t+1, \\rho +n-t-1)}{\\beta (\\theta ,\\rho )} v(T) \\\\$ = TN: T= t (+t, +n-t)(,)v(T) - TN: T= N (n-t) (+t+1, +n-t-1)(,) v(T) = n(+n,)(, )v(N) - n(+1,+n-1)(, )v() + TN: T= , T= N t (+t, +n-t) - (n-t) (+t+1, +n-t-1)(,)v(T) = n(+n,)(,)v(N) - n+n-1(,+n)(, )v() + TN: T= N, T= [ t - (n-t) (+t)+n-t-1 ] (+t, +n-t)(,)v(T) = n(+n,)(, )v(N) + TN: T= N t(+-1)-n+n-t-1 (+t, +n-t)(,)v(T).","$$ Also by Eq.","(REF ), the aggregate value of the unemployed labor $\\sum \\limits _{i\\in \\mathbb {N}} \\lambda _i [v]$ is $\\begin{array}{rcl}&&\\sum \\limits _{i\\in \\mathbb {N}} \\sum \\limits _{T\\subseteq \\mathbb {N}: T\\ni i} \\frac{\\beta (\\theta +t-1, \\rho +n-t+1)}{\\beta (\\theta ,\\rho )}v(T)- \\sum \\limits _{i\\in \\mathbb {N}} \\sum \\limits _{Z \\subseteq \\mathbb {N} \\setminus \\overline{i}} \\frac{\\beta (\\theta +|Z|, \\rho +n-|Z|)}{\\beta (\\theta ,\\rho )} v(Z) \\\\\\end{array}&\\stackrel{T=Z}{=}& \\sum \\limits _{T\\subseteq \\mathbb {N}: T\\ne \\emptyset }\\sum \\limits _{i\\in T} \\frac{\\beta (\\theta +t-1, \\rho +n-t+1)}{\\beta (\\theta ,\\rho )}v(T)- \\sum \\limits _{T\\subseteq \\mathbb {N}: T \\ne \\mathbb {N}}\\sum \\limits _{i\\in \\mathbb {N} \\setminus T} \\frac{\\beta (\\theta +t, \\rho +n-t)}{\\beta (\\theta ,\\rho )} v(T) \\\\$ = TN: T= t(+t-1, +n-t+1)(,)v(T) - TN: T =N(n-t)(+t, +n-t)(,) v(T) = n(+n-1, +1)(,)v(N) - n(, +n)(,) v() +TN: T= , T =N t(+t-1, +n-t+1)-(n-t)(+t, +n-t)(,)v(T) = n+n-1(+n, )(,)v(N) - n(, +n)(,) v() + TN: T= , T= N [ t(+n-t)+t-1 - (n-t) ] (+t, +n-t)(,) v(T) = TN: T= t(+-1)-n(-1)+t-1 (+t, +n-t)(,) v(T) - n(, +n)(,) v().","$$ We can re-write Eq.","(REF ) and Eq.","(REF ) as a linear system of unknowns $(\\theta , \\rho )$ : $\\left\\lbrace \\begin{array}{rcl}(1-\\tau )(\\rho +n-s-1)&=&s(\\theta +\\rho -1)-n\\theta , \\\\\\end{array}(\\tau -\\delta )(\\theta +s-1)&=&s(\\theta +\\rho -1)-n(\\theta -1).\\right.$ As a consequence, the symbolic solution of $(\\theta , \\rho )$ is unique.","Let us assume Eq.","(REF ) and verify that it satisfies both Eq.","(REF ) and Eq.","(REF ) by the following identities, some of which are used in other proofs: $\\begin{array}{rcl}\\end{array}\\theta +s&=&\\frac{n^2 \\omega \\Delta _1 + n\\Delta _2 +\\Delta _3 }{n\\Delta + \\Delta _3}+ \\frac{n\\omega (n\\Delta + \\Delta _3)}{n\\Delta + \\Delta _3}=\\frac{n^2 \\omega + n\\Delta _6 + \\Delta _3}{n\\Delta + \\Delta _3}, \\\\$ +s-1 = n2 + n6 + 3n+ 3 - n+ 3n+ 3 = n2 + n (6-) n+ 3 = n2 + n (- 1) n+ 3, += n2 1 + n2 +3 n+ 3 + n2 (1-) 1 + n4 + 3n+ 3 = n2 1 + n 5 + 23n+ 3, +- 1 = n2 1 + n 5 + 23n+ 3 - n+ 3n+ 3 = n2 1 + n ( - 2 - 2 + 3- 1) + 3 n+ 3, ++ n = n2 1 + n 5 + 23n+ 3 + n(n+ 3)n+ 3 = n2 + n 8 + 23n+ 3, ++ n + 1 = n2 + n 8 + 23n+ 3 +n+ 3n+ 3 = n2 + n 9 + 33 n+ 3, ++ n - 1 = n2 + n 8 + 23n+ 3 - n+ 3n+ 3 = n2 + n (8 - ) + 3 n+ 3, ++ n - 2 = n2 + n 8 + 23n+ 3 - 2(n+ 3)n+ 3 = n2 + n (8-2) n+ 3, +n-s = n2 (1-) 1 + n4 + 3n+ 3 + n(1-)(n+ 3)n+ 3 = n2 (1-) + n 7 + 3n+ 3, +n-s - 1 = n2 (1-) + n 7 + 3n+ 3 -n+ 3n+ 3 = n2 (1-)+ n (1-)(-) n+ 3.","$Thus,$ rcl s(+- 1)-n= n[n2 1 + n ( - 2 - 2 + 3- 1) + 3] n+ 3 - n(n2 1 + n2 +3)n+ 3 = n2[(- 2 - 2 + 3- 1) -2] + n(-1)3 n+ 3 = n2(1-)(1-) + n(-1)3 n+ 3, s(+- 1)-n(-1) = [s(+- 1)-n] + n = n2(1-)(1-) + n(-1)3 n+ 3 + n(n+ 3)n+ 3 = n2 (-) + n 3 n+ 3.","$Therefore,$ rclcl s(+- 1)-n+n-s - 1 = n2(1-)(1-)+ n(-1)3n2 (1-)+ n (1-)(-) = 1-, s(+- 1)-n(-1)+s-1 = n2 (-) + n 3n2 + n (- 1) = - , $which are equivalent to Eq.","(\\ref {eq:tax_rate_def}) and Eq.","(\\ref {eq:tax_rate_c}), respectively.$ For any integer $z \\ge 0$ , by the proof of Lemma REF , $\\begin{array}{rcl}\\frac{\\theta +s+z}{\\theta +\\rho +n+z}\\end{array}&=&\\frac{\\frac{n^2\\omega +n\\Delta _6+\\Delta _3}{n\\Delta +\\Delta _3}+\\frac{z(n\\Delta +\\Delta _3)}{n\\Delta +\\Delta _3}}{\\frac{n^2+n\\Delta _8+2\\Delta _3}{n\\Delta +\\Delta _3}+\\frac{z(n\\Delta +\\Delta _3)}{n\\Delta +\\Delta _3}} \\\\$ = n2+n(6+z)+(1+z)3 n2+n(8+z)+(2+z)3 , as n. $$ As a function of $\\eta $ , the characteristic function of $\\tilde{p}_{n,\\omega }$ (e.g., Johnson et al.","1995, Chapter 21) is $\\mathbb {E} \\left[e^{i \\eta \\tilde{p}_{n,\\omega }} \\right]= 1 + \\sum \\limits _{k=1}^\\infty \\frac{(i \\eta )^k}{k!","}\\prod \\limits _{z=0}^{k-1} \\frac{\\theta +s+z}{\\theta +\\rho +n+z}$ where $i$ is the unit imaginary number, i.e., $i^2=-1$ .","We let $n\\rightarrow \\infty $ , $\\begin{array}{rcl}\\lim \\limits _{n\\rightarrow \\infty } \\mathbb {E}[e^{i \\eta \\tilde{p}_{n,\\omega }}]&=&1 + \\sum \\limits _{k=1}^\\infty \\frac{(i \\eta )^k}{k!}","\\lim \\limits _{n\\rightarrow \\infty }\\prod \\limits _{z=0}^{k-1}\\frac{\\theta +s+z}{\\theta +\\rho +n+z}\\\\\\end{array}&=&1 + \\sum \\limits _{k=1}^\\infty \\frac{(i \\eta \\omega )^k}{k!}","\\\\$ = (i ).","$Therefore, as $ $, $ pn,$ converges in distribution to the degenerate distribution with mass at $$,which has the characteristic function $ (i )$.$$$ As $\\tilde{p}_{n,\\omega }$ has a Beta distribution with parameters $(\\theta +s, \\rho +n-s)$ (see Section REF ), its variance is $\\frac{(\\theta +s)(\\rho +n-s)}{(\\theta +\\rho +n)^2(\\theta +\\rho +n+1)}$ (e.g., Gupta and Nadarajah 2004, page 35).","By the proof of Lemma REF , the variance of $\\sqrt{n} \\tilde{p}_{n,\\omega }$ is $\\begin{array}{rcl}&&\\frac{n \\frac{n^2\\omega +n\\Delta _6+\\Delta _3}{n\\Delta +\\Delta _3}\\frac{n^2(1-\\omega )+n\\Delta _7+\\Delta _3}{n\\Delta +\\Delta _3}}{ \\left(\\frac{n^2+n\\Delta _8+2\\Delta _3}{n\\Delta +\\Delta _3}\\right)^2\\frac{n^2+n\\Delta _9+3\\Delta _3}{n\\Delta +\\Delta _3}} \\\\\\end{array}&=&\\frac{n(n\\Delta +\\Delta _3)(n^2\\omega +n\\Delta _6+\\Delta _3)\\left[n^2(1-\\omega )+n\\Delta _7+\\Delta _3 \\right]}{\\left(n^2+n\\Delta _8+2\\Delta _3 \\right)^2\\left(n^2+n\\Delta _9+3\\Delta _3 \\right)} \\\\$ = (+3n) [ (1+6n+3n2 ) ] [ (1-) (1+7n(1-)+3n2(1-) ) ] (1+ 8n+23n2)2 (1+ 9n+ 33n2 ) = (1-) (+3n) [ 1+ 6n + 7n(1-) - 28n-9n + O(1n2 ) ] = (1-)+ (1-)n [ 3 + ( 6 + 71- - 28 - 9) ] + O(1n2 ) (1-) (+--1), as n .","To minimize $\\lim \\limits _{n\\rightarrow \\infty }\\mathrm {VAR}(\\sqrt{n} \\tilde{p}_{n,\\omega })= \\omega (1-\\omega )\\Delta $ while $\\lim \\limits _{n\\rightarrow \\infty } \\mathrm {VAR}(\\sqrt{n} \\tilde{p}_{n,\\omega } )\\ge 0$ , we have to set $\\Delta = 0$ .","Thus, $\\mathop {\\mathrm {argmin}}\\limits _{\\tau } \\lim \\limits _{n\\rightarrow \\infty }\\mathrm {VAR} \\left(\\sqrt{n} \\tilde{p}_{n,\\omega } \\right) = 1-\\omega +\\delta \\omega .$ However, when applying $\\tau = 1-\\omega + \\delta \\omega $ to a labor market with finite $n$ , we have $\\Delta = 0$ , $\\Delta _1 = 1$ and $\\Delta _3 = -\\omega (1-\\omega ) (1-\\delta )^2$ .","And Lemma REF reduces to $\\left\\lbrace \\begin{array}{rcl}\\theta &=&\\frac{n^2 \\omega + n \\Delta _2 - \\omega (1-\\omega ) (1-\\delta )^2}{-\\omega (1-\\omega ) (1-\\delta )^2}, \\\\\\end{array}\\rho &=&\\frac{n^2 (1-\\omega )+n\\Delta _4-\\omega (1-\\omega ) (1-\\delta )^2}{-\\omega (1-\\omega ) (1-\\delta )^2},\\right.$ which converge to $-\\infty $ as $n\\rightarrow \\infty $ .","In theory, therefore, for a large but finite $n$ , we need to choose $\\tau $ to be $1-\\omega +\\delta \\omega $ plus a small positive number to ensure that $\\theta >0$ and $\\rho >0$ .","To estimate the small positive number, let us first try the higher-order approximation of $\\tau = 1 - \\omega + \\delta \\omega + \\frac{c}{n},$ for some constant $c > 0$ .","Then $ \\begin{array}{lcl}\\Delta &=&\\frac{c}{n}, \\\\\\end{array}\\Delta _3&=&\\left[\\omega (1-\\delta ) - \\frac{c}{n} \\right]\\left[-(1-\\omega ) (1-\\delta ) - \\frac{c}{n} \\right] \\\\$ = -(1-)(1-)2 + O(1n ).","By the next to the last step in (REF ), $\\mathrm {VAR} \\left(\\sqrt{n} \\tilde{p}_{n,\\omega } \\Bigm \\vert \\tau = 1 - \\omega + \\delta \\omega + \\frac{c}{n} \\right)=\\frac{\\omega (1-\\omega ) c - \\omega ^2 (1-\\omega )^2 (1-\\delta )^2}{n} + O\\left(\\frac{1}{n^2} \\right).$ To minimize the above variance, we let $c = \\omega (1-\\omega )(1-\\delta )^2$ .","Thus, $\\tau = 1 - \\omega +\\delta \\omega + \\frac{\\omega (1-\\omega )(1-\\delta )^2}{n}$ is a higher-order approximation for $1-\\omega +\\delta \\omega $ .","Further high-order approximations, if necessary, could be found similarly.","When $\\tau = 1 - \\omega +\\delta \\omega + \\frac{c}{n}$ , let us use Eq.","(REF ) to calculate $ \\begin{array}{rcl}n\\Delta +\\Delta _3&=&c + \\left[\\omega (1-\\delta ) - \\frac{c}{n} \\right] \\left[-(1-\\omega ) (1-\\delta ) - \\frac{c}{n} \\right] \\\\\\end{array}&=&\\frac{(1-2\\omega ) (1-\\delta ) c}{n} + \\frac{c^2}{n^2}$ which is negative when $\\omega >.5$ and $n\\ge \\left| \\frac{\\omega (1-\\omega )(1-\\delta )}{1-2\\omega }\\right|$ .","As the numerators of $\\theta $ and $\\rho $ in Lemma REF are both positive for a large $n$ , $\\theta $ and $\\rho $ are negative when $n$ is large and $\\omega >.5$ .","Let us make another try at $\\tau = 1-\\omega +\\delta \\omega +\\frac{2c}{n}$ .","Similar to Eq.","(REF ), $\\begin{array}{rcl}n\\Delta +\\Delta _3&=&2c + \\left[\\omega (1-\\delta ) - \\frac{2c}{n} \\right] \\left[-(1-\\omega ) (1-\\delta ) - \\frac{2c}{n} \\right] \\\\\\end{array}&=&c + \\frac{2(1-2\\omega ) (1-\\delta ) c}{n} + \\frac{4c^2}{n^2}.$ $In this case, both $ $ and $$ are larger than $ 1n$ when $ n$ is large.Note from the last step of Eq.","(\\ref {eq:asym_var}) that $ VAR (n pn, )$ is an increasing function of $$ when $ n$ is large; thus,the small positive number to be added to $ 1-+$ is between $ cn$and $ 2cn$.","In practice, however,as the number is too small for a large $ n$,there is no necessity to exactly calculate it and add it to $ 1-+$.$ Let $\\tilde{\\tau }_n = \\mathop {\\mathrm {argmin}}\\limits _{\\tau } \\mathrm {VAR} \\left(\\sqrt{n} \\tilde{p}_{n,\\omega } \\bigm \\vert \\theta , \\rho \\ge \\frac{1}{n} \\right)$ .","We add the restriction $\\theta , \\rho \\ge \\frac{1}{n}$ to ensure the existence of $\\tilde{\\tau }_n$ .","As $\\mathrm {VAR} \\left(\\sqrt{n} \\tilde{p}_{n,\\omega } \\right)$ is an increasing function of $\\tau $ when $n$ is large, $\\tilde{\\tau }_n$ is unique and less than $1 - \\omega + \\delta \\omega + \\frac{2c}{n}$ when $n$ is large.","By Eq.","(REF ), $\\mathrm {VAR}(\\sqrt{n} \\tilde{p}_{n,\\omega } \\Bigm \\vert \\tau = \\tilde{\\tau }_n) = \\omega (1-\\omega ) (\\omega + \\tilde{\\tau }_n-\\delta \\omega -1) + O \\left(\\frac{1}{n} \\right).$ Finally, we use the relation $0\\le \\mathrm {VAR}\\left(\\sqrt{n} \\tilde{p}_{n,\\omega } \\Bigm \\vert \\tau = \\tilde{\\tau }_n \\right)\\le \\mathrm {VAR} \\left(\\sqrt{n} \\tilde{p}_{n,\\omega } \\Bigm \\vert \\tau = 1 - \\omega + \\delta \\omega + \\frac{2c}{n} \\right)$ to get $0\\le \\omega (1-\\omega ) (\\omega + \\tilde{\\tau }_n-\\delta \\omega -1) + O \\left(\\frac{1}{n} \\right)\\le O \\left(\\frac{1}{n} \\right).$ Letting $n \\rightarrow \\infty $ in the above inequality, we get $\\omega (1-\\omega ) (\\omega + \\tilde{\\tau }_n-\\delta \\omega -1) = O(\\frac{1}{n})$ , i.e., $\\lim \\limits _{n\\rightarrow \\infty } \\mathop {\\mathrm {argmin}}\\limits _{\\tau } \\mathrm {VAR} \\left(\\sqrt{n} \\tilde{p}_{n,\\omega } \\bigm \\vert \\theta , \\rho \\ge \\frac{1}{n} \\right) = 1 - \\omega + \\delta \\omega $ .","$$ In this proof, we constantly apply the identities in the proof of Lemma REF .","Let $\\mu _n = \\frac{\\theta +s}{\\theta +\\rho +n}$ be the mean of $\\tilde{p}_{n,\\omega }$ , and let $\\sigma _n^2$ be the variance of $\\tilde{p}_{n,\\omega }$ .","The lower semivariance of $\\tilde{p}_{n,\\omega }$ is calculated as $\\sigma ^2_{n-} = \\int _0^{\\mu _n} (x - \\mu _n)^2 \\frac{x^{\\theta +s-1}(1-x)^{\\rho +n-s-1}}{\\beta ( \\theta +s, \\rho +n-s)} \\mathrm {d} x.$ We apply Chebychev's inequality in terms of the lower semivariance (e.g., Berck and Hihn 1982) to get $\\mathbb {P} \\left(\\tilde{p}_{n,\\omega } \\le \\mu _n - a_n \\sigma _{n-} \\right)\\le \\frac{1}{a_n^2}, \\quad \\forall \\ a_n>0.$ By the proof of Theorem REF , $\\tilde{p}_{n,\\omega }$ has the variance $\\sigma _n^2 = \\frac{\\omega (1-\\omega )\\Delta }{n}+O(\\frac{1}{n^2})$ .","We let $a_n = \\frac{\\sqrt{\\frac{\\omega (1-\\omega )\\Delta }{n}}}{\\sigma _{n-}}$ , then $\\mathbb {P} \\left(\\tilde{p}_{n,\\omega } \\le \\mu _n - \\sqrt{\\frac{\\omega (1-\\omega )\\Delta }{n}} \\right)\\le \\frac{n\\sigma ^2_{n-}}{\\omega (1-\\omega )\\Delta }.\\qquad \\mathrm {(A.6)}$ Let $\\kappa _n = \\frac{\\theta +s-1}{\\theta +\\rho +n-2}$ and $\\varepsilon _n$ be the mode and median of $\\tilde{p}_{n,\\omega }$ , respectively.","The mode $\\kappa _n$ maximizes the density function $\\frac{x^{\\theta +s-1}(1-x)^{\\rho +n-s-1}}{\\beta (\\theta +s, \\rho +n-s)}, 0 < x < 1$ .","As the median lies between the mean and mode, we have $\\begin{array}{rcl}|\\mu _n - \\varepsilon _n|&\\le &|\\mu _n - \\kappa _n |=\\left| \\frac{\\theta +s}{\\theta +\\rho +n} - \\frac{\\theta +s-1}{\\theta +\\rho +n-2} \\right| \\\\\\end{array}&=&\\left| \\frac{n+\\rho -\\theta -2s}{(\\theta +\\rho +n)(\\theta +\\rho +n-2)} \\right| \\\\$ | +s(++n)(++n-2) | + | n+-s(++n)(++n-2) | = O(1n).","$In the following lower-bound estimation of Eq.","(\\ref {eq:lower1se}), we useGamma function, denoted by $ ()$,and its Stirling^{\\prime }s approximation $ (z+1)2z (z(1) )z$.$$\\begin{array}{rcl}&&\\mathbb {P} \\left(\\tilde{p}_{n,\\omega } \\le \\mu _n - \\sqrt{\\frac{\\omega (1-\\omega )\\Delta }{n}} \\right) \\\\\\end{array}&=&\\mathbb {P} \\left(\\tilde{p}_{n,\\omega } \\le \\varepsilon _n \\right)- \\int \\limits _{\\mu _n - \\sqrt{\\frac{\\omega (1-\\omega )\\Delta }{n}}}^{\\varepsilon _n}\\frac{x^{\\theta +s-1}(1-x)^{\\rho +n-s-1}}{\\beta (\\theta +s, \\rho +n-s)} \\mathrm {d} x \\\\$ 12 - | n-(n- (1-)n ) | n+s-1(1-n)+n-s-1(+s,+n-s) = 12 - | (1-)n + O(1n) | (+s-1++n-2)+s-1 (1-+s-1++n-2)+n-s-1(+s) (+n-s)(++n) = 12 - | (1-)n + O(1n) | (+s-1)+s-1(+s) (+n-s-1)+n-s-1(+n-s)(++n-2)++n-2(++n-1) (++n-1) 12 - (1-)n (+s-1)2(+s-1) (+n-s-1)2(+n-s-1) (++n-2)(++n-1) 2(++n-2) = 12 - (1-)n (++n-1) ++n-22(+s-1)(+n-s-1) 12 - (1-)n n n 2n n(1-) = 12 - 12.","$$ Finally, we re-write Eq.","(REF ) as $\\left(\\frac{1}{2}-\\frac{1}{\\sqrt{2\\pi }} \\right)\\omega (1-\\omega )\\Delta + O \\left(\\frac{1}{\\sqrt{n}} \\right)\\le n \\sigma ^2_{n-}\\le n \\sigma _n^2=\\omega (1-\\omega )\\Delta + O \\left(\\frac{1}{n} \\right).$ Letting $n\\rightarrow \\infty $ , we get $\\left(\\frac{1}{2}-\\frac{1}{\\sqrt{2\\pi }} \\right)\\omega (1-\\omega )\\Delta \\le \\liminf \\limits _{n\\rightarrow \\infty } n\\sigma _{n-}^2\\le \\limsup \\limits _{n\\rightarrow \\infty } n\\sigma _{n-}^2\\le \\omega (1-\\omega )\\Delta .$ Therefore, $\\Delta =0$ minimizes the limit of lower semivariance of $\\sqrt{n} \\tilde{p}_{n,\\omega }$ .","We can apply similar arguments to the upper semivariance of $\\sqrt{n} \\tilde{p}_{n,\\omega }$ .","$$ Note that $\\Delta _6-\\omega \\Delta _8 = (1-\\tau )(2\\omega -1)+\\omega ^2 (\\delta -1).$ By the proof of Lemma REF , $\\begin{array}{rcl}\\mathbb {E} [\\tilde{p}_{n,\\omega }]&=&\\frac{\\theta +s}{\\theta +\\rho +n}=\\frac{n^2\\omega + n \\Delta _6 + \\Delta _3}{n^2 + n\\Delta _8 + 2\\Delta _3}\\\\\\end{array}&=&\\omega + \\frac{n (\\Delta _6-\\omega \\Delta _8) + (1-2\\omega ) \\Delta _3}{n^2 + n\\Delta _8 + 2\\Delta _3} \\\\$ = + (1-)(2-1)+2 (-1)n + O(1n2).","$$ In the above approximation, the mean reacts negatively with an increasing $\\tau $ , when $n$ is finitely large and $\\omega >.5$ .","To maximize the mean, we thus minimize $\\tau \\in (1-\\omega +\\delta \\omega )$ such that $\\theta >0$ and $\\rho >0$ .","Particularly, using the proof of Theorem REF , we can make it smaller than $1-\\omega +\\delta \\omega + \\frac{2c}{n}$ when $n$ is large enough, i.e.","$1-\\omega +\\delta \\omega <\\mathop {\\mathrm {argmax}}\\limits _{\\tau } \\mathrm {MEAN}\\left(\\tilde{p}_{n,\\omega } \\bigm \\vert \\theta ,\\rho \\ge \\frac{1}{n} \\right)\\le 1-\\omega +\\delta \\omega + \\frac{2c}{n}.$ Finally, let $n\\rightarrow \\infty $ in the above inequalities to get $\\lim \\limits _{n\\rightarrow \\infty } \\mathop {\\mathrm {argmax}}\\limits _{\\tau } \\mathrm {MEAN}\\left(\\tilde{p}_{n,\\omega } \\bigm \\vert \\theta ,\\rho \\ge \\frac{1}{n} \\right)=1 -\\omega +\\delta \\omega .$ for any $\\omega \\in (.5,1)$ .","$$ When $n$ is large, $\\theta , \\rho \\ge \\frac{1}{n}$ implies $\\tau (\\omega , \\delta ) \\in (1-\\omega +\\delta \\omega , 1)$ .","By the proof of Lemma REF , both $\\theta +s\\rightarrow \\infty $ and $\\rho +n-s\\rightarrow \\infty $ as $n\\rightarrow \\infty $ .","Applying Stirling's formula, Johnson et al.","(1995, page 219) derive the following approximation for the ratio of the variance and the squared MAD around the mean: $\\lim \\limits _{\\theta +s\\rightarrow \\infty , \\rho +n-s \\rightarrow \\infty }\\frac{\\left(\\mathbb {E} \\left[ \\left| \\tilde{p}_{n,\\omega } - \\mathbb {E} (\\tilde{p}_{n,\\omega }) \\right| \\right] \\right)^2}{ \\mathrm {VAR} (\\tilde{p}_{n,\\omega })}= \\frac{2}{\\pi }.$ Thus, minimizing the MAD around the mean is equivalent to minimizing the variance of $\\tilde{p}_{n,\\omega }$ when $n$ is large.","By Theorem REF , we have proved Theorem REF .","$$ For any $i, j\\in \\mathbb {N}$ and $i\\ne j$ , if $i$ uniformly outperforms $j$ and they have equal employment opportunity, then $\\begin{array}{rcl}\\gamma _j[v]&=&\\sum \\limits _{T\\subseteq \\mathbb {N}: j\\in T}\\mathbb {P} (\\mathbf {S}=T) [v(T) - v(T\\setminus \\overline{j})] \\\\\\end{array}&=&\\sum \\limits _{T\\subseteq \\mathbb {N}: j\\in T, i\\in T}\\mathbb {P} (\\mathbf {S}=T) [v(T) - v(T\\setminus \\overline{j})] \\\\&&+\\sum \\limits _{T\\subseteq \\mathbb {N}: j\\in T, i\\notin T}\\mathbb {P} (\\mathbf {S}=T) [v(T) - v(T\\setminus \\overline{j})] \\\\$ TN: jT, iT P (S=T) [v(T) - v(Ti)] + TN: jT, iT P (S=T) [v(T) - v(Tj)] Z=Tj= TN: jT, iT P (S=T) [v(T) - v(Ti)] + ZN: jZ, iZ P (S=Zj) [v(Z j) - v(Z)] TN: jT, iT P (S=T) [v(T) - v(Ti)] + ZN: jZ, iZ P (S=Zi) [v(Z i) - v(Z)] T =Zi = TN: jT, iT P (S=T) [v(T) - v(Ti)] + TN: jT, iT P (S=T) [v(T) - v(Ti)] = TN: iT P (S=T) [v(T) - v(Ti)] = i[v].","$$ Neither the Beta-Binomial distribution nor Eq.","(REF ) is required in the proof.","Also, the equality of employment opportunity is not required for other players in $\\mathbb {N}$ , except that $\\mathbb {P} (\\mathbf {S}=Z\\cup \\overline{i})=\\mathbb {P} (\\mathbf {S}=Z\\cup \\overline{j})$ for any $Z\\subseteq \\mathbb {N} \\setminus \\overline{i} \\setminus \\overline{j}$ .","This identity implies that $i$ and $j$ have equal chance to be hired by $Z$ , when both are unemployed.","Similar arguments can be used to prove $\\lambda _j[v] \\le \\lambda _i[v]$ .","In this case, we use the identity $\\mathbb {P} (\\mathbf {S}=Z\\setminus \\overline{i})=\\mathbb {P} (\\mathbf {S}=Z\\setminus \\overline{j})$ for any $Z\\subseteq \\mathbb {N}$ such that $i,j \\in Z$ .","This identity implies both $i$ and $j$ have equal opportunity to be laid off from $Z$ , when both are employed in $Z$ .","$$ We distribute $v(\\mathbf {S})$ to $\\mathbb {N}$ : $(1-\\tau ) v(\\mathbf {S})$ to $\\mathbf {S}$ and $(\\tau -\\delta )v(\\mathbf {S})$ to $\\mathbb {N}\\setminus \\mathbf {S}$ .","When the employed individuals are symmetric in $v$ , Theorem REF claims that each employed person receives $\\frac{(1-\\tau ) v(\\mathbf {S})}{|\\mathbf {S}| }$ as his or her employment benefits.","Similarly, any unemployed person receives $\\frac{(\\tau -\\delta ) v(\\mathbf {S})}{n - |\\mathbf {S}| }$ as his or her unemployment welfare.","Finally, $\\frac{(1-\\tau ) v(\\mathbf {S})}{|\\mathbf {S}| }= \\frac{(\\tau -\\delta ) v(\\mathbf {S})}{n - |\\mathbf {S}| }$ is equivalent to $\\frac{(1-\\tau ) }{\\omega }= \\frac{(\\tau -\\delta )}{1 - \\omega },$ which itself is equivalent to $\\tau = 1 - \\omega +\\delta \\omega .$ $$"]],"string":"[\n [\n \"A Dichotomous Analysis of Unemployment Welfare\"\n ],\n [\n \"Abstract In an economy which could not accommodate the full employment of its labor force, it employs some labor but does not employ others.\",\n \"The bipartition of the labor force is random, and we characterize it by an axiom of equal employment opportunity.\",\n \"We value each employed individual by his or her marginal contribution to the production function; we also value each unemployed individual by the potential marginal contribution the person would make if the market hired the individual.\",\n \"We then use the aggregate individual value to distribute the net production to the unemployment welfare and the employment benefits.\",\n \"Using real-time balanced-budget rule as a constraint and policy stability as an objective, we derive a scientific formula which describes a fair, debt-free, and asymptotic risk-free tax rate for any given unemployment rate and national spending level.\",\n \"The tax rate minimizes the asymptotic mean, variance, semi-variance, and mean absolute deviation of the underlying posterior unemployment rate.\",\n \"The allocation rule stimulates employment and boosts productivity.\",\n \"Under some symmetry assumptions, we even find that an unemployed person should enjoy equivalent employment benefits, and the tax rate goes with this welfare equality.\",\n \"The tool employed is the cooperative game theory in which we assume many players.\",\n \"The players are randomly bipartitioned, and the payoff varies with the partition.\",\n \"One could apply the fair distribution rule and valuation approach to other profit-sharing or cost-sharing situations with these characteristics.\",\n \"This framework is open to alternative identification strategies and other forms of equal opportunity axiom.\"\n ],\n [\n \"INTRODUCTION\",\n \"The problem with which we are concerned relates to the following typical situation: consider an economy which could not achieve full employment of its labor force, and therefore some people are employed, and others are not.\",\n \"As the employed receives wages and employment benefits (e.g., pension, health insurance, social security, education allowances, paid vacation), should the unemployed receive some unemployment welfare?\",\n \"If YES, how much is fair?\",\n \"In a specific jurisdiction system, the term “unemployment welfare\\\" here may also mean “unemployment benefits,\\\" “unemployment insurance,\\\" or even “unemployment compensation.\\\"\",\n \"In an advanced economy, the answer to the first question is likely YES.\",\n \"This paper answers the second question by justifying a fair share of unemployment welfare for the unemployed and deriving a fair tax rate for the employed.\",\n \"Fair unemployment welfare and a fair tax rate are among the most fundamental topics of our society.\",\n \"The fair-division problem arises in various real-world settings.\",\n \"For a simple motivating example, let us consider a $k$ -out-of-$n$ redundant system in engineering which has $n$ identical components, any $k$ of which being in good condition makes the system work properly.\",\n \"When valuing the importance of each component (either working or standby), one may intuitively claim that these components should be equally valued.\",\n \"A very similar situation occurs in a simple majority voting where not all the voters would support the proposal to vote; thus, the proposal could be passed or failed.\",\n \"Nevertheless, voters are supposed to have the same voting power no matter what they support and for whom they vote.\",\n \"For another example, in the health insurance industry, not all of the policyholders are ill and use the insurance to cover their medical expenses.\",\n \"The question is how to fairly share the total medical cost among both the ill policyholders and the non-ill ones.\",\n \"In a labor market, we have a similar, but more complicated situation: on the one hand, the market could not hire all of its labor force even though everyone in the market would like to be employed; on the other hand, the participants in the market have heterogeneous performance in the production.\",\n \"There are four common features in these examples: a coalition of players with cooperative nature, a random bipartition of the players, a payoff associated with the partition, and an objective to share the payoff with all the players.\",\n \"This paper derives a solution for situations with these features.\",\n \"In the $k$ -out-of-$n$ redundant system and the simple majority voting, we expect equality of outcome.\",\n \"We face a few challenges to deal with when fairly distributing the welfare and benefits, both generated by the employed.\",\n \"First of all, fairness may be an abstract but vague concept.\",\n \"We believe that “fairness\\\" is bound with the equality of employment opportunity, not with the equality of outcome, nor with the equality of productivity.\",\n \"Furthermore, we also believe that everyone in the labor market could contribute in some way; but the opportunities are limited.\",\n \"Thus, unemployment is not a fault of the unemployed, nor a flaw of the labor market, but a self-adjustment mechanism toward the efficiency of the market.\",\n \"Secondly, we attempt to apply a taxation policy to the labor market which operates in an ever-changing economy and with ever-changing productivity.\",\n \"For it to be useful, the tax and division rule should be not only fair to all people but also be able to cope with the uncertainty and sustainability in generating and distributing the net production.\",\n \"Ideally, it should balance the account of value generated and that of value paid in each employment contingency.\",\n \"Thirdly, in a perfect world, a fair tax rate should depend only on observed data, to avoid any excessively political bargaining and costly strategic voting.\",\n \"One major issue, however, is the non-observability of the heterogeneous-agent production function in all employment scenarios at all the time.\",\n \"Another data issue is the desynchronization between the unemployment rate and the tax rate; the former is high-frequency data while the latter has a lower frequency.\",\n \"Often in a yearly time-frame, policymakers determine the tax rate after observing most monthly unemployment rates.\",\n \"Vast literature (e.g., Kornhauser 1995; Fleurbaey and Maniquet 2006) from various aspects has studied the fairness in taxation and unemployment payments.\",\n \"In particular, Shapley (1953) proposes an influential axiom of fairness to develop a fair-division method, called the Shapley value, which is widely used in distributing employment compensation and welfare (see, for example, Moulin 2004; Devicienti 2010; Giorgi and Guandalini 2018; Krawczyk and Platkowski 2018).\",\n \"Beneath the pillars of the Shapley value and the Shapley axiom, however, are two underlying assumptions: players' unanimous participation in the production, and distributor's complete information about the production function.\",\n \"Recently, Hu (2002, 2006, 2018) relax the unanimity assumption and generalize the Shapley value, using some non-informative probability distributions for the dichotomy or bipartition of the players.\",\n \"In particular, Hu (2018) proposes using a Beta-Binomial distribution to address the equality of opportunity.\",\n \"This current research also capitalizes on the Beta-Binomial distributions.\",\n \"Furthermore, we do not assume the complete information about the production function.\",\n \"We only need its value at one observation which does occur.\",\n \"The advantage of our approach is twofold.\",\n \"On the one hand, we provide a game-theoretical micro-foundation for a fair-division solution to distribute the unemployment welfare and employment benefits.\",\n \"One can apply the solution concept to many similar situations without substantive alternations, and one may also extend the framework using other identification schemes, rather than the tax policy stability or unemployment rate minimization, detailed in Section .\",\n \"On the other hand, the fair tax rate we provide is simple enough to be used in practice.\",\n \"It relies only on the unemployment rate and a reserved portion of production, which is not for personal use.\",\n \"The total unemployment welfare depends only on the tax rate and the observed production.\",\n \"We attempt to immunize our solution from any unnecessary randomness, hypotheticals, ambiguity, and latency.\",\n \"These include, but are not limited to, the competitive and cooperative features of the labor market, endogenous employment search behavior, non-linear schedule of tax rates, the exact sizes of the labor market and time-varying unemployment population.\",\n \"With this simplicity in hand, a certain level of abstraction is necessary and any application of the fair solution should accommodate to the concrete reality.\",\n \"We organize the remainder of the paper as follows.\",\n \"Section applies the framework of dichotomous valuation (or simply, “D-value\\\") in Hu (2002, 2006, 2018) to value each person in the labor market, assuming equal employment opportunity.\",\n \"The two sides of D-value depend on two unknown parameters and are aggregated separately for the unemployed and the employed labor.\",\n \"Next, Section formulates a set of fair divisions of the net production, using the aggregate components of D-value.\",\n \"In this section, we base the set of fair tax rates on two accounting identities for a balanced budget.\",\n \"In Section , by maximizing the stability of the tax rate or minimizing the expected posterior unemployment rate, we identify a specific relation between the fair tax rate and the unemployment rate.\",\n \"The particular solution is robust under a few other criteria.\",\n \"Section lists three other applications (namely, voting power, health insurance, and highway toll) of the distribution rule, and Section suggests several ways to extend this framework.\",\n \"The account is self-contained, and the proofs are in the Appendix.\"\n ],\n [\n \"DICHOTOMOUS VALUATION\",\n \"Before our formal discussions, let us introduce a few notations.\",\n \"For a general economy, we assume that its labor force consists of the employed labor and the unemployed labor, ignoring any part-time labor.\",\n \"Let $\\\\mathbb {N} = \\\\lbrace 1, 2,\\\\cdots ,n\\\\rbrace $ denote the set of people in the labor force, indexed as $1,2,..., n$ , and let $\\\\mathbf {S} \\\\subseteq \\\\mathbb {N}$ denote the random subset of the employed labor in $\\\\mathbb {N}$ .\",\n \"For any subset $T$ of $\\\\mathbb {N}$ , let $|T|$ denote its cardinality; for notational simplicity, we often use $n$ for $|\\\\mathbb {N}|$ , $t$ for $|T|$ , and $s$ for $|\\\\mathbf {S}|$ .\",\n \"Let us also write the employment rate as $\\\\omega =\\\\frac{s}{n}$ , which is one minus the unemployment rate.\",\n \"Besides, we employ the vinculum (overbar) in naming a singleton set; for example, “$\\\\overline{i}$ \\\" is for the singleton set $\\\\lbrace i \\\\rbrace $ .\",\n \"Also, we use “$\\\\setminus $ \\\" for set subtraction, “$\\\\cup $ \\\" for set union, and $\\\\beta (\\\\cdot , \\\\cdot )$ for a two-parameter Beta function.\",\n \"The Appendix defines $\\\\Delta $ , $\\\\Delta _1, \\\\cdots , \\\\Delta _9$ as shorthand notations.\"\n ],\n [\n \"Equality of Employment Opportunity\",\n \"Equal employment opportunity is widely acknowledged and is the starting point or axiom for us to study fairness and welfare.\",\n \"In the United States, for example, equal employment opportunity has been enacted to prohibit federal contractors from discriminating against employees by race, sex, creed, religion, color, or national origin since President Lyndon Johnson signed Executive Order 11246 in 1965.\",\n \"In the literatureBesides the justification from equality of opportunity, unemployment welfare could also come from other considerations (e.g., Sandmo 1998, Tzannatos and Roddis 1998, Vodopivec 2004).\",\n \"These include, but are not limited to, social protection, insurance of income flow, poverty prevention, the efficiency of the labor market, and political considerations.\",\n \"In this paper, we focus solely on equal employment opportunity., there are abundant qualitative descriptions, informal or formal, about the equal opportunity (e.g., Friedman and Friedman 1990; Roemer 1998; Rawls 1999).\",\n \"In this section, we introduce a quantitative and probabilistic version of equal opportunity whereby the employment opportunity is assumed equitable for all persons in the labor force.\",\n \"We assume three layers of uncertainty for the random subset $\\\\mathbf {S}$ while maintaining the equality of employment opportunity.\",\n \"In the first layer, the employment size $|\\\\mathbf {S}|$ follows a binomial distribution with parameters $(n, p)$ .\",\n \"When independence is assumed, $p$ is the probability of any given person being employed.\",\n \"In the second layer, the unknown parameter $p$ has a prior Beta distribution with hyperparameters $(\\\\theta , \\\\rho )$ , where $\\\\theta >0$ and $\\\\rho >0$ are to be determined by estimation or specification.\",\n \"Thus, the joint probability density of $p\\\\in (0,1)$ and $|\\\\mathbf {S}|=s$ is $ \\\\frac{p^{\\\\theta -1}(1-p)^{\\\\rho -1}}{\\\\beta (\\\\theta , \\\\rho )}\\\\left(\\\\begin{array}{c}n \\\\\\\\s\\\\end{array}\\\\right)p^s (1-p)^{n-s}=\\\\frac{n!}{s!(n-s)!\",\n \"}\\\\frac{p^{\\\\theta +s-1}(1-p)^{\\\\rho +n-s-1}}{\\\\beta (\\\\theta , \\\\rho )}.$ Eq.\",\n \"(REF ) implies the following marginal probability density for $|\\\\mathbf {S}|=s$ : $ \\\\begin{array}{rcl}\\\\mathbb {P}(|\\\\mathbf {S}| = s)& = &\\\\mathop {{\\\\int _0^1}}\\\\frac{n!}{s!(n-s)!\",\n \"}\\\\frac{p^{\\\\theta +s-1}(1-p)^{\\\\rho +n-s-1}}{\\\\beta (\\\\theta , \\\\rho )} \\\\mathrm {d} p \\\\\\\\&=&\\\\frac{n!}{s!(n-s)!\",\n \"}\\\\frac{\\\\beta (\\\\theta +s, \\\\rho +n-s)}{\\\\beta (\\\\theta , \\\\rho )}, \\\\hspace{28.45274pt} s=0,1,\\\\cdots ,n.\\\\end{array}$ In the third layer, for any given employment size $s$ , all subsets of size $s$ have the same probability of being $\\\\mathbf {S}$ .\",\n \"As there are $\\\\frac{n!}{s!(n-s)!\",\n \"}$ subsets of size $s$ in $\\\\mathbb {N}$ , the probability for the employment scenario $\\\\mathbf {S}=T$ is $ \\\\mathbb {P}(\\\\mathbf {S}=T)=\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\frac{\\\\mathbb {P}(|\\\\mathbf {S}| = s)}{\\\\frac{n!}{s!(n-s)!}}\",\n \"= \\\\frac{\\\\beta (\\\\theta +s,\\\\rho +n-s)}{\\\\beta (\\\\theta ,\\\\rho )},\\\\quad &\\\\mathrm {if} \\\\ t=s;\\\\\\\\\\\\end{array}0, & \\\\mathrm {otherwise}.\\\\right.\",\n \"\\\\\\\\$ Clearly, the equality of employment opportunity is implied in the assumed triple-layered uncertainty.\",\n \"Furthermore, equal opportunity is also assumed for all coalitions of the same size.\",\n \"By Eq.\",\n \"(REF ) and Eq.\",\n \"(REF ), the posterior density function of $p$ given $|\\\\mathbf {S}|=s$ is $\\\\frac{\\\\frac{n!}{s!(n-s)!\",\n \"}\\\\frac{p^{\\\\theta +s-1}(1-p)^{\\\\rho +n-s-1}}{\\\\beta (\\\\theta , \\\\rho )}}{\\\\frac{n!}{s!(n-s)!\",\n \"}\\\\frac{\\\\beta (\\\\theta +s, \\\\rho +n-s)}{\\\\beta (\\\\theta , \\\\rho )}}=\\\\frac{p^{\\\\theta +s-1}(1-p)^{\\\\rho +n-s-1}}{\\\\beta (\\\\theta +s, \\\\rho +n-s)}.$ Thus, the posterior employment rate follows a Beta distribution with parameters $(\\\\theta +s, \\\\rho +n-s)$ .\",\n \"In the following, let us use $\\\\tilde{p}_{n,\\\\omega }$ to denote the posterior employment rate given the observance of $|\\\\mathbf {S}|=n \\\\omega $ .\",\n \"In contrast, $p$ is the unobservable prior employment rate, and $\\\\omega $ is the observable employment rate.\"\n ],\n [\n \"Value of the Employed and the Unemployed\",\n \"For any $T\\\\subseteq \\\\mathbb {N}$ , we use a heterogeneous-agent production function $v(T)$ to measure the aggregate productivity when $\\\\mathbf {S}=T$ .\",\n \"We assume the net-profit production function $v(T)$ excludes the labor cost which compensates the time and efforts devoted by the employed labor in producing $v(T)$ .\",\n \"To isolate the added value by the labor force alone, we also assume that $v(T)$ excludes the cost of consumed physical and financial resources.\",\n \"Both the labor cost and the resource cost are exempt from taxation in a firm.\",\n \"Thus, without loss of generality, we may assume that $v(\\\\emptyset )=0$ for the empty set $\\\\emptyset $ .\",\n \"However, $v(T)$ does not necessarily increase with $T$ nor with its size $|T|$ .\",\n \"To retain its labor or to minimize its labor turnover, a firm would share part of its net profit with its employees.\",\n \"To keep things simple, we use the term “employment benefits\\\" to denote the employees' profit-sharing part in $v(T)$ , in contrast to the term “unemployment welfare.\\\"\",\n \"Let us formally introduce two components of the D-value.\",\n \"For any $i \\\\in \\\\mathbb {N}$ , to analyze its marginal effect on the value generating process, we consider two jointly exhaustive and mutually exclusive events: Event 1: $i \\\\in {\\\\mathbf {S}}$ .\",\n \"Then, $i$ 's marginal effect is $v({\\\\mathbf {S}}) - v( {\\\\mathbf {S}} \\\\setminus \\\\overline{i})$ , called marginal gain, in that he or she contributes $v({\\\\mathbf {S}}) - v( {\\\\mathbf {S}} \\\\setminus \\\\overline{i})$ to the production, due to his or her existence in ${\\\\mathbf {S}}$ .\",\n \"The expected marginal gain is $ \\\\gamma _i[v] \\\\stackrel{\\\\mbox{def}}{=} \\\\mathbb {E} \\\\left[v(\\\\mathbf {S}) - v(\\\\mathbf {S} \\\\setminus \\\\overline{i}) \\\\right].$ In the above, “def\\\" is for definition and “$\\\\mathbb {E}$ \\\" for expectation under the probability distribution specified by Eq.\",\n \"(REF ).\",\n \"Event 2: $i \\\\notin {\\\\mathbf {S}}$ .\",\n \"Then, $i$ 's marginal effect is $v({\\\\mathbf {S}}\\\\cup \\\\overline{i}) - v({\\\\mathbf {S}})$ in that ${\\\\mathbf {S}}$ faces a marginal loss $v({\\\\mathbf {S}}\\\\cup \\\\overline{i}) - v({\\\\mathbf {S}})$ , due to $i$ 's absence from the employed labor force ${\\\\mathbf {S}}$ .\",\n \"In other words, the person would increase the production by $v({\\\\mathbf {S}}\\\\cup \\\\overline{i} ) - v({\\\\mathbf {S}})$ if the market included him or her in $\\\\mathbf {S}$ .\",\n \"The expected marginal loss is then $\\\\lambda _i[v] \\\\stackrel{\\\\mbox{def}}{=}\\\\mathbb {E} \\\\left[v({\\\\mathbf {S}}\\\\cup \\\\overline{i}) - v({\\\\mathbf {S}}) \\\\right].$ We let $\\\\gamma _i[v]$ be the employment benefits $i$ receives when he or she is employed, and let $\\\\lambda _i[v]$ be the unemployment welfare $i$ receives when he or she is unemployed.\",\n \"Note that, even if $i$ always remains employed, both $\\\\mathbf {S}$ and $\\\\mathbf {S}\\\\setminus \\\\overline{i}$ change daily, if not hourly; thus $i$ 's marginal gain is not a constant.\",\n \"Similarly, even if $i$ remains unemployed for a while, $\\\\mathbf {S}$ , $\\\\mathbf {S}\\\\cup \\\\overline{i}$ , and $i$ 's marginal loss are not constant.\",\n \"To account for this uncertainty, we have already taken expectations in Eq.\",\n \"(REF ) and Eq.\",\n \"(REF ) when defining $\\\\gamma _i[v]$ and $\\\\lambda _i[v]$ .\",\n \"A few key points worth mentioning to help us understand the profit-sharing strategy.\",\n \"First, in addition to receiving employment benefits, the employed labor also receives the reimbursement for labor cost, which compensates its human capital usage in generating $v(\\\\mathbf {S})$ .\",\n \"Human capital also accumulates in prior employment and pre-employment education.\",\n \"Labor, physical, and financial costs are not part of the generated value $v(\\\\mathbf {S})$ .\",\n \"The unemployed labor, however, only receives unemployment welfare.\",\n \"Secondly, if $i \\\\in \\\\mathbf {S}$ , then $v(\\\\mathbf {S} \\\\setminus \\\\overline{i})$ is not observable while we observe $v( \\\\mathbf {S})$ .\",\n \"Similarly, when $j \\\\notin \\\\mathbf {S}$ , we cannot observe both $v(\\\\mathbf {S} \\\\cup \\\\overline{j})$ and $v( \\\\mathbf {S})$ simultaneously.\",\n \"Thus, we need to transform the aggregate marginals into observable forms, such as those in Theorem REF .\",\n \"Thirdly, the aggregate employment benefits $\\\\sum \\\\limits _{i\\\\in \\\\mathbf {S}} \\\\left[v(\\\\mathbf {S})-v(\\\\mathbf {S}\\\\setminus \\\\overline{i}) \\\\right]$ is not necessarily equal to $v(\\\\mathbf {S})$ , the value collectively generated by $\\\\mathbf {S}$ .\",\n \"Thus, we distribute some of the surpluses $v(\\\\mathbf {S}) - \\\\sum \\\\limits _{i\\\\in \\\\mathbf {S}}\\\\left[v(\\\\mathbf {S})-v(\\\\mathbf {S}\\\\setminus \\\\overline{i}) \\\\right]$ to the unemployed labor $\\\\mathbb {N}\\\\setminus \\\\mathbf {S}$ .\",\n \"The distribution is not through personal giving but government taxation and unemployment payment systems.\",\n \"This distribution channel also appeals to us for the aggregate benefits and aggregate welfare at the national level, as stated in Theorem REF .\",\n \"The aggregate components of the D-value are $\\\\begin{array}{rcl}\\\\end{array}\\\\sum \\\\limits _{i\\\\in \\\\mathbb {N}} \\\\gamma _i [v]&=&\\\\mathbb {E} \\\\left[ \\\\sum \\\\limits _{i\\\\in \\\\mathbf {S}} \\\\left( v(\\\\mathbf {S}) - v(\\\\mathbf {S}\\\\setminus \\\\overline{i}) \\\\right) \\\\right] \\\\\\\\$ = n(+n,)(, )v(N) + TN: T= N t(+-1)-n+n-t-1 (+t, +n-t)(,)v(T), iN i [v] = E [ i NS (v(Si) - v(S) ) ] = TN:T= t(+-1)-n(-1)+t-1 (+t, +n-t)(,) v(T) - n(,+n)(,)v().\"\n ],\n [\n \"ACCOUNTING IDENTITIES FOR A BALANCED BUDGET\",\n \"By Eq.\",\n \"(REF ), the expected production is $ \\\\mathbb {E} \\\\left[ v(\\\\mathbf {S}) \\\\right]= \\\\sum \\\\limits _{T\\\\subseteq \\\\mathbb {N}} \\\\frac{\\\\beta (\\\\theta +t, \\\\rho +n-t)}{\\\\beta (\\\\theta ,\\\\rho )} v(T).$ Out of the $2^n$ employment scenarios for $\\\\mathbf {S}$ , we observe only one.\",\n \"Let us consider this particular scenario at $\\\\mathbf {S}=T$ , which occurs with probability $\\\\frac{\\\\beta (\\\\theta +t, \\\\rho +n-t)}{\\\\beta (\\\\theta ,\\\\rho )}$ and which generates the value $v(T)$ .\",\n \"After the production, we face a challenge of fairly dividing $v(T)$ between $T$ and $\\\\mathbb {N}\\\\setminus T$ , whom both have an entitlement to $v(T)$ .\",\n \"Our division rule should fully respect the entitlement claim: each employed person receives his or her expected marginal gain, and each unemployed person receives his or her expected marginal loss.\",\n \"As a contrast, the Shapley value distributes $v(\\\\mathbb {N})$ to all players in $\\\\mathbb {N}$ ; Hu(2018) formulates solutions to divide $\\\\mathbb {E} \\\\left[ v(\\\\mathbf {S}) \\\\right]$ and $v(\\\\mathbb {N})-\\\\mathbb {E} \\\\left[ v(\\\\mathbf {S}) \\\\right]$ .\",\n \"Besides $\\\\gamma _i [v]$ and $\\\\lambda _i [v]$ , we should reserve a portion of $v(T)$ for the common good of the economy and society.\"\n ],\n [\n \"A Real-Time Balanced Budget Rule\",\n \"As noted above, we purposely divide the net production $v(T)$ into three components.\",\n \"The first one is for employment benefits.\",\n \"We compare the coefficients of $v(T)$ in Eq.\",\n \"(REF ) and that of $\\\\mathbb {E} \\\\left[ \\\\sum \\\\limits _{i\\\\in \\\\mathbf {S}} \\\\left(v(\\\\mathbf {S}) - v(\\\\mathbf {S}\\\\setminus \\\\overline{i}) \\\\right) \\\\right]$ in Eq.\",\n \"(REF ), ignoring the probability density for the scenario; the employed labor $T$ should retain $\\\\frac{t (\\\\theta +\\\\rho -1)-n\\\\theta }{\\\\rho +n-t-1} v(T)$ as their employment benefits.\",\n \"The rest, $\\\\left[1 - \\\\frac{t(\\\\theta +\\\\rho -1)-n\\\\theta }{\\\\rho +n-t-1} \\\\right] v(T)$ , pays to the government as “tax\\\"In practice, not all unemployment welfare comes from the taxation system.\",\n \"In the United States, for example, it is funded by a compulsory governmental insurance system, which manages the collection and payment accounts.\",\n \"However, the contribution to and distribution from the insurance system are de facto of a type of payroll tax.\",\n \"In Australia, unemployment welfare, as part of social security benefits, is funded through the taxation system..\",\n \"Besides, we assume both $\\\\lambda _i[v]$ and $\\\\gamma _i[v]$ are tax-exempt in order to avoid any double taxation.\",\n \"Thus, we define the tax rate $\\\\tau $ as $\\\\tau \\\\stackrel{\\\\mathrm {def}}{=} 1 - \\\\frac{s(\\\\theta +\\\\rho -1)-n\\\\theta }{\\\\rho +n-s-1},\\\\qquad s=1,\\\\cdots , n-1.$ For the time being, the rate relies on the labor market size $n$ and the employment rate $\\\\omega = \\\\frac{|\\\\mathbf {S}|}{n}$ , but not on the production $v(\\\\mathbf {S})$ .\",\n \"In this definition, we exclude two extreme but unlikely cases at $\\\\omega =0$ and $\\\\omega =1$ .In reality, the two cases have zero probability to occur, even though the probability model (REF ) gives a very slight chance.\",\n \"Secondly, we compare the coefficients of $v(T)$ in Eq.\",\n \"(REF ) and that of $\\\\mathbb {E} \\\\left[ \\\\sum \\\\limits _{i \\\\in \\\\mathbb {N}\\\\setminus \\\\mathbf {S}} \\\\left(v(\\\\mathbf {S}\\\\cup \\\\overline{i}) - v(\\\\mathbf {S}) \\\\right) \\\\right]$ in Eq.\",\n \"(REF ); the unemployed labor $\\\\mathbb {N}\\\\setminus T$ should claim $\\\\frac{t(\\\\theta +\\\\rho -1) - n(\\\\theta -1)}{\\\\theta +t-1} v(T)$ from the tax revenue $\\\\tau v(T)$ as its unemployment welfare.\",\n \"Thirdly, we assume that a reserved proportion, $\\\\delta v(T)$ , is not individually and not directly distributed to the labor force $\\\\mathbb {N}$ .\",\n \"As a consequence, the tax rate $\\\\tau $ includes both the reserve ratio $\\\\delta $ and the proportion to the unemployment welfare, i.e., $ \\\\tau \\\\equiv \\\\delta + \\\\frac{s(\\\\theta +\\\\rho -1)-n(\\\\theta -1)}{\\\\theta +s-1}.$ The reserve $\\\\delta v(T)$ is meant to serve the general public's interests and to have broad appeal, rather than the individual needs.\",\n \"More specifically, $\\\\delta v(T)$ includes, but is not limited to, the payments to the population who are not in the labor force, to the corporate equity earnings not used for employment benefits, to the public administration and national defense, to the public welfare, to the tax deficit from the past, to the future development, and so on.\",\n \"By admitting the corporate equity earnings to the reserve $\\\\delta v(T)$ , we have deliberately ignored firm-specific re-distribution processes of the net corporate earnings, and have purposely avoided the associated corporate income taxation.\",\n \"In practice, government expenditureincluding that in all levels of government but excluding all unemployment welfare payments.\",\n \"and corporate earnings are two major components of the reserve $\\\\delta v(T)$ .\",\n \"When the government has other sources of tax revenue, a pro rata share of its expenditure will come from $\\\\delta v(T)$ .\",\n \"Besides, the government could implement a countercyclical fiscal policy by adjusting its spending level in $\\\\delta v(T)$ such that it counteracts the ratio of the corporate earnings to the production.\",\n \"Indeed, by Eq.\",\n \"(REF ), $\\\\tau $ automatically reacts positively to the change of the corporate earning ratio, other things remaining the same.\",\n \"Put in another way, the real-time balanced budget rule implied in Eq.\",\n \"(REF ) and Eq.\",\n \"(REF ) forbids any borrowing between different labor market scenarios or forbids any inter-temporal borrowing.\",\n \"Thus, this sustainable taxation policy meets the needs of the present market scenario without compromising the ability of future market scenarios to meet their own needs.\",\n \"In practice, however, it is challenging, if not impossible, to enforce or enact the balanced budget rule at the labor market scenario level or on a real-time basis.\",\n \"In the United States, for example, the employment rate $\\\\omega $ changes daily and is recorded monthly by the Bureau of Labor Statistics; as a policy variable, the tax rate $\\\\tau $ changes yearly.\",\n \"Striving for the balanced budget rule to the highest degree, one could minimize the variance of the employment rate $\\\\omega $ within a yearly time frame.\",\n \"In a perfect situation, the employment rate closely follows a degenerate probability distribution and remains almost constant within the year.\",\n \"We discuss the variance minimization in Section .\",\n \"In contrast to the exhaustive distribution of the net production in the national accounts, a household could still maximize its utility through inter-temporal borrowing, saving, lending, and consuming.\"\n ],\n [\n \"Sets of Feasible Solutions\",\n \"Although the terms “tax rate\\\" and “tax rule\\\" have been used loosely and interchangeably about $\\\\tau $ , we must distinguish between them to have an accurate discussion.\",\n \"As a tax rule or tax policy, $\\\\tau $ is a function of $(n, \\\\omega , \\\\delta )$ , subject to the equality of employment opportunity and the balance of budget as specified in Eq.\",\n \"(REF ) and Eq.\",\n \"(REF ).\",\n \"In contrast, as a tax rate, $\\\\tau $ is merely the value of the function at a specific $(n,\\\\omega , \\\\delta )$ .\",\n \"For a given triple of $(n, \\\\omega , \\\\delta )$ , there are three indeterminates $(\\\\theta , \\\\rho , \\\\tau )$ in the system of two equations Eq.\",\n \"(REF ) and Eq.\",\n \"(REF ).\",\n \"Let $\\\\Omega _{n,\\\\omega ,\\\\delta }$ denote the set of all feasible combinations of $(\\\\theta , \\\\rho , \\\\tau )$ which satisfy both Eq.\",\n \"(REF ) and Eq.\",\n \"(REF ): $\\\\Omega _{n,\\\\omega ,\\\\delta }\\\\stackrel{\\\\mathrm {def}}{=} \\\\ \\\\left\\\\lbrace (\\\\theta ,\\\\rho , \\\\tau ) \\\\left|\\\\begin{array}{l}\\\\tau = 1 - \\\\frac{n \\\\omega (\\\\theta +\\\\rho -1)-n\\\\theta }{\\\\rho +n- n \\\\omega -1}, \\\\\\\\\\\\tau = \\\\delta + \\\\frac{n \\\\omega (\\\\theta +\\\\rho -1)-n(\\\\theta -1)}{\\\\theta + n \\\\omega -1}, \\\\\\\\0 \\\\le \\\\tau \\\\le 1,\\\\ \\\\theta >0,\\\\ \\\\rho >0.\\\\end{array}\\\\right.\\\\right\\\\rbrace .$ From now on, a “fair tax rate\\\" could mean a solution of $\\\\tau $ in $\\\\Omega _{n,\\\\omega ,\\\\delta }$ ; or it could be a limit of tax rates which satisfy Eq.\",\n \"(REF ) and Eq.\",\n \"(REF ).\",\n \"In general, for a reasonable $\\\\delta $ , a finite $n$ , and a $\\\\omega \\\\in (0,1)$ , there are infinitely many solutions in $\\\\Omega _{n,\\\\omega ,\\\\delta }$ .\",\n \"In this case, we need one more restriction to solve $(\\\\theta ,\\\\rho , \\\\tau )$ uniquely.\",\n \"To do so, one could capitalize on the statistical relation between $\\\\omega $ and $(\\\\theta , \\\\rho )$ : the prior mean and mode of $\\\\omega $ are $\\\\frac{\\\\theta }{\\\\theta +\\\\rho }$ and $\\\\frac{\\\\theta -1}{\\\\theta +\\\\rho -2}$ , respectively (e.g., Johnson et al.\",\n \"1995, Chapter 21).\",\n \"Alongside this direction, for example, we could set $\\\\frac{\\\\theta }{\\\\theta +\\\\rho }$ (or $\\\\frac{\\\\theta -1}{\\\\theta +\\\\rho -2}$ ) to be the historical average (or mode, respectively) of $\\\\omega $ in the previous year.\",\n \"Alternatively, we could set $\\\\frac{\\\\theta }{\\\\theta +\\\\rho }$ or $\\\\frac{\\\\theta -1}{\\\\theta +\\\\rho -2}$ to be a target employment rate or the natural employment rate.\",\n \"However, this type of identification schemes requires additional input (e.g., historical average, target employment rate, or natural employment rate).\",\n \"Furthermore, to derive a unique solution from $\\\\Omega _{n,\\\\omega ,\\\\delta }$ or its boundary, we should base our deviation on observed data only.\",\n \"One concern in the input $(n, \\\\omega , \\\\delta )$ is the size of the labor market $n$ .\",\n \"Even though the size $n$ is not random in our model of equal employment opportunity, it is likely a time-varying latent variable in that there is no clear cut between entry to and exit from the labor market; and many depressed unemployed persons may not actively seek new positions.\",\n \"In practice, $n$ changes daily whereas $\\\\tau $ most likely changes yearly.\",\n \"No matter how it changes and how much latent it is, however, we are confident that $n$ is a large number in the general economy.\",\n \"Thus, in contrast to Eq.\",\n \"(REF ) and Eq.\",\n \"(REF ), we seek a fair tax rule which is valid for all large $n$ 's but is not specific to a particular $n$ .\",\n \"As a result, the tax rule we are targeting should not involve $n$ , and we may write it as $\\\\tau (\\\\omega , \\\\delta ): (0,1) \\\\times (0,1) \\\\rightarrow (0,1)$ .\",\n \"To visualize the solution set $\\\\Omega _{n,\\\\omega ,\\\\delta }$ , let us write $(\\\\theta ,\\\\rho )$ in terms of $(n, \\\\delta , \\\\tau , \\\\omega )$ using Lemma REF in the Appendix.\",\n \"Figure REF (a-b) plots the feasible solution sets for $n=10000$ , $\\\\delta =.1$ , and any $\\\\omega \\\\in (0,1)$ .\",\n \"Note that there is a straight line which dramatically raises both $\\\\theta $ and $\\\\rho $ .\",\n \"From Figure REF (c-d), we observe that both $\\\\theta $ and $\\\\rho $ drop sharply on the other side of the straight line.\",\n \"As a matter of fact, any point on the other side of the straight line does not represent a fair solution, owing to the positivity requirement of $\\\\theta $ and $\\\\rho $ .\",\n \"Actually, the straight line has a tax rule $\\\\tau (\\\\omega , \\\\delta ) = 1-\\\\omega + \\\\delta \\\\omega $ .\",\n \"This linear relation between $\\\\tau $ and $\\\\omega $ can be seen from Lemma REF : on the one hand, when $\\\\Delta \\\\equiv \\\\omega +\\\\tau -\\\\delta \\\\omega -1 > 0$ , both $\\\\theta =\\\\frac{n^2 \\\\omega \\\\Delta _1 + n\\\\Delta _2 +\\\\Delta _3 }{n\\\\Delta + \\\\Delta _3}$ and $\\\\rho = \\\\frac{n^2 (1-\\\\omega ) \\\\Delta _1 + n\\\\Delta _4 + \\\\Delta _3}{n\\\\Delta + \\\\Delta _3}$ increase to $+\\\\infty $ as $n\\\\rightarrow \\\\infty $ , due to the positivity of $\\\\Delta _1 \\\\equiv \\\\delta \\\\omega -\\\\omega -\\\\tau +2> 0$ ; on the other hand, when $\\\\Delta <0$ , both $\\\\theta $ and $\\\\rho $ decrease to $-\\\\infty $ as $n\\\\rightarrow \\\\infty $ .\",\n \"Thus, the singular line is expressed as $\\\\Delta =0$ , or $\\\\tau (\\\\omega , \\\\delta ) = 1-\\\\omega +\\\\delta \\\\omega $ .\",\n \"Furthermore, for any fair tax rule on the side of $\\\\Delta >0$ , the posterior employment rate $\\\\tilde{p}_{n,\\\\omega }$ has a degenerated limit distribution as $n\\\\rightarrow \\\\infty $ .\",\n \"This is claimed in Theorem REF .\",\n \"The next section answers a related question: which tax rule makes the distribution convergence fastest.\",\n \"The answer happens to be the solution on the singular line.\",\n \"Figure: Solutions of Eq.\",\n \"() and Eq.\",\n \"() for n=10000,δ=.1,ω∈(0,1)n=10000,\\\\ \\\\delta =.1,\\\\ \\\\omega \\\\in (0,1).For any fair tax rule $\\\\tau (\\\\omega , \\\\delta ) \\\\in (1-\\\\omega +\\\\delta \\\\omega ,1)$ , as $n\\\\rightarrow \\\\infty $ , $\\\\tilde{p}_{n,\\\\omega }$ converges in distribution to the degenerate probability distribution with mass at $\\\\omega $ .\"\n ],\n [\n \"AN ASYMPTOTIC RISK-FREE TAX RATE\",\n \"In this section, we derive the limit fair tax rule $\\\\tau (\\\\omega , \\\\delta ) =1-\\\\omega +\\\\delta \\\\omega $ from a few different angles.\",\n \"At a minimum, a good tax rule should not discourage employment incentives and productivity, such as detailed in Theorem REF and REF .\",\n \"On the other end, we expect a good rule to be robust and optimal under multiple criteria.\",\n \"We study five criteria in Theorem REF , REF , REF , REF , and REF , any of which uniquely identifies the solution.\",\n \"They either minimize the employment market risk or maximize the employment expectation, under a given market capacity and a budget balance constraint.\",\n \"First, we should heavily exploit the observed market behavior.\",\n \"While an efficient labor market stimulates productivity $v(\\\\mathbf {S})$ , a higher employment rate does not imply higher productivity and vice versa — the production function $v(\\\\mathbf {S})$ does not necessarily increase with the employment size $|\\\\mathbf {S}|$ .\",\n \"Acemoglu and Shimer (2000) find that a moderate level of unemployment could boost productivity by improving the quality of jobs.\",\n \"Indeed, it fosters peer pressure in producing $v(\\\\mathbf {S})$ , allows workers to move on from declining firms, and enables rising companies and the economy as a whole to optimally respond to external shocks.\",\n \"Therefore, our tax rule would not merely target a higher expectation of employment rate, but grounds its assumption in the observed market behavior, and lets the market itself respond with a higher employment rate, to the maximum extent permitted by the budget rule and the market capacity.\",\n \"Secondly, from a statistical viewpoint, our tax rule relies on a realization of $p$ , not on the uncertainty of the realized value.\",\n \"A natural way to factor out this uncertainty is to study the posterior rate $\\\\tilde{p}_{n,\\\\omega }$ , where both $s$ and $\\\\omega $ are no longer random.\",\n \"As Theorem REF indicates, $\\\\omega $ is informative and indicative in revealing the central tendency of the posterior labor market, delineated by $\\\\tilde{p}_{n,\\\\omega }$ .\",\n \"Asymptotic dispersion of $\\\\tilde{p}_{n,\\\\omega }$ and the market's response to the tax rule $\\\\tau (\\\\omega , \\\\delta )$ are among other essential ingredients in the complete profile of $\\\\tilde{p}_{n,\\\\omega }$ .\",\n \"Thus, we set optimal criteria to minimize dispersion measures or maximize the expected market response.\"\n ],\n [\n \"Zero Asymptotic Posterior Variance\",\n \"Tax rate stability creates the right environment for a balance of payments, reduces the uncertainty of the labor market, and creates confidence in technology and human capital investment.\",\n \"As a function of $\\\\omega $ , $\\\\tau $ migrates the risk from the unemployment rate to the tax rate.\",\n \"With the absence of other exogenous shocks, indeed, stability in the tax rate is equivalent to that in the unemployment rate.\",\n \"The same argument is also valid when the exogenous shocks are orthogonal to that in $\\\\omega $ .\",\n \"When we use the variance of $\\\\tilde{p}_{n,\\\\omega }$ to measure its instability, Theorem REF states that the tax rule $\\\\tau (\\\\omega , \\\\delta ) = 1 - \\\\omega + \\\\delta \\\\omega $ minimizes the asymptotic variance of $\\\\tilde{p}_{n,\\\\omega }$ .\",\n \"Furthermore, it is also the limit of variance-minimizing tax rules for finite labor markets.\",\n \"We add the restriction “$\\\\theta , \\\\rho \\\\ge \\\\frac{1}{n}$ \\\" in the finite labor markets to ensure the positivity of the hyperparameters $(\\\\theta ,\\\\rho )$ .\",\n \"As $n\\\\rightarrow \\\\infty $ , the tax rule $\\\\tau (\\\\omega , \\\\delta ) = 1 - \\\\omega + \\\\delta \\\\omega $ minimizes the limit variance of $\\\\sqrt{n} \\\\tilde{p}_{n,\\\\omega }$ , i.e., $\\\\mathop {\\\\mathrm {argmin}}\\\\limits _{\\\\tau } \\\\lim \\\\limits _{n\\\\rightarrow \\\\infty } \\\\mathrm {VAR} \\\\left(\\\\sqrt{n} \\\\tilde{p}_{n,\\\\omega } \\\\right)= 1 -\\\\omega +\\\\delta \\\\omega .$ Besides, $\\\\lim \\\\limits _{n\\\\rightarrow \\\\infty } \\\\mathop {\\\\mathrm {argmin}}\\\\limits _{\\\\tau } \\\\mathrm {VAR} \\\\left(\\\\sqrt{n} \\\\tilde{p}_{n,\\\\omega } \\\\bigm \\\\vert \\\\theta , \\\\rho \\\\ge \\\\frac{1}{n} \\\\right)= 1 -\\\\omega +\\\\delta \\\\omega .$ Furthermore, both the minimum limit variance and the limit minimum variance are zero.\",\n \"We provide a few comments to help clarify any potential misunderstandings about the theorem.\",\n \"First, a stable $\\\\sqrt{n} \\\\tilde{p}_{n,\\\\omega }$ is achieved at $n=\\\\infty $ where the limit variance is zero.\",\n \"However, $\\\\tilde{p}_{n,\\\\omega }$ is still exposed to exogenous shocks, such as those studied in Pissarides (1992) and Blanchard (2000).\",\n \"Secondly, it is worth emphasizing that $1-\\\\omega +\\\\delta \\\\omega $ is the limit tax rule as $n\\\\rightarrow \\\\infty $ .\",\n \"For a large but finite $n$ , a small positive number could be added to $1-\\\\omega +\\\\delta \\\\omega $ to ensure the positivitySee details in the proof of Theorem REF .\",\n \"of $\\\\theta $ and $\\\\rho $ .\",\n \"That small positive number, however, is negligible; thus, we can practically use the rule $\\\\tau (\\\\omega , \\\\delta ) = 1-\\\\omega +\\\\delta \\\\omega $ without any addition.\",\n \"Moreover, a higher-order approximation $\\\\tau =1-\\\\omega +\\\\delta \\\\omega +\\\\frac{\\\\omega (1-\\\\omega )(1-\\\\delta )^2}{n}$ could be an excellent alternativeSee the proof of Theorem REF for details.\",\n \"to $\\\\tau =1-\\\\omega +\\\\delta \\\\omega $ .\",\n \"Thirdly, with a zero or near zero variance in the unemployment rate, labor mobility means one layoff and one new hire should almost coincide to ensure the total employment size $|\\\\mathbf {S}|$ remains nearly constant.\",\n \"It also means that the sizes of employment and labor market change proportionally so that their ratio remains unchanged.\",\n \"Lastly, though the posterior distribution is skewed, the tax rule minimizes both the overall risk and the one-sided risk of $\\\\sqrt{n} \\\\tilde{p}_{n,\\\\omega }$ , as stated in Theorem REF .\",\n \"In particular, a policymaker's concern is on the downside risk only.\",\n \"As $n\\\\rightarrow \\\\infty $ , the tax rule $\\\\tau (\\\\omega , \\\\delta ) = 1 - \\\\omega + \\\\delta \\\\omega $ minimizes both the limit lower semivariance and the limit upper semivariance of $\\\\sqrt{n} \\\\tilde{p}_{n,\\\\omega }$ .\"\n ],\n [\n \"Consistency and Robustness\",\n \"The above fair tax rule also captures several striking features of the labor market.\",\n \"In the first place, it is the policymaker's best response to the market to stimulate employment within the framework of Eq.\",\n \"(REF ) and Eq.\",\n \"(REF ).\",\n \"In the second place, we can also derive it by minimizing statistical dispersion measures other than the posterior variance or semivariance.\",\n \"Meanwhile, it helps mitigate income inequality.\",\n \"The rule $\\\\tau (\\\\omega , \\\\delta ) = 1 - \\\\omega + \\\\delta \\\\omega $ is an effective taxation strategy to maximally boost the employment size without breaking the opportunity equality and the budget balance.\",\n \"For an economic policymaker, one primary concern is the forward-looking employment profile $\\\\tilde{p}_{n,\\\\omega }$ .\",\n \"By Theorem REF , the mean of $\\\\tilde{p}_{n,\\\\omega }$ converges to $\\\\omega $ as $n\\\\rightarrow \\\\infty $ for any fair rule $\\\\tau (\\\\omega , \\\\delta ) \\\\in (1-\\\\omega +\\\\delta \\\\omega ,1)$ .\",\n \"When $n$ is finitely large, they respond adversely to an increasing $\\\\tau $ (cf Theorem REF ).\",\n \"Consequently, to maximize the posterior mean, we should minimize the tax rate $\\\\tau $ while still maintaining the conditions $\\\\tau (\\\\omega , \\\\delta ) \\\\in (1-\\\\omega +\\\\delta \\\\omega ,1)$ , $\\\\theta >0$ and $\\\\rho >0$ .\",\n \"Thus, the limit of fair tax rates which maximize the mean of $\\\\tilde{p}_{n,\\\\omega }$ would be $1-\\\\omega +\\\\delta \\\\omega $ .\",\n \"As a remark, the condition $\\\\omega >.5$ in Theorem REF is satisfied in the general economy.\",\n \"For any $\\\\omega \\\\in (.5,1)$ and a finitely large $n$ , the mean of $\\\\tilde{p}_{n,\\\\omega }$ reacts negatively to an increasing tax rate $\\\\tau \\\\in (1-\\\\omega +\\\\delta \\\\omega ,1)$ .\",\n \"As a consequence, $\\\\lim \\\\limits _{n\\\\rightarrow \\\\infty } \\\\mathop {\\\\mathrm {argmax}}\\\\limits _{\\\\tau } \\\\mathrm {MEAN}\\\\left(\\\\tilde{p}_{n,\\\\omega } \\\\bigm \\\\vert \\\\theta , \\\\rho \\\\ge \\\\frac{1}{n} \\\\right)=1 -\\\\omega +\\\\delta \\\\omega .$ Furthermore, the tax rule also minimizes the mean absolute deviation (thereafter, MAD) from the mean as $n \\\\rightarrow \\\\infty $ .\",\n \"For a Beta distribution, especially with large parameters, MAD is a more robust measure of statistical dispersion than the variance.\",\n \"The MAD around the mean for the posterior $\\\\tilde{p}_{n,\\\\omega }$ is (e.g., Gupta and Nadarajah 2004, page 37): $\\\\mathbb {E} \\\\left[ \\\\left|\\\\tilde{p}_{n,\\\\omega } - \\\\mathbb {E} (\\\\tilde{p}_{n,\\\\omega }) \\\\right| \\\\right]=\\\\frac{2(\\\\theta +s)^{\\\\theta +s} (\\\\rho +n-s)^{\\\\rho +n-s}}{\\\\beta (\\\\theta +s, \\\\rho +n-s)(\\\\theta +\\\\rho +n)^{\\\\theta +\\\\rho +n}}.$ In the next theorem, we identify the same tax rule by minimizing the asymptotic MAD.\",\n \"As $n\\\\rightarrow \\\\infty $ , the tax rule $\\\\tau (\\\\omega , \\\\delta ) = 1 - \\\\omega + \\\\delta \\\\omega $ minimizes the MAD of $n \\\\tilde{p}_{n,\\\\omega }$ around the mean, i.e.\",\n \"$\\\\lim \\\\limits _{n\\\\rightarrow \\\\infty } \\\\mathop {\\\\mathrm {argmin}}\\\\limits _{\\\\tau }\\\\mathbb {E} \\\\left[n \\\\left|\\\\tilde{p}_{n,\\\\omega } - \\\\mathbb {E} (\\\\tilde{p}_{n,\\\\omega }) \\\\right| \\\\biggm \\\\vert \\\\theta , \\\\rho \\\\ge \\\\frac{1}{n} \\\\right]= 1 -\\\\omega +\\\\delta \\\\omega .$ Besides, $\\\\mathop {\\\\mathrm {argmin}}\\\\limits _{\\\\tau } \\\\lim \\\\limits _{n\\\\rightarrow \\\\infty }\\\\mathbb {E} \\\\left[n \\\\left|\\\\tilde{p}_{n,\\\\omega } - \\\\mathbb {E} (\\\\tilde{p}_{n,\\\\omega }) \\\\right| \\\\biggm \\\\vert \\\\theta , \\\\rho \\\\ge \\\\frac{1}{n} \\\\right]= 1 -\\\\omega +\\\\delta \\\\omega .$\"\n ],\n [\n \"Equality of Outcome with Symmetric Production\",\n \"To see the relation between productivity and the tax rate, we introduce a partial ordering in the labor market.\",\n \"For any $i, j\\\\in \\\\mathbb {N}$ with $i\\\\ne j$ , we say $i$ uniformly outperforms $j$ in $v$ if $v(T\\\\cup \\\\overline{i}) - v(T) \\\\ge v(T\\\\cup \\\\overline{j}) - v(T)$ for any $T\\\\subseteq \\\\mathbb {N} \\\\setminus \\\\overline{i} \\\\setminus \\\\overline{j}$ ; and $v(T)-v(T\\\\setminus \\\\overline{i}) \\\\ge v(T)-v(T \\\\setminus \\\\overline{j})$ for any $T\\\\subseteq \\\\mathbb {N}$ with $i,j \\\\in T$ .\",\n \"With these two inequality conditions, $i$ has higher marginal productivity than $j$ in all comparable employment contingencies $-$ either both employed or both unemployed.\",\n \"As productivity is highly valued in Eq.\",\n \"(REF ) and Eq.\",\n \"(REF ), $j$ should receive fewer employment benefits and less unemployment welfare than $i$ does.\",\n \"This is formally claimed in Theorem REF .\",\n \"Besides, the theorem does not require the Beta-Binomial distribution, as long as $i$ and $j$ alone have the same chance of being employed; other players in the labor market may have unequal employment opportunities.\",\n \"The theorem is valid for all fair tax rules, including the special one $\\\\tau (\\\\omega , \\\\delta )=1-\\\\omega +\\\\delta \\\\omega $ .\",\n \"If $i\\\\in \\\\mathbb {N}$ uniformly outperforms $j\\\\in \\\\mathbb {N}$ in $v$ and they have equal employment opportunity, then $\\\\gamma _i[v] \\\\ge \\\\gamma _j[v]$ and $\\\\lambda _i[v] \\\\ge \\\\lambda _j[v]$ .\",\n \"We say $i, j\\\\in \\\\mathbb {N}$ are symmetric in the production function $v$ if they uniformly outperform each other.\",\n \"By Theorem REF , $\\\\lambda _i[v] =\\\\lambda _j[v]$ and $\\\\gamma _i[v] = \\\\gamma _j[v]$ if $i$ and $j$ are symmetric and they have equal employment opportunity.\",\n \"In other words, they should receive the same amount of unemployment welfare if both are unemployed; they should also receive the same amount of employment benefits if both are employed.\",\n \"Without any further analysis of or any prior knowledge about the production function, symmetry among the unemployed (or the employed) could be a reasonable a priori assumption to distribute the unemployment welfare (or employment benefits, respectively).\",\n \"For example, the Cobb-Douglas production function is symmetric among all employed persons in the labor market.\",\n \"Besides, if we assume symmetry among the employed labor and also assume symmetry among the unemployed labor, then the tax rule $\\\\tau (\\\\omega , \\\\delta ) = 1 - \\\\omega + \\\\delta \\\\omega $ eliminates the income inequality, when the income is either employment benefits or unemployment welfare.\",\n \"Theorem REF affirms this equality of outcome, but an employed individual and an unemployed one may not be symmetric in $v$ .\",\n \"Furthermore, the theorem does not restrict the size of $n$ and the specific probability distribution for the equal employment opportunity.\",\n \"In the $k$ -out-of-$n$ redundant system mentioned in Section , accordingly, the $n$ components (either working or standby) are equally important if they have equal quality.\",\n \"Section offers a few alternative probability distributions for the equal employment opportunity.\",\n \"Assume equal employment opportunity in $\\\\mathbb {N}$ .\",\n \"If all employed individuals are symmetric in $v$ and all unemployed individuals are also symmetric in $v$ , then $\\\\tau (\\\\omega , \\\\delta ) = 1 - \\\\omega + \\\\delta \\\\omega $ if and only if an unemployed person's unemployment welfare equals an employee's employment benefits.\"\n ],\n [\n \"Labor Cost\",\n \"We could use the symmetry to calibrate the labor cost, which is excluded from $v({\\\\mathbf {S}})$ .\",\n \"For example, let the labor cost for $i \\\\in {\\\\mathbf {S}}$ be the minimum wage requirement from all $j \\\\in \\\\mathbb {N}\\\\setminus {\\\\mathbf {S}}$ , who either are symmetric with $i$ or uniformly outperforms $i$ in $v$ .\",\n \"That is, some $j$ from $\\\\mathbb {N}\\\\setminus {\\\\mathbf {S}}$ can do $i$ 's work without compromising the production $v$ .\",\n \"The minimum wage is called reservation wage, below which $j$ is unwilling to work.\",\n \"At this minimum market replacement cost, ${\\\\mathbf {S}}$ can switch $i$ with someone else from $\\\\mathbb {N} \\\\setminus {\\\\mathbf {S}}$ without sacrificing its net profit.\",\n \"In fact, $j$ does not need to be symmetric with or outperform $i$ as long as the probability of $v({\\\\mathbf {S}}\\\\cup \\\\overline{j}\\\\setminus \\\\overline{i}) \\\\ge v({\\\\mathbf {S}}\\\\cup \\\\overline{i}\\\\setminus \\\\overline{j})$ is significantly large and the person has a willingness to accept (e.g., Horowitz and McConnell 2003).\",\n \"The probability of such $\\\\mathbf {S}$ could be difficult for job interviewers to approximate and estimate.\",\n \"At the other end, in order to avoid creating an undue disincentive to work, $j$ 's unemployment welfare must be less than the reservation wage plus employment benefit.\",\n \"Moreover, to ensure every one of $\\\\mathbb {N}$ is in the labor market, the welfare is necessary to bind to the incentive-compatibility constraint.\",\n \"This incentive requirement places a lower bound for the labor cost.\"\n ],\n [\n \"OTHER APPLICATIONS\",\n \"In an abstract sense, the above account is a fair-division solution for the following game-theoretic setting: there are a large number of players; the players are randomly divided into two groups; the payoff comes from one group.\",\n \"A wide range of applications falls into this type of games.\",\n \"In this section, we analyze three applications other than labor markets to show how to use the formula derived in the last three sections.\"\n ],\n [\n \"Voting Power\",\n \"In a voting game (e.g., Shapley 1962), $v: 2^\\\\mathbb {N} \\\\rightarrow \\\\lbrace 0, 1\\\\rbrace $ is a monotonically increasing set function.\",\n \"Let $\\\\mathbf {S}$ denote the random subset of voters who vote for the proposal.\",\n \"The proposal passes when $v(\\\\mathbf {S})=1$ ; otherwise, it blocks when $v(\\\\mathbf {S})=0$ .\",\n \"However, $v$ should not mean “production\\\" or alike.\",\n \"No matter the outcome, Hu (2006) describes $\\\\gamma _i [v]$ as $i$ 's probability of turning a blocked result to a passed one, and $\\\\lambda _i [v]$ the probability of turning a passed result to being blocked.\",\n \"Thus, the sum of $\\\\lambda _i [v]$ and $\\\\gamma _i [v]$ quantifies $i$ 's power in the game.\",\n \"The ratio $\\\\delta $ plays a role in some circumstances.\",\n \"Let us consider, for example, $10\\\\%$ of the voters approve just a proposal before a referendum voting on the proposal, and assume the number of other voters' support ballots follows a Beta-Binomial distribution.\",\n \"Many voting games are symmetric.\",\n \"In these cases, the equality of outcome becomes an egalitarian allocation of power.\"\n ],\n [\n \"Health Insurance\",\n \"Health insurance has two types of policyholders: some are ill and use the insurance to cover their medical expenses; others are healthy and do not use the insurance.\",\n \"Let $\\\\mathbf {S}$ denote the random set of ill policyholders, $v(\\\\mathbf {S})$ be the total medical expenses with copays deducted, and $\\\\tilde{\\\\delta }v(\\\\mathbf {S})$ be the surcharge paid to the insurance company.\",\n \"Let $\\\\delta = - \\\\tilde{\\\\delta }$ .\",\n \"Then the total expenses except for the copays, $(1 - \\\\delta ) v(\\\\mathbf {S})$ , are billed to all insurance policyholders.\",\n \"If $\\\\tau =1-\\\\omega +\\\\delta \\\\omega $ and $v$ is symmetric among the two types of policyholders, respectively, then by the equality of outcome, the cost to buy the insurance policy would be $\\\\frac{(1-\\\\delta ) \\\\mathbb {E} \\\\left[ v(\\\\mathbf {S}) \\\\right]}{n}$ per policyholder.\",\n \"We take expectation on $\\\\frac{(1-\\\\delta ) v(\\\\mathbf {S})}{n}$ because the policyholders pay it upfront.\",\n \"On the contrary, the unemployment welfare and employment benefits payments come after the production.\",\n \"In this example, patients pay the predetermined copays.\",\n \"In the labor market studied before, the labor costs are exempt from the distribution of the net production; they, however, have an indirect effect on $\\\\tau $ through the corporate earning ratio and $\\\\delta $ .\",\n \"In the next example, we use the equality of outcome to derive the same type of payments as copays and labor cost.\"\n ],\n [\n \"Highway Toll\",\n \"The I-66 highway inside the Capital Beltway of the Washington metropolitan area has enforced a dynamic toll rule during rush hours: a carpool driver pays no toll, but a solo driver pays a dynamic toll fee, say $\\\\xi (n, \\\\omega )$ .\",\n \"Here $n$ is the number of cars in the segment of the highway, and $\\\\omega $ is the percentage of solo drivers in the traffic.\",\n \"Let $g(n)$ be the average carpool driver's cost in the traffic when the traffic volume is $n$ cars.\",\n \"It is likely a nonlinear increasing function of $n$ .\",\n \"An excellent choice of $g(n)$ is the expected driving time in hours multiplied by the average hourly pay rate, plus expenses on gas and vehicle depreciation.\",\n \"Also, let $\\\\mathbf {S}$ denote the random set of solo drivers.\",\n \"Then, $v(\\\\mathbf {S}) = ng(n) - n(1-\\\\omega ) g\\\\left(n(1-\\\\omega ) \\\\right) - n\\\\omega \\\\xi (n, \\\\omega ) $ is the total cost of over-traffic generated by the solo drivers, with toll fees deducted.\",\n \"The production function $v$ is symmetric among all solo drivers and also symmetric among all carpool drivers.\",\n \"By the equality of outcome, each driver shares the same cost $\\\\frac{v(\\\\mathbf {S})}{n}$ .\",\n \"As a carpool driver pays no toll, his or her shared cost should exactly offset the extra cost caused by the solo drivers, which is $g(n) - g \\\\left(n(1-\\\\omega ) \\\\right)$ .\",\n \"Finally, equation $\\\\frac{v(\\\\mathbf {S})}{n}=g(n) - g\\\\left(n(1-\\\\omega ) \\\\right)$ implies $\\\\xi (n, \\\\omega ) = g\\\\left(n(1-\\\\omega ) \\\\right).$ In this example, an administration surcharge $\\\\delta $ may apply; the carpooled passengers are free-riders, but their costs are exempt from $v(\\\\mathbf {S})$ .\"\n ],\n [\n \"CONCLUSIONS\",\n \"In this paper, a fair-division solution is proposed to allocate the unemployment welfare in an economy, where the heterogeneous-agent production function is almost unknown.\",\n \"We interpret “fairness\\\" as equal employment opportunity and model it by a Beta-Binomial probability distribution.\",\n \"Our “sustainability\\\" is meant to be free of debt and free of surplus in the taxation budget.\",\n \"To justify the value of the unemployed labor, we capitalize on the D-value concept in Hu (2018).\",\n \"The D-value specifies how much of the net production to be retained with the employed labor, and what portion to be distributed to the unemployed.\",\n \"Finally, we postulate that the labor market is static and identify a sustainable tax policy by minimizing the asymptotic variance of the posterior employment rate.\",\n \"The policy can also be uniquely determined by minimizing the asymptotic posterior mean of the unemployment rate, or minimizing the downside risk of the posterior employment rate, or minimizing the posterior mean absolute deviation.\",\n \"Surprisingly, the tax rule is not only simple enough for practical use but also motivates the unemployed to seek employment and the employed to improve productivity.\",\n \"One could extend this framework in several ways.\",\n \"One way is to re-specify the probability distribution of equal employment opportunity, for example, by any of the following re-specifications.\",\n \"First, we can replace the two-parameter Beta distribution with a four-parameter one or a Beta rectangular distribution.\",\n \"Secondly, we can let $\\\\theta $ and $\\\\rho $ be some functions of other unknown parameters.\",\n \"Thirdly, we could substitute the Beta-Binomial distribution with a Dirichlet-Multinomial distribution or a Beta-Geometric distribution.\",\n \"Fourthly, we could randomize $p$ without involving $(\\\\theta , \\\\rho )$ , by generating two independent three-parameter Gamma random variables $X$ and $Y$ .\",\n \"Then, the ratio $X/(X+Y)$ is a Beta random variable.\",\n \"In any of the four cases, however, we need additional identification restrictions to figure out a $\\\\tau (\\\\omega , \\\\delta )$ precisely.\",\n \"From other angles, we could apply other identification schemes or other objective functions to find a unique fair tax rule.\",\n \"From a statistical viewpoint, one could try the maximum likelihood estimation using the twelve months' data prior to determining the policy tax rate, or minimize the ex-ante risk of $\\\\omega $ , or apply the statistical methods mentioned in Section REF .\",\n \"From an economic viewpoint, one could minimize the Gini coefficient of the Beta distribution of $\\\\tilde{p}_{n,\\\\omega }$ .\",\n \"From a strategic game-theoretical viewpoint, one could seek a bargaining solution from the feasible solution set $\\\\Omega _{n,\\\\omega ,\\\\delta }$ , which may be particularly useful when $n$ is small.\",\n \"Finally, a policymaker could treat the reserve ratio $\\\\delta $ as endogenous, for example, letting it be an increasing function of $\\\\omega $ .\",\n \"He or she could also place a heavier weight on the marginal gain than on the marginal loss, in order to stimulate employment.\",\n \"The simple static model, however, ignores several important aspects of a real labor market.\",\n \"First, it does not capture the dynamic features of the income inequality, nor its rational response to the tax rule.\",\n \"Secondly, while preventing the fungibility of borrowing funds from the future reduces the risk that a government administration piles up its national debt, it impairs that administration's ability (especially, monetary policy) to intervene in the economy.\",\n \"The government, however, can still moderately stimulate the economy during a recession by adjusting the reserve ratio $\\\\delta $ .\",\n \"Thirdly, the postulation of labor market efficiency does neglect the recent development of the incomplete-market theory (e.g., Magill and Quinzii 1996).\",\n \"Fourthly, a multi-criteria objective function may be a viable alternative to the minimum-variance one, especially when there is a high unemployment rate $1-\\\\omega $ or a large $\\\\delta $ .\",\n \"Lastly, a single tax rule $\\\\tau (\\\\omega , \\\\delta )$ could have overly simplified the complexity of the taxation system, which is also affected by other determinants.\",\n \"These are just a few challenges our framework introduces, which require further development.\",\n \"In summary, the fair and sustainable tax policy studied here has a solid theoretical underpinning, together with simplicity in practical use, consistency with productivity and employment incentives, and robustness to similar objectives.\",\n \"When applying this framework to a real fair-division problem, one should also consider the benefits of alternative probability distributions for equal opportunity, alternative objective functions, alternative restrictions, and dynamic thinking.\"\n ],\n [\n \"APPENDIX\",\n \"Our focus in this paper is to study the relationship between the fair tax rate $\\\\tau $ and the employment rate $\\\\omega $ when the labor market size $n$ is large.\",\n \"To analyze the limit behavior of $\\\\tau $ and $\\\\omega $ for a large $n$ , we only need the relevant asymptotic approximations.\",\n \"We say two functions $f(n)=O \\\\left(g(n) \\\\right)$ if $\\\\limsup \\\\limits _{n\\\\rightarrow \\\\infty } \\\\left|\\\\frac{f(n)}{g(n)} \\\\right| < \\\\infty $ ; and we say $f(n) \\\\approx g(n)$ if $\\\\lim \\\\limits _{n\\\\rightarrow \\\\infty } \\\\frac{f(n)}{g(n)} =1$ .\",\n \"For simplicity, let us denote the following shorthands: $\\\\begin{array}{lcl}\\\\Delta &\\\\equiv &\\\\omega + \\\\tau - \\\\delta \\\\omega - 1, \\\\\\\\\\\\end{array}\\\\Delta _1&\\\\equiv &\\\\delta \\\\omega - \\\\omega -\\\\tau + 2\\\\equiv 1 - \\\\Delta , \\\\\\\\$ 2- 2 - 2 + 2 + -1, 3(1- )(-) = --+ 2, 4- + + - 2+ 2 - 2 + - 2 + 3-1, 5- + - 2 + - 2 + 4-2 2+4, 6-+ +-1 2 + 3, 7-- + 2+ - 1 4 + (1-) 3, 8-- + + 3-2 3+5, 9 -2-+2+ 4-3 + 8.\",\n \"$$ The following lemma, to be used in other proofs, re-writes Eq.\",\n \"(REF ) and Eq.\",\n \"(REF ).\",\n \"In terms of $(n,\\\\delta ,\\\\tau , \\\\omega )$ , we can solve $(\\\\theta ,\\\\rho )$ from Eq.\",\n \"(REF ) and Eq.\",\n \"(REF ) as $ \\\\left\\\\lbrace \\\\begin{array}{l}\\\\theta =\\\\frac{n^2 \\\\omega \\\\Delta _1 + n\\\\Delta _2 +\\\\Delta _3 }{n\\\\Delta + \\\\Delta _3}, \\\\\\\\\\\\end{array}\\\\rho =\\\\frac{n^2 (1-\\\\omega ) \\\\Delta _1 + n\\\\Delta _4 + \\\\Delta _3}{n\\\\Delta + \\\\Delta _3}.\\\\right.\\\\qquad \\\\mathrm {(A.1)}$ In this proof, we use the following relation about Beta functions: $\\\\left\\\\lbrace \\\\begin{array}{lcll}\\\\beta (x-1, y+1) &=& \\\\frac{y}{x-1} \\\\beta (x, y), &x>1,\\\\ y>0; \\\\\\\\\\\\beta (x+1, y-1) &=& \\\\frac{x}{y-1} \\\\beta (x,y), &x>0,\\\\ y>1.\",\n \"\\\\\\\\\\\\end{array}\\\\right.$ First, the expected aggregate marginal gain and loss are: $\\\\begin{array}{rcl}\\\\mathbb {E}\\\\left[ \\\\sum \\\\limits _{i\\\\in \\\\mathbf {S}} \\\\left( v(\\\\mathbf {S})-v(\\\\mathbf {S}\\\\setminus \\\\overline{i}) \\\\right) \\\\right]\\\\end{array}&=&\\\\sum \\\\limits _{T\\\\subseteq \\\\mathbb {N}} \\\\mathbb {P}(\\\\mathbf {S}=T) \\\\sum \\\\limits _{i\\\\in T} \\\\left[ v(T)-v(T\\\\setminus \\\\overline{i}) \\\\right] \\\\\\\\$ = i N TN: i T P(S=T) [ v(T)-v(Ti) ] = i N i[v], E[ i NS ( v(Si)-v(S) ) ] = TN P(S=T) i NT [ v(Ti)-v(T) ] = i N T N: i T P(S=T) [ v(Ti)-v(T) ] = i N i[v].\",\n \"$$ Next, by Eq.\",\n \"(REF )$-$ (REF ), we re-write the expected marginal gain and loss as: $ \\\\begin{array}{rcl}\\\\end{array}\\\\gamma _i [v]&=& \\\\sum \\\\limits _{T\\\\subseteq \\\\mathbb {N}: T\\\\ni i} \\\\frac{\\\\beta (\\\\theta +t,\\\\rho +n-t)}{\\\\beta (\\\\theta ,\\\\rho )}[v(T)-v(T\\\\setminus \\\\overline{i})] \\\\\\\\$ Z=Ti= TN: Ti (+t,+n-t)(,) v(T) - ZN i (+|Z|+1, +n-|Z|-1)(,) v(Z), i [v] =ZNi (+|Z|,+n-|Z|)(,)[v(Zi) -v(Z)] T=Zi=TN: Ti (+t-1,+n-t+1)(,) v(T) - ZNi (+|Z|,+n-|Z|)(,) v(Z).\",\n \"By Eq.\",\n \"(REF ), the aggregate value of the employed labor $\\\\sum \\\\limits _{i \\\\in \\\\mathbb {N}} \\\\gamma _i [v]$ is $\\\\begin{array}{rcl}&&\\\\sum \\\\limits _{i\\\\in \\\\mathbb {N}} \\\\sum \\\\limits _{T\\\\subseteq \\\\mathbb {N}: T\\\\ni i} \\\\frac{\\\\beta (\\\\theta +t, \\\\rho +n-t)}{\\\\beta (\\\\theta ,\\\\rho )}v(T)- \\\\sum \\\\limits _{i\\\\in \\\\mathbb {N}} \\\\sum \\\\limits _{Z\\\\subseteq \\\\mathbb {N} \\\\setminus \\\\overline{i}} \\\\frac{\\\\beta (\\\\theta +|Z|+1, \\\\rho +n-|Z|-1)}{\\\\beta (\\\\theta ,\\\\rho )} v(Z) \\\\\\\\\\\\end{array}&\\\\stackrel{T=Z}{=}& \\\\sum \\\\limits _{T\\\\subseteq \\\\mathbb {N}: T\\\\ne \\\\emptyset }\\\\sum \\\\limits _{i\\\\in T} \\\\frac{\\\\beta (\\\\theta +t, \\\\rho +n-t)}{\\\\beta (\\\\theta ,\\\\rho )}v(T)- \\\\sum \\\\limits _{T\\\\subseteq \\\\mathbb {N}: T\\\\ne \\\\mathbb {N}} \\\\sum \\\\limits _{i \\\\in \\\\mathbb {N} \\\\setminus T} \\\\frac{\\\\beta (\\\\theta +t+1, \\\\rho +n-t-1)}{\\\\beta (\\\\theta ,\\\\rho )} v(T) \\\\\\\\$ = TN: T= t (+t, +n-t)(,)v(T) - TN: T= N (n-t) (+t+1, +n-t-1)(,) v(T) = n(+n,)(, )v(N) - n(+1,+n-1)(, )v() + TN: T= , T= N t (+t, +n-t) - (n-t) (+t+1, +n-t-1)(,)v(T) = n(+n,)(,)v(N) - n+n-1(,+n)(, )v() + TN: T= N, T= [ t - (n-t) (+t)+n-t-1 ] (+t, +n-t)(,)v(T) = n(+n,)(, )v(N) + TN: T= N t(+-1)-n+n-t-1 (+t, +n-t)(,)v(T).\",\n \"$$ Also by Eq.\",\n \"(REF ), the aggregate value of the unemployed labor $\\\\sum \\\\limits _{i\\\\in \\\\mathbb {N}} \\\\lambda _i [v]$ is $\\\\begin{array}{rcl}&&\\\\sum \\\\limits _{i\\\\in \\\\mathbb {N}} \\\\sum \\\\limits _{T\\\\subseteq \\\\mathbb {N}: T\\\\ni i} \\\\frac{\\\\beta (\\\\theta +t-1, \\\\rho +n-t+1)}{\\\\beta (\\\\theta ,\\\\rho )}v(T)- \\\\sum \\\\limits _{i\\\\in \\\\mathbb {N}} \\\\sum \\\\limits _{Z \\\\subseteq \\\\mathbb {N} \\\\setminus \\\\overline{i}} \\\\frac{\\\\beta (\\\\theta +|Z|, \\\\rho +n-|Z|)}{\\\\beta (\\\\theta ,\\\\rho )} v(Z) \\\\\\\\\\\\end{array}&\\\\stackrel{T=Z}{=}& \\\\sum \\\\limits _{T\\\\subseteq \\\\mathbb {N}: T\\\\ne \\\\emptyset }\\\\sum \\\\limits _{i\\\\in T} \\\\frac{\\\\beta (\\\\theta +t-1, \\\\rho +n-t+1)}{\\\\beta (\\\\theta ,\\\\rho )}v(T)- \\\\sum \\\\limits _{T\\\\subseteq \\\\mathbb {N}: T \\\\ne \\\\mathbb {N}}\\\\sum \\\\limits _{i\\\\in \\\\mathbb {N} \\\\setminus T} \\\\frac{\\\\beta (\\\\theta +t, \\\\rho +n-t)}{\\\\beta (\\\\theta ,\\\\rho )} v(T) \\\\\\\\$ = TN: T= t(+t-1, +n-t+1)(,)v(T) - TN: T =N(n-t)(+t, +n-t)(,) v(T) = n(+n-1, +1)(,)v(N) - n(, +n)(,) v() +TN: T= , T =N t(+t-1, +n-t+1)-(n-t)(+t, +n-t)(,)v(T) = n+n-1(+n, )(,)v(N) - n(, +n)(,) v() + TN: T= , T= N [ t(+n-t)+t-1 - (n-t) ] (+t, +n-t)(,) v(T) = TN: T= t(+-1)-n(-1)+t-1 (+t, +n-t)(,) v(T) - n(, +n)(,) v().\",\n \"$$ We can re-write Eq.\",\n \"(REF ) and Eq.\",\n \"(REF ) as a linear system of unknowns $(\\\\theta , \\\\rho )$ : $\\\\left\\\\lbrace \\\\begin{array}{rcl}(1-\\\\tau )(\\\\rho +n-s-1)&=&s(\\\\theta +\\\\rho -1)-n\\\\theta , \\\\\\\\\\\\end{array}(\\\\tau -\\\\delta )(\\\\theta +s-1)&=&s(\\\\theta +\\\\rho -1)-n(\\\\theta -1).\\\\right.$ As a consequence, the symbolic solution of $(\\\\theta , \\\\rho )$ is unique.\",\n \"Let us assume Eq.\",\n \"(REF ) and verify that it satisfies both Eq.\",\n \"(REF ) and Eq.\",\n \"(REF ) by the following identities, some of which are used in other proofs: $\\\\begin{array}{rcl}\\\\end{array}\\\\theta +s&=&\\\\frac{n^2 \\\\omega \\\\Delta _1 + n\\\\Delta _2 +\\\\Delta _3 }{n\\\\Delta + \\\\Delta _3}+ \\\\frac{n\\\\omega (n\\\\Delta + \\\\Delta _3)}{n\\\\Delta + \\\\Delta _3}=\\\\frac{n^2 \\\\omega + n\\\\Delta _6 + \\\\Delta _3}{n\\\\Delta + \\\\Delta _3}, \\\\\\\\$ +s-1 = n2 + n6 + 3n+ 3 - n+ 3n+ 3 = n2 + n (6-) n+ 3 = n2 + n (- 1) n+ 3, += n2 1 + n2 +3 n+ 3 + n2 (1-) 1 + n4 + 3n+ 3 = n2 1 + n 5 + 23n+ 3, +- 1 = n2 1 + n 5 + 23n+ 3 - n+ 3n+ 3 = n2 1 + n ( - 2 - 2 + 3- 1) + 3 n+ 3, ++ n = n2 1 + n 5 + 23n+ 3 + n(n+ 3)n+ 3 = n2 + n 8 + 23n+ 3, ++ n + 1 = n2 + n 8 + 23n+ 3 +n+ 3n+ 3 = n2 + n 9 + 33 n+ 3, ++ n - 1 = n2 + n 8 + 23n+ 3 - n+ 3n+ 3 = n2 + n (8 - ) + 3 n+ 3, ++ n - 2 = n2 + n 8 + 23n+ 3 - 2(n+ 3)n+ 3 = n2 + n (8-2) n+ 3, +n-s = n2 (1-) 1 + n4 + 3n+ 3 + n(1-)(n+ 3)n+ 3 = n2 (1-) + n 7 + 3n+ 3, +n-s - 1 = n2 (1-) + n 7 + 3n+ 3 -n+ 3n+ 3 = n2 (1-)+ n (1-)(-) n+ 3.\",\n \"$Thus,$ rcl s(+- 1)-n= n[n2 1 + n ( - 2 - 2 + 3- 1) + 3] n+ 3 - n(n2 1 + n2 +3)n+ 3 = n2[(- 2 - 2 + 3- 1) -2] + n(-1)3 n+ 3 = n2(1-)(1-) + n(-1)3 n+ 3, s(+- 1)-n(-1) = [s(+- 1)-n] + n = n2(1-)(1-) + n(-1)3 n+ 3 + n(n+ 3)n+ 3 = n2 (-) + n 3 n+ 3.\",\n \"$Therefore,$ rclcl s(+- 1)-n+n-s - 1 = n2(1-)(1-)+ n(-1)3n2 (1-)+ n (1-)(-) = 1-, s(+- 1)-n(-1)+s-1 = n2 (-) + n 3n2 + n (- 1) = - , $which are equivalent to Eq.\",\n \"(\\\\ref {eq:tax_rate_def}) and Eq.\",\n \"(\\\\ref {eq:tax_rate_c}), respectively.$ For any integer $z \\\\ge 0$ , by the proof of Lemma REF , $\\\\begin{array}{rcl}\\\\frac{\\\\theta +s+z}{\\\\theta +\\\\rho +n+z}\\\\end{array}&=&\\\\frac{\\\\frac{n^2\\\\omega +n\\\\Delta _6+\\\\Delta _3}{n\\\\Delta +\\\\Delta _3}+\\\\frac{z(n\\\\Delta +\\\\Delta _3)}{n\\\\Delta +\\\\Delta _3}}{\\\\frac{n^2+n\\\\Delta _8+2\\\\Delta _3}{n\\\\Delta +\\\\Delta _3}+\\\\frac{z(n\\\\Delta +\\\\Delta _3)}{n\\\\Delta +\\\\Delta _3}} \\\\\\\\$ = n2+n(6+z)+(1+z)3 n2+n(8+z)+(2+z)3 , as n. $$ As a function of $\\\\eta $ , the characteristic function of $\\\\tilde{p}_{n,\\\\omega }$ (e.g., Johnson et al.\",\n \"1995, Chapter 21) is $\\\\mathbb {E} \\\\left[e^{i \\\\eta \\\\tilde{p}_{n,\\\\omega }} \\\\right]= 1 + \\\\sum \\\\limits _{k=1}^\\\\infty \\\\frac{(i \\\\eta )^k}{k!\",\n \"}\\\\prod \\\\limits _{z=0}^{k-1} \\\\frac{\\\\theta +s+z}{\\\\theta +\\\\rho +n+z}$ where $i$ is the unit imaginary number, i.e., $i^2=-1$ .\",\n \"We let $n\\\\rightarrow \\\\infty $ , $\\\\begin{array}{rcl}\\\\lim \\\\limits _{n\\\\rightarrow \\\\infty } \\\\mathbb {E}[e^{i \\\\eta \\\\tilde{p}_{n,\\\\omega }}]&=&1 + \\\\sum \\\\limits _{k=1}^\\\\infty \\\\frac{(i \\\\eta )^k}{k!}\",\n \"\\\\lim \\\\limits _{n\\\\rightarrow \\\\infty }\\\\prod \\\\limits _{z=0}^{k-1}\\\\frac{\\\\theta +s+z}{\\\\theta +\\\\rho +n+z}\\\\\\\\\\\\end{array}&=&1 + \\\\sum \\\\limits _{k=1}^\\\\infty \\\\frac{(i \\\\eta \\\\omega )^k}{k!}\",\n \"\\\\\\\\$ = (i ).\",\n \"$Therefore, as $ $, $ pn,$ converges in distribution to the degenerate distribution with mass at $$,which has the characteristic function $ (i )$.$$$ As $\\\\tilde{p}_{n,\\\\omega }$ has a Beta distribution with parameters $(\\\\theta +s, \\\\rho +n-s)$ (see Section REF ), its variance is $\\\\frac{(\\\\theta +s)(\\\\rho +n-s)}{(\\\\theta +\\\\rho +n)^2(\\\\theta +\\\\rho +n+1)}$ (e.g., Gupta and Nadarajah 2004, page 35).\",\n \"By the proof of Lemma REF , the variance of $\\\\sqrt{n} \\\\tilde{p}_{n,\\\\omega }$ is $\\\\begin{array}{rcl}&&\\\\frac{n \\\\frac{n^2\\\\omega +n\\\\Delta _6+\\\\Delta _3}{n\\\\Delta +\\\\Delta _3}\\\\frac{n^2(1-\\\\omega )+n\\\\Delta _7+\\\\Delta _3}{n\\\\Delta +\\\\Delta _3}}{ \\\\left(\\\\frac{n^2+n\\\\Delta _8+2\\\\Delta _3}{n\\\\Delta +\\\\Delta _3}\\\\right)^2\\\\frac{n^2+n\\\\Delta _9+3\\\\Delta _3}{n\\\\Delta +\\\\Delta _3}} \\\\\\\\\\\\end{array}&=&\\\\frac{n(n\\\\Delta +\\\\Delta _3)(n^2\\\\omega +n\\\\Delta _6+\\\\Delta _3)\\\\left[n^2(1-\\\\omega )+n\\\\Delta _7+\\\\Delta _3 \\\\right]}{\\\\left(n^2+n\\\\Delta _8+2\\\\Delta _3 \\\\right)^2\\\\left(n^2+n\\\\Delta _9+3\\\\Delta _3 \\\\right)} \\\\\\\\$ = (+3n) [ (1+6n+3n2 ) ] [ (1-) (1+7n(1-)+3n2(1-) ) ] (1+ 8n+23n2)2 (1+ 9n+ 33n2 ) = (1-) (+3n) [ 1+ 6n + 7n(1-) - 28n-9n + O(1n2 ) ] = (1-)+ (1-)n [ 3 + ( 6 + 71- - 28 - 9) ] + O(1n2 ) (1-) (+--1), as n .\",\n \"To minimize $\\\\lim \\\\limits _{n\\\\rightarrow \\\\infty }\\\\mathrm {VAR}(\\\\sqrt{n} \\\\tilde{p}_{n,\\\\omega })= \\\\omega (1-\\\\omega )\\\\Delta $ while $\\\\lim \\\\limits _{n\\\\rightarrow \\\\infty } \\\\mathrm {VAR}(\\\\sqrt{n} \\\\tilde{p}_{n,\\\\omega } )\\\\ge 0$ , we have to set $\\\\Delta = 0$ .\",\n \"Thus, $\\\\mathop {\\\\mathrm {argmin}}\\\\limits _{\\\\tau } \\\\lim \\\\limits _{n\\\\rightarrow \\\\infty }\\\\mathrm {VAR} \\\\left(\\\\sqrt{n} \\\\tilde{p}_{n,\\\\omega } \\\\right) = 1-\\\\omega +\\\\delta \\\\omega .$ However, when applying $\\\\tau = 1-\\\\omega + \\\\delta \\\\omega $ to a labor market with finite $n$ , we have $\\\\Delta = 0$ , $\\\\Delta _1 = 1$ and $\\\\Delta _3 = -\\\\omega (1-\\\\omega ) (1-\\\\delta )^2$ .\",\n \"And Lemma REF reduces to $\\\\left\\\\lbrace \\\\begin{array}{rcl}\\\\theta &=&\\\\frac{n^2 \\\\omega + n \\\\Delta _2 - \\\\omega (1-\\\\omega ) (1-\\\\delta )^2}{-\\\\omega (1-\\\\omega ) (1-\\\\delta )^2}, \\\\\\\\\\\\end{array}\\\\rho &=&\\\\frac{n^2 (1-\\\\omega )+n\\\\Delta _4-\\\\omega (1-\\\\omega ) (1-\\\\delta )^2}{-\\\\omega (1-\\\\omega ) (1-\\\\delta )^2},\\\\right.$ which converge to $-\\\\infty $ as $n\\\\rightarrow \\\\infty $ .\",\n \"In theory, therefore, for a large but finite $n$ , we need to choose $\\\\tau $ to be $1-\\\\omega +\\\\delta \\\\omega $ plus a small positive number to ensure that $\\\\theta >0$ and $\\\\rho >0$ .\",\n \"To estimate the small positive number, let us first try the higher-order approximation of $\\\\tau = 1 - \\\\omega + \\\\delta \\\\omega + \\\\frac{c}{n},$ for some constant $c > 0$ .\",\n \"Then $ \\\\begin{array}{lcl}\\\\Delta &=&\\\\frac{c}{n}, \\\\\\\\\\\\end{array}\\\\Delta _3&=&\\\\left[\\\\omega (1-\\\\delta ) - \\\\frac{c}{n} \\\\right]\\\\left[-(1-\\\\omega ) (1-\\\\delta ) - \\\\frac{c}{n} \\\\right] \\\\\\\\$ = -(1-)(1-)2 + O(1n ).\",\n \"By the next to the last step in (REF ), $\\\\mathrm {VAR} \\\\left(\\\\sqrt{n} \\\\tilde{p}_{n,\\\\omega } \\\\Bigm \\\\vert \\\\tau = 1 - \\\\omega + \\\\delta \\\\omega + \\\\frac{c}{n} \\\\right)=\\\\frac{\\\\omega (1-\\\\omega ) c - \\\\omega ^2 (1-\\\\omega )^2 (1-\\\\delta )^2}{n} + O\\\\left(\\\\frac{1}{n^2} \\\\right).$ To minimize the above variance, we let $c = \\\\omega (1-\\\\omega )(1-\\\\delta )^2$ .\",\n \"Thus, $\\\\tau = 1 - \\\\omega +\\\\delta \\\\omega + \\\\frac{\\\\omega (1-\\\\omega )(1-\\\\delta )^2}{n}$ is a higher-order approximation for $1-\\\\omega +\\\\delta \\\\omega $ .\",\n \"Further high-order approximations, if necessary, could be found similarly.\",\n \"When $\\\\tau = 1 - \\\\omega +\\\\delta \\\\omega + \\\\frac{c}{n}$ , let us use Eq.\",\n \"(REF ) to calculate $ \\\\begin{array}{rcl}n\\\\Delta +\\\\Delta _3&=&c + \\\\left[\\\\omega (1-\\\\delta ) - \\\\frac{c}{n} \\\\right] \\\\left[-(1-\\\\omega ) (1-\\\\delta ) - \\\\frac{c}{n} \\\\right] \\\\\\\\\\\\end{array}&=&\\\\frac{(1-2\\\\omega ) (1-\\\\delta ) c}{n} + \\\\frac{c^2}{n^2}$ which is negative when $\\\\omega >.5$ and $n\\\\ge \\\\left| \\\\frac{\\\\omega (1-\\\\omega )(1-\\\\delta )}{1-2\\\\omega }\\\\right|$ .\",\n \"As the numerators of $\\\\theta $ and $\\\\rho $ in Lemma REF are both positive for a large $n$ , $\\\\theta $ and $\\\\rho $ are negative when $n$ is large and $\\\\omega >.5$ .\",\n \"Let us make another try at $\\\\tau = 1-\\\\omega +\\\\delta \\\\omega +\\\\frac{2c}{n}$ .\",\n \"Similar to Eq.\",\n \"(REF ), $\\\\begin{array}{rcl}n\\\\Delta +\\\\Delta _3&=&2c + \\\\left[\\\\omega (1-\\\\delta ) - \\\\frac{2c}{n} \\\\right] \\\\left[-(1-\\\\omega ) (1-\\\\delta ) - \\\\frac{2c}{n} \\\\right] \\\\\\\\\\\\end{array}&=&c + \\\\frac{2(1-2\\\\omega ) (1-\\\\delta ) c}{n} + \\\\frac{4c^2}{n^2}.$ $In this case, both $ $ and $$ are larger than $ 1n$ when $ n$ is large.Note from the last step of Eq.\",\n \"(\\\\ref {eq:asym_var}) that $ VAR (n pn, )$ is an increasing function of $$ when $ n$ is large; thus,the small positive number to be added to $ 1-+$ is between $ cn$and $ 2cn$.\",\n \"In practice, however,as the number is too small for a large $ n$,there is no necessity to exactly calculate it and add it to $ 1-+$.$ Let $\\\\tilde{\\\\tau }_n = \\\\mathop {\\\\mathrm {argmin}}\\\\limits _{\\\\tau } \\\\mathrm {VAR} \\\\left(\\\\sqrt{n} \\\\tilde{p}_{n,\\\\omega } \\\\bigm \\\\vert \\\\theta , \\\\rho \\\\ge \\\\frac{1}{n} \\\\right)$ .\",\n \"We add the restriction $\\\\theta , \\\\rho \\\\ge \\\\frac{1}{n}$ to ensure the existence of $\\\\tilde{\\\\tau }_n$ .\",\n \"As $\\\\mathrm {VAR} \\\\left(\\\\sqrt{n} \\\\tilde{p}_{n,\\\\omega } \\\\right)$ is an increasing function of $\\\\tau $ when $n$ is large, $\\\\tilde{\\\\tau }_n$ is unique and less than $1 - \\\\omega + \\\\delta \\\\omega + \\\\frac{2c}{n}$ when $n$ is large.\",\n \"By Eq.\",\n \"(REF ), $\\\\mathrm {VAR}(\\\\sqrt{n} \\\\tilde{p}_{n,\\\\omega } \\\\Bigm \\\\vert \\\\tau = \\\\tilde{\\\\tau }_n) = \\\\omega (1-\\\\omega ) (\\\\omega + \\\\tilde{\\\\tau }_n-\\\\delta \\\\omega -1) + O \\\\left(\\\\frac{1}{n} \\\\right).$ Finally, we use the relation $0\\\\le \\\\mathrm {VAR}\\\\left(\\\\sqrt{n} \\\\tilde{p}_{n,\\\\omega } \\\\Bigm \\\\vert \\\\tau = \\\\tilde{\\\\tau }_n \\\\right)\\\\le \\\\mathrm {VAR} \\\\left(\\\\sqrt{n} \\\\tilde{p}_{n,\\\\omega } \\\\Bigm \\\\vert \\\\tau = 1 - \\\\omega + \\\\delta \\\\omega + \\\\frac{2c}{n} \\\\right)$ to get $0\\\\le \\\\omega (1-\\\\omega ) (\\\\omega + \\\\tilde{\\\\tau }_n-\\\\delta \\\\omega -1) + O \\\\left(\\\\frac{1}{n} \\\\right)\\\\le O \\\\left(\\\\frac{1}{n} \\\\right).$ Letting $n \\\\rightarrow \\\\infty $ in the above inequality, we get $\\\\omega (1-\\\\omega ) (\\\\omega + \\\\tilde{\\\\tau }_n-\\\\delta \\\\omega -1) = O(\\\\frac{1}{n})$ , i.e., $\\\\lim \\\\limits _{n\\\\rightarrow \\\\infty } \\\\mathop {\\\\mathrm {argmin}}\\\\limits _{\\\\tau } \\\\mathrm {VAR} \\\\left(\\\\sqrt{n} \\\\tilde{p}_{n,\\\\omega } \\\\bigm \\\\vert \\\\theta , \\\\rho \\\\ge \\\\frac{1}{n} \\\\right) = 1 - \\\\omega + \\\\delta \\\\omega $ .\",\n \"$$ In this proof, we constantly apply the identities in the proof of Lemma REF .\",\n \"Let $\\\\mu _n = \\\\frac{\\\\theta +s}{\\\\theta +\\\\rho +n}$ be the mean of $\\\\tilde{p}_{n,\\\\omega }$ , and let $\\\\sigma _n^2$ be the variance of $\\\\tilde{p}_{n,\\\\omega }$ .\",\n \"The lower semivariance of $\\\\tilde{p}_{n,\\\\omega }$ is calculated as $\\\\sigma ^2_{n-} = \\\\int _0^{\\\\mu _n} (x - \\\\mu _n)^2 \\\\frac{x^{\\\\theta +s-1}(1-x)^{\\\\rho +n-s-1}}{\\\\beta ( \\\\theta +s, \\\\rho +n-s)} \\\\mathrm {d} x.$ We apply Chebychev's inequality in terms of the lower semivariance (e.g., Berck and Hihn 1982) to get $\\\\mathbb {P} \\\\left(\\\\tilde{p}_{n,\\\\omega } \\\\le \\\\mu _n - a_n \\\\sigma _{n-} \\\\right)\\\\le \\\\frac{1}{a_n^2}, \\\\quad \\\\forall \\\\ a_n>0.$ By the proof of Theorem REF , $\\\\tilde{p}_{n,\\\\omega }$ has the variance $\\\\sigma _n^2 = \\\\frac{\\\\omega (1-\\\\omega )\\\\Delta }{n}+O(\\\\frac{1}{n^2})$ .\",\n \"We let $a_n = \\\\frac{\\\\sqrt{\\\\frac{\\\\omega (1-\\\\omega )\\\\Delta }{n}}}{\\\\sigma _{n-}}$ , then $\\\\mathbb {P} \\\\left(\\\\tilde{p}_{n,\\\\omega } \\\\le \\\\mu _n - \\\\sqrt{\\\\frac{\\\\omega (1-\\\\omega )\\\\Delta }{n}} \\\\right)\\\\le \\\\frac{n\\\\sigma ^2_{n-}}{\\\\omega (1-\\\\omega )\\\\Delta }.\\\\qquad \\\\mathrm {(A.6)}$ Let $\\\\kappa _n = \\\\frac{\\\\theta +s-1}{\\\\theta +\\\\rho +n-2}$ and $\\\\varepsilon _n$ be the mode and median of $\\\\tilde{p}_{n,\\\\omega }$ , respectively.\",\n \"The mode $\\\\kappa _n$ maximizes the density function $\\\\frac{x^{\\\\theta +s-1}(1-x)^{\\\\rho +n-s-1}}{\\\\beta (\\\\theta +s, \\\\rho +n-s)}, 0 < x < 1$ .\",\n \"As the median lies between the mean and mode, we have $\\\\begin{array}{rcl}|\\\\mu _n - \\\\varepsilon _n|&\\\\le &|\\\\mu _n - \\\\kappa _n |=\\\\left| \\\\frac{\\\\theta +s}{\\\\theta +\\\\rho +n} - \\\\frac{\\\\theta +s-1}{\\\\theta +\\\\rho +n-2} \\\\right| \\\\\\\\\\\\end{array}&=&\\\\left| \\\\frac{n+\\\\rho -\\\\theta -2s}{(\\\\theta +\\\\rho +n)(\\\\theta +\\\\rho +n-2)} \\\\right| \\\\\\\\$ | +s(++n)(++n-2) | + | n+-s(++n)(++n-2) | = O(1n).\",\n \"$In the following lower-bound estimation of Eq.\",\n \"(\\\\ref {eq:lower1se}), we useGamma function, denoted by $ ()$,and its Stirling^{\\\\prime }s approximation $ (z+1)2z (z(1) )z$.$$\\\\begin{array}{rcl}&&\\\\mathbb {P} \\\\left(\\\\tilde{p}_{n,\\\\omega } \\\\le \\\\mu _n - \\\\sqrt{\\\\frac{\\\\omega (1-\\\\omega )\\\\Delta }{n}} \\\\right) \\\\\\\\\\\\end{array}&=&\\\\mathbb {P} \\\\left(\\\\tilde{p}_{n,\\\\omega } \\\\le \\\\varepsilon _n \\\\right)- \\\\int \\\\limits _{\\\\mu _n - \\\\sqrt{\\\\frac{\\\\omega (1-\\\\omega )\\\\Delta }{n}}}^{\\\\varepsilon _n}\\\\frac{x^{\\\\theta +s-1}(1-x)^{\\\\rho +n-s-1}}{\\\\beta (\\\\theta +s, \\\\rho +n-s)} \\\\mathrm {d} x \\\\\\\\$ 12 - | n-(n- (1-)n ) | n+s-1(1-n)+n-s-1(+s,+n-s) = 12 - | (1-)n + O(1n) | (+s-1++n-2)+s-1 (1-+s-1++n-2)+n-s-1(+s) (+n-s)(++n) = 12 - | (1-)n + O(1n) | (+s-1)+s-1(+s) (+n-s-1)+n-s-1(+n-s)(++n-2)++n-2(++n-1) (++n-1) 12 - (1-)n (+s-1)2(+s-1) (+n-s-1)2(+n-s-1) (++n-2)(++n-1) 2(++n-2) = 12 - (1-)n (++n-1) ++n-22(+s-1)(+n-s-1) 12 - (1-)n n n 2n n(1-) = 12 - 12.\",\n \"$$ Finally, we re-write Eq.\",\n \"(REF ) as $\\\\left(\\\\frac{1}{2}-\\\\frac{1}{\\\\sqrt{2\\\\pi }} \\\\right)\\\\omega (1-\\\\omega )\\\\Delta + O \\\\left(\\\\frac{1}{\\\\sqrt{n}} \\\\right)\\\\le n \\\\sigma ^2_{n-}\\\\le n \\\\sigma _n^2=\\\\omega (1-\\\\omega )\\\\Delta + O \\\\left(\\\\frac{1}{n} \\\\right).$ Letting $n\\\\rightarrow \\\\infty $ , we get $\\\\left(\\\\frac{1}{2}-\\\\frac{1}{\\\\sqrt{2\\\\pi }} \\\\right)\\\\omega (1-\\\\omega )\\\\Delta \\\\le \\\\liminf \\\\limits _{n\\\\rightarrow \\\\infty } n\\\\sigma _{n-}^2\\\\le \\\\limsup \\\\limits _{n\\\\rightarrow \\\\infty } n\\\\sigma _{n-}^2\\\\le \\\\omega (1-\\\\omega )\\\\Delta .$ Therefore, $\\\\Delta =0$ minimizes the limit of lower semivariance of $\\\\sqrt{n} \\\\tilde{p}_{n,\\\\omega }$ .\",\n \"We can apply similar arguments to the upper semivariance of $\\\\sqrt{n} \\\\tilde{p}_{n,\\\\omega }$ .\",\n \"$$ Note that $\\\\Delta _6-\\\\omega \\\\Delta _8 = (1-\\\\tau )(2\\\\omega -1)+\\\\omega ^2 (\\\\delta -1).$ By the proof of Lemma REF , $\\\\begin{array}{rcl}\\\\mathbb {E} [\\\\tilde{p}_{n,\\\\omega }]&=&\\\\frac{\\\\theta +s}{\\\\theta +\\\\rho +n}=\\\\frac{n^2\\\\omega + n \\\\Delta _6 + \\\\Delta _3}{n^2 + n\\\\Delta _8 + 2\\\\Delta _3}\\\\\\\\\\\\end{array}&=&\\\\omega + \\\\frac{n (\\\\Delta _6-\\\\omega \\\\Delta _8) + (1-2\\\\omega ) \\\\Delta _3}{n^2 + n\\\\Delta _8 + 2\\\\Delta _3} \\\\\\\\$ = + (1-)(2-1)+2 (-1)n + O(1n2).\",\n \"$$ In the above approximation, the mean reacts negatively with an increasing $\\\\tau $ , when $n$ is finitely large and $\\\\omega >.5$ .\",\n \"To maximize the mean, we thus minimize $\\\\tau \\\\in (1-\\\\omega +\\\\delta \\\\omega )$ such that $\\\\theta >0$ and $\\\\rho >0$ .\",\n \"Particularly, using the proof of Theorem REF , we can make it smaller than $1-\\\\omega +\\\\delta \\\\omega + \\\\frac{2c}{n}$ when $n$ is large enough, i.e.\",\n \"$1-\\\\omega +\\\\delta \\\\omega <\\\\mathop {\\\\mathrm {argmax}}\\\\limits _{\\\\tau } \\\\mathrm {MEAN}\\\\left(\\\\tilde{p}_{n,\\\\omega } \\\\bigm \\\\vert \\\\theta ,\\\\rho \\\\ge \\\\frac{1}{n} \\\\right)\\\\le 1-\\\\omega +\\\\delta \\\\omega + \\\\frac{2c}{n}.$ Finally, let $n\\\\rightarrow \\\\infty $ in the above inequalities to get $\\\\lim \\\\limits _{n\\\\rightarrow \\\\infty } \\\\mathop {\\\\mathrm {argmax}}\\\\limits _{\\\\tau } \\\\mathrm {MEAN}\\\\left(\\\\tilde{p}_{n,\\\\omega } \\\\bigm \\\\vert \\\\theta ,\\\\rho \\\\ge \\\\frac{1}{n} \\\\right)=1 -\\\\omega +\\\\delta \\\\omega .$ for any $\\\\omega \\\\in (.5,1)$ .\",\n \"$$ When $n$ is large, $\\\\theta , \\\\rho \\\\ge \\\\frac{1}{n}$ implies $\\\\tau (\\\\omega , \\\\delta ) \\\\in (1-\\\\omega +\\\\delta \\\\omega , 1)$ .\",\n \"By the proof of Lemma REF , both $\\\\theta +s\\\\rightarrow \\\\infty $ and $\\\\rho +n-s\\\\rightarrow \\\\infty $ as $n\\\\rightarrow \\\\infty $ .\",\n \"Applying Stirling's formula, Johnson et al.\",\n \"(1995, page 219) derive the following approximation for the ratio of the variance and the squared MAD around the mean: $\\\\lim \\\\limits _{\\\\theta +s\\\\rightarrow \\\\infty , \\\\rho +n-s \\\\rightarrow \\\\infty }\\\\frac{\\\\left(\\\\mathbb {E} \\\\left[ \\\\left| \\\\tilde{p}_{n,\\\\omega } - \\\\mathbb {E} (\\\\tilde{p}_{n,\\\\omega }) \\\\right| \\\\right] \\\\right)^2}{ \\\\mathrm {VAR} (\\\\tilde{p}_{n,\\\\omega })}= \\\\frac{2}{\\\\pi }.$ Thus, minimizing the MAD around the mean is equivalent to minimizing the variance of $\\\\tilde{p}_{n,\\\\omega }$ when $n$ is large.\",\n \"By Theorem REF , we have proved Theorem REF .\",\n \"$$ For any $i, j\\\\in \\\\mathbb {N}$ and $i\\\\ne j$ , if $i$ uniformly outperforms $j$ and they have equal employment opportunity, then $\\\\begin{array}{rcl}\\\\gamma _j[v]&=&\\\\sum \\\\limits _{T\\\\subseteq \\\\mathbb {N}: j\\\\in T}\\\\mathbb {P} (\\\\mathbf {S}=T) [v(T) - v(T\\\\setminus \\\\overline{j})] \\\\\\\\\\\\end{array}&=&\\\\sum \\\\limits _{T\\\\subseteq \\\\mathbb {N}: j\\\\in T, i\\\\in T}\\\\mathbb {P} (\\\\mathbf {S}=T) [v(T) - v(T\\\\setminus \\\\overline{j})] \\\\\\\\&&+\\\\sum \\\\limits _{T\\\\subseteq \\\\mathbb {N}: j\\\\in T, i\\\\notin T}\\\\mathbb {P} (\\\\mathbf {S}=T) [v(T) - v(T\\\\setminus \\\\overline{j})] \\\\\\\\$ TN: jT, iT P (S=T) [v(T) - v(Ti)] + TN: jT, iT P (S=T) [v(T) - v(Tj)] Z=Tj= TN: jT, iT P (S=T) [v(T) - v(Ti)] + ZN: jZ, iZ P (S=Zj) [v(Z j) - v(Z)] TN: jT, iT P (S=T) [v(T) - v(Ti)] + ZN: jZ, iZ P (S=Zi) [v(Z i) - v(Z)] T =Zi = TN: jT, iT P (S=T) [v(T) - v(Ti)] + TN: jT, iT P (S=T) [v(T) - v(Ti)] = TN: iT P (S=T) [v(T) - v(Ti)] = i[v].\",\n \"$$ Neither the Beta-Binomial distribution nor Eq.\",\n \"(REF ) is required in the proof.\",\n \"Also, the equality of employment opportunity is not required for other players in $\\\\mathbb {N}$ , except that $\\\\mathbb {P} (\\\\mathbf {S}=Z\\\\cup \\\\overline{i})=\\\\mathbb {P} (\\\\mathbf {S}=Z\\\\cup \\\\overline{j})$ for any $Z\\\\subseteq \\\\mathbb {N} \\\\setminus \\\\overline{i} \\\\setminus \\\\overline{j}$ .\",\n \"This identity implies that $i$ and $j$ have equal chance to be hired by $Z$ , when both are unemployed.\",\n \"Similar arguments can be used to prove $\\\\lambda _j[v] \\\\le \\\\lambda _i[v]$ .\",\n \"In this case, we use the identity $\\\\mathbb {P} (\\\\mathbf {S}=Z\\\\setminus \\\\overline{i})=\\\\mathbb {P} (\\\\mathbf {S}=Z\\\\setminus \\\\overline{j})$ for any $Z\\\\subseteq \\\\mathbb {N}$ such that $i,j \\\\in Z$ .\",\n \"This identity implies both $i$ and $j$ have equal opportunity to be laid off from $Z$ , when both are employed in $Z$ .\",\n \"$$ We distribute $v(\\\\mathbf {S})$ to $\\\\mathbb {N}$ : $(1-\\\\tau ) v(\\\\mathbf {S})$ to $\\\\mathbf {S}$ and $(\\\\tau -\\\\delta )v(\\\\mathbf {S})$ to $\\\\mathbb {N}\\\\setminus \\\\mathbf {S}$ .\",\n \"When the employed individuals are symmetric in $v$ , Theorem REF claims that each employed person receives $\\\\frac{(1-\\\\tau ) v(\\\\mathbf {S})}{|\\\\mathbf {S}| }$ as his or her employment benefits.\",\n \"Similarly, any unemployed person receives $\\\\frac{(\\\\tau -\\\\delta ) v(\\\\mathbf {S})}{n - |\\\\mathbf {S}| }$ as his or her unemployment welfare.\",\n \"Finally, $\\\\frac{(1-\\\\tau ) v(\\\\mathbf {S})}{|\\\\mathbf {S}| }= \\\\frac{(\\\\tau -\\\\delta ) v(\\\\mathbf {S})}{n - |\\\\mathbf {S}| }$ is equivalent to $\\\\frac{(1-\\\\tau ) }{\\\\omega }= \\\\frac{(\\\\tau -\\\\delta )}{1 - \\\\omega },$ which itself is equivalent to $\\\\tau = 1 - \\\\omega +\\\\delta \\\\omega .$ $$\"\n ]\n]"},"id":{"kind":"string","value":"1808.08563"}}},{"rowIdx":285,"cells":{"text":{"kind":"list like","value":[["Saliency Detection via Bidirectional Absorbing Markov Chain"],["Abstract Traditional saliency detection via Markov chain only considers boundaries nodes.","However, in addition to boundaries cues, background prior and foreground prior cues play a complementary role to enhance saliency detection.","In this paper, we propose an absorbing Markov chain based saliency detection method considering both boundary information and foreground prior cues.","The proposed approach combines both boundaries and foreground prior cues through bidirectional Markov chain.","Specifically, the image is first segmented into superpixels and four boundaries nodes (duplicated as virtual nodes) are selected.","Subsequently, the absorption time upon transition node's random walk to the absorbing state is calculated to obtain foreground possibility.","Simultaneously, foreground prior as the virtual absorbing nodes is used to calculate the absorption time and obtain the background possibility.","Finally, two obtained results are fused to obtain the combined saliency map using cost function for further optimization at multi-scale.","Experimental results demonstrate the outperformance of our proposed model on 4 benchmark datasets as compared to 17 state-of-the-art methods."],["Introduction","Saliency detection aims to effectively highlight the most important pixels in an image.","It helps to reduce computing costs and has widely been used in various computer vision applications, such as image segmentation [1], image retrieval [2], object detection [3], [4], object recognition [5], image adaptation [6], and video segmentation [7], [8].","Saliency detection could be summarized in three methods: bottom-up methods [9], [10], [11], top-down methods [12], [13] and mixed methods [14], [15], [16].","The top-down methods are driven by tasks and could be used in object detection tasks.","The authors in [17] proposed a top-down method that jointly learns a conditional random field and a discriminative dictionary.","Top-down methods could be applied to address complex and special tasks but they lack versatility.","The bottom-up methods are driven by data, such as color, light, texture and other basic features.","Itti et al [18] proposed a saliency method by using these basic features.","It could be effectively used for real-time systems.","The mixed methods are considered both bottom-up and top-down methods.","In this paper, we focus on the bottom-up methods, the proposed method is based on the properties of Markov model, there are many works based on Markov model, such as [19], [20].","Traditional saliency detection via Markov chain [21] is based on Marov model as well, but it only consider boundaries nodes.","However, in addition to boundaries cues, background prior and foreground prior cues play a complementary role to enhance saliency detection.","We consider four boundaries information and the foreground prior saliency object, using absorbing Markov chain, namely, both boundary absorbing and foreground prior are considered to get background and foreground possibility.","In addition, we further optimize our model by fusing these two possibilities, and exploite multi-scale processing.","Fig.1 demonstrates and compares the results of our proposed method with the traditional saliency detection absorbing Markov chain (MC) method [21], where the outperformance of our method is evident.","Figure: Comparison of the proposed method with the ground truth and MC method."],["Related works","There are existing many studies on saliency detection in the past decades according to the resent surveys [22], [23], [24].","And the proposed model ia based on absorbing Markov chain belonging to bottom-up method, therefore, in this section, we mainly focus on the traditional models and other newly models.","The traditional models which belong to bottom-up methods, have the features of unconscious, fast, data driven, low-level feature driven, which means without any prior knowledge, bottom-up saliency detection can achieve the goal of finding the important regions in a image.","The earliest researchers Itti and Koch model [18], [25], which compute saliency maps though low-level features, such as texture, orientation, intensity, and color contrast.","From then on, multitudinous traditional saliency models have been appeared and acquired outperformances.","Some methods based on pixel [26], [27], [28], Some methods based on superpixel [29], [30], and others based on multi-scale [31], [32], [33].","Jiang et al [21] propose saliency detection model by random walk via absorbing Markov Chain where absorbing nodes are duplicated from the four boundaries, and compute absorbing time from the transient nodes to absorbing nodes, obtain the final saliency maps.","Considering the importance of the transition probability matrix, Zhang et al [34] based on their aforementioned work propose a learnt transition probability matrix to improve the perfermance.","There are some other work based on Markov chain.","Zhang et al [35] propose an approach to detection salient objects by exporing both patch-level and object-level cues via absorbing Markov chain.","Zhang et al [36] present a data-driven salient region detection model based on absorbing Markov chain via multi-feature.","Zhu et al [37] integrate boundary connectivity based on geodesic distances into a cost function to get the final optimized saliency map.","Li et al [38] propose a saliency optimization scheme by considering both the foreground appearance and background prior.","Resent, due to the great developing of deep learning, there are numerous saliency detection models [11], [15], [39], [40] based on deep learning, which obtain outperformance than the traditional methods.","However, deep learning based methods need much dates to train, costing much time on computation, which makes the proceeding of saliency detection much complex than before.","In this work, based on the traditional method, we introduce bidirectional absorbing Markov chain to this kinds of work to get excellent performance in saliency detection."],["Fundamentals of absorbing Markov chain","In absorbing Markov chain, the transition matrix $P$ is primitive [41], by definition, state $i$ is absorbing when $P(i,i)=1$ , and $P(i,j)=0$ for all $i \\ne j$ .","If the Markov chain satisfies the following two conditions, it means there is at least one or more absorbing states in the Markov chain.","In every state, it is possible to go to an absorbing state in a finite number of steps(not necessarily in one step), then we call it absorbing Markov chain.","In an absorbing Markov chain, if a state is not a absorbing state, it is called transient state.","An absorbing chain has $m$ absorbing states and $n$ transient states, the transfer matrix $P$ can be written as: $P\\rightarrow \\left(\\begin{array}{cc}Q&R\\\\0&I\\\\\\end{array}\\right),$ where $Q$ is a n-by-n matrix, giving transient probabilities between any transient states, $R$ is a nonzero n-by-m matrix giving these probabilities from transient state to any absorbing state, 0 is a m-by-n zero matrix and $I$ is the m-by-m identity matrix.","For an absorbing chain $P$ , all the transient states can achieve absorbing states in one or more steps, we can write the expected number of times $N(i,j)$ (which means the transient state moves from $i$ state to the $j$ state), its standard form is written as: $N=(I-Q)^{-1},$ namely, the matrix $N$ with invertible matrix, where $n_{ij}$ denotes the average transfer times between transient state $i$ to transient state $j$ .","Supposing $c=[1,1,\\cdot \\cdot \\cdot ,1]_{1\\times n}^{N}$ , the absorbed time for each transient state can be expressed as: $z=N\\times c.$"],["The proposed approach","To obtain more robust and accurate saliency maps, we propose a method via bidirectional absorbing Markov chain.","This section explains the procedure to find the saliency area in an image in two orientations.","Simple linear iterative clustering(SLIC) algorithm [42] has been used to get the superpixels.","The pipeline is explained below: Figure: The processing of our proposed method"],["Graph construction","The SLIC algorithm is used to split the image into different pitches of superpixels.","Afterwards, two kinds of graphs $G^1(V^1,E^1)$ and $G^2(V^2,E^2)$ are constructed, see Figure REF for detail, Figure: The graph construction.","(a) The superpixels in the yellow area are the duplicated absorbing nodes.","(b) The foreground cues are obtained and regarded as absorbing nodes which are duplicated with red area."],["Graph construction","where $G^1$ represents the graph of boundary absorbing process and $G^2$ represents the graph of foreground prior absorbing process.","In each of the graphs, $V^1, V^2$ represent the graph nodes and $E^1,E^2$ represent the edges between any nodes in the graphs.","For the process of boundary absorbing, superpixels around the four boundaries as the virtual nodes are duplicated.","For the process of foreground prior absorbing, superpixels from the regions (calculated by the foreground prior) are duplicated.","There are two kinds of nodes in both graphs, transient nodes (superpixels) and absorbing nodes (duplicated nodes).","The nodes in these two graphs constitute following three properties: (1) The nodes (including transient or absorbing) are associated with each other when superpixels in the image are adjacent nodes or have the same neighbors.","And also boundary nodes (superpixels on the boundary of image) are fully connected with each other to reduce the geodesic distance between similar superpixels.","(2) Any pair of absorbing nodes (which are duplicated from the boundaries or foreground nodes) are not connected (3) The nodes, which are duplicated from the four boundaries or foreground prior nodes, are also connected with original duplicated nodes.","In this paper, the weight $w_{ij}$ of the edges is defined as $w_{ij}=e^{-\\frac{\\left\\Vert x_{i}-x_{j} \\right\\Vert }{\\sigma ^{^{2}}}}, i,j \\in V^1 \\ \\text{or} \\ i,j \\in V^2$ where $\\sigma $ is the constant parameter to adjust the strength of the weights in CIELAB color space.","Then we can get the affinity matrix $A$ $a_{ij}={\\left\\lbrace \\begin{array}{ll}w_{ij}, &\\text{ if $j\\in M(i)$ \\quad $1 \\le i \\le j$} \\\\1, &\\text{ if $i=j$ } \\\\0, &\\text{ otherwise},\\end{array}\\right.","}$ where $M(i)$ is a nodes set, in which the nodes are all connected to nodes $i$ .","The diagonal matrix is given as: $D = diag(\\sum _{j}a_{ij})$ , and the obtained transient matrix is calculated as: $P = D^{-1} \\times A.$"],["Saliency detection model","Following the aforementioned procedures, the initial image is transformed into superpixels, now two kinds of absorbing nodes for saliency detection are required.","Firstly, we choose boundary nodes and foreground prior nodes to duplicate as absorbing nodes and obtain the absorbed times of transient nodes as foreground possibility and background possibility.","Secondly, we use a cost function to optimize two possibility results together and obtain saliency results of all transient nodes."],["Absorb Markov chain via boundary nodes","In normal conditions, four boundaries of an image rarely have salient objects.","Therefore, boundary nodes are assumed as background, and four boundaries nodes set $H^1$ are duplicated as absorbing nodes set $D^1$ , $H^1,D^1 \\subset V^1$ .","The graph $G^1$ is constructed and absorbed time $z$ is calculated via Eq.REF .","Finally, foreground possibility of transient nodes $z^f = \\bar{z}(i) \\quad i=1,2,\\cdot \\cdot \\cdot ,n,$ is obtained, and $\\bar{z}$ denotes the normalizing the absorbed time vector."],["Absorb Markov chain via foreground prior nodes","We use boundary connectivity to get the foreground prior $\\textbf {f}_i$ without using the down-top method [37].","$f_i = \\sum ^N_{j=1} (1 - \\mathrm {exp}\\big (-\\frac{BC_j^2}{2\\sigma _b^2}\\big ))d_a(i,j)\\mathrm {exp}\\big (-\\frac{d_s^2(i,j)}{2\\sigma _s^2}\\big )$ where $d_a(i,j)$ and $d_s(i,j)$ denote the CIELAB color feature distance and spatial distance respectively between superpixel $i$ and $j$ , the boundary connectivity (BC) of superpixel $i$ is defined as $BC_i = \\frac{\\sum _{j\\in \\mathcal {H}}w_{ij}}{\\sqrt{\\sum ^N_{j=1}w_{ij}}}$ in Fig.","REF , $\\sigma _b = 1 $ , $\\sigma _s = 0.25 $ .","$\\mathcal {H}$ denotes the boundary area of image and $w_{ij}$ is the similarity between nodes $i$ and $j$ .","$N$ is the number of superpixels.","Afterwards, nodes ($\\lbrace i|f_i > avg(f)\\rbrace $ ) with high level values are selected to get a set $H^2$ , which are duplicated as absorbing nodes set $D^2$ , $H^2,D^2 \\subset V^2$ .","The graph $G^2$ is constructed and absorbed time $z$ is calculated using Eq.REF .","Finally, the background possibility of transient nodes $z^b = \\bar{z}(i) \\quad i=1,2,\\cdot \\cdot \\cdot ,n,$ is obtained, where $\\bar{z}$ denotes the absorbed time vector normalization.","Figure: An illustrative example of boundary connectivity.","(a) input image (b) the superpixels of input image (c) the superpixels of similarity in each pitches (d) an illustrative example of boundary connectivity."],["Saliency Optimization","In order to combine different cues, this paper has used the optimization model presented in [37], which fused background possibility and foreground possibility for final saliency map.","It is defined as $\\sum _{i=1}^{N}z^b_{i}s_{i}^{2}+\\sum _{i=1}^{N}z^f_{i}(s_{i}-1)^{2}+\\sum _{i,j}w_{ij}(s_{i}-s_{j})^{2}$ where the first term defines superpixel $i$ with large background probability $z^b$ to obtain a small value $s_i$ (close to 0).","The second term encourages a superpixel $i$ with large foreground probability $z^f$ to obtain a large value $s_i$ (close to 1).","The third term defines the smoothness to acquire continuous saliency values.","In this work, the used super-pixel numbers $N$ are 200, 250, 300 in the superpixel element, and the final saliency map is given as: $\\textbf {S} = \\sum _h{S^h}$ at each scale, where $h = 1, 2, 3$ .","The algorithm of our proposed method is summarized in Algorithm REF .","[htb] Saliency detection by bidirectional Markov chain.","[1] An image and required parameters as the input image.","Using SLIC method to segment the input image into superpixels, construct two graphs $G^1(V^1,E^1)$ and $G^2(V^2,E^2)$ ; Duplicate the four boundaries nodes and foreground prior nodes(which is computed by Eq.","REF ) as the absorbing nodes on graphs $G^1(V^1,E^1)$ and $G^2(V^2,E^2)$ , respectively; Compute the transient matrix $P$ with Eg.","REF on both graphs; Obtain the absorbed time $z^f$ and $z^b$ with Eq.","REF , respectively; Optimize the model with the Eq.","REF ; Compute final saliency values with $\\textbf {S} = \\sum _{h=1}^{3}\\textbf {S}^h$ ; Output a saliency map $\\textbf {S}$ with the same size as the input image."],["Experiments","The proposed method is evaluated on four benchmark datasets ASD [43], CSSD [31], ECSSD [31] and SED [44].","ASD dataset is a subset of the MSRA dataset, which contains 1000 images with accurate human-labeled ground truth.","CSSD dataset, namely complex scene saliency detection contains 200 complex images.","ECSSD dataset, an extension of CSSD dataset contains 1000 images and has accurate human-labeled ground truth.","SED dataset has two parts, SED1 and SED2, images in SED1 contains one object, and images in SED2 contains two objects, in total they contain 200 images.","We compare our model with 17 different state-of-the-art saliency detection algorithms: CA [45], FT [43], SEG [46], BM [47], SWD [48], SF [49], GCHC [50], LMLC [51], HS [31], PCA [30], DSR [52], MC [21], MR [10], MS [53], RBD [37], RR [54], MST [55].","The tuning parameters in the proposed algorithm is the edge weight $\\sigma ^2=0.1$ that controls the strength of weight between a pair of nodes.","In the following results of the experiments, we show the evaluation of our proposed saliency models based on the aforementioned datasets comparing with the best works.","In addition, we also give some limitation about our model and analysis the reason."],["Evaluation of the proposed model","The precision-recall curves and F-measure are used as performance metrics.","The precision is defined as the ratio of salient pixels correctly assigned to all the pixels of extracted regions.","The recall is defined as the ratio of detected salient pixels to the ground-truth number.","Which can be fomulated as, $Precision = \\frac{TP}{TP+FP},Recall = \\frac{TP}{TP+FN}£¬$ where $TP$ , $FP$ and $FN$ represent the true positive, false positive and false negative, respectively.","A PR curve is obtained by the threshold sliding from 0 to 255 to get the difference between the saliency map (which is calculated) and ground truth(which is labeled manually).","F-measure is calculated by the weighted average between the precision values and recall values, which can be regarded as overall performance measurement, given as: $F_{\\beta } = \\frac{(1+\\beta ^{2})Precision \\times Recall}{\\beta ^{2}Precision + Recall},$ we set $\\beta ^{2} = 0.3$ to stress precision more than recall.","PR-curves and the F-measure curves are shown in Figure REF - REF , where the outperformance of our proposed method as compared to 17 state-of-the-art methods is evident.","Fig.REF presets visual comparisons selected from four datasets.","It can be seen that the proposed method achieved best saliency results as compared to the state-of-the-art methods.","Figure: PR-curves and F-measure curves comparing with different methods on ASD dataset.Figure: PR-curves and F-value curves comparing with different methods on ECSSD dataset.Figure: PR-curves and F-measure curves comparing with different methods on CSSD dataset.Figure: PR-curves and F-measure curves comparing with different methods on SED dataset.Figure: Examples of output saliency maps results using different algorithms on the ASD, CSSD, ECSSD and SED datasets"],["Failure cases analysis","In this work, the idea of bidirectional absorbing Markov chain is first proposed.","Although the proposed method is effective for most images on the four datasets.","However, if the appearances of four boundaries and the foreground prior are similar to each other, the performance is not obviously, which is shown in Figure REF .","Figure: Examples of our failure examples.","(a) Input images.","(b) Ground truth.","(c) Saliency maps."],["Conclusion","In this paper, a bidirectional absorbing Markov chain based saliency detection method is proposed considering both boundary information and foreground prior cues.","A novel optimization model is developed to combine both background and foreground possibilities, acquired through bidirectional absorbing Markov chain.","The proposed approach outperformed 17 different state-of-the-art methods over four benchmark datasets, which demonstrate the superiority of our proposed approach.","In future, we intend to apply our proposed saliency detection algorithm to problems such as multi-pose lipreading and audio-visual speech recognition."],["Acknowledgments","This work was supported by China Scholarship Council, the National Natural Science Foundation of China (No.913203002), the Pilot Project of Chinese Academy of Sciences (No.XDA08040109).","Prof. Amir Hussain and Dr. Ahsan Adeel were supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant No.EP/M026981/1."]],"string":"[\n [\n \"Saliency Detection via Bidirectional Absorbing Markov Chain\"\n ],\n [\n \"Abstract Traditional saliency detection via Markov chain only considers boundaries nodes.\",\n \"However, in addition to boundaries cues, background prior and foreground prior cues play a complementary role to enhance saliency detection.\",\n \"In this paper, we propose an absorbing Markov chain based saliency detection method considering both boundary information and foreground prior cues.\",\n \"The proposed approach combines both boundaries and foreground prior cues through bidirectional Markov chain.\",\n \"Specifically, the image is first segmented into superpixels and four boundaries nodes (duplicated as virtual nodes) are selected.\",\n \"Subsequently, the absorption time upon transition node's random walk to the absorbing state is calculated to obtain foreground possibility.\",\n \"Simultaneously, foreground prior as the virtual absorbing nodes is used to calculate the absorption time and obtain the background possibility.\",\n \"Finally, two obtained results are fused to obtain the combined saliency map using cost function for further optimization at multi-scale.\",\n \"Experimental results demonstrate the outperformance of our proposed model on 4 benchmark datasets as compared to 17 state-of-the-art methods.\"\n ],\n [\n \"Introduction\",\n \"Saliency detection aims to effectively highlight the most important pixels in an image.\",\n \"It helps to reduce computing costs and has widely been used in various computer vision applications, such as image segmentation [1], image retrieval [2], object detection [3], [4], object recognition [5], image adaptation [6], and video segmentation [7], [8].\",\n \"Saliency detection could be summarized in three methods: bottom-up methods [9], [10], [11], top-down methods [12], [13] and mixed methods [14], [15], [16].\",\n \"The top-down methods are driven by tasks and could be used in object detection tasks.\",\n \"The authors in [17] proposed a top-down method that jointly learns a conditional random field and a discriminative dictionary.\",\n \"Top-down methods could be applied to address complex and special tasks but they lack versatility.\",\n \"The bottom-up methods are driven by data, such as color, light, texture and other basic features.\",\n \"Itti et al [18] proposed a saliency method by using these basic features.\",\n \"It could be effectively used for real-time systems.\",\n \"The mixed methods are considered both bottom-up and top-down methods.\",\n \"In this paper, we focus on the bottom-up methods, the proposed method is based on the properties of Markov model, there are many works based on Markov model, such as [19], [20].\",\n \"Traditional saliency detection via Markov chain [21] is based on Marov model as well, but it only consider boundaries nodes.\",\n \"However, in addition to boundaries cues, background prior and foreground prior cues play a complementary role to enhance saliency detection.\",\n \"We consider four boundaries information and the foreground prior saliency object, using absorbing Markov chain, namely, both boundary absorbing and foreground prior are considered to get background and foreground possibility.\",\n \"In addition, we further optimize our model by fusing these two possibilities, and exploite multi-scale processing.\",\n \"Fig.1 demonstrates and compares the results of our proposed method with the traditional saliency detection absorbing Markov chain (MC) method [21], where the outperformance of our method is evident.\",\n \"Figure: Comparison of the proposed method with the ground truth and MC method.\"\n ],\n [\n \"Related works\",\n \"There are existing many studies on saliency detection in the past decades according to the resent surveys [22], [23], [24].\",\n \"And the proposed model ia based on absorbing Markov chain belonging to bottom-up method, therefore, in this section, we mainly focus on the traditional models and other newly models.\",\n \"The traditional models which belong to bottom-up methods, have the features of unconscious, fast, data driven, low-level feature driven, which means without any prior knowledge, bottom-up saliency detection can achieve the goal of finding the important regions in a image.\",\n \"The earliest researchers Itti and Koch model [18], [25], which compute saliency maps though low-level features, such as texture, orientation, intensity, and color contrast.\",\n \"From then on, multitudinous traditional saliency models have been appeared and acquired outperformances.\",\n \"Some methods based on pixel [26], [27], [28], Some methods based on superpixel [29], [30], and others based on multi-scale [31], [32], [33].\",\n \"Jiang et al [21] propose saliency detection model by random walk via absorbing Markov Chain where absorbing nodes are duplicated from the four boundaries, and compute absorbing time from the transient nodes to absorbing nodes, obtain the final saliency maps.\",\n \"Considering the importance of the transition probability matrix, Zhang et al [34] based on their aforementioned work propose a learnt transition probability matrix to improve the perfermance.\",\n \"There are some other work based on Markov chain.\",\n \"Zhang et al [35] propose an approach to detection salient objects by exporing both patch-level and object-level cues via absorbing Markov chain.\",\n \"Zhang et al [36] present a data-driven salient region detection model based on absorbing Markov chain via multi-feature.\",\n \"Zhu et al [37] integrate boundary connectivity based on geodesic distances into a cost function to get the final optimized saliency map.\",\n \"Li et al [38] propose a saliency optimization scheme by considering both the foreground appearance and background prior.\",\n \"Resent, due to the great developing of deep learning, there are numerous saliency detection models [11], [15], [39], [40] based on deep learning, which obtain outperformance than the traditional methods.\",\n \"However, deep learning based methods need much dates to train, costing much time on computation, which makes the proceeding of saliency detection much complex than before.\",\n \"In this work, based on the traditional method, we introduce bidirectional absorbing Markov chain to this kinds of work to get excellent performance in saliency detection.\"\n ],\n [\n \"Fundamentals of absorbing Markov chain\",\n \"In absorbing Markov chain, the transition matrix $P$ is primitive [41], by definition, state $i$ is absorbing when $P(i,i)=1$ , and $P(i,j)=0$ for all $i \\\\ne j$ .\",\n \"If the Markov chain satisfies the following two conditions, it means there is at least one or more absorbing states in the Markov chain.\",\n \"In every state, it is possible to go to an absorbing state in a finite number of steps(not necessarily in one step), then we call it absorbing Markov chain.\",\n \"In an absorbing Markov chain, if a state is not a absorbing state, it is called transient state.\",\n \"An absorbing chain has $m$ absorbing states and $n$ transient states, the transfer matrix $P$ can be written as: $P\\\\rightarrow \\\\left(\\\\begin{array}{cc}Q&R\\\\\\\\0&I\\\\\\\\\\\\end{array}\\\\right),$ where $Q$ is a n-by-n matrix, giving transient probabilities between any transient states, $R$ is a nonzero n-by-m matrix giving these probabilities from transient state to any absorbing state, 0 is a m-by-n zero matrix and $I$ is the m-by-m identity matrix.\",\n \"For an absorbing chain $P$ , all the transient states can achieve absorbing states in one or more steps, we can write the expected number of times $N(i,j)$ (which means the transient state moves from $i$ state to the $j$ state), its standard form is written as: $N=(I-Q)^{-1},$ namely, the matrix $N$ with invertible matrix, where $n_{ij}$ denotes the average transfer times between transient state $i$ to transient state $j$ .\",\n \"Supposing $c=[1,1,\\\\cdot \\\\cdot \\\\cdot ,1]_{1\\\\times n}^{N}$ , the absorbed time for each transient state can be expressed as: $z=N\\\\times c.$\"\n ],\n [\n \"The proposed approach\",\n \"To obtain more robust and accurate saliency maps, we propose a method via bidirectional absorbing Markov chain.\",\n \"This section explains the procedure to find the saliency area in an image in two orientations.\",\n \"Simple linear iterative clustering(SLIC) algorithm [42] has been used to get the superpixels.\",\n \"The pipeline is explained below: Figure: The processing of our proposed method\"\n ],\n [\n \"Graph construction\",\n \"The SLIC algorithm is used to split the image into different pitches of superpixels.\",\n \"Afterwards, two kinds of graphs $G^1(V^1,E^1)$ and $G^2(V^2,E^2)$ are constructed, see Figure REF for detail, Figure: The graph construction.\",\n \"(a) The superpixels in the yellow area are the duplicated absorbing nodes.\",\n \"(b) The foreground cues are obtained and regarded as absorbing nodes which are duplicated with red area.\"\n ],\n [\n \"Graph construction\",\n \"where $G^1$ represents the graph of boundary absorbing process and $G^2$ represents the graph of foreground prior absorbing process.\",\n \"In each of the graphs, $V^1, V^2$ represent the graph nodes and $E^1,E^2$ represent the edges between any nodes in the graphs.\",\n \"For the process of boundary absorbing, superpixels around the four boundaries as the virtual nodes are duplicated.\",\n \"For the process of foreground prior absorbing, superpixels from the regions (calculated by the foreground prior) are duplicated.\",\n \"There are two kinds of nodes in both graphs, transient nodes (superpixels) and absorbing nodes (duplicated nodes).\",\n \"The nodes in these two graphs constitute following three properties: (1) The nodes (including transient or absorbing) are associated with each other when superpixels in the image are adjacent nodes or have the same neighbors.\",\n \"And also boundary nodes (superpixels on the boundary of image) are fully connected with each other to reduce the geodesic distance between similar superpixels.\",\n \"(2) Any pair of absorbing nodes (which are duplicated from the boundaries or foreground nodes) are not connected (3) The nodes, which are duplicated from the four boundaries or foreground prior nodes, are also connected with original duplicated nodes.\",\n \"In this paper, the weight $w_{ij}$ of the edges is defined as $w_{ij}=e^{-\\\\frac{\\\\left\\\\Vert x_{i}-x_{j} \\\\right\\\\Vert }{\\\\sigma ^{^{2}}}}, i,j \\\\in V^1 \\\\ \\\\text{or} \\\\ i,j \\\\in V^2$ where $\\\\sigma $ is the constant parameter to adjust the strength of the weights in CIELAB color space.\",\n \"Then we can get the affinity matrix $A$ $a_{ij}={\\\\left\\\\lbrace \\\\begin{array}{ll}w_{ij}, &\\\\text{ if $j\\\\in M(i)$ \\\\quad $1 \\\\le i \\\\le j$} \\\\\\\\1, &\\\\text{ if $i=j$ } \\\\\\\\0, &\\\\text{ otherwise},\\\\end{array}\\\\right.\",\n \"}$ where $M(i)$ is a nodes set, in which the nodes are all connected to nodes $i$ .\",\n \"The diagonal matrix is given as: $D = diag(\\\\sum _{j}a_{ij})$ , and the obtained transient matrix is calculated as: $P = D^{-1} \\\\times A.$\"\n ],\n [\n \"Saliency detection model\",\n \"Following the aforementioned procedures, the initial image is transformed into superpixels, now two kinds of absorbing nodes for saliency detection are required.\",\n \"Firstly, we choose boundary nodes and foreground prior nodes to duplicate as absorbing nodes and obtain the absorbed times of transient nodes as foreground possibility and background possibility.\",\n \"Secondly, we use a cost function to optimize two possibility results together and obtain saliency results of all transient nodes.\"\n ],\n [\n \"Absorb Markov chain via boundary nodes\",\n \"In normal conditions, four boundaries of an image rarely have salient objects.\",\n \"Therefore, boundary nodes are assumed as background, and four boundaries nodes set $H^1$ are duplicated as absorbing nodes set $D^1$ , $H^1,D^1 \\\\subset V^1$ .\",\n \"The graph $G^1$ is constructed and absorbed time $z$ is calculated via Eq.REF .\",\n \"Finally, foreground possibility of transient nodes $z^f = \\\\bar{z}(i) \\\\quad i=1,2,\\\\cdot \\\\cdot \\\\cdot ,n,$ is obtained, and $\\\\bar{z}$ denotes the normalizing the absorbed time vector.\"\n ],\n [\n \"Absorb Markov chain via foreground prior nodes\",\n \"We use boundary connectivity to get the foreground prior $\\\\textbf {f}_i$ without using the down-top method [37].\",\n \"$f_i = \\\\sum ^N_{j=1} (1 - \\\\mathrm {exp}\\\\big (-\\\\frac{BC_j^2}{2\\\\sigma _b^2}\\\\big ))d_a(i,j)\\\\mathrm {exp}\\\\big (-\\\\frac{d_s^2(i,j)}{2\\\\sigma _s^2}\\\\big )$ where $d_a(i,j)$ and $d_s(i,j)$ denote the CIELAB color feature distance and spatial distance respectively between superpixel $i$ and $j$ , the boundary connectivity (BC) of superpixel $i$ is defined as $BC_i = \\\\frac{\\\\sum _{j\\\\in \\\\mathcal {H}}w_{ij}}{\\\\sqrt{\\\\sum ^N_{j=1}w_{ij}}}$ in Fig.\",\n \"REF , $\\\\sigma _b = 1 $ , $\\\\sigma _s = 0.25 $ .\",\n \"$\\\\mathcal {H}$ denotes the boundary area of image and $w_{ij}$ is the similarity between nodes $i$ and $j$ .\",\n \"$N$ is the number of superpixels.\",\n \"Afterwards, nodes ($\\\\lbrace i|f_i > avg(f)\\\\rbrace $ ) with high level values are selected to get a set $H^2$ , which are duplicated as absorbing nodes set $D^2$ , $H^2,D^2 \\\\subset V^2$ .\",\n \"The graph $G^2$ is constructed and absorbed time $z$ is calculated using Eq.REF .\",\n \"Finally, the background possibility of transient nodes $z^b = \\\\bar{z}(i) \\\\quad i=1,2,\\\\cdot \\\\cdot \\\\cdot ,n,$ is obtained, where $\\\\bar{z}$ denotes the absorbed time vector normalization.\",\n \"Figure: An illustrative example of boundary connectivity.\",\n \"(a) input image (b) the superpixels of input image (c) the superpixels of similarity in each pitches (d) an illustrative example of boundary connectivity.\"\n ],\n [\n \"Saliency Optimization\",\n \"In order to combine different cues, this paper has used the optimization model presented in [37], which fused background possibility and foreground possibility for final saliency map.\",\n \"It is defined as $\\\\sum _{i=1}^{N}z^b_{i}s_{i}^{2}+\\\\sum _{i=1}^{N}z^f_{i}(s_{i}-1)^{2}+\\\\sum _{i,j}w_{ij}(s_{i}-s_{j})^{2}$ where the first term defines superpixel $i$ with large background probability $z^b$ to obtain a small value $s_i$ (close to 0).\",\n \"The second term encourages a superpixel $i$ with large foreground probability $z^f$ to obtain a large value $s_i$ (close to 1).\",\n \"The third term defines the smoothness to acquire continuous saliency values.\",\n \"In this work, the used super-pixel numbers $N$ are 200, 250, 300 in the superpixel element, and the final saliency map is given as: $\\\\textbf {S} = \\\\sum _h{S^h}$ at each scale, where $h = 1, 2, 3$ .\",\n \"The algorithm of our proposed method is summarized in Algorithm REF .\",\n \"[htb] Saliency detection by bidirectional Markov chain.\",\n \"[1] An image and required parameters as the input image.\",\n \"Using SLIC method to segment the input image into superpixels, construct two graphs $G^1(V^1,E^1)$ and $G^2(V^2,E^2)$ ; Duplicate the four boundaries nodes and foreground prior nodes(which is computed by Eq.\",\n \"REF ) as the absorbing nodes on graphs $G^1(V^1,E^1)$ and $G^2(V^2,E^2)$ , respectively; Compute the transient matrix $P$ with Eg.\",\n \"REF on both graphs; Obtain the absorbed time $z^f$ and $z^b$ with Eq.\",\n \"REF , respectively; Optimize the model with the Eq.\",\n \"REF ; Compute final saliency values with $\\\\textbf {S} = \\\\sum _{h=1}^{3}\\\\textbf {S}^h$ ; Output a saliency map $\\\\textbf {S}$ with the same size as the input image.\"\n ],\n [\n \"Experiments\",\n \"The proposed method is evaluated on four benchmark datasets ASD [43], CSSD [31], ECSSD [31] and SED [44].\",\n \"ASD dataset is a subset of the MSRA dataset, which contains 1000 images with accurate human-labeled ground truth.\",\n \"CSSD dataset, namely complex scene saliency detection contains 200 complex images.\",\n \"ECSSD dataset, an extension of CSSD dataset contains 1000 images and has accurate human-labeled ground truth.\",\n \"SED dataset has two parts, SED1 and SED2, images in SED1 contains one object, and images in SED2 contains two objects, in total they contain 200 images.\",\n \"We compare our model with 17 different state-of-the-art saliency detection algorithms: CA [45], FT [43], SEG [46], BM [47], SWD [48], SF [49], GCHC [50], LMLC [51], HS [31], PCA [30], DSR [52], MC [21], MR [10], MS [53], RBD [37], RR [54], MST [55].\",\n \"The tuning parameters in the proposed algorithm is the edge weight $\\\\sigma ^2=0.1$ that controls the strength of weight between a pair of nodes.\",\n \"In the following results of the experiments, we show the evaluation of our proposed saliency models based on the aforementioned datasets comparing with the best works.\",\n \"In addition, we also give some limitation about our model and analysis the reason.\"\n ],\n [\n \"Evaluation of the proposed model\",\n \"The precision-recall curves and F-measure are used as performance metrics.\",\n \"The precision is defined as the ratio of salient pixels correctly assigned to all the pixels of extracted regions.\",\n \"The recall is defined as the ratio of detected salient pixels to the ground-truth number.\",\n \"Which can be fomulated as, $Precision = \\\\frac{TP}{TP+FP},Recall = \\\\frac{TP}{TP+FN}£¬$ where $TP$ , $FP$ and $FN$ represent the true positive, false positive and false negative, respectively.\",\n \"A PR curve is obtained by the threshold sliding from 0 to 255 to get the difference between the saliency map (which is calculated) and ground truth(which is labeled manually).\",\n \"F-measure is calculated by the weighted average between the precision values and recall values, which can be regarded as overall performance measurement, given as: $F_{\\\\beta } = \\\\frac{(1+\\\\beta ^{2})Precision \\\\times Recall}{\\\\beta ^{2}Precision + Recall},$ we set $\\\\beta ^{2} = 0.3$ to stress precision more than recall.\",\n \"PR-curves and the F-measure curves are shown in Figure REF - REF , where the outperformance of our proposed method as compared to 17 state-of-the-art methods is evident.\",\n \"Fig.REF presets visual comparisons selected from four datasets.\",\n \"It can be seen that the proposed method achieved best saliency results as compared to the state-of-the-art methods.\",\n \"Figure: PR-curves and F-measure curves comparing with different methods on ASD dataset.Figure: PR-curves and F-value curves comparing with different methods on ECSSD dataset.Figure: PR-curves and F-measure curves comparing with different methods on CSSD dataset.Figure: PR-curves and F-measure curves comparing with different methods on SED dataset.Figure: Examples of output saliency maps results using different algorithms on the ASD, CSSD, ECSSD and SED datasets\"\n ],\n [\n \"Failure cases analysis\",\n \"In this work, the idea of bidirectional absorbing Markov chain is first proposed.\",\n \"Although the proposed method is effective for most images on the four datasets.\",\n \"However, if the appearances of four boundaries and the foreground prior are similar to each other, the performance is not obviously, which is shown in Figure REF .\",\n \"Figure: Examples of our failure examples.\",\n \"(a) Input images.\",\n \"(b) Ground truth.\",\n \"(c) Saliency maps.\"\n ],\n [\n \"Conclusion\",\n \"In this paper, a bidirectional absorbing Markov chain based saliency detection method is proposed considering both boundary information and foreground prior cues.\",\n \"A novel optimization model is developed to combine both background and foreground possibilities, acquired through bidirectional absorbing Markov chain.\",\n \"The proposed approach outperformed 17 different state-of-the-art methods over four benchmark datasets, which demonstrate the superiority of our proposed approach.\",\n \"In future, we intend to apply our proposed saliency detection algorithm to problems such as multi-pose lipreading and audio-visual speech recognition.\"\n ],\n [\n \"Acknowledgments\",\n \"This work was supported by China Scholarship Council, the National Natural Science Foundation of China (No.913203002), the Pilot Project of Chinese Academy of Sciences (No.XDA08040109).\",\n \"Prof. Amir Hussain and Dr. Ahsan Adeel were supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant No.EP/M026981/1.\"\n ]\n]"},"id":{"kind":"string","value":"1808.08393"}}},{"rowIdx":286,"cells":{"text":{"kind":"list like","value":[["Causes of Effects via a Bayesian Model Selection Procedure"],["Abstract In causal inference, and specifically in the \\textit{Causes of Effects} problem, one is interested in how to use statistical evidence to understand causation in an individual case, and so how to assess the so-called {\\em probability of causation} (PC).","The answer relies on the potential responses, which can incorporate information about what would have happened to the outcome as we had observed a different value of the exposure.","However, even given the best possible statistical evidence for the association between exposure and outcome, we can typically only provide bounds for the PC.","Dawid et al.","(2016) highlighted some fundamental conditions, namely, exogeneity, comparability, and sufficiency, required to obtain such bounds, based on experimental data.","The aim of the present paper is to provide methods to find, in specific cases, the best subsample of the reference dataset to satisfy such requirements.","To this end, we introduce a new variable, expressing the desire to be exposed or not, and we set the question up as a model selection problem.","The best model will be selected using the marginal probability of the responses and a suitable prior proposal over the model space.","An application in the educational field is presented."],["Introduction","The Causes of Effects (CoE) problem concerns the study of individual causation and explicitly refers to something happened to a well identified individual.","This nuance of causation has received less attention than the study of the Effects of Causes (EoC), also called the general causation problem.","Actually, in EoC, the aim is the prediction of an outcome after the realization of an alleged cause; in CoE, instead, we want to evaluate the Probability of Causation (PC), i.e.","the probability the observed outcome would not have been realized if the alleged cause had not been made effective, despite the fact that the cause and the outcome were already observed.","Since we always refer causation to a specific individual, for the sake of simplicity, we call her Ann.","A simplified CoE question is the following.","“Ann had a headache and decided to take aspirin.","Her headache went away.","Was that caused by the aspirin?” In CoE, to evaluate the probability of causation, formally defined in Section , the relevant questions are: How might one use experimental and/or observational data, gathered on a reference group to which Ann belongs?","How would one find the characteristics shared with Ann by the group of individuals from which the data came?","These problems, also called the “Group to individual” (G2i) issue in forensic science, have generated a large debate in legal circles (see Faigman et al.","(2014) [10]).","The issue was extensively studied in the statistical literature from a technical and a philosophical point of view, reaching different and somewhat related results.","Essential references are Dawid (2000; 2016), [6], [3] and Pearl (2009; 2015) [14], [15].","Even if their approaches follow different routes, they agree about the use of counterfactuals and potential outcomes originating from Neyman ([12]) and re-introduced in the modern literature by Rubin (1974) ([17])Rubin's approach has been mainly used to solve EoC-type problems: for this reason it will be not considered in this paper..","The need for counterfactuals in CoE is well illustrated by the question: “What would be the probability of the response if no treatment had been provided to the individual in whom we are interested?” Since only the treatment can be assigned to the individual, the matter concerns a counterfactual, i.e.","an event which assumes something different from what happened.","Depending on the assumptions, different results emerge in the forms of a precise probability or bounds (see Section ).","Although not as specific as one would ideally like, such bounds can be of use.","In particular, if the lower bound exceeds one-half, then, in civil cases, we can infer causality “on the balance of probabilities”.","To evaluate the PC, we follow the approach of Dawid et al.","(2016) [6], that identified and detailed three fundamental conditions used to estimate from the data upper and lower bounds for the probability of causation.","A related problem is the choice of the reference population.","More specifically, how to find that group, from among those obtainable from partitioning the randomized experiment sample according to Ann's characteristics, which best fulfills the fundamental conditions.","How much information to take into account and how to select the best comparison group is a tricky issue, also because the choice of the reference class can affect significantly the conclusions of the inference.","The main aim of the present paper is to make the fundamental conditions operative by means of empirical testing of such underlying assumptions, in a way that comes up with the “best” comparison group.","We set the question up as a Bayesian model selection problem, where each model specifies a particular choice (more or less detailed) for the characteristics to be included, shared by Ann and the other individuals participating in the study.","The best model will be selected considering the marginal probabilities of the response and a satisfactory prior proposal in the model space.","To this end, we introduce a new variable that expresses, for each individual in the study, the desire to receive the treatment or not.","This variable allows introducing the fundamental conditions in the model selection procedure.","Such a method can have various applications in sociology and education, as well as in medicine.","We present an example in the field of education, where we investigate the relation between success in a test and whether or not a hint was received (taking into account the student's preference for receiving or not receiving the hint).","The structure of this paper is as follows.","We first introduce the notation and we define formally the Probability of Causation (PC).","After the review of some results in the CoE literature, see Section , we provide the assumptions we require, Section .","Then, in Section , we detail how to find the reference sample suitable for evaluating CoE by a model selection procedure.","After presenting the application in Section , we draw some final conclusions."],["Notation","We first specify the notation we need to introduce our approach.","Given data obtained from an ideal large randomized study concerning $n$ individuals, drawn from a population to which Ann belongs, assume that the study records the outcome $R=\\lbrace 0,1\\rbrace $ of a treatment $T=\\lbrace 0,1\\rbrace $ , which is supposed large enough that the sampling variability of the estimates is negligible.","Let $H$ be a large set of variables $H=\\lbrace H_1,\\dots ,H_k\\rbrace $ , characteristic of both Ann and the individuals participating in the study, where we assume that each $H_j$ , $j\\in 1,\\dots ,k$ is discrete (or even dichotomous).","We denote by $H^{A}$ the value that the set of variables $H$ has for Ann.","Since Ann not only takes the aspirin but also expresses her desire to take it, we introduce a variable $E=\\lbrace 0,1\\rbrace $ (for Ann $E=1$ ) expressing such a desire for each individual in the study.","This variable, first proposed as an unobservable by Dawid (2011), is here considered observable.","We also introduce the potential variables $R_T=\\lbrace 0,1\\rbrace $ , so that, if the triple $(T,R_0,R_1)$ were observed contemporaneously, it would be easy to solve the CoE problem by stating the probability of causation (PC), defined as $PC_A& \\stackrel{\\triangle }{=}& \\Pr (R_0=0|H^A,T=1,R_1=1) \\nonumber \\\\&=& \\dfrac{\\Pr (R_0=0,R_1=1|H^A, T=1)}{\\Pr (R_1=1|H^A,T=1)}.$ Unfortunately, $R_0,R_1$ are not jointly estimable from the data, and consequently $PC_A$ cannot be evaluated without some further assumptions."],["Some results from the literature","In epidemiology, $PC_A$ has often been expressed by the quantity referred to as the excess risk ratio (see for instance [16]) $ERR &=&\\dfrac{\\Pr (R_1=1|H^A)-\\Pr (R_0=1|H^A)}{\\Pr (R_1=1|H^A)}$ and sometime evaluated in terms of the observational risk ratio, $ORR&=& 1- 1/\\dfrac{\\Pr (R=1|H^A, T=1)}{\\Pr (R=1|H^A, T=0)}.$ The quantity (REF ), which plays an important role in the developments we consider, can be evaluated by using the data for the treated and untreated individuals coming from a randomized study or from observational data.","The choice between these two sources of data depends on the assumptions.","We review the following three contributions.","a) Pearl, 2000 (Theorem 9.2.14) [13], showed that, under Exogeneity ($R_0,R_1\\hspace{4.2679pt}\\perp \\hspace{-11.09654pt}\\perp \\hspace{4.2679pt}T |H^A$ , i.e.","the potential outcomes $(R_0, R_1)$ have the same joint distribution, among both treated and untreated study subjects sharing the same background information $H^A$ as Ann) and Monotonicity ($R_0=1 \\rightarrow R_1=1$ , i.e.","if Ann were to recover if untreated, she would certainly recover if treated), $PC_A$ is identified and equal to ORR, evaluated on observational data.","This result is remarkable since it ends up with a precise probability.","At the same time, these assumptions are not easily defensible: Exogeneity, also called Strong Ignorability, is reasonable for data coming from a randomized study but it is considered weak for observational data.","Monotonicity, also called No Prevention, is apparently reasonable (the treatment cannot be obtain worse results than the placebo) but for some individuals, for example those allergic to a medical treatment, it may not hold.","In any case, this latter assumption can not usually be verified.","b) Tian and Pearl, 2000 [18] demonstrated that, relaxing Exogeneity but retaining Monotonicity, the evaluation of the probability of causation can be obtained in a more refined form as $PC_A &=&\\dfrac{\\Pr (R^{obs}=1|T=1,H^A)-\\Pr (R^{obs}=0|T=1, H^A)}{\\Pr (R^{obs}=1|T=1, H^A)} + \\nonumber \\\\&+& \\dfrac{\\Pr (R^{obs}1|T=0, H^A)-\\Pr (R^{exp}=1|T=0,H^A)}{\\Pr (R^{obs}=1,T=1|H^A)} $ where $obs$ and $exp$ specify the source of the data (observational or experimental).","This result points out that CoE has both an experimental and an observational nature.","Data from a randomized experiment amount for no confounding, i.e.","the desirable Exogeneity property can be assumed quite safely.","At the same time, Ann made a choice to receive the treatment, i.e.","she was not forced to receive it.","Hence, a difference between $\\Pr (R^{obs}=1| T=0,H^A)$ and $\\Pr (R^{exp}=1|T=0,H^A)$ is plausible and must be taken into account.","For instance, if Ann's disease is at an advanced stage, she has little will to be treated because she perceives that her survival is almost independent of the treatment.","Since expression (REF ) points out the double nature of CoE, its computation requires having data from two different surveys, and this can be problematic.","c) Dawid et al.","(2016) consider the possibility of evaluating CoE by using only data coming from a randomized experiment.","The authors proceed in two steps.","First they derive bounds for the probability of causation based on the potential outcomes and the constraints implied in their joint distribution.","Interestingly, the relevant $PC_A$ lower bound is equal to ERR (see (REF )).","$ PC_A > \\max \\lbrace 0,1-1/RR_A\\rbrace $ where $RR_A$ , the risk ratio, is $RR_A=\\frac{\\Pr (R_1=1|H^A,T=1)}{\\Pr (R_0=1|H^A,T=1)}.$ Specifically, a large lower bound is the minimum probability of observing the response opposite to that actually observed if the treatment were not provided, so demonstrating that a different story would have been possible if the treatment had not been applied.","The question is: “Even accepting working with bounds, what cautions must be taken to estimate $\\Pr (R_1=1|H^A,T=1)$ and $\\Pr (R_0=1|H^A,T=1)$ from experimental data?” To answer the question, the authors detailed three conditions, called the fundamental conditions, to be assumed so as to estimate upper and lower bounds for $PC_A$ based on the marginal probabilities of the response.","The three conditions are the following: Exogeneity: Already defined in Section .","Comparability: Ann's potential response, $R^A_1$ , is comparable with those of the treated subjects having the same background characteristics $H^A$ as Ann.","Sufficiency: Ann's potential response $R^A_0$ and those of the untreated subjects, all having the same background characteristics $H^A$ as Ann, are comparable.","While Exogeneity follows directly from randomization, the other two conditions deserve careful reasoning, so the authors restrict their approach to when we can make good arguments for the acceptability of these fundamental conditions."],["Validating the fundamental conditions","Our proposal to evaluate the CoE consists in finding, among all the possible groups of individuals differing from the specification of $H$ , that one that “best fits” the conditions of Comparability and Sufficiency.","The problem of validating the assumptions is turned into the search for the most suitable group of experimental data supporting the fundamental conditions.","To take into account the experimental and observational nature of the CoE, we consider as observed the variable $E$ , (see Section ), the decision to receive the treatment.","For Ann, this variable provides some indirect information about her state of health, in the light of which it would no longer be appropriate to consider her similar to individuals in a pure experimental study for which this information was not available.","For this reason we extend our experimental data to include the desire of the individuals in the sample to be treated or not.","We believe that it is possible to get this information from people who have accepted a randomized treatment and we also believe that this practice is much less troublesome than to have a double survey of the same population, as in Tian and Pearl (2000) ([18]) (see Section ).","In this extended scenario, Comparability means that, conditional on my knowledge of the pre-treatment characteristics of Ann and the trial subjects, I regard Ann's potential response as comparable with those of the sub-group identified by ($T=1,E=1$ ) having characteristics $H^A$ .","In the same framework, the Sufficiency condition refers to the counterfactual scenario in which Ann was not treated.","In this case we do not have information about Ann's response nor the information concerning her will to receive the treatment or not.","Apparently, for $T=0$ , we could imagine that Ann did not desire receive the treatment (so $E=0$ ) but it might also be possible that she did not have the drug available, but her wish was to receive it (so $E=1$ ).","Our concern is to find the specification of $H$ that makes irrelevant the influence of $E$ on the responses in the untreated group.","If we can obtain reasonable support for the condition $R_0 \\hspace{4.2679pt}\\perp \\hspace{-11.09654pt}\\perp \\hspace{4.2679pt}E | H^A$ , i.e.","if $\\Pr (R_0=1|H^A, E=1)= \\Pr (R_0=1|H^A, E=0),$ it would be possible to estimate (REF ) by $\\Pr (R=1|H^A, T=0)$ using the data of the untreated."],["Model selection","To perform a selection from models characterized by different $H$ , we need to compute the marginal probabilities of the observed responses of Ann and the individuals participating in the study.","We assume a relevant effect of the treatment that can be modelled using partial exchangeability among treated and untreated individuals.","Furthermore the Comparability condition establishes that we are not able to distinguish between Ann and the group of treated individuals who desire to receive the treatment (identified by $(T=1, E=1)$ ), sharing with Ann the same characteristics $H^A$ , as it concerns the uncertainty of their responses to the treatment.","Since Comparability focusses on the group with the same characteristics as Ann, we don't need to detail the responses of the remaining individuals in the treated group, and we model them as exchangeable.","The Sufficiency condition requires that in the untreated group, individuals sharing Ann's characteristics (despite the fact that for some of them $E=0$ and for others $E=1$ ) are considered exchangeable.","Also here we are not interested in distinguishing between the remaining individuals in the untreated group (those not sharing Ann's characteristics), so that the untreated group is modeled as partially exchangeable.","Of course according to what characteristics are included in $H$ , different individuals in the randomized sample will be compared with Ann.","We have $2^k$ ways of selecting a set of characteristics from $H$ .","Let $J$ be one of these choices, identified as a subset of $\\lbrace 1,\\ldots ,K\\rbrace $ .","Each choice of $J$ induces a partition of the sample (treated and untreated), which defines a model $M_j$ .","Then, by assuming partial exchangeability and by using de Finetti's representation theorem inside every specified exchangeable group, we can evaluate the probability of observing Ann and the group of responses, induced by different subsets of $H$ .","In this way we turn the issue of finding the group most supporting the fundamental conditions into a model selection problem, solved, as usual, by computing the marginal probability of the data conditionally on different instantiations of $H$ .","Restricted to the treated group, we denote by $A_{1,1}$ the set of individuals who desire to take the treatment and share the same characteristics as Ann, with $\\bar{A}_{1,1}$ its complement, and with $\\mathbf {r}_{A_{1,1}}$ , $\\mathbf {r}_{\\bar{A}_{1,1}}$ the corresponding vector of responses, while $r_A$ denotes Ann's response.","In the untreated group, let $A_{0,e}$ be the sets of individuals considered, $e=\\lbrace 0,1\\rbrace $ , $A_0=A_{0,0}\\cup A_{0,1}$ and $\\bar{A}_{0}$ its complement.","We extend these notations in the obvious way to the vectors of responses and to the mixing parameters.","We have: $& \\Pr (r_A,\\mathbf {r}_{A_{1,1}}, \\mathbf {r}_{\\bar{A}_{1,1}}, \\mathbf {r}_{A_{0,1}}, \\mathbf {r}_{A_{0,0}} , \\mathbf {r}_{\\bar{A}_{0}}) |M_J)= \\\\& \\int _{\\Theta }\\Pr (r_A,\\mathbf {r}_{A_{1,1}}, \\mathbf {r}_{\\bar{A}_{1,1}}, \\mathbf {r}_{A_{0,1}}, \\mathbf {r}_{A_{0,0}} , \\mathbf {r}_{\\bar{A}_{0}} \\mid \\theta _{A_{1,1}},\\theta _{\\bar{A}_{1,1}}, \\theta _{A_{0}},\\theta _{\\bar{A}_{0}}, M_j)\\cdot \\\\& \\cdot \\pi (\\theta _{A_{1,1}},\\theta _{\\bar{A}_{1,1}}, \\theta _{A_{0}},\\theta _{\\bar{A}_{0}})d\\theta _{A_{1,1}}d\\theta _{\\bar{A}_{1,1}}d\\theta _{A_{0}}d\\theta _{\\bar{A}_{0}}= \\\\&= \\int _{\\Theta }\\Pr (r_A,\\mathbf {r}_{A_{1,1}}, \\mathbf {r}_{\\bar{A}_{1,1}}, \\mathbf {r}_{A_{0,1}}, \\mathbf {r}_{A_{0,0}} , \\mathbf {r}_{\\bar{A}_{0}} \\mid \\theta _{A_{1,1}},\\theta _{\\bar{A}_{1,1}}, \\theta _{A_{0}},\\theta _{\\bar{A}_{0}}, M_j)\\cdot \\\\& \\cdot \\pi (\\theta _{A_{1,1}})\\pi (\\theta _{\\bar{A}_{1,1}})\\pi (\\theta _{A_{0}})\\pi (\\theta _{\\bar{A}_{0}})d\\theta _{A_{1,1}}d\\theta _{\\bar{A}_{1,1}}d\\theta _{A_{0}}d\\theta _{\\bar{A}_{0}}= \\\\& \\int _{\\Theta _{A_{1,1}}} \\Pr (r_A,\\mathbf {r}_{A_{1,1}}\\mid \\theta _{A_{1,1}},M_J)\\pi (\\theta _{A_{1,1}})d\\theta _{A_{1,1}} \\cdot \\int _{\\Theta _{\\bar{A}_{1,1}}} \\Pr (\\mathbf {r}_{\\bar{A}_{1,1}}\\mid \\theta _{\\bar{A}_{1,1}},M_J)\\pi (\\theta _{\\bar{A}_{1,1}})d\\theta _{\\bar{A}_{1,1}}\\cdot \\\\ \\nonumber &\\cdot \\int _{\\Theta _{A_{0}}} \\Pr (\\mathbf {r}_{A_{0,1}}, \\mathbf {r}_{A_{0,0}}\\mid \\theta _{A_{0}},M_J)\\pi (\\theta _{A_{0}})d\\theta _{A_{0}}\\cdot \\int _{\\Theta _{\\bar{A}_{0}}} \\Pr (\\mathbf {r}_{\\bar{A}_{0}}\\mid \\theta _{\\bar{A}_{0}},M_J)\\pi (\\theta _{\\bar{A}_{0}})d\\theta _{\\bar{A}_{0}}$ By the de Finetti's Representation Theorem, the conditional probabilities of the responses are a mixture of a binomial model and a mixture distribution over the corresponding parameters $\\theta $ s that are assumed independent of each other.","As a consequence the overall integral easily factorizes.","Now we provide expressions for the above integrals.","All the details are in Appendix ().","We have: $\\int _{\\Theta _{A_{1,1}}} \\Pr (r_A,\\mathbf {r}_{A_{1,1}}\\mid \\theta _{A_{1,1}},M_J)\\pi (\\theta _{A_{1,1}})d\\theta _{A_{1,1}}=\\frac{x_{A_{1,1}}+1}{n_{A_{1,1}}+2} \\cdot \\frac{1}{n_{A_{1,1}}+1}$ where $x_{A_{1,1}}$ is the number of successes in the group $A_{1,1}$ and $n_{A_{1,1}}=|A_{1,1}|$ .","The notation is extended in the obvious way in the other groups.","$\\int _{\\Theta _{\\bar{A}_{1,1}}} \\Pr (\\mathbf {r}_{\\bar{A}_{1,1}}\\mid \\theta _{\\bar{A}_{1,1}},M_J)\\pi (\\theta _{\\bar{A}_{1,1}})d\\theta _{\\bar{A}_{1,1}}=\\frac{1}{n_{\\bar{A}_{1,1}}+1}$ $\\int _{\\Theta _{A_{0}}} \\Pr (\\mathbf {r}_{A_{0,1}}, \\mathbf {r}_{A_{0,0}}\\mid \\theta _{A_{0}},M_J)\\pi (\\theta _{A_{0}})d\\theta _{A_{0}} &=\\dbinom{n_{A_{0,0}}}{x_{A_{0,0}}}\\dbinom{n_{A_{0,1}}}{x_{A_{0,1}}}\\dbinom{n_{A_{0,0}}+n_{A_{0,1}}}{x_{A_{0,0}}+x_{A_{0,1}}}^{-1}\\cdot \\\\& \\cdot \\frac{1}{n_{A_{0,0}}+n_{A_{0,1}}+1}$ $\\int _{\\Theta _{\\bar{A}_{0}}} \\Pr (\\mathbf {r}_{\\bar{A}_{0}}\\mid \\theta _{\\bar{A}_{0}},M_J)\\pi (\\theta _{\\bar{A}_{0}})d\\theta _{\\bar{A}_{0}}= \\frac{1}{n_{\\bar{A}_{0}}+1}$ Readers might recognize the conditional Irving–Fisher exact test as part of (REF ).","The result is not surprising since we are looking for the set of $H$ making $E$ irrelevant, and thus supporting the hypothesis of no-difference between the success ratio in the two groups.","An illustration of the behavior of the hypergeometric for $n_{A_{0,0}}=n_{A_{0,1}}=10$ is given in Figure REF .","High support to the model is achieved when the number of successes $x_{A_{0,0}}$ and $x_{A_{0,1}}$ is almost the same in the two groups.","Figure: Hypergeometric behaviour for n A 0,0 =n A 0,1 =10n_{A_{0,0}}=n_{A_{0,1}}=10 and different values for x A 0,0 x_{A_{0,0}} and x A 0,1 x_{A_{0,1}}"],["Prior and posterior in the model space","The goal is to evaluate the posterior probability of $M_J$ given the responses observed for Ann and the sample.","The overall marginal likelihood of $M_j$ is the product of (REF ), (REF ), (REF ) and (REF ) $& \\Pr (r_A,\\mathbf {r}_{A_{1,1}}, \\mathbf {r}_{\\bar{A}_{1,1}}, \\mathbf {r}_{A_{0,1}}, \\mathbf {r}_{A_{0,0}} , \\mathbf {r}_{\\bar{A}_{0}}) |M_J)=\\frac{x_{A_{1,1}}+1}{n_{A_{1,1}}+2} \\cdot \\frac{1}{n_{A_{1,1}}+1}\\cdot \\frac{1}{n_{\\bar{A}_{1,1}}+1}\\\\ \\nonumber \\cdot & \\dbinom{n_{A_{0,0}}}{x_{A_{0,0}}}\\dbinom{n_{A_{0,1}}}{x_{A_{0,1}}}\\dbinom{n_{A_{0,0}}+n_{A_{0,1}}}{x_{A_{0,0}}+x_{A_{0,1}}}^{-1}\\frac{1}{n_{A_{0,0}}+n_{A_{0,1}}+1} \\cdot \\frac{1}{n_{\\bar{A}_{0}}+1}$ Concerning the prior over the space of models, the simplest choice is to consider a uniform distribution $ \\Pr (M_J)=\\frac{1}{2^k}.$ Another choice is the one proposed by Chen and Chen, (2008) [2].","They give the same prior probability (equal to $\\frac{1}{k+1}$ ) for all models sharing the same number of characteristics $k$ .","In this way, for the generic model $M_J$ , we have $\\Pr (M_J)= \\frac{1}{k+1}\\binom{k}{|M_J|}^{-1}\\cdot I(|M_J|\\le k/2)$ where the search spans all models including at most $k/2$ characteristics.","This last choice favors model selection according to Occam's razor principle: the fewer characteristics employed, the more probable is the model.","This rationale is reasonably objective.","Combining (REF ) or (REF ) with (REF ), we get the required posterior."],["Computational issues","If the model size becomes huge, we may not be in a position to evaluate the normalizing constant of the posterior distribution, but we can establish an MCMC to make an inference about the variable $M_J$ .","Essentially a Metropolis–Hastings would suffice, the acceptance ratio being given by (REF ), evaluated for two different elements of $M_J$ , taking into account the probability of proposing a new model."],["Application","We carried out an experiment at the University of Florence, School of Engineering, Fall 2017.","We asked 161 students to solve a simple probabilistic question and we provided randomly a hint (the treatment $T$ ).","Table (REF ) presents the students' background information included in the analysis (the characteristics $H$ ).","Before the test, we asked the students if they wished to be helped or not (the desire variable $E$ ).","Table: List of student characteristics included in the experimentWe wish to investigate whether there is a causal relation between the hint and the ability to solve the question, for those students who desired to receive a hint.","We had 8 of these cases.","The corresponding risk ratio (obtained considering the model best fitting the fundamental conditions) lies in the interval $[1.73, 2.43]$ and in one case it exceeds 2 ($RR=2.43$ ), which is a clue for there being a causal relation.","In all the other cases, the causal relation is not strongly supported, since $RR$ is close but does not reach 2 (see, in Figure (REF )), the left side of each sub-picture).","In the right side of each sub-picture in Figure(REF ) we illustrate how the model selection procedure proposes models respecting the fundamental conditions.","For the model with the highest probability, the red dot indicates, in the $x-$ axis, the success ratio for treated and, in the $y-$ axis, the ratio between the success ratio for untreated with $E=1$ and with $E=0$ , respectively.","These are the main forces driving the marginal probability for the responses, as shown in (REF ).","Ideally, comparability and sufficiency are mostly supported by the highest possible success ratio among the treated and by a ratio between untreated with E=0 and E=1 approximately equal 1.","As is apparent, the selected model achieves a good compromise between these requirements.","The set of variables selected in the 8 cases are shown in Table (REF ).","Note that, overall, these models include only 5 characteristics, and all of them include the educational status of the family.","In 4 out of the 8 cases, the same model, including family education, $1^\\text{first}$ University registration (a proxy for understanding whether the student failed earlier in their educational career) and previous exposure to statistical training was selected.","Figure: Educational causation.Risk Ratio and posterior probability for each explored model (left side).Success ratio for treated vs relative success ratio for untreated with E=1E=1 and E=0E=0 (right side).In the picture there is an example concerning four students who succeed.Figure: Forensic causation.Risk Ratio and posterior probability for each explored model (left side).Success ratio for treated vs relative success ratio for untreated with E=1E=1 and E=0E=0 (right side).In the picture there is an example concerning four students who DID NOT succeed.Table: Educational causation.","For the 8 students who desired the hint, got it, and succeed, there are reported the characteristics HH selected by the model that best supported the fundamental conditions and the Risk RatioAs a result of our experiment we also have that among the students who asked for and received the hint, 24 did not succeed.","We can suppose that some of them claimed that it was the hint which caused their failure.","In this case, the $RR$ lies in the interval $[0.86,1.60]$ , an example for four students is in Figure (REF ).","It is not conclusive that there is a causal relation between the hint and the failure since all the models for all the considered students provided values of RR much smaller than 2.","In a civil trial this would not suggest to a judge that compensation be awarded."],["Conclusions","We introduced a typical Causes of Effects problem by means of an archetypical example considering Ann and the effect of an aspirin on her headache.","We have proposed a possible solution to make operational the choice of variables to include, so as to validate the fundamental assumptions underlying the assessment of Ann's probability of causation.","We assume it is possible to take a randomized sample from Ann's population where, as usual, $T$ is assigned following a randomized protocol and $E$ (this is a novelty) is a question asking the members of the sample about their preference to be treated or not.","In the evaluation of $RR_A$ (see REF ), an extreme position would be to include all the subjects participating in the experiment so that simply belonging to the reference population would make the individuals in the sample similar to Ann.","On the other hand, the choice could be to find the persons most similar to Ann, i.e.","those matching all the available characteristics.","Clearly neither of these positions is safe: the former ignores some characteristics of Ann which could be very influential on her reaction to the headache after taking aspirin.","The latter greatly reduces the number of individuals to be employed in the estimation, so producing a very unstable inference.","Our approach takes a sensible middle course and provide a sensible different causal inference for individuals experiencing the same treatment.","Interestingly, this is exactly the aim of Precision Medicine (see Mesko (2017), [11]) which looks for different medical interventions for a group of individuals sharing some relevant (for the reaction between treatment and outcome) characteristics.","The next step will be to extend the method to observational studies, to make possible in a wider range of cases the evaluation of the $PC_A$ for Causes of Effects problems."],["Appendix","Now we detail the computation of the integrals (REF ), (REF ), (REF ) and (REF ).","We always assume a no informative prior for all $\\pi (\\cdot )=Beta(1,1)$ .","We start with (REF ).","$& \\int _{\\Theta _{A_{1,1}}} \\Pr (r_A,\\mathbf {r}_{A_{1,1}}\\mid \\theta _{A_{1,1}},M_J)\\pi (\\theta _{A_{1,1}})d\\theta _{A_{1,1}}= \\nonumber \\\\&= \\int _{\\Theta _{A_{1,1}}} \\Pr (r_A | \\mathbf {r}_{A_{1,1}},\\theta _{A_{1,1}},M_J)\\underbrace{\\Pr (\\mathbf {r}_{A_{1,1}}|\\theta _{A_{1,1}},M_J)\\pi (\\theta _{A_{1,1}}|M_J)}_{\\text{numerator of the }\\theta _{A_{1,1}} \\text{ updating}}d \\theta _{A_{1,1}} \\nonumber \\\\&= \\underbrace{\\int _{\\Theta _{A_{1,1}}} \\Pr (r_A|\\theta _{A_{1,1}},M_J)\\pi (\\theta _{A_{1,1}}|\\mathbf {r}_{A_{1,1}},M_J)d\\theta _{A_{1,1}}}_{\\dfrac{x_{A_{1,1}}+1}{n_{A_{1,1}}+2}}\\cdot \\nonumber \\\\& \\cdot \\underbrace{\\int _{\\Theta _{A_{1,1}}}\\Pr (\\mathbf {r}_{A_{1,1}}|\\theta _{A_{1,1}},M_J)\\Pr (\\theta _{A_{1,1}}|M_J)d\\theta _{A_{1,1}} }_{\\dbinom{n_{A_{1,1}}}{x_{A_{1,1}}}\\dfrac{\\Gamma (1+1)}{\\Gamma (1)\\Gamma (1)} \\dfrac{\\Gamma (1+x_{A_{1,1}}) \\Gamma (n_{A_{1,1}}+ 1 -x_{A_{1,1}})}{\\Gamma (n_{A_{1,1}}+ 1 +1)}} \\nonumber \\\\ & =\\frac{x_{A_{1,1}}+1}{n_{A_{1,1}}+2} \\cdot \\frac{1}{n_{A_{1,1}}+1}.","\\nonumber $ For (REF ) we have $&\\int _{\\Theta _{\\bar{A}_{1,1}}} \\Pr (\\mathbf {r}_{\\bar{A}_{1,1}}\\mid \\theta _{\\bar{A}_{1,1}},M_J)\\pi (\\theta _{\\bar{A}_{1,1}})d\\theta _{\\bar{A}_{1,1}}= \\\\& \\underbrace{\\int _{\\Theta _{\\bar{A}_{1,1}}}\\Pr (\\mathbf {r}_{\\bar{A}_{1,1}}|\\theta _{\\bar{A}_{1,1}},M_J)\\Pr (\\theta _{\\bar{A}_{1,1}}|M_J)d\\theta _{\\bar{A}_{1,1}} }_{\\dbinom{n_{\\bar{A}_{1,1}}}{x_{\\bar{A}_{1,1}}}\\dfrac{\\Gamma (1+1)}{\\Gamma (1)\\Gamma (1)} \\dfrac{\\Gamma (1+x_{\\bar{A}_{1,1}}) \\Gamma (n_{\\bar{A}_{1,1}}+ 1 -x_{\\bar{A}_{1,1}})}{\\Gamma (n_{\\bar{A}_{1,1}}+ 1 +1)}} \\nonumber \\\\& =\\frac{1}{n_{\\bar{A}_{1,1}}+1}$ We now compute (REF ) $&\\int _{\\Theta _{A_{0}}} \\Pr (\\mathbf {r}_{A_{0,1}}, \\mathbf {r}_{A_{0,0}}\\mid \\theta _{A_{0}},M_J)\\pi (\\theta _{A_{0}},M_J)d\\theta _{A_{0}}= \\\\&= \\int _{\\Theta _{A_{0}}} \\Pr (\\mathbf {r}_{A_{0,0}},|\\theta _{A_{0}},\\mathbf {r}_{A_{0,1}},M_J)\\underbrace{\\Pr (\\mathbf {r}_{A_{0,1}}|\\theta _{A_{0}}, M_J)\\pi (\\theta _{A_{0}}|M_J)}_{\\text{numerator of the }\\theta _{A_{0}} \\text{ updating}}d\\theta _{A_{0}} \\nonumber \\\\&= \\int _{\\Theta _{A_{0}}} \\Pr (\\mathbf {r}_{A_{0,0}}|\\theta _{A_{0}},M_J)\\pi (\\theta _{A_{0}}| \\mathbf {r}_{A_{0,1}},M_J)d\\theta _{A_{0}} \\int _{\\Theta _{A_{0}}}\\Pr (\\mathbf {r}_{A_{0,1}}|\\theta _{A_{0}},M_J)\\pi (\\theta _{A_{0}}| M_J)d\\theta _{A_{0}} \\nonumber \\\\&= \\dbinom{n_{A_{0,0}}}{x_{A_{0,0}}} \\dfrac{\\Gamma (n_{A_{0,1}}+2)}{\\Gamma (x_{A_{0,1}}+1)\\Gamma (n_{A_{0,1}}- x_{A_{0,1}}+1)} \\dfrac{\\Gamma (x_{A_{0,1} } +1 + x_{A_{0,0}})\\Gamma (n_{A_{0,0}}+ n_{A_{0,1}}- x_{A_{0,1} }+1- x_{A_{0,0}}) }{\\Gamma (n_{A_{0,0}} + n_{A_{0,1}}+2)} \\nonumber \\\\&\\cdot \\dbinom{n_{A_{0,1}}}{x_{A_{0,1}}}\\dfrac{\\Gamma (1+1)}{\\Gamma (1)\\Gamma (1)} \\dfrac{\\Gamma (1+x_{A_{0,1}}) \\Gamma (n_{A_{0,1}}+ 1 -x_{A_{0,1} })}{\\Gamma (n_{A_{0,1}} + 1 +1)} \\nonumber \\\\& = \\dbinom{n_{A_{0,0}}}{x_{A_{0,0}}}\\dbinom{n_{A_{0,1}}}{x_{A_{0,1}}}\\dbinom{n_{A_{0,0}}+n_{A_{0,1}}}{x_{A_{0,0}}+x_{A_{0,1}}}^{-1}\\frac{1}{n_{A_{0,0}}+n_{A_{0,1}}+1.","}$ Concerning the last term, (REF ), with a similar computation as for (REF ), it is straightforward to see that $\\int _{\\Theta _{\\bar{A}_{0}}} \\Pr (\\mathbf {r}_{\\bar{A}_{0}}\\mid \\theta _{\\bar{A}_{0}},M_J)\\pi (\\theta _{\\bar{A}_{0}})d\\theta _{\\bar{A}_{0}}= \\frac{1}{n_{\\bar{A}_{0}}+1}.$ Acknowledgement:The second author was supported by the project GESTA of the Fondazione di Sardegna and Regione Autonoma di Sardegna"]],"string":"[\n [\n \"Causes of Effects via a Bayesian Model Selection Procedure\"\n ],\n [\n \"Abstract In causal inference, and specifically in the \\\\textit{Causes of Effects} problem, one is interested in how to use statistical evidence to understand causation in an individual case, and so how to assess the so-called {\\\\em probability of causation} (PC).\",\n \"The answer relies on the potential responses, which can incorporate information about what would have happened to the outcome as we had observed a different value of the exposure.\",\n \"However, even given the best possible statistical evidence for the association between exposure and outcome, we can typically only provide bounds for the PC.\",\n \"Dawid et al.\",\n \"(2016) highlighted some fundamental conditions, namely, exogeneity, comparability, and sufficiency, required to obtain such bounds, based on experimental data.\",\n \"The aim of the present paper is to provide methods to find, in specific cases, the best subsample of the reference dataset to satisfy such requirements.\",\n \"To this end, we introduce a new variable, expressing the desire to be exposed or not, and we set the question up as a model selection problem.\",\n \"The best model will be selected using the marginal probability of the responses and a suitable prior proposal over the model space.\",\n \"An application in the educational field is presented.\"\n ],\n [\n \"Introduction\",\n \"The Causes of Effects (CoE) problem concerns the study of individual causation and explicitly refers to something happened to a well identified individual.\",\n \"This nuance of causation has received less attention than the study of the Effects of Causes (EoC), also called the general causation problem.\",\n \"Actually, in EoC, the aim is the prediction of an outcome after the realization of an alleged cause; in CoE, instead, we want to evaluate the Probability of Causation (PC), i.e.\",\n \"the probability the observed outcome would not have been realized if the alleged cause had not been made effective, despite the fact that the cause and the outcome were already observed.\",\n \"Since we always refer causation to a specific individual, for the sake of simplicity, we call her Ann.\",\n \"A simplified CoE question is the following.\",\n \"“Ann had a headache and decided to take aspirin.\",\n \"Her headache went away.\",\n \"Was that caused by the aspirin?” In CoE, to evaluate the probability of causation, formally defined in Section , the relevant questions are: How might one use experimental and/or observational data, gathered on a reference group to which Ann belongs?\",\n \"How would one find the characteristics shared with Ann by the group of individuals from which the data came?\",\n \"These problems, also called the “Group to individual” (G2i) issue in forensic science, have generated a large debate in legal circles (see Faigman et al.\",\n \"(2014) [10]).\",\n \"The issue was extensively studied in the statistical literature from a technical and a philosophical point of view, reaching different and somewhat related results.\",\n \"Essential references are Dawid (2000; 2016), [6], [3] and Pearl (2009; 2015) [14], [15].\",\n \"Even if their approaches follow different routes, they agree about the use of counterfactuals and potential outcomes originating from Neyman ([12]) and re-introduced in the modern literature by Rubin (1974) ([17])Rubin's approach has been mainly used to solve EoC-type problems: for this reason it will be not considered in this paper..\",\n \"The need for counterfactuals in CoE is well illustrated by the question: “What would be the probability of the response if no treatment had been provided to the individual in whom we are interested?” Since only the treatment can be assigned to the individual, the matter concerns a counterfactual, i.e.\",\n \"an event which assumes something different from what happened.\",\n \"Depending on the assumptions, different results emerge in the forms of a precise probability or bounds (see Section ).\",\n \"Although not as specific as one would ideally like, such bounds can be of use.\",\n \"In particular, if the lower bound exceeds one-half, then, in civil cases, we can infer causality “on the balance of probabilities”.\",\n \"To evaluate the PC, we follow the approach of Dawid et al.\",\n \"(2016) [6], that identified and detailed three fundamental conditions used to estimate from the data upper and lower bounds for the probability of causation.\",\n \"A related problem is the choice of the reference population.\",\n \"More specifically, how to find that group, from among those obtainable from partitioning the randomized experiment sample according to Ann's characteristics, which best fulfills the fundamental conditions.\",\n \"How much information to take into account and how to select the best comparison group is a tricky issue, also because the choice of the reference class can affect significantly the conclusions of the inference.\",\n \"The main aim of the present paper is to make the fundamental conditions operative by means of empirical testing of such underlying assumptions, in a way that comes up with the “best” comparison group.\",\n \"We set the question up as a Bayesian model selection problem, where each model specifies a particular choice (more or less detailed) for the characteristics to be included, shared by Ann and the other individuals participating in the study.\",\n \"The best model will be selected considering the marginal probabilities of the response and a satisfactory prior proposal in the model space.\",\n \"To this end, we introduce a new variable that expresses, for each individual in the study, the desire to receive the treatment or not.\",\n \"This variable allows introducing the fundamental conditions in the model selection procedure.\",\n \"Such a method can have various applications in sociology and education, as well as in medicine.\",\n \"We present an example in the field of education, where we investigate the relation between success in a test and whether or not a hint was received (taking into account the student's preference for receiving or not receiving the hint).\",\n \"The structure of this paper is as follows.\",\n \"We first introduce the notation and we define formally the Probability of Causation (PC).\",\n \"After the review of some results in the CoE literature, see Section , we provide the assumptions we require, Section .\",\n \"Then, in Section , we detail how to find the reference sample suitable for evaluating CoE by a model selection procedure.\",\n \"After presenting the application in Section , we draw some final conclusions.\"\n ],\n [\n \"Notation\",\n \"We first specify the notation we need to introduce our approach.\",\n \"Given data obtained from an ideal large randomized study concerning $n$ individuals, drawn from a population to which Ann belongs, assume that the study records the outcome $R=\\\\lbrace 0,1\\\\rbrace $ of a treatment $T=\\\\lbrace 0,1\\\\rbrace $ , which is supposed large enough that the sampling variability of the estimates is negligible.\",\n \"Let $H$ be a large set of variables $H=\\\\lbrace H_1,\\\\dots ,H_k\\\\rbrace $ , characteristic of both Ann and the individuals participating in the study, where we assume that each $H_j$ , $j\\\\in 1,\\\\dots ,k$ is discrete (or even dichotomous).\",\n \"We denote by $H^{A}$ the value that the set of variables $H$ has for Ann.\",\n \"Since Ann not only takes the aspirin but also expresses her desire to take it, we introduce a variable $E=\\\\lbrace 0,1\\\\rbrace $ (for Ann $E=1$ ) expressing such a desire for each individual in the study.\",\n \"This variable, first proposed as an unobservable by Dawid (2011), is here considered observable.\",\n \"We also introduce the potential variables $R_T=\\\\lbrace 0,1\\\\rbrace $ , so that, if the triple $(T,R_0,R_1)$ were observed contemporaneously, it would be easy to solve the CoE problem by stating the probability of causation (PC), defined as $PC_A& \\\\stackrel{\\\\triangle }{=}& \\\\Pr (R_0=0|H^A,T=1,R_1=1) \\\\nonumber \\\\\\\\&=& \\\\dfrac{\\\\Pr (R_0=0,R_1=1|H^A, T=1)}{\\\\Pr (R_1=1|H^A,T=1)}.$ Unfortunately, $R_0,R_1$ are not jointly estimable from the data, and consequently $PC_A$ cannot be evaluated without some further assumptions.\"\n ],\n [\n \"Some results from the literature\",\n \"In epidemiology, $PC_A$ has often been expressed by the quantity referred to as the excess risk ratio (see for instance [16]) $ERR &=&\\\\dfrac{\\\\Pr (R_1=1|H^A)-\\\\Pr (R_0=1|H^A)}{\\\\Pr (R_1=1|H^A)}$ and sometime evaluated in terms of the observational risk ratio, $ORR&=& 1- 1/\\\\dfrac{\\\\Pr (R=1|H^A, T=1)}{\\\\Pr (R=1|H^A, T=0)}.$ The quantity (REF ), which plays an important role in the developments we consider, can be evaluated by using the data for the treated and untreated individuals coming from a randomized study or from observational data.\",\n \"The choice between these two sources of data depends on the assumptions.\",\n \"We review the following three contributions.\",\n \"a) Pearl, 2000 (Theorem 9.2.14) [13], showed that, under Exogeneity ($R_0,R_1\\\\hspace{4.2679pt}\\\\perp \\\\hspace{-11.09654pt}\\\\perp \\\\hspace{4.2679pt}T |H^A$ , i.e.\",\n \"the potential outcomes $(R_0, R_1)$ have the same joint distribution, among both treated and untreated study subjects sharing the same background information $H^A$ as Ann) and Monotonicity ($R_0=1 \\\\rightarrow R_1=1$ , i.e.\",\n \"if Ann were to recover if untreated, she would certainly recover if treated), $PC_A$ is identified and equal to ORR, evaluated on observational data.\",\n \"This result is remarkable since it ends up with a precise probability.\",\n \"At the same time, these assumptions are not easily defensible: Exogeneity, also called Strong Ignorability, is reasonable for data coming from a randomized study but it is considered weak for observational data.\",\n \"Monotonicity, also called No Prevention, is apparently reasonable (the treatment cannot be obtain worse results than the placebo) but for some individuals, for example those allergic to a medical treatment, it may not hold.\",\n \"In any case, this latter assumption can not usually be verified.\",\n \"b) Tian and Pearl, 2000 [18] demonstrated that, relaxing Exogeneity but retaining Monotonicity, the evaluation of the probability of causation can be obtained in a more refined form as $PC_A &=&\\\\dfrac{\\\\Pr (R^{obs}=1|T=1,H^A)-\\\\Pr (R^{obs}=0|T=1, H^A)}{\\\\Pr (R^{obs}=1|T=1, H^A)} + \\\\nonumber \\\\\\\\&+& \\\\dfrac{\\\\Pr (R^{obs}1|T=0, H^A)-\\\\Pr (R^{exp}=1|T=0,H^A)}{\\\\Pr (R^{obs}=1,T=1|H^A)} $ where $obs$ and $exp$ specify the source of the data (observational or experimental).\",\n \"This result points out that CoE has both an experimental and an observational nature.\",\n \"Data from a randomized experiment amount for no confounding, i.e.\",\n \"the desirable Exogeneity property can be assumed quite safely.\",\n \"At the same time, Ann made a choice to receive the treatment, i.e.\",\n \"she was not forced to receive it.\",\n \"Hence, a difference between $\\\\Pr (R^{obs}=1| T=0,H^A)$ and $\\\\Pr (R^{exp}=1|T=0,H^A)$ is plausible and must be taken into account.\",\n \"For instance, if Ann's disease is at an advanced stage, she has little will to be treated because she perceives that her survival is almost independent of the treatment.\",\n \"Since expression (REF ) points out the double nature of CoE, its computation requires having data from two different surveys, and this can be problematic.\",\n \"c) Dawid et al.\",\n \"(2016) consider the possibility of evaluating CoE by using only data coming from a randomized experiment.\",\n \"The authors proceed in two steps.\",\n \"First they derive bounds for the probability of causation based on the potential outcomes and the constraints implied in their joint distribution.\",\n \"Interestingly, the relevant $PC_A$ lower bound is equal to ERR (see (REF )).\",\n \"$ PC_A > \\\\max \\\\lbrace 0,1-1/RR_A\\\\rbrace $ where $RR_A$ , the risk ratio, is $RR_A=\\\\frac{\\\\Pr (R_1=1|H^A,T=1)}{\\\\Pr (R_0=1|H^A,T=1)}.$ Specifically, a large lower bound is the minimum probability of observing the response opposite to that actually observed if the treatment were not provided, so demonstrating that a different story would have been possible if the treatment had not been applied.\",\n \"The question is: “Even accepting working with bounds, what cautions must be taken to estimate $\\\\Pr (R_1=1|H^A,T=1)$ and $\\\\Pr (R_0=1|H^A,T=1)$ from experimental data?” To answer the question, the authors detailed three conditions, called the fundamental conditions, to be assumed so as to estimate upper and lower bounds for $PC_A$ based on the marginal probabilities of the response.\",\n \"The three conditions are the following: Exogeneity: Already defined in Section .\",\n \"Comparability: Ann's potential response, $R^A_1$ , is comparable with those of the treated subjects having the same background characteristics $H^A$ as Ann.\",\n \"Sufficiency: Ann's potential response $R^A_0$ and those of the untreated subjects, all having the same background characteristics $H^A$ as Ann, are comparable.\",\n \"While Exogeneity follows directly from randomization, the other two conditions deserve careful reasoning, so the authors restrict their approach to when we can make good arguments for the acceptability of these fundamental conditions.\"\n ],\n [\n \"Validating the fundamental conditions\",\n \"Our proposal to evaluate the CoE consists in finding, among all the possible groups of individuals differing from the specification of $H$ , that one that “best fits” the conditions of Comparability and Sufficiency.\",\n \"The problem of validating the assumptions is turned into the search for the most suitable group of experimental data supporting the fundamental conditions.\",\n \"To take into account the experimental and observational nature of the CoE, we consider as observed the variable $E$ , (see Section ), the decision to receive the treatment.\",\n \"For Ann, this variable provides some indirect information about her state of health, in the light of which it would no longer be appropriate to consider her similar to individuals in a pure experimental study for which this information was not available.\",\n \"For this reason we extend our experimental data to include the desire of the individuals in the sample to be treated or not.\",\n \"We believe that it is possible to get this information from people who have accepted a randomized treatment and we also believe that this practice is much less troublesome than to have a double survey of the same population, as in Tian and Pearl (2000) ([18]) (see Section ).\",\n \"In this extended scenario, Comparability means that, conditional on my knowledge of the pre-treatment characteristics of Ann and the trial subjects, I regard Ann's potential response as comparable with those of the sub-group identified by ($T=1,E=1$ ) having characteristics $H^A$ .\",\n \"In the same framework, the Sufficiency condition refers to the counterfactual scenario in which Ann was not treated.\",\n \"In this case we do not have information about Ann's response nor the information concerning her will to receive the treatment or not.\",\n \"Apparently, for $T=0$ , we could imagine that Ann did not desire receive the treatment (so $E=0$ ) but it might also be possible that she did not have the drug available, but her wish was to receive it (so $E=1$ ).\",\n \"Our concern is to find the specification of $H$ that makes irrelevant the influence of $E$ on the responses in the untreated group.\",\n \"If we can obtain reasonable support for the condition $R_0 \\\\hspace{4.2679pt}\\\\perp \\\\hspace{-11.09654pt}\\\\perp \\\\hspace{4.2679pt}E | H^A$ , i.e.\",\n \"if $\\\\Pr (R_0=1|H^A, E=1)= \\\\Pr (R_0=1|H^A, E=0),$ it would be possible to estimate (REF ) by $\\\\Pr (R=1|H^A, T=0)$ using the data of the untreated.\"\n ],\n [\n \"Model selection\",\n \"To perform a selection from models characterized by different $H$ , we need to compute the marginal probabilities of the observed responses of Ann and the individuals participating in the study.\",\n \"We assume a relevant effect of the treatment that can be modelled using partial exchangeability among treated and untreated individuals.\",\n \"Furthermore the Comparability condition establishes that we are not able to distinguish between Ann and the group of treated individuals who desire to receive the treatment (identified by $(T=1, E=1)$ ), sharing with Ann the same characteristics $H^A$ , as it concerns the uncertainty of their responses to the treatment.\",\n \"Since Comparability focusses on the group with the same characteristics as Ann, we don't need to detail the responses of the remaining individuals in the treated group, and we model them as exchangeable.\",\n \"The Sufficiency condition requires that in the untreated group, individuals sharing Ann's characteristics (despite the fact that for some of them $E=0$ and for others $E=1$ ) are considered exchangeable.\",\n \"Also here we are not interested in distinguishing between the remaining individuals in the untreated group (those not sharing Ann's characteristics), so that the untreated group is modeled as partially exchangeable.\",\n \"Of course according to what characteristics are included in $H$ , different individuals in the randomized sample will be compared with Ann.\",\n \"We have $2^k$ ways of selecting a set of characteristics from $H$ .\",\n \"Let $J$ be one of these choices, identified as a subset of $\\\\lbrace 1,\\\\ldots ,K\\\\rbrace $ .\",\n \"Each choice of $J$ induces a partition of the sample (treated and untreated), which defines a model $M_j$ .\",\n \"Then, by assuming partial exchangeability and by using de Finetti's representation theorem inside every specified exchangeable group, we can evaluate the probability of observing Ann and the group of responses, induced by different subsets of $H$ .\",\n \"In this way we turn the issue of finding the group most supporting the fundamental conditions into a model selection problem, solved, as usual, by computing the marginal probability of the data conditionally on different instantiations of $H$ .\",\n \"Restricted to the treated group, we denote by $A_{1,1}$ the set of individuals who desire to take the treatment and share the same characteristics as Ann, with $\\\\bar{A}_{1,1}$ its complement, and with $\\\\mathbf {r}_{A_{1,1}}$ , $\\\\mathbf {r}_{\\\\bar{A}_{1,1}}$ the corresponding vector of responses, while $r_A$ denotes Ann's response.\",\n \"In the untreated group, let $A_{0,e}$ be the sets of individuals considered, $e=\\\\lbrace 0,1\\\\rbrace $ , $A_0=A_{0,0}\\\\cup A_{0,1}$ and $\\\\bar{A}_{0}$ its complement.\",\n \"We extend these notations in the obvious way to the vectors of responses and to the mixing parameters.\",\n \"We have: $& \\\\Pr (r_A,\\\\mathbf {r}_{A_{1,1}}, \\\\mathbf {r}_{\\\\bar{A}_{1,1}}, \\\\mathbf {r}_{A_{0,1}}, \\\\mathbf {r}_{A_{0,0}} , \\\\mathbf {r}_{\\\\bar{A}_{0}}) |M_J)= \\\\\\\\& \\\\int _{\\\\Theta }\\\\Pr (r_A,\\\\mathbf {r}_{A_{1,1}}, \\\\mathbf {r}_{\\\\bar{A}_{1,1}}, \\\\mathbf {r}_{A_{0,1}}, \\\\mathbf {r}_{A_{0,0}} , \\\\mathbf {r}_{\\\\bar{A}_{0}} \\\\mid \\\\theta _{A_{1,1}},\\\\theta _{\\\\bar{A}_{1,1}}, \\\\theta _{A_{0}},\\\\theta _{\\\\bar{A}_{0}}, M_j)\\\\cdot \\\\\\\\& \\\\cdot \\\\pi (\\\\theta _{A_{1,1}},\\\\theta _{\\\\bar{A}_{1,1}}, \\\\theta _{A_{0}},\\\\theta _{\\\\bar{A}_{0}})d\\\\theta _{A_{1,1}}d\\\\theta _{\\\\bar{A}_{1,1}}d\\\\theta _{A_{0}}d\\\\theta _{\\\\bar{A}_{0}}= \\\\\\\\&= \\\\int _{\\\\Theta }\\\\Pr (r_A,\\\\mathbf {r}_{A_{1,1}}, \\\\mathbf {r}_{\\\\bar{A}_{1,1}}, \\\\mathbf {r}_{A_{0,1}}, \\\\mathbf {r}_{A_{0,0}} , \\\\mathbf {r}_{\\\\bar{A}_{0}} \\\\mid \\\\theta _{A_{1,1}},\\\\theta _{\\\\bar{A}_{1,1}}, \\\\theta _{A_{0}},\\\\theta _{\\\\bar{A}_{0}}, M_j)\\\\cdot \\\\\\\\& \\\\cdot \\\\pi (\\\\theta _{A_{1,1}})\\\\pi (\\\\theta _{\\\\bar{A}_{1,1}})\\\\pi (\\\\theta _{A_{0}})\\\\pi (\\\\theta _{\\\\bar{A}_{0}})d\\\\theta _{A_{1,1}}d\\\\theta _{\\\\bar{A}_{1,1}}d\\\\theta _{A_{0}}d\\\\theta _{\\\\bar{A}_{0}}= \\\\\\\\& \\\\int _{\\\\Theta _{A_{1,1}}} \\\\Pr (r_A,\\\\mathbf {r}_{A_{1,1}}\\\\mid \\\\theta _{A_{1,1}},M_J)\\\\pi (\\\\theta _{A_{1,1}})d\\\\theta _{A_{1,1}} \\\\cdot \\\\int _{\\\\Theta _{\\\\bar{A}_{1,1}}} \\\\Pr (\\\\mathbf {r}_{\\\\bar{A}_{1,1}}\\\\mid \\\\theta _{\\\\bar{A}_{1,1}},M_J)\\\\pi (\\\\theta _{\\\\bar{A}_{1,1}})d\\\\theta _{\\\\bar{A}_{1,1}}\\\\cdot \\\\\\\\ \\\\nonumber &\\\\cdot \\\\int _{\\\\Theta _{A_{0}}} \\\\Pr (\\\\mathbf {r}_{A_{0,1}}, \\\\mathbf {r}_{A_{0,0}}\\\\mid \\\\theta _{A_{0}},M_J)\\\\pi (\\\\theta _{A_{0}})d\\\\theta _{A_{0}}\\\\cdot \\\\int _{\\\\Theta _{\\\\bar{A}_{0}}} \\\\Pr (\\\\mathbf {r}_{\\\\bar{A}_{0}}\\\\mid \\\\theta _{\\\\bar{A}_{0}},M_J)\\\\pi (\\\\theta _{\\\\bar{A}_{0}})d\\\\theta _{\\\\bar{A}_{0}}$ By the de Finetti's Representation Theorem, the conditional probabilities of the responses are a mixture of a binomial model and a mixture distribution over the corresponding parameters $\\\\theta $ s that are assumed independent of each other.\",\n \"As a consequence the overall integral easily factorizes.\",\n \"Now we provide expressions for the above integrals.\",\n \"All the details are in Appendix ().\",\n \"We have: $\\\\int _{\\\\Theta _{A_{1,1}}} \\\\Pr (r_A,\\\\mathbf {r}_{A_{1,1}}\\\\mid \\\\theta _{A_{1,1}},M_J)\\\\pi (\\\\theta _{A_{1,1}})d\\\\theta _{A_{1,1}}=\\\\frac{x_{A_{1,1}}+1}{n_{A_{1,1}}+2} \\\\cdot \\\\frac{1}{n_{A_{1,1}}+1}$ where $x_{A_{1,1}}$ is the number of successes in the group $A_{1,1}$ and $n_{A_{1,1}}=|A_{1,1}|$ .\",\n \"The notation is extended in the obvious way in the other groups.\",\n \"$\\\\int _{\\\\Theta _{\\\\bar{A}_{1,1}}} \\\\Pr (\\\\mathbf {r}_{\\\\bar{A}_{1,1}}\\\\mid \\\\theta _{\\\\bar{A}_{1,1}},M_J)\\\\pi (\\\\theta _{\\\\bar{A}_{1,1}})d\\\\theta _{\\\\bar{A}_{1,1}}=\\\\frac{1}{n_{\\\\bar{A}_{1,1}}+1}$ $\\\\int _{\\\\Theta _{A_{0}}} \\\\Pr (\\\\mathbf {r}_{A_{0,1}}, \\\\mathbf {r}_{A_{0,0}}\\\\mid \\\\theta _{A_{0}},M_J)\\\\pi (\\\\theta _{A_{0}})d\\\\theta _{A_{0}} &=\\\\dbinom{n_{A_{0,0}}}{x_{A_{0,0}}}\\\\dbinom{n_{A_{0,1}}}{x_{A_{0,1}}}\\\\dbinom{n_{A_{0,0}}+n_{A_{0,1}}}{x_{A_{0,0}}+x_{A_{0,1}}}^{-1}\\\\cdot \\\\\\\\& \\\\cdot \\\\frac{1}{n_{A_{0,0}}+n_{A_{0,1}}+1}$ $\\\\int _{\\\\Theta _{\\\\bar{A}_{0}}} \\\\Pr (\\\\mathbf {r}_{\\\\bar{A}_{0}}\\\\mid \\\\theta _{\\\\bar{A}_{0}},M_J)\\\\pi (\\\\theta _{\\\\bar{A}_{0}})d\\\\theta _{\\\\bar{A}_{0}}= \\\\frac{1}{n_{\\\\bar{A}_{0}}+1}$ Readers might recognize the conditional Irving–Fisher exact test as part of (REF ).\",\n \"The result is not surprising since we are looking for the set of $H$ making $E$ irrelevant, and thus supporting the hypothesis of no-difference between the success ratio in the two groups.\",\n \"An illustration of the behavior of the hypergeometric for $n_{A_{0,0}}=n_{A_{0,1}}=10$ is given in Figure REF .\",\n \"High support to the model is achieved when the number of successes $x_{A_{0,0}}$ and $x_{A_{0,1}}$ is almost the same in the two groups.\",\n \"Figure: Hypergeometric behaviour for n A 0,0 =n A 0,1 =10n_{A_{0,0}}=n_{A_{0,1}}=10 and different values for x A 0,0 x_{A_{0,0}} and x A 0,1 x_{A_{0,1}}\"\n ],\n [\n \"Prior and posterior in the model space\",\n \"The goal is to evaluate the posterior probability of $M_J$ given the responses observed for Ann and the sample.\",\n \"The overall marginal likelihood of $M_j$ is the product of (REF ), (REF ), (REF ) and (REF ) $& \\\\Pr (r_A,\\\\mathbf {r}_{A_{1,1}}, \\\\mathbf {r}_{\\\\bar{A}_{1,1}}, \\\\mathbf {r}_{A_{0,1}}, \\\\mathbf {r}_{A_{0,0}} , \\\\mathbf {r}_{\\\\bar{A}_{0}}) |M_J)=\\\\frac{x_{A_{1,1}}+1}{n_{A_{1,1}}+2} \\\\cdot \\\\frac{1}{n_{A_{1,1}}+1}\\\\cdot \\\\frac{1}{n_{\\\\bar{A}_{1,1}}+1}\\\\\\\\ \\\\nonumber \\\\cdot & \\\\dbinom{n_{A_{0,0}}}{x_{A_{0,0}}}\\\\dbinom{n_{A_{0,1}}}{x_{A_{0,1}}}\\\\dbinom{n_{A_{0,0}}+n_{A_{0,1}}}{x_{A_{0,0}}+x_{A_{0,1}}}^{-1}\\\\frac{1}{n_{A_{0,0}}+n_{A_{0,1}}+1} \\\\cdot \\\\frac{1}{n_{\\\\bar{A}_{0}}+1}$ Concerning the prior over the space of models, the simplest choice is to consider a uniform distribution $ \\\\Pr (M_J)=\\\\frac{1}{2^k}.$ Another choice is the one proposed by Chen and Chen, (2008) [2].\",\n \"They give the same prior probability (equal to $\\\\frac{1}{k+1}$ ) for all models sharing the same number of characteristics $k$ .\",\n \"In this way, for the generic model $M_J$ , we have $\\\\Pr (M_J)= \\\\frac{1}{k+1}\\\\binom{k}{|M_J|}^{-1}\\\\cdot I(|M_J|\\\\le k/2)$ where the search spans all models including at most $k/2$ characteristics.\",\n \"This last choice favors model selection according to Occam's razor principle: the fewer characteristics employed, the more probable is the model.\",\n \"This rationale is reasonably objective.\",\n \"Combining (REF ) or (REF ) with (REF ), we get the required posterior.\"\n ],\n [\n \"Computational issues\",\n \"If the model size becomes huge, we may not be in a position to evaluate the normalizing constant of the posterior distribution, but we can establish an MCMC to make an inference about the variable $M_J$ .\",\n \"Essentially a Metropolis–Hastings would suffice, the acceptance ratio being given by (REF ), evaluated for two different elements of $M_J$ , taking into account the probability of proposing a new model.\"\n ],\n [\n \"Application\",\n \"We carried out an experiment at the University of Florence, School of Engineering, Fall 2017.\",\n \"We asked 161 students to solve a simple probabilistic question and we provided randomly a hint (the treatment $T$ ).\",\n \"Table (REF ) presents the students' background information included in the analysis (the characteristics $H$ ).\",\n \"Before the test, we asked the students if they wished to be helped or not (the desire variable $E$ ).\",\n \"Table: List of student characteristics included in the experimentWe wish to investigate whether there is a causal relation between the hint and the ability to solve the question, for those students who desired to receive a hint.\",\n \"We had 8 of these cases.\",\n \"The corresponding risk ratio (obtained considering the model best fitting the fundamental conditions) lies in the interval $[1.73, 2.43]$ and in one case it exceeds 2 ($RR=2.43$ ), which is a clue for there being a causal relation.\",\n \"In all the other cases, the causal relation is not strongly supported, since $RR$ is close but does not reach 2 (see, in Figure (REF )), the left side of each sub-picture).\",\n \"In the right side of each sub-picture in Figure(REF ) we illustrate how the model selection procedure proposes models respecting the fundamental conditions.\",\n \"For the model with the highest probability, the red dot indicates, in the $x-$ axis, the success ratio for treated and, in the $y-$ axis, the ratio between the success ratio for untreated with $E=1$ and with $E=0$ , respectively.\",\n \"These are the main forces driving the marginal probability for the responses, as shown in (REF ).\",\n \"Ideally, comparability and sufficiency are mostly supported by the highest possible success ratio among the treated and by a ratio between untreated with E=0 and E=1 approximately equal 1.\",\n \"As is apparent, the selected model achieves a good compromise between these requirements.\",\n \"The set of variables selected in the 8 cases are shown in Table (REF ).\",\n \"Note that, overall, these models include only 5 characteristics, and all of them include the educational status of the family.\",\n \"In 4 out of the 8 cases, the same model, including family education, $1^\\\\text{first}$ University registration (a proxy for understanding whether the student failed earlier in their educational career) and previous exposure to statistical training was selected.\",\n \"Figure: Educational causation.Risk Ratio and posterior probability for each explored model (left side).Success ratio for treated vs relative success ratio for untreated with E=1E=1 and E=0E=0 (right side).In the picture there is an example concerning four students who succeed.Figure: Forensic causation.Risk Ratio and posterior probability for each explored model (left side).Success ratio for treated vs relative success ratio for untreated with E=1E=1 and E=0E=0 (right side).In the picture there is an example concerning four students who DID NOT succeed.Table: Educational causation.\",\n \"For the 8 students who desired the hint, got it, and succeed, there are reported the characteristics HH selected by the model that best supported the fundamental conditions and the Risk RatioAs a result of our experiment we also have that among the students who asked for and received the hint, 24 did not succeed.\",\n \"We can suppose that some of them claimed that it was the hint which caused their failure.\",\n \"In this case, the $RR$ lies in the interval $[0.86,1.60]$ , an example for four students is in Figure (REF ).\",\n \"It is not conclusive that there is a causal relation between the hint and the failure since all the models for all the considered students provided values of RR much smaller than 2.\",\n \"In a civil trial this would not suggest to a judge that compensation be awarded.\"\n ],\n [\n \"Conclusions\",\n \"We introduced a typical Causes of Effects problem by means of an archetypical example considering Ann and the effect of an aspirin on her headache.\",\n \"We have proposed a possible solution to make operational the choice of variables to include, so as to validate the fundamental assumptions underlying the assessment of Ann's probability of causation.\",\n \"We assume it is possible to take a randomized sample from Ann's population where, as usual, $T$ is assigned following a randomized protocol and $E$ (this is a novelty) is a question asking the members of the sample about their preference to be treated or not.\",\n \"In the evaluation of $RR_A$ (see REF ), an extreme position would be to include all the subjects participating in the experiment so that simply belonging to the reference population would make the individuals in the sample similar to Ann.\",\n \"On the other hand, the choice could be to find the persons most similar to Ann, i.e.\",\n \"those matching all the available characteristics.\",\n \"Clearly neither of these positions is safe: the former ignores some characteristics of Ann which could be very influential on her reaction to the headache after taking aspirin.\",\n \"The latter greatly reduces the number of individuals to be employed in the estimation, so producing a very unstable inference.\",\n \"Our approach takes a sensible middle course and provide a sensible different causal inference for individuals experiencing the same treatment.\",\n \"Interestingly, this is exactly the aim of Precision Medicine (see Mesko (2017), [11]) which looks for different medical interventions for a group of individuals sharing some relevant (for the reaction between treatment and outcome) characteristics.\",\n \"The next step will be to extend the method to observational studies, to make possible in a wider range of cases the evaluation of the $PC_A$ for Causes of Effects problems.\"\n ],\n [\n \"Appendix\",\n \"Now we detail the computation of the integrals (REF ), (REF ), (REF ) and (REF ).\",\n \"We always assume a no informative prior for all $\\\\pi (\\\\cdot )=Beta(1,1)$ .\",\n \"We start with (REF ).\",\n \"$& \\\\int _{\\\\Theta _{A_{1,1}}} \\\\Pr (r_A,\\\\mathbf {r}_{A_{1,1}}\\\\mid \\\\theta _{A_{1,1}},M_J)\\\\pi (\\\\theta _{A_{1,1}})d\\\\theta _{A_{1,1}}= \\\\nonumber \\\\\\\\&= \\\\int _{\\\\Theta _{A_{1,1}}} \\\\Pr (r_A | \\\\mathbf {r}_{A_{1,1}},\\\\theta _{A_{1,1}},M_J)\\\\underbrace{\\\\Pr (\\\\mathbf {r}_{A_{1,1}}|\\\\theta _{A_{1,1}},M_J)\\\\pi (\\\\theta _{A_{1,1}}|M_J)}_{\\\\text{numerator of the }\\\\theta _{A_{1,1}} \\\\text{ updating}}d \\\\theta _{A_{1,1}} \\\\nonumber \\\\\\\\&= \\\\underbrace{\\\\int _{\\\\Theta _{A_{1,1}}} \\\\Pr (r_A|\\\\theta _{A_{1,1}},M_J)\\\\pi (\\\\theta _{A_{1,1}}|\\\\mathbf {r}_{A_{1,1}},M_J)d\\\\theta _{A_{1,1}}}_{\\\\dfrac{x_{A_{1,1}}+1}{n_{A_{1,1}}+2}}\\\\cdot \\\\nonumber \\\\\\\\& \\\\cdot \\\\underbrace{\\\\int _{\\\\Theta _{A_{1,1}}}\\\\Pr (\\\\mathbf {r}_{A_{1,1}}|\\\\theta _{A_{1,1}},M_J)\\\\Pr (\\\\theta _{A_{1,1}}|M_J)d\\\\theta _{A_{1,1}} }_{\\\\dbinom{n_{A_{1,1}}}{x_{A_{1,1}}}\\\\dfrac{\\\\Gamma (1+1)}{\\\\Gamma (1)\\\\Gamma (1)} \\\\dfrac{\\\\Gamma (1+x_{A_{1,1}}) \\\\Gamma (n_{A_{1,1}}+ 1 -x_{A_{1,1}})}{\\\\Gamma (n_{A_{1,1}}+ 1 +1)}} \\\\nonumber \\\\\\\\ & =\\\\frac{x_{A_{1,1}}+1}{n_{A_{1,1}}+2} \\\\cdot \\\\frac{1}{n_{A_{1,1}}+1}.\",\n \"\\\\nonumber $ For (REF ) we have $&\\\\int _{\\\\Theta _{\\\\bar{A}_{1,1}}} \\\\Pr (\\\\mathbf {r}_{\\\\bar{A}_{1,1}}\\\\mid \\\\theta _{\\\\bar{A}_{1,1}},M_J)\\\\pi (\\\\theta _{\\\\bar{A}_{1,1}})d\\\\theta _{\\\\bar{A}_{1,1}}= \\\\\\\\& \\\\underbrace{\\\\int _{\\\\Theta _{\\\\bar{A}_{1,1}}}\\\\Pr (\\\\mathbf {r}_{\\\\bar{A}_{1,1}}|\\\\theta _{\\\\bar{A}_{1,1}},M_J)\\\\Pr (\\\\theta _{\\\\bar{A}_{1,1}}|M_J)d\\\\theta _{\\\\bar{A}_{1,1}} }_{\\\\dbinom{n_{\\\\bar{A}_{1,1}}}{x_{\\\\bar{A}_{1,1}}}\\\\dfrac{\\\\Gamma (1+1)}{\\\\Gamma (1)\\\\Gamma (1)} \\\\dfrac{\\\\Gamma (1+x_{\\\\bar{A}_{1,1}}) \\\\Gamma (n_{\\\\bar{A}_{1,1}}+ 1 -x_{\\\\bar{A}_{1,1}})}{\\\\Gamma (n_{\\\\bar{A}_{1,1}}+ 1 +1)}} \\\\nonumber \\\\\\\\& =\\\\frac{1}{n_{\\\\bar{A}_{1,1}}+1}$ We now compute (REF ) $&\\\\int _{\\\\Theta _{A_{0}}} \\\\Pr (\\\\mathbf {r}_{A_{0,1}}, \\\\mathbf {r}_{A_{0,0}}\\\\mid \\\\theta _{A_{0}},M_J)\\\\pi (\\\\theta _{A_{0}},M_J)d\\\\theta _{A_{0}}= \\\\\\\\&= \\\\int _{\\\\Theta _{A_{0}}} \\\\Pr (\\\\mathbf {r}_{A_{0,0}},|\\\\theta _{A_{0}},\\\\mathbf {r}_{A_{0,1}},M_J)\\\\underbrace{\\\\Pr (\\\\mathbf {r}_{A_{0,1}}|\\\\theta _{A_{0}}, M_J)\\\\pi (\\\\theta _{A_{0}}|M_J)}_{\\\\text{numerator of the }\\\\theta _{A_{0}} \\\\text{ updating}}d\\\\theta _{A_{0}} \\\\nonumber \\\\\\\\&= \\\\int _{\\\\Theta _{A_{0}}} \\\\Pr (\\\\mathbf {r}_{A_{0,0}}|\\\\theta _{A_{0}},M_J)\\\\pi (\\\\theta _{A_{0}}| \\\\mathbf {r}_{A_{0,1}},M_J)d\\\\theta _{A_{0}} \\\\int _{\\\\Theta _{A_{0}}}\\\\Pr (\\\\mathbf {r}_{A_{0,1}}|\\\\theta _{A_{0}},M_J)\\\\pi (\\\\theta _{A_{0}}| M_J)d\\\\theta _{A_{0}} \\\\nonumber \\\\\\\\&= \\\\dbinom{n_{A_{0,0}}}{x_{A_{0,0}}} \\\\dfrac{\\\\Gamma (n_{A_{0,1}}+2)}{\\\\Gamma (x_{A_{0,1}}+1)\\\\Gamma (n_{A_{0,1}}- x_{A_{0,1}}+1)} \\\\dfrac{\\\\Gamma (x_{A_{0,1} } +1 + x_{A_{0,0}})\\\\Gamma (n_{A_{0,0}}+ n_{A_{0,1}}- x_{A_{0,1} }+1- x_{A_{0,0}}) }{\\\\Gamma (n_{A_{0,0}} + n_{A_{0,1}}+2)} \\\\nonumber \\\\\\\\&\\\\cdot \\\\dbinom{n_{A_{0,1}}}{x_{A_{0,1}}}\\\\dfrac{\\\\Gamma (1+1)}{\\\\Gamma (1)\\\\Gamma (1)} \\\\dfrac{\\\\Gamma (1+x_{A_{0,1}}) \\\\Gamma (n_{A_{0,1}}+ 1 -x_{A_{0,1} })}{\\\\Gamma (n_{A_{0,1}} + 1 +1)} \\\\nonumber \\\\\\\\& = \\\\dbinom{n_{A_{0,0}}}{x_{A_{0,0}}}\\\\dbinom{n_{A_{0,1}}}{x_{A_{0,1}}}\\\\dbinom{n_{A_{0,0}}+n_{A_{0,1}}}{x_{A_{0,0}}+x_{A_{0,1}}}^{-1}\\\\frac{1}{n_{A_{0,0}}+n_{A_{0,1}}+1.\",\n \"}$ Concerning the last term, (REF ), with a similar computation as for (REF ), it is straightforward to see that $\\\\int _{\\\\Theta _{\\\\bar{A}_{0}}} \\\\Pr (\\\\mathbf {r}_{\\\\bar{A}_{0}}\\\\mid \\\\theta _{\\\\bar{A}_{0}},M_J)\\\\pi (\\\\theta _{\\\\bar{A}_{0}})d\\\\theta _{\\\\bar{A}_{0}}= \\\\frac{1}{n_{\\\\bar{A}_{0}}+1}.$ Acknowledgement:The second author was supported by the project GESTA of the Fondazione di Sardegna and Regione Autonoma di Sardegna\"\n ]\n]"},"id":{"kind":"string","value":"1808.08440"}}},{"rowIdx":287,"cells":{"text":{"kind":"list like","value":[["The Eulerian distribution on involutions is indeed $\\gamma$-positive"],["Abstract Let $\\mathcal I_n$ and $\\mathcal J_n$ denote the set of involutions and fixed-point free involutions of $\\{1, \\dots, n\\}$, respectively, and let $\\text{des}(\\pi)$ denote the number of descents of the permutation $\\pi$.","We prove a conjecture of Guo and Zeng which states that $I_n(t) := \\sum_{\\pi \\in \\mathcal I_n} t^{\\text{des}(\\pi)}$ is $\\gamma$-positive for $n \\ge 1$ and $J_{2n}(t) := \\sum_{\\pi \\in \\mathcal J_{2n}} t^{\\text{des}(\\pi)}$ is $\\gamma$-positive for $n \\ge 9$.","We also prove that the number of $(3412, 3421)$-avoiding permutations with $m$ double descents and $k$ descents is equal to the number of separable permutations with $m$ double descents and $k$ descents."],["Introduction","A polynomial $p(t) = a_rt^r + a_{r+1}t^{r+1}+ \\cdots + a_st^s$ is called palindromic of center $\\frac{n}{2}$ if $n = r + s$ and $a_{r+i} = a_{s-i}$ for $0 \\le i \\le \\frac{n}{2} - r$ .","A palindromic polynomial can be written uniquely [2] as $p(t) = \\sum _{k=r}^{\\lfloor \\frac{n}{2}\\rfloor }\\gamma _k t^k(1 + t)^{n - 2k},$ and it is called $\\gamma $ -positive if $\\gamma _k \\ge 0$ for each $k$ .","The $\\gamma $ -positivity of a palindromic polynomial implies unimodality of its coefficients (i.e., the coefficients $a_i$ satisfy $a_r \\le a_{r+1} \\le \\cdots \\le a_{\\lfloor n/2\\rfloor } \\ge a_{\\lfloor n/2\\rfloor + 1} \\ge \\cdots \\ge a_s$ ).","Let $\\mathfrak {S}_n$ be the set of all permutations of $[n] = \\lbrace 1, 2, \\dots , n\\rbrace $ .","For $\\pi \\in \\mathfrak {S}_n$ , the descent set of $\\pi $ is $\\operatorname{Des}(\\pi ) = \\lbrace i \\in [n-1] : \\pi (i) > \\pi (i+1)\\rbrace ,$ and the descent number is $\\operatorname{des}(\\pi ) = \\#\\operatorname{Des}(\\pi )$ .","The double descent set is $\\operatorname{DD}(\\pi ) = \\lbrace i \\in [n] : \\pi (i-1) > \\pi (i) > \\pi (i+1) \\rbrace $ where $\\pi (0) = \\pi (n+1) = \\infty $ , and we define $\\operatorname{dd}(\\pi ) = \\#\\operatorname{DD}(\\pi )$ .","Finally, a permutation $\\pi $ is said to avoid a permutation $\\sigma $ (henceforth called a pattern) if $\\pi $ does not contain a subsequence (not necessarily consecutive) with the same relative order as $\\sigma $ .","We let $\\mathfrak {S}_n(\\sigma _1, \\dots , \\sigma _r)$ denote the set of permutations in $\\mathfrak {S}_n$ avoiding the patterns $\\sigma _1,\\dots , \\sigma _r$ .","The descent polynomial $A_n(t) = \\sum _{\\pi \\in \\mathfrak {S}_n} t^{\\operatorname{des}(\\pi )}$ is called the Eulerian polynomial, and we have the following remarkable fact, which implies that $A_n(t)$ is $\\gamma $ -positive.","Theorem 1.1 (Foata–Schützenberger [4]) For $n \\ge 1$ , $A_n(t) = \\sum _{k = 0}^{\\lfloor \\frac{n-1}{2} \\rfloor }\\gamma _{n,k}t^k(1+t)^{n-1-2k},$ where $\\gamma _{n,k} = \\# \\lbrace \\pi \\in \\mathfrak {S}_n :\\operatorname{dd}(\\pi ) = 0, \\operatorname{des}(\\pi ) = k\\rbrace $ .","Similarly, let $I_n$ be the set of all involutions in $\\mathfrak {S}_n$ , and let $J_n$ be the set of all fixed-point free involutions in $\\mathfrak {S}_n$ .","Define $I_n(t) = \\sum _{\\pi \\in I_n} t^{\\operatorname{des}(\\pi )}, \\quad J_n(t) = \\sum _{\\pi \\in J_n} t^{\\operatorname{des}(\\pi )}.$ Note that $J_n(t) = 0$ for $n$ odd.","Strehl [15] first showed that the polynomials $I_n(t)$ and $J_{2n}(t)$ are palindromic.","Guo and Zeng [6] proved that $I_n(t)$ and $J_{2n}(t)$ are unimodal and conjectured that they are in fact $\\gamma $ -positive.","Our first two theorems, which we prove in Sections and , confirm their conjectures.","thmmIn For $n \\ge 1$ , the polynomial $I_n(t)$ is $\\gamma $ -positive.","thmmJn For $n \\ge 9$ , the polynomial $J_{2n}(t)$ is $\\gamma $ -positive.","Theorem REF and many variations of it have been proved by many methods, see for example [8], [13].","One of these methods uses the Modified Foata–Strehl (MFS) action on $\\mathfrak {S}_n$ [1], [11], [12].","In fact it follows from [1] that the same property holds for all subsets of $\\mathfrak {S}_n$ which are invariant under the MFS action.","The $(2413, 3142)$ -avoiding permutations are the separable permutations, which are permutations that can be built from the trivial permutation through direct sums and skew sums [7].","These are not invariant under the MFS action.","However, in 2017, Fu, Lin, and Zeng [5], using a bijection with di-sk trees, and Lin [9], using an algebraic approach, proved that the separable permutations also satisfy the following theorem.","Theorem 1.2 ([5]) For $n \\ge 1$ , $\\sum _{\\pi \\in \\mathfrak {S}_n(2413, 3142)} t^{\\operatorname{des}(\\pi )} = \\sum _{k = 0}^{\\lfloor \\frac{n-1}{2}\\rfloor } \\gamma _{n,k}^S t^k(1+t)^{n-1-2k},$ where $\\gamma _{n,k}^S = \\#\\lbrace \\pi \\in \\mathfrak {S}_n(2413, 3142) : \\operatorname{dd}(\\pi ) = 0, \\operatorname{des}(\\pi ) = k\\rbrace $ .","Two sets of patterns $\\Pi _1$ and $\\Pi _2$ are $\\operatorname{des}$ -Wilf equivalent if $\\sum _{\\pi \\in \\mathfrak {S}_n(\\Pi _1)} t^{\\operatorname{des}(\\pi )} =\\sum _{\\pi \\in \\mathfrak {S}_n(\\Pi _2)} t^{\\operatorname{des}(\\pi )},$ and are $\\operatorname{Des}$ -Wilf equivalent if $\\sum _{\\pi \\in \\mathfrak {S}_n(\\Pi _1)} \\prod _{i \\in \\operatorname{Des}(\\pi )}t_i= \\sum _{\\pi \\in \\mathfrak {S}_n(\\Pi _2)} \\prod _{i \\in \\operatorname{Des}(\\pi )}t_i.$ We also say that the permutation classes $\\mathfrak {S}_n(\\Pi _1)$ and $\\mathfrak {S}_n(\\Pi _2)$ are $\\operatorname{des}$ -Wilf or $\\operatorname{Des}$ -Wilf equivalent.","In 2018, Lin and Kim [10] determined all permutation classes avoiding two patterns of length 4 which are $\\operatorname{des}$ -Wilf equivalent to the separable permutations, all of which are $\\operatorname{Des}$ -Wilf equivalent to each other but not to the separable permutations.","One such class is the $(3412, 3421)$ -avoiding permutations, which is invariant under the MFS action.","A byproduct of this is that the number of $(3412, 3421)$ -avoiding permutations with no double descents and $k$ descents is also equal to $\\gamma _{n,k}^S$ .","In Section , we prove the following more general fact.","thmmdddes For $n \\ge 1$ , $\\sum _{\\pi \\in \\mathfrak {S}_n(3412, 3421)} x^{\\operatorname{des}(\\pi )}y^{\\operatorname{dd}(\\pi )} = \\sum _{\\pi \\in \\mathfrak {S}_n(2413, 3142)} x^{\\operatorname{des}(\\pi )}y^{\\operatorname{dd}(\\pi )}.$"],["Proof of the $\\gamma $ -positivity of {{formula:e9bf8e0e-d54f-4ead-ae10-133b4eddd1e0}}","In this section we prove Theorem , restated below for clarity.","Let the $\\gamma $ -expansion of $I_n(t)$ be $I_n(t) = \\sum _{k=0}^{\\lfloor \\frac{n-1}{2}\\rfloor }a_{n,k} t^k(1+t)^{n-2k-1}.$ We have the following recurrence relation for the coefficients $a_{n,k}$ .","Theorem 2.1 ([6]) For $n \\ge 3$ and $k \\ge 0$ , $na_{n,k} = &(k+1)a_{n-1,k} + (2n-4k)a_{n-1,k-1}+ [k(k+2) + n-1]a_{n-2,k} \\\\&+ [(k-1)(4n-8k-14) + 2n-8]a_{n-2,k-1} \\\\&+ 4(n-2k)(n-2k+1)a_{n-2,k-2},$ where $a_{n,k} = 0$ if $k < 0$ or $k > (n-1)/2$ .","* We will prove by induction on $n$ the slightly stronger claim that $a_{n,k} \\ge 0$ for $n \\ge 1$ , $k \\ge 0$ , and $a_{n,k} \\ge \\frac{2}{n} a_{n-1,k-1}$ if $n = 2k + 1$ and $k \\ge 4$ .","Assume the claim is true whenever the first index is less than $m$ .","We want to prove the claim for all $a_{m,k}$ .","If $m \\le 2000$ , we can check the claim directly (this has been done by the author using Sage).","Thus, we may assume that $m > 2000$ .","If $m \\ge 2k + 3$ , then all the coefficients in the recursion are nonnegative, so we are done by induction.","Thus, assume that $(m, k) = (2n+1, n)$ or $(2n+2, n)$ with $n \\ge 1000$ .","Case 1: $(m,k) = (2n+2, n)$ .","We wish to show that $a_{2n+2,n} \\ge 0$ .","We apply the recurrence in Theorem REF , noting that $a_{2n, n} = 0$ since $n > (2n-1)/2$ , to get $(2n+2)a_{2n+2, n} &= (n+1)a_{2n+1,n} + 4a_{2n+1,n-1}+ 24a_{2n,n-2} - (2n-2)a_{2n,n-1} \\\\&\\ge 4a_{2n+1,n-1} + 24a_{2n,n-2} - 2na_{2n,n-1} \\\\&\\ge 4a_{2n+1, n-1} + 24a_{2n,n-2} - na_{2n-1, n-1}- 4a_{2n-1, n-2} \\\\& \\quad - 24a_{2n-2, n-3}, $ where the last inequality comes from applying the recurrence once again to obtain $2n a_{2n,n-1} \\le na_{2n-1, n-1} + 4a_{2n-1, n-2}+ 24a_{2n-2, n-3}.$ Note that since $2n+1 \\ge 2(n-1) + 3$ , when we apply the recurrence relation for $a_{2n+1, n-1}$ all terms in the sum are positive.","We drop all terms but the $a_{2n-1, n-1}$ term and multiply by $4/(2n+1)$ to get $4a_{2n+1,n-1} &\\ge \\frac{4}{2n+1} \\left[ (n-1)(n+1) + 2n \\right] a_{2n-1,n-1} \\ge na_{2n-1, n-1}.$ Similarly, since $2n \\ge 2(n-2) + 3$ , when we use the recursion to calculate $a_{2n, n-2}$ , we can drop all terms but the $a_{2n-1, n-2}$ term, which after multiplying by $12/(2n)$ gives $12 a_{2n, n-2} \\ge \\frac{12}{2n}[(n-1) a_{2n-1, n-2}]\\ge 4 a_{2n-1, n-2}.$ Alternatively, we could have dropped all but the $a_{2n-2, n-3}$ term to get $12a_{2n, n-2} &\\ge \\frac{12}{2n}[ (n-3)\\cdot 2 + 4n - 8]a_{2n-2, n-3} \\ge 24a_{n-2, n-3}.$ Plugging the previous three inequalities into $(\\dagger )$ proves that $a_{2n+2, n} \\ge 0$ , as desired.","Case 2: $(m,k) = (2n+1, n)$ .","We want to show $(2n+1)a_{2n+1,n} \\ge 2a_{2n, n-1}$ .","By the recurrence relation, we have $(2n+1)a_{2n+1,n} = 2a_{2n, n-1} + 8a_{2n-1, n-2}- (6n-4)a_{2n-1, n-1}.$ Thus, it suffices to show that $8a_{2n-1,n-2} - (6n-3)a_{2n-1,n-1} \\ge 0$ .","Note that $8a_{2n-1,n-2} - &(6n-3)a_{2n-1,n-1} \\\\ &\\ge 8a_{2n-1,n-2} - 6a_{2n-2, n-2} - 24a_{2n-3, n-3}$ because, by applying the same recurrence relation for $a_{2n-1, n-1}$ and dropping the $-(6n-10)a_{2n-3, n-2}$ term, which is negative, we see that $(6n-3)a_{2n-1,n-1} &= 3(2n-1)a_{2n-1, n-1}\\\\&\\le 3(2a_{2n-2, n-2} + 8a_{2n-3, n-3})\\\\&= 6a_{2n-2, n-2} + 24a_{2n-3, n-3}.$ Multiplying the right hand side of $(\\ast )$ by $2n-1$ and applying the recurrence relation for $a_{2n-1, n-2}$ we get $(2n-1) \\cdot (\\ast ) &= 8(2n-1)a_{2n-1, n-2} - (12n-6)a_{2n-2, n-2}- (48n - 24)a_{2n-3, n-3} \\\\&= 8[(n-1)a_{2n-2, n-2} + 6a_{2n-2, n-3} + (n^2 - 2)a_{2n-3, n-2}\\\\&\\quad + (2n-4) a_{2n-3, n-3} + 48a_{2n-3, n-4}] \\\\&\\quad - (12n - 6)a_{2n-2, n-2} - (48n - 24)a_{2n-3, n-3} \\\\&= 48a_{2n-2,n-3} + (8n^2-16)a_{2n-3,n-2}+ 384a_{2n-3,n-4}\\\\&\\quad - (4n+2)a_{2n-2,n-2} - (32n+8)a_{2n-3,n-3}.","$ Now, since $2n-2\\ge 2(n-3) + 3$ , we use the recursion to calculate $a_{2n-2, n-3}$ , drop some terms, and multiply by $48/(2n-2)$ to get $48 a_{2n-2,n-3} \\ge \\frac{48}{2n-2}[(n-4)\\cdot 2 + 4n-12] a_{2n-4,n-4}\\ge 120 a_{2n-4, n-4}.$ Now we apply the recursion for $a_{2n-2, n-2}$ and multiply by $(4n+2)/(2n-2)$ , which is less than 5, to obtain $(4n + 2)a_{2n-2, n-2}& = \\frac{4n+2}{2n-2}[(n-1)a_{2n-3,n-2} + 4a_{2n-3,n-3} \\\\ &\\quad + 24a_{2n-4, n-4} - (2n-6)a_{2n-4, n-3}] \\\\&< (5n-5)a_{2n-3,n-2} + 20a_{2n-3, n-3} + 120a_{2n-4, n-4} \\\\&\\quad - (10n-30)a_{2n-4,n-3}.$ We substitute the above two bounds on $48a_{2n-2, n-3}$ and $(4n+2)a_{2n-2, n-2}$ for the corresponding terms in $(\\ast \\ast )$ to get $(\\ast \\ast ) &\\ge 120 a_{2n-4, n-4} + (8n^2-16)a_{2n-3,n-2}+ 384a_{2n-3,n-4}\\\\&\\quad - (5n-5)a_{2n-3,n-2} - 20a_{2n-3, n-3} - 120a_{2n-4, n-4} \\\\&\\quad + (10n-30)a_{2n-4,n-3} - (32n + 8)a_{2n-3, n-3} \\\\&= (8n^2 - 5n -11)a_{2n-3, n-2} + 384 a_{2n-3, n-4}+ (10n- 30)a_{2n-4, n-3} \\\\&\\quad - (32n + 28)a_{2n-3, n-3}.","$ Since $2n-3 = 2(n-2) + 1$ , by the $a_{2k+1, k} \\ge \\frac{2}{2k+1}a_{2k, k-1}$ part of the induction hypothesis, we have $(8n^2-5n-11)a_{2n-3,n-2} \\ge \\frac{2(8n^2-5n-11)}{2n-3}a_{2n-4, n-3}\\ge (8n + 6) a_{2n-4, n-3}.$ Plugging the previous inequality into $(\\ast \\ast \\ast )$ gives $(\\ast \\ast \\ast ) &\\ge (18n - 24)a_{2n-4, n-3}+ 384a_{2n-3, n-4} - (32n + 28)a_{2n-3,n-3}.$ Since $2n-3 \\ge 2(n-4) \\ge 3$ , when apply the recursion for $a_{2n-3, n-4}$ , we can drop the $a_{2n-5, n-6}$ term which is positive, to get (after multiplying by $384/(2n-3)$ ), $384a_{2n-3,n-4} &\\ge \\frac{384}{2n-3} [(n-3)a_{2n-4,n-4} + 10a_{2n-4,n-5} \\\\&\\quad + (n^2-4n + 4)a_{2n-5, n-4} + (10n-44)a_{2n-5,n-5}].$ Apply the recursion for $a_{2n-3, n-3}$ directly and multiply by $32n+28/(2n-3)$ to get $(32n + 28)a_{2n-3,n-3}&= \\frac{32n+28}{2n-3}[(n-2)a_{2n-4,n-3} + 6a_{2n-4,n-4} \\\\&\\quad + (n^2-2n-1) a_{2n-5,n-3} + (2n-6)a_{2n-5,n-4}\\\\&\\quad + 48a_{2n-5, n-5} ].$ Now, we will check that each of the terms in the expansion for $(32n + 28)a_{2n-3, n-3}$ is less than one of the terms in the expansion of $(18n-24)a_{2n-4,n-3}+ 384a_{2n-3,n-4}$ .","We have $(32n+28)/(2n-3) \\le 17$ , and we see that $17\\cdot (n-2)a_{2n-4,n-3} &\\le (18n-24)a_{2n-4,n-3}, \\\\17\\cdot 6a_{2n-4,n-4} &\\le \\frac{384(n-3)}{2n-3}a_{2n-4,n-4},\\\\17\\cdot (2n-6)a_{2n-5,n-4} &\\le \\frac{68(n^2-4n+4)}{2n-3}a_{2n-5,n-4}, \\\\17\\cdot 48a_{2n-5,n-5} &\\le \\frac{384(10n-44)}{2n-3}a_{2n-5,n-5}.$ Now, it suffices to show $17\\cdot (n^2 - 2n - 1)a_{2n-5,n-3}\\le \\frac{316(n^2-4n+4)}{2n-3}a_{2n-5,n-4}.$ It suffices to show that $9a_{2n-5,n-4} \\ge na_{2n-5,n-3}.$ By the recurrence relation we have $na_{2n-5,n-3} &\\le \\frac{n}{(2n-5)}(2a_{2n-6,n-4} + 8a_{2n-7,n-5}) \\\\4.5 a_{2n-5,n-4} &\\ge \\frac{4.5(n-3)}{(2n-5)}a_{2n-6,n-4} \\\\4.5 a_{2n-5,n-4} &\\ge \\frac{4.5(2n-8)}{(2n-5)} a_{2n-7,n-5}.$ Combining these gives the desired inequality."],["Proof of the $\\gamma $ -positivity of {{formula:5ef23ba2-5ba5-4aff-8556-9c14ddca32a6}}","In this section we prove Theorem , restated below.","Let the $\\gamma $ -expansion of $J_{2n}(t)$ be $J_{2n}(t) = \\sum _{k=1}^n b_{2n,k} t^k(1+t)^{2n-2k}.$ We have the following recurrence relation for the coefficients $b_{2n,k}$ .","Theorem 3.1 ([6]) For $n\\ge 2$ and $k\\ge 1$ , we have $2nb_{2n,k} &= [k(k+1) + 2n-2]b_{2n-2,k} +[2+2(k-1)(4n-4k-3)]b_{2n-2,k-1} \\\\&\\quad + 8(n-k+1)(2n-2k+1)b_{2n-2,k-2},$ where $b_{2n,k} = 0$ if $k < 1$ or $k > n$ .","* We will prove by induction on $n$ the slightly stronger claim that for $b_{2n,k} \\ge 0$ for $n \\ge 9, k \\ge 1$ , and $b_{2n, n} \\ge b_{2n-2,n-1}$ for $n \\ge 11$ .","Assume the claim is true whenever the first index is less than $m$ .","We want to prove the claim for all $b_{m, k}$ .","If $m \\le 2000$ , we can check the claim directly (this has been checked using Sage).","Thus, we may assume $m > 2000$ .","If $m > 2k$ , then all of the coefficients in the recursion are nonnegative, so we are done by induction.","Thus, we can assume that $(m,k) = (2n, n)$ with $n > 1000$ .","By the recurrence relation, we have $2nb_{2n,n} = 8b_{2n-2,n-2} - (6n-8)b_{2n-2,n-1}.$ We want to show that $8b_{2n-2,n-2} - (8n-8)b_{2n-2,n-1} \\ge 0$ .","We have $8b_{2n-2,n-2} - &(8n-8)b_{2n-2,n-1} \\\\&= 8b_{2n-2,n-2} - 32b_{2n-4, n-3}+ 4(6n-14)b_{2n-4, n-2}.$ Multiplying by $(2n-2)/8$ , it suffices to show $(2n-2) b_{2n-2,n-2} - (8n-8) b_{2n-4,n-3} + (6n^2 - 20n + 14)b_{2n-4,n-2} \\ge 0.$ By expanding $(2n-2)b_{2n-2, n-2}$ using the recursion, we find that the above is equivalent to $&(7n^2 - 21n + 12)b_{2n-4, n-2}+ 48b_{2n-4, n-4} - (6n-4)b_{2n-4,n-3}\\ge 0.","$ By the induction hypothesis, $(7n^2-21n+12)b_{2n-4,n-2} \\ge (7n^2-21n+12)b_{2n-6,n-3}.$ By the recurrence in Theorem REF , $(2n-4)_{2n-4,n-4} &\\ge (n^2 - 5n + 6)b_{2n-6, n-4}+ (10n - 48)b_{2n-6,n-5}.$ Multiplying by $48/(2n-4)$ yields $48b_{2n-4,n-4} &\\ge \\frac{48}{2n-4}[ (n^2 - 5n + 6)b_{2n-6, n-4}+ (10n - 48)b_{2n-6,n-5} ].$ Also, multiplying the recurrence for $b_{2n-4, n-3}$ by $(6n-4)/(2n-4)$ yields $(6n-4)b_{2n-4,n-3} &= \\frac{6n-4}{2n-4}[(n^2 -3n)b_{2n-6,n-3}+ (2n - 6)b_{2n-6,n-4} \\\\&\\qquad \\qquad \\qquad + 48b_{2n-6,n-5} ].$ We check that each term in this sum is less that one of the terms in the expansion of $(7n^2-21n+12)b_{2n-6,n-3} + 48b_{2n-4,n-4}$ .","We have $(6n-4)/(2n-4) \\le 4$ and $4(n^2-3n)b_{2n-6,n-3} &\\le (7n^2-21n+12)b_{2n-6,n-3} \\\\4(2n-6)b_{2n-6,n-4} &\\le \\frac{48(n^2-5n+6)}{2n-4}b_{2n-6,n-4} \\\\4\\cdot 48 b_{2n-6,n-5} &\\le \\frac{48(10n-48)}{2n-4}b_{2n-6,n-5}.$ Thus $(\\ddagger )$ is true, as desired."],["$(3412, 3421)$ -avoiding permutations and\nseparable permutations","In this section we prove Theorem , restated below.","* For convenience we define the following variants of the double descent set.","Let $\\operatorname{DD}_0(\\pi ) = \\lbrace i \\in [n] : \\pi (i-1) > \\pi (i) > \\pi (i+1)\\rbrace $ where $\\pi (0) = 0$ , $\\pi (n+1) = \\infty $ , and $\\operatorname{DD}_\\infty (\\pi ) = \\lbrace i \\in [n] : \\pi (i-1) > \\pi (i) > \\pi (i+1)\\rbrace $ where $\\pi (0) = \\infty $ , $\\pi (n+1) = 0$ .","Similarly define $\\operatorname{dd}_0(\\pi )$ and $\\operatorname{dd}_\\infty (\\pi )$ .","Finally, let $\\operatorname{des}^{\\prime }(\\pi ) = \\#(\\operatorname{Des}(\\pi ) \\setminus \\lbrace n-1\\rbrace ),\\quad \\operatorname{dd}^{\\prime }(\\pi ) = \\#(\\operatorname{DD}(\\pi ) \\setminus \\lbrace n - 1\\rbrace ).$ Let $\\mathfrak {S}_n^1 = \\mathfrak {S}_n(2413, 3142)$ and $\\mathfrak {S}_n^2 = \\mathfrak {S}_n(3412, 3421)$ .","For $i = 1, 2$ , define $S_i(x,y,z) &= \\sum _{n \\ge 1} \\sum _{\\pi \\in \\mathfrak {S}_n^i}x^{\\operatorname{des}(\\pi )}y^{\\operatorname{dd}(\\pi )}z^n.$ Moreover, define $F_1(x,y,z) &= \\sum _{n \\ge 1} \\sum _{\\pi \\in \\mathfrak {S}_n^1}x^{\\operatorname{des}(\\pi )}y^{\\operatorname{dd}_0(\\pi )}z^n \\\\R_1(x,y,z) &= \\sum _{n \\ge 1} \\sum _{\\pi \\in \\mathfrak {S}_n^1}x^{\\operatorname{des}(\\pi )}y^{\\operatorname{dd}_\\infty (\\pi )}z^n \\\\T_2(x,y,z) &= \\sum _{n \\ge 1} \\sum _{\\pi \\in \\mathfrak {S}_n^2}x^{\\operatorname{des}^{\\prime }(\\pi )} y^{\\operatorname{dd}^{\\prime }(\\pi )}.$ We will also use $S_i$ , $F_1$ , etc.","to denote $S_i(x, y,z)$ , $F_1(x, y, z)$ , etc.","The proof of the following lemma is very similar to the proof of [9], so it is omitted.","The essence of the proof is Stankova's block decomposition [14].","Lemma 4.1 We have the system of equations $S_1 &= z + (z + xyz)S_1 + \\frac{2xzS_1^2}{1 - xR_1F_1} +\\frac{xzS_1^2(F_1 + xR_1)}{1 - xR_1F_1}, \\\\F_1 &= z + (xzS_1 + zF_1) + \\frac{2xzF_1S_1}{1 - xR_1F_1} +\\frac{xzF_1S_1(F_1 + xR_1)}{1-xR_1F_1}, \\\\R_1 &= yz + zS_1 + xyzR_1 + \\frac{2xzR_1S_1}{1-xR_1F_1}+ \\frac{xzR_1S_1(F_1 + xR_1)}{1 - xR_1F_1}.$ Combining the first equation multiplied by $F_1$ and the second equation multiplied by $S_1$ , and combining the first equation multiplied by $R_1$ and the third equation multiplied by $S_1$ , respectively, gives us $ F_1 = \\frac{S_1+xS_1^2}{1+xyS_1}, \\quad R_1 = \\frac{yS_1 + S_1^2}{1 + S_1}.","$ Plugging these values into the first equation and expanding yields the following.","Corollary 4.2 We have $S_1(x,y,z) = xS_1^3(x,y,z) + xzS_1^2(x,y,z) + (z + xyz)S_1(x,y,z) + z.$ We will show that $S_2$ satisfies the same equation.","Lemma 4.3 We have the system of equations $S_2 &= z + zS_2 + (xy - x)zS_2 + xT_2S_2, \\\\T_2 &= z + (x - xy)z^2 + zS_2 + (xyz - 2xz + z)T_2 + xT_2^2.$ By considering the position of $n$ , we see that every permutation $\\pi \\in \\mathfrak {S}_n^2$ can be uniquely written as either $\\pi _1 n$ where $\\pi _1 \\in \\mathfrak {S}_{n-1}^2$ or $\\pi _1 \\ast \\pi _2$ where $\\pi _1\\in \\mathfrak {S}_{k}^2$ , $\\pi _2 \\in \\mathfrak {S}_{n-k}^2$ , $1 \\le k \\le n - 1$ , and $\\pi _1 \\ast \\pi _2 = AnB$ where $A&= \\pi _1(1) \\cdots \\pi _1(k-1) \\\\B&= (\\pi _2(1) + \\ell )\\cdots (\\pi _2(j-1) + \\ell ) \\pi _1(k) (\\pi _2(j+1) + \\ell )\\cdots (\\pi _2(n-k) + \\ell ),$ where $\\pi _2(j) = 1$ and $\\ell = k - 1$ .","Furthermore, $\\begin{aligned}[c]\\operatorname{des}(\\pi _1 n) &= \\operatorname{des}(\\pi _1) \\\\\\operatorname{dd}(\\pi _1 n) &= \\operatorname{dd}(\\pi _1) \\\\\\operatorname{des}^{\\prime }(\\pi _1 n) &= \\operatorname{des}(\\pi _1)\\\\\\operatorname{dd}^{\\prime }(\\pi _1 n) &= \\operatorname{dd}(\\pi _1)\\end{aligned}\\hspace{28.45274pt}\\begin{aligned}\\operatorname{des}(\\pi _1 \\ast \\pi _2) &= \\operatorname{des}^{\\prime }(\\pi _1) + \\operatorname{des}(\\pi _2) + 1 \\\\\\operatorname{dd}(\\pi _1 \\ast \\pi _2) &= \\operatorname{dd}^{\\prime }(\\pi _1) + \\operatorname{dd}(\\pi _2) \\\\\\operatorname{des}^{\\prime }(\\pi _1 \\ast \\pi _2) &= \\operatorname{des}^{\\prime }(\\pi _1) + \\operatorname{des}^{\\prime }(\\pi _2) + 1\\\\\\operatorname{dd}^{\\prime }(\\pi _1 \\ast \\pi _2) &= \\operatorname{dd}^{\\prime }(\\pi _1) + \\operatorname{dd}^{\\prime }(\\pi _2),\\end{aligned}$ with the exceptions $\\operatorname{dd}(1 \\ast \\pi _2)= \\operatorname{dd}(\\pi _2) + 1$ , $\\operatorname{des}^{\\prime }(\\pi _1 \\ast 1) = \\operatorname{des}^{\\prime }(\\pi _1)$ , $\\operatorname{dd}^{\\prime }(1 \\ast \\pi _2) = \\operatorname{dd}^{\\prime }(\\pi _2) + 1$ , and $\\operatorname{des}^{\\prime }(1 \\ast \\pi _2) = \\operatorname{des}^{\\prime }(\\pi _2)$ if $n \\le 2$ .","With the initial conditions $S_2(x, y, z) = z + \\cdots ,\\quad T_2(x, y, z) = z + 2z^2 + \\cdots ,$ the above implies the stated equations.","Solving the equations in Lemma REF shows that $S_2$ satisfies the same equation as $S_1$ ."],["Concluding remarks and open problems","Our proofs of the $\\gamma $ -positivity of $I_n(t)$ and $J_{2n}(t)$ are purely computational.","Guo and Zeng first suggested the following question.","Problem 5.1 (Guo–Zeng [6]) Give a combinatorial interpretation of the coefficients $a_{n,k}$ .","Dilks [3] conjectured the following $q$ -analog of the $\\gamma $ -positivity of $I_n(t)$ .","Here $\\operatorname{maj}(\\pi )$ denotes the major index of $\\pi $ , which is the sum of the descents of $\\pi $ .","Conjecture 5.2 (Dilks [3]) For $n \\ge 1$ , $\\sum _{\\pi \\in I_n} t^{\\operatorname{des}(\\pi )}q^{\\operatorname{maj}(\\pi )} = \\sum _{k = 0}^{\\lfloor \\frac{n-1}{2}\\rfloor } \\gamma _{n,k}^{(I)} t^k q^{\\binom{k+1}{2}} \\prod _{i = k + 1}^{n - 1 - k} (1 + tq^i),$ where $\\gamma _{n,k}^{(I)}(q) \\in \\mathbb {N}[q]$ .","Since the $(3412, 3421)$ -avoiding permutations are invariant under the MFS action, it would be interesting to find a combinatorial proof of Theorem , since this would lead to a group action on $\\mathfrak {S}_n(2413, 3142)$ such that each orbit contains exactly one element of $\\lbrace \\pi \\in \\mathfrak {S}_2(2413, 3142) : \\operatorname{dd}(\\pi ) = 0, \\operatorname{des}(\\pi ) = k \\rbrace $ (cf.","[5]).","Problem 5.3 Give a bijection between $(3412, 3421)$ -avoiding permutations with $m$ double descents and $k$ descents and separable permutations with $m$ double descents and $k$ descents.","Note that there does not exist a bijection preserving descent sets because the separable permutations are not $\\operatorname{Des}$ -Wilf equivalent to any permutation classes avoiding two patterns.","Finally, Lin [9] proved that the only permutations $\\sigma $ of length 4 which satisfy $\\sum _{\\pi \\in \\mathfrak {S}_n(\\sigma , \\sigma ^r)} t^{\\operatorname{des}(\\pi )}= \\sum _{k = 0}^{\\lfloor \\frac{n-1}{2}\\rfloor }\\gamma _{n,k}t^k(1 + t)^{n-1-2k}$ where $\\gamma _{n,k} = \\#\\lbrace \\pi \\in \\mathfrak {S}_n(\\sigma , \\sigma ^r): \\operatorname{dd}(\\pi ) = 0, \\operatorname{des}(\\pi ) = k\\rbrace $ are the permutations $\\sigma = 2413$ , 3142, 1342, 2431.","Here $\\sigma ^r$ denotes the reverse of $\\sigma $ .","We can similarly ask the following.","Problem 5.4 Which permutations $\\sigma $ of length $\\ell \\ge 6$ satisfy the above property?","Remark 5.5 For $\\ell = 5$ , the answer to Problem REF is $\\sigma = 13254$ , 15243, 15342, 23154, 25143 and their reverses.","We have verified using Sage that these are the only permutations which satisfy the property for $n = 5, 6, 7$ , and these permutation classes are all invariant under the MFS action because in these patterns, every index $i \\in [5]$ is either a valley or a peak."],["Acknowledgements","This research was conducted at the University of Minnesota Duluth REU and was supported by NSF / DMS grant 1650947 and NSA grant H98230-18-1-0010.","I would like to thank Joe Gallian for suggesting the problem, and Brice Huang for many careful comments on the paper."]],"string":"[\n [\n \"The Eulerian distribution on involutions is indeed $\\\\gamma$-positive\"\n ],\n [\n \"Abstract Let $\\\\mathcal I_n$ and $\\\\mathcal J_n$ denote the set of involutions and fixed-point free involutions of $\\\\{1, \\\\dots, n\\\\}$, respectively, and let $\\\\text{des}(\\\\pi)$ denote the number of descents of the permutation $\\\\pi$.\",\n \"We prove a conjecture of Guo and Zeng which states that $I_n(t) := \\\\sum_{\\\\pi \\\\in \\\\mathcal I_n} t^{\\\\text{des}(\\\\pi)}$ is $\\\\gamma$-positive for $n \\\\ge 1$ and $J_{2n}(t) := \\\\sum_{\\\\pi \\\\in \\\\mathcal J_{2n}} t^{\\\\text{des}(\\\\pi)}$ is $\\\\gamma$-positive for $n \\\\ge 9$.\",\n \"We also prove that the number of $(3412, 3421)$-avoiding permutations with $m$ double descents and $k$ descents is equal to the number of separable permutations with $m$ double descents and $k$ descents.\"\n ],\n [\n \"Introduction\",\n \"A polynomial $p(t) = a_rt^r + a_{r+1}t^{r+1}+ \\\\cdots + a_st^s$ is called palindromic of center $\\\\frac{n}{2}$ if $n = r + s$ and $a_{r+i} = a_{s-i}$ for $0 \\\\le i \\\\le \\\\frac{n}{2} - r$ .\",\n \"A palindromic polynomial can be written uniquely [2] as $p(t) = \\\\sum _{k=r}^{\\\\lfloor \\\\frac{n}{2}\\\\rfloor }\\\\gamma _k t^k(1 + t)^{n - 2k},$ and it is called $\\\\gamma $ -positive if $\\\\gamma _k \\\\ge 0$ for each $k$ .\",\n \"The $\\\\gamma $ -positivity of a palindromic polynomial implies unimodality of its coefficients (i.e., the coefficients $a_i$ satisfy $a_r \\\\le a_{r+1} \\\\le \\\\cdots \\\\le a_{\\\\lfloor n/2\\\\rfloor } \\\\ge a_{\\\\lfloor n/2\\\\rfloor + 1} \\\\ge \\\\cdots \\\\ge a_s$ ).\",\n \"Let $\\\\mathfrak {S}_n$ be the set of all permutations of $[n] = \\\\lbrace 1, 2, \\\\dots , n\\\\rbrace $ .\",\n \"For $\\\\pi \\\\in \\\\mathfrak {S}_n$ , the descent set of $\\\\pi $ is $\\\\operatorname{Des}(\\\\pi ) = \\\\lbrace i \\\\in [n-1] : \\\\pi (i) > \\\\pi (i+1)\\\\rbrace ,$ and the descent number is $\\\\operatorname{des}(\\\\pi ) = \\\\#\\\\operatorname{Des}(\\\\pi )$ .\",\n \"The double descent set is $\\\\operatorname{DD}(\\\\pi ) = \\\\lbrace i \\\\in [n] : \\\\pi (i-1) > \\\\pi (i) > \\\\pi (i+1) \\\\rbrace $ where $\\\\pi (0) = \\\\pi (n+1) = \\\\infty $ , and we define $\\\\operatorname{dd}(\\\\pi ) = \\\\#\\\\operatorname{DD}(\\\\pi )$ .\",\n \"Finally, a permutation $\\\\pi $ is said to avoid a permutation $\\\\sigma $ (henceforth called a pattern) if $\\\\pi $ does not contain a subsequence (not necessarily consecutive) with the same relative order as $\\\\sigma $ .\",\n \"We let $\\\\mathfrak {S}_n(\\\\sigma _1, \\\\dots , \\\\sigma _r)$ denote the set of permutations in $\\\\mathfrak {S}_n$ avoiding the patterns $\\\\sigma _1,\\\\dots , \\\\sigma _r$ .\",\n \"The descent polynomial $A_n(t) = \\\\sum _{\\\\pi \\\\in \\\\mathfrak {S}_n} t^{\\\\operatorname{des}(\\\\pi )}$ is called the Eulerian polynomial, and we have the following remarkable fact, which implies that $A_n(t)$ is $\\\\gamma $ -positive.\",\n \"Theorem 1.1 (Foata–Schützenberger [4]) For $n \\\\ge 1$ , $A_n(t) = \\\\sum _{k = 0}^{\\\\lfloor \\\\frac{n-1}{2} \\\\rfloor }\\\\gamma _{n,k}t^k(1+t)^{n-1-2k},$ where $\\\\gamma _{n,k} = \\\\# \\\\lbrace \\\\pi \\\\in \\\\mathfrak {S}_n :\\\\operatorname{dd}(\\\\pi ) = 0, \\\\operatorname{des}(\\\\pi ) = k\\\\rbrace $ .\",\n \"Similarly, let $I_n$ be the set of all involutions in $\\\\mathfrak {S}_n$ , and let $J_n$ be the set of all fixed-point free involutions in $\\\\mathfrak {S}_n$ .\",\n \"Define $I_n(t) = \\\\sum _{\\\\pi \\\\in I_n} t^{\\\\operatorname{des}(\\\\pi )}, \\\\quad J_n(t) = \\\\sum _{\\\\pi \\\\in J_n} t^{\\\\operatorname{des}(\\\\pi )}.$ Note that $J_n(t) = 0$ for $n$ odd.\",\n \"Strehl [15] first showed that the polynomials $I_n(t)$ and $J_{2n}(t)$ are palindromic.\",\n \"Guo and Zeng [6] proved that $I_n(t)$ and $J_{2n}(t)$ are unimodal and conjectured that they are in fact $\\\\gamma $ -positive.\",\n \"Our first two theorems, which we prove in Sections and , confirm their conjectures.\",\n \"thmmIn For $n \\\\ge 1$ , the polynomial $I_n(t)$ is $\\\\gamma $ -positive.\",\n \"thmmJn For $n \\\\ge 9$ , the polynomial $J_{2n}(t)$ is $\\\\gamma $ -positive.\",\n \"Theorem REF and many variations of it have been proved by many methods, see for example [8], [13].\",\n \"One of these methods uses the Modified Foata–Strehl (MFS) action on $\\\\mathfrak {S}_n$ [1], [11], [12].\",\n \"In fact it follows from [1] that the same property holds for all subsets of $\\\\mathfrak {S}_n$ which are invariant under the MFS action.\",\n \"The $(2413, 3142)$ -avoiding permutations are the separable permutations, which are permutations that can be built from the trivial permutation through direct sums and skew sums [7].\",\n \"These are not invariant under the MFS action.\",\n \"However, in 2017, Fu, Lin, and Zeng [5], using a bijection with di-sk trees, and Lin [9], using an algebraic approach, proved that the separable permutations also satisfy the following theorem.\",\n \"Theorem 1.2 ([5]) For $n \\\\ge 1$ , $\\\\sum _{\\\\pi \\\\in \\\\mathfrak {S}_n(2413, 3142)} t^{\\\\operatorname{des}(\\\\pi )} = \\\\sum _{k = 0}^{\\\\lfloor \\\\frac{n-1}{2}\\\\rfloor } \\\\gamma _{n,k}^S t^k(1+t)^{n-1-2k},$ where $\\\\gamma _{n,k}^S = \\\\#\\\\lbrace \\\\pi \\\\in \\\\mathfrak {S}_n(2413, 3142) : \\\\operatorname{dd}(\\\\pi ) = 0, \\\\operatorname{des}(\\\\pi ) = k\\\\rbrace $ .\",\n \"Two sets of patterns $\\\\Pi _1$ and $\\\\Pi _2$ are $\\\\operatorname{des}$ -Wilf equivalent if $\\\\sum _{\\\\pi \\\\in \\\\mathfrak {S}_n(\\\\Pi _1)} t^{\\\\operatorname{des}(\\\\pi )} =\\\\sum _{\\\\pi \\\\in \\\\mathfrak {S}_n(\\\\Pi _2)} t^{\\\\operatorname{des}(\\\\pi )},$ and are $\\\\operatorname{Des}$ -Wilf equivalent if $\\\\sum _{\\\\pi \\\\in \\\\mathfrak {S}_n(\\\\Pi _1)} \\\\prod _{i \\\\in \\\\operatorname{Des}(\\\\pi )}t_i= \\\\sum _{\\\\pi \\\\in \\\\mathfrak {S}_n(\\\\Pi _2)} \\\\prod _{i \\\\in \\\\operatorname{Des}(\\\\pi )}t_i.$ We also say that the permutation classes $\\\\mathfrak {S}_n(\\\\Pi _1)$ and $\\\\mathfrak {S}_n(\\\\Pi _2)$ are $\\\\operatorname{des}$ -Wilf or $\\\\operatorname{Des}$ -Wilf equivalent.\",\n \"In 2018, Lin and Kim [10] determined all permutation classes avoiding two patterns of length 4 which are $\\\\operatorname{des}$ -Wilf equivalent to the separable permutations, all of which are $\\\\operatorname{Des}$ -Wilf equivalent to each other but not to the separable permutations.\",\n \"One such class is the $(3412, 3421)$ -avoiding permutations, which is invariant under the MFS action.\",\n \"A byproduct of this is that the number of $(3412, 3421)$ -avoiding permutations with no double descents and $k$ descents is also equal to $\\\\gamma _{n,k}^S$ .\",\n \"In Section , we prove the following more general fact.\",\n \"thmmdddes For $n \\\\ge 1$ , $\\\\sum _{\\\\pi \\\\in \\\\mathfrak {S}_n(3412, 3421)} x^{\\\\operatorname{des}(\\\\pi )}y^{\\\\operatorname{dd}(\\\\pi )} = \\\\sum _{\\\\pi \\\\in \\\\mathfrak {S}_n(2413, 3142)} x^{\\\\operatorname{des}(\\\\pi )}y^{\\\\operatorname{dd}(\\\\pi )}.$\"\n ],\n [\n \"Proof of the $\\\\gamma $ -positivity of {{formula:e9bf8e0e-d54f-4ead-ae10-133b4eddd1e0}}\",\n \"In this section we prove Theorem , restated below for clarity.\",\n \"Let the $\\\\gamma $ -expansion of $I_n(t)$ be $I_n(t) = \\\\sum _{k=0}^{\\\\lfloor \\\\frac{n-1}{2}\\\\rfloor }a_{n,k} t^k(1+t)^{n-2k-1}.$ We have the following recurrence relation for the coefficients $a_{n,k}$ .\",\n \"Theorem 2.1 ([6]) For $n \\\\ge 3$ and $k \\\\ge 0$ , $na_{n,k} = &(k+1)a_{n-1,k} + (2n-4k)a_{n-1,k-1}+ [k(k+2) + n-1]a_{n-2,k} \\\\\\\\&+ [(k-1)(4n-8k-14) + 2n-8]a_{n-2,k-1} \\\\\\\\&+ 4(n-2k)(n-2k+1)a_{n-2,k-2},$ where $a_{n,k} = 0$ if $k < 0$ or $k > (n-1)/2$ .\",\n \"* We will prove by induction on $n$ the slightly stronger claim that $a_{n,k} \\\\ge 0$ for $n \\\\ge 1$ , $k \\\\ge 0$ , and $a_{n,k} \\\\ge \\\\frac{2}{n} a_{n-1,k-1}$ if $n = 2k + 1$ and $k \\\\ge 4$ .\",\n \"Assume the claim is true whenever the first index is less than $m$ .\",\n \"We want to prove the claim for all $a_{m,k}$ .\",\n \"If $m \\\\le 2000$ , we can check the claim directly (this has been done by the author using Sage).\",\n \"Thus, we may assume that $m > 2000$ .\",\n \"If $m \\\\ge 2k + 3$ , then all the coefficients in the recursion are nonnegative, so we are done by induction.\",\n \"Thus, assume that $(m, k) = (2n+1, n)$ or $(2n+2, n)$ with $n \\\\ge 1000$ .\",\n \"Case 1: $(m,k) = (2n+2, n)$ .\",\n \"We wish to show that $a_{2n+2,n} \\\\ge 0$ .\",\n \"We apply the recurrence in Theorem REF , noting that $a_{2n, n} = 0$ since $n > (2n-1)/2$ , to get $(2n+2)a_{2n+2, n} &= (n+1)a_{2n+1,n} + 4a_{2n+1,n-1}+ 24a_{2n,n-2} - (2n-2)a_{2n,n-1} \\\\\\\\&\\\\ge 4a_{2n+1,n-1} + 24a_{2n,n-2} - 2na_{2n,n-1} \\\\\\\\&\\\\ge 4a_{2n+1, n-1} + 24a_{2n,n-2} - na_{2n-1, n-1}- 4a_{2n-1, n-2} \\\\\\\\& \\\\quad - 24a_{2n-2, n-3}, $ where the last inequality comes from applying the recurrence once again to obtain $2n a_{2n,n-1} \\\\le na_{2n-1, n-1} + 4a_{2n-1, n-2}+ 24a_{2n-2, n-3}.$ Note that since $2n+1 \\\\ge 2(n-1) + 3$ , when we apply the recurrence relation for $a_{2n+1, n-1}$ all terms in the sum are positive.\",\n \"We drop all terms but the $a_{2n-1, n-1}$ term and multiply by $4/(2n+1)$ to get $4a_{2n+1,n-1} &\\\\ge \\\\frac{4}{2n+1} \\\\left[ (n-1)(n+1) + 2n \\\\right] a_{2n-1,n-1} \\\\ge na_{2n-1, n-1}.$ Similarly, since $2n \\\\ge 2(n-2) + 3$ , when we use the recursion to calculate $a_{2n, n-2}$ , we can drop all terms but the $a_{2n-1, n-2}$ term, which after multiplying by $12/(2n)$ gives $12 a_{2n, n-2} \\\\ge \\\\frac{12}{2n}[(n-1) a_{2n-1, n-2}]\\\\ge 4 a_{2n-1, n-2}.$ Alternatively, we could have dropped all but the $a_{2n-2, n-3}$ term to get $12a_{2n, n-2} &\\\\ge \\\\frac{12}{2n}[ (n-3)\\\\cdot 2 + 4n - 8]a_{2n-2, n-3} \\\\ge 24a_{n-2, n-3}.$ Plugging the previous three inequalities into $(\\\\dagger )$ proves that $a_{2n+2, n} \\\\ge 0$ , as desired.\",\n \"Case 2: $(m,k) = (2n+1, n)$ .\",\n \"We want to show $(2n+1)a_{2n+1,n} \\\\ge 2a_{2n, n-1}$ .\",\n \"By the recurrence relation, we have $(2n+1)a_{2n+1,n} = 2a_{2n, n-1} + 8a_{2n-1, n-2}- (6n-4)a_{2n-1, n-1}.$ Thus, it suffices to show that $8a_{2n-1,n-2} - (6n-3)a_{2n-1,n-1} \\\\ge 0$ .\",\n \"Note that $8a_{2n-1,n-2} - &(6n-3)a_{2n-1,n-1} \\\\\\\\ &\\\\ge 8a_{2n-1,n-2} - 6a_{2n-2, n-2} - 24a_{2n-3, n-3}$ because, by applying the same recurrence relation for $a_{2n-1, n-1}$ and dropping the $-(6n-10)a_{2n-3, n-2}$ term, which is negative, we see that $(6n-3)a_{2n-1,n-1} &= 3(2n-1)a_{2n-1, n-1}\\\\\\\\&\\\\le 3(2a_{2n-2, n-2} + 8a_{2n-3, n-3})\\\\\\\\&= 6a_{2n-2, n-2} + 24a_{2n-3, n-3}.$ Multiplying the right hand side of $(\\\\ast )$ by $2n-1$ and applying the recurrence relation for $a_{2n-1, n-2}$ we get $(2n-1) \\\\cdot (\\\\ast ) &= 8(2n-1)a_{2n-1, n-2} - (12n-6)a_{2n-2, n-2}- (48n - 24)a_{2n-3, n-3} \\\\\\\\&= 8[(n-1)a_{2n-2, n-2} + 6a_{2n-2, n-3} + (n^2 - 2)a_{2n-3, n-2}\\\\\\\\&\\\\quad + (2n-4) a_{2n-3, n-3} + 48a_{2n-3, n-4}] \\\\\\\\&\\\\quad - (12n - 6)a_{2n-2, n-2} - (48n - 24)a_{2n-3, n-3} \\\\\\\\&= 48a_{2n-2,n-3} + (8n^2-16)a_{2n-3,n-2}+ 384a_{2n-3,n-4}\\\\\\\\&\\\\quad - (4n+2)a_{2n-2,n-2} - (32n+8)a_{2n-3,n-3}.\",\n \"$ Now, since $2n-2\\\\ge 2(n-3) + 3$ , we use the recursion to calculate $a_{2n-2, n-3}$ , drop some terms, and multiply by $48/(2n-2)$ to get $48 a_{2n-2,n-3} \\\\ge \\\\frac{48}{2n-2}[(n-4)\\\\cdot 2 + 4n-12] a_{2n-4,n-4}\\\\ge 120 a_{2n-4, n-4}.$ Now we apply the recursion for $a_{2n-2, n-2}$ and multiply by $(4n+2)/(2n-2)$ , which is less than 5, to obtain $(4n + 2)a_{2n-2, n-2}& = \\\\frac{4n+2}{2n-2}[(n-1)a_{2n-3,n-2} + 4a_{2n-3,n-3} \\\\\\\\ &\\\\quad + 24a_{2n-4, n-4} - (2n-6)a_{2n-4, n-3}] \\\\\\\\&< (5n-5)a_{2n-3,n-2} + 20a_{2n-3, n-3} + 120a_{2n-4, n-4} \\\\\\\\&\\\\quad - (10n-30)a_{2n-4,n-3}.$ We substitute the above two bounds on $48a_{2n-2, n-3}$ and $(4n+2)a_{2n-2, n-2}$ for the corresponding terms in $(\\\\ast \\\\ast )$ to get $(\\\\ast \\\\ast ) &\\\\ge 120 a_{2n-4, n-4} + (8n^2-16)a_{2n-3,n-2}+ 384a_{2n-3,n-4}\\\\\\\\&\\\\quad - (5n-5)a_{2n-3,n-2} - 20a_{2n-3, n-3} - 120a_{2n-4, n-4} \\\\\\\\&\\\\quad + (10n-30)a_{2n-4,n-3} - (32n + 8)a_{2n-3, n-3} \\\\\\\\&= (8n^2 - 5n -11)a_{2n-3, n-2} + 384 a_{2n-3, n-4}+ (10n- 30)a_{2n-4, n-3} \\\\\\\\&\\\\quad - (32n + 28)a_{2n-3, n-3}.\",\n \"$ Since $2n-3 = 2(n-2) + 1$ , by the $a_{2k+1, k} \\\\ge \\\\frac{2}{2k+1}a_{2k, k-1}$ part of the induction hypothesis, we have $(8n^2-5n-11)a_{2n-3,n-2} \\\\ge \\\\frac{2(8n^2-5n-11)}{2n-3}a_{2n-4, n-3}\\\\ge (8n + 6) a_{2n-4, n-3}.$ Plugging the previous inequality into $(\\\\ast \\\\ast \\\\ast )$ gives $(\\\\ast \\\\ast \\\\ast ) &\\\\ge (18n - 24)a_{2n-4, n-3}+ 384a_{2n-3, n-4} - (32n + 28)a_{2n-3,n-3}.$ Since $2n-3 \\\\ge 2(n-4) \\\\ge 3$ , when apply the recursion for $a_{2n-3, n-4}$ , we can drop the $a_{2n-5, n-6}$ term which is positive, to get (after multiplying by $384/(2n-3)$ ), $384a_{2n-3,n-4} &\\\\ge \\\\frac{384}{2n-3} [(n-3)a_{2n-4,n-4} + 10a_{2n-4,n-5} \\\\\\\\&\\\\quad + (n^2-4n + 4)a_{2n-5, n-4} + (10n-44)a_{2n-5,n-5}].$ Apply the recursion for $a_{2n-3, n-3}$ directly and multiply by $32n+28/(2n-3)$ to get $(32n + 28)a_{2n-3,n-3}&= \\\\frac{32n+28}{2n-3}[(n-2)a_{2n-4,n-3} + 6a_{2n-4,n-4} \\\\\\\\&\\\\quad + (n^2-2n-1) a_{2n-5,n-3} + (2n-6)a_{2n-5,n-4}\\\\\\\\&\\\\quad + 48a_{2n-5, n-5} ].$ Now, we will check that each of the terms in the expansion for $(32n + 28)a_{2n-3, n-3}$ is less than one of the terms in the expansion of $(18n-24)a_{2n-4,n-3}+ 384a_{2n-3,n-4}$ .\",\n \"We have $(32n+28)/(2n-3) \\\\le 17$ , and we see that $17\\\\cdot (n-2)a_{2n-4,n-3} &\\\\le (18n-24)a_{2n-4,n-3}, \\\\\\\\17\\\\cdot 6a_{2n-4,n-4} &\\\\le \\\\frac{384(n-3)}{2n-3}a_{2n-4,n-4},\\\\\\\\17\\\\cdot (2n-6)a_{2n-5,n-4} &\\\\le \\\\frac{68(n^2-4n+4)}{2n-3}a_{2n-5,n-4}, \\\\\\\\17\\\\cdot 48a_{2n-5,n-5} &\\\\le \\\\frac{384(10n-44)}{2n-3}a_{2n-5,n-5}.$ Now, it suffices to show $17\\\\cdot (n^2 - 2n - 1)a_{2n-5,n-3}\\\\le \\\\frac{316(n^2-4n+4)}{2n-3}a_{2n-5,n-4}.$ It suffices to show that $9a_{2n-5,n-4} \\\\ge na_{2n-5,n-3}.$ By the recurrence relation we have $na_{2n-5,n-3} &\\\\le \\\\frac{n}{(2n-5)}(2a_{2n-6,n-4} + 8a_{2n-7,n-5}) \\\\\\\\4.5 a_{2n-5,n-4} &\\\\ge \\\\frac{4.5(n-3)}{(2n-5)}a_{2n-6,n-4} \\\\\\\\4.5 a_{2n-5,n-4} &\\\\ge \\\\frac{4.5(2n-8)}{(2n-5)} a_{2n-7,n-5}.$ Combining these gives the desired inequality.\"\n ],\n [\n \"Proof of the $\\\\gamma $ -positivity of {{formula:5ef23ba2-5ba5-4aff-8556-9c14ddca32a6}}\",\n \"In this section we prove Theorem , restated below.\",\n \"Let the $\\\\gamma $ -expansion of $J_{2n}(t)$ be $J_{2n}(t) = \\\\sum _{k=1}^n b_{2n,k} t^k(1+t)^{2n-2k}.$ We have the following recurrence relation for the coefficients $b_{2n,k}$ .\",\n \"Theorem 3.1 ([6]) For $n\\\\ge 2$ and $k\\\\ge 1$ , we have $2nb_{2n,k} &= [k(k+1) + 2n-2]b_{2n-2,k} +[2+2(k-1)(4n-4k-3)]b_{2n-2,k-1} \\\\\\\\&\\\\quad + 8(n-k+1)(2n-2k+1)b_{2n-2,k-2},$ where $b_{2n,k} = 0$ if $k < 1$ or $k > n$ .\",\n \"* We will prove by induction on $n$ the slightly stronger claim that for $b_{2n,k} \\\\ge 0$ for $n \\\\ge 9, k \\\\ge 1$ , and $b_{2n, n} \\\\ge b_{2n-2,n-1}$ for $n \\\\ge 11$ .\",\n \"Assume the claim is true whenever the first index is less than $m$ .\",\n \"We want to prove the claim for all $b_{m, k}$ .\",\n \"If $m \\\\le 2000$ , we can check the claim directly (this has been checked using Sage).\",\n \"Thus, we may assume $m > 2000$ .\",\n \"If $m > 2k$ , then all of the coefficients in the recursion are nonnegative, so we are done by induction.\",\n \"Thus, we can assume that $(m,k) = (2n, n)$ with $n > 1000$ .\",\n \"By the recurrence relation, we have $2nb_{2n,n} = 8b_{2n-2,n-2} - (6n-8)b_{2n-2,n-1}.$ We want to show that $8b_{2n-2,n-2} - (8n-8)b_{2n-2,n-1} \\\\ge 0$ .\",\n \"We have $8b_{2n-2,n-2} - &(8n-8)b_{2n-2,n-1} \\\\\\\\&= 8b_{2n-2,n-2} - 32b_{2n-4, n-3}+ 4(6n-14)b_{2n-4, n-2}.$ Multiplying by $(2n-2)/8$ , it suffices to show $(2n-2) b_{2n-2,n-2} - (8n-8) b_{2n-4,n-3} + (6n^2 - 20n + 14)b_{2n-4,n-2} \\\\ge 0.$ By expanding $(2n-2)b_{2n-2, n-2}$ using the recursion, we find that the above is equivalent to $&(7n^2 - 21n + 12)b_{2n-4, n-2}+ 48b_{2n-4, n-4} - (6n-4)b_{2n-4,n-3}\\\\ge 0.\",\n \"$ By the induction hypothesis, $(7n^2-21n+12)b_{2n-4,n-2} \\\\ge (7n^2-21n+12)b_{2n-6,n-3}.$ By the recurrence in Theorem REF , $(2n-4)_{2n-4,n-4} &\\\\ge (n^2 - 5n + 6)b_{2n-6, n-4}+ (10n - 48)b_{2n-6,n-5}.$ Multiplying by $48/(2n-4)$ yields $48b_{2n-4,n-4} &\\\\ge \\\\frac{48}{2n-4}[ (n^2 - 5n + 6)b_{2n-6, n-4}+ (10n - 48)b_{2n-6,n-5} ].$ Also, multiplying the recurrence for $b_{2n-4, n-3}$ by $(6n-4)/(2n-4)$ yields $(6n-4)b_{2n-4,n-3} &= \\\\frac{6n-4}{2n-4}[(n^2 -3n)b_{2n-6,n-3}+ (2n - 6)b_{2n-6,n-4} \\\\\\\\&\\\\qquad \\\\qquad \\\\qquad + 48b_{2n-6,n-5} ].$ We check that each term in this sum is less that one of the terms in the expansion of $(7n^2-21n+12)b_{2n-6,n-3} + 48b_{2n-4,n-4}$ .\",\n \"We have $(6n-4)/(2n-4) \\\\le 4$ and $4(n^2-3n)b_{2n-6,n-3} &\\\\le (7n^2-21n+12)b_{2n-6,n-3} \\\\\\\\4(2n-6)b_{2n-6,n-4} &\\\\le \\\\frac{48(n^2-5n+6)}{2n-4}b_{2n-6,n-4} \\\\\\\\4\\\\cdot 48 b_{2n-6,n-5} &\\\\le \\\\frac{48(10n-48)}{2n-4}b_{2n-6,n-5}.$ Thus $(\\\\ddagger )$ is true, as desired.\"\n ],\n [\n \"$(3412, 3421)$ -avoiding permutations and\\nseparable permutations\",\n \"In this section we prove Theorem , restated below.\",\n \"* For convenience we define the following variants of the double descent set.\",\n \"Let $\\\\operatorname{DD}_0(\\\\pi ) = \\\\lbrace i \\\\in [n] : \\\\pi (i-1) > \\\\pi (i) > \\\\pi (i+1)\\\\rbrace $ where $\\\\pi (0) = 0$ , $\\\\pi (n+1) = \\\\infty $ , and $\\\\operatorname{DD}_\\\\infty (\\\\pi ) = \\\\lbrace i \\\\in [n] : \\\\pi (i-1) > \\\\pi (i) > \\\\pi (i+1)\\\\rbrace $ where $\\\\pi (0) = \\\\infty $ , $\\\\pi (n+1) = 0$ .\",\n \"Similarly define $\\\\operatorname{dd}_0(\\\\pi )$ and $\\\\operatorname{dd}_\\\\infty (\\\\pi )$ .\",\n \"Finally, let $\\\\operatorname{des}^{\\\\prime }(\\\\pi ) = \\\\#(\\\\operatorname{Des}(\\\\pi ) \\\\setminus \\\\lbrace n-1\\\\rbrace ),\\\\quad \\\\operatorname{dd}^{\\\\prime }(\\\\pi ) = \\\\#(\\\\operatorname{DD}(\\\\pi ) \\\\setminus \\\\lbrace n - 1\\\\rbrace ).$ Let $\\\\mathfrak {S}_n^1 = \\\\mathfrak {S}_n(2413, 3142)$ and $\\\\mathfrak {S}_n^2 = \\\\mathfrak {S}_n(3412, 3421)$ .\",\n \"For $i = 1, 2$ , define $S_i(x,y,z) &= \\\\sum _{n \\\\ge 1} \\\\sum _{\\\\pi \\\\in \\\\mathfrak {S}_n^i}x^{\\\\operatorname{des}(\\\\pi )}y^{\\\\operatorname{dd}(\\\\pi )}z^n.$ Moreover, define $F_1(x,y,z) &= \\\\sum _{n \\\\ge 1} \\\\sum _{\\\\pi \\\\in \\\\mathfrak {S}_n^1}x^{\\\\operatorname{des}(\\\\pi )}y^{\\\\operatorname{dd}_0(\\\\pi )}z^n \\\\\\\\R_1(x,y,z) &= \\\\sum _{n \\\\ge 1} \\\\sum _{\\\\pi \\\\in \\\\mathfrak {S}_n^1}x^{\\\\operatorname{des}(\\\\pi )}y^{\\\\operatorname{dd}_\\\\infty (\\\\pi )}z^n \\\\\\\\T_2(x,y,z) &= \\\\sum _{n \\\\ge 1} \\\\sum _{\\\\pi \\\\in \\\\mathfrak {S}_n^2}x^{\\\\operatorname{des}^{\\\\prime }(\\\\pi )} y^{\\\\operatorname{dd}^{\\\\prime }(\\\\pi )}.$ We will also use $S_i$ , $F_1$ , etc.\",\n \"to denote $S_i(x, y,z)$ , $F_1(x, y, z)$ , etc.\",\n \"The proof of the following lemma is very similar to the proof of [9], so it is omitted.\",\n \"The essence of the proof is Stankova's block decomposition [14].\",\n \"Lemma 4.1 We have the system of equations $S_1 &= z + (z + xyz)S_1 + \\\\frac{2xzS_1^2}{1 - xR_1F_1} +\\\\frac{xzS_1^2(F_1 + xR_1)}{1 - xR_1F_1}, \\\\\\\\F_1 &= z + (xzS_1 + zF_1) + \\\\frac{2xzF_1S_1}{1 - xR_1F_1} +\\\\frac{xzF_1S_1(F_1 + xR_1)}{1-xR_1F_1}, \\\\\\\\R_1 &= yz + zS_1 + xyzR_1 + \\\\frac{2xzR_1S_1}{1-xR_1F_1}+ \\\\frac{xzR_1S_1(F_1 + xR_1)}{1 - xR_1F_1}.$ Combining the first equation multiplied by $F_1$ and the second equation multiplied by $S_1$ , and combining the first equation multiplied by $R_1$ and the third equation multiplied by $S_1$ , respectively, gives us $ F_1 = \\\\frac{S_1+xS_1^2}{1+xyS_1}, \\\\quad R_1 = \\\\frac{yS_1 + S_1^2}{1 + S_1}.\",\n \"$ Plugging these values into the first equation and expanding yields the following.\",\n \"Corollary 4.2 We have $S_1(x,y,z) = xS_1^3(x,y,z) + xzS_1^2(x,y,z) + (z + xyz)S_1(x,y,z) + z.$ We will show that $S_2$ satisfies the same equation.\",\n \"Lemma 4.3 We have the system of equations $S_2 &= z + zS_2 + (xy - x)zS_2 + xT_2S_2, \\\\\\\\T_2 &= z + (x - xy)z^2 + zS_2 + (xyz - 2xz + z)T_2 + xT_2^2.$ By considering the position of $n$ , we see that every permutation $\\\\pi \\\\in \\\\mathfrak {S}_n^2$ can be uniquely written as either $\\\\pi _1 n$ where $\\\\pi _1 \\\\in \\\\mathfrak {S}_{n-1}^2$ or $\\\\pi _1 \\\\ast \\\\pi _2$ where $\\\\pi _1\\\\in \\\\mathfrak {S}_{k}^2$ , $\\\\pi _2 \\\\in \\\\mathfrak {S}_{n-k}^2$ , $1 \\\\le k \\\\le n - 1$ , and $\\\\pi _1 \\\\ast \\\\pi _2 = AnB$ where $A&= \\\\pi _1(1) \\\\cdots \\\\pi _1(k-1) \\\\\\\\B&= (\\\\pi _2(1) + \\\\ell )\\\\cdots (\\\\pi _2(j-1) + \\\\ell ) \\\\pi _1(k) (\\\\pi _2(j+1) + \\\\ell )\\\\cdots (\\\\pi _2(n-k) + \\\\ell ),$ where $\\\\pi _2(j) = 1$ and $\\\\ell = k - 1$ .\",\n \"Furthermore, $\\\\begin{aligned}[c]\\\\operatorname{des}(\\\\pi _1 n) &= \\\\operatorname{des}(\\\\pi _1) \\\\\\\\\\\\operatorname{dd}(\\\\pi _1 n) &= \\\\operatorname{dd}(\\\\pi _1) \\\\\\\\\\\\operatorname{des}^{\\\\prime }(\\\\pi _1 n) &= \\\\operatorname{des}(\\\\pi _1)\\\\\\\\\\\\operatorname{dd}^{\\\\prime }(\\\\pi _1 n) &= \\\\operatorname{dd}(\\\\pi _1)\\\\end{aligned}\\\\hspace{28.45274pt}\\\\begin{aligned}\\\\operatorname{des}(\\\\pi _1 \\\\ast \\\\pi _2) &= \\\\operatorname{des}^{\\\\prime }(\\\\pi _1) + \\\\operatorname{des}(\\\\pi _2) + 1 \\\\\\\\\\\\operatorname{dd}(\\\\pi _1 \\\\ast \\\\pi _2) &= \\\\operatorname{dd}^{\\\\prime }(\\\\pi _1) + \\\\operatorname{dd}(\\\\pi _2) \\\\\\\\\\\\operatorname{des}^{\\\\prime }(\\\\pi _1 \\\\ast \\\\pi _2) &= \\\\operatorname{des}^{\\\\prime }(\\\\pi _1) + \\\\operatorname{des}^{\\\\prime }(\\\\pi _2) + 1\\\\\\\\\\\\operatorname{dd}^{\\\\prime }(\\\\pi _1 \\\\ast \\\\pi _2) &= \\\\operatorname{dd}^{\\\\prime }(\\\\pi _1) + \\\\operatorname{dd}^{\\\\prime }(\\\\pi _2),\\\\end{aligned}$ with the exceptions $\\\\operatorname{dd}(1 \\\\ast \\\\pi _2)= \\\\operatorname{dd}(\\\\pi _2) + 1$ , $\\\\operatorname{des}^{\\\\prime }(\\\\pi _1 \\\\ast 1) = \\\\operatorname{des}^{\\\\prime }(\\\\pi _1)$ , $\\\\operatorname{dd}^{\\\\prime }(1 \\\\ast \\\\pi _2) = \\\\operatorname{dd}^{\\\\prime }(\\\\pi _2) + 1$ , and $\\\\operatorname{des}^{\\\\prime }(1 \\\\ast \\\\pi _2) = \\\\operatorname{des}^{\\\\prime }(\\\\pi _2)$ if $n \\\\le 2$ .\",\n \"With the initial conditions $S_2(x, y, z) = z + \\\\cdots ,\\\\quad T_2(x, y, z) = z + 2z^2 + \\\\cdots ,$ the above implies the stated equations.\",\n \"Solving the equations in Lemma REF shows that $S_2$ satisfies the same equation as $S_1$ .\"\n ],\n [\n \"Concluding remarks and open problems\",\n \"Our proofs of the $\\\\gamma $ -positivity of $I_n(t)$ and $J_{2n}(t)$ are purely computational.\",\n \"Guo and Zeng first suggested the following question.\",\n \"Problem 5.1 (Guo–Zeng [6]) Give a combinatorial interpretation of the coefficients $a_{n,k}$ .\",\n \"Dilks [3] conjectured the following $q$ -analog of the $\\\\gamma $ -positivity of $I_n(t)$ .\",\n \"Here $\\\\operatorname{maj}(\\\\pi )$ denotes the major index of $\\\\pi $ , which is the sum of the descents of $\\\\pi $ .\",\n \"Conjecture 5.2 (Dilks [3]) For $n \\\\ge 1$ , $\\\\sum _{\\\\pi \\\\in I_n} t^{\\\\operatorname{des}(\\\\pi )}q^{\\\\operatorname{maj}(\\\\pi )} = \\\\sum _{k = 0}^{\\\\lfloor \\\\frac{n-1}{2}\\\\rfloor } \\\\gamma _{n,k}^{(I)} t^k q^{\\\\binom{k+1}{2}} \\\\prod _{i = k + 1}^{n - 1 - k} (1 + tq^i),$ where $\\\\gamma _{n,k}^{(I)}(q) \\\\in \\\\mathbb {N}[q]$ .\",\n \"Since the $(3412, 3421)$ -avoiding permutations are invariant under the MFS action, it would be interesting to find a combinatorial proof of Theorem , since this would lead to a group action on $\\\\mathfrak {S}_n(2413, 3142)$ such that each orbit contains exactly one element of $\\\\lbrace \\\\pi \\\\in \\\\mathfrak {S}_2(2413, 3142) : \\\\operatorname{dd}(\\\\pi ) = 0, \\\\operatorname{des}(\\\\pi ) = k \\\\rbrace $ (cf.\",\n \"[5]).\",\n \"Problem 5.3 Give a bijection between $(3412, 3421)$ -avoiding permutations with $m$ double descents and $k$ descents and separable permutations with $m$ double descents and $k$ descents.\",\n \"Note that there does not exist a bijection preserving descent sets because the separable permutations are not $\\\\operatorname{Des}$ -Wilf equivalent to any permutation classes avoiding two patterns.\",\n \"Finally, Lin [9] proved that the only permutations $\\\\sigma $ of length 4 which satisfy $\\\\sum _{\\\\pi \\\\in \\\\mathfrak {S}_n(\\\\sigma , \\\\sigma ^r)} t^{\\\\operatorname{des}(\\\\pi )}= \\\\sum _{k = 0}^{\\\\lfloor \\\\frac{n-1}{2}\\\\rfloor }\\\\gamma _{n,k}t^k(1 + t)^{n-1-2k}$ where $\\\\gamma _{n,k} = \\\\#\\\\lbrace \\\\pi \\\\in \\\\mathfrak {S}_n(\\\\sigma , \\\\sigma ^r): \\\\operatorname{dd}(\\\\pi ) = 0, \\\\operatorname{des}(\\\\pi ) = k\\\\rbrace $ are the permutations $\\\\sigma = 2413$ , 3142, 1342, 2431.\",\n \"Here $\\\\sigma ^r$ denotes the reverse of $\\\\sigma $ .\",\n \"We can similarly ask the following.\",\n \"Problem 5.4 Which permutations $\\\\sigma $ of length $\\\\ell \\\\ge 6$ satisfy the above property?\",\n \"Remark 5.5 For $\\\\ell = 5$ , the answer to Problem REF is $\\\\sigma = 13254$ , 15243, 15342, 23154, 25143 and their reverses.\",\n \"We have verified using Sage that these are the only permutations which satisfy the property for $n = 5, 6, 7$ , and these permutation classes are all invariant under the MFS action because in these patterns, every index $i \\\\in [5]$ is either a valley or a peak.\"\n ],\n [\n \"Acknowledgements\",\n \"This research was conducted at the University of Minnesota Duluth REU and was supported by NSF / DMS grant 1650947 and NSA grant H98230-18-1-0010.\",\n \"I would like to thank Joe Gallian for suggesting the problem, and Brice Huang for many careful comments on the paper.\"\n ]\n]"},"id":{"kind":"string","value":"1808.08481"}}},{"rowIdx":288,"cells":{"text":{"kind":"list like","value":[["Dialkali-Metal Monochalcogenide Semiconductors with High Mobility and\n Tunable Magnetism"],["Abstract The discovery of archetypal two-dimensional (2D) materials provides enormous opportunities in both fundamental breakthroughs and device applications, as evident by the research booming in graphene, atomically thin transition-metal chalcogenides, and few-layer black phosphorous in the past decade.","Here, we report a new, large family of semiconducting dialkali-metal monochalcogenides (DMMCs) with an inherent A$_{2}$X monolayer structure, in which two alkali sub-monolayers form hexagonal close packing and sandwich the triangular chalcogen atomic plane.","Such unique lattice structure leads to extraordinary physical properties, such as good dynamical and thermal stability, visible to near-infrared light energy gap, high electron mobility (e.g.","$1.87\\times10^{4}$ cm$^{2}$V$^{-1}$S$^{-1}$ in K$_{2}$O).","Most strikingly, DMMC monolayers (MLs) host extended van Hove singularities near the valence band (VB) edge, which can be readily accessed by moderate hole doping of $\\sim1.0\\times10^{13}$ cm$^{-2}$.","Once the critical points are reached, DMMC MLs undergo spontaneous ferromagnetic transition when the top VBs become fully spin-polarized by strong exchange interactions.","Such gate tunable magnetism in DMMC MLs are promising for exploring novel device concepts in spintronics, electronics and optoelectronics."],["Introduction","Although the critical importance of dimensionality in determining the extraordinary physical properties of low-dimension systems has long been recognized since Richard Feyman, the groundbreaking experiments on graphene [1], [2], [3] provide a fascinating platform for exploring exotic phenomena in a rather simple hexagonal lattice.","After the gold rush of graphene, the search of new two-dimensional (2D) systems have received unparalleled attention in the hope of novel physical properties and prototype functionalities.","However, despite the flourishing of 2D materials [4], [5], [6], [7], [8], [9], [10], the majority of 2D researches remain heavily focusing on the archetypal systems of graphene [11], transition-metal dichalcogenides (TMDCs) [12], [13], [14], [15], and few-layer black phosphorus (BP) [16], [17].","These 2D paradigms share the common features of weak interlayer coupling by van der Waals interactions, good thermal stability at room temperature, and high charge carrier mobility.","Nevertheless, very recently, there are emergent 2D materials with tempting physical properties beyond the aforementioned systems, such as the InSe family [18], [19], 2D ferromagnetic van der Waals crystals [20], [21], binary maingroup compounds with the BP-type puckering lattice [22], [23], [24], and the huge family of carbide and nitride based transition metal MXenes, in which M represents transition metal cation and X is C or N anion, respectively [8], [25].","Here, by using ab initio density functional calculations, we discover a large family of dialkali-metal monochalcogenides (DMMCs) with excellent dynamical and thermal stability, visible to near-infrared light energy gap, and very high electron mobility of exceeding $1.0\\times 10^{3}\\,\\textrm {cm}^{2}\\textrm {V}^{-1}\\textrm {s}^{-1}$ .","With the same lattice structure of 1T-TMDCs, DMMCs have the inherent layer-by-layer structure with very weak interlayer coupling, due to the intrinsic +1 oxidation state of alkali cations.","All these features make DMMCs a unique choice beyond TMDCs for fundamental studies of 2D systems and for developing potential electronic and optoelectronic devices.","Most fascinatingly, all DMMC monolayers (MLs) host extended singularity points in the density of states near the Mexican-hat shaped valence band maximum (VBM).","Using hole doping in the range of $\\sim 1.0\\times 10^{13}- 2.5\\times 10^{13}$ cm$^{-2}$ via electrostatic or liquid ion gating, spontaneous ferromagnetic transitions, which lead to half-metallic character with full spin polarized top valence bands, can be triggered by strong exchange interactions in these 2D systems.","Such gate tunable magnetism and half metallicity in DMMC MLs may pave new routes in novel device concepts for spintronics.","Dialkali-metal monochalcogenides form a large family with sixteen compounds in the general formula of A$_{2}$ X, where A represents an alkali atom (Na, K, Rb or Cs) and X is a chalcogen anion of O, S, Se or Te.","The synthesis of bulk dicesium monoxide (Cs$_{2}$ O) was first reported in 1955 [26] with the space group $R\\bar{3}m$ (No.","166), which has the same lattice structures as the well-known 1T-TMDCs [27] as displayed in Fig.","REF a and REF a.","By taking van der Waals (vdW) corrections into account, the optimized lattice constants of bulk Cs$_{2}$ O are $a=4.23$ Å, $b=4.23$ Å, and $c=19.88$ Å, respectively, which are in excellent agreement with the experimental values [26].","In the 2D limit, the monolayer of DMMCs has the space group $P\\bar{3}m1$ (No.","164).","The primitive unit cell consists of two alkali cations and one group-VIa anions, making A$_{2}$ X stoichiometry.","The metal-chalcogen ratio in DMMC MLs is distinct from the well-known TMDC MLs, in which the unit cell includes two chalcogenide anions and one metal cation.","For instance, in Cs$_{2}$ O ML, the centering O atom is surrounded by six Cs cations, forming a distinctive O-Cs octahedron, as shown in Fig.","REF a.","Such lattice structure is rooted in strong intralayer O-Cs ionic bonding, as revealed by the analysis of electron localization functions (ELFs).","As shown in Fig.","REF b, the electron densities are highly localized around Cs and O atoms with negligible inter-atom distribution, reflecting ionic bonding and electron donation from Cs to O atoms.","The calculated oxidation state of O anions, as represented by the Hirshfeld charge, is nearly identical to that of BaO, also confirming the dianionic character of oxygen in Cs$_{2}$ O.","No accumulation of electron density is observed between Cs atoms, suggesting no chemical bond between them.","The A-X octahedral geometry centring the chalcogen dianion is general observed for the whole family, as presented in Fig.","REF b for Na$_{2}$ O ML and in Supplementary Information (SI) Figure S1-S4 for the other 14 DMMC MLs.","Due to the intrinsic full oxidation state (+1) of Cs, the interlayer coupling between Cs sub-MLs is very weak, which only introduces marginal changes to the in-plane lattice constants.","The optimized $a$ and $b$ of Cs$_{2}$ O ML (4.26 Å) are almost the same as the experimental bulk value.","Comparing with the paradigmatic 2D systems of MoS$_{2}$ and graphite, we find that there are even less interlayer electron density localization in Cs$_{2}$ O as a result of dominant in-plane ionic bonding, which also means micromechanical exfoliation of Cs$_{2}$ O crystals can be readily achieved (Fig.","REF a).","Indeed, we calculate the exfoliation energy of Cs$_{2}$ O ML to be $\\sim 0.19$ J/m$^{2}$ (Fig.","REF b), which is surprisingly lower than the value of 0.33 J/m$^{2}$ for graphene [28].","It has to been emphasized here that the lower exfoliation energy of bulk Cs$_{2}$ O, compared with MoS$_{2}$ ($\\sim $ 0.42 J/m$^{2}$ ) and graphite, is consistently obtained by different vdW correction methods, which is summarized in SI Table S1.","To assess the dynamical stability, which is also crucial for the micro-exfoliation of Cs$_{2}$ O ML, we have computed the phonon dispersion with the finite displacement method.","As shown in Fig.","REF c, the good stability of ML Cs$_{2}$ O is evident by the positive values of all phonon modes.","We further check the room-temperature (RT) thermal stability of Cs$_{2}$ O ML by first-principle molecular dynamics simulations.","The fluctuation of energy and temperature as a function of time are plotted in Fig.","REF d. After running 4000 steps (10 ps), the 3$\\times $ 3$\\times $ 1 trigonal lattice is well sustained, and the free energy of the supercell converges.","Excellent dynamical and thermal stability have also been validated for the other DMMC MLs (See SI Figure S5 and S6).","Thus, it is feasible to experimentally exfoliate DMMC MLs for device fabrications and applications."],["Electronic Properties","We now elucidate the electronic structures and fundamental physical properties of DMMC MLs using two representative examples of Cs$_{2}$ O and Na$_{2}$ O MLs.","As shown in Fig.","REF a, Na$_{2}$ O ML is a 2D semiconductor with a direct bandgap of $\\sim $ 1.99 eV at the $\\Gamma $ point.","Noticeably, the VBM of Na$_{2}$ O ML is mainly contributed by O-2$p$ orbitals (its projection weight is not shown in the energy dispersion for clarity) hybridized with very low weight of Na-2$p$ (weighted by green triangles), while the CBM is dominated by Na-3$s$ (red squares).","The electronic band structure undergoes a pronounced change when Na is replaced by Cs, whose strongly delocalized 5$p$ orbitals induce significant density of states (DOS) away from the $\\Gamma $ point, which is still dominated by O-2$p$ .","As the energy of Cs-5$p$ orbitals is $\\sim 0.4$ eV higher than the 2$p$ orbitals of O, the inter $p$ -orbital hybridization reverses the energy levels of VB1 and VB2 along the $\\Gamma -M$ direction.","The resulting Cs$_{2}$ O ML is an indirect semiconductor with multiple VBMs centering the $M$ points (Fig.","REF b).","In general, for heavier cations of K, Rb and Cs, the inter $p$ -orbital hybridization with chalcogen anions is prevailing, shifting VBMs from $\\Gamma $ to the $M$ point.","As a result, except for Na$_{2}$ X (X=O, S, Se and Te), all DMMC MLs are indirect semiconductors.","The inter $p$ -orbital hybridization induced direct to indirect bandgap transition from Na$_{2}$ X to other A$_{2}$ X MLs has also been confirmed by the PBE method, as shown in SI Table S2 and SI Figure S7.","As summarized in Fig.","REF c, the energy gaps of DMMC MLs range from 1 eV to 3 eV, covering the near-infrared (K$_{2}$ O, Rb$_{2}$ O and Cs$_{2}$ O MLs) and visible light regions, which are highly desirable for optoelectronic device applications.","To evaluate the optical performance, the absorption coefficients of Cs$_{2}$ O and Na$_{2}$ O MLs are directly compared with bulk silicon [29].","As shown in Fig.","REF d, Na$_{2}$ O ML shows absorption attenuation above $\\sim 625$ nm, in consistent with the direct bandgap of 1.99 eV.","Strikingly, indirect-bandgap Cs$_{2}$ O ML exhibits unrivaled cyan-to-red light absorption efficiency, which is nearly constant over the whole visible light wavelengths.","Such unusual findings not only predict Cs$_{2}$ O MLs to be extraordinary optoelectronic materials, but also are rooted in the enormous, non-differentiable DOS in the vicinity of VBM, i.e.","van Hove singularity.","As shown in Fig.","REF a and REF b, the top VBs of DMMC MLs, dominated by the localized $p$ orbitals of chalcogen anions, are distinctive by very flat dispersion in the momentum space.","The inter $p$ -orbital hybridization in Cs$_{2}$ O ML drastically reduces the VB energy dispersion, which effectively enhances DOS near the VBM and creates multiple saddle points in the vicinity of $\\Gamma $ , $K$ and $M$ .","This leads to a prominent extended van Hove singularity in the density of states [30].","It is also noteworthy that the inter $p$ -orbital hybridization also greatly enhances DOS near the CBM, which contributes significantly to the extraordinary light absorption characteristics of Cs$_{2}$ O ML.","Charge carrier mobility is another critical parameter for the device performance of 2D material-based devices.","As summarized in Table REF as well as in SI Table S3 and S4, the mobility of DMMC MLs are largely asymmetric between electrons and holes due to the drastic difference in the energy dispersion between the CBs and VBs, producing highly asymmetric effective mass.","Noticeably, K$_{2}$ O ML has the highest electron mobility of 1.87$\\times 10^{4}$ cm$^{2}$ V$^{-1}$ S$^{-1}$ along the $y$ direction.","Such electron mobility is comparable to the hole mobility of few-layer BP [31], [17] and order of magnitude larger than that of MoS${_2}$ atomic layers [13].","Large electron mobility in DMMC MLs may hold a great promise for applications in high-performance electronics.","We have noticed that with the same trigonal structure ($R\\bar{3}m$ , No.166), monolayer Tl$_{2}$ O is predicted to be a semiconductor with a direct bandgap and highly anisotropic charge carrier mobilities up to $4.3\\times 10^{3}\\,\\textrm {cm}^{2}\\textrm {V}^{-1}\\textrm {s}^{-1}$ [32].","This is not surprising since Tl is one of the pseudo-alkali metals, in close resemblance to K both in the oxidation state and ionic radius."],["Tunable Magnetism by Hole doping.","The aforementioned van Hove singularities in the top VB of DMMC MLs are associated with inherent electronic instability, when hole doping pushes the Fermi level approaching the divergent point.","Taking Cs$_{2}$ O as an example, we find that the system undergoes a spontaneous ferromagnetic phase transition with a moderate hole doping of $n\\sim 1.0\\times 10^{13}$ cm$^{-2}$ .","In Fig.","REF a, we plot the magnetic moment and the spin polarization energy per doped hole carrier as a function of $n$ .","The latter represents the energy difference between the non-spin-polarized phase and the ferromagnetic phase.","Clearly, the top VB becomes fully spin polarized by the critical hole doping, as manifested by the constant magnetic moment of 1 $\\mu _{B}$ per carrier above $1.0\\times 10^{13}$ cm$^{-2}$ .","The spin polarization energy, which becomes -3.11 meV per carrier for a minimal hoel doping, monotonically decreases as a function of $n$ , approving that the doping-induced ferromagnetic ordering is energy favourable.","Such spontaneous ferromagnetic ordering can be well understood by the Stoner model, which has been adopted to explain the so-called “$d^{0}$ ferromagnetism” in hole doped nitrides and oxides [33].","In this picture, ferromagnetism can spontaneously appear when the Stoner criterion $U*g(E_{f})>$ 1 is satisfied, where $U$ is the exchange interaction strength and $g(E_{f})$ is the DOS at the Fermi energy of non-magnetic state.","In such condition, the energy gain by exchange interactions exceeds the loss in kinetic energy, and hence the system would favor a ferromagnetic ground state.","We have confirmed that the $U*g(E_{f})$ of Cs$_{2}$ O is always larger than one above the critical doping level, indicating that the huge DOS associated with the van Hove singularity plays a key role in the occurrence of ferromagnetism.","In Fig.","REF b and REF c, we plot the spin-polarized DOS of Cs$_{2}$ O with a doping level of $n=5.1\\times 10^{14}$ cm$^{-2}$ .","Compared with the undoped case (Fig.","REF b), the corresponding spin splitting process causes a significant energy shit of 0.1 eV between the spin-up and spin-down bands, leaving the Fermi level cutting through only one spin channel.","Such half-metal state allows fully spin-polarized transport, which is crucial for spintronics applications.","It is fascinating that this half-metallic ferromagnetism can be reversibly switched on and off in a $\\sim $ 1 nm thick atomic layer without extrinsic dopant and defects.","We have found that except for Na$_{2}$ Te, the ferromagnetic transition can be triggered in all DMMC MLs by hole doping, ranging from $\\sim 1.0\\times 10^{13}$ cm$^{-2}$ for Cs$_{2}$ O to $8.0\\times 10^{15}$ cm$^{-2}$ for Na$_{2}$ Se (see SI Figure S8-S11).","The special case of Na$_{2}$ Te can be understood by insufficient DOS near the VBM, which is dominated by the 5$p$ orbitals of Te.","Due to much lower electron negativity, the 5$p$ orbitals of Te are more delocalized than the other chalcogen elements.","Consequently, the high dispersion of Te-5$p$ reduce DOS near the VBM of Na$_{2}$ Te (SI Fig.","S8), invalidating the Stoner criterion.","We further evaluated the doping dependent Curie temperature ($T_{c}$ ) of DMMC MLs using the Heisenberg model.","As shown in Fig.","REF d, using $n=5.0\\times 10^{14}$ cm$^{-2}$ as a reference, which can be readily induced by ionic liquid gating or lithium glass gating [16], [34], RT ferromagnetism is achievable in K$_{2}$ O and Na$_{2}$ O MLs (SI Figure S12).","For Cs$_{2}$ O, the corresponding $T_{c}$ is $\\sim 200$ K, which is nearly five times higher than the recently reported ferromagnetic CrI$_{3}$ and Cr$_{2}$ Ge$_{2}$ Te$_{6}$ [20], [21].","In summary, we have reported a new family of 1T-TMDC structured dialkali-metal monochalcogenides (DMMCs), which show fascinating physical properties and great device application promises in the 2D limit.","Distinctively, DMMC MLs are characterized by large electron mobility, e.g.","$1.87\\times 10^{4}$ cm$^{2}$ V$^{-1}$ S$^{-1}$ for K$_{2}$ O and $5.42\\times 10^{3}$ cm$^{2}$ V$^{-1}$ S$^{-1}$ for Na$_{2}$ O, and by very flat VBs with rather weak energy dispersion.","The latter are responsible for the formation of extended van Hove singularities with extremely high DOS just below the VB edge of DMMC MLs.","By introducing moderate hole doping, these DMMC MLs are subjected to strong electronic instabilities induced by electron-electron interaction, which ultimately leads to spontaneous ferromagnetic phase transitions where the top VBs becomes fully spin-polarized.","Using heavy doping methods of ion liquid and solid gating, we can readily increase the $T_{c}$ and switch on and off the magnetism of DMMC MLs even at room temperature.","Although alkali compounds are not very stable in air by reacting with ambient moisture, the excellent dynamical and thermal stability allows DMMC MLs to be exfoliated and encapsulated in glove box environment, which is now routine research facilities for studying air-sensitive 2D materials [16], [17], [19], [20], [21].","It is also feasible to grow DMMC MLs by molecular beam epitaxy or by chemical vapor deposition, with abundant options of single crystal substrates of 1T- or 2H-TMDCs.","By forming heterostructure with high electron affinity main-group MDCs, such as SnSe$_{2}$ , DMMC MLs may become effectively hole doped by interfacial charge transferring [24], which would provide the opportunity for in-situ studying the gate-tunable ferromagnetism in DMMC MLs by scanning probe microscope.","Figure: (a) Structure of DMMC MLs.","The upper inset is the first Brillouin zone.","Each chalcogen anion is surrounded by six alkali cations, forming XA 6 _{6} octahedron.","(b-c) Normalized electron localization function (ELF) of Cs 2 _{2}O and Na 2 _{2}O, respectively.","ELF = 1 (red) and 0 (blue) indicate accumulated and vanishing electron densities, respectively.","The 2D ELF plane is defined by the green line in Fig.","1a.Figure: Stability and Micromechanical Exfoliation.","(a) Interlayer differential charge density of Cs 2 _{2}O bulk, graphite and MoS 2 _{2} bulk.","(b) Calculated exfoliation energy of Cs 2 _{2}O ML, in comparison with graphene (blue triangles).","(c) The phonon spectrum of Cs 2 _{2}O ML.","(d) First-principle molecular dynamics simulations of Cs 2 _{2}O ML.","The thermal stability of the supercell is evident by the convergence in the free energy.Figure: Electronic Properties.","Band structures and the corresponding DOS of Na 2 _{2}O ML (a) and Cs 2 _{2}O ML (b).","The contributions of alkali ss-orbital and pp-orbital are indicated by red and green circles, respectively.","The insets are the spatial distribution of the wave-functions for the VBM and CBM.","(c) Energy gaps of all DMMC MLs.","Strong inter pp-orbital hybridization leads to the transition from direct bandgaps in Na 2 _{2}X to indirect gaps in other DMMC MLs.","The gap values of DMMC MLs are mainly determined by the electron affinity of the anions.","The results have been confirmed by both the HSE06 and PBE methods.","(d) Extraordinary light absorbance of Cs 2 _{2}O and Na 2 _{2}O MLs, in comparison with silicon.Figure: Tunable Magnetism by Hole Doping.","(a) Magnetic moment and spin polarization energy per carrier of Cs 2 _{2}O ML.","Remarkably, spontaneous ferromagnetic phase transition is trigger by a moderate hole doping of 1.28×10 13 1.28\\times 10^{13} cm -2 ^{-2}.","(b) Spin-resolved DOS of Cs 2 _{2}O ML at n=5×10 14 n=5\\times 10^{14} cm -2 ^{-2}.","The inset shows the spin density distribution, apparently locating at O atoms.","(c) Spin-splitted top valence bands in the vicinity of MM points in Cs 2 _{2}O ML (n=5×10 14 n=5\\times 10^{14} cm -2 ^{-2}).","(d) Curie temperature verusverus hole doping density, showing the feasibility of RT ferromagnetism by heavy hole doping.","Note that antiferromagnetic ordering in DMMC MLs is energetically higher in free energy than the ferromagnetic state.Table: Mobility and effective mass along the xx and yy transport directions in four Cs 2 _{2}X MLs and K 2 _{2}O.","We calculated the carrier mobility using a phonon-limited scattering model, which includes deformation potential (E 1 _{1}) and elastic modulus (C 2D _{2D}) in the propagation direction of the longitudinal acoustic wave.","Noticeably, K 2 _{2}O has the highest electron mobility.This work is supported by the National Key R&D Program of the MOST of China (Grant Nos.","2016YFA0300204 and 2017YFA0303002), and the National Science Foundation of China (Grant Nos.","11574264 and 61574123).","Y.Z.","acknowledges the funding support from the Fundamental Research Funds for the Central Universities and the Thousand Talents Plan."],["Competing financial interests:","The authors declare no competing financial interests."]],"string":"[\n [\n \"Dialkali-Metal Monochalcogenide Semiconductors with High Mobility and\\n Tunable Magnetism\"\n ],\n [\n \"Abstract The discovery of archetypal two-dimensional (2D) materials provides enormous opportunities in both fundamental breakthroughs and device applications, as evident by the research booming in graphene, atomically thin transition-metal chalcogenides, and few-layer black phosphorous in the past decade.\",\n \"Here, we report a new, large family of semiconducting dialkali-metal monochalcogenides (DMMCs) with an inherent A$_{2}$X monolayer structure, in which two alkali sub-monolayers form hexagonal close packing and sandwich the triangular chalcogen atomic plane.\",\n \"Such unique lattice structure leads to extraordinary physical properties, such as good dynamical and thermal stability, visible to near-infrared light energy gap, high electron mobility (e.g.\",\n \"$1.87\\\\times10^{4}$ cm$^{2}$V$^{-1}$S$^{-1}$ in K$_{2}$O).\",\n \"Most strikingly, DMMC monolayers (MLs) host extended van Hove singularities near the valence band (VB) edge, which can be readily accessed by moderate hole doping of $\\\\sim1.0\\\\times10^{13}$ cm$^{-2}$.\",\n \"Once the critical points are reached, DMMC MLs undergo spontaneous ferromagnetic transition when the top VBs become fully spin-polarized by strong exchange interactions.\",\n \"Such gate tunable magnetism in DMMC MLs are promising for exploring novel device concepts in spintronics, electronics and optoelectronics.\"\n ],\n [\n \"Introduction\",\n \"Although the critical importance of dimensionality in determining the extraordinary physical properties of low-dimension systems has long been recognized since Richard Feyman, the groundbreaking experiments on graphene [1], [2], [3] provide a fascinating platform for exploring exotic phenomena in a rather simple hexagonal lattice.\",\n \"After the gold rush of graphene, the search of new two-dimensional (2D) systems have received unparalleled attention in the hope of novel physical properties and prototype functionalities.\",\n \"However, despite the flourishing of 2D materials [4], [5], [6], [7], [8], [9], [10], the majority of 2D researches remain heavily focusing on the archetypal systems of graphene [11], transition-metal dichalcogenides (TMDCs) [12], [13], [14], [15], and few-layer black phosphorus (BP) [16], [17].\",\n \"These 2D paradigms share the common features of weak interlayer coupling by van der Waals interactions, good thermal stability at room temperature, and high charge carrier mobility.\",\n \"Nevertheless, very recently, there are emergent 2D materials with tempting physical properties beyond the aforementioned systems, such as the InSe family [18], [19], 2D ferromagnetic van der Waals crystals [20], [21], binary maingroup compounds with the BP-type puckering lattice [22], [23], [24], and the huge family of carbide and nitride based transition metal MXenes, in which M represents transition metal cation and X is C or N anion, respectively [8], [25].\",\n \"Here, by using ab initio density functional calculations, we discover a large family of dialkali-metal monochalcogenides (DMMCs) with excellent dynamical and thermal stability, visible to near-infrared light energy gap, and very high electron mobility of exceeding $1.0\\\\times 10^{3}\\\\,\\\\textrm {cm}^{2}\\\\textrm {V}^{-1}\\\\textrm {s}^{-1}$ .\",\n \"With the same lattice structure of 1T-TMDCs, DMMCs have the inherent layer-by-layer structure with very weak interlayer coupling, due to the intrinsic +1 oxidation state of alkali cations.\",\n \"All these features make DMMCs a unique choice beyond TMDCs for fundamental studies of 2D systems and for developing potential electronic and optoelectronic devices.\",\n \"Most fascinatingly, all DMMC monolayers (MLs) host extended singularity points in the density of states near the Mexican-hat shaped valence band maximum (VBM).\",\n \"Using hole doping in the range of $\\\\sim 1.0\\\\times 10^{13}- 2.5\\\\times 10^{13}$ cm$^{-2}$ via electrostatic or liquid ion gating, spontaneous ferromagnetic transitions, which lead to half-metallic character with full spin polarized top valence bands, can be triggered by strong exchange interactions in these 2D systems.\",\n \"Such gate tunable magnetism and half metallicity in DMMC MLs may pave new routes in novel device concepts for spintronics.\",\n \"Dialkali-metal monochalcogenides form a large family with sixteen compounds in the general formula of A$_{2}$ X, where A represents an alkali atom (Na, K, Rb or Cs) and X is a chalcogen anion of O, S, Se or Te.\",\n \"The synthesis of bulk dicesium monoxide (Cs$_{2}$ O) was first reported in 1955 [26] with the space group $R\\\\bar{3}m$ (No.\",\n \"166), which has the same lattice structures as the well-known 1T-TMDCs [27] as displayed in Fig.\",\n \"REF a and REF a.\",\n \"By taking van der Waals (vdW) corrections into account, the optimized lattice constants of bulk Cs$_{2}$ O are $a=4.23$ Å, $b=4.23$ Å, and $c=19.88$ Å, respectively, which are in excellent agreement with the experimental values [26].\",\n \"In the 2D limit, the monolayer of DMMCs has the space group $P\\\\bar{3}m1$ (No.\",\n \"164).\",\n \"The primitive unit cell consists of two alkali cations and one group-VIa anions, making A$_{2}$ X stoichiometry.\",\n \"The metal-chalcogen ratio in DMMC MLs is distinct from the well-known TMDC MLs, in which the unit cell includes two chalcogenide anions and one metal cation.\",\n \"For instance, in Cs$_{2}$ O ML, the centering O atom is surrounded by six Cs cations, forming a distinctive O-Cs octahedron, as shown in Fig.\",\n \"REF a.\",\n \"Such lattice structure is rooted in strong intralayer O-Cs ionic bonding, as revealed by the analysis of electron localization functions (ELFs).\",\n \"As shown in Fig.\",\n \"REF b, the electron densities are highly localized around Cs and O atoms with negligible inter-atom distribution, reflecting ionic bonding and electron donation from Cs to O atoms.\",\n \"The calculated oxidation state of O anions, as represented by the Hirshfeld charge, is nearly identical to that of BaO, also confirming the dianionic character of oxygen in Cs$_{2}$ O.\",\n \"No accumulation of electron density is observed between Cs atoms, suggesting no chemical bond between them.\",\n \"The A-X octahedral geometry centring the chalcogen dianion is general observed for the whole family, as presented in Fig.\",\n \"REF b for Na$_{2}$ O ML and in Supplementary Information (SI) Figure S1-S4 for the other 14 DMMC MLs.\",\n \"Due to the intrinsic full oxidation state (+1) of Cs, the interlayer coupling between Cs sub-MLs is very weak, which only introduces marginal changes to the in-plane lattice constants.\",\n \"The optimized $a$ and $b$ of Cs$_{2}$ O ML (4.26 Å) are almost the same as the experimental bulk value.\",\n \"Comparing with the paradigmatic 2D systems of MoS$_{2}$ and graphite, we find that there are even less interlayer electron density localization in Cs$_{2}$ O as a result of dominant in-plane ionic bonding, which also means micromechanical exfoliation of Cs$_{2}$ O crystals can be readily achieved (Fig.\",\n \"REF a).\",\n \"Indeed, we calculate the exfoliation energy of Cs$_{2}$ O ML to be $\\\\sim 0.19$ J/m$^{2}$ (Fig.\",\n \"REF b), which is surprisingly lower than the value of 0.33 J/m$^{2}$ for graphene [28].\",\n \"It has to been emphasized here that the lower exfoliation energy of bulk Cs$_{2}$ O, compared with MoS$_{2}$ ($\\\\sim $ 0.42 J/m$^{2}$ ) and graphite, is consistently obtained by different vdW correction methods, which is summarized in SI Table S1.\",\n \"To assess the dynamical stability, which is also crucial for the micro-exfoliation of Cs$_{2}$ O ML, we have computed the phonon dispersion with the finite displacement method.\",\n \"As shown in Fig.\",\n \"REF c, the good stability of ML Cs$_{2}$ O is evident by the positive values of all phonon modes.\",\n \"We further check the room-temperature (RT) thermal stability of Cs$_{2}$ O ML by first-principle molecular dynamics simulations.\",\n \"The fluctuation of energy and temperature as a function of time are plotted in Fig.\",\n \"REF d. After running 4000 steps (10 ps), the 3$\\\\times $ 3$\\\\times $ 1 trigonal lattice is well sustained, and the free energy of the supercell converges.\",\n \"Excellent dynamical and thermal stability have also been validated for the other DMMC MLs (See SI Figure S5 and S6).\",\n \"Thus, it is feasible to experimentally exfoliate DMMC MLs for device fabrications and applications.\"\n ],\n [\n \"Electronic Properties\",\n \"We now elucidate the electronic structures and fundamental physical properties of DMMC MLs using two representative examples of Cs$_{2}$ O and Na$_{2}$ O MLs.\",\n \"As shown in Fig.\",\n \"REF a, Na$_{2}$ O ML is a 2D semiconductor with a direct bandgap of $\\\\sim $ 1.99 eV at the $\\\\Gamma $ point.\",\n \"Noticeably, the VBM of Na$_{2}$ O ML is mainly contributed by O-2$p$ orbitals (its projection weight is not shown in the energy dispersion for clarity) hybridized with very low weight of Na-2$p$ (weighted by green triangles), while the CBM is dominated by Na-3$s$ (red squares).\",\n \"The electronic band structure undergoes a pronounced change when Na is replaced by Cs, whose strongly delocalized 5$p$ orbitals induce significant density of states (DOS) away from the $\\\\Gamma $ point, which is still dominated by O-2$p$ .\",\n \"As the energy of Cs-5$p$ orbitals is $\\\\sim 0.4$ eV higher than the 2$p$ orbitals of O, the inter $p$ -orbital hybridization reverses the energy levels of VB1 and VB2 along the $\\\\Gamma -M$ direction.\",\n \"The resulting Cs$_{2}$ O ML is an indirect semiconductor with multiple VBMs centering the $M$ points (Fig.\",\n \"REF b).\",\n \"In general, for heavier cations of K, Rb and Cs, the inter $p$ -orbital hybridization with chalcogen anions is prevailing, shifting VBMs from $\\\\Gamma $ to the $M$ point.\",\n \"As a result, except for Na$_{2}$ X (X=O, S, Se and Te), all DMMC MLs are indirect semiconductors.\",\n \"The inter $p$ -orbital hybridization induced direct to indirect bandgap transition from Na$_{2}$ X to other A$_{2}$ X MLs has also been confirmed by the PBE method, as shown in SI Table S2 and SI Figure S7.\",\n \"As summarized in Fig.\",\n \"REF c, the energy gaps of DMMC MLs range from 1 eV to 3 eV, covering the near-infrared (K$_{2}$ O, Rb$_{2}$ O and Cs$_{2}$ O MLs) and visible light regions, which are highly desirable for optoelectronic device applications.\",\n \"To evaluate the optical performance, the absorption coefficients of Cs$_{2}$ O and Na$_{2}$ O MLs are directly compared with bulk silicon [29].\",\n \"As shown in Fig.\",\n \"REF d, Na$_{2}$ O ML shows absorption attenuation above $\\\\sim 625$ nm, in consistent with the direct bandgap of 1.99 eV.\",\n \"Strikingly, indirect-bandgap Cs$_{2}$ O ML exhibits unrivaled cyan-to-red light absorption efficiency, which is nearly constant over the whole visible light wavelengths.\",\n \"Such unusual findings not only predict Cs$_{2}$ O MLs to be extraordinary optoelectronic materials, but also are rooted in the enormous, non-differentiable DOS in the vicinity of VBM, i.e.\",\n \"van Hove singularity.\",\n \"As shown in Fig.\",\n \"REF a and REF b, the top VBs of DMMC MLs, dominated by the localized $p$ orbitals of chalcogen anions, are distinctive by very flat dispersion in the momentum space.\",\n \"The inter $p$ -orbital hybridization in Cs$_{2}$ O ML drastically reduces the VB energy dispersion, which effectively enhances DOS near the VBM and creates multiple saddle points in the vicinity of $\\\\Gamma $ , $K$ and $M$ .\",\n \"This leads to a prominent extended van Hove singularity in the density of states [30].\",\n \"It is also noteworthy that the inter $p$ -orbital hybridization also greatly enhances DOS near the CBM, which contributes significantly to the extraordinary light absorption characteristics of Cs$_{2}$ O ML.\",\n \"Charge carrier mobility is another critical parameter for the device performance of 2D material-based devices.\",\n \"As summarized in Table REF as well as in SI Table S3 and S4, the mobility of DMMC MLs are largely asymmetric between electrons and holes due to the drastic difference in the energy dispersion between the CBs and VBs, producing highly asymmetric effective mass.\",\n \"Noticeably, K$_{2}$ O ML has the highest electron mobility of 1.87$\\\\times 10^{4}$ cm$^{2}$ V$^{-1}$ S$^{-1}$ along the $y$ direction.\",\n \"Such electron mobility is comparable to the hole mobility of few-layer BP [31], [17] and order of magnitude larger than that of MoS${_2}$ atomic layers [13].\",\n \"Large electron mobility in DMMC MLs may hold a great promise for applications in high-performance electronics.\",\n \"We have noticed that with the same trigonal structure ($R\\\\bar{3}m$ , No.166), monolayer Tl$_{2}$ O is predicted to be a semiconductor with a direct bandgap and highly anisotropic charge carrier mobilities up to $4.3\\\\times 10^{3}\\\\,\\\\textrm {cm}^{2}\\\\textrm {V}^{-1}\\\\textrm {s}^{-1}$ [32].\",\n \"This is not surprising since Tl is one of the pseudo-alkali metals, in close resemblance to K both in the oxidation state and ionic radius.\"\n ],\n [\n \"Tunable Magnetism by Hole doping.\",\n \"The aforementioned van Hove singularities in the top VB of DMMC MLs are associated with inherent electronic instability, when hole doping pushes the Fermi level approaching the divergent point.\",\n \"Taking Cs$_{2}$ O as an example, we find that the system undergoes a spontaneous ferromagnetic phase transition with a moderate hole doping of $n\\\\sim 1.0\\\\times 10^{13}$ cm$^{-2}$ .\",\n \"In Fig.\",\n \"REF a, we plot the magnetic moment and the spin polarization energy per doped hole carrier as a function of $n$ .\",\n \"The latter represents the energy difference between the non-spin-polarized phase and the ferromagnetic phase.\",\n \"Clearly, the top VB becomes fully spin polarized by the critical hole doping, as manifested by the constant magnetic moment of 1 $\\\\mu _{B}$ per carrier above $1.0\\\\times 10^{13}$ cm$^{-2}$ .\",\n \"The spin polarization energy, which becomes -3.11 meV per carrier for a minimal hoel doping, monotonically decreases as a function of $n$ , approving that the doping-induced ferromagnetic ordering is energy favourable.\",\n \"Such spontaneous ferromagnetic ordering can be well understood by the Stoner model, which has been adopted to explain the so-called “$d^{0}$ ferromagnetism” in hole doped nitrides and oxides [33].\",\n \"In this picture, ferromagnetism can spontaneously appear when the Stoner criterion $U*g(E_{f})>$ 1 is satisfied, where $U$ is the exchange interaction strength and $g(E_{f})$ is the DOS at the Fermi energy of non-magnetic state.\",\n \"In such condition, the energy gain by exchange interactions exceeds the loss in kinetic energy, and hence the system would favor a ferromagnetic ground state.\",\n \"We have confirmed that the $U*g(E_{f})$ of Cs$_{2}$ O is always larger than one above the critical doping level, indicating that the huge DOS associated with the van Hove singularity plays a key role in the occurrence of ferromagnetism.\",\n \"In Fig.\",\n \"REF b and REF c, we plot the spin-polarized DOS of Cs$_{2}$ O with a doping level of $n=5.1\\\\times 10^{14}$ cm$^{-2}$ .\",\n \"Compared with the undoped case (Fig.\",\n \"REF b), the corresponding spin splitting process causes a significant energy shit of 0.1 eV between the spin-up and spin-down bands, leaving the Fermi level cutting through only one spin channel.\",\n \"Such half-metal state allows fully spin-polarized transport, which is crucial for spintronics applications.\",\n \"It is fascinating that this half-metallic ferromagnetism can be reversibly switched on and off in a $\\\\sim $ 1 nm thick atomic layer without extrinsic dopant and defects.\",\n \"We have found that except for Na$_{2}$ Te, the ferromagnetic transition can be triggered in all DMMC MLs by hole doping, ranging from $\\\\sim 1.0\\\\times 10^{13}$ cm$^{-2}$ for Cs$_{2}$ O to $8.0\\\\times 10^{15}$ cm$^{-2}$ for Na$_{2}$ Se (see SI Figure S8-S11).\",\n \"The special case of Na$_{2}$ Te can be understood by insufficient DOS near the VBM, which is dominated by the 5$p$ orbitals of Te.\",\n \"Due to much lower electron negativity, the 5$p$ orbitals of Te are more delocalized than the other chalcogen elements.\",\n \"Consequently, the high dispersion of Te-5$p$ reduce DOS near the VBM of Na$_{2}$ Te (SI Fig.\",\n \"S8), invalidating the Stoner criterion.\",\n \"We further evaluated the doping dependent Curie temperature ($T_{c}$ ) of DMMC MLs using the Heisenberg model.\",\n \"As shown in Fig.\",\n \"REF d, using $n=5.0\\\\times 10^{14}$ cm$^{-2}$ as a reference, which can be readily induced by ionic liquid gating or lithium glass gating [16], [34], RT ferromagnetism is achievable in K$_{2}$ O and Na$_{2}$ O MLs (SI Figure S12).\",\n \"For Cs$_{2}$ O, the corresponding $T_{c}$ is $\\\\sim 200$ K, which is nearly five times higher than the recently reported ferromagnetic CrI$_{3}$ and Cr$_{2}$ Ge$_{2}$ Te$_{6}$ [20], [21].\",\n \"In summary, we have reported a new family of 1T-TMDC structured dialkali-metal monochalcogenides (DMMCs), which show fascinating physical properties and great device application promises in the 2D limit.\",\n \"Distinctively, DMMC MLs are characterized by large electron mobility, e.g.\",\n \"$1.87\\\\times 10^{4}$ cm$^{2}$ V$^{-1}$ S$^{-1}$ for K$_{2}$ O and $5.42\\\\times 10^{3}$ cm$^{2}$ V$^{-1}$ S$^{-1}$ for Na$_{2}$ O, and by very flat VBs with rather weak energy dispersion.\",\n \"The latter are responsible for the formation of extended van Hove singularities with extremely high DOS just below the VB edge of DMMC MLs.\",\n \"By introducing moderate hole doping, these DMMC MLs are subjected to strong electronic instabilities induced by electron-electron interaction, which ultimately leads to spontaneous ferromagnetic phase transitions where the top VBs becomes fully spin-polarized.\",\n \"Using heavy doping methods of ion liquid and solid gating, we can readily increase the $T_{c}$ and switch on and off the magnetism of DMMC MLs even at room temperature.\",\n \"Although alkali compounds are not very stable in air by reacting with ambient moisture, the excellent dynamical and thermal stability allows DMMC MLs to be exfoliated and encapsulated in glove box environment, which is now routine research facilities for studying air-sensitive 2D materials [16], [17], [19], [20], [21].\",\n \"It is also feasible to grow DMMC MLs by molecular beam epitaxy or by chemical vapor deposition, with abundant options of single crystal substrates of 1T- or 2H-TMDCs.\",\n \"By forming heterostructure with high electron affinity main-group MDCs, such as SnSe$_{2}$ , DMMC MLs may become effectively hole doped by interfacial charge transferring [24], which would provide the opportunity for in-situ studying the gate-tunable ferromagnetism in DMMC MLs by scanning probe microscope.\",\n \"Figure: (a) Structure of DMMC MLs.\",\n \"The upper inset is the first Brillouin zone.\",\n \"Each chalcogen anion is surrounded by six alkali cations, forming XA 6 _{6} octahedron.\",\n \"(b-c) Normalized electron localization function (ELF) of Cs 2 _{2}O and Na 2 _{2}O, respectively.\",\n \"ELF = 1 (red) and 0 (blue) indicate accumulated and vanishing electron densities, respectively.\",\n \"The 2D ELF plane is defined by the green line in Fig.\",\n \"1a.Figure: Stability and Micromechanical Exfoliation.\",\n \"(a) Interlayer differential charge density of Cs 2 _{2}O bulk, graphite and MoS 2 _{2} bulk.\",\n \"(b) Calculated exfoliation energy of Cs 2 _{2}O ML, in comparison with graphene (blue triangles).\",\n \"(c) The phonon spectrum of Cs 2 _{2}O ML.\",\n \"(d) First-principle molecular dynamics simulations of Cs 2 _{2}O ML.\",\n \"The thermal stability of the supercell is evident by the convergence in the free energy.Figure: Electronic Properties.\",\n \"Band structures and the corresponding DOS of Na 2 _{2}O ML (a) and Cs 2 _{2}O ML (b).\",\n \"The contributions of alkali ss-orbital and pp-orbital are indicated by red and green circles, respectively.\",\n \"The insets are the spatial distribution of the wave-functions for the VBM and CBM.\",\n \"(c) Energy gaps of all DMMC MLs.\",\n \"Strong inter pp-orbital hybridization leads to the transition from direct bandgaps in Na 2 _{2}X to indirect gaps in other DMMC MLs.\",\n \"The gap values of DMMC MLs are mainly determined by the electron affinity of the anions.\",\n \"The results have been confirmed by both the HSE06 and PBE methods.\",\n \"(d) Extraordinary light absorbance of Cs 2 _{2}O and Na 2 _{2}O MLs, in comparison with silicon.Figure: Tunable Magnetism by Hole Doping.\",\n \"(a) Magnetic moment and spin polarization energy per carrier of Cs 2 _{2}O ML.\",\n \"Remarkably, spontaneous ferromagnetic phase transition is trigger by a moderate hole doping of 1.28×10 13 1.28\\\\times 10^{13} cm -2 ^{-2}.\",\n \"(b) Spin-resolved DOS of Cs 2 _{2}O ML at n=5×10 14 n=5\\\\times 10^{14} cm -2 ^{-2}.\",\n \"The inset shows the spin density distribution, apparently locating at O atoms.\",\n \"(c) Spin-splitted top valence bands in the vicinity of MM points in Cs 2 _{2}O ML (n=5×10 14 n=5\\\\times 10^{14} cm -2 ^{-2}).\",\n \"(d) Curie temperature verusverus hole doping density, showing the feasibility of RT ferromagnetism by heavy hole doping.\",\n \"Note that antiferromagnetic ordering in DMMC MLs is energetically higher in free energy than the ferromagnetic state.Table: Mobility and effective mass along the xx and yy transport directions in four Cs 2 _{2}X MLs and K 2 _{2}O.\",\n \"We calculated the carrier mobility using a phonon-limited scattering model, which includes deformation potential (E 1 _{1}) and elastic modulus (C 2D _{2D}) in the propagation direction of the longitudinal acoustic wave.\",\n \"Noticeably, K 2 _{2}O has the highest electron mobility.This work is supported by the National Key R&D Program of the MOST of China (Grant Nos.\",\n \"2016YFA0300204 and 2017YFA0303002), and the National Science Foundation of China (Grant Nos.\",\n \"11574264 and 61574123).\",\n \"Y.Z.\",\n \"acknowledges the funding support from the Fundamental Research Funds for the Central Universities and the Thousand Talents Plan.\"\n ],\n [\n \"Competing financial interests:\",\n \"The authors declare no competing financial interests.\"\n ]\n]"},"id":{"kind":"string","value":"1808.08461"}}},{"rowIdx":289,"cells":{"text":{"kind":"list like","value":[["Context-Free Session Types for Applied Pi-Calculus"],["Abstract We present a binary session type system using context-free session types to a version of the applied pi-calculus of Abadi et.","al.","where only base terms, constants and channels can be sent.","Session types resemble process terms from BPA and we use a version of bisimulation equivalence to characterize type equivalence.","We present a quotiented type system defined on type equivalence classes for which type equivalence is built into the type system.","Both type systems satisfy general soundness properties; this is established by an appeal to a generic session type system for psi-calculi."],["Introduction","Binary session types [6] describe the protocol followed by the two ends of a communication medium, in which messages are passed.","A sound type system of this kind guarantees that a well-typed process does not exhibit communication errors at runtime.","Session types have traditionally been used to describe linear interaction between partners [10], but later type systems are able to distinguish between linear and unlimited channel usages.","In particular Vasconcelos has proposed session types with $\\textsf {lin}/\\textsf {un}$ qualifiers that describe linear interaction as well as shared resources [10].","Context-free session types introduced by [9] are more descriptive than the regular types described in previous type systems [6], [10] in that they allow full sequential composition of types.","Because of this, session types can now describe protocols that cannot be captured in the regular session types, such as transmitting complex composite data structures.","Many binary session type systems are concerned with languages that use the selection and branching constructs introduced in [6] in addition to the normal input and output constructs.","These constructs are synchronisation operations, where two processes synchronise on a channel, and are similar to method invocations found in object-oriented programming.","A branching process $l \\rhd \\lbrace l_1 :P_1, \\ldots , l_n : P_n \\rbrace $ continues as process $P_k$ together with $P$ if label $l_k$ is selected on channel $c$ using the selection $c \\lhd l_k.P$ .","In this article we consider a session type system for a version of the applied pi-calculus, due to Abadi et.","al.","[1].","This is an extension of the pi-calculus [8] with terms and extended processes and in our case extended further with selection and branching.","Our version is a “low-level” version in that allows one to build composite terms but only allows for the communication of names and nullary function symbols.","In this way, the resulting version is close to the versions of the pi-calculus used for encoding composite terms [8].","Our session type system combine ideas from the type systems from [9] and [10] into a type system in which session types are context-free and use $\\textsf {lin}/\\textsf {un}$ qualifiers in order to distinguish between linear and unbounded resources.","The resulting type system uses types that are essentially process terms from a variant of the BPA process calculus [4].","In Section we present our version of the applied pi-calculus and in Section we define the syntax and semantics of our session type system.","We then prove the soundess of our type system by using the general results about psi-calculi from [7].","This is done by showing that the applied pi-calculus is a psi-calculus, and that our type system is a instance of the generic type system presented by Hüttel in [7].","Lastly we present type equivalence between types by introducing a notion of type bisimulation for endpoint types.","By considering equivalence classes under type bisimilarity we get a new quotiented type system whose types are equivalence classess.","It then follows from the theorems in [7] that the general results for our first type system also hold for the quotiented type system."],["Applied pi-Calculus","We consider a “low-level” version of the applied pi-calculus [1] in which composite terms are allowed but only the transmission of simple data is possible: Only names $n$ , constants represented by functions $f_0$ with arity 0 and functions that evaluate to values of base types can be transmitted.","We use the notation $\\widetilde{M}$ to represent the sequence $M_1,\\ldots ,M_i$ and $\\widetilde{x}$ to represent the sequence of variables $x_1,\\ldots ,x_i$ .","We always assume that our processes are specified relative to a family of parameterized agent definitions that are on the form $N(\\widetilde{x})\\overset{\\underset{\\mathrm {def}}{}}{=}A$ and that every agent variable $N$ occurring in a process has a corresponding definition.","The formation rules for processes $P$ and extended processes $A$ can be seen below in ().","Note that we distinguish between variables ranged over by $x, y \\ldots $ and names ranged over by $m,n, \\ldots $ .","We let $a$ range over the union of these sets.","We extend the syntax of processes with branching $c \\rhd \\lbrace l_1:P_1,\\ldots ,l_k:P_k \\rbrace $ and selection $c \\lhd l.P$ where $l_1, \\ldots l_k$ are taken from a set of labels.","Extended processes extend processes with the ability to use active substitutions of the form $\\lbrace {M}{x}\\rbrace $ that instantiate variables.","P ::= 0P1 P2 !P ( n)P ( n) P if M1=M2 then P1 else P2 n(x).P nu .P c l.P c { l1:P1,...,lk:Pk } N(M) u ::= n x f0 M ::= n x f(M) A ::= P A1 A2 (n)A ( x)A {Mx} The notion of structural congruence extends that of the usual pi-calculus [8].","The following two further axioms that are particular to the applied pi-calculus are of particular importance, as they show the role played by active substitutions.","$ P \\mid \\lbrace {M}{x}\\rbrace \\equiv P\\lbrace {M}{x}\\rbrace \\mid \\lbrace {M}{x}\\rbrace & & & \\mathbf {0}\\equiv (\\nu \\, x) \\lbrace {M}{x}\\rbrace $ Together with the axioms of [8] they allow us to factor out composite terms such that they only occur in active substitutions.","For instance, we have that $\\texttt {if }M_1 = M_2\\texttt { then }P\\texttt { else }Q\\equiv \\nu {x}\\nu {y}(\\texttt {if }x=y\\texttt { then }P\\texttt { else }Q \\mid \\lbrace {M_1}{x}\\rbrace \\mid \\lbrace {M_2}{y}\\rbrace )$ .","We can therefore use a term by only mentioning the variable associated with it.","In our version of the applied pi-calculus we very directly make use of this.","Our semantics consists of reduction semantics and a labelled operational semantics that extend those of [1] with rules for branching and selection.","Reductions are of the form $P \\rightarrow P^{\\prime }$ , while transitions are of the form $P \\xrightarrow{} P^{\\prime }$ where the label $\\alpha $ is given by $ \\alpha ::= c \\lhd l \\mid c \\rhd l \\mid a(x) \\mid \\overline{a}x \\mid \\tau $ and $c \\lhd l$ is the co-label of $c \\rhd l$ .","The rules defining reductions and labelled transitions are found in Tables REF and REF , respectively.","Table: Reduction rulesTable: Labelled transition rules"],["A context-free session type system","We now present the syntax and semantics of our type system."],["Session types","Session types describe the communication protocol followed by a channel.","Consider writing a process that transmits a binary tree where each internal node containsan integer.","We want to transmit this tree by sending only base types (i.e.","the integers) on a channel.","The tree data type can be described with the grammar below.","$\\textsf {Tree} ::= (\\textsf {Int}, \\textsf {Tree}, \\textsf {Tree})\\ \\text{ ǀ\\ }\\ \\textsf {Leaf}$ Using the regular types introduced in [10], the type for a channel transmitting such a data structure could be the recursive type $\\mu z.\\oplus \\lbrace \\textsf {Leaf}: \\textsf {lin }\\textsf {skip}\\quad \\textsf {Node}:\\textsf {lin }!\\textsf {Int}.z\\rbrace $ .","In this type $z$ is a type variable that is recursively defined.","The type describes how a single node is transmitted, if it is a leaf node we do not do anything, if it is an internal node then the integer value is transmitted with the output type $!\\textsf {Int}$ and then the sub-trees of the node are transmitted with a recursive call.","However, if we use this regular session type, we are not able to guarantee that the tree structure is preserved.","The reason is that the session type describes that a list of nodes are being sent, but not the position in the tree of each node.","On the other hand, if we use the context-free session type disciple introduced by Thiemann and Vasconcelos [9], we can specify the preservation of tree structures by using types such as $\\mu z.\\oplus \\lbrace \\textsf {Leaf}: \\textsf {skip}\\quad \\textsf {Node}:!\\textsf {Int};z;z\\rbrace $ .","Using sequential composition with the $\\_;\\_$ operator, we can specify a protocol that will guarantee that the tree structure by first sending the left sub-tree and then the right sub-tree.","Introducing a sequential operator can introduce challenges for typing a calculus, as the following example shows.","If we were to reuse a channel by sending an integer after transmitting a tree, the type would be $\\mu z.\\oplus \\lbrace \\textsf {Leaf}: \\textsf {skip}\\quad \\textsf {Node}:!\\textsf {Int};z;z\\rbrace ;!\\textsf {Int}$ .","${\\Gamma (c)=\\oplus \\lbrace l_i: T_{E_i}\\rbrace _{i\\in I}}{\\Gamma \\vdash \\text{select } l_i \\text{ on } c}$ When typing rules are created for a calculus, it is often defined on the structure of types and terms.","An example of such a typing rule for a select statement is shown in (REF ), which says that a select statement is well typed if the select operation is performed on a channel with a select type.","If however the channel has a type as shown before, the select rule cannot be used, as the channel has the type of a sequential composition, and inside the sequential composition we have a recursive type.","We require an equirecursive treatment of types, which allows us to expand the type to $\\oplus \\lbrace \\textsf {Leaf}: \\textsf {skip}\\quad \\textsf {Node}: !\\textsf {Int};\\mu z.\\oplus \\lbrace \\textsf {Leaf}:\\textsf {skip}\\quad \\textsf {Node}: !\\textsf {Int};z;z\\rbrace ;\\mu z.\\oplus \\lbrace \\textsf {Leaf}: \\textsf {skip}\\quad \\textsf {Node}: !\\textsf {Int};z;z\\rbrace \\rbrace ;!\\textsf {Int}$ .","This is achieved by unfolding a recursive type $\\mu z.T$ to $T$ where all occurrences of $z$ in $T$ are replaced with the original $\\mu z.T$ .","So now we are left with a sequential composition with a select type and output type.","In order to transform this into a select type, we need a distributive law that allows us to move the sequential composition inside the select type, in order to obtain the type $\\oplus \\lbrace \\textsf {Leaf}:\\textsf {skip};!\\textsf {Int}\\quad \\textsf {Node}: !\\textsf {Int};\\mu z.\\oplus \\lbrace \\textsf {Leaf}: \\textsf {skip}\\quad \\textsf {Node}: !\\textsf {Int};z;z\\rbrace ;\\mu z.\\oplus \\lbrace \\textsf {Leaf}:\\textsf {skip}\\quad \\textsf {Node}: !\\textsf {Int};z;z\\rbrace ;!\\textsf {Int}\\rbrace $ .","Such a rule does exist, we will introduce a method to prove that these types exhibit the same behaviour in Section when we introduce type equivalence."],["The language of types","We denote the set of all types by $\\mathcal {T}$ ; the types $T\\in \\mathcal {T}$ are described by the formation rules in (REF ).","These session types are a modified version of the context-free session types presented by [9].","The modifications made to the types are that we allow input and output session types to transmit other session types to allow sending channels in other channels, and finally that we introduce the lin and un qualifiers.","We let $B$ range over a set of base types.","p ::= skip ?T !T &{li:TEi} {li:TEi} q ::= lin un TE ::= q p z z.TE TE1; TE2 T ::= S B TE S ::= (TE1, TE2) From the formation rules we can see that a session type $S$ is a pair of endpoint types $T_{E_1}$ and $T_{E_2}$ .","Endpoint types describe one end of channel, and are the types that evolve when a channel is used.","The qualifiers lin and un from [10] are used to describe a linear interaction between two partners and a unrestricted shared resources respectively.","An example of an un type could be a server, that a lot of processes have access to.","To ensure that no communication errors occur if multiple processes can read from a channel concurrently, we require that the type must never change behaviour.","This means that an un type must be the same before and after a transition."],["Transitions for types","We now present the transition rules for endpoint types, which describe how the types can evolve when an action is performed.","We use an annotated reduction semantics to describe the behaviour of our types.","Our labels are generated by the grammar in (REF ) where we have select, branch, input and output actions.","The transitions are of the form $T_E\\xrightarrow{}T_E^{\\prime }$ and are shown in Table REF .","We let $T_E\\lbrace {y}{x}\\rbrace $ be the endpoint type $T_E$ where all free occurrences of $x$ has been replaced with $y$ .","$\\lambda ::= !T_1 \\mid ?T_1 \\mid \\rhd \\, l \\mid \\lhd \\, l$ In the transition rules, we use the function $Q$ defined in (REF ) to find the qualifier of compound types such as recursive types or sequential composition types.","$\\begin{split}{Q}(q\\ p) &\\overset{\\underset{\\mathrm {def}}{}}{=}q \\\\{Q}(T_{E_1};T_{E_2}) &\\overset{\\underset{\\mathrm {def}}{}}{=}{Q}(T_{E_1}) \\\\{Q}(\\mu z.T_E) &\\overset{\\underset{\\mathrm {def}}{}}{=}{Q}(T_E)\\end{split}$ A relation $\\sqsubseteq =\\lbrace (\\textsf {lin},\\textsf {un}),(\\textsf {un},\\textsf {un}),(\\textsf {lin},\\textsf {lin})\\rbrace $ is defined for qualifiers in [10].","We also follow the definition of $q(T)$ and $q(\\Gamma )$ from that of [10].","In short $\\textsf {lin}(\\Gamma )$ is always satisfied, and $\\textsf {un}(\\Gamma )$ is satisfied iff all elements in $\\Gamma $ are unrestricted.","The type system contains the sequential operator $\\_;\\_$ as well as choice operators; select $\\&\\lbrace \\ldots \\rbrace $ and branch $\\oplus \\lbrace \\ldots \\rbrace $ .","This is very similar to Basic Process Algebra (BPA)[4] that contains the sequential operator $\\cdot $ and nondeterministic choice operator $+$ .","We also have recursive types, which corresponds to variables with recursive definitions in BPA.","A BPA expression is guarded if all recursive variables on the right hand side is preceded by an action $\\lambda $ [4].","Similarly we say that a type is guarded if every recursion variable is preceded by an input or output.","These similarities with BPA will become very important, as we can describe types as BPA expressions and by showing results about these expression, we can in turn show results about our types.","Table: Annotated reduction semantics for types"],["Typing rules","We now present a type system for our version of the applied pi-calculus in Table REF .","In this type system, the type judgements for processes are on the form $\\Gamma \\vdash P$ meaning that the process $P$ is well typed in context $\\Gamma $ .","The judgments for terms are on the form $\\Gamma \\vdash M:T$ meaning that the term $M$ is well typed with type $T$ in context $\\Gamma $ .","Lastly the judgments for extended processes are on the form $\\Gamma \\vdash _A A$ meaning that the process $A$ is well typed in context $\\Gamma $ .","We type processes in a type context $\\Gamma $ which contains types for the variables, names and functions symbols of a process.","We follow the definition of a type context from [10]: $\\emptyset $ is the empty context, $\\Gamma , x: T$ is the context equal to $\\Gamma $ except that $x$ has the type $T$ in the new context.","This operation is only defined when $x\\notin \\text{dom}(\\Gamma )$ .","Table REF shows the typing rules for processes.","We use the context split $\\circ $ and context update $+$ operations from [10].","The context split operation is used to split a context into two constituents.","A maximum of two processes must have access to a given linear session type; a context either contains the entire session type $S=(T_{E_1},T_{E_2})$ , or a single endpoint type $T_{E}$ .","When splitting a context into two, we can pass $S$ to either context, or one endpoint to each context.","If the context only has an endpoint type, the endpoint type can only be passed on to one of the two contexts.","This way we ensure that each lin endpoint of a channel is known in exactly one context.","Names of unrestricted type can be shared among all contexts.","The context update operation updates the type of a channel.","The $\\Gamma , x:T$ operation is only defined when $x\\notin \\text{dom}(\\Gamma )$ .","The $+$ operation $\\Gamma =\\Gamma _1+\\Gamma _2$ uses the type of $x$ in $\\Gamma _2$ to update the type in $\\Gamma _1$ .","So if $\\Gamma _1(x)=T_1$ and $\\Gamma _2(x)=T_2$ then $\\Gamma (x)=T_2$ .","The two operations are used in (Input) and (Output) where we must split our context into two, to type each endpoint of a linear channel in its own context.","Table: Typing rules for Extended ProcessesTable: Context split for Applied π\\pi -calculus, based on Table: Context update for Applied π\\pi -calculus from Table: Typing rules for termsTable: Typing rules for processes"],["Duality of types","The notion of type duality is central to session type systems; it expresses that the protocols followed by the two endpoints of a name must be opposites: If one end transmits a value, the other end must receive it.","We denote the dual of an endpoint type $\\overline{T_E}$ and define duality below, following [9].","$\\overline{q\\ \\textsf {skip}}&\\overset{\\underset{\\mathrm {def}}{}}{=}q\\ \\textsf {skip} && \\overline{q\\ \\&\\lbrace l_1: T_{E_1}, \\ldots , l_k:T_{E_k}\\rbrace }\\overset{\\underset{\\mathrm {def}}{}}{=}q\\ \\oplus \\lbrace l_1: \\overline{T_{E_1}}, \\ldots , l_k:\\overline{T_{E_k}}\\rbrace \\\\\\overline{q\\ ?T}&\\overset{\\underset{\\mathrm {def}}{}}{=}q\\ !T && \\overline{q\\ \\oplus \\lbrace l_1: T_{E_1}, \\ldots , l_k:T_{E_k}\\rbrace }\\overset{\\underset{\\mathrm {def}}{}}{=}q\\ \\&\\lbrace l_1: \\overline{T_{E_1}}, \\ldots , l_k:\\overline{T_{E_k}}\\rbrace \\\\\\overline{q\\ !T}&\\overset{\\underset{\\mathrm {def}}{}}{=}q\\ ?T && \\overline{\\mu z.T_E} \\overset{\\underset{\\mathrm {def}}{}}{=}\\mu z.\\overline{T_E} \\\\\\overline{T_{E_1};T_{E_2}}&\\overset{\\underset{\\mathrm {def}}{}}{=}\\overline{T_{E_1}};\\overline{T_{E_2}} && \\overline{z} \\overset{\\underset{\\mathrm {def}}{}}{=}z$ A session type $S = (T_{E_1},T_{E_2})$ is balanced iff its endpoint types are dual, that is, if $\\overline{T_{E_1}} = T_{E_2}$ .","A type context $\\Gamma $ is balanced iff every session type in the range of $\\Gamma $ is balanced.","We use the notation $\\Gamma \\vdash _\\text{bal}P$ to describe that a process $P$ is well typed in a balanced context $\\Gamma $ ."],["Applied pi-calculus as a psi-calculus instance","Psi-calculus is a general framework for process calculi.","In this section we show that the applied pi-calculus is an instance of it, and this will then be used in Section to obtain results about our type system.","An instance of the psi-calculus framework contains the seven elements [3] given in Table REF .","Table: Elements of psi-calculiThe syntax of an instance of the psi-calculus is described by the formation rules in (REF ) that generalize those of the pi-calculus.","The input and output constructions allows to use arbitrary terms as channels, and the input construction allows for matching on a pattern $(\\lambda \\tilde{x})N$ ; the pattern variables in $\\tilde{x}$ are bound to the subterms that match.","The other syntactic constructs are similar to those of the pi-calculus.","The case construct is a generic case of the if-construct; we generalize match conditions $M_1=M_2$ to allow any $\\phi \\in \\textbf {C}$ as a condition.","Lastly in psi-calculi we have a concept of assertions $\\in \\textbf {A}$ .","These generalize the notion of active substition found in the applied pi calculus.","$P :=\\ &\\mathbf {0}\\mid \\overline{M}N.P \\mid \\underline{M}(\\lambda \\widetilde{x})N.P \\mid \\textbf {case}\\ \\phi _1:P_1[]\\ldots [] \\phi _n: P_n \\\\\\text{ ǀ\\ }\\ &(\\nu a)P \\mid P\\text{ǀ}Q \\mid !P \\mid \\llparenthesis \\rrparenthesis \\mid M l.P \\mid M \\triangleright \\lbrace l_1:P_1,\\ldots ,l_k:P_k \\rbrace $ The structural operational semantics of psi-calculi has transitions of the form $\\Psi \\blacktriangleright P\\xrightarrow{}P^{\\prime }$ where the labels $\\alpha $ are given by the formation rules below [7].","::= M l | M l | M(va)N | KN | (va)@ (vb)(MNK) | (va)@ (MlN) The first four actions correspond to the actions we know from the applied pi-calculus: selection, branch, output and input.","The last two action are internal $\\tau $ -actions that correspond to internal actions that are either an input/output-exchange or a branch/select-exchange.","It follows from[3] that the standard pi-calculus is an instance of the psi-calculus.","Below we do the same for the instance APi that shows that the applied pi-calculus is also an instance of the psi-calculus.","In the definition $\\mathcal {N}$ is the set of all names and $\\mathcal {F}$ is the set of all function names.","$\\textbf {T} &\\overset{\\underset{\\mathrm {def}}{}}{=}\\mathcal {N} \\cup \\lbrace f(M_1, \\ldots M_k) \\text{ ǀ\\ } f \\in \\mathcal {F}, M_i \\in \\textbf {T}\\rbrace & &\\textbf {C} && \\overset{\\underset{\\mathrm {def}}{}}{=}\\lbrace M=N \\text{ ǀ\\ } M,N\\in \\textbf {T}\\rbrace \\\\\\textbf {A} &\\overset{\\underset{\\mathrm {def}}{}}{=}\\lbrace 1\\rbrace & &{\\hspace{-1.66656pt}\\cdot }&&\\overset{\\underset{\\mathrm {def}}{}}{=}\\lbrace ((n, n), n=n) \\text{ ǀ\\ } n \\in \\mathcal {N} \\rbrace \\\\\\otimes &\\overset{\\underset{\\mathrm {def}}{}}{=}\\lbrace ((_1, _2), 1) \\text{ ǀ\\ } \\in \\textbf {A} \\rbrace & &\\textbf {1} & &\\overset{\\underset{\\mathrm {def}}{}}{=}1 \\\\\\vdash &\\overset{\\underset{\\mathrm {def}}{}}{=}\\lbrace (1, M=M) \\text{ ǀ\\ } M\\in \\textbf {T}\\rbrace $"],["Properties of our type system","A generic binary session type system for psi-calculi was presented in [7].","The intention is that any existing binary session type systems for process calculi can be captured as special instances of the generic system, as long it satisfies four specific requirements.","We already know that our applied pi-calculus is a simple psi-calculus and now establish that our type system fulfils the requirements of [7].","This will then allow us to obtain the usual results for binary type systems as a simple corollary of the theorems for the generic system."],["Transition structure","Type transitions in our type system are an instance of the generic type transitions in [7].","Both the generic type transitions and our type transition consist of send, receive, branch and select.","The syntax of type transitions is uniform across the two articles as they are both generated by the grammar in (REF ).","This illustrates that there is a one-to-one correspondence between the type transitions in the two type systems.","$\\lambda ::= l \\text{ ǀ\\ } \\triangleright l \\text{ ǀ\\ } !T \\text{ ǀ\\ } ?T$"],["Revisiting duality","Duality of types in our pi-calculus is defined on the structure of types, as seen in Section REF .","In [7] duality is defined on type transitions, where (REF ) holds for dual types (we use $\\underline{\\,\\cdot \\,}$ to denote duality defined on type transitions).","$T_E \\xrightarrow{} T_E^{\\prime } \\Leftrightarrow \\underline{T_E} \\xrightarrow{} \\underline{T_E^{\\prime }}$ In (REF ) and () we see the duality of type transitions, as presented by [7].","$\\overline{!T_1} = ?T_2 && \\overline{?T_1} = !T_2 \\\\\\overline{l} = \\triangleright l && \\overline{\\triangleright l} = l$ We now show that the duality defined in Section REF upholds the property in (REF ).","Lemma 1 $\\underline{T_E} =\\overline{T_E}$ By induction in the structure of types."],["Checking requirements","In the type system presented in [7], type judgements are relative to a type context and an assignment and therefore of the form $\\Gamma ,\\vdash {J}$ , where the judgment body is either ${J}$ is either a term typing $M : T$ or $P$ , the statement that process $P$ is well-typed.","We write $\\Gamma ,\\vdash _\\text{min} {J}$ , if $\\Gamma ^{\\prime },^{\\prime }\\lnot \\vdash _\\text{min} {J}$ for every smaller $\\Gamma ^{\\prime }$ and $^{\\prime }$ .","For each requirement presented in [7] we show that it is satisfied in our type system."],["Requirement 1:","If $\\Gamma _1, _1 \\vdash _\\text{min}{J} $ and $\\Gamma _2, _2 \\vdash _\\text{min} {J}$ then $\\Gamma _1 = \\Gamma _2 $ and $_1 \\simeq _2$ .","As we only have the assertion 1 in type judgements, $_1\\simeq _2$ is trivially fulfilled as both are 1.","Let $\\Gamma _1,\\Gamma _2$ be type contexts such that $\\Gamma _1, \\textbf {1} \\vdash _{\\text{min}}{J}$ and $\\Gamma _2, \\textbf {1}\\vdash _{\\text{min}}{J}$ .","Assume that $\\Gamma _1 \\ne \\Gamma _2$ then without loss of generality there exists an $x$ such that $x\\in \\text{dom}(\\Gamma _1)$ and $x\\notin \\text{dom}(\\Gamma _2)$ .","Because judgments must be well-formed we know that $fn({J})\\subseteq \\text{dom}(\\Gamma _1)$ and $fn({J})\\subseteq \\text{dom}(\\Gamma _2)$ , hence $x\\notin fn({J})$ .","Let $\\Gamma _1^{\\prime }$ be defined as $\\Gamma _1$ except $x\\notin \\text{dom}(\\Gamma _1^{\\prime })$ , then $\\Gamma _1^{\\prime },\\textbf {1}\\vdash {J}$ and $\\Gamma _1^{\\prime }<\\Gamma _1$ thus $\\Gamma _1,\\textbf {1}\\nvdash _{\\text{min}}{J}$ .","This is a contradiction, hence our assumption is wrong and $\\Gamma _1=\\Gamma _2$ , which means that the requirement is fulfilled."],["Requirement 2:","If $\\Gamma , \\vdash M:T@c$ then $\\Gamma (c)=T_E$ for some endpoint type $T_E$ .","In our calculus, the only terms that can be used as channels are names.","The $(\\nu n)P$ construct uses a channel constructor $n$ to declare a channel with a session type $\\Gamma , \\vdash n: S@n$ .","$S$ is a session type $S=(T_{E_1}, T_{E_2})$ for some endpoint types.","When type checking $n$ , only one endpoint is present in the context $\\Gamma $ , in which case $\\Gamma (n)=T_{E_1}$ or $\\Gamma (n)=T_{E_2}$ , meaning that the requirement is satisfied."],["Requirement 3:","Suppose $\\widetilde{N}\\in \\textsc {match}(M, \\widetilde{x}, X)$ , $\\Gamma , \\vdash M$ and $\\Gamma _1+\\widetilde{x}:\\widetilde{T},_1\\vdash _{\\text{min}}X:\\widetilde{T}\\rightarrow U$ .","Then there exist $\\Gamma _{2i},_{2i}$ such that $\\Gamma _{2i}, _{2i}\\vdash _{\\text{min}}N_i:T_i$ for all $1\\le i \\le \\vert \\widetilde{x}\\vert =n$ .","In our calculus, the only possible match is $M \\in MATCH(M, x, x)$ .","As $\\vert \\widetilde{x}\\vert =1$ the requirement becomes $\\Gamma _{2},_{2}\\vdash _{\\text{min}}M : T$ .","From the requirement that $M$ is well typed, this is trivially fulfilled."],["Requirement 4:","If ${\\hspace{2.77771pt}\\text{ǀ}\\hspace{-5.0pt}=}M {\\hspace{-1.66656pt}\\cdot }K$ and $\\Gamma , \\vdash M:S$ then $\\Gamma , \\vdash K:S$ .","If ${\\hspace{2.77771pt}\\text{ǀ}\\hspace{-5.0pt}=}M {\\hspace{-1.66656pt}\\cdot }K$ and $\\Gamma , \\vdash M:T$ then $\\Gamma , \\vdash K:\\overline{T}$ .","In Section () we defined ${\\hspace{-1.66656pt}\\cdot }$ as $=$ , meaning that two channels are equal if it is the same name.","The first part of the requirement becomes: If ${\\hspace{2.77771pt}\\text{ǀ}\\hspace{-5.0pt}=}n = n$ and $\\Gamma ,\\vdash n:S$ then $\\Gamma , \\vdash n:S$ which is trivially fulfilled.","The second part happens when only one end of a channel is in the context, with an endpoint type, then the other end of the channel must have a dual endpoint type.","If ${\\hspace{2.77771pt}\\text{ǀ}\\hspace{-5.0pt}=}n = c$ and $\\Gamma , \\vdash n:T_E$ then $\\Gamma , \\vdash c:\\overline{T_E}$ .","A channel has a balanced session type $S=(T_E, \\overline{T_E})$ for some endpoint type.","If $n$ and $c$ have endpoint types and are the same channel, then they must also have types dual of each other.","If the types are not unrestricted then they can only evolve into other types dual of each other, due to the requirement that only two processes have access to the channels and that if $T_E\\xrightarrow{}T_E^{\\prime }\\Longleftrightarrow \\underline{T_E}\\xrightarrow{}\\underline{T_E^{\\prime }}$ we know that the types will stay dual of each other.","If the types are unrestricted then the requirement that whenever $T_E\\xrightarrow{}T_E^{\\prime }$ for some $\\lambda $ then $T_E=T_E^{\\prime }$ , ensures that the types will stay dual of each other.","In this section we discuss the results that we obtain by showing that our type system is an instance of the generic type system.","Hüttel presents and proves two main theorems for the generic type system, that we will use to show results about our type system as well.","The first theorem, which is about well typed $\\tau $ -actions from [7] follows below: Theorem 1 (Well-typed $\\tau $ -actions, Theorem 9 of [7]) Suppose we have $_0 \\blacktriangleright P\\xrightarrow{}P^{\\prime }$ , where $\\alpha $ is a $\\tau $ -action and that $\\Gamma ,\\vdash _{\\normalfont \\textsf {bal}}P$ and $\\le _0$ then for some $^{\\prime }\\le $ and $\\Gamma ^{\\prime }\\le \\Gamma $ we have $\\Gamma ^{\\prime },^{\\prime }\\vdash _{{\\normalfont \\text{min}}}\\alpha :(T@c,U)$ .","This theorem says that if a process $P$ can make a $\\tau $ -action and become $P^{\\prime }$ , and that $P$ is well typed in an environment where channels have pairs of dual endpoint types, then the action is also well-typed.","So internal synchronisation in a well-typed process is well typed as well.","The second theorem is about fidelity and follows below: Theorem 2 (Fidelity, Theorem 10 of [7]) Suppose we have $_0\\blacktriangleright P \\xrightarrow{}P^{\\prime }$ , where $\\alpha $ is a $\\tau $ -action and that $\\Gamma ,\\vdash _{\\normalfont \\textsf {bal}}P$ .","Then for some $\\Gamma ^{\\prime }\\le \\Gamma $ and for some $^{\\prime }\\le $ we have $\\Gamma ^{\\prime },^{\\prime }\\vdash _{\\normalfont \\textsf {min}}\\alpha :(T@c,U)$ and $\\Gamma \\pm (\\alpha ,(T@c,U)),^{\\prime }\\vdash _{\\normalfont \\textsf {bal}}\\circeq P^{\\prime }$ .","This theorem states that when an action performed in a well typed process and the action is well typed, which is guaranteed by the previous theorem, then the resulting process after the $\\tau $ -action is also well typed in an updated type environment.This result gives us the property that if a process is well typed, then it will never experience communication errors.","The theorem also tells us that processes evolve according to the types prescription."],["Type equivalence","In this section we discuss type equivalence in our type system and show how this leads to a new session type system."],["Why type equivalence matters","First we motivate type equivalence by expressing an example from [9], in our applied pi-calculus and our type system.","Recall the example from Section REF where the goal was to transmit a binary tree while preserving the tree structure.","Example 1 Consider the parameterised agent $S$ below.","The parameter $t$ is the tree to be transmitted, $c$ is the channel the tree is sent on, and $w$ is a channel used for initiating the transition of the right sub-tree after the transmission of the left sub-tree has finished.","The projection functions fst, snd and thrd are used to access the elements of a tuple.","$S(t, c, w) \\eqdef$ if $t=\\textsf{Leaf}$ then $c\\< \\textsf{Leaf}.\\o{w}\\l c \\r.\\nil$ else $c\\<\\textsf{Node}.$ $\\o{c}\\l\\textsf{fst(t)}\\r.$ $(\\nu n)(S(\\textsf{snd}(t), c, n)|n(x).S(\\textsf{thrd}(t), c, w))$ If we analyse the $S$ process, the endpoint type of $c$ is $T_C$ , which is the same type as was described in Section REF .","The type describes how a single node is transmitted, if it is a leaf node we do not do anything, if it is internal node then the value is transmitted and then the sub-trees of the node.","$T_C = \\mu z.\\oplus\\{$ $\\textsf{Leaf}: \\lin \\textsf{skip} $ $\\textsf{Node}: \\lin \\textsf{!Int};z;z$ $\\}$ The process below receives the transmitted tree preserving the original structure.","Through similar analysis as on the sending end, we can confirm that the endpoint type of $c$ in this process is $\\overline{T_C}$ .","$R(c, w) \\eqdef c\\>\\{$ $\\textsf{Leaf}: \\o{w}\\l c\\r.\\nil$ $\\textsf{Node}: (\\nu n)(c(x).R(c, n) | n(x).R(c, w))$ $\\}$ We can now create a process $P$ that transmits a tree on the $c$ channel.","$P=S(\\textsf {Tree}(1, \\textsf {Leaf}, \\textsf {Leaf}), c, w) \\text{ ǀ\\ } R(c, w)$ With an addition to $P$ , we can create a process $P^{\\prime }$ that reuses $c$ for transmitting an integer after transmitting the tree.","$P^{\\prime }=\\left[S(\\textsf {Tree}(1, \\textsf {Leaf}, \\textsf {Leaf}), c, w) \\text{ ǀ\\ } R(c, v)\\right]\\text{ ǀ\\ }\\left[v(c^{\\prime }).c^{\\prime }(x).\\mathbf {0}\\text{ ǀ\\ }w(c^{\\prime \\prime }).\\overline{c^{\\prime \\prime }}\\langle 1\\rangle .\\mathbf {0}\\right]$ The type of $c$ after expanding the recursive type is now $(T_C^{\\prime }, \\overline{T_C^{\\prime }})$ where $T_C' = \\lin \\oplus\\{$ $\\textsf{Leaf}: \\lin \\textsf{skip} $ $\\textsf{Node}: \\lin \\textsf{!Int};$ $\\mu z.\\oplus\\{\\textsf{Leaf}: \\lin \\textsf{skip}\\quad\\textsf{Node}:\\lin \\textsf{!Int};z; z\\};$ $\\mu z.\\oplus\\{\\textsf{Leaf}:\\lin \\textsf{skip}\\quad\\textsf{Node}:\\lin \\textsf{!Int};z;z\\}$ $\\};\\lin !\\textsf{Int}$ The last step is what motivates type equivalence; we would like to have a distributive law that allows the output to be moved into the select type, as illustrated below.","$T_C'' = \\lin \\oplus\\{$ $\\textsf{Leaf}: \\lin \\textsf{skip};\\lin !\\textsf{Int} $ $\\textsf{Node}: \\lin \\textsf{!Int};$ $\\mu z.\\oplus\\{\\textsf{Leaf}: \\lin \\textsf{skip}\\quad\\textsf{Node}:\\lin \\textsf{!Int};z;z\\};$ $\\mu z.\\oplus\\{\\textsf{Leaf}: \\lin \\textsf{skip}\\quad\\textsf{Node}:\\lin \\textsf{!Int};z;z\\};$ $\\lin !\\textsf{Int}$ $\\}$ As the typing rules in our applied pi-calculus are defined on the type transitions instead of the structure of a type, the distributive law is not essential in our calculus, since we require that the type exhibit specific behaviour instead of having a specific structure, but the example shows how type equivalence can be used to tell if two types can be used interchangeably.","In the next section we will introduce a method for checking if two types are equivalent, and we will return to Example REF , to check that it is indeed the case that $T_C^{\\prime \\prime }$ is equivalent to $T_C^{\\prime }$ ."],["Type bisimilarity","As described in Section REF , our types are highly reminiscent of BPA.","For BPA expressions, bisimulation is used to prove that two processes exhibit the same behaviour.","We now extend bisimulation to work on types as well.","The definition follows from the definition of bisimulation in [2].","Definition 1 (Type bisimulation) A binary relation $\\mathcal {R}$ between endpoint types is a type bisimulation iff whenever $T_{E_1} \\mathcal {R}\\ T_{E_2}$ : $Q(T_{E_1})=Q(T_{E_2})$ $\\forall \\lambda $ if $T_{E_1} \\xrightarrow{} T_{E_1}^{\\prime } $ then $\\exists T_{E_2}^{\\prime }$ such that $ T_{E_2} \\xrightarrow{} T_{E_2}^{\\prime }$ and $T_{E_1}^{\\prime } \\mathcal {R}\\ T_{E_2}^{\\prime }$ $\\forall \\lambda $ if $T_{E_2} \\xrightarrow{} T_{E_2}^{\\prime } $ then $\\exists T_{E_1}^{\\prime }$ such that $ T_{E_1} \\xrightarrow{} T_{E_1}^{\\prime }$ and $T_{E_1}^{\\prime } \\mathcal {R}\\ T_{E_2}^{\\prime }$ We write that $T_{E_1}\\sim T_{E_2}$ if $T_{E_1} \\mathcal {R} T_{E_2}$ for some type bisimulation and then say that $T_{E_1}$ and $T_{E_2}$ are type bisimilar.","Type bisimilar endpoint types will exhibit the same behaviour.","In other words, for a type $T_{E}$ , any type $T_E^{\\prime }$ where $T_E\\sim T_E^{\\prime }$ , $T_E^{\\prime }$ can be used instead of $T_E$ , without introducing communication errors.","Example 2 (A distributive law) Consider the types $T_C^{\\prime }$ and $T_C^{\\prime \\prime }$ from Example REF .","We can use type bisimulation to show that these two types describe the same communication behaviour on a channel.","To do so, we must provide a bisimulation that shows that $T_C^{\\prime }$ and $T_C^{\\prime \\prime }$ are type bisimilar.","Let $R$ be a relation over endpoint types.","Let $R$ be the symmetric closure of $\\lbrace (T_C^{\\prime },T_C^{\\prime \\prime }), ((\\textsf {lin }!\\textsf {int};r;r);\\textsf {lin }!\\textsf {int},\\linebreak [0]\\textsf {lin }!\\textsf {int};r;r;\\textsf {lin }!\\textsf {int})\\rbrace \\cup \\lbrace (T_E,T_E) \\text{ ǀ\\ } \\forall T_E\\in \\mathcal {T}\\rbrace $ where $r$ is the term $\\mu z.\\oplus \\lbrace \\textsf {Leaf}: \\textsf {lin }\\textsf {skip}\\quad \\textsf {Node}:\\textsf {lin }\\textsf {!Int};z;z\\rbrace $ .","In fact, we can generalise this result and prove the distributive law for both select and branch.","Lemma 2 Let $\\star $ be $\\oplus $ or $\\&$ .","Then $q\\ \\star \\lbrace l_i: T_{E_i}\\rbrace _{i\\in I};T_E\\sim q\\ \\star \\lbrace l_i: T_{E_i};T_E\\rbrace _{i\\in I}$ Let $R$ be a relation between types.","We must show that $R$ is a bisimulation of $q\\ \\star \\lbrace l_i: T_{E_i}\\rbrace _{i\\in I};T_E$ and $q\\ \\star \\lbrace l_i: T_{E_i};T_E\\rbrace _{i\\in I}$ .","Define $R$ as the symmetric closure of $\\lbrace (q\\ \\star \\lbrace l_i: T_{E_i}\\rbrace _{i\\in I};T_E,q\\ \\star \\lbrace l_i: T_{E_i};T_E\\rbrace _{i\\in I})\\rbrace \\cup \\lbrace (T_E,T_E) \\text{ ǀ\\ } \\forall T_E\\in \\mathcal {T}\\rbrace $ From the first requirement for a type bisimulation we have that $Q(q\\ \\star \\lbrace l_i: T_{E_i}\\rbrace _{i\\in I};T_E) = Q(q\\ \\star \\lbrace l_i: T_{E_i};T_E\\rbrace _{i\\in I}))$ , this is trivially fulfilled as $q=q$ .","By the (Seq) rule we have that the transitions of a sequential compositions are those of the left-hand side of the operator.","So the transitions of $q\\ \\star \\lbrace l_i: T_{E_i}\\rbrace _{i\\in I};T_E$ are $q\\ \\star \\lbrace l_i: T_{E_i}\\rbrace _{i\\in I};T_E\\xrightarrow{} T_{E_i};T_E$ , where $\\diamond \\in \\lbrace \\lhd ,\\rhd \\rbrace $ .","The available transitions for $q\\ \\star \\lbrace l_i: T_{E_i};T_E\\rbrace _{i\\in I}$ are $q\\ \\star \\lbrace l_i: T_{E_i};T_E\\rbrace _{i\\in I}\\xrightarrow{} T_{E_i};T_E$ .","Since $R$ is reflexive, we have that $(T_{E_i};T_E, T_{E_i};T_E)\\in R$ .","So any transition taken by one of the types can be matched by the other to end up with syntactically equivalent types.","Since the types are syntactically equivalent, all further transitions can be matched by any of the two types, and all further type pairs will be in $R$ .","This means that $R$ is a type bisimulation, and that $q\\ \\star \\lbrace l_i: T_{E_i}\\rbrace _{i\\in I};T_E \\sim q\\ \\star \\lbrace l_i: T_{E_i};T_E\\rbrace _{i\\in I}$ ."],["A quotiented session type system","We now define a quotiented session type system whose types are equivalence classes of session types from the the already existing type system.","We define transitions between equivalence classes instead of endpoint types as follows.","Definition 2 (Equivalence Classes) Let $\\vert T_E \\vert $ be the equivalence class of $T_E$ given by: $\\vert T_E \\vert = \\lbrace T_E^{\\prime }\\in \\mathcal {T} \\text{ ǀ\\ } T_E^{\\prime } \\sim T_E\\rbrace $ We denote a transition between equivalence classes with action $\\lambda $ as $\\vert T_{E_1}\\vert \\xrightarrow{} \\vert T_{E_2}\\vert $ .","Lemma 3 $\\vert T_E \\vert \\xrightarrow{} \\vert T_E^{\\prime } \\vert $ iff $\\forall T_{E_1} \\in \\vert T_E \\vert \\ \\exists T_{E_1}^{\\prime } \\in \\vert T_E^{\\prime } \\vert $ such that $T_{E_1} \\xrightarrow{} T_{E_1}^{\\prime }$ By the properties of type bisimilarity that states that two bisimilar types will evolve to bisimilar types for all possible transitions.","We use Lemma REF to define a new type system that is defined on equivalence classes of types from the previous type system.","In Section REF the transitions of endpoint types were of the form $T_E\\xrightarrow{}T_E^{\\prime }$ .","In the new type system the transitions are of the form $\\vert T_E \\vert \\xrightarrow{}\\vert T_E^{\\prime }\\vert $ .","The transition rules from Table REF still apply to the new type system, but where all endpoint types $T_E$ have been replaced with $\\vert T_E \\vert $ .","For example, the (Select) rule from Table REF would in the new type system be (REF ).","We also expand the $Q$ function on equivalence classes to be the result of applying $Q$ to any witness of the equivalence class.","$\\textsc {(Select)}\\qquad {q \\sqsubseteq Q(\\vert T_{E_k} \\vert )}{\\vert q\\ \\oplus \\lbrace l_i : \\vert T_{E_i} \\vert \\rbrace _{i\\in I} \\vert \\xrightarrow{} \\vert T_{E_k}\\vert }$ The typing rules in the new type system would also be the rules from Tables REF , REF and REF , where endpoint types $T_E$ have been replaced with equivalence classes $\\vert T_E \\vert $ .","In REF the (Input) rule from Table REF has been changed to fit the new type system.","$\\textsc {(Input)} \\qquad {\\Gamma _1 \\vdash n:\\vert T_E \\vert & \\vert T_E \\vert \\xrightarrow{} \\vert T_E^{\\prime } \\vert & \\Gamma _2, x:T + n: \\vert T_E^{\\prime } \\vert \\vdash P}{\\Gamma _1 \\circ \\Gamma _2 \\vdash n(x).P }$ The generic type system from [7] depends on the transitions available to types.","We have already shown that the previous type system is an instance of generic type systems.","From Lemma REF , we can see that equivalence classes have the exact same transitions as the old endpoint types had.","From these two results we can see that the type system defined on equivalence classes is an instance of the generic type system as well, allowing us to retain previously obtained results of fidelity and well typed internal actions from Section .","In conclusion, this gives us a type system where type equivalence is a trivial property, since two endpoint types of the old type system would be the same type in the new type system."],["Conclusion","In this paper we have considered a “low-level” applied pi-calculus that allows composite terms to be built but only allows for passing names and nullary function symbols.","In this setting, we introduced a type system for the applied pi-calculus based on the work on context-free session types by Thiemann and Vasconcelos in [9] and on the work on qualified session types by Vasconcelos in [10].","The type system is a context-free session type system with qualifiers.","The type system is an instance of the psi-calculus type system introduced in [7], and this allows us to establish a fidelity result about the type system that ensures that a well typed process continues to be well typed, and without communication errors, until termination.","The type system has a notion of type equivalence defined by introducing type bisimilarity.","Using it, we then get a type system of equivalence classes, for which type equivalence is a natural part of the type system itself.","The current focus is to deal with the decidability of type equivalence.","In [9] Thiemann and Vasconcelos show the decidability of type equivalence using a transformation from their types to guarded BPA expressions, for which bisimilarity is decidable.","As already discussed in this article, our types are very close to BPA, in which case we should be able to achieve the same results about decidability more directly.","In [5] an algorithm is presented for deciding bisimilarity for normed context free processes in polynomial time, and we conjecture that this algorithm can be adapted to our setting for checking type bisimilarity.","Here, it would be important to find a characterization of the class of applied pi processes that can be typed using normed session types only."]],"string":"[\n [\n \"Context-Free Session Types for Applied Pi-Calculus\"\n ],\n [\n \"Abstract We present a binary session type system using context-free session types to a version of the applied pi-calculus of Abadi et.\",\n \"al.\",\n \"where only base terms, constants and channels can be sent.\",\n \"Session types resemble process terms from BPA and we use a version of bisimulation equivalence to characterize type equivalence.\",\n \"We present a quotiented type system defined on type equivalence classes for which type equivalence is built into the type system.\",\n \"Both type systems satisfy general soundness properties; this is established by an appeal to a generic session type system for psi-calculi.\"\n ],\n [\n \"Introduction\",\n \"Binary session types [6] describe the protocol followed by the two ends of a communication medium, in which messages are passed.\",\n \"A sound type system of this kind guarantees that a well-typed process does not exhibit communication errors at runtime.\",\n \"Session types have traditionally been used to describe linear interaction between partners [10], but later type systems are able to distinguish between linear and unlimited channel usages.\",\n \"In particular Vasconcelos has proposed session types with $\\\\textsf {lin}/\\\\textsf {un}$ qualifiers that describe linear interaction as well as shared resources [10].\",\n \"Context-free session types introduced by [9] are more descriptive than the regular types described in previous type systems [6], [10] in that they allow full sequential composition of types.\",\n \"Because of this, session types can now describe protocols that cannot be captured in the regular session types, such as transmitting complex composite data structures.\",\n \"Many binary session type systems are concerned with languages that use the selection and branching constructs introduced in [6] in addition to the normal input and output constructs.\",\n \"These constructs are synchronisation operations, where two processes synchronise on a channel, and are similar to method invocations found in object-oriented programming.\",\n \"A branching process $l \\\\rhd \\\\lbrace l_1 :P_1, \\\\ldots , l_n : P_n \\\\rbrace $ continues as process $P_k$ together with $P$ if label $l_k$ is selected on channel $c$ using the selection $c \\\\lhd l_k.P$ .\",\n \"In this article we consider a session type system for a version of the applied pi-calculus, due to Abadi et.\",\n \"al.\",\n \"[1].\",\n \"This is an extension of the pi-calculus [8] with terms and extended processes and in our case extended further with selection and branching.\",\n \"Our version is a “low-level” version in that allows one to build composite terms but only allows for the communication of names and nullary function symbols.\",\n \"In this way, the resulting version is close to the versions of the pi-calculus used for encoding composite terms [8].\",\n \"Our session type system combine ideas from the type systems from [9] and [10] into a type system in which session types are context-free and use $\\\\textsf {lin}/\\\\textsf {un}$ qualifiers in order to distinguish between linear and unbounded resources.\",\n \"The resulting type system uses types that are essentially process terms from a variant of the BPA process calculus [4].\",\n \"In Section we present our version of the applied pi-calculus and in Section we define the syntax and semantics of our session type system.\",\n \"We then prove the soundess of our type system by using the general results about psi-calculi from [7].\",\n \"This is done by showing that the applied pi-calculus is a psi-calculus, and that our type system is a instance of the generic type system presented by Hüttel in [7].\",\n \"Lastly we present type equivalence between types by introducing a notion of type bisimulation for endpoint types.\",\n \"By considering equivalence classes under type bisimilarity we get a new quotiented type system whose types are equivalence classess.\",\n \"It then follows from the theorems in [7] that the general results for our first type system also hold for the quotiented type system.\"\n ],\n [\n \"Applied pi-Calculus\",\n \"We consider a “low-level” version of the applied pi-calculus [1] in which composite terms are allowed but only the transmission of simple data is possible: Only names $n$ , constants represented by functions $f_0$ with arity 0 and functions that evaluate to values of base types can be transmitted.\",\n \"We use the notation $\\\\widetilde{M}$ to represent the sequence $M_1,\\\\ldots ,M_i$ and $\\\\widetilde{x}$ to represent the sequence of variables $x_1,\\\\ldots ,x_i$ .\",\n \"We always assume that our processes are specified relative to a family of parameterized agent definitions that are on the form $N(\\\\widetilde{x})\\\\overset{\\\\underset{\\\\mathrm {def}}{}}{=}A$ and that every agent variable $N$ occurring in a process has a corresponding definition.\",\n \"The formation rules for processes $P$ and extended processes $A$ can be seen below in ().\",\n \"Note that we distinguish between variables ranged over by $x, y \\\\ldots $ and names ranged over by $m,n, \\\\ldots $ .\",\n \"We let $a$ range over the union of these sets.\",\n \"We extend the syntax of processes with branching $c \\\\rhd \\\\lbrace l_1:P_1,\\\\ldots ,l_k:P_k \\\\rbrace $ and selection $c \\\\lhd l.P$ where $l_1, \\\\ldots l_k$ are taken from a set of labels.\",\n \"Extended processes extend processes with the ability to use active substitutions of the form $\\\\lbrace {M}{x}\\\\rbrace $ that instantiate variables.\",\n \"P ::= 0P1 P2 !P ( n)P ( n) P if M1=M2 then P1 else P2 n(x).P nu .P c l.P c { l1:P1,...,lk:Pk } N(M) u ::= n x f0 M ::= n x f(M) A ::= P A1 A2 (n)A ( x)A {Mx} The notion of structural congruence extends that of the usual pi-calculus [8].\",\n \"The following two further axioms that are particular to the applied pi-calculus are of particular importance, as they show the role played by active substitutions.\",\n \"$ P \\\\mid \\\\lbrace {M}{x}\\\\rbrace \\\\equiv P\\\\lbrace {M}{x}\\\\rbrace \\\\mid \\\\lbrace {M}{x}\\\\rbrace & & & \\\\mathbf {0}\\\\equiv (\\\\nu \\\\, x) \\\\lbrace {M}{x}\\\\rbrace $ Together with the axioms of [8] they allow us to factor out composite terms such that they only occur in active substitutions.\",\n \"For instance, we have that $\\\\texttt {if }M_1 = M_2\\\\texttt { then }P\\\\texttt { else }Q\\\\equiv \\\\nu {x}\\\\nu {y}(\\\\texttt {if }x=y\\\\texttt { then }P\\\\texttt { else }Q \\\\mid \\\\lbrace {M_1}{x}\\\\rbrace \\\\mid \\\\lbrace {M_2}{y}\\\\rbrace )$ .\",\n \"We can therefore use a term by only mentioning the variable associated with it.\",\n \"In our version of the applied pi-calculus we very directly make use of this.\",\n \"Our semantics consists of reduction semantics and a labelled operational semantics that extend those of [1] with rules for branching and selection.\",\n \"Reductions are of the form $P \\\\rightarrow P^{\\\\prime }$ , while transitions are of the form $P \\\\xrightarrow{} P^{\\\\prime }$ where the label $\\\\alpha $ is given by $ \\\\alpha ::= c \\\\lhd l \\\\mid c \\\\rhd l \\\\mid a(x) \\\\mid \\\\overline{a}x \\\\mid \\\\tau $ and $c \\\\lhd l$ is the co-label of $c \\\\rhd l$ .\",\n \"The rules defining reductions and labelled transitions are found in Tables REF and REF , respectively.\",\n \"Table: Reduction rulesTable: Labelled transition rules\"\n ],\n [\n \"A context-free session type system\",\n \"We now present the syntax and semantics of our type system.\"\n ],\n [\n \"Session types\",\n \"Session types describe the communication protocol followed by a channel.\",\n \"Consider writing a process that transmits a binary tree where each internal node containsan integer.\",\n \"We want to transmit this tree by sending only base types (i.e.\",\n \"the integers) on a channel.\",\n \"The tree data type can be described with the grammar below.\",\n \"$\\\\textsf {Tree} ::= (\\\\textsf {Int}, \\\\textsf {Tree}, \\\\textsf {Tree})\\\\ \\\\text{ ǀ\\\\ }\\\\ \\\\textsf {Leaf}$ Using the regular types introduced in [10], the type for a channel transmitting such a data structure could be the recursive type $\\\\mu z.\\\\oplus \\\\lbrace \\\\textsf {Leaf}: \\\\textsf {lin }\\\\textsf {skip}\\\\quad \\\\textsf {Node}:\\\\textsf {lin }!\\\\textsf {Int}.z\\\\rbrace $ .\",\n \"In this type $z$ is a type variable that is recursively defined.\",\n \"The type describes how a single node is transmitted, if it is a leaf node we do not do anything, if it is an internal node then the integer value is transmitted with the output type $!\\\\textsf {Int}$ and then the sub-trees of the node are transmitted with a recursive call.\",\n \"However, if we use this regular session type, we are not able to guarantee that the tree structure is preserved.\",\n \"The reason is that the session type describes that a list of nodes are being sent, but not the position in the tree of each node.\",\n \"On the other hand, if we use the context-free session type disciple introduced by Thiemann and Vasconcelos [9], we can specify the preservation of tree structures by using types such as $\\\\mu z.\\\\oplus \\\\lbrace \\\\textsf {Leaf}: \\\\textsf {skip}\\\\quad \\\\textsf {Node}:!\\\\textsf {Int};z;z\\\\rbrace $ .\",\n \"Using sequential composition with the $\\\\_;\\\\_$ operator, we can specify a protocol that will guarantee that the tree structure by first sending the left sub-tree and then the right sub-tree.\",\n \"Introducing a sequential operator can introduce challenges for typing a calculus, as the following example shows.\",\n \"If we were to reuse a channel by sending an integer after transmitting a tree, the type would be $\\\\mu z.\\\\oplus \\\\lbrace \\\\textsf {Leaf}: \\\\textsf {skip}\\\\quad \\\\textsf {Node}:!\\\\textsf {Int};z;z\\\\rbrace ;!\\\\textsf {Int}$ .\",\n \"${\\\\Gamma (c)=\\\\oplus \\\\lbrace l_i: T_{E_i}\\\\rbrace _{i\\\\in I}}{\\\\Gamma \\\\vdash \\\\text{select } l_i \\\\text{ on } c}$ When typing rules are created for a calculus, it is often defined on the structure of types and terms.\",\n \"An example of such a typing rule for a select statement is shown in (REF ), which says that a select statement is well typed if the select operation is performed on a channel with a select type.\",\n \"If however the channel has a type as shown before, the select rule cannot be used, as the channel has the type of a sequential composition, and inside the sequential composition we have a recursive type.\",\n \"We require an equirecursive treatment of types, which allows us to expand the type to $\\\\oplus \\\\lbrace \\\\textsf {Leaf}: \\\\textsf {skip}\\\\quad \\\\textsf {Node}: !\\\\textsf {Int};\\\\mu z.\\\\oplus \\\\lbrace \\\\textsf {Leaf}:\\\\textsf {skip}\\\\quad \\\\textsf {Node}: !\\\\textsf {Int};z;z\\\\rbrace ;\\\\mu z.\\\\oplus \\\\lbrace \\\\textsf {Leaf}: \\\\textsf {skip}\\\\quad \\\\textsf {Node}: !\\\\textsf {Int};z;z\\\\rbrace \\\\rbrace ;!\\\\textsf {Int}$ .\",\n \"This is achieved by unfolding a recursive type $\\\\mu z.T$ to $T$ where all occurrences of $z$ in $T$ are replaced with the original $\\\\mu z.T$ .\",\n \"So now we are left with a sequential composition with a select type and output type.\",\n \"In order to transform this into a select type, we need a distributive law that allows us to move the sequential composition inside the select type, in order to obtain the type $\\\\oplus \\\\lbrace \\\\textsf {Leaf}:\\\\textsf {skip};!\\\\textsf {Int}\\\\quad \\\\textsf {Node}: !\\\\textsf {Int};\\\\mu z.\\\\oplus \\\\lbrace \\\\textsf {Leaf}: \\\\textsf {skip}\\\\quad \\\\textsf {Node}: !\\\\textsf {Int};z;z\\\\rbrace ;\\\\mu z.\\\\oplus \\\\lbrace \\\\textsf {Leaf}:\\\\textsf {skip}\\\\quad \\\\textsf {Node}: !\\\\textsf {Int};z;z\\\\rbrace ;!\\\\textsf {Int}\\\\rbrace $ .\",\n \"Such a rule does exist, we will introduce a method to prove that these types exhibit the same behaviour in Section when we introduce type equivalence.\"\n ],\n [\n \"The language of types\",\n \"We denote the set of all types by $\\\\mathcal {T}$ ; the types $T\\\\in \\\\mathcal {T}$ are described by the formation rules in (REF ).\",\n \"These session types are a modified version of the context-free session types presented by [9].\",\n \"The modifications made to the types are that we allow input and output session types to transmit other session types to allow sending channels in other channels, and finally that we introduce the lin and un qualifiers.\",\n \"We let $B$ range over a set of base types.\",\n \"p ::= skip ?T !T &{li:TEi} {li:TEi} q ::= lin un TE ::= q p z z.TE TE1; TE2 T ::= S B TE S ::= (TE1, TE2) From the formation rules we can see that a session type $S$ is a pair of endpoint types $T_{E_1}$ and $T_{E_2}$ .\",\n \"Endpoint types describe one end of channel, and are the types that evolve when a channel is used.\",\n \"The qualifiers lin and un from [10] are used to describe a linear interaction between two partners and a unrestricted shared resources respectively.\",\n \"An example of an un type could be a server, that a lot of processes have access to.\",\n \"To ensure that no communication errors occur if multiple processes can read from a channel concurrently, we require that the type must never change behaviour.\",\n \"This means that an un type must be the same before and after a transition.\"\n ],\n [\n \"Transitions for types\",\n \"We now present the transition rules for endpoint types, which describe how the types can evolve when an action is performed.\",\n \"We use an annotated reduction semantics to describe the behaviour of our types.\",\n \"Our labels are generated by the grammar in (REF ) where we have select, branch, input and output actions.\",\n \"The transitions are of the form $T_E\\\\xrightarrow{}T_E^{\\\\prime }$ and are shown in Table REF .\",\n \"We let $T_E\\\\lbrace {y}{x}\\\\rbrace $ be the endpoint type $T_E$ where all free occurrences of $x$ has been replaced with $y$ .\",\n \"$\\\\lambda ::= !T_1 \\\\mid ?T_1 \\\\mid \\\\rhd \\\\, l \\\\mid \\\\lhd \\\\, l$ In the transition rules, we use the function $Q$ defined in (REF ) to find the qualifier of compound types such as recursive types or sequential composition types.\",\n \"$\\\\begin{split}{Q}(q\\\\ p) &\\\\overset{\\\\underset{\\\\mathrm {def}}{}}{=}q \\\\\\\\{Q}(T_{E_1};T_{E_2}) &\\\\overset{\\\\underset{\\\\mathrm {def}}{}}{=}{Q}(T_{E_1}) \\\\\\\\{Q}(\\\\mu z.T_E) &\\\\overset{\\\\underset{\\\\mathrm {def}}{}}{=}{Q}(T_E)\\\\end{split}$ A relation $\\\\sqsubseteq =\\\\lbrace (\\\\textsf {lin},\\\\textsf {un}),(\\\\textsf {un},\\\\textsf {un}),(\\\\textsf {lin},\\\\textsf {lin})\\\\rbrace $ is defined for qualifiers in [10].\",\n \"We also follow the definition of $q(T)$ and $q(\\\\Gamma )$ from that of [10].\",\n \"In short $\\\\textsf {lin}(\\\\Gamma )$ is always satisfied, and $\\\\textsf {un}(\\\\Gamma )$ is satisfied iff all elements in $\\\\Gamma $ are unrestricted.\",\n \"The type system contains the sequential operator $\\\\_;\\\\_$ as well as choice operators; select $\\\\&\\\\lbrace \\\\ldots \\\\rbrace $ and branch $\\\\oplus \\\\lbrace \\\\ldots \\\\rbrace $ .\",\n \"This is very similar to Basic Process Algebra (BPA)[4] that contains the sequential operator $\\\\cdot $ and nondeterministic choice operator $+$ .\",\n \"We also have recursive types, which corresponds to variables with recursive definitions in BPA.\",\n \"A BPA expression is guarded if all recursive variables on the right hand side is preceded by an action $\\\\lambda $ [4].\",\n \"Similarly we say that a type is guarded if every recursion variable is preceded by an input or output.\",\n \"These similarities with BPA will become very important, as we can describe types as BPA expressions and by showing results about these expression, we can in turn show results about our types.\",\n \"Table: Annotated reduction semantics for types\"\n ],\n [\n \"Typing rules\",\n \"We now present a type system for our version of the applied pi-calculus in Table REF .\",\n \"In this type system, the type judgements for processes are on the form $\\\\Gamma \\\\vdash P$ meaning that the process $P$ is well typed in context $\\\\Gamma $ .\",\n \"The judgments for terms are on the form $\\\\Gamma \\\\vdash M:T$ meaning that the term $M$ is well typed with type $T$ in context $\\\\Gamma $ .\",\n \"Lastly the judgments for extended processes are on the form $\\\\Gamma \\\\vdash _A A$ meaning that the process $A$ is well typed in context $\\\\Gamma $ .\",\n \"We type processes in a type context $\\\\Gamma $ which contains types for the variables, names and functions symbols of a process.\",\n \"We follow the definition of a type context from [10]: $\\\\emptyset $ is the empty context, $\\\\Gamma , x: T$ is the context equal to $\\\\Gamma $ except that $x$ has the type $T$ in the new context.\",\n \"This operation is only defined when $x\\\\notin \\\\text{dom}(\\\\Gamma )$ .\",\n \"Table REF shows the typing rules for processes.\",\n \"We use the context split $\\\\circ $ and context update $+$ operations from [10].\",\n \"The context split operation is used to split a context into two constituents.\",\n \"A maximum of two processes must have access to a given linear session type; a context either contains the entire session type $S=(T_{E_1},T_{E_2})$ , or a single endpoint type $T_{E}$ .\",\n \"When splitting a context into two, we can pass $S$ to either context, or one endpoint to each context.\",\n \"If the context only has an endpoint type, the endpoint type can only be passed on to one of the two contexts.\",\n \"This way we ensure that each lin endpoint of a channel is known in exactly one context.\",\n \"Names of unrestricted type can be shared among all contexts.\",\n \"The context update operation updates the type of a channel.\",\n \"The $\\\\Gamma , x:T$ operation is only defined when $x\\\\notin \\\\text{dom}(\\\\Gamma )$ .\",\n \"The $+$ operation $\\\\Gamma =\\\\Gamma _1+\\\\Gamma _2$ uses the type of $x$ in $\\\\Gamma _2$ to update the type in $\\\\Gamma _1$ .\",\n \"So if $\\\\Gamma _1(x)=T_1$ and $\\\\Gamma _2(x)=T_2$ then $\\\\Gamma (x)=T_2$ .\",\n \"The two operations are used in (Input) and (Output) where we must split our context into two, to type each endpoint of a linear channel in its own context.\",\n \"Table: Typing rules for Extended ProcessesTable: Context split for Applied π\\\\pi -calculus, based on Table: Context update for Applied π\\\\pi -calculus from Table: Typing rules for termsTable: Typing rules for processes\"\n ],\n [\n \"Duality of types\",\n \"The notion of type duality is central to session type systems; it expresses that the protocols followed by the two endpoints of a name must be opposites: If one end transmits a value, the other end must receive it.\",\n \"We denote the dual of an endpoint type $\\\\overline{T_E}$ and define duality below, following [9].\",\n \"$\\\\overline{q\\\\ \\\\textsf {skip}}&\\\\overset{\\\\underset{\\\\mathrm {def}}{}}{=}q\\\\ \\\\textsf {skip} && \\\\overline{q\\\\ \\\\&\\\\lbrace l_1: T_{E_1}, \\\\ldots , l_k:T_{E_k}\\\\rbrace }\\\\overset{\\\\underset{\\\\mathrm {def}}{}}{=}q\\\\ \\\\oplus \\\\lbrace l_1: \\\\overline{T_{E_1}}, \\\\ldots , l_k:\\\\overline{T_{E_k}}\\\\rbrace \\\\\\\\\\\\overline{q\\\\ ?T}&\\\\overset{\\\\underset{\\\\mathrm {def}}{}}{=}q\\\\ !T && \\\\overline{q\\\\ \\\\oplus \\\\lbrace l_1: T_{E_1}, \\\\ldots , l_k:T_{E_k}\\\\rbrace }\\\\overset{\\\\underset{\\\\mathrm {def}}{}}{=}q\\\\ \\\\&\\\\lbrace l_1: \\\\overline{T_{E_1}}, \\\\ldots , l_k:\\\\overline{T_{E_k}}\\\\rbrace \\\\\\\\\\\\overline{q\\\\ !T}&\\\\overset{\\\\underset{\\\\mathrm {def}}{}}{=}q\\\\ ?T && \\\\overline{\\\\mu z.T_E} \\\\overset{\\\\underset{\\\\mathrm {def}}{}}{=}\\\\mu z.\\\\overline{T_E} \\\\\\\\\\\\overline{T_{E_1};T_{E_2}}&\\\\overset{\\\\underset{\\\\mathrm {def}}{}}{=}\\\\overline{T_{E_1}};\\\\overline{T_{E_2}} && \\\\overline{z} \\\\overset{\\\\underset{\\\\mathrm {def}}{}}{=}z$ A session type $S = (T_{E_1},T_{E_2})$ is balanced iff its endpoint types are dual, that is, if $\\\\overline{T_{E_1}} = T_{E_2}$ .\",\n \"A type context $\\\\Gamma $ is balanced iff every session type in the range of $\\\\Gamma $ is balanced.\",\n \"We use the notation $\\\\Gamma \\\\vdash _\\\\text{bal}P$ to describe that a process $P$ is well typed in a balanced context $\\\\Gamma $ .\"\n ],\n [\n \"Applied pi-calculus as a psi-calculus instance\",\n \"Psi-calculus is a general framework for process calculi.\",\n \"In this section we show that the applied pi-calculus is an instance of it, and this will then be used in Section to obtain results about our type system.\",\n \"An instance of the psi-calculus framework contains the seven elements [3] given in Table REF .\",\n \"Table: Elements of psi-calculiThe syntax of an instance of the psi-calculus is described by the formation rules in (REF ) that generalize those of the pi-calculus.\",\n \"The input and output constructions allows to use arbitrary terms as channels, and the input construction allows for matching on a pattern $(\\\\lambda \\\\tilde{x})N$ ; the pattern variables in $\\\\tilde{x}$ are bound to the subterms that match.\",\n \"The other syntactic constructs are similar to those of the pi-calculus.\",\n \"The case construct is a generic case of the if-construct; we generalize match conditions $M_1=M_2$ to allow any $\\\\phi \\\\in \\\\textbf {C}$ as a condition.\",\n \"Lastly in psi-calculi we have a concept of assertions $\\\\in \\\\textbf {A}$ .\",\n \"These generalize the notion of active substition found in the applied pi calculus.\",\n \"$P :=\\\\ &\\\\mathbf {0}\\\\mid \\\\overline{M}N.P \\\\mid \\\\underline{M}(\\\\lambda \\\\widetilde{x})N.P \\\\mid \\\\textbf {case}\\\\ \\\\phi _1:P_1[]\\\\ldots [] \\\\phi _n: P_n \\\\\\\\\\\\text{ ǀ\\\\ }\\\\ &(\\\\nu a)P \\\\mid P\\\\text{ǀ}Q \\\\mid !P \\\\mid \\\\llparenthesis \\\\rrparenthesis \\\\mid M l.P \\\\mid M \\\\triangleright \\\\lbrace l_1:P_1,\\\\ldots ,l_k:P_k \\\\rbrace $ The structural operational semantics of psi-calculi has transitions of the form $\\\\Psi \\\\blacktriangleright P\\\\xrightarrow{}P^{\\\\prime }$ where the labels $\\\\alpha $ are given by the formation rules below [7].\",\n \"::= M l | M l | M(va)N | KN | (va)@ (vb)(MNK) | (va)@ (MlN) The first four actions correspond to the actions we know from the applied pi-calculus: selection, branch, output and input.\",\n \"The last two action are internal $\\\\tau $ -actions that correspond to internal actions that are either an input/output-exchange or a branch/select-exchange.\",\n \"It follows from[3] that the standard pi-calculus is an instance of the psi-calculus.\",\n \"Below we do the same for the instance APi that shows that the applied pi-calculus is also an instance of the psi-calculus.\",\n \"In the definition $\\\\mathcal {N}$ is the set of all names and $\\\\mathcal {F}$ is the set of all function names.\",\n \"$\\\\textbf {T} &\\\\overset{\\\\underset{\\\\mathrm {def}}{}}{=}\\\\mathcal {N} \\\\cup \\\\lbrace f(M_1, \\\\ldots M_k) \\\\text{ ǀ\\\\ } f \\\\in \\\\mathcal {F}, M_i \\\\in \\\\textbf {T}\\\\rbrace & &\\\\textbf {C} && \\\\overset{\\\\underset{\\\\mathrm {def}}{}}{=}\\\\lbrace M=N \\\\text{ ǀ\\\\ } M,N\\\\in \\\\textbf {T}\\\\rbrace \\\\\\\\\\\\textbf {A} &\\\\overset{\\\\underset{\\\\mathrm {def}}{}}{=}\\\\lbrace 1\\\\rbrace & &{\\\\hspace{-1.66656pt}\\\\cdot }&&\\\\overset{\\\\underset{\\\\mathrm {def}}{}}{=}\\\\lbrace ((n, n), n=n) \\\\text{ ǀ\\\\ } n \\\\in \\\\mathcal {N} \\\\rbrace \\\\\\\\\\\\otimes &\\\\overset{\\\\underset{\\\\mathrm {def}}{}}{=}\\\\lbrace ((_1, _2), 1) \\\\text{ ǀ\\\\ } \\\\in \\\\textbf {A} \\\\rbrace & &\\\\textbf {1} & &\\\\overset{\\\\underset{\\\\mathrm {def}}{}}{=}1 \\\\\\\\\\\\vdash &\\\\overset{\\\\underset{\\\\mathrm {def}}{}}{=}\\\\lbrace (1, M=M) \\\\text{ ǀ\\\\ } M\\\\in \\\\textbf {T}\\\\rbrace $\"\n ],\n [\n \"Properties of our type system\",\n \"A generic binary session type system for psi-calculi was presented in [7].\",\n \"The intention is that any existing binary session type systems for process calculi can be captured as special instances of the generic system, as long it satisfies four specific requirements.\",\n \"We already know that our applied pi-calculus is a simple psi-calculus and now establish that our type system fulfils the requirements of [7].\",\n \"This will then allow us to obtain the usual results for binary type systems as a simple corollary of the theorems for the generic system.\"\n ],\n [\n \"Transition structure\",\n \"Type transitions in our type system are an instance of the generic type transitions in [7].\",\n \"Both the generic type transitions and our type transition consist of send, receive, branch and select.\",\n \"The syntax of type transitions is uniform across the two articles as they are both generated by the grammar in (REF ).\",\n \"This illustrates that there is a one-to-one correspondence between the type transitions in the two type systems.\",\n \"$\\\\lambda ::= l \\\\text{ ǀ\\\\ } \\\\triangleright l \\\\text{ ǀ\\\\ } !T \\\\text{ ǀ\\\\ } ?T$\"\n ],\n [\n \"Revisiting duality\",\n \"Duality of types in our pi-calculus is defined on the structure of types, as seen in Section REF .\",\n \"In [7] duality is defined on type transitions, where (REF ) holds for dual types (we use $\\\\underline{\\\\,\\\\cdot \\\\,}$ to denote duality defined on type transitions).\",\n \"$T_E \\\\xrightarrow{} T_E^{\\\\prime } \\\\Leftrightarrow \\\\underline{T_E} \\\\xrightarrow{} \\\\underline{T_E^{\\\\prime }}$ In (REF ) and () we see the duality of type transitions, as presented by [7].\",\n \"$\\\\overline{!T_1} = ?T_2 && \\\\overline{?T_1} = !T_2 \\\\\\\\\\\\overline{l} = \\\\triangleright l && \\\\overline{\\\\triangleright l} = l$ We now show that the duality defined in Section REF upholds the property in (REF ).\",\n \"Lemma 1 $\\\\underline{T_E} =\\\\overline{T_E}$ By induction in the structure of types.\"\n ],\n [\n \"Checking requirements\",\n \"In the type system presented in [7], type judgements are relative to a type context and an assignment and therefore of the form $\\\\Gamma ,\\\\vdash {J}$ , where the judgment body is either ${J}$ is either a term typing $M : T$ or $P$ , the statement that process $P$ is well-typed.\",\n \"We write $\\\\Gamma ,\\\\vdash _\\\\text{min} {J}$ , if $\\\\Gamma ^{\\\\prime },^{\\\\prime }\\\\lnot \\\\vdash _\\\\text{min} {J}$ for every smaller $\\\\Gamma ^{\\\\prime }$ and $^{\\\\prime }$ .\",\n \"For each requirement presented in [7] we show that it is satisfied in our type system.\"\n ],\n [\n \"Requirement 1:\",\n \"If $\\\\Gamma _1, _1 \\\\vdash _\\\\text{min}{J} $ and $\\\\Gamma _2, _2 \\\\vdash _\\\\text{min} {J}$ then $\\\\Gamma _1 = \\\\Gamma _2 $ and $_1 \\\\simeq _2$ .\",\n \"As we only have the assertion 1 in type judgements, $_1\\\\simeq _2$ is trivially fulfilled as both are 1.\",\n \"Let $\\\\Gamma _1,\\\\Gamma _2$ be type contexts such that $\\\\Gamma _1, \\\\textbf {1} \\\\vdash _{\\\\text{min}}{J}$ and $\\\\Gamma _2, \\\\textbf {1}\\\\vdash _{\\\\text{min}}{J}$ .\",\n \"Assume that $\\\\Gamma _1 \\\\ne \\\\Gamma _2$ then without loss of generality there exists an $x$ such that $x\\\\in \\\\text{dom}(\\\\Gamma _1)$ and $x\\\\notin \\\\text{dom}(\\\\Gamma _2)$ .\",\n \"Because judgments must be well-formed we know that $fn({J})\\\\subseteq \\\\text{dom}(\\\\Gamma _1)$ and $fn({J})\\\\subseteq \\\\text{dom}(\\\\Gamma _2)$ , hence $x\\\\notin fn({J})$ .\",\n \"Let $\\\\Gamma _1^{\\\\prime }$ be defined as $\\\\Gamma _1$ except $x\\\\notin \\\\text{dom}(\\\\Gamma _1^{\\\\prime })$ , then $\\\\Gamma _1^{\\\\prime },\\\\textbf {1}\\\\vdash {J}$ and $\\\\Gamma _1^{\\\\prime }<\\\\Gamma _1$ thus $\\\\Gamma _1,\\\\textbf {1}\\\\nvdash _{\\\\text{min}}{J}$ .\",\n \"This is a contradiction, hence our assumption is wrong and $\\\\Gamma _1=\\\\Gamma _2$ , which means that the requirement is fulfilled.\"\n ],\n [\n \"Requirement 2:\",\n \"If $\\\\Gamma , \\\\vdash M:T@c$ then $\\\\Gamma (c)=T_E$ for some endpoint type $T_E$ .\",\n \"In our calculus, the only terms that can be used as channels are names.\",\n \"The $(\\\\nu n)P$ construct uses a channel constructor $n$ to declare a channel with a session type $\\\\Gamma , \\\\vdash n: S@n$ .\",\n \"$S$ is a session type $S=(T_{E_1}, T_{E_2})$ for some endpoint types.\",\n \"When type checking $n$ , only one endpoint is present in the context $\\\\Gamma $ , in which case $\\\\Gamma (n)=T_{E_1}$ or $\\\\Gamma (n)=T_{E_2}$ , meaning that the requirement is satisfied.\"\n ],\n [\n \"Requirement 3:\",\n \"Suppose $\\\\widetilde{N}\\\\in \\\\textsc {match}(M, \\\\widetilde{x}, X)$ , $\\\\Gamma , \\\\vdash M$ and $\\\\Gamma _1+\\\\widetilde{x}:\\\\widetilde{T},_1\\\\vdash _{\\\\text{min}}X:\\\\widetilde{T}\\\\rightarrow U$ .\",\n \"Then there exist $\\\\Gamma _{2i},_{2i}$ such that $\\\\Gamma _{2i}, _{2i}\\\\vdash _{\\\\text{min}}N_i:T_i$ for all $1\\\\le i \\\\le \\\\vert \\\\widetilde{x}\\\\vert =n$ .\",\n \"In our calculus, the only possible match is $M \\\\in MATCH(M, x, x)$ .\",\n \"As $\\\\vert \\\\widetilde{x}\\\\vert =1$ the requirement becomes $\\\\Gamma _{2},_{2}\\\\vdash _{\\\\text{min}}M : T$ .\",\n \"From the requirement that $M$ is well typed, this is trivially fulfilled.\"\n ],\n [\n \"Requirement 4:\",\n \"If ${\\\\hspace{2.77771pt}\\\\text{ǀ}\\\\hspace{-5.0pt}=}M {\\\\hspace{-1.66656pt}\\\\cdot }K$ and $\\\\Gamma , \\\\vdash M:S$ then $\\\\Gamma , \\\\vdash K:S$ .\",\n \"If ${\\\\hspace{2.77771pt}\\\\text{ǀ}\\\\hspace{-5.0pt}=}M {\\\\hspace{-1.66656pt}\\\\cdot }K$ and $\\\\Gamma , \\\\vdash M:T$ then $\\\\Gamma , \\\\vdash K:\\\\overline{T}$ .\",\n \"In Section () we defined ${\\\\hspace{-1.66656pt}\\\\cdot }$ as $=$ , meaning that two channels are equal if it is the same name.\",\n \"The first part of the requirement becomes: If ${\\\\hspace{2.77771pt}\\\\text{ǀ}\\\\hspace{-5.0pt}=}n = n$ and $\\\\Gamma ,\\\\vdash n:S$ then $\\\\Gamma , \\\\vdash n:S$ which is trivially fulfilled.\",\n \"The second part happens when only one end of a channel is in the context, with an endpoint type, then the other end of the channel must have a dual endpoint type.\",\n \"If ${\\\\hspace{2.77771pt}\\\\text{ǀ}\\\\hspace{-5.0pt}=}n = c$ and $\\\\Gamma , \\\\vdash n:T_E$ then $\\\\Gamma , \\\\vdash c:\\\\overline{T_E}$ .\",\n \"A channel has a balanced session type $S=(T_E, \\\\overline{T_E})$ for some endpoint type.\",\n \"If $n$ and $c$ have endpoint types and are the same channel, then they must also have types dual of each other.\",\n \"If the types are not unrestricted then they can only evolve into other types dual of each other, due to the requirement that only two processes have access to the channels and that if $T_E\\\\xrightarrow{}T_E^{\\\\prime }\\\\Longleftrightarrow \\\\underline{T_E}\\\\xrightarrow{}\\\\underline{T_E^{\\\\prime }}$ we know that the types will stay dual of each other.\",\n \"If the types are unrestricted then the requirement that whenever $T_E\\\\xrightarrow{}T_E^{\\\\prime }$ for some $\\\\lambda $ then $T_E=T_E^{\\\\prime }$ , ensures that the types will stay dual of each other.\",\n \"In this section we discuss the results that we obtain by showing that our type system is an instance of the generic type system.\",\n \"Hüttel presents and proves two main theorems for the generic type system, that we will use to show results about our type system as well.\",\n \"The first theorem, which is about well typed $\\\\tau $ -actions from [7] follows below: Theorem 1 (Well-typed $\\\\tau $ -actions, Theorem 9 of [7]) Suppose we have $_0 \\\\blacktriangleright P\\\\xrightarrow{}P^{\\\\prime }$ , where $\\\\alpha $ is a $\\\\tau $ -action and that $\\\\Gamma ,\\\\vdash _{\\\\normalfont \\\\textsf {bal}}P$ and $\\\\le _0$ then for some $^{\\\\prime }\\\\le $ and $\\\\Gamma ^{\\\\prime }\\\\le \\\\Gamma $ we have $\\\\Gamma ^{\\\\prime },^{\\\\prime }\\\\vdash _{{\\\\normalfont \\\\text{min}}}\\\\alpha :(T@c,U)$ .\",\n \"This theorem says that if a process $P$ can make a $\\\\tau $ -action and become $P^{\\\\prime }$ , and that $P$ is well typed in an environment where channels have pairs of dual endpoint types, then the action is also well-typed.\",\n \"So internal synchronisation in a well-typed process is well typed as well.\",\n \"The second theorem is about fidelity and follows below: Theorem 2 (Fidelity, Theorem 10 of [7]) Suppose we have $_0\\\\blacktriangleright P \\\\xrightarrow{}P^{\\\\prime }$ , where $\\\\alpha $ is a $\\\\tau $ -action and that $\\\\Gamma ,\\\\vdash _{\\\\normalfont \\\\textsf {bal}}P$ .\",\n \"Then for some $\\\\Gamma ^{\\\\prime }\\\\le \\\\Gamma $ and for some $^{\\\\prime }\\\\le $ we have $\\\\Gamma ^{\\\\prime },^{\\\\prime }\\\\vdash _{\\\\normalfont \\\\textsf {min}}\\\\alpha :(T@c,U)$ and $\\\\Gamma \\\\pm (\\\\alpha ,(T@c,U)),^{\\\\prime }\\\\vdash _{\\\\normalfont \\\\textsf {bal}}\\\\circeq P^{\\\\prime }$ .\",\n \"This theorem states that when an action performed in a well typed process and the action is well typed, which is guaranteed by the previous theorem, then the resulting process after the $\\\\tau $ -action is also well typed in an updated type environment.This result gives us the property that if a process is well typed, then it will never experience communication errors.\",\n \"The theorem also tells us that processes evolve according to the types prescription.\"\n ],\n [\n \"Type equivalence\",\n \"In this section we discuss type equivalence in our type system and show how this leads to a new session type system.\"\n ],\n [\n \"Why type equivalence matters\",\n \"First we motivate type equivalence by expressing an example from [9], in our applied pi-calculus and our type system.\",\n \"Recall the example from Section REF where the goal was to transmit a binary tree while preserving the tree structure.\",\n \"Example 1 Consider the parameterised agent $S$ below.\",\n \"The parameter $t$ is the tree to be transmitted, $c$ is the channel the tree is sent on, and $w$ is a channel used for initiating the transition of the right sub-tree after the transmission of the left sub-tree has finished.\",\n \"The projection functions fst, snd and thrd are used to access the elements of a tuple.\",\n \"$S(t, c, w) \\\\eqdef$ if $t=\\\\textsf{Leaf}$ then $c\\\\< \\\\textsf{Leaf}.\\\\o{w}\\\\l c \\\\r.\\\\nil$ else $c\\\\<\\\\textsf{Node}.$ $\\\\o{c}\\\\l\\\\textsf{fst(t)}\\\\r.$ $(\\\\nu n)(S(\\\\textsf{snd}(t), c, n)|n(x).S(\\\\textsf{thrd}(t), c, w))$ If we analyse the $S$ process, the endpoint type of $c$ is $T_C$ , which is the same type as was described in Section REF .\",\n \"The type describes how a single node is transmitted, if it is a leaf node we do not do anything, if it is internal node then the value is transmitted and then the sub-trees of the node.\",\n \"$T_C = \\\\mu z.\\\\oplus\\\\{$ $\\\\textsf{Leaf}: \\\\lin \\\\textsf{skip} $ $\\\\textsf{Node}: \\\\lin \\\\textsf{!Int};z;z$ $\\\\}$ The process below receives the transmitted tree preserving the original structure.\",\n \"Through similar analysis as on the sending end, we can confirm that the endpoint type of $c$ in this process is $\\\\overline{T_C}$ .\",\n \"$R(c, w) \\\\eqdef c\\\\>\\\\{$ $\\\\textsf{Leaf}: \\\\o{w}\\\\l c\\\\r.\\\\nil$ $\\\\textsf{Node}: (\\\\nu n)(c(x).R(c, n) | n(x).R(c, w))$ $\\\\}$ We can now create a process $P$ that transmits a tree on the $c$ channel.\",\n \"$P=S(\\\\textsf {Tree}(1, \\\\textsf {Leaf}, \\\\textsf {Leaf}), c, w) \\\\text{ ǀ\\\\ } R(c, w)$ With an addition to $P$ , we can create a process $P^{\\\\prime }$ that reuses $c$ for transmitting an integer after transmitting the tree.\",\n \"$P^{\\\\prime }=\\\\left[S(\\\\textsf {Tree}(1, \\\\textsf {Leaf}, \\\\textsf {Leaf}), c, w) \\\\text{ ǀ\\\\ } R(c, v)\\\\right]\\\\text{ ǀ\\\\ }\\\\left[v(c^{\\\\prime }).c^{\\\\prime }(x).\\\\mathbf {0}\\\\text{ ǀ\\\\ }w(c^{\\\\prime \\\\prime }).\\\\overline{c^{\\\\prime \\\\prime }}\\\\langle 1\\\\rangle .\\\\mathbf {0}\\\\right]$ The type of $c$ after expanding the recursive type is now $(T_C^{\\\\prime }, \\\\overline{T_C^{\\\\prime }})$ where $T_C' = \\\\lin \\\\oplus\\\\{$ $\\\\textsf{Leaf}: \\\\lin \\\\textsf{skip} $ $\\\\textsf{Node}: \\\\lin \\\\textsf{!Int};$ $\\\\mu z.\\\\oplus\\\\{\\\\textsf{Leaf}: \\\\lin \\\\textsf{skip}\\\\quad\\\\textsf{Node}:\\\\lin \\\\textsf{!Int};z; z\\\\};$ $\\\\mu z.\\\\oplus\\\\{\\\\textsf{Leaf}:\\\\lin \\\\textsf{skip}\\\\quad\\\\textsf{Node}:\\\\lin \\\\textsf{!Int};z;z\\\\}$ $\\\\};\\\\lin !\\\\textsf{Int}$ The last step is what motivates type equivalence; we would like to have a distributive law that allows the output to be moved into the select type, as illustrated below.\",\n \"$T_C'' = \\\\lin \\\\oplus\\\\{$ $\\\\textsf{Leaf}: \\\\lin \\\\textsf{skip};\\\\lin !\\\\textsf{Int} $ $\\\\textsf{Node}: \\\\lin \\\\textsf{!Int};$ $\\\\mu z.\\\\oplus\\\\{\\\\textsf{Leaf}: \\\\lin \\\\textsf{skip}\\\\quad\\\\textsf{Node}:\\\\lin \\\\textsf{!Int};z;z\\\\};$ $\\\\mu z.\\\\oplus\\\\{\\\\textsf{Leaf}: \\\\lin \\\\textsf{skip}\\\\quad\\\\textsf{Node}:\\\\lin \\\\textsf{!Int};z;z\\\\};$ $\\\\lin !\\\\textsf{Int}$ $\\\\}$ As the typing rules in our applied pi-calculus are defined on the type transitions instead of the structure of a type, the distributive law is not essential in our calculus, since we require that the type exhibit specific behaviour instead of having a specific structure, but the example shows how type equivalence can be used to tell if two types can be used interchangeably.\",\n \"In the next section we will introduce a method for checking if two types are equivalent, and we will return to Example REF , to check that it is indeed the case that $T_C^{\\\\prime \\\\prime }$ is equivalent to $T_C^{\\\\prime }$ .\"\n ],\n [\n \"Type bisimilarity\",\n \"As described in Section REF , our types are highly reminiscent of BPA.\",\n \"For BPA expressions, bisimulation is used to prove that two processes exhibit the same behaviour.\",\n \"We now extend bisimulation to work on types as well.\",\n \"The definition follows from the definition of bisimulation in [2].\",\n \"Definition 1 (Type bisimulation) A binary relation $\\\\mathcal {R}$ between endpoint types is a type bisimulation iff whenever $T_{E_1} \\\\mathcal {R}\\\\ T_{E_2}$ : $Q(T_{E_1})=Q(T_{E_2})$ $\\\\forall \\\\lambda $ if $T_{E_1} \\\\xrightarrow{} T_{E_1}^{\\\\prime } $ then $\\\\exists T_{E_2}^{\\\\prime }$ such that $ T_{E_2} \\\\xrightarrow{} T_{E_2}^{\\\\prime }$ and $T_{E_1}^{\\\\prime } \\\\mathcal {R}\\\\ T_{E_2}^{\\\\prime }$ $\\\\forall \\\\lambda $ if $T_{E_2} \\\\xrightarrow{} T_{E_2}^{\\\\prime } $ then $\\\\exists T_{E_1}^{\\\\prime }$ such that $ T_{E_1} \\\\xrightarrow{} T_{E_1}^{\\\\prime }$ and $T_{E_1}^{\\\\prime } \\\\mathcal {R}\\\\ T_{E_2}^{\\\\prime }$ We write that $T_{E_1}\\\\sim T_{E_2}$ if $T_{E_1} \\\\mathcal {R} T_{E_2}$ for some type bisimulation and then say that $T_{E_1}$ and $T_{E_2}$ are type bisimilar.\",\n \"Type bisimilar endpoint types will exhibit the same behaviour.\",\n \"In other words, for a type $T_{E}$ , any type $T_E^{\\\\prime }$ where $T_E\\\\sim T_E^{\\\\prime }$ , $T_E^{\\\\prime }$ can be used instead of $T_E$ , without introducing communication errors.\",\n \"Example 2 (A distributive law) Consider the types $T_C^{\\\\prime }$ and $T_C^{\\\\prime \\\\prime }$ from Example REF .\",\n \"We can use type bisimulation to show that these two types describe the same communication behaviour on a channel.\",\n \"To do so, we must provide a bisimulation that shows that $T_C^{\\\\prime }$ and $T_C^{\\\\prime \\\\prime }$ are type bisimilar.\",\n \"Let $R$ be a relation over endpoint types.\",\n \"Let $R$ be the symmetric closure of $\\\\lbrace (T_C^{\\\\prime },T_C^{\\\\prime \\\\prime }), ((\\\\textsf {lin }!\\\\textsf {int};r;r);\\\\textsf {lin }!\\\\textsf {int},\\\\linebreak [0]\\\\textsf {lin }!\\\\textsf {int};r;r;\\\\textsf {lin }!\\\\textsf {int})\\\\rbrace \\\\cup \\\\lbrace (T_E,T_E) \\\\text{ ǀ\\\\ } \\\\forall T_E\\\\in \\\\mathcal {T}\\\\rbrace $ where $r$ is the term $\\\\mu z.\\\\oplus \\\\lbrace \\\\textsf {Leaf}: \\\\textsf {lin }\\\\textsf {skip}\\\\quad \\\\textsf {Node}:\\\\textsf {lin }\\\\textsf {!Int};z;z\\\\rbrace $ .\",\n \"In fact, we can generalise this result and prove the distributive law for both select and branch.\",\n \"Lemma 2 Let $\\\\star $ be $\\\\oplus $ or $\\\\&$ .\",\n \"Then $q\\\\ \\\\star \\\\lbrace l_i: T_{E_i}\\\\rbrace _{i\\\\in I};T_E\\\\sim q\\\\ \\\\star \\\\lbrace l_i: T_{E_i};T_E\\\\rbrace _{i\\\\in I}$ Let $R$ be a relation between types.\",\n \"We must show that $R$ is a bisimulation of $q\\\\ \\\\star \\\\lbrace l_i: T_{E_i}\\\\rbrace _{i\\\\in I};T_E$ and $q\\\\ \\\\star \\\\lbrace l_i: T_{E_i};T_E\\\\rbrace _{i\\\\in I}$ .\",\n \"Define $R$ as the symmetric closure of $\\\\lbrace (q\\\\ \\\\star \\\\lbrace l_i: T_{E_i}\\\\rbrace _{i\\\\in I};T_E,q\\\\ \\\\star \\\\lbrace l_i: T_{E_i};T_E\\\\rbrace _{i\\\\in I})\\\\rbrace \\\\cup \\\\lbrace (T_E,T_E) \\\\text{ ǀ\\\\ } \\\\forall T_E\\\\in \\\\mathcal {T}\\\\rbrace $ From the first requirement for a type bisimulation we have that $Q(q\\\\ \\\\star \\\\lbrace l_i: T_{E_i}\\\\rbrace _{i\\\\in I};T_E) = Q(q\\\\ \\\\star \\\\lbrace l_i: T_{E_i};T_E\\\\rbrace _{i\\\\in I}))$ , this is trivially fulfilled as $q=q$ .\",\n \"By the (Seq) rule we have that the transitions of a sequential compositions are those of the left-hand side of the operator.\",\n \"So the transitions of $q\\\\ \\\\star \\\\lbrace l_i: T_{E_i}\\\\rbrace _{i\\\\in I};T_E$ are $q\\\\ \\\\star \\\\lbrace l_i: T_{E_i}\\\\rbrace _{i\\\\in I};T_E\\\\xrightarrow{} T_{E_i};T_E$ , where $\\\\diamond \\\\in \\\\lbrace \\\\lhd ,\\\\rhd \\\\rbrace $ .\",\n \"The available transitions for $q\\\\ \\\\star \\\\lbrace l_i: T_{E_i};T_E\\\\rbrace _{i\\\\in I}$ are $q\\\\ \\\\star \\\\lbrace l_i: T_{E_i};T_E\\\\rbrace _{i\\\\in I}\\\\xrightarrow{} T_{E_i};T_E$ .\",\n \"Since $R$ is reflexive, we have that $(T_{E_i};T_E, T_{E_i};T_E)\\\\in R$ .\",\n \"So any transition taken by one of the types can be matched by the other to end up with syntactically equivalent types.\",\n \"Since the types are syntactically equivalent, all further transitions can be matched by any of the two types, and all further type pairs will be in $R$ .\",\n \"This means that $R$ is a type bisimulation, and that $q\\\\ \\\\star \\\\lbrace l_i: T_{E_i}\\\\rbrace _{i\\\\in I};T_E \\\\sim q\\\\ \\\\star \\\\lbrace l_i: T_{E_i};T_E\\\\rbrace _{i\\\\in I}$ .\"\n ],\n [\n \"A quotiented session type system\",\n \"We now define a quotiented session type system whose types are equivalence classes of session types from the the already existing type system.\",\n \"We define transitions between equivalence classes instead of endpoint types as follows.\",\n \"Definition 2 (Equivalence Classes) Let $\\\\vert T_E \\\\vert $ be the equivalence class of $T_E$ given by: $\\\\vert T_E \\\\vert = \\\\lbrace T_E^{\\\\prime }\\\\in \\\\mathcal {T} \\\\text{ ǀ\\\\ } T_E^{\\\\prime } \\\\sim T_E\\\\rbrace $ We denote a transition between equivalence classes with action $\\\\lambda $ as $\\\\vert T_{E_1}\\\\vert \\\\xrightarrow{} \\\\vert T_{E_2}\\\\vert $ .\",\n \"Lemma 3 $\\\\vert T_E \\\\vert \\\\xrightarrow{} \\\\vert T_E^{\\\\prime } \\\\vert $ iff $\\\\forall T_{E_1} \\\\in \\\\vert T_E \\\\vert \\\\ \\\\exists T_{E_1}^{\\\\prime } \\\\in \\\\vert T_E^{\\\\prime } \\\\vert $ such that $T_{E_1} \\\\xrightarrow{} T_{E_1}^{\\\\prime }$ By the properties of type bisimilarity that states that two bisimilar types will evolve to bisimilar types for all possible transitions.\",\n \"We use Lemma REF to define a new type system that is defined on equivalence classes of types from the previous type system.\",\n \"In Section REF the transitions of endpoint types were of the form $T_E\\\\xrightarrow{}T_E^{\\\\prime }$ .\",\n \"In the new type system the transitions are of the form $\\\\vert T_E \\\\vert \\\\xrightarrow{}\\\\vert T_E^{\\\\prime }\\\\vert $ .\",\n \"The transition rules from Table REF still apply to the new type system, but where all endpoint types $T_E$ have been replaced with $\\\\vert T_E \\\\vert $ .\",\n \"For example, the (Select) rule from Table REF would in the new type system be (REF ).\",\n \"We also expand the $Q$ function on equivalence classes to be the result of applying $Q$ to any witness of the equivalence class.\",\n \"$\\\\textsc {(Select)}\\\\qquad {q \\\\sqsubseteq Q(\\\\vert T_{E_k} \\\\vert )}{\\\\vert q\\\\ \\\\oplus \\\\lbrace l_i : \\\\vert T_{E_i} \\\\vert \\\\rbrace _{i\\\\in I} \\\\vert \\\\xrightarrow{} \\\\vert T_{E_k}\\\\vert }$ The typing rules in the new type system would also be the rules from Tables REF , REF and REF , where endpoint types $T_E$ have been replaced with equivalence classes $\\\\vert T_E \\\\vert $ .\",\n \"In REF the (Input) rule from Table REF has been changed to fit the new type system.\",\n \"$\\\\textsc {(Input)} \\\\qquad {\\\\Gamma _1 \\\\vdash n:\\\\vert T_E \\\\vert & \\\\vert T_E \\\\vert \\\\xrightarrow{} \\\\vert T_E^{\\\\prime } \\\\vert & \\\\Gamma _2, x:T + n: \\\\vert T_E^{\\\\prime } \\\\vert \\\\vdash P}{\\\\Gamma _1 \\\\circ \\\\Gamma _2 \\\\vdash n(x).P }$ The generic type system from [7] depends on the transitions available to types.\",\n \"We have already shown that the previous type system is an instance of generic type systems.\",\n \"From Lemma REF , we can see that equivalence classes have the exact same transitions as the old endpoint types had.\",\n \"From these two results we can see that the type system defined on equivalence classes is an instance of the generic type system as well, allowing us to retain previously obtained results of fidelity and well typed internal actions from Section .\",\n \"In conclusion, this gives us a type system where type equivalence is a trivial property, since two endpoint types of the old type system would be the same type in the new type system.\"\n ],\n [\n \"Conclusion\",\n \"In this paper we have considered a “low-level” applied pi-calculus that allows composite terms to be built but only allows for passing names and nullary function symbols.\",\n \"In this setting, we introduced a type system for the applied pi-calculus based on the work on context-free session types by Thiemann and Vasconcelos in [9] and on the work on qualified session types by Vasconcelos in [10].\",\n \"The type system is a context-free session type system with qualifiers.\",\n \"The type system is an instance of the psi-calculus type system introduced in [7], and this allows us to establish a fidelity result about the type system that ensures that a well typed process continues to be well typed, and without communication errors, until termination.\",\n \"The type system has a notion of type equivalence defined by introducing type bisimilarity.\",\n \"Using it, we then get a type system of equivalence classes, for which type equivalence is a natural part of the type system itself.\",\n \"The current focus is to deal with the decidability of type equivalence.\",\n \"In [9] Thiemann and Vasconcelos show the decidability of type equivalence using a transformation from their types to guarded BPA expressions, for which bisimilarity is decidable.\",\n \"As already discussed in this article, our types are very close to BPA, in which case we should be able to achieve the same results about decidability more directly.\",\n \"In [5] an algorithm is presented for deciding bisimilarity for normed context free processes in polynomial time, and we conjecture that this algorithm can be adapted to our setting for checking type bisimilarity.\",\n \"Here, it would be important to find a characterization of the class of applied pi processes that can be typed using normed session types only.\"\n ]\n]"},"id":{"kind":"string","value":"1808.08648"}}},{"rowIdx":290,"cells":{"text":{"kind":"list like","value":[["IIIDYT at IEST 2018: Implicit Emotion Classification With Deep\n Contextualized Word Representations"],["Abstract In this paper we describe our system designed for the WASSA 2018 Implicit Emotion Shared Task (IEST), which obtained 2$^{\\text{nd}}$ place out of 26 teams with a test macro F1 score of $0.710$.","The system is composed of a single pre-trained ELMo layer for encoding words, a Bidirectional Long-Short Memory Network BiLSTM for enriching word representations with context, a max-pooling operation for creating sentence representations from said word vectors, and a Dense Layer for projecting the sentence representations into label space.","Our official submission was obtained by ensembling 6 of these models initialized with different random seeds.","The code for replicating this paper is available at https://github.com/jabalazs/implicit_emotion."],["Introduction","Although the definition of emotion is still debated among the scientific community, the automatic identification and understanding of human emotions by machines has long been of interest in computer science.","It has usually been assumed that emotions are triggered by the interpretation of a stimulus event according to its meaning.","As language usually reflects the emotional state of an individual, it is natural to study human emotions by understanding how they are reflected in text.","We see that many words indeed have affect as a core part of their meaning, for example, dejected and wistful denote some amount of sadness, and are thus associated with sadness.","On the other hand, some words are associated with affect even though they do not denote affect.","For example, failure and death describe concepts that are usually accompanied by sadness and thus they denote some amount of sadness.","In this context, the task of automatically recognizing emotions from text has recently attracted the attention of researchers in Natural Language Processing.","This task is usually formalized as the classification of words, phrases, or documents into predefined discrete emotion categories or dimensions.","Some approaches have aimed at also predicting the degree to which an emotion is expressed in text [11].","In light of this, the WASSA 2018 Implicit Emotion Shared Task (IEST) [10] was proposed to help find ways to automatically learn the link between situations and the emotion they trigger.","The task consisted in predicting the emotion of a word excluded from a tweet.","Removed words, or trigger-words, included the terms “sad”, “happy”, “disgusted”, “surprised”, “angry”, “afraid” and their synonyms, and the task was to predict the emotion they conveyed, specifically sadness, joy, disgust, surprise, anger and fear.","From a machine learning perspective, this problem can be seen as sentence classification, in which the goal is to classify a sentence, or in particular a tweet, into one of several categories.","In the case of IEST, the problem is specially challenging since tweets contain informal language, the heavy usage of emoji, hashtags and username mentions.","In this paper we describe our system designed for IEST, which obtained the second place out of 30 teams.","Our system did not require manual feature engineering and only minimal use of external data.","Concretely, our approach is composed of a single pre-trained ELMo layer for encoding words [13], a Bidirectional Long-Short Memory Network (BiLSTM) [7], [6], for enriching word representations with context, a max-pooling operation for creating sentence representations from said word vectors, and finally a Dense Layer for projecting the sentence representations into label space.","To the best of our knowledge, our system, which we plan to release, is the first to utilize ELMo for emotion recognition.","Figure: Proposed architecture."],["Preprocessing","As our model is purely character-based, we performed little data preprocessing.","Table REF shows the special tokens found in the datasets, and how we substituted them.","Table: Preprocessing substitutions.Furthermore, we tokenized the text using a variation of the twokenize.pyhttps://github.com/myleott/ark-twokenize-py script, a Python port of the original Twokenize.java [5].","Concretely, we created an emoji-aware version of it by incorporating knowledge from an emoji database,https://github.com/carpedm20/emoji/blob/e7bff32/emoji/unicode_codes.py which we slightly modified for avoiding conflict with emoji sharing unicode codes with common glyphs used in Twitter,For example, the hashtag emoji is composed by the unicode code points U+23 U+FE0F U+20E3, which include U+23, the same code point for the # glyph.","and for making it compatible with Python 3."],["Architecture","Figure REF summarizes our proposed architecture.","Our input is based on Embeddings from Language Models (ELMo) by [13].","These are character-based word representations allowing the model to avoid the “unknown token” problem.","ELMo uses a set of convolutional neural networks to extract features from character embeddings, and builds word vectors from them.","These are then fed to a multi-layer Bidirectional Language Model (BiLM) which returns context-sensitive vectors for each input word.","We used a single-layer BiLSTM as context fine-tuner [7], [6], on top of the ELMo embeddings, and then aggregated the hidden states it returned by using max-pooling, which has been shown to perform well on sentence classification tasks [3].","Finally, we used a single-layer fully-connected network for projecting the pooled BiLSTM output into a vector corresponding to the label logits for each predicted class."],["Implementation Details and Hyperparameters"," ELMo Layer: We used the official AllenNLP implementation of the ELMo modelhttps://allenai.github.io/allennlp-docs/api/allennlp.modules.elmo.html, with the official weights pre-trained on the 1 Billion Word Language Model Benchmark, which contains about 800M tokens of news crawl data from WMT 2011 [2].","Dimensionalities: By default the ELMo layer outputs a 1024-dimensional vector, which we then feed to a BiLSTM with output size 2048, resulting in a 4096-dimensional vector when concatenating forward and backward directions for each word of the sequenceA BiLSTM is composed of two separate LSTMs that read the input in opposite directions and whose outputs are concatenated at the hidden dimension.","This results in a vector double the dimension of the input for each time step.. After max-pooling the BiLSTM output over the sequence dimension, we obtain a single 4096-dimensional vector corresponding to the tweet representation.","This representation is finally fed to a single-layer fully-connected network with input size 4096, 512 hidden units, output size 6, and a ReLU nonlinearity after the hidden layer.","The output of the dense layer is a 6-dimensional logit vector for each input example.","Loss Function: As this corresponds to a multiclass classification problem (predicting a single class for each example, with more than 2 classes to choose from), we used the Cross-Entropy Loss as implemented in PyTorch [12].","Optimization: We optimized the model with Adam [9], using default hyperparameters ($\\beta _1=0.9$ , $\\beta _2=0.999$ , $\\epsilon =10^{-8}$ ), following a slanted triangular learning rate schedule [8], also with default hyperparameters ($cut\\_frac=0.1$ , $ratio=32$ ), and a maximum learning rate $\\eta _{max}=0.001$ , over $T=23,970$ iterationsThis number is obtained by multiplying the number of epochs (10), times the total number of batches, which for the training dataset corresponds to 2396 batches of 64 elements, and 1 batch of 39 elements, hence $2397\\times 10=23,970$ .. Regularization: we used a dropout layer [14], with probability of 0.5 after both the ELMo and the hidden fully-connected layer, and another one with probability of 0.1 after the max-pooling aggregation layer.","We also reshuffled the training examples between epochs, resulting in a different batch for each iteration.","Model Selection: To choose the best hyperparameter configuration we measured the classification accuracy on the validation (trial) set."],["Ensembles","Once we found the best-performing configuration we trained 10 models using different random seeds, and tried averaging the output class probabilities of all their possible $\\sum _{k=1}^{9}{\\binom{9}{k}}=511$ combinations.","As Figure REF shows, we empirically found that a specific combination of 6 models yielded the best results ($70.52\\%$ ), providing evidence for the fact that using a number of independent classifiers equal to the number of class labels provides the best results when doing average ensembling [1].","Figure: Effect of the number of ensembled models on validation performance."],["Experiments and Analyses","We performed several experiments to gain insights on how the proposed model's performance interacts with the shared task's data.","We performed an ablation study to see how some of the main hyperparameters affect performance, and an analysis of tweets containing hashtags and emoji to understand how these two types of tokens help the model predict the trigger-word's emotion.","We also observed the effects of varying the amount of data used for training the model to evaluate whether it would be worthwhile to gather more training data."],["Ablation Study","We performed an ablation study on a single model having obtained 69.23% accuracy on the validation set.","Results are summarized in Table REF .","Table: *We can observe that the architectural choice that had the greatest impact on our model was the ELMo layer, providing a $3.71\\%$ boost in performance as compared to using GloVe pre-trained word embeddings.","We can further see that emoji also contributed significantly to the model's performance.","In Section REF we give some pointers to understanding why this is so.","Additionally, we tried using the concatenation of the max-pooled, average-pooled and last hidden states of the BiLSTM as the sentence representation, following [8], but found out that this impacted performance negatively.","We hypothesize this is due to tweets being too short for needing such a rich representation.","Also, the size of the concatenated vector was $4096\\times 3=12,288$ , which probably could not be properly exploited by the 512-dimensional fully-connected layer.","Using a greater BiLSTM hidden size did not help the model, probably because of the reason mentioned earlier; the fully-connected layer was not big or deep enough to exploit the additional information.","Similarly, using a smaller hidden size neither helped.","We found that using 50-dimensional part-of-speech embeddings slightly improved results, which implies that better fine-tuning this hyperparameter, or using a better POS tagger could yield an even better performance.","Regarding optimization strategies, we also tried using SGD with different learning rates and a step-wise learning rate schedule as described by [4], but we found that doing this did not improve performance.","Finally, Figure REF shows the effect of using different dropout probabilities.","We can see that having higher dropout after the word-representation layer and the fully-connected network's hidden layer, while having a low dropout after the sentence encoding layer yielded better results overall.","Figure: *"],["Error Analysis","Figure REF shows the confusion matrix of a single model evaluated on the test set, and Table REF the corresponding classification report.","In general, we confirm what klinger2018iest report: anger was the most difficult class to predict, followed by surprise, whereas joy, fear, and disgust are the better performing ones.","To observe whether any particular pattern arose from the sentence representations encoded by our model, we projected them into 3d space through Principal Component Analysis (PCA), and were surprised to find that 2 clearly defined clusters emerged (see Figure REF ), one containing the majority of datapoints, and another containing joy tweets exclusively.","Upon further exploration we also found that the smaller cluster was composed only by tweets containing the pattern un __TRIGGERWORD__, and further, that all of them were correctly classified.","It is also worth mentioning that there are 5827 tweets in the training set with this pattern.","Of these, 5822 (99.9%) correspond to the label joy.","We observe a similar trend on the test set; 1115 of the 1116 tweets having the un __TRIGGERWORD__ pattern correspond to joy tweets.","We hypothesize this is the reason why the model learned this pattern as a strong discriminating feature.","Finally, the only tweet in the test set that contained this pattern and did not belong to the joy class, originally had unsurprised as its triggerwordWe manually searched for the original tweet., and unsurprisingly, was misclassified.","Figure: Confusion Matrix (Test Set).Table: Classification Report (Test Set)."],["Effect of the Amount of Training Data","As Figure REF shows, increasing the amount of data with which our model was trained consistently increased validation accuracy and validation macro F1 score.","The trend suggests that the proposed model is expressive enough to learn from more data, and is not overfitting the training set.","Figure: Effect of the amount of training data on classification performance."],["Effect of Emoji and Hashtags","Table REF shows the overall effect of hashtags and emoji on classification performance.","Tweets containing emoji seem to be easier for the model to classify than those without.","Hashtags also have a positive effect on classification performance, however it is less significant.","This implies that emoji, and hashtags in a smaller degree, provide tweets with a context richer in sentiment information, allowing the model to better guess the emotion of the trigger-word.","Table: *Table REF shows the effect specific emoji have on classification performance.","It is clear some emoji strongly contribute to improving prediction quality.","The most interesting ones are mask, rage, and cry, which significantly increase accuracy.","Further, contrary to intuition, the sob emoji contributes less than cry, despite representing a stronger emotion.","This is probably due to sob being used for depicting a wider spectrum of emotions.","Finally, not all emoji are beneficial for this task.","When removing sweat_smile and confused accuracy increased, probably because they represent emotions other than the ones being predicted.","Figure: 3d Projection of the Test Sentence Representations."],["Conclusions and Future Work","We described the model that got second place in the WASSA 2018 Implicit Emotion Shared Task.","Despite its simplicity, and low amount of dependencies on libraries and external features, it performed almost as well as the system that obtained the first place.","Our ablation study revealed that our hyperparameters were indeed quite well-tuned for the task, which agrees with the good results obtained in the official submission.","However, the ablation study also showed that increased performance can be obtained by incorporating POS embeddings as additional inputs.","Further experiments are required to accurately measure the impact that this additional input may have on the results.","We also think the performance can be boosted by making the architecture more complex, concretely, by using a BiLSTM with multiple layers and skip connections in a way akin to [13], or by making the fully-connected network bigger and deeper.","We also showed that, what was probably an annotation artifact, the un __TRIGGERWORD__ pattern, resulted in increased performance for the joy label.","This pattern was probably originated by a heuristic naïvely replacing the ocurrence of happy by the trigger-word indicator.","We think the dataset could be improved by replacing the word unhappy, in the original examples, by __TRIGGERWORD__ instead of un __TRIGGERWORD__, and labeling it as sad, or angry, instead of joy.","Finally, our studies regarding the importance of hashtags and emoji in the classification showed that both of them seem to contribute significantly to the performance, although in different measures."],["Acknowledgements","We thank the anonymous reviewers for their reviews and suggestions.","The first author is partially supported by the Japanese Government MEXT Scholarship."]],"string":"[\n [\n \"IIIDYT at IEST 2018: Implicit Emotion Classification With Deep\\n Contextualized Word Representations\"\n ],\n [\n \"Abstract In this paper we describe our system designed for the WASSA 2018 Implicit Emotion Shared Task (IEST), which obtained 2$^{\\\\text{nd}}$ place out of 26 teams with a test macro F1 score of $0.710$.\",\n \"The system is composed of a single pre-trained ELMo layer for encoding words, a Bidirectional Long-Short Memory Network BiLSTM for enriching word representations with context, a max-pooling operation for creating sentence representations from said word vectors, and a Dense Layer for projecting the sentence representations into label space.\",\n \"Our official submission was obtained by ensembling 6 of these models initialized with different random seeds.\",\n \"The code for replicating this paper is available at https://github.com/jabalazs/implicit_emotion.\"\n ],\n [\n \"Introduction\",\n \"Although the definition of emotion is still debated among the scientific community, the automatic identification and understanding of human emotions by machines has long been of interest in computer science.\",\n \"It has usually been assumed that emotions are triggered by the interpretation of a stimulus event according to its meaning.\",\n \"As language usually reflects the emotional state of an individual, it is natural to study human emotions by understanding how they are reflected in text.\",\n \"We see that many words indeed have affect as a core part of their meaning, for example, dejected and wistful denote some amount of sadness, and are thus associated with sadness.\",\n \"On the other hand, some words are associated with affect even though they do not denote affect.\",\n \"For example, failure and death describe concepts that are usually accompanied by sadness and thus they denote some amount of sadness.\",\n \"In this context, the task of automatically recognizing emotions from text has recently attracted the attention of researchers in Natural Language Processing.\",\n \"This task is usually formalized as the classification of words, phrases, or documents into predefined discrete emotion categories or dimensions.\",\n \"Some approaches have aimed at also predicting the degree to which an emotion is expressed in text [11].\",\n \"In light of this, the WASSA 2018 Implicit Emotion Shared Task (IEST) [10] was proposed to help find ways to automatically learn the link between situations and the emotion they trigger.\",\n \"The task consisted in predicting the emotion of a word excluded from a tweet.\",\n \"Removed words, or trigger-words, included the terms “sad”, “happy”, “disgusted”, “surprised”, “angry”, “afraid” and their synonyms, and the task was to predict the emotion they conveyed, specifically sadness, joy, disgust, surprise, anger and fear.\",\n \"From a machine learning perspective, this problem can be seen as sentence classification, in which the goal is to classify a sentence, or in particular a tweet, into one of several categories.\",\n \"In the case of IEST, the problem is specially challenging since tweets contain informal language, the heavy usage of emoji, hashtags and username mentions.\",\n \"In this paper we describe our system designed for IEST, which obtained the second place out of 30 teams.\",\n \"Our system did not require manual feature engineering and only minimal use of external data.\",\n \"Concretely, our approach is composed of a single pre-trained ELMo layer for encoding words [13], a Bidirectional Long-Short Memory Network (BiLSTM) [7], [6], for enriching word representations with context, a max-pooling operation for creating sentence representations from said word vectors, and finally a Dense Layer for projecting the sentence representations into label space.\",\n \"To the best of our knowledge, our system, which we plan to release, is the first to utilize ELMo for emotion recognition.\",\n \"Figure: Proposed architecture.\"\n ],\n [\n \"Preprocessing\",\n \"As our model is purely character-based, we performed little data preprocessing.\",\n \"Table REF shows the special tokens found in the datasets, and how we substituted them.\",\n \"Table: Preprocessing substitutions.Furthermore, we tokenized the text using a variation of the twokenize.pyhttps://github.com/myleott/ark-twokenize-py script, a Python port of the original Twokenize.java [5].\",\n \"Concretely, we created an emoji-aware version of it by incorporating knowledge from an emoji database,https://github.com/carpedm20/emoji/blob/e7bff32/emoji/unicode_codes.py which we slightly modified for avoiding conflict with emoji sharing unicode codes with common glyphs used in Twitter,For example, the hashtag emoji is composed by the unicode code points U+23 U+FE0F U+20E3, which include U+23, the same code point for the # glyph.\",\n \"and for making it compatible with Python 3.\"\n ],\n [\n \"Architecture\",\n \"Figure REF summarizes our proposed architecture.\",\n \"Our input is based on Embeddings from Language Models (ELMo) by [13].\",\n \"These are character-based word representations allowing the model to avoid the “unknown token” problem.\",\n \"ELMo uses a set of convolutional neural networks to extract features from character embeddings, and builds word vectors from them.\",\n \"These are then fed to a multi-layer Bidirectional Language Model (BiLM) which returns context-sensitive vectors for each input word.\",\n \"We used a single-layer BiLSTM as context fine-tuner [7], [6], on top of the ELMo embeddings, and then aggregated the hidden states it returned by using max-pooling, which has been shown to perform well on sentence classification tasks [3].\",\n \"Finally, we used a single-layer fully-connected network for projecting the pooled BiLSTM output into a vector corresponding to the label logits for each predicted class.\"\n ],\n [\n \"Implementation Details and Hyperparameters\",\n \" ELMo Layer: We used the official AllenNLP implementation of the ELMo modelhttps://allenai.github.io/allennlp-docs/api/allennlp.modules.elmo.html, with the official weights pre-trained on the 1 Billion Word Language Model Benchmark, which contains about 800M tokens of news crawl data from WMT 2011 [2].\",\n \"Dimensionalities: By default the ELMo layer outputs a 1024-dimensional vector, which we then feed to a BiLSTM with output size 2048, resulting in a 4096-dimensional vector when concatenating forward and backward directions for each word of the sequenceA BiLSTM is composed of two separate LSTMs that read the input in opposite directions and whose outputs are concatenated at the hidden dimension.\",\n \"This results in a vector double the dimension of the input for each time step.. After max-pooling the BiLSTM output over the sequence dimension, we obtain a single 4096-dimensional vector corresponding to the tweet representation.\",\n \"This representation is finally fed to a single-layer fully-connected network with input size 4096, 512 hidden units, output size 6, and a ReLU nonlinearity after the hidden layer.\",\n \"The output of the dense layer is a 6-dimensional logit vector for each input example.\",\n \"Loss Function: As this corresponds to a multiclass classification problem (predicting a single class for each example, with more than 2 classes to choose from), we used the Cross-Entropy Loss as implemented in PyTorch [12].\",\n \"Optimization: We optimized the model with Adam [9], using default hyperparameters ($\\\\beta _1=0.9$ , $\\\\beta _2=0.999$ , $\\\\epsilon =10^{-8}$ ), following a slanted triangular learning rate schedule [8], also with default hyperparameters ($cut\\\\_frac=0.1$ , $ratio=32$ ), and a maximum learning rate $\\\\eta _{max}=0.001$ , over $T=23,970$ iterationsThis number is obtained by multiplying the number of epochs (10), times the total number of batches, which for the training dataset corresponds to 2396 batches of 64 elements, and 1 batch of 39 elements, hence $2397\\\\times 10=23,970$ .. Regularization: we used a dropout layer [14], with probability of 0.5 after both the ELMo and the hidden fully-connected layer, and another one with probability of 0.1 after the max-pooling aggregation layer.\",\n \"We also reshuffled the training examples between epochs, resulting in a different batch for each iteration.\",\n \"Model Selection: To choose the best hyperparameter configuration we measured the classification accuracy on the validation (trial) set.\"\n ],\n [\n \"Ensembles\",\n \"Once we found the best-performing configuration we trained 10 models using different random seeds, and tried averaging the output class probabilities of all their possible $\\\\sum _{k=1}^{9}{\\\\binom{9}{k}}=511$ combinations.\",\n \"As Figure REF shows, we empirically found that a specific combination of 6 models yielded the best results ($70.52\\\\%$ ), providing evidence for the fact that using a number of independent classifiers equal to the number of class labels provides the best results when doing average ensembling [1].\",\n \"Figure: Effect of the number of ensembled models on validation performance.\"\n ],\n [\n \"Experiments and Analyses\",\n \"We performed several experiments to gain insights on how the proposed model's performance interacts with the shared task's data.\",\n \"We performed an ablation study to see how some of the main hyperparameters affect performance, and an analysis of tweets containing hashtags and emoji to understand how these two types of tokens help the model predict the trigger-word's emotion.\",\n \"We also observed the effects of varying the amount of data used for training the model to evaluate whether it would be worthwhile to gather more training data.\"\n ],\n [\n \"Ablation Study\",\n \"We performed an ablation study on a single model having obtained 69.23% accuracy on the validation set.\",\n \"Results are summarized in Table REF .\",\n \"Table: *We can observe that the architectural choice that had the greatest impact on our model was the ELMo layer, providing a $3.71\\\\%$ boost in performance as compared to using GloVe pre-trained word embeddings.\",\n \"We can further see that emoji also contributed significantly to the model's performance.\",\n \"In Section REF we give some pointers to understanding why this is so.\",\n \"Additionally, we tried using the concatenation of the max-pooled, average-pooled and last hidden states of the BiLSTM as the sentence representation, following [8], but found out that this impacted performance negatively.\",\n \"We hypothesize this is due to tweets being too short for needing such a rich representation.\",\n \"Also, the size of the concatenated vector was $4096\\\\times 3=12,288$ , which probably could not be properly exploited by the 512-dimensional fully-connected layer.\",\n \"Using a greater BiLSTM hidden size did not help the model, probably because of the reason mentioned earlier; the fully-connected layer was not big or deep enough to exploit the additional information.\",\n \"Similarly, using a smaller hidden size neither helped.\",\n \"We found that using 50-dimensional part-of-speech embeddings slightly improved results, which implies that better fine-tuning this hyperparameter, or using a better POS tagger could yield an even better performance.\",\n \"Regarding optimization strategies, we also tried using SGD with different learning rates and a step-wise learning rate schedule as described by [4], but we found that doing this did not improve performance.\",\n \"Finally, Figure REF shows the effect of using different dropout probabilities.\",\n \"We can see that having higher dropout after the word-representation layer and the fully-connected network's hidden layer, while having a low dropout after the sentence encoding layer yielded better results overall.\",\n \"Figure: *\"\n ],\n [\n \"Error Analysis\",\n \"Figure REF shows the confusion matrix of a single model evaluated on the test set, and Table REF the corresponding classification report.\",\n \"In general, we confirm what klinger2018iest report: anger was the most difficult class to predict, followed by surprise, whereas joy, fear, and disgust are the better performing ones.\",\n \"To observe whether any particular pattern arose from the sentence representations encoded by our model, we projected them into 3d space through Principal Component Analysis (PCA), and were surprised to find that 2 clearly defined clusters emerged (see Figure REF ), one containing the majority of datapoints, and another containing joy tweets exclusively.\",\n \"Upon further exploration we also found that the smaller cluster was composed only by tweets containing the pattern un __TRIGGERWORD__, and further, that all of them were correctly classified.\",\n \"It is also worth mentioning that there are 5827 tweets in the training set with this pattern.\",\n \"Of these, 5822 (99.9%) correspond to the label joy.\",\n \"We observe a similar trend on the test set; 1115 of the 1116 tweets having the un __TRIGGERWORD__ pattern correspond to joy tweets.\",\n \"We hypothesize this is the reason why the model learned this pattern as a strong discriminating feature.\",\n \"Finally, the only tweet in the test set that contained this pattern and did not belong to the joy class, originally had unsurprised as its triggerwordWe manually searched for the original tweet., and unsurprisingly, was misclassified.\",\n \"Figure: Confusion Matrix (Test Set).Table: Classification Report (Test Set).\"\n ],\n [\n \"Effect of the Amount of Training Data\",\n \"As Figure REF shows, increasing the amount of data with which our model was trained consistently increased validation accuracy and validation macro F1 score.\",\n \"The trend suggests that the proposed model is expressive enough to learn from more data, and is not overfitting the training set.\",\n \"Figure: Effect of the amount of training data on classification performance.\"\n ],\n [\n \"Effect of Emoji and Hashtags\",\n \"Table REF shows the overall effect of hashtags and emoji on classification performance.\",\n \"Tweets containing emoji seem to be easier for the model to classify than those without.\",\n \"Hashtags also have a positive effect on classification performance, however it is less significant.\",\n \"This implies that emoji, and hashtags in a smaller degree, provide tweets with a context richer in sentiment information, allowing the model to better guess the emotion of the trigger-word.\",\n \"Table: *Table REF shows the effect specific emoji have on classification performance.\",\n \"It is clear some emoji strongly contribute to improving prediction quality.\",\n \"The most interesting ones are mask, rage, and cry, which significantly increase accuracy.\",\n \"Further, contrary to intuition, the sob emoji contributes less than cry, despite representing a stronger emotion.\",\n \"This is probably due to sob being used for depicting a wider spectrum of emotions.\",\n \"Finally, not all emoji are beneficial for this task.\",\n \"When removing sweat_smile and confused accuracy increased, probably because they represent emotions other than the ones being predicted.\",\n \"Figure: 3d Projection of the Test Sentence Representations.\"\n ],\n [\n \"Conclusions and Future Work\",\n \"We described the model that got second place in the WASSA 2018 Implicit Emotion Shared Task.\",\n \"Despite its simplicity, and low amount of dependencies on libraries and external features, it performed almost as well as the system that obtained the first place.\",\n \"Our ablation study revealed that our hyperparameters were indeed quite well-tuned for the task, which agrees with the good results obtained in the official submission.\",\n \"However, the ablation study also showed that increased performance can be obtained by incorporating POS embeddings as additional inputs.\",\n \"Further experiments are required to accurately measure the impact that this additional input may have on the results.\",\n \"We also think the performance can be boosted by making the architecture more complex, concretely, by using a BiLSTM with multiple layers and skip connections in a way akin to [13], or by making the fully-connected network bigger and deeper.\",\n \"We also showed that, what was probably an annotation artifact, the un __TRIGGERWORD__ pattern, resulted in increased performance for the joy label.\",\n \"This pattern was probably originated by a heuristic naïvely replacing the ocurrence of happy by the trigger-word indicator.\",\n \"We think the dataset could be improved by replacing the word unhappy, in the original examples, by __TRIGGERWORD__ instead of un __TRIGGERWORD__, and labeling it as sad, or angry, instead of joy.\",\n \"Finally, our studies regarding the importance of hashtags and emoji in the classification showed that both of them seem to contribute significantly to the performance, although in different measures.\"\n ],\n [\n \"Acknowledgements\",\n \"We thank the anonymous reviewers for their reviews and suggestions.\",\n \"The first author is partially supported by the Japanese Government MEXT Scholarship.\"\n ]\n]"},"id":{"kind":"string","value":"1808.08672"}}},{"rowIdx":291,"cells":{"text":{"kind":"list like","value":[["On free subgroups in maximal subgroups of skew linear groups"],["Abstract The study of the existence of free groups in skew linear groups have been begun since the last decades of the 20-th century.","The starting point is the theorem of Tits (1972), now often is referred as Tits' Alternative, stating that every finitely generated subgroup of the general linear group $\\GL_n(F)$ over a field $F$ either contains a non-cyclic free subgroup or it is solvable-by-finite.","In this paper, we study the existence of non-cyclic free subgroups in maximal subgroups of an almost subnormal subgroup of the general skew linear group over a locally finite division ring."],["all"]],"string":"[\n [\n \"On free subgroups in maximal subgroups of skew linear groups\"\n ],\n [\n \"Abstract The study of the existence of free groups in skew linear groups have been begun since the last decades of the 20-th century.\",\n \"The starting point is the theorem of Tits (1972), now often is referred as Tits' Alternative, stating that every finitely generated subgroup of the general linear group $\\\\GL_n(F)$ over a field $F$ either contains a non-cyclic free subgroup or it is solvable-by-finite.\",\n \"In this paper, we study the existence of non-cyclic free subgroups in maximal subgroups of an almost subnormal subgroup of the general skew linear group over a locally finite division ring.\"\n ],\n [\n \"all\"\n ]\n]"},"id":{"kind":"string","value":"1808.08453"}}},{"rowIdx":292,"cells":{"text":{"kind":"list like","value":[["Black holes and higher depth mock modular forms"],["Abstract By enforcing invariance under S-duality in type IIB string theory compactified on a Calabi-Yau threefold, we derive modular properties of the generating function of BPS degeneracies of D4-D2-D0 black holes in type IIA string theory compactified on the same space.","Mathematically, these BPS degeneracies are the generalized Donaldson-Thomas invariants counting coherent sheaves with support on a divisor $\\cal D$, at the large volume attractor point.","For $\\cal D$ irreducible, this function is closely related to the elliptic genus of the superconformal field theory obtained by wrapping M5-brane on $\\cal D$ and is therefore known to be modular.","Instead, when $\\cal D$ is the sum of $n$ irreducible divisors ${\\cal D}_i$, we show that the generating function acquires a modular anomaly.","We characterize this anomaly for arbitrary $n$ by providing an explicit expression for a non-holomorphic modular completion in terms of generalized error functions.","As a result, the generating function turns out to be a (mixed) mock modular form of depth $n-1$."],["Introduction and summary","The degeneracies of BPS black holes in string vacua with extended supersymmetry possess remarkable modular properties, which have been instrumental in recent progress on explaining the statistical origin of the Bekenstein-Hawking entropy in [1] and many subsequent works.","Namely, the indices $\\Omega (\\gamma )$ counting — with sign — microstates of BPS black holes with electromagnetic charge $\\gamma $ may often be collected into a suitable generating function which exhibits modular invariance, providing powerful constraints on its Fourier coefficients and enabling direct access to their asymptotic growth.","When the black holes can be realized as black strings wrapped on a circle, a natural candidate for such a generating function is the elliptic genus of the superconformal field theory supported by the black string, which is modular invariant by construction [2], [3], [4].","Equivalently, one may consider the partition function of the effective three-dimensional gravity living on the near-horizon geometry of the black string [5].","In most cases however, the BPS indices depend not only on the charge $\\gamma $ but also on the moduli $z^a$ at spatial infinity, due to the wall-crossing phenomenon: some of the BPS bound states with total charge $\\gamma $ only exist in a certain chamber in moduli space, and decay as the moduli are varied across `walls of marginal stability' which delimit this chamber.","At strong coupling where the black hole description is accurate, this phenomenon has a transparent interpretation in terms of the (dis)appearance of multi-centered black hole configurations, which can be used to derive a universal wall-crossing formula [6], [7], [8], [9].","In the case of four-dimensional string vacua with $\\mathcal {N}=4$ supersymmetry, where the BPS index is sensitive only to single-centered 1/4-BPS black holes and to bound states of two 1/2-BPS black holes, the resulting moduli dependence is reflected in poles in the generating function, requiring a suitable choice of contour for extracting the Fourier coefficients in a given chamber [10], [11], [12].","Upon subtracting contributions of two-centered bound states, the generating function of single-centered indices is no longer modular in the usual sense but it transforms as a mock Jacobi form with specific `shadow' — a property which is almost as constraining as standard modular invariance [13].","In four-dimensional string vacua with $\\mathcal {N}=2$ supersymmetry, the situation is much more complicated, firstly due to the fact that the moduli space of scalars receives quantum corrections, and secondly due to BPS bound states potentially involving an arbitrary number of constituents, resulting in an extremely intricate pattern of walls of marginal stability.","Thus, it does not seem plausible that a single generating function may capture the BPS indices $\\Omega (\\gamma ,z^a)$ — which are known in this context as generalized Donaldson-Thomas (DT) invariants — in all chambers.","Nevertheless, modular invariance is still expected to constrain them.","In particular, D4-D2-D0 black holes in type IIA string theory compactified on a generic compact Calabi-Yau (CY) threefold $\\mathfrak {Y}$ can be lifted to an M5-brane wrapped on a divisor $\\mathcal {D}\\subset \\mathfrak {Y}$ [2].","If the divisor $\\mathcal {D}$ labelled by the D4-brane charge $p^a$ is irreducible, the indices $\\Omega (\\gamma ,z^a)$ are independent of the moduli of $\\mathfrak {Y}$ , at least in the limit where the volume of $\\mathfrak {Y}$ is scaled to infinity, and their generating function is known to be a holomorphic (vector valued) modular form of weight $-{1\\over 2}\\, b_2(\\mathfrak {Y})-1$ [3], [4], [6], [14].","This includes the case of vertical, rank 1 D4-D2-D0 branes in K3-fibered Calabi-Yau threefolds [15], [16], [17].","But if the divisor $\\mathcal {D}$ is a sum of $n$ effective divisors $\\mathcal {D}_i$ , the indices $\\Omega (\\gamma ,z^a)$ do depend on the Kähler moduli $z^a$ , even in the large volume limit.","In general however, the black string SCFT is supposed to capture the states associated to a single $AdS_3$ throat, while for generic values of the moduli multiple $AdS_3$ throats can contribute [18], [19].","It is thus natural to consider the modular properties of the DT invariants $\\Omega (\\gamma ,z^a)$ at the large volume attractor pointHere $p^a$ and $q_a$ are D4 and D2-brane charges, respectively, and the index of $q_a$ is raised with help of the inverse of the metric $\\kappa _{ab}=\\kappa _{abc}p^a$ with $\\kappa _{abc}$ being the triple intersection numbers on $H_4(\\mathfrak {Y},{Z})$ .","$z^a_\\infty (\\gamma )= \\lim _{\\lambda \\rightarrow +\\infty }\\left(-q^a+I\\lambda p^a\\right),$ where only a single $AdS_3$ throat is allowed [20].","Following [21] we denote these invariants by $\\Omega ^{\\rm MSW}(\\gamma )=\\Omega (\\gamma ,z^a_\\infty (\\gamma ))$ and call them Maldacena-Strominger-Witten (MSW) invariants.","As we discuss in Section , the DT invariants $\\Omega (\\gamma ,z^a)$ can be recovered from the MSW invariants $\\Omega ^{\\rm MSW}(\\gamma )$ by using a version of the split attractor flow conjecture [22], [6] developed in [23], which we call the flow tree formula.","The case where $\\mathcal {D}$ is the sum of two irreducible divisors was first considered in [20], [24], [25], and studied more recently in [14], [26].","In that case, the generating function of MSW invariants turns out to be a mock modular form, with a specific non-holomorphic completion obtained by smoothing out the sign functions entering in the bound state contributions, recovering the prescription of [27], [20].","The goal of this paper is to extend this result to the general case where $\\mathcal {D}$ is the sum of $n$ irreducible divisors $\\mathcal {D}_i$ , where $n$ can be arbitrarily large.","In such generic situation, we characterize the modular properties of the generating function $h_{p,\\mu }$ of MSW invariants and find an explicit expression for its non-holomorphic completion $\\widehat{h}_{p,\\mu }$ in terms of the generating functions $h_{p_i,\\mu _i}$ associated to the $n$ constituents, multiplied by certain iterated integrals introduced in [28], [29], which generalize the usual error function appearing when $n=2$ .","This result implies that in this case $h_{p,\\mu }$ is a (mixed, vector valued) mock modular form of depth $n-1$ , in the sense that the antiholomorphic derivative of its modular completion is itself a linear combination of modular completions of mock modular forms of lower depth, times antiholomorphic modular forms (with the depth 1 case reducing to the standard mock modular forms introduced in [27], [30], and the depth 0 case to usual weakly holomorphic modular forms; see [31] for a more precise definition).","In order to establish this result, we follow the same strategy as in our earlier works [14], [26] and analyze D3-D1-D(-1) instanton corrections to the metric on the hypermultiplet moduli space $\\mathcal {M}_H$ in type IIB string theory compactified on $\\mathfrak {Y}$ , at arbitrary order in the instanton expansion.","After reducing on a circle and T-dualizing, this moduli space is identical to the vector multiplet moduli space in type IIA string theory compactified on $\\mathfrak {Y}\\times S_1$ , where it receives instanton corrections from D4-D2-D0 black holes winding around the circle.","In either case, each instanton contribution is weighted by the same generalized DT invariant $\\Omega (\\gamma )$ which counts the number of BPS black hole microstates with electromagnetic charge $\\gamma $ .","The modular properties of the generalized DT invariants are fixed by requiring that the quaternion-Kähler (QK) metric on $\\mathcal {M}_H$ admits an isometric action of $SL(2,{Z})$ , which comes from S-duality in type IIB, or equivalently from large diffeomorphisms of the torus appearing when viewing type IIA/$\\mathfrak {Y}\\times S_1$ as M-theory on $\\mathfrak {Y}\\times T^2$ [32].","This QK metric is most efficiently encoded in the complex contact structure on the associated twistor space, a ${C}P^1$ -bundle over $\\mathcal {M}_H$ [33], [34].","The latter is specified by a set of gluing conditions determined by the DT invariants [32], [35], which can in turn be expressed in terms of the MSW invariants using the tree flow formula.","An important quantity appearing in this twistorial formulationA familiarity with the twistorial formulation is not required for this work.","Here we use only two equations relevant for the twistorial description of D-instantons: the integral equation (REF ) for certain Darboux coordinates, which appears also in the study of four-dimensional $N=2$ gauge theory on a circle [36], and the expression for the contact potential (REF ) in terms of these Darboux coordinates.","These two equations lead to (REF ) and (REF ), respectively, which can be taken as the starting point of our analysis.","Note that in the context of gauge theories the contact potential can be interpreted as a supersymmetric index [37].","is the so called contact potential $e^\\phi $ , a real function on $\\mathcal {M}_H$ related to the Kähler potential on the twistor space, and afforded by the existence of a continuous isometry unbroken by D-instantons.","On the type IIB side, $e^\\phi $ can be identified with the four-dimensional dilaton $1/g_4^2$ .","When expressed in terms of the ten-dimensional axio-dilaton $\\tau =c_0+I/g_s$ (or in terms of the modulus of the 2-torus on the M-theory side), it becomes a complicated function having a classical contribution, a one-loop correction, and a series of instanton corrections expressed as contour integrals on the ${C}P^1$ fiber.","The importance of the contact potential stems from the fact that it must be a non-holomorphic modular form of weight $(-\\frac{1}{2},-\\frac{1}{2})$ in the variable $\\tau $ in order for $\\mathcal {M}_H$ to admit an isometric action of $SL(2,{Z})$ [32].","This requirement imposes very non-trivial constraints on the instanton contributions to $e^\\phi $ , which can be used to deduce the modular properties of generating functions of DT invariants, at each order in the instanton expansion.","This strategy was used in [14] at two-instanton order to characterize the modular behavior of the generating function of MSW invariants in the case of a divisor equal to the sum of $n=2$ irreducible divisors.","In this paper we generalize this result to all $n$ , by analyzing the instanton expansion to all orders.","Below we summarize the main steps in our analysis and our main results.","First, we show that the contact potential $e^\\phi $ in the large volume limit, where the Kähler parameters of the CY are sent to infinity, can be expressed (see (REF )) through another (complex valued) function $\\mathcal {G}$ (REF ) on the moduli space, which we call the instanton generating function.","The expansion of $\\mathcal {G}$ in powers of DT invariants is governed by a sum over unrooted trees decorated by charges $\\gamma _i$ (see (REF ) and (REF )).","The modularity of the contact potential requires $\\mathcal {G}$ to transform as a modular form of weight $(-\\tfrac{3}{2},\\tfrac{1}{2})$ .","After expressing the DT invariants $\\Omega (\\gamma ,z^a)$ through the moduli independent MSW invariants $\\Omega ^{\\rm MSW}(\\gamma )$ using the tree flow formula of [23], and expanding $\\mathcal {G}$ in powers of $\\Omega ^{\\rm MSW}(\\gamma )$ , each order in this expansion can be decomposed into a sum of products of certain indefinite theta series and of holomorphic generating functions $h_{p_i,\\mu _i}$ of the invariants $\\Omega ^{\\rm MSW}(\\gamma )$ (see (REF )), similarly to the usual decomposition of standard Jacobi forms.","Thus, the modular properties of the indefinite theta series $\\vartheta _{{p},{\\mu }}\\bigl (\\Phi ^{{\\rm tot}}_{n},n-2\\bigr )$ are tied with the modular properties of the generating functions $h_{p_i,\\mu _i}$ , in order for $\\mathcal {G}$ to be modular.","In order for an indefinite theta series $\\vartheta _{{p},{\\mu }}\\bigl (\\Phi ,\\lambda )$ to be modular, its kernel $\\Phi $ must satisfy a certain differential equation (REF ), which we call Vignéras' equation [38].","By construction, the kernels $\\Phi ^{\\rm tot}_n$ appearing in our problem are given by iterated contour integrals along the ${C}P^1$ fiber of the twistor space, multiplied by so-called `tree indices' $g_{{\\rm tr},n}$ coming from the expression of $\\Omega (\\gamma ,z^a)$ in terms of $\\Omega ^{\\rm MSW}(\\gamma )$ .","We evaluate the twistorial integrals in terms of the generalized error functions introduced in [28], [29], and show that the resulting kernels satisfy Vignéras' equation away from certain loci where they have discontinuities.","Furthermore, we prove that the discontinuities corresponding to walls of marginal stability cancel between the integrals and the tree indices.","But there are additional discontinuities coming from certain moduli independent contributions to the tree index.","They spoil Vignéras' equation at the multi-instanton level so that the theta series $\\vartheta _{{p},{\\mu }}\\bigl (\\Phi ^{{\\rm tot}}_{n},n-2\\bigr )$ are not modular.","In turn, this implies that the holomorphic generating functions $h_{p_i,\\mu _i}$ are not modular either.","However, we show that one can complete $h_{p,\\mu }$ into a non-holomorphic modular form $\\widehat{h}_{p,\\mu }(\\tau )$ , by adding to it a series of corrections proportional to products of $h_{p_i,\\mu _i}$ with the same total D4-brane charge $p=\\sum _i p_i$ (see (REF )).","The non-holomorphic functions $R_n$ entering the completion are determined by the condition that the expansion of $\\mathcal {G}$ rewritten in terms of $\\widehat{h}_{p,\\mu }$ gives rise to a non-anomalous, modular theta series.","Equivalently, one can work with functions $\\widehat{g}_n$ appearing in the expansion (REF ) of the holomorphic generating function of DT invariants $h^{\\rm DT}_{p,q}$ in powers of the non-holomorphic functions $\\widehat{h}_{p_i,\\mu _i}$ .","Imposing the conditions for modularity, we find that $\\widehat{g}_n$ can be represented in an iterative form (REF ), or more explicitly as a sum (REF ) over rooted trees with valency $\\ge 3$ at each vertex (known as Schröder trees), where $g^{(0)}_n$ are certain locally polynomial functions defined in (REF ), while $\\mathcal {E}_n$ are smooth solutions of Vignéras' equation constructed in terms of generalized error functions.","The functions $R_n$ are similarly given by a sum over Schröder trees (REF ) in terms of exponentially decreasing and non-decreasing parts of $\\mathcal {E}_n$ .","These equations represent the main technical result of this work.","For $n\\le 5$ our general formulas can be drastically simplified.","In particular, the main building blocks, the functions $g^{(0)}_n$ and $\\mathcal {E}_n$ , can be written as a sum over a suitable subset of flow trees, as in (REF ).","In addition, we show that $g^{(0)}_n$ has a natural extension including the refinement parameter conjugate to the spin $J_3$ of the black hole.","Figure: Various types of trees arising in this work.T n,m {T}_{n,m} denotes the set of unrooted trees with nn vertices and mm marks distributed between the vertices.T n ≡T n,0 {T}_n\\!\\equiv {T}_{n,0} comprises the trees without marks.T n r {T}_n^{\\rm r} is the set of rooted trees with nn vertices.T n af {T}_n^{\\rm af} is the set of attractor flow trees with nn leaves (of which we only draw the different topologies).T n S {T}_n^{\\rm S} denotes the set of Schröder trees with nn leaves.In addition, an important rôle is played by the sets T n,m ℓ {T}_{n,m}^\\ell and T n ℓ =T n,0 ℓ {T}_n^\\ell ={T}_{n,0}^\\ell of unrooted, labelled, marked trees which are obtained from T n,m {T}_{n,m} and T n {T}_{n} by assigning different labels and markings to the verticesin such way that a vertex with m 𝔳 m_\\mathfrak {v} marks is decorated by 2m 𝔳 +12m_\\mathfrak {v}+1 labels.At the end of this lengthy analysis, we thus find a modular completion of the generating functions $h_{p,\\mu }$ of MSW invariants for an arbitrary divisor, i.e.","decomposable into a sum of any number of irreducible divisors.","The result is expressed through sums of products of generalized error functions labelled by trees of various types.","For the reader's convenience, in Fig.","REF we display the various trees which appear in our construction, up to $n=4$ .","Since generalized error functions are known to be related to iterated Eichler integrals [14], [39], which occur in the non-holomorphic completion of mock modular forms, we loosely refer to the generating functions of MSW invariants as higher depth mock modular forms, although we do not spell out the precise meaning of this notion.","An unexpected byproduct of our analysis is an interesting combinatorial identity, relating rooted trees to the binomial coefficients, which plays a rôle in our derivation of the modular completion.","Since we are not aware of such an identity in the mathematical literature,We were informed by Karen Yeats that the special case $m=n-1$ appears in [40].","The denominator in (REF ) is sometimes known as the tree factorial $T^{\\prime }!$ , see (REF ).","we state it here as a theorem whose proof can be found in appendix .","Theorem 1 Let $V_T$ be the set of vertices of a rooted ordered tree $T$ and $n_v(T)$ is the number of descendants of vertex $v$ inside $T$ plus 1 (alternatively, the number of vertices of the subtree in $T$ with the root being the vertex $v$ and the leaves being the leaves of $T$ ).","Then for a rooted tree $T$ with $n$ vertices and $m0$ at each vertex.","This condition will hold automatically for the case of our interest, namely, D4-D2-D0 black holes in the large volume limit, so we do not impose it explicitly.","$\\gamma _{L(v)R(v)}\\, \\,{\\rm Im}\\,\\bigl [ Z_{\\gamma _{L(v)}}\\bar{Z}_{\\gamma _{R(v)}}(z^a_{p(v)})\\bigr ]>0$ is satisfied for all $v\\in V_T$ , which ensures admissibility of the flow tree $T$ , i.e.","the existence of the corresponding nested bound state.","Importantly, the sign of $\\,{\\rm Im}\\,\\bigl [ Z_{\\gamma _{L(v)}}\\bar{Z}_{\\gamma _{R(v)}}(z^a_{p(v)})\\bigr ]$ entering (REF ) can be computed recursively in terms of asymptotic data, without evaluating the attractor flow along the edges [23].","More precisely, the signs depend only on the stability parameters (also known as Fayet-Iliopoulos parameters) $c_i(z^a) = \\,{\\rm Im}\\,\\bigl [ Z_{\\gamma _i}\\bar{Z}_\\gamma (z^a)\\bigr ]\\, .$ Note that due to $\\gamma =\\sum _{i=1}^n \\gamma _i$ , these parameters satisfy $\\sum _{i=1}^n c_i=0$ .","Accordingly, we shall often denote $g_{{\\rm tr},n}(\\lbrace \\gamma _i\\rbrace ,z^a)$ by $g_{{\\rm tr},n}(\\lbrace \\gamma _i,c_i\\rbrace )$ , and always assume that $\\lbrace c_i\\rbrace $ are generic real parameters subject to the condition $\\sum _{i=1}^n c_i=0$ , such that no proper subset of them sums up to zero.","The second factor in (REF ) is independent of the moduli, and given by $\\kappa (T) \\equiv (-1)^{n-1} \\prod _{v\\in V_T} \\kappa ( \\gamma _{L(v)R(v)}),\\qquad \\kappa (x)=(-1)^x\\, x.$ This is simply the product of the BPS indices of the nested two-centered solutions associated with the tree $T$ .","Note that the signs of $\\Delta (T)$ and $\\kappa (T)$ separately depend on the choice of ordering $(\\gamma _{L(v)},\\gamma _{R(v)})$ at each vertex, but their product is independent of that choice.","Sometimes, it is useful to consider the refined BPS index $\\Omega (\\gamma ,z^a,y)$ which carries additional dependence on the fugacity $y$ conjugate to the spin $J_3$ .","All the above equations remain valid in this case as well (except for the definition of the rational invariant (REF ), which must be slightly modified, see e.g.","[8]), but now the function $\\kappa (x)$ appearing in (REF ) becomes a symmetric Laurent polynomial in $y$ $\\kappa (x)=(-1)^x \\, \\frac{ y^x-y^{-x}}{y-y^{-1}}\\, ,$ reducing to $(-1)^x x$ in the unrefined limit $y\\rightarrow 1$ .","It is easy to see that the `flow tree formula' (REF ) is consistent with the primitive wall-crossing formula [6], [7] $\\Delta \\bar{\\Omega }(\\gamma _L+\\gamma _R) = -{\\rm sgn}(\\gamma _{LR})\\, \\kappa (\\gamma _{LR})\\, \\bar{\\Omega }(\\gamma _L,z^a)\\, \\bar{\\Omega }(\\gamma _R,z^a) ,$ which gives the jump of the BPS index due to the decay of bound states after crossing the wall defined by a pair of primitiveHere, by primitive we mean that all charges with non-zero index in the two-dimensional lattice spanned by $\\gamma _L$ and $\\gamma _R$ are linear combinations $N_L \\gamma _L + N_R\\gamma _R$ with coefficients $N_L, N_R$ of the same sign.","charges $\\gamma _L$ and $\\gamma _R$ .","To this end, it suffices to consider all flow trees which start with the splitting $\\gamma \\rightarrow \\gamma _L+\\gamma _R$ at the root of the tree.","It is also consistent with the general wall-crossing formula of [41], provided the tree index is computed for a small generic perturbation of the DSZ matrix $\\gamma _{ij}$ [23].","Finally, it is useful to note that, assuming that all charges $\\gamma _i$ are distinct, the sum over splittings and flow trees in (REF ) can be generated by iterating the quadratic equation [23] $\\bar{\\Omega }(\\gamma ,z^a) &= &\\bar{\\Omega }_*(\\gamma )-{1\\over 2}\\sum _{\\begin{array}{c}\\gamma =\\gamma _L+\\gamma _R\\\\\\langle \\gamma _L,\\gamma _R \\rangle \\ne 0\\end{array}}\\Delta _{\\gamma _L\\gamma _R}^z \\, \\kappa (\\gamma _{LR})\\,\\bar{\\Omega }(\\gamma _L,z^a_{LR})\\, \\bar{\\Omega }(\\gamma _R,z^a_{LR}),$ where $z^a_{LR}$ is the point where the attractor flow of charge $\\gamma $ crosses the wall of marginal stability $\\,{\\rm Im}\\,\\bigl [Z_{\\gamma _L}\\bar{Z}_{\\gamma _R}(z^a_{LR})\\bigr ]=0$ ."],["Partial tree index","While the representation of the BPS index based on attractor flows is useful for many purposes, it produces a sum of products of sign functions depending on non-linear combinations of DSZ products $\\gamma _{ij}$ , which are very difficult to work with.","A solution to overcome this problem was found in [23].","The key idea is to introduce a refined index with a fugacity $y$ conjugate to angular momentum, and represent it as $g_{{\\rm tr},n}(\\lbrace \\gamma _i,c_i\\rbrace ,y)= \\frac{(-1)^{n-1+\\sum _{i 0,\\qquad (r p^2)> 0,\\qquad k_a p^a > 0,$ for all effective divisors $r^a \\gamma _a \\in H_4^+(\\mathfrak {Y},{Z})$ and effective curves $k_a \\gamma ^a \\in H_2^+(\\mathfrak {Y},{Z})$ , where $\\gamma _a$ denotes irreducible divisors giving an integer basis of $ \\Lambda =H_4(\\mathfrak {Y},{Z})$ , dual to the basis $\\gamma ^a$ of $\\Lambda ^*=H_2(\\mathfrak {Y},{Z})$ .","The charge $p^a$ induces a quadratic form $\\kappa _{ab}=\\kappa _{abc} p^c$ on $\\Lambda \\otimes {R}\\simeq {R}^{b_2}$ of signature $(1,b_2-1)$ .","This quadratic form allows to embed $\\Lambda $ into $\\Lambda ^*$ , but the map $\\epsilon ^a \\mapsto \\kappa _{ab} \\epsilon ^b$ is in general not surjective, the quotient $\\Lambda ^*/\\Lambda $ being a finite group of order $|\\,{\\rm det}\\, \\kappa _{ab}|$ .","The holomorphic central charge, governing the mass of BPS states, is given by $Z_\\gamma (z^a)=q_\\Lambda X^\\Lambda (z^a) -p^\\Lambda F_\\Lambda (z^a),$ where $X^\\Lambda (z^a)=(1,z^a)$ are the special coordinates and $F_\\Lambda =\\partial _{X^\\Lambda } F(X)$ is the derivative of the holomorphic prepotential $F(X)$ on $\\mathcal {M}_\\mathcal {K}$ .","In the large volume limit $t^a\\rightarrow \\infty $ , the prepotential reduces to the classical cubic contribution $F(X)\\approx F^{\\rm cl}(X)=-\\kappa _{abc}\\, \\frac{ X^a X^b X^c}{6X^0}\\, ,$ and the central charge can be approximated as $Z_\\gamma \\approx -{1\\over 2}\\, (pt^2)+I\\left( q_a t^a+(pbt)\\right)+q_0+q_a b^a+{1\\over 2}\\,(pb^2).$ Note, in particular, that it always has a large negative real part.","Another useful observation is that both quantities appearing in the definition of $\\Delta _{\\gamma _L\\gamma _R}^z$ (REF ) are independent of the last component $q_0$ of the charge vector.","Indeed, $\\begin{split}\\langle \\gamma ,\\gamma ^{\\prime }\\rangle =&\\, q_a p^{\\prime a}-q^{\\prime }_ap^a,\\\\\\,{\\rm Im}\\,\\bigl [ Z_{\\gamma }\\bar{Z}_{\\gamma ^{\\prime }}\\bigr ]=&\\, -{1\\over 2}\\,\\Bigl ((p^{\\prime } t^2)(q_{a} +(pb)_a)t^a - (p t^2) (q^{\\prime }_{a}+(p^{\\prime }b)_a) t^a\\Bigr ).\\end{split}$ The BPS index $\\bar{\\Omega }(\\gamma ,z^a)$ counting D4-D2-D0 black holes is given mathematically by the generalized Donaldson-Thomas invariant, which countsMore precisely, the generalized DT invariant computes the weighted Euler characteristic of the moduli space of semi-stable coherent sheaves [42]; in this context, the DSZ product $\\langle \\gamma ,\\gamma ^{\\prime }\\rangle $ coincides with the antisymmetrized Euler form.","semi-stable coherent sheaves supported on a divisor $\\mathcal {D}$ in the homology class $p^a\\gamma _a$ , with first and second Chern numbers determined by $(q_a,q_0)$ .","An important property of these invariants is that they are unchanged under a combined integer shift of the Kalb-Ramond field, $b^a\\mapsto b^a +\\epsilon ^a$ , and a spectral flow transformation acting on the D2 and D0 charges $q_a \\mapsto q_a - \\kappa _{abc}p^b\\epsilon ^c,\\qquad q_0 \\mapsto q_0 - \\epsilon ^a q_a + \\frac{1}{2}\\, (p\\epsilon \\epsilon ).$ The shift of $b^a$ is important since the DT invariants are only piecewise constant as functions of the complexified Kähler moduli $z^a=b^a+It^a$ due to wall-crossing.","In contrast, the MSW invariants ${\\bar{\\Omega }}^{\\rm MSW}(\\gamma )$ , defined as the generalized DT invariants $\\bar{\\Omega }(\\gamma ,z^a)$ evaluated at their respective large volume attractor point (REF ), are by construction independent of the moduli, and therefore invariant under the spectral flow (REF ).","As a result, they only depend on $p^a, \\mu _a$ and $\\hat{q}_{0}$ , where we traded the electric charges $(q_a,q_0)$ for $(\\epsilon ^a, \\mu _a,\\hat{q}_0)$ .","The latter comprise the spectral flow parameter $\\epsilon ^a$ , the residue class $\\mu _a\\in \\Lambda ^*/\\Lambda $ defined by the decomposition $q_a = \\mu _a + \\frac{1}{2}\\, \\kappa _{abc} p^b p^c + \\kappa _{abc} p^b \\epsilon ^c,\\qquad \\epsilon ^a\\in \\Lambda \\, ,$ and the invariant charge ($\\kappa ^{ab}$ is the inverse of $\\kappa _{ab}$ ) $\\hat{q}_0 \\equiv q_0 -\\frac{1}{2}\\, \\kappa ^{ab} q_a q_b\\, ,$ which is invariant under (REF ).","This allows to write ${\\bar{\\Omega }}^{\\rm MSW}(\\gamma )=\\bar{\\Omega }_{p,\\mu }( \\hat{q}_0)$ .","An important fact is that the invariant charge $\\hat{q}_0$ is bounded from above by $\\hat{q}_0^{\\rm max}=\\tfrac{1}{24}((p^3)+c_{2,a}p^a)$ .","This allows to define two generating functions $h^{\\rm DT}_{p,q}(\\tau ,z^a) &=& \\sum _{\\hat{q}_0 \\le \\hat{q}_0^{\\rm max}}\\bar{\\Omega }(\\gamma ,z^a)\\,{\\bf e}\\!\\left( -\\hat{q}_0 \\tau \\right),\\\\h_{p,\\mu }(\\tau ) &=& \\sum _{\\hat{q}_0 \\le \\hat{q}_0^{\\rm max}}\\bar{\\Omega }_{p,\\mu }(\\hat{q}_0)\\,{\\bf e}\\!\\left( -\\hat{q}_0 \\tau \\right),$ where we used notation ${\\bf e}\\!\\left( x\\right)=e^{2\\pi Ix}$ .","Whereas the generating function of DT invariants $h^{\\rm DT}_{p,q}$ depends on the full electric charge $q_a$ and depends on the moduli $z^a$ in a piecewise constant fashion, the generating function of MSW invariants $h_{p,\\mu }(\\tau )$ , due to the spectral flow symmetry, depends only on the residue class $\\mu _a$ .","This generating function will be the central object of interest in this paper, and our main goal will be to understand its behavior under modular transformations of $\\tau $ .","In general, the MSW invariants ${\\bar{\\Omega }}^{\\rm MSW}(\\gamma )$ are distinct from the attractor moduli $\\bar{\\Omega }_\\star (\\gamma )$ , since the latter coincide with the generalized DT invariants $\\bar{\\Omega }(\\gamma ,z^a)$ evaluated at the true attractor point $z^a_*(\\gamma )$ for the charge $\\gamma $ , while the former are the generalized DT invariants evaluated at the large volume attractor point $z^a_\\infty (\\gamma )$ defined in (REF ).","Nevertheless, we claim that in the large volume limit $t^a\\rightarrow \\infty $ , the tree flow formula reviewed in the previous subsections still allows to express $\\bar{\\Omega }(\\gamma ,z^a)$ in terms of the MSW invariants, namely $\\bar{\\Omega }(\\gamma ,z^a) =\\sum _{\\sum _{i=1}^n \\gamma _i=\\gamma }g_{{\\rm tr},n}(\\lbrace \\gamma _i\\rbrace , z^a)\\,\\prod _{i=1}^n {\\bar{\\Omega }}^{\\rm MSW}(\\gamma _i) \\qquad (\\,{\\rm Im}\\,z^a\\rightarrow \\infty )$ The point is that the only walls of marginal stability which extend to infinite volume are those where the constituents carry no D6-brane charge, and that non-trivial bound states involving constituents with D4-brane charge are ruled out at the large volume attractor point, similarly to the usual attractor chamber.","Since the r.h.s.","of (REF ) is consistent with wall-crossing in the infinite volume limit and agrees with the left-hand side at $z^a=z^a_\\infty (\\gamma )$ , it must therefore hold everywhere at large volume.","Of course, some of the states contributing to ${\\bar{\\Omega }}^{\\rm MSW}(\\gamma _i)$ may have some substructure, e.g.","be realized as D6-$\\overline{\\rm D6}$ bound states, but this structure cannot be probed in the large volume limit.","Importantly, since the quantities (REF ) entering in the definition of the tree index are independent of the D0-brane charge $q_0$ , the flow tree formula (REF ) may be rewritten as a relation between the generating functions, $h^{\\rm DT}_{p,q}(\\tau ,z^a) =\\sum _{\\sum _{i=1}^n \\check{\\gamma }_i=\\check{\\gamma }}g_{{\\rm tr},n}(\\lbrace \\check{\\gamma }_i\\rbrace ,z^a)\\,e^{\\pi I\\tau Q_n(\\lbrace \\check{\\gamma }_i\\rbrace )}\\prod _{i=1}^n h_{p_i,\\mu _i}(\\tau ),$ where $\\check{\\gamma }=(p^a,q_a)$ denotes the projection of the charge vector $\\gamma $ on $H_4\\oplus H_2$ , and the phase proportional to $Q_n(\\lbrace \\check{\\gamma }_i\\rbrace )= \\kappa ^{ab}q_a q_b-\\sum _{i=1}^n\\kappa _i^{ab}q_{i,a} q_{i,b}$ appears due to the quadratic term in the definition (REF ) of the invariant charge $\\hat{q}_0$ .","In this section, we switch to the dual setupOur preference for the type IIB set-up is merely for consistency with our earlier works on hypermultiplet moduli spaces in $d=4$ Calabi-Yau vacua.","The same considerations apply verbatim, with minor changes of wording, to the vector multiplet moduli space in type IIA string theory compactified on $\\mathfrak {Y}\\times S^1$ , which is more directly related to the counting of D4-D2-D0 black holes in four dimensions.","of type IIB string theory compactified on the same CY manifold $\\mathfrak {Y}$ .","The DT invariants, describing the BPS degeneracies of D4-D2-D0 black holes in type IIA, now appear as coefficients in front of the D3-D1-D(-1) instanton effects affecting the metric on the hypermultiplet moduli space $\\mathcal {M}_H$ .","The main idea of our approach is that these instanton effects are strongly constrained by demanding that $\\mathcal {M}_H$ admits an isometric action of the type IIB S-duality $SL(2,{Z})$ .","This constraint uniquely fixes the modular behavior of the generating functions introduced in the previous section.","Here we recall the twistorial construction of D-instanton corrections to the hypermultiplet metric, describe the action of S-duality, and analyze the instanton expansion of a particular function on $\\mathcal {M}_H$ known as contact potential."],["$\\mathcal {M}_H$ and twistorial description of instantons ","The moduli space of four-dimensional $N=2$ supergravity is a direct product of vector and hypermultiplet moduli spaces, $\\mathcal {M}_V\\times \\mathcal {M}_H$ .","The former is a (projective) special Kähler manifold, whereas the latter is a quaternion-Kähler (QK) manifold.","In type IIB string theory compactified on a CY threefold $\\mathfrak {Y}$ , $\\mathcal {M}_H$ is a space of real dimension $4b_2(\\mathfrak {Y})+4$ , which is fibered over the complexified Kähler moduli space $\\mathcal {M}_\\mathcal {K}(\\mathfrak {Y})$ of dimension $2b_2(\\mathfrak {Y})$ .","In addition to the Kähler moduli $z^a=b^a+It^a$ , it describes the dynamics of the ten-dimensional axio-dilaton $\\tau =c^0+I/g_s$ , the Ramond-Ramond (RR) scalars $c^a,\\tilde{c}_a,\\tilde{c}_0$ , corresponding to periods of the RR 2-form, 4-form and 6-form on a basis of $H^{\\rm even}(\\mathfrak {Y},{Z})$ , and finally, the NS-axion $\\psi $ , dual to the Kalb-Ramond two-form $B$ in four dimensions.","At tree-level, the QK metric on $\\mathcal {M}_H$ is obtained from the Kähler moduli space $\\mathcal {M}_{\\mathcal {K}}$ via the $c$ -map construction [52], [53] and thus is completely determined by the holomorphic prepotential $F(X)$ .","But this metric receives $g_s$ -corrections, both perturbative and non-perturbative.","The latter can be of two types: either from Euclidean D-branes wrapping even dimensional cycles on $\\mathfrak {Y}$ , or from NS5-branes wrapped around the whole $\\mathfrak {Y}$ .","In this paper we shall be interested only in the effects of D3-D1-D(-1) instantons, and ignore the effects of NS5 and D5-instantons, which are subleading in the large volume limit.","Since NS5-instantons only mix with D5-instantons under S-duality, this truncation does not spoil modular invariance [21].","The most concise way to describe the D-instanton corrections is to consider type IIA string theory compactified on the mirror CY threefold ${\\widehat{\\mathfrak {Y}}}$ and use the twistor formalism for quaternionic geometries [33], [34].","In this approach the metric is encoded in the complex contact structure on the twistor space, a ${C}P^1$ -bundle over $\\mathcal {M}_H$ .","The D-instanton corrected contact structure has been constructed to all orders in the instanton expansion in [32], [35], and an explicit expression for the metric has been derived recently in [54], [55].","Here we will present only those elements of the construction which are relevant for the subsequent analysis, and refer to reviews [56], [57] for more details.","The crucial point is that, locally, the contact structure is determined by a set of holomorphic Darboux coordinates $(\\xi ^\\Lambda ,\\tilde{\\xi }_\\Lambda ,\\alpha )$ on the twistor space, considered as functions of coordinates on $\\mathcal {M}_H$ and of the stereographic coordinate $t$ on the ${C}P^1$ fiber, so that the contact one-form takes the canonical form $\\mathrm {d}\\alpha +\\tilde{\\xi }_\\Lambda \\mathrm {d}\\xi ^\\Lambda $ .","Although all Darboux coordinates are important for recovering the metric, for the purposes of this paper the coordinate $\\alpha $ is irrelevant.","Therefore, we consider only $\\xi ^\\Lambda $ and $\\tilde{\\xi }_\\Lambda $ which can be conveniently packaged into holomorphic Fourier modes $\\mathcal {X}_\\gamma = {\\bf e}\\!\\left( p^\\Lambda \\tilde{\\xi }_\\Lambda -q_\\Lambda \\xi ^\\Lambda \\right)$ labelled by a charge vector $\\gamma =(p^\\Lambda ,q_\\Lambda )$ .","At tree level, the Darboux coordinates (multiplied by $t$ ) are known to be simple quadratic polynomials in $t$ so that $\\mathcal {X}_\\gamma $ take the formThe superscript `sf' stands for `semi-flat', which refers to the flatness of the classical geometry in the directions along the torus fibers parametrized by $\\zeta ^\\Lambda ,\\tilde{\\zeta }_\\Lambda $ .","$\\mathcal {X}^{\\rm sf}_\\gamma (t)={\\bf e}\\!\\left( \\frac{\\tau _2}{2}\\left(\\bar{Z}_\\gamma (\\bar{u}^a)\\,t-\\frac{Z_\\gamma (u^a)}{t}\\right)+p^\\Lambda \\tilde{\\zeta }_\\Lambda - q_\\Lambda \\zeta ^\\Lambda \\right),$ where $Z_\\gamma (u^a)$ is the central charge (REF ), now expressed in terms of the complex structure moduli $u^a$ of the CY threefold ${\\widehat{\\mathfrak {Y}}}$ mirror to $\\mathfrak {Y}$ , $\\zeta ^\\Lambda $ and $\\tilde{\\zeta }_\\Lambda $ are periods of the RR 3-form in the type IIA formulation, and $\\tau _2=g_s^{-1}$ is the inverse ten-dimensional string coupling.","At the non-perturbative level, this expression gets modified and the Darboux coordinates are determined by the integral equation $\\mathcal {X}_\\gamma (t) = \\mathcal {X}^{\\rm sf}_\\gamma (t)\\, {\\bf e}\\!\\left( \\frac{1}{8\\pi ^2}\\sum _{\\gamma ^{\\prime }} \\sigma _{\\gamma ^{\\prime }} \\, \\bar{\\Omega }(\\gamma ^{\\prime })\\, \\langle \\gamma ,\\gamma ^{\\prime }\\rangle \\int _{\\ell _{\\gamma ^{\\prime }} }\\frac{\\mathrm {d}t^{\\prime }}{t^{\\prime }}\\, \\frac{t+t^{\\prime }}{t-t^{\\prime }}\\,\\mathcal {X}_{\\gamma ^{\\prime }}(t^{\\prime })\\right),$ where the sum goes over all charges labelling cycles wrapped by D-branes, $\\bar{\\Omega }(\\gamma ^{\\prime })=\\bar{\\Omega }(\\gamma ^{\\prime },z^a)$ is the corresponding rational Donaldson-Thomas invariant, $\\ell _\\gamma = \\lbrace t\\in {C}P^1\\ :\\ \\ Z_\\gamma /t\\in I{R}^-\\rbrace $ is the so called BPS ray, a contour on ${C}P^1$ extending from $t=0$ to $t=\\infty $ along the direction fixed by the central charge, and $\\sigma _\\gamma $ is a quadratic refinement of the DSZ product on the charge lattice $\\Gamma $ , i.e.","a sign factor satisfying the defining relation $\\sigma _{\\gamma _1}\\sigma _{\\gamma _2}=(-1)^{\\langle \\gamma _1,\\gamma _2\\rangle }\\sigma _{\\gamma _1+\\gamma _2},\\qquad \\forall \\gamma _1,\\gamma _2\\in \\Gamma .$ The system of integral equations (REF ) can be solved iteratively by first substituting $\\mathcal {X}_{\\gamma ^{\\prime }}(t^{\\prime })$ on the r.h.s.","with its zero-th order value $\\mathcal {X}^{\\rm sf}_{\\gamma ^{\\prime }}(t^{\\prime })$ in the weak coupling limit $\\tau _2\\rightarrow \\infty $ , computing the leading correction from the integral and iterating this process.","This produces an asymptotic series at weak coupling, in powers of the DT invariants $\\bar{\\Omega }(\\gamma )$ .","Using the saddle point method, it is easy to check that the coefficient of each monomial $\\prod _i \\bar{\\Omega }(\\gamma _i)$ is suppressed by a factor $e^{-\\pi \\tau _2 \\sum _i |Z_{\\gamma _i}|}$ , corresponding to an $n$ -instanton effect [36], [58].","Note that multi-instanton effects become of the same order as one-instanton effects on walls of marginal stability where the phases of $Z_{\\gamma _i}$ become aligned, and that the wall-crossing formula ensures that the QK metric on $\\mathcal {M}_H$ is smooth across the walls.","[36], [35]."],["D3-instantons in the large volume limit","The above construction of D-instantons is adapted to the type IIA formulation because the equation (REF ) defines the Darboux coordinates in terms of the type IIA fields appearing explicitly in the tree level expression (REF ).","To pass to the mirror dual type IIB formulation, one should apply the mirror map, a coordinate transformation from the type IIA to the type IIB physical fields.","This transformation was determined in the classical limit in [59], but it also receives instanton corrections.","In order to fix the form of these corrections, we require that the metric on $\\mathcal {M}_H$ carries an isometric action of S-duality group $SL(2,{Z})$ of type IIB string theory, which acts on the type IIB fields by an element $g={\\scriptsize \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}}$ in the following way $\\begin{split}&\\tau \\mapsto \\frac{a \\tau +b}{c \\tau + d} \\, ,\\qquad t^a \\mapsto |c\\tau +d| \\,t^a,\\qquad \\begin{pmatrix} c^a \\\\ b^a \\end{pmatrix} \\mapsto \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}\\begin{pmatrix} c^a \\\\ b^a \\end{pmatrix} ,\\\\&\\qquad \\tilde{c}_a\\mapsto \\tilde{c}_a - c_{2,a} \\varepsilon (g) ,\\qquad \\begin{pmatrix} \\tilde{c}_0 \\\\ \\psi \\end{pmatrix} \\mapsto \\begin{pmatrix} d & -c \\\\ -b & a \\end{pmatrix}\\begin{pmatrix} \\tilde{c}_0 \\\\ \\psi \\end{pmatrix},\\end{split}$ where $\\varepsilon (g)$ is the logarithm of the multiplier system of the Dedekind eta function [51].","For this purpose, one uses the fact that any isometric action on a quaternion-Kähler manifold (preserving the quaternionic structure) can be lifted to a holomorphic contact transformation on twistor space.","In the present case, $SL(2,{Z})$ acts on the fiber coordinate $t$ by a fractional-linear transformation with $\\tau $ -dependent coefficients.","This transformation takes a much simpler form when formulated in terms of another coordinate $z$ on ${C}P^1$ (not to be confused with the Kähler moduli $z^a$ ), which is related to $t$ by a Cayley transformation, $z =\\frac{t+I}{t-I}\\, .$ Then the action of $SL(2,{Z})$ on the fiber is given by a simple phase rotationActually, this is true only when five-brane instanton corrections are ignored.","Otherwise, the lift also gets a non-trivial deformation [60].","$z\\mapsto \\frac{c\\bar{\\tau }+d}{|c\\tau +d|}\\, z\\, .$ Using the holomorphy constraint for the $SL(2,{Z})$ action on the twistor space, quantum corrections to the classical mirror map were computed in [61], [62], [21], [26], in the large volume limit where the Kähler moduli are taken to be large, $t^a\\rightarrow \\infty $ .","In this limit, one finds $\\begin{split}u^a=&\\,b^a+It^a-\\frac{I}{2\\tau _2}{\\sum _{\\gamma \\in \\Gamma _+}}p^a\\left[\\int _{\\ell _{\\gamma }}\\mathrm {d}z\\,(1-z)\\,H_{\\gamma }+\\int _{\\ell _{-\\gamma }}\\frac{\\mathrm {d}z}{z^3} (1-z)\\,H_{-\\gamma }\\right]\\\\\\zeta ^0=&\\,\\tau _1,\\qquad \\zeta ^a=-(c^a-\\tau _1 b^a) -3\\sum _{\\gamma \\in \\Gamma _+} p^a\\,{\\rm Re}\\,\\int _{\\ell _{\\gamma }} \\mathrm {d}z\\, z\\,H_{\\gamma },\\\\\\tilde{\\zeta }_a =&\\,\\tilde{c}_a+ \\frac{1}{2}\\, \\kappa _{abc} \\,b^b (c^c - \\tau _1 b^c)+\\kappa _{abc}t^b \\sum _{\\gamma \\in \\Gamma _+} p^c\\,{\\rm Im}\\,\\int _{\\ell _{\\gamma }} \\mathrm {d}z\\,H_{\\gamma },\\end{split}$ where we introduced the convenient notationThe functions $H_\\gamma $ have a simple geometric meaning [32], [63]: they generate contact transformations (i.e.","preserving the contact structure) relating the Darboux coordinates living on patches separated by BPS rays.","In fact, these functions together with the contours $\\ell _\\gamma $ are the fundamental data fixing the contact structure on the twistor space.","$H_\\gamma (t)= \\frac{\\bar{\\Omega }(\\gamma )}{(2\\pi )^2}\\,\\sigma _\\gamma \\mathcal {X}_\\gamma (t)\\, .$ Similar results are known for $\\tilde{\\zeta }_0$ and the NS-axion dual to the $B$ -field, but will not be needed in this paper.","Note that the integral contributions to the mirror map are written in terms of the coordinate $z$ (REF ).","The reason for using this variable is that, in the large volume limit, the integrals along BPS rays $\\ell _\\gamma $ in (REF ) are dominated by the saddle point [21] $z^{\\prime }_\\gamma \\approx -I\\,\\frac{(q_a+(pb)_a)\\,t^a}{(pt^2)}\\, ,$ for $(pt^2)>0$ , and $z^{\\prime }_{-\\gamma }=1/z^{\\prime }_\\gamma $ in the opposite case.","This shows that all integrands can be expanded in Fourier series either around $z=0$ or $z=\\infty $ , keeping constant $t^a z$ or $t^a/z$ , respectively.","This allows to extract the leading order in the large volume limit in a simple way.","Let us therefore evaluate the combined limit $t^a\\rightarrow \\infty $ , $z\\rightarrow 0$ of the system of integral equations (REF ), assuming that only D3-D1-D(-1) instantons contribute.","As a first step, we rewrite the tree level expression (REF ) in terms of the type IIB fields.","To this end, we restrict the charge $\\gamma $ to lie in the cone $\\Gamma _+$ , take the central charge as in (REF ) with the cubicThe other contributions to the prepotential, representing perturbative $\\alpha ^{\\prime }$ -corrections and worldsheet instantons, combine with D(-1) and D1-instantons, but are irrelevant for our discussion of D3-instantons in the large volume limit.","prepotential (REF ), and substitute the mirror map (REF ).","Furthermore, we change the coordinate $t$ to $z$ and take the combined limit.","In this way one finds $\\mathcal {X}^{\\rm sf}_\\gamma (z)=\\mathcal {X}^{\\rm cl}_\\gamma (z)\\exp \\left[2\\pi \\sum _{\\gamma ^{\\prime }\\in \\Gamma _+} (tpp^{\\prime })\\int _{\\ell _{\\gamma ^{\\prime }}} \\mathrm {d}z^{\\prime }\\,H_{\\gamma ^{\\prime }}\\right],$ where $\\mathcal {X}^{\\rm cl}_\\gamma (z) ={\\bf e}\\!\\left( - \\hat{q}_0\\tau \\right)\\mathcal {X}^{(\\theta )}_{p,q}(z)$ is the classical part of the Darboux coordinates which we represented as a product of two factors: exponential of the invariant charge (REF ) and the remaining $q_0$ -independent exponential $\\mathcal {X}^{(\\theta )}_{p,q}(z) = e^{-S^{\\rm cl}_p}\\,{\\bf e}\\!\\left( - \\frac{\\tau }{2}\\,(q+b)^2+c^a (q_a +\\textstyle {1\\over 2}(pb)_a)+I\\tau _2 (pt^2)(z^2-2zz_\\gamma )\\right),$ with $S^{\\rm cl}_p$ being the leading part of the Euclidean D3-brane action in the large volume limit given by $S^{\\rm cl}_p= \\pi \\tau _2(pt^2) - 2\\pi Ip^a \\tilde{c}_a$ .","Next, we can approximate $\\frac{\\mathrm {d}t^{\\prime }}{t^{\\prime }}\\, \\frac{t+t^{\\prime }}{t-t^{\\prime }}=\\frac{2\\mathrm {d}z^{\\prime }}{1-z^{\\prime 2}}\\, \\frac{1-zz^{\\prime }}{z-z^{\\prime }}\\approx \\left\\lbrace \\begin{array}{cc}\\frac{2\\mathrm {d}z^{\\prime }}{z-z^{\\prime }}\\, , \\qquad & \\gamma ^{\\prime }\\in \\Gamma _+,\\\\\\frac{2(1-zz^{\\prime })\\mathrm {d}z^{\\prime }}{z^{\\prime 3}}\\, , \\qquad & \\gamma ^{\\prime }\\in -\\Gamma _+.\\end{array} \\right.$ This shows that the contribution of $\\gamma ^{\\prime }\\in -\\Gamma _+$ is suppressed comparing to $\\gamma ^{\\prime }\\in \\Gamma _+$ and therefore can be neglected.","As a result, the system of integral equations (REF ) in the large volume limit where only D3-D1-D(-1) instantons contribute reduces to the following system of integral equations for $H_\\gamma $ , $H_\\gamma (z)=H^{\\rm cl}_\\gamma (z) \\, \\exp \\left[\\sum _{\\gamma ^{\\prime }\\in \\Gamma _+}\\int _{\\ell _{\\gamma ^{\\prime }}}\\mathrm {d}z^{\\prime }\\, K_{\\gamma \\gamma ^{\\prime }}(z,z^{\\prime })\\,H_{\\gamma ^{\\prime }}(z^{\\prime })\\right].$ Here $H^{\\rm cl}_\\gamma $ is the classical limit of $H_\\gamma $ , i.e.","the function (REF ) with $\\mathcal {X}_\\gamma $ replaced by $\\mathcal {X}^{\\rm cl}_\\gamma $ , the integration kernel is now $K_{\\gamma _1\\gamma _2}(z_1,z_2)=2\\pi \\left((tp_1p_2)+\\frac{I\\langle \\gamma _1,\\gamma _2\\rangle }{z_1-z_2}\\right),$ and the BPS ray $\\ell _{\\gamma }$ effectively extends from $z^{\\prime }=-\\infty $ to $z^{\\prime }=+\\infty $ , going through the saddle point (REF ) [21].","Below we shall need a perturbative solution of the integral equation (REF ).","Applying the iterative procedure outlined below (REF ), or equivalently using the Lagrange inversion theorem, such solution can be written as a sum over rooted trees [36], $H_{\\gamma _1}(z_1) = H^{\\rm cl}_{\\gamma _1}(z_1) \\, \\sum _{n=1}^{\\infty } \\left(\\prod _{i=2}^n \\sum _{\\gamma _i\\in \\Gamma _+}\\,\\int _{\\ell _{\\gamma _i}} \\mathrm {d}z_i\\, H^{\\rm cl}_{\\gamma _i}(z_i) \\right) \\sum _{\\mathcal {T}\\in {T}_n^{\\rm r}}\\,\\frac{\\mathcal {A}(\\mathcal {T})}{|{\\rm Aut}(\\mathcal {T})|}\\, ,$ where ${T}_n^{\\rm r}$ is the set of rooted trees with $n$ vertices and $\\mathcal {A}(\\mathcal {T}) = \\prod _{e\\in E_{\\mathcal {T}}}K_{\\gamma _{s(e)},\\gamma _{t(e)}}(z_{s(e)},z_{t(e)}).$ A rooted treeWe will use calligraphic letters $\\mathcal {T}$ for trees where charges $\\gamma _i$ are assigned to vertices to distinguish them from rooted trees $T$ where the charges are assigned to leaves (hence $T$ has always more than $n$ vertices).","Similarly, we will use notations $\\mathfrak {v}$ and $v$ for vertices of these two types of trees, respectively.","Note also that whereas $V_\\mathcal {T}$ denotes the set of all vertices, $V_T$ does not includes the leaves.","An example of trees of the latter type are attractor flow trees.","$\\mathcal {T}$ consists of $n$ vertices joined by directed edges so that the root vertex has only outgoing edges, whereas all other vertices have one incoming edge and an arbitrary number of outgoing ones.","We label the vertices of $\\mathcal {T}$ by $\\mathfrak {v}=1,\\dots , n$ in an arbitrary fashion, except for the root which is labelled by $\\mathfrak {v}=1$ .","The symmetry factor $|{\\rm Aut}(\\mathcal {T})|$ is the order of the symmetry group which permutes the labels $2,\\dots , n$ without changing the topology of the tree.","Each vertex is decorated by a charge vector $\\gamma _\\mathfrak {v}$ and a complex variable $z_\\mathfrak {v}\\in {C}P^1$ .","We denote the set of edges by $E_{\\mathcal {T}}$ , the set of vertices by $V_{\\mathcal {T}}$ , and the source and target vertex of an edge $e$ by $s(e)$ and $t(e)$ , respectively.","Unpacking these notations, we get, at the few leading orders, $H_{\\gamma _1} &= & H^{\\rm cl}_{\\gamma _1} + \\sum _{\\gamma _2} K_{12} H^{\\rm cl}_{\\gamma _1} \\,H^{\\rm cl}_{\\gamma _2}+ \\sum _{\\gamma _2,\\gamma _3} \\left( \\tfrac{1}{2}\\, K_{12} K_{13} + K_{12} K_{23} \\right)H^{\\rm cl}_{\\gamma _1}\\, H^{\\rm cl}_{\\gamma _2}\\, H^{\\rm cl}_{\\gamma _3}\\\\& + & \\sum _{\\gamma _2,\\gamma _3,\\gamma _4} \\!\\!\\!\\left( \\tfrac{1}{6}\\, K_{12} K_{13} K_{14}+ \\tfrac{1}{2}\\, K_{12} K_{23} K_{24} + K_{12} K_{13} K_{24} + K_{12} K_{23} K_{34} \\right)H^{\\rm cl}_{\\gamma _1} \\, H^{\\rm cl}_{\\gamma _2}\\, H^{\\rm cl}_{\\gamma _3}\\, H^{\\rm cl}_{\\gamma _4}+ \\dots \\nonumber $ where we omitted the integrals and denoted $K_{ij}=K_{\\gamma _i,\\gamma _j}(z_i,z_j)$ .","The expansion (REF ) is effectively a multi-instanton expansion in powers of the DT invariants $\\bar{\\Omega }(\\gamma _i)$ , which is asymptotic to the exact solution to (REF ) in the weak coupling limit $\\tau _2\\rightarrow \\infty $ ."],["From the contact potential to the instanton generating function","Recall that our goal is to derive constraints imposed by S-duality on the DT invariants $\\bar{\\Omega }(\\gamma )$ appearing as coefficients in the multi-instanton expansion.","To achieve this goal, rather than studying the full metric on $\\mathcal {M}_H$ , it suffices to consider a suitable function on this moduli space which has a non-trivial dependence on $\\bar{\\Omega }(\\gamma )$ and specified transformations under S-duality.","There is a natural candidate with the above properties: the so-called contact potential $e^{\\phi }$ , a real function which is well-defined on any quaternion-Kähler manifold with a continuous isometry [34].","Furthermore, there is a general expression for the contact potential in terms of Penrose-type integrals on the ${C}P^1$ fiber.","In the present case, the required isometry is the shift of the NS-axion, which survives all quantum corrections as long as NS5-instantons are switched off.","The contact potential is then given by the exact formula [32] $e^{\\phi } = \\frac{I\\tau _2^2}{16}\\left(\\bar{u}^\\Lambda F_\\Lambda - u^\\Lambda \\bar{F}_\\Lambda \\right)-\\frac{\\chi _{{\\widehat{\\mathfrak {Y}}}}}{192\\pi }+\\frac{I\\tau _2}{16}\\,\\sum _\\gamma \\int _{\\ell _\\gamma } \\frac{\\text{d}t}{t} \\left( t^{-1} Z_\\gamma (u^a)-t\\bar{Z}_\\gamma (\\bar{u}^a)\\right) H_\\gamma ,$ where $\\chi _{{\\widehat{\\mathfrak {Y}}}}$ is the Euler characteristic of ${\\widehat{\\mathfrak {Y}}}$ .","This formula indeed captures contribution from D-instantons due to the last term proportional to $H_\\gamma $ .","On the other hand, in the classical, large volume limit one finds $e^\\phi =\\frac{\\tau _2^2}{12}(t^3)$ , which shows that the contact potential can be identified with the four-dimensional dilaton and in this approximation behaves as a modular form of weight $(-\\tfrac{1}{2}, -\\tfrac{1}{2})$ under S-duality transformations (REF ).","In fact, one can show [32] that $SL(2,{Z})$ preserves the contact structure, i.e.","it is an isometry of $\\mathcal {M}_H$ , only if the full non-perturbative contact potential transforms in this way, $e^\\phi \\mapsto \\frac{e^\\phi }{|c\\tau +d|}\\, .$ Furthermore, since S-duality acts by rescaling the Kähler moduli $t^a$ and by a phase rotation of the fiber coordinate $z$ (see (REF )), it preserves each order in the expansion around the large volume limit.","This implies that the large volume limit of the D3-instanton contribution to $e^\\phi $ , which we denote by $(e^\\phi )_{\\rm D3}$ , must itself transform as (REF ).","It is this condition that we shall exploit to derive modularity constraints on the DT invariants.","To make this condition more explicit, let us extract the D3-instanton contribution to the function (REF ).","The procedure is the same as the one used to get (REF ), and we relegate the details of the calculation to appendix .","The result can be written in a concise way using the complex function defined by $\\mathcal {G}&=& \\sum _{\\gamma \\in \\Gamma _+}\\int _{\\ell _{\\gamma }} \\mathrm {d}z\\, H_{\\gamma }(z)-{1\\over 2}\\,\\sum _{\\gamma _1,\\gamma _2\\in \\Gamma _+}\\int _{\\ell _{\\gamma _1}}\\mathrm {d}z_1\\,\\int _{\\ell _{\\gamma _2}} \\mathrm {d}z_2\\, K_{\\gamma _1\\gamma _2}(z_1,z_2)\\,H_{\\gamma _1}(z_1)H_{\\gamma _2}(z_2)$ and the Maass raising operator $\\mathcal {D}_{\\mathfrak {h}} = \\frac{1}{2\\pi I}\\left(\\partial _\\tau +\\frac{\\mathfrak {h}}{2I\\tau _2}+ \\frac{It^a}{4\\tau _2}\\, \\partial _{t^a}\\right),$ which maps modular functions of weight $(\\mathfrak {h},\\bar{\\mathfrak {h}})$ to modular functions of weight $(\\mathfrak {h}+2,\\bar{\\mathfrak {h}})$ .","Then one has (generalizing [14] to all orders in the instanton expansion) $(e^\\phi )_{\\rm D3}=\\frac{\\tau _2}{2}\\,{\\rm Re}\\,\\left(\\mathcal {D}_{-\\frac{3}{2}}\\mathcal {G}\\right)+\\frac{1}{32\\pi ^2}\\,\\kappa _{abc}t^c\\partial _{\\tilde{c}_a}\\mathcal {G}\\partial _{\\tilde{c}_b}\\overline{\\mathcal {G}}.$ It is immediate to see that $(e^\\phi )_{\\rm D3}$ transforms under S-duality as (REF ) provided the function $\\mathcal {G}$ transforms as a modular form of weight $(-\\tfrac{3}{2},\\tfrac{1}{2})$ .","In order to derive the implications of this fact, we need to express $\\mathcal {G}$ in terms of the generalized DT invariants.","For this purpose, we substitute the multi-instanton expansion (REF ) into (REF ).","We claim that the result takes the simple form $\\mathcal {G}=\\sum _{n=1}^\\infty \\left[\\prod _{i=1}^{n} \\sum _{\\gamma _i\\in \\Gamma _+}\\int _{\\ell _{\\gamma _i}}\\mathrm {d}z_i\\, H^{\\rm cl}_{\\gamma _i}(z_i) \\right]\\mathcal {G}_n(\\lbrace \\gamma _i,z_i\\rbrace ),$ where $\\mathcal {G}_n(\\lbrace \\gamma _i,z_i\\rbrace )$ is now a sum over unrooted trees with $n$ vertices, $\\mathcal {G}_n(\\lbrace \\gamma _i,z_i\\rbrace )=\\sum _{\\mathcal {T}\\in \\, {T}_n} \\frac{\\mathcal {A}(\\mathcal {T})}{|{\\rm Aut}(\\mathcal {T})|}=\\frac{1}{n!","}\\sum _{\\mathcal {T}\\in \\, {T}_n^\\ell } \\mathcal {A}(\\mathcal {T}),$ and in the second equality we rewrote the result as a sum over unrooted labelled trees.The number of such trees is $|{T}_n^\\ell |=n^{n-2}=\\lbrace 1,1,3,16,125,1296,\\dots \\rbrace $ for $n\\ge 1$ .","Such trees also appear in the Joyce-Song wall-crossing formula [42], [43] and are conveniently labelled by their Prüfer code.","To see why this is the case, observe that under the action of the Euler operator $\\hat{D}=H^{\\rm cl}\\partial _{H^{\\rm cl}}$ rescaling all functions $H^{\\rm cl}_\\gamma $ , the function $\\mathcal {G}$ maps to the first term in (REF ), which we denote by $\\mathcal {F}$ .","Namely, $\\hat{D}\\cdot \\mathcal {G}=\\mathcal {F},\\qquad \\mathcal {F}\\equiv \\sum _{\\gamma \\in \\Gamma _+}\\int _{\\ell _{\\gamma }} \\mathrm {d}z\\, H_{\\gamma },$ as can be verified with the help of the integral equation (REF ).","The multi-instanton expansion of $\\mathcal {F}$ follows immediately from (REF ), $\\mathcal {F}= \\sum _{n=1}^{\\infty } \\left(\\prod _{i=1}^n \\sum _{\\gamma _i\\in \\Gamma _+}\\,\\int _{\\ell _{\\gamma _i}} \\mathrm {d}z_i\\, H^{\\rm cl}_{\\gamma _i}(z_i) \\right) \\sum _{\\mathcal {T}\\in {T}_n^{\\rm r}}\\,\\frac{\\mathcal {A}(\\mathcal {T})}{|{\\rm Aut}(\\mathcal {T})|}\\, .$ Integrating back the action of the derivative operator $\\hat{D}$ , we see that the sum over rooted trees in (REF ) turns into the sum over unrooted trees in (REF ).","At the first few orders we get, using the same shorthand notation as in (REF ), $\\begin{split}\\mathcal {G}= &\\sum _{\\gamma } H^{\\rm cl}_\\gamma + \\frac{1}{2} \\sum _{\\gamma _1,\\gamma _2}K_{12}H^{\\rm cl}_{\\gamma _1} \\,H^{\\rm cl}_{\\gamma _2}+ \\frac{1}{2} \\sum _{\\gamma _1,\\gamma _2,\\gamma _3}K_{12}K_{23}H^{\\rm cl}_{\\gamma _1}\\, H^{\\rm cl}_{\\gamma _2}\\, H^{\\rm cl}_{\\gamma _3}\\\\&+\\sum _{\\gamma _1,\\gamma _2,\\gamma _3,\\gamma _4}\\left(\\frac{1}{6} \\, K_{12} K_{13} K_{14} + \\frac{1}{2}\\,K_{12} K_{23} K_{34} \\right)H^{\\rm cl}_{\\gamma _1}\\, H^{\\rm cl}_{\\gamma _2}\\, H^{\\rm cl}_{\\gamma _3}\\, H^{\\rm cl}_{\\gamma _4}+\\dots \\end{split}$ The simplicity of the expansion (REF ), and the relation (REF ) to the contact potential, show that the function $\\mathcal {G}$ is very natural and, in some sense, more fundamentalIn [14], it was noticed that the function $\\mathcal {G}$ , denoted by $\\tilde{\\mathcal {F}}$ in that reference and computed at second order in the multi-instanton expansion, could be obtained from the seemingly simpler function $\\mathcal {F}$ by halving the coefficient of its second order contribution.","Now we see that this ad hoc prescription is the consequence of going from rooted to unrooted trees, as a result of adding the second term in (REF ).","than the naive instanton sum $\\mathcal {F}$ .","We shall henceforth refer to $\\mathcal {G}$ as the `instanton generating function'.","In the following we shall postulate that $\\mathcal {G}$ transforms as a modular form of weight $(-\\frac{3}{2},\\frac{1}{2})$ , and analyze the consequences of this requirement for the DT invariants."],["Theta series decomposition and modularity","In this section, we use the spectral flow symmetry to decompose the instanton generating function $\\mathcal {G}$ into a sum of indefinite theta series multiplied by holomorphic generating functions of MSW invariants.","We then study the modular properties of these indefinite theta series, and identify the origin of the modular anomaly."],["Factorisation","To derive modularity constraints on the DT invariants, we need to perform a theta series decomposition of the generating function $\\mathcal {G}$ defined in (REF ).","To this end, let us make use of the fact noticed in (REF ) that the DSZ products $\\langle \\gamma ,\\gamma ^{\\prime }\\rangle $ and hence the kernels (REF ) do not depend on the $q_0$ charge.","Choosing the quadratic refinement $\\sigma _\\gamma $ as in (REF ), which is also $q_0$ -independent, and using the factorization (REF ) of $\\mathcal {X}^{\\rm cl}_\\gamma $ , one can rewrite the expansion (REF ) as follows $\\mathcal {G}=\\sum _{n=1}^\\infty \\frac{1}{(2\\pi )^{2n}}\\left[\\prod _{i=1}^{n}\\sum _{p_i,q_i}\\sigma _{p_i,q_i}h^{\\rm DT}_{p_i,q_i}\\int _{\\ell _{\\gamma _i}}\\mathrm {d}z_i\\, \\mathcal {X}^{(\\theta )}_{p_i,q_i}(z_i) \\right]\\mathcal {G}_n(\\lbrace \\gamma _i,z_i\\rbrace ),$ where the sum over the invariant charges $\\hat{q}_{i,0}$ gave rise to the generating functions of DT invariants defined in (REF ).","This is not yet the desired form because these generating functions depend non-trivially on the remaining electric charges $q_{i,a}$ .","If it were not for this dependence, the sum over $q_{i,a}$ would produce certain non-Gaussian theta series, and at each order we would have a product of this theta series and $n$ generating functions.","Then the modular properties of the theta series would dictate the modular properties of the generating function.","Such a theta series decomposition can be achieved by expressing the DT invariants in terms of the MSW invariants, for which the dependence on electric charges $q_{i,a}$ reduces to the dependence on the residue classes $\\mu _{i,a}$ due to the spectral flow symmetry.","Substituting the expansion (REF ) of $h^{\\rm DT}_{p,q}$ in terms of $h_{p,\\mu }$ , the expansion (REF ) of the function $\\mathcal {G}$ can be brought to the following factorized form $\\mathcal {G}=\\sum _{n=1}^\\infty \\frac{2^{-\\frac{n}{2}}}{\\pi \\sqrt{2\\tau _2}}\\left[\\prod _{i=1}^{n}\\sum _{p_i,\\mu _i}\\sigma _{p_i}h_{p_i,\\mu _i}\\right]e^{-S^{\\rm cl}_p}\\vartheta _{{p},{\\mu }}\\bigl (\\Phi ^{{\\rm tot}}_{n},n-2\\bigr ),$ where $\\vartheta _{{p},{\\mu }}$ is a theta series (REF ) with parameter $\\lambda =n-2$ , whose kernel has the following structure $\\Phi ^{{\\rm tot}}_n({x})=\\,{\\rm Sym}\\, \\left\\lbrace \\sum _{n_1+\\cdots n_m=n \\atop n_k\\ge 1} \\Phi ^{\\scriptscriptstyle \\,\\int }_m({x}^{\\prime })\\prod _{k=1}^m \\Phi ^{\\,g}_{n_k}(x_{j_{k-1}+1},\\dots ,x_{j_k})\\right\\rbrace .$ Here the sum runs over ordered partitions of $n$ , whose number of parts is denoted by $m$ , and we adopted notations from (REF ) for indices $j_k$ .","The argument ${x}$ of the kernel encodes the electric components of the charges (shifted by the B-field and rescaled by $\\sqrt{2\\tau _2}$ ) and lives in a vector space $\\left(\\oplus _{i=1}^n \\Lambda _i\\right) \\otimes {R}$ of dimension $d=nb_2$ , given by $b_2$ copies of the lattice $\\Lambda $ , where the $i$ -th copy $\\Lambda _i$ carries the bilinear form $\\kappa _{i,ab}=\\kappa _{abc}p_i^c$ of signature $(1,b_2-1)$ .","Therefore, $\\vartheta _{{p},{\\mu }}$ is an indefinite theta series associated to the bilinear form given explicitly in (REF ), which has signature $(n,n(b_2-1))$ .","Finally, the kernel (REF ) is constructed from two other functions.","The first, $\\Phi ^{\\scriptscriptstyle \\,\\int }_n$ , is the iterated integral of the coefficient $\\mathcal {G}_n$ in the expansion (REF ), $\\Phi ^{\\scriptscriptstyle \\,\\int }_n({x}) =\\left(\\frac{\\sqrt{2\\tau _2}}{2\\pi }\\right)^{n-1}\\left[\\prod _{i=1}^n\\int _{\\ell _{\\gamma _i}}\\frac{\\mathrm {d}z_i}{2\\pi } \\, W_{p_i}(x_i,z_i)\\right]\\mathcal {G}_n(\\lbrace \\gamma _i,z_i\\rbrace ),$ weighted with the Gaussian measure factor $W_{p}(x,z)=e^{-2\\pi \\tau _2 z^2(pt^2)-2\\pi I\\sqrt{2\\tau _2}\\, z\\, (pxt)}$ coming from the $z$ -dependent part of (REF ).","Although this function is written in terms of $\\mathcal {G}_n$ depending on full electromagnetic charge vectors $\\gamma _i$ , it is actually independent of their $q_0$ components.","Indeed, using the result (REF ), it can be rewritten as $\\Phi ^{\\scriptscriptstyle \\,\\int }_n({x}) = \\frac{1}{n!","}\\left[\\prod _{i=1}^n\\int _{\\ell _{\\gamma _i}}\\frac{\\mathrm {d}z_i}{2\\pi } \\, W_{p_i}(x_i,z_i)\\right]\\sum _{\\mathcal {T}\\in \\, {T}_n^\\ell }\\prod _{e\\in E_\\mathcal {T}} \\hat{K}_{s(e) t(e)},$ where we introduced a rescaled version of the kernel (REF ) $\\hat{K}_{ij}(z_i,z_j)=\\left(\\left(\\sqrt{2\\tau _2}\\,t+I\\,\\frac{x_i-x_j}{z_i-z_j}\\right)p_ip_j\\right).$ Note that $\\tau _2$ and $t^a$ appear only in the modular invariant combination $\\sqrt{2\\tau _2}\\, t^a$ .","In (REF ) this function appears with the argument ${x}^{\\prime }$ and carries a dependence on ${p}^{\\prime }$ (not indicated explicitly) which are both $mb_2$ -dimensional vectors with components (cf.","(REF )) $p^{\\prime a}_k=\\sum _{i=j_{k-1}+1}^{j_k}p^a_i,\\qquad x^{\\prime a}_k=\\kappa ^{\\prime ab}_k\\sum _{i=j_{k-1}+1}^{j_k} \\kappa _{i,bc} x^c_i,$ where $\\kappa ^{\\prime }_{k,ab}=\\kappa _{abc}p^{\\prime c}_k$ .","The second function, $\\Phi ^{\\,g}_{n}$ , appears due to the expansion of DT invariants in terms of the MSW invariants and is given by a suitably rescaled tree index $\\Phi ^{\\,g}_{n}({x})=\\frac{\\sigma _{\\gamma }(\\sqrt{2\\tau _2})^{n-1}}{\\prod _{i=1}^n\\sigma _{\\gamma _i}}\\,g_{{\\rm tr},n}(\\lbrace \\gamma _i,c_i\\rbrace ).$ It is also written in terms of functions depending on the full electromagnetic charge vectors $\\gamma _i$ (with $\\gamma =\\gamma _1+\\cdots \\gamma _n$ ).","However, using (REF ), all quadratic refinements can be expressed through $(-1)^{\\langle \\gamma _i,\\gamma _j\\rangle }$ which cancel the corresponding sign factors in the tree index (see (REF )).","Furthermore, as was noticed in the end of section REF , the tree index is independent of the $q_0$ components of the charge vectors.","Therefore, it can be written as a function of $p_i^a$ , $\\mu _{i,a}$ and $x_i^a=\\sqrt{2\\tau _2}(\\kappa _i^{ab} q_{i,b}+b^a)$ .","Then, since after cancelling the sign factors, $g_{{\\rm tr},n}$ becomes a homogeneous function of degree $n-1$ in the D2-brane charge $q_{i,a}$ , all factors of $\\sqrt{2\\tau _2}$ in (REF ) cancel as well."],["Modularity and Vignéras' equation","As explained in appendix REF , the theta series $\\vartheta _{{p},{\\mu }}\\bigl (\\Phi ,\\lambda \\bigr )$ is a vector-valued modular form of weight $(\\lambda +d/2,0)$ provided the kernel $\\Phi $ satisfies Vignéras' equation (REF ) — along with certain growth conditions which we expect to be automatically satisfied for the kernels of interest in this work.","In our case $\\lambda =n-2$ and the dimension of the lattice is $d=nb_2(\\mathfrak {Y})$ so that the expected weight of the theta series is $(2(n-1)+nb_2/2,0)$ .","It is consistent with weight $(-\\tfrac{3}{2},\\tfrac{1}{2})$ of $\\mathcal {G}$ given in (REF ) only if $h_{p,\\mu }$ is a vector-valued holomorphic modular form of weight $(-b_2/2-1,0)$ .","However for this to be true, the kernel $\\Phi ^{{\\rm tot}}_n$ ought to satisfy Vignéras' equation.","Let us examine whether or not this is the case.","To this end, we first consider the kernel $\\Phi ^{\\scriptscriptstyle \\,\\int }_n$ (REF ).","In appendix we evaluate explicitly the iterated integrals defining this kernel.","To present the final result, let us introduce the following $d$ -dimensional vectors ${v}_{ij}, {u}_{ij}$ : $\\begin{split}({v}_{ij})_k^a=&\\, \\delta _{ki} p_j^a-\\delta _{kj} p_i^a \\qquad \\qquad \\quad \\ \\mbox{such that} \\quad {v}_{ij}\\cdot {x}=(p_ip_j(x_i-x_j)),\\\\({u}_{ij})_k^a=&\\, \\delta _{ki}(p_jt^2)t^a-\\delta _{kj} (p_it^2)t^a \\quad \\mbox{such that} \\quad {u}_{ij}\\cdot {x}=(p_jt^2)(p_ix_it)-(p_it^2)(p_jx_jt),\\end{split}$ where $k$ labels the copy in $\\oplus _{k=1}^n \\Lambda _k$ , $a=1,\\dots , b_2$ , and the bilinear form is given in (REF ).","The first scalar product ${v}_{ij}\\cdot {x}$ corresponds to the DSZ product $\\langle \\gamma _i,\\gamma _j\\rangle $ , whereas the second product ${u}_{ij}\\cdot {x}$ corresponds to $-2\\,{\\rm Im}\\,[Z_{\\gamma _i}\\bar{Z}_{\\gamma _j}]$ (REF ), both rescaled by $\\sqrt{2\\tau _2}$ and expressed in terms of $x_i^a$ .","From these vectors we can construct two sets of vectors which are assigned to the edges of an unrooted labelled tree $\\mathcal {T}$ , such as the trees appearing in (REF ) and (REF ).","Namely, ${v}_e=\\sum _{i\\in V_{\\mathcal {T}_e^s}}\\sum _{j\\in V_{\\mathcal {T}_e^t}}{v}_{ij},\\qquad {u}_e=\\sum _{i\\in V_{\\mathcal {T}_e^s}}\\sum _{j\\in V_{\\mathcal {T}_e^t}}{u}_{ij},$ where $\\mathcal {T}_e^s$ , $\\mathcal {T}_e^t$ are the two disconnected trees obtained from the tree $\\mathcal {T}$ by removing the edge $e$ .","Then the kernel $\\Phi ^{\\scriptscriptstyle \\,\\int }_n$ can be expressed as followsBoth vectors ${u}_e$ and ${v}_{s(e) t(e)}$ depend on the choice of orientation of the edge $e$ , but this ambiguity is cancelled in the function $\\widetilde{\\Phi }^M_{n-1}$ .","$\\Phi ^{\\scriptscriptstyle \\,\\int }_n({x})=\\frac{\\Phi ^{\\scriptscriptstyle \\,\\int }_1(x)}{2^{n-1} n!","}\\sum _{\\mathcal {T}\\in \\, {T}_n^\\ell }\\widetilde{\\Phi }^M_{n-1}(\\lbrace {u}_e\\rbrace , \\lbrace {v}_{s(e) t(e)}\\rbrace ;{x}).$ Here the first factor is simply a Gaussian $\\Phi ^{\\scriptscriptstyle \\,\\int }_1(x)=\\frac{e^{-\\frac{\\pi (pxt)^2}{(pt^2)}}}{2\\pi \\sqrt{2\\tau _2(pt^2)}}$ which ensures the suppression along the direction of the total charge in the charge lattice.","In the second factor one sums over unrooted labelled trees $\\mathcal {T}$ with $n$ vertices, with summand given by a function $\\widetilde{\\Phi }^M_n$ defined as in (REF ), upon replacing $\\Phi _n^E$ by $\\Phi _n^M$ in that expression and setting $m=n$ .","Both $\\Phi _n^E$ and $\\Phi _n^M$ are the so-called generalized (complementary) error functions introduced in [28] and further studied in [29], whose definitions are recalled in (REF ), () and (REF ).","The functions $\\widetilde{\\Phi }^M_{n-1}$ in (REF ) depend on two sets of $n-1$ $d$ -dimensional vectors: the vectors in the first set are given by ${u}_e$ defined above, whereas the vectors ${v}_{s(e) t(e)}$ in the second set coincide with ${v}_{ij}$ for $i$ and $j$ corresponding to the source and target vertices of edge $e$ of the labelled tree.","The remarkable property of the generalized error functions $\\Phi ^M_{n-1}$ and their uplifted versions $\\widetilde{\\Phi }^M_{n-1}$ is that, away from certain loci where these functions are discontinuous, they satisfy Vignéras' equation for $\\lambda =0$ and $n-1$ , respectively.","Given that $\\Phi ^{\\scriptscriptstyle \\,\\int }_1$ is also a solution for $\\lambda =-1$ , and the vector ${t}=(t^a,\\dots ,t^a)$ (such that ${t}\\cdot {x}=(pxt)$ ) is orthogonal to all vectors ${u}_e$ and ${v}_{s(e) t(e)}$ , the kernel (REF ) satisfies this equation for $\\lambda =n-2$ .","However, as mentioned above, it fails to do so on the loci where it is discontinuous.","These discontinuities arise due to dependence of the integration contours $\\ell _\\gamma $ on moduli and electric charges.","Of course, since the integrands are meromorphic functions, the integrals do not depend on deformations of the contours provided they do not cross the poles.","But this is exactly what happens when two BPS rays, say $\\ell _\\gamma $ and $\\ell _{\\gamma ^{\\prime }}$ , exchange their positions, which in turn takes place when the phases of the corresponding central charges $Z_\\gamma $ and $Z_{\\gamma ^{\\prime }}$ align, as follows from (REF ).","The loci where such alignment takes place are nothing else but the walls of marginal stability.","This point will play an important rôle in the next subsection since it makes it possible to recombine the discontinuities of the generalized error functions with discontinuities of the tree indices.","We now turn to the action of Vignéras' operator on $\\Phi ^{\\,g}_{n}$ .","To this end, it is convenient to use the representation of the tree index as a sum over attractor flow trees (REF ).","Let us assign a $nb_2$ -dimensional vector $\\tilde{{v}}_{v}$ to each vertex $v$ of a flow tree.","Denoting by $\\mathcal {I}_v$ the set of leaves which are descendants of vertex $v$ , we set $\\tilde{{v}}_v=\\sum _{i\\in \\mathcal {I}_{L(v)}}\\sum _{j\\in \\mathcal {I}_{R(v)}}{v}_{ij}.$ With these definitions the kernel (REF ) can be written as $\\Phi ^{\\,g}_n({x}) &=&(-1)^{n-1}\\sum _{T\\in {T}_n^{\\rm af}}\\prod _{v\\in V_T}(\\tilde{{v}}_v,{x})\\,\\Delta _{\\gamma _{L(v)}\\gamma _{R(v)}}^{z_{p(v)}}.$ The factors $\\Delta _{\\gamma _L\\gamma _R}^z$ are locally constant and therefore, away from the loci where they are discontinuous, the action of Vignéras operator reduces to its action on the scalar products $(\\tilde{{v}}_v,{x})$ .","For a single such factor one finds $V_\\lambda (\\tilde{{v}}_v,{x})=(\\tilde{{v}}_v,{x})V_{\\lambda -1}+2\\tilde{{v}}_v\\cdot \\partial _{x}.$ The crucial observation is that all vectors $\\tilde{{v}}_v$ appearing in the product (REF ) for a single tree are mutually orthogonal $(\\tilde{{v}}_v,\\tilde{{v}}_{v^{\\prime }})=0$ , which is clear because $\\langle \\gamma _{L(v)},\\gamma _{R(v)}\\rangle $ is antisymmetric in charges $\\gamma _{L(v)},\\gamma _{R(v)}$ , whereas the factors associated with vertices which are not descendants of $v$ either depend on their sum or do not depend on them at all.","Therefore, one obtains $V_\\lambda \\Phi ^{\\,g}_{n_k}({x})=\\Phi ^{\\,g}_{n_k}({x})V_{\\lambda -n_k+1}+2(-1)^{n_k-1}\\sum _{T\\in {T}_{n_k}^{\\rm af}}\\Delta (T)\\sum _{v\\in V_T}\\left[\\prod _{v^{\\prime }\\in V_T\\setminus \\lbrace v\\rbrace }(\\tilde{{v}}_{v^{\\prime }},{x})\\right]\\tilde{{v}}_v\\cdot \\partial _{x}.$ Let us now evaluate the action of $V_{n-2}$ on the full kernel $\\Phi ^{{\\rm tot}}_n$ .","Applying the result (REF ), we observe that the second term vanishes on the other factors in (REF ) due to the same reason that they either do not depend (in the case of $\\Phi ^{\\,g}_{n_{k^{\\prime }}}$ , $k^{\\prime }\\ne k$ ) or depend (in the case of $\\Phi ^{\\scriptscriptstyle \\,\\int }_{m}$ ) only on the sum of charges entering $\\Phi ^{\\,g}_{n_k}$ .","Therefore, one finds that, away from discontinuities of generalized error functions and $\\Delta (T)$, one has $V_{n-2}\\cdot \\Phi ^{{\\rm tot}}_n=0$ ."],["Discontinuities and the anomaly","Let us now turn to the discontinuities of $\\Phi ^{{\\rm tot}}_n$ which we ignored so far and which spoil Vignéras' equation and hence modularity of the theta series.","There are three potential sources of such discontinuities: walls of marginal stability — at these loci $\\Phi ^{\\scriptscriptstyle \\,\\int }_{m}$ are discontinuous due to exchange of integration contours and $\\Phi ^{\\,g}_{n_k}$ jump due to factors $\\Delta _{\\gamma _{L(v)}\\gamma _{R(v)}}^{z_{p(v)}}$ assigned to the root vertices of attractor flow trees; `fake walls' — these are loci in the moduli space where $\\,{\\rm Im}\\,\\bigl [ Z_{\\gamma _{L(v)}}\\bar{Z}_{\\gamma _{R(v)}}(z^a_{p(v)})\\bigr ]=0$ , and hence the corresponding $\\Delta $ -factor jumps, where $v$ is not a root vertex — they correspond to walls of marginal stability for the intermediate bound states appearing in the attractor flow; moduli independent loci where $(\\tilde{{v}}_v,{x})=0$ — at these loci the factors $\\Delta _{\\gamma _{L(v)}\\gamma _{R(v)}}^{z_{p(v)}}$ and hence $\\Phi ^{\\,g}_{n_k}$ are discontinuous due to the second term in (REF ).","Remarkably, the two effects due to the non-trivial charge and moduli dependence of the DT invariants and the exchange of contours cancel each other and the function $\\mathcal {G}$ turns out to be smooth at loci of the first type.","This is expected because the whole construction of D-instantons has been designed to make the resulting metric on the moduli space smooth across these loci, which required the cancellation of these discontinuities [36], [35].","Moreover, in [37] it was proven that the contact potential is also smooth, which indicates that the function $\\mathcal {G}$ must be smooth as well.","In appendix we present an explicit proof of this fact based on the representation in terms of trees.","Furthermore, in [23] it was shown that the discontinuities across `fake walls' cancel in the sum over flow trees as well.","In fact, this cancellation is explicit in the representation of the partial tree index given by the recursive formula (REF ) where the signs responsible for such `fake discontinuities' do not arise at all.","As a result, it remains to consider only the discontinuities of the third type corresponding to the moduli independent loci.","It is straightforward to check that already for $n=2$ these discontinuities are indeed present and do spoil modularity of the theta series.","For small $n$ one can explicitly evaluate the anomaly in Vignéras' equation.","It is given by a series of terms proportional to $\\delta (\\tilde{{v}}_v,{x})$ .","Note that no derivatives of delta functions or products of two delta functions arise despite the presence of the second derivative in Vignéras' operator.","This is because each ${\\rm sgn}(\\tilde{{v}}_v,{x})$ from $\\Delta (T)$ is multiplied by $(\\tilde{{v}}_v,{x})$ from $\\kappa (T)$ in (REF ) and one gets a non-vanishing result only if the second order derivative operator acts on both factors.","In particular, this implies that the anomaly is completely characterized by the action of $V_\\lambda $ on $\\Phi ^{\\,g}_{n}$ ."],["Modular completion","Since the theta series $\\vartheta _{{p},{\\mu }}(\\Phi ^{{\\rm tot}}_{n})$ are not modular for $n\\ge 2$ , the analysis of the previous section implies that the generating function $h_{p,\\mu }$ of the MSW invariants is not modular either whenever the divisor $\\mathcal {D}=p^a\\gamma _a$ is the sum of $n\\ge 2$ irreducible divisors.","However, its modular anomaly has a definite structure.","In particular, in [14] it was shown that for $n=2$ , $h_{p,\\mu }$ must be a vector-valued mixed mock modular form, i.e.","it has a non-holomorphic completion $\\widehat{h}_{p,\\mu }$ constructed in a specific way from a set of holomorphic modular forms and their Eichler integrals [30], [13].","In this section we generalize this result for arbitrary $n$ , i.e.","for any degree of reducibility of the divisor."],["Completion of the generating function","Let us recall the notations $\\check{\\gamma }=(p^a,q_a)$ and $Q_n$ from (REF ), and decompose the electric component $q_a$ using spectral flow as in (REF ).","Then we define $\\widehat{h}_{p,\\mu }(\\tau )= h_{p,\\mu }(\\tau )+\\sum _{n=2}^\\infty \\sum _{\\sum _{i=1}^n \\check{\\gamma }_i=\\check{\\gamma }}R_n(\\lbrace \\check{\\gamma }_i\\rbrace ,\\tau _2)\\, e^{\\pi I\\tau Q_n(\\lbrace \\check{\\gamma }_i\\rbrace )}\\prod _{i=1}^n h_{p_i,\\mu _i}(\\tau ).$ We are looking for non-holomorphic functions $R_n$ , exponentially suppressed for large $\\tau _2$ , such that $\\widehat{h}_{p,\\mu }$ transforms as a modular form.","The condition for this to be true can be found along the same lines as before: one needs to rewrite the expansion of the function $\\mathcal {G}$ as a series in $\\widehat{h}_{p,\\mu }$ and require that at each order the coefficient is given by a modular covariant theta series.","For such a theta series decomposition to be possible however, it is important that $\\widehat{h}_{p,\\mu }$ be invariant under the spectral flow, which implies that the functions $R_n$ be independent of the spectral flow parameter $\\epsilon ^a$ in the decomposition (REF ) of the total charge $\\check{\\gamma }$ .As a result, this parameter can be fixed to zero so that the sum over the D2-brane charges $q_{i,a}$ is restricted to those which satisfy the constraint $\\sum _{i=1}^n q_{i,a}=\\mu _a+\\frac{1}{2} \\kappa _{abc}p^b p^c$ .","This condition will be an important consistency requirement on our construction.","Rather than inverting (REF ) and substituting the result into (REF ), we can consider the generating function of DT invariants $h^{\\rm DT}_{p,\\mu }$ and, as a first step, rewrite it as a series in $\\widehat{h}_{p,\\mu }$ .","Denoting the coefficient of such expansion by $\\widehat{g}_n(\\lbrace \\check{\\gamma }_i\\rbrace ,z^a)$ (with $\\widehat{g}_1\\equiv 1$ ), we get $\\begin{split}h^{\\rm DT}_{p,q}(\\tau ,z^a)=&\\,\\sum _{\\sum _{i=1}^n \\check{\\gamma }_i=\\check{\\gamma }}\\widehat{g}_n(\\lbrace \\check{\\gamma }_i\\rbrace ;z^a, \\tau _2) \\,e^{\\pi I\\tau Q_n(\\lbrace \\check{\\gamma }_i\\rbrace )}\\prod _{i=1}^n \\widehat{h}_{p_i,\\mu _i}(\\tau ).\\end{split}$ Comparing with (REF ), we see that the coefficients $\\widehat{g}_n$ are a direct analogue of the tree index $g_{{\\rm tr},n}$ .In fact, they also depend on $z^a$ only through the stability parameters (REF ), so we shall often denote them by $\\widehat{g}_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace ;\\tau _2)$ .","The expansion of $\\mathcal {G}$ in terms of $\\widehat{h}_{p,\\mu }$ is then simply obtained by replacing $g_{{\\rm tr},n}$ by $\\widehat{g}_n$ in (REF ), which affects the kernel of the corresponding theta series.","Our first problem is to express these coefficients in terms of the functions $R_n$ .","The result can be nicely formulated using so-called Schröder trees, which are rooted ordered trees such that all vertices (except the leaves) have at least 2 children (see Fig.","REF ).","Their vertices are decorated by charges in such way that the leaves have charges $\\gamma _i$ , whereas the charges at other vertices are defined by the inductive rule $\\gamma _v=\\sum _{v^{\\prime }\\in {\\rm Ch}(v)}\\gamma _{v^{\\prime }}$ where ${\\rm Ch}(v)$ is the set of children of vertex $v$ .","Note that these trees should not be confused with flow trees, since they are not restricted to be binary nor do they carry moduli at the vertices.","We denote the set of Schröder treesThe number of Schröder trees with $n$ leaves is the $n-1$ -th super-Catalan number, $|{T}_n^{\\rm S}|=\\lbrace 1,1, 3,11,45,197,903,\\dots \\rbrace $ for $n\\ge 1$ (sequence A001003 on the Online Encyclopedia of Integer Sequences).","with $n$ leaves by ${T}_n^{\\rm S}$ .","Figure: An example of Schröder tree contributing to W 8 W_8.","Near each vertex we showed the corresponding factorusing the shorthand notation γ i+j =γ i +γ j \\gamma _{i+j}=\\gamma _i+\\gamma _j.Let us also introduce a convenient notation: for any set of functions $f_n(\\lbrace \\check{\\gamma }_i\\rbrace )$ depending on $n$ charges and a given Schröder tree $T$ , we set $f_{v}\\equiv f_{k_v}(\\lbrace \\check{\\gamma }_{v^{\\prime }}\\rbrace )$ where $v^{\\prime }\\in {\\rm Ch}(v)$ and $k_v$ is their number.","Using these notations, the expression of $\\widehat{g}_n$ in terms of $R_n$ can be encoded into a recursive equation provided by the following proposition, whose proof we relegate to appendix : Proposition 1 The coefficients $\\widehat{g}_n$ are determined recursively by the following equation $\\widehat{g}_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace )=-{1\\over 2}\\,{\\rm Sym}\\, \\left\\lbrace \\sum _{\\ell =1}^{n-1}\\Delta _{\\gamma _L^\\ell \\gamma _R^\\ell }^z\\,\\kappa (\\gamma _{LR}^\\ell )\\,\\widehat{g}_\\ell (\\lbrace \\check{\\gamma }_i,c_i^{(\\ell )}\\rbrace _{i=1}^\\ell )\\,\\widehat{g}_{n-\\ell }(\\lbrace \\check{\\gamma }_i,c_i^{(\\ell )}\\rbrace _{i=\\ell +1}^n)\\right\\rbrace +W_n(\\lbrace \\check{\\gamma }_i\\rbrace ),$ where $\\gamma _L^\\ell =\\sum _{i=1}^\\ell \\gamma _i$ , $\\gamma _R^\\ell =\\sum _{i=\\ell +1}^n\\gamma _i$ , $\\gamma _{LR}^\\ell =\\langle \\gamma _L^\\ell ,\\gamma _R^\\ell \\rangle $ , $c_i^{(\\ell )}$ are the stability parameters at the point where the attractor flow crosses the wall for the decay $\\gamma \\rightarrow \\gamma _L^\\ell +\\gamma _R^\\ell $ (cf.","(REF )), and $W_n$ are functions given by the sum over Schröder trees $W_n(\\lbrace \\check{\\gamma }_i\\rbrace )= \\,{\\rm Sym}\\, \\left\\lbrace \\sum _{T\\in {T}_n^{\\rm S}}(-1)^{n_T}\\prod _{v\\in V_T} R_{v}\\right\\rbrace ,$ with $n_T$ being the number of vertices of the tree $T$ (excluding the leaves).","The same functions $W_n$ also provide the inverse formula to (REF ), namely $h_{p,\\mu }(\\tau )= \\widehat{h}_{p,\\mu }(\\tau )+\\sum _{n=2}^\\infty \\sum _{\\sum _{i=1}^n \\check{\\gamma }_i=\\check{\\gamma }}W_n(\\lbrace \\check{\\gamma }_i\\rbrace )\\, e^{\\pi I\\tau Q_n(\\lbrace \\check{\\gamma }_i\\rbrace )}\\prod _{i=1}^n \\widehat{h}_{p_i,\\mu _i}(\\tau ).$ What are the conditions on $\\widehat{g}_n$ for the corresponding theta series to be modular?","Let us denote by $\\Phi ^{\\,\\widehat{g}}_{n}$ the kernel defined by $\\widehat{g}_n$ analogously to (REF ), namely $\\Phi ^{\\,\\widehat{g}}_{n}({x})=\\frac{\\sigma _{\\gamma }(\\sqrt{2\\tau _2})^{n-1}}{\\prod _{i=1}^n\\sigma _{\\gamma _i}}\\,\\widehat{g}_n(\\lbrace \\gamma _i,c_i\\rbrace ).$ Then the above analysis implies that the modularity requires $\\Phi ^{\\,\\widehat{g}}_{n}$ to satisfy Vignéras equation away from walls of marginal stability, whereas at these walls its discontinuities should follow the same pattern as before in order to cancel the discontinuities from the contour exchange in $\\Phi ^{\\scriptscriptstyle \\,\\int }_m$ .","Thus, the completion should smoothen out the discontinuities from the moduli independent signs ${\\rm sgn}(\\tilde{{v}}_v,{x})$ , but otherwise leave the action of Vignéras' operator unchanged.","Formally, this means that $\\Phi ^{\\,\\widehat{g}}_{n}$ must satisfy the following equation $V_{n-1} \\cdot \\Phi ^{\\,\\widehat{g}}_{n}=\\,{\\rm Sym}\\, \\sum _{\\ell =1}^{n-1}\\Bigl [{u}_\\ell ^2\\,\\Delta _{n,\\ell }^{\\widehat{g}} \\,\\delta ^{\\prime }({u}_\\ell \\cdot {x})+ 2{u}_\\ell \\cdot \\partial _{x}\\Delta _{n,\\ell }^{\\widehat{g}} \\,\\delta ({u}_\\ell \\cdot {x})\\Bigr ],\\qquad \\Delta _{n,\\ell }^{\\widehat{g}}=\\frac{1}{2}\\,({v}_\\ell ,{x})\\,\\Phi ^{\\,\\widehat{g}}_\\ell \\,\\Phi ^{\\,\\widehat{g}}_{n-\\ell },$ where we introduced two sets of vectors constructed from the vectors (REF ), ${v}_\\ell =\\sum _{i=1}^\\ell \\sum _{j=\\ell +1}^n{v}_{ij}\\,,\\qquad {u}_\\ell =\\sum _{i=1}^\\ell \\sum _{j=\\ell +1}^n{u}_{ij}\\,.$ Note that $({v}_\\ell ,{x})$ and $({u}_\\ell ,{x})$ correspond to the quantities $-\\Gamma _{n\\ell }$ and $-S_\\ell $ (REF ), respectively.","To solve the above constraints, let us consider the following iterative ansatz $\\widehat{g}_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace )=g^{(0)}_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace )-\\,{\\rm Sym}\\, \\left\\lbrace \\sum _{n_1+\\cdots +n_m= n\\atop n_k\\ge 1, \\ m1}W_m(\\lbrace \\check{\\gamma }^{\\prime }_k\\rbrace )\\prod _{k=1}^m \\mathcal {E}_{n_k}(\\check{\\gamma }_{j_{k-1}+1},\\dots ,\\check{\\gamma }_{j_{k}})\\right\\rbrace .$ Furthermore, provided the functions $\\mathcal {E}_n$ depend on electric charges $q_{i,a}$ only through the DSZ products $\\gamma _{ij}$ and their kernels $\\Phi ^{\\,\\mathcal {E}}_n({x})$ defined as in (REF ) are smooth solutions of Vignéras' equation, $V_{n-1}\\cdot \\Phi ^{\\,\\mathcal {E}}_n=0,$ then the ansatz (REF ) also satisfies the modularity constraint (REF ).","This proposition allows in principle to fix all unknown functions.","Indeed, the recursive relation (REF ) determines all $g^{(0)}_n$ .","Then equations (REF ) and (REF ) give the two parts of $\\mathcal {E}_n=\\mathcal {E}^{(0)}_n+\\mathcal {E}^{(+)}_n$ in terms of $g^{(0)}_n$ and $W_n$ .","At this point the latter are still undetermined and are defined in terms of the unknown functions $R_n$ .","The additional constraint that $\\mathcal {E}_n$ should satisfy Vignéras' equation links together $\\mathcal {E}^{(0)}_n$ and $\\mathcal {E}^{(+)}_n$ and thus establishes a relation between $W_n$ and $g^{(0)}_n$ .","Lastly, inverting (REF ) generates a solution for $R_n$ .","We end this discussion by making an observation which will become relevant in the next subsection: by comparing (REF ) and (REF ), it is clear that $\\widehat{g}_n$ must agree with the tree index $g_{{\\rm tr},n}$ in the limit $\\tau _2\\rightarrow \\infty $ .","In particular, $g_{{\\rm tr},n}$ satisfies a similar ansatz as (REF ), upon replacing $\\mathcal {E}_n$ by its non-decaying part: $g_{{\\rm tr},n}(\\lbrace \\check{\\gamma }_i,c_i\\rbrace )=g^{(0)}_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace )-\\,{\\rm Sym}\\, \\left\\lbrace \\sum _{n_1+\\cdots +n_m= n\\atop n_k\\ge 1, \\ m1$ .","In this case more than one pair of joint edges collapse.","The representation (REF ) in terms of rooted ternary trees is obtained by collapsing $m$ pairs of edges successively, where the coefficient $1/T!$ takes into account that such procedure leads to an overcounting of different assignments of labels.","Figure: Three unrooted trees constructed from the same three subtrees.Unfortunately, the function (REF ) cannot yet be taken as an ansatz for $g^{(0)}_n$ because it depends non-trivially on the modulus $\\tau _2$ , which is in tension with the observation made at the end of the previous subsection.","Therefore, we shall modify the function (REF ) into a function which is still a smooth solution of Vignéras' equation, but whose large ${x}$ limit is independent of $\\tau _2$ .","Taking cue from the structure of (REF ), we defineThe term $m=0$ in (REF ) reduces to the original function (REF ), while the terms with $m>0$ are the afore-mentioned modification.","$\\widetilde{\\Phi }^{\\,\\mathcal {E}}_n({x})=\\frac{1}{2^{n-1} n!","}\\sum _{m=0}^{[(n-1)/2]}\\sum _{\\mathcal {T}\\in \\, {T}_{n-2m,m}^\\ell }\\left[\\prod _{\\mathfrak {v}\\in V_\\mathcal {T}}\\mathcal {D}_{m_\\mathfrak {v}}(\\lbrace \\check{\\gamma }_{\\mathfrak {v},s}\\rbrace )\\right]\\widetilde{\\Phi }^E_{n-2m-1}(\\lbrace {u}_e\\rbrace , \\lbrace {v}_{s(e) t(e)}\\rbrace ;{x}),$ where $\\mathcal {D}_{m}(\\lbrace \\check{\\gamma }_s\\rbrace )=\\sum _{\\mathcal {T}^{\\prime }\\in \\, {T}_{2m+1}^\\ell } a_{\\mathcal {T}^{\\prime }}\\prod _{e\\in E_{\\mathcal {T}^{\\prime }}}\\mathcal {D}({v}_{s(e)t(v)}).$ The coefficients $a_\\mathcal {T}$ are rational numbers which depend (up to a sign determined by the orientation of edges) only on the topology of the tree.","We fix them by requiring that they satisfy the following system of equations $a_{\\hat{\\mathcal {T}}_1}+a_{\\hat{\\mathcal {T}}_2}-a_{\\hat{\\mathcal {T}}_3}=a_{\\mathcal {T}_1}a_{\\mathcal {T}_2}a_{\\mathcal {T}_3},$ where $\\mathcal {T}_r$ ($r=1,2,3$ ) are arbitrary unrooted trees with marked vertex $\\mathfrak {v}_r$ and $\\hat{\\mathcal {T}}_r$ are the three trees constructed from $\\mathcal {T}_r$ as shown in Fig.","REF .For these equations, it is important to take into account the orientation of the edges: a change of orientation of an edge flips the sign of $a_\\mathcal {T}$ .","The equations (REF ) are written assuming the orientation shown in Fig.","REF , namely $e_1=(\\mathfrak {v}_2,\\mathfrak {v}_3)$ , $e_2=(\\mathfrak {v}_1,\\mathfrak {v}_3)$ , $e_3=(\\mathfrak {v}_1,\\mathfrak {v}_2)$ .","This system of equations is imposed in order to ensure certain properties of the operators (REF ) which are crucial for the cancellation of $\\tau _2$ -dependent terms in the asymptotics of $\\widetilde{\\Phi }^{\\,\\mathcal {E}}_n$ (see (REF )).","It is readily seen that the system (REF ) fixes all coefficients $a_\\mathcal {T}$ recursively starting from $a_\\bullet =1$ , $a_{\\bullet \\!\\mbox{-}\\!\\bullet }=0$ and going to trees with $n\\ge 3$ vertices.","(Starting from $n=7$ , the system (REF ) is overdetermined, but it can be checked that it does have a unique solution, with $a_\\mathcal {T}=0$ for trees with even number of vertices.)","For a generic tree, in appendix we prove the following Proposition 4 For a tree $\\mathcal {T}$ with $n$ vertices the coefficient $a_\\mathcal {T}$ is given recursively by $a_\\mathcal {T}=\\frac{1}{n}\\sum _{\\mathfrak {v}\\in V_\\mathcal {T}} \\epsilon _\\mathfrak {v}\\prod _{s=1}^{n_\\mathfrak {v}} a_{\\mathcal {T}_s(\\mathfrak {v})},$ where $n_\\mathfrak {v}$ is the valency of the vertex $\\mathfrak {v}$ , $\\mathcal {T}_s(\\mathfrak {v})$ are the trees obtained from $\\mathcal {T}$ by removing this vertex, and $\\epsilon _\\mathfrak {v}$ is the sign determined by the choice of orientation of edges, $\\epsilon _\\mathfrak {v}=(-1)^{n_\\mathfrak {v}^+}$ with $n_\\mathfrak {v}^+$ being the number of incoming edges at the vertex.","(See Fig.","REF for an example.)","Figure: An example of calculation of the coefficient a 𝒯 a_\\mathcal {T} for a tree with 7 vertices.We indicated in red the only vertices which produce non-vanishing contributions to the sum over vertices.For other vertices at least one of the trees 𝒯 s (𝔳)\\mathcal {T}_s(\\mathfrak {v}) has even number of vertices and hence vanishing coefficient.Let us now return to the function (REF ), which is now fully specified given the prescription (REF ) for the coefficients $a_\\mathcal {T}$ .","Similarly to (REF ) (which coincides with the $m=0$ contribution in (REF )), it is a solution of Vignéras' equation for $\\lambda =n-1$ .","We claim that in the large ${x}$ asymptotics of this function, all $\\tau _2$ -dependent contributions, coming from the mutual action of derivative operators $\\mathcal {D}$ , cancel.","More precisely, the asymptotics is given by the following Proposition 5 $\\lim _{{x}\\rightarrow \\infty }\\widetilde{\\Phi }^{\\,\\mathcal {E}}_n({x})=\\frac{1}{2^{n-1} n!","}\\sum _{m=0}^{[(n-1)/2]}\\sum _{\\mathcal {T}\\in \\, {T}_{n-2m,m}^\\ell }\\prod _{\\mathfrak {v}\\in V_\\mathcal {T}}\\mathcal {V}_{m_\\mathfrak {v}}(\\lbrace \\check{\\gamma }_{\\mathfrak {v},s}\\rbrace )\\prod _{e\\in E_{\\mathcal {T}}}({v}_{s(e) t(e)},{x})\\,{\\rm sgn}({u}_e,{x}),$ where $\\mathcal {V}_{m}(\\lbrace \\check{\\gamma }_{s}\\rbrace )=\\sum _{\\mathcal {T}\\in \\, {T}_{2m+1}^\\ell } a_\\mathcal {T}\\prod _{e\\in E_\\mathcal {T}}({v}_{s(e)t(v)},{x}).$ Importantly the function (REF ) is locally a homogeneous polynomial of degree $n-1$ .","Therefore, when written in terms of charges, the $\\tau _2$ -dependence factorizes and is cancelled after the same rescaling as in (REF ).","This naturally leads us to the following ansatz for the function $g^{(0)}_n$ , which we prove in appendix : Proposition 6 The function $g^{(0)}_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace )=\\frac{(-1)^{n-1+\\sum _{i0$ for $i=1,\\dots ,n-1$ and $c_n=-\\sum _{i=1}^{n-1} c_i<0$ , one finds numerically that all coefficients $b_n$ with $n$ even vanish,This follows from the fact that under the permutation $c_i\\mapsto c_{n-i+1}$ one has $S_i\\mapsto -S_{n-i}$ and therefore $F^{\\rm (ref)}_n$ flips sign.","The deeper reason for this is that the terms which are products of even and odd number of signs cannot mix.","In contrast, sign identities such as (REF ) can decrease the number of signs in a product by even number.","Thus, a constant can appear in $F^{\\rm (ref)}_n$ only for $n$ odd.","The same fact ensures the vanishing of the coefficients $a_\\mathcal {T}$ in (REF ) for trees with even number of vertices.","while those with $n$ odd are given by $b_1=1,\\quad b_3=-\\frac{1}{3},\\quad b_5=\\frac{2}{15},\\quad b_7=-\\frac{17}{315},\\quad b_9=\\frac{62}{2835},\\quad b_{11}=-\\frac{1382}{155925},\\quad \\dots $ We observe that these coefficients coincide with the first coefficients in the Taylor series of $\\tanh x = x - \\frac{1}{3} x^3+ \\frac{2}{15} x^5 +\\dots $ .","We therefore conjecture that this identification continues to hold in general, so that $b_n$ is expressed in terms of the Bernoulli number $B_{n+1}$ through $b_{n-1}=\\frac{2^{n} (2^{n} - 1)}{n!","}\\, B_{n} .$ Using the iterative equation (REF ), the constraint (REF ) and proceeding by induction, it is easy to see that $g^{\\rm (ref)}_n$ defined by (REF ) and its first $n-2$ derivatives with respect to $y$ vanish at $y=1$ .","This ensures that $g^{\\rm (ref)}_n$ is smooth and its limit $y\\rightarrow 1$ is well-defined.","Conjecture 3 The function $g^{\\rm (ref)}_n$ , defined by (REF ), (REF ) and (REF ), reproduces the function $g^{(0)}_n$ (REF ) in the unrefined limit, $\\lim _{y\\rightarrow 1}g^{\\rm (ref)}_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace ,y)=g^{(0)}_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace ).$ We have checked this conjecture numerically up to $n=6$ evaluating the limit $y\\rightarrow 1$ using l'Hôpital's rule.","Unfortunately, the new representation of $g^{(0)}_n$ obtained in this way is not helpful for the purposes of this work because it does not lead to any new representation for the completion.","Nevertheless, the comparison of (REF ) and (REF ) shows that the sum over partitions and the simple form of products of sign functions characterizing the representation discussed in the previous subsection find their origin in the formula for $F^{\\rm (ref)}_n$ .","Furthermore, this construction strongly suggests that the refinement may be compatible with S-duality, and that a suitably defined generating function of refined DT invariants may also possess interesting modular properties, which can even be simpler than those of the usual DT invariants."],["Discussion","In this paper we studied the modular properties of the generating function $h_{p,\\mu }(\\tau )$ of MSW invariants encoding BPS degeneracies of D4-D2-D0 black holes in Type IIA string theory on a Calabi-Yau threefold, with fixed D4-charge $p^a$ , D2-brane charge (up to spectral flow) $\\mu _a$ and invariant D0-brane charge $\\hat{q}_0$ (defined in (REF )) conjugate to the modular parameter $\\tau $ .","These properties follow from demanding that the vector multiplet moduli space in $D=3$ (or the hypermultiplet moduli space in the dual type IIB picture) admits an isometric action of $SL(2,{Z})$ .","Our main result is an explicit formula for the non-holomorphic modular completion $\\widehat{h}_{p,\\mu }$ (REF ), where the functions $R_n(\\lbrace \\gamma _i\\rbrace ,\\tau _2)$ are given by Eq.","(REF ), with $\\mathcal {E}_v=\\mathcal {E}_v^{(0)}+\\mathcal {E}_v^{(+)}$ specified by (REF ) and (REF ).","This result applies for D4-branes wrapping a general effective divisor which may be the sum of an arbitrary number $n$ of irreducible divisors.","The existence of such completion was not guaranteed.","We arrived at this result by cancelling the modular anomaly of an indefinite theta series of signature $(n, n b_2(\\mathfrak {Y})-n)$ with a complicated kernel (REF ).","In particular, Vignéras' equation encoding this anomaly involves Kähler moduli in a non-trivial way through the walls of marginal stability, whereas the functions $R_n$ in our ansatz (REF ) can only depend (non-holomorphically) on the modular parameter $\\tau $ and on the charges $\\gamma _i$ .","Note also that the transformation (REF ) of the contact potential, which was the starting point of our analysis, is only a necessary condition for the existence of an isometric action of $SL(2,{Z})$ on $\\mathcal {M}_H$ .","To demonstrate that this condition is sufficient, one would have to construct suitable complex Darboux coordinates on the twistor space such that, after expressing them in terms of the completed generating function $\\widehat{h}_{p,\\mu }$ found in this paper, $SL(2,{Z})$ acts on these coordinates by a complex contact transformation.","Such coordinates were constructed at the two-instanton level in [26], and it is an interesting challenge to extend this construction to arbitrary $n$ .","The structure of the completion suggests that, for a divisor decomposable into a sum of $n$ effective divisors, the holomorphic generating function $h_{p,\\mu }$ is a mixed vector valued mock modular form of higher depth, equal to $n-1$ .","Such objects have recently appeared in various mathematical and physical contexts [39], [64], [65], [66], [31] and correspond to holomorphic functions whose completion is constructed from period (or Eichler) integrals of mock modular forms of smaller depth (with depth 0 mock modular forms being synonymous with ordinary modular forms).","For instance, in the case of standard mock modular forms of depth one, the completion is given by an Eichler integral of a modular function [27].","In our case this structure follows from the fact that the completion $\\widehat{h}_{p,\\mu }$ is built from the generalized error functions which have a representation in terms of iterated period integrals [28], [39].","In order to make manifest the mixed mock modular nature of $h_{p,\\mu }$ , one should represent the antiholomorphic derivative of its completion $\\widehat{h}_{p,\\mu }$ , computed in (REF ), as a sum of products of completions of mixed mock modular forms of lower depth and anti-holomorphic modular forms.","This is a non-trivial task which involves the technique of lattice factorization for indefinite theta series, which we leave for future research.","An important caveat in our construction is that we did not establish the convergence of the various generating functions and indefinite theta series.","Technically, we constructed a theta series $\\vartheta _{{p},{\\mu }}\\bigl (\\widehat{\\Phi }^{{\\rm tot}}_{n},n-2\\bigr )$ whose kernel satisfies Vignéras' equation (REF ), but we did not demonstrate that it decays as required by Vignéras' criterium.","(For $n=2$ this is easy to show [14] since the kernel is a product of an exponentially suppressed factor and a difference of two error functions defined by two positive vectors.)","In fact, convergence issues already arise when expressing the generating function of DT invariants in terms of MSW invariants in (REF ), and are related to the problem of convergence of the BPS black hole partition function.","The latter was proven to converge in the large volume limit for the case involving up to $n\\le 3$ centers in [50].","We hope that the results for the tree index obtained in [23] will allow to extend this result to arbitrary $n$ .","We note that many of the complications in our construction originate from the occurrence of polynomial factors in the kernel of theta series, which in turn can be traced to the factors of DSZ products $\\gamma _{ij}$ in the wall-crossing formula.","Presumably, most of these complications would disappear if one could find a generalization of the contact potential involving the refined DT invariants, so that all these prefactors can be traded for explicit powers of the fugacity $y$ , as in section REF .","It is possible that twistorial topological strings [67] may provide the appropriate framework for finding such a refinement, at least in the context of string vacua on non-compact Calabi-Yau threefolds.","On the physics side, we expect that our results will have important implications for the physics of BPS black holes.","Indeed, the mock modular property of the generating function $h_{p,\\mu }$ will affect the growth of its Fourier coefficients [68], and should be taken into account when performing precision tests of the microscopic origin of the Bekenstein-Hawking entropy.","A natural question is to understand the origin of the non-holomorphic correction terms $R_n$ , in terms of the quantum mechanics of $n$ -centered black holes [37], [69].","Another outstanding question is to understand the physical significance of the instanton generating function $\\mathcal {G}$ defined in (REF ).","Its modular properties are exactly those expected from the elliptic genus of a superconformal field theory, except for the fact that it is not holomorphic in $\\tau $ .","It is natural to expect that this non-holomorphy can be traced to the existence of a continuum of states with a non-trivial spectral asymmetry between bosons and fermions [70].","It would be very interesting to understand the origin of this continuum in terms of the worldvolume theory of an M5-brane wrapped on a reducible divisor.","Moreover, for special $\\mathcal {N}=2$ theories in $D=4$ obtained by circle compactification of an $\\mathcal {N}=1$ theory in $D=5$ , $\\mathcal {G}$ is closely related to the index considered in [37].","It would be interesting to understand the physical origin of its modular invariance in this context.","Finally, we expect that the structure found in the context of generalized DT-invariants on Calabi-Yau threefolds will also arise in the study of other types of BPS invariants where higher depth mock modular forms are expected to occur, such as Vafa-Witten invariants and Donaldson invariants of four-folds with $b_2^+=1$ [28], [64].","In particular, it would be interesting to determine the modular completion of the generating function of Vafa–Witten invariants on the complex projective plane computed for all ranks in [71], generalizing the rank 3 case studied in [64], and compare with the completion found in the present paper.","The authors are grateful to Sibasish Banerjee and Jan Manschot for useful discussions and collaboration on [14], [28], [26] which paved the way for the present work, to Karen Yeats for useful communication about the combinatorics of rooted trees, and to Don Zagier for providing a proof of the conjecture (REF ).","The hospitality and financial support of the Theoretical Physics Department of CERN, where this work was initiated, is also gratefully acknowledged.","The research of BP is supported in part by French state funds managed by the Agence Nationale de la Recherche (ANR) in the context of the LABEX ILP (ANR-11-IDEX-0004-02, ANR-10- LABX-63).","toc"],["Proof of the Theorem","In this appendix we prove Theorem REF presented in the Introduction.","The crucial ingredient is provided by the following LemmaWe were informed that this lemma appears as an exercise in [72].","Lemma 1 The number of ways of labelling the vertices of a rooted ordered tree $T$ with increasing labels is given by $N_T=n_T!","\\prod _{v\\in V_T}\\frac{1}{n_v(T)}\\, ,$ where $n_T$ is the number of vertices of the tree $T$ .","First, let us find a recursive relation between the numbers $N_T$ .","Namely, consider a vertex $v$ and the set of its children ${\\rm Ch}(v)$ .","Denote by $T_v$ the tree rooted at $v$ and with leaves coinciding with leaves of $T$ , for which one has $n_{T_v}=n_v$ .","Then it is clear that $N_{T_v}=\\frac{\\left(\\sum _{v^{\\prime }\\in {\\rm Ch}(v)}n_{v^{\\prime }}\\right)!","}{\\prod _{v^{\\prime }\\in {\\rm Ch}(v)}n_{v^{\\prime }}!","}\\prod _{v^{\\prime }\\in {\\rm Ch}(v)}N_{T_{v^{\\prime }}}\\, .$ From this relation, one easily shows by recursion that (REF ) follows.","Indeed, assuming that it holds for the subtrees $T_{v^{\\prime }}$ and taking into account that $\\sum _{v^{\\prime }\\in {\\rm Ch}(v)}n_{v^{\\prime }}=n_v-1$ , the r.h.s.","can be rewritten as $(n_v-1)!\\prod _{v^{\\prime }\\in {\\rm Ch}(v)}\\prod _{v^{\\prime \\prime }\\in V_{T_{v^{\\prime }}}}\\frac{1}{n_{v^{\\prime \\prime }}}=n_v!\\prod _{v^{\\prime }\\in V_{T_v}}\\frac{1}{n_{v^{\\prime }}},$ which coincides with (REF ) when $v$ is the root of $T$ .","Given this Lemma, we can now rewrite the l.h.s.","of (REF ) as $\\sum _{T^{\\prime }\\subset T}\\frac{N_{T^{\\prime }}}{m!","}\\prod _{v\\in V_{T^{\\prime }}}n_v(T)=\\frac{n!","}{N_{T}}\\sum _{T^{\\prime }\\subset T}\\frac{N_{T^{\\prime }}}{m!","}\\prod _{T_r\\subset T\\setminus T^{\\prime }}\\frac{N_{T_r}}{n_{T_r}!","},$ where the last product goes over the subtrees which complement $T^{\\prime }$ to the full tree $T$ .","On the other hand, it is easy to check that the relation (REF ) is the special case $m=1$ of a more general relation $N_{T}=\\frac{\\left(\\sum _{T_r\\subset T\\setminus T^{\\prime }}n_{T_r}\\right)!","}{\\prod _{T_r\\subset T\\setminus T^{\\prime }}n_{T_r}!","}\\sum _{T^{\\prime }\\subset T}N_{T^{\\prime }}\\prod _{T_r\\subset T\\setminus T^{\\prime }}N_{T_r}.$ where the sum runs over subtrees $T^{\\prime }$ with $m$ vertices.","Taking into account that $\\sum _{T_r\\subset T\\setminus T^{\\prime }}n_{T_r}=n-m$ and substituting the resulting expression into (REF ), one recovers the binomial coefficient as stated in the Theorem.","toc"],["D3-instanton contribution to the contact potential","To evaluate $(e^\\phi )_{\\rm D3}$ , we first replace in (REF ) the full prepotential $F$ by its classical, cubic part $F^{\\rm cl}$ (REF ) and take into account that the sum over $\\gamma \\in -\\Gamma _+$ is complex conjugate to the sum over $\\gamma \\in \\Gamma _+$ .","This gives $e^{\\phi } \\approx \\frac{\\tau _2^2}{12}\\,((\\,{\\rm Im}\\,u)^3)+\\frac{I\\tau _2}{8}\\,\\sum _{\\gamma \\in \\Gamma _+}\\,{\\rm Re}\\,\\int _{\\ell _\\gamma } \\frac{\\text{d}t}{t} \\left( t^{-1} Z_\\gamma (u^a)-t\\bar{Z}_\\gamma (\\bar{u}^a)\\right) H_\\gamma .$ Next, we substitute the quantum corrected mirror map (REF ), change the integration variable $t$ to $z$ , and take the combined limit $t^a\\rightarrow \\infty $ , $z\\rightarrow 0$ .","Keeping only the leading contributions, one obtains $(e^\\phi )_{\\rm D3} &=&-\\frac{\\tau _2}{2}\\sum _{\\gamma \\in \\Gamma _+}\\,{\\rm Re}\\,\\int _{\\ell _{\\gamma }}\\mathrm {d}z\\left[\\hat{q}_0+{1\\over 2}\\, (q+b)^2+2(pt^2)zz_\\gamma -\\frac{3}{2}\\, z^2(pt^2)\\right]H_{\\gamma }\\nonumber \\\\&&-\\frac{1}{4}\\sum _{\\gamma _1,\\gamma _2\\in \\Gamma _+}(tp_1p_2)\\left(\\,{\\rm Re}\\,\\int _{\\ell _{\\gamma _1}}\\!\\!\\mathrm {d}z_1\\, H_{\\gamma _1}\\right)\\left( \\,{\\rm Re}\\,\\int _{\\ell _{\\gamma _2}}\\!\\!\\mathrm {d}z_2\\, H_{\\gamma _2}\\right).$ To further simplify this expression, we note that $\\begin{split}0=&\\, \\frac{1}{4\\pi }\\sum _{\\gamma \\in \\Gamma _+}\\int _{\\ell _{\\gamma }}\\mathrm {d}z\\,\\partial _z\\left( z\\,H_\\gamma \\right)\\\\=&\\,\\sum _{\\gamma \\in \\Gamma _+}\\int _{\\ell _{\\gamma }}\\mathrm {d}z \\left[\\frac{1}{4\\pi }+\\tau _2(pt^2) (zz_\\gamma - z^2)-\\frac{I}{4}\\sum _{\\gamma ^{\\prime }\\in \\Gamma _+}\\langle \\gamma ,\\gamma ^{\\prime }\\rangle \\int _{\\ell _{\\gamma ^{\\prime }}} \\frac{\\mathrm {d}z^{\\prime }}{z-z^{\\prime }}\\, H_{\\gamma ^{\\prime }}\\right]H_\\gamma ,\\end{split}$ where we used the integral equation (REF ).","Multiplying this identity by 3/4 and adding its real part to (REF ), one finds $(e^\\phi )_{\\rm D3} &=&\\frac{\\tau _2}{2}\\sum _{\\gamma \\in \\Gamma _+}\\,{\\rm Re}\\,\\int _{\\ell _{\\gamma }}\\mathrm {d}z\\, a_{\\gamma ,-\\frac{3}{2}}(z)\\,H_{\\gamma }\\nonumber \\\\&&-\\frac{1}{4}\\sum _{\\gamma _1,\\gamma _2\\in \\Gamma _+}\\left[(tp_1p_2)\\left(\\,{\\rm Re}\\,\\int _{\\ell _{\\gamma _1}}\\!\\!\\mathrm {d}z_1\\, H_{\\gamma _1}\\right)\\left( \\,{\\rm Re}\\,\\int _{\\ell _{\\gamma _2}}\\!\\!\\mathrm {d}z_2\\, H_{\\gamma _2}\\right)\\right.\\\\&&\\left.\\qquad +\\frac{3}{4}\\,\\,{\\rm Re}\\,\\left(\\int _{\\ell _{\\gamma _1}}\\!\\!\\mathrm {d}z_1\\, H_{\\gamma _1}\\int _{\\ell _{\\gamma _2}}\\!\\!\\mathrm {d}z_2\\, H_{\\gamma _2} \\,\\frac{I\\langle \\gamma ,\\gamma ^{\\prime }\\rangle }{z-z^{\\prime }}\\right)\\right],\\nonumber $ where we introduced $a_{\\gamma ,\\mathfrak {h}}(z)=-\\left(\\hat{q}_0+{1\\over 2}\\,(q+b)^2+\\frac{1}{2}\\, (pt^2)zz_\\gamma +\\frac{\\mathfrak {h}}{4\\pi \\tau _2}\\right).$ The meaning of this function is actually very simple: it gives the action of the modular covariant derivative operator $\\mathcal {D}_{\\mathfrak {h}}$ (REF ) on the classical part of the Darboux coordinate (REF ), $\\mathcal {D}_{\\mathfrak {h}}\\mathcal {X}^{\\rm cl}_{\\gamma } =a_{\\gamma ,\\mathfrak {h}}(z)\\mathcal {X}^{\\rm cl}_{\\gamma }.$ Combining this fact with equation (REF ), it is easy to check that the function (REF ) satisfies, for any value of the weight $\\mathfrak {h}$ , $\\mathcal {D}_{\\mathfrak {h}}\\mathcal {G}&=& \\sum _{\\gamma \\in \\Gamma _+}\\int _{\\ell _{\\gamma }} \\mathrm {d}z\\,a_{\\gamma ,\\mathfrak {h}}(z)\\, H_{\\gamma }\\\\&&+\\frac{1}{8\\tau _2}\\sum _{\\gamma _1,\\gamma _2\\in \\Gamma _+}\\int _{\\ell _{\\gamma _1}}\\mathrm {d}z_1\\,\\int _{\\ell _{\\gamma _2}} \\mathrm {d}z_2\\left[(tp_1p_2)+2\\mathfrak {h}\\left((tp_1p_2)+\\frac{I\\langle \\gamma _1,\\gamma _2\\rangle }{z_1-z_2} \\right)\\right]H_{\\gamma _1}(z_1)H_{\\gamma _2}(z_2),\\nonumber \\\\\\partial _{\\tilde{c}_a}\\mathcal {G}&=& 2\\pi I\\sum _{\\gamma \\in \\Gamma _+}p^a \\int _{\\ell _{\\gamma }} \\mathrm {d}z\\, H_{\\gamma },$ which allows to rewrite (REF ) exactly as in (REF ).","toc"],["Smoothness of the instanton generating function","In this appendix we prove the smoothness of the function $\\mathcal {G}$ across walls of marginal stability.","The starting point is the representation (REF ) where the potential discontinuities are hidden in the kernel of the theta series (REF ).","Given that $\\Phi ^{\\scriptscriptstyle \\,\\int }_{m}$ is represented as a sum over unrooted labelled trees (REF ), whereas $\\Phi ^{\\,g}_{n_k}$ appear as sums over flow trees (REF ), the kernel $\\Phi ^{{\\rm tot}}_{n}$ can be viewed as a sum over `blooming trees' which are unrooted trees with a flow tree (the `flower') growing from each vertex.","Then the idea is that the discontinuities due to flow trees (i.e.","due to DT invariants) of a blooming tree with $m$ vertices are cancelled by the discontinuities due to exchange of integration contours in $\\Phi ^{\\scriptscriptstyle \\,\\int }_{m+1}$ corresponding to blooming trees with $m+1$ vertices.","Figure: Combination of trees showing cancellation of discontinuities across the wall of marginal stabilitycorresponding to the decay γ 𝔳 →γ L +γ R \\gamma _\\mathfrak {v}\\rightarrow \\gamma _L+\\gamma _R.","The parts corresponding to attractor flow trees are drown in blue.Following this idea, let us consider a tree which has an attractor flow tree $T_\\mathfrak {v}$ growing from a vertex $\\mathfrak {v}$ (see Fig.","REF ).","We denote by $\\mathcal {T}_i$ , $i=1,\\dots ,n_\\mathfrak {v}$ , the blooming subtrees connected to this vertex and by $T_{L}$ and $T_{R}$ the two parts (which may be trivial) of $T_\\mathfrak {v}$ with the total charges $\\gamma _L$ and $\\gamma _R$ so that $\\gamma _\\mathfrak {v}=\\gamma _L+\\gamma _R$ .","Together with the contribution of this tree, we consider the contributions of the trees obtained by splitting the vertex $\\mathfrak {v}$ into two vertices $\\mathfrak {v}_L$ and $\\mathfrak {v}_R$ connected by an edge, carrying charges $\\gamma _L$ and $\\gamma _R$ and all possible allocations of the subtrees $\\mathcal {T}_i$ to these two vertices.","Different allocations are accounted for by the sum over permutations, whilethe weight $\\frac{1}{\\ell !","(n_\\mathfrak {v}-\\ell )!","}$ takes into account the fact that permutations between subtrees connected to one vertex are redundant.","The attractor flow trees $T_{L}$ and $T_{R}$ are then connected to $\\mathfrak {v}_L$ and $\\mathfrak {v}_R$ , respectively, as shown in Fig.","REF .","The contribution corresponding to the first blooming tree has a discontinuity at the wall of marginal stability for the bound state $\\gamma _L+\\gamma _R$ and originating from the factor $\\Delta ^{z}_{\\gamma _L\\gamma _R}$ assigned to the root vertex of $T_\\mathfrak {v}$ .","The other contributions have discontinuities at the same wall due to the exchange of the contours $\\ell _{\\gamma _L}$ and $\\ell _{\\gamma _R}$ for the integrals assigned to $\\mathfrak {v}_L$ and $\\mathfrak {v}_R$ , respectively.","They are given by the residues at the pole of the integration kernel $K_{\\gamma _L\\gamma _R}$ .","It is clear that the structure of all jumps is very similar since different subtrees produce essentially the same weights.","Let us analyze what differences may arise.","First, the contributions of the flow trees $T_{L}$ and $T_{R}$ could differ in the two cases because they have different starting points for the attractor flows: for the trees on the right side of Fig.","REF this is $z^a\\in \\mathcal {M}_K$ , whereas for the tree on the left this is the point on the wall for $\\gamma _L+\\gamma _R$ reached by the flow from $z^a$ .","But we are evaluating the discontinuity exactly on the wall where the two points coincide.","Thus, the contributions are the same.","Although the subtrees $\\mathcal {T}_i$ give rise to the same contributions for all blooming trees shown in Fig.","REF , the contributions of the edges connecting them to either $\\mathfrak {v}$ or $\\mathfrak {v}_L$ , $\\mathfrak {v}_R$ are not exactly the same.","Each of them contributes the factor given by the kernel (REF ).","After taking the residue, the $z$ -dependence of these kernels for the trees shown on the left and the right sides of the picture is identical.","However, their charge dependence is different: for the tree on the left they depend on $\\gamma _\\mathfrak {v}$ , whereas for the trees on the right they depend on $\\gamma _L$ or $\\gamma _R$ , depending on which vertex they are connected to.","But it is easy to see that the sums over $\\ell $ and permutations produce the standard binomial expansion of a single product of kernels which all depend on $\\gamma _L+\\gamma _R=\\gamma _\\mathfrak {v}$ and thus coinciding with the contribution of the tree on the left.","Finally, one should take into account that the discontinuity of $\\Delta ^{z}_{\\gamma _L\\gamma _R}$ in the first contribution gives the factor $\\langle \\gamma _L,\\gamma _R\\rangle $ .","But exactly the same factor arises as the residue of $K_{\\gamma _L\\gamma _R}$ corresponding to the additional edge.","Thus, it remains only to check that all numerical factors work out correctly.","Leaving aside the factors which are common to both contributions, we have $-\\frac{(-1)^{\\langle \\gamma _L,\\gamma _R\\rangle }}{2}\\,\\frac{\\sigma _{\\gamma _\\mathfrak {v}}}{(2\\pi )^2}\\, \\frac{2m}{ m!","}+(2\\pi I)(-2\\pi I)\\,\\frac{\\sigma _{\\gamma _L}\\sigma _{\\gamma _R}}{(2\\pi )^4}\\,\\frac{m(m+1)}{(m+1)!","}.$ Here $-\\frac{(-1)^{\\langle \\gamma _L,\\gamma _R\\rangle }}{2}$ comes from the factor $-\\Delta ^{z}_{\\gamma _L\\gamma _R}$ in $T_\\mathfrak {v}$ , the factors with quadratic refinement are due to functions $H_\\gamma $ (REF ) assigned to $\\mathfrak {v}$ or $\\mathfrak {v}_L$ , $\\mathfrak {v}_R$ , $(2\\pi I)$ is the standard weight of the residue, $(-2\\pi I)$ is the residue of $K_{\\gamma _L\\gamma _R}$ (REF ), and factorials are the weights of the trees in the expansion (REF ).","Finally, the factors $2m$ and $m(m+1)$ arise due to the freedom to relabel charges assigned to the marked vertices: on the left these are vertex $\\mathfrak {v}$ and the two children of the root in $T_\\mathfrak {v}$ , whereas on the right these are $\\mathfrak {v}_L$ and $\\mathfrak {v}_R$ .","It is immediate to check that all these numerical weights cancel, which ensures that the function $\\mathcal {G}$ is continuous across walls of marginal stability.","Moreover, in this cancellation the condition that we sit on the wall was used only in locally constant factors.","Therefore, this reasoning proves not only that $\\mathcal {G}$ is continuous, but that it is actually smooth around these loci.","toc"],["Vignéras' theorem","Let $\\mathbf {\\Lambda }$ be a $d$ -dimensional lattice equipped with a bilinear form $({x},{y})\\equiv {x}\\cdot {y}$ , where ${x},{y}\\in \\mathbf {\\Lambda }\\otimes {R}$ , such that its associated quadratic form has signature $(n,d-n)$ and is integer valued, i.e.","${k}^2\\equiv {k}\\cdot {k}\\in {Z}$ for ${k}\\in \\mathbf {\\Lambda }$ .","Furthermore, let ${p}\\in \\mathbf {\\Lambda }$ be a characteristic vector (such that ${k}\\cdot ({k}+ {p})\\in 2{Z}$ , $\\forall \\,{k}\\in \\mathbf {\\Lambda }$ ), ${\\mu }\\in \\mathbf {\\Lambda }^*/\\mathbf {\\Lambda }$ a glue vector, and $\\lambda $ an arbitrary integer.","We consider the following family of theta series $\\vartheta _{{p},{\\mu }}(\\Phi ,\\lambda ;\\tau , {b}, {c})=\\tau _2^{-\\lambda /2}\\!\\!\\!\\!\\sum _{{{k}}\\in \\mathbf {\\Lambda }+{\\mu }+{1\\over 2}{p}}\\!\\!","(-1)^{{k}\\cdot {p}}\\,\\Phi (\\sqrt{2\\tau _2}({k}+{b}))\\, {\\bf e}\\!\\left( - \\tfrac{\\tau }{2}\\,({k}+{b})^2+{c}\\cdot ({k}+\\textstyle {1\\over 2}{b})\\right)$ defined by a kernel $\\Phi ({x})$ such that the function $f({x})\\equiv \\Phi ({x})\\, e^{\\tfrac{\\pi }{2}\\,{x}^2}\\in L^1(\\mathbf {\\Lambda }\\otimes {R})$ so that the sum is absolutely convergent.","Irrespective of the choice of this kernel and of the parameter $\\lambda $ , any such theta series satisfies the following elliptic properties $\\begin{split}\\vartheta _{{p},{\\mu }}\\left({\\Phi } ,\\lambda ; \\tau , {b}+{k},{c}\\right) =&(-1)^{{k}\\cdot {p}}\\,{\\bf e}\\!\\left( -\\textstyle {1\\over 2}\\, {c}\\cdot {k}\\right) \\vartheta _{{p},{\\mu }}\\left({\\Phi } ,\\lambda ; \\tau , {b},{c}\\right),\\\\\\vphantom{A^A \\over A_A}\\vartheta _{{p},{\\mu }}\\left({\\Phi }, \\lambda ; \\tau , {b},{c}+{k}\\right)=&(-1)^{{k}\\cdot {p}}\\,{\\bf e}\\!\\left( \\textstyle {1\\over 2}\\, {b}\\cdot {k}\\right) \\vartheta _{{p},{\\mu }}\\left({\\Phi } ,\\lambda ; \\tau , {b},{c}\\right).\\end{split}$ Now let us require that in addition the kernel satisfies the following two conditions: Let $D({x})$ be any differential operator of order $\\le 2$ , and $R({x})$ any polynomial of degree $\\le 2$ .","Then $f({x})$ defined above must be such that $f({x})$ , $D({x})f({x})$ and $R({x})f({x})\\in L^2(\\mathbf {\\Lambda }\\otimes {R})\\bigcap L^1(\\mathbf {\\Lambda }\\otimes {R})$ .","$\\Phi ({x})$ must satisfy $V_\\lambda \\cdot \\Phi ({x})=0,\\qquad V_\\lambda = \\partial _{{x}}^2 + 2\\pi \\left( {x}\\cdot \\partial _{{x}} - \\lambda \\right) .$ Then in [38] it was proven that the theta series (REF ) transforms as a vector-valued modular form of weight $(\\lambda +d/2,0)$ (see Theorem 2.1 in [28] for the detailed transformation under $\\tau \\rightarrow -1/\\tau $ ).","We refer to $V_\\lambda $ as Vignéras' operator.","The simplest example is the Siegel theta series for which the kernel is $\\Phi ({x})=e^{-\\pi {x}_+^2}$ where ${x}_+$ is the projection of ${x}$ on a fixed positive plane of dimension $n$ .","This kernel is annihilated by $V_{-n}$ .","In this paper we apply the Vignéras' theorem to the case of $\\mathbf {\\Lambda }=\\oplus _{i=1}^n \\Lambda _i$ .","Thus, the charges appearing in the description of the theta series (REF ) are of the type ${k}=(k_1^a,\\dots ,k_n^a)$ , whereas the vectors ${b}$ and ${c}$ are taken with $i$ -independent components, namely, ${b}_i^a=b^a$ , ${c}_i^a=c^a$ for $i=1,\\dots , n$ .","The lattices $\\Lambda _i$ carry the bilinear forms $\\kappa _{i,ab}=\\kappa _{abc}p_i^c$ which are all of signature $(1,b_2-1)$ .","This induces a natural bilinear form on $\\mathbf {\\Lambda }$ : ${x}\\cdot {y}=\\sum _{i=1}^n (p_ix_iy_i).$ Note also that the sign factor $(-1)^{{k}\\cdot {p}}$ in (REF ) can be identified with the quadratic refinement provided we choose the latter as $\\sigma _\\gamma =\\sigma _{p,q}\\equiv {\\bf e}\\!\\left( {1\\over 2}\\, p^a q_a\\right)\\sigma _p,\\qquad \\sigma _p={\\bf e}\\!\\left( {1\\over 2}\\, A_{ab}p^ap^b\\right).$ The matrix $A_{ab}$ , satisfying $A_{ab} p^p - \\frac{1}{2}\\, \\kappa _{abc} p^b p^c\\in {Z}\\quad \\text{for}\\ \\forall p^a\\in {Z}\\, ,$ appears due to the non-trivial quantization of charges on the type IIB side (REF ) and can be used to perform a symplectic rotation to identify them with mirror dual integer charges on the type IIA side [51].","It is easy to check that the quadratic refinement (REF ) satisfies (REF )."],["Generalized error functions","An important class of solutions of Vignéras' equation is given by the error function and its generalizations constructed in [28] and further elaborated in [29].","Let us take $M_1(u)&=&-\\mbox{sgn}(u)\\, \\text{Erfc}(|u|\\sqrt{\\pi }) = \\frac{I}{\\pi } \\int _{\\ell }\\frac{\\mathrm {d}z}{z}\\,e^{-\\pi z^2 -2\\pi Iz u},\\\\E_1(u)&=&\\mbox{sgn}(u)+M_1(u)\\nonumber \\\\&=& \\operatorname{Erf}(u\\sqrt{\\pi })= \\int _{{R}} \\mathrm {d}u^{\\prime } \\, e^{-\\pi (u-u^{\\prime })^2} {\\rm sgn}(u^{\\prime }),$ where the contour $\\ell ={R}-Iu$ runs parallel to the real axis through the saddle point at $z=-Iu$ .","Then, given a vector with a positive norm ${v}^2>0$ so that $|{v}|=\\sqrt{{v}^2}$ , we define $\\Phi _1^E({v};{x})=E_1\\left(\\frac{{v}\\cdot {x}}{|{v}|}\\right),\\qquad \\Phi _1^M({v};{x})=M_1\\left(\\frac{{v}\\cdot {x}}{|{v}|}\\right).$ It is easy to check that the first function is a smooth solution of (REF ) with $\\lambda =0$ , whereas the second is exponentially suppressed at large ${x}$ and also solves the same equation, but only away from the locus ${v}\\cdot {x}=0$ where it has a discontinuity.","Generalizing the integral representations (REF ) and (), we define the generalized (complementary) error functions $M_n(\\mathcal {M};\\mathbb {u})&=&\\left(\\frac{I}{\\pi }\\right)^n |\\,{\\rm det}\\, \\mathcal {M}|^{-1} \\int _{{R}^n-I\\mathbb {u}}\\mathrm {d}^n z\\,\\frac{e^{-\\pi \\mathbb {z}^{\\rm tr} \\mathbb {z}-2\\pi I\\mathbb {z}^{\\rm tr} \\mathbb {u}}}{\\prod (\\mathcal {M}^{-1}\\mathbb {z})},\\\\E_n(\\mathcal {M};\\mathbb {u})&=& \\int _{{R}^n} \\mathrm {d}\\mathbb {u}^{\\prime } \\, e^{-\\pi (\\mathbb {u}-\\mathbb {u}^{\\prime })^{\\rm tr}(\\mathbb {u}-\\mathbb {u}^{\\prime })} {\\rm sgn}(\\mathcal {M}^{\\rm tr} \\mathbb {u}^{\\prime }),$ where $\\mathbb {z}=(z_1,\\dots ,z_n)$ and $\\mathbb {u}=(u_1,\\dots ,u_n)$ are $n$ -dimensional vectors, $\\mathcal {M}$ is $n\\times n$ matrix of parameters, and we used the shorthand notations $\\prod \\mathbb {z}=\\prod _{i=1}^n z_i$ and ${\\rm sgn}(\\mathbb {u})=\\prod _{i=1}^n {\\rm sgn}(u_i)$ .","The detailed properties of these functions can be found in [29].","Here we mention only a few: $M_n$ are exponentially suppressed for large $\\mathbb {u}$ as $M_n\\sim \\frac{(-1)^n}{\\pi ^n}|\\,{\\rm det}\\, \\mathcal {M}|^{-1}\\,\\frac{e^{-\\pi \\mathbb {u}^{\\rm tr} \\mathbb {u}}}{\\prod (\\mathcal {M}^{-1}\\mathbb {u})}$ , whereas $E_n$ are locally constant for large $\\mathbb {u}$ as $E_n\\sim {\\rm sgn}(\\mathcal {M}^{\\rm tr} \\mathbb {u})$ .","More generally, $E_n$ can be expressed as a linear combination of $M_k$ , $k=0,\\dots ,n$ , multiplied by $n-k$ sign functions, generalizing the first relation in () (see Eq.","(REF ) below for a precise statement).","From () it follows that every identity between products of sign functions implies an identity between generalized error functions $E_n$ .","Moreover, expanding the $E_n$ functions in terms of $M_k$ 's and sign functions, one obtains similar identities for functions $M_n$ .","For instance, the identity $(\\mbox{sgn}(x_1)+\\mbox{sgn}(x_2))\\,\\mbox{sgn}(x_1+x_2)=1+\\mbox{sgn}(x_1)\\,\\mbox{sgn}(x_2)$ implies $\\begin{split}E_2((\\mathbb {v}_1,\\mathbb {v}_1+\\mathbb {v}_2);\\mathbb {u})+E_2((\\mathbb {v}_2,\\mathbb {v}_1+\\mathbb {v}_2);\\mathbb {u})=&\\, 1+E_2((\\mathbb {v}_1,\\mathbb {v}_2);\\mathbb {u}),\\\\M_2((\\mathbb {v}_1,\\mathbb {v}_1+\\mathbb {v}_2);\\mathbb {u})+M_2((\\mathbb {v}_2,\\mathbb {v}_1+\\mathbb {v}_2);\\mathbb {u})=&\\, M_2((\\mathbb {v}_1,\\mathbb {v}_2);\\mathbb {u}),\\end{split}$ where $\\mathbb {v}_1,\\mathbb {v}_2$ are two-dimensional vectors used to encode the $2\\times 2$ matrix of parameters.","The main reason to introduce these functions is that, similarly to the usual error and complementary error functions, they can be used to produce solutions of Vignéras' equation on ${R}^{n,d-n}$ .","To write them down, let us consider $d\\times n$ matrix $\\mathcal {V}$ which can be viewed as a collection of $n$ vectors, $\\mathcal {V}=({v}_1,\\dots ,{v}_n)$ .","We assume that these vectors span a positive definite subspace, i.e.","$\\mathcal {V}^{\\rm tr}\\cdot \\mathcal {V}$ is a positive definite matrix.","Let $\\mathcal {B}$ be $n\\times d$ matrix whose rows define an orthonormal basis for this subspace.","Then we define the boosted generalized error functions $\\Phi _n^M(\\mathcal {V};{x})=M_n(\\mathcal {B}\\cdot \\mathcal {V};\\mathcal {B}\\cdot {x}),\\qquad \\Phi _n^E(\\mathcal {V};{x})=E_n(\\mathcal {B}\\cdot \\mathcal {V};\\mathcal {B}\\cdot {x}).$ It can be shown that both these functions satisfy Vignéras' equation (for $\\Phi _n^M$ one should stay away from its discontinuities, i.e.","loci where ${\\rm sgn}((\\mathcal {B}\\cdot \\mathcal {V})^{-1}\\mathcal {B}\\cdot {x})=0$ ).","Moreover, they are symmetric under permutation of the vectors ${v}_i$ .","Since at large ${x}$ one has $\\Phi _n^E\\sim {\\rm sgn}(\\mathcal {V}^{\\rm tr}\\cdot {x})=\\prod _{i=1}^n {\\rm sgn}({v}_i\\cdot {x})$ , one can think about this function as providing the modular completion for (indefinite) theta series with kernel given by a product of signs.","The relation between functions $E_n$ and $M_n$ mentioned above implies a similar relation between the functions (REF ).","For generic $n$ , it takes the following formThis relation reduces to Eq.","(3.53) and (6.18) in [28] for $n=2,3$ , and follows from Eq.","(63) in [29] for any $n$ .","$\\Phi _n^E(\\mathcal {V};{x})=\\sum _{\\mathcal {I}\\subseteq {Z}_n}\\Phi _{|\\mathcal {I}|}^M(\\lbrace {v}_i\\rbrace _{i\\in \\mathcal {I}};{x})\\prod _{j\\in {Z}_n\\setminus \\mathcal {I}}{\\rm sgn}({v}_{j\\perp \\mathcal {I}},{x}),$ where the sum goes over all possible subsets (including the empty set) of the set ${Z}_{n}=\\lbrace 1,\\dots ,n\\rbrace $ , $|\\mathcal {I}|$ is the cardinality of $\\mathcal {I}$ , and ${v}_{j\\perp \\mathcal {I}}$ denotes the projection of ${v}_j$ orthogonal to the subspace spanned by $\\lbrace {v}_i\\rbrace _{i\\in \\mathcal {I}}$ .","The cardinality $|\\mathcal {I}|$ can also be interpreted as the number of directions in ${R}^d$ along which the corresponding contribution has an exponential fall off.","Finally, note that a solution of Vignéras' equation can be uplifted to a solution of the same equation with $\\lambda $ shifted to $\\lambda +1$ by acting with the differential operator $\\mathcal {D}({v})={v}\\cdot \\left({x}+\\frac{1}{2\\pi }\\,\\partial _{x}\\right),$ which realizes the action of the covariant derivative raising the holomorphic weight by 1.","In particular, we can construct solutions with $\\lambda =m$ which behave for large ${x}$ as products of $n$ sign functions.","To this end, it is enough to act on $\\Phi _n^E$ by this operator $m$ times.","Thus, we define the uplifted boosted error function $\\widetilde{\\Phi }_{n,m}^E(\\mathcal {V},\\tilde{\\mathcal {V}};{x})=\\left[\\prod _{i=1}^m \\mathcal {D}(\\tilde{{v}}_i)\\right]\\Phi _n^E(\\mathcal {V};{x}),$ where $\\tilde{\\mathcal {V}}=(\\tilde{{v}}_1,\\dots ,\\tilde{{v}}_m)$ encodes the vectors contracted with the covariant derivatives.","Since the operators $\\mathcal {D}(\\tilde{{v}}_i)$ commute, (REF ) is invariant under independent permutations of the vectors ${v}_i$ and $\\tilde{{v}}_i$ .","In the case where all $\\tilde{{v}}_i$ are mutually orthogonal, one finds the following asymptotics at large ${x}$ $\\lim _{{x}\\rightarrow \\infty }\\widetilde{\\Phi }_{n,m}^E(\\mathcal {V},\\tilde{\\mathcal {V}};{x})= \\prod _{i=1}^m (\\tilde{{v}}_i,{x})\\prod _{j=1}^n {\\rm sgn}({v}_j,{x}).$ Note that the derivative $\\partial _{x}$ in (REF ) does not act on sign functions since $\\Phi _n^E$ is smooth and all discontinuities due to signs are guaranteed to cancel.","Similarly to (REF ), we can also define $\\widetilde{\\Phi }_{n,m}^M(\\mathcal {V},\\tilde{\\mathcal {V}};{x})$ where the action of derivatives on the discontinuities of $\\Phi _n^M$ is ignored as well.","In the particular case $m=n$ , we will omit the second label and simply write $\\widetilde{\\Phi }_{n}^E$ or $\\widetilde{\\Phi }_{n}^M$ .","toc"],["Twistorial integrals and generalized error functions","In this appendix we evaluate the kernels $\\Phi ^{\\scriptscriptstyle \\,\\int }_n$ (REF ) and show that they can be expressed through the generalized error functions introduced in appendix REF .","To this end, let us note the following identity $I\\,\\mathcal {D}({v}_{ij})\\,\\frac{W_{p_i}(x_i,z_i)\\,W_{p_j}(x_j,z_j)}{z_i-z_j}= \\hat{K}_{ij}\\, W_{p_i}(x_i,z_i)\\,W_{p_j}(x_j,z_j).$ By virtue of this relation, the kernel can be represented as $\\Phi ^{\\scriptscriptstyle \\,\\int }_n({x})=\\frac{1}{n!","}\\sum _{\\mathcal {T}\\in \\, {T}_n^\\ell }\\left[\\prod _{e\\in E_\\mathcal {T}}\\mathcal {D}({v}_{s(e) t(e)}) \\right]\\Phi _\\mathcal {T}({x}),$ where $\\Phi _\\mathcal {T}({x}) =\\frac{I^{n-1}}{(2\\pi )^n}\\left[\\prod _{i=1}^n\\int _{\\ell _{\\gamma _i}}\\mathrm {d}z_i \\, W_{p_i}(x_i,z_i)\\right]\\frac{1}{\\prod _{e\\in E_\\mathcal {T}}\\left(z_{s(e)}-z_{t(e)}\\right)}\\, .$ One may think that the representation (REF ) misses contributions from the covariant derivatives acting on each other.","However, such contributions are proportional to the scalar products of two vectors ${v}_{s(e) t(e)}$ and are non-vanishing provided the two edges have a common vertex.","If this is the case, the two edges generate the following factor $\\frac{(p_{\\mathfrak {v}_1}p_{\\mathfrak {v}_2}p_{\\mathfrak {v}_{3}})}{(z_{\\mathfrak {v}_1}-z_{\\mathfrak {v}_3})(z_{\\mathfrak {v}_2}-z_{\\mathfrak {v}_{3}})}\\, ,$ where $\\mathfrak {v}_1,\\mathfrak {v}_2,\\mathfrak {v}_3$ are the tree vertices joint by the edges and $\\mathfrak {v}_{3}=e_1\\cap e_2$ .","The crucial observation is that if we pick up 3 subtrees, each with a marked vertex, there are exactly 3 ways to form a labelled tree out of them by joining the marked vertices as shown in Fig.","REF on page REF .","Each subtree contributes the same factor in all 3 cases, whereas the joining edges and the sum over trees give rise to the vanishing factor $\\frac{(p_1p_2p_3)}{(z_1-z_3)(z_2-z_3)}+\\frac{(p_1p_2p_3)}{(z_1-z_2)(z_1-z_3)}-\\frac{(p_1p_2p_3)}{(z_2-z_3)(z_1-z_2)}=0.$ This ensures that no additional contributions arise and thereby proves (REF ).","Note that for this proof it was crucial that inequivalent labelled trees enter the sum with the same weight.","Then let us do the change of variables $z_i=z^{\\prime }-\\frac{I(pxt)}{\\sqrt{2\\tau _2}(pt^2)}+\\sum _{\\alpha =1}^{n-1} e_i^\\alpha z^{\\prime }_\\alpha ,$ where $x^a=\\kappa ^{ab}\\sum \\kappa _{i,bc} x^c_i$ (cf.","(REF )), and $e_i^\\alpha $ are such that $\\sum _{i=1}^n \\mathfrak {p}_i e_i^\\alpha =0,\\qquad \\sum _{i=1}^n \\mathfrak {p}_i e_i^\\alpha e_i^\\beta =0, \\quad \\alpha \\ne \\beta $ and we introduced the convenient notation $\\mathfrak {p}_i=(p_it^2)$ .","Labeling the $n-1$ edges of the tree by the same index $\\alpha $ , one can rewrite the function (REF ) in the new variables as $\\Phi _\\mathcal {T}({x}) =\\frac{I^{n-1} \\sqrt{\\Delta }}{(2\\pi )^n}\\,e^{-\\frac{\\pi (pxt)^2}{(pt^2)}}\\int \\mathrm {d}z^{\\prime } \\, e^{-2\\pi \\tau _2(pt^2)z^{\\prime 2}}\\prod _{\\alpha =1}^{n-1}\\int \\frac{\\mathrm {d}z^{\\prime }_\\alpha \\,e^{-2\\pi \\tau _2\\, \\Delta _\\alpha (z^{\\prime }_\\alpha )^2 -2\\pi I\\sqrt{2\\tau _2}\\, w^\\alpha z^{\\prime }_\\alpha }}{\\sum _{\\beta =1}^{n-1}\\left(e^\\beta _{s(\\alpha )}-e^\\beta _{t(\\alpha )}\\right)z^{\\prime }_\\beta }\\, ,$ where $w^\\alpha =\\sum _{i=1}^n (p_ix_i t) e_i^\\alpha ,\\qquad \\Delta _\\alpha =\\sum _{i=1}^n \\mathfrak {p}_i (e_i^\\alpha )^2,\\qquad \\Delta =\\frac{\\mathfrak {p}\\prod _{\\alpha =1}^{n-1} \\Delta _\\alpha }{\\prod _{i=1}^{n}\\mathfrak {p}_i}\\, .$ The integral over $z^{\\prime }$ is Gaussian and is easily evaluated.","In the remaining integrals, rescaling the integration variables by $\\sqrt{2\\tau _2\\Delta _\\alpha }$ , one recognizes the generalized error functions (REF ).","Thus, one obtains $\\Phi _\\mathcal {T}({x}) =\\frac{1}{2^{n-1}}\\,\\frac{\\sqrt{\\Delta }|\\,{\\rm det}\\, \\mathcal {M}|}{\\prod _{\\alpha =1}^{n-1} \\Delta _\\alpha }\\,\\Phi ^{\\scriptscriptstyle \\,\\int }_1(x)\\, M_{n-1}\\left(\\mathcal {M};\\left\\lbrace \\frac{w^\\alpha }{\\sqrt{\\Delta _\\alpha }}\\right\\rbrace \\right).$ where we used the function $\\Phi ^{\\scriptscriptstyle \\,\\int }_1$ (REF ) and introduced the matrix $\\mathcal {M}$ such that $\\mathcal {M}^{-1}_{\\alpha \\beta }=(\\Delta _\\alpha \\Delta _\\beta )^{-1/2}\\left(e^\\beta _{s(\\alpha )}-e^\\beta _{t(\\alpha )}\\right).$ A simple solution to the conditions (REF ) may be constructed as follows.","Let $T$ be a rooted ordered binary tree with $n$ leaves decorated by $\\gamma _i$ , $i=1\\dots n$ .","As usual for such trees, other vertices $v$ carry charges given by the sum of charges of their children, i.e.","$\\gamma _v=\\sum _{i\\in \\mathcal {I}_v}\\gamma _i$ where $\\mathcal {I}_v$ is the set of leaves which are descendants of $v$ .","There are $n-1$ such vertices which we label by index $\\alpha $ .","Then we can choose $e^\\alpha _i=\\sum _{j\\in \\mathcal {I}_{L(v_\\alpha )}}\\sum _{k\\in \\mathcal {I}_{R(v_\\alpha )}} \\left(\\delta _{ij}\\,\\mathfrak {p}_k-\\delta _{ik}\\,\\mathfrak {p}_j\\right),$ which satisfy (REF ) as can be easily checked.","For this choice (cf.","(REF )) $\\begin{split}w^\\alpha =&\\, (\\tilde{{u}}_{\\alpha },{x}),\\quad \\mbox{where}\\quad \\tilde{{u}}_\\alpha =\\sum _{i\\in \\mathcal {I}_{L(v_\\alpha )}}\\sum _{j\\in \\mathcal {I}_{R(v_\\alpha )}}{u}_{ij},\\\\\\Delta _\\alpha =&\\,\\tilde{{u}}_\\alpha ^2=\\mathfrak {p}_{v_\\alpha } \\mathfrak {p}_{L(v_\\alpha )}\\mathfrak {p}_{R(v_\\alpha )}.\\end{split}$ Note that the vectors $\\tilde{{u}}_\\alpha $ are mutually orthogonal.","In principle, any rooted binary tree $T$ is suitable for the above construction.","However, given the unrooted tree $\\mathcal {T}$ , there is a simple (but non-unique) choice of $T$ which simplifies the resulting matrix $\\mathcal {M}$ .","To define it, let us construct a partially increasing family of subtrees of $\\mathcal {T}$ , such that two members in this family are either disjoint, or contained in one another.","Moreover, we require that the largest subtree is $\\mathcal {T}$ itself, while each subtree containing more than one vertex is obtained by joining two smaller subtrees along an edge of $\\mathcal {T}$ .","Any such family contains $2n-1$ subtrees $\\mathcal {T}_{\\hat{\\alpha }}$ labelled by $\\hat{\\alpha }=1,\\dots ,2n-1$ .","Among them, $n-1$ subtrees, which we label by $\\alpha =\\hat{\\alpha }=1,\\dots ,n-1$ , contain several vertices, while the remaining $n$ subtrees with label $\\hat{\\alpha }=n,\\dots ,2n-1$ have only one vertex.","For each subtree $T_\\alpha $ , we denote by $e_\\alpha $ the edge of $\\mathcal {T}$ which is used to reconstruct $T_\\alpha $ from two smaller subtrees $\\mathcal {T}_{\\alpha _L}, \\mathcal {T}_{\\alpha _R}$ .","From this data, we construct a rooted binary tree $T$ with $n-1$ vertices in one-to-one correspondence with the subtrees $\\mathcal {T}_\\alpha $ and $n$ leaves in one-to-one correspondence with the one-vertex subtrees $\\mathcal {T}_{\\hat{\\alpha }}$ with $\\hat{\\alpha }\\ge n$ .","In this correspondence, the two children of a vertex associated to $\\mathcal {T}_{\\alpha }$ are the vertices associated to the two subtrees $\\mathcal {T}_{\\alpha _L}, \\mathcal {T}_{\\alpha _R}$ .","The ordering at each vertex is defined to be such that the subtree containing the source/target vertex of the corresponding edge $e_\\alpha $ is on the left/right.The orientation of the edges of $\\mathcal {T}$ is fixed already in (REF ), but the full kernel $\\Phi ^{\\scriptscriptstyle \\,\\int }_n$ does not depend on its choice.","Of course, this construction is not unique since there are many ways to decompose $\\mathcal {T}$ into such a set of subtrees (see Fig.","REF ).","Figure: An example of an unrooted labelled tree with 4 vertices and two choices of decompositions into subtrees withthe corresponding rooted binary trees.","The edges are labelled e 1 ,e 2 ,e 3 e_1,e_2,e_3 from left to rightalong 𝒯\\mathcal {T}.Applying the above construction to this particular choice of rooted tree, one finds $\\sqrt{\\Delta _\\alpha \\Delta _\\beta }\\,\\mathcal {M}^{-1}_{\\alpha \\beta }=\\left\\lbrace \\begin{array}{ll}\\mathfrak {p}_{v_\\alpha }, \\quad & \\alpha =\\beta ,\\\\\\epsilon _{\\alpha \\beta } \\mathfrak {p}_{L(v_\\beta )}, \\quad & e_\\alpha \\cap \\mathcal {T}_{\\beta _R} \\ne \\emptyset ,\\ e_\\alpha \\nsubseteq \\mathcal {T}_\\beta ,\\\\\\epsilon _{\\alpha \\beta } \\mathfrak {p}_{R(v_\\beta )}, \\quad & e_\\alpha \\cap \\mathcal {T}_{\\beta _L} \\ne \\emptyset ,\\ e_\\alpha \\nsubseteq \\mathcal {T}_\\beta ,\\\\0, & e_\\alpha \\cap \\mathcal {T}_\\beta = \\emptyset \\ \\mbox{ or }\\ e_\\alpha \\subset \\mathcal {T}_\\beta ,\\end{array}\\right.$ where $\\epsilon _{\\alpha \\beta }=-1$ if the orientations of $e_\\alpha $ and $e_\\beta $ on the path joining them are the same and $+1$ otherwise.","This result shows that the matrix $\\mathcal {M}^{-1}$ turns out to be triangular which makes it much simpler to find its inverse.","On the basis of (REF ), below we will prove the following Lemma 2 One has $\\mathcal {B}\\cdot \\mathcal {V}=\\mathcal {M}$ and $\\mathcal {B}\\cdot {x}=\\left\\lbrace \\frac{w^\\alpha }{\\sqrt{\\Delta _\\alpha }}\\right\\rbrace $ provided $\\begin{split}\\mathcal {B}=&\\, \\left(\\frac{\\tilde{{u}}_1}{\\sqrt{\\Delta _1}},\\dots , \\frac{\\tilde{{u}}_{n-1}}{\\sqrt{\\Delta _{n-1}}}\\right)^{\\rm tr},\\\\\\mathcal {V}=&\\, \\mathfrak {p}^{-1}\\Bigl (\\sqrt{\\Delta _1}{u}_1,\\dots , \\sqrt{\\Delta _{n-1}}{u}_{n-1}\\Bigr ),\\end{split}$ where the vectors ${u}_\\alpha $ are defined in (REF ).","Moreover, the vectors $\\tilde{{u}}_\\alpha $ form an orthogonal basis in the subspace spanned by ${u}_\\alpha $ .","This lemma allows to reexpress the kernel $\\Phi _\\mathcal {T}$ (REF ) in terms of the boosted generalized error function $\\Phi ^M_{n-1}$ (REF ).","It is important that its argument $\\mathcal {V}$ does not depend on the choice of the binary rooted tree $T$ , but only on the unrooted tree $\\mathcal {T}$ .","In addition, the function actually does not depend on the normalization of the vectors composing $\\mathcal {V}$ .","Given also that the determinant of $\\mathcal {M}$ is found to be $|\\,{\\rm det}\\, \\mathcal {M}|=\\prod _{\\alpha =1}^{n-1}\\frac{ \\Delta _\\alpha }{\\mathfrak {p}_{v_\\alpha }}=\\prod _{i=1}^{n}\\mathfrak {p}_i\\prod _{v\\in V_{T}\\setminus {\\lbrace v_0\\rbrace }}\\mathfrak {p}_v=\\sqrt{\\mathfrak {p}^{-1}\\,\\prod _{i=1}^{n}\\mathfrak {p}_i\\,\\prod _{\\alpha =1}^{n-1} \\Delta _\\alpha },$ so that the prefactor in (REF ) cancels, one arrives at $\\Phi _\\mathcal {T}({x}) =\\frac{1}{2^{n-1}}\\,\\Phi ^{\\scriptscriptstyle \\,\\int }_1(x)\\, \\Phi ^M_{n-1}(\\lbrace {u}_e\\rbrace ;{x}).$ Finally, since the differential operator in (REF ) commutes with functions of $x$ due to the orthogonality of ${v}_{ij}$ and ${t}$ , one can write the kernel $\\Phi ^{\\scriptscriptstyle \\,\\int }_n$ as $\\Phi ^{\\scriptscriptstyle \\,\\int }_n({x})=\\frac{\\Phi ^{\\scriptscriptstyle \\,\\int }_1(x)}{2^{n-1} n!","}\\sum _{\\mathcal {T}\\in \\, {T}_n^\\ell }\\left[\\prod _{e\\in E_\\mathcal {T}}\\mathcal {D}( {v}_{s(e) t(e)})\\right]\\Phi ^M_{n-1}(\\lbrace {u}_\\alpha \\rbrace ;{x}).$ which is the same as (REF )."],["Proof of Lemma 2","We start by proving that the vectors $\\tilde{{u}}_\\alpha $ form an orthogonal basis in the subspace spanned by ${u}_\\alpha $ .","Since the orthogonality is ensured by construction based on a rooted binary tree, it remains to show that any vector ${u}_\\alpha $ can be decomposed as a linear combination of $\\tilde{{u}}_\\alpha $ .","To this end, we show that the determinant of the Gram matrix constructed from the set of vectors $\\lbrace \\tilde{{u}}_\\alpha \\rbrace _{\\alpha =1}^{n-1}\\cup \\lbrace {u}_\\beta \\rbrace $ vanishes.","This requires to calculate the scalar product $(\\tilde{{u}}_\\alpha , {u}_\\beta )$ which can be done using $({u}_{ij},{u}_{kl})=\\left\\lbrace \\begin{array}{ll}\\mathfrak {p}_i\\mathfrak {p}_j\\mathfrak {p}_{i+j},\\qquad & i=k,\\ j=l,\\\\\\mathfrak {p}_i\\mathfrak {p}_j\\mathfrak {p}_{l},\\qquad & i=k,\\ j\\ne l,\\\\0,\\qquad & i,j\\ne k,l.\\end{array}\\right.$ Summing $i,j,k,l$ over appropriate subsets, it is immediate to see that $(\\tilde{{u}}_\\alpha , {u}_\\beta )=0$ if $\\mathcal {T}_{\\alpha }\\subset \\mathcal {T}_\\beta ^s$ or $\\mathcal {T}_\\beta ^t$ , which are the two trees obtained by dividing the tree $\\mathcal {T}$ into two parts by cutting the edge $e_\\alpha $ .","In other words, it is non-vanishing only if $e_\\beta \\subseteq \\mathcal {T}_\\alpha $ .","Then there are two cases which give $(\\tilde{{u}}_\\alpha , {u}_\\beta )=\\left\\lbrace \\begin{array}{ll}\\mathfrak {p}_\\alpha ^L\\,\\mathfrak {p}_\\alpha ^R\\,\\mathfrak {p}, \\quad &\\alpha =\\beta ,\\\\\\mathfrak {p}_{\\alpha \\beta }^{Ls}\\,\\mathfrak {p}_\\alpha ^R\\,\\mathfrak {p}_\\beta ^t-\\mathfrak {p}_{\\alpha \\beta }^{Lt}\\,\\mathfrak {p}_\\alpha ^R\\,\\mathfrak {p}_\\beta ^s+\\mathfrak {p}_{\\alpha }^{L}\\,\\mathfrak {p}_\\alpha ^R\\,\\mathfrak {p}_\\beta ^s=\\mathfrak {p}_{\\alpha \\beta }^{Ls}\\,\\mathfrak {p}_\\alpha ^R\\,\\mathfrak {p},\\quad & e_\\beta \\subset \\mathcal {T}_\\alpha ,\\end{array}\\right.$ where we introduced $\\mathfrak {p}_\\alpha ^L=\\sum _{i\\in \\mathcal {I}_{L(v_\\alpha )}}\\mathfrak {p}_i,\\qquad \\mathfrak {p}_{\\beta }^{s}=\\sum _{i\\in V_{\\mathcal {T}_\\beta ^s}}\\mathfrak {p}_i,\\qquad \\mathfrak {p}_{\\alpha \\beta }^{Ls}=\\sum _{i\\in \\mathcal {I}_{L(v_\\alpha )}\\cap V_{\\mathcal {T}_\\beta ^s}}\\mathfrak {p}_i,\\qquad \\mathfrak {p}_{\\alpha \\beta }^{st}=\\sum _{i\\in V_{\\mathcal {T}_\\alpha ^s}\\cap V_{\\mathcal {T}_\\beta ^t}}\\mathfrak {p}_i,$ and similarly for variables with labels $R$ and $t$ .","In (REF ) in the second case we assumed that the orientation of edges is such that $e_\\beta \\subset \\mathcal {T}_\\alpha ^s$ and $e_\\alpha \\subset \\mathcal {T}_\\beta ^t$ .","If this is not the case, one should replace $s$ by $t$ , $L$ by $R$ and flip the sign for each change of orientation.","Below we use the same assumption, but the computation can easily be generalized to a more general situation.","Given the result (REF ), $(\\tilde{{u}}_\\alpha , \\tilde{{u}}_\\beta )=\\Delta _\\alpha \\delta _{\\alpha \\beta }$ and $({u}_\\beta , {u}_\\beta )=\\mathfrak {p}_\\beta ^s\\,\\mathfrak {p}_\\beta ^t\\,\\mathfrak {p}$ , the determinant of the Gram matrix is easily found to be $\\,{\\rm det}\\, {\\rm Gram}(\\tilde{{u}}_1,\\cdots , \\tilde{{u}}_{n-1},{u}_\\beta )=\\mathfrak {p}\\prod _{\\alpha =1}^{n-1}\\Delta _\\alpha \\left[\\mathfrak {p}_\\beta ^s\\,\\mathfrak {p}_\\beta ^t-\\mathfrak {p}\\sum _{T_\\alpha \\supseteq e_\\beta }\\frac{(\\mathfrak {p}_{\\alpha \\beta }^{Ls})^2\\,\\mathfrak {p}_\\alpha ^R}{\\mathfrak {p}_{v_\\alpha }\\mathfrak {p}_\\alpha ^L } \\right].$ Note that the subtrees $\\mathcal {T}_\\alpha $ containing the edge $e_\\beta $ form an ordered set so that the sum in the square brackets goes over $\\alpha _\\ell $ , $\\ell =1,\\dots , m$ , such that $\\mathcal {T}_{\\alpha _\\ell }\\subset \\mathcal {T}_{\\alpha _{\\ell +1}}$ .","The first element of this set $\\alpha _1=\\beta $ , whereas the last corresponds to the total tree, $\\mathcal {T}_{\\alpha _m}=\\mathcal {T}$ .","Due to $\\mathfrak {p}_{\\alpha _m}^L=\\mathfrak {p}_{\\alpha _m}^s$ , $\\mathfrak {p}_{\\alpha _m}^R=\\mathfrak {p}_{\\alpha _m}^t$ and $\\mathfrak {p}_{\\alpha _m\\beta }^{Ls}=\\mathfrak {p}_\\beta ^s$ , the first term in the square brackets together with the term in the sum corresponding to $\\alpha _m$ gives $\\mathfrak {p}_\\beta ^s\\left(\\mathfrak {p}_\\beta ^t -\\frac{\\mathfrak {p}_\\beta ^s\\,\\mathfrak {p}_{\\alpha _m}^t}{\\mathfrak {p}_{\\alpha _m}^s}\\right)=\\frac{\\mathfrak {p}\\,\\mathfrak {p}_\\beta ^s\\,\\mathfrak {p}_{\\alpha _m\\beta }^{st}}{\\mathfrak {p}_{\\alpha _m}^s},$ where we have used that $\\mathfrak {p}_\\beta ^t=\\mathfrak {p}_{\\alpha _m\\beta }^{st}+\\mathfrak {p}_{\\alpha _m}^t$ and $\\mathfrak {p}_{\\alpha _m}^s=\\mathfrak {p}_{\\alpha _m\\beta }^{st}+\\mathfrak {p}_\\beta ^s$ .","Thus, the expression in the square brackets in (REF ) becomes $\\frac{\\mathfrak {p}}{\\mathfrak {p}_{\\alpha _m}^s}\\left[\\mathfrak {p}_\\beta ^s\\,\\mathfrak {p}_{\\alpha _m\\beta }^{st}-\\mathfrak {p}_{\\alpha _m}^s\\sum _{\\ell =1}^{m-1}\\frac{(\\mathfrak {p}_{\\alpha _\\ell \\beta }^{Ls})^2\\,\\mathfrak {p}_{\\alpha _\\ell }^R}{\\mathfrak {p}_{v_{\\alpha _\\ell }}\\mathfrak {p}_{\\alpha _\\ell }^L }\\right].$ The new expression in the square brackets is exactly the same as in (REF ) where tree $\\mathcal {T}$ was replaced by subtree $\\mathcal {T}_{\\alpha _m}^s=\\mathcal {T}_{\\alpha _{m-1}}$ .","Thus, one can repeat the above manipulation until one exhausts all terms in the sum.","As a result, the determinant of the Gram matrix turns out to be proportional to $\\mathfrak {p}_{\\alpha _1\\beta }^{st}$ .","But since $\\alpha _1=\\beta $ , this quantity, and hence the whole determinant, trivially vanish.","Next, we prove that (REF ) is consistent with $\\mathcal {B}\\cdot {x}=\\left\\lbrace \\frac{w^\\alpha }{\\sqrt{\\Delta _\\alpha }}\\right\\rbrace $ and $\\mathcal {B}\\cdot \\mathcal {V}=\\mathcal {M}$ .","The first relation is a direct consequence of (REF ).","The second relation requires to show that $(\\tilde{{u}}_\\alpha , {u}_\\beta )=\\mathfrak {p}\\,\\Delta _\\alpha ^{1/2}\\Delta _\\beta ^{-1/2}\\mathcal {M}_{\\alpha \\beta }$ or equivalently $\\sum _{\\gamma }(\\tilde{{u}}_\\alpha , {u}_\\gamma ) \\sqrt{\\Delta _\\gamma \\Delta _\\beta }\\mathcal {M}_{\\gamma \\beta }^{-1}=\\mathfrak {p}\\,\\Delta _\\alpha \\delta _{\\alpha \\beta }$ .","Using the result (REF ) for the matrix $\\mathcal {M}_{\\gamma \\beta }^{-1}$ , this relation can be written explicitly as $\\mathfrak {p}_{v_\\beta }(\\tilde{{u}}_\\alpha , {u}_\\beta )+\\left(\\sum _{e_\\gamma \\in E_\\beta ^R}\\epsilon _{\\gamma \\beta } \\mathfrak {p}_{L(v_\\beta )}+\\sum _{e_\\gamma \\in E_\\beta ^L}\\epsilon _{\\gamma \\beta } \\mathfrak {p}_{R(v_\\beta )}\\right)(\\tilde{{u}}_\\alpha , {u}_\\gamma )=\\mathfrak {p}\\,\\Delta _\\alpha \\delta _{\\alpha \\beta },$ where $E_{\\beta }^{L}=\\lbrace e\\in E_\\mathcal {T}:\\ e\\cap \\mathcal {T}_{\\beta _L}\\ne \\emptyset , \\ e\\nsubseteq \\mathcal {T}_\\beta \\rbrace $ and similarly for $E_{\\beta }^{R}$ .","Consider first the case $\\alpha =\\beta $ .","From (REF ), it immediately follows that the first term gives $\\mathfrak {p}\\,\\Delta _\\alpha $ .","On the other hand, the second contribution sums over edges for which $\\mathcal {T}_\\alpha \\subseteq \\mathcal {T}_\\gamma ^s$ or $\\mathcal {T}_\\gamma ^t$ , and as noted above (REF ) this leads to vanishing of the scalar product.","Thus, in this case the relation (REF ) indeed holds.","Let us now show that it holds as well for $\\alpha \\ne \\beta $ .","To this end, one should consider several options.","If $\\mathcal {T}_\\alpha \\cap \\mathcal {T}_\\beta =\\emptyset $ , then $\\mathcal {T}_\\alpha \\subset \\mathcal {T}_\\beta ^s$ or $\\mathcal {T}_\\beta ^t$ which implies vanishing of the first term.","But the second term vanishes as well because the conditions $e_\\gamma \\subseteq \\mathcal {T}_\\alpha $ and $e_\\gamma \\cap \\mathcal {T}_\\beta \\ne \\emptyset $ are inconsistent with $\\mathcal {T}_\\alpha \\cap \\mathcal {T}_\\beta =\\emptyset $ .","Similarly, if $\\mathcal {T}_\\alpha \\subset \\mathcal {T}_\\beta $ , one has $\\mathcal {T}_\\alpha \\subset \\mathcal {T}_\\beta ^s$ or $T_\\beta ^t$ which again leads to the vanishing of the first term, whereas the vanishing of the second is a consequence of that $e_\\gamma \\subseteq \\mathcal {T}_\\alpha $ implies $e_\\gamma \\subset \\mathcal {T}_\\beta $ so that the sum over $e_\\gamma $ is empty.","It remains to consider the case $\\mathcal {T}_\\beta \\subset \\mathcal {T}_\\alpha $ .","It is clear that $\\mathcal {T}_\\beta \\subset \\mathcal {T}_\\alpha ^s$ or $\\mathcal {T}_\\alpha ^t$ .","Without loss of generality, let us assume that $\\mathcal {T}_\\beta \\subset \\mathcal {T}_\\alpha ^s$ and $e_\\alpha \\subset \\mathcal {T}_\\beta ^t$ .","Then according to (REF ), the first term gives $\\mathfrak {p}_{v_\\beta }\\,\\mathfrak {p}_{\\alpha \\beta }^{Ls}\\,\\mathfrak {p}_\\alpha ^R\\,\\mathfrak {p}$ .","If instead we have chosen the orientation such that $e_\\alpha \\subset \\mathcal {T}_\\beta ^s$ , then we would find $-\\mathfrak {p}_{v_\\beta }\\,\\mathfrak {p}_{\\alpha \\beta }^{Lt}\\,\\mathfrak {p}_\\alpha ^R\\,\\mathfrak {p}$ .","Similar results are obtained for each term in the sum of the second contribution.","Again without loss of generality we assume that for all relevant edges $e_\\gamma $ one has $e_\\alpha \\subset \\mathcal {T}_\\gamma ^t$ , otherwise one flips their orientation.","Then the l.h.s.","of (REF ) is proportional to $\\mathfrak {p}_{v_\\beta }\\,\\mathfrak {p}_{\\alpha \\beta }^{Ls}+\\sum _{e_\\gamma \\in E_{\\beta }^{R}}\\epsilon _{\\gamma \\beta } \\mathfrak {p}_\\beta ^L\\,\\mathfrak {p}_{\\alpha \\gamma }^{Ls}+\\sum _{e_\\gamma \\in E_{\\beta }^{L}}\\epsilon _{\\gamma \\beta } \\mathfrak {p}_\\beta ^R \\,\\mathfrak {p}_{\\alpha \\gamma }^{Ls}.$ Let $e_\\alpha ^\\star \\in E_{\\beta }^{R}$ is the edge belonging to the path from $\\mathcal {T}_\\beta $ to $e_\\alpha $ (which may coincide with $e_\\alpha $ ).","Then our choice of orientation implies that $\\epsilon _{\\gamma \\beta }=-1$ for $e_\\gamma \\in E_\\beta ^L\\cup \\lbrace e_\\alpha ^\\star \\rbrace $ and $\\epsilon _{\\gamma \\beta }=1$ for $e_\\gamma \\in E_\\beta ^R\\setminus \\lbrace e_\\alpha ^\\star \\rbrace $ .","Furthermore, one has $\\mathfrak {p}_{\\alpha \\gamma _\\alpha ^\\star }^{Ls}=\\mathfrak {p}_{v_\\beta }+\\sum _{e_\\gamma \\in E_\\beta ^L\\cup E_\\beta ^R\\setminus \\lbrace e_\\alpha ^\\star \\rbrace }\\mathfrak {p}_{\\alpha \\gamma }^{Ls},\\qquad \\mathfrak {p}_{\\alpha \\beta }^{Ls}=\\mathfrak {p}_\\beta ^L+\\sum _{e_\\gamma \\in E_\\beta ^L}\\mathfrak {p}_{\\alpha \\gamma }^{Ls}.$ As a result, one finds $(\\mathfrak {p}_\\beta ^L+\\mathfrak {p}_\\beta ^R)\\left(\\mathfrak {p}_\\beta ^L+\\sum _{e_\\gamma \\in E_\\beta ^L}\\mathfrak {p}_{\\alpha \\gamma }^{Ls}\\right)-\\mathfrak {p}_\\beta ^L \\,\\mathfrak {p}_{\\alpha \\gamma _\\alpha ^\\star }^{Ls}+\\sum _{e_\\gamma \\in E_{\\beta }^{R}\\setminus \\lbrace e_\\alpha ^\\star \\rbrace } \\mathfrak {p}_\\beta ^L\\,\\mathfrak {p}_{\\alpha \\gamma }^{Ls}-\\sum _{e_\\gamma \\in E_{\\beta }^{L}}\\mathfrak {p}_\\beta ^R \\,\\mathfrak {p}_{\\alpha \\gamma }^{Ls}=0.$ This completes the proof of the required statement (REF ).","toc"],["Proofs of propositions","In this appendix we prove several propositions which we stated in the main text."],["Proposition ","To prove the recursive equation (REF ), let us note that the tree index $g_{{\\rm tr},n}$ satisfies a very similar equation (cf.","(REF ) or (REF )) which can be seen as the origin of its representation (REF ) in terms of attractor flow trees.","The only difference is the absence of the last term in (REF ).","Therefore, it is easy to see that this equation implies a simple relation between $\\widehat{g}_n$ and $g_{{\\rm tr},n}$ $\\widehat{g}_n(\\lbrace \\check{\\gamma }_i\\rbrace ,z^a)=\\sum _{n_1+\\cdots +n_m= n\\atop n_k\\ge 1}g_{{\\rm tr},m}(\\lbrace \\check{\\gamma }^{\\prime }_k\\rbrace ,z^a)\\prod _{k=1}^m W_{n_k}(\\check{\\gamma }_{j_{k-1}+1},\\dots ,\\check{\\gamma }_{j_{k}}),$ where as usual we use notations from (REF ).","Next, we substitute this relation into the expansion (REF ).","The result can be represented in the following form $h^{\\rm DT}_{p,q}=\\sum _{\\sum _{i=1}^n \\check{\\gamma }_i=\\check{\\gamma }}g_{{\\rm tr},n}(\\lbrace \\check{\\gamma }_i\\rbrace ,z^a)\\,e^{\\pi I\\tau Q_n(\\lbrace \\check{\\gamma }_i\\rbrace )}\\prod _{i=1}^n h^R_{p_i,\\mu _i}(\\tau ),$ where we introduced $\\begin{split}h^{R}_{p,\\mu }=&\\, \\sum _{\\sum _{i=1}^n \\check{\\gamma }_i=\\check{\\gamma }}W_n(\\lbrace \\check{\\gamma }_i\\rbrace )\\,e^{\\pi I\\tau Q_n(\\lbrace \\check{\\gamma }_i\\rbrace )}\\prod _{i=1}^n \\widehat{h}_{p_i,\\mu _i}(\\tau )\\\\=&\\, \\sum _{\\sum _{i=1}^n \\check{\\gamma }_i=\\check{\\gamma }}\\left[\\sum _{T\\in {T}_n^{\\rm S}}(-1)^{n_T}\\prod _{v\\in V_T} R_{v}\\right]\\,e^{\\pi I\\tau Q_n(\\lbrace \\check{\\gamma }_i\\rbrace )}\\\\&\\, \\times \\prod _{i=1}^n \\left(h_{p_i,\\mu _i}+\\sum _{n_i=2}^\\infty \\sum _{\\sum _{j=1}^{n_i} \\check{\\gamma }^{\\prime }_j=\\check{\\gamma }}R_{n_i}(\\lbrace \\check{\\gamma }^{\\prime }_j\\rbrace )\\, e^{\\pi I\\tau Q_{n_i}(\\lbrace \\check{\\gamma }^{\\prime }_j\\rbrace )}\\prod _{j_i=1}^{n_i} h_{p^{\\prime }_{j_i},\\mu ^{\\prime }_{j_i}}\\right)\\end{split}$ and in the last relation we used the definition of $W_n$ (REF ) and the expansion of $\\widehat{h}_{p,\\mu }$ (REF ).","The crucial observation is that if one picks up a factor $R_{n_i}$ from the second line of (REF ), appearing due to the expansion of $\\widehat{h}_{p_i,\\mu _i}$ , and combines it with the contribution of a tree $T$ from the first line, one finds the opposite of the contribution of another tree obtained from $T$ by adding $n_i$ children to its $i$ th leaf.","As a result, all such contributions cancel and the function (REF ) reduces to the trivial term $h^{R}_{p,\\mu }=h_{p,\\mu }.$ Substituting this into (REF ), it gives back the original expansion (REF ) of the generating function of DT invariants, which proves the recursive equation (REF ).","Finally, let us evaluate (REF ) at the attractor point $z^a_\\infty (\\gamma )$ .","At this point the DT invariants coincide with MSW invariants so that the l.h.s.","becomes the generating function $h_{p,\\mu }$ .","Meanwhile, all factors $\\Delta _{\\gamma _L\\gamma _R}^z$ vanish at the attractor point and $\\widehat{g}_n$ reduces to $W_n$ .","As a result, the relation (REF ) reproduces (REF ), which completes the proof of the proposition.","We prove Proposition REF by induction.","For $n=2$ the recursive relation (REF ) reads $g^{(0)}_2(\\check{\\gamma }_1,\\check{\\gamma }_2;c_1)-g^{(0)}_2(\\check{\\gamma }_1,\\check{\\gamma }_2;\\beta _{21})=-\\frac{1}{4}\\, \\bigl (\\mbox{sgn}(c_1)-\\mbox{sgn}(\\beta _{21}) \\bigr )\\, \\kappa (\\gamma _{12}),$ where we took into account that $S_1=c_1$ and $\\Gamma _{21}=\\beta _{21}=-\\gamma _{12}$ .","Since $g^{(0)}_2$ is supposed to have discontinuities only at walls of marginal stability, it must not involve signs of DSZ products.","Therefore, we are led to take $g^{(0)}_2(\\check{\\gamma }_1,\\check{\\gamma }_2;c_1)=-\\frac{1}{4}\\, \\mbox{sgn}(c_1) \\kappa (\\gamma _{12}).$ Then (REF ) and (REF ) imply $\\mathcal {E}_2=\\frac{1}{4}\\,\\mbox{sgn}(\\gamma _{12})\\,\\kappa (\\gamma _{12})+R_2,$ so that the ansatz (REF ) reads $\\widehat{g}_2(\\check{\\gamma }_1,\\check{\\gamma }_2;c_1)=g^{(0)}_2(\\check{\\gamma }_1,\\check{\\gamma }_2;c_1)-\\mathcal {E}_2(\\check{\\gamma }_1,\\check{\\gamma }_2)=-\\frac{1}{4}\\,\\Bigl [\\mbox{sgn}(c_1)+\\mbox{sgn}(\\gamma _{12})\\Bigr ]\\,\\kappa (\\gamma _{12})-R_2(\\check{\\gamma }_1,\\check{\\gamma }_2),$ which reproduces the recursive equation (REF ).","Furthermore, in appendix REF it is shown that there is a smooth solution of Vignéras' equation which asymptotes the function $({v},{x}){\\rm sgn}({v},{x})$ , coinciding with the (rescaled) first term in $\\mathcal {E}_2$ for ${v}={v}_{12}$ (REF ).","It is given by $\\widetilde{\\Phi }_1^E({v},{v};{x})={v}\\cdot \\left({x}+\\frac{1}{2\\pi }\\,\\partial _{x}\\right)\\operatorname{Erf}\\left(\\frac{\\sqrt{\\pi }{v}\\cdot {x}}{|{v}|}\\right),$ which corresponds to the following choice of $R_2$ [14] $R_2=\\frac{(-1)^{\\gamma _{12}}}{8\\pi }\\, |\\gamma _{12}|\\,\\beta _{\\frac{3}{2}}\\!\\left({\\frac{2\\tau _2\\gamma _{12}^2 }{(pp_1p_2)}}\\right),$ where $\\beta _{\\frac{3}{2}}(x^2)=2|x|^{-1}e^{-\\pi x^2}-2\\pi \\text{Erfc}(\\sqrt{\\pi } |x|)$ .","Note that the resulting $\\mathcal {E}_2$ depends on the electric charges only through the DSZ product $\\gamma _{12}$ .","Finally, it is immediate to see that the kernel $\\Phi ^{\\,\\widehat{g}}_2$ (REF ) satisfies (REF ).","Now we assume that (REF ) is consistent with the ansatz (REF ) for all orders up to $n-1$ and check it at order $n$ .","Denoting the second term in (REF ) by $\\widehat{g}^{(+)}_n$ and substituting this ansatz into the r.h.s.","of (REF ), one finds $\\qquad \\qquad \\,{\\rm Sym}\\, \\left\\lbrace \\sum _{\\ell =1}^{n-1}g_2^\\ell \\,\\Bigl [g^{(0)}_\\ell \\,g^{(0)}_{n-\\ell }-\\widehat{g}^{(+)}_\\ell \\,\\widehat{g}_{n-\\ell }-\\widehat{g}_\\ell \\,\\widehat{g}^{(+)}_{n-\\ell }-\\widehat{g}^{(+)}_\\ell \\,\\widehat{g}^{(+)}_{n-\\ell }\\Bigr ]_{c_i\\rightarrow c_i^{(\\ell )}}\\right\\rbrace +W_n,$ where $g_2^\\ell =-{1\\over 2}\\,\\Delta _{\\gamma _L^\\ell \\gamma _R^\\ell }^z\\,\\kappa (\\gamma _{LR}^\\ell )=\\frac{1}{4}\\, \\bigl (\\mbox{sgn}(S_\\ell )-\\mbox{sgn}(\\Gamma _{n\\ell })\\bigr )\\,\\kappa (\\Gamma _{n\\ell }).$ In the first term proportional to $g^{(0)}_\\ell \\,g^{(0)}_{n-\\ell }$ , one can apply the relation (REF ), which together with (REF ) gives $g^{(0)}_n-\\mathcal {E}^{(0)}_n$ .","The other terms in the sum over $\\ell $ can be reorganized as follows $\\begin{split}&\\,-\\,{\\rm Sym}\\, \\left\\lbrace \\sum _{\\ell =1}^{n-1} g_2^\\ell \\sum _{n_1+\\cdots +n_m= n\\atop n_k\\ge 1, \\ m1$ (following from $\\ell \\in \\lbrace j_k\\rbrace $ in the previous representation).","In square brackets one recognizes the first contribution from the r.h.s.","of (REF ) with $n$ replaced by $m1$ in the remaining sum with $\\widehat{g}_{m}$ .","As a result, (REF ) becomes equivalent to the r.h.s.","of (REF ), which proves the consistency of this ansatz with the recursive equation.","Finally, let us show that the ansatz satisfies the modularity constraint (REF ).","The crucial observation is that the vectors ${v}_{ij}$ and ${u}_{ij}$ (REF ) satisfy $({v}_{i+j,k},{v}_{ij})=({u}_{i+j,k},{v}_{ij})=0,$ where we abused notation and denoted ${v}_{i+j,k}={v}_{ik}+{v}_{jk}$ , etc.","These orthogonality relations, together with the assumption that $\\mathcal {E}_n$ depend on electric charges only through the DSZ products $\\gamma _{ij}\\sim ({v}_{ij},{x})$ , imply factorization of the action of Vignéras' operator on the kernel corresponding to the second term in (REF ).","Indeed, all contributions of $\\partial _{x}^2$ where two derivatives act on different factors vanish and the action reduces to the sum of terms where Vignéras' operator acts on one of the factors.","But since it is supposed to vanish on $\\Phi ^{\\,\\mathcal {E}}_n$ , one obtains the simple result $V_{n-1} \\cdot \\Phi ^{\\,\\widehat{g}}_{n}=V_{n-1} \\cdot \\Phi ^{\\,g^{(0)}}_{n}-\\,{\\rm Sym}\\, \\left\\lbrace \\sum _{n_1+\\cdots +n_m= n\\atop n_k\\ge 1, \\ m0$ through $\\mathcal {V}_{m}$ .","The idea is to replace the factors $\\mathcal {V}_{m_r}$ by the operators $\\mathcal {D}_{m_r}$ .","Then one can realize that, extracting from the resulting function the terms homogeneous in ${x}$ of order $2(m-2)$ (which have two factors of $(p^3)$ ), one obtains the result for $2\\mathcal {V}_{m,2}$ .","Proceeding in this way, one arrives at the representation very similar to (REF ), $\\mathcal {V}_{m,k}(\\lbrace \\check{\\gamma }_{s}\\rbrace )=\\frac{1}{(2\\pi )^k}\\,{\\rm Sym}\\, \\left\\lbrace \\sum _{\\sum \\limits _{r=1}^{2k+1}m_r=m-k}\\!\\!\\!\\!","C(\\lbrace m_r\\rbrace )\\,\\mathcal {P}_{k}(\\lbrace p^{\\prime }_{r}\\rbrace )\\prod _{r=1}^{2k+1}\\mathcal {V}_{m_r}(\\check{\\gamma }_{j_{r-1}+1},\\dots ,\\check{\\gamma }_{j_{r}})\\right\\rbrace .$ Substituting it into (REF ) and using the expression (REF ) for the factors $\\mathcal {P}_{m}$ through the sum over rooted ternary trees, one can recombine all these sums in the following way $\\begin{split}\\mathcal {V}^{\\rm tot}_m=&\\,\\,{\\rm Sym}\\, \\Biggl \\lbrace \\sum _{k=0}^{m}\\sum _{\\sum \\limits _{r=1}^{2k+1}m_r=m-k}\\!\\!\\!\\!\\!\\!","C(\\lbrace m_r\\rbrace )\\sum _{T\\in \\, {T}_{2k+1}^{(3)}(\\lbrace \\check{\\gamma }^{\\prime }_r\\rbrace )}\\frac{1}{T!","}\\sum _{T^{\\prime }\\subseteq T} (-1)^{n_{T^{\\prime }}}\\prod _{v\\in V_{T^{\\prime }}}\\frac{n_v(T)}{n_v(T^{\\prime })}\\\\&\\,\\times \\prod _{v\\in V_{T}}\\frac{(p_{d_1(v)}p_{d_2(v)}p_{d_3(v)})}{2\\pi }\\prod _{r=1}^{2k+1}\\mathcal {V}_{m_r}(\\check{\\gamma }_{j_{r-1}+1},\\dots ,\\check{\\gamma }_{j_{r}})\\Biggr \\rbrace ,\\end{split}$ where the sum over $T^{\\prime }$ is the sum over subtrees of $T$ having the same root.","In terms of the variables appearing in (REF ), one can identify $k=m_0+\\sum _r k_r$ and $n_{T^{\\prime }}=m_0$ .","Thus, $T^{\\prime }$ is the tree appearing in the decomposition of $\\mathcal {P}_{m_0}$ , whereas $T$ is its union with $2m_0+1$ trees $T_r$ appearing in the decomposition of $\\mathcal {P}_{k_r}$ , i.e.","$T=T^{\\prime }\\cup \\left(\\cup _r T_r\\right)$ where $T_r$ are the trees rooted at leaves of $T^{\\prime }$ , Finally, we took into account that for such trees one has $T!=T^{\\prime }!\\,\\prod _r T_r!\\,\\prod _{v\\in V_{T^{\\prime }}}\\frac{n_v(T)}{n_v(T^{\\prime })}\\,.$ Remarkably, the sum over subtrees in (REF ) factorizes and for a fixed number of vertices $n_{T^{\\prime }}=m_0$ is subject to Theorem REF where the rôle of the trees is played by rooted ternary trees $T$ and $T^{\\prime }$ after stripping out their leaves.","As a result, one finds for $k>0$ $\\sum _{T^{\\prime }\\subseteq T} (-1)^{n_{T^{\\prime }}}\\prod _{v\\in V_{T^{\\prime }}}\\frac{n_v(T)}{n_v(T^{\\prime })}=\\sum _{m_0=0}^k\\frac{(-1)^{m_0}k!}{m_0!(k-m_0)!","}=(1-1)^k = 0.$ Thus, the only non-vanishing contribution is the one with $k=0$ which coincides with $\\mathcal {V}_m$ .","This is what we had to show and therefore completes the proof of the proposition.","For simplicity, let us first show that the recursive equation (REF ) is satisfied by the contribution to $g^{(0)}_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace )$ given by the trees without any marks, i.e.","by the function $g^{\\star }_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace )=\\frac{(-1)^{n-1+\\sum _{i1$ (from the sum in (REF )) is cancelled by the contribution of another tree obtained from $T$ by adding $n_k$ children to its $k$ th leaf and the same splitting but with $n_k$ replaced by $1+\\cdots +1$ (repeated $n_k$ times).","The only contribution which survives is the one generated by the tree with a single vertex and $n$ leaves and the splitting with all $n_k=1$ .","It is given by $\\mathcal {E}^{(+)}_n$ , which verifies the constraint and proves the proposition.","Our starting point to prove the proposition is the formula $\\partial _{\\bar{\\tau }}\\widehat{h}_{p,\\mu }(\\tau )=\\frac{I}{2}\\sum _{n=2}^\\infty \\sum _{\\sum _{i=1}^n \\check{\\gamma }_i=\\check{\\gamma }}\\partial _{\\tau _2}R_n(\\lbrace \\check{\\gamma }_i\\rbrace ,\\tau _2)\\, e^{\\pi I\\tau Q_n(\\lbrace \\check{\\gamma }_i\\rbrace )}\\prod _{i=1}^n h_{p_i,\\mu _i}(\\tau ).$ Substituting $\\partial _{\\tau _2}R_n$ following from (REF ) and the inverse formula (REF ) expressing $h_{p,\\mu }$ in terms of the completion, with the functions $W_n$ found in (REF ), one arrives at the result (REF ) where the functions $\\mathcal {J}_n$ are given by $\\mathcal {J}_n= \\frac{I}{2}\\,{\\rm Sym}\\, \\left\\lbrace \\sum _{T\\in {T}_n^{\\rm S}}(-1)^{n_T-1} \\partial _{\\tau _2}\\mathcal {E}_{v_0}\\sum _{T^{\\prime }\\subseteq T}\\prod _{v\\in V_{T^{\\prime }}\\setminus {\\lbrace v_0\\rbrace }}\\mathcal {E}^{(0)}_{v}\\prod _{v\\in L_{T^{\\prime }}}\\mathcal {E}^{(+)}_{v}\\prod _{v\\in V_T\\setminus (V_{T^{\\prime }}\\cup L_{T^{\\prime }})}\\mathcal {E}_{v}\\right\\rbrace ,$ where the second sum goes over all subtrees $T^{\\prime }$ of $T$ containing its root and $L_{T^{\\prime }}$ is the set of their leaves.","Here the subtree $T^{\\prime }$ corresponds to the tree in the formula (REF ) for $R_n$ , whereas the subtrees starting from its leaves correspond to the trees in the expression (REF ) for $W_n$ .","The sum over subtrees can be evaluated in the same way as the sum over decompositions into subtrees in (REF ): the two contributions differing only by whether the vertex $v$ belongs to $V_{T^{\\prime }}$ or $L_{T^{\\prime }}$ combine into the factor $\\mathcal {E}_v$ assigned to this vertex.","Performing this recombination for all vertices of the tree $T$ , one obtains the expression (REF ), which proves the proposition.","Our goal is to prove that in the expression $g_{{\\rm tr},n}= \\,{\\rm Sym}\\, \\left\\lbrace \\sum _{T\\in {T}_n^{\\rm S}}(-1)^{n_T-1} \\left(g^{(0)}_{v_0}-\\mathcal {E}^{(0)}_{v_0}\\right)\\prod _{v\\in V_T\\setminus {\\lbrace v_0\\rbrace }}\\mathcal {E}^{(0)}_{v}\\right\\rbrace ,$ following from (REF ), all contributions due to marked trees cancel leaving only contributions of trees without marks, i.e with $m=0$ .","There are several possible situations which we need to analyze.","Figure: Combination of two Schröder trees ensuring the cancellation of contributions generated by marked trees.First, let us consider the contributions generated by non-trivial marked trees, i.e.","trees having more than one vertex and at least one mark.","Let us focus on the contribution corresponding to a vertex $\\mathfrak {v}$ with $m_\\mathfrak {v}>0$ marks of a tree $\\mathcal {T}$ , which appears in the sum over marked unrooted trees living at a vertex $v$ of a Schröder tree $T$ .","Let $k=n_v$ be the number of children of the vertex $v$ and $\\gamma _i$ ($i=1,\\dots ,k$ ) their charges so that $\\gamma _s$ ($s=1,\\dots ,2m_\\mathfrak {v}+1$ ) are the charges labelling the marked vertex $\\mathfrak {v}$ .","Note that $k\\ge 2m_\\mathfrak {v}+2$ because the tree $\\mathcal {T}$ has at least one additional vertex except $\\mathfrak {v}$ .","Then the contribution we described is cancelled by the contribution coming from another Schröder tree, which is obtained from $T$ by adding an edge connecting the vertex $v$ to a new vertex $v^{\\prime }$ , whose children are the $2m_\\mathfrak {v}+1$ children of $v$ in $T$ carrying charges $\\gamma _s$ (see Fig.","REF ).The new tree is of Schröder type because its vertex $v$ has $k-2m_{\\mathfrak {v}}\\ge 2$ children and vertex $v^{\\prime }$ has $2m_\\mathfrak {v}+1\\ge 3$ children.","Indeed, choosing the same tree $\\mathcal {T}$ as before in the sum over marked trees at vertex $v$ , but now with $m_\\mathfrak {v}=0$ , and in the sum at vertex $v^{\\prime }$ the trivial tree having one vertex and $m_\\mathfrak {v}$ marks, one gets exactly the same contribution as before, but now with an opposite sign due to the presence of an additional vertex in the Schröder tree.","Thus, all contributions from non-trivial marked trees are cancelled.","As a result, we remain only with the contributions generated by trivial marked trees, i.e.","having only one vertex and $m_\\mathfrak {v}$ marks.","One has to distinguish two cases: either the corresponding vertex $v$ of the Schröder tree is the root or not.","In the former case, this contribution is trivially cancelled in the difference $g^{(0)}_{v_0}-\\mathcal {E}^{(0)}_{v_0}$ in (REF ).","In the latter case, this is precisely the contribution used above to cancel the contributions from non-trivial marked trees.","This exhausts all possibilities and we arrive at the formula (REF ).","toc toc"],["Explicit results up to 4th order","In this appendix we provide explicit expressions for various functions appearing in our construction up to the forth order.","To write them down, we will use the shorthand notation $\\gamma _{i+j}=\\gamma _i+\\gamma _j$ , $c_{i+j}=c_i+c_j$ , etc.","as well as indicate the arguments of functions through their indices, for instance, $\\mathcal {E}_{i_1\\cdots i_n}=\\mathcal {E}_n(\\check{\\gamma }_{i_1},\\dots ,\\check{\\gamma }_{i_n})$ .","These expressions are obtained by simple substitutions using the results found in the main text and the sets of trees shown in Fig.","REF .","For $n=2$ they all agree with the results of [14].","The results (REF ) and (REF ) generate the following expansions $h^{\\rm DT}_{p,q}&=&\\widehat{h}_{p,\\mu }+\\sum _{\\check{\\gamma }_1+\\check{\\gamma }_2=\\check{\\gamma }}\\left[g^{(0)}_{12}-\\mathcal {E}_{12}\\right]e^{\\pi I\\tau Q_2(\\lbrace \\check{\\gamma }_i\\rbrace )}\\widehat{h}_{p_1,\\mu _1}\\widehat{h}_{p_2,\\mu _2}\\nonumber \\\\&& +\\sum _{\\sum _{i=1}^3 \\check{\\gamma }_i=\\check{\\gamma }}\\left[g^{(0)}_{123}-\\mathcal {E}_{123}-2\\left(g^{(0)}_{1+2,3}-\\mathcal {E}_{1+2,3}\\right)\\mathcal {E}_{12}\\right] e^{\\pi I\\tau Q_3(\\lbrace \\check{\\gamma }_i\\rbrace )}\\prod _{i=1}^3 \\widehat{h}_{p_i,\\mu _i}\\\\&&+\\sum _{\\sum _{i=1}^4 \\check{\\gamma }_i=\\check{\\gamma }}\\left[ g^{(0)}_{1234}-\\mathcal {E}_{1234}-2\\left(g^{(0)}_{1+2+3,4}-\\mathcal {E}_{1+2+3,4}\\right)\\left(\\mathcal {E}_{123} -2\\mathcal {E}_{1+2,3}\\mathcal {E}_{12}\\right)\\right.\\nonumber \\\\&& \\left.\\qquad -3\\left(g^{(0)}_{1+2,34}-\\mathcal {E}_{1+2,34}\\right)\\mathcal {E}_{12}+\\left(g^{(0)}_{1+2,3+4}-\\mathcal {E}_{1+2,3+4}\\right)\\mathcal {E}_{12}\\mathcal {E}_{34}\\right]e^{\\pi I\\tau Q_4(\\lbrace \\check{\\gamma }_i\\rbrace )}\\prod _{i=1}^4 \\widehat{h}_{p_i,\\mu _i}+\\cdots ,\\nonumber $ $\\widehat{h}_{p,q}&=&h_{p,\\mu }+\\sum _{\\check{\\gamma }_1+\\check{\\gamma }_2=\\check{\\gamma }}\\mathcal {E}^{(+)}_{12}\\,e^{\\pi I\\tau Q_2(\\lbrace \\check{\\gamma }_i\\rbrace )}h_{p_1,\\mu _1}h_{p_2,\\mu _2}\\nonumber \\\\&& +\\sum _{\\sum _{i=1}^3 \\check{\\gamma }_i=\\check{\\gamma }}\\Bigl [\\mathcal {E}^{(+)}_{123}-2\\mathcal {E}^{(+)}_{1+2,3}\\mathcal {E}^{(0)}_{12}\\Bigr ] \\,e^{\\pi I\\tau Q_3(\\lbrace \\check{\\gamma }_i\\rbrace )}\\prod _{i=1}^3 h_{p_i,\\mu _i}\\\\&&+\\sum _{\\sum _{i=1}^4 \\check{\\gamma }_i=\\check{\\gamma }}\\Bigl [\\mathcal {E}^{(+)}_{1234}-2\\mathcal {E}^{(+)}_{1+2+3,4}\\left(\\mathcal {E}^{(0)}_{123} -2\\mathcal {E}^{(0)}_{1+2,3}\\mathcal {E}^{(0)}_{12}\\right)-3\\mathcal {E}^{(+)}_{1+2,34}\\mathcal {E}^{(0)}_{12}\\Bigr .\\nonumber \\\\&& \\Bigl .","\\qquad +\\mathcal {E}^{(+)}_{1+2,3+4}\\mathcal {E}^{(0)}_{12}\\mathcal {E}^{(0)}_{34}\\Bigr ]\\, e^{\\pi I\\tau Q_4(\\lbrace \\check{\\gamma }_i\\rbrace )}\\prod _{i=1}^4 h_{p_i,\\mu _i}+\\cdots ,\\nonumber $ where the functions $g^{(0)}_n$ and $\\mathcal {E}_n$ can be read off from (REF ), (REF ) and (REF ), $\\begin{split}g^{(0)}_2=&\\, \\frac{(-1)^{1+\\gamma _{12}}}{4}\\, \\gamma _{12}\\,\\mbox{sgn}(c_1),\\\\g^{(0)}_3=&\\, \\frac{(-1)^{1+\\gamma _{12}+\\gamma _{1+2,3}}}{8}\\,\\,{\\rm Sym}\\, \\biggl \\lbrace \\gamma _{12}\\,\\gamma _{23}\\,\\mbox{sgn}(c_1)\\, \\mbox{sgn}(c_3)+\\frac{1}{3}\\,\\gamma _{12}\\,\\gamma _{23} \\biggr \\rbrace ,\\\\g^{(0)}_4=&\\, \\frac{(-1)^{\\gamma _{12}+\\gamma _{1+2,3}+\\gamma _{1+2+3,4}}}{16}\\,\\,{\\rm Sym}\\, \\biggl \\lbrace \\gamma _{12}\\gamma _{23}\\gamma _{34}\\,\\mbox{sgn}(c_1)\\mbox{sgn}(c_{1+2})\\mbox{sgn}(c_4)\\\\&\\, \\qquad -\\frac{1}{3}\\,\\gamma _{12}\\gamma _{23}\\gamma _{24}\\,\\mbox{sgn}(c_1)\\mbox{sgn}(c_3)\\mbox{sgn}(c_4)-\\frac{1}{3}\\, \\gamma _{12}\\gamma _{23} \\gamma _{1+2+3,4}\\mbox{sgn}(c_4)\\biggr \\rbrace ,\\end{split}$ $\\mathcal {E}_{12} &=&\\frac{(-1)^{\\gamma _{12}}}{4\\sqrt{2\\tau _2}}\\,\\widetilde{\\Phi }_{1}^E\\bigl ({v}_{12},{v}_{12}\\bigr )\\nonumber \\\\&=&\\frac{(-1)^{\\gamma _{12}}}{4}\\left[\\gamma _{12}\\, E_1\\left(\\frac{\\sqrt{2\\tau _2}\\gamma _{12} }{\\sqrt{(pp_1p_2)}}\\right)+\\frac{\\sqrt{(pp_1p_2)}}{\\pi \\sqrt{2\\tau _2}}\\, e^{-\\frac{2\\pi \\tau _2\\gamma _{12}^2 }{(pp_1p_2)}} \\right] ,\\nonumber \\\\\\mathcal {E}_{123} &=&\\frac{(-1)^{\\gamma _{12}+\\gamma _{1+2,3}}}{8}\\,\\,{\\rm Sym}\\, \\left\\lbrace \\frac{1}{2\\tau _2}\\,\\widetilde{\\Phi }_{2}^E\\bigl (({v}_{1,2+3},{v}_{1+2,3}),({v}_{12},{v}_{23})\\bigr )\\right.\\nonumber \\\\&&\\left.","\\qquad -\\frac{1}{3}\\left(\\gamma _{12}\\gamma _{23}-\\frac{(p_1p_2p_3)}{4\\pi \\tau _2} \\right)\\right\\rbrace ,\\\\\\mathcal {E}_{1234} &=&\\frac{(-1)^{\\gamma _{12}+\\gamma _{1+2,3}+\\gamma _{1+2+3,4}}}{16\\sqrt{2\\tau _2}}\\,\\,{\\rm Sym}\\, \\biggl \\lbrace \\frac{1}{2\\tau _2}\\biggl (\\widetilde{\\Phi }_{3}^E\\bigl (({v}_{1,2+3+4},{v}_{1+2,3+4},{v}_{1+2+3,4}),({v}_{12},{v}_{23},{v}_{34})\\bigr )\\nonumber \\\\&&+\\frac{1}{3}\\,\\widetilde{\\Phi }_{3}^E\\bigl (({v}_{1,2+3+4},{v}_{1+2+4,3},{v}_{1+2+3,4}),({v}_{12},{v}_{23},{v}_{24})\\bigr )\\biggr )\\nonumber \\\\&& -\\frac{1}{3}\\left(\\gamma _{12}\\gamma _{23}-\\frac{(p_1p_2p_3)}{4\\pi \\tau _2}\\right)\\widetilde{\\Phi }_{1}^E\\bigl ({v}_{1+2+3,4},{v}_{1+2+3,4}\\bigr )\\biggr \\rbrace ,\\nonumber $ where all generalized error functions are evaluated at ${x}=\\sqrt{2\\tau _2}({q}+{b})$ .","The last terms appearing in the above quantities for $n=3$ and $n=4$ correspond to contributions of trees with one mark ($m=1$ ).","In fact, at these orders these results can be rewritten in a simpler form, which coincides with the representations (REF ) and (REF ) (in the latter formula one should drop the sum over partitions and take $d_T=d_n$ ): $g^{(0)}_3&=& \\frac{(-1)^{1+\\gamma _{12}+\\gamma _{1+2,3}}}{12}\\,\\,{\\rm Sym}\\, \\Bigl \\lbrace \\gamma _{12}\\,\\gamma _{1+2,3}\\,\\mbox{sgn}(c_1)\\, \\mbox{sgn}(c_3) \\Bigr \\rbrace ,\\\\g^{(0)}_4&=&\\textstyle {\\frac{(-1)^{\\gamma _{12}+\\gamma _{1+2,3}+\\gamma _{1+2+3,4}}}{96}}\\,{\\rm Sym}\\, \\!\\Bigl \\lbrace \\!","\\Bigl (2\\,\\gamma _{23}\\gamma _{1,2+3}\\gamma _{1+2+3,4}+\\gamma _{12}\\gamma _{34}\\gamma _{1+2,3+4}\\Bigr )\\mbox{sgn}(c_1)\\mbox{sgn}(c_{1+2})\\mbox{sgn}(c_4)\\Bigr \\rbrace ,\\nonumber $ $\\mathcal {E}_{123} &=&\\frac{(-1)^{\\gamma _{12}+\\gamma _{1+2,3}}}{24\\tau _2}\\,\\,{\\rm Sym}\\, \\left\\lbrace \\widetilde{\\Phi }_{2}^E\\bigl (({v}_{1,2+3},{v}_{1+2,3}),({v}_{12},{v}_{1+2,3})\\bigr )\\right\\rbrace ,\\\\\\mathcal {E}_{1234} &=&\\frac{(-1)^{\\gamma _{12}+\\gamma _{1+2,3}+\\gamma _{1+2+3,4}}}{96(2\\tau _2)^{3/2}}\\,\\,{\\rm Sym}\\, \\biggl \\lbrace 2\\,\\widetilde{\\Phi }_{3}^E\\bigl (({v}_{1,2+3+4},{v}_{1+2,3+4},{v}_{1+2+3,4}),({v}_{23},{v}_{1,2+3},{v}_{1+2+3,4})\\bigr )\\nonumber \\\\&&+\\widetilde{\\Phi }_{3}^E\\bigl (({v}_{1,2+3+4},{v}_{1+2,3+4},{v}_{1+2+3,4}),({v}_{12},{v}_{34},{v}_{1+2,3+4})\\bigr )\\biggr )\\biggr \\rbrace .\\nonumber $ The simplest way to prove the equality of the two representations of $g^{(0)}_n$ is to expand the DSZ products appearing in (REF ) into elementary $\\gamma _{ij}$ 's and then, using symmetrization, bring all their products to the form appearing in (REF ).","These products are multiplied by combinations of sign functions which can be recombined with the help of the identity (REF ).","As an example, let us perform these manipulations for $n=3$ : $\\begin{split}&\\,\\,{\\rm Sym}\\, \\Bigl \\lbrace \\gamma _{12}\\,\\gamma _{1+2,3}\\,\\mbox{sgn}(c_1)\\, \\mbox{sgn}(c_3) \\Bigr \\rbrace =\\,{\\rm Sym}\\, \\Bigl \\lbrace \\gamma _{12}\\,\\gamma _{23}\\bigl (\\mbox{sgn}(c_1)-\\mbox{sgn}(c_2) \\bigr )\\,\\mbox{sgn}(c_3)\\Bigr \\rbrace \\\\=&\\,\\,{\\rm Sym}\\, \\Bigl \\lbrace \\gamma _{12}\\,\\gamma _{23}\\Bigl (\\mbox{sgn}(c_1)\\,\\mbox{sgn}(c_3)+{1\\over 2}\\, \\mbox{sgn}(c_{1+3})\\bigl (\\mbox{sgn}(c_1)+\\mbox{sgn}(c_3) \\bigr )\\Bigr )\\Bigr \\rbrace \\\\=&\\,{1\\over 2}\\,\\,{\\rm Sym}\\, \\Bigl \\lbrace \\gamma _{12}\\,\\gamma _{23}\\Bigl (3\\, \\mbox{sgn}(c_1)\\,\\mbox{sgn}(c_3)+1\\Bigr )\\Bigr \\rbrace ,\\end{split}$ where we used that $c_2=-(c_1+c_3)$ .","This identity then shows the equality of the two forms of $g^{(0)}_3$ given above.","For $g^{(0)}_4$ the manipulations are very similar, but a bit more cumbersome.","The equality of the two forms of $\\mathcal {E}_n$ follows from the equality of their asymptotics $\\mathcal {E}^{(0)}_n$ , which is in turn ensured by the same identities as for $g^{(0)}_n$ .","Finally, we provide expressions for the kernels $\\widehat{\\Phi }^{{\\rm tot}}_n$ of the indefinite theta series appearing in the expansion (REF ) of $\\mathcal {G}$ in powers of $\\widehat{h}_{p,\\mu }$ .","For the first two orders, one has $\\begin{split}\\widehat{\\Phi }^{{\\rm tot}}_1=&\\,\\Phi ^{\\scriptscriptstyle \\,\\int }_1,\\\\\\widehat{\\Phi }^{{\\rm tot}}_2=&\\,\\Phi ^{\\scriptscriptstyle \\,\\int }_2+\\Phi ^{\\scriptscriptstyle \\,\\int }_1 \\Phi ^{\\,\\widehat{g}}_2=\\frac{1}{4}\\, \\Phi ^{\\scriptscriptstyle \\,\\int }_1\\left(\\widetilde{\\Phi }_1^E({u}_{12},{v}_{12})-\\widetilde{\\Phi }_1^E({v}_{12},{v}_{12})\\right),\\end{split}$ where $\\Phi ^{\\scriptscriptstyle \\,\\int }_1(x)$ is defined in (REF ).","At the next order, $\\widehat{\\Phi }^{{\\rm tot}}_3=\\Phi ^{\\scriptscriptstyle \\,\\int }_3+2\\,{\\rm Sym}\\, \\Bigl \\lbrace \\Phi ^{\\scriptscriptstyle \\,\\int }_2(x_{1+2},x_3)\\Phi ^{\\,\\widehat{g}}_2(x_1,x_2) \\Bigr \\rbrace +\\Phi ^{\\scriptscriptstyle \\,\\int }_1 \\Phi ^{\\,\\widehat{g}}_3.$ To get an explicit expression in terms of smooth solutions of Vignéras' equation, one should use the relation (REF ).","Applying it to the case $n=2$ with $\\mathcal {V}=({u}_{1,2+3},{u}_{1+2,3})$ , using the orthogonality properties ${u}_{(1,2+3)\\perp (1+2,3)}={u}_{12},\\qquad {u}_{(1+2,3)\\perp (1,2+3)}={u}_{23},$ and acting by the operator $\\mathcal {D}({v}_{12})\\mathcal {D}({v}_{23})$ , one can show that $&&\\,{\\rm Sym}\\, \\Bigl \\lbrace \\widetilde{\\Phi }^E_{2}\\bigl (({u}_{1,2+3},{u}_{1+2,3}), ({v}_{12},{v}_{23})\\bigr )\\Bigr \\rbrace =\\,{\\rm Sym}\\, \\biggl \\lbrace \\widetilde{\\Phi }^M_{2}\\bigl (({u}_{1,2+3},{u}_{1+2,3}), ({v}_{12},{v}_{23})\\bigr )\\nonumber \\\\&& \\qquad +({v}_{12},{x})\\,\\mbox{sgn}({u}_{12},{x})\\, \\widetilde{\\Phi }^M_1\\bigl ({u}_{1+2,3},{v}_{1+2,3}\\bigr )+({v}_{12},{x})\\,({v}_{23},{x})\\,\\mbox{sgn}({u}_{1,2+3},{x})\\,\\mbox{sgn}({u}_{1+2,3},{x})\\nonumber \\\\&& \\qquad -\\frac{(p_1p_2p_3)}{6\\pi }\\biggr \\rbrace .$ This result allows to obtain the following representation for the kernel $\\widehat{\\Phi }^{{\\rm tot}}_3&=&\\frac{1}{8}\\,\\Phi ^{\\scriptscriptstyle \\,\\int }_1 \\,{\\rm Sym}\\, \\biggl \\lbrace \\widetilde{\\Phi }^E_{2}\\bigl (({u}_{1,2+3},{u}_{1+2,3}), ({v}_{12},{v}_{23})\\bigr )- \\widetilde{\\Phi }^E_{2}\\bigl (({v}_{1,2+3},{v}_{1+2,3}), ({v}_{12},{v}_{23})\\bigr )\\nonumber \\\\&&-\\left(\\widetilde{\\Phi }_1^E({u}_{1+2,3},{v}_{1+2,3})-\\widetilde{\\Phi }_1^E({v}_{1+2,3},{v}_{1+2,3})\\right)\\widetilde{\\Phi }^E_1({v}_{12},{v}_{12})\\biggr \\rbrace .$ Using the identity between functions $\\widetilde{\\Phi }^E_{2}$ implied by the identity (REF ), the kernel can also be rewritten as $\\widehat{\\Phi }^{{\\rm tot}}_3&=&\\frac{1}{8}\\,\\Phi ^{\\scriptscriptstyle \\,\\int }_1 \\,{\\rm Sym}\\, \\biggl \\lbrace \\frac{2}{3}\\, \\Bigl (\\widetilde{\\Phi }^E_{2}\\bigl (({u}_{1,2+3},{u}_{1+2,3}), ({v}_{12},{v}_{1+2,3})\\bigr )- \\widetilde{\\Phi }^E_{2}\\bigl (({v}_{1,2+3},{v}_{1+2,3}), ({v}_{12},{v}_{1+2,3})\\bigr )\\Bigr )\\nonumber \\\\&&-\\left(\\widetilde{\\Phi }_1^E({u}_{1+2,3},{v}_{1+2,3})-\\widetilde{\\Phi }_1^E({v}_{1+2,3},{v}_{1+2,3})\\right)\\widetilde{\\Phi }^E_1({v}_{12},{v}_{12})\\biggr \\rbrace ,$ so that the vectors appearing in the second argument of $\\widetilde{\\Phi }_2^E$ are now mutually orthogonal.","For $n=4$ , the kernel is given by $\\begin{split}\\widehat{\\Phi }^{{\\rm tot}}_4=&\\, \\Phi ^{\\scriptscriptstyle \\,\\int }_4+\\,{\\rm Sym}\\, \\Bigl \\lbrace 3\\Phi ^{\\scriptscriptstyle \\,\\int }_3(x_{1+2},x_3,x_4)\\Phi ^{\\,\\widehat{g}}_2(x_1,x_2)+\\Phi ^{\\scriptscriptstyle \\,\\int }_2(x_{1+2},x_{3+4})\\Phi ^{\\,\\widehat{g}}_2(x_1,x_2)\\Phi ^{\\,\\widehat{g}}_2(x_3,x_4)\\\\&\\,+2\\Phi ^{\\scriptscriptstyle \\,\\int }_2(x_{1+2+3},x_4)\\Phi ^{\\,\\widehat{g}}_3(x_1,x_2,x_3)\\Bigr \\rbrace +\\Phi ^{\\scriptscriptstyle \\,\\int }_1 \\Phi ^{\\,\\widehat{g}}_4.\\end{split}$ Proceeding in the same way as for $n=3$ , obtaining a generalization of (REF ) to $n=4$ , one arrives at $\\widehat{\\Phi }^{{\\rm tot}}_4&=&\\frac{1}{16}\\, \\Phi ^{\\scriptscriptstyle \\,\\int }_1\\,{\\rm Sym}\\, \\biggl \\lbrace \\widetilde{\\Phi }^E_{3}\\bigl (({u}_{1,2+3+4},{u}_{1+2,3+4},{u}_{1+2+3,4}), ({v}_{12},{v}_{23},{v}_{34})\\bigr )\\nonumber \\\\&&\\qquad -\\widetilde{\\Phi }^E_{3}\\bigl (({v}_{1,2+3+4},{v}_{1+2,3+4},{v}_{1+2+3,4}), ({v}_{12},{v}_{23},{v}_{34})\\bigr )\\nonumber \\\\&&\\quad +\\frac{1}{3}\\left(\\widetilde{\\Phi }^E_{3}\\bigl (({u}_{1,2+3+4},{u}_{1+2+4,3},{u}_{1+2+3,4}), ({v}_{12},{v}_{24},{v}_{34})\\bigr )\\right.\\nonumber \\\\&&\\qquad \\left.-\\widetilde{\\Phi }^E_{3}\\bigl (({v}_{1,2+3+4},{v}_{1+2+4,3},{v}_{1+2+3,4}), ({v}_{12},{v}_{24},{v}_{34})\\bigr )\\right)\\nonumber \\\\&&-\\left(\\widetilde{\\Phi }^E_{2}\\bigl (({u}_{1+2,3+4},{u}_{1+2+3,4}), ({v}_{1+2,3},{v}_{34})\\bigr )-\\widetilde{\\Phi }^E_{2}\\bigl (({v}_{1+2,3+4},{v}_{1+2+3,4}), ({v}_{1+2,3},{v}_{34})\\bigr )\\right.\\nonumber \\\\&&\\quad +{1\\over 2}\\left(\\widetilde{\\Phi }^E_{2}\\bigl (({u}_{1+2+4,3},{u}_{1+2+3,4}), ({v}_{1+2,3},{v}_{1+2,4})\\bigr )\\right.\\\\&&\\quad \\qquad \\left.\\left.-\\widetilde{\\Phi }^E_{2}\\bigl (({v}_{1+2+4,3},{v}_{1+2+3,4}), ({v}_{1+2,3},{v}_{1+2,4})\\bigr )\\right)\\right)\\widetilde{\\Phi }^E_1({v}_{12},{v}_{12})\\nonumber \\\\&&-\\left(\\widetilde{\\Phi }_1^E({u}_{1+2+3,4},{v}_{1+2+3,4})-\\widetilde{\\Phi }_1^E({v}_{1+2+3,4},{v}_{1+2+3,4})\\right)\\widetilde{\\Phi }^E_{2}\\bigl (({v}_{1,2+3},{v}_{1+2,3}), ({v}_{12},{v}_{23})\\bigr )\\nonumber \\\\&&+\\left(\\widetilde{\\Phi }_1^E({u}_{1+2+3,4},{v}_{1+2+3,4})-\\widetilde{\\Phi }_1^E({v}_{1+2+3,4},{v}_{1+2+3,4})\\right)\\widetilde{\\Phi }^E_1({v}_{12},{v}_{12})\\widetilde{\\Phi }^E_1({v}_{1+2,3},{v}_{1+2,3})\\nonumber \\\\&&+\\frac{1}{4}\\left(\\widetilde{\\Phi }_1^E({u}_{1+2,3+4},{v}_{1+2,3+4})-\\widetilde{\\Phi }_1^E({v}_{1+2,3+4},{v}_{1+2,3+4})\\right)\\widetilde{\\Phi }^E_1({v}_{12},{v}_{12})\\widetilde{\\Phi }^E_1({v}_{34},{v}_{34})\\biggr \\rbrace .\\nonumber $ It is possible also to rewrite this expression in terms of generalized error functions $\\widetilde{\\Phi }^E_n(\\mathcal {V},\\tilde{\\mathcal {V}})$ where the vectors entering the second argument are mutually orthogonal, as in (REF ).","The reader can easily guess the result by comparing (REF ) and (REF ).","The explicit results for $\\widehat{\\Phi }^{{\\rm tot}}_n$ , $n\\le 4$ , presented above are the basis for the conjectural formula (REF ).","Note that all terms in these expressions have the sum of ranks of the generalized error functions equal to $n-1$ .","This shows that all contributions due to trees with non-zero number of marks cancel in the sum over Schröder trees.","toc"],["Index of notations","lp10cml Symbol Description Appears/defined in $\\mathcal {A}(\\mathcal {T}) $ contribution of tree $\\mathcal {T}$ to the integrand of the multi-instanton expansion of $H_\\gamma $ and $\\mathcal {G}$ (REF ) $a_\\mathcal {T}$ coefficient of unrooted labelled tree $\\mathcal {T}$ in $\\mathcal {D}_{m}(\\lbrace \\check{\\gamma }_s\\rbrace )$ (REF ), (REF ) $\\beta _{k\\ell }$ DSZ product $\\langle \\gamma _1+\\dots +\\gamma _k,\\gamma _\\ell \\rangle $ (REF ) $b_2=b_2(\\mathfrak {Y})$ second Betti number of $\\mathfrak {Y}$ p. REF $b^a=\\,{\\rm Re}\\,(z^a)$ periods of the Kalb-Ramond field p. REF $b_n$ rational coefficients in the expansion of $F^{\\rm (ref)}_n$ (REF ), (REF ) $c_i$ stability parameters (REF ) $c_i^{(\\ell )}$ stability parameters after attractor flow (REF ) $c_{2,a}$ components of the second Chern class of $\\mathfrak {Y}$ (REF ) $d=n b_2$ dimension of the lattice $\\mathbf {\\Lambda }=\\oplus _{i=1}^n \\Lambda _i$ p. REF $d_n,d_T$ rational weights in the representation of $g^{(0)}_n$ via flow tree (REF ),(REF ) $\\Delta (T)$ , $\\Delta _{\\gamma _L\\gamma _R}^z$ sign factors assigned to attractor flow tree $T$ (REF ) $\\mathcal {D}_{\\mathfrak {h}}$ Maass raising operator (REF ) $\\mathcal {D}({v})$ modular-covariant derivative contracted with vector ${v}$ (REF ) $\\mathcal {D}_{m}(\\lbrace \\check{\\gamma }_s\\rbrace ) $ derivative operator assigning weight to vertices with $m$ marks (REF ) $E_n(\\mathcal {M};\\mathbb {u})$ generalized error function on ${R}^n$ () $\\mathcal {E}_n=\\mathcal {E}^{(0)}_n+\\mathcal {E}^{(+)}_n$ function encoding the modular completion (REF ), (REF ) $\\phi $ (logarithm of) contact potential on $\\mathcal {M}_H$ (REF ) $\\Phi _n^E$ , $\\Phi _n^M$ boosted (complementary) error functions (REF ) $\\widetilde{\\Phi }_{n,m}^E$ , $\\widetilde{\\Phi }_{n,m}^M$ uplifted boosted error functions in the kernel of $\\mathcal {V}_ m$ (denoted by $\\widetilde{\\Phi }^E_n$ , $\\widetilde{\\Phi }^M_n$ when $n=m$ ) (REF ) $\\Phi _\\mathcal {T}$ contribution of tree $\\mathcal {T}$ to $\\Phi ^{\\scriptscriptstyle \\,\\int }_n$ (REF ), (REF ) $\\Phi ^{\\scriptscriptstyle \\,\\int }_n$ kernel defined by twistorial integrals (REF ), (REF ) $\\Phi ^{\\,\\mathcal {E}}_n$ kernel corresponding to function $\\mathcal {E}_n$ (REF ) $\\widetilde{\\Phi }^{\\,\\mathcal {E}}_n$ kernel promoting $g^{(0)}_n$ to a solution of Vignéras' equation (REF ) $\\Phi ^{\\,g}_{n}$ kernel corresponding to the tree index (REF ), (REF ) $\\Phi ^{\\,\\widehat{g}}_{n}$ kernel corresponding to completed tree index $\\widehat{g}_n$ (REF ) $\\Phi ^{{\\rm tot}}_n$ total kernel in the expansion of $\\mathcal {G}$ in terms of $h_{p,\\mu }$ (REF ) $\\widehat{\\Phi }^{{\\rm tot}}_n$ total kernel in the expansion of $\\mathcal {G}$ in terms of $\\widehat{h}_{p,\\mu }$ (REF ) $F(X)$ holomorphic prepotential p. REF $F_{{\\rm tr},n} (\\lbrace \\gamma _{i}\\rbrace ,z^a)$ partial tree index (REF ) $F^{\\rm (ref)}_{n} (\\lbrace c_i\\rbrace )$ partial contribution in $g^{\\rm (ref)}_n$ (REF ) $\\mathcal {F}$ image of $\\mathcal {G}$ under Euler operator (REF ) $\\Gamma $ charge lattice inside $H_{\\rm even}(\\mathfrak {Y},{Q})$ (REF ) $\\Gamma _+$ positive cone in the charge lattice (REF ) $\\Gamma _{k\\ell }, \\Gamma _e$ sums of DSZ products (REF ), (REF ) $\\gamma = (0,p^a,q_a,q_0)$ charge vector of a generic D4 (or D3) brane (REF ) $\\check{\\gamma }=(p^a,q_a)$ projection of $\\gamma $ on $H_4(\\mathfrak {Y},\\mathbb {Q})\\oplus H_2(\\mathfrak {Y},\\mathbb {Q})$ (REF ) $\\gamma _{ij}=\\langle \\gamma _i,\\gamma _j\\rangle $ Dirac-Schwinger-Zwanziger product, or Euler pairing (REF ) $g_{{\\rm tr},n}(\\lbrace \\gamma _i\\rbrace ,z^a)$ tree index, also denoted by $g_{{\\rm tr},n}(\\lbrace \\gamma _i,c_i\\rbrace )$ (REF ) $\\widehat{g}_n(\\lbrace \\check{\\gamma }_i\\rbrace ,z^a,\\tau _2)$ completed tree index, also denoted by $\\widehat{g}_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace )$ (REF ), (REF ) $g^{(0)}_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace )$ seed term in recursion for $\\widehat{g}_n$ (REF ) $g^{\\rm (ref)}_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace ,y)$ refined version of $g^{(0)}_n$ (REF ) $\\mathcal {G}$ instanton generating function (REF ) $\\mathcal {G}_n(\\lbrace \\gamma _i,z_i\\rbrace )$ integrand in the $n$ -instanton contribution to $\\mathcal {G}$ (REF ) $G_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace ;\\tau _2)$ large ${x}$ limit of $\\widetilde{\\Phi }_n({x})$ (REF ), (REF ) $h^{\\rm DT}_{p,q}(\\tau ,z^a)$ generating function of DT invariants (REF ) $h_{p,\\mu }(\\tau ) $ generating function of MSW invariants () $\\widehat{h}_{p,\\mu }(\\tau )$ modular completion of $h_{p,\\mu }(\\tau )$ (REF ) $H_\\gamma (z)$ generator of contact transformation across $\\ell _\\gamma $ (REF ) $H^{\\rm cl}_\\gamma (z)$ classical, large volume limit of $H_\\gamma (z)$ (REF ) $\\kappa (T)$ weight of attractor flow tree $T$ (REF ) $\\kappa (x)$ BPS index of two-centered solutions (REF ) $\\kappa _{abc}$ intersection numbers on a fixed basis of $H_4(\\mathfrak {Y})$ p. REF $K_{\\gamma _1\\gamma _2}(z_1,z_2)$ integration kernel in large volume limit (REF ) $\\hat{K}_{ij}(z_i,z_j)$ rescaled integration kernel in large volume limit (REF ) $\\mathcal {J}_n(\\lbrace \\check{\\gamma }_i\\rbrace ,\\tau _2)$ coefficient in the formula for the shadow of $\\widehat{h}_{p,\\mu }$ (REF ) $\\Lambda =H_4(\\mathfrak {Y},{Z})$ lattice equipped with quadratic form $\\kappa _{ab}=\\kappa _{abc} p^a$ p. REF $\\Lambda _i=H_4(\\mathfrak {Y},{Z})$ lattice equipped with quadratic form $\\kappa _{i,ab}=\\kappa _{abc} p_i^a$ p. REF $\\mathbf {\\Lambda }$ lattice $\\mathbf {\\Lambda }=\\oplus _{i=1}^n \\Lambda _i$ spanned by $n$ D1-brane charges p. REF $\\lambda $ eigenvalue under Vignéras' operator (REF ) $\\ell _\\gamma $ BPS ray on the twistor fiber, or its large volume limit (REF ), p. REF $\\mu _a$ residue class of $q_a$ modulo spectral flow (REF ) $M_n(\\mathcal {M};\\mathbb {u})$ generalized complementary error function on ${R}^n$ (REF ) $\\mathcal {M}_{\\alpha \\beta }$ matrix of parameters in the generalized error functions (REF ), () $\\mathcal {M}_{\\mathcal {K}}(\\mathfrak {Y})$ complexified Kähler moduli space of $\\mathfrak {Y}$ p. REF $\\mathcal {M}_H$ hypermultiplet moduli space in IIB/$\\mathfrak {Y}$ , or vector multiplet moduli space in IIA/$(\\mathfrak {Y}\\times S^1)=$ M/$(\\mathfrak {Y}\\times T^2)$ p. $n_T$ number of vertices of rooted tree $T$ excluding the leaves (REF ) $n_v(T)$ number of descendants of the vertex $v$ in $T$ plus one (REF ) $n_\\mathfrak {v}$ valency of vertex $\\mathfrak {v}$ of an unrooted tree (REF ) $p^a$ homology class of the divisor wrapped by the D3-brane (REF ) $\\mathcal {P}_{m}(\\lbrace p_s\\rbrace )$ weight of a vertex with $m$ marks in $G_n$ (REF ) $\\hat{q}_0$ invariant D0-brane charge (REF ) $Q_n(\\lbrace \\check{\\gamma }_i\\rbrace )$ difference of quadratic forms for constituents and the total charge (REF ) $R_n(\\lbrace \\check{\\gamma }_i\\rbrace ,\\tau _2)$ non-holomorphic correction in the completion $\\widehat{h}_{p,\\mu }(\\tau )$ (REF ), (REF ) $\\sigma _\\gamma $ quadratic refinement (REF ), (REF ) $S_k, S_e$ sums of stability parameters (REF ), (REF ) $S^{\\rm cl}_p$ classical action of a D3-instanton in large volume limit (REF ) $\\vartheta _{{p},{\\mu }}\\bigl (\\Phi ,\\lambda )$ indefinite theta series with kernel $\\Phi $ (REF ) $\\tau =\\tau _1+I\\tau _2$ 4D axio-dilaton in IIA, or torus modulus in M theory p. REF $t$ complex coordinate on the twistor fiber p. REF $t^a=\\,{\\rm Im}\\,(z^a)$ Kähler moduli on $\\mathfrak {Y}$ p. REF $T$ rooted tree with charges assigned to the leaves footnote REF $\\mathcal {T}$ tree with charges assigned to vertices footnote REF ${T}_n, {T}_n^\\ell $ set of unrooted (labelled) trees with $n$ vertices (REF ) ${T}_{n,m}, {T}_{n,m}^\\ell $ set of unrooted (labelled) trees with $n$ vertices and $m$ marks (REF ) ${T}_n^{\\rm r}$ set of rooted trees with $n$ vertices (REF ) ${T}_n^{\\rm af}$ set of attractor flow trees with $n$ leaves (REF ) ${T}_n^{\\rm S}$ set of Schröder trees with $n$ leaves (REF ) ${T}_{2m+1}^{(3)}$ set of rooted ternary trees with $n$ leaves (REF ) $u^\\Lambda =(1,u^a)$ complex structure moduli of mirror threefold ${\\widehat{\\mathfrak {Y}}}$ p. REF ${u}_{ij}$ , ${u}_e$ , ${u}_\\ell $ vectors in ${R}^d$ associated to $-2\\,{\\rm Im}\\,[Z_{\\gamma _i}\\bar{Z}_{\\gamma _j}]$ , $-S_e$ and $-S_\\ell $ (REF ), (REF ), (REF ) ${v}_{ij}$ , ${v}_e$ , ${v}_\\ell $ vectors in ${R}^d$ associated to $\\langle \\gamma _i,\\gamma _j\\rangle $ , $\\Gamma _e$ and $-\\Gamma _{n\\ell }$ (REF ), (REF ), (REF ) $V_T$ set of vertices of rooted tree $T$ excluding leaves p. REF $V_\\lambda $ Vignéras' operator (REF ) $\\mathcal {V}_m$ weight of a vertex with $m$ marks in large ${x}$ limit of $\\widetilde{\\Phi }^{\\,\\mathcal {E}}_n$ (REF ) $\\tilde{\\mathcal {V}}_m$ weight of a vertex with $m$ marks in $g^{(0)}_n$ (REF ) $W_n(\\lbrace \\check{\\gamma }_i\\rbrace ,\\tau _2)$ coefficient in the formula for $h_{p,\\mu }$ in terms of $\\widehat{h}_{p_i,\\mu _i}$ (REF ) $\\mathcal {X}_\\gamma $ holomorphic Fourier modes on the twistor space of $\\mathcal {M}_H$ (REF ) $\\mathcal {X}^{\\rm sf}_\\gamma $ semi-flat limit of $\\mathcal {X}_\\gamma $ (REF ) $\\mathcal {X}^{\\rm cl}_\\gamma $ classical limit of $\\mathcal {X}_\\gamma $ (REF ) $\\mathcal {X}^{(\\theta )}_{p,q}$ $\\hat{q}_0$ -independent part of $\\mathcal {X}^{\\rm cl}_\\gamma $ (REF ) ${x}$ $d$ -dimensional vector, argument of kernels of theta series p. REF $z$ coordinate on the twistor fiber, after Cayley transf.","(REF ) $z_\\gamma $ saddle point on twistor fiber (REF ) $z^a=b^a+It^a$ complexified Kähler moduli of $\\mathfrak {Y}$ p. REF $z^a_*(\\gamma )$ attractor moduli for charge $\\gamma $ p. REF $z^a_\\infty (\\gamma )$ large volume attractor point for charge $\\gamma $ (REF ) $Z_\\gamma (z^a)$ central charge (REF ) $\\Omega (\\gamma ,z^a)$ generalized Donaldson-Thomas invariant p. REF $\\bar{\\Omega }(\\gamma ,z^a)$ rational DT invariant (REF ) $\\bar{\\Omega }_*(\\gamma )$ attractor index p. REF ${\\bar{\\Omega }}^{\\rm MSW}(\\gamma )$ MSW invariant, also denoted by $\\bar{\\Omega }_{p,\\mu }( \\hat{q}_0)$ p. REF toc"]],"string":"[\n [\n \"Black holes and higher depth mock modular forms\"\n ],\n [\n \"Abstract By enforcing invariance under S-duality in type IIB string theory compactified on a Calabi-Yau threefold, we derive modular properties of the generating function of BPS degeneracies of D4-D2-D0 black holes in type IIA string theory compactified on the same space.\",\n \"Mathematically, these BPS degeneracies are the generalized Donaldson-Thomas invariants counting coherent sheaves with support on a divisor $\\\\cal D$, at the large volume attractor point.\",\n \"For $\\\\cal D$ irreducible, this function is closely related to the elliptic genus of the superconformal field theory obtained by wrapping M5-brane on $\\\\cal D$ and is therefore known to be modular.\",\n \"Instead, when $\\\\cal D$ is the sum of $n$ irreducible divisors ${\\\\cal D}_i$, we show that the generating function acquires a modular anomaly.\",\n \"We characterize this anomaly for arbitrary $n$ by providing an explicit expression for a non-holomorphic modular completion in terms of generalized error functions.\",\n \"As a result, the generating function turns out to be a (mixed) mock modular form of depth $n-1$.\"\n ],\n [\n \"Introduction and summary\",\n \"The degeneracies of BPS black holes in string vacua with extended supersymmetry possess remarkable modular properties, which have been instrumental in recent progress on explaining the statistical origin of the Bekenstein-Hawking entropy in [1] and many subsequent works.\",\n \"Namely, the indices $\\\\Omega (\\\\gamma )$ counting — with sign — microstates of BPS black holes with electromagnetic charge $\\\\gamma $ may often be collected into a suitable generating function which exhibits modular invariance, providing powerful constraints on its Fourier coefficients and enabling direct access to their asymptotic growth.\",\n \"When the black holes can be realized as black strings wrapped on a circle, a natural candidate for such a generating function is the elliptic genus of the superconformal field theory supported by the black string, which is modular invariant by construction [2], [3], [4].\",\n \"Equivalently, one may consider the partition function of the effective three-dimensional gravity living on the near-horizon geometry of the black string [5].\",\n \"In most cases however, the BPS indices depend not only on the charge $\\\\gamma $ but also on the moduli $z^a$ at spatial infinity, due to the wall-crossing phenomenon: some of the BPS bound states with total charge $\\\\gamma $ only exist in a certain chamber in moduli space, and decay as the moduli are varied across `walls of marginal stability' which delimit this chamber.\",\n \"At strong coupling where the black hole description is accurate, this phenomenon has a transparent interpretation in terms of the (dis)appearance of multi-centered black hole configurations, which can be used to derive a universal wall-crossing formula [6], [7], [8], [9].\",\n \"In the case of four-dimensional string vacua with $\\\\mathcal {N}=4$ supersymmetry, where the BPS index is sensitive only to single-centered 1/4-BPS black holes and to bound states of two 1/2-BPS black holes, the resulting moduli dependence is reflected in poles in the generating function, requiring a suitable choice of contour for extracting the Fourier coefficients in a given chamber [10], [11], [12].\",\n \"Upon subtracting contributions of two-centered bound states, the generating function of single-centered indices is no longer modular in the usual sense but it transforms as a mock Jacobi form with specific `shadow' — a property which is almost as constraining as standard modular invariance [13].\",\n \"In four-dimensional string vacua with $\\\\mathcal {N}=2$ supersymmetry, the situation is much more complicated, firstly due to the fact that the moduli space of scalars receives quantum corrections, and secondly due to BPS bound states potentially involving an arbitrary number of constituents, resulting in an extremely intricate pattern of walls of marginal stability.\",\n \"Thus, it does not seem plausible that a single generating function may capture the BPS indices $\\\\Omega (\\\\gamma ,z^a)$ — which are known in this context as generalized Donaldson-Thomas (DT) invariants — in all chambers.\",\n \"Nevertheless, modular invariance is still expected to constrain them.\",\n \"In particular, D4-D2-D0 black holes in type IIA string theory compactified on a generic compact Calabi-Yau (CY) threefold $\\\\mathfrak {Y}$ can be lifted to an M5-brane wrapped on a divisor $\\\\mathcal {D}\\\\subset \\\\mathfrak {Y}$ [2].\",\n \"If the divisor $\\\\mathcal {D}$ labelled by the D4-brane charge $p^a$ is irreducible, the indices $\\\\Omega (\\\\gamma ,z^a)$ are independent of the moduli of $\\\\mathfrak {Y}$ , at least in the limit where the volume of $\\\\mathfrak {Y}$ is scaled to infinity, and their generating function is known to be a holomorphic (vector valued) modular form of weight $-{1\\\\over 2}\\\\, b_2(\\\\mathfrak {Y})-1$ [3], [4], [6], [14].\",\n \"This includes the case of vertical, rank 1 D4-D2-D0 branes in K3-fibered Calabi-Yau threefolds [15], [16], [17].\",\n \"But if the divisor $\\\\mathcal {D}$ is a sum of $n$ effective divisors $\\\\mathcal {D}_i$ , the indices $\\\\Omega (\\\\gamma ,z^a)$ do depend on the Kähler moduli $z^a$ , even in the large volume limit.\",\n \"In general however, the black string SCFT is supposed to capture the states associated to a single $AdS_3$ throat, while for generic values of the moduli multiple $AdS_3$ throats can contribute [18], [19].\",\n \"It is thus natural to consider the modular properties of the DT invariants $\\\\Omega (\\\\gamma ,z^a)$ at the large volume attractor pointHere $p^a$ and $q_a$ are D4 and D2-brane charges, respectively, and the index of $q_a$ is raised with help of the inverse of the metric $\\\\kappa _{ab}=\\\\kappa _{abc}p^a$ with $\\\\kappa _{abc}$ being the triple intersection numbers on $H_4(\\\\mathfrak {Y},{Z})$ .\",\n \"$z^a_\\\\infty (\\\\gamma )= \\\\lim _{\\\\lambda \\\\rightarrow +\\\\infty }\\\\left(-q^a+I\\\\lambda p^a\\\\right),$ where only a single $AdS_3$ throat is allowed [20].\",\n \"Following [21] we denote these invariants by $\\\\Omega ^{\\\\rm MSW}(\\\\gamma )=\\\\Omega (\\\\gamma ,z^a_\\\\infty (\\\\gamma ))$ and call them Maldacena-Strominger-Witten (MSW) invariants.\",\n \"As we discuss in Section , the DT invariants $\\\\Omega (\\\\gamma ,z^a)$ can be recovered from the MSW invariants $\\\\Omega ^{\\\\rm MSW}(\\\\gamma )$ by using a version of the split attractor flow conjecture [22], [6] developed in [23], which we call the flow tree formula.\",\n \"The case where $\\\\mathcal {D}$ is the sum of two irreducible divisors was first considered in [20], [24], [25], and studied more recently in [14], [26].\",\n \"In that case, the generating function of MSW invariants turns out to be a mock modular form, with a specific non-holomorphic completion obtained by smoothing out the sign functions entering in the bound state contributions, recovering the prescription of [27], [20].\",\n \"The goal of this paper is to extend this result to the general case where $\\\\mathcal {D}$ is the sum of $n$ irreducible divisors $\\\\mathcal {D}_i$ , where $n$ can be arbitrarily large.\",\n \"In such generic situation, we characterize the modular properties of the generating function $h_{p,\\\\mu }$ of MSW invariants and find an explicit expression for its non-holomorphic completion $\\\\widehat{h}_{p,\\\\mu }$ in terms of the generating functions $h_{p_i,\\\\mu _i}$ associated to the $n$ constituents, multiplied by certain iterated integrals introduced in [28], [29], which generalize the usual error function appearing when $n=2$ .\",\n \"This result implies that in this case $h_{p,\\\\mu }$ is a (mixed, vector valued) mock modular form of depth $n-1$ , in the sense that the antiholomorphic derivative of its modular completion is itself a linear combination of modular completions of mock modular forms of lower depth, times antiholomorphic modular forms (with the depth 1 case reducing to the standard mock modular forms introduced in [27], [30], and the depth 0 case to usual weakly holomorphic modular forms; see [31] for a more precise definition).\",\n \"In order to establish this result, we follow the same strategy as in our earlier works [14], [26] and analyze D3-D1-D(-1) instanton corrections to the metric on the hypermultiplet moduli space $\\\\mathcal {M}_H$ in type IIB string theory compactified on $\\\\mathfrak {Y}$ , at arbitrary order in the instanton expansion.\",\n \"After reducing on a circle and T-dualizing, this moduli space is identical to the vector multiplet moduli space in type IIA string theory compactified on $\\\\mathfrak {Y}\\\\times S_1$ , where it receives instanton corrections from D4-D2-D0 black holes winding around the circle.\",\n \"In either case, each instanton contribution is weighted by the same generalized DT invariant $\\\\Omega (\\\\gamma )$ which counts the number of BPS black hole microstates with electromagnetic charge $\\\\gamma $ .\",\n \"The modular properties of the generalized DT invariants are fixed by requiring that the quaternion-Kähler (QK) metric on $\\\\mathcal {M}_H$ admits an isometric action of $SL(2,{Z})$ , which comes from S-duality in type IIB, or equivalently from large diffeomorphisms of the torus appearing when viewing type IIA/$\\\\mathfrak {Y}\\\\times S_1$ as M-theory on $\\\\mathfrak {Y}\\\\times T^2$ [32].\",\n \"This QK metric is most efficiently encoded in the complex contact structure on the associated twistor space, a ${C}P^1$ -bundle over $\\\\mathcal {M}_H$ [33], [34].\",\n \"The latter is specified by a set of gluing conditions determined by the DT invariants [32], [35], which can in turn be expressed in terms of the MSW invariants using the tree flow formula.\",\n \"An important quantity appearing in this twistorial formulationA familiarity with the twistorial formulation is not required for this work.\",\n \"Here we use only two equations relevant for the twistorial description of D-instantons: the integral equation (REF ) for certain Darboux coordinates, which appears also in the study of four-dimensional $N=2$ gauge theory on a circle [36], and the expression for the contact potential (REF ) in terms of these Darboux coordinates.\",\n \"These two equations lead to (REF ) and (REF ), respectively, which can be taken as the starting point of our analysis.\",\n \"Note that in the context of gauge theories the contact potential can be interpreted as a supersymmetric index [37].\",\n \"is the so called contact potential $e^\\\\phi $ , a real function on $\\\\mathcal {M}_H$ related to the Kähler potential on the twistor space, and afforded by the existence of a continuous isometry unbroken by D-instantons.\",\n \"On the type IIB side, $e^\\\\phi $ can be identified with the four-dimensional dilaton $1/g_4^2$ .\",\n \"When expressed in terms of the ten-dimensional axio-dilaton $\\\\tau =c_0+I/g_s$ (or in terms of the modulus of the 2-torus on the M-theory side), it becomes a complicated function having a classical contribution, a one-loop correction, and a series of instanton corrections expressed as contour integrals on the ${C}P^1$ fiber.\",\n \"The importance of the contact potential stems from the fact that it must be a non-holomorphic modular form of weight $(-\\\\frac{1}{2},-\\\\frac{1}{2})$ in the variable $\\\\tau $ in order for $\\\\mathcal {M}_H$ to admit an isometric action of $SL(2,{Z})$ [32].\",\n \"This requirement imposes very non-trivial constraints on the instanton contributions to $e^\\\\phi $ , which can be used to deduce the modular properties of generating functions of DT invariants, at each order in the instanton expansion.\",\n \"This strategy was used in [14] at two-instanton order to characterize the modular behavior of the generating function of MSW invariants in the case of a divisor equal to the sum of $n=2$ irreducible divisors.\",\n \"In this paper we generalize this result to all $n$ , by analyzing the instanton expansion to all orders.\",\n \"Below we summarize the main steps in our analysis and our main results.\",\n \"First, we show that the contact potential $e^\\\\phi $ in the large volume limit, where the Kähler parameters of the CY are sent to infinity, can be expressed (see (REF )) through another (complex valued) function $\\\\mathcal {G}$ (REF ) on the moduli space, which we call the instanton generating function.\",\n \"The expansion of $\\\\mathcal {G}$ in powers of DT invariants is governed by a sum over unrooted trees decorated by charges $\\\\gamma _i$ (see (REF ) and (REF )).\",\n \"The modularity of the contact potential requires $\\\\mathcal {G}$ to transform as a modular form of weight $(-\\\\tfrac{3}{2},\\\\tfrac{1}{2})$ .\",\n \"After expressing the DT invariants $\\\\Omega (\\\\gamma ,z^a)$ through the moduli independent MSW invariants $\\\\Omega ^{\\\\rm MSW}(\\\\gamma )$ using the tree flow formula of [23], and expanding $\\\\mathcal {G}$ in powers of $\\\\Omega ^{\\\\rm MSW}(\\\\gamma )$ , each order in this expansion can be decomposed into a sum of products of certain indefinite theta series and of holomorphic generating functions $h_{p_i,\\\\mu _i}$ of the invariants $\\\\Omega ^{\\\\rm MSW}(\\\\gamma )$ (see (REF )), similarly to the usual decomposition of standard Jacobi forms.\",\n \"Thus, the modular properties of the indefinite theta series $\\\\vartheta _{{p},{\\\\mu }}\\\\bigl (\\\\Phi ^{{\\\\rm tot}}_{n},n-2\\\\bigr )$ are tied with the modular properties of the generating functions $h_{p_i,\\\\mu _i}$ , in order for $\\\\mathcal {G}$ to be modular.\",\n \"In order for an indefinite theta series $\\\\vartheta _{{p},{\\\\mu }}\\\\bigl (\\\\Phi ,\\\\lambda )$ to be modular, its kernel $\\\\Phi $ must satisfy a certain differential equation (REF ), which we call Vignéras' equation [38].\",\n \"By construction, the kernels $\\\\Phi ^{\\\\rm tot}_n$ appearing in our problem are given by iterated contour integrals along the ${C}P^1$ fiber of the twistor space, multiplied by so-called `tree indices' $g_{{\\\\rm tr},n}$ coming from the expression of $\\\\Omega (\\\\gamma ,z^a)$ in terms of $\\\\Omega ^{\\\\rm MSW}(\\\\gamma )$ .\",\n \"We evaluate the twistorial integrals in terms of the generalized error functions introduced in [28], [29], and show that the resulting kernels satisfy Vignéras' equation away from certain loci where they have discontinuities.\",\n \"Furthermore, we prove that the discontinuities corresponding to walls of marginal stability cancel between the integrals and the tree indices.\",\n \"But there are additional discontinuities coming from certain moduli independent contributions to the tree index.\",\n \"They spoil Vignéras' equation at the multi-instanton level so that the theta series $\\\\vartheta _{{p},{\\\\mu }}\\\\bigl (\\\\Phi ^{{\\\\rm tot}}_{n},n-2\\\\bigr )$ are not modular.\",\n \"In turn, this implies that the holomorphic generating functions $h_{p_i,\\\\mu _i}$ are not modular either.\",\n \"However, we show that one can complete $h_{p,\\\\mu }$ into a non-holomorphic modular form $\\\\widehat{h}_{p,\\\\mu }(\\\\tau )$ , by adding to it a series of corrections proportional to products of $h_{p_i,\\\\mu _i}$ with the same total D4-brane charge $p=\\\\sum _i p_i$ (see (REF )).\",\n \"The non-holomorphic functions $R_n$ entering the completion are determined by the condition that the expansion of $\\\\mathcal {G}$ rewritten in terms of $\\\\widehat{h}_{p,\\\\mu }$ gives rise to a non-anomalous, modular theta series.\",\n \"Equivalently, one can work with functions $\\\\widehat{g}_n$ appearing in the expansion (REF ) of the holomorphic generating function of DT invariants $h^{\\\\rm DT}_{p,q}$ in powers of the non-holomorphic functions $\\\\widehat{h}_{p_i,\\\\mu _i}$ .\",\n \"Imposing the conditions for modularity, we find that $\\\\widehat{g}_n$ can be represented in an iterative form (REF ), or more explicitly as a sum (REF ) over rooted trees with valency $\\\\ge 3$ at each vertex (known as Schröder trees), where $g^{(0)}_n$ are certain locally polynomial functions defined in (REF ), while $\\\\mathcal {E}_n$ are smooth solutions of Vignéras' equation constructed in terms of generalized error functions.\",\n \"The functions $R_n$ are similarly given by a sum over Schröder trees (REF ) in terms of exponentially decreasing and non-decreasing parts of $\\\\mathcal {E}_n$ .\",\n \"These equations represent the main technical result of this work.\",\n \"For $n\\\\le 5$ our general formulas can be drastically simplified.\",\n \"In particular, the main building blocks, the functions $g^{(0)}_n$ and $\\\\mathcal {E}_n$ , can be written as a sum over a suitable subset of flow trees, as in (REF ).\",\n \"In addition, we show that $g^{(0)}_n$ has a natural extension including the refinement parameter conjugate to the spin $J_3$ of the black hole.\",\n \"Figure: Various types of trees arising in this work.T n,m {T}_{n,m} denotes the set of unrooted trees with nn vertices and mm marks distributed between the vertices.T n ≡T n,0 {T}_n\\\\!\\\\equiv {T}_{n,0} comprises the trees without marks.T n r {T}_n^{\\\\rm r} is the set of rooted trees with nn vertices.T n af {T}_n^{\\\\rm af} is the set of attractor flow trees with nn leaves (of which we only draw the different topologies).T n S {T}_n^{\\\\rm S} denotes the set of Schröder trees with nn leaves.In addition, an important rôle is played by the sets T n,m ℓ {T}_{n,m}^\\\\ell and T n ℓ =T n,0 ℓ {T}_n^\\\\ell ={T}_{n,0}^\\\\ell of unrooted, labelled, marked trees which are obtained from T n,m {T}_{n,m} and T n {T}_{n} by assigning different labels and markings to the verticesin such way that a vertex with m 𝔳 m_\\\\mathfrak {v} marks is decorated by 2m 𝔳 +12m_\\\\mathfrak {v}+1 labels.At the end of this lengthy analysis, we thus find a modular completion of the generating functions $h_{p,\\\\mu }$ of MSW invariants for an arbitrary divisor, i.e.\",\n \"decomposable into a sum of any number of irreducible divisors.\",\n \"The result is expressed through sums of products of generalized error functions labelled by trees of various types.\",\n \"For the reader's convenience, in Fig.\",\n \"REF we display the various trees which appear in our construction, up to $n=4$ .\",\n \"Since generalized error functions are known to be related to iterated Eichler integrals [14], [39], which occur in the non-holomorphic completion of mock modular forms, we loosely refer to the generating functions of MSW invariants as higher depth mock modular forms, although we do not spell out the precise meaning of this notion.\",\n \"An unexpected byproduct of our analysis is an interesting combinatorial identity, relating rooted trees to the binomial coefficients, which plays a rôle in our derivation of the modular completion.\",\n \"Since we are not aware of such an identity in the mathematical literature,We were informed by Karen Yeats that the special case $m=n-1$ appears in [40].\",\n \"The denominator in (REF ) is sometimes known as the tree factorial $T^{\\\\prime }!$ , see (REF ).\",\n \"we state it here as a theorem whose proof can be found in appendix .\",\n \"Theorem 1 Let $V_T$ be the set of vertices of a rooted ordered tree $T$ and $n_v(T)$ is the number of descendants of vertex $v$ inside $T$ plus 1 (alternatively, the number of vertices of the subtree in $T$ with the root being the vertex $v$ and the leaves being the leaves of $T$ ).\",\n \"Then for a rooted tree $T$ with $n$ vertices and $m0$ at each vertex.\",\n \"This condition will hold automatically for the case of our interest, namely, D4-D2-D0 black holes in the large volume limit, so we do not impose it explicitly.\",\n \"$\\\\gamma _{L(v)R(v)}\\\\, \\\\,{\\\\rm Im}\\\\,\\\\bigl [ Z_{\\\\gamma _{L(v)}}\\\\bar{Z}_{\\\\gamma _{R(v)}}(z^a_{p(v)})\\\\bigr ]>0$ is satisfied for all $v\\\\in V_T$ , which ensures admissibility of the flow tree $T$ , i.e.\",\n \"the existence of the corresponding nested bound state.\",\n \"Importantly, the sign of $\\\\,{\\\\rm Im}\\\\,\\\\bigl [ Z_{\\\\gamma _{L(v)}}\\\\bar{Z}_{\\\\gamma _{R(v)}}(z^a_{p(v)})\\\\bigr ]$ entering (REF ) can be computed recursively in terms of asymptotic data, without evaluating the attractor flow along the edges [23].\",\n \"More precisely, the signs depend only on the stability parameters (also known as Fayet-Iliopoulos parameters) $c_i(z^a) = \\\\,{\\\\rm Im}\\\\,\\\\bigl [ Z_{\\\\gamma _i}\\\\bar{Z}_\\\\gamma (z^a)\\\\bigr ]\\\\, .$ Note that due to $\\\\gamma =\\\\sum _{i=1}^n \\\\gamma _i$ , these parameters satisfy $\\\\sum _{i=1}^n c_i=0$ .\",\n \"Accordingly, we shall often denote $g_{{\\\\rm tr},n}(\\\\lbrace \\\\gamma _i\\\\rbrace ,z^a)$ by $g_{{\\\\rm tr},n}(\\\\lbrace \\\\gamma _i,c_i\\\\rbrace )$ , and always assume that $\\\\lbrace c_i\\\\rbrace $ are generic real parameters subject to the condition $\\\\sum _{i=1}^n c_i=0$ , such that no proper subset of them sums up to zero.\",\n \"The second factor in (REF ) is independent of the moduli, and given by $\\\\kappa (T) \\\\equiv (-1)^{n-1} \\\\prod _{v\\\\in V_T} \\\\kappa ( \\\\gamma _{L(v)R(v)}),\\\\qquad \\\\kappa (x)=(-1)^x\\\\, x.$ This is simply the product of the BPS indices of the nested two-centered solutions associated with the tree $T$ .\",\n \"Note that the signs of $\\\\Delta (T)$ and $\\\\kappa (T)$ separately depend on the choice of ordering $(\\\\gamma _{L(v)},\\\\gamma _{R(v)})$ at each vertex, but their product is independent of that choice.\",\n \"Sometimes, it is useful to consider the refined BPS index $\\\\Omega (\\\\gamma ,z^a,y)$ which carries additional dependence on the fugacity $y$ conjugate to the spin $J_3$ .\",\n \"All the above equations remain valid in this case as well (except for the definition of the rational invariant (REF ), which must be slightly modified, see e.g.\",\n \"[8]), but now the function $\\\\kappa (x)$ appearing in (REF ) becomes a symmetric Laurent polynomial in $y$ $\\\\kappa (x)=(-1)^x \\\\, \\\\frac{ y^x-y^{-x}}{y-y^{-1}}\\\\, ,$ reducing to $(-1)^x x$ in the unrefined limit $y\\\\rightarrow 1$ .\",\n \"It is easy to see that the `flow tree formula' (REF ) is consistent with the primitive wall-crossing formula [6], [7] $\\\\Delta \\\\bar{\\\\Omega }(\\\\gamma _L+\\\\gamma _R) = -{\\\\rm sgn}(\\\\gamma _{LR})\\\\, \\\\kappa (\\\\gamma _{LR})\\\\, \\\\bar{\\\\Omega }(\\\\gamma _L,z^a)\\\\, \\\\bar{\\\\Omega }(\\\\gamma _R,z^a) ,$ which gives the jump of the BPS index due to the decay of bound states after crossing the wall defined by a pair of primitiveHere, by primitive we mean that all charges with non-zero index in the two-dimensional lattice spanned by $\\\\gamma _L$ and $\\\\gamma _R$ are linear combinations $N_L \\\\gamma _L + N_R\\\\gamma _R$ with coefficients $N_L, N_R$ of the same sign.\",\n \"charges $\\\\gamma _L$ and $\\\\gamma _R$ .\",\n \"To this end, it suffices to consider all flow trees which start with the splitting $\\\\gamma \\\\rightarrow \\\\gamma _L+\\\\gamma _R$ at the root of the tree.\",\n \"It is also consistent with the general wall-crossing formula of [41], provided the tree index is computed for a small generic perturbation of the DSZ matrix $\\\\gamma _{ij}$ [23].\",\n \"Finally, it is useful to note that, assuming that all charges $\\\\gamma _i$ are distinct, the sum over splittings and flow trees in (REF ) can be generated by iterating the quadratic equation [23] $\\\\bar{\\\\Omega }(\\\\gamma ,z^a) &= &\\\\bar{\\\\Omega }_*(\\\\gamma )-{1\\\\over 2}\\\\sum _{\\\\begin{array}{c}\\\\gamma =\\\\gamma _L+\\\\gamma _R\\\\\\\\\\\\langle \\\\gamma _L,\\\\gamma _R \\\\rangle \\\\ne 0\\\\end{array}}\\\\Delta _{\\\\gamma _L\\\\gamma _R}^z \\\\, \\\\kappa (\\\\gamma _{LR})\\\\,\\\\bar{\\\\Omega }(\\\\gamma _L,z^a_{LR})\\\\, \\\\bar{\\\\Omega }(\\\\gamma _R,z^a_{LR}),$ where $z^a_{LR}$ is the point where the attractor flow of charge $\\\\gamma $ crosses the wall of marginal stability $\\\\,{\\\\rm Im}\\\\,\\\\bigl [Z_{\\\\gamma _L}\\\\bar{Z}_{\\\\gamma _R}(z^a_{LR})\\\\bigr ]=0$ .\"\n ],\n [\n \"Partial tree index\",\n \"While the representation of the BPS index based on attractor flows is useful for many purposes, it produces a sum of products of sign functions depending on non-linear combinations of DSZ products $\\\\gamma _{ij}$ , which are very difficult to work with.\",\n \"A solution to overcome this problem was found in [23].\",\n \"The key idea is to introduce a refined index with a fugacity $y$ conjugate to angular momentum, and represent it as $g_{{\\\\rm tr},n}(\\\\lbrace \\\\gamma _i,c_i\\\\rbrace ,y)= \\\\frac{(-1)^{n-1+\\\\sum _{i 0,\\\\qquad (r p^2)> 0,\\\\qquad k_a p^a > 0,$ for all effective divisors $r^a \\\\gamma _a \\\\in H_4^+(\\\\mathfrak {Y},{Z})$ and effective curves $k_a \\\\gamma ^a \\\\in H_2^+(\\\\mathfrak {Y},{Z})$ , where $\\\\gamma _a$ denotes irreducible divisors giving an integer basis of $ \\\\Lambda =H_4(\\\\mathfrak {Y},{Z})$ , dual to the basis $\\\\gamma ^a$ of $\\\\Lambda ^*=H_2(\\\\mathfrak {Y},{Z})$ .\",\n \"The charge $p^a$ induces a quadratic form $\\\\kappa _{ab}=\\\\kappa _{abc} p^c$ on $\\\\Lambda \\\\otimes {R}\\\\simeq {R}^{b_2}$ of signature $(1,b_2-1)$ .\",\n \"This quadratic form allows to embed $\\\\Lambda $ into $\\\\Lambda ^*$ , but the map $\\\\epsilon ^a \\\\mapsto \\\\kappa _{ab} \\\\epsilon ^b$ is in general not surjective, the quotient $\\\\Lambda ^*/\\\\Lambda $ being a finite group of order $|\\\\,{\\\\rm det}\\\\, \\\\kappa _{ab}|$ .\",\n \"The holomorphic central charge, governing the mass of BPS states, is given by $Z_\\\\gamma (z^a)=q_\\\\Lambda X^\\\\Lambda (z^a) -p^\\\\Lambda F_\\\\Lambda (z^a),$ where $X^\\\\Lambda (z^a)=(1,z^a)$ are the special coordinates and $F_\\\\Lambda =\\\\partial _{X^\\\\Lambda } F(X)$ is the derivative of the holomorphic prepotential $F(X)$ on $\\\\mathcal {M}_\\\\mathcal {K}$ .\",\n \"In the large volume limit $t^a\\\\rightarrow \\\\infty $ , the prepotential reduces to the classical cubic contribution $F(X)\\\\approx F^{\\\\rm cl}(X)=-\\\\kappa _{abc}\\\\, \\\\frac{ X^a X^b X^c}{6X^0}\\\\, ,$ and the central charge can be approximated as $Z_\\\\gamma \\\\approx -{1\\\\over 2}\\\\, (pt^2)+I\\\\left( q_a t^a+(pbt)\\\\right)+q_0+q_a b^a+{1\\\\over 2}\\\\,(pb^2).$ Note, in particular, that it always has a large negative real part.\",\n \"Another useful observation is that both quantities appearing in the definition of $\\\\Delta _{\\\\gamma _L\\\\gamma _R}^z$ (REF ) are independent of the last component $q_0$ of the charge vector.\",\n \"Indeed, $\\\\begin{split}\\\\langle \\\\gamma ,\\\\gamma ^{\\\\prime }\\\\rangle =&\\\\, q_a p^{\\\\prime a}-q^{\\\\prime }_ap^a,\\\\\\\\\\\\,{\\\\rm Im}\\\\,\\\\bigl [ Z_{\\\\gamma }\\\\bar{Z}_{\\\\gamma ^{\\\\prime }}\\\\bigr ]=&\\\\, -{1\\\\over 2}\\\\,\\\\Bigl ((p^{\\\\prime } t^2)(q_{a} +(pb)_a)t^a - (p t^2) (q^{\\\\prime }_{a}+(p^{\\\\prime }b)_a) t^a\\\\Bigr ).\\\\end{split}$ The BPS index $\\\\bar{\\\\Omega }(\\\\gamma ,z^a)$ counting D4-D2-D0 black holes is given mathematically by the generalized Donaldson-Thomas invariant, which countsMore precisely, the generalized DT invariant computes the weighted Euler characteristic of the moduli space of semi-stable coherent sheaves [42]; in this context, the DSZ product $\\\\langle \\\\gamma ,\\\\gamma ^{\\\\prime }\\\\rangle $ coincides with the antisymmetrized Euler form.\",\n \"semi-stable coherent sheaves supported on a divisor $\\\\mathcal {D}$ in the homology class $p^a\\\\gamma _a$ , with first and second Chern numbers determined by $(q_a,q_0)$ .\",\n \"An important property of these invariants is that they are unchanged under a combined integer shift of the Kalb-Ramond field, $b^a\\\\mapsto b^a +\\\\epsilon ^a$ , and a spectral flow transformation acting on the D2 and D0 charges $q_a \\\\mapsto q_a - \\\\kappa _{abc}p^b\\\\epsilon ^c,\\\\qquad q_0 \\\\mapsto q_0 - \\\\epsilon ^a q_a + \\\\frac{1}{2}\\\\, (p\\\\epsilon \\\\epsilon ).$ The shift of $b^a$ is important since the DT invariants are only piecewise constant as functions of the complexified Kähler moduli $z^a=b^a+It^a$ due to wall-crossing.\",\n \"In contrast, the MSW invariants ${\\\\bar{\\\\Omega }}^{\\\\rm MSW}(\\\\gamma )$ , defined as the generalized DT invariants $\\\\bar{\\\\Omega }(\\\\gamma ,z^a)$ evaluated at their respective large volume attractor point (REF ), are by construction independent of the moduli, and therefore invariant under the spectral flow (REF ).\",\n \"As a result, they only depend on $p^a, \\\\mu _a$ and $\\\\hat{q}_{0}$ , where we traded the electric charges $(q_a,q_0)$ for $(\\\\epsilon ^a, \\\\mu _a,\\\\hat{q}_0)$ .\",\n \"The latter comprise the spectral flow parameter $\\\\epsilon ^a$ , the residue class $\\\\mu _a\\\\in \\\\Lambda ^*/\\\\Lambda $ defined by the decomposition $q_a = \\\\mu _a + \\\\frac{1}{2}\\\\, \\\\kappa _{abc} p^b p^c + \\\\kappa _{abc} p^b \\\\epsilon ^c,\\\\qquad \\\\epsilon ^a\\\\in \\\\Lambda \\\\, ,$ and the invariant charge ($\\\\kappa ^{ab}$ is the inverse of $\\\\kappa _{ab}$ ) $\\\\hat{q}_0 \\\\equiv q_0 -\\\\frac{1}{2}\\\\, \\\\kappa ^{ab} q_a q_b\\\\, ,$ which is invariant under (REF ).\",\n \"This allows to write ${\\\\bar{\\\\Omega }}^{\\\\rm MSW}(\\\\gamma )=\\\\bar{\\\\Omega }_{p,\\\\mu }( \\\\hat{q}_0)$ .\",\n \"An important fact is that the invariant charge $\\\\hat{q}_0$ is bounded from above by $\\\\hat{q}_0^{\\\\rm max}=\\\\tfrac{1}{24}((p^3)+c_{2,a}p^a)$ .\",\n \"This allows to define two generating functions $h^{\\\\rm DT}_{p,q}(\\\\tau ,z^a) &=& \\\\sum _{\\\\hat{q}_0 \\\\le \\\\hat{q}_0^{\\\\rm max}}\\\\bar{\\\\Omega }(\\\\gamma ,z^a)\\\\,{\\\\bf e}\\\\!\\\\left( -\\\\hat{q}_0 \\\\tau \\\\right),\\\\\\\\h_{p,\\\\mu }(\\\\tau ) &=& \\\\sum _{\\\\hat{q}_0 \\\\le \\\\hat{q}_0^{\\\\rm max}}\\\\bar{\\\\Omega }_{p,\\\\mu }(\\\\hat{q}_0)\\\\,{\\\\bf e}\\\\!\\\\left( -\\\\hat{q}_0 \\\\tau \\\\right),$ where we used notation ${\\\\bf e}\\\\!\\\\left( x\\\\right)=e^{2\\\\pi Ix}$ .\",\n \"Whereas the generating function of DT invariants $h^{\\\\rm DT}_{p,q}$ depends on the full electric charge $q_a$ and depends on the moduli $z^a$ in a piecewise constant fashion, the generating function of MSW invariants $h_{p,\\\\mu }(\\\\tau )$ , due to the spectral flow symmetry, depends only on the residue class $\\\\mu _a$ .\",\n \"This generating function will be the central object of interest in this paper, and our main goal will be to understand its behavior under modular transformations of $\\\\tau $ .\",\n \"In general, the MSW invariants ${\\\\bar{\\\\Omega }}^{\\\\rm MSW}(\\\\gamma )$ are distinct from the attractor moduli $\\\\bar{\\\\Omega }_\\\\star (\\\\gamma )$ , since the latter coincide with the generalized DT invariants $\\\\bar{\\\\Omega }(\\\\gamma ,z^a)$ evaluated at the true attractor point $z^a_*(\\\\gamma )$ for the charge $\\\\gamma $ , while the former are the generalized DT invariants evaluated at the large volume attractor point $z^a_\\\\infty (\\\\gamma )$ defined in (REF ).\",\n \"Nevertheless, we claim that in the large volume limit $t^a\\\\rightarrow \\\\infty $ , the tree flow formula reviewed in the previous subsections still allows to express $\\\\bar{\\\\Omega }(\\\\gamma ,z^a)$ in terms of the MSW invariants, namely $\\\\bar{\\\\Omega }(\\\\gamma ,z^a) =\\\\sum _{\\\\sum _{i=1}^n \\\\gamma _i=\\\\gamma }g_{{\\\\rm tr},n}(\\\\lbrace \\\\gamma _i\\\\rbrace , z^a)\\\\,\\\\prod _{i=1}^n {\\\\bar{\\\\Omega }}^{\\\\rm MSW}(\\\\gamma _i) \\\\qquad (\\\\,{\\\\rm Im}\\\\,z^a\\\\rightarrow \\\\infty )$ The point is that the only walls of marginal stability which extend to infinite volume are those where the constituents carry no D6-brane charge, and that non-trivial bound states involving constituents with D4-brane charge are ruled out at the large volume attractor point, similarly to the usual attractor chamber.\",\n \"Since the r.h.s.\",\n \"of (REF ) is consistent with wall-crossing in the infinite volume limit and agrees with the left-hand side at $z^a=z^a_\\\\infty (\\\\gamma )$ , it must therefore hold everywhere at large volume.\",\n \"Of course, some of the states contributing to ${\\\\bar{\\\\Omega }}^{\\\\rm MSW}(\\\\gamma _i)$ may have some substructure, e.g.\",\n \"be realized as D6-$\\\\overline{\\\\rm D6}$ bound states, but this structure cannot be probed in the large volume limit.\",\n \"Importantly, since the quantities (REF ) entering in the definition of the tree index are independent of the D0-brane charge $q_0$ , the flow tree formula (REF ) may be rewritten as a relation between the generating functions, $h^{\\\\rm DT}_{p,q}(\\\\tau ,z^a) =\\\\sum _{\\\\sum _{i=1}^n \\\\check{\\\\gamma }_i=\\\\check{\\\\gamma }}g_{{\\\\rm tr},n}(\\\\lbrace \\\\check{\\\\gamma }_i\\\\rbrace ,z^a)\\\\,e^{\\\\pi I\\\\tau Q_n(\\\\lbrace \\\\check{\\\\gamma }_i\\\\rbrace )}\\\\prod _{i=1}^n h_{p_i,\\\\mu _i}(\\\\tau ),$ where $\\\\check{\\\\gamma }=(p^a,q_a)$ denotes the projection of the charge vector $\\\\gamma $ on $H_4\\\\oplus H_2$ , and the phase proportional to $Q_n(\\\\lbrace \\\\check{\\\\gamma }_i\\\\rbrace )= \\\\kappa ^{ab}q_a q_b-\\\\sum _{i=1}^n\\\\kappa _i^{ab}q_{i,a} q_{i,b}$ appears due to the quadratic term in the definition (REF ) of the invariant charge $\\\\hat{q}_0$ .\",\n \"In this section, we switch to the dual setupOur preference for the type IIB set-up is merely for consistency with our earlier works on hypermultiplet moduli spaces in $d=4$ Calabi-Yau vacua.\",\n \"The same considerations apply verbatim, with minor changes of wording, to the vector multiplet moduli space in type IIA string theory compactified on $\\\\mathfrak {Y}\\\\times S^1$ , which is more directly related to the counting of D4-D2-D0 black holes in four dimensions.\",\n \"of type IIB string theory compactified on the same CY manifold $\\\\mathfrak {Y}$ .\",\n \"The DT invariants, describing the BPS degeneracies of D4-D2-D0 black holes in type IIA, now appear as coefficients in front of the D3-D1-D(-1) instanton effects affecting the metric on the hypermultiplet moduli space $\\\\mathcal {M}_H$ .\",\n \"The main idea of our approach is that these instanton effects are strongly constrained by demanding that $\\\\mathcal {M}_H$ admits an isometric action of the type IIB S-duality $SL(2,{Z})$ .\",\n \"This constraint uniquely fixes the modular behavior of the generating functions introduced in the previous section.\",\n \"Here we recall the twistorial construction of D-instanton corrections to the hypermultiplet metric, describe the action of S-duality, and analyze the instanton expansion of a particular function on $\\\\mathcal {M}_H$ known as contact potential.\"\n ],\n [\n \"$\\\\mathcal {M}_H$ and twistorial description of instantons \",\n \"The moduli space of four-dimensional $N=2$ supergravity is a direct product of vector and hypermultiplet moduli spaces, $\\\\mathcal {M}_V\\\\times \\\\mathcal {M}_H$ .\",\n \"The former is a (projective) special Kähler manifold, whereas the latter is a quaternion-Kähler (QK) manifold.\",\n \"In type IIB string theory compactified on a CY threefold $\\\\mathfrak {Y}$ , $\\\\mathcal {M}_H$ is a space of real dimension $4b_2(\\\\mathfrak {Y})+4$ , which is fibered over the complexified Kähler moduli space $\\\\mathcal {M}_\\\\mathcal {K}(\\\\mathfrak {Y})$ of dimension $2b_2(\\\\mathfrak {Y})$ .\",\n \"In addition to the Kähler moduli $z^a=b^a+It^a$ , it describes the dynamics of the ten-dimensional axio-dilaton $\\\\tau =c^0+I/g_s$ , the Ramond-Ramond (RR) scalars $c^a,\\\\tilde{c}_a,\\\\tilde{c}_0$ , corresponding to periods of the RR 2-form, 4-form and 6-form on a basis of $H^{\\\\rm even}(\\\\mathfrak {Y},{Z})$ , and finally, the NS-axion $\\\\psi $ , dual to the Kalb-Ramond two-form $B$ in four dimensions.\",\n \"At tree-level, the QK metric on $\\\\mathcal {M}_H$ is obtained from the Kähler moduli space $\\\\mathcal {M}_{\\\\mathcal {K}}$ via the $c$ -map construction [52], [53] and thus is completely determined by the holomorphic prepotential $F(X)$ .\",\n \"But this metric receives $g_s$ -corrections, both perturbative and non-perturbative.\",\n \"The latter can be of two types: either from Euclidean D-branes wrapping even dimensional cycles on $\\\\mathfrak {Y}$ , or from NS5-branes wrapped around the whole $\\\\mathfrak {Y}$ .\",\n \"In this paper we shall be interested only in the effects of D3-D1-D(-1) instantons, and ignore the effects of NS5 and D5-instantons, which are subleading in the large volume limit.\",\n \"Since NS5-instantons only mix with D5-instantons under S-duality, this truncation does not spoil modular invariance [21].\",\n \"The most concise way to describe the D-instanton corrections is to consider type IIA string theory compactified on the mirror CY threefold ${\\\\widehat{\\\\mathfrak {Y}}}$ and use the twistor formalism for quaternionic geometries [33], [34].\",\n \"In this approach the metric is encoded in the complex contact structure on the twistor space, a ${C}P^1$ -bundle over $\\\\mathcal {M}_H$ .\",\n \"The D-instanton corrected contact structure has been constructed to all orders in the instanton expansion in [32], [35], and an explicit expression for the metric has been derived recently in [54], [55].\",\n \"Here we will present only those elements of the construction which are relevant for the subsequent analysis, and refer to reviews [56], [57] for more details.\",\n \"The crucial point is that, locally, the contact structure is determined by a set of holomorphic Darboux coordinates $(\\\\xi ^\\\\Lambda ,\\\\tilde{\\\\xi }_\\\\Lambda ,\\\\alpha )$ on the twistor space, considered as functions of coordinates on $\\\\mathcal {M}_H$ and of the stereographic coordinate $t$ on the ${C}P^1$ fiber, so that the contact one-form takes the canonical form $\\\\mathrm {d}\\\\alpha +\\\\tilde{\\\\xi }_\\\\Lambda \\\\mathrm {d}\\\\xi ^\\\\Lambda $ .\",\n \"Although all Darboux coordinates are important for recovering the metric, for the purposes of this paper the coordinate $\\\\alpha $ is irrelevant.\",\n \"Therefore, we consider only $\\\\xi ^\\\\Lambda $ and $\\\\tilde{\\\\xi }_\\\\Lambda $ which can be conveniently packaged into holomorphic Fourier modes $\\\\mathcal {X}_\\\\gamma = {\\\\bf e}\\\\!\\\\left( p^\\\\Lambda \\\\tilde{\\\\xi }_\\\\Lambda -q_\\\\Lambda \\\\xi ^\\\\Lambda \\\\right)$ labelled by a charge vector $\\\\gamma =(p^\\\\Lambda ,q_\\\\Lambda )$ .\",\n \"At tree level, the Darboux coordinates (multiplied by $t$ ) are known to be simple quadratic polynomials in $t$ so that $\\\\mathcal {X}_\\\\gamma $ take the formThe superscript `sf' stands for `semi-flat', which refers to the flatness of the classical geometry in the directions along the torus fibers parametrized by $\\\\zeta ^\\\\Lambda ,\\\\tilde{\\\\zeta }_\\\\Lambda $ .\",\n \"$\\\\mathcal {X}^{\\\\rm sf}_\\\\gamma (t)={\\\\bf e}\\\\!\\\\left( \\\\frac{\\\\tau _2}{2}\\\\left(\\\\bar{Z}_\\\\gamma (\\\\bar{u}^a)\\\\,t-\\\\frac{Z_\\\\gamma (u^a)}{t}\\\\right)+p^\\\\Lambda \\\\tilde{\\\\zeta }_\\\\Lambda - q_\\\\Lambda \\\\zeta ^\\\\Lambda \\\\right),$ where $Z_\\\\gamma (u^a)$ is the central charge (REF ), now expressed in terms of the complex structure moduli $u^a$ of the CY threefold ${\\\\widehat{\\\\mathfrak {Y}}}$ mirror to $\\\\mathfrak {Y}$ , $\\\\zeta ^\\\\Lambda $ and $\\\\tilde{\\\\zeta }_\\\\Lambda $ are periods of the RR 3-form in the type IIA formulation, and $\\\\tau _2=g_s^{-1}$ is the inverse ten-dimensional string coupling.\",\n \"At the non-perturbative level, this expression gets modified and the Darboux coordinates are determined by the integral equation $\\\\mathcal {X}_\\\\gamma (t) = \\\\mathcal {X}^{\\\\rm sf}_\\\\gamma (t)\\\\, {\\\\bf e}\\\\!\\\\left( \\\\frac{1}{8\\\\pi ^2}\\\\sum _{\\\\gamma ^{\\\\prime }} \\\\sigma _{\\\\gamma ^{\\\\prime }} \\\\, \\\\bar{\\\\Omega }(\\\\gamma ^{\\\\prime })\\\\, \\\\langle \\\\gamma ,\\\\gamma ^{\\\\prime }\\\\rangle \\\\int _{\\\\ell _{\\\\gamma ^{\\\\prime }} }\\\\frac{\\\\mathrm {d}t^{\\\\prime }}{t^{\\\\prime }}\\\\, \\\\frac{t+t^{\\\\prime }}{t-t^{\\\\prime }}\\\\,\\\\mathcal {X}_{\\\\gamma ^{\\\\prime }}(t^{\\\\prime })\\\\right),$ where the sum goes over all charges labelling cycles wrapped by D-branes, $\\\\bar{\\\\Omega }(\\\\gamma ^{\\\\prime })=\\\\bar{\\\\Omega }(\\\\gamma ^{\\\\prime },z^a)$ is the corresponding rational Donaldson-Thomas invariant, $\\\\ell _\\\\gamma = \\\\lbrace t\\\\in {C}P^1\\\\ :\\\\ \\\\ Z_\\\\gamma /t\\\\in I{R}^-\\\\rbrace $ is the so called BPS ray, a contour on ${C}P^1$ extending from $t=0$ to $t=\\\\infty $ along the direction fixed by the central charge, and $\\\\sigma _\\\\gamma $ is a quadratic refinement of the DSZ product on the charge lattice $\\\\Gamma $ , i.e.\",\n \"a sign factor satisfying the defining relation $\\\\sigma _{\\\\gamma _1}\\\\sigma _{\\\\gamma _2}=(-1)^{\\\\langle \\\\gamma _1,\\\\gamma _2\\\\rangle }\\\\sigma _{\\\\gamma _1+\\\\gamma _2},\\\\qquad \\\\forall \\\\gamma _1,\\\\gamma _2\\\\in \\\\Gamma .$ The system of integral equations (REF ) can be solved iteratively by first substituting $\\\\mathcal {X}_{\\\\gamma ^{\\\\prime }}(t^{\\\\prime })$ on the r.h.s.\",\n \"with its zero-th order value $\\\\mathcal {X}^{\\\\rm sf}_{\\\\gamma ^{\\\\prime }}(t^{\\\\prime })$ in the weak coupling limit $\\\\tau _2\\\\rightarrow \\\\infty $ , computing the leading correction from the integral and iterating this process.\",\n \"This produces an asymptotic series at weak coupling, in powers of the DT invariants $\\\\bar{\\\\Omega }(\\\\gamma )$ .\",\n \"Using the saddle point method, it is easy to check that the coefficient of each monomial $\\\\prod _i \\\\bar{\\\\Omega }(\\\\gamma _i)$ is suppressed by a factor $e^{-\\\\pi \\\\tau _2 \\\\sum _i |Z_{\\\\gamma _i}|}$ , corresponding to an $n$ -instanton effect [36], [58].\",\n \"Note that multi-instanton effects become of the same order as one-instanton effects on walls of marginal stability where the phases of $Z_{\\\\gamma _i}$ become aligned, and that the wall-crossing formula ensures that the QK metric on $\\\\mathcal {M}_H$ is smooth across the walls.\",\n \"[36], [35].\"\n ],\n [\n \"D3-instantons in the large volume limit\",\n \"The above construction of D-instantons is adapted to the type IIA formulation because the equation (REF ) defines the Darboux coordinates in terms of the type IIA fields appearing explicitly in the tree level expression (REF ).\",\n \"To pass to the mirror dual type IIB formulation, one should apply the mirror map, a coordinate transformation from the type IIA to the type IIB physical fields.\",\n \"This transformation was determined in the classical limit in [59], but it also receives instanton corrections.\",\n \"In order to fix the form of these corrections, we require that the metric on $\\\\mathcal {M}_H$ carries an isometric action of S-duality group $SL(2,{Z})$ of type IIB string theory, which acts on the type IIB fields by an element $g={\\\\scriptsize \\\\begin{pmatrix} a & b \\\\\\\\ c & d \\\\end{pmatrix}}$ in the following way $\\\\begin{split}&\\\\tau \\\\mapsto \\\\frac{a \\\\tau +b}{c \\\\tau + d} \\\\, ,\\\\qquad t^a \\\\mapsto |c\\\\tau +d| \\\\,t^a,\\\\qquad \\\\begin{pmatrix} c^a \\\\\\\\ b^a \\\\end{pmatrix} \\\\mapsto \\\\begin{pmatrix} a & b \\\\\\\\ c & d \\\\end{pmatrix}\\\\begin{pmatrix} c^a \\\\\\\\ b^a \\\\end{pmatrix} ,\\\\\\\\&\\\\qquad \\\\tilde{c}_a\\\\mapsto \\\\tilde{c}_a - c_{2,a} \\\\varepsilon (g) ,\\\\qquad \\\\begin{pmatrix} \\\\tilde{c}_0 \\\\\\\\ \\\\psi \\\\end{pmatrix} \\\\mapsto \\\\begin{pmatrix} d & -c \\\\\\\\ -b & a \\\\end{pmatrix}\\\\begin{pmatrix} \\\\tilde{c}_0 \\\\\\\\ \\\\psi \\\\end{pmatrix},\\\\end{split}$ where $\\\\varepsilon (g)$ is the logarithm of the multiplier system of the Dedekind eta function [51].\",\n \"For this purpose, one uses the fact that any isometric action on a quaternion-Kähler manifold (preserving the quaternionic structure) can be lifted to a holomorphic contact transformation on twistor space.\",\n \"In the present case, $SL(2,{Z})$ acts on the fiber coordinate $t$ by a fractional-linear transformation with $\\\\tau $ -dependent coefficients.\",\n \"This transformation takes a much simpler form when formulated in terms of another coordinate $z$ on ${C}P^1$ (not to be confused with the Kähler moduli $z^a$ ), which is related to $t$ by a Cayley transformation, $z =\\\\frac{t+I}{t-I}\\\\, .$ Then the action of $SL(2,{Z})$ on the fiber is given by a simple phase rotationActually, this is true only when five-brane instanton corrections are ignored.\",\n \"Otherwise, the lift also gets a non-trivial deformation [60].\",\n \"$z\\\\mapsto \\\\frac{c\\\\bar{\\\\tau }+d}{|c\\\\tau +d|}\\\\, z\\\\, .$ Using the holomorphy constraint for the $SL(2,{Z})$ action on the twistor space, quantum corrections to the classical mirror map were computed in [61], [62], [21], [26], in the large volume limit where the Kähler moduli are taken to be large, $t^a\\\\rightarrow \\\\infty $ .\",\n \"In this limit, one finds $\\\\begin{split}u^a=&\\\\,b^a+It^a-\\\\frac{I}{2\\\\tau _2}{\\\\sum _{\\\\gamma \\\\in \\\\Gamma _+}}p^a\\\\left[\\\\int _{\\\\ell _{\\\\gamma }}\\\\mathrm {d}z\\\\,(1-z)\\\\,H_{\\\\gamma }+\\\\int _{\\\\ell _{-\\\\gamma }}\\\\frac{\\\\mathrm {d}z}{z^3} (1-z)\\\\,H_{-\\\\gamma }\\\\right]\\\\\\\\\\\\zeta ^0=&\\\\,\\\\tau _1,\\\\qquad \\\\zeta ^a=-(c^a-\\\\tau _1 b^a) -3\\\\sum _{\\\\gamma \\\\in \\\\Gamma _+} p^a\\\\,{\\\\rm Re}\\\\,\\\\int _{\\\\ell _{\\\\gamma }} \\\\mathrm {d}z\\\\, z\\\\,H_{\\\\gamma },\\\\\\\\\\\\tilde{\\\\zeta }_a =&\\\\,\\\\tilde{c}_a+ \\\\frac{1}{2}\\\\, \\\\kappa _{abc} \\\\,b^b (c^c - \\\\tau _1 b^c)+\\\\kappa _{abc}t^b \\\\sum _{\\\\gamma \\\\in \\\\Gamma _+} p^c\\\\,{\\\\rm Im}\\\\,\\\\int _{\\\\ell _{\\\\gamma }} \\\\mathrm {d}z\\\\,H_{\\\\gamma },\\\\end{split}$ where we introduced the convenient notationThe functions $H_\\\\gamma $ have a simple geometric meaning [32], [63]: they generate contact transformations (i.e.\",\n \"preserving the contact structure) relating the Darboux coordinates living on patches separated by BPS rays.\",\n \"In fact, these functions together with the contours $\\\\ell _\\\\gamma $ are the fundamental data fixing the contact structure on the twistor space.\",\n \"$H_\\\\gamma (t)= \\\\frac{\\\\bar{\\\\Omega }(\\\\gamma )}{(2\\\\pi )^2}\\\\,\\\\sigma _\\\\gamma \\\\mathcal {X}_\\\\gamma (t)\\\\, .$ Similar results are known for $\\\\tilde{\\\\zeta }_0$ and the NS-axion dual to the $B$ -field, but will not be needed in this paper.\",\n \"Note that the integral contributions to the mirror map are written in terms of the coordinate $z$ (REF ).\",\n \"The reason for using this variable is that, in the large volume limit, the integrals along BPS rays $\\\\ell _\\\\gamma $ in (REF ) are dominated by the saddle point [21] $z^{\\\\prime }_\\\\gamma \\\\approx -I\\\\,\\\\frac{(q_a+(pb)_a)\\\\,t^a}{(pt^2)}\\\\, ,$ for $(pt^2)>0$ , and $z^{\\\\prime }_{-\\\\gamma }=1/z^{\\\\prime }_\\\\gamma $ in the opposite case.\",\n \"This shows that all integrands can be expanded in Fourier series either around $z=0$ or $z=\\\\infty $ , keeping constant $t^a z$ or $t^a/z$ , respectively.\",\n \"This allows to extract the leading order in the large volume limit in a simple way.\",\n \"Let us therefore evaluate the combined limit $t^a\\\\rightarrow \\\\infty $ , $z\\\\rightarrow 0$ of the system of integral equations (REF ), assuming that only D3-D1-D(-1) instantons contribute.\",\n \"As a first step, we rewrite the tree level expression (REF ) in terms of the type IIB fields.\",\n \"To this end, we restrict the charge $\\\\gamma $ to lie in the cone $\\\\Gamma _+$ , take the central charge as in (REF ) with the cubicThe other contributions to the prepotential, representing perturbative $\\\\alpha ^{\\\\prime }$ -corrections and worldsheet instantons, combine with D(-1) and D1-instantons, but are irrelevant for our discussion of D3-instantons in the large volume limit.\",\n \"prepotential (REF ), and substitute the mirror map (REF ).\",\n \"Furthermore, we change the coordinate $t$ to $z$ and take the combined limit.\",\n \"In this way one finds $\\\\mathcal {X}^{\\\\rm sf}_\\\\gamma (z)=\\\\mathcal {X}^{\\\\rm cl}_\\\\gamma (z)\\\\exp \\\\left[2\\\\pi \\\\sum _{\\\\gamma ^{\\\\prime }\\\\in \\\\Gamma _+} (tpp^{\\\\prime })\\\\int _{\\\\ell _{\\\\gamma ^{\\\\prime }}} \\\\mathrm {d}z^{\\\\prime }\\\\,H_{\\\\gamma ^{\\\\prime }}\\\\right],$ where $\\\\mathcal {X}^{\\\\rm cl}_\\\\gamma (z) ={\\\\bf e}\\\\!\\\\left( - \\\\hat{q}_0\\\\tau \\\\right)\\\\mathcal {X}^{(\\\\theta )}_{p,q}(z)$ is the classical part of the Darboux coordinates which we represented as a product of two factors: exponential of the invariant charge (REF ) and the remaining $q_0$ -independent exponential $\\\\mathcal {X}^{(\\\\theta )}_{p,q}(z) = e^{-S^{\\\\rm cl}_p}\\\\,{\\\\bf e}\\\\!\\\\left( - \\\\frac{\\\\tau }{2}\\\\,(q+b)^2+c^a (q_a +\\\\textstyle {1\\\\over 2}(pb)_a)+I\\\\tau _2 (pt^2)(z^2-2zz_\\\\gamma )\\\\right),$ with $S^{\\\\rm cl}_p$ being the leading part of the Euclidean D3-brane action in the large volume limit given by $S^{\\\\rm cl}_p= \\\\pi \\\\tau _2(pt^2) - 2\\\\pi Ip^a \\\\tilde{c}_a$ .\",\n \"Next, we can approximate $\\\\frac{\\\\mathrm {d}t^{\\\\prime }}{t^{\\\\prime }}\\\\, \\\\frac{t+t^{\\\\prime }}{t-t^{\\\\prime }}=\\\\frac{2\\\\mathrm {d}z^{\\\\prime }}{1-z^{\\\\prime 2}}\\\\, \\\\frac{1-zz^{\\\\prime }}{z-z^{\\\\prime }}\\\\approx \\\\left\\\\lbrace \\\\begin{array}{cc}\\\\frac{2\\\\mathrm {d}z^{\\\\prime }}{z-z^{\\\\prime }}\\\\, , \\\\qquad & \\\\gamma ^{\\\\prime }\\\\in \\\\Gamma _+,\\\\\\\\\\\\frac{2(1-zz^{\\\\prime })\\\\mathrm {d}z^{\\\\prime }}{z^{\\\\prime 3}}\\\\, , \\\\qquad & \\\\gamma ^{\\\\prime }\\\\in -\\\\Gamma _+.\\\\end{array} \\\\right.$ This shows that the contribution of $\\\\gamma ^{\\\\prime }\\\\in -\\\\Gamma _+$ is suppressed comparing to $\\\\gamma ^{\\\\prime }\\\\in \\\\Gamma _+$ and therefore can be neglected.\",\n \"As a result, the system of integral equations (REF ) in the large volume limit where only D3-D1-D(-1) instantons contribute reduces to the following system of integral equations for $H_\\\\gamma $ , $H_\\\\gamma (z)=H^{\\\\rm cl}_\\\\gamma (z) \\\\, \\\\exp \\\\left[\\\\sum _{\\\\gamma ^{\\\\prime }\\\\in \\\\Gamma _+}\\\\int _{\\\\ell _{\\\\gamma ^{\\\\prime }}}\\\\mathrm {d}z^{\\\\prime }\\\\, K_{\\\\gamma \\\\gamma ^{\\\\prime }}(z,z^{\\\\prime })\\\\,H_{\\\\gamma ^{\\\\prime }}(z^{\\\\prime })\\\\right].$ Here $H^{\\\\rm cl}_\\\\gamma $ is the classical limit of $H_\\\\gamma $ , i.e.\",\n \"the function (REF ) with $\\\\mathcal {X}_\\\\gamma $ replaced by $\\\\mathcal {X}^{\\\\rm cl}_\\\\gamma $ , the integration kernel is now $K_{\\\\gamma _1\\\\gamma _2}(z_1,z_2)=2\\\\pi \\\\left((tp_1p_2)+\\\\frac{I\\\\langle \\\\gamma _1,\\\\gamma _2\\\\rangle }{z_1-z_2}\\\\right),$ and the BPS ray $\\\\ell _{\\\\gamma }$ effectively extends from $z^{\\\\prime }=-\\\\infty $ to $z^{\\\\prime }=+\\\\infty $ , going through the saddle point (REF ) [21].\",\n \"Below we shall need a perturbative solution of the integral equation (REF ).\",\n \"Applying the iterative procedure outlined below (REF ), or equivalently using the Lagrange inversion theorem, such solution can be written as a sum over rooted trees [36], $H_{\\\\gamma _1}(z_1) = H^{\\\\rm cl}_{\\\\gamma _1}(z_1) \\\\, \\\\sum _{n=1}^{\\\\infty } \\\\left(\\\\prod _{i=2}^n \\\\sum _{\\\\gamma _i\\\\in \\\\Gamma _+}\\\\,\\\\int _{\\\\ell _{\\\\gamma _i}} \\\\mathrm {d}z_i\\\\, H^{\\\\rm cl}_{\\\\gamma _i}(z_i) \\\\right) \\\\sum _{\\\\mathcal {T}\\\\in {T}_n^{\\\\rm r}}\\\\,\\\\frac{\\\\mathcal {A}(\\\\mathcal {T})}{|{\\\\rm Aut}(\\\\mathcal {T})|}\\\\, ,$ where ${T}_n^{\\\\rm r}$ is the set of rooted trees with $n$ vertices and $\\\\mathcal {A}(\\\\mathcal {T}) = \\\\prod _{e\\\\in E_{\\\\mathcal {T}}}K_{\\\\gamma _{s(e)},\\\\gamma _{t(e)}}(z_{s(e)},z_{t(e)}).$ A rooted treeWe will use calligraphic letters $\\\\mathcal {T}$ for trees where charges $\\\\gamma _i$ are assigned to vertices to distinguish them from rooted trees $T$ where the charges are assigned to leaves (hence $T$ has always more than $n$ vertices).\",\n \"Similarly, we will use notations $\\\\mathfrak {v}$ and $v$ for vertices of these two types of trees, respectively.\",\n \"Note also that whereas $V_\\\\mathcal {T}$ denotes the set of all vertices, $V_T$ does not includes the leaves.\",\n \"An example of trees of the latter type are attractor flow trees.\",\n \"$\\\\mathcal {T}$ consists of $n$ vertices joined by directed edges so that the root vertex has only outgoing edges, whereas all other vertices have one incoming edge and an arbitrary number of outgoing ones.\",\n \"We label the vertices of $\\\\mathcal {T}$ by $\\\\mathfrak {v}=1,\\\\dots , n$ in an arbitrary fashion, except for the root which is labelled by $\\\\mathfrak {v}=1$ .\",\n \"The symmetry factor $|{\\\\rm Aut}(\\\\mathcal {T})|$ is the order of the symmetry group which permutes the labels $2,\\\\dots , n$ without changing the topology of the tree.\",\n \"Each vertex is decorated by a charge vector $\\\\gamma _\\\\mathfrak {v}$ and a complex variable $z_\\\\mathfrak {v}\\\\in {C}P^1$ .\",\n \"We denote the set of edges by $E_{\\\\mathcal {T}}$ , the set of vertices by $V_{\\\\mathcal {T}}$ , and the source and target vertex of an edge $e$ by $s(e)$ and $t(e)$ , respectively.\",\n \"Unpacking these notations, we get, at the few leading orders, $H_{\\\\gamma _1} &= & H^{\\\\rm cl}_{\\\\gamma _1} + \\\\sum _{\\\\gamma _2} K_{12} H^{\\\\rm cl}_{\\\\gamma _1} \\\\,H^{\\\\rm cl}_{\\\\gamma _2}+ \\\\sum _{\\\\gamma _2,\\\\gamma _3} \\\\left( \\\\tfrac{1}{2}\\\\, K_{12} K_{13} + K_{12} K_{23} \\\\right)H^{\\\\rm cl}_{\\\\gamma _1}\\\\, H^{\\\\rm cl}_{\\\\gamma _2}\\\\, H^{\\\\rm cl}_{\\\\gamma _3}\\\\\\\\& + & \\\\sum _{\\\\gamma _2,\\\\gamma _3,\\\\gamma _4} \\\\!\\\\!\\\\!\\\\left( \\\\tfrac{1}{6}\\\\, K_{12} K_{13} K_{14}+ \\\\tfrac{1}{2}\\\\, K_{12} K_{23} K_{24} + K_{12} K_{13} K_{24} + K_{12} K_{23} K_{34} \\\\right)H^{\\\\rm cl}_{\\\\gamma _1} \\\\, H^{\\\\rm cl}_{\\\\gamma _2}\\\\, H^{\\\\rm cl}_{\\\\gamma _3}\\\\, H^{\\\\rm cl}_{\\\\gamma _4}+ \\\\dots \\\\nonumber $ where we omitted the integrals and denoted $K_{ij}=K_{\\\\gamma _i,\\\\gamma _j}(z_i,z_j)$ .\",\n \"The expansion (REF ) is effectively a multi-instanton expansion in powers of the DT invariants $\\\\bar{\\\\Omega }(\\\\gamma _i)$ , which is asymptotic to the exact solution to (REF ) in the weak coupling limit $\\\\tau _2\\\\rightarrow \\\\infty $ .\"\n ],\n [\n \"From the contact potential to the instanton generating function\",\n \"Recall that our goal is to derive constraints imposed by S-duality on the DT invariants $\\\\bar{\\\\Omega }(\\\\gamma )$ appearing as coefficients in the multi-instanton expansion.\",\n \"To achieve this goal, rather than studying the full metric on $\\\\mathcal {M}_H$ , it suffices to consider a suitable function on this moduli space which has a non-trivial dependence on $\\\\bar{\\\\Omega }(\\\\gamma )$ and specified transformations under S-duality.\",\n \"There is a natural candidate with the above properties: the so-called contact potential $e^{\\\\phi }$ , a real function which is well-defined on any quaternion-Kähler manifold with a continuous isometry [34].\",\n \"Furthermore, there is a general expression for the contact potential in terms of Penrose-type integrals on the ${C}P^1$ fiber.\",\n \"In the present case, the required isometry is the shift of the NS-axion, which survives all quantum corrections as long as NS5-instantons are switched off.\",\n \"The contact potential is then given by the exact formula [32] $e^{\\\\phi } = \\\\frac{I\\\\tau _2^2}{16}\\\\left(\\\\bar{u}^\\\\Lambda F_\\\\Lambda - u^\\\\Lambda \\\\bar{F}_\\\\Lambda \\\\right)-\\\\frac{\\\\chi _{{\\\\widehat{\\\\mathfrak {Y}}}}}{192\\\\pi }+\\\\frac{I\\\\tau _2}{16}\\\\,\\\\sum _\\\\gamma \\\\int _{\\\\ell _\\\\gamma } \\\\frac{\\\\text{d}t}{t} \\\\left( t^{-1} Z_\\\\gamma (u^a)-t\\\\bar{Z}_\\\\gamma (\\\\bar{u}^a)\\\\right) H_\\\\gamma ,$ where $\\\\chi _{{\\\\widehat{\\\\mathfrak {Y}}}}$ is the Euler characteristic of ${\\\\widehat{\\\\mathfrak {Y}}}$ .\",\n \"This formula indeed captures contribution from D-instantons due to the last term proportional to $H_\\\\gamma $ .\",\n \"On the other hand, in the classical, large volume limit one finds $e^\\\\phi =\\\\frac{\\\\tau _2^2}{12}(t^3)$ , which shows that the contact potential can be identified with the four-dimensional dilaton and in this approximation behaves as a modular form of weight $(-\\\\tfrac{1}{2}, -\\\\tfrac{1}{2})$ under S-duality transformations (REF ).\",\n \"In fact, one can show [32] that $SL(2,{Z})$ preserves the contact structure, i.e.\",\n \"it is an isometry of $\\\\mathcal {M}_H$ , only if the full non-perturbative contact potential transforms in this way, $e^\\\\phi \\\\mapsto \\\\frac{e^\\\\phi }{|c\\\\tau +d|}\\\\, .$ Furthermore, since S-duality acts by rescaling the Kähler moduli $t^a$ and by a phase rotation of the fiber coordinate $z$ (see (REF )), it preserves each order in the expansion around the large volume limit.\",\n \"This implies that the large volume limit of the D3-instanton contribution to $e^\\\\phi $ , which we denote by $(e^\\\\phi )_{\\\\rm D3}$ , must itself transform as (REF ).\",\n \"It is this condition that we shall exploit to derive modularity constraints on the DT invariants.\",\n \"To make this condition more explicit, let us extract the D3-instanton contribution to the function (REF ).\",\n \"The procedure is the same as the one used to get (REF ), and we relegate the details of the calculation to appendix .\",\n \"The result can be written in a concise way using the complex function defined by $\\\\mathcal {G}&=& \\\\sum _{\\\\gamma \\\\in \\\\Gamma _+}\\\\int _{\\\\ell _{\\\\gamma }} \\\\mathrm {d}z\\\\, H_{\\\\gamma }(z)-{1\\\\over 2}\\\\,\\\\sum _{\\\\gamma _1,\\\\gamma _2\\\\in \\\\Gamma _+}\\\\int _{\\\\ell _{\\\\gamma _1}}\\\\mathrm {d}z_1\\\\,\\\\int _{\\\\ell _{\\\\gamma _2}} \\\\mathrm {d}z_2\\\\, K_{\\\\gamma _1\\\\gamma _2}(z_1,z_2)\\\\,H_{\\\\gamma _1}(z_1)H_{\\\\gamma _2}(z_2)$ and the Maass raising operator $\\\\mathcal {D}_{\\\\mathfrak {h}} = \\\\frac{1}{2\\\\pi I}\\\\left(\\\\partial _\\\\tau +\\\\frac{\\\\mathfrak {h}}{2I\\\\tau _2}+ \\\\frac{It^a}{4\\\\tau _2}\\\\, \\\\partial _{t^a}\\\\right),$ which maps modular functions of weight $(\\\\mathfrak {h},\\\\bar{\\\\mathfrak {h}})$ to modular functions of weight $(\\\\mathfrak {h}+2,\\\\bar{\\\\mathfrak {h}})$ .\",\n \"Then one has (generalizing [14] to all orders in the instanton expansion) $(e^\\\\phi )_{\\\\rm D3}=\\\\frac{\\\\tau _2}{2}\\\\,{\\\\rm Re}\\\\,\\\\left(\\\\mathcal {D}_{-\\\\frac{3}{2}}\\\\mathcal {G}\\\\right)+\\\\frac{1}{32\\\\pi ^2}\\\\,\\\\kappa _{abc}t^c\\\\partial _{\\\\tilde{c}_a}\\\\mathcal {G}\\\\partial _{\\\\tilde{c}_b}\\\\overline{\\\\mathcal {G}}.$ It is immediate to see that $(e^\\\\phi )_{\\\\rm D3}$ transforms under S-duality as (REF ) provided the function $\\\\mathcal {G}$ transforms as a modular form of weight $(-\\\\tfrac{3}{2},\\\\tfrac{1}{2})$ .\",\n \"In order to derive the implications of this fact, we need to express $\\\\mathcal {G}$ in terms of the generalized DT invariants.\",\n \"For this purpose, we substitute the multi-instanton expansion (REF ) into (REF ).\",\n \"We claim that the result takes the simple form $\\\\mathcal {G}=\\\\sum _{n=1}^\\\\infty \\\\left[\\\\prod _{i=1}^{n} \\\\sum _{\\\\gamma _i\\\\in \\\\Gamma _+}\\\\int _{\\\\ell _{\\\\gamma _i}}\\\\mathrm {d}z_i\\\\, H^{\\\\rm cl}_{\\\\gamma _i}(z_i) \\\\right]\\\\mathcal {G}_n(\\\\lbrace \\\\gamma _i,z_i\\\\rbrace ),$ where $\\\\mathcal {G}_n(\\\\lbrace \\\\gamma _i,z_i\\\\rbrace )$ is now a sum over unrooted trees with $n$ vertices, $\\\\mathcal {G}_n(\\\\lbrace \\\\gamma _i,z_i\\\\rbrace )=\\\\sum _{\\\\mathcal {T}\\\\in \\\\, {T}_n} \\\\frac{\\\\mathcal {A}(\\\\mathcal {T})}{|{\\\\rm Aut}(\\\\mathcal {T})|}=\\\\frac{1}{n!\",\n \"}\\\\sum _{\\\\mathcal {T}\\\\in \\\\, {T}_n^\\\\ell } \\\\mathcal {A}(\\\\mathcal {T}),$ and in the second equality we rewrote the result as a sum over unrooted labelled trees.The number of such trees is $|{T}_n^\\\\ell |=n^{n-2}=\\\\lbrace 1,1,3,16,125,1296,\\\\dots \\\\rbrace $ for $n\\\\ge 1$ .\",\n \"Such trees also appear in the Joyce-Song wall-crossing formula [42], [43] and are conveniently labelled by their Prüfer code.\",\n \"To see why this is the case, observe that under the action of the Euler operator $\\\\hat{D}=H^{\\\\rm cl}\\\\partial _{H^{\\\\rm cl}}$ rescaling all functions $H^{\\\\rm cl}_\\\\gamma $ , the function $\\\\mathcal {G}$ maps to the first term in (REF ), which we denote by $\\\\mathcal {F}$ .\",\n \"Namely, $\\\\hat{D}\\\\cdot \\\\mathcal {G}=\\\\mathcal {F},\\\\qquad \\\\mathcal {F}\\\\equiv \\\\sum _{\\\\gamma \\\\in \\\\Gamma _+}\\\\int _{\\\\ell _{\\\\gamma }} \\\\mathrm {d}z\\\\, H_{\\\\gamma },$ as can be verified with the help of the integral equation (REF ).\",\n \"The multi-instanton expansion of $\\\\mathcal {F}$ follows immediately from (REF ), $\\\\mathcal {F}= \\\\sum _{n=1}^{\\\\infty } \\\\left(\\\\prod _{i=1}^n \\\\sum _{\\\\gamma _i\\\\in \\\\Gamma _+}\\\\,\\\\int _{\\\\ell _{\\\\gamma _i}} \\\\mathrm {d}z_i\\\\, H^{\\\\rm cl}_{\\\\gamma _i}(z_i) \\\\right) \\\\sum _{\\\\mathcal {T}\\\\in {T}_n^{\\\\rm r}}\\\\,\\\\frac{\\\\mathcal {A}(\\\\mathcal {T})}{|{\\\\rm Aut}(\\\\mathcal {T})|}\\\\, .$ Integrating back the action of the derivative operator $\\\\hat{D}$ , we see that the sum over rooted trees in (REF ) turns into the sum over unrooted trees in (REF ).\",\n \"At the first few orders we get, using the same shorthand notation as in (REF ), $\\\\begin{split}\\\\mathcal {G}= &\\\\sum _{\\\\gamma } H^{\\\\rm cl}_\\\\gamma + \\\\frac{1}{2} \\\\sum _{\\\\gamma _1,\\\\gamma _2}K_{12}H^{\\\\rm cl}_{\\\\gamma _1} \\\\,H^{\\\\rm cl}_{\\\\gamma _2}+ \\\\frac{1}{2} \\\\sum _{\\\\gamma _1,\\\\gamma _2,\\\\gamma _3}K_{12}K_{23}H^{\\\\rm cl}_{\\\\gamma _1}\\\\, H^{\\\\rm cl}_{\\\\gamma _2}\\\\, H^{\\\\rm cl}_{\\\\gamma _3}\\\\\\\\&+\\\\sum _{\\\\gamma _1,\\\\gamma _2,\\\\gamma _3,\\\\gamma _4}\\\\left(\\\\frac{1}{6} \\\\, K_{12} K_{13} K_{14} + \\\\frac{1}{2}\\\\,K_{12} K_{23} K_{34} \\\\right)H^{\\\\rm cl}_{\\\\gamma _1}\\\\, H^{\\\\rm cl}_{\\\\gamma _2}\\\\, H^{\\\\rm cl}_{\\\\gamma _3}\\\\, H^{\\\\rm cl}_{\\\\gamma _4}+\\\\dots \\\\end{split}$ The simplicity of the expansion (REF ), and the relation (REF ) to the contact potential, show that the function $\\\\mathcal {G}$ is very natural and, in some sense, more fundamentalIn [14], it was noticed that the function $\\\\mathcal {G}$ , denoted by $\\\\tilde{\\\\mathcal {F}}$ in that reference and computed at second order in the multi-instanton expansion, could be obtained from the seemingly simpler function $\\\\mathcal {F}$ by halving the coefficient of its second order contribution.\",\n \"Now we see that this ad hoc prescription is the consequence of going from rooted to unrooted trees, as a result of adding the second term in (REF ).\",\n \"than the naive instanton sum $\\\\mathcal {F}$ .\",\n \"We shall henceforth refer to $\\\\mathcal {G}$ as the `instanton generating function'.\",\n \"In the following we shall postulate that $\\\\mathcal {G}$ transforms as a modular form of weight $(-\\\\frac{3}{2},\\\\frac{1}{2})$ , and analyze the consequences of this requirement for the DT invariants.\"\n ],\n [\n \"Theta series decomposition and modularity\",\n \"In this section, we use the spectral flow symmetry to decompose the instanton generating function $\\\\mathcal {G}$ into a sum of indefinite theta series multiplied by holomorphic generating functions of MSW invariants.\",\n \"We then study the modular properties of these indefinite theta series, and identify the origin of the modular anomaly.\"\n ],\n [\n \"Factorisation\",\n \"To derive modularity constraints on the DT invariants, we need to perform a theta series decomposition of the generating function $\\\\mathcal {G}$ defined in (REF ).\",\n \"To this end, let us make use of the fact noticed in (REF ) that the DSZ products $\\\\langle \\\\gamma ,\\\\gamma ^{\\\\prime }\\\\rangle $ and hence the kernels (REF ) do not depend on the $q_0$ charge.\",\n \"Choosing the quadratic refinement $\\\\sigma _\\\\gamma $ as in (REF ), which is also $q_0$ -independent, and using the factorization (REF ) of $\\\\mathcal {X}^{\\\\rm cl}_\\\\gamma $ , one can rewrite the expansion (REF ) as follows $\\\\mathcal {G}=\\\\sum _{n=1}^\\\\infty \\\\frac{1}{(2\\\\pi )^{2n}}\\\\left[\\\\prod _{i=1}^{n}\\\\sum _{p_i,q_i}\\\\sigma _{p_i,q_i}h^{\\\\rm DT}_{p_i,q_i}\\\\int _{\\\\ell _{\\\\gamma _i}}\\\\mathrm {d}z_i\\\\, \\\\mathcal {X}^{(\\\\theta )}_{p_i,q_i}(z_i) \\\\right]\\\\mathcal {G}_n(\\\\lbrace \\\\gamma _i,z_i\\\\rbrace ),$ where the sum over the invariant charges $\\\\hat{q}_{i,0}$ gave rise to the generating functions of DT invariants defined in (REF ).\",\n \"This is not yet the desired form because these generating functions depend non-trivially on the remaining electric charges $q_{i,a}$ .\",\n \"If it were not for this dependence, the sum over $q_{i,a}$ would produce certain non-Gaussian theta series, and at each order we would have a product of this theta series and $n$ generating functions.\",\n \"Then the modular properties of the theta series would dictate the modular properties of the generating function.\",\n \"Such a theta series decomposition can be achieved by expressing the DT invariants in terms of the MSW invariants, for which the dependence on electric charges $q_{i,a}$ reduces to the dependence on the residue classes $\\\\mu _{i,a}$ due to the spectral flow symmetry.\",\n \"Substituting the expansion (REF ) of $h^{\\\\rm DT}_{p,q}$ in terms of $h_{p,\\\\mu }$ , the expansion (REF ) of the function $\\\\mathcal {G}$ can be brought to the following factorized form $\\\\mathcal {G}=\\\\sum _{n=1}^\\\\infty \\\\frac{2^{-\\\\frac{n}{2}}}{\\\\pi \\\\sqrt{2\\\\tau _2}}\\\\left[\\\\prod _{i=1}^{n}\\\\sum _{p_i,\\\\mu _i}\\\\sigma _{p_i}h_{p_i,\\\\mu _i}\\\\right]e^{-S^{\\\\rm cl}_p}\\\\vartheta _{{p},{\\\\mu }}\\\\bigl (\\\\Phi ^{{\\\\rm tot}}_{n},n-2\\\\bigr ),$ where $\\\\vartheta _{{p},{\\\\mu }}$ is a theta series (REF ) with parameter $\\\\lambda =n-2$ , whose kernel has the following structure $\\\\Phi ^{{\\\\rm tot}}_n({x})=\\\\,{\\\\rm Sym}\\\\, \\\\left\\\\lbrace \\\\sum _{n_1+\\\\cdots n_m=n \\\\atop n_k\\\\ge 1} \\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_m({x}^{\\\\prime })\\\\prod _{k=1}^m \\\\Phi ^{\\\\,g}_{n_k}(x_{j_{k-1}+1},\\\\dots ,x_{j_k})\\\\right\\\\rbrace .$ Here the sum runs over ordered partitions of $n$ , whose number of parts is denoted by $m$ , and we adopted notations from (REF ) for indices $j_k$ .\",\n \"The argument ${x}$ of the kernel encodes the electric components of the charges (shifted by the B-field and rescaled by $\\\\sqrt{2\\\\tau _2}$ ) and lives in a vector space $\\\\left(\\\\oplus _{i=1}^n \\\\Lambda _i\\\\right) \\\\otimes {R}$ of dimension $d=nb_2$ , given by $b_2$ copies of the lattice $\\\\Lambda $ , where the $i$ -th copy $\\\\Lambda _i$ carries the bilinear form $\\\\kappa _{i,ab}=\\\\kappa _{abc}p_i^c$ of signature $(1,b_2-1)$ .\",\n \"Therefore, $\\\\vartheta _{{p},{\\\\mu }}$ is an indefinite theta series associated to the bilinear form given explicitly in (REF ), which has signature $(n,n(b_2-1))$ .\",\n \"Finally, the kernel (REF ) is constructed from two other functions.\",\n \"The first, $\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_n$ , is the iterated integral of the coefficient $\\\\mathcal {G}_n$ in the expansion (REF ), $\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_n({x}) =\\\\left(\\\\frac{\\\\sqrt{2\\\\tau _2}}{2\\\\pi }\\\\right)^{n-1}\\\\left[\\\\prod _{i=1}^n\\\\int _{\\\\ell _{\\\\gamma _i}}\\\\frac{\\\\mathrm {d}z_i}{2\\\\pi } \\\\, W_{p_i}(x_i,z_i)\\\\right]\\\\mathcal {G}_n(\\\\lbrace \\\\gamma _i,z_i\\\\rbrace ),$ weighted with the Gaussian measure factor $W_{p}(x,z)=e^{-2\\\\pi \\\\tau _2 z^2(pt^2)-2\\\\pi I\\\\sqrt{2\\\\tau _2}\\\\, z\\\\, (pxt)}$ coming from the $z$ -dependent part of (REF ).\",\n \"Although this function is written in terms of $\\\\mathcal {G}_n$ depending on full electromagnetic charge vectors $\\\\gamma _i$ , it is actually independent of their $q_0$ components.\",\n \"Indeed, using the result (REF ), it can be rewritten as $\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_n({x}) = \\\\frac{1}{n!\",\n \"}\\\\left[\\\\prod _{i=1}^n\\\\int _{\\\\ell _{\\\\gamma _i}}\\\\frac{\\\\mathrm {d}z_i}{2\\\\pi } \\\\, W_{p_i}(x_i,z_i)\\\\right]\\\\sum _{\\\\mathcal {T}\\\\in \\\\, {T}_n^\\\\ell }\\\\prod _{e\\\\in E_\\\\mathcal {T}} \\\\hat{K}_{s(e) t(e)},$ where we introduced a rescaled version of the kernel (REF ) $\\\\hat{K}_{ij}(z_i,z_j)=\\\\left(\\\\left(\\\\sqrt{2\\\\tau _2}\\\\,t+I\\\\,\\\\frac{x_i-x_j}{z_i-z_j}\\\\right)p_ip_j\\\\right).$ Note that $\\\\tau _2$ and $t^a$ appear only in the modular invariant combination $\\\\sqrt{2\\\\tau _2}\\\\, t^a$ .\",\n \"In (REF ) this function appears with the argument ${x}^{\\\\prime }$ and carries a dependence on ${p}^{\\\\prime }$ (not indicated explicitly) which are both $mb_2$ -dimensional vectors with components (cf.\",\n \"(REF )) $p^{\\\\prime a}_k=\\\\sum _{i=j_{k-1}+1}^{j_k}p^a_i,\\\\qquad x^{\\\\prime a}_k=\\\\kappa ^{\\\\prime ab}_k\\\\sum _{i=j_{k-1}+1}^{j_k} \\\\kappa _{i,bc} x^c_i,$ where $\\\\kappa ^{\\\\prime }_{k,ab}=\\\\kappa _{abc}p^{\\\\prime c}_k$ .\",\n \"The second function, $\\\\Phi ^{\\\\,g}_{n}$ , appears due to the expansion of DT invariants in terms of the MSW invariants and is given by a suitably rescaled tree index $\\\\Phi ^{\\\\,g}_{n}({x})=\\\\frac{\\\\sigma _{\\\\gamma }(\\\\sqrt{2\\\\tau _2})^{n-1}}{\\\\prod _{i=1}^n\\\\sigma _{\\\\gamma _i}}\\\\,g_{{\\\\rm tr},n}(\\\\lbrace \\\\gamma _i,c_i\\\\rbrace ).$ It is also written in terms of functions depending on the full electromagnetic charge vectors $\\\\gamma _i$ (with $\\\\gamma =\\\\gamma _1+\\\\cdots \\\\gamma _n$ ).\",\n \"However, using (REF ), all quadratic refinements can be expressed through $(-1)^{\\\\langle \\\\gamma _i,\\\\gamma _j\\\\rangle }$ which cancel the corresponding sign factors in the tree index (see (REF )).\",\n \"Furthermore, as was noticed in the end of section REF , the tree index is independent of the $q_0$ components of the charge vectors.\",\n \"Therefore, it can be written as a function of $p_i^a$ , $\\\\mu _{i,a}$ and $x_i^a=\\\\sqrt{2\\\\tau _2}(\\\\kappa _i^{ab} q_{i,b}+b^a)$ .\",\n \"Then, since after cancelling the sign factors, $g_{{\\\\rm tr},n}$ becomes a homogeneous function of degree $n-1$ in the D2-brane charge $q_{i,a}$ , all factors of $\\\\sqrt{2\\\\tau _2}$ in (REF ) cancel as well.\"\n ],\n [\n \"Modularity and Vignéras' equation\",\n \"As explained in appendix REF , the theta series $\\\\vartheta _{{p},{\\\\mu }}\\\\bigl (\\\\Phi ,\\\\lambda \\\\bigr )$ is a vector-valued modular form of weight $(\\\\lambda +d/2,0)$ provided the kernel $\\\\Phi $ satisfies Vignéras' equation (REF ) — along with certain growth conditions which we expect to be automatically satisfied for the kernels of interest in this work.\",\n \"In our case $\\\\lambda =n-2$ and the dimension of the lattice is $d=nb_2(\\\\mathfrak {Y})$ so that the expected weight of the theta series is $(2(n-1)+nb_2/2,0)$ .\",\n \"It is consistent with weight $(-\\\\tfrac{3}{2},\\\\tfrac{1}{2})$ of $\\\\mathcal {G}$ given in (REF ) only if $h_{p,\\\\mu }$ is a vector-valued holomorphic modular form of weight $(-b_2/2-1,0)$ .\",\n \"However for this to be true, the kernel $\\\\Phi ^{{\\\\rm tot}}_n$ ought to satisfy Vignéras' equation.\",\n \"Let us examine whether or not this is the case.\",\n \"To this end, we first consider the kernel $\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_n$ (REF ).\",\n \"In appendix we evaluate explicitly the iterated integrals defining this kernel.\",\n \"To present the final result, let us introduce the following $d$ -dimensional vectors ${v}_{ij}, {u}_{ij}$ : $\\\\begin{split}({v}_{ij})_k^a=&\\\\, \\\\delta _{ki} p_j^a-\\\\delta _{kj} p_i^a \\\\qquad \\\\qquad \\\\quad \\\\ \\\\mbox{such that} \\\\quad {v}_{ij}\\\\cdot {x}=(p_ip_j(x_i-x_j)),\\\\\\\\({u}_{ij})_k^a=&\\\\, \\\\delta _{ki}(p_jt^2)t^a-\\\\delta _{kj} (p_it^2)t^a \\\\quad \\\\mbox{such that} \\\\quad {u}_{ij}\\\\cdot {x}=(p_jt^2)(p_ix_it)-(p_it^2)(p_jx_jt),\\\\end{split}$ where $k$ labels the copy in $\\\\oplus _{k=1}^n \\\\Lambda _k$ , $a=1,\\\\dots , b_2$ , and the bilinear form is given in (REF ).\",\n \"The first scalar product ${v}_{ij}\\\\cdot {x}$ corresponds to the DSZ product $\\\\langle \\\\gamma _i,\\\\gamma _j\\\\rangle $ , whereas the second product ${u}_{ij}\\\\cdot {x}$ corresponds to $-2\\\\,{\\\\rm Im}\\\\,[Z_{\\\\gamma _i}\\\\bar{Z}_{\\\\gamma _j}]$ (REF ), both rescaled by $\\\\sqrt{2\\\\tau _2}$ and expressed in terms of $x_i^a$ .\",\n \"From these vectors we can construct two sets of vectors which are assigned to the edges of an unrooted labelled tree $\\\\mathcal {T}$ , such as the trees appearing in (REF ) and (REF ).\",\n \"Namely, ${v}_e=\\\\sum _{i\\\\in V_{\\\\mathcal {T}_e^s}}\\\\sum _{j\\\\in V_{\\\\mathcal {T}_e^t}}{v}_{ij},\\\\qquad {u}_e=\\\\sum _{i\\\\in V_{\\\\mathcal {T}_e^s}}\\\\sum _{j\\\\in V_{\\\\mathcal {T}_e^t}}{u}_{ij},$ where $\\\\mathcal {T}_e^s$ , $\\\\mathcal {T}_e^t$ are the two disconnected trees obtained from the tree $\\\\mathcal {T}$ by removing the edge $e$ .\",\n \"Then the kernel $\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_n$ can be expressed as followsBoth vectors ${u}_e$ and ${v}_{s(e) t(e)}$ depend on the choice of orientation of the edge $e$ , but this ambiguity is cancelled in the function $\\\\widetilde{\\\\Phi }^M_{n-1}$ .\",\n \"$\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_n({x})=\\\\frac{\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_1(x)}{2^{n-1} n!\",\n \"}\\\\sum _{\\\\mathcal {T}\\\\in \\\\, {T}_n^\\\\ell }\\\\widetilde{\\\\Phi }^M_{n-1}(\\\\lbrace {u}_e\\\\rbrace , \\\\lbrace {v}_{s(e) t(e)}\\\\rbrace ;{x}).$ Here the first factor is simply a Gaussian $\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_1(x)=\\\\frac{e^{-\\\\frac{\\\\pi (pxt)^2}{(pt^2)}}}{2\\\\pi \\\\sqrt{2\\\\tau _2(pt^2)}}$ which ensures the suppression along the direction of the total charge in the charge lattice.\",\n \"In the second factor one sums over unrooted labelled trees $\\\\mathcal {T}$ with $n$ vertices, with summand given by a function $\\\\widetilde{\\\\Phi }^M_n$ defined as in (REF ), upon replacing $\\\\Phi _n^E$ by $\\\\Phi _n^M$ in that expression and setting $m=n$ .\",\n \"Both $\\\\Phi _n^E$ and $\\\\Phi _n^M$ are the so-called generalized (complementary) error functions introduced in [28] and further studied in [29], whose definitions are recalled in (REF ), () and (REF ).\",\n \"The functions $\\\\widetilde{\\\\Phi }^M_{n-1}$ in (REF ) depend on two sets of $n-1$ $d$ -dimensional vectors: the vectors in the first set are given by ${u}_e$ defined above, whereas the vectors ${v}_{s(e) t(e)}$ in the second set coincide with ${v}_{ij}$ for $i$ and $j$ corresponding to the source and target vertices of edge $e$ of the labelled tree.\",\n \"The remarkable property of the generalized error functions $\\\\Phi ^M_{n-1}$ and their uplifted versions $\\\\widetilde{\\\\Phi }^M_{n-1}$ is that, away from certain loci where these functions are discontinuous, they satisfy Vignéras' equation for $\\\\lambda =0$ and $n-1$ , respectively.\",\n \"Given that $\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_1$ is also a solution for $\\\\lambda =-1$ , and the vector ${t}=(t^a,\\\\dots ,t^a)$ (such that ${t}\\\\cdot {x}=(pxt)$ ) is orthogonal to all vectors ${u}_e$ and ${v}_{s(e) t(e)}$ , the kernel (REF ) satisfies this equation for $\\\\lambda =n-2$ .\",\n \"However, as mentioned above, it fails to do so on the loci where it is discontinuous.\",\n \"These discontinuities arise due to dependence of the integration contours $\\\\ell _\\\\gamma $ on moduli and electric charges.\",\n \"Of course, since the integrands are meromorphic functions, the integrals do not depend on deformations of the contours provided they do not cross the poles.\",\n \"But this is exactly what happens when two BPS rays, say $\\\\ell _\\\\gamma $ and $\\\\ell _{\\\\gamma ^{\\\\prime }}$ , exchange their positions, which in turn takes place when the phases of the corresponding central charges $Z_\\\\gamma $ and $Z_{\\\\gamma ^{\\\\prime }}$ align, as follows from (REF ).\",\n \"The loci where such alignment takes place are nothing else but the walls of marginal stability.\",\n \"This point will play an important rôle in the next subsection since it makes it possible to recombine the discontinuities of the generalized error functions with discontinuities of the tree indices.\",\n \"We now turn to the action of Vignéras' operator on $\\\\Phi ^{\\\\,g}_{n}$ .\",\n \"To this end, it is convenient to use the representation of the tree index as a sum over attractor flow trees (REF ).\",\n \"Let us assign a $nb_2$ -dimensional vector $\\\\tilde{{v}}_{v}$ to each vertex $v$ of a flow tree.\",\n \"Denoting by $\\\\mathcal {I}_v$ the set of leaves which are descendants of vertex $v$ , we set $\\\\tilde{{v}}_v=\\\\sum _{i\\\\in \\\\mathcal {I}_{L(v)}}\\\\sum _{j\\\\in \\\\mathcal {I}_{R(v)}}{v}_{ij}.$ With these definitions the kernel (REF ) can be written as $\\\\Phi ^{\\\\,g}_n({x}) &=&(-1)^{n-1}\\\\sum _{T\\\\in {T}_n^{\\\\rm af}}\\\\prod _{v\\\\in V_T}(\\\\tilde{{v}}_v,{x})\\\\,\\\\Delta _{\\\\gamma _{L(v)}\\\\gamma _{R(v)}}^{z_{p(v)}}.$ The factors $\\\\Delta _{\\\\gamma _L\\\\gamma _R}^z$ are locally constant and therefore, away from the loci where they are discontinuous, the action of Vignéras operator reduces to its action on the scalar products $(\\\\tilde{{v}}_v,{x})$ .\",\n \"For a single such factor one finds $V_\\\\lambda (\\\\tilde{{v}}_v,{x})=(\\\\tilde{{v}}_v,{x})V_{\\\\lambda -1}+2\\\\tilde{{v}}_v\\\\cdot \\\\partial _{x}.$ The crucial observation is that all vectors $\\\\tilde{{v}}_v$ appearing in the product (REF ) for a single tree are mutually orthogonal $(\\\\tilde{{v}}_v,\\\\tilde{{v}}_{v^{\\\\prime }})=0$ , which is clear because $\\\\langle \\\\gamma _{L(v)},\\\\gamma _{R(v)}\\\\rangle $ is antisymmetric in charges $\\\\gamma _{L(v)},\\\\gamma _{R(v)}$ , whereas the factors associated with vertices which are not descendants of $v$ either depend on their sum or do not depend on them at all.\",\n \"Therefore, one obtains $V_\\\\lambda \\\\Phi ^{\\\\,g}_{n_k}({x})=\\\\Phi ^{\\\\,g}_{n_k}({x})V_{\\\\lambda -n_k+1}+2(-1)^{n_k-1}\\\\sum _{T\\\\in {T}_{n_k}^{\\\\rm af}}\\\\Delta (T)\\\\sum _{v\\\\in V_T}\\\\left[\\\\prod _{v^{\\\\prime }\\\\in V_T\\\\setminus \\\\lbrace v\\\\rbrace }(\\\\tilde{{v}}_{v^{\\\\prime }},{x})\\\\right]\\\\tilde{{v}}_v\\\\cdot \\\\partial _{x}.$ Let us now evaluate the action of $V_{n-2}$ on the full kernel $\\\\Phi ^{{\\\\rm tot}}_n$ .\",\n \"Applying the result (REF ), we observe that the second term vanishes on the other factors in (REF ) due to the same reason that they either do not depend (in the case of $\\\\Phi ^{\\\\,g}_{n_{k^{\\\\prime }}}$ , $k^{\\\\prime }\\\\ne k$ ) or depend (in the case of $\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_{m}$ ) only on the sum of charges entering $\\\\Phi ^{\\\\,g}_{n_k}$ .\",\n \"Therefore, one finds that, away from discontinuities of generalized error functions and $\\\\Delta (T)$, one has $V_{n-2}\\\\cdot \\\\Phi ^{{\\\\rm tot}}_n=0$ .\"\n ],\n [\n \"Discontinuities and the anomaly\",\n \"Let us now turn to the discontinuities of $\\\\Phi ^{{\\\\rm tot}}_n$ which we ignored so far and which spoil Vignéras' equation and hence modularity of the theta series.\",\n \"There are three potential sources of such discontinuities: walls of marginal stability — at these loci $\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_{m}$ are discontinuous due to exchange of integration contours and $\\\\Phi ^{\\\\,g}_{n_k}$ jump due to factors $\\\\Delta _{\\\\gamma _{L(v)}\\\\gamma _{R(v)}}^{z_{p(v)}}$ assigned to the root vertices of attractor flow trees; `fake walls' — these are loci in the moduli space where $\\\\,{\\\\rm Im}\\\\,\\\\bigl [ Z_{\\\\gamma _{L(v)}}\\\\bar{Z}_{\\\\gamma _{R(v)}}(z^a_{p(v)})\\\\bigr ]=0$ , and hence the corresponding $\\\\Delta $ -factor jumps, where $v$ is not a root vertex — they correspond to walls of marginal stability for the intermediate bound states appearing in the attractor flow; moduli independent loci where $(\\\\tilde{{v}}_v,{x})=0$ — at these loci the factors $\\\\Delta _{\\\\gamma _{L(v)}\\\\gamma _{R(v)}}^{z_{p(v)}}$ and hence $\\\\Phi ^{\\\\,g}_{n_k}$ are discontinuous due to the second term in (REF ).\",\n \"Remarkably, the two effects due to the non-trivial charge and moduli dependence of the DT invariants and the exchange of contours cancel each other and the function $\\\\mathcal {G}$ turns out to be smooth at loci of the first type.\",\n \"This is expected because the whole construction of D-instantons has been designed to make the resulting metric on the moduli space smooth across these loci, which required the cancellation of these discontinuities [36], [35].\",\n \"Moreover, in [37] it was proven that the contact potential is also smooth, which indicates that the function $\\\\mathcal {G}$ must be smooth as well.\",\n \"In appendix we present an explicit proof of this fact based on the representation in terms of trees.\",\n \"Furthermore, in [23] it was shown that the discontinuities across `fake walls' cancel in the sum over flow trees as well.\",\n \"In fact, this cancellation is explicit in the representation of the partial tree index given by the recursive formula (REF ) where the signs responsible for such `fake discontinuities' do not arise at all.\",\n \"As a result, it remains to consider only the discontinuities of the third type corresponding to the moduli independent loci.\",\n \"It is straightforward to check that already for $n=2$ these discontinuities are indeed present and do spoil modularity of the theta series.\",\n \"For small $n$ one can explicitly evaluate the anomaly in Vignéras' equation.\",\n \"It is given by a series of terms proportional to $\\\\delta (\\\\tilde{{v}}_v,{x})$ .\",\n \"Note that no derivatives of delta functions or products of two delta functions arise despite the presence of the second derivative in Vignéras' operator.\",\n \"This is because each ${\\\\rm sgn}(\\\\tilde{{v}}_v,{x})$ from $\\\\Delta (T)$ is multiplied by $(\\\\tilde{{v}}_v,{x})$ from $\\\\kappa (T)$ in (REF ) and one gets a non-vanishing result only if the second order derivative operator acts on both factors.\",\n \"In particular, this implies that the anomaly is completely characterized by the action of $V_\\\\lambda $ on $\\\\Phi ^{\\\\,g}_{n}$ .\"\n ],\n [\n \"Modular completion\",\n \"Since the theta series $\\\\vartheta _{{p},{\\\\mu }}(\\\\Phi ^{{\\\\rm tot}}_{n})$ are not modular for $n\\\\ge 2$ , the analysis of the previous section implies that the generating function $h_{p,\\\\mu }$ of the MSW invariants is not modular either whenever the divisor $\\\\mathcal {D}=p^a\\\\gamma _a$ is the sum of $n\\\\ge 2$ irreducible divisors.\",\n \"However, its modular anomaly has a definite structure.\",\n \"In particular, in [14] it was shown that for $n=2$ , $h_{p,\\\\mu }$ must be a vector-valued mixed mock modular form, i.e.\",\n \"it has a non-holomorphic completion $\\\\widehat{h}_{p,\\\\mu }$ constructed in a specific way from a set of holomorphic modular forms and their Eichler integrals [30], [13].\",\n \"In this section we generalize this result for arbitrary $n$ , i.e.\",\n \"for any degree of reducibility of the divisor.\"\n ],\n [\n \"Completion of the generating function\",\n \"Let us recall the notations $\\\\check{\\\\gamma }=(p^a,q_a)$ and $Q_n$ from (REF ), and decompose the electric component $q_a$ using spectral flow as in (REF ).\",\n \"Then we define $\\\\widehat{h}_{p,\\\\mu }(\\\\tau )= h_{p,\\\\mu }(\\\\tau )+\\\\sum _{n=2}^\\\\infty \\\\sum _{\\\\sum _{i=1}^n \\\\check{\\\\gamma }_i=\\\\check{\\\\gamma }}R_n(\\\\lbrace \\\\check{\\\\gamma }_i\\\\rbrace ,\\\\tau _2)\\\\, e^{\\\\pi I\\\\tau Q_n(\\\\lbrace \\\\check{\\\\gamma }_i\\\\rbrace )}\\\\prod _{i=1}^n h_{p_i,\\\\mu _i}(\\\\tau ).$ We are looking for non-holomorphic functions $R_n$ , exponentially suppressed for large $\\\\tau _2$ , such that $\\\\widehat{h}_{p,\\\\mu }$ transforms as a modular form.\",\n \"The condition for this to be true can be found along the same lines as before: one needs to rewrite the expansion of the function $\\\\mathcal {G}$ as a series in $\\\\widehat{h}_{p,\\\\mu }$ and require that at each order the coefficient is given by a modular covariant theta series.\",\n \"For such a theta series decomposition to be possible however, it is important that $\\\\widehat{h}_{p,\\\\mu }$ be invariant under the spectral flow, which implies that the functions $R_n$ be independent of the spectral flow parameter $\\\\epsilon ^a$ in the decomposition (REF ) of the total charge $\\\\check{\\\\gamma }$ .As a result, this parameter can be fixed to zero so that the sum over the D2-brane charges $q_{i,a}$ is restricted to those which satisfy the constraint $\\\\sum _{i=1}^n q_{i,a}=\\\\mu _a+\\\\frac{1}{2} \\\\kappa _{abc}p^b p^c$ .\",\n \"This condition will be an important consistency requirement on our construction.\",\n \"Rather than inverting (REF ) and substituting the result into (REF ), we can consider the generating function of DT invariants $h^{\\\\rm DT}_{p,\\\\mu }$ and, as a first step, rewrite it as a series in $\\\\widehat{h}_{p,\\\\mu }$ .\",\n \"Denoting the coefficient of such expansion by $\\\\widehat{g}_n(\\\\lbrace \\\\check{\\\\gamma }_i\\\\rbrace ,z^a)$ (with $\\\\widehat{g}_1\\\\equiv 1$ ), we get $\\\\begin{split}h^{\\\\rm DT}_{p,q}(\\\\tau ,z^a)=&\\\\,\\\\sum _{\\\\sum _{i=1}^n \\\\check{\\\\gamma }_i=\\\\check{\\\\gamma }}\\\\widehat{g}_n(\\\\lbrace \\\\check{\\\\gamma }_i\\\\rbrace ;z^a, \\\\tau _2) \\\\,e^{\\\\pi I\\\\tau Q_n(\\\\lbrace \\\\check{\\\\gamma }_i\\\\rbrace )}\\\\prod _{i=1}^n \\\\widehat{h}_{p_i,\\\\mu _i}(\\\\tau ).\\\\end{split}$ Comparing with (REF ), we see that the coefficients $\\\\widehat{g}_n$ are a direct analogue of the tree index $g_{{\\\\rm tr},n}$ .In fact, they also depend on $z^a$ only through the stability parameters (REF ), so we shall often denote them by $\\\\widehat{g}_n(\\\\lbrace \\\\check{\\\\gamma }_i,c_i\\\\rbrace ;\\\\tau _2)$ .\",\n \"The expansion of $\\\\mathcal {G}$ in terms of $\\\\widehat{h}_{p,\\\\mu }$ is then simply obtained by replacing $g_{{\\\\rm tr},n}$ by $\\\\widehat{g}_n$ in (REF ), which affects the kernel of the corresponding theta series.\",\n \"Our first problem is to express these coefficients in terms of the functions $R_n$ .\",\n \"The result can be nicely formulated using so-called Schröder trees, which are rooted ordered trees such that all vertices (except the leaves) have at least 2 children (see Fig.\",\n \"REF ).\",\n \"Their vertices are decorated by charges in such way that the leaves have charges $\\\\gamma _i$ , whereas the charges at other vertices are defined by the inductive rule $\\\\gamma _v=\\\\sum _{v^{\\\\prime }\\\\in {\\\\rm Ch}(v)}\\\\gamma _{v^{\\\\prime }}$ where ${\\\\rm Ch}(v)$ is the set of children of vertex $v$ .\",\n \"Note that these trees should not be confused with flow trees, since they are not restricted to be binary nor do they carry moduli at the vertices.\",\n \"We denote the set of Schröder treesThe number of Schröder trees with $n$ leaves is the $n-1$ -th super-Catalan number, $|{T}_n^{\\\\rm S}|=\\\\lbrace 1,1, 3,11,45,197,903,\\\\dots \\\\rbrace $ for $n\\\\ge 1$ (sequence A001003 on the Online Encyclopedia of Integer Sequences).\",\n \"with $n$ leaves by ${T}_n^{\\\\rm S}$ .\",\n \"Figure: An example of Schröder tree contributing to W 8 W_8.\",\n \"Near each vertex we showed the corresponding factorusing the shorthand notation γ i+j =γ i +γ j \\\\gamma _{i+j}=\\\\gamma _i+\\\\gamma _j.Let us also introduce a convenient notation: for any set of functions $f_n(\\\\lbrace \\\\check{\\\\gamma }_i\\\\rbrace )$ depending on $n$ charges and a given Schröder tree $T$ , we set $f_{v}\\\\equiv f_{k_v}(\\\\lbrace \\\\check{\\\\gamma }_{v^{\\\\prime }}\\\\rbrace )$ where $v^{\\\\prime }\\\\in {\\\\rm Ch}(v)$ and $k_v$ is their number.\",\n \"Using these notations, the expression of $\\\\widehat{g}_n$ in terms of $R_n$ can be encoded into a recursive equation provided by the following proposition, whose proof we relegate to appendix : Proposition 1 The coefficients $\\\\widehat{g}_n$ are determined recursively by the following equation $\\\\widehat{g}_n(\\\\lbrace \\\\check{\\\\gamma }_i,c_i\\\\rbrace )=-{1\\\\over 2}\\\\,{\\\\rm Sym}\\\\, \\\\left\\\\lbrace \\\\sum _{\\\\ell =1}^{n-1}\\\\Delta _{\\\\gamma _L^\\\\ell \\\\gamma _R^\\\\ell }^z\\\\,\\\\kappa (\\\\gamma _{LR}^\\\\ell )\\\\,\\\\widehat{g}_\\\\ell (\\\\lbrace \\\\check{\\\\gamma }_i,c_i^{(\\\\ell )}\\\\rbrace _{i=1}^\\\\ell )\\\\,\\\\widehat{g}_{n-\\\\ell }(\\\\lbrace \\\\check{\\\\gamma }_i,c_i^{(\\\\ell )}\\\\rbrace _{i=\\\\ell +1}^n)\\\\right\\\\rbrace +W_n(\\\\lbrace \\\\check{\\\\gamma }_i\\\\rbrace ),$ where $\\\\gamma _L^\\\\ell =\\\\sum _{i=1}^\\\\ell \\\\gamma _i$ , $\\\\gamma _R^\\\\ell =\\\\sum _{i=\\\\ell +1}^n\\\\gamma _i$ , $\\\\gamma _{LR}^\\\\ell =\\\\langle \\\\gamma _L^\\\\ell ,\\\\gamma _R^\\\\ell \\\\rangle $ , $c_i^{(\\\\ell )}$ are the stability parameters at the point where the attractor flow crosses the wall for the decay $\\\\gamma \\\\rightarrow \\\\gamma _L^\\\\ell +\\\\gamma _R^\\\\ell $ (cf.\",\n \"(REF )), and $W_n$ are functions given by the sum over Schröder trees $W_n(\\\\lbrace \\\\check{\\\\gamma }_i\\\\rbrace )= \\\\,{\\\\rm Sym}\\\\, \\\\left\\\\lbrace \\\\sum _{T\\\\in {T}_n^{\\\\rm S}}(-1)^{n_T}\\\\prod _{v\\\\in V_T} R_{v}\\\\right\\\\rbrace ,$ with $n_T$ being the number of vertices of the tree $T$ (excluding the leaves).\",\n \"The same functions $W_n$ also provide the inverse formula to (REF ), namely $h_{p,\\\\mu }(\\\\tau )= \\\\widehat{h}_{p,\\\\mu }(\\\\tau )+\\\\sum _{n=2}^\\\\infty \\\\sum _{\\\\sum _{i=1}^n \\\\check{\\\\gamma }_i=\\\\check{\\\\gamma }}W_n(\\\\lbrace \\\\check{\\\\gamma }_i\\\\rbrace )\\\\, e^{\\\\pi I\\\\tau Q_n(\\\\lbrace \\\\check{\\\\gamma }_i\\\\rbrace )}\\\\prod _{i=1}^n \\\\widehat{h}_{p_i,\\\\mu _i}(\\\\tau ).$ What are the conditions on $\\\\widehat{g}_n$ for the corresponding theta series to be modular?\",\n \"Let us denote by $\\\\Phi ^{\\\\,\\\\widehat{g}}_{n}$ the kernel defined by $\\\\widehat{g}_n$ analogously to (REF ), namely $\\\\Phi ^{\\\\,\\\\widehat{g}}_{n}({x})=\\\\frac{\\\\sigma _{\\\\gamma }(\\\\sqrt{2\\\\tau _2})^{n-1}}{\\\\prod _{i=1}^n\\\\sigma _{\\\\gamma _i}}\\\\,\\\\widehat{g}_n(\\\\lbrace \\\\gamma _i,c_i\\\\rbrace ).$ Then the above analysis implies that the modularity requires $\\\\Phi ^{\\\\,\\\\widehat{g}}_{n}$ to satisfy Vignéras equation away from walls of marginal stability, whereas at these walls its discontinuities should follow the same pattern as before in order to cancel the discontinuities from the contour exchange in $\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_m$ .\",\n \"Thus, the completion should smoothen out the discontinuities from the moduli independent signs ${\\\\rm sgn}(\\\\tilde{{v}}_v,{x})$ , but otherwise leave the action of Vignéras' operator unchanged.\",\n \"Formally, this means that $\\\\Phi ^{\\\\,\\\\widehat{g}}_{n}$ must satisfy the following equation $V_{n-1} \\\\cdot \\\\Phi ^{\\\\,\\\\widehat{g}}_{n}=\\\\,{\\\\rm Sym}\\\\, \\\\sum _{\\\\ell =1}^{n-1}\\\\Bigl [{u}_\\\\ell ^2\\\\,\\\\Delta _{n,\\\\ell }^{\\\\widehat{g}} \\\\,\\\\delta ^{\\\\prime }({u}_\\\\ell \\\\cdot {x})+ 2{u}_\\\\ell \\\\cdot \\\\partial _{x}\\\\Delta _{n,\\\\ell }^{\\\\widehat{g}} \\\\,\\\\delta ({u}_\\\\ell \\\\cdot {x})\\\\Bigr ],\\\\qquad \\\\Delta _{n,\\\\ell }^{\\\\widehat{g}}=\\\\frac{1}{2}\\\\,({v}_\\\\ell ,{x})\\\\,\\\\Phi ^{\\\\,\\\\widehat{g}}_\\\\ell \\\\,\\\\Phi ^{\\\\,\\\\widehat{g}}_{n-\\\\ell },$ where we introduced two sets of vectors constructed from the vectors (REF ), ${v}_\\\\ell =\\\\sum _{i=1}^\\\\ell \\\\sum _{j=\\\\ell +1}^n{v}_{ij}\\\\,,\\\\qquad {u}_\\\\ell =\\\\sum _{i=1}^\\\\ell \\\\sum _{j=\\\\ell +1}^n{u}_{ij}\\\\,.$ Note that $({v}_\\\\ell ,{x})$ and $({u}_\\\\ell ,{x})$ correspond to the quantities $-\\\\Gamma _{n\\\\ell }$ and $-S_\\\\ell $ (REF ), respectively.\",\n \"To solve the above constraints, let us consider the following iterative ansatz $\\\\widehat{g}_n(\\\\lbrace \\\\check{\\\\gamma }_i,c_i\\\\rbrace )=g^{(0)}_n(\\\\lbrace \\\\check{\\\\gamma }_i,c_i\\\\rbrace )-\\\\,{\\\\rm Sym}\\\\, \\\\left\\\\lbrace \\\\sum _{n_1+\\\\cdots +n_m= n\\\\atop n_k\\\\ge 1, \\\\ m1}W_m(\\\\lbrace \\\\check{\\\\gamma }^{\\\\prime }_k\\\\rbrace )\\\\prod _{k=1}^m \\\\mathcal {E}_{n_k}(\\\\check{\\\\gamma }_{j_{k-1}+1},\\\\dots ,\\\\check{\\\\gamma }_{j_{k}})\\\\right\\\\rbrace .$ Furthermore, provided the functions $\\\\mathcal {E}_n$ depend on electric charges $q_{i,a}$ only through the DSZ products $\\\\gamma _{ij}$ and their kernels $\\\\Phi ^{\\\\,\\\\mathcal {E}}_n({x})$ defined as in (REF ) are smooth solutions of Vignéras' equation, $V_{n-1}\\\\cdot \\\\Phi ^{\\\\,\\\\mathcal {E}}_n=0,$ then the ansatz (REF ) also satisfies the modularity constraint (REF ).\",\n \"This proposition allows in principle to fix all unknown functions.\",\n \"Indeed, the recursive relation (REF ) determines all $g^{(0)}_n$ .\",\n \"Then equations (REF ) and (REF ) give the two parts of $\\\\mathcal {E}_n=\\\\mathcal {E}^{(0)}_n+\\\\mathcal {E}^{(+)}_n$ in terms of $g^{(0)}_n$ and $W_n$ .\",\n \"At this point the latter are still undetermined and are defined in terms of the unknown functions $R_n$ .\",\n \"The additional constraint that $\\\\mathcal {E}_n$ should satisfy Vignéras' equation links together $\\\\mathcal {E}^{(0)}_n$ and $\\\\mathcal {E}^{(+)}_n$ and thus establishes a relation between $W_n$ and $g^{(0)}_n$ .\",\n \"Lastly, inverting (REF ) generates a solution for $R_n$ .\",\n \"We end this discussion by making an observation which will become relevant in the next subsection: by comparing (REF ) and (REF ), it is clear that $\\\\widehat{g}_n$ must agree with the tree index $g_{{\\\\rm tr},n}$ in the limit $\\\\tau _2\\\\rightarrow \\\\infty $ .\",\n \"In particular, $g_{{\\\\rm tr},n}$ satisfies a similar ansatz as (REF ), upon replacing $\\\\mathcal {E}_n$ by its non-decaying part: $g_{{\\\\rm tr},n}(\\\\lbrace \\\\check{\\\\gamma }_i,c_i\\\\rbrace )=g^{(0)}_n(\\\\lbrace \\\\check{\\\\gamma }_i,c_i\\\\rbrace )-\\\\,{\\\\rm Sym}\\\\, \\\\left\\\\lbrace \\\\sum _{n_1+\\\\cdots +n_m= n\\\\atop n_k\\\\ge 1, \\\\ m1$ .\",\n \"In this case more than one pair of joint edges collapse.\",\n \"The representation (REF ) in terms of rooted ternary trees is obtained by collapsing $m$ pairs of edges successively, where the coefficient $1/T!$ takes into account that such procedure leads to an overcounting of different assignments of labels.\",\n \"Figure: Three unrooted trees constructed from the same three subtrees.Unfortunately, the function (REF ) cannot yet be taken as an ansatz for $g^{(0)}_n$ because it depends non-trivially on the modulus $\\\\tau _2$ , which is in tension with the observation made at the end of the previous subsection.\",\n \"Therefore, we shall modify the function (REF ) into a function which is still a smooth solution of Vignéras' equation, but whose large ${x}$ limit is independent of $\\\\tau _2$ .\",\n \"Taking cue from the structure of (REF ), we defineThe term $m=0$ in (REF ) reduces to the original function (REF ), while the terms with $m>0$ are the afore-mentioned modification.\",\n \"$\\\\widetilde{\\\\Phi }^{\\\\,\\\\mathcal {E}}_n({x})=\\\\frac{1}{2^{n-1} n!\",\n \"}\\\\sum _{m=0}^{[(n-1)/2]}\\\\sum _{\\\\mathcal {T}\\\\in \\\\, {T}_{n-2m,m}^\\\\ell }\\\\left[\\\\prod _{\\\\mathfrak {v}\\\\in V_\\\\mathcal {T}}\\\\mathcal {D}_{m_\\\\mathfrak {v}}(\\\\lbrace \\\\check{\\\\gamma }_{\\\\mathfrak {v},s}\\\\rbrace )\\\\right]\\\\widetilde{\\\\Phi }^E_{n-2m-1}(\\\\lbrace {u}_e\\\\rbrace , \\\\lbrace {v}_{s(e) t(e)}\\\\rbrace ;{x}),$ where $\\\\mathcal {D}_{m}(\\\\lbrace \\\\check{\\\\gamma }_s\\\\rbrace )=\\\\sum _{\\\\mathcal {T}^{\\\\prime }\\\\in \\\\, {T}_{2m+1}^\\\\ell } a_{\\\\mathcal {T}^{\\\\prime }}\\\\prod _{e\\\\in E_{\\\\mathcal {T}^{\\\\prime }}}\\\\mathcal {D}({v}_{s(e)t(v)}).$ The coefficients $a_\\\\mathcal {T}$ are rational numbers which depend (up to a sign determined by the orientation of edges) only on the topology of the tree.\",\n \"We fix them by requiring that they satisfy the following system of equations $a_{\\\\hat{\\\\mathcal {T}}_1}+a_{\\\\hat{\\\\mathcal {T}}_2}-a_{\\\\hat{\\\\mathcal {T}}_3}=a_{\\\\mathcal {T}_1}a_{\\\\mathcal {T}_2}a_{\\\\mathcal {T}_3},$ where $\\\\mathcal {T}_r$ ($r=1,2,3$ ) are arbitrary unrooted trees with marked vertex $\\\\mathfrak {v}_r$ and $\\\\hat{\\\\mathcal {T}}_r$ are the three trees constructed from $\\\\mathcal {T}_r$ as shown in Fig.\",\n \"REF .For these equations, it is important to take into account the orientation of the edges: a change of orientation of an edge flips the sign of $a_\\\\mathcal {T}$ .\",\n \"The equations (REF ) are written assuming the orientation shown in Fig.\",\n \"REF , namely $e_1=(\\\\mathfrak {v}_2,\\\\mathfrak {v}_3)$ , $e_2=(\\\\mathfrak {v}_1,\\\\mathfrak {v}_3)$ , $e_3=(\\\\mathfrak {v}_1,\\\\mathfrak {v}_2)$ .\",\n \"This system of equations is imposed in order to ensure certain properties of the operators (REF ) which are crucial for the cancellation of $\\\\tau _2$ -dependent terms in the asymptotics of $\\\\widetilde{\\\\Phi }^{\\\\,\\\\mathcal {E}}_n$ (see (REF )).\",\n \"It is readily seen that the system (REF ) fixes all coefficients $a_\\\\mathcal {T}$ recursively starting from $a_\\\\bullet =1$ , $a_{\\\\bullet \\\\!\\\\mbox{-}\\\\!\\\\bullet }=0$ and going to trees with $n\\\\ge 3$ vertices.\",\n \"(Starting from $n=7$ , the system (REF ) is overdetermined, but it can be checked that it does have a unique solution, with $a_\\\\mathcal {T}=0$ for trees with even number of vertices.)\",\n \"For a generic tree, in appendix we prove the following Proposition 4 For a tree $\\\\mathcal {T}$ with $n$ vertices the coefficient $a_\\\\mathcal {T}$ is given recursively by $a_\\\\mathcal {T}=\\\\frac{1}{n}\\\\sum _{\\\\mathfrak {v}\\\\in V_\\\\mathcal {T}} \\\\epsilon _\\\\mathfrak {v}\\\\prod _{s=1}^{n_\\\\mathfrak {v}} a_{\\\\mathcal {T}_s(\\\\mathfrak {v})},$ where $n_\\\\mathfrak {v}$ is the valency of the vertex $\\\\mathfrak {v}$ , $\\\\mathcal {T}_s(\\\\mathfrak {v})$ are the trees obtained from $\\\\mathcal {T}$ by removing this vertex, and $\\\\epsilon _\\\\mathfrak {v}$ is the sign determined by the choice of orientation of edges, $\\\\epsilon _\\\\mathfrak {v}=(-1)^{n_\\\\mathfrak {v}^+}$ with $n_\\\\mathfrak {v}^+$ being the number of incoming edges at the vertex.\",\n \"(See Fig.\",\n \"REF for an example.)\",\n \"Figure: An example of calculation of the coefficient a 𝒯 a_\\\\mathcal {T} for a tree with 7 vertices.We indicated in red the only vertices which produce non-vanishing contributions to the sum over vertices.For other vertices at least one of the trees 𝒯 s (𝔳)\\\\mathcal {T}_s(\\\\mathfrak {v}) has even number of vertices and hence vanishing coefficient.Let us now return to the function (REF ), which is now fully specified given the prescription (REF ) for the coefficients $a_\\\\mathcal {T}$ .\",\n \"Similarly to (REF ) (which coincides with the $m=0$ contribution in (REF )), it is a solution of Vignéras' equation for $\\\\lambda =n-1$ .\",\n \"We claim that in the large ${x}$ asymptotics of this function, all $\\\\tau _2$ -dependent contributions, coming from the mutual action of derivative operators $\\\\mathcal {D}$ , cancel.\",\n \"More precisely, the asymptotics is given by the following Proposition 5 $\\\\lim _{{x}\\\\rightarrow \\\\infty }\\\\widetilde{\\\\Phi }^{\\\\,\\\\mathcal {E}}_n({x})=\\\\frac{1}{2^{n-1} n!\",\n \"}\\\\sum _{m=0}^{[(n-1)/2]}\\\\sum _{\\\\mathcal {T}\\\\in \\\\, {T}_{n-2m,m}^\\\\ell }\\\\prod _{\\\\mathfrak {v}\\\\in V_\\\\mathcal {T}}\\\\mathcal {V}_{m_\\\\mathfrak {v}}(\\\\lbrace \\\\check{\\\\gamma }_{\\\\mathfrak {v},s}\\\\rbrace )\\\\prod _{e\\\\in E_{\\\\mathcal {T}}}({v}_{s(e) t(e)},{x})\\\\,{\\\\rm sgn}({u}_e,{x}),$ where $\\\\mathcal {V}_{m}(\\\\lbrace \\\\check{\\\\gamma }_{s}\\\\rbrace )=\\\\sum _{\\\\mathcal {T}\\\\in \\\\, {T}_{2m+1}^\\\\ell } a_\\\\mathcal {T}\\\\prod _{e\\\\in E_\\\\mathcal {T}}({v}_{s(e)t(v)},{x}).$ Importantly the function (REF ) is locally a homogeneous polynomial of degree $n-1$ .\",\n \"Therefore, when written in terms of charges, the $\\\\tau _2$ -dependence factorizes and is cancelled after the same rescaling as in (REF ).\",\n \"This naturally leads us to the following ansatz for the function $g^{(0)}_n$ , which we prove in appendix : Proposition 6 The function $g^{(0)}_n(\\\\lbrace \\\\check{\\\\gamma }_i,c_i\\\\rbrace )=\\\\frac{(-1)^{n-1+\\\\sum _{i0$ for $i=1,\\\\dots ,n-1$ and $c_n=-\\\\sum _{i=1}^{n-1} c_i<0$ , one finds numerically that all coefficients $b_n$ with $n$ even vanish,This follows from the fact that under the permutation $c_i\\\\mapsto c_{n-i+1}$ one has $S_i\\\\mapsto -S_{n-i}$ and therefore $F^{\\\\rm (ref)}_n$ flips sign.\",\n \"The deeper reason for this is that the terms which are products of even and odd number of signs cannot mix.\",\n \"In contrast, sign identities such as (REF ) can decrease the number of signs in a product by even number.\",\n \"Thus, a constant can appear in $F^{\\\\rm (ref)}_n$ only for $n$ odd.\",\n \"The same fact ensures the vanishing of the coefficients $a_\\\\mathcal {T}$ in (REF ) for trees with even number of vertices.\",\n \"while those with $n$ odd are given by $b_1=1,\\\\quad b_3=-\\\\frac{1}{3},\\\\quad b_5=\\\\frac{2}{15},\\\\quad b_7=-\\\\frac{17}{315},\\\\quad b_9=\\\\frac{62}{2835},\\\\quad b_{11}=-\\\\frac{1382}{155925},\\\\quad \\\\dots $ We observe that these coefficients coincide with the first coefficients in the Taylor series of $\\\\tanh x = x - \\\\frac{1}{3} x^3+ \\\\frac{2}{15} x^5 +\\\\dots $ .\",\n \"We therefore conjecture that this identification continues to hold in general, so that $b_n$ is expressed in terms of the Bernoulli number $B_{n+1}$ through $b_{n-1}=\\\\frac{2^{n} (2^{n} - 1)}{n!\",\n \"}\\\\, B_{n} .$ Using the iterative equation (REF ), the constraint (REF ) and proceeding by induction, it is easy to see that $g^{\\\\rm (ref)}_n$ defined by (REF ) and its first $n-2$ derivatives with respect to $y$ vanish at $y=1$ .\",\n \"This ensures that $g^{\\\\rm (ref)}_n$ is smooth and its limit $y\\\\rightarrow 1$ is well-defined.\",\n \"Conjecture 3 The function $g^{\\\\rm (ref)}_n$ , defined by (REF ), (REF ) and (REF ), reproduces the function $g^{(0)}_n$ (REF ) in the unrefined limit, $\\\\lim _{y\\\\rightarrow 1}g^{\\\\rm (ref)}_n(\\\\lbrace \\\\check{\\\\gamma }_i,c_i\\\\rbrace ,y)=g^{(0)}_n(\\\\lbrace \\\\check{\\\\gamma }_i,c_i\\\\rbrace ).$ We have checked this conjecture numerically up to $n=6$ evaluating the limit $y\\\\rightarrow 1$ using l'Hôpital's rule.\",\n \"Unfortunately, the new representation of $g^{(0)}_n$ obtained in this way is not helpful for the purposes of this work because it does not lead to any new representation for the completion.\",\n \"Nevertheless, the comparison of (REF ) and (REF ) shows that the sum over partitions and the simple form of products of sign functions characterizing the representation discussed in the previous subsection find their origin in the formula for $F^{\\\\rm (ref)}_n$ .\",\n \"Furthermore, this construction strongly suggests that the refinement may be compatible with S-duality, and that a suitably defined generating function of refined DT invariants may also possess interesting modular properties, which can even be simpler than those of the usual DT invariants.\"\n ],\n [\n \"Discussion\",\n \"In this paper we studied the modular properties of the generating function $h_{p,\\\\mu }(\\\\tau )$ of MSW invariants encoding BPS degeneracies of D4-D2-D0 black holes in Type IIA string theory on a Calabi-Yau threefold, with fixed D4-charge $p^a$ , D2-brane charge (up to spectral flow) $\\\\mu _a$ and invariant D0-brane charge $\\\\hat{q}_0$ (defined in (REF )) conjugate to the modular parameter $\\\\tau $ .\",\n \"These properties follow from demanding that the vector multiplet moduli space in $D=3$ (or the hypermultiplet moduli space in the dual type IIB picture) admits an isometric action of $SL(2,{Z})$ .\",\n \"Our main result is an explicit formula for the non-holomorphic modular completion $\\\\widehat{h}_{p,\\\\mu }$ (REF ), where the functions $R_n(\\\\lbrace \\\\gamma _i\\\\rbrace ,\\\\tau _2)$ are given by Eq.\",\n \"(REF ), with $\\\\mathcal {E}_v=\\\\mathcal {E}_v^{(0)}+\\\\mathcal {E}_v^{(+)}$ specified by (REF ) and (REF ).\",\n \"This result applies for D4-branes wrapping a general effective divisor which may be the sum of an arbitrary number $n$ of irreducible divisors.\",\n \"The existence of such completion was not guaranteed.\",\n \"We arrived at this result by cancelling the modular anomaly of an indefinite theta series of signature $(n, n b_2(\\\\mathfrak {Y})-n)$ with a complicated kernel (REF ).\",\n \"In particular, Vignéras' equation encoding this anomaly involves Kähler moduli in a non-trivial way through the walls of marginal stability, whereas the functions $R_n$ in our ansatz (REF ) can only depend (non-holomorphically) on the modular parameter $\\\\tau $ and on the charges $\\\\gamma _i$ .\",\n \"Note also that the transformation (REF ) of the contact potential, which was the starting point of our analysis, is only a necessary condition for the existence of an isometric action of $SL(2,{Z})$ on $\\\\mathcal {M}_H$ .\",\n \"To demonstrate that this condition is sufficient, one would have to construct suitable complex Darboux coordinates on the twistor space such that, after expressing them in terms of the completed generating function $\\\\widehat{h}_{p,\\\\mu }$ found in this paper, $SL(2,{Z})$ acts on these coordinates by a complex contact transformation.\",\n \"Such coordinates were constructed at the two-instanton level in [26], and it is an interesting challenge to extend this construction to arbitrary $n$ .\",\n \"The structure of the completion suggests that, for a divisor decomposable into a sum of $n$ effective divisors, the holomorphic generating function $h_{p,\\\\mu }$ is a mixed vector valued mock modular form of higher depth, equal to $n-1$ .\",\n \"Such objects have recently appeared in various mathematical and physical contexts [39], [64], [65], [66], [31] and correspond to holomorphic functions whose completion is constructed from period (or Eichler) integrals of mock modular forms of smaller depth (with depth 0 mock modular forms being synonymous with ordinary modular forms).\",\n \"For instance, in the case of standard mock modular forms of depth one, the completion is given by an Eichler integral of a modular function [27].\",\n \"In our case this structure follows from the fact that the completion $\\\\widehat{h}_{p,\\\\mu }$ is built from the generalized error functions which have a representation in terms of iterated period integrals [28], [39].\",\n \"In order to make manifest the mixed mock modular nature of $h_{p,\\\\mu }$ , one should represent the antiholomorphic derivative of its completion $\\\\widehat{h}_{p,\\\\mu }$ , computed in (REF ), as a sum of products of completions of mixed mock modular forms of lower depth and anti-holomorphic modular forms.\",\n \"This is a non-trivial task which involves the technique of lattice factorization for indefinite theta series, which we leave for future research.\",\n \"An important caveat in our construction is that we did not establish the convergence of the various generating functions and indefinite theta series.\",\n \"Technically, we constructed a theta series $\\\\vartheta _{{p},{\\\\mu }}\\\\bigl (\\\\widehat{\\\\Phi }^{{\\\\rm tot}}_{n},n-2\\\\bigr )$ whose kernel satisfies Vignéras' equation (REF ), but we did not demonstrate that it decays as required by Vignéras' criterium.\",\n \"(For $n=2$ this is easy to show [14] since the kernel is a product of an exponentially suppressed factor and a difference of two error functions defined by two positive vectors.)\",\n \"In fact, convergence issues already arise when expressing the generating function of DT invariants in terms of MSW invariants in (REF ), and are related to the problem of convergence of the BPS black hole partition function.\",\n \"The latter was proven to converge in the large volume limit for the case involving up to $n\\\\le 3$ centers in [50].\",\n \"We hope that the results for the tree index obtained in [23] will allow to extend this result to arbitrary $n$ .\",\n \"We note that many of the complications in our construction originate from the occurrence of polynomial factors in the kernel of theta series, which in turn can be traced to the factors of DSZ products $\\\\gamma _{ij}$ in the wall-crossing formula.\",\n \"Presumably, most of these complications would disappear if one could find a generalization of the contact potential involving the refined DT invariants, so that all these prefactors can be traded for explicit powers of the fugacity $y$ , as in section REF .\",\n \"It is possible that twistorial topological strings [67] may provide the appropriate framework for finding such a refinement, at least in the context of string vacua on non-compact Calabi-Yau threefolds.\",\n \"On the physics side, we expect that our results will have important implications for the physics of BPS black holes.\",\n \"Indeed, the mock modular property of the generating function $h_{p,\\\\mu }$ will affect the growth of its Fourier coefficients [68], and should be taken into account when performing precision tests of the microscopic origin of the Bekenstein-Hawking entropy.\",\n \"A natural question is to understand the origin of the non-holomorphic correction terms $R_n$ , in terms of the quantum mechanics of $n$ -centered black holes [37], [69].\",\n \"Another outstanding question is to understand the physical significance of the instanton generating function $\\\\mathcal {G}$ defined in (REF ).\",\n \"Its modular properties are exactly those expected from the elliptic genus of a superconformal field theory, except for the fact that it is not holomorphic in $\\\\tau $ .\",\n \"It is natural to expect that this non-holomorphy can be traced to the existence of a continuum of states with a non-trivial spectral asymmetry between bosons and fermions [70].\",\n \"It would be very interesting to understand the origin of this continuum in terms of the worldvolume theory of an M5-brane wrapped on a reducible divisor.\",\n \"Moreover, for special $\\\\mathcal {N}=2$ theories in $D=4$ obtained by circle compactification of an $\\\\mathcal {N}=1$ theory in $D=5$ , $\\\\mathcal {G}$ is closely related to the index considered in [37].\",\n \"It would be interesting to understand the physical origin of its modular invariance in this context.\",\n \"Finally, we expect that the structure found in the context of generalized DT-invariants on Calabi-Yau threefolds will also arise in the study of other types of BPS invariants where higher depth mock modular forms are expected to occur, such as Vafa-Witten invariants and Donaldson invariants of four-folds with $b_2^+=1$ [28], [64].\",\n \"In particular, it would be interesting to determine the modular completion of the generating function of Vafa–Witten invariants on the complex projective plane computed for all ranks in [71], generalizing the rank 3 case studied in [64], and compare with the completion found in the present paper.\",\n \"The authors are grateful to Sibasish Banerjee and Jan Manschot for useful discussions and collaboration on [14], [28], [26] which paved the way for the present work, to Karen Yeats for useful communication about the combinatorics of rooted trees, and to Don Zagier for providing a proof of the conjecture (REF ).\",\n \"The hospitality and financial support of the Theoretical Physics Department of CERN, where this work was initiated, is also gratefully acknowledged.\",\n \"The research of BP is supported in part by French state funds managed by the Agence Nationale de la Recherche (ANR) in the context of the LABEX ILP (ANR-11-IDEX-0004-02, ANR-10- LABX-63).\",\n \"toc\"\n ],\n [\n \"Proof of the Theorem\",\n \"In this appendix we prove Theorem REF presented in the Introduction.\",\n \"The crucial ingredient is provided by the following LemmaWe were informed that this lemma appears as an exercise in [72].\",\n \"Lemma 1 The number of ways of labelling the vertices of a rooted ordered tree $T$ with increasing labels is given by $N_T=n_T!\",\n \"\\\\prod _{v\\\\in V_T}\\\\frac{1}{n_v(T)}\\\\, ,$ where $n_T$ is the number of vertices of the tree $T$ .\",\n \"First, let us find a recursive relation between the numbers $N_T$ .\",\n \"Namely, consider a vertex $v$ and the set of its children ${\\\\rm Ch}(v)$ .\",\n \"Denote by $T_v$ the tree rooted at $v$ and with leaves coinciding with leaves of $T$ , for which one has $n_{T_v}=n_v$ .\",\n \"Then it is clear that $N_{T_v}=\\\\frac{\\\\left(\\\\sum _{v^{\\\\prime }\\\\in {\\\\rm Ch}(v)}n_{v^{\\\\prime }}\\\\right)!\",\n \"}{\\\\prod _{v^{\\\\prime }\\\\in {\\\\rm Ch}(v)}n_{v^{\\\\prime }}!\",\n \"}\\\\prod _{v^{\\\\prime }\\\\in {\\\\rm Ch}(v)}N_{T_{v^{\\\\prime }}}\\\\, .$ From this relation, one easily shows by recursion that (REF ) follows.\",\n \"Indeed, assuming that it holds for the subtrees $T_{v^{\\\\prime }}$ and taking into account that $\\\\sum _{v^{\\\\prime }\\\\in {\\\\rm Ch}(v)}n_{v^{\\\\prime }}=n_v-1$ , the r.h.s.\",\n \"can be rewritten as $(n_v-1)!\\\\prod _{v^{\\\\prime }\\\\in {\\\\rm Ch}(v)}\\\\prod _{v^{\\\\prime \\\\prime }\\\\in V_{T_{v^{\\\\prime }}}}\\\\frac{1}{n_{v^{\\\\prime \\\\prime }}}=n_v!\\\\prod _{v^{\\\\prime }\\\\in V_{T_v}}\\\\frac{1}{n_{v^{\\\\prime }}},$ which coincides with (REF ) when $v$ is the root of $T$ .\",\n \"Given this Lemma, we can now rewrite the l.h.s.\",\n \"of (REF ) as $\\\\sum _{T^{\\\\prime }\\\\subset T}\\\\frac{N_{T^{\\\\prime }}}{m!\",\n \"}\\\\prod _{v\\\\in V_{T^{\\\\prime }}}n_v(T)=\\\\frac{n!\",\n \"}{N_{T}}\\\\sum _{T^{\\\\prime }\\\\subset T}\\\\frac{N_{T^{\\\\prime }}}{m!\",\n \"}\\\\prod _{T_r\\\\subset T\\\\setminus T^{\\\\prime }}\\\\frac{N_{T_r}}{n_{T_r}!\",\n \"},$ where the last product goes over the subtrees which complement $T^{\\\\prime }$ to the full tree $T$ .\",\n \"On the other hand, it is easy to check that the relation (REF ) is the special case $m=1$ of a more general relation $N_{T}=\\\\frac{\\\\left(\\\\sum _{T_r\\\\subset T\\\\setminus T^{\\\\prime }}n_{T_r}\\\\right)!\",\n \"}{\\\\prod _{T_r\\\\subset T\\\\setminus T^{\\\\prime }}n_{T_r}!\",\n \"}\\\\sum _{T^{\\\\prime }\\\\subset T}N_{T^{\\\\prime }}\\\\prod _{T_r\\\\subset T\\\\setminus T^{\\\\prime }}N_{T_r}.$ where the sum runs over subtrees $T^{\\\\prime }$ with $m$ vertices.\",\n \"Taking into account that $\\\\sum _{T_r\\\\subset T\\\\setminus T^{\\\\prime }}n_{T_r}=n-m$ and substituting the resulting expression into (REF ), one recovers the binomial coefficient as stated in the Theorem.\",\n \"toc\"\n ],\n [\n \"D3-instanton contribution to the contact potential\",\n \"To evaluate $(e^\\\\phi )_{\\\\rm D3}$ , we first replace in (REF ) the full prepotential $F$ by its classical, cubic part $F^{\\\\rm cl}$ (REF ) and take into account that the sum over $\\\\gamma \\\\in -\\\\Gamma _+$ is complex conjugate to the sum over $\\\\gamma \\\\in \\\\Gamma _+$ .\",\n \"This gives $e^{\\\\phi } \\\\approx \\\\frac{\\\\tau _2^2}{12}\\\\,((\\\\,{\\\\rm Im}\\\\,u)^3)+\\\\frac{I\\\\tau _2}{8}\\\\,\\\\sum _{\\\\gamma \\\\in \\\\Gamma _+}\\\\,{\\\\rm Re}\\\\,\\\\int _{\\\\ell _\\\\gamma } \\\\frac{\\\\text{d}t}{t} \\\\left( t^{-1} Z_\\\\gamma (u^a)-t\\\\bar{Z}_\\\\gamma (\\\\bar{u}^a)\\\\right) H_\\\\gamma .$ Next, we substitute the quantum corrected mirror map (REF ), change the integration variable $t$ to $z$ , and take the combined limit $t^a\\\\rightarrow \\\\infty $ , $z\\\\rightarrow 0$ .\",\n \"Keeping only the leading contributions, one obtains $(e^\\\\phi )_{\\\\rm D3} &=&-\\\\frac{\\\\tau _2}{2}\\\\sum _{\\\\gamma \\\\in \\\\Gamma _+}\\\\,{\\\\rm Re}\\\\,\\\\int _{\\\\ell _{\\\\gamma }}\\\\mathrm {d}z\\\\left[\\\\hat{q}_0+{1\\\\over 2}\\\\, (q+b)^2+2(pt^2)zz_\\\\gamma -\\\\frac{3}{2}\\\\, z^2(pt^2)\\\\right]H_{\\\\gamma }\\\\nonumber \\\\\\\\&&-\\\\frac{1}{4}\\\\sum _{\\\\gamma _1,\\\\gamma _2\\\\in \\\\Gamma _+}(tp_1p_2)\\\\left(\\\\,{\\\\rm Re}\\\\,\\\\int _{\\\\ell _{\\\\gamma _1}}\\\\!\\\\!\\\\mathrm {d}z_1\\\\, H_{\\\\gamma _1}\\\\right)\\\\left( \\\\,{\\\\rm Re}\\\\,\\\\int _{\\\\ell _{\\\\gamma _2}}\\\\!\\\\!\\\\mathrm {d}z_2\\\\, H_{\\\\gamma _2}\\\\right).$ To further simplify this expression, we note that $\\\\begin{split}0=&\\\\, \\\\frac{1}{4\\\\pi }\\\\sum _{\\\\gamma \\\\in \\\\Gamma _+}\\\\int _{\\\\ell _{\\\\gamma }}\\\\mathrm {d}z\\\\,\\\\partial _z\\\\left( z\\\\,H_\\\\gamma \\\\right)\\\\\\\\=&\\\\,\\\\sum _{\\\\gamma \\\\in \\\\Gamma _+}\\\\int _{\\\\ell _{\\\\gamma }}\\\\mathrm {d}z \\\\left[\\\\frac{1}{4\\\\pi }+\\\\tau _2(pt^2) (zz_\\\\gamma - z^2)-\\\\frac{I}{4}\\\\sum _{\\\\gamma ^{\\\\prime }\\\\in \\\\Gamma _+}\\\\langle \\\\gamma ,\\\\gamma ^{\\\\prime }\\\\rangle \\\\int _{\\\\ell _{\\\\gamma ^{\\\\prime }}} \\\\frac{\\\\mathrm {d}z^{\\\\prime }}{z-z^{\\\\prime }}\\\\, H_{\\\\gamma ^{\\\\prime }}\\\\right]H_\\\\gamma ,\\\\end{split}$ where we used the integral equation (REF ).\",\n \"Multiplying this identity by 3/4 and adding its real part to (REF ), one finds $(e^\\\\phi )_{\\\\rm D3} &=&\\\\frac{\\\\tau _2}{2}\\\\sum _{\\\\gamma \\\\in \\\\Gamma _+}\\\\,{\\\\rm Re}\\\\,\\\\int _{\\\\ell _{\\\\gamma }}\\\\mathrm {d}z\\\\, a_{\\\\gamma ,-\\\\frac{3}{2}}(z)\\\\,H_{\\\\gamma }\\\\nonumber \\\\\\\\&&-\\\\frac{1}{4}\\\\sum _{\\\\gamma _1,\\\\gamma _2\\\\in \\\\Gamma _+}\\\\left[(tp_1p_2)\\\\left(\\\\,{\\\\rm Re}\\\\,\\\\int _{\\\\ell _{\\\\gamma _1}}\\\\!\\\\!\\\\mathrm {d}z_1\\\\, H_{\\\\gamma _1}\\\\right)\\\\left( \\\\,{\\\\rm Re}\\\\,\\\\int _{\\\\ell _{\\\\gamma _2}}\\\\!\\\\!\\\\mathrm {d}z_2\\\\, H_{\\\\gamma _2}\\\\right)\\\\right.\\\\\\\\&&\\\\left.\\\\qquad +\\\\frac{3}{4}\\\\,\\\\,{\\\\rm Re}\\\\,\\\\left(\\\\int _{\\\\ell _{\\\\gamma _1}}\\\\!\\\\!\\\\mathrm {d}z_1\\\\, H_{\\\\gamma _1}\\\\int _{\\\\ell _{\\\\gamma _2}}\\\\!\\\\!\\\\mathrm {d}z_2\\\\, H_{\\\\gamma _2} \\\\,\\\\frac{I\\\\langle \\\\gamma ,\\\\gamma ^{\\\\prime }\\\\rangle }{z-z^{\\\\prime }}\\\\right)\\\\right],\\\\nonumber $ where we introduced $a_{\\\\gamma ,\\\\mathfrak {h}}(z)=-\\\\left(\\\\hat{q}_0+{1\\\\over 2}\\\\,(q+b)^2+\\\\frac{1}{2}\\\\, (pt^2)zz_\\\\gamma +\\\\frac{\\\\mathfrak {h}}{4\\\\pi \\\\tau _2}\\\\right).$ The meaning of this function is actually very simple: it gives the action of the modular covariant derivative operator $\\\\mathcal {D}_{\\\\mathfrak {h}}$ (REF ) on the classical part of the Darboux coordinate (REF ), $\\\\mathcal {D}_{\\\\mathfrak {h}}\\\\mathcal {X}^{\\\\rm cl}_{\\\\gamma } =a_{\\\\gamma ,\\\\mathfrak {h}}(z)\\\\mathcal {X}^{\\\\rm cl}_{\\\\gamma }.$ Combining this fact with equation (REF ), it is easy to check that the function (REF ) satisfies, for any value of the weight $\\\\mathfrak {h}$ , $\\\\mathcal {D}_{\\\\mathfrak {h}}\\\\mathcal {G}&=& \\\\sum _{\\\\gamma \\\\in \\\\Gamma _+}\\\\int _{\\\\ell _{\\\\gamma }} \\\\mathrm {d}z\\\\,a_{\\\\gamma ,\\\\mathfrak {h}}(z)\\\\, H_{\\\\gamma }\\\\\\\\&&+\\\\frac{1}{8\\\\tau _2}\\\\sum _{\\\\gamma _1,\\\\gamma _2\\\\in \\\\Gamma _+}\\\\int _{\\\\ell _{\\\\gamma _1}}\\\\mathrm {d}z_1\\\\,\\\\int _{\\\\ell _{\\\\gamma _2}} \\\\mathrm {d}z_2\\\\left[(tp_1p_2)+2\\\\mathfrak {h}\\\\left((tp_1p_2)+\\\\frac{I\\\\langle \\\\gamma _1,\\\\gamma _2\\\\rangle }{z_1-z_2} \\\\right)\\\\right]H_{\\\\gamma _1}(z_1)H_{\\\\gamma _2}(z_2),\\\\nonumber \\\\\\\\\\\\partial _{\\\\tilde{c}_a}\\\\mathcal {G}&=& 2\\\\pi I\\\\sum _{\\\\gamma \\\\in \\\\Gamma _+}p^a \\\\int _{\\\\ell _{\\\\gamma }} \\\\mathrm {d}z\\\\, H_{\\\\gamma },$ which allows to rewrite (REF ) exactly as in (REF ).\",\n \"toc\"\n ],\n [\n \"Smoothness of the instanton generating function\",\n \"In this appendix we prove the smoothness of the function $\\\\mathcal {G}$ across walls of marginal stability.\",\n \"The starting point is the representation (REF ) where the potential discontinuities are hidden in the kernel of the theta series (REF ).\",\n \"Given that $\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_{m}$ is represented as a sum over unrooted labelled trees (REF ), whereas $\\\\Phi ^{\\\\,g}_{n_k}$ appear as sums over flow trees (REF ), the kernel $\\\\Phi ^{{\\\\rm tot}}_{n}$ can be viewed as a sum over `blooming trees' which are unrooted trees with a flow tree (the `flower') growing from each vertex.\",\n \"Then the idea is that the discontinuities due to flow trees (i.e.\",\n \"due to DT invariants) of a blooming tree with $m$ vertices are cancelled by the discontinuities due to exchange of integration contours in $\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_{m+1}$ corresponding to blooming trees with $m+1$ vertices.\",\n \"Figure: Combination of trees showing cancellation of discontinuities across the wall of marginal stabilitycorresponding to the decay γ 𝔳 →γ L +γ R \\\\gamma _\\\\mathfrak {v}\\\\rightarrow \\\\gamma _L+\\\\gamma _R.\",\n \"The parts corresponding to attractor flow trees are drown in blue.Following this idea, let us consider a tree which has an attractor flow tree $T_\\\\mathfrak {v}$ growing from a vertex $\\\\mathfrak {v}$ (see Fig.\",\n \"REF ).\",\n \"We denote by $\\\\mathcal {T}_i$ , $i=1,\\\\dots ,n_\\\\mathfrak {v}$ , the blooming subtrees connected to this vertex and by $T_{L}$ and $T_{R}$ the two parts (which may be trivial) of $T_\\\\mathfrak {v}$ with the total charges $\\\\gamma _L$ and $\\\\gamma _R$ so that $\\\\gamma _\\\\mathfrak {v}=\\\\gamma _L+\\\\gamma _R$ .\",\n \"Together with the contribution of this tree, we consider the contributions of the trees obtained by splitting the vertex $\\\\mathfrak {v}$ into two vertices $\\\\mathfrak {v}_L$ and $\\\\mathfrak {v}_R$ connected by an edge, carrying charges $\\\\gamma _L$ and $\\\\gamma _R$ and all possible allocations of the subtrees $\\\\mathcal {T}_i$ to these two vertices.\",\n \"Different allocations are accounted for by the sum over permutations, whilethe weight $\\\\frac{1}{\\\\ell !\",\n \"(n_\\\\mathfrak {v}-\\\\ell )!\",\n \"}$ takes into account the fact that permutations between subtrees connected to one vertex are redundant.\",\n \"The attractor flow trees $T_{L}$ and $T_{R}$ are then connected to $\\\\mathfrak {v}_L$ and $\\\\mathfrak {v}_R$ , respectively, as shown in Fig.\",\n \"REF .\",\n \"The contribution corresponding to the first blooming tree has a discontinuity at the wall of marginal stability for the bound state $\\\\gamma _L+\\\\gamma _R$ and originating from the factor $\\\\Delta ^{z}_{\\\\gamma _L\\\\gamma _R}$ assigned to the root vertex of $T_\\\\mathfrak {v}$ .\",\n \"The other contributions have discontinuities at the same wall due to the exchange of the contours $\\\\ell _{\\\\gamma _L}$ and $\\\\ell _{\\\\gamma _R}$ for the integrals assigned to $\\\\mathfrak {v}_L$ and $\\\\mathfrak {v}_R$ , respectively.\",\n \"They are given by the residues at the pole of the integration kernel $K_{\\\\gamma _L\\\\gamma _R}$ .\",\n \"It is clear that the structure of all jumps is very similar since different subtrees produce essentially the same weights.\",\n \"Let us analyze what differences may arise.\",\n \"First, the contributions of the flow trees $T_{L}$ and $T_{R}$ could differ in the two cases because they have different starting points for the attractor flows: for the trees on the right side of Fig.\",\n \"REF this is $z^a\\\\in \\\\mathcal {M}_K$ , whereas for the tree on the left this is the point on the wall for $\\\\gamma _L+\\\\gamma _R$ reached by the flow from $z^a$ .\",\n \"But we are evaluating the discontinuity exactly on the wall where the two points coincide.\",\n \"Thus, the contributions are the same.\",\n \"Although the subtrees $\\\\mathcal {T}_i$ give rise to the same contributions for all blooming trees shown in Fig.\",\n \"REF , the contributions of the edges connecting them to either $\\\\mathfrak {v}$ or $\\\\mathfrak {v}_L$ , $\\\\mathfrak {v}_R$ are not exactly the same.\",\n \"Each of them contributes the factor given by the kernel (REF ).\",\n \"After taking the residue, the $z$ -dependence of these kernels for the trees shown on the left and the right sides of the picture is identical.\",\n \"However, their charge dependence is different: for the tree on the left they depend on $\\\\gamma _\\\\mathfrak {v}$ , whereas for the trees on the right they depend on $\\\\gamma _L$ or $\\\\gamma _R$ , depending on which vertex they are connected to.\",\n \"But it is easy to see that the sums over $\\\\ell $ and permutations produce the standard binomial expansion of a single product of kernels which all depend on $\\\\gamma _L+\\\\gamma _R=\\\\gamma _\\\\mathfrak {v}$ and thus coinciding with the contribution of the tree on the left.\",\n \"Finally, one should take into account that the discontinuity of $\\\\Delta ^{z}_{\\\\gamma _L\\\\gamma _R}$ in the first contribution gives the factor $\\\\langle \\\\gamma _L,\\\\gamma _R\\\\rangle $ .\",\n \"But exactly the same factor arises as the residue of $K_{\\\\gamma _L\\\\gamma _R}$ corresponding to the additional edge.\",\n \"Thus, it remains only to check that all numerical factors work out correctly.\",\n \"Leaving aside the factors which are common to both contributions, we have $-\\\\frac{(-1)^{\\\\langle \\\\gamma _L,\\\\gamma _R\\\\rangle }}{2}\\\\,\\\\frac{\\\\sigma _{\\\\gamma _\\\\mathfrak {v}}}{(2\\\\pi )^2}\\\\, \\\\frac{2m}{ m!\",\n \"}+(2\\\\pi I)(-2\\\\pi I)\\\\,\\\\frac{\\\\sigma _{\\\\gamma _L}\\\\sigma _{\\\\gamma _R}}{(2\\\\pi )^4}\\\\,\\\\frac{m(m+1)}{(m+1)!\",\n \"}.$ Here $-\\\\frac{(-1)^{\\\\langle \\\\gamma _L,\\\\gamma _R\\\\rangle }}{2}$ comes from the factor $-\\\\Delta ^{z}_{\\\\gamma _L\\\\gamma _R}$ in $T_\\\\mathfrak {v}$ , the factors with quadratic refinement are due to functions $H_\\\\gamma $ (REF ) assigned to $\\\\mathfrak {v}$ or $\\\\mathfrak {v}_L$ , $\\\\mathfrak {v}_R$ , $(2\\\\pi I)$ is the standard weight of the residue, $(-2\\\\pi I)$ is the residue of $K_{\\\\gamma _L\\\\gamma _R}$ (REF ), and factorials are the weights of the trees in the expansion (REF ).\",\n \"Finally, the factors $2m$ and $m(m+1)$ arise due to the freedom to relabel charges assigned to the marked vertices: on the left these are vertex $\\\\mathfrak {v}$ and the two children of the root in $T_\\\\mathfrak {v}$ , whereas on the right these are $\\\\mathfrak {v}_L$ and $\\\\mathfrak {v}_R$ .\",\n \"It is immediate to check that all these numerical weights cancel, which ensures that the function $\\\\mathcal {G}$ is continuous across walls of marginal stability.\",\n \"Moreover, in this cancellation the condition that we sit on the wall was used only in locally constant factors.\",\n \"Therefore, this reasoning proves not only that $\\\\mathcal {G}$ is continuous, but that it is actually smooth around these loci.\",\n \"toc\"\n ],\n [\n \"Vignéras' theorem\",\n \"Let $\\\\mathbf {\\\\Lambda }$ be a $d$ -dimensional lattice equipped with a bilinear form $({x},{y})\\\\equiv {x}\\\\cdot {y}$ , where ${x},{y}\\\\in \\\\mathbf {\\\\Lambda }\\\\otimes {R}$ , such that its associated quadratic form has signature $(n,d-n)$ and is integer valued, i.e.\",\n \"${k}^2\\\\equiv {k}\\\\cdot {k}\\\\in {Z}$ for ${k}\\\\in \\\\mathbf {\\\\Lambda }$ .\",\n \"Furthermore, let ${p}\\\\in \\\\mathbf {\\\\Lambda }$ be a characteristic vector (such that ${k}\\\\cdot ({k}+ {p})\\\\in 2{Z}$ , $\\\\forall \\\\,{k}\\\\in \\\\mathbf {\\\\Lambda }$ ), ${\\\\mu }\\\\in \\\\mathbf {\\\\Lambda }^*/\\\\mathbf {\\\\Lambda }$ a glue vector, and $\\\\lambda $ an arbitrary integer.\",\n \"We consider the following family of theta series $\\\\vartheta _{{p},{\\\\mu }}(\\\\Phi ,\\\\lambda ;\\\\tau , {b}, {c})=\\\\tau _2^{-\\\\lambda /2}\\\\!\\\\!\\\\!\\\\!\\\\sum _{{{k}}\\\\in \\\\mathbf {\\\\Lambda }+{\\\\mu }+{1\\\\over 2}{p}}\\\\!\\\\!\",\n \"(-1)^{{k}\\\\cdot {p}}\\\\,\\\\Phi (\\\\sqrt{2\\\\tau _2}({k}+{b}))\\\\, {\\\\bf e}\\\\!\\\\left( - \\\\tfrac{\\\\tau }{2}\\\\,({k}+{b})^2+{c}\\\\cdot ({k}+\\\\textstyle {1\\\\over 2}{b})\\\\right)$ defined by a kernel $\\\\Phi ({x})$ such that the function $f({x})\\\\equiv \\\\Phi ({x})\\\\, e^{\\\\tfrac{\\\\pi }{2}\\\\,{x}^2}\\\\in L^1(\\\\mathbf {\\\\Lambda }\\\\otimes {R})$ so that the sum is absolutely convergent.\",\n \"Irrespective of the choice of this kernel and of the parameter $\\\\lambda $ , any such theta series satisfies the following elliptic properties $\\\\begin{split}\\\\vartheta _{{p},{\\\\mu }}\\\\left({\\\\Phi } ,\\\\lambda ; \\\\tau , {b}+{k},{c}\\\\right) =&(-1)^{{k}\\\\cdot {p}}\\\\,{\\\\bf e}\\\\!\\\\left( -\\\\textstyle {1\\\\over 2}\\\\, {c}\\\\cdot {k}\\\\right) \\\\vartheta _{{p},{\\\\mu }}\\\\left({\\\\Phi } ,\\\\lambda ; \\\\tau , {b},{c}\\\\right),\\\\\\\\\\\\vphantom{A^A \\\\over A_A}\\\\vartheta _{{p},{\\\\mu }}\\\\left({\\\\Phi }, \\\\lambda ; \\\\tau , {b},{c}+{k}\\\\right)=&(-1)^{{k}\\\\cdot {p}}\\\\,{\\\\bf e}\\\\!\\\\left( \\\\textstyle {1\\\\over 2}\\\\, {b}\\\\cdot {k}\\\\right) \\\\vartheta _{{p},{\\\\mu }}\\\\left({\\\\Phi } ,\\\\lambda ; \\\\tau , {b},{c}\\\\right).\\\\end{split}$ Now let us require that in addition the kernel satisfies the following two conditions: Let $D({x})$ be any differential operator of order $\\\\le 2$ , and $R({x})$ any polynomial of degree $\\\\le 2$ .\",\n \"Then $f({x})$ defined above must be such that $f({x})$ , $D({x})f({x})$ and $R({x})f({x})\\\\in L^2(\\\\mathbf {\\\\Lambda }\\\\otimes {R})\\\\bigcap L^1(\\\\mathbf {\\\\Lambda }\\\\otimes {R})$ .\",\n \"$\\\\Phi ({x})$ must satisfy $V_\\\\lambda \\\\cdot \\\\Phi ({x})=0,\\\\qquad V_\\\\lambda = \\\\partial _{{x}}^2 + 2\\\\pi \\\\left( {x}\\\\cdot \\\\partial _{{x}} - \\\\lambda \\\\right) .$ Then in [38] it was proven that the theta series (REF ) transforms as a vector-valued modular form of weight $(\\\\lambda +d/2,0)$ (see Theorem 2.1 in [28] for the detailed transformation under $\\\\tau \\\\rightarrow -1/\\\\tau $ ).\",\n \"We refer to $V_\\\\lambda $ as Vignéras' operator.\",\n \"The simplest example is the Siegel theta series for which the kernel is $\\\\Phi ({x})=e^{-\\\\pi {x}_+^2}$ where ${x}_+$ is the projection of ${x}$ on a fixed positive plane of dimension $n$ .\",\n \"This kernel is annihilated by $V_{-n}$ .\",\n \"In this paper we apply the Vignéras' theorem to the case of $\\\\mathbf {\\\\Lambda }=\\\\oplus _{i=1}^n \\\\Lambda _i$ .\",\n \"Thus, the charges appearing in the description of the theta series (REF ) are of the type ${k}=(k_1^a,\\\\dots ,k_n^a)$ , whereas the vectors ${b}$ and ${c}$ are taken with $i$ -independent components, namely, ${b}_i^a=b^a$ , ${c}_i^a=c^a$ for $i=1,\\\\dots , n$ .\",\n \"The lattices $\\\\Lambda _i$ carry the bilinear forms $\\\\kappa _{i,ab}=\\\\kappa _{abc}p_i^c$ which are all of signature $(1,b_2-1)$ .\",\n \"This induces a natural bilinear form on $\\\\mathbf {\\\\Lambda }$ : ${x}\\\\cdot {y}=\\\\sum _{i=1}^n (p_ix_iy_i).$ Note also that the sign factor $(-1)^{{k}\\\\cdot {p}}$ in (REF ) can be identified with the quadratic refinement provided we choose the latter as $\\\\sigma _\\\\gamma =\\\\sigma _{p,q}\\\\equiv {\\\\bf e}\\\\!\\\\left( {1\\\\over 2}\\\\, p^a q_a\\\\right)\\\\sigma _p,\\\\qquad \\\\sigma _p={\\\\bf e}\\\\!\\\\left( {1\\\\over 2}\\\\, A_{ab}p^ap^b\\\\right).$ The matrix $A_{ab}$ , satisfying $A_{ab} p^p - \\\\frac{1}{2}\\\\, \\\\kappa _{abc} p^b p^c\\\\in {Z}\\\\quad \\\\text{for}\\\\ \\\\forall p^a\\\\in {Z}\\\\, ,$ appears due to the non-trivial quantization of charges on the type IIB side (REF ) and can be used to perform a symplectic rotation to identify them with mirror dual integer charges on the type IIA side [51].\",\n \"It is easy to check that the quadratic refinement (REF ) satisfies (REF ).\"\n ],\n [\n \"Generalized error functions\",\n \"An important class of solutions of Vignéras' equation is given by the error function and its generalizations constructed in [28] and further elaborated in [29].\",\n \"Let us take $M_1(u)&=&-\\\\mbox{sgn}(u)\\\\, \\\\text{Erfc}(|u|\\\\sqrt{\\\\pi }) = \\\\frac{I}{\\\\pi } \\\\int _{\\\\ell }\\\\frac{\\\\mathrm {d}z}{z}\\\\,e^{-\\\\pi z^2 -2\\\\pi Iz u},\\\\\\\\E_1(u)&=&\\\\mbox{sgn}(u)+M_1(u)\\\\nonumber \\\\\\\\&=& \\\\operatorname{Erf}(u\\\\sqrt{\\\\pi })= \\\\int _{{R}} \\\\mathrm {d}u^{\\\\prime } \\\\, e^{-\\\\pi (u-u^{\\\\prime })^2} {\\\\rm sgn}(u^{\\\\prime }),$ where the contour $\\\\ell ={R}-Iu$ runs parallel to the real axis through the saddle point at $z=-Iu$ .\",\n \"Then, given a vector with a positive norm ${v}^2>0$ so that $|{v}|=\\\\sqrt{{v}^2}$ , we define $\\\\Phi _1^E({v};{x})=E_1\\\\left(\\\\frac{{v}\\\\cdot {x}}{|{v}|}\\\\right),\\\\qquad \\\\Phi _1^M({v};{x})=M_1\\\\left(\\\\frac{{v}\\\\cdot {x}}{|{v}|}\\\\right).$ It is easy to check that the first function is a smooth solution of (REF ) with $\\\\lambda =0$ , whereas the second is exponentially suppressed at large ${x}$ and also solves the same equation, but only away from the locus ${v}\\\\cdot {x}=0$ where it has a discontinuity.\",\n \"Generalizing the integral representations (REF ) and (), we define the generalized (complementary) error functions $M_n(\\\\mathcal {M};\\\\mathbb {u})&=&\\\\left(\\\\frac{I}{\\\\pi }\\\\right)^n |\\\\,{\\\\rm det}\\\\, \\\\mathcal {M}|^{-1} \\\\int _{{R}^n-I\\\\mathbb {u}}\\\\mathrm {d}^n z\\\\,\\\\frac{e^{-\\\\pi \\\\mathbb {z}^{\\\\rm tr} \\\\mathbb {z}-2\\\\pi I\\\\mathbb {z}^{\\\\rm tr} \\\\mathbb {u}}}{\\\\prod (\\\\mathcal {M}^{-1}\\\\mathbb {z})},\\\\\\\\E_n(\\\\mathcal {M};\\\\mathbb {u})&=& \\\\int _{{R}^n} \\\\mathrm {d}\\\\mathbb {u}^{\\\\prime } \\\\, e^{-\\\\pi (\\\\mathbb {u}-\\\\mathbb {u}^{\\\\prime })^{\\\\rm tr}(\\\\mathbb {u}-\\\\mathbb {u}^{\\\\prime })} {\\\\rm sgn}(\\\\mathcal {M}^{\\\\rm tr} \\\\mathbb {u}^{\\\\prime }),$ where $\\\\mathbb {z}=(z_1,\\\\dots ,z_n)$ and $\\\\mathbb {u}=(u_1,\\\\dots ,u_n)$ are $n$ -dimensional vectors, $\\\\mathcal {M}$ is $n\\\\times n$ matrix of parameters, and we used the shorthand notations $\\\\prod \\\\mathbb {z}=\\\\prod _{i=1}^n z_i$ and ${\\\\rm sgn}(\\\\mathbb {u})=\\\\prod _{i=1}^n {\\\\rm sgn}(u_i)$ .\",\n \"The detailed properties of these functions can be found in [29].\",\n \"Here we mention only a few: $M_n$ are exponentially suppressed for large $\\\\mathbb {u}$ as $M_n\\\\sim \\\\frac{(-1)^n}{\\\\pi ^n}|\\\\,{\\\\rm det}\\\\, \\\\mathcal {M}|^{-1}\\\\,\\\\frac{e^{-\\\\pi \\\\mathbb {u}^{\\\\rm tr} \\\\mathbb {u}}}{\\\\prod (\\\\mathcal {M}^{-1}\\\\mathbb {u})}$ , whereas $E_n$ are locally constant for large $\\\\mathbb {u}$ as $E_n\\\\sim {\\\\rm sgn}(\\\\mathcal {M}^{\\\\rm tr} \\\\mathbb {u})$ .\",\n \"More generally, $E_n$ can be expressed as a linear combination of $M_k$ , $k=0,\\\\dots ,n$ , multiplied by $n-k$ sign functions, generalizing the first relation in () (see Eq.\",\n \"(REF ) below for a precise statement).\",\n \"From () it follows that every identity between products of sign functions implies an identity between generalized error functions $E_n$ .\",\n \"Moreover, expanding the $E_n$ functions in terms of $M_k$ 's and sign functions, one obtains similar identities for functions $M_n$ .\",\n \"For instance, the identity $(\\\\mbox{sgn}(x_1)+\\\\mbox{sgn}(x_2))\\\\,\\\\mbox{sgn}(x_1+x_2)=1+\\\\mbox{sgn}(x_1)\\\\,\\\\mbox{sgn}(x_2)$ implies $\\\\begin{split}E_2((\\\\mathbb {v}_1,\\\\mathbb {v}_1+\\\\mathbb {v}_2);\\\\mathbb {u})+E_2((\\\\mathbb {v}_2,\\\\mathbb {v}_1+\\\\mathbb {v}_2);\\\\mathbb {u})=&\\\\, 1+E_2((\\\\mathbb {v}_1,\\\\mathbb {v}_2);\\\\mathbb {u}),\\\\\\\\M_2((\\\\mathbb {v}_1,\\\\mathbb {v}_1+\\\\mathbb {v}_2);\\\\mathbb {u})+M_2((\\\\mathbb {v}_2,\\\\mathbb {v}_1+\\\\mathbb {v}_2);\\\\mathbb {u})=&\\\\, M_2((\\\\mathbb {v}_1,\\\\mathbb {v}_2);\\\\mathbb {u}),\\\\end{split}$ where $\\\\mathbb {v}_1,\\\\mathbb {v}_2$ are two-dimensional vectors used to encode the $2\\\\times 2$ matrix of parameters.\",\n \"The main reason to introduce these functions is that, similarly to the usual error and complementary error functions, they can be used to produce solutions of Vignéras' equation on ${R}^{n,d-n}$ .\",\n \"To write them down, let us consider $d\\\\times n$ matrix $\\\\mathcal {V}$ which can be viewed as a collection of $n$ vectors, $\\\\mathcal {V}=({v}_1,\\\\dots ,{v}_n)$ .\",\n \"We assume that these vectors span a positive definite subspace, i.e.\",\n \"$\\\\mathcal {V}^{\\\\rm tr}\\\\cdot \\\\mathcal {V}$ is a positive definite matrix.\",\n \"Let $\\\\mathcal {B}$ be $n\\\\times d$ matrix whose rows define an orthonormal basis for this subspace.\",\n \"Then we define the boosted generalized error functions $\\\\Phi _n^M(\\\\mathcal {V};{x})=M_n(\\\\mathcal {B}\\\\cdot \\\\mathcal {V};\\\\mathcal {B}\\\\cdot {x}),\\\\qquad \\\\Phi _n^E(\\\\mathcal {V};{x})=E_n(\\\\mathcal {B}\\\\cdot \\\\mathcal {V};\\\\mathcal {B}\\\\cdot {x}).$ It can be shown that both these functions satisfy Vignéras' equation (for $\\\\Phi _n^M$ one should stay away from its discontinuities, i.e.\",\n \"loci where ${\\\\rm sgn}((\\\\mathcal {B}\\\\cdot \\\\mathcal {V})^{-1}\\\\mathcal {B}\\\\cdot {x})=0$ ).\",\n \"Moreover, they are symmetric under permutation of the vectors ${v}_i$ .\",\n \"Since at large ${x}$ one has $\\\\Phi _n^E\\\\sim {\\\\rm sgn}(\\\\mathcal {V}^{\\\\rm tr}\\\\cdot {x})=\\\\prod _{i=1}^n {\\\\rm sgn}({v}_i\\\\cdot {x})$ , one can think about this function as providing the modular completion for (indefinite) theta series with kernel given by a product of signs.\",\n \"The relation between functions $E_n$ and $M_n$ mentioned above implies a similar relation between the functions (REF ).\",\n \"For generic $n$ , it takes the following formThis relation reduces to Eq.\",\n \"(3.53) and (6.18) in [28] for $n=2,3$ , and follows from Eq.\",\n \"(63) in [29] for any $n$ .\",\n \"$\\\\Phi _n^E(\\\\mathcal {V};{x})=\\\\sum _{\\\\mathcal {I}\\\\subseteq {Z}_n}\\\\Phi _{|\\\\mathcal {I}|}^M(\\\\lbrace {v}_i\\\\rbrace _{i\\\\in \\\\mathcal {I}};{x})\\\\prod _{j\\\\in {Z}_n\\\\setminus \\\\mathcal {I}}{\\\\rm sgn}({v}_{j\\\\perp \\\\mathcal {I}},{x}),$ where the sum goes over all possible subsets (including the empty set) of the set ${Z}_{n}=\\\\lbrace 1,\\\\dots ,n\\\\rbrace $ , $|\\\\mathcal {I}|$ is the cardinality of $\\\\mathcal {I}$ , and ${v}_{j\\\\perp \\\\mathcal {I}}$ denotes the projection of ${v}_j$ orthogonal to the subspace spanned by $\\\\lbrace {v}_i\\\\rbrace _{i\\\\in \\\\mathcal {I}}$ .\",\n \"The cardinality $|\\\\mathcal {I}|$ can also be interpreted as the number of directions in ${R}^d$ along which the corresponding contribution has an exponential fall off.\",\n \"Finally, note that a solution of Vignéras' equation can be uplifted to a solution of the same equation with $\\\\lambda $ shifted to $\\\\lambda +1$ by acting with the differential operator $\\\\mathcal {D}({v})={v}\\\\cdot \\\\left({x}+\\\\frac{1}{2\\\\pi }\\\\,\\\\partial _{x}\\\\right),$ which realizes the action of the covariant derivative raising the holomorphic weight by 1.\",\n \"In particular, we can construct solutions with $\\\\lambda =m$ which behave for large ${x}$ as products of $n$ sign functions.\",\n \"To this end, it is enough to act on $\\\\Phi _n^E$ by this operator $m$ times.\",\n \"Thus, we define the uplifted boosted error function $\\\\widetilde{\\\\Phi }_{n,m}^E(\\\\mathcal {V},\\\\tilde{\\\\mathcal {V}};{x})=\\\\left[\\\\prod _{i=1}^m \\\\mathcal {D}(\\\\tilde{{v}}_i)\\\\right]\\\\Phi _n^E(\\\\mathcal {V};{x}),$ where $\\\\tilde{\\\\mathcal {V}}=(\\\\tilde{{v}}_1,\\\\dots ,\\\\tilde{{v}}_m)$ encodes the vectors contracted with the covariant derivatives.\",\n \"Since the operators $\\\\mathcal {D}(\\\\tilde{{v}}_i)$ commute, (REF ) is invariant under independent permutations of the vectors ${v}_i$ and $\\\\tilde{{v}}_i$ .\",\n \"In the case where all $\\\\tilde{{v}}_i$ are mutually orthogonal, one finds the following asymptotics at large ${x}$ $\\\\lim _{{x}\\\\rightarrow \\\\infty }\\\\widetilde{\\\\Phi }_{n,m}^E(\\\\mathcal {V},\\\\tilde{\\\\mathcal {V}};{x})= \\\\prod _{i=1}^m (\\\\tilde{{v}}_i,{x})\\\\prod _{j=1}^n {\\\\rm sgn}({v}_j,{x}).$ Note that the derivative $\\\\partial _{x}$ in (REF ) does not act on sign functions since $\\\\Phi _n^E$ is smooth and all discontinuities due to signs are guaranteed to cancel.\",\n \"Similarly to (REF ), we can also define $\\\\widetilde{\\\\Phi }_{n,m}^M(\\\\mathcal {V},\\\\tilde{\\\\mathcal {V}};{x})$ where the action of derivatives on the discontinuities of $\\\\Phi _n^M$ is ignored as well.\",\n \"In the particular case $m=n$ , we will omit the second label and simply write $\\\\widetilde{\\\\Phi }_{n}^E$ or $\\\\widetilde{\\\\Phi }_{n}^M$ .\",\n \"toc\"\n ],\n [\n \"Twistorial integrals and generalized error functions\",\n \"In this appendix we evaluate the kernels $\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_n$ (REF ) and show that they can be expressed through the generalized error functions introduced in appendix REF .\",\n \"To this end, let us note the following identity $I\\\\,\\\\mathcal {D}({v}_{ij})\\\\,\\\\frac{W_{p_i}(x_i,z_i)\\\\,W_{p_j}(x_j,z_j)}{z_i-z_j}= \\\\hat{K}_{ij}\\\\, W_{p_i}(x_i,z_i)\\\\,W_{p_j}(x_j,z_j).$ By virtue of this relation, the kernel can be represented as $\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_n({x})=\\\\frac{1}{n!\",\n \"}\\\\sum _{\\\\mathcal {T}\\\\in \\\\, {T}_n^\\\\ell }\\\\left[\\\\prod _{e\\\\in E_\\\\mathcal {T}}\\\\mathcal {D}({v}_{s(e) t(e)}) \\\\right]\\\\Phi _\\\\mathcal {T}({x}),$ where $\\\\Phi _\\\\mathcal {T}({x}) =\\\\frac{I^{n-1}}{(2\\\\pi )^n}\\\\left[\\\\prod _{i=1}^n\\\\int _{\\\\ell _{\\\\gamma _i}}\\\\mathrm {d}z_i \\\\, W_{p_i}(x_i,z_i)\\\\right]\\\\frac{1}{\\\\prod _{e\\\\in E_\\\\mathcal {T}}\\\\left(z_{s(e)}-z_{t(e)}\\\\right)}\\\\, .$ One may think that the representation (REF ) misses contributions from the covariant derivatives acting on each other.\",\n \"However, such contributions are proportional to the scalar products of two vectors ${v}_{s(e) t(e)}$ and are non-vanishing provided the two edges have a common vertex.\",\n \"If this is the case, the two edges generate the following factor $\\\\frac{(p_{\\\\mathfrak {v}_1}p_{\\\\mathfrak {v}_2}p_{\\\\mathfrak {v}_{3}})}{(z_{\\\\mathfrak {v}_1}-z_{\\\\mathfrak {v}_3})(z_{\\\\mathfrak {v}_2}-z_{\\\\mathfrak {v}_{3}})}\\\\, ,$ where $\\\\mathfrak {v}_1,\\\\mathfrak {v}_2,\\\\mathfrak {v}_3$ are the tree vertices joint by the edges and $\\\\mathfrak {v}_{3}=e_1\\\\cap e_2$ .\",\n \"The crucial observation is that if we pick up 3 subtrees, each with a marked vertex, there are exactly 3 ways to form a labelled tree out of them by joining the marked vertices as shown in Fig.\",\n \"REF on page REF .\",\n \"Each subtree contributes the same factor in all 3 cases, whereas the joining edges and the sum over trees give rise to the vanishing factor $\\\\frac{(p_1p_2p_3)}{(z_1-z_3)(z_2-z_3)}+\\\\frac{(p_1p_2p_3)}{(z_1-z_2)(z_1-z_3)}-\\\\frac{(p_1p_2p_3)}{(z_2-z_3)(z_1-z_2)}=0.$ This ensures that no additional contributions arise and thereby proves (REF ).\",\n \"Note that for this proof it was crucial that inequivalent labelled trees enter the sum with the same weight.\",\n \"Then let us do the change of variables $z_i=z^{\\\\prime }-\\\\frac{I(pxt)}{\\\\sqrt{2\\\\tau _2}(pt^2)}+\\\\sum _{\\\\alpha =1}^{n-1} e_i^\\\\alpha z^{\\\\prime }_\\\\alpha ,$ where $x^a=\\\\kappa ^{ab}\\\\sum \\\\kappa _{i,bc} x^c_i$ (cf.\",\n \"(REF )), and $e_i^\\\\alpha $ are such that $\\\\sum _{i=1}^n \\\\mathfrak {p}_i e_i^\\\\alpha =0,\\\\qquad \\\\sum _{i=1}^n \\\\mathfrak {p}_i e_i^\\\\alpha e_i^\\\\beta =0, \\\\quad \\\\alpha \\\\ne \\\\beta $ and we introduced the convenient notation $\\\\mathfrak {p}_i=(p_it^2)$ .\",\n \"Labeling the $n-1$ edges of the tree by the same index $\\\\alpha $ , one can rewrite the function (REF ) in the new variables as $\\\\Phi _\\\\mathcal {T}({x}) =\\\\frac{I^{n-1} \\\\sqrt{\\\\Delta }}{(2\\\\pi )^n}\\\\,e^{-\\\\frac{\\\\pi (pxt)^2}{(pt^2)}}\\\\int \\\\mathrm {d}z^{\\\\prime } \\\\, e^{-2\\\\pi \\\\tau _2(pt^2)z^{\\\\prime 2}}\\\\prod _{\\\\alpha =1}^{n-1}\\\\int \\\\frac{\\\\mathrm {d}z^{\\\\prime }_\\\\alpha \\\\,e^{-2\\\\pi \\\\tau _2\\\\, \\\\Delta _\\\\alpha (z^{\\\\prime }_\\\\alpha )^2 -2\\\\pi I\\\\sqrt{2\\\\tau _2}\\\\, w^\\\\alpha z^{\\\\prime }_\\\\alpha }}{\\\\sum _{\\\\beta =1}^{n-1}\\\\left(e^\\\\beta _{s(\\\\alpha )}-e^\\\\beta _{t(\\\\alpha )}\\\\right)z^{\\\\prime }_\\\\beta }\\\\, ,$ where $w^\\\\alpha =\\\\sum _{i=1}^n (p_ix_i t) e_i^\\\\alpha ,\\\\qquad \\\\Delta _\\\\alpha =\\\\sum _{i=1}^n \\\\mathfrak {p}_i (e_i^\\\\alpha )^2,\\\\qquad \\\\Delta =\\\\frac{\\\\mathfrak {p}\\\\prod _{\\\\alpha =1}^{n-1} \\\\Delta _\\\\alpha }{\\\\prod _{i=1}^{n}\\\\mathfrak {p}_i}\\\\, .$ The integral over $z^{\\\\prime }$ is Gaussian and is easily evaluated.\",\n \"In the remaining integrals, rescaling the integration variables by $\\\\sqrt{2\\\\tau _2\\\\Delta _\\\\alpha }$ , one recognizes the generalized error functions (REF ).\",\n \"Thus, one obtains $\\\\Phi _\\\\mathcal {T}({x}) =\\\\frac{1}{2^{n-1}}\\\\,\\\\frac{\\\\sqrt{\\\\Delta }|\\\\,{\\\\rm det}\\\\, \\\\mathcal {M}|}{\\\\prod _{\\\\alpha =1}^{n-1} \\\\Delta _\\\\alpha }\\\\,\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_1(x)\\\\, M_{n-1}\\\\left(\\\\mathcal {M};\\\\left\\\\lbrace \\\\frac{w^\\\\alpha }{\\\\sqrt{\\\\Delta _\\\\alpha }}\\\\right\\\\rbrace \\\\right).$ where we used the function $\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_1$ (REF ) and introduced the matrix $\\\\mathcal {M}$ such that $\\\\mathcal {M}^{-1}_{\\\\alpha \\\\beta }=(\\\\Delta _\\\\alpha \\\\Delta _\\\\beta )^{-1/2}\\\\left(e^\\\\beta _{s(\\\\alpha )}-e^\\\\beta _{t(\\\\alpha )}\\\\right).$ A simple solution to the conditions (REF ) may be constructed as follows.\",\n \"Let $T$ be a rooted ordered binary tree with $n$ leaves decorated by $\\\\gamma _i$ , $i=1\\\\dots n$ .\",\n \"As usual for such trees, other vertices $v$ carry charges given by the sum of charges of their children, i.e.\",\n \"$\\\\gamma _v=\\\\sum _{i\\\\in \\\\mathcal {I}_v}\\\\gamma _i$ where $\\\\mathcal {I}_v$ is the set of leaves which are descendants of $v$ .\",\n \"There are $n-1$ such vertices which we label by index $\\\\alpha $ .\",\n \"Then we can choose $e^\\\\alpha _i=\\\\sum _{j\\\\in \\\\mathcal {I}_{L(v_\\\\alpha )}}\\\\sum _{k\\\\in \\\\mathcal {I}_{R(v_\\\\alpha )}} \\\\left(\\\\delta _{ij}\\\\,\\\\mathfrak {p}_k-\\\\delta _{ik}\\\\,\\\\mathfrak {p}_j\\\\right),$ which satisfy (REF ) as can be easily checked.\",\n \"For this choice (cf.\",\n \"(REF )) $\\\\begin{split}w^\\\\alpha =&\\\\, (\\\\tilde{{u}}_{\\\\alpha },{x}),\\\\quad \\\\mbox{where}\\\\quad \\\\tilde{{u}}_\\\\alpha =\\\\sum _{i\\\\in \\\\mathcal {I}_{L(v_\\\\alpha )}}\\\\sum _{j\\\\in \\\\mathcal {I}_{R(v_\\\\alpha )}}{u}_{ij},\\\\\\\\\\\\Delta _\\\\alpha =&\\\\,\\\\tilde{{u}}_\\\\alpha ^2=\\\\mathfrak {p}_{v_\\\\alpha } \\\\mathfrak {p}_{L(v_\\\\alpha )}\\\\mathfrak {p}_{R(v_\\\\alpha )}.\\\\end{split}$ Note that the vectors $\\\\tilde{{u}}_\\\\alpha $ are mutually orthogonal.\",\n \"In principle, any rooted binary tree $T$ is suitable for the above construction.\",\n \"However, given the unrooted tree $\\\\mathcal {T}$ , there is a simple (but non-unique) choice of $T$ which simplifies the resulting matrix $\\\\mathcal {M}$ .\",\n \"To define it, let us construct a partially increasing family of subtrees of $\\\\mathcal {T}$ , such that two members in this family are either disjoint, or contained in one another.\",\n \"Moreover, we require that the largest subtree is $\\\\mathcal {T}$ itself, while each subtree containing more than one vertex is obtained by joining two smaller subtrees along an edge of $\\\\mathcal {T}$ .\",\n \"Any such family contains $2n-1$ subtrees $\\\\mathcal {T}_{\\\\hat{\\\\alpha }}$ labelled by $\\\\hat{\\\\alpha }=1,\\\\dots ,2n-1$ .\",\n \"Among them, $n-1$ subtrees, which we label by $\\\\alpha =\\\\hat{\\\\alpha }=1,\\\\dots ,n-1$ , contain several vertices, while the remaining $n$ subtrees with label $\\\\hat{\\\\alpha }=n,\\\\dots ,2n-1$ have only one vertex.\",\n \"For each subtree $T_\\\\alpha $ , we denote by $e_\\\\alpha $ the edge of $\\\\mathcal {T}$ which is used to reconstruct $T_\\\\alpha $ from two smaller subtrees $\\\\mathcal {T}_{\\\\alpha _L}, \\\\mathcal {T}_{\\\\alpha _R}$ .\",\n \"From this data, we construct a rooted binary tree $T$ with $n-1$ vertices in one-to-one correspondence with the subtrees $\\\\mathcal {T}_\\\\alpha $ and $n$ leaves in one-to-one correspondence with the one-vertex subtrees $\\\\mathcal {T}_{\\\\hat{\\\\alpha }}$ with $\\\\hat{\\\\alpha }\\\\ge n$ .\",\n \"In this correspondence, the two children of a vertex associated to $\\\\mathcal {T}_{\\\\alpha }$ are the vertices associated to the two subtrees $\\\\mathcal {T}_{\\\\alpha _L}, \\\\mathcal {T}_{\\\\alpha _R}$ .\",\n \"The ordering at each vertex is defined to be such that the subtree containing the source/target vertex of the corresponding edge $e_\\\\alpha $ is on the left/right.The orientation of the edges of $\\\\mathcal {T}$ is fixed already in (REF ), but the full kernel $\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_n$ does not depend on its choice.\",\n \"Of course, this construction is not unique since there are many ways to decompose $\\\\mathcal {T}$ into such a set of subtrees (see Fig.\",\n \"REF ).\",\n \"Figure: An example of an unrooted labelled tree with 4 vertices and two choices of decompositions into subtrees withthe corresponding rooted binary trees.\",\n \"The edges are labelled e 1 ,e 2 ,e 3 e_1,e_2,e_3 from left to rightalong 𝒯\\\\mathcal {T}.Applying the above construction to this particular choice of rooted tree, one finds $\\\\sqrt{\\\\Delta _\\\\alpha \\\\Delta _\\\\beta }\\\\,\\\\mathcal {M}^{-1}_{\\\\alpha \\\\beta }=\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\mathfrak {p}_{v_\\\\alpha }, \\\\quad & \\\\alpha =\\\\beta ,\\\\\\\\\\\\epsilon _{\\\\alpha \\\\beta } \\\\mathfrak {p}_{L(v_\\\\beta )}, \\\\quad & e_\\\\alpha \\\\cap \\\\mathcal {T}_{\\\\beta _R} \\\\ne \\\\emptyset ,\\\\ e_\\\\alpha \\\\nsubseteq \\\\mathcal {T}_\\\\beta ,\\\\\\\\\\\\epsilon _{\\\\alpha \\\\beta } \\\\mathfrak {p}_{R(v_\\\\beta )}, \\\\quad & e_\\\\alpha \\\\cap \\\\mathcal {T}_{\\\\beta _L} \\\\ne \\\\emptyset ,\\\\ e_\\\\alpha \\\\nsubseteq \\\\mathcal {T}_\\\\beta ,\\\\\\\\0, & e_\\\\alpha \\\\cap \\\\mathcal {T}_\\\\beta = \\\\emptyset \\\\ \\\\mbox{ or }\\\\ e_\\\\alpha \\\\subset \\\\mathcal {T}_\\\\beta ,\\\\end{array}\\\\right.$ where $\\\\epsilon _{\\\\alpha \\\\beta }=-1$ if the orientations of $e_\\\\alpha $ and $e_\\\\beta $ on the path joining them are the same and $+1$ otherwise.\",\n \"This result shows that the matrix $\\\\mathcal {M}^{-1}$ turns out to be triangular which makes it much simpler to find its inverse.\",\n \"On the basis of (REF ), below we will prove the following Lemma 2 One has $\\\\mathcal {B}\\\\cdot \\\\mathcal {V}=\\\\mathcal {M}$ and $\\\\mathcal {B}\\\\cdot {x}=\\\\left\\\\lbrace \\\\frac{w^\\\\alpha }{\\\\sqrt{\\\\Delta _\\\\alpha }}\\\\right\\\\rbrace $ provided $\\\\begin{split}\\\\mathcal {B}=&\\\\, \\\\left(\\\\frac{\\\\tilde{{u}}_1}{\\\\sqrt{\\\\Delta _1}},\\\\dots , \\\\frac{\\\\tilde{{u}}_{n-1}}{\\\\sqrt{\\\\Delta _{n-1}}}\\\\right)^{\\\\rm tr},\\\\\\\\\\\\mathcal {V}=&\\\\, \\\\mathfrak {p}^{-1}\\\\Bigl (\\\\sqrt{\\\\Delta _1}{u}_1,\\\\dots , \\\\sqrt{\\\\Delta _{n-1}}{u}_{n-1}\\\\Bigr ),\\\\end{split}$ where the vectors ${u}_\\\\alpha $ are defined in (REF ).\",\n \"Moreover, the vectors $\\\\tilde{{u}}_\\\\alpha $ form an orthogonal basis in the subspace spanned by ${u}_\\\\alpha $ .\",\n \"This lemma allows to reexpress the kernel $\\\\Phi _\\\\mathcal {T}$ (REF ) in terms of the boosted generalized error function $\\\\Phi ^M_{n-1}$ (REF ).\",\n \"It is important that its argument $\\\\mathcal {V}$ does not depend on the choice of the binary rooted tree $T$ , but only on the unrooted tree $\\\\mathcal {T}$ .\",\n \"In addition, the function actually does not depend on the normalization of the vectors composing $\\\\mathcal {V}$ .\",\n \"Given also that the determinant of $\\\\mathcal {M}$ is found to be $|\\\\,{\\\\rm det}\\\\, \\\\mathcal {M}|=\\\\prod _{\\\\alpha =1}^{n-1}\\\\frac{ \\\\Delta _\\\\alpha }{\\\\mathfrak {p}_{v_\\\\alpha }}=\\\\prod _{i=1}^{n}\\\\mathfrak {p}_i\\\\prod _{v\\\\in V_{T}\\\\setminus {\\\\lbrace v_0\\\\rbrace }}\\\\mathfrak {p}_v=\\\\sqrt{\\\\mathfrak {p}^{-1}\\\\,\\\\prod _{i=1}^{n}\\\\mathfrak {p}_i\\\\,\\\\prod _{\\\\alpha =1}^{n-1} \\\\Delta _\\\\alpha },$ so that the prefactor in (REF ) cancels, one arrives at $\\\\Phi _\\\\mathcal {T}({x}) =\\\\frac{1}{2^{n-1}}\\\\,\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_1(x)\\\\, \\\\Phi ^M_{n-1}(\\\\lbrace {u}_e\\\\rbrace ;{x}).$ Finally, since the differential operator in (REF ) commutes with functions of $x$ due to the orthogonality of ${v}_{ij}$ and ${t}$ , one can write the kernel $\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_n$ as $\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_n({x})=\\\\frac{\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_1(x)}{2^{n-1} n!\",\n \"}\\\\sum _{\\\\mathcal {T}\\\\in \\\\, {T}_n^\\\\ell }\\\\left[\\\\prod _{e\\\\in E_\\\\mathcal {T}}\\\\mathcal {D}( {v}_{s(e) t(e)})\\\\right]\\\\Phi ^M_{n-1}(\\\\lbrace {u}_\\\\alpha \\\\rbrace ;{x}).$ which is the same as (REF ).\"\n ],\n [\n \"Proof of Lemma 2\",\n \"We start by proving that the vectors $\\\\tilde{{u}}_\\\\alpha $ form an orthogonal basis in the subspace spanned by ${u}_\\\\alpha $ .\",\n \"Since the orthogonality is ensured by construction based on a rooted binary tree, it remains to show that any vector ${u}_\\\\alpha $ can be decomposed as a linear combination of $\\\\tilde{{u}}_\\\\alpha $ .\",\n \"To this end, we show that the determinant of the Gram matrix constructed from the set of vectors $\\\\lbrace \\\\tilde{{u}}_\\\\alpha \\\\rbrace _{\\\\alpha =1}^{n-1}\\\\cup \\\\lbrace {u}_\\\\beta \\\\rbrace $ vanishes.\",\n \"This requires to calculate the scalar product $(\\\\tilde{{u}}_\\\\alpha , {u}_\\\\beta )$ which can be done using $({u}_{ij},{u}_{kl})=\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\mathfrak {p}_i\\\\mathfrak {p}_j\\\\mathfrak {p}_{i+j},\\\\qquad & i=k,\\\\ j=l,\\\\\\\\\\\\mathfrak {p}_i\\\\mathfrak {p}_j\\\\mathfrak {p}_{l},\\\\qquad & i=k,\\\\ j\\\\ne l,\\\\\\\\0,\\\\qquad & i,j\\\\ne k,l.\\\\end{array}\\\\right.$ Summing $i,j,k,l$ over appropriate subsets, it is immediate to see that $(\\\\tilde{{u}}_\\\\alpha , {u}_\\\\beta )=0$ if $\\\\mathcal {T}_{\\\\alpha }\\\\subset \\\\mathcal {T}_\\\\beta ^s$ or $\\\\mathcal {T}_\\\\beta ^t$ , which are the two trees obtained by dividing the tree $\\\\mathcal {T}$ into two parts by cutting the edge $e_\\\\alpha $ .\",\n \"In other words, it is non-vanishing only if $e_\\\\beta \\\\subseteq \\\\mathcal {T}_\\\\alpha $ .\",\n \"Then there are two cases which give $(\\\\tilde{{u}}_\\\\alpha , {u}_\\\\beta )=\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\mathfrak {p}_\\\\alpha ^L\\\\,\\\\mathfrak {p}_\\\\alpha ^R\\\\,\\\\mathfrak {p}, \\\\quad &\\\\alpha =\\\\beta ,\\\\\\\\\\\\mathfrak {p}_{\\\\alpha \\\\beta }^{Ls}\\\\,\\\\mathfrak {p}_\\\\alpha ^R\\\\,\\\\mathfrak {p}_\\\\beta ^t-\\\\mathfrak {p}_{\\\\alpha \\\\beta }^{Lt}\\\\,\\\\mathfrak {p}_\\\\alpha ^R\\\\,\\\\mathfrak {p}_\\\\beta ^s+\\\\mathfrak {p}_{\\\\alpha }^{L}\\\\,\\\\mathfrak {p}_\\\\alpha ^R\\\\,\\\\mathfrak {p}_\\\\beta ^s=\\\\mathfrak {p}_{\\\\alpha \\\\beta }^{Ls}\\\\,\\\\mathfrak {p}_\\\\alpha ^R\\\\,\\\\mathfrak {p},\\\\quad & e_\\\\beta \\\\subset \\\\mathcal {T}_\\\\alpha ,\\\\end{array}\\\\right.$ where we introduced $\\\\mathfrak {p}_\\\\alpha ^L=\\\\sum _{i\\\\in \\\\mathcal {I}_{L(v_\\\\alpha )}}\\\\mathfrak {p}_i,\\\\qquad \\\\mathfrak {p}_{\\\\beta }^{s}=\\\\sum _{i\\\\in V_{\\\\mathcal {T}_\\\\beta ^s}}\\\\mathfrak {p}_i,\\\\qquad \\\\mathfrak {p}_{\\\\alpha \\\\beta }^{Ls}=\\\\sum _{i\\\\in \\\\mathcal {I}_{L(v_\\\\alpha )}\\\\cap V_{\\\\mathcal {T}_\\\\beta ^s}}\\\\mathfrak {p}_i,\\\\qquad \\\\mathfrak {p}_{\\\\alpha \\\\beta }^{st}=\\\\sum _{i\\\\in V_{\\\\mathcal {T}_\\\\alpha ^s}\\\\cap V_{\\\\mathcal {T}_\\\\beta ^t}}\\\\mathfrak {p}_i,$ and similarly for variables with labels $R$ and $t$ .\",\n \"In (REF ) in the second case we assumed that the orientation of edges is such that $e_\\\\beta \\\\subset \\\\mathcal {T}_\\\\alpha ^s$ and $e_\\\\alpha \\\\subset \\\\mathcal {T}_\\\\beta ^t$ .\",\n \"If this is not the case, one should replace $s$ by $t$ , $L$ by $R$ and flip the sign for each change of orientation.\",\n \"Below we use the same assumption, but the computation can easily be generalized to a more general situation.\",\n \"Given the result (REF ), $(\\\\tilde{{u}}_\\\\alpha , \\\\tilde{{u}}_\\\\beta )=\\\\Delta _\\\\alpha \\\\delta _{\\\\alpha \\\\beta }$ and $({u}_\\\\beta , {u}_\\\\beta )=\\\\mathfrak {p}_\\\\beta ^s\\\\,\\\\mathfrak {p}_\\\\beta ^t\\\\,\\\\mathfrak {p}$ , the determinant of the Gram matrix is easily found to be $\\\\,{\\\\rm det}\\\\, {\\\\rm Gram}(\\\\tilde{{u}}_1,\\\\cdots , \\\\tilde{{u}}_{n-1},{u}_\\\\beta )=\\\\mathfrak {p}\\\\prod _{\\\\alpha =1}^{n-1}\\\\Delta _\\\\alpha \\\\left[\\\\mathfrak {p}_\\\\beta ^s\\\\,\\\\mathfrak {p}_\\\\beta ^t-\\\\mathfrak {p}\\\\sum _{T_\\\\alpha \\\\supseteq e_\\\\beta }\\\\frac{(\\\\mathfrak {p}_{\\\\alpha \\\\beta }^{Ls})^2\\\\,\\\\mathfrak {p}_\\\\alpha ^R}{\\\\mathfrak {p}_{v_\\\\alpha }\\\\mathfrak {p}_\\\\alpha ^L } \\\\right].$ Note that the subtrees $\\\\mathcal {T}_\\\\alpha $ containing the edge $e_\\\\beta $ form an ordered set so that the sum in the square brackets goes over $\\\\alpha _\\\\ell $ , $\\\\ell =1,\\\\dots , m$ , such that $\\\\mathcal {T}_{\\\\alpha _\\\\ell }\\\\subset \\\\mathcal {T}_{\\\\alpha _{\\\\ell +1}}$ .\",\n \"The first element of this set $\\\\alpha _1=\\\\beta $ , whereas the last corresponds to the total tree, $\\\\mathcal {T}_{\\\\alpha _m}=\\\\mathcal {T}$ .\",\n \"Due to $\\\\mathfrak {p}_{\\\\alpha _m}^L=\\\\mathfrak {p}_{\\\\alpha _m}^s$ , $\\\\mathfrak {p}_{\\\\alpha _m}^R=\\\\mathfrak {p}_{\\\\alpha _m}^t$ and $\\\\mathfrak {p}_{\\\\alpha _m\\\\beta }^{Ls}=\\\\mathfrak {p}_\\\\beta ^s$ , the first term in the square brackets together with the term in the sum corresponding to $\\\\alpha _m$ gives $\\\\mathfrak {p}_\\\\beta ^s\\\\left(\\\\mathfrak {p}_\\\\beta ^t -\\\\frac{\\\\mathfrak {p}_\\\\beta ^s\\\\,\\\\mathfrak {p}_{\\\\alpha _m}^t}{\\\\mathfrak {p}_{\\\\alpha _m}^s}\\\\right)=\\\\frac{\\\\mathfrak {p}\\\\,\\\\mathfrak {p}_\\\\beta ^s\\\\,\\\\mathfrak {p}_{\\\\alpha _m\\\\beta }^{st}}{\\\\mathfrak {p}_{\\\\alpha _m}^s},$ where we have used that $\\\\mathfrak {p}_\\\\beta ^t=\\\\mathfrak {p}_{\\\\alpha _m\\\\beta }^{st}+\\\\mathfrak {p}_{\\\\alpha _m}^t$ and $\\\\mathfrak {p}_{\\\\alpha _m}^s=\\\\mathfrak {p}_{\\\\alpha _m\\\\beta }^{st}+\\\\mathfrak {p}_\\\\beta ^s$ .\",\n \"Thus, the expression in the square brackets in (REF ) becomes $\\\\frac{\\\\mathfrak {p}}{\\\\mathfrak {p}_{\\\\alpha _m}^s}\\\\left[\\\\mathfrak {p}_\\\\beta ^s\\\\,\\\\mathfrak {p}_{\\\\alpha _m\\\\beta }^{st}-\\\\mathfrak {p}_{\\\\alpha _m}^s\\\\sum _{\\\\ell =1}^{m-1}\\\\frac{(\\\\mathfrak {p}_{\\\\alpha _\\\\ell \\\\beta }^{Ls})^2\\\\,\\\\mathfrak {p}_{\\\\alpha _\\\\ell }^R}{\\\\mathfrak {p}_{v_{\\\\alpha _\\\\ell }}\\\\mathfrak {p}_{\\\\alpha _\\\\ell }^L }\\\\right].$ The new expression in the square brackets is exactly the same as in (REF ) where tree $\\\\mathcal {T}$ was replaced by subtree $\\\\mathcal {T}_{\\\\alpha _m}^s=\\\\mathcal {T}_{\\\\alpha _{m-1}}$ .\",\n \"Thus, one can repeat the above manipulation until one exhausts all terms in the sum.\",\n \"As a result, the determinant of the Gram matrix turns out to be proportional to $\\\\mathfrak {p}_{\\\\alpha _1\\\\beta }^{st}$ .\",\n \"But since $\\\\alpha _1=\\\\beta $ , this quantity, and hence the whole determinant, trivially vanish.\",\n \"Next, we prove that (REF ) is consistent with $\\\\mathcal {B}\\\\cdot {x}=\\\\left\\\\lbrace \\\\frac{w^\\\\alpha }{\\\\sqrt{\\\\Delta _\\\\alpha }}\\\\right\\\\rbrace $ and $\\\\mathcal {B}\\\\cdot \\\\mathcal {V}=\\\\mathcal {M}$ .\",\n \"The first relation is a direct consequence of (REF ).\",\n \"The second relation requires to show that $(\\\\tilde{{u}}_\\\\alpha , {u}_\\\\beta )=\\\\mathfrak {p}\\\\,\\\\Delta _\\\\alpha ^{1/2}\\\\Delta _\\\\beta ^{-1/2}\\\\mathcal {M}_{\\\\alpha \\\\beta }$ or equivalently $\\\\sum _{\\\\gamma }(\\\\tilde{{u}}_\\\\alpha , {u}_\\\\gamma ) \\\\sqrt{\\\\Delta _\\\\gamma \\\\Delta _\\\\beta }\\\\mathcal {M}_{\\\\gamma \\\\beta }^{-1}=\\\\mathfrak {p}\\\\,\\\\Delta _\\\\alpha \\\\delta _{\\\\alpha \\\\beta }$ .\",\n \"Using the result (REF ) for the matrix $\\\\mathcal {M}_{\\\\gamma \\\\beta }^{-1}$ , this relation can be written explicitly as $\\\\mathfrak {p}_{v_\\\\beta }(\\\\tilde{{u}}_\\\\alpha , {u}_\\\\beta )+\\\\left(\\\\sum _{e_\\\\gamma \\\\in E_\\\\beta ^R}\\\\epsilon _{\\\\gamma \\\\beta } \\\\mathfrak {p}_{L(v_\\\\beta )}+\\\\sum _{e_\\\\gamma \\\\in E_\\\\beta ^L}\\\\epsilon _{\\\\gamma \\\\beta } \\\\mathfrak {p}_{R(v_\\\\beta )}\\\\right)(\\\\tilde{{u}}_\\\\alpha , {u}_\\\\gamma )=\\\\mathfrak {p}\\\\,\\\\Delta _\\\\alpha \\\\delta _{\\\\alpha \\\\beta },$ where $E_{\\\\beta }^{L}=\\\\lbrace e\\\\in E_\\\\mathcal {T}:\\\\ e\\\\cap \\\\mathcal {T}_{\\\\beta _L}\\\\ne \\\\emptyset , \\\\ e\\\\nsubseteq \\\\mathcal {T}_\\\\beta \\\\rbrace $ and similarly for $E_{\\\\beta }^{R}$ .\",\n \"Consider first the case $\\\\alpha =\\\\beta $ .\",\n \"From (REF ), it immediately follows that the first term gives $\\\\mathfrak {p}\\\\,\\\\Delta _\\\\alpha $ .\",\n \"On the other hand, the second contribution sums over edges for which $\\\\mathcal {T}_\\\\alpha \\\\subseteq \\\\mathcal {T}_\\\\gamma ^s$ or $\\\\mathcal {T}_\\\\gamma ^t$ , and as noted above (REF ) this leads to vanishing of the scalar product.\",\n \"Thus, in this case the relation (REF ) indeed holds.\",\n \"Let us now show that it holds as well for $\\\\alpha \\\\ne \\\\beta $ .\",\n \"To this end, one should consider several options.\",\n \"If $\\\\mathcal {T}_\\\\alpha \\\\cap \\\\mathcal {T}_\\\\beta =\\\\emptyset $ , then $\\\\mathcal {T}_\\\\alpha \\\\subset \\\\mathcal {T}_\\\\beta ^s$ or $\\\\mathcal {T}_\\\\beta ^t$ which implies vanishing of the first term.\",\n \"But the second term vanishes as well because the conditions $e_\\\\gamma \\\\subseteq \\\\mathcal {T}_\\\\alpha $ and $e_\\\\gamma \\\\cap \\\\mathcal {T}_\\\\beta \\\\ne \\\\emptyset $ are inconsistent with $\\\\mathcal {T}_\\\\alpha \\\\cap \\\\mathcal {T}_\\\\beta =\\\\emptyset $ .\",\n \"Similarly, if $\\\\mathcal {T}_\\\\alpha \\\\subset \\\\mathcal {T}_\\\\beta $ , one has $\\\\mathcal {T}_\\\\alpha \\\\subset \\\\mathcal {T}_\\\\beta ^s$ or $T_\\\\beta ^t$ which again leads to the vanishing of the first term, whereas the vanishing of the second is a consequence of that $e_\\\\gamma \\\\subseteq \\\\mathcal {T}_\\\\alpha $ implies $e_\\\\gamma \\\\subset \\\\mathcal {T}_\\\\beta $ so that the sum over $e_\\\\gamma $ is empty.\",\n \"It remains to consider the case $\\\\mathcal {T}_\\\\beta \\\\subset \\\\mathcal {T}_\\\\alpha $ .\",\n \"It is clear that $\\\\mathcal {T}_\\\\beta \\\\subset \\\\mathcal {T}_\\\\alpha ^s$ or $\\\\mathcal {T}_\\\\alpha ^t$ .\",\n \"Without loss of generality, let us assume that $\\\\mathcal {T}_\\\\beta \\\\subset \\\\mathcal {T}_\\\\alpha ^s$ and $e_\\\\alpha \\\\subset \\\\mathcal {T}_\\\\beta ^t$ .\",\n \"Then according to (REF ), the first term gives $\\\\mathfrak {p}_{v_\\\\beta }\\\\,\\\\mathfrak {p}_{\\\\alpha \\\\beta }^{Ls}\\\\,\\\\mathfrak {p}_\\\\alpha ^R\\\\,\\\\mathfrak {p}$ .\",\n \"If instead we have chosen the orientation such that $e_\\\\alpha \\\\subset \\\\mathcal {T}_\\\\beta ^s$ , then we would find $-\\\\mathfrak {p}_{v_\\\\beta }\\\\,\\\\mathfrak {p}_{\\\\alpha \\\\beta }^{Lt}\\\\,\\\\mathfrak {p}_\\\\alpha ^R\\\\,\\\\mathfrak {p}$ .\",\n \"Similar results are obtained for each term in the sum of the second contribution.\",\n \"Again without loss of generality we assume that for all relevant edges $e_\\\\gamma $ one has $e_\\\\alpha \\\\subset \\\\mathcal {T}_\\\\gamma ^t$ , otherwise one flips their orientation.\",\n \"Then the l.h.s.\",\n \"of (REF ) is proportional to $\\\\mathfrak {p}_{v_\\\\beta }\\\\,\\\\mathfrak {p}_{\\\\alpha \\\\beta }^{Ls}+\\\\sum _{e_\\\\gamma \\\\in E_{\\\\beta }^{R}}\\\\epsilon _{\\\\gamma \\\\beta } \\\\mathfrak {p}_\\\\beta ^L\\\\,\\\\mathfrak {p}_{\\\\alpha \\\\gamma }^{Ls}+\\\\sum _{e_\\\\gamma \\\\in E_{\\\\beta }^{L}}\\\\epsilon _{\\\\gamma \\\\beta } \\\\mathfrak {p}_\\\\beta ^R \\\\,\\\\mathfrak {p}_{\\\\alpha \\\\gamma }^{Ls}.$ Let $e_\\\\alpha ^\\\\star \\\\in E_{\\\\beta }^{R}$ is the edge belonging to the path from $\\\\mathcal {T}_\\\\beta $ to $e_\\\\alpha $ (which may coincide with $e_\\\\alpha $ ).\",\n \"Then our choice of orientation implies that $\\\\epsilon _{\\\\gamma \\\\beta }=-1$ for $e_\\\\gamma \\\\in E_\\\\beta ^L\\\\cup \\\\lbrace e_\\\\alpha ^\\\\star \\\\rbrace $ and $\\\\epsilon _{\\\\gamma \\\\beta }=1$ for $e_\\\\gamma \\\\in E_\\\\beta ^R\\\\setminus \\\\lbrace e_\\\\alpha ^\\\\star \\\\rbrace $ .\",\n \"Furthermore, one has $\\\\mathfrak {p}_{\\\\alpha \\\\gamma _\\\\alpha ^\\\\star }^{Ls}=\\\\mathfrak {p}_{v_\\\\beta }+\\\\sum _{e_\\\\gamma \\\\in E_\\\\beta ^L\\\\cup E_\\\\beta ^R\\\\setminus \\\\lbrace e_\\\\alpha ^\\\\star \\\\rbrace }\\\\mathfrak {p}_{\\\\alpha \\\\gamma }^{Ls},\\\\qquad \\\\mathfrak {p}_{\\\\alpha \\\\beta }^{Ls}=\\\\mathfrak {p}_\\\\beta ^L+\\\\sum _{e_\\\\gamma \\\\in E_\\\\beta ^L}\\\\mathfrak {p}_{\\\\alpha \\\\gamma }^{Ls}.$ As a result, one finds $(\\\\mathfrak {p}_\\\\beta ^L+\\\\mathfrak {p}_\\\\beta ^R)\\\\left(\\\\mathfrak {p}_\\\\beta ^L+\\\\sum _{e_\\\\gamma \\\\in E_\\\\beta ^L}\\\\mathfrak {p}_{\\\\alpha \\\\gamma }^{Ls}\\\\right)-\\\\mathfrak {p}_\\\\beta ^L \\\\,\\\\mathfrak {p}_{\\\\alpha \\\\gamma _\\\\alpha ^\\\\star }^{Ls}+\\\\sum _{e_\\\\gamma \\\\in E_{\\\\beta }^{R}\\\\setminus \\\\lbrace e_\\\\alpha ^\\\\star \\\\rbrace } \\\\mathfrak {p}_\\\\beta ^L\\\\,\\\\mathfrak {p}_{\\\\alpha \\\\gamma }^{Ls}-\\\\sum _{e_\\\\gamma \\\\in E_{\\\\beta }^{L}}\\\\mathfrak {p}_\\\\beta ^R \\\\,\\\\mathfrak {p}_{\\\\alpha \\\\gamma }^{Ls}=0.$ This completes the proof of the required statement (REF ).\",\n \"toc\"\n ],\n [\n \"Proofs of propositions\",\n \"In this appendix we prove several propositions which we stated in the main text.\"\n ],\n [\n \"Proposition \",\n \"To prove the recursive equation (REF ), let us note that the tree index $g_{{\\\\rm tr},n}$ satisfies a very similar equation (cf.\",\n \"(REF ) or (REF )) which can be seen as the origin of its representation (REF ) in terms of attractor flow trees.\",\n \"The only difference is the absence of the last term in (REF ).\",\n \"Therefore, it is easy to see that this equation implies a simple relation between $\\\\widehat{g}_n$ and $g_{{\\\\rm tr},n}$ $\\\\widehat{g}_n(\\\\lbrace \\\\check{\\\\gamma }_i\\\\rbrace ,z^a)=\\\\sum _{n_1+\\\\cdots +n_m= n\\\\atop n_k\\\\ge 1}g_{{\\\\rm tr},m}(\\\\lbrace \\\\check{\\\\gamma }^{\\\\prime }_k\\\\rbrace ,z^a)\\\\prod _{k=1}^m W_{n_k}(\\\\check{\\\\gamma }_{j_{k-1}+1},\\\\dots ,\\\\check{\\\\gamma }_{j_{k}}),$ where as usual we use notations from (REF ).\",\n \"Next, we substitute this relation into the expansion (REF ).\",\n \"The result can be represented in the following form $h^{\\\\rm DT}_{p,q}=\\\\sum _{\\\\sum _{i=1}^n \\\\check{\\\\gamma }_i=\\\\check{\\\\gamma }}g_{{\\\\rm tr},n}(\\\\lbrace \\\\check{\\\\gamma }_i\\\\rbrace ,z^a)\\\\,e^{\\\\pi I\\\\tau Q_n(\\\\lbrace \\\\check{\\\\gamma }_i\\\\rbrace )}\\\\prod _{i=1}^n h^R_{p_i,\\\\mu _i}(\\\\tau ),$ where we introduced $\\\\begin{split}h^{R}_{p,\\\\mu }=&\\\\, \\\\sum _{\\\\sum _{i=1}^n \\\\check{\\\\gamma }_i=\\\\check{\\\\gamma }}W_n(\\\\lbrace \\\\check{\\\\gamma }_i\\\\rbrace )\\\\,e^{\\\\pi I\\\\tau Q_n(\\\\lbrace \\\\check{\\\\gamma }_i\\\\rbrace )}\\\\prod _{i=1}^n \\\\widehat{h}_{p_i,\\\\mu _i}(\\\\tau )\\\\\\\\=&\\\\, \\\\sum _{\\\\sum _{i=1}^n \\\\check{\\\\gamma }_i=\\\\check{\\\\gamma }}\\\\left[\\\\sum _{T\\\\in {T}_n^{\\\\rm S}}(-1)^{n_T}\\\\prod _{v\\\\in V_T} R_{v}\\\\right]\\\\,e^{\\\\pi I\\\\tau Q_n(\\\\lbrace \\\\check{\\\\gamma }_i\\\\rbrace )}\\\\\\\\&\\\\, \\\\times \\\\prod _{i=1}^n \\\\left(h_{p_i,\\\\mu _i}+\\\\sum _{n_i=2}^\\\\infty \\\\sum _{\\\\sum _{j=1}^{n_i} \\\\check{\\\\gamma }^{\\\\prime }_j=\\\\check{\\\\gamma }}R_{n_i}(\\\\lbrace \\\\check{\\\\gamma }^{\\\\prime }_j\\\\rbrace )\\\\, e^{\\\\pi I\\\\tau Q_{n_i}(\\\\lbrace \\\\check{\\\\gamma }^{\\\\prime }_j\\\\rbrace )}\\\\prod _{j_i=1}^{n_i} h_{p^{\\\\prime }_{j_i},\\\\mu ^{\\\\prime }_{j_i}}\\\\right)\\\\end{split}$ and in the last relation we used the definition of $W_n$ (REF ) and the expansion of $\\\\widehat{h}_{p,\\\\mu }$ (REF ).\",\n \"The crucial observation is that if one picks up a factor $R_{n_i}$ from the second line of (REF ), appearing due to the expansion of $\\\\widehat{h}_{p_i,\\\\mu _i}$ , and combines it with the contribution of a tree $T$ from the first line, one finds the opposite of the contribution of another tree obtained from $T$ by adding $n_i$ children to its $i$ th leaf.\",\n \"As a result, all such contributions cancel and the function (REF ) reduces to the trivial term $h^{R}_{p,\\\\mu }=h_{p,\\\\mu }.$ Substituting this into (REF ), it gives back the original expansion (REF ) of the generating function of DT invariants, which proves the recursive equation (REF ).\",\n \"Finally, let us evaluate (REF ) at the attractor point $z^a_\\\\infty (\\\\gamma )$ .\",\n \"At this point the DT invariants coincide with MSW invariants so that the l.h.s.\",\n \"becomes the generating function $h_{p,\\\\mu }$ .\",\n \"Meanwhile, all factors $\\\\Delta _{\\\\gamma _L\\\\gamma _R}^z$ vanish at the attractor point and $\\\\widehat{g}_n$ reduces to $W_n$ .\",\n \"As a result, the relation (REF ) reproduces (REF ), which completes the proof of the proposition.\",\n \"We prove Proposition REF by induction.\",\n \"For $n=2$ the recursive relation (REF ) reads $g^{(0)}_2(\\\\check{\\\\gamma }_1,\\\\check{\\\\gamma }_2;c_1)-g^{(0)}_2(\\\\check{\\\\gamma }_1,\\\\check{\\\\gamma }_2;\\\\beta _{21})=-\\\\frac{1}{4}\\\\, \\\\bigl (\\\\mbox{sgn}(c_1)-\\\\mbox{sgn}(\\\\beta _{21}) \\\\bigr )\\\\, \\\\kappa (\\\\gamma _{12}),$ where we took into account that $S_1=c_1$ and $\\\\Gamma _{21}=\\\\beta _{21}=-\\\\gamma _{12}$ .\",\n \"Since $g^{(0)}_2$ is supposed to have discontinuities only at walls of marginal stability, it must not involve signs of DSZ products.\",\n \"Therefore, we are led to take $g^{(0)}_2(\\\\check{\\\\gamma }_1,\\\\check{\\\\gamma }_2;c_1)=-\\\\frac{1}{4}\\\\, \\\\mbox{sgn}(c_1) \\\\kappa (\\\\gamma _{12}).$ Then (REF ) and (REF ) imply $\\\\mathcal {E}_2=\\\\frac{1}{4}\\\\,\\\\mbox{sgn}(\\\\gamma _{12})\\\\,\\\\kappa (\\\\gamma _{12})+R_2,$ so that the ansatz (REF ) reads $\\\\widehat{g}_2(\\\\check{\\\\gamma }_1,\\\\check{\\\\gamma }_2;c_1)=g^{(0)}_2(\\\\check{\\\\gamma }_1,\\\\check{\\\\gamma }_2;c_1)-\\\\mathcal {E}_2(\\\\check{\\\\gamma }_1,\\\\check{\\\\gamma }_2)=-\\\\frac{1}{4}\\\\,\\\\Bigl [\\\\mbox{sgn}(c_1)+\\\\mbox{sgn}(\\\\gamma _{12})\\\\Bigr ]\\\\,\\\\kappa (\\\\gamma _{12})-R_2(\\\\check{\\\\gamma }_1,\\\\check{\\\\gamma }_2),$ which reproduces the recursive equation (REF ).\",\n \"Furthermore, in appendix REF it is shown that there is a smooth solution of Vignéras' equation which asymptotes the function $({v},{x}){\\\\rm sgn}({v},{x})$ , coinciding with the (rescaled) first term in $\\\\mathcal {E}_2$ for ${v}={v}_{12}$ (REF ).\",\n \"It is given by $\\\\widetilde{\\\\Phi }_1^E({v},{v};{x})={v}\\\\cdot \\\\left({x}+\\\\frac{1}{2\\\\pi }\\\\,\\\\partial _{x}\\\\right)\\\\operatorname{Erf}\\\\left(\\\\frac{\\\\sqrt{\\\\pi }{v}\\\\cdot {x}}{|{v}|}\\\\right),$ which corresponds to the following choice of $R_2$ [14] $R_2=\\\\frac{(-1)^{\\\\gamma _{12}}}{8\\\\pi }\\\\, |\\\\gamma _{12}|\\\\,\\\\beta _{\\\\frac{3}{2}}\\\\!\\\\left({\\\\frac{2\\\\tau _2\\\\gamma _{12}^2 }{(pp_1p_2)}}\\\\right),$ where $\\\\beta _{\\\\frac{3}{2}}(x^2)=2|x|^{-1}e^{-\\\\pi x^2}-2\\\\pi \\\\text{Erfc}(\\\\sqrt{\\\\pi } |x|)$ .\",\n \"Note that the resulting $\\\\mathcal {E}_2$ depends on the electric charges only through the DSZ product $\\\\gamma _{12}$ .\",\n \"Finally, it is immediate to see that the kernel $\\\\Phi ^{\\\\,\\\\widehat{g}}_2$ (REF ) satisfies (REF ).\",\n \"Now we assume that (REF ) is consistent with the ansatz (REF ) for all orders up to $n-1$ and check it at order $n$ .\",\n \"Denoting the second term in (REF ) by $\\\\widehat{g}^{(+)}_n$ and substituting this ansatz into the r.h.s.\",\n \"of (REF ), one finds $\\\\qquad \\\\qquad \\\\,{\\\\rm Sym}\\\\, \\\\left\\\\lbrace \\\\sum _{\\\\ell =1}^{n-1}g_2^\\\\ell \\\\,\\\\Bigl [g^{(0)}_\\\\ell \\\\,g^{(0)}_{n-\\\\ell }-\\\\widehat{g}^{(+)}_\\\\ell \\\\,\\\\widehat{g}_{n-\\\\ell }-\\\\widehat{g}_\\\\ell \\\\,\\\\widehat{g}^{(+)}_{n-\\\\ell }-\\\\widehat{g}^{(+)}_\\\\ell \\\\,\\\\widehat{g}^{(+)}_{n-\\\\ell }\\\\Bigr ]_{c_i\\\\rightarrow c_i^{(\\\\ell )}}\\\\right\\\\rbrace +W_n,$ where $g_2^\\\\ell =-{1\\\\over 2}\\\\,\\\\Delta _{\\\\gamma _L^\\\\ell \\\\gamma _R^\\\\ell }^z\\\\,\\\\kappa (\\\\gamma _{LR}^\\\\ell )=\\\\frac{1}{4}\\\\, \\\\bigl (\\\\mbox{sgn}(S_\\\\ell )-\\\\mbox{sgn}(\\\\Gamma _{n\\\\ell })\\\\bigr )\\\\,\\\\kappa (\\\\Gamma _{n\\\\ell }).$ In the first term proportional to $g^{(0)}_\\\\ell \\\\,g^{(0)}_{n-\\\\ell }$ , one can apply the relation (REF ), which together with (REF ) gives $g^{(0)}_n-\\\\mathcal {E}^{(0)}_n$ .\",\n \"The other terms in the sum over $\\\\ell $ can be reorganized as follows $\\\\begin{split}&\\\\,-\\\\,{\\\\rm Sym}\\\\, \\\\left\\\\lbrace \\\\sum _{\\\\ell =1}^{n-1} g_2^\\\\ell \\\\sum _{n_1+\\\\cdots +n_m= n\\\\atop n_k\\\\ge 1, \\\\ m1$ (following from $\\\\ell \\\\in \\\\lbrace j_k\\\\rbrace $ in the previous representation).\",\n \"In square brackets one recognizes the first contribution from the r.h.s.\",\n \"of (REF ) with $n$ replaced by $m1$ in the remaining sum with $\\\\widehat{g}_{m}$ .\",\n \"As a result, (REF ) becomes equivalent to the r.h.s.\",\n \"of (REF ), which proves the consistency of this ansatz with the recursive equation.\",\n \"Finally, let us show that the ansatz satisfies the modularity constraint (REF ).\",\n \"The crucial observation is that the vectors ${v}_{ij}$ and ${u}_{ij}$ (REF ) satisfy $({v}_{i+j,k},{v}_{ij})=({u}_{i+j,k},{v}_{ij})=0,$ where we abused notation and denoted ${v}_{i+j,k}={v}_{ik}+{v}_{jk}$ , etc.\",\n \"These orthogonality relations, together with the assumption that $\\\\mathcal {E}_n$ depend on electric charges only through the DSZ products $\\\\gamma _{ij}\\\\sim ({v}_{ij},{x})$ , imply factorization of the action of Vignéras' operator on the kernel corresponding to the second term in (REF ).\",\n \"Indeed, all contributions of $\\\\partial _{x}^2$ where two derivatives act on different factors vanish and the action reduces to the sum of terms where Vignéras' operator acts on one of the factors.\",\n \"But since it is supposed to vanish on $\\\\Phi ^{\\\\,\\\\mathcal {E}}_n$ , one obtains the simple result $V_{n-1} \\\\cdot \\\\Phi ^{\\\\,\\\\widehat{g}}_{n}=V_{n-1} \\\\cdot \\\\Phi ^{\\\\,g^{(0)}}_{n}-\\\\,{\\\\rm Sym}\\\\, \\\\left\\\\lbrace \\\\sum _{n_1+\\\\cdots +n_m= n\\\\atop n_k\\\\ge 1, \\\\ m0$ through $\\\\mathcal {V}_{m}$ .\",\n \"The idea is to replace the factors $\\\\mathcal {V}_{m_r}$ by the operators $\\\\mathcal {D}_{m_r}$ .\",\n \"Then one can realize that, extracting from the resulting function the terms homogeneous in ${x}$ of order $2(m-2)$ (which have two factors of $(p^3)$ ), one obtains the result for $2\\\\mathcal {V}_{m,2}$ .\",\n \"Proceeding in this way, one arrives at the representation very similar to (REF ), $\\\\mathcal {V}_{m,k}(\\\\lbrace \\\\check{\\\\gamma }_{s}\\\\rbrace )=\\\\frac{1}{(2\\\\pi )^k}\\\\,{\\\\rm Sym}\\\\, \\\\left\\\\lbrace \\\\sum _{\\\\sum \\\\limits _{r=1}^{2k+1}m_r=m-k}\\\\!\\\\!\\\\!\\\\!\",\n \"C(\\\\lbrace m_r\\\\rbrace )\\\\,\\\\mathcal {P}_{k}(\\\\lbrace p^{\\\\prime }_{r}\\\\rbrace )\\\\prod _{r=1}^{2k+1}\\\\mathcal {V}_{m_r}(\\\\check{\\\\gamma }_{j_{r-1}+1},\\\\dots ,\\\\check{\\\\gamma }_{j_{r}})\\\\right\\\\rbrace .$ Substituting it into (REF ) and using the expression (REF ) for the factors $\\\\mathcal {P}_{m}$ through the sum over rooted ternary trees, one can recombine all these sums in the following way $\\\\begin{split}\\\\mathcal {V}^{\\\\rm tot}_m=&\\\\,\\\\,{\\\\rm Sym}\\\\, \\\\Biggl \\\\lbrace \\\\sum _{k=0}^{m}\\\\sum _{\\\\sum \\\\limits _{r=1}^{2k+1}m_r=m-k}\\\\!\\\\!\\\\!\\\\!\\\\!\\\\!\",\n \"C(\\\\lbrace m_r\\\\rbrace )\\\\sum _{T\\\\in \\\\, {T}_{2k+1}^{(3)}(\\\\lbrace \\\\check{\\\\gamma }^{\\\\prime }_r\\\\rbrace )}\\\\frac{1}{T!\",\n \"}\\\\sum _{T^{\\\\prime }\\\\subseteq T} (-1)^{n_{T^{\\\\prime }}}\\\\prod _{v\\\\in V_{T^{\\\\prime }}}\\\\frac{n_v(T)}{n_v(T^{\\\\prime })}\\\\\\\\&\\\\,\\\\times \\\\prod _{v\\\\in V_{T}}\\\\frac{(p_{d_1(v)}p_{d_2(v)}p_{d_3(v)})}{2\\\\pi }\\\\prod _{r=1}^{2k+1}\\\\mathcal {V}_{m_r}(\\\\check{\\\\gamma }_{j_{r-1}+1},\\\\dots ,\\\\check{\\\\gamma }_{j_{r}})\\\\Biggr \\\\rbrace ,\\\\end{split}$ where the sum over $T^{\\\\prime }$ is the sum over subtrees of $T$ having the same root.\",\n \"In terms of the variables appearing in (REF ), one can identify $k=m_0+\\\\sum _r k_r$ and $n_{T^{\\\\prime }}=m_0$ .\",\n \"Thus, $T^{\\\\prime }$ is the tree appearing in the decomposition of $\\\\mathcal {P}_{m_0}$ , whereas $T$ is its union with $2m_0+1$ trees $T_r$ appearing in the decomposition of $\\\\mathcal {P}_{k_r}$ , i.e.\",\n \"$T=T^{\\\\prime }\\\\cup \\\\left(\\\\cup _r T_r\\\\right)$ where $T_r$ are the trees rooted at leaves of $T^{\\\\prime }$ , Finally, we took into account that for such trees one has $T!=T^{\\\\prime }!\\\\,\\\\prod _r T_r!\\\\,\\\\prod _{v\\\\in V_{T^{\\\\prime }}}\\\\frac{n_v(T)}{n_v(T^{\\\\prime })}\\\\,.$ Remarkably, the sum over subtrees in (REF ) factorizes and for a fixed number of vertices $n_{T^{\\\\prime }}=m_0$ is subject to Theorem REF where the rôle of the trees is played by rooted ternary trees $T$ and $T^{\\\\prime }$ after stripping out their leaves.\",\n \"As a result, one finds for $k>0$ $\\\\sum _{T^{\\\\prime }\\\\subseteq T} (-1)^{n_{T^{\\\\prime }}}\\\\prod _{v\\\\in V_{T^{\\\\prime }}}\\\\frac{n_v(T)}{n_v(T^{\\\\prime })}=\\\\sum _{m_0=0}^k\\\\frac{(-1)^{m_0}k!}{m_0!(k-m_0)!\",\n \"}=(1-1)^k = 0.$ Thus, the only non-vanishing contribution is the one with $k=0$ which coincides with $\\\\mathcal {V}_m$ .\",\n \"This is what we had to show and therefore completes the proof of the proposition.\",\n \"For simplicity, let us first show that the recursive equation (REF ) is satisfied by the contribution to $g^{(0)}_n(\\\\lbrace \\\\check{\\\\gamma }_i,c_i\\\\rbrace )$ given by the trees without any marks, i.e.\",\n \"by the function $g^{\\\\star }_n(\\\\lbrace \\\\check{\\\\gamma }_i,c_i\\\\rbrace )=\\\\frac{(-1)^{n-1+\\\\sum _{i1$ (from the sum in (REF )) is cancelled by the contribution of another tree obtained from $T$ by adding $n_k$ children to its $k$ th leaf and the same splitting but with $n_k$ replaced by $1+\\\\cdots +1$ (repeated $n_k$ times).\",\n \"The only contribution which survives is the one generated by the tree with a single vertex and $n$ leaves and the splitting with all $n_k=1$ .\",\n \"It is given by $\\\\mathcal {E}^{(+)}_n$ , which verifies the constraint and proves the proposition.\",\n \"Our starting point to prove the proposition is the formula $\\\\partial _{\\\\bar{\\\\tau }}\\\\widehat{h}_{p,\\\\mu }(\\\\tau )=\\\\frac{I}{2}\\\\sum _{n=2}^\\\\infty \\\\sum _{\\\\sum _{i=1}^n \\\\check{\\\\gamma }_i=\\\\check{\\\\gamma }}\\\\partial _{\\\\tau _2}R_n(\\\\lbrace \\\\check{\\\\gamma }_i\\\\rbrace ,\\\\tau _2)\\\\, e^{\\\\pi I\\\\tau Q_n(\\\\lbrace \\\\check{\\\\gamma }_i\\\\rbrace )}\\\\prod _{i=1}^n h_{p_i,\\\\mu _i}(\\\\tau ).$ Substituting $\\\\partial _{\\\\tau _2}R_n$ following from (REF ) and the inverse formula (REF ) expressing $h_{p,\\\\mu }$ in terms of the completion, with the functions $W_n$ found in (REF ), one arrives at the result (REF ) where the functions $\\\\mathcal {J}_n$ are given by $\\\\mathcal {J}_n= \\\\frac{I}{2}\\\\,{\\\\rm Sym}\\\\, \\\\left\\\\lbrace \\\\sum _{T\\\\in {T}_n^{\\\\rm S}}(-1)^{n_T-1} \\\\partial _{\\\\tau _2}\\\\mathcal {E}_{v_0}\\\\sum _{T^{\\\\prime }\\\\subseteq T}\\\\prod _{v\\\\in V_{T^{\\\\prime }}\\\\setminus {\\\\lbrace v_0\\\\rbrace }}\\\\mathcal {E}^{(0)}_{v}\\\\prod _{v\\\\in L_{T^{\\\\prime }}}\\\\mathcal {E}^{(+)}_{v}\\\\prod _{v\\\\in V_T\\\\setminus (V_{T^{\\\\prime }}\\\\cup L_{T^{\\\\prime }})}\\\\mathcal {E}_{v}\\\\right\\\\rbrace ,$ where the second sum goes over all subtrees $T^{\\\\prime }$ of $T$ containing its root and $L_{T^{\\\\prime }}$ is the set of their leaves.\",\n \"Here the subtree $T^{\\\\prime }$ corresponds to the tree in the formula (REF ) for $R_n$ , whereas the subtrees starting from its leaves correspond to the trees in the expression (REF ) for $W_n$ .\",\n \"The sum over subtrees can be evaluated in the same way as the sum over decompositions into subtrees in (REF ): the two contributions differing only by whether the vertex $v$ belongs to $V_{T^{\\\\prime }}$ or $L_{T^{\\\\prime }}$ combine into the factor $\\\\mathcal {E}_v$ assigned to this vertex.\",\n \"Performing this recombination for all vertices of the tree $T$ , one obtains the expression (REF ), which proves the proposition.\",\n \"Our goal is to prove that in the expression $g_{{\\\\rm tr},n}= \\\\,{\\\\rm Sym}\\\\, \\\\left\\\\lbrace \\\\sum _{T\\\\in {T}_n^{\\\\rm S}}(-1)^{n_T-1} \\\\left(g^{(0)}_{v_0}-\\\\mathcal {E}^{(0)}_{v_0}\\\\right)\\\\prod _{v\\\\in V_T\\\\setminus {\\\\lbrace v_0\\\\rbrace }}\\\\mathcal {E}^{(0)}_{v}\\\\right\\\\rbrace ,$ following from (REF ), all contributions due to marked trees cancel leaving only contributions of trees without marks, i.e with $m=0$ .\",\n \"There are several possible situations which we need to analyze.\",\n \"Figure: Combination of two Schröder trees ensuring the cancellation of contributions generated by marked trees.First, let us consider the contributions generated by non-trivial marked trees, i.e.\",\n \"trees having more than one vertex and at least one mark.\",\n \"Let us focus on the contribution corresponding to a vertex $\\\\mathfrak {v}$ with $m_\\\\mathfrak {v}>0$ marks of a tree $\\\\mathcal {T}$ , which appears in the sum over marked unrooted trees living at a vertex $v$ of a Schröder tree $T$ .\",\n \"Let $k=n_v$ be the number of children of the vertex $v$ and $\\\\gamma _i$ ($i=1,\\\\dots ,k$ ) their charges so that $\\\\gamma _s$ ($s=1,\\\\dots ,2m_\\\\mathfrak {v}+1$ ) are the charges labelling the marked vertex $\\\\mathfrak {v}$ .\",\n \"Note that $k\\\\ge 2m_\\\\mathfrak {v}+2$ because the tree $\\\\mathcal {T}$ has at least one additional vertex except $\\\\mathfrak {v}$ .\",\n \"Then the contribution we described is cancelled by the contribution coming from another Schröder tree, which is obtained from $T$ by adding an edge connecting the vertex $v$ to a new vertex $v^{\\\\prime }$ , whose children are the $2m_\\\\mathfrak {v}+1$ children of $v$ in $T$ carrying charges $\\\\gamma _s$ (see Fig.\",\n \"REF ).The new tree is of Schröder type because its vertex $v$ has $k-2m_{\\\\mathfrak {v}}\\\\ge 2$ children and vertex $v^{\\\\prime }$ has $2m_\\\\mathfrak {v}+1\\\\ge 3$ children.\",\n \"Indeed, choosing the same tree $\\\\mathcal {T}$ as before in the sum over marked trees at vertex $v$ , but now with $m_\\\\mathfrak {v}=0$ , and in the sum at vertex $v^{\\\\prime }$ the trivial tree having one vertex and $m_\\\\mathfrak {v}$ marks, one gets exactly the same contribution as before, but now with an opposite sign due to the presence of an additional vertex in the Schröder tree.\",\n \"Thus, all contributions from non-trivial marked trees are cancelled.\",\n \"As a result, we remain only with the contributions generated by trivial marked trees, i.e.\",\n \"having only one vertex and $m_\\\\mathfrak {v}$ marks.\",\n \"One has to distinguish two cases: either the corresponding vertex $v$ of the Schröder tree is the root or not.\",\n \"In the former case, this contribution is trivially cancelled in the difference $g^{(0)}_{v_0}-\\\\mathcal {E}^{(0)}_{v_0}$ in (REF ).\",\n \"In the latter case, this is precisely the contribution used above to cancel the contributions from non-trivial marked trees.\",\n \"This exhausts all possibilities and we arrive at the formula (REF ).\",\n \"toc toc\"\n ],\n [\n \"Explicit results up to 4th order\",\n \"In this appendix we provide explicit expressions for various functions appearing in our construction up to the forth order.\",\n \"To write them down, we will use the shorthand notation $\\\\gamma _{i+j}=\\\\gamma _i+\\\\gamma _j$ , $c_{i+j}=c_i+c_j$ , etc.\",\n \"as well as indicate the arguments of functions through their indices, for instance, $\\\\mathcal {E}_{i_1\\\\cdots i_n}=\\\\mathcal {E}_n(\\\\check{\\\\gamma }_{i_1},\\\\dots ,\\\\check{\\\\gamma }_{i_n})$ .\",\n \"These expressions are obtained by simple substitutions using the results found in the main text and the sets of trees shown in Fig.\",\n \"REF .\",\n \"For $n=2$ they all agree with the results of [14].\",\n \"The results (REF ) and (REF ) generate the following expansions $h^{\\\\rm DT}_{p,q}&=&\\\\widehat{h}_{p,\\\\mu }+\\\\sum _{\\\\check{\\\\gamma }_1+\\\\check{\\\\gamma }_2=\\\\check{\\\\gamma }}\\\\left[g^{(0)}_{12}-\\\\mathcal {E}_{12}\\\\right]e^{\\\\pi I\\\\tau Q_2(\\\\lbrace \\\\check{\\\\gamma }_i\\\\rbrace )}\\\\widehat{h}_{p_1,\\\\mu _1}\\\\widehat{h}_{p_2,\\\\mu _2}\\\\nonumber \\\\\\\\&& +\\\\sum _{\\\\sum _{i=1}^3 \\\\check{\\\\gamma }_i=\\\\check{\\\\gamma }}\\\\left[g^{(0)}_{123}-\\\\mathcal {E}_{123}-2\\\\left(g^{(0)}_{1+2,3}-\\\\mathcal {E}_{1+2,3}\\\\right)\\\\mathcal {E}_{12}\\\\right] e^{\\\\pi I\\\\tau Q_3(\\\\lbrace \\\\check{\\\\gamma }_i\\\\rbrace )}\\\\prod _{i=1}^3 \\\\widehat{h}_{p_i,\\\\mu _i}\\\\\\\\&&+\\\\sum _{\\\\sum _{i=1}^4 \\\\check{\\\\gamma }_i=\\\\check{\\\\gamma }}\\\\left[ g^{(0)}_{1234}-\\\\mathcal {E}_{1234}-2\\\\left(g^{(0)}_{1+2+3,4}-\\\\mathcal {E}_{1+2+3,4}\\\\right)\\\\left(\\\\mathcal {E}_{123} -2\\\\mathcal {E}_{1+2,3}\\\\mathcal {E}_{12}\\\\right)\\\\right.\\\\nonumber \\\\\\\\&& \\\\left.\\\\qquad -3\\\\left(g^{(0)}_{1+2,34}-\\\\mathcal {E}_{1+2,34}\\\\right)\\\\mathcal {E}_{12}+\\\\left(g^{(0)}_{1+2,3+4}-\\\\mathcal {E}_{1+2,3+4}\\\\right)\\\\mathcal {E}_{12}\\\\mathcal {E}_{34}\\\\right]e^{\\\\pi I\\\\tau Q_4(\\\\lbrace \\\\check{\\\\gamma }_i\\\\rbrace )}\\\\prod _{i=1}^4 \\\\widehat{h}_{p_i,\\\\mu _i}+\\\\cdots ,\\\\nonumber $ $\\\\widehat{h}_{p,q}&=&h_{p,\\\\mu }+\\\\sum _{\\\\check{\\\\gamma }_1+\\\\check{\\\\gamma }_2=\\\\check{\\\\gamma }}\\\\mathcal {E}^{(+)}_{12}\\\\,e^{\\\\pi I\\\\tau Q_2(\\\\lbrace \\\\check{\\\\gamma }_i\\\\rbrace )}h_{p_1,\\\\mu _1}h_{p_2,\\\\mu _2}\\\\nonumber \\\\\\\\&& +\\\\sum _{\\\\sum _{i=1}^3 \\\\check{\\\\gamma }_i=\\\\check{\\\\gamma }}\\\\Bigl [\\\\mathcal {E}^{(+)}_{123}-2\\\\mathcal {E}^{(+)}_{1+2,3}\\\\mathcal {E}^{(0)}_{12}\\\\Bigr ] \\\\,e^{\\\\pi I\\\\tau Q_3(\\\\lbrace \\\\check{\\\\gamma }_i\\\\rbrace )}\\\\prod _{i=1}^3 h_{p_i,\\\\mu _i}\\\\\\\\&&+\\\\sum _{\\\\sum _{i=1}^4 \\\\check{\\\\gamma }_i=\\\\check{\\\\gamma }}\\\\Bigl [\\\\mathcal {E}^{(+)}_{1234}-2\\\\mathcal {E}^{(+)}_{1+2+3,4}\\\\left(\\\\mathcal {E}^{(0)}_{123} -2\\\\mathcal {E}^{(0)}_{1+2,3}\\\\mathcal {E}^{(0)}_{12}\\\\right)-3\\\\mathcal {E}^{(+)}_{1+2,34}\\\\mathcal {E}^{(0)}_{12}\\\\Bigr .\\\\nonumber \\\\\\\\&& \\\\Bigl .\",\n \"\\\\qquad +\\\\mathcal {E}^{(+)}_{1+2,3+4}\\\\mathcal {E}^{(0)}_{12}\\\\mathcal {E}^{(0)}_{34}\\\\Bigr ]\\\\, e^{\\\\pi I\\\\tau Q_4(\\\\lbrace \\\\check{\\\\gamma }_i\\\\rbrace )}\\\\prod _{i=1}^4 h_{p_i,\\\\mu _i}+\\\\cdots ,\\\\nonumber $ where the functions $g^{(0)}_n$ and $\\\\mathcal {E}_n$ can be read off from (REF ), (REF ) and (REF ), $\\\\begin{split}g^{(0)}_2=&\\\\, \\\\frac{(-1)^{1+\\\\gamma _{12}}}{4}\\\\, \\\\gamma _{12}\\\\,\\\\mbox{sgn}(c_1),\\\\\\\\g^{(0)}_3=&\\\\, \\\\frac{(-1)^{1+\\\\gamma _{12}+\\\\gamma _{1+2,3}}}{8}\\\\,\\\\,{\\\\rm Sym}\\\\, \\\\biggl \\\\lbrace \\\\gamma _{12}\\\\,\\\\gamma _{23}\\\\,\\\\mbox{sgn}(c_1)\\\\, \\\\mbox{sgn}(c_3)+\\\\frac{1}{3}\\\\,\\\\gamma _{12}\\\\,\\\\gamma _{23} \\\\biggr \\\\rbrace ,\\\\\\\\g^{(0)}_4=&\\\\, \\\\frac{(-1)^{\\\\gamma _{12}+\\\\gamma _{1+2,3}+\\\\gamma _{1+2+3,4}}}{16}\\\\,\\\\,{\\\\rm Sym}\\\\, \\\\biggl \\\\lbrace \\\\gamma _{12}\\\\gamma _{23}\\\\gamma _{34}\\\\,\\\\mbox{sgn}(c_1)\\\\mbox{sgn}(c_{1+2})\\\\mbox{sgn}(c_4)\\\\\\\\&\\\\, \\\\qquad -\\\\frac{1}{3}\\\\,\\\\gamma _{12}\\\\gamma _{23}\\\\gamma _{24}\\\\,\\\\mbox{sgn}(c_1)\\\\mbox{sgn}(c_3)\\\\mbox{sgn}(c_4)-\\\\frac{1}{3}\\\\, \\\\gamma _{12}\\\\gamma _{23} \\\\gamma _{1+2+3,4}\\\\mbox{sgn}(c_4)\\\\biggr \\\\rbrace ,\\\\end{split}$ $\\\\mathcal {E}_{12} &=&\\\\frac{(-1)^{\\\\gamma _{12}}}{4\\\\sqrt{2\\\\tau _2}}\\\\,\\\\widetilde{\\\\Phi }_{1}^E\\\\bigl ({v}_{12},{v}_{12}\\\\bigr )\\\\nonumber \\\\\\\\&=&\\\\frac{(-1)^{\\\\gamma _{12}}}{4}\\\\left[\\\\gamma _{12}\\\\, E_1\\\\left(\\\\frac{\\\\sqrt{2\\\\tau _2}\\\\gamma _{12} }{\\\\sqrt{(pp_1p_2)}}\\\\right)+\\\\frac{\\\\sqrt{(pp_1p_2)}}{\\\\pi \\\\sqrt{2\\\\tau _2}}\\\\, e^{-\\\\frac{2\\\\pi \\\\tau _2\\\\gamma _{12}^2 }{(pp_1p_2)}} \\\\right] ,\\\\nonumber \\\\\\\\\\\\mathcal {E}_{123} &=&\\\\frac{(-1)^{\\\\gamma _{12}+\\\\gamma _{1+2,3}}}{8}\\\\,\\\\,{\\\\rm Sym}\\\\, \\\\left\\\\lbrace \\\\frac{1}{2\\\\tau _2}\\\\,\\\\widetilde{\\\\Phi }_{2}^E\\\\bigl (({v}_{1,2+3},{v}_{1+2,3}),({v}_{12},{v}_{23})\\\\bigr )\\\\right.\\\\nonumber \\\\\\\\&&\\\\left.\",\n \"\\\\qquad -\\\\frac{1}{3}\\\\left(\\\\gamma _{12}\\\\gamma _{23}-\\\\frac{(p_1p_2p_3)}{4\\\\pi \\\\tau _2} \\\\right)\\\\right\\\\rbrace ,\\\\\\\\\\\\mathcal {E}_{1234} &=&\\\\frac{(-1)^{\\\\gamma _{12}+\\\\gamma _{1+2,3}+\\\\gamma _{1+2+3,4}}}{16\\\\sqrt{2\\\\tau _2}}\\\\,\\\\,{\\\\rm Sym}\\\\, \\\\biggl \\\\lbrace \\\\frac{1}{2\\\\tau _2}\\\\biggl (\\\\widetilde{\\\\Phi }_{3}^E\\\\bigl (({v}_{1,2+3+4},{v}_{1+2,3+4},{v}_{1+2+3,4}),({v}_{12},{v}_{23},{v}_{34})\\\\bigr )\\\\nonumber \\\\\\\\&&+\\\\frac{1}{3}\\\\,\\\\widetilde{\\\\Phi }_{3}^E\\\\bigl (({v}_{1,2+3+4},{v}_{1+2+4,3},{v}_{1+2+3,4}),({v}_{12},{v}_{23},{v}_{24})\\\\bigr )\\\\biggr )\\\\nonumber \\\\\\\\&& -\\\\frac{1}{3}\\\\left(\\\\gamma _{12}\\\\gamma _{23}-\\\\frac{(p_1p_2p_3)}{4\\\\pi \\\\tau _2}\\\\right)\\\\widetilde{\\\\Phi }_{1}^E\\\\bigl ({v}_{1+2+3,4},{v}_{1+2+3,4}\\\\bigr )\\\\biggr \\\\rbrace ,\\\\nonumber $ where all generalized error functions are evaluated at ${x}=\\\\sqrt{2\\\\tau _2}({q}+{b})$ .\",\n \"The last terms appearing in the above quantities for $n=3$ and $n=4$ correspond to contributions of trees with one mark ($m=1$ ).\",\n \"In fact, at these orders these results can be rewritten in a simpler form, which coincides with the representations (REF ) and (REF ) (in the latter formula one should drop the sum over partitions and take $d_T=d_n$ ): $g^{(0)}_3&=& \\\\frac{(-1)^{1+\\\\gamma _{12}+\\\\gamma _{1+2,3}}}{12}\\\\,\\\\,{\\\\rm Sym}\\\\, \\\\Bigl \\\\lbrace \\\\gamma _{12}\\\\,\\\\gamma _{1+2,3}\\\\,\\\\mbox{sgn}(c_1)\\\\, \\\\mbox{sgn}(c_3) \\\\Bigr \\\\rbrace ,\\\\\\\\g^{(0)}_4&=&\\\\textstyle {\\\\frac{(-1)^{\\\\gamma _{12}+\\\\gamma _{1+2,3}+\\\\gamma _{1+2+3,4}}}{96}}\\\\,{\\\\rm Sym}\\\\, \\\\!\\\\Bigl \\\\lbrace \\\\!\",\n \"\\\\Bigl (2\\\\,\\\\gamma _{23}\\\\gamma _{1,2+3}\\\\gamma _{1+2+3,4}+\\\\gamma _{12}\\\\gamma _{34}\\\\gamma _{1+2,3+4}\\\\Bigr )\\\\mbox{sgn}(c_1)\\\\mbox{sgn}(c_{1+2})\\\\mbox{sgn}(c_4)\\\\Bigr \\\\rbrace ,\\\\nonumber $ $\\\\mathcal {E}_{123} &=&\\\\frac{(-1)^{\\\\gamma _{12}+\\\\gamma _{1+2,3}}}{24\\\\tau _2}\\\\,\\\\,{\\\\rm Sym}\\\\, \\\\left\\\\lbrace \\\\widetilde{\\\\Phi }_{2}^E\\\\bigl (({v}_{1,2+3},{v}_{1+2,3}),({v}_{12},{v}_{1+2,3})\\\\bigr )\\\\right\\\\rbrace ,\\\\\\\\\\\\mathcal {E}_{1234} &=&\\\\frac{(-1)^{\\\\gamma _{12}+\\\\gamma _{1+2,3}+\\\\gamma _{1+2+3,4}}}{96(2\\\\tau _2)^{3/2}}\\\\,\\\\,{\\\\rm Sym}\\\\, \\\\biggl \\\\lbrace 2\\\\,\\\\widetilde{\\\\Phi }_{3}^E\\\\bigl (({v}_{1,2+3+4},{v}_{1+2,3+4},{v}_{1+2+3,4}),({v}_{23},{v}_{1,2+3},{v}_{1+2+3,4})\\\\bigr )\\\\nonumber \\\\\\\\&&+\\\\widetilde{\\\\Phi }_{3}^E\\\\bigl (({v}_{1,2+3+4},{v}_{1+2,3+4},{v}_{1+2+3,4}),({v}_{12},{v}_{34},{v}_{1+2,3+4})\\\\bigr )\\\\biggr )\\\\biggr \\\\rbrace .\\\\nonumber $ The simplest way to prove the equality of the two representations of $g^{(0)}_n$ is to expand the DSZ products appearing in (REF ) into elementary $\\\\gamma _{ij}$ 's and then, using symmetrization, bring all their products to the form appearing in (REF ).\",\n \"These products are multiplied by combinations of sign functions which can be recombined with the help of the identity (REF ).\",\n \"As an example, let us perform these manipulations for $n=3$ : $\\\\begin{split}&\\\\,\\\\,{\\\\rm Sym}\\\\, \\\\Bigl \\\\lbrace \\\\gamma _{12}\\\\,\\\\gamma _{1+2,3}\\\\,\\\\mbox{sgn}(c_1)\\\\, \\\\mbox{sgn}(c_3) \\\\Bigr \\\\rbrace =\\\\,{\\\\rm Sym}\\\\, \\\\Bigl \\\\lbrace \\\\gamma _{12}\\\\,\\\\gamma _{23}\\\\bigl (\\\\mbox{sgn}(c_1)-\\\\mbox{sgn}(c_2) \\\\bigr )\\\\,\\\\mbox{sgn}(c_3)\\\\Bigr \\\\rbrace \\\\\\\\=&\\\\,\\\\,{\\\\rm Sym}\\\\, \\\\Bigl \\\\lbrace \\\\gamma _{12}\\\\,\\\\gamma _{23}\\\\Bigl (\\\\mbox{sgn}(c_1)\\\\,\\\\mbox{sgn}(c_3)+{1\\\\over 2}\\\\, \\\\mbox{sgn}(c_{1+3})\\\\bigl (\\\\mbox{sgn}(c_1)+\\\\mbox{sgn}(c_3) \\\\bigr )\\\\Bigr )\\\\Bigr \\\\rbrace \\\\\\\\=&\\\\,{1\\\\over 2}\\\\,\\\\,{\\\\rm Sym}\\\\, \\\\Bigl \\\\lbrace \\\\gamma _{12}\\\\,\\\\gamma _{23}\\\\Bigl (3\\\\, \\\\mbox{sgn}(c_1)\\\\,\\\\mbox{sgn}(c_3)+1\\\\Bigr )\\\\Bigr \\\\rbrace ,\\\\end{split}$ where we used that $c_2=-(c_1+c_3)$ .\",\n \"This identity then shows the equality of the two forms of $g^{(0)}_3$ given above.\",\n \"For $g^{(0)}_4$ the manipulations are very similar, but a bit more cumbersome.\",\n \"The equality of the two forms of $\\\\mathcal {E}_n$ follows from the equality of their asymptotics $\\\\mathcal {E}^{(0)}_n$ , which is in turn ensured by the same identities as for $g^{(0)}_n$ .\",\n \"Finally, we provide expressions for the kernels $\\\\widehat{\\\\Phi }^{{\\\\rm tot}}_n$ of the indefinite theta series appearing in the expansion (REF ) of $\\\\mathcal {G}$ in powers of $\\\\widehat{h}_{p,\\\\mu }$ .\",\n \"For the first two orders, one has $\\\\begin{split}\\\\widehat{\\\\Phi }^{{\\\\rm tot}}_1=&\\\\,\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_1,\\\\\\\\\\\\widehat{\\\\Phi }^{{\\\\rm tot}}_2=&\\\\,\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_2+\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_1 \\\\Phi ^{\\\\,\\\\widehat{g}}_2=\\\\frac{1}{4}\\\\, \\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_1\\\\left(\\\\widetilde{\\\\Phi }_1^E({u}_{12},{v}_{12})-\\\\widetilde{\\\\Phi }_1^E({v}_{12},{v}_{12})\\\\right),\\\\end{split}$ where $\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_1(x)$ is defined in (REF ).\",\n \"At the next order, $\\\\widehat{\\\\Phi }^{{\\\\rm tot}}_3=\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_3+2\\\\,{\\\\rm Sym}\\\\, \\\\Bigl \\\\lbrace \\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_2(x_{1+2},x_3)\\\\Phi ^{\\\\,\\\\widehat{g}}_2(x_1,x_2) \\\\Bigr \\\\rbrace +\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_1 \\\\Phi ^{\\\\,\\\\widehat{g}}_3.$ To get an explicit expression in terms of smooth solutions of Vignéras' equation, one should use the relation (REF ).\",\n \"Applying it to the case $n=2$ with $\\\\mathcal {V}=({u}_{1,2+3},{u}_{1+2,3})$ , using the orthogonality properties ${u}_{(1,2+3)\\\\perp (1+2,3)}={u}_{12},\\\\qquad {u}_{(1+2,3)\\\\perp (1,2+3)}={u}_{23},$ and acting by the operator $\\\\mathcal {D}({v}_{12})\\\\mathcal {D}({v}_{23})$ , one can show that $&&\\\\,{\\\\rm Sym}\\\\, \\\\Bigl \\\\lbrace \\\\widetilde{\\\\Phi }^E_{2}\\\\bigl (({u}_{1,2+3},{u}_{1+2,3}), ({v}_{12},{v}_{23})\\\\bigr )\\\\Bigr \\\\rbrace =\\\\,{\\\\rm Sym}\\\\, \\\\biggl \\\\lbrace \\\\widetilde{\\\\Phi }^M_{2}\\\\bigl (({u}_{1,2+3},{u}_{1+2,3}), ({v}_{12},{v}_{23})\\\\bigr )\\\\nonumber \\\\\\\\&& \\\\qquad +({v}_{12},{x})\\\\,\\\\mbox{sgn}({u}_{12},{x})\\\\, \\\\widetilde{\\\\Phi }^M_1\\\\bigl ({u}_{1+2,3},{v}_{1+2,3}\\\\bigr )+({v}_{12},{x})\\\\,({v}_{23},{x})\\\\,\\\\mbox{sgn}({u}_{1,2+3},{x})\\\\,\\\\mbox{sgn}({u}_{1+2,3},{x})\\\\nonumber \\\\\\\\&& \\\\qquad -\\\\frac{(p_1p_2p_3)}{6\\\\pi }\\\\biggr \\\\rbrace .$ This result allows to obtain the following representation for the kernel $\\\\widehat{\\\\Phi }^{{\\\\rm tot}}_3&=&\\\\frac{1}{8}\\\\,\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_1 \\\\,{\\\\rm Sym}\\\\, \\\\biggl \\\\lbrace \\\\widetilde{\\\\Phi }^E_{2}\\\\bigl (({u}_{1,2+3},{u}_{1+2,3}), ({v}_{12},{v}_{23})\\\\bigr )- \\\\widetilde{\\\\Phi }^E_{2}\\\\bigl (({v}_{1,2+3},{v}_{1+2,3}), ({v}_{12},{v}_{23})\\\\bigr )\\\\nonumber \\\\\\\\&&-\\\\left(\\\\widetilde{\\\\Phi }_1^E({u}_{1+2,3},{v}_{1+2,3})-\\\\widetilde{\\\\Phi }_1^E({v}_{1+2,3},{v}_{1+2,3})\\\\right)\\\\widetilde{\\\\Phi }^E_1({v}_{12},{v}_{12})\\\\biggr \\\\rbrace .$ Using the identity between functions $\\\\widetilde{\\\\Phi }^E_{2}$ implied by the identity (REF ), the kernel can also be rewritten as $\\\\widehat{\\\\Phi }^{{\\\\rm tot}}_3&=&\\\\frac{1}{8}\\\\,\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_1 \\\\,{\\\\rm Sym}\\\\, \\\\biggl \\\\lbrace \\\\frac{2}{3}\\\\, \\\\Bigl (\\\\widetilde{\\\\Phi }^E_{2}\\\\bigl (({u}_{1,2+3},{u}_{1+2,3}), ({v}_{12},{v}_{1+2,3})\\\\bigr )- \\\\widetilde{\\\\Phi }^E_{2}\\\\bigl (({v}_{1,2+3},{v}_{1+2,3}), ({v}_{12},{v}_{1+2,3})\\\\bigr )\\\\Bigr )\\\\nonumber \\\\\\\\&&-\\\\left(\\\\widetilde{\\\\Phi }_1^E({u}_{1+2,3},{v}_{1+2,3})-\\\\widetilde{\\\\Phi }_1^E({v}_{1+2,3},{v}_{1+2,3})\\\\right)\\\\widetilde{\\\\Phi }^E_1({v}_{12},{v}_{12})\\\\biggr \\\\rbrace ,$ so that the vectors appearing in the second argument of $\\\\widetilde{\\\\Phi }_2^E$ are now mutually orthogonal.\",\n \"For $n=4$ , the kernel is given by $\\\\begin{split}\\\\widehat{\\\\Phi }^{{\\\\rm tot}}_4=&\\\\, \\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_4+\\\\,{\\\\rm Sym}\\\\, \\\\Bigl \\\\lbrace 3\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_3(x_{1+2},x_3,x_4)\\\\Phi ^{\\\\,\\\\widehat{g}}_2(x_1,x_2)+\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_2(x_{1+2},x_{3+4})\\\\Phi ^{\\\\,\\\\widehat{g}}_2(x_1,x_2)\\\\Phi ^{\\\\,\\\\widehat{g}}_2(x_3,x_4)\\\\\\\\&\\\\,+2\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_2(x_{1+2+3},x_4)\\\\Phi ^{\\\\,\\\\widehat{g}}_3(x_1,x_2,x_3)\\\\Bigr \\\\rbrace +\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_1 \\\\Phi ^{\\\\,\\\\widehat{g}}_4.\\\\end{split}$ Proceeding in the same way as for $n=3$ , obtaining a generalization of (REF ) to $n=4$ , one arrives at $\\\\widehat{\\\\Phi }^{{\\\\rm tot}}_4&=&\\\\frac{1}{16}\\\\, \\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_1\\\\,{\\\\rm Sym}\\\\, \\\\biggl \\\\lbrace \\\\widetilde{\\\\Phi }^E_{3}\\\\bigl (({u}_{1,2+3+4},{u}_{1+2,3+4},{u}_{1+2+3,4}), ({v}_{12},{v}_{23},{v}_{34})\\\\bigr )\\\\nonumber \\\\\\\\&&\\\\qquad -\\\\widetilde{\\\\Phi }^E_{3}\\\\bigl (({v}_{1,2+3+4},{v}_{1+2,3+4},{v}_{1+2+3,4}), ({v}_{12},{v}_{23},{v}_{34})\\\\bigr )\\\\nonumber \\\\\\\\&&\\\\quad +\\\\frac{1}{3}\\\\left(\\\\widetilde{\\\\Phi }^E_{3}\\\\bigl (({u}_{1,2+3+4},{u}_{1+2+4,3},{u}_{1+2+3,4}), ({v}_{12},{v}_{24},{v}_{34})\\\\bigr )\\\\right.\\\\nonumber \\\\\\\\&&\\\\qquad \\\\left.-\\\\widetilde{\\\\Phi }^E_{3}\\\\bigl (({v}_{1,2+3+4},{v}_{1+2+4,3},{v}_{1+2+3,4}), ({v}_{12},{v}_{24},{v}_{34})\\\\bigr )\\\\right)\\\\nonumber \\\\\\\\&&-\\\\left(\\\\widetilde{\\\\Phi }^E_{2}\\\\bigl (({u}_{1+2,3+4},{u}_{1+2+3,4}), ({v}_{1+2,3},{v}_{34})\\\\bigr )-\\\\widetilde{\\\\Phi }^E_{2}\\\\bigl (({v}_{1+2,3+4},{v}_{1+2+3,4}), ({v}_{1+2,3},{v}_{34})\\\\bigr )\\\\right.\\\\nonumber \\\\\\\\&&\\\\quad +{1\\\\over 2}\\\\left(\\\\widetilde{\\\\Phi }^E_{2}\\\\bigl (({u}_{1+2+4,3},{u}_{1+2+3,4}), ({v}_{1+2,3},{v}_{1+2,4})\\\\bigr )\\\\right.\\\\\\\\&&\\\\quad \\\\qquad \\\\left.\\\\left.-\\\\widetilde{\\\\Phi }^E_{2}\\\\bigl (({v}_{1+2+4,3},{v}_{1+2+3,4}), ({v}_{1+2,3},{v}_{1+2,4})\\\\bigr )\\\\right)\\\\right)\\\\widetilde{\\\\Phi }^E_1({v}_{12},{v}_{12})\\\\nonumber \\\\\\\\&&-\\\\left(\\\\widetilde{\\\\Phi }_1^E({u}_{1+2+3,4},{v}_{1+2+3,4})-\\\\widetilde{\\\\Phi }_1^E({v}_{1+2+3,4},{v}_{1+2+3,4})\\\\right)\\\\widetilde{\\\\Phi }^E_{2}\\\\bigl (({v}_{1,2+3},{v}_{1+2,3}), ({v}_{12},{v}_{23})\\\\bigr )\\\\nonumber \\\\\\\\&&+\\\\left(\\\\widetilde{\\\\Phi }_1^E({u}_{1+2+3,4},{v}_{1+2+3,4})-\\\\widetilde{\\\\Phi }_1^E({v}_{1+2+3,4},{v}_{1+2+3,4})\\\\right)\\\\widetilde{\\\\Phi }^E_1({v}_{12},{v}_{12})\\\\widetilde{\\\\Phi }^E_1({v}_{1+2,3},{v}_{1+2,3})\\\\nonumber \\\\\\\\&&+\\\\frac{1}{4}\\\\left(\\\\widetilde{\\\\Phi }_1^E({u}_{1+2,3+4},{v}_{1+2,3+4})-\\\\widetilde{\\\\Phi }_1^E({v}_{1+2,3+4},{v}_{1+2,3+4})\\\\right)\\\\widetilde{\\\\Phi }^E_1({v}_{12},{v}_{12})\\\\widetilde{\\\\Phi }^E_1({v}_{34},{v}_{34})\\\\biggr \\\\rbrace .\\\\nonumber $ It is possible also to rewrite this expression in terms of generalized error functions $\\\\widetilde{\\\\Phi }^E_n(\\\\mathcal {V},\\\\tilde{\\\\mathcal {V}})$ where the vectors entering the second argument are mutually orthogonal, as in (REF ).\",\n \"The reader can easily guess the result by comparing (REF ) and (REF ).\",\n \"The explicit results for $\\\\widehat{\\\\Phi }^{{\\\\rm tot}}_n$ , $n\\\\le 4$ , presented above are the basis for the conjectural formula (REF ).\",\n \"Note that all terms in these expressions have the sum of ranks of the generalized error functions equal to $n-1$ .\",\n \"This shows that all contributions due to trees with non-zero number of marks cancel in the sum over Schröder trees.\",\n \"toc\"\n ],\n [\n \"Index of notations\",\n \"lp10cml Symbol Description Appears/defined in $\\\\mathcal {A}(\\\\mathcal {T}) $ contribution of tree $\\\\mathcal {T}$ to the integrand of the multi-instanton expansion of $H_\\\\gamma $ and $\\\\mathcal {G}$ (REF ) $a_\\\\mathcal {T}$ coefficient of unrooted labelled tree $\\\\mathcal {T}$ in $\\\\mathcal {D}_{m}(\\\\lbrace \\\\check{\\\\gamma }_s\\\\rbrace )$ (REF ), (REF ) $\\\\beta _{k\\\\ell }$ DSZ product $\\\\langle \\\\gamma _1+\\\\dots +\\\\gamma _k,\\\\gamma _\\\\ell \\\\rangle $ (REF ) $b_2=b_2(\\\\mathfrak {Y})$ second Betti number of $\\\\mathfrak {Y}$ p. REF $b^a=\\\\,{\\\\rm Re}\\\\,(z^a)$ periods of the Kalb-Ramond field p. REF $b_n$ rational coefficients in the expansion of $F^{\\\\rm (ref)}_n$ (REF ), (REF ) $c_i$ stability parameters (REF ) $c_i^{(\\\\ell )}$ stability parameters after attractor flow (REF ) $c_{2,a}$ components of the second Chern class of $\\\\mathfrak {Y}$ (REF ) $d=n b_2$ dimension of the lattice $\\\\mathbf {\\\\Lambda }=\\\\oplus _{i=1}^n \\\\Lambda _i$ p. REF $d_n,d_T$ rational weights in the representation of $g^{(0)}_n$ via flow tree (REF ),(REF ) $\\\\Delta (T)$ , $\\\\Delta _{\\\\gamma _L\\\\gamma _R}^z$ sign factors assigned to attractor flow tree $T$ (REF ) $\\\\mathcal {D}_{\\\\mathfrak {h}}$ Maass raising operator (REF ) $\\\\mathcal {D}({v})$ modular-covariant derivative contracted with vector ${v}$ (REF ) $\\\\mathcal {D}_{m}(\\\\lbrace \\\\check{\\\\gamma }_s\\\\rbrace ) $ derivative operator assigning weight to vertices with $m$ marks (REF ) $E_n(\\\\mathcal {M};\\\\mathbb {u})$ generalized error function on ${R}^n$ () $\\\\mathcal {E}_n=\\\\mathcal {E}^{(0)}_n+\\\\mathcal {E}^{(+)}_n$ function encoding the modular completion (REF ), (REF ) $\\\\phi $ (logarithm of) contact potential on $\\\\mathcal {M}_H$ (REF ) $\\\\Phi _n^E$ , $\\\\Phi _n^M$ boosted (complementary) error functions (REF ) $\\\\widetilde{\\\\Phi }_{n,m}^E$ , $\\\\widetilde{\\\\Phi }_{n,m}^M$ uplifted boosted error functions in the kernel of $\\\\mathcal {V}_ m$ (denoted by $\\\\widetilde{\\\\Phi }^E_n$ , $\\\\widetilde{\\\\Phi }^M_n$ when $n=m$ ) (REF ) $\\\\Phi _\\\\mathcal {T}$ contribution of tree $\\\\mathcal {T}$ to $\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_n$ (REF ), (REF ) $\\\\Phi ^{\\\\scriptscriptstyle \\\\,\\\\int }_n$ kernel defined by twistorial integrals (REF ), (REF ) $\\\\Phi ^{\\\\,\\\\mathcal {E}}_n$ kernel corresponding to function $\\\\mathcal {E}_n$ (REF ) $\\\\widetilde{\\\\Phi }^{\\\\,\\\\mathcal {E}}_n$ kernel promoting $g^{(0)}_n$ to a solution of Vignéras' equation (REF ) $\\\\Phi ^{\\\\,g}_{n}$ kernel corresponding to the tree index (REF ), (REF ) $\\\\Phi ^{\\\\,\\\\widehat{g}}_{n}$ kernel corresponding to completed tree index $\\\\widehat{g}_n$ (REF ) $\\\\Phi ^{{\\\\rm tot}}_n$ total kernel in the expansion of $\\\\mathcal {G}$ in terms of $h_{p,\\\\mu }$ (REF ) $\\\\widehat{\\\\Phi }^{{\\\\rm tot}}_n$ total kernel in the expansion of $\\\\mathcal {G}$ in terms of $\\\\widehat{h}_{p,\\\\mu }$ (REF ) $F(X)$ holomorphic prepotential p. REF $F_{{\\\\rm tr},n} (\\\\lbrace \\\\gamma _{i}\\\\rbrace ,z^a)$ partial tree index (REF ) $F^{\\\\rm (ref)}_{n} (\\\\lbrace c_i\\\\rbrace )$ partial contribution in $g^{\\\\rm (ref)}_n$ (REF ) $\\\\mathcal {F}$ image of $\\\\mathcal {G}$ under Euler operator (REF ) $\\\\Gamma $ charge lattice inside $H_{\\\\rm even}(\\\\mathfrak {Y},{Q})$ (REF ) $\\\\Gamma _+$ positive cone in the charge lattice (REF ) $\\\\Gamma _{k\\\\ell }, \\\\Gamma _e$ sums of DSZ products (REF ), (REF ) $\\\\gamma = (0,p^a,q_a,q_0)$ charge vector of a generic D4 (or D3) brane (REF ) $\\\\check{\\\\gamma }=(p^a,q_a)$ projection of $\\\\gamma $ on $H_4(\\\\mathfrak {Y},\\\\mathbb {Q})\\\\oplus H_2(\\\\mathfrak {Y},\\\\mathbb {Q})$ (REF ) $\\\\gamma _{ij}=\\\\langle \\\\gamma _i,\\\\gamma _j\\\\rangle $ Dirac-Schwinger-Zwanziger product, or Euler pairing (REF ) $g_{{\\\\rm tr},n}(\\\\lbrace \\\\gamma _i\\\\rbrace ,z^a)$ tree index, also denoted by $g_{{\\\\rm tr},n}(\\\\lbrace \\\\gamma _i,c_i\\\\rbrace )$ (REF ) $\\\\widehat{g}_n(\\\\lbrace \\\\check{\\\\gamma }_i\\\\rbrace ,z^a,\\\\tau _2)$ completed tree index, also denoted by $\\\\widehat{g}_n(\\\\lbrace \\\\check{\\\\gamma }_i,c_i\\\\rbrace )$ (REF ), (REF ) $g^{(0)}_n(\\\\lbrace \\\\check{\\\\gamma }_i,c_i\\\\rbrace )$ seed term in recursion for $\\\\widehat{g}_n$ (REF ) $g^{\\\\rm (ref)}_n(\\\\lbrace \\\\check{\\\\gamma }_i,c_i\\\\rbrace ,y)$ refined version of $g^{(0)}_n$ (REF ) $\\\\mathcal {G}$ instanton generating function (REF ) $\\\\mathcal {G}_n(\\\\lbrace \\\\gamma _i,z_i\\\\rbrace )$ integrand in the $n$ -instanton contribution to $\\\\mathcal {G}$ (REF ) $G_n(\\\\lbrace \\\\check{\\\\gamma }_i,c_i\\\\rbrace ;\\\\tau _2)$ large ${x}$ limit of $\\\\widetilde{\\\\Phi }_n({x})$ (REF ), (REF ) $h^{\\\\rm DT}_{p,q}(\\\\tau ,z^a)$ generating function of DT invariants (REF ) $h_{p,\\\\mu }(\\\\tau ) $ generating function of MSW invariants () $\\\\widehat{h}_{p,\\\\mu }(\\\\tau )$ modular completion of $h_{p,\\\\mu }(\\\\tau )$ (REF ) $H_\\\\gamma (z)$ generator of contact transformation across $\\\\ell _\\\\gamma $ (REF ) $H^{\\\\rm cl}_\\\\gamma (z)$ classical, large volume limit of $H_\\\\gamma (z)$ (REF ) $\\\\kappa (T)$ weight of attractor flow tree $T$ (REF ) $\\\\kappa (x)$ BPS index of two-centered solutions (REF ) $\\\\kappa _{abc}$ intersection numbers on a fixed basis of $H_4(\\\\mathfrak {Y})$ p. REF $K_{\\\\gamma _1\\\\gamma _2}(z_1,z_2)$ integration kernel in large volume limit (REF ) $\\\\hat{K}_{ij}(z_i,z_j)$ rescaled integration kernel in large volume limit (REF ) $\\\\mathcal {J}_n(\\\\lbrace \\\\check{\\\\gamma }_i\\\\rbrace ,\\\\tau _2)$ coefficient in the formula for the shadow of $\\\\widehat{h}_{p,\\\\mu }$ (REF ) $\\\\Lambda =H_4(\\\\mathfrak {Y},{Z})$ lattice equipped with quadratic form $\\\\kappa _{ab}=\\\\kappa _{abc} p^a$ p. REF $\\\\Lambda _i=H_4(\\\\mathfrak {Y},{Z})$ lattice equipped with quadratic form $\\\\kappa _{i,ab}=\\\\kappa _{abc} p_i^a$ p. REF $\\\\mathbf {\\\\Lambda }$ lattice $\\\\mathbf {\\\\Lambda }=\\\\oplus _{i=1}^n \\\\Lambda _i$ spanned by $n$ D1-brane charges p. REF $\\\\lambda $ eigenvalue under Vignéras' operator (REF ) $\\\\ell _\\\\gamma $ BPS ray on the twistor fiber, or its large volume limit (REF ), p. REF $\\\\mu _a$ residue class of $q_a$ modulo spectral flow (REF ) $M_n(\\\\mathcal {M};\\\\mathbb {u})$ generalized complementary error function on ${R}^n$ (REF ) $\\\\mathcal {M}_{\\\\alpha \\\\beta }$ matrix of parameters in the generalized error functions (REF ), () $\\\\mathcal {M}_{\\\\mathcal {K}}(\\\\mathfrak {Y})$ complexified Kähler moduli space of $\\\\mathfrak {Y}$ p. REF $\\\\mathcal {M}_H$ hypermultiplet moduli space in IIB/$\\\\mathfrak {Y}$ , or vector multiplet moduli space in IIA/$(\\\\mathfrak {Y}\\\\times S^1)=$ M/$(\\\\mathfrak {Y}\\\\times T^2)$ p. $n_T$ number of vertices of rooted tree $T$ excluding the leaves (REF ) $n_v(T)$ number of descendants of the vertex $v$ in $T$ plus one (REF ) $n_\\\\mathfrak {v}$ valency of vertex $\\\\mathfrak {v}$ of an unrooted tree (REF ) $p^a$ homology class of the divisor wrapped by the D3-brane (REF ) $\\\\mathcal {P}_{m}(\\\\lbrace p_s\\\\rbrace )$ weight of a vertex with $m$ marks in $G_n$ (REF ) $\\\\hat{q}_0$ invariant D0-brane charge (REF ) $Q_n(\\\\lbrace \\\\check{\\\\gamma }_i\\\\rbrace )$ difference of quadratic forms for constituents and the total charge (REF ) $R_n(\\\\lbrace \\\\check{\\\\gamma }_i\\\\rbrace ,\\\\tau _2)$ non-holomorphic correction in the completion $\\\\widehat{h}_{p,\\\\mu }(\\\\tau )$ (REF ), (REF ) $\\\\sigma _\\\\gamma $ quadratic refinement (REF ), (REF ) $S_k, S_e$ sums of stability parameters (REF ), (REF ) $S^{\\\\rm cl}_p$ classical action of a D3-instanton in large volume limit (REF ) $\\\\vartheta _{{p},{\\\\mu }}\\\\bigl (\\\\Phi ,\\\\lambda )$ indefinite theta series with kernel $\\\\Phi $ (REF ) $\\\\tau =\\\\tau _1+I\\\\tau _2$ 4D axio-dilaton in IIA, or torus modulus in M theory p. REF $t$ complex coordinate on the twistor fiber p. REF $t^a=\\\\,{\\\\rm Im}\\\\,(z^a)$ Kähler moduli on $\\\\mathfrak {Y}$ p. REF $T$ rooted tree with charges assigned to the leaves footnote REF $\\\\mathcal {T}$ tree with charges assigned to vertices footnote REF ${T}_n, {T}_n^\\\\ell $ set of unrooted (labelled) trees with $n$ vertices (REF ) ${T}_{n,m}, {T}_{n,m}^\\\\ell $ set of unrooted (labelled) trees with $n$ vertices and $m$ marks (REF ) ${T}_n^{\\\\rm r}$ set of rooted trees with $n$ vertices (REF ) ${T}_n^{\\\\rm af}$ set of attractor flow trees with $n$ leaves (REF ) ${T}_n^{\\\\rm S}$ set of Schröder trees with $n$ leaves (REF ) ${T}_{2m+1}^{(3)}$ set of rooted ternary trees with $n$ leaves (REF ) $u^\\\\Lambda =(1,u^a)$ complex structure moduli of mirror threefold ${\\\\widehat{\\\\mathfrak {Y}}}$ p. REF ${u}_{ij}$ , ${u}_e$ , ${u}_\\\\ell $ vectors in ${R}^d$ associated to $-2\\\\,{\\\\rm Im}\\\\,[Z_{\\\\gamma _i}\\\\bar{Z}_{\\\\gamma _j}]$ , $-S_e$ and $-S_\\\\ell $ (REF ), (REF ), (REF ) ${v}_{ij}$ , ${v}_e$ , ${v}_\\\\ell $ vectors in ${R}^d$ associated to $\\\\langle \\\\gamma _i,\\\\gamma _j\\\\rangle $ , $\\\\Gamma _e$ and $-\\\\Gamma _{n\\\\ell }$ (REF ), (REF ), (REF ) $V_T$ set of vertices of rooted tree $T$ excluding leaves p. REF $V_\\\\lambda $ Vignéras' operator (REF ) $\\\\mathcal {V}_m$ weight of a vertex with $m$ marks in large ${x}$ limit of $\\\\widetilde{\\\\Phi }^{\\\\,\\\\mathcal {E}}_n$ (REF ) $\\\\tilde{\\\\mathcal {V}}_m$ weight of a vertex with $m$ marks in $g^{(0)}_n$ (REF ) $W_n(\\\\lbrace \\\\check{\\\\gamma }_i\\\\rbrace ,\\\\tau _2)$ coefficient in the formula for $h_{p,\\\\mu }$ in terms of $\\\\widehat{h}_{p_i,\\\\mu _i}$ (REF ) $\\\\mathcal {X}_\\\\gamma $ holomorphic Fourier modes on the twistor space of $\\\\mathcal {M}_H$ (REF ) $\\\\mathcal {X}^{\\\\rm sf}_\\\\gamma $ semi-flat limit of $\\\\mathcal {X}_\\\\gamma $ (REF ) $\\\\mathcal {X}^{\\\\rm cl}_\\\\gamma $ classical limit of $\\\\mathcal {X}_\\\\gamma $ (REF ) $\\\\mathcal {X}^{(\\\\theta )}_{p,q}$ $\\\\hat{q}_0$ -independent part of $\\\\mathcal {X}^{\\\\rm cl}_\\\\gamma $ (REF ) ${x}$ $d$ -dimensional vector, argument of kernels of theta series p. REF $z$ coordinate on the twistor fiber, after Cayley transf.\",\n \"(REF ) $z_\\\\gamma $ saddle point on twistor fiber (REF ) $z^a=b^a+It^a$ complexified Kähler moduli of $\\\\mathfrak {Y}$ p. REF $z^a_*(\\\\gamma )$ attractor moduli for charge $\\\\gamma $ p. REF $z^a_\\\\infty (\\\\gamma )$ large volume attractor point for charge $\\\\gamma $ (REF ) $Z_\\\\gamma (z^a)$ central charge (REF ) $\\\\Omega (\\\\gamma ,z^a)$ generalized Donaldson-Thomas invariant p. REF $\\\\bar{\\\\Omega }(\\\\gamma ,z^a)$ rational DT invariant (REF ) $\\\\bar{\\\\Omega }_*(\\\\gamma )$ attractor index p. REF ${\\\\bar{\\\\Omega }}^{\\\\rm MSW}(\\\\gamma )$ MSW invariant, also denoted by $\\\\bar{\\\\Omega }_{p,\\\\mu }( \\\\hat{q}_0)$ p. REF toc\"\n ]\n]"},"id":{"kind":"string","value":"1808.08479"}}},{"rowIdx":293,"cells":{"text":{"kind":"list like","value":[["Massive gravity with Lorentz symmetry breaking: black holes as heat\n engines"],["Abstract In extended phase space, a static black hole in massive gravity is studied as a holographic heat engine.","In the massive gravity theory considered, the graviton gain a mass due to Lorentz symmetry breaking.","Exact efficiency formula is obtained for a rectangle engine cycle for the black hole considered.","The efficiency is computed by varying two parameters in the theory, the scalar charge Q and $\\lambda$.","The efficiency is compared with the Carnot efficiency for the heat engine.","It is observed that when Q and $\\lambda$ are increased that the efficiency for the rectangle cycle increases.","When compared to the Schwarzschild AdS black hole, the efficiency for the rectangle cycle is larger for the Massive gravity black hole."],[" Introduction","Black holes in anti-de Sitter space as a thermodynamical system has attracted lot of attention in a variety of contexts: when the negative cosmological constant is taken as the thermodynamical pressure of the black hole with the relation $ P = - \\frac{\\Lambda }{ 8 \\pi }$ , the resulting thermodynamics lead to interesting features.","In this extended phase space, the first law of thermodynamics is modified by a term $V dP$ and the mass $M$ of the black hole is treated as the enthalpy rather than the internal energy $E$ of the black hole [1] [2].","Many black holes in the context of extended phase space has demonstrated Van der Waals type phase transitions between small and large black holes.","Due to the large number of work published related to this topic we will mention few here: [3] [4] [5] [6] [7][8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18][19].","There is a nice review on Black hole chemistry written by Kubiznak et.al.","which gives a comprehensive summary of the interesting thermodynamical features of black holes with a cosmological constant [20].","In classical thermodynamics, there are four basic thermodynamical processes.","They are isothermal, adiabatic, isobaric, and isochoric processes.","In each of these processes the thermodynamical quantities temperature, entropy, pressure, and, volume are kept constant respectively.","In a heat engine, a thermodynamical cycle is chosen which consist of aforementioned processes.","For example, the Carnot cycle, which has the highest efficiency, have two isothermal and two adiabatic processes in the cycle.","Brayton cycle has two isobaric and two adiabatic processes.","In this paper, the goal is to study a black hole in massive gravity as a heat engine.","The idea that black holes could be used as a working substance in a heat engine was first presented by Johnson [21].","In that paper, charged black hole in AdS space in $D = 4$ was presented as an example.","In an extension of that work, Johnson presented the Born-Infeld AdS black hole as a heat engine in [22].","In [23], heat engines from dilatonic Born-Infeld black holes were analyzed.","Black holes in conformal gravity as heat engine was presented by Xu et.al.","in [24].","Charged BTZ black hole in 2+1 dimensions as a heat engine was studied by Mo et.al.","in [25].","Effects of dark energy on the efficiency of heat engines of AdS black holes were analyzed in detail by Liu and Meng in [26].","Class of black holes in massive gravity as heat engines were discussed by Hendi et.al.","[27].","The first study of rotating black holes as holographic heat engines were done by Henniger et.al.","[28].","Heat engines are defined for space-times that are not black holes as well: Johnson [29] studied Taub-Bolt space-time as an example of a heat engine and compared with the analogous Schwarzschild black hole as a heat engine.","Massive gravity theories have become popular as a mean of explaining accelerated expansion of the universe without having to introduce a component of “dark energy.” There are many theories of massive gravity in the literature.","There are large volume of work related to theories in massive gravity [30] [31] [32] [33]: here we will mention two nice reviews on the subject by de Rham [34] and Hinterbichler [35].","In this paper we will consider a massive gravity theory where the graviton acquire a mass due to the Lorenz symmetry breaking.","Here, Higgs mechanism for gravity is introduced with space-time depending scalar fields that are coupled to gravity [36].","The resulting theory will exhibit modified gravitational interactions at large scale.","These models are free from ghosts and tachyonic instabilities around curved space as well as in flat space [37] [38].","A review of Lorentz violating massive gravity theory can be found in [39] [40].","The details of the theory is discussed in section 2.","The paper is organized as follows: in section 2, the black hole in massive gravity is introduced.","Thermodynamics and phase transitions of the massive gravity black hole are discussed in section 3.","Black hole as a heat engine with a rectangle cycle is introduced in section 4 and in section 5 the efficiencies are computed.","Finally the conclusion is given in section 6."],[" Introduction to AdS black holes in massive gravity with Lorentz symmetry breaking","The action for the massive gravity theory considered in this paper given by, $S = \\int d^4 x \\sqrt{-g } \\left[ - \\frac{1}{ 16 \\pi } \\mathcal {R} + \\Omega ^4 \\mathcal {F}( X, W^{ij}) \\right]$ Here $\\mathcal {F}$ is a function of our scalar fields $\\phi ^{\\mu }$ .","$\\phi ^{\\mu }$ are minimally coupled to gravity by covariant derivatives.","$\\mathcal {F}$ depends on two combinations of the scalar fields given by $X$ and $W$ defined in terms of the scalar fields as, $X = \\frac{\\partial ^{\\mu } \\phi ^0 \\partial _{\\mu } \\phi ^0 }{ \\Omega ^4}$ $W^{i j} = \\frac{\\partial ^{\\mu } \\phi ^i \\partial _{\\mu } \\phi ^j}{ \\Omega ^4} - \\frac{\\partial ^{\\mu } \\phi ^i \\partial _{\\mu } \\phi ^0 \\partial ^{\\nu } \\phi ^j \\partial _{\\nu } \\phi ^0 }{ \\Omega ^8 X}$ The constant $\\Omega $ has dimensions of mass: it is in the order of $ \\sqrt{ m_g M_{pl}}$ where $m_g$ is the graviton mass and $M_{pl}$ the Plank mass [39] [36] [37] [38].","The scalar fields $\\phi ^0, \\phi ^i$ are responsible for breaking Lorentz symmetry spontaneously when they acquire a vacuum expectation value.","More details on the theory can be found in [39] [36] [37] [38] [40].","A class of black hole solutions to the above action was derived in [41] and [42].","$ ds^2 = - f(r) dt^2 + \\frac{ dr^2}{ f(r)} + r^2 ( d \\theta ^2 + sin^2 \\theta d \\phi ^2)$ where, $ f(r) = 1 - \\frac{ 2 M}{ r} - \\gamma \\frac{ Q^2}{r^{\\lambda }} - \\frac{\\Lambda r^2}{3}$ Here, the parameter $Q$ represents a scalar charge related to massive gravity and $\\gamma = \\pm 1$ .","The constant $\\lambda $ in is an integration constant and is positive.","$\\Lambda $ is the cosmological constant.","A detailed description of this black hole is given by Fernando in [43].","Since for $\\lambda <1$ , the ADM mass become divergent, such solutions will not be considered.","When $\\lambda >1$ , for $ r \\rightarrow \\infty $ the metric approaches the Schwarzschild-AdS black hole with a finite mass $M$ ; hence we will choose $\\lambda >1$ in the rest of the paper.","When $ \\gamma = 1$ , the space-time of the black hole is very similar to the Schwarzschild-AdS black hole and for $\\gamma = -1$ , the geometry is similar to the Reissner-Nordstrom-AdS charged black hole.","There are several works related to the black holes given above.","P-V criticality and phase transitions of masive gravity black holes with a negative cosmological constant were presented by Fernando in [43] [44].","Thermodynamics for $\\Lambda =0$ case were studied in [45] [46].","Stability and quasi normal modes were studied in [47][48] [49] [50]."],["Thermodynamics and phase transitions","In this section we will derive the thermodynamical quantities and present the phase transitions that could occur in massive gravity black holes."],[" Defining the thermodynamical quantities","The temperature of the black hole could be obtained by using the definition of Hawking temperature: it is based on the surface gravity $\\kappa $ at the outer horizon, $r_h$ .","It is given by, $T_H = \\frac{ \\kappa }{ 2 \\pi } = \\frac{ 1}{ 4 \\pi } \\left| \\frac{ df(r)}{ dr} \\right|_ { r = r_h} = \\frac{1}{ 4 \\pi } \\left(\\frac{ 2 M}{ r_h^2} + \\frac{ \\gamma Q^2 \\lambda }{ r_h^{ \\lambda + 1}} - \\frac{ 2 \\Lambda r_h}{ 3}\\right)$ Since $ f(r_h) = 0$ , the mass of the black hole could be written as, $M = r_h \\left( \\frac{1}{2} - \\frac{ \\gamma Q^2}{ 2 r_h^{( \\lambda )} }- \\frac{ r_h^2 \\Lambda }{6} \\right)$ The value of the mass $M$ could be substituted to eq$(\\ref {tempe1})$ and rewrite temperature in terms of $r_h, Q,$ and $\\lambda $ as, $T = \\frac{ 1 }{ 4 \\pi } \\left( \\frac{1}{ r_h} - r_h \\Lambda + \\frac{ \\gamma ( \\lambda -1)Q^2}{ r_h^{ \\lambda +1} } \\right)$ The conjugate quantity for temperature, the entropy S is given by the area law, $ S = \\pi r_h^2$ for this black hole.","As discussed in the introduction, in the extended phase space, the pressure $P$ is given by, $P = -\\frac{ \\Lambda }{ 8 \\pi }$ The conjugate quantity for P, the volume $V$ , is given by, $\\frac{ 4 \\pi r_h^3}{3}$ .","The scalar potential which is conjugated to the scalar charge $Q$ is given by, $\\Phi = - \\frac{ \\gamma Q}{ r_h^{ \\lambda -1}}$ One could rewrite the temperature in terms of $S$ and $P$ as follows: $T_H = \\frac{1}{ 4 \\sqrt{ \\pi S}} \\left( 1 + 8 P S + \\gamma ( \\lambda - 1) Q^2 \\left(\\frac{\\pi }{ S}\\right)^{\\lambda /2} \\right)$ The temperature is plotted against the entropy $S$ in Fig.$(\\ref {temp1})$ and Fig.$(\\ref {temp2})$ .","For $\\gamma =1$ , the temperature has a minimum: hence black holes cannot exist below this minimum for a given $\\Lambda $ value.","For $\\gamma =-1$ , the temperature could have an inflection point.","Figure: The figure shows TT vs SS for γ=1 \\gamma = 1.","Here λ=2.305,P=0.905\\lambda = 2.305, P = 0.905, and Q=0.115 Q = 0.115.Figure: The figure shows TT vs SS for γ=-1 \\gamma =-1.","Here λ=2.215\\lambda = 2.215, and P=0.05 P = 0.05.","The top graph is at Q=0Q=0 and the rest have Q=0.128,0.222,0.5Q = 0.128, 0.222, 0.5"],[" First law in the extended phase space","In the extended phase space, the mass of the black hole M is not considered as the internal energy as it is usually done.","Instead, it is considered as the enthalpy H. Hence, $ M = H = U + P V$ .","Hence, the first law of the given black hole is given by, $dM = T dS + \\Phi dQ + V dP$ It is possible to combine the thermodynamical quantities, $M, P, V, S, T, \\Phi $ , and $Q$ to obtain the Smarr formula as, $M = 2 T S + \\frac{\\lambda }{2} \\Phi Q - 2 P V$ which also could be obtained using the scaling argument presented by Kastor et.al [1].","When $\\lambda =2$ , the Smarr formula simplifies to the one for the Reissner-Nordstrom-AdS black hole obtained by Kubiznak and Mann in [3]."],[" Behavior of pressure","In eq.$(\\ref {tempe1})$ , one could substitute $ \\Lambda = - 8 \\pi P$ and rearrange the equation to obtain $P$ as a function of $r_h$ and $T$ as, $P = - \\frac{ 1}{ 8 \\pi r_h^2} + \\frac{ T}{ 2 r_h} + \\frac{ Q^2 \\gamma ( 1 - \\lambda )}{ 8 \\pi r_h^{ 2 + \\lambda }}$ Since the black hole radius $r_h$ is given by, $r_h = \\left(\\frac{ 3 V}{4 \\pi }\\right)^{1/3}$ one could rewrite $P$ in terms of $V$ as, $P = - \\frac{ 1}{ 8 \\pi } \\left( \\frac{ 4 \\pi }{ 3 V} \\right) ^ {2/3} + \\frac{T}{2} \\left( \\frac{ 4 \\pi }{ 3 V} \\right) ^ {1/3} +\\frac{ Q^2 \\gamma ( 1 - \\lambda )}{ 8 \\pi } \\left( \\frac{ 4 \\pi }{ 3 V} \\right) ^ {(\\lambda + 2)/3}$ The pressure is plotted vs V as in Fig.$(\\ref {pvsr2})$ and Fig.$(\\ref {pvsr1})$ .","For $\\gamma =1$ , there is a maximum pressure given by, $P_{max} = \\frac{ 4 \\pi r_h T ( 1 + \\lambda ) - \\lambda }{ 8 \\pi r_h^2 ( 2 + \\lambda )}$ Since $r_h$ and $V$ are related, $P_{max}$ also could be rewritten in terms of $V_m$ at which the pressure is maximum as, $P_{max} = \\frac{ \\left( (4 \\pi )^{2/3} ( 3 V_m)^{1/3} T ( 1 + \\lambda ) - \\lambda \\right)}{ 2 ( 2 + \\lambda ) (4 \\pi )^{1/3} ( 3 V_m)^{2/3}}$ Hence when the $P > P_{max}$ black holes dose not exist.","This implies that there is a maximum value of the cosmological constant that the black holes could exist for a given horizon radius (or volume).","For $\\gamma =-1$ , the behavior is quite different from what is of $\\gamma =1$ .","There are critical points as demonstrated from Fig$(\\ref {pvsr1})$ .","Figure: The figure shows PP vs r h r_h for γ=1 \\gamma =1 for varying temperature.","Here λ=2.865\\lambda = 2.865, and Q=0.334 Q = 0.334.","For large temperature the peak is higher.Figure: The figure shows PP vs VV for γ=-1 \\gamma = -1 for varying temperature.","Here λ=1.98\\lambda = 1.98, and Q=0.094 Q = 0.094."],[" Phase transitions","For $\\gamma = -1$ , there are phase transitions between small and large black holes.","These phase transitions are first order and are similar to Van der Waals phase transitions between gas and liquid under constant temperature.","A thorough analysis of the phase transitions were discussed in the paper by the current author in [43].","There is a critical temperature $T_c$ at which the temperature which $P$ vs $V$ curve has an inflection point.","At that point, $\\frac{ \\partial P}{ \\partial V } = \\frac{ \\partial ^2 P}{ \\partial V^2 } =0$ The inflection point occur at the volume $V_c$ given by, $V_c = \\frac{ 4 \\pi r_{hc}^3}{ 3}$ where $r_{hc} = \\frac{1}{2} \\left( Q^2 ( \\lambda ^2 - 1) ( \\lambda +2 ) 2 ^{\\lambda -1}\\right)^{1/\\lambda }$ The corresponding $T_c$ and $P_c$ are given by, $T_c = \\frac{ \\lambda }{ \\pi ( \\lambda +1)} \\left( Q^2 ( \\lambda ^2 - 1) ( \\lambda +2 ) 2 ^{\\lambda -1}\\right)^{-1/\\lambda }$ $P_c = \\frac{ \\lambda }{ 2 \\pi ( \\lambda +2)} \\left( Q^2 ( \\lambda ^2 - 1) ( \\lambda +2 ) 2 ^{\\lambda -1}\\right)^{-2/\\lambda }$"],[" Specific heat capacities","There are two different heat capacities for a thermodynamical system: heat capacity at constant pressure, $C_P$ , and heat capacity at constant volume, $C_V$ .","They are given by, $C_P = T \\left.","\\frac{\\partial S}{ \\partial T}\\right|_P = \\frac{ 2 S \\left( 8 P S^{\\frac{2 + \\lambda }{2}} + S ^{\\frac{\\lambda }{2}} + \\pi ^{\\frac{\\lambda }{2}} Q^2 \\gamma ( -1 + \\lambda )\\right) }{ \\left( 8 P S^{\\frac{2 + \\lambda }{2}} - S ^{\\frac{\\lambda }{2}} - \\pi ^{\\frac{\\lambda }{2}} Q^2 \\gamma ( -1 + \\lambda ^2)\\right) }$ $C_V = T \\left.","\\frac{\\partial S}{ \\partial T}\\right|_V = 0$ $C_V$ is zero because the entropy $S$ is proportional to $V$ ."],[" Black hole as a heat engine","Since the black hole is considered as a thermodynamical system with a $P dV$ term, one could extract mechanical work from the black hole.","The given black hole has an equation of state which clearly articulate the relation between $P$ and $V$ .","Net work can be extracted from a given cycle in state space.","If the net heat input is $Q_H$ , net heat out put is $Q_C$ and the net work out put from the system is $W_{net}$ , the relation between them are, $Q_H = W_{net} + Q_C$ ."],["Carnot cycle efficiency","If the given thermodynamical cycle is given by two isothermal and two isentropic paths, then the efficiency of the heat engine is given by, $\\eta _c = \\frac{ W_{net}}{ Q_H} = \\left( 1 - \\frac{ T_C}{T_H} \\right)$"],[" Efficiency of a cycle with two isochoric and two isobaric paths","In this paper, we focus on a thermodynamical path consisting of two constant pressure and two constant volume paths as shown in Fig.$(\\ref {pvcycle})$ .","The heat supplied along the isobaric path $1 \\rightarrow 2$ which is the higher temperature path is given by, $Q_{1 \\rightarrow 2} = Q_H = \\int _1^2 d Q = \\int _1^2 T dS$ Since $S = \\pi r_h^2$ , eq.$(\\ref {heat1})$ can be rewritten as, $Q_H = \\int _1^2 T_H d ( \\pi r_h^2) = 2 \\pi \\int _{r_1}^{r_2} T_H(r_h) r_h dr_h$ Since the pressure $P$ is constant along the path $1 \\rightarrow 2$ , and $Q, \\lambda $ are also kept constant, $T_H$ becomes a function of $r_h$ along $1 \\rightarrow 2$ .","Hence the integral in eq.$(\\ref {integral})$ could be computed to be, $Q_H = \\left( \\frac{r_h}{2} - \\frac{ \\gamma Q^2}{ 2 r_h^{( \\lambda -1 )} }- \\frac{ 4 \\pi r_h^3 P}{3} \\right)^{r_2}_{r_1} = M(r_2) - M(r_1)$ Similarly the net heat out put from the system could be calculated along the path $4 \\rightarrow 3$ as, $Q_{ 3 \\rightarrow 4} = Q_{C} = M(r_3) - M(r_4)$ Therefore, the efficiency of the the heat engine for this particular cycle is, $\\eta = \\frac{W_{net}}{ Q_H} = 1 - \\frac{ (M(r_3) - M(r_4) )}{ (M(r_2) - M(r_1) )}$ The above formula was obtained by Johnson [22] by using enthalpy and the first law of thermodynamics.","It is also noted that since volume is a function of entropy in this black hole, isochoric path also represents adiabatic path.","Figure: The figure shows PP vs VV for γ=-1 \\gamma = -1 for varying temperature.","Here λ=1.98\\lambda = 1.98, and Q=0.094 Q = 0.094.","The maximum pressure, P 1 =P 2 =0.925P_1 = P_2 = 0.925 and minimum pressure, P 3 =P 4 =0.503P_3 = P_4 = 0.503."],["Efficiency of the heat engine for varying parameters of the black hole","The goal of this section is to compute the efficiency of the black hole heat engine and compare to the Carnot efficiency $\\eta _C$ .","We will restrict the black holes with $\\gamma = -1$ for the rest of the paper.","There are two parts to this computation: first we will change $Q$ for three values and perform the calculations, and second we will change $\\lambda $ and perform the computations.","In both cases, $V$ (or $r_h$ ) and the $P$ for the coordinates 1 and 4 are kept constant.","Then $V$ (or $r_h$ ) for coordinates 2 and 3 are varied to observe how the efficiency varies.","Before we proceed, some cautionary remarks are in order: for a given mass, the horizon radius $r_h$ is possible only if the parameters of the black hole are chosen appropriately.","To clarify this further we have shown an example in the Fig.$(\\ref {frvsmass})$ of the function $f(r)$ whose roots lead to the horizon radius $r_h$ .","In this example, when the mass is low, there are no horizons.","Only for a mass greater than a critical value that the horizon radius exists.","On the other hand, we also would like to make sure that for all the points on the chosen thermodynamical cycle that there is a real mass value.","In Fig.$(\\ref {mvsr})$ , $M$ is plotted vs $r_h$ for various values of $P$ .","The values of the pressures are chosen so that they are between the maximum and the minimum pressure of the cycle considered.","It is clear that the mass $M$ is positive for all values of $P$ chosen.","However, it is interesting to note that the mass has a minimum value for a given $P$ .","We did a spot check to make sure that for the range of $r_2$ chosen, that $M(r_2), M(r_4) > 0$ .","Figure: The figure shows f(r)f(r) vs rr for γ=-1 \\gamma = -1 for varying mass.","Here λ=4,Λ=-0.2\\lambda = 4, \\Lambda = -0.2, and Q=0.3 Q = 0.3.Figure: The figure shows MM vs r h r_h for γ=-1 \\gamma = -1 for varying pressure.","Here λ=1.98\\lambda = 1.98, and Q=0.094 Q = 0.094.","The values of the pressures are chosen so that they are between the maximum and the minimum pressures of the cycle, P 1 =P 2 =0.925P_1 = P_2 = 0.925 and P 3 =P 4 =0.503P_3 = P_4 = 0.503 respectively."],["Efficiency for varying scalar charge $Q$","In this section, the scalar charge $Q$ will be varied for three values.","Here, $P_1, P_4, r_4= r_1$ will be kept constant.","In the thermodynamical cycle in Fig.$(\\ref {pvcycle})$ , the smallest temperature is at $T_4$ .","The cycle is chosen such that it does not include the region where the phase transition occur.","Due to this reason, the critical temperature $T_c$ needs to be smaller than $T_4$ for all chosen $Q$ values.","$T_c$ is computed by eq.$(\\ref {tc})$ .","$T_4$ is computed as, $T_4 = \\frac{ 1 }{ 4 \\pi } \\left( \\frac{1}{ r_4} + 8 \\pi P_4 r_4 + \\frac{ \\gamma ( \\lambda -1)Q^2}{ r_4^{ \\lambda +1} } \\right)$ $T_c$ and $T_4$ are plotted together in the same graph in Fig.$(\\ref {tct4})$ for varying $Q$ .","For large $Q$ , $T_c > T_4$ .","Hence one need to choose the correct range of $Q$ values to make sure that $T_c < T_4$ .","Figure: The figure shows T c ,T 4 T_c, T_4 vs QQ for γ=-1 \\gamma = -1.","Here λ=1.98\\lambda = 1.98, and P 4 =0.509P_4 = 0.509.When $Q$ is fixed, $M(r_3)$ and $M(r_2)$ will depend only on $r_3 = r_2, P_1$ and $P_4$ .","Now, $\\eta , \\eta _C$ are calculated using eq.$(\\ref {eta})$ and plotted for 3 different values of $Q$ in Fig.$(\\ref {etavsq})$ .","Figure: The figure shows η,η c \\eta , \\eta _c vs r 2 r_2 for γ=-1 \\gamma = -1 for varying scalar charge QQ.","Here λ=1.98\\lambda = 1.98Figure: The figure shows η η c \\frac{\\eta }{\\eta _c} vs r 2 r_2 for γ=-1 \\gamma = -1 for varying scalar charge QQ.","Here λ=1.98\\lambda = 1.98"],["Efficiency for varying $\\lambda $","In this section, the value of $\\lambda $ is varied for three values.","Once again, $P_1, P_4, r_4= r_1$ will be kept constant.","It is also important to make sure that $T_c < T_4$ for chosen values of $\\lambda $ .","In Fig.$(\\ref {tct4lambda})$ , $T_c, T_4$ are plotted against $\\lambda $ .","It is clear that there is a value $\\lambda $ that $T_c > T_4$ .","Hence values of $\\lambda $ are chosen such that $T_c < T_4$ .","Figure: The figure shows T c ,T 4 T_c, T_4 vs λ\\lambda for γ=-1 \\gamma = -1.","Here Q=0.094Q = 0.094, and P 4 =0.509P_4 = 0.509.Figure: The figure shows η,η c \\eta , \\eta _c vs r 2 r_2 for γ=-1 \\gamma = -1 for varying λ\\lambda .","Here Q=0.094Q = 0.094Figure: The figure shows η η c \\frac{\\eta }{\\eta _c} vs r 2 r_2 for γ=-1 \\gamma = -1 for varying λ\\lambda .","Here Q=0.094Q = 0.094"],[" Conclusion","In this paper, we have studied black holes in massive gravity with a negative cosmological constant.","in the extended phase space.","Here the black hole has a pressure given by, $P = -\\frac{\\Lambda }{ 8 \\pi }$ .","There are two types of black holes for the values of $\\gamma $ in the theory.","Thermodynamical behavior differ significantly for $\\gamma =1$ and $\\gamma =-1$ .","For $\\gamma =1$ , the black hole behaves similar to the Schwarzschild-anti-de Sitter black hole: the pressure has a maximum and the temperature has a minimum.","For $\\gamma =-1$ , the black hole exhibits phase transitions for certain range of temperatures: for higher temperatures, the black holes behave like an ideal gas.","Phase transitions are between large black holes and small black holes.","A detailed analysis of the black holes in this context is published by the current author in [43].","The main goal of this paper is to study the massive gravity black hole as a heat engine.","The thermodynamical cycle for the heat engine considered here is a rectangle in $P, V$ space with two isobaric and two isochoric processes.","The efficiency of the heat engine taking black hole as the working substance is computed for the rectangle cycle as well as for the Carnot cycle by varying $Q, \\lambda , r_2$ .","When $Q$ is increased, the efficiency for the rectangle cycle increases, but, the Carnot efficiency decreases.","The ratio $\\frac{\\eta }{\\eta _c}$ decreases when $Q$ is increased.","Hence to achieve better efficiency, a higher $Q$ is appropriate.","Larger the volume $V_2$ (or $r_2$ ), higher the $\\eta $ of the rectangle cycle.","When $Q =0$ , we get the Schwarzschild AdS black hole.","Hence from the graphs it can be concluded that the Schwarzschild AdS black hole has a smaller efficiency for the rectangle cycle compared with the massive gravity black hole.","When $\\lambda $ is increased, the efficiency of the rectangle cycle as the Carnot cycle increases.","Both efficiencies increase with volume $V_2$ (or $r_2$ ).","The ratio $\\eta /\\eta _c$ decreases with $\\lambda $ .","The Reissner-Nordstrom AdS (RNAdS) black hole has the metric with $f(r) = 1 - \\frac{2 M}{r} + \\frac{Q_e^2}{r^2} - \\Lambda r^2$ .","When $Q_e = Q$ and $\\lambda =2$ , the massive gravity black hole and the RNAdS black hole are the same.","Hence one can conclude that for $\\lambda >2$ with $ Q_e = Q$ , the massive gravity black hole will have higher efficiency.","When $ 1 \\le \\lambda < 2 $ , massive gravity black hole will have lower efficiency."]],"string":"[\n [\n \"Massive gravity with Lorentz symmetry breaking: black holes as heat\\n engines\"\n ],\n [\n \"Abstract In extended phase space, a static black hole in massive gravity is studied as a holographic heat engine.\",\n \"In the massive gravity theory considered, the graviton gain a mass due to Lorentz symmetry breaking.\",\n \"Exact efficiency formula is obtained for a rectangle engine cycle for the black hole considered.\",\n \"The efficiency is computed by varying two parameters in the theory, the scalar charge Q and $\\\\lambda$.\",\n \"The efficiency is compared with the Carnot efficiency for the heat engine.\",\n \"It is observed that when Q and $\\\\lambda$ are increased that the efficiency for the rectangle cycle increases.\",\n \"When compared to the Schwarzschild AdS black hole, the efficiency for the rectangle cycle is larger for the Massive gravity black hole.\"\n ],\n [\n \" Introduction\",\n \"Black holes in anti-de Sitter space as a thermodynamical system has attracted lot of attention in a variety of contexts: when the negative cosmological constant is taken as the thermodynamical pressure of the black hole with the relation $ P = - \\\\frac{\\\\Lambda }{ 8 \\\\pi }$ , the resulting thermodynamics lead to interesting features.\",\n \"In this extended phase space, the first law of thermodynamics is modified by a term $V dP$ and the mass $M$ of the black hole is treated as the enthalpy rather than the internal energy $E$ of the black hole [1] [2].\",\n \"Many black holes in the context of extended phase space has demonstrated Van der Waals type phase transitions between small and large black holes.\",\n \"Due to the large number of work published related to this topic we will mention few here: [3] [4] [5] [6] [7][8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18][19].\",\n \"There is a nice review on Black hole chemistry written by Kubiznak et.al.\",\n \"which gives a comprehensive summary of the interesting thermodynamical features of black holes with a cosmological constant [20].\",\n \"In classical thermodynamics, there are four basic thermodynamical processes.\",\n \"They are isothermal, adiabatic, isobaric, and isochoric processes.\",\n \"In each of these processes the thermodynamical quantities temperature, entropy, pressure, and, volume are kept constant respectively.\",\n \"In a heat engine, a thermodynamical cycle is chosen which consist of aforementioned processes.\",\n \"For example, the Carnot cycle, which has the highest efficiency, have two isothermal and two adiabatic processes in the cycle.\",\n \"Brayton cycle has two isobaric and two adiabatic processes.\",\n \"In this paper, the goal is to study a black hole in massive gravity as a heat engine.\",\n \"The idea that black holes could be used as a working substance in a heat engine was first presented by Johnson [21].\",\n \"In that paper, charged black hole in AdS space in $D = 4$ was presented as an example.\",\n \"In an extension of that work, Johnson presented the Born-Infeld AdS black hole as a heat engine in [22].\",\n \"In [23], heat engines from dilatonic Born-Infeld black holes were analyzed.\",\n \"Black holes in conformal gravity as heat engine was presented by Xu et.al.\",\n \"in [24].\",\n \"Charged BTZ black hole in 2+1 dimensions as a heat engine was studied by Mo et.al.\",\n \"in [25].\",\n \"Effects of dark energy on the efficiency of heat engines of AdS black holes were analyzed in detail by Liu and Meng in [26].\",\n \"Class of black holes in massive gravity as heat engines were discussed by Hendi et.al.\",\n \"[27].\",\n \"The first study of rotating black holes as holographic heat engines were done by Henniger et.al.\",\n \"[28].\",\n \"Heat engines are defined for space-times that are not black holes as well: Johnson [29] studied Taub-Bolt space-time as an example of a heat engine and compared with the analogous Schwarzschild black hole as a heat engine.\",\n \"Massive gravity theories have become popular as a mean of explaining accelerated expansion of the universe without having to introduce a component of “dark energy.” There are many theories of massive gravity in the literature.\",\n \"There are large volume of work related to theories in massive gravity [30] [31] [32] [33]: here we will mention two nice reviews on the subject by de Rham [34] and Hinterbichler [35].\",\n \"In this paper we will consider a massive gravity theory where the graviton acquire a mass due to the Lorenz symmetry breaking.\",\n \"Here, Higgs mechanism for gravity is introduced with space-time depending scalar fields that are coupled to gravity [36].\",\n \"The resulting theory will exhibit modified gravitational interactions at large scale.\",\n \"These models are free from ghosts and tachyonic instabilities around curved space as well as in flat space [37] [38].\",\n \"A review of Lorentz violating massive gravity theory can be found in [39] [40].\",\n \"The details of the theory is discussed in section 2.\",\n \"The paper is organized as follows: in section 2, the black hole in massive gravity is introduced.\",\n \"Thermodynamics and phase transitions of the massive gravity black hole are discussed in section 3.\",\n \"Black hole as a heat engine with a rectangle cycle is introduced in section 4 and in section 5 the efficiencies are computed.\",\n \"Finally the conclusion is given in section 6.\"\n ],\n [\n \" Introduction to AdS black holes in massive gravity with Lorentz symmetry breaking\",\n \"The action for the massive gravity theory considered in this paper given by, $S = \\\\int d^4 x \\\\sqrt{-g } \\\\left[ - \\\\frac{1}{ 16 \\\\pi } \\\\mathcal {R} + \\\\Omega ^4 \\\\mathcal {F}( X, W^{ij}) \\\\right]$ Here $\\\\mathcal {F}$ is a function of our scalar fields $\\\\phi ^{\\\\mu }$ .\",\n \"$\\\\phi ^{\\\\mu }$ are minimally coupled to gravity by covariant derivatives.\",\n \"$\\\\mathcal {F}$ depends on two combinations of the scalar fields given by $X$ and $W$ defined in terms of the scalar fields as, $X = \\\\frac{\\\\partial ^{\\\\mu } \\\\phi ^0 \\\\partial _{\\\\mu } \\\\phi ^0 }{ \\\\Omega ^4}$ $W^{i j} = \\\\frac{\\\\partial ^{\\\\mu } \\\\phi ^i \\\\partial _{\\\\mu } \\\\phi ^j}{ \\\\Omega ^4} - \\\\frac{\\\\partial ^{\\\\mu } \\\\phi ^i \\\\partial _{\\\\mu } \\\\phi ^0 \\\\partial ^{\\\\nu } \\\\phi ^j \\\\partial _{\\\\nu } \\\\phi ^0 }{ \\\\Omega ^8 X}$ The constant $\\\\Omega $ has dimensions of mass: it is in the order of $ \\\\sqrt{ m_g M_{pl}}$ where $m_g$ is the graviton mass and $M_{pl}$ the Plank mass [39] [36] [37] [38].\",\n \"The scalar fields $\\\\phi ^0, \\\\phi ^i$ are responsible for breaking Lorentz symmetry spontaneously when they acquire a vacuum expectation value.\",\n \"More details on the theory can be found in [39] [36] [37] [38] [40].\",\n \"A class of black hole solutions to the above action was derived in [41] and [42].\",\n \"$ ds^2 = - f(r) dt^2 + \\\\frac{ dr^2}{ f(r)} + r^2 ( d \\\\theta ^2 + sin^2 \\\\theta d \\\\phi ^2)$ where, $ f(r) = 1 - \\\\frac{ 2 M}{ r} - \\\\gamma \\\\frac{ Q^2}{r^{\\\\lambda }} - \\\\frac{\\\\Lambda r^2}{3}$ Here, the parameter $Q$ represents a scalar charge related to massive gravity and $\\\\gamma = \\\\pm 1$ .\",\n \"The constant $\\\\lambda $ in is an integration constant and is positive.\",\n \"$\\\\Lambda $ is the cosmological constant.\",\n \"A detailed description of this black hole is given by Fernando in [43].\",\n \"Since for $\\\\lambda <1$ , the ADM mass become divergent, such solutions will not be considered.\",\n \"When $\\\\lambda >1$ , for $ r \\\\rightarrow \\\\infty $ the metric approaches the Schwarzschild-AdS black hole with a finite mass $M$ ; hence we will choose $\\\\lambda >1$ in the rest of the paper.\",\n \"When $ \\\\gamma = 1$ , the space-time of the black hole is very similar to the Schwarzschild-AdS black hole and for $\\\\gamma = -1$ , the geometry is similar to the Reissner-Nordstrom-AdS charged black hole.\",\n \"There are several works related to the black holes given above.\",\n \"P-V criticality and phase transitions of masive gravity black holes with a negative cosmological constant were presented by Fernando in [43] [44].\",\n \"Thermodynamics for $\\\\Lambda =0$ case were studied in [45] [46].\",\n \"Stability and quasi normal modes were studied in [47][48] [49] [50].\"\n ],\n [\n \"Thermodynamics and phase transitions\",\n \"In this section we will derive the thermodynamical quantities and present the phase transitions that could occur in massive gravity black holes.\"\n ],\n [\n \" Defining the thermodynamical quantities\",\n \"The temperature of the black hole could be obtained by using the definition of Hawking temperature: it is based on the surface gravity $\\\\kappa $ at the outer horizon, $r_h$ .\",\n \"It is given by, $T_H = \\\\frac{ \\\\kappa }{ 2 \\\\pi } = \\\\frac{ 1}{ 4 \\\\pi } \\\\left| \\\\frac{ df(r)}{ dr} \\\\right|_ { r = r_h} = \\\\frac{1}{ 4 \\\\pi } \\\\left(\\\\frac{ 2 M}{ r_h^2} + \\\\frac{ \\\\gamma Q^2 \\\\lambda }{ r_h^{ \\\\lambda + 1}} - \\\\frac{ 2 \\\\Lambda r_h}{ 3}\\\\right)$ Since $ f(r_h) = 0$ , the mass of the black hole could be written as, $M = r_h \\\\left( \\\\frac{1}{2} - \\\\frac{ \\\\gamma Q^2}{ 2 r_h^{( \\\\lambda )} }- \\\\frac{ r_h^2 \\\\Lambda }{6} \\\\right)$ The value of the mass $M$ could be substituted to eq$(\\\\ref {tempe1})$ and rewrite temperature in terms of $r_h, Q,$ and $\\\\lambda $ as, $T = \\\\frac{ 1 }{ 4 \\\\pi } \\\\left( \\\\frac{1}{ r_h} - r_h \\\\Lambda + \\\\frac{ \\\\gamma ( \\\\lambda -1)Q^2}{ r_h^{ \\\\lambda +1} } \\\\right)$ The conjugate quantity for temperature, the entropy S is given by the area law, $ S = \\\\pi r_h^2$ for this black hole.\",\n \"As discussed in the introduction, in the extended phase space, the pressure $P$ is given by, $P = -\\\\frac{ \\\\Lambda }{ 8 \\\\pi }$ The conjugate quantity for P, the volume $V$ , is given by, $\\\\frac{ 4 \\\\pi r_h^3}{3}$ .\",\n \"The scalar potential which is conjugated to the scalar charge $Q$ is given by, $\\\\Phi = - \\\\frac{ \\\\gamma Q}{ r_h^{ \\\\lambda -1}}$ One could rewrite the temperature in terms of $S$ and $P$ as follows: $T_H = \\\\frac{1}{ 4 \\\\sqrt{ \\\\pi S}} \\\\left( 1 + 8 P S + \\\\gamma ( \\\\lambda - 1) Q^2 \\\\left(\\\\frac{\\\\pi }{ S}\\\\right)^{\\\\lambda /2} \\\\right)$ The temperature is plotted against the entropy $S$ in Fig.$(\\\\ref {temp1})$ and Fig.$(\\\\ref {temp2})$ .\",\n \"For $\\\\gamma =1$ , the temperature has a minimum: hence black holes cannot exist below this minimum for a given $\\\\Lambda $ value.\",\n \"For $\\\\gamma =-1$ , the temperature could have an inflection point.\",\n \"Figure: The figure shows TT vs SS for γ=1 \\\\gamma = 1.\",\n \"Here λ=2.305,P=0.905\\\\lambda = 2.305, P = 0.905, and Q=0.115 Q = 0.115.Figure: The figure shows TT vs SS for γ=-1 \\\\gamma =-1.\",\n \"Here λ=2.215\\\\lambda = 2.215, and P=0.05 P = 0.05.\",\n \"The top graph is at Q=0Q=0 and the rest have Q=0.128,0.222,0.5Q = 0.128, 0.222, 0.5\"\n ],\n [\n \" First law in the extended phase space\",\n \"In the extended phase space, the mass of the black hole M is not considered as the internal energy as it is usually done.\",\n \"Instead, it is considered as the enthalpy H. Hence, $ M = H = U + P V$ .\",\n \"Hence, the first law of the given black hole is given by, $dM = T dS + \\\\Phi dQ + V dP$ It is possible to combine the thermodynamical quantities, $M, P, V, S, T, \\\\Phi $ , and $Q$ to obtain the Smarr formula as, $M = 2 T S + \\\\frac{\\\\lambda }{2} \\\\Phi Q - 2 P V$ which also could be obtained using the scaling argument presented by Kastor et.al [1].\",\n \"When $\\\\lambda =2$ , the Smarr formula simplifies to the one for the Reissner-Nordstrom-AdS black hole obtained by Kubiznak and Mann in [3].\"\n ],\n [\n \" Behavior of pressure\",\n \"In eq.$(\\\\ref {tempe1})$ , one could substitute $ \\\\Lambda = - 8 \\\\pi P$ and rearrange the equation to obtain $P$ as a function of $r_h$ and $T$ as, $P = - \\\\frac{ 1}{ 8 \\\\pi r_h^2} + \\\\frac{ T}{ 2 r_h} + \\\\frac{ Q^2 \\\\gamma ( 1 - \\\\lambda )}{ 8 \\\\pi r_h^{ 2 + \\\\lambda }}$ Since the black hole radius $r_h$ is given by, $r_h = \\\\left(\\\\frac{ 3 V}{4 \\\\pi }\\\\right)^{1/3}$ one could rewrite $P$ in terms of $V$ as, $P = - \\\\frac{ 1}{ 8 \\\\pi } \\\\left( \\\\frac{ 4 \\\\pi }{ 3 V} \\\\right) ^ {2/3} + \\\\frac{T}{2} \\\\left( \\\\frac{ 4 \\\\pi }{ 3 V} \\\\right) ^ {1/3} +\\\\frac{ Q^2 \\\\gamma ( 1 - \\\\lambda )}{ 8 \\\\pi } \\\\left( \\\\frac{ 4 \\\\pi }{ 3 V} \\\\right) ^ {(\\\\lambda + 2)/3}$ The pressure is plotted vs V as in Fig.$(\\\\ref {pvsr2})$ and Fig.$(\\\\ref {pvsr1})$ .\",\n \"For $\\\\gamma =1$ , there is a maximum pressure given by, $P_{max} = \\\\frac{ 4 \\\\pi r_h T ( 1 + \\\\lambda ) - \\\\lambda }{ 8 \\\\pi r_h^2 ( 2 + \\\\lambda )}$ Since $r_h$ and $V$ are related, $P_{max}$ also could be rewritten in terms of $V_m$ at which the pressure is maximum as, $P_{max} = \\\\frac{ \\\\left( (4 \\\\pi )^{2/3} ( 3 V_m)^{1/3} T ( 1 + \\\\lambda ) - \\\\lambda \\\\right)}{ 2 ( 2 + \\\\lambda ) (4 \\\\pi )^{1/3} ( 3 V_m)^{2/3}}$ Hence when the $P > P_{max}$ black holes dose not exist.\",\n \"This implies that there is a maximum value of the cosmological constant that the black holes could exist for a given horizon radius (or volume).\",\n \"For $\\\\gamma =-1$ , the behavior is quite different from what is of $\\\\gamma =1$ .\",\n \"There are critical points as demonstrated from Fig$(\\\\ref {pvsr1})$ .\",\n \"Figure: The figure shows PP vs r h r_h for γ=1 \\\\gamma =1 for varying temperature.\",\n \"Here λ=2.865\\\\lambda = 2.865, and Q=0.334 Q = 0.334.\",\n \"For large temperature the peak is higher.Figure: The figure shows PP vs VV for γ=-1 \\\\gamma = -1 for varying temperature.\",\n \"Here λ=1.98\\\\lambda = 1.98, and Q=0.094 Q = 0.094.\"\n ],\n [\n \" Phase transitions\",\n \"For $\\\\gamma = -1$ , there are phase transitions between small and large black holes.\",\n \"These phase transitions are first order and are similar to Van der Waals phase transitions between gas and liquid under constant temperature.\",\n \"A thorough analysis of the phase transitions were discussed in the paper by the current author in [43].\",\n \"There is a critical temperature $T_c$ at which the temperature which $P$ vs $V$ curve has an inflection point.\",\n \"At that point, $\\\\frac{ \\\\partial P}{ \\\\partial V } = \\\\frac{ \\\\partial ^2 P}{ \\\\partial V^2 } =0$ The inflection point occur at the volume $V_c$ given by, $V_c = \\\\frac{ 4 \\\\pi r_{hc}^3}{ 3}$ where $r_{hc} = \\\\frac{1}{2} \\\\left( Q^2 ( \\\\lambda ^2 - 1) ( \\\\lambda +2 ) 2 ^{\\\\lambda -1}\\\\right)^{1/\\\\lambda }$ The corresponding $T_c$ and $P_c$ are given by, $T_c = \\\\frac{ \\\\lambda }{ \\\\pi ( \\\\lambda +1)} \\\\left( Q^2 ( \\\\lambda ^2 - 1) ( \\\\lambda +2 ) 2 ^{\\\\lambda -1}\\\\right)^{-1/\\\\lambda }$ $P_c = \\\\frac{ \\\\lambda }{ 2 \\\\pi ( \\\\lambda +2)} \\\\left( Q^2 ( \\\\lambda ^2 - 1) ( \\\\lambda +2 ) 2 ^{\\\\lambda -1}\\\\right)^{-2/\\\\lambda }$\"\n ],\n [\n \" Specific heat capacities\",\n \"There are two different heat capacities for a thermodynamical system: heat capacity at constant pressure, $C_P$ , and heat capacity at constant volume, $C_V$ .\",\n \"They are given by, $C_P = T \\\\left.\",\n \"\\\\frac{\\\\partial S}{ \\\\partial T}\\\\right|_P = \\\\frac{ 2 S \\\\left( 8 P S^{\\\\frac{2 + \\\\lambda }{2}} + S ^{\\\\frac{\\\\lambda }{2}} + \\\\pi ^{\\\\frac{\\\\lambda }{2}} Q^2 \\\\gamma ( -1 + \\\\lambda )\\\\right) }{ \\\\left( 8 P S^{\\\\frac{2 + \\\\lambda }{2}} - S ^{\\\\frac{\\\\lambda }{2}} - \\\\pi ^{\\\\frac{\\\\lambda }{2}} Q^2 \\\\gamma ( -1 + \\\\lambda ^2)\\\\right) }$ $C_V = T \\\\left.\",\n \"\\\\frac{\\\\partial S}{ \\\\partial T}\\\\right|_V = 0$ $C_V$ is zero because the entropy $S$ is proportional to $V$ .\"\n ],\n [\n \" Black hole as a heat engine\",\n \"Since the black hole is considered as a thermodynamical system with a $P dV$ term, one could extract mechanical work from the black hole.\",\n \"The given black hole has an equation of state which clearly articulate the relation between $P$ and $V$ .\",\n \"Net work can be extracted from a given cycle in state space.\",\n \"If the net heat input is $Q_H$ , net heat out put is $Q_C$ and the net work out put from the system is $W_{net}$ , the relation between them are, $Q_H = W_{net} + Q_C$ .\"\n ],\n [\n \"Carnot cycle efficiency\",\n \"If the given thermodynamical cycle is given by two isothermal and two isentropic paths, then the efficiency of the heat engine is given by, $\\\\eta _c = \\\\frac{ W_{net}}{ Q_H} = \\\\left( 1 - \\\\frac{ T_C}{T_H} \\\\right)$\"\n ],\n [\n \" Efficiency of a cycle with two isochoric and two isobaric paths\",\n \"In this paper, we focus on a thermodynamical path consisting of two constant pressure and two constant volume paths as shown in Fig.$(\\\\ref {pvcycle})$ .\",\n \"The heat supplied along the isobaric path $1 \\\\rightarrow 2$ which is the higher temperature path is given by, $Q_{1 \\\\rightarrow 2} = Q_H = \\\\int _1^2 d Q = \\\\int _1^2 T dS$ Since $S = \\\\pi r_h^2$ , eq.$(\\\\ref {heat1})$ can be rewritten as, $Q_H = \\\\int _1^2 T_H d ( \\\\pi r_h^2) = 2 \\\\pi \\\\int _{r_1}^{r_2} T_H(r_h) r_h dr_h$ Since the pressure $P$ is constant along the path $1 \\\\rightarrow 2$ , and $Q, \\\\lambda $ are also kept constant, $T_H$ becomes a function of $r_h$ along $1 \\\\rightarrow 2$ .\",\n \"Hence the integral in eq.$(\\\\ref {integral})$ could be computed to be, $Q_H = \\\\left( \\\\frac{r_h}{2} - \\\\frac{ \\\\gamma Q^2}{ 2 r_h^{( \\\\lambda -1 )} }- \\\\frac{ 4 \\\\pi r_h^3 P}{3} \\\\right)^{r_2}_{r_1} = M(r_2) - M(r_1)$ Similarly the net heat out put from the system could be calculated along the path $4 \\\\rightarrow 3$ as, $Q_{ 3 \\\\rightarrow 4} = Q_{C} = M(r_3) - M(r_4)$ Therefore, the efficiency of the the heat engine for this particular cycle is, $\\\\eta = \\\\frac{W_{net}}{ Q_H} = 1 - \\\\frac{ (M(r_3) - M(r_4) )}{ (M(r_2) - M(r_1) )}$ The above formula was obtained by Johnson [22] by using enthalpy and the first law of thermodynamics.\",\n \"It is also noted that since volume is a function of entropy in this black hole, isochoric path also represents adiabatic path.\",\n \"Figure: The figure shows PP vs VV for γ=-1 \\\\gamma = -1 for varying temperature.\",\n \"Here λ=1.98\\\\lambda = 1.98, and Q=0.094 Q = 0.094.\",\n \"The maximum pressure, P 1 =P 2 =0.925P_1 = P_2 = 0.925 and minimum pressure, P 3 =P 4 =0.503P_3 = P_4 = 0.503.\"\n ],\n [\n \"Efficiency of the heat engine for varying parameters of the black hole\",\n \"The goal of this section is to compute the efficiency of the black hole heat engine and compare to the Carnot efficiency $\\\\eta _C$ .\",\n \"We will restrict the black holes with $\\\\gamma = -1$ for the rest of the paper.\",\n \"There are two parts to this computation: first we will change $Q$ for three values and perform the calculations, and second we will change $\\\\lambda $ and perform the computations.\",\n \"In both cases, $V$ (or $r_h$ ) and the $P$ for the coordinates 1 and 4 are kept constant.\",\n \"Then $V$ (or $r_h$ ) for coordinates 2 and 3 are varied to observe how the efficiency varies.\",\n \"Before we proceed, some cautionary remarks are in order: for a given mass, the horizon radius $r_h$ is possible only if the parameters of the black hole are chosen appropriately.\",\n \"To clarify this further we have shown an example in the Fig.$(\\\\ref {frvsmass})$ of the function $f(r)$ whose roots lead to the horizon radius $r_h$ .\",\n \"In this example, when the mass is low, there are no horizons.\",\n \"Only for a mass greater than a critical value that the horizon radius exists.\",\n \"On the other hand, we also would like to make sure that for all the points on the chosen thermodynamical cycle that there is a real mass value.\",\n \"In Fig.$(\\\\ref {mvsr})$ , $M$ is plotted vs $r_h$ for various values of $P$ .\",\n \"The values of the pressures are chosen so that they are between the maximum and the minimum pressure of the cycle considered.\",\n \"It is clear that the mass $M$ is positive for all values of $P$ chosen.\",\n \"However, it is interesting to note that the mass has a minimum value for a given $P$ .\",\n \"We did a spot check to make sure that for the range of $r_2$ chosen, that $M(r_2), M(r_4) > 0$ .\",\n \"Figure: The figure shows f(r)f(r) vs rr for γ=-1 \\\\gamma = -1 for varying mass.\",\n \"Here λ=4,Λ=-0.2\\\\lambda = 4, \\\\Lambda = -0.2, and Q=0.3 Q = 0.3.Figure: The figure shows MM vs r h r_h for γ=-1 \\\\gamma = -1 for varying pressure.\",\n \"Here λ=1.98\\\\lambda = 1.98, and Q=0.094 Q = 0.094.\",\n \"The values of the pressures are chosen so that they are between the maximum and the minimum pressures of the cycle, P 1 =P 2 =0.925P_1 = P_2 = 0.925 and P 3 =P 4 =0.503P_3 = P_4 = 0.503 respectively.\"\n ],\n [\n \"Efficiency for varying scalar charge $Q$\",\n \"In this section, the scalar charge $Q$ will be varied for three values.\",\n \"Here, $P_1, P_4, r_4= r_1$ will be kept constant.\",\n \"In the thermodynamical cycle in Fig.$(\\\\ref {pvcycle})$ , the smallest temperature is at $T_4$ .\",\n \"The cycle is chosen such that it does not include the region where the phase transition occur.\",\n \"Due to this reason, the critical temperature $T_c$ needs to be smaller than $T_4$ for all chosen $Q$ values.\",\n \"$T_c$ is computed by eq.$(\\\\ref {tc})$ .\",\n \"$T_4$ is computed as, $T_4 = \\\\frac{ 1 }{ 4 \\\\pi } \\\\left( \\\\frac{1}{ r_4} + 8 \\\\pi P_4 r_4 + \\\\frac{ \\\\gamma ( \\\\lambda -1)Q^2}{ r_4^{ \\\\lambda +1} } \\\\right)$ $T_c$ and $T_4$ are plotted together in the same graph in Fig.$(\\\\ref {tct4})$ for varying $Q$ .\",\n \"For large $Q$ , $T_c > T_4$ .\",\n \"Hence one need to choose the correct range of $Q$ values to make sure that $T_c < T_4$ .\",\n \"Figure: The figure shows T c ,T 4 T_c, T_4 vs QQ for γ=-1 \\\\gamma = -1.\",\n \"Here λ=1.98\\\\lambda = 1.98, and P 4 =0.509P_4 = 0.509.When $Q$ is fixed, $M(r_3)$ and $M(r_2)$ will depend only on $r_3 = r_2, P_1$ and $P_4$ .\",\n \"Now, $\\\\eta , \\\\eta _C$ are calculated using eq.$(\\\\ref {eta})$ and plotted for 3 different values of $Q$ in Fig.$(\\\\ref {etavsq})$ .\",\n \"Figure: The figure shows η,η c \\\\eta , \\\\eta _c vs r 2 r_2 for γ=-1 \\\\gamma = -1 for varying scalar charge QQ.\",\n \"Here λ=1.98\\\\lambda = 1.98Figure: The figure shows η η c \\\\frac{\\\\eta }{\\\\eta _c} vs r 2 r_2 for γ=-1 \\\\gamma = -1 for varying scalar charge QQ.\",\n \"Here λ=1.98\\\\lambda = 1.98\"\n ],\n [\n \"Efficiency for varying $\\\\lambda $\",\n \"In this section, the value of $\\\\lambda $ is varied for three values.\",\n \"Once again, $P_1, P_4, r_4= r_1$ will be kept constant.\",\n \"It is also important to make sure that $T_c < T_4$ for chosen values of $\\\\lambda $ .\",\n \"In Fig.$(\\\\ref {tct4lambda})$ , $T_c, T_4$ are plotted against $\\\\lambda $ .\",\n \"It is clear that there is a value $\\\\lambda $ that $T_c > T_4$ .\",\n \"Hence values of $\\\\lambda $ are chosen such that $T_c < T_4$ .\",\n \"Figure: The figure shows T c ,T 4 T_c, T_4 vs λ\\\\lambda for γ=-1 \\\\gamma = -1.\",\n \"Here Q=0.094Q = 0.094, and P 4 =0.509P_4 = 0.509.Figure: The figure shows η,η c \\\\eta , \\\\eta _c vs r 2 r_2 for γ=-1 \\\\gamma = -1 for varying λ\\\\lambda .\",\n \"Here Q=0.094Q = 0.094Figure: The figure shows η η c \\\\frac{\\\\eta }{\\\\eta _c} vs r 2 r_2 for γ=-1 \\\\gamma = -1 for varying λ\\\\lambda .\",\n \"Here Q=0.094Q = 0.094\"\n ],\n [\n \" Conclusion\",\n \"In this paper, we have studied black holes in massive gravity with a negative cosmological constant.\",\n \"in the extended phase space.\",\n \"Here the black hole has a pressure given by, $P = -\\\\frac{\\\\Lambda }{ 8 \\\\pi }$ .\",\n \"There are two types of black holes for the values of $\\\\gamma $ in the theory.\",\n \"Thermodynamical behavior differ significantly for $\\\\gamma =1$ and $\\\\gamma =-1$ .\",\n \"For $\\\\gamma =1$ , the black hole behaves similar to the Schwarzschild-anti-de Sitter black hole: the pressure has a maximum and the temperature has a minimum.\",\n \"For $\\\\gamma =-1$ , the black hole exhibits phase transitions for certain range of temperatures: for higher temperatures, the black holes behave like an ideal gas.\",\n \"Phase transitions are between large black holes and small black holes.\",\n \"A detailed analysis of the black holes in this context is published by the current author in [43].\",\n \"The main goal of this paper is to study the massive gravity black hole as a heat engine.\",\n \"The thermodynamical cycle for the heat engine considered here is a rectangle in $P, V$ space with two isobaric and two isochoric processes.\",\n \"The efficiency of the heat engine taking black hole as the working substance is computed for the rectangle cycle as well as for the Carnot cycle by varying $Q, \\\\lambda , r_2$ .\",\n \"When $Q$ is increased, the efficiency for the rectangle cycle increases, but, the Carnot efficiency decreases.\",\n \"The ratio $\\\\frac{\\\\eta }{\\\\eta _c}$ decreases when $Q$ is increased.\",\n \"Hence to achieve better efficiency, a higher $Q$ is appropriate.\",\n \"Larger the volume $V_2$ (or $r_2$ ), higher the $\\\\eta $ of the rectangle cycle.\",\n \"When $Q =0$ , we get the Schwarzschild AdS black hole.\",\n \"Hence from the graphs it can be concluded that the Schwarzschild AdS black hole has a smaller efficiency for the rectangle cycle compared with the massive gravity black hole.\",\n \"When $\\\\lambda $ is increased, the efficiency of the rectangle cycle as the Carnot cycle increases.\",\n \"Both efficiencies increase with volume $V_2$ (or $r_2$ ).\",\n \"The ratio $\\\\eta /\\\\eta _c$ decreases with $\\\\lambda $ .\",\n \"The Reissner-Nordstrom AdS (RNAdS) black hole has the metric with $f(r) = 1 - \\\\frac{2 M}{r} + \\\\frac{Q_e^2}{r^2} - \\\\Lambda r^2$ .\",\n \"When $Q_e = Q$ and $\\\\lambda =2$ , the massive gravity black hole and the RNAdS black hole are the same.\",\n \"Hence one can conclude that for $\\\\lambda >2$ with $ Q_e = Q$ , the massive gravity black hole will have higher efficiency.\",\n \"When $ 1 \\\\le \\\\lambda < 2 $ , massive gravity black hole will have lower efficiency.\"\n ]\n]"},"id":{"kind":"string","value":"1808.08595"}}},{"rowIdx":294,"cells":{"text":{"kind":"list like","value":[["On Realizability Of Gauss Diagrams And Constructions Of Meanders"],["Abstract The problem of which Gauss diagram can be realized by plane curves is an old one and has been solved in several ways.","In this paper, we present a direct approach to this problem.","We show that needed conditions for realizability of a Gauss diagram can be interpreted as follows \"the number of exits = the number of entrances\" and the sufficient condition is based on Jordan curve Theorem.","We give a matrix approach of realization of Gauss diagrams and then we present an algorithm to construct meanders"],["Introduction","In the earliest time of the Knot Theory C.F.","Gauss defined the chord diagram (= Gauss diagram).","C.F.","Gauss [3] observed that if a chord diagram can be realized by a plane curve, then every chord is crossed only by an even number of chords, but that this condition is not sufficient.","The aim of this paper is to present a direct approach to the problem of which Gauss diagram can be realized by knots.","This problem is an old one, and has been solved in several ways.","In 1936, M. Dehn [2] found a sufficient algorithmic solution based on the existence of a touch Jordan curve which is the image of a transformation of the knot diagram by successive splits replacing all the crossings.","A long time after in 1976, L. Lovasz and M.L.","Marx [5] found a second necessary condition and finally during the same year, R.C.","Read and P. Rosenstiehl [7] found the third condition which allowed the set of these three conditions to be sufficient.","The last characterization is based on the tripartition of graphs into cycles, cocycles and bicycles.","In [8] the notation of oriented chord diagram was introduced and it was showed that these diagrams classify cellular generic curves on oriented surfaces.","As a corollary a simple combinatorial classification of plane generic curves was derived, and the problem of realizability of these diagrams was also solved.","However all these ways are indirect; they rest upon deep and nontrivial auxiliary construction.","There is a natural question: whether one can arrive at these conditions in a more direct and natural fashion?","We believe that the conditions for realizability of a Gauss diagram (by some plane curve) should be obtained in a natural manner; they should be deduced from an intrinsic structure of the curve.","In this paper, we suggest an approach, which satisfies the above principle.","We use the fact that every Gauss diagram $\\mathfrak {G}$ defines a (virtual) plane curve ${C}(\\mathfrak {G})$ (see [4]), and the following simple ideas: (1) For every chord of a Gauss diagram $\\mathfrak {G}$ , we can associate a closed path along the curve ${C}(\\mathfrak {G})$ .","(2) For every two non-intersecting chords of a Gauss diagram $\\mathfrak {G}$ , we can associate two closed paths along the curve ${C}(\\mathfrak {G})$ such that every chord crosses both of those chords correspondences to the point of intersection of the paths.","(3) If a Gauss diagram $\\mathfrak {G}$ is realizable (say by a plane curve ${C}(\\mathfrak {G})$ ), then for every closed path (say) ${P}$ along ${C}(\\mathfrak {G})$ we can associate a coloring another part of ${C}(\\mathfrak {G})$ into two colors (roughly speaking we get “inner” and “outer” sides of ${P}$ cf.","Jordan curve Theorem).","If a Gauss diagram is not realizable then ([4]) it defines a virtual plane curve ${C}(\\mathfrak {G})$ .","We shall show that there exists a closed path along ${C}(\\mathfrak {G})$ for which we cannot associate a well-defined coloring of ${C}(\\mathfrak {G})$ , i.e., ${C}(\\mathfrak {G})$ contains a path is colored into two colors.","Using these ideas we solve the problem of which Gauss diagram can be realized by knots.","We then give a matrix approach of realization of Gauss diagrams and then we present an algorithm to construct meanders."],["Preliminaries","Recall that classically, a knot is defined as an embedding of the circle $S^1$ into $\\mathbb {R}^3$ , or equivalently into the 3-sphere $S^3$ , i.e., a knot is a closed curve embedded on $\\mathbb {R}^3$ (or $S^3$ ) without intersecting itself, up to ambient isotopy.","The projection of a knot onto a 2-manifold is considered with all multiple points are transversal double with will be call crossing points (or shortly crossings).","Such a projection is called the shadow by the knots theorists [1], [9], following [8] we shall also call these projections as plane curves.","A knot diagram is a generic immersion of a circle $S^1$ to a plane $\\mathbb {R}^2$ enhanced by information on overpasses and underpasses at double points."],["Gauss Diagrams","A generic immersion of a circle to a plane is characterized by its Gauss diagram [6].","Figure: The plane curve and its Gauss diagram are shown.Definition 1.1 The Gauss diagram is the immersing circle with the preimages of each double point connected with a chord.","On the other words, this natation can be defined as follows.","Let us walk on a path along the plane curve until returning back to the origin and then generate a word $W$ which is the sequence of the crossings in the order we meet them on the path.","$W$ is a double occurrence word.","If we put the labels of the crossing on a circle in the order of the word $W$ and if we join by a chord all pairs of identical labels then we obtain a chord diagram (=Gauss diagram) of the plane curve (see Figure REF ).","A virtual knot diagram [4] is a generic immersion of the circle into the plane, with double points divided into real crossing points and virtual crossing points, with the real crossing points enhanced by information on overpasses and underpasses (as for classical knot diagrams).","At a virtual crossing the branches are not divided into an overpass and an underpass.","The Gauss diagram of a virtual knot is constructed in the same way as for a classical knot, but all virtual crossings are disregarded.","Theorem 1.2 [4] A Gauss diagram defines a virtual knot diagram up to virtual moves.","Arguing similarly as in the real knot case, one can define a shadow of the virtual knot (see Figure REF ).","Figure: The chord diagram and the shadow of the virtual knot are shown.","Here xx and yy are the virtual crossing points."],["Conway's Smoothing","We frequently use the following notations.","Let $K$ be a knot, ${C}$ its shadow and $\\mathfrak {G}$ the Gauss diagram of ${C}$ .","For every crossing $c$ of ${C}$ we denote by $\\mathfrak {c}$ the corresponding chord of $\\mathfrak {G}$ .","If a Gauss diagram $\\mathfrak {G}$ contains a chord $\\mathfrak {c}$ then we write $\\mathfrak {c}\\in \\mathfrak {G}$ .","We denote by $\\mathfrak {c}_0$ , $\\mathfrak {c}_1$ the endpoints of every chord $\\mathfrak {c}\\in \\mathfrak {G}$ .","We shall also consider every chord $\\mathfrak {c} \\in \\mathfrak {G}$ together with one of two arcs are between its endpoints, and a chosen arc is denoted by $\\mathfrak {c}_0\\mathfrak {c}_1$ .","Further, $\\mathfrak {c}_\\times $ denotes the set of all chords cross the chord $\\mathfrak {c}$ and $\\mathfrak {c}_\\parallel $ denotes the set of all chords do not cross the chord $\\mathfrak {c}$ .","We put $\\mathfrak {c} \\notin \\mathfrak {c}_\\times $ , and $\\mathfrak {c} \\in \\mathfrak {c}_\\parallel $ .","Throughout this paper we consider Gauss diagrams such that $\\mathfrak {c}_\\times \\ne \\varnothing $ for every $\\mathfrak {c \\in G}$ .","As well known, John Conway introduced a “surgical” operation on knots, called smoothing, consists in eliminating the crossing by interchanging the strands (Figure REF ).","Figure: The Conway smoothing the crossings are shown.We aim to specialize a Conway smoothing a crossing of a plane curve to an operation on chords of the corresponding Gauss diagram.","Let $K$ be a knot, ${C}$ its shadow, and $\\mathfrak {G}$ the Gauss diagram of ${C}$ .","Take a crossing point $c$ of ${C}$ and let $D_c$ be a small disk centered at $c$ such that $D_c\\cap {C}$ does not contain another crossings of ${C}$ .","Denote by $\\partial D_c$ the boundary of $D_c$ .","Starting from $c$ , let us walk on a path along the curve ${C}$ until returning back to $c$ .","Denote this path by ${L}_c$ and let $c c_a^lc_z^lc$ be the sequence of the points in the order we meet them on ${L}_c$ , where $ \\lbrace c_a^l,c_z^l\\rbrace = {L}_c \\cap \\partial D_c$ .","After returning back to $c$ let us keep walking along the curve ${C}$ in the same direction as before until returning back to $c$ .","Denote the corresponding path by ${R}_c$ and let $cc_a^rc_z^rc$ be the sequence of the points in the order we meet them on ${R}_c$ , where $\\lbrace c_a^r, c_z^r\\rbrace = \\partial D_c \\cap {R}_c$ .","Figure: The Conway smoothing the crossing cc and the chord 𝔠\\mathfrak {c} are shown.Next, let us delete the inner side of $D_c \\cap {C}$ and attach $c_l^a$ to $c_r^a$ , and $c^r_z$ to $c^l_z$ .","We thus get the new plane curve $\\widehat{{C}}_c$ (see Figure REF ).","It is easy to see that this curve is the shadow of the knot, which is obtained from $K$ by Conway's smoothing the crossing $c$ .","Let $\\widehat{\\mathfrak {G}}_\\mathfrak {c}$ be the Gauss diagram of $\\widehat{{C}}_c$ .","We shall say that the Gauss diagram $\\widehat{\\mathfrak {G}}_\\mathfrak {c}$ is obtained from the Gauss diagram $\\mathfrak {G}$ by Conway's smoothing the chord $\\mathfrak {c}$ .","As an immediate consequence of the preceding discussion, we get the following proposition.","Proposition 1.1 Let $\\mathfrak {G}$ be a Gauss diagram and $\\mathfrak {c}$ be its arbitrary chord.","Then $\\widehat{\\mathfrak {G}}_{\\mathfrak {c}}$ is obtained from $\\mathfrak {G}$ as follows: (1) delete the chord $\\mathfrak {c}$ , (2) if two chords $\\mathfrak {a}, \\mathfrak {b} \\in \\mathfrak {c}_\\times $ intersect (resp.","do not intersected) in $\\mathfrak {G}$ then they do not intersect in $\\mathfrak {\\widehat{G}_c}$ (resp.","intersected), (3) another chords keep their positions.","Indeed, let $W$ be the word which is the sequence of the crossings in the order we meet them on the curve ${C}$ .","Since $\\mathfrak {c}_\\times \\ne \\varnothing $ , $W$ can be written as follows $W = W_1 c W_2 c W_3$ , where $W_1,W_2,W_3$ are subwords of $W$ and at least one of $W_1,W_3$ is not empty.","Define $W_2^R$ as the reversal of the word $W_2$ .","Then, from the preceding discussion, the word $\\widehat{W}_c: = W_1 W_2^R W_3$ gives $\\widehat{\\mathfrak {G}}_{\\mathfrak {c}}$ (see Figure REF ) and the statement follows."],["Partitions of Gauss Diagrams","In this section we introduce notations, whose importance will become clear as we proceed.","Definition 2.1 Let $\\mathfrak {G}$ be a Gauss diagram and $\\mathfrak {a}$ a chord of $\\mathfrak {G}$ .","A $C$ -contour, denoted $C(\\mathfrak {a})$ , consists of the chord $\\mathfrak {a}$ , a chosen arc $\\mathfrak {a}_0\\mathfrak {a}_1$ , and all chords of $\\mathfrak {G}$ such that all their endpoints lie on the arc $\\mathfrak {a}_0\\mathfrak {a}_1$ .","We call a chord from the set $\\mathfrak {a}_\\times $ the door chord of the $C$ -contour $C(\\mathfrak {a})$.","Let us consider a plane curve ${C}:S^1 \\rightarrow \\mathbb {R}^2$ and let $\\mathfrak {G}$ be its Gauss diagram.","Every chord $\\mathfrak {c \\in G}$ correspondences to the crossing $c$ of ${C}$ .","Thus for every $C$ -contour $C(\\mathfrak {c})$ , we can associate a closed path ${C}(c)$ along the curve ${C}$ .","We call ${C}(c)$ the loop of the curve ${C}$ .","It is obviously that there is the one-to-one correspondence between self-intersection points of ${C}(c)$ and all chords from $C(\\mathfrak {c})$ .","Figure: Every CC-contour of the Gauss diagram correspondences to the closed path along the plane curve and vise versa.","We see that the chord 6 correspondences to the self-intersection point 6 of the dotted loop.Example 2.2 In Figure REF the plane curve ${C}$ and its Gauss diagram are shown.","Consider the (cyan) $C$ -contour $C(5)$ .","We see that ${C}(5)$ is the closed path along the curve.","It is the self-intersecting path and we see that the crossing 6 correspondences to the chord from the set $5_\\parallel $ .","Further, the red closed path ${C}(0)$ correspondences to the red $C$ -contour $C(0)$ .","Definition 2.3 Let $\\mathfrak {G}$ be a Gauss diagram, $\\mathfrak {a},\\mathfrak {b}$ its intersecting chords.","An $X$ -contour, denoted $X(\\mathfrak {a},\\mathfrak {b})$ , consists of two non-intersecting arcs $\\mathfrak {a}_0\\mathfrak {b}_0$ , $\\mathfrak {a}_1\\mathfrak {b}_1$ and all chords of $\\mathfrak {G}$ such that all their endpoints lie on $\\mathfrak {a}_0\\mathfrak {b}_0$ or on $\\mathfrak {a}_1\\mathfrak {b}_1$ .","A chord is called the door chord of the $X$ -contour $X(\\mathfrak {a},\\mathfrak {b})$ if only one of its endpoints belongs to $X(\\mathfrak {a},\\mathfrak {b})$ .","We say that the $X$ -contour $X(\\mathfrak {a},\\mathfrak {b})$ is non-degenerate if it has at least one door chord, and it does not contain all chords of $\\mathfrak {G}$ .","Example 2.4 Let us consider the Gauss diagram in Figure REF.","The black chords are the door chords of the black $X$ -contour $X(1,3)$ .","We see that the door chords correspondence to “entrances” and “exits” of the black closed path along the curve.","We also see that this $X$ -contour is non-degenerate.","The previous Example implies a partition of a Gauss diagram (resp.","a plane curve) into two parts.","Definition 2.5 (An $X$ -contour coloring) Given a Gauss diagram $\\mathfrak {G}$ and its an $X$ -contour.","Let us walk along the circle of $\\mathfrak {G}$ in a chosen direction and color all arcs of $\\mathfrak {G}$ until returning back to the origin as follows: (1) we don't colors the arcs of the $X$ -contour, (2) we use only two different colors, (3) we change a color whenever we meet an endpoint of a door chord.","Similarly, one can define a $C$ -contour coloring of a Gauss diagram $\\mathfrak {G}$ .","Remark 2.6 Let $\\mathfrak {G}$ be a Gauss diagram and ${C}$ the corresponding (may be virtual) plane curve, i.e., $\\mathfrak {G}$ determines the curve ${C}$ .","For every $X$ -contour $X(\\mathfrak {a},\\mathfrak {b})$ in $\\mathfrak {G}$ , we can associate the closed path along the curve ${C}$ .","We call this path the ${X}$ -contour and denote by ${X}(a,b)$ .","Similarly one can define door crossing for ${X}(a,b)$ .","Further, for the $X(\\mathfrak {a},\\mathfrak {b})$ -contour coloring of $\\mathfrak {G}$ , we can associate ${X}(a,b)$ -contour coloring of the curve ${C}$.","Next, let $\\mathfrak {G}$ be a realizable Gauss diagram determines the plane curve ${C}$ and let $X(\\mathfrak {a},\\mathfrak {b})$ be an $X$ -contour of $\\mathfrak {G}$ such that ${X}(a,b)$ is the non-self-intersecting path (= the Jordan curve).","Then the ${X}(a,b)$ -contour coloring of ${C}$ divides the curve ${C}$ into two colored parts, cf.","Jordan curve Theorem.","Figure: For the X(1,3)X(1,3)-contour coloring of the Gauss diagram, we associate the plane curve coloring.","We see that the X{X}-contour X(1,3){X}(1,3) (= black loop) divides the plane curve into two parts."],["The Even and The Sufficient Conditions","If a Gauss diagram can be realized by a plane curve we then say that this Gauss diagram is realizable, and non-realizable otherwise.","So, in this section, we give a criterion allowing verification and comprehension of whether a given Gauss diagram is realizable or not.","Moreover, we give an explanation allowing comprehension of why the needed condition is not sufficient for realizability of Gauss diagrams."],["The Even Condition","Proposition 3.1 Let ${C}:S^1 \\rightarrow \\mathbb {R}^2$ be a plane curve and $\\mathfrak {G}$ its Gauss diagram.","Then (1) $|\\mathfrak {a}_\\times \\cap \\mathfrak {b}_\\times | \\equiv 0 \\bmod 2$ for every two non-interesting chords $\\mathfrak {a,b \\in G}$ , (2) $|\\mathfrak {c}_\\times | \\equiv 0 \\bmod 2$ for every chord $\\mathfrak {c \\in G}$ .","Let $\\mathfrak {a},\\mathfrak {b}\\in \\mathfrak {G}$ be two non-intersecting chords of $\\mathfrak {G}$ .","Take two $C$ -contours $C(\\mathfrak {a})$ , $C(\\mathfrak {b})$ such that their arcs $\\mathfrak {a_0a_1}$ , $\\mathfrak {b_0b_1}$ do not intersect.","It is obvious that for the loops ${C}(a)$ , ${C}(b)$ , we can associate the one-to-one correspondence between the set ${C}(a) \\cap {C}(b)$ and the set $\\mathfrak {a}_\\times \\cap \\mathfrak {b}_\\times $ .","Because, by Proposition REF , all chord from the set $\\mathfrak {a}_\\times \\cap \\mathfrak {b}_\\times $ keep their positions in Gauss diagram $\\mathfrak {\\widehat{G}_c}$ (= Conway's smoothing the chord $\\mathfrak {c}$ ) for every $\\mathfrak {c} \\in \\mathfrak {a}_\\parallel \\cap \\mathfrak {b}_\\parallel \\setminus \\lbrace \\mathfrak {a},\\mathfrak {b}\\rbrace $ , it is sufficient to prove the statement in the case $\\mathfrak {a}_\\parallel \\cap \\mathfrak {b}_\\parallel = \\lbrace \\mathfrak {a},\\mathfrak {b}\\rbrace $ , i.e., the loops ${C}(a)$ , ${C}(b)$ are non-self-intersecting loops (= the Jordan curves).","From Jordan curve Theorem, it follows that the loop ${C}(a)$ divides the curve ${C}$ into two regions, say, ${I}$ and ${O}$ .","Assume that $b\\in {O}$ and let us walk along the loop ${C}(b)$ .","We say that an intersection point $c \\in {C}(a) \\cap {C}(b)$ is the entrance (resp.","the exit) if we shall be in the region ${I}$ (resp.","${O}$ ) after meeting $c$ with respect to our walk.","Since a number of entrances has to be equal to the number of exits, then $|\\mathfrak {a}_\\times \\cap \\mathfrak {b}_\\times | \\equiv 0\\bmod 2$ .","Arguing similarly, we prove $|\\mathfrak {c}_\\times | \\equiv 0 \\bmod 2$ for every chord $\\mathfrak {c}$ .","As an immediate consequence of Proposition REF we get the following.","Corollary 3.1 (The Even Condition) If a Gauss diagram is realizable then the number of all chords that cross a both of non-intersecting chords and every chord is even (including zero).","We conclude this subsection with an explanation why the even condition is not sufficient for realizability of Gauss diagrams.","Roughly speaking, from the proof of Proposition REF it follows that every plane curve can be obtained by attaching its loops to each other by given points.","Conversely, if a Gauss diagram satisfies the even condition then it may be non-releasible.","Indeed, when we attach a loop, say, ${C}(b)$ to a loop ${C}(a)$ , where $\\mathfrak {b} \\in \\mathfrak {a}_\\parallel $ , by given points (=elements of the set $\\mathfrak {a}_\\times \\cap \\mathfrak {b}_\\times $ ) then the loop ${C}(b)$ can be self-intersected curve, which means that we get new crossings (= virtual crossings), see Figure REF .","To be more precisely, we have the following proposition.","Proposition 3.2 Let $\\mathfrak {G}$ be a non-realizable Gauss diagram which satisfies the even condition.","Let $\\mathfrak {G}$ defines a virtual plane curve ${C}$ (up to virtual moves).","There exist two non-intersecting chords $\\mathfrak {a,b \\in G}$ such that there are paths $c \\rightarrow x \\rightarrow d$ , $e \\rightarrow x \\rightarrow f$ on a loop ${C}(b)$ , where $\\mathfrak {c,d,e,f \\in a_\\times \\cap b_\\times }$ are different chords and $x$ is a virtual crossing of ${C}$ .","Let $\\mathfrak {a,b \\in G}$ be two non-intersecting chords.","Take non-intersecting $C$ -contours $C(\\mathfrak {a})$ , $C(\\mathfrak {b})$ .","Hence we may say that the loop ${C}(b)$ attaches to the loop ${C}(a)$ by the given points $p_1,\\ldots ,p_n $ , where $\\lbrace \\mathfrak {p_1,\\ldots ,p_n}\\rbrace = \\mathfrak {a_\\times \\cap b_\\times }$ .","Since $\\mathfrak {G}$ is not realizable and satisfies the even condition then a virtual crossing may arise only as a self-intersecting point of, say, the loop ${C}(b)$ .","Indeed, when we attach ${C}(b)$ to ${C}(a)$ by $p_1,\\ldots ,p_n $ we may get self-interesting points, say, $q_1,\\ldots ,q_m$ of the loop ${C}(b)$ .","If $\\mathfrak {G}$ contains all chords $\\mathfrak {q_1},\\ldots ,\\mathfrak {q}_m$ for every such chords $\\mathfrak {a,b}$ , then $\\mathfrak {G}$ is realizable.","Thus, a virtual crossing $x$ does not belong to $\\lbrace p_1,\\ldots , p_n\\rbrace = {C}(a) \\cap {C}(b)$ for some non-intersecting chords $\\mathfrak {a,b \\in G}$ .","Then we get two paths $c \\rightarrow x \\rightarrow d$ , $e \\rightarrow x \\rightarrow f$ , where $c,d,e,f \\in {C}(a) \\cap {C}(b)$ are different chords, as claimed.","Figure: It shows that the even condition is not sufficient for realizability of the Gauss diagrams.","We see that the plane curve can be obtained by attaching the white-black loop to the dotted one by the points 3,4,5,63,4,5,6, and thus the dotted loop has to have “new” crossings (= self-intersections) x 1 ,x 2 ,x 3 x_1,x_2,x_3."],["The Sufficient Condition","Definition 3.1 Let $\\mathfrak {G}$ be a Gauss diagram (not necessarily realizable) and $X(\\mathfrak {a,b})$ its $X$ -contour.","Take the $X$ -contour coloring of $\\mathfrak {G}$ .","A chord of $\\mathfrak {G}$ is called colorful for $X(\\mathfrak {a,b})$ if its endpoints are in arcs which have different colors.","Similarly, one can define a colorful chord for a $C$ -contour $C(\\mathfrak {a})$ of $\\mathfrak {G}$ .","Example 3.2 Let us consider the Gauss diagram, which is shown in Figure REF .","One can easy check that this Gauss diagram is not realizable.","Let us consider the orange $X$ -contour $X(1,3)$ and the $X(1,3)$ -coloring of $\\mathfrak {G}$ .","The chord with the endpoints 5 is colorful for the $X$ -contour $X(1,3)$ .","It is interesting to consider the corresponding coloring of the virtual plane curve: one can think that we forget to change color when we cross the gray loop, i.e., the gray loop “does not divide” the curve into two parts.","We shall show that this observation is typical for every non-realizable Gauss diagram.","Figure: This Gauss diagram satisfies even condition but is non-realizable.","There are colorful chords (e.g.","the chord with endpoints 5).We have seen that if a Gauss diagram is realizable then there is no colorful chord, with respect to every $X$ -contour.","We shall show that it is sufficient condition for realizability of a Gauss diagram.","Proposition 3.3 Let $\\mathfrak {G}$ be a non-realizable Gauss diagram but satisfy the even condition.","Then there exists an $X$ -contour and a colorful chord for this $X$ -contour.","By Theorem REF , $\\mathfrak {G}$ defines a virtual curve ${C}$ (= the shadow of a virtual knot diagram) up to virtual moves.","Starting from a crossing, say, $o$ , let us walk along ${C}$ till we meet the first virtual crossing, say, $x$ .","Next, let us keep walking in the same direction till we meet the first real crossing, say $d$ .","Denote this path by ${P}$ .","Just for convinces, let us put the labels, say, $x_0,x_1$ of the virtual crossing $x$ on the circle of $\\mathfrak {G}$ in the order we meet them on ${P}$ .","We can take two real crossings, say, $a,b$ such that the path $a \\rightarrow x \\rightarrow b$ does not contain another real crossings and $\\mathfrak {a,b} \\in \\mathfrak {o}_\\times .$ Indeed, from Proposition REF it follows that for a chord $\\mathfrak {o}\\in \\mathfrak {G}$ we can find $n \\ge 1$ chords $\\mathfrak {o}^1,\\ldots ,\\mathfrak {o}^n$ such that, for every $1\\le i \\le n$ , we have: (1) the chords $\\mathfrak {o}, \\mathfrak {o}^i$ do not intersect, (2) the loop ${C}(o)$ contains the following paths $a^i \\rightarrow x \\rightarrow b^i$ , $c^i \\rightarrow x \\rightarrow d^i$ where $\\mathfrak {a}^i,\\mathfrak {b}^i,\\mathfrak {c}^i,\\mathfrak {d}^i \\in \\mathfrak {o}_\\times \\cap \\mathfrak {o}^i_\\times $ are different chords.","Thus, for some $1\\le i,j \\le n$ we have an arc, say, $\\mathfrak {a}^i_1\\mathfrak {b}^j_1$ contains only one of $x_0$ or $x_1$ , no endpoints of another chords, and $\\mathfrak {a}^i,\\mathfrak {b}^j \\in \\mathfrak {o}_\\times $ .","Denote this arc by $\\mathfrak {a_1b_1}$ and assume that $x_0$ lies on $\\mathfrak {a_1b_1}$ .","We next construct an $X$ -contour contains the arc $\\mathfrak {a_1b_1}$ .","To do so, we have to consider the following two cases: Case 1.","The chords $\\mathfrak {a,b}$ intersect.","Figure: Since the curve C{C} is determined up to virtual moves then the cyan line has to cross the olive line.Take the $X$ -contour $X(\\mathfrak {a,b})$ which contains the arc $\\mathfrak {a_1b_1}$ .","It is easy to see that this $X$ -contour has at least one real door chord, because its another arc $\\mathfrak {a_0b_0}$ has no virtual crossing thus it has to have at least one real crossing (see Figure REF).","Case 2.","The chords $\\mathfrak {a,b}$ do not intersect.","Figure: The “general” position of chords 𝔞,𝔟,𝔠,𝔡\\mathfrak {a,b,c,d} is shown.","Our walk along the path P{P} correspondences to 𝔬 0 →𝔬 ˜ 0 →𝔟 0 →⋯→𝔡 1 \\mathfrak {o_0 \\rightarrow \\widetilde{o}_0 \\rightarrow b_0\\rightarrow \\cdots \\rightarrow \\mathfrak {d_1}}.Let us walk along the circle of $\\mathfrak {G}$ in the direction $\\mathfrak {b_1} \\rightarrow x_0 \\rightarrow \\mathfrak {a}_1$ (Figure REF ) till we meet the first chord, say, $\\widetilde{\\mathfrak {o}}$ such that $\\mathfrak {a,b,o} \\in \\widetilde{\\mathfrak {o}}_\\times $ .","If we cannot find such chord we set $\\widetilde{\\mathfrak {o}}:=\\mathfrak {o}$ .","Take the $X$ -contour $X(\\widetilde{\\mathfrak {o}},\\mathfrak {b})$ contains the arc $\\mathfrak {a_1b_1}$ .","This $X$ -contour is non-degenerate.","Indeed, let $\\widetilde{\\mathfrak {o}}:=\\mathfrak {o}$ and endpoints of all chords, which start in the arc $\\mathfrak {b_1}x_1\\mathfrak {d}_1$ , lie on the arc $\\mathfrak {o}_0\\mathfrak {b_0}$ .","We then get the situation is shown in Figure REF .","Figure: The dotted blue line gives another path from cc to dd.It follows that we can take another path ${P}$ without the virtual crossing $x$ .","This contradiction implies that the $X$ -contour $X(\\mathfrak {o,b})$ is non-degenerate.","So, we have a non-degenerate ${X}$ -contour of the curve ${C}$ such that $x$ is its virtual door.","If the $X$ -contour of $\\mathfrak {G}$ contains another chords such that all their endpoints lie on an arc of this $X$ -contour, then Proposition REF implies that Conway's smoothing all such chords does not change the ${X}$ -contour coloring of ${P}$ .","Thus we may assume that this ${X}$ -contour is the closed curve without self-intersections.","Hence, by Jordan curve Theorem, it divides ${P}$ into two parts.","Let us color these parts into two different colors.","If the crossing $x$ would be a real door of this ${X}$ -contour, then by Definition REF , we can take such ${X}$ -contour coloring of ${P}$ as before.","But since $x$ is not real door then after meeting it at the second time we do not change the color and thus we get an intersection point of two lines which have different colors, i.e., the corresponding chord is colorful.","This completes the proof.","Lemma 3.1 Let $\\mathfrak {G}$ be a Gauss diagram.","Consider a $C(\\mathfrak {a})$ -contour coloring of $\\mathfrak {G}$ for some chord $\\mathfrak {a\\in G}$ .","If there exists a colorful chord for the $C$ -contour $C(\\mathfrak {a})$ then the diagram $\\mathfrak {G}$ does not satisfy the even condition.","Indeed, let $\\mathfrak {b}$ be a colorful chord for the $C$ -contour $C(\\mathfrak {a})$ .","First note that, the chord $\\mathfrak {b}$ cannot cross $\\mathfrak {a}$ because otherwise $\\mathfrak {b}$ should be a door chord of the $C$ -contour $C(\\mathfrak {a})$ .","Next, if the chord $\\mathfrak {b}$ is colorful then it crosses an odd number of door chords of $C(\\mathfrak {a})$ .","Hence $|\\mathfrak {a}_\\times \\cap \\mathfrak {b}_\\times | \\equiv 1 \\bmod 2$ , as claimed.","Figure: The Gauss diagram does not satisfy the even condition; both the chords 1, 6 are crossed by only one chord 2.Example 3.3 Let us consider the Gauss diagram which is shown in Figure REF .","We have the orange $C$ -contour $C(1)$ and the $C(1)$ -contour coloring of the Gauss diagram and the corresponding (virtual) curve.","We see that there are two chords (namely 5 and 6) which are colorful and $5_\\times \\cap 1_\\times = 6_\\times \\cap 1_\\times = \\lbrace 2\\rbrace $ .","Lemma 3.2 Let $\\mathfrak {G}$ be a Gauss diagram and $\\mathfrak {a}, \\mathfrak {b} \\in \\mathfrak {G}$ be its intersecting chords.","Suppose that there exists a colorful chord $\\mathfrak {c}$ for an $X$ -contour $X(\\mathfrak {a},\\mathfrak {b})$ .","Then there exists a $C$ -contour of the Gauss diagram $\\widehat{\\mathfrak {G}}_\\mathfrak {b}$ (= Conway's smoothing the chord $\\mathfrak {b}$ ) such that the chord $\\mathfrak {c}$ is colorful for this $C$ -contour in $\\widehat{\\mathfrak {G}}_\\mathfrak {b}$ .","Indeed, consider the Gauss diagram $\\widehat{\\mathfrak {G}}_\\mathfrak {b}$ .","From Proposition REF it follows that after Conway's smoothing the chord $\\mathfrak {b}$ , the chord $\\mathfrak {c}$ does not intersect $\\mathfrak {a}$ and intersects the same door chords of the $X$ -contour $X(\\mathfrak {a},\\mathfrak {b})$ as in $\\mathfrak {G}$ .","Further, let us consider the $C$ -contour $C(\\mathfrak {a})$ in $\\widehat{\\mathfrak {G}}_\\mathfrak {b}$ such that it does not contain the chord $\\mathfrak {c}$ .","By Proposition REF , the chord $\\mathfrak {a}$ crosses in $\\widehat{\\mathfrak {G}}_\\mathfrak {b}$ only the chord that are door chords of the $X$ -contour $X(\\mathfrak {a},\\mathfrak {b})$ .","Hence, by Definition REF , we may take the $C(\\mathfrak {a})$ -contour coloring of $\\widehat{\\mathfrak {G}}_\\mathfrak {b}$ such that $\\mathfrak {c}$ is the colorful chord for this $C$ -contour.","Proposition 3.4 Let a Gauss diagram $\\mathfrak {G}$ satisfy the even condition.","Then $\\mathfrak {G}$ is realizable if and only if $\\widehat{\\mathfrak {G}}_\\mathfrak {c}$ satisfies the even condition for every chord $\\mathfrak {c} \\in \\mathfrak {G}$ , $\\widehat{\\mathfrak {G}}_\\mathfrak {c}$ .","Indeed, let $\\mathfrak {G}$ be a non-realizable Gauss diagram and let $\\mathfrak {G}$ satisfy the even condition.","By Proposition REF , there exists a colorful chord (say $\\mathfrak {c}$ ) for a $X$ -counter $X(\\mathfrak {a},\\mathfrak {b})$ of $\\mathfrak {G}$ .","By Lemma REF , the chord $\\mathfrak {c}$ is the colorful chord in $\\widehat{\\mathfrak {G}}_\\mathfrak {b}$ .","Hence from Lemma REF it follows that $\\widehat{\\mathfrak {G}}_\\mathfrak {b}$ does not satisfy the even condition, and the statement follows.","We can summarize our results in the following theorem.","Theorem 3.4 A Gauss diagram $\\mathfrak {G}$ is realizable if and only if the following conditions hold: (1) the number of all chords that cross a both of non-intersecting chords and every chord is even (including zero), (2) for every chord $\\mathfrak {c} \\in \\mathfrak {G}$ the Gauss diagram $\\widehat{\\mathfrak {G}}_\\mathfrak {c}$ (= Conway's smoothing the chord $\\mathfrak {c}$ ) also satisfies the above condition."],["Matrixes of Gauss Diagrams","In this section we are going to translate into matrix language the conditions of realizability of Gauss diagrams.","We consider all matrixes over the field $\\mathbb {Z}_2$ .","Definition 4.1 Given a Gauss diagram $\\mathfrak {G}$ contains $n$ chords, say, $\\mathfrak {c}_1,\\ldots ,\\mathfrak {c}_n$ .","Construct a $n \\times n$ matrix $\\mathsf {M}(\\mathfrak {G}) = (\\mathsf {m}_{i,j})_{1 \\le i,j\\le n}$ as follows: (1) put $\\mathsf {m}_{ii} = 0$ , $1 \\le i \\le n$ , (2) $\\mathsf {m}_{ij} = 1$ if the chords $\\mathfrak {c}_i$ , $\\mathfrak {c}_j$ intersect and $\\mathsf {m}_{ij} = 0$ in otherwise.","It is obviously that $\\mathsf {M}(\\mathfrak {G})$ is a symmetric matrix with respect to the main diagonal.","Next, let $\\mathsf {M}(\\mathfrak {G})$ be a $n\\times n$ matrix of a Gauss diagram $\\mathfrak {G}$ .","Let $\\cup _{1 \\le i \\le n}\\lbrace \\mathsf {m}_i = (\\mathsf {m}_{i1}, \\ldots , \\mathsf {m}_{in})\\rbrace $ be all its strings.","Define a scalar product of strings as follows: $\\langle \\mathsf {m}_i, \\mathsf {m}_j \\rangle :=\\mathsf {m}_{i1}\\mathsf {m}_{j1} + \\cdots + \\mathsf {m}_{in}\\mathsf {m}_{jn}.$ It is clear that we can define “scalar product” of any number of strings as follows $\\langle \\mathsf {m}_{i_1}, \\ldots , \\mathsf {m}_{i_k} \\rangle : = \\mathsf {m}_{i_11} \\cdots \\mathsf {m}_{i_k1} + \\cdots + \\mathsf {m}_{i_1n} \\cdots \\mathsf {m}_{i_kn}.$ Lemma 4.1 Let $\\mathfrak {G}$ be a Gauss diagram contains $n$ chords $\\mathfrak {c}_1,\\ldots ,\\mathfrak {c}_n$ .","Consider the corresponding matrix $\\mathsf {M}(\\mathfrak {G})$ .","Let the chords $\\mathfrak {c}_1,\\ldots ,\\mathfrak {c}_n$ correspondence to the strings $\\mathsf {m}_1,\\ldots ,\\mathsf {m}_n$ , respectively.","$| {\\mathfrak {c}_{i_1}}_\\times \\cap \\cdots \\cap {\\mathfrak {c}_{i_k}}_\\times | = \\langle \\mathsf {m}_{i_1}, \\ldots , \\mathsf {m}_{i_k} \\rangle ,$ for every $\\lbrace i_1,\\ldots ,i_k\\rbrace \\subseteq \\lbrace 1,2,\\ldots ,n\\rbrace $ .","Indeed, let $\\mathsf {m}_{i_1t} \\cdots \\mathsf {m}_{i_kt} \\ne 0$ , for some $1 \\le t \\le n$ , then the chord $\\mathfrak {c}_t$ intersect chords $\\mathfrak {c}_{i_1}, \\ldots , \\mathfrak {c}_{i_k}$ .","This completes the proof.","Theorem 4.2 Let $\\mathfrak {G}$ be a realizable Gauss diagram contains $n$ chords.","Then its matrix $\\mathsf {M}(\\mathfrak {G})$ satisfies the following conditions: (1) $\\langle \\mathsf {m}_i, \\mathsf {m}_i \\rangle \\equiv 0 \\bmod (2)$ , $1 \\le i \\le n$ , (2) $\\langle \\mathsf {m}_i, \\mathsf {m}_j \\rangle \\equiv 0 \\bmod (2)$ , if the corresponding chords do not intersect, (3) $\\langle \\mathsf {m}_i, \\mathsf {m}_j \\rangle + \\langle \\mathsf {m}_i, \\mathsf {m}_k \\rangle + \\langle \\mathsf {m}_j, \\mathsf {m}_k \\rangle \\equiv 1 \\bmod (2)$ , if the corresponding chords pairwise intersect.","The first two conditions follow from the first condition of Theorem REF and Lemma REF .","Assume that $\\mathfrak {G}$ does not contain three pairwise intersecting chords.","Then $\\mathfrak {G}$ is realizable if and only if the first condition of Theorem REF holds and we get the first two conditions for $\\mathsf {M}(\\mathfrak {G})$ .","Next, let $\\mathfrak {a},\\mathfrak {b},\\mathfrak {c}$ be three pairwise intersecting chords.","Consider the following sets of chords (see Figure REF ): $&A: = \\mathfrak {a}_\\times \\setminus \\lbrace \\mathfrak {b}_\\times , \\mathfrak {c}_\\times \\rbrace ,\\\\& B:= \\mathfrak {b}_\\times \\cap \\mathfrak {c}_\\times \\setminus \\left\\lbrace \\lbrace \\mathfrak {a}\\rbrace , \\mathfrak {a}_\\times \\right\\rbrace ).","$ Figure: The sets AA, BB are roughly shown.By Proposition REF , chords $\\mathfrak {b}, \\mathfrak {c}$ does not intersect in $\\widehat{\\mathfrak {G}}_\\mathfrak {a}$ , and the set $\\mathfrak {b}_\\times \\cap \\mathfrak {c}$ of chords in $\\widehat{\\mathfrak {G}}_\\mathfrak {a}$ is equal to the set $A \\cup B$ .","Hence, by Theorem REF , $|A| + |B| \\equiv 0 \\bmod (2).$ Further, using Lemma REF , we obtain: $&|A| = \\langle \\mathfrak {a}, \\mathfrak {a} \\rangle - \\langle \\mathfrak {a}, \\mathfrak {b} \\rangle - \\langle \\mathfrak {a}, \\mathfrak {c} \\rangle - \\langle \\mathfrak {a}, \\mathfrak {b}, \\mathfrak {c} \\rangle ,\\\\&|B| = \\langle \\mathfrak {b}, \\mathfrak {c} \\rangle -1- \\langle \\mathfrak {a}, \\mathfrak {b}, \\mathfrak {c} \\rangle .$ It follows that $|A| + |B| & =& \\langle \\mathfrak {a}, \\mathfrak {a} \\rangle - \\langle \\mathfrak {a}, \\mathfrak {b} \\rangle - \\langle \\mathfrak {a}, \\mathfrak {c} \\rangle - \\langle \\mathfrak {a}, \\mathfrak {b}, \\mathfrak {c} \\rangle \\\\&&+ \\langle \\mathfrak {b}, \\mathfrak {c} \\rangle -1- \\langle \\mathfrak {a}, \\mathfrak {b}, \\mathfrak {c} \\rangle \\\\&\\equiv & \\langle \\mathfrak {a}, \\mathfrak {b} \\rangle + \\langle \\mathfrak {a}, \\mathfrak {c} \\rangle + \\langle \\mathfrak {b}, \\mathfrak {c} \\rangle \\\\&\\equiv & 1 \\bmod (2),$ because, as we have already discussed, $\\langle \\mathfrak {a}, \\mathfrak {a} \\rangle \\equiv 0 \\bmod (2)$ .","As claimed.","Remark 4.3 The number of matrix satisfy the above conditions is bigger than a number of realizable Gauss diagrams.","The reason is the following observation.","A matrix which satisfies the above conditions “knows” only intersections but don't know positions of chords.","However there is a very important sort of plane curves (=meanderes) such that there is a one-to-one correspondence between Gauss diagrams of these curves and the corresponding matrixes."],["Thurston Generators of Braid Groups","We will use as generators for $B_n$ the set of positive crossings, that is, the crossings between two (necessary adjacent) strands, with the front strand having a positive slope.","We denote these generators by $\\sigma _1, \\ldots , \\sigma _{n-1}$ .","These generators are subject to the following relations: ${\\left\\lbrace \\begin{array}{ll}\\sigma _i\\sigma _j = \\sigma _j\\sigma _i, \\mbox{ if } |i-j| >1,\\\\\\sigma _i\\sigma _{i+1}\\sigma _i = \\sigma _{i+1}\\sigma _i\\sigma _{i+1}.\\end{array}\\right.","}$ One obvious invariant of an isotopy of a braid is the permutation it induces on the order of the strands: given a braid $B$ , the strands define a map $p(B)$ from the top set of endpoints to the bottom set of endpoints, which we interpret as a permutation of $\\lbrace 1, \\ldots , n\\rbrace $ .","In this way we get a homomorphism $p:B_n \\rightarrow \\mathfrak {S}_n$ , where $\\mathfrak {S}_n$ is the symmetric group.","The generator $\\sigma _i$ is mapped to the transposition $s_i = (i,i+1)$ .","We denote by $S_n = \\lbrace s_1, \\ldots , s_{n-1}\\rbrace $ the set of generators for the symmetric group $\\mathfrak {S}_n$ .","Now we want to define an inverse map $p^{-1}:\\mathfrak {S}_n \\rightarrow B_n$ .","To this end, we need the following definition [10] Definition 5.1 Let $S = \\lbrace s_1, \\ldots , s_{n-1}\\rbrace $ be the set of generators for $\\mathfrak {S}_n$ .","Each permutation $\\pi $ gives rise to a total order relation $\\le _\\pi $ on $\\lbrace 1, \\ldots ,n\\rbrace $ with $i \\le _\\pi j$ if $\\pi (i) < \\pi (j)$ .","We set $R_\\pi : = \\lbrace (i,j) \\in \\lbrace 1, \\ldots , n\\rbrace \\times \\lbrace 1,\\ldots ,n\\rbrace |i \\pi (j)\\rbrace .$ Lemma 5.1 [10] A set $R$ of pairs $(i,j)$ , with $i \\pi (2),\\end{array}\\right.}","\\quad {\\left\\lbrace \\begin{array}{ll}1<4, \\\\ \\pi (1) > \\pi (4),\\end{array}\\right.}","\\quad {\\left\\lbrace \\begin{array}{ll}1<6, \\\\ \\pi (1) > \\pi (6),\\end{array}\\right.}","\\quad {\\left\\lbrace \\begin{array}{ll}2<4, \\\\ \\pi (2) > \\pi (4),\\end{array}\\right.","}\\\\& {\\left\\lbrace \\begin{array}{ll}3<4,\\\\ \\pi (3) > \\pi (4),\\end{array}\\right.}","\\quad {\\left\\lbrace \\begin{array}{ll}3<5,\\\\ \\pi (3) > \\pi (5),\\end{array}\\right.}","\\quad {\\left\\lbrace \\begin{array}{ll}3<6,\\\\ \\pi (3) > \\pi (6), \\end{array}\\right.}","\\quad {\\left\\lbrace \\begin{array}{ll}5<6,\\\\ \\pi (5) > \\pi (6).\\end{array}\\right.","}$ hence $R(\\pi ) = \\lbrace (1,2),(1,4),(1,6),(2,4),(3,4),(3,5),(3,6),(5,6)\\rbrace ,$ and we get a non-repeating braid is shown in Figure REF .","Figure: The simple braid R π R_\\pi is shown."],["Meanders and its Gauss Diagrams","In this section we deal with closed meanders.","We show that any closed meander defines a special sort of Gauss diagrams (= Gauss diagrams of meanders) and then we will see that these diagrams are completely determined by its matrixes (= meander matrix) i.e., there is a one-to-one correspondence between meander matrixes satisfy the conditions of Theorem REF .","It allows us to describe an algorithm constructs all closed meanders.","Definition 6.1 A (closed) meander is a planar configuration consisting of a simple closed curve and on orientied line, that cross finitely many times and intersect only transversally.","Two meanders are equivalent if there exists a homeomorphism of the plane that maps one to the other.","Figure: An Example of a closed meander.Definition 6.2 A matrix $\\mathbf {M} = (\\mathbf {m}_{ij}) \\in \\mathsf {Mat}_{n+1}(\\mathbb {Z}_2)$ is called meander matrix if the following conditions hold: (1) its main diagonal contains only 0, (2) $\\mathbf {M}$ is symmetric, (3) it has at leas one string $\\mathbf {m}_i = (\\mathbf {m}_{i1}, \\ldots , \\mathbf {m}_{i, n+1})$ with $\\mathbf {m}_{ik} = 1$ for every $1 \\le k\\le n+1$ , $k \\ne i$ , (4) $\\mathbf {M}$ satisfies the conditions of Theorem REF , (5) if $\\mathbf {m}_{ij}=1$ and $\\mathbf {m}_{jk} = 1$ then $\\mathbf {m}_{ik}=1$ for all $1 \\le i,j,k \\le n+1$ , (6) if $\\mathbf {m}_{ik}=1$ , then $\\mathbf {m}_{ij}=1$ or $\\mathbf {m}_{jk}=1$ for every $j$ with $1 \\le i \\equiv 1 \\bmod (2),$ for any $j>i$ .","Indeed, by Lemma REF , $\\mathbf {m}_{ij}=1$ for every $j>i$ in the matrix $\\mathbf {M}$ , and then by condition (3) of Theorem REF , $\\bigl <\\mathsf {0,i} \\bigr > + \\bigl < \\mathsf {0,j} \\bigr > + \\bigl <\\mathsf {i,j} \\bigr > \\equiv 1 \\bmod (2)$ .","We thus get $\\bigl <\\mathsf {0,i} \\bigr > + \\bigl < \\mathsf {0,j} \\bigr > + \\bigl <\\mathsf {i,j} \\bigr >= \\bigl < \\mathsf {i,j}\\bigr >,$ because of $\\bigl <\\mathsf {0,i} \\bigr > = \\bigl <\\mathsf {i,i} \\bigr > - 1$ , $1 \\le i \\le n$ , as claimed.","Theorem 7.3 Let $\\mathbf {M} \\in \\mathsf {Mat}_{n}(\\mathbb {Z}_2)$ be a symmetric matrix with zero main diagonal.","$\\mathbf {M}$ is a meander matrix if and only if the following hold: (1) there is a string $\\mathbf {m}_i$ such that $\\mathbf {m}_{ij}=1$ if $j \\ne i$ and $\\mathbf {m}_{ii}=0$ , (2) if $\\mathbf {m}_{ij} = 1$ , $\\mathbf {m}_{jk} =1$ , then $\\mathbf {m}_{ik}=1$ , for all $i,j,k \\in \\lbrace 1,2, \\ldots , n\\rbrace $ , (3) if $\\mathbf {m}_{ik}=1$ then $\\mathbf {m}_{ij}=1$ or $\\mathbf {m}_{jk}=1$ for all $i \\le j \\le k$ , (4) for every $1 \\le i,j\\le n$ we have $\\bigl < \\mathbf {m}_i, \\mathbf {m}_j \\bigr > &\\equiv & 0 \\bmod (2), \\qquad \\mbox{if $i=j$ or $\\mathbf {m}_{ij} = 0$},\\\\\\bigl < \\mathbf {m}_i, \\mathbf {m}_j \\bigr > &\\equiv & 1 \\bmod (2), \\qquad \\mbox{if $\\mathbf {m}_{ij} = 1$}.$ By Definition REF and Proposition REF , it is sufficient to prove that condition (3) of Theorem REF is equivalent to the last condition.","It is clear that an equality $\\bigl < \\mathbf {m}_i, \\mathbf {m}_i \\bigr > \\equiv 0 \\bmod (2)$ implies that only an even number (including zero) of chords intersect chord correspondences to string $\\mathbf {m}_i$ .","Next, an equality $\\mathbf {m}_{ij}=0$ implies that chord $i$ do not intersect chord $j$ , and an equality $\\mathbf {m}_{ij}=0$ implies that only even number of chords intersect both of chords $i$ , $j$ .","We thus get the first two conditions of Theorem REF .","Further, by Lemma REF , from the third condition of Theorem REF it follows (REF ).","Assume now that (REF ) holds and consider three pairwise intersecting chords, say $i,j,k$ .","Then $\\bigl < \\mathsf {m}_i, \\mathsf {m}_j \\bigr > \\equiv 1 \\bmod (2)$ , $\\bigl < \\mathbf {m}_i, \\mathbf {m}_k \\bigr > \\equiv 1 \\bmod (2)$ , and $\\bigl < \\mathbf {m}_j, \\mathbf {m}_k \\bigr > \\equiv 1 \\bmod (2)$ and we complete the proof.","From this Theorem it follows the following meanders construction algorithm.","REQUIRE an even number $N$ , $S_1 = \\lbrace 1,3,\\ldots , N-1\\rbrace $ , $S_0 = \\lbrace 2,4, \\ldots ,N\\rbrace $ ; ENSURE $n_1,\\ldots , n_N \\in S_1 \\cup S_0$ ; 1. take a $(N+1) \\times (N+1)$ tableau with empty cells and fill the main diagonal by zeros; 2. number strings from left to right, and number columns from top to bottom by numbers $0,1, \\ldots , N$ ; 3.","$\\Delta $ -fill string with number 0; 4. choose a string with an odd number $n\\in S_1$ and $\\Delta $ -fill it and column with the same number; 5.","$i=1$ ; 6.","PRINT $n$ ; 6.","$S_i: = S_i \\setminus \\lbrace n\\rbrace $ ; 7.","IF $S_i = \\varnothing $ THEN GOTO 13 ELSE GOTO 8; 8 $i:=i+1 \\bmod (2)$ ; 9. choose a string (a column) with number $m \\in S_i$ ; 10.","IF the string (the column) can be $\\Delta $ -filled THEN GOTO 11 ELSE choose another $m^{\\prime } \\in S_i\\setminus \\lbrace m\\rbrace $ GOTO 10; 11. using (REF ), () get a system of equations for empty cells; 12.","IF the system can by solved THEN GOTO 6 ELSE choose a string (column) with another number $m^{\\prime } \\in S_i\\setminus \\lbrace m\\rbrace $ GOTO 10; 13.","END Example 7.4 Let us construct a $9\\times 9$ meander matrix.","(0) Take $9\\times 9$ tableau and partially fill it as follows Table: NO_CAPTION (1) Choose a string and column 5 and $\\Delta $ -fill them Table: NO_CAPTION By(), ${\\left\\lbrace \\begin{array}{ll}\\bigl < 5, 6 \\bigr > = 1 + (6,7) + (6,8) \\equiv 1 \\bmod (2),\\\\\\bigl < 5, 7 \\bigr > = 1 + (6,7) + (7,8) \\equiv 1 \\bmod (2),\\\\\\bigl < 5, 8 \\bigr > = 1 + (6,8) + (7,8) \\equiv 1 \\bmod (2),\\end{array}\\right.","}$ it implies that $(6,7) = (6,8) = (7,8) = {a} \\in \\lbrace 0,1\\rbrace ,$ and we thus get Table: NO_CAPTION (2) We have to choose a string (column) with an even number.","(i) Take a string (column) 2, we then get $(1,2) = 0, (2,3) = (2,4) =0, (2,6) = (2,7) = (2,8)=1,$ it follows that $\\bigl <\\mathsf {1,2} \\bigr > \\equiv 0 \\bmod (2)$ .","We have $\\bigl <\\mathsf {1,2} \\bigr > &=& 1 + (1,3) + (1,4) \\\\&&+ (1,6)+ (1,7) + (1,8) \\equiv 0 \\bmod (2),$ and we do not get any contradiction, hence this string may be chosen.","(ii) Similarly, by the straightforward verification, one can easy verify that strings 4, 6 can be chosen.","(iii) Take string 8.","We get $(1,8) = (2,8) = (3,8) = (4,8) = 0= {a} = 0,$ and hence $\\bigl <\\mathsf {1,8} \\bigr > \\equiv 0 \\bmod (2)$ , but the tableau implies that $\\bigl <\\mathsf {1,8} \\bigr > = 1$ .","Therefore this string cannot be chosen in this step.","(3) Chose string 2.","We then get Table: NO_CAPTION by (), ${\\left\\lbrace \\begin{array}{ll}\\bigl < \\mathsf {2,3} \\bigr > = 1 + (3,4) + (3,6) + (3,7) + (3,8) \\equiv 1 \\bmod (2),\\\\\\bigl < \\mathsf {2,4} \\bigr > = 1 + (3,4) + (4,6) + (4,7) + (4,8) \\equiv 1 \\bmod (2),\\\\\\bigl < \\mathsf {2,6} \\bigr > = 1 + (3,6) + (4,6) + {a} + {a} \\equiv 1 \\bmod (2),\\\\\\bigl < \\mathsf {2,7} \\bigr > = 1 + (3,7) + (4,7) + {a} + {a} \\equiv 1 \\bmod (2),\\\\\\bigl < \\mathsf {2,8} \\bigr > = 1 + (3,8) + (4,8) + {a} + {a} \\equiv 1 \\bmod (2),\\end{array}\\right.","}$ and we then obtain $& (3,6) = (4,6) = b \\in \\lbrace 0,1\\rbrace \\\\& (3,7) = (4,7) = c \\in \\lbrace 0,1\\rbrace \\\\& (3,8) = (4,8) = d \\in \\lbrace 0,1\\rbrace \\\\& (3,4) = b+c+d \\bmod (2).$ Next, using the condition of Theorem REF , $\\bigl < \\mathsf {3,3} \\bigr > \\equiv 0 \\bmod (2)$ , we get $(1,3) = 0$ .","Similarly, one can get $(1,4) = 0$ .","Using $\\bigl < \\mathsf {6,6} \\bigr > \\equiv \\bigl < \\mathsf {7,7} \\bigr > \\equiv \\bigl < \\mathsf {8,8} \\bigr > \\equiv 0 \\bmod (2)$ we get $(1,6) = (1,7) = (1,8) = 1$ .","We thus have Table: NO_CAPTION (4) We have to chose a string with an odd number $1,3$ or 7.","Since string 4 has not been chosen and $(1,4)=0$ then string 1 cannot be $\\Delta $ -filled.","Next, if we chose string 7 we then get $(6,7) = 0$ , $(7,8) = 1$ , but $(6,7) = (7,8) = a$ , we then get a contradiction.","Further, one can easy verify that string 3 can be chosen.","(5) Take string 3.","It follows that $b=c=d=1$ and we then get Table: NO_CAPTION We see that strings $1,4$ are automatically $\\Delta $ -filled.","So, we have two possibilities: 1) $a=0$ , and 2) $a = 1$ .","These cases correspondence to the following possibilities for our meander (see Figure REF ).","Figure: The meanders which are correspondence to case a=0a=0 (in the top) and a=1a=1 (in the bottom)."]],"string":"[\n [\n \"On Realizability Of Gauss Diagrams And Constructions Of Meanders\"\n ],\n [\n \"Abstract The problem of which Gauss diagram can be realized by plane curves is an old one and has been solved in several ways.\",\n \"In this paper, we present a direct approach to this problem.\",\n \"We show that needed conditions for realizability of a Gauss diagram can be interpreted as follows \\\"the number of exits = the number of entrances\\\" and the sufficient condition is based on Jordan curve Theorem.\",\n \"We give a matrix approach of realization of Gauss diagrams and then we present an algorithm to construct meanders\"\n ],\n [\n \"Introduction\",\n \"In the earliest time of the Knot Theory C.F.\",\n \"Gauss defined the chord diagram (= Gauss diagram).\",\n \"C.F.\",\n \"Gauss [3] observed that if a chord diagram can be realized by a plane curve, then every chord is crossed only by an even number of chords, but that this condition is not sufficient.\",\n \"The aim of this paper is to present a direct approach to the problem of which Gauss diagram can be realized by knots.\",\n \"This problem is an old one, and has been solved in several ways.\",\n \"In 1936, M. Dehn [2] found a sufficient algorithmic solution based on the existence of a touch Jordan curve which is the image of a transformation of the knot diagram by successive splits replacing all the crossings.\",\n \"A long time after in 1976, L. Lovasz and M.L.\",\n \"Marx [5] found a second necessary condition and finally during the same year, R.C.\",\n \"Read and P. Rosenstiehl [7] found the third condition which allowed the set of these three conditions to be sufficient.\",\n \"The last characterization is based on the tripartition of graphs into cycles, cocycles and bicycles.\",\n \"In [8] the notation of oriented chord diagram was introduced and it was showed that these diagrams classify cellular generic curves on oriented surfaces.\",\n \"As a corollary a simple combinatorial classification of plane generic curves was derived, and the problem of realizability of these diagrams was also solved.\",\n \"However all these ways are indirect; they rest upon deep and nontrivial auxiliary construction.\",\n \"There is a natural question: whether one can arrive at these conditions in a more direct and natural fashion?\",\n \"We believe that the conditions for realizability of a Gauss diagram (by some plane curve) should be obtained in a natural manner; they should be deduced from an intrinsic structure of the curve.\",\n \"In this paper, we suggest an approach, which satisfies the above principle.\",\n \"We use the fact that every Gauss diagram $\\\\mathfrak {G}$ defines a (virtual) plane curve ${C}(\\\\mathfrak {G})$ (see [4]), and the following simple ideas: (1) For every chord of a Gauss diagram $\\\\mathfrak {G}$ , we can associate a closed path along the curve ${C}(\\\\mathfrak {G})$ .\",\n \"(2) For every two non-intersecting chords of a Gauss diagram $\\\\mathfrak {G}$ , we can associate two closed paths along the curve ${C}(\\\\mathfrak {G})$ such that every chord crosses both of those chords correspondences to the point of intersection of the paths.\",\n \"(3) If a Gauss diagram $\\\\mathfrak {G}$ is realizable (say by a plane curve ${C}(\\\\mathfrak {G})$ ), then for every closed path (say) ${P}$ along ${C}(\\\\mathfrak {G})$ we can associate a coloring another part of ${C}(\\\\mathfrak {G})$ into two colors (roughly speaking we get “inner” and “outer” sides of ${P}$ cf.\",\n \"Jordan curve Theorem).\",\n \"If a Gauss diagram is not realizable then ([4]) it defines a virtual plane curve ${C}(\\\\mathfrak {G})$ .\",\n \"We shall show that there exists a closed path along ${C}(\\\\mathfrak {G})$ for which we cannot associate a well-defined coloring of ${C}(\\\\mathfrak {G})$ , i.e., ${C}(\\\\mathfrak {G})$ contains a path is colored into two colors.\",\n \"Using these ideas we solve the problem of which Gauss diagram can be realized by knots.\",\n \"We then give a matrix approach of realization of Gauss diagrams and then we present an algorithm to construct meanders.\"\n ],\n [\n \"Preliminaries\",\n \"Recall that classically, a knot is defined as an embedding of the circle $S^1$ into $\\\\mathbb {R}^3$ , or equivalently into the 3-sphere $S^3$ , i.e., a knot is a closed curve embedded on $\\\\mathbb {R}^3$ (or $S^3$ ) without intersecting itself, up to ambient isotopy.\",\n \"The projection of a knot onto a 2-manifold is considered with all multiple points are transversal double with will be call crossing points (or shortly crossings).\",\n \"Such a projection is called the shadow by the knots theorists [1], [9], following [8] we shall also call these projections as plane curves.\",\n \"A knot diagram is a generic immersion of a circle $S^1$ to a plane $\\\\mathbb {R}^2$ enhanced by information on overpasses and underpasses at double points.\"\n ],\n [\n \"Gauss Diagrams\",\n \"A generic immersion of a circle to a plane is characterized by its Gauss diagram [6].\",\n \"Figure: The plane curve and its Gauss diagram are shown.Definition 1.1 The Gauss diagram is the immersing circle with the preimages of each double point connected with a chord.\",\n \"On the other words, this natation can be defined as follows.\",\n \"Let us walk on a path along the plane curve until returning back to the origin and then generate a word $W$ which is the sequence of the crossings in the order we meet them on the path.\",\n \"$W$ is a double occurrence word.\",\n \"If we put the labels of the crossing on a circle in the order of the word $W$ and if we join by a chord all pairs of identical labels then we obtain a chord diagram (=Gauss diagram) of the plane curve (see Figure REF ).\",\n \"A virtual knot diagram [4] is a generic immersion of the circle into the plane, with double points divided into real crossing points and virtual crossing points, with the real crossing points enhanced by information on overpasses and underpasses (as for classical knot diagrams).\",\n \"At a virtual crossing the branches are not divided into an overpass and an underpass.\",\n \"The Gauss diagram of a virtual knot is constructed in the same way as for a classical knot, but all virtual crossings are disregarded.\",\n \"Theorem 1.2 [4] A Gauss diagram defines a virtual knot diagram up to virtual moves.\",\n \"Arguing similarly as in the real knot case, one can define a shadow of the virtual knot (see Figure REF ).\",\n \"Figure: The chord diagram and the shadow of the virtual knot are shown.\",\n \"Here xx and yy are the virtual crossing points.\"\n ],\n [\n \"Conway's Smoothing\",\n \"We frequently use the following notations.\",\n \"Let $K$ be a knot, ${C}$ its shadow and $\\\\mathfrak {G}$ the Gauss diagram of ${C}$ .\",\n \"For every crossing $c$ of ${C}$ we denote by $\\\\mathfrak {c}$ the corresponding chord of $\\\\mathfrak {G}$ .\",\n \"If a Gauss diagram $\\\\mathfrak {G}$ contains a chord $\\\\mathfrak {c}$ then we write $\\\\mathfrak {c}\\\\in \\\\mathfrak {G}$ .\",\n \"We denote by $\\\\mathfrak {c}_0$ , $\\\\mathfrak {c}_1$ the endpoints of every chord $\\\\mathfrak {c}\\\\in \\\\mathfrak {G}$ .\",\n \"We shall also consider every chord $\\\\mathfrak {c} \\\\in \\\\mathfrak {G}$ together with one of two arcs are between its endpoints, and a chosen arc is denoted by $\\\\mathfrak {c}_0\\\\mathfrak {c}_1$ .\",\n \"Further, $\\\\mathfrak {c}_\\\\times $ denotes the set of all chords cross the chord $\\\\mathfrak {c}$ and $\\\\mathfrak {c}_\\\\parallel $ denotes the set of all chords do not cross the chord $\\\\mathfrak {c}$ .\",\n \"We put $\\\\mathfrak {c} \\\\notin \\\\mathfrak {c}_\\\\times $ , and $\\\\mathfrak {c} \\\\in \\\\mathfrak {c}_\\\\parallel $ .\",\n \"Throughout this paper we consider Gauss diagrams such that $\\\\mathfrak {c}_\\\\times \\\\ne \\\\varnothing $ for every $\\\\mathfrak {c \\\\in G}$ .\",\n \"As well known, John Conway introduced a “surgical” operation on knots, called smoothing, consists in eliminating the crossing by interchanging the strands (Figure REF ).\",\n \"Figure: The Conway smoothing the crossings are shown.We aim to specialize a Conway smoothing a crossing of a plane curve to an operation on chords of the corresponding Gauss diagram.\",\n \"Let $K$ be a knot, ${C}$ its shadow, and $\\\\mathfrak {G}$ the Gauss diagram of ${C}$ .\",\n \"Take a crossing point $c$ of ${C}$ and let $D_c$ be a small disk centered at $c$ such that $D_c\\\\cap {C}$ does not contain another crossings of ${C}$ .\",\n \"Denote by $\\\\partial D_c$ the boundary of $D_c$ .\",\n \"Starting from $c$ , let us walk on a path along the curve ${C}$ until returning back to $c$ .\",\n \"Denote this path by ${L}_c$ and let $c c_a^lc_z^lc$ be the sequence of the points in the order we meet them on ${L}_c$ , where $ \\\\lbrace c_a^l,c_z^l\\\\rbrace = {L}_c \\\\cap \\\\partial D_c$ .\",\n \"After returning back to $c$ let us keep walking along the curve ${C}$ in the same direction as before until returning back to $c$ .\",\n \"Denote the corresponding path by ${R}_c$ and let $cc_a^rc_z^rc$ be the sequence of the points in the order we meet them on ${R}_c$ , where $\\\\lbrace c_a^r, c_z^r\\\\rbrace = \\\\partial D_c \\\\cap {R}_c$ .\",\n \"Figure: The Conway smoothing the crossing cc and the chord 𝔠\\\\mathfrak {c} are shown.Next, let us delete the inner side of $D_c \\\\cap {C}$ and attach $c_l^a$ to $c_r^a$ , and $c^r_z$ to $c^l_z$ .\",\n \"We thus get the new plane curve $\\\\widehat{{C}}_c$ (see Figure REF ).\",\n \"It is easy to see that this curve is the shadow of the knot, which is obtained from $K$ by Conway's smoothing the crossing $c$ .\",\n \"Let $\\\\widehat{\\\\mathfrak {G}}_\\\\mathfrak {c}$ be the Gauss diagram of $\\\\widehat{{C}}_c$ .\",\n \"We shall say that the Gauss diagram $\\\\widehat{\\\\mathfrak {G}}_\\\\mathfrak {c}$ is obtained from the Gauss diagram $\\\\mathfrak {G}$ by Conway's smoothing the chord $\\\\mathfrak {c}$ .\",\n \"As an immediate consequence of the preceding discussion, we get the following proposition.\",\n \"Proposition 1.1 Let $\\\\mathfrak {G}$ be a Gauss diagram and $\\\\mathfrak {c}$ be its arbitrary chord.\",\n \"Then $\\\\widehat{\\\\mathfrak {G}}_{\\\\mathfrak {c}}$ is obtained from $\\\\mathfrak {G}$ as follows: (1) delete the chord $\\\\mathfrak {c}$ , (2) if two chords $\\\\mathfrak {a}, \\\\mathfrak {b} \\\\in \\\\mathfrak {c}_\\\\times $ intersect (resp.\",\n \"do not intersected) in $\\\\mathfrak {G}$ then they do not intersect in $\\\\mathfrak {\\\\widehat{G}_c}$ (resp.\",\n \"intersected), (3) another chords keep their positions.\",\n \"Indeed, let $W$ be the word which is the sequence of the crossings in the order we meet them on the curve ${C}$ .\",\n \"Since $\\\\mathfrak {c}_\\\\times \\\\ne \\\\varnothing $ , $W$ can be written as follows $W = W_1 c W_2 c W_3$ , where $W_1,W_2,W_3$ are subwords of $W$ and at least one of $W_1,W_3$ is not empty.\",\n \"Define $W_2^R$ as the reversal of the word $W_2$ .\",\n \"Then, from the preceding discussion, the word $\\\\widehat{W}_c: = W_1 W_2^R W_3$ gives $\\\\widehat{\\\\mathfrak {G}}_{\\\\mathfrak {c}}$ (see Figure REF ) and the statement follows.\"\n ],\n [\n \"Partitions of Gauss Diagrams\",\n \"In this section we introduce notations, whose importance will become clear as we proceed.\",\n \"Definition 2.1 Let $\\\\mathfrak {G}$ be a Gauss diagram and $\\\\mathfrak {a}$ a chord of $\\\\mathfrak {G}$ .\",\n \"A $C$ -contour, denoted $C(\\\\mathfrak {a})$ , consists of the chord $\\\\mathfrak {a}$ , a chosen arc $\\\\mathfrak {a}_0\\\\mathfrak {a}_1$ , and all chords of $\\\\mathfrak {G}$ such that all their endpoints lie on the arc $\\\\mathfrak {a}_0\\\\mathfrak {a}_1$ .\",\n \"We call a chord from the set $\\\\mathfrak {a}_\\\\times $ the door chord of the $C$ -contour $C(\\\\mathfrak {a})$.\",\n \"Let us consider a plane curve ${C}:S^1 \\\\rightarrow \\\\mathbb {R}^2$ and let $\\\\mathfrak {G}$ be its Gauss diagram.\",\n \"Every chord $\\\\mathfrak {c \\\\in G}$ correspondences to the crossing $c$ of ${C}$ .\",\n \"Thus for every $C$ -contour $C(\\\\mathfrak {c})$ , we can associate a closed path ${C}(c)$ along the curve ${C}$ .\",\n \"We call ${C}(c)$ the loop of the curve ${C}$ .\",\n \"It is obviously that there is the one-to-one correspondence between self-intersection points of ${C}(c)$ and all chords from $C(\\\\mathfrak {c})$ .\",\n \"Figure: Every CC-contour of the Gauss diagram correspondences to the closed path along the plane curve and vise versa.\",\n \"We see that the chord 6 correspondences to the self-intersection point 6 of the dotted loop.Example 2.2 In Figure REF the plane curve ${C}$ and its Gauss diagram are shown.\",\n \"Consider the (cyan) $C$ -contour $C(5)$ .\",\n \"We see that ${C}(5)$ is the closed path along the curve.\",\n \"It is the self-intersecting path and we see that the crossing 6 correspondences to the chord from the set $5_\\\\parallel $ .\",\n \"Further, the red closed path ${C}(0)$ correspondences to the red $C$ -contour $C(0)$ .\",\n \"Definition 2.3 Let $\\\\mathfrak {G}$ be a Gauss diagram, $\\\\mathfrak {a},\\\\mathfrak {b}$ its intersecting chords.\",\n \"An $X$ -contour, denoted $X(\\\\mathfrak {a},\\\\mathfrak {b})$ , consists of two non-intersecting arcs $\\\\mathfrak {a}_0\\\\mathfrak {b}_0$ , $\\\\mathfrak {a}_1\\\\mathfrak {b}_1$ and all chords of $\\\\mathfrak {G}$ such that all their endpoints lie on $\\\\mathfrak {a}_0\\\\mathfrak {b}_0$ or on $\\\\mathfrak {a}_1\\\\mathfrak {b}_1$ .\",\n \"A chord is called the door chord of the $X$ -contour $X(\\\\mathfrak {a},\\\\mathfrak {b})$ if only one of its endpoints belongs to $X(\\\\mathfrak {a},\\\\mathfrak {b})$ .\",\n \"We say that the $X$ -contour $X(\\\\mathfrak {a},\\\\mathfrak {b})$ is non-degenerate if it has at least one door chord, and it does not contain all chords of $\\\\mathfrak {G}$ .\",\n \"Example 2.4 Let us consider the Gauss diagram in Figure REF.\",\n \"The black chords are the door chords of the black $X$ -contour $X(1,3)$ .\",\n \"We see that the door chords correspondence to “entrances” and “exits” of the black closed path along the curve.\",\n \"We also see that this $X$ -contour is non-degenerate.\",\n \"The previous Example implies a partition of a Gauss diagram (resp.\",\n \"a plane curve) into two parts.\",\n \"Definition 2.5 (An $X$ -contour coloring) Given a Gauss diagram $\\\\mathfrak {G}$ and its an $X$ -contour.\",\n \"Let us walk along the circle of $\\\\mathfrak {G}$ in a chosen direction and color all arcs of $\\\\mathfrak {G}$ until returning back to the origin as follows: (1) we don't colors the arcs of the $X$ -contour, (2) we use only two different colors, (3) we change a color whenever we meet an endpoint of a door chord.\",\n \"Similarly, one can define a $C$ -contour coloring of a Gauss diagram $\\\\mathfrak {G}$ .\",\n \"Remark 2.6 Let $\\\\mathfrak {G}$ be a Gauss diagram and ${C}$ the corresponding (may be virtual) plane curve, i.e., $\\\\mathfrak {G}$ determines the curve ${C}$ .\",\n \"For every $X$ -contour $X(\\\\mathfrak {a},\\\\mathfrak {b})$ in $\\\\mathfrak {G}$ , we can associate the closed path along the curve ${C}$ .\",\n \"We call this path the ${X}$ -contour and denote by ${X}(a,b)$ .\",\n \"Similarly one can define door crossing for ${X}(a,b)$ .\",\n \"Further, for the $X(\\\\mathfrak {a},\\\\mathfrak {b})$ -contour coloring of $\\\\mathfrak {G}$ , we can associate ${X}(a,b)$ -contour coloring of the curve ${C}$.\",\n \"Next, let $\\\\mathfrak {G}$ be a realizable Gauss diagram determines the plane curve ${C}$ and let $X(\\\\mathfrak {a},\\\\mathfrak {b})$ be an $X$ -contour of $\\\\mathfrak {G}$ such that ${X}(a,b)$ is the non-self-intersecting path (= the Jordan curve).\",\n \"Then the ${X}(a,b)$ -contour coloring of ${C}$ divides the curve ${C}$ into two colored parts, cf.\",\n \"Jordan curve Theorem.\",\n \"Figure: For the X(1,3)X(1,3)-contour coloring of the Gauss diagram, we associate the plane curve coloring.\",\n \"We see that the X{X}-contour X(1,3){X}(1,3) (= black loop) divides the plane curve into two parts.\"\n ],\n [\n \"The Even and The Sufficient Conditions\",\n \"If a Gauss diagram can be realized by a plane curve we then say that this Gauss diagram is realizable, and non-realizable otherwise.\",\n \"So, in this section, we give a criterion allowing verification and comprehension of whether a given Gauss diagram is realizable or not.\",\n \"Moreover, we give an explanation allowing comprehension of why the needed condition is not sufficient for realizability of Gauss diagrams.\"\n ],\n [\n \"The Even Condition\",\n \"Proposition 3.1 Let ${C}:S^1 \\\\rightarrow \\\\mathbb {R}^2$ be a plane curve and $\\\\mathfrak {G}$ its Gauss diagram.\",\n \"Then (1) $|\\\\mathfrak {a}_\\\\times \\\\cap \\\\mathfrak {b}_\\\\times | \\\\equiv 0 \\\\bmod 2$ for every two non-interesting chords $\\\\mathfrak {a,b \\\\in G}$ , (2) $|\\\\mathfrak {c}_\\\\times | \\\\equiv 0 \\\\bmod 2$ for every chord $\\\\mathfrak {c \\\\in G}$ .\",\n \"Let $\\\\mathfrak {a},\\\\mathfrak {b}\\\\in \\\\mathfrak {G}$ be two non-intersecting chords of $\\\\mathfrak {G}$ .\",\n \"Take two $C$ -contours $C(\\\\mathfrak {a})$ , $C(\\\\mathfrak {b})$ such that their arcs $\\\\mathfrak {a_0a_1}$ , $\\\\mathfrak {b_0b_1}$ do not intersect.\",\n \"It is obvious that for the loops ${C}(a)$ , ${C}(b)$ , we can associate the one-to-one correspondence between the set ${C}(a) \\\\cap {C}(b)$ and the set $\\\\mathfrak {a}_\\\\times \\\\cap \\\\mathfrak {b}_\\\\times $ .\",\n \"Because, by Proposition REF , all chord from the set $\\\\mathfrak {a}_\\\\times \\\\cap \\\\mathfrak {b}_\\\\times $ keep their positions in Gauss diagram $\\\\mathfrak {\\\\widehat{G}_c}$ (= Conway's smoothing the chord $\\\\mathfrak {c}$ ) for every $\\\\mathfrak {c} \\\\in \\\\mathfrak {a}_\\\\parallel \\\\cap \\\\mathfrak {b}_\\\\parallel \\\\setminus \\\\lbrace \\\\mathfrak {a},\\\\mathfrak {b}\\\\rbrace $ , it is sufficient to prove the statement in the case $\\\\mathfrak {a}_\\\\parallel \\\\cap \\\\mathfrak {b}_\\\\parallel = \\\\lbrace \\\\mathfrak {a},\\\\mathfrak {b}\\\\rbrace $ , i.e., the loops ${C}(a)$ , ${C}(b)$ are non-self-intersecting loops (= the Jordan curves).\",\n \"From Jordan curve Theorem, it follows that the loop ${C}(a)$ divides the curve ${C}$ into two regions, say, ${I}$ and ${O}$ .\",\n \"Assume that $b\\\\in {O}$ and let us walk along the loop ${C}(b)$ .\",\n \"We say that an intersection point $c \\\\in {C}(a) \\\\cap {C}(b)$ is the entrance (resp.\",\n \"the exit) if we shall be in the region ${I}$ (resp.\",\n \"${O}$ ) after meeting $c$ with respect to our walk.\",\n \"Since a number of entrances has to be equal to the number of exits, then $|\\\\mathfrak {a}_\\\\times \\\\cap \\\\mathfrak {b}_\\\\times | \\\\equiv 0\\\\bmod 2$ .\",\n \"Arguing similarly, we prove $|\\\\mathfrak {c}_\\\\times | \\\\equiv 0 \\\\bmod 2$ for every chord $\\\\mathfrak {c}$ .\",\n \"As an immediate consequence of Proposition REF we get the following.\",\n \"Corollary 3.1 (The Even Condition) If a Gauss diagram is realizable then the number of all chords that cross a both of non-intersecting chords and every chord is even (including zero).\",\n \"We conclude this subsection with an explanation why the even condition is not sufficient for realizability of Gauss diagrams.\",\n \"Roughly speaking, from the proof of Proposition REF it follows that every plane curve can be obtained by attaching its loops to each other by given points.\",\n \"Conversely, if a Gauss diagram satisfies the even condition then it may be non-releasible.\",\n \"Indeed, when we attach a loop, say, ${C}(b)$ to a loop ${C}(a)$ , where $\\\\mathfrak {b} \\\\in \\\\mathfrak {a}_\\\\parallel $ , by given points (=elements of the set $\\\\mathfrak {a}_\\\\times \\\\cap \\\\mathfrak {b}_\\\\times $ ) then the loop ${C}(b)$ can be self-intersected curve, which means that we get new crossings (= virtual crossings), see Figure REF .\",\n \"To be more precisely, we have the following proposition.\",\n \"Proposition 3.2 Let $\\\\mathfrak {G}$ be a non-realizable Gauss diagram which satisfies the even condition.\",\n \"Let $\\\\mathfrak {G}$ defines a virtual plane curve ${C}$ (up to virtual moves).\",\n \"There exist two non-intersecting chords $\\\\mathfrak {a,b \\\\in G}$ such that there are paths $c \\\\rightarrow x \\\\rightarrow d$ , $e \\\\rightarrow x \\\\rightarrow f$ on a loop ${C}(b)$ , where $\\\\mathfrak {c,d,e,f \\\\in a_\\\\times \\\\cap b_\\\\times }$ are different chords and $x$ is a virtual crossing of ${C}$ .\",\n \"Let $\\\\mathfrak {a,b \\\\in G}$ be two non-intersecting chords.\",\n \"Take non-intersecting $C$ -contours $C(\\\\mathfrak {a})$ , $C(\\\\mathfrak {b})$ .\",\n \"Hence we may say that the loop ${C}(b)$ attaches to the loop ${C}(a)$ by the given points $p_1,\\\\ldots ,p_n $ , where $\\\\lbrace \\\\mathfrak {p_1,\\\\ldots ,p_n}\\\\rbrace = \\\\mathfrak {a_\\\\times \\\\cap b_\\\\times }$ .\",\n \"Since $\\\\mathfrak {G}$ is not realizable and satisfies the even condition then a virtual crossing may arise only as a self-intersecting point of, say, the loop ${C}(b)$ .\",\n \"Indeed, when we attach ${C}(b)$ to ${C}(a)$ by $p_1,\\\\ldots ,p_n $ we may get self-interesting points, say, $q_1,\\\\ldots ,q_m$ of the loop ${C}(b)$ .\",\n \"If $\\\\mathfrak {G}$ contains all chords $\\\\mathfrak {q_1},\\\\ldots ,\\\\mathfrak {q}_m$ for every such chords $\\\\mathfrak {a,b}$ , then $\\\\mathfrak {G}$ is realizable.\",\n \"Thus, a virtual crossing $x$ does not belong to $\\\\lbrace p_1,\\\\ldots , p_n\\\\rbrace = {C}(a) \\\\cap {C}(b)$ for some non-intersecting chords $\\\\mathfrak {a,b \\\\in G}$ .\",\n \"Then we get two paths $c \\\\rightarrow x \\\\rightarrow d$ , $e \\\\rightarrow x \\\\rightarrow f$ , where $c,d,e,f \\\\in {C}(a) \\\\cap {C}(b)$ are different chords, as claimed.\",\n \"Figure: It shows that the even condition is not sufficient for realizability of the Gauss diagrams.\",\n \"We see that the plane curve can be obtained by attaching the white-black loop to the dotted one by the points 3,4,5,63,4,5,6, and thus the dotted loop has to have “new” crossings (= self-intersections) x 1 ,x 2 ,x 3 x_1,x_2,x_3.\"\n ],\n [\n \"The Sufficient Condition\",\n \"Definition 3.1 Let $\\\\mathfrak {G}$ be a Gauss diagram (not necessarily realizable) and $X(\\\\mathfrak {a,b})$ its $X$ -contour.\",\n \"Take the $X$ -contour coloring of $\\\\mathfrak {G}$ .\",\n \"A chord of $\\\\mathfrak {G}$ is called colorful for $X(\\\\mathfrak {a,b})$ if its endpoints are in arcs which have different colors.\",\n \"Similarly, one can define a colorful chord for a $C$ -contour $C(\\\\mathfrak {a})$ of $\\\\mathfrak {G}$ .\",\n \"Example 3.2 Let us consider the Gauss diagram, which is shown in Figure REF .\",\n \"One can easy check that this Gauss diagram is not realizable.\",\n \"Let us consider the orange $X$ -contour $X(1,3)$ and the $X(1,3)$ -coloring of $\\\\mathfrak {G}$ .\",\n \"The chord with the endpoints 5 is colorful for the $X$ -contour $X(1,3)$ .\",\n \"It is interesting to consider the corresponding coloring of the virtual plane curve: one can think that we forget to change color when we cross the gray loop, i.e., the gray loop “does not divide” the curve into two parts.\",\n \"We shall show that this observation is typical for every non-realizable Gauss diagram.\",\n \"Figure: This Gauss diagram satisfies even condition but is non-realizable.\",\n \"There are colorful chords (e.g.\",\n \"the chord with endpoints 5).We have seen that if a Gauss diagram is realizable then there is no colorful chord, with respect to every $X$ -contour.\",\n \"We shall show that it is sufficient condition for realizability of a Gauss diagram.\",\n \"Proposition 3.3 Let $\\\\mathfrak {G}$ be a non-realizable Gauss diagram but satisfy the even condition.\",\n \"Then there exists an $X$ -contour and a colorful chord for this $X$ -contour.\",\n \"By Theorem REF , $\\\\mathfrak {G}$ defines a virtual curve ${C}$ (= the shadow of a virtual knot diagram) up to virtual moves.\",\n \"Starting from a crossing, say, $o$ , let us walk along ${C}$ till we meet the first virtual crossing, say, $x$ .\",\n \"Next, let us keep walking in the same direction till we meet the first real crossing, say $d$ .\",\n \"Denote this path by ${P}$ .\",\n \"Just for convinces, let us put the labels, say, $x_0,x_1$ of the virtual crossing $x$ on the circle of $\\\\mathfrak {G}$ in the order we meet them on ${P}$ .\",\n \"We can take two real crossings, say, $a,b$ such that the path $a \\\\rightarrow x \\\\rightarrow b$ does not contain another real crossings and $\\\\mathfrak {a,b} \\\\in \\\\mathfrak {o}_\\\\times .$ Indeed, from Proposition REF it follows that for a chord $\\\\mathfrak {o}\\\\in \\\\mathfrak {G}$ we can find $n \\\\ge 1$ chords $\\\\mathfrak {o}^1,\\\\ldots ,\\\\mathfrak {o}^n$ such that, for every $1\\\\le i \\\\le n$ , we have: (1) the chords $\\\\mathfrak {o}, \\\\mathfrak {o}^i$ do not intersect, (2) the loop ${C}(o)$ contains the following paths $a^i \\\\rightarrow x \\\\rightarrow b^i$ , $c^i \\\\rightarrow x \\\\rightarrow d^i$ where $\\\\mathfrak {a}^i,\\\\mathfrak {b}^i,\\\\mathfrak {c}^i,\\\\mathfrak {d}^i \\\\in \\\\mathfrak {o}_\\\\times \\\\cap \\\\mathfrak {o}^i_\\\\times $ are different chords.\",\n \"Thus, for some $1\\\\le i,j \\\\le n$ we have an arc, say, $\\\\mathfrak {a}^i_1\\\\mathfrak {b}^j_1$ contains only one of $x_0$ or $x_1$ , no endpoints of another chords, and $\\\\mathfrak {a}^i,\\\\mathfrak {b}^j \\\\in \\\\mathfrak {o}_\\\\times $ .\",\n \"Denote this arc by $\\\\mathfrak {a_1b_1}$ and assume that $x_0$ lies on $\\\\mathfrak {a_1b_1}$ .\",\n \"We next construct an $X$ -contour contains the arc $\\\\mathfrak {a_1b_1}$ .\",\n \"To do so, we have to consider the following two cases: Case 1.\",\n \"The chords $\\\\mathfrak {a,b}$ intersect.\",\n \"Figure: Since the curve C{C} is determined up to virtual moves then the cyan line has to cross the olive line.Take the $X$ -contour $X(\\\\mathfrak {a,b})$ which contains the arc $\\\\mathfrak {a_1b_1}$ .\",\n \"It is easy to see that this $X$ -contour has at least one real door chord, because its another arc $\\\\mathfrak {a_0b_0}$ has no virtual crossing thus it has to have at least one real crossing (see Figure REF).\",\n \"Case 2.\",\n \"The chords $\\\\mathfrak {a,b}$ do not intersect.\",\n \"Figure: The “general” position of chords 𝔞,𝔟,𝔠,𝔡\\\\mathfrak {a,b,c,d} is shown.\",\n \"Our walk along the path P{P} correspondences to 𝔬 0 →𝔬 ˜ 0 →𝔟 0 →⋯→𝔡 1 \\\\mathfrak {o_0 \\\\rightarrow \\\\widetilde{o}_0 \\\\rightarrow b_0\\\\rightarrow \\\\cdots \\\\rightarrow \\\\mathfrak {d_1}}.Let us walk along the circle of $\\\\mathfrak {G}$ in the direction $\\\\mathfrak {b_1} \\\\rightarrow x_0 \\\\rightarrow \\\\mathfrak {a}_1$ (Figure REF ) till we meet the first chord, say, $\\\\widetilde{\\\\mathfrak {o}}$ such that $\\\\mathfrak {a,b,o} \\\\in \\\\widetilde{\\\\mathfrak {o}}_\\\\times $ .\",\n \"If we cannot find such chord we set $\\\\widetilde{\\\\mathfrak {o}}:=\\\\mathfrak {o}$ .\",\n \"Take the $X$ -contour $X(\\\\widetilde{\\\\mathfrak {o}},\\\\mathfrak {b})$ contains the arc $\\\\mathfrak {a_1b_1}$ .\",\n \"This $X$ -contour is non-degenerate.\",\n \"Indeed, let $\\\\widetilde{\\\\mathfrak {o}}:=\\\\mathfrak {o}$ and endpoints of all chords, which start in the arc $\\\\mathfrak {b_1}x_1\\\\mathfrak {d}_1$ , lie on the arc $\\\\mathfrak {o}_0\\\\mathfrak {b_0}$ .\",\n \"We then get the situation is shown in Figure REF .\",\n \"Figure: The dotted blue line gives another path from cc to dd.It follows that we can take another path ${P}$ without the virtual crossing $x$ .\",\n \"This contradiction implies that the $X$ -contour $X(\\\\mathfrak {o,b})$ is non-degenerate.\",\n \"So, we have a non-degenerate ${X}$ -contour of the curve ${C}$ such that $x$ is its virtual door.\",\n \"If the $X$ -contour of $\\\\mathfrak {G}$ contains another chords such that all their endpoints lie on an arc of this $X$ -contour, then Proposition REF implies that Conway's smoothing all such chords does not change the ${X}$ -contour coloring of ${P}$ .\",\n \"Thus we may assume that this ${X}$ -contour is the closed curve without self-intersections.\",\n \"Hence, by Jordan curve Theorem, it divides ${P}$ into two parts.\",\n \"Let us color these parts into two different colors.\",\n \"If the crossing $x$ would be a real door of this ${X}$ -contour, then by Definition REF , we can take such ${X}$ -contour coloring of ${P}$ as before.\",\n \"But since $x$ is not real door then after meeting it at the second time we do not change the color and thus we get an intersection point of two lines which have different colors, i.e., the corresponding chord is colorful.\",\n \"This completes the proof.\",\n \"Lemma 3.1 Let $\\\\mathfrak {G}$ be a Gauss diagram.\",\n \"Consider a $C(\\\\mathfrak {a})$ -contour coloring of $\\\\mathfrak {G}$ for some chord $\\\\mathfrak {a\\\\in G}$ .\",\n \"If there exists a colorful chord for the $C$ -contour $C(\\\\mathfrak {a})$ then the diagram $\\\\mathfrak {G}$ does not satisfy the even condition.\",\n \"Indeed, let $\\\\mathfrak {b}$ be a colorful chord for the $C$ -contour $C(\\\\mathfrak {a})$ .\",\n \"First note that, the chord $\\\\mathfrak {b}$ cannot cross $\\\\mathfrak {a}$ because otherwise $\\\\mathfrak {b}$ should be a door chord of the $C$ -contour $C(\\\\mathfrak {a})$ .\",\n \"Next, if the chord $\\\\mathfrak {b}$ is colorful then it crosses an odd number of door chords of $C(\\\\mathfrak {a})$ .\",\n \"Hence $|\\\\mathfrak {a}_\\\\times \\\\cap \\\\mathfrak {b}_\\\\times | \\\\equiv 1 \\\\bmod 2$ , as claimed.\",\n \"Figure: The Gauss diagram does not satisfy the even condition; both the chords 1, 6 are crossed by only one chord 2.Example 3.3 Let us consider the Gauss diagram which is shown in Figure REF .\",\n \"We have the orange $C$ -contour $C(1)$ and the $C(1)$ -contour coloring of the Gauss diagram and the corresponding (virtual) curve.\",\n \"We see that there are two chords (namely 5 and 6) which are colorful and $5_\\\\times \\\\cap 1_\\\\times = 6_\\\\times \\\\cap 1_\\\\times = \\\\lbrace 2\\\\rbrace $ .\",\n \"Lemma 3.2 Let $\\\\mathfrak {G}$ be a Gauss diagram and $\\\\mathfrak {a}, \\\\mathfrak {b} \\\\in \\\\mathfrak {G}$ be its intersecting chords.\",\n \"Suppose that there exists a colorful chord $\\\\mathfrak {c}$ for an $X$ -contour $X(\\\\mathfrak {a},\\\\mathfrak {b})$ .\",\n \"Then there exists a $C$ -contour of the Gauss diagram $\\\\widehat{\\\\mathfrak {G}}_\\\\mathfrak {b}$ (= Conway's smoothing the chord $\\\\mathfrak {b}$ ) such that the chord $\\\\mathfrak {c}$ is colorful for this $C$ -contour in $\\\\widehat{\\\\mathfrak {G}}_\\\\mathfrak {b}$ .\",\n \"Indeed, consider the Gauss diagram $\\\\widehat{\\\\mathfrak {G}}_\\\\mathfrak {b}$ .\",\n \"From Proposition REF it follows that after Conway's smoothing the chord $\\\\mathfrak {b}$ , the chord $\\\\mathfrak {c}$ does not intersect $\\\\mathfrak {a}$ and intersects the same door chords of the $X$ -contour $X(\\\\mathfrak {a},\\\\mathfrak {b})$ as in $\\\\mathfrak {G}$ .\",\n \"Further, let us consider the $C$ -contour $C(\\\\mathfrak {a})$ in $\\\\widehat{\\\\mathfrak {G}}_\\\\mathfrak {b}$ such that it does not contain the chord $\\\\mathfrak {c}$ .\",\n \"By Proposition REF , the chord $\\\\mathfrak {a}$ crosses in $\\\\widehat{\\\\mathfrak {G}}_\\\\mathfrak {b}$ only the chord that are door chords of the $X$ -contour $X(\\\\mathfrak {a},\\\\mathfrak {b})$ .\",\n \"Hence, by Definition REF , we may take the $C(\\\\mathfrak {a})$ -contour coloring of $\\\\widehat{\\\\mathfrak {G}}_\\\\mathfrak {b}$ such that $\\\\mathfrak {c}$ is the colorful chord for this $C$ -contour.\",\n \"Proposition 3.4 Let a Gauss diagram $\\\\mathfrak {G}$ satisfy the even condition.\",\n \"Then $\\\\mathfrak {G}$ is realizable if and only if $\\\\widehat{\\\\mathfrak {G}}_\\\\mathfrak {c}$ satisfies the even condition for every chord $\\\\mathfrak {c} \\\\in \\\\mathfrak {G}$ , $\\\\widehat{\\\\mathfrak {G}}_\\\\mathfrak {c}$ .\",\n \"Indeed, let $\\\\mathfrak {G}$ be a non-realizable Gauss diagram and let $\\\\mathfrak {G}$ satisfy the even condition.\",\n \"By Proposition REF , there exists a colorful chord (say $\\\\mathfrak {c}$ ) for a $X$ -counter $X(\\\\mathfrak {a},\\\\mathfrak {b})$ of $\\\\mathfrak {G}$ .\",\n \"By Lemma REF , the chord $\\\\mathfrak {c}$ is the colorful chord in $\\\\widehat{\\\\mathfrak {G}}_\\\\mathfrak {b}$ .\",\n \"Hence from Lemma REF it follows that $\\\\widehat{\\\\mathfrak {G}}_\\\\mathfrak {b}$ does not satisfy the even condition, and the statement follows.\",\n \"We can summarize our results in the following theorem.\",\n \"Theorem 3.4 A Gauss diagram $\\\\mathfrak {G}$ is realizable if and only if the following conditions hold: (1) the number of all chords that cross a both of non-intersecting chords and every chord is even (including zero), (2) for every chord $\\\\mathfrak {c} \\\\in \\\\mathfrak {G}$ the Gauss diagram $\\\\widehat{\\\\mathfrak {G}}_\\\\mathfrak {c}$ (= Conway's smoothing the chord $\\\\mathfrak {c}$ ) also satisfies the above condition.\"\n ],\n [\n \"Matrixes of Gauss Diagrams\",\n \"In this section we are going to translate into matrix language the conditions of realizability of Gauss diagrams.\",\n \"We consider all matrixes over the field $\\\\mathbb {Z}_2$ .\",\n \"Definition 4.1 Given a Gauss diagram $\\\\mathfrak {G}$ contains $n$ chords, say, $\\\\mathfrak {c}_1,\\\\ldots ,\\\\mathfrak {c}_n$ .\",\n \"Construct a $n \\\\times n$ matrix $\\\\mathsf {M}(\\\\mathfrak {G}) = (\\\\mathsf {m}_{i,j})_{1 \\\\le i,j\\\\le n}$ as follows: (1) put $\\\\mathsf {m}_{ii} = 0$ , $1 \\\\le i \\\\le n$ , (2) $\\\\mathsf {m}_{ij} = 1$ if the chords $\\\\mathfrak {c}_i$ , $\\\\mathfrak {c}_j$ intersect and $\\\\mathsf {m}_{ij} = 0$ in otherwise.\",\n \"It is obviously that $\\\\mathsf {M}(\\\\mathfrak {G})$ is a symmetric matrix with respect to the main diagonal.\",\n \"Next, let $\\\\mathsf {M}(\\\\mathfrak {G})$ be a $n\\\\times n$ matrix of a Gauss diagram $\\\\mathfrak {G}$ .\",\n \"Let $\\\\cup _{1 \\\\le i \\\\le n}\\\\lbrace \\\\mathsf {m}_i = (\\\\mathsf {m}_{i1}, \\\\ldots , \\\\mathsf {m}_{in})\\\\rbrace $ be all its strings.\",\n \"Define a scalar product of strings as follows: $\\\\langle \\\\mathsf {m}_i, \\\\mathsf {m}_j \\\\rangle :=\\\\mathsf {m}_{i1}\\\\mathsf {m}_{j1} + \\\\cdots + \\\\mathsf {m}_{in}\\\\mathsf {m}_{jn}.$ It is clear that we can define “scalar product” of any number of strings as follows $\\\\langle \\\\mathsf {m}_{i_1}, \\\\ldots , \\\\mathsf {m}_{i_k} \\\\rangle : = \\\\mathsf {m}_{i_11} \\\\cdots \\\\mathsf {m}_{i_k1} + \\\\cdots + \\\\mathsf {m}_{i_1n} \\\\cdots \\\\mathsf {m}_{i_kn}.$ Lemma 4.1 Let $\\\\mathfrak {G}$ be a Gauss diagram contains $n$ chords $\\\\mathfrak {c}_1,\\\\ldots ,\\\\mathfrak {c}_n$ .\",\n \"Consider the corresponding matrix $\\\\mathsf {M}(\\\\mathfrak {G})$ .\",\n \"Let the chords $\\\\mathfrak {c}_1,\\\\ldots ,\\\\mathfrak {c}_n$ correspondence to the strings $\\\\mathsf {m}_1,\\\\ldots ,\\\\mathsf {m}_n$ , respectively.\",\n \"$| {\\\\mathfrak {c}_{i_1}}_\\\\times \\\\cap \\\\cdots \\\\cap {\\\\mathfrak {c}_{i_k}}_\\\\times | = \\\\langle \\\\mathsf {m}_{i_1}, \\\\ldots , \\\\mathsf {m}_{i_k} \\\\rangle ,$ for every $\\\\lbrace i_1,\\\\ldots ,i_k\\\\rbrace \\\\subseteq \\\\lbrace 1,2,\\\\ldots ,n\\\\rbrace $ .\",\n \"Indeed, let $\\\\mathsf {m}_{i_1t} \\\\cdots \\\\mathsf {m}_{i_kt} \\\\ne 0$ , for some $1 \\\\le t \\\\le n$ , then the chord $\\\\mathfrak {c}_t$ intersect chords $\\\\mathfrak {c}_{i_1}, \\\\ldots , \\\\mathfrak {c}_{i_k}$ .\",\n \"This completes the proof.\",\n \"Theorem 4.2 Let $\\\\mathfrak {G}$ be a realizable Gauss diagram contains $n$ chords.\",\n \"Then its matrix $\\\\mathsf {M}(\\\\mathfrak {G})$ satisfies the following conditions: (1) $\\\\langle \\\\mathsf {m}_i, \\\\mathsf {m}_i \\\\rangle \\\\equiv 0 \\\\bmod (2)$ , $1 \\\\le i \\\\le n$ , (2) $\\\\langle \\\\mathsf {m}_i, \\\\mathsf {m}_j \\\\rangle \\\\equiv 0 \\\\bmod (2)$ , if the corresponding chords do not intersect, (3) $\\\\langle \\\\mathsf {m}_i, \\\\mathsf {m}_j \\\\rangle + \\\\langle \\\\mathsf {m}_i, \\\\mathsf {m}_k \\\\rangle + \\\\langle \\\\mathsf {m}_j, \\\\mathsf {m}_k \\\\rangle \\\\equiv 1 \\\\bmod (2)$ , if the corresponding chords pairwise intersect.\",\n \"The first two conditions follow from the first condition of Theorem REF and Lemma REF .\",\n \"Assume that $\\\\mathfrak {G}$ does not contain three pairwise intersecting chords.\",\n \"Then $\\\\mathfrak {G}$ is realizable if and only if the first condition of Theorem REF holds and we get the first two conditions for $\\\\mathsf {M}(\\\\mathfrak {G})$ .\",\n \"Next, let $\\\\mathfrak {a},\\\\mathfrak {b},\\\\mathfrak {c}$ be three pairwise intersecting chords.\",\n \"Consider the following sets of chords (see Figure REF ): $&A: = \\\\mathfrak {a}_\\\\times \\\\setminus \\\\lbrace \\\\mathfrak {b}_\\\\times , \\\\mathfrak {c}_\\\\times \\\\rbrace ,\\\\\\\\& B:= \\\\mathfrak {b}_\\\\times \\\\cap \\\\mathfrak {c}_\\\\times \\\\setminus \\\\left\\\\lbrace \\\\lbrace \\\\mathfrak {a}\\\\rbrace , \\\\mathfrak {a}_\\\\times \\\\right\\\\rbrace ).\",\n \"$ Figure: The sets AA, BB are roughly shown.By Proposition REF , chords $\\\\mathfrak {b}, \\\\mathfrak {c}$ does not intersect in $\\\\widehat{\\\\mathfrak {G}}_\\\\mathfrak {a}$ , and the set $\\\\mathfrak {b}_\\\\times \\\\cap \\\\mathfrak {c}$ of chords in $\\\\widehat{\\\\mathfrak {G}}_\\\\mathfrak {a}$ is equal to the set $A \\\\cup B$ .\",\n \"Hence, by Theorem REF , $|A| + |B| \\\\equiv 0 \\\\bmod (2).$ Further, using Lemma REF , we obtain: $&|A| = \\\\langle \\\\mathfrak {a}, \\\\mathfrak {a} \\\\rangle - \\\\langle \\\\mathfrak {a}, \\\\mathfrak {b} \\\\rangle - \\\\langle \\\\mathfrak {a}, \\\\mathfrak {c} \\\\rangle - \\\\langle \\\\mathfrak {a}, \\\\mathfrak {b}, \\\\mathfrak {c} \\\\rangle ,\\\\\\\\&|B| = \\\\langle \\\\mathfrak {b}, \\\\mathfrak {c} \\\\rangle -1- \\\\langle \\\\mathfrak {a}, \\\\mathfrak {b}, \\\\mathfrak {c} \\\\rangle .$ It follows that $|A| + |B| & =& \\\\langle \\\\mathfrak {a}, \\\\mathfrak {a} \\\\rangle - \\\\langle \\\\mathfrak {a}, \\\\mathfrak {b} \\\\rangle - \\\\langle \\\\mathfrak {a}, \\\\mathfrak {c} \\\\rangle - \\\\langle \\\\mathfrak {a}, \\\\mathfrak {b}, \\\\mathfrak {c} \\\\rangle \\\\\\\\&&+ \\\\langle \\\\mathfrak {b}, \\\\mathfrak {c} \\\\rangle -1- \\\\langle \\\\mathfrak {a}, \\\\mathfrak {b}, \\\\mathfrak {c} \\\\rangle \\\\\\\\&\\\\equiv & \\\\langle \\\\mathfrak {a}, \\\\mathfrak {b} \\\\rangle + \\\\langle \\\\mathfrak {a}, \\\\mathfrak {c} \\\\rangle + \\\\langle \\\\mathfrak {b}, \\\\mathfrak {c} \\\\rangle \\\\\\\\&\\\\equiv & 1 \\\\bmod (2),$ because, as we have already discussed, $\\\\langle \\\\mathfrak {a}, \\\\mathfrak {a} \\\\rangle \\\\equiv 0 \\\\bmod (2)$ .\",\n \"As claimed.\",\n \"Remark 4.3 The number of matrix satisfy the above conditions is bigger than a number of realizable Gauss diagrams.\",\n \"The reason is the following observation.\",\n \"A matrix which satisfies the above conditions “knows” only intersections but don't know positions of chords.\",\n \"However there is a very important sort of plane curves (=meanderes) such that there is a one-to-one correspondence between Gauss diagrams of these curves and the corresponding matrixes.\"\n ],\n [\n \"Thurston Generators of Braid Groups\",\n \"We will use as generators for $B_n$ the set of positive crossings, that is, the crossings between two (necessary adjacent) strands, with the front strand having a positive slope.\",\n \"We denote these generators by $\\\\sigma _1, \\\\ldots , \\\\sigma _{n-1}$ .\",\n \"These generators are subject to the following relations: ${\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\sigma _i\\\\sigma _j = \\\\sigma _j\\\\sigma _i, \\\\mbox{ if } |i-j| >1,\\\\\\\\\\\\sigma _i\\\\sigma _{i+1}\\\\sigma _i = \\\\sigma _{i+1}\\\\sigma _i\\\\sigma _{i+1}.\\\\end{array}\\\\right.\",\n \"}$ One obvious invariant of an isotopy of a braid is the permutation it induces on the order of the strands: given a braid $B$ , the strands define a map $p(B)$ from the top set of endpoints to the bottom set of endpoints, which we interpret as a permutation of $\\\\lbrace 1, \\\\ldots , n\\\\rbrace $ .\",\n \"In this way we get a homomorphism $p:B_n \\\\rightarrow \\\\mathfrak {S}_n$ , where $\\\\mathfrak {S}_n$ is the symmetric group.\",\n \"The generator $\\\\sigma _i$ is mapped to the transposition $s_i = (i,i+1)$ .\",\n \"We denote by $S_n = \\\\lbrace s_1, \\\\ldots , s_{n-1}\\\\rbrace $ the set of generators for the symmetric group $\\\\mathfrak {S}_n$ .\",\n \"Now we want to define an inverse map $p^{-1}:\\\\mathfrak {S}_n \\\\rightarrow B_n$ .\",\n \"To this end, we need the following definition [10] Definition 5.1 Let $S = \\\\lbrace s_1, \\\\ldots , s_{n-1}\\\\rbrace $ be the set of generators for $\\\\mathfrak {S}_n$ .\",\n \"Each permutation $\\\\pi $ gives rise to a total order relation $\\\\le _\\\\pi $ on $\\\\lbrace 1, \\\\ldots ,n\\\\rbrace $ with $i \\\\le _\\\\pi j$ if $\\\\pi (i) < \\\\pi (j)$ .\",\n \"We set $R_\\\\pi : = \\\\lbrace (i,j) \\\\in \\\\lbrace 1, \\\\ldots , n\\\\rbrace \\\\times \\\\lbrace 1,\\\\ldots ,n\\\\rbrace |i \\\\pi (j)\\\\rbrace .$ Lemma 5.1 [10] A set $R$ of pairs $(i,j)$ , with $i \\\\pi (2),\\\\end{array}\\\\right.}\",\n \"\\\\quad {\\\\left\\\\lbrace \\\\begin{array}{ll}1<4, \\\\\\\\ \\\\pi (1) > \\\\pi (4),\\\\end{array}\\\\right.}\",\n \"\\\\quad {\\\\left\\\\lbrace \\\\begin{array}{ll}1<6, \\\\\\\\ \\\\pi (1) > \\\\pi (6),\\\\end{array}\\\\right.}\",\n \"\\\\quad {\\\\left\\\\lbrace \\\\begin{array}{ll}2<4, \\\\\\\\ \\\\pi (2) > \\\\pi (4),\\\\end{array}\\\\right.\",\n \"}\\\\\\\\& {\\\\left\\\\lbrace \\\\begin{array}{ll}3<4,\\\\\\\\ \\\\pi (3) > \\\\pi (4),\\\\end{array}\\\\right.}\",\n \"\\\\quad {\\\\left\\\\lbrace \\\\begin{array}{ll}3<5,\\\\\\\\ \\\\pi (3) > \\\\pi (5),\\\\end{array}\\\\right.}\",\n \"\\\\quad {\\\\left\\\\lbrace \\\\begin{array}{ll}3<6,\\\\\\\\ \\\\pi (3) > \\\\pi (6), \\\\end{array}\\\\right.}\",\n \"\\\\quad {\\\\left\\\\lbrace \\\\begin{array}{ll}5<6,\\\\\\\\ \\\\pi (5) > \\\\pi (6).\\\\end{array}\\\\right.\",\n \"}$ hence $R(\\\\pi ) = \\\\lbrace (1,2),(1,4),(1,6),(2,4),(3,4),(3,5),(3,6),(5,6)\\\\rbrace ,$ and we get a non-repeating braid is shown in Figure REF .\",\n \"Figure: The simple braid R π R_\\\\pi is shown.\"\n ],\n [\n \"Meanders and its Gauss Diagrams\",\n \"In this section we deal with closed meanders.\",\n \"We show that any closed meander defines a special sort of Gauss diagrams (= Gauss diagrams of meanders) and then we will see that these diagrams are completely determined by its matrixes (= meander matrix) i.e., there is a one-to-one correspondence between meander matrixes satisfy the conditions of Theorem REF .\",\n \"It allows us to describe an algorithm constructs all closed meanders.\",\n \"Definition 6.1 A (closed) meander is a planar configuration consisting of a simple closed curve and on orientied line, that cross finitely many times and intersect only transversally.\",\n \"Two meanders are equivalent if there exists a homeomorphism of the plane that maps one to the other.\",\n \"Figure: An Example of a closed meander.Definition 6.2 A matrix $\\\\mathbf {M} = (\\\\mathbf {m}_{ij}) \\\\in \\\\mathsf {Mat}_{n+1}(\\\\mathbb {Z}_2)$ is called meander matrix if the following conditions hold: (1) its main diagonal contains only 0, (2) $\\\\mathbf {M}$ is symmetric, (3) it has at leas one string $\\\\mathbf {m}_i = (\\\\mathbf {m}_{i1}, \\\\ldots , \\\\mathbf {m}_{i, n+1})$ with $\\\\mathbf {m}_{ik} = 1$ for every $1 \\\\le k\\\\le n+1$ , $k \\\\ne i$ , (4) $\\\\mathbf {M}$ satisfies the conditions of Theorem REF , (5) if $\\\\mathbf {m}_{ij}=1$ and $\\\\mathbf {m}_{jk} = 1$ then $\\\\mathbf {m}_{ik}=1$ for all $1 \\\\le i,j,k \\\\le n+1$ , (6) if $\\\\mathbf {m}_{ik}=1$ , then $\\\\mathbf {m}_{ij}=1$ or $\\\\mathbf {m}_{jk}=1$ for every $j$ with $1 \\\\le i \\\\equiv 1 \\\\bmod (2),$ for any $j>i$ .\",\n \"Indeed, by Lemma REF , $\\\\mathbf {m}_{ij}=1$ for every $j>i$ in the matrix $\\\\mathbf {M}$ , and then by condition (3) of Theorem REF , $\\\\bigl <\\\\mathsf {0,i} \\\\bigr > + \\\\bigl < \\\\mathsf {0,j} \\\\bigr > + \\\\bigl <\\\\mathsf {i,j} \\\\bigr > \\\\equiv 1 \\\\bmod (2)$ .\",\n \"We thus get $\\\\bigl <\\\\mathsf {0,i} \\\\bigr > + \\\\bigl < \\\\mathsf {0,j} \\\\bigr > + \\\\bigl <\\\\mathsf {i,j} \\\\bigr >= \\\\bigl < \\\\mathsf {i,j}\\\\bigr >,$ because of $\\\\bigl <\\\\mathsf {0,i} \\\\bigr > = \\\\bigl <\\\\mathsf {i,i} \\\\bigr > - 1$ , $1 \\\\le i \\\\le n$ , as claimed.\",\n \"Theorem 7.3 Let $\\\\mathbf {M} \\\\in \\\\mathsf {Mat}_{n}(\\\\mathbb {Z}_2)$ be a symmetric matrix with zero main diagonal.\",\n \"$\\\\mathbf {M}$ is a meander matrix if and only if the following hold: (1) there is a string $\\\\mathbf {m}_i$ such that $\\\\mathbf {m}_{ij}=1$ if $j \\\\ne i$ and $\\\\mathbf {m}_{ii}=0$ , (2) if $\\\\mathbf {m}_{ij} = 1$ , $\\\\mathbf {m}_{jk} =1$ , then $\\\\mathbf {m}_{ik}=1$ , for all $i,j,k \\\\in \\\\lbrace 1,2, \\\\ldots , n\\\\rbrace $ , (3) if $\\\\mathbf {m}_{ik}=1$ then $\\\\mathbf {m}_{ij}=1$ or $\\\\mathbf {m}_{jk}=1$ for all $i \\\\le j \\\\le k$ , (4) for every $1 \\\\le i,j\\\\le n$ we have $\\\\bigl < \\\\mathbf {m}_i, \\\\mathbf {m}_j \\\\bigr > &\\\\equiv & 0 \\\\bmod (2), \\\\qquad \\\\mbox{if $i=j$ or $\\\\mathbf {m}_{ij} = 0$},\\\\\\\\\\\\bigl < \\\\mathbf {m}_i, \\\\mathbf {m}_j \\\\bigr > &\\\\equiv & 1 \\\\bmod (2), \\\\qquad \\\\mbox{if $\\\\mathbf {m}_{ij} = 1$}.$ By Definition REF and Proposition REF , it is sufficient to prove that condition (3) of Theorem REF is equivalent to the last condition.\",\n \"It is clear that an equality $\\\\bigl < \\\\mathbf {m}_i, \\\\mathbf {m}_i \\\\bigr > \\\\equiv 0 \\\\bmod (2)$ implies that only an even number (including zero) of chords intersect chord correspondences to string $\\\\mathbf {m}_i$ .\",\n \"Next, an equality $\\\\mathbf {m}_{ij}=0$ implies that chord $i$ do not intersect chord $j$ , and an equality $\\\\mathbf {m}_{ij}=0$ implies that only even number of chords intersect both of chords $i$ , $j$ .\",\n \"We thus get the first two conditions of Theorem REF .\",\n \"Further, by Lemma REF , from the third condition of Theorem REF it follows (REF ).\",\n \"Assume now that (REF ) holds and consider three pairwise intersecting chords, say $i,j,k$ .\",\n \"Then $\\\\bigl < \\\\mathsf {m}_i, \\\\mathsf {m}_j \\\\bigr > \\\\equiv 1 \\\\bmod (2)$ , $\\\\bigl < \\\\mathbf {m}_i, \\\\mathbf {m}_k \\\\bigr > \\\\equiv 1 \\\\bmod (2)$ , and $\\\\bigl < \\\\mathbf {m}_j, \\\\mathbf {m}_k \\\\bigr > \\\\equiv 1 \\\\bmod (2)$ and we complete the proof.\",\n \"From this Theorem it follows the following meanders construction algorithm.\",\n \"REQUIRE an even number $N$ , $S_1 = \\\\lbrace 1,3,\\\\ldots , N-1\\\\rbrace $ , $S_0 = \\\\lbrace 2,4, \\\\ldots ,N\\\\rbrace $ ; ENSURE $n_1,\\\\ldots , n_N \\\\in S_1 \\\\cup S_0$ ; 1. take a $(N+1) \\\\times (N+1)$ tableau with empty cells and fill the main diagonal by zeros; 2. number strings from left to right, and number columns from top to bottom by numbers $0,1, \\\\ldots , N$ ; 3.\",\n \"$\\\\Delta $ -fill string with number 0; 4. choose a string with an odd number $n\\\\in S_1$ and $\\\\Delta $ -fill it and column with the same number; 5.\",\n \"$i=1$ ; 6.\",\n \"PRINT $n$ ; 6.\",\n \"$S_i: = S_i \\\\setminus \\\\lbrace n\\\\rbrace $ ; 7.\",\n \"IF $S_i = \\\\varnothing $ THEN GOTO 13 ELSE GOTO 8; 8 $i:=i+1 \\\\bmod (2)$ ; 9. choose a string (a column) with number $m \\\\in S_i$ ; 10.\",\n \"IF the string (the column) can be $\\\\Delta $ -filled THEN GOTO 11 ELSE choose another $m^{\\\\prime } \\\\in S_i\\\\setminus \\\\lbrace m\\\\rbrace $ GOTO 10; 11. using (REF ), () get a system of equations for empty cells; 12.\",\n \"IF the system can by solved THEN GOTO 6 ELSE choose a string (column) with another number $m^{\\\\prime } \\\\in S_i\\\\setminus \\\\lbrace m\\\\rbrace $ GOTO 10; 13.\",\n \"END Example 7.4 Let us construct a $9\\\\times 9$ meander matrix.\",\n \"(0) Take $9\\\\times 9$ tableau and partially fill it as follows Table: NO_CAPTION (1) Choose a string and column 5 and $\\\\Delta $ -fill them Table: NO_CAPTION By(), ${\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\bigl < 5, 6 \\\\bigr > = 1 + (6,7) + (6,8) \\\\equiv 1 \\\\bmod (2),\\\\\\\\\\\\bigl < 5, 7 \\\\bigr > = 1 + (6,7) + (7,8) \\\\equiv 1 \\\\bmod (2),\\\\\\\\\\\\bigl < 5, 8 \\\\bigr > = 1 + (6,8) + (7,8) \\\\equiv 1 \\\\bmod (2),\\\\end{array}\\\\right.\",\n \"}$ it implies that $(6,7) = (6,8) = (7,8) = {a} \\\\in \\\\lbrace 0,1\\\\rbrace ,$ and we thus get Table: NO_CAPTION (2) We have to choose a string (column) with an even number.\",\n \"(i) Take a string (column) 2, we then get $(1,2) = 0, (2,3) = (2,4) =0, (2,6) = (2,7) = (2,8)=1,$ it follows that $\\\\bigl <\\\\mathsf {1,2} \\\\bigr > \\\\equiv 0 \\\\bmod (2)$ .\",\n \"We have $\\\\bigl <\\\\mathsf {1,2} \\\\bigr > &=& 1 + (1,3) + (1,4) \\\\\\\\&&+ (1,6)+ (1,7) + (1,8) \\\\equiv 0 \\\\bmod (2),$ and we do not get any contradiction, hence this string may be chosen.\",\n \"(ii) Similarly, by the straightforward verification, one can easy verify that strings 4, 6 can be chosen.\",\n \"(iii) Take string 8.\",\n \"We get $(1,8) = (2,8) = (3,8) = (4,8) = 0= {a} = 0,$ and hence $\\\\bigl <\\\\mathsf {1,8} \\\\bigr > \\\\equiv 0 \\\\bmod (2)$ , but the tableau implies that $\\\\bigl <\\\\mathsf {1,8} \\\\bigr > = 1$ .\",\n \"Therefore this string cannot be chosen in this step.\",\n \"(3) Chose string 2.\",\n \"We then get Table: NO_CAPTION by (), ${\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\bigl < \\\\mathsf {2,3} \\\\bigr > = 1 + (3,4) + (3,6) + (3,7) + (3,8) \\\\equiv 1 \\\\bmod (2),\\\\\\\\\\\\bigl < \\\\mathsf {2,4} \\\\bigr > = 1 + (3,4) + (4,6) + (4,7) + (4,8) \\\\equiv 1 \\\\bmod (2),\\\\\\\\\\\\bigl < \\\\mathsf {2,6} \\\\bigr > = 1 + (3,6) + (4,6) + {a} + {a} \\\\equiv 1 \\\\bmod (2),\\\\\\\\\\\\bigl < \\\\mathsf {2,7} \\\\bigr > = 1 + (3,7) + (4,7) + {a} + {a} \\\\equiv 1 \\\\bmod (2),\\\\\\\\\\\\bigl < \\\\mathsf {2,8} \\\\bigr > = 1 + (3,8) + (4,8) + {a} + {a} \\\\equiv 1 \\\\bmod (2),\\\\end{array}\\\\right.\",\n \"}$ and we then obtain $& (3,6) = (4,6) = b \\\\in \\\\lbrace 0,1\\\\rbrace \\\\\\\\& (3,7) = (4,7) = c \\\\in \\\\lbrace 0,1\\\\rbrace \\\\\\\\& (3,8) = (4,8) = d \\\\in \\\\lbrace 0,1\\\\rbrace \\\\\\\\& (3,4) = b+c+d \\\\bmod (2).$ Next, using the condition of Theorem REF , $\\\\bigl < \\\\mathsf {3,3} \\\\bigr > \\\\equiv 0 \\\\bmod (2)$ , we get $(1,3) = 0$ .\",\n \"Similarly, one can get $(1,4) = 0$ .\",\n \"Using $\\\\bigl < \\\\mathsf {6,6} \\\\bigr > \\\\equiv \\\\bigl < \\\\mathsf {7,7} \\\\bigr > \\\\equiv \\\\bigl < \\\\mathsf {8,8} \\\\bigr > \\\\equiv 0 \\\\bmod (2)$ we get $(1,6) = (1,7) = (1,8) = 1$ .\",\n \"We thus have Table: NO_CAPTION (4) We have to chose a string with an odd number $1,3$ or 7.\",\n \"Since string 4 has not been chosen and $(1,4)=0$ then string 1 cannot be $\\\\Delta $ -filled.\",\n \"Next, if we chose string 7 we then get $(6,7) = 0$ , $(7,8) = 1$ , but $(6,7) = (7,8) = a$ , we then get a contradiction.\",\n \"Further, one can easy verify that string 3 can be chosen.\",\n \"(5) Take string 3.\",\n \"It follows that $b=c=d=1$ and we then get Table: NO_CAPTION We see that strings $1,4$ are automatically $\\\\Delta $ -filled.\",\n \"So, we have two possibilities: 1) $a=0$ , and 2) $a = 1$ .\",\n \"These cases correspondence to the following possibilities for our meander (see Figure REF ).\",\n \"Figure: The meanders which are correspondence to case a=0a=0 (in the top) and a=1a=1 (in the bottom).\"\n ]\n]"},"id":{"kind":"string","value":"1808.08542"}}},{"rowIdx":295,"cells":{"text":{"kind":"list like","value":[["Advances in Computational Methods for Phylogenetic Networks in the\n Presence of Hybridization"],["Abstract Phylogenetic networks extend phylogenetic trees to allow for modeling reticulate evolutionary processes such as hybridization.","They take the shape of a rooted, directed, acyclic graph, and when parameterized with evolutionary parameters, such as divergence times and population sizes, they form a generative process of molecular sequence evolution.","Early work on computational methods for phylogenetic network inference focused exclusively on reticulations and sought networks with the fewest number of reticulations to fit the data.","As processes such as incomplete lineage sorting (ILS) could be at play concurrently with hybridization, work in the last decade has shifted to computational approaches for phylogenetic network inference in the presence of ILS.","In such a short period, significant advances have been made on developing and implementing such computational approaches.","In particular, parsimony, likelihood, and Bayesian methods have been devised for estimating phylogenetic networks and associated parameters using estimated gene trees as data.","Use of those inference methods has been augmented with statistical tests for specific hypotheses of hybridization, like the D-statistic.","Most recently, Bayesian approaches for inferring phylogenetic networks directly from sequence data were developed and implemented.","In this chapter, we survey such advances and discuss model assumptions as well as methods' strengths and limitations.","We also discuss parallel efforts in the population genetics community aimed at inferring similar structures.","Finally, we highlight major directions for future research in this area."],["Introduction","Hybridization is often defined as reproduction between members of genetically distinct populations [4].","This process could occur in various spatial contexts, and could have impacts on speciation and differentiation [2], [3], [74], [75], [99], [1], [76].","Furthermore, increasing evidence as to the adaptive role of hybridization has been documented, for example, in humans [94], macaques [111], [6], [88], mice [107], [66], butterflies [114], [135], and mosquitoes [32], [119].","Hybridization is “generically\" used to contain two different processes: hybrid speciation and introgression [31].","In the case of hybrid speciation, a new population made of the hybrid individuals forms as a separate and distinct lineage from either of its two parental populations.In this chapter, we do not make a distinction between species, population, or sub-population.","The modeling assumptions and algorithmic techniques underlying all the methods we describe here neither require nor make use of such a distinction.","Introgression, or introgressive hybridization, on the other hand, describes the incorporation of genetic material into the genome of a population via interbreeding and backcrossing, yet without creating a new population [45].","As Harrison and Larson noted [45], introgression is a relative term: alleles at some loci introgress with respect to alleles at other loci within the same genomes.","From a genomic perspective, and as the basis for detection of hybridization, the general view is that in the case of hybrid speciation, regions derived from either of the parental ancestries of a hybrid species would be common across the genomes, whereas in the case of introgression, regions derived from introgression would be rare across the genomes [31].","Fig.","REF illustrates both hybridization scenarios.","Figure: Hybrid speciation and introgression.","(a) A phylogenetic tree on three taxa, A, B, and C, and a gene tree within itsbranches.","Genetic material is inherited from ancestors to descendants and it is expected that loci across the genome would have the shown gene tree.","(b) A hybrid speciation scenario depicted by a phylogenetic network,where B is a hybrid population that is distinct from its parental species.","Shown within the branches of the network are two gene trees, both of which are assumed to be very commonacross the genome.","(c) An introgression scenario.","Through hybridization and backcrossing, genetic material from (an ancestor of) C is incorporated into the genomes of individuals in (an ancestor of) B.","The introgressedgenetic material would have the gene tree shown in red, and the majority of loci in B's genomes would have the gene tree shown in blue.","Incomplete lineage sorting would complicate all threescenarios by giving rise to loci with other possible gene trees and by changing the distribution of the various gene trees.A major caveat to the aforementioned general view is that, along with hybridization, other evolutionary processes could also be at play, which significantly complicates the identification of hybrid species and their parental ancestries.","Chief among those processes are incomplete lineage sorting (ILS) and gene duplication and loss.","Indeed, various studies have highlighted the importance of accounting for ILS when attempting to detect hybridization based on patterns of gene tree incongruence [114], [77], [127], [32], [16], [94], [119], [135], [90].","Furthermore, gene duplication and loss are very common across all branches of the Tree of Life.","While the main focus of this chapter is on modeling and inferring hybridization, a discussion of how ILS is accounted for is also provided since recent developments have made great strides in modeling hybridization and ILS simultaneously.","While signatures of gene duplication and loss are ubiquitous in genomic data sets, we do not include a discussion of these two processes in this chapter since methods that account for them in the context of phylogenetic networks are currently lacking.","When hybridization occurs, the evolutionary history of the set of species is best modeled by a phylogenetic network, which extends the phylogenetic tree model by allowing for “horizontal\" edges to denote hybridization and to facilitate modeling bi-parental inheritance of genetic material.","Fig.","REF shows two phylogenetic networks that model hybrid speciation and introgression.","It is very important to note, though, that from the perspective of existing models, both phylogenetic networks are topologically identical.","This issue highlights two important issues that must be thought about carefully when interpreting a phylogenetic network.","First, neither the phylogenetic network nor the method underlying its inference distinguish between hybrid speciation and introgression.","This distinction is a matter of interpretation by the user.","For example, the phylogenetic network in Fig.","REF (c) could be redrawn, without changing the model or any of its properties, so that the introgression is from (an ancestor of) A to (an ancestor of) B, in which case the “red\" gene tree would be expected to appear with much higher frequency than the “blue\" gene tree.","In other words, the way a phylogenetic network is drawn could convey different messages about the evolutionary history that is not inherent in the model or the inference methods.","This issue was importantly highlighted with respect to data analysis in [119] (Figure 7 therein).","Second, the phylogenetic network does not by itself encode any specific backbone species tree that introgressions could be interpreted with respect to.","This, too, is a matter of interpretation by the user.","This is why, for example, Clark and Messer [16] recently argued that “perhaps we should dispense with the tree and acknowledge that these genomes are best described by a network.\"","Furthermore, recent studies demonstrated the limitations of inferring a species tree “despite hybridization\" [106], [138].","With the availability of data from multiple genomic regions, and increasingly often from whole genomes, a wide array of methods for inferring species trees, mainly based on the multispecies coalescent (MSC) model [20], have been developed [67], [81], [68].","Building on these methods, and often extending them in novel ways, the development of computational methods for inferring phylogenetic networks from genome-wide data has made great strides in recent years.","Fig.","REF summarizes the general approaches that most phylogenetic network inference methods have followed in terms of the data they utilize, the model they employ, and the inferences they make.","Figure: Phylogenetic network inference process and approaches.","The process of phylogenetic network inference starts with collecting the genomicdata and identifying the orthology groups of unlinked loci.","Multiple sequence alignments or single bi-allelic markers corresponding to the unlinked loci arethen obtained; phylogenetic network inference methods use one of these two types of data.","In two-step inference methods, gene trees are first estimated for theindividual loci from the sequence alignment data, and these gene tree estimates are used as the input data for network inference.","If incomplete lineage sorting (ILS)is not accounted for, a smallest displaying network of the gene tree estimates is sought.","If ILS is accounted for, parsimony inference based on the MDC (minimizingdeep coalescences) criterion, a maximum likelihood estimate (MLE), a maximum a posteriori (MAP) estimate, or samples of the posterior distribution can beobtained.","In the direct inference approach, whether based on sequence alignment or bi-allelic marker data, a MAP estimate or samples of the posterior distributioncan be obtained directly from the data.","The two parsimony methods consider only the topologies of the gene tree estimates as input (i.e., they ignore gene tree branch lengths) and return as output phylogenetic network topologies.","The likelihood and Bayesian methods that take gene tree estimates as input can operate on gene tree topologies alone or gene trees with branch lengths as well.","Both methods estimate phylogenetic network topologies along with branch lengths (in coalescent units) and inheritance probabilities.","The direct inference methods estimate the phylogenetic network along with its associated parameters.The overarching goal of this chapter is to review the existing methods for inferring phylogenetic networks in the presence of hybridization,We emphasize hybridization (in eukaryotic species) here since processes such as horizontal gene transfer in microbial organisms result in reticulate evolutionary histories, but the applicability of methods we describe in this chapter has not been investigated or explored in such a domain.","describe their strengths and limitations, and highlight major directions for future research in this area.","All the methods discussed hereafter make use of multi-locus data, where a locus in this context refers to a segment of genome present across the individuals and species sampled for a given study and related through common descent.","A locus can be of varying length, coding or non-coding, and can be either functional or non-functional.","Therefore, the use of the term “gene trees\" is only historical; we use it to mean the evolutionary history of an individual locus, regardless of whether the locus overlaps with a coding region or not.","Care must be taken with increasingly long loci spanning hundreds or thousands of contiguous basepairs, however, as many methods assume a locus has not been affected by recombination.","Multi-locus methods are fairly popular because the model fits several types of reduced representation genomic data sets commonly generated to study biological systems.","Reduced representation refers to capturing many segments scattered throughout a genome, but only covering a fraction of the total genome sequence [37].","Reduced representation data sets which have been used with multi-locus methods include RAD-seq and genotyping by sequencing (GBS), which capture loci of roughly 100 bp associated with palindromic restriction enzyme recognition sites [30].","Another family of techniques often applied to studies of deeper time scales, sequence capture, extracts conserved sequences using probes complementary to targetted exons or ultraconserved elements [41].","Sequence capture can also be performed in silico when whole genomes are available [52].","The rest of the chapter is organized as follows.","We begin in Section 2 by defining terminology for the non-biologist, and give a very brief review of phylogenetic trees and their likelihood.","In Section 3, we describe the earliest, and simplest from a modeling perspective, approaches to inferring parsimonious phylogenetic networks from gene tree topologies by utilizing their incongruence as the signal for hybridization.","To account for ILS, we describe in Section 4 the multispecies network coalescent, or MSNC, which is the core model for developing statistical approaches to phylogenetic network inference while accounting for ILS simultaneously with hybridization.","In Sections 5 and 6 we describe the maximum likelihood and Bayesian methods for inferring phylogenetic networks from multi-locus data.","In Section 7, we briefly discuss an approach aimed at detecting hybridization by using phylogenetic invariants.","This approach does not explicitly build a phylogenetic network.","In Section 8 we briefly discuss the efforts for developing methods for phylogenetic network inference that took place in parallel in the population genetics community (they are often referred to as “admixture graphs\" in the population genetics literature).","In Section 9, we summarize the available software for phylogenetic network inference, discuss the data that these methods use, and then list some of the limitations of these methods in practice.","We conclude with final remarks and directions for future research in Section 10."],["Background for Non-biologists","In this section we define the biological terminology used throughout the chapter so that it is accessible for non-biologists.","We also provide a brief review of phylogenetic trees and their likelihood, which is the basis for maximum likelihood and Bayesian inference of phylogenetic trees from molecular sequence data.","Excellent books that cover mathematical and computational aspects of phylogenetic inference include [29], [102], [34], [117], [110]."],["Terminology","As we mentioned above, hybridization is reproduction between two members of genetically distinct populations, or species (Fig.","REF ).","Diploid species (e.g., humans) have two copies of each genome.","Aside from a few unusual organisms such as parthenogenic species, one copy will be maternal in origin and the other paternal.","When the hybrid individual (or $F_1$ ) is also diploid this process is called homoploid hybridization.","Figure: Hybridization, recombination, and the generation of a mosaic genome.","Diploid individual a from species A and individual b from species B mate, resulting ina diploid hybrid individual with one copy of its genome inherited from parent a and the other copy inherited from parent b.","A recombination event results in the “swapping\" of entire regionsbetween two copies of the genome.","After multiple generations in which more recombination happens, the genome becomes a mosaic.","Walking across the genomefrom left to right, the color switches back and forth between red and blue, where switches happen at recombination breakpoints.","Shown are four different loci.","Loci 1 and 3 are not appropriatefor tree inference since they span recombination breakpoints and, thus, include segments that have different evolutionary histories.","Loci 2 and 4 are the “ideal\" loci for analyses by methodsdescribed in this chapter.While each of the two copies of the genome in the hybrid individual traces its evolution back to precisely one of the two parents, this picture becomes much more complex after several rounds of recombination.","Recombination is the swapping of a stretch of DNA between the two copies of the genome.","Mathematically, if the two copies of the genome are given by strings $u$ and $v$ (for DNA, the alphabet for the strings is $\\lbrace A,C,T,G\\rbrace $ ), then recombination results in two strings $u^{\\prime }=u_1u_2u_3$ and $v^{\\prime }=v_1v_2v_3$ , where $u_1$ , $u_2$ , $u_3$ , $v_1$ , $v_2$ , and $v_3$ are all strings over the same alphabet, and $u_1v_2u_3=u$ and $v_1u_2v_3=v$ ; that is, substrings $u_2$ and $v_2$ were swapped.","Observe that when this happens, $u_1$ and $u_3$ in the copy $u^{\\prime }$ are inherited from one parent, and $v_2$ , also in the copy $u^{\\prime }$ , is inherited from a different parent.","A similar scenario happens in copy $v^{\\prime }$ of the diploid genome.","This picture gets further complicated due to backcrossing, which is the mating between the hybrid individual, or one of its descendants, with an individual in one of the parental species.","For example, consider a scenario in which descendants of the hybrid individual in Fig.","REF repeatedly mate with individuals from species A.","After several generations, it is expected that the genomes of the hybrid individuals become more similar to the genomes of individuals in species A, and less similar to the genomes of individuals in species B (using the illustration of Fig.","REF , the two copies of the genome would have much more red in them than blue).","Most models and methods for phylogenetic inference assume the two copies of a diploid genome are known separately and often only one of them is used to represent the corresponding individual.","However, it is important to note that knowing the two copies separately is not a trivial task.","Sequencing technologies produce data on both copies simultaneously, and separating them into their constituent copies is a well-studied computational problem known as genome phasing.","Biologists often focus on certain regions within the genomes for phylogenetic inference.","If we consider the genome to be represented as a string $w$ over the alphabet $\\lbrace A,C,T,G\\rbrace $ , then a locus is simply a substring of $w$ given by the start and end positions of the substring in $w$ .","The size of a locus can range anywhere from a single position in the genome to a (contiguous) stretch of 1 million or more positions in the genome.","As we discussed above, when recombination happens, an individual copy of the genome would have segments with different ancestries (the blue and red regions in Fig.","REF ).","A major assumption underlying phylogenetic tree inference is that the sequence data of a locus used for inference has evolved down a single tree.","Therefore, the more recombination which has occured within a locus over its evolutionary history (limited to the history connecting the species being studied), the less suitable it will be for phylogenetic inference.","Conversely, loci with low recombination rates may be more suitable in terms of avoiding intra-locus recombination, although such loci are more susceptible to linked selection [44]."],["Phylogenetic Trees and Their Likelihood","An unrooted binary phylogenetic tree $T$ on set ${{X}}$ of taxa (e.g., ${{X}}=\\lbrace humans,chimp,gorilla\\rbrace $ ) is a binary tree whose leaves are bijectively labeled by the elements of ${{X}}$ .","That is, if $|{{X}}|=n$ , then $T$ has $n$ leaf nodes and $n-2$ non-leaf (internal) nodes (each leaf node has degree 1 and each internal node has degree 3).","A rooted binary phylogenetic tree is a directed binary tree with a single node designated as the root and all edges are directed away from the root.","For $n$ taxa, a rooted binary tree has $n$ leaves and $n-1$ internal nodes (each leaf node has in-degree 1 and out-degree 0; each internal node except for the root has in-degree 1 and out-degree 2; the root has in-degree 0 and out-degree 2).","Modern methods for phylogenetic tree inference make use of molecular sequence data, such as DNA sequences, obtained from individuals within the species of interest.","The sequences are assumed to have evolved from a common ancestral sequence (we say the sequences are homologous) according to a model of evolution that specifies the rates at which the various mutational events could occur (Fig.","REF ).","Figure: Sequence evolution on a tree.","At the top is the ancestral sequence for a certain locus in the genome of an individual.","Through cell division and DNA replication, thissequence is inherited from parent to children.","However, mutations could alter the inherited sequences.","Boxes indicate letters that were deleted due to mutation.","Letters in blueindicate substitutions (a mutation that alters the state of the nucleotide).","The letter in green has mutated more than once during its evolutionary history.","With respect to sequenceU 1 U_1, sequence U 2 U_2 has two deletions at the 3rd and 8th positions, and a substitution (C to G) at the 5th position.","With respect to sequence U 1 U_1, sequenceS 3 S_3 has 7 deletions at positions 3–9, and substitutions at the first two positions (A to T, and A to C).","Sequences S 1 S_1, S 2 S_2, S 3 S_3, and S 4 S_4, with the boxes and colorsunknown, are often the data forphylogenetic inference.","That is, the four sequences used as data here are: AAATGTTAA, AAAGGTTAA, TCAA, and TCCACGAA.For example, to infer a phylogenetic tree $T$ on set ${{X}}=\\lbrace X_1,\\ldots ,X_n\\rbrace $ of taxa, the sequence $S_1$ of a certain locus is obtained from the genome of an individual in species $X_1$ , the sequence $S_2$ of a certain locus is obtained from the genome of an individual in species $X_2$ , and so on until $n$ sequences $S_1,\\ldots ,S_n$ are obtained.","To perform phylogenetic tree inference, the $n$ sequences must satisfy two important conditions (see Fig.","REF ): The sequences are homologous: The obtained sequences must have evolved down a single tree from a single sequence in an individual in an ancestral species.","Two sequences are homologous if they evolved from a common ancestor, including in the presence of events such as duplication.","Two homologous sequences are orthologs if they evolved from a common ancestor solely by means of DNA replication and speciation events.","Two homologous sequences are paralogs if their common ancestor had duplicated to give rise to the two sequences.","The sequences are aligned: While the obtained homologous “raw\" sequences might be of different lengths due to events such as insertions and deletions, the sequences must be made to be the same length before phylogenetic inference is conducted so that positional homology is established.","Intuitively, positional homology is the (evolutionary) correspondence among sites across the $n$ sequences.","That is, the sequences must be made of the same length so that the $i$ -th site in all of them had evolved from a single site in the sequence that is ancestral to all of them.","Figure: From homologous sequences to a phylogenetic tree.","Identifying homologous sequences across genomes is the first step towards a phylogenetic analysis.","The homologoussequences, once identified, are not necessarily of the same length, due to insertions and deletions.","Multiple sequence alignment is performed on the homologous sequences and the resultis sequences of the same length where boxes indicate deleted nucleotides.","Finally, a phylogenetic tree is constructed on the aligned sequences.Identifying homologous sequences across genomes is not an easy task; see, for example, [84] for a recent review of methods for homology detection.","Multiple sequence alignment is also a hard computational problem, with a wide array of heuristics and computer programs currently available for it; see, for example, [13] for a recent review.","We are now in position to define a basic version of the Phylogeny Inference Problem: Input: Set $S=\\lbrace S_1,\\ldots ,S_n\\rbrace $ of homologous sequences, where sequence $S_i$ is obtained from taxon $X_i$ , and the $n$ sequences are aligned.","Output: A phylogenetic tree $T$ on set ${{X}}$ of taxa such that $T$ is optimal, given the sequences, with respect to some criterion $\\Phi $ .","The books we cited above give a great survey of the various criterion that $\\Phi $ could take, as well as algorithms and heuristics for inferring optimal trees under the different criteria.","Here, we focus on the main criterion in statistical phylogenetic inference, namely likelihood.","We will make two assumptions when defining the likelihood that are (1) sites are identically and independently distributed, and (2) following a DNA replication event, the two resulting sequences continue to evolve independently of each other.","To define the likelihood of a tree $T$ , we first assign lengths $\\lambda :E(T) \\rightarrow {\\mathbb {R}}^+$ to its branches, so that $\\lambda (b)$ is the length of branch $b$ in units of expected number of mutations per site per generation.","Furthermore, we need a model of sequence evolution ${\\cal M}$ .","Most models of sequence evolution are Markov processes where the probability of observing a sequence $S$ at node $u$ depends only on the sequence at $u$ 's parent, the length of the branch that links $u$ to its parent, and the parameters of the model of sequence evolution.","If we denote by $p^{(i)}_{uv}(t)$ the probability that the $i$ -th nucleotide in the sequence at node $u$ evolves into the $i$ -th nucleotide in the sequence at node $v$ over time $t$ (measured in units of expected number of mutations as well), then the likelihood of a tree $T$ and its branch lengths $\\lambda $ is $L(T,\\lambda |S) = P(S|T,\\lambda ) = \\prod _{i} \\left(\\sum _R \\left(p(root^{(i)}) \\cdot \\prod _{b=(u,v)\\in E(T)} p^{(i)}_{uv}(\\lambda _b) \\right) \\right).$ Here, the outer product is taken over all sites $i$ in the sequences; i.e., if each of the $n$ sequences is of length $m$ , then $1 \\le i \\le m$ .","The summation is taken over $R$ , which is the set of all possible labelings of the internal nodes of $T$ with sequences of length $m$ .","Inside the summation, $p(root^{(i)})$ gives the stationary distribution of the nucleotides at position $i$ .","The likelihood as given by Eq.","(REF ) is computed in polynomial time in $m$ and $n$ using Felsenstein's “pruning\" algorithm [27].","Finally, the maximum likelihood estimate for solving the Phylogeny Inference Problem is given by $(T^*,\\lambda ^*) \\leftarrow {\\rm argmax}_{(T,\\lambda )} L(T,\\lambda |S).$ Computing the maximum likelihood estimate from a set $S$ of sequences is NP-hard [15], [100].","However, much progress has been made in terms of developing heuristics that scale up to thousands of taxa while achieving high accuracy, e.g., [109]."],["From Humble Beginnings: Smallest Displaying Networks","Early work and, still, much effort in the community has focused on inferring the topology of a phylogenetic network from a set of gene tree topologies estimated for the individual loci in a data set.","In this section, we discuss parsimony approaches to inferring phylogenetic network topologies from sets of gene trees."],["The Topology of a Phylogenetic Network","As discussed above, a reticulate, i.e., non-treelike, evolutionary history that arises in the presence of processes such as hybridization and horizontal gene transfer is best represented by a phylogenetic network.","A phylogenetic ${{X}}$ -network (Fig.","REF ), or ${{X}}$ -network for short, $\\Psi $ is a rooted, directed, acyclic graph (rDAG) with set of nodes $V(\\Psi )=\\lbrace r\\rbrace \\cup V_L \\cup V_T \\cup V_N $ , where $indeg(r)=0$ ($r$ is the root of $\\Psi $ ); $\\forall {v \\in V_L}$ , $indeg(v)=1$ and $outdeg(v)=0$ ($V_L$ are the external tree nodes, or leaves, of $\\Psi $ ); $\\forall {v \\in V_T}$ , $indeg(v)=1$ and $outdeg(v) \\ge 2$ ($V_T$ are the internal tree nodes of $\\Psi $ ); and, $\\forall {v \\in V_N}$ , $indeg(v) = 2$ and $outdeg(v) = 1$ ($V_N$ are the reticulation nodes of $\\Psi $ ).","For binary phylogenetic networks, the out-degree of the root and every internal tree node is 2.","The network's set of edges, denoted by $E(\\Psi ) \\subseteq V \\times V$ is bipartitioned into reticulation edges, whose heads are reticulation nodes, and tree edges, whose heads are tree nodes (internal or external).","Finally, the leaves of $\\Psi $ are bijectively labeled by the leaf-labeling function $\\ell :V_L \\rightarrow {{X}}$ .","Figure: An example of a phylogenetic network Ψ\\Psi with a single reticulation event.","This network is made up of leaf nodes V L ={v 1 ,v 2 ,v 3 ,v 4 }V_L = \\lbrace v_1,v_2,v_3,v_4\\rbrace , internal tree nodes V T ={u 1 ,u 2 ,u 3 }V_T=\\lbrace u_1,u_2,u_3\\rbrace , reticulation nodes V N ={h}V_N=\\lbrace h\\rbrace , and the root rr.","The nodes are connected by branches belonging to the set of phylogenetic network edges E(Ψ)E(\\Psi ).","The branches are:(r,u 1 )(r,u_1), (r,u 2 )(r,u_2), (u 1 ,h)(u_1,h), (u 2 ,h)(u_2,h), (h,u 3 )(h,u_3), (u 1 ,v 1 )(u_1,v_1), (u 2 ,v 4 )(u_2,v_4), (u 3 ,v 2 )(u_3,v_2), and (u 3 ,v 3 )(u_3,v_3).","The leaves are labeled by set X={A,B,C,D}{{X}}=\\lbrace A,B,C,D\\rbrace of taxa: ℓ(v 1 )=A\\ell (v_1)=A, ℓ(v 2 )=B\\ell (v_2)=B, ℓ(v 3 )=C\\ell (v_3)=C, and ℓ(v 4 )=D\\ell (v_4)=D."],["Inferring Smallest Displaying Networks","Early work on phylogenetic networks focused on the problem of identifying a network with the fewest number of reticulation nodes that summarizes all gene trees in the input.","More formally, let $\\Psi $ be a phylogenetic network.","We say that $\\Psi $ displays phylogenetic tree $t$ if $t$ can be obtained from $\\Psi $ by repeatedly applying the following operations until they are not applicable: For a reticulation node $h$ with two incoming edges $e_1=(u_1,h)$ and $e_2=(u_2,h)$ , remove one of the two edges.","For a node $u$ with a single parent $v$ and a single child $w$ , remove the two edges $(v,u)$ and $(u,w)$ , and add edge $(v,w)$ .","The set of all trees displayed by the phylogenetic network is ${\\cal T}(\\Psi )=\\lbrace t: \\; \\Psi \\; {\\rm displays} \\; t\\rbrace .$ For example, for the phylogenetic network $\\Psi $ of Fig.","REF , we have ${\\cal T}(\\Psi )=\\lbrace T_1,T_2\\rbrace $ , where $T_1=((A,(B,C)),D)$ and $T_2=(A,((B,C),D))$ .","Using this definition, the earliest phylogenetic network inference problem was defined as follows: Input: A set ${\\cal G}=\\lbrace g_1,g_2,\\ldots ,g_m\\rbrace $ of gene trees, where $g_i$ is a gene tree for locus $i$ .","Output: A phylogenetic network $\\Psi $ with the smallest number of reticulation nodes such that ${\\cal G} \\subseteq {\\cal T}(\\Psi )$ .","This problem is NP-hard [116] and methods were developed for solving it and variations thereof, some of which are heuristics [115], [123], [89], [124].","Furthermore, the view of a phylogenetic network in terms of the set of trees it displays was used for pursuing other questions in this domain.","For example, the topological difference between two networks could be quantified in terms of the topological differences among their displayed trees [83].","The parsimony and likelihood criteria were extended to the case of phylogenetic networks based on the assumption that each site (or, locus) has evolved down one of the trees displayed by the network [80], [54], [53], [56], [55], [57].","The concepts of character compatibility and perfect phylogeny were also extended to phylogenetic networks based on the notion of displayed trees [116], [82], [60].","Furthermore, questions related to distinguishability of phylogenetic networks based on their displayed trees have been pursued [59] and relationships between networks and trees have been established in terms of this definition [33], [134].","However, the computational complexity of this problem notwithstanding, the problem formulation could be deficient with respect to practical applications.","For one thing, solving the aforementioned problem only yields the topology of a phylogenetic network, but no other parameters.","In practice, biologists would be interested in divergence times, population parameters, and some quantification of the amount of introgression in the genomes.","These quantities are not recoverable under the given formulation.","Moreover, for the biologist seeking to analyze her data with respect to hypotheses of reticulate evolutionary events, solving the aforementioned problem could result in misleading evolutionary scenarios for at least three reasons.","First, the smallest number of reticulations required in a phylogenetic network to display all trees in the input could be arbitrarily far from the true (unknown) number of reticulations.","One reason for this phenomenon is the occurrence of reticulations between sister taxa, which would not be detectable from gene tree topologies alone.","Second, a smallest set of reticulations could be very different from the actual reticulation events that took place.","Third, and probably most importantly, some or even all of gene tree incongruence in an empirical data set could have nothing to do with reticulation.","For example, hidden paralogy and/or incomplete lineage sorting could also give rise to incongruence in gene trees.","When such phenomena are at play, seeking a smallest phylogenetic network that displays all the trees in the input is the wrong approach and might result in an overly complex network that is very far from the true evolutionary history.","To address all these issues, the community has shifted its attention in the last decade toward statistical approaches that view phylogenetic networks in terms of a probability distribution on gene trees that could encompass a variety of evolutionary processes, including incomplete lineage sorting."],["Phylogenetic Networks as Summaries of Trees","Before we turn our attention to these statistical approaches, it is worth contrasting smallest phylogenetic networks that display all trees in the input to the concept of consensus trees.","In the domain of phylogenetic trees, consensus trees have played an important role in compactly summarizing sets of trees.","For example, the strict consensus tree contains only the clusters that are present in the input set of trees, and nothing else.","The majority-rule consensus tree contains only the clusters that appear in at least 50% of the input trees.","When there is incongruence in the set of trees, these consensus trees are most often non-binary trees (contain “soft polytomies\") such that each of the input trees can be obtained as a binary resolution of the consensus tree.","Notice that while the consensus tree could be resolved to yield each tree in the input, there is no guarantee in most cases that it cannot also be resolved to generate trees that are not in the input.","Smallest phylogenetic networks that display all trees in the input could also be viewed as summaries of the trees, but instead of removing clusters that are not present in some trees in the input, they display all clusters that are present in all trees in the input.","Similarly to consensus trees, a smallest phylogenetic network could also display trees not in the input (which is the reason why we use $\\subseteq $ , rather than $=$ , in the problem formulation above).","These issues are illustrated in Fig.","REF .","Figure: Consensus trees and phylogenetic networks as two contrasting summary methods.","(a)-(c) Three (input) gene trees whose summary is sought.","(d) The strict consensus of the input trees.","(e) The 70% majority-rule consensus of the input trees.","(f) A smallest phylogenetic network that displays all three trees iin the input.","The strict consensus could be resolved to yield 15 different binary trees, only three of which are in the input.","The majority-rule consensus tree could beresolved to yield three possible trees, two of which are the trees in (a) and (d), but the third, which is (((a,b),d),c), is not in the input.","Furthermore, the tree in panel (b)is not included in the summary provided by the majority-rule consensus.","The phylogenetic network displays four trees, three of which are the input trees, and thefourth is ((a,(b,c)),d), which is not in the input.As discussed above, ILS is another process that could cause gene trees to be incongruent with each other and complicates the inference of phylogenetic networks since incongruence due to ILS should not induce additional reticulation nodes.","Before we move on to discuss statistical approaches that account for ILS in a principled probabilistic manner under the coalescent, we describe an extension of the minimizing deep coalescences, or, MDC, criterion [73], [72], [112], to phylogenetic networks, which was devised in [125]."],["A Step Towards More Complexity: Minimizing Deep Coalescences","Let $\\Psi $ be a phylogenetic network and consider node $u \\in V(\\Psi )$ .","We denote by $B_u \\subseteq V(\\Psi )$ the set of nodes in $\\Psi $ that are below node $u$ (that is, the set of nodes that are reachable from the root of $\\Psi $ via at least one path that goes through node $u$ ).","A coalescent history of a gene tree $g$ and a species (phylogenetic) network $\\Psi $ as a function $h:V(g) \\rightarrow V(\\Psi )$ such that the following conditions hold: if $w$ is a leaf in $g$ , then $h(w)$ is the leaf in $\\Psi $ with the same label (in the case of multiple alleles, $h(w)$ is the leaf in $\\Psi $ with the label of the species from which the allele labeling leaf $w$ in $g$ is sampled); and, if $w$ is a node in $g_v$ , then $h(w)$ is a node in $B_{h(v)}$ .","Given a phylogenetic network $\\Psi $ and a gene tree $g$ , we denote by $H_{\\Psi }(g)$ the set of all coalescent histories of gene tree $g$ within the branches of phylogenetic network $\\Psi $ .","Given a coalescent history $h$ , the number of extra lineages arising from $h$ on a branch $b=(u,v)$ in phylogenetic network $\\Psi $ is the number of gene tree lineages exiting branch $b$ from below node $u$ toward the root, minus one.","Finally, $XL(\\Psi ,h)$ is defined as the sum of the numbers of extra lineages arising from $h$ on all branches $b \\in E(\\Psi )$ .","Using coalescent histories, the minimum number of extra lineages required to reconcile gene tree $g$ within the branches of $\\Psi $ , denoted by $XL(\\Psi ,g)$ is given by $XL(\\Psi ,g)=\\min _{h \\in H_{\\Psi }(g)}XL(\\Psi ,h).$ Under the MDC (minimizing deep coalescence) criterion, the optimal coalescent history refers to the one that results in the fewest number of extra lineages [73], [112], and thus, $XL(\\Psi , g) = \\sum _{e \\in E(\\Psi )}[k_e(g)-1]$ where $k_e(g)$ is the number of extra lineages on edge $e$ of $\\Psi $ in the optimal coalescent history of gene tree $g$ .","Figure: The MDC criterion on phylogenetic networks.","A phylogenetic network and coalescent histories within its branches of the three gene trees in Fig.","(a)-(c).","The highlighted branch that separates the hybridization event from the MRCA of B and C has two extra lineages arising from the threeshown coalescent histories.","All other branches have 0 extra lineages.","Therefore, the total number of extra lineages in this case is 2.A connection between extra lineages and the displayed trees of a phylogenetic network is given by the following observation.","Observation 1 If gene tree $g$ is displayed by phylogenetic network $\\Psi $ , then $XL(\\Psi ,g)=0$ .","The implication of this observation is that if one seeks the phylogenetic network that minimizes the number of extra lineages, the problem can be trivially solvable by finding an overly complex network that displays every tree in the input.","Therefore, inferring a phylogenetic network $\\Psi $ from a collection of gene tree topologies ${{G}}$ based on the MDC criterion is more appropriately defined by $\\hat{\\Psi }(m) = \\operatornamewithlimits{argmin}_{\\Psi (m)} \\left( \\sum _{g \\in {{G}}} XL(\\Psi (m),g) \\right),$ where we write $\\Psi (m)$ to denote a phylogenetic network with $m$ reticulation nodes.","While the number of reticulations $m$ is unknown and is often a quantity of interest, there is a trade-off between the number of reticulation nodes and number of extra lineages in a network: Reticulation edges can be added to reduce the number of extra lineages.","Observing this reduction in the number of extra lineages could provide a mechanism to determine when to stop adding reticulations to the network [125].","In the previous section, we focused on two parsimony formulations for inferring a phylogenetic network from a collection of input gene tree topologies: The first seeks a network with the fewest number of reticulations that displays each of the input gene trees, and the second seeks a network that does not have to display every gene tree in the input, but must minimize the number of “extra lineages\" that could arise within a given number of reticulations.","Both formulations result in phylogenetic network topologies alone and make use of only the gene tree topologies.","In this section, we introduce the multispecies network coalescent, or MSNC [120], as a generative process that extends the popular multispecies coalescent, or MSC [20], that is the basis for most multi-locus species tree inference methods.","The MSNC allows for the coalescent to operate within the branches of a phylogenetic network by viewing a set of populations—extant and ancestral—glued together by a rooted, directed, acyclic graph structure."],["Parameterizing the Network's Topology","In addition to the topology of a phylogenetic network $\\Psi $ , as given by Definition REF above, the nodes and edges are parameterized as follows.","Associated with the nodes are divergence/reticulation times, $\\tau : V(\\Psi ) \\rightarrow {\\mathbb {R}}^+$ , where $\\tau (u)$ is the divergence time associated with tree node $u$ and $\\tau (v)$ is the reticulation time associated with reticulation node $v$ .","All leaf nodes $u$ in the network have $\\tau (u)=0$ .","Furthermore, if $u$ is on a path from the root of the network to a node $v$ , then $\\tau (u) \\ge \\tau (v)$ .","Associated with the edges are population mutation rate parameters, $\\theta : E(\\Psi ) \\rightarrow {\\mathbb {R}}^+$ , where $\\theta _b=4N_b \\mu $ is the population mutation rate associated with edge $b$ , $N_b$ is the effective population size associated with edge $b$ , and $\\mu $ is the mutation rate per site per generation.","Divergence times associated with nodes in the phylogenetic network could be measured in units of years, generations, or coalescent units.","Branch lengths in gene trees are often given in units of expected number of mutations per site.","The following rules are used to convert back and forth between these units: Given divergence time $\\tau $ in units of expected number of mutations per site, mutation rate per site per generation $\\mu $ and the number of generations per year $g$ , $\\tau / (\\mu g)$ represents divergence times in units of years.","Given population size parameter $\\theta $ in units of population mutation rate per site, $2\\tau / \\theta $ represents divergence times in coalescent units.","In addition to the divergence times and population size parameters, the reticulation edges of the network are associated with inheritance probabilities.","For every reticulation node $u \\in V_N$ , let $left(u)$ and $right(u)$ be the “left\" and “right\" edges incoming into node $u$ , respectively (which of the two edges is labeled left and which is labeled right is arbitrary).","Let $E_R \\subseteq E(\\Psi )$ be the set of reticulation edges in the network.","The inheritance probabilities are a function $\\gamma : E_R \\rightarrow [0,1]$ such that for every reticulation node $u \\in V_N$ , $\\gamma (left(u))+\\gamma (right(u)) = 1$ .","In the literature, $\\gamma $ is sometimes described as a vector $\\Gamma $ ."],["The Multispecies Network Coalescent and Gene Tree Distributions","As an orthologous, non-recombining genomic region from a set ${{X}}$ of species evolves within the branches of the species phylogeny of ${{X}}$ , the genealogy of this region, also called the gene tree, can be viewed as a discrete random variable whose values are all possible gene tree topologies on the set of genomic regions.","When the gene tree branch lengths are also taken into account, the random variable becomes continuous.","Yu et al.","[126] gave the probability mass function (pmf) for this discrete random variable given the phylogenetic network $\\Psi $ and an additional parameter $\\Gamma $ that contains the inheritance probabilities associated with reticulation nodes, which we now describe briefly.","The parameters $\\Psi $ and $\\Gamma $ specify the multispecies network coalescent, or MSNC (Fig.","REF ), and allow for a full derivation of the mass and density functions of gene trees when the evolutionary history of species involves both ILS and reticulation [126], [127].","This is a generalization of the multispecies coalescent, which describes the embedding and distribution of gene trees within a species tree without any reticulate nodes [20].","Figure: Layers of the multispecies network coalescent model.","A phylogenetic network describes the relationship between species (top).","The MSNC describes the distribution of gene trees within the network, in which alleles from the same species can have different topologies and inheritance histories due to reticulation and/or ILS (middle).","Some kind of mutation process occurs along the gene trees, resulting in observed differences between alleles in the present, which vary between genes based on their individual trees.It is important to note that a single reticulation edge between two nodes does not mean a single hybridization event.","Rather, a reticulation edge abstracts a continuous epoch of repeated gene flow between the two species, as illustrated in Fig.","REF .","Figure: Reticulation edges as abstractions of gene flow epochs.","(Left) An epoch of gene flow from one population to another with migration rate α\\alpha per generation.","(Right) A phylogenetic networkwith a single reticulation edge that abstracts the gene flow epoch, with inheritance probability γ\\gamma .The two models shown in Fig.","REF were referred to as the “gene flow\" model (left) and “intermixture\" model (right) of hybridization in [70].","While the “gene flow” model is used by the IM family of methods [47] to incorporate admixture, the MSNC adopts the “intermixture” model.","In this model, the $\\gamma $ inheritance probabilities indicate the ratio of genetic materials of a hybrid coming from its two parents.","This means that unlinked loci from a hybrid species will have independent evolutionary histories, and will have evolved through the “left” or “right” parent with some probability $\\gamma $ and $1 - \\gamma $ respectively.","The performance of phylogenetic network inference on data simulated under the gene flow model was demonstrated in [121].","Wen and Nakhleh [121] derived the density function of the probability of gene trees given a phylogenetic network, with its topology, divergence/migration times, population mutation rates and inheritance probabilities.","The divergence/migration times are in units of expected number of mutations per site, and population mutation rates are in units of population mutation rate per site.","Based on the MSNC, and by integrating out all possible gene trees, Zhu et al.","[137] developed an algorithm to compute the probability of a bi-allelic genetic marker given a phylogenetic network."],["Maximum Likelihood Inference of Phylogenetic Networks","Phylogenetic networks are more complicated than a tree with some reticulation edges.","The gene tree topology with highest mass probability may not be one of the backbone trees of the network with 4 or more taxa [138].","Also, not all networks can be obtained by simply adding edges between the original edges of a tree [33].","Therefore inferring phylogenetic networks is not a trivial extension of methods to infer species trees.","Most phylogenetic network methods [125], [127], [120], [121], [133], [137], [136], [128] sample from whole-network space rather than simply adding reticulations to a backbone tree.","As such methods walk the space from one phylogenetic network to another, the point estimate or posterior distribution of networks is not tree-based, does not return or imply a backbone tree, and can include networks which cannot be described by merely adding reticulate branches between tree branches."],["Inference","Sequential inference was initially developed to estimate species trees under the multispecies coalescent [78], [69], and in recent years has been extended to species networks [129], [127], [128], [104], [120].","These methods follow a two-step approach, where the first step is to estimate gene trees from multiple sequence alignments.","The second step is to estimate a species tree or network from the distribution of estimated gene tree topologies.","As described above and illustrated in Fig.","REF , two key requirements have facilitated the development of several methods for phylogenetic inference from multi-locus data, including those that follow the two-step approach.","One key requirement for current methods is that the segments are the result of speciation and not gene duplication, that is, sequences from different individuals and species are orthologs and not paralogs.","Meeting this requirement ensures that the nodes in each gene tree represent coalescent events and can be fit to a coalescent model within each species network branch.","A model which accounts for gene duplication and loss in addition to coalescent events has been developed to reconcile gene family trees with a fixed species tree [98], [132].","The most recent implementation of this model can also use it to estimate the species tree [21], but this model has yet to be extended to work with species networks.","A second key requirement is that the evolutionary history of the locus can be accurately modeled using a single tree.","Recombination or gene fusion should not have occurred within a locus, otherwise a gene network would be required to model that locus, breaking the MSNC model of gene trees within a species network [130].","Because of this requirement, multi-locus methods should be used with short contiguous sequences.","The results of a previous study on mammal phylogenetics suggest that individual exons are an appropriate target sequence [101].","Under these two key requirements, each gene tree is considered to be a valid and independent sample from an underlying distribution of gene trees conditional on some unobserved species network.","Of course this assumption in sequential inference is violated as the gene trees are only estimates.","Particular methods may be more or less sensitive to gene tree estimation errors.","For species trees, methods which infer unrooted species trees (e.g.","ASTRAL [78]) appear to be more robust relative to methods which infer rooted species trees (e.g.","MP-EST [69]).","This is because unrooted methods take unrooted gene trees as input and do not rely on correct rooting of the gene trees [103].","An estimate $\\hat{g}$ of the true gene tree is typically made using phylogenetic likelihood (see Section 2).","The phylogenetic likelihood of the sequence alignments can be combined sequentially or simultaneously with the MSNC probability densities of the gene trees to estimate a species network from sequence data.","Given gene trees where each node represents a coalescent event, the probability densities and masses of those gene trees given a species network can be calculated [127].","This can be based on the topologies and node heights of the gene trees, or based on the topologies alone (see Section REF ).","Maximum likelihood (ML) methods seek a phylogenetic network (along with its parameters) that maximizes some likelihood function.","In a coalescent context, these methods search for the species network which maximizes the likelihood of observing a sample of gene trees given the proposed species network.","The sample of gene trees can include branch lengths, in which case the likelihood is derived from the time intervals between successive coalescent events [127].","In the absence of branch lengths, the likelihood is derived from the probability mass of each gene tree topology [126], [127].","This probability is marginalized over every coalescent history $h$ , which is all the ways for a gene tree to follow the reticulate branching of the network: $P(g|\\Psi ,\\Gamma ,\\theta ) = \\sum _{h \\in H_\\Psi (g)}P(h|\\Psi ,\\Gamma ,\\theta )$ and the ML species network is therefore: $\\hat{\\Psi } = \\operatornamewithlimits{argmax}_\\Psi {\\prod _{g \\in {{G}}}{P(g|\\Psi ,\\Gamma ,\\theta )}}.$ ML inference of species networks has been implemented as the InferNetwork_ML command in PhyloNet [127], which identifies the ML species network up to a maximum number of reticulations.","Similar to our discussion of the MDC criterion above, absent any explicit stopping criterion or a penalty term in the likelihood function, obtaining an ML estimate according to Eq.","(REF ) can result in overly complex phylogenetic networks since adding more reticulations often improves the likelihood of the resulting network.","Therefore, it is important to parameterize the search by the number of reticulations sought, $m$ , and solve $\\hat{\\Psi }(m) = \\operatornamewithlimits{argmax}_{\\Psi (m)}{\\prod _{g \\in {{G}}}{P(g|\\Psi (m),\\Gamma ,\\theta )}},$ where the value $m$ is experimented with by observing the improvement in the likelihood for varying values of $m$ (for example, maximum likelihood inference of phylogenetic networks in PhyloNet implements information criteria, such as AIC and BIC, for this purpose [127]).","The computational cost of the likelihood calculation increases with larger species networks and gene trees.","Not only does this increase the number of branches and coalescent times, but as more reticulations are added many more possible coalescent histories exist to be summed over.","Even with one reticulation edge attached onto a tree, the difficulty of the problem is exponential to tree cases.","The computational complexity of the likelihood calculation is highly related to the size of the set of all coalescent histories of a gene tree conciliated in a network.","Zhu et al.","[136] proposed an algorithm to compute the number of coalescent histories of a gene tree for a network, and demonstrated that the number can grow exponentially after adding merely one reticulation edge to a species tree.","Figure: Running time of computing the likelihood of a phylogenetic network given gene tree topologies.","150 1-reticulation phylogenetic networks with 5 species and 4 individuals per species was used, and the data consisted of 10,000 bi-allelic markers.","The networks varied in terms of the diameter of a reticulation node (the number of edges on the cycle in the underlyingundirected graph) and the number of taxa (leaves) under the reticulation nodes.To show how running time of likelihood computation varies in a network with a single reticulation, we generated 150 random 1-reticulation networks with 5 taxa, then simulated 10,000 bi-allelic markers with 4 individuals per species.","When a reticulation node exists in a phylogenetic network, this will induce a cycle in the unrooted equivalent of the acyclic rooted network.","The “diameter” is the length of that cycle, and we ran the likelihood computation in [137] and summarized the maximum running time according to values of diameter and number of taxa under reticulation.","Fig.","(REF ) shows that the complexity of the likelihood computation is highly related to the structure of the network.","The running time in the worst case is hundreds of times slower than that of the best case.","A faster way to estimate a species network is to calculate a pseudolikelihood instead of a full likelihood.","The InferNetwork_MPL command in PhyloNet implements a maximum pseudo-likelihood (MPL) method for species networks.","This method is based on rooted triples, which is akin to the MP-EST method for species tree inference [128], [69].","Unlike phylogenetic trees, a given phylogenetic network is not necessarily uniquely distinguished by its induced set of rooted triples.","Therefore this method cannot distinguish the correct network when other networks induce the same sets of rooted triples [128].","However, it is much more scalable than ML methods in terms of the number of taxa [46].","Another MPL method, SNaQ, is available as part of the PhyloNetworks software [104], [105].","SNaQ is based on unrooted quartets, akin to the ASTRAL method for species tree inference [78].","It is even more scalable than InferNetwork_MPL [46], but can only infer level-1 networks (Fig.","REF ).","Figure: Level-1 network definition.","Reticulation nodes induce cycles in the (undirected graphs underlying the) phylogenetic networks.","The edges of the cycles are highlighted with red lines.","(a) A level-1 network is one where no edge of the network is shared by two or more cycles.","(b) A non-level-1 network is one where at least one edge is shared by at least two cycles (the shared edge in this case is the one inside the blue circle)."],["Bayesian Inference of Phylogenetic Networks","Maximum likelihood estimation of phylogenetic networks, as described in the previous section, has three main limitations: As discussed, without penalizing the complexity of the phylogenetic network, the ML estimate could be an overly complex network with many false reticulations.","The inference results in a single point estimate that does not allow for assessing confidence in the inferred network.","The formulation does not allow for making use of the sequence data directly, but is based on gene tree estimates.","One way of addressing these limitations is to adopt Bayesian inference where an estimate of the posterior distribution on networks is sought directly from the sequence data of the individual loci, and where the prior distribution on phylogenetic networks accounts for model complexity in a principled manner.","Before we describe the work on Bayesian inference, it is important to note that while maximum likelihood estimation is not satisfactory, we cannot say that Bayesian estimation is without challenges.","Such methods like [121], [133], [137] are based on reversible-jump MCMC [38] with varying numbers of parameters.","Mixing problems arise when they involve dimension changing moves: adding a reticulation and removing a reticulation.","This is because while walking over the space of phylogenetic networks, these methods jump between probability spaces of different models.","Therefore moves should be carefully designed to account for mixing issues."],["Probability Distributions Over Species Networks","It is useful to define probability distributions over species trees or networks without reference to sequence data or gene trees.","Among other uses, these probability distributions can be applied as prior distributions in Bayesian inference.","The two most common types of prior distributions used for species trees are birth-death tree priors and compound priors.","Both types have been extended to create probability distributions over species networks.","As their name implies, birth-death tree priors combine a rate of birth with a rate of death.","These are the rates at which one lineage splits into two, and one lineage ceases to exist respectively [36].","In the context of species trees these rates are more informatively called speciation and extinction.","When the extinction rate is set to zero, this is known as a Yule prior [131].","Birth-death tree priors have been extended to support incomplete sampling in the present, and sampling-through-time [108].","A birth-death prior for species networks has been developed, called the birth-hybridization prior.","This prior combines a rate of birth (or speciation) with a rate of hybridization, which is the rate at which two lineages merge into one.","This model does not include a rate of extinction [133].","All birth-death tree priors induce a uniform probability distribution over ranked tree topologies, regardless of the rates of speciation and extinction.","This means that birth-death priors favor symmetric over asymmetric trees, as symmetric trees have more possible ranked histories.","Empirical trees generally have more asymmetric shapes than predicted by birth-death models [43].","For the birth-hybridization prior the probability distribution over network topologies is not invariant to the hybridization rate, which when set to zero reduces to the Yule model, and any topology containing a reticulation will have zero probability.","All birth-death priors are generative, as is the birth-hybridization prior.","This means that not only can these distributions be used as priors for Bayesian inference, but they can be used to simulate trees and networks.","These simulated distributions can then be used for ABC inference, which is used for models which are difficult to implement using MCMC.","They can also be used for posterior predictive checks, which is an absolute measure of model goodness of fit [35].","While birth-death priors induce a probability distribution over topologies and branch lengths, compound tree and network priors are constructed from separate distributions on both.","Typically compound tree priors combine a uniform distribution over unranked tree topologies, favoring more asymmetric trees.","Empirical trees generally have more symmetric shapes than predicted by this distribution [43].","Then a continuous distribution such as gamma can be applied to branch lengths or node heights.","Compound priors are used for network inference by adding a third distribution describing the number of reticulations [121].","A Poisson distribution is a natural fit for this parameter as it describes a probability on non-negative integers.","The probability distribution for each network topology can still be uniform for all networks given $k$ reticulations.","Unlike birth-death priors, compound priors are not generative, so it is not straightforward to simulate trees or networks from those distributions.","The most obvious way to simulate such trees and networks would be running an existing Markov Chain Monte Carlo sampler without any data, and subsampling states from the chain at a low enough frequency to ensure independence between samples."],["Sampling the Posterior Distribution","The ML species network with $k + 1$ reticulations will always have a higher likelihood than the ML network with $k$ reticulations.","For this reason, some threshold of significance must be applied to estimate the number of reticulations.","This threshold may be arbitrary or it may be theoretically based, for example the Akaike information criterion (AIC) and Bayesian information criterion (BIC) measures of relative fit [10].","In contrast, Bayesian methods of species network inference are able to naturally model the probability distribution over species networks including the number of reticulations by using a prior (see Section REF ).","In a Bayesian model, the posterior probability of a species network $P(\\Psi )$ is proportional to the likelihood of the gene trees $P(G|\\Psi ,\\Gamma ,\\theta )$ , multiplied by the prior on the network and other parameters of the model $P(\\Psi ,\\Gamma ,\\theta )$ , and marginalized over all possible values of $\\Gamma $ and $\\theta $ : $P(\\Psi ) \\propto \\iint P(G|\\Psi ,\\theta ) \\cdot P(\\Psi ,\\Gamma ,\\theta ) \\mathop {}\\!\\mathrm {d}\\Gamma \\mathop {}\\!\\mathrm {d}\\theta .$ When a decaying prior is used on the number of reticulations or on the rate of hybridization, the prior probability of species networks with large numbers of reticulations will be very low, and so will the posterior probability (Fig.","REF ).","Figure: Bayesian inference of the number of reticulations.","In this example the topology of the network Ψ\\Psi is fixed, the true number of reticulations is 1, and the likelihood is calculated for a topology with no reticulations, the true reticulation, and additional reticulations, with maximum likelihood branch lengths.","The posterior probability was normalized to sum to 1, although as this is not integrated over branch lengths, the typical Bayesian posterior probability might be a bit different.Bayesian methods for phylogenetic inference typically use the Metropolis-Hastings Markov Chain Monte Carlo (MCMC) algorithm to estimate the posterior distribution of trees or networks.","MCMC is a random walk where each step depends on the previous state, and it is flexible enough to be used for implementing extremely complex models such as species network inference with relative ease.","Bayesian estimation of species networks from gene trees is implemented in the PhyloNet command MCMC_GT.","The posterior probability of a species network is equal to the integral in equation REF multiplied by a normalizing constant $Z$ known as the marginal likelihood.","In the case of sequential multilocus inference, this constant is equal to $Z = P(G)^{-1}$ The marginal likelihood is usually intractable to calculate, but MCMC sidesteps this calculation by sampling topologies and other parameters with frequencies proportional to their probability mass or density.","The posterior probability of a species network $\\Psi $ can therefore be approximated as the proportion of steps in the MCMC chain where the network topology $\\Psi _i$ at the end of the step $i$ is equal to $\\Psi $ .","The value of any particular parameter, for example an inheritance probability $\\gamma $ for a given reticulation node $v$ , can be estimated by averaging its value over the set of $X$ steps where the state includes that parameter.","In this case it is averaged over the states where the species network includes that node, i.e.","the set $X = \\lbrace i: v \\in \\Psi _i\\rbrace $ : $E(\\gamma ) = \\frac{1}{|X|} \\sum ^i_{X} \\gamma _i$ Bayesian inference has also enabled the inference of species trees and networks directly from multilocus sequence data.","Instead of first estimating individual gene trees from multiple sequence alignments, these methods jointly infer the gene trees and species network using an application of Bayes' rule: $P(G,\\Psi ) \\propto \\iint P(D|G) \\cdot P(G|\\Psi ,\\Gamma ,\\theta ) \\cdot P(\\Psi ,\\Gamma ,\\theta ) \\mathop {}\\!\\mathrm {d}\\Gamma \\mathop {}\\!\\mathrm {d}\\theta .$ Here $P(D|G)$ is the likelihood of the data over all gene trees.","In practice this is the sum of phylogenetic likelihoods $\\sum _i P(d_i|g_i)$ for every sequence alignment $d$ and associated gene tree $g$ .","As with sequential Bayesian inference of species trees and networks, the use of MCMC avoids the calculation of the marginal likelihood, which for joint inference can be expressed as $Z = P(D)^{-1}$ .","Joint Bayesian inference was first developed for species trees, and now has several popular implementations including StarBEAST2 [86] and BPP [96].","Joint Bayesian inference of species networks has been implemented independently as the PhyloNet command MCMC_SEQ, and as the BEAST2 package “SpeciesNetwork” [121], [133].","These two methods are broadly similar in their model and implementation, with a few notable differences.","MCMC_SEQ uses a compound prior on the species network, whereas SpeciesNetwork has a birth-hybridization prior (see Section REF ).","SpeciesNetwork is able to use any of the protein and nucleotide substitution models available in BEAST2.","MCMC_SEQ can be used with any nested GTR model but with fixed rates and base frequencies.","So the rates (e.g.","the transition/transversion ratio for HKY) must be estimated before running the analysis, or Jukes-Cantor is used where all rates and base frequencies are equal."],["Inference Under MSC vs. MSNC When Hybridization Is Present","We simulated 128 loci on the phylogenetic network of Fig.","REF (a).","The program ms [48] was used to simulate 128 gene trees on the network, and each gene tree was used to simulate a sequence alignment of 500 sites using the program Seq-gen [95] under the GTR model and $\\theta = 0.036$ for the population mutation rate.","The exact command used was: seq-gen -mgtr -s$0.018$ -f$0.2112,0.2888,0.2896,0.2104$ -r$0.2173,\\\\0.9798,0.2575,0.1038,1,0.2070$ -l500 We then ran both StarBEAST and MCMC_SEQ, as inference methods under the MSC and MSNC models, respectively, for $6\\times 10^7$ iterations each.","The results are shown in Fig.","REF .","Figure: Inference under the MSC and MSNC when the evolutionary history involves hybridization.","(a) The true phylogenetic network with the shown inheritance probabilities and branch lengths (in coalescent units).","(b) The MPP (maximum a posteriori probability) species tree estimated under the MSC by StarBEAST (frequency of 94% in the 95%95\\% credible set) with the average divergence times.","(c) The MPP phylogenetic network along with the inheritance probabilities estimated under the MSNC by MCMC_SEQ (the only network topology in the 95%95\\% credible set).","The scale bar of divergence times represents 1 coalescent unit for (a-c).","(d) The coalescent times of the MRCAs of (C,G), (A,Q), (A,C), (Q,R) from co-estimated gene trees inferred by StarBEAST (green) and MCMC_SEQ (blue).A few observations are in order.","First, while StarBEAST is not designed to deal with hybridization, it inferred the tree topology (Fig.","REF (b)) that is obtainable by removing the two hybridization events (the two arrows) from the true phylogenetic network (the backbone tree).","Second, MCMC_SEQ identified the true phylogenetic network as the one with the highest posterior (Fig.","REF (c)).","Furthermore, the estimated inheritance probabilities are very close to the true ones.","Third, and most interestingly, since StarBEAST does not account for hybridization, it accounts for all heterogeneity across loci as being caused by incomplete lineage sorting (ILS) by underestimating all branch lengths (that is, “squashing\" the divergence times so as to explain the heterogeneity by ILS).","Indeed, Fig.","REF (d) shows that the minimum coalescent times of the co-estimated gene trees by StarBEAST force the divergence times in the inferred species tree to be very low.","MCMC_SEQ, on the other hand, accurately estimates the branch lengths of the inferred phylogenetic network since networks differentiate between divergence and hybridization times.","For example, Fig.","REF (d) shows that the coalescent times of clade (C,G) across all co-estimated gene trees is a continuum with a minimum value around 2, which defines the divergence time of these two taxa in the phylogenetic network.","MCMC_SEQ clearly identifies two groups of coalescent times for each of the two clades (A,C) and (Q,R): The lower group of coalescent times correspond to hybridization, while the upper group of coalescent times correspond to the coalescences above the respective MRCAs of the clades.","We also note that the minimum value of coalescent times corresponding to (Q,R) is larger than that corresponding to (A,Q), which correctly reflects the fact that hybridization from R to Q happened before hybridization from C to A, as indicated in the true phylogenetic network.","Finally, for clade (A,Q), three groups of coalescence times are identified by MCMC_SEQ, which makes sense since there are three common ancestors of A and Q in the network: at the MRCA of (A,Q) in the case of no hybridization involving either of the two taxa, at the MRCA of (A,Q,L,R) in the case of the hybridization involving Q, and at the root of the network in the case of the hybridization involving A.","More thorough analysis and comparison of inferences under the MSC and MSNC can be found in [121].","These results illustrate the power of using a phylogenetic network inference method when hybridization is involved.","In particular, if hybridization had occurred, and the practitioner did not suspect it and ran StarBEAST instead, they would get wrong inferences.","In this case, the errors all have to do with the divergence time estimates.","However, the topology of the inferred tree could be wrong as well, depending on the hybridization scenarios."],["Phylogenetic Invariants Methods","The focus of this chapter up to this point has largely been on the MSC and MSNC models.","A parallel effort has been led to detect reticulate evolution by using the notion of phylogenetic invariants [12], [64].","Phylogenetic invariants are polynomial relationships satisfied by frequencies of site patterns at the taxa labeling the leaves of a phylogenetic tree (and given a model of sequence evolution).","Invariants that are predictive of particular tree topologies could then be used for inferring the tree topology by focusing on the space of site patterns rather than the space of tree topologies [28].","As Felsenstein wrote in his book, “invariants are worth attention, not for what they do for us now, but what they might lead to in the future.\"","With the availability of whole-genome data and, consequently, the ability to obtain better estimates of site frequencies, the future is here.","Indeed, methods like SVDQuartets [14] use phylogenetic invariants to estimate species trees under the MSC model.","A detailed discussion of phylogenetic invariants in general is beyond the scope of this manuscript.","Interested readers should consult the excellent exposition on the subject in Felsenstein's seminal book (Chapter 22 in [28]).","In this section, we briefly review phylogenetic invariants-based methods for detecting reticulation, starting with the most commonly used one, known as the $D$ -statistic or the “ABBA-BABA\" test.","The $D$ -Statistic [39] is a widely known and frequently applied statistical test for inferring reticulate evolution events.","The power of the test to infer reticulate evolution derives from the likelihood calculations of the MSNC.","Despite this, the test itself is simple to calculate and formalize.","The $D$ -Statistic is given by $\\frac{N_{ABBA} - N_{BABA}}{N_{ABBA} + N_{BABA}}$ To calculate these quantities, we are given as input the four taxon tree including outgroup of Fig.","REF and a sequence alignment of the genomes of P1, P2, P3, and O.","Given this alignment, $N_{ABBA}$ is calculated as the number of occurrences of single sites in the alignment where P1 and O have the same letter and P2 and P3 have the same letter, but these two letters are not the same i.e.","CTTC or GCCG.","Similarly, $N_{BABA}$ can be calculated as the number of occurrences in the alignment where the letters of $P1 = P3$ and $P2 = O$ with no other equalities between letters.","Upon calculating the $D$ -Statistic, a significant deviation away from a value of 0 gives evidence for reticulate evolution.","As shown in Fig.","REF , a strong positive value implies introgression between P2 and P3 while a strong negative value implies introgression between P1 and P3.","No such conclusions can be made from a D value very close to 0.","The crux of the theory behind the $D$ -Statistic lies in the expectation of the probabilities of discordant gene trees given the overall phylogeny of Fig.","REF .","If we remove the two reticulation events in Fig.","REF we end up with a species tree $\\Psi $ .","Figure: The four-taxon tree topology used for the DD-Statistic.","Significant deviations away from a value of 0 of the DD-statistic (Eq.","()) support introgression between P3 and either P1 or P2.","As shown, a significant positive value supports introgression between P2 and P3.","A significant negative value supports introgression between P1 and P3.Given $\\Psi $ , the two gene trees whose topologies disagree with that of the species tree are equally probable under the MSC.","Figure: The three scenarios of probabilities of the two gene trees that are discordant with the species tree in the case of a single reticulation event between P3 and one of the othertwo (in-group) species.","If the evolutionary history of the species is a tree (Ψ)(\\Psi ), the two discordant gene trees are equally probable.","However, if the evolutionary history of the speciesis non-treelike, as given by phylogenetic networks Ψ 1 \\Psi _1 and Ψ 2 \\Psi _2, then the probabilities of the two discordant gene trees are unequal in different ways.On the other hand, when a reticulation between P1 and P3 occurs, this results in an increase in the probability of the discordant gene tree that groups P1 and P3 as sister taxa, as compared to the other discordant gene tree.","Similarly, when a reticulation between P2 and P3 occurs, this results in an increase in the probability of the discordant gene tree that groups P2 and P3 as sister taxa.","These three scenarios are illustrated in Fig.","REF .","Assuming an infinite sites model of sequence evolution, the frequencies of gene trees ((P1,(P2,P3)),O) and (((P1,P3),P2),O) directly correlate with the values $N_{ABBA}$ and $N_{BABA}$ , respectively, explaining the rationale behind Eq.","(REF ).","To apply the $D$ -statistic, frequencies of the $ABBA$ and $BABA$ site patterns are counted across an alignment of four genomes, the value of Eq.","(REF ) is calculated, and deviation from 0 is assessed for statistical significance.","A significant deviation is taken as evidence of introgression.","Since the introduction of the $D$ -Statistic, work has been done to extend this framework.","Recently, the software package HyDe [5] was introduced with several extensions including handling multiple individuals from four populations as well as identifying individual hybrids in a population based on the method of [62].","In HyDe, higher numbers of individuals are handled through calculating statistics on all permutations of quartets of the individuals.","Another recent extension to move the $D$ -Statistic beyond four taxa is the DFOIL framework introduced by [91].","In it, we see the same derivation used in the $D$ -Statistic on a particular five-taxon tree.","This derivation includes isolating gene trees whose probabilities go from equal to unequal when going from the tree case to the network case as well as converting these gene trees to corresponding site patterns to count in an alignment.","Finally, Elworth et al.","recently devised a heuristic, $D_{GEN}$ , for automatically deriving phylogenetic invariants for detecting hybridization in more general cases than the $D$ -Statistic and DFOIL can handle [25].","The rationale behind the approach of Elworth et al.","is that invariants could be derived by computing the probabilities of gene trees under a given species tree (e.g., using the method of [19]) and then computing the probabilities of the same trees under the same species tree with any reticulation scenarios added to it (using the method of [126]), and contrasting the two to identify sets of gene trees whose equal probabilities under the tree model get violated under the network model.","The $D$ -Statistic is very simple to implement and understand, and it can be calculated on 4-genome alignments very efficiently, making it an appealing choice of a test for detecting introgression.","Indeed, applications of the $D$ -Statistic are widespread in the literature, reporting on introgression in ancient hominids [39], [22], butterflies [114], bears [63], and sparrows [23], just to name a few.","However, it is important to note here that the derivation of the $D$ -Statistic (and its extensions) relies on many assumptions that can easily be violated in practice.","One major cause of such a violation is that the mathematics behind the $D$ -Statistic relies on the coalescent which makes many simplifying assumptions about the evolutionary model and processes taking place.","Of course, this shortcoming applies as well to all phylogenetic inference methods that employ the MSC or MSNC models.","A second cause of such a violation is that in practice, more than a single reticulation could have taken place and ignoring those could result in erroneous inferences [25].","A third violation stems from the way the $D$ -Statistic is applied.We especially thank David Morrison for requesting that we highlight this issue.","In propositional logic, the statement “If $p$ , then $q$ \" and its converse “If $q$ , then $p$ \" are logically not equivalent.","That is, if one is true, it is not necessarily the case the other is.","Looking back at Fig.","REF , the statement illustrated by the figure is: If there is no reticulation (i.e., the species phylogeny is a tree), then the probabilities of the two discordant trees are equal.","The converse (if the probabilities of the two discordant trees are equal, then the species phylogeny is a tree) does not follow logically.","However, this is how the test is used in practice.","In all fairness, though, this logical fallacy is commonplace in inferences in biology, including when inferring species trees and networks under the MSC and MSNC, respectively.","The fallacy is always dealt with by resorting to the “simplest possible explanation\" argument.","For example, why various scenarios could have given rise to equal frequencies of the frequencies of the $ABBA$ and $BABA$ site patterns, the species tree scenario is considered the simplest such possible explanation and is invoked as such.","Last but not least, Peter [92] recently provided a review and elegant connections between the $D$ -Statistic and a family of statistics known as the $F$ -statistics."],["Phylogenetic Networks in the Population Genetics Community","The population genetics community has long adopted rooted, directed acyclic graphs as a model of evolutionary histories, typically of individuals within a single species.","Ancestral recombination graphs, or ARGs, were introduced [49], [40] to model the evolutionary history of a set of genomic sequences in terms of the coalescence and recombination events that occurred since their most recent common ancestor.","Statistical methods for inference of ARGs from genome-wide data have also been developed [97].","Gusfield's recent book [42] discusses algorithmic and combinatorial aspects of ARGs.","However, while ARGs take the shape of a phylogenetic network as defined above, they are aimed at modeling recombination and methods for their inference are generally not applicable to hybridization detection.","Efforts in the population genetics community that are aimed at modeling admixture and gene flow are more relevant to hybridization detection.","Here we discuss one of the most popular methods in this domain, namely TreeMix [93].","In population genetics, the counterparts to species trees and phylogenetic networks are population trees and admixture graphs, respectively.","The difference between these models boils down to what labels their leaves: If the leaves are labeled by different species, then the models are called species trees/networks, and if the leaves are labeled by different sub-populations of the same species, then the models are called population trees and admixture graphs.","Of course it is not always easy to identify whether a species or sub-populations have been delimited, and hence what particular tree/network should be called in that case, as species and populations may exist on a continuum [18]."],["TreeMix","TreeMix [93] models the evolution of a set of SNPs where the input data consists of allele frequencies of these SNPs in a set of populations whose evolutionary history is given by a population tree (in the case of no migration) or an admixture graph (in the case where migration is included).","The basis of the model used in TreeMix is in the notion of modeling drift over time as a diffusion process, where an original allele frequency of $x_1$ of a given SNP undergoes drift by an amount $c$ to give rise to a new allele frequency $x_2$ [11], [85], [17], as given by $x_2 = x_1 + N(0,c \\cdot x_1[1-x_1]).$ It is worth noting here that, as pointed out in [93], $c ~= t/2N_e$ for drift over small time scales where the time scale is on the same order of the effective population size [85].","When there are multiple populations under the effects of drift that evolved down a tree, the drift processes become linked and can no longer be described with independent Gaussian additions.","This process is modeled with a covariance matrix derived from the amounts of drift occurring along the branches of the evolutionary tree.","Finally, to incorporate reticulate evolution into the model one only needs to alter this covariance matrix based on the rate of gene flow along reticulate edges in the admixture graph.","In its current implementation, the authors of TreeMix assume the evolutionary history of the sampled, extant populations is very close to a tree-like structure.","Based on this assumption, the search for a maximum likelihood admixture graph proceeds by first estimating a rooted tree, and then adding migration events one at a time until they are no longer statistically significant (however, as the authors point out, they “prefer to stop adding migration events well before this point so that the result graph remains interpretable.\").","Clearly, adding additional edges connecting the edges of a tree in this way will infer a tree-based network, which is a more limited class of networks compared with phylogenetic networks [33]."],["Data, Methods, and Software","Given the interest in reticulate evolution from both theoreticians and empirical researchers, it is perhaps unsurprising that software to infer hybridization has proliferated in recent years.","Such methods have been developed for a variety of data types, including multilocus data, SNP matrices, and whole genomes (Table ).","Some of these methods are able to infer a phylogenetic network, whereas others infer introgression between species tree lineages.","With the exception of TreeMix and MixMapper, methods which infer networks are not constructed around a backbone tree, and so do not assume that tree-like evolution is the dominant process.","Regardless of the input and output, most of these methods allow for ILS in addition to hybridization (Table ), which is necessary to infer phylogenetic networks representing reticulate evolution in biological systems where ILS is a possibility.","Likelihood (including Bayesian) methods incorporate the possibility or effect of ILS into the likelihood function.","Maximum parsimony methods that minimize deep coalescences, for example InferNetwork_MP, essentially attempt to infer the tree or network that minimizes the quantity of ILS, but do not necessarily eliminate all genetic discordance.","Methods which do not allow for ILS will instead infer phylogenetic networks representing conflicting signals [9].","Reticulation is one such conflicting signal, but so is ILS, so reticulate branches in these networks should not be blindly interpreted as necessarily representing introgression or hybridization.","Common methods to infer hybridization Table: NO_CAPTION"],["Limitations","The biggest limitation of methods to infer introgression and hybridization, including species network methods, is scalability.","Methods which infer a species network directly from multilocus sequences have only been used with a handful of taxa, and less than 200 loci.","A systematic study of the species tree method StarBEAST found that the number of loci used has a power law relationship with a large exponent with the required computational time, making inference using thousands of loci intractable [87].","Although no systematic study of computational performance has been conducted for equivalent species network methods such as MCMC_SEQ, anecdotally they suffer from similar scaling issues.","Methods which scale better than direct multilocus inference have been developed, but they are no silver bullet.","Species networks can be estimated directly from unlinked biallelic markers by integrating over all possible gene trees for each marker, which avoids having to sequentially or jointly estimate gene trees.","Biallelic methods make the inference of species trees and networks from thousands of markers possible, at the cost of using less informative markers.","Pseudolikelihood inference has been developed for both biallelic and multilocus methods [136], [128].","This reduces the computational cost of computing the likelihood of a species network as the number of taxa increases, and enabled the reanalysis of an empirical data set with 1070 genes from 23 species [128].","The ABBA-BABA test and similar phylogenetic invariant methods are capable of analyzing an enormous depth of data (whole genomes), but can be limited in taxonomic breadth based on hard limits of four or five taxa for the D-Statistic and $D_{FOIL}$ , respectively, or by computational requirements for the case of $D_{GEN}$ (Table ).","In addition, the D-Statistic and $D_{FOIL}$ are limited to testing a specific hypothesis for introgression given a fixed species tree topology of a specific shape.","This can be understood as a trade off, where the flexibility of species network methods is sacrificed for the ability to use more data.","Beyond scalability, another present limitation is visualizing or summarizing posterior or bootstrap distributions of networks.","Methods have been developed to visualize whole distributions of trees, or summarize a distribution as a single tree.","Equivalent tools for networks are underdeveloped, leaving researchers to report the topology or set of topologies with the highest posterior or bootstrap support."],["Conclusions and Future Directions","Great strides have been made over the past decade in the inference of evolutionary histories in the presence of hybridization and other processes, most notably incomplete lineage sorting.","Species networks can now be inferred directly from species-level data which do not assume any kind of backbone tree, and instead put reticulate evolution on an equal basis with speciation.","To some extent the development of species network methods have recapitulated the development of species tree methods, starting with maximum parsimony and transitioning to likelihood methods, both maximum likelihood and Bayesian.","To improve computational performance and enable the analysis of large data sets, pseudo-likelihood species network methods have been developed, inspired by similar species tree methods.","Phylogenetic invariant methods such as the ABBA-BABA test are able to test for reticulate evolution across whole genomes, uncovering chromosomal inversions and other features associated with hybridization and introgression.","Last but not least, the population genetics community has long been interested in and developing methods for phylogenetic networks mainly to model the evolution of sub-populations in the presence of admixture and gene flow.","In this chapter, we surveyed the recent computational developments in the field and listed computer software programs that enable reticulate evolutionary analyses for the study of hybridization and introgression, and generally to infer more accurate evolutionary histories of genes and species.","Empirical biologists feel constrained by the computational performance of existing species network methods.","For species trees, phylogenetic invariant methods can be combined with quartet reconciliation to infer large species trees from genomic data, as in SVDquartets [14].","For networks, phylogenetic invariant methods to identify the true network with a limited number of edges need to be developed, as do methods to reconcile the resulting subnets.","Even for species trees, Bayesian methods have practical limitations in terms of the amount of data they can be used with.","Bayesian methods for trees and networks, with few exceptions, have been built on Markov chain Monte Carlo (MCMC).","This technique is inherently serial and hence unsuited to modern workstations, which contain many CPU and GPU cores working in parallel.","It is important to continue to explore other Bayesian algorithms which work in parallel such as sequential Monte Carlo [7], or algorithms which are orders of magnitude faster than MCMC such as variational Bayes [118].","Phylogenetic methods for species tree inference have a huge head start on methods for species network inference.","Not only is the problem of species network inference much more complicated, but species tree methods have been in development for much longer.","For example, MDC for species trees was first described in 1997, and extended to phylogenetic networks 14 years later [71], [130].","In this light the progress made is remarkable.","However, as evolutionary biology is moving towards data sets containing whole genomes for hundreds or even thousands of taxa, methods developers must focus on improving the scalability of their methods without sacrificing accuracy so that the full potential of this data may be realized.","While preliminary studies exist of the performance of the different methods for phylogenetic network inference [58], more thorough studies are needed to assess the accuracy as well as computational requirements of the different methods.","Last but not least, it is important to highlight that all the development described above excludes processes such as gene duplication and loss, and so may be susceptible to errors and artifacts which can be present in data such as hidden paralogy.","Furthermore, the multispecies network coalescent already has its own population-genetic assumptions, almost all of which are not necessarily realistic for analyses in practice.","Accounting for these is a major next step (though it is important to point out that these have not been fully explored in the context of species tree inference either), but the mathematical complexity will most likely add, extensively, to the computational complexity of the inference step.","Luay Nakhleh started working on phylogenetic networks in 2002 in a close collaboration with Tandy Warnow, Bernard M.E.","Moret, and C. Randal Linder.","At the time, their focus was on phylogenetic networks in terms of displaying trees.","This focus led to work on inference of smallest phylogenetic networks that display a given set of trees, as well as on comparing networks in terms of their displayed trees.","While the approaches pursued at the time were basic, they were foundational in terms of pursuing more sophisticated models and approaches by Nakhleh and his group.","Therefore, we would like to acknowledge the role that Bernard played in the early days of (explicit) phylogenetic networks.","The authors would also like to acknowledge James Mallet, Craig Moritz, David Morrison, and Mike Steel for extensive discussions and detailed comments that helped us significantly improve this chapter.","The authors thank Matthew Hahn and Kelley Harris for their discussion of the definition of phylogenetic invariant methods.","This work was partially supported by NSF grants DBI-1355998, CCF-1302179, CCF-1514177, CCF-1800723, and DMS-1547433."]],"string":"[\n [\n \"Advances in Computational Methods for Phylogenetic Networks in the\\n Presence of Hybridization\"\n ],\n [\n \"Abstract Phylogenetic networks extend phylogenetic trees to allow for modeling reticulate evolutionary processes such as hybridization.\",\n \"They take the shape of a rooted, directed, acyclic graph, and when parameterized with evolutionary parameters, such as divergence times and population sizes, they form a generative process of molecular sequence evolution.\",\n \"Early work on computational methods for phylogenetic network inference focused exclusively on reticulations and sought networks with the fewest number of reticulations to fit the data.\",\n \"As processes such as incomplete lineage sorting (ILS) could be at play concurrently with hybridization, work in the last decade has shifted to computational approaches for phylogenetic network inference in the presence of ILS.\",\n \"In such a short period, significant advances have been made on developing and implementing such computational approaches.\",\n \"In particular, parsimony, likelihood, and Bayesian methods have been devised for estimating phylogenetic networks and associated parameters using estimated gene trees as data.\",\n \"Use of those inference methods has been augmented with statistical tests for specific hypotheses of hybridization, like the D-statistic.\",\n \"Most recently, Bayesian approaches for inferring phylogenetic networks directly from sequence data were developed and implemented.\",\n \"In this chapter, we survey such advances and discuss model assumptions as well as methods' strengths and limitations.\",\n \"We also discuss parallel efforts in the population genetics community aimed at inferring similar structures.\",\n \"Finally, we highlight major directions for future research in this area.\"\n ],\n [\n \"Introduction\",\n \"Hybridization is often defined as reproduction between members of genetically distinct populations [4].\",\n \"This process could occur in various spatial contexts, and could have impacts on speciation and differentiation [2], [3], [74], [75], [99], [1], [76].\",\n \"Furthermore, increasing evidence as to the adaptive role of hybridization has been documented, for example, in humans [94], macaques [111], [6], [88], mice [107], [66], butterflies [114], [135], and mosquitoes [32], [119].\",\n \"Hybridization is “generically\\\" used to contain two different processes: hybrid speciation and introgression [31].\",\n \"In the case of hybrid speciation, a new population made of the hybrid individuals forms as a separate and distinct lineage from either of its two parental populations.In this chapter, we do not make a distinction between species, population, or sub-population.\",\n \"The modeling assumptions and algorithmic techniques underlying all the methods we describe here neither require nor make use of such a distinction.\",\n \"Introgression, or introgressive hybridization, on the other hand, describes the incorporation of genetic material into the genome of a population via interbreeding and backcrossing, yet without creating a new population [45].\",\n \"As Harrison and Larson noted [45], introgression is a relative term: alleles at some loci introgress with respect to alleles at other loci within the same genomes.\",\n \"From a genomic perspective, and as the basis for detection of hybridization, the general view is that in the case of hybrid speciation, regions derived from either of the parental ancestries of a hybrid species would be common across the genomes, whereas in the case of introgression, regions derived from introgression would be rare across the genomes [31].\",\n \"Fig.\",\n \"REF illustrates both hybridization scenarios.\",\n \"Figure: Hybrid speciation and introgression.\",\n \"(a) A phylogenetic tree on three taxa, A, B, and C, and a gene tree within itsbranches.\",\n \"Genetic material is inherited from ancestors to descendants and it is expected that loci across the genome would have the shown gene tree.\",\n \"(b) A hybrid speciation scenario depicted by a phylogenetic network,where B is a hybrid population that is distinct from its parental species.\",\n \"Shown within the branches of the network are two gene trees, both of which are assumed to be very commonacross the genome.\",\n \"(c) An introgression scenario.\",\n \"Through hybridization and backcrossing, genetic material from (an ancestor of) C is incorporated into the genomes of individuals in (an ancestor of) B.\",\n \"The introgressedgenetic material would have the gene tree shown in red, and the majority of loci in B's genomes would have the gene tree shown in blue.\",\n \"Incomplete lineage sorting would complicate all threescenarios by giving rise to loci with other possible gene trees and by changing the distribution of the various gene trees.A major caveat to the aforementioned general view is that, along with hybridization, other evolutionary processes could also be at play, which significantly complicates the identification of hybrid species and their parental ancestries.\",\n \"Chief among those processes are incomplete lineage sorting (ILS) and gene duplication and loss.\",\n \"Indeed, various studies have highlighted the importance of accounting for ILS when attempting to detect hybridization based on patterns of gene tree incongruence [114], [77], [127], [32], [16], [94], [119], [135], [90].\",\n \"Furthermore, gene duplication and loss are very common across all branches of the Tree of Life.\",\n \"While the main focus of this chapter is on modeling and inferring hybridization, a discussion of how ILS is accounted for is also provided since recent developments have made great strides in modeling hybridization and ILS simultaneously.\",\n \"While signatures of gene duplication and loss are ubiquitous in genomic data sets, we do not include a discussion of these two processes in this chapter since methods that account for them in the context of phylogenetic networks are currently lacking.\",\n \"When hybridization occurs, the evolutionary history of the set of species is best modeled by a phylogenetic network, which extends the phylogenetic tree model by allowing for “horizontal\\\" edges to denote hybridization and to facilitate modeling bi-parental inheritance of genetic material.\",\n \"Fig.\",\n \"REF shows two phylogenetic networks that model hybrid speciation and introgression.\",\n \"It is very important to note, though, that from the perspective of existing models, both phylogenetic networks are topologically identical.\",\n \"This issue highlights two important issues that must be thought about carefully when interpreting a phylogenetic network.\",\n \"First, neither the phylogenetic network nor the method underlying its inference distinguish between hybrid speciation and introgression.\",\n \"This distinction is a matter of interpretation by the user.\",\n \"For example, the phylogenetic network in Fig.\",\n \"REF (c) could be redrawn, without changing the model or any of its properties, so that the introgression is from (an ancestor of) A to (an ancestor of) B, in which case the “red\\\" gene tree would be expected to appear with much higher frequency than the “blue\\\" gene tree.\",\n \"In other words, the way a phylogenetic network is drawn could convey different messages about the evolutionary history that is not inherent in the model or the inference methods.\",\n \"This issue was importantly highlighted with respect to data analysis in [119] (Figure 7 therein).\",\n \"Second, the phylogenetic network does not by itself encode any specific backbone species tree that introgressions could be interpreted with respect to.\",\n \"This, too, is a matter of interpretation by the user.\",\n \"This is why, for example, Clark and Messer [16] recently argued that “perhaps we should dispense with the tree and acknowledge that these genomes are best described by a network.\\\"\",\n \"Furthermore, recent studies demonstrated the limitations of inferring a species tree “despite hybridization\\\" [106], [138].\",\n \"With the availability of data from multiple genomic regions, and increasingly often from whole genomes, a wide array of methods for inferring species trees, mainly based on the multispecies coalescent (MSC) model [20], have been developed [67], [81], [68].\",\n \"Building on these methods, and often extending them in novel ways, the development of computational methods for inferring phylogenetic networks from genome-wide data has made great strides in recent years.\",\n \"Fig.\",\n \"REF summarizes the general approaches that most phylogenetic network inference methods have followed in terms of the data they utilize, the model they employ, and the inferences they make.\",\n \"Figure: Phylogenetic network inference process and approaches.\",\n \"The process of phylogenetic network inference starts with collecting the genomicdata and identifying the orthology groups of unlinked loci.\",\n \"Multiple sequence alignments or single bi-allelic markers corresponding to the unlinked loci arethen obtained; phylogenetic network inference methods use one of these two types of data.\",\n \"In two-step inference methods, gene trees are first estimated for theindividual loci from the sequence alignment data, and these gene tree estimates are used as the input data for network inference.\",\n \"If incomplete lineage sorting (ILS)is not accounted for, a smallest displaying network of the gene tree estimates is sought.\",\n \"If ILS is accounted for, parsimony inference based on the MDC (minimizingdeep coalescences) criterion, a maximum likelihood estimate (MLE), a maximum a posteriori (MAP) estimate, or samples of the posterior distribution can beobtained.\",\n \"In the direct inference approach, whether based on sequence alignment or bi-allelic marker data, a MAP estimate or samples of the posterior distributioncan be obtained directly from the data.\",\n \"The two parsimony methods consider only the topologies of the gene tree estimates as input (i.e., they ignore gene tree branch lengths) and return as output phylogenetic network topologies.\",\n \"The likelihood and Bayesian methods that take gene tree estimates as input can operate on gene tree topologies alone or gene trees with branch lengths as well.\",\n \"Both methods estimate phylogenetic network topologies along with branch lengths (in coalescent units) and inheritance probabilities.\",\n \"The direct inference methods estimate the phylogenetic network along with its associated parameters.The overarching goal of this chapter is to review the existing methods for inferring phylogenetic networks in the presence of hybridization,We emphasize hybridization (in eukaryotic species) here since processes such as horizontal gene transfer in microbial organisms result in reticulate evolutionary histories, but the applicability of methods we describe in this chapter has not been investigated or explored in such a domain.\",\n \"describe their strengths and limitations, and highlight major directions for future research in this area.\",\n \"All the methods discussed hereafter make use of multi-locus data, where a locus in this context refers to a segment of genome present across the individuals and species sampled for a given study and related through common descent.\",\n \"A locus can be of varying length, coding or non-coding, and can be either functional or non-functional.\",\n \"Therefore, the use of the term “gene trees\\\" is only historical; we use it to mean the evolutionary history of an individual locus, regardless of whether the locus overlaps with a coding region or not.\",\n \"Care must be taken with increasingly long loci spanning hundreds or thousands of contiguous basepairs, however, as many methods assume a locus has not been affected by recombination.\",\n \"Multi-locus methods are fairly popular because the model fits several types of reduced representation genomic data sets commonly generated to study biological systems.\",\n \"Reduced representation refers to capturing many segments scattered throughout a genome, but only covering a fraction of the total genome sequence [37].\",\n \"Reduced representation data sets which have been used with multi-locus methods include RAD-seq and genotyping by sequencing (GBS), which capture loci of roughly 100 bp associated with palindromic restriction enzyme recognition sites [30].\",\n \"Another family of techniques often applied to studies of deeper time scales, sequence capture, extracts conserved sequences using probes complementary to targetted exons or ultraconserved elements [41].\",\n \"Sequence capture can also be performed in silico when whole genomes are available [52].\",\n \"The rest of the chapter is organized as follows.\",\n \"We begin in Section 2 by defining terminology for the non-biologist, and give a very brief review of phylogenetic trees and their likelihood.\",\n \"In Section 3, we describe the earliest, and simplest from a modeling perspective, approaches to inferring parsimonious phylogenetic networks from gene tree topologies by utilizing their incongruence as the signal for hybridization.\",\n \"To account for ILS, we describe in Section 4 the multispecies network coalescent, or MSNC, which is the core model for developing statistical approaches to phylogenetic network inference while accounting for ILS simultaneously with hybridization.\",\n \"In Sections 5 and 6 we describe the maximum likelihood and Bayesian methods for inferring phylogenetic networks from multi-locus data.\",\n \"In Section 7, we briefly discuss an approach aimed at detecting hybridization by using phylogenetic invariants.\",\n \"This approach does not explicitly build a phylogenetic network.\",\n \"In Section 8 we briefly discuss the efforts for developing methods for phylogenetic network inference that took place in parallel in the population genetics community (they are often referred to as “admixture graphs\\\" in the population genetics literature).\",\n \"In Section 9, we summarize the available software for phylogenetic network inference, discuss the data that these methods use, and then list some of the limitations of these methods in practice.\",\n \"We conclude with final remarks and directions for future research in Section 10.\"\n ],\n [\n \"Background for Non-biologists\",\n \"In this section we define the biological terminology used throughout the chapter so that it is accessible for non-biologists.\",\n \"We also provide a brief review of phylogenetic trees and their likelihood, which is the basis for maximum likelihood and Bayesian inference of phylogenetic trees from molecular sequence data.\",\n \"Excellent books that cover mathematical and computational aspects of phylogenetic inference include [29], [102], [34], [117], [110].\"\n ],\n [\n \"Terminology\",\n \"As we mentioned above, hybridization is reproduction between two members of genetically distinct populations, or species (Fig.\",\n \"REF ).\",\n \"Diploid species (e.g., humans) have two copies of each genome.\",\n \"Aside from a few unusual organisms such as parthenogenic species, one copy will be maternal in origin and the other paternal.\",\n \"When the hybrid individual (or $F_1$ ) is also diploid this process is called homoploid hybridization.\",\n \"Figure: Hybridization, recombination, and the generation of a mosaic genome.\",\n \"Diploid individual a from species A and individual b from species B mate, resulting ina diploid hybrid individual with one copy of its genome inherited from parent a and the other copy inherited from parent b.\",\n \"A recombination event results in the “swapping\\\" of entire regionsbetween two copies of the genome.\",\n \"After multiple generations in which more recombination happens, the genome becomes a mosaic.\",\n \"Walking across the genomefrom left to right, the color switches back and forth between red and blue, where switches happen at recombination breakpoints.\",\n \"Shown are four different loci.\",\n \"Loci 1 and 3 are not appropriatefor tree inference since they span recombination breakpoints and, thus, include segments that have different evolutionary histories.\",\n \"Loci 2 and 4 are the “ideal\\\" loci for analyses by methodsdescribed in this chapter.While each of the two copies of the genome in the hybrid individual traces its evolution back to precisely one of the two parents, this picture becomes much more complex after several rounds of recombination.\",\n \"Recombination is the swapping of a stretch of DNA between the two copies of the genome.\",\n \"Mathematically, if the two copies of the genome are given by strings $u$ and $v$ (for DNA, the alphabet for the strings is $\\\\lbrace A,C,T,G\\\\rbrace $ ), then recombination results in two strings $u^{\\\\prime }=u_1u_2u_3$ and $v^{\\\\prime }=v_1v_2v_3$ , where $u_1$ , $u_2$ , $u_3$ , $v_1$ , $v_2$ , and $v_3$ are all strings over the same alphabet, and $u_1v_2u_3=u$ and $v_1u_2v_3=v$ ; that is, substrings $u_2$ and $v_2$ were swapped.\",\n \"Observe that when this happens, $u_1$ and $u_3$ in the copy $u^{\\\\prime }$ are inherited from one parent, and $v_2$ , also in the copy $u^{\\\\prime }$ , is inherited from a different parent.\",\n \"A similar scenario happens in copy $v^{\\\\prime }$ of the diploid genome.\",\n \"This picture gets further complicated due to backcrossing, which is the mating between the hybrid individual, or one of its descendants, with an individual in one of the parental species.\",\n \"For example, consider a scenario in which descendants of the hybrid individual in Fig.\",\n \"REF repeatedly mate with individuals from species A.\",\n \"After several generations, it is expected that the genomes of the hybrid individuals become more similar to the genomes of individuals in species A, and less similar to the genomes of individuals in species B (using the illustration of Fig.\",\n \"REF , the two copies of the genome would have much more red in them than blue).\",\n \"Most models and methods for phylogenetic inference assume the two copies of a diploid genome are known separately and often only one of them is used to represent the corresponding individual.\",\n \"However, it is important to note that knowing the two copies separately is not a trivial task.\",\n \"Sequencing technologies produce data on both copies simultaneously, and separating them into their constituent copies is a well-studied computational problem known as genome phasing.\",\n \"Biologists often focus on certain regions within the genomes for phylogenetic inference.\",\n \"If we consider the genome to be represented as a string $w$ over the alphabet $\\\\lbrace A,C,T,G\\\\rbrace $ , then a locus is simply a substring of $w$ given by the start and end positions of the substring in $w$ .\",\n \"The size of a locus can range anywhere from a single position in the genome to a (contiguous) stretch of 1 million or more positions in the genome.\",\n \"As we discussed above, when recombination happens, an individual copy of the genome would have segments with different ancestries (the blue and red regions in Fig.\",\n \"REF ).\",\n \"A major assumption underlying phylogenetic tree inference is that the sequence data of a locus used for inference has evolved down a single tree.\",\n \"Therefore, the more recombination which has occured within a locus over its evolutionary history (limited to the history connecting the species being studied), the less suitable it will be for phylogenetic inference.\",\n \"Conversely, loci with low recombination rates may be more suitable in terms of avoiding intra-locus recombination, although such loci are more susceptible to linked selection [44].\"\n ],\n [\n \"Phylogenetic Trees and Their Likelihood\",\n \"An unrooted binary phylogenetic tree $T$ on set ${{X}}$ of taxa (e.g., ${{X}}=\\\\lbrace humans,chimp,gorilla\\\\rbrace $ ) is a binary tree whose leaves are bijectively labeled by the elements of ${{X}}$ .\",\n \"That is, if $|{{X}}|=n$ , then $T$ has $n$ leaf nodes and $n-2$ non-leaf (internal) nodes (each leaf node has degree 1 and each internal node has degree 3).\",\n \"A rooted binary phylogenetic tree is a directed binary tree with a single node designated as the root and all edges are directed away from the root.\",\n \"For $n$ taxa, a rooted binary tree has $n$ leaves and $n-1$ internal nodes (each leaf node has in-degree 1 and out-degree 0; each internal node except for the root has in-degree 1 and out-degree 2; the root has in-degree 0 and out-degree 2).\",\n \"Modern methods for phylogenetic tree inference make use of molecular sequence data, such as DNA sequences, obtained from individuals within the species of interest.\",\n \"The sequences are assumed to have evolved from a common ancestral sequence (we say the sequences are homologous) according to a model of evolution that specifies the rates at which the various mutational events could occur (Fig.\",\n \"REF ).\",\n \"Figure: Sequence evolution on a tree.\",\n \"At the top is the ancestral sequence for a certain locus in the genome of an individual.\",\n \"Through cell division and DNA replication, thissequence is inherited from parent to children.\",\n \"However, mutations could alter the inherited sequences.\",\n \"Boxes indicate letters that were deleted due to mutation.\",\n \"Letters in blueindicate substitutions (a mutation that alters the state of the nucleotide).\",\n \"The letter in green has mutated more than once during its evolutionary history.\",\n \"With respect to sequenceU 1 U_1, sequence U 2 U_2 has two deletions at the 3rd and 8th positions, and a substitution (C to G) at the 5th position.\",\n \"With respect to sequence U 1 U_1, sequenceS 3 S_3 has 7 deletions at positions 3–9, and substitutions at the first two positions (A to T, and A to C).\",\n \"Sequences S 1 S_1, S 2 S_2, S 3 S_3, and S 4 S_4, with the boxes and colorsunknown, are often the data forphylogenetic inference.\",\n \"That is, the four sequences used as data here are: AAATGTTAA, AAAGGTTAA, TCAA, and TCCACGAA.For example, to infer a phylogenetic tree $T$ on set ${{X}}=\\\\lbrace X_1,\\\\ldots ,X_n\\\\rbrace $ of taxa, the sequence $S_1$ of a certain locus is obtained from the genome of an individual in species $X_1$ , the sequence $S_2$ of a certain locus is obtained from the genome of an individual in species $X_2$ , and so on until $n$ sequences $S_1,\\\\ldots ,S_n$ are obtained.\",\n \"To perform phylogenetic tree inference, the $n$ sequences must satisfy two important conditions (see Fig.\",\n \"REF ): The sequences are homologous: The obtained sequences must have evolved down a single tree from a single sequence in an individual in an ancestral species.\",\n \"Two sequences are homologous if they evolved from a common ancestor, including in the presence of events such as duplication.\",\n \"Two homologous sequences are orthologs if they evolved from a common ancestor solely by means of DNA replication and speciation events.\",\n \"Two homologous sequences are paralogs if their common ancestor had duplicated to give rise to the two sequences.\",\n \"The sequences are aligned: While the obtained homologous “raw\\\" sequences might be of different lengths due to events such as insertions and deletions, the sequences must be made to be the same length before phylogenetic inference is conducted so that positional homology is established.\",\n \"Intuitively, positional homology is the (evolutionary) correspondence among sites across the $n$ sequences.\",\n \"That is, the sequences must be made of the same length so that the $i$ -th site in all of them had evolved from a single site in the sequence that is ancestral to all of them.\",\n \"Figure: From homologous sequences to a phylogenetic tree.\",\n \"Identifying homologous sequences across genomes is the first step towards a phylogenetic analysis.\",\n \"The homologoussequences, once identified, are not necessarily of the same length, due to insertions and deletions.\",\n \"Multiple sequence alignment is performed on the homologous sequences and the resultis sequences of the same length where boxes indicate deleted nucleotides.\",\n \"Finally, a phylogenetic tree is constructed on the aligned sequences.Identifying homologous sequences across genomes is not an easy task; see, for example, [84] for a recent review of methods for homology detection.\",\n \"Multiple sequence alignment is also a hard computational problem, with a wide array of heuristics and computer programs currently available for it; see, for example, [13] for a recent review.\",\n \"We are now in position to define a basic version of the Phylogeny Inference Problem: Input: Set $S=\\\\lbrace S_1,\\\\ldots ,S_n\\\\rbrace $ of homologous sequences, where sequence $S_i$ is obtained from taxon $X_i$ , and the $n$ sequences are aligned.\",\n \"Output: A phylogenetic tree $T$ on set ${{X}}$ of taxa such that $T$ is optimal, given the sequences, with respect to some criterion $\\\\Phi $ .\",\n \"The books we cited above give a great survey of the various criterion that $\\\\Phi $ could take, as well as algorithms and heuristics for inferring optimal trees under the different criteria.\",\n \"Here, we focus on the main criterion in statistical phylogenetic inference, namely likelihood.\",\n \"We will make two assumptions when defining the likelihood that are (1) sites are identically and independently distributed, and (2) following a DNA replication event, the two resulting sequences continue to evolve independently of each other.\",\n \"To define the likelihood of a tree $T$ , we first assign lengths $\\\\lambda :E(T) \\\\rightarrow {\\\\mathbb {R}}^+$ to its branches, so that $\\\\lambda (b)$ is the length of branch $b$ in units of expected number of mutations per site per generation.\",\n \"Furthermore, we need a model of sequence evolution ${\\\\cal M}$ .\",\n \"Most models of sequence evolution are Markov processes where the probability of observing a sequence $S$ at node $u$ depends only on the sequence at $u$ 's parent, the length of the branch that links $u$ to its parent, and the parameters of the model of sequence evolution.\",\n \"If we denote by $p^{(i)}_{uv}(t)$ the probability that the $i$ -th nucleotide in the sequence at node $u$ evolves into the $i$ -th nucleotide in the sequence at node $v$ over time $t$ (measured in units of expected number of mutations as well), then the likelihood of a tree $T$ and its branch lengths $\\\\lambda $ is $L(T,\\\\lambda |S) = P(S|T,\\\\lambda ) = \\\\prod _{i} \\\\left(\\\\sum _R \\\\left(p(root^{(i)}) \\\\cdot \\\\prod _{b=(u,v)\\\\in E(T)} p^{(i)}_{uv}(\\\\lambda _b) \\\\right) \\\\right).$ Here, the outer product is taken over all sites $i$ in the sequences; i.e., if each of the $n$ sequences is of length $m$ , then $1 \\\\le i \\\\le m$ .\",\n \"The summation is taken over $R$ , which is the set of all possible labelings of the internal nodes of $T$ with sequences of length $m$ .\",\n \"Inside the summation, $p(root^{(i)})$ gives the stationary distribution of the nucleotides at position $i$ .\",\n \"The likelihood as given by Eq.\",\n \"(REF ) is computed in polynomial time in $m$ and $n$ using Felsenstein's “pruning\\\" algorithm [27].\",\n \"Finally, the maximum likelihood estimate for solving the Phylogeny Inference Problem is given by $(T^*,\\\\lambda ^*) \\\\leftarrow {\\\\rm argmax}_{(T,\\\\lambda )} L(T,\\\\lambda |S).$ Computing the maximum likelihood estimate from a set $S$ of sequences is NP-hard [15], [100].\",\n \"However, much progress has been made in terms of developing heuristics that scale up to thousands of taxa while achieving high accuracy, e.g., [109].\"\n ],\n [\n \"From Humble Beginnings: Smallest Displaying Networks\",\n \"Early work and, still, much effort in the community has focused on inferring the topology of a phylogenetic network from a set of gene tree topologies estimated for the individual loci in a data set.\",\n \"In this section, we discuss parsimony approaches to inferring phylogenetic network topologies from sets of gene trees.\"\n ],\n [\n \"The Topology of a Phylogenetic Network\",\n \"As discussed above, a reticulate, i.e., non-treelike, evolutionary history that arises in the presence of processes such as hybridization and horizontal gene transfer is best represented by a phylogenetic network.\",\n \"A phylogenetic ${{X}}$ -network (Fig.\",\n \"REF ), or ${{X}}$ -network for short, $\\\\Psi $ is a rooted, directed, acyclic graph (rDAG) with set of nodes $V(\\\\Psi )=\\\\lbrace r\\\\rbrace \\\\cup V_L \\\\cup V_T \\\\cup V_N $ , where $indeg(r)=0$ ($r$ is the root of $\\\\Psi $ ); $\\\\forall {v \\\\in V_L}$ , $indeg(v)=1$ and $outdeg(v)=0$ ($V_L$ are the external tree nodes, or leaves, of $\\\\Psi $ ); $\\\\forall {v \\\\in V_T}$ , $indeg(v)=1$ and $outdeg(v) \\\\ge 2$ ($V_T$ are the internal tree nodes of $\\\\Psi $ ); and, $\\\\forall {v \\\\in V_N}$ , $indeg(v) = 2$ and $outdeg(v) = 1$ ($V_N$ are the reticulation nodes of $\\\\Psi $ ).\",\n \"For binary phylogenetic networks, the out-degree of the root and every internal tree node is 2.\",\n \"The network's set of edges, denoted by $E(\\\\Psi ) \\\\subseteq V \\\\times V$ is bipartitioned into reticulation edges, whose heads are reticulation nodes, and tree edges, whose heads are tree nodes (internal or external).\",\n \"Finally, the leaves of $\\\\Psi $ are bijectively labeled by the leaf-labeling function $\\\\ell :V_L \\\\rightarrow {{X}}$ .\",\n \"Figure: An example of a phylogenetic network Ψ\\\\Psi with a single reticulation event.\",\n \"This network is made up of leaf nodes V L ={v 1 ,v 2 ,v 3 ,v 4 }V_L = \\\\lbrace v_1,v_2,v_3,v_4\\\\rbrace , internal tree nodes V T ={u 1 ,u 2 ,u 3 }V_T=\\\\lbrace u_1,u_2,u_3\\\\rbrace , reticulation nodes V N ={h}V_N=\\\\lbrace h\\\\rbrace , and the root rr.\",\n \"The nodes are connected by branches belonging to the set of phylogenetic network edges E(Ψ)E(\\\\Psi ).\",\n \"The branches are:(r,u 1 )(r,u_1), (r,u 2 )(r,u_2), (u 1 ,h)(u_1,h), (u 2 ,h)(u_2,h), (h,u 3 )(h,u_3), (u 1 ,v 1 )(u_1,v_1), (u 2 ,v 4 )(u_2,v_4), (u 3 ,v 2 )(u_3,v_2), and (u 3 ,v 3 )(u_3,v_3).\",\n \"The leaves are labeled by set X={A,B,C,D}{{X}}=\\\\lbrace A,B,C,D\\\\rbrace of taxa: ℓ(v 1 )=A\\\\ell (v_1)=A, ℓ(v 2 )=B\\\\ell (v_2)=B, ℓ(v 3 )=C\\\\ell (v_3)=C, and ℓ(v 4 )=D\\\\ell (v_4)=D.\"\n ],\n [\n \"Inferring Smallest Displaying Networks\",\n \"Early work on phylogenetic networks focused on the problem of identifying a network with the fewest number of reticulation nodes that summarizes all gene trees in the input.\",\n \"More formally, let $\\\\Psi $ be a phylogenetic network.\",\n \"We say that $\\\\Psi $ displays phylogenetic tree $t$ if $t$ can be obtained from $\\\\Psi $ by repeatedly applying the following operations until they are not applicable: For a reticulation node $h$ with two incoming edges $e_1=(u_1,h)$ and $e_2=(u_2,h)$ , remove one of the two edges.\",\n \"For a node $u$ with a single parent $v$ and a single child $w$ , remove the two edges $(v,u)$ and $(u,w)$ , and add edge $(v,w)$ .\",\n \"The set of all trees displayed by the phylogenetic network is ${\\\\cal T}(\\\\Psi )=\\\\lbrace t: \\\\; \\\\Psi \\\\; {\\\\rm displays} \\\\; t\\\\rbrace .$ For example, for the phylogenetic network $\\\\Psi $ of Fig.\",\n \"REF , we have ${\\\\cal T}(\\\\Psi )=\\\\lbrace T_1,T_2\\\\rbrace $ , where $T_1=((A,(B,C)),D)$ and $T_2=(A,((B,C),D))$ .\",\n \"Using this definition, the earliest phylogenetic network inference problem was defined as follows: Input: A set ${\\\\cal G}=\\\\lbrace g_1,g_2,\\\\ldots ,g_m\\\\rbrace $ of gene trees, where $g_i$ is a gene tree for locus $i$ .\",\n \"Output: A phylogenetic network $\\\\Psi $ with the smallest number of reticulation nodes such that ${\\\\cal G} \\\\subseteq {\\\\cal T}(\\\\Psi )$ .\",\n \"This problem is NP-hard [116] and methods were developed for solving it and variations thereof, some of which are heuristics [115], [123], [89], [124].\",\n \"Furthermore, the view of a phylogenetic network in terms of the set of trees it displays was used for pursuing other questions in this domain.\",\n \"For example, the topological difference between two networks could be quantified in terms of the topological differences among their displayed trees [83].\",\n \"The parsimony and likelihood criteria were extended to the case of phylogenetic networks based on the assumption that each site (or, locus) has evolved down one of the trees displayed by the network [80], [54], [53], [56], [55], [57].\",\n \"The concepts of character compatibility and perfect phylogeny were also extended to phylogenetic networks based on the notion of displayed trees [116], [82], [60].\",\n \"Furthermore, questions related to distinguishability of phylogenetic networks based on their displayed trees have been pursued [59] and relationships between networks and trees have been established in terms of this definition [33], [134].\",\n \"However, the computational complexity of this problem notwithstanding, the problem formulation could be deficient with respect to practical applications.\",\n \"For one thing, solving the aforementioned problem only yields the topology of a phylogenetic network, but no other parameters.\",\n \"In practice, biologists would be interested in divergence times, population parameters, and some quantification of the amount of introgression in the genomes.\",\n \"These quantities are not recoverable under the given formulation.\",\n \"Moreover, for the biologist seeking to analyze her data with respect to hypotheses of reticulate evolutionary events, solving the aforementioned problem could result in misleading evolutionary scenarios for at least three reasons.\",\n \"First, the smallest number of reticulations required in a phylogenetic network to display all trees in the input could be arbitrarily far from the true (unknown) number of reticulations.\",\n \"One reason for this phenomenon is the occurrence of reticulations between sister taxa, which would not be detectable from gene tree topologies alone.\",\n \"Second, a smallest set of reticulations could be very different from the actual reticulation events that took place.\",\n \"Third, and probably most importantly, some or even all of gene tree incongruence in an empirical data set could have nothing to do with reticulation.\",\n \"For example, hidden paralogy and/or incomplete lineage sorting could also give rise to incongruence in gene trees.\",\n \"When such phenomena are at play, seeking a smallest phylogenetic network that displays all the trees in the input is the wrong approach and might result in an overly complex network that is very far from the true evolutionary history.\",\n \"To address all these issues, the community has shifted its attention in the last decade toward statistical approaches that view phylogenetic networks in terms of a probability distribution on gene trees that could encompass a variety of evolutionary processes, including incomplete lineage sorting.\"\n ],\n [\n \"Phylogenetic Networks as Summaries of Trees\",\n \"Before we turn our attention to these statistical approaches, it is worth contrasting smallest phylogenetic networks that display all trees in the input to the concept of consensus trees.\",\n \"In the domain of phylogenetic trees, consensus trees have played an important role in compactly summarizing sets of trees.\",\n \"For example, the strict consensus tree contains only the clusters that are present in the input set of trees, and nothing else.\",\n \"The majority-rule consensus tree contains only the clusters that appear in at least 50% of the input trees.\",\n \"When there is incongruence in the set of trees, these consensus trees are most often non-binary trees (contain “soft polytomies\\\") such that each of the input trees can be obtained as a binary resolution of the consensus tree.\",\n \"Notice that while the consensus tree could be resolved to yield each tree in the input, there is no guarantee in most cases that it cannot also be resolved to generate trees that are not in the input.\",\n \"Smallest phylogenetic networks that display all trees in the input could also be viewed as summaries of the trees, but instead of removing clusters that are not present in some trees in the input, they display all clusters that are present in all trees in the input.\",\n \"Similarly to consensus trees, a smallest phylogenetic network could also display trees not in the input (which is the reason why we use $\\\\subseteq $ , rather than $=$ , in the problem formulation above).\",\n \"These issues are illustrated in Fig.\",\n \"REF .\",\n \"Figure: Consensus trees and phylogenetic networks as two contrasting summary methods.\",\n \"(a)-(c) Three (input) gene trees whose summary is sought.\",\n \"(d) The strict consensus of the input trees.\",\n \"(e) The 70% majority-rule consensus of the input trees.\",\n \"(f) A smallest phylogenetic network that displays all three trees iin the input.\",\n \"The strict consensus could be resolved to yield 15 different binary trees, only three of which are in the input.\",\n \"The majority-rule consensus tree could beresolved to yield three possible trees, two of which are the trees in (a) and (d), but the third, which is (((a,b),d),c), is not in the input.\",\n \"Furthermore, the tree in panel (b)is not included in the summary provided by the majority-rule consensus.\",\n \"The phylogenetic network displays four trees, three of which are the input trees, and thefourth is ((a,(b,c)),d), which is not in the input.As discussed above, ILS is another process that could cause gene trees to be incongruent with each other and complicates the inference of phylogenetic networks since incongruence due to ILS should not induce additional reticulation nodes.\",\n \"Before we move on to discuss statistical approaches that account for ILS in a principled probabilistic manner under the coalescent, we describe an extension of the minimizing deep coalescences, or, MDC, criterion [73], [72], [112], to phylogenetic networks, which was devised in [125].\"\n ],\n [\n \"A Step Towards More Complexity: Minimizing Deep Coalescences\",\n \"Let $\\\\Psi $ be a phylogenetic network and consider node $u \\\\in V(\\\\Psi )$ .\",\n \"We denote by $B_u \\\\subseteq V(\\\\Psi )$ the set of nodes in $\\\\Psi $ that are below node $u$ (that is, the set of nodes that are reachable from the root of $\\\\Psi $ via at least one path that goes through node $u$ ).\",\n \"A coalescent history of a gene tree $g$ and a species (phylogenetic) network $\\\\Psi $ as a function $h:V(g) \\\\rightarrow V(\\\\Psi )$ such that the following conditions hold: if $w$ is a leaf in $g$ , then $h(w)$ is the leaf in $\\\\Psi $ with the same label (in the case of multiple alleles, $h(w)$ is the leaf in $\\\\Psi $ with the label of the species from which the allele labeling leaf $w$ in $g$ is sampled); and, if $w$ is a node in $g_v$ , then $h(w)$ is a node in $B_{h(v)}$ .\",\n \"Given a phylogenetic network $\\\\Psi $ and a gene tree $g$ , we denote by $H_{\\\\Psi }(g)$ the set of all coalescent histories of gene tree $g$ within the branches of phylogenetic network $\\\\Psi $ .\",\n \"Given a coalescent history $h$ , the number of extra lineages arising from $h$ on a branch $b=(u,v)$ in phylogenetic network $\\\\Psi $ is the number of gene tree lineages exiting branch $b$ from below node $u$ toward the root, minus one.\",\n \"Finally, $XL(\\\\Psi ,h)$ is defined as the sum of the numbers of extra lineages arising from $h$ on all branches $b \\\\in E(\\\\Psi )$ .\",\n \"Using coalescent histories, the minimum number of extra lineages required to reconcile gene tree $g$ within the branches of $\\\\Psi $ , denoted by $XL(\\\\Psi ,g)$ is given by $XL(\\\\Psi ,g)=\\\\min _{h \\\\in H_{\\\\Psi }(g)}XL(\\\\Psi ,h).$ Under the MDC (minimizing deep coalescence) criterion, the optimal coalescent history refers to the one that results in the fewest number of extra lineages [73], [112], and thus, $XL(\\\\Psi , g) = \\\\sum _{e \\\\in E(\\\\Psi )}[k_e(g)-1]$ where $k_e(g)$ is the number of extra lineages on edge $e$ of $\\\\Psi $ in the optimal coalescent history of gene tree $g$ .\",\n \"Figure: The MDC criterion on phylogenetic networks.\",\n \"A phylogenetic network and coalescent histories within its branches of the three gene trees in Fig.\",\n \"(a)-(c).\",\n \"The highlighted branch that separates the hybridization event from the MRCA of B and C has two extra lineages arising from the threeshown coalescent histories.\",\n \"All other branches have 0 extra lineages.\",\n \"Therefore, the total number of extra lineages in this case is 2.A connection between extra lineages and the displayed trees of a phylogenetic network is given by the following observation.\",\n \"Observation 1 If gene tree $g$ is displayed by phylogenetic network $\\\\Psi $ , then $XL(\\\\Psi ,g)=0$ .\",\n \"The implication of this observation is that if one seeks the phylogenetic network that minimizes the number of extra lineages, the problem can be trivially solvable by finding an overly complex network that displays every tree in the input.\",\n \"Therefore, inferring a phylogenetic network $\\\\Psi $ from a collection of gene tree topologies ${{G}}$ based on the MDC criterion is more appropriately defined by $\\\\hat{\\\\Psi }(m) = \\\\operatornamewithlimits{argmin}_{\\\\Psi (m)} \\\\left( \\\\sum _{g \\\\in {{G}}} XL(\\\\Psi (m),g) \\\\right),$ where we write $\\\\Psi (m)$ to denote a phylogenetic network with $m$ reticulation nodes.\",\n \"While the number of reticulations $m$ is unknown and is often a quantity of interest, there is a trade-off between the number of reticulation nodes and number of extra lineages in a network: Reticulation edges can be added to reduce the number of extra lineages.\",\n \"Observing this reduction in the number of extra lineages could provide a mechanism to determine when to stop adding reticulations to the network [125].\",\n \"In the previous section, we focused on two parsimony formulations for inferring a phylogenetic network from a collection of input gene tree topologies: The first seeks a network with the fewest number of reticulations that displays each of the input gene trees, and the second seeks a network that does not have to display every gene tree in the input, but must minimize the number of “extra lineages\\\" that could arise within a given number of reticulations.\",\n \"Both formulations result in phylogenetic network topologies alone and make use of only the gene tree topologies.\",\n \"In this section, we introduce the multispecies network coalescent, or MSNC [120], as a generative process that extends the popular multispecies coalescent, or MSC [20], that is the basis for most multi-locus species tree inference methods.\",\n \"The MSNC allows for the coalescent to operate within the branches of a phylogenetic network by viewing a set of populations—extant and ancestral—glued together by a rooted, directed, acyclic graph structure.\"\n ],\n [\n \"Parameterizing the Network's Topology\",\n \"In addition to the topology of a phylogenetic network $\\\\Psi $ , as given by Definition REF above, the nodes and edges are parameterized as follows.\",\n \"Associated with the nodes are divergence/reticulation times, $\\\\tau : V(\\\\Psi ) \\\\rightarrow {\\\\mathbb {R}}^+$ , where $\\\\tau (u)$ is the divergence time associated with tree node $u$ and $\\\\tau (v)$ is the reticulation time associated with reticulation node $v$ .\",\n \"All leaf nodes $u$ in the network have $\\\\tau (u)=0$ .\",\n \"Furthermore, if $u$ is on a path from the root of the network to a node $v$ , then $\\\\tau (u) \\\\ge \\\\tau (v)$ .\",\n \"Associated with the edges are population mutation rate parameters, $\\\\theta : E(\\\\Psi ) \\\\rightarrow {\\\\mathbb {R}}^+$ , where $\\\\theta _b=4N_b \\\\mu $ is the population mutation rate associated with edge $b$ , $N_b$ is the effective population size associated with edge $b$ , and $\\\\mu $ is the mutation rate per site per generation.\",\n \"Divergence times associated with nodes in the phylogenetic network could be measured in units of years, generations, or coalescent units.\",\n \"Branch lengths in gene trees are often given in units of expected number of mutations per site.\",\n \"The following rules are used to convert back and forth between these units: Given divergence time $\\\\tau $ in units of expected number of mutations per site, mutation rate per site per generation $\\\\mu $ and the number of generations per year $g$ , $\\\\tau / (\\\\mu g)$ represents divergence times in units of years.\",\n \"Given population size parameter $\\\\theta $ in units of population mutation rate per site, $2\\\\tau / \\\\theta $ represents divergence times in coalescent units.\",\n \"In addition to the divergence times and population size parameters, the reticulation edges of the network are associated with inheritance probabilities.\",\n \"For every reticulation node $u \\\\in V_N$ , let $left(u)$ and $right(u)$ be the “left\\\" and “right\\\" edges incoming into node $u$ , respectively (which of the two edges is labeled left and which is labeled right is arbitrary).\",\n \"Let $E_R \\\\subseteq E(\\\\Psi )$ be the set of reticulation edges in the network.\",\n \"The inheritance probabilities are a function $\\\\gamma : E_R \\\\rightarrow [0,1]$ such that for every reticulation node $u \\\\in V_N$ , $\\\\gamma (left(u))+\\\\gamma (right(u)) = 1$ .\",\n \"In the literature, $\\\\gamma $ is sometimes described as a vector $\\\\Gamma $ .\"\n ],\n [\n \"The Multispecies Network Coalescent and Gene Tree Distributions\",\n \"As an orthologous, non-recombining genomic region from a set ${{X}}$ of species evolves within the branches of the species phylogeny of ${{X}}$ , the genealogy of this region, also called the gene tree, can be viewed as a discrete random variable whose values are all possible gene tree topologies on the set of genomic regions.\",\n \"When the gene tree branch lengths are also taken into account, the random variable becomes continuous.\",\n \"Yu et al.\",\n \"[126] gave the probability mass function (pmf) for this discrete random variable given the phylogenetic network $\\\\Psi $ and an additional parameter $\\\\Gamma $ that contains the inheritance probabilities associated with reticulation nodes, which we now describe briefly.\",\n \"The parameters $\\\\Psi $ and $\\\\Gamma $ specify the multispecies network coalescent, or MSNC (Fig.\",\n \"REF ), and allow for a full derivation of the mass and density functions of gene trees when the evolutionary history of species involves both ILS and reticulation [126], [127].\",\n \"This is a generalization of the multispecies coalescent, which describes the embedding and distribution of gene trees within a species tree without any reticulate nodes [20].\",\n \"Figure: Layers of the multispecies network coalescent model.\",\n \"A phylogenetic network describes the relationship between species (top).\",\n \"The MSNC describes the distribution of gene trees within the network, in which alleles from the same species can have different topologies and inheritance histories due to reticulation and/or ILS (middle).\",\n \"Some kind of mutation process occurs along the gene trees, resulting in observed differences between alleles in the present, which vary between genes based on their individual trees.It is important to note that a single reticulation edge between two nodes does not mean a single hybridization event.\",\n \"Rather, a reticulation edge abstracts a continuous epoch of repeated gene flow between the two species, as illustrated in Fig.\",\n \"REF .\",\n \"Figure: Reticulation edges as abstractions of gene flow epochs.\",\n \"(Left) An epoch of gene flow from one population to another with migration rate α\\\\alpha per generation.\",\n \"(Right) A phylogenetic networkwith a single reticulation edge that abstracts the gene flow epoch, with inheritance probability γ\\\\gamma .The two models shown in Fig.\",\n \"REF were referred to as the “gene flow\\\" model (left) and “intermixture\\\" model (right) of hybridization in [70].\",\n \"While the “gene flow” model is used by the IM family of methods [47] to incorporate admixture, the MSNC adopts the “intermixture” model.\",\n \"In this model, the $\\\\gamma $ inheritance probabilities indicate the ratio of genetic materials of a hybrid coming from its two parents.\",\n \"This means that unlinked loci from a hybrid species will have independent evolutionary histories, and will have evolved through the “left” or “right” parent with some probability $\\\\gamma $ and $1 - \\\\gamma $ respectively.\",\n \"The performance of phylogenetic network inference on data simulated under the gene flow model was demonstrated in [121].\",\n \"Wen and Nakhleh [121] derived the density function of the probability of gene trees given a phylogenetic network, with its topology, divergence/migration times, population mutation rates and inheritance probabilities.\",\n \"The divergence/migration times are in units of expected number of mutations per site, and population mutation rates are in units of population mutation rate per site.\",\n \"Based on the MSNC, and by integrating out all possible gene trees, Zhu et al.\",\n \"[137] developed an algorithm to compute the probability of a bi-allelic genetic marker given a phylogenetic network.\"\n ],\n [\n \"Maximum Likelihood Inference of Phylogenetic Networks\",\n \"Phylogenetic networks are more complicated than a tree with some reticulation edges.\",\n \"The gene tree topology with highest mass probability may not be one of the backbone trees of the network with 4 or more taxa [138].\",\n \"Also, not all networks can be obtained by simply adding edges between the original edges of a tree [33].\",\n \"Therefore inferring phylogenetic networks is not a trivial extension of methods to infer species trees.\",\n \"Most phylogenetic network methods [125], [127], [120], [121], [133], [137], [136], [128] sample from whole-network space rather than simply adding reticulations to a backbone tree.\",\n \"As such methods walk the space from one phylogenetic network to another, the point estimate or posterior distribution of networks is not tree-based, does not return or imply a backbone tree, and can include networks which cannot be described by merely adding reticulate branches between tree branches.\"\n ],\n [\n \"Inference\",\n \"Sequential inference was initially developed to estimate species trees under the multispecies coalescent [78], [69], and in recent years has been extended to species networks [129], [127], [128], [104], [120].\",\n \"These methods follow a two-step approach, where the first step is to estimate gene trees from multiple sequence alignments.\",\n \"The second step is to estimate a species tree or network from the distribution of estimated gene tree topologies.\",\n \"As described above and illustrated in Fig.\",\n \"REF , two key requirements have facilitated the development of several methods for phylogenetic inference from multi-locus data, including those that follow the two-step approach.\",\n \"One key requirement for current methods is that the segments are the result of speciation and not gene duplication, that is, sequences from different individuals and species are orthologs and not paralogs.\",\n \"Meeting this requirement ensures that the nodes in each gene tree represent coalescent events and can be fit to a coalescent model within each species network branch.\",\n \"A model which accounts for gene duplication and loss in addition to coalescent events has been developed to reconcile gene family trees with a fixed species tree [98], [132].\",\n \"The most recent implementation of this model can also use it to estimate the species tree [21], but this model has yet to be extended to work with species networks.\",\n \"A second key requirement is that the evolutionary history of the locus can be accurately modeled using a single tree.\",\n \"Recombination or gene fusion should not have occurred within a locus, otherwise a gene network would be required to model that locus, breaking the MSNC model of gene trees within a species network [130].\",\n \"Because of this requirement, multi-locus methods should be used with short contiguous sequences.\",\n \"The results of a previous study on mammal phylogenetics suggest that individual exons are an appropriate target sequence [101].\",\n \"Under these two key requirements, each gene tree is considered to be a valid and independent sample from an underlying distribution of gene trees conditional on some unobserved species network.\",\n \"Of course this assumption in sequential inference is violated as the gene trees are only estimates.\",\n \"Particular methods may be more or less sensitive to gene tree estimation errors.\",\n \"For species trees, methods which infer unrooted species trees (e.g.\",\n \"ASTRAL [78]) appear to be more robust relative to methods which infer rooted species trees (e.g.\",\n \"MP-EST [69]).\",\n \"This is because unrooted methods take unrooted gene trees as input and do not rely on correct rooting of the gene trees [103].\",\n \"An estimate $\\\\hat{g}$ of the true gene tree is typically made using phylogenetic likelihood (see Section 2).\",\n \"The phylogenetic likelihood of the sequence alignments can be combined sequentially or simultaneously with the MSNC probability densities of the gene trees to estimate a species network from sequence data.\",\n \"Given gene trees where each node represents a coalescent event, the probability densities and masses of those gene trees given a species network can be calculated [127].\",\n \"This can be based on the topologies and node heights of the gene trees, or based on the topologies alone (see Section REF ).\",\n \"Maximum likelihood (ML) methods seek a phylogenetic network (along with its parameters) that maximizes some likelihood function.\",\n \"In a coalescent context, these methods search for the species network which maximizes the likelihood of observing a sample of gene trees given the proposed species network.\",\n \"The sample of gene trees can include branch lengths, in which case the likelihood is derived from the time intervals between successive coalescent events [127].\",\n \"In the absence of branch lengths, the likelihood is derived from the probability mass of each gene tree topology [126], [127].\",\n \"This probability is marginalized over every coalescent history $h$ , which is all the ways for a gene tree to follow the reticulate branching of the network: $P(g|\\\\Psi ,\\\\Gamma ,\\\\theta ) = \\\\sum _{h \\\\in H_\\\\Psi (g)}P(h|\\\\Psi ,\\\\Gamma ,\\\\theta )$ and the ML species network is therefore: $\\\\hat{\\\\Psi } = \\\\operatornamewithlimits{argmax}_\\\\Psi {\\\\prod _{g \\\\in {{G}}}{P(g|\\\\Psi ,\\\\Gamma ,\\\\theta )}}.$ ML inference of species networks has been implemented as the InferNetwork_ML command in PhyloNet [127], which identifies the ML species network up to a maximum number of reticulations.\",\n \"Similar to our discussion of the MDC criterion above, absent any explicit stopping criterion or a penalty term in the likelihood function, obtaining an ML estimate according to Eq.\",\n \"(REF ) can result in overly complex phylogenetic networks since adding more reticulations often improves the likelihood of the resulting network.\",\n \"Therefore, it is important to parameterize the search by the number of reticulations sought, $m$ , and solve $\\\\hat{\\\\Psi }(m) = \\\\operatornamewithlimits{argmax}_{\\\\Psi (m)}{\\\\prod _{g \\\\in {{G}}}{P(g|\\\\Psi (m),\\\\Gamma ,\\\\theta )}},$ where the value $m$ is experimented with by observing the improvement in the likelihood for varying values of $m$ (for example, maximum likelihood inference of phylogenetic networks in PhyloNet implements information criteria, such as AIC and BIC, for this purpose [127]).\",\n \"The computational cost of the likelihood calculation increases with larger species networks and gene trees.\",\n \"Not only does this increase the number of branches and coalescent times, but as more reticulations are added many more possible coalescent histories exist to be summed over.\",\n \"Even with one reticulation edge attached onto a tree, the difficulty of the problem is exponential to tree cases.\",\n \"The computational complexity of the likelihood calculation is highly related to the size of the set of all coalescent histories of a gene tree conciliated in a network.\",\n \"Zhu et al.\",\n \"[136] proposed an algorithm to compute the number of coalescent histories of a gene tree for a network, and demonstrated that the number can grow exponentially after adding merely one reticulation edge to a species tree.\",\n \"Figure: Running time of computing the likelihood of a phylogenetic network given gene tree topologies.\",\n \"150 1-reticulation phylogenetic networks with 5 species and 4 individuals per species was used, and the data consisted of 10,000 bi-allelic markers.\",\n \"The networks varied in terms of the diameter of a reticulation node (the number of edges on the cycle in the underlyingundirected graph) and the number of taxa (leaves) under the reticulation nodes.To show how running time of likelihood computation varies in a network with a single reticulation, we generated 150 random 1-reticulation networks with 5 taxa, then simulated 10,000 bi-allelic markers with 4 individuals per species.\",\n \"When a reticulation node exists in a phylogenetic network, this will induce a cycle in the unrooted equivalent of the acyclic rooted network.\",\n \"The “diameter” is the length of that cycle, and we ran the likelihood computation in [137] and summarized the maximum running time according to values of diameter and number of taxa under reticulation.\",\n \"Fig.\",\n \"(REF ) shows that the complexity of the likelihood computation is highly related to the structure of the network.\",\n \"The running time in the worst case is hundreds of times slower than that of the best case.\",\n \"A faster way to estimate a species network is to calculate a pseudolikelihood instead of a full likelihood.\",\n \"The InferNetwork_MPL command in PhyloNet implements a maximum pseudo-likelihood (MPL) method for species networks.\",\n \"This method is based on rooted triples, which is akin to the MP-EST method for species tree inference [128], [69].\",\n \"Unlike phylogenetic trees, a given phylogenetic network is not necessarily uniquely distinguished by its induced set of rooted triples.\",\n \"Therefore this method cannot distinguish the correct network when other networks induce the same sets of rooted triples [128].\",\n \"However, it is much more scalable than ML methods in terms of the number of taxa [46].\",\n \"Another MPL method, SNaQ, is available as part of the PhyloNetworks software [104], [105].\",\n \"SNaQ is based on unrooted quartets, akin to the ASTRAL method for species tree inference [78].\",\n \"It is even more scalable than InferNetwork_MPL [46], but can only infer level-1 networks (Fig.\",\n \"REF ).\",\n \"Figure: Level-1 network definition.\",\n \"Reticulation nodes induce cycles in the (undirected graphs underlying the) phylogenetic networks.\",\n \"The edges of the cycles are highlighted with red lines.\",\n \"(a) A level-1 network is one where no edge of the network is shared by two or more cycles.\",\n \"(b) A non-level-1 network is one where at least one edge is shared by at least two cycles (the shared edge in this case is the one inside the blue circle).\"\n ],\n [\n \"Bayesian Inference of Phylogenetic Networks\",\n \"Maximum likelihood estimation of phylogenetic networks, as described in the previous section, has three main limitations: As discussed, without penalizing the complexity of the phylogenetic network, the ML estimate could be an overly complex network with many false reticulations.\",\n \"The inference results in a single point estimate that does not allow for assessing confidence in the inferred network.\",\n \"The formulation does not allow for making use of the sequence data directly, but is based on gene tree estimates.\",\n \"One way of addressing these limitations is to adopt Bayesian inference where an estimate of the posterior distribution on networks is sought directly from the sequence data of the individual loci, and where the prior distribution on phylogenetic networks accounts for model complexity in a principled manner.\",\n \"Before we describe the work on Bayesian inference, it is important to note that while maximum likelihood estimation is not satisfactory, we cannot say that Bayesian estimation is without challenges.\",\n \"Such methods like [121], [133], [137] are based on reversible-jump MCMC [38] with varying numbers of parameters.\",\n \"Mixing problems arise when they involve dimension changing moves: adding a reticulation and removing a reticulation.\",\n \"This is because while walking over the space of phylogenetic networks, these methods jump between probability spaces of different models.\",\n \"Therefore moves should be carefully designed to account for mixing issues.\"\n ],\n [\n \"Probability Distributions Over Species Networks\",\n \"It is useful to define probability distributions over species trees or networks without reference to sequence data or gene trees.\",\n \"Among other uses, these probability distributions can be applied as prior distributions in Bayesian inference.\",\n \"The two most common types of prior distributions used for species trees are birth-death tree priors and compound priors.\",\n \"Both types have been extended to create probability distributions over species networks.\",\n \"As their name implies, birth-death tree priors combine a rate of birth with a rate of death.\",\n \"These are the rates at which one lineage splits into two, and one lineage ceases to exist respectively [36].\",\n \"In the context of species trees these rates are more informatively called speciation and extinction.\",\n \"When the extinction rate is set to zero, this is known as a Yule prior [131].\",\n \"Birth-death tree priors have been extended to support incomplete sampling in the present, and sampling-through-time [108].\",\n \"A birth-death prior for species networks has been developed, called the birth-hybridization prior.\",\n \"This prior combines a rate of birth (or speciation) with a rate of hybridization, which is the rate at which two lineages merge into one.\",\n \"This model does not include a rate of extinction [133].\",\n \"All birth-death tree priors induce a uniform probability distribution over ranked tree topologies, regardless of the rates of speciation and extinction.\",\n \"This means that birth-death priors favor symmetric over asymmetric trees, as symmetric trees have more possible ranked histories.\",\n \"Empirical trees generally have more asymmetric shapes than predicted by birth-death models [43].\",\n \"For the birth-hybridization prior the probability distribution over network topologies is not invariant to the hybridization rate, which when set to zero reduces to the Yule model, and any topology containing a reticulation will have zero probability.\",\n \"All birth-death priors are generative, as is the birth-hybridization prior.\",\n \"This means that not only can these distributions be used as priors for Bayesian inference, but they can be used to simulate trees and networks.\",\n \"These simulated distributions can then be used for ABC inference, which is used for models which are difficult to implement using MCMC.\",\n \"They can also be used for posterior predictive checks, which is an absolute measure of model goodness of fit [35].\",\n \"While birth-death priors induce a probability distribution over topologies and branch lengths, compound tree and network priors are constructed from separate distributions on both.\",\n \"Typically compound tree priors combine a uniform distribution over unranked tree topologies, favoring more asymmetric trees.\",\n \"Empirical trees generally have more symmetric shapes than predicted by this distribution [43].\",\n \"Then a continuous distribution such as gamma can be applied to branch lengths or node heights.\",\n \"Compound priors are used for network inference by adding a third distribution describing the number of reticulations [121].\",\n \"A Poisson distribution is a natural fit for this parameter as it describes a probability on non-negative integers.\",\n \"The probability distribution for each network topology can still be uniform for all networks given $k$ reticulations.\",\n \"Unlike birth-death priors, compound priors are not generative, so it is not straightforward to simulate trees or networks from those distributions.\",\n \"The most obvious way to simulate such trees and networks would be running an existing Markov Chain Monte Carlo sampler without any data, and subsampling states from the chain at a low enough frequency to ensure independence between samples.\"\n ],\n [\n \"Sampling the Posterior Distribution\",\n \"The ML species network with $k + 1$ reticulations will always have a higher likelihood than the ML network with $k$ reticulations.\",\n \"For this reason, some threshold of significance must be applied to estimate the number of reticulations.\",\n \"This threshold may be arbitrary or it may be theoretically based, for example the Akaike information criterion (AIC) and Bayesian information criterion (BIC) measures of relative fit [10].\",\n \"In contrast, Bayesian methods of species network inference are able to naturally model the probability distribution over species networks including the number of reticulations by using a prior (see Section REF ).\",\n \"In a Bayesian model, the posterior probability of a species network $P(\\\\Psi )$ is proportional to the likelihood of the gene trees $P(G|\\\\Psi ,\\\\Gamma ,\\\\theta )$ , multiplied by the prior on the network and other parameters of the model $P(\\\\Psi ,\\\\Gamma ,\\\\theta )$ , and marginalized over all possible values of $\\\\Gamma $ and $\\\\theta $ : $P(\\\\Psi ) \\\\propto \\\\iint P(G|\\\\Psi ,\\\\theta ) \\\\cdot P(\\\\Psi ,\\\\Gamma ,\\\\theta ) \\\\mathop {}\\\\!\\\\mathrm {d}\\\\Gamma \\\\mathop {}\\\\!\\\\mathrm {d}\\\\theta .$ When a decaying prior is used on the number of reticulations or on the rate of hybridization, the prior probability of species networks with large numbers of reticulations will be very low, and so will the posterior probability (Fig.\",\n \"REF ).\",\n \"Figure: Bayesian inference of the number of reticulations.\",\n \"In this example the topology of the network Ψ\\\\Psi is fixed, the true number of reticulations is 1, and the likelihood is calculated for a topology with no reticulations, the true reticulation, and additional reticulations, with maximum likelihood branch lengths.\",\n \"The posterior probability was normalized to sum to 1, although as this is not integrated over branch lengths, the typical Bayesian posterior probability might be a bit different.Bayesian methods for phylogenetic inference typically use the Metropolis-Hastings Markov Chain Monte Carlo (MCMC) algorithm to estimate the posterior distribution of trees or networks.\",\n \"MCMC is a random walk where each step depends on the previous state, and it is flexible enough to be used for implementing extremely complex models such as species network inference with relative ease.\",\n \"Bayesian estimation of species networks from gene trees is implemented in the PhyloNet command MCMC_GT.\",\n \"The posterior probability of a species network is equal to the integral in equation REF multiplied by a normalizing constant $Z$ known as the marginal likelihood.\",\n \"In the case of sequential multilocus inference, this constant is equal to $Z = P(G)^{-1}$ The marginal likelihood is usually intractable to calculate, but MCMC sidesteps this calculation by sampling topologies and other parameters with frequencies proportional to their probability mass or density.\",\n \"The posterior probability of a species network $\\\\Psi $ can therefore be approximated as the proportion of steps in the MCMC chain where the network topology $\\\\Psi _i$ at the end of the step $i$ is equal to $\\\\Psi $ .\",\n \"The value of any particular parameter, for example an inheritance probability $\\\\gamma $ for a given reticulation node $v$ , can be estimated by averaging its value over the set of $X$ steps where the state includes that parameter.\",\n \"In this case it is averaged over the states where the species network includes that node, i.e.\",\n \"the set $X = \\\\lbrace i: v \\\\in \\\\Psi _i\\\\rbrace $ : $E(\\\\gamma ) = \\\\frac{1}{|X|} \\\\sum ^i_{X} \\\\gamma _i$ Bayesian inference has also enabled the inference of species trees and networks directly from multilocus sequence data.\",\n \"Instead of first estimating individual gene trees from multiple sequence alignments, these methods jointly infer the gene trees and species network using an application of Bayes' rule: $P(G,\\\\Psi ) \\\\propto \\\\iint P(D|G) \\\\cdot P(G|\\\\Psi ,\\\\Gamma ,\\\\theta ) \\\\cdot P(\\\\Psi ,\\\\Gamma ,\\\\theta ) \\\\mathop {}\\\\!\\\\mathrm {d}\\\\Gamma \\\\mathop {}\\\\!\\\\mathrm {d}\\\\theta .$ Here $P(D|G)$ is the likelihood of the data over all gene trees.\",\n \"In practice this is the sum of phylogenetic likelihoods $\\\\sum _i P(d_i|g_i)$ for every sequence alignment $d$ and associated gene tree $g$ .\",\n \"As with sequential Bayesian inference of species trees and networks, the use of MCMC avoids the calculation of the marginal likelihood, which for joint inference can be expressed as $Z = P(D)^{-1}$ .\",\n \"Joint Bayesian inference was first developed for species trees, and now has several popular implementations including StarBEAST2 [86] and BPP [96].\",\n \"Joint Bayesian inference of species networks has been implemented independently as the PhyloNet command MCMC_SEQ, and as the BEAST2 package “SpeciesNetwork” [121], [133].\",\n \"These two methods are broadly similar in their model and implementation, with a few notable differences.\",\n \"MCMC_SEQ uses a compound prior on the species network, whereas SpeciesNetwork has a birth-hybridization prior (see Section REF ).\",\n \"SpeciesNetwork is able to use any of the protein and nucleotide substitution models available in BEAST2.\",\n \"MCMC_SEQ can be used with any nested GTR model but with fixed rates and base frequencies.\",\n \"So the rates (e.g.\",\n \"the transition/transversion ratio for HKY) must be estimated before running the analysis, or Jukes-Cantor is used where all rates and base frequencies are equal.\"\n ],\n [\n \"Inference Under MSC vs. MSNC When Hybridization Is Present\",\n \"We simulated 128 loci on the phylogenetic network of Fig.\",\n \"REF (a).\",\n \"The program ms [48] was used to simulate 128 gene trees on the network, and each gene tree was used to simulate a sequence alignment of 500 sites using the program Seq-gen [95] under the GTR model and $\\\\theta = 0.036$ for the population mutation rate.\",\n \"The exact command used was: seq-gen -mgtr -s$0.018$ -f$0.2112,0.2888,0.2896,0.2104$ -r$0.2173,\\\\\\\\0.9798,0.2575,0.1038,1,0.2070$ -l500 We then ran both StarBEAST and MCMC_SEQ, as inference methods under the MSC and MSNC models, respectively, for $6\\\\times 10^7$ iterations each.\",\n \"The results are shown in Fig.\",\n \"REF .\",\n \"Figure: Inference under the MSC and MSNC when the evolutionary history involves hybridization.\",\n \"(a) The true phylogenetic network with the shown inheritance probabilities and branch lengths (in coalescent units).\",\n \"(b) The MPP (maximum a posteriori probability) species tree estimated under the MSC by StarBEAST (frequency of 94% in the 95%95\\\\% credible set) with the average divergence times.\",\n \"(c) The MPP phylogenetic network along with the inheritance probabilities estimated under the MSNC by MCMC_SEQ (the only network topology in the 95%95\\\\% credible set).\",\n \"The scale bar of divergence times represents 1 coalescent unit for (a-c).\",\n \"(d) The coalescent times of the MRCAs of (C,G), (A,Q), (A,C), (Q,R) from co-estimated gene trees inferred by StarBEAST (green) and MCMC_SEQ (blue).A few observations are in order.\",\n \"First, while StarBEAST is not designed to deal with hybridization, it inferred the tree topology (Fig.\",\n \"REF (b)) that is obtainable by removing the two hybridization events (the two arrows) from the true phylogenetic network (the backbone tree).\",\n \"Second, MCMC_SEQ identified the true phylogenetic network as the one with the highest posterior (Fig.\",\n \"REF (c)).\",\n \"Furthermore, the estimated inheritance probabilities are very close to the true ones.\",\n \"Third, and most interestingly, since StarBEAST does not account for hybridization, it accounts for all heterogeneity across loci as being caused by incomplete lineage sorting (ILS) by underestimating all branch lengths (that is, “squashing\\\" the divergence times so as to explain the heterogeneity by ILS).\",\n \"Indeed, Fig.\",\n \"REF (d) shows that the minimum coalescent times of the co-estimated gene trees by StarBEAST force the divergence times in the inferred species tree to be very low.\",\n \"MCMC_SEQ, on the other hand, accurately estimates the branch lengths of the inferred phylogenetic network since networks differentiate between divergence and hybridization times.\",\n \"For example, Fig.\",\n \"REF (d) shows that the coalescent times of clade (C,G) across all co-estimated gene trees is a continuum with a minimum value around 2, which defines the divergence time of these two taxa in the phylogenetic network.\",\n \"MCMC_SEQ clearly identifies two groups of coalescent times for each of the two clades (A,C) and (Q,R): The lower group of coalescent times correspond to hybridization, while the upper group of coalescent times correspond to the coalescences above the respective MRCAs of the clades.\",\n \"We also note that the minimum value of coalescent times corresponding to (Q,R) is larger than that corresponding to (A,Q), which correctly reflects the fact that hybridization from R to Q happened before hybridization from C to A, as indicated in the true phylogenetic network.\",\n \"Finally, for clade (A,Q), three groups of coalescence times are identified by MCMC_SEQ, which makes sense since there are three common ancestors of A and Q in the network: at the MRCA of (A,Q) in the case of no hybridization involving either of the two taxa, at the MRCA of (A,Q,L,R) in the case of the hybridization involving Q, and at the root of the network in the case of the hybridization involving A.\",\n \"More thorough analysis and comparison of inferences under the MSC and MSNC can be found in [121].\",\n \"These results illustrate the power of using a phylogenetic network inference method when hybridization is involved.\",\n \"In particular, if hybridization had occurred, and the practitioner did not suspect it and ran StarBEAST instead, they would get wrong inferences.\",\n \"In this case, the errors all have to do with the divergence time estimates.\",\n \"However, the topology of the inferred tree could be wrong as well, depending on the hybridization scenarios.\"\n ],\n [\n \"Phylogenetic Invariants Methods\",\n \"The focus of this chapter up to this point has largely been on the MSC and MSNC models.\",\n \"A parallel effort has been led to detect reticulate evolution by using the notion of phylogenetic invariants [12], [64].\",\n \"Phylogenetic invariants are polynomial relationships satisfied by frequencies of site patterns at the taxa labeling the leaves of a phylogenetic tree (and given a model of sequence evolution).\",\n \"Invariants that are predictive of particular tree topologies could then be used for inferring the tree topology by focusing on the space of site patterns rather than the space of tree topologies [28].\",\n \"As Felsenstein wrote in his book, “invariants are worth attention, not for what they do for us now, but what they might lead to in the future.\\\"\",\n \"With the availability of whole-genome data and, consequently, the ability to obtain better estimates of site frequencies, the future is here.\",\n \"Indeed, methods like SVDQuartets [14] use phylogenetic invariants to estimate species trees under the MSC model.\",\n \"A detailed discussion of phylogenetic invariants in general is beyond the scope of this manuscript.\",\n \"Interested readers should consult the excellent exposition on the subject in Felsenstein's seminal book (Chapter 22 in [28]).\",\n \"In this section, we briefly review phylogenetic invariants-based methods for detecting reticulation, starting with the most commonly used one, known as the $D$ -statistic or the “ABBA-BABA\\\" test.\",\n \"The $D$ -Statistic [39] is a widely known and frequently applied statistical test for inferring reticulate evolution events.\",\n \"The power of the test to infer reticulate evolution derives from the likelihood calculations of the MSNC.\",\n \"Despite this, the test itself is simple to calculate and formalize.\",\n \"The $D$ -Statistic is given by $\\\\frac{N_{ABBA} - N_{BABA}}{N_{ABBA} + N_{BABA}}$ To calculate these quantities, we are given as input the four taxon tree including outgroup of Fig.\",\n \"REF and a sequence alignment of the genomes of P1, P2, P3, and O.\",\n \"Given this alignment, $N_{ABBA}$ is calculated as the number of occurrences of single sites in the alignment where P1 and O have the same letter and P2 and P3 have the same letter, but these two letters are not the same i.e.\",\n \"CTTC or GCCG.\",\n \"Similarly, $N_{BABA}$ can be calculated as the number of occurrences in the alignment where the letters of $P1 = P3$ and $P2 = O$ with no other equalities between letters.\",\n \"Upon calculating the $D$ -Statistic, a significant deviation away from a value of 0 gives evidence for reticulate evolution.\",\n \"As shown in Fig.\",\n \"REF , a strong positive value implies introgression between P2 and P3 while a strong negative value implies introgression between P1 and P3.\",\n \"No such conclusions can be made from a D value very close to 0.\",\n \"The crux of the theory behind the $D$ -Statistic lies in the expectation of the probabilities of discordant gene trees given the overall phylogeny of Fig.\",\n \"REF .\",\n \"If we remove the two reticulation events in Fig.\",\n \"REF we end up with a species tree $\\\\Psi $ .\",\n \"Figure: The four-taxon tree topology used for the DD-Statistic.\",\n \"Significant deviations away from a value of 0 of the DD-statistic (Eq.\",\n \"()) support introgression between P3 and either P1 or P2.\",\n \"As shown, a significant positive value supports introgression between P2 and P3.\",\n \"A significant negative value supports introgression between P1 and P3.Given $\\\\Psi $ , the two gene trees whose topologies disagree with that of the species tree are equally probable under the MSC.\",\n \"Figure: The three scenarios of probabilities of the two gene trees that are discordant with the species tree in the case of a single reticulation event between P3 and one of the othertwo (in-group) species.\",\n \"If the evolutionary history of the species is a tree (Ψ)(\\\\Psi ), the two discordant gene trees are equally probable.\",\n \"However, if the evolutionary history of the speciesis non-treelike, as given by phylogenetic networks Ψ 1 \\\\Psi _1 and Ψ 2 \\\\Psi _2, then the probabilities of the two discordant gene trees are unequal in different ways.On the other hand, when a reticulation between P1 and P3 occurs, this results in an increase in the probability of the discordant gene tree that groups P1 and P3 as sister taxa, as compared to the other discordant gene tree.\",\n \"Similarly, when a reticulation between P2 and P3 occurs, this results in an increase in the probability of the discordant gene tree that groups P2 and P3 as sister taxa.\",\n \"These three scenarios are illustrated in Fig.\",\n \"REF .\",\n \"Assuming an infinite sites model of sequence evolution, the frequencies of gene trees ((P1,(P2,P3)),O) and (((P1,P3),P2),O) directly correlate with the values $N_{ABBA}$ and $N_{BABA}$ , respectively, explaining the rationale behind Eq.\",\n \"(REF ).\",\n \"To apply the $D$ -statistic, frequencies of the $ABBA$ and $BABA$ site patterns are counted across an alignment of four genomes, the value of Eq.\",\n \"(REF ) is calculated, and deviation from 0 is assessed for statistical significance.\",\n \"A significant deviation is taken as evidence of introgression.\",\n \"Since the introduction of the $D$ -Statistic, work has been done to extend this framework.\",\n \"Recently, the software package HyDe [5] was introduced with several extensions including handling multiple individuals from four populations as well as identifying individual hybrids in a population based on the method of [62].\",\n \"In HyDe, higher numbers of individuals are handled through calculating statistics on all permutations of quartets of the individuals.\",\n \"Another recent extension to move the $D$ -Statistic beyond four taxa is the DFOIL framework introduced by [91].\",\n \"In it, we see the same derivation used in the $D$ -Statistic on a particular five-taxon tree.\",\n \"This derivation includes isolating gene trees whose probabilities go from equal to unequal when going from the tree case to the network case as well as converting these gene trees to corresponding site patterns to count in an alignment.\",\n \"Finally, Elworth et al.\",\n \"recently devised a heuristic, $D_{GEN}$ , for automatically deriving phylogenetic invariants for detecting hybridization in more general cases than the $D$ -Statistic and DFOIL can handle [25].\",\n \"The rationale behind the approach of Elworth et al.\",\n \"is that invariants could be derived by computing the probabilities of gene trees under a given species tree (e.g., using the method of [19]) and then computing the probabilities of the same trees under the same species tree with any reticulation scenarios added to it (using the method of [126]), and contrasting the two to identify sets of gene trees whose equal probabilities under the tree model get violated under the network model.\",\n \"The $D$ -Statistic is very simple to implement and understand, and it can be calculated on 4-genome alignments very efficiently, making it an appealing choice of a test for detecting introgression.\",\n \"Indeed, applications of the $D$ -Statistic are widespread in the literature, reporting on introgression in ancient hominids [39], [22], butterflies [114], bears [63], and sparrows [23], just to name a few.\",\n \"However, it is important to note here that the derivation of the $D$ -Statistic (and its extensions) relies on many assumptions that can easily be violated in practice.\",\n \"One major cause of such a violation is that the mathematics behind the $D$ -Statistic relies on the coalescent which makes many simplifying assumptions about the evolutionary model and processes taking place.\",\n \"Of course, this shortcoming applies as well to all phylogenetic inference methods that employ the MSC or MSNC models.\",\n \"A second cause of such a violation is that in practice, more than a single reticulation could have taken place and ignoring those could result in erroneous inferences [25].\",\n \"A third violation stems from the way the $D$ -Statistic is applied.We especially thank David Morrison for requesting that we highlight this issue.\",\n \"In propositional logic, the statement “If $p$ , then $q$ \\\" and its converse “If $q$ , then $p$ \\\" are logically not equivalent.\",\n \"That is, if one is true, it is not necessarily the case the other is.\",\n \"Looking back at Fig.\",\n \"REF , the statement illustrated by the figure is: If there is no reticulation (i.e., the species phylogeny is a tree), then the probabilities of the two discordant trees are equal.\",\n \"The converse (if the probabilities of the two discordant trees are equal, then the species phylogeny is a tree) does not follow logically.\",\n \"However, this is how the test is used in practice.\",\n \"In all fairness, though, this logical fallacy is commonplace in inferences in biology, including when inferring species trees and networks under the MSC and MSNC, respectively.\",\n \"The fallacy is always dealt with by resorting to the “simplest possible explanation\\\" argument.\",\n \"For example, why various scenarios could have given rise to equal frequencies of the frequencies of the $ABBA$ and $BABA$ site patterns, the species tree scenario is considered the simplest such possible explanation and is invoked as such.\",\n \"Last but not least, Peter [92] recently provided a review and elegant connections between the $D$ -Statistic and a family of statistics known as the $F$ -statistics.\"\n ],\n [\n \"Phylogenetic Networks in the Population Genetics Community\",\n \"The population genetics community has long adopted rooted, directed acyclic graphs as a model of evolutionary histories, typically of individuals within a single species.\",\n \"Ancestral recombination graphs, or ARGs, were introduced [49], [40] to model the evolutionary history of a set of genomic sequences in terms of the coalescence and recombination events that occurred since their most recent common ancestor.\",\n \"Statistical methods for inference of ARGs from genome-wide data have also been developed [97].\",\n \"Gusfield's recent book [42] discusses algorithmic and combinatorial aspects of ARGs.\",\n \"However, while ARGs take the shape of a phylogenetic network as defined above, they are aimed at modeling recombination and methods for their inference are generally not applicable to hybridization detection.\",\n \"Efforts in the population genetics community that are aimed at modeling admixture and gene flow are more relevant to hybridization detection.\",\n \"Here we discuss one of the most popular methods in this domain, namely TreeMix [93].\",\n \"In population genetics, the counterparts to species trees and phylogenetic networks are population trees and admixture graphs, respectively.\",\n \"The difference between these models boils down to what labels their leaves: If the leaves are labeled by different species, then the models are called species trees/networks, and if the leaves are labeled by different sub-populations of the same species, then the models are called population trees and admixture graphs.\",\n \"Of course it is not always easy to identify whether a species or sub-populations have been delimited, and hence what particular tree/network should be called in that case, as species and populations may exist on a continuum [18].\"\n ],\n [\n \"TreeMix\",\n \"TreeMix [93] models the evolution of a set of SNPs where the input data consists of allele frequencies of these SNPs in a set of populations whose evolutionary history is given by a population tree (in the case of no migration) or an admixture graph (in the case where migration is included).\",\n \"The basis of the model used in TreeMix is in the notion of modeling drift over time as a diffusion process, where an original allele frequency of $x_1$ of a given SNP undergoes drift by an amount $c$ to give rise to a new allele frequency $x_2$ [11], [85], [17], as given by $x_2 = x_1 + N(0,c \\\\cdot x_1[1-x_1]).$ It is worth noting here that, as pointed out in [93], $c ~= t/2N_e$ for drift over small time scales where the time scale is on the same order of the effective population size [85].\",\n \"When there are multiple populations under the effects of drift that evolved down a tree, the drift processes become linked and can no longer be described with independent Gaussian additions.\",\n \"This process is modeled with a covariance matrix derived from the amounts of drift occurring along the branches of the evolutionary tree.\",\n \"Finally, to incorporate reticulate evolution into the model one only needs to alter this covariance matrix based on the rate of gene flow along reticulate edges in the admixture graph.\",\n \"In its current implementation, the authors of TreeMix assume the evolutionary history of the sampled, extant populations is very close to a tree-like structure.\",\n \"Based on this assumption, the search for a maximum likelihood admixture graph proceeds by first estimating a rooted tree, and then adding migration events one at a time until they are no longer statistically significant (however, as the authors point out, they “prefer to stop adding migration events well before this point so that the result graph remains interpretable.\\\").\",\n \"Clearly, adding additional edges connecting the edges of a tree in this way will infer a tree-based network, which is a more limited class of networks compared with phylogenetic networks [33].\"\n ],\n [\n \"Data, Methods, and Software\",\n \"Given the interest in reticulate evolution from both theoreticians and empirical researchers, it is perhaps unsurprising that software to infer hybridization has proliferated in recent years.\",\n \"Such methods have been developed for a variety of data types, including multilocus data, SNP matrices, and whole genomes (Table ).\",\n \"Some of these methods are able to infer a phylogenetic network, whereas others infer introgression between species tree lineages.\",\n \"With the exception of TreeMix and MixMapper, methods which infer networks are not constructed around a backbone tree, and so do not assume that tree-like evolution is the dominant process.\",\n \"Regardless of the input and output, most of these methods allow for ILS in addition to hybridization (Table ), which is necessary to infer phylogenetic networks representing reticulate evolution in biological systems where ILS is a possibility.\",\n \"Likelihood (including Bayesian) methods incorporate the possibility or effect of ILS into the likelihood function.\",\n \"Maximum parsimony methods that minimize deep coalescences, for example InferNetwork_MP, essentially attempt to infer the tree or network that minimizes the quantity of ILS, but do not necessarily eliminate all genetic discordance.\",\n \"Methods which do not allow for ILS will instead infer phylogenetic networks representing conflicting signals [9].\",\n \"Reticulation is one such conflicting signal, but so is ILS, so reticulate branches in these networks should not be blindly interpreted as necessarily representing introgression or hybridization.\",\n \"Common methods to infer hybridization Table: NO_CAPTION\"\n ],\n [\n \"Limitations\",\n \"The biggest limitation of methods to infer introgression and hybridization, including species network methods, is scalability.\",\n \"Methods which infer a species network directly from multilocus sequences have only been used with a handful of taxa, and less than 200 loci.\",\n \"A systematic study of the species tree method StarBEAST found that the number of loci used has a power law relationship with a large exponent with the required computational time, making inference using thousands of loci intractable [87].\",\n \"Although no systematic study of computational performance has been conducted for equivalent species network methods such as MCMC_SEQ, anecdotally they suffer from similar scaling issues.\",\n \"Methods which scale better than direct multilocus inference have been developed, but they are no silver bullet.\",\n \"Species networks can be estimated directly from unlinked biallelic markers by integrating over all possible gene trees for each marker, which avoids having to sequentially or jointly estimate gene trees.\",\n \"Biallelic methods make the inference of species trees and networks from thousands of markers possible, at the cost of using less informative markers.\",\n \"Pseudolikelihood inference has been developed for both biallelic and multilocus methods [136], [128].\",\n \"This reduces the computational cost of computing the likelihood of a species network as the number of taxa increases, and enabled the reanalysis of an empirical data set with 1070 genes from 23 species [128].\",\n \"The ABBA-BABA test and similar phylogenetic invariant methods are capable of analyzing an enormous depth of data (whole genomes), but can be limited in taxonomic breadth based on hard limits of four or five taxa for the D-Statistic and $D_{FOIL}$ , respectively, or by computational requirements for the case of $D_{GEN}$ (Table ).\",\n \"In addition, the D-Statistic and $D_{FOIL}$ are limited to testing a specific hypothesis for introgression given a fixed species tree topology of a specific shape.\",\n \"This can be understood as a trade off, where the flexibility of species network methods is sacrificed for the ability to use more data.\",\n \"Beyond scalability, another present limitation is visualizing or summarizing posterior or bootstrap distributions of networks.\",\n \"Methods have been developed to visualize whole distributions of trees, or summarize a distribution as a single tree.\",\n \"Equivalent tools for networks are underdeveloped, leaving researchers to report the topology or set of topologies with the highest posterior or bootstrap support.\"\n ],\n [\n \"Conclusions and Future Directions\",\n \"Great strides have been made over the past decade in the inference of evolutionary histories in the presence of hybridization and other processes, most notably incomplete lineage sorting.\",\n \"Species networks can now be inferred directly from species-level data which do not assume any kind of backbone tree, and instead put reticulate evolution on an equal basis with speciation.\",\n \"To some extent the development of species network methods have recapitulated the development of species tree methods, starting with maximum parsimony and transitioning to likelihood methods, both maximum likelihood and Bayesian.\",\n \"To improve computational performance and enable the analysis of large data sets, pseudo-likelihood species network methods have been developed, inspired by similar species tree methods.\",\n \"Phylogenetic invariant methods such as the ABBA-BABA test are able to test for reticulate evolution across whole genomes, uncovering chromosomal inversions and other features associated with hybridization and introgression.\",\n \"Last but not least, the population genetics community has long been interested in and developing methods for phylogenetic networks mainly to model the evolution of sub-populations in the presence of admixture and gene flow.\",\n \"In this chapter, we surveyed the recent computational developments in the field and listed computer software programs that enable reticulate evolutionary analyses for the study of hybridization and introgression, and generally to infer more accurate evolutionary histories of genes and species.\",\n \"Empirical biologists feel constrained by the computational performance of existing species network methods.\",\n \"For species trees, phylogenetic invariant methods can be combined with quartet reconciliation to infer large species trees from genomic data, as in SVDquartets [14].\",\n \"For networks, phylogenetic invariant methods to identify the true network with a limited number of edges need to be developed, as do methods to reconcile the resulting subnets.\",\n \"Even for species trees, Bayesian methods have practical limitations in terms of the amount of data they can be used with.\",\n \"Bayesian methods for trees and networks, with few exceptions, have been built on Markov chain Monte Carlo (MCMC).\",\n \"This technique is inherently serial and hence unsuited to modern workstations, which contain many CPU and GPU cores working in parallel.\",\n \"It is important to continue to explore other Bayesian algorithms which work in parallel such as sequential Monte Carlo [7], or algorithms which are orders of magnitude faster than MCMC such as variational Bayes [118].\",\n \"Phylogenetic methods for species tree inference have a huge head start on methods for species network inference.\",\n \"Not only is the problem of species network inference much more complicated, but species tree methods have been in development for much longer.\",\n \"For example, MDC for species trees was first described in 1997, and extended to phylogenetic networks 14 years later [71], [130].\",\n \"In this light the progress made is remarkable.\",\n \"However, as evolutionary biology is moving towards data sets containing whole genomes for hundreds or even thousands of taxa, methods developers must focus on improving the scalability of their methods without sacrificing accuracy so that the full potential of this data may be realized.\",\n \"While preliminary studies exist of the performance of the different methods for phylogenetic network inference [58], more thorough studies are needed to assess the accuracy as well as computational requirements of the different methods.\",\n \"Last but not least, it is important to highlight that all the development described above excludes processes such as gene duplication and loss, and so may be susceptible to errors and artifacts which can be present in data such as hidden paralogy.\",\n \"Furthermore, the multispecies network coalescent already has its own population-genetic assumptions, almost all of which are not necessarily realistic for analyses in practice.\",\n \"Accounting for these is a major next step (though it is important to point out that these have not been fully explored in the context of species tree inference either), but the mathematical complexity will most likely add, extensively, to the computational complexity of the inference step.\",\n \"Luay Nakhleh started working on phylogenetic networks in 2002 in a close collaboration with Tandy Warnow, Bernard M.E.\",\n \"Moret, and C. Randal Linder.\",\n \"At the time, their focus was on phylogenetic networks in terms of displaying trees.\",\n \"This focus led to work on inference of smallest phylogenetic networks that display a given set of trees, as well as on comparing networks in terms of their displayed trees.\",\n \"While the approaches pursued at the time were basic, they were foundational in terms of pursuing more sophisticated models and approaches by Nakhleh and his group.\",\n \"Therefore, we would like to acknowledge the role that Bernard played in the early days of (explicit) phylogenetic networks.\",\n \"The authors would also like to acknowledge James Mallet, Craig Moritz, David Morrison, and Mike Steel for extensive discussions and detailed comments that helped us significantly improve this chapter.\",\n \"The authors thank Matthew Hahn and Kelley Harris for their discussion of the definition of phylogenetic invariant methods.\",\n \"This work was partially supported by NSF grants DBI-1355998, CCF-1302179, CCF-1514177, CCF-1800723, and DMS-1547433.\"\n ]\n]"},"id":{"kind":"string","value":"1808.08662"}}},{"rowIdx":296,"cells":{"text":{"kind":"list like","value":[["Parametrized Measuring and Club Guessing"],["Abstract We introduce Strong Measuring, a maximal strengthening of J. T. Moore's Measuring principle, which asserts that every collection of fewer than continuum many closed bounded subsets of $\\omega_1$ is measured by some club subset of $\\omega_1$.","The consistency of Strong Measuring with the negation of CH is shown, solving an open problem from about parametrized measuring principles.","Specifically, we prove that Strong Measuring follows from MRP together with Martin's Axiom for $\\sigma$-centered forcings, as well as from BPFA.","We also consider strong versions of Measuring in the absence of the Axiom of Choice."],["Background","We review some background material and notation which is needed for understanding the paper.","Let $\\mathfrak {c}$ denote the cardinality of the continuum $2^\\omega $ .","A set $S \\subseteq [\\omega ]^\\omega $ is a splitting family if for any infinite set $x \\subseteq \\omega $ , there exists $A \\in S$ such that $A$ splits $x$ in the sense that both $x \\cap A$ and $x \\setminus A$ are infinite.","The splitting number $\\mathfrak {s}$ is the least cardinality of some splitting family.","Given functions $f, g : \\omega \\rightarrow \\omega $ , we say that $g$ dominates $f$ if for all $n < \\omega $ , $f(n) < g(n)$ .","We say that $g$ eventually dominates $f$ if there is some $m<\\omega $ such that $f(n)m$ .","A family $B \\subseteq \\omega ^\\omega $ is bounded if there exists a function $g \\in \\omega ^\\omega $ which eventually dominates every member of $B$ , and otherwise it is unbounded.","The bounding number $\\mathfrak {b}$ is the least cardinality of some unbounded family.","Both cardinal characteristics $\\mathfrak {s}$ and $\\mathfrak {b}$ are uncountable.","Let $\\mathbb {P}$ be a forcing poset.","A set $X \\subseteq \\mathbb {P}$ is centered if every finite subset of $X$ has a lower bound.","We say that $\\mathbb {P}$ is $\\sigma $ -centered if it is a union of countably many centered sets.","Martin's Axiom for $\\sigma $ -centered forcings ($\\textsf {MA}$ ($\\sigma $ -centered)) is the statement that for any $\\sigma $ -centered forcing $\\mathbb {P}$ and any collection of fewer than $\\mathfrak {c}$ many dense subsets of $\\mathbb {P}$ , there exists a filter on $\\mathbb {P}$ which meets each dense set in the collection.","More generally, let $\\mathfrak {m}$ ($\\sigma $ -centered) be the least cardinality of a collection of dense subsets of some $\\sigma $ -centered forcing poset for which there does not exist a filter which meets each dense set in the collection.","Note that $\\textsf {MA}$ ($\\sigma $ -centered) is equivalent to the statement that $\\mathfrak {m}$ ($\\sigma $ -centered) equals $\\mathfrak {c}$ .","The Bounded Proper Forcing Axiom (BPFA) is the statement that whenever $\\mathbb {P}$ is a proper forcing and $\\langle A_i : i < \\omega _1 \\rangle $ is a sequence of maximal antichains of $\\mathbb {P}$ each of size at most $\\omega _1$ , then there exists a filter on $\\mathbb {P}$ which meets each $A_i$ ([9]).","We note that BPFA implies $\\mathfrak {c} = \\omega _2$ ([12]).","It easily follows that BPFA implies Martin's Axiom, and in particular, implies $\\textsf {MA}$ ($\\sigma $ -centered).","The forcing axiom BPFA is equivalent to the statement that for any proper forcing poset $\\mathbb {P}$ and any $\\Sigma _1$ statement $\\Phi $ with a parameter from $H(\\omega _2)$ , if $\\Phi $ holds in a generic extension by $\\mathbb {P}$ , then $\\Phi $ holds in the ground model ([7]).","An open stationary set mapping for an uncountable set $X$ and regular cardinal $\\theta > \\omega _1$ is a function $\\Sigma $ whose domain is the collection of all countable elementary substructures $M$ of $H(\\theta )$ with $X \\in M$ , such that for all such $M$ , $\\Sigma (M)$ is an open, $M$ -stationary subset of $[X]^\\omega $ .","By open we mean in the Ellentuck topology on $[X]^\\omega $ , and $M$ -stationary means meeting every club subset of $[X]^\\omega $ which is a member of $M$ (see [12] for the complete details).","In this article, we are only concerned with these ideas in the simplest case that $X = \\omega _1$ and for each $M \\in \\mathrm {dom}(\\Sigma )$ , $\\Sigma (M) \\subseteq \\omega _1$ .","In this case, being open is equivalent to being open in the topology on $\\omega _1$ with basis the collection of all open intervals of ordinals, and being $M$ -stationary is equivalent to meeting every club subset of $\\omega _1$ in $M$ .","For an open stationary set mapping $\\Sigma $ for $X$ and $\\theta $ , a $\\Sigma $ -reflecting sequence is an $\\in $ -increasing and continuous sequence $\\langle M_i : i < \\omega _1 \\rangle $ of countable elementary substructures of $H(\\theta )$ containing $X$ as a member satisfying that for all limit ordinals $\\delta < \\omega _1$ , there exists $\\beta < \\delta $ so that for all $\\beta \\le \\xi < \\delta $ , $M_\\xi \\cap X \\in \\Sigma (M_\\delta )$ .","The Mapping Reflection Principle (MRP) is the statement that for any open stationary set mapping $\\Sigma $ , there exists a $\\Sigma $ -reflecting sequence.","We will use the fact that for any open stationary set mapping $\\Sigma $ , there exists a proper forcing which adds a $\\Sigma $ -reflecting sequence ([12]).","Consequently, MRP follows from PFA."],["Parametrized Measuring and Club Guessing","Let $X$ and $Y$ be countable subsets of $\\omega _1$ with the same supremum $\\delta $ .","We say that $X$ measures $Y$ if there exists $\\beta < \\delta $ such that $X \\setminus \\beta $ is either contained in, or disjoint from, $Y$ .","Measuring is the statement that for any sequence $\\langle c_\\alpha : \\alpha \\in \\omega _1 \\cap \\mathrm {Lim}\\rangle $ , where each $c_\\alpha $ is a closed and cofinal subset of $\\alpha $ , there exists a club $D \\subseteq \\omega _1$ such that for all limit points $\\alpha $ of $D$ , $D \\cap \\alpha $ measures $c_\\alpha $ .","The next two results are due to J. T. Moore ([8]).","Theorem 2.1 MRP implies Measuring.","Theorem 2.2 BPFA implies Measuring.","We now describe parametrized forms of measuring which were introduced in [2].","Let $\\vec{\\mathcal {C}} = \\langle \\mathcal {C}_\\alpha : \\alpha \\in \\omega _1 \\cap \\textrm {Lim} \\rangle $ be a sequence such that each $\\mathcal {C}_\\alpha $ is a collection of closed and cofinal subsets of $\\alpha $ .","A club $D \\subseteq \\omega _1$ is said to measure $\\vec{\\mathcal {C}}$ if for all $\\alpha \\in \\lim (D)$ and all $c \\in \\mathcal {C}_\\alpha $ , $D \\cap \\alpha $ measures $c$ .","Definition 2.3 For a cardinal $\\kappa $ , let Measuring$_{< \\kappa }$ denote the statement that whenever $\\vec{\\mathcal {C}} = \\langle \\mathcal {C}_\\alpha : \\alpha \\in \\omega _1 \\cap \\textrm {Lim} \\rangle $ is a sequence such that each $\\mathcal {C}_\\alpha $ is a collection of fewer than $\\kappa $ many closed and cofinal subsets of $\\alpha $ , then there exists a club $D \\subseteq \\omega _1$ which measures $\\vec{\\mathcal {C}}$ .","For a cardinal $\\lambda $ , let $\\textsf {Measuring}_\\lambda $ denote $\\textsf {Measuring}_{< \\lambda ^+}$ .","Observe that the principle Measuring is the same as $\\textsf {Measuring}_1$ .","If $\\kappa < \\lambda $ , then clearly Measuring$_{< \\lambda }$ implies Measuring$_{<\\kappa }$ .","It is easy to see that Measuring$_{\\mathfrak {c}}$ is false.","Definition 2.4 Strong Measuring is the statement that Measuring$_{<\\mathfrak {c}}$ holds.","Since the intersection of countably many clubs in $\\omega _1$ is club, Measuring easily implies Measuring$_\\omega $ .","In particular, Measuring together with CH implies Strong Measuring.","We will prove in Section 3 the consistency of Strong Measuring together with $\\lnot \\textsf {CH}$ .","We also observe at the end of that section that Measuring does not imply Measuring$_{\\omega _1}$ .","Proposition 2.5 ([2]) Measuring$_{\\mathfrak {s}}$ is false.","Fix a splitting family $S$ of cardinality $\\mathfrak {s}$ .","For each limit ordinal $\\alpha < \\omega _1$ , fix a function $f_\\alpha : \\omega \\rightarrow \\alpha $ which is increasing and cofinal in $\\alpha $ .","For each $A \\in S$ , let $c_{\\alpha ,A} = \\bigcup \\lbrace (f_\\alpha (n),f_\\alpha (n+1)] : n \\in A \\rbrace $ , which is clearly closed and cofinal in $\\alpha $ .","Let $\\mathcal {C}_\\alpha := \\lbrace c_{\\alpha ,A} : A \\in S \\rbrace $ .","Then $\\vec{\\mathcal {C}} := \\langle \\mathcal {C}_\\alpha : \\alpha \\in \\omega _1 \\cap \\mathrm {Lim}\\rangle $ is a sequence such that for each $\\alpha $ , $\\mathcal {C}_\\alpha $ is a collection of at most $\\mathfrak {s}$ many closed and cofinal subsets of $\\alpha $ .","Let $D \\subseteq \\omega _1$ be a club.","Fix $\\alpha \\in \\lim (D)$ .","We will show that there exists a member of $\\mathcal {C}_\\alpha $ which $D \\cap \\alpha $ does not measure.","Define $x := \\lbrace n < \\omega : D \\cap (f_\\alpha (n),f_{\\alpha }(n+1)] \\ne \\emptyset \\rbrace $ .","Since $\\alpha \\in \\lim (D)$ , $x$ is infinite.","As $S$ is a splitting family, we can fix $A \\in S$ which splits $x$ .","So both $x \\cap A$ and $x \\setminus A$ are infinite.","We claim that $D \\cap \\alpha $ does not measure $c_{\\alpha ,A}$ .","Suppose for a contradiction that for some $\\beta < \\alpha $ , $(D \\cap \\alpha ) \\setminus \\beta $ is either a subset of, or disjoint from, $c_{\\alpha ,A}$ .","Since $A \\cap x$ is infinite, we can fix $n \\in A \\cap x$ such that $f_\\alpha (n) > \\beta $ .","Then $n \\in x$ implies that $D \\cap (f_\\alpha (n),f_{\\alpha }(n+1)] \\ne \\emptyset $ , and $n \\in A$ implies that $(f_{\\alpha }(n),f_{\\alpha }(n+1)] \\subseteq c_{\\alpha ,A}$ .","It follows that $(D \\cap \\alpha ) \\setminus \\beta $ meets $c_{\\alpha ,A}$ .","By the choice of $\\beta $ , this implies that $(D \\cap \\alpha ) \\setminus \\beta $ is a subset of $c_{\\alpha ,A}$ .","But $x \\setminus A$ is also infinite, so we can fix $m \\in x \\setminus A$ such that $f_{\\alpha }(m) > \\beta $ .","Then $m \\in x$ implies that $D \\cap (f_\\alpha (m),f_\\alpha (m+1)] \\ne \\emptyset $ , and $m \\notin A$ implies that $(f_{\\alpha }(m),f_{\\alpha }(m+1)]$ is disjoint from $c_{\\alpha ,A}$ .","Thus, there is a member of $(D \\cap \\alpha ) \\setminus \\beta $ which is not in $c_{\\alpha ,A}$ , which is a contradiction.","We will prove later in this section that Measuring$_{\\mathfrak {b}}$ is also false.","We now turn to parametrized club guessing.","We recall some standard definitions.","Consider a sequence $\\vec{L} = \\langle L_\\alpha : \\alpha \\in \\omega _1 \\cap \\mathrm {Lim}\\rangle $ , where each $L_\\alpha $ is a cofinal subset of $\\alpha $ with order type $\\omega $ (that is, a ladder system).","We say that $\\vec{L}$ is a club guessing sequence, weak club guessing sequence, or very weak club guessing sequence, respectively, if for every club $D \\subseteq \\omega _1$ , there exists a limit ordinal $\\alpha < \\omega _1$ such that: $L_\\alpha \\subseteq D$ , $L_\\alpha \\setminus D$ is finite, or $L_\\alpha \\cap D$ is infinite, respectively.","We say that Club Guessing, Weak Club Guessing, or Very Weak Club Guessing holds, respectively, if there exists a club guessing sequence, a weak club guessing sequence, or a very weak club guessing sequence, respectively.","It is well known that Measuring implies the failure of Very Weak Club Guessing (see Proposition 2.8 below).","Definition 2.6 Let $\\vec{\\mathcal {L}} = \\langle \\mathcal {L}_\\alpha : \\alpha \\in \\omega _1 \\cap \\mathrm {Lim}\\rangle $ be a sequence where each $\\mathcal {L}_\\alpha $ is a non–empty collection of cofinal subsets of $\\alpha $ with order type $\\omega $ .","The sequence $\\vec{\\mathcal {L}}$ is said to be a club guessing sequence, weak club guessing sequence, or very weak club guessing sequence, respectively, if for every club $D \\subseteq \\omega _1$ , there exists a limit ordinal $\\alpha < \\omega _1$ and some $L \\in \\mathcal {L}_\\alpha $ such that: $L \\subseteq D$ , $L \\setminus D$ is finite, or $L \\cap D$ is infinite, respectively.","Definition 2.7 For a cardinal $\\kappa $ , let CG$_{< \\kappa }$ , WCG$_{<\\kappa }$ , and VWCG$_{<\\kappa }$ , respectively, be the statements that there exists a club guessing sequence, weak club guessing sequence, or very weak club guessing sequence $\\langle \\mathcal {L}_\\alpha : \\alpha \\in \\omega _1 \\cap \\mathrm {Lim}\\rangle $ , respectively, such that for each $\\alpha $ , $|\\mathcal {L}_\\alpha | < \\kappa $ .","Let CG$_{\\kappa }$ , WCG$_{\\kappa }$ , and VWCG$_{\\kappa }$ denote the statements CG$_{< \\kappa ^+}$ , WCG$_{< \\kappa ^+}$ , and VWCG$_{< \\kappa ^+}$ , respectively.","Clearly, if $\\kappa < \\lambda $ , then CG$_{< \\kappa }$ implies CG$_{<\\lambda }$ , and similarly with WCG and VWCG.","Observe that Club Guessing, Weak Club Guessing, and Very Weak Club Guessing are equivalent to CG$_{1}$ , WCG$_{1}$ , and VWCG$_{1}$ , respectively.","Obviously, CG$_{\\mathfrak {c}}$ is true.","The weakest forms of club guessing principles which are not provable in ZFC are when the index is $< \\mathfrak {c}$ .","Proposition 2.8 For any cardinal $\\kappa \\ge 2$ , Measuring$_{< \\kappa }$ implies the failure of VWCG$_{< \\kappa }$ .","Suppose for a contradiction that Measuring$_{< \\kappa }$ and VWCG$_{<\\kappa }$ both hold.","Fix a very weak club guessing sequence $\\vec{\\mathcal {L}} = \\langle \\mathcal {L}_\\alpha : \\alpha \\in \\omega _1 \\cap \\mathrm {Lim}\\rangle $ such that each $\\mathcal {L}_\\alpha $ has cardinality less than $\\kappa $ .","Observe that for each $\\alpha $ , every member of $\\mathcal {L}_\\alpha $ is vacuously a closed subset of $\\alpha $ since it has order type $\\omega $ .","By Measuring$_{< \\kappa }$ , there exists a club $D \\subseteq \\omega _1$ which measures $\\vec{\\mathcal {L}}$ .","Let $E$ be the club set of indecomposable limit ordinals $\\alpha >\\omega $ in $\\lim (D)$ such that $\\mathrm {ot}(D \\cap \\alpha ) = \\alpha $ .","Since $\\vec{\\mathcal {L}}$ is a very weak club guessing sequence, there exists a limit ordinal $\\alpha $ and $L \\in \\mathcal {L}_\\alpha $ such that $L \\cap E$ is infinite.","In particular, $\\alpha $ is a limit point of $E$ , and hence of $D$ .","Since $D$ measures $\\vec{\\mathcal {L}}$ and $L \\in \\mathcal {L}_\\alpha $ , $D \\cap \\alpha $ measures $L$ .","So we can fix $\\beta < \\alpha $ such that $(D \\cap \\alpha ) \\setminus \\beta $ is either a subset of, or disjoint from, $L$ .","Now $L \\cap E$ , and hence $L \\cap D$ , is infinite.","As $L$ has order type $\\omega $ , this implies that $L \\cap D$ is cofinal in $\\alpha $ .","By the choice of $\\beta $ , $(D \\cap \\alpha ) \\setminus \\beta $ must be a subset of $L$ .","But since $\\alpha \\in E$ , $\\mathrm {ot}(D \\cap \\alpha ) = \\alpha $ and $\\alpha $ is indecomposable, which implies that $\\mathrm {ot}((D \\cap \\alpha ) \\setminus \\beta ) = \\alpha $ .","As $\\alpha > \\omega $ , this is impossible since $(D \\cap \\alpha ) \\setminus \\beta $ is a subset of $L$ and $L$ has order type $\\omega $ .","In particular, since Strong Measuring is consistent, so is the failure of VWCG$_{< \\mathfrak {c}}$ .","(The consistency of $\\lnot \\textsf {VWCG}_{< \\mathfrak {c}}$ together with $\\mathfrak {c}$ arbitrarily large was previously shown in [4].)","Proposition 2.9 (Hrušák [5]) VWCG$_{\\mathfrak {b}}$ is true.","Fix an unbounded family $\\lbrace r_\\alpha : \\alpha < \\mathfrak {b} \\rbrace $ in $\\omega ^\\omega $ .","For each limit ordinal $\\delta < \\omega _1$ , fix a cofinal subset $C_\\delta $ of $\\delta $ with order type $\\omega $ and a bijection $h_\\delta : \\omega \\rightarrow \\delta $ .","Let $C_\\delta (n)$ denote the $n$ -th member of $C_\\delta $ for all $n < \\omega $ .","For all limit ordinals $\\delta < \\omega _1$ and $\\alpha < \\mathfrak {b}$ , define $A_\\delta ^\\alpha := C_\\delta \\cup \\bigcup \\lbrace h_\\delta [r_\\alpha (n)] \\setminus C_\\delta (n) : n < \\omega \\rbrace .$ It is easy to check that for all $\\delta $ and $\\alpha $ , $A_\\delta ^\\alpha $ has order type $\\omega $ and $\\sup (A_\\delta ^\\alpha ) = \\delta $ .","Given a club $C \\subseteq \\omega _1$ , let $\\delta $ be a limit point of $C$ and let $g_{C, \\delta } : \\omega \\rightarrow \\omega $ be the function given by $g_{C, \\delta }(n) =\\min \\lbrace m < \\omega :h_\\delta (m) \\in C \\setminus C_\\delta (n) \\rbrace .$ Now let $\\alpha < \\mathfrak {b}$ be such that $r_\\alpha (n) > g_{C, \\delta }(n)$ for infinitely many $n$ .","It then follows that $|A_\\delta ^\\alpha \\cap C| = \\omega $ .","By Propositions 2.8 and 2.9, the following is immediate.","Corollary 2.10 Measuring$_{\\mathfrak {b}}$ is false.","An obvious question is whether the parametrized versions of club guessing are actually the same as the usual ones.","We conclude this section by showing that they are not.","Recall that a forcing poset $\\mathbb {P}$ is $\\omega ^\\omega $ -bounding if every function in $\\omega ^\\omega \\cap V^\\mathbb {P}$ is dominated by a function in $\\omega ^\\omega \\cap V$ .","Lemma 2.11 (Hrušák) Assume that VWCG fails.","Let $\\mathbb {P}$ be any $\\omega _1$ -c.c., $\\omega ^\\omega $ -bounding forcing.","Then $\\mathbb {P}$ forces that VWCG fails.","Since $\\mathbb {P}$ is $\\omega _1$ -c.c.","and $\\omega ^\\omega $ -bounding, a standard argument shows that whenever $p \\in \\mathbb {P}$ and $p$ forces that $\\dot{b} \\in \\omega ^\\omega $ , then there exists a function $b^* \\in \\omega ^\\omega $ such that $p$ forces that $b^*$ dominates $\\dot{b}$ .","Let us show that whenever $p \\in \\mathbb {P}$ , $\\delta < \\omega _1$ , and $p$ forces that $\\dot{X}$ is a cofinal subset of $\\delta $ of order type $\\omega $ , then there exists a set $Y$ with order type $\\omega $ such that $p$ forces that $\\dot{X} \\subseteq Y$ .","To see this, fix a bijection $f : \\omega \\rightarrow \\delta $ and a strictly increasing sequence $\\langle \\alpha _n : n < \\omega \\rangle $ cofinal in $\\alpha $ with $\\alpha _0 = 0$ .","We claim that there exists a $\\mathbb {P}$ -name $\\dot{b}$ for a function from $\\omega $ to $\\omega $ such that $p$ forces that for all $n < \\omega $ , $\\dot{b}(n)$ is the least $m < \\omega $ such that $\\dot{X} \\cap [\\alpha _n,\\alpha _{n+1}) \\subseteq f[m]$ .","This is true since $p$ forces that $\\dot{X}$ has order type $\\omega $ and hence that $\\dot{X} \\cap [\\alpha _n,\\alpha _{n+1})$ is finite for all $n < \\omega $ .","Fix a function $b^* : \\omega \\rightarrow \\omega $ such that $p$ forces that $b^*$ dominates $\\dot{b}$ .","Now let $Y := \\bigcup \\lbrace f[b^*(n)] \\cap [\\alpha _n,\\alpha _{n+1}) : n < \\omega \\rbrace .$ It is easy to check that $Y$ has order type $\\omega $ and $p$ forces that $\\dot{X} \\subseteq Y$ .","Now we are ready to prove the proposition.","So suppose that $p \\in \\mathbb {P}$ forces that $\\langle \\dot{X}_\\alpha : \\alpha \\in \\omega _1 \\cap \\mathrm {Lim}\\rangle $ is a very weak club guessing sequence.","By the previous paragraph, for each limit ordinal $\\alpha < \\omega _1$ we can fix a cofinal subset $Y_\\alpha $ of $\\alpha $ with order type $\\omega $ such that $p$ forces that $\\dot{X}_\\alpha \\subseteq Y_\\alpha $ .","We claim that $\\langle Y_\\alpha : \\alpha \\in \\omega _1 \\cap \\mathrm {Lim}\\rangle $ is a very weak club guessing sequence in the ground model, which completes the proof.","So consider a club $C \\subseteq \\omega _1$ .","Then $C$ is still a club in $V^\\mathbb {P}$ .","Fix $q \\le p$ and a limit ordinal $\\alpha < \\omega _1$ such that $q$ forces that $\\dot{X}_\\alpha \\cap C$ is infinite.","Then clearly $q$ forces that $Y_\\alpha \\cap C$ is infinite, so in fact, $Y_\\alpha \\cap C$ is infinite.","Proposition 2.12 It is consistent that $\\lnot \\textsf {VWCG}$ and $\\textsf {CG}_{\\omega _1}$ both hold.","Let $V$ be a model in which CH holds and VWCG fails.","Such a model was shown to exist by Shelah [13].","Let $\\mathbb {P}$ be an $\\omega _1$ -c.c., $\\omega ^\\omega $ -bounding forcing poset which adds at least $\\omega _2$ many reals; for example, random real forcing with product measure is such a forcing.","We claim that in $V^\\mathbb {P}$ , CG$_{\\omega _1}$ holds but VWCG fails.","By Lemma 2.11, VWCG is false in $V^\\mathbb {P}$ .","In $V$ , define $\\vec{\\mathcal {L}} = \\langle \\mathcal {L}_\\alpha : \\alpha \\in \\omega _1 \\cap \\mathrm {Lim}\\rangle $ by letting $\\mathcal {L}_\\alpha $ be the collection of all cofinal subsets of $\\alpha $ with order type $\\omega $ .","Since CH holds, the cardinality of each $\\mathcal {L}_\\alpha $ is $\\omega _1$ .","If $C$ is a club subset of $\\omega _1$ in $V^\\mathbb {P}$ , then since $\\mathbb {P}$ is $\\omega _1$ -c.c., there is a club $D \\subseteq \\omega _1$ in $V$ such that $D \\subseteq C$ .","In $V$ , fix $d \\subseteq D$ with order type $\\omega $ , and let $\\alpha := \\sup (d)$ .","Then $d \\in \\mathcal {L}_\\alpha $ and $d \\subseteq C$ .","Thus, $\\vec{\\mathcal {L}}$ witnesses that CG$_{\\omega _1}$ holds in $V^\\mathbb {P}$ ."],["The Consistency of Strong Measuring and $\\lnot \\textsf {CH}$","As we previously mentioned, Measuring is equivalent to Measuring$_\\omega $ , and therefore under CH, Measuring is equivalent to Strong Measuring.","In this section we establish the consistency of Strong Measuring with the negation of CH.","More precisely, we will prove that MRP together with MA($\\sigma $ -centered) implies Strong Measuring, and BPFA implies Strong Measuring.","Recall that both MRP and BPFA imply that $\\mathfrak {c} = \\omega _2$ ([12]).","A set $M$ is suitable if for some regular cardinal $\\theta > \\omega _1$ , $M$ is a countable elementary substructure of $H(\\theta )$ .","We will follow the conventions introduced in Section 1 that the properties “open” and “$M$ -stationary” refer to open and $M$ -stationary subsets of $\\omega _1$ (where $\\omega _1$ is considered as a subspace of $[\\omega _1]^\\omega $ ).","Proposition 3.1 Assume that $M$ is suitable.","Let $\\delta := M \\cap \\omega _1$ .","Suppose that $\\mathcal {Y}$ is a collection of open subsets of $\\delta $ such that for any finite set $a \\subseteq \\mathcal {Y}$ , $\\bigcap a$ is $M$ -stationary.","Then there exists a $\\sigma $ -centered forcing $\\mathbb {P}$ and a collection $\\mathcal {D}$ of dense subsets of $\\mathbb {P}$ of size at most $|\\mathcal {Y}| + \\omega $ such that whenever $G$ is a filter on $\\mathbb {P}$ in some outer model $W$ of $V$ with $\\omega _1^V = \\omega _1^W$ which meets each member of $\\mathcal {D}$ , then there exists a set $z \\subseteq \\delta $ in $W$ which is open, $M$ -stationary, and satisfies that for all $X \\in \\mathcal {Y}$ , $z \\setminus X$ is bounded in $\\delta $ .","Define a forcing poset $\\mathbb {P}$ to consist of conditions which are pairs $(x,a)$ , where $x$ is an open and bounded subset of $\\delta $ in $M$ and $a$ is a finite subset of $\\mathcal {Y}$ .","Let $(y,b) \\le (x,a)$ if $y$ is an end-extension of $x$ , $a \\subseteq b$ , and $y \\setminus x \\subseteq \\bigcap a$ .","Since $M$ is countable, there are only countably many possibilities for the first component of a condition.","If $(x,a_0), \\ldots , (x,a_n)$ are finitely many conditions with the same first component, then easily $(x,a_0 \\cup \\ldots \\cup a_n)$ is a condition in $\\mathbb {P}$ which is below each of the conditions $(x,a_0), \\ldots , (x,a_n)$ .","It follows that $\\mathbb {P}$ is $\\sigma $ -centered.","For each $X \\in \\mathcal {Y}$ , let $D_X$ denote the set of conditions $(x,a)$ such that $X \\in a$ .","Observe that $D_X$ is dense.","For every club $C$ of $\\omega _1$ which is a member of $M$ , let $E_C$ denote the set of conditions $(x,a)$ such that $x \\cap C$ is non–empty.","We claim that $E_C$ is dense.","Let $(x,a)$ be a condition.","Since $\\bigcap a$ is $M$ -stationary and $\\lim (C) \\setminus (\\sup (x)+1)$ is a club subset of $\\omega _1$ in $M$ , we can find a limit ordinal $\\alpha $ in $C \\cap (\\bigcap a)$ which is in the interval $(\\sup (x),\\delta )$ .","Since $\\alpha \\in \\bigcap a$ and $\\bigcap a$ is open, we can find $\\beta < \\gamma < \\delta $ such that $\\alpha \\in (\\beta ,\\gamma ) \\subseteq \\bigcap a$ .","As $\\sup (x) + 1 < \\alpha $ , without loss of generality $\\sup (x) < \\beta $ .","By elementarity, the interval $b := (\\beta ,\\gamma )$ is in $M$ .","It follows that $(x \\cup b,a)$ is a condition, $x \\cup b$ end-extends $x$ , and $(x \\cup b) \\setminus x = b \\subseteq \\bigcap a$ .","Thus, $(x \\cup b,a) \\le (x,a)$ , and since $\\alpha \\in C$ , $(x \\cup b,a) \\in E_C$ .","Let $\\mathcal {D}$ denote the collection of all dense sets of the form $D_X$ where $X \\in \\mathcal {Y}$ , or $E_C$ where $C$ is a club subset of $\\omega _1$ belonging to $M$ .","Then $|\\mathcal {D}| \\le |\\mathcal {Y}| + \\omega $ .","Let $G$ be a filter on $\\mathbb {P}$ in some outer model $W$ with $\\omega _1^V = \\omega _1^W$ which meets each dense set in $\\mathcal {D}$ .","Define $z := \\bigcup \\lbrace x : \\exists a \\ (x,a) \\in G \\rbrace $ .","Note that since $z$ is a union of open sets, it is open (using the fact that being open is absolute between $V$ and $W$ ).","For each club $C \\subseteq \\omega _1$ which lies in $M$ , there exists a condition $(x,a)$ which belongs to $G \\cap E_C$ , and thus $x \\cap C \\ne \\emptyset $ .","Therefore, $z \\cap C \\ne \\emptyset $ .","Hence, $z$ is $M$ -stationary.","It remains to show that for all $X \\in \\mathcal {Y}$ , $z \\setminus X$ is bounded in $\\delta $ .","Consider $X \\in \\mathcal {Y}$ .","Then we can fix $(x,a) \\in G \\cap D_X$ , which means that $X \\in a$ .","Now the definition of the ordering on $\\mathbb {P}$ together with the fact that $G$ is a filter easily implies that $z \\setminus x \\subseteq X$ .","Therefore, $z \\setminus X \\subseteq x$ , and hence $z \\setminus X$ is bounded in $\\delta $ .","Corollary 3.2 Assume that $M$ is suitable.","Let $\\delta := M \\cap \\omega _1$ .","Suppose that $\\mathcal {Y}$ is a collection of less than $\\mathfrak {m}$ ($\\sigma $ -centered) many open subsets of $\\delta $ such that for any finite set $a \\subseteq \\mathcal {Y}$ , $\\bigcap a$ is $M$ -stationary.","Then there exists a set $z \\subseteq \\delta $ which is open, $M$ -stationary, and satisfies that for all $X \\in \\mathcal {Y}$ , $z \\setminus X$ is bounded in $\\delta $ .","Fix a $\\sigma $ -centered forcing $\\mathbb {P}$ and a collection $\\mathcal {D}$ of dense subsets of $\\mathbb {P}$ of size at most $|\\mathcal {Y}| + \\omega $ as described in Proposition 3.1.","Since $\\mathfrak {m}$ ($\\sigma $ -centered) is uncountable, $| \\mathcal {D} | < \\mathfrak {m}(\\textrm {$$-centered})$ .","Hence, there exists a filter $G$ on $\\mathbb {P}$ which meets each dense set in $\\mathcal {D}$ .","By Proposition 3.1, there exists a set $z \\subseteq \\delta $ which is open, $M$ -stationary, and satisfies that for all $X \\in \\mathcal {Y}$ , $z \\setminus X$ is bounded in $\\delta $ .","Proposition 3.3 Let $\\vec{\\mathcal {C}} = \\langle \\mathcal {C}_\\alpha : \\alpha \\in \\omega _1 \\cap \\textrm {Lim} \\rangle $ be a sequence such that each $\\mathcal {C}_\\alpha $ is a collection of less than $\\mathfrak {m}$ ($\\sigma $ -centered) many closed and cofinal subsets of $\\alpha $ .","Then there exists an open stationary set mapping $\\Sigma $ such that, if $W$ is any outer model with the same $\\omega _1$ in which there exists a $\\Sigma $ -reflecting sequence, then there exists in $W$ a club subset of $\\omega _1$ which measures $\\vec{\\mathcal {C}}$ .","For each limit ordinal $\\alpha < \\omega _1$ , let $\\mathcal {D}_\\alpha := \\lbrace \\alpha \\setminus c : c \\in \\mathcal {C}_\\alpha \\rbrace $ .","Observe that each $\\mathcal {D}_\\alpha $ is a collection of fewer than $\\mathfrak {m}$ ($\\sigma $ -centered) many open subsets of $\\alpha $ .","We will define $\\Sigma $ to have domain the collection of all countable elementary substructures $M$ of $H(\\omega _2)$ .","Consider such an $M$ and we define $\\Sigma (M)$ .","Note that $M$ is suitable.","Let $\\delta := M \\cap \\omega _1$ .","We consider two cases.","In the first case, there does not exist a member of $\\mathcal {D}_\\delta $ which is $M$ -stationary.","Define $\\Sigma (M) = \\delta $ , which is clearly open and $M$ -stationary.","In the second case, there exists some member of $\\mathcal {D}_\\delta $ which is $M$ -stationary.","A straightforward application of Zorn's lemma implies that there exists a non–empty set ${\\mathcal {Y}}_M \\subseteq \\mathcal {D}_\\delta $ such that for any $a \\in [{\\mathcal {Y}}_M]^{<\\omega }$ , $\\bigcap a$ is $M$ -stationary, and moreover, $\\mathcal {Y}_M$ is a maximal subset of $\\mathcal {D}_\\delta $ with this property.","Since $\\mathcal {Y}_M \\subseteq \\mathcal {D}_\\delta $ , $|\\mathcal {Y}_M| < \\mathfrak {m}(\\textrm {$$-centered})$ .","So the collection $\\mathcal {Y}_M$ satisfies the assumptions of Corollary 3.2.","It follows that there exists a set $z_M \\subseteq \\delta $ which is open, $M$ -stationary, and satisfies that for all $X \\in \\mathcal {Y}_M$ , $z_M \\setminus X$ is bounded in $\\delta $ .","Now define $\\Sigma (M) := z_M$ .","This completes the definition of $\\Sigma $ .","Consider an outer model $W$ of $V$ with the same $\\omega _1$ , and assume that in $W$ there exists a $\\Sigma $ -reflecting sequence $\\langle M_\\delta : \\delta < \\omega _1 \\rangle $ .","Let $\\alpha _\\delta := M_\\delta \\cap \\omega _1$ for all $\\delta < \\omega _1$ .","Let $D$ be the club set of $\\delta < \\omega _1$ such that $\\alpha _\\delta = \\delta $ .","We claim that $D$ measures $\\vec{\\mathcal {C}}$ .","Consider $\\delta \\in \\lim (D)$ .","Then $\\delta = \\alpha _\\delta = M_\\delta \\cap \\omega _1$ .","Let $M := M_\\delta $ .","We first claim that if $c \\in \\mathcal {C}_\\delta $ and $\\delta \\setminus c$ is not $M$ -stationary, then for some $\\beta < \\delta $ , $(D \\cap \\delta ) \\setminus \\beta \\subseteq c$ .","Fix a club subset $E$ of $\\omega _1$ in $M$ which is disjoint from $\\delta \\setminus c$ .","By the continuity of the $\\Sigma $ -reflecting sequence, there exists $\\beta < \\delta $ such that $E \\in M_\\beta $ .","We claim that $(D \\cap \\delta ) \\setminus \\beta \\subseteq c$ .","Let $\\xi \\in (D \\cap \\delta ) \\setminus \\beta $ .","Then $E \\in M_\\xi $ , and hence by elementarity, $\\xi = M_\\xi \\cap \\omega _1 \\in E$ .","Since $E$ is disjoint from $\\delta \\setminus c$ , $\\xi \\in c$ .","We split the argument according to the two cases in the definition of $\\Sigma (M)$ .","In the first case, there does not exist a member of $\\mathcal {D}_\\delta $ which is $M$ -stationary.","Consider $c \\in \\mathcal {C}_\\delta $ .","Then $\\delta \\setminus c$ is not $M$ -stationary.","By the previous paragraph, there exists $\\beta < \\delta $ such that $(D \\cap \\delta ) \\setminus \\beta \\subseteq c$ .","In the second case, there exists a member of $\\mathcal {D}_\\delta $ which is $M$ -stationary.","Consider $c \\in \\mathcal {C}_\\delta $ .","Then $X := \\delta \\setminus c \\in \\mathcal {D}_\\delta $ .","We consider two possibilities.","First, assume that $X$ is in $\\mathcal {Y}_M$ .","By the choice of $\\mathcal {Y}_M$ and $z_M$ , we know that $z_M \\setminus X$ is bounded in $\\delta $ .","So fix $\\beta _0 < \\delta $ so that $z_M \\setminus \\beta _0 \\subseteq X$ .","By the definition of being a $\\Sigma $ -reflecting sequence, there exists $\\beta _1 < \\delta $ so that for all $\\beta _1 \\le \\xi < \\delta $ , $M_\\xi \\cap \\omega _1 \\in \\Sigma (M) = z_M$ .","Let $\\beta := \\max \\lbrace \\beta _1, \\beta _2 \\rbrace $ .","Consider $\\xi \\in (D \\cap \\delta ) \\setminus \\beta $ .","Then $\\xi \\ge \\beta _1$ implies that $\\xi = M_\\xi \\cap \\omega _1 \\in z_M$ .","So $\\xi \\in z_M \\setminus \\beta _0 \\subseteq X = \\delta \\setminus c$ .","Secondly, assume that $X$ is not in $\\mathcal {Y}_M$ .","By the maximality of $\\mathcal {Y}_M$ , there exists a set $a \\in [\\mathcal {Y}_M]^{<\\omega }$ such that $X \\cap \\bigcap a$ is not $M$ -stationary.","Fix a club $E$ in $M$ which is disjoint from $X \\cap \\bigcap a$ .","By the continuity of the $\\Sigma $ -reflecting sequence, there exists $\\beta < \\delta $ such that $E \\in M_\\beta $ .","Consider $\\xi \\in (D \\cap \\delta ) \\setminus \\beta $ .","Then $E \\in M_\\xi $ , which implies that $\\xi = M_\\xi \\cap \\omega _1 \\in E$ .","Thus, $\\xi $ is not in $X \\cap \\bigcap a$ .","On the other hand, letting $a = \\lbrace X_0, \\ldots , X_n \\rbrace $ , for each $i \\le n$ the previous paragraph implies that there exists $\\beta _i < \\delta $ such that $(D \\cap \\delta ) \\setminus \\beta _i \\subseteq X_i$ .","Let $\\beta ^*$ be an ordinal in $\\delta $ which is larger than $\\beta $ and $\\beta _i$ for all $i \\le n$ .","Consider $\\xi \\in (D \\cap \\delta ) \\setminus \\beta ^*$ .","Then by the choice of $\\beta $ , $\\xi \\notin X \\cap \\bigcap a$ .","By the choice of the $\\beta _i$ 's, $\\xi \\in \\bigcap a$ .","Therefore, $\\xi \\notin X = \\delta \\setminus c$ , which means that $\\xi \\in c$ .","Thus, $(D \\cap \\delta ) \\setminus \\beta ^* \\subseteq c$ .","Corollary 3.4 Assume MRP and MA($\\sigma $ -centered).","Then Strong Measuring holds.","Let $\\vec{\\mathcal {C}} = \\langle \\mathcal {C}_\\alpha :\\alpha \\in \\omega _1 \\cap \\textrm {Lim} \\rangle $ be a sequence such that each $\\mathcal {C}_\\alpha $ is a collection of fewer than $\\mathfrak {c}$ many closed and cofinal subsets of $\\alpha $ .","We claim that there exists a club subset of $\\omega _1$ which measures $\\vec{\\mathcal {C}}$ .","By MA($\\sigma $ -centered), $\\mathfrak {m}$ ($\\sigma $ -centered) equals $\\mathfrak {c}$ .","So each $\\mathcal {C}_\\alpha $ has size less than $\\mathfrak {m}$ ($\\sigma $ -centered).","By Proposition 3.3, there exists an open stationary set mapping $\\Sigma $ such that, if $W$ is any outer model with the same $\\omega _1$ in which there exists a $\\Sigma $ -reflecting sequence, then there exists in $W$ a club subset of $\\omega _1$ which measures $\\vec{\\mathcal {C}}$ .","Applying MRP, there exists a $\\Sigma $ -reflecting sequence in $V$ .","Thus, in $V$ there exists a club subset of $\\omega _1$ which measures $\\vec{\\mathcal {C}}$ .","Corollary 3.5 Assume BPFA.","Then Strong Measuring holds.","Let $\\vec{\\mathcal {C}} = \\langle \\mathcal {C}_\\alpha :\\alpha \\in \\omega _1 \\cap \\textrm {Lim} \\rangle $ be a sequence such that each $\\mathcal {C}_\\alpha $ is a collection of fewer than $\\mathfrak {c} = \\omega _2$ many closed and cofinal subsets of $\\alpha $ .","We claim that there exists a club subset of $\\omega _1$ which measures $\\vec{\\mathcal {C}}$ .","Since $\\mathfrak {c} = \\omega _2$ , $\\vec{\\mathcal {C}}$ is a member of $H(\\omega _2)$ .","Thus, the existence of a club subset of $\\omega _2$ which measures $\\vec{\\mathcal {C}}$ is expressible as a $\\Sigma _1$ statement involving a parameter in $H(\\omega _2)$ .","By BPFA, it suffices to show that there exists a proper forcing which forces that such a club exists.","Now BPFA implies Martin's Axiom, and in particular, that $\\mathfrak {m}$ ($\\sigma $ -centered) is equal to $\\mathfrak {c}$ .","So each $\\mathcal {C}_\\alpha $ has size less than $\\mathfrak {m}$ ($\\sigma $ -centered).","By Proposition 3.3, there exists an open stationary set mapping $\\Sigma $ such that, if $W$ is any outer model with the same $\\omega _1$ in which there exists a $\\Sigma $ -reflecting sequence, then there exists in $W$ a club subset of $\\omega _1$ which measures $\\vec{\\mathcal {C}}$ .","By [12], there exists a proper forcing $\\mathbb {P}$ which adds a $\\Sigma $ -reflecting sequence, so in $V^\\mathbb {P}$ there is a club subset of $\\omega _1$ which measures $\\vec{\\mathcal {C}}$ .","We now sketch a proof that MRP alone does not imply Strong Measuring.","In particular, Measuring does not imply Strong Measuring.","Start with a model of CH in which there exists a supercompact cardinal $\\kappa $ .","Construct a forcing iteration $\\mathbb {P}$ in the standard way to obtain a model of MRP.","To do this, fix a Laver function $f : \\kappa \\rightarrow V_\\kappa $ .","Then define a countable support forcing iteration $\\langle \\mathbb {P}_\\alpha , \\dot{\\mathbb {Q}}_\\beta : \\alpha \\le \\kappa , \\beta < \\kappa \\rangle $ as follows.","Given $\\mathbb {P}_\\alpha $ , consider $f(\\alpha )$ .","If $f(\\alpha )$ happens to be a $\\mathbb {P}_\\alpha $ -name for some open stationary set mapping, then let $\\dot{\\mathbb {Q}}_\\alpha $ be a $\\mathbb {P}_\\alpha $ -name for a proper forcing which adds an $f(\\alpha )$ -reflecting sequence.","Otherwise let $\\dot{\\mathbb {Q}}_\\alpha $ be a $\\mathbb {P}_\\alpha $ -name for $Col(\\omega _1,\\omega _2)$ .","Now define $\\mathbb {P}:= \\mathbb {P}_\\kappa $ .","Arguments similar to those in the standard construction of a model of PFA can be used to show that $\\mathbb {P}$ forces MRP.","The forcing for adding a $\\Sigma $ -reflecting sequence for a given open stationary set mapping does not add reals ([12]).","In particular, it is vacuously $\\omega ^\\omega $ -bounding.","The property of being proper and $\\omega ^\\omega $ -bounding is preserved under countable support forcing iterations ([1]), so $\\mathbb {P}$ is also $\\omega ^\\omega $ -bounding.","In particular, $V \\cap \\omega ^\\omega $ is an unbounded family in $V^\\mathbb {P}$ , and it has size $\\omega _1$ since CH holds in $V$ .","It follows that the bounding number $\\mathfrak {b}$ is equal to $\\omega _1$ .","But by Corollary 2.9, Measuring$_{\\mathfrak {b}}$ is false.","So $\\mathbb {P}$ forces that $\\textsf {Measuring}_{\\omega _1}$ is false.","As $\\mathfrak {c} = \\omega _2$ in $V^\\mathbb {P}$ , Strong Measuring fails in $V^\\mathbb {P}$ .","We also note that Strong Measuring plus $\\mathfrak {c} = \\omega _2$ is consistent with the existence of an $\\omega _1$ -Suslin tree.","Namely, both the forcing for adding a $\\Sigma $ -reflecting sequence for a given open stationary set mapping $\\Sigma $ , as well as any $\\sigma $ -centered forcing, preserve Suslin trees ([11]).","And the property of being proper and preserving a given Suslin tree is preserved under countable support forcings iterations ([10]).","So starting with a model in which there exists an $\\omega _1$ -Suslin tree $S$ and a supercompact cardinal $\\kappa $ , we can iterate forcing similar to the argument in the preceding paragraphs to produce a model of MA($\\sigma $ -centered) plus MRP in which $S$ is an $\\omega _1$ -Suslin tree.","By Corollary 3.4, Strong Measuring holds in that model."],["Measuring Without the Axiom of Choice","Another natural way to strengthen $\\operatorname{\\textsf {Measuring}}$ is to allow, in the sequence to be measured, not just closed sets, but also sets of higher Borel complexity.","This line of strengthenings of $\\operatorname{\\textsf {Measuring}}$ was also considered in [2].","For completeness, we are including here the corresponding observations.","The version of $\\operatorname{\\textsf {Measuring}}$ where one considers sequences $\\vec{X}=\\langle X_\\alpha \\,:\\,\\alpha \\in \\omega _1\\cap \\mathrm {Lim}\\rangle $ , with each $X_\\alpha $ an open subset of $\\alpha $ in the order topology, is of course equivalent to $\\operatorname{\\textsf {Measuring}}$ .","A natural next step would therefore be to consider sequences in which each $X_\\alpha $ is a countable union of closed sets.","This is obviously the same as allowing each $X_\\alpha $ to be an arbitrary subset of $\\alpha $ .","Let us call the corresponding statement $\\operatorname{\\textsf {Measuring}}^\\ast $ : Definition 4.1 $\\operatorname{\\textsf {Measuring}}^\\ast $ holds if and only if for every sequence $\\vec{X}=\\langle X_\\alpha \\,:\\,\\alpha \\in \\omega _1\\cap \\mathrm {Lim}\\rangle $ , if $X_\\alpha \\subseteq \\alpha $ for each $\\alpha $ , then there is some club $D\\subseteq \\omega _1$ such that for every limit point $\\delta \\in D$ of $D$ , $D\\cap \\delta $ measures $X_\\delta $ .","It is easy to see that $\\operatorname{\\textsf {Measuring}}^\\ast $ is false in $\\textsf {ZFC}$ .","In fact, given a stationary and co-stationary $S\\subseteq \\omega _1$ , there is no club of $\\omega _1$ measuring $\\vec{X}=\\langle S\\cap \\alpha \\,:\\,\\alpha \\in \\omega _1\\cap \\mathrm {Lim}\\rangle $ .","The reason is that if $D$ is any club of $\\omega _1$ , then both $D\\cap S\\cap \\delta $ and $(D\\cap \\delta )\\setminus S$ are cofinal subsets of $\\delta $ for each $\\delta $ in the club of limit points in $\\omega _1$ of both $D\\cap S$ and $D\\setminus S$ .","The status of $\\operatorname{\\textsf {Measuring}}^\\ast $ is more interesting in the absence of the Axiom of Choice.","Let $\\mathcal {C}_{\\omega _1}=\\lbrace X\\subseteq \\omega _1\\,:\\,C \\subseteq X\\mbox{ for some club $C$ of $\\omega _1$}\\rbrace $ .","Observation 4.2 ($\\textsf {ZF}$ + $\\mathcal {C}_{\\omega _1}$ is a normal filter on $\\omega _1$ ) Suppose $\\vec{X}=\\langle X_\\delta \\,:\\,\\delta \\in \\omega _1\\cap \\mathrm {Lim}\\rangle $ is such that $X_\\delta \\subseteq \\delta $ for each $\\delta $ .","For each club $C\\subseteq \\omega _1$ , there is some $\\delta \\in C$ such that $C\\cap X_\\delta \\ne \\emptyset $ , and there is some $\\delta \\in C$ such that $(C\\cap \\delta )\\setminus X_\\delta \\ne \\emptyset $ .","Then there is a stationary and co-stationary subset of $\\omega _1$ definable from $\\vec{X}$ .","We have two possible cases.","The first case is that in which for all $\\alpha <\\omega _1$ , either $W_\\alpha ^0=\\lbrace \\delta <\\omega _1\\,:\\,\\alpha \\notin X_\\delta \\rbrace $ is in $\\mathcal {C}_{\\omega _1}$ , or $W_\\alpha ^1=\\lbrace \\delta <\\omega _1\\,:\\,\\alpha \\in X_\\delta \\rbrace $ is in $\\mathcal {C}_{\\omega _1}$ .","For each $\\alpha <\\omega _1$ , let $W_\\alpha $ be $W_\\alpha ^\\epsilon $ for the unique $\\epsilon \\in \\lbrace 0, 1\\rbrace $ such that $W_\\alpha ^\\epsilon \\in \\mathcal {C}_{\\omega _1}$ , and let $W^\\ast =\\Delta _{\\alpha <\\omega _1} W_\\alpha \\in \\mathcal {C}_{\\omega _1}$ .","Then $X_{\\delta _0}=X_{\\delta _1}\\cap \\delta _0$ for all $\\delta _0<\\delta _1$ in $W^\\ast $ .","It then follows, by (2), that $S=\\bigcup _{\\delta \\in W^\\ast }X_\\delta $ , which of course is definable from $\\vec{C}$ , is a stationary and co-stationary subset of $\\omega _1$ .","Indeed, suppose $C\\subseteq \\omega _1$ is a club, and let us fix a club $D\\subseteq W^\\ast $ .","There is then some $\\delta \\in C\\cap D$ and some $\\alpha \\in C\\cap D\\cap X_\\delta $ .","But then $\\alpha \\in S$ since $\\delta \\in W^\\ast $ and $\\alpha \\in W^\\ast \\cap X_\\delta $ .","There is also some $\\delta \\in C\\cap D$ and some $\\alpha \\in C\\cap D$ such that $\\alpha \\notin X_\\delta $ , which implies that $\\alpha \\notin S$ by a symmetrical argument, using the fact that $X_{\\delta _0}=X_{\\delta _1}\\cap \\delta _0$ for all $\\delta _0<\\delta _1$ in $W^\\ast $ .","The second possible case is that there is some $\\alpha <\\omega _1$ with the property that both $W^0_\\alpha $ and $W^1_\\alpha $ are stationary subsets of $\\omega _1$ .","But now we can let $S$ be $W^0_\\alpha $ , where $\\alpha $ is first such that $W^0_\\alpha $ is stationary and co-stationary.","It is worth comparing the above observation with Solovay's classic result that an $\\omega _1$ –sequence of pairwise disjoint stationary subsets of $\\omega _1$ is definable from any given ladder system on $\\omega _1$ (working in the same theory).","Corollary 4.3 ($\\textsf {ZF}$ + $\\mathcal {C}_{\\omega _1}$ is a normal filter on $\\omega _1$ ) The following are equivalent.","$\\mathcal {C}_{\\omega _1}$ is an ultrafilter on $\\omega _1$ ; $\\operatorname{\\textsf {Measuring}}^\\ast $ ; For every sequence $\\langle X_\\alpha \\,:\\,\\alpha \\in \\omega _1\\cap \\mathrm {Lim}\\rangle $ , if $X_\\alpha \\subseteq \\alpha $ for each $\\alpha $ , then there is a club $C\\subseteq \\omega _1$ such that either $C\\cap \\delta \\subseteq X_\\delta $ for every $\\delta \\in C$ , or $C\\cap X_\\delta =\\emptyset $ for every $\\delta \\in C$ .","(3) trivially implies (2), and by the observation (1) implies (3).","Finally, to see that (2) implies (1), note that the argument right after the definition of $\\operatorname{\\textsf {Measuring}}^\\ast $ uses only $\\textsf {ZF}$ together with the regularity of $\\omega _1$ and the negation of (1).","In particular, the strong form of $\\operatorname{\\textsf {Measuring}}^\\ast $ given by (3) in the above observation follows from $\\textsf {ZF}$ together with the Axiom of Determinacy.","We finish this digression into set theory without the Axiom of Choice by observing that any attempt to parametrize $\\operatorname{\\textsf {Measuring}}^\\ast $ , in the same vein as we did with Measuring, gives rise to principles vacuously equivalent to $\\operatorname{\\textsf {Measuring}}^\\ast $ itself, at least when the parametrization is done with the alephs.This was pointed out by Asaf Karagila.","Specifically, given an aleph $\\kappa $ , let us define $\\operatorname{\\textsf {Measuring}}^*_\\kappa $ as the statement that for every sequence $\\langle \\mathcal {X}_\\alpha \\,:\\,\\alpha \\in \\omega _1\\cap \\mathrm {Lim}\\rangle $ , if each $\\mathcal {X}_\\alpha $ is a set of cardinality at most $\\kappa $ consisting of subsets of $\\alpha $ , then there is a club $D\\subseteq \\omega _1$ such that for every limit point $\\delta \\in D$ of $D$ , $D\\cap \\delta $ measures $X$ for all $X\\in \\mathcal {X}_\\delta $ .","Then $\\operatorname{\\textsf {Measuring}}^\\ast _{\\omega }$ is clearly equivalent to $\\operatorname{\\textsf {Measuring}}^\\ast $ under $\\textsf {ZF}$ together with the normality of $\\mathcal {C}_{\\omega _1}$ and the Axiom of Choice for countable families of subsets of $\\omega _1$ (which of course follows from the Axiom of Choice for countable families of sets of reals, and therefore also from $\\textsf {ZF} + \\textsf {AD}$ ).","On the other hand, working in $\\textsf {ZF}$ + $\\mathcal {C}_{\\omega _1}$ is a normal filter on $\\omega _1$ , we have that $\\operatorname{\\textsf {Measuring}}^\\ast _{\\omega _1}$ follows vacuously from $\\operatorname{\\textsf {Measuring}}^\\ast $ simply because under $\\operatorname{\\textsf {Measuring}}^\\ast $ there is no sequence $\\langle \\mathcal {X}_\\alpha \\,:\\,\\alpha \\in \\omega _1\\cap \\mathrm {Lim}\\rangle $ as in the definition of $\\operatorname{\\textsf {Measuring}}^\\ast _{\\omega _1}$ and such that $\\vert \\mathcal {X}_\\alpha \\vert =\\omega _1$ for some $\\alpha $ ; indeed, $\\operatorname{\\textsf {Measuring}}^\\ast $ implies, over this base theory, that $\\mathcal {C}_{\\omega _1}$ is an ultrafilter (Corollary REF ), and if $\\mathcal {C}_{\\omega _1}$ is an ultrafilter then there is no $\\omega _1$ -sequence of distinct reals, whereas the existence of a family of size $\\omega _1$ consisting of subsets of some fixed countable ordinal clearly implies that there is such a sequence.","We conclude the article with two natural questions.","Question 4.4 Is Measuring$_{\\mathfrak {p}}$ false?","Question 4.5 Are Measuring and Strong Measuring equivalent statements assuming Martin's Axiom?"]],"string":"[\n [\n \"Parametrized Measuring and Club Guessing\"\n ],\n [\n \"Abstract We introduce Strong Measuring, a maximal strengthening of J. T. Moore's Measuring principle, which asserts that every collection of fewer than continuum many closed bounded subsets of $\\\\omega_1$ is measured by some club subset of $\\\\omega_1$.\",\n \"The consistency of Strong Measuring with the negation of CH is shown, solving an open problem from about parametrized measuring principles.\",\n \"Specifically, we prove that Strong Measuring follows from MRP together with Martin's Axiom for $\\\\sigma$-centered forcings, as well as from BPFA.\",\n \"We also consider strong versions of Measuring in the absence of the Axiom of Choice.\"\n ],\n [\n \"Background\",\n \"We review some background material and notation which is needed for understanding the paper.\",\n \"Let $\\\\mathfrak {c}$ denote the cardinality of the continuum $2^\\\\omega $ .\",\n \"A set $S \\\\subseteq [\\\\omega ]^\\\\omega $ is a splitting family if for any infinite set $x \\\\subseteq \\\\omega $ , there exists $A \\\\in S$ such that $A$ splits $x$ in the sense that both $x \\\\cap A$ and $x \\\\setminus A$ are infinite.\",\n \"The splitting number $\\\\mathfrak {s}$ is the least cardinality of some splitting family.\",\n \"Given functions $f, g : \\\\omega \\\\rightarrow \\\\omega $ , we say that $g$ dominates $f$ if for all $n < \\\\omega $ , $f(n) < g(n)$ .\",\n \"We say that $g$ eventually dominates $f$ if there is some $m<\\\\omega $ such that $f(n)m$ .\",\n \"A family $B \\\\subseteq \\\\omega ^\\\\omega $ is bounded if there exists a function $g \\\\in \\\\omega ^\\\\omega $ which eventually dominates every member of $B$ , and otherwise it is unbounded.\",\n \"The bounding number $\\\\mathfrak {b}$ is the least cardinality of some unbounded family.\",\n \"Both cardinal characteristics $\\\\mathfrak {s}$ and $\\\\mathfrak {b}$ are uncountable.\",\n \"Let $\\\\mathbb {P}$ be a forcing poset.\",\n \"A set $X \\\\subseteq \\\\mathbb {P}$ is centered if every finite subset of $X$ has a lower bound.\",\n \"We say that $\\\\mathbb {P}$ is $\\\\sigma $ -centered if it is a union of countably many centered sets.\",\n \"Martin's Axiom for $\\\\sigma $ -centered forcings ($\\\\textsf {MA}$ ($\\\\sigma $ -centered)) is the statement that for any $\\\\sigma $ -centered forcing $\\\\mathbb {P}$ and any collection of fewer than $\\\\mathfrak {c}$ many dense subsets of $\\\\mathbb {P}$ , there exists a filter on $\\\\mathbb {P}$ which meets each dense set in the collection.\",\n \"More generally, let $\\\\mathfrak {m}$ ($\\\\sigma $ -centered) be the least cardinality of a collection of dense subsets of some $\\\\sigma $ -centered forcing poset for which there does not exist a filter which meets each dense set in the collection.\",\n \"Note that $\\\\textsf {MA}$ ($\\\\sigma $ -centered) is equivalent to the statement that $\\\\mathfrak {m}$ ($\\\\sigma $ -centered) equals $\\\\mathfrak {c}$ .\",\n \"The Bounded Proper Forcing Axiom (BPFA) is the statement that whenever $\\\\mathbb {P}$ is a proper forcing and $\\\\langle A_i : i < \\\\omega _1 \\\\rangle $ is a sequence of maximal antichains of $\\\\mathbb {P}$ each of size at most $\\\\omega _1$ , then there exists a filter on $\\\\mathbb {P}$ which meets each $A_i$ ([9]).\",\n \"We note that BPFA implies $\\\\mathfrak {c} = \\\\omega _2$ ([12]).\",\n \"It easily follows that BPFA implies Martin's Axiom, and in particular, implies $\\\\textsf {MA}$ ($\\\\sigma $ -centered).\",\n \"The forcing axiom BPFA is equivalent to the statement that for any proper forcing poset $\\\\mathbb {P}$ and any $\\\\Sigma _1$ statement $\\\\Phi $ with a parameter from $H(\\\\omega _2)$ , if $\\\\Phi $ holds in a generic extension by $\\\\mathbb {P}$ , then $\\\\Phi $ holds in the ground model ([7]).\",\n \"An open stationary set mapping for an uncountable set $X$ and regular cardinal $\\\\theta > \\\\omega _1$ is a function $\\\\Sigma $ whose domain is the collection of all countable elementary substructures $M$ of $H(\\\\theta )$ with $X \\\\in M$ , such that for all such $M$ , $\\\\Sigma (M)$ is an open, $M$ -stationary subset of $[X]^\\\\omega $ .\",\n \"By open we mean in the Ellentuck topology on $[X]^\\\\omega $ , and $M$ -stationary means meeting every club subset of $[X]^\\\\omega $ which is a member of $M$ (see [12] for the complete details).\",\n \"In this article, we are only concerned with these ideas in the simplest case that $X = \\\\omega _1$ and for each $M \\\\in \\\\mathrm {dom}(\\\\Sigma )$ , $\\\\Sigma (M) \\\\subseteq \\\\omega _1$ .\",\n \"In this case, being open is equivalent to being open in the topology on $\\\\omega _1$ with basis the collection of all open intervals of ordinals, and being $M$ -stationary is equivalent to meeting every club subset of $\\\\omega _1$ in $M$ .\",\n \"For an open stationary set mapping $\\\\Sigma $ for $X$ and $\\\\theta $ , a $\\\\Sigma $ -reflecting sequence is an $\\\\in $ -increasing and continuous sequence $\\\\langle M_i : i < \\\\omega _1 \\\\rangle $ of countable elementary substructures of $H(\\\\theta )$ containing $X$ as a member satisfying that for all limit ordinals $\\\\delta < \\\\omega _1$ , there exists $\\\\beta < \\\\delta $ so that for all $\\\\beta \\\\le \\\\xi < \\\\delta $ , $M_\\\\xi \\\\cap X \\\\in \\\\Sigma (M_\\\\delta )$ .\",\n \"The Mapping Reflection Principle (MRP) is the statement that for any open stationary set mapping $\\\\Sigma $ , there exists a $\\\\Sigma $ -reflecting sequence.\",\n \"We will use the fact that for any open stationary set mapping $\\\\Sigma $ , there exists a proper forcing which adds a $\\\\Sigma $ -reflecting sequence ([12]).\",\n \"Consequently, MRP follows from PFA.\"\n ],\n [\n \"Parametrized Measuring and Club Guessing\",\n \"Let $X$ and $Y$ be countable subsets of $\\\\omega _1$ with the same supremum $\\\\delta $ .\",\n \"We say that $X$ measures $Y$ if there exists $\\\\beta < \\\\delta $ such that $X \\\\setminus \\\\beta $ is either contained in, or disjoint from, $Y$ .\",\n \"Measuring is the statement that for any sequence $\\\\langle c_\\\\alpha : \\\\alpha \\\\in \\\\omega _1 \\\\cap \\\\mathrm {Lim}\\\\rangle $ , where each $c_\\\\alpha $ is a closed and cofinal subset of $\\\\alpha $ , there exists a club $D \\\\subseteq \\\\omega _1$ such that for all limit points $\\\\alpha $ of $D$ , $D \\\\cap \\\\alpha $ measures $c_\\\\alpha $ .\",\n \"The next two results are due to J. T. Moore ([8]).\",\n \"Theorem 2.1 MRP implies Measuring.\",\n \"Theorem 2.2 BPFA implies Measuring.\",\n \"We now describe parametrized forms of measuring which were introduced in [2].\",\n \"Let $\\\\vec{\\\\mathcal {C}} = \\\\langle \\\\mathcal {C}_\\\\alpha : \\\\alpha \\\\in \\\\omega _1 \\\\cap \\\\textrm {Lim} \\\\rangle $ be a sequence such that each $\\\\mathcal {C}_\\\\alpha $ is a collection of closed and cofinal subsets of $\\\\alpha $ .\",\n \"A club $D \\\\subseteq \\\\omega _1$ is said to measure $\\\\vec{\\\\mathcal {C}}$ if for all $\\\\alpha \\\\in \\\\lim (D)$ and all $c \\\\in \\\\mathcal {C}_\\\\alpha $ , $D \\\\cap \\\\alpha $ measures $c$ .\",\n \"Definition 2.3 For a cardinal $\\\\kappa $ , let Measuring$_{< \\\\kappa }$ denote the statement that whenever $\\\\vec{\\\\mathcal {C}} = \\\\langle \\\\mathcal {C}_\\\\alpha : \\\\alpha \\\\in \\\\omega _1 \\\\cap \\\\textrm {Lim} \\\\rangle $ is a sequence such that each $\\\\mathcal {C}_\\\\alpha $ is a collection of fewer than $\\\\kappa $ many closed and cofinal subsets of $\\\\alpha $ , then there exists a club $D \\\\subseteq \\\\omega _1$ which measures $\\\\vec{\\\\mathcal {C}}$ .\",\n \"For a cardinal $\\\\lambda $ , let $\\\\textsf {Measuring}_\\\\lambda $ denote $\\\\textsf {Measuring}_{< \\\\lambda ^+}$ .\",\n \"Observe that the principle Measuring is the same as $\\\\textsf {Measuring}_1$ .\",\n \"If $\\\\kappa < \\\\lambda $ , then clearly Measuring$_{< \\\\lambda }$ implies Measuring$_{<\\\\kappa }$ .\",\n \"It is easy to see that Measuring$_{\\\\mathfrak {c}}$ is false.\",\n \"Definition 2.4 Strong Measuring is the statement that Measuring$_{<\\\\mathfrak {c}}$ holds.\",\n \"Since the intersection of countably many clubs in $\\\\omega _1$ is club, Measuring easily implies Measuring$_\\\\omega $ .\",\n \"In particular, Measuring together with CH implies Strong Measuring.\",\n \"We will prove in Section 3 the consistency of Strong Measuring together with $\\\\lnot \\\\textsf {CH}$ .\",\n \"We also observe at the end of that section that Measuring does not imply Measuring$_{\\\\omega _1}$ .\",\n \"Proposition 2.5 ([2]) Measuring$_{\\\\mathfrak {s}}$ is false.\",\n \"Fix a splitting family $S$ of cardinality $\\\\mathfrak {s}$ .\",\n \"For each limit ordinal $\\\\alpha < \\\\omega _1$ , fix a function $f_\\\\alpha : \\\\omega \\\\rightarrow \\\\alpha $ which is increasing and cofinal in $\\\\alpha $ .\",\n \"For each $A \\\\in S$ , let $c_{\\\\alpha ,A} = \\\\bigcup \\\\lbrace (f_\\\\alpha (n),f_\\\\alpha (n+1)] : n \\\\in A \\\\rbrace $ , which is clearly closed and cofinal in $\\\\alpha $ .\",\n \"Let $\\\\mathcal {C}_\\\\alpha := \\\\lbrace c_{\\\\alpha ,A} : A \\\\in S \\\\rbrace $ .\",\n \"Then $\\\\vec{\\\\mathcal {C}} := \\\\langle \\\\mathcal {C}_\\\\alpha : \\\\alpha \\\\in \\\\omega _1 \\\\cap \\\\mathrm {Lim}\\\\rangle $ is a sequence such that for each $\\\\alpha $ , $\\\\mathcal {C}_\\\\alpha $ is a collection of at most $\\\\mathfrak {s}$ many closed and cofinal subsets of $\\\\alpha $ .\",\n \"Let $D \\\\subseteq \\\\omega _1$ be a club.\",\n \"Fix $\\\\alpha \\\\in \\\\lim (D)$ .\",\n \"We will show that there exists a member of $\\\\mathcal {C}_\\\\alpha $ which $D \\\\cap \\\\alpha $ does not measure.\",\n \"Define $x := \\\\lbrace n < \\\\omega : D \\\\cap (f_\\\\alpha (n),f_{\\\\alpha }(n+1)] \\\\ne \\\\emptyset \\\\rbrace $ .\",\n \"Since $\\\\alpha \\\\in \\\\lim (D)$ , $x$ is infinite.\",\n \"As $S$ is a splitting family, we can fix $A \\\\in S$ which splits $x$ .\",\n \"So both $x \\\\cap A$ and $x \\\\setminus A$ are infinite.\",\n \"We claim that $D \\\\cap \\\\alpha $ does not measure $c_{\\\\alpha ,A}$ .\",\n \"Suppose for a contradiction that for some $\\\\beta < \\\\alpha $ , $(D \\\\cap \\\\alpha ) \\\\setminus \\\\beta $ is either a subset of, or disjoint from, $c_{\\\\alpha ,A}$ .\",\n \"Since $A \\\\cap x$ is infinite, we can fix $n \\\\in A \\\\cap x$ such that $f_\\\\alpha (n) > \\\\beta $ .\",\n \"Then $n \\\\in x$ implies that $D \\\\cap (f_\\\\alpha (n),f_{\\\\alpha }(n+1)] \\\\ne \\\\emptyset $ , and $n \\\\in A$ implies that $(f_{\\\\alpha }(n),f_{\\\\alpha }(n+1)] \\\\subseteq c_{\\\\alpha ,A}$ .\",\n \"It follows that $(D \\\\cap \\\\alpha ) \\\\setminus \\\\beta $ meets $c_{\\\\alpha ,A}$ .\",\n \"By the choice of $\\\\beta $ , this implies that $(D \\\\cap \\\\alpha ) \\\\setminus \\\\beta $ is a subset of $c_{\\\\alpha ,A}$ .\",\n \"But $x \\\\setminus A$ is also infinite, so we can fix $m \\\\in x \\\\setminus A$ such that $f_{\\\\alpha }(m) > \\\\beta $ .\",\n \"Then $m \\\\in x$ implies that $D \\\\cap (f_\\\\alpha (m),f_\\\\alpha (m+1)] \\\\ne \\\\emptyset $ , and $m \\\\notin A$ implies that $(f_{\\\\alpha }(m),f_{\\\\alpha }(m+1)]$ is disjoint from $c_{\\\\alpha ,A}$ .\",\n \"Thus, there is a member of $(D \\\\cap \\\\alpha ) \\\\setminus \\\\beta $ which is not in $c_{\\\\alpha ,A}$ , which is a contradiction.\",\n \"We will prove later in this section that Measuring$_{\\\\mathfrak {b}}$ is also false.\",\n \"We now turn to parametrized club guessing.\",\n \"We recall some standard definitions.\",\n \"Consider a sequence $\\\\vec{L} = \\\\langle L_\\\\alpha : \\\\alpha \\\\in \\\\omega _1 \\\\cap \\\\mathrm {Lim}\\\\rangle $ , where each $L_\\\\alpha $ is a cofinal subset of $\\\\alpha $ with order type $\\\\omega $ (that is, a ladder system).\",\n \"We say that $\\\\vec{L}$ is a club guessing sequence, weak club guessing sequence, or very weak club guessing sequence, respectively, if for every club $D \\\\subseteq \\\\omega _1$ , there exists a limit ordinal $\\\\alpha < \\\\omega _1$ such that: $L_\\\\alpha \\\\subseteq D$ , $L_\\\\alpha \\\\setminus D$ is finite, or $L_\\\\alpha \\\\cap D$ is infinite, respectively.\",\n \"We say that Club Guessing, Weak Club Guessing, or Very Weak Club Guessing holds, respectively, if there exists a club guessing sequence, a weak club guessing sequence, or a very weak club guessing sequence, respectively.\",\n \"It is well known that Measuring implies the failure of Very Weak Club Guessing (see Proposition 2.8 below).\",\n \"Definition 2.6 Let $\\\\vec{\\\\mathcal {L}} = \\\\langle \\\\mathcal {L}_\\\\alpha : \\\\alpha \\\\in \\\\omega _1 \\\\cap \\\\mathrm {Lim}\\\\rangle $ be a sequence where each $\\\\mathcal {L}_\\\\alpha $ is a non–empty collection of cofinal subsets of $\\\\alpha $ with order type $\\\\omega $ .\",\n \"The sequence $\\\\vec{\\\\mathcal {L}}$ is said to be a club guessing sequence, weak club guessing sequence, or very weak club guessing sequence, respectively, if for every club $D \\\\subseteq \\\\omega _1$ , there exists a limit ordinal $\\\\alpha < \\\\omega _1$ and some $L \\\\in \\\\mathcal {L}_\\\\alpha $ such that: $L \\\\subseteq D$ , $L \\\\setminus D$ is finite, or $L \\\\cap D$ is infinite, respectively.\",\n \"Definition 2.7 For a cardinal $\\\\kappa $ , let CG$_{< \\\\kappa }$ , WCG$_{<\\\\kappa }$ , and VWCG$_{<\\\\kappa }$ , respectively, be the statements that there exists a club guessing sequence, weak club guessing sequence, or very weak club guessing sequence $\\\\langle \\\\mathcal {L}_\\\\alpha : \\\\alpha \\\\in \\\\omega _1 \\\\cap \\\\mathrm {Lim}\\\\rangle $ , respectively, such that for each $\\\\alpha $ , $|\\\\mathcal {L}_\\\\alpha | < \\\\kappa $ .\",\n \"Let CG$_{\\\\kappa }$ , WCG$_{\\\\kappa }$ , and VWCG$_{\\\\kappa }$ denote the statements CG$_{< \\\\kappa ^+}$ , WCG$_{< \\\\kappa ^+}$ , and VWCG$_{< \\\\kappa ^+}$ , respectively.\",\n \"Clearly, if $\\\\kappa < \\\\lambda $ , then CG$_{< \\\\kappa }$ implies CG$_{<\\\\lambda }$ , and similarly with WCG and VWCG.\",\n \"Observe that Club Guessing, Weak Club Guessing, and Very Weak Club Guessing are equivalent to CG$_{1}$ , WCG$_{1}$ , and VWCG$_{1}$ , respectively.\",\n \"Obviously, CG$_{\\\\mathfrak {c}}$ is true.\",\n \"The weakest forms of club guessing principles which are not provable in ZFC are when the index is $< \\\\mathfrak {c}$ .\",\n \"Proposition 2.8 For any cardinal $\\\\kappa \\\\ge 2$ , Measuring$_{< \\\\kappa }$ implies the failure of VWCG$_{< \\\\kappa }$ .\",\n \"Suppose for a contradiction that Measuring$_{< \\\\kappa }$ and VWCG$_{<\\\\kappa }$ both hold.\",\n \"Fix a very weak club guessing sequence $\\\\vec{\\\\mathcal {L}} = \\\\langle \\\\mathcal {L}_\\\\alpha : \\\\alpha \\\\in \\\\omega _1 \\\\cap \\\\mathrm {Lim}\\\\rangle $ such that each $\\\\mathcal {L}_\\\\alpha $ has cardinality less than $\\\\kappa $ .\",\n \"Observe that for each $\\\\alpha $ , every member of $\\\\mathcal {L}_\\\\alpha $ is vacuously a closed subset of $\\\\alpha $ since it has order type $\\\\omega $ .\",\n \"By Measuring$_{< \\\\kappa }$ , there exists a club $D \\\\subseteq \\\\omega _1$ which measures $\\\\vec{\\\\mathcal {L}}$ .\",\n \"Let $E$ be the club set of indecomposable limit ordinals $\\\\alpha >\\\\omega $ in $\\\\lim (D)$ such that $\\\\mathrm {ot}(D \\\\cap \\\\alpha ) = \\\\alpha $ .\",\n \"Since $\\\\vec{\\\\mathcal {L}}$ is a very weak club guessing sequence, there exists a limit ordinal $\\\\alpha $ and $L \\\\in \\\\mathcal {L}_\\\\alpha $ such that $L \\\\cap E$ is infinite.\",\n \"In particular, $\\\\alpha $ is a limit point of $E$ , and hence of $D$ .\",\n \"Since $D$ measures $\\\\vec{\\\\mathcal {L}}$ and $L \\\\in \\\\mathcal {L}_\\\\alpha $ , $D \\\\cap \\\\alpha $ measures $L$ .\",\n \"So we can fix $\\\\beta < \\\\alpha $ such that $(D \\\\cap \\\\alpha ) \\\\setminus \\\\beta $ is either a subset of, or disjoint from, $L$ .\",\n \"Now $L \\\\cap E$ , and hence $L \\\\cap D$ , is infinite.\",\n \"As $L$ has order type $\\\\omega $ , this implies that $L \\\\cap D$ is cofinal in $\\\\alpha $ .\",\n \"By the choice of $\\\\beta $ , $(D \\\\cap \\\\alpha ) \\\\setminus \\\\beta $ must be a subset of $L$ .\",\n \"But since $\\\\alpha \\\\in E$ , $\\\\mathrm {ot}(D \\\\cap \\\\alpha ) = \\\\alpha $ and $\\\\alpha $ is indecomposable, which implies that $\\\\mathrm {ot}((D \\\\cap \\\\alpha ) \\\\setminus \\\\beta ) = \\\\alpha $ .\",\n \"As $\\\\alpha > \\\\omega $ , this is impossible since $(D \\\\cap \\\\alpha ) \\\\setminus \\\\beta $ is a subset of $L$ and $L$ has order type $\\\\omega $ .\",\n \"In particular, since Strong Measuring is consistent, so is the failure of VWCG$_{< \\\\mathfrak {c}}$ .\",\n \"(The consistency of $\\\\lnot \\\\textsf {VWCG}_{< \\\\mathfrak {c}}$ together with $\\\\mathfrak {c}$ arbitrarily large was previously shown in [4].)\",\n \"Proposition 2.9 (Hrušák [5]) VWCG$_{\\\\mathfrak {b}}$ is true.\",\n \"Fix an unbounded family $\\\\lbrace r_\\\\alpha : \\\\alpha < \\\\mathfrak {b} \\\\rbrace $ in $\\\\omega ^\\\\omega $ .\",\n \"For each limit ordinal $\\\\delta < \\\\omega _1$ , fix a cofinal subset $C_\\\\delta $ of $\\\\delta $ with order type $\\\\omega $ and a bijection $h_\\\\delta : \\\\omega \\\\rightarrow \\\\delta $ .\",\n \"Let $C_\\\\delta (n)$ denote the $n$ -th member of $C_\\\\delta $ for all $n < \\\\omega $ .\",\n \"For all limit ordinals $\\\\delta < \\\\omega _1$ and $\\\\alpha < \\\\mathfrak {b}$ , define $A_\\\\delta ^\\\\alpha := C_\\\\delta \\\\cup \\\\bigcup \\\\lbrace h_\\\\delta [r_\\\\alpha (n)] \\\\setminus C_\\\\delta (n) : n < \\\\omega \\\\rbrace .$ It is easy to check that for all $\\\\delta $ and $\\\\alpha $ , $A_\\\\delta ^\\\\alpha $ has order type $\\\\omega $ and $\\\\sup (A_\\\\delta ^\\\\alpha ) = \\\\delta $ .\",\n \"Given a club $C \\\\subseteq \\\\omega _1$ , let $\\\\delta $ be a limit point of $C$ and let $g_{C, \\\\delta } : \\\\omega \\\\rightarrow \\\\omega $ be the function given by $g_{C, \\\\delta }(n) =\\\\min \\\\lbrace m < \\\\omega :h_\\\\delta (m) \\\\in C \\\\setminus C_\\\\delta (n) \\\\rbrace .$ Now let $\\\\alpha < \\\\mathfrak {b}$ be such that $r_\\\\alpha (n) > g_{C, \\\\delta }(n)$ for infinitely many $n$ .\",\n \"It then follows that $|A_\\\\delta ^\\\\alpha \\\\cap C| = \\\\omega $ .\",\n \"By Propositions 2.8 and 2.9, the following is immediate.\",\n \"Corollary 2.10 Measuring$_{\\\\mathfrak {b}}$ is false.\",\n \"An obvious question is whether the parametrized versions of club guessing are actually the same as the usual ones.\",\n \"We conclude this section by showing that they are not.\",\n \"Recall that a forcing poset $\\\\mathbb {P}$ is $\\\\omega ^\\\\omega $ -bounding if every function in $\\\\omega ^\\\\omega \\\\cap V^\\\\mathbb {P}$ is dominated by a function in $\\\\omega ^\\\\omega \\\\cap V$ .\",\n \"Lemma 2.11 (Hrušák) Assume that VWCG fails.\",\n \"Let $\\\\mathbb {P}$ be any $\\\\omega _1$ -c.c., $\\\\omega ^\\\\omega $ -bounding forcing.\",\n \"Then $\\\\mathbb {P}$ forces that VWCG fails.\",\n \"Since $\\\\mathbb {P}$ is $\\\\omega _1$ -c.c.\",\n \"and $\\\\omega ^\\\\omega $ -bounding, a standard argument shows that whenever $p \\\\in \\\\mathbb {P}$ and $p$ forces that $\\\\dot{b} \\\\in \\\\omega ^\\\\omega $ , then there exists a function $b^* \\\\in \\\\omega ^\\\\omega $ such that $p$ forces that $b^*$ dominates $\\\\dot{b}$ .\",\n \"Let us show that whenever $p \\\\in \\\\mathbb {P}$ , $\\\\delta < \\\\omega _1$ , and $p$ forces that $\\\\dot{X}$ is a cofinal subset of $\\\\delta $ of order type $\\\\omega $ , then there exists a set $Y$ with order type $\\\\omega $ such that $p$ forces that $\\\\dot{X} \\\\subseteq Y$ .\",\n \"To see this, fix a bijection $f : \\\\omega \\\\rightarrow \\\\delta $ and a strictly increasing sequence $\\\\langle \\\\alpha _n : n < \\\\omega \\\\rangle $ cofinal in $\\\\alpha $ with $\\\\alpha _0 = 0$ .\",\n \"We claim that there exists a $\\\\mathbb {P}$ -name $\\\\dot{b}$ for a function from $\\\\omega $ to $\\\\omega $ such that $p$ forces that for all $n < \\\\omega $ , $\\\\dot{b}(n)$ is the least $m < \\\\omega $ such that $\\\\dot{X} \\\\cap [\\\\alpha _n,\\\\alpha _{n+1}) \\\\subseteq f[m]$ .\",\n \"This is true since $p$ forces that $\\\\dot{X}$ has order type $\\\\omega $ and hence that $\\\\dot{X} \\\\cap [\\\\alpha _n,\\\\alpha _{n+1})$ is finite for all $n < \\\\omega $ .\",\n \"Fix a function $b^* : \\\\omega \\\\rightarrow \\\\omega $ such that $p$ forces that $b^*$ dominates $\\\\dot{b}$ .\",\n \"Now let $Y := \\\\bigcup \\\\lbrace f[b^*(n)] \\\\cap [\\\\alpha _n,\\\\alpha _{n+1}) : n < \\\\omega \\\\rbrace .$ It is easy to check that $Y$ has order type $\\\\omega $ and $p$ forces that $\\\\dot{X} \\\\subseteq Y$ .\",\n \"Now we are ready to prove the proposition.\",\n \"So suppose that $p \\\\in \\\\mathbb {P}$ forces that $\\\\langle \\\\dot{X}_\\\\alpha : \\\\alpha \\\\in \\\\omega _1 \\\\cap \\\\mathrm {Lim}\\\\rangle $ is a very weak club guessing sequence.\",\n \"By the previous paragraph, for each limit ordinal $\\\\alpha < \\\\omega _1$ we can fix a cofinal subset $Y_\\\\alpha $ of $\\\\alpha $ with order type $\\\\omega $ such that $p$ forces that $\\\\dot{X}_\\\\alpha \\\\subseteq Y_\\\\alpha $ .\",\n \"We claim that $\\\\langle Y_\\\\alpha : \\\\alpha \\\\in \\\\omega _1 \\\\cap \\\\mathrm {Lim}\\\\rangle $ is a very weak club guessing sequence in the ground model, which completes the proof.\",\n \"So consider a club $C \\\\subseteq \\\\omega _1$ .\",\n \"Then $C$ is still a club in $V^\\\\mathbb {P}$ .\",\n \"Fix $q \\\\le p$ and a limit ordinal $\\\\alpha < \\\\omega _1$ such that $q$ forces that $\\\\dot{X}_\\\\alpha \\\\cap C$ is infinite.\",\n \"Then clearly $q$ forces that $Y_\\\\alpha \\\\cap C$ is infinite, so in fact, $Y_\\\\alpha \\\\cap C$ is infinite.\",\n \"Proposition 2.12 It is consistent that $\\\\lnot \\\\textsf {VWCG}$ and $\\\\textsf {CG}_{\\\\omega _1}$ both hold.\",\n \"Let $V$ be a model in which CH holds and VWCG fails.\",\n \"Such a model was shown to exist by Shelah [13].\",\n \"Let $\\\\mathbb {P}$ be an $\\\\omega _1$ -c.c., $\\\\omega ^\\\\omega $ -bounding forcing poset which adds at least $\\\\omega _2$ many reals; for example, random real forcing with product measure is such a forcing.\",\n \"We claim that in $V^\\\\mathbb {P}$ , CG$_{\\\\omega _1}$ holds but VWCG fails.\",\n \"By Lemma 2.11, VWCG is false in $V^\\\\mathbb {P}$ .\",\n \"In $V$ , define $\\\\vec{\\\\mathcal {L}} = \\\\langle \\\\mathcal {L}_\\\\alpha : \\\\alpha \\\\in \\\\omega _1 \\\\cap \\\\mathrm {Lim}\\\\rangle $ by letting $\\\\mathcal {L}_\\\\alpha $ be the collection of all cofinal subsets of $\\\\alpha $ with order type $\\\\omega $ .\",\n \"Since CH holds, the cardinality of each $\\\\mathcal {L}_\\\\alpha $ is $\\\\omega _1$ .\",\n \"If $C$ is a club subset of $\\\\omega _1$ in $V^\\\\mathbb {P}$ , then since $\\\\mathbb {P}$ is $\\\\omega _1$ -c.c., there is a club $D \\\\subseteq \\\\omega _1$ in $V$ such that $D \\\\subseteq C$ .\",\n \"In $V$ , fix $d \\\\subseteq D$ with order type $\\\\omega $ , and let $\\\\alpha := \\\\sup (d)$ .\",\n \"Then $d \\\\in \\\\mathcal {L}_\\\\alpha $ and $d \\\\subseteq C$ .\",\n \"Thus, $\\\\vec{\\\\mathcal {L}}$ witnesses that CG$_{\\\\omega _1}$ holds in $V^\\\\mathbb {P}$ .\"\n ],\n [\n \"The Consistency of Strong Measuring and $\\\\lnot \\\\textsf {CH}$\",\n \"As we previously mentioned, Measuring is equivalent to Measuring$_\\\\omega $ , and therefore under CH, Measuring is equivalent to Strong Measuring.\",\n \"In this section we establish the consistency of Strong Measuring with the negation of CH.\",\n \"More precisely, we will prove that MRP together with MA($\\\\sigma $ -centered) implies Strong Measuring, and BPFA implies Strong Measuring.\",\n \"Recall that both MRP and BPFA imply that $\\\\mathfrak {c} = \\\\omega _2$ ([12]).\",\n \"A set $M$ is suitable if for some regular cardinal $\\\\theta > \\\\omega _1$ , $M$ is a countable elementary substructure of $H(\\\\theta )$ .\",\n \"We will follow the conventions introduced in Section 1 that the properties “open” and “$M$ -stationary” refer to open and $M$ -stationary subsets of $\\\\omega _1$ (where $\\\\omega _1$ is considered as a subspace of $[\\\\omega _1]^\\\\omega $ ).\",\n \"Proposition 3.1 Assume that $M$ is suitable.\",\n \"Let $\\\\delta := M \\\\cap \\\\omega _1$ .\",\n \"Suppose that $\\\\mathcal {Y}$ is a collection of open subsets of $\\\\delta $ such that for any finite set $a \\\\subseteq \\\\mathcal {Y}$ , $\\\\bigcap a$ is $M$ -stationary.\",\n \"Then there exists a $\\\\sigma $ -centered forcing $\\\\mathbb {P}$ and a collection $\\\\mathcal {D}$ of dense subsets of $\\\\mathbb {P}$ of size at most $|\\\\mathcal {Y}| + \\\\omega $ such that whenever $G$ is a filter on $\\\\mathbb {P}$ in some outer model $W$ of $V$ with $\\\\omega _1^V = \\\\omega _1^W$ which meets each member of $\\\\mathcal {D}$ , then there exists a set $z \\\\subseteq \\\\delta $ in $W$ which is open, $M$ -stationary, and satisfies that for all $X \\\\in \\\\mathcal {Y}$ , $z \\\\setminus X$ is bounded in $\\\\delta $ .\",\n \"Define a forcing poset $\\\\mathbb {P}$ to consist of conditions which are pairs $(x,a)$ , where $x$ is an open and bounded subset of $\\\\delta $ in $M$ and $a$ is a finite subset of $\\\\mathcal {Y}$ .\",\n \"Let $(y,b) \\\\le (x,a)$ if $y$ is an end-extension of $x$ , $a \\\\subseteq b$ , and $y \\\\setminus x \\\\subseteq \\\\bigcap a$ .\",\n \"Since $M$ is countable, there are only countably many possibilities for the first component of a condition.\",\n \"If $(x,a_0), \\\\ldots , (x,a_n)$ are finitely many conditions with the same first component, then easily $(x,a_0 \\\\cup \\\\ldots \\\\cup a_n)$ is a condition in $\\\\mathbb {P}$ which is below each of the conditions $(x,a_0), \\\\ldots , (x,a_n)$ .\",\n \"It follows that $\\\\mathbb {P}$ is $\\\\sigma $ -centered.\",\n \"For each $X \\\\in \\\\mathcal {Y}$ , let $D_X$ denote the set of conditions $(x,a)$ such that $X \\\\in a$ .\",\n \"Observe that $D_X$ is dense.\",\n \"For every club $C$ of $\\\\omega _1$ which is a member of $M$ , let $E_C$ denote the set of conditions $(x,a)$ such that $x \\\\cap C$ is non–empty.\",\n \"We claim that $E_C$ is dense.\",\n \"Let $(x,a)$ be a condition.\",\n \"Since $\\\\bigcap a$ is $M$ -stationary and $\\\\lim (C) \\\\setminus (\\\\sup (x)+1)$ is a club subset of $\\\\omega _1$ in $M$ , we can find a limit ordinal $\\\\alpha $ in $C \\\\cap (\\\\bigcap a)$ which is in the interval $(\\\\sup (x),\\\\delta )$ .\",\n \"Since $\\\\alpha \\\\in \\\\bigcap a$ and $\\\\bigcap a$ is open, we can find $\\\\beta < \\\\gamma < \\\\delta $ such that $\\\\alpha \\\\in (\\\\beta ,\\\\gamma ) \\\\subseteq \\\\bigcap a$ .\",\n \"As $\\\\sup (x) + 1 < \\\\alpha $ , without loss of generality $\\\\sup (x) < \\\\beta $ .\",\n \"By elementarity, the interval $b := (\\\\beta ,\\\\gamma )$ is in $M$ .\",\n \"It follows that $(x \\\\cup b,a)$ is a condition, $x \\\\cup b$ end-extends $x$ , and $(x \\\\cup b) \\\\setminus x = b \\\\subseteq \\\\bigcap a$ .\",\n \"Thus, $(x \\\\cup b,a) \\\\le (x,a)$ , and since $\\\\alpha \\\\in C$ , $(x \\\\cup b,a) \\\\in E_C$ .\",\n \"Let $\\\\mathcal {D}$ denote the collection of all dense sets of the form $D_X$ where $X \\\\in \\\\mathcal {Y}$ , or $E_C$ where $C$ is a club subset of $\\\\omega _1$ belonging to $M$ .\",\n \"Then $|\\\\mathcal {D}| \\\\le |\\\\mathcal {Y}| + \\\\omega $ .\",\n \"Let $G$ be a filter on $\\\\mathbb {P}$ in some outer model $W$ with $\\\\omega _1^V = \\\\omega _1^W$ which meets each dense set in $\\\\mathcal {D}$ .\",\n \"Define $z := \\\\bigcup \\\\lbrace x : \\\\exists a \\\\ (x,a) \\\\in G \\\\rbrace $ .\",\n \"Note that since $z$ is a union of open sets, it is open (using the fact that being open is absolute between $V$ and $W$ ).\",\n \"For each club $C \\\\subseteq \\\\omega _1$ which lies in $M$ , there exists a condition $(x,a)$ which belongs to $G \\\\cap E_C$ , and thus $x \\\\cap C \\\\ne \\\\emptyset $ .\",\n \"Therefore, $z \\\\cap C \\\\ne \\\\emptyset $ .\",\n \"Hence, $z$ is $M$ -stationary.\",\n \"It remains to show that for all $X \\\\in \\\\mathcal {Y}$ , $z \\\\setminus X$ is bounded in $\\\\delta $ .\",\n \"Consider $X \\\\in \\\\mathcal {Y}$ .\",\n \"Then we can fix $(x,a) \\\\in G \\\\cap D_X$ , which means that $X \\\\in a$ .\",\n \"Now the definition of the ordering on $\\\\mathbb {P}$ together with the fact that $G$ is a filter easily implies that $z \\\\setminus x \\\\subseteq X$ .\",\n \"Therefore, $z \\\\setminus X \\\\subseteq x$ , and hence $z \\\\setminus X$ is bounded in $\\\\delta $ .\",\n \"Corollary 3.2 Assume that $M$ is suitable.\",\n \"Let $\\\\delta := M \\\\cap \\\\omega _1$ .\",\n \"Suppose that $\\\\mathcal {Y}$ is a collection of less than $\\\\mathfrak {m}$ ($\\\\sigma $ -centered) many open subsets of $\\\\delta $ such that for any finite set $a \\\\subseteq \\\\mathcal {Y}$ , $\\\\bigcap a$ is $M$ -stationary.\",\n \"Then there exists a set $z \\\\subseteq \\\\delta $ which is open, $M$ -stationary, and satisfies that for all $X \\\\in \\\\mathcal {Y}$ , $z \\\\setminus X$ is bounded in $\\\\delta $ .\",\n \"Fix a $\\\\sigma $ -centered forcing $\\\\mathbb {P}$ and a collection $\\\\mathcal {D}$ of dense subsets of $\\\\mathbb {P}$ of size at most $|\\\\mathcal {Y}| + \\\\omega $ as described in Proposition 3.1.\",\n \"Since $\\\\mathfrak {m}$ ($\\\\sigma $ -centered) is uncountable, $| \\\\mathcal {D} | < \\\\mathfrak {m}(\\\\textrm {$$-centered})$ .\",\n \"Hence, there exists a filter $G$ on $\\\\mathbb {P}$ which meets each dense set in $\\\\mathcal {D}$ .\",\n \"By Proposition 3.1, there exists a set $z \\\\subseteq \\\\delta $ which is open, $M$ -stationary, and satisfies that for all $X \\\\in \\\\mathcal {Y}$ , $z \\\\setminus X$ is bounded in $\\\\delta $ .\",\n \"Proposition 3.3 Let $\\\\vec{\\\\mathcal {C}} = \\\\langle \\\\mathcal {C}_\\\\alpha : \\\\alpha \\\\in \\\\omega _1 \\\\cap \\\\textrm {Lim} \\\\rangle $ be a sequence such that each $\\\\mathcal {C}_\\\\alpha $ is a collection of less than $\\\\mathfrak {m}$ ($\\\\sigma $ -centered) many closed and cofinal subsets of $\\\\alpha $ .\",\n \"Then there exists an open stationary set mapping $\\\\Sigma $ such that, if $W$ is any outer model with the same $\\\\omega _1$ in which there exists a $\\\\Sigma $ -reflecting sequence, then there exists in $W$ a club subset of $\\\\omega _1$ which measures $\\\\vec{\\\\mathcal {C}}$ .\",\n \"For each limit ordinal $\\\\alpha < \\\\omega _1$ , let $\\\\mathcal {D}_\\\\alpha := \\\\lbrace \\\\alpha \\\\setminus c : c \\\\in \\\\mathcal {C}_\\\\alpha \\\\rbrace $ .\",\n \"Observe that each $\\\\mathcal {D}_\\\\alpha $ is a collection of fewer than $\\\\mathfrak {m}$ ($\\\\sigma $ -centered) many open subsets of $\\\\alpha $ .\",\n \"We will define $\\\\Sigma $ to have domain the collection of all countable elementary substructures $M$ of $H(\\\\omega _2)$ .\",\n \"Consider such an $M$ and we define $\\\\Sigma (M)$ .\",\n \"Note that $M$ is suitable.\",\n \"Let $\\\\delta := M \\\\cap \\\\omega _1$ .\",\n \"We consider two cases.\",\n \"In the first case, there does not exist a member of $\\\\mathcal {D}_\\\\delta $ which is $M$ -stationary.\",\n \"Define $\\\\Sigma (M) = \\\\delta $ , which is clearly open and $M$ -stationary.\",\n \"In the second case, there exists some member of $\\\\mathcal {D}_\\\\delta $ which is $M$ -stationary.\",\n \"A straightforward application of Zorn's lemma implies that there exists a non–empty set ${\\\\mathcal {Y}}_M \\\\subseteq \\\\mathcal {D}_\\\\delta $ such that for any $a \\\\in [{\\\\mathcal {Y}}_M]^{<\\\\omega }$ , $\\\\bigcap a$ is $M$ -stationary, and moreover, $\\\\mathcal {Y}_M$ is a maximal subset of $\\\\mathcal {D}_\\\\delta $ with this property.\",\n \"Since $\\\\mathcal {Y}_M \\\\subseteq \\\\mathcal {D}_\\\\delta $ , $|\\\\mathcal {Y}_M| < \\\\mathfrak {m}(\\\\textrm {$$-centered})$ .\",\n \"So the collection $\\\\mathcal {Y}_M$ satisfies the assumptions of Corollary 3.2.\",\n \"It follows that there exists a set $z_M \\\\subseteq \\\\delta $ which is open, $M$ -stationary, and satisfies that for all $X \\\\in \\\\mathcal {Y}_M$ , $z_M \\\\setminus X$ is bounded in $\\\\delta $ .\",\n \"Now define $\\\\Sigma (M) := z_M$ .\",\n \"This completes the definition of $\\\\Sigma $ .\",\n \"Consider an outer model $W$ of $V$ with the same $\\\\omega _1$ , and assume that in $W$ there exists a $\\\\Sigma $ -reflecting sequence $\\\\langle M_\\\\delta : \\\\delta < \\\\omega _1 \\\\rangle $ .\",\n \"Let $\\\\alpha _\\\\delta := M_\\\\delta \\\\cap \\\\omega _1$ for all $\\\\delta < \\\\omega _1$ .\",\n \"Let $D$ be the club set of $\\\\delta < \\\\omega _1$ such that $\\\\alpha _\\\\delta = \\\\delta $ .\",\n \"We claim that $D$ measures $\\\\vec{\\\\mathcal {C}}$ .\",\n \"Consider $\\\\delta \\\\in \\\\lim (D)$ .\",\n \"Then $\\\\delta = \\\\alpha _\\\\delta = M_\\\\delta \\\\cap \\\\omega _1$ .\",\n \"Let $M := M_\\\\delta $ .\",\n \"We first claim that if $c \\\\in \\\\mathcal {C}_\\\\delta $ and $\\\\delta \\\\setminus c$ is not $M$ -stationary, then for some $\\\\beta < \\\\delta $ , $(D \\\\cap \\\\delta ) \\\\setminus \\\\beta \\\\subseteq c$ .\",\n \"Fix a club subset $E$ of $\\\\omega _1$ in $M$ which is disjoint from $\\\\delta \\\\setminus c$ .\",\n \"By the continuity of the $\\\\Sigma $ -reflecting sequence, there exists $\\\\beta < \\\\delta $ such that $E \\\\in M_\\\\beta $ .\",\n \"We claim that $(D \\\\cap \\\\delta ) \\\\setminus \\\\beta \\\\subseteq c$ .\",\n \"Let $\\\\xi \\\\in (D \\\\cap \\\\delta ) \\\\setminus \\\\beta $ .\",\n \"Then $E \\\\in M_\\\\xi $ , and hence by elementarity, $\\\\xi = M_\\\\xi \\\\cap \\\\omega _1 \\\\in E$ .\",\n \"Since $E$ is disjoint from $\\\\delta \\\\setminus c$ , $\\\\xi \\\\in c$ .\",\n \"We split the argument according to the two cases in the definition of $\\\\Sigma (M)$ .\",\n \"In the first case, there does not exist a member of $\\\\mathcal {D}_\\\\delta $ which is $M$ -stationary.\",\n \"Consider $c \\\\in \\\\mathcal {C}_\\\\delta $ .\",\n \"Then $\\\\delta \\\\setminus c$ is not $M$ -stationary.\",\n \"By the previous paragraph, there exists $\\\\beta < \\\\delta $ such that $(D \\\\cap \\\\delta ) \\\\setminus \\\\beta \\\\subseteq c$ .\",\n \"In the second case, there exists a member of $\\\\mathcal {D}_\\\\delta $ which is $M$ -stationary.\",\n \"Consider $c \\\\in \\\\mathcal {C}_\\\\delta $ .\",\n \"Then $X := \\\\delta \\\\setminus c \\\\in \\\\mathcal {D}_\\\\delta $ .\",\n \"We consider two possibilities.\",\n \"First, assume that $X$ is in $\\\\mathcal {Y}_M$ .\",\n \"By the choice of $\\\\mathcal {Y}_M$ and $z_M$ , we know that $z_M \\\\setminus X$ is bounded in $\\\\delta $ .\",\n \"So fix $\\\\beta _0 < \\\\delta $ so that $z_M \\\\setminus \\\\beta _0 \\\\subseteq X$ .\",\n \"By the definition of being a $\\\\Sigma $ -reflecting sequence, there exists $\\\\beta _1 < \\\\delta $ so that for all $\\\\beta _1 \\\\le \\\\xi < \\\\delta $ , $M_\\\\xi \\\\cap \\\\omega _1 \\\\in \\\\Sigma (M) = z_M$ .\",\n \"Let $\\\\beta := \\\\max \\\\lbrace \\\\beta _1, \\\\beta _2 \\\\rbrace $ .\",\n \"Consider $\\\\xi \\\\in (D \\\\cap \\\\delta ) \\\\setminus \\\\beta $ .\",\n \"Then $\\\\xi \\\\ge \\\\beta _1$ implies that $\\\\xi = M_\\\\xi \\\\cap \\\\omega _1 \\\\in z_M$ .\",\n \"So $\\\\xi \\\\in z_M \\\\setminus \\\\beta _0 \\\\subseteq X = \\\\delta \\\\setminus c$ .\",\n \"Secondly, assume that $X$ is not in $\\\\mathcal {Y}_M$ .\",\n \"By the maximality of $\\\\mathcal {Y}_M$ , there exists a set $a \\\\in [\\\\mathcal {Y}_M]^{<\\\\omega }$ such that $X \\\\cap \\\\bigcap a$ is not $M$ -stationary.\",\n \"Fix a club $E$ in $M$ which is disjoint from $X \\\\cap \\\\bigcap a$ .\",\n \"By the continuity of the $\\\\Sigma $ -reflecting sequence, there exists $\\\\beta < \\\\delta $ such that $E \\\\in M_\\\\beta $ .\",\n \"Consider $\\\\xi \\\\in (D \\\\cap \\\\delta ) \\\\setminus \\\\beta $ .\",\n \"Then $E \\\\in M_\\\\xi $ , which implies that $\\\\xi = M_\\\\xi \\\\cap \\\\omega _1 \\\\in E$ .\",\n \"Thus, $\\\\xi $ is not in $X \\\\cap \\\\bigcap a$ .\",\n \"On the other hand, letting $a = \\\\lbrace X_0, \\\\ldots , X_n \\\\rbrace $ , for each $i \\\\le n$ the previous paragraph implies that there exists $\\\\beta _i < \\\\delta $ such that $(D \\\\cap \\\\delta ) \\\\setminus \\\\beta _i \\\\subseteq X_i$ .\",\n \"Let $\\\\beta ^*$ be an ordinal in $\\\\delta $ which is larger than $\\\\beta $ and $\\\\beta _i$ for all $i \\\\le n$ .\",\n \"Consider $\\\\xi \\\\in (D \\\\cap \\\\delta ) \\\\setminus \\\\beta ^*$ .\",\n \"Then by the choice of $\\\\beta $ , $\\\\xi \\\\notin X \\\\cap \\\\bigcap a$ .\",\n \"By the choice of the $\\\\beta _i$ 's, $\\\\xi \\\\in \\\\bigcap a$ .\",\n \"Therefore, $\\\\xi \\\\notin X = \\\\delta \\\\setminus c$ , which means that $\\\\xi \\\\in c$ .\",\n \"Thus, $(D \\\\cap \\\\delta ) \\\\setminus \\\\beta ^* \\\\subseteq c$ .\",\n \"Corollary 3.4 Assume MRP and MA($\\\\sigma $ -centered).\",\n \"Then Strong Measuring holds.\",\n \"Let $\\\\vec{\\\\mathcal {C}} = \\\\langle \\\\mathcal {C}_\\\\alpha :\\\\alpha \\\\in \\\\omega _1 \\\\cap \\\\textrm {Lim} \\\\rangle $ be a sequence such that each $\\\\mathcal {C}_\\\\alpha $ is a collection of fewer than $\\\\mathfrak {c}$ many closed and cofinal subsets of $\\\\alpha $ .\",\n \"We claim that there exists a club subset of $\\\\omega _1$ which measures $\\\\vec{\\\\mathcal {C}}$ .\",\n \"By MA($\\\\sigma $ -centered), $\\\\mathfrak {m}$ ($\\\\sigma $ -centered) equals $\\\\mathfrak {c}$ .\",\n \"So each $\\\\mathcal {C}_\\\\alpha $ has size less than $\\\\mathfrak {m}$ ($\\\\sigma $ -centered).\",\n \"By Proposition 3.3, there exists an open stationary set mapping $\\\\Sigma $ such that, if $W$ is any outer model with the same $\\\\omega _1$ in which there exists a $\\\\Sigma $ -reflecting sequence, then there exists in $W$ a club subset of $\\\\omega _1$ which measures $\\\\vec{\\\\mathcal {C}}$ .\",\n \"Applying MRP, there exists a $\\\\Sigma $ -reflecting sequence in $V$ .\",\n \"Thus, in $V$ there exists a club subset of $\\\\omega _1$ which measures $\\\\vec{\\\\mathcal {C}}$ .\",\n \"Corollary 3.5 Assume BPFA.\",\n \"Then Strong Measuring holds.\",\n \"Let $\\\\vec{\\\\mathcal {C}} = \\\\langle \\\\mathcal {C}_\\\\alpha :\\\\alpha \\\\in \\\\omega _1 \\\\cap \\\\textrm {Lim} \\\\rangle $ be a sequence such that each $\\\\mathcal {C}_\\\\alpha $ is a collection of fewer than $\\\\mathfrak {c} = \\\\omega _2$ many closed and cofinal subsets of $\\\\alpha $ .\",\n \"We claim that there exists a club subset of $\\\\omega _1$ which measures $\\\\vec{\\\\mathcal {C}}$ .\",\n \"Since $\\\\mathfrak {c} = \\\\omega _2$ , $\\\\vec{\\\\mathcal {C}}$ is a member of $H(\\\\omega _2)$ .\",\n \"Thus, the existence of a club subset of $\\\\omega _2$ which measures $\\\\vec{\\\\mathcal {C}}$ is expressible as a $\\\\Sigma _1$ statement involving a parameter in $H(\\\\omega _2)$ .\",\n \"By BPFA, it suffices to show that there exists a proper forcing which forces that such a club exists.\",\n \"Now BPFA implies Martin's Axiom, and in particular, that $\\\\mathfrak {m}$ ($\\\\sigma $ -centered) is equal to $\\\\mathfrak {c}$ .\",\n \"So each $\\\\mathcal {C}_\\\\alpha $ has size less than $\\\\mathfrak {m}$ ($\\\\sigma $ -centered).\",\n \"By Proposition 3.3, there exists an open stationary set mapping $\\\\Sigma $ such that, if $W$ is any outer model with the same $\\\\omega _1$ in which there exists a $\\\\Sigma $ -reflecting sequence, then there exists in $W$ a club subset of $\\\\omega _1$ which measures $\\\\vec{\\\\mathcal {C}}$ .\",\n \"By [12], there exists a proper forcing $\\\\mathbb {P}$ which adds a $\\\\Sigma $ -reflecting sequence, so in $V^\\\\mathbb {P}$ there is a club subset of $\\\\omega _1$ which measures $\\\\vec{\\\\mathcal {C}}$ .\",\n \"We now sketch a proof that MRP alone does not imply Strong Measuring.\",\n \"In particular, Measuring does not imply Strong Measuring.\",\n \"Start with a model of CH in which there exists a supercompact cardinal $\\\\kappa $ .\",\n \"Construct a forcing iteration $\\\\mathbb {P}$ in the standard way to obtain a model of MRP.\",\n \"To do this, fix a Laver function $f : \\\\kappa \\\\rightarrow V_\\\\kappa $ .\",\n \"Then define a countable support forcing iteration $\\\\langle \\\\mathbb {P}_\\\\alpha , \\\\dot{\\\\mathbb {Q}}_\\\\beta : \\\\alpha \\\\le \\\\kappa , \\\\beta < \\\\kappa \\\\rangle $ as follows.\",\n \"Given $\\\\mathbb {P}_\\\\alpha $ , consider $f(\\\\alpha )$ .\",\n \"If $f(\\\\alpha )$ happens to be a $\\\\mathbb {P}_\\\\alpha $ -name for some open stationary set mapping, then let $\\\\dot{\\\\mathbb {Q}}_\\\\alpha $ be a $\\\\mathbb {P}_\\\\alpha $ -name for a proper forcing which adds an $f(\\\\alpha )$ -reflecting sequence.\",\n \"Otherwise let $\\\\dot{\\\\mathbb {Q}}_\\\\alpha $ be a $\\\\mathbb {P}_\\\\alpha $ -name for $Col(\\\\omega _1,\\\\omega _2)$ .\",\n \"Now define $\\\\mathbb {P}:= \\\\mathbb {P}_\\\\kappa $ .\",\n \"Arguments similar to those in the standard construction of a model of PFA can be used to show that $\\\\mathbb {P}$ forces MRP.\",\n \"The forcing for adding a $\\\\Sigma $ -reflecting sequence for a given open stationary set mapping does not add reals ([12]).\",\n \"In particular, it is vacuously $\\\\omega ^\\\\omega $ -bounding.\",\n \"The property of being proper and $\\\\omega ^\\\\omega $ -bounding is preserved under countable support forcing iterations ([1]), so $\\\\mathbb {P}$ is also $\\\\omega ^\\\\omega $ -bounding.\",\n \"In particular, $V \\\\cap \\\\omega ^\\\\omega $ is an unbounded family in $V^\\\\mathbb {P}$ , and it has size $\\\\omega _1$ since CH holds in $V$ .\",\n \"It follows that the bounding number $\\\\mathfrak {b}$ is equal to $\\\\omega _1$ .\",\n \"But by Corollary 2.9, Measuring$_{\\\\mathfrak {b}}$ is false.\",\n \"So $\\\\mathbb {P}$ forces that $\\\\textsf {Measuring}_{\\\\omega _1}$ is false.\",\n \"As $\\\\mathfrak {c} = \\\\omega _2$ in $V^\\\\mathbb {P}$ , Strong Measuring fails in $V^\\\\mathbb {P}$ .\",\n \"We also note that Strong Measuring plus $\\\\mathfrak {c} = \\\\omega _2$ is consistent with the existence of an $\\\\omega _1$ -Suslin tree.\",\n \"Namely, both the forcing for adding a $\\\\Sigma $ -reflecting sequence for a given open stationary set mapping $\\\\Sigma $ , as well as any $\\\\sigma $ -centered forcing, preserve Suslin trees ([11]).\",\n \"And the property of being proper and preserving a given Suslin tree is preserved under countable support forcings iterations ([10]).\",\n \"So starting with a model in which there exists an $\\\\omega _1$ -Suslin tree $S$ and a supercompact cardinal $\\\\kappa $ , we can iterate forcing similar to the argument in the preceding paragraphs to produce a model of MA($\\\\sigma $ -centered) plus MRP in which $S$ is an $\\\\omega _1$ -Suslin tree.\",\n \"By Corollary 3.4, Strong Measuring holds in that model.\"\n ],\n [\n \"Measuring Without the Axiom of Choice\",\n \"Another natural way to strengthen $\\\\operatorname{\\\\textsf {Measuring}}$ is to allow, in the sequence to be measured, not just closed sets, but also sets of higher Borel complexity.\",\n \"This line of strengthenings of $\\\\operatorname{\\\\textsf {Measuring}}$ was also considered in [2].\",\n \"For completeness, we are including here the corresponding observations.\",\n \"The version of $\\\\operatorname{\\\\textsf {Measuring}}$ where one considers sequences $\\\\vec{X}=\\\\langle X_\\\\alpha \\\\,:\\\\,\\\\alpha \\\\in \\\\omega _1\\\\cap \\\\mathrm {Lim}\\\\rangle $ , with each $X_\\\\alpha $ an open subset of $\\\\alpha $ in the order topology, is of course equivalent to $\\\\operatorname{\\\\textsf {Measuring}}$ .\",\n \"A natural next step would therefore be to consider sequences in which each $X_\\\\alpha $ is a countable union of closed sets.\",\n \"This is obviously the same as allowing each $X_\\\\alpha $ to be an arbitrary subset of $\\\\alpha $ .\",\n \"Let us call the corresponding statement $\\\\operatorname{\\\\textsf {Measuring}}^\\\\ast $ : Definition 4.1 $\\\\operatorname{\\\\textsf {Measuring}}^\\\\ast $ holds if and only if for every sequence $\\\\vec{X}=\\\\langle X_\\\\alpha \\\\,:\\\\,\\\\alpha \\\\in \\\\omega _1\\\\cap \\\\mathrm {Lim}\\\\rangle $ , if $X_\\\\alpha \\\\subseteq \\\\alpha $ for each $\\\\alpha $ , then there is some club $D\\\\subseteq \\\\omega _1$ such that for every limit point $\\\\delta \\\\in D$ of $D$ , $D\\\\cap \\\\delta $ measures $X_\\\\delta $ .\",\n \"It is easy to see that $\\\\operatorname{\\\\textsf {Measuring}}^\\\\ast $ is false in $\\\\textsf {ZFC}$ .\",\n \"In fact, given a stationary and co-stationary $S\\\\subseteq \\\\omega _1$ , there is no club of $\\\\omega _1$ measuring $\\\\vec{X}=\\\\langle S\\\\cap \\\\alpha \\\\,:\\\\,\\\\alpha \\\\in \\\\omega _1\\\\cap \\\\mathrm {Lim}\\\\rangle $ .\",\n \"The reason is that if $D$ is any club of $\\\\omega _1$ , then both $D\\\\cap S\\\\cap \\\\delta $ and $(D\\\\cap \\\\delta )\\\\setminus S$ are cofinal subsets of $\\\\delta $ for each $\\\\delta $ in the club of limit points in $\\\\omega _1$ of both $D\\\\cap S$ and $D\\\\setminus S$ .\",\n \"The status of $\\\\operatorname{\\\\textsf {Measuring}}^\\\\ast $ is more interesting in the absence of the Axiom of Choice.\",\n \"Let $\\\\mathcal {C}_{\\\\omega _1}=\\\\lbrace X\\\\subseteq \\\\omega _1\\\\,:\\\\,C \\\\subseteq X\\\\mbox{ for some club $C$ of $\\\\omega _1$}\\\\rbrace $ .\",\n \"Observation 4.2 ($\\\\textsf {ZF}$ + $\\\\mathcal {C}_{\\\\omega _1}$ is a normal filter on $\\\\omega _1$ ) Suppose $\\\\vec{X}=\\\\langle X_\\\\delta \\\\,:\\\\,\\\\delta \\\\in \\\\omega _1\\\\cap \\\\mathrm {Lim}\\\\rangle $ is such that $X_\\\\delta \\\\subseteq \\\\delta $ for each $\\\\delta $ .\",\n \"For each club $C\\\\subseteq \\\\omega _1$ , there is some $\\\\delta \\\\in C$ such that $C\\\\cap X_\\\\delta \\\\ne \\\\emptyset $ , and there is some $\\\\delta \\\\in C$ such that $(C\\\\cap \\\\delta )\\\\setminus X_\\\\delta \\\\ne \\\\emptyset $ .\",\n \"Then there is a stationary and co-stationary subset of $\\\\omega _1$ definable from $\\\\vec{X}$ .\",\n \"We have two possible cases.\",\n \"The first case is that in which for all $\\\\alpha <\\\\omega _1$ , either $W_\\\\alpha ^0=\\\\lbrace \\\\delta <\\\\omega _1\\\\,:\\\\,\\\\alpha \\\\notin X_\\\\delta \\\\rbrace $ is in $\\\\mathcal {C}_{\\\\omega _1}$ , or $W_\\\\alpha ^1=\\\\lbrace \\\\delta <\\\\omega _1\\\\,:\\\\,\\\\alpha \\\\in X_\\\\delta \\\\rbrace $ is in $\\\\mathcal {C}_{\\\\omega _1}$ .\",\n \"For each $\\\\alpha <\\\\omega _1$ , let $W_\\\\alpha $ be $W_\\\\alpha ^\\\\epsilon $ for the unique $\\\\epsilon \\\\in \\\\lbrace 0, 1\\\\rbrace $ such that $W_\\\\alpha ^\\\\epsilon \\\\in \\\\mathcal {C}_{\\\\omega _1}$ , and let $W^\\\\ast =\\\\Delta _{\\\\alpha <\\\\omega _1} W_\\\\alpha \\\\in \\\\mathcal {C}_{\\\\omega _1}$ .\",\n \"Then $X_{\\\\delta _0}=X_{\\\\delta _1}\\\\cap \\\\delta _0$ for all $\\\\delta _0<\\\\delta _1$ in $W^\\\\ast $ .\",\n \"It then follows, by (2), that $S=\\\\bigcup _{\\\\delta \\\\in W^\\\\ast }X_\\\\delta $ , which of course is definable from $\\\\vec{C}$ , is a stationary and co-stationary subset of $\\\\omega _1$ .\",\n \"Indeed, suppose $C\\\\subseteq \\\\omega _1$ is a club, and let us fix a club $D\\\\subseteq W^\\\\ast $ .\",\n \"There is then some $\\\\delta \\\\in C\\\\cap D$ and some $\\\\alpha \\\\in C\\\\cap D\\\\cap X_\\\\delta $ .\",\n \"But then $\\\\alpha \\\\in S$ since $\\\\delta \\\\in W^\\\\ast $ and $\\\\alpha \\\\in W^\\\\ast \\\\cap X_\\\\delta $ .\",\n \"There is also some $\\\\delta \\\\in C\\\\cap D$ and some $\\\\alpha \\\\in C\\\\cap D$ such that $\\\\alpha \\\\notin X_\\\\delta $ , which implies that $\\\\alpha \\\\notin S$ by a symmetrical argument, using the fact that $X_{\\\\delta _0}=X_{\\\\delta _1}\\\\cap \\\\delta _0$ for all $\\\\delta _0<\\\\delta _1$ in $W^\\\\ast $ .\",\n \"The second possible case is that there is some $\\\\alpha <\\\\omega _1$ with the property that both $W^0_\\\\alpha $ and $W^1_\\\\alpha $ are stationary subsets of $\\\\omega _1$ .\",\n \"But now we can let $S$ be $W^0_\\\\alpha $ , where $\\\\alpha $ is first such that $W^0_\\\\alpha $ is stationary and co-stationary.\",\n \"It is worth comparing the above observation with Solovay's classic result that an $\\\\omega _1$ –sequence of pairwise disjoint stationary subsets of $\\\\omega _1$ is definable from any given ladder system on $\\\\omega _1$ (working in the same theory).\",\n \"Corollary 4.3 ($\\\\textsf {ZF}$ + $\\\\mathcal {C}_{\\\\omega _1}$ is a normal filter on $\\\\omega _1$ ) The following are equivalent.\",\n \"$\\\\mathcal {C}_{\\\\omega _1}$ is an ultrafilter on $\\\\omega _1$ ; $\\\\operatorname{\\\\textsf {Measuring}}^\\\\ast $ ; For every sequence $\\\\langle X_\\\\alpha \\\\,:\\\\,\\\\alpha \\\\in \\\\omega _1\\\\cap \\\\mathrm {Lim}\\\\rangle $ , if $X_\\\\alpha \\\\subseteq \\\\alpha $ for each $\\\\alpha $ , then there is a club $C\\\\subseteq \\\\omega _1$ such that either $C\\\\cap \\\\delta \\\\subseteq X_\\\\delta $ for every $\\\\delta \\\\in C$ , or $C\\\\cap X_\\\\delta =\\\\emptyset $ for every $\\\\delta \\\\in C$ .\",\n \"(3) trivially implies (2), and by the observation (1) implies (3).\",\n \"Finally, to see that (2) implies (1), note that the argument right after the definition of $\\\\operatorname{\\\\textsf {Measuring}}^\\\\ast $ uses only $\\\\textsf {ZF}$ together with the regularity of $\\\\omega _1$ and the negation of (1).\",\n \"In particular, the strong form of $\\\\operatorname{\\\\textsf {Measuring}}^\\\\ast $ given by (3) in the above observation follows from $\\\\textsf {ZF}$ together with the Axiom of Determinacy.\",\n \"We finish this digression into set theory without the Axiom of Choice by observing that any attempt to parametrize $\\\\operatorname{\\\\textsf {Measuring}}^\\\\ast $ , in the same vein as we did with Measuring, gives rise to principles vacuously equivalent to $\\\\operatorname{\\\\textsf {Measuring}}^\\\\ast $ itself, at least when the parametrization is done with the alephs.This was pointed out by Asaf Karagila.\",\n \"Specifically, given an aleph $\\\\kappa $ , let us define $\\\\operatorname{\\\\textsf {Measuring}}^*_\\\\kappa $ as the statement that for every sequence $\\\\langle \\\\mathcal {X}_\\\\alpha \\\\,:\\\\,\\\\alpha \\\\in \\\\omega _1\\\\cap \\\\mathrm {Lim}\\\\rangle $ , if each $\\\\mathcal {X}_\\\\alpha $ is a set of cardinality at most $\\\\kappa $ consisting of subsets of $\\\\alpha $ , then there is a club $D\\\\subseteq \\\\omega _1$ such that for every limit point $\\\\delta \\\\in D$ of $D$ , $D\\\\cap \\\\delta $ measures $X$ for all $X\\\\in \\\\mathcal {X}_\\\\delta $ .\",\n \"Then $\\\\operatorname{\\\\textsf {Measuring}}^\\\\ast _{\\\\omega }$ is clearly equivalent to $\\\\operatorname{\\\\textsf {Measuring}}^\\\\ast $ under $\\\\textsf {ZF}$ together with the normality of $\\\\mathcal {C}_{\\\\omega _1}$ and the Axiom of Choice for countable families of subsets of $\\\\omega _1$ (which of course follows from the Axiom of Choice for countable families of sets of reals, and therefore also from $\\\\textsf {ZF} + \\\\textsf {AD}$ ).\",\n \"On the other hand, working in $\\\\textsf {ZF}$ + $\\\\mathcal {C}_{\\\\omega _1}$ is a normal filter on $\\\\omega _1$ , we have that $\\\\operatorname{\\\\textsf {Measuring}}^\\\\ast _{\\\\omega _1}$ follows vacuously from $\\\\operatorname{\\\\textsf {Measuring}}^\\\\ast $ simply because under $\\\\operatorname{\\\\textsf {Measuring}}^\\\\ast $ there is no sequence $\\\\langle \\\\mathcal {X}_\\\\alpha \\\\,:\\\\,\\\\alpha \\\\in \\\\omega _1\\\\cap \\\\mathrm {Lim}\\\\rangle $ as in the definition of $\\\\operatorname{\\\\textsf {Measuring}}^\\\\ast _{\\\\omega _1}$ and such that $\\\\vert \\\\mathcal {X}_\\\\alpha \\\\vert =\\\\omega _1$ for some $\\\\alpha $ ; indeed, $\\\\operatorname{\\\\textsf {Measuring}}^\\\\ast $ implies, over this base theory, that $\\\\mathcal {C}_{\\\\omega _1}$ is an ultrafilter (Corollary REF ), and if $\\\\mathcal {C}_{\\\\omega _1}$ is an ultrafilter then there is no $\\\\omega _1$ -sequence of distinct reals, whereas the existence of a family of size $\\\\omega _1$ consisting of subsets of some fixed countable ordinal clearly implies that there is such a sequence.\",\n \"We conclude the article with two natural questions.\",\n \"Question 4.4 Is Measuring$_{\\\\mathfrak {p}}$ false?\",\n \"Question 4.5 Are Measuring and Strong Measuring equivalent statements assuming Martin's Axiom?\"\n ]\n]"},"id":{"kind":"string","value":"1808.08596"}}},{"rowIdx":297,"cells":{"text":{"kind":"list like","value":[["Interferometric Observations of Cyanopolyynes toward the G28.28-0.36\n High-Mass Star-Forming Region"],["Abstract We have carried out interferometric observations of cyanopolyynes, HC$_{3}$N, HC$_{5}$N, and HC$_{7}$N, in the 36 GHz band toward the G28.28$-$0.36 high-mass star-forming region using the Karl G. Jansky Very Large Array (VLA) Ka-band receiver.","The spatial distributions of HC$_{3}$N and HC$_{5}$N are obtained.","HC$_{5}$N emission is coincident with a 450 $\\mu$m dust continuum emission and this clump with a diameter of $\\sim 0.2$ pc is located at the east position from the 6.7 GHz methanol maser by $\\sim 0.15$ pc.","HC$_{7}$N is tentatively detected toward the clump.","The HC$_{3}$N : HC$_{5}$N : HC$_{7}$N column density ratios are estimated at 1.0 : $\\sim 0.3$ : $\\sim 0.2$ at an HC$_{7}$N peak position.","We discuss possible natures of the 450 $\\mu$m continuum clump associated with the cyanopolyynes.","The 450 $\\mu$m continuum clump seems to contain deeply embedded low- or intermediate-mass protostellar cores, and the most possible formation mechanism of the cyanopolyynes is the warm carbon chain chemistry (WCCC) mechanism.","In addition, HC$_{3}$N and compact HC$_{5}$N emission is detected at the edge of the 4.5 $\\mu$m emission, which possibly implies that such emission is the shock origin."],["Introduction","Cyanopolyynes (HC$_{2n+1}$ N, $n=1,2,3,...$ ) are one of the representative carbon-chain species.","In low-mass star-forming regions, carbon-chain molecules are known as early-type species; they are abundant in young starless cores and deficient in star-forming cores [31], [15].","In contrast to the general picture, cyanoacetylene (HC$_{3}$ N), the shortest member of cyanopolyynes, is detected from various regions such as infrared dark clouds [28], molecular outflows [1], protoplanetary disks [23], [4], and comets [22] and it is interesting to trace cyanopolyyne chemistry for better understanding of the molecular evolution during star/planet formation process.","Cyanopolyynes attract astrobiological as well as astrochemical interests.","Since they contain the nitrile bond (–C$\\equiv $ N), cyanopolyynes have been suggested as possible intermediates in the synthesis of simple amino acids [11], [5].","Saturated complex organic molecules (COMs), consisting of more than six atoms with rich hydrogen atoms, are abundant around protostars.","Such chemistry is known as hot core in high-mass star-forming regions and hot corino in low-mass star-forming regions.","In addition to hot corino, around a few low-mass protostars, carbon-chain molecules are formed from CH$_{4}$ evaporated from dust grains, which is known as warm carbon chain chemistry [27].","Progress in observational studies of carbon-chain molecules in high-mass star-forming regions has been slower, compared to low-mass star-forming regions.","Regarding hot cores, HC$_{5}$ N has been detected in chemically rich sources, Orion KL [9] and Sgr B2 [2], while only a tentative detection of HC$_{7}$ N in Orion KL was reported [10].","[6] performed a chemical network simulation and suggested that cyanopolyynes could be formed in a hot core from C$_{2}$ H$_{2}$ evaporated from grain mantles.","Motivated by the chemical network simulation, [13] carried out survey observations of HC$_{5}$ N toward 79 hot cores associated with the 6.7 GHz methanol masers and reported its detection in 35 sources.","However, the association with the maser is questionable, because they used a large beam (0.95) and a low-excitation energy line ($J=12-11$ ; $E_{\\rm u}/k = 10.0$ K), which can be excited even in dark clouds.","[33] carried out observations of long cyanopolyynes (HC$_{5}$ N and HC$_{7}$ N) toward four massive young stellar objects, where [13] had reported the HC$_{5}$ N detection, using the Green Bank 100-m and the Nobeyama 45-m radio telescopes, and detected high-excitation energy lines ($E_{\\rm u}/k \\approx 100$ K) of HC$_{5}$ N. The detection of such lines means that HC$_{5}$ N exists at least in the warm gas, not in cold molecular clouds ($T_{\\rm {kin}} \\simeq 10$ K).","[34] found that the G28.28$-$ 0.36 high-mass star-forming region is a particular cyanopolyyne-rich source with less COMs compared with other sources.","Hence, G28.28$-$ 0.36 is considered to be a good target region to study the cyanopolyyne chemistry around massive young stellar objects (MYSOs).","Using the Nobeyama 45-m radio telescope, [35] investigated the main formation mechanism of HC$_{3}$ N in G28.28$-$ 0.36 from its $^{13}$ C isotopic fractionation.","The reaction of “C$_{2}$ H$_{2}$ + CN\" was proposed as the main formation pathway of HC$_{3}$ N, which is consistent with the chemical network simulation conducted by [6] and the WCCC model [14].","Figure: Spitzer's IRAC 3.6 μ\\mu m image toward G28.28--0.36.","The open circle and cross indicate the 6.7 GHz methanol maser and ultracompact H2 (UCH2) region , respectively.In this paper, we carried out interferometric observations of cyanopolyynes (HC$_{3}$ N, HC$_{5}$ N, and HC$_{7}$ N) toward the G28.28$-$ 0.36 high-mass star-forming region ($d = 3$ kpc) with the Karl G. Jansky Very Large Array (VLA).","Figure REF shows the Spitzer IRAC 3.6 $\\mu $ m imagehttp://sha.ipac.caltech.edu/applications/Spitzer/SHA/ toward the region.","G28.28$-$ 0.36 is classified as an Extended Green Object (EGO) source [8] from the Spitzer Galactic Legacy Infrared Mid-Plane Survey Extraordinaire [3].","In Figure REF , the open circle and cross indicate the 6.7 GHz methanol maser [7] and ultracompact H2 (UCH2) region [36], respectively.","The 6.7 GHz maser is considered to give us the exact position of MYSOs [37].","A UCH2 region seems to heat the environment.","As shown in Figure REF , the ring structure around the UCH2 region is suggestive of expanding motion and on-going massive star formation.","We describe the observational details and data analyses in Section .","The resultant images and spectra of cyanopolyynes are presented in Section .","We compare the spatial distributions of cyanopolyynes with the infrared images and discuss possible formation mechanisms in Section ."],["Observations","The observations of G28.28$-$ 0.36 using the VLA Ka-band receiver were carried out in the C configuration with the 27 $\\times $ 25-m antennas on March 20th, 2016 (Proposal ID = 16A-084, PI; Kotomi Taniguchi).","The field of view (FoV) is $\\sim $ 604.","Four spectral windows of the correlator were set at our target lines summarized in Table .","All of these target lines were simultaneously observed.","The channel separation of the correlator is 0.5 km s$^{-1}$ .","The angular resolutions and Position Angles (PA) for each line are summarized in Table .","The phase reference center was set at ($\\alpha _{2000}$ , $\\delta _{2000}$ ) = (18$^{\\rm h}$ 44$^{\\rm m}$ 133, -0418030), the 6.7 GHz methanol maser position.","The pointing source is J1832$-$ 1035 at ($\\alpha _{2000}$ , $\\delta _{2000}$ ) = (18$^{\\rm h}$ 32$^{\\rm m}$ 20836, $-$ 1035112).","The absolute flux density calibration and the bandpass calibration were conducted by observing 3C286 at ($\\alpha _{2000}$ , $\\delta _{2000}$ ) = (13$^{\\rm h}$ 31$^{\\rm m}$ 0828798, +3030329589).","The gain/phase calibration was conducted by observing J1851+0035 at ($\\alpha _{2000}$ , $\\delta _{2000}$ ) = (18$^{\\rm h}$ 51$^{\\rm m}$ 467217, +003532414).","cccccc Summary of target lines 0pt Species Transition Rest Frequency $E_{\\rm {u}}/k$ Angular PA (GHz) (K) Resolution (deg) HC$_{3}$ N $J=4-3$ 36.39232 4.4 084 $\\times $ 063 -9.92 HC$_{5}$ N $J=14-13$ 37.276994 13.4 081 $\\times $ 063 -11.04 HC$_{7}$ N $J=33-32$ 37.22349 30.4 082 $\\times $ 063 -10.20 CH$_{3}$ CN $J_{\\rm K} =2_{0} - 1_{0}$ 36.7954747 2.6 083 $\\times $ 064 -11.22 Rest frequencies are taken from the Cologne Database for Molecular Spectroscopy [21] and the Jet Propulsion Laboratory catalog [24].","We conducted data reduction using the Common Astronomy Software Application [20].","We used the VLA calibration pipelinehttps://science.nrao.edu/facilities/vla/data-processing/pipeline provided by the National Radio Astronomical Observatory (NRAO) to perform basic flagging and calibration.","The data cubes were imaged using the CLEAN task.","Natural weighting was applied.","The pixel size and image size are $0.2$ and $1000 \\times 1000$ pixels.","After the CLEAN, we smoothed the cube using the “imsmooth\" command, applying $1\\times 1$ and the position angle of 0 with the gaussian kernel.","The spatial resolution of 1 of the resultant images corresponds to $\\sim 0.015$ pc.","The $1 \\sigma $ values are approximately 0.6, 0.7, 0.7, and 0.6 mK for HC$_{3}$ N, HC$_{5}$ N, HC$_{7}$ N, and CH$_{3}$ CN, respectively.","We made the moment zero images of HC$_{3}$ N and HC$_{5}$ N using the “immoments\" task in CASA."],["Results","Figure REF shows the moment zero images of (a) HC$_{3}$ N and (b) HC$_{5}$ N in G28.28$-$ 0.36.","The velocity components in the range $V_{\\rm {LSR}} = 47.5 - 51.5$ km s$^{-1}$ were integrated in these moment zero images.","The spatial distribution of HC$_{3}$ N is more extended than that of HC$_{5}$ N, because of their excitation energies of the observed lines (Table ).","The observed HC$_{3}$ N line has lower excitation energy ($E_{\\rm {u}}/k = 4.4$ K) than that of HC$_{5}$ N ($E_{\\rm {u}}/k = 13.4$ K), and colder envelopes could be traced by HC$_{3}$ N. Figure: Moment zero images of (a) HC 3 _{3}N and (b) HC 5 _{5}N obtained with the VLA, including the data above 2σ2 \\sigma .","The contour levels are 0.2, 0.4, 0.6, and 0.8 of their peak levels, where the peak intensities are 27.9 and 13.5 (mK ·\\cdot km s -1 ^{-1}) for (a) HC 3 _{3}N and (b) HC 5 _{5}N, respectively.","The rms noise levels are 1.8 and 1.3 (mK ·\\cdot km s -1 ^{-1}) in the images of HC 3 _{3}N and HC 5 _{5}N, respectively.","The orange open circle and magenta cross indicate the 6.7 GHz methanol maser and UCH2 region , respectively.","The filled circles at the bottom left corner represent the angular resolution of these images (1).Regarding HC$_{7}$ N, the signal-to-noise ratio is low, which precludes determination of its spatial distribution.","The bottom panel of Figure REF shows HC$_{7}$ N spectra, as well as HC$_{3}$ N and HC$_{5}$ N spectra, observed toward its four peak positions A$-$ D indicated in the panel (b) of Figure REF .","In order to improve the signal-to-noise ratio, we applied the 1 uvtaper for HC$_{7}$ N data.","The intensities of these spectra are estimated within $1.5 $ regions, which corresponds to the spatial resolution of HC$_{7}$ N with the uvtaper.","HC$_{7}$ N is detected around the regions where HC$_{5}$ N is detected as shown in the panel (b) of Figure REF .","CH$_{3}$ CN emission is undetected at the rms noise level of 0.6 mK.","Figure: HC 3 _{3}N, HC 5 _{5}N, and HC 7 _{7}N spectra at four HC 7 _{7}N peak positions labeled as A--D, denoting in the panel (b) of Figure .","The gray vertical lines show the systemic velocity of the G28.28-0.36 MYSO (49 km s -1 ^{-1}).","The rms noise levels are 0.16, 0.20, and 0.12 K for HC 3 _{3}N, HC 5 _{5}N, and HC 7 _{7}N spectra, respectively."],["Comparisons of Cyanopolyyne Ratios","We derived the column densities of HC$_{3}$ N, HC$_{5}$ N, and HC$_{7}$ N at Position A (Figure REF ) assuming the local thermodynamic equilibrium (LTE).","We use the following formulae [12]: $ \\tau = - {\\mathrm {ln}} \\left[1- \\frac{T_{\\rm b} }{\\left\\lbrace J(T_{\\rm {ex}}) - J(T_{\\rm {bg}}) \\right\\rbrace } \\right],$ where $ J(T) = \\frac{h\\nu }{k}\\Bigl \\lbrace \\exp \\Bigl (\\frac{h\\nu }{kT}\\Bigr ) -1\\Bigr \\rbrace ^{-1},$ and $ N = \\tau \\frac{3h\\Delta v}{8\\pi ^3}\\sqrt{\\frac{\\pi }{4\\mathrm {ln}2}}Q\\frac{1}{\\mu ^2}\\frac{1}{J_{\\rm {lower}}+1}\\exp \\Bigl (\\frac{E_{\\rm {lower}}}{kT_{\\rm {ex}}}\\Bigr )\\Bigl \\lbrace 1-\\exp \\Bigl (-\\frac{h\\nu }{kT_{\\rm {ex}}}\\Bigr )\\Bigr \\rbrace ^{-1}.$ In Equation (REF ), $\\tau $ and $T_{\\rm b}$ denote the optical depth and brightness temperature, respectively.","$T_{\\rm {ex}}$ and $T_{\\rm {bg}}$ are the excitation temperature and the cosmic microwave background temperature ($\\simeq 2.73$ K), respectively.","$J$ ($T$ ) in Equation (REF ) is the effective temperature equivalent to that in the Rayleigh-Jeans law.","In Equation (REF ), N, $\\Delta v$ , $Q$ , $\\mu $ , and $E_{\\rm {lower}}$ denote the column density, line width (FWHM), partition function, permanent electric dipole moment, and energy of the lower rotational energy level, respectively.","The brightness temperatures and line widths are obtained by the gaussian fitting of spectra.","Figure REF in Appendix shows the fitting results for each spectra, and the obtained spectral line parameters and permanent electric dipole moments of each species are summarized in Table .","We derived the column densities assuming the excitation temperatures of 15, 20, 30, and 50 K, respectively.","The column densities of cyanopolyynes and the HC$_{3}$ N : HC$_{5}$ N : HC$_{7}$ N ratios at each excitation temperature are summarized in Table REF .","The uncertainties in the excitation temperatures do not significantly affect the derived column densities of HC$_{5}$ N and HC$_{7}$ N, and their derived column densities agree with each other within their the standard deviation errors.","On the other hand, the uncertainties in the excitation temperatures bring larger differences in the HC$_{3}$ N column density.","ccccc Column densities of cyanopolyynes at Position A 0pt $T_{\\rm {ex}}$ $N$ (HC$_{3}$ N) $N$ (HC$_{5}$ N) $N$ (HC$_{7}$ N) HC$_{3}$ N : HC$_{5}$ N : HC$_{7}$ N (K) ($\\times 10^{14}$ cm$^{-2}$ ) ($\\times 10^{13}$ cm$^{-2}$ ) ($\\times 10^{13}$ cm$^{-2}$ ) 15 $1.36 \\pm 0.14$ $4.4 \\pm 0.9$ $4.0 \\pm 1.2$ 1.00 : 0.32 : 0.29 20 $1.47 \\pm 0.15$ $4.3 \\pm 0.9$ $3.1 \\pm 0.9$ 1.00 : 0.29 : 0.21 30 $1.84 \\pm 0.19$ $4.9 \\pm 1.1$ $2.6 \\pm 0.8$ 1.00 : 0.26 : 0.14 50 $2.7 \\pm 0.3$ $6.5 \\pm 1.4$ $2.8 \\pm 0.8$ 1.00 : 0.24 : 0.11 The errors represent the standard deviation.","The errors of column densities are derived from uncertainties of the gaussian fitting (see Table in Appendix).","The HC$_{3}$ N : HC$_{5}$ N : HC$_{7}$ N ratios at Position A are derived to be 1.0 : $\\sim 0.3$ : $\\sim 0.2$ .","The ratios in L1527, which is one of the WCCC sources, are 1.0 : $0.3-0.6$ : $\\sim 0.1$ [25], [26].","The tendency of the ratios at Position A seems to be similar to that in L1527."],["Comparison of Spatial Distributions between Cyanopolyynes and 450 $\\mu $ m Dust Continuum","Figure REF shows 450 $\\mu $ m dust continuum images overlaid by the black contours of moment zero images of (a) HC$_{3}$ N and (b) HC$_{5}$ N, respectively.","The 450 $\\mu $ m data, which are available from the James Clerk Maxwell Telescope (JCMT) Science Archivehttp://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/en/jcmt/index.html, were obtained with the SCUBA installed on the JCMT.","The main beam size of the SCUBA is $7.9$ at 450 $\\mu $ m, corresponding to $\\sim 0.11$ pc.","Three strong 450 $\\mu $ m continuum emission peaks can be recognized; the UCH2 position, the north-west position from the UCH2, and the east position from the 6.7 GHz methanol maser.","The spatial distribution of HC$_{5}$ N is consistent with the 450 $\\mu $ m continuum peak of the east position from the 6.7 GHz methanol maser (hereafter Cyanopolyyne-rich clump, panel (b) of Figure REF ).","The spatial distribution of HC$_{3}$ N seems to surround the Cyanopolyyne-rich clump (panel (a) of Figure REF ), not only at the Cyanopolyyne-rich clump.","HC$_{3}$ N emission is also located at the west position of the 6.7 GHz methanol maser position, or the edge of the 450 $\\mu $ m continuum.","Small HC$_{5}$ N emission region is seen at the same edge of the 450 $\\mu $ m continuum.","We will briefly discuss a possible origin of this emission region in Section REF .","Figure: 450 μ\\mu m continuum image obtained with the SCUBA installed on the JCMT overlaid by black contours of (a) HC 3 _{3}N and (b) HC 5 _{5}N moment zero images, the same ones as in Figure .","The contour levels are 0.2, 0.4, 0.6, and 0.8 of their peak intensities 27.9 and 13.5 (mK ·\\cdot km s -1 ^{-1}) for (a) HC 3 _{3}N and (b) HC 5 _{5}N, respectively.","The blue filled circle and blue cross indicate the 6.7 GHz methanol maser and UCH2 region, respectively.","The A–D positions are indicated as the green filled circles.","The filed white circles at the bottom left corner represent the angular resolution of the images of HC 3 _{3}N and HC 5 _{5}N (1)."],["Possible Nature of the 450 $\\mu $ m Continuum Peak Position associated with Cyanopolyynes","We examine four possible types of objects of the Cyanopolyyne-rich clump (Section REF ); a hot core, a starless clump, low- or intermediate-mass protostellar core(s), and a photo-dominated region (PDR) driven by the associated UCH2 region.","Figure REF shows the Spitzer 3-color (3.6 $\\mu $ m, 4.5 $\\mu $ m, 8.0 $\\mu $ m) imageshttp://atlasgal.mpifr-bonn.mpg.de/cgi-bin/ATLASGAL$_{-}$ DATABASE.cgi, overlaid by yellow contours showing (b) 450 $\\mu $ m continuum, (c) HC$_{3}$ N moment zero image, and (d) HC$_{5}$ N moment zero image, respectively.","In panel (b), the green contours show the 8.3 mm continuum emission obtained simultaneously with cyanopolyynes by the VLA.","The 8.3 mm continuum peak is compact and well consistent with the UCH2 region.","At the Cyanopolyyne-rich clump, no point source can be recognized from the Spitzer image, which suggests that no massive young protostar is currently present at the clump position.","Therefore, the possibility of hot core is unrealistic.","We derived the average column density of H$_{2}$ , $N$ (H$_{2}$ ), of the Cyanopolyyne-rich clump from the 450 $\\mu $ m continuum data using the following formula [30]: $ N({\\rm {H}}_2) = 2.02 \\times 10^{20} {\\rm {cm}}^{-2} \\Bigl (e^{1.439(\\lambda /{\\rm {mm}})^{-1}(T/10 {\\rm {K}})^{-1}}-1\\Bigr )\\Bigl (\\frac{\\kappa _{\\nu }}{0.01\\: {\\rm {cm}}^{2}\\: {\\rm {g}}^{-1}}\\Bigr )^{-1}\\Bigl (\\frac{S_{\\nu }^{\\rm {beam}}}{{\\rm {mJy}}\\: {\\rm {beam}}^{-1}}\\Bigr )\\Bigl (\\frac{\\theta _{\\rm {HPBW}}}{10\\: {\\rm {arcsec}}}\\Bigr )^{-2}\\Bigl (\\frac{\\lambda }{{\\rm {mm}}}\\Bigr )^{3}.$ We estimated the continuum flux ($S_{\\nu }^{\\rm {beam}}$ ) toward the Cyanopolyyne-rich clump within a size of 14, and the flux intensity is $9.7 \\times 10^{3}$ mJy beam$^{-1}$ .","We assumed that $\\kappa _{\\nu }$ is 0.0619 cm$^{2}$ g$^{-1}$ at $\\lambda = 450$ $\\mu $ m. $T$ is the dust temperature, and we assumed it to be $30 \\pm 15$ K. The derived $N$ (H$_{2}$ ) value is ($2.8^{+8.1}_{-1.3}$ )$\\times 10^{22}$ cm$^{-2}$ , which is similar to the typical value of massive clumps forming young star clusters [29].","The value is an average for the clump and then it can be considered as the lower limits for cores.","The derived $N$ (H$_{2}$ ) value for the Cyanopolyyne-rich clump is higher than the threshold value for star formation cores, $N$ (H$_{2}$ ) $\\approx 9 \\times 10^{21}$ cm$^{-2}$ [32].","Hence, the Cyanopolyyne-rich clump is considered to contain deeply embedded low- or intermediate-mass protostellar core(s), and is not a starless clump.","In case that a low- or intermediate-mass protostar has been born, the WCCC mechanism is likely to work; cyanopolyynes are formed from CH$_{4}$ evaporated from grain mantles in the lukewarm gas ($T = 20-30$ K).","This may be supported by similar HC$_{3}$ N : HC$_{5}$ N : HC$_{7}$ N ratios between this clump and L1527 (Section REF ).","In order to investigate the possibility of the PDR, we compare the fractional abundances of cyanopolyynes at the Cyanopolyyne-rich clump with those derived from a PDR chemical network simulation [19].","We derived the fractional abundances, $X$ ($a$ )$= N$ ($a$ )$/N$ (H$_{2}$ ), of cyanopolyynes at Position A as summarized in Table REF .","Recent chemical network simulation about the Horsehead nebula, which is one of the most studied PDRs, estimated the HC$_{3}$ N and HC$_{5}$ N abundances [19].","The PDR and Core positions are positions with a visual extinctions ($A_{v}$ ) of $\\sim 2$ mag and $\\sim 10-20$ mag, respectively.","We summarize the fractional abundances of these two positions in Table REF .","The $X$ (HC$_{3}$ N) and $X$ (HC$_{5}$ N) values at the Cyanopolyyne-rich clump are significantly higher than those in PDR position by three order of magnitudes and more than four order of magnitudes, respectively.","Moreover, the HC$_{3}$ N and HC$_{5}$ N fractional abundances at the Cyanopolyyne-rich clump are still higher than those in Core position in the PDR model by a factor of $\\sim 55$ and $\\sim 80$ , respectively.","Therefore, cyanopolyynes at the Cyanopolyyne-rich clump in the G28.28$-$ 0.36 high-mass star-forming region are not formed by the PDR chemistry.","If we take large uncertainties in deriving $N$ (H$_{2}$ ) values into consideration, the differences between the Cyanopolyyne-rich clump and the PDR models are plausible.","However, we cannot completely exclude any effects from the UCHII region such as UV radiation.","lccccc Comparison of fractional abundances of cyanopolyynes 0pt Source Method $X$ (HC$_{3}$ N) $X$ (HC$_{5}$ N) $X$ (HC$_{7}$ N) Referencea ($\\times 10^{-9}$ ) ($\\times 10^{-9}$ ) ($\\times 10^{-9}$ ) Cyanopolyyne-rich clump Observation $6.6^{+5.5}_{-4.9}$ $1.7^{+1.5}_{-1.3}$ $0.9^{+0.8}_{-0.7}$ 1 Horsehead nebula (PDR position) Simulation $3.1 \\times 10^{-3}$ $4.3 \\times 10^{-5}$ ... 2 Horsehead nebula (Core position) Simulation 0.12 0.022 ... 2 L1527 (WCCC) Observation 0.43 – 0.96 0.05 – 0.24 $\\sim 0.05$ 3,4,5 The errors represent the standard deviation.","a(1) This work; (2) [19]; (3) [17]; (4) [25]; (5) [26] We also compare the fractional abundances of HC$_{3}$ N, HC$_{5}$ N, and HC$_{7}$ N at the Cyanopolyyne-rich clump to those in L1527.","These fractional abundances at the Cyanopolyyne-rich clump are higher than those in L1527 by a factor of $7-15$ , $7-37$ , and $\\sim 17$ , respectively.","Hence, the cyanopolyynes at the Cyanopolyyne-rich clump seem to be more abundant compared to L1527, even if we take uncertainties in fractional abundances into account.","It cannot be excluded that several low-mass protostars are concentrated within a small region (e.g., $\\sim 0.02$ pc corresponding to $1.5$ at the distance of 3 kpc).","As another interpretation, these results may imply that there is an intermediate-mass protostar because an intermediate-mass protostar heats its wider surroundings compared to a low-mass protostar; the size of lukewarm envelopes will be larger and the column densities of carbon-chain species in such lukewarm envelopes will increase.","This could explain the results that the cyanopolyynes at the Cyanopolyyne-rich clump are more abundant than those in L1527.","In Figure REF (d), compact HC$_{5}$ N emission region is located at the west position of the 6.7 GHz methanol maser position.","This position corresponds to the edge of the 4.5 $\\mu $ m emission, which seems to trace shock regions [8].","Figure REF shows spectra of HC$_{3}$ N and HC$_{5}$ N at the shock region.","The line widths (FWHM) derived from the Gaussian fit are $2.7 \\pm 0.2$ and $2.6 \\pm 0.2$ km s$^{-1}$ for HC$_{3}$ N and HC$_{5}$ N, respectively.","These line widths at the shock region are wider those at the Cyanopolyyne-rich clump, $2.05 \\pm 0.16$ and $1.4 \\pm 0.2$ km s$^{-1}$ for HC$_{3}$ N and HC$_{5}$ N, respectively.","These results probably support that the cyanopolyynes are the shock origin.","Molecules evaporated from grain mantles such as C$_{2}$ H$_{2}$ may be parent species of the cyanopolyynes.","It is still unclear whether HC$_{5}$ N is formed and can survive in shock regions, but this may be the first observational result showing that HC$_{5}$ N can be enhanced in shock regions.","Figure: Spectra of HC 3 _{3}N and HC 5 _{5}N at the shock region.","The red lines show the Gaussian fitting results."],["Comparison with Previous Single-Dish Telescope Observations","[33] reported the detection of high-excitation-energy ($E_{\\rm {u}}/k \\simeq 100$ K) lines of HC$_{5}$ N with the Nobeyama 45-m radio telescope (HPBW = 18) toward the methanol maser position, which is considered to be a MYSO position [37].","The significant HC$_{5}$ N peak is not seen at the methanol maser position in the VLA map.","This is probably caused by the different excitation energies of the observed lines.","The lines observed with the VLA have low excitation energies (Table ), and preferably trace lower-temperature regions such as low- or intermediate-mass protostellar cores or envelopes.","On the other hand, HC$_{5}$ N emission observed with the Nobeyama 45-m telescope seems to come from higher-temperature regions closer to the methanol maser or the MYSO.","In these higher-temperature regions, HC$_{5}$ N should be highly excited and low excitation energy lines observed with the VLA should be weak.","In contrast, the hotter components detected with the Nobeyama 45-m telescope were not detected with the VLA, because the low-excitation-energy lines are not suitable tracers of the hot components.","Combining with the VLA and Nobeyama 45-m telescope results, both the MYSO associated with the methanol maser and low- or intermediate-mass protostellar core(s) at the Cyanopolyyne-rich clump appear to be rich in cyanopolyynes.","It is still unclear why the G28.28$-$ 0.36 high-mass star-forming region is a cyanopolyyne-rich/COMs-poor source [34].","Studies about chemical diversity among high-mass star-forming regions will become a key to our understanding of massive star formation processes.","We need the high-spatial-resolution and higher-frequency band observations in order to investigate the spatial resolution of higher temperature components of cyanopolyynes to confirm that cyanopolyynes exist at the hot core position."],["Conclusions","We have carried out interferometric observations of cyanopolyynes (HC$_{3}$ N, HC$_{5}$ N, and HC$_{7}$ N) toward the G28.28$-$ 0.36 high-mass star-forming region with the VLA Ka-band.","We obtained the moment zero images of HC$_{3}$ N and HC$_{5}$ N and tentatively detected HC$_{7}$ N. The spatial distributions of HC$_{3}$ N and HC$_{5}$ N are consistent with the 450 $\\mu $ m dust continuum clump, i.e., the Cyanopolyyne-rich clump.","The HC$_{3}$ N : HC$_{5}$ N : HC$_{7}$ N column density ratios are estimated at 1.0 : $\\sim 0.3$ : $\\sim 0.2$ at Position A.","The Cyanopolyyne-rich clump seems to contain deeply embedded low- or intermediate-mass protostellar core(s).","The most probable formation mechanism of the cyanopolyynes at the Cyanopolyyne-rich clump is the WCCC mechanism.","We possibly found the HC$_{3}$ N and HC$_{5}$ N emission in the shock region.","We express our sincere thanks and appreciate for the staff of the National Radio Astronomy Observatory.","The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. KT appreciates support from a Granting-Aid for Science Research of Japan (17J03516).","Karl G. Jansky Very Large Array (VLA) Common Astronomy Software Applications (CASA)"],["Gaussian Fitting Results and Spectral Line Parameters","We show the gaussian fitting results for the spectra of HC$_{3}$ N, HC$_{5}$ N, and HC$_{7}$ N at Position A in Figure REF .","The obtained spectral line parameters are summarized in Table .","Figure: Spectra of HC 3 _{3}N, HC 5 _{5}N, and HC 7 _{7}N at Position A obtained with the VLA.","The best gaussian fits are shown overlaid in red.ccccc Spectral line parameters at Position A 0pt Species $\\mu $ $T_{\\rm {b}}$ FWHM $\\int T_{\\mathrm {b}}dv$ (Debye) (K) (km s$^{-1}$ ) (K km s$^{-1}$ ) HC$_{3}$ N 3.73 $5.4 \\pm 0.4$ $2.05 \\pm 0.16$ $11.7 \\pm 1.2$ HC$_{5}$ N 4.33 $2.3 \\pm 0.3$ $1.4 \\pm 0.2$ $3.5 \\pm 0.8$ HC$_{7}$ N 4.82 $1.0 \\pm 0.2$ $1.3 \\pm 0.3$ $1.3 \\pm 0.4$ The errors represent the standard deviation.","Values of their permanent electric dipole moments are taken from the Cologne Database for Molecular Spectroscopy [21]."]],"string":"[\n [\n \"Interferometric Observations of Cyanopolyynes toward the G28.28-0.36\\n High-Mass Star-Forming Region\"\n ],\n [\n \"Abstract We have carried out interferometric observations of cyanopolyynes, HC$_{3}$N, HC$_{5}$N, and HC$_{7}$N, in the 36 GHz band toward the G28.28$-$0.36 high-mass star-forming region using the Karl G. Jansky Very Large Array (VLA) Ka-band receiver.\",\n \"The spatial distributions of HC$_{3}$N and HC$_{5}$N are obtained.\",\n \"HC$_{5}$N emission is coincident with a 450 $\\\\mu$m dust continuum emission and this clump with a diameter of $\\\\sim 0.2$ pc is located at the east position from the 6.7 GHz methanol maser by $\\\\sim 0.15$ pc.\",\n \"HC$_{7}$N is tentatively detected toward the clump.\",\n \"The HC$_{3}$N : HC$_{5}$N : HC$_{7}$N column density ratios are estimated at 1.0 : $\\\\sim 0.3$ : $\\\\sim 0.2$ at an HC$_{7}$N peak position.\",\n \"We discuss possible natures of the 450 $\\\\mu$m continuum clump associated with the cyanopolyynes.\",\n \"The 450 $\\\\mu$m continuum clump seems to contain deeply embedded low- or intermediate-mass protostellar cores, and the most possible formation mechanism of the cyanopolyynes is the warm carbon chain chemistry (WCCC) mechanism.\",\n \"In addition, HC$_{3}$N and compact HC$_{5}$N emission is detected at the edge of the 4.5 $\\\\mu$m emission, which possibly implies that such emission is the shock origin.\"\n ],\n [\n \"Introduction\",\n \"Cyanopolyynes (HC$_{2n+1}$ N, $n=1,2,3,...$ ) are one of the representative carbon-chain species.\",\n \"In low-mass star-forming regions, carbon-chain molecules are known as early-type species; they are abundant in young starless cores and deficient in star-forming cores [31], [15].\",\n \"In contrast to the general picture, cyanoacetylene (HC$_{3}$ N), the shortest member of cyanopolyynes, is detected from various regions such as infrared dark clouds [28], molecular outflows [1], protoplanetary disks [23], [4], and comets [22] and it is interesting to trace cyanopolyyne chemistry for better understanding of the molecular evolution during star/planet formation process.\",\n \"Cyanopolyynes attract astrobiological as well as astrochemical interests.\",\n \"Since they contain the nitrile bond (–C$\\\\equiv $ N), cyanopolyynes have been suggested as possible intermediates in the synthesis of simple amino acids [11], [5].\",\n \"Saturated complex organic molecules (COMs), consisting of more than six atoms with rich hydrogen atoms, are abundant around protostars.\",\n \"Such chemistry is known as hot core in high-mass star-forming regions and hot corino in low-mass star-forming regions.\",\n \"In addition to hot corino, around a few low-mass protostars, carbon-chain molecules are formed from CH$_{4}$ evaporated from dust grains, which is known as warm carbon chain chemistry [27].\",\n \"Progress in observational studies of carbon-chain molecules in high-mass star-forming regions has been slower, compared to low-mass star-forming regions.\",\n \"Regarding hot cores, HC$_{5}$ N has been detected in chemically rich sources, Orion KL [9] and Sgr B2 [2], while only a tentative detection of HC$_{7}$ N in Orion KL was reported [10].\",\n \"[6] performed a chemical network simulation and suggested that cyanopolyynes could be formed in a hot core from C$_{2}$ H$_{2}$ evaporated from grain mantles.\",\n \"Motivated by the chemical network simulation, [13] carried out survey observations of HC$_{5}$ N toward 79 hot cores associated with the 6.7 GHz methanol masers and reported its detection in 35 sources.\",\n \"However, the association with the maser is questionable, because they used a large beam (0.95) and a low-excitation energy line ($J=12-11$ ; $E_{\\\\rm u}/k = 10.0$ K), which can be excited even in dark clouds.\",\n \"[33] carried out observations of long cyanopolyynes (HC$_{5}$ N and HC$_{7}$ N) toward four massive young stellar objects, where [13] had reported the HC$_{5}$ N detection, using the Green Bank 100-m and the Nobeyama 45-m radio telescopes, and detected high-excitation energy lines ($E_{\\\\rm u}/k \\\\approx 100$ K) of HC$_{5}$ N. The detection of such lines means that HC$_{5}$ N exists at least in the warm gas, not in cold molecular clouds ($T_{\\\\rm {kin}} \\\\simeq 10$ K).\",\n \"[34] found that the G28.28$-$ 0.36 high-mass star-forming region is a particular cyanopolyyne-rich source with less COMs compared with other sources.\",\n \"Hence, G28.28$-$ 0.36 is considered to be a good target region to study the cyanopolyyne chemistry around massive young stellar objects (MYSOs).\",\n \"Using the Nobeyama 45-m radio telescope, [35] investigated the main formation mechanism of HC$_{3}$ N in G28.28$-$ 0.36 from its $^{13}$ C isotopic fractionation.\",\n \"The reaction of “C$_{2}$ H$_{2}$ + CN\\\" was proposed as the main formation pathway of HC$_{3}$ N, which is consistent with the chemical network simulation conducted by [6] and the WCCC model [14].\",\n \"Figure: Spitzer's IRAC 3.6 μ\\\\mu m image toward G28.28--0.36.\",\n \"The open circle and cross indicate the 6.7 GHz methanol maser and ultracompact H2 (UCH2) region , respectively.In this paper, we carried out interferometric observations of cyanopolyynes (HC$_{3}$ N, HC$_{5}$ N, and HC$_{7}$ N) toward the G28.28$-$ 0.36 high-mass star-forming region ($d = 3$ kpc) with the Karl G. Jansky Very Large Array (VLA).\",\n \"Figure REF shows the Spitzer IRAC 3.6 $\\\\mu $ m imagehttp://sha.ipac.caltech.edu/applications/Spitzer/SHA/ toward the region.\",\n \"G28.28$-$ 0.36 is classified as an Extended Green Object (EGO) source [8] from the Spitzer Galactic Legacy Infrared Mid-Plane Survey Extraordinaire [3].\",\n \"In Figure REF , the open circle and cross indicate the 6.7 GHz methanol maser [7] and ultracompact H2 (UCH2) region [36], respectively.\",\n \"The 6.7 GHz maser is considered to give us the exact position of MYSOs [37].\",\n \"A UCH2 region seems to heat the environment.\",\n \"As shown in Figure REF , the ring structure around the UCH2 region is suggestive of expanding motion and on-going massive star formation.\",\n \"We describe the observational details and data analyses in Section .\",\n \"The resultant images and spectra of cyanopolyynes are presented in Section .\",\n \"We compare the spatial distributions of cyanopolyynes with the infrared images and discuss possible formation mechanisms in Section .\"\n ],\n [\n \"Observations\",\n \"The observations of G28.28$-$ 0.36 using the VLA Ka-band receiver were carried out in the C configuration with the 27 $\\\\times $ 25-m antennas on March 20th, 2016 (Proposal ID = 16A-084, PI; Kotomi Taniguchi).\",\n \"The field of view (FoV) is $\\\\sim $ 604.\",\n \"Four spectral windows of the correlator were set at our target lines summarized in Table .\",\n \"All of these target lines were simultaneously observed.\",\n \"The channel separation of the correlator is 0.5 km s$^{-1}$ .\",\n \"The angular resolutions and Position Angles (PA) for each line are summarized in Table .\",\n \"The phase reference center was set at ($\\\\alpha _{2000}$ , $\\\\delta _{2000}$ ) = (18$^{\\\\rm h}$ 44$^{\\\\rm m}$ 133, -0418030), the 6.7 GHz methanol maser position.\",\n \"The pointing source is J1832$-$ 1035 at ($\\\\alpha _{2000}$ , $\\\\delta _{2000}$ ) = (18$^{\\\\rm h}$ 32$^{\\\\rm m}$ 20836, $-$ 1035112).\",\n \"The absolute flux density calibration and the bandpass calibration were conducted by observing 3C286 at ($\\\\alpha _{2000}$ , $\\\\delta _{2000}$ ) = (13$^{\\\\rm h}$ 31$^{\\\\rm m}$ 0828798, +3030329589).\",\n \"The gain/phase calibration was conducted by observing J1851+0035 at ($\\\\alpha _{2000}$ , $\\\\delta _{2000}$ ) = (18$^{\\\\rm h}$ 51$^{\\\\rm m}$ 467217, +003532414).\",\n \"cccccc Summary of target lines 0pt Species Transition Rest Frequency $E_{\\\\rm {u}}/k$ Angular PA (GHz) (K) Resolution (deg) HC$_{3}$ N $J=4-3$ 36.39232 4.4 084 $\\\\times $ 063 -9.92 HC$_{5}$ N $J=14-13$ 37.276994 13.4 081 $\\\\times $ 063 -11.04 HC$_{7}$ N $J=33-32$ 37.22349 30.4 082 $\\\\times $ 063 -10.20 CH$_{3}$ CN $J_{\\\\rm K} =2_{0} - 1_{0}$ 36.7954747 2.6 083 $\\\\times $ 064 -11.22 Rest frequencies are taken from the Cologne Database for Molecular Spectroscopy [21] and the Jet Propulsion Laboratory catalog [24].\",\n \"We conducted data reduction using the Common Astronomy Software Application [20].\",\n \"We used the VLA calibration pipelinehttps://science.nrao.edu/facilities/vla/data-processing/pipeline provided by the National Radio Astronomical Observatory (NRAO) to perform basic flagging and calibration.\",\n \"The data cubes were imaged using the CLEAN task.\",\n \"Natural weighting was applied.\",\n \"The pixel size and image size are $0.2$ and $1000 \\\\times 1000$ pixels.\",\n \"After the CLEAN, we smoothed the cube using the “imsmooth\\\" command, applying $1\\\\times 1$ and the position angle of 0 with the gaussian kernel.\",\n \"The spatial resolution of 1 of the resultant images corresponds to $\\\\sim 0.015$ pc.\",\n \"The $1 \\\\sigma $ values are approximately 0.6, 0.7, 0.7, and 0.6 mK for HC$_{3}$ N, HC$_{5}$ N, HC$_{7}$ N, and CH$_{3}$ CN, respectively.\",\n \"We made the moment zero images of HC$_{3}$ N and HC$_{5}$ N using the “immoments\\\" task in CASA.\"\n ],\n [\n \"Results\",\n \"Figure REF shows the moment zero images of (a) HC$_{3}$ N and (b) HC$_{5}$ N in G28.28$-$ 0.36.\",\n \"The velocity components in the range $V_{\\\\rm {LSR}} = 47.5 - 51.5$ km s$^{-1}$ were integrated in these moment zero images.\",\n \"The spatial distribution of HC$_{3}$ N is more extended than that of HC$_{5}$ N, because of their excitation energies of the observed lines (Table ).\",\n \"The observed HC$_{3}$ N line has lower excitation energy ($E_{\\\\rm {u}}/k = 4.4$ K) than that of HC$_{5}$ N ($E_{\\\\rm {u}}/k = 13.4$ K), and colder envelopes could be traced by HC$_{3}$ N. Figure: Moment zero images of (a) HC 3 _{3}N and (b) HC 5 _{5}N obtained with the VLA, including the data above 2σ2 \\\\sigma .\",\n \"The contour levels are 0.2, 0.4, 0.6, and 0.8 of their peak levels, where the peak intensities are 27.9 and 13.5 (mK ·\\\\cdot km s -1 ^{-1}) for (a) HC 3 _{3}N and (b) HC 5 _{5}N, respectively.\",\n \"The rms noise levels are 1.8 and 1.3 (mK ·\\\\cdot km s -1 ^{-1}) in the images of HC 3 _{3}N and HC 5 _{5}N, respectively.\",\n \"The orange open circle and magenta cross indicate the 6.7 GHz methanol maser and UCH2 region , respectively.\",\n \"The filled circles at the bottom left corner represent the angular resolution of these images (1).Regarding HC$_{7}$ N, the signal-to-noise ratio is low, which precludes determination of its spatial distribution.\",\n \"The bottom panel of Figure REF shows HC$_{7}$ N spectra, as well as HC$_{3}$ N and HC$_{5}$ N spectra, observed toward its four peak positions A$-$ D indicated in the panel (b) of Figure REF .\",\n \"In order to improve the signal-to-noise ratio, we applied the 1 uvtaper for HC$_{7}$ N data.\",\n \"The intensities of these spectra are estimated within $1.5 $ regions, which corresponds to the spatial resolution of HC$_{7}$ N with the uvtaper.\",\n \"HC$_{7}$ N is detected around the regions where HC$_{5}$ N is detected as shown in the panel (b) of Figure REF .\",\n \"CH$_{3}$ CN emission is undetected at the rms noise level of 0.6 mK.\",\n \"Figure: HC 3 _{3}N, HC 5 _{5}N, and HC 7 _{7}N spectra at four HC 7 _{7}N peak positions labeled as A--D, denoting in the panel (b) of Figure .\",\n \"The gray vertical lines show the systemic velocity of the G28.28-0.36 MYSO (49 km s -1 ^{-1}).\",\n \"The rms noise levels are 0.16, 0.20, and 0.12 K for HC 3 _{3}N, HC 5 _{5}N, and HC 7 _{7}N spectra, respectively.\"\n ],\n [\n \"Comparisons of Cyanopolyyne Ratios\",\n \"We derived the column densities of HC$_{3}$ N, HC$_{5}$ N, and HC$_{7}$ N at Position A (Figure REF ) assuming the local thermodynamic equilibrium (LTE).\",\n \"We use the following formulae [12]: $ \\\\tau = - {\\\\mathrm {ln}} \\\\left[1- \\\\frac{T_{\\\\rm b} }{\\\\left\\\\lbrace J(T_{\\\\rm {ex}}) - J(T_{\\\\rm {bg}}) \\\\right\\\\rbrace } \\\\right],$ where $ J(T) = \\\\frac{h\\\\nu }{k}\\\\Bigl \\\\lbrace \\\\exp \\\\Bigl (\\\\frac{h\\\\nu }{kT}\\\\Bigr ) -1\\\\Bigr \\\\rbrace ^{-1},$ and $ N = \\\\tau \\\\frac{3h\\\\Delta v}{8\\\\pi ^3}\\\\sqrt{\\\\frac{\\\\pi }{4\\\\mathrm {ln}2}}Q\\\\frac{1}{\\\\mu ^2}\\\\frac{1}{J_{\\\\rm {lower}}+1}\\\\exp \\\\Bigl (\\\\frac{E_{\\\\rm {lower}}}{kT_{\\\\rm {ex}}}\\\\Bigr )\\\\Bigl \\\\lbrace 1-\\\\exp \\\\Bigl (-\\\\frac{h\\\\nu }{kT_{\\\\rm {ex}}}\\\\Bigr )\\\\Bigr \\\\rbrace ^{-1}.$ In Equation (REF ), $\\\\tau $ and $T_{\\\\rm b}$ denote the optical depth and brightness temperature, respectively.\",\n \"$T_{\\\\rm {ex}}$ and $T_{\\\\rm {bg}}$ are the excitation temperature and the cosmic microwave background temperature ($\\\\simeq 2.73$ K), respectively.\",\n \"$J$ ($T$ ) in Equation (REF ) is the effective temperature equivalent to that in the Rayleigh-Jeans law.\",\n \"In Equation (REF ), N, $\\\\Delta v$ , $Q$ , $\\\\mu $ , and $E_{\\\\rm {lower}}$ denote the column density, line width (FWHM), partition function, permanent electric dipole moment, and energy of the lower rotational energy level, respectively.\",\n \"The brightness temperatures and line widths are obtained by the gaussian fitting of spectra.\",\n \"Figure REF in Appendix shows the fitting results for each spectra, and the obtained spectral line parameters and permanent electric dipole moments of each species are summarized in Table .\",\n \"We derived the column densities assuming the excitation temperatures of 15, 20, 30, and 50 K, respectively.\",\n \"The column densities of cyanopolyynes and the HC$_{3}$ N : HC$_{5}$ N : HC$_{7}$ N ratios at each excitation temperature are summarized in Table REF .\",\n \"The uncertainties in the excitation temperatures do not significantly affect the derived column densities of HC$_{5}$ N and HC$_{7}$ N, and their derived column densities agree with each other within their the standard deviation errors.\",\n \"On the other hand, the uncertainties in the excitation temperatures bring larger differences in the HC$_{3}$ N column density.\",\n \"ccccc Column densities of cyanopolyynes at Position A 0pt $T_{\\\\rm {ex}}$ $N$ (HC$_{3}$ N) $N$ (HC$_{5}$ N) $N$ (HC$_{7}$ N) HC$_{3}$ N : HC$_{5}$ N : HC$_{7}$ N (K) ($\\\\times 10^{14}$ cm$^{-2}$ ) ($\\\\times 10^{13}$ cm$^{-2}$ ) ($\\\\times 10^{13}$ cm$^{-2}$ ) 15 $1.36 \\\\pm 0.14$ $4.4 \\\\pm 0.9$ $4.0 \\\\pm 1.2$ 1.00 : 0.32 : 0.29 20 $1.47 \\\\pm 0.15$ $4.3 \\\\pm 0.9$ $3.1 \\\\pm 0.9$ 1.00 : 0.29 : 0.21 30 $1.84 \\\\pm 0.19$ $4.9 \\\\pm 1.1$ $2.6 \\\\pm 0.8$ 1.00 : 0.26 : 0.14 50 $2.7 \\\\pm 0.3$ $6.5 \\\\pm 1.4$ $2.8 \\\\pm 0.8$ 1.00 : 0.24 : 0.11 The errors represent the standard deviation.\",\n \"The errors of column densities are derived from uncertainties of the gaussian fitting (see Table in Appendix).\",\n \"The HC$_{3}$ N : HC$_{5}$ N : HC$_{7}$ N ratios at Position A are derived to be 1.0 : $\\\\sim 0.3$ : $\\\\sim 0.2$ .\",\n \"The ratios in L1527, which is one of the WCCC sources, are 1.0 : $0.3-0.6$ : $\\\\sim 0.1$ [25], [26].\",\n \"The tendency of the ratios at Position A seems to be similar to that in L1527.\"\n ],\n [\n \"Comparison of Spatial Distributions between Cyanopolyynes and 450 $\\\\mu $ m Dust Continuum\",\n \"Figure REF shows 450 $\\\\mu $ m dust continuum images overlaid by the black contours of moment zero images of (a) HC$_{3}$ N and (b) HC$_{5}$ N, respectively.\",\n \"The 450 $\\\\mu $ m data, which are available from the James Clerk Maxwell Telescope (JCMT) Science Archivehttp://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/en/jcmt/index.html, were obtained with the SCUBA installed on the JCMT.\",\n \"The main beam size of the SCUBA is $7.9$ at 450 $\\\\mu $ m, corresponding to $\\\\sim 0.11$ pc.\",\n \"Three strong 450 $\\\\mu $ m continuum emission peaks can be recognized; the UCH2 position, the north-west position from the UCH2, and the east position from the 6.7 GHz methanol maser.\",\n \"The spatial distribution of HC$_{5}$ N is consistent with the 450 $\\\\mu $ m continuum peak of the east position from the 6.7 GHz methanol maser (hereafter Cyanopolyyne-rich clump, panel (b) of Figure REF ).\",\n \"The spatial distribution of HC$_{3}$ N seems to surround the Cyanopolyyne-rich clump (panel (a) of Figure REF ), not only at the Cyanopolyyne-rich clump.\",\n \"HC$_{3}$ N emission is also located at the west position of the 6.7 GHz methanol maser position, or the edge of the 450 $\\\\mu $ m continuum.\",\n \"Small HC$_{5}$ N emission region is seen at the same edge of the 450 $\\\\mu $ m continuum.\",\n \"We will briefly discuss a possible origin of this emission region in Section REF .\",\n \"Figure: 450 μ\\\\mu m continuum image obtained with the SCUBA installed on the JCMT overlaid by black contours of (a) HC 3 _{3}N and (b) HC 5 _{5}N moment zero images, the same ones as in Figure .\",\n \"The contour levels are 0.2, 0.4, 0.6, and 0.8 of their peak intensities 27.9 and 13.5 (mK ·\\\\cdot km s -1 ^{-1}) for (a) HC 3 _{3}N and (b) HC 5 _{5}N, respectively.\",\n \"The blue filled circle and blue cross indicate the 6.7 GHz methanol maser and UCH2 region, respectively.\",\n \"The A–D positions are indicated as the green filled circles.\",\n \"The filed white circles at the bottom left corner represent the angular resolution of the images of HC 3 _{3}N and HC 5 _{5}N (1).\"\n ],\n [\n \"Possible Nature of the 450 $\\\\mu $ m Continuum Peak Position associated with Cyanopolyynes\",\n \"We examine four possible types of objects of the Cyanopolyyne-rich clump (Section REF ); a hot core, a starless clump, low- or intermediate-mass protostellar core(s), and a photo-dominated region (PDR) driven by the associated UCH2 region.\",\n \"Figure REF shows the Spitzer 3-color (3.6 $\\\\mu $ m, 4.5 $\\\\mu $ m, 8.0 $\\\\mu $ m) imageshttp://atlasgal.mpifr-bonn.mpg.de/cgi-bin/ATLASGAL$_{-}$ DATABASE.cgi, overlaid by yellow contours showing (b) 450 $\\\\mu $ m continuum, (c) HC$_{3}$ N moment zero image, and (d) HC$_{5}$ N moment zero image, respectively.\",\n \"In panel (b), the green contours show the 8.3 mm continuum emission obtained simultaneously with cyanopolyynes by the VLA.\",\n \"The 8.3 mm continuum peak is compact and well consistent with the UCH2 region.\",\n \"At the Cyanopolyyne-rich clump, no point source can be recognized from the Spitzer image, which suggests that no massive young protostar is currently present at the clump position.\",\n \"Therefore, the possibility of hot core is unrealistic.\",\n \"We derived the average column density of H$_{2}$ , $N$ (H$_{2}$ ), of the Cyanopolyyne-rich clump from the 450 $\\\\mu $ m continuum data using the following formula [30]: $ N({\\\\rm {H}}_2) = 2.02 \\\\times 10^{20} {\\\\rm {cm}}^{-2} \\\\Bigl (e^{1.439(\\\\lambda /{\\\\rm {mm}})^{-1}(T/10 {\\\\rm {K}})^{-1}}-1\\\\Bigr )\\\\Bigl (\\\\frac{\\\\kappa _{\\\\nu }}{0.01\\\\: {\\\\rm {cm}}^{2}\\\\: {\\\\rm {g}}^{-1}}\\\\Bigr )^{-1}\\\\Bigl (\\\\frac{S_{\\\\nu }^{\\\\rm {beam}}}{{\\\\rm {mJy}}\\\\: {\\\\rm {beam}}^{-1}}\\\\Bigr )\\\\Bigl (\\\\frac{\\\\theta _{\\\\rm {HPBW}}}{10\\\\: {\\\\rm {arcsec}}}\\\\Bigr )^{-2}\\\\Bigl (\\\\frac{\\\\lambda }{{\\\\rm {mm}}}\\\\Bigr )^{3}.$ We estimated the continuum flux ($S_{\\\\nu }^{\\\\rm {beam}}$ ) toward the Cyanopolyyne-rich clump within a size of 14, and the flux intensity is $9.7 \\\\times 10^{3}$ mJy beam$^{-1}$ .\",\n \"We assumed that $\\\\kappa _{\\\\nu }$ is 0.0619 cm$^{2}$ g$^{-1}$ at $\\\\lambda = 450$ $\\\\mu $ m. $T$ is the dust temperature, and we assumed it to be $30 \\\\pm 15$ K. The derived $N$ (H$_{2}$ ) value is ($2.8^{+8.1}_{-1.3}$ )$\\\\times 10^{22}$ cm$^{-2}$ , which is similar to the typical value of massive clumps forming young star clusters [29].\",\n \"The value is an average for the clump and then it can be considered as the lower limits for cores.\",\n \"The derived $N$ (H$_{2}$ ) value for the Cyanopolyyne-rich clump is higher than the threshold value for star formation cores, $N$ (H$_{2}$ ) $\\\\approx 9 \\\\times 10^{21}$ cm$^{-2}$ [32].\",\n \"Hence, the Cyanopolyyne-rich clump is considered to contain deeply embedded low- or intermediate-mass protostellar core(s), and is not a starless clump.\",\n \"In case that a low- or intermediate-mass protostar has been born, the WCCC mechanism is likely to work; cyanopolyynes are formed from CH$_{4}$ evaporated from grain mantles in the lukewarm gas ($T = 20-30$ K).\",\n \"This may be supported by similar HC$_{3}$ N : HC$_{5}$ N : HC$_{7}$ N ratios between this clump and L1527 (Section REF ).\",\n \"In order to investigate the possibility of the PDR, we compare the fractional abundances of cyanopolyynes at the Cyanopolyyne-rich clump with those derived from a PDR chemical network simulation [19].\",\n \"We derived the fractional abundances, $X$ ($a$ )$= N$ ($a$ )$/N$ (H$_{2}$ ), of cyanopolyynes at Position A as summarized in Table REF .\",\n \"Recent chemical network simulation about the Horsehead nebula, which is one of the most studied PDRs, estimated the HC$_{3}$ N and HC$_{5}$ N abundances [19].\",\n \"The PDR and Core positions are positions with a visual extinctions ($A_{v}$ ) of $\\\\sim 2$ mag and $\\\\sim 10-20$ mag, respectively.\",\n \"We summarize the fractional abundances of these two positions in Table REF .\",\n \"The $X$ (HC$_{3}$ N) and $X$ (HC$_{5}$ N) values at the Cyanopolyyne-rich clump are significantly higher than those in PDR position by three order of magnitudes and more than four order of magnitudes, respectively.\",\n \"Moreover, the HC$_{3}$ N and HC$_{5}$ N fractional abundances at the Cyanopolyyne-rich clump are still higher than those in Core position in the PDR model by a factor of $\\\\sim 55$ and $\\\\sim 80$ , respectively.\",\n \"Therefore, cyanopolyynes at the Cyanopolyyne-rich clump in the G28.28$-$ 0.36 high-mass star-forming region are not formed by the PDR chemistry.\",\n \"If we take large uncertainties in deriving $N$ (H$_{2}$ ) values into consideration, the differences between the Cyanopolyyne-rich clump and the PDR models are plausible.\",\n \"However, we cannot completely exclude any effects from the UCHII region such as UV radiation.\",\n \"lccccc Comparison of fractional abundances of cyanopolyynes 0pt Source Method $X$ (HC$_{3}$ N) $X$ (HC$_{5}$ N) $X$ (HC$_{7}$ N) Referencea ($\\\\times 10^{-9}$ ) ($\\\\times 10^{-9}$ ) ($\\\\times 10^{-9}$ ) Cyanopolyyne-rich clump Observation $6.6^{+5.5}_{-4.9}$ $1.7^{+1.5}_{-1.3}$ $0.9^{+0.8}_{-0.7}$ 1 Horsehead nebula (PDR position) Simulation $3.1 \\\\times 10^{-3}$ $4.3 \\\\times 10^{-5}$ ... 2 Horsehead nebula (Core position) Simulation 0.12 0.022 ... 2 L1527 (WCCC) Observation 0.43 – 0.96 0.05 – 0.24 $\\\\sim 0.05$ 3,4,5 The errors represent the standard deviation.\",\n \"a(1) This work; (2) [19]; (3) [17]; (4) [25]; (5) [26] We also compare the fractional abundances of HC$_{3}$ N, HC$_{5}$ N, and HC$_{7}$ N at the Cyanopolyyne-rich clump to those in L1527.\",\n \"These fractional abundances at the Cyanopolyyne-rich clump are higher than those in L1527 by a factor of $7-15$ , $7-37$ , and $\\\\sim 17$ , respectively.\",\n \"Hence, the cyanopolyynes at the Cyanopolyyne-rich clump seem to be more abundant compared to L1527, even if we take uncertainties in fractional abundances into account.\",\n \"It cannot be excluded that several low-mass protostars are concentrated within a small region (e.g., $\\\\sim 0.02$ pc corresponding to $1.5$ at the distance of 3 kpc).\",\n \"As another interpretation, these results may imply that there is an intermediate-mass protostar because an intermediate-mass protostar heats its wider surroundings compared to a low-mass protostar; the size of lukewarm envelopes will be larger and the column densities of carbon-chain species in such lukewarm envelopes will increase.\",\n \"This could explain the results that the cyanopolyynes at the Cyanopolyyne-rich clump are more abundant than those in L1527.\",\n \"In Figure REF (d), compact HC$_{5}$ N emission region is located at the west position of the 6.7 GHz methanol maser position.\",\n \"This position corresponds to the edge of the 4.5 $\\\\mu $ m emission, which seems to trace shock regions [8].\",\n \"Figure REF shows spectra of HC$_{3}$ N and HC$_{5}$ N at the shock region.\",\n \"The line widths (FWHM) derived from the Gaussian fit are $2.7 \\\\pm 0.2$ and $2.6 \\\\pm 0.2$ km s$^{-1}$ for HC$_{3}$ N and HC$_{5}$ N, respectively.\",\n \"These line widths at the shock region are wider those at the Cyanopolyyne-rich clump, $2.05 \\\\pm 0.16$ and $1.4 \\\\pm 0.2$ km s$^{-1}$ for HC$_{3}$ N and HC$_{5}$ N, respectively.\",\n \"These results probably support that the cyanopolyynes are the shock origin.\",\n \"Molecules evaporated from grain mantles such as C$_{2}$ H$_{2}$ may be parent species of the cyanopolyynes.\",\n \"It is still unclear whether HC$_{5}$ N is formed and can survive in shock regions, but this may be the first observational result showing that HC$_{5}$ N can be enhanced in shock regions.\",\n \"Figure: Spectra of HC 3 _{3}N and HC 5 _{5}N at the shock region.\",\n \"The red lines show the Gaussian fitting results.\"\n ],\n [\n \"Comparison with Previous Single-Dish Telescope Observations\",\n \"[33] reported the detection of high-excitation-energy ($E_{\\\\rm {u}}/k \\\\simeq 100$ K) lines of HC$_{5}$ N with the Nobeyama 45-m radio telescope (HPBW = 18) toward the methanol maser position, which is considered to be a MYSO position [37].\",\n \"The significant HC$_{5}$ N peak is not seen at the methanol maser position in the VLA map.\",\n \"This is probably caused by the different excitation energies of the observed lines.\",\n \"The lines observed with the VLA have low excitation energies (Table ), and preferably trace lower-temperature regions such as low- or intermediate-mass protostellar cores or envelopes.\",\n \"On the other hand, HC$_{5}$ N emission observed with the Nobeyama 45-m telescope seems to come from higher-temperature regions closer to the methanol maser or the MYSO.\",\n \"In these higher-temperature regions, HC$_{5}$ N should be highly excited and low excitation energy lines observed with the VLA should be weak.\",\n \"In contrast, the hotter components detected with the Nobeyama 45-m telescope were not detected with the VLA, because the low-excitation-energy lines are not suitable tracers of the hot components.\",\n \"Combining with the VLA and Nobeyama 45-m telescope results, both the MYSO associated with the methanol maser and low- or intermediate-mass protostellar core(s) at the Cyanopolyyne-rich clump appear to be rich in cyanopolyynes.\",\n \"It is still unclear why the G28.28$-$ 0.36 high-mass star-forming region is a cyanopolyyne-rich/COMs-poor source [34].\",\n \"Studies about chemical diversity among high-mass star-forming regions will become a key to our understanding of massive star formation processes.\",\n \"We need the high-spatial-resolution and higher-frequency band observations in order to investigate the spatial resolution of higher temperature components of cyanopolyynes to confirm that cyanopolyynes exist at the hot core position.\"\n ],\n [\n \"Conclusions\",\n \"We have carried out interferometric observations of cyanopolyynes (HC$_{3}$ N, HC$_{5}$ N, and HC$_{7}$ N) toward the G28.28$-$ 0.36 high-mass star-forming region with the VLA Ka-band.\",\n \"We obtained the moment zero images of HC$_{3}$ N and HC$_{5}$ N and tentatively detected HC$_{7}$ N. The spatial distributions of HC$_{3}$ N and HC$_{5}$ N are consistent with the 450 $\\\\mu $ m dust continuum clump, i.e., the Cyanopolyyne-rich clump.\",\n \"The HC$_{3}$ N : HC$_{5}$ N : HC$_{7}$ N column density ratios are estimated at 1.0 : $\\\\sim 0.3$ : $\\\\sim 0.2$ at Position A.\",\n \"The Cyanopolyyne-rich clump seems to contain deeply embedded low- or intermediate-mass protostellar core(s).\",\n \"The most probable formation mechanism of the cyanopolyynes at the Cyanopolyyne-rich clump is the WCCC mechanism.\",\n \"We possibly found the HC$_{3}$ N and HC$_{5}$ N emission in the shock region.\",\n \"We express our sincere thanks and appreciate for the staff of the National Radio Astronomy Observatory.\",\n \"The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. KT appreciates support from a Granting-Aid for Science Research of Japan (17J03516).\",\n \"Karl G. Jansky Very Large Array (VLA) Common Astronomy Software Applications (CASA)\"\n ],\n [\n \"Gaussian Fitting Results and Spectral Line Parameters\",\n \"We show the gaussian fitting results for the spectra of HC$_{3}$ N, HC$_{5}$ N, and HC$_{7}$ N at Position A in Figure REF .\",\n \"The obtained spectral line parameters are summarized in Table .\",\n \"Figure: Spectra of HC 3 _{3}N, HC 5 _{5}N, and HC 7 _{7}N at Position A obtained with the VLA.\",\n \"The best gaussian fits are shown overlaid in red.ccccc Spectral line parameters at Position A 0pt Species $\\\\mu $ $T_{\\\\rm {b}}$ FWHM $\\\\int T_{\\\\mathrm {b}}dv$ (Debye) (K) (km s$^{-1}$ ) (K km s$^{-1}$ ) HC$_{3}$ N 3.73 $5.4 \\\\pm 0.4$ $2.05 \\\\pm 0.16$ $11.7 \\\\pm 1.2$ HC$_{5}$ N 4.33 $2.3 \\\\pm 0.3$ $1.4 \\\\pm 0.2$ $3.5 \\\\pm 0.8$ HC$_{7}$ N 4.82 $1.0 \\\\pm 0.2$ $1.3 \\\\pm 0.3$ $1.3 \\\\pm 0.4$ The errors represent the standard deviation.\",\n \"Values of their permanent electric dipole moments are taken from the Cologne Database for Molecular Spectroscopy [21].\"\n ]\n]"},"id":{"kind":"string","value":"1808.08435"}}},{"rowIdx":298,"cells":{"text":{"kind":"list like","value":[["Predicting a solar cycle before its onset using a flux transport dynamo\n model"],["Abstract We begin with a review of the predictions for cycle~24 before its onset.","After summarizing the basics of the flux transport dynamo model, we discuss how this model had been used to make a successful prediction of cycle~24, on the assumption that the irregularities of the solar cycle arise due to the fluctuations in the Babcock--Leighton mechanism.","We point out that fluctuations in the meridional circulation can be another cause of irregularities in the cycle."],["Introduction","Let us begin with a disclaimer.","This review will focus on the physics of predicting solar cycles from dynamo models and will refrain from presenting any detailed prediction for the upcoming cycle 25, which is nowadays becoming a hot topic of research.","Figure: Different predictions of the strength of cycle 24, adopted from Pesnell (2008).The two predictions based on theoretical dynamo models (Dikpati et al.","2006; Choudhuriet al.","2007) are indicated by arrows.","The horizontalline added by us indicates the actual peak strength of cycle 24 reached around April 2014.Now that we know what the present cycle 24 has been like, let us take a look at the many predictions of cycle 24 before its onset.","Pesnell (2008) produced a plot combining all the different predictions of the peak sunspot number of cycle 24.","Figure 1 is adopted from this plot, indicating the two predictions based on theoretical dynamo models.","The first theoretical prediction by Dikpati et al.","(2006) was that cycle 24 would be a very strong cycle, whereas the other prediction by Choudhuri et al.","(2007) was that it would be a fairly weak cycle.","All the other predictions shown in Figure 1 were based on various precursors and empirical projections.","We can see that the predictions covered almost the entire range of possible values of the peak sunspot number from $\\approx $ 40 to $\\approx $ 190.","The horizontal line indicates the actual peak sunspot number of cycle 24 and was added by us while preparing this presentation.","It is clear that Choudhuri et al.","(2007) predicted the cycle 24 peak almost correctly.","If several people make several predictions covering the entire possible range, then somebody's prediction has got to come out right!","Were Choudhuri et al.","(2007) simply the lucky persons whose prediction accidentally turned out to be correct?","Or did they get it correct because they figured out the correct physics for making such predictions?","We would like to argue that they figured out the correct physics partially, but not fully.","Their success in predicting cycle 24 was due to a combination of intuition and luck.","In a classic work, Parker (1955) envisaged that the solar cycle is produced by an oscillation between the toroidal and poloidal magnetic fields of the Sun.","Sunspots form out of the toroidal field due to magnetic buoyancy and provide an indication of the strength of the toroidal field.","On the other hand, the magnetic fields in the polar regions of the Sun are a manifestation of the poloidal field.","We now know that there is truly an oscillation between these two field components.","The polar (i.e.","poloidal) field becomes strongest around the time when the sunspot number (i.e.","toroidal field) has its lowest value and vice versa.","Svalgaard et al.","(2005) and Schatten (2005), whose predictions for cycle 24 are included in Figure 1, suggested that the polar field at the beginning of a cycle is a good precursor for the strength of the cycle and used the weak polar field at the beginning of cycle 24 to predict essentially the same low value of the cycle peak that was predicted from the theoretical dynamo calculations of Choudhuri et al.","(2007).","While discussing the physics of cycle prediction, we need to address the question of why the polar field at the beginning of a cycle acts as such a good precursor of the cycle strength."],["Basics of flux transport dynamo model","The flux transport dynamo model, which started being developed in the 1990s (Wang et al.","1991; Choudhuri et al.","1995; Durney 1995; Dikpati & Charbonneau 1999; Nandy & Choudhuri 2002) and has been recently reviewed by several authors (Choudhuri 2011, 2014; Charbonneau 2014; Karak et al.","2014a), has emerged as an attractive theoretical model of the solar cycle.","We describe the basics of this model in this Section before getting into the question of cycle prediction in the next Section.","In order to have an oscillation between the toroidal and poloidal magnetic components, we need to have mechanisms to generate each component from the other.","The generation of the toroidal component from the poloidal component due to stretching by differential rotation is a straightforward process.","To complete the loop, we need a mechanism to generate the poloidal component back from the toroidal component.","The flux transport dynamo model assumes that this is achieved by the Babcock–Leighton (BL) mechanism (Babcock 1961; Leighton 1964), in which the toroidal field first gives rise to bipolar sunspots which emerge with a tilt due to the action of the Coriolis force (D'Silva & Choudhuri 1993) and then the poloidal field arises from the decay of these tilted bipolar sunspots (Hazra et al.","2017).","A dynamo model based just on the idea of toroidal field generation by differential rotation and poloidal field generation by the BL mechanism gives a poleward propagating dynamo wave—in accordance with what is called the Parker–Yoshimura sign rule (Parker 1955; Yoshimura 1975).","In order to explain the observation that sunspots appear increasingly at lower latitudes with the progress of the solar cycle, we need an extra mechanism to reverse the direction of the dynamo wave and make it propagate equatorward.","Choudhuri et al.","(1995) realized that this extra mechanism is provided by the meridional circulation (MC) of the Sun, which is observed to be poleward near the surface of the Sun.","In order to avoid mass piling up in the polar region, there has to be a subsurface reverse flow towards the equator.","The majority of flux transport dynamo models assume an one-cell MC with the equatorward flow at the bottom of the convection zone.","However, Hazra et al.","(2014) have shown that the flux transport model can work even with a multi-cell MC structure as long as there is an equatorward flow at the bottom of the convection zone.","The toroidal field produced there due to the strong differential rotation discovered by helioseismology is advected by this flow equatorward, making sure that sunspots appear at lower latitudes with the progress of the cycle.","The poloidal field generated by the BL mechanism near the surface is advected poleward by the poleward MC there—in accordance with observational data.","Perhaps, at this stage, it is a good idea to state clearly what we mean by flux transport dynamo.","We would refer to a solar dynamo model as a flux transport dynamo model if it satisfies the following characteristics: (i) the Babcock–Leighton (BL) mechanism for generating the poloidal field at the solar surface is included in the model; and (ii) the meridional circulation (MC) plays an important role by transporting the poloidal field poleward near the surface and the toroidal field equatorward at the bottom of the convection zone.","While different authors in the past might have meant slightly different things by flux transport dynamo, we believe that the definition we give here has come to be regarded as the universally accepted definition of flux transport dynamo at the present time.","Detailed theoretical calculations based on the flux transport dynamo model are broadly in agreement with observational data pertaining to the solar cycle.","A flux transport dynamo code based on mean field equations tends to give a periodic solution, unless something special is done to make the solution irregular.","After our brief summary of the flux transport dynamo model, we come to the question as to what causes irregularities of the cycle and whether an understanding of that will help us in predicting future cycles."],["Fluctuations in the Babcock–Leighton (BL) mechanism","It has been known for some time that fluctuations in the poloidal field generation mechanism can cause irregularities in the cycle (Choudhuri 1992).","Choudhuri et al.","(2007) suggested that the fluctuations in the BL mechanism would be the main source of irregularities in the dynamo process.","It is not difficult to understand how the fluctuations in the BL mechanism may arise.","This mechanism depends on the tilts of bipolar sunspot pairs—a pair with a larger tilt making a more significant contribution to the poloidal field.","The average tilt of sunspot pairs is given by Joy's law.","However, one finds a distribution of tilt angles around this average (Stenflo & Kosovichev 2012)—presumably caused by the effect of the turbulence in the solar convection zone through which flux tubes rise to form bipolar sunspot pairs (Longcope & Choudhuri 2002).","Figure: A sketch explaining how the correlation between the polar field at theend of a cycle and the strength of the next cycle arises.","From Jiang et al.","(2007).Let us first come to the question how the observed correlation between the polar field at the beginning of a solar cycle and the strength of the cycle arises.","Refer to Fig.","2 taken from Jiang et al.","(2007), who provided an explanation of this.","The BL mechanism produces poloidal field at C in mid-latitudes near the surface.","This poloidal field will simultaneously be advected poleward by the meridional circulation (MC) to produce the polar field at P at the beginning of the next cycle and will also diffuse to T at the bottom of the convection zone, where it will act as the seed of the next cycle.","If the fluctuations in the BL mechanism make the poloidal field produced at C in a cycle stronger than the usual, then both the polar field at P at the beginning of the next cycle and the seed of the next cycle at T will be stronger than usual.","On the other hand, if the poloidal field produced at C is weaker than the usual, then the opposite of this will happen.","We believe that this is how the correlation between the polar field at the beginning of a cycle and the strength of the cycle arises.","It may be pointed out that the turbulent diffusion has to be sufficiently high to ensure that the poloidal field diffuses from C to T in time of the order of 5–10 yr, in order to produce the required correlation.","Choudhuri et al.","(2007) used a value of turbulent diffusivity on the basis of mixing length arguments, which made the correlation come out beautifully.","However, Dikpati et al.","(2006) used a rather unrealistically low value of turbulent diffusion and the diffusion time across the convection zone in their model is of the order of 200 yr. Their model would not give a correlation between the polar field at the beginning of a cycle and the strength of the cycle.","It may be mentioned that solar dynamo models with high and low diffusivities have very different characteristics (Jiang et al.","2007; Yeates et al.","2008).","There is enough evidence now that models with higher diffusivity are closer to reality.","Higher diffusivity helps in explaining such observational features as the dipolar parity (Chatterjee et al.","2004; Hotta & Yokoyama 2010) and the lack of hemispheric asymmetry (Chatterjee & Choudhuri 2006; Goel & Choudhuri 2009).","In order to model actual cycles, one needs to incorporate the actual fluctuations in the BL mechanism into the code.","Choudhuri et al.","(2007) devised a scheme of figuring out actual fluctuations of the BL mechanism from the observational data of the poloidal fields and then incorporating these in the dynamo code.","Since such data are available only from the 1970s, actual cycles could be modelled only from that time.","Choudhuri et al.","(2007) succeeded in modelling cycles 21–23 reasonably well and cycle 24 was predicted to be a weak cycle.","Their prediction of cycle 24 was a robust prediction, since they had incorporated the weakness of the polar field at the beginning of cycle 24 in their model and the high diffusivity of their model would make this correlated with the strength of cycle 24.","As we have already pointed out, this prediction has been borne out triumphantly—making this the first successful prediction of a solar cycle based on a theoretical dynamo model in the history of this subject."],["Fluctuations in the meridional circulation (MC)","When Choudhuri et al.","(2007) made their prediction, it was not realized that the meridional circulation (MC) probably has occasional large fluctuations and these also may produce irregularities in the solar cycle.","These fluctuations are different from the periodic variation of MC with the solar cycle, presumably due to the Lorentz force of the dynamo-generated magnetic field (Hazra & Choudhuri 2017).","It has been known for some time that the strength of MC determines the period of the dynamo—faster MC making cycles shorter and vice versa (Dikpati & Charbonneau 1999).","Although we have MC data only for about two decades, we have data pertaining to durations of cycles for about two centuries and these data indicate that there have fluctuations in MC with correlation time of the order of a few decades (Karak & Choudhuri 2011).","What kinds of irregularities will the fluctuating MC introduce?","Suppose MC has slowed down, making cycles longer.","Then diffusion will have more time to act, making cycles weaker.","We may thus expect longer cycles to be weaker and shorter cycles stronger.","Since a shorter cycle would imply a faster rise time of the cycle, we may expect a correlation between the rise time and the strength of a cycle.","Such a correlation is found in the observational data and is known as the Waldmeier effect.","Karak & Choudhuri (2011) gave an explanation of this effect by introducing fluctuations in MC in their model.","On the basis of such studies, we conclude that there are two important sources of irregularities in the solar cycle—fluctuations in the BL mechanism and fluctuations in MC.","Choudhuri et al.","(2007) made their prediction of cycle 24 on the basis of the assumption that the irregularities are produced by fluctuations in the BL mechanism alone.","Presumably their prediction was so successful because there had not been large fluctuations in MC in the last few years.","Now that we know the fluctuations of MC to be another factor introducing irregularities in solar cycles, we need to develop cycle prediction methods taking this into account.","We are presently working on this problem.","It may be mentioned that one big challenge in this field is to develop a theory of grand minima like the Maunder minimum in the 17th century.","While it has been shown that grand minima can be induced by fluctuations in the BL mechanism alone (Choudhuri & Karak 2009) or by fluctuations in MC alone (Karak 2010), we need both of these to develop a comprehensive theory of grand minima (Choudhuri & Karak 2012; Karak & Choudhuri 2013)."],["Conclusion","Within the last few years, the flux transport dynamo model has emerged as an attractive model for explaining the solar cycle and there is increasing evidence that other solar-like stars also may have similar dynamos working inside them (Karak et al.","2014b; Choudhuri 2017).","It is important that we understand how the irregularities in the cycle arise, since such an understanding may enable us to predict a future cycle before its onset.","It appears that fluctuations in the BL mechanism and fluctuations in MC are the two main sources of irregularities in the solar cycle.","Before the beginning of cycle 24, the role of MC fluctuations was not generally appreciated.","The successful theoretical prediction of Choudhuri et al.","(2007) was based on the assumption that irregularities in the solar cycle are caused only by fluctuations in the BL mechanism.","With the realization that MC fluctuations also can introduce additional irregularities, it is necessary to develop prediction methods taking this into account.","Acknowledgments.","I thank Gopal Hazra for help in preparing the manuscript.","My research is supported by DST through a J.C. Bose Fellowship."]],"string":"[\n [\n \"Predicting a solar cycle before its onset using a flux transport dynamo\\n model\"\n ],\n [\n \"Abstract We begin with a review of the predictions for cycle~24 before its onset.\",\n \"After summarizing the basics of the flux transport dynamo model, we discuss how this model had been used to make a successful prediction of cycle~24, on the assumption that the irregularities of the solar cycle arise due to the fluctuations in the Babcock--Leighton mechanism.\",\n \"We point out that fluctuations in the meridional circulation can be another cause of irregularities in the cycle.\"\n ],\n [\n \"Introduction\",\n \"Let us begin with a disclaimer.\",\n \"This review will focus on the physics of predicting solar cycles from dynamo models and will refrain from presenting any detailed prediction for the upcoming cycle 25, which is nowadays becoming a hot topic of research.\",\n \"Figure: Different predictions of the strength of cycle 24, adopted from Pesnell (2008).The two predictions based on theoretical dynamo models (Dikpati et al.\",\n \"2006; Choudhuriet al.\",\n \"2007) are indicated by arrows.\",\n \"The horizontalline added by us indicates the actual peak strength of cycle 24 reached around April 2014.Now that we know what the present cycle 24 has been like, let us take a look at the many predictions of cycle 24 before its onset.\",\n \"Pesnell (2008) produced a plot combining all the different predictions of the peak sunspot number of cycle 24.\",\n \"Figure 1 is adopted from this plot, indicating the two predictions based on theoretical dynamo models.\",\n \"The first theoretical prediction by Dikpati et al.\",\n \"(2006) was that cycle 24 would be a very strong cycle, whereas the other prediction by Choudhuri et al.\",\n \"(2007) was that it would be a fairly weak cycle.\",\n \"All the other predictions shown in Figure 1 were based on various precursors and empirical projections.\",\n \"We can see that the predictions covered almost the entire range of possible values of the peak sunspot number from $\\\\approx $ 40 to $\\\\approx $ 190.\",\n \"The horizontal line indicates the actual peak sunspot number of cycle 24 and was added by us while preparing this presentation.\",\n \"It is clear that Choudhuri et al.\",\n \"(2007) predicted the cycle 24 peak almost correctly.\",\n \"If several people make several predictions covering the entire possible range, then somebody's prediction has got to come out right!\",\n \"Were Choudhuri et al.\",\n \"(2007) simply the lucky persons whose prediction accidentally turned out to be correct?\",\n \"Or did they get it correct because they figured out the correct physics for making such predictions?\",\n \"We would like to argue that they figured out the correct physics partially, but not fully.\",\n \"Their success in predicting cycle 24 was due to a combination of intuition and luck.\",\n \"In a classic work, Parker (1955) envisaged that the solar cycle is produced by an oscillation between the toroidal and poloidal magnetic fields of the Sun.\",\n \"Sunspots form out of the toroidal field due to magnetic buoyancy and provide an indication of the strength of the toroidal field.\",\n \"On the other hand, the magnetic fields in the polar regions of the Sun are a manifestation of the poloidal field.\",\n \"We now know that there is truly an oscillation between these two field components.\",\n \"The polar (i.e.\",\n \"poloidal) field becomes strongest around the time when the sunspot number (i.e.\",\n \"toroidal field) has its lowest value and vice versa.\",\n \"Svalgaard et al.\",\n \"(2005) and Schatten (2005), whose predictions for cycle 24 are included in Figure 1, suggested that the polar field at the beginning of a cycle is a good precursor for the strength of the cycle and used the weak polar field at the beginning of cycle 24 to predict essentially the same low value of the cycle peak that was predicted from the theoretical dynamo calculations of Choudhuri et al.\",\n \"(2007).\",\n \"While discussing the physics of cycle prediction, we need to address the question of why the polar field at the beginning of a cycle acts as such a good precursor of the cycle strength.\"\n ],\n [\n \"Basics of flux transport dynamo model\",\n \"The flux transport dynamo model, which started being developed in the 1990s (Wang et al.\",\n \"1991; Choudhuri et al.\",\n \"1995; Durney 1995; Dikpati & Charbonneau 1999; Nandy & Choudhuri 2002) and has been recently reviewed by several authors (Choudhuri 2011, 2014; Charbonneau 2014; Karak et al.\",\n \"2014a), has emerged as an attractive theoretical model of the solar cycle.\",\n \"We describe the basics of this model in this Section before getting into the question of cycle prediction in the next Section.\",\n \"In order to have an oscillation between the toroidal and poloidal magnetic components, we need to have mechanisms to generate each component from the other.\",\n \"The generation of the toroidal component from the poloidal component due to stretching by differential rotation is a straightforward process.\",\n \"To complete the loop, we need a mechanism to generate the poloidal component back from the toroidal component.\",\n \"The flux transport dynamo model assumes that this is achieved by the Babcock–Leighton (BL) mechanism (Babcock 1961; Leighton 1964), in which the toroidal field first gives rise to bipolar sunspots which emerge with a tilt due to the action of the Coriolis force (D'Silva & Choudhuri 1993) and then the poloidal field arises from the decay of these tilted bipolar sunspots (Hazra et al.\",\n \"2017).\",\n \"A dynamo model based just on the idea of toroidal field generation by differential rotation and poloidal field generation by the BL mechanism gives a poleward propagating dynamo wave—in accordance with what is called the Parker–Yoshimura sign rule (Parker 1955; Yoshimura 1975).\",\n \"In order to explain the observation that sunspots appear increasingly at lower latitudes with the progress of the solar cycle, we need an extra mechanism to reverse the direction of the dynamo wave and make it propagate equatorward.\",\n \"Choudhuri et al.\",\n \"(1995) realized that this extra mechanism is provided by the meridional circulation (MC) of the Sun, which is observed to be poleward near the surface of the Sun.\",\n \"In order to avoid mass piling up in the polar region, there has to be a subsurface reverse flow towards the equator.\",\n \"The majority of flux transport dynamo models assume an one-cell MC with the equatorward flow at the bottom of the convection zone.\",\n \"However, Hazra et al.\",\n \"(2014) have shown that the flux transport model can work even with a multi-cell MC structure as long as there is an equatorward flow at the bottom of the convection zone.\",\n \"The toroidal field produced there due to the strong differential rotation discovered by helioseismology is advected by this flow equatorward, making sure that sunspots appear at lower latitudes with the progress of the cycle.\",\n \"The poloidal field generated by the BL mechanism near the surface is advected poleward by the poleward MC there—in accordance with observational data.\",\n \"Perhaps, at this stage, it is a good idea to state clearly what we mean by flux transport dynamo.\",\n \"We would refer to a solar dynamo model as a flux transport dynamo model if it satisfies the following characteristics: (i) the Babcock–Leighton (BL) mechanism for generating the poloidal field at the solar surface is included in the model; and (ii) the meridional circulation (MC) plays an important role by transporting the poloidal field poleward near the surface and the toroidal field equatorward at the bottom of the convection zone.\",\n \"While different authors in the past might have meant slightly different things by flux transport dynamo, we believe that the definition we give here has come to be regarded as the universally accepted definition of flux transport dynamo at the present time.\",\n \"Detailed theoretical calculations based on the flux transport dynamo model are broadly in agreement with observational data pertaining to the solar cycle.\",\n \"A flux transport dynamo code based on mean field equations tends to give a periodic solution, unless something special is done to make the solution irregular.\",\n \"After our brief summary of the flux transport dynamo model, we come to the question as to what causes irregularities of the cycle and whether an understanding of that will help us in predicting future cycles.\"\n ],\n [\n \"Fluctuations in the Babcock–Leighton (BL) mechanism\",\n \"It has been known for some time that fluctuations in the poloidal field generation mechanism can cause irregularities in the cycle (Choudhuri 1992).\",\n \"Choudhuri et al.\",\n \"(2007) suggested that the fluctuations in the BL mechanism would be the main source of irregularities in the dynamo process.\",\n \"It is not difficult to understand how the fluctuations in the BL mechanism may arise.\",\n \"This mechanism depends on the tilts of bipolar sunspot pairs—a pair with a larger tilt making a more significant contribution to the poloidal field.\",\n \"The average tilt of sunspot pairs is given by Joy's law.\",\n \"However, one finds a distribution of tilt angles around this average (Stenflo & Kosovichev 2012)—presumably caused by the effect of the turbulence in the solar convection zone through which flux tubes rise to form bipolar sunspot pairs (Longcope & Choudhuri 2002).\",\n \"Figure: A sketch explaining how the correlation between the polar field at theend of a cycle and the strength of the next cycle arises.\",\n \"From Jiang et al.\",\n \"(2007).Let us first come to the question how the observed correlation between the polar field at the beginning of a solar cycle and the strength of the cycle arises.\",\n \"Refer to Fig.\",\n \"2 taken from Jiang et al.\",\n \"(2007), who provided an explanation of this.\",\n \"The BL mechanism produces poloidal field at C in mid-latitudes near the surface.\",\n \"This poloidal field will simultaneously be advected poleward by the meridional circulation (MC) to produce the polar field at P at the beginning of the next cycle and will also diffuse to T at the bottom of the convection zone, where it will act as the seed of the next cycle.\",\n \"If the fluctuations in the BL mechanism make the poloidal field produced at C in a cycle stronger than the usual, then both the polar field at P at the beginning of the next cycle and the seed of the next cycle at T will be stronger than usual.\",\n \"On the other hand, if the poloidal field produced at C is weaker than the usual, then the opposite of this will happen.\",\n \"We believe that this is how the correlation between the polar field at the beginning of a cycle and the strength of the cycle arises.\",\n \"It may be pointed out that the turbulent diffusion has to be sufficiently high to ensure that the poloidal field diffuses from C to T in time of the order of 5–10 yr, in order to produce the required correlation.\",\n \"Choudhuri et al.\",\n \"(2007) used a value of turbulent diffusivity on the basis of mixing length arguments, which made the correlation come out beautifully.\",\n \"However, Dikpati et al.\",\n \"(2006) used a rather unrealistically low value of turbulent diffusion and the diffusion time across the convection zone in their model is of the order of 200 yr. Their model would not give a correlation between the polar field at the beginning of a cycle and the strength of the cycle.\",\n \"It may be mentioned that solar dynamo models with high and low diffusivities have very different characteristics (Jiang et al.\",\n \"2007; Yeates et al.\",\n \"2008).\",\n \"There is enough evidence now that models with higher diffusivity are closer to reality.\",\n \"Higher diffusivity helps in explaining such observational features as the dipolar parity (Chatterjee et al.\",\n \"2004; Hotta & Yokoyama 2010) and the lack of hemispheric asymmetry (Chatterjee & Choudhuri 2006; Goel & Choudhuri 2009).\",\n \"In order to model actual cycles, one needs to incorporate the actual fluctuations in the BL mechanism into the code.\",\n \"Choudhuri et al.\",\n \"(2007) devised a scheme of figuring out actual fluctuations of the BL mechanism from the observational data of the poloidal fields and then incorporating these in the dynamo code.\",\n \"Since such data are available only from the 1970s, actual cycles could be modelled only from that time.\",\n \"Choudhuri et al.\",\n \"(2007) succeeded in modelling cycles 21–23 reasonably well and cycle 24 was predicted to be a weak cycle.\",\n \"Their prediction of cycle 24 was a robust prediction, since they had incorporated the weakness of the polar field at the beginning of cycle 24 in their model and the high diffusivity of their model would make this correlated with the strength of cycle 24.\",\n \"As we have already pointed out, this prediction has been borne out triumphantly—making this the first successful prediction of a solar cycle based on a theoretical dynamo model in the history of this subject.\"\n ],\n [\n \"Fluctuations in the meridional circulation (MC)\",\n \"When Choudhuri et al.\",\n \"(2007) made their prediction, it was not realized that the meridional circulation (MC) probably has occasional large fluctuations and these also may produce irregularities in the solar cycle.\",\n \"These fluctuations are different from the periodic variation of MC with the solar cycle, presumably due to the Lorentz force of the dynamo-generated magnetic field (Hazra & Choudhuri 2017).\",\n \"It has been known for some time that the strength of MC determines the period of the dynamo—faster MC making cycles shorter and vice versa (Dikpati & Charbonneau 1999).\",\n \"Although we have MC data only for about two decades, we have data pertaining to durations of cycles for about two centuries and these data indicate that there have fluctuations in MC with correlation time of the order of a few decades (Karak & Choudhuri 2011).\",\n \"What kinds of irregularities will the fluctuating MC introduce?\",\n \"Suppose MC has slowed down, making cycles longer.\",\n \"Then diffusion will have more time to act, making cycles weaker.\",\n \"We may thus expect longer cycles to be weaker and shorter cycles stronger.\",\n \"Since a shorter cycle would imply a faster rise time of the cycle, we may expect a correlation between the rise time and the strength of a cycle.\",\n \"Such a correlation is found in the observational data and is known as the Waldmeier effect.\",\n \"Karak & Choudhuri (2011) gave an explanation of this effect by introducing fluctuations in MC in their model.\",\n \"On the basis of such studies, we conclude that there are two important sources of irregularities in the solar cycle—fluctuations in the BL mechanism and fluctuations in MC.\",\n \"Choudhuri et al.\",\n \"(2007) made their prediction of cycle 24 on the basis of the assumption that the irregularities are produced by fluctuations in the BL mechanism alone.\",\n \"Presumably their prediction was so successful because there had not been large fluctuations in MC in the last few years.\",\n \"Now that we know the fluctuations of MC to be another factor introducing irregularities in solar cycles, we need to develop cycle prediction methods taking this into account.\",\n \"We are presently working on this problem.\",\n \"It may be mentioned that one big challenge in this field is to develop a theory of grand minima like the Maunder minimum in the 17th century.\",\n \"While it has been shown that grand minima can be induced by fluctuations in the BL mechanism alone (Choudhuri & Karak 2009) or by fluctuations in MC alone (Karak 2010), we need both of these to develop a comprehensive theory of grand minima (Choudhuri & Karak 2012; Karak & Choudhuri 2013).\"\n ],\n [\n \"Conclusion\",\n \"Within the last few years, the flux transport dynamo model has emerged as an attractive model for explaining the solar cycle and there is increasing evidence that other solar-like stars also may have similar dynamos working inside them (Karak et al.\",\n \"2014b; Choudhuri 2017).\",\n \"It is important that we understand how the irregularities in the cycle arise, since such an understanding may enable us to predict a future cycle before its onset.\",\n \"It appears that fluctuations in the BL mechanism and fluctuations in MC are the two main sources of irregularities in the solar cycle.\",\n \"Before the beginning of cycle 24, the role of MC fluctuations was not generally appreciated.\",\n \"The successful theoretical prediction of Choudhuri et al.\",\n \"(2007) was based on the assumption that irregularities in the solar cycle are caused only by fluctuations in the BL mechanism.\",\n \"With the realization that MC fluctuations also can introduce additional irregularities, it is necessary to develop prediction methods taking this into account.\",\n \"Acknowledgments.\",\n \"I thank Gopal Hazra for help in preparing the manuscript.\",\n \"My research is supported by DST through a J.C. Bose Fellowship.\"\n ]\n]"},"id":{"kind":"string","value":"1808.08543"}}},{"rowIdx":299,"cells":{"text":{"kind":"list like","value":[["Scale Drift Correction of Camera Geo-Localization using Geo-Tagged\n Images"],["Abstract Camera geo-localization from a monocular video is a fundamental task for video analysis and autonomous navigation.","Although 3D reconstruction is a key technique to obtain camera poses, monocular 3D reconstruction in a large environment tends to result in the accumulation of errors in rotation, translation, and especially in scale: a problem known as scale drift.","To overcome these errors, we propose a novel framework that integrates incremental structure from motion (SfM) and a scale drift correction method utilizing geo-tagged images, such as those provided by Google Street View.","Our correction method begins by obtaining sparse 6-DoF correspondences between the reconstructed 3D map coordinate system and the world coordinate system, by using geo-tagged images.","Then, it corrects scale drift by applying pose graph optimization over Sim(3) constraints and bundle adjustment.","Experimental evaluations on large-scale datasets show that the proposed framework not only sufficiently corrects scale drift, but also achieves accurate geo-localization in a kilometer-scale environment."],["Introduction","Camera geo-localization from a monocular video in a kilometer-scale environment is a essential technology for AR, video analysis, and autonomous navigation.","To achieve accurate geo-localization, 3D reconstruction from a video is a key technique.","Incremental structure from motion (SfM) and visual simultaneous localization and mapping (visual SLAM) achieve large-scale 3D reconstructions by simultaneously localizing camera poses with six degrees-of-freedom (6-DoF) and reconstructing a 3D environment map [1], [2].","Unlike for a stereo camera, an absolute scale of the real world cannot be derived using a single observation from a monocular camera.","Although it is possible to estimate an environment's relative scale from a series of monocular observations, errors in the relative scale estimation accumulate over time, and this is referred to as scale drift [3], [4].","For an accurate geo-localization not affected by scale drift ,prior information in a geographic information system (GIS) has been utilized in previous studies.","For example, point clouds, 3D models, building footprints, and road maps have been proven to be efficient for correcting reconstructed 3D maps [5], [6], [7], [8], [9].","However, these priors are only available in limited situations, e.g., in an area that is observed in advance, or in an environment consisting of simply-shaped buildings.","Therefore, there is a good chance that other GIS information can help to extend the area in which a 3D map can be corrected.","Hence, in this paper, motivated by the recent availability of massive public repositories of geo-tagged images taken all over the world, we propose a novel framework for correcting the scale drift of monocular 3D reconstruction by utilizing geo-tagged images, such as those in Google Street View [10], and achieve accurate camera geo-localization.","Owing to the high coverage of Google Street View, our proposal is more scalable than those in previous studies.","The proposed framework integrates incremental SfM and a scale drift correction method utilizing geo-tagged images.","Our correction method begins by computing 6-DoF correspondences between the reconstructed 3D map coordinate system and the world coordinate system, by using geo-tagged images.","Owing to significant differences in illumination, viewpoint, and the environment resulting from differences in time, it tends to be difficult to acquire correspondences between video frames and geo-tagged images (Fig.","REF ).","Therefore, a new correction method that can deal with the large scale drift of a 3D map using a limited number of correspondences is required.","Bundle adjustment with constraints of global position information, which represents one of the most important correction methods, cannot be applied directly.","This is because bundle adjustment tends to get stuck in a local minimum when starting from a 3D map including large errors [4].","Hence, the proposed correction method consists of two coarse-to-fine steps: pose graph optimization over Sim(3) constraints, and bundle adjustment.","In these steps, our key idea is to extend the pose graph optimization method proposed for the loop closure technique of monocular SLAM [4], such that it incorporates the correspondences between the 3D map coordinate system and the world coordinate system.","This step corrects the large errors, and enables bundle adjustment to obtain precise results.","After implementing this framework, we conducted experiments to evaluate the proposal.","The contributions of this work are as follows.","First, we propose a novel framework for camera geo-localization that can correct scale drift by utilizing geo-tagged images.","Second, we extend the pose graph optimization approach to dealing with scale drift using a limited number of correspondences to geo-tags.","Finally, we validate the effectiveness of the proposal through experimental evaluations on kilometer-scale datasets.","Incremental SfM and visual SLAM are important approaches to reconstructing 3D maps from monocular videos.","Klein et al.","proposed PTAM for small AR workspaces [11].","Mur-Artal et al.","developed ORB-SLAM, which can reconstruct large-scale outdoor environments [2].","For accurate 3D reconstruction, the loop closure technique has commonly been employed in recent SLAM approaches [4], [2].","Loop closure deals with errors that accumulate between two camera poses that occur at the same location, i.e., when the camera trajectory forms a loop.","Lu and Milios [12] formulated this technique as a pose graph optimization problem, and Strasdat et al.","[4] extended pose graph optimization to deal with scale drift for monocular visual SLAM.","It is certain that loop closure can significantly improve 3D maps, but this is only effective if a loop exists in the video."],["Geo-registration of Reconstructions","Correcting reconstructed 3D maps by using geo-referenced information has been regarded as a geo-registration problem.","Kaminsky et al.","proposed a method that aligns 3D reconstructions to 2D aerial images [13].","Wendel et al.","used an overhead digital surface model (DSM) for the geo-registration of 3D maps [14].","Similar to our work, Wang et al.","used Google Street View geo-tagged images and a Google Earth 3D model for the geo-registration of reconstructed 3D maps [15].","However, because all these methods focus on estimating a best-fitting similarity transformation to geo-referenced information, they only correct the global scale in terms of 3D map correction.","Methods for geo-registration using non-linear transformations have also been proposed.","To integrate GPS information, Lhuillier et al.","proposed incremental SfM using bundle adjustment with constraints from GPS [16], and Rehder et al.","formulated a global pose estimation problem using stereo visual odometry, inertial measurements, and infrequent GPS information as a 6-DoF pose graph optimization problem [17].","In terms of correcting camera poses using sparse global information, Rehder's method is similar to our pose graph optimization approach.","However, our 7-DoF pose graph optimization differs in focusing on scale drift resulting from monocular 3D reconstruction, and in utilizing geo-tagged images.","In addition to GPS information, various kinds of reference data have been used for the non-linear geo-registration or geo-localization of a video, such as point clouds [5], [6], 3D models [7], building footprints [8], and road maps [9].","In this paper, we address a method that introduces geo-tagged images to the non-linear geo-registration of 3D maps."],["Proposed Method","Fig.","REF provides a flowchart of the proposed framework, which is roughly divided into three parts.","The first part is incremental SfM, and is described in Sec.","REF .","The second part computes 6-DoF correspondences between the 3D map coordinate system and the world coordinate system (as defined below), by making use of geo-tagged images (Sec.","REF ).","The third part then uses the correspondences to correct the scale drift of the 3D map, by applying pose graph optimization over Sim(3) constraints (Sec.","REF ) and bundle adjustment (Sec.","REF ) incrementally.","The initialization of the scale drift correction method is described in Sec.","REF ."],["World Coordinate System","In this paper, the world coordinates are represented by 3D coordinates $(x, y, z)$ , where the $xz$ -plane corresponds to the Universal Transverse Mercator (UTM) coordinate system, which is an orthogonal coordinate system using meters, and $y$ corresponds to the height from the ground in meters.","The UTM coordinates can be converted into latitude and longitude if necessary."],["Incremental SfM ","As large-scale incremental SfM, we use ORB-SLAM [2] (with no real-time constraints).","This is one of the best-performing monocular SLAM systems.","Frames that are important for 3D reconstruction are selected as keyframes by ORB-SLAM.","Every time a new keyframe is selected, our correction method is performed, and the 3D map reconstructed up to that point is corrected.","In the 3D reconstruction, we identify 3D map points and their corresponding 2D keypoints in the keyframes (collectively denoted by $C_{\\scalebox {0.7}{map-kf}}$ ).","Our proposed framework does not depend on a certain 3D reconstruction method, and can be applied to the other monocular 3D reconstruction methods, such as incremental SfM and feature-based visual SLAM."],["Obtaining Correspondences between 3D Map and World Coordinates ","Here, we describe the second part of the proposed method, which uses geo-tagged images to compute a 6-DoF correspondence, $C_{\\scalebox {0.7}{map-world}}$ , between the 3D map and world coordinate system.","For this purpose, we modify Agarwal's method [18] to integrate it into ORB-SLAM.","This part consists of the following four steps: geo-tagged image collection, similar geo-tagged image retrieval, keypoint matching, and geo-tagged image localization.","Geo-tagged Image Collection.","Google Street View [10] is a browsable street-level GIS, which is one of the largest repositories of global geo-tagged images (i.e., images and their associated geo-tags).","All images are high-resolution RGB panorama images, containing highly accurate world positions [19].","We make use of this data by converting each panorama image into eight rectilinear images with the same field-of-view as our input video, with eight horizontal directions.","Note that because each geo-tag has a position and rotation in the world coordinates, we can obtain the 6-DoF correspondences between the 3D map coordinate system and world coordinate system if geo-tagged images are localized in the 3D map coordinate system.","Similar Geo-tagged Image Retrieval.","When a new keyframe is selected, we retrieve the top-$k$ similar geo-tagged images.","The retrieval system employs a bag-of-words approach based on SIFT descriptors [18].","Keypoint Matching.","Given the pairs of keyframes and retrieved geo-tagged images, we detect ORB keypoints [20] from the pairs and perform keypoint matching.","Because the matching between video frames and Google Street View images tends to include many outliers [21], we use a virtual line descriptor (kVLD) [22], which can reject outliers by using a graph matching method even when inlier rate is around 10 Geo-tagged Image Localization.","To compute $C_{\\scalebox {0.7}{map-world}}$ , we first compute 3D-to-2D correspondences $C_{\\scalebox {0.7}{map-geo}}$ between 3D map points and their corresponding 2D keypoints in geo-tagged images.","In particular, we obtain $C_{\\scalebox {0.7}{map-geo}}$ by combining the 2D keypoint matches (computed in the previous step) with the correspondences $C_{\\scalebox {0.7}{map-kf}}$ between 3D map points and their corresponding 2D keypoints in keyframes (computed in 3D reconstruction).","Then, we obtain the 6-DoF camera poses of geo-tagged images in the 3D map coordinate system by minimizing the re-projection errors of $C_{\\scalebox {0.7}{map-geo}}$ , using the LM algorithm.","Finally, we obtain $C_{\\scalebox {0.7}{map-world}}$ by combining the camera poses of geo-tagged images and 6-DoF camera poses of the associated geo-tags.","Figure: An example of the proposed pose graph optimization.","This optimization maintains overall relative poses, except for gradual scale changes, and keeps camera poses of geo-tagged images close to the positions of the corresponding geo-tags."],["Initialization (INIT) ","As the initialization, two kinds of linear transformations are performed on the 3D map, because the positions and scales of the 3D map coordinates and world coordinates are significantly different.","Initialization is applied once, when the $i$ -th geo-tagged image is localized.","We set $i = 4$ .","Given the first to $i$ -th $C_{\\scalebox {0.7}{map-world}}$ , the first transformation assumes that all camera poses are approximately located in one plane, and rotates the 3D map to align that plane to the world $xz$ -plane.","The best-fitting plane can be estimated by a principal component analysis.","Next, we estimate the best-fitting transformation matrix given by Eq.","REF , which transforms a point in the 3D map coordinate system $\\mathbf {p}_{\\scalebox {0.6}{SLAM},k}$ to be closer to a corresponding point in the world coordinate system $\\mathbf {p}_{world,k}$ ($\\mathbf {p}_{\\scalebox {0.6}{SLAM},k}$ and $\\mathbf {p}_{world,k}$ are denoted using a homogeneous representation): $\\mathbf {A} =\\begin{bmatrix}s * \\cos (\\theta ) & 0 & -s * \\sin (\\theta ) & a\\\\0 & s & 0 & 1 \\\\s * \\sin (\\theta ) & 0 & s * \\cos (\\theta ) & b\\\\0 & 0 & 0 & 1\\end{bmatrix}$ Using the first to $i$ -th $C_{\\scalebox {0.7}{map-world}}$ , we estimate the four matrix parameters $[a, b, s, \\theta ]$ by minimizing the following cost using RANSAC [23] and the Levenberg-Marquart (LM) algorithm: $E =\\sum _{k \\in 1,2...i} \\Vert \\mathbf {p}_{world,k} - \\mathbf {A} \\mathbf {p}_{\\scalebox {0.6}{SLAM},k} \\Vert ^{2}$ The camera poses of the geo-tagged images in $C_{\\scalebox {0.7}{map-world}}$ , keyframes, and 3D map point can then be transformed using the resulting matrix."],["Pose Graph Optimization over Sim(3) Constraints (PGO)","We correct the 3D map focusing on scale drift by using the newest three of $C_{\\scalebox {0.7}{map-world}}$ .","This correction is performed every time a new $C_{\\scalebox {0.7}{map-world}}$ is found after initialization.","Then, we propose a graph-based non-linear optimization method (pose graph optimization) on Lie manifolds, which simultaneously corrects the scale drift and aligns the 3D map with the world coordinates.","Notation.","A 3D rigid body transformation $\\mathbf {G} \\in \\mathrm {SE}(3)$ and a 3D similarity transformation $\\mathbf {S} \\in \\mathrm {Sim}(3)$ are defined by Eq.","REF , where $\\mathbf {R} \\in \\mathrm {SO}(3)$ , $\\mathbf {t} \\in \\mathbb {R}^3$ , and $s \\in \\mathbb {R}^{+}$ .","Here, $\\mathrm {SO}(3)$ , $\\mathrm {SE}(3)$ , and $\\mathrm {Sim}(3)$ are Lie groups, and $\\mathfrak {so}(3)$ , $\\mathfrak {se}(3)$ , and $\\mathfrak {sim}(3)$ are their corresponding Lie algebras.","A Lie group can be transformed into a Lie algebra using its exponential map, and the inverse transformation is defined by the inverse logarithm map.","Each Lie algebra is represented by a vector of its coefficients.","For example, $\\mathfrak {sim}(3)$ is represented as the seven-vector $\\xi = (\\omega _{1},\\omega _{2},\\omega _{3}, \\sigma , \\nu _{1}, \\nu _{2}, \\nu _{3})^{\\mathrm {T}} = (\\omega , \\sigma , \\nu )^{\\mathrm {T}}$ , and the exponential map $\\exp _{\\mathrm {Sim}(3)}$ and logarithm map $\\log _{\\mathrm {Sim}(3)}$ are defined as in Eq.","REF and Eq.","REF , respectively, where $\\mathbf {W}$ is a term similar to Rodriguez's formula.","Further details of $\\mathrm {Sim}(3)$ are given in [4].","$\\mathbf {G}=\\begin{bmatrix}\\mathbf {R} & \\mathbf {t}\\\\\\mathbf {0} & 1\\end{bmatrix}&\\qquad \\mathbf {S}=\\begin{bmatrix}s \\mathbf {R} & \\mathbf {t}\\\\\\mathbf {0} & 1\\end{bmatrix}$ $\\begin{split}\\exp _{\\mathrm {Sim}(3)}(\\xi )&=\\begin{bmatrix}e^{\\sigma } \\exp _{\\mathrm {SO}(3)}(\\omega ) & \\mathbf {W} \\nu \\\\\\mathbf {0} & 1\\end{bmatrix} = \\mathbf {S}\\end{split}$ $\\log _{\\mathrm {Sim}(3)}(\\mathbf {S}) = {\\exp _{\\mathrm {Sim}(3)}}^{-1}(\\mathbf {S}) = \\xi $ Proposed pose graph optimization.","In a general pose graph optimization approach [12], [17], camera poses and relative transformations between two camera poses are represented as elements of $\\mathrm {SE}(3)$ .","However, in our approach, 6-DoF camera poses and relative transformations are converted into 7-DoF camera poses, represented by elements of $\\mathrm {Sim}(3)$ .","This is achieved by leaving the rotation $R$ and translation $\\mathbf {t}$ of a camera pose unchanged, and setting the scale $s$ to 1.","The idea that camera poses and relative pose constraints can be handled in $\\mathrm {Sim}(3)$ was proposed by Strasdat et al.","[4], for dealing with the scale drift problem in monocular SLAM.","In this paper, we introduce 7-DoF pose graph optimization, which has previously only been used in the context of loop closure, to correct 3D reconstruction by utilizing sparse correspondences between two coordinate systems.","Our pose graph contains two kinds of nodes and three kinds of edges, as follows (see Fig.","REF ): Node $\\mathbf {S}_{n} \\in \\mathrm {Sim}(3)$ , where $n \\in C_{1} $ : the camera pose of the $n^{th}$ keyframe.","Node $\\mathbf {S}_{m} \\in \\mathrm {Sim}(3)$ , where $m \\in C_{2}$ : the camera pose of the $m^{th}$ geo-tagged image.","Edge $\\mathbf {e}_{1_{i, j}}$ , where $(i, j) \\in C_{3}$ : the relative pose constraint between the $i^{th}$ and $j^{th}$ keyframes.","(Eq.","REF ) Edge $\\textbf {e}_{2_{k,l}}$ , where $(k, l) \\in C_{4}$ : the relative pose constraint between the $k^{th}$ keyframe and the $l^{th}$ geo-tagged image.","(Eq. )","Edge $\\textbf {e}_{3_{m}}$ , where $m \\in C_{2}$ : the distance error between the position of the $m^{th}$ geo-tagged image and the world position $\\mathbf {y}_{m}$ of the corresponding geo-tag.","(Eq. )","$\\textbf {e}_{1_{i,j}} = \\log _{\\mathrm {Sim}(3)}(\\Delta \\mathbf {S}_{i,j} \\cdot \\mathbf {S}_{i} \\cdot \\mathbf {S}_{j}^{-1}) \\in \\mathbb {R}^7 \\\\\\textbf {e}_{2_{k,l}} = \\log _{\\mathrm {Sim}(3)}(\\Delta \\mathbf {S}_{k,l} \\cdot \\mathbf {S}_{k} \\cdot \\mathbf {S}_{l}^{-1}) \\in \\mathbb {R}^7\\\\\\textbf {e}_{3_{m}} = \\mathrm {trans}(\\mathbf {S}_{m}) - \\mathbf {y}_{m} \\in \\mathbb {R}^3$ where $\\mathrm {trans}(\\mathbf {S}) \\equiv (\\mathbf {S}_{1,4}, \\mathbf {S}_{2,4}, \\mathbf {S}_{3,4})^{\\mathrm {T}}$ .","Here, $N$ is the total number of keyframes, and $M$ is the total number of geo-tagged images that have correspondences to keyframes.","The set $C_{1}$ contains all the keyframes positioned between the two that have the newest and the third newest $C_{\\scalebox {0.7}{map-world}}$ .","The set $C_{2}$ contains the newest three of $C_{\\scalebox {0.7}{map-world}}$ .","The set $C_{3}$ contains the pairs of keyframes that observe the same 3D map point in 3D reconstruction, and $C_{4}$ contains pairs of keyframes and their corresponding geo-tagged images.","Finally, $\\Delta \\mathbf {S}_{i,j}$ is the converted $\\mathrm {Sim}(3)$ relative transformation between $\\mathbf {S}_{i}$ and $\\mathbf {S}_{j}$ , which is calculated before the optimization and remains fixed during the optimization.","Note that we newly introduced the nodes $\\mathbf {S}_{m}$ , edges $\\textbf {e}_{2_{k,l}}$ , and edges $\\textbf {e}_{3_{m}}$ to Strasdat's pose graph optimization.","Minimizing $\\textbf {e}_{1_{i, j}}$ and $\\textbf {e}_{2_{k,l}}$ suppresses changes in the relative transformations between camera poses, with the exception of gradual scale changes.","Minimizing $\\textbf {e}_{3_{m}}$ keeps the positions of the geo-tagged images close to the positions obtained from the associated geo-tags.","Our overall cost function $E_{PGO}$ is defined as follows: $\\begin{split}E_{PGO}(\\bigl \\lbrace \\mathbf {S}_{i} \\bigl \\rbrace _{i \\in C_{1} \\cup C_{2}})&= \\lambda _{1}\\sum _{(i,j) \\in C_{3}} \\textbf {e}_{1_{i,j}}^{\\mathrm {T}} \\textbf {e}_{1_{i,j}} \\\\&+\\lambda _{2}\\sum _{(k,l) \\in C_{4}} \\textbf {e}_{2_{k,l}}^{\\mathrm {T}} \\textbf {e}_{2_{k,l}} +\\lambda _{3}\\sum _{m \\in C_{2}} \\textbf {e}_{3_{m}}^{\\mathrm {T}} \\textbf {e}_{3_{m}}\\end{split}$ The corrected camera poses of keyframes $\\mathbf {S}_{n}$ and geo-tagged images $\\mathbf {S}_{m}$ are obtained by minimizing the cost function $E_{PGO}$ on Lie manifolds using the LM algorithm.","Following this optimization, we also reflect this correction in the 3D map points, as in [4]."],["Bundle Adjustment (BA) ","Following the pose graph optimization, we refine the 3D reconstruction by applying bundle adjustment with the constraints of the geo-tagged images.","Bundle adjustment is a classic method that jointly refines the 3D structure and camera poses (and camera intrinsic parameters) by minimizing the total re-projection errors.","Each re-projection error $\\mathbf {r}_{i,j}$ between the $i^{th}$ 3D point and $j^{th}$ camera is defined as: $\\mathbf {r}_{i,j} = \\mathbf {x_{i}} - \\pi (\\mathbf {R}_{j} \\mathbf {X}_{i} + \\mathbf {t}_{j})$ $\\pi (\\mathbf {p}) = [f_{x} \\dfrac{\\mathbf {p}_{x}}{\\mathbf {p}_{z}} + c_{x},~f_{y} \\dfrac{\\mathbf {p}_{y}}{\\mathbf {p}_{z}} + c_{y}]^{\\mathrm {T}}$ where $\\mathbf {X}_{i}$ is a 3D point and $\\mathbf {x}_{i}$ is the 2D observation of that 3D point; $\\mathbf {R}_{j}$ and $\\mathbf {t}_{j}$ are the rotation and translation of the $j^{th}$ camera pose, respectively; $\\mathbf {p} = [\\mathbf {p}_{x}, \\mathbf {p}_{y}, \\mathbf {p}_{z}]^{\\mathrm {T}}$ is a 3D point; $\\pi (\\cdot ): \\mathbb {R}^3 \\mapsto \\mathbb {R}^2$ is the projection function; $(f_{x}, f_{y})$ is the focal length; and $(c_{x}, c_{y})$ is the center of projection.","To incorporate global position information of geo-tagged images with bundle adjustment, we add a penalty term corresponding to the constraint for a geo-tagged image [16].","The total cost function with this constraint is given by: $\\begin{split}E_{BA}(\\bigl \\lbrace \\mathbf {X}_{i} \\bigl \\rbrace _{i \\in C_{5}}, \\bigl \\lbrace \\mathbf {T}_{j} \\bigl \\rbrace _{j \\in C_{1}}) = \\sum _{(i,j) \\in C_{\\scalebox {0.7}{map-kf}}} \\rho (\\mathbf {r}_{i,j}^{\\mathrm {T}} \\mathbf {r}_{i,j}) + \\lambda \\sum _{m \\in C_{3}} \\Vert \\mathbf {t}_{m} - \\mathbf {y}_{m} \\Vert ^{2}\\end{split}$ where $\\mathbf {T}$ is a camera pose of a keyframe represented as an element of $\\mathrm {SE}(3)$ , $\\rho $ is the Huber robust cost function, $C_{5}$ consists of map points observed by keyframes in $C_{1}$ , and $C_{1}$ and $C_{3}$ are defined in Sec.","REF .","Both the positions of 3D points and the camera poses of keyframes are optimized by minimizing the cost function on Lie manifolds using the LM algorithm.","This step can potentially correct the 3D map more precisely when it starts from a reasonably good 3D map."],["Experiments","In this section, we evaluate the proposed method on the Málaga dataset [24], using geo-tagged images obtained from Google Street View.","We also investigate the performance of pose graph optimization and bundle adjustment using the KITTI Dataset [25]."],["Implementation","We obtained geo-tagged images from Google Street View at intervals of 5 m within the area where the video was captured.","We set the cost function weights to $\\lambda _{1} = \\lambda _{2} = 1.0 \\times {10}^{5}$ and $\\lambda _{3} = 1.0$ , and we employed the g2o library [26] for the implementation of the pose graph optimization and bundle adjustment."],["Performance of the Proposed Method ","To verify the practical effectiveness of the proposed method, we evaluate it on the Málaga dataset using geo-tagged images obtained from Google Street View.","The Málaga Stereo and Laser Urban Data Set (the Málaga dataset) [24]—a large-scale video dataset that captures Street-View-usable areas—is employed in this experiment.","The Málaga dataset contains a driving video captured at a resolution of 1024 $\\times $ 768 at 20 fps in a Spanish urban area.","We extracted two video clips (video 1 and video 2) from the video, and used these for the evaluation.","The two video clips contain no loops, and their trajectories are over 1 km long.","All frames in the videos contain inaccurate GPS positions, which are sometimes confirmed to contain errors of more than 10 m. Because of the inaccuracies, we manually assigned the ground truth positions to some selected keyframes by referring to the videos, inaccurate GPS positions, and Google Street View 3D Map.","Fig.","REF presents an example of inaccurate GPS data and our assigned ground truth.","Because the ground truth positions are assigned by taking into account the lane from which the video was taken, the errors in the ground truth are considered to be within 2 m, and these errors are sufficiently small for this experiment.","Figure: The left figure shows an example of inaccurate GPS data (brown dots) and manually assigned ground truth positions (back crosses) on Google Maps.","Although we use Google Maps to visualize the results clearly, the shapes of roads are not sufficiently accurate.","Our ground truth positions are always assigned in the appropriate lane of the road, as seen in the satellite image (white crosses in the center figure).","The right figure shows an example of a video frame captured at the left of the two ground truth positions in the left figure.We evaluated the proposed method on the two videos by comparing the proposal and a baseline method that uses a similarity transformation (like a part of [15]).","For the baseline method, we apply the initialization (INIT: described in Sec.","REF ) without applying pose graph optimization and bundle adjustment.","We did not employ a global similarity transformation as a baseline because it cannot be applied until the end of the whole 3D reconstruction.","To evaluate the proposed method quantitatively, we considered the average (Ave) and standard deviation (SD) of 2D distances between the ground truth positions and corresponding keyframe positions in the UTM coordinate system (in meters).","Table REF presents the quantitative results, and Fig.","REF visualizes the results on Google Maps.","As is clearly shown in these results, the baseline results accumulate scale errors, resulting in large errors of over 50 m. This is because the trajectories of these videos are long (greater than 1 km) and contain no loops.","The proposed method sufficiently corrects scale drift, and significantly improves the 3D map by using geo-tagged images.","In (b) and (e) of the visualized results, the 3D map points corrected using the proposed method are projected onto Google Maps, and it is shown that the 3D map points are correctly aligned to the map.","To visualize all the correspondences between the 3D map coordinate system and the world coordinate system used in the proposal, we present the correspondences between the positions of geo-tagged images transformed by initialization and the positions of the corresponding geo-tags.","These correspondences are employed incrementally for the correction.","Table: Results of our proposed method on the Málaga dataset using Google Street View.Figure: Results of our proposed method visualized on Google Maps.","Top: results on video 1.","Bottom: results on video 2.","In (a) and (d), red and blue dots—which appear like lines—indicate the positions of keyframes corrected using a global similarity transformation (INIT) and our proposed method (Ours), respectively.","In (b) and (e), 3D map points corrected by our method are depicted by green dots.","(c) and (f) show all of the employed correspondences between the positions of geo-tagged images transformed using a global similarity transformation (green crosses) and the positions of the corresponding geo-tags (red pin icons).","The correspondences are applied incrementally for scale drift correction in our proposed method."],["Performance of PGO and BA","To investigate the performance of the pose graph optimization and the bundle adjustment in our proposed method, we evaluated the performance using different combinations of these when varying the interval of $C_{\\scalebox {0.7}{map-world}}$ .","Through the previous experiment, we found that the geo-tag location information of Google Street View and the manually assigned ground truths of the Málaga dataset occasionally had errors of several meters.","In this experiment, we control the interval of $C_{\\scalebox {0.7}{map-world}}$ , and use high-accuracy ground truths and geo-tags by using the KITTI dataset.","The odometry benchmark of KITTI dataset [25] contains 11 sequences of stereo videos and precise location information obtained from RTK-GPS/IMU, and unfortunately Google Street View is not available in Germany where this dataset was captured.","The experiment was conducted on two sequences, which include the largest and second-largest errors when applying ORB-SLAM: sequences 02 and 08 (containing 4660 and 4047 frames, respectively).","The left images of the stereo videos are used as input, and pairs of a right image and location information are identified as geo-tagged images.","All the location information associated with keyframes is used as the ground truth.","In this experiment with KITTI dataset, we can compare the performances of correction methods accurately for the following reasons: geo-tag information and ground truths are sufficiently precise (open sky localization errors of RTK-GPS/IMU $<$ 5 cm); and errors in geo-tagged image localization are sufficiently small, because keypoint matching between corresponding left and right images performs very well.","For the comparison, we present the results of the methods employing the initialization + the pose graph optimization (INIT+PGO), and initialization + the bundle adjustment (INIT+BA).","The correction method of INIT + BA is the same as [16], which is often used with a GPS location information.","Ours includes the initialization, the pose graph optimization and the bundle adjustment.","We changed the interval of geo-tagged images from 100 frames to 500 frames.","For an equal initialization, we set geo-tagged images in the interval of 50 frames from the first to the 200th frame.","Table: Results of the experiments on the KITTI dataset: sequences 02 and 08.","Values denote average 2D errors between ground truth positions and the corresponding keyframe positions [m].","Ours consists of INIT, PGO, and BA.Figure: Results of the experiment on the KITTI dataset when the interval of geo-tagged images is 300 frames.","Keyframe trajectories estimated by INIT+BA, INIT+PGO, and Ours are visualized.Fig.","REF visualizes the ground truth and keyframe trajectories estimated by INIT+BA, INIT+PGO, and Ours when the interval of geo-tagged images is 300 frames.","Table REF presents the quantitative results of the experiment, where the values represent the average 2D errors between ground truth positions and the corresponding keyframe positions in the UTM coordinate system (in meters).","Moreover, we report the errors of the global linear transformation on the sequence 02 and 08 by aligning the keyframe trajectory obtained by ORB-SLAM with ground truths through a similarity transformation: $20.15$ and $25.12$ , respectively.","The results show that bundle adjustment with geo-tag constraints, which is typically employed in the fusion of 3D reconstruction and GPS information [16], is not suitable when the interval of $C_{\\scalebox {0.7}{map-world}}$ is large.","It can also be seen that Ours (the combination of initialization, pose graph optimization, and bundle adjustment) often estimates the keyframe positions more accurately than any other method."],["Scale Drift Correction","To confirm that scale drift is corrected incrementally, we visualize the change in scale factor of the proposed method on the KITTI dataset sequences 02 and 08.","Fig.","REF shows that ORB-SLAM with the initialization accumulates scale errors, and our method can keep the scale factor around 1.","Figure: Change in scale factor of the proposed method on the KITTI dataset sequences 02 and 08."],["Conclusion","In this paper, we propose a novel framework for camera geo-localization that can correct scale drift by utilizing massive public repositories of geo-tagged images, such as those provided by Google Street View.","By virtue of the expansion of such repositories, this framework can be applied in many countries around the world, without requiring the user to observe an environment.","The framework integrates incremental SfM and a scale drift correction method utilizing geo-tagged images.","In the correction method, we first acquire sparse 6-DoF correspondences between the 3D map coordinate system and the world coordinate system by using geo-tagged images.","Then, we apply pose graph optimization over $\\mathrm {Sim}(3)$ constraints and bundle adjustment.","Our experiments on large-scale datasets show that the proposed framework sufficiently improves the 3D map by using geo-tagged images.","Note that our framework not only corrects the scale drift of 3D reconstruction, but also accurately geo-localizes a video.","Our results are no less accurate than those of mobile devices (between 5 and 8.5 m) that use a cellular network and low-cost GPS [27], and those using monocular video and road network maps [9] (8.1 m in the KITTI sequence 02 and 45 m in sequence 08).","This implies that geo-localization using geo-tagged images is sufficiently useful compared with methods using other GIS information."],["Acknowledgment","This work was partially supported by VTEC laboratories Inc."]],"string":"[\n [\n \"Scale Drift Correction of Camera Geo-Localization using Geo-Tagged\\n Images\"\n ],\n [\n \"Abstract Camera geo-localization from a monocular video is a fundamental task for video analysis and autonomous navigation.\",\n \"Although 3D reconstruction is a key technique to obtain camera poses, monocular 3D reconstruction in a large environment tends to result in the accumulation of errors in rotation, translation, and especially in scale: a problem known as scale drift.\",\n \"To overcome these errors, we propose a novel framework that integrates incremental structure from motion (SfM) and a scale drift correction method utilizing geo-tagged images, such as those provided by Google Street View.\",\n \"Our correction method begins by obtaining sparse 6-DoF correspondences between the reconstructed 3D map coordinate system and the world coordinate system, by using geo-tagged images.\",\n \"Then, it corrects scale drift by applying pose graph optimization over Sim(3) constraints and bundle adjustment.\",\n \"Experimental evaluations on large-scale datasets show that the proposed framework not only sufficiently corrects scale drift, but also achieves accurate geo-localization in a kilometer-scale environment.\"\n ],\n [\n \"Introduction\",\n \"Camera geo-localization from a monocular video in a kilometer-scale environment is a essential technology for AR, video analysis, and autonomous navigation.\",\n \"To achieve accurate geo-localization, 3D reconstruction from a video is a key technique.\",\n \"Incremental structure from motion (SfM) and visual simultaneous localization and mapping (visual SLAM) achieve large-scale 3D reconstructions by simultaneously localizing camera poses with six degrees-of-freedom (6-DoF) and reconstructing a 3D environment map [1], [2].\",\n \"Unlike for a stereo camera, an absolute scale of the real world cannot be derived using a single observation from a monocular camera.\",\n \"Although it is possible to estimate an environment's relative scale from a series of monocular observations, errors in the relative scale estimation accumulate over time, and this is referred to as scale drift [3], [4].\",\n \"For an accurate geo-localization not affected by scale drift ,prior information in a geographic information system (GIS) has been utilized in previous studies.\",\n \"For example, point clouds, 3D models, building footprints, and road maps have been proven to be efficient for correcting reconstructed 3D maps [5], [6], [7], [8], [9].\",\n \"However, these priors are only available in limited situations, e.g., in an area that is observed in advance, or in an environment consisting of simply-shaped buildings.\",\n \"Therefore, there is a good chance that other GIS information can help to extend the area in which a 3D map can be corrected.\",\n \"Hence, in this paper, motivated by the recent availability of massive public repositories of geo-tagged images taken all over the world, we propose a novel framework for correcting the scale drift of monocular 3D reconstruction by utilizing geo-tagged images, such as those in Google Street View [10], and achieve accurate camera geo-localization.\",\n \"Owing to the high coverage of Google Street View, our proposal is more scalable than those in previous studies.\",\n \"The proposed framework integrates incremental SfM and a scale drift correction method utilizing geo-tagged images.\",\n \"Our correction method begins by computing 6-DoF correspondences between the reconstructed 3D map coordinate system and the world coordinate system, by using geo-tagged images.\",\n \"Owing to significant differences in illumination, viewpoint, and the environment resulting from differences in time, it tends to be difficult to acquire correspondences between video frames and geo-tagged images (Fig.\",\n \"REF ).\",\n \"Therefore, a new correction method that can deal with the large scale drift of a 3D map using a limited number of correspondences is required.\",\n \"Bundle adjustment with constraints of global position information, which represents one of the most important correction methods, cannot be applied directly.\",\n \"This is because bundle adjustment tends to get stuck in a local minimum when starting from a 3D map including large errors [4].\",\n \"Hence, the proposed correction method consists of two coarse-to-fine steps: pose graph optimization over Sim(3) constraints, and bundle adjustment.\",\n \"In these steps, our key idea is to extend the pose graph optimization method proposed for the loop closure technique of monocular SLAM [4], such that it incorporates the correspondences between the 3D map coordinate system and the world coordinate system.\",\n \"This step corrects the large errors, and enables bundle adjustment to obtain precise results.\",\n \"After implementing this framework, we conducted experiments to evaluate the proposal.\",\n \"The contributions of this work are as follows.\",\n \"First, we propose a novel framework for camera geo-localization that can correct scale drift by utilizing geo-tagged images.\",\n \"Second, we extend the pose graph optimization approach to dealing with scale drift using a limited number of correspondences to geo-tags.\",\n \"Finally, we validate the effectiveness of the proposal through experimental evaluations on kilometer-scale datasets.\",\n \"Incremental SfM and visual SLAM are important approaches to reconstructing 3D maps from monocular videos.\",\n \"Klein et al.\",\n \"proposed PTAM for small AR workspaces [11].\",\n \"Mur-Artal et al.\",\n \"developed ORB-SLAM, which can reconstruct large-scale outdoor environments [2].\",\n \"For accurate 3D reconstruction, the loop closure technique has commonly been employed in recent SLAM approaches [4], [2].\",\n \"Loop closure deals with errors that accumulate between two camera poses that occur at the same location, i.e., when the camera trajectory forms a loop.\",\n \"Lu and Milios [12] formulated this technique as a pose graph optimization problem, and Strasdat et al.\",\n \"[4] extended pose graph optimization to deal with scale drift for monocular visual SLAM.\",\n \"It is certain that loop closure can significantly improve 3D maps, but this is only effective if a loop exists in the video.\"\n ],\n [\n \"Geo-registration of Reconstructions\",\n \"Correcting reconstructed 3D maps by using geo-referenced information has been regarded as a geo-registration problem.\",\n \"Kaminsky et al.\",\n \"proposed a method that aligns 3D reconstructions to 2D aerial images [13].\",\n \"Wendel et al.\",\n \"used an overhead digital surface model (DSM) for the geo-registration of 3D maps [14].\",\n \"Similar to our work, Wang et al.\",\n \"used Google Street View geo-tagged images and a Google Earth 3D model for the geo-registration of reconstructed 3D maps [15].\",\n \"However, because all these methods focus on estimating a best-fitting similarity transformation to geo-referenced information, they only correct the global scale in terms of 3D map correction.\",\n \"Methods for geo-registration using non-linear transformations have also been proposed.\",\n \"To integrate GPS information, Lhuillier et al.\",\n \"proposed incremental SfM using bundle adjustment with constraints from GPS [16], and Rehder et al.\",\n \"formulated a global pose estimation problem using stereo visual odometry, inertial measurements, and infrequent GPS information as a 6-DoF pose graph optimization problem [17].\",\n \"In terms of correcting camera poses using sparse global information, Rehder's method is similar to our pose graph optimization approach.\",\n \"However, our 7-DoF pose graph optimization differs in focusing on scale drift resulting from monocular 3D reconstruction, and in utilizing geo-tagged images.\",\n \"In addition to GPS information, various kinds of reference data have been used for the non-linear geo-registration or geo-localization of a video, such as point clouds [5], [6], 3D models [7], building footprints [8], and road maps [9].\",\n \"In this paper, we address a method that introduces geo-tagged images to the non-linear geo-registration of 3D maps.\"\n ],\n [\n \"Proposed Method\",\n \"Fig.\",\n \"REF provides a flowchart of the proposed framework, which is roughly divided into three parts.\",\n \"The first part is incremental SfM, and is described in Sec.\",\n \"REF .\",\n \"The second part computes 6-DoF correspondences between the 3D map coordinate system and the world coordinate system (as defined below), by making use of geo-tagged images (Sec.\",\n \"REF ).\",\n \"The third part then uses the correspondences to correct the scale drift of the 3D map, by applying pose graph optimization over Sim(3) constraints (Sec.\",\n \"REF ) and bundle adjustment (Sec.\",\n \"REF ) incrementally.\",\n \"The initialization of the scale drift correction method is described in Sec.\",\n \"REF .\"\n ],\n [\n \"World Coordinate System\",\n \"In this paper, the world coordinates are represented by 3D coordinates $(x, y, z)$ , where the $xz$ -plane corresponds to the Universal Transverse Mercator (UTM) coordinate system, which is an orthogonal coordinate system using meters, and $y$ corresponds to the height from the ground in meters.\",\n \"The UTM coordinates can be converted into latitude and longitude if necessary.\"\n ],\n [\n \"Incremental SfM \",\n \"As large-scale incremental SfM, we use ORB-SLAM [2] (with no real-time constraints).\",\n \"This is one of the best-performing monocular SLAM systems.\",\n \"Frames that are important for 3D reconstruction are selected as keyframes by ORB-SLAM.\",\n \"Every time a new keyframe is selected, our correction method is performed, and the 3D map reconstructed up to that point is corrected.\",\n \"In the 3D reconstruction, we identify 3D map points and their corresponding 2D keypoints in the keyframes (collectively denoted by $C_{\\\\scalebox {0.7}{map-kf}}$ ).\",\n \"Our proposed framework does not depend on a certain 3D reconstruction method, and can be applied to the other monocular 3D reconstruction methods, such as incremental SfM and feature-based visual SLAM.\"\n ],\n [\n \"Obtaining Correspondences between 3D Map and World Coordinates \",\n \"Here, we describe the second part of the proposed method, which uses geo-tagged images to compute a 6-DoF correspondence, $C_{\\\\scalebox {0.7}{map-world}}$ , between the 3D map and world coordinate system.\",\n \"For this purpose, we modify Agarwal's method [18] to integrate it into ORB-SLAM.\",\n \"This part consists of the following four steps: geo-tagged image collection, similar geo-tagged image retrieval, keypoint matching, and geo-tagged image localization.\",\n \"Geo-tagged Image Collection.\",\n \"Google Street View [10] is a browsable street-level GIS, which is one of the largest repositories of global geo-tagged images (i.e., images and their associated geo-tags).\",\n \"All images are high-resolution RGB panorama images, containing highly accurate world positions [19].\",\n \"We make use of this data by converting each panorama image into eight rectilinear images with the same field-of-view as our input video, with eight horizontal directions.\",\n \"Note that because each geo-tag has a position and rotation in the world coordinates, we can obtain the 6-DoF correspondences between the 3D map coordinate system and world coordinate system if geo-tagged images are localized in the 3D map coordinate system.\",\n \"Similar Geo-tagged Image Retrieval.\",\n \"When a new keyframe is selected, we retrieve the top-$k$ similar geo-tagged images.\",\n \"The retrieval system employs a bag-of-words approach based on SIFT descriptors [18].\",\n \"Keypoint Matching.\",\n \"Given the pairs of keyframes and retrieved geo-tagged images, we detect ORB keypoints [20] from the pairs and perform keypoint matching.\",\n \"Because the matching between video frames and Google Street View images tends to include many outliers [21], we use a virtual line descriptor (kVLD) [22], which can reject outliers by using a graph matching method even when inlier rate is around 10 Geo-tagged Image Localization.\",\n \"To compute $C_{\\\\scalebox {0.7}{map-world}}$ , we first compute 3D-to-2D correspondences $C_{\\\\scalebox {0.7}{map-geo}}$ between 3D map points and their corresponding 2D keypoints in geo-tagged images.\",\n \"In particular, we obtain $C_{\\\\scalebox {0.7}{map-geo}}$ by combining the 2D keypoint matches (computed in the previous step) with the correspondences $C_{\\\\scalebox {0.7}{map-kf}}$ between 3D map points and their corresponding 2D keypoints in keyframes (computed in 3D reconstruction).\",\n \"Then, we obtain the 6-DoF camera poses of geo-tagged images in the 3D map coordinate system by minimizing the re-projection errors of $C_{\\\\scalebox {0.7}{map-geo}}$ , using the LM algorithm.\",\n \"Finally, we obtain $C_{\\\\scalebox {0.7}{map-world}}$ by combining the camera poses of geo-tagged images and 6-DoF camera poses of the associated geo-tags.\",\n \"Figure: An example of the proposed pose graph optimization.\",\n \"This optimization maintains overall relative poses, except for gradual scale changes, and keeps camera poses of geo-tagged images close to the positions of the corresponding geo-tags.\"\n ],\n [\n \"Initialization (INIT) \",\n \"As the initialization, two kinds of linear transformations are performed on the 3D map, because the positions and scales of the 3D map coordinates and world coordinates are significantly different.\",\n \"Initialization is applied once, when the $i$ -th geo-tagged image is localized.\",\n \"We set $i = 4$ .\",\n \"Given the first to $i$ -th $C_{\\\\scalebox {0.7}{map-world}}$ , the first transformation assumes that all camera poses are approximately located in one plane, and rotates the 3D map to align that plane to the world $xz$ -plane.\",\n \"The best-fitting plane can be estimated by a principal component analysis.\",\n \"Next, we estimate the best-fitting transformation matrix given by Eq.\",\n \"REF , which transforms a point in the 3D map coordinate system $\\\\mathbf {p}_{\\\\scalebox {0.6}{SLAM},k}$ to be closer to a corresponding point in the world coordinate system $\\\\mathbf {p}_{world,k}$ ($\\\\mathbf {p}_{\\\\scalebox {0.6}{SLAM},k}$ and $\\\\mathbf {p}_{world,k}$ are denoted using a homogeneous representation): $\\\\mathbf {A} =\\\\begin{bmatrix}s * \\\\cos (\\\\theta ) & 0 & -s * \\\\sin (\\\\theta ) & a\\\\\\\\0 & s & 0 & 1 \\\\\\\\s * \\\\sin (\\\\theta ) & 0 & s * \\\\cos (\\\\theta ) & b\\\\\\\\0 & 0 & 0 & 1\\\\end{bmatrix}$ Using the first to $i$ -th $C_{\\\\scalebox {0.7}{map-world}}$ , we estimate the four matrix parameters $[a, b, s, \\\\theta ]$ by minimizing the following cost using RANSAC [23] and the Levenberg-Marquart (LM) algorithm: $E =\\\\sum _{k \\\\in 1,2...i} \\\\Vert \\\\mathbf {p}_{world,k} - \\\\mathbf {A} \\\\mathbf {p}_{\\\\scalebox {0.6}{SLAM},k} \\\\Vert ^{2}$ The camera poses of the geo-tagged images in $C_{\\\\scalebox {0.7}{map-world}}$ , keyframes, and 3D map point can then be transformed using the resulting matrix.\"\n ],\n [\n \"Pose Graph Optimization over Sim(3) Constraints (PGO)\",\n \"We correct the 3D map focusing on scale drift by using the newest three of $C_{\\\\scalebox {0.7}{map-world}}$ .\",\n \"This correction is performed every time a new $C_{\\\\scalebox {0.7}{map-world}}$ is found after initialization.\",\n \"Then, we propose a graph-based non-linear optimization method (pose graph optimization) on Lie manifolds, which simultaneously corrects the scale drift and aligns the 3D map with the world coordinates.\",\n \"Notation.\",\n \"A 3D rigid body transformation $\\\\mathbf {G} \\\\in \\\\mathrm {SE}(3)$ and a 3D similarity transformation $\\\\mathbf {S} \\\\in \\\\mathrm {Sim}(3)$ are defined by Eq.\",\n \"REF , where $\\\\mathbf {R} \\\\in \\\\mathrm {SO}(3)$ , $\\\\mathbf {t} \\\\in \\\\mathbb {R}^3$ , and $s \\\\in \\\\mathbb {R}^{+}$ .\",\n \"Here, $\\\\mathrm {SO}(3)$ , $\\\\mathrm {SE}(3)$ , and $\\\\mathrm {Sim}(3)$ are Lie groups, and $\\\\mathfrak {so}(3)$ , $\\\\mathfrak {se}(3)$ , and $\\\\mathfrak {sim}(3)$ are their corresponding Lie algebras.\",\n \"A Lie group can be transformed into a Lie algebra using its exponential map, and the inverse transformation is defined by the inverse logarithm map.\",\n \"Each Lie algebra is represented by a vector of its coefficients.\",\n \"For example, $\\\\mathfrak {sim}(3)$ is represented as the seven-vector $\\\\xi = (\\\\omega _{1},\\\\omega _{2},\\\\omega _{3}, \\\\sigma , \\\\nu _{1}, \\\\nu _{2}, \\\\nu _{3})^{\\\\mathrm {T}} = (\\\\omega , \\\\sigma , \\\\nu )^{\\\\mathrm {T}}$ , and the exponential map $\\\\exp _{\\\\mathrm {Sim}(3)}$ and logarithm map $\\\\log _{\\\\mathrm {Sim}(3)}$ are defined as in Eq.\",\n \"REF and Eq.\",\n \"REF , respectively, where $\\\\mathbf {W}$ is a term similar to Rodriguez's formula.\",\n \"Further details of $\\\\mathrm {Sim}(3)$ are given in [4].\",\n \"$\\\\mathbf {G}=\\\\begin{bmatrix}\\\\mathbf {R} & \\\\mathbf {t}\\\\\\\\\\\\mathbf {0} & 1\\\\end{bmatrix}&\\\\qquad \\\\mathbf {S}=\\\\begin{bmatrix}s \\\\mathbf {R} & \\\\mathbf {t}\\\\\\\\\\\\mathbf {0} & 1\\\\end{bmatrix}$ $\\\\begin{split}\\\\exp _{\\\\mathrm {Sim}(3)}(\\\\xi )&=\\\\begin{bmatrix}e^{\\\\sigma } \\\\exp _{\\\\mathrm {SO}(3)}(\\\\omega ) & \\\\mathbf {W} \\\\nu \\\\\\\\\\\\mathbf {0} & 1\\\\end{bmatrix} = \\\\mathbf {S}\\\\end{split}$ $\\\\log _{\\\\mathrm {Sim}(3)}(\\\\mathbf {S}) = {\\\\exp _{\\\\mathrm {Sim}(3)}}^{-1}(\\\\mathbf {S}) = \\\\xi $ Proposed pose graph optimization.\",\n \"In a general pose graph optimization approach [12], [17], camera poses and relative transformations between two camera poses are represented as elements of $\\\\mathrm {SE}(3)$ .\",\n \"However, in our approach, 6-DoF camera poses and relative transformations are converted into 7-DoF camera poses, represented by elements of $\\\\mathrm {Sim}(3)$ .\",\n \"This is achieved by leaving the rotation $R$ and translation $\\\\mathbf {t}$ of a camera pose unchanged, and setting the scale $s$ to 1.\",\n \"The idea that camera poses and relative pose constraints can be handled in $\\\\mathrm {Sim}(3)$ was proposed by Strasdat et al.\",\n \"[4], for dealing with the scale drift problem in monocular SLAM.\",\n \"In this paper, we introduce 7-DoF pose graph optimization, which has previously only been used in the context of loop closure, to correct 3D reconstruction by utilizing sparse correspondences between two coordinate systems.\",\n \"Our pose graph contains two kinds of nodes and three kinds of edges, as follows (see Fig.\",\n \"REF ): Node $\\\\mathbf {S}_{n} \\\\in \\\\mathrm {Sim}(3)$ , where $n \\\\in C_{1} $ : the camera pose of the $n^{th}$ keyframe.\",\n \"Node $\\\\mathbf {S}_{m} \\\\in \\\\mathrm {Sim}(3)$ , where $m \\\\in C_{2}$ : the camera pose of the $m^{th}$ geo-tagged image.\",\n \"Edge $\\\\mathbf {e}_{1_{i, j}}$ , where $(i, j) \\\\in C_{3}$ : the relative pose constraint between the $i^{th}$ and $j^{th}$ keyframes.\",\n \"(Eq.\",\n \"REF ) Edge $\\\\textbf {e}_{2_{k,l}}$ , where $(k, l) \\\\in C_{4}$ : the relative pose constraint between the $k^{th}$ keyframe and the $l^{th}$ geo-tagged image.\",\n \"(Eq. )\",\n \"Edge $\\\\textbf {e}_{3_{m}}$ , where $m \\\\in C_{2}$ : the distance error between the position of the $m^{th}$ geo-tagged image and the world position $\\\\mathbf {y}_{m}$ of the corresponding geo-tag.\",\n \"(Eq. )\",\n \"$\\\\textbf {e}_{1_{i,j}} = \\\\log _{\\\\mathrm {Sim}(3)}(\\\\Delta \\\\mathbf {S}_{i,j} \\\\cdot \\\\mathbf {S}_{i} \\\\cdot \\\\mathbf {S}_{j}^{-1}) \\\\in \\\\mathbb {R}^7 \\\\\\\\\\\\textbf {e}_{2_{k,l}} = \\\\log _{\\\\mathrm {Sim}(3)}(\\\\Delta \\\\mathbf {S}_{k,l} \\\\cdot \\\\mathbf {S}_{k} \\\\cdot \\\\mathbf {S}_{l}^{-1}) \\\\in \\\\mathbb {R}^7\\\\\\\\\\\\textbf {e}_{3_{m}} = \\\\mathrm {trans}(\\\\mathbf {S}_{m}) - \\\\mathbf {y}_{m} \\\\in \\\\mathbb {R}^3$ where $\\\\mathrm {trans}(\\\\mathbf {S}) \\\\equiv (\\\\mathbf {S}_{1,4}, \\\\mathbf {S}_{2,4}, \\\\mathbf {S}_{3,4})^{\\\\mathrm {T}}$ .\",\n \"Here, $N$ is the total number of keyframes, and $M$ is the total number of geo-tagged images that have correspondences to keyframes.\",\n \"The set $C_{1}$ contains all the keyframes positioned between the two that have the newest and the third newest $C_{\\\\scalebox {0.7}{map-world}}$ .\",\n \"The set $C_{2}$ contains the newest three of $C_{\\\\scalebox {0.7}{map-world}}$ .\",\n \"The set $C_{3}$ contains the pairs of keyframes that observe the same 3D map point in 3D reconstruction, and $C_{4}$ contains pairs of keyframes and their corresponding geo-tagged images.\",\n \"Finally, $\\\\Delta \\\\mathbf {S}_{i,j}$ is the converted $\\\\mathrm {Sim}(3)$ relative transformation between $\\\\mathbf {S}_{i}$ and $\\\\mathbf {S}_{j}$ , which is calculated before the optimization and remains fixed during the optimization.\",\n \"Note that we newly introduced the nodes $\\\\mathbf {S}_{m}$ , edges $\\\\textbf {e}_{2_{k,l}}$ , and edges $\\\\textbf {e}_{3_{m}}$ to Strasdat's pose graph optimization.\",\n \"Minimizing $\\\\textbf {e}_{1_{i, j}}$ and $\\\\textbf {e}_{2_{k,l}}$ suppresses changes in the relative transformations between camera poses, with the exception of gradual scale changes.\",\n \"Minimizing $\\\\textbf {e}_{3_{m}}$ keeps the positions of the geo-tagged images close to the positions obtained from the associated geo-tags.\",\n \"Our overall cost function $E_{PGO}$ is defined as follows: $\\\\begin{split}E_{PGO}(\\\\bigl \\\\lbrace \\\\mathbf {S}_{i} \\\\bigl \\\\rbrace _{i \\\\in C_{1} \\\\cup C_{2}})&= \\\\lambda _{1}\\\\sum _{(i,j) \\\\in C_{3}} \\\\textbf {e}_{1_{i,j}}^{\\\\mathrm {T}} \\\\textbf {e}_{1_{i,j}} \\\\\\\\&+\\\\lambda _{2}\\\\sum _{(k,l) \\\\in C_{4}} \\\\textbf {e}_{2_{k,l}}^{\\\\mathrm {T}} \\\\textbf {e}_{2_{k,l}} +\\\\lambda _{3}\\\\sum _{m \\\\in C_{2}} \\\\textbf {e}_{3_{m}}^{\\\\mathrm {T}} \\\\textbf {e}_{3_{m}}\\\\end{split}$ The corrected camera poses of keyframes $\\\\mathbf {S}_{n}$ and geo-tagged images $\\\\mathbf {S}_{m}$ are obtained by minimizing the cost function $E_{PGO}$ on Lie manifolds using the LM algorithm.\",\n \"Following this optimization, we also reflect this correction in the 3D map points, as in [4].\"\n ],\n [\n \"Bundle Adjustment (BA) \",\n \"Following the pose graph optimization, we refine the 3D reconstruction by applying bundle adjustment with the constraints of the geo-tagged images.\",\n \"Bundle adjustment is a classic method that jointly refines the 3D structure and camera poses (and camera intrinsic parameters) by minimizing the total re-projection errors.\",\n \"Each re-projection error $\\\\mathbf {r}_{i,j}$ between the $i^{th}$ 3D point and $j^{th}$ camera is defined as: $\\\\mathbf {r}_{i,j} = \\\\mathbf {x_{i}} - \\\\pi (\\\\mathbf {R}_{j} \\\\mathbf {X}_{i} + \\\\mathbf {t}_{j})$ $\\\\pi (\\\\mathbf {p}) = [f_{x} \\\\dfrac{\\\\mathbf {p}_{x}}{\\\\mathbf {p}_{z}} + c_{x},~f_{y} \\\\dfrac{\\\\mathbf {p}_{y}}{\\\\mathbf {p}_{z}} + c_{y}]^{\\\\mathrm {T}}$ where $\\\\mathbf {X}_{i}$ is a 3D point and $\\\\mathbf {x}_{i}$ is the 2D observation of that 3D point; $\\\\mathbf {R}_{j}$ and $\\\\mathbf {t}_{j}$ are the rotation and translation of the $j^{th}$ camera pose, respectively; $\\\\mathbf {p} = [\\\\mathbf {p}_{x}, \\\\mathbf {p}_{y}, \\\\mathbf {p}_{z}]^{\\\\mathrm {T}}$ is a 3D point; $\\\\pi (\\\\cdot ): \\\\mathbb {R}^3 \\\\mapsto \\\\mathbb {R}^2$ is the projection function; $(f_{x}, f_{y})$ is the focal length; and $(c_{x}, c_{y})$ is the center of projection.\",\n \"To incorporate global position information of geo-tagged images with bundle adjustment, we add a penalty term corresponding to the constraint for a geo-tagged image [16].\",\n \"The total cost function with this constraint is given by: $\\\\begin{split}E_{BA}(\\\\bigl \\\\lbrace \\\\mathbf {X}_{i} \\\\bigl \\\\rbrace _{i \\\\in C_{5}}, \\\\bigl \\\\lbrace \\\\mathbf {T}_{j} \\\\bigl \\\\rbrace _{j \\\\in C_{1}}) = \\\\sum _{(i,j) \\\\in C_{\\\\scalebox {0.7}{map-kf}}} \\\\rho (\\\\mathbf {r}_{i,j}^{\\\\mathrm {T}} \\\\mathbf {r}_{i,j}) + \\\\lambda \\\\sum _{m \\\\in C_{3}} \\\\Vert \\\\mathbf {t}_{m} - \\\\mathbf {y}_{m} \\\\Vert ^{2}\\\\end{split}$ where $\\\\mathbf {T}$ is a camera pose of a keyframe represented as an element of $\\\\mathrm {SE}(3)$ , $\\\\rho $ is the Huber robust cost function, $C_{5}$ consists of map points observed by keyframes in $C_{1}$ , and $C_{1}$ and $C_{3}$ are defined in Sec.\",\n \"REF .\",\n \"Both the positions of 3D points and the camera poses of keyframes are optimized by minimizing the cost function on Lie manifolds using the LM algorithm.\",\n \"This step can potentially correct the 3D map more precisely when it starts from a reasonably good 3D map.\"\n ],\n [\n \"Experiments\",\n \"In this section, we evaluate the proposed method on the Málaga dataset [24], using geo-tagged images obtained from Google Street View.\",\n \"We also investigate the performance of pose graph optimization and bundle adjustment using the KITTI Dataset [25].\"\n ],\n [\n \"Implementation\",\n \"We obtained geo-tagged images from Google Street View at intervals of 5 m within the area where the video was captured.\",\n \"We set the cost function weights to $\\\\lambda _{1} = \\\\lambda _{2} = 1.0 \\\\times {10}^{5}$ and $\\\\lambda _{3} = 1.0$ , and we employed the g2o library [26] for the implementation of the pose graph optimization and bundle adjustment.\"\n ],\n [\n \"Performance of the Proposed Method \",\n \"To verify the practical effectiveness of the proposed method, we evaluate it on the Málaga dataset using geo-tagged images obtained from Google Street View.\",\n \"The Málaga Stereo and Laser Urban Data Set (the Málaga dataset) [24]—a large-scale video dataset that captures Street-View-usable areas—is employed in this experiment.\",\n \"The Málaga dataset contains a driving video captured at a resolution of 1024 $\\\\times $ 768 at 20 fps in a Spanish urban area.\",\n \"We extracted two video clips (video 1 and video 2) from the video, and used these for the evaluation.\",\n \"The two video clips contain no loops, and their trajectories are over 1 km long.\",\n \"All frames in the videos contain inaccurate GPS positions, which are sometimes confirmed to contain errors of more than 10 m. Because of the inaccuracies, we manually assigned the ground truth positions to some selected keyframes by referring to the videos, inaccurate GPS positions, and Google Street View 3D Map.\",\n \"Fig.\",\n \"REF presents an example of inaccurate GPS data and our assigned ground truth.\",\n \"Because the ground truth positions are assigned by taking into account the lane from which the video was taken, the errors in the ground truth are considered to be within 2 m, and these errors are sufficiently small for this experiment.\",\n \"Figure: The left figure shows an example of inaccurate GPS data (brown dots) and manually assigned ground truth positions (back crosses) on Google Maps.\",\n \"Although we use Google Maps to visualize the results clearly, the shapes of roads are not sufficiently accurate.\",\n \"Our ground truth positions are always assigned in the appropriate lane of the road, as seen in the satellite image (white crosses in the center figure).\",\n \"The right figure shows an example of a video frame captured at the left of the two ground truth positions in the left figure.We evaluated the proposed method on the two videos by comparing the proposal and a baseline method that uses a similarity transformation (like a part of [15]).\",\n \"For the baseline method, we apply the initialization (INIT: described in Sec.\",\n \"REF ) without applying pose graph optimization and bundle adjustment.\",\n \"We did not employ a global similarity transformation as a baseline because it cannot be applied until the end of the whole 3D reconstruction.\",\n \"To evaluate the proposed method quantitatively, we considered the average (Ave) and standard deviation (SD) of 2D distances between the ground truth positions and corresponding keyframe positions in the UTM coordinate system (in meters).\",\n \"Table REF presents the quantitative results, and Fig.\",\n \"REF visualizes the results on Google Maps.\",\n \"As is clearly shown in these results, the baseline results accumulate scale errors, resulting in large errors of over 50 m. This is because the trajectories of these videos are long (greater than 1 km) and contain no loops.\",\n \"The proposed method sufficiently corrects scale drift, and significantly improves the 3D map by using geo-tagged images.\",\n \"In (b) and (e) of the visualized results, the 3D map points corrected using the proposed method are projected onto Google Maps, and it is shown that the 3D map points are correctly aligned to the map.\",\n \"To visualize all the correspondences between the 3D map coordinate system and the world coordinate system used in the proposal, we present the correspondences between the positions of geo-tagged images transformed by initialization and the positions of the corresponding geo-tags.\",\n \"These correspondences are employed incrementally for the correction.\",\n \"Table: Results of our proposed method on the Málaga dataset using Google Street View.Figure: Results of our proposed method visualized on Google Maps.\",\n \"Top: results on video 1.\",\n \"Bottom: results on video 2.\",\n \"In (a) and (d), red and blue dots—which appear like lines—indicate the positions of keyframes corrected using a global similarity transformation (INIT) and our proposed method (Ours), respectively.\",\n \"In (b) and (e), 3D map points corrected by our method are depicted by green dots.\",\n \"(c) and (f) show all of the employed correspondences between the positions of geo-tagged images transformed using a global similarity transformation (green crosses) and the positions of the corresponding geo-tags (red pin icons).\",\n \"The correspondences are applied incrementally for scale drift correction in our proposed method.\"\n ],\n [\n \"Performance of PGO and BA\",\n \"To investigate the performance of the pose graph optimization and the bundle adjustment in our proposed method, we evaluated the performance using different combinations of these when varying the interval of $C_{\\\\scalebox {0.7}{map-world}}$ .\",\n \"Through the previous experiment, we found that the geo-tag location information of Google Street View and the manually assigned ground truths of the Málaga dataset occasionally had errors of several meters.\",\n \"In this experiment, we control the interval of $C_{\\\\scalebox {0.7}{map-world}}$ , and use high-accuracy ground truths and geo-tags by using the KITTI dataset.\",\n \"The odometry benchmark of KITTI dataset [25] contains 11 sequences of stereo videos and precise location information obtained from RTK-GPS/IMU, and unfortunately Google Street View is not available in Germany where this dataset was captured.\",\n \"The experiment was conducted on two sequences, which include the largest and second-largest errors when applying ORB-SLAM: sequences 02 and 08 (containing 4660 and 4047 frames, respectively).\",\n \"The left images of the stereo videos are used as input, and pairs of a right image and location information are identified as geo-tagged images.\",\n \"All the location information associated with keyframes is used as the ground truth.\",\n \"In this experiment with KITTI dataset, we can compare the performances of correction methods accurately for the following reasons: geo-tag information and ground truths are sufficiently precise (open sky localization errors of RTK-GPS/IMU $<$ 5 cm); and errors in geo-tagged image localization are sufficiently small, because keypoint matching between corresponding left and right images performs very well.\",\n \"For the comparison, we present the results of the methods employing the initialization + the pose graph optimization (INIT+PGO), and initialization + the bundle adjustment (INIT+BA).\",\n \"The correction method of INIT + BA is the same as [16], which is often used with a GPS location information.\",\n \"Ours includes the initialization, the pose graph optimization and the bundle adjustment.\",\n \"We changed the interval of geo-tagged images from 100 frames to 500 frames.\",\n \"For an equal initialization, we set geo-tagged images in the interval of 50 frames from the first to the 200th frame.\",\n \"Table: Results of the experiments on the KITTI dataset: sequences 02 and 08.\",\n \"Values denote average 2D errors between ground truth positions and the corresponding keyframe positions [m].\",\n \"Ours consists of INIT, PGO, and BA.Figure: Results of the experiment on the KITTI dataset when the interval of geo-tagged images is 300 frames.\",\n \"Keyframe trajectories estimated by INIT+BA, INIT+PGO, and Ours are visualized.Fig.\",\n \"REF visualizes the ground truth and keyframe trajectories estimated by INIT+BA, INIT+PGO, and Ours when the interval of geo-tagged images is 300 frames.\",\n \"Table REF presents the quantitative results of the experiment, where the values represent the average 2D errors between ground truth positions and the corresponding keyframe positions in the UTM coordinate system (in meters).\",\n \"Moreover, we report the errors of the global linear transformation on the sequence 02 and 08 by aligning the keyframe trajectory obtained by ORB-SLAM with ground truths through a similarity transformation: $20.15$ and $25.12$ , respectively.\",\n \"The results show that bundle adjustment with geo-tag constraints, which is typically employed in the fusion of 3D reconstruction and GPS information [16], is not suitable when the interval of $C_{\\\\scalebox {0.7}{map-world}}$ is large.\",\n \"It can also be seen that Ours (the combination of initialization, pose graph optimization, and bundle adjustment) often estimates the keyframe positions more accurately than any other method.\"\n ],\n [\n \"Scale Drift Correction\",\n \"To confirm that scale drift is corrected incrementally, we visualize the change in scale factor of the proposed method on the KITTI dataset sequences 02 and 08.\",\n \"Fig.\",\n \"REF shows that ORB-SLAM with the initialization accumulates scale errors, and our method can keep the scale factor around 1.\",\n \"Figure: Change in scale factor of the proposed method on the KITTI dataset sequences 02 and 08.\"\n ],\n [\n \"Conclusion\",\n \"In this paper, we propose a novel framework for camera geo-localization that can correct scale drift by utilizing massive public repositories of geo-tagged images, such as those provided by Google Street View.\",\n \"By virtue of the expansion of such repositories, this framework can be applied in many countries around the world, without requiring the user to observe an environment.\",\n \"The framework integrates incremental SfM and a scale drift correction method utilizing geo-tagged images.\",\n \"In the correction method, we first acquire sparse 6-DoF correspondences between the 3D map coordinate system and the world coordinate system by using geo-tagged images.\",\n \"Then, we apply pose graph optimization over $\\\\mathrm {Sim}(3)$ constraints and bundle adjustment.\",\n \"Our experiments on large-scale datasets show that the proposed framework sufficiently improves the 3D map by using geo-tagged images.\",\n \"Note that our framework not only corrects the scale drift of 3D reconstruction, but also accurately geo-localizes a video.\",\n \"Our results are no less accurate than those of mobile devices (between 5 and 8.5 m) that use a cellular network and low-cost GPS [27], and those using monocular video and road network maps [9] (8.1 m in the KITTI sequence 02 and 45 m in sequence 08).\",\n \"This implies that geo-localization using geo-tagged images is sufficiently useful compared with methods using other GIS information.\"\n ],\n [\n \"Acknowledgment\",\n \"This work was partially supported by VTEC laboratories Inc.\"\n 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[
"Big Data Privacy Context: Literature Effects On Secure Informational\n Assets"
],
[
"Abstract This article's objective is the identification of research opportunities in the current big data privacy domain, evaluating literature effects on secure informational assets.",
"Until now, no study has analyzed such relation.",
"Its results can foster science, technologies and businesses.",
"To achieve these objectives, a big data privacy Systematic Literature Review (SLR) is performed on the main scientific peer reviewed journals in Scopus database.",
"Bibliometrics and text mining analysis complement the SLR.",
"This study provides support to big data privacy researchers on: most and least researched themes, research novelty, most cited works and authors, themes evolution through time and many others.",
"In addition, TOPSIS and VIKOR ranks were developed to evaluate literature effects versus informational assets indicators.",
"Secure Internet Servers (SIS) was chosen as decision criteria.",
"Results show that big data privacy literature is strongly focused on computational aspects.",
"However, individuals, societies, organizations and governments face a technological change that has just started to be investigated, with growing concerns on law and regulation aspects.",
"TOPSIS and VIKOR Ranks differed in several positions and the only consistent country between literature and SIS adoption is the United States.",
"Countries in the lowest ranking positions represent future research opportunities."
],
[
"Big Data and privacy studies promote market excitement due to its perceived potential in research, business economy and social activities [26], [41], [35].",
"Secure Internet Servers (SIS) are key data storage elements in big data's value chain [9], [51].",
"When individual's privacy enters the equation, frictional and also controversial effects show off like data misuse, user's overexposure, data breaches and many others [21], [36], [68], [6].",
"One of these effects lays on gaps between current big data privacy theory and its practical indicators on key informational assets adoption like SIS.",
"However, no study has evaluated big data privacy relation through Systematic Literature Review (SLR), bibliometrics, text mining approach and multi-criteria decision making methods (MCDM) rankings."
],
[
"This study main objective is the identification of research opportunities in the current big data privacy domain, giving a decision support alternative to researchers.",
"First, the SLR provides the basis for bibliometrics mapping.",
"Second, a theme and text mining analysis is performed over selected documents.",
"Finally, the ranking on paper-country-SIS is performed with Technique for Order Preferences by Similarity to an Ideal Solution (TOPSIS) and the VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR)."
],
[
"Big data privacy social sciences studies are focused on: users' concerns, awareness, self-management, self-disclosure, personality traits, privacy preservation.",
"Variables as gender, age, education, and their relations have been explored but not exhausted [32], [7].",
"The computational studies focus on encryption algorithms and informational security.",
"Questions like “how did they get my name?\"",
"[14] could be investigated with the support of research production versus informational assets indicators providing insights on big data privacy maturity level."
],
[
"Data is contextual, stored in servers, and desired by many [38], [45], [62].",
"SIS are one of the data storage and transmission alternatives in the big data value chain.",
"They are gateways to “personally identifiable information\", which makes them privacy violations targets just as safes in banks.",
"They also reflects the investment level in Privacy Enhancing Technologies (PETs), used to protect stored data from unauthorized access.",
"World Bank's SIS indicator from 2002 to 2015, in table REF , provides a incipient, but valid, approximation to countries' big data privacy concerns.",
"[54]."
],
[
"Servers' violations have consequences like: impersonation, record misuse, patents theft and many other types of frauds.",
"These effects impact economy and countries' sovereignty [50], [13].",
"The first privacy computational studies alert for encryptions and control at servers level [72], [1], [77] and on social impacts [47] highlighting the relevance of stronger privacy regulations."
],
[
"The 21st century computer was defined as “an invisible technology aiming current world's improvement, transparently enhancing human (inter)actions\" [75].",
"People are drown by sensors embedded in almost everywhere: cars, wearables and in-home appliances monitoring technologies.",
"Physical barriers are over [34] and monitoring efforts are transparent [28].",
"Ubiquitous computing became big data's necessary environmental condition [18], [40], [12]."
],
[
"In this study, section 2 presents the literature review that will provide SLR inputs for the search string.",
"Section 3 describes the research method.",
"Section 4 describes SLR results and TOPSIS and VIKOR ranks, eliciting research opportunities.",
"Finally, section 5 discusses conclusions, limitations and implications."
],
[
"Privacy is conceived through many lenses.",
"According to James F. Stephen, it is impossible to define privacy clearly.",
"But, its violations are easy to point out.",
"The intrusion of a stranger in someone's liberty be it by coercion, persuasion or even by the law are examples of privacy violations [64].",
"One of the first formal attempts to defy privacy is seen in Bentham's Inspection House.",
"The Panopticon was designed as a place where an individual was set under custody and fully exposed [5].",
"Its violations and dilemmas were first analyzed in 1890, when photography was the new technological phenomenon, and its use by journalists was ethically questioned.",
"Privacy was defined as the “right to be let alone\"[74]."
],
[
"Other definitions are: “claim of individuals, or groups, or institutions to determine for themselves when, how and to what extent information about them is communicated to others\"[76]; “the condition of not having undocumented personal knowledge about one possessed by others\";“A person's privacy is diminished exactly to the degree that others possess this kind of knowledge about him.",
"Documented information consists on information that is found in the public record or is publicly available\" [49]; “not simply the absence of information about us in the minds of others, rather is the control we have over information about ourselves\" [22], and “the ability of the individual to control the terms under which personal information is acquired and used\"[15]."
],
[
"Privacy is also defined in terms of protection from intrusion and information gathering, with individual control as choice, consent and correction [66].",
"Can also be stated as “a right to control access to places, locations, and personal information along with use and control rights to these goods\"[43].Other privacy perspectives are: combination of secrecy, anonymity, and solitude [25], and still physical access, decisional, physiological and informational elements[65], [27].",
".",
"Privacy is conceived as a value that, when present at some level, improves society relations in each and every term [76], [60] [31]; as a right that ought to be protected [74]; as a need to ensure liberty and autonomy[52], [53]."
],
[
"Privacy can also relate to culture.This raises some questionings on its importance among all people, on what is inherently private or merely social conventions [69].",
"Privacy definitions compilation revealed an overlapping among: (1) the right to be let alone; (2) limited access to the self; (3) secrecy; (4) control of personal information;(5) person-hood; and (6) intimacy [60].",
"Privacy seems to be about everything, and therefore it is a vague concept.",
"Still, a privacy taxonomy is based on informational processing, dissemination and violations[61].",
"In all cases, privacy faces limited protections by the law [74], [2]."
],
[
"Big data definitions have evolved in time and perspectives, from 3V's (Volume, Velocity and Variety), passing through 4V's (Veracity), 5V's(Variability), 6V's(Value) [24] definitions.",
"Big data analytics operates on statistical methods and semantics extraction processes from both structured and unstructured captured data."
],
[
"When it comes to privacy, third parties data sharing and accessibility have a growing potential as investigation field [35].",
"Theories such as Communication Privacy Management(CPM) [10] and Privacy Calculus(PC) [19] are examples of data sharing studies which unveil new research opportunities focused on user self-exposure aspects."
],
[
"Both analytics and big data's capacities are associated in many definitions: automation, search, aggregation and cross geo-referentiation of massive data volumes [37], [6].",
"Big data applications intersect economical, strategical, security and consumer welfare domains [8], [35], [39], [79], highlighting ethics as one of the most critical aspects [80], [57], [55], [31], [58], [67]."
],
[
"Privacy threats affect not only law and computer science but social, psychology, economics and media studies [14], [30], [79].",
"Data breaches generate economical effects that cannot be ignored [63]."
],
[
"Literature have documented data records selling and information misuse practices [15].",
"These effects are negative among individuals, ranging from value destruction to rights violations, potentially jeopardizing big data's environment.",
"People get surprised when they discovered about their data being used with unexpected purposes [14].",
"Complaints are mainly related to uninformed, not-consented and out of context data usage, generating data-context distortions and information abuse [46]."
],
[
"Big data privacy also seen as a surveillance dilemma [68], [37], [71].",
"Big data constitutes a technological evolution with exponential scale effects, environmentally, constituted by ubiquitous computing [40], [79] in a need to preserve individuals' privacy.",
"If such balance is not reached, this environment will be jeopardized."
],
[
"Privacy literature reviews are non-systematic, many of them cannot be reproduced[78], [59].",
"A few studies presented a systematic literature reviews related to security and privacy for big data, like [44].",
"Hopefully, big data research count on Systematic Literature Reviews (SLR)[23], [73], [9].",
"Just a few studies are focused on classical bibliometrics indicators.",
"However, there have been no big data privacy literature reviews providing research production analysis and practical effects evaluation on computational assets."
],
[
"This research is based on SLR method [33], added to a literature mapping exploring bibliometrics indicators.",
"This work's objectives are: identify literature gaps ; analyze research themes and its evolution trough time; evaluate research opportunities per country.",
"The first two objectives are covered by the SLR with support of bibliometrics mapping and text mining.",
"To evaluate research opportunities per country, TOPSIS and VIKOR were applied."
],
[
"This approach provides a precise, concise, technically reproducible and transparent evidence summary around a knowledge domain.",
"The literature mapping elicits themes' evolution through time.",
"Some of the mapped relations are: Most productive authors, countries, most related keywords, most cited authors, current research efforts and starting ones, themes' concentration areas and coupling relations."
],
[
"Scopus was chosen as database because of its availability, broadness and reliability [42].",
"A peer-reviewed research paper database, such as Scopus, provides a consistent platform to disseminate scientific investigation results, fostering research opportunities and trends."
],
[
"Chosen query parameters were “Title-Abstract-Keywords\"; limited to “Articles\" and “Conference papers\", written in English.",
"“Privacy\" and “Big data\" queries intersect directly constraining the result from 2002 to 2016.",
"Research string terms were chosen based on literature selection exposed in section 2.",
"Exclusion criteria were: inaccessible, non-authored, and/or redundant documents."
],
[
"Search strings were constructed from Privacy and big data definition terms.",
"First,privacy query revealed 83,657 publications (80,256 in English) while “Big Data\" returned 27,111 document results (26,076 in English).",
"Search strings such as “Priv*\" included private funds and other themes that are out of the research scope.",
"The intersection query retrieved 338 articles.",
"The final string, which consists on the conjunction of privacy and big data search strings, returned 262 documents:"
],
[
"TITLE-ABS-KEY ( ( privacy AND ( encrypt* OR crypt* OR auth* OR signature OR steganograph* OR anonymization ) AND ( protect* OR secre* OR confident* OR \"Polic*\" OR control OR \"self-management\" OR preserv* OR hid* ) ) ) ) AND ( TITLE-ABS-KEY ( ( ( \"Big Data\" OR \"Ubiquitous Computing\" OR \"Penetrate Computing\" ) AND ( \"Informed Consent\" OR disclos* OR expos* OR shar* OR distribut* OR dissemination OR \"Data Exchange\" OR \"Data Trade\" ) ) ) ) AND ( LIMIT-TO ( DOCTYPE , \"cp\" ) OR LIMIT-TO ( DOCTYPE , \"ar\" ) OR LIMIT-TO ( DOCTYPE , \"cr\" ) ) AND ( LIMIT-TO ( LANGUAGE , \"English\" ) ) AND ( LIMIT-TO ( SRCTYPE , \"p\" ) OR LIMIT-TO ( SRCTYPE , \"j\" ) )."
],
[
"In this sample, 13 documents had no authors and were excluded from the analysis.",
"The final publication database has 226 publications.",
"Articles with indexes 17, 199 and 249 had DOI errors.",
"Thus, were not retrievable from Scopus.",
"Articles 51 and 52 are redundant and so are 261 and 262.",
"Articles 30, 53, 106 and 107, 120, 139, 157, 198, 216, 233, 241, 255, 250, 254, 256 were not accessible either.",
"Article 147 had only abstract available, which was removed from corpus.",
"Article 253 was a talk in a book chapter, and only one page was retrievable."
],
[
"First, the documents went through bibliometrics mapping and analysis with the support of VOSViewer [70].",
"Second, cluster and content analysis where employed to evaluate the key terms and their quantitative relevance.",
"Hierarchical clusterization employed the k-means method.",
"On clusterization algorithms word stemming was included due to semantic aspects and theme's extraction objectives.",
"So, words such as “privacy\" versus “private\" were also treated as a single keyword and resolved using wildcards, e.g.",
"“priva*\"."
],
[
"Text mining and classification were based on article contents inspection through Tf - IDf matrix.",
"All algorithms were operationalized in R language [56] with support of“tm\" [20] and “bibliometrix\" [3] packages.",
"Terms relations, by similarities or distances, revealed concentration areas and relational gaps that were also confirmed by bibliometrics keyword analysis."
],
[
"TOPSIS and VIKOR methods were chosen to check between practical literature effects on SIS.",
"Both TOPSIS and VIKOR are widely considered as MCDM options[4].",
"Both methods could be applied to rank alternatives, propose a solution to the research question, having the decision criteria weights under decision-maker's discretion.",
"Data convexity is not mandatory."
],
[
"TOPSIS purely employs analytical methods based on applying Euclidean distance functions on normalized vectors of positive (outputs) and negative (inputs) criteria[11].",
"VIKOR determines the compromise ranking list, the trade-off solution, and the weight stability intervals of the obtained compromise solution [48]."
],
[
"TOPSIS and VIKOR focus on ranking alternatives selection in the presence of conflicting criteria.",
"VIKOR provides a maximum “group utility\" for the “majority\" and a minimum of an individual regret for the “opponent\", its ranking index is based on the particular measure of “closeness\" to the ideal solution.",
"TOPSIS rank has the “shortest distance\" to the ideal solution, which is the best level for all considered attributes, and the “farthest distance\" from the “negative-ideal\" solution, which is the one with worst attributed values.",
"So, TOPSIS returns two “reference\" points, but it does not consider the relative importance of the distances from these points."
],
[
"TOPSIS and VIKOR proposal analysis are applied to generate rankings presented here as a publication impact alternative measure for big data privacy research and also point countries' consistency between papers' impact and SIS installed capacity.",
"Ranks provide a supported decision mechanism to big data privacy researchers."
],
[
"The percentage of non-cited-publications (NCP) was assumed as the negative ideal while all other criteria are considered positive.",
"These ranks reveal a new approach on publication's relevance, differently than classical bibliometrics.",
"Rankings provide big data privacy research indicator."
],
[
"As presented on Figure REF , the privacy and big data intersection had its first increase in 2008, with significant growth from 2013 to 2014, experiencing a Annual Percentage Growth Rate (APGR) of 18.614% in all subject areas, having APGR of 16.644% in computer science and 24.573% in non-computer science areas."
],
[
"In the same period, privacy research had APGR of 21.762% in all subject areas, 23.721% in computer science alone and 17.035% in non-computer science domains.",
"Big data in all fields had 36.403%, 34.670% in computer science and 53.781% in non-computer science areas.",
"These rates reveals that other areas, different from computer science, turned their attention to big data.",
"Big data privacy have taken other subject areas attention since 2005.",
"Figure: Big Data Privacy Themes And Research Evolution Per Year"
],
[
"Most productive authors are Chinese, associated to American institutions.",
"Liu with 7 articles followed by Chen, Ma and Zhang with 5 articles.",
"Most cited authors are Agrawal and Srinkant with 7 citations, and Weiser with 5 citations.",
"Author's production shows Liu, and Zhang among most productive and cited authors, as seen in Figure REF .",
"Figure: Bibliometrics Research Production Indicators"
],
[
"Figure REF shows Keyword Co-occurrence graph.",
"Its edges reveal that“data mining\" cannot reach “access control\" directly.",
"All minimum paths connecting these nodes have privacy or security related keywords, i.e.",
"access control\" can only be achieved if security as privacy aspects are considered.",
"These relations indicate a new research challenge when non-digital aspects, such as “shoulder surfing\", and other off-line information gathering techniques are present.",
"Figure: Top 20 Keywords Co-Ocurrences"
],
[
"Most related keywords (Table REF ), returned Hadoop and MapReduce, not present in SLR search string.",
"This indicates a potential relation between storage and file access technologies to privacy and to big data's Volume dimension.",
"Table: Most Related Keywords"
],
[
"Figure REF shows that initial research was focused on “access control policy\", “schema\",“Proxy re-encryption\".",
"Access control policy defines which users or groups have permissions to access information.",
"The proxy-re-encryption is a encryption process where a third-parties alter the previous encrypted cyphertext.",
"These cryptosystems depends on “schemes\" and are relevant to protect user keys.",
"All of these terms are related to SIS."
],
[
"Research production evolved to “pervasive computing\", “authentication\", “privacy Protection\", “google\", “context aware resource management\".",
"Subjects like “authentication\" relates to “servers\" and also to “access control\", since the former depends on the later.",
"As a research theme deployment, “Privacy\", “anonymity\", “access control\" “homomorphic encryption\", “biometrics\" became more relevant."
],
[
"Keywords like “secure cloud computing\", “incremental conceptual cluster\" appeared as emerging research trends.",
"Terms such as“law and regulation\" are also under investigation and reasons are mainly because of ubiquitous computing effects.",
"Other topics like \"shoulder surfing\" also called the attention because it is not related to big data itself, but as a information gathering off-line practice.",
"Figure: All Keywords Co-Ocurrences by Average Association Strength per YearFigure: Top 100 Keywords Co-Ocurrences by Association Strength"
],
[
"Keyword co-occurrence by association strength Figure REF revealed research focus evolution from sensors, passing through “access control\" and finally reaching big data and privacy aspects.",
"Such evolution is in conformity with what is perceived in [40] when he states that privacy is never sufficient when computers are everywhere."
],
[
"Keyword co-occurrence by association strength also showed the themes' evolution from 2006 to 2016.",
"The first stage research revealed concentrated efforts on: “sensors\", “wireless networks\", “wireless sensor networks\", “context aware\", “semantics\", “computer privacy\" and “ubiquitous computing\" as the most relevant among all of them."
],
[
"Research evolved from “ubiquitous computing\" to “access control\" and “access control schemes\" followed by “scalability\",“location\" and “data storage systems\".",
"The third research stage gathers “data privacy\", “sensitive information\", “cryptography\", “anonymization\", reaching “big data\", “cloud computing\", “data handling\" and “data mining\"."
],
[
"Research themes started by sensors and networks, than evolved to scalability, storage and access issues.",
"All these aspects are ubiquitous computing pillars.",
"Later, works have focused on cryptography, privacy and security.",
"Finally reaching “big data\", “data mining\", distributed and cloud computing caught researchers' attention.",
"Privacy preservation aspects highlights the current research."
],
[
"FigureREF , revealed 8 clusters, with minimum of one document per country and minimum one citation per document, with average normalized citations method.",
"Canada, Germany and Saudi Arabia are leading countries in this metric.",
"India, China and United States are leaders in research production.",
"India leads in author's with most recent publications."
],
[
"Figure REF shows that bibliometrics coupling per documents, represented as (Author,Year), has a network of 100 references listed only once where 71 nodes and 10 clusters network.",
"Canny's “Collaborative Filtering with Privacy\" and Al-Muhtadi's “Routing through the mist: privacy preserving communication in ubiquitous computing environments\" as most referenced articles in privacy intersection with big data research domains.",
"These documents reinforce the SIS's role as a critical element in big data privacy research."
],
[
"Al-Muhtadi's work alerts to ubiquitous computing surveillance potential and proposes a “mist\" between routers.",
"Canny's work defines a server-based collaborative filtering systems to protect people from monopolies.",
"In this model, users control all of their log data.",
"Users can compute a public “aggregate”` of all of their data without exposing individual users’ data.",
"This model is based on homomorphic encryption with verification schemes distributed to all users.",
"This is one of the first works to be proposed for untrusted servers.",
"Both works propose privacy preservation through anonymization.",
"Fabian's work on multi-cloud storage and sharing architecture is a natural evolution from both.",
"This work focus on medical record anonymization shared among an cloud server array."
],
[
"Figure REF shows three clusters, all related to storage, encryption and information security.",
"In the first one, in red, literature mentions hadoop, mapreduce, privacy preserving and anonymization.",
"The green cluster relates to privacy preserving, privacy enhancement, anonymization, information classification.",
"The blue one represents theme's convergence relation.",
"Anonymization, privacy preservation are challenges in computer security.",
"Figure: Co-Citations - with a minimum of 3 co-citations"
],
[
"Table REF shows several country research indicators.",
"Countries research production, citations per paper and non-cited papers measure how relevant papers are and their impact.",
"It presents research production frame against its practical effects and investments such as SIS per country.",
"These differences reveal research opportunities on publication relevance, currently measured by citations.",
"When these results are inserted on SIS adoption, there is an approximation between literature and its practical effects."
],
[
"On citations per country the United States leads, followed by China, India and South Korea.",
"United States also leads in average Citations per paper (CPP) indicator, followed by Canada and Germany.",
"On SIS adoption rank results are different, with Switzerland leading with 3102 SIS, and China, India and Pakistan on the last positions."
],
[
"North American research is the most diversified and cited among all countries.",
"It covers privacy awareness and preserving, surveillance and its economical effects privacy meta-data protection, network privacy architectures.",
"Other topics are e-government policies, privacy usability challenges, self-disclosure, health, anonymization, geo-privacy, trust building and sensor networks."
],
[
"China has the second place in publications and in TOPSIS evaluation, and third in citations, with only 10 SIS.",
"However, China comes in second on publications.",
"Internet in China is strongly regulated [29], indicating that government controls its SIS.",
"This condition reflects some potential difficulties on big data privacy research.",
"Chinese publications relate to privacy preserving, trust building, authentication protocols, anonymity, encryption scalability and efficiency."
],
[
"South Korea is forth in publications; second in citations, VIKOR and third in TOPSIS with 2320 SIS.",
"Both Countries offer good research opportunities.",
"South Korea's high SIS indicator would be explained by the companies' contribution in GDP, such as Samsung.",
"Korean research production is one of the most diversified among all countries in the ranking.",
"It varies from authentication and encryption schemes to information policies and e-government.",
"User behavior aspects are rarely investigated."
],
[
"India is third in publications, forth in TOPSIS, last in VIKOR.",
"India has the 19th position in SIS indicator.",
"It is interesting to notice that many of Indian researchers are associated to American institutions, instead of Indian universities.",
"This contributes to United Stated leading position.",
"Big data privacy literature practical effects in India's are not as expressive as in other countries.",
"Its research is concentrated on privacy preserving, anonymization algorithms, cloud computing, Internet of Things, wireless networks, health and trust building.",
"User privacy awareness has not been investigated yet."
],
[
"Australian research is focused on privacy preserving, cloud, green and ubiquitous computing.",
"Surveillance, trust building and big data sharing integrate the research production.",
"This literature has close relation to the Canadian and Asian on privacy preservation."
],
[
"Canada's production is concentrated on privacy preserving through encryption algorithms and anonymization, access control and identity hiding schemes.",
"Italian publications are on trust building in pervasive computing, anonymous mining, privacy preserving.",
"Privacy law and regulations have not been investigated so far."
],
[
"United Kingdom's research focus varies from intrusion detection, Radio Frequency Identification secure based protocols, privacy systems for context aware and ubiquitous computing, wearables, transparency, ethics and health care.",
"Research has interesting aspects because it focus from ethics to digital attacks countermeasures like intrusion detection systems.",
"User's behavior and regulations were not explored in their production yet."
],
[
"French research is diversified and related to trust building, cryptography, biometrics, cloud and ubiquitous computing.",
"Literature is marked by transparency on social mining and sharing.",
"The trust building proposed by these works are in conformity with privacy by design fundamentals.",
"Trust related papers are the most cited among the French research production."
],
[
"German studies include wearables, willingness to share, big data privacy health care management, social mining, trust in ubiquitous computing.",
"These studies are mostly focused on building and ensuring trust.",
"Their fundamentals are aligned to privacy by design principles [16]."
],
[
"Japanese papers focus on privacy preserving aspects and challenges.",
"These studies may integrate a new framework that addresses changing business needs and fresh concerns over breaches of personal data.",
"Asian are marked by a strong privacy preservation bias.",
"Regulation development and privacy breaches effects are research opportunities in the country as well as in the Asian region."
],
[
"Singapore's research production is on privacy preserving.",
"Singapore enacted the Personal Data Protection Act (PDPA) in 2012, and in 2014 it became fully operational.",
"PDPA's effects on corporate governance and data protection practices are also research opportunities in the country and region."
],
[
"Malaysian publications are diversified from soft computing, privacy concerns on health records and intrusion detection.",
"Malaysia invests in high end engineering research development.",
"Privacy and data protection by design principles are also an opportunity with potential positive effects on country's strategy [17]."
],
[
"Saudi Arabia presents works focused essentially on cloud computing frameworks and ubiquitous computing.",
"Papers application domains are health and multimedia, with utility driven to policy making.",
"These studies take into consideration current frameworks security flaws on their analysis.",
"Ubiquitous computing plays a key role on Saudi Arabia diversification strategy: from oil to big data.",
"Thus, further big data privacy research is a need."
],
[
"Spanish papers vary from anonymization to PETs, passing through surveillance and authorization.",
"Interesting to notice that user privacy awareness was absent as theme.",
"This is also an aspect to be investigated as research opportunity."
],
[
"Norwegian works are related to anonymization and privacy preserving.",
"These works present as fundamentals some of the principles present in [16] like transparency and control.",
"Further developments can be derived from these works, contributing to privacy by design philosophy through the proposed frameworks."
],
[
"Pakistan's research production has a strong focus on protocols, frameworks and signature schemes to build trust.",
"This country has no privacy laws nor policies.",
"This affects country's social and economical development.",
"This is also a research opportunity."
],
[
"Switzerland appears 18th in publications, 4th in citations per paper, 8th in TOPSIS and 9th in VIKOR, and leads with 3102 SIS.",
"It is interesting to notice that: Switzerland is considered a financial center, it has a high SIS indicator, and also a high Citation per Paper indicator.",
"Swiss citations per paper is the forth highest in the rankings.",
"A new research opportunity is on measuring if and how financial organizations influence affects this relation."
],
[
"Brazilian research production focus on cloud computing security and privacy framework development.",
"These efforts complement the PETs works and also protocols.",
"Brazilian research did not focus on big data social aspects yet.",
"Big data privacy effects on human behavior may become a prolific opportunity, like anti-fraud detection and also government transparency to fight corruption."
],
[
"Netherlands is in 20th place regarding number of publications, but 5th in TOPSIS and 7th in VIKOR.",
"Difference between TOPSIS and VIKOR proposed rankings can be due to normalization method applied in these methods.",
"Netherlands' high relation between Citations per Papers and SIS indicator is a research opportunity on how this relation is established.",
"Netherlands' research production is mainly focused on information security.",
"There are still several applications to be covered, specially in ubiquitous computing and trust building."
],
[
"There is a discrepancy between number of publications and SIS on countries like Brazil, Spain, Malaysia, Saudi Arabia and United Kingdom.",
"These countries have a small research production, less than 10 publications in the analyzed period, and represent new big data privacy local relations to explore.",
"The small publication number indicates research venues, unexplored local opportunities.",
"Big data privacy questions, specially in law and regulation are still concealed.",
"Furthermore, both analysis help in identifying the publication efficiency and effects on SIS implementations and data breaches.",
"Table: Bibliometric Indicators Ranking Compared To TOPSIS And VIKORPub: Number of Publications; Cites: Number of Citations CPP: Citations per paper; Std.Dev: Citations Standard Deviation Max.Cites: Maximum Citations NCP: percentage of Non-Cited Papers T.s:TOPSIS Score; T.r:TOPSIS Ranking V.s:VIKOR Score; V.r:VIKOR Ranking Pub.Sis: Publications/SIS SIS: Secure Internet Servers in 2015 Source:http://data.worldbank.org/indicator/IT.NET.SECR.P6?view=chart retrieved in 13/12/2016."
],
[
"This study consists on a RSL, bibliometrics mapping and text mining analysis on big data privacy research evaluation.",
"TOPSIS and VIKOR MCDM Methods were employed to evaluate research practical effects, identifying new research rankings and opportunities."
],
[
"Privacy in big data is richly represented in the computer science domain, but non-computer science areas have started to investigate it.",
"This study identified “access control\", anonymization, authentication and PETs as recent concentration areas and also “ubiquitous computing\" as a necessary environmental condition to big data.",
"Non-computer science studies are concentrated on privacy perservation, trust building and privacy self-management.",
"Computer Science studies are focused on encryption, anonymization, storage, cloud computing and data mining."
],
[
"The TOPSIS and VIKOR Rankings revealed that United States leads on research impact and on the applying literature practical effects, which are represented by SIS.",
"SIS ranking per country was the chosen criteria because it is an worldwide accepted computational asset indicator available from the World Bank.",
"Another reason was SIS technical essence, which is secure data storage and transmission."
],
[
"Rankings revealed that countries like Brazil, Spain may represent new opportunities according to both rankings.",
"Saudi Arabia and United Kingdom, India, Japan and Pakistan according to VIKOR ranks.",
"It is interesting to notice that Asia and Europe have research bias, driven to ethical aspects and privacy preserving, while United States drives efforts towards encryption, storage, and technical frameworks.",
"Arabian countries investigate themes related to their economical growth.",
"Latin countries like Brazil have just started to research big data privacy.",
"Countries with a incipient research production is prolific in investigation opportunities because too little is known out their reality and matters."
],
[
"Results may vary if inclusion and exclusion criteria are changed.",
"Ranking may also change according to chosen MCDM method, criteria and weights adopted by decision makers.",
"Since there was no previous study relating research production and MCDM methods, this work adds a contribution on a structured process where researchers should focus their efforts.",
"SIS can provide non-exhaustive, but still relevant, measure of privacy concern per country.",
"It is massively present in computer science research production and represents a key factor in data protection and application services."
],
[
"The whole process had to be documented, including intermediary results to avoid inconsistency.",
"Data retrieval depended on Scopus' search engine technical structure.",
"Article's classification by publishers is a biased process and another recognized limitation.",
"Documents exclusively available in other bases such as Web of Science and DBLP are excluded from the sample.",
"Data extraction processed was limited to available articles and pdf conversion readability.",
"Since each publisher has its own text template, data cleaning and text mining processes had increased in complexity.",
"Text mining was performed on English-only article corpus.",
"Such limitations can be surpassed with the addition of other languages' dictionaries, improving semantic broadness."
],
[
"Future studies should target on big data privacy cultural aspects.",
"User behavior, laws and regulations, and visual privacy are interesting topics that appeared on this analysis.",
"Studies related to data breaches and practice versus theory evaluation on privacy governance would also be an interesting field to explore.",
"Too little is known about privacy law and regulation causes and effects on people, organizations and government.",
"These studies should be evaluated on their “intention to inform and evaluate\" big data privacy practical effects.",
"Would be desirable that these studies describe the big data privacy implications versus measurable protection practices, their benefits to policymakers, planners, researchers and citizens."
]
] | 1808.08537 |
[
[
"On the Stability of the Cauchy Problem of Timoshenko Thermoelastic\n Systems with Past History: Cattaneo and Fourier Law"
],
[
"Abstract In this paper, we investigate the decay properties of the thermoelastic Timoshenko system with past history in the whole space where the thermal effects are given by Cattaneo and Fourier laws.",
"We obtain that both systems, Timoshenko-Fourier and Timoshenko-Cattaneo, have the same rate of decay $(1 + t)^{-1/8}$ and satisfy the regularity-loss type property.",
"Moreover, for the Cattaneo case, we show that the decay rate depends of a new condition on the wave speed of propagation $\\chi_{0,\\tau}$.",
"This new condition has been recently introduced to study the asymptotic behavior in bounded domains, see for instance [5] and [27].",
"We found that this number also plays an important role in unbounded situation, affecting the decay rate of the solution."
],
[
"Introduction",
"In the literature concerning Timoshenko systems, the stability nature of solutions have a relationship with the wave speeds of propagation, essentially, when these speeds are equal or different.",
"In this context, denoting by $\\chi _0$ the difference of propagation's speed (see equation (REF )), we investigate how the decay rates of solutions of thermoelastic Timoshenko systems, depend of $\\chi _0$ , in particular when the thermal effects are given by Cattaneo and Fourier law, both with additional history terms.",
"As we can see from the references, the constant $\\chi _0$ will play an important role in the characterization of asymptotic behavior of solutions.",
"Recalling that Cattaneo's law is a hyperbolic heat model implying that the temperature has a finite speed of propagation, we can observe the impact of this heat model on the stability of Timoshenko systems.",
"Being more specific, recently in [5], [30], the authors proved that Cattaneo's law modifies the stability number $\\chi _0$ when the model is formulated in bounded domains.",
"Since this hyperbolic model generates dissipative thermal effects weaker than the parabolic Fourier model, the authors introduce a new stability number $\\chi _{0,\\tau }$ (see equation (REF )), which generalizes the previous one $\\chi _0$ in the sense that, when $\\tau = 0$ , Cattaneo's law turns into the Fourier law and the conditions over the new number $\\chi _{0,\\tau }$ are equivalent to the old stability number $\\chi _0$ .",
"In this line of research, our goal in this paper is to investigate the relation between $\\chi _{0,\\tau }$ and $\\chi _{0}$ with the decay rates of the solution of Timoshenko system posed in the whole real line.",
"In fact, we consider the Cauchy problem of the Timoshenko system with the heat conduction described by the Cattaneo and Fourier law with a history term, given by $\\left\\lbrace \\begin{tabular}{l l}\\rho _1\\varphi _{tt}-k \\left( \\varphi _x - \\psi \\right)_x = 0 & in (0,\\infty ) \\times \\mathbb {R}, \\\\\\rho _2\\psi _{tt}-b\\psi _{xx} +m\\displaystyle \\int _0^{\\infty }g(s)\\psi _{xx}(t-s,x)ds - k\\left( \\varphi _x - \\psi \\right)+\\delta \\theta _{x} =0 & in (0,\\infty ) \\times \\mathbb {R}, \\\\\\rho _3\\theta _{t} +q_x +\\delta \\psi _{xt}=0 & in (0,\\infty ) \\times \\mathbb {R},\\\\\\tau q_t+\\beta q +\\theta _x=0 & in (0,\\infty ) \\times \\mathbb {R}.\\end{tabular} \\right.$ where $b, k, m, \\delta , \\beta , \\rho _1, \\rho _2, \\rho _3, $ and $\\tau $ are positive constants; $\\psi (t,x)$ has been assigned on $(-\\infty ,0] \\times \\mathbb {R}$ , with initial data $\\begin{tabular}{l l l }\\varphi (0,\\cdot )=\\varphi _0(\\cdot ), & \\psi (s,\\cdot )=\\psi _0(s,\\cdot ), & \\theta (0,\\cdot )=\\theta _0(\\cdot ),\\quad \\forall s\\in (-\\infty ,0]\\\\\\varphi _t(0,\\cdot )=\\varphi _1(\\cdot ), & \\psi _t(0,\\cdot )=\\psi _1(\\cdot ), & q(0,\\cdot )=q_0(\\cdot ),\\end{tabular}$ with $\\phi , \\psi , \\theta $ and $q$ denoting the transversal displacement, the rotation angle of the beam, the temperature and the heat flow, respectively.",
"The integral term represents a history term with kernel $g$ satisfying the following hypotheses: $g(\\cdot )$ is a non negative function.",
"There exist positive constants $k_1$ and $k_2$ , such that, $-k_1 g(s) \\le g^{\\prime }(s) \\le -k_2 g(s).$ $a:=b-b_0>0$ , where $b_0=\\int _0^{\\infty }g(s)ds$ .",
"For this system, in bounded domains $(0,L)$ , see for instance [5]; the associated stability number is given by $\\chi _{0,\\tau }=\\left( \\tau -\\frac{\\rho _1}{\\rho _3 k}\\right)\\left( \\rho _2 -\\frac{b \\rho _1}{k}\\right) - \\frac{\\tau \\rho _1 \\delta ^2}{\\rho _3 k}.$ As mentioned in several references, condition (REF ) is more mathematical than physical, because is not realistic to assume that the propagation speeds associated to system (REF ) will satisfy condition (REF ).",
"Note that, when $\\tau =0$ , the system (REF ) has a thermal effect given by the Fourier law ($q= -\\widetilde{\\beta }\\theta _x$ ).",
"Indeed, formally the Timoshenko-Cattaneo system (REF ) is reduced to the following Timoshenko-Fourier system $\\left\\lbrace \\begin{tabular}{l l}\\rho _1\\varphi _{tt}-k \\left( \\varphi _x - \\psi \\right)_x = 0 & in (0,\\infty ) \\times \\mathbb {R}, \\\\\\rho _2\\psi _{tt}-b\\psi _{xx} +m\\displaystyle \\int _0^{\\infty }g(s)\\psi _{xx}(t-s,x)ds - k\\left( \\varphi _x - \\psi \\right)+\\delta \\theta _{x} =0 & in (0,\\infty ) \\times \\mathbb {R}, \\\\\\rho _3\\theta _{t} -\\widetilde{\\beta }\\theta _{xx} +\\delta \\psi _{xt}=0 & in (0,\\infty ) \\times \\mathbb {R},\\end{tabular} \\right.$ with initial data $\\begin{tabular}{l l l }\\varphi (0,\\cdot )=\\varphi _0(\\cdot ), & \\psi (s,\\cdot )=\\psi _0(s,\\cdot ), & \\theta (0,\\cdot )=\\theta _0(\\cdot ),\\quad \\forall s\\in (-\\infty ,0],\\\\\\varphi _t(0,\\cdot )=\\varphi _1(\\cdot ), & \\psi _t(0,\\cdot )=\\psi _1(\\cdot ), &\\end{tabular}$ and stability number given by $\\chi _{0}= \\rho _2 -\\frac{b \\rho _1}{k}.$ As we said previously, the main purpose of this article is to investigate the relationship between damping terms, the stability numbers $\\chi _{0,\\tau }, \\chi _0$ and their influence on the decay rate of solutions of systems (REF )-(REF ) and (REF )-(REF ), respectively.",
"Let us start by giving some references on Timoshenko systems.",
"The original Timoshenko system was first introduced by Timoshenko [34], [35] and describes the vibration of a beam taking into account the transversal displacement and the rotational angle of the beam filaments.",
"An initial boundary value problem associated to (REF ) and (REF ) under hypotheses $(H_1), (H_2), (H_3)$ was considered by Fernández Sare and Racke in [6].",
"They prove that the energy of the solution for the Timoshenko-Cattaneo model with history does not decay exponentially as $t\\rightarrow \\infty $ if $\\chi _0=0$ , while for the Timoshenko-Fourier system the energy decays exponentially if and only if $\\chi _0=0$ .",
"This result has been recently improved by Fatori et al in [5] where, for the Cattaneo's case, the exponential stability is obtained if and only if a new condition on the wave speeds of propagation is satisfied, i.e, the energy of solution of a IBVP Timoshenko-Cattaneo decay exponentially if and only if $\\chi _{0,\\tau }=0$ , where $\\chi _{0,\\tau }$ is given by (REF ).",
"Furthermore, if $\\chi _{0,\\tau }\\ne 0$ , they prove that the energy decays polynomially with rate $t^{-\\frac{1}{2}}$ .",
"There are many other references on Timoshenko systems in bounded domains with interesting results.",
"In particular, the problem of stability for Timoshenko-type systems in bounded domains has received much attention in the last years, and quite a number of results concerning uniform and asymptotic decay of energy have been established, see for instance [1], [2], [8], [9], [13], [16], [17], [18], [20], [21], [22], [23], [31] and references therein.",
"As a matter of fact, in bounded domains the proofs of stability results for Timoshenko systems are based on Poincaré inequalities and boundary conditions of the systems.",
"In this paper we are specially interested in the unbounded situation: when the system is formulated in the whole space $\\mathbb {R}$ .",
"This kind of problem has been considered in recent papers because it exhibits the regularity-loss phenomenon that usually appears in the pure Cauchy problems; see for instance [10], [11], [25], [36] and references therein.",
"Roughly speaking, a decay rate of solution is of regularity-loss type when it is obtained only by assuming some additional regularity on the initial conditions.",
"In this direction, we can mentioned some recent results on stabilization of Cauchy Timoshenko systems.",
"For instance, in Ide-Haramoto-Kawashima [11], Ide-Kawashima [12] and Racke-Houari [24], [25], the authors consider Timoshenko systems with normalized coefficients proving that the assumptions $b=1$ or $b\\ne 1$ play decisive roles in showing whether or not the decay estimates of solutions are of regularity-loss type.",
"For Cauchy problems associated to Timoshenko systems in thermoelasticity, as far as we know, the decay rate of solutions has been first studied by Said-Houari and Kasimov in [28], [29].",
"In particular, the authors proved in [29] that the Timoshenko system couppled with Cattaneo or Fourier law have the same rate of decay, this is, the solutions $W= (\\varphi _t,\\psi _t,a\\psi _x,\\varphi _x-\\psi ,\\theta )^T$ decay with the rate: $\\Vert \\partial _x^k W(t)\\Vert _{L^2} \\le C(1+t)^{-\\frac{1}{12}-\\frac{k}{6}}\\Vert W_0\\Vert _{L^1}+Ce ^{-ct}\\Vert \\partial _x^{k+l}W_0\\Vert _{L^2}$ for $a=1$ and $\\Vert \\partial _x^k W(t)\\Vert _{L^2} \\le C(1+t)^{-\\frac{1}{12}-\\frac{k}{6}}\\Vert W_0\\Vert _{L^1}+C(1+t)^{-\\frac{l}{2}}\\Vert \\partial _x^{k+l}W_0\\Vert _{L^2}$ for $a\\ne 1$ , $k=1,2,...,s-l$ .",
"In [28], considering an additional frictional damping $\\lambda \\psi _t(x,t)$ in the second equation, they obtain the same decay estimates with optimal rates $(1 + t)^{-\\frac{1}{4}-\\frac{k}{2}}$ .",
"More recently, Khader and Said-Houari in [14] studied the Cauchy problem for the Timoshenko system with the Gurtin-Pipkin thermal law: $\\left\\lbrace \\begin{tabular}{l l}\\varphi _{tt}-\\left( \\varphi _x - \\psi \\right)_x = 0 & in (0,\\infty ) \\times \\mathbb {R}, \\\\\\psi _{tt}-a^2\\psi _{xx} - \\left( \\varphi _x - \\psi \\right)+\\delta \\theta _{x} =0 & in (0,\\infty ) \\times \\mathbb {R}, \\\\\\theta _t-\\frac{1}{\\beta }\\displaystyle \\int _0^{\\infty }g(s)\\theta _{xx}(t-s,x)ds+ \\delta \\psi _{tx}=0 & in (0,\\infty ) \\times \\mathbb {R},\\end{tabular} \\right.$ where the memory kernel $g(s)$ is a convex summable function on $[0,\\infty )$ with total mass equal to 1.",
"They proved that the rate of decay depends of the number $\\alpha _g:=\\left((g(0))^{-1}\\beta -1\\right)\\left(1-a^2\\right)-(g(0))^{-1}\\delta ^2\\beta $ , which also controls the rate in bounded domains, see [4].",
"Additionally, we can cite some recent papers studying more general beam models posed in the whole real line, [7], [15], [26], [27], [32], [33].",
"The main goal of this paper is to investigate the decay rate of the Cauchy problems (REF ) and (REF ).",
"We prove that the same number $\\chi _{0,\\tau }$ defined in (REF ), which controls the behavior of the solution in bounded domains [5], also plays an important role in unbounded domains and affects the decay rate of solutions, see Theorems REF and REF below.",
"More precisely, we show that the respective solutions of Timoshenko-Cattaneo and Timoshenko-Fourier with history term, are of regularity-loss type and decay slowly with the rate $(1+ t)^{-\\frac{1}{8}}$ in the $L^2$ -norm.",
"Our proofs are based on some estimates for the Fourier image of the solution, Plancherel Theorem, as well as on a suitable linear combination of series associated to energy estimates.",
"Here, the decay rate $(1 + t)^{-\\frac{1}{8}}$ will be obtained by taking regular initial data $U_0 \\in H^s(\\mathbb {R})$ , for some $s\\in \\mathbb {R}$ .",
"This regularity loss comes from the analysis of the Fourier image, $\\hat{U}(\\xi , t)$ , of the solution $U(\\xi , t)$ .",
"In fact, we will obtain the estimate $\\left| \\hat{U}(\\xi , t)\\right|^2 \\le C e^{-\\beta \\rho (\\xi )t}\\left| \\hat{U}(\\xi , 0)\\right|^2,$ where $C, \\beta $ are positive constants and $\\rho (\\xi ) = {\\left\\lbrace \\begin{array}{ll}\\dfrac{\\xi ^4}{\\left(1+\\xi ^2\\right)^3}, & \\text{if $\\chi _{0,\\tau } = 0$ (resp, $\\chi _{0} = 0$)}, \\\\\\\\\\dfrac{\\xi ^4}{\\left(1+\\xi ^2\\right)^4}, &\\text{if $\\chi _{0,\\tau } \\ne 0$ (resp, $\\chi _{0} \\ne 0$)}.\\end{array}\\right.",
"}$ As we will see, the decay estimates for Timoshenko-Cattaneo and Timoshenko-Fourier, depend on the properties of the function $\\rho (\\xi )$ .",
"In fact, this function $\\rho (\\xi )$ behaves like $\\xi ^4$ in the low frequency region $(|\\xi | \\le 1)$ and like $\\xi ^{-2}$ near infinity, whenever $\\chi _{0,\\tau } = 0$ (resp, $\\chi _{0} = 0$ ).",
"Otherwise, if $\\chi _{0,\\tau } \\ne 0$ (resp, $\\chi _{0} \\ne 0$ ), the function $\\rho (\\xi )$ behaves also like $\\xi ^4$ in the low frequency region but like $\\xi ^{-4}$ near infinity, which means that the dissipation in the hight frequency region is very weak and produces the regularity loss phenomenon.",
"It is known that this regularity loss causes some difficulties in the nonlinear cases, see for example [11], [12] for more details.",
"This paper is organized as follows.",
"Section is dedicated to state the problems.",
"In section , we will present the energy method in the Fourier space and the construction of the Lyapunov functionals.",
"The main results, Theorems REF and REF are formulated in Section ."
],
[
"Setting of the Problem",
"In order to establish the decay rates of the Timoshenko systems (REF ) and (REF ), we have to transform the original problems to a first-order (in variable $t$ ) systems, defining new variables.",
"Then, we apply the energy method in the Fourier space to prove some point wise estimates which will help in the proof of the decay estimates."
],
[
"The Cattaneo Model",
"We consider the Timoshenko system with history and Cattaneo law.",
"Using the change of variable, introduced in [3], $\\eta (t,s,x):=\\psi (t,x)-\\psi (t-s,x),\\qquad (t,x) \\in (0,\\infty ) \\times \\mathbb {R}, \\quad s\\ge 0,$ the system (REF ), can be rewritten as $\\left\\lbrace \\begin{tabular}{l l}\\rho _1\\varphi _{tt}-k \\left( \\varphi _x - \\psi \\right)_x = 0 & in (0,\\infty ) \\times \\mathbb {R}, \\\\\\rho _2\\psi _{tt}-a\\psi _{xx} - m \\displaystyle \\int _0^{\\infty }g(s)\\eta _{xx}(s)ds - k\\left( \\varphi _x - \\psi \\right)+\\delta \\theta _{x} =0 & in (0,\\infty ) \\times \\mathbb {R}, \\\\\\rho _3\\theta _{t} +q_x +\\delta \\psi _{xt}=0 & in (0,\\infty ) \\times \\mathbb {R},\\\\\\tau q_t+\\beta q +\\theta _x=0 & in (0,\\infty ) \\times \\mathbb {R}, \\\\\\eta _t+\\eta _s -\\psi _t=0 & in (0,\\infty ) \\times \\mathbb {R}, \\\\\\eta (\\cdot ,0,\\cdot )=0 & in (0,\\infty ) \\times \\mathbb {R},\\end{tabular} \\right.$ where $a = a(b,g)$ is a positive constant given by $(H_3)$ and operator $T\\eta =-\\eta _s$ is the usual operator defined in problems with history terms, see for instance [22], [6] and references therein.",
"Here, the last two equations of system (REF ) are obtained differentiating equation (REF ).",
"We define also the initial data $\\begin{tabular}{l l l}\\varphi (0,\\cdot )=\\varphi _0(\\cdot ), & \\psi (0,\\cdot )=\\psi _0(\\cdot ), & \\theta (0,\\cdot )=\\theta _0(\\cdot ), \\\\\\varphi _t(0,\\cdot )=\\varphi _1(\\cdot ), & \\psi _t(0,\\cdot )=\\psi _1(\\cdot ), & q(0,\\cdot )=q_0(\\cdot ), \\\\\\end{tabular}\\\\\\eta (0,s,\\cdot )= \\psi _0(0,\\cdot )-\\psi _0(-s,\\cdot ).$ Furthermore, we can rewrite the system (REF ) by considering the following change of variables $u=\\varphi _t, \\qquad z=\\psi _x, \\qquad y=\\psi _t, \\qquad v=\\varphi _x-\\psi .$ Then, (REF ) takes the form $\\left\\lbrace \\begin{tabular}{l}v_t-u_x+y=0, \\\\\\rho _1u_t-kv_x=0, \\\\z_t-y_x =0, \\\\\\rho _2y_t -az_x-m\\displaystyle \\int _0^{\\infty }g(s)\\eta _{xx}(s)ds -kv+\\delta \\theta _x=0,\\\\\\rho _3\\theta _t+q_x+\\delta y_x=0,\\\\\\tau q_t +\\beta q+\\theta _x=0, \\\\\\eta _t+\\eta _s-y=0,\\\\\\eta (\\cdot ,0,\\cdot )=0\\end{tabular}\\right.$ Now, we define the solution of (REF ) by the vector $U$ , which is given by $U(t,x)=(v,u,z,y,\\theta ,q,\\eta )^{T}.$ The initial condition can be written as $U_0(x)=U(0,x)=(v_0,u_0,z_0,y_0,\\theta _0,q_0,\\eta _0)^{T},$ where $u_0=\\varphi _1, z_0=\\psi _{0,x}, y_0=\\psi _1$ , $v_0=\\varphi _{0,x}-\\psi _0$ and $\\eta _0=\\eta (0,s,\\cdot )$ which is defined, as usual, in the history space $L^2_g(\\mathbb {R}^+,H^1(\\mathbb {R}))$ , endowed with the norm $||\\eta ||^2:=\\int _{\\mathbb {R}}\\int _0^{\\infty }g(s)|\\eta _x(s)|^2dsdx.$"
],
[
"The Fourier Model",
"Similarly to Section REF , we consider the Timoshenko system (REF ) with history and the Fourier law, i.e, when $\\tau = 0$ .",
"Indeed, we can eliminate $q$ easily and obtain the following differential equation for $\\theta $ : $\\rho _3 \\theta _t - \\tilde{\\beta } \\theta _{xx} +\\delta \\psi _{xt} = 0,$ where $\\tilde{\\beta }=\\beta ^{-1}>0$ .",
"Then, introducing $\\eta $ as in the previous subsection, we have the differential equations $\\left\\lbrace \\begin{tabular}{l l}\\rho _1\\varphi _{tt}-k \\left( \\varphi _x - \\psi \\right)_x = 0 & in (0,\\infty ) \\times \\mathbb {R}, \\\\\\rho _2\\psi _{tt}-a\\psi _{xx} - m \\displaystyle \\int _0^{\\infty }g(s)\\eta _{xx}(s)ds - k\\left( \\varphi _x - \\psi \\right)+\\delta \\theta _{x} =0 & in (0,\\infty ) \\times \\mathbb {R}, \\\\\\rho _3\\theta _{t} -\\tilde{\\beta }\\theta _{xx} +\\delta \\psi _{xt}=0 & in (0,\\infty ) \\times \\mathbb {R},\\\\\\eta _t+\\eta _s -\\psi _t=0 & in (0,\\infty ) \\times \\mathbb {R}.",
"\\\\\\eta (\\cdot ,0,\\cdot )=0 & in (0,\\infty ) \\times \\mathbb {R}.\\end{tabular} \\right.$ with initial data $\\begin{tabular}{l l l }\\varphi (0,\\cdot )=\\varphi _0(\\cdot ), & \\psi (0,\\cdot )=\\psi _0(\\cdot ), & \\theta (0,\\cdot )=\\theta _0(\\cdot ), \\\\\\varphi _t(0,\\cdot )=\\varphi _1(\\cdot ), & \\psi _t(0,\\cdot )=\\psi _1(\\cdot ), & \\eta (0,s,\\cdot )= \\psi (0,\\cdot )-\\psi (-s,\\cdot ).\\end{tabular}$ As in the previous section, we can rewrite the system as a fist-order system, by defining the following variables $u=\\varphi _t, \\qquad z=\\psi _x, \\qquad y=\\psi _t, \\qquad v=\\varphi _x-\\psi .$ Then, (REF ) takes the form, $\\left\\lbrace \\begin{tabular}{l}v_t-u_x+y=0, \\\\\\rho _1u_t-kv_x=0, \\\\z_t-y_x =0, \\\\\\rho _2y_t -az_x-m\\displaystyle \\int _0^{\\infty }g(s)\\eta _{xx}(s)ds -kv+\\delta \\theta _x=0,\\\\\\rho _3\\theta _t-\\tilde{\\beta }\\theta _{xx}+\\delta y_x=0,\\\\\\eta _t+\\eta _s-y=0.\\end{tabular}\\right.$ We define the vector solution $V$ of the system (REF ), as $V(t,x)=(v,u,z,y,\\theta ,\\eta )^{T}.$ Thus, the initial condition can be written $V_0(x)=V(x,0)=(v_0,u_0,z_0,y_0,\\theta _0,\\eta _0)^{T},$ where $u_0=\\varphi _1, z_0=\\psi _{0,x}, y_0=\\psi _1$ , $v_0=\\varphi _{0,x}-\\psi _0$ and $\\eta _0=\\eta (0,s,\\cdot )$ defined in the history space given in the Cattaneo's version."
],
[
"The energy method in the frequency space",
"This section is devoted to showing the relationship between the rate of decay of solutions and the new condition (see [5]) $\\chi _{0,\\tau }=\\left( \\tau -\\frac{\\rho _1}{\\rho _3 k}\\right)\\left( \\rho _2 -\\frac{b \\rho _1}{k}\\right) - \\frac{\\tau \\rho _1 \\delta ^2}{\\rho _3 k}.$ For this reason, we will discuss two cases: the case where $\\chi _{0,\\tau }=0$ and the case where $\\chi _{0,\\tau }\\ne 0$ .",
"Moreover, for the Timoshenko-Fourier model, i,e., when $\\tau =0$ , we consider the usual wave speeds propagation given by $\\chi _{0}= \\rho _2 -\\frac{b \\rho _1}{k}.$ In each case, we use a delicate energy method to build appropriate Lyapunov functionals in the Fourier space."
],
[
"The Timoshenko-Cattaneo Law",
"We consider the Fourier image of the Timoshenko-Cattaneo model with history and we show that the heat damping induced by Cattaneo law and the past history are strong enough to stabilize the whole system.",
"Thus, taking Fourier Transform in (REF ), we obtain the following integro-differential system: $&\\hat{v}_t-i\\xi \\hat{u}+\\hat{y}=0, \\\\&\\rho _1\\hat{u}_t-ik\\xi \\hat{v}=0, \\\\&\\hat{z}_t-i\\xi \\hat{y} =0, \\\\&\\rho _2\\hat{y}_t -ia\\xi \\hat{z}+m\\xi ^2\\displaystyle \\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds -k\\hat{v}+i\\delta \\xi \\hat{\\theta }=0,\\\\&\\rho _3\\hat{\\theta }_t+i\\xi \\hat{q}+i\\delta \\xi \\hat{y}=0,\\\\&\\tau \\hat{q}_t +\\beta \\hat{q}+i\\xi \\hat{\\theta }=0, \\\\&\\hat{\\eta }_t+\\hat{\\eta }_s-\\hat{y}=0.",
"$ Here, the solution vector and initial data are given by $\\hat{U}(\\xi ,t)=(\\hat{v},\\hat{u},\\hat{z},\\hat{y},\\hat{\\theta },\\hat{q},\\hat{\\eta })^{T}$ and $\\hat{U}(\\xi ,0)=\\hat{U}_0(\\xi )$ , respectively.",
"The energy functional associated to the above system is defined as: $\\hat{E}\\left( \\xi ,t\\right) = \\rho _1|\\hat{u}|^{2}+\\rho _2|\\hat{y}|^{2}+\\rho _3 |\\hat{\\theta }|^{2}+k|\\hat{v} |^{2}+a|\\hat{z} |^{2}+\\tau |\\hat{q}|^{2} +m\\xi ^2\\int _{0}^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds.$ Lemma 3.1 The energy (REF ) satisfies the following estimate: $\\frac{d}{dt}\\hat{E}(\\xi ,t)\\le -2\\beta |\\hat{q}|^2 -k_1m\\xi ^2\\int _0^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds,$ where the constant $k_1>0$ is given by $(H_2)$ .",
"Multiplying (REF ) by $k\\overline{\\hat{v}}$ , () by $\\overline{\\hat{u}}$ , () by $a\\overline{\\hat{z}}$ , () by $\\overline{\\hat{y}}$ , () by $\\overline{\\hat{\\theta }}$ and () by $\\overline{\\hat{q}}$ , adding and taking real part, it follows that $\\frac{1}{2}\\frac{d}{dt}\\left\\lbrace \\rho _1|\\hat{u}|^{2}+\\rho _2|\\hat{y}|^{2}+\\rho _3 |\\hat{\\theta }|^{2}+k|\\hat{v} |^{2}+a|\\hat{z} |^{2}+\\tau |\\hat{q}|^{2}\\right\\rbrace =-\\beta |\\hat{q}|^2 -Re\\left(m \\xi ^2\\int _0^{\\infty } g(s)\\eta (s)\\overline{\\hat{y}}ds\\right).$ On the other hand, taking the conjugate of equation (), multiplying the resulting equation by $g(s)\\hat{\\eta }(t,s,x)$ and integrating with respect to $s$ , we obtain $Re\\left(\\int _0^{\\infty }g(s)\\hat{\\eta }(s)\\overline{\\hat{y}}ds\\right)=\\frac{1}{2}\\frac{d}{dt}\\int _0^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds +\\frac{1}{2}\\int _0^{\\infty }g(s)\\frac{d}{ds}|\\hat{\\eta }(s)|^2ds.$ Integrating by parts the last term in the left hand side of (REF ), we have $Re\\left( \\int _0^{\\infty }g(s)\\hat{\\eta }(s)\\overline{\\hat{y}}ds\\right)=\\frac{1}{2}\\frac{d}{dt}\\int _0^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds -\\frac{1}{2}\\int _0^{\\infty }g^{\\prime }(s)|\\hat{\\eta }(s)|^2ds.$ Plugging the above equation in (REF ), it follows that $\\frac{d}{dt}\\hat{E}(\\xi ,t)=-2\\beta |\\hat{q}|^2 +m\\xi ^2\\int _0^{\\infty }g^{\\prime }(s)|\\eta (s)|^2ds.$ Using $(H_2)$ , we obtain (REF ).",
"With this energy dissipation in hands, the following questions arise: Does $\\hat{E}(t) \\rightarrow 0$ as $t \\rightarrow \\infty $ ?",
"If it is the case, can we find the decay rate of $\\hat{E}(t)$ ?",
"The following Theorem provides a positive answer establishing the exponential decay of the integro-differential system (REF )-().",
"This result is a fundamental ingredient in the proof of our main results.",
"Theorem 3.2 Let $\\chi _{0,\\tau }=\\left( \\tau -\\frac{\\rho _1}{\\rho _3 k}\\right)\\left( \\rho _2 -\\frac{b \\rho _1}{k}\\right) - \\frac{\\tau \\rho _1 \\delta ^2}{\\rho _3 k}.$ Then, for any $t \\ge 0$ and $\\xi \\in \\mathbb {R}$ , the energy of system (REF )-() satisfies $\\hat{E}(\\xi ,t) \\le C e^{-\\lambda \\rho (\\xi )}\\hat{E}(0,\\xi ),$ where $C, \\lambda $ are positive constants and the function $\\rho (\\cdot )$ is given by $\\rho (\\xi ) = {\\left\\lbrace \\begin{array}{ll}\\dfrac{\\xi ^4}{(1+\\xi ^2)^{3}} & \\text{if $\\chi _{0,\\tau }=0$}, \\\\\\\\\\dfrac{\\xi ^4}{(1+\\xi ^2)^{4}} & \\text{if $\\chi _{0,\\tau }\\ne 0$}.\\end{array}\\right.",
"}$ Following the ideas contain in [29], we construct some functionals to capture the dissipation of all the components of the vector solution.",
"These functionals allow us to build an appropriate Lyapunov functional equivalent to the energy.",
"The proof of Theorem REF is based on the following lemmas: Lemma 3.3 Consider the functional $J_1(\\xi ,t)=-\\tau \\rho _2Re\\left(\\hat{v}\\overline{\\hat{y}}\\right)-\\frac{a\\tau \\rho _1}{k}Re\\left(\\hat{z}\\overline{\\hat{u}}\\right)+\\frac{\\delta \\rho _1}{k}\\left(\\tau + \\frac{1}{\\delta ^2}\\left(\\rho _2-\\frac{b\\rho _1}{k}+\\tau b_0\\rho _3\\right)\\right)Re(\\hat{\\theta }\\overline{\\hat{u}}) \\\\- \\frac{\\tau }{\\delta }\\left(\\rho _2-\\frac{b\\rho _1}{k}+\\tau b_0\\rho _3\\right) Re(\\hat{v}\\overline{\\hat{q}})$ Then, for any $\\varepsilon >0$ , $J_1$ satisfies $\\frac{d}{dt}J_1(\\xi ,t) +\\tau k(1-\\varepsilon )|\\hat{v}|^2 \\le \\tau \\rho _2|\\hat{y}|^2 +C(\\varepsilon )\\xi ^4\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2 \\\\+\\chi _{0,\\tau } Re\\left(i\\xi \\overline{\\hat{u}}\\hat{y}\\right)+\\frac{1}{\\delta }\\left( \\chi _{0,\\tau }+\\tau b_0\\rho _3\\left(\\tau -\\frac{\\rho _1}{\\rho _3k}\\right) \\right) Re(i\\xi \\hat{q}\\overline{\\hat{u}}) \\\\+\\frac{\\tau }{\\delta }\\left(\\rho _2-\\frac{b\\rho _1}{k} + \\tau b_0 \\rho _3\\right) Re(\\hat{y}\\overline{\\hat{q}}) +C(\\varepsilon )|\\hat{q}|^2.$ where $C(\\varepsilon )$ is a positive constant and $\\chi _{0,\\tau }$ is given by (REF ).",
"Multiplying (REF ) by $-\\rho _2 \\overline{\\hat{y}}$ and taking real part, we obtain $-\\rho _2Re\\left(\\hat{v}_t\\overline{\\hat{y}}\\right) + \\rho _2Re\\left(i\\xi \\hat{u}\\overline{\\hat{y}}\\right)-\\rho _2|\\hat{y}|^2=0.$ Multiplying () by $- \\overline{\\hat{v}}$ and taking real part, it follows that $-\\rho _2Re\\left(\\hat{y}_t\\overline{\\hat{v}}\\right) + aRe\\left(i\\xi \\hat{z}\\overline{\\hat{v}}\\right)+k|\\hat{v}|^2-Re\\left(m\\xi ^2\\overline{\\hat{v}}\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right)-\\delta Re\\left(i\\xi \\hat{\\theta }\\overline{\\hat{v}}\\right)=0.$ Adding the above identities, $-\\rho _2\\frac{d}{dt}Re\\left(\\hat{v}\\overline{\\hat{y}}\\right)+k|\\hat{v}|^2=\\rho _2|\\hat{y}|^2 - aRe\\left(i\\xi \\hat{z}\\overline{\\hat{v}}\\right)+Re\\left(m\\xi ^2\\overline{\\hat{v}}\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right) -\\rho _2Re\\left(i\\xi \\hat{u}\\overline{\\hat{y}}\\right)+\\delta Re\\left(i\\xi \\hat{\\theta }\\overline{\\hat{v}}\\right).$ On the other hand, multiplying () by $-\\frac{a}{k}\\overline{\\hat{z}}$ , () by $-\\frac{a\\rho _1}{k}\\overline{\\hat{u}}$ , adding the results and taking real part, it follows that $-\\frac{a\\rho _1}{k}\\frac{d}{dt}Re\\left(\\hat{z}\\overline{\\hat{u}}\\right) = -\\frac{a\\rho _1}{k}Re\\left(i\\xi \\hat{y}\\overline{\\hat{u}}\\right)-aRe\\left(i\\xi \\hat{v}\\overline{\\hat{z}}\\right).$ Moreover, multiplying () by $\\frac{\\delta }{k}\\overline{\\hat{\\theta }}$ and taking real part, $\\frac{\\delta \\rho _1}{k}Re(\\hat{u}_t\\overline{\\hat{\\theta }})-\\delta Re(i\\xi \\hat{v}\\overline{\\hat{\\theta }})=0.$ Next, multiplying () by $\\frac{\\delta \\rho _1}{\\rho _3 k}\\overline{\\hat{u}}$ and taking real part, $\\frac{\\delta \\rho _1}{k}Re(\\hat{\\theta }_t\\overline{\\hat{u}})+\\frac{\\delta \\rho _1}{\\rho _3 k}Re(i\\xi \\hat{q}\\overline{\\hat{u}})+\\frac{\\delta ^2\\rho _1}{\\rho _3 k}Re(i\\xi \\hat{y}\\overline{\\hat{u}})=0.$ Adding the above identities, we obtain $\\frac{\\delta \\rho _1}{k}\\frac{d}{dt}Re(\\hat{\\theta }\\overline{\\hat{u}})= -\\frac{\\delta \\rho _1}{\\rho _3 k}Re(i\\xi \\hat{q}\\overline{\\hat{u}})-\\frac{\\delta ^2\\rho _1}{\\rho _3 k}Re(i\\xi \\hat{y}\\overline{\\hat{u}})+\\delta Re(i\\xi \\hat{v}\\overline{\\hat{\\theta }}).$ Furthermore, multiplying () by $-\\overline{\\hat{v}}$ and taking real part, $-\\tau Re(\\hat{q}_t\\overline{\\hat{v}})-\\beta Re(\\hat{q}\\overline{\\hat{v}})-Re(i\\xi \\hat{\\theta }\\overline{\\hat{v}})=0.$ Multiplying, (REF ) by $-\\tau \\overline{\\hat{q}}$ and taking real part, $-\\tau Re(\\hat{v}_t\\overline{\\hat{q}})+\\tau Re(i\\xi \\hat{u}\\overline{\\hat{q}})-\\tau Re(\\hat{y}\\overline{\\hat{q}})=0.$ Adding the above identities, it follows that $- \\tau \\frac{d}{dt} Re(\\hat{v}\\overline{\\hat{q}})=-\\tau Re(i\\xi \\hat{u}\\overline{\\hat{q}})+\\tau Re(\\hat{y}\\overline{\\hat{q}})+\\beta Re(\\hat{q}\\overline{\\hat{v}})+Re(i\\xi \\hat{\\theta }\\overline{\\hat{v}}).$ Now, computing (REF )+(REF )+(REF ), we have $\\frac{d}{dt}\\left\\lbrace -\\rho _2Re\\left(\\hat{v}\\overline{\\hat{y}}\\right)-\\frac{a\\rho _1}{k}Re\\left(\\hat{z}\\overline{\\hat{u}}\\right) +\\frac{\\delta \\rho _1}{k}Re(\\hat{\\theta }\\overline{\\hat{u}})\\right\\rbrace +k|\\hat{v}|^2=\\rho _2|\\hat{y}|^2 +Re\\left(m\\xi ^2\\overline{\\hat{v}}\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right) \\\\+\\left(\\rho _2 - \\frac{a\\rho _1}{k}-\\frac{\\delta ^2\\rho _1}{\\rho _3 k}\\right) Re\\left(i\\xi \\overline{\\hat{u}}\\hat{y}\\right) - \\frac{\\delta \\rho _1}{\\rho _3 k}Re(i\\xi \\hat{q}\\overline{\\hat{u}}).$ Computing $\\tau (\\ref {e11}) + \\tau (\\ref {e12}) + \\underbrace{\\left(\\tau + \\frac{1}{\\delta ^2} \\left( \\rho _2-\\frac{b\\rho _1}{k} +\\tau b_0 \\rho _3\\right)\\right)}_{\\Gamma }(\\ref {n1})$ , we find that $\\frac{d}{dt}\\left\\lbrace -\\tau \\rho _2Re\\left(\\hat{v}\\overline{\\hat{y}}\\right)-\\frac{a\\tau \\rho _1}{k}Re\\left(\\hat{z}\\overline{\\hat{u}}\\right)+ \\frac{\\Gamma \\delta \\rho _1}{k}Re(\\hat{\\theta }\\overline{\\hat{u}})\\right\\rbrace +\\tau k|\\hat{v}|^2= \\tau \\rho _2|\\hat{y}|^2 + \\tau m Re\\left(\\xi ^2\\overline{\\hat{v}}\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right) \\\\+ \\left[\\tau \\left(\\rho _2 - \\frac{a\\rho _1}{k}\\right)-\\frac{\\delta ^2\\rho _1}{\\rho _3 k} \\Gamma \\right] Re\\left(i\\xi \\overline{\\hat{u}}\\hat{y}\\right)+\\tau \\delta Re(i\\xi \\hat{\\theta }\\overline{\\hat{v}})- \\frac{\\delta \\rho _1}{\\rho _3 k} \\Gamma Re(i\\xi \\hat{q}\\overline{\\hat{u}}) + \\Gamma \\delta Re(i\\xi \\overline{\\hat{\\theta }}\\hat{v}).$ Hence, $\\frac{d}{dt}\\left\\lbrace -\\tau \\rho _2Re\\left(\\hat{v}\\overline{\\hat{y}}\\right)-\\frac{a\\tau \\rho _1}{k}Re\\left(\\hat{z}\\overline{\\hat{u}}\\right)+ \\frac{\\Gamma \\delta \\rho _1}{k}Re(\\hat{\\theta }\\overline{\\hat{u}})\\right\\rbrace +\\tau k|\\hat{v}|^2= \\tau \\rho _2|\\hat{y}|^2 + \\tau m Re\\left(\\xi ^2\\overline{\\hat{v}}\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right) \\\\+\\left[\\left(\\tau -\\frac{\\rho _1}{\\rho _3 k}\\right)\\left(\\rho _2 - \\frac{b\\rho _1}{k}\\right)-\\frac{\\delta ^2\\rho _1}{\\rho _3 k} \\tau \\right] Re\\left(i\\xi \\overline{\\hat{u}}\\hat{y}\\right)+\\delta \\left(\\tau -\\Gamma \\right) Re(i\\xi \\hat{\\theta }\\overline{\\hat{v}})- \\frac{\\delta \\rho _1}{\\rho _3 k}\\Gamma Re(i\\xi \\hat{q}\\overline{\\hat{u}}).$ Multiplying (REF ) by $\\Gamma _1=- \\delta (\\tau -\\Gamma )$ and adding the result to (REF ), it follows that $\\frac{d}{dt}\\left\\lbrace -\\tau \\rho _2Re\\left(\\hat{v}\\overline{\\hat{y}}\\right)-\\frac{a\\tau \\rho _1}{k}Re\\left(\\hat{z}\\overline{\\hat{u}}\\right) +\\frac{\\Gamma \\delta \\rho _1}{k}Re(\\hat{\\theta }\\overline{\\hat{u}}) - \\tau \\Gamma _1 Re(\\hat{v}\\overline{\\hat{q}})\\right\\rbrace +\\tau k|\\hat{v}|^2 \\\\= \\tau \\rho _2|\\hat{y}|^2 + \\tau m Re\\left(\\xi ^2\\overline{\\hat{v}}\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right)+\\left[\\left(\\tau -\\frac{\\rho _1}{\\rho _3 k}\\right)\\left(\\rho _2 - \\frac{a\\rho _1}{k}\\right)-\\frac{\\delta ^2\\rho _1}{\\rho _3 k} \\tau \\right] Re\\left(i\\xi \\overline{\\hat{u}}\\hat{y}\\right)\\\\+\\delta \\left(\\tau -\\Gamma \\right) Re(i\\xi \\hat{\\theta }\\overline{\\hat{v}})- \\frac{\\delta \\rho _1}{\\rho _3 k} \\Gamma Re(i\\xi \\hat{q}\\overline{\\hat{u}}) -\\Gamma _1\\tau Re(i\\xi \\hat{u}\\overline{\\hat{q}})+\\Gamma _1\\tau Re(\\hat{y}\\overline{\\hat{q}})\\\\+\\Gamma _1\\beta Re(\\hat{q}\\overline{\\hat{v}})+\\Gamma _1 Re(i\\xi \\hat{\\theta }\\overline{\\hat{v}}).$ Hence, $\\frac{d}{dt}J_1(\\xi ,t) +\\tau k|\\hat{v}|^2= \\tau \\rho _2|\\hat{y}|^2 + \\tau m Re\\left(\\xi ^2\\overline{\\hat{v}}\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right) \\\\+\\chi _{0,\\tau } Re\\left(i\\xi \\overline{\\hat{u}}\\hat{y}\\right)+\\left( -\\delta \\tau (\\tau -\\Gamma ) - \\frac{\\delta \\rho _1\\Gamma }{\\rho _3 k}\\right) Re(i\\xi \\hat{q}\\overline{\\hat{u}}) \\\\-\\delta \\tau (\\tau -\\Gamma ) Re(\\hat{y}\\overline{\\hat{q}}) -\\delta \\beta (\\tau -\\Gamma ) Re(\\hat{q}\\overline{\\hat{v}}).$ Applying Young inequality, (REF ) follows.",
"Lemma 3.4 Consider the functional $J_2(\\xi ,t)=\\rho _1Re\\left(i\\xi \\hat{v}\\overline{\\hat{u}}\\right)+ \\rho _2Re\\left(i\\xi \\hat{y}\\overline{\\hat{z}}\\right)+\\delta \\tau Re(i\\xi \\hat{z}\\overline{\\hat{q}})$ For any $\\varepsilon >0$ , the estimate $\\frac{d}{dt}J_2(\\xi ,t) +\\rho _1(1-\\varepsilon )\\xi ^2|\\hat{u}|^2+a(1-\\varepsilon )\\xi ^2|\\hat{z}|^2\\le C(\\varepsilon )(1+\\xi ^2)|\\hat{v}|^2 +C(\\varepsilon )(1+\\xi ^2)|\\hat{y}|^2 \\\\+C(\\varepsilon )(1+\\xi ^2)|\\hat{q}|^2 +C(\\varepsilon )\\xi ^4\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2$ is satisfied.",
"Multiplying (REF ) by $i\\rho _1\\xi \\overline{\\hat{u}}$ and taking real part, $\\rho _1Re\\left(i\\xi \\hat{v}_t\\overline{\\hat{u}}\\right) +\\rho _1\\xi ^2|\\hat{u}|^2+\\rho _1Re\\left(i\\xi \\hat{y}\\overline{\\hat{u}}\\right)=0.$ Multiplying () by $-i\\xi \\overline{\\hat{v}}$ and taking real part, $-\\rho _1Re\\left(i\\xi \\hat{u}_t\\overline{\\hat{v}}\\right) -k\\xi ^2|\\hat{v}|^2=0.$ Adding the above identities, we obtain $\\rho _1\\frac{d}{dt}Re\\left(i\\xi \\hat{v}\\overline{\\hat{u}}\\right)+\\rho _1\\xi ^2|\\hat{u}|^2=k\\xi ^2|\\hat{v}|^2 -\\rho _1Re\\left(i\\xi \\hat{y}\\overline{\\hat{u}}\\right).$ Moreover, multiplying () by $-i\\rho _2\\xi \\overline{\\hat{y}}$ and taking real part, $-\\rho _2Re\\left(i\\xi \\hat{z}_t\\overline{\\hat{y}}\\right) -\\rho _2\\xi ^2|\\hat{y}|^2=0.$ Multiplying () by $i\\xi \\overline{\\hat{z}}$ and taking real part, $\\rho _2Re\\left(i\\xi \\hat{y}_t\\overline{\\hat{z}}\\right) +a\\xi ^2|\\hat{z}|^2 -kRe\\left(i\\xi \\hat{v}\\overline{\\hat{z}}\\right)+m Re\\left(i\\xi ^3\\overline{\\hat{z}}\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right) -\\delta Re\\left(\\xi ^2\\hat{\\theta }\\overline{\\hat{z}}\\right)=0.$ Adding the above identities, $\\rho _2\\frac{d}{dt}Re\\left(i\\xi \\hat{y}\\overline{\\hat{z}}\\right) +a\\xi ^2|\\hat{z}|^2= \\rho _2\\xi ^2|\\hat{y}|^2+kRe\\left(i\\xi \\hat{v}\\overline{\\hat{z}}\\right)-m Re\\left(i\\xi ^3\\overline{\\hat{z}}\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right) +\\delta Re\\left(\\xi ^2\\hat{\\theta }\\overline{\\hat{z}}\\right).$ Multiplying () by $-i\\delta \\xi \\overline{\\hat{z}}$ and taking real part, $-\\delta \\tau Re(i\\xi \\hat{q}_t\\overline{\\hat{z}})-\\beta \\delta Re (i\\xi \\hat{q}\\overline{\\hat{z}})+\\delta Re(\\xi ^2\\hat{\\theta }\\overline{\\hat{z}})=0.$ Multiplying () by $i\\delta \\tau \\xi \\overline{\\hat{q}}$ and taking real part, $\\delta \\tau Re(i\\xi \\hat{z}_t\\overline{\\hat{q}})+\\delta \\tau Re(\\xi ^2\\hat{y}\\overline{\\hat{q}})=0.$ Adding the above identities, we obtain $\\delta \\tau \\frac{d}{dt} Re(i\\xi \\hat{z}\\overline{\\hat{q}}) = -\\delta \\tau Re(\\xi ^2\\hat{y}\\overline{\\hat{q}}) + \\beta \\delta Re (i\\xi \\hat{q}\\overline{\\hat{z}})-\\delta Re(\\xi ^2\\hat{\\theta }\\overline{\\hat{z}}).$ Computing (REF ) $+$ (REF ) $+$ (REF ), we find that $\\frac{d}{dt}\\left\\lbrace \\rho _1Re\\left(i\\xi \\hat{v}\\overline{\\hat{u}}\\right)+ \\rho _2Re\\left(i\\xi \\hat{y}\\overline{\\hat{z}}\\right)+\\delta \\tau Re(i\\xi \\hat{z}\\overline{\\hat{q}}) \\right\\rbrace +a\\xi ^2|\\hat{z}|^2+\\rho _1\\xi ^2|\\hat{u}|^2= \\rho _2\\xi ^2|\\hat{y}|^2+kRe\\left(i\\xi \\hat{v}\\overline{\\hat{z}}\\right) \\\\-m Re\\left(i\\xi ^3\\overline{\\hat{z}}\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right)-\\delta \\tau Re(\\xi ^2\\hat{y}\\overline{\\hat{q}}) + \\beta \\delta Re (i\\xi \\hat{q}\\overline{\\hat{z}})+ k\\xi ^2|\\hat{v}|^2 -\\rho _1Re\\left(i\\xi \\hat{y}\\overline{\\hat{u}}\\right).$ Hence, $\\frac{d}{dt}J_2(\\xi ,t)+a\\xi ^2|\\hat{z}|^2+\\rho _1\\xi ^2|\\hat{u}|^2 \\le \\rho _2\\xi ^2|\\hat{y}|^2+k|\\xi ||\\hat{v}||\\hat{z}| \\\\+m|\\xi |^3|\\hat{z}|\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|+\\delta \\tau \\xi ^2|\\hat{y}||\\hat{q}| + \\beta \\delta |\\xi ||\\hat{q}||\\hat{z}|+ k\\xi ^2|\\hat{v}|^2 +\\rho _1|\\xi ||\\hat{y}||\\hat{u}|$ applying Young's inequality, we obtain (REF ).",
"Lemma 3.5 Consider the functional $J_3(\\xi ,t)=-\\rho _2Re\\left( \\xi ^2 \\overline{\\hat{y}}\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right).$ Then, for any $\\varepsilon >0$ , the following estimate $\\frac{d}{dt}J_3(\\xi ,t) +\\rho _2b_0(1-\\varepsilon )\\xi ^2|\\hat{y}|^2 \\le C(\\varepsilon ) \\xi ^2 \\int _0^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds +a |\\xi |^3 |\\hat{z}|\\left| \\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right| \\\\+m\\xi ^4\\left| \\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2 +k\\xi ^2 |\\hat{v}| \\left| \\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right| +\\delta |\\xi |^3 |\\hat{\\theta }| \\left| \\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|$ holds.",
"Multiplying () by $-\\xi ^2g(s) \\overline{\\hat{\\eta }}$ and taking the integration with respect to $s$ for the real parts, it follows that $-\\rho _2 Re\\left(\\xi ^2 \\hat{y}_t\\int _0^{\\infty }g(s)\\overline{\\hat{\\eta }}(s)ds\\right) + a Re\\left(i \\xi ^3 \\hat{z}\\int _0^{\\infty }g(s)\\overline{\\hat{\\eta }}(s)ds\\right) \\\\-m Re\\left( \\xi ^4 \\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\int _0^{\\infty }g(s)\\overline{\\hat{\\eta }}(s)ds\\right) + k Re\\left( \\xi ^2 \\hat{v}\\int _0^{\\infty }g(s)\\overline{\\hat{\\eta }}(s)ds\\right) \\\\- \\delta Re\\left(i \\xi ^3 \\hat{\\theta }\\int _0^{\\infty }g(s)\\overline{\\hat{\\eta }}(s)ds\\right)=0.$ Multiplying () by $-\\rho _2\\xi ^2g(s) \\overline{\\hat{y}}$ and taking the integration with respect to $s$ for the real parts, it follows that $-\\rho _2 Re\\left(\\xi ^2 \\overline{\\hat{y}}\\int _0^{\\infty }g(s)\\hat{\\eta }_t(s)ds\\right) -\\rho _2 Re\\left(\\xi ^2 \\overline{\\hat{y}}\\int _0^{\\infty }g(s)\\hat{\\eta }_s(s)ds\\right)+\\rho _2\\xi ^2\\int _0^{\\infty }g(s)ds |\\hat{y}|^2=0.$ Adding the above identities, we obtain that $\\frac{d}{dt}K_3(\\xi ,t) +\\rho _2\\xi ^2b_0|\\hat{y}|^2 = \\rho _2Re\\left( \\xi ^2 \\overline{\\hat{y}}\\int _0^{\\infty }g(s)\\hat{\\eta }_s(s)ds\\right) -aRe\\left(i \\xi ^3 \\hat{z}\\int _0^{\\infty }g(s)\\overline{\\hat{\\eta }}(s)ds\\right) \\\\+m\\xi ^4\\left| \\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2 - k Re\\left( \\xi ^2 \\hat{v}\\int _0^{\\infty }g(s)\\overline{\\hat{\\eta }}(s)ds\\right) +\\delta Re\\left(i \\xi ^3 \\hat{\\theta }\\int _0^{\\infty }g(s)\\overline{\\hat{\\eta }}(s)ds\\right).$ Now, integrating by parts the first term on the right-hand side of the above equality, we get $\\frac{d}{dt}K_3(\\xi ,t) +\\rho _2\\xi ^2b_0|\\hat{y}|^2 = -\\rho _2Re\\left( \\xi ^2 \\overline{\\hat{y}}\\int _0^{\\infty }g^{\\prime }(s)\\hat{\\eta }(s)ds\\right) -aRe\\left(i \\xi ^3 \\hat{z}\\int _0^{\\infty }g(s)\\overline{\\hat{\\eta }}(s)ds\\right) \\\\+m\\xi ^4\\left| \\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2 - k Re\\left( \\xi ^2 \\hat{v}\\int _0^{\\infty }g(s)\\overline{\\hat{\\eta }}(s)ds\\right) +\\delta Re\\left(i \\xi ^3 \\hat{\\theta }\\int _0^{\\infty }g(s)\\overline{\\hat{\\eta }}(s)ds\\right).$ Hence, $\\frac{d}{dt}K_3(\\xi ,t) +\\rho _2\\xi ^2b_0|\\hat{y}|^2 \\le \\rho _2 \\xi ^2|\\hat{y}| \\left| \\int _0^{\\infty }g^{\\prime }(s)\\hat{\\eta }(s)ds\\right| +a |\\xi ^3| |\\hat{z}|\\left| \\int _0^{\\infty }g(s)\\overline{\\hat{\\eta }}(s)ds\\right| \\\\+m\\xi ^4\\left| \\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2 +k\\xi ^2 |\\hat{v}| \\left| \\int _0^{\\infty }g(s)\\overline{\\hat{\\eta }}(s)ds\\right| +\\delta |\\xi |^3 |\\hat{\\theta }| \\left| \\int _0^{\\infty }g(s)\\overline{\\hat{\\eta }}(s)ds\\right|.$ Young inequality yields $\\frac{d}{dt}K_3(\\xi ,t) +\\rho _2\\xi ^2b_0(1-\\varepsilon )|\\hat{y}|^2 \\le C(\\varepsilon ) \\xi ^2 \\int _0^{\\infty }g^{\\prime }(s)|\\hat{\\eta }(s)|^2ds +a |\\xi ^3| |\\hat{z}|\\left| \\int _0^{\\infty }g(s)\\overline{\\hat{\\eta }}(s)ds\\right| \\\\+m\\xi ^4\\left| \\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2 +k\\xi ^2 |\\hat{v}| \\left| \\int _0^{\\infty }g(s)\\overline{\\hat{\\eta }}(s)ds\\right| +\\delta |\\xi |^3 |\\hat{\\theta }| \\left| \\int _0^{\\infty }g(s)\\overline{\\hat{\\eta }}(s)ds\\right|.$ Using the hypothesis on $g^{\\prime }$ , the result follows.",
"Lemma 3.6 Consider the functional $J_4(\\xi ,t)=\\tau \\rho _3Re\\left(i\\xi \\hat{\\theta }\\overline{\\hat{q}}\\right).$ For any $\\varepsilon >0$ , the following estimate $\\frac{d}{dt}J_4(\\xi ,t) +\\rho _3(1 -\\varepsilon )\\xi ^2|\\hat{\\theta }|^2 \\le \\tau \\delta \\xi ^2|\\hat{y}||\\hat{q}|+C(\\varepsilon )(1+\\xi ^2)|\\hat{q}|^2$ holds.",
"Multiplying () by $i\\tau \\xi \\overline{\\hat{q}}$ and taking real part, $\\tau \\rho _3Re\\left(i\\xi \\hat{\\theta }_t\\overline{\\hat{q}}\\right)- \\tau \\xi ^2|\\hat{q}| ^2-\\tau \\delta Re\\left(\\xi ^2\\hat{y}\\overline{\\hat{q}}\\right)=0.$ Multiplying () by $-i\\rho _3\\xi \\overline{\\hat{\\theta }}$ and taking real part, $-\\tau \\rho _3Re\\left(i\\xi \\hat{q}_t\\overline{\\hat{\\theta }}\\right)-\\beta \\rho _3Re\\left(i\\xi \\hat{q}\\overline{\\hat{\\theta }}\\right) +\\rho _3\\xi ^2|\\hat{\\theta }| ^2=0.$ Adding up the above identities, it follows that $\\frac{d}{dt}J_4(\\xi ,t) +\\rho _3\\xi ^2|\\hat{\\theta }| ^2\\le \\tau \\xi ^2|\\hat{q}| ^2+\\tau \\delta \\xi ^2|\\hat{y}||\\hat{q}| +\\beta \\rho _3|\\xi ||\\hat{q}||\\hat{\\theta }|.$ Applying Young's inequality, the result follows.",
"[ Proof of Theorem REF] In order to make the proof clear, we will consider several cases: I.",
"Case $\\chi _{0,\\tau }=0$ :.",
"Consider the expressions $\\lambda _1\\xi ^2J_1(\\xi ,t), \\quad \\lambda _2\\dfrac{\\xi ^2}{1+\\xi ^2}J_2(\\xi ,t), \\quad \\lambda _3J_3(\\xi ,t)$ where $\\lambda _1$ , $\\lambda _2$ and $\\lambda _3$ are positive constants to be fixed later.",
"Thus, Lemmas REF and REF imply that $\\frac{d}{dt}\\left\\lbrace \\lambda _1\\xi ^2 J_1(\\xi ,t)+\\lambda _2\\dfrac{\\xi ^2}{1+\\xi ^2}J_2(\\xi ,t) \\right\\rbrace + k\\left[\\lambda _1\\tau (1-\\varepsilon )-C(\\varepsilon )\\lambda _2\\right]\\xi ^2|\\hat{v}|^2 +\\rho _1(1-2\\varepsilon )\\lambda _2\\dfrac{\\xi ^4}{1+\\xi ^2}|\\hat{u}|^2 \\\\+a(1-\\varepsilon )\\lambda _2\\dfrac{\\xi ^4}{1+\\xi ^2}|\\hat{z}|^2 \\\\\\le C(\\varepsilon ,\\lambda _1,\\lambda _2)\\xi ^2|\\hat{y}|^2 +C(\\varepsilon ,\\lambda _1,\\lambda _2)\\xi ^6\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2+C(\\varepsilon ,\\lambda _1,\\lambda _2)\\xi ^2(1+\\xi ^2)|\\hat{q}|^2.$ Furthermore, Applying Young's inequality in equation (REF ) of Lemma REF , it follows that $\\frac{d}{dt}\\lambda _3J_3(\\xi ,t) +\\rho _2\\lambda _3b_0 (1 -\\varepsilon )\\xi ^2|\\hat{y}|^2\\le C(\\varepsilon ,\\lambda _3)\\xi ^2\\int _0^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds+ a\\lambda _2\\varepsilon \\frac{\\xi ^4}{1+\\xi ^2}|\\hat{z}|^2 \\\\+ C(\\varepsilon ,\\lambda _2,\\lambda _3)\\xi ^2(1+\\xi ^2)\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2 +\\lambda _3m\\xi ^4\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2+k\\lambda _1\\tau \\varepsilon \\xi ^2|\\hat{v}|^2 \\\\+C(\\varepsilon ,\\lambda _1,\\lambda _3)\\xi ^2\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2+ \\lambda _3\\delta |\\xi |^3|\\hat{\\theta }|\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|.$ Thus, $\\frac{d}{dt}\\lambda _3J_3(\\xi ,t) +\\rho _2\\lambda _3b_0 (1 -\\varepsilon )\\xi ^2|\\hat{y}|^2\\le C(\\varepsilon ,\\lambda _3)\\xi ^2\\int _0^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds+ a\\lambda _2\\varepsilon \\frac{\\xi ^4}{1+\\xi ^2}|\\hat{z}|^2 +k\\lambda _1\\tau \\varepsilon \\xi ^2|\\hat{v}|^2\\\\+ C(\\varepsilon ,\\lambda _1,\\lambda _2,\\lambda _3)\\xi ^2(1+\\xi ^2)\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2 + \\lambda _3\\delta |\\xi |^3|\\hat{\\theta }|\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|.$ Computing (REF ) $+$ (REF ), we obtain $\\frac{d}{dt}\\left\\lbrace \\lambda _1\\xi ^2 J_1(\\xi ,t)+\\lambda _2\\dfrac{\\xi ^2}{1+\\xi ^2}J_2(\\xi ,t)+\\lambda _3J_3(\\xi ,t) \\right\\rbrace + k\\left[\\lambda _1\\tau (1-2\\varepsilon )-C(\\varepsilon )\\lambda _2\\right]\\xi ^2|\\hat{v}|^2 +\\rho _1(1-2\\varepsilon )\\lambda _2\\dfrac{\\xi ^4}{1+\\xi ^2}|\\hat{u}|^2 \\\\+a(1-2\\varepsilon )\\lambda _2\\dfrac{\\xi ^4}{1+\\xi ^2}|\\hat{z}|^2 +\\rho _2\\left[\\lambda _3b_0 (1 -\\varepsilon )-C(\\varepsilon ,\\lambda _1,\\lambda _2)\\right]\\xi ^2|\\hat{y}|^2 \\\\\\le C(\\varepsilon ,\\lambda _3)\\xi ^2\\int _0^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds +C(\\varepsilon ,\\lambda _1,\\lambda _2,\\lambda _3)\\xi ^2(1+\\xi ^2)^2\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2 \\\\+C(\\varepsilon ,\\lambda _1,\\lambda _2,\\lambda _3)\\xi ^2(1+\\xi ^2)|\\hat{q}|^2 + \\lambda _3\\delta |\\xi |^3|\\hat{\\theta }|\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|.$ Now, consider the following functional $\\mathcal {L}_1(\\xi ,t)= \\lambda _1\\xi ^2 J_1(\\xi ,t)+\\lambda _2\\dfrac{\\xi ^2}{1+\\xi ^2}J_2(\\xi ,t)+\\lambda _3J_3(\\xi ,t)+ J_4(\\xi ,t).$ From Lemma REF and using Young inequality, it yields that $\\frac{d}{dt} \\mathcal {L}_1(\\xi ,t) + k\\left[\\lambda _1\\tau (1-2\\varepsilon )-C(\\varepsilon )\\lambda _2\\right]\\xi ^2|\\hat{v}|^2 +\\rho _1(1-2\\varepsilon )\\lambda _2\\dfrac{\\xi ^4}{1+\\xi ^2}|\\hat{u}|^2 \\\\+a(1-2\\varepsilon )\\lambda _2\\dfrac{\\xi ^4}{1+\\xi ^2}|\\hat{z}|^2 +\\rho _2\\left[\\lambda _3b_0 (1 -2\\varepsilon )-C(\\varepsilon ,\\lambda _1,\\lambda _2)\\right]\\xi ^2|\\hat{y}|^2 +\\rho _3(1 -2\\varepsilon )\\xi ^2|\\hat{\\theta }|^2\\\\\\le C(\\varepsilon ,\\lambda _3)\\xi ^2\\int _0^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds +C(\\varepsilon ,\\lambda _1,\\lambda _2,\\lambda _3)\\xi ^2(1+\\xi ^2)^2\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2 \\\\+C(\\varepsilon ,\\lambda _1,\\lambda _2,\\lambda _3)(1+\\xi ^2)^2|\\hat{q}|^2.$ Now, using the following inequality: $\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2 &= \\left|\\int _0^{\\infty }g^{\\frac{1}{2}}(s)g^{\\frac{1}{2}}(s)\\hat{\\eta }(s)ds\\right|^2 \\\\&\\le \\left|\\left(\\int _0^{\\infty }g(s)ds\\right)^{\\frac{1}{2}}\\left(\\int _0^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds\\right)^{\\frac{1}{2}}\\right|^2 \\\\&= b_0 \\int _0^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds ,$ we obtain that $\\frac{d}{dt} \\mathcal {L}_1(\\xi ,t) + k\\left[\\lambda _1\\tau (1-2\\varepsilon )-C(\\varepsilon )\\lambda _2\\right]\\xi ^2|\\hat{v}|^2 +\\rho _1(1-2\\varepsilon )\\lambda _2\\dfrac{\\xi ^4}{1+\\xi ^2}|\\hat{u}|^2 \\\\+a(1-2\\varepsilon )\\lambda _2\\dfrac{\\xi ^4}{1+\\xi ^2}|\\hat{z}|^2 +\\rho _2\\left[\\lambda _3b_0 (1 -2\\varepsilon )-C(\\varepsilon ,\\lambda _1,\\lambda _2)\\right]\\xi ^2|\\hat{y}|^2 +\\rho _3(1 -2\\varepsilon )\\xi ^2|\\hat{\\theta }|^2\\\\\\le C_1(1+b_0)\\xi ^2(1+\\xi ^2)^2\\int _0^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds +C_1(1+\\xi ^2)^2|\\hat{q}|^2,$ where $C_1$ is a positive constant that depends on $\\varepsilon $ and $\\lambda _j$ for $j=1,2,3$ .",
"Then, we can choose the constants to make all the coefficients in the right side in (REF ) positive.",
"First, let us fix $\\varepsilon $ such that $\\varepsilon < \\frac{1}{2}.$ Thus, we can take first choose $\\lambda _2>0$ and $\\lambda _1 >\\frac{C(\\varepsilon )\\lambda _2}{\\tau (1-2\\varepsilon )}, \\quad \\lambda _3 >\\frac{C(\\varepsilon ,\\lambda _1,\\lambda _2)}{b_0(1-2\\varepsilon )}.$ Then, from (REF ) and some trivial inequalities such as $\\frac{\\xi ^2}{1+\\xi ^2}\\le 1 \\quad \\text{and} \\quad \\frac{1}{1+\\xi ^2}\\le 1,$ we can deduce the existence of a positive constant $M_1$ such that $\\frac{d}{dt}\\mathcal {L}_1(\\xi ,t) \\le -M_1 \\frac{\\xi ^4}{1+\\xi ^2}\\left\\lbrace k|\\hat{v}|^2 +\\rho _1|\\hat{u}|^2+a|\\hat{z}|^2 + \\rho _3|\\hat{\\theta }|^2 +\\rho _2|\\hat{y}|^2\\right\\rbrace \\\\+ C_1(1+b_0)\\xi ^2(1+\\xi ^2)^2\\int _0^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds +C_1(1+\\xi ^2)^2|\\hat{q}|^2.$ Finally, we define the following Lyapunov functional: $\\mathcal {L}(\\xi ,t) = \\mathcal {L}_1(\\xi ,t,t)+N(1+\\xi ^2)^{2}\\hat{E}(\\xi ,t),$ where $N$ is a positive constant to be fixed later.",
"Note that the definition of $\\mathcal {L}_1$ together with inequality (REF ), imply that $\\left|\\mathcal {L}_1(\\xi ,t)\\right| &\\le M_2 \\left\\lbrace |J_1(\\xi ,t)|+|J_2(\\xi ,t)|+|J_3(\\xi ,t)|+ |J_4(\\xi ,t)| \\right\\rbrace \\\\& \\le M_2 (1+\\xi ^2)^{ 2}\\hat{E}(\\xi ,t).$ Hence, we obtain $(N-M_2)(1+\\xi ^2)^{2}\\hat{E}(\\xi ,t) \\le \\mathcal {L}(\\xi ,t)\\le (N+M_2)(1+\\xi ^2)^{2} \\hat{E}(\\xi ,t).$ On the other hand, taking the derivative of $\\mathcal {L}$ with respect to $t$ and using the estimates (REF ) and Lemma REF , it follows that $\\frac{d}{dt}\\mathcal {L}(\\xi ,t) \\le -M_1 \\frac{\\xi ^4}{1+\\xi ^2}\\left\\lbrace k|\\hat{v}|^2 +\\rho _1|\\hat{u}|^2+a|\\hat{z}|^2 + \\rho _3|\\hat{\\theta }|^2 +\\rho _2|\\hat{y}|^2\\right\\rbrace \\\\-\\left(2N\\beta -C_1\\right)(1+\\xi ^2)^{2}|\\hat{q}|^2 -\\left( k_1Nm - C_1 (1+b_0)\\right)(1+\\xi ^2)^{2}\\xi ^2\\int _0^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds.$ Now, choosing $N$ such that $N\\ge \\max \\left\\lbrace M_2,\\dfrac{C_1}{2\\beta }, \\dfrac{C_1(1+b_0)}{k_1m}\\right\\rbrace $ and using the inequality $(1+\\xi ^2)^{2}\\ge \\dfrac{\\xi ^4}{1+\\xi ^2}$ , there exists a positive constant $M_3$ such that $\\frac{d}{dt}\\mathcal {L}(\\xi ,t) \\le - M_3\\frac{\\xi ^4}{1+\\xi ^2} \\hat{E}(\\xi ,t).$ Estimate $(\\ref {e29})$ implies that $\\frac{d}{dt}\\mathcal {L}(\\xi ,t) \\le - \\Gamma \\frac{\\xi ^4}{(1+\\xi ^2)^{3}} \\mathcal {L}(\\xi ,t)$ where $\\Gamma =\\dfrac{M_3}{N+M_2}$ .",
"By Gronwall's inequality, it follows that $\\mathcal {L}(\\xi ,t) \\le e^{-\\Gamma \\rho (\\xi )t}\\mathcal {L}(\\xi ,0), \\qquad \\rho (\\xi )=\\frac{\\xi ^4}{(1+\\xi ^2)^{3}}.$ Again by using $(\\ref {e29})$ , we have that $\\hat{E}(\\xi ,t) \\le C e^{-\\Gamma \\rho (\\xi )t}\\hat{E}(\\xi ,0), \\quad \\text{where}\\quad C= \\frac{N+M_2}{N-M_2}> 0.$ II.",
"Case $\\chi _{0,\\tau }\\ne 0$ : Similar to previous case, we introduce positive constants $\\gamma _1$ , $\\gamma _2$ , $\\gamma _3$ and $\\gamma _4$ that will be fixed later.",
"Next, we estimate the following terms by applying Young's inequality, $\\left| \\chi _{0,\\tau } Re(i\\xi \\hat{u}\\overline{\\hat{y}})\\right| &\\le \\frac{\\rho _1\\gamma _2\\varepsilon }{2\\gamma _1}\\frac{\\xi ^2}{1+\\xi ^2}|\\hat{u}|^2+C(\\varepsilon ,\\gamma _1,\\gamma _2)(1+\\xi ^2)|\\hat{y}|^2, \\\\\\frac{1}{\\delta }\\left| \\left(\\chi _{0,\\tau } +\\tau b_0 \\rho _3 \\left( \\tau - \\frac{\\rho _1}{\\rho _3 k}\\right)\\right)Re(i\\xi \\hat{q}\\overline{\\hat{u}})\\right| &\\le \\frac{\\rho _1\\gamma _2\\varepsilon }{2\\gamma _1}\\frac{\\xi ^2}{1+\\xi ^2}|\\hat{u}|^2+C(\\varepsilon ,\\gamma _1,\\gamma _2)(1+\\xi ^2)|\\hat{q}|^2.$ Hence, from Lemma REF , we can rewrite (REF ) as $\\frac{d}{dt}J_1(\\xi ,t) + k\\tau (1-\\varepsilon )|\\hat{v}|^2 \\le \\frac{\\rho _1\\gamma _2\\varepsilon }{\\gamma _1}\\frac{\\xi ^2}{1+\\xi ^2}|\\hat{u}|^2+C(\\varepsilon ,\\gamma _1,\\gamma _2)(1+\\xi ^2)|\\hat{y}|^2 \\\\+C(\\varepsilon ,\\gamma _1,\\gamma _2)(1+\\xi ^2)|\\hat{q}|^2+C(\\varepsilon )\\xi ^4\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2.$ Thus, the Lemma REF and (REF ) imply that $\\frac{d}{dt}\\left\\lbrace \\gamma _1\\frac{\\xi ^2}{1+\\xi ^2} J_1(\\xi ,t)+\\gamma _2\\dfrac{\\xi ^2}{(1+\\xi ^2)^2}J_2(\\xi ,t) \\right\\rbrace + k\\left[\\gamma _1\\tau (1-\\varepsilon )-C(\\varepsilon )\\gamma _2\\right]\\frac{\\xi ^2}{1+\\xi ^2}|\\hat{v}|^2 +\\rho _1(1-2\\varepsilon )\\gamma _2\\dfrac{\\xi ^4}{(1+\\xi ^2)^2}|\\hat{u}|^2 \\\\+a(1-\\varepsilon )\\gamma _2\\dfrac{\\xi ^4}{(1+\\xi ^2)^2}|\\hat{z}|^2 \\\\\\le C(\\varepsilon ,\\gamma _1,\\gamma _2)\\xi ^2|\\hat{y}|^2+C(\\varepsilon ,\\gamma _1,\\gamma _2)\\xi ^2|\\hat{q}|^2+C(\\varepsilon ,\\gamma _1)\\frac{\\xi ^6}{1+\\xi ^2}\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2 \\\\+C(\\varepsilon ,\\gamma _2)\\frac{\\xi ^2}{1+\\xi ^2}|\\hat{y}|^2 +C(\\varepsilon ,\\gamma _2)\\frac{\\xi ^2}{1+\\xi ^2}|\\hat{q}|^2+C(\\varepsilon ,\\gamma _2)\\frac{\\xi ^6}{(1+\\xi ^2)^2}\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2 \\\\\\le C(\\varepsilon ,\\gamma _1,\\gamma _2)\\xi ^2|\\hat{y}|^2+C(\\varepsilon ,\\gamma _1,\\gamma _2)\\xi ^2|\\hat{q}|^2+C(\\varepsilon ,\\gamma _1,\\gamma _2)(1+\\xi ^2)\\xi ^2\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2.$ Furthermore, applying Young's inequality in the equation (REF ) of the Lemma REF , it follows that $\\frac{d}{dt}\\gamma _3J_3(\\xi ,t) +\\rho _2\\gamma _3b_0 (1 -\\varepsilon )\\xi ^2|\\hat{y}|^2\\le C(\\varepsilon ,\\gamma _3)\\xi ^2\\int _0^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds+ a\\gamma _2\\varepsilon \\frac{\\xi ^4}{(1+\\xi ^2)^2}|\\hat{z}|^2 \\\\+ C(\\varepsilon ,\\gamma _2,\\gamma _3)\\xi ^2(1+\\xi ^2)^2\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2 +\\gamma _3m\\xi ^4\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2+k\\gamma _1\\tau \\varepsilon \\frac{\\xi ^2}{1+\\xi ^2}|\\hat{v}|^2 \\\\+C(\\varepsilon ,\\gamma _1,\\gamma _3)\\xi ^2(1+\\xi ^2)\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2+ \\gamma _3\\delta |\\xi |^3|\\hat{\\theta }|\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|.$ Thus, $\\frac{d}{dt}\\gamma _3J_3(\\xi ,t) +\\rho _2\\gamma _3b_0 (1 -\\varepsilon )\\xi ^2|\\hat{y}|^2\\le C(\\varepsilon ,\\gamma _3)\\xi ^2\\int _0^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds+ a\\gamma _2\\varepsilon \\frac{\\xi ^4}{(1+\\xi ^2)^2}|\\hat{z}|^2 +k\\gamma _1\\tau \\varepsilon \\xi ^2(1+\\xi ^2)|\\hat{v}|^2\\\\+ C(\\varepsilon ,\\gamma _1,\\gamma _2,\\gamma _3)\\xi ^2(1+\\xi ^2)^2\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2 + \\gamma _3\\delta |\\xi |^3|\\hat{\\theta }|\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|.$ Adding (REF ) and (REF ), we obtain $\\frac{d}{dt}\\left\\lbrace \\gamma _1\\frac{\\xi ^2}{1+\\xi ^2} J_1(\\xi ,t)+\\gamma _2\\dfrac{\\xi ^2}{(1+\\xi ^2)^2}J_2(\\xi ,t) + \\gamma _3J_3(\\xi ,t) \\right\\rbrace + k\\left[\\gamma _1\\tau (1-2\\varepsilon )-C(\\varepsilon )\\gamma _2\\right]\\frac{\\xi ^2}{1+\\xi ^2}|\\hat{v}|^2 \\\\+\\rho _1(1-2\\varepsilon )\\gamma _2\\dfrac{\\xi ^4}{(1+\\xi ^2)^2}|\\hat{u}|^2 +a(1-2\\varepsilon )\\gamma _2\\dfrac{\\xi ^4}{(1+\\xi ^2)^2}|\\hat{z}|^2+\\rho _2\\left[\\gamma _3b_0(1 -\\varepsilon )- C(\\varepsilon ,\\gamma _1,\\gamma _2)\\right]\\xi ^2|\\hat{y}|^2 \\\\\\le C(\\varepsilon ,\\gamma _3)\\xi ^2\\int _0^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds+ C(\\varepsilon ,\\gamma _1,\\gamma _2,\\gamma _3)\\xi ^2(1+\\xi ^2)^2\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2 \\\\ + \\gamma _3\\delta |\\xi |^3|\\hat{\\theta }|\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|+C(\\varepsilon ,\\gamma _1,\\gamma _2,\\gamma _3)\\xi ^2|\\hat{q}|^2.$ Now, consider the following functional $\\mathcal {L}_2(\\xi ,t)= \\gamma _1\\xi ^2 J_1(\\xi ,t)+\\gamma _2\\dfrac{\\xi ^2}{1+\\xi ^2}J_2(\\xi ,t)+\\gamma _3J_5(\\xi ,t) + J_4(\\xi ,t).$ From Lemma REF , applying Young inequality to (REF ) and (REF ), together with (REF ), we have $\\frac{d}{dt}\\left\\lbrace \\gamma _1\\frac{\\xi ^2}{1+\\xi ^2} J_1(\\xi ,t)+\\gamma _2\\dfrac{\\xi ^2}{(1+\\xi ^2)^2}J_2(\\xi ,t) + \\gamma _3J_3(\\xi ,t) \\right\\rbrace + k\\left[\\gamma _1\\tau (1-2\\varepsilon )-C(\\varepsilon )\\gamma _2\\right]\\frac{\\xi ^2}{1+\\xi ^2}|\\hat{v}|^2 \\\\+\\rho _1(1-2\\varepsilon )\\gamma _2\\dfrac{\\xi ^4}{(1+\\xi ^2)^2}|\\hat{u}|^2 +a(1-2\\varepsilon )\\gamma _2\\dfrac{\\xi ^4}{(1+\\xi ^2)^2}|\\hat{z}|^2+\\rho _2\\left[\\gamma _3b_0(1 -2\\varepsilon )- C(\\varepsilon ,\\gamma _1,\\gamma _2)\\right]\\xi ^2|\\hat{y}|^2 \\\\ + \\rho _3(1-2\\varepsilon )\\xi ^2|\\hat{\\theta }|^2\\le C_1(1+b_0)\\xi ^2(1+\\xi ^2)^2\\int _0^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds + C_1(1+\\xi ^2)|\\hat{q}|^2,$ where $C_1$ is a positive constant that depends on $\\varepsilon $ and $\\gamma _j$ for $j=1,2,3$ .",
"In order to make all the coefficients in the right side in (REF ) positive, we have to choose appropriate constant $\\gamma _i$ .",
"First, let us fix $\\varepsilon $ such that $\\varepsilon < \\frac{1}{2}.$ Thus, we can take any $\\gamma _2 >0$ and $\\gamma _1 >\\frac{C(\\varepsilon )\\gamma _2}{\\tau (1-2\\varepsilon )}, \\quad \\gamma _3 >\\frac{C(\\varepsilon ,\\gamma _1,\\gamma _2)}{b_0(1-2\\varepsilon )}.$ Then, from (REF ) and (REF ), we can deduce the existence of a positive constant $M_1$ such that $\\frac{d}{dt}\\mathcal {L}_2(\\xi ,t) \\le -M_1 \\frac{\\xi ^4}{(1+\\xi ^2)^2}\\left\\lbrace k|\\hat{v}|^2 +\\rho _1|\\hat{u}|^2+a|\\hat{z}|^2 +\\rho _2|\\hat{y}|^2+\\rho _3|\\hat{\\theta }|^2\\right\\rbrace \\\\+ C_1(1+\\xi ^2)^2\\xi ^2\\int _0^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds +C_1(1+\\xi ^2)^2|\\hat{q}|^2.$ Finally, we define the following Lyapunov functional: $\\mathcal {L}(\\xi ,t) = \\mathcal {L}_2(\\xi ,t,t)+N(1+\\xi ^2)^2\\hat{E}(\\xi ,t),$ where $N$ is a positive constant to be fixed later.",
"Note that the definition of $\\mathcal {L}_2$ together with inequality (REF ) imply that $\\left|\\mathcal {L}_2(\\xi ,t)\\right| &\\le M_2 \\left\\lbrace |J_1(\\xi ,t)|+|J_2(\\xi ,t)|+|J_3(\\xi ,t)|+ |J_4(\\xi ,t)|\\right\\rbrace \\\\& \\le M_2 (1+\\xi ^2)^2\\hat{E}(\\xi ,t).$ Hence, we obtain $(N-M_2)(1+\\xi ^2)^2\\hat{E}(\\xi ,t) \\le \\mathcal {L}(\\xi ,t)\\le (N+M_2)(1+\\xi ^2)^2 \\hat{E}(\\xi ,t).$ On the other hand, taking the derivative of $\\mathcal {L}$ with respect to $t$ and using the estimates (REF ) and Lemma REF , it follows that $\\frac{d}{dt}\\mathcal {L}(\\xi ,t) \\le -M_1 \\frac{\\xi ^4}{(1+\\xi ^2)^2}\\left\\lbrace k|\\hat{v}|^2 +\\rho _1|\\hat{u}|^2+a|\\hat{z}|^2 + \\rho _2|\\hat{y}|^2+\\rho _3|\\hat{\\theta }|^2\\right\\rbrace \\\\-\\left(2N\\beta -C_1\\right)(1+\\xi ^2)^2|\\hat{q}|^2 -\\left( Nk_1 - C_1 (1+b_0)\\right)(1+\\xi ^2)^2\\xi ^2\\int _0^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds.$ Now, choosing $N$ such that $N\\ge \\max \\left\\lbrace M_2,\\dfrac{C_1}{2\\beta }, \\dfrac{C_1(1+b_0)}{k_1m}\\right\\rbrace $ and using the inequality $(1+\\xi ^2)^2\\ge \\dfrac{\\xi ^4}{(1+\\xi ^2)^2}$ , there exists a positive constant $M_3$ such that $\\frac{d}{dt}\\mathcal {L}(\\xi ,t) \\le - M_3\\frac{\\xi ^4}{(1+\\xi ^2)^2} \\hat{E}(\\xi ,t).$ $(\\ref {e36})$ implies that $\\frac{d}{dt}\\mathcal {L}(\\xi ,t) \\le - \\Gamma \\frac{\\xi ^4}{(1+\\xi ^2)^4} \\mathcal {L}(\\xi ,t)$ where $\\Gamma =\\dfrac{M_3}{N+M_2}$ .",
"By Gronwall's inequality, it follows that $\\mathcal {L}(\\xi ,t) \\le \\mathcal {L}(\\xi ,0) e^{-\\Gamma \\rho (\\xi )t}, \\qquad \\rho (\\xi )=\\frac{\\xi ^4}{(1+\\xi ^2)^4},$ again by using $(\\ref {e36})$ , we have that $\\hat{E}(\\xi ,t) \\le C \\hat{E}(\\xi ,0) e^{-\\Gamma \\rho (\\xi )t}, \\quad \\text{where}\\quad C= \\frac{N+M_2}{N-M_2}> 0.$"
],
[
"The Timoshenko-Fourier Law",
"Similarly to the previous case, taking Fourier transform in (REF ), we obtain the following integro differential system $&\\hat{v}_t-i\\xi \\hat{u}+\\hat{y}=0, \\\\&\\rho _1\\hat{u}_t-ik\\xi \\hat{v}=0, \\\\&\\hat{z}_t-i\\xi \\hat{y} =0, \\\\&\\rho _2\\hat{y}_t -ia\\xi \\hat{z}+m\\xi ^2\\displaystyle \\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds -k\\hat{v}+i\\delta \\xi \\hat{\\theta }=0,\\\\&\\rho _3\\hat{\\theta }_t+\\tilde{\\beta }\\xi ^2\\hat{\\theta }+i\\delta \\xi \\hat{y}=0,\\\\&\\hat{\\eta }_t+\\hat{\\eta }_s-\\hat{y}=0, $ where the solution vector and initial data are given by $\\hat{V}(\\xi ,t)=(\\hat{v},\\hat{u},\\hat{z},\\hat{y},\\hat{\\theta },\\hat{q},\\hat{\\eta })^{T}$ and $\\hat{V}(\\xi ,0)=\\hat{V}_0(\\xi )$ , respectively.",
"Furthermore, the energy functional associated to the above system is defined as $\\hat{\\mathcal {E}}\\left(t, \\xi \\right) = \\rho _1|\\hat{u}|^{2}+\\rho _2|\\hat{y}|^{2}+\\rho _3 |\\hat{\\theta }|^{2}+k|\\hat{v} |^{2}+a|\\hat{z} |^{2} +m\\xi ^2\\int _{0}^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds.$ Lemma 3.7 The energy of the system (REF )-(), satisfies $\\frac{d}{dt}\\hat{\\mathcal {E}}(\\xi ,t)\\le -2\\tilde{\\beta }\\xi ^2 |\\hat{\\theta }|^2 -k_1 m \\xi ^2\\int _0^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds,$ where the constant $k_1>0$ is given by $(H_2)$ .",
"Multiplying (REF ) by $k\\overline{\\hat{v}}$ , () by $\\overline{\\hat{u}}$ , () by $a\\overline{\\hat{z}}$ , () by $\\overline{\\hat{y}}$ and () by $\\overline{\\hat{\\theta }}$ , adding and taking real part, it follows that $\\frac{1}{2}\\frac{d}{dt}\\left\\lbrace \\rho _1|\\hat{u}|^{2}+\\rho _2|\\hat{y}|^{2}+\\rho _3 |\\hat{\\theta }|^{2}+k|\\hat{v} |^{2}+a|\\hat{z} |^{2}\\right\\rbrace =-\\tilde{\\beta }\\xi ^2 |\\hat{\\theta }|^2 -Re\\left(m \\xi ^2\\int _0^{\\infty } g(s)\\eta (s)\\overline{\\hat{y}}ds\\right).$ On the other hand, taking the conjugate of equation (), multiplying the resulting equation by $g(s)\\hat{\\eta }(t,s,x)$ and integrating with respect to $s$ , we obtain $\\int _0^{\\infty }g(s)\\hat{\\eta }(s)\\overline{\\hat{y}}ds=\\frac{1}{2}\\frac{d}{dt}\\int _0^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds +\\frac{1}{2}\\int _0^{\\infty }g(s)\\frac{d}{ds}|\\hat{\\eta }(s)|^2ds.$ Integrating by parts the last term in the left side of (REF ) and substituting in (REF ), it follows that $\\frac{d}{dt}\\hat{E}(\\xi ,t)= -2\\tilde{\\beta }\\xi ^2 |\\hat{\\theta }|^2 -m\\xi ^2\\int _0^{\\infty }g^{\\prime }(s)|\\eta (s)|^2ds.$ By using $(H_2)$ , we obtain (REF ).",
"Since the energy of the system (REF )-() is dissipative, we expect the exponential decay like in the previous subsection.",
"The principal result of this subsection reads as follows: Theorem 3.8 Let $\\chi _{0}= \\left(\\rho _2-\\dfrac{b\\rho _1}{k}\\right).$ Then, for any $t \\ge 0$ and $\\xi \\in \\mathbb {R}$ , we obtain the following decay rates for the energy of the system (REF )-(): $\\hat{\\mathcal {E}}(\\xi ,t) \\le C e^{-\\lambda \\rho (\\xi )}\\hat{\\mathcal {E}}(0,\\xi ),$ where $C, \\lambda $ are positive constants and the function $\\rho (\\cdot )$ is given by $\\rho (\\xi ) = {\\left\\lbrace \\begin{array}{ll}\\dfrac{\\xi ^4}{(1+\\xi ^2)^3} & \\text{if $\\chi _{0}=0$}, \\\\\\\\\\dfrac{\\xi ^4}{(1+\\xi ^2)^4} & \\text{if $\\chi _{0}\\ne 0$}.\\end{array}\\right.",
"}$ Similarly to for Cattaneo's law, we need establish some preliminary results.",
"Lemma 3.9 Consider the functional $K_1(\\xi ,t)=-\\rho _2Re(\\hat{v}\\overline{\\hat{y}})-\\frac{a\\rho _1}{k}Re(\\hat{z}\\overline{\\hat{u}}) +\\frac{b_0\\rho _1\\rho _3}{k\\delta }Re(\\hat{\\theta }\\overline{\\hat{u}}).$ Then, for any $\\varepsilon >0$ , $K_1$ satisfies $\\frac{d}{dt}K_1(\\xi ,t) + k(1-\\varepsilon )|\\hat{v}|^2 \\le \\rho _2 |\\hat{y}|^2 + \\chi _{0} Re\\left(i\\xi \\overline{\\hat{u}}\\hat{y}\\right)+C(\\varepsilon )\\xi ^2|\\hat{\\theta }|^2+C(\\varepsilon )\\xi ^4\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2+ \\frac{b_0\\tilde{\\beta }\\rho _1}{k \\delta } \\xi ^2|\\hat{\\theta }||\\hat{u}|,$ for $t\\ge 0$ , where $C(\\varepsilon )$ is a positive constant and $\\chi _{0}= \\left(\\rho _2-\\dfrac{b\\rho _1}{k}\\right)$ .",
"Multiplying (REF ) by $-\\rho _2 \\overline{\\hat{y}}$ and taking real part, we obtain $-\\rho _2Re\\left(\\hat{v}_t\\overline{\\hat{y}}\\right) + \\rho _2Re\\left(i\\xi \\hat{u}\\overline{\\hat{y}}\\right)-\\rho _2|\\hat{y}|^2=0.$ Multiplying () by $- \\overline{\\hat{v}}$ and taking real part, it follows that $-\\rho _2Re\\left(\\hat{y}_t\\overline{\\hat{v}}\\right) + aRe\\left(i\\xi \\hat{z}\\overline{\\hat{v}}\\right)+k|\\hat{v}|^2- mRe\\left(\\xi ^2\\overline{\\hat{v}}\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right)-\\delta Re\\left(i\\xi \\hat{\\theta }\\overline{\\hat{v}}\\right)=0.$ Adding the above identities, $-\\rho _2\\frac{d}{dt}Re\\left(\\hat{v}\\overline{\\hat{y}}\\right)+k|\\hat{v}|^2=\\rho _2|\\hat{y}|^2 - aRe\\left(i\\xi \\hat{z}\\overline{\\hat{v}}\\right)+ m Re\\left(\\xi ^2\\overline{\\hat{v}}\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right) -\\rho _2Re\\left(i\\xi \\hat{u}\\overline{\\hat{y}}\\right)+\\delta Re\\left(i\\xi \\hat{\\theta }\\overline{\\hat{v}}\\right).$ On the other hand, multiplying () by $-\\frac{a}{k}\\overline{\\hat{z}}$ , () by $-\\frac{a\\rho _1}{k}\\overline{\\hat{u}}$ , adding the results and taking real part, it follows that $-\\frac{a\\rho _1}{k}\\frac{d}{dt}Re\\left(\\hat{z}\\overline{\\hat{u}}\\right) = -\\frac{a\\rho _1}{k}Re\\left(i\\xi \\hat{y}\\overline{\\hat{u}}\\right)-aRe\\left(i\\xi \\hat{v}\\overline{\\hat{z}}\\right).$ Adding (REF ) and (REF ), we have $\\frac{d}{dt}\\left\\lbrace -\\rho _2Re\\left(\\hat{v}\\overline{\\hat{y}}\\right) -\\frac{a\\rho _1}{k}Re\\left(\\hat{z}\\overline{\\hat{u}}\\right) \\right\\rbrace +k|\\hat{v}|^2 = \\rho _2|\\hat{y}|^2 + m Re\\left(\\xi ^2\\overline{\\hat{v}}\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right) \\\\+\\left(\\rho _2-\\frac{a\\rho _1}{k}\\right) Re\\left(i\\xi \\hat{y}\\overline{\\hat{u}}\\right)+\\delta Re\\left(i\\xi \\hat{\\theta }\\overline{\\hat{v}}\\right).$ Moreover, multiplying () by $\\frac{\\delta }{k}\\overline{\\hat{\\theta }}$ and taking real part, $\\frac{\\delta \\rho _1}{k}Re(\\hat{u}_t\\overline{\\hat{\\theta }})-\\delta Re(i\\xi \\hat{v}\\overline{\\hat{\\theta }})=0.$ Next, multiplying () by $\\frac{\\delta \\rho _1}{\\rho _3 k}\\overline{\\hat{u}}$ and taking real part, $\\frac{\\delta \\rho _1}{k}Re(\\hat{\\theta }_t\\overline{\\hat{u}})+\\frac{\\delta \\tilde{\\beta }\\rho _1}{\\rho _3 k}Re(\\xi ^2\\hat{\\theta }\\overline{\\hat{u}})+\\frac{\\delta ^2\\rho _1}{\\rho _3 k}Re(i\\xi \\hat{y}\\overline{\\hat{u}})=0.$ Adding the above identities, we obtain $\\frac{\\delta \\rho _1}{k}\\frac{d}{dt}Re(\\hat{\\theta }\\overline{\\hat{u}})= -\\frac{\\delta \\tilde{\\beta }\\rho _1}{\\rho _3 k}Re(\\xi ^2\\hat{\\theta }\\overline{\\hat{u}})-\\frac{\\delta ^2\\rho _1}{\\rho _3 k}Re(i\\xi \\hat{y}\\overline{\\hat{u}})+\\delta Re(i\\xi \\hat{v}\\overline{\\hat{\\theta }}).$ Computing $(\\ref {eqqq1})+\\frac{b_0\\rho _3}{\\delta ^2}(\\ref {eqqq2})$ , it follows that $\\frac{d}{dt}\\left\\lbrace -\\rho _2Re\\left(\\hat{v}\\overline{\\hat{y}}\\right) -\\frac{a\\rho _1}{k}Re\\left(\\hat{z}\\overline{\\hat{u}}\\right)+ \\frac{b_0\\rho _1\\rho _3}{k\\delta }Re(\\hat{\\theta }\\overline{\\hat{u}})\\right\\rbrace +k|\\hat{v}|^2 = \\rho _2|\\hat{y}|^2 + m Re\\left(\\xi ^2\\overline{\\hat{v}}\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right) \\\\+\\left(\\rho _2-\\frac{a\\rho _1}{k}-\\frac{b_0\\rho _1}{ k}\\right) Re\\left(i\\xi \\overline{\\hat{u}}\\hat{y}\\right)+\\left(\\delta -\\frac{b_0\\rho _3}{\\delta }\\right) Re\\left(i\\xi \\hat{\\theta }\\overline{\\hat{v}}\\right) -\\frac{b_0\\tilde{\\beta }\\rho _1}{k \\delta }Re(\\xi ^2\\hat{\\theta }\\overline{\\hat{u}}).$ Then, $\\frac{d}{dt}K_1(\\xi ,t) +k|\\hat{v}|^2 \\le \\rho _2|\\hat{y}|^2 + m \\xi ^2|\\hat{v}|\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right| +\\chi _0 Re\\left(i\\xi \\overline{\\hat{u}}\\hat{y}\\right)+\\left|\\delta -\\frac{b_0\\rho _3}{\\delta }\\right| |\\xi | |\\hat{\\theta }||\\hat{v}| +\\frac{b_0\\tilde{\\beta }\\rho _1}{k \\delta } \\xi ^2|\\hat{\\theta }||\\hat{u}|.$ Applying Young's inequality, (REF ) follows.",
"Lemma 3.10 Consider the functional $K_2(\\xi ,t)=\\rho _1Re\\left(i\\xi \\overline{\\hat{u}}\\hat{v}\\right) +\\rho _2Re\\left(i\\xi \\hat{y}\\overline{\\hat{z}}\\right).$ For any $\\varepsilon >0$ , the estimate $\\frac{d}{dt}K_2(\\xi ,t) +\\rho _1(1-\\varepsilon )\\xi ^2|\\hat{u}|^2+a(1-\\varepsilon )\\xi ^2|\\hat{z}|^2\\le C(\\varepsilon )(1+\\xi ^2)|\\hat{v}|^2 +C(\\varepsilon )(1+\\xi ^2)|\\hat{y}|^2 \\\\+C(\\varepsilon )\\xi ^2|\\hat{\\theta }|^2 +C(\\varepsilon )\\xi ^4\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2$ is satisfied.",
"Multiplying (REF ) by $i\\rho _1\\xi \\overline{\\hat{u}}$ and taking real part, $\\rho _1Re\\left(i\\xi \\hat{v}_t\\overline{\\hat{u}}\\right) +\\rho _1\\xi ^2|\\hat{u}|^2+\\rho _1Re\\left(i\\xi \\hat{y}\\overline{\\hat{u}}\\right)=0.$ Multiplying () by $-i\\xi \\overline{\\hat{v}}$ and taking real part, $-\\rho _1Re\\left(i\\xi \\hat{u}_t\\overline{\\hat{v}}\\right) -k\\xi ^2|\\hat{v}|^2=0.$ Adding the above identities, we obtain $\\rho _1\\frac{d}{dt}Re\\left(i\\xi \\hat{v}\\overline{\\hat{u}}\\right)+\\rho _1\\xi ^2|\\hat{u}|^2=k\\xi ^2|\\hat{v}|^2 -\\rho _1Re\\left(i\\xi \\hat{y}\\overline{\\hat{u}}\\right).$ Moreover, multiplying () by $-i\\rho _2\\xi \\overline{\\hat{y}}$ and taking real part, $-\\rho _2Re\\left(i\\xi \\hat{z}_t\\overline{\\hat{y}}\\right) -\\rho _2\\xi ^2|\\hat{y}|^2=0.$ Multiplying () by $i\\xi \\overline{\\hat{z}}$ and taking real part, $\\rho _2Re\\left(i\\xi \\hat{y}_t\\overline{\\hat{z}}\\right) +a\\xi ^2|\\hat{z}|^2 -kRe\\left(i\\xi \\hat{v}\\overline{\\hat{z}}\\right)+ m Re\\left(i\\xi ^3\\overline{\\hat{z}}\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right) -\\delta Re\\left(\\xi ^2\\hat{\\theta }\\overline{\\hat{z}}\\right)=0.$ Adding the above identities, $\\rho _2\\frac{d}{dt}Re\\left(i\\xi \\hat{y}\\overline{\\hat{z}}\\right) +a\\xi ^2|\\hat{z}|^2= \\rho _2\\xi ^2|\\hat{y}|^2+kRe\\left(i\\xi \\hat{v}\\overline{\\hat{z}}\\right)- m Re\\left(i\\xi ^3\\overline{\\hat{z}}\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right) +\\delta Re\\left(\\xi ^2\\hat{\\theta }\\overline{\\hat{z}}\\right).$ Therefore, computing (REF ) $+$ (REF ), it follows that $\\frac{d}{dt}K_2(\\xi ,t) +\\rho _1\\xi ^2|\\hat{u}|^2 +a\\xi ^2|\\hat{z}|^2\\le \\rho _2\\xi ^2|\\hat{y}|^2+k\\xi ^2|\\hat{v}|^2+k|\\xi ||\\hat{v}|\\hat{z}| \\\\- m |\\xi |^3|\\hat{z}|\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right| +\\delta \\xi ^2|\\hat{\\theta }||\\hat{z}|+\\rho _1|\\xi ||\\hat{y}||\\hat{u}|.$ Applying Young's inequality, (REF ) holds.",
"Lemma 3.11 Consider the functional $K_3(\\xi ,t)=-\\rho _2Re\\left( \\xi ^2 \\overline{\\hat{y}}\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right).$ Then, for any $\\varepsilon >0$ , the following estimate $\\frac{d}{dt}K_3(\\xi ,t) +\\rho _2b_0(1-\\varepsilon )\\xi ^2|\\hat{y}|^2 \\le C(\\varepsilon ) \\xi ^2 \\int _0^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds +a |\\xi |^3 |\\hat{z}|\\left| \\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right| \\\\+m\\xi ^4\\left| \\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2 +k\\xi ^2 |\\hat{v}| \\left| \\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right| +\\delta |\\xi |^3 |\\hat{\\theta }| \\left| \\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|.$ holds.",
"Proceeding as proof of Lemma (REF ), we obtain (REF ).",
"Indeed, we have to multiply () by $-\\xi ^2g(s) \\overline{\\hat{\\eta }}$ and () by $-\\rho _2\\xi ^2g(s) \\overline{\\hat{y}}$ , next we take the integration with respect to $s$ for the real parts.",
"We omit the details.",
"[Proof of Theorem REF] As Theorem REF , we will consider several cases: I.",
"Case $\\chi _0=0$ : Consider the expressions $\\zeta _1\\xi ^2K_1(\\xi ,t), \\quad \\zeta _2\\dfrac{\\xi ^2}{1+\\xi ^2}K_2(\\xi ,t), \\quad \\zeta _3K_3(\\xi ,t),$ where $\\zeta _1$ , $\\zeta _2$ and $\\zeta _3$ are positive constants to be fixed later.",
"Lemmas REF and REF imply that $\\frac{d}{dt}\\left\\lbrace \\zeta _1\\xi ^2 K_1(\\xi ,t)+\\zeta _2\\dfrac{\\xi ^2}{1+\\xi ^2}K_2(\\xi ,t) \\right\\rbrace + k\\left[\\zeta _1(1-\\varepsilon )-C(\\varepsilon )\\zeta _2\\right]\\xi ^2|\\hat{v}|^2 +\\rho _1(1-\\varepsilon )\\zeta _2\\dfrac{\\xi ^4}{1+\\xi ^2}|\\hat{u}|^2 \\\\+a(1-\\varepsilon )\\zeta _2\\dfrac{\\xi ^4}{1+\\xi ^2}|\\hat{z}|^2\\le C(\\varepsilon ,\\zeta _1,\\zeta _2)\\xi ^2|\\hat{y}|^2 +C(\\varepsilon ,\\zeta _1,\\zeta _2)\\xi ^4|\\hat{\\theta }|^2 \\\\\\frac{b_0\\tilde{\\beta }\\rho _1}{k\\delta }\\zeta _1\\xi ^4|\\hat{\\theta }||\\hat{u}|+C(\\varepsilon ,\\zeta _1,\\zeta _2)\\xi ^6\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2.$ By Young's inequality, $\\frac{d}{dt}\\left\\lbrace \\zeta _1\\xi ^2 K_1(\\xi ,t)+\\zeta _2\\dfrac{\\xi ^2}{1+\\xi ^2}K_2(\\xi ,t) \\right\\rbrace + k\\left[\\zeta _1(1-\\varepsilon )-C(\\varepsilon )\\zeta _2\\right]\\xi ^2|\\hat{v}|^2 +\\rho _1(1-2\\varepsilon )\\zeta _2\\dfrac{\\xi ^4}{1+\\xi ^2}|\\hat{u}|^2 \\\\+a(1-\\varepsilon )\\zeta _2\\dfrac{\\xi ^4}{1+\\xi ^2}|\\hat{z}|^2\\le C(\\varepsilon ,\\zeta _1,\\zeta _2)\\xi ^2|\\hat{y}|^2 +C(\\varepsilon ,\\zeta _1,\\zeta _2)\\xi ^4(1+\\xi ^2)|\\hat{\\theta }|^2\\\\+C(\\varepsilon ,\\zeta _1,\\zeta _2)\\xi ^6\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2.$ Further, applying Young's inequality in (REF ) in the Lemma REF , it follows that $\\frac{d}{dt}\\zeta _3K_3(\\xi ,t) +\\rho _2\\zeta _3b_0 (1 -\\varepsilon )\\xi ^2|\\hat{y}|^2\\le C(\\varepsilon ,\\zeta _3)\\xi ^2\\int _0^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds+ a\\zeta _2\\varepsilon \\frac{\\xi ^4}{1+\\xi ^2}|\\hat{z}|^2 \\\\+ C(\\varepsilon ,\\zeta _2,\\zeta _3)\\xi ^2(1+\\xi ^2)\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2 +\\zeta _3m\\xi ^4\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2+k\\zeta _1\\varepsilon \\xi ^2|\\hat{v}|^2 \\\\+C(\\varepsilon ,\\zeta _1,\\zeta _3)\\xi ^2\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2+ \\zeta _3\\delta |\\xi |^3|\\hat{\\theta }|\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|.$ Thus, $\\frac{d}{dt}\\zeta _3K_3(\\xi ,t) +\\rho _2\\zeta _3b_0 (1 -\\varepsilon )\\xi ^2|\\hat{y}|^2\\le C(\\varepsilon ,\\zeta _3)\\xi ^2\\int _0^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds+ a\\zeta _2\\varepsilon \\frac{\\xi ^4}{1+\\xi ^2}|\\hat{z}|^2 +k\\zeta _1\\varepsilon \\xi ^2|\\hat{v}|^2\\\\+ C(\\varepsilon ,\\zeta _1,\\zeta _2,\\zeta _3)\\xi ^2(1+\\xi ^2)\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2 + C(\\varepsilon )\\xi ^4|\\hat{\\theta }|^2.$ Computing (REF ) $+$ (REF ), we obtain $\\frac{d}{dt}\\left\\lbrace \\zeta _1\\xi ^2 K_1(\\xi ,t)+\\zeta _2\\dfrac{\\xi ^2}{1+\\xi ^2}K_2(\\xi ,t)+\\zeta _3K_3(\\xi ,t) \\right\\rbrace + k\\left[\\zeta _1(1-2\\varepsilon )-C(\\varepsilon )\\zeta _2\\right]\\xi ^2|\\hat{v}|^2\\\\+\\rho _1(1-2\\varepsilon )\\zeta _2\\dfrac{\\xi ^4}{1+\\xi ^2}|\\hat{u}|^2+a(1-2\\varepsilon )\\zeta _2\\dfrac{\\xi ^4}{1+\\xi ^2}|\\hat{z}|^2 +\\rho _2\\left[\\zeta _3b_0(1 -\\varepsilon )-C(\\varepsilon ,\\zeta _1,\\zeta _2)\\right]\\xi ^2|\\hat{y}|^2 \\\\\\le C(\\varepsilon ,\\zeta _3)\\xi ^2\\int _0^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds +C(\\varepsilon ,\\zeta _1,\\zeta _2,\\zeta _3)\\xi ^2(1+\\xi ^2)^2\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2 \\\\+ C(\\varepsilon ,\\zeta _1,\\zeta _2,\\zeta _3)\\xi ^2(1+\\xi ^2)^2|\\hat{\\theta }|^2.$ From inequality (REF ), we conclude that $\\frac{d}{dt}\\left\\lbrace \\zeta _1\\xi ^2 K_1(\\xi ,t)+\\zeta _2\\dfrac{\\xi ^2}{1+\\xi ^2}K_2(\\xi ,t)+\\zeta _3K_3(\\xi ,t) \\right\\rbrace + k\\left[\\zeta _1(1-2\\varepsilon )-C(\\varepsilon )\\zeta _2\\right]\\xi ^2|\\hat{v}|^2\\\\+\\rho _1(1-2\\varepsilon )\\zeta _2\\dfrac{\\xi ^4}{1+\\xi ^2}|\\hat{u}|^2+a(1-2\\varepsilon )\\zeta _2\\dfrac{\\xi ^4}{1+\\xi ^2}|\\hat{z}|^2 +\\rho _2\\left[\\zeta _3b_0(1 -\\varepsilon )-C(\\varepsilon ,\\zeta _1,\\zeta _2)\\right]\\xi ^2|\\hat{y}|^2 \\\\\\le C(\\varepsilon ,\\zeta _1,\\zeta _2,\\zeta _3)(1+b_0)\\xi ^2(1+\\xi ^2)^2 \\int _0^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds + C(\\varepsilon ,\\zeta _1,\\zeta _2,\\zeta _3)\\xi ^2(1+\\xi ^2)^2|\\hat{\\theta }|^2.$ In order to make all coefficients in the right-hand side in (REF ) positive, we have to choose appropriate constant $\\zeta _i$ .",
"First, let us fix $\\varepsilon $ , such that $\\varepsilon < \\frac{1}{2}.$ Thus, we can take any $\\zeta _2 >0$ and $\\zeta _1 >\\frac{C(\\varepsilon )\\zeta _2}{1-2\\varepsilon }, \\quad \\zeta _3 >\\frac{C(\\varepsilon ,\\zeta _1,\\zeta _2)}{b_0(1-\\varepsilon )}.$ Then, from (REF ) and the estimate $\\frac{\\xi ^2}{1+\\xi ^2}\\le 1$ , we can deduce the existence of a positive constant $M_1$ such that $\\frac{d}{dt}\\mathcal {Q}_1(\\xi ,t) \\le -M_1 \\frac{\\xi ^4}{1+\\xi ^2}\\left\\lbrace k|\\hat{v}|^2 +\\rho _1|\\hat{u}|^2+a|\\hat{z}|^2 +\\rho _2|\\hat{y}|^2\\right\\rbrace \\\\ +C_1(1+b_0)(1+\\xi ^2)^2\\xi ^2\\int _0^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds+C_1(1+\\xi ^2)^2\\xi ^2|\\hat{\\theta }|^2,$ where $\\mathcal {Q}_1(\\xi ,t)= \\zeta _1\\xi ^2 K_1(\\xi ,t)+\\zeta _2\\dfrac{\\xi ^2}{1+\\xi ^2}K_2(\\xi ,t)+\\zeta _3K_3(\\xi ,t)$ and $C_1$ is a positive constant that depends of $\\varepsilon $ and $\\zeta _j$ for $j=1,2,3$ .",
"Finally, we define the following Lyapunov functional: $\\mathcal {Q}(\\xi ,t) = \\mathcal {Q}_1(\\xi ,t,t)+N(1+\\xi ^2)^2\\hat{\\mathcal {E}}(\\xi ,t),$ where $N$ is a positive constant to be fixed later.",
"Note that the definition of $\\mathcal {Q}_1$ together with (REF ) imply that $\\left|\\mathcal {Q}_1(\\xi ,t)\\right| &\\le M_2 \\left\\lbrace \\xi ^2|K_1(\\xi ,t)|+|K_2(\\xi ,t)|+|K_3(\\xi ,t)|\\right\\rbrace \\\\& \\le M_2 (1+\\xi ^2)\\hat{\\mathcal {E}}(\\xi ,t).$ Hence, we obtain $(N-M_2)(1+\\xi ^2)^2\\hat{\\mathcal {E}}(\\xi ,t) \\le \\mathcal {Q}(\\xi ,t)\\le (N+M_2)(1+\\xi ^2)^2 \\hat{\\mathcal {E}}(\\xi ,t).$ On the other hand, taking the derivative of $\\mathcal {Q}$ with respect to $t$ and using the estimates (REF ) and Lemma REF , it follows that $\\frac{d}{dt}\\mathcal {Q}(\\xi ,t) \\le -M_1 \\frac{\\xi ^4}{1+\\xi ^2}\\left\\lbrace k|\\hat{v}|^2 +\\rho _1|\\hat{u}|^2+a|\\hat{z}|^2 + \\rho _2|\\hat{y}|^2\\right\\rbrace -\\left(2N\\tilde{\\beta }-C_1\\right)(1+\\xi ^2)^2\\xi ^2|\\hat{\\theta }|^2 \\\\-\\left( Nk_1m - C_1(1+ b_0)\\right)(1+\\xi ^2)^2\\xi ^2\\int _0^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds.$ Now, choosing $N$ such that $N\\ge \\max \\left\\lbrace M_2,\\dfrac{C_1}{2\\tilde{\\beta }}, \\dfrac{C_1(1+b_0)}{k_1 m}\\right\\rbrace $ and using the inequalities $(1+\\xi ^2)^2 \\ge \\dfrac{\\xi ^4}{1+\\xi ^2}$ and $(1+\\xi ^2)^2 \\ge \\dfrac{\\xi ^2}{1+\\xi ^2}$ , there exists a positive constant $M_3$ such that $\\frac{d}{dt}\\mathcal {Q}(\\xi ,t) \\le - M_3\\frac{\\xi ^4}{1+\\xi ^2} \\hat{\\mathcal {E}}(\\xi ,t).$ Estimate $(\\ref {eq29+})$ implies that $\\frac{d}{dt}\\mathcal {Q}(\\xi ,t) \\le - \\Gamma \\frac{\\xi ^4}{(1+\\xi ^2)^3} \\mathcal {Q}(\\xi ,t)$ where $\\Gamma =\\dfrac{M_3}{(N+M_2)}$ .",
"By Gronwall's inequality, it follows that $\\mathcal {Q}(\\xi ,t) \\le e^{-\\Gamma \\rho (\\xi )t}\\mathcal {Q}(\\xi ,0), \\quad \\rho (\\xi )=\\frac{\\xi ^4}{(1+\\xi ^2)^3}.$ Again by using $(\\ref {eq29+})$ , we have that $\\hat{\\mathcal {E}}(\\xi ,t) \\le C e^{-\\Gamma \\rho (\\xi )t}\\hat{\\mathcal {E}}(\\xi ,0), \\quad \\text{where}\\quad C= \\frac{N+M_2}{N-M_2}> 0.$ II.",
"Case $\\chi _0 \\ne 0$: Similar to previous case, we introduce positive constants $\\kappa _1$ , $\\kappa _2$ and $\\kappa _3$ that will be fixed later.",
"Next, we estimate the following term by applying Young's inequality, ${\\left\\lbrace \\begin{array}{ll}\\left| \\chi _0 Re(i\\xi \\hat{u}\\overline{\\hat{y}})\\right| \\le \\dfrac{\\rho _1 \\kappa _2\\varepsilon }{2\\kappa _1} \\dfrac{\\xi ^2}{1+\\xi ^2}|\\hat{u}|^2+C(\\varepsilon ,\\kappa _1,\\kappa _2)(1+\\xi ^2)|\\hat{y}|^2, \\\\\\\\\\dfrac{b_0\\beta \\rho _1}{k \\delta } \\xi ^2|\\hat{\\theta }||\\hat{u}|\\le \\dfrac{\\rho _1 \\kappa _2\\varepsilon }{2\\kappa _1} \\dfrac{\\xi ^2}{1+\\xi ^2}|\\hat{u}|^2+C(\\varepsilon ,\\kappa _1,\\kappa _2)(1+\\xi ^2)\\xi ^2|\\hat{\\theta }|^2,\\end{array}\\right.",
"}$ Hence, (REF ) can be written as $\\frac{d}{dt}K_1(\\xi ,t) + k(1-\\varepsilon )|\\hat{v}|^2 \\le \\frac{\\rho _1 \\kappa _2\\varepsilon }{\\kappa _1} \\frac{\\xi ^2}{1+\\xi ^2}|\\hat{u}|^2+C(\\varepsilon ,\\kappa _1,\\kappa _2)(1+\\xi ^2)|\\hat{y}|^2 \\\\+C(\\varepsilon , \\kappa _1,\\kappa _2)(1+\\xi ^2)\\xi ^2|\\hat{\\theta }|^2+C(\\varepsilon )\\xi ^4\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2.$ Thus, inequalities (REF ) and (REF ) imply that $\\frac{d}{dt}\\left\\lbrace \\kappa _1\\frac{\\xi ^2}{1+\\xi ^2} K_1(\\xi ,t)+\\kappa _2\\dfrac{\\xi ^2}{(1+\\xi ^2)^2}K_2(\\xi ,t) \\right\\rbrace + k\\left[\\kappa _1(1-\\varepsilon )-C(\\varepsilon )\\kappa _2\\right]\\frac{\\xi ^2}{1+\\xi ^2}|\\hat{v}|^2 \\\\+\\rho _1(1-2\\varepsilon )\\kappa _2\\dfrac{\\xi ^4}{(1+\\xi ^2)^2}|\\hat{u}|^2+a(1-\\varepsilon )\\kappa _2\\dfrac{\\xi ^4}{(1+\\xi ^2)^2}|\\hat{z}|^2 \\\\\\le C(\\varepsilon ,\\kappa _1,\\kappa _2)\\xi ^2|\\hat{y}|^2+C(\\varepsilon ,\\kappa _1,\\kappa _2)\\xi ^4|\\hat{\\theta }|^2+C(\\varepsilon , \\kappa _1,\\kappa _2)\\xi ^4\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2$ Furthermore, applying Young's inequality in (REF ) in the Lemma REF , it follows that $\\frac{d}{dt}\\kappa _3K_3(\\xi ,t) +\\rho _2\\kappa _3b_0 (1 -\\varepsilon )\\xi ^2|\\hat{y}|^2\\le C(\\varepsilon ,\\kappa _3)\\xi ^2\\int _0^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds+ a\\kappa _2\\varepsilon \\frac{\\xi ^4}{(1+\\xi ^2)^2}|\\hat{z}|^2 \\\\+ C(\\varepsilon ,\\kappa _2,\\kappa _3)\\xi ^2(1+\\xi ^2)^2\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2 +\\kappa _3m\\xi ^4\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2+k\\kappa _1\\varepsilon \\frac{\\xi ^2}{1+\\xi ^2}|\\hat{v}|^2 \\\\+C(\\varepsilon ,\\kappa _1,\\kappa _3)\\xi ^2(1+\\xi ^2)\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2+ \\kappa _3\\delta |\\xi |^3|\\hat{\\theta }|\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|.$ Thus, $\\frac{d}{dt}\\kappa _3K_3(\\xi ,t) +\\rho _2\\kappa _3b_0 (1 -\\varepsilon )\\xi ^2|\\hat{y}|^2\\le C(\\varepsilon ,\\kappa _3)\\xi ^2\\int _0^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds+ a\\kappa _2\\varepsilon \\frac{\\xi ^4}{(1+\\xi ^2)^2}|\\hat{z}|^2 +k\\kappa _1\\varepsilon \\frac{\\xi ^2}{1+\\xi ^2}|\\hat{v}|^2\\\\+ C(\\varepsilon ,\\kappa _1,\\kappa _2,\\kappa _3)\\xi ^2(1+\\xi ^2)^2\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2 + C(\\varepsilon )\\xi ^4|\\hat{\\theta }|^2$ Computing (REF ) $+$ (REF ), we obtain $\\frac{d}{dt}\\left\\lbrace \\kappa _1\\frac{\\xi ^2}{1+\\xi ^2} K_1(\\xi ,t)+\\kappa _2\\dfrac{\\xi ^2}{(1+\\xi ^2)^2}K_2(\\xi ,t) + \\kappa _3K_3(\\xi ,t) \\right\\rbrace + k\\left[\\kappa _1(1-2\\varepsilon )-C(\\varepsilon )\\kappa _2\\right]\\frac{\\xi ^2}{1+\\xi ^2}|\\hat{v}|^2 \\\\+\\rho _1(1-2\\varepsilon )\\kappa _2\\dfrac{\\xi ^4}{(1+\\xi ^2)^2}|\\hat{u}|^2 +a(1-2\\varepsilon )\\kappa _2\\dfrac{\\xi ^4}{(1+\\xi ^2)^2}|\\hat{z}|^2+\\rho _2\\left[\\kappa _3b_0(1 -\\varepsilon )- C(\\varepsilon ,\\kappa _1,\\kappa _2)\\right]\\xi ^2|\\hat{y}|^2 \\\\\\le C(\\varepsilon ,\\kappa _3)\\xi ^2\\int _0^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds+ C(\\varepsilon ,\\kappa _1,\\kappa _2,\\kappa _3)\\xi ^4|\\hat{\\theta }|^2+ C(\\varepsilon , \\kappa _1,\\kappa _2,\\kappa _3)\\xi ^2(1+\\xi ^2)^2\\left|\\int _0^{\\infty }g(s)\\hat{\\eta }(s)ds\\right|^2.$ From inequality (REF ), we conclude that $\\frac{d}{dt}\\left\\lbrace \\kappa _1\\frac{\\xi ^2}{1+\\xi ^2} K_1(\\xi ,t)+\\kappa _2\\dfrac{\\xi ^2}{(1+\\xi ^2)^2}K_2(\\xi ,t) + \\kappa _3K_3(\\xi ,t) \\right\\rbrace + k\\left[\\kappa _1(1-2\\varepsilon )-C(\\varepsilon )\\kappa _2\\right]\\frac{\\xi ^2}{1+\\xi ^2}|\\hat{v}|^2 \\\\+\\rho _1(1-2\\varepsilon )\\kappa _2\\dfrac{\\xi ^4}{(1+\\xi ^2)^2}|\\hat{u}|^2 +a(1-2\\varepsilon )\\kappa _2\\dfrac{\\xi ^4}{(1+\\xi ^2)^2}|\\hat{z}|^2+\\rho _2\\left[\\kappa _3b_0(1 -\\varepsilon )- C(\\varepsilon ,\\kappa _1,\\kappa _2)\\right]\\xi ^2|\\hat{y}|^2 \\\\\\le C(\\varepsilon , \\kappa _1,\\kappa _2,\\kappa _3)(1+b_0)\\xi ^2(1+\\xi ^2)^2\\int _0^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds+ C(\\varepsilon ,\\kappa _1,\\kappa _2,\\kappa _3)\\xi ^2(1+\\xi ^2)^2|\\hat{\\theta }|^2.$ In order to make all coefficients in the right-hand side in (REF ) positive, we have to choose appropriate constant $\\kappa _i$ .",
"First, let us fix $\\varepsilon $ , such that $\\varepsilon < \\frac{1}{2}.$ Thus, we can take $\\kappa _2>0$ and $\\kappa _1 >\\frac{C(\\varepsilon )\\kappa _2}{1-2\\varepsilon }, \\quad \\kappa _3 >\\frac{C(\\varepsilon ,\\kappa _1,\\kappa _2)}{b_0(1-\\varepsilon )}.$ Then, from (REF ) and the estimate $\\frac{\\xi ^2}{1+\\xi ^2}\\le 1$ , we can deduce the existence of a positive constant $M_1$ such that $\\frac{d}{dt}\\mathcal {Q}_1(\\xi ,t) \\le -M_1 \\frac{\\xi ^4}{(1+\\xi ^2)^2}\\left\\lbrace k|\\hat{v}|^2 +\\rho _1|\\hat{u}|^2+a|\\hat{z}|^2 +\\rho _2|\\hat{y}|^2\\right\\rbrace \\\\+ C_1(1+b_0)(1+\\xi ^2)^2\\xi ^2\\int _0^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds+C_1(1+\\xi ^2)^2\\xi ^2|\\hat{\\theta }|^2,$ where $\\mathcal {Q}_1(\\xi ,t)= \\kappa _1\\frac{\\xi ^2}{1+\\xi ^2} K_1(\\xi ,t)+\\kappa _2\\dfrac{\\xi ^2}{(1+\\xi ^2)^2}K_2(\\xi ,t)+\\kappa _3K_3(\\xi ,t)$ and $C_1$ is a positive constant that depends of $\\varepsilon $ and $\\kappa _j$ for $j=1,2,3$ .",
"Finally, we define the following Lyapunov functional: $\\mathcal {Q}(\\xi ,t) = \\mathcal {Q}_1(\\xi ,t,t)+N(1+\\xi ^2)^2\\hat{\\mathcal {E}}(\\xi ,t),$ where $N$ is a positive constant to be fixed later.",
"Note that the definition of $\\mathcal {Q}_1$ together with the inequality (REF ) imply that $\\left|\\mathcal {Q}_1(\\xi ,t)\\right| & \\le M_2 (1+\\xi ^2)^2\\hat{\\mathcal {E}}(\\xi ,t)$ for some positive constant $M_2$ .",
"Hence, we obtain $(N-M_2)(1+\\xi ^2)^2\\hat{\\mathcal {E}}(\\xi ,t) \\le \\mathcal {Q}(\\xi ,t)\\le (N+M_2)(1+\\xi ^2)^2 \\hat{\\mathcal {E}}(\\xi ,t).$ On the other hand, taking the derivative of $\\mathcal {Q}$ with respect to $t$ and using the estimates (REF ) and Lemma REF , it follows that $\\frac{d}{dt}\\mathcal {Q}(\\xi ,t) \\le -M_1 \\frac{\\xi ^4}{(1+\\xi ^2)^2}\\left\\lbrace k|\\hat{v}|^2 +\\rho _1|\\hat{u}|^2+a|\\hat{z}|^2 + \\rho _2|\\hat{y}|^2\\right\\rbrace -\\left(2N\\tilde{\\beta }-C_1\\right)(1+\\xi ^2)^2\\xi ^2|\\hat{\\theta }|^2 \\\\-\\left( Nk_1m - C_1 (1+b_0)\\right)(1+\\xi ^2)^2\\xi ^2\\int _0^{\\infty }g(s)|\\hat{\\eta }(s)|^2ds.$ Now, choosing $N$ such that $N\\ge \\max \\left\\lbrace M_2,\\dfrac{C_1}{2\\tilde{\\beta }}, \\dfrac{C_1(1+b_0)}{k_1 m}\\right\\rbrace $ and noting that $(1+\\xi ^2)^2 \\ge \\dfrac{\\xi ^4}{(1+\\xi ^2)^2}$ and $(1+\\xi ^2)^2 \\ge \\dfrac{\\xi ^2}{(1+\\xi ^2)^2}$ , there exists a positive constant $M_3$ such that $\\frac{d}{dt}\\mathcal {Q}(\\xi ,t) \\le - M_3\\frac{\\xi ^4}{(1+\\xi ^2)^2} \\hat{\\mathcal {E}}(\\xi ,t).$ Estimate $(\\ref {eq36})$ implies that $\\frac{d}{dt}\\mathcal {Q}(\\xi ,t) \\le - \\Gamma \\frac{\\xi ^4}{(1+\\xi ^2)^4} \\mathcal {Q}(\\xi ,t)$ where $\\Gamma =\\dfrac{M_3}{(N+M_2)}$ .",
"By Gronwall's inequality, it follows that $\\mathcal {Q}(\\xi ,t) \\le e^{-\\Gamma \\rho (\\xi )t}\\mathcal {Q}(\\xi ,0), \\quad \\rho (\\xi )=\\frac{\\xi ^4}{(1+\\xi ^2)^4},$ again by using $(\\ref {e29})$ , we have that $\\hat{\\mathcal {E}}(\\xi ,t) \\le C e^{-\\Gamma \\rho (\\xi )t}\\hat{\\mathcal {E}}(\\xi ,0), \\quad \\text{where}\\quad C= \\frac{N+M_2}{N-M_2}> 0.$"
],
[
"The decay estimates",
"In this section, we establish the decay rates of solutions $U(x, t)$ , $V(x, t)$ of systems (REF )-(REF ) and (REF )-(REF ), respectively.",
"By using the energy inequalities in the Fourier space, we show that the decay rates depend of condition $\\chi _{0,\\tau }=0$ or $\\chi _{0,\\tau }\\ne 0$ (resp.",
"$\\chi _{0}=0$ or $\\chi _{0}\\ne 0$ ).",
"In any case, the regularity loss phenomenon is present.",
"This first main result reads as follows: Theorem 4.1 Let $s$ be a nonnegative integer and $\\chi _{0,\\tau }=\\left( \\tau -\\frac{\\rho _1}{\\rho _3 k}\\right)\\left( \\rho _2 -\\frac{b \\rho _1}{k}\\right) - \\frac{\\tau \\rho _1 \\delta ^2}{\\rho _3 k}.$ Suppose that $U_0 \\in \\mathbb {H}^s(\\mathbb {R})\\cap \\mathbb {L}^1(\\mathbb {R})$ , where $\\mathbb {H}^s(\\mathbb {R}):=\\left[H^s(\\mathbb {R})\\right]^6\\times L^2_g(\\mathbb {R};H^{1+s}(\\mathbb {R}))\\qquad and \\qquad \\mathbb {L}^1(\\mathbb {R}):=\\left[L^1(\\mathbb {R})\\right]^6\\times L^2_g(\\mathbb {R};W^{1,1}(\\mathbb {R})).$ Then, the solution $U$ of the system (REF ), satisfies the following decay estimates, If $\\chi _{0,\\tau } =0$ , then $\\Vert \\partial ^k_xU(t)\\Vert _2\\le C(1+t)^{-\\frac{1}{8}-\\frac{k}{4}}\\Vert U_0\\Vert _1 + C (1+t)^{-\\frac{l}{2}}\\Vert \\partial _x^{k+l}U_0\\Vert _2, \\quad t\\ge 0.$ If $\\chi _{0,\\tau }\\ne 0$ , then $\\Vert \\partial ^k_xU(t)\\Vert _2\\le C(1+t)^{-\\frac{1}{8}-\\frac{k}{4}}\\Vert U_0\\Vert _1 + C(1+t)^{-\\frac{l}{4}}\\Vert \\partial _x^{k+l}U_0\\Vert _2,\\quad t\\ge 0.$ where $k+l \\le s$ , $C$ and $c$ are two positive constants.",
"Applying the Plancherel's identity, we have $\\Vert \\partial ^k_xU(t)\\Vert _2^2 = \\Vert (i\\xi )^k\\hat{U}(t)\\Vert _2^2 = \\int _{\\mathbb {R}}|\\xi |^{2k}\\left|\\hat{U}(\\xi ,t)\\right|^2 d\\xi .$ It is easy to see that $c_1\\left|\\hat{U}(\\xi ,t)\\right|^2 \\le \\hat{E}(\\xi ,t) \\le c_2\\left|\\hat{U}(\\xi ,t)\\right|^2,$ for some positive constant $c_1$ and $c_2$ .",
"Thus, it follows that $\\Vert \\partial ^k_xU(t)\\Vert _2^2 \\le \\frac{1}{c_1} \\int _{\\mathbb {R}}|\\xi |^{2k}\\hat{E}(\\xi ,t) d\\xi .$ From Theorems REF and (REF ), there exist a postie constant $M>0$ , such that $\\Vert \\partial ^k_xU(t)\\Vert _2^2 &\\le M\\int _{\\mathbb {R}}|\\xi |^{2k}e^{-\\lambda \\rho (\\xi )t}\\left|\\hat{U}(0,\\xi )\\right|^2 d\\xi \\\\&\\le M \\underbrace{\\int _{\\left\\lbrace |\\xi |\\le 1\\right\\rbrace }|\\xi |^{2k}e^{-\\lambda \\rho (\\xi )t}\\left|\\hat{U}_0(\\xi )\\right|^2 d\\xi }_{I_1} + \\underbrace{ M\\int _{\\left\\lbrace |\\xi |\\ge 1\\right\\rbrace }|\\xi |^{2k}e^{-\\lambda \\rho (\\xi )t}\\left|\\hat{U}_0(\\xi )\\right|^2 d\\xi }_{I_2}$ Case $\\chi _{0,\\tau } = 0$ : It is not difficult to see that the function $\\rho (\\cdot )$ satisfies $\\left\\lbrace \\begin{tabular}{l c l}\\rho (\\xi ) \\ge \\frac{1}{8}\\xi ^4 & if & |\\xi |\\le 1, \\\\\\\\\\rho (\\xi ) \\ge \\frac{1}{8}\\xi ^{-2} & if & |\\xi |\\ge 1.\\end{tabular}\\right.$ Thus, we estimate $I_1$ as follows, $I_1 \\le M \\Vert \\hat{U_0}\\Vert _{L^\\infty }^2 \\int _{|\\xi |\\le 1}|\\xi |^{2k}e^{-\\frac{\\lambda }{8}\\xi ^4t} d\\xi \\le C_1\\Vert \\hat{U_0}\\Vert _{L^\\infty }^2\\left( 1+t\\right)^{-\\frac{1}{4}(1+2k)} \\le C_1\\left( 1+t\\right)^{-\\frac{1}{4}(1+2k)} \\Vert U_0\\Vert _{L^1}^2.$ On the other hand, by using the second inequality in (REF ), we obtain $I_2 &\\le M\\int _{|\\xi |\\ge 1}|\\xi |^{2k}e^{-\\frac{\\lambda }{8}\\xi ^{-2}t}\\left|\\hat{U_0}(\\xi )\\right|^2d\\xi \\le M \\sup _{|\\xi | \\ge 1}\\lbrace |\\xi |^{-2l}e^{-\\frac{\\lambda }{8} \\xi ^{-2}t} \\rbrace \\int _{\\mathbb {R}}|\\xi |^{2(k+l)}\\left|\\hat{U_0}^2(\\xi )\\right|^2d\\xi \\\\&\\le C_2 (1+t)^{-l} \\Vert \\partial _x^{k+l}U_0\\Vert _2^2.$ Combining the estimates of $I_1$ and $I_2$ , we obtain $(\\ref {e38})$ .",
"Case $\\chi _{0,\\tau } \\ne 0$ : In this cases, the function $\\rho (\\cdot )$ satisfies $\\left\\lbrace \\begin{tabular}{l c l}\\rho (\\xi ) \\ge \\frac{1}{16}\\xi ^4 & if & |\\xi |\\le 1 \\\\\\\\\\rho (\\xi ) \\ge \\frac{1}{16}\\xi ^{-4} & if & |\\xi |\\ge 1\\end{tabular}\\right.$ Thus, we estimate $I_1$ as following, $I_1 \\le M \\Vert \\hat{U_0}\\Vert _{L^\\infty }^2 \\int _{|\\xi |\\le 1}|\\xi |^{2k}e^{-\\frac{\\lambda }{16}\\xi ^4t} d\\xi \\le C_1\\Vert \\hat{V_0}\\Vert _{L^\\infty }^2\\left( 1+t\\right)^{-\\frac{1}{4}(1+2k)} \\le C_1\\left( 1+t\\right)^{-\\frac{1}{4}(1+2k)} \\Vert U_0\\Vert _{L^1}^2.$ Moreover, by using the second inequality in (REF ), it follows that $I_2 &\\le M\\int _{|\\xi |\\ge 1}|\\xi |^{2k}e^{-\\frac{\\lambda }{16}\\xi ^{-4}t}\\left|\\hat{U_0}(\\xi )\\right|^2d\\xi \\le M \\sup _{|\\xi | \\ge 1}\\lbrace |\\xi |^{-2l}e^{-\\frac{\\lambda }{16} \\xi ^{-4}t} \\rbrace \\int _{\\mathbb {R}}|\\xi |^{2(k+l)}\\left|\\hat{U_0}^2(\\xi )\\right|^2d\\xi \\\\&\\le C_2 (1+t)^{-\\frac{l}{2}} \\Vert \\partial _x^{k+l}U_0\\Vert _2^2.$ Combining the estimates of $I_1$ and $I_2$ , we obtain $(\\ref {e39})$ .",
"Similar to the proof of Theorem REF , we establish decay estimates of the solution $V(x,t)$ of Timoshenko-Fourier system (REF )-(REF ).",
"The proof of next theorem is carried out by the same technique as that of Theorem REF .",
"Therefore, we omit it.",
"Theorem 4.2 Let $s$ be a nonnegative integer and $\\chi _{0}=\\left( \\rho _2 -\\frac{b \\rho _1}{k}\\right).$ Suppose that $V_0 \\in \\mathbb {H}^s(\\mathbb {R})\\cap \\mathbb {L}^1(\\mathbb {R})$ , where $\\mathbb {H}^s(\\mathbb {R}):=\\left[H^s(\\mathbb {R})\\right]^5\\times L^2_g(\\mathbb {R};H^{1+s}(\\mathbb {R}))\\qquad and \\qquad \\mathbb {L}^1(\\mathbb {R}):=\\left[L^1(\\mathbb {R})\\right]^5\\times L^2_g(\\mathbb {R};W^{1,1}(\\mathbb {R})).$ Then, the solution $V$ of the system (REF ), satisfies the following decay estimates, If $\\chi _{0} =0$ , then $\\Vert \\partial ^k_xV(t)\\Vert _2\\le C_1(1+t)^{-\\frac{1}{8}-\\frac{k}{4}}\\Vert V_0\\Vert _1 + C_2(1+t)^{-\\frac{l}{2}}\\Vert \\partial _x^{k+l}V_0\\Vert _2, \\quad t\\ge 0.$ If $\\chi _{0}\\ne 0$ , then $\\Vert \\partial ^k_xV(t)\\Vert _2\\le C_1(1+t)^{-\\frac{1}{8}-\\frac{k}{4}}\\Vert V_0\\Vert _1 + C_2(1+t)^{-\\frac{l}{4}}\\Vert \\partial _x^{k+l}V_0\\Vert _2, \\quad t\\ge 0.$ where $k+l \\le s$ , $C_1,C_2$ are two positive constants."
],
[
"Acknowledgments",
"The first author was partially supported by Facultad de Ciencias Exactas y Naturales, Unversidad Nacional de Colombia Sede Manizales, under project number 45511."
]
] | 1808.08499 |
[
[
"Dirac spinors and their application to Bianchi-I space-times in 5\n dimensions"
],
[
"Abstract We consider a five-dimensional Einstein--Cartan spacetime upon which Dirac spinor fields can be defined.",
"Dirac spinor fields in five and four dimensions share many features, like the fact that both are described by four-component spinor fields, but they are also characterized by strong differences, like the fact that in five dimensions we do not have the possibility to project on left-handed and right-handed chiral parts unlike what happens in the four-dimensional instance: we conduct a polar decomposition of the spinorial fields, so to highlight all similarities and discrepancies.",
"As an application of spinor fields in five dimensions, we study Bianchi-I spacetimes, verifying whether the Dirac fields in five dimensions can give rise to inflation or dark-energy dominated cosmological eras or not."
],
[
"Introduction",
"The spin-$\\frac{1}{2}$ spinor field is the only type of spinor field that we have observed so far in nature and quite possibly one of the most fundamental fields we can define in general: according to the well-known Lounesto classification [1], it may represent a regular spinor, like a Dirac field (describing massive-charged particles), or it may represent a singular spinor, either of Weyl type (describing massless particles) or of Majorana type (describing neutral particles).",
"Particles such as the neutrinos may be Majorana fields, while charged leptons and quarks are Weyl or Dirac according to whether they are considered before or after symmetry breaking in the standard model of particle physics.",
"One key property of all these types of spinor fields is the fact that they are rather sensitive to the dimension of the space in which they live: so for instance, in three dimensions, Dirac spinors are the well known Pauli spinors, and they are described by a two components complex (column) fields; in four dimensions, however, Dirac spinors are described by a four components complex (column) fields.",
"The differences arise from the fact that Dirac spinor fields are defined in terms of an underlying structure known as Clifford algebra, and it is this algebra that is sensitive to the dimension: so for example, in three dimensions the Clifford algebra is built up in terms of three mutually anti-commuting matrices that can be taken to be the $2\\times 2$ Pauli matrices, while in four dimensions the Clifford algebra must contain a fourth matrix anti-commuting with the other three matrices and there is no way to do this unless the Pauli matrices are extended to the $4\\times 4$ Dirac matrices.",
"Hence, in three and four dimensions, the corresponding spinor fields have two and four complex components, respectively.",
"Nevertheless, spinor fields defined in a given odd dimension preserves the spinorial structure of the co-dimension one spacetime, i.e., spinor fields defined in two dimensions have the same number of components as the spinor fields defined in three dimensions, as well as spinor fields defined four dimensions have the same number of components as the spinor fields defined in five dimensions.",
"And therefore, taking the standard definition of spinor field as the four-dimensional one, the five-dimensional space is somewhat special, because among all different spacetimes, it is the only one for which the spinorial structure is unchanged.",
"As a consequence, it becomes interesting to ask what would change in the spinorial structure if the four-dimensional spinor were not defined in four dimensions but obtained as the result of a dimensional reduction from a higher-dimensional spacetime, and among all of them the five-dimensional spacetime is a perfect starting place.",
"In this paper we are going to do precisely this: we will give the five-dimensional definition of spinors and eventually reduce to the four-dimensional standard case, and in the process we shall stress on the analogies and differences between the two approaches.",
"As an application of our study, we investigate cosmological scenarios arising from the presence of Dirac fields in five-dimensional Bianchi-I Universes.",
"In cosmology, spinor fields have been largely studied, both minimally and non-minimally coupled to gravity (for example, see references [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17] and references therein); in general, non-minimal coupling or self-interaction potentials are seen to be necessary for the Dirac fields to generate inflationary or dark-energy dominated eras.",
"Here we consider Dirac fields minimally coupled to five-dimensional Einstein-Cartan gravity, without self-interaction potential.",
"More in particular, relying on the additional dimensional degree of freedom, we want to verify whether a contraction in the fourth spatial dimension corresponds to a (possibly accelerated) expansion in the remaining other three dimensions, thus giving rise to a four-dimensional expanding Universe.",
"As we shall see, the non-diagonal part of the Einstein-like equations, together with the anisotropy on the fourth spatial dimension, impose stricter constraints than those appearing in four dimensions.",
"This reduces the admissible forms of the spinor fields and simplifies the resulting field equations.",
"We give explicit solutions of the dynamical equations, showing that Dirac fields in five dimensions can not be considered as the source of accelerated expansions of our Universe, because at the most they are seen to generate Friedmann eras.",
"Let us consider a five-dimensional manifold $\\pi : Q \\rightarrow M$ , fibered over a four-dimensional spacetime $M$ , and allowing a metric tensor of signature $(1,-1,-1,-1,-1)$ .",
"For our purposes, we do not need to assume that the fibration $\\pi : Q \\rightarrow M$ has any particular structure (for example, we do not require $\\pi : Q \\rightarrow M$ to be a principal fiber bundle).",
"Introducing a fünfbein $e^\\mu =e^\\mu _i\\,dx^i$ defined on the manifold $Q$ , the metric tensor can be expressed as $g=\\eta _{\\mu \\nu }\\,e^\\mu \\otimes e^\\nu $ where $\\eta _{\\mu \\nu }=\\eta ^{\\mu \\nu }={\\rm diag}(1,-1,-1,-1,-1)$ .",
"Greek and Latin indices run from zero to four: Latin indices label local coordinates on the manifold $Q$ , while Greek indices label elements of local orthonormal frames and co-frames undergoing five-dimensional Lorentz transformations.",
"Writing $e_\\mu =e^i_\\mu \\partial _{i}$ as the dual frame of $e^\\mu $ we have $e^j_\\mu e^\\mu _i= \\delta ^j_i\\\\e^j_\\mu e^\\nu _j= \\delta ^\\nu _\\mu $ as the duality relations allowing us to get the fünfbein from the dual fünfbein.",
"The assignment of a metric-compatible linear connection $\\Gamma _{ij}{}^{h}$ on $Q$ induces a corresponding spin-connection defined as $\\omega ^{\\;\\;\\mu }_{i\\;\\;\\;\\nu } = \\Gamma _{ij}^{\\;\\;\\;h} e^\\mu _h e^j_\\nu - e^j_\\nu \\frac{\\partial {e^\\mu _j}}{\\partial {x^i}}\\ \\ \\ \\ \\mathrm {with}\\ \\ \\ \\ \\omega _i^{\\;\\;\\mu \\nu } =-\\omega _i^{\\;\\;\\nu \\mu }$ according to the requirement that the covariant derivative applied to the fünfbein and the metric be zero identically.",
"The simultaneous introduction of a fünfbein and a spin-connection generates corresponding torsion and curvature tensors expressed in local coordinates as $T^{\\;\\;\\mu }_{ij} =\\partial _i e^\\mu _j-\\partial _j e^\\mu _i+\\omega ^{\\;\\;\\mu }_{i\\;\\;\\;\\nu }e^\\nu _{j}-\\omega ^{\\;\\;\\mu }_{j\\;\\;\\;\\nu }e^\\nu _{i}$ $R_{ij}^{\\;\\;\\;\\;\\mu \\nu } =\\partial _i\\omega _{j}^{\\;\\;\\mu \\nu }-\\partial _j{\\omega _{i}^{\\;\\;\\mu \\nu }}+\\omega ^{\\;\\;\\mu }_{i\\;\\;\\;\\lambda }\\omega _{j}^{\\;\\;\\lambda \\nu }-\\omega ^{\\;\\;\\mu }_{j\\;\\;\\;\\lambda }\\omega _{i}^{\\;\\;\\lambda \\nu }$ and by contraction, from equation (REF ) we derive the expressions of the Ricci tensor $R^i_{\\;j}:=R_{\\sigma \\mu }^{\\phantom{\\sigma \\mu }\\sigma \\lambda }e^{i}_{\\lambda }e^\\mu _j$ and the Ricci scalar $R= R_{ij}^{\\phantom{ij}\\mu \\nu } e^i_\\mu e^j_\\nu $ , while no contraction will be considered for torsion since it will be taken to be completely antisymmetric (the reasons of this will become clear later on in the development of the theory).",
"Following a standard procedure, we decompose the linear connection $\\Gamma _{ij}^{\\;\\;\\;h}$ into the Levi-Civita connection $\\tilde{\\Gamma }_{ij}^{\\;\\;\\;h}$ (associated with the metric $g$ ) plus torsional contributions so that $\\nabla _{i}A_{j} = \\tilde{\\nabla }_{i}A_{j} - \\frac{1}{2}T^{\\phantom{ij}h}_{ij}A_{h}$ where $\\tilde{\\nabla }_{i}$ is the Levi-Civita covariant derivative and where the total antisymmetry of torsion was used.",
"Analogously, given the antisymmetry of torsion, the Ricci tensor and Ricci scalar are decomposed as $R_{ij}=\\tilde{R}_{ij}+\\frac{1}{2} \\tilde{\\nabla }_p T_{ji}^{\\;\\;\\;p}-\\frac{1}{4}T_{pi}^{\\;\\;\\;q} T_{jq}^{\\;\\;\\;p}$ $R=\\tilde{R}-\\frac{1}{4}T_{qpr}T^{qpr}$ with $\\tilde{R}_{ij}$ and $\\tilde{R}$ respectively the Ricci tensor and Ricci scalar of the spin connection $\\tilde{\\omega }_{j}^{\\;\\;\\mu \\nu }$ which is associated with the Levi-Civita connection.",
"For the matter content, we employ representations of the Clifford algebra given in terms of five Dirac matrices $\\gamma ^{\\mu }$ satisfying $\\left\\lbrace \\gamma _{\\mu },\\gamma _{\\nu }\\right\\rbrace = 2 \\eta _{\\mu \\nu } \\mathbb {I}$ like in the four-dimensional case.",
"We recall that in four as well as five dimensions, the minimal size of these Dirac matrices is $4\\times 4$ although in the five-dimensional situation they will no longer be block-diagonal, even in chiral representation, as we shall see in a while.",
"Defining $S_{\\mu \\nu } = \\frac{1}{8} \\left[\\gamma _{\\mu },\\gamma _{\\nu }\\right]$ it is possible to verify that they satisfy the commutation relationships of the Lorentz algebra, although in five dimensions there no longer is a non-identity matrix commuting with all generators: this shows that no representation can be reducible, compatibly with the fact that we cannot find block-diagonal representations.",
"We also have the identities $\\gamma _{\\mu }\\gamma _{\\nu }\\gamma _{\\lambda } = \\eta _{\\nu \\lambda }\\gamma _{\\mu }-\\eta _{\\mu \\lambda }\\gamma _{\\nu }+\\eta _{\\mu \\nu }\\gamma _{\\lambda }-\\frac{1}{2}\\epsilon _{\\mu \\nu \\lambda \\alpha \\beta }\\gamma ^{\\alpha }\\gamma ^{\\beta }$ as well as the contractions $&\\gamma ^{\\mu }S^{\\alpha \\beta }\\gamma _{\\mu }= S^{\\alpha \\beta }\\\\&\\lbrace \\gamma ^\\mu ,S^{\\nu \\lambda }\\rbrace = -\\epsilon ^{\\mu \\nu \\lambda \\tau \\rho }S_{\\tau \\rho }\\\\&[\\gamma ^\\mu ,S^{\\nu \\lambda }]= \\frac{1}{2}(\\gamma ^\\lambda \\eta ^{\\mu \\nu }-\\gamma ^\\nu \\eta ^{\\mu \\lambda })$ with $\\epsilon ^{\\mu \\nu \\lambda \\tau \\rho }$ denoting the Levi-Civita completely antisymmetric five-dimensional pseudo-tensor.",
"It is important to notice that the introduction of this tensor makes the set of formulas sensitive to the dimension of the space, as for example the first two of these identities would be very different for the standard number of four dimensions, although equation (REF ) remains unchanged, as it should be, since it is what ensures the correct transformation law of the Dirac matrices, and so the Lorentzian structure.",
"An explicit choice of the Dirac matrices is the chiral representation $\\begin{pmatrix}0 & \\mathbb {I}\\\\\\mathbb {I} & 0\\end{pmatrix} = \\gamma ^{0}\\quad \\begin{pmatrix}0 & \\sigma ^\\mathcal {A}\\\\-\\sigma ^\\mathcal {A} & 0\\end{pmatrix} = \\gamma ^\\mathcal {A}\\quad \\begin{pmatrix}i\\mathbb {I} & 0\\\\0 &-i\\mathbb {I}\\end{pmatrix} = \\gamma ^{4}$ where the sigmas ($\\mathcal {A}=1,2,3$ ) are the Pauli matrices.",
"The complex Lorentz transformation laws are given by $\\Lambda = e^{S^{ab}\\theta _{ab}}$ and they amount to $\\begin{aligned}\\Lambda _{R12}= \\begin{pmatrix}e^{i\\frac{\\theta }{2}} & 0 & 0 & 0 \\\\0 & e^{-i\\frac{\\theta }{2}} & 0 & 0 \\\\0 & 0 & e^{i\\frac{\\theta }{2}} & 0 \\\\0 & 0 & 0 & e^{-i\\frac{\\theta }{2}}\\end{pmatrix}&\\Lambda _{R34}= \\begin{pmatrix}\\cos {\\frac{\\theta }{2}} & 0 & i\\sin {\\frac{\\theta }{2}} & 0 \\\\0 & \\cos {\\frac{\\theta }{2}} & 0 & -i\\sin {\\frac{\\theta }{2}} \\\\i\\sin {\\frac{\\theta }{2}} & 0 & \\cos {\\frac{\\theta }{2}} & 0 \\\\0 & -i\\sin {\\frac{\\theta }{2}} & 0 & \\cos {\\frac{\\theta }{2}}\\end{pmatrix}\\\\\\Lambda _{R31}= \\begin{pmatrix}\\cos {\\frac{\\theta }{2}} & \\sin {\\frac{\\theta }{2}} & 0 & 0 \\\\-\\sin {\\frac{\\theta }{2}} & \\cos {\\frac{\\theta }{2}} & 0 & 0 \\\\0 & 0 & \\cos {\\frac{\\theta }{2}} & \\sin {\\frac{\\theta }{2}} \\\\0 & 0 & -\\sin {\\frac{\\theta }{2}} & \\cos {\\frac{\\theta }{2}}\\end{pmatrix}&\\Lambda _{R42}= \\begin{pmatrix}\\cos {\\frac{\\theta }{2}} & 0 & 0 & \\sin {\\frac{\\theta }{2}} \\\\0 & \\cos {\\frac{\\theta }{2}} & -\\sin {\\frac{\\theta }{2}} & 0 \\\\0 & \\sin {\\frac{\\theta }{2}} & \\cos {\\frac{\\theta }{2}} & 0 \\\\-\\sin {\\frac{\\theta }{2}} & 0 & 0 & \\cos {\\frac{\\theta }{2}}\\end{pmatrix}\\\\\\Lambda _{R23}= \\begin{pmatrix}\\cos {\\frac{\\theta }{2}} & i\\sin {\\frac{\\theta }{2}} & 0 & 0 \\\\i\\sin {\\frac{\\theta }{2}} & \\cos {\\frac{\\theta }{2}} & 0 & 0 \\\\0 & 0 & \\cos {\\frac{\\theta }{2}} & i\\sin {\\frac{\\theta }{2}} \\\\0 & 0 & i\\sin {\\frac{\\theta }{2}} & \\cos {\\frac{\\theta }{2}}\\end{pmatrix}&\\Lambda _{R14}= \\begin{pmatrix}\\cos {\\frac{\\theta }{2}} & 0 & 0 & i\\sin {\\frac{\\theta }{2}} \\\\0 & \\cos {\\frac{\\theta }{2}} & i\\sin {\\frac{\\theta }{2}} & 0 \\\\0 & i\\sin {\\frac{\\theta }{2}} & \\cos {\\frac{\\theta }{2}} & 0 \\\\i\\sin {\\frac{\\theta }{2}} & 0 & 0 & \\cos {\\frac{\\theta }{2}}\\end{pmatrix}\\end{aligned}$ as the rotations in all possible pairs of planes with $\\theta $ angles and $\\begin{aligned}\\Lambda _{B1}= \\begin{pmatrix}\\cosh {\\frac{\\zeta }{2}} & \\sinh {\\frac{\\zeta }{2}} & 0 & 0 \\\\\\sinh {\\frac{\\zeta }{2}} & \\cosh {\\frac{\\zeta }{2}} & 0 & 0 \\\\0 & 0 & \\cosh {\\frac{\\zeta }{2}} & -\\sinh {\\frac{\\zeta }{2}} \\\\0 & 0 & -\\sinh {\\frac{\\zeta }{2}} & \\cosh {\\frac{\\zeta }{2}}\\end{pmatrix}&\\Lambda _{B2}= \\begin{pmatrix}\\cosh {\\frac{\\zeta }{2}} & i\\sinh {\\frac{\\zeta }{2}} & 0 & 0 \\\\-i\\sinh {\\frac{\\zeta }{2}} & \\cosh {\\frac{\\zeta }{2}} & 0 & 0 \\\\0 & 0 & \\cosh {\\frac{\\zeta }{2}} & -i\\sinh {\\frac{\\zeta }{2}} \\\\0 & 0 & i\\sinh {\\frac{\\zeta }{2}} & \\cosh {\\frac{\\zeta }{2}}\\end{pmatrix}\\\\\\Lambda _{B3}= \\begin{pmatrix}e^{-\\frac{\\zeta }{2}} & 0 & 0 & 0 \\\\0 & e^{\\frac{\\zeta }{2}} & 0 & 0 \\\\0 & 0 & e^{\\frac{\\zeta }{2}} & 0 \\\\0 & 0 & 0 & e^{-\\frac{\\zeta }{2}}\\end{pmatrix}&\\Lambda _{B4}= \\begin{pmatrix}\\cosh {\\frac{\\zeta }{2}} & 0 & i\\sinh {\\frac{\\zeta }{2}} & 0 \\\\0 & \\cosh {\\frac{\\zeta }{2}} & 0 & i\\sinh {\\frac{\\zeta }{2}} \\\\-i\\sinh {\\frac{\\zeta }{2}} & 0 & \\cosh {\\frac{\\zeta }{2}} & 0 \\\\0 & -i\\sinh {\\frac{\\zeta }{2}} & 0 & \\cosh {\\frac{\\zeta }{2}}\\end{pmatrix}\\end{aligned}$ as the boosts along all axes with $\\zeta $ rapidities, and where from the identity (REF ) in the form $\\Lambda \\gamma ^{\\nu }\\Lambda ^{-1}\\Lambda ^{\\mu }_{\\phantom{\\mu }\\nu }=\\gamma ^{\\mu }$ it is possible to check the form of the matrices $\\Lambda ^{\\mu }_{\\phantom{\\mu }\\nu }$ yielding the expressions for the real representation of the generic transformation of the Lorentz group.",
"Dirac spinor fields in five dimensions are defined exactly like in the four-dimensional counterpart, and so in terms of a column of four complex scalar fields indicated with $\\psi $ in general, although now they will be characterized by different transformation properties, as it is clear from the fact that, despite all matrices on the left are the standard four-dimensional ones with block-diagonal form, all matrices written on the right are those involving the fifth dimension without block-diagonal form even in chiral representation: as a consequence of this non-reducible structure there will always be some mixing between the left-handed and the right-handed parts, which cannot therefore be defined as separate projections.",
"Because it is still $\\gamma _{0}\\gamma _{\\mathcal {A}}^{\\dagger }\\gamma _{0}\\!=\\!\\gamma _{\\mathcal {A}}$ for $\\mathcal {A}=1,2,3$ but now it is also $\\gamma _{0}\\gamma _{4}^{\\dagger }\\gamma _{0}\\!=\\!\\gamma _{4}$ then we have $\\gamma _{0}\\gamma _{\\mu }^{\\dagger }\\gamma _{0}\\!=\\!\\gamma _{\\mu }$ in general: as a consequence $\\gamma _{0}S_{\\mu \\nu }^{\\dagger }\\gamma _{0}\\!=\\!-S_{\\mu \\nu }$ and thus $\\gamma _{0} \\Lambda ^{\\dagger }\\gamma _{0}\\!=\\!\\Lambda ^{-1}$ ensuring that the conjugation is $\\bar{\\psi } = \\psi ^{\\dagger }\\gamma _{0}$ also in 5 dimensions.",
"The Dirac spinorial bilinears are defined as $&4i\\bar{\\psi }S^{\\mu \\nu }\\psi = M^{\\mu \\nu }\\\\&\\bar{\\psi }\\gamma ^{\\mu }\\psi = U^{\\mu }\\\\&\\bar{\\psi }\\psi =\\Phi $ so to turn out all real and transforming with $\\Lambda ^{\\mu }_{\\phantom{\\mu }\\nu }$ as the real representation of the Lorentz group.",
"They verify $&M_{\\mu \\nu }U^{\\mu }=0\\\\&\\frac{1}{4}M_{\\mu \\nu }M^{\\mu \\nu }=U_{\\mu }U^{\\mu }=\\Phi ^{2}$ as a direct substitution would straightforwardly show and which are called Fierz identities.",
"The spinorial covariant derivative is defined as in the usual four-dimensional framework.",
"In particular, equation (REF ) gives the possibility to see what is the form of the Fock-Ivanenko coefficients in the spinorial covariant derivative $D_{i}\\psi = \\partial _{i}\\psi + \\omega _{i}^{\\phantom{i}\\mu \\nu }S_{\\mu \\nu }\\psi $ defined in this way so that once applied to the gamma matrices it vanishes identically.",
"We now proceed to discuss a way to categorize spinors according to the idea of the Lounesto classification [1] (a complementary but similar classification is done by Cavalcanti in reference [2]).",
"In the standard case of four dimensions, we have two main classes according to whether the scalar and pseudo-scalar are both zero or not, but in the five-dimensional case the pseudo-scalar has become the fifth component of the five-vector $U^{\\mu }$ and because there exist transformations mixing it with the other components then requiring its vanishing does not make sense covariantly; however, it still makes sense to require the vanishing of the scalar, which is unchanged: thus we have the two cases, given by either $\\Phi = 0$ or not.",
"As in references [1], [2], we shall call these cases singular when $\\Phi = 0$ , and regular in the most general circumstance where this constraint does not hold.",
"In the four-dimensional situation, it has been shown in reference [3] that it is always possible to find Lorentz transformations bringing the spinor in the most general case in the following form $\\psi = \\phi \\begin{pmatrix}e^{i\\frac{\\beta }{2}}\\\\0\\\\e^{-i\\frac{\\beta }{2}}\\\\0\\end{pmatrix}$ where $\\phi $ and $\\beta $ are real scalars or, in the special case in which both scalar and pseudo-scalar vanish, in the form $\\psi = e^{i\\xi } \\begin{pmatrix}\\cos {\\frac{\\theta }{2}}\\\\0\\\\0\\\\\\sin {\\frac{\\theta }{2}}\\end{pmatrix}$ where $\\xi $ and $\\theta $ are real scalars, and should be set to zero, $\\xi =0$ , if the spinor is also charged.",
"These results are general, obtained only through use of the specific form of the Lorentz transformations.",
"In the five-dimensional situation, Lorentz transformations are changed and therefore we should expect that even the very same analysis would furnish somewhat different results on the spinorial structure: we will now see how this polar decomposition would change.",
"As above, we split the regular and singular situations in what follows.",
"In the case of regular spinor we have that no constraint is given: the Fierz orthogonal identity, equation (), tells us that $M_{\\mu \\nu }M^{\\mu \\nu } = U_{\\mu }U^{\\mu } > 0$ and thus, in particular, that $U^{\\mu }$ is time-like with the consequence that it is always possible to employ four boosts to remove one by one all its spatial components; then, employing rotations we can remove components of the $M^{\\mu \\nu }$ tensor.",
"Because the fifth component of the five-vector $U^{\\mu }$ is what in the four-dimensional case would be the pseudo-scalar, the analysis essentially reduces to that of the four-dimensional case plus the additional restriction of vanishing of the pseudo-scalar.",
"This means that the most general spinor can always be Lorentz-transformed into the form $\\psi = \\phi \\begin{pmatrix}1\\\\0\\\\1\\\\0\\end{pmatrix}$ in terms of a single scalar function.",
"Thus regular spinors in five dimensions are simpler than in the four-dimensional case.",
"Not so, and indeed the opposite, for singular spinors.",
"For singular spinors we have the constraint $\\Phi = 0$ identically: the Fierz identity () tells that $M_{\\mu \\nu }M^{\\mu \\nu } = U_{\\mu }U^{\\mu } =0$ and in particular $U^{0}U^{0} - \\sum _{\\mathcal {A}=1}^3{ U^{\\mathcal {A}}U^{\\mathcal {A}}} = U^{4}U^{4}> 0$ so that we can employ three boosts to set $U^{\\mathcal {A}}=0$ ($\\mathcal {A}=1,2,3$ ) identically; we still have all rotations at our disposal to align $M^{0\\mathcal {A}}$ and $M^{\\mathcal {A}4}$ along the third axis, so that we get $\\psi = \\phi \\begin{pmatrix}e^{i\\frac{\\pi }{4}}\\\\0\\\\e^{-i\\frac{\\pi }{4}}\\\\0\\end{pmatrix}$ in terms of a single scalar function.",
"Thus singular spinors in five dimensions are more complex than in the four-dimensional case, and indeed they are not singular at all.",
"So, whereas in four dimensions the real scalar degrees of freedom of regular spinors are two while for singular spinors are zero, in five dimensions the real scalar degrees of freedom for both regular and singular spinors are just one.",
"As a matter of fact, it is even possible to write both in the same manner as $\\psi = \\phi \\begin{pmatrix}e^{i p \\frac{\\pi }{4}}\\\\0\\\\e^{- i p \\frac{\\pi }{4}}\\\\0\\end{pmatrix}$ with $p = 0$ for regular and $p = 1$ for singular spinors.",
"With the spinor in equation (REF ), we get that the spinorial covariant derivative is $D_{i}\\psi = \\left( \\partial _{i}\\ln {\\phi }\\mathbb {I} + \\omega _{i}^{\\phantom{i}\\mu \\nu }S_{\\mu \\nu }\\right) \\psi $ so that the spinorial covariant derivative actually acts as a local matrix operator."
],
[
"Dynamical equations",
"We consider Einstein-Cartan gravity coupled to a Dirac field in five dimensions.",
"The Lagrangian function of the theory is then given by ${L} = R + \\frac{i}{2}\\left(\\bar{\\psi }\\gamma ^iD_{i}\\psi - D_{i}\\bar{\\psi }\\gamma ^{i}\\psi \\right) - m\\bar{\\psi }\\psi $ where $\\gamma ^i = \\gamma ^\\mu e_\\mu ^i$ and $m$ is the mass of the spinor field.",
"Upon variations, the field equations are the Einstein gravitational field equations $R_{ij} - \\frac{1}{2}Rg_{ij} = \\Sigma _{ij}$ the Sciama-Kibble torsional field equations $T_{tsi} = S_{tsi}$ and the Dirac spinor field equations $i\\gamma ^hD_h\\psi -m\\psi =0$ where $\\Sigma _{ij}$ and $S_{ijh}$ are the energy and the spin density tensors respectively expressed as $\\Sigma _{ij}=\\frac{i}{4}\\left(\\bar{\\psi }\\gamma _{i}D_{j}\\psi -D_{j}\\bar{\\psi }\\gamma _{i}\\psi \\right)$ and $S_{ijh}=\\frac{i}{2}\\bar{\\psi }\\left\\lbrace \\gamma _{h},S_{ij}\\right\\rbrace \\psi =-\\frac{1}{8}\\epsilon _{ijhab}M^{ab}$ in which the complete antisymmetry of the Dirac spin density tensor clarifies why we considered only a totally antisymmetric torsion without loss of generality.",
"Moreover, we recall that the Dirac spinor field equations imply the conservation laws $\\nabla _i\\Sigma ^{ij}= T^{jik}\\Sigma _{ik}+\\frac{1}{2}S_{pqi}R^{pqij}\\\\\\nabla _h S^{ijh}= \\Sigma ^{ji}-\\Sigma ^{ij}$ where the antisymmetry of spin and torsion has been systematically used [6].",
"Inserting equations (REF ) into the decomposition (REF ), we can rewrite equation (REF ) in the form $\\tilde{R}_{ij}-\\frac{1}{2}\\tilde{R} g_{ij}+\\frac{1}{2} \\tilde{\\nabla }_p S_{ji}^{\\;\\;\\;p}-\\frac{1}{4}S_{pi}^{\\;\\;\\;q} S_{jq}^{\\;\\;\\;p}+\\frac{1}{8} S_{qpr} S^{qpr}g_{ij} = \\Sigma _{ij}$ and it is an easy matter to verify that the antisymmetric part of equation (REF ) amounts to the conservation laws (REF ).",
"Therefore the significant part of equation (REF ) reduces to the symmetric one and reads as $\\tilde{R}_{ij}-\\frac{1}{2}\\tilde{R} g_{ij}=\\frac{i}{4}\\left[\\bar{\\psi }\\gamma _{(i}\\tilde{D}_{j)}\\psi -\\tilde{D}_{(j}\\bar{\\psi }\\gamma _{i)}\\psi \\right]+\\frac{3}{32}\\Phi ^{2}g_{ij}$ where $\\tilde{D}_i$ denotes the spinorial covariant derivative induced by the Levi-Civita connection.",
"In detail, equations (REF ) are deduced by making use of the identities $\\Sigma _{ij} = \\frac{i}{4}\\left[\\bar{\\psi }\\gamma _i\\tilde{D}_j\\psi -(\\tilde{D}_j\\bar{\\psi })\\gamma _i\\psi \\right] -\\frac{1}{4}S_{pi}^{\\;\\;\\;q} S_{jq}^{\\;\\;\\;p}\\\\S_{hi}^{\\;\\;\\;p}S_{jp}^{\\;\\;\\;h} =\\frac{1}{64}(\\bar{\\psi }[\\gamma ^\\alpha ,\\gamma ^\\beta ]\\psi )(\\bar{\\psi }[\\gamma _\\alpha ,\\gamma _\\beta ]\\psi )g_{ij}-\\frac{1}{32}(\\bar{\\psi }[\\gamma _i,\\gamma _p]\\psi )(\\bar{\\psi }[\\gamma _j,\\gamma ^p]\\psi )\\\\S_{hqp}S^{hqp} = \\frac{3}{32}(\\bar{\\psi }[\\gamma ^\\alpha ,\\gamma ^\\beta ]\\psi )(\\bar{\\psi }\\gamma _\\alpha \\gamma _\\beta \\psi )\\\\\\left(\\bar{\\psi }\\left[\\gamma _\\alpha ,\\gamma _\\beta \\right]\\psi \\right)\\left[\\gamma ^\\alpha ,\\gamma ^\\beta \\right]\\psi =-16\\left(\\bar{\\psi }\\psi \\right)\\psi \\\\\\left(\\bar{\\psi }\\left[\\gamma _\\alpha ,\\gamma _\\beta \\right]\\psi \\right)\\left(\\bar{\\psi }\\left[\\gamma ^\\alpha ,\\gamma ^\\beta \\right]\\psi \\right) =-16\\left(\\bar{\\psi }\\psi \\right)^2$ Analogously, Dirac equation, i.e.",
"equation (REF ), can be worked out by using the decomposition $D_i \\psi = \\tilde{D}_i \\psi +\\frac{1}{8}T_{ijh}\\gamma ^h\\gamma ^j\\psi $ and then expressed in the final form $i \\gamma ^h \\tilde{D}_h \\psi - \\left( \\frac{3}{8}\\Phi + m \\right) \\psi =0$ where the non-linearity has been translated into a simple correction to the mass term."
],
[
"Polar equations",
"By employing the polar form of the spinor field, it is possible to provide a corresponding polar form of the spinor field equations: this has been done in [4], where all Gordon decompositions of the polar form of Dirac field equations were found, and in [5], where we isolated the sub-set of Gordon decompositions that imply the polar form of Dirac field equations.",
"Therefore, the combined results of [4], [5] can be used to see that the Dirac field equations in polar form can equivalently be written in terms of two real vector field equations.",
"When the same procedure is done in the 5 dimensional case however, we obtain a single real vector equation $\\partial _{s}\\Phi +\\tilde{\\omega }^{h}_{\\phantom{h}hs}\\Phi +\\frac{1}{2}\\epsilon _{hijps}\\tilde{\\omega }^{hij}U^{p}=0$ giving first-order derivatives of the only degree of freedom in terms of the spin connection.",
"Therefore, in the four-dimensional case, the Dirac spinor equations, accounting for 8 real equations, can be decomposed in two real vector equations, accounting for a corresponding number of 8 real equations, while in the five-dimensional case, the Dirac spinor equations, accounting for 8 real equations, can be decomposed in one real vector equation, accounting for 5 real equations, with 3 Dirac equations converting into constraints.",
"We refer the manifold $\\pi : Q\\rightarrow M$ to local fiber coordinates $x^{A},x^4$ (with capital Latin letters running from zero to three), where $x^{A}$ are coordinates on $M$ .",
"Local fiber coordinate transformations are of the form $\\left\\lbrace \\begin{aligned}&\\bar{x}^{A}= \\bar{x}^{A}(x^{B})\\\\&\\bar{x}^4= \\bar{x}^4 (x^{B},x^4)\\end{aligned}\\right.$ and in particular, whenever the subset of 1-forms $e^\\Psi =e^\\Psi _{A}(x^{B})\\,dx^{A}$ (with capital Greek letters running from zero to three) results to be the pull-back of a tetrads field on $M$ , then the quantity $\\pi ^*(\\bar{g}) := g+e^4\\otimes e^4$ defines a metric tensor $\\bar{g}$ on $M$ .",
"Such a construction is invariant under the action of the sub-group of $SO(1,4)$ consisting of the matrices of the form $\\Lambda ^\\Psi _{\\;\\;\\Phi } (\\pi (x))=\\begin{pmatrix}\\Lambda ^{\\Psi }_{\\;\\;{\\Phi }} (\\pi (x)) & 0\\\\0 & 1\\end{pmatrix}$ with $\\Lambda ^{\\Psi }_{\\;\\;{\\Phi }} (\\pi (x)) \\in SO (1,3)$ , $\\forall x\\in Q$ .",
"Under simultaneous coordinate and Lorentz transformations, equations (REF ) and (REF ), the spin-connection coefficients undergo the transformation laws $\\bar{\\omega }_A^{\\;\\;\\Psi \\Phi } = \\Lambda ^\\Psi _{\\;\\;\\Sigma }\\Lambda ^\\Phi _{\\;\\;\\Omega }\\frac{\\partial {x^B}}{\\partial {\\bar{x}^A}}\\omega _B^{\\;\\;\\Sigma \\Omega }-\\Lambda _\\Sigma ^{\\;\\;\\Omega }\\frac{\\partial {\\Lambda ^\\Psi _{\\;\\;\\Omega }}}{\\partial {x^B}}\\frac{\\partial {x^B}}{\\partial {\\bar{x}^A}}\\eta ^{\\Sigma \\Phi }+\\Lambda ^\\Psi _{\\;\\;\\Sigma }\\Lambda ^\\Phi _{\\;\\;\\Omega }\\frac{\\partial {x^4}}{\\partial {\\bar{x}^A}}\\omega _4^{\\;\\;\\Sigma \\Omega }\\\\\\bar{\\omega }_A^{\\;\\;\\Psi 4} = \\Lambda ^\\Psi _{\\;\\;\\Phi }\\frac{\\partial {x^B}}{\\partial {\\bar{x}^A}}\\omega _B^{\\;\\;\\Phi 4}+\\Lambda ^\\Psi _{\\;\\;\\Phi }\\frac{\\partial {x^4}}{\\partial {\\bar{x}^A}}\\omega _4^{\\;\\;\\Phi 4}\\\\\\bar{\\omega }_4^{\\;\\;\\Psi 4} = \\Lambda ^\\Psi _{\\;\\;\\Phi }\\frac{\\partial {x^4}}{\\partial {\\bar{x}^4}}\\omega _4^{\\;\\;\\Phi 4}\\\\\\bar{\\omega }_4^{\\;\\;\\Psi \\Phi } = \\Lambda ^\\Psi _{\\;\\;\\Sigma }\\Lambda ^\\Phi _{\\;\\;\\Omega }\\frac{\\partial {x^4}}{\\partial {\\bar{x}^4}}\\omega _4^{\\;\\;\\Sigma \\Omega }$ showing that, limited to the transformation subgroups (REF ) and (REF ), a spin-connection $\\omega _A^{\\;\\;\\Psi \\Phi }$ on $M$ can be always lifted to a corresponding spin-connection on $Q$ by setting $\\omega _A^{\\;\\;\\Psi 4} =\\omega _4^{\\;\\;\\Psi 4} = \\omega _4^{\\;\\;\\Psi \\Phi } =0$ .",
"Conversely, given a spin-connection $\\omega _i^{\\;\\;\\mu \\nu }$ on $Q$ (independent of the $x^4$ coordinate), the subset of coefficients $\\omega _A^{\\;\\;\\Psi \\Phi }$ defines a spin-connection on $M$ provided that the group of coordinate transformations is reduced to $\\left\\lbrace \\begin{aligned}\\bar{x}^{A}= \\bar{x}^{A}(x^{B})\\\\\\bar{x}^4= \\bar{x}^4 (x^4)\\end{aligned}\\right.$ consistent with the trivial fibration $Q=M\\times U$ being $U$ a generic one-dimensional manifold."
],
[
"The prototypical case of the Bianchi-I models",
"Let us consider a Bianchi type I metric of the form $ds^2=dt^2-a^2(t)\\,dx^2-b^2(t)\\,dy^2-c^2(t)\\,dz^2-d^2(t)\\,du^2$ identifying $x^0 =t$ , $x^1 =x$ , $x^2 =y$ , $x^3 =z$ and $x^4= u$ for simplicity; the components of the fünfbein associated with the metric (REF ) are expressed as $e^\\mu _0=\\delta ^\\mu _0, \\quad e^\\mu _1=a(t) \\delta ^\\mu _1, \\quad e^\\mu _2=b(t) \\delta ^\\mu _2, \\quad e^\\mu _3=c(t) \\delta ^\\mu _3, \\quad e^\\mu _4=d(t) \\delta ^\\mu _4,$ with $e^0_\\mu =\\delta ^0_\\mu , \\quad e^1_\\mu =\\frac{1}{a(t)}\\delta ^1_\\mu , \\quad e^2_\\mu =\\frac{1}{b(t)}\\delta ^2_\\mu , \\quad e^3_\\mu =\\frac{1}{c(t)}\\delta ^3_\\mu \\quad e^4_\\mu =\\frac{1}{d(t)}\\delta ^4_\\mu ,$ where $\\mu =0,1,2,3,4$ as in general.",
"The non-null components of the Levi-Civita connection associated with the metric (REF ) are $\\begin{aligned}\\tilde{\\Gamma }_{10}^{\\;\\;\\;1}& = \\frac{\\dot{a}}{a}, &\\tilde{\\Gamma }_{20}^{\\;\\;\\;2}& = \\frac{\\dot{b}}{b}, &\\tilde{\\Gamma }_{30}^{\\;\\;\\;3}& = \\frac{\\dot{c}}{c}, &\\tilde{\\Gamma }_{40}^{\\;\\;\\;4}& = \\frac{\\dot{d}}{d}\\\\\\tilde{\\Gamma }_{11}^{\\;\\;\\;0}& = a{\\dot{a}}, &\\tilde{\\Gamma }_{22}^{\\;\\;\\;0}& = b{\\dot{b}}, &\\tilde{\\Gamma }_{33}^{\\;\\;\\;0}& = c{\\dot{c}}, &\\tilde{\\Gamma }_{44}^{\\;\\;\\;0}& = d{\\dot{d}}\\end{aligned}$ and consequently we also have that $\\tilde{\\Omega }_1=\\frac{1}{2}{\\dot{a}}\\gamma ^1\\gamma ^0, \\quad \\tilde{\\Omega }_2=\\frac{1}{2}{\\dot{b}}\\gamma ^2\\gamma ^0, \\quad \\tilde{\\Omega }_3=\\frac{1}{2}{\\dot{c}}\\gamma ^3\\gamma ^0, \\quad \\tilde{\\Omega }_4=\\frac{1}{2}{\\dot{d}}\\gamma ^4\\gamma ^0$ are the non-zero coefficients of the spinorial connection needed to construct the spinorial covariant derivative in the form $\\tilde{D}_i\\psi =\\partial _i\\psi -\\tilde{\\Omega }_i\\psi $ .",
"With the spinorial covariant derivative and () the Dirac spinor field equations are $\\dot{\\psi }+\\frac{\\dot{\\tau }}{2\\tau }\\psi +im\\gamma ^0\\psi +\\frac{3i}{8} (\\bar{\\psi }\\psi )\\gamma ^0\\psi =0$ $\\dot{\\bar{\\psi }}+\\frac{\\dot{\\tau }}{2\\tau }\\bar{\\psi }-im\\bar{\\psi }\\gamma ^0-\\frac{3i}{8} (\\bar{\\psi }\\psi )\\bar{\\psi }\\gamma ^0=0$ where we have defined $\\tau = abcd$ as the volume element.",
"Multiplying equation (REF ) by $\\bar{\\psi }$ and equation (REF ) by $\\psi $ and summing the results, we obtain $\\frac{d}{dt} (\\tau \\bar{\\psi }\\psi )=0$ yielding $\\bar{\\psi }\\psi =\\frac{C}{\\tau }$ where $C$ is a suitable integration constant.",
"Analogously, evaluating the Einstein equations (REF ) for the metric (REF ) and using again equation (), we get $\\frac{\\dot{a}}{a}\\frac{\\dot{b}}{b}+\\frac{\\dot{b}}{b}\\frac{\\dot{c}}{c}+\\frac{\\dot{a}}{a}\\frac{\\dot{c}}{c}+\\frac{\\dot{a}}{a}\\frac{\\dot{d}}{d}+\\frac{\\dot{b}}{b}\\frac{\\dot{d}}{d}+\\frac{\\dot{c}}{c}\\frac{\\dot{d}}{d}=+\\frac{m}{2}\\bar{\\psi }\\psi + \\frac{9}{32}\\left(\\bar{\\psi }\\psi \\right)^2$ $\\frac{\\ddot{b}}{b}+\\frac{\\ddot{c}}{c}+\\frac{\\ddot{d}}{d}+\\frac{\\dot{b}}{b}\\frac{\\dot{c}}{c}+\\frac{\\dot{b}}{b}\\frac{\\dot{d}}{d}+\\frac{\\dot{c}}{c}\\frac{\\dot{d}}{d}= \\frac{3}{32}\\left(\\bar{\\psi }\\psi \\right)^2$ $\\frac{\\ddot{a}}{a}+\\frac{\\ddot{c}}{c}+\\frac{\\ddot{d}}{d}+\\frac{\\dot{a}}{a}\\frac{\\dot{c}}{c}+\\frac{\\dot{a}}{a}\\frac{\\dot{d}}{d}+\\frac{\\dot{c}}{c}\\frac{\\dot{d}}{d}= \\frac{3}{32}\\left(\\bar{\\psi }\\psi \\right)^2$ $\\frac{\\ddot{a}}{a}+\\frac{\\ddot{b}}{b}+\\frac{\\ddot{d}}{d}+\\frac{\\dot{a}}{a}\\frac{\\dot{b}}{b}+\\frac{\\dot{a}}{a}\\frac{\\dot{d}}{d}+\\frac{\\dot{b}}{b}\\frac{\\dot{d}}{d}= \\frac{3}{32}\\left(\\bar{\\psi }\\psi \\right)^2$ $\\frac{\\ddot{a}}{a}+\\frac{\\ddot{b}}{b}+\\frac{\\ddot{c}}{c}+\\frac{\\dot{a}}{a}\\frac{\\dot{b}}{b}+\\frac{\\dot{b}}{b}\\frac{\\dot{c}}{c}+\\frac{\\dot{a}}{a}\\frac{\\dot{c}}{c}= \\frac{3}{32}\\left(\\bar{\\psi }\\psi \\right)^2$ together with the conditions $\\Sigma _{(12)}=0\\quad \\Rightarrow \\quad a \\dot{b}-b \\dot{a}=0 \\quad \\cup \\quad \\bar{\\psi }\\gamma ^4\\gamma ^3\\psi =0$ $\\Sigma _{(23)}=0\\quad \\Rightarrow \\quad c \\dot{b}-b \\dot{c}=0 \\quad \\cup \\quad \\bar{\\psi }\\gamma ^4\\gamma ^1\\psi =0$ $\\Sigma _{(13)}=0\\quad \\Rightarrow \\quad a \\dot{c}-c \\dot{a}=0 \\quad \\cup \\quad \\bar{\\psi }\\gamma ^4\\gamma ^2\\psi =0$ $\\Sigma _{(14)}=0\\quad \\Rightarrow \\quad a \\dot{d}-d \\dot{a} =0 \\quad \\cup \\quad \\bar{\\psi }\\gamma ^2\\gamma ^3\\psi =0$ $\\Sigma _{(24)}=0\\quad \\Rightarrow \\quad b \\dot{d}-d \\dot{b} =0 \\quad \\cup \\quad \\bar{\\psi }\\gamma ^1\\gamma ^3\\psi =0$ $\\Sigma _{(34)}=0\\quad \\Rightarrow \\quad c \\dot{d}-d \\dot{c} =0 \\quad \\cup \\quad \\bar{\\psi }\\gamma ^1\\gamma ^2\\psi =0$ with equations $\\Sigma _{0{\\cal A}}=0$ (${\\cal A}=1,2,3,4$ ) being automatically satisfied identities.",
"Subtracting equations (REF ) from (REF ), (REF ) from (REF ) and (REF ) from (REF ), we obtain the relations $\\frac{a}{b}=X_1 e^{Y_1\\int {\\frac{dt}{\\tau }}}$ $\\frac{b}{c}=X_2 e^{Y_2\\int {\\frac{dt}{\\tau }}}$ $\\frac{c}{d}=X_3 e^{Y_3\\int {\\frac{dt}{\\tau }}}$ where $X$ and $Y$ are suitable integration constants.",
"A linear combination of equations (REF )-(REF ) gives $3\\ddot{\\tau }/\\tau =2m\\bar{\\psi }\\psi + \\frac{3}{2}\\left(\\bar{\\psi }\\psi \\right)^2$ Together with equations (REF ), the equation (REF ) accounts for the dynamics of the metric (REF ), while equation (REF ) plays the role of a constraint on the initial data."
],
[
"Compatibility with polar form",
"In the previous section we have established the fact that even a most general spinor field can always be reduced, by employing Lorentz transformations down to the polar form (REF ).",
"In this section we have studied specific types of universes with metric (REF ): despite the polar form can always be achieved, the Dirac spinor is a field, so the needed Lorentz transformation is local, and this in general produces additional contributions in the fünfbein, which means that choosing the fünfbein cannot be done in general.",
"Since we have done so, it is necessary at this point to spend time to check whether or not our choices are all compatible.",
"And if yes, what other constraints can be implemented.",
"Our goal is to make sure that the polar form be compatible with restrictions (REF ).",
"We consider now the standard representation of the gamma matrices so that, writing the generic spinor according to the expression $\\psi =\\begin{pmatrix}\\psi _1\\\\\\psi _2\\\\\\psi _3\\\\\\psi _4\\end{pmatrix},$ the restrictions become $\\bar{\\psi }\\gamma ^4\\gamma ^3\\psi =0 \\quad \\Longleftrightarrow \\quad \\left(-\\psi _1^*\\psi _1+\\psi ^*_2\\psi _2-\\psi ^*_3\\psi _3+\\psi ^*_4\\psi _4\\right)= 0,$ $\\bar{\\psi }\\gamma ^4\\gamma ^1\\psi =0 \\quad \\Longleftrightarrow \\quad \\left(\\psi _1^*\\psi _2+\\psi ^*_2\\psi _1+\\psi ^*_3\\psi _4+\\psi ^*_4\\psi _3\\right)= 0,$ $\\bar{\\psi }\\gamma ^4\\gamma ^2\\psi =0 \\quad \\Longleftrightarrow \\quad \\left(\\psi _1^*\\psi _2-\\psi ^*_2\\psi _1+\\psi ^*_3\\psi _4-\\psi ^*_4\\psi _3\\right)= 0,$ $\\bar{\\psi }\\gamma ^2\\gamma ^3\\psi =0 \\quad \\Longleftrightarrow \\quad \\left(\\psi _1^*\\psi _2+\\psi ^*_2\\psi _1-\\psi ^*_3\\psi _4-\\psi ^*_4\\psi _3\\right)= 0,$ $\\bar{\\psi }\\gamma ^1\\gamma ^3\\psi =0 \\quad \\Longleftrightarrow \\quad \\left(\\psi _1^*\\psi _2-\\psi ^*_2\\psi _1-\\psi ^*_3\\psi _4+\\psi ^*_4\\psi _3\\right)= 0,$ $\\bar{\\psi }\\gamma ^1\\gamma ^2\\psi =0 \\quad \\Longleftrightarrow \\quad \\left(-\\psi _1^*\\psi _1+\\psi ^*_2\\psi _2+\\psi ^*_3\\psi _3-\\psi ^*_4\\psi _4\\right)= 0,$ It seems physically meaningful requiring that the scale factor of the fourth spatial dimension differs from the other ones.",
"This means, therefore, that constraints in equations (REF )-(REF ) have to be always imposed.",
"General solutions of Dirac equations (REF ) are given in the form $\\psi =\\frac{1}{\\sqrt{2\\tau }}\\begin{pmatrix}A_1 e^{-i\\left(mt+\\frac{3C}{8}\\int {\\frac{dt}{\\tau }}\\right)}\\\\A_2 e^{-i\\left(mt+\\frac{3C}{8}\\int {\\frac{dt}{\\tau }}\\right)}\\\\A_3 e^{i\\left(mt+\\frac{3C}{8}\\int {\\frac{dt}{\\tau }}\\right)}\\\\A_4 e^{i\\left(mt+\\frac{3C}{8}\\int {\\frac{dt}{\\tau }}\\right)}\\end{pmatrix},$ where the $A_{j}=r_{j}e^{i\\theta _{j}}$ with $j=1,2,3,4$ are four complex integration constants.",
"Equations (REF )-(REF ) yield $r_1r_2\\cos (\\theta _2-\\theta _1)= r_3r_4\\cos (\\theta _4-\\theta _3),\\\\r_1r_2\\sin (\\theta _2-\\theta _1)= r_3r_4\\sin (\\theta _4-\\theta _3),\\\\r_2^2+r_3^2= r_1^2+r_4^2,$ eventually giving either $\\begin{pmatrix}A_1\\\\A_2\\\\A_3\\\\A_4\\end{pmatrix}=\\begin{pmatrix}re^{i\\theta }\\\\0\\\\re^{i\\varphi }\\\\0\\end{pmatrix}\\quad \\text{or}\\quad \\begin{pmatrix}A_1\\\\A_2\\\\A_3\\\\A_4\\end{pmatrix} =\\begin{pmatrix}0\\\\re^{i\\theta }\\\\0\\\\re^{i\\varphi }\\end{pmatrix},$ where $r\\ge 0$ , $\\theta $ and $\\varphi $ are arbitrary real numbers, and also $\\begin{pmatrix}A_1\\\\A_2\\\\A_3\\\\A_4\\end{pmatrix} =\\begin{pmatrix}qe^{i\\theta _1}\\\\re^{i\\theta _2}\\\\qe^{i\\theta _3}\\\\re^{i\\theta _4}\\end{pmatrix},$ where $q\\ge 0$ and $r\\ge 0$ are arbitrary, while the phases satisfy $\\theta _2-\\theta _1=\\theta _4-\\theta _3+2k\\pi $ being $k$ an integer number.",
"However, in any case, all the admissible solutions of Dirac equations (REF ) with constraints (REF )-(REF ) satisfy $\\Phi =0$ necessarily.",
"We must now check compatibility with the polar form.",
"The condition $\\Phi =0$ implies that we are treating singular spinor fields, and for these, Fierz identities in equations (REF ) and () tell $M_{ik}U^{i}=0\\\\M_{ab}M^{ab} = U_{a}U^{a} = 0$ identically; we have already discussed how in this situation it is always possible to boost into the rest frame, the one for which $U^{A}=0$ identically, and in this frame Fierz identities $M_{40}=0\\\\M^{0A}U^{0}=M^{4A}U^{4}\\\\2M_{0A}M^{0A}+M_{AB}M^{AB}+2M_{4A}M^{4A} = 0\\\\U^{0}U^{0} = U^{4}U^{4}$ can be further re-arranged into $|M^{0A}|=|M^{4A}|\\\\|U^{0}| = |U^{4}|\\\\M_{40}=0\\\\M^{AB} = 0$ identically in this frame.",
"Notice that equation (REF ) is equivalent to requiring the validity of constraints (REF )-(REF ) which we know should always be imposed.",
"This establishes the compatibility of the isotropy constraints with the polar form.",
"In the maximally isotropic case $a=b=c$ the constraints (REF )-(REF ) are automatically satisfied.",
"The solution of Dirac spinor field equations satisfying the remaining constraints (REF )-(REF ) is given by equation (REF ) together with (REF ) and (REF ), which imply the further condition $\\bar{\\psi }\\psi =0$ ($C=0$ ).",
"This simplify the Einstein-like equations.",
"In particular equation (REF ) reduces to $\\ddot{\\tau }=0$ .",
"Therefore, we distinguish $\\tau =\\beta $ and $\\tau =\\alpha t+\\beta $ as the only two admissible sub-cases.",
"If $\\tau =\\beta $ from equations (REF ), we deduce $a(t)=\\left(\\beta X\\right)^{\\frac{1}{4}}e^{\\frac{Y}{4\\beta }t}$ and $d(t)=\\beta ^{\\frac{1}{4}}X^{-\\frac{3}{4}}e^{-\\frac{3Y}{4\\beta }t}$ from which we have $\\frac{\\dot{a}}{a}=\\frac{Y}{4\\beta } \\qquad \\text{and} \\qquad \\frac{\\dot{d}}{d}=-\\frac{3Y}{4\\beta }$ which can be inserted into (REF ) giving $Y=0$ necessarily, which amounts to a five-dimensional flat spacetime filled by a constant Dirac field.",
"This is an unphysical solution.",
"If $\\tau =\\alpha t+\\beta $ , we have $a(t)=X^{\\frac{1}{4}}\\left(\\alpha t+\\beta \\right)^{\\frac{Y+1}{4}}$ and $d(t)=X^{-\\frac{3}{4}}\\left(\\alpha t+\\beta \\right)^{\\frac{-3Y+1}{4}}$ implying $\\frac{\\dot{a}}{a}=\\frac{\\alpha \\left(Y+1\\right)}{4\\left(\\alpha t+\\beta \\right)} \\qquad {\\rm and} \\qquad \\frac{\\dot{d}}{d} =\\frac{\\alpha \\left(-3Y+1\\right)}{4\\left(\\alpha t+\\beta \\right)}$ and it is a straightforward matter to see that $Y=1$ and $Y=-1$ are the only initial data consistent with (REF ): for $Y=-1$ the scale factor $a$ is constant, while the scale factor $d$ expands, so the resulting cosmological scenario is not acceptable.",
"For $Y=1$ , $d$ contracts and $a$ undergoes a decelerated expansion, the associated four-dimensional spacetime could describe a Friedmann era of our Universe, but it can represent neither a Universe undergoing inflation nor a Universe characterized by a dark energy era."
],
[
"Partially isotropic case",
"In the partially isotropic case as for instance if $a=b\\ne c$ , we have that also constraints (REF ) and (REF ) have to be imposed.",
"It is easy to see that the admissible solutions for the Dirac field are now given by equations (REF ) and (REF ).",
"If $\\tau =\\beta $ , from equations (REF ) and renaming some integration constants, we get the identities $a(t)=b(t)=X_1\\left(\\frac{\\beta }{X_1^2X_2}\\right)^{\\frac{1}{4}}e^{\\left(\\frac{1}{2}Y_1-\\frac{1}{4}Y_2\\right)t} \\\\c(t)= X_2\\left(\\frac{\\beta }{X_1^2X_2}\\right)^{\\frac{1}{4}}e^{\\left(-\\frac{1}{2}Y_1+\\frac{3}{4}Y_2\\right)t}\\\\d(t)= \\left(\\frac{\\beta }{X_1^2X_2}\\right)^{\\frac{1}{4}}e^{\\left(-\\frac{1}{2}Y_1-\\frac{1}{4}Y_2\\right)t}$ which imply the relations $\\frac{\\dot{a}}{a}=\\frac{\\dot{b}}{b}=\\frac{1}{2}Y_1-\\frac{1}{4}Y_2, \\quad \\frac{\\dot{c}}{c}=-\\frac{1}{2}Y_1+\\frac{3}{4}Y_2 \\quad {\\rm and} \\quad \\frac{\\dot{d}}{d}=-\\frac{1}{2}Y_1-\\frac{1}{4}Y_2.$ Inserting these into (REF ) it is easy to see that the only admissible solution corresponds to $Y_1 =Y_2 =0$ representing a five-dimensional flat spacetime.",
"If $\\tau =\\alpha t+\\beta $ the factor scales of the metric (REF ) are expressed as $a(t)=b(t)= X_1\\left(\\frac{1}{X_1^2X_2}\\right)^{\\frac{1}{4}}\\left(\\alpha t+\\beta \\right)^{\\frac{1}{4}\\left(1+2Y_1-Y_2\\right)}\\\\c(t)= X_2\\left(\\frac{1}{X_1^2X_2}\\right)^{\\frac{1}{4}}\\left(\\alpha t+\\beta \\right)^{\\frac{1}{4}\\left(1-2Y_1+3Y_2\\right)}\\\\d(t)= \\left(\\frac{1}{X_1^2X_2}\\right)^{\\frac{1}{4}}\\left(\\alpha t+\\beta \\right)^{\\frac{1}{4}\\left(1-2Y_1-Y_2\\right)}$ and therefore we have the relations $\\frac{\\dot{a}}{a}=\\frac{\\dot{b}}{b}=\\frac{\\alpha \\left(1+2Y_1-Y_2\\right)}{4\\left(\\alpha t+\\beta \\right)}\\\\\\frac{\\dot{c}}{c}=\\frac{\\alpha \\left(1-2Y_1+3Y_2\\right)}{4\\left(\\alpha t+\\beta \\right)}\\\\\\frac{\\dot{d}}{d}=\\frac{\\alpha \\left(1-2Y_1-Y_2\\right)}{4\\left(\\alpha t+\\beta \\right)},$ which can be inserted into equation (REF ) yielding a constraint for the initial data $4Y_1^2-4Y_1Y_2-3-4Y_2+3Y_2^2 = 0.$ In order to obtain accelerated expansion for the scale factors $a(t)$ and $c(t)$ as well as contraction for $d(t)$ we should have solutions of equation (REF ) satisfying the conditions $1+2Y_1-Y_2>4, \\quad 1-2Y_1+3Y_2>4, \\quad {\\rm and} \\quad 1-2Y_1-Y_2<0,$ but once again it is straightforward to verify that equation (REF ), subjected to the constraints (REF ), has no solutions (indeed, the system of equations (REF ) admits solutions only for $Y_2>3$ while equation (REF ) possesses real solutions only for $\\frac{2-\\sqrt{10}}{2}\\le Y_2 \\le \\frac{2+\\sqrt{10}}{2} <3$ ).",
"Instead, if we require that the scale factors $a(t)$ and $c(t)$ expand but not necessarily accelerating, namely if we only impose $1+2Y_1-Y_2>0, \\quad 1-2Y_1+3Y_2>0, \\quad {\\rm and} \\quad 1-2Y_1-Y_2<0,$ it is easily seen that solutions do exist.",
"For example, the pair $Y_2=1$ and $Y_1=\\frac{1+\\sqrt{5}}{2}$ solves simultaneously equations (REF ) and (REF )."
],
[
"Totally anisotropic case",
"In the totally anisotropic situation we have $a\\ne b$ , $a\\ne c$ , $b\\ne c$ and therefore also the constraint (REF ) have to be imposed.",
"However, solutions (REF ) and (REF ) are clearly not compatible with (REF ).",
"Therefore, the totally anisotropic case is not viable."
],
[
"Conclusion",
"In this paper, we have considered a Dirac spinor field in a five-dimensional background, and proceeded to reduce such set-up to a four-dimensional spacetime.",
"We have seen that it is always possible to choose a frame in which the spinor field can be written-without loss of generality-in the form shown in equation (REF ), so that we also have $\\Phi = \\bar{\\psi }\\psi = 2\\phi ^{2}(1 - p)$ with $p=0$ and $p=1$ designing regular and singular spinors respectively.",
"The corresponding decomposition of the Dirac spinor field equation is $\\partial _{s}\\ln {\\phi } + \\frac{1}{2}\\tilde{\\omega }^{h}_{\\phantom{h}hs}+\\frac{1}{4}\\epsilon _{hijps}\\tilde{\\omega }^{hij}U^{p}/\\Phi =0,$ giving first-order derivatives of $\\phi $ in terms of the spin connection, and reducing to $\\epsilon _{hijps}\\tilde{\\omega }^{hij}U^{p}=0$ as a constraint over the spin connection in the case of singular spinor fields.",
"Therefore, a four-dimensional spinor field obtained as a reduction of a five-dimensional spinor field differs from a genuine four-dimensional spinor field for the fact that the former has only one degree of freedom, the module, while the latter has in general two degrees of freedom, the module and the so-called Yvon-Takabayashi angle.",
"Additionally, in the former case the number of spinor field equations (that is 8) does not match the number of field equations after the polar decomposition (which are only 5).",
"As a consequence of this circumstance, we conclude that an initially five-dimensional spinor field later reduced to a four-dimensional spinor field is always more constrained than a genuine four-dimensional spinor field.",
"Eventually, we have studied the case of five-dimensional geometry of Bianchi-I anisotropic Universes, investigating whether a contraction of the fourth spatial scale factor can give rise to a possibly accelerated expansion of the reduced four-dimensional Universe.",
"Due to the further constraints that the spinor field have to satisfy in five-dimensions, we found that in the maximally and partially isotropic cases, the reduced four-dimensional Universe can-at the most-experience a Friedmann expansion, but accelerated phases are not allowed.",
"In the totally anisotropic case, solutions do not exist.",
"Once more we want to highlight that, the behaviour of anisotropic universes defined in five dimensions and later reduced to four dimensions, is different from what is obtained for anisotropic universes in 4 dimensions [7], [8], [9].",
"The underlying theory, as well as the example of the Bianchi-I Universes, indicates that an initially five-dimensional Universe with spinors later reduced to a four-dimensional Universe with spinors is more constrained than a genuine four-dimensional Universe with spinors."
]
] | 1808.08536 |
[
[
"Malliavin regularity and weak approximation of semilinear SPDE with\n L\\'evy noise"
],
[
"Abstract We investigate the weak order of convergence for space-time discrete approximations of semilinear parabolic stochastic evolution equations driven by additive square-integrable L\\'evy noise.",
"To this end, the Malliavin regularity of the solution is analyzed and recent results on refined Malliavin-Sobolev spaces from the Gaussian setting are extended to a Poissonian setting.",
"For a class of path-dependent test functions, we obtain that the weak rate of convergence is twice the strong rate."
],
[
"Introduction",
"Stochastic partial differential equations (SPDE) with Lévy noise occur in various applications, ranging from environmental pollution models [19] to the statistical theory of turbulence [6], to mention only two examples.",
"In the context of the numerical approximation of the solution processes of such equations, the quantity of interest is typically the expected value of some functional of the solution and one is thus interested in the weak convergence rate of the considered numerical scheme.",
"While the weak convergence analysis for numerical approximations of SPDE with Gaussian noise is meanwhile relatively far developed, see, e.g., [1], [2], [3], [7], [8], [9], [10], [11], [12], [14], [15], [16], [17], [18], [23], [30], available results for non-Gaussian Lévy noise have been restricted to linear equations so far [4], [5], [21], [25].",
"In this article, we analyze for the first time the weak convergence rate of numerical approximations for a class of semi-linear SPDE with non-Gaussian Lévy noise.",
"We consider equations of the type $\\,\\mathrm {d}X(t)+AX(t)\\,\\mathrm {d}t=F(X(t))\\,\\mathrm {d}t+\\,\\mathrm {d}L(t),\\quad t\\in [0,T],\\quad X(0)=X_0,$ where $X$ takes values in a separable real Hilbert space $H$ and $A\\colon D(A)\\subset H\\rightarrow H$ is an unbounded linear operator such that $-A$ generates an analytic semigroup $(S(t))_{t\\geqslant 0}\\subset \\mathcal {L}(H)$ .",
"By $\\dot{H}^\\rho $ , $\\rho \\in {R}$ , we denote the smoothness spaces associated to $A$ via $\\dot{H}^{\\rho }=D(A^{\\frac{\\rho }{2}})$ , see Subsection REF for details.",
"The driving Lévy process $L=(L(t))_{t\\in [0,T]}$ is assumed to be $\\dot{H}^{\\beta -1}$ -valued for some regularity parameter $\\beta \\in (0,1]$ , square-integrable with mean zero, and of pure jump type.",
"The nonlinearity $F\\colon H\\rightarrow \\dot{H}^{\\beta -1}$ is supposed to satisfy suitable Lipschitz conditions.",
"The precise assumptions are stated in Subsection REF and REF .",
"We remark that for a strong convergence analysis one could allow $F$ to be only $\\dot{H}^{\\beta -2}$ -valued, but to obtain a weak convergence rate which is twice the strong rate we need to assume more than that.",
"Our main example for the abstract equation (REF ) is the semilinear heat equation $\\left\\lbrace \\begin{aligned}&\\dot{u}(t,\\xi )-\\Delta _\\xi \\;\\!",
"u(t,\\xi )= f(u(t,\\xi ))+\\dot{\\eta }(t,\\xi ),\\quad &&(t,\\xi )\\in [0,T]\\times \\mathcal {O},\\\\&u(t,\\xi )=0, &&(t,\\xi )\\in [0,T]\\times \\partial \\mathcal {O},\\\\&u(0,\\xi )=u_0(\\xi ), &&\\xi \\in \\mathcal {O}.\\end{aligned}\\right.$ Here $\\mathcal {O}\\subset R^{d}$ is an open, bounded, convex, polygonal/polyhedral domain, $d\\in \\lbrace 1,2,3\\rbrace $ , $f\\colon {R}\\rightarrow {R}$ is twice continuously differentiable with bounded derivatives, and $\\dot{\\eta }$ is an impulsive space-time noise, cf.",
"Example REF .",
"The discretization in space is performed by a standard finite element method and in time by an implicit Euler method, cf.",
"Subsection REF .",
"Several approaches to analyzing the weak error of numerical approximations of SPDE can be found in the literature.",
"We follow the the approach from [1], [2], [4], [5], [22], which is based on duality principles in Malliavin calculus.",
"We remark that Malliavin calculus for Poisson or Lévy noise is fundamentally different from that for Gaussian noise.",
"Our analysis heavily relies on the results on Hilbert space-valued Poisson Malliavin calculus from [4].",
"Following the ideas in [24], [28], the Malliavin derivative in [4] is in fact a finite difference operator $D\\colon L^0(\\Omega ;H)\\rightarrow L^0(\\Omega \\times [0,T]\\times U;H),$ where $(\\Omega ,\\mathcal {F},{P})$ is the underlying probability space and $U=\\dot{H}^{\\beta -1}$ is the state space of the Lévy process $L$ , endowed with the Borel-$\\sigma $ -algebra $\\mathcal {B}(U)$ and the Lévy measure $\\nu $ of $L$ .",
"Starting with the operator (REF ), one can in a second step define Malliavin-Sobolev-type spaces as classes of $H$ -valued random variables satisfying certain integrability properties together with their Malliavin derivatives, cf.",
"Subsections REF and REF .",
"In this article, we extend the strategy for semilinear SPDE from [1], [2] to Poisson noise and analyze the weak approximation error in a framework of Gelfand triples of refined Malliavin-Sobolev spaces ${M}^{1,p,q}(H)\\subset L^2(\\Omega ;H)\\subset ({M}^{1,p,q}(H))^*$ , see Subsection REF for the definition of these spaces.",
"We first investigate in Section the Malliavin regularity of the mild solution $X=(X(t))_{t\\geqslant 0}$ to Eq.",
"(REF ).",
"We start by proving in Proposition REF that the Malliavin derivative $DX(t)$ of $X(t)$ satisfies for all $t\\in [0,T]$ the equality $\\begin{aligned}D_{s,x}X(t)&={1}_{s\\leqslant t}\\cdot \\int _s^tS(t-r)\\big [F\\big (X(r)+D_{s,x}X(r)\\big )-F\\big (X(r)\\big )\\big ]\\,\\mathrm {d}r\\\\&\\quad +{1}_{s\\leqslant t}\\cdot S(t-s)x\\end{aligned}$ ${P}\\otimes \\mathrm {d}s\\otimes \\nu (\\mathrm {d}x)$ -almost everywhere on $\\Omega \\times [0,T]\\times U$ .",
"The terms on the right hand side are understood to be zero for $s>t$ .",
"Based on this equality we derive in Proposition REF and REF suitable integrability and time regularity properties of $DX(t)$ by using Gronwall-type arguments.",
"The regularity results from Section are then used in Section for the analysis of the weak error ${E}[f(\\tilde{X}_{h,k})-f(X)]$ , where $X_{h,k}=(\\tilde{X}_{h,k}(t))_{t\\in [0,T]}$ , $h,k\\in (0,1)$ , are time interpolated numerical approximations of $X$ .",
"We use a standard finite element method with maximal mesh size $h$ for the discretization in space and an implicit Euler method with step size $k$ for the discretization in time.",
"For finite Borel measures $\\mu _1,\\ldots ,\\mu _n$ on $[0,T]$ , we consider path-dependent functionals $f\\colon L^1([0,T],\\sum _{i=1}^n\\mu _i;H)\\rightarrow {R}$ of the form $f(x)=\\varphi \\big (\\int _{[0,T]}x(t)\\,\\mu _1(\\mathrm {d}t),\\ldots ,\\int _{[0,T]}x(t)\\,\\mu _n(\\mathrm {d}t)\\big )$ , where $\\varphi \\colon \\bigoplus _{i=1}^n\\!H\\rightarrow {R}$ is assumed to be Fréchet differentiable with globally Lipschitz continuous derivative mapping $\\varphi ^{\\prime }\\colon \\bigoplus _{i=1}^n\\!H\\rightarrow \\mathcal {L}\\big (\\bigoplus _{i=1}^n\\!H,{R}\\big )$ .",
"Our main result, Theorem REF , states that for all $\\gamma \\in [0,\\beta )$ there exists a finite constant $C$ such that $|{E}[f(\\tilde{X}_{h,k})-f(X)]|\\leqslant C\\,(h^{2\\gamma }+k^\\gamma ),\\quad h,k\\in (0,1).$ For the considered class of test functions, the weak rate of convergence is thus twice the strong rate.",
"The idea of the proof is to exploit the Malliavin regularity of $X$ and $\\tilde{X}_{h,k}$ in order to estimate the weak error $|{E}[f(\\tilde{X}_{h,k})-f(X)]|$ in terms of the norm of the error $\\tilde{X}_{h,k}(t)-X(t)$ in the dual space $({M}^{1,p,q}(H))^*$ , for suitable exponents $p,q\\in [2,\\infty )$ .",
"As an exemplary application, we consider in Corollary REF the approximation of covariances $\\mathrm {Cov}(\\langle X(t_1),\\psi _1\\rangle ,\\langle X(t_2),\\psi _2\\rangle )$ , $t_1,t_2\\in [0,T]$ , $\\psi _1,\\psi _1\\in H$ of the solution process.",
"We remark that weak error estimates for SPDE involving path-dependent functionals have been derived so far only in [1], [4], [10].",
"Our setting allows for integral-type functionals as well as for functionals of the form $f(x)=\\varphi (x(t_1),\\ldots ,x(t_n))$ , where $x=(x(t))_{t\\in [0,T]}$ is an $H$ -valued path, $0\\leqslant t_1\\leqslant \\ldots \\leqslant t_n\\leqslant T$ , and $\\varphi \\colon \\bigoplus _{j=1}^n\\!H\\rightarrow {R}$ .",
"The paper is organized as follows: In Section we collect some general notation (Subsection REF ), introduce the precise assumptions on the Lévy process $L$ (Subsection REF ), and review fundamental concepts and results from Hilbert space-valued Poisson Malliavin calculus (Subsection REF ).",
"Section is concerned with the Malliavin regularity of the mild solution $X$ to Eq.",
"(REF ).",
"Here we first describe in detail our assumptions on the considered equation (Subsection REF ) before we analyse the regularity of $X$ (Subsection REF ) and derive some auxiliary results concerning refined Malliavin-Sobolev spaces (Subsection REF ).",
"The weak convergence analysis is found in Section , where we present the numerical scheme and our main result (Subsection REF ), analyze the regularity of the approximation process (Subsection REF ) as well as convergence in negative order Malliavin-Sobolev spaces (Subsection REF ), and finally prove the main result by combining the results previously collected (Subsection REF .)"
],
[
"General notation",
"If $(U,\\Vert \\cdot \\Vert _U,\\langle \\cdot ,\\cdot \\rangle _U)$ and $(V,\\Vert \\cdot \\Vert _V,\\langle \\cdot ,\\cdot \\rangle _V)$ are separable real Hilbert spaces, we denote by ${\\mathcal {L}}(U,V)$ and ${\\mathcal {L}}_2(U,V)\\subset {\\mathcal {L}}(U,V)$ the spaces of bounded linear operators and Hilbert-Schmidt operators from $U$ to $V$ , respectively.",
"By $\\mathcal {C}^1(U,V)$ we denote the space of Fréchet differentiable functions $f\\colon U\\rightarrow V$ with continuous derivative $f^{\\prime }\\colon U\\rightarrow \\mathcal {L}(U,V)$ .",
"In the special case $V={R}$ we identify $\\mathcal {L}(U,{R})$ with $U$ via the Riesz isomorphism and consider $f^{\\prime }$ as a $U$ -valued mapping.",
"The Lipschitz spaces $\\operatorname{Lip}^0(U,V)&:=\\lbrace f\\in \\mathcal {C}(U,V):|f|_{\\operatorname{Lip}^0(U,V)}<\\infty \\rbrace ,\\\\\\operatorname{Lip}^1(U,V)&:=\\lbrace f\\in \\mathcal {C}^1(U,V):|f|_{\\operatorname{Lip}^0(U,V)}+|f|_{\\text{Lip}^1(U,V)}<\\infty \\rbrace ,$ are defined in terms of the semi-norms $|f|_{\\operatorname{Lip}^0(U,V)}&:=\\sup \\Big (\\Big \\lbrace \\tfrac{\\Vert f(x)-f(y)\\Vert _V}{\\Vert x-y\\Vert _U}:x,y\\in U,\\,x\\ne y\\Big \\rbrace \\cup \\lbrace 0\\rbrace \\Big ),\\\\|f|_{\\operatorname{Lip}^1(U,V)}&:=\\sup \\Big (\\Big \\lbrace \\tfrac{\\Vert f^{\\prime }(x)-f^{\\prime }(y)\\Vert _{\\mathcal {L}(U,V)}}{\\Vert x-y\\Vert _U}:x,y\\in U,\\,x\\ne y\\Big \\rbrace \\cup \\lbrace 0\\rbrace \\Big ),$ compare, e.g., [11].",
"We also use the norm $\\Vert f\\Vert _{\\operatorname{Lip}^0(U,V)}:=\\Vert f(0)\\Vert _V+|f|_{\\operatorname{Lip}^0(U,V)}$ .",
"If $(S,\\mathcal {S},m)$ is a $\\sigma $ -finite measure space and $(X,\\Vert \\cdot \\Vert _X)$ is a Banach space, we denote by $L^0(S;X):=L^0(S,\\mathcal {S},m;X)$ the space of (equivalence classes of) strongly $\\mathcal {S}$ -measurable functions $f\\colon S\\rightarrow X$ .",
"As usual, we identify functions which coincide $m$ -almost everywhere.",
"The space $L^0(S;X)$ is endowed with the topology of local convergence in measure.",
"For $p\\in [1,\\infty ]$ , we denote by $L^p(S;X):=L^p(S,\\mathcal {S},m;X)$ the subspace of $L^0(S;X)$ consisting of all (equivalence classes of) strongly $\\mathcal {S}$ -measurable mappings $f\\colon S\\rightarrow X$ such that $\\Vert f\\Vert _{L^p(S;X)}:=\\big (\\int _S\\Vert f(s)\\Vert _X^p\\,m(\\mathrm {d}s)\\big )^{1/p}<\\infty $ if $p\\in [1,\\infty )$ and $\\Vert f\\Vert _{L^\\infty (S;X)}:=\\operatorname{ess\\,sup}_{s\\in S}\\Vert f(s)\\Vert _X<\\infty $ if $p=\\infty $ .",
"By $\\lambda $ we denote one-dimensional Lebesgue measure and we sometimes also write $\\lambda (\\mathrm {d}t)$ , $\\mathrm {d}t$ , $\\lambda (\\mathrm {d}s)$ , $\\mathrm {d}s$ etc.",
"in place of $\\lambda $ to improve readability."
],
[
"Lévy processes and Poisson random measures",
"Here we describe in detail the setting concerning the driving process $L$ in Eq.",
"(REF ).",
"Our standard reference for Hilbert space-valued Lévy processes is [27].",
"Assumption 2.1 The following setting is considered throughout the article.",
"[leftmargin=7mm] $(\\Omega ,\\mathcal {F},{{P}})$ is a complete probability space.",
"The $\\sigma $ -algebra $\\mathcal {F}$ coincides with the ${{P}}$ -completion of the $\\sigma $ -algebra $\\sigma (L(t):t\\in [0,T])$ generated by the Lévy process $L$ introduced below.",
"$L=(L(t))_{t\\in [0,T]}$ is a Lévy process defined on $(\\Omega ,\\mathcal {F},{{P}})$ , taking values in a separable real Hilbert space $(U,\\Vert \\cdot \\Vert _U,\\langle \\cdot ,\\cdot \\rangle _U)$ .",
"Here $T\\in (0,\\infty )$ is fixed.",
"We assume that $L$ is square-integrable with mean zero, i.e., $L(t)\\in L^2(\\Omega ;U)$ and ${E}( L(t))=0$ , and that the Gaussian part of $L$ is zero.",
"$(H,\\Vert \\cdot \\Vert ,\\langle \\cdot ,\\cdot \\rangle )$ is a further separable real Hilbert space.",
"The jump intensity measure (Lévy measure) $\\nu \\colon \\mathcal {B}(U)\\rightarrow [0,\\infty ]$ of a general $U$ -valued Lévy process $L$ satisfies $\\nu (\\lbrace 0\\rbrace )=0$ and $\\int _U\\min (\\Vert x\\Vert _U^2,1)\\,\\nu (\\mathrm {d}x)<\\infty $ , cf. [27].",
"Due to our square integrability assumption on $L$ we additionally have $|\\nu |_2 := \\Big (\\int _{U}\\Vert y\\Vert _U^2\\,\\nu (\\mathrm {d}y)\\Big )^{\\frac{1}{2}}<\\infty ,$ see, e.g., [27].",
"As a further consequence of our assumptions on $L$ , the characteristic function of $L(t)$ is of given by ${E}e^{i\\langle x,L(t)\\rangle _U}=\\exp \\Big (-t\\int _U\\big (1-e^{i\\langle x,y\\rangle _U}+i\\langle x,y\\rangle _U\\big )\\,\\nu (\\mathrm {d}y)\\Big ),\\quad x\\in U,$ cf. [27].",
"Conversely, every $U$ -valued Lévy process $L$ satisfying (REF ) and (REF ) is square-integrable with mean zero and vanishing Gaussian part.",
"We always consider a fixed càdlàg (right continuous with left limits) modification of $L$ .",
"The jumps of $L$ determine a Poisson random measure on $\\mathcal {B}([0,T]\\times U)$ as follows: For $(\\omega ,t)\\in \\Omega \\times (0,T]$ we denote by $\\Delta L(t)(\\omega ):=L(t)(\\omega )-\\lim _{s\\nearrow t}L(s)(\\omega )\\in U$ the jump of a trajectory of $L$ at time $t$ .",
"Then $N(\\omega ):=\\sum _{t\\in (0,T]:\\Delta L(t)(\\omega )\\ne 0}\\delta _{(t,\\Delta L(t)(\\omega ))},\\quad \\omega \\in \\Omega ,$ defines a Poisson random measure $N$ on $\\mathcal {B}([0,T]\\times U)$ with intensity measure $\\lambda \\otimes \\nu $ , where $\\delta _{(t,x)}$ denotes Dirac measure at $(t,x)\\in [0,T]\\times U$ and $\\nu $ is the Lévy measure of $L$ .",
"This follows, e.g., from Theorem 6.5 in [27] together with Theorems 4.9, 4.15, 4.23 and Lemma 4.25 therein.",
"It the context of Poisson Malliavin calculus it is useful to consider $N$ as a random variable with values in the space $\\mathbf {N}=\\mathbf {N}([0,T]\\times U)$ of all $\\sigma $ -finite ${N}_0\\cup \\lbrace +\\infty \\rbrace $ -valued measures on $\\mathcal {B}([0,T]\\times U)$ .",
"It is endowed with the $\\sigma $ -algebra $\\mathcal {N}=\\mathcal {N}([0,T]\\times U)$ generated by the mappings $\\mathbf {N}\\ni \\mu \\mapsto \\mu (B)\\in {N}_0\\cup \\lbrace +\\infty \\rbrace $ , $B\\in \\mathcal {B}([0,T]\\times U)$ .",
"We now list some important notation used in the present context.",
"Notation 2.2 The following notation is used throughout the article.",
"[leftmargin=7mm] $\\nu \\colon \\mathcal {B}(U)\\rightarrow [0,\\infty ]$ and $(U_0,\\Vert \\cdot \\Vert _{U_0},\\langle \\cdot ,\\cdot \\rangle _{U_0})$ are the Lévy measure and the reproducing kernel Hilbert space of $L$ , respectively; cf. [27].",
"$N\\colon \\Omega \\rightarrow \\mathbf {N}$ is the Poisson random measure (Poisson point process) on $[0,T]\\times U$ determined by the jumps of $L$ as specified in Eq.",
"(REF ) above.",
"The compensated Poisson random measure is denoted by $\\tilde{N}:=N-\\lambda \\otimes \\nu $ , i.e., $\\tilde{N}(B)=N(B)-(\\lambda \\otimes \\nu )(B)$ for all $B\\in \\mathcal {B}([0,T]\\times U)$ with $(\\lambda \\otimes \\nu )(B)<\\infty $ $(\\mathcal {F}_t)_{t\\in [0,T]}$ is the filtration given by $\\mathcal {F}_t:=\\bigcap _{u\\in (t,T]}\\tilde{\\mathcal {F}}_u,$ where $\\tilde{\\mathcal {F}}_u$ is the ${P}$ -completion of $\\sigma (L(s):s\\in [0,u])$ .",
"For $p\\in \\lbrace 0\\rbrace \\cup [1,\\infty ]$ set $L^p(\\Omega ;H):=L^p(\\Omega ,\\mathcal {F},{P};H)$ and $L^p(\\Omega \\times [0,T]\\times U;H):=$ $L^p(\\Omega \\times [0,T]\\times U,\\mathcal {F}\\otimes \\mathcal {B}([0,T]\\times U),{P}\\otimes \\lambda \\otimes \\nu ;H)$ .",
"Moreover, $\\mathcal {P}_T\\subset \\mathcal {F}\\otimes \\mathcal {B}([0,T])$ denotes the $\\sigma $ -algebra of predictable sets w.r.t.",
"to $(\\mathcal {F}_t)_{t\\in [0,T]}$ and we further set $L^2_{\\operatorname{pr}}(\\Omega \\times [0,T]\\times U; H):=L^2\\big (\\Omega \\times [0,T]\\times U,\\mathcal {P}_T\\otimes \\mathcal {B}(U),{{P}}\\otimes \\lambda \\otimes \\nu ;H\\big ).$ We end this section by recalling some basics on stochastic integration w.r.t.",
"$L$ and $\\tilde{N}$ , cf. [27].",
"The $H$ -valued $L^2$ stochastic integral $\\int _0^T\\Phi (s)\\,\\mathrm {d}L(s)$ w.r.t.",
"$L$ is defined for all $\\Phi \\in L^2_{\\operatorname{pr}}(\\Omega \\times [0,T];\\mathcal {L}_2(U_0,H)):=L^2(\\Omega \\times [0,T],\\mathcal {P}_T,{P}\\otimes \\lambda ;\\mathcal {L}_2(U_0,H))$ , and we have the Itô isometry ${E}\\big \\Vert \\int _0^T\\Phi (s)\\,\\mathrm {d}L(s)\\big \\Vert ^2=\\int _0^T{E}\\Vert \\Phi (s)\\Vert _{\\mathcal {L}_2(U_0,H)}^2\\mathrm {d}s$ .",
"The $H$ -valued $L^2$ stochastic integral $\\int _0^T\\int _U\\Phi (s,x)\\,\\tilde{N}(\\mathrm {d}s,\\mathrm {d}x)$ w.r.t.",
"$\\tilde{N}$ is defined for all $\\Phi \\in L^2_{\\operatorname{pr}}(\\Omega \\times [0,T]\\times U;H)$ , and here it holds that ${E}\\big \\Vert \\int _0^T\\int _U\\Phi (s,x)\\,\\tilde{N}(\\mathrm {d}s,\\mathrm {d}x)\\big \\Vert ^2=\\int _0^T\\int _U{E}\\Vert \\Phi (s,x)\\Vert ^2\\nu (\\mathrm {d}x)\\mathrm {d}s$ .",
"As usual, we set $\\int _0^t\\Phi (s)\\,\\mathrm {d}L(s):=\\int _0^T{1}_{(0,t]}(s)\\Phi (s)\\,\\mathrm {d}L(s)$ and $\\int _0^t\\int _U\\Phi (s,x)\\,\\tilde{N}(\\mathrm {d}s,\\mathrm {d}x):=\\int _0^T\\int _U{1}_{(0,t]}(s)\\Phi (s,x)\\,\\tilde{N}(\\mathrm {d}s,\\mathrm {d}x)$ , $t\\in [0,T]$ .",
"A useful property shown in [21] is the following: There exists an isometric embedding $\\kappa \\colon L^2_{\\operatorname{pr}}(\\Omega \\times [0,T];\\mathcal {L}_2(U_0,H))\\rightarrow L^2_{\\operatorname{pr}}(\\Omega \\times [0,T]\\times U;H)$ such that $\\int _0^T\\Phi (s)\\,\\mathrm {d}L(s)=\\int _0^T\\int _U \\Phi (s)x\\,\\tilde{N}(\\mathrm {d}s,\\mathrm {d}x)$ for $\\Phi \\in L^2_{\\operatorname{pr}}(\\Omega \\times [0,T];\\mathcal {L}_2(U_0,H))$ , where we set $\\Phi (s)x:=\\kappa (\\Phi )(s,x)$ to simplify notation."
],
[
"Poisson-Malliavin calculus in Hilbert space",
"In this subsection we collect some concepts and results from Hilbert space-valued Poisson Malliavin calculus.",
"We refer to [4] and the references therein for a more detailed exposition.",
"While in the Gaussian case the Malliavin derivative is a differential operator, one possible analogue in the Poisson case is a finite difference operator $D\\colon L^0(\\Omega ;H)\\rightarrow L^0(\\Omega \\times [0,T]\\times U;H)$ defined as follows.",
"Recall that $\\mathcal {F}$ is the ${P}$ -completion of the $\\sigma $ -algebra generated by the Lévy process $L$ , which coincides with the ${P}$ -completion of the $\\sigma $ -algebra generated by the Poisson random measure $N$ .",
"This and the factorization theorem from measure theory imply that for every random variable $F\\colon \\Omega \\rightarrow H$ there exists a $\\mathcal {N}$ -$\\mathcal {B}(H)$ -measurable function $f\\colon \\mathbf {N}\\rightarrow H$ , called a representative of $F$ , such that $F=f(N)$ ${P}$ -almost surely.",
"In this situation we set $\\varepsilon ^+_{t,x}F:=f(N+\\delta _{(t,x)})$ , where $\\delta _{(t,x)}$ denotes Dirac measure at $(t,x)\\in [0,T]\\times U$ .",
"As a consequence of Mecke's formula, this definition is ${P}\\otimes \\mathrm {d}t\\otimes \\nu (\\mathrm {d}x)$ -almost everywhere independent of the choice of the representative $f$ , so that $F\\mapsto \\big (\\varepsilon ^+_{t,x}F\\big )$ is well-defined as a mapping from $L^0(\\Omega ;H)$ to $L^0(\\Omega \\times [0,T]\\times U;H)$ , cf. [4].",
"The difference operator $D\\colon L^0(\\Omega ;H)\\rightarrow L^0(\\Omega \\times [0,T]\\times U;H),\\;F\\mapsto DF=\\big (D_{t,x}F\\big )$ is then defined by $D_{t,x}F:=\\varepsilon ^+_{t,x}F-F,\\quad (t,x)\\in [0,T]\\times U.$ The Malliavin-Sobolev space $D^{1,2}(H)$ consists of all $F\\in L^2(\\Omega ;H)$ satisfying $DF\\in L^2(\\Omega \\times [0,T]\\times U; H)$ .",
"In Subsection REF we introduce refined Malliavin-Sobolev spaces ${M}^{1,p,q}(H)$ , $p,q\\in (1,\\infty ]$ .",
"The following basic lemmata are taken from [4].",
"Lemma 2.3 Let $F\\in L^0(\\Omega ;H)$ and $h$ be a measurable mapping from $H$ to another separable real Hilbert space $V$ .",
"Then it holds that $D h(F)=h(F+D F)-h(F).$ Lemma 2.4 Let $t\\in [0,T]$ and $F\\colon \\Omega \\rightarrow H$ be $\\mathcal {F}_t$ -$\\mathcal {B}(H)$ -measurable.",
"Then the equality $D_{s,x}F=0$ holds ${P}\\otimes \\mathrm {d}s\\otimes \\nu (\\mathrm {d}x)$ -almost everywhere on $\\Omega \\times (t,T]\\times U$ .",
"The next result is a special case of the general duality formula in [4].",
"It is crucial for our approach to weak error analysis for Lévy driven SPDE.",
"Proposition 2.5 (Duality formula) For all $F\\in {D}^{1,2}(H)$ and $\\Phi \\in L^{2}_{\\operatorname{pr}}(\\Omega \\times [0,T]\\times U;H)$ we have ${E}\\,\\Big \\langle F,\\int _0^T\\int _U\\Phi (t,x)\\,\\tilde{N}(\\mathrm {d}t,\\mathrm {d}x)\\Big \\rangle ={{E}}\\int _0^T\\int _U\\big \\langle D_{t,x}F,\\,\\Phi (t,x)\\big \\rangle \\,\\nu (\\mathrm {d}x)\\,\\mathrm {d}t.$ Before we proceed with two further important results, we need to discuss the application of $D$ on stochastic processes.",
"Remark 2.6 (Difference operator for stochastic processes) One can define in a analogous way as above for stochastic processes a further difference operator $D$ mapping $X\\in L^0(\\Omega \\times [0,T];H)$ to $DX=\\big (D_{s,x}X(t)\\big )_{t\\in [0,T],(s,x)\\in [0,T]\\times U}\\in L^0\\big (\\Omega \\times [0,T]\\times [0,T]\\times U;H\\big )$ , see [4].",
"Then it holds for $\\lambda $ -almost all $t\\in [0,T]$ that $D_{s,x}X(t)=D_{s,x}(X(t))\\quad {P}\\otimes \\mathrm {d}s\\otimes \\nu (\\mathrm {d}x)\\text{-a.e.",
"},$ where $D(X(t))=\\big (D_{s,x}(X(t))\\big )_{(s,x)\\in [0,T]\\times U}\\in L^0(\\Omega \\times [0,T]\\times U;H)$ is for fixed $t$ the Malliavin derivative of the random variable $F=X(t)$ as introduced above.",
"We will, however, typically encounter the situation where $X=(X(t))_{t\\in [0,T]}$ is not given as an equivalence class of stochastic processes but as a single stochastic process with $X(t)$ being specifically defined for every $t\\in [0,T]$ .",
"If $X$ is not only $\\mathcal {F}\\otimes \\mathcal {B}([0,T])$ -measurable but also stochastically continuous or piecewise stochastically continuous, then there exists a ${P}\\otimes \\mathrm {d}t\\otimes \\mathrm {d}s\\otimes \\nu (\\mathrm {d}x)$ -version of $DX=\\big (D_{s,x}X(t)\\big )_{t\\in [0,T],(s,x)\\in [0,T]\\times U}$ such that (REF ) holds for every $t\\in [0,T]$ , cf. [4].",
"We also use a further analogously defined difference operator $D$ mapping $\\Phi \\in L^0(\\Omega \\times [0,T]\\times U;H)$ to $D\\Phi =\\big (D_{s,x}\\Phi (t,y)\\big )_{(t,y),(s,x)\\in [0,T]\\times U}\\in L^0(\\Omega \\times ([0,T]\\times U)^2;H)$ in such a way that for $\\lambda \\otimes \\nu $ -almost all $(t,y)\\in [0,T]\\times U$ we have $D_{s,x}\\Phi (t,y)=D_{s,x}(\\Phi (t,y))$ ${P}\\otimes \\mathrm {d}s\\otimes \\nu (\\mathrm {d}x)$ -a.e., cf. [4].",
"In the regularity analysis of SPDEs it is important to know how $D$ acts on Lebesgue integrals and stochastic integrals.",
"For this purpose we recall the following results.",
"The first one is taken from [4], the second is a special case of [4] combined with [4].",
"Proposition 2.7 (Malliavin derivative of time integrals) Let $X\\colon \\Omega \\times [0,T]\\rightarrow H$ be a stochastic process which is $\\mathcal {F}\\otimes \\mathcal {B}([0,T])$ -measurable and piecewise stochastically continuous, let $\\mu $ be a $\\sigma $ -finite Borel-measure on $[0,T]$ , and assume that $X$ belongs to $L^1([0,T],\\mu ;L^p(\\Omega ;H))$ for some $p>1$ .",
"Consider a fixed version of $DX=(D_{s,x}X(t))_{t\\in [0,T],(s,x)\\in [0,T]\\times U}$ such that for all $t\\in [0,T]$ the identity $D_{s,x}X(t)=D_{s,x}(X(t))$ holds ${P}\\otimes \\mathrm {d}s\\otimes \\nu (\\mathrm {d}x)$ -almost everywhere, cf.",
"Remark REF .",
"Then, for all $B\\in \\mathcal {B}(U)$ with $\\nu (B)<\\infty $ we have ${E}\\Big [\\int _{[0,T]}\\int _B\\int _{[0,T]}\\Vert D_{s,x}X(t)\\Vert \\,\\mu (\\mathrm {d}t)\\,\\nu (\\mathrm {d}x)\\,\\mathrm {d}s\\Big ]<\\infty ,$ so that the integral $\\int _{[0,T]} D_{s,x}X(t)\\,\\mu (\\mathrm {d}t)$ is defined ${P}\\otimes \\mathrm {d}s\\otimes \\nu (\\mathrm {d}x)$ -almost everywhere on $\\Omega \\times [0,T]\\times U$ as an $H$ -valued Bochner integral.",
"Moreover, the equality $D_{s,x}\\int _{[0,T]} X(t)\\,\\mu (\\mathrm {d}t)=\\int _{[0,T]} D_{s,x}X(t)\\,\\mu (\\mathrm {d}t)$ holds ${P}\\otimes \\mathrm {d}s\\otimes \\nu (\\mathrm {d}x)$ -almost everywhere on $\\Omega \\times [0,T]\\times U$ .",
"Proposition 2.8 (Malliavin derivative of stochastic integrals) Let $\\Phi \\in L^{2}_{\\operatorname{pr}}(\\Omega \\times [0,T]\\times U;H)$ .",
"Then the derivative $D\\Phi \\in L^0\\big (\\Omega \\times ([0,T]\\times U)^2;H\\big )$ has a $\\mathcal {P}_T\\otimes \\mathcal {B}(U)\\otimes \\mathcal {B}([0,T]\\times U)$ -measurable version, i.e., the mapping $D\\Phi \\colon \\Omega \\times ([0,T]\\times U)^2\\rightarrow H,\\;(\\omega ,t,y,s,x)\\mapsto D_{s,x}\\Phi (\\omega ,t,y)$ has a ${{P}}\\otimes (\\lambda \\otimes \\nu )^{\\otimes 2}$ -version which is $\\mathcal {P}_T\\otimes \\mathcal {B}(U)\\otimes \\mathcal {B}([0,T]\\times U)$ -measurable.",
"If moreover ${{E}}\\int _0^T\\int _U\\Vert D_{s,x}\\Phi (t,y)\\Vert ^2\\,\\nu (\\mathrm {d}y)\\,\\mathrm {d}t<\\infty $ for $\\lambda \\otimes \\nu $ -almost all $(s,x)\\in [0,T]\\times U$ , then the equality $D_{s,x}\\int _0^T\\int _U\\Phi (t,y)\\,\\tilde{N}(\\mathrm {d}t,\\mathrm {d}y)=\\int _0^T\\int _UD_{s,x}\\Phi (t,y)\\,\\tilde{N}(\\mathrm {d}t,\\mathrm {d}y)+\\Phi (s,x)$ holds ${P}\\otimes \\mathrm {d}s\\otimes \\nu (\\mathrm {d}x)$ -almost everywhere on $\\Omega \\times [0,T]\\times U$ ."
],
[
"Assumptions on the considered equation",
"We next state the precise assumptions on the operator $A$ , the driving noise $L$ , the nonlinearity $F$ , and the initial value $X_0$ in Eq.",
"(REF ).",
"Assumption 3.1 In addition to Assumption REF , suppose that the following holds: [leftmargin=8mm] (i) The operator $A\\colon D(A)\\subset H\\rightarrow H$ is densely defined, linear, self-adjoint, positive definite and has a compact inverse.",
"In particular, $-A$ is the generator of an analytic semigroup of contractions, which we denote by $(S(t))_{t\\geqslant 0}\\subset \\mathcal {L}(H)$ .",
"The spaces $\\dot{H}^\\rho $ , $\\rho \\in {{R}}$ , are defined for $\\rho \\geqslant 0$ as $\\dot{H}^\\rho := D(A^{\\frac{\\rho }{2}})$ with norm $\\Vert \\cdot \\Vert _{\\dot{H}^\\rho }:=\\Vert A^{\\frac{\\rho }{2}}\\cdot \\Vert $ and for $\\rho <0$ as the closure of $H$ w.r.t.",
"the analogously defined $\\Vert \\cdot \\Vert _{\\dot{H}^\\rho }$ -norm.",
"(ii) For some $\\beta \\in (0,1]$ , the state space $U$ of the Lévy process $L=(L(t))_{t\\in [0,T]}$ in Assumption REF is given by $U=\\dot{H}^{\\beta -1}$ .",
"(iii) For some $\\delta \\in [1-\\beta ,2)$ , the drift function $F\\colon H\\rightarrow \\dot{H}^{\\beta -1}$ belongs to the class $\\operatorname{Lip}^0(H,\\dot{H}^{\\beta -1})\\cap \\operatorname{Lip}^1(H,\\dot{H}^{-\\delta })$ .",
"(iv) The initial value $X_0$ is an element of the space $\\dot{H}^{2\\beta }$ .",
"It is well known that, under Assumption REF (i), there exist constants $C_\\rho \\in [0,\\infty )$ (independent of $t$ ) such that $\\big \\Vert A^\\frac{\\rho }{2}S(t)\\big \\Vert _{{\\mathcal {L}}(H)}&\\leqslant C_\\rho \\, t^{-\\frac{\\rho }{2}},\\quad t>0, \\ \\rho \\geqslant 0,\\\\\\big \\Vert A^{-\\frac{\\rho }{2}}(S(t)-\\mathrm {id}_H)\\big \\Vert _{{\\mathcal {L}}(H)}&\\leqslant C_\\rho \\, t^{\\frac{\\rho }{2}},\\quad \\;\\;t\\geqslant 0, \\ \\rho \\in (0,2],$ see, e.g., [26].",
"Concerning Assumption REF (iii), let us remark that Lipschitz continuity of the derivative $F^{\\prime }$ of $F$ is needed for the weak convergence analysis in Section .",
"Assuming $F\\in \\operatorname{Lip}^1(H,H)$ is sufficient for the analysis, compare, e.g., [2].",
"In applications to SPDE this assumption is not satisfactory as the most important type of nonlinear drift, the Nemytskii type drift, typically does not satisfy the assumption.",
"By assuming that $F^{\\prime }$ is Lipschitz continuous only as a mapping into the larger space $\\dot{H}^{-\\delta }$ , for suitable $\\delta $ , the Sobolev embedding theorem can be used to prove that Nemytskii type nonlinearities are in fact included in $d\\in \\lbrace 1,2,3\\rbrace $ space dimensions.",
"More precisely this holds for $\\delta >\\frac{d}{2}$ , compare [29].",
"Example 3.2 For $d\\in \\lbrace 1,2,3\\rbrace $ let $\\mathcal {O}\\subset {R}^d$ be an open, bounded, convex, poly-gonal/polyhedral domain and set $H:=L^2(\\mathcal {O})$ .",
"Our standard example for $A$ is a second order elliptic partial differential operator with zero Dirichlet boundary condition of the form $Au:=-\\nabla \\cdot (a\\nabla u)+cu,\\; u\\in D(A):=H^1_0(\\mathcal {O})\\cap H^2(\\mathcal {O}),$ with bounded and sufficiently smooth coefficients $a,c\\colon \\mathcal {O}\\rightarrow {R}$ such that $a(\\xi )\\geqslant \\theta >0$ and $c(\\xi )\\geqslant 0$ for all $\\xi \\in \\mathcal {O}$ .",
"Here $H^1_0(\\mathcal {O})$ and $H^2(\\mathcal {O})$ are the classical $L^2$ -Sobolev spaces of order one with zero Dirichlet boundary condition and of order two, respectively.",
"As an example for the drift function $F$ we consider the Nemytskii type nonlinearity given by $(F(x))(\\xi )=f(x(\\xi ))$ , $x\\in L^2(\\mathcal {O})$ , $\\xi \\in \\mathcal {O}$ , where $f\\colon {R}\\rightarrow {R}$ is twice continuously differentiable with bounded first and second derivative.",
"In this situation, Assumption REF (iii) is fullfilled for $\\delta >\\frac{d}{2}$ , compare [29].",
"Concrete examples for the Lévy process $L$ can be found in [21].",
"By a mild solution to Eq.",
"(REF ) we mean an $(\\mathcal {F}_t)_{t\\in [0,T]}$ -predictable stochastic process $X\\colon \\Omega \\times [0,T] \\rightarrow H$ such that $\\sup _{t\\in [0,T]}\\Vert X(t)\\Vert _{L^2(\\Omega ;H)} < \\infty ,$ and such that for all $t\\in [0,T]$ it holds ${{P}}$ -almost surely that $X(t)=S(t)X_0+\\int _0^tS(t-s)F(X(s))\\,\\mathrm {d}s+\\int _0^tS(t-s)\\,\\mathrm {d}L(s).$ Under Assumption REF there exists a unique (up to modification) mild solution $X$ to Eq.",
"(REF ).",
"This follows, e.g., from a straightforward modification of the proof of [27], where slightly different assumptions are used.",
"Moreover, this solution is mean-square continuous, i.e., $X\\in \\mathcal {C}([0,T],L^2(\\Omega ;H))$ , which can be seen by using standard arguments analogous to those used in the Gaussian case."
],
[
"Regularity results for the solution process",
"We are now ready to analyze the Malliavin regularity of the mild solution to Eq.",
"(REF ).",
"Proposition 3.3 Let Assumption REF hold, let $X=(X(t))_{t\\in [0,T]}$ be the mild solution to Eq.",
"(REF ), and consider a fixed version of $DX=(D_{s,x}X(t))_{t\\in [0,T],(s,x)\\in [0,T]\\times U}$ such that for all $t\\in [0,T]$ the identity $D_{s,x}X(t)=D_{s,x}(X(t))$ holds ${P}\\otimes \\mathrm {d}s\\otimes \\nu (\\mathrm {d}x)$ -almost everywhere, cf.",
"Remark REF .",
"Then for all $t\\in [0,T]$ and all $B\\in \\mathcal {B}(U)$ with $\\nu (B)<\\infty $ we have ${E}\\int _0^T\\int _B\\int _0^t \\Big \\Vert S(t-r)\\Big [F\\big (X(r)+D_{s,x}X(r)\\big )-F\\big (X(r)\\big )\\Big ]\\Big \\Vert \\,\\mathrm {d}r\\,\\nu (\\mathrm {d}x)\\,\\mathrm {d}s<\\infty ,$ so that for all $t\\in [0,T]$ the integral $\\int _0^tS(t-r)\\big [F\\big (X(r)+D_{s,x}X(r)\\big )-F\\big (X(r)\\big )\\big ]\\,\\mathrm {d}r$ is defined ${P}\\otimes \\mathrm {d}s\\otimes \\nu (\\mathrm {d}x)$ -almost everywhere on $\\Omega \\times [0,T]\\times U$ as an $H$ -valued Bochner integral.",
"Moreover, for all $t\\in [0,T]$ the equality (REF ) holds ${P}\\otimes \\mathrm {d}s\\otimes \\nu (\\mathrm {d}x)$ -almost everywhere on $\\Omega \\times [0,T]\\times U$ .",
"We fix $t\\in [0,T]$ and apply the difference operator $D\\colon L^0(\\Omega ;H)\\rightarrow L^0(\\Omega \\times [0,T]\\times U;H)$ to the single terms in (REF ).",
"As the initial value $X_0$ is deterministic, it is clear that $D_{s,x}(S(t)X_0)=0$ ${P}\\otimes \\mathrm {d}s\\otimes \\nu (\\mathrm {d}x)$ -almost everywhere on $\\Omega \\times [0,T]\\times U$ .",
"Next, observe that by (REF ), the linear growth of $F$ and (REF ) we have $\\begin{aligned}&\\int _0^t\\big \\Vert S(t-r)F(X(r))\\big \\Vert _{L^2(\\Omega ;H)}\\mathrm {d}r\\\\&\\leqslant C_{1-\\beta }\\Vert F\\Vert _{\\operatorname{Lip}(H,\\dot{H}^{\\beta -1})}\\int _0^t (t-r)^{\\frac{\\beta -1}{2}}\\big (1+\\Vert X(r)\\Vert _{L^2(\\Omega ;H)}\\big )\\,\\mathrm {d}r\\\\&\\leqslant C_{1-\\beta }\\Vert F\\Vert _{\\operatorname{Lip}(H,\\dot{H}^{\\beta -1})}\\frac{2}{\\beta +1}T^{\\frac{\\beta +1}{2}}\\,\\big (1+\\sup _{t\\in [0,T]}\\Vert X(t)\\Vert _{L^2(\\Omega ;H)}\\big )\\,<\\infty .\\end{aligned}$ Proposition REF thus implies (REF ) and that the equality $D_{s,x}\\int _0^t S(t-r)F(X(r))\\,\\mathrm {d}r=\\int _0^t D_{s,x}\\big (S(t-r)F(X(r))\\big )\\,\\mathrm {d}r$ holds ${P}\\otimes \\mathrm {d}s\\otimes \\nu (\\mathrm {d}x)$ -a.e.",
"on $\\Omega \\times [0,T]\\times U$ .",
"Hereby we consider a version of $\\big (D_{s,x}\\big (S(t-r)F(X(r))\\big )\\big )_{r\\in [0,t),(s,x)\\in [0,T]\\times U}$ which is $\\mathcal {F}\\otimes \\mathcal {B}([0,t))\\otimes \\mathcal {B}([0,T]\\times U)$ -measurable, cf.",
"Remark REF .",
"Using also Lemma REF and Lemma REF , we obtain $\\begin{aligned}&D_{s,x}\\int _0^t S(t-r)F(X(r))\\,\\mathrm {d}r\\\\&={1}_{s\\leqslant t}\\cdot \\int _s^tS(t-r)\\big [F\\big (X(r)+D_{s,x}X(r)\\big )-F\\big (X(r)\\big )\\big ]\\,\\mathrm {d}r\\end{aligned}$ ${P}\\otimes \\mathrm {d}s\\otimes \\nu (\\mathrm {d}x)$ -a.e.",
"on $\\Omega \\times [0,T]\\times U$ .",
"Finally, the identity $\\int _0^t S(t-r)\\,\\mathrm {d}L(r)=\\int _0^t\\int _U S(t-r)x\\,\\tilde{N}(\\mathrm {d}r,\\mathrm {d}x)$ and the commutation relation in Proposition REF yield $D_{s,x}\\int _0^t S(t-r)\\,\\mathrm {d}L(r)={1}_{s\\leqslant t}\\cdot S(t-s)x$ ${P}\\otimes \\mathrm {d}s\\otimes \\nu (\\mathrm {d}x)$ -a.e.",
"on $\\Omega \\times [0,T]\\times U$ .",
"Summing up, we have shown that (REF ) holds for every fixed $t\\in [0,T]$ as an equality in $L^0(\\Omega \\times [0,T]\\times U;H)$ .",
"The refined Malliavin-Sobolev spaces introduced next and the subsequent regularity results have Gaussian counterparts in [1], [2].",
"Definition 3.4 (Refined Sobolev-Malliavin spaces) Consider the setting described in Subsections REF and REF .",
"For $p,q\\in (1,\\infty ]$ we define ${M}^{1,p,q}(H)$ as the space consisting of all $F\\in L^p(\\Omega ;H)$ such that $DF\\in L^p(\\Omega ;L^q([0,T];L^2(U;H)))$ .",
"It is equipped with the seminorm $|F|_{{M}^{1,p,q}(H)}: =\\Vert DF\\Vert _{L^p(\\Omega ;L^q([0,T];L^2(U;H)))}$ and norm $\\Vert F\\Vert _{{M}^{1,p,q}(H)}:=\\Big (\\Vert F\\Vert _{L^p(\\Omega ;H)}^p+|F|_{{M}^{1,p,q}(H)}^p\\Big )^\\frac{1}{p}.$ For $p,p^{\\prime },q,q^{\\prime }\\in (1,\\infty )$ such that $\\frac{1}{p}+\\frac{1}{p^{\\prime }}=\\frac{1}{q}+\\frac{1}{q^{\\prime }}=1$ , the space ${M}^{-1,p^{\\prime },q^{\\prime }}(H)$ is defined as the (topological) dual space of ${M}^{1,p,q}(H)$ .",
"Arguing as in [4] one finds that ${M}^{1,p,q}(H)$ is a Banach space for all $p,q\\in (1,\\infty )$ .",
"If additionally $p\\in [2,\\infty )$ , then ${M}^{1,p,q}(H)$ is continuously embedded in $L^2(\\Omega ;H)$ .",
"This embedding is dense according to [4].",
"In this situation we will use the Gelfand triple ${M}^{1,p,q}(H)\\subset L^2(\\Omega ;H)\\subset {M}^{-1,p^{\\prime },q^{\\prime }}(H)$ .",
"Proposition 3.5 (Regularity I) Let Assumption REF hold.",
"Depending on the value of $\\beta \\in (0,1]$ , we assume either that $q\\in (1,\\tfrac{2}{1-\\beta })$ if $\\beta \\in (0,1)$ or $q=\\infty $ if $\\beta =1$ .",
"Then it holds that $\\sup _{t\\in [0,T]}|X(t)|_{{M}^{1,\\infty ,q}(H)}<\\infty .$ As a consequence, we also have $\\sup _{t\\in [0,T]}\\Vert X(t)\\Vert _{{M}^{1,2,q}(H)}<\\infty $ .",
"We consider a fixed version of $DX=(D_{s,x}X(t))_{t\\in [0,T],(s,x)\\in [0,T]\\times U}$ such that for all $t\\in [0,T]$ the identity $D_{s,x}X(t)=D_{s,x}(X(t))$ holds ${P}\\otimes \\mathrm {d}s\\otimes \\nu (\\mathrm {d}x)$ -almost everywhere, cf.",
"Remark REF .",
"As a consequence of Proposition REF , the smoothing property (REF ), the fact that $U=\\dot{H}^{\\beta -1}$ and the Lipschitz continuity of $F$ , we know that for all $t\\in [0,T]$ the estimate $\\begin{aligned}\\Vert D_{s,x}X(t)\\Vert &\\leqslant {1}_{s\\leqslant t}\\cdot C_{1-\\beta }|F|_{\\operatorname{Lip}^0(H,\\dot{H}^{\\beta -1})}\\int _s^t(t-r)^{\\frac{\\beta -1}{2}}\\Vert D_{s,x}X(r)\\Vert \\,\\mathrm {d}r\\\\&\\quad +{1}_{s\\leqslant t}\\cdot C_{1-\\beta }\\Vert x\\Vert _{U}(t-s)^{\\frac{\\beta -1}{2}}\\end{aligned}$ holds ${P}\\otimes \\mathrm {d}s\\otimes \\nu (\\mathrm {d}x)$ -almost everywhere on $\\Omega \\times [0,T]\\times U$ .",
"Moreover, Proposition REF and $(\\ref {eq:XL2})$ imply that $\\int _0^T\\Vert D_{s,x}X(t)\\Vert \\,\\mathrm {d}t<\\infty $ ${P}\\otimes \\mathrm {d}s\\otimes \\nu (\\mathrm {d}x)$ -almost everywhere on $\\Omega \\times [0,T]\\times U$ .",
"In order to be able to apply the generalized Gronwall Lemma REF , we construct a new version of $(D_{s,x}X(t))_{t\\in [0,T],(s,x)\\in [0,T]\\times U}$ such that the estimates (REF ) and (REF ) hold everywhere on $\\Omega \\times [0,T]\\times [0,T]\\times U$ and $\\Omega \\times [0,T]\\times U$ , respectively.",
"For this purpose, let $A\\in \\mathcal {F}\\otimes \\mathcal {B}([0,T])\\otimes \\mathcal {B}([0,T]\\times U)$ be the set consisting of all $(\\omega ,t,s,x)\\in \\Omega \\times [0,T]\\times [0,T]\\times U$ for which (REF ) holds.",
"Let $B\\in \\mathcal {F}\\otimes \\mathcal {B}([0,T]\\times U)$ be the set consisting of all $(\\omega ,s,x)\\in \\Omega \\times [0,T]\\times U$ for which (REF ) holds $\\mathrm {d}t$ -almost everywhere on $[0,T]$ .",
"Finally, let $C\\in \\mathcal {F}\\otimes \\mathcal {B}([0,T]\\times U)$ be the set consisting of all $(\\omega ,s,x)\\in \\Omega \\times [0,T]\\times U$ for which (REF ) holds.",
"Let $\\Gamma \\colon \\Omega \\times [0,T]\\times [0,T]\\times U\\rightarrow H$ be defined by $\\Gamma :={1}_{A\\cap \\pi ^{-1}(B\\cap C)}DX$ , where $\\pi \\colon \\Omega \\times [0,T]\\times [0,T]\\times U\\rightarrow \\Omega \\times [0,T]\\times U$ is the coordinate projection given by $\\pi (\\omega ,t,s,x):=(\\omega ,s,x)$ .",
"Note that for all $t\\in [0,T]$ the identity $\\Gamma (\\cdot ,t,\\cdot ,\\cdot )=DX(t)$ holds ${P}\\otimes \\lambda \\otimes \\nu $ -almost everywhere on $\\Omega \\times [0,T]\\times U$ .",
"We choose $\\Gamma $ as our new version of $DX$ and henceforth write $DX=(D_{s,x}X(t))_{t\\in [0,T],(s,x)\\in [0,T]\\times U}$ instead of $\\Gamma $ to simplify notation.",
"Observe that for this new version the estimates (REF ) and (REF ) hold indeed everywhere on $\\Omega \\times [0,T]\\times [0,T]\\times U$ and $\\Omega \\times [0,T]\\times U$ , respectively.",
"The generalized Gronwall Lemma REF thus implies that there exists a constant $C=C\\big (C_{1-\\beta }|F|_{\\operatorname{Lip}^0(H,\\dot{H}^{\\beta -1})},T,\\beta \\big )\\in [0,\\infty )$ such that the estimate $\\Vert D_{s,x}X(t)\\Vert \\leqslant {1}_{s\\leqslant t}\\,C\\,C_{1-\\beta }\\Vert x\\Vert _U(t-s)^{\\frac{\\beta -1}{2}}$ holds everywhere on $\\Omega \\times [0,T]\\times [0,T]\\times U$ .",
"Assume the case where $\\beta \\in (0,1)$ , $q\\in (1,\\frac{2}{1-\\beta })$ and consider the version of $DX=(D_{s,x}X(t))_{t\\in [0,T],(s,x)\\in [0,T]\\times U}$ constructed above.",
"Integration of (REF ) yields $&\\sup _{t\\in [0,T]}\\Big [\\int _0^T\\Big (\\int _U\\Vert D_{s,x}X(t)\\Vert ^2\\nu (\\mathrm {d}x)\\Big )^{\\frac{q}{2}}\\,\\mathrm {d}s\\Big ]^{\\frac{1}{q}}\\\\&\\leqslant C\\,C_{1-\\beta }\\,|\\nu |_2\\sup _{t\\in [0,T]}\\Big [\\int _0^t(t-s)^{q\\cdot \\frac{\\beta -1}{2}}\\,\\mathrm {d}s\\Big ]^{\\frac{1}{q}}\\leqslant C\\,C_{1-\\beta }\\,|\\nu |_2\\frac{1}{(q\\cdot \\frac{\\beta -1}{2}+1)^{\\frac{1}{q}}}T^{\\frac{\\beta -1}{2}+\\frac{1}{q}},$ which implies (REF ).",
"The case where $\\beta =1$ and $q=\\infty $ is treated similarly.",
"Finally, the second assertion in Proposition REF follows from (REF ) and (REF ).",
"Proposition 3.6 (Negative norm inequality) Consider the setting described in Subsections REF and REF .",
"Let $p^{\\prime },q^{\\prime }\\in (1,2]$ .",
"For predictable integrands $\\Phi \\in L_{\\mathrm {pr}}^2(\\Omega \\times [0,T]\\times U;H)$ it holds that $\\Big \\Vert \\int _0^T \\int _U\\Phi (t,y)\\tilde{N}(\\mathrm {d}t, \\mathrm {d}y)\\Big \\Vert _{{M}^{-1,p^{\\prime },q^{\\prime }}(H)}\\leqslant \\Vert \\Phi \\Vert _{L^{p^{\\prime }}(\\Omega ;L^{q^{\\prime }}([0,T];L^2(U;H)))}.$ Let $p,q\\in [2,\\infty )$ satisfy $\\tfrac{1}{p}+\\tfrac{1}{p^{\\prime }}=\\tfrac{1}{q}+\\tfrac{1}{q^{\\prime }}=1$ .",
"By the duality formula from Proposition REF , duality in the Gelfand triple ${M}^{1,p,q}(H)\\subset L^2(\\Omega ;H)\\subset {M}^{-1,p^{\\prime },q^{\\prime }}(H)$ , and by the Hölder inequality it holds that $&\\Big \\Vert \\int _0^T \\int _U\\Phi (t,y)\\tilde{N}(\\mathrm {d}t, \\mathrm {d}y)\\Big \\Vert _{{M}^{-1,p^{\\prime },q^{\\prime }}(H)}\\\\&\\quad =\\sup _{Z\\in {M}^{1,p,q}(H)\\setminus \\lbrace 0\\rbrace }\\frac{\\big \\langle Z,\\int _0^T \\int _U\\Phi (t,y)\\tilde{N}(\\mathrm {d}t, \\mathrm {d}y)\\big \\rangle _{L^2(\\Omega ;H)}}{\\Vert Z\\Vert _{{M}^{1,p,q}(H)}}\\\\&\\quad =\\sup _{Z\\in {M}^{1,p,q}(H)\\setminus \\lbrace 0\\rbrace }\\frac{\\langle D Z,\\Phi \\rangle _{L^2(\\Omega \\times [0,T]\\times U;H)}}{\\Vert Z\\Vert _{{M}^{1,p,q}(H)}}\\\\&\\quad \\leqslant \\sup _{Z\\in {M}^{1,p,q}(H)\\setminus \\lbrace 0\\rbrace }\\frac{\\Vert D Z\\Vert _{L^{p}(\\Omega ;L^{q}([0,T];L^2(U;H)))}\\Vert \\Phi \\Vert _{L^{p^{\\prime }}(\\Omega ;L^{q^{\\prime }}([0,T];L^2(U;H)))}}{\\Vert Z\\Vert _{{M}^{1,p,q}(H)}}\\\\&\\quad \\leqslant \\Vert \\Phi \\Vert _{L^{p^{\\prime }}(\\Omega ;L^{q^{\\prime }}([0,T];L^2(U;H)))}.$ Proposition 3.7 (Regularity II) Let Assumption REF hold and $X=(X(t))_{t\\in [0,T]}$ be the mild solution to Eq.",
"(REF ).",
"For all $\\gamma \\in [0,\\beta )$ and $q^{\\prime }=\\tfrac{2}{1+\\gamma }$ there exist a constant $C\\in [0,\\infty )$ such that $\\Vert X(t_2)-X(t_1)\\Vert _{{M}^{-1,2,q^{\\prime }}(H)}\\leqslant C |t_2 - t_1|^{\\gamma },\\quad t_1,t_2\\in [0,T].$ Let $0\\leqslant t_1\\leqslant t_2\\leqslant T$ .",
"Representing the increment $X(t_2)-X(t_1)$ via (REF ), taking norms and using the continuous embedding $L^2(\\Omega ;H)\\subset {M}^{-1,2,q^{\\prime }}(H)$ , we obtain $&\\Vert X(t_2) - X(t_1)\\Vert _{{M}^{-1,2,q^{\\prime }}(H)}\\leqslant \\Vert (S(t_2-t_1)-\\mathrm {id}_H)A^{-\\gamma }S(t_1) A^{\\gamma }X_0\\Vert \\\\&\\quad +\\Big \\Vert \\int _0^{t_1}(S(t_2-t_1)-\\mathrm {id}_H)A^{-\\gamma }A^\\gamma S(t_1-s)A^{\\frac{1-\\beta }{2}}A^{\\frac{\\beta -1}{2}}F(X(s))\\,\\mathrm {d}s\\Big \\Vert _{L^2(\\Omega ;H)}\\\\&\\quad +\\Big \\Vert \\int _{t_1}^{t_2}S(t_2-s))A^{\\frac{1-\\beta }{2}}A^{\\frac{\\beta -1}{2}}F(X(s))\\,\\mathrm {d}s\\Big \\Vert _{L^2(\\Omega ;H)}\\\\&\\quad +\\Big \\Vert \\int _0^{t_1}\\int _{\\dot{H}^{\\beta -1}}(S(t_2-t_1)-\\mathrm {id}_H)A^{-\\gamma }A^{\\gamma }S(t_1-s)x\\,\\tilde{N}(\\mathrm {d}s,\\mathrm {d}x)\\Big \\Vert _{{M}^{-1,2,q^{\\prime }}(H)}\\\\&\\quad +\\Big \\Vert \\int _{t_1}^{t_2}\\int _{\\dot{H}^{\\beta -1}}S(t_2-s)A^{\\frac{1-\\beta }{2}}A^{\\frac{\\beta -1}{2}}x\\,\\tilde{N}(\\mathrm {d}s,\\mathrm {d}x)\\Big \\Vert _{{M}^{-1,2,q^{\\prime }}(H)}.$ Further, from (REF ), (), (REF ), the linear growth of $F$ and the negative norm inequality in Proposition REF we obtain $\\begin{split}&\\Vert X(t_2) - X(t_1)\\Vert _{{M}^{-1,2,q^{\\prime }}(H)}\\,\\leqslant \\,C_0 C_{2\\gamma }\\Vert X_0\\Vert _{\\dot{H}^{2\\gamma }}(t_2-t_1)^\\gamma \\\\&+C_{2\\gamma +1-\\beta }\\int _0^{t_1}(t_1-s)^{-\\gamma -\\frac{1-\\beta }{2}}\\mathrm {d}s\\,C_{2\\gamma }\\,(t_2-t_1)^\\gamma \\\\&\\quad \\cdot \\Vert F\\Vert _{\\operatorname{Lip}^0(H,\\dot{H}^{\\beta -1})}\\big (1+\\sup _{t\\in [0,T]}\\Vert X(t)\\Vert _{L^2(\\Omega ;H)}\\big )\\\\&+C_{1-\\beta }\\int _{t_1}^{t_2}(t_2-s)^{-\\frac{1-\\beta }{2}}\\,\\mathrm {d}s\\,\\Vert F\\Vert _{\\operatorname{Lip}^0(H,\\dot{H}^{\\beta -1})}\\big (1+\\sup _{t\\in [0,T]}\\Vert X(t)\\Vert _{L^2(\\Omega ;H)}\\big )\\\\&+\\Big [\\int _0^{t_1}\\Big (\\int _{\\dot{H}^{\\beta -1}}\\big \\Vert (S(t_2-t_1)-\\mathrm {id}_H)A^{-\\gamma }A^{\\gamma }S(t_1-s)A^{\\frac{1-\\beta }{2}}A^{\\frac{\\beta -1}{2}}x\\big \\Vert ^2\\nu (\\mathrm {d}x)\\Big )^{\\frac{q^{\\prime }}{2}}\\,\\mathrm {d}s\\Big ]^{\\frac{1}{q^{\\prime }}}\\\\&+\\Big [\\int _{t_1}^{t_2}\\Big (\\int _{\\dot{H}^{\\beta -1}}\\big \\Vert S(t_2-s)A^{\\frac{1-\\beta }{2}}A^{\\frac{\\beta -1}{2}}x\\big \\Vert ^2\\nu (\\mathrm {d}x)\\Big )^{\\frac{q^{\\prime }}{2}}\\,\\mathrm {d}s\\Big ]^{\\frac{1}{q^{\\prime }}}.\\end{split}$ Note that $\\gamma +\\frac{1-\\beta }{2}<1$ and thus the integral in the second term on the right hand side of (REF ) is bounded by $(-\\gamma +\\frac{1+\\beta }{2})^{-1}T^{-\\gamma +\\frac{1+\\beta }{2}}$ .",
"The integral in the third term on the right hand side of (REF ) satisfies $\\int _{t_1}^{t_2}(t_2-s)^{-\\frac{1-\\beta }{2}}\\,\\mathrm {d}s=\\frac{2}{1+\\beta }(t_2-t_1)^{\\frac{1+\\beta }{2}}\\lesssim (t_2-t_1)^\\gamma .$ The fourth term on the right hand side of (REF ) can be estimated by $C_{2\\gamma }\\,(t_2-t_1)^\\gamma \\,C_{2\\gamma + (1-\\beta )}\\big (\\int _0^{t_1}(t_1-s)^{-\\frac{2}{1+\\gamma } \\frac{2\\gamma +1-\\beta }{2}}\\,\\mathrm {d}s\\big )^{\\frac{1+\\gamma }{2}}\\,|\\nu |_2\\lesssim (t_2-t_1)^\\gamma .$ The latter integral is bounded by $\\int _0^{T}s^{-\\frac{2}{1+\\gamma } \\frac{2\\gamma +1-\\beta }{2}}\\,\\mathrm {d}s$ , which is finite since $\\frac{2}{1+\\gamma } \\frac{2\\gamma +1-\\beta }{2}=\\frac{1 + \\gamma -(\\beta -\\gamma )}{1+\\gamma }<1.$ Finally, the the last term (REF ) is bounded by $C_{1-\\beta }\\,\\big (\\int _{t_1}^{t_2}(t_2-s)^{\\frac{2}{1+\\gamma }\\frac{\\beta -1}{2}}\\,\\mathrm {d}s\\big )^{\\frac{1+\\gamma }{2}}|\\nu |_2=\\big (\\frac{\\beta +\\gamma }{1+\\gamma }\\big )^{-\\frac{1+\\gamma }{2}}C_{1-\\beta }|\\nu |_2(t_2-t_1)^{\\frac{\\beta +\\gamma }{2}}\\lesssim (t_2-t_1)^\\gamma .$ This completes the proof."
],
[
"Auxiliary results on refined Malliavin Sobolev spaces",
"In the sequel, we consider the setting described in Subsection REF and REF .",
"Lemma 3.8 Let $p,q\\in (1,\\infty )$ , let $V_1$ , $V_2$ be separable real Hilbert spaces, and let $\\varphi \\colon H\\rightarrow \\mathcal {L}(V_1,V_2)$ be a bounded function belonging to the class $\\operatorname{Lip}^0(H,\\mathcal {L}(V_1,V_2))$ .",
"For all $Y\\in L^0(\\Omega ;H)$ satisfying $DY\\in L^\\infty (\\Omega ;L^q([0,T];L^2(U;H)))$ and all $Z\\in {M}^{1,p,q}(V_1)$ it holds that $\\varphi (Y)Z\\in {M}^{1,p,q}(V_2)$ and $\\Vert \\varphi (Y)Z\\Vert _{{M}^{1,p,q}(V_2)}&\\leqslant 2^{\\frac{1}{p}}\\Big (2\\sup _{x\\in H}\\Vert \\varphi (x)\\Vert _{\\mathcal {L}(V_1,V_2)} + |\\varphi |_{\\operatorname{Lip}^0(H,\\mathcal {L}(V_1,V_2))}|Y|_{{M}^{1,\\infty ,q}(H)}\\Big )\\\\&\\quad \\cdot \\Vert Z\\Vert _{{M}^{1,p,q}(V_1)}.$ Take $Y$ and $Z$ as in the statement and observe that $\\Vert \\varphi (Y)Z\\Vert _{L^{p}(\\Omega ;V_2)}\\leqslant \\sup _{x\\in H}\\Vert \\varphi (x)\\Vert _{\\mathcal {L}(V_1,V_2)}\\Vert Z\\Vert _{L^p(\\Omega ;V_1)}.$ Next, due to the definition of the difference operator $D$ in Subsection REF and the identities $D_{t,y}Y = \\varepsilon _{t,y}^+Y-Y$ and $\\varepsilon _{t,y}^+Y=Y + D_{t,y}Y$ it holds ${P}\\otimes \\mathrm {d}t\\otimes \\nu (\\mathrm {d}y)$ –almost everywhere on $\\Omega \\times [0,T]\\times U$ that $D_{t,y}(\\varphi (Y)Z)=\\varphi (\\varepsilon _{t,y}^+Y)\\,\\varepsilon _{t,y}^+Z - \\varphi (Y)Z=\\varphi (Y+D_{t,y}Y)\\,D_{t,y}Z + (\\varphi (Y + D_{t,y}Y)-\\varphi (Y))Z.$ As a consequence, we obtain $\\begin{aligned}\\big \\Vert D\\big (\\varphi (Y)&Z\\big )\\big \\Vert _{L^p(\\Omega ;L^q([0,T];L^2(U;V_2)))}\\leqslant \\Big (\\sup _{x\\in H}\\Vert \\varphi (x)\\Vert _{\\mathcal {L}(V_1,V_2)}\\\\&\\quad +|\\varphi |_{\\operatorname{Lip}^0(H,\\mathcal {L}(V_1,V_2))}\\Vert DY\\Vert _{L^{\\infty }(\\Omega ;L^q([0,T];L^2(U;H)))}\\Big )\\Vert Z\\Vert _{{M}^{1,p,q}(V_1)}.\\end{aligned}$ Combining (REF ) and (REF ) finishes the proof.",
"Proposition 3.9 (Local Lipschitz bound) Let $p^{\\prime },q^{\\prime }\\in (1,2]$ , $q\\in [2,\\infty )$ be such that $\\frac{1}{q}+\\frac{1}{q^{\\prime }}=1$ , let $V$ be a separable real Hilbert space, and $\\psi \\in \\operatorname{Lip}^1(H,V)$ .",
"Then there exists $C\\in [0,\\infty )$ such that for all $Y_1,Y_2\\in L^2(\\Omega ;H)$ with $DY_1,DY_2\\in L^\\infty (\\Omega ;L^{q}([0,T];L^2(U;H)))$ it holds that $&\\Vert \\psi (Y_1)-\\psi (Y_2)\\Vert _{{M}^{-1,p^{\\prime },q^{\\prime }}(V)}\\\\&\\quad \\leqslant 4\\Big (|\\psi |_{\\operatorname{Lip}^0(H,V)}+|\\psi |_{\\operatorname{Lip}^1(H,V)}\\sum _{i=1}^2|Y_i|_{{M}^{1,\\infty ,q}(H)}\\Big )\\Vert Y_1-Y_2\\Vert _{{M}^{-1,p^{\\prime },q^{\\prime }}(H)}.$ Let $p=p^{\\prime }/(p^{\\prime }-1)$ .",
"Due to the fundamental theorem of calculus and the structure of the Gelfand triple ${M}^{1,p,q}(V)\\subset L^2(\\Omega ;V)\\subset {M}^{-1,p^{\\prime },q^{\\prime }}(V)$ , it holds that $&\\Vert \\psi (Y_1)-\\psi (Y_2)\\Vert _{{M}^{-1,p^{\\prime },q^{\\prime }}(V)}=\\Big \\Vert \\int _0^1\\psi ^{\\prime }\\big (Y_2+\\lambda (Y_1-Y_2)\\big )(Y_1-Y_2)\\,\\mathrm {d}\\lambda \\Big \\Vert _{{M}^{-1,p^{\\prime },q^{\\prime }}(V)}\\\\&=\\sup _{\\begin{array}{c}Z\\in {M}^{1,p,q}(V)\\\\ \\Vert Z\\Vert _{{M}^{1,p,q}(V)}=1\\end{array}}\\Big \\langle Z, \\int _0^1\\psi ^{\\prime }\\big (Y_2+\\lambda (Y_1-Y_2)\\big )(Y_1-Y_2)\\,\\mathrm {d}\\lambda \\Big \\rangle _{L^2(\\Omega ;V)}\\\\&\\leqslant \\sup _{\\begin{array}{c}Z\\in {M}^{1,p,q}(V)\\\\ \\Vert Z\\Vert _{{M}^{1,p,q}(V)}=1\\end{array}}\\int _0^1\\big \\Vert \\big [\\psi ^{\\prime }\\big (Y_2+\\lambda (Y_1-Y_2)\\big )\\big ]^*Z\\big \\Vert _{{M}^{1,p,q}(H)}\\,\\mathrm {d}\\lambda \\,\\Vert Y_1-Y_2\\Vert _{{M}^{-1,p^{\\prime },q^{\\prime }}(H)},$ where for $x\\in H$ we denote by $[\\psi ^{\\prime }(x)]^*\\in \\mathcal {L}(V,H)$ the Hilbert space adjoint of $\\psi ^{\\prime }(x)\\in \\mathcal {L}(H,V)$ .",
"Note that the mapping $\\varphi \\colon H\\rightarrow \\mathcal {L}(V,H)$ defined by $\\varphi (x):=[\\psi ^{\\prime }(x)]^*$ , $x\\in H$ , is bounded and belongs to the class $\\operatorname{Lip}^0(H,\\mathcal {L}(V,H))$ .",
"We have $\\sup _{x\\in H}\\Vert \\varphi (x)\\Vert _{\\mathcal {L}(V,H)}\\leqslant |\\psi |_{\\operatorname{Lip}^0(H,V)}$ and $|\\varphi |_{\\operatorname{Lip}^0(H,\\mathcal {L}(V,H))}\\leqslant |\\psi |_{\\operatorname{Lip}^1(H,V)}$ .",
"An application of Lemma REF with $V_1=V$ , $V_2=H$ thus yields the assertion.",
"Lemma 3.10 Let $p^{\\prime },q^{\\prime }\\in (1,2]$ , $F\\in L^2(\\Omega ;H)$ and $S\\in {\\mathcal {L}}(H)$ .",
"It holds that $\\Vert SF\\Vert _{{M}^{-1,p^{\\prime },q^{\\prime }}(H)}\\leqslant \\Vert S\\Vert _{{\\mathcal {L}}(H)}\\Vert F\\Vert _{{M}^{-1,p^{\\prime },q^{\\prime }}(H)}$ .",
"Let $p,q\\in [2,\\infty )$ satisfy $\\tfrac{1}{p}+\\tfrac{1}{p^{\\prime }} = \\tfrac{1}{q} + \\tfrac{1}{q^{\\prime }} =1$ .",
"For notational convenience let $B={M}^{1,p,q}(H)$ and hence $B^*={M}^{-1,p^{\\prime },q^{\\prime }}(H)$ .",
"Assuming without loss of generality that $\\Vert S\\Vert _{\\mathcal {L}(H)}>0$ , it holds that $\\Vert SF\\Vert _{B^*}&=\\sup _{\\Vert Z\\Vert _{B}=1}\\langle SF,Z \\rangle _{L^2(\\Omega ;H)}=\\Vert S^*\\Vert _{{\\mathcal {L}}(H)}\\sup _{\\Vert Z\\Vert _{B}=1}\\Big \\langle F,\\frac{S^*Z}{\\Vert S^*\\Vert _{{\\mathcal {L}}(H)}} \\Big \\rangle _{L^2(\\Omega ;H)}\\\\&\\leqslant \\Vert S^*\\Vert _{{\\mathcal {L}}(H)}\\sup _{\\Vert Z\\Vert _{B}=1}\\langle F,Z \\rangle _{L^2(\\Omega ;H)}=\\Vert S\\Vert _{{\\mathcal {L}}(H)}\\Vert F\\Vert _{B^*}$"
],
[
"The main result and an application",
"Here we describe the numerical space-time discretization scheme for Eq.",
"(REF ) and formulate our main result on weak convergence in Theorem REF .",
"For the sake of comparability, we also state a corresponding strong convergence result in Proposition REF .",
"An application of Theorem REF to covariance convergence in presented in Corollary REF , see [20] for related results.",
"Assumption 4.1 (Discretization) For the spatial discretization we use a family $(V_h)_{h\\in (0,1)}$ of finite dimensional subspaces of $H$ and linear operators $A_h\\colon V_h\\rightarrow V_h$ that serve as discretizations of $A$ .",
"By $P_h\\colon H\\rightarrow V_h$ we denote the orthogonal projectors w.r.t.",
"the inner product in $H$ .",
"For the discretization in time we use a linearly implicit Euler scheme with uniform grid $t_m=km$ , $m\\in \\lbrace 0,\\ldots ,M\\rbrace $ , where $k\\in (0,1)$ is the stepsize and $M=M_k\\in {N}$ is determined by $t_M\\leqslant T< t_M+k$ .",
"The operators $S_{h,k}:=(\\mathrm {id}_{V_h}+kA_h)^{-1}P_h$ thus serve as discretizations of $S(k)$ , and $E_{h,k}^m:=S_{h,k}^m-S(t_m)$ are the corresponding error operators.",
"We assume that there are constants $D_\\rho ,\\,D_{\\rho ,\\sigma }\\in [0,\\infty )$ (independent of $h$ , $k$ , $m$ ) such that, $&\\Vert A_h^{\\frac{\\rho }{2}}S_{h,k}^m \\Vert _{{\\mathcal {L}}(H)} + \\Vert S_{h,k}^m A^{\\frac{\\min (\\rho ,1)}{2}} \\Vert _{{\\mathcal {L}}(H)}\\leqslant D_\\rho \\,t_m^{-\\frac{\\rho }{2}},\\quad \\rho \\geqslant 0,\\\\&\\Vert E_{h,k}^m A^{\\frac{\\rho }{2}}\\Vert _{{\\mathcal {L}}(H)}\\leqslant D_{\\rho ,\\sigma } \\,t_m^{-\\frac{\\rho +\\sigma }{2}}\\big (h^\\sigma + k^{\\frac{\\sigma }{2}}\\big ),\\quad \\sigma \\in [0,2],\\ \\rho \\in [-\\sigma ,\\min (1,2-\\sigma )],$ for all $h,k\\in (0,1)$ and $m\\in \\lbrace 1,\\ldots ,M\\rbrace $ .",
"Example 4.2 In the situation of Example REF , the spaces $V_h$ can be chosen as standard finite element spaces consisting of continuous, piecewise linear functions w.r.t.",
"regular triangulations of $\\mathcal {O}$ , with maximal mesh size bounded by $h$ .",
"See, e.g., [2] for a proof of the estimates (REF ), () in this case.",
"For $h,k\\in (0,1)$ and $M=M_k\\in {{N}}$ the approximation $(X^m_{h,k})_{m\\in \\lbrace 0,\\ldots ,M\\rbrace }$ of the mild solution $(X(t))_{t\\in [0,T]}$ to Eq.",
"(REF ) is defined recursively by $X^0_{h,k}=P_hX_0$ and $X^m_{h,k}=S^m_{h,k}X_0+k\\sum _{j=0}^{m-1}S^{m-j}_{h,k}F(X^j_{h,k})+\\sum _{j=0}^{m-1}S^{m-j}_{h,k}(L(t_{j+1})-L(t_j)),$ $m\\in \\lbrace 1,\\dots ,M\\rbrace $ .",
"By $(\\tilde{X}_{h,k}(t))_{t\\in [0,T]}$ we denote the piecewise constant interpolation of $(X_{h,k}^m)_{m\\in \\lbrace 0,\\ldots ,M\\rbrace }$ which is defined as $\\tilde{X}_{h,k}(t)=\\sum _{m=0}^{M-1}\\mathbf {1}_{[t_{m},t_{m+1})}(t)X_{h,k}^m+\\mathbf {1}_{[t_M,T]}(t)X^M_{h,k}.$ The following strong convergence result can be proven analogously to the Gaussian case, cf. [1].",
"Proposition 4.3 (Strong convergence) Let Assumption REF hold, let $(X(t))_{t\\in [0,T]}$ be the mild solution to Eq.",
"(REF ) and $(\\tilde{X}_{h,k}(t))_{t\\in [0,T]}$ be its discretization given by (REF ), (REF ).",
"Then, for every $\\gamma \\in [0,\\beta )$ there exists a constant $C\\in [0,\\infty )$ , which does not depend on $h,k$ , such that $\\sup _{t\\in [0,T]}\\Vert X(t)-\\tilde{X}_{h,k}(t)\\Vert _{L^2(\\Omega ;H)}\\leqslant C(h^\\gamma +k^{\\frac{\\gamma }{2}}),\\quad h,k\\in (0,1).$ For the weak convergence we consider path dependent functionals as specified by the next assumption.",
"In the related work [1] functionals of the form $f(x)=\\prod _{i=1}^n\\varphi _i\\big (\\int _{[0,T]}x(t)\\,\\mu _i(\\mathrm {d}t)\\big )$ , with $\\varphi _1,\\dots ,\\varphi _n$ being twice differentiable with polynomially growing derivatives of some fixed but arbitrary degree, and $\\mu _1,\\dots ,\\mu _n$ being finite Borel measures on $[0,T]$ , were considered for equations with Gaussian noise.",
"Here we generalize by removing the product structure, but we only allow for quadratically growing test functions.",
"The reason for the latter restrition is that the solution to our equation has in general only finite moments up to order two while solutions to equations with Gaussian noise have all moments finite.",
"Assumption 4.4 (Test function $f$ ) Let $n\\in {N}$ and $\\varphi \\colon \\bigoplus _{i=1}^n\\!H\\rightarrow {R}$ be Fréchet differentiable with globally Lipschitz continuous derivative mapping $\\varphi ^{\\prime }\\colon \\bigoplus _{i=1}^n\\!H\\rightarrow \\mathcal {L}\\big (\\bigoplus _{i=1}^n\\!H,{R}\\big )$ .",
"Let $\\mu _1,\\ldots ,\\mu _n$ be finite Borel-measures on $[0,T]$ .",
"The functional $f\\colon L^1([0,T],\\sum _{i=1}^n\\mu _i;H)\\rightarrow {R}$ is given by $f(x):=\\varphi \\Big (\\int _{[0,T]}x(t)\\,\\mu _1(\\mathrm {d}t),\\ldots ,\\int _{[0,T]}x(t)\\,\\mu _n(\\mathrm {d}t)\\Big ).$ Observe that $X,\\,\\tilde{X}_{h,k}\\in L^2(\\Omega ;L^1([0,T],\\sum _{i=1}^n\\mu _i;H))$ due to (REF ) and, e.g., the estimate (REF ) below.",
"In particular, the random variables $f(X)$ , $f(\\tilde{X}_{h,k})$ are defined and integrable.",
"We next state our main result on weak convergence.",
"The proof is postponed to Subsections REF –REF .",
"Note that the obtained weak rate of convergence is twice the strong rate from Propostion REF .",
"Theorem 4.5 (Weak convergence) Let Assumption REF , REF and REF hold.",
"Let $X=(X(t))_{t\\in [0,T]}$ be the mild solution to Eq.",
"(REF ) and $\\tilde{X}_{h,k}=(\\tilde{X}_{h,k}(t))_{t\\in [0,T]}$ be its discretization given by (REF ), (REF ).",
"Then, for every $\\gamma \\in [0,\\beta )$ there exists a constant $C\\in [0,\\infty )$ , which does not depend on $h,k$ , such that the weak error estimate (REF ) holds.",
"Corollary 4.6 (Covariance convergence) Consider the setting of Theorem REF .",
"For all $\\gamma \\in [0,\\beta )$ , $t_1,t_2\\in (0,T]$ and $\\phi _1,\\phi _2\\in H$ there exists a constant $C\\in [0,\\infty )$ , which does not depend on $h,k$ , such that $\\big |\\mathrm {Cov}\\big (\\big \\langle X(t_1),\\phi _1\\big \\rangle ,\\big \\langle X(t_2),\\phi _2\\big \\rangle \\big )-\\mathrm {Cov}\\big (\\big \\langle \\tilde{X}_{h,k}(t_1),\\phi _1&\\big \\rangle ,\\big \\langle \\tilde{X}_{h,k}(t_2),\\phi _2\\big \\rangle \\big )\\big |\\\\&\\leqslant C\\,(h^{2\\gamma }+k^\\gamma ),\\quad h,k\\in (0,1).$ For random variables $Y_1,Y_2,Z_1,Z_2\\in L^2(\\Omega ;H)$ and vectors $\\phi _1,\\phi _2\\in H$ it holds that $\\begin{split}&\\mathrm {Cov}\\big (\\langle Y_1,\\phi _1 \\rangle ,\\langle Y_2,\\phi _2 \\rangle \\big )-\\mathrm {Cov}\\big (\\langle Z_1,\\phi _1 \\rangle ,\\langle Z_2,\\phi _2 \\rangle \\big )\\\\&={{E}}\\big [\\langle Y_1,\\phi _1 \\rangle \\langle Y_2,\\phi _2 \\rangle -\\langle Z_1,\\phi _1 \\rangle \\langle Z_2,\\phi _2 \\rangle \\big ]-{{E}}\\big [\\langle Y_1,\\phi _1 \\rangle -\\langle Z_1,\\phi _1\\rangle \\big ]\\,{{E}}\\langle Y_2,\\phi _2\\rangle \\\\&\\quad -{{E}}\\langle Z_1,\\phi _1\\rangle \\,{{E}}\\big [\\langle Y_2,\\phi _2\\rangle -\\langle Z_2,\\phi _2\\rangle \\big ]\\end{split}$ We consider the Borel measure $\\mu :=\\delta _{t_1}+\\delta _{t_2}$ on $[0,T]$ as well as the functionals $f_i\\colon L^2([0,T],\\mu ;H)\\rightarrow {R}$ , $i\\in \\lbrace 1,2,3\\rbrace $ , given by $f_1(x):=\\langle x(t_1),\\phi _1\\rangle \\langle x(t_2),\\phi _2\\rangle $ , $f_2(x):=\\langle x(t_1),\\phi _1\\rangle $ , $f_3(x):=\\langle x(t_2),\\phi _2\\rangle $ .",
"These functionals satisfy Assumption REF .",
"From (REF ) with $Y_1 = X(t_1)$ , $Y_2 = X(t_2)$ , $Z_1=\\tilde{X}_{h,k}(t_1)$ and $Z_2=\\tilde{X}_{h,k}(t_2)$ we obtain $&\\big |\\mathrm {Cov}\\big (\\langle X(t_1),\\phi _1\\rangle ,\\langle X(t_2),\\phi _2\\rangle \\big )-\\mathrm {Cov}\\big (\\langle \\tilde{X}_{h,k}(t_1),\\phi _1\\rangle ,\\langle \\tilde{X}_{h,k}(t_2),\\phi _2\\rangle \\big )\\big |\\\\&\\leqslant \\big |{{E}}\\big [f_1(X)-f_1(\\tilde{X}_{h,k})\\big ]\\big |+\\Vert \\phi _2\\Vert \\sup _{t\\in [0,T]}\\Vert X(t)\\Vert _{L^2(\\Omega ;H)}\\big |{{E}}\\big [f_2(X)-f_2(\\tilde{X}_{h,k})\\big ]\\big |\\\\&\\quad +\\Vert \\phi _1\\Vert \\sup _{h,k\\in (0,1)}\\Vert \\tilde{X}_{h,k}(t_1)\\Vert _{L^2(\\Omega ;H)}\\big |{{E}}\\big [f_3(X)-f_3(\\tilde{X}_{h,k})\\big ]\\big |$ Three applications of Theorem REF together with (REF ) and the estimate (REF ) below complete the proof."
],
[
"A regularity result for the discrete solution",
"Here we prove an analogue of Proposition REF for the discrete solution.",
"It has Gaussian counterparts in [1] and [2].",
"Proposition 4.7 Let Assumption REF and REF hold.",
"Depending on the value of $\\beta \\in (0,1]$ , we assume either that $q\\in (1,\\tfrac{2}{1-\\beta })$ if $\\beta \\in (0,1)$ or $q=\\infty $ if $\\beta =1$ .",
"Then, $\\sup _{h,k\\in (0,1)}\\sup _{m\\in \\lbrace 0,\\dots ,M_k\\rbrace }\\Vert X^m_{h,k}\\Vert _{{M}^{1,2,q}(H)}<\\infty .$ By a classical Gronwall argument based on Lemma REF , it holds that $\\sup _{h,k\\in (0,1)}\\sup _{m\\in \\lbrace 0,\\dots ,M_k\\rbrace }\\Vert X^m_{h,k}\\Vert _{L^2(\\Omega ;H)}<\\infty .$ Up to some straightforward modifications, the proof of (REF ) is analogous to that of [2] in the Gaussian case and is therefore omitted.",
"Next, we rewrite the scheme (REF ) in the form $X^m_{h,k}=S^m_{h,k}X_0+k\\sum _{j=0}^{m-1}S^{m-j}_{h,k}F(X^j_{h,k})+\\sum _{j=0}^{m-1}\\int _0^T\\int _U{1}_{(t_j,t_{j+1}]}(s)\\,S^{m-j}_{h,k}x\\,\\tilde{N}(\\mathrm {d}s,\\mathrm {d}x),$ $m\\in \\lbrace 1,\\ldots ,M\\rbrace $ .",
"Applying the difference operator $D$ on the single terms in this equation and taking into account Lemma REF , Lemma REF and Proposition REF , we obtain $\\begin{aligned}D_{s,x}X^m_{h,k}&=k\\sum _{j=\\lceil s\\rceil _k}^{m-1}S^{m-j}_{h,k}\\big [F(X_{h,k}^j+D_{s,x}X_{h,k}^j)-F(X_{h,k}^j)\\big ]\\\\&\\quad +\\sum _{j=0}^{m-1}1_{(t_j,t_{j+1}]}(s)\\,S^{m-j}_{h,k}x\\end{aligned}$ holding ${P}\\otimes \\mathrm {d}s\\otimes \\nu (\\mathrm {d}x)$ -almost everywhere on $\\Omega \\times [0,T]\\times U$ .",
"Here we denote for $s\\in [0,T]$ by $\\lceil s\\rceil _k$ is the smallest number $i\\in {{N}}$ such that $ik\\geqslant s$ .",
"According to Lemma REF , the identity $D_{s,x}X_{h,k}^j=0$ holds ${P}\\otimes \\mathrm {d}s\\otimes \\nu (\\mathrm {d}x)$ -almost everywhere on $\\Omega \\times (t_j,T]\\times U$ .",
"Taking norms in (REF ) yields $\\begin{aligned}&\\Vert D_{s,x}X^m_{h,k}\\Vert _{L^\\infty (\\Omega ;L^q([0,T];L^2(U;H)))}\\\\&\\leqslant k\\sum _{j=0}^{m-1}\\Big \\Vert S^{m-j}_{h,k}\\big [F(X_{h,k}^j+DX_{h,k}^j)-F(X_{h,k}^j)\\big ]\\Big \\Vert _{L^\\infty (\\Omega ;L^q([0,T];L^2(U;H)))}\\\\&\\quad +\\Big \\Vert (s,x)\\mapsto \\sum _{j=0}^{m-1} 1_{(t_j,t_{j+1}]}(s)\\,S^{m-j}_{h,k}x\\Big \\Vert _{L^q([0,T];L^2(U;H))}.\\end{aligned}$ Using the estimate (REF ) and the Lipschitz assumption on $F$ , we obtain $\\begin{aligned}& \\Big \\Vert S^{m-j}_{h,k}\\big [F(X_{h,k}^j+DX_{h,k}^j)-F(X_{h,k}^j)\\big ]\\Big \\Vert _{L^\\infty (\\Omega ;L^q([0,T];L^2(U;H)))}\\\\&\\quad \\leqslant D_{1-\\beta }\\,t_{m-j}^{\\frac{\\beta -1}{2}}|F|_{\\operatorname{Lip}(H,\\dot{H}^{\\beta -1})}\\Vert DX_{h,k}^j\\Vert _{L^\\infty (\\Omega ;L^q([0,T];L^2(U,H)))}.\\end{aligned}$ Concerning the second term in (REF ) we apply the estimate (REF ) together with the identity $U=\\dot{H}^{\\beta -1}$ and observe that $\\begin{aligned}&\\Big \\Vert (s,x)\\mapsto \\sum _{j=0}^{m-1} 1_{(t_j,t_{j+1}]}(s)\\,S^{m-j}_{h,k}x\\Big \\Vert _{L^q([0,T];L^2(U;H))}\\\\&\\quad =\\Big (\\int _0^{T}\\Big (\\int _U\\Big \\Vert \\sum _{j=0}^{m-1} 1_{(t_j,t_{j+1}]}(s)\\,S^{m-j}_{h,k}x\\Big \\Vert ^2\\nu (\\mathrm {d}x)\\Big )^\\frac{q}{2}\\,\\mathrm {d}s\\Big )^{\\frac{1}{q}}\\\\&\\quad \\leqslant D_{1-\\beta }\\,|\\nu |_2\\Big (k\\sum _{j=0}^{m-1}t_{m-j}^{\\frac{q(\\beta -1)}{2}}\\Big )^{\\frac{1}{q}}\\leqslant D_{1-\\beta }\\,|\\nu |_2\\Big (\\int _0^T(T-r)^{\\frac{q(\\beta -1)}{2}}\\mathrm {d}r\\Big )^{\\frac{1}{q}}<\\infty .\\end{aligned}$ The penultimate inequality follows by approximating the sum by a Riemann integral and observing that the singularity is integrable.",
"From (REF ), (REF ) and (REF ) we conclude that for all $m\\in \\lbrace 1,\\dots M_k\\rbrace $ , uniformly in $h,k\\in (0,1)$ , $\\Vert DX_{h,k}^m\\Vert _{L^\\infty (\\Omega ;L^q([0,T];L^2(U,H)))}\\lesssim 1+k \\sum _{j=0}^{m-1}t_{m-j}^{\\frac{\\beta -1}{2}}\\Vert DX_{h,k}^j\\Vert _{L^\\infty (\\Omega ;L^q([0,T];L^2(U,H)))}.$ By induction we obtain that $DX_{h,k}^m\\in L^{\\infty }(\\Omega ;L^q([0,T];L^2(U;H)))$ for all $m$ , so that (REF ) and an application of the discrete Gronwall Lemma REF yield the uniform bound (REF )."
],
[
"Convergence in negative order spaces",
"The following crucial result has Gaussian counterparts in [1] and [2].",
"Lemma 4.8 Let Assumption REF and REF hold, let $(X(t))_{t\\in [0,T]}$ be the mild solution to Eq.",
"(REF ) and $(\\tilde{X}_{h,k}(t))_{t\\in [0,T]}$ be its discretization given by (REF ), (REF ).",
"Then, for every $\\gamma \\in [0,\\beta )$ and $q^{\\prime }=\\tfrac{2}{1+\\gamma }$ there exists a constant $C\\in [0,\\infty )$ , which does not depend on $h,k$ , such that $\\sup _{t\\in [0,T]}\\Vert \\tilde{X}_{h,k}(t)-X(t)\\Vert _{{M}^{-1,2,q^{\\prime }}(H)}\\leqslant C\\,\\big (h^{2\\gamma }+k^\\gamma \\big ),\\quad h,k\\in (0,1).$ For notational convenience we introduce the piecewise continuous error mapping $\\tilde{E}_{h,k}\\colon [0,T)\\rightarrow {\\mathcal {L}}(H)$ given by $\\tilde{E}_{h,k}(t):=S_{h,k}^m-S(t)$ for $t\\in [t_{m-1},t_m)$ , so that $X_{h,k}^m - X(t_m)&=E_{h,k}^mX_0+\\int _0^{t_m}\\tilde{E}_{h,k}(t_m-s)F(\\tilde{X}_{h,k}(s))\\,\\mathrm {d}s\\\\&\\quad +\\int _0^{t_m}S(t_m-s)\\big (F(\\tilde{X}_{h,k}(s))-F(X(s))\\big )\\,\\mathrm {d}s\\\\&\\quad +\\int _0^{t_m}\\int _{\\dot{H}^{\\beta -1}}\\tilde{E}_{h,k}(t_m-s)x\\,\\tilde{N}(\\mathrm {d}s,\\mathrm {d}x).$ Taking norms and using the continuous embedding $L^2(\\Omega ;H)\\subset {M}^{-1,2,q^{\\prime }}(H)$ as well as Minkowski's integral inequality yields $\\begin{aligned}&\\big \\Vert X_{h,k}^m - X(t_m)\\big \\Vert _{{M}^{-1,2,q^{\\prime }}(H)}\\\\&\\quad \\leqslant \\Vert E_{h,k}^mX_0\\Vert +\\int _0^{t_m}\\big \\Vert \\tilde{E}_{h,k}(t_m-s)F(\\tilde{X}_{h,k}(s))\\big \\Vert _{L^2(\\Omega ;H)}\\,\\mathrm {d}s\\\\&\\qquad +\\int _0^{t_m}\\big \\Vert S(t_m-s)\\big (F(\\tilde{X}_{h,k}(s))-F(X(s))\\big )\\big \\Vert _{{M}^{-1,2,q^{\\prime }}(H)}\\,\\mathrm {d}s\\\\&\\qquad +\\Big \\Vert \\int _0^{t_m}\\int _{\\dot{H}^{\\beta -1}}\\tilde{E}_{h,k}(t_m-s)x\\,\\tilde{N}(\\mathrm {d}s,\\mathrm {d}x)\\Big \\Vert _{{M}^{-1,2,q^{\\prime }}(H)}.\\end{aligned}$ We estimate the terms on the right hand side separately.",
"To this end, note that the error estimate () extends to the piecewise continuous error mapping $\\tilde{E}_{h,k}$ .",
"Indeed, as a consequence of the identity $\\tilde{E}_{h,k}(t)=E_{h,k}^m+(S(t_m)-S(t))$ , $t\\in [t_{m-1},t_m)$ , and the estimates (REF ), (), (), we have $\\big \\Vert \\tilde{E}_{h,k}(t)A^{\\frac{\\rho }{2}}\\big \\Vert _{{\\mathcal {L}}(H)}\\leqslant (D_{\\rho ,\\sigma }+C_\\sigma C_{\\sigma +\\rho })\\,t^{-\\frac{\\rho +\\sigma }{2}}\\big (h^\\sigma +k^{\\frac{\\sigma }{2}}\\big ),$ holding for $\\sigma \\in [0,2]$ , $\\rho \\in [-\\sigma ,\\min (1,2-\\sigma )]$ and $h,k\\in (0,1)$ , $t\\in (0,T]$ .",
"Concerning the first two terms on the right hand side of (REF ) we observe that (), (REF ), and the linear growth of $F$ yield $\\begin{aligned}&\\Vert E_{h,k}^mX_0\\Vert +\\int _0^{t_m}\\big \\Vert \\tilde{E}_{h,k}(t_m-s)F(\\tilde{X}_{h,k}(s))\\big \\Vert _{L^2(\\Omega ;H)}\\,\\mathrm {d}s\\\\&\\leqslant D_{-2\\gamma ,2\\gamma }\\Vert X_0\\Vert _{\\dot{H}^{2\\gamma }}\\big (h^{2\\gamma } + k^\\gamma \\big )+(D_{1-\\beta ,2\\gamma }+C_{2\\gamma }C_{2\\gamma +1-\\beta })\\frac{T^{\\frac{1+\\beta }{2}-\\gamma }}{(1+\\beta )/2-\\gamma }\\\\&\\quad \\cdot \\Vert F\\Vert _{\\operatorname{Lip}^0(H,\\dot{H}^{\\beta -1})}\\big (1+\\sup _{t\\in [0,T]}\\Vert X(t)\\Vert _{L^2(\\Omega ;H)}\\big )\\big (h^{2\\gamma } + k^\\gamma \\big ).\\end{aligned}$ Next, we use Lemma REF , (REF ) and Proposition REF to estimate the third term on the right hand side of (REF ) from above by $\\begin{aligned}&\\int _0^{t_m}\\big \\Vert S(t_m-s)A^{\\frac{\\delta }{2}}\\big \\Vert _{{\\mathcal {L}}(H)}\\big \\Vert A^{-\\frac{\\delta }{2}}\\big (F(\\tilde{X}_{h,k}(s))-F(X(s))\\big )\\big \\Vert _{{M}^{-1,2,q^{\\prime }}(H)}\\,\\mathrm {d}s\\\\&\\quad \\leqslant C_{\\delta }K\\sum _{i=0}^{m-1}\\int _{t_i}^{t_{i+1}}(t_m-s)^{-\\frac{\\delta }{2}}\\big \\Vert X_{h,k}^i-X(t_i)\\big \\Vert _{{M}^{-1,2,q^{\\prime }}(H)}\\,\\mathrm {d}s\\\\&\\qquad +C_{\\delta }K\\sum _{i=0}^{m-1}\\int _{t_i}^{t_{i+1}}(t_m-s)^{-\\frac{\\delta }{2}}\\big \\Vert X(t_i)-X(s)\\big \\Vert _{{M}^{-1,2,q^{\\prime }}(H)}\\,\\mathrm {d}s,\\end{aligned}$ where $K$ is, by Proposition REF , Proposition REF and Lemma REF , the finite constant $K&=4\\Big (|F|_{\\operatorname{Lip}^0(H,\\dot{H}^{-\\delta })}+ |F|_{\\operatorname{Lip}^1(H,\\dot{H}^{-\\delta })}\\sup _{s\\in [0,T]}|X(s)|_{{M}^{1,\\infty ,q}(H)}\\\\& \\qquad + |F|_{\\operatorname{Lip}^1(H,\\dot{H}^{-\\delta })} \\sup _{h,k\\in (0,1)}\\sup _{m\\in \\lbrace 0,\\ldots ,M_k\\rbrace }|X_{h,k}^m|_{{M}^{1,\\infty ,q}(H)}\\Big )<\\infty .$ The terms on the right hand side of (REF ) can be estimated as follows: We have $&\\sum _{i=0}^{m-1}\\int _{t_i}^{t_{i+1}}(t_m-s)^{-\\frac{\\delta }{2}}\\big \\Vert X_{h,k}^i-X(t_i)\\big \\Vert _{{M}^{-1,2,q^{\\prime }}(H)}\\,\\mathrm {d}s\\\\&\\leqslant k\\sum _{i=0}^{m-2}t_{m-i-1}^{-\\frac{\\delta }{2}}\\big \\Vert X_{h,k}^i-X(t_i)\\big \\Vert _{{M}^{-1,2,q^{\\prime }}(H)}+\\frac{k^{1-\\frac{\\delta }{2}}}{1-\\frac{\\delta }{2}}\\big \\Vert X_{h,k}^{m-1}-X(t_{m-1})\\big \\Vert _{{M}^{-1,2,q^{\\prime }}(H)}.$ Since for all $m\\in \\lbrace 2,3,\\ldots \\rbrace $ it holds that $\\max _{i\\in \\lbrace 0,1,\\dots , m-2\\rbrace }\\big (t_{m-i-1}^{-\\frac{\\delta }{2}} \\cdot t_{m-i}^{\\frac{\\delta }{2}} \\big )=$$\\max _{i\\in \\lbrace 0,1,\\dots , m-2\\rbrace }\\big ((m-i-1)^{-\\frac{\\delta }{2}}\\cdot (m-i)^{\\frac{\\delta }{2}}\\big )=2^{\\frac{\\delta }{2}},$ we obtain for all $m\\in {{N}}=\\lbrace 1,2,\\ldots \\rbrace $ $\\begin{aligned}&\\sum _{i=0}^{m-1}\\int _{t_i}^{t_{i+1}}(t_m-s)^{-\\frac{\\delta }{2}}\\big \\Vert X_{h,k}^i-X(t_i)\\big \\Vert _{{M}^{-1,2,q^{\\prime }}(H)}\\,\\mathrm {d}s\\\\&\\leqslant 2^{\\frac{\\delta }{2}}k\\sum _{i=0}^{m-2}t_{m-i}^{-\\frac{\\delta }{2}}\\big \\Vert X_{h,k}^i-X(t_i)\\big \\Vert _{{M}^{-1,2,q^{\\prime }}(H)}+\\frac{k^{1-\\frac{\\delta }{2}}}{1-\\frac{\\delta }{2}}\\big \\Vert X_{h,k}^{m-1}-X(t_{m-1})\\big \\Vert _{{M}^{-1,2,q^{\\prime }}(H)}\\\\&\\leqslant \\frac{2^{\\frac{\\delta }{2}}k}{1-\\frac{\\delta }{2}}\\sum _{i=0}^{m-1}t_{m-i}^{-\\frac{\\delta }{2}}\\big \\Vert X_{h,k}^i-X(t_i)\\big \\Vert _{{M}^{-1,2,q^{\\prime }}(H)}.\\end{aligned}$ Moreover, by the Hölder continuity of Proposition REF it holds $\\begin{aligned}\\sum _{i=0}^{m-1}\\int _{t_i}^{t_{i+1}}(t_m-s)^{-\\frac{\\delta }{2}}\\big \\Vert X(t_i)-X(s)\\big \\Vert _{{M}^{-1,2,q^{\\prime }}(H)}\\,\\mathrm {d}s\\leqslant Ck^{\\gamma }\\int _{0}^{t_m}(t_m-s)^{-\\frac{\\delta }{2}}\\,\\mathrm {d}s\\lesssim k^{\\gamma }.\\end{aligned}$ Concerning the fourth term on the right hand side of (REF ), note that the negative norm inequality in Proposition REF yields $\\begin{aligned}&\\Big \\Vert \\int _0^{t_m}\\int _{\\dot{H}^{\\beta -1}}\\tilde{E}_{h,k}(t_m-s)x\\,\\tilde{N}(\\mathrm {d}s,\\mathrm {d}x)\\Big \\Vert _{{M}^{-1,2,q^{\\prime }}(H)}\\\\&\\qquad \\leqslant \\Big [\\int _0^{t_m}\\Big (\\int _{\\dot{H}^{\\beta -1}}\\Vert \\tilde{E}_{h,k}(t_m-s)x\\Vert ^{2}\\,\\nu (\\mathrm {d}x)\\Big )^{\\frac{q^{\\prime }}{2}}\\,\\mathrm {d}s\\Big ]^{\\frac{1}{q^{\\prime }}}\\\\&\\qquad \\leqslant D_{1-\\beta ,2\\gamma }\\,|\\nu |_2\\,\\big (h^{2\\gamma } + k^\\gamma \\big )\\Big (\\int _0^{t_m}(t_m-s)^{\\frac{2}{1+\\gamma }\\frac{\\beta -1-2\\gamma }{2}}\\,\\mathrm {d}s\\Big )^{\\frac{1+\\gamma }{2}}\\lesssim h^{2\\gamma } + k^\\gamma ,\\end{aligned}$ where the last integral is finite due to (REF ).",
"Combining the estimates (REF )–(REF ) yields $&\\big \\Vert X_{h,k}^m - X(t_m)\\big \\Vert _{{M}^{-1,2,q^{\\prime }}(H)}\\lesssim h^{2\\gamma } + k^\\gamma +k\\sum _{i=0}^{m-1}t_{m-i}^{-\\frac{\\delta }{2}}\\big \\Vert X_{h,k}^i-X(t_i)\\big \\Vert _{{M}^{-1,2,q^{\\prime }}(H)}.$ The discrete Gronwall Lemma REF thus implies that there exists $C^{\\prime }\\in [0,\\infty )$ , which does not depend on $h,k$ , such that $\\sup _{m\\in \\lbrace 1,\\dots ,M_k\\rbrace }\\Vert X^m_{h,k}-X(t_m)\\Vert _{{M}^{-1,2,q^{\\prime }}(H)}\\leqslant C^{\\prime }\\big (h^{2\\gamma }+k^\\gamma \\big )$ for all $h,k\\in (0,1).$ This and Proposition REF imply the claimed assertion."
],
[
"Proof of the main result",
"We are finally prepared to prove the weak convergence result in Theorem REF .",
"Recall from Subsection REF that the processes $X=(X(t))_{t\\in [0,T]}$ and $\\tilde{X}_{h,k}=(\\tilde{X}_{h,k}(t))_{t\\in [0,T]}$ belong to $L^2(\\Omega ;L^1([0,T],\\sum _{i=1}^n\\mu _i;H))$ .",
"To simplify notation, we introduce the $(\\bigoplus _{i=1}^n\\!H)$ -valued random variables $Y=(Y^{(1)},\\ldots ,Y^{(n)})$ , $\\tilde{Y}_{h,k}=(\\tilde{Y}_{h,k}^{(1)},\\ldots ,\\tilde{Y}_{h,k}^{(n)})$ and $\\Phi _{h,k}=(\\Phi ^{(1)}_{h,k},\\ldots ,\\Phi ^{(n)}_{h,k})$ defined by $\\begin{aligned}Y^{(i)}&:=\\int _{[0,T]}X(t)\\,\\mu _i(\\mathrm {d}t),\\quad \\tilde{Y}^{(i)}_{h,k}:=\\int _{[0,T]}\\tilde{X}_{h,k}(t)\\,\\mu _i(\\mathrm {d}t),\\\\\\Phi ^{(i)}_{h,k}&:=\\int _0^1\\partial _i\\varphi \\big ((1-\\theta )Y+\\theta \\tilde{Y}_{h,k}\\big )\\,\\mathrm {d}\\theta .\\end{aligned}$ Here we denote for $x=(x^{(1)},\\ldots ,x^{(n)})\\in \\bigoplus _{j=1}^n\\!H$ by $\\partial _i\\varphi (x)=\\frac{\\partial }{\\partial x^{(i)}}\\varphi (x)$ the Fréchet derivative of $\\varphi $ w.r.t.",
"the $i$ -th coordinate of $x$ , considered as an element of $H$ via the Riesz isomorphism $\\mathcal {L}(H,{R})\\equiv H$ .",
"Moreover, we set set $q:=\\tfrac{2}{1-\\gamma }$ and $q^{\\prime }:=\\tfrac{2}{1+\\gamma }$ .",
"Using the notation above, the fundamental theorem of calculus, and duality in the Gelfand triple ${M}^{1,2,q}(H)\\subset L^2(\\Omega ;H)\\subset {M}^{-1,2,q^{\\prime }}(H)$ , we represent and estimate the weak error as follows: $\\begin{aligned}&\\big |{E}\\big [f(\\tilde{X}_{h,k})-f(X)\\big ]\\big |=\\big |{E}\\big [\\varphi (\\tilde{Y}_{h,k})-\\varphi (Y)\\big ]\\big |=\\Big |{E}\\sum _{i=1}^n\\big \\langle \\Phi ^{(i)}_{h,k},\\tilde{Y}^{(i)}_{h,k}-Y^{(i)}\\big \\rangle \\Big |\\\\&\\quad =\\Big |\\sum _{i=1}^n\\int _{[0,T]}{E}\\big \\langle \\Phi ^{(i)}_{h,k},\\tilde{X}_{h,k}(t)-X(t)\\big \\rangle \\,\\mu _i(\\mathrm {d}t)\\Big |\\\\&\\quad \\leqslant \\sum _{i=1}^n\\mu _i([0,T])\\,\\big \\Vert \\Phi ^{(i)}_{h,k}\\big \\Vert _{{M}^{1,2,q}(H)}\\sup _{t\\in [0,T]}\\big \\Vert \\tilde{X}_{h,k}(t)-X(t)\\big \\Vert _{{M}^{-1,2,q^{\\prime }}(H)}\\end{aligned}$ The assertion of Theorem REF now follows from (REF ) together with Lemma REF and Lemma REF below.",
"Lemma 4.9 Let Assumption REF , REF and REF hold.",
"Let $(X(t))_{t\\in [0,T]}$ be the mild solution to Eq.",
"(REF ), $(\\tilde{X}_{h,k}(t))_{t\\in [0,T]}$ be its discretization given by (REF ), (REF ), and let $\\Phi ^{(i)}_{h,k}$ , $i\\in \\lbrace 1,\\ldots ,n\\rbrace $ , $h,k\\in (0,1)$ be the $H$ -valued random variables defined by (REF ).",
"For all $\\gamma \\in [0,\\beta )$ and $q=\\tfrac{2}{1-\\gamma }$ it holds that $\\max _{i\\in \\lbrace 1,\\ldots ,n\\rbrace }\\sup _{h,k\\in (0,1)}\\big \\Vert \\Phi ^{(i)}_{h,k}\\big \\Vert _{{M}^{1,2,q}(H)}<\\infty .$ First note that the linear growth of $\\partial _i\\varphi \\colon \\bigoplus _{j=1}^n\\!H\\rightarrow H$ , the estimates (REF ), (REF ), and the fact that $\\mu _i([0,T])<\\infty $ imply for all $i\\in \\lbrace 1,\\ldots ,n\\rbrace $ that $\\sup _{h,k\\in (0,1)}\\big \\Vert \\Phi ^{(i)}_{h,k}\\big \\Vert _{L^2(\\Omega ;H)}<\\infty $ .",
"It remains to check that $\\sup _{h,k\\in (0,1)}\\big |\\Phi ^{(i)}_{h,k}\\big |_{{M}^{1,2,q}(H)}$ is finite.",
"The chain rule from Lemma REF , applied to the function $h\\colon (\\bigoplus _{j=1}^n\\!H)\\oplus (\\bigoplus _{j=1}^n\\!H)\\rightarrow H,\\,(y,\\tilde{y})\\mapsto \\int _0^1\\partial _i\\varphi \\big ((1-\\theta )y+\\theta \\tilde{y}\\big )\\,\\mathrm {d}\\theta $ , yields for all $i\\in \\lbrace 1,\\ldots ,n\\rbrace $ $&D_{s,x}\\Phi ^{(i)}_{h,k}=D_{s,x}\\int _0^1\\partial _i\\varphi \\big ((1-\\theta )Y+\\theta \\tilde{Y}_{h,k}\\big )\\,\\mathrm {d}\\theta \\\\&=\\int _0^1\\!\\Big [\\partial _i\\varphi \\Big (\\!",
"(1-\\theta )\\big (Y+D_{s,x}Y\\big )\\!+\\theta \\big (\\tilde{Y}_{h,k}+D_{s,x}\\tilde{Y}_{h,k}\\big )\\Big )\\!-\\partial _i\\varphi \\big ((1-\\theta )Y+\\theta \\tilde{Y}_{h,k}\\big )\\Big ]\\mathrm {d}\\theta $ ${P}\\otimes \\mathrm {d}s\\otimes \\nu (\\mathrm {d}x)$ -almost everywhere on $\\Omega \\times [0,T]\\times U$ .",
"This, the global Lipschitz continuity of $\\partial _i\\varphi \\colon \\bigoplus _{j=1}^n\\!H\\rightarrow H$ , and Proposition REF imply $\\big \\Vert D_{s,x}\\Phi ^{(i)}_{h,k}\\big \\Vert &\\leqslant |\\varphi |_{\\operatorname{Lip}^1(\\bigoplus _{j=1}^n\\!H;{R})}\\Big (\\big \\Vert D_{s,x}Y^{(i)}\\big \\Vert +\\big \\Vert D_{s,x}\\tilde{Y}^{(i)}_{h,k}\\big \\Vert \\Big )\\\\&\\leqslant |\\varphi |_{\\operatorname{Lip}^1(\\bigoplus _{j=1}^n\\!H;{R})}\\int _{[0,T]}\\Big (\\big \\Vert D_{s,x}X(t)\\big \\Vert +\\big \\Vert D_{s,x}\\tilde{X}_{h,k}(t)\\big \\Vert \\Big )\\mu _i(\\mathrm {d}t)$ Iterated integration w.r.t.",
"$\\nu (\\mathrm {d}x)$ , $\\mathrm {d}s$ , ${P}$ , and three applications of Minkowski's integral inequality lead to $\\big |\\Phi ^{(i)}_{h,k}\\big |_{{M}^{1,2,q}(H)}&\\leqslant |\\varphi |_{\\operatorname{Lip}^1(\\bigoplus _{j=1}^n\\!H;{R})}\\int _{[0,T]}\\!\\Big (|X(t)|_{{M}^{1,2,q}(H)}+\\big | \\tilde{X}_{h,k}(t)\\big |_{{M}^{1,2,q}(H)}\\Big )\\mu _i(\\mathrm {d}t)\\\\$ The estimates (REF ), (REF ) and the assumption that $\\mu _i([0,T])<\\infty $ thus imply for all $i\\in \\lbrace 1,\\ldots ,n\\rbrace $ the finiteness of $\\sup _{h,k\\in (0,1)}\\big |\\Phi ^{(i)}_{h,k}\\big |_{{M}^{1,2,q}(H)}$ ."
],
[
"appendix",
"Kristin Kirchner, Raphael Kruse, Annika Lang and Stig Larsson are gratefully acknowledged for participating in early discussion regarding this work and [4].",
"In this section we state two versions of Gronwall's lemma.",
"The first one follows from the arguments in the proof of [13] together with the standard version of Gronwall's lemma for measurable functions.",
"The second one is a slight modification of [22], compare also [13].",
"Lemma 1.1 (Generalized Gronwall lemma) Let $T\\in (0,\\infty )$ and $\\phi \\colon \\lbrace (t,s):0\\leqslant s\\leqslant t\\leqslant T\\rbrace \\rightarrow [0,\\infty )$ be a Borel measurable function satisfying $\\int _s^T\\phi (r,s)\\,\\mathrm {d}r<\\infty $ for all $s\\in [0,T]$ .",
"If $\\phi (t,s)\\leqslant A\\,(t-s)^{-1+\\alpha }+B\\int _s^t(t-r)^{-1+\\beta }\\phi (r,s)\\,\\mathrm {d}r,\\quad 0\\leqslant s\\leqslant t\\leqslant T,$ for some constants $A,B\\in [0,\\infty )$ , $\\alpha ,\\beta \\in (0,\\infty )$ , then there exists a constant $C=C(B,T,\\alpha ,\\beta )\\in [0,\\infty )$ such that $\\phi (t,s)\\leqslant C\\,A\\,(t-s)^{-1+\\alpha },\\;0\\leqslant s\\leqslant t\\leqslant T.$ Lemma 1.2 (Discrete Gronwall lemma) Let $T\\in (0,\\infty )$ , $k\\in (0,1)$ and $M=M_k\\in {N}$ be such that $M k\\leqslant T<(M+1)k$ , and set $t_m:=mk$ , $m\\in \\lbrace 0,\\ldots ,M\\rbrace $ .",
"Let $(\\phi _i)_{i=0}^{M}$ be a sequence of nonnegative real numbers.",
"If $\\phi _m\\leqslant A + B\\,k\\sum _{i=0}^{m-1}t_{m-i}^{-1+\\beta }\\phi _i,\\quad m\\in \\lbrace 0,\\dots ,M\\rbrace ,$ for some constants $A,B\\in [0,\\infty )$ , $\\beta \\in (0,1]$ , then there exists a constant $C=C(B,T,\\beta )\\in [0,\\infty )$ such that $\\phi _m\\leqslant C\\,A$ , $m\\in \\lbrace 0,\\dots ,M\\rbrace $ ."
]
] | 1808.08574 |
[
[
"Bernstein-Bezier weight-adjusted discontinuous Galerkin methods for wave\n propagation in heterogeneous media"
],
[
"Abstract This paper presents an efficient discontinuous Galerkin method to simulate wave propagation in heterogeneous media with sub-cell variations.",
"This method is based on a weight-adjusted discontinuous Galerkin method (WADG), which achieves high order accuracy for arbitrary heterogeneous media.",
"However, the computational cost of WADG grows rapidly with the order of approximation.",
"In this work, we propose a Bernstein-B\\'ezier weight-adjusted discontinuous Galerkin method (BBWADG) to address this cost.",
"By approximating sub-cell heterogeneities by a fixed degree polynomial, the main steps of WADG can be expressed as polynomial multiplication and $L^2$ projection, which we carry out using fast Bernstein algorithms.",
"The proposed approach reduces the overall computational complexity from $O(N^{2d})$ to $O(N^{d+1})$ in $d$ dimensions.",
"Numerical experiments illustrate the accuracy of the proposed approach, and computational experiments for a GPU implementation of BBWADG verify that this theoretical complexity is achieved in practice."
],
[
"Introduction",
"Efficient and accurate simulations of wave propagation are central to applications in seismology, where heterogeneities arise from the presence of different geological structures in the subsurface.",
"Accurate and efficient numerical methods for wave problems are becoming more and more important as the demand for solutions of large-scale problems increases.",
"This paper presents an efficient discontinuous Galerkin (DG) method for wave equations in heterogeneous media with sub-cell variations.",
"DG methods combine advantages of the finite volume method and the finite element method, which providing high order accuracy and addressing complex geometries through the use of unstructured meshes.",
"These methods are straightforward to parallelize and can be accelerated by taking advantage of high performance architectures such as Graphics Processing Units (GPUs) [2].",
"High order methods are especially attractive for wave propagation problems.",
"The simulation of wave propagation is observed to be more robust to grid distortion at high orders than at low orders [3], [4], and numerical dispersion and dissipation errors are small for high order approximations [5].",
"The goal of this work is to address two issues related to high order DG methods for wave propagation: computational cost at high orders and accurate resolution of media with sub-cell heterogeneities.",
"Nodal DG methods, which are popular implementations of DG for wave propagation problems [6], have a high computational complexity with respect to the order of approximation.",
"We aim to reduce this computational complexity using Bernstein polynomials [7].",
"Bernstein polynomials have been previously utilized by Ainsworth at el.",
"[8] and Kirby [9] to reduce computational costs associated with high order continuous finite element methods on simplices.",
"More recent work has exploited properties of Bernstein polynomials for DG methods.",
"For example, Kirby introduced a fast algorithm in [10] to invert the local mass matrix in DG schemes by exploiting a recursive block structure blackpresent under a Bernstein basis.",
"blackChan and Warburton later introduced a Bernstein-Bézier discontinuous Galerkin (BBDG) method based on the “strong” DG formulation [11].",
"In contrast to the approach of Kirby [10], the use of the “strong” formulation avoids explicitly introducing a mass matrix inverse, and instead formulates the DG formulation in terms of differentiation and lifting matrices.",
"BBDG exploits the facts that, in $d$ dimensions, blackthe derivative and the lift matrices can be recast as a combination of sparse matrices.",
"By exploiting this structure, the right-hand side of BBDG can be evaluated in $O(N^d)$ operations blackper element.",
"blackIn comparison, the dense linear algebra of nodal DG methods generally blackresults in a computational complexity of $O(N^{2d})$ blackper element.",
"A separate challenge in the simulation of wave propagation is the blackapproximation of media heterogeneities.",
"High order finite difference methods are widely used [12] in practice, but face challenges for complex geometries and non-smooth media [13].",
"The spectral element method (SEM) [14] provides one alternative to explicit high order finite difference methods.",
"SEM produces a diagonal global mass matrix, making it well-suited for explicit time-stepping, and can accommodate both complex geometries (through unstructured meshes) and discontinuous media.",
"However, SEM is restricted to quadrilateral and hexahedral meshes, which are less geometrically flexible than tetrahedral meshes.",
"Several modifications have been proposed to extend SEM to triangular and tetrahedral meshes, but they require non-standard approximation spaces and do not support arbitrarily high order approximations [15].",
"An alternative to triangular and tetrahedral SEM are high order DG methods.",
"High order DG methods can accommodate unstructured triangular and tetrahedral meshes, and naturally result in a block-diagonal global mass matrix, making them amenable to explicit time-stepping schemes and complex geometries.",
"However, in most DG implementations for heterogeneous media, the discretization is based on the assumption that wavespeed is piecewise constant over each element [16].",
"Fewer DG methods address the case when wavespeed varies within an element.",
"Castro et al.",
"[17] addressed sub-element variations in wavespeed by recasting the wave equation as a new linear hyperbolic PDE with variable coefficients and source terms, which are non-zero in the presence of sub-element variations in wavespeed.",
"However, this method introduces additional source terms and stiffness matrices with variable coefficients, resulting in a more complex formulation.",
"Additionally, semi-discrete energy stability is not guaranteed.",
"Mercerat and Glinsky [18] proposed instead replacing the mass matrix by a weighted mass matrix, where the wavespeed acts as a weight function.",
"The weighted mass matrix is obtained by introducing a set of quadrature points for the material approximation and computing integrals for entries of the mass matrix through quadrature rules.",
"This modification does not require new stiffness matrices or source terms, and can be shown to be energy stable and high order accurate.",
"However, because the wavespeed varies from element to element, each local weighted mass matrix is different.",
"Thus, one needs to store inverses of weighted mass matrices over each element for time-explicit schemes, which significantly increases storage costs.",
"Because GPUs have limited memory, these high storage costs restrict the problem sizes that can be run on a single GPU.",
"Moreover, increased storage costs lead to more data movement, which is becoming increasingly expensive compared to the cost of floating point operations [19].",
"blackTo address these storage costs, we utilize a weight-adjusted approximation of the weighted mass matrix, whose inverse can be applied in a low-storage manner.",
"The idea of a weight-adjusted approximation to a weighted mass matrix was first introduced as “reverse numerical integration” in [20], though it was not analyzed in detail.",
"The idea was independently reintroduced and analyzed by Chan et al.",
"in [1].",
"The key idea is to approximate the weighted $L^2$ inner product using an equivalent weight-adjusted inner product, which produces provably high order accurate and energy stable DG methods with low storage requirements.",
"Since WADG only modifies the local mass matrix, it maintains much of the structure of DG methods and is able to reuse existing DG implementations.",
"The main blackcomputational step of WADG is the computation of a quadrature-based polynomial $L^2$ projection.",
"However, the implementation of the quadrature-based $L^2$ projection in WADG requires $O(N^{2d})$ operations, while complexity of BBDG is only $O(N^d)$ .",
"Hence, combining BBDG with WADG would result in the cost of the quadrature-based $L^2$ projection dominating the implementation at high polynomial degrees.",
"The goal of this work is to reduce the computational complexity of WADG at high orders of approximation, which we do using Bernstein bases.",
"We develop an efficient algorithm to implement the polynomial $L^2$ projection in terms of Bernstein coefficients, which leads to a Bernstein-Bézier WADG (BBWADG) method.",
"The main idea is to decompose the projection operator into a combination of degree elevation operators.",
"Due to the sparsity of the one-degree elevation matrices, the $L^2$ projection can be applied in $O(N^{d+1})$ operations, reducing the complexity of right-hand evaluation from $O(N^6)$ to $O(N^4)$ in three dimensions.",
"The paper is organized as follows.",
"In Section , we review the weight-adjusted DG discretization of the first order acoustic and elastic wave equations in heterogeneous media.",
"Section introduces a Bernstein-Bézier DG method and its fast implementation.",
"In Section , we propose a Bernstein-Bézier weight-adjusted DG method, based on an algorithm to efficiently apply the polynomial $L^2$ projection under Bernstein bases.",
"Section presents numerical validation and verification."
],
[
"Mathematical notation",
"In this paper, we focus on wave problems in three dimensions since BBWADG can reduce the computational complexity by two orders.",
"In contrast, only one order of complexity can be reduced in two dimensions.",
"We assume the physical domain $\\Omega $ is well approximated by a triangulation $\\Omega _h$ consisting of $K$ non-overlapping elements $D^k$ .",
"The reference tetrahedron is defined as follows $\\widehat{D}=\\lbrace \\left( r,s,t\\right)\\ge -1; r+s+t\\le -1\\rbrace .$ We assume that each element $D^k$ is the image of the reference element $\\widehat{D}$ under an affine mapping $\\mathbf {\\Phi }^k$ $\\mathbf {x}=\\mathbf {\\Phi }^k\\widehat{\\mathbf {x}},\\qquad \\mathbf {x}\\in D^k,\\ \\ \\widehat{\\mathbf {x}}\\in \\widehat{D},$ where $\\mathbf {x}=\\left(x,y,z\\right)$ are physical coordinates on the $k$ th element and $\\widehat{\\mathbf {x}}=\\left(r,s,t\\right)$ are coordinates on the reference element.",
"Over each element $D^k$ , we define the approximation space $V_h\\left(D^k\\right)$ as $V_h\\left(D^k\\right)=\\mathbf {\\Phi }^k\\circ V_h\\left(\\widehat{D}\\right),$ where $V_h\\left(\\widehat{D}\\right)$ is a polynomial approximation space of degree $N$ on the reference element.",
"For the reference tetrahedron, $V_h\\left(\\widehat{D}\\right)$ is defined as follows $V_h\\left(\\widehat{D}\\right)=P^N\\left(\\widehat{D}\\right)=\\big \\lbrace r^i s^j t^k,\\ \\ 0\\le i+j+k\\le N\\big \\rbrace .$ In three dimensions, Bernstein polynomials on a tetrahedron are expressed using barycentric coordinates.",
"The barycentric coordinates for the reference tetrahedron are given as $\\lambda _0=-\\frac{\\left(1+r+s+t\\right)}{2},\\ \\ \\ \\lambda _1=\\frac{\\left(1+r\\right)}{2},\\ \\ \\ \\lambda _2=\\frac{\\left(1+s\\right)}{2},\\ \\ \\ \\lambda _3=\\frac{\\left(1+t\\right)}{2}.$ The $N$ th degree Bernstein basis is simply as a scaling of the barycentric monomials $B^N_{ijkl}=C^N_{ijkl}\\lambda _0^i\\lambda _1^j\\lambda _2^k\\lambda _3^l,\\qquad C^N_{ijkl}=\\frac{N!}{i!j!k!l!",
"},\\qquad i+j+k+l=N,$ which forms a nonnegative partition of unity.",
"For simplicity, we introduce the multi-index $\\mathbf {\\alpha }=\\left(\\alpha _0,\\dots ,\\alpha _d\\right)$ to denote the tuple of barycentric indices $\\left(i,j,k,l\\right)$ .",
"We define the order of a multi-index as $|\\mathbf {\\alpha }|:=\\sum _{i=0}^{d}\\alpha _i.$ We take $\\mathbf {\\alpha }\\le \\mathbf {\\beta }$ to mean that $\\alpha _j\\le \\beta _j,\\ \\forall j=0,\\dots ,d$ ."
],
[
"Weight-adjusted Discontinuous Galerkin methods",
"blackThe following sections introduce weight-adjusted DG discretizations of acoustic and elastic wave equations."
],
[
"Acoustic wave equation",
"We consider a first order velocity-pressure formulation of the acoustic wave equation given as $\\begin{split}\\frac{1}{c^2}\\frac{\\partial p}{\\partial \\tau }+\\nabla \\cdot \\mathbf {u}=0,\\\\\\frac{\\partial \\mathbf {u}}{\\partial \\tau }+\\nabla p = 0,\\end{split}$ where $p$ is the acoustic pressure, $\\mathbf {u}\\in \\mathbb {R}^d$ is the vector of velocities in each coordinate direction and $c$ is the wavespeed.",
"We assume that (REF ) is posed over time $\\tau \\in [0,T)$ on the physical domain $\\Omega $ with boundary $\\partial \\Omega $ , and the wavespeed is bounded by $0<c_{\\textmd {min}}\\le c(\\mathbf {x})\\le c_{\\textmd {max}}<\\infty .$ We define the jump across element interfaces as $[\\![p]\\!",
"]=p^+-p, \\qquad [\\!",
"[\\mathbf {u}]\\!",
"]=\\mathbf {u}^+-\\mathbf {u},$ where $p^+, \\mathbf {u}^+$ and $p, \\mathbf {u}$ are the neighboring and local traces of the solution over each interface, respectively.",
"The average across an element interface is denoted by $\\lbrace \\!\\lbrace p\\rbrace \\!\\rbrace =\\frac{1}{2}\\left(p^++p\\right),\\qquad \\lbrace \\!\\lbrace \\mathbf {u}\\rbrace \\!\\rbrace =\\frac{1}{2}\\left(\\mathbf {u}^++\\mathbf {u}\\right).$ We discretize the acoustic wave equation (REF ) in space using a strong formulation and choose penalty fluxes as $\\mathbf {u}^*=\\lbrace \\!\\lbrace \\mathbf {u}\\rbrace \\!\\rbrace -\\frac{\\tau _p}{2}[\\![p]\\!",
"]\\mathbf {n},\\qquad p^*=\\lbrace \\!\\lbrace p\\rbrace \\!\\rbrace -\\frac{\\tau _u}{2}[\\!",
"[\\mathbf {u}]\\!",
"]\\cdot \\mathbf {n},$ where $\\mathbf {n}$ is the outward unit normal vector on $D^k$ .",
"The corresponding semi-discrete formulation is given as follows $\\begin{split}\\int _{D^k}\\frac{1}{c^2}\\frac{\\partial p^k_h}{\\partial \\tau }\\phi d\\mathbf {x}&=-\\int _{D^k}\\nabla \\cdot \\mathbf {u}^k_h\\phi d\\mathbf {x}+\\int _{\\partial D^k} \\frac{1}{2}\\left(\\tau _p[\\![p_h^k]\\!",
"]-\\mathbf {n}\\cdot [\\!",
"[\\mathbf {u}^k_h]\\!",
"]\\right)\\phi d\\mathbf {x},\\\\\\int _{D^k}\\frac{\\partial \\left(\\mathbf {u}^k_h\\right)_i}{\\partial \\tau }\\mathbf {\\psi }_i d\\mathbf {x}&=-\\int _{D^k} \\frac{\\partial p^k_h}{\\partial \\mathbf {x}_i}\\mathbf {\\psi }_i d\\mathbf {x}+\\int _{\\partial D^k} \\frac{1}{2}\\left(\\tau _u[\\!",
"[\\mathbf {u}^k_h]\\!",
"]\\cdot \\mathbf {n}-[\\![p^k_h]\\!",
"]\\right)\\mathbf {\\psi }_i\\mathbf {n}_id\\mathbf {x},\\end{split}$ where $\\phi ,\\mathbf {\\psi }$ are test functions and $\\tau _p,\\tau _u\\ge 0$ are penalty parameters.",
"We define the mass matrix $\\mathbf {M}$ and the face mass matrix $\\mathbf {M}_f$ on $\\widehat{D}$ as $\\left(\\mathbf {M}\\right)_{ij}=\\int _{\\widehat{D}}\\phi _i(\\widehat{\\mathbf {x}})\\phi _j(\\widehat{\\mathbf {x}}) d\\widehat{\\mathbf {x}},\\qquad \\left(\\mathbf {M}_f\\right)_{ij}=\\int _{f_{\\widehat{D}}}\\phi _i(\\widehat{\\mathbf {x}})\\phi _j(\\widehat{\\mathbf {x}})d\\widehat{\\mathbf {x}}.$ where $f_{\\widehat{D}}$ is a face of the reference element $\\widehat{D}$ and $\\lbrace \\phi _i\\rbrace _{i=1}^{N_p}$ is an $N$ th degree polynomial basis on $\\widehat{D}$ .",
"Through an affine mapping $\\mathbf {\\Phi }^k$ , we can map the local operators on $D^k$ to the reference operators $\\mathbf {M}^k=J^k\\mathbf {M},\\qquad \\mathbf {M}^{k}_f=J^{k}_f\\mathbf {M}_f,$ where $J^k$ is the determinant of the volume Jacobian and $J^{k}_f$ is the determinant of the face Jacobian for $f$ .",
"Similarly, the weighted mass matrix $\\mathbf {M}^k_{w}$ on $D^k$ are given by $\\left(\\mathbf {M}^k_{w}\\right)_{ij}=J^k\\int _{\\widehat{D}}w\\left(\\mathbf {\\Phi }^k\\widehat{\\mathbf {x}}\\right)\\phi _i\\left(\\widehat{\\mathbf {x}}\\right)\\phi _j\\left(\\widehat{\\mathbf {x}}\\right)d\\widehat{\\mathbf {x}}.$ The stiffness matrix on $D^k$ with respect to $x$ is defined as $\\left(\\mathbf {S}^k_x\\right)_{ij}=\\int _{D^k}\\varphi _i\\frac{\\partial \\varphi _j}{\\partial x}d\\mathbf {x},$ and $\\mathbf {S}^k_y, \\mathbf {S}^k_z$ are defined similarly with respect to $y$ and $z$ .",
"Through chain rule, we can express stiffness matrices on $D^k$ in terms of the reference stiffness matrices $\\mathbf {S}_1, \\mathbf {S}_2, \\mathbf {S}_3$ with respect to reference coordinates $r,s$ and $t$ , respectively.",
"Then, the semi-discrete formulation (REF ) can be written as $\\begin{split}\\mathbf {M}^k_{1/c^2}\\frac{d\\mathbf {p}}{d\\tau }&=-J^k\\sum _{i=1}^{d}\\left(\\mathbf {G}^k_{i1}\\mathbf {S}_1+\\mathbf {G}^k_{i2}\\mathbf {S}_2+\\mathbf {G}^k_{i3}\\mathbf {S}_3\\right)\\mathbf {U}_i+\\sum _{f=1}^{N_{\\textmd {faces}}}J^{k}_f\\mathbf {M}_fF_p,\\\\J^k\\mathbf {M}\\frac{d\\mathbf {U}_i}{d\\tau }&=-J^k\\left(\\mathbf {G}^k_{i1}\\mathbf {S}_1+\\mathbf {G}^k_{i2}\\mathbf {S}_2+\\mathbf {G}^k_{i3}\\mathbf {S}_3\\right)\\mathbf {p}+\\sum _{f=1}^{N_{\\textmd {faces}}}J^{k}_f\\mathbf {n}_i\\mathbf {M}_fF_u,\\end{split}$ where $\\mathbf {p}$ and $\\mathbf {U}_i$ are degrees of freedom for $p$ and $\\mathbf {u}_i$ , and $\\mathbf {G}^k$ is the matrix of geometric factors $r_x,s_x,t_x$ , etc.",
"The flux terms $F_p,F_u$ are defined such that $\\begin{split}\\left(\\mathbf {M}_fF_p(\\mathbf {p},\\mathbf {p}^+,\\mathbf {U},\\mathbf {U}^+)\\right)_j &= \\int _{f_{\\widehat{D}}}\\frac{1}{2}\\left(\\tau _p [\\![p]\\!",
"]-\\mathbf {n}\\cdot [\\!",
"[\\mathbf {u}]\\!",
"]\\right)\\phi _jd\\widehat{\\mathbf {x}},\\\\\\left(\\mathbf {n}_i\\mathbf {M}_fF_u(\\mathbf {p},\\mathbf {p}^+,\\mathbf {U},\\mathbf {U}^+)\\right)_j &= \\int _{f_{\\widehat{D}}}\\frac{1}{2}\\left(\\tau _u [\\!",
"[\\mathbf {u}]\\!]-[\\![p]\\!",
"]\\right)\\mathbf {\\psi }_j\\mathbf {n}_id\\widehat{\\mathbf {x}}.\\end{split}$ Inverting $\\mathbf {M}^k_{1/c^2}$ and $\\mathbf {M}$ in (REF ) produces a system of ODEs that can be solved by time-explicit methods.",
"blackWhen the wavespeed $c^2$ is approximated by a constant over each element, $\\mathbf {M}^k_{1/c^2} = \\frac{J^k}{c^2}\\mathbf {M}$ , and $\\left(\\mathbf {M}^k_{1/c^2} \\right)^{-1} = \\frac{c^2}{J^k}\\mathbf {M}^{-1}$ .",
"Thus, to apply $\\left(\\mathbf {M}^k_{1/c^2} \\right)^{-1}$ , we need only store values of $J^k, c^2$ over each element and a single reference mass matrix inverse $\\mathbf {M}^{-1}$ over the entire mesh.blackIn practice, the reference inverse mass matrix is incorporated into the definition of differentiation and lifting matrices on the reference element..",
"However, inverses of weighted mass matrices are blackdistinct from element to element if the wavespeed possesses sub-element variations.",
"Typical implementations precompute and store these weighted mass matrix inverese, which significantly increases the storage cost of blackhigh order DG schemes.",
"To address this issue, a weight-adjusted discontinuous Galerkin (WADG) is proposed in [21], [1], which is energy stable and high order accurate for sufficiently regular weight functions.",
"WADG approximates the weighted mass matrix by a weight-adjusted approximation $\\widetilde{\\mathbf {M}}^k_w$ given as $\\mathbf {M}^k_w\\approx \\widetilde{\\mathbf {M}}_w^k=\\mathbf {M}^k\\left(\\mathbf {M}^k_{1/w}\\right)^{-1}\\mathbf {M}^k.$ Plugging above expression into (REF ), we obtain the semi-discrete WADG discretization of (REF ) as follows $\\begin{split}&\\frac{d\\mathbf {p}}{d\\tau }=-\\left(\\mathbf {M}^k\\right)^{-1}\\mathbf {M}^k_{c^2}\\left(\\sum _{i=1}^{d}\\sum _{j=1}^d\\mathbf {G}_{ij}^k\\mathbf {D}_j\\mathbf {U}_i+\\sum _{f=1}^{N_{\\textmd {faces}}}\\frac{J^k_f}{J^k}\\mathbf {L}^fF_p\\right),\\\\&\\frac{d\\mathbf {U}_i}{d\\tau }=-\\left(\\mathbf {G}_{i1}^k\\mathbf {D}_1+\\mathbf {G}^k_{i2}\\mathbf {D}_2+\\mathbf {G}^k_{i3}\\mathbf {D}_3\\right)\\mathbf {p}+\\sum _{f=1}^{N_{\\textmd {faces}}}\\frac{J^k_f}{J^k}\\mathbf {n}_i\\mathbf {L}^fF_u,\\end{split}$ where $\\mathbf {D}_i=\\mathbf {M}^{-1}\\mathbf {S}_i$ are derivative operators with respect to reference coordinates $r,s,t$ , $\\mathbf {L}^f=\\mathbf {M}^{-1}\\mathbf {M}_f$ are lift operators over faces."
],
[
"Elastic wave equation",
"blackThe weight-adjusted approach can be extended to matrix-valued weights, which appear in symmetrized first order velocity-stress formulations of the elastic wave equation [22].",
"Let $\\rho $ be the density and $\\mathbf {C}$ be the symmetric matrix form of constitutive tensor relating stress and strain.",
"The first-order elastic wave equations are given by $\\begin{split}\\rho \\frac{\\partial \\mathbf {v}}{\\partial \\tau }&=\\sum _{i=1}^{d}\\mathbf {A}_i^T\\frac{\\partial \\mathbf {\\sigma }}{\\partial \\mathbf {x}_i},\\\\\\mathbf {C}^{-1}\\frac{\\partial \\mathbf {\\sigma }}{\\partial \\tau }&=\\sum _{i=1}^{d} \\mathbf {A}_i\\frac{\\partial \\mathbf {v}}{\\partial \\mathbf {x}_i},\\end{split}$ where $\\mathbf {v}$ is the velocity and $\\mathbf {\\sigma }$ is a vector consisting of unique entries of the symmetric stress tensor.",
"The matrices $\\mathbf {A}_i$ are given as $\\mathbf {A}_1=\\begin{pmatrix}1&0&0\\\\0&0&0\\\\0&0&0\\\\0&0&0\\\\0&0&1\\\\0&1&0\\end{pmatrix},\\qquad \\mathbf {A}_2=\\begin{pmatrix}0&0&0\\\\0&1&0\\\\0&0&0\\\\0&0&1\\\\0&0&0\\\\1&0&0\\end{pmatrix},\\qquad \\mathbf {A}_3=\\begin{pmatrix}0&0&0\\\\0&0&0\\\\0&0&1\\\\0&1&0\\\\1&0&0\\\\0&0&0\\end{pmatrix}.$ For isotropic media, $\\mathbf {C}$ is given by $\\mathbf {C}=\\begin{pmatrix}2\\mu +\\lambda &\\lambda &\\lambda &\\\\\\lambda &2\\mu +\\lambda &\\lambda &\\\\\\lambda &\\lambda &2\\mu +\\lambda &\\\\&&&&\\mu \\mathbf {I}^{3\\times 3}\\end{pmatrix},$ where $\\mu ,\\lambda $ are Lamé parameters.",
"We note that $\\mathbf {A}_i$ are blackspatially constant independently of media heterogeneities.",
"blackWe can construct a semi-discrete DG scheme for elasticity analogous to the formulation (REF ) for the acoustic wave equation $\\begin{split}&\\left(\\rho \\frac{\\partial \\mathbf {v}}{\\partial \\tau },\\mathbf {w}\\right)_{L^2(D^k)}=\\left(\\sum _{i=1}^{d}\\mathbf {A}_i^T\\frac{\\partial \\mathbf {\\sigma }}{\\partial \\mathbf {x}_i},\\mathbf {w}\\right)_{L^2(D^k)}+\\Bigg \\langle \\frac{1}{2}\\mathbf {A}_n^T[\\!",
"[\\mathbf {\\sigma }]\\!",
"]+\\frac{\\tau _v}{2}\\mathbf {A}_n^T\\mathbf {A}_n[\\!",
"[\\mathbf {v}]\\!",
"],\\mathbf {w}\\Bigg \\rangle _{L^2(\\partial D^k)},\\\\&\\left(\\mathbf {C}^{-1}\\frac{\\partial \\mathbf {\\sigma }}{\\partial \\tau },\\mathbf {q}\\right)_{L^2(D^k)}=\\left(\\sum _{i=1}^{d}\\mathbf {A}_i\\frac{\\partial \\mathbf {v}}{\\partial \\mathbf {x}_i},\\mathbf {q}\\right)_{L^2(D^k)}+\\Bigg \\langle \\frac{1}{2}\\mathbf {A}_n[\\!",
"[\\mathbf {v}]\\!",
"]+\\frac{\\tau _{\\sigma }}{2}\\mathbf {A}_n\\mathbf {A}_n^T[\\!",
"[\\mathbf {\\sigma }]\\!",
"],\\mathbf {q}\\Bigg \\rangle _{L^2(\\partial D^k)},\\end{split}$ where $(\\cdot ,\\cdot )_{L(D^k)}$ and $\\langle \\cdot ,\\cdot \\rangle _{L(D^k)}$ denote the $L^2$ inner product on $D^k$ and $\\partial D^k$ , respectively.",
"blackThe presence of $\\mathbf {C}^{-1}$ on the left-hand side produces a matrix-valued mass matrix $\\mathbf {M}_{\\mathbf {C}^{-1}}$ involving the constitutive stress tensor $\\mathbf {C}$ $\\mathbf {M}_{\\mathbf {C}^{-1}} = \\begin{bmatrix}\\mathbf {M}_{\\mathbf {C}^{-1}_{11}} & \\ldots &\\mathbf {M}_{\\mathbf {C}^{-1}_{1d}}\\\\\\vdots & \\ddots & \\vdots \\\\\\mathbf {M}_{\\mathbf {C}^{-1}_{d1}} & \\ldots &\\mathbf {M}_{\\mathbf {C}^{-1}_{dd}}\\\\\\end{bmatrix},$ where $\\mathbf {C}^{-1}_{ij}$ denotes the $ij$ th entry of $\\mathbf {C}^{-1}$ and $\\mathbf {M}_{\\mathbf {C}^{-1}_{ij}}$ denotes the scalar weighted mass matrix with weight $\\mathbf {C}^{-1}_{ij}$ .",
"The matrix $\\mathbf {M}_{\\mathbf {C}^{-1}}$ can be understood as the matrix-weighted analogue of the scalar wavespeed-weighted mass matrix $\\mathbf {M}_{1/c^2}$ which appeared for the acoustic wave equation.",
"black The inverse of $\\mathbf {M}_{\\mathbf {C}^{-1}}$ can be approximated by the inverse of a matrix-weighted weight-adjusted mass matrix $\\mathbf {M}^{-1}_{\\mathbf {C}^{-1}}\\approx \\left(\\mathbf {I}\\otimes \\mathbf {M}^{-1}\\right) \\mathbf {M}_{\\mathbf {C}} \\left(\\mathbf {I}\\otimes \\mathbf {M}^{-1}\\right),$ where $\\otimes $ denotes the Kronecker product.",
"We note that this approximation can be applied in terms of scalar weight-adjusted mass matrix inverses.",
"Incorporating this approximation yields the following WADG scheme for the elastic wave equations (REF ) $\\begin{split}\\frac{\\partial \\mathbf {V}}{\\partial \\tau }& \\!= \\!\\left( \\!\\mathbf {I} \\!\\otimes \\!\\left(\\mathbf {M}^k\\right)^{-1} \\!\\right) \\!\\mathbf {M}^k_{\\rho ^{-1} \\!\\mathbf {I}} \\!\\left(\\sum _{i=1}^{d}\\sum _{j=1}^{d}\\mathbf {G}^k_{ij}\\left(\\mathbf {A}_i^T \\!\\otimes \\!",
"\\mathbf {D}_j\\right) \\!\\mathbf {\\Sigma } \\!+ \\!\\frac{J^k_f}{J^k}\\sum _{f=1}^{N_{\\textmd {faces}}} \\!\\left(\\mathbf {I} \\!\\otimes \\!\\mathbf {L}^f\\right) \\!\\mathbf {F}_v \\!\\right),\\\\\\frac{\\partial \\mathbf {\\Sigma }}{\\partial \\tau }& \\!= \\!\\left( \\!\\mathbf {I} \\!\\otimes \\!\\left(\\mathbf {M}^k\\right)^{-1} \\!\\right) \\!\\mathbf {M}^k_{\\mathbf {C}} \\!\\left(\\sum _{i=1}^{d}\\sum _{j=1}^{d}\\mathbf {G}^k_{ij}\\left(\\mathbf {A}_i \\!\\otimes \\!",
"\\mathbf {D}_j\\right) \\!\\mathbf {V} \\!+ \\!\\frac{J^k_f}{J^k}\\sum _{f=1}^{N_{\\textmd {faces}}} \\!\\left(\\mathbf {I} \\!\\otimes \\!\\mathbf {L}^f\\right) \\!\\mathbf {F}_\\sigma \\!\\right),\\end{split}$ where $\\mathbf {V},\\mathbf {\\Sigma }$ are constructed by concatenating $\\mathbf {\\Sigma }_i,\\mathbf {V}_i$ into single vectors, respectively, and $\\mathbf {F}_v, \\mathbf {F}_{\\sigma }$ are vectors representing the velocity and stress numerical fluxes.",
"blackWe note that this formulation is energy stable and high order accurate for elastic wave propagation in either isotropic or aniostropic heterogeneous media [21]."
],
[
"Quadrature-based implementation",
"In practice, weight-adjusted mass matrrix inverses are applied in a matrix-free fashion using sufficiently accurate quadrature rules.",
"blackWe follow [1] and use simplicial quadratures which are exact for polynomials of degree $2N+1$ [23].",
"Let $\\widehat{\\mathbf {x}}_i,\\widehat{\\mathbf {w}}_i$ denote the quadrature points and weights on the reference element.",
"We define the interpolation matrix $\\mathbf {V}_q$ as $\\left(\\mathbf {V}_q\\right)_{ij}=\\phi _j\\left(\\widehat{\\mathbf {x}}_i\\right),$ whose columns consist of values of basis functions at quadrature points.",
"On each element $D^k$ , we have $\\mathbf {M}^k=J^k\\mathbf {M}=J^k\\mathbf {V}_q^T\\textmd {diag}\\left(\\widehat{\\mathbf {w}}\\right)\\mathbf {V}_q,\\ \\ \\ \\mathbf {M}^k_{c^2}=J^k\\mathbf {V}_q^T\\textmd {diag}\\left(\\mathbf {d}\\right)\\mathbf {V}_q,\\ \\ \\ \\mathbf {d}_i=\\frac{\\widehat{\\mathbf {w}}_i}{c^2\\left(\\mathbf {\\Phi }^k\\widehat{\\mathbf {x}}_i\\right)}$ where $\\mathbf {\\Phi }^k\\widehat{\\mathbf {x}}_i$ are quadrature points on $D^k$ and black $c^2\\left(\\mathbf {\\Phi }^k\\widehat{\\mathbf {x}}\\right)$ denote the values of the wavespeed at quadrature points.",
"blackEvaluating the right hand side of (REF ) and (REF ) requires applying the product of an unweighted mass matrix and weighted mass matrix, such as $\\left(\\mathbf {M}^k\\right)^{-1}\\mathbf {M}^k_{c^2}$ .",
"This can be done using quadrature-based matrices as follows $\\left(\\mathbf {M}^k\\right)^{-1}\\mathbf {M}^k_{c^2} = \\mathbf {P}_q\\textmd {diag}\\left(\\frac{1}{c^2\\left(\\mathbf {\\Phi }^k\\widehat{\\mathbf {x}}\\right)}\\right)\\mathbf {V}_q,$ where $\\mathbf {P}_q=\\mathbf {M}^{-1}\\mathbf {V}_q^T\\textmd {diag}\\left(\\widehat{\\mathbf {w}}\\right)$ is a quadrature-based polynomial $L^2$ projection operator on the reference element.",
"Moreover, since $\\mathbf {P}_q, \\mathbf {V}_q$ are reference operators, the implementation of (REF ) requires only $O\\left(N^d\\right)$ storage for values of the wavespeed black$c^2\\left(\\mathbf {\\Phi }^k\\widehat{\\mathbf {x}}\\right)$ at quadrature points for each element.",
"In contrast, blackstoring full weighted mass matrix inverses or factorizations requires $O\\left(N^{2d}\\right)$ storage on each element.",
"For example, in three dimensions, the number of quadrature points on one element, scales with $O(N_p)=O(N^3)$ , while size of the weighted mass matrix inverse is $O(N_p)\\times O(N_p)$ , implying an $O(N^6)$ storage requirement."
],
[
"Bernstein-Bézier DG methods",
"In this section, we review how to use Bernstein-Bézier polynomial bases to construct efficient high order DG methods.",
"blackFor nodal DG methods, the numerical fluxes can be computed in terms of the difference between blackdegrees of freedom at face nodes on two neighboring elements.",
"blackThis is also true of the Bernstein basis, since Bernstein polynomials share a geometrical decomposition with vertex, edge, face and interior nodes in the sense that edge basis functions vanish at vertices, face basis functions vanish at vertices and edges, and interior basis functions vanish at vertices, edges, and faces [7].",
"Hence, the value of a Bernstein polynomial on one face is determined by basis functions associated with that face only, blackand the jumps of polynomial solutions under Bernstein bases across element interfaces can be computed similarly using node-to-node connectivity maps and degrees of freedom corresponding to face points on two neighboring elements.",
"Evaluating the DG formulation (REF ) requires applying derivative and lift operators.",
"These steps can be accelerated using properties of Bernstein polynomials.",
"Let $\\mathbf {D}^i$ be the Bernstein derivative operator with respect to $i$ th barycentric coordinate.",
"blackDifferentiation matrices with respect to reference coordinates can be expressed as a linear combination of barycentric differentiation matrices $\\mathbf {D}^i$ .",
"It can be shown that each row of black$\\mathbf {D}^i$ has at most $d+1$ non-zeros in $d$ dimensions [8], [9], such that blackthe sparse application of barycentric Bernstein differentiation matrices requires only $O(N^d)$ operations.",
"In contrast, nodal derivative operators are blackgenerally dense matrices blackof size $N_p\\times N_p$ , which require $O(N^{2d})$ operations to apply.",
"For a Bernstein lift operator $\\mathbf {L}^f$ , it was observed in [11] that black$\\mathbf {L}^f$ can be factorized as $\\mathbf {L}^f=\\mathbf {E}^f_L\\mathbf {L}_0,$ where $\\mathbf {E}^f_L$ is the face reduction matrix and $\\mathbf {L}_0$ is a sparse black$N_p^f\\times N_p^f$ matrix, where $N_p^f$ is the number of degrees of freedom in the $N$ th degree polynomial space on a single face.",
"Moreover, $\\mathbf {L}_0$ has no more than seven nonzeros per row (independent of $N$ ).blackExplicit expressions for $\\mathbf {E}^f_L$ and $\\mathbf {L}_0$ can be found in [11].",
"blackThe fixed bandwidth of the matrix $\\mathbf {L}_0$ blackimplies that it can be applied blackin $O(N^{d-1})$ operations.",
"The face reduction operator $\\mathbf {E}^f_L$ can be further expanded as product of one-degree reduction operators.",
"Application of $\\mathbf {E}^f_L$ requires applying $N$ triangular one-degree reduction operators, each of which costs $O(N^{d-1})$ to apply.",
"Hence, the total cost of the implementation of the lift matrix $\\mathbf {L}^f$ is $O(N^d)$ in $d$ dimensions.",
"In contrast, the lift matrices under a nodal basis have size $N_p\\times N_p^f$ and cost $O(N^{2d-1})$ to apply.",
"blackTo summarize, the overall cost of evaluating the DG right-hand side is $O(N^d)$ per element in $d$ dimensions under a Bernstein basis.",
"Since the number of degreees of freedom grows as $O(N^d)$ , this complexity is optimal."
],
[
"A fast implementation of weight-adjusted DG methods",
"blackWhile the evaluation of the BBDG right-hand side requires only $O(N^d)$ operations per element, this is true only if media is homogeneous (piecewise constant) over each element.",
"Sub-element heterogeneities can be incorporated using WADG and numerical quadrature as discusssed in Section REF .",
"However, blackbecause quadrature-based WADG involves dense matrix-vector products, the cost generally scales as $O(N^{2d})$ in $d$ dimensions.",
"Thus, naively utilizing WADG to address sub-cell heterogeneities results in a computational complexity of $O(N^{2d})$ per element, which will dominate the $O(N^d)$ complexity of BBDG and negate any gains in computational efficiency.",
"To address this, we propose a Bernstein-Bézier weight-adjusted discontinuous Galerkin (BBWADG) method blackbased on a polynomial approximation of media heterogeneities.",
"We first note that the evaluation of the DG right-hand side yields a polynomial of degree $N$ .",
"Let $u(x)$ denote this $N$ polynomial, and let $\\mathbf {u}$ denote its coefficients in some basis.",
"WADG involves applying (REF ) to $\\mathbf {u}$ to compute $\\mathbf {P}_q\\textmd {diag}\\left(\\frac{1}{c^2\\left(\\mathbf {\\Phi }^k\\widehat{\\mathbf {x}}\\right)}\\right)\\mathbf {V}_q\\mathbf {u}.$ Since $\\mathbf {P}_q$ is a quadrature-based discretization of the $L^2$ projection operator, this is simply a quadrature-based $L^2$ projection of $u/c^2$ onto polynomials of degree $N$ .",
"blackSuppose now that $1/c^2$ is a polynomial of degree $M$ .",
"Then, the main steps of WADG are equivalent to computing $u/c^2$ , which is a polynomial of degree $M+N$ , and projecting this polynomial onto degree $N$ polynomials.",
"These two steps correspond to polynomial multiplication and polynomial $L^2$ projection, both of which can be performed efficiently under Bernstein bases.",
"The resulting algorithms require $O(N^{d+1})$ operations per element in $d$ dimensions.",
"blackIn practice, we construct a polynomial approximation of $1/c^2$ using a quadrature-based $L^2$ projection of the true wavespeed.",
"Since the wavespeed does not generally change during a simulation, this approximation can be computed and stored once in a pre-processing step so that it does not affect the computational cost of the solver.",
"blackThe remainder of this section describes efficient algorithms for computing the polynomial multiplication and polynomial $L^2$ projection of two Bernstein polynomials.",
"This section is separated into four parts: in Section REF , we explain how to compute the product of two Bernstein polynomials as a higher degree Bernstein polynomial.",
"blackWe introduce Bernstein degree elevation matrices in Section REF , which are then used in Section REF blackto construct a representation of the polynomial $L^2$ projection matrix which can be evaluated in $O(N^{d+1})$ operations.",
"Finally, we present a GPU-accelerated algorithm of the Bernstein polynomial $L^2$ projection in Section REF ."
],
[
"Bernstein polynomial multiplication",
"blackEfficient algorithms exist for the multiplication of two Bernstein polynomials based on discrete convolutions [24].",
"We describe a sparse matrix-based implementation here, which is simpler to implement on GPUs.",
"Let $B^{N}_{\\mathbf {\\alpha }}$ and $B^{M}_{\\mathbf {\\beta }}$ be any two Bernstein basis functions of degree $N$ and $M$ respectively.",
"Their product is $\\begin{split}B^{N}_{\\mathbf {\\alpha }}B^{M}_{\\mathbf {\\beta }}=\\frac{\\binom{\\mathbf {\\alpha }+\\mathbf {\\beta }}{\\mathbf {\\alpha }}}{\\binom{N+M}{N}}B^{N+M}_{\\mathbf {\\alpha }+\\mathbf {\\beta }},\\end{split}$ which is a Bernstein basis function of degree $N+M$ up to a scaling.",
"This observation can be used to efficiently compute the product of two Bernstein polynomials.",
"Let $f({\\mathbf {x}})$ and $g({\\mathbf {x}})$ be two Bernstein polynomials of degree $N$ and $M$ respectively with representations $f({\\mathbf {x}})=\\sum _{|\\mathbf {\\alpha }|=N}a_{\\mathbf {\\alpha }}B^N_{\\mathbf {\\alpha }}(\\mathbf {x}),\\qquad g({\\mathbf {x}})=\\sum _{|\\mathbf {\\beta }|=M}b_{\\mathbf {\\beta }} B^M_{\\mathbf {\\beta }}(\\mathbf {x}).$ Then, $h(\\mathbf {x})=f(\\mathbf {x})g(\\mathbf {x})$ is a Bernstein polynomial of degree $N+M$ .",
"We first blackillustrate polynomial multiplication for the $M=1$ blackcase, such that $g(\\mathbf {x})$ is a linear polynomial.",
"Let $\\mathbf {e}_j$ denote the canonical vector such that $g(\\mathbf {x})=\\sum _{j=0}^{d}b_jB^1_{\\mathbf {e}_j}(\\mathbf {x})$ .",
"Then, the product of $f,g$ is $\\begin{split}h(\\mathbf {x})&=\\sum _{j=0}^{d}\\sum _{|\\mathbf {\\alpha }|=N}a_{\\mathbf {\\alpha }}b_jB^N_{\\mathbf {\\alpha }}(\\mathbf {x})B^1_{\\mathbf {e}_j}(\\mathbf {x})\\\\&=\\sum _{j=0}^{d}\\sum _{|\\mathbf {\\alpha }|=N}a_{\\mathbf {\\alpha }}b_j\\frac{\\alpha _{j}+1}{N+1}B^{N+1}_{\\mathbf {\\alpha }+\\mathbf {e}_j}(\\mathbf {x}).\\end{split}$ Let $\\mathbf {\\gamma }$ be a multi-index and $c_{\\mathbf {\\gamma }}$ denote the coefficient of $B^{N+1}_{\\mathbf {\\gamma }}$ in the expression for $h$ in (REF ).",
"Then $c_{\\mathbf {\\gamma }}$ can be computed as $c_{\\mathbf {\\gamma }}=\\sum _{j=0}^{d}a_{\\mathbf {\\gamma }-\\mathbf {e}_j}b_j\\frac{\\gamma _{j}}{N+1},$ where we set the coefficient to be zero if the corresponding multi-index $\\mathbf {\\gamma }-\\mathbf {e}_j$ has negative components.",
"Hence, for the case $M=1$ , the Bernstein coefficients of $h(\\mathbf {x})$ can be expressed as a linear combination of at most $d+1$ products of coefficients for $f(\\mathbf {x})$ and coefficients for $g(\\mathbf {x})$ .",
"This, in turn, can be efficiently computed using sparse matrix operations, as illustrated in Fig.",
"REF .",
"Figure: Visualization of Bernstein polynomial multiplication for M=1M=1We now consider the blackmore general case of arbitrrary $M$ .",
"We are interested in computing the product $h(\\mathbf {x})=f(\\mathbf {x})g(\\mathbf {x})$ , where $f(\\mathbf {x})\\in P^N$ and $g(\\mathbf {x})\\in P^M$ .",
"We have the following $\\begin{split}h(\\mathbf {x}) &=\\sum _{|\\mathbf {\\beta }|=M}\\sum _{|\\mathbf {\\alpha }|=N}a_{\\mathbf {\\alpha }}b_{\\mathbf {\\beta }}B^N_{\\mathbf {\\alpha }}(\\mathbf {x})B^M_{\\mathbf {\\beta }}(\\mathbf {x})\\\\&=\\sum _{|\\mathbf {\\beta }|=M}\\sum _{|\\mathbf {\\alpha }|=N}a_{\\mathbf {\\alpha }}b_{\\mathbf {\\beta }}\\frac{\\binom{\\mathbf {\\alpha }+\\mathbf {\\beta }}{\\mathbf {\\alpha }}}{\\binom{N+M}{N}}B^{N+M}_{\\mathbf {\\alpha }+\\mathbf {\\beta }}(\\mathbf {x}).\\end{split}$ Hence, the coefficient $c_{\\mathbf {\\gamma }}$ of $B^{N+M}_{\\mathbf {\\gamma }}$ in $h(\\mathbf {x})$ can be computed as $c_{\\mathbf {\\gamma }}=\\sum _{|\\mathbf {\\beta }|=M}a_{\\mathbf {\\gamma }-\\mathbf {\\beta }}b_{\\mathbf {\\beta }}\\frac{\\binom{\\mathbf {\\gamma }}{\\mathbf {\\beta }}}{\\binom{N+M}{N}}.$ As in (REF ), the coefficient $c_{\\mathbf {\\gamma }}$ is set to be zero if the corresponding multi-index $\\mathbf {\\gamma }-\\mathbf {\\beta }$ has negative components.",
"Hence, $c_{\\mathbf {\\gamma }}$ can be written as a combination of at most $M_p$ products of coefficients from $f$ and $h$ , where $M_p$ is the dimension of the $M$ th degree polynomial space.",
"blackAs in the $M=1$ case, the multiplication of two arbitrary Bernstein polynomials can be implemented efficiently using sparse matrix multiplications.",
"We can also determine the computational complexity of Bernstein polynomial multiplication from the expression (REF ) for the product of two Bernstein polynomials.",
"We summarize this in the following theorem: Theorem 5.1 The multiplication of two Bernstein polynomials of degree $N$ and $M$ can be performed in $O\\left( \\left( M N \\right)^d\\right)$ operations.",
"For fixed $M$ , polynomial multiplication requires $O(N^d)$ operations."
],
[
"Bernstein degree elevation operators",
"blackIn this section, we introduce degree elevation matrices, which are used within algorithms for polynomial $L^2$ projection in Section REF ..",
"Degree elevation refers to the representation of a lower degree polynomial in a high degree polynomial basis.",
"It can be shown that the $d$ -dimensional Bernstein polynomial of degree $N-1$ can be expressed as a linear combination of no more than $d+1$ Bernstein polynomials of degree $N$ [10].",
"For example, a basis function $B^{N-1}_{\\mathbf {\\alpha }}$ can be written as $\\begin{split}B^{N-1}_{\\mathbf {\\alpha }}=\\sum _{j=0}^{d}\\frac{\\alpha _j+1}{N}B^{N}_{\\mathbf {\\alpha }+\\mathbf {e}_j},\\end{split}$ where $\\mathbf {e}_j$ is the $j$ th canonical vector [9].",
"This property can be used to construct degree elevation matrices under the Bernstein basis.",
"Let $\\mathbf {E}^N_{N-i}$ denote the degree elevation operator, which evaluates a polynomial of degree $N-i$ as a degree $N$ polynomials on a triangle.",
"From (REF ), we know that the one-degree elevation matrix $\\mathbf {E}^N_{N-1}$ is sparse, and only contains at most $d+1$ non-zero entries per row independently of the degree $N$ .",
"Let $\\mathbf {\\alpha }$ denote the multi-index for the row corresponding to the basis function $B^{N-1}_{\\mathbf {\\alpha }}$ .",
"Then, the non-zero values and column indices $\\mathbf {\\beta }$ of $\\mathbf {E}^N_{N-1}$ are $\\left(\\mathbf {E}^N_{N-1}\\right)_{\\mathbf {\\alpha },\\mathbf {\\beta }}=\\frac{\\alpha _j+1}{N},\\ \\ \\ \\ \\mathbf {\\beta }=\\mathbf {\\alpha }+\\mathbf {e}_j,\\ \\ \\ j=1,\\dots ,d.$ The degree elevation matrix $\\mathbf {E}^N_{N-i}$ between arbitrary degrees can be expressed as the product of one-degree elevation matrices $\\mathbf {E}^N_{N-i}=\\mathbf {E}^{N}_{N-1}\\mathbf {E}^{N-1}_{N-2}\\cdots \\mathbf {E}^{N-i+1}_{N-i}.$ We also refer to the transpose of the degree elevation operator $\\left(\\mathbf {E}^N_{N-1}\\right)^T$ as the degree reduction operator."
],
[
"Bernstein polynomial $L^2$ projection",
"blackRecall that the two steps of BBWADG are polynomial multiplication and polynomial $L^2$ projection.",
"The first step was discussed in Section REF , and we discuss the second step in this section.",
"We introduce an efficient method of computing the $L^2$ projection of a polynomial to a lower degree polynomial under a Bernstein basis.",
"This approach is based on a representation of the polynomial projection matrix in terms of sparse one-degree elevation matrices.",
"The blackrepresentation of the polynomial $L^2$ projection matrix using degree elevation matrices is based on two observations.",
"The first observation is that the polynomial $L^2$ projection operator is rectangular diagonal under a modal (orthogonal) basis.",
"These modal basis functions [25], [26], [27], [28] are hierarchical and $L^2$ orthogonal, such that $\\Big (L_{\\mathbf {\\gamma }},L_{\\mathbf {\\sigma }}\\Big )={\\left\\lbrace \\begin{array}{ll}\\text{$1$,} &{\\mathbf {\\gamma }=\\mathbf {\\sigma }},\\\\[2ex]\\text{$0$,} &{\\textmd {otherwise,}}\\end{array}\\right.",
"},\\qquad L_{\\mathbf {\\gamma }}\\in P^{|\\mathbf {\\gamma }|},$ where $\\mathbf {\\gamma }$ and $\\mathbf {\\sigma }$ are $d$ -dimensional multi-indices.",
"For simplicity, we assume the hierarchical modal basis functions are arranged in ascending order blackwith respect to $|\\mathbf {\\gamma }|$ .",
"The second observation is that the outer product of the degree elevation matrix and its transpose is diagonal under a modal basis.",
"We wish to represent the polynomial $L^2$ projection matrix as a linear combination of these outer products.",
"We recall some results from [11], which will be used in this proof.",
"Lemma 5.2 (Lemma A.2 in [11]) Suppose $p\\in P^N(\\widehat{D})$ .",
"Let $\\mathbf {T}$ be the transformation matrix mapping model coefficients to Bernstein coefficients such that $p=\\sum _{|\\mathbf {\\gamma }|\\le N}c_{\\mathbf {\\gamma }}^LL_{\\mathbf {\\gamma }}=\\sum _{|\\mathbf {\\alpha }|=N}c_{\\mathbf {\\alpha }}^BB_{\\mathbf {\\alpha }}^N,\\ \\ \\ \\ \\mathbf {c}^B=\\mathbf {T}\\mathbf {c}^L,$ where $L_{\\mathbf {\\gamma }},B_{\\mathbf {\\alpha }}^N$ are modal and Bernstein polynomials, respectively.",
"Define $\\widetilde{\\mathbf {D}}$ as $\\widetilde{\\mathbf {D}}=\\mathbf {T}^{-1}_{N-i}\\left(\\mathbf {E}^N_{N-i}\\right)^T\\mathbf {T}_N$ Suppose $0\\le k\\le N$ , and let $\\lambda _k^N, \\lambda _k^{N-i}$ be the distinct eigenvalues of $\\mathbf {M}_N$ and $\\mathbf {M}_{N-i}$ , respectively.",
"The entries of $\\widetilde{\\mathbf {D}}$ are $\\widetilde{\\mathbf {D}}_{\\mathbf {\\nu ,\\gamma }}={\\left\\lbrace \\begin{array}{ll}\\text{$\\lambda _{|\\mathbf {\\gamma }|}^{N-i}/\\lambda _{|\\mathbf {\\gamma }|}^N$,}&{\\mathbf {\\nu }=\\mathbf {\\gamma }},\\\\[2ex]\\text{$0$,} &{\\textmd {otherwise,}}\\end{array}\\right.}",
"\\ \\ \\ \\ \\ \\ \\widetilde{\\mathbf {D}}\\in \\mathbb {R}^{(N-i)_p,N_p}$ where $N_p,(N-i)_p$ are the dimensions of the space of polynomials of total degree $N$ and $N-i$ , respectively.",
"Corollary 1 (Corollary A.3 in [11]) Under a transformation to a modal basis, $\\mathbf {E}^N_{N-i}(\\mathbf {E}_{N-i}^N)^T$ is diagonal, with entries $\\left(\\mathbf {T}_N^{-1}\\mathbf {E}^N_{N-i}\\left(\\mathbf {E}^N_{N-i}\\right)^T\\mathbf {T}_N\\right)_{\\mathbf {\\gamma },\\mathbf {\\gamma }}={\\left\\lbrace \\begin{array}{ll}\\text{$0$,}&{|\\mathbf {\\gamma }|>(N-i)},\\\\[2ex]\\text{$\\lambda _{|\\mathbf {\\gamma }|}^{N-i}/\\lambda _{|\\mathbf {\\gamma }|}^{N}$,}&{|\\mathbf {\\gamma }|\\le (N-i)}.\\end{array}\\right.",
"}$ A straightforward extension of Corollary REF gives the following corollary: Corollary 2 Under a transformation to a modal basis, $\\left\\lbrace \\mathbf {E}^N_{N-i}(\\mathbf {E}_{N-i}^N)^T\\right\\rbrace ^N_{i=0}$ is a basis for any $\\mathbf {D}$ such that $\\mathbf {D}=\\begin{pmatrix}d_0 & & &\\\\&d_1\\mathbf {I}_1& &\\\\& & \\ddots &\\\\& & & d_N\\mathbf {I}_N\\end{pmatrix},$ where $\\mathbf {I}_i$ is the identity matrix of dimension $(i_p-(i-1)_p)\\times (i_p-(i-1)_p)$ .",
"Let $\\mathbf {P}^{N+M}_{N}$ denote the Bernstein polynomial $L^2$ projection operator from the polynomial space of degree $N+M$ to the polynomial space of degree $N$ .",
"By transforming to a modal basis, we observe that the projection operator should be a diagonal rectangular matrix with diagonal entries equal to one, i.e., $\\mathbf {T}_N^{-1}\\left(\\mathbf {P}^{N+M}_N\\right)\\mathbf {T}_{N+M}=\\left(\\begin{array}{c|c}\\mathbf {I}& \\mathbf {0}\\end{array}\\right)$ where $\\mathbf {T}_N,\\mathbf {T}_{N+M}$ are basis transformation matrices between Bernstein and modal bases of degree $N$ and $N+M$ respectively.",
"Based on this observation, we have the following theorem: Theorem 5.3 There exist $c_j$ , $0\\le j\\le N$ , such that $\\mathbf {P}^{N+M}_N=\\sum _{j=0}^{N} c_j\\mathbf {E}^N_{N-j}\\left(\\mathbf {E}^N_{N-j}\\right)^T\\left(\\mathbf {E}^{N+M}_N\\right)^T.$ From Lemma REF , we know that $\\mathbf {T}^{-1}_{N}\\left(\\mathbf {E}^{N+M}_N\\right)^T\\mathbf {T}_{N+M}=\\left(\\begin{array}{cccc|ccc}\\frac{\\lambda _0^{N}}{\\lambda _0^{N\\!+\\!M}}&& &&0&\\cdots &0\\\\&\\frac{\\lambda _1^{N}}{\\lambda _1^{N\\!+\\!M}}\\mathbf {I}_1& &&\\vdots &&\\vdots \\\\& &\\ddots &&\\vdots &&\\vdots \\\\&& &\\frac{\\lambda _N^{N}}{\\lambda _N^{N\\!+\\!M}}\\mathbf {I}_N&0&\\cdots &0\\\\\\end{array}\\right),$ which is a rectangular diagonal diagonal matrix.",
"By Corollary REF , there exist $c_j,\\ 0\\le j\\le N$ , such that $\\sum _{j=0}^{N} c_j\\mathbf {T}^{-1}_N\\mathbf {E}^N_{N-j}\\left(\\mathbf {E}^N_{N-j}\\right)^T\\mathbf {T}_N=\\begin{pmatrix}\\frac{\\lambda _0^{N\\!+\\!M}}{\\lambda _0^{N}} & & &\\\\&\\frac{\\lambda _1^{N\\!+\\!M}}{\\lambda _1^{N}}\\mathbf {I}_1& &\\\\& & \\ddots &\\\\& & & \\frac{\\lambda _N^{N\\!+\\!M}}{\\lambda _N^{N}}\\mathbf {I}_N\\end{pmatrix}.$ Then, we obtain $\\sum _{j=0}^{N} c_j\\mathbf {T}_N^{-1}\\mathbf {E}^N_{N-j}\\left(\\mathbf {E}^N_{N-j}\\right)^T\\left(\\mathbf {E}^{N+M}_N\\right)^T\\mathbf {T}_{N+M}=\\mathbf {T}_N^{-1}\\left(\\mathbf {P}^{N+M}_N\\right)\\mathbf {T}_{N+M}.$ Multiplying $\\mathbf {T}_N$ and $\\mathbf {T}_{N+M}^{-1}$ from left and right hand side, respectively, gives $\\mathbf {P}^{N+M}_N=\\sum _{j=0}^{N} c_j\\mathbf {E}^N_{N-j}\\left(\\mathbf {E}^N_{N-j}\\right)^T\\left(\\mathbf {E}^{N+M}_N\\right)^T.$ In practice, these coefficient $c_j$ can be computed by solving a linear system.",
"Table REF shows values of $c_j$ for several combinations of degree $N$ and $M$ in three dimensions.",
"Table: Coefficients c j c_j for the Bernstein polynomial projection matrix 𝐏 N M+N \\mathbf {P}^{M+N}_N for different choices of degree NN and MM."
],
[
"A note on fast mass matrix inversion",
"It should be noted that the approach described in Theorem REF is in fact applicable to matrices beyond the polynomial projection matrix.",
"For example, since the Bernstein mass matrix is diagonal under a modal basis [10], the inverse Bernstein mass matrix can also be represented as a combination of degree elevation matrices.",
"We start with an interesting observation in the proof of Lemma REF (see [11]): Lemma 5.4 Let $\\mathbf {M}_N$ be the Bernstein mass matrix of degree $N$ .",
"Under a transformation to a modal basis, the inverse $\\mathbf {M}^{-1}_N$ is diagonal given by $\\mathbf {T}_N^{-1}\\mathbf {M}^{-1}_N\\mathbf {T}_N=\\begin{pmatrix}\\frac{1}{\\lambda ^N_0} & & &\\\\&\\frac{1}{\\lambda ^N_1}\\mathbf {I}_1& &\\\\& & \\ddots &\\\\& & & \\frac{1}{\\lambda ^N_N}\\mathbf {I}_N\\end{pmatrix},$ where $\\lambda ^{N}_j$ is the $j$ th distinct eigenvalue of $\\mathbf {M}_N$ .",
"Applying Corollary REF to (REF ) directly, we obtain the following theorem: Theorem 5.5 There exist $c_j$ , $0\\le j\\le N$ , such that, the inverse of Bernstein mass matrix can be written as $\\mathbf {M}^{-1}_N= \\sum _{j=0}^{N}c_j\\mathbf {E}^N_{N-j}\\left(\\mathbf {E}^{N}_{N-j}\\right)^T.$ Using (REF ), the inverse of a Bernstein mass matrix can be represented as a linear combination of sparse Bernstein degree elevation matrices.",
"Thus, we can apply $\\mathbf {M}^{-1}_N$ using the expression (REF ), which requires $O(N^4)$ operations in 3D.",
"Since WADG requires only applications of $\\mathbf {V}_q$ and $\\mathbf {P}_q = \\mathbf {M}^{-1}\\mathbf {V}_q^T\\mathbf {W}$ , by combining fast mass matrix inversion with efficient $O(N^4)$ algorithms for evaluating Bernstein polynomials at quadrature points [8], it is possible to implement 3D quadrature-based WADG in $O(N^4)$ total operations.Fast Bernstein mass matrix inversion could also be performed using the algorithm described in [10].",
"However, as noted by Kirby, this approach is more involved and may be difficult to implement efficiently on GPUs.",
"In light of these results, one may then ask why we bother with the strategy presented in Section , which involves both approximation of the weight function and specialized algorithms for polynomial multiplication and polynomial $L^2$ projection.",
"The answer lies in the nature of the coefficients $c_j$ .",
"We observe that, when representing the Bernstein mass matrix inverse using (REF ), the coefficients $c_j$ are highly oscillatory with large positive and negative components (see Table REF ), which can result in significant numerical roundoff in the application of $\\mathbf {M}^{-1}$ using (REF ).",
"In contrast, the coefficients used to represent the Bernstein polynomial projection matrix are much less oscillatory (see Table REF ) and result in less roundoff error.",
"We can estimate sensitivity of black(REF ) and (REF ) to roundoff by computing $\\sum _{j=0}^N |c_j|$ blackIn the context of numerical quadrature with negative weights, the quantity (REF ) is referred to as the condition number of a quadrature rule [29].",
"For $N = 7$ , the value of (REF ) is approximately $1.67\\times 10^7$ for $\\mathbf {M}^{-1}$ .",
"In contrast, for $N= 7$ , the value of (REF ) for $\\mathbf {P}^{M+N}_N$ is approximately $14.53$ for $M = 1$ and $41.35$ for $M = 2$ .",
"blackWe also investigated roundoff errors numerically by computing the difference between $\\mathbf {M}^{-1}\\mathbf {b} - \\mathbf {e}$ (where $\\mathbf {M}^{-1}$ is computed using backslash in Matlab) and the quantity $\\sum _{j=0}^{N}c_j\\mathbf {E}^N_{N-j}\\left(\\mathbf {E}^{N}_{N-j}\\right)^T\\mathbf {b} - \\mathbf {e}.$ Here, $\\mathbf {e}$ is the vector of all ones and $\\mathbf {b} = \\mathbf {M}\\mathbf {e}$ .",
"In the absence of roundoff errors, both quantities should be zero.",
"However, for all $N$ , the roundoff error in applying $\\mathbf {M}^{-1}$ using (REF ) is larger than the roundoff error incurred when using Matlab's backslash directly.",
"Since the Bernstein mass matrix $\\mathbf {M}$ is already known to become highly ill-conditioned as $N$ increases [8], [30], these numerical experiments suggest that evaluating $\\mathbf {M}^{-1}$ using (REF ) is impractical for large $N$ .",
"Table: Coefficient c j c_j for 𝐌 -1 \\mathbf {M}^{-1} represented using () for different orders NN."
],
[
"GPU algorithms",
"In this section, we describe GPU-accelerated algorithms for Bernstein polynomial multiplication and polynomial $L^2$ projection."
],
[
"Polynomial multiplication",
"blackFor polynomial multiplication, we aim to compute Bernstein coefficients of of the product $h(\\mathbf {x})=f(\\mathbf {x})g(\\mathbf {x})$ , where $f(\\mathbf {x}),g(\\mathbf {x})$ are Bernstein polynomials of degree $N$ and degree $M$ , respectively.",
"From (REF ), we observe that each coefficient of $h(\\mathbf {x})$ is a linear combination of at most $M_p$ products of coefficients from $f$ and $g$ as follows black $\\begin{split}c_{\\mathbf {\\gamma }} &= \\sum _{|\\mathbf {\\beta }|=M}a_{\\mathbf {\\gamma }-\\mathbf {\\beta }}b_{\\mathbf {\\beta }}\\frac{\\binom{\\mathbf {\\gamma }}{\\mathbf {\\beta }}}{\\binom{N+M}{N}} = \\sum _{|\\mathbf {\\beta }|=M}a_{\\mathbf {\\gamma }-\\mathbf {\\beta }}b_{\\mathbf {\\beta }} \\ell _{\\mathbf {\\beta }},\\end{split}$ where $a_{\\mathbf {\\gamma }-\\mathbf {\\beta }}$ and $b_{\\mathbf {\\beta }}$ are coefficients of $f$ and $g$ , respectively.",
"In our implementation, we store the coefficients $\\ell _{\\mathbf {\\beta }}$ in some sparse matrix, where the row and column indices correspond to the multi-indices $\\mathbf {\\gamma }$ and $\\mathbf {\\beta }$ , respectively.",
"Each thread will load non-zero entries in a row of this matrix along with the corresponding coefficients $b_j$ and $a_{\\mathbf {\\gamma }-\\mathbf {e}_j}$ , compute one of the coefficients $c_{\\mathbf {\\gamma }}$ , and store the result into shared memory."
],
[
"Polynomial $L^2$ projection",
"blackWe now introduce an algorithm to evaluate the polynomial $L^2$ projection based on (REF ).",
"Unfortunately, it is difficult to directly evaluate (REF ) in a low-complexity fashion.",
"This is because the degree elevation matrices $\\mathbf {E}^N_{N-j}$ transition from sparse to dense matrices as $j$ increases.",
"Instead, we evaluate (REF ) using an equivalent reformulation.",
"By plugging (REF ) into (REF ), we can derive a “telescoping form” for $\\mathbf {P}^{M+N}_N$ involving sparse one-degree elevation matrices $\\mathbf {P}^{N\\!",
"+\\!",
"M}_N\\!",
"&=\\!",
"\\left(\\!",
"c_0\\mathbf {I}\\!",
"+\\!",
"\\mathbf {E}^N_{N\\!",
"-\\!",
"1}\\!",
"\\left(\\!",
"c_1\\mathbf {I}\\!",
"+\\!",
"\\mathbf {E}^{N\\!",
"-\\!",
"1}_{N\\!",
"-\\!",
"2}\\!",
"\\left(\\!",
"c_2\\mathbf {I}\\!",
"+\\!",
"\\cdots \\!",
"\\right)\\!",
"\\left(\\!",
"\\mathbf {E}^{N\\!",
"-\\!",
"1}_{N\\!",
"-\\!",
"2}\\!",
"\\right)^T\\right)\\!",
"\\left(\\!",
"\\mathbf {E}^N_{N\\!",
"-\\!",
"1}\\!",
"\\right)^T\\right)\\!",
"\\left(\\!",
"\\mathbf {E}^{N\\!",
"+\\!",
"1}_N\\!",
"\\right)^T\\!",
"\\cdots \\!",
"\\left(\\!",
"\\mathbf {E}^{N\\!",
"+\\!",
"M}_{N\\!",
"+\\!",
"M\\!",
"-\\!",
"1}\\!",
"\\right)^T \\\\&= \\widetilde{\\mathbf {P}}_N \\left(\\!",
"\\mathbf {E}^{N\\!",
"+\\!",
"1}_N\\!",
"\\right)^T\\!",
"\\cdots \\!",
"\\left(\\!",
"\\mathbf {E}^{N\\!",
"+\\!",
"M}_{N\\!",
"+\\!",
"M\\!",
"-\\!",
"1}\\!",
"\\right)^T\\nonumber $ where we have defined $\\widetilde{\\mathbf {P}}_N = \\left(\\!",
"c_0\\mathbf {I}\\!",
"+\\!",
"\\mathbf {E}^N_{N\\!",
"-\\!",
"1}\\!",
"\\left(\\!",
"c_1\\mathbf {I}\\!",
"+\\!",
"\\mathbf {E}^{N\\!",
"-\\!",
"1}_{N\\!",
"-\\!",
"2}\\!",
"\\left(\\!",
"c_2\\mathbf {I}\\!",
"+\\!",
"\\cdots \\!",
"\\right)\\!",
"\\left(\\!",
"\\mathbf {E}^{N\\!",
"-\\!",
"1}_{N\\!",
"-\\!",
"2}\\!",
"\\right)^T\\right)\\!",
"\\left(\\!",
"\\mathbf {E}^N_{N\\!",
"-\\!",
"1}\\!",
"\\right)^T\\right)\\!$ .",
"We next provide an algorithm to efficiently evaluate this telescoping expression on GPUs.",
"blackThe first step in applying $\\mathbf {P}^{N+M}_N$ is to apply the product of degree reduction matrices $\\left(\\!",
"\\mathbf {E}^{N\\!",
"+\\!",
"1}_N\\!",
"\\right)^T\\!",
"\\cdots \\!",
"\\left(\\!",
"\\mathbf {E}^{N\\!",
"+\\!",
"M}_{N\\!",
"+\\!",
"M\\!",
"-\\!",
"1}\\!",
"\\right)^T$ .",
"Since each of these matrices is sparse and requires $O(N^d)$ operations to apply, this step has an overall computational complexity of $O(N^d)$ for fixed $M$ .",
"blackThe next step applies $\\widetilde{\\mathbf {P}}_N$ to the degree-reduced result.",
"We separate the application of $\\widetilde{\\mathbf {P}}_N$ into two parts.",
"The first part applies the one-degree reduction matrices in a “downward” sweep, while the second applies the one-degree elevation matrices in an “upward” sweep (see Fig.",
"REF for an illustration).",
"Both the application of degree elevation or reduction operators and accumulate results during each step simultaneously.",
"blackWe briefly describe our GPU implementation used to apply $\\widetilde{\\mathbf {P}}_N$ .",
"Let $\\mathbf {p}$ be some vector to which we will apply $\\widetilde{\\mathbf {P}}_N$ .",
"In the first step, we set $\\mathbf {p}_s=\\mathbf {p}$ , then compute the product of $\\left(\\mathbf {E}^i_{i-1}\\right)^T$ and the matrix-vector product $\\mathbf {p}_s$ stored in shared memory.",
"More specifically, each thread computes the dot product of a sparse row of $\\left(\\mathbf {E}^i_{i-1}\\right)^T$ with the vector $\\mathbf {p}_s$ .",
"The resulting output vector $\\mathbf {q}_s$ will be stored in another shared memory array and transfered to $\\mathbf {p}_s$ after all threads complete their computation.",
"At the same time, $\\mathbf {q}_s$ will be scaled by the constant black$c_{\\mathbf {\\gamma }}$ in (REF ) and stored in thread-local register memory.",
"For the second part, we compute the product of $\\mathbf {E}^i_{i-1}$ and the vector $\\mathbf {p}_s$ in shared memory, and accumulate results with the values in register memory during each step.",
"More specifically, each thread computes the dot product of a sparse row of $\\mathbf {E}^i_{i-1}$ with $\\mathbf {p}_s$ , and the result will be added to the corresponding value in register memory.",
"After the accumulation, the values in register memory will be transfered to $\\mathbf {p}_s$ in shared memory, which will be used in the next step.",
"Figure: Illustration of GPU algorithm for the polynomial L 2 L^2 projectionIn our algorithm, the multiplication of two Bernstein polynomials can be blackcomputed in $O(N^d)$ operations.",
"For the polynomial $L^2$ projection, each application of $\\mathbf {E}^i_{i-1}$ or $\\left(\\mathbf {E}^i_{i-1}\\right)^T$ requires $O(N^d)$ operations.",
"We need to apply $N$ one-degree elevation operators and $N+M$ one-degree reduction operators, resulting in a total asymptotic complexity of $O(N^{d+1})$ blackfor fixed $M$ .",
"This reduces the computational complexity of the projection step in WADG from $O(N^{6})$ to $O(N^{4})$ in three dimensions."
],
[
"Numerical results",
"In this section, we examine the accuracy and performance of BBWADG.",
"For blackclarity, we refer to WADG as the quadrature-based weight-adjusted discontinuous Galerkin method.",
"This section is divided into four parts: in Section REF , we discuss accuracy of BBWADG using the method of manufactured solutions; In Section REF , we test BBWADG for wavespeed with different frequencies; in Section REF , we present runtime comparisons between BBWADG and WADG; in Section REF , we present results which quantify the computational efficiency of BBWADG."
],
[
"Convergence for heterogeneous media",
"In this section, we investigate the convergence of BBWADG to manufactured solutions.",
"In two dimensions, we assume that the pressure $p(x,y,\\tau )$ is of the form $p\\left(x,y,\\tau \\right) = \\sin \\left(\\pi x\\right)\\sin \\left(\\pi y\\right)\\cos \\left(\\pi \\tau \\right).$ We take the corresponding velocity vector as follows $\\mathbf {u}=\\begin{pmatrix}u\\\\v\\end{pmatrix}=\\begin{pmatrix}-\\cos (\\pi x)\\sin (\\pi y)\\sin (\\pi \\tau )\\\\-\\sin (\\pi x)\\cos (\\pi y)\\sin (\\pi \\tau )\\end{pmatrix}.$ Because this is not a solution of the acoustic wave equation in heterogeneous media, we utilize the method of manufactured solutions and add a source term $f(x,y,\\tau )$ for which $p(x,y,\\tau )$ is a solution.",
"Plugging $p,\\mathbf {u}$ into (REF ), we obtain the source term $f$ $\\begin{split}f(x,y,\\tau ) & = \\frac{1}{c^2(x,y)}\\frac{\\partial p}{\\partial \\tau }+\\nabla \\cdot \\mathbf {u}\\\\&= -\\frac{1}{c^2(x,y)}\\pi \\sin (\\pi x)\\sin (\\pi y)\\sin (\\pi \\tau )+2\\pi \\sin (\\pi x)\\sin (\\pi y)\\sin (\\pi \\tau )\\\\&= \\left(2-\\frac{1}{c^2(x,y)}\\right)\\pi \\sin (\\pi x)\\sin (\\pi y)\\sin (\\pi \\tau ).\\end{split}$ Similarly, in three dimensions, we assume the pressure $p(x,y,z,\\tau )$ satisfies $p\\left(x,y,z,\\tau \\right) = \\sin \\left(\\pi x\\right)\\sin \\left(\\pi y\\right)\\sin \\left(\\pi z\\right)\\cos \\left(\\pi \\tau \\right).$ We can compute the corresponding velocity vector as follows $\\mathbf {u}=\\begin{pmatrix}u\\\\v\\\\w\\end{pmatrix}=\\begin{pmatrix}-\\cos (\\pi x)\\sin \\left(\\pi y\\right)\\sin (\\pi z)\\sin (\\pi \\tau )\\\\[.5ex]-\\sin (\\pi x)\\cos \\left(\\pi y\\right)\\sin (\\pi z)\\sin (\\pi \\tau )\\\\[.5ex]-\\sin (\\pi x)\\sin \\left(\\pi y\\right)\\cos (\\pi z)\\sin (\\pi \\tau )\\end{pmatrix}.$ Plugging $p,\\mathbf {u}$ into (REF ), we obtain the source term $f$ $\\begin{split}f(x,y,z,\\tau ) & = \\frac{1}{c^2(x,y,z)}\\frac{\\partial p}{\\partial \\tau }+\\nabla \\cdot \\mathbf {u}\\\\&= \\left(3-\\frac{1}{c^2(x,y,z)}\\right)\\pi \\sin (\\pi x)\\sin \\left(\\pi y\\right)\\sin (\\pi z)\\sin (\\pi \\tau ).\\end{split}$ In numerical experiments, we choose the wavespeed as $c^2(x,y,z) = 1+\\frac{1}{2}\\sin (\\pi x)\\sin (\\pi y)$ for two dimensions and $c^2(x,y,z) = 1+\\frac{1}{2}\\sin (\\pi x)\\sin (\\pi y)\\sin \\left(\\pi z\\right).$ for three dimensions.",
"In BBWADG, we project $c^2$ onto a polynomial space of degree $M$ in $L^2$ sense.",
"Fig.",
"REF and Fig.",
"REF show the convergence of BBWADG and WADG to the manufactured solution under mesh refinement.",
"The 3D uniform meshes used in our experiments are generated by GMSH [31].",
"From these plots, we observe that the convergence rate of BBWADG is $O(h^r)$ , where $r=2$ when $M=0$ and $r=\\min \\lbrace N+1,M+3\\rbrace $ when $M\\ge 1$ .",
"We note that rates of convergence only observed when $c^2$ is approximated using the polynomial $L^2$ projection onto $P^M$ .",
"For other approximations (e.g.",
"piecewise linear interpolation), the convergence rates are $O(h^M)$ in general.",
"It should be noted that these rates of convergence are better than those suggested by an initial error analysis.",
"It is straightforward to extend the error analysis of [1], [21] to accomodate approximations of $c^2 \\in P^M$ .",
"However, this extension predicts that, when $c^2$ is approximated using $L^2$ projection onto degree $M$ polynomials, the $L^2$ error should converge at a rate of $O(h^{M+1})$ .",
"This rate is observed only for $M=0$ , and the source of the discrepancy between the predicted and observed rates for $M> 0$ is presently unclear to the authors.",
"blackIncreasing from $M=0$ to $M=1$ increases the observed rate of convergence by 2 orders.",
"In contrast, increasing $M$ further only increases the observed rate of convergence by one order for each degree past $M=1$ .",
"blackFor this reason, $M=1$ may be an attractive choice for practical computations, since it provides a larger improvement in terms of rates of convergence relative to the increase in computational cost.",
"Figure: Convergence under mesh refinement (2D)Figure: Convergence under mesh refinement (3D)."
],
[
"Wavespeed with different frequencies",
"blackSince the accuracy of the polynomial approximation of the wavespeed depends on $M$ , we examine how the error depends on the approximability of $c^2$ .",
"We test BBWADG using the following wavespeeds $\\begin{split}c^2(x,y) &= 1+\\frac{1}{2}\\sin (k\\pi x)\\sin (k\\pi y)\\qquad \\textmd {(2D)},\\\\c^2(x,y,z) &= 1+\\frac{1}{2}\\sin (k\\pi x)\\sin (k\\pi y)\\sin (k\\pi z)\\qquad \\textmd {(3D)},\\end{split}$ with different frequencies $k$ .",
"blackHowever, the manufactured solution remains the same independently of $k$ .",
"blackThis experiment is intended to show how the error depends on the approximability of the wavespeed.",
"For higher $k$ , $c^2$ is more oscillatory and harder to approximate; thus, we expect that the error should increase as $k$ increases, despite the fact that the exact solution is independent of $k$ .",
"Figure: Convergence of L 2 L^2 error when approximating wavespeeds given by ().We compute $L^2$ errors on a fixed mesh for various choices of $k$ , choose $N=7$ and a uniform mesh with $h=0.0625$ for 2D experiments, and choose $N=6$ and a uniform mesh with $h=0.0833$ for 3D experiments.",
"From Fig.",
"REF , we observe that, for a fixed $M$ , the accuracy of the method blackdoes indeed depend on the frequency of wavespeed: the lower frequency is (or the smaller $k$ is), the smaller the error, despite the fact that the solution remains the same for all $k$ ."
],
[
"Runtime comparisons",
"In this section, we present runtime comparisons between BBWADG and quadrature-based WADG blackfor $M=1$ and $M=2$ .",
"In Section REF , we showed that the computational complexity of BBWADG is $O(N^{d+1})$ blackfor a fixed $M$ .",
"In this section, we will verify that this complexity is observed in practice, blackthough the constant depends on $M$ .",
"All results are run on an Nvidia GTX 980 GPU, and the solvers are implemented in the Open Concurrent Compute Abstraction framework (OCCA) [32] for clarity and portability."
],
[
"Computational implementation",
"A time-explicit DG scheme consists of the evaluation of the right hand side and the solution update.",
"Its implementation is typically divided into three kernels.",
"A volume kernel, which evaluates contributions to the right hand side resulting from volume terms in (REF ).",
"Specifically, the volume kernel evaluates derivatives of local solutions over each element.",
"A surface kernel, which evaluates numerical fluxes and contributions to the right hand side resulting from the surface terms in (REF ).",
"More specifically, the surface kernel computes numerical fluxes and applies the lift matrix.",
"An update kernel, which updates the solution in time.",
"We use a low-storage 4th order Runge-Kutta method [33] in this thesis.",
"We adopt the same volume and surface kernels from [11].",
"BBWADG and WADG are implemented within the update kernel by modifying the right hand side computed in the volume and surface kernels."
],
[
"Acoustic wave equations",
"In this experiment, we apply both BBWADG and WADG to the acoustic wave equation (REF ).",
"blackRuntimes for the update kernels are given in Fig.",
"REF .",
"blackThe case of $M=1$ is denoted by BBWADG-1, while $M=2$ is denoted by BBWADG-2.",
"Figure: Per-element runtimes of update kernels using BBWADG and WADG on a mesh of 7854 elements (acoustic).Table: Achieved speedup for M=1M=1Table: Achieved speedup for M=2M=2From Fig.",
"REF , we observe that BBWADG is more expensive than WADG for low orders $(N < 4)$ .",
"However, runtime of the WADG update kernel increases more rapidly with $N$ , displaying an asymptotic complexity of $O(N^6)$ .",
"On the other hand, the runtime of the BBWADG update kernel increases more slowly and displays a complexity of $O(N^4)$ as proven in Section REF .",
"Table REF displays observed speedups of BBWADG over WADG.",
"We find that for $N=7$ , the BBWADG update kernel for $M=1$ achieves a 3.6 times speedup over the WADG update kernel.",
"For $N=8$ , we observe an unexpected over 30 times speedup.",
"However, we should note that this result is due to the use of different quadratures between $N=7$ and $N=8$ .",
"We choose a tetrahedral quadrature from Xiao and Gimbutas [23] blackwhich is exact for degree $2N+1$ polynomials for $N\\le 7$ .",
"blackFor $N = 8$ , this implies that the quadrature rule should be exact for polynomials of degree 17.",
"However, the publicly available quadrature rules are only exact up to degree 15 polynomials.",
"Because optimized quadrature points were not publicly available for $N>7$ , we blackswitch to a collapsed coordinate quadrature [34] for $N>7$ (see Fig.",
"REF ).",
"Since the construction of quadrature points is different, one should not compare results for degrees $N\\le 7$ with degrees $N>7$ .",
"Table REF shows observed speedups for $M=2$ .",
"We observe that the BBWADG update kernel for $M=2$ is slower than WADG until $N = 5$ .",
"This is due to several reasons.",
"blackFirst, increasing from $M=1$ to $M=2$ does not change the overall computational complexity with respect to $N$ , but it does change the constant, which scales as $O(M^d)$ .",
"blackSecondly, for $M=1$ , since we know a-priori that the sparse matrices involved in polynomial multiplication contain only $d+1 = 4$ nonzeros per row in 3D, we can store such matrices using float4 and int4 data structures, which have a slightly faster access time on GPUs [11].",
"This convenient storage structure is not available for $M=2$ .",
"Figure: Visualization of the quadrature points on the reference triangle"
],
[
"Elastic wave equation",
"In this experiment, we compute runtimes for both BBWADG and WADG applied the elastic wave equations (REF ).",
"Computational results for $M=1$ and $M=2$ are presented in Fig.",
"REF .",
"Figure: Per-element runtimes for update kernels using BBWADG and WADG on a mesh of 7854 elements (elastic).We observe that the runtime behaves similarly to the acoustic case.",
"The runtime of the BBWADG update kernel increases roughly as $O(N^4)$ up to $N = 8$ , with about a 2.2 times speedup achieved for $M=1$ and $N=7$ .",
"However, the runtime of BBWADG increases more rapidly than $O(N^4)$ for $N > 8$ .",
"We expect that this is due to GPU occupancy/memory effects.",
"blackThe application of $\\widetilde{\\mathbf {P}}_N$ described in Section REF requires storage of $(N+1)$ intermediate values per thread for each application of a scalar weight-adjusted inverse mass matrix.",
"For the scalar acoustic wave equation, this additional storage is negligible, as only a single weight-adjusted inverse mass matrix is applied per element.",
"However, for the elastic wave equation, we apply a matrix-weighted weight-adjusted inverse mass matrix, which is computed by applying multiple scalar weight-adjusted inverse mass matrices and combining the results.",
"For elastic wave propagation in 3D, this increases the per-thread memory cost by a factor of 6 (corresponding to each of the six components of the elastic stress tensor), resulting in significant register pressure and reduced GPU occupancy.. blackThis additional storage can be decreased by processing fewer components simultaneously; however, processing fewer components simultaneously also reduces data reuse and temporal locality.",
"It is not immediately clear whether this approach will result in an overall lower runtime, and will be the subject of future investigation.",
"Figure: Achieved bandwidth (GB/s) for update kernels using BBWADG and WADG."
],
[
"Performance analysis",
"In this section, we present computational results for BBWADG with $M=1$ and WADG.",
"Fig.",
"REF and Fig.",
"REF show the profiled computational performance and bandwidth of the BBWADG and WADG update kernels.",
"From Fig.",
"REF , we observe that the bandwidth of the WADG update kernel decreases steadily as $N$ increases.",
"In comparison, the BBWADG update kernel sustains a near-constant bandwidth as $N$ increases.",
"From Fig.",
"REF , we can see that, for all $N$ , the BBWADG update kernel achieves a lower computational performance compared to the WADG update kernel.",
"These results are similar to those achieved for BBDG with piecewise constant wavespeeds [11].",
"Figure: Achieved computational performance (TFLOPS/s) for update kernels using BBWADG and WADG."
],
[
"Conclusion and future work",
"In this paper, we present a Bernstein-Bézier discontinuous Galerkin (BBWADG) method to simulate acoustic and elastic wave propagation in heterogeneous media blackbased on a polynomial approximation of sub-cell heterogeneities and fast algorithms for Bernstein polynomial multiplication and $L^2$ projection.",
"black The resulting solver inherits the advantages of WADG (provable energy stability, high order accuracy) while reducing the computational complexity of the update kernel from $O(N^{2d})$ to $O(N^{d+1})$ in $d$ dimensions.",
"Moreover, this implementation reuses the BBDG volume and surface kernels from [11], both of which can be applied in $O(N^d)$ operations.",
"Thus, the total computational complexity of the BBWADG solver is $O(N^{d+1})$ per timestep blackfor a fixed polynomial approximation of sub-cell media heterogeneities.",
"Due to its low computational complexity, BBWADG offers advantages in simulating wave propagation in heterogeneous media using higher order approximations.",
"These properties make BBWADG promising for accurate and efficient simulation of large-scale wave propagation problems."
],
[
"Acknowledgments",
"The authors acknowledge the support of the National Science Foundation under awards DMS-1719818 and DMS-1712639."
]
] | 1808.08645 |
[
[
"Privacy in Internet of Things: from Principles to Technologies"
],
[
"Abstract Ubiquitous deployment of low-cost smart devices and widespread use of high-speed wireless networks have led to the rapid development of the Internet of Things (IoT).",
"IoT embraces countless physical objects that have not been involved in the traditional Internet and enables their interaction and cooperation to provide a wide range of IoT applications.",
"Many services in the IoT may require a comprehensive understanding and analysis of data collected through a large number of physical devices that challenges both personal information privacy and the development of IoT.",
"Information privacy in IoT is a broad and complex concept as its understanding and perception differ among individuals and its enforcement requires efforts from both legislation as well as technologies.",
"In this paper, we review the state-of-the-art principles of privacy laws, the architectures for IoT and the representative privacy enhancing technologies (PETs).",
"We analyze how legal principles can be supported through a careful implementation of privacy enhancing technologies (PETs) at various layers of a layered IoT architecture model to meet the privacy requirements of the individuals interacting with IoT systems.",
"We demonstrate how privacy legislation maps to privacy principles which in turn drives the design of necessary privacy enhancing technologies to be employed in the IoT architecture stack."
],
[
"Introduction",
"Ubiquitous deployment of low-cost smart devices and widespread use of high-speed wireless networks have led to the rapid development of Internet of Things (IoT).",
"IoT embraces countless physical objects embedded with Radio Frequency Identification (RFID) tags, sensors and actuators that have not been involved in the traditional Internet and enables their interaction and cooperation through both traditional as well as IoT-specific communication protocols [1], [2].",
"Gartner [3] estimates that around 20.4 billion `things' will be connected by the year 2020.",
"These pervasive and heterogeneous devices that interact with the physical and digital worlds have the potential to significantly enhance the quality of life for individuals interacting with the IoT.",
"With smart home and wearable devices, users obtain seamless and customized services from digital housekeepers, doctors and fitness instructors [4].",
"Smart building and smart city applications provide an increased awareness of the surroundings and offer greater convenience and benefits to the users [5], [6].",
"Many services offered by IoT may require a comprehensive understanding of user interests and preferences, behavior patterns and thinking models.",
"For instance, in the Christmas special episode of the British series `Black Mirror', the soul of a woman is copied to serve as the controller of her smart home, which can wake up the woman with her favorite music and breakfast as the copy knows her as no one else can [7].",
"Such a digital copy, which could be hard to create in the traditional Internet, is relatively easier to be generated in the IoT era.",
"While some individuals prefer the convenience of the services, some others may be concerned about their personal data being shared [8].",
"In 2013, the IEEE Internet of Things survey showed that 46% of respondents consider privacy concerns as the biggest challenge for IoT adoption [9].",
"Large scale data collection in the IoT poses significant privacy challenges and may hamper the further development and adoption by privacy-conscious individuals [10].",
"Information privacy is a broad and complex notion as its understanding and perception differ among individuals and its enforcement requires efforts from both legislation and technologies [5], [11].",
"Privacy laws help to enforce compliance and accountability of privacy protection and make privacy protection a necessity for every service provider [11].",
"Privacy enhancing technologies (PETs) on the other hand support the underlying principles guided by privacy laws that enable privacy protection strategies to be implemented in engineering [12], [13].",
"In this paper, we study the privacy protection problem in IoT through a comprehensive review by jointly considering three key dimensions, namely the state-of-the-art principles of privacy laws, architectures for the IoT system and representative privacy enhancing technologies (PETs).",
"Based on an extensive analysis along these three dimensions, we show that IoT privacy protection requires significant support from both privacy enhancing technologies (PETs) and their enforcement through privacy legislation.",
"We analyze how legal principles can be supported through a careful implementation of various privacy enhancing technologies (PETs) at various layers of a layered IoT architecture model to meet the privacy requirements of the individuals interacting with IoT systems.",
"Our study is focused on providing a broader understanding of the state-of-the-art principles in privacy legislation associated with the design of relevant privacy enhancing technologies (PETs) and on demonstrating how privacy legislation maps to privacy principles which in turn drives the design of necessary privacy enhancing technologies to be employed in the IoT architecture stack.",
"We organize the paper in the following manner.",
"In Section , we analyze the principles of privacy laws and present the privacy-by-design strategies that can adopt the general principles to engineering practice.",
"In Section , we introduce the IoT system using a layered reference architecture and describe the functionalities and enabling technologies of each layer.",
"We discuss how privacy-by-design strategies can be integrated into the reference architecture.",
"In Section to Section , we introduce the state-of-the-art privacy enhancing technologies (PETs), analyze their suitability for privacy-by-design strategies and discuss the pros and cons of their use and implementation in each IoT layer.",
"In Section , we discuss privacy issues in IoT applications.",
"Finally, we present the related work in Section and conclude in Section ."
],
[
"Privacy",
"Privacy is a complex and a subjective notion as its understanding and perception differ among individuals.",
"In this section, we review the definitions of privacy in the past, introduce the privacy laws and analyze the state-of-the-art privacy legislation.",
"We then introduce the privacy-by-design (PbD) strategies that facilitate the design of privacy-preserving systems satisfying the legal principles."
],
[
"Definition",
"As far back as the thirteenth century, when the eavesdroppers were claimed to be guilty, the notion of media privacy had come into being [14].",
"Then, with the technical and social development, the notion of privacy successively shifted to territorial (eighteenth century), communication (1930s), and bodily privacy (1940s) [11].",
"Finally, in the 1960s, it was the rise of electronic data processing that brought into being the notion of information privacy (or data privacy) that has achieved lasting prominence until now.",
"In 1890, Warren and Brandeis defined privacy as `the right to be let alone' in their famous article `The Right to Privacy' [15].",
"After that, many privacy definitions have been emerging unceasingly, but the one proposed by Alan Westin in his book `Privacy and Freedom' has become the base of several modern data privacy principles and law [11].",
"Westin defined privacy as `the claim of individuals, groups, or institutions to determine for themselves when, how, and to what extent information about them is communicated to others' [16], which mainly emphasized the control of the data subjects over their data.",
"The authors in [10] argued that Westin's definition was too general for the IoT area and they proposed a more focused one that defines the IoT privacy as the threefold guarantee including `awareness of privacy risks imposed by smart things and services surrounding the data subject; individual control over the collection and processing of personal information by the surrounding smart things; awareness and control of subsequent use and dissemination of personal information by those entities to any entity outside the subject’s personal control sphere'."
],
[
"Legislation",
"Privacy laws form a critical foundation in the design of any privacy-preserving system.",
"As the cornerstone of most modern privacy laws and policies, the Fair Information Practices (FIPs) are a set of internationally recognized practices to protect individual information privacy [17].",
"The code of FIPs was born out of a report from the Department of Health, Education & Welfare (HEW) [18] in 1973 and then adopted by the US Privacy Act of 1974, the most famous privacy legislation in the early stage.",
"The original HEW FIPs consist of five principles that can be summarized as [18]: [innerleftmargin=1.6pt] No secret systems of personal data.",
"Ability for individuals to find out what is in the record, and how it is used.",
"Ability for individuals to prevent secondary use.",
"Ability to correct or amend records.",
"Data must be secure from misuse.",
"However, as a federal law, the US Privacy Act of 1974 only works with the federal government.",
"There is no general information privacy legislation that covers all states and areas [19].",
"As a result, the FIPs always act as the guideline of the various privacy laws and regulations ranging from different organizations (e.g., Stanford University [20], Department of Homeland Security [21]) to different areas (e.g., HIPAA [22], COPPA [23]).",
"In 1980, based on the core HEW FIPs, the Organization for Economic Cooperation and Development (OECD) adopted the Guidelines on the Protection of Personal Privacy and Transborder Flows of Personal Data [24].",
"It is considered a historical milestone as it represented the first internationally-agreed upon privacy protection [19].",
"The eight principles extended from the five basic FIPs have been the foundation of most EU privacy laws later.",
"They can be summarized as: [innerleftmargin=1.6pt] Collection Limitation: Collection should be lawful, fair and with knowledge or consent of the data subject.",
"Data Quality: Personal data should be purpose-relevant, accurate, complete and kept up-to-date.",
"Purpose Specification: Purposes should be specified earlier than collection and complied with.",
"Use Limitation: Personal data should not be disclosed, made available or used for non-specified purposes.",
"Security Safeguards: Personal data should be protected by reasonable security safeguards.",
"Openness: There should be a general policy of openness about developments, practices and policies with respect to personal data.",
"Individual Participation: An individual should have the right to access his data, be timely informed on data collection, be given disputable reason for denied lawful request and challenge his data to have the data erased, rectified, completed or amended.",
"Accountability: A data controller should be accountable for complying with measures which give effect to the principles stated above.",
"Although the OECD guidelines achieved worldwide recognition, it was nonbinding.",
"It was not until 1995 that the EU passed Directive 95/46/EC [25] and the OECD guidelines were incorporated into an influential privacy law for the first time.",
"Unlike the US, the EU dedicated to enforcing the omnibus privacy laws to comprehensively protect individual data in its member countries through not only the principles, but the restriction on the data transference with non-EU countries, which in turn has influenced the development of the privacy laws in the non-EU countries and the appearance of the data exchange regulations such as the Safe Harbor [26] and its later replacement, the EU-US Privacy Shield framework [27].",
"Recently, as the successor of Directive 95/46/EC, the General Data Protection Regulation (GDPR) [28] was adopted by the EU in 2016 and it has come into force in 2018.",
"In the GDPR, most principles are covered by the Article 5, including `lawfulness, fairness and transparency', `purpose limitation', `data minimization', `accuracy', `storage limitation', `integrity and confidentiality' and `accountability'.",
"Its key changes in terms of the principles, compared with the Directive 95/46/EC, include six aspects [28], [29]: [leftmargin=*] Consent: The GDPR is more strict with consents.",
"A consent should be graspable, distinguishable and easy to be withdrawn.",
"Breach Notification: The GDPR makes the breach notification mandatory.",
"The notification should be sent within 72 hours after being aware of the breach.",
"Right to Access: The first right mentioned in the OECD `individual participation' principle is strengthened in the GDPR.",
"Right to be Forgotten: In the Article 17, data is required to be erased when the personal data are no longer necessary in relation to the purposes or the consent is withdrawn.",
"Data Portability: In the Article 20, a data subject has the right to receive his uploaded data in a machine-readable format and transmit it to another data controller.",
"Privacy by Design: The privacy by design is finally integrated into the privacy legal framework.",
"As claimed in the Article 25, `the controller shall, both at the time of the determination of the means for processing and at the time of the processing itself, implement appropriate technical and organizational measures', which asks the privacy to be taken into account at the design stage, rather than as an add-on function.",
"Figure: IoT architecture and data flow , , ,"
],
[
"Privacy by design",
"The notion of privacy by design (PbD) [5], [13], [14], namely embedding the privacy measures and privacy enhancing technologies (PETs) directly into the design of software or system, is not new.",
"As early as 2001, Langheinrich [14] proposed six principles to guide the PbD in the ubiquitous systems, including notice, choice and consent, proximity and locality, anonymity and pseudonymity, security, and access and recourse.",
"However, the PbD has never been extensively used in engineering.",
"The main reason for its rare adoption is that most engineers either neglect the importance of privacy or refuse their responsibility on it , .",
"A privacy law may also need support from technologies.",
"In IoT, the enforcement of each principle in a privacy law may need to be supported by a set of technologies (e.g., PETs) in one or multiple layers.",
"Here, the principles in the laws are usually described with very general and broad terms [5] that makes it hard for engineers to properly implement them in the system design.",
"Also, the availability of so many technologies makes the engineers' job of mapping technologies to principles difficult.",
"Therefore, we need the PbD to take the role as an adaptation layer between laws and technologies to translate legal principles to more engineer-friendly principles that can facilitate the system design.",
"In , Spiekermann and Cranor divided the technologies into two types of approaches to enable privacy in engineering, namely `privacy-by-architecture' and `privacy-by-policy'.",
"The privacy-by-architecture approaches can protect higher-level privacy through technologies enabling data minimization and local processing but is rarely adopted because of the lack of legal enforcement at that time and its conflict with the business interests.",
"In contrast, the privacy-by-policy approaches protect only the bottom-line privacy through technologies supporting the notice and choice principles when the privacy-by-architecture technologies are not implemented.",
"The authors argued that the privacy-by-policy technologies become less important when rigorous minimization has been guaranteed by the privacy-by-architecture technologies.",
"Based on the two approaches, in 2014, Hoepman [13] proposed eight privacy design strategies, including four data-oriented strategies and four process-oriented strategies that roughly match the privacy-by-architecture and privacy-by-policy classification [12], [13]: [innerleftmargin=1.6pt] Data-oriented strategies: Minimize: The amount of processed personal data should be restricted to the minimal amount possible.",
"Hide: Any personal data, and their interrelationships, should be hidden from plain view.",
"Separate: Personal data should be processed in a distributed fashion, in separate compartments whenever possible.",
"Aggregate: Personal data should be processed at the highest level of aggregation and with the least possible detail in which it is (still) useful.",
"Process-oriented strategies: Inform: Data subjects should be adequately informed whenever personal data is processed.",
"Control: Data subjects should be provided agency over the processing of their personal data.",
"Enforce: A privacy policy compatible with legal requirements should be in place and should be enforced.",
"Demonstrate: Be able to demonstrate compliance with the privacy policy and any applicable legal requirements.",
"These strategies proposed by Hoepman not only inherit and develop the two engineering privacy approaches proposed by Spiekermann and Cranor, but also support the legal principles and PbD enforcement of the GDPR [13].",
"As a good combination point between legal principles and privacy enhancing technologies(PETs), these privacy design strategies have been widely accepted by recent work on privacy to fill the gap between legislation and engineering [5], [12], .",
"Therefore, we also adopt the eight privacy design strategies in this paper and study their relevant IoT layers (Section REF ) and enabling PETs (Section to Section ) in the context of IoT privacy.",
"Several existing Internet-of-Things systems are designed using a layered architecture [30], [31], [32], .",
"In an IoT system, data is usually collected by end devices, transmitted through communication networks, processed by local/remote servers and finally provided to various applications.",
"Thus, private data as it flows through multiple layers of the architecture stack, needs privacy protection at all layers.",
"Here, implementing proper privacy design strategies based on the roles of the layers in the lifecycle of the data is important.",
"Otherwise, techniques implemented at a specific layer may become either insufficient (privacy is breached at other layers) or redundant (privacy has been protected by techniques implemented at other layers).",
"In this section, we introduce the reference IoT architecture adopted in this study and present the IoT privacy protection framework that shows how to integrate the privacy design strategies in the layered IoT architecture."
],
[
"Reference IoT architecture",
"In general, the number of layers proposed for the architecture of IoT varies considerably.",
"After reviewing a number of existing IoT architectures, we adopt a four-layer architecture as the reference IoT architecture in this paper, which consists of perception layer, networking layer, middleware layer and application layer.",
"The adoption of the four-layer reference architecture in our study has two key benefits.",
"First, the importance of each layer in the four-layer architecture has been recognized by most existing architectures as the four-layer architecture allows a comprehensive view of privacy in IoT.",
"As shown in Table REF , several existing architectures , [32], [31], , , include all the four layers, either as separate layers or integrated layers.",
"Second, as the four-layer architecture is the most fine-grained model among all the candidate architectures, it allows a detailed and fine-grained analysis of privacy protection at different layers and avoids possible lack of differentiation when the layers are not distinct as in , [32], [31], , .",
"Figure: IoT privacy protection framework , , , As the lowest layer of the architecture (Fig.",
"REF ), perception layer works as the base of entire Internet of Things.",
"It bridges the gap between physical world and digital world by making innumerable physical entities identifiable (e.g., RFIDs ), perceptible (e.g., sensors [1]) and controllable (e.g., actuators ) to enable deep interaction between physical and digital worlds , .",
"The networking layer plays a pivotal role to link the perception layer and middleware layer so that sensed data and corresponding commands can be seamlessly transmitted between the two layers.",
"Unlike the traditional Internet, the vast number of heterogeneous power-limited devices in the perception layer and the various application scenarios in the application layer create a vital need for communication technologies that support low energy consumption, low latency, high data rate and high capacity.",
"Main techniques supporting IoT networking layer include ZigBee , Bluetooth 5 , Wi-Fi HaLow and 5th generation mobile networks .",
"The middleware layer works as the `brain' of IoT to process the numerous data received from lower layers.",
"To cope with the interoperability of the heterogeneous physical devices , , the device abstraction component semantically describes the resources with a consistent language such as the eXtensible Markup Language (XML), Resource Description Framework (RDF) or Web Ontology Language (OWL) , , .",
"Based on that, resources are made discoverable through the resource discovery component by using Semantic Annotations for WSDL and XML Schema (SAWSDL) or simply key words .",
"Then, if needed, multiple resources can be composed through the composition component [1], to enhance their functionality.",
"After that, received data could be stored (storage component) in either cloud or databases and kept available to be queried.",
"Different computational and analytical units can be combined to form the processing component.",
"Here, security of data, namely their confidentiality, integrity, availability, and non-repudiation need to be well protected.",
"If data can make its owner either identified or identifiable, privacy enhancement technologies (PETs) are necessary to protect privacy so that privacy principles required by the laws can be satisfied.",
"As the highest layer of the architecture, the application layer contains various IoT applications that have been widely studied in past literature [1], [4].",
"Depending on the scenarios that the private data is collected, different applications may encounter different privacy issues."
],
[
"IoT privacy protection framework",
"In this section, we study the integration of privacy design strategies in the layered architecture by introducing an IoT privacy protection framework.",
"From the viewpoint of the IoT architecture, data is collected by devices at perception layer and transmitted to middleware layer through networking layer, which makes data move away from the control of data subjects [8], [10].",
"We can apply the notion of personal sphere [10], to assist interpretation.",
"A personal sphere consists of a set of personal devices and a gateway, both trusted by data subjects.",
"In some cases, gateway can be assisted by a more powerful trusted third party (TTP).",
"Data collected by these personal devices has to be passed to the gateway and/or the TTP to be processed before being transmitted to the data controllers that data subjects distrust.",
"Such a personal sphere is quite important to implement the four data-oriented privacy design strategies because it offers a reliable platform to make the raw data minimized, hidden, separated and aggregated.",
"As pointed out by Spiekermann and Cranor , once sensitive information in data has been adequately constrained through PETs such as homomorphic encryption and $k$ -anonymization , the privacy-by-policy approaches, namely the four process-oriented strategies become less important.",
"In IoT, such a personal sphere can be created when data is actively collected, as shown in Fig.",
"REF .",
"For example, smart appliances and home router form an indoor personal sphere while wearable devices and smartphones compose an outdoor personal sphere.",
"The trusted local gateway and/or remote TTP is a critical element in the system, which allows data subjects to launch proper PETs to process data with the four data-oriented strategies.",
"Due to the invisibility of numerous IoT devices at perception layer, personal data may be sensed by untrusted devices outside the personal sphere and the data subjects may be completely unaware of the collection , , .",
"Such a passive collection makes data subjects lose control over their personal data at the earliest stage and provides no trusted platform to implement the four data-oriented privacy design strategies, as shown in Fig.",
"REF .",
"It is therefore the four process-oriented strategies that can play a more important role by promoting the power of data subjects when raw data is obtained by data controllers [13], .",
"Specifically, the inform strategy and control strategy enhance the interaction between data subjects and their data while the enforce strategy and demonstrate strategy force data controllers to comply with privacy policy and further require the compliance to be verifiable.",
"As it is the remote data controllers that should offer proper PETs to support the four process-oriented strategies, these system-level strategies are primarily implemented at middleware layer, with the assistance of other layers.",
"It is worth mentioning that we do not mean active collection only needs data-oriented strategies and passive collection only requires process-oriented strategies.",
"In both cases, all the strategies are required to jointly work to support the legal principles.",
"For example, although the single minimize strategy is hard to be fulfilled in the passive collection, its implementation can be enforced and verified by process-oriented strategies.",
"In the next three sections, we present and evaluate PETs implemented at the perception layer, networking layer and middleware layer respectively."
],
[
"Privacy at perception Layer",
"We evaluate and compare the anonymization-based PETs and perturbation-based PETs that help to implement the Minimize and Aggregate strategies and present the encryption-based PETs that implements the Hide strategy.",
"These PETs are primarily implemented at the perception layer (e.g., local personal gateway, trusted edge server), but they can also be implemented at the middleware layer using a trusted third party (TTP).",
"It is worth noting that the Separate strategy is naturally achieved by local processing in the perception layer."
],
[
"Anonymization and perturbation",
"Both anonymization and perturbation techniques can fulfill the Minimize and Aggregate strategies by reducing the released information and increasing the granularity.",
"The main difference between them is that the results of the anonymization are generalized while the results of the perturbation are with noises.",
"In this section, we evaluate the representative anonymization and perturbation privacy paradigms, namely the $k$ -anonymity and differential privacy respectively, in terms of their practicability in IoT by analyzing their performance under the following IoT specific challenges [5]: [leftmargin=*] Large data volume: The gateways may control thousands of sensors that collect massive data.",
"Streaming data processing: In some real-time IoT applications (e.g., traffic control), data may be seamlessly collected to form streams.",
"Lightweight computation: Since the gateways (e.g., router and phone) are still resource-constrained devices, algorithms are expected to have low complexity.",
"Decentralized computation: In the IoT applications such as smart grid, the personal data may be collected by untrusted entities.",
"Decentralization data aggregation may be employed under such scenarios.",
"Composability: The privacy should still be guaranteed after the data uploaded to the middleware layer is combined with other data.",
"Personalization: For most personal service IoT applications, each customer has different privacy understanding and requirements and there is a natural need for personalized solutions.",
"The traditional privacy-persevering data publication (PPDP) schemes typically involve four entities, namely data subject, data curator, data user and data attacker .",
"The data curator collects data from data subjects, processes the collected data and releases the privacy-preserving dataset to the data users.",
"Usually, the collected data related to a data subject can be classified into four categories, namely explicit identifiers (e.g., names, SSN), quasi-identifiers (e.g., age, gender), sensitive attributes and non-sensitive attributes .",
"In IoT, unlike the traditional Internet that requires all the records to be typed in, the identifiers are usually input through RFID tags and cameras.",
"For example, vehicles can be identified by E-ZPass through RFID and individuals can be identified through RFID-enabled smart cards in shopping malls .",
"The sensitive and non-sensitive attributes are usually collected by sensors.",
"As a candidate PPDP approach, anonymization aims to cut off the connection between each record and its corresponding data subject so that the sensitive attributes cannot be linked with specific individuals .",
"Obviously, the explicit identifiers should be removed before publication for the privacy purpose.",
"However, in 2000, Sweeney found that 87% of US citizens can be uniquely re-identified by combining three quasi-identifiers, namely [ZIP, gender, date of birth] .",
"This linking attack has motivated the researchers to devise stronger anonymization paradigms including $k$ -anonymity , $l$ -diversity and $t$ -closeness , where $k$ -anonymity requires each quasi-identifier group to appear at least $k$ times in the dataset.",
"We next discuss the use of anonymization in the context of IoT: Large volume: The performance of anonymization algorithms may be affected by the dimensions of both rows and columns in the table, so the anonymization scheme is expected to be scalable for datasets with millions of records and multi-dimensional attributes.",
"For the former, spatial indexing has been proved to be a good solution to handle numerous records in a dataset , .",
"One attribute can be efficiently $k$ -anonymized through $B^{+}$ tree indexing and the $R^{+}$ tree indexing can be implemented to effectively generate non-overlapping partitions for tables with 100 million records and nine attributes .",
"However, as analyzed by , the $k$ -anonymity algorithms may work well for tables with a small number of attributes (e.g., 10) but not the ones with a large number of attributes (e.g., 50).",
"The increasing number of attributes makes the number of combinations of dimensions exponentially increased and results in unacceptable information loss.",
"Therefore, how to enhance the utility of $k$ -anonymized datasets with a large number of attributes is still an open issue for future research.",
"An anonymization method for the sparse high-dimensional binary dataset with low information loss was proposed in , but there were no effective schemes for non-binary datasets.",
"Streaming data: There have been several strategies to anonymize data streams , .",
"In CASTLE , a set of clusters of tuples are maintained and each incoming tuple in a stream is grouped into a cluster and generalized to the same level of other tuples in the cluster.",
"Each tuple maintains a delay constraint $\\delta $ and must be sent out before the deadline to make the processing real-time.",
"At the end of $\\delta $ , if the cluster containing that tuple has at least $k$ members, all the tuples within it can be released.",
"Otherwise, a cluster satisfying the $k$ requirement can be generated through a merge and split technique for the tuple and the information loss during the process can be minimized.",
"In SKY , a top-down specialization tree is maintained and each incoming tuple is mapped to one node in the tree based on its attributes.",
"Each node can be a work node or a candidate node depending on whether there have been at least $k$ tuples generalized and output from it.",
"If the incoming node is mapped to a work node, it can be directly generalized and released.",
"Otherwise, it has to wait for other arriving tuples at the node during the time $\\delta $ or be generalized and released through the parent node at the end of $\\delta $ .",
"Lightweight: It has been proved that the optimal $k$ -anonymity aiming to anonymize a table with minimum suppressed cells is NP-hard even when the attribute values are ternary , , .",
"The complexity of approximate algorithms for k-Anonymity has been reduced from $O(k \\log k)$ to $O(k)$ and later to $O(\\log k)$ .",
"Collaboration: Anonymization techniques can be implemented in a distributed manner.",
"That is, multiple entities can locally anonymize their own table to make the integrated table $k$ -anonymous without revealing any additional information during the process.",
"Several SMC protocols have been proposed to solve this problem , .",
"In , a top-down specialization scheme was proposed to support joint anonymization between two parties.",
"Specifically, the two parties first generalize their local table to the root.",
"Then, in each iteration, they find the local specialization maximizing the ratio between information gain and privacy loss (IGPL) over the local table.",
"The party with a higher IGPL wins the game in this iteration, applies its local specialization over its local table and then instructs the grouping in the table of the other party.",
"For the same objective, a scheme based on cryptography was proposed in .",
"Each party locally generalizes the local table and then jointly determines whether the integrated table is $k$ -anonymous.",
"If not, each party then generalizes its local table to the next layer and repeats the two steps.",
"Composability: As shown in [5], the $k$ -anonymity does not offer composability.",
"That is, two $k$ -anonymous datasets cannot guarantee their joint dataset is $k^{\\prime }$ -anonymous ($k^{\\prime }>1$ ).",
"Because of this, the integration of multiple $k$ -anonymous datasets in the middleware layer can be a significant challenge.",
"Personalization: Most anonymization algorithms assume that all the record owners have same privacy preference.",
"Therefore, less-anonymization can put privacy in risk but over-anonymization increases the information loss.",
"To solve this, Xiao et al.",
"organize the sensitive attributes in a top-down taxonomy tree and allow each record owner to indicate a guarding node.",
"That is, the sensitive attribute of a specific record owner should be generalized to at least the guarding node in the taxonomy tree and the adversary has little opportunity to link the record owner with the child nodes of the guarding node that carry fine-grained information.",
"Their algorithm first runs common $k$ -anonymity algorithms over the quasi-identifiers and then generalizes the sensitive attribute through the taxonomy tree based on the claimed guarding nodes.",
"Recently, Xu et al.",
"argued that the generalization of sensitive attributes results in information loss and they allow the record owners to claim the expected value of $k$ .",
"Their algorithm first achieves $k_{min}$ -anonymity over the entire dataset, where $k_{min}$ is the minimum expected $k$ value, namely the most strict privacy requirement.",
"Then, based on the data structure called d-dimensional quasi-attribute generalization lattice, some quasi-attributes can be merged to match the lower values of $k$ expected by some record owners."
],
[
"Differential privacy",
"Differential privacy is a classical privacy definition that makes very conservative assumptions about the adversary's background knowledge and bounds the allowable error in a quantified manner.",
"In general, differential privacy is designed to protect a single individual's privacy by considering adjacent data sets which differ only in one record.",
"Before presenting the formal definition of $\\epsilon $ -differential privacy, we first define the notion of adjacent datasets in the context of differential privacy.",
"A data set $D$ can be considered as a subset of records from the universe $U$ , represented by $D\\in \\mathbb {N}^{|U|}$ , where $\\mathbb {N}$ stands for the non-negative set and $D_i$ is the number of element $i$ in $\\mathbb {N}$ .",
"For example, if $U=\\lbrace a,b,c\\rbrace $ , $D=\\lbrace a,b,c\\rbrace $ can be represented as $\\lbrace 1,1,1\\rbrace $ as it contains each element of $U$ once.",
"Similarly, $D^{\\prime }=\\lbrace a,c\\rbrace $ can be represented as $\\lbrace 1,0,1\\rbrace $ as it does not contain $b$ .",
"Based on this representation, it is appropriate to use $\\mathit {l}_1$ distance (Manhattan distance) to measure the distance between data sets.",
"Definition 1 (Data set Distance) The $l_1$ distance between two data sets $D_1$ and $D_2$ is defined as $||D_1-D_2||_1$ , which is calculated by: $||D_1-D_2||_1=\\sum _{i=1}^{|U|}|D_{1_i}-D_{2_i}|$ The manhattan distance between the datasets leads us the notion of adjacent data sets as follows.",
"Definition 2 (Adjacent Data set) Two data sets $D_1$ , $D_2$ are adjacent data sets of each other if $||D_1-D_2||_1=1$.",
"Based on the notion of adjacent datasets defined above, differential privacy can be defined formally as follows.",
"In general, $\\epsilon $ -differential privacy is designed to protect the privacy between adjacent data sets which differ only in one record.",
"Definition 3 (Differential privacy ) A randomized algorithm $\\mathcal {A}$ guarantees $\\epsilon $ -differential privacy if for all adjacent datasets $D_1$ and $D_2$ differing by at most one record, and for all possible results $\\mathcal {S}\\subseteq Range(\\mathcal {A})$ , $Pr[\\mathcal {A}(D_1)=\\mathcal {S}]\\le e^{\\epsilon }\\times Pr[\\mathcal {A}(D_2)=\\mathcal {S}]$ where the probability space is over the randomness of $\\mathcal {A}$ .",
"Many randomized algorithms have been proposed to guarantee differential privacy, such as the Laplace Mechanism, the Gaussian Mechanism and the Exponential Mechanism.",
"Given a data set $D$ , a function $f$ and the budget $\\epsilon $ , the Laplace Mechanism first calculates the actual $f(D)$ and then perturbs this true answer by adding a noise.",
"The noise is calculated based on a Laplace random variable, with the variance $\\lambda =\\triangle f /\\epsilon $ , where $\\triangle f$ is the $\\mathit {l}_1$ sensitivity.",
"We next analyze differential privacy in terms of the challenges in the context of IoT: Table: Evaluation of kk-anonymization and differential privacy ( Good Not enough Poor)Large volume: The large volume of data is naturally not a problem for differential privacy as the perturbation is usually implemented over the statistical value of the collected data.",
"Streaming data: There have been many works on applying differential privacy over streaming data since 2010 , .",
"The data stream was assumed to be a bitstream, where each bit can be either 1 or 0 representing if an event was happening or not at each timestamp.",
"Mechanisms were proposed to protect either the event-level or user-level differential privacy, depending on whether a single event or all the events related to a single user can be hidden by the injected noise.",
"The early works focused on event-level privacy.",
"In , a counter was set to report the accumulated 1s in the data stream at each timestamp and each update value can be added with a $Lap(\\frac{1}{\\epsilon })$ noise to guarantee the differential privacy.",
"Furthermore, for a sparse stream with few 1s, an update can be set to happen only after the number of 1s has been accumulated over a threshold.",
"Later in , the noise error was reduced through using a binary tree data structure.",
"Specifically, the nodes in the binary tree, except the leaf nodes, represent the sums of sections of consecutive bits in the stream and the Laplace noises were added to these nodes, instead of the leaf nodes.",
"This scheme can effectively reduce the noise error from $O(T)$ to $O((\\log T)^{1.5})$ , where $T$ denotes the number of timestamps, namely the length of the stream.",
"In , the user-level privacy was supported and the noise error in this work was suppressed through sampling.",
"Lightweight: The complexity of differential privacy algorithms is quite variable on a case-by-case manner.",
"If both the sensitivity and budget allocation are fixed, the complexity can be very low, as only one value is required to be sampled from a random distribution with fixed variance.",
"However, in the cases that the sensitivity or budget allocation has to be calculated on the fly, the complexity will increase.",
"Collaboration: Differential privacy for data aggregation is usually guaranteed by noises added through Laplace mechanism .",
"A simple solution for this is to make the data aggregator directly aggregate the raw data received from data subjects and then add noise to it.",
"However, in some scenarios such as smart metering, the aggregator (electricity supplier) may be untrusted and may require the data subjects (smart meters) to locally add noise to perturb its raw data and then send the perturbed data to the aggregator so that the raw data is protected from the aggregator and the aggregated noise automatically satisfies the Laplace Mechanism.",
"This distributed implementation of Laplace Mechanism, also known as Distributed Perturbation Laplace Algorithm (DLPA), has recently received attention from privacy researchers.",
"The base of DLPA is the infinite divisibility feature of Laplace distribution that allows the noise sampled from Laplace distribution (central noise) to be the sum of $n$ other random variables (local noises).",
"The local noise can still follow the Laplace distribution .",
"However, since a Laplace distributed random variable can be simulated by two gamma distributed random variables and four normal distributed random variables, the local noise can also follow the gamma distribution or Gaussian distribution .",
"In , the three schemes were compared and the Laplace distributed local noise was shown to be more efficient in terms of local noise generation.",
"Composability: Differential privacy offers strong composability: Theorem 1 (Composition theorem ) Let $\\mathcal {A}_i$ be $\\epsilon _i$ -differential private algorithms applied to independent datasets $D_i$ for $i \\in [1,k]$ .",
"Then their combination $\\mathcal {A}_{\\sum _{i=1}^k}$ is $max(\\epsilon _i)$ -differential private.",
"In the middleware layer, multiple independent differentially private outputs can be combined and their integration still satisfies differential privacy.",
"Differential privacy also satisfies the post-processing theorem, which further enhances its flexibility in the middleware layer.",
"Theorem 2 (Post-processing ) Let $\\mathcal {A}$ be a $\\epsilon $ -differentially private algorithm and $g$ be an arbitrary function.",
"Then $g(\\mathcal {A})$ is also $\\epsilon $ -differentially private.",
"Personalization: In traditional differential privacy, the parameter $\\epsilon $ is usually set globally for all the record owners.",
"Recently, several works try to make it personalized.",
"In , two solutions were proposed, based on sampling and Exponential Mechanism respectively.",
"The first approach non-uniformly samples the records from the dataset with the inclusion probabilities related to the preferred privacy preferences (values of $\\epsilon $ ).",
"For each record, if the expected $\\epsilon $ is smaller than a threshold $t$ , it may only be selected with a probability related to the $\\epsilon $ .",
"Otherwise, the record will be selected.",
"Then, any $t$ -differentially private mechanism can be applied to the sampled dataset.",
"Their second approach is inspired by the Exponential Mechanism.",
"Unlike the traditional Exponential Mechanism, to take personalization into account, the probability of each possible output values is computed based on the personalized privacy preferences (values of $\\epsilon $ )."
],
[
"Anonymization vs. Differential privacy",
"To sum up, as shown in Table REF , both the techniques have similar features in terms of their support for streaming data, collaboration and personalization.",
"Anonymization techniques are difficult to scale for datasets with many attributes while the complexity of differential privacy algorithms varies case by case.",
"It is the composability feature that makes differential privacy a clear winner.",
"Due to lack of composability, the operability and utility of the data protected by the $k$ -anonymization paradigm are significantly constrained in the middleware layer."
],
[
"Encryption",
"Encryption techniques are not only the fundamental building block of security, but also the foundation of a large number of PETs in privacy.",
"With respect to the eight privacy design strategies, encryption is the most direct supporter of the Hide strategy, which also satisfies the `security safeguards' requirement of privacy laws.",
"Therefore, in terms of IoT privacy, the role of encryption is twofold.",
"On one hand, the commonly used cryptographic primitives, such as AES and RSA , protect the security of every IoT layer so that the adversaries are prevented from easily compromising the confidentiality and integrity of data in IoT devices.",
"From this perspective, the personal data is confined to a safe zone without being disclosed to unknown parties, thus also protecting the privacy of the data subject as the control over the data is enhanced.",
"On the other hand, in IoT, the middleware may not be trusted or trustworthy but it is an indispensable stakeholder in most IoT applications.",
"Hence, PETs such as homomorphic encryption , searchable encryption and SMC are required to make the middleware work without accessing the private information.",
"Here, lightweight cryptography that can support encryption over devices with low capacity becomes a critical element in protecting IoT privacy.",
"In this section, to comprehensively review the current state of work in this area, we first go through the real capacity of various types of IoT devices in the perception layer and evaluate the implementation of commonly used cryptographic primitives over them to see when and where lightweight cryptography is required.",
"Then, we review the candidate lightweight solutions in each area of cryptography and present the NIST General Design Considerations .",
"Finally, we discuss the PETs aiming to blind the middleware and their performance over IoT devices.",
"The capacity of IoT devices: The types of IoT edge devices in the perception layer range from resource-rich devices such as computers and smartphones to resource-constrained devices such as embedded systems, RFID and sensors.",
"For the resource-rich devices, the traditional cryptographic primitives work well for the encryption tasks.",
"Thus, the lightweight cryptography techniques are mainly required by the resource-constrained devices that can not support traditional cryptographic primitives.",
"This also requires the resource-rich devices in the middleware layer to adopt them in order to decrypt the data encrypted using lightweight cryptography techniques.",
"Most IoT embedded systems and intelligent systems are enabled by the 8-bit, 16-bit or 32-bit microcontrollers (MCUs) with highly restricted random-access memory (RAM) as low as 64 bytes RAM (e.g., NXP RS08KA, $0.399) .",
"The RFID and sensor devices are usually more cost-sensitive and they employ the use of application specific integrated circuit (ASIC) .",
"Therefore, in hardware, the price of these devices is proportional to the area of ASIC in silicon, measured by the gate equivalents (GE), namely the ratio between the area of ASIC and the area of a two-input NAND gate .",
"The implementation of lightweight cryptography techniques over such devices has to meet several stringent conditions, including under 2000 GE to achieve low-cost, under 50 cycles for obtaining low-latency and less than $10\\frac{\\mu W}{MHz}$ average power usage for meeting low-energy requirements .",
"Traditional cryptographic primitives over constrained devices: Most traditional commonly-used cryptographic primitives face severe challenges in the constrained environment.",
"The AES-128 may be the most suitable lightweight block cipher because of its low number of rounds and small key size.",
"In [2], AES-128 was tested over several MCUs and smart cards and achieved $1.58ms$ execution time and 0.6kB RAM consumption over the MSP microcontrollers.",
"The results show that AES works well for most MCUs, but not the ones with ultra-low RAM (e.g., NXP RS08KA).",
"In terms of hash functions, the SHA-2 is acceptable to implement the cryptographic schemes requiring a few hash functions over the MSP microcontrollers with tens to hundreds of milliseconds execution time and 0.1kB RAM.",
"However, as illustrated by Ideguchi et al.",
", the SHA-3 candidates cannot be supported by the low-cost 8-bit microcontrollers with 64 byte RAM.",
"In the NIST competition, the lowest number of GE required by the SHA-3 is still 9200 .",
"Also, both the RSA for asymmetric encryption and elliptic curve point multiplication for ECDH and ECDSA schemes were found to be too high-cost for even the MSP microcontrollers [2].",
"Attribute-Based Encryption in IoT: Attribute-Based Encryption (ABE) is a promising mechanism to implement fine-grained access control over encrypted data.",
"With ABE, an access policy can be enforced during data encryption, which only allows authorized users with the desired attributes (e.g., age, gender) to decrypt the data.",
"Depending on whether the access policy is associated with the key or ciphertext, Key-Policy ABE (KP-ABE) and Ciphertext-Policy ABE (CP-ABE) were proposed, respectively.",
"Although ABE looks like the desired approach to secure data communication and storage in IoT with flexible access control, its implementation in IoT may encounter three main challenges.",
"First, current IoT applications only need IoT devices to encrypt data using public keys and hence, key management may not be a significant issue.",
"However, future autonomous IoT devices would require direct device-to-device communication with each other requiring different secret keys from the attribute authority (AA) based on their attributes to decrypt data.",
"In such cases, the AA may become a bottleneck for issuing secret keys and we will need techniques to distribute secret keys in a scalable and efficient manner.",
"Potential solutions for this include Hierarchical ABE (HABE) and decentralizing multi-authority ABE (DMA-ABE) .",
"In short, the HABE scheme manages the workflow in a hierarchical structure with each domain authority serving a set of domain users, whereas the DMA-ABE scheme decentralizes the single centralized AA to multiple AAs.",
"Second, when an access policy needs to be updated, due to the limited storage space of IoT devices, the re-encryption of the data based on the new policy is hard to be operated locally.",
"A solution for this has been proposed by Huang et al.",
", which designs a set of policy updating algorithms that allow the re-encryption to be operated at untrusted remote servers without breaching the privacy of the encrypted data.",
"The third and perhaps the greatest challenge is the issue of limited resources in IoT devices.",
"It has been demonstrated that most classical CP-ABE schemes can hardly fit the smartphone devices and IoT devices such as Intel Edison board and Raspberry Pi , , .",
"To solve this, the most common approach is to outsource the most consuming operations of ABE to powerful nodes in the network , .",
"In case that such powerful nodes are not available, Yao et al.",
"proposed a lightweight no-pairing ECC-based ABE scheme to reduce the power consumption.",
"Lightweight cryptographic candidates: As can be seen, most traditional cryptographic primitives are not applicable over resource-constrained devices.",
"Hence, IoT privacy creates a critical need for lightweight cryptographic solutions.",
"A non-exhaustive list of lightweight cryptographic candidates can be found in .",
"The design of lightweight block ciphers, based on the classification in , consists of the Substitution-Permutation Networks (SPN) family and Feistel Networks family.",
"The SPN-based schemes usually apply the S-boxes and P-boxes to perform confusion and diffusion respectively and can be roughly divided into three categories, namely the AES-like schemes (e.g., KLEIN ), schemes with Bit-Sliced S-Boxes (e.g., PRIDE ) and other schemes (e.g., PRESENT ).",
"The schemes based on the Feistel Networks split the input block into two sides, permute one with the other and then swap them.",
"They can be designed to only use modular Addition, Rotation and XOR (e.g., RC5 ) or not (e.g., DESLX ).",
"These lightweight schemes usually apply smaller block sizes lower than 128 bits as AES or simpler rounds without S-boxes or with smaller S-boxes to reduce the resource requirements .",
"The lightweight hash functions are designed based on either the Merkle-Damgård or P-Sponge and T-Sponge.",
"The existing lightweight hash functions such as PHOTON and SPONGENT have already been able to achieve under 2000 GE with $0.18 \\mu m$ technology for 128 digest size.",
"In terms of lightweight stream ciphers, the Grain , MICKEY and Trivium have stood out since 2008.",
"In addition, recently, the NIST published its report on lightweight cryptography and recommended the General Design Considerations for the future design: [innerleftmargin=1.6pt] Security strength: The security strength should be at least 112 bits.",
"Flexibility: Algorithms should be executable over an assortment of platforms and should be configurable on a single platform.",
"Low overhead for multiple functions: Multiple functions (such as encryption and decryption) should share the same logic.",
"Ciphertext expansion: The size of the ciphertext should not be significantly longer than the plaintext.",
"Side channel and fault attacks: Algorithms should be resilient to the side channel and fault attacks.",
"Limits on the number of plaintext-ciphertext pairs: The number of plaintext/ciphertext pairs processed should be limited by an upper bound.",
"Related-key attacks: Algorithms should be resilient to the related-key attacks, where the relationship between multiple unknown keys is used by the adversary.",
"Middleware-blinding PETs in IoT: The homomorphic encryption , as the most fundamental building block of the Middleware-blinding PETs, is a suite of cryptographic techniques that enable the decrypted results of computations over the ciphertext to match the results of computation over the plaintext.",
"Its characteristics make it the best solution for outsourcing private data to untrusted parties to get their service without compromising privacy, which refers to blinding the middleware in IoT domain.",
"Homomorphic encryption was proposed as early as 1978 but it was not until the year 2009 that the first plausible solution of the fully homomorphic encryption was proposed by Craig Gentry .",
"Unlike the partially homomorphic cryptosystems such as the ones based on Paillier cryptosystem that support a small number of operations, the fully homomorphic encryption can enable both addition and multiplication operations over ciphertexts and therefore arbitrary computations.",
"However, although the efficiency of the fully homomorphic encryption has been significantly improved, it is still too time-consuming for most applications.",
"Therefore, in many cases, the partially homomorphic cryptosystems are still the preferred solution.",
"IoT can benefit a lot from the homomorphic encryption as well as the secure multi-party computation (SMC) schemes in the context of service discovery, data retrieval, data sharing and data outsourcing.",
"Although most of the applications interact closely with the middleware layer, the encryption of private data is usually implemented in the perception layer.",
"In [2], the Paillier's partially homomorphic scheme was tested and the results showed that the scheme is still heavy for the resource-constrained devices."
],
[
"Privacy at Networking Layer",
"In this section, we discuss the secure communication and anonymous communication in the networking layer that support the Hide and Minimize strategies respectively."
],
[
"Secure communication",
"In the traditional Internet with TCP/IP stack, the communication is usually secured by either IPsec in the network layer or TLS in the transport layer.",
"In the context of IoT, due to numerous devices with constrained power, the protocol stack has to be adapted to support the transmission of IPv6 over IEEE 802.15.4 PHY and MAC, which is enabled by the adoption of 6LoWPAN as an adaptation layer between them.",
"A reference IoT protocol stack is shown in Fig.",
"REF , which is mainly based on the IETF LLN protocol stack .",
"Above the network layer, TCP and UDP in the transport layer support different IoT application layer protocols, such as Message Queue Telemetry Transport (MQTT) and Constrained Application Protocol (CoAP) , respectively.",
"In terms of security, as pointed out by RFC 4944 and other literature , , the AES-based security modes provided by the IEEE 802.15.4 that can support confidentiality, data authenticity and integrity, have some shortcomings.",
"That is, the IEEE 802.15.4 only provides hop-by-hop security that requires all nodes in the path to be trusted without host authentication and key management.",
"It may be acceptable for isolated WSNs, but not for the Internet-integrated WSNs when the messages have to travel over an IP network.",
"Therefore, security mechanisms are required to be implemented in the higher layers to provide end-to-end security.",
"Like the traditional Internet, the potential options include the IPsec in the network layer and the TLS/DTLS in the transport layer, where TLS and Datagram TLS (DTLS) support TCP and UDP, respectively.",
"The TLS/DTLS solution is the default security option of most common IoT application protocols.",
"For example, the MQTT Version 3.1.1 claimed that it should be the implementer's responsibility to handle security issues and then recommended the TLS and registered TSP port 8883 for MQTT TLS communication.",
"In contrast, the CoAP is secured by DTLS as it transmits messages over the unreliable but simpler UDP .",
"The various security modes allow the devices to have either a list of pre-shared symmetric keys or a pair of asymmetric keys with or without an X.509 certificate.",
"In addition to DTLS, the CoRE working group also proposed a draft for using CoAP with IPsec .",
"The adoption of IPsec can make use of the built-in link-layer encryption hardware and perform transparently towards the application layer.",
"However, due to its well-known issues with using the firewalls and Network Address Translation (NAT), the IPsec is not always available.",
"In addition, the configuration and management of IPsec in IoT is very difficult due to the huge number of heterogeneous devices .",
"Figure: Reference IoT protocol stack , , , Based on the IETF protocol stack, there are some other IoT protocol stacks proposed by other standardization bodies and industry alliances.",
"We briefly review some representatives among them.",
"The Thread stack adopts 6LoWPAN to support IPv6 and leverages DTLS to secure UDP.",
"The Thread stack has been widely adopted for connecting home devices and applications.",
"The IPSO Alliance argued that using standardized protocols (e.g., IETF stack) may fail to ensure interoperability at the application layer.",
"They proposed the IPSO Smart Objects, an object model that provides high-level interoperability between applications and devices.",
"The core idea is to leverage the open Mobile Alliance Lightweight Specification (OMA LWM2M) on top of CoAP to enable device management operations such as bootstrapping and firmware updates.",
"Again, DTLS is in charge of security.",
"The Industrial Internet of Things (IIoT) was proposed by the Industrial Internet Consortium (IIC), with the aim to connect industrial objects to enterprise systems and business processes .",
"Its reference architecture adopts DDSI-RTPS /CoAP for UDP and MQTT/HTTP for TCP, respectively.",
"Therefore, its security requires both TLS and DTLS."
],
[
"Anonymous communication",
"The end-to-end security provided by either IPsec or TLS/DTLS can only hide the content of the messages, but not the meta-data, such as the identity (e.g., IP) of the two sides or the time, frequency and amount of the communications.",
"Therefore, PETs enabling anonymous communication are required to handle the privacy problem due to the disclosure of meta-data, especially the identity of the initiator of the communication.",
"For example, when health data or smart home data has to be sent to the middleware layer to get some service, it is better to make the data subject anonymous so that the personal health condition or living habits cannot be easily linked to the data subject.",
"Such an objective can be achieved through the implementation of the anonymization and perturbation mechanisms in the perception layer, but the anonymous communication makes it also possible to handle in the networking layer.",
"Figure: Tor over IoT , The communication can be anonymized through the Proxy, the Virtual Private Network (VPN) and the onion router (Tor) [12], .",
"Among them, Tor is considered an important anonymous communication PET because of its strong attack resilience .",
"We show a potential Tor-based anonymous communication framework in Fig.",
"REF .",
"An IoT node, either a device or a gateway, wants to communicate with the middleware to get service without revealing its identity (e.g., IP address).",
"For this purpose, instead of directly communicating with the middleware, the IoT node can first connect with the Tor network to anonymize itself.",
"The Tor network is a distributed network with thousands of volunteers all around the world performing as the onion routers .",
"Its scale, as monitored by the torstatus website, is around 7000-8000 nodes in 2018 .",
"To process the request of the IoT node, Tor will build a path (circuit) formed by one entry node, one or multiple intermediate nodes and one exit node.",
"The raw package sent by the IoT node is then encrypted by the public keys of the nodes on the path one by one, from the entry node to the exit node, forming a layered structure, just like an onion.",
"Each node on the path, on receiving a package from its predecessor, should decrypt one layer of the package with its private key, learn the IP of its successor and transmit the decrypted package to the successor.",
"Each node on the path only knows the IP of its predecessor and successor and hence, the IP address of the IoT node is only revealed to the entry node and the middleware only knows the IP address of the exit node.",
"The implementation of Tor over smart home was evaluated in in which, Tail, a subproject of Tor, was set up to be the central smart home gateway passed by all the outgoing data packages generated by the appliances.",
"The results showed that Tor works well for multimedia transmission (smart TV) but not the voice-over-Internet protocol application such as Skype, due to the short time-to-live duration of UDP packets.",
"This work demonstrated the practicability of Tor in IoT.",
"However, several key challenges still need to be addressed.",
"First, the access point to the Tor network should be designed to make it available to the capacity-constrained IoT devices.",
"Second, as Tor does not support UDP, for the devices unable to encapsulate the UDP into TCP packets, mechanisms are required to enable UDP transmission over Tor.",
"Third, the affordability of the Tor network in terms of the massive data generated by the billions of IoT nodes should be evaluated.",
"Figure: Notification in the WSN"
],
[
"Privacy at Middleware Layer",
"In this section, we present the interaction-enhancing PETs fulfilling Inform and Control strategies and discuss the compliance-enhancing PETs enabling Enforce and Demonstrate strategies.",
"We evaluate existing middlewares on their support for these four process-oriented strategies."
],
[
"Interaction-enhancing techniques",
"The main objective of interaction-enhancing techniques is to break the isolation between data subjects and their data so that data subjects can track the status of their data (Inform strategy) and also remotely control their data (Control strategy).",
"The GDPR [28] requires data subjects to get notification both before and after the data collection.",
"Before the data collection, in addition to the data collection notification itself, data subjects should also be notified more information such as identity and contact details of data collector and purpose of the processing (Article 13).",
"After the data collection, Inform strategy can be combined with Control strategy to assist data subjects to safeguard their rights, such as the right of access (Article 14), right to rectification or erasure of personal data and restriction of processing (Article 15) and right to know the personal data breach (Article 30).",
"In the traditional Internet, Inform strategy is easy to be implemented because it is the data subjects who actively determine whether to click the link to enter a website.",
"The PETs such as the P3P aim to assist the end users with little privacy knowledge or with no patience to quickly understand the privacy condition of the visiting websites in an automatic and usable manner .",
"Specifically, the privacy policies provided by most websites are both long and obscure with dense legalese, which makes the visitors hard to understand how their private data such as browsing history is handled.",
"The P3P solved this problem by providing both a computer-readable format for websites to standardize the privacy policies and a protocol for the web browsers to understand the privacy policies and automatically process them based on the pre-determined privacy preference.",
"Unfortunately, things become harder in IoT.",
"Unlike the traditional Internet where the end users can easily interact with the websites through static web browsers, it is essential to figure out how to effectively build the communication between data subjects and data controllers in dynamic IoT scenarios to enable Inform and Control strategies.",
"To build such a communication for active collection is not hard.",
"An example is the privacy coach , a phone application to help end users decide whether to buy products with RFID tags by actively reading RFID tags to learn corresponding privacy policies.",
"However, to do the same thing for passive collection is more challenging.",
"Consider the example in Fig.",
"REF where an individual quickly passes a WSN area, the gateway has to quickly and actively get connected with the personal phone to notify the data collection, get the consent and leave information for future notifications.",
"All these should be completed within a short period of time before the communication is disconnected.",
"Figure: Central control platformFor Control strategy, the main challenge is not how to technically implement the actions such as revision and deletion but how to design a centralized platform to simplify the control of data subjects when there are multiple data controllers.",
"In active collection, each data subject can actively upload private data for different data controllers to a common personal space in cloud to simplify the tracking and control of their data , .",
"In passive collection, as personal data of a data subject may be passively uploaded by data controllers to different storage places, a centralized user control platform is required, such as the one in Fig.",
"REF .",
"A data subject, after login, should be able to check the list of his/her personal data collected by different data controllers.",
"Each data controller, after collecting the data, should report the collection to this central platform, link its database to the platform and provide APIs to allow the authorized data subjects to control their data.",
"The format of a report should contain identity of the data collector, description of collection purpose, collected data and a list of possible actions that can be made by data subjects.",
"Then, data subjects can remotely revise or delete their data."
],
[
"Compliance-enhancing techniques",
"The goal of compliance-enhancing techniques is to enforce and demonstrate compliance with the privacy policy.",
"The Enforce and Demonstrate strategies are highly related.",
"First, the Enforce strategy requires a privacy policy compatible with laws to be in place and a set of proper PETs to technically enforce it in engineering so that a data controller has the ability to comply with privacy laws.",
"We require the Demonstrate strategy here to enforce it so that the data controllers can technically prove their compliance.",
"As the first step, a privacy policy should be in place to guide the processing of private data.",
"By considering personalization, this privacy policy can be replaced by a privacy preference in many cases to also reflect personal privacy demands.",
"Such a privacy preference should be in place during the entire lifecycle of the personal data .",
"That is, even if the personal data is disseminated from the initial data controller to the others, the privacy preference of the original data subject should be simultaneously transmitted along with the data.",
"In other words, the privacy preference should be stuck to the corresponding data in the complicated middleware layer, which can be supported by the PET.",
"Such a scheme was named sticky policy .",
"The privacy model of sticky policy requires data to be first encrypted by data subjects.",
"Then, the encrypted data and the sticky policy are sent to the data controller while the decryption key is sent to a Trusted Third Party (TTP).",
"Any party who wants to decrypt the data, including the initial data controller and later ones, should submit a request to the TTP with the sticky policy and credentials.",
"The TTP will then check the integrity and trustworthiness of them to decide whether the decrypted key can be given.",
"During the whole process, data subjects can join or check the decision making through the TTP.",
"To sum up, the privacy preference must also flow along with the data and its existence should be enforced and monitored by the TTP.",
"A similar approach was proposed in , where the data is encrypted by the data subjects at their gateways and attached with semantic data handling annotations as the privacy preference.",
"After the privacy preference is in place, the PETs that can fulfill the privacy preference are required.",
"To make it automatic, the sticky policy is recommended to be used as machine-readable semantic annotations that can be parsed by the middleware to configure the corresponding PETs.",
"The implementation of the policy can be supported by access control mechanisms [5].",
"In terms of purpose limitation, the mechanism proposed in require the data requesters to declare their purpose of usage and the range of required data so that the current data controller is able to compare the declaration with the sticky annotations to make decisions.",
"Another choice is the Hippocratic database .",
"As a database designed to fulfill the Fair Information Practices [17] and especially the purpose limitation, the Hippocratic database requires the queries to be tagged with a purpose and only access the columns and tuples matching the purpose.",
"Finally, the most common solution to verify the compliance is the audit mechanism.",
"That is, any interaction with private data should either be pre-checked or logged for later inspection.",
"An example of pre-checking is the sticky policy , where data requesters must first submit the sticky policy and credentials to the TTP and accept the inspection of TTP about their environment.",
"An audit approach using the log was proposed in , where personal data is encrypted in personal sphere by a gateway and then stored in a cloud platform.",
"The cloud platform offers a database abstraction layer that can log every access of a data controller to the data with detailed information such as the access time and purpose.",
"Next, the data subject should verify that the usage of the data complies with the privacy preference.",
"However, even with the log information and available source code of the service, data subjects may not have the expertise to audit it.",
"Therefore, a trusted auditor is deployed to verify the data usage in the service implementation by checking the source code."
],
[
"Evaluation of existing middlewares",
"Currently, only a few middlewares support privacy protection.",
"Among 61 middlewares reviewed by a recent survey , only eight of them were labeled to support privacy.",
"We evaluate their performance over the inform, control, enforce and demonstrate strategies.",
"As can be seen in the first part of Table REF , among the eight middlewares, only the IrisNet mentioned the importance of the enforcement of privacy policies.",
"In the second part of Table REF , we present middlewares reviewed by another recent survey .",
"In Xively, permission is not required for data collection and sharing, but users are allowed to review, update or change their data in the account, which satisfies the control strategy.",
"Similar to Xively, the Paraimpu middleware tries to support user privacy according to the privacy laws.",
"Both Xively and Paraimpu have the privacy policy, but the details on the enforcement are not clearly presented.",
"The Webinos middleware can meet the three strategies in terms of protecting user privacy.",
"In Webinos, applications require permission to access the private data.",
"The private data is processed and stored in a local Personal Zone Proxy (PZP) and a remote Personal Zone Hub (PZH) so the users can fully control their data.",
"Besides, through the eXtensible Access Control Markup Language (XACML) and the Webinos policy enforcement framework, users can define fine-grained access control policies that will be enforced by the PZP and PZH to mediate every access to a Webinos API.",
"Additionally, we have reviewed some other IoT middlewares and software frameworks regarding their adoption of the inform, control, enforce and demonstrate strategies.",
"The results are shown as the third part of Table REF .",
"The OpenHAB is a software framework designed for managing home automation systems.",
"It makes all the devices and data stay in the local network and provides a single channel to enter the local network.",
"It allows users to decide automation rules and has the ability to enforce the rules.",
"It provides logging information for user-defined rules.",
"Therefore, it satisfies all the four strategies.",
"The AllJoyn is a software framework aimed to create dynamic proximal networks by enhancing interoperability among devices and applications across manufacturers.",
"Such proximal networks can make private data stay inside the local network and therefore has the potential to satisfy all the four strategies.",
"The middleware based on NetwOrked Smart objects (NOS) extracts privacy information from incoming data as part of security metadata at the Analysis layer, which is then used to annotate the data at the Data Annotation layer.",
"It requires users to actively register and input private information to annotate their data.",
"Further, the privacy protection can be enforced by the Integration layer and thus, the NOS-based middleware satisfies the three strategies.",
"In summary, we found that not all middlewares emphasize privacy protection.",
"Although the recent middlewares have better protection than the previous ones, there are still privacy requirements that may be implemented at the middleware layer through PbD privacy strategies."
],
[
"Privacy at Application Layer",
"The unprecedented proximity between physical and digital worlds facilitated by IoT creates a huge number of applications [1], [4].",
"Different IoT applications may face different kinds of privacy risks as data collected in IoT applications may contain sensitive information related to the users.",
"For instance, in smart home applications, religious beliefs of users may be inferred from smart refrigerators and similarly, daily schedules of users may be inferred from smart lamps.",
"In automobile driving applications, dozens of internal sensors monitor data related to vehicle speed and seatbelt usage that can be used by insurance companies to determine insurance premium for the users.",
"In healthcare and fitness applications, wearable devices may collect data that may reflect users' health information .",
"Similarly in smart meters, by applying energy disaggregation over the power usage data, it may be possible to learn when and how a home appliance was used by the residents .",
"In general, many of the application-level privacy risks can be handled at lower layers of the IoT architecture stack using PETs presented in Section to Section .",
"For example, software frameworks such as OpenHAB can make smart home a personal sphere so that data can be securely stored locally and any interaction with the data can be examined and logged.",
"As another example, differential privacy mechanisms , can be applied to perturb the smart meter data , , where the injected noises can be added by an in-home device.",
"However, it is important to ensure that the PETs employed to achieve the privacy goals does not adversely affect the utility of the target IoT application.",
"For example, perturbation PETs such as differential privacy when applied to healthcare data that require high accuracy to be retained, the resulting perturbed data may not retain the desirable clinical efficacy and as a result, it may lead to lower application utility .",
"In such cases, a cross-layer understanding of the impact of the employed PETs on the application-level utility is critical in determining the privacy-utility tradeoffs while designing the applications."
],
[
"Related work",
"Research on privacy in IoT has become an important topic in the recent years.",
"A number of surveys have summarized various challenges and potential solutions for privacy in IoT.",
"Roman et al.",
"analyzed the features and challenges of security and privacy in distributed Internet of Things.",
"The authors mentioned that data management and privacy can get immediate benefit from distributed IoTs as every entity in distributed IoTs has more control over the data it generates and processes.",
"In , the authors discussed several types of PETs and focused on building a heterogeneous and differentiated legal framework that can handle the features of IoT including globality, verticality, ubiquity and technicity.",
"Fink et al.",
"reviewed the challenges of privacy in IoT from both technical and legal standpoints.",
"Ziegeldorf et al.",
"[10] discussed the threats and challenges of privacy in IoT by first introducing the privacy definitions, reference models and legislation and reviewed the evolution of techniques and features for IoT.",
"In both and , security risks, challenges and promising techniques were presented in a layered IoT architecture but the discussion on privacy protection is limited to the techniques related to security problems.",
"Although most of the existing surveys review privacy in IoT from either a technical standpoint or a legal standpoint, to the best of our knowledge, none of the existing surveys analyzed the IoT privacy problem through a systematic fine-grained analysis of the privacy principles and techniques implemented at different layers of the IoT architecture stack.",
"In this paper, we study the privacy protection problem in IoT through a comprehensive review of the state-of-the-art by jointly considering three key dimensions, namely the state-of-the-art principles of privacy laws, architecture of the IoT system and representative privacy enhancing technologies (PETs).",
"Our work differentiates itself by its unique analysis of how legal principles can be supported through a careful implementation of various privacy enhancing technologies (PETs) at various layers of a layered IoT architecture model to meet the privacy requirements of the individuals interacting with the IoT systems."
],
[
"Conclusion",
"The fast proliferation of low-cost smart sensing devices and the widespread deployment of high-speed wireless networks have resulted in the rapid emergence of the Internet-of-things.",
"In this paper, we study the privacy protection problem in IoT through a comprehensive review of the state-of-the-art by jointly considering three key dimensions, namely the architecture of the IoT system, state-of-the-art principles of privacy laws and representative privacy enhancing technologies (PETs).",
"We analyze, evaluate and compare various PETs that can be deployed at different layers of a layered IoT architecture to meet the privacy requirements of the individuals interacting with the IoT systems.",
"Our analysis has shown that while many existing PETs (e.g., differential privacy, Tor) demonstrate a great potential for use in the IoT, the adoption of these techniques requires a careful consideration of the unique features associated with the IoT, including the use of heterogeneous power-limited devices and the massive need for streaming data flow.",
"We expect this study to provide a broader understanding of the state-of-the-art principles in privacy legislation associated with the design of relevant privacy enhancing technologies (PETs) and how privacy legislation maps to privacy principles which in turn drives the design of necessary privacy enhancing technologies to be employed in the IoT architecture stack.",
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] | 1808.08443 |
[
[
"Data Motifs: A Lens Towards Fully Understanding Big Data and AI\n Workloads"
],
[
"Abstract The complexity and diversity of big data and AI workloads make understanding them difficult and challenging.",
"This paper proposes a new approach to modelling and characterizing big data and AI workloads.",
"We consider each big data and AI workload as a pipeline of one or more classes of units of computation performed on different initial or intermediate data inputs.",
"Each class of unit of computation captures the common requirements while being reasonably divorced from individual implementations, and hence we call it a data motif.",
"For the first time, among a wide variety of big data and AI workloads, we identify eight data motifs that take up most of the run time of those workloads, including Matrix, Sampling, Logic, Transform, Set, Graph, Sort and Statistic.",
"We implement the eight data motifs on different software stacks as the micro benchmarks of an open-source big data and AI benchmark suite ---BigDataBench 4.0 (publicly available from http://prof.ict.ac.cn/BigDataBench), and perform comprehensive characterization of those data motifs from perspective of data sizes, types, sources, and patterns as a lens towards fully understanding big data and AI workloads.",
"We believe the eight data motifs are promising abstractions and tools for not only big data and AI benchmarking, but also domain-specific hardware and software co-design."
],
[
"Introduction",
"The complexity and diversity of big data and AI workloads make understanding them difficult and challenging.",
"First, modern big data and AI workloads expand and change very fast, and it is impossible to create a new benchmark or proxy for every possible workload.",
"Second, several fundamental changes, i.e., end of Dennard scaling, ending of Moore's Law, Amdahl's Law and its implications for ending \"Easy\" multicore era, indicate only hardware-centric path left is Domain-specific Architectures [24].",
"To achieve higher efficiency, we need tailor the architecture to characteristics of a domain of applications [24].",
"However, the first step is to understand Big Data and AI workloads.",
"Third, whatever early in the architecture design process or later in the system evaluation, it is time-consuming to run a comprehensive benchmark suite.",
"The complex software stacks of the modern workloads aggravate this issue.",
"The modern big data or AI benchmark suites [41], [18] are too huge to run on simulators and hence challenge time-constrained simulation and even make it impossible.",
"Fourth, too complex workloads raise challenges in both reproducibility and interpretability of performance data in benchmarking systems.",
"Identifying abstractions of time-consuming units of computation is an important step toward fully understanding complex workloads.",
"Much previous work [11], [10], [12], [6], [37] has illustrated the importance of abstracting workloads in corresponding domains.",
"TPC-C [10] is a successful benchmark built on the basis of frequently-appearing operations in the OLTP domain.",
"HPCC [33] adopts a similar method to design a benchmark suite for high performance computing.",
"National Research Council proposes seven major tasks in massive data analysis [14], while they are macroscopical definition of problems from the perspective of mathematics.",
"Unfortunately, to the best of our knowledge, none of previous work has identified time-consuming classes of unit of computation in big data and AI workloads.",
"Also, identifying abstractions of time-consuming units of computation is an important step toward domain-specific hardware and software co-design.",
"Straightforwardly, we can tailor the architecture to characteristics of an application, several applications, or even a domain of applications [24].",
"The past witnesses the success of neural network processors for machine learning [28], [9], GPUs for graphics, virtual reality [35], and programmable network switches and interfaces [24].",
"Moreover, if we can identify abstractions of time-consuming units of computation in Big Data and AI workloads and design domain-specific hardware and software system for them, our target will be much general-purpose.",
"Meanwhile, optimizing most time-consuming units of computation other than many algorithms case by case on different hardware or software systems will be much efficient.",
"In this paper, we propose a new approach to modelling and characterizing big data and AI workloads.",
"We consider each big data and AI workload as a pipeline of one or more classes of unit of computation performed on different initial or intermediate data inputs, each of which captures the common requirements while being reasonably divorced from individual implementations [6].",
"We call this abstraction a data motif.",
"Significantly different from the traditional kernels, a data motif's behaviors are affected by the sizes, patterns, types, and sources of different data inputs; Moreover, it reflects not only computation patterns, memory access patterns, but also disk and network I/O patterns.",
"After thoroughly analyzing a majority of workloads in five typical big data application domains (search engine, social network, e-commerce, multimedia and bioinformatics), we identify eight data motifs that take up most of run time, including Matrix, Sampling, Logic, Transform, Set, Graph, Sort and Statistic.",
"We found the combinations of one or more data motifs with different weights in terms of runtime can describe most of big data and AI workloads we investigated [19].",
"Considering various data inputs—text, sequence, graph, matrix and image data—with different data types and distributions, we implement eight data motifs on different software stacks, including Hadoop [1], Spark [46], TensorFlow [5] and POSIX-thread (Pthread) [8].",
"For big data, the implemented data motifs include sort (Sort), wordcount (Statistics), grep (Set), MD5 hash (Logic), matrix multiplication (Matrix), random sampling (Sampling), graph traversal (Graph) and FFT transformation (Transform), while for AI, we implement 2-dimensional convolution (Transform), max pooling (Sampling), average pooling (Sampling), ReLU activation (Logic), sigmoid activation (Matrix), tanh activation (Matrix), fully connected (Matrix), and element-wise multiplication (Matrix), which are frequently-used computation in neural network modelling.",
"We release the implemented data motifs as the micro benchmarks of an open-source big data and AI benchmark suite — BigDataBench.",
"In the rest of paper, we use the big data motifs to indicate the motif implementations for big data, and use the AI motifs to indicate the motif implementations for AI.",
"Just like relation algebra in database, the data motifs are promising fundamental concepts and tools for benchmarking, designing, measuring, and optimizing big data and AI systems.",
"Based on the data motifs, we build the fourth version of BigDataBench [20], including micro benchmarks, each of which is a data motif, and component benchmarks, each of which is a combination of several data motifs, and end-to-end application benchmarks, each of which is a combination of component benchmarks.",
"Also, we build the proxy benchmarks [19] for big data and AI workloads, which has a speedup up to 1000 times in terms of runtime and a micro-architectural data accuracy of more than 90%.",
"In this paper, as the first step, we call attention to performing comprehensive characterization of those data motifs from perspective of data sizes, types, sources, and patterns as a lens towards fully understanding big data and AI workloads.",
"On a typical state-of-practice processor: Intel Xeon E5-2620 V3, we comprehensively characterize all data motif implementations and identify their bottlenecks.",
"Our contributions are five-fold as follows: We identify eight data motifs through profiling a wide variety of big data and AI workloads.",
"We provide diverse data motif implementations on the software stacks of Hadoop, Spark, TensorFlow, Pthread.",
"From the system and micro-architecture perspectives, we comprehensively characterize the behaviors of data motifs and identify their bottlenecks.",
"We find that these data motifs cover a wide variety of performance space, from the perspectives of system and micro-architecture behaviors.",
"Moreover, the behavior of each motif is not only influenced by its algorithm, but also largely affected by the type, source, size, and pattern of input data.",
"From the system aspect, we find that some AI motifs like convolution, fully-connected are CPU-intensive, while the other AI motifs are not CPU-intensive, such as Relu, Sigmoid used as activation layer.",
"Further, the AI motifs have little pressure on disk I/O, since they load a batch (e.g.",
"128 images) from disk every iteration.",
"From the micro-architecture aspect, we find that these motifs show various computation and memory access patterns, exploiting different parallelism degrees of ILP and MLP.",
"With the data size expanding, the percentage of frontend bound decreases while the backend bound increases.",
"The rest of the paper is organized as follows.",
"Section 2 illustrates the motivation of identifying data motifs.",
"Section 3 introduces data motif identification methodology.",
"Section 4 performs system and micro-architecture evaluations on the data motif implementations.",
"In Section 5, we report the data impact on the data motifs' behaviors from perspectives of data size, data pattern, data type and data source.",
"Section 6 introduces the related work.",
"Finally, we draw a conclusion in Section 7."
],
[
"Motivation",
"We take two examples to explain why we should call attention to performing comprehensive characterization of those data motifs."
],
[
"SIFT [32] is a typical workload for feature extraction, and widely used to detect local features of input images.",
"Fig.",
"REF shows the computation dependency graph and run time breakdown of SIFT workload.",
"In total, SIFT involves five data motifs.",
"Gaussian filters $G(x,y,\\partial )$ with different space scale factors $\\partial $ are used to generate a group of image scale spaces, through the convolution with the input image.",
"Image pyramid is to downsample these image scale spaces.",
"DOG image means difference-of-Gaussian image, which is produced by matrix subtraction of adjacent image scale spaces in image pyramid.",
"After that, every point in one DOG scale space would sort with eight adjacent points in the same scale space and points in adjacent two scale spaces, to find the key points in the image.",
"Through profiling, we find that computes descirptors, finds keypoints and builds gaussian pyramid are three main time-consuming parts of the SIFT workload.",
"Furthermore, we analyze those three parts and find they consist of several classes of unit of computation, like Matrix, Sampling, Transform, Sort and Statistics, summing up to 83.23% of the total SIFT run time."
],
[
"AlexNet [30] is a representative and widely-used convolutional neural network in deep learning.",
"In total, it has eight layers, including five convolutional layers and three fully connected layers.",
"We profile one iteration of the AlexNet workload (implemented with TensorFlow) using TensorBoard toolkit.",
"Fig.",
"REF presents its computation dependency graph and run time breakdown.",
"For each operator, we report its run time and its percentage of the total run time, such as 6.57 ms and 1.35% for the first convolution operator.",
"We find that each iteration involves Transform (conv2d), Sampling (max pooling, dropout), Statistics (normalization), and Matrix (fully connected).",
"Among them, matrix and transform computations occupy a large proportion—48.87% and 36.91%, respectively.",
"Through the above analysis, we have the following observation.",
"Though big data and AI workloads are very complex and fast-changing, we can consider them as a pipeline of one or more fundamental classes of unit of computation performed on different initial or intermediate data inputs.",
"Those classes of unit of computation, which we call data motifs, occupy most of the run time of the workloads, so we should pay more attention to them.",
"In the next section, we will investigate more extensive big data and AI workloads, and elaborate the design of data motifs."
],
[
"Methodology",
"Data motifs are frequently-appearing classes of unit of computation handling different data inputs.",
"In this section, we illustrate how to identify data motifs from big data and AI workloads, and illustrate our data motif implementations.",
"Table: The Importance of Eight Data motifs in Big Data and AI workloads."
],
[
"Motif Identification Methodology",
"Fig.",
"REF overviews the methodology of motif identification.",
"We first single out a broad spectrum of big data and AI workloads through investigating five typical application domains (search engine, social network, e-commerce, multimedia, and bioinformatics) and representative algorithms in four processing techniques (machine learning, data mining, computer vision and natural language processing).",
"Then we conduct algorithmic analysis and profiling analysis on these workloads.",
"We profile the workload to analyze the computation dependency graph and run time breakdown, to find and correlate the hotspot functions to the code segments.",
"Combing with algorithmic analysis, we decompose the workload into a pipeline of units of computation and focus on the input/intermediate data as well.",
"Then we summarize the frequently-appearing and time-consuming units as data motifs.",
"We repeat this procedure on forty workloads with a broad spectrum to guarantee the representativeness of our data motifs.",
"According to the units of computation pipeline and run time breakdown, we finalize eight big data and AI motifs, which are essential computations that take up most of run time.",
"Table REF shows the importance of eight data motifs in a majority of big data and AI workloads.",
"Note that previous work [23] has identified four basic units of computation in online service, including get, put, post, delete.",
"We don't include those four in our motif set."
],
[
"Eight Data Motifs",
"In this subsection, we summarize eight data motifs that frequently appear in big data and AI workloads.",
"Matrix In big data and AI workloads, many problems involve matrix computations, such as vector-vector, matrix-vector and matrix-matrix operations.",
"Sampling Sampling plays an essential role in big data and AI processing, which selects a subset samples according to certain statistical population.",
"It can be used to obtain an approximate solution when one problem cannot be solved by deterministic method.",
"Logic We name computations performing bit manipulation as logic computations, such as hash, data compression and encryption.",
"Transform The transform computations here mean the conversion from the original domain (such as time) to another domain (such as frequency).",
"Common transform computations include discrete fourier transform (DFT), discrete cosine transform (DCT) and wavelet transform.",
"Set In mathematics, Set means a collection of distinct objects.",
"Likewise, the concept of Set is widely used in computer science.",
"Set is also the foundation of relational algebra [34].",
"In addition, similarity analysis of two data sets involves set computations, such as Jaccard similarity.",
"Furthermore, fuzzy set and rough set play very important roles in computer science.",
"Graph A lot of applications involve graphs, with nodes representing entities and edges representing dependencies.",
"Graph computation is notorious for having irregular memory access patterns.",
"Sort Sort is widely used in many areas.",
"Jim Gray thought sort is the core of modern databases [6], which shows its fundamentality.",
"Statistics Statistic computations are used to obtain the summary information through statistical computations, such as counting and probability statistics."
],
[
"Data Motif Implementations",
"Data motifs are the fundamental components of big data and AI workloads, which is of great significance for evaluation, considering the complexity and diversity of big data and AI workloads.",
"We provide the data motif implementations for big data and AI separately, according to their computation specialties.",
"For the big data motif implementations, we provide Hadoop [1], Spark [46], and Pthreads [8] implementations.",
"These data motifs include sort, wordcount, grep, MD5 hash, matrix multiplication, random sampling, graph traversal and FFT transformation.",
"For the AI motifs, we provide TensorFlow [5] and Pthread implementations, including 2-dimensional convolution, max pooling, average pooling, relu activation, sigmoid activation, tanh activation, fully connected (matmul), and element-wise multiply.",
"We consider the impact of data input from the perspectives of type, source, size, and pattern.",
"Among them, data type includes structure, un-structured, and semi-structured data.",
"Data source indicates the data storage format, including text, sequence, graph, matrix, and image data.",
"Data pattern includes the data distribution, data sparsity, et al.",
"As for data size, we provide big data generators for text, sequence, graph and matrix data to fulfill different size requirements."
],
[
"Characterization",
"In this section, we evaluate data motifs with various software stacks from the perspectives of both system and architecture behaviors."
],
[
"Experiment Setups",
"We deploy a three-node cluster, with one master node and two slave nodes.",
"They are connected using 1Gb Ethernet network.",
"Each node is equipped with two Intel Xeon E5-2620 V3 (Haswell) processors, and each processor has six physical out-of-order cores.",
"The memory of each node is 64 GB.",
"The operating system, software stacks and gcc versions are as follows: CentOS 7.2 (with kernel 4.1.13); JDK 1.8.0_65; Hadoop 2.7.1; Spark 1.5.2; TensorFlow 1.0; GCC 4.8.5.",
"The data motifs implemented with Pthread are compiled using \"-O2\" option for optimization.",
"The hardware and software details are listed in Table REF .",
"Since Pthread is a multi-thread programming model, we evaluate both the TensorFlow and Pthread implementations of AI motifs on one node for apple-to-apple comparison.",
"Table: Configuration Details of Xeon E5-2620 V3"
],
[
"Experiment Methodology",
"We evaluate eight big data motifs implemented with Hadoop, Spark, and eight AI data motifs implemented with TensorFlow and Pthread.",
"Note that we use the optimal configurations for each software stack, according to the cluster scale and memory size.",
"The data configuration and selected metrics are listed as follows.",
"Data Configuration To evaluate the impacts of data input comprehensively, we evaluate the data motifs with three data sizes: Small, Medium, and Large.",
"We choose the Large data size according to the memory capacity of the cluster so as to fully utilize the memory resources, and the other two are chosen for comparison.",
"For the graph motif, Small, Medium, Large is $2^{22}$ , $2^{24}$ and $2^{26}$ -vertex, respectively.",
"For the matrix motif, we use 100, 1K and 10K two-dimensional matrix data with the same distribution and sparsity.",
"For the transform motif, we use 16384, 32768 and 65536 two-dimension matrix data.",
"For the other big data motifs, we use 1, 10 and 100 GB wikipedia text data, respectively.",
"For the AI motifs, we use three configurations in terms of input tensor sizes and channels.",
"They are (224*224,64), (112*112,128) and (56*56,256).",
"Among them, the first value indicates the dimension of input tensor, the second value indicates the channels, and all of them use 128 as batch size.",
"We choose these three configurations because they are widely used in neural network models [39].",
"Note that the dimension for all input tensors is 224 for Large configuration, 112 for Medium configuration and 56 for Small configuration.",
"For the Pthread-version AI motifs, we use 1K, 10K, 100K images from ImageNet [15].",
"In the following subsections, we characterize the system and micro-architectural behaviors of data motifs with the Large data size.",
"In Section , we will analyze the impact of data input on characteristics with all data sizes.",
"System and Micro-architecture Metrics We characterize the system and micro-architectural behaviors [40] of the data motifs, which are significant for design and optimization [36].",
"For system evaluation, we report the metrics of CPU utilization, I/O Wait, disk I/O bandwidth, and network I/O bandwidth.",
"The system metrics are collected through the proc file system.",
"For micro-architectural evaluation, we use the Top-Down analysis method [44], which categorizes the pipeline slots into four categories, including retiring, bad speculation, frontend bound and backend bound.",
"Among them, retiring represents the useful work, which means the issued micro operations (uops) eventually get retired.",
"Bad speculation represents the pipeline is blocked due to incorrect speculations.",
"Frontend bound represents the stalls due to frontend, which undersupplies uops to the backend.",
"Backend bound represents the stalls due to backend, which is a lack of required resources for new uops [4].",
"We use Perf [3], a Linux profiling tool, to collect the hardware events referring to the Intel Developerś Manual [22] and pmu-tools [4]."
],
[
"System Evaluation",
"Fig.",
"REF presents the CPU utilization and I/O Wait of all data motifs.",
"We find that Hadoop motifs have higher CPU utilization than Spark motifs, and suffer from less I/O Wait than Spark motifs do.",
"Particularly, Hadoop motifs take 80 percent CPU time.",
"The I/O Waits of AI data motifs are extremely lower than that of big data motifs.",
"For deep neural networks, even the total input data is large, the input layer loads a batch from disk every iteration, so data loading size from disk by the input layer occupies a very small proportion comparing to intermediate data, and thus introduces little disk I/O requests.",
"Pthread motifs have less CPU utilization and I/O Wait in general, because Pthread motifs have less memory allocation and relocation operations than counterparts using other stacks.",
"Moreover, the data loading time overlaps the processing time since computation is simple, except that Pthread Matmul has almost 100% CPU utilization because of its high computation complexity and CPU-intensive characteristics.",
"TensorFlow motifs, such as AvgPool, Conv, Matmul, Maxpool, and Multiply, have taken most of CPU time, because these five motifs are CPU-intensive.",
"Nevertheless, we also find that the other AI motifs are not that CPU-intensive, such as Relu, Sigmoid, and Tanh.",
"Fig REF presents the network bandwidth and disk I/O bandwidth.",
"For AI motifs, most of them (e.g.",
"matmul, relu, pooling, activation) are executed in the hidden layers, and the intermediate states of hidden layers are stored in the memory.",
"That is to say, the hidden layers consume the most resources of computation and memory storage, while the disk I/O for input layer is relatively minor.",
"Our evaluation confirms this observation.",
"Meanwhile, as mentioned in Section 4.1, we evaluate both the TensorFlow and Pthread implementations of AI motifs on one node for apple-to-apple comparison.",
"So we do not report the I/O behaviors of AI motifs.",
"We find that for all big data motifs, Spark stack has much larger network I/O pressure than that of Hadoop stack, because Spark stack has more data shuffles, so it needs transferring data from one node to another one frequently.",
"Five of the eight Spark implementations have smaller disk I/O pressure than that of Hadoop, because Spark targets in-memory computing.",
"Except Spark Matmul, Spark MD5 and Spark WordCount have larger disk I/O pressure than that of Hadoop counterparts.",
"Their disk I/O read sector numbers are nearly equal, while the write sector numbers are much larger."
],
[
"Micro-architecture Evaluation",
"To better understand the data motifs, we analyze their performance and micro-architectural characteristics.",
"Execution Performance The execution performance indicates the overall running efficiency of the workloads [29].",
"We use the instruction level parallelism (ILP) and memory level parallelism (MLP) to reflect the execution performance.",
"Among them, ILP measures the number of instructions that can be executed simultaneously.",
"Here we use the retired instructions per cycle (IPC) to measure ILP.",
"MLP indicates the parallelism degree that memory accesses can be generated and executed [21].",
"MLP is computed through dividing L1D_PEND_MISS.PENDING by L1D_PEND_MISS.PENDING_CYCLES [4].",
"Fig.",
"REF presents the ILP and MLP of all data motifs.",
"We find that these motifs cover a wide range of ILP and MLP behaviors, reflecting distinct computation and memory access patterns.",
"For example, TensorFlow Multiply does element-wise multiplications and has high MLP (5.27) but extremely low ILP (0.15).",
"This is because that its computation is simple and has little data dependencies, so it generates many concurrent data loads, thus incurs a large amount of data cache misses.",
"Also, max pooling and average pooling have high MLP.",
"The MLP of average pooling is lower than max pooling, because average computation involves many divide operations, and thus suffers from more stalls due to the delay of divider unit.",
"The software stack changes workload's computation and memory access patterns, which is also found in previous work [25].",
"For example, both Hadoop FFT and Spark FFT are based on cooley-tukey algorithm [13], while they have different parallelism degrees.",
"Spark FFT is more memory-intensive and has higher MLP.",
"Figure: The Uppermost Level Breakdown of Data Motifs.The Uppermost Level Breakdown Fig.",
"REF shows the uppermost level breakdown of all data motifs we evaluated.",
"We find that these motifs have different pipeline bottlenecks.",
"For Hadoop motifs, they suffer from notable stalls due to frontend bound and bad speculation.",
"Moreover, Hadoop motifs reflect nearly consistent bottlenecks, indicating the Hadoop stack impacts workload behaviors more than other stacks like Spark and TensorFlow.",
"For Spark motifs, which mainly compute in memory, they suffer from a higher percentage of backend bound than that of Hadoop counterparts.",
"Spark Grep, Sample and Sort suffer from more frontend bound and their percentages of backend bound are smaller than the others.",
"The AI data motifs face different bottlenecks both on TensorFlow and Pthreads.",
"Conv and Matmul have the highest IPC (about 2.2) and retiring percentages (about 50% on TensorFlow).",
"Max pooling, average pooling, and multiply have extremely low retiring percentages, which has been illustrated in above.",
"However, activation operation like ReLU, sigmoid and tanh suffer from more frontend bound than backend bound.",
"For AI data motifs implemented with Pthread, their main bottleneck is backend bound.",
"They suffer from little frontend and bad speculation stalls.",
"Figure: The Frontend Breakdown of Data Motifs.Figure: The Frontend Latency Breakdown of Data Motifs.Frontend Bound Frontend bound can be split into frontend latency bound and frontend bandwidth bound.",
"Among them, latency bound means the frontend delivers no uops to the backend, while bandwidth bound means delivering insufficient uops comparing to the theoretical value.",
"Fig.",
"REF presents the frontend breakdown of the data motifs.",
"We find that the main reason that incurs the frontend stalls is latency bound for almost all motifs that suffer from severe frontend bound.",
"We further investigate the reasons for the frontend latency bound and frontend bandwidth bound, respectively.",
"Generally, the frontend latency bound are incurred by six reasons, including icache miss, itlb miss, branch resteers, DSB (Decoded Stream Buffer) switches, LCP (Length Changing Prefix), and MS (microcode sequencer) switches.",
"Among them, icache miss and itlb miss are instruction cache miss and instruction tlb miss.",
"Branch resteers means the delays to obtain the correct instructions, such as the delays due to branch misprediction.",
"LCP measures the stalls when decoding the instructions with a length changing prefix.",
"Generally, uops comes from three places, including the decoded uops cache (DSB), legacy decode pipeline (MITE) and microcode sequencer (MS).",
"DSB switches record the stalls caused by switching from the DSB to MITE.",
"MS switches measure the penalty of switching to MS unit.",
"As for latency bandwidth bound, there are mainly two reasons: the inefficiency of MITE pipeline and the inefficient utilization of DSB cache.",
"Additionally, LSD represents the stalls due to waiting the uops from the loop stream detector [2].",
"Fig.",
"REF lists the latency and bandwidth bound breakdown of all data motifs.",
"For almost all data motifs, branch resteers is a main reason for the high percentage of frontend bound, except Spark Matmul and Relu, Sigmoid, Tanh on TensorFlow.",
"For these three activation functions, nearly 60% frontend bound is due to instruction cache miss.",
"On average, big data motifs implemented with Hadoop and Spark suffer from more icache misses than AI data motifs.",
"Moreover, MS switch is another significant factor that incurs frontend latency bound.",
"Because big data and AI systems use many CISC instructions that cannot be decoded by default decoder, so they must be decoded by MS unit, and results in performance penalties.",
"Figure: The Backend Bound Breakdown of Data Motifs.Figure: The Backend Core Bound Breakdown of Data Motifs.Backend Bound Fig REF presents the backend bound breakdown of data motifs, which are split into backend memory bound and backend core bound.",
"Backend memory bound is mainly caused by the data movement delays among different memory hierarchies.",
"Backend core bound is mainly caused by the lack of hardware resources (e.g.",
"divider unit) or port under-utilization because of instruction dependencies and execution unit overloading.",
"We find that more than half of these data motifs suffer from more backend memory bound than core bound.",
"However, for each software stack, there is at least one data motif that suffers from equal percentages of core bound or even more percentages of core bound than memory bound, such as Hadoop WordCount, Spark MD5, TensorFlow Conv and Pthread AvgPool.",
"Fig.",
"REF shows the core bound breakdown.",
"We find that TensorFlow AvgPool and Hadoop WordCount suffer from significantly long latency of divider unit.",
"While for Spark MD5 and TensorFlow Conv, which has the highest percentage of backend core bound, mainly suffer from the stalls due to port under-utilization.",
"As for backend memory bound, we find that DRAM memory bound is much severe than level 1, 2, and 3 cache bound for almost all big data and AI motifs, indicating that the memory wall [42] still exists and needs to be optimized.",
"Figure: Linkage Distance of Data Motifs."
],
[
"Impact of Data Input",
"In this section, we evaluate the impact of data input on system and micro-architecture behaviors from the perspectives of size, source, type, and pattern.",
"For type and pattern evaluation, we use Sort and FFT as an example, respectively."
],
[
"Impact of Data Size",
"Based on all sixty metrics spanning system and micro-architecture we evaluated in Section , we conduct a coarse-grained similarity analysis using PCA (Principal Component Analysis) [27] and hierarchical clustering [26] methods on three data size configurations.",
"Fig.",
"REF presents the linkage distance of all data motifs, which indicates the similarity of system and micro-architecture behaviors.",
"Note that the smaller the linkage distance, the more similar the behaviors.",
"We find that data motifs with small data size are more likely to be clustered together.",
"A small data size will not fully utilize the system and hardware resources, hence that they tend to reflect similar behaviors.",
"However, for the motif that is computation intensive and has high computation complexity, even with the large data set, it will be clustered together with small data set.",
"For example, FFTs with three data size configurations are clustered together for both Hadoop and Spark version.",
"AI Motifs with TensorFlow implementations are also greatly affected by the input data size.",
"However, they reflect distinct behaviors with big data motifs implemented with Hadoop and Spark, with the least linkage distance of 6.71.",
"Figure: Impact of Data Size on I/O Behaviors.Impact of Data Size on I/O Behaviors We evaluate the impact of data size on I/O behaviors using the fully distributed Hadoop and Spark motif implementations.",
"Using the I/O bandwidth of Small data size as baseline, we normalize the I/O bandwidth of Medium and Large data size, as illustrated in Fig.",
"REF .",
"The bold black horizontal line in Fig.",
"REF shows the equal line with the small input.",
"That is to say, the value higher than the line means larger I/O bandwidth than the value of the small input.",
"Here we do not report the performance data of the AI motifs because the disk I/O behavior is little in neural network modelling, which we have illustrated in Subsection REF .",
"We find that almost for all data motifs, their I/O behaviors are sensitive to the data size.",
"When the data size large enough, the whole data can not be stored in memory, then the data have to be swapped in and swapped out frequently, and hence put great pressure on disk I/O access.",
"Modern big data and AI systems adopt a distributed manner, with the data storing on an distributed file system, such as HDFS [38], the data shuffling or data unbalance will generate a large amount of network I/O.",
"Figure: Impact of Data Size on Pipeline Efficiency.Impact of Data Size on Pipeline Efficiency We further measure the impact of data size on pipeline efficiency.",
"As shown in Fig.",
"REF , we find that with the data size increases, the percentage of frontend bound decrease, while the percentage of backend bound increase.",
"For example, Spark Matmul with large input size decrease nearly 20% of frontend bound and increase more than 30% of backend bound.",
"As the data size increase, the high-speed cache and even memory are unable to hold all of them, and further incur many data cache misses, resulting in large penalties due to memory hierarchy."
],
[
"Impact of Data Pattern",
"Data pattern and data distribution impact the workload performance significantly [43], [45].",
"To evaluate the impact of data pattern on the motifs, we use two different patterns of dense matrix and sparse matrix, to run FFT motif as an example.",
"The matrix sparsity indicates the ratio of zero value among all matrix elements.",
"With different sparsity, the data access patterns vary, and thus reflect different behaviors.",
"We use two 16384$\\times $ 16384 matrixes as the input for the FFT motif, with the one having 10% sparsity and the other one 90% sparsity.",
"Fig.",
"REF shows the impact of data pattern on the data motifs from system (Fig.",
"REF ) and micro-architecture perspectives(Fig.",
"REF ).",
"We find that using the matrix with high sparsity, the network I/O and disk I/O are nearly half of the values using the dense matrix, and the major page fault per second is almost the same.",
"Spark motifs suffer from more I/O pressure than Hadoop motifs.",
"As for pipeline bottlenecks, sparse data input incurs more frontend stalls while less backend stalls."
],
[
"Impact of Data Type and Source",
"Data types and sources are of great significance for read and write efficiency [17], considering their storage format and targeted scenarios, such as the supports for splitable files and compression level.",
"To evaluate the impact of the data type and source on system and micro-architecture behaviors, we use two different data types for Sort motif, with the same data size of 10 GB.",
"Two types are un-structured wikipedia text data and semi-structured sequence data.",
"Wikipedia text file is laid out in lines and each line records an article content.",
"Sequence files are flat files that consist of key and value pairs, stored in binary format.",
"Fig.",
"REF lists the impact of data type on data motifs from the system (Fig.",
"REF ) and micro-architecture aspects (Fig.",
"REF ).",
"We find that the difference between using text type and sequence type ranges from 1.12 times to 7.29 times from the system aspects.",
"Using text data type, the CPU utilization is lower than using sequence data, which indicates that using sequence data has better performance.",
"Moreover, both Hadoop Sort and Spark Sort suffer from more major page faults and further impact the execution performance, because of page loads from disk.",
"Note that we use the major page fault number per second in Fig.",
"REF and the total number during the running process is about 100 to 200.",
"Even with the same amount of data size, their network I/O and disk I/O bandwidth still have a great difference.",
"We find that the sequence format have larger requirements for I/O bandwidth than the text format.",
"From the micro-architecture aspect (Fig.",
"REF ), Sort with different data types reflect different percentages of pipeline bottlenecks.",
"With the text format, backend bound bottleneck is more severe, especially backend memory bound, which indicates that they waste more cycles to wait for the data from cache or memory."
],
[
"Related Work",
"Our big data and AI motifs are inspired by previous successful abstractions in other application scenarios.",
"The set concept in relational algebra [11] abstracted five primitive and fundamental operators, setting off a wave of relational database research.",
"The set abstraction is the basis of relational algebra and theoretical foundation of database.",
"Phil Colella [12] identified seven motifs of numerical methods which he thought would be important for the next decade.",
"Based on that, a multidisciplinary group of Berkeley researchers proposed 13 motifs which were highly abstractions of parallel computing, capturing the computation and communication patterns of a great mass of applications [6].",
"National Research Council proposed seven major tasks in massive data analysis [14], which they called giants.",
"These seven giants are macroscopical definition of problems in massive data analysis from the perspective of mathematics, while our eight classes of motifs are main time-consuming units of computation in the Big Data and AI workloads.",
"Application kernels [7], [16] also aim at scaling down the run time of the real applications, while preserving the main characteristics of the workload.",
"Consisting of the major function of the application, Kernel tries to cover the bottleneck of the real application.",
"But kernel is still hard to understand the complex and diversity big data and AI workloads [7], [31].",
"Other than that, kernel mainly focuses on the CPU and memory behaviors, and pays little attention to the I/O, which is also important for many real applications, especially in an era of data explosion."
],
[
"Conclusions",
"In this paper, we answer what are abstractions of time-consuming units of computation in big data and AI workloads.",
"We identify eight data motifs among a wide variety of big data and AI workloads, including Matrix, Sampling, Logic, Transform, Set, Graph, Sort and Statistic computations.",
"We found the combinations of one or more data motifs with different weights in terms of runtime can describe most of big data and AI workloads we investigated [19].",
"We implement the data motifs for big data and AI separately, including the big data motif implementations using Hadoop, Spark, Pthreads, and the AI data motif implementations using TensorFlow, Pthreads, considering the impact of data type, data source, data size, and data pattern.",
"We release them as the micro benchmarks of an open-source Big Data and AI benchmark suite—BigDataBench, publicly available from http://prof.ict.ac.cn/BigDataBench.",
"From the system and micro-architecture perspectives, we comprehensively characterize the behaviors of data motifs and identify their bottlenecks.",
"Further, we measure the impact of data type, data source, data pattern and data size on their behaviors.",
"We find that these data motifs cover a wide variety of performance space, from the perspectives of system and micro-architecture behaviors.",
"Moreover, the behavior of each data motif is not only influenced by its algorithm, but also largely affected by the type, source, size, and pattern of input data.",
"We believe our work is an important step toward not only Big Data and AI benchmarking, but also domain-specific hardware and software co-design."
],
[
"Acknowledgements",
"This work is supported by the National Key Research and Development Plan of China (Grant No.",
"2016YFB1000600 and 2016YFB1000601).",
"The authors are very grateful to anonymous reviewers for their insightful feedback and Dr. Zhen Jia for his valuable suggestions."
],
[
"Abstract",
"The artifact contains our big data and AI motif implementations on Hadoop, Spark, Pthreads, and TensorFlow stacks.",
"It can support the characterization results in Chapter four and Chapter five of our PACT 2018 paper Data Motifs: A Lens Towards Fully Understanding Big Data and AI Workloads.",
"To validate the results, deploy the experiment environment and profile the benchmarks."
],
[
"Artifact check-list (meta-information)",
" Program: Data motif implementations Compilation: GCC 4.8.5; Python 2.7.5; Java 1.8.0_65 Data set: generated by BigDataBench Run-time environment: CentOS 7.2, Linux Kernel 4.1.13 with Perf tool Hardware: Processor supporting Top-Down analysis, above Sandy Bridge series, and the performance events corresponding to the processor Run-time state: Disable Hyper-Threading Execution: root user or users that can execute sudo without password Output: the system and micro-architecture profiling results Experiment: Deploy the data motifs and corresponding software stacks; run benchmarks; profile using perf; output the results Workflow frameworks used?",
"No Publicly available?",
": Yes"
],
[
"How delivered",
"The data motifs are the micro benchmarks of BigDataBench 4.0—an open source big data and AI benchmark suite.",
"Download link: http://prof.ict.ac.cn/bdb_uploads/bdb_4/pact2018.tar.gz All the related files are under the \"pact2018\" directory, please refer to README for detailed description.",
"Note that to obtain accurate performance data, the user should make sure there is no other motif running before run a motif.",
"The running scripts we provide suit for our cluster environment, like the node ip/hostname and port number, if you download and use it in your cluster environment, you need to modify the scripts to suit for your environment."
],
[
"Hardware dependencies",
"The data motifs can be run on all processors that can deploy Hadoop, Spark, TensorFlow and Pthread stacks.",
"However, for Top-Down analysis, due to the performance counter limitations, we suggest the Intel Xeon processors, above Sandy Bridge series.",
"Also, user need to find the performance counters corresponding to specific processor.",
"We have provided profiling scripts for Xeon E5-2620 V3 (Haswell) processor."
],
[
"Software dependencies",
"JDK 1.8.0_65; Hadoop 2.7.1; Spark 1.5.2; TensorFlow 1.0; GCC 4.8.5."
],
[
"Data sets",
"We provide data generators for text, sequence, graph, and matrix data.",
"Users can find the data generation method in the README file or BigDataBench user manual.",
"The generation parameter used in our paper for the graph motif is 22 (Small), 24 (Medium), 26 (Large), respectively.",
"User need to install Hadoop, Spark, GCC and TensorFlow.",
"The install details can be found in the User Manual of BigDataBench.",
"We provide \"Makefile\" for pthread motifs.",
"For all data motifs, we provide running scripts in our package."
],
[
"Experiment workflow",
"Before profiling system and micro-architecture metrics of one motif, users should make sure there is no other motif/workload running."
],
[
"Data generation",
"We provide text, graph, matrix, and sequence data generators under data-generator directory.",
"To generate large, medium, small data used in our paper, we provide a script \"data-generator.sh\".",
"Make sure hadoop is running, because the script upload the generated data to HDFS.",
"The script running command: #sh data-generator.sh <format> <datasize> Note that <format> can be text, seq, graph or matrix, and <datasize> can be large, medium or small.",
"Also, the generators support generate other data size the user needed.",
"Graph data generation: #cd $pact2018/data-generator/genGraphData #./genGraph.sh <log2_vertex> For example, ./genGraph.sh 26 for 26-vertex graph data.",
"Matrix data generation: #cd $pact2018/data-generator/genMatrixData For floating-point data: #./generate-matrix.sh <row_num> <colum_num> <sparsity> For integer data: #./generate-matrix-int <row_num> <colum_num> <sparsity> The sparsity means \"sparsity\" percentage elements are zero.",
"Text data generation: #cd $pact2018/data-generator/genTextData #./genText.sh <size> Note that the parameter size means \"size\" gigabytes text data.",
"Sequence data generation: Transfer the wiki text data to sequence data, so user should generate text data first and put it on HDFS, for example, \"wiki-10G\" data are on HDFS.",
"#cd $pact2018/data-generator/genSeqData #./sort-transfer.sh <size>"
],
[
"Run the workloads.",
"We provide running scripts for all workloads.",
"During the running process, the profiling scripts are started to sample the system and architecture metrics.",
"For Hadoop motifs: 1) Under pact2018 directory 2) Start Hadoop: #./start-hadoop.sh 3) Choose one Hadoop motif: #./run-hadoop.sh motif datasize Note that datasize parameter can be \"large\", \"medium\" or \"small\", means using large/medium/small data size,respectively.",
"For example: #./run-hadoop.sh graph large For Spark motifs: 1) Under pact2018 directory 2) Start Spark: #./start-spark.sh 3) Choose one Spark motif: #./run-spark.sh motif datasize Note that datasize parameter can be \"large\", \"medium\" or \"small\", means using large/medium/small data size,respectively.",
"For example: #./run-spark.sh graph large For TensorFlow motifs: 1) Under pact2018 directory 2) Choose one TensorFlow motif: #./run-tensorflow.sh motif datasize Note that datasize parameter can be \"large\", \"medium\" or \"small\", means using large/medium/small data size,respectively.",
"For example: #./run-tensorflow.sh relu large For Pthread motifs: 1) Under pact2018 directory 2) Choose one Pthread motif: #./run-pthread.sh motif datasize Note that datasize parameter can be \"large\", \"medium\" or \"small\", means using large/medium/small data size,respectively.",
"For example: #./run-pthread.sh relu large The sampling results of system and micro-architecture metrics are under \"result\" directory.",
"We provide processing scripts for computing the result and plot the figures.",
"Please refer to \"README\" file for the details."
],
[
"Process the metric data and plot the figures",
"We provide processing scripts and figure plotting scripts to generate the figures used in the paper.",
"Note that the sampling results are saved under \"result\" directory when test finished.",
"1) Compute the performance data and save them in an excel file.",
"#python lsdata.py result result_new 1 Parameter \"result\" means the input directory which contains the sampling results; Parameter \"result_new\" means the output excel file name and the output file is result_new.xls.",
"2) Plot the figures and save them as png image format #python plot.py result_new.xls Parameter \"result_new.xls\" is the excel file generated by the first step.",
"After running the command, several png files will be generated.",
"In addition, \"pact-AE.txt\" is generated for linkage distance analysis.",
"3) Linkage distance computing #$pact2018/Linkage-Distance #python hiclust_wiht_newpca.py pact-AE.txt Parameter \"pact-AE.txt\" is the text file generated by the second step.",
"After running the command, a png file will be generated under the Linkage-Distance directory, which is used as Figure 12 in our paper."
],
[
"Evaluation and expected result",
"To evaluate the system and micro-architecture performance of data motifs, users need to run those motifs and profile them.",
"These data motifs should reflect similar characteristics like figures in Chapter 4 and Chapter 5.",
"Our profiling scripts sample the performance data every 1 second during the whole motif runtime, and the performance data possibly vary within a slightly variation for each run."
],
[
"Experiment customization",
"Users can run these data motifs for different benchmarking purpose, e.g.",
"software stack comparison, different aspects of system and architecture characterizations.",
"Also, the data motifs can be deployed on different processors and cluster scales."
],
[
"Notes",
"For the artifact evaluation, since every motifs need to run three times for collecting dozens of performance events, it may cost several weeks to profiling all motifs, which is too expensive for the artifact evaluation.",
"So we provide the profiling scripts and the profiling data used in our paper, which are suit for our Haswell processor configurations.",
"Since the platform configurations of software (e.g.",
"Hadoop/Spark configuration) and hardware (e.g.",
"memory capacity, BIOS configuration) may be different, so the performance data may be different on another platform."
]
] | 1808.08512 |
[
[
"Numerical renormalization group method for entanglement negativity at\n finite temperature"
],
[
"Abstract We develop a numerical method to compute the negativity, an entanglement measure for mixed states, between the impurity and the bath in quantum impurity systems at finite temperature.",
"We construct a thermal density matrix by using the numerical renormalization group (NRG), and evaluate the negativity by implementing the NRG approximation that reduces computational cost exponentially.",
"We apply the method to the single-impurity Kondo model and the single-impurity Anderson model.",
"In the Kondo model, the negativity exhibits a power-law scaling at temperature much lower than the Kondo temperature and a sudden death at high temperature.",
"In the Anderson model, the charge fluctuation of the impurity contribute to the negativity even at zero temperature when the on-site Coulomb repulsion of the impurity is finite, while at low temperature the negativity between the impurity spin and the bath exhibits the same power-law scaling behavior as in the Kondo model."
],
[
"Introduction",
"Entanglement is a truly non-classical correlation [1], [2], [3], which often appears in many-body systems at macroscopic scale [4], [5], [6].",
"It can be quantified by various entanglement measures [1], [2], [3], and useful to understand many-body phenomena such as topological order [7], [8] and quantum criticality [9].",
"The Kondo effect, a many-body pheonomenon in quantum impurity systems induced by the bath electrons screening the impurity [10], involves the entanglement between the impurity and the bath electrons.",
"This impurity-bath entanglement provides a quantum information perspective on quantum impurity systems [11], [12], [13], [14], [15], [16], [17].",
"For quantum impurity systems, entanglement at finite temperature can provide new information in comparison with zero-temperature entanglement of ground states.",
"For example, the impurity-bath entanglement exhibits power-law scaling in the Kondo regime, and its power exponent differs between the Fermi liquid in the single-channel Kondo model and the non-Fermi liquid in the two-channel Kondo model [14].",
"Despite the importance, the impurity-bath entanglement has not been computed exactly at finite temperature [14] due to the following difficulty.",
"While pure quantum states (e.g., ground states) contain no classical correlation, mixed states such as thermal states generally have both quantum entanglement and classical correlation [1], [2], [3].",
"These two different types of correlations are not easily distinguishable; the entanglement quantification for mixed states is NP hard [18], [19].",
"For example, computation of the entanglement of formation (EoF) [20], a mixed-state generalization of the entanglement entropy, generally requires heavy optimization.",
"Therefore a practical choice of an entanglement measure for thermal states is the entanglement negativity [21], [22], [23], as the negativity can be computed exactly (although it cannot detect the bound entanglement [24]).",
"The negativity $\\mathcal {N}$ between a subsystem $A$ and its complementary $B$ is $\\mathcal {N}$ () = $\\mathrm {Tr}$ |TA| - $\\mathrm {Tr}$ , where $\\rho $ is the density matrix of a target system, $\\rho ^{T_A}$ is the partial transpose of $\\rho $ with respect to the subsystem $A$ , $\\mathrm {Tr}\\, |\\rho ^{T_A}|$ is the sum of the singular values of $\\rho ^{T_A}$ , and $\\mathrm {Tr}\\, \\rho $ is the trace of $\\rho $ .",
"To quantify the impurity-bath entanglement, one assigns $A$ the impurity and $B$ the bath.",
"$\\mathcal {N}(\\rho )$ is computable as long as $\\mathrm {Tr}\\, |\\rho ^{T_A}|$ is.",
"Due to this computational advantage, the negativity has been widely used to study entanglement in many-body systems at finite temperature [25], [26], [27], [28], [29], [30], [31], [32], [33].",
"The numerical computation of the negativity $N(\\rho )$ , however, becomes difficult, as the size of $\\rho $ becomes larger.",
"The difficulty appears for quantum impurity systems at finite temperature because of the following reasons.",
"First, the Kondo cloud [34], [35] is a macroscopic object whose size exponentially increases with decreasing Kondo coupling strength.",
"Second, quantum impurity systems are generally gapless, so their thermal density matrix involves many eigenstates and has high rank.",
"In this paper, we develop a numerical renormalization group (NRG) [36], [37] method to compute the entanglement negativity between the impurity and the bath of quantum impurity models at finite temperature.",
"We construct the thermal density matrix in the complete basis set of the energy eigenstates, and then evaluate the negativity, by applying the NRG approximation, which has been originally introduced to obtain impurity correlation fuctions [38], [39], [40].",
"Employing the method, we compute the temperature dependence of the negativity in the single-impurity Kondo model (SIKM) and the single-impurity Anderson model (SIAM), the simplest models exhibiting the Kondo effect.",
"In the SIKM, the negativity exhibits a universal quadratic temperature dependence in the Kondo regime at low temperature, the Kondo crossover at intermediate temperature, and a sudden death [41] at high temperature.",
"In the SIAM, both the spin and charge degrees of freedom at the impurity affect the negativity.",
"The impurity spin behaves in the same way as in the SIKM, while the charge fluctuation remain even at zero temperature as long as the on-site Coulomb repulsion at the impurity is finite.",
"To show this, we compute the negativity between the total degrees of freedom of the impurity and the bath, and the negativity between the spin degree of freedom of the impurity and the bath.",
"The former depends on the Coulomb repulsion strength, and the latter shows the same quadratic scaling as in the SIKM.",
"Finally, we demonstrate that our method is sufficiently accurate by computing and analyzing its errors for the example of the SIKM.",
"This paper is organized as follows.",
"In Sec.",
", we explain how to construct a thermal density matrix of an impurity problem by the NRG, and the NRG approximation.",
"We apply the NRG approximation to the impurity-bath negativity and propose how to compute the negativity in Sec. .",
"We compute the negativity for the SIKM in Sec.",
", and the SIAM in Sec. .",
"We estimate and analyze the errors in our method in Sec. .",
"Conclusion is given in Sec.",
"."
],
[
"Numerical Renormalization Group",
"The NRG is a powerful non-perturbative method to solve quantum impurity systems.",
"It provides an efficient way to construct a thermal density matrix by using a complete basis of many-body energy eigenstates [42], [39], over a wide range of temperature, in the thermodyamic limit.",
"In this section, we provide model Hamiltonians, notations, and brief introduction to the NRG including the NRG approximation."
],
[
"Model Hamiltonian",
"In this work, we apply the NRG to two paradigmatic impurity models, the SIKM and the SIAM.",
"The SIKM describes a spin-$1/2$ impurity interacting with the bath of conduction electrons, $H^\\mathrm {SIKM} = J\\vec{S}_\\mathrm {d} \\cdot \\vec{s}_0 + \\sum _{\\mu } \\int \\mathrm {d}\\epsilon \\, \\epsilon \\, c^{\\dagger }_{\\epsilon \\mu }c_{\\epsilon \\mu }.$ Here $J > 0$ is the coupling strength, $\\vec{S}_\\mathrm {d}$ the impurity spin, $c_{\\epsilon \\mu }$ the operator annihilating a bath electron of spin $\\mu = {\\uparrow }, {\\downarrow }$ and energy $\\epsilon $ , $\\vec{s}_0 = \\int \\mathrm {d}\\epsilon \\int \\mathrm {d}\\epsilon ^{\\prime } \\sum _{\\mu \\mu ^{\\prime }} c_{\\epsilon \\mu }^\\dagger [\\vec{\\sigma }]_{\\mu \\mu ^{\\prime }} c_{\\epsilon ^{\\prime }\\mu ^{\\prime }} / 2$ the spin of the bath electron at the impurity site, and $\\vec{\\sigma }$ the vector of the Pauli matrices.",
"We consider the bath of constant density of states within $[-D,D]$ .",
"We set the half-bandwidth $D \\equiv 1$ as the energy unit, and set $\\hbar = k_\\mathrm {B} = 1$ henceforth.",
"On the other hand, the SIAM contains a fermionic site with local repulsive Coulomb interaction at the impurity, $\\begin{aligned}H^\\mathrm {SIAM} &=\\sum _{\\mu } \\epsilon _\\mathrm {d} n_{\\mathrm {d}\\mu } + Un_{\\mathrm {d}\\uparrow }n_{\\mathrm {d}\\downarrow } +\\sum _{\\mu } \\int \\mathrm {d}\\epsilon \\, \\epsilon \\, c^{\\dagger }_{\\epsilon \\mu }c_{\\epsilon \\mu } \\\\& \\quad + \\sum _\\mu \\int \\mathrm {d} \\epsilon \\, \\sqrt{\\frac{\\Gamma (\\epsilon )}{\\pi }} (d_{\\mu }^\\dagger c_{\\epsilon \\mu } + c_{\\epsilon \\mu }^\\dagger d_{\\mu }).\\end{aligned}$ Here $d_\\mu $ annihilates a spin-$\\mu $ particle at the impurity, $n_{d\\mu } \\equiv d_{\\mu }^\\dagger d_{\\mu }$ is the number operator, $\\epsilon _\\mathrm {d}$ the on-site energy at the impurity, $U$ the Coulomb interaction strength, and $\\Gamma (\\epsilon )$ the hybridization function.",
"Throughout this work, we consider $\\epsilon _\\mathrm {d} = -U/2$ to make the impurity half-filled $\\langle n_{d\\mu } \\rangle = 1/2$ , and the constant hybridization function $\\Gamma (\\epsilon ) = \\Gamma \\Theta ( D- |\\epsilon |)$ which relates to the constant density of states within $[-D,D]$ .",
"Despite different type of impurities, both the SIKM and the SIAM can exhibit the Kondo effect.",
"It is natural since the SIKM can be derived from the SIAM as the low-energy effective Hamiltonian, via the Schrieffer-Wolff transformation [10]."
],
[
"Thermal density matrix",
"The NRG starts with the logarithmic discretization of the bath.",
"The bath of energy interval $[-1,1]$ is discretized by a logarithmic energy grid $\\pm \\Lambda ^{-k+z}$ for $k = 1, 2, \\cdots $ , where $\\Lambda > 1$ is a discretization parameter and $z = 0, \\tfrac{1}{n_z} \\cdots , 1 - \\tfrac{1}{n_z}$ is the discretization shift [43], [44].",
"Then the impurity model is mapped onto the so-called Wilson chain where the bath degrees of freedom lie along a tight-binding chain and the impurity couples to one end of the chain.",
"The models in Eqs.",
"(REF ) and (REF ) are mapped onto the chain Hamiltonians, H$\\mathrm {SIKM}$ N = J S$\\mathrm {d}$ s0 + HN$\\mathrm {bath}$ , H$\\mathrm {SIAM}$ N = $\\mathrm {d}$ n$\\mathrm {d}$ + Un$\\mathrm {d}$ n$\\mathrm {d}$ + HN$\\mathrm {bath}$ + 2 (df0 + f0d), where $H^\\mathrm {bath}_N = \\sum _\\mu \\sum _{n = 1}^N t_{n} f_{n-1,\\mu }^\\dagger f_{n\\mu } + \\text{H.c.}$ is the bath Hamiltonian of the chain length $N+1$ , $f_{n\\mu }$ annihilates a spin-$\\mu $ particle at site $n \\in [0,N]$ , and $\\vec{s}_0$ is the spin operator at site 0 next to the impurity.",
"Due to the logarithmic discretization, the hopping amplitudes decay exponentially as $t_n \\sim \\Lambda ^{-n/2}$ .",
"In practice, we consider the chain of a finite $N$ such that its lowest energy scale $\\sim \\Lambda ^{-N/2}$ is smaller than any other physical energy scales such as the system temperature $T$ .",
"The Fock space of the chain is spanned by the basis $\\lbrace | s_\\mathrm {d} \\rangle \\otimes | s_0 \\rangle \\otimes \\cdots \\otimes | s_N \\rangle \\rbrace $ , where $| s_\\mathrm {d} \\rangle $ is the impurity state and $| s_n \\rangle $ is the state of a bath site $n$ .",
"Since the Fock space dimension of the chain scales as $O (d^N)$ (here $d = 4$ is the dimension of each bath site for the single-channel problems considered in this work), it is hard to exactly diagonalize the chain with large $N$ .",
"By taking advantage of the exponential decay of the hopping amplitudes, one can construct the complete basis of the energy eigenstates by using the iterative diagonalization [42], [38].",
"In the $n$ th iterative diagonalization step, one obtains a set of energy eigenstates in an energy window $[E_{n 1}^K, E_{n i_\\textrm {max}}^D]$ for a short chain composed of sites from the impurity to site $n$ , where $E_{n 1}^K$ and $E_{n i_\\textrm {max}}^D$ are the lowest and highest energies of the set.",
"The energy level spacing between these eigenstates is of the order of $t_n \\sim \\Lambda ^{-n/2}$ .",
"Then, one separates the set into two subsets, the “discarded” energy eigenstates $\\lbrace | E_{n i}^D \\rangle \\rbrace $ and the “kept” eigenstates $\\lbrace | E_{n i}^K \\rangle \\rbrace $ , by energy.",
"Here these eigenstates are indexed by a common index $i$ such that their corresponding energy eigenvalues are in increasing order; the kept states are within energy window $[E_{n 1}^K, E_{n N_{\\mathrm {tr}}}^K]$ , while the discarded states are in $[E_{n, N_\\mathrm {tr} + 1}^D, E_{n,i_{\\max }}^D ]$ , where $N_\\mathrm {tr}$ is the number of the kept states and $i_{\\max }$ is the number of total states at a given iteration $n$ .",
"One typically takes $E_\\mathrm {tr} \\equiv (E_{n, N_\\mathrm {tr}+1}^D - E_{n, 1}^K)/\\Lambda ^{-n/2} \\gtrsim 7$ [40].",
"In the $(n+1)$ th diagonalization step, one constructs the Hilbert space $\\lbrace | E_{n i}^K \\rangle \\otimes | s_{n+1} \\rangle \\rbrace $ and diagonalize the Hamiltonian for a longer chain composed of the short chain and the next site $n+1$ .",
"One iterates these processes until one reaches the last site $N$ .",
"At the last iteration, all the eigenstates are discarded.",
"The discarded states $\\lbrace | E_{ni}^D \\rangle \\rbrace $ decouple from the states of the sites $n^{\\prime } > n$ , $\\lbrace | s_{n+1} \\rangle \\otimes \\cdots \\otimes | s_{N} \\rangle \\rbrace $ , which we call the environment states of $\\lbrace | E_{ni}^D \\rangle \\rbrace $ .",
"The whole Fock space can be constructed by the complete basis states of $\\Big \\lbrace | E_{ni\\vec{s}}^D \\rangle \\equiv | E_{ni}^D \\rangle \\otimes | s_{n+1} \\rangle \\otimes \\cdots \\otimes | s_N \\rangle \\Big | n=n_0, n_0 + 1, \\cdots N \\Big \\rbrace ,$ where $n_0$ is the earliest iteration at which the Hilbert space truncation happens.",
"These basis states can be used as the approximate eigenstates of the full Hamiltonian (the whole Wilson chain), and $E_{ni}^D$ provides an approximate eigenenergy.",
"Based on energy scale separation, the approximation error $\\delta E_{ni}^D$ for each energy $E_{ni}^D$ , which originates from neglecting its coupling to the environment states, is estimated by $\\delta E_{ni}^D / E_{ni}^D \\sim t_{n+1} / E_{n,N_\\mathrm {tr}+1}^D \\sim 1 / E_\\mathrm {tr} \\sqrt{\\Lambda } \\ll 1$ .",
"Therefore, for large enough $\\Lambda $ and $E_\\mathrm {tr}$ , the basis states in Eq.",
"(REF ) are efficient description of energy eigenstates, since the total number $O (N_\\mathrm {tr} N)$ of $\\lbrace |E_{ni}^D \\rangle \\rbrace $ is much smaller than $O (d^N)$ .",
"Using the complete basis states in Eq.",
"(REF ), one writes the thermal density matrix $\\rho _T$ at temperature $T$ as $\\begin{aligned}\\rho _T &= \\sum _{n=n_0}^N\\sum _{i\\vec{s}} \\frac{e^{-E_{ni}^D / T}}{Z} | E_{ni\\vec{s}}^D \\rangle \\!\\langle E_{ni\\vec{s}}^D | = \\sum _{n=n_0}^N \\mathcal {R}_n,\\end{aligned} \\\\\\mathcal {R}_n \\equiv \\rho _n^D \\otimes I_{n+1} \\otimes \\cdots \\otimes I_{N}, \\\\\\rho _n^D = \\sum _i \\frac{d^{N-n} e^{-E_{ni}^D/T}}{Z} | E_{ni}^D \\rangle \\!\\langle E_{ni}^D |, $ where $I_n = \\sum _{s_n} | s_n \\rangle \\!\\langle s_n | / d$ is the identity with normalization $\\mathrm {Tr}\\, I_n = 1$ , and $Z$ is the partition function."
],
[
"NRG approximation of correlation functions",
"The complete basis $\\lbrace | E_{ni\\vec{s}}^D \\rangle \\rbrace $ provides the systematic way of computing various physical properties.",
"One needs to use the NRG approximation [39], [40], to reduce the cost of computing matrix elements $\\langle E_{ni\\vec{s}}^D | \\mathcal {O} | E_{n^{\\prime }i^{\\prime }\\vec{s}^{\\prime }}^D \\rangle $ of an operator $\\mathcal {O}$ .",
"Since we will apply the NRG approximation to compute negativity in Sec.",
", we here briefly explain the NRG approximation for computing the impurity correlation function.",
"By using the complete basis, the impurity correlation function can be expressed in the Lehmann representation $\\begin{aligned}\\mathcal {A} (\\omega ) &\\equiv \\frac{1}{\\pi } \\mathrm {Im} \\int _{-\\infty }^{\\infty } \\mathrm {d} t \\, e^{i\\omega t} i \\Theta (t) \\mathrm {Tr}\\left(\\rho _T [ \\mathcal {O} (t), \\mathcal {O}^\\dagger ]_\\pm \\right) \\\\&= \\sum _{n n^{\\prime } i i^{\\prime } \\vec{s} \\vec{s}^{\\prime }} A_{(n i \\vec{s}),(n^{\\prime } i^{\\prime } \\vec{s}^{\\prime })} \\, \\delta (\\omega - \\omega _{(ni),(n^{\\prime }i^{\\prime })}),\\end{aligned}$ $\\begin{aligned}A_{(n i \\vec{s}),(n^{\\prime } i^{\\prime } \\vec{s}^{\\prime })} &= | \\langle E_{ni\\vec{s}}^D | \\mathcal {O} | E_{n^{\\prime }i^{\\prime }\\vec{s}^{\\prime }}^D \\rangle |^2 ( \\rho _{n i \\vec{s}} \\pm \\rho _{n^{\\prime }i^{\\prime }\\vec{s}^{\\prime }}), \\\\\\rho _{n i \\vec{s}} &= \\langle E_{ni\\vec{s}}^D | \\rho _T | E_{ni\\vec{s}}^D \\rangle = e^{- E_{ni}^D / T} / Z , \\\\\\omega _{(ni),(n^{\\prime }i^{\\prime })} &= E_{n^{\\prime }i^{\\prime }}^D - E_{ni}^D,\\end{aligned}$ where $\\mathcal {O}$ is the local operator acting on the impurity and $+$ ($-$ ) in $\\pm $ is for a fermionic (bosonic) operator $\\mathcal {O}$ .",
"Direct calculation of Eq.",
"(REF ) is impractical, since the number of matrix elements $A_{(n i \\vec{s}),(n^{\\prime } i^{\\prime } \\vec{s}^{\\prime })}$ is $O(N_\\textrm {tr}^2 d^{2N})$ .",
"To make the calculation feasible, one applies the NRG appriximation, with which the number is significantly reduced to $O(N_\\textrm {tr}^2 N)$ .",
"The approximation is accurate within the intrisic error of the NRG that the inaccuracy of the energies $E_{ni\\vec{s}}^D$ is estimated as $\\delta E^D_n \\sim \\Lambda ^{-(n+1)/2}$ .",
"We now explain the NRG approximation.",
"In the calculation of Eq.",
"(REF ), one applies the identity of $\\sum _{n^{\\prime } >n; i \\vec{s}} | E^D_{n^{\\prime } i \\vec{s}} \\rangle \\langle E^D_{n^{\\prime } i \\vec{s}} |= \\sum _{i^{\\prime } \\vec{s}^{\\prime }} |E^K_{n i^{\\prime } \\vec{s}^{\\prime }} \\rangle \\langle E^K_{n i^{\\prime } \\vec{s}^{\\prime }}|$ and approximately treats $|E^K_{n i^{\\prime } \\vec{s}^{\\prime }} \\rangle $ as an eigenstate of the full Hamiltonian, although $|E^K_{n i^{\\prime } \\vec{s}^{\\prime }} \\rangle $ is an eigenstate of the NRG chain with incomplete chain length $n+1$ .",
"As a result, an energy differences $\\omega _{(ni),(n^{\\prime }i^{\\prime })} = E_{n^{\\prime }i^{\\prime }}^D - E_{ni}^D$ is replaced by $E_{n i^{\\prime \\prime }}^K - E_{ni}^D$ if $n^{\\prime } > n$ or by $E_{n^{\\prime }i^{\\prime }}^D - E_{n^{\\prime }i^{\\prime \\prime }}^K$ if $n^{\\prime } < n$ .",
"The error in $\\omega _{(ni),(n^{\\prime }i^{\\prime })}$ , i.e., $\\delta \\omega _{(ni),(n^{\\prime }i^{\\prime })} \\sim \\Lambda ^{-(n+1)/2}$ due to this replacement, is comparable with the error of the Hilbert space truncation $\\sim \\delta E_{ni}^D \\sim \\Lambda ^{-(n+1)/2}$ .",
"The NRG approximation simplifies the summation in Eq.",
"(REF ) without inducing further numerical error: Only the matrix elements $ \\langle E_{ni\\vec{s}}^X | \\mathcal {O} | E_{n i^{\\prime }\\vec{s}^{\\prime }}^{X^{\\prime }} \\rangle $ diagonal in $n$ remain in the subsequent steps as $\\langle E_{ni}^X | \\mathcal {O} | E_{n i^{\\prime }}^{X^{\\prime }} \\rangle \\delta _{\\vec{s} \\vec{s}^{\\prime }}$ , which removes the sum over $\\sum _{\\vec{s}\\vec{s}^{\\prime }}$ and reduces the computation cost to $O(N_\\textrm {tr}^2 N)$ mentioned above.",
"Then $\\mathcal {A} (\\omega )$ becomes $\\mathcal {A} (\\omega ) \\approx \\sum _{n X X^{\\prime } i i^{\\prime }}^{(X, X^{\\prime }) \\ne (K, K)} \\tilde{A}_{n,(Xi) ,(X^{\\prime }i^{\\prime })} \\, \\delta (\\omega - \\tilde{\\omega }_{n,(Xi),(X^{\\prime }i^{\\prime })}),$ $\\begin{aligned}\\tilde{A}_{n,(Xi) ,(X^{\\prime }i^{\\prime })} &= | \\langle E_{ni}^X | \\mathcal {O} | E_{ni^{\\prime }}^{X^{\\prime }} \\rangle |^2 ( \\rho _{n i}^X \\pm \\rho _{n i^{\\prime }}^{X^{\\prime }}), \\\\\\rho _{n i}^X &= \\langle E_{ni}^X | \\rho _n^X | E_{ni}^X \\rangle , \\\\\\tilde{\\omega }_{n,(Xi),(X^{\\prime }i^{\\prime })} &= E_{ni^{\\prime }}^{X^{\\prime }} - E_{ni}^X ,\\end{aligned}$ where $(X,X^{\\prime }) = (D,D)$ , $(D,K)$ , or $(K,D)$ ; the case $(X,X^{\\prime }) = (K,K)$ is excluded to avoid double counting.",
"The density matrix $\\rho _n^K$ nK = $\\mathrm {Tr}$ n+1, , N [ n' > nN Rn' ], is introduced in the calculation; $\\mathcal {R}_{n^{\\prime }}$ is defined in Eq.",
"() and $\\mathrm {Tr}_{n+1, \\cdots , N} (\\cdot ) \\equiv \\sum _{s_{n+1}, \\cdots , s_{N}} \\langle s_{N} | \\otimes \\cdots \\otimes \\langle s_{n+1} | (\\cdot ) | s_{n+1} \\rangle \\otimes \\cdots \\otimes | s_{N} \\rangle $ .",
"Summarizing, consider a contribution to the spectral function, which involves the eigenstates $| E_{ni\\vec{s}}^D \\rangle $ and $| E_{n^{\\prime }i^{\\prime }\\vec{s}^{\\prime }}^D \\rangle $ from different iterations $n$ and $n^{\\prime } (> n)$ .",
"The NRG approximation neglects the detailed information of the later sites $n^{\\prime } > n$ by tracing them out.",
"Then the contribution is simplified to the approximated one involving the discarded and kept states at the same iteration, say $| E_{ni}^D \\rangle $ and $| E_{ni^{\\prime }}^K \\rangle $ .",
"As long as the energy scale separation $1/\\sqrt{\\Lambda }E_\\mathrm {tr} \\ll 1$ holds by appropriately choosing parameters ($\\Lambda $ , $N_\\mathrm {tr}$ , and/or $E_\\mathrm {tr}$ ), the result obtained after the NRG approximation is accurate; for example, the impurity spectral function at $\\omega = 0$ and $T = 0$ satisfies the Friedel sum rule within sub-1% error [39].",
"The NRG approximation is equivalent to the replacements of $\\mathcal {R}_n$ by $\\rho _n^D$ and $\\sum _{n^{\\prime } > n}^N \\mathcal {R}_{n^{\\prime }}$ by $\\rho _n^K$ in the calculation, $\\begin{gathered}\\mathcal {R}_n = \\rho _n^D \\otimes I_{n+1} \\otimes \\cdots \\otimes I_{N} \\rightarrow \\rho _n^D,\\\\\\sum _{n^{\\prime } > n}^N \\mathcal {R}_{n^{\\prime }} = \\sum _{n^{\\prime } > n}^N \\rho _{n^{\\prime }}^D \\otimes I_{n^{\\prime }+1} \\otimes \\cdots \\otimes I_{N} \\rightarrow \\rho _n^K.\\end{gathered}$ Here, the information of sites $n^{\\prime } > n$ is traced out.",
"This is in parallel to that $|E^D_{n i \\vec{s}} \\rangle $ and $|E^K_{n i \\vec{s}} \\rangle $ are approximately treated as an eigenstate of the full Hamiltonian.",
"We apply these replacements for computing $\\mathcal {N}$ below.",
"We propose how to compute the negativity $\\mathcal {N}$ in Eq.",
"() that quantifies the impurity-bath entanglement of the thermal density matrix $\\rho _T$ in Eq.",
"(REF ).",
"$\\mathcal {N} (\\rho _T)$ is computed in two steps, taking partial transpose on $\\rho _T$ to get $\\rho ^{T_A}_T$ and then diagonalizing $\\rho ^{T_A}_T$ .",
"However, one cannot compute $\\mathcal {N}$ directly applying these two steps, since the environment states $I_{n+1} \\otimes \\cdots \\otimes I_{N}$ in Eq.",
"(REF ) make the dimension of $\\rho _T$ exponentially large $\\sim O(d^N)$ .",
"We overcome this difficulty by utilizing the NRG approximation."
],
[
"NRG approximation of negativity",
"To start with, we decompose the expression of $\\mathcal {N} (\\rho _T)$ .",
"$\\mathcal {N}(\\rho _T) = \\mathcal {N} \\bigg ( \\sum _{n=n_0}^N \\mathcal {R}_n \\bigg ) = \\sum _{n=n_0}^N \\mathcal {N} (\\mathcal {R}_n) - \\sum _{n=n_0}^N \\delta _n, \\\\\\delta _n \\equiv \\mathcal {N} (\\mathcal {R}_n) + \\mathcal {N} \\bigg ( \\sum _{n^{\\prime } > n}^N \\mathcal {R}_{n^{\\prime }} \\bigg ) - \\mathcal {N} \\bigg ( \\mathcal {R}_n + \\sum _{n^{\\prime } > n}^N \\mathcal {R}_{n^{\\prime }} \\bigg ).$ In Eq.",
"(REF ), $\\mathcal {N}(\\rho _T)$ has two parts, $\\sum _n \\mathcal {N}(\\mathcal {R}_n)$ and $\\sum _n \\delta _n$ .",
"The first part $\\sum _n \\mathcal {N}(\\mathcal {R}_n)$ is the sum of the entanglement in each density matrix $\\mathcal {R}_n$ , and the second $\\sum _n \\delta _n$ counts contribution from mixtures of different $\\mathcal {R}_n$ 's.",
"Due to the convexity of the negativity [22], [23], $\\delta _n \\ge 0$ is guaranteed.",
"Equations (REF ) and () are exact, given construction of density matrix $\\rho _T$ .",
"One can derive the expression in Eq.",
"(REF ), applying the definition of $\\delta _n$ in Eq.",
"() recursively: (i) Start from the iteration step $n_0$ at which the first Hilbert space truncation happens during the iterative diagonalization.",
"Using Eq.",
"(REF ) and the definition of $\\delta _{n = n_0}$ , one decomposes the negativity $\\mathcal {N}(\\rho _T)$ as $\\mathcal {N}(\\rho _T) = \\mathcal {N} \\left( \\mathcal {R}_{n_0} \\right) - \\delta _{n_0} + \\mathcal {N} \\left( \\sum _{n^{\\prime } > n_0}^{N} \\mathcal {R}_{n^{\\prime }} \\right) .$ (ii) Next, we use an inductive argument.",
"Suppose that one can decompose the negativity $\\mathcal {N}(\\rho _T)$ as $\\mathcal {N}(\\rho _T) = \\sum _{n^{\\prime } = n_0}^n \\mathcal {N} (\\mathcal {R}_{n^{\\prime }}) - \\sum _{n^{\\prime }=n_0}^n \\delta _{n^{\\prime }} + \\mathcal {N} \\bigg (\\sum _{n^{\\prime } > n}^N \\mathcal {R}_{n^{\\prime }} \\bigg ) .$ Then, one decomposes Eq.",
"(REF ) by rewriting the last term in its right hand side using $\\delta _{n+1}$ (cf. Eq. ()).",
"$\\mathcal {N}(\\rho _T) = \\sum _{n^{\\prime } = n_0}^{n+1} \\mathcal {N} (\\mathcal {R}_{n^{\\prime }}) - \\sum _{n^{\\prime }=n_0}^{n+1} \\delta _{n^{\\prime }} + \\mathcal {N} \\bigg (\\sum _{n^{\\prime } > n+1}^N \\mathcal {R}_{n^{\\prime }} \\bigg ) .$ Notice that Eq.",
"(REF ) remains in the same form as the index $n$ increases to $n+1$ .",
"By induction, one obtains Eq.",
"(REF ).",
"Now we apply the NRG approximation to compute $\\delta _n$ .",
"The second and third terms on the right hand side of Eq.",
"() involve the density matrices $\\mathcal {R}_{n^{\\prime }}$ from different iterations $n^{\\prime } (> n)$ .",
"As done in the correlation functions (see Sec.",
"REF or Eq.",
"(REF )), we trace out the later sites $n^{\\prime } > n$ for the arguments $\\sum _{n^{\\prime }>n}^N \\mathcal {R}_{n^{\\prime }}$ and $\\mathcal {R}_n + \\sum _{n^{\\prime }>n}^N \\mathcal {R}_{n^{\\prime }}$ .",
"Accordingly we have N(Rn) N(nD), n $\\mathcal {N}$ ( nD ) + $\\mathcal {N}$ (nK) - $\\mathcal {N}$ ( nD + nK) n[0].",
"The superscript $[0]$ indicates that the NRG approximation is applied to $\\delta _n$ .",
"Then, the negativity $\\mathcal {N}(\\rho _T)$ is computed using $\\mathcal {N}(\\rho _n^D)$ , $\\mathcal {N}(\\rho _n^K)$ , and $\\mathcal {N}(\\rho _n^D + \\rho _n^K)$ .",
"The dimension of the matrices $\\rho _n^D$ , $\\rho _n^K$ , $\\rho _n^D + \\rho _n^K$ is independent of $N$ and less than or equal to $O(d N_\\textrm {tr})$ , which is exponentially smaller than the dimension $\\sim O(d^N)$ of $\\rho _T$ .",
"This reduction of the matrix size makes computation of $\\mathcal {N}$ feasible.",
"As we will discuss in Sec.",
", the error generated by the NRG approximation in Eq.",
"(REF ) is smaller than or comparable to the intrinsic error of the NRG in computing $\\mathcal {N}$ ."
],
[
"Constructing impurity-bath bipartite basis",
"To compute $\\mathcal {N}(\\rho _n^D)$ , $\\mathcal {N}(\\rho _n^K)$ , and $\\mathcal {N}(\\rho _n^D + \\rho _n^K)$ , one needs to represent the eigenstates $\\lbrace | E_{ni}^X \\rangle \\rbrace $ $(X=D,K)$ in the bipartite basis of the impurity and the bath as $| E_{ni}^X \\rangle $ j, s$\\mathrm {d}$ [TnX]s$\\mathrm {d}$ ,j,i $| s_\\mathrm {d} \\rangle $ $| \\phi _{nj} \\rangle $ .",
"Here $| s_\\mathrm {d} \\rangle $ is the impurity state, $| \\phi _{nj} \\rangle $ is the bath state satisfying $| \\phi _{nj} \\rangle \\in \\mathrm {span}\\lbrace | s_0 \\rangle \\otimes \\cdots \\otimes | s_n \\rangle \\rbrace $ , $\\langle \\phi _{nj} | \\phi _{nj^{\\prime }} \\rangle = \\delta _{jj^{\\prime }}$ , and $T_n^X$ is the “coefficient” tensor whose element is $[T_n^{X}]_{s_\\mathrm {d},j,i} = \\left( \\langle s_\\mathrm {d} | \\otimes \\langle \\phi _{nj} | \\right) | E_{ni}^X \\rangle .$ Given coefficient tensor $T_n^X$ , we express the states $\\rho _n^X$ in the basis of $\\lbrace | s_\\mathrm {d} \\rangle \\otimes | \\phi _{nj} \\rangle \\rbrace $ , to take the partial transpose $(\\rho _n^X)^{T_A}$ with respect to $\\lbrace | s_\\mathrm {d} \\rangle \\rbrace $ .",
"Then we evaluate $\\mathrm {Tr}\\, | (\\rho _n^X)^{T_A} |$ by obtaining the singular value decomposition (or equivalently, eigendecomposition) of $(\\rho _n^X)^{T_A}$ .",
"It is the same for the sum $\\rho _n^D + \\rho _n^K$ .",
"We iteratively construct $T_n^X$ from $T_{n-1}^{K}$ and $U_n^X$ , where $U_n^X$ is a left-unitary matrix which relates the eigenstates at iterations $n-1$ and $n$ , $\\begin{gathered}\\phantom{.}",
"[U_n^X]_{s_n,k,i} \\equiv \\left( \\langle s_n | \\otimes \\langle E_{n-1,k}^K | \\right) | E_{ni}^X \\rangle ,\\\\\\sum _{s_n,k} [U_n^{X}]^*_{s_n,k,i} [U_n^{X^{\\prime }}]_{s_n,k,i^{\\prime }} = \\delta _{XX^{\\prime }} \\delta _{ii^{\\prime }} ,\\end{gathered}$ where $X, X^{\\prime } = D, K$ .",
"We construct these matrices $T_n^X$ and $U_n^X$ during the standard NRG iterative diagonalization.",
"We start the iterative construction from $T_0^X$ with the bath state $| \\phi _{0,j=s_0} \\rangle \\equiv | s_{0} \\rangle $ , $[T_0^{X}]_{s_\\mathrm {d},s_0,i} \\equiv \\left( \\langle s_\\mathrm {d} | \\otimes \\langle s_0 | \\right) | E_{0i}^X \\rangle .$ Then consider an iteration $n$ , and suppose we know $T_{n-1}^K$ at the earlier iteration $n-1$ .",
"We first obtain $U_n^X$ which diagonalizes the Hamiltonian at the current iteration $n$ .",
"Then we construct the matrix $Q_n^X$ in terms of $T_{n-1}^{K}$ and $U_n^X$ as $\\begin{aligned}\\phantom{.",
"}[Q_n^X]_{(j^{\\prime },s_n),(s_\\mathrm {d},i)} &\\equiv \\left( \\langle s_\\mathrm {d} | \\otimes \\langle \\phi _{n-1,j^{\\prime }} | \\otimes \\langle s_n | \\right) | E_{ni}^X \\rangle \\\\&= \\sum _{k} [T_{n-1}^{K}]_{s_\\mathrm {d},j^{\\prime },k} [U_n^X]_{s_n, k, i}.\\end{aligned}$ To ensure the orthonormality of $\\lbrace | \\phi _{nj} \\rangle \\rbrace $ , we perform the singular value decomposition as $[Q_n^K + Q_n^D]_{(j^{\\prime }, s_n), (s_\\mathrm {d}, i)} = \\sum _{j} [V_L]_{(j^{\\prime }, s_n), j} [\\Sigma V_R^\\dagger ]_{j, (s_\\mathrm {d}, i)},$ where $V_L$ and $V_R$ are unitary matrices, $\\Sigma $ is the diagonal matrix of non-zero singular values, and $Q_n^K$ and $Q_n^D$ act on disjoint set of column indices $(s_\\mathrm {d}, i)$ .",
"Based on its unitarity, we assign $V_L$ as the matrix which defines the mapping from $\\lbrace |\\phi _{nj} \\rangle \\rbrace $ to $\\lbrace | \\phi _{n-1,j^{\\prime }} \\rangle \\otimes | s_n \\rangle \\rbrace $ such that $[V_L]_{(j^{\\prime }, s_n), j} = ( \\langle \\phi _{n-1,j^{\\prime }} | \\otimes \\langle s_n | ) |\\phi _{nj} \\rangle $ .",
"Hence we construct the desired tensor $T_n^X$ , [TnX]s$\\mathrm {d}$ ,j,i = j',sn [VL]*j,(j',sn) [QnX](j',sn),(s$\\mathrm {d}$ ,i) .",
"Note that $V_L$ is left-unitary; the multiplication of non-square $V_L^\\dagger $ in Eq.",
"(REF ) indicates the truncation of the bath Hilbert space.",
"After this iterative construction, the dimension of the bath space spanned by $\\lbrace | \\phi _{nj} \\rangle \\rbrace $ for a single $n$ scales as $O (d_\\mathrm {imp} N_\\mathrm {tr})$ ; the maximum number of non-zero singular values in the decomposition of Eq.",
"(REF ) is $O(d_\\mathrm {imp} N_\\mathrm {tr})$ .",
"Thus the matrix form of $\\rho _n^D + \\rho _n^K$ in the basis of $\\lbrace | s_\\mathrm {d} \\rangle \\otimes | \\phi _{nj} \\rangle \\rbrace $ has dimension $O (d_\\mathrm {imp}^2 N_\\mathrm {tr})$ .",
"The computational cost of evaluating the singular value decomposition of $(\\rho _n^D + \\rho _n^K)^{T_A}$ , which is the most computationally demanding part in computing the negativity, is the cube of the matrix dimension, i.e., $O (d_\\mathrm {imp}^6 N_\\mathrm {tr}^3)$ .",
"This estimation indicates that the cost of computing the negativity for the SIAM ($d_\\mathrm {imp} = 4$ ) will be 64 times larger than that for the SIKM ($d_\\mathrm {imp} = 2$ ) if the other numerical parameters are the same."
],
[
"Symmetry",
"Quantum impurity systems possess various symmetries such as $\\mathrm {U}(1)$ charge symmetry and $\\mathrm {SU}(2)$ spin symmetry.",
"The NRG exploits these symmetries to reduce the computational cost and to increase the numerical accuracy [45], [40], [46].",
"For example, a thermal density matrix $\\rho _T$ possesses the symmetries of its Hamiltonian, hence, it can be computed and represented efficiently in a block diagonal form whose blocks are labelled by the eigenvalues of the operators corresponding to the symmetries.",
"Unfortunately however, the symmetries cannot be fully exploited in computing the negativity.",
"Partial transpose can destroy the block diagonal form of the thermal density matrix $\\rho _T$ ; that is, a symmetry operator $Q$ satisfying $[Q, H] = 0$ commutes with $\\rho _T$ , but not necessarily with $\\rho _T^{T_A}$ .",
"For example, the SIKM has $\\mathrm {U}(1) \\times \\mathrm {U}(1)$ symmetry conserving spin-up charge (the corresponding symmetry operator is the spin-up particle number operator $Q_\\uparrow $ ) and spin-down charge ($Q_\\downarrow $ ).",
"Consider a nonzero matrix element $\\rho _{(\\Uparrow \\phi ), (\\Downarrow \\phi ^{\\prime })}$ of a density matrix $\\rho $ , where $| {\\Uparrow } \\rangle $ and $| {\\Downarrow } \\rangle $ are impurity spin states.",
"Both $| {\\Uparrow } \\rangle \\otimes | \\phi \\rangle $ and $| {\\Downarrow } \\rangle \\otimes | \\phi ^{\\prime } \\rangle $ have the same eigenvalues $(q_\\uparrow , q_\\downarrow )$ of $(Q_\\uparrow , Q_\\downarrow )$ .",
"After partial transpose, the matrix element $\\rho _{(\\Uparrow \\phi ), (\\Downarrow \\phi ^{\\prime })}$ is relocated to the position indexed by ${(\\Downarrow \\phi ), (\\Uparrow \\phi ^{\\prime })}$ , where $| {\\Downarrow } \\rangle \\otimes | \\phi \\rangle $ has an eigenvalues $(q_\\uparrow -1, q_\\downarrow +1)$ and $| {\\Uparrow } \\rangle \\otimes | \\phi ^{\\prime } \\rangle $ has an eigenvalues $(q_\\uparrow +1, q_\\downarrow -1)$ .",
"Therefore, to make $\\rho _T^{T_A}$ block-diagonal, one should resort to the weaker symmetry, i.e., the total charge conservation, leading to larger block size.",
"Even worse, for the SIAM, $\\rho _T^{T_A}$ does not respect even the total charge conservation, since the partial transpose on the impurity Hilbert space mixes up the blocks with different charges.",
"Since Hamiltonian symmetries may not be useful for computing $\\rho _T^{T_A}$ , we choose small $N_\\mathrm {tr} \\gtrsim 100$ to treat the SIKM and the SIAM within a practical cost.",
"We choose large $\\Lambda = 10$ to ensure energy scale separation with this small $N_\\mathrm {tr}$ .",
"Such large $\\Lambda = 10$ can yield accurate values of static, i.e., frequency-independent quantities; for example, impurity contributions, obtained with $\\Lambda = 10$ , to magnetic susceptibility or to specific heat agree with the Bethe ansatz result within a few $\\%$ [47].",
"We will show in Sec.",
"that our result of the negativity, obtained with small $N_\\mathrm {tr} \\gtrsim 100$ and large $\\Lambda = 10$ , is also sufficiently accurate."
],
[
"Negativity in the Kondo Model",
"We apply the method developed in the previous section to the SIKM.",
"In Fig.",
"REF , we compute the temperature dependence of the negativity $\\mathcal {N}$ that quantifies the impurity-bath entanglement in the SIKM.",
"The negativity $\\mathcal {N}$ exhibits a universal Kondo behavior at low temperature $T \\ll T_\\mathrm {K}$ , shows a thermal crossover around $T = T_\\mathrm {K}$ , and vanishes at high temperature $T \\gg T_\\mathrm {K}$ .",
"Here the Kondo temperature is defined as $T_\\mathrm {K}= \\sqrt{J/2D}e^{-2D/J}$ .",
"We first explain the universal behavior of the negativity $\\mathcal {N}$ at low temperature $T \\lesssim T_\\mathrm {K}$ .",
"The curves $\\mathcal {N} (T / T_\\mathrm {K})$ of different $J$ 's lie on top of each other.",
"At the strong-coupling fixed point of $T = 0^+$ , the impurity and the bath are entangled to form the Kondo spin singlet, as indicated by the maximal negativity $\\mathcal {N}=1$ .",
"At $T \\ll T_\\mathrm {K}$ , the negativity $\\mathcal {N}$ follows the power-law scaling $\\mathcal {N} \\simeq 1 - a_{\\mathcal {N},\\mathrm {1CK}} (T/T_\\mathrm {K})^2 ,$ where a coefficient $a_{\\mathcal {N},\\mathrm {1CK}} > 0$ is order $O(1)$ , as shown in Fig.",
"REF (b).",
"This quadratic dependence originates from the low-energy excitation of the Fermi-liquid quasiparticles [14], which can be confirmed by using the bosonization.",
"(See App.",
"for the details.)",
"The behavior of the negativity $\\mathcal {N}$ at $T \\lesssim T_\\mathrm {K}$ is consistent with that of the EoF [14] quantifying the impurity-bath entanglement in the SIKM.",
"Next we explain the behavior of the negativity $\\mathcal {N}$ at high temperature $T \\gtrsim T_\\mathrm {K}$ .",
"As $T$ increases from $0^+$ , the negativity $\\mathcal {N}$ exhibits the thermal crossover around Kondo temperature $T_\\mathrm {K}$ .",
"At high temperature $T \\gg T_\\mathrm {K}$ , the impurity and the bath are weakly correlated, having small negativity $\\mathcal {N} \\ll 1$ at the local-moment fixed point.",
"The negativity $\\mathcal {N}$ suffers sudden death [41] (within numerical noise) at $T = T_\\mathrm {SD} \\sim J$ [see Fig.",
"REF (c)], that is, $\\mathcal {N}$ is finite at $T < T_\\mathrm {SD}$ , while it vanishes at $T \\ge T_\\mathrm {SD}$ .",
"One can understand the linear dependence of $T_\\mathrm {SD}$ vs. $J$ from a minimal model $H_{N=0}^\\mathrm {SIKM}$ [see Eq.",
"(REF )].",
"$H_{N=0}^\\mathrm {SIKM}$ is composed of the impurity and only the nearest bath site, which describes the $T \\rightarrow \\infty $ limit of the Wilson chain since the effective chain length scales as $\\sim -2 \\log _\\Lambda T$ [39], [40].",
"We analytically show in App.",
"that the minimal model $H_{N=0}^\\mathrm {SIKM}$ exhibits the entanglement sudden death in terms of both the negativity and the EoF at $T = J / \\ln 3$ .",
"This provides the underlying mechanism of the linear dependence of $T_\\mathrm {SD}$ vs. $J$ .",
"Note that the entanglement sudden death also appears in other many-body systems at finite temperature [31], [32], [33]."
],
[
"Negativity in the Anderson Model",
"We next study the negativity between the impurity and the bath in the SIAM.",
"As the Anderson impurity has both spin and charge fluctuations, the negativity can be affected by the both.",
"In Fig.",
"REF we show the negativity $\\mathcal {N}$ between the whole degrees (spin and charge) of freedom of the impurity and the bath.",
"The negativity $\\mathcal {N}$ depends on $U$ , reflecting the dependence of the SIAM on $U$ .",
"The negativity $\\mathcal {N}$ has a different value at zero temperature $T=0^+$ .",
"Moreover, $\\mathcal {N}$ exhibits a crossover around $T=T_\\mathrm {SC}$ for any value of $U$ and another crossover around $T=T_\\mathrm {LM}$ for large U (e.g., $U=20\\Gamma $ ).",
"At zero temperature $T=0^+$ , the negativity $\\mathcal {N}$ in Fig.",
"REF (b) decreases with increasing $U$ , has a value 1 for $U \\rightarrow \\infty $ , and 3 for $U=0$ .",
"It happens since the charge fluctuation at the impurity is not completely suppressed (i.e., there is a finite probability that the impurity is empty or doubly occupied) for finite $U$ even at $T=0^+$ .",
"One can understand the $U$ -dependence of the negativity $\\mathcal {N}(T = 0^+)$ in the two limits of $U \\rightarrow \\infty $ and $U = 0$ as follows.",
"In the limit of $U \\rightarrow \\infty $ , the ground state of the SIAM is the Kondo singlet, since the SIAM reduces to the SIKM at low temperature [10].",
"Therefore, for $U \\rightarrow \\infty $ , the SIAM has the same value $\\mathcal {N} (T = 0^+) = 1$ as the SIKM.",
"In the limit of $U=0$ , the SIAM is equivalent to two copies of the resonant level model of spinless fermions, where each copy corresponds to the electron system of each spin.",
"Because of $\\epsilon _\\mathrm {d} = -U/2 = 0$ , the ground state of each copy is a Bell state, which is an equal-weight superposition of a state with the empty resonant level and the other state with the filled resonant level.",
"So the ground state of the SIAM at $U=0$ is a tensor product of two Bell states.",
"The negativity of this tensor product is 3, which can be understood using the logarithmic negativity.",
"The logarithmic negativity $\\log _2 (\\mathcal {N}+1)$ is a monotone function of the negativity $\\mathcal {N}$ , and the logarithmic negativity is additive though not convex [23].",
"Each Bell state has the logarithmic negativity $\\log _2 (\\mathcal {N}+1) = \\log _2 (1+1) = 1$ .",
"Due to the additivity, the logarithmic negativity is 2 for the tensor product of the two Bell states.",
"$\\log _2 (\\mathcal {N}+1) = 2$ means that for $U=0$ , the SIAM has the negativity $\\mathcal {N}(T=0^+) = 3$ .",
"At finite temperature $T$ , the negativity $\\mathcal {N}$ shows two kinks, one around $T=T_\\mathrm {SC}$ and another around $T=T_\\mathrm {LM}$ which indicate crossovers.",
"The crossover around $T = T_\\mathrm {SC}$ occurs for any value of $U$ , while the crossover around $T = T_\\mathrm {LM}$ appears only for sufficiently large $U$ (as for $U = 20\\Gamma $ ).",
"In Fig.",
"REF , we show that the crossovers correspond to those of the impurity entropy $S_\\mathrm {imp} \\equiv S_\\mathrm {tot}-S_\\mathrm {bath}$ , where $S_\\mathrm {tot}$ ($S_\\mathrm {bath}$ ) is the entropy of the impurity-bath system (of the bath only) [37].",
"The plateaus in $S_\\mathrm {imp}$ imply the fixed points in the SIAM, and the slanted lines connecting adjacent plateaus represent crossovers between the fixed points.",
"In the curve for $U=20\\Gamma $ in Fig.",
"REF (c), we observe three plateaus of $S_\\mathrm {imp}$ which have been interpreted as different fixed points: The plateau at the highest $T$ means the free-orbital fixed point, where the charge degree of freedom of the impurity is not frozen and the spin degree of freedom of the impurity is weakly correlated to the bath.",
"The intermediate plateau indicates the local-moment fixed point where the charge degree of freedom becomes frozen (i.e., only the singly occupied impurity states involve in the fixed-point Hamiltonian) for large $U$ and the spin degree of freedom is still weakly correlated to the bath.",
"$S_\\mathrm {imp}$ does not show clearly the intermediate plateau if $U/\\Gamma $ is not sufficiently large (e.g., when $U / \\Gamma =10$ and 5).",
"The plateau at the lowest $T$ corresponds to the strong-coupling fixed point in which the spin degrees of freedom of the impurity is strongly entangled with the bath, similarly to the strong-coupling fixed point in the SIKM.",
"In Fig.",
"REF (c), $T=T_\\mathrm {SC}$ is located at the end of the plateau for the strong-coupling fixed point for all values of $U$ , and $T = T_\\mathrm {LM}$ is located at the end of the intermediate plateau (the local-moment fixed point) of the $S_\\mathrm {imp}$ only for $U=20\\Gamma $ .",
"The comparison between $\\mathcal {N}$ and $S_\\mathrm {imp}$ shows that $\\mathcal {N}$ captures the fixed points and the crossovers between them.",
"Note that the dependence of $\\mathcal {N}(T = 0^+)$ vs. $U$ is not contradictory to the interpretation of the local-moment and strong-coupling fixed points.",
"The impurity states away from single occupation are not forbidden in these two fixed points; they merely do not participate in the effective Hamiltonian of these fixed points.",
"Thus the NRG result of the ground state, which includes the empty and doubly occupied impurity states, is consistent with the interpretation of the fixed points.",
"Figure: (Color online)(a) Temperature (TT) dependence of the negativity 𝒩 s \\mathcal {N}_s quantifying the entanglement between the impurity spin and the bath in the SIAM.Contrary to 𝒩\\mathcal {N} in Fig.",
", 𝒩 s \\mathcal {N}_s shows the same behavior as the negativity 𝒩\\mathcal {N} of the SIKM shown in Fig.",
"(a).At zero temperature T=0 + T = 0^+, 𝒩 s \\mathcal {N}_s is independent of UU,and around T=T LM T = T_\\mathrm {LM}, 𝒩 s \\mathcal {N}_s does not exhibit any kink.",
"(b) At low temperature T≪T K T \\ll T_\\mathrm {K}, 𝒩 s \\mathcal {N}_s has a quadratic dependence on TT,similarly to the negativity 𝒩\\mathcal {N} of the SIKM in Fig. (b).",
"(c) The probability 〈P n d =1 〉= Tr ρ s \\langle P_{n_d = 1} \\rangle = \\mathrm {Tr}\\rho _s that the impurity is singly occupied,as a function of U/ΓU/\\Gamma .",
"It increases as UU increases.Here the Kondo temperature T K T_\\mathrm {K} defined in Fig.",
"is used.Next we focus on the effect of the spin fluctuation on the entanglement between the impurity and the bath.",
"In Fig.",
"REF we compute the negativity $\\mathcal {N}_s$ between the spin degree of freedom of the impurity and the bath, after projecting out the doubly occupied and empty impurity states.",
"The negativity $\\mathcal {N}_s$ shows the same behavior as the negativity $\\mathcal {N}$ in the SIKM.",
"That is, $\\mathcal {N}_s$ is defined as $\\mathcal {N}$ s $\\mathcal {N}$ (s / $\\mathrm {Tr}$ s ), where $\\rho _s \\equiv P_{n_d = 1} \\rho _T P_{n_d = 1}$ , $\\rho _T$ the thermal density matrix in Eq.",
"(REF ), and $P_{n_d = 1}$ the projector onto the subspace in which the impurity is half-filled, i.e., $n_d = \\sum _\\mu n_{d \\mu } = 1$ .",
"The doubly occupied and empty impurity states are projected out by applying the projector $P_{n_d = 1}$ , so only the spin degree of freedom of the impurity remain.",
"Therefore, $\\mathcal {N}_s = 1$ means that the impurity spin and the bath are maximally entangled, as in the SIKM case.",
"The negativity $\\mathcal {N}_s(T=0^+)=1$ is independent of $U$ , which is due to the Kondo spin singlet formed by the impurity spin and the bath near the strong coupling fixed point.",
"At low temperature $T \\ll T_\\mathrm {K}$ near the strong-coupling fixed point, the negativity $\\mathcal {N}_s$ in Fig.",
"REF (b) shows a universal quadratic scaling behavior $\\mathcal {N}_s \\simeq 1 - a_{\\mathcal {N},\\mathrm {1CK}} (T/T_\\mathrm {K})^2$ .",
"This scaling behavior is the same as that of the impurity-bath negativity $\\mathcal {N}$ of the SIKM in Fig.",
"REF (b).",
"Moreover, $\\mathcal {N}_s$ has no kink around $T=T_\\mathrm {LM}$ , since the crossover around $T=T_\\mathrm {LM}$ , occuring between the local-moment fixed point and the free-orbital fixed point, involves only the change in charge fluctuations.",
"It is natural that $\\mathcal {N}_s$ in the SIAM shows the same behavior as $\\mathcal {N}$ in the SIKM at low temperature, since the SIKM can be obtained from the SIAM by restricting the impurity to be half-filled or suppressing charge fluctuations.",
"In contrast, the impurity-bath negativity $\\mathcal {N}$ of the SIAM does not show the low-temperature universal scaling because the charge fluctuation of the impurity does not participate in the universal Kondo physics.",
"In addition, we characterize the degree of the charge fluctuation at the impurity by using the probability $\\langle P_{n_d = 1} \\rangle = \\mathrm {Tr}\\rho _s$ of the single occupancy at the impurity, in Fig.",
"REF (c).",
"The single occupancy probability $\\langle P_{n_d = 1} \\rangle $ increases as $U$ increases, since the charge fluctuation gets suppressed.",
"It is consistent with the $U$ dependence of the $\\mathcal {N}(T=0^+)$ of the SIAM in Fig.",
"REF (b).",
"In the limit $U \\rightarrow \\infty $ , the charge fluctuation is completely suppressed to compel the impurity to be half-filled, so $\\mathcal {N}(T=0^+)=1$ and $\\langle P_{n_d=1} \\rangle = 1$ .",
"In the opposite limit $U = 0$ , the ground state is equivalent to the tensor product of two Bell states as discussed before.",
"In this case, $\\langle P_{n_d = 1} \\rangle = 1/2$ , since the ground state can be represented as an equal superposition of the four state vectors whose impurity states are fully occupied, spin-up, spin-down, and empty, respectively."
],
[
"Error analysis",
"We analyze the errors in the negativity calculation subject to the NRG method.",
"For the SIKM, for example, we investigate how the computed value of $\\mathcal {N}$ depends on the NRG approximation, the truncation in the iterative diagonalization, and the logarithmic discretization.",
"We first estimate how the NRG approximation affects the value of $\\mathcal {N}$ .",
"Under the NRG approximation in Eq.",
"(REF ), we replace $\\mathcal {R}_n$ and $\\delta _n$ by $\\rho _n^D$ and $\\delta _n^{[0]}$ , respectively, where the information of the chain site $n^{\\prime } > n$ is traced out.",
"This approximation can be improved by replacing $\\mathcal {R}_n$ and $\\delta _n$ by $\\rho _n^D$ and $\\delta _n^{[k]}$ , respectively, where the information of the chain site $n^{\\prime } > n +k $ is traced out.",
"The expression of $\\delta _n^{[k]}$ is n[k] $\\mathcal {N}$ ( $\\mathrm {Tr}$ n+k+1, , N [ $\\mathcal {R}$ n ] ) + $\\mathcal {N}$ ( $\\mathrm {Tr}$ n+k+1, , N [ n'>nN $\\mathcal {R}$ n' ] ) - $\\mathcal {N}$ ( $\\mathrm {Tr}$ n+k+1, , N [ $\\mathcal {R}$ n + n'>nN $\\mathcal {R}$ n' ] ) = $\\mathcal {N}$ ( nD In+1 In+k ) + $\\mathcal {N}$ ( n'>nn+k n'D In'+1 In+k + n+kK ) - $\\mathcal {N}$ ( n'=nn+k n'D In'+1 In+k + n+kK ), where $k = 0, 1, 2, \\cdots $ .",
"For $k = 0$ , Eq.",
"() reduces to Eq.",
"(REF ).",
"For larger $k$ , less information is traced out so that $\\mathcal {N}$ can be computed more precisely, however, the computation cost rapidly increases; as $k \\rightarrow \\infty $ , the calculation becomes exact within the NRG method.",
"Note that the replacement of $\\mathcal {R}_n$ by $\\rho _n^D$ is not affected although less information is traced out, because $\\mathcal {N}$ ($\\mathcal {R}$ n) = $\\mathcal {N}$ ($\\mathrm {Tr}$ n+k+1, , N [ $\\mathcal {R}$ n ] ) = $\\mathcal {N}$ (nD) .",
"In Fig.",
"REF , we show the magnitudes of $\\delta _n^{[0]}$ and of the deviations $\\delta _n^{[k]} - \\delta _n^{[0]}$ for $k = 1, 2$ .",
"In Fig.",
"REF , we display $|\\mathcal {N} (k) - \\mathcal {N} (k = 0)|$ for $k=1,2$ , where $\\mathcal {N}(k)$ is the computation of $\\mathcal {N}$ with the approximation of replacing $\\delta _n$ by $\\delta _n^{[k]}$ .",
"$|\\mathcal {N} (k =1,2) - \\mathcal {N} (k = 0)|$ is at most $O(10^{-3})$ for $T \\gtrsim T_\\mathrm {K}$ , and scale as $\\sim 10^{-3} \\times (T / T_\\mathrm {K})^2$ for $T \\ll T_\\mathrm {K}$ , showing that $|\\mathcal {N} (k =1,2) - \\mathcal {N} (k = 0)|$ is negligibly small.",
"These verify that the NRG approximation of $\\delta _n \\rightarrow \\delta _n^{[0]}$ is already good enough.",
"We next check the change of $\\mathcal {N}$ with varying an NRG parameter $N_\\textrm {tr}$ , the number of the kept states in each iteration step.",
"As shown in Fig.",
"REF , the change is negligible, showing that $\\mathcal {N}$ is almost independent of $N_\\textrm {tr}$ .",
"We notice that the change is comparable with $|\\mathcal {N} (k =1,2) - \\mathcal {N} (k = 0)|$ .",
"This is natural, since both of choosing smaller $N_\\textrm {tr}$ and smaller $k$ lead to common errors due to neglecting the information of a later part of the NRG chain.",
"This observation suggests that the amount of errors in computing $\\mathcal {N}$ due to the NRG approximation can be estimated by the change $\\mathcal {N}$ with varying $N_\\textrm {tr}$ .",
"This will provide a practical approach to estimate the errors due to the NRG approximation in general systems such as the multi-channel Kondo model, where the direct calculations of $\\delta _n^{[k]}$ ($k > 0$ ) are hardly feasible.",
"We also check the change of $\\mathcal {N}$ with varying the NRG discretization parameter $\\Lambda $ .",
"The change is also negligible in comparison with $\\mathcal {N}$ .",
"Note that the change of $\\mathcal {N}$ with $\\Lambda $ is larger than that with $N_\\textrm {tr}$ and $k$ .",
"It is because different values of $\\Lambda $ yield different discretized Hamiltonians.",
"The accuracy of our computation of $\\mathcal {N}$ can be also tested at $T > T_\\textrm {K}$ .",
"In this temperature range, the relevant length (less than 7) of the Wilson chain is so short that $\\mathcal {N}$ can be computed exactly by diagonalizing the whole NRG chain.",
"Figure REF (b) shows that our computation of $\\mathcal {N}$ with the NRG approximation is almost identical to the values obtained by the exact diagonalization.",
"All the above observations demonstrate that our computation of $\\mathcal {N}$ with the NRG approximation is sufficiently accurate."
],
[
"Conclusion",
"We develop the NRG method for computing the negativity $\\mathcal {N}$ quantifying an impurity-bath entanglement in a quantum impurity system at finite temperature, and apply it to the SIKM and the SIAM.",
"For the SIKM, the $T$ -dependence of $\\mathcal {N}$ shows the universal power-law scaling at low temperature, and the sudden death at high temperature.",
"For the SIAM, $\\mathcal {N}$ is affected by both the spin and charge fluctuations at the impurity.",
"The spin fluctuation causes $\\mathcal {N}$ to show a universal power-law scaling behavior similar to the SIKM.",
"The negativity $\\mathcal {N}$ depends on $U$ even at zero temperature, indicating that the charge fluctuation survives even near the strong-coupling fixed point for finite $U$ .",
"Since the error due to the NRG approximation is smaller than the other artifacts intrinsic to the NRG, our computation of $\\mathcal {N}$ is sufficiently accurate.",
"In this sense, the current scheme for computing the negativity is advantageous over the earlier one for the EoF [14]: The latter could only provide the lower and upper bounds of entanglement, and the interval between these bounds can exceed the intrinsic errors in the NRG.",
"We anticipate that our method will be applicable to general quantum impurity systems in various situations and reveal entanglement perspective in understanding them.",
"We thank A. Weichselbaum for fruitful discussion.",
"H.-S.S. and J.S.",
"are supported by Korea NRF (Grant Nos.",
"2015R1A2A1A15051869 and 2016R1A5A1008184).",
"S.-S.B.L.",
"acknowledges support from the Alexander von Humboldt Foundation and the Carl Friedrich von Siemens Foundation."
],
[
"Scaling behavior at low tmperature",
"We derive the scaling behavior of the impurity-bath negativity in Eq.",
"(REF ) for the SIKM at low $T\\ll T_{\\mathrm {K}}$ using the bosonization.",
"This scaling behavior originates from the low-energy excitations of the Fermi-liquid quasiparticles in the SIKM.",
"We set the thermal density matrix $\\rho = \\sum _i w_i |E_i\\rangle \\langle E_i|$ in terms of the energy eigenstate $|E_i\\rangle $ of the SIKM with energy $E_i$ and the Boltzmann factor $w_i$ of $|E_i\\rangle $ satisfying $\\sum _i w_i = 1$ .",
"$\\rho $ can be approximated by the eigenstates $\\lbrace |E_i \\rangle \\rbrace $ satisfying $E_i \\sim T$ , because $w_i$ decreases exponentially in $E_i / T$ while state degeneracy increases algebraically in $E_i$ .",
"To compute $\\mathcal {N}$ , we represent $\\rho $ in a bipartite basis of $\\lbrace |\\mu \\rangle \\otimes |\\phi _{i\\eta }\\rangle \\rbrace $ , where $\\lbrace |\\mu \\rangle \\rbrace $ ($\\lbrace |\\phi _{i\\eta }\\rangle \\rbrace $ ) is the orthonormal impurity (bath) basis.",
"Using the bosonization [48] and the effective theory near the strong-coupling fixed point [10], we represent the eigenstate $|E_i\\rangle $ as [14] |Ei = 12= , |( |i+ |i) , where $\\langle E_i | E_{i^{\\prime }}\\rangle = \\delta _{ii^{\\prime }}$ and $\\langle \\phi _{i\\eta }| \\phi _{i^{\\prime }\\eta ^{\\prime }}\\rangle = \\delta _{ii^{\\prime }}\\delta _{\\eta \\eta ^{\\prime }}$ .",
"$\\lbrace |\\chi _{i\\eta }\\rangle \\rbrace $ are bath states of $|\\chi _{i\\eta }\\rangle \\in \\mathrm {span}\\lbrace | \\phi _{i\\eta } \\rangle \\rbrace $ , satisfying $\\langle \\chi _{i\\eta }| \\phi _{i\\eta }\\rangle = 0$ , and $\\sqrt{ \\langle \\chi _{i\\eta }| \\chi _{i^{\\prime }\\eta ^{\\prime }}\\rangle } \\sim \\langle \\chi _{i\\eta }|\\phi _{i^{\\prime }\\eta ^{\\prime }}\\rangle \\sim O(T/T_\\mathrm {K})$ .",
"The latter relation is due to the Fermi-liquid behavior of the SIKM at low $T$ , and it determines the scaling exponent of the negativity.",
"Applying Eq.",
"(), we write the density matrix $\\rho $ as = ii',',,' = , [](, i, ), (', i',') |'| |ii''| , whose element is [](, i, ), (', i',') = jwj2 [ ij + i | j] [ ji''' + j'| i''] .",
"To obtain the negativity using Eq.",
"(), we need to compute $\\mathrm {Tr}|\\rho ^{T_A}|$ , where $\\rho ^{T_A}$ is TA = ii',',,' = , [](, i, ), (', i',') |'| |ii''|.",
"$\\mathrm {Tr}|\\rho ^{T_A}|$ , the sum of the singular values $\\sigma _{\\mu i \\eta }$ of $\\rho ^{T_A}$ , equals the sum of the square root of the singular values $\\sigma _{\\mu i \\eta }^2$ of $(\\rho ^{T_A})^2$ .",
"We compute the singular values of $(\\rho ^{T_A})^2$ , since they are easier to be estimated.",
"Using the facts that (i) the leading order and the next leading order of the diagonal terms of $(\\rho ^{T_A})^2$ are $O(1)$ and $O(T^2 / T^2_\\textrm {K})$ , respectively, (ii) the leading order of the off-diagonal terms of $(\\rho ^{T_A})^2$ are $O(T / T_\\textrm {K})$ , and (iii) $T / T_\\textrm {K} \\ll 1$ , we compute the singular values $\\sigma _{\\mu i \\eta }^2$ of $(\\rho ^{T_A})^2$ and find i = ci + c'i (T/TK)2 + , where $c_{\\mu i \\eta }$ and $c^{\\prime }_{\\mu i \\eta }$ are coefficients of order O(1).",
"Then, the impurity-bath negativity $\\mathcal {N}(\\rho )$ is obtained as $\\mathcal {N}$ () = Tr|TA| - $\\mathrm {Tr}$ = i i - 1 = c + a' (T/TK)2, where $c$ and $a^{\\prime }$ are constants.",
"Using the property of the SIKM that $\\mathcal {N}=1$ at $T=0$ and it cannot increase with increasing $T$ , we obtain Eq.",
"(REF ) at low $T \\ll T_{\\mathrm {K}}$ , N 1 - a$\\mathcal {N}$ ,$\\mathrm {1CK}$ (T/TK)2 , where a coefficient $a_{\\mathcal {N},\\mathrm {1CK}} >0$ is $O(1)$ ."
],
[
"Sudden death in the Impurity-Bath Entanglement",
"Here we explain the linear dependence of the sudden death temperature $T_\\mathrm {SD} \\sim J$ in the SIKM result of Fig.",
"REF (c), by considering the Wilson chain with only one bath site, i.e., $N = 0$ , as a minimal model.",
"For this minimal model, both the negativity and the EoF yields the same sudden death temperature $T_\\mathrm {SD} = J / \\ln 3$ .",
"Note that there is no bound entanglement at $T_\\mathrm {SD}$ , as the EoF, which can detect any bound entanglement, vanishes at $T_\\mathrm {SD}$ .",
"The energy eigenvalues and eigenstates of the Hamiltonian $H_{N=0}^\\mathrm {SIKM}$ are given by: $\\begin{tabular}{| c | c |}\\hline \\,\\,\\, Eigenvalue \\,\\,\\, & Eigenstate \\\\\\hline -3J/4 & \\,\\,\\, (| {\\Uparrow } \\rangle | {\\downarrow } \\rangle - | {\\Downarrow } \\rangle | {\\uparrow } \\rangle )/\\sqrt{2} \\,\\,\\, \\\\\\hline {3}{*}{J/4} & | {\\Uparrow } \\rangle | {\\uparrow } \\rangle \\\\& | {\\Downarrow } \\rangle | {\\downarrow } \\rangle \\\\& (| {\\Uparrow } \\rangle | {\\downarrow } \\rangle + | {\\Downarrow } \\rangle | {\\uparrow } \\rangle )/\\sqrt{2} \\\\\\hline {4}{*}{0} & | {\\Uparrow } \\rangle | {\\uparrow }{\\downarrow } \\rangle \\\\& | {\\Uparrow } \\rangle | 0 \\rangle \\\\& | {\\Downarrow } \\rangle | {\\uparrow }{\\downarrow } \\rangle \\\\& | {\\Downarrow } \\rangle | 0 \\rangle \\\\\\hline \\end{tabular}$ Here $| {\\Uparrow } \\rangle $ and $| {\\Downarrow } \\rangle $ are the impurity spin state, and $| 0 \\rangle $ , $| {\\uparrow } \\rangle $ , $| {\\downarrow } \\rangle $ , and $| {\\uparrow }{\\downarrow } \\rangle $ indicate the empty, spin-up, spin-down, and doubly occupied states of the electron bath site, respectively.",
"Then we construct the thermal density matrix $\\rho _{0}^\\mathrm {SIKM} = e^{-H_{N=0}^\\mathrm {SIKM}/T} / \\mathrm {Tr}\\, e^{-H_{N=0}^\\mathrm {SIKM}/T}$ based on the eigendecomposition above.",
"First, for the negativity, one can directly apply Eq.",
"() to the $\\rho ^\\mathrm {SIKM}_0$ to obtain N(SIKM0) = max(1 - 3e-J/T1 + 4e-3J/4T + 3e-J/T, 0 ) .",
"The negativity $\\mathcal {N}(\\rho ^\\mathrm {SIKM}_0)$ suffers sudden death at $T_\\mathrm {SD} = J/\\ln 3$ .",
"On the other hand, the EoF is defined as an optimization problem, $\\mathcal {E}_\\mathrm {F}(\\rho ) \\equiv \\inf _{\\lbrace p_i, | \\psi _i \\rangle \\rbrace } \\sum _i p_i \\, \\mathcal {E}_\\mathrm {E} (| \\psi _i \\rangle ),$ where $\\mathcal {E}_\\mathrm {E} (| \\psi _i \\rangle ) = - \\mathrm {Tr}\\rho _{iA} \\log _2 \\rho _{iA}$ is the entanglement entropy of $| \\psi _i \\rangle $ , and $\\rho _{iA} = \\mathrm {Tr}_B | \\psi _i \\rangle \\!\\langle \\psi _i |$ is the reduced density matrix in which the bath $B$ is traced out.",
"That is, the EoF for a mixed state $\\rho $ is the infimum of the weighted sum of the entanglement entropy, $\\sum _i p_i \\mathcal {E}_\\mathrm {E} (| \\psi _i \\rangle )$ , over all possible pure-state decomposition $\\rho = \\sum _i p_i | \\psi _i \\rangle \\!\\langle \\psi _i |$ .",
"Here $| \\psi _i \\rangle $ 's are normalized, i.e., $\\langle \\psi _i | \\psi _i \\rangle = 1$ , but do not need to be orthogonal to each other.",
"As mentioned in Sec.",
", there is no general solution of Eq.",
"(REF ).",
"But fortunately for $\\rho _0^\\mathrm {SIKM}$ , there exists an analytic solution, which we will derive by the following steps.",
"(i) The density matrix $\\rho _0^\\mathrm {SIKM}$ can be decomposed into a block diagonal form, $\\rho _0^\\mathrm {SIKM} = \\rho _1 + \\rho _2,$ where $\\rho _1 \\in \\mathcal {H}_1 \\equiv \\mathrm {span}\\lbrace | {\\Uparrow } \\rangle , | {\\Downarrow } \\rangle \\rbrace \\otimes \\mathrm {span}\\lbrace | {\\uparrow } \\rangle , | {\\downarrow } \\rangle \\rbrace $ and $\\rho _2 \\in \\mathcal {H}_2 \\equiv \\mathrm {span}\\lbrace | {\\Uparrow } \\rangle , | {\\Downarrow } \\rangle \\rbrace \\otimes \\mathrm {span}\\lbrace | 0 \\rangle , | {\\uparrow }{\\downarrow } \\rangle \\rbrace $ .",
"The bath site is half filled in the subspace $\\mathcal {H}_1$ , while empty or doubly occupied in $\\mathcal {H}_2$ .",
"In other words, $\\mathcal {H}_2$ is spanned by the energy eigenstates with zero eigenvalues, and $\\mathcal {H}_1$ by the rest.",
"(ii) Consider a pure state $| \\varphi \\rangle = c_1 | \\varphi _1 \\rangle + c_2 | \\varphi _2 \\rangle $ for arbitrary normalized states $| \\varphi _1 \\rangle \\in \\mathcal {H}_1$ and $| \\varphi _2 \\rangle \\in \\mathcal {H}_2$ , where $c_1$ and $c_2$ are complex numbers satisfying $|c_1|^2 + |c_2|^2 = 1$ .",
"Since the bath states of $| \\varphi _1 \\rangle $ and $| \\varphi _2 \\rangle $ are orthogonal by construction, we have $\\mathrm {Tr}_B | \\varphi \\rangle \\!\\langle \\varphi | = |c_1|^2 \\mathrm {Tr}_B | \\varphi _1 \\rangle \\!\\langle \\varphi _1 | + |c_2|^2 \\mathrm {Tr}_B | \\varphi _2 \\rangle \\!\\langle \\varphi _2 | .$ Then the concavity of the von Neumann entropy leads to an inequality $\\mathcal {E}_E(| \\varphi \\rangle ) \\ge |c_1|^2 \\mathcal {E}_E(| \\varphi _1 \\rangle ) + |c_2|^2 \\mathcal {E}_E(| \\varphi _2 \\rangle ) .$ Based on the block diagonal form in Eq.",
"(REF ) and this concavity, we find a restriction to the optimal pure-state decomposition $\\rho _0^\\mathrm {SIKM} = \\sum _i p_i^\\mathrm {op} | \\psi _i^\\mathrm {op} \\rangle \\!\\langle \\psi _i^\\mathrm {op} |$ , which provides $\\mathcal {E}_\\mathrm {F} (\\rho _0^\\mathrm {SIKM}) = \\sum _i p_i^\\mathrm {op} \\mathcal {E}_\\mathrm {E} (| \\psi _i^\\mathrm {op} \\rangle )$ : Each state $| \\psi _i^\\mathrm {op} \\rangle $ should be in either $\\mathcal {H}_1$ or $\\mathcal {H}_2$ , not a superposition of a state in $\\mathcal {H}_1$ and another in $\\mathcal {H}_2$ .",
"(It can be proven by contradiction.)",
"Therefore, the EoF reduces to $\\begin{aligned}\\mathcal {E}_\\mathrm {F} (\\rho _0^\\mathrm {SIKM}) &= \\mathcal {E}_\\mathrm {F} (\\rho _1) + \\mathcal {E}_\\mathrm {F} (\\rho _2) \\\\&= \\mathcal {E}_\\mathrm {F} (\\rho _1) \\\\&= \\mathrm {Tr}\\, \\rho _1 \\cdot \\mathcal {E}_\\mathrm {F} (\\rho _1 / \\mathrm {Tr}\\, \\rho _1)\\end{aligned}$ where at the second equality we used $\\mathcal {E}_\\mathrm {F} (\\rho _2) = 0$ since $\\rho _2$ is the mixture of product states [see Eq.",
"(REF )], and at the last equality we pulled out the normalization factor $\\mathrm {Tr}\\, \\rho _1 = \\frac{ e^{3J/4T} + 3 e^{J/4T} }{ e^{3J/4T} + 3 e^{J/4T} + 4} ,$ for convenience below.",
"(iii) We can regard $\\rho _1$ as the state of two qubits; now we can use the concurrence [49] to derive the EoF of the normalized state $\\rho _1 / \\mathrm {Tr}\\, \\rho _1$ , $\\mathcal {E}_\\mathrm {F} \\left( \\frac{\\rho _1}{\\mathrm {Tr}\\, \\rho _1} \\right) = h \\left( \\frac{1 + \\sqrt{1 - \\mathcal {C}^2}}{2} \\right), $ where $h(x) = -x \\log _2 x - (1-x) \\log _2 (1-x)$ and $\\mathcal {C}$ is the concurrence of $\\rho _1 / \\mathrm {Tr}\\, \\rho _1$ .",
"Here the right-hand side expression of Eq.",
"(REF ) is a monotonically increasing function of $\\mathcal {C}$ .",
"The concurrence is given by $\\mathcal {C} = \\mathrm {max} \\Big ( \\frac{e^{J/T} - 3}{e^{J/T} + 3}, \\, 0 \\Big )$ which indicates that $\\mathcal {E}_\\mathrm {F} (\\rho _1 / \\mathrm {Tr}\\, \\rho _1)$ , and $\\mathcal {E}_\\mathrm {F}(\\rho _0^\\mathrm {SIKM})$ also, suffer the sudden death at $T_\\mathrm {SD} = J / \\log 3$ .",
"Both the negativity and the EoF yield the same $T_\\mathrm {SD}$ , which means that there is no bound entanglement.",
"It is natural, since the entanglement of $\\rho _0^\\mathrm {SIKM}$ is contributed only from $\\rho _1$ that can be regarded as a two-qubit state, and there is no bound entanglment for two qubits in general."
]
] | 1808.08506 |
[
[
"Consensus-Before-Talk: Distributed Dynamic Spectrum Access via\n Distributed Spectrum Ledger Technology"
],
[
"Abstract This paper proposes Consensus-Before-Talk (CBT), a spectrum etiquette architecture leveraged by distributed ledger technology (DLT).",
"In CBT, secondary users' spectrum access requests reach a consensus in a distributed way, thereby enabling collision-free distributed dynamic spectrum access.",
"To achieve this consensus, the secondary users need to pay for the extra request exchanging delays.",
"Incorporating the consensus delay, the end-to-end latency under CBT is investigated.",
"Both the latency analysis and numerical evaluation validate that the proposed CBT achieves the lower end-to-end latency particularly under severe secondary user traffic, compared to the Listen-Before-Talk (LBT) benchmark scheme."
],
[
"Introduction",
"Unlicensed spectrum bands are envisaged to be at the cusp of the collapse, due to the unprecedented overuse by a huge number of WiFi devices [1] as well as by the contribution from cellular devices such as licensed assisted access (LAA) [2], [3].",
"Their number of access requests may become too large to be supported by traditional unlicensed spectrum etiquettes that include Carrier Sensing Multiple Access with Collision Avoidance (CSMA/CA) [4], [5] and Listen-Before-Talk (LBT) [6], [7], [8] for standalone WiFi and WiFi-cellular coexistence scenarios, respectively.",
"Furthermore, even with the unlicensed spectrum, a cellular-grade latency guarantee is expected to be demanded, in order for WiFi connections to provide seamless WiFi-to-cellular experiences.",
"A compelling example could be an ultra-reliable and low-latency communication (URLLC) scenario where a target latency constraint should be ensured anytime anywhere [9], [10], [11], regardless of the connection types.",
"This void is difficult to be filled by the traditional spectrum etiquettes that commonly incur random back-off delays due to collisions [4], [5], as illustrated in Fig. 1-a.",
"In order to resolve the aforementioned spectrum etiquette problems, by leveraging distributed ledger technology (DLT) we propose a novel unlicensed spectrum etiquette, Consensus-Before-Talk (CBT).",
"In CBT, the unlicensed users' spectrum requests are first come to a consensus in a distributed way, yielding a distributed spectrum ledger (DSL).",
"The DSL, stored at each user, contains a consensual sequence of the spectrum requests.",
"This sequence is ordered by a pre-defined consensus policy, thereby enabling the distributed dynamic spectrum access of the users without any collision, as shown in Fig. 1-b.",
"To enjoy this benefit, CBT needs to pay for the extra consensus latency.",
"For the purpose of minimizing this latency without suffering from severe interference, inspired by the Hashgraph algorithm [12], the users in CBT exchange their spectrum access requests using a gossip protocol, and achieve their consensus by the local computation at each user.",
"In this paper, the resulting end-to-end CBT latency is investigated, and its effectiveness is highlighted by comparing it with a benchmark LBT scheme via analysis and numerical evaluations, followed by the discussion for possible extensions to the proposed CBT.",
"Figure: An illustration of (a) listen-before-talk (LBT) and (b) the proposed consensus-before-talk (CBT) spectrum etiquettes."
],
[
"Related Works",
"Towards enabling distributed DSA, unlicensed spectrum etiquettes have been suggested, which rely on the transmitter-side information [4], [5], [6], [7], [8] or on the receiver-side information [13], [14].",
"These approaches are commonly rooted in random carrier-sensing techniques, and thus are still insufficient for supporting a large number of users with a strict latency guarantee due to collisions.",
"With the aid central management, as used in Citizens Broadband Radio Service (CBRS) [15], [16], [17] and Licensed Shared Access (LSA) [18], [19], one may partly control the spectrum access requests, thereby ameliorating the collision problem.",
"Nevertheless, due to the central controller's complexity, it is difficult to cope with dynamic user traffic, motivating this research.",
"Towards implementing DLT, distributed consensus algorithms have been studied under both large-scale [20], [21] and small-scale systems [22], [23], [24], with the scalability in terms of the number of participating users.",
"In general, large-scale consensus algorithms suffer from too long consensus latency, e.g., avg.",
"10 min.",
"in Blockchain [20].",
"On the contrary, small-scale consensus algorithms, such as Byzantine fault tolerant tolerance (BFT) schemes, guarantee a fast consensus latency.",
"Nonetheless, they cannot further increase the participating users, e.g., up to a few dozen participants [22], [23], [24], because of the transaction exchange delays before starting the consensus process.",
"To resolve this latency-scalability trade-off, directed acyclic graph (DAG) based solutions have been propose [25], [26], [12].",
"As opposed to the conventional Blockchain structure, each transaction in a DAG based algorithm is connected to more than one transactions so that their connections become tangled.",
"Since a DAG has a topological ordering upon the vertices of the graph, the transaction order can readily be retrieved, thereby preventing double-spending problems.",
"Furthermore, from the perspective of the security guarantee, Blockchain has a single-chained structure, and tries to guarantee its security level by inserting dummy computation, i.e.",
"Proof-of-Work (PoW), in-between the vertices, i.e., blocks.",
"On the contrary, a DAG based solution has the vertices multi-dimensionally connected to the other vertices, and this tangled structure automatically guarantees its security level, which can be beneficial for saving power.",
"Recently, one of the DAG based solutions, the Hashgraph algorithm [12] has been proposed, which minimizes the BFT algorithm's transaction exchange bottleneck via a local consensus protocol, thereby allowing the algorithm to support far more participating users.",
"To be specific, the Hashgraph algorithm employs a random gossip algorithm as a means of disseminating transactions.",
"Generally, a gossip algorithm randomly selects the source-destination pairs, and spreads information in an asynchronous fashion.",
"When it comes to its BFT application, on the one hand, it is efficient for quickly disseminating information without suffering from severe interference.",
"On the other hand, its asynchronous dissemination may incur stragglers, and obstructs completing the dissemination, prerequisite to BFT consensus, thus bringing about too long BFT consensus latency.",
"On this account, the Hashgraph algorithm exploits an asynchronous BFT consensus protocol that operates simultaneously with the asynchronous gossip dissemination.",
"Inspired by this protocol, in this paper we design a local consensus algorithm for CBT."
],
[
"Consensus-Before-Talk (CBT) Architecture",
"In this section, we describe the propose CBT architecture.",
"For the sake of convenience, following the convention in DSA, hereafter we consider primary and secondary users, and focus primarily on the secondary users' spectrum access.",
"With non-zero primary users, it may correspond to a WiFi-cellular coexistence scenario; otherwise, it can be interpreted as a stand-alone WiFi scenario.",
"As shown in Fig.",
"2, the CBT architecture comprises: a spectrum access transaction (SAT), a distributed spectrum ledger (DSL), and a consensus policy module.",
"In CBT, a secondary user's access request is encapsulated in SAT and exchanged with all the other secondary users.",
"For each received SAT, the secondary user initiates a consensus protocol with a pre-defined consensus policy.",
"Once it reaches the consensus with all the other secondary users, the SAT is verified, and is stored in the secondary user's local DSL.",
"Each component of the said consensus process is detailed in the following subsections."
],
[
"Spectrum Access Transaction (SAT)",
"Each secondary user generates a single SAT when the user requests spectrum access.",
"At the SAT, the user records its generation timestamp and the corresponding digital signature created by a public-key algorithm, e.g., Rivest-Shamir-Adleman (RSA) cryptosystem [27].",
"Then, the SAT is exchanged with all the other secondary users by using a gossip protocol.",
"For each received SAT, the secondary user first verifies the digital signature by the public key of the SAT generator, and then adds the record of its verified timestamp.",
"As a result, each SAT contains (i) a single generated timestamp and (ii) verified timestamps cumulatively recorded by the secondary users.",
"Note that any verified timestamp is larger than the generated timestamp, due to the propagation delays."
],
[
"Distributed Spectrum Ledger (DSL)",
"Each secondary user possesses a DSL.",
"As shown in Fig.",
"2, the DSL consists of spectrum access queue (SAQ), spectrum access history (SAH), a consensus policy module, and a header.",
"The SAQ is a queue of the SATs, and is managed by the consensus policy module that enables the consensus process and adjusts the scheduling priority of the consensus SATs.",
"The arrival rate of the SAQ is determined by the secondary user traffic as well as by the gossip and consensus delays in CBT.",
"The service rate of the SAQ is set as the number of maximum accessible secondary users for a unit timespan that is hereafter given as a number $\\mu $ of time slots.",
"The maximum accessible amount is determined by the primary user access information stored in the header.",
"This information is periodically updated by an external spectrum sensing entity, as done in Citizens Broadband Radio Service (CBRS) [15], [16], [17] and Licensed Shared Access (LSA) [18], [19].",
"The served SATs are kept stored on the SAH that can affect the consensus policy as detailed next.",
"Figure: An illustration of a Distributed Spectrum Ledger (DSL) that contains spectrum access queue (SAQ) and spectrum access history (SAH)."
],
[
"Consensus Policy",
"The consensus algorithm of CBT is based on practical Byzantine fault tolerance [23], thus ensuring the following conditions: Termination (Liveliness) – All SATs will be eventually known by all the secondary users; Validity (Correctness) – Invalid SATs cannot be validated by the secondary users; and Agreement (Consistency) – Two secondary users should not have disagreement on the validity and the time order of SATs.",
"For simplicity, we henceforth assume that all the secondary users are honest, while neglecting Byzantine users that obstruct the consensus process.",
"The impact of the Byzantine users on CBT is to be elaborated in Section .",
"In CBT, the consensus objective is to enable secondary users to follow a pre-defined scheduling rule in a distributed way.",
"To this end, at first each secondary user's spectrum access request becomes associated with its verified timestamps recorded in its corresponding SAT.",
"Such SATs are then exchanged and verified by all secondary users.",
"After the verification, the accumulated verified timestamps reach a consensus, following a pre-defined consensus algorithm stored in the consensus policy module within each DSL.",
"Motivated by the gossip-of-gossip protocol in Hashgraph [12], the CBT consensus algorithm is locally operated at each secondary user.",
"To elaborate, as exemplified in Fig.",
"3, consider user 1 generated an SAT, and the SAT is propagated through the following order: users 2$\\rightarrow $ 3$\\rightarrow $ 1$\\rightarrow $ 2$\\rightarrow $ 3.",
"Denoting as $t_{i}(i)$ and $t_{i}(j)$ with $j\\ne i$ the generated timestamp of user $i$ and its verified timestamp by user $j$ , respectively, the generated SAT's consensus timestamp $\\hat{t}_i(j)$ at user $j$ is given as $\\hat{t}_i(j)=\\sum _{k\\ne j} t_{i}(k) /n$ , where $n$ is the number of secondary users.",
"Here, the consensus timestamp calculation at each user neglects its own verified timestamp, in order to avoid any selfish manipulation.",
"So long as $n$ is sufficiently large, the consensus timestamps for different users become almost identical.",
"Thus, each user can independently and locally calculate its own consensus time, while achieving global consensus.",
"The SAT containing the consensual timestamp is then stored in the SAQ of each user, according to a pre-defined scheduling rule.",
"Two possible scheduling examples are described as below.",
"First verified, first served – The spectrum access priority is determined by the verification timestamp order.",
"For instance, if $\\hat{t}_1(j)\\le \\hat{t}_2(j)$ , then $\\mathrm {SAT}_1$ is placed prior to $\\mathrm {SAT}_2$ in the SAQ of user $j$ .",
"Fairness guarantee – Exploiting the SAH, one can maximize the fairness by prioritizing the access requests from the least served users.",
"In a similar way, one may avoid selfish users occupying excessive spectrum bands by first counting their number of access requests in the SAH and then adjusting their next SAQ priority.",
"Figure: An example of the consensus algorithm in CBT.",
"For the SAT generated by user 1, it is propagated to users 2→\\rightarrow 3→\\rightarrow 1→\\rightarrow 2→\\rightarrow 3.",
"Afterwards, each user jj independently computes its local consensus timestamp t ^ 1 (j)\\hat{t}_1(j)."
],
[
"End-to-End Latency Analysis",
"Following the CBT architecture proposed in Sect.",
"II, in this section we derive an analytic expression of the end-to-end latency under CBT, as well as under its benchmark LBT.",
"The end-to-end latency is determined as the average delay from a secondary user's access request generation to its successful access, to be formally defined with a proper network model in the following subsections."
],
[
"Network model",
"The network under study consists of a number $n$ of secondary users that are sharing the frequency-time resource with primary users.",
"During a unit time span, set as $\\mu $ time slots, a number $n_r\\le n$ of the secondary users request their spectrum access.",
"Assuming the primary user traffic information is known from a database server as done in [15], [16], [17], [18], [19], these secondary users can access up to a number $n_v$ of vacant resource blocks during the unit time span.",
"For the sake of convenience, we consider each user access consumes a single resource block, and focus only the case $n_r\\le n_v$ with a uniformly randomly selected secondary user, referred to as a typical user.",
"The typical user generates its access request at time $t_0\\in (i \\mu , (i+1)\\mu ]$ with $i>0$ .",
"For CBT, the typical user's end-to-end latency $T_\\text{CBT}$ incorporates the consensus delay.",
"For LBT, on the contrary, the end-to-end latency $T_\\text{LBT}$ includes all the back-off delays until the first successful access."
],
[
"LBT Latency",
"The end-to-end latency under the benchmark LBT scheme is evaluated as follows.",
"We consider a large number of secondary users having limited listening coverages, leading to collisions due to their hidden node problem [4], [5] and simultaneous spectrum access attempts.",
"Each access collision consumes a constant back-off delay set identically as the unit time span, i.e., $\\mu $ time slots.",
"Assuming that each secondary user uniformly randomly selects its access resource block out of the total $n_v$ blocks, the typical user avoids any collision with probability $(1-1/n_v)^{n_r - 1}$ .",
"Due to the backed-off secondary uses, at $t=i\\mu $ , an average number $\\bar{n}_{r,i}$ of secondary users attempt spectrum access.",
"Its sequence $[\\bar{n}_{r,i}]$ is an increasing sequence that satisfies the following relation.",
"$\\bar{n}_{r,i+1} = n_{r} + \\bar{n}_{r,i} \\left(1 - \\left(1 - 1/n_v\\right)^{\\bar{n}_{r,i} - 1} \\right),$ where the initial value equals $\\bar{n}_{r,1} = n_r$ .",
"According to the fixed-point theorem [28], if $n_r$ satisfies the condition $n_r \\le -\\frac{1}{e (1-1/n_v)\\log (1-1/n_v)},$ then as $i \\rightarrow \\infty $ , $[\\bar{n}_{r,i}]$ converges to a certain fixed point $\\hat{n}_r$ that is the smallest root of the equation $x(1-1/n_v)^{x-1} - n_r = 0$ .",
"In this case, for some $\\bar{n}_{r,i+1}$ , the access success probability becomes $(1-1/n_v)^{\\bar{n}_{r,i+1} - 1}$ .",
"By supposing that the number of access failure follows a geometric distribution with the parameter given by the access success probability, the typical user's average aggregate back-off delay until the first access success is given as $ T_\\mathrm {LBT} = \\left( \\frac{1}{\\left(1- 1/ n_v\\right)^{\\hat{n}_r-1}} - 1 \\right)\\mu + \\frac{\\mu }{2} ,$ where ${\\mu }/{2}$ comes from the average waiting time for the first access attempt after the generation of an access request.",
"On the other hand, if $n_r$ does not satisfy the condition (REF ), then $[\\bar{n}_{r,i}]$ diverges as $i \\rightarrow \\infty $ .",
"It implies that the number of collisions keep increasing over time, and eventually any secondary user access becomes unavailable.",
"Finally, combining this with (REF ) and (REF ), we obtain the typical user's end-to-end latency under LBT as: $T_{\\mathrm {LBT}} ={\\left\\lbrace \\begin{array}{ll}\\frac{\\mu }{\\left(1- 1/ n_v\\right)^{\\hat{n}_r-1}} - \\frac{1}{2}\\mu & \\mathrm {if\\ (\\ref {eq:converge})}\\\\\\infty & \\text{otherwise}.\\end{array}\\right.",
"}$"
],
[
"CBT Latency",
"Assuming the delays incurred by the local consensus time stamp calculations addressed in Sect.",
"II-C are negligibly small, the consensus latency for the typical user is given by the delay brought by the typical user's disseminating its SAT to all the other secondary users.",
"For the SAT dissemination, we consider a push gossip algorithm [29], where the SAT is transmitted to a randomly chosen target user, regardless of whether the target user has already received the SAT or not.",
"During the SAT dissemination process, we neglect their interference and collisions in that the gossip algorithm can easily mitigate the concurrent transmissions within a small region.",
"With the push gossip algorithm at time $t \\ge t_0$ , our focus is to derive the dissemination delay $t-t_0$ such that the average number $n_s(t)$ of users who received the typical user's SAT becomes the entire $n-1$ users.",
"To this end, following [30], [31], $n_s(t)$ is given by a logistic difference equation: $n_s(t+1) = n_s(t) + \\phi n_s(t)\\left(1-\\frac{n_s(t)}{n} \\right).",
"$ where $\\phi $ denotes the number of receivers that can be concurrently connected to a single transmitter; e.g., $\\phi =1$ implies a one-to-one pairwise communication, while $\\phi >1$ indicates one-to-many broadcast communication.",
"Applying the inverse of the Euler's approximation, (REF ) in discrete time domain is recast as the following differential equation in continuous time domain: $\\frac{d}{dt}n_s(t) = \\phi n_s(t) \\left(1-\\frac{n_s(t)}{n}\\right),$ with the initial condition $n_s(t_0) = 1$ .",
"By solving (REF ), we obtain $n_s(t) = \\frac{n}{1+(n-1)e^{- \\phi (t - t_0)}}.$ In (REF ), it reads $\\lim _{t\\rightarrow \\infty } n_s(t)= n$ , thus asymptotically guaranteeing the termination condition in Sect. II-C.",
"In order to derive non-asymptotic delay, we suppose the dissemination of an SAT becomes completed at $t$ if a fraction $\\gamma =n_s(t)/n<1$ of the users receive the SAT, where the target gossip success proportion $\\gamma $ is set as the value close to 1, e.g., 0.999.",
"Then, the corresponding dissemination delay $t-t_0$ of the typical user's SAT equals $t-t_0 = \\frac{1}{\\phi }\\log \\left(\\frac{1+(n-1)\\gamma }{1-\\gamma } \\right).",
"$ Note that (REF ) is the SAT dissemination delay.",
"As addressed in Sect.",
"II-C, this does not guarantee to achieve the consensus, which requires one additional round of the dissemination.",
"Incorporating this, we finally obtain the end-to-end latency under CBT: $T_{\\mathrm {CBT}} = \\frac{2 n_r}{\\phi }\\log \\left(\\frac{1+(n-1)\\gamma }{1-\\gamma } \\right) + \\frac{\\mu }{2},$ where the first multiplication term $n_r$ in (REF ) is because there exist $n_r$ spectrum access requesting secondary users for every $\\mu $ time slots.",
"The term ${\\mu }/{2}$ comes from the average waiting time of the SAQ."
],
[
"Simulation results",
"In this section we numerically evaluate the effectiveness of the proposed CBT.",
"A random push gossip protocol is used for CBT in all simulations and we assume that all communications during gossip algorithm is pairwise, i.e., $\\phi = 1$ .",
"All simulation results are the averaged output of 10,000 iterations.",
"Fig.",
"REF describes the time required to disseminate a SAT to the secondary users, i.e., gossip delay, versus the fraction of secondary users who received SAT.",
"The number of secondary users is fixed $n = 1,000$ and only a single SAT is generated in the network, If the fraction $\\gamma = 0.99$ , it means that 990 secondary users received the SAT.",
"First of all, the figure shows that the gossip algorithm analyzed in REF is precise in some degree, when $\\gamma <0.996$ .",
"On the other hand, if $\\gamma \\ge 0.996$ , the gap between the simulation and analysis increase and the analysis of gossip delay tends to diverge eventually.",
"From the simulation, the gossip delay of a complete gossiping, i.e., $\\gamma = 1$ is around $14.5$ .",
"The delay is almost same as the gossip delay obtained by analysis when $\\gamma = 0.999$ .",
"Therefore, we fix $\\gamma = 0.999$ , i.e., the 99.9% of the secondary users successfully exchange all the SATs during $\\mu $ time slots, in the rest of the experiments.",
"Fig.",
"REF shows the latency performance comparison between LBT and the proposed CBT protocol, with respect to the number of secondary access requests in $\\mu $ time slots, that is $n_{r}$ .",
"The latency is normalized by $\\mu = 1,000$ , $5,000$ and $10,000$ , and the number of secondary users and the number of vacant resource blocks in every $\\mu $ time slots is fixed to $n = 1,000$ and $n_v = 100$ , respectively.",
"Note that $\\mu $ is closely related to the ratio between the size of SAT and the information size communicated via a single access by a primary or secondary user.",
"Clearly, the normalized latency of LBT is independent of $\\mu $ , however, the normalized latency becomes smaller as $\\mu $ increases in the proposed CBT.",
"For $\\mu = 5,000$ and $10,000$ , the normalized latency of CBT is small compared to that of LBT.",
"On the other hand, when $\\mu = 1,000$ , LBT outperforms CBT, which means that the consensus process in CBT requires more than $1,000$ time slots.",
"However, the latency of LBT increases exponentially as $n_r$ increases and near $n_r = 34$ , there is a crossing point and CBT outperforms LBT.",
"For $n_r \\ge 36$ , the normalized latency of LBT diverges, since from (REF ), LBT cannot serve more than 36 secondary accesses in $\\mu $ time slots due to collisions between the secondary access requests.",
"Fig.",
"REF shows the latency performance comparison between LBT and CBT, with respect to the number of secondary users in the network.",
"In order to show the case when there exists a crossing point on the latency performance of LBT and CBT as the number of secondary users increases, we fix the numbers $n_r = 10$ and $\\mu = 2,500$ in this experiment.",
"In the figure, it is shown that the normalized latency of CBT increases logarithmically with respect to the increase of the number of secondary users, while the latency of LBT remains unchanged and unaffected by the increase of the secondary users.",
"This is because the required time for a consensus process in CBT is largely dependent on the number of participating users in the consensus process, while LBT does not have this kind of process.",
"This scalability issue will be discussed in the later section and in the future works.",
"Figure: Gossip delay versus the gossip success proportion.Figure: Latency normalized with μ=1,000\\mu = 1,000, 5,0005,000 and 10,00010,000 versus the number of secondary access requests in μ\\mu time slots for LBT and the proposed CBT.Figure: Latency normalized with μ=2,500\\mu = 2,500 versus the number of secondary users for LBT and CBT."
],
[
"Discussion",
"In order to further improve the proposed CBT spectrum etiquette architecture, this section describes (i) how many users are need to participate in, (ii) how to exchange their SATs, and (iii) how to calculate the consensus timestamps, as follows."
],
[
"Consensus Participation – Direct vs. Representative",
"Reaching a consensus with many users are more tolerant to the various attacks compared to reaching a consensus with small number of users.",
"However, if there is too many nodes in the consensus process, it will cause a critical latency problem, since the time required to reach a consensus increases as the number of nodes increases.",
"Hopefully, if there are a set of nodes, so called representative nodes, that can operate a consensus process as a representative, the secondary users can have a common result on the spectrum scheduling in a short period time.",
"In other words, depending on how many of the representative nodes are chosen, there will be a trade-off between the consensus latency and the security performance.",
"Therefore, for a given required condition for latency and security, we will investigate and discuss about the optimal number of representative consensus nodes in our future works."
],
[
"Gossip Protocol – Push vs. Pull",
"The gossip algorithm considered in this paper is push gossip.",
"In push gossip, the users who have the gossip message randomly selects the receivers and pushes the message to the target.",
"As the time goes by, the number of users who does not have the message will become smaller, however, disseminating speed of the gossip message will slow down since the receivers are randomly selected among all the users and the portion of the users who have not yet got the message becomes smaller.",
"On the other hand, in pull gossip, the users who have the message makes use of a side information obtained from the users who have not yet got the message.",
"In other words, the users who have not yet got the message requests the gossip message from the message holders.",
"This makes the message holders to disseminate the message by choosing the random users among the smaller set of users.",
"However, in order to deliver the side information, it costs extra communication resources.",
"Therefore in order to implement an efficient gossip algorithm in our structure, mixing two types of gossip algorithm can be one solution.",
"For example, at the beginning, the gossip message holders use push gossip to disseminate the message.",
"After some time, the disseminating speed will be slowed down and then the gossip algorithm can be changed to pull gossip to speed up the dissemination."
],
[
"Consensus Timestamp Calculation – Mean vs. Median",
"In the previous section, we assumed that all the secondary users are honest.",
"However, in the real world environment, a selfish secondary can be even more selfish so that they cheat during the consensus process.",
"For example, Alice and Bob generate $\\mathrm {SAT}_a$ and $\\mathrm {SAT}_b$ , respectively, at time $t = 0$ and $\\mathrm {SAT}_a$ is verified by Bob, Carol and David at $t = 2$ .",
"Meanwhile, $\\mathrm {SAT}_b$ is verified by Carol and David at $t =1$ and if Alice is a cheater and selfish so that she wants to put her transaction into the queue before that of Bob, she can simply delay verifying $\\mathrm {SAT}_b$ and stamp time at, say $t = 98$ .",
"Then the computed average timestamp of $\\mathrm {SAT}_a$ and $\\mathrm {SAT}_b$ will be 2 and 50, respectively, which puts $\\mathrm {SAT}_a$ in a higher priority.",
"Meanwhile, rather than taking the average timestamp as the representative value for the distributed consensus, using the median timestamp in the consensus process is more tolerant to Byzantine attacks.",
"In the above example, if the median timestamp is taken to decide access priority of the users, then Bob accesses before Alice since the median timestamp of $\\mathrm {SAT}_a$ and $\\mathrm {SAT}_b$ is 2 and 1, respectively, which is tolerant to the selfish attack by Alice."
],
[
"Conclusion",
"This paper proposes a CBT spectrum etiquette based on the distributed spectrum ledger technology.",
"We introduced the structure and mechanism of CBT and also analyzed it from a technical point of view.",
"Specifically, a latency performance is compared with the conventional LBT and showed that under high secondary traffic environment, the proposed CBT performs better since it avoids collisions via distributed consensus based spectrum scheduling."
],
[
"Acknowledgment",
"This work was supported in part by the Academy of Finland project CARMA, and 6Genesis Flagship (grant no.",
"318927), in part by the INFOTECH project NOOR, in part by the Kvantum Institute strategic project SAFARI, and in part by the Korea Electric Power Corporation (grant no.",
"R17XA05-63)."
]
] | 1808.08356 |
[
[
"Semantic-Unit-Based Dilated Convolution for Multi-Label Text\n Classification"
],
[
"Abstract We propose a novel model for multi-label text classification, which is based on sequence-to-sequence learning.",
"The model generates higher-level semantic unit representations with multi-level dilated convolution as well as a corresponding hybrid attention mechanism that extracts both the information at the word-level and the level of the semantic unit.",
"Our designed dilated convolution effectively reduces dimension and supports an exponential expansion of receptive fields without loss of local information, and the attention-over-attention mechanism is able to capture more summary relevant information from the source context.",
"Results of our experiments show that the proposed model has significant advantages over the baseline models on the dataset RCV1-V2 and Ren-CECps, and our analysis demonstrates that our model is competitive to the deterministic hierarchical models and it is more robust to classifying low-frequency labels."
],
[
"Introduction",
"Multi-label text classification refers to assigning multiple labels for a given text, which can be applied to a number of important real-world applications.",
"One typical example is that news on the website often requires labels with the purpose of the improved quality of search and recommendation so that the users can find the preferred information with high efficiency with less disturbance of the irrelevant information.",
"As a significant task of natural language processing, a number of methods have been applied and have gradually achieved satisfactory performance.",
"For instance, a series of methods based on machine learning have been extensively utilized in the industries, such as Binary Relevance [3].",
"BR treats the task as multiple single-label classifications and can achieve satisfactory performance.",
"With the development of Deep Learning, neural methods are applied to this task and achieved improvements [31], [20], [2].",
"However, these methods cannot model the internal correlations among labels.",
"To capture such correlations, the following work, including ML-DT [5], Rank-SVM [6], LP [27], ML-KNN [32], CC [23], attempt to capture the relationship, which though demonstrated improvements yet simply captured low-order correlations.",
"A milestone in this field is the application of sequence-to-sequence learning to multi-label text classification [21].",
"Sequence-to-sequence learning is about the transformation from one type of sequence to another type of sequence, whose most common architecture is the attention-based sequence-to-sequence (Seq2Seq) model.",
"The attention-based Seq2Seq [26] model is initially designed for neural machine translation (NMT) [1], [19].",
"Seq2Seq is able to encode a given source text and decode the representation for a new sequence to approximate the target text, and with the attention mechanism, the decoder is competent in extracting vital source-side information to improve the quality of decoding.",
"Multi-label text classification can be regarded as the prediction of the target label sequence given a source text, which can be modeled by the Seq2Seq.",
"Moreover, it is able to model the high-order correlations among the source text as well as those among the label sequence with deep recurrent neural networks (RNN).",
"Nevertheless, we study the attention-based Seq2Seq model for multi-label text classification [21] and find that the attention mechanism does not play a significant role in this task as it does in other NLP tasks, such as NMT and abstractive summarization.",
"In Section , we demonstrate the results of our ablation study, which show that the attention mechanism cannot improve the performance of the Seq2Seq model.",
"We hypothesize that compared with neural machine translation, the requirements for neural multi-label text classification are different.",
"The conventional attention mechanism extracts the word-level information from the source context, which makes little contribution to a classification task.",
"For text classification, human does not assign texts labels simply based on the word-level information but usually based on their understanding of the salient meanings in the source text.",
"For example, regarding the text “The young boys are playing basketball with great excitement and apparently they enjoy the fun of competition”, it can be found that there are two salient ideas, which are “game of the young” and “happiness of basketball game”, which we call “semantic units” of the text.",
"The semantic units, instead of word-level information, mainly determine that the text can be classified into the target categories “youth” and “sports”.",
"Semantic units construct the semantic meaning of the whole text.",
"To assign proper labels for text, the model should capture the core semantic units of the source text, the higher-level information compared with word-level information, and then assign the text labels based on its understanding of the semantic units.",
"However, it is difficult to extract information from semantic units as the conventional attention mechanism focuses on extracting word-level information, which contains redundancy and irrelevant details.",
"In order to capture semantic units in the source text, we analyze the texts and find that the semantic units are often wrapped in phrases or sentences, connecting with other units with the help of contexts.",
"Inspired by the idea of global encoding for summarization [18], we utilize the power of convolutional neural networks (CNN) to capture local interaction among words and generate representations of information of higher levels than the word, such as phrase or sentence.",
"Moreover, to tackle the problem of long-term dependency, we design a multi-level dilated convolution for text to capture local correlation and long-term dependency without loss of coverage as we do not apply any form of pooling or strided convolution.",
"Based on the annotations generated by our designed module and those by the original recurrent neural networks, we implement our hybrid attention mechanism with the purpose of capturing information at different levels, and furthermore, it can extract word-level information from the source context based on its attention on the semantic units.",
"In brief, our contributions are illustrated below: We analyze that the conventional attention mechanism is not useful for multi-label text classification, and we propose a novel model with multi-level dilated convolution to capture semantic units in the source text.",
"Experimental results demonstrate that our model outperforms the baseline models and achieves the state-of-the-art performance on the dataset RCV1-v2 and Ren-CECps, and our model is competitive with the hierarchical models with the deterministic setting of sentence or phrase.",
"Our analysis shows that compared with the conventional Seq2Seq model, our model with effective information extracted from the source context can better predict the labels of low frequency, and it is less influenced by the prior distribution of the label sequence."
],
[
"Attention-based Seq2Seq for Multi-label Text Classification",
"As illustrated below, multi-label text classification has the potential to be regarded as a task of sequence prediction, as long as there are certain correlation patterns in the label data.",
"Owing to the correlations among labels, it is possible to improve the performance of the model in this task by assigning certain label permutations for the label sequence and maximizing subset accuracy, which means that the label permutation and the corresponding attention-based Seq2Seq method are competent in learning the label classification and the label correlations.",
"By maximizing the subset accuracy, the model can improve the performance of classification with the assistance of the information about the label correlations.",
"Regarding label permutation, a straightforward method is to reorder the label data in accordance with the descending order by frequency, which shows satisfactory effects [4].",
"Multi-label text classification can be regarded as a Seq2Seq learning task, which is formulated as below.",
"Given a source text $x = \\lbrace x_1, ..., x_i, ..., x_n\\rbrace $ and a target label sequence $y = \\lbrace y_1, ..., y_i, ..., y_m\\rbrace $ , the Seq2Seq model learns to approximate the probability $P(y|x) = \\prod _{t=1}^{m}P(y_t|y_{<t}, x)$ , where $P(y_t|y_{<t}, x)$ is computed by the Seq2Seq model, which is commonly based on recurrent neural network (RNN).",
"The encoder, which is bidirectional Long Short-Term Memory (LSTM) [9], encodes the source text $x$ from both directions and generates the source annotations $h$ , where the annotations from both directions at each time step are concatenated ($h_i\\!=\\!",
"[\\overrightarrow{h_i}; \\overleftarrow{h_i}]$ ).",
"To be specific, the computations of $\\overrightarrow{h_i}$ and $\\overleftarrow{h_i}$ are illustrated below: $\\overrightarrow{{h_{i}}} = {LSTM}({x_{i}}, \\overrightarrow{{h_{i-1}}},{C_{i-1}}) \\\\\\overleftarrow{{h_{i}}} = {LSTM}({x_{i}}, \\overleftarrow{{h_{i-1}}}, {C_{i-1}}) $ We implement a unidirectional LSTM decoder to generate labels sequentially.",
"At each time step $t$ , the decoder generates a label $y_{t}$ by sampling from a distribution of the target label set $P_{vocab}$ until sampling the token representing the end of sentence, where: $P_{vocab} &= g(y_{t-1}, c_{t},s_{t-1})$ where $g(\\cdot )$ refers to non-linear functions including the LSTM decoder, the attention mechanism as well as the softmax function for prediction.",
"The attention mechanism generates $c_t$ as shown in the following: $c_{t} &= \\sum ^{n}_{i=1} \\alpha _{t,i}h_{i}\\\\\\alpha _{t,i} &= \\frac{exp(e_{t,i})}{\\sum _{j=1}^{n}exp(e_{t,j})}\\\\e_{t,i} &= s_{t-1}^{\\top }W_{a}h_{i}$"
],
[
"Problem",
"As we analyze the effects of the attention mechanism in multi-label text classification, we find that it contributes little to the improvement of the model's performance.",
"To verify the effects of the attention mechanism, we conduct an ablation test to compare the performance of the Seq2Seq model without the attention mechanism and the attention-based SeqSeq model on the multi-label text classification dataset RCV1-v2, which is introduced in detail in Section .",
"As is shown in Table REF , the Seq2Seq models with and without the attention mechanism demonstrate similar performances on the RCV1-v2 according to their scores of micro-${\\rm F_1}$ , a significant evaluation metric for multi-label text classification.",
"This can be a proof that the conventional attention mechanism does not play a significant role in the improvement of the Seq2Seq model's performance.",
"We hypothesize that the conventional attention mechanism does not meet the requirements of multi-label text classification.",
"A common sense for such a classification task is that the classification should be based on the salient ideas of the source text.",
"The semantic units, instead of word-level information, mainly determine that the text can be classified into the target categories “youth” and “sports”.",
"For each of a variety of texts, there are always semantic units that construct the semantic meaning of the whole text.",
"Regarding an automatic system for multi-label text classification, the system should be able to extract the semantic units in the source text for better performance in classification.",
"Therefore, we propose our model to tackle this problem."
],
[
"Proposed Method",
"In the following, we introduce our proposed modules to improve the conventional Seq2Seq model for multi-label text classification.",
"In general, it contains two components: multi-level dilated convolution (MDC) as well as hybrid attention mechanism."
],
[
"Multi-level Dilated Convolution",
"On top of the representations generated by the original encoder, which is an LSTM in our model, we apply the multi-layer convolutional neural networks to generate representations of semantic units by capturing local correlations and long-term dependencies among words.",
"To be specific, our CNN is a three-layer one-dimensional CNN.",
"Following the previous work [11] on CNN for NLP, we use one-dimensional convolution with the number of channels equal to the number of units of the hidden layer, so that the information at each dimension of a representation vector will not be disconnected as 2-dimension convolution does.",
"Besides, as we are to capture semantic units in the source text instead of higher-level word representations, there is no need to use padding for the convolution.",
"A special design for the CNN is the implementation of dilated convolution.",
"Dilation has become popular in semantic segmentation in computer vision in recent years [30], [28], and it has been introduced to the fields of NLP [10] and speech processing [22].",
"Dilated convolution refers to convolution inserted with “holes” so that it is able to remove the negative effects such as information loss caused by common down-sampling methods, such as max-pooling and strided convolution.",
"Besides, it is able to expand the receptive fields at the exponential level without increasing the number of parameters.",
"Thus, it becomes possible for dilated convolution to capture longer-term dependency.",
"Furthermore, with the purpose of avoiding gridding effects caused by dilation (e.g., the dilated segments of the convolutional kernel can cause missing of vital local correlation and break the continuity between word representations), we implement a multi-level dilated convolution with different dilation rates at different levels, where the dilation rates are hyperparameters in our model.",
"Instead of using the same dilation rate or dilation rates with the common factor, which can cause gridding effects, we apply multi-level dilated convolution with different dilation rates, such as [1,2,3].",
"Following [28], for $N$ layers of 1-dimension convolution with kernel size $K$ with dilation rates $[r_1, ..., r_N]$ , the maximum distance between two nonzero values is $max(M_{i+1} - 2r_i, M_{i+1}-2(M_{i+1}-r_{i}), r_{i})$ with $M_N = r_N$ , and the goal is $M_2 \\le K$ .",
"In our experiments, we set the dilation rates to [1, 2, 3] and $K$ to 3, and we have $M_2 = 2$ .",
"The implementations can avoid the gridding effects and allows the top layer to access information between longer distance without loss of coverage.",
"Moreover, as there may be irrelevant information to the semantic units at a long distance, we carefully design the dilation rates to [1, 2, 3] based on the performance in validation, instead of others such as [2, 5, 9], so that the top layer will not process the information among overlong distance and reduce the influence of unrelated information.",
"Therefore, our model can generate semantic unit representations from the information at phrase level with small dilation rates and those at sentence level with large dilation rates."
],
[
"Hybrid Attention",
"As we have annotations from the RNN encoder and semantic unit representations from the MDC, we design two types of attention mechanism to evaluate the effects of information of different levels.",
"One is the common attention mechanism, which attends to the semantic unit representations instead of the source word annotations as the conventional does, the other is our designed hybrid attention mechanism to incorporate information of the two levels.",
"The idea of hybrid attention is motivated by memory networks [25] and multi-step attention [8].",
"It can be regarded as the attention mechanism with multiple “hops”, with the first hop attending to the higher-level semantic unit information and the second hop attending to the lower-level word unit information based on the decoding and the first attention to the semantic units.",
"Details are presented below.",
"For the output of the decoder at each time step, it not only attends to the source annotations from the RNN encoder as it usually does but also attends to the semantic unit representations from the MDC.",
"In our model, the decoder output first pays attention to the semantic unit representations from the MDC to figure out the most relevant semantic units and generates a new representation based on the attention.",
"Next, the new representation with both the information from the decoding process as well as the attention to the semantic units attends to the source annotations from the LSTM encoder, so it can extract word-level information from the source text with the guidance of the semantic units, mitigating the problem of irrelevance and redundancy.",
"To be specific, for the source annotations from the LSTM encoder $h = \\lbrace h_1, ..., h_i, ..., h_n\\rbrace $ and the semantic unit representations $g = \\lbrace g_1, ..., g_i, ..., g_m\\rbrace $ , the decoder output $s_t$ first attends to the semantic unit representations $g$ and generates a new representation $s^{\\prime }_t$ .",
"Then the new representation $s^{\\prime }_t$ attends to the source annotations $h$ and generates another representation $\\tilde{s}_t$ following the identical attention mechanism as mentioned above.",
"In the final step, the model generates $o_t$ for the prediction of $y_t$ , where: $o_t &= s^{\\prime }_t \\oplus \\tilde{s}_t$ For comparison, we also propose another type of attention called “additive attention”, whose experimental results are in the ablation test.",
"In this mechanism, instead of paying attention to the two types of representations step by step as mentioned above, the output of the LSTM decoder $s_t$ attends to the semantic unit representations $g$ and the source annotations $h$ respectively to generate $s^{\\prime }_t$ and $\\tilde{s}_t$ , which are finally added element-wisely for the final output $o_t$ ."
],
[
"Experiment Setup",
"In the following, we introduce the datasets and our experiment settings as well as the baseline models that we compare with."
],
[
"Datasets and Preprocessing",
"Reuters Corpus Volume I (RCV1-v2)http://www.ai.mit.edu/projects/jmlr/papers/volume5/lewis04a/lyrl2004_rcv1v2_README.htm: The dataset [15] consists of more than 800k manually categorized newswire stories by Reuters Ltd. for research purpose, where each story is assigned with multiple topics.",
"The total number of topics is 103.",
"To be specific, the training set contains around 802414 samples, while the development set and test set contain 1000 samples respectively.",
"We filter the samples whose lengths are over 500 words in the dataset, which removes about 0.5% of the samples in the training, development and test sets.",
"The vocabulary size is set to 50k words.",
"Numbers as well as out-of-vocabulary words are masked by special tokens “#” and “UNK”.",
"For label permutation, we apply the descending order by frequency following [12].",
"Ren-CECps: The dataset is a sentence corpus collected from Chinese blogs, annotated with 8 emotion tags anger, anxiety, expect, hate, joy, love, sorrow and surprise as well as 3 polarity tags positive, negative and neutral.",
"The dataset contains 35096 sentences for multi-label text classification.",
"We apply preprocessing for the data similar to that for the RCV1-v2, which are filtering samples of over 500 words, setting the vocabulary size to 50k and applying the descending order by frequency for label permutation."
],
[
"Experiment Settings",
"We implement our experiments in PyTorch on an NVIDIA 1080Ti GPU.",
"In the experiments, the batch size is set to 64, and the embedding size and the number of units of hidden layers are 512.",
"We use Adam optimizer [13] with the default setting $\\beta _{1}=0.9$ , $\\beta _{2}=0.999$ and $\\epsilon =1\\times 10^{-8}$ .",
"The learning rate is initialized to $0.0003$ based on the performance on the development set, and it is halved after every epoch of training.",
"Gradient clipping is applied with the range [-10, 10].",
"Following the previous studies [32], [4], we choose hamming loss and micro-${\\rm F_1}$ score to evaluate the performance of our model.",
"Hamming loss refers to the fraction of incorrect prediction [24], and micro-${\\rm F_1}$ score refers to the weighted average ${\\rm F_1}$ score.",
"For reference, the micro-precision as well as micro-recall scores are also reported.",
"To be specific, the computations of Hamming Loss (HL) micro-${\\rm F_1}$ score are illustrated below: $HL &= \\frac{1}{L} \\sum \\mathbb {I}(y \\ne \\hat{y})\\\\microF_1 &= \\frac{\\sum _{j=1}^{L}2tp_j}{\\sum _{j=1}^{L}2tp_j+fp_j+fn_j}$ where $tp_j$ , $fp_j$ and $fn_j$ refer to the number of true positive examples, false positive examples and false negative examples respectively."
],
[
"Baseline Models",
"In the following, we introduce the baseline models for comparison for both datasets.",
"Binary Relevance (BR) [3] transforms the MLC task into multiple single-label classification problems.",
"Classifier Chains (CC) [23] transforms the MLC task into a chain of binary classification problems to model the correlations between labels.",
"Label Powerset (LP) [27] creates one binary classifier for every label combination attested in the training set.",
"CNN [12] uses multiple convolution kernels to extract text feature, which is then input to the linear transformation layer followed by a sigmoid function to output the probability distribution over the label space.",
"CNN-RNN [4] utilizes CNN and RNN to capture both global and local textual semantics and model label correlations.",
"S2S and S2S+Attn [26], [1] are our implementation of the RNN-based sequence-to-sequence models without and with the attention mechanism respectively."
],
[
"Results and Discussion",
"In the following sections, we report the results of our experiments on the RCV1-v2 and Ren-CECps.",
"Moreover, we conduct an ablation test and the comparison with models with hierarchical models with the deterministic setting of sentence or phrase, to illustrate that our model with learnable semantic units possesses a clear advantage over the baseline models.",
"Furthermore, we demonstrate that the higher-level representations are useful for the prediction of labels of low frequency in the dataset so that it can ensure that the model is not strictly learning the prior distribution of the label sequence."
],
[
"Results",
"We present the results of our implementations of our model as well as the baselines on the RCV1-v2 on Table REF .",
"From the results of the conventional baselines, it can be found that the classical methods for multi-label text classification still own competitiveness compared with the machine-learning-based and even deep-learning-based methods, instead of the Seq2Seq-based models.",
"Regarding the Seq2Seq model, both the S2S and the S2S+Attn achieve improvements on the dataset, compared with the baselines above.",
"However, as mentioned previously, the attention mechanism does not play a significant role in the Seq2Seq model for multi-label text classification.",
"By contrast, our proposed mechanism, which is label-classification-oriented, can take both the information of semantic units and that of word units into consideration.",
"Our proposed model achieves the best performance in the evaluation of Hamming loss and micro-${\\rm F_1}$ score, which reduces 9.8% of Hamming loss and improves 1.3% of micro-${\\rm F_1}$ score, in comparison with the S2S+Attn.",
"We also present the results of our experiments on Ren-CECps.",
"Similar to the models' performance on the RCV1-v2, the conventional baselines except for Seq2Seq models achieve lower performance on the evaluation of micro-$\\rm {F_1}$ score compared with the Seq2Seq models.",
"Moreover, the S2S and the S2S+Attn still achieve similar performance on micro-$\\rm {F_1}$ on this dataset, and our proposed model achieves the best performance with the improvement of 0.009 micro-$\\rm {F_1}$ score.",
"An interesting finding is that the Seq2Seq models do not possess an advantage over the conventional baselines on the evaluation of Hamming Loss.",
"We observe that there are fewer labels in the Ren-CECps than in the RCV1-v2 (11 and 103).",
"As our label data are reordered according to the descending order of label frequency, the Seq2Seq model is inclined to learn the frequency distribution, which is similar to a long-tailed distribution.",
"However, regarding the low-frequency labels with only a few samples, their amounts are similar, whose distributions are much more uniform than that of the whole label data.",
"It is more difficult for the Seq2Seq model to classify them correctly while the model is approximating the long-tailed distribution compared with the conventional baselines.",
"As Hamming loss reflects the average incorrect prediction, the errors in classifying into low-frequency labels will lead to a sharper increase in Hamming Loss, in comparison with micro-$\\rm {F_1}$ score."
],
[
"Ablation Test",
"To evaluate the effects of our proposed modules, we present an ablation test for our model.",
"We remove certain modules to control variables so that their effects can be fairly compared.",
"To be specific, besides the evaluation of the conventional attention mechanism mentioned in Section , we evaluate the effects of hybrid attention and its modules.",
"We demonstrate the performance of five models with different attention implementation for comparison, which are model without attention, one with only attention to the source annotations from LSTM, one with only attention to the semantic unit representations from the MDC, one with the attention to both the source annotations and semantic unit representations (additive) and hybrid attention, respectively.",
"Therefore, the effects of each of our proposed modules, including MDC and hybrid attention, can be evaluated individually without the influence of the other modules.",
"Results in Table REF reflect that our model still performs the best in comparison with models with the other types of attention mechanism.",
"Except for the insignificant effect of the conventional attention mechanism mentioned above, it can be found that the high-level representations generated by the MDC contribute much to the performance of the Seq2Seq model for multi-label text classification, which improves about 0.9 micro-$\\rm F_1$ score.",
"Moreover, simple additive attention mechanism, which is equivalent to the element-wise addition of the representations of MDC and those of the conventional mechanism, achieves similar performance to the single MDC, which also demonstrates that conventional attention mechanism in this task makes little contribution.",
"As to our proposed hybrid attention, which is a relatively complex combination of the two mechanisms, can improve the performance of MDC.",
"This shows that although conventional attention mechanism for word-level information does not influence the performance of the SeqSeq model significantly, the hybrid attention which extracts word-level information based on the generated high-level semantic information can provide some information about important details that are relevant to the most contributing semantic units."
],
[
"Comparison with the Hierarchical Models",
"Another method that can extract high-level representations is a heuristic method that manually annotates sentences or phrases first and applies a hierarchical model for high-level representations.",
"To be specific, the method does not only apply an RNN encoder to the word representations but also to sentence representations.",
"In our reimplementation, we regard the representation from the LSTM encoder at the time step of the end of each sentence as the sentence representation, and we implement another LSTM on top of the original encoder that receives sentence representations as input so that the whole encoder can be hierarchical.",
"We implement the experiment on the dataset RCV1-v2.",
"As there is no sentence marker in the dataset RCV1-v2, we set a sentence boundary for the source text and we apply a hierarchical model to generate sentence representations.",
"Compared with our proposed MDC, the hierarchical model for the high-level representations is relatively deterministic since the sentences or phrases are predefined manually.",
"However, our proposed MDC learns the high-level representations through dilated convolution, which is not restricted by the manually-annotated boundaries.",
"Through the evaluation, we expect to see if our model with multi-level dilated convolution as well as hybrid attention can achieve similar or better performance than the hierarchical model.",
"Moreover, we note that the number of parameters of the hierarchical model is more than that of our model, which are 47.24M and 45.13M respectively.",
"Therefore, it is obvious that our model does not possess the advantage of parameter number in the comparison.",
"We present the results of the evaluation on Table REF , where it can be found that our model with fewer parameters still outperforms the hierarchical model with the deterministic setting of sentence or phrase.",
"Moreover, in order to alleviate the influence of the deterministic sentence boundary, we compare the performance of different hierarchical models with different boundaries, which sets the boundaries at the end of every 5, 10, 15 and 20 words respectively.",
"The results in Table REF show that the hierarchical models achieve similar performances, which are all higher than the performances of the baselines.",
"This shows that high-level representations can contribute to the performance of the Seq2Seq model on the multi-label text classification task.",
"Furthermore, as these performances are no better than that of our proposed model, it can reflect that the learnable high-level representations can contribute more than deterministic sentence-level representations, as it can be more flexible to represent information of diverse levels, instead of fixed phrase or sentence level.",
"Figure: Micro-F 1 \\rm {F_1} scores of our model and the baseline on the evaluation of labels of different frequency.",
"The x-axis refers to the ranking of the most frequent label in the labels for classification, and the y-axis refers to the micro-F 1 \\rm {F_1} score performance."
],
[
"Error Analysis",
"Another finding in our experiments is that the model's performance on low-frequency label classification is lower than that on high-frequency label classification.",
"This problem is also reflected in our report of the experimental results on the Ren-CECps.",
"The decrease in performance is reasonable since the model is sensitive to the amount of data, especially on small datasets such as Ren-CECps.",
"We also hypothesize that this error comes from the essence of the Seq2Seq model.",
"As the frequency of our label data is similar to a long-tailed distribution and the data are organized by descending order of label frequency, the Seq2Seq model is inclined to model the distribution.",
"As the frequency distribution of the low-frequency labels is relatively uniform, it is much harder for it to model the distribution.",
"In contrast, as our model is capable of capturing deeper semantic information for the label classification, we believe that it is more robust to the classification of low-frequency labels with the help of the information from multiple levels.",
"We remove the top 10, 20, 30, 40, 50 and 60 most frequent labels subsequently, and we evaluate the performance of our model and the baseline Seq2Seq model on the classification of these labels.",
"Figure REF shows the results of the models on label data of different frequency.",
"It is obvious that although the performances of both models decrease with the decrease of the label frequency, our model continues to perform better than the baseline on all levels of label frequency.",
"In addition, the gap between the performances of the two models continues to increase with the decrease of label frequency, demonstrating our model's advantage over the baseline on classifying low-frequency labels."
],
[
"Related Work",
"The current models for the multi-label classification task can be classified into three categories: problem transformation methods, algorithm adaptation methods, and neural network models.",
"Problem transformation methods decompose the multi-label classification task into multiple single-label learning tasks.",
"The BR algorithm [3] builds a separate classifier for each label, causing the label correlations to be ignored.",
"In order to model label correlations, Label Powerset (LP) [27] creates one binary classifier for every label combination attested in the training set and Classifier Chains (CC) [23] connects all classifiers in a chain through feature space.",
"Algorithm adaptation methods adopt specific learning algorithms to the multi-label classification task without requiring problem transformations.",
"[5] constructed decision tree based on multi-label entropy to perform classification.",
"[6] adopted a Support Vector Machine (SVM) like learning system to handle multi-label problem.",
"[32] utilized the $k$ -nearest neighbor algorithm and maximum a posteriori principle to determine the label set of each sample.",
"[7] made ranking among labels by utilizing pairwise comparison.",
"[16] used joint learning predictions as features.",
"Recent studies of multi-label text classification have turned to the application of neural networks, which have achieved great success in natural language processing.",
"[31] implemented the fully-connected neural networks with pairwise ranking loss function.",
"[20] changed the ranking loss function to the cross-entropy loss to better the training.",
"[14] proposed a novel neural network initialization method to treat some neurons as dedicated neurons to model label correlations.",
"[4] incorporated CNN and RNN so as to capture both global and local semantic information and model high-order label correlations.",
"[21] proposed to generate labels sequentially, and [29], [17] both adopted the Seq2Seq, one with a novel decoder and one with a soft loss function respectively."
],
[
"Conclusion",
"In this study, we propose our model based on the multi-level dilated convolution and the hybrid attention mechanism, which can extract both the semantic-unit-level information and word-level information.",
"Experimental results demonstrate that our proposed model can significantly outperform the baseline models.",
"Moreover, the analyses reflect that our model is competitive with the deterministic hierarchical models and it is more robust to classifying the low-frequency labels than the baseline."
],
[
"Acknowledgements",
"This work was supported in part by National Natural Science Foundation of China (No.",
"61673028) and the National Thousand Young Talents Program.",
"Xu Sun is the corresponding author of this paper."
]
] | 1808.08561 |
[
[
"Exploring the Applications of Faster R-CNN and Single-Shot Multi-box\n Detection in a Smart Nursery Domain"
],
[
"Abstract The ultimate goal of a baby detection task concerns detecting the presence of a baby and other objects in a sequence of 2D images, tracking them and understanding the semantic contents of the scene.",
"Recent advances in deep learning and computer vision offer various powerful tools in general object detection and can be applied to a baby detection task.",
"In this paper, the Faster Region-based Convolutional Neural Network and the Single-Shot Multi-Box Detection approaches are explored.",
"They are the two state-of-the-art object detectors based on the region proposal tactic and the multi-box tactic.",
"The presence of a baby in the scene obtained from these detectors, tested using different pre-trained models, are discussed.",
"This study is important since the behaviors of these detectors in a baby detection task using different pre-trained models are still not well understood.",
"This exploratory study reveals many useful insights into the applications of these object detectors in the smart nursery domain."
],
[
"Introduction",
"Smart nursery is a niche market that has been quickly growing in the past decades.",
"This is attributed to the economic pressure in our modern lifestyle where both parents often opt to work to increase their income.",
"Baby monitoring gadgets are emerged as parenting tools in the urban lifestyle.",
"It provides a means for parents to monitor the well-being of their babies when they have to attend to other chores and cannot be present in the same physical space.",
"The common functionalities of baby monitoring products are movement detection, breathing detection, remote visual and audio monitoring.",
"Contemporary visual and audio monitoring devices provide the means to transmit visual and audio information.",
"Imagine if the monitoring gadget could report a baby's activities whether the baby is sitting, crawling, sleeping face up/down, etc., then this will be a very useful value-added alert functionality.",
"A summary of the baby's activities could also give valuable information for the parents to evaluate the development of the babies.",
"To date, this kind of application still does not exist.",
"With recent advances in artificial intelligence, visual and audio processing, we attempt to explore this area, starting from the detection functionality.",
"Object detection, localization, tracking and recognition are important prerequisites to the scenic understanding task which is still an open research problem.",
"There are many open challenges from the following issues: variants introduced by transformations e.g., translation, scaling, and rotation; variants introduced by changes in lighting conditions, and colors; variants introduced by occlusion; and variants introduced by deformation in shape.",
"These issues are the main challenges for researchers in computer vision community.",
"With recent advances in convolutional neural networks [1], the last few years have seen many new robust techniques devised to handle the challenges mentioned above.",
"Region-based Convolutional Neural Networks (R-CNN) [2], Fast R-CNN [3], Faster R-CNN [4], and Single-Shot Multi-box Detector (SSD) [5] are examples of state-of-the-art visual object detection algorithms.",
"In this paper, we explore the applications of the Faster R-CNN and the SSD both of which are the latest state-of-the-art detectors.",
"Both techniques are employed to detect the presence of a baby from a video footage obtained from YoutubeYoutube standard license for fair use of public content.",
"The rest of the materials in the paper are organized into the following sections; Section 2: Overview of the object detection task; Section 3: Exploratory study and Discussion; and Section 4: Conclusion."
],
[
"Overview of the Object Detection Task",
"In order to detect whether an instance of a semantic object is in a 2D image, one could create a classification model trained with features extracted from positive and negative classes and use the trained model to classify a given image.",
"This feature-based approach requires that the training images must be in the same nature as the testing images.",
"Variations in position, color, lighting conditions, size, etc., greatly affect the performance of the model.",
"Object detection and tracking process are commonly fine-tuned to the problem at hand.",
"This is usually dictated by the characteristics of the target.",
"Detecting and tracking vehicles, require different parameter tuning than detecting pedestrians, persons face, or a baby, etc.",
"Different targets require different detailed process due to the variations introduced by the environmental conditions e.g., target size, change in appearance.",
"Feature-based techniques exploit the discriminative features of the objects derived from 2D pixel information in an image [6].",
"color, and intensity are the primitives where other structures could be derived from e.g., edge, contour, color histogram, and intensity gradient, etc.",
"Intensity gradient peaks are robust to illumination variations, they have a good repeatability and provide good descriptions of local appearances described as points.",
"Detectors such as Kanade-Lucas-Tomasi (KLT) corner detectors [7], Scale Invariant Feature Transform (SIFT) [8] and Speeded Up Robust Features (SURF) [9] successfully exploit this aspect of the feature-based technique.",
"Features such as points and gradient may be used to describe local structure.",
"Composite structures such as Haar cascade and Histogram of Gradient (HOG) are successful descriptors that effectively describe the appearances of objects.",
"Haar feature effectively abstracts human face through the abstraction of high/low intensity pattern of the human face e.g., forehead is the brighter region as compared to the eyes; HOG feature effectively abstracts a human figure using histogram of gradient, etc.",
"To increase the flexibility of the traditional feature-based approach, one could create a model based on a cropped version of the semantic object in the image and search for the object in a test image by scanning through it using different window sizes.",
"This is plausible but the flexibility to deal with the variant of scale, rotation, deformation etc., comes with computing cost since the process requires many computation passes to handle each different variation."
],
[
"Faster Region-based Convolutional Neural Network",
"Although it is plausible to find an instance of an object in a 2D image by sliding different shapes and sizes of search windows over the entire image, it is expensive to exhaustively scan through the image this way.",
"In [10], the authors propose a concept known as selective search (SS) where the region of interests in the image is proposed based on the similarity of local appearance.",
"Figure: Top pane: the Faster R-CNN approach (figure is adapted from ).",
"Bottom pane: the Single Shot Multibox Detection approach (figure is adapted from ).Instead of performing exhaustive sliding window search, R-CNN and Fast R-CNN employ SS to provide over 2,000 proposed regions (per image) for its object detection search.",
"Two thousand regions may seem a lot but this is just a fraction of the amount required by the exhaustive sliding window search alternative.",
"The combination of SS and CNN allows instances of interested objects to be allocated in almost real-time (R-CNN can process one image in fifty seconds [2] and Fast R-CNN can process one image in two seconds).",
"Faster R-CNN is an improvement of R-CNN and Fast R-CNN.",
"Both R-CNN and Fast R-CNN have external region proposal process which requires a considerable amount of processing time.",
"Faster R-CNN further improves the processing pipeline by embedding the region proposal network after the CNN (see Figure REF ).",
"The region proposal process takes less than 10 ms to process the region proposal task and the Faster R-CNN offers the processing speed up to five frames per second which is good enough for some real life applications.",
"In this study, the experiments with Faster R-CNN are carried out using the following pre-trained detection models obtained from Google Tensorflowhttps://github.com/tensorflow/models/blob/master/research/object_detection/: Inception and ResNet.",
"These models are trained on the Microsoft Common Object Content (COCO) datasethttp://cocodataset.org/."
],
[
"Single-Shot Multi-Box Detection",
"Leveraging on established deep convolutional neural network classifiers such as VGG network [1], the authors in [5] propose Single-Shot Multi-Box Detection (SSD) for an object detection task.",
"SSD offers a novel object detection approach which separates itself from previous region-based approaches (R-CNN, Fast R-CNN and Faster R-CNN).",
"SSD incorporates class prediction and bounding box prediction in a single process, hence, the detection speed of SSD is significantly faster than the Faster R-CNN approach.",
"The SSD architecture consists of a base convolutional neural networkBase convolutional neural network refers to a CNN network without classification layers.",
"followed by the multi-box convolutional layers (see Figure REF , bottom pane).",
"The base VGG and multi-box layers predict (i) the presence of object class instances and (ii) their bounding box locations.",
"There are six prediction layers decreasing in feature-map size which allows the SSD to handle object detection in multi-scale.",
"The total six layer feature maps with the sizes: $38^2, 19^2, 10^2, 5^2, 3^2$ , and $1^2$ , is implemented with the following numbers of default boxes: 4, 6, 6, 6, 4, and 4.",
"Hence there are 8732 detections per class i.e., $38^2 \\times 4 + 19^2 \\times 6 + 10^2 \\times 6 + 5^2 \\times 6 + 3^2 \\times 4 + 1^2 \\times 4$ (see Figure REF , bottom pane).",
"Among these detections, many will be unlikely candidates with very small probability and they will be removed using the Non-Maximum Suppression (NMS).",
"Experiments with SSD are carried out using the base VGG network with the pre-trained PASCAL Visual Object Classes (VOC) modelhttp://host.robots.ox.ac.uk/pascal/VOC/voc2007/ and the recent MobileNet implementation [11] from Google Tensorflow with the pre-trained COCO dataset model.",
"Figure: Input to our object detection system is image frames (each frame should be at least 300x300 pixels 2 ^2).",
"The top most row shows examples of negative input and the three bottom rows shows a typical input.",
"The target object (baby) may be in various poses and orientations, and may be displayed full body or with occlusion.",
"The environment may have different lighting conditions."
],
[
"Exploratory Study and Discussion",
"Due to the non-existence of any standard baby monitoring benchmark dataset, we prepare our test data using in-house footage as well as by editing video clips of baby (babies are less than twelve months old) found on Youtube.",
"These Youtube clips are utilized under the Youtube's standard fair use license.",
"These clips are shot and posted by parents.",
"Hence, these clips are shot with unprofessional setting, with different lighting environments, various camera angles and with camera movement.",
"This imperfection provides reasonable variants for us to evaluate the usage of pre-trained detector in the actual environment.",
"Figure REF provides examples of the test images used in this paper."
],
[
"Evaluation Criteria",
"We are interested to know the behaviors of the pre-trained Faster R-CNN and the SSD detectors in detecting the presence of a baby in a video sequence.",
"Since the Faster R-CNN and the SSD detectors are capable of detecting more than one class, in this setup, the evaluation is based on the correct detection and classification of the baby in the scene but not for other objects.",
"In other words, a frame is classified as a true positive (TP) if there is a baby in the scene and the detector has correctly detected the baby; as a true negative (TN) if a baby is not in the scene and the detector does not output any detection of a baby; as a false positive (FP) if a baby is not in the scene but the detector wrongly reports the detection; and as a false negative (FN) if a baby is in the scene but the detector does not output any detection of a baby, or wrongly labels the detected baby.",
"Figure REF displays examples of TP, TN, FP and FN cases.",
"We calculate the sensitivity, the specificity and the accuracy using the equations below: Figure: Representative examples of (top row) true positive and true negative cases; (middle row) false positive and false negative cases.",
"Bottom row: the detectors may suggest many bounding box for the same object.",
"This is considered as a correct classification.",
"This detection behavior is, however, not preferable since it complicates the precision of the localization task.$Sensitivity = \\frac{\\mbox{True positive}}{\\mbox{True positive + False negative}} $ $Specificity = \\frac{\\mbox{True negative}}{\\mbox{True negtive + False positive}} $ $Accuracy = \\frac{\\mbox{True positive + True negative}}{\\mbox{True positive + True negtive + False positive + False negative}}$ Each test clip is manually prepared using the content from the Youtube video clips.",
"It is decided to have 100 positive examples and 50 negative examples in each test clip.",
"The mp4 movies and image sequences of the test clips can be accessed from this url https://sites.google.com/view/eventanalysis/home/datasets"
],
[
"Discussion",
"We employ both Faster R-CNN and SSD to detect the presence of a baby.",
"The detectors are chosen as they are the representative of the current state-of-the-art detectors.",
"The results from this exploratory study are tabulated in Table REF .",
"The results show a reasonable detection accuracy from all modelsPre-trained models: faster_rcnn_inception_v2_coco, faster_rcnn_ResNet101_coco, ssd_mobilenet_v2_coco and ssd300_mAP_77.43_v2.",
"The accuracy of Faster R-CNN in detecting the presence of baby appears to outperform the performance from SSD.",
"However, from our observations (see Figure REF ), SSD is faster and gives higher quality bounding box than Faster R-CNN.",
"Table: Summary of sensitivity, specificity and average accuracy of the Faster R-CNN and the SSD detectors.",
"Microsoft COCO is the training dataset for all models except the SSD VGG model which uses the VOC dataset.Accurate detection, localization and tracking enhance the performance of the recognition system which in turn enable accurate scenic understanding.",
"This work investigates the performance of the two state-of-the-art detectors using various pre-trained CNN architectures: Inception model, ResNet model, MobileNet model and VGG model; with two detection approaches: Faster R-CNN and SSD.",
"Figure REF shows the representative detection examples from these models.",
"We would like to highlight the following points to the readers: Figure: Comparing behaviors of Faster R-CNN (columns one and two) and SSD (columns three and four), it is apparent that the Faster R-CNN predicts more bounding boxes of an object more than the SDD does.",
"This is due to the non-maximum supression mechanism in SSD while the Faster R-CNN relies on the region proposal mechanism and does not provide the combining mechanism in the final stage."
],
[
"Quality of the bounding box prediction :",
"SSD appears to provide appropriate bounding boxes for its predictions while Faster R-CNN appears to offer many bounding boxes overlapping the interested regions (see Figure REF ).",
"Many bounding boxes are not desired since this will complicate the localization task."
],
[
"Quality of the class detection :",
"Faster R-CNN provides a better baby detection accuracy (the best is 97.5% with our dataset) while the accuracy from the SSD is only 86.1%.",
"This level of accuracy could be enhanced with various tactics such as (i) spatial and temporal continuity constraints on the detected baby; and (ii) ensemble learning where prediction from various models may be combined.",
"These are important future work areas."
],
[
"The gap between detection and activity recognition :",
"Detection only detects an instance of a class but does not provide any distinction among instances of the same class.",
"If we are interested in a counting task, e.g., counting the number of babies occupying a room, accurate detection is enough for the task.",
"However, if we would like to collect their detailed profiles such as their activities, then the ID-tracking and activity recognition functionalities must be appended to the pipeline.",
"ID-tracking enables each object to be tracked anonymously.",
"Activity recognition provides detailed activity labels to the tracked objects.",
"Common activity recognition techniques are based on motion descriptors derived from 2D image sequences such as Histograms of Oriented Optical Flow (HOOF) [12].",
"Improvement of speed for the detection-recognition pipeline is also an important future work area."
],
[
"The gap between detection and scenic understanding :",
"In order to provide an appropriate summary of the visual content, accurate object detection alone is also not enough since relationships between detected components play a crucial role in scenic understanding.",
"To date, this is still an open research area.",
"From our analysis, we categorize the approaches toward bridging the gap between object detection and scenic understanding into two main categories (i) a knowledge-based framework and (ii) an encoder-decoder framework.",
"The knowledge-based framework attempts to explicitly construct objects and their relationships in the scene.",
"For example, in a babywatching task, if a baby, a teddy bear and small Lego pieces are detected in the scene, then relationships between these objects can be represented as stereotyped situations [13] where various inferences may be drawn from them.",
"External knowledge is commonly injected into the knowledge system.",
"This provides an effective inference process but requires a hand-crafted knowledge encoding which is a well-known bottleneck in the knowledge engineering task.",
"The encoder-decoder framework is leveraged from recent advances in deep learning.",
"A deep CNN is a common encoder and a long short-term memory (LSTM) network [14] is a typical decoder.",
"An image frame is encoded into a feature vector which can be decoded into appropriate natural language descriptions of the given frame [15].",
"The encoder-decoder framework does not explicitly encode knowledge in the fashion employed in the knowledge-based approach.",
"Hence, further processing steps are required to transform an image caption of each frame into a coherent scenic understanding.",
"The goal of object detection is to detect an instance of semantic objects.",
"The approach has been constantly evolving in the past four decades.",
"In the early days, the process was considered successful if the main object in the scene had been correctly identified.",
"Later, localization has become a priority and it is normally provided as a bounding box over the object.",
"With the progress in deep CNN [5], [16], [17], [18], accurate detection and localization have been witnessed.",
"In this paper, we carried out experiments in the babywatching domain using state-of-the-art detectors.",
"The results obtained from these detectors based on the recent pre-trained models from the Tensorflow detection model zoo have been carefully analyzed and discussed.",
"This reveals new insights and many potential future research in this domain that could bridge the gap between object detection and scenic understanding."
],
[
"Acknowledgments",
"We would like to thank the Centre for Innovative Engineering (Universiti Teknologi Brunei), NICT (Japan) and CLEALINKTECHNOLOGY (Japan) for their supports given to this work."
]
] | 1808.08675 |
[
[
"Groupoids and Relative Internality"
],
[
"Abstract In a stable theory, a stationary type $q \\in S(A)$ internal to a family of partial types $\\mathcal{P}$ over $A$ gives rise to a type-definable group, called its binding group.",
"This group is isomorphic to the group $\\mathrm{Aut}(q/\\mathcal{P},A)$ of permutations of the set of realizations of $q$, induced by automorphisms of the monster model, fixing $\\mathcal{P} \\cup A$ pointwise.",
"In this paper, we investigate families of internal types varying uniformly, what we will call relative internality.",
"We prove that the binding groups also vary uniformly, and are the isotropy groups of a natural type-definable groupoid (and even more).",
"We then investigate how properties of this groupoid are related to properties of the type.",
"In particular, we obtain internality criteria for certain 2-analysable types, and a sufficient condition for a type to preserve internality."
],
[
"Introduction",
"In geometric stability theory, the notion of internality plays a central role, as a tool to understand the fine structure of definable sets.",
"More recently, it has been developed outside of stable theories, using only stable embeddedness.",
"In this paper, we will restrict ourselves to the stable context.",
"More precisely, our basic setup will be the following: we work in a monster model $\\mathbb {M}$ of a stable theory $T$ , eliminating imaginaries, and is given a family of partial types $\\mathcal {P}$ , all over some algebraically closed set of parameters $A$ .",
"A tuple $c$ is said to be a realization of $\\mathcal {P}$ if it is a realization of some partial type in $\\mathcal {P}$ .",
"We will often also write $\\mathcal {P}$ for the set of realizations of $\\mathcal {P}$ in $\\mathbb {M}$ .",
"A stationary type $q \\in S(A)$ is said to be $\\mathcal {P}$ -internal if there are $B \\supseteq A$ , a realization $a$ of $q$ , independent of $B$ over $A$ , and a tuple $\\overline{c}$ of realizations of $\\mathcal {P}$ (i.e.",
"each realizing some type in $\\mathcal {P}$ ) such that $a \\in \\operatorname{dcl}(\\overline{c},B)$ .",
"It is said to be almost $\\mathcal {P}$ -internal if $a \\in \\operatorname{acl}(\\overline{c},B)$ instead.",
"The important part of this definition is the introduction of the new parameters $B$ .",
"The following result, which is Theorem 7.4.8 in [9], produces a type-definable group action from this configuration: Theorem 1.1 Let $\\mathbb {M}$ be the monster model of a stable theory $T$ , eliminating imaginaries.",
"Suppose $q\\in S(A)$ is internal to a family of types $\\mathcal {P}$ over $A$ , an algebraically closed set of parameters.",
"Then there are an $A$ -type-definable group $G$ and an $A$ -definable group action of $G$ on the set of realizations of $q$ , which is naturally isomorphic (as a group action), to the group $\\operatorname{Aut}(q/\\mathcal {P},A)$ of permutations of the set of realizations of $q$ , induced by automorphisms of $\\mathbb {M}$ fixing $\\mathcal {P}\\cup A$ pointwise.",
"This is the result that we will generalize in this paper.",
"The group arising in this theorem is called the binding group of $q$ over $\\mathcal {P}$ , and was first introduced by Zilber.",
"At our level of generality, its existence was proved by Hrushovski.",
"Our proof will follow very closely the proof given in [9], with a few minor adjustments.",
"The binding group of $q$ over $\\mathcal {P}$ will often be denoted by $\\operatorname{Aut}(q/\\mathcal {P},A)$ .",
"It encodes the dependence on the extra parameters $B$ .",
"For example, if $B = A$ , the binding group is trivial.",
"For a more modern treatment of these binding groups, outside of stable theories, and with definable sets instead of types, we refer the reader to [7] and [6].",
"Recall that $\\mathbb {M}$ is a monster model of a stable theory $T$ , eliminating imaginaries.",
"If $\\Phi $ is a partial type, we will denote $\\Phi (\\mathbb {M})$ the set of its realizations in $\\mathbb {M}$ .",
"Again, let $\\mathcal {P}$ be a family of partial types over $A$ algebraically closed.",
"Suppose there is a type $q \\in S(A)$ , and an $A$ -definable function $\\pi $ , whose domain contains $q(\\mathbb {M})$ .",
"Definition 1.2 The type $q$ is said to be relatively $\\mathcal {P}$ -internal via $\\pi $ if for any $a \\models q$ , the type $\\operatorname{tp}(a/\\pi (a)A)$ is stationary and $\\mathcal {P}$ -internal.",
"Denote this type $q _{\\pi (a)}$ .",
"From this configuration, we can define a groupoid $\\mathcal {G}$ (see section for a definition of groupoid).",
"Its objects are realizations of $\\pi (q)$ , and for any $a,b \\models q$ , the set of morphisms $\\operatorname{Mor}(\\pi (a),\\pi (b))$ consists of bijections from $q _{\\pi (a)}(\\mathbb {M})$ to $q _{\\pi (b)}(\\mathbb {M})$ , induced by automorphisms of $\\mathbb {M}$ fixing $\\mathcal {P}\\cup A$ pointwise, and taking $\\pi (a)$ to $\\pi (b)$ .",
"In particular, the isotropy groups $\\operatorname{Mor}(\\pi (a), \\pi (a))$ are the binding groups $\\operatorname{Aut}(q_{\\pi (a)}/\\mathcal {P},A)$ , hence are type-definable over $A\\pi (a)$ by Theorem REF .",
"This groupoid acts naturally on the set of realizations of $q$ .",
"By that we mean, setting $X = \\lbrace (\\sigma , a) \\in \\operatorname{Mor}(\\mathcal {G}) \\times q(\\mathbb {M}): \\operatorname{dom}(\\sigma ) = \\pi (a) \\rbrace $ , that we have a map $X \\rightarrow q(\\mathbb {M})$ , satisfying the obvious group action-like axioms.",
"In our case, the action is given by $(\\sigma ,a) \\rightarrow \\sigma (a)$ .",
"We obtain the following generalization of the type-definability of binding groups: Theorem 1.3 The groupoid $\\mathcal {G}$ is isomorphic to an $A$ -type-definable groupoid, and its natural groupoid action on realizations of $q$ is $A$ -definable.",
"In particular, the binding groups are uniformly type-definable, and are the isotropy groups of the type-definable groupoid.",
"This groupoid arises because the group $\\operatorname{Aut}(q/\\mathcal {P})$ , even when $q$ is not internal, still acts definably on the fibers of $\\pi $ , but its global action is not definable.",
"The new and interesting fact is that all these local fiber actions are uniformly definable, and come together to form a type-definable groupoid.",
"In [7] and [6], internality is considered in a different context, and from some internal sorts, a definable groupoid is constructed.",
"It arises for different reasons, and we will compare the two groupoids in the next paragraphs.",
"In these two papers, the authors fix a monster model $\\mathbb {U}$ of some theory $T$ , eliminating imaginaries, and a monster model $\\mathbb {U}^{\\prime }$ of some theory $T^{\\prime }$ , with $\\mathbb {U} \\subset \\mathbb {U}^{\\prime }$ .",
"They assume that $\\mathbb {U}$ is stably embedded in $\\mathbb {U}^{\\prime }$ , and $\\mathbb {U}^{\\prime }$ is internal to $\\mathbb {U}$ with only one new sort, called $S$ .",
"Under these assumptions, they construct a $*$ -definable (over $\\emptyset $ ) connected groupoid $\\mathcal {G}^{\\prime }$ , in $(\\mathbb {U}^{\\prime } )^{eq}$ , with one distinguished object $a$ and a full $*$ -definable (over $\\emptyset $ ) subgroupoid $\\mathcal {G}$ in $\\mathbb {U}$ , such that $\\operatorname{Mor}_{\\mathcal {G}^{\\prime }}(a,a)$ acts definably on $\\mathbb {U}^{\\prime }$ , and this action is isomorphic to $\\operatorname{Aut}(\\mathbb {U}^{\\prime } /\\mathbb {U})$ acting on $S$ .",
"Note that in [6], the groupoid constructed is actually proven to be $\\emptyset $ -definable, under some mild additional assumption.",
"The starting point of their proof is the following observation: since $\\mathbb {U}^{\\prime }$ is internal to $\\mathbb {U}$ , there is a $b$ -definable set $O_b$ in $\\mathbb {U}$ , and a $c$ -definable bijection $f_c: S \\rightarrow O_b$ .",
"Roughly speaking, the idea is now to allow these parameters $b$ and $c$ to vary, the $b$ yielding objects of a groupoid, and the $f_c$ morphisms between objects.",
"Therefore, this groupoid will encode the non-canonicity of the parameters $b$ and $c$ used to witness internality.",
"The groupoid constructed in the present paper, however, encodes the fact that some maps are partially definable, but not globally definable.",
"Comparing these papers and ours, some questions arise.",
"First, one can define relative internality in this different context, and it would be interesting to see if a groupoid witnessing it could be contructed there.",
"Second, the groupoids obtained from internality live in the sort $\\mathbb {U}$ , and this is used to obtain a correspondence between certain groupoids in $\\mathbb {U}$ and internal generalised imaginary sorts of $\\mathbb {U}$ .",
"In our setup, this would be equivalent to our groupoid $\\mathcal {G}$ living in $\\mathcal {P}^{eq}$ .",
"As it will become clear from the proof, our groupoid does not live in $\\mathcal {P}^{eq}$ .",
"It would be desirable to identify some object in $\\mathcal {P}^{eq}$ coming from relative internality.",
"In section , we will discuss some obstruction to this.",
"The rest of the paper will explore different properties of the groupoids arising from relative internality, and how they relate to the type $q$ .",
"Mostly, we will seek to link some properties of $\\mathcal {G}$ to $\\mathcal {P}$ -internality, or almost $\\mathcal {P}$ -internality, of the type $q$ .",
"One motivation for this is to be able to determine when an analysable type is in fact internal.",
"Recall that a type $q$ is said to be $\\mathcal {P}$ -analysable in n steps if for any $a \\models q$ there are $a_n = a, a_{n-1}, \\cdots ,a_1$ such that $a_{i} \\in \\operatorname{dcl}(a_{i+1})$ and $\\operatorname{tp}(a_{i+1}/a_i)$ is $\\mathcal {P}$ -internal for all $i$ .",
"Therefore, if $q$ is relatively $\\mathcal {P}$ -internal via $\\pi $ , and the type $\\pi (q)$ is $\\mathcal {P}$ -internal, we see that $q$ is $\\mathcal {P}$ -analysable in two steps.",
"The question of which analysable types are actually internal is connected with the Canonical Base Property, which is a property of finite U-rank theories.",
"Introduced in [10], it is a model theoretic translation of a result in complex geometry, and has some attractive consequences (see [1], [11]).",
"It states that for any tuple $a,b$ , if $b = \\operatorname{Cb}(\\operatorname{stp}(a/b))$ , then $\\operatorname{tp}(b/a)$ is almost internal to the family $\\mathcal {P}$ of nonmodular U-rank one types.",
"But it is proven in [1] that this type is always $\\mathcal {P}$ -analysable.",
"Therefore, the Canonical Base Property boils down to the collapse of an analysable type into an internal one.",
"In this paper, we will expose two properties of groupoids implying that a relatively internal type is internal.",
"The first one, retractability, was introduced in [5], and is related to 3-uniqueness.",
"Here, it will imply that the 2-analysable type is a product of two weakly orthogonal types, one of which is $\\mathcal {P}$ -internal.",
"The second needs the construction of a Delta groupoid, which adds simplicial data to the groupoid.",
"We define a notion of collapsing for Delta groupoids, which turns out to be equivalent, if $\\pi (q)$ is $\\mathcal {P}$ -internal, to $\\mathcal {P}$ -internality of the type $q$ .",
"Finally, in [8], a strengthening of internality, called being Moishezon, or preserving internality in later papers, was introduced, again motivated by properties of compact complex manifolds.",
"A criterion for when an internal type preserves internality was proved in this paper, but under the assumption that the ambient theory has the Canonical Base Property.",
"Here, we prove a criterion for preserving internality in terms of Delta groupoids, valid in any superstable theory.",
"The paper is organized as follow: in section 2, we recall some results concerning internality and stable theories that will be used frequently, and say a few words about groupoids.",
"In section 3, we construct the type-definable groupoid of Theorem REF .",
"In section 4, we define retractability for a type-definable groupoid, and explore the consequences of this property.",
"In Section 5, we define Delta groupoids, introduce the notion of collapsing, and link it with internality and preservation of internality.",
"Before we start, let us give a few conventions and notations.",
"As stated before, we will work throughout in the monster model $\\mathbb {M}$ of a stable theory, eliminating imaginaries.",
"Theorem REF is stated over any small parameter set $A$ , but we will work, without loss of generality, over the empty set.",
"We also assume that $\\operatorname{acl}(\\emptyset ) = \\emptyset $ .",
"Recall that if $\\mathcal {P}$ is a family of partial types over the empty set, by a realization of $\\mathcal {P}$ , we mean a tuple $c$ realizing some partial type in $\\mathcal {P}$ .",
"We will often write $\\mathcal {P}$ for the set of realizations of the family of partial type $\\mathcal {P}$ in $\\mathbb {M}$ , since no confusion could arise from this.",
"Finally, recall that if $\\Phi $ is any partial type, we will denote $\\Phi (\\mathbb {M})$ the set of its realizations in $\\mathbb {M}$ .",
"We assume familiarity with stability theory and geometric stability theory, for which [9] is a good reference.",
"I would like to thank my advisor, Anand Pillay, for giving me regular input and suggestions during the writing of this paper.",
"I would also like to thank Levon Haykazyan, Rahim Moosa, and Omar Léon Sánchez for discussing the subject of this paper with me.",
"Finally, I am grateful to my referee, whose comments and suggestions lead to a substantial improvement of this paper."
],
[
"Preliminaries",
"Internality of a type $q$ to a family of partial types $\\mathcal {P}$ is equivalent to: there exists a set of parameters $B$ such that for any $a \\models q$ , there are $c_1, \\cdots , c_n$ realizing $\\mathcal {P}$ satisfying $a \\in \\operatorname{dcl}(c_1, \\cdots ,c_n, B)$ .",
"Moreover, the parameters $B$ can be taken as realizations of $q$ , as the following, which is Lemma 7.4.2 from [9], shows: Lemma 2.1 Let $A$ be a small set of parameters.",
"Suppose $\\mathcal {P}$ is a family of partial types over $A$ , and $q$ is a $\\mathcal {P}$ -internal stationary type over $A$ .",
"Then there exist a partial $A$ -definable function $f(y_1, \\cdots , y_m, z_1, \\cdots ,z_n)$ , a sequence $a_1, \\cdots , a_m$ of realizations of $q$ , and a sequence $\\Psi _1 ,\\cdots ,\\Psi _n$ of partial types in $\\mathcal {P}$ , such that for any $a$ realizing $q$ , there are $c_i$ realizing $\\Psi _i$ , for $i = 1 \\cdots n$ , such that $a = f(a_1, \\cdots , a_m, c_1, \\cdots ,c_n)$ .",
"The tuple $a_1, \\cdots , a_m$ obtained in this lemma is called a fundamental system of solutions for $q$ .",
"In fact, we define, for any type $q$ : Definition 2.2 If $q$ is $\\mathcal {P}$ -internal, a tuple $\\overline{a}$ of realizations of $q$ is said to be a fundamental system of solutions for $q$ if for any $b \\models q$ , we have $b \\in \\operatorname{dcl}(\\overline{a}, \\mathcal {P})$ .",
"If $q$ has a fundamental system consisting of only one realization, it is said to be a fundamental type.",
"The following fact will be used implicitly throughout the article: Fact 2.3 If $q$ is internal to $\\mathcal {P}$ , and $r \\in S(\\emptyset )$ is the type of a fundamental system of solutions for $q$ , then the binding groups $\\operatorname{Aut}(q/\\mathcal {P})$ and $\\operatorname{Aut}(r/\\mathcal {P})$ are $\\emptyset $ -definably isomorphic.",
"By Lemma REF , any internal type has a fundamental system of solutions.",
"Remark 2.4 By inspecting the proof (in [9]) of the previous lemma, one notices that the tuples $a_1, \\cdots , a_n$ are independent realizations of $q$ .",
"This will be useful in section 5.",
"The following two facts will be useful to us: Fact 2.5 For any family of partial types $\\mathcal {P}$ and tuple $a$ , we have $\\operatorname{tp}(a/\\operatorname{dcl}(a) \\cap \\mathcal {P}) \\models \\operatorname{tp}(a/\\mathcal {P})$ .",
"For a proof of this, see Claim II of the proof of Theorem 7.4.8 in [9].",
"Fact 2.6 If $\\mathcal {P}$ is a family of partial types, for any two tuples $a$ and $b$ , we have $\\operatorname{tp}(a/\\mathcal {P}) = \\operatorname{tp}(b/\\mathcal {P})$ if and only if there is an automorphism of $\\mathbb {M}$ , fixing $\\mathcal {P}$ , and taking $a$ to $b$ .",
"This can be proven adapting the proof of Lemma 10.1.5 in [12].",
"The main algebraic objects considered in this paper are groupoids.",
"We recall their definition: Definition 2.7 A groupoid $\\mathcal {G}$ is a non-empty category such that every morphism is invertible.",
"Therefore, a groupoid consists of two sets: a set of objects $\\operatorname{Ob}(\\mathcal {G})$ , and a set of morphisms $\\operatorname{Mor}(\\mathcal {G})$ .",
"These are equipped with the partial composition on morphisms, and the domain and codomain maps.",
"Moreover, for each object $a$ , there is an identity map $\\operatorname{id}_a \\in \\operatorname{Mor}(a,a)$ .",
"Groupoids generalize groups.",
"Indeed, every object of a groupoid $\\mathcal {G}$ gives rise to the group $\\operatorname{Mor}(a,a)$ , called the isotropy group of $a$ .",
"But we also have the extra morphisms $\\operatorname{Mor}(a,b)$ , for any $a,b \\in \\operatorname{Ob}(\\mathcal {G})$ .",
"Remark that a group is then exactly a groupoid with only one object.",
"The set $\\operatorname{Mor}(a,b)$ could be empty if $a \\ne b$ .",
"This will actually have some meaningful model-theoretic content, and we can define: Definition 2.8 If $\\mathcal {G}$ is a groupoid and $a \\in \\operatorname{Ob}(\\mathcal {G})$ , then the connected component of $a$ is the set $\\lbrace b \\in \\operatorname{Ob}(\\mathcal {G}): \\operatorname{Mor}(a,b) \\ne \\emptyset \\rbrace $ .",
"A groupoid is connected if it has only one connected component, and totally disconnected if the connected component of any object is itself.",
"Since we are interested in definable, or type-definable objects, we need to define these notions for groupoids.",
"Definition 2.9 A groupoid $\\mathcal {G}$ is definable if the sets $\\operatorname{Ob}(\\mathcal {G})$ and $\\operatorname{Mor}(\\mathcal {G})$ are definable, and the composition, domain, codomain and inverse maps are definable.",
"It is type-definable is these sets and maps are type-definable."
],
[
"The construction of a groupoid",
"Let $q \\in S(\\emptyset )$ , a family of partial types $\\mathcal {P}$ over $\\emptyset $ , and an $\\emptyset $ -definable function $\\pi $ , whose domain contains $q(\\mathbb {M})$ , such that $q$ is relatively $\\mathcal {P}$ -internal via $\\pi $ .",
"Remark that for any $a \\models q$ , the type $\\operatorname{tp}(a/\\pi (a))$ is implied by $q(x) \\cup \\lbrace \\pi (x) = \\pi (a) \\rbrace $ .",
"We will denote this type $q_{\\pi (a)}$ .",
"To ease notation, if $\\overline{a}$ is a tuple of realizations of $q$ with same image under $\\pi $ , we will denote $\\pi (\\overline{a})$ their common image.",
"Recall that there is a groupoid $\\mathcal {G}$ , whose objects are given by $\\pi (q)(\\mathbb {M})$ , and morphisms $\\operatorname{Mor}(\\pi (a),\\pi (b))$ by the set of bijections from $q _{\\pi (a)}(\\mathbb {M})$ to $q _{\\pi (b)}(\\mathbb {M})$ , induced by automorphisms of $\\mathbb {M}$ fixing $\\mathcal {P}$ pointwise, and taking $\\pi (a)$ to $\\pi (b)$ .",
"Our goal is to prove this groupoid, as well as its action on realizations of $q$ (see in the introduction for a definition of a groupoid action) are $\\emptyset $ -type-definable.",
"We now start the proof, which follows closely the proof of Theorem 7.4.8 from [9]: First note that the objects are the $\\emptyset $ -type-definable set $\\pi (q)$ .",
"So what we have to show is that the set of morphisms is $\\emptyset $ -type-definable, as well as domain and codomain maps, and composition.",
"Note that since each $\\pi $ -fiber is $\\mathcal {P}$ -internal, we can apply Lemma REF to any of them, so each type $q_{\\pi (a)}$ has a fundamental system of solution.",
"The first step of the proof is to show that these fundamental systems can be chosen uniformly, in the following sense: Claim 3.1 There exist a type $r$ over $\\emptyset $ , a partial $\\emptyset $ -definable function $f(y, z_1, \\cdots ,z_n)$ , a sequence $\\Psi _1 ,\\cdots ,\\Psi _n$ of partial types in $\\mathcal {P}$ .",
"These satisfy that for each $\\pi (a) \\models \\pi (q)$ , there is $\\overline{a}\\models r$ such that $\\pi (a) = \\pi (\\overline{a})$ , and for any other $a^{\\prime } \\models q_{\\pi (a)}$ , there are $c_i$ realizing $\\Psi _i$ , for $i = 1 \\cdots n$ , with $a^{\\prime } = f(\\overline{a}, c_1, \\cdots ,c_n)$ .",
"Let $\\pi (a)$ be a realization of $\\pi (q)$ .",
"Applying Lemma REF to $q_{\\pi (a)}$ yields a partial $\\pi (a)$ -definable function $f(y_1, \\cdots , y_m, z_1, \\cdots ,z_n)$ , a sequence $a_1, \\cdots , a_m$ of realizations of $q_{\\pi (a)}$ , and a sequence $\\Psi _1 ,\\cdots ,\\Psi _n$ of partial types in $\\mathcal {P}$ , such that $q_{\\pi (a)} \\subset f(\\overline{a}, \\Psi _1(\\mathbb {M}), \\cdots , \\Psi _n(\\mathbb {M}))$ .",
"Denote $\\overline{a}= (a_1, \\cdots , a_m)$ , and $r = \\operatorname{tp}(\\overline{a}/\\emptyset )$ .",
"Remark that since $\\pi (\\overline{a}) = \\pi (a) \\in \\operatorname{dcl}(\\overline{a})$ , the function $f$ is actually $\\emptyset $ -definable.",
"By invariance, we see that $f,r$ and $\\Psi _1, \\cdots ,\\Psi _n$ satisfy the required properties.",
"We will now fix $r, f$ be as in Claim REF , and $\\Phi (\\overline{x}) = \\Psi (x_1) \\cup \\cdots \\cup \\Psi (x_n)$ .",
"Fix $\\pi (a)$ , $\\pi (b)$ and a realization $\\overline{a}$ of $r$ in $\\pi ^{-1}(\\pi (a))$ .",
"Consider the set $X = \\lbrace (\\overline{a},\\overline{b}): \\operatorname{tp}(\\overline{a}) = \\operatorname{tp}(\\overline{b}) = r, \\operatorname{tp}(\\overline{a}/\\mathcal {P}) = \\operatorname{tp}(\\overline{b}/\\mathcal {P}) \\rbrace $ , it is the set we will use to encode morphisms.",
"We have: Claim 3.2 The set $X$ is $\\emptyset $ -type-definable.",
"Fact REF yields that $\\operatorname{tp}(\\overline{a}/\\operatorname{dcl}(\\overline{a}) \\cap \\mathcal {P}) \\models \\operatorname{tp}(\\overline{a}/\\mathcal {P})$ .",
"Consider the set $\\lbrace \\lambda _i(\\overline{x}): i \\in I \\rbrace $ of partial $\\emptyset $ -definable functions defined at $\\overline{a}$ with values in $\\mathcal {P}$ (and these are the same at every realization of $r$ ).",
"Then $\\operatorname{tp}(\\overline{a}/\\mathcal {P}) = \\operatorname{tp}(\\overline{b}/\\mathcal {P})$ if and only if $\\lambda _i(\\overline{a}) = \\lambda _i(\\overline{b})$ for all $i \\in I$ .",
"Therefore $X = \\lbrace (\\overline{a},\\overline{b}): \\operatorname{tp}(\\overline{a}) = \\operatorname{tp}(\\overline{b}) = r, \\lambda _i(\\overline{a}) = \\lambda _i(\\overline{b}) \\text{ for all } i \\in I \\rbrace $ , which is an $\\emptyset $ -type-definable set.",
"Let $r_{\\overline{a}} = \\operatorname{tp}(\\overline{a}/\\mathcal {P})$ .",
"We then have the following: Claim 3.3 The map from $\\operatorname{Mor}(\\pi (a), \\pi (b))$ to $r_{\\overline{a}}(\\mathbb {M}) \\cap \\lbrace \\overline{x}: \\pi (\\overline{x}) = \\pi (b) \\rbrace $ taking $\\sigma $ to $\\sigma (\\overline{a})$ is a bijection.",
"First injectivity: suppose $\\sigma (\\overline{a}) = \\tau (\\overline{a})$ .",
"Every element of $\\pi ^{-1}(\\pi (a)) \\cap q(\\mathbb {M})$ is written as $f(\\overline{a},c)$ , for some $c \\models \\Phi $ , and $\\sigma (f(\\overline{a},c)) = f(\\sigma (\\overline{a}),\\sigma (c)) = f(\\tau (a),c) = \\tau (f(\\overline{a},c))$ , so $\\tau = \\sigma $ .",
"For surjectivity, given $\\overline{b}\\models r_{\\overline{a}}$ , since $\\overline{a}$ and $\\overline{b}$ have the same type over $\\mathcal {P}$ , by Fact REF , there is an automorphism of the monster model, fixing $\\mathcal {P}$ , and taking $\\overline{a}$ to $\\overline{b}$ .",
"The restriction of this automorphism to $q_{\\pi (a)}(\\mathbb {M})$ belongs to $\\operatorname{Mor}(\\pi (a), \\pi (b))$ .",
"By Claim REF , for any $(\\overline{a},\\overline{b}) \\in X$ , there is a unique $\\sigma \\in \\operatorname{Mor}(\\pi (\\overline{a}), \\pi (\\overline{b}))$ such that $\\sigma (\\overline{a}) = \\overline{b}$ .",
"And for any $\\overline{a}\\models r$ and $\\sigma \\in \\operatorname{Mor}(\\pi (\\overline{a}), \\pi (\\overline{b}))$ , we also have $(\\overline{a},\\sigma (\\overline{a})) \\in X$ .",
"However, this correspondence may not be injective: for a $\\sigma \\in \\operatorname{Mor}(\\pi (a),\\pi (b))$ , there are multiple elements of $X$ corresponding to it.",
"We will solve this problem with an equivalence relation.",
"Claim 3.4 There is a formula $\\psi (\\overline{x}_1,\\overline{x}_2,y,z)$ such that for any $\\sigma \\in \\operatorname{Mor}(\\mathcal {G})$ , any $\\overline{a}\\models r$ , any $a \\models q$ such that $\\operatorname{dom}(\\sigma ) = \\pi (a) = \\pi (\\overline{a})$ and any $b$ , we have $\\models \\psi (\\overline{a},\\sigma (\\overline{a}),a,b)$ if and only if $b = \\sigma (a)$ .",
"By the proofs of Claim REF and Claim REF , if $\\overline{a},\\overline{b}$ realise $r$ , with $\\lambda _i(\\overline{a}) = \\lambda _i(\\overline{b})$ for all $i$ , and $c_1,c_2$ realise the partial type $\\Phi $ of Claim REF , then $f(\\overline{a},c_1) = f(\\overline{a},c_2)$ if and only if $f(\\overline{b},c_1) = f(\\overline{b},c_2)$ (and these are well defined).",
"By compactness, there is a formula $\\theta (w)$ and a finite subset $J \\subset I$ such that the previous property is true replacing $\\Phi $ by $\\theta $ and $I$ by $J$ .",
"Let the formula $\\psi (\\overline{x}_1,\\overline{x}_2,y,z)$ be $\\exists w (f(\\overline{x}_1,w) = y \\wedge f(\\overline{x}_2,w) = z \\wedge \\theta (w))$ .",
"We now check that this formula works.",
"Suppose first that $\\overline{a},\\sigma (\\overline{a}),a,b$ satisfy it.",
"Then there is $c \\models \\theta (w)$ such that $f(\\overline{a},c) = a$ and $f(\\sigma (\\overline{a}),c) = b$ .",
"But as $a \\models q$ , there is also $d \\models \\Phi $ such that $f(\\overline{a}, d) = a$ .",
"Since $d \\models \\Phi $ , it is a realization of $\\mathcal {P}$ , hence $\\sigma (a) = \\sigma (f(\\overline{a},d)) = f(\\sigma (\\overline{a}),d)$ .",
"So $f(\\overline{a}, c) = a = f(\\overline{a},d)$ , the tuple $d$ is a realization of $\\Phi $ , and $f(\\sigma (\\overline{a},d)) = \\sigma (a)$ .",
"By choice of $\\psi $ , this implies $b = f(\\sigma (\\overline{a}),c) = f(\\sigma (\\overline{a}),d) = \\sigma (a)$ .",
"Conversely, suppose that $b = \\sigma (a)$ .",
"Since $\\overline{a}\\models r$ and $\\operatorname{dom}(\\sigma ) = \\pi (a) = \\pi (\\overline{a})$ , there is $c \\models \\Phi $ such that $f(\\overline{a},c) = a$ .",
"Therefore $b = \\sigma (f(\\overline{a},c)) = f(\\sigma (\\overline{a}), c)$ , so we can take $c$ to be the $w$ of the formula.",
"Now define an equivalence relation $E$ on $X$ as $(\\overline{a}_1,\\overline{b}_1)E(\\overline{a}_2,\\overline{b}_2)$ if and only if $\\pi (\\overline{a}_1) = \\pi (\\overline{a}_2)$ and for some $\\sigma \\in \\operatorname{Mor}(\\mathcal {G})$ , we have $\\sigma (\\overline{a}_1) = \\overline{b}_1$ and $\\sigma (\\overline{a}_2) = \\overline{b}_2$ .",
"So $(\\overline{a}_1,\\overline{b}_1)E(\\overline{a}_2,\\overline{b}_2)$ if and only if the tuples $(\\overline{a}_1,\\overline{b}_1)$ and $(\\overline{a}_2,\\overline{b}_2)$ represent the same morphism $\\sigma $ .",
"Then the following is true: Claim 3.5 $E$ is relatively $\\emptyset $ -definable on $X \\times X$ .",
"Recall that we denote $\\operatorname{tp}(a/\\pi (a))$ , for $a \\models q$ , by $q_{\\pi (a)}$ .",
"We also denote $r_{\\pi (a)} = \\operatorname{tp}(\\overline{a}/\\pi (\\overline{a}))$ , for some $\\overline{a}\\models r$ with $\\pi (\\overline{a}) = \\pi (a)$ .",
"We first show that $(\\overline{a}_1,\\overline{b}_1)E(\\overline{a}_2,\\overline{b}_2)$ if and only if $\\pi (\\overline{a}_1) = \\pi (\\overline{a}_2)$ and for any $a \\models q_{\\pi (\\overline{a}_1)}|_{\\overline{a}_1,\\overline{a}_2,\\overline{b}_1,\\overline{b}_2}$ we have $\\forall z \\psi (\\overline{a}_1,\\overline{b}_1,a,z) \\leftrightarrow \\psi (\\overline{a}_2,\\overline{b}_2,a,z)$ .",
"The left to right direction is immediate.",
"So assume that the right-hand condition holds.",
"There are $\\sigma ,\\tau \\in \\operatorname{Mor}(\\mathcal {G})$ with $\\sigma (\\overline{a}_1) = \\overline{b}_1$ and $\\tau (\\overline{a}_2) = \\overline{b}_2$ .",
"Let $\\overline{a}_3 \\models r_{\\pi (\\overline{a}_1)}$ be independent from $\\overline{a}_1,\\overline{a}_2,\\overline{b}_1,\\overline{b}_2$ .",
"If we let $\\overline{a}_3 = (a_{3,1}, \\cdots , a_{3,n})$ , then by independence, we have $\\forall z \\psi (\\overline{a}_1,\\overline{b}_1,a_{3,i},z) \\leftrightarrow \\psi (\\overline{a}_2,\\overline{b}_2,a_{3,i},z)$ , for all $1 \\le i \\le n$ .",
"Hence $\\sigma (\\overline{a}_3) = \\tau (\\overline{a}_3)$ .",
"Let $a^{\\prime }$ be any realization of $q_{\\pi (\\overline{a}_1)}$ .",
"Then $a^{\\prime } = f(\\overline{a}_3,c)$ for some $c$ , since $\\overline{a}_3$ is a realization of $r_{\\pi (\\overline{a}_1)}$ .",
"So $\\sigma (a^{\\prime }) = \\sigma (f(\\overline{a}_3,c)) = f(\\sigma (\\overline{a}_3),c) = f(\\tau (\\overline{a}_3),c) = \\tau (a^{\\prime })$ .",
"This is true for any realization $a^{\\prime }$ of $q_{\\pi (\\overline{a}_1)}$ , so $\\tau = \\sigma $ .",
"Notice that the right-hand condition is equivalent to a formula over $\\pi (\\overline{a}_1)$ because the stationary type $q_{\\pi (\\overline{a}_1)}$ is definable over $\\pi (\\overline{a}_1)$ .",
"So if we fix $\\pi (a) \\models \\pi (q)$ , there is a formula $\\theta _{\\pi (a)}(z_1,t_1,z_2,t_2,y)$ over $\\emptyset $ such that for any $\\overline{a}_1, \\overline{a}_2 \\models r_{\\pi (a)}$ , and any $\\overline{b}_1,\\overline{b}_2$ , we have $(\\overline{a}_1,\\overline{b}_1)E(\\overline{a}_2,\\overline{b}_2)$ if and only if $\\theta _{\\pi (a)}(\\overline{a}_1,\\overline{b}_1,\\overline{a}_2,\\overline{b}_2,\\pi (a))$ .",
"A priori, this formula $\\theta _{\\pi (a)}$ depends on $\\pi (a)$ , hence we cannot yet conclude that $E$ is relatively definable, let alone relatively $\\emptyset $ -definable.",
"However, if we can prove that for any $a,b \\models q$ the formulas $\\theta _{\\pi (a)}(z_1,t_1,z_2,t_2,\\pi (a))$ and $\\theta _{\\pi (b)}(z_1,t_1,z_2,t_2,\\pi (a))$ are equivalent, we would get relative $\\emptyset $ -definability.",
"Note that the formula $\\theta _{\\pi (a)}$ we obtained is a defining scheme for a formula in the stationary type $q_{\\pi (a)} = \\operatorname{tp}(a/\\pi (a))$ .",
"We will use this to show the desired equivalence.",
"Let $\\pi (a),\\pi (b) \\models \\pi (q)$ , and $\\sigma $ an automorphism such that $\\sigma (\\pi (a)) = \\pi (b)$ .",
"Let $\\phi (x,y,z)$ be a formula over $\\emptyset $ .",
"Since $q_{\\pi (a)}$ and $q_{\\pi (b)}$ are definable and stationary, there are defining schemes $\\theta _{\\pi (a)}(z,\\pi (a))$ (respectively $\\theta _{\\pi (b)}(z,\\pi (b))$ ) for $\\phi (x,y,z)$ and $q_{\\pi (a)}$ (respectively $q_{\\pi (b)}$ ), and the formulas $\\theta _{\\pi (-)}(z,y)$ are over the empty set.",
"Now, let $\\overline{c}$ be a tuple, and $a^{\\prime }$ a realization of $q_{\\pi (a)}\\vert _{\\overline{c}}$ , the unique non-forking extension of $q_{\\pi (a)}$ to $\\lbrace \\pi (a),\\overline{c}\\rbrace $ .",
"Then: $\\theta _{\\pi (a)}(\\overline{c},\\pi (a)) &\\Leftrightarrow \\phi (x,\\pi (a),\\overline{c}) \\in q_{\\pi (a)}\\vert _{\\overline{c}} \\\\& \\Leftrightarrow \\models \\phi (a^{\\prime },\\pi (a),\\overline{c}) \\\\& \\Leftrightarrow \\models \\phi (\\sigma (a^{\\prime }), \\pi (b), \\sigma (\\overline{c})) \\\\& \\Leftrightarrow \\phi (x,\\pi (b),\\sigma (\\overline{c})) \\in q_{\\pi (b)}|_{\\sigma (\\overline{c})} \\text{ because } \\sigma (a^{\\prime }) \\models q_{\\pi (b)}|_{\\sigma (\\overline{c})} \\\\& \\Leftrightarrow \\theta _{\\pi (b)}(\\sigma (\\overline{c}),\\pi (b)) \\\\& \\Leftrightarrow \\theta _{\\pi (b)}(\\overline{c}, \\pi (a))$ Applying this to the formula $\\forall w \\psi (z_1,t_1,x,w) \\leftrightarrow (\\psi (z_2,t_2,x,w) \\wedge \\pi (z_1) = y = \\pi (z_2))$ and the $q_{\\pi (a)}$ , where $z = (z_1,t_1,z_2,t_2)$ , we obtain, for any $\\pi (a) ,\\pi (b)$ realizations of $\\pi (q)$ , that $\\models \\theta _{\\pi (a)}(\\overline{a}_1,\\overline{b}_1,\\overline{a}_2,\\overline{b}_2,\\pi (a))$ if and only if $\\models \\theta _{\\pi (b)}(\\overline{a}_1,\\overline{b}_1,\\overline{a}_2,\\overline{b}_2,\\pi (a))$ .",
"Therefore, we can fix $\\pi (b)$ , and use the formula $\\theta _{\\pi (b)}$ to obtain for any $\\pi (a) \\models \\pi (q)$ , for any $\\overline{a}_1, \\overline{a}_2 \\models r_{\\pi (a)}$ and any $\\overline{b}_1,\\overline{b}_2$ , that $(\\overline{a}_1,\\overline{b}_1)E(\\overline{a}_2,\\overline{b}_2)$ if and only if $\\theta _{\\pi (b)}(\\overline{a}_1,\\overline{b}_1,\\overline{a}_2,\\overline{b}_2,\\pi (a))$ .",
"So $\\theta _{\\pi (b)}$ is the formula defining $E$ .",
"Hence we obtain an $\\emptyset $ -type-definable set $X/E$ .",
"But we had, by Claim REF , a map from $X$ to $\\operatorname{Mor}(\\mathcal {G})$ .",
"And $(\\overline{a}_1,\\overline{b}_1)E(\\overline{a}_2,\\overline{b}_2)$ if and only if they have the same image under this map.",
"Therefore we have obtained a bijection from $X/E$ to $\\operatorname{Mor}(\\mathcal {G})$ .",
"Notice that this also yields $\\emptyset $ -definability of domain and codomain: since the maps are represented by elements in the fibers, we can just take images under $\\pi $ of any of their representant.",
"We can, using this coding for morphisms of the groupoid, prove that the groupoip action is relatively $\\emptyset $ -definable.",
"If $\\sigma \\in \\operatorname{Mor}(\\pi (a),\\pi (b))$ , we can pick any representant $(\\overline{a}, \\sigma (\\overline{a}))$ .",
"Then $\\sigma (a)$ is the unique tuple satisfying $\\psi (\\overline{a},\\sigma (\\overline{a}),a,z)$ .",
"Since this does not depend on the representant we pick, we obtain that $\\sigma (a) \\in \\operatorname{dcl}(\\sigma ,a)$ (and the formula witnessing it is uniform in $\\sigma $ and $a$ ).",
"This yields that the groupoid action is relatively $\\emptyset $ -definable.",
"To finish the proof, we need to construct the composition in an $\\emptyset $ -definable way.",
"Claim 3.6 The composition of $\\operatorname{Mor}(\\mathcal {G})$ is definable.",
"Let $\\sigma ,\\tau ,\\mu \\in \\operatorname{Mor}(\\mathcal {G})$ .",
"Let $\\overline{a},\\overline{b},\\overline{c}\\models r$ , with $\\pi (\\overline{a}) = \\operatorname{dom}(\\sigma ), \\pi (\\overline{b}) = \\operatorname{dom}(\\tau )$ and $\\pi (\\overline{c}) = \\operatorname{dom}(\\sigma )$ .",
"We will show that the equality $\\tau \\circ \\sigma = \\mu $ holds if and only if $\\operatorname{dom}(\\sigma ) = \\operatorname{dom}(\\mu ), \\operatorname{cod}(\\tau ) = \\operatorname{cod}(\\mu ),\\operatorname{cod}(\\sigma ) = \\operatorname{dom}(\\tau ) $ and for any $a \\models q_{\\pi (\\overline{a})}|_{\\overline{a},\\overline{b},\\overline{c},\\sigma ,\\tau ,\\mu }$ , we have: $\\forall z \\psi (\\overline{c},\\mu (\\overline{c}),a,z) \\leftrightarrow \\exists u (\\psi (\\overline{a},\\sigma (\\overline{a}),a,u) \\wedge (\\psi (\\overline{b},\\tau (\\overline{b}),u,z))) $ The left to right direction is again immediate.",
"For the right to left direction, we can proceed as in Claim REF , and assume that the right-hand side holds.",
"Pick $\\overline{a}_2 \\models r_{\\pi (\\overline{a})}|_{\\sigma ,\\tau ,\\mu ,\\overline{a},\\overline{b},\\overline{c}}$ , then, as was done in Claim REF , we obtain $\\mu (\\overline{a}_2) = \\tau \\circ \\sigma (\\overline{a}_2)$ .",
"But any $a^{\\prime } \\models q_{\\pi (\\overline{a})}$ is equal to $f(\\overline{a}_2,c)$ for some $c$ tuple of realizations of $\\mathcal {P}$ .",
"So we get $\\mu (a^{\\prime }) = f(\\mu (\\overline{a}_2),c) = f(\\tau \\circ \\sigma (\\overline{a}_2),c) = \\tau \\circ \\sigma (a^{\\prime })$ .",
"So $\\mu = \\tau \\circ \\sigma $ .",
"Note that since the type $q_{\\pi (\\overline{a})}$ is stationary and definable, the right-hand side condition is equivalent to a formula over $\\operatorname{dom}(\\sigma ) = \\pi (\\overline{a})$ .",
"Moreover, the truth of this formula does not depend on the representants of $\\sigma ,\\tau $ and $\\mu $ that we pick.",
"Therefore it only depends on $\\sigma ,\\tau $ and $\\mu $ .",
"Hence, if we fix $\\pi (a)$ , we obtain a formula $\\theta _{\\pi (a)}(x,y,z)$ over $\\pi (a)$ such that for all $\\sigma ,\\tau ,\\mu \\in \\operatorname{Mor}(\\mathcal {G})$ with $\\operatorname{dom}(\\sigma ) = \\operatorname{dom}(\\mu ) = \\pi (a)$ , we have $\\theta _{\\pi (a)}(\\sigma ,\\tau ,\\mu )$ if and only if $\\mu = \\tau \\circ \\sigma $ .",
"Again, this formula is a defining scheme for $q_{\\pi (a)}$ .",
"We can apply the proof of Claim REF to this situation, to get a formula $\\theta $ over $\\emptyset $ such that for all $\\sigma ,\\tau ,\\mu \\in \\operatorname{Mor}(\\mathcal {G})$ , $\\mu = \\tau \\circ \\sigma $ if and only if $\\theta (\\sigma ,\\tau ,\\mu )$ .",
"So the composition in $\\mathcal {G}$ is relatively $\\emptyset $ -definable.",
"This finishes the proof: we have obtained a type-definable groupoid, and we already saw that its natural action on $q(\\mathbb {M})$ is relatively $\\emptyset $ -definable.",
"We will denote this groupoid $\\mathcal {G}(q,\\pi /\\mathcal {P})$ , or just $\\mathcal {G}$ when it is clear what type and projection are considered.",
"Here is a first connection between the groupoid $\\mathcal {G}(q, \\pi / \\mathcal {P})$ and the type $q$ .",
"In section , we defined the connected component of a groupoid.",
"An easy consequence of Fact REF is that the connected components of $\\mathcal {G}(q,\\pi /\\mathcal {P})$ correspond to the orbits of $\\pi (q)$ under $\\operatorname{Aut}(\\pi (q)/\\mathcal {P})$ (even if this group is not type-definable).",
"What if $q$ is $\\mathcal {P}$ -internal?",
"Our theorem specializes in the following way: we can pick $e \\in \\operatorname{dcl}(\\emptyset )$ , and set $\\pi (a) = e$ for all $a \\models q$ .",
"We obtain a groupoid with only one object, that is, a group, which is just the type-definable binding group of $q$ over $\\mathcal {P}$ .",
"In the internal case, the type-definable group $\\operatorname{Aut}(q/\\mathcal {P})$ can be shown (see [9]) to be definably isomorphic to a type-definable group in $\\mathcal {P}^{eq}$ , possibly using some extra parameters.",
"In particular, the group $\\operatorname{Aut}(q/\\mathcal {P})$ is internal to $\\mathcal {P}$ .",
"One would hope that in our context, the groupoid $\\mathcal {G}(q,\\pi /\\mathcal {P})$ is also $\\mathcal {P}$ -internal.",
"This result, proved with the help of Omar Léon Sánchez, shows that it is unfortunately not the case: Proposition 3.7 If $\\mathcal {G}$ is internal to $\\mathcal {P}$ and connected, then $q$ is internal to $\\mathcal {P}$ .",
"By internality assumption, there is a set of parameters $B$ such that $\\operatorname{Mor}(\\mathcal {G}) \\subset \\operatorname{dcl}(\\mathcal {P},B)$ .",
"Let $a$ and $b$ be any realizations of $q$ , and let $\\overline{a}$ be a fundamental system of solutions for $\\operatorname{tp}(a/\\pi (a))$ .",
"Since $\\mathcal {G}$ is connected, there is $\\sigma \\in \\operatorname{Mor}(\\pi (a),\\pi (b))$ .",
"Moreover, the tuple $\\overline{b}= \\sigma (\\overline{a})$ is a fundamental system of solutions for $\\operatorname{tp}(b/\\pi (b))$ .",
"Therefore, there is $\\overline{d}\\in \\mathcal {P}$ such that $b = f(\\overline{b},\\overline{d}) = f(\\sigma (\\overline{a}),\\overline{d})$ .",
"But $\\sigma \\in \\operatorname{dcl}(\\mathcal {P},B)$ , therefore $b \\in \\operatorname{dcl}(\\mathcal {P},B,\\overline{a})$ .",
"Note that this result is also true if one replace connected by boundedly many connected components.",
"Some connectedness assumption is required to make this proof work.",
"In some cases, the groupoid associated to a non-internal type might actually be internal as well.",
"Indeed, it is proven in [6], that the groupoids associated to certain relatively internal definable sets are internal to the base set.",
"The context in which these objects are studied in this paper is slightly different from ours, and it would be interesting to see which results can transfer, in one way or the other.",
"Remark 3.8 If we assume that $\\pi (q)$ is $\\mathcal {P}$ -internal, Theorem REF yields the type-definable binding group $\\operatorname{Aut}(\\pi (q)/\\mathcal {P})$ .",
"If we moreover assume that $\\pi (q)$ is fundamental (see Definition REF ), we obtain a definable functor $\\Pi : \\mathcal {G}(q,\\pi /\\mathcal {P}) \\rightarrow \\operatorname{Aut}(\\pi (q)/\\mathcal {P})$ .",
"Indeed, we can send the morphism represented by $(\\overline{a},\\overline{b})$ to the one represented by $(\\pi (\\overline{a}), \\pi (\\overline{b}))$ .",
"We will see in the fifth section that this generalizes to the case of $\\pi (q)$ not fundamental, after introducing Delta groupoids."
],
[
"Retractability",
"In this section, we consider retractability, which was introduced in [5].",
"There, it was used to study groupoids arising from internality, and was linked to 3-amalgamation in stable theories.",
"Interestingly, it has some meaningful content in the context of our paper as well.",
"Definition 4.1 An $\\emptyset $ -type-definable groupoid $\\mathcal {G}$ is retractable if it is connected and there exist an $\\emptyset $ -definable partial function $g(x,y)=g_{x,y}$ such that for all $a,b$ objects of $\\mathcal {G}$ , we have $g_{a,b} \\in \\operatorname{Mor}(a,b)$ .",
"Moreover, we require the compatibility condition that $g(b,c) \\circ g(a,b) = g(a,c)$ for all objects $a,b,c$ (note that this implies $g_{a,a} = \\operatorname{id}_a$ and $g_{a,b}^{-1} = g_{b,a}$ for all $a,b$ ).",
"The following was proved in [5], but we include their proof here for completeness: Remark 4.2 An equivalent definition of retractability is given by: there exist an $\\emptyset $ -type-definable group $G$ , and a full, faithfull $\\emptyset $ -definable functor $F: \\mathcal {G}\\rightarrow G$ .",
"If we have such a functor $F: \\mathcal {G}\\rightarrow G$ , we can take $g_{a,b} = F^{-1}(\\lbrace \\operatorname{id}_{G} \\rbrace ) \\cap \\operatorname{Mor}(a,b)$ , which is a singleton because $F$ is full and faithfull.",
"The compatiblity condition is easily checked, and this is definable uniformly in $(a,b)$ .",
"If $\\mathcal {G}$ is retractable, then we can construct a relation $E$ on $\\operatorname{Mor}(\\mathcal {G})$ as follows: if $\\sigma \\in \\operatorname{Mor}(a,b)$ and $\\tau \\in \\operatorname{Mor}(c,d)$ , then $\\sigma E \\tau $ if and only if $\\tau = g_{b,d} \\circ \\sigma \\circ g_{c,a}$ .",
"By the compatibility condition, this is an equivalence relation, and it is $\\emptyset $ -definable.",
"Now consider $G = \\operatorname{Mor}(\\mathcal {G}) /E$ , and $F: \\mathcal {G}\\rightarrow G$ the quotient map.",
"The groupoid law of $\\mathcal {G}$ goes down to a group law on $G$ .",
"Indeed, if we want to compose $\\sigma \\in \\operatorname{Mor}(a,b)$ and $\\tau \\in \\operatorname{Mor}(c,d)$ in $G$ , notice that $\\tau E g_{d,a} \\circ \\tau $ , so we can define $F(\\sigma ) \\circ F(\\tau ) = F(\\sigma \\circ g_{d,a} \\circ \\tau )$ .",
"Again by the compatibility condition, this is well defined.",
"Finally, it is easy to derive the group axioms from the groupoid axioms of $\\mathcal {G}$ .",
"We are still working with a family of partial types $\\mathcal {P}$ over the empty set, a type $q$ , and an $\\emptyset $ -definable function $\\pi $ such that $q$ is relatively $\\mathcal {P}$ -internal via $\\pi $ .",
"Let $\\mathcal {G}= \\mathcal {G}(q,\\pi /\\mathcal {P})$ .",
"Recall that we denote, for $a \\models q$ , the type $\\operatorname{tp}(a/\\pi (a))$ by $q_{\\pi (a)}$ .",
"In a stable theory, if $p$ and $q$ are stationary types over some fixed set of parameters $A$ , the type of $(a,b)$ , with $a \\models p$ and $b \\models q$ independent over $A$ , is unique.",
"We denote this type $p \\otimes q$ .",
"Proposition 4.3 If $\\mathcal {G}$ is retractable, then there is a complete type $p \\in S(\\emptyset )$ , internal to $\\mathcal {P}$ , weakly orthogonal to $\\pi (q)$ , and an $\\emptyset $ -definable bijection between $q(\\mathbb {M})$ and $p \\otimes \\pi (q) (\\mathbb {M})$ .",
"We consider the $\\emptyset $ -definable relation $x E y \\Leftrightarrow g_{\\pi (x),\\pi (y)}(x) = y$ .",
"The compatibility condition of retractability implies that this is an equivalence relation.",
"Let $\\rho $ be the quotient map, then $\\rho (q)$ is a complete type over the empty set, and it will be the type $p$ of the proposition.",
"There is an $\\emptyset $ -definable function $s: q(\\mathbb {M}) \\rightarrow \\rho (q)(\\mathbb {M}) \\times \\pi (q)(\\mathbb {M})$ sending $x$ to $(\\rho (x),\\pi (x))$ .",
"Since $q$ is a complete type, $s(q(\\mathbb {M}))$ is the set of realizations of a complete type, denoted $s(q)$ .",
"But the function $s$ is bijective.",
"Indeed, notice that each $E$ -class has exactly one element in each fiber of $\\pi $ : each class has at least one element in a given fiber because $\\mathcal {G}$ is connected, and no more than one because $g_{\\pi (a), \\pi (a)} = \\operatorname{id}_a$ .",
"Therefore we can send $(\\rho (a),\\pi (b))$ to the unique element both in the $\\pi (b)$ fiber and in the $E$ -class of $a$ , to obtain an inverse of $s$ .",
"So $\\rho (q)(\\mathbb {M}) \\times \\pi (q)(\\mathbb {M})$ is in $\\emptyset $ -definable bijection with a complete type, hence is itself a complete type over the empty set.",
"In particular $\\pi (q)$ and $\\rho (q)$ are weakly orthogonal, so $\\rho (q)(\\mathbb {M}) \\times \\pi (q)(\\mathbb {M}) = \\rho (q) \\otimes \\pi (q) (\\mathbb {M})$ .",
"We denote $p = \\rho (q)$ .",
"We now just need to prove that $p$ is $\\mathcal {P}$ -internal.",
"Each $E$ -class has a unique representant in each $\\pi $ -fiber.",
"Therefore, fixing $a \\models q$ , we have $p(\\mathbb {M}) \\subset \\operatorname{dcl}(q_{\\pi (a)}(\\mathbb {M}))$ .",
"But by internality of the fibers, we get $q_{\\pi (a)}(\\mathbb {M}) \\subset \\operatorname{dcl}(\\overline{a}, \\mathcal {P})$ , for some tuple $\\overline{a}$ .",
"This yields $p(\\mathbb {M}) \\subset \\operatorname{dcl}(\\overline{a},\\mathcal {P})$ .",
"Corollary 4.4 If $\\mathcal {G}$ is retractable and $\\pi (q)$ is $\\mathcal {P}$ -internal, then $q$ is $\\mathcal {P}$ -internal.",
"Retractability yields a functor $F: \\mathcal {G}\\rightarrow G$ , but one could ask if it has any consequence on the group $\\operatorname{Aut}(q/\\mathcal {P})$ .",
"As it turns out, it does: Proposition 4.5 If $\\mathcal {G}$ is retractable, there is a morphism $R : \\operatorname{Aut}(q/\\mathcal {P}) \\rightarrow G$ , which is surjective.",
"We use the functor $F: \\mathcal {G}\\rightarrow G$ .",
"For $\\sigma \\in \\operatorname{Aut}(q/\\mathcal {P})$ , note that the restriction of $\\sigma $ to $q_{\\pi (a)}(\\mathbb {M})$ is an element of $\\operatorname{Mor}(\\pi (a), \\sigma (\\pi (a)))$ .",
"We denote it by $\\sigma \\vert _{\\pi (a)}$ .",
"We can then set $R(\\sigma ) = F(\\sigma \\vert _{\\pi (a)})$ .",
"Let us show that $R$ is a surjective morphism $R : \\operatorname{Aut}(q/\\mathcal {P}) \\rightarrow G$ .",
"First, we need to prove that $R$ is well defined.",
"To do so, we need to show that for any $b$ , we have $\\sigma \\vert _{\\pi (b)} = g_{\\sigma (\\pi (a)), \\sigma (\\pi (b))} \\circ \\sigma \\vert _{\\pi (a)} \\circ g_{\\pi (b), \\pi (a)}$ , by definition of $F$ .",
"Pick any $x$ with $\\pi (x) = \\sigma (\\pi (a))$ .",
"Since $g_{\\_,\\_}$ is an uniformly $\\emptyset $ -definable family of partial functions, we have $g_{\\sigma (\\pi (a)), \\sigma (\\pi (b))}(x) = y$ if and only if $g_{\\pi (a),\\pi (b)} (\\sigma ^{-1}(x)) = \\sigma ^{-1}(y)$ , for any $y$ .",
"Applying $\\sigma $ to the second equality, we get, for all $y$ , that $g_{\\sigma (\\pi (a)), \\sigma (\\pi (b))}(x) = y$ if and only if $\\sigma (g_{\\pi (a), \\pi (b)}(\\sigma ^{-1}(x))) = y$ , which yields that $\\sigma \\vert _{\\pi (b)} \\circ g_{\\pi (a), \\pi (b)} \\circ \\sigma \\vert _{\\pi (a)}^{-1} = g_{\\sigma (\\pi (a)), \\sigma (\\pi (b))}$ , what we wanted.",
"Therefore we have a well defined map $R: \\operatorname{Aut}(q/\\mathcal {P}) \\rightarrow G$ .",
"It is a morphism because: $R(\\sigma \\circ \\tau ) & = F((\\sigma \\circ \\tau ) \\vert _{\\pi (a)}) \\\\& = F(\\sigma \\vert _{\\tau (\\pi (a))} \\circ \\tau \\vert _{\\pi (a)}) \\\\& = F(\\sigma \\vert _{\\tau (\\pi (a))}) \\circ F(\\tau \\vert _{\\pi (a)}))\\\\& = R(\\sigma ) \\circ M(\\tau )$ For surjectivity, by fullness of $F$ , it is enough to prove that for $\\sigma \\in \\operatorname{Mor}(\\pi (a), \\pi (a))$ , there is $\\tau \\in \\operatorname{Aut}(q/\\mathcal {P})$ restricting to $\\sigma $ .",
"This is true by definition of $\\operatorname{Mor}(\\pi (a), \\pi (a))$ .",
"Proposition 4.6 The group $G$ witnessing retractability is relatively $\\emptyset $ -definably isomorphic to $\\operatorname{Aut}(p/\\mathcal {P})$ , the binding group of $p$ over $\\mathcal {P}$ (where $p$ is the type of Proposition REF ).",
"Recall that $\\operatorname{Mor}(\\mathcal {G})$ is given by $X/E$ , where $X$ is an $\\emptyset $ -type-definable set, and $E$ is an $\\emptyset $ -definable equivalence relation.",
"Moreover, the type-definable set $X$ is composed of pairs of realizations of $r$ , the type introduced in the proof of Theorem REF .",
"In the proof Proposition REF , we constructed an $\\emptyset $ -definable quotient map $\\rho : q(\\mathbb {M}) \\rightarrow p(\\mathbb {M})$ .",
"The type $p = \\rho (q)$ is $\\mathcal {P}$ -internal, hence its binding group $\\operatorname{Aut}(p/\\mathcal {P})$ is similarly given by the type $r^{\\prime }$ of a fundamental system of solutions, an $\\emptyset $ -type-definable set $X^{\\prime }$ and an $\\emptyset $ -definable equivalence relation $E^{\\prime }$ .",
"We can assume that $r^{\\prime } = \\rho (r)$ .",
"For any $\\overline{a}\\models r$ , this allows us to define a group morphism: $P_{\\pi (\\overline{a})} : \\operatorname{Aut}(\\operatorname{tp}(\\overline{a}/\\pi (\\overline{a}))/\\mathcal {P}) & \\rightarrow \\operatorname{Aut}(p/\\mathcal {P}) \\\\\\sigma = (\\overline{a}, \\sigma (\\overline{a}))/E & \\rightarrow (\\rho (\\overline{a}), \\rho (\\sigma (\\overline{a})))/E^{\\prime }$ and by construction of $\\rho $ , this is an isomorphism.",
"It is relatively $\\overline{a}$ -definable.",
"We are also given, by the retractability assumption, a relatively $\\emptyset $ -definable full and faithfull functor $F : \\operatorname{Mor}(\\mathcal {G}) \\rightarrow G$ .",
"By restriction this yields, for any $\\overline{a}\\models r$ , a relatively $\\pi (\\overline{a})$ -definable group isomorphism $F_{\\pi (\\overline{a})} : \\operatorname{Aut}(\\operatorname{tp}(\\overline{a}/\\pi (\\overline{a}))/\\mathcal {P}) \\rightarrow G$ .",
"Hence, for any $\\overline{a}\\models r$ , the groups $G$ and $\\operatorname{Aut}(p/\\mathcal {P})$ are relatively $\\overline{a}$ -definably isomorphic via the composition $P_{\\pi (\\overline{a})} \\circ F_{\\pi (\\overline{a})}^{-1}$ .",
"To complete the proof, we need to show that this morphism is actually relatively $\\emptyset $ -definable.",
"To do so, it is enough (via a compactness argument) to prove that the graph of $P_{\\pi (\\overline{a})} \\circ F_{\\pi (\\overline{a})}^{-1}$ is fixed by any automorphism of $\\mathbb {M}$ .",
"Claim 4.7 For any $\\overline{a},\\overline{b}\\models r$ and $g \\in G$ , we have $P_{\\pi (\\overline{b})} \\circ F_{\\pi (\\overline{b})}^{-1}(g) = P_{\\pi (\\overline{a})} \\circ F_{\\pi (\\overline{a})}^{-1} (g)$ .",
"By the proof of Proposition REF , if $\\overline{a},\\overline{b}$ are realizations of $r$ and $g \\in G$ , then there is $\\sigma \\in \\operatorname{Aut}(q/\\mathcal {P})$ such that $F_{\\pi (\\overline{a})}^{-1}(g)$ is the restriction of $\\sigma $ to $\\operatorname{Aut}(\\operatorname{tp}(\\overline{a}/\\pi (\\overline{a}))/\\mathcal {P})$ and $F_{\\pi (\\overline{b})}^{-1}(g)$ is the restriction of $\\sigma $ to $\\operatorname{Aut}(\\operatorname{tp}(\\overline{b}/\\pi (\\overline{b}))/\\mathcal {P})$ .",
"Hence $F_{\\pi (\\overline{a})}^{-1}(g) = (\\overline{a},\\sigma (\\overline{a}))/E$ and $F_{\\pi (\\overline{b})}^{-1}(g) = (g_{\\pi (\\overline{a}), \\pi (\\overline{b})}(\\overline{a}), \\sigma (g_{\\pi (\\overline{a}), \\pi (\\overline{b})}(\\overline{a})))/E$ , as $g_{\\pi (\\overline{a}), \\pi (\\overline{b})}(\\overline{a}) \\models r$ .",
"We then obtain: $P_{\\pi (\\overline{b})} \\circ F_{\\pi (\\overline{b})}^{-1}(g) & = P_{\\pi (\\overline{b})}((g_{\\pi (\\overline{a}), \\pi (\\overline{b})}(\\overline{a}), \\sigma (g_{\\pi (\\overline{a}), \\pi (\\overline{b})}(\\overline{a})))/E) \\\\& = (\\rho (g_{\\pi (\\overline{a}), \\pi (\\overline{b})}(\\overline{a})), \\rho (\\sigma (g_{\\pi (\\overline{a}), \\pi (\\overline{b})}(\\overline{a}))))/E^{\\prime } \\\\& = (\\rho (g_{\\pi (\\overline{a}), \\pi (\\overline{b})}(\\overline{a})), \\sigma (\\rho (g_{\\pi (\\overline{a}), \\pi (\\overline{b})}(\\overline{a}))))/E^{\\prime } \\\\& = (\\rho (\\overline{a}), \\sigma (\\rho (\\overline{a})))/E^{\\prime }\\text{ by definition of } \\rho \\\\& = P_{\\pi (\\overline{a})}(\\overline{a},\\sigma (\\overline{a})/E) \\\\& = P_{\\pi (\\overline{a})} \\circ F_{\\pi (\\overline{a})}^{-1} (g)$ Now let $g \\in G$ , let $(g, P_{\\pi (\\overline{a})} \\circ F_{\\pi (\\overline{a})}^{-1}(g))$ be in the graph of $P_{\\pi (\\overline{a})} \\circ F_{\\pi (\\overline{a})}^{-1}$ , and let $\\mu $ be an automorphism of $\\mathbb {M}$ .",
"We want to show that $\\mu (g, P_{\\pi (\\overline{a})} \\circ F_{\\pi (\\overline{a})}^{-1}(g))$ is also in the graph of $P_{\\pi (\\overline{a})} \\circ F_{\\pi (\\overline{a})}^{-1}$ .",
"Claim 4.8 We have $\\mu (F_{\\pi (\\overline{a})}) = F_{\\pi (\\mu (\\overline{a}))}$ .",
"This is because the maps $F_{\\pi (\\overline{a})}^{-1}$ are uniformly $\\pi (\\overline{a})$ -definable.",
"Claim 4.9 For any $\\sigma \\in \\operatorname{Aut}(\\operatorname{tp}(\\overline{a}/\\pi (\\overline{a}))/\\mathcal {P})$ , we have $\\mu (P_{\\pi (\\overline{a})}(\\sigma )) = P_{\\pi (\\mu (\\overline{a}))}(\\mu (\\sigma ))$ .",
"The set $\\operatorname{Mor}(\\mathcal {G})$ is $\\emptyset $ -type-definable, hence for any $\\sigma \\in \\operatorname{Aut}(\\operatorname{tp}(\\overline{a}/\\pi (\\overline{a}))/\\mathcal {P})$ we have $\\mu (\\sigma ) = \\tau \\in \\operatorname{Mor}(\\mathcal {G})$ .",
"In particular, we obtain $\\mu (\\sigma (\\overline{a})) = \\tau (\\mu (\\overline{a}))$ , which yields: $\\mu (P_{\\pi (\\overline{a})}(\\sigma )) & = \\mu (P_{\\pi (\\overline{a})}((\\overline{a},\\sigma (\\overline{a}))/E)) \\\\& = (\\rho (\\mu (\\overline{a})),\\rho (\\mu (\\sigma (\\overline{a}))))/E^{\\prime } \\\\& = (\\rho (\\mu (\\overline{a})),\\rho (\\tau (\\mu (\\overline{a}))))/E^{\\prime } \\\\& = P_{\\pi (\\mu (\\overline{a}))}((\\mu (\\overline{a})), \\tau (\\mu (\\overline{a}))/E) \\\\& = P_{\\pi (\\mu (\\overline{a}))}(\\tau ) \\\\& = P_{\\pi (\\mu (\\overline{a}))}(\\mu (\\sigma ))$ Putting everything together, we obtain: $\\mu (P_{\\pi (\\overline{a})} \\circ F_{\\pi (\\overline{a})}^{-1}(g)) & = \\mu (P_{\\pi (\\overline{a})}) \\circ \\mu (F_{\\pi (\\overline{a})})^{-1}(\\mu (g)) \\\\& = P_{\\pi (\\mu (\\overline{a}))} \\circ F_{\\pi (\\mu (\\overline{a}))}(\\mu (g)) \\text{ by Claims } \\ref {unifmor2} \\text{ and } \\ref {unifmor3}\\\\& = P_{\\pi (\\overline{a})} \\circ F_{\\pi (\\overline{a})}^{-1}(\\mu (g)) \\text{ by Claim }\\ref {unifmor1}$ so $(\\mu (g),P_{\\pi (\\overline{a})} \\circ F_{\\pi (\\overline{a})}^{-1}(\\mu (g)))$ belongs to the graph of $P_{\\pi (\\overline{a})} \\circ F_{\\pi (\\overline{a})}^{-1}$ , what we needed to prove.",
"If $\\pi (q)$ is $\\mathcal {P}$ -internal, it has an $\\emptyset $ -type-definable binding group $\\operatorname{Aut}(\\pi (q)/\\mathcal {P})$ , and we have: Theorem 4.10 If $\\mathcal {G}$ is retractable and $\\pi (q)$ is $\\mathcal {P}$ -internal and fundamental, then $q$ is $\\mathcal {P}$ -internal and $\\operatorname{Aut}(q/\\mathcal {P})$ is $\\emptyset $ -definably isomorphic to $G \\times \\operatorname{Aut}(\\pi (q)/\\mathcal {P})$ .",
"We know from Corollary REF that $q$ is internal.",
"Let $\\overline{a}$ be a fundamental system of solutions for $q$ .",
"Recall that there are two $\\emptyset $ -definable quotient maps $\\pi : q(\\mathbb {M}) \\rightarrow \\pi (q)(\\mathbb {M})$ and $\\rho : q(\\mathbb {M}) \\rightarrow p(\\mathbb {M}) = \\rho (q)(\\mathbb {M}) $ .",
"The tuples $\\pi (\\overline{a})$ and $\\rho (\\overline{a})$ are fundamental systems of solutions for $\\pi (q)$ and $\\rho (q)$ .",
"As was done in Proposition REF , we can use this to construct two $\\overline{a}$ -definable surjective group morphisms $\\overline{\\pi } : \\operatorname{Aut}(q/\\mathcal {P}) \\rightarrow \\operatorname{Aut}(\\pi (q)/\\mathcal {P})$ and $\\overline{\\rho } : \\operatorname{Aut}(q/\\mathcal {P}) \\rightarrow \\operatorname{Aut}(\\rho (q)/\\mathcal {P})$ .",
"Using techniques similar to the ones in Proposition REF , we can prove that these two morphisms are $\\emptyset $ -definable.",
"Hence we have produced two $\\emptyset $ -definable group morphisms $\\overline{\\pi } : \\operatorname{Aut}(q/\\mathcal {P}) \\rightarrow \\operatorname{Aut}(\\pi (q)/\\mathcal {P})$ and $\\overline{\\rho } : \\operatorname{Aut}(q/\\mathcal {P}) \\rightarrow \\operatorname{Aut}(\\rho (q)/\\mathcal {P})$ , both surjective.",
"To obtain the desired isomorphism, it would be enough to prove that $\\ker (\\overline{\\pi }) \\cap \\ker (\\overline{\\rho }) = \\operatorname{id}$ and that any element of $\\operatorname{Aut}(q/\\mathcal {P})$ can be written as the product of an element of $\\ker (\\overline{\\rho })$ and an element of $\\ker (\\overline{\\pi })$ .",
"Suppose that $\\sigma \\in \\ker (\\overline{\\pi }) \\cap \\ker (\\overline{\\rho })$ , and let $a \\models q$ .",
"Then $\\sigma $ fixes $\\pi ^{-1} \\lbrace \\pi (a) \\rbrace $ setwise.",
"But $\\sigma \\in \\ker (\\overline{\\rho })$ , hence must fix $\\pi ^{-1} \\lbrace \\pi (a) \\rbrace $ pointwise.",
"Since this is true for any $a \\models q$ , we conclude that $\\sigma = \\operatorname{id}$ .",
"Let $\\sigma $ be any morphism in $\\operatorname{Aut}(q/\\mathcal {P})$ and $a \\models q$ .",
"Consider $g_{\\pi (a), \\pi (\\sigma (a))} \\in \\mathcal {G}$ .",
"It extends to an automorphism $\\tau \\in \\operatorname{Aut}(q/\\mathcal {P})$ by Fact REF , which has to belong to $\\ker (\\overline{\\rho })$ .",
"We can write $\\sigma = \\tau \\circ \\tau ^{-1} \\circ \\sigma $ , so we only need to prove that $\\tau ^{-1} \\circ \\sigma \\in \\ker (\\overline{\\pi })$ .",
"But $\\pi (\\tau ^{-1} \\circ \\sigma (a)) = \\pi (a)$ and $\\pi (q)$ is fundamental, so this implies $\\overline{\\pi }(\\tau ^{-1} \\circ \\sigma ) = \\operatorname{id}$ .",
"Remark 4.11 The assumption that $\\pi (q)$ is fundamental seems necessary for this proof to go through.",
"We still do not know if this theorem is valid without that assumption.",
"So retractability of the groupoid gives a lot more than just internality of the type.",
"In fact, internality does not imply retractability, even if the groupoid is connected.",
"Example 4.12 Consider the two sorted structure $\\mathbb {M}= (G,X, \\mathcal {L}_G, *)$ with one sort being a connected stable group $G$ in the language $\\mathcal {L}_G$ , and the other sort being a principal homogeneous space $X$ for $G$ , with group action $*$ .",
"We will work in $\\mathbb {M}^{eq}$ .",
"One can quickly prove that the sort $X$ has only one 1-type $q$ over $\\emptyset $ , and that this type is stationary and internal to $G$ , with binding group isomorphic to $G$ .",
"Assume that there is an $\\emptyset $ -definable normal subgroup $H$ of $G$ , such that the short exact sequence: $1 \\rightarrow H \\rightarrow G \\rightarrow G/H \\rightarrow 1$ does not definably split.",
"The group action of $G$ on $X$ defines an equivalence relation $E$ , where the class of an element $a \\in X$ is its orbit $H * a$ .",
"Hence, we can define a map $\\pi : X \\rightarrow X/E$ , sending $a \\in X$ to $H * a$ .",
"This is $\\emptyset $ -definable, we have $\\operatorname{Aut}(\\pi (q)/G) \\cong G/H$ and for any $a$ , that $\\operatorname{Aut}(\\operatorname{tp}(a/\\pi (a)) = H$ .",
"The type $q$ is relatively $G$ -internal via $\\pi $ , yielding a groupoid $\\mathcal {G}$ .",
"Since $G/H$ acts transitively on $\\pi (q)$ , this groupoid is connected.",
"Moreover, we have a definable short exact sequence: $1 \\rightarrow H \\rightarrow G \\rightarrow G/H \\rightarrow 1$ which, by assumption, is not definably split.",
"However, if $\\mathcal {G}$ was retractable, our previous work implies that this sequence would be definably split.",
"Hence $\\mathcal {G}$ is not retractable, even though $q$ is $G$ -internal.",
"In the next section, we will introduce a necessary and sufficient condition for internality, using Delta groupoids."
],
[
"Delta groupoids and collapsing",
"In this section, we are again working with a family of partial types $\\mathcal {P}$ over the empty set, a type $q \\in S(\\emptyset )$ , and an $\\emptyset $ -definable function $\\pi $ such that $q$ is relatively $\\mathcal {P}$ -internal via $\\pi $ .",
"We have obtained a groupoid from this relatively internal type.",
"But since $q$ is stationary, for any $n$ , we can form the product of $q$ with itself $n$ -times, denoted $q^{(n)}$ .",
"We still have an $\\emptyset $ -definable projection map $\\pi ^{(n)}$ , given by applying $\\pi $ on each coordinate, and its fibers are $\\mathcal {P}$ -internal too.",
"All we are missing to get relative internality and apply Theorem REF is that the type of a fiber be stationary.",
"This is an easy application of forking calculus.",
"Fact 5.1 For any $n$ and $(a_1, \\cdots ,a_n) \\models q^{(n)}$ , $\\operatorname{tp}(a_1, \\cdots , a_n/\\pi (a_1), \\cdots , \\pi (a_n))$ is stationary.",
"Hence for each $n \\ge 1$ , the type $q^{(n)}$ , together with the map $\\pi ^{(n)}$ , satisfies the assumptions of Theorem REF .",
"We therefore obtain a sequence $\\mathcal {G}_n$ of $\\emptyset $ -type-definable groupoids.",
"Our first groupoid $\\mathcal {G}$ , associated to $q$ and $\\pi $ , becomes $\\mathcal {G}_1$ in this new notation.",
"Recall that $\\mathcal {G}$ was constructed using a type $r$ , which corresponds to a fundamental system of solutions of the type $q_{\\pi (a)} = \\operatorname{tp}(a/\\pi (a))$ , for some (any) $a \\models q$ .",
"Morphisms of $\\mathcal {G}$ were then obtained as elements of $X/E$ , where $X$ is an $\\emptyset $ -type-definable subset of $r(\\mathbb {M})^2$ , and $E$ is a relatively $\\emptyset $ -definable equivalence relation on $X \\times X$ .",
"Notice that for any $\\overline{a}\\models q^{(n)}$ , the type $\\operatorname{tp}(\\overline{a}/\\pi (\\overline{a}))$ has a fundamental solution that is a realization of $r^{(n)}$ .",
"Hence, for each $n$ , the type $r^{(n)}$ will play for $q^{(n)}$ and $\\pi ^{(n)}$ the same role as $r$ for $q$ and $\\pi $ .",
"Thus we obtain, for each $n$ , an $\\emptyset $ -type-definable subset $X_n \\subset r^{(n)}(\\mathbb {M})^2$ and a relatively $\\emptyset $ -definable equivalence relation $E_n$ on $X_n \\times X_n$ , such that $\\operatorname{Mor}(\\mathcal {G}_n)$ is given by $X_n / E_n$ .",
"This yields $\\emptyset $ -definable functors between the $\\mathcal {G}_n$ .",
"To see this, let us introduce some notation: if $\\overline{a}= (a_1, \\cdots , a_n)$ is a tuple, then for any $1 \\le i \\le n$ , we denote $\\overline{a}^{\\wedge i} = (a_1,\\cdots , \\hat{a_i}, \\cdots a_n)$ where the hat means the corresponding coordinate has been removed.",
"Now, if $n>1$ , an element $\\sigma $ of $\\operatorname{Mor}(\\mathcal {G}_n)$ corresponds to the $E_n$ -class of $(\\overline{a}, \\overline{b}) = ((a_1, \\cdots , a_n), (b_1, \\cdots , b_n))$ , where $\\overline{a}$ and $\\overline{b}$ are realizations of $r^{(n)}$ .",
"For any $1 \\le i \\le n $ , we can then send $(\\overline{a}, \\overline{b})/E_n$ to $(\\overline{a}^{\\wedge i},\\overline{b}^{\\wedge i})/E_{n-1}$ .",
"This is well defined, as $(\\overline{a}^{\\wedge i} ,\\overline{b}^{\\wedge i}) \\in X_{n-1}^2$ , and $\\emptyset $ -definable.",
"For each $n>1$ and each $1 \\le i \\le n$ , we hence obtain $\\emptyset $ -definable maps: $\\partial _i^n : \\operatorname{Mor}(\\mathcal {G}_n) & \\rightarrow \\operatorname{Mor}(\\mathcal {G}_{n-1}) \\\\(\\overline{a},\\overline{b})/E_n & \\rightarrow (\\overline{a}^{\\wedge i} ,\\overline{b}^{\\wedge i})/E_{n-1}$ and by setting $\\partial _i^n(\\pi (\\overline{a})) = \\pi (\\overline{a}^{\\wedge i})$ , we can easily check that each $\\partial _i^n$ is an $\\emptyset $ -definable functor from $\\mathcal {G}_{n}$ to $\\mathcal {G}_{n-1}$ .",
"These functors have a clear interpretation as restrictions of partial automorphisms.",
"Indeed, if $\\overline{a}= (a_1, \\cdots ,a_n) \\models r^{(n)}$ and $\\overline{b}= (b_1, \\cdots ,b_n) \\models r^{(n)}$ , then an element $\\sigma $ of $\\operatorname{Hom}(\\mathcal {G}_n)(\\pi (\\overline{a}),\\pi (\\overline{b}))$ is a bijection: $\\sigma : \\operatorname{tp}(\\overline{a}/\\pi (\\overline{a}))(\\mathbb {M}) \\rightarrow \\operatorname{tp}(\\overline{b}/\\pi (\\overline{b}))(\\mathbb {M})$ which is the restriction of an automorphism of $\\mathbb {M}$ fixing $\\mathcal {P}$ pointwise.",
"The element $\\partial _i^n(\\sigma )$ of $\\mathcal {G}_{n-1}$ is then the restriction of $\\sigma $ to a bijection: $\\partial _i^n(\\sigma ) : \\operatorname{tp}(\\overline{a}^{\\wedge i}/\\pi (\\overline{a}^{\\wedge i}))(\\mathbb {M}) \\rightarrow \\operatorname{tp}(\\overline{b}^ {\\wedge i }/\\pi (\\overline{b}^ {\\wedge i }))(\\mathbb {M})$ which still is the restriction of the same global automorphism.",
"Remark 5.2 If we assume that $\\pi (q)$ is $\\mathcal {P}$ -internal, then by Remark REF , there is some $n$ such that any $n$ independent realizations of $\\pi (q)$ form a fundamental system of solutions.",
"Therefore we obtain, as was done in Remark REF , a definable functor $F: \\mathcal {G}_n \\rightarrow \\operatorname{Aut}(\\pi (q)^{(n)}/\\mathcal {P})$ .",
"We are now ready to define the algebraic structure of interest, which will be an $\\emptyset $ -type-definable Delta groupoid.",
"Definition 5.3 A Delta groupoid is the following data: For every integer $n \\in \\mathbb {N}\\backslash \\lbrace 0 \\rbrace $ , a groupoid $\\mathcal {G}_n$ For every integer $n \\in \\mathbb {N}\\backslash \\lbrace 0,1 \\rbrace $ , and every $i \\in \\lbrace 0, \\cdots ,n \\rbrace $ , a groupoid morphism (that is, a functor) $\\partial _i^n: \\mathcal {G}_{n} \\rightarrow \\mathcal {G}_{n-1}$ , called a face map subject to the following condition: $\\partial _i^{n} \\circ \\partial _j^{n+1} = \\partial _{j-1}^n \\circ \\partial _i^{n+1}$ for all $i<j \\le n$ and $n \\ge 1$ .",
"Note that this definition, while adapted to our purpose, is not the one usually given in the simplicial homotopy literature.",
"The interested reader can find an alternative category-theoretic definition in [4].",
"Definition 5.4 A Delta groupoid $\\mathcal {G}$ is $\\emptyset $ -type-definable if every groupoid $\\mathcal {G}_n$ is $\\emptyset $ -type-definable, and all the face maps are $\\emptyset $ -type-definable.",
"The previously defined groupoids $\\mathcal {G}_n$ and maps $\\partial $ are then easily checked to form an $\\emptyset $ -type-definable Delta groupoid.",
"We will denote it by $\\mathcal {G}$ (the previously constructed groupoid now becomes $\\mathcal {G}_1$ ).",
"Remark that the $\\mathcal {G}_n$ are not uniformly type-definable (they do not even live in the same sorts).",
"Notation If $\\overline{a}\\models q^{n}$ for some $n$ , then the type $\\operatorname{tp}(\\overline{a}/\\pi (\\overline{a}))$ is $\\mathcal {P}$ -internal, and we will denote $G_{\\pi (\\overline{a})}$ its binding group.",
"It is $\\operatorname{Mor}(\\pi (\\overline{a}),\\pi (\\overline{a}))$ in $\\mathcal {G}_n$ .",
"Using the Delta groupoid structure, the data of the $G_{\\pi (\\overline{a})}$ can be formed into a projective system of type-definable groups.",
"Indeed, we can take our directed set to be $\\lbrace \\pi (\\overline{a}): \\pi (\\overline{a}) \\models \\pi ^n(q^{(n)}) \\text{ for some } n \\rbrace $ , with $(\\pi (a_1), \\cdots ,\\pi (a_n)) \\le (\\pi (b_1), \\cdots ,\\pi (b_m))$ if and only if $n \\le m$ and $\\pi (a_i) = \\pi (b_i)$ for all $i \\le n$ .",
"If $\\pi (\\overline{a}) \\le \\pi (\\overline{b})$ , the restriction map $G_{\\pi (\\overline{b})} \\rightarrow G_{\\pi (\\overline{a})}$ is definable, as it is a composition of face maps.",
"These maps, together with the $G_{\\pi (\\overline{a})}$ , are easily checked to form a projective system.",
"In particular, we obtain the projective limit $\\varprojlim G_{\\pi (\\overline{a})}$ .",
"Definition 5.5 The Delta groupoid $\\mathcal {G}$ is said to collapse if there is a tuple $\\overline{a}$ of independent realizations of $q$ such that for any $\\overline{b}\\ge \\overline{a}$ , the map $G_{\\pi (\\overline{b})} \\rightarrow G_{\\pi (\\overline{a})}$ is injective.",
"It is said to almost collapse if the maps $G_{\\pi (\\overline{b})} \\rightarrow G_{\\pi (\\overline{a})}$ have finite kernel instead.",
"These maps $G_{\\pi (\\overline{b})} \\rightarrow G_{\\pi (\\overline{a})}$ are not necessarily surjective, but some will be if $\\pi (q)$ is $\\mathcal {P}$ -internal: Remark 5.6 If $\\pi (q)$ is $\\mathcal {P}$ -internal, then there is $m \\in \\mathbb {N}$ such that for all $n \\ge m$ , all $\\pi (\\overline{a}) \\models q^{(n)}$ and $\\pi (\\overline{a}) \\le \\pi (\\overline{b})$ , the map $G_{\\pi (\\overline{b})} \\rightarrow G_{\\pi (\\overline{a})}$ is surjective.",
"Let $\\overline{a}_0 \\models q^{(m)}$ be such that $\\pi (\\overline{a}_0)$ is a fundamental system of solutions for $\\pi (q)$ (such an $\\overline{a}_0$ exist by Remark REF ).",
"Then any $m$ independent realizations of $\\pi (q)$ will be a fundamental system a solutions.",
"Hence for any $n \\ge m$ and any $\\overline{a}\\models q^{(n)}$ , the tuple $\\pi (\\overline{a})$ is a fundamental system of solutions for $\\pi (q)$ .",
"Fix $\\overline{a}\\models q^{(n)}$ for $n \\ge m$ and $\\pi (\\overline{b}) \\ge \\pi (\\overline{a})$ , consider the map $G_{\\pi (\\overline{b})} \\rightarrow G_{\\pi (\\overline{a})}$ .",
"Let $\\sigma \\in G_{\\pi (\\overline{a})}$ , it is the restriction to $\\operatorname{tp}(\\overline{a}/\\pi (\\overline{a}))(\\mathbb {M})$ of an automorphism $\\tilde{\\sigma }$ of $\\mathbb {M}$ .",
"But $\\pi (\\overline{a})$ is a fundamental system of solutions for $\\pi (q)$ , and $\\tilde{\\sigma }$ fixes $\\pi (\\overline{a})$ .",
"Hence $\\tilde{\\sigma }$ fixes $\\pi (q)(\\mathbb {M})$ , and in particular fixes $\\pi (\\overline{b})$ .",
"Therefore $\\tilde{\\sigma }$ restricts to an element of $G_{\\pi (\\overline{b})}$ , and the image of this element under $G_{\\pi (\\overline{b})} \\rightarrow G_{\\pi (\\overline{a})}$ has to be $\\sigma $ .",
"We will now prove a very useful equivalent condition.",
"Lemma 5.7 The Delta groupoid associated to $q, \\pi $ and $\\mathcal {P}$ collapses (respectively almost collapses) if and only if there is a tuple $\\overline{a}$ of independent realizations of $r$ such that for any (some) $b \\models q$ , independent of $\\overline{a}$ , we have $b \\in \\operatorname{dcl}(\\overline{a}, \\pi (b), \\mathcal {P})$ (respectively $b \\in \\operatorname{acl}(\\overline{a}, \\pi (b), \\mathcal {P})$ ).",
"We will only prove the equivalence for collapsing, the other equivalence being proved in a similar way.",
"Suppose first that there is a tuple $\\overline{a}$ of realizations of $r$ such that for any $b \\models q$ independent of $\\overline{a}$ , we have $b \\in \\operatorname{dcl}(\\overline{a}, \\pi (b), \\mathcal {P})$ .",
"Let $\\pi (\\overline{b}) > \\pi (\\overline{a})$ , these are tuples of independent realizations of $\\pi (q)$ , we want to prove that $G_{\\pi (\\overline{b})} \\rightarrow G_{\\pi (\\overline{a})}$ is injective.",
"The type $\\operatorname{tp}(\\overline{b}/\\pi (\\overline{b}))$ is $\\mathcal {P}$ -internal, hence it has a fundamental system of solutions $(b_1, \\cdots ,b_n)$ .",
"Each of these $b_i$ is either in $\\pi ^{-1}(\\pi (\\overline{a}))$ , and hence in $\\operatorname{dcl}(\\overline{a}, \\mathcal {P})$ , or $\\pi (b_i)$ is independent of $\\pi (\\overline{a})$ over $\\emptyset $ , and we can then assume $b_i$ to be independent of $\\overline{a}$ over $\\emptyset $ .",
"In this second case, the assumption yields $b_i \\in \\operatorname{dcl}(\\overline{a}, \\pi (b_i), \\mathcal {P})$ .",
"Hence we obtain $b_i \\in \\operatorname{dcl}(\\overline{a}, \\pi (b_i), \\mathcal {P})$ for all $i$ , so $\\operatorname{tp}(\\overline{b}/\\pi (\\overline{b}))(\\mathbb {M}) \\subset \\operatorname{dcl}(\\overline{a}, \\pi (\\overline{b}), \\mathcal {P})$ .",
"Now let $\\sigma \\in G_{\\pi (\\overline{b}))}$ be such that its image under $G_{\\pi (\\overline{b})} \\rightarrow G_{\\pi (\\overline{a})}$ is the identity.",
"Then it has to fix $\\overline{a}$ , and it fixes $\\pi (\\overline{b})$ and $\\mathcal {P}$ too.",
"Hence it has to fix $\\operatorname{tp}(\\overline{b}/\\pi (\\overline{b}))(\\mathbb {M})$ , so it is the identity of $G_{\\pi (\\overline{b})}$ .",
"For the other implication, suppose that the Delta groupoid collapses.",
"Hence there is a tuple $\\overline{a}$ of independent realizations of $q$ such that for any $\\pi (\\overline{b}) \\ge \\pi (\\overline{a})$ , the map $G_{\\pi (\\overline{b})} \\rightarrow G_{\\pi (\\overline{a})}$ is injective.",
"The type $\\operatorname{tp}(\\overline{a}/\\pi (\\overline{a}))$ is internal, and it has a fundamental system of solutions, which can be taken to be a tuple of independent realizations of $r$ .",
"From now on, we replace $\\overline{a}$ by this tuple.",
"We need to prove that for any $b \\models q$ independent of $\\overline{a}$ , we have $b \\in \\operatorname{dcl}(\\overline{a}, \\pi (b), \\mathcal {P})$ .",
"To do so, it is enough, by Fact REF , to prove that any automorphism $\\sigma $ of $\\mathbb {M}$ fixing $\\overline{a}, \\pi (b)$ and $\\mathcal {P}$ pointwise has to fix $b$ .",
"So consider such an automorphism $\\sigma $ .",
"It restricts to $\\sigma \\in G_{\\pi (b)\\pi (\\overline{a})}$ , as it fixes $\\pi (b)$ and $\\pi (\\overline{a})$ .",
"But it also fixes $\\overline{a}$ , which is a fundamental system of solutions for $\\operatorname{tp}(\\overline{a}/\\pi (\\overline{a}))$ .",
"Hence, its image under the map $G_{\\pi (\\overline{a})\\pi (b)} \\rightarrow \\pi (\\overline{a})$ is the identity, so by collapse assumption, it is itself the identity in $G_{\\pi (\\overline{a})\\pi (b)}$ , and in particular fixes $b$ .",
"Note that we needed the independence assumption in order for the group $G_{\\pi (\\overline{a})\\pi (b)}$ to be in the Delta groupoid.",
"However, if the type $q$ has finite weight (see [9] Chapter 1, Subsection 4.4 for a definition of weight), we obtain: Proposition 5.8 If the type $q$ has finite weight, the Delta groupoid associated to $q, \\pi $ and $\\mathcal {P}$ collapses (respectively almost collapses) if and only if there is a tuple $\\overline{a}$ of independent realizations of $r$ such that for any $b \\models q$ , we have $b \\in \\operatorname{dcl}(\\overline{a}, \\pi (b), \\mathcal {P})$ (respectively $b \\in \\operatorname{acl}(\\overline{a}, \\pi (b), \\mathcal {P})$ ).",
"Again, we will only prove the equivalence for collapsing, the other equivalence being proved in a similar way.",
"The right to left direction is an immediate consequence of Lemma REF (and does not require superstability), so we only need to prove the left to right direction.",
"Assume that the Delta groupoid collapses, and let $\\overline{a}$ be a tuple of independent realizations of $r$ such that for all $b \\models q$ independent of $\\overline{a}$ over $\\emptyset $ , we have $b \\in \\operatorname{dcl}(\\overline{a}, \\pi (b), \\mathcal {P})$ , it exists by Lemma REF .",
"Pick a Morley sequence $(\\overline{a}_i)_{i \\in \\mathbb {N}}$ in $\\operatorname{tp}(\\overline{a}/\\emptyset )$ .",
"Because the type $q$ has finite weight there is $n \\in \\mathbb {N}$ such that for any $b \\models q$ , there is $i \\le n$ such that $b$ and $\\overline{a}_i$ are independent over the empty set.",
"Let $\\sigma $ be an automorphism of $\\mathbb {M}$ such that $\\sigma (\\overline{a}_i) = \\overline{a}$ , we then have that $\\sigma (b)$ is a realization of $q$ , independent of $\\overline{a}$ .",
"Therefore $\\sigma (b) \\in \\operatorname{dcl}(\\overline{a}, \\pi (\\sigma (b)), \\mathcal {P})$ by Lemma REF .",
"Applying $\\sigma ^{-1}$ , we obtain $b \\in \\operatorname{dcl}(\\overline{a}_i, \\pi (b), \\mathcal {P})$ .",
"Hence, picking $\\overline{\\alpha } = (\\overline{a}_1 , \\cdots , \\overline{a}_n)$ , for any $b \\models q$ , we have $b \\in \\operatorname{dcl}(\\overline{\\alpha }, \\pi (b), \\mathcal {P})$ .",
"Remark 5.9 Recall that in a superstable theory, any type has finite weight.",
"Hence, this proposition is true for any type in a superstable theory.",
"We also can prove the following proposition, which is similar to what can be obtained for internal types: Proposition 5.10 Let $q \\in S(\\emptyset )$ be relatively $\\mathcal {P}$ -internal via the $\\emptyset $ -definable function $\\pi $ .",
"Suppose that there is a tuple $e \\in \\mathbb {M}$ such that for all $a \\models q$ , we have $a \\in \\operatorname{acl}(\\pi (a),e,\\mathcal {P})$ .",
"Then the Delta groupoid $\\mathcal {G}$ associated to $q$ and $\\pi $ almost collapses.",
"Let $a$ be a realization of $q$ , independent from $e$ over the empty set.",
"By assumption, there is a tuple $c$ of realizations of $\\mathcal {P}$ such that $a \\in \\operatorname{acl}(e, \\pi (a), c)$ .",
"Consider $\\operatorname{tp}(ac/\\operatorname{acl}(e))$ , it is a stationary type, let $d$ be its canonical base.",
"Pick $(a_i c_i)_{i \\in \\mathbb {N}}$ , a Morley sequence in $\\operatorname{tp}(ac/\\operatorname{acl}(e))$ , which we can assume to be independent from $ac$ over $e$ .",
"We know that $ac \\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss }\\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $d acl(e) $, and from this and the assumption, forking calculus yields $ a acl((a),c,d)$.",
"But $ d acl((ai ci)1 i n)$ for some $ n$, hence $ a acl((a), (ai ci)1 i n, c)$, so $ a acl((a), (ai)1 i n , P)$.$ Now let $a^{\\prime } \\models q$ , independent from $(a_i)_{1 \\le i \\le n}$ over the empty set.",
"Since $a$ is independent from $e$ over the empty set, and independent over $e$ of the sequence $(a_i)_{i \\in \\mathbb {N}}$ , we have that $a$ is independent from $(a_i)_{i \\in \\mathbb {N}}$ over the empty set.",
"Since $q = \\operatorname{tp}(a/\\emptyset )$ is stationary, this implies that $\\operatorname{tp}(a/(a_i)_{i \\in \\mathbb {N}}) = \\operatorname{tp}(a^{\\prime }/(a_i)_{i \\in \\mathbb {N}})$ , hence $a^{\\prime } \\in \\operatorname{acl}(\\pi (a^{\\prime }), (a_i)_{1 \\le i \\le n} , \\mathcal {P})$ .",
"By Lemma REF , this implies that $\\mathcal {G}$ almost collapses.",
"As a corollary of Lemma REF , we obtain the following test for internality: Corollary 5.11 The type $q$ is internal (respectively almost internal) to $\\mathcal {P}$ if and only if and only if the Delta groupoid $\\mathcal {G}$ collapses (respectively almost collapses) and $\\pi (q)$ is internal (respectively almost internal) to $\\mathcal {P}$ .",
"Once again, we will only treat the case of internality and collapse.",
"Suppose first that $q$ is internal to $\\mathcal {P}$ .",
"We immediately get that $\\pi (q)$ is internal as well.",
"It also yields a fundamental system of solutions, denote it $\\overline{a}$ , which we can pick as a tuple of independent realizations of $q$ .",
"Moreover, we can extend $\\overline{a}$ into a tuple of independent realizations of $r$ .",
"If we now pick any $b \\models q$ , we have $b \\in \\operatorname{dcl}(\\overline{a}, \\mathcal {P})$ , hence also $b \\in \\operatorname{dcl}(\\overline{a}, \\pi (b), \\mathcal {P})$ , so $\\mathcal {G}$ collapses by Lemma REF .",
"For the other implication, assume that $\\mathcal {G}$ collapses and $\\pi (q)$ is $\\mathcal {P}$ -internal.",
"As a consequence of Lemma REF , the type $q$ is internal to the family of types $\\mathcal {P}\\cup \\lbrace \\pi (q) \\rbrace $ .",
"But because $\\pi (q)$ is $\\mathcal {P}$ -internal, this implies that $q$ itself is $\\mathcal {P}$ -internal (see [9], Remark 7.4.3).",
"Notice that even without any internality assumption, there is always a surjective morphism $\\operatorname{Aut}(q/\\mathcal {P}) \\rightarrow \\operatorname{Aut}(\\pi (q)/\\mathcal {P})$ .",
"If we assume $\\pi (q)$ is $\\mathcal {P}$ -internal, then the target group is $\\emptyset $ -type-definable.",
"Corollary 5.12 If the type $q$ is internal to $\\mathcal {P}$ , then there is a definable (possibly over some extra parameters) short exact sequence: $1 \\rightarrow \\varprojlim G_{\\pi (\\overline{a})} \\rightarrow \\operatorname{Aut}(q/\\mathcal {P}) \\rightarrow \\operatorname{Aut}(\\pi (q)/\\mathcal {P}) \\rightarrow 1$ and the groups and morphisms are internal to $\\mathcal {P}$ .",
"Set $H = \\ker (\\operatorname{Aut}(q/\\mathcal {P}) \\rightarrow \\operatorname{Aut}(\\pi (q)/\\mathcal {P}))$ .",
"Then we have a short exact sequence: $1 \\rightarrow H \\rightarrow \\operatorname{Aut}(q/\\mathcal {P}) \\rightarrow \\operatorname{Aut}(\\pi (q)/\\mathcal {P}) \\rightarrow 1$ Every group in this sequence is type-definable.",
"Moreover, the left arrow is just inclusion, so is $\\emptyset $ -definable.",
"As for the right arrow, if $\\sigma \\in \\operatorname{Aut}(q/\\mathcal {P})$ is represented by $(\\overline{a},\\sigma (\\overline{a}))$ , we can simply send it to $(\\pi (\\overline{a}),\\pi (\\sigma (\\overline{a})))$ , so the right arrow is definable.",
"The groups and morphisms are internal to $\\mathcal {P}$ .",
"So all we need to do to finish the proof is show that $\\varprojlim G_{\\pi (\\overline{a})}$ is definably isomorphic to $H$ .",
"Since $q$ is $\\mathcal {P}$ -internal the Delta groupoid associated to $q, \\pi $ and $\\mathcal {P}$ collapses.",
"By Corollary REF there is a tuple $\\overline{b}$ of realizations of $q$ such that $G_{\\pi (\\overline{c})} \\rightarrow G_{\\pi (\\overline{b})}$ is injective for any $\\pi (\\overline{c}) \\ge \\pi (\\overline{b})$ .",
"Moreover, since $\\pi (q)$ is $\\mathcal {P}$ -internal, we can also assume, by Remark REF , that these maps are isomorphisms, hence $\\varprojlim G_{\\pi (\\overline{a})} = G_{\\pi (\\overline{b})}$ .",
"By extending $\\overline{b}$ we can assume both that $\\overline{b}$ is a fundamental system of solutions for $q$ and $\\pi (\\overline{b})$ is a fundamental system of solutions for $\\pi (q)$ .",
"We can then define a morphism $G_{\\pi (\\overline{b})} \\rightarrow \\operatorname{Aut}(q/\\mathcal {P})$ by sending $\\sigma \\in G_{\\pi (b)}$ to $\\overline{(\\overline{b}, \\sigma (\\overline{b}))}$ , this is well-defined because $\\overline{b}$ is a fundamental system for $q$ .",
"It is a relatively $\\overline{b}$ -definable map, and it is injective, again because $\\overline{b}$ is a fundamental system for $q$ .",
"But $\\pi (\\overline{b})$ is a fundamental system for $\\pi (q)$ , so the image of this map is contained in $H = \\ker (\\operatorname{Aut}(q/\\mathcal {P}) \\rightarrow \\operatorname{Aut}(\\pi (q)/\\mathcal {P}))$ .",
"Finally, if $\\sigma \\in H$ , then it has to fix $\\pi (\\overline{b})$ , and hence restricts to an element of $G_{\\pi (\\overline{b})}$ , which yields surjectivity of $G_{\\pi (\\overline{b})} \\rightarrow H$ .",
"The splitting of the short exact sequence we obtained has, in some cases, nice consequences: Proposition 5.13 Suppose $q$ is $\\mathcal {P}$ -internal and $\\pi (q)$ is fundamental.",
"If the short exact sequence: $1 \\rightarrow \\varprojlim G_{\\overline{a}} \\rightarrow \\operatorname{Aut}(q/\\mathcal {P}) \\rightarrow \\operatorname{Aut}(\\pi (q)/\\mathcal {P}) \\rightarrow 1$ is definably split and $\\mathcal {G}_1$ is connected, then $\\mathcal {G}_1$ is retractable.",
"Since $\\pi (q)$ is fundamental, an element of $\\operatorname{Aut}(\\pi (q)/\\mathcal {P})$ is then defined as the class of $(\\pi (a), \\pi (b))$ , for $\\pi (a),\\pi (b)$ two realizations of $\\pi (q)$ .",
"Let $s$ be a section of the short exact sequence.",
"We can then define $g_{\\pi (a), \\pi (b)} = s((\\pi (a),\\pi (b))/E^{\\prime })$ , where $E^{\\prime }$ is the equivalence relation used to define $\\operatorname{Aut}(\\pi (q)/\\mathcal {P})$ .",
"This is uniformly $\\emptyset $ -definable, and the compatibility condition is easily checked.",
"We hence obtain a partial converse to Theorem REF : Theorem 5.14 Suppose $q$ is $\\mathcal {P}$ -internal.",
"Assume $\\mathcal {G}_1$ is connected, and $\\pi (q)$ is fundamental.",
"Then $\\mathcal {G}_1$ is retractable if and only if the short exact sequence: $1 \\rightarrow \\varprojlim G_{\\overline{a}} \\rightarrow \\operatorname{Aut}(q/\\mathcal {P}) \\rightarrow \\operatorname{Aut}(\\pi (q)/\\mathcal {P}) \\rightarrow 1$ is definably split.",
"We have seen that internality of $q$ can be read from the collapse of the Delta groupoid.",
"It is also linked to the following notion, first introduced in [8]: Definition 5.15 Suppose $q \\in S(d)$ is a stationary type, and $\\mathcal {P}$ is a family of partial types, over the empty set.",
"We say that $q(x)$ preserves internality to $\\mathcal {P}$ if whenever $a \\models q$ and $c$ are such that $\\operatorname{tp}(d/c)$ is almost $\\mathcal {P}$ -internal, then $\\operatorname{tp}(a/c)$ is also almost $\\mathcal {P}$ -internal.",
"We want to obtain a sufficient condition for $\\operatorname{tp}(a/d)$ to preserve internality.",
"Note that by setting $c=d$ , we get that preserving internality implies almost internality.",
"Remark that if $\\operatorname{tp}(a/d) $ is $\\mathcal {P}$ -internal and stationary, then we can consider the type $p = \\operatorname{tp}(ad/\\emptyset )$ , and the projection $\\pi $ on the $d$ -coordinate.",
"This is a projection with $\\mathcal {P}$ -internal fibers, so yields an $\\emptyset $ -type-definable Delta groupoid $\\mathcal {G}_{p}$ .",
"Intuitively, collapse of the groupoid associated to $\\pi $ and $q$ means that the only thing missing for $q$ to be $\\mathcal {P}$ -internal is for $\\pi (q)$ to be $\\mathcal {P}$ -internal.",
"Therefore, the following result appears quite natural: Proposition 5.16 Suppose $q = \\operatorname{tp}(a/d)$ is $\\mathcal {P}$ -internal and stationary.",
"Let $p = \\operatorname{tp}(ad/\\emptyset )$ .",
"If the Delta groupoid $\\mathcal {G}$ associated to $p$ and the projection $\\pi $ on the $d$ -coordinate almost collapses, then $\\operatorname{tp}(a/d)$ preserves internality to $\\mathcal {P}$ .",
"Recall that we assume $\\emptyset = \\operatorname{acl}(\\emptyset )$ , hence $p$ is stationary.",
"Lemma REF implies the existence of a tuple $\\overline{e}$ of realizations of $p$ , independent from $ad$ over $\\emptyset $ , such that $ad \\in \\operatorname{acl}(\\overline{e}, d, \\mathcal {P})$ .",
"Taking a realization of $\\operatorname{tp}(\\overline{e}/ad)$ independent from $c$ over $ad$ , we can assume that $\\overline{e}$ is independent from $adc$ over $\\emptyset $ .",
"Now, the type $\\operatorname{tp}(d/c)$ is almost $\\mathcal {P}$ -internal, hence there is a tuple $\\overline{d}$ of realizations of $\\operatorname{tp}(d/c)$ , independent from $d$ over $c$ , such that $d \\in \\operatorname{acl}(\\overline{d}, c , \\mathcal {P})$ .",
"We can assume, without loss of generality, that $\\overline{d}$ is independent from $ad \\overline{e}$ over $c$ .",
"Forking calculus yields that $\\overline{d}e$ is independent from $ad$ over $c$ .",
"But $ad \\in \\operatorname{acl}(\\overline{e},d , \\mathcal {P})$ and $d \\in \\operatorname{acl}(\\overline{d}, c ,\\mathcal {P})$ , so $ad \\in \\operatorname{acl}(\\overline{e}, \\overline{d}, c , \\mathcal {P})$ .",
"Hence $\\operatorname{tp}(ad/c)$ is almost $\\mathcal {P}$ -internal.",
"The converse to this proposition is likely to be false.",
"Indeed, suppose $\\operatorname{tp}(a/d)$ is $\\mathcal {P}$ -internal and stationary, but for any tuple $c$ , the type $\\operatorname{tp}(d/c)$ is almost $\\mathcal {P}$ -internal if and only if it is algebraic.",
"This implies that $\\operatorname{tp}(a/d)$ preserves internality to $\\mathcal {P}$ , but should not imply that the groupoid associated to $\\operatorname{tp}(ad/\\emptyset )$ collapses.",
"The construction given on top of page 4 of [8] is a good candidate for a counterexample.",
"It would be interesting to find a necessary and sufficient condition, in terms of Delta groupoids, for a type to preserve internality.",
"In the literature, examples of types preserving internality appear in [1], [2] and [8].",
"A potential direction for future work is to examine, for each of these examples, if the collapse of a Delta groupoid is involved or not."
]
] | 1808.08323 |
[
[
"Detecting Outliers in Data with Correlated Measures"
],
[
"Abstract Advances in sensor technology have enabled the collection of large-scale datasets.",
"Such datasets can be extremely noisy and often contain a significant amount of outliers that result from sensor malfunction or human operation faults.",
"In order to utilize such data for real-world applications, it is critical to detect outliers so that models built from these datasets will not be skewed by outliers.",
"In this paper, we propose a new outlier detection method that utilizes the correlations in the data (e.g., taxi trip distance vs. trip time).",
"Different from existing outlier detection methods, we build a robust regression model that explicitly models the outliers and detects outliers simultaneously with the model fitting.",
"We validate our approach on real-world datasets against methods specifically designed for each dataset as well as the state of the art outlier detectors.",
"Our outlier detection method achieves better performances, demonstrating the robustness and generality of our method.",
"Last, we report interesting case studies on some outliers that result from atypical events."
],
[
"Introduction",
"With the development in sensor technology, increasing amount of data collected from sensors become publicly available.",
"Analyzing such data could benefit many applications such as smart city, transportation, and sustainability.",
"For example, New York City (NYC) has released a massive taxi data set including information such as pickup and dropoff locations and time, trip cost, and trip distance.",
"Such data have been used for studies such as characterizing urban dynamics [21], detecting events in city [34], and estimating travel time [28].",
"In these large-scale sensor datasets, there could be a significant amount of outliers due to sensor malfunction or human operation faults.",
"For example, in NYC taxi data, we have observed trips with extremely long moving distances but unreasonably low trip fares.",
"There are also trips with short displacements between pickup and dropoff locations but have a long trip distance.",
"In a recent work on travel time estimation [28], Wang et al.",
"found that such outliers in the original datasets can break effective travel time estimation methods.",
"Figure: Taxi trip example: suspicious outlying tripThere have been many methods proposed in literature on outlier detection [10].",
"Typical outlier detection methods define a sample as an outlier if it significantly deviates from other data samples.",
"However, such definition may not apply in our case.",
"Consider an example shown in Figure REF .",
"There could be many interpretations of what is an outlier in this figure.",
"One possibility is point A is an outlier while points B and C are more likely to be labeled as normal points based on the spatial proximity of every datum to its neighbors.",
"However, another possibility is sample A could be a long but normal trip because the ratio between travel distance and L2 distance between end points is within the normal range.",
"On the other hand, sample B and sample C, even though being closer to other data samples, could be outliers.",
"Sample B could be a trip with detour because the travel distance is much longer than L2 distance between end points.",
"Sample C has a nearly zero L2 distance (i.e., the same pickup and dropoff locations), which could be an outlier due to sensor malfunction.",
"Figure: Biased Model IllustrationMotivated by the observations on real-world data, we detect outliers based on empirical correlations of attributes, which is close to the contextual outlier detection proposed by Song et.",
"al. [24].",
"For example, we expect correlations between attributes trip time and trip distance in taxi data, and between voltage and temperature in CPU sensor data.",
"If the attributes of a data sample significantly deviate from expected correlations, this data sample is likely to be an anomaly.",
"Domain experts can specify correlation templates so that the definition of an outlier can be customized to the application.",
"We propose a robust regression model that explicitly models the non-outliers and outliers.",
"We feed the algorithm domain knowledge about correlations (e.g., the fact that trip time should be predictable from trip distance & time of day) and it learns how to model them (e.g., how to predict trip time from trip distance and time of day).",
"The model is robust (so outliers do not skew the model parameters) and automatically generates a probability for each data sample being as an outlier and also automatically generates a cut-off threshold on probabilities for outliers.",
"In literature, there is a series of contextual outlier detection methods that use the correlation between contextual attributes and behavioral attributes to detect outliers [24], [14], [19].",
"One problem with contextual outlier detection is that outliers can bias a model that is learned from noisy data.",
"To the best of our knowledge, prior work on contextual outlier detection did not consider this issue.",
"The biased model could end up marking outliers as non-outliers and non-outliers as outliers.",
"Take fig:bias as an example.",
"The blue line indicates a model that would have been learned if it was trained on clean data.",
"However, because clean data is not available, contextual outlier detection trains on noisy data.",
"The red line shows the result.",
"To address this problem, we propose a regression model that explicitly models for outliers and non-outliers.",
"We conduct experiments on four real-world datasets and demonstrate the effectiveness of our proposed method with comparison to classical regression methods and five existing outlier detection algorithms.",
"With the help of our model, the root cause of outliers can be identified.",
"For example, in the taxi dataset, we found that many outliers are from sensors produced by a certain manufacturer.",
"We report case studies to support our detected outliers and provide insights into the data that can then be used to study new phenomena or devise ways to improve sensors reliability.",
"In summary, our key contributions are: [leftmargin=*,topsep=2pt,itemsep=1pt] We propose an outlier detection method that utilizes correlations between attributes.",
"Such correlations can be specified by domain experts depending on the application.",
"Different from existing work, our method is a robust regression model that explicitly considers outliers and automatically learns the probability for a data sample being an outlier.",
"It intrinsically generates the thresholds for classification while being robust to parameter skewed by outliers, which is a common problem with other approaches.",
"We conduct rigorous experiments on real-world datasets.",
"For these datasets with missing ground truth, human annotation system is used to obtain labels.",
"We design the machine learning task to show that outliers may bias the model trained on unsanitized dataset.",
"We also inject synthetic outliers to validate the model's robustness to different types of outliers.",
"We compare our approach against five recent outlier detectors (including other contextual outlier detection algorithms).",
"Our method significantly outperformed competing methods and continues to perform well even in extremely noisy datasets (which are common in big data obtained from sensor measurements).",
"The rest of the paper is organized as follows.",
"Related work is discussed in sec:related.",
"sec:problem describes the system overview.",
"We present our outlier model in sec:noise.",
"We then empirically evaluate our methods in sec:exp.",
"We present conclusions in sec:conclusion."
],
[
"Related Work",
"We outline the progress related to two categories: unsupervised outlier detection for numerical datasets and contextual outlier detection."
],
[
"Unsupervised Outlier Detection",
"Typical unsupervised outlier detection methods aim to find data samples that are significantly different from other samples.",
"Yamanishi et al.",
"[32] assume that data is generated from an underlying statistical distribution.",
"The notion of outlier is captured by a strong deviation from the presumed data dependent probabilistic distribution.",
"In distance-based outlier work [17], [22], [25], they measure the distance of a data point to its neighbors.",
"The assumption is that normal objects have a dense neighborhood, thus the outlier is the one furthest from its neighbors.",
"Similar approaches using the spatial proximity are density-based [20], [8], [16].",
"These works adopt the concept of neighbors by measuring the density around a given datum as well as its neighborhood.",
"Breunig et al.",
"[8] introduce a local outlier factor (LOF) for each object in the dataset, indicating its degree of outlierness.",
"The outlier factor is local in the sense that the degree depends on how isolated the object is with respect to only neighboring points.",
"These outlier algorithms consider different characteristics and properties of anomalous objects in a dataset.",
"These outlying properties can vary largely on the type of data and the application domain for which the algorithm is being developed.",
"However, all these studies do not consider the outlying behavior with respect to a given context, assuming every attribute contributes equally to the feature vector."
],
[
"Contextual Outlier Detection",
"Another line of works related to our correlation templates is contextual/conditional outlier detection where one set of attributes defines the context and the other set is examined for unusual behaviors.",
"Song et al.",
"[24] propose conditional anomaly detection that takes into account the user-specified environmental variables.",
"Hong et al.",
"[14] model the data distribution by multivariate function and transform the output space into a new unconditional space.",
"Lang et al.",
"[19] model the relationship of behavioral attributes and contextual attributes from local perspectives (i.e., contextual neighbors) as well as global perspectives.",
"However, none of these works build their models under the awareness/assumption of outlier and thus the training process is limited to clean data.",
"There are also contextual outlier detection for graphs [27], [29] and categorical data [26].",
"Valko et al.",
"[27] proposed a non-parametric graph-based algorithm to detect conditional anomalies.",
"However it assumes the labeled training set is available.",
"Wang et al.",
"[29] address the problem of detecting contextual outliers in graphs using random walk.",
"Tang et al.",
"[26] identify contextual outliers on categorical relational data by leveraging data cube computation techniques.",
"But they are not applicable to numerical data used in our work."
],
[
"Notations and System Overview",
"A dataset $\\operatorname{\\mathcal {I}}$ is a collection of $n$ records $\\lbrace \\vec{z}_1, \\dots , \\vec{z}_n\\rbrace $ where each $\\vec{z}_i$ has $m$ attributes $\\vec{z}_i[1], \\dots , \\vec{z}_i[m]$ .",
"A correlation template is a pair $(j, S)$ where $j$ is a behavior attribute and $S$ $\\subseteq {1,\\dots , m}$ is a set of contextual attributes.",
"This means that the value $\\vec{z}_i[j]$ can be predicted from attributes $\\vec{z}_i[s]$ for $s \\in S$ .",
"To avoid heavy use of sub-subscripts, we will also use the following renaming.",
"For a correlation $(j, S)$ , we set $y_i$ to be $\\vec{z}_i[j]$ and $\\vec{x}_i$ to be the vector of the attribute values in $S$ (i.e.",
"$\\vec{x}_i = [\\vec{z}_i[s] \\text{ for } s \\in S]$ ).",
"An overview of the outlier detector, called $\\texttt {Doc} $ , is shown in fig:SysName-detector.",
"It contains an outlier detector that flags suspicious records.",
"The inputs to the outlier detector are $\\operatorname{\\mathcal {I}}$ and a set $Corr$ of $C$ correlation templates $Corr= \\lbrace (j_c, S_c)\\rbrace _{c=1}^{C}$ .",
"In different applications, some attributes $j$ are usually associated with outlier behavior; but if its relevant attributes $S$ are not specified by domain experts, the system will take the rest of attributes as $S$ , serving as the context of the behavior.",
"In the outlier detector, a filter is a model that learns how to predict $\\vec{z}[j]$ from the $\\vec{z}[s]$ for $s\\in S$ .",
"The goal of each filter is to assign a score $t_i$ to every record indicating its estimated probability that the record is an outlier (this is described in Section ).",
"Higher score implies its higher probability of being an outlier.",
"The expected number of outliers $K$ is the sum of these scores $t_i$ , and the top $K$ records are flagged as outliers by the filter.",
"When using multiple filters, a record is marked as an outlier if at least one filter marks it as an outlier.",
"We average outlier scores returned from multiple filters as an overall outlier score of a record.",
"The result is a dataset $\\tilde{\\operatorname{\\mathcal {I}}}$ in which every record $\\vec{z}_i$ has a flag $\\ell _i$ indicating whether it should be considered an outlier ($\\ell _i=1$ ) or not ($\\ell _i=0$ ).",
"The summary of notations is in tab:notation.",
"Figure: Doc System Overview Table: Notations"
],
[
"Outlier Detector",
"The job of the outlier detector is to take each correlation template $(j, S)$ and learn a model that, for each record $\\vec{z}_i$ , can predict $\\vec{z}_i[j]$ from the attributes $\\vec{z}_i[s]$ for $s\\in S$ .",
"It then assigns an outlier score $t_i$ to each record $\\vec{z}_i$ .",
"This score is the estimated probability that the record is an outlier and is based on how much the actual value $\\vec{z}_i$ deviates from its prediction.",
"We do this by modeling the prediction error as a mixture of light-tailed distributions (for non-outliers) and heavy-tailed distributions (for outliers).",
"Similar noise mixtures are used in robust statistics [9], [23], [30], [15], and typically M-estimators or MCMC inference are used to find model parameters.",
"Instead, we specifically use a variant of expectation-maximization (EM) [12] because it produces variables that, as explained in Section REF , can be interpreted as outlier probabilities $t_i$ .",
"Indeed, we are more interested in these $t_i$ than in the model parameters themselves.",
"We provide an algorithm for linear models in Section REF .",
"Linear models are popular because they are not as restrictive as they initially seem – features can be transformed (e.g., by taking logs, square roots, etc.)",
"so that they have an approximately linear relationship with the target.",
"The ideas from Section REF can be extended to more complex models, such as generalized linear models, and learned with variations of the expectation-maximization framework (EM) [12].",
"Assuming records are independent, the expected number of outliers $K$ is the sum of the outlier probabilities of each record: $K=\\lfloor \\sum _{i=1}^n t_i\\rfloor $ .",
"This means we can take the records with the top $K$ outlier probabilities and flag them as outliers.",
"Since the system can accept many correlation templates as input, it will be learning many models, and a record is labeled as an outlier if any of these models flag it as an outlier."
],
[
"Outlier Data Modeling",
"For the purpose of simplicity and clearness, we use the following renaming in this section.",
"For a correlation $(j, S)$ , we set $y_i$ to be $\\vec{z}_i[j]$ and $\\vec{x}_i$ to be the vector of the attribute values in $S$ (i.e.",
"$\\vec{x}_i = [\\vec{z}_i[s] \\text{ for } s \\in S]$ ).",
"Linear models have a weight vector $\\vec{w}$ , a noise random variable $\\epsilon _i$ , and the functional form $y_i = \\vec{w} \\cdot \\vec{x}_i + \\epsilon _i$ The noise distribution $\\epsilon _i$ for record $i$ is modeled as follows.",
"We assume that there is a probability $p$ that a data point is an outlier.",
"Hence, the error $\\epsilon _i$ is modeled as a mixture distribution – with probability $1-p$ it is a zero mean Gaussian with unknown variance $\\sigma ^2$ , and with probability $p$ it is a Cauchy random variable.",
"Note that the Gaussian distribution has probability density $f_G(\\epsilon _i; {\\sigma }^2) = \\frac{1}{\\sqrt{2\\pi {\\sigma }^2} } \\; \\text{exp} ( -\\frac{\\epsilon _i^2}{2\\sigma ^2} )$ The Cauchy distribution with scale parameter $b$ is a heavy-tailed distribution with undefined mean and variance, hence it is ideal for modeling outliers.",
"It is equivalent to the Student's t distribution with 1 degree of freedom [18].",
"A sample $\\epsilon _i$ from this distribution can be obtained by first sampling a value $\\tau _i$ from the Gamma(0.5, b) distribution then sampling $\\epsilon _i$ from the Gaussian(0, 1/$\\tau _i$ ) distribution [6].",
"The probability of this joint sampling is $ f_{C}(\\epsilon _i, \\tau _i; b) = \\frac{b^{0.5}}{\\Gamma (0.5)} {\\tau _i}^{0.5-1} e^{-b\\tau _i} \\frac{\\sqrt{\\tau _i}}{\\sqrt{2 \\pi }} \\text{exp} ( -\\frac{ \\tau _i \\epsilon _i^2}{2}) $ Given $\\vec{x}_i$ and $y_i$ , we introduce a latent indicator $\\chi _i$ to denote where the error of $\\vec{x}_i$ comes: $\\chi _i &= {\\left\\lbrace \\begin{array}{ll}1 & \\text{ if the error of $\\vec{x}_i$ is generated from the Cauchy}\\\\0 & \\text{ if the error of $\\vec{x}_i$ is generated from the Gaussian}\\end{array}\\right.",
"}$ The expected value of $\\chi _i$ is denoted by $t_i$ and is automatically computed by the EM algorithm.",
"With the model parameters $\\vec{w}$ and unknown noise parameters $\\sigma ^2$ (variance of non-outliers), $p$ (outlier probability), $b$ (scale parameter of outlier distribution), the likelihood function is: $\\begin{aligned}&L(\\vec{w}, {\\sigma }^2, p, b, \\vec{\\chi }, \\vec{\\tau })\\\\&=\\prod _{i=1}^{n} \\left[ (1-p) \\frac{1}{\\sqrt{2\\pi {\\sigma }^2} } \\; \\text{exp} ( -\\frac{( y_i - \\vec{w} \\cdot \\vec{x_i})^2}{2\\sigma ^2} )\\right]^{1-\\chi _i} \\times \\\\&\\left[ p \\frac{b^{0.5}}{\\Gamma (0.5)} {\\tau _i}^{0.5-1} e^{-b\\tau _i} \\frac{\\sqrt{\\tau _i}}{\\sqrt{2 \\pi }} \\text{exp} ( -\\frac{ \\tau _i( y_i - \\vec{w} \\cdot \\vec{x_i} )^2}{2} ) \\right]^{\\chi _i} \\\\\\end{aligned}$ We iteratively update the estimates of $\\sigma ^2$ , $p$ , $b$ , $\\tau _i$ and $t_i$ (the expected value of $\\chi _i$ ) using the EM framework as described below.",
"Note that the scale parameter $b$ of the Cauchy distribution cannot be estimated using maximum likelihood, so we update it using the interquartile range (the standard technique for Cauchy [5]) as explained below."
],
[
"Model Parameters Learning",
"We employ EM algorithm [12] to solve the above likelihood function $L$ .",
"We iteratively update parameters so we add a superscript $^{(k)}$ to parameters to denote their values at the $k^{\\text{th}}$ iteration.",
"The E and M steps are described next."
],
[
"E step",
" $\\tau _i$ update: In eq:likelihood, $\\tau _i$ only appears in $e^{-\\tau _i(b+0.5(y_i - \\vec{w} \\cdot \\vec{x_i} )^2)}$ (after cancellation), which shows that $\\tau _i$ (conditioned on the rest of the variables) follows exponential distribution.",
"The conditional expected value of $\\tau _i$ is $\\frac{1}{ b+0.5(y_i - \\vec{w} \\cdot \\vec{x_i} )^2 }$ By replacing $\\tau _i$ with this expectation, the likelihood function $L$ in eq:likelihood is reduced to $\\begin{aligned}&L(\\vec{w}, {\\sigma }^2, p, b, \\vec{\\chi } ) \\\\&= \\prod _{i=1}^{n} \\left[ (1-p) \\frac{1}{\\sqrt{2\\pi {\\sigma }^2} } \\; \\text{exp} ( -\\frac{( y_i - \\vec{w} \\cdot \\vec{x_i})^2}{2\\sigma ^2} )\\right]^{1-\\chi _i} \\times \\\\& \\left[ p \\frac{\\sqrt{b}}{\\sqrt{\\pi }} e^{-1} \\frac{1}{\\sqrt{2 \\pi }} \\right]^{\\chi _i}\\end{aligned}$ $t_i$ update (here $\\text{sigmoid}(z) = \\frac{1}{1+e^{-z}}$ ): $\\begin{aligned}\\hspace{-8.5359pt}&t_i^{(k+1)} =\\\\\\hspace{-8.5359pt}& \\text{sigmoid} ( \\text{log}(\\frac{p^{(k)}}{1-p^{(k)}})+ 0.5 \\text{log}(\\frac{b^{(k)} {\\sigma ^{2}}^{(k)}}{\\pi e^2}) +\\frac{ (y_i -\\vec{w}^{(k)} \\cdot \\vec{x_i} )^2 }{2 {\\sigma ^{2}}^{(k)}} )\\end{aligned}$ $b$ update: $b^{(k+1)} = \\frac{1}{Median(\\vec{\\xi })}$ where $\\vec{\\xi }$ is the vector of absolute error $| y_i - \\vec{w}^{(k)} \\cdot \\vec{x_i} | $ for the top $K$ records with highest $t_i^{(k)}$ values (note $K=\\lfloor \\sum _{j=1}^n t_j^{(k)}\\rfloor $ )."
],
[
"M step",
"For each iteration before convergence, we update the estimated outlier probability $p$ , the variance of non-outliers $\\sigma ^2$ , and the coefficients $\\vec{w}$ .",
"The updated parameters are listed below.",
"$p$ update: $\\quad $ $p^{(k+1)}= \\frac{1}{n} \\sum _{i=1}^n {t_i}^{(k+1)}$ $\\sigma ^2$ update: ${\\sigma ^{2}}^{(k+1)} = \\frac{\\sum _{i=1}^n (1-{t_i}^{(k+1)})(y_i - \\vec{w}^{(k)} \\cdot \\vec{x_i})^2}{n- \\sum _{i=1}^{n} {t_i}^{(k+1)}}$ $\\vec{w}$ update: $\\vec{w}^{(k+1)}$ is the solution to the weighted least square problem where we give each ($y_i$ , $\\vec{x_i}$ ) a weight (1-$t_i^{(k+1)}$ ).",
"Specifically, this weight (1-$t_i^{(k+1)}$ ) tells how much the model should rely on this datum.",
"Thus, if one example has its $t_i^{(k+1)}$ as 1 (with probability 1 as an outlier), then it does not contribute to the $\\vec{w}^{(k+1)}$ update coefficients.",
"The update for $\\vec{w}^{(k+1)}$ is a weighted least squares update: $\\vec{w}^{(k+1)} \\leftarrow \\left[{X^{\\star }}^TX^{\\star }\\right]^{-1} {X^{\\star }}^T \\vec{y^\\star }$ where $X^{\\star } = VX$ , $\\vec{y^\\star } = V\\vec{y}$ and $V$ is a $n$ by $n$ diagonal matrix with $V_{ii} = \\sqrt{1-t_i^{(k+1)}}$ .",
"The algorithm will terminate when parameters $\\vec{w}, \\sigma , p, b$ converge.",
"Since each iteration involves finding the median absolute error in $b$ update, the time complexity is $O(n \\log n \\cdot T)$ where $T$ is the number of iterations."
],
[
"Outlier Labeling",
"As mentioned before, every filter model assigns to every record $i$ a score $t_i$ indicating an estimated probability that it is an outlier and an estimated fraction of outliers $p$ .",
"The filter then labels a record an outlier if it has one of the top $K$ values of $t_i$ where $K=\\lfloor \\sum _{i=1}^n t_i\\rfloor \\approx p\\times n$ .",
"Each filter model identifies different types of outliers.",
"After the data pass through $i$ filters, each record receives $i$ labels $\\ell _1, \\ell _2, \\cdots \\ell _i$ from $i$ filters where $\\ell _i = 1$ indicates it is an outlier flagged by filter $i$ and $\\ell _i = 0$ otherwise.",
"At the end, we add this record to the outlier set if $\\ell _1 \\vee \\ell _2 \\vee \\cdots \\vee \\ell _i = 1$ ."
],
[
"Experiments",
"The outlier detector was implemented using MapReduce.",
"The rest of experiments used a machine with 2.00GHz Intel(R) Xeon(R) CPU and 48 GB RAM."
],
[
"Datasets",
"We apply our filtering model to four real-world unlabeled datasets.",
"We assume that records in ElNino and Houses datasets are not corrupted which is also an assumption in [27], [19], so we inject synthetic outliers.",
"Other datasets such as Bodyfat and Algae used in [27] exhibit correlations between attributes.",
"However, we do not consider them in our experiments because the data size is too small.",
"A large-scale 22GB public New York City taxi dataset is collected from more than 14,000 taxis, which contains $173,179,771$ taxi trips from 01/01/2013 to 12/31/2013.",
"Each record is a trip with attributes: medallion number (anonymized), hack license (anonymized), vendor, rate code, store and forward flag, pickup location, pickup datetime, drop off location, dropoff datetime, passenger count, payment type, trip time, trip distance, fare amount, tips, tax, tolls, surcharge, and total amount.",
"We use the subset of 143,540,889 trips which are within the Manhattan borough (the boundary is queried from wikimapia.org).",
"We examine the outlying behavior in the trip time, distance, and fare."
],
[
"Intel Lab Sensor",
"This is a public Intel sensor dataset containing a log of about 2.3 million readings from 54 sensors deployed in the Intel Berkeley Research lab between 02/28/2004 to 04/05/2004.",
"Each record is a sensor reading with date, time, sequence number, sensor id, temperature ($$ C), humidity, light, voltage, and the coordinates of sensors' location.",
"We consider two behavioral attributes as humidity and temperature."
],
[
"ElNino",
"This dataset is from UCI repository with 93,935 records after removing records with missing values.",
"These readings are collected from buoys positioned around equatorial Pacific.",
"The sea surface temperature is used as behavior variable while the rest of the oceanographic and meteorological variables are contextual variables."
],
[
"Houses",
"This dataset is from UCI repository with 20,640 observations on the housing in California.",
"The house price is used as behavioral attributes and other variables such as median income, housing median age, total rooms, etc.",
"are contextual attributes."
],
[
"Initial Parameters Setting and Sensitivity",
"We describe how we decide the initial value for $p$ , $b$ , $\\sigma ^2$ , and $\\vec{w}$ used in the outlier detector.",
"[leftmargin=*,topsep=2pt,itemsep=1pt] $p = 0.05$ .",
"We start with 5% as initial value.",
"We observed that the final converged value is not very sensitive to initial settings.",
"Figure REF gives an example of how $p$ changes in the iterations on NYC taxi dataset, with different starting points, to converge to approximately the same value 0.114.",
"${\\sigma ^2} = 1$ .",
"As ${\\sigma ^2}$ represents the variance of non-outliers, it is preferred to initially be a small number.",
"In NYC taxi data, Figure REF shows the convergence path of $\\sigma ^2$ with different starting values in $ [0.5, 2]$ .",
"$\\vec{w} = (1, 0, \\cdots , 0)$ .",
"Let the feature variable be a $j$ dimension vector $\\vec{x} = (\\vec{x}[1], \\vec{x}[2], \\cdots , \\vec{x}[j])$ and the target variable $y$ .",
"Suppose there is a linear relationship between variables $\\vec{x}[1]$ and $y$ .",
"The initial coefficient for $\\vec{x}[1]$ is set to be 1, i.e., $\\vec{w}[1] = 1$ .",
"Others are initialized as 0.",
"$b = \\pi e^2$ .",
"The $b$ value only affects $t_i$ .",
"We choose this setting so that the term $0.5 \\text{log}(\\frac{b \\sigma ^2}{\\pi e^2})$ in the $t_i$ update in Equation REF equals 0, and thus each data point's $t_i$ value in the beginning of the algorithm is dominated by the error ($y_i - \\vec{w} \\cdot \\vec{x}_i$ ).",
"Figure: Parameter sensitivity of one filter used for sensor dataset."
],
[
"Outlier Detection Baselines",
"We evaluate Doc against the state-of-the-art algorithms including traditional outlier detection, contextual outlier detection, regression models and methods specifically designed for outliers in taxi data.",
"[leftmargin=*,topsep=2pt,itemsep=1pt] density-based method.",
"A widely referenced density-based algorithm LOF [8] outlier mining.",
"We implement this method and adopt the commonly used settings for neighbor parameter $k = 10$ .",
"distance-based method.",
"A recent distance-based outlier detection algorithm with sampling [25].",
"We use the provided code and the default sample size $s = 20$ .",
"OLS.",
"The linear regression with ordinary least square estimation.",
"The outlier score of record $i$ is its Cook's distance $\\mathcal {D}_i$ = $\\frac{e_{i}^2}{s^{2}p} \\left[ \\frac{h_{i}}{(1-h_{i})^2} \\right]$ where $e_i$ is the error of the $i$ th record, $s^{2}$ is the mean squared error of the ordinary linear regression model, $p$ is the dimension of feature vector $\\vec{x_i}$ and the leverage of record $i$ is $h_i = \\vec{x}_i [X^T X]^{-1} {\\vec{x}_i}^{~T}$ .",
"GBT.",
"The gradient boosting tree regression model [13].",
"We select parameters from a validation set.",
"The outlier score is defined as the absolute difference between the predicted value and the true value.",
"CAD [24].",
"Conditional Anomaly Detection.",
"A Gaussian mixture model $U$ is used to model contextual attributes $x$ where $U_i$ denotes the $i$ -th component.",
"Another Gaussian mixture model $V$ is used to model behavioral attributes $y$ with $V_j$ .",
"Then, a mapping function $p(V_j | U_i)$ is used to compute the probability of $V_j$ being selected under the condition that its contextual variables are generated from $U_j$ .",
"We set the number of Gaussian components as 30.",
"The outlier score is defined as an inverse of the probability computed from this approach.",
"ROCOD [19].",
"Robust Contextual Outlier Detection.",
"An ensemble of local expected behavior and global expected behavior is used to detect outliers.",
"For the local behavior, a neighbor-based locality sensitive hashing is used to locate contextual neighbors and an average of neighbors' behavior attribute is considered as expected local estimation.",
"A linear ridge regression or non-linear tree regression is chosen to model the global expected behavior.",
"We chose the non-linear model as its global estimation because it is the best performance in Houses and Elnino datasets reported in their work.",
"The outlier score is computed as the absolute value of a weighted average of global and local estimates minus the true value.",
"SOD [33].",
"Smarter Outlier Detection.",
"A method specifically designed for taxi dataset.",
"SOD works by snapping the pickup and dropoff locations to the nearest street segments.",
"The trips which fail to be mapped to the street are considered type I outliers.",
"Next, it computes the shortest path distance and compares that to actual trip distance to detect outliers (called type II outliers).",
"It is worth noting that our outlier filtering model can also employ road network for detecting outliers by simply using the shortest path distance as an input feature.",
"However, we do not do this so that we can give SOD an advantage, while seeing how other features in the data can be used to detect anomalies."
],
[
"Experiments on Intel Sensor Data",
"We describe the filters and confirm detected outliers with Scorpion [31], which also uses the sensor data for evaluation."
],
[
"Sensor filters",
"In this dataset, the temperature is correlated with voltage and humidity.",
"We use 2 filters below.",
"The first filter marks 5.2% of total records as outliers ($p = 0.052$ ) while the second filter marks 11.4% ($p = 0.114$ ).",
"Among these marked records, 44% are captured by both filters.",
"Note that sensor's age refers to the days or weeks since these sensors were deployed.",
"log(humidity) = $w_{1}$ $\\times $ log(temperature) + $\\vec{w_a} \\cdot \\vec{a_d}$ + $\\beta _1$ , where $\\vec{a_d}$ is sensor's age measured in days.",
"log(temperature) = $w_{2}$ $\\times $ log(voltage) + $\\vec{w_a^{\\prime }} \\cdot \\vec{a_w}$ + $\\beta _1^{\\prime }$ , where $\\vec{a_w}$ is sensor's age measured in weeks."
],
[
"Method for Comparison",
"Because the dataset does not contain ground truth, we validate our detected Intel sensor outliers with findings of Wu and Madden in the Scorpion system [31] where they, using domain knowledge, manually identify one type of outliers.",
"The problematic sensors claimed in Scorpion are temperature readings $\\in $ (90C, 122C) generated from sensor 15 and sensor 18 and they account for 5.6% of records in the whole dataset.",
"Approximately 11% of records are flagged by our system Doc, including manually identified outliers in the Scorpion paper.",
"While those manual annotations provide some ground truth (i.e.",
"have high precision), they may not have flagged all outliers (i.e.",
"recall is unknown).",
"We also compare with linear regression model with ordinary least squares estimatation (OLS).",
"We apply the Cook's distance ($\\mathcal {D}$ ) to estimate the influence, or the combination of leverage and residual values, of each record.",
"Points with large Cook's distance are considered to have further examination.",
"We flag outliers as points with $\\mathcal {D} > 4/n$ where n is the number of observations [7].",
"The result shows that 4.13% of records are flagged as outliers by OLS.",
"We do not choose other outlier detection methods listed in sec:baselines because none of them provides a threshold in outlier score for users to flag outliers."
],
[
"Evaluation Metric",
"We validate flagged outliers by machine learning tasks.",
"In cases where ground truth is missing, it is customary to divide data into training/testing sets.",
"We run our outlier detector on the training set and use the flagged training records to modify the training data (i.e., remove or downweight records suspected of being outliers).",
"Then we build various machine learning models on the modified training data.",
"The goal is to compare these accuracy of the models on a common testing set.",
"The main intuition is that uncaught outliers will degrade the training of the models and thus hurt testing accuracy; better outlier detection algorithms are therefore more likely to result in good training datasets that yield models to perform better on testing data.",
"To follow this intuition, we use 5-fold cross validation and design prediction tasks with linear and non-linear regression models.",
"The evaluation metrics are mean absolute error (MAE) and mean relative error (MRE).",
"Since there are also anomalous records in the testing data, we also use the median absolute error (MedAE) and median relative error (MedRE).",
"In tab:predictiontasks, we employ linear regression (LR), support vector regression with quadratic error function (SVR), and decision tree regression (DTR) and we put all attributes as features for the prediction task1 – predicting temperature where temperature is the variable involved in 2 outlier detecting filters.",
"We train the models on four different training sets – all training set, all training set minus Scorpion outliers, all training set minus Doc outliers, and all weighted training set.",
"Note that we use the scikit-learn implementation for the model LR, SVR and DTR."
],
[
"Results",
"Results are presented in tab:predictiontasks.",
"In Task1 with models LR and SVR, removing our detected outliers from training set or down-weighting those outliers results in lower error.",
"We also conduct the Paired Student's t-test and Wilcoxon signed rank test to show that it is statistically significant that the MAE of our modified training set (i.e., training set minus our detected outliers) is lower than the MAE of Scorpion's modified training set.",
"Note that with the DTR model, using all the training data gets the lower mean absolute and relative error.",
"However, because the testing data does contain outliers, the mean can be skewed by them.",
"The median errors (MedAE, MedRE) are more robust measures of performance and show that taking out outliers in training set (or downweighting them) leads to more accurate prediction.",
"Table: Performance with/without Outlier"
],
[
"Case Study",
"57.2% of flagged outliers are associated with anomalous temperature reading in Week 3.",
"Doc observes a general sensor's malfunction pattern as it is unlikely to be real temperature in the lab – fifty five out of total fifty eight sensors exhibit that the temperature reading is increasing until it reaches around 122C and it keeps generating 122C or above in Week 3 (as shown in fig:temp).",
"However, on Week 4, almost all sensors generate temperature $\\in $ [122.15, 175.68) – hence that is the norm in the data for that week.",
"This is a very common pattern in the data that 92% of records produced on week 4 generates temperature $\\in $ [122.15, 175.68).",
"Hence they are classified as normal by Doc.",
"Note that Scorpion refers to sensors generating high temperature as problematic sensors.",
"In figure REF , we see that there is a decreasing trend in voltage for this batch of sensors and this helps to justify the fact that records with voltage $\\ge $ 2.8 in Week 4 are identified as outliers by Doc.",
"Figure: The average temperature readings sequence from 2/28/2004 to 4/5/2004Figure: The average voltage readings sequence from 2/28/2004 to 4/5/2004First we describe five filters we used to detect outliers.",
"We validate the results with human-annotated trips and compare with a method called Smarter Outlier Detection (SOD) [33], which was specifically designed for this dataset.",
"In Section REF we also compare against the state of the art outlier detection algorithms."
],
[
"Taxi filters",
"We used the following filtering models (where $w_i$ is the coefficient and $\\beta _i$ is offset).",
"Trip time = $w_{1}$$\\times $ (dropoff time-pickup time)+$\\beta _1$ Fare = $w_2$$\\times $ (total amount-tips-tax-toll-surcharge)+$\\beta _2$ log(Trip time) = $w_3$$\\times $ log(Fare)+$\\beta _3$ log(Trip distance) = $w_4$$\\times $ log(L2)+$\\beta _4$ log(Trip time) = $w_5$ $\\times $ log(L2) + $\\vec{w_t} \\cdot \\vec{ts} + \\beta _5$ , where $\\vec{ts}$ is the vector of 24-dimension temporal features described below.",
"Filter REF and REF encode what should be functional dependencies.",
"However, they may differ due to software bugs, data entry errors, or device miscalibration.",
"For example, trip time might be recorded by the taxi meter while pickup and drop-off times might be recorded by a gps unit with a separate clock.",
"In Filters REF , REF , REF , trip time/distance/fare/L2-displacement are all positively correlated and we expect their variance to grow proportionally with the length of a trip.",
"For this reason, we use logs (so that multiplicative error becomes additive error).",
"Also, trip time may depend on the time of day (e.g., rush hour), so we include those components in Filter REF .",
"We partition time of day into 2-hour time slots and separate out weekends from weekdays (this giving $12 \\times 2=24$ temporal features).",
"We note that filters REF and REF did not have overlap in the records they flagged as outliers, thus showing that correlated sensor readings can fail in different ways."
],
[
"Taxi Outlier Detection Method Comparison",
"In total, Doc flagged 7% of records as outliers (the dataset is known to be noisy).",
"The code for SOD was not available, so we reproduced it with a different software package.http://project-osrm.org We discarded trips whose end points are not on roads.",
"The dataset can be categorized into four disjoint sets: $M\\widetilde{S}$ - records flagged as outliers by Doc but not SOD, $\\widetilde{M}S$ - records flagged as outliers by SOD but not Doc, $MS$ - records flagged as outliers by both methods, and $\\widetilde{M}\\widetilde{S}$ - records not flagged by any method."
],
[
"Evaluation Metric",
"We designed a human labeling system for experienced taxi riders to determine outlier trips and to provide reasons to support their judgements.",
"We provided the labelers with the taxi fare rate information from the NYC Taxi & Limousine Commission.",
"Each trip is labeled by three people and we take the majority votes as the ground truth.",
"To provide a quantitative evaluation, each time the labeling webpage randomly selects 10 trips from each of the 4 sets for a person to label."
],
[
"Results",
"In all, 6517 trips were labeled and the results are shown in tab:4sets.",
"In set $M\\widetilde{S}$ , 92% of trips were labeled by humans as outliers and these are consistent with our approach.",
"In set $\\widetilde{M}S$ , humans only labeled 16% of the records as outliers.",
"Thus, when Doc and SOD disagreed, humans tended to agree with the classification provided by $\\texttt {Doc} $ .",
"For $MS$ and $\\widetilde{M}\\widetilde{S}$ , both our method and SOD get the accuracy of 98% and 94% respectively.",
"Table: 4 Sets of Labeled TripsWe use the following evaluation criteria for overall outlier detection performance: detection rate ($DR = \\frac{TP}{TP+FN}$ , i.e., fraction of outliers that are successfully detected as outliers), false positive rate ($FPR = \\frac{FP}{FP+TN}$ , i.e., fraction of normal records that are predicted to be outlier), precision ($Precision = \\frac{TP}{TP+FP}$ , i.e., fraction of detected outlier that are real outlier), true negative rate ($TNR = \\frac{TN}{TN+FP}$ , i.e., fraction of non-outlier that are detected as non-outlier).",
"Note that we use the labeled sampled trips to estimate the ground truth statistics for entire dataset.",
"The estimation approach is as follow: suppose, in set $i$ , the total number of trips is $u$ ; the number of sampled trips labeled is $v$ ; out of labeled trips $v$ the outliers account for $\\phi $ %.",
"Hence $u \\times \\phi $ is the estimated outliers for set $i$ .",
"The evaluation on both the labeled trips (denoted as on labeled) and estimated results for all trips (denoted as on all) are presented in tab:detectionres.",
"From the results it is clear that Doc achieves much better detection rates for slightly larger false positive rates.",
"Table: Outlier Detection Performance"
],
[
"Outlier Detection Methods Comparison",
"We evaluate Doc against traditional outlier detection and contextual outlier detection approaches described in sec:baselines.",
"We also adopt the following statistical-based method as baseline.",
"statistical-based method.",
"Since we observe some detour trips in the taxi data.",
"We fit the ratio of travel distance and L2 distance between end points into Gaussian distribution.",
"The outlier score of point $x$ is $1-p(x)$ where $p(x)$ is the gaussian density function.",
"Evaluation metrics.",
"We randomly sampled 22,463 trips as input data to these outlier detectors which give the outlier rank to every trip.",
"The labeling process uses the same type of methodology mentioned in sec:taxioutlier.",
"Given the outlier rank of the trips, we first select every 5 trips to be labeled (i.e., 5th, 10th $\\dots $ ).",
"Thus we get a rough idea of the approximate number of outliers in this sampled dataset.",
"We label the top 1400 trips for each method and use the following metrics for evaluation.",
"$\\text{Precision $@ \\kappa $} = \\frac{ \\text{\\# trips whose rank $\\le \\kappa $ and label = Outlier} }{\\kappa }$ fig:precisionatn shows that our method outperforms others.",
"The top outlier trips detected by density-based and distance-based methods are long trips such as trips from upper to lower manhattan.",
"Even though they have less neighbors than shorter trips, their trip information is considered as reasonable by the labeled outcome.",
"In contrast, our top outlier trips are mainly from device error and thus it could be obviously identified by people.",
"For the linear and non-linear regression model as well as the existing contextual outlier detection (CAD & ROCOD), they can identify extreme outliers in their top 100 outliers.",
"But their precision drops with more false positives which is due to the biased prediction.",
"We find that trips with rank greater than 1200 are mostly labeled as non-outliers.",
"Hence the precision $@ \\kappa $ drops around $\\kappa = 1000$ .",
"Figure: Precision at κ\\kappa"
],
[
"Case Study",
"We describe the anomalous trips and interesting findings.",
"First, we point out the payment systems and trip time tracking systems provided by Creative Mobile Technologies (CMT) are programmed differently from those provided by Verifone (VTS).",
"We expect that the travel time should be consistent with the duration of dropoff time subtracted by pickup time.",
"Our identified outliers show that this is not always the case and the discrepancies are almost always associated with vendor CMT.",
"Similarly, the sum of cost fields (i.e.,Tip, Tax, Surcharge, Toll, Fare) should equal to total amount (Total).",
"Those inconsistent cost related fields are all produced by the VTS payment system.",
"Second, we identify a type of outliers ranked lower as compared to those extreme corrupted records.",
"These records contain trip fare $<$ $3.",
"The metered fare regulated by Taxi & Limousine Commission (TLC) initially charges $2.5 once a passenger gets in taxis and plus $0.5 per 0.2 mile or $0.5 per minute in slow traffic.",
"We believe these records are outliers as they are even less than minimum taxi fare.",
"Last, 1% of detected anomalous records are trips with almost the same GPS coordinates from pickup to dropoff location.",
"To investigate this further, we found the park-cemetery manhattan neighborhood and bridges linking manhattan to nearby boroughs (where gps signal might be weak) are highly correlated with these trips.",
"For the Elnino and Houses datasets, we inject synthetic outliers into the original clean data.",
"One perturbation scheme used in [19], [24] is that they first randomly select a sample $\\vec{z_i} = (\\vec{x_i}, y_i)$ then, from $k$ data points of the entire dataset, select another sample $\\vec{z_j} = (\\vec{x_j}, y_j)$ where the difference between $y_i$ and $y_j$ is maximized.",
"This new data point $(\\vec{x_i}, y_j)$ is added as an outlier.",
"We do not follow this scheme for several reasons.",
"First, swapping the attribute values may not always obtain desired outliers.",
"It is likely that most of the swaps could result in normal data.",
"Second, as we observe many extreme outliers in the real-world datasets, swapping values between samples in a clean data is less likely to produce this extreme difference between $y_i$ and $y_j$ .",
"Here we present another way to generate outliers and we explore different types of outliers where we give controls to where and how many outliers are injected or its degree of outlierness."
],
[
"Perturbation Scheme",
"To inject $q \\times N$ outliers into a dataset with $N$ data samples, we randomly select $q \\times N$ records $\\vec{z_i} = (\\vec{x_i}, y_i)$ to be perturbed.",
"Let $y_i$ be the target attribute for perturbation.",
"Let $\\vec{x_i}$ be the rest of attributes.",
"For all selected records, a random number from (0, $\\alpha $ ) is added up to $y_i$ as $y_i^{~\\prime }$ .",
"Then we add new sample $\\vec{z}^{~\\prime } = (\\vec{x_i}, y_i^{~\\prime })$ into the original dataset and flag it as outlier.",
"Note that original $N$ data samples are flagged as non-outlier.",
"In the experiments, we standardized the target attribute to range (18, 30) which are the min and max value of the behavioral attribute in Elnino dataset.",
"Set $\\alpha $ as 50 by default."
],
[
"Evaluation Metric",
"Since all these outlier detection approaches considered in sec:baselines give rank to each record according to outlier score, the Precision-Recall curve (PRC) is obtained by Precision $@ \\kappa $ and Recall $@ \\kappa $ for all possible $\\kappa $ where the first $\\kappa $ ranked records are determined to be outlier.",
"The evaluation metric we use here is the Area Under the Curve (AUC) of the Precision-Recall curve instead of the Receiver Operating Characteristic (ROC) as it is less informative in imbalanced class problem [11]."
],
[
"Results",
"As up to 6% of records in Sensor dataset are flagged as outliers due to sensor malfunction, we vary the perturbation ratio $q$ from 0.01 to 0.15 to see if our model is robust in the presence of a large fractions of anomalies.",
"The performance in terms of AUC is shown in the following tables.",
"tab:auc presents the results when we perturb behavioral attributes to generate outliers.",
"$\\texttt {Doc} $ consistently perform the best and its difference compared to other methods becomes significant when more outliers are involved ($q > 0.05$ ).",
"Another type of synthetic outliers is produced by adding noise to contextual attributes.",
"To see how it affects the performance, we select features with highest Pearson correlation to behavioral attribute for perturbation.",
"In tab:auccontextual, we observe that a small fraction of outliers in contextual attribute could hurt the performance considerably for the other methods, especially for the tree-based approaches such as ROCOD and GBT on these two datasets.",
"However, our method is robust and resistant to the fraction of outliers.",
"We next investigate degree of outlierness of the injected anomalies.",
"As $\\alpha $ increases, larger magnitude of noise will have more chance to be added to the original value.",
"Consequently, there are more extreme outliers and our performance is increased as expected in tab:aucsize.",
"Table: AUC of Precision Recall Curve w.r.t different fractions of synthetic outliers in behavioral attributeTable: AUC of Precision Recall Curve w.r.t different fractions of synthetic outliers in contextual attribute Table: AUC of Precision Recall Curve w.r.t degree of outlierness α\\alpha in contextual attribute"
],
[
"Conclusions",
"Motivated by a real-world problem, we develop a system Doc which aims to detect outliers and explicitly considers outliers effect in modeling.",
"It is a robust outlier detector as compared to the existing algorithms built on all the data records where their model parameters are skewed by outliers.",
"Our method could potentially facilitate the public or research use of large-scale data collected from a network of sensors."
],
[
"Acknowledgments",
"The work was partially supported by NSF grants 1054389, 1544455, 1702760, 1652525.",
"The views in this paper are those of the authors, and do not necessarily represent the funding institutes."
]
] | 1808.08640 |
[
[
"The Maslov Index and the Spectral Flow - revisited"
],
[
"Abstract We give an elementary proof of a celebrated theorem of Cappell, Lee and Miller which relates the Maslov index of a pair of paths of Lagrangian subspaces to the spectral flow of an associated path of selfadjoint first-order operators.",
"We particularly pay attention to the continuity of the latter path of operators, where we consider the gap-metric on the set of all closed operators on a Hilbert space.",
"Finally, we obtain from Cappell, Lee and Miller's theorem a spectral flow formula for linear Hamiltonian systems which generalises a recent result of Hu and Portaluri."
],
[
"Introduction",
"Let $\\langle \\cdot ,\\cdot \\rangle $ be the Euclidean scalar product on $\\mathbb {R}^{2n}$ and $\\omega _0(\\cdot ,\\cdot )=\\langle J\\cdot ,\\cdot \\rangle $ the standard symplectic form, where $J=\\begin{pmatrix}0&-I_n\\\\I_n&0\\end{pmatrix}$ and $I_n$ denotes the identity matrix.",
"Let us recall that an $n$ -dimensional subspace $L\\subset \\mathbb {R}^{2n}$ is called Lagrangian if the restriction of $\\omega _0$ to $L\\times L$ vanishes.",
"The set $\\Lambda (n)$ of all Lagrangian subspaces in $\\mathbb {R}^{2n}$ is called the Lagrangian Grassmannian.",
"It can be regarded as a submanifold of the Grassmannian $G_n(\\mathbb {R}^{2n})$ and so it has a canonical topology.",
"In what follows, we denote by $I$ the unit interval $[0,1]$ .",
"The Maslov index $\\mu _{Mas}(\\gamma _1,\\gamma _2)$ assigns to any pair of paths $\\gamma _1, \\gamma _2:I\\rightarrow \\Lambda (n)$ an integer which, roughly speaking, is the total number of non-trivial intersections of the Lagrangian spaces $\\gamma _1(\\lambda )$ and $\\gamma _2(\\lambda )$ whilst the parameter $\\lambda $ travels along the interval $I$ .",
"There are several different approaches to the Maslov index and here we just want to mention [1], [4], [6], [8], [17] and [19], which is far from being exhaustive.",
"Cappell, Lee and Miller introduced in [5] four different ways to define the Maslov index and showed that they are all equivalent.",
"They first construct the Maslov index geometrically by using a stratification of $\\Lambda (n)$ and intersection theory from differential topology following [8].",
"Their approach also yields a uniqueness theorem for the Maslov index characterising this invariant uniquely by six axioms.",
"The uniqueness theorem is then used to show that the Maslov index can alternatively be defined by determinant line bundles, $\\eta $ -invariants and the spectral flow, respectively.",
"In this paper we focus on the latter invariant and aim to give a more elementary proof of the equality of the Maslov index and the spectral flow of a path of operators as introduced by Cappell, Lee and Miller in [5].",
"Let us first recall that the spectral flow is a homotopy invariant for paths of selfadjoint Fredholm operators that was invented by Atiyah, Patodi and Singer in [2], and since then has been used in various different settings (see e.g.",
"[21]).",
"The spectrum of a selfadjoint Fredholm operator consists only of eigenvalues of finite multiplicity in a neighbourhood of $0\\in \\mathbb {R}$ and, roughly speaking, the spectral flow of a path of such operators is the net number of eigenvalues crossing 0 whilst the parameter of the path travels along the interval.",
"Let us now consider for a pair of paths $(\\gamma _1,\\gamma _2)$ in $\\Lambda (n)$ the differential operators $\\mathcal {A}_\\lambda :\\mathcal {D}(\\mathcal {A}_\\lambda )\\subset L^2(I,\\mathbb {R}^{2n})\\rightarrow L^2(I,\\mathbb {R}^{2n}),\\quad (\\mathcal {A}_\\lambda u)(t)=Ju^{\\prime }(t),$ where $\\mathcal {D}(\\mathcal {A}_\\lambda )=\\lbrace u\\in H^1(I,\\mathbb {R}^{2n}):\\, u(0)\\in \\gamma _1(\\lambda ), u(1)\\in \\gamma _2(\\lambda )\\rbrace .$ By an elementary computation, $\\mathcal {A}_\\lambda $ is symmetric, and it is also not difficult to see that it actually is a selfadjoint Fredholm operator.",
"Note that the kernel of $\\mathcal {A}_\\lambda $ is isomorphic to $\\gamma _1(\\lambda )\\cap \\gamma _2(\\lambda )$ , which suggests that the spectral flow of the path $\\mathcal {A}=\\lbrace \\mathcal {A}_\\lambda \\rbrace _{\\lambda \\in I}$ is related to the Maslov index of the pair $(\\gamma _1,\\gamma _2)$ .",
"As we already mentioned above, their equality is one of the main achievements of [5].",
"However, before we formulate this as a theorem, we want to highlight a further issue related to this problem.",
"Above, we have spoken about paths of differential operators and so tacitly assumed continuity.",
"Note that the family (REF ) has the non-constant domains (REF ) and so continuity is a non-trivial problem.",
"There are different metrics on spaces of unbounded selfadjoint Fredholm operators on a Hilbert space $H$ and we recommend [11] for an exhaustive discussion (see also [20]).",
"A classical approach is to transform unbounded selfadjoint operators $T$ by functional calculus to the bounded selfadjoint operators $(I_H+T^2)^{-\\frac{1}{2}}\\in \\mathcal {L}(H),$ and to use the operator norm on $\\mathcal {L}(H)$ for introducing a distance between unbounded operators.",
"Actually, Atiyah, Patodi and Singer defined the spectral flow in [2] for bounded selfadjoint Fredholm operators and applied it to paths of differential operators by using (REF ).",
"However, checking continuity along these lines is tedious, if possible at all (see e.g.",
"[13]), and it seems that the continuity of families of unbounded operators has sometimes been ignored in the literature.",
"Every (generally unbounded) selfadjoint operator on a Hilbert space is closed, and there is a canonical metric on the set of all closed operators which is called the gap-metric (see §IV.2 in Kato's monograph [10]).",
"It was shown in [13] (see also [11]) that every path of selfadjoint Fredholm operators that is mapped to a continuous path of bounded operators under (REF ) is also continuous with respect to the gap-metric.",
"Finally, Booss-Bavnbek, Lesch and Phillips constructed in [3] the spectral flow for paths of selfadjoint Fredholm operators in this more general setting.",
"The main result of this paper now reads as follows (see [5]).",
"Theorem 1.1 If $(\\gamma _1,\\gamma _2)$ is a pair of paths in $\\Lambda (n)$ , then the family of differential operators (REF ) is continuous with respect to the gap-metric and $\\operatorname{sf}(\\mathcal {A})=\\mu _{Mas}(\\gamma _1,\\gamma _2).$ Let us make a few comments on our proof.",
"Firstly, we want to emphasise that we prove the gap-continuity of the family (REF ) from first principles just by elementary estimates and standard facts about orthogonal projections that can all be found in the monograph [10].",
"Secondly, our proof of the spectral flow formula in Theorem REF is surprisingly simple.",
"We assume at first that $\\gamma _1(0)\\cap \\gamma _2(0)=\\gamma _1(1)\\cap \\gamma _2(1)=\\lbrace 0\\rbrace $ and show that the Maslov index can be characterised in this case by three axioms.",
"This uniqueness theorem needs nothing else than the elementary properties of the Maslov index and the fact that the fundamental group of $\\Lambda (n)$ is infinitely cyclic, which was known already from Arnold's classical paper [1].",
"Two of our axioms are trivially satisfied for the spectral flow of (REF ), and the remaining one only requires the computation of the spectra of two simple examples of differential operators as in (REF ).",
"The general case when (REF ) is not assumed, can easily be obtained from the previous case by a simple conjugation by a path of invertible operators.",
"After a brief recapitulation of the Maslov index in Section REF , and the gap-metric and spectral flow in Section REF , we explain all this in detail in Section where we prove Theorem REF .",
"Throughout the paper, we aim our presentation to be rather self-contained, and we will just use some well-known facts from Kato [10].",
"Finally, we review a recent spectral flow formula for linear Hamiltonian systems by Hu and Portaluri from [9], which they call a new index theory on bounded domains.",
"Firstly, we note that the considered families of Hamiltonian systems are continuous with respect to the gap-metric, which follows easily from our approach to Cappell, Lee and Miller's Theorem.",
"Secondly, we obtain a spectral flow formula in this setting by a conjugation from Cappell, Lee and Miller, and we explain that our result actually is a generalisation of Hu and Portaluri's Theorem."
],
[
"The Maslov Index",
"The aim of this section is to briefly recall the definition of the Maslov index, where we follow [16].",
"Let $\\operatorname{Sp}(2n,\\mathbb {R})$ denote the group of symplectic matrices on $\\mathbb {R}^{2n}$ , i.e., those $A\\in M(2n,\\mathbb {R})$ satisfying $A^TJA=J$ or, alternatively, which preserve $\\omega _0$ .",
"If we identify $\\mathbb {R}^{2n}$ with $\\mathbb {C}^n$ by $(x_1,\\ldots ,x_{2n})\\mapsto (x_1,\\ldots ,x_n)+i(x_{n+1},\\ldots , x_{2n})$ then the standard hermitian scalar product on $\\mathbb {C}^n$ is $\\langle x,y\\rangle _{\\mathbb {C}}=\\langle x,y\\rangle -i\\omega _0(x,y).$ Hence each unitary matrix $U\\in U(n)$ preserves $\\omega _0$ and so we can regard $U(n)$ as a subset of $\\operatorname{Sp}(2n,\\mathbb {R})$ .",
"Also, the orthogonal matrices $O(n)$ can be seen as a subgroup of $U(n)$ by complexification.",
"Then $O(n)$ consists exactly of those $A\\in U(n)$ which leave $\\mathbb {R}^n\\times \\lbrace 0\\rbrace $ invariant.",
"Obviously, $AL\\in \\Lambda (n)$ if $L\\in \\Lambda (n)$ and $A\\in \\operatorname{Sp}(2n,\\mathbb {R})$ , and it can be shown that the restriction of this action to $U(n)\\times \\Lambda (n)\\rightarrow \\Lambda (n)$ is transitive.",
"As the stabiliser subgroup of $\\mathbb {R}^n\\times \\lbrace 0\\rbrace \\in \\Lambda (n)$ is $O(n)$ , we see that there is a diffeomorphism $U(n)/O(n)\\simeq \\Lambda (n),\\quad A\\mapsto A(\\mathbb {R}^n\\times \\lbrace 0\\rbrace ).$ Let us now consider the map $d:U(n)\\rightarrow S^1$ , $d(A)=\\det ^2(A)$ , which descends to the quotient by $\\overline{d}:U(n)/O(n)\\rightarrow S^1,\\quad A\\cdot O(n)\\mapsto \\det {^2}(A).$ Note that $\\ker (d)/O(n)\\hookrightarrow U(n)/O(n)\\xrightarrow{} S^1$ is a fibre bundle, and it is not difficult to see that $\\ker (d)/O(n)\\simeq SU(n)/SO(n)$ , where the latter space is simply connected.",
"It follows from the long exact sequence of a fibre bundle that the induced map $\\overline{d}_\\ast :\\pi _1(U(n)/O(n))\\rightarrow \\pi _1(S^1)\\cong \\mathbb {Z}$ is an isomorphism.",
"Consequently, we obtain from (REF ) an isomorphism $\\mu _{Mas}:\\pi _1(\\Lambda (n))\\rightarrow \\mathbb {Z},$ which is the Maslov index for closed paths in $\\Lambda (n)$ .",
"Roughly speaking, given an arbitrary $L_0\\in \\Lambda (n)$ , the Maslov index counts the total number of intersections of a loop in $\\Lambda (n)$ with $L_0$ .",
"This is independent of the particular choice of $L_0$ , which however is no longer the case if we extend the definition to non closed paths in $\\Lambda (n)$ as follows.",
"We fix $L_0\\in \\Lambda (n)$ and note at first that $L_0$ yields a stratification $\\Lambda (n)=\\bigcup ^n_{k=0}\\Lambda _k(L_0),$ where $\\Lambda _k(L_0)=\\lbrace L\\in \\Lambda (n):\\, \\dim (L\\cap L_0)=k\\rbrace .$ From the fact that $\\Lambda _0(L_0)$ is contractible (see e.g.",
"[16]) and the long exact sequence of homology, we see that the inclusion induces an isomorphism $H_1(\\Lambda (n))\\rightarrow H_1(\\Lambda (n),\\Lambda _0(L_0)).$ Also, as $\\pi _1(\\Lambda (n))$ is abelian, $H_1(\\Lambda (n))$ is isomorphic to $\\pi _1(\\Lambda (n))$ and so we obtain a sequence of isomorphisms $H_1(\\Lambda (n),\\Lambda _0(L_0))\\rightarrow H_1(\\Lambda (n))\\rightarrow \\pi _1(\\Lambda (n))\\rightarrow \\pi _1(U(n)/O(n))\\rightarrow \\mathbb {Z}.$ Finally, every path in $\\Lambda (n)$ having endpoints in $\\Lambda _0(L_0)$ canonically yields an element in$H_1(\\Lambda (n),\\Lambda _0(L_0))$ .",
"The Maslov index of the path is the integer obtained from the sequence of isomorphisms (REF ).",
"Let us note from the very definition the following three properties of the Maslov index: (i) If $\\gamma _1,\\gamma _2$ are homotopic by a homotopy having endpoints in $\\Lambda _0(L_0)$ , then $\\mu _{Mas}(\\gamma _1,L_0)=\\mu _{Mas}(\\gamma _2,L_0).$ (ii) If $\\gamma _1, \\gamma _2$ are such that $\\gamma _1(1)=\\gamma _2(0)$ , then $\\mu _{Mas}(\\gamma _1\\ast \\gamma _2,L_0)=\\mu _{Mas}(\\gamma _1,L_0)+\\mu _{Mas}(\\gamma _2,L_0).$ (iii) If $\\gamma (\\lambda )\\in \\Lambda _0(L_0)$ for all $\\lambda \\in I$ , then $\\mu _{Mas}(\\gamma ,L_0)=0$ .",
"Let us point out that (iii) also follows from (i) and (ii) independently of the construction.",
"The Maslov index can easily be generalised to a pair of paths in $\\Lambda (n)$ .",
"To this aim let us call a pair of paths $(\\gamma _1,\\gamma _2)$ admissible if $\\gamma _1(0)\\cap \\gamma _2(0)=\\gamma _1(1)\\cap \\gamma _2(1)=\\lbrace 0\\rbrace .$ In what follows we consider $\\mathbb {R}^{2n}\\times \\mathbb {R}^{2n}$ as a symplectic space with respect to the symplectic form $(-\\omega _0)\\times \\omega _0$ .",
"Note that the diagonal $\\Delta $ is in $\\Lambda (2n)$ , as well as $L_1\\times L_2$ for any $L_1,L_2\\in \\Lambda (n)$ .",
"Moreover, $L_1\\cap L_2\\ne \\lbrace 0\\rbrace $ if and only if $(L_1\\times L_2)\\cap \\Delta \\ne \\lbrace 0\\rbrace $ .",
"Hence it is natural to define the Maslov index for a pair $(\\gamma _1,\\gamma _2)$ of admissible paths in $\\Lambda (n)$ as $\\mu _{Mas}(\\gamma _1,\\gamma _2)=\\mu _{Mas}(\\gamma _1\\times \\gamma _2,\\Delta ).$ Note that the basic properties which we previously mentioned carry over immediately, i.e., (i') $\\mu _{Mas}(\\gamma _1,\\gamma _2)=0$ if $\\gamma _1(\\lambda )\\cap \\gamma _2(\\lambda )=\\lbrace 0\\rbrace $ for all $\\lambda \\in I$ .",
"(ii') $\\mu _{Mas}(\\gamma _1\\ast \\gamma _3,\\gamma _2\\ast \\gamma _4)=\\mu _{Mas}(\\gamma _1,\\gamma _2)+\\mu _{Mas}(\\gamma _3,\\gamma _4)$ if $\\gamma _{1}(1)=\\gamma _{3}(0)$ and $\\gamma _{2}(1)=\\gamma _{4}(0)$ .",
"(iii') $\\mu _{Mas}(\\gamma _1,\\gamma _2)=\\mu _{Mas}(\\gamma _3,\\gamma _4)$ if $\\gamma _1\\simeq \\gamma _3$ and $\\gamma _2\\simeq \\gamma _4$ are homotopic by a homotopy through admissible pairs.",
"Also, it is not difficult to see from the construction of the Maslov index that (iv') $\\mu _{Mas}(\\gamma _1,\\gamma _2)=\\mu _{Mas}(\\gamma _1,L_0)$ in case that $\\gamma _2(\\lambda )=L_0$ for some $L_0\\in \\Lambda (n)$ and all $\\lambda \\in I$ , (v') $\\mu _{Mas}(\\gamma _1,\\gamma _2)=-\\mu _{Mas}(\\gamma _2,\\gamma _1)$ for any admissible pair $(\\gamma _1,\\gamma _2)$ .",
"Finally, let us define the Maslov index for a non-admissible pair of paths.",
"It is important to note that in this case there are different definitions in the literature.",
"Here we follow [5], and note that given $L_1, L_2\\in \\Lambda (n)$ such that $L_1\\cap L_2\\ne \\lbrace 0\\rbrace $ , there is $\\varepsilon >0$ such that $e^{\\Theta J}L_2\\in \\Lambda (n)$ and $L_1\\cap e^{\\Theta J}L_2=\\lbrace 0\\rbrace $ for all $0<|\\Theta |\\le \\varepsilon $ .",
"We define the Maslov index as $\\mu _{Mas}(\\gamma _1,\\gamma _2)=\\mu _{Mas}(\\gamma _1,e^{-\\Theta J}\\gamma _2),$ where $\\Theta $ is such that $\\gamma _1(0)\\cap e^{-\\Theta ^{\\prime } J}\\gamma _2(0)=\\gamma _1(1)\\cap e^{-\\Theta ^{\\prime } J}\\gamma _2(1)=\\lbrace 0\\rbrace $ for all $0<|\\Theta ^{\\prime }|\\le \\Theta $ .",
"By the homotopy invariance, it is clear that this definition does not depend on the choice of $\\Theta $ .",
"Also, it coincides with the previous definition in case that the pair of paths is admissible."
],
[
"The Paths $\\gamma _{nor}$ and {{formula:dc9a6f34-310c-42d1-847a-bfcf0b8a57b8}}",
"The aim of this section is to compute the Maslov index for two elementary paths that will also become important in our proof of Theorem REF below.",
"The examples also show that (REF ) is very convenient to obtain paths in $\\Lambda (n)$ with a given Maslov index.",
"Let us first consider the path $[0,1]\\ni \\lambda \\mapsto A(\\lambda )=\\operatorname{diag}(e^{i\\pi \\lambda },1,\\ldots ,1)\\in U(n)$ and its projection $\\overline{A}(\\lambda ):=A(\\lambda )\\cdot O(n)$ to the quotient $U(n)/O(n)$ .",
"Note that$A(0)\\operatorname{diag}(-1,1,\\ldots ,1)=A(1)$ and so $\\overline{A}$ is a closed curve.",
"Also, as $\\det ^2(A(\\lambda ))=e^{2\\pi i\\lambda }$ , we see that the Maslov index of the corresponding path in $\\Lambda (n)$ is 1.",
"Using the identification $\\mathbb {C}^n\\cong \\mathbb {R}^{2n}$ , it is readily seen that $\\gamma _{nor}(\\lambda ):=A(\\lambda )(\\mathbb {R}^n\\times \\lbrace 0\\rbrace )=\\mathbb {R}(\\cos (\\pi \\lambda )e_1+\\sin (\\pi \\lambda )e_{n+1})+\\sum ^{n}_{j=2}{\\mathbb {R}e_j}\\in \\Lambda (n).$ Hence we have found a path $\\gamma _{nor}$ such that $\\gamma _{nor}(0)=\\gamma _{nor}(1)=\\mathbb {R}^n\\times \\lbrace 0\\rbrace $ and $\\mu _{Mas}(\\gamma _{nor})=1$ .",
"Let us now consider $[0,1]\\ni \\lambda \\mapsto B(\\lambda )=\\operatorname{diag}(-ie^{ i\\pi \\lambda },i,\\ldots ,i)\\in U(n)$ and note that again the projection $\\overline{B}$ to $U(n)/O(n)$ is a closed path and $\\det ^2(B(\\lambda ))=(-1)^{n}e^{2\\pi i\\lambda }$ .",
"Hence $\\gamma ^{\\prime }_{nor}(\\lambda ):=B(\\lambda )(\\mathbb {R}^n\\times \\lbrace 0\\rbrace )=\\mathbb {R}(\\sin (\\pi \\lambda )e_1-\\cos (\\pi \\lambda )e_{n+1})+\\sum ^{2n}_{j=n+2}{\\mathbb {R}e_j}\\in \\Lambda (n)$ is such that $\\gamma ^{\\prime }_{nor}(0)=\\gamma ^{\\prime }_{nor}(1)=\\lbrace 0\\rbrace \\times \\mathbb {R}^n$ and $\\mu _{Mas}(\\gamma ^{\\prime }_{nor})=1$ ."
],
[
"The Gap-Metric and the Spectral Flow",
"Our first aim of this section is to recall the definition of the gap-metric, where we follow Kato's monograph [10].",
"Let $H$ be a real Hilbert space and let $G(H)$ denote the set of all closed subspaces of $H$ .",
"For every $U\\in G(H)$ there is a unique orthogonal projection $P_U$ onto $U$ which is a bounded operator on $H$ .",
"We set $d_G(U,V)=\\Vert P_U-P_V\\Vert ,\\quad U,V\\in G(H),$ and note that this is obviously a metric on $G(H)$ .",
"The distance between two non-trivial subspaces $U,V\\in G(H)$ can also be obtained as follows.",
"Let $S_U$ denote the unit sphere in $U$ and $d(u,V)=\\inf _{v\\in V}{\\Vert u-v\\Vert }$ .",
"Then for $\\delta (U,V)=\\sup _{u\\in S_U} d(u,V)$ , $d_G(U,V)=\\max \\lbrace \\delta (U,V),\\delta (V,U)\\rbrace ,$ which explains why $d_G(U,V)$ is called the gap between $U$ and $V$ .",
"We now consider operators $T:\\mathcal {D}(T)\\subset H\\rightarrow H$ which we assume to be defined on a dense domain $\\mathcal {D}(T)$ .",
"Let us recall that $T$ is called closed if its graph $\\operatorname{graph}(T)$ is closed in $H\\times H$ .",
"If we denote by $\\mathcal {C}(H)$ the set of all closed operators, then the gap-metric on $H\\times H$ induces a metric on $\\mathcal {C}(H)$ by $d_G(T,S)=d_G(\\operatorname{graph}(T),\\operatorname{graph}(S)), \\quad S,T\\in \\mathcal {C}(H).$ As the adjoint of a densely defined operator is closed, every selfadjoint operator on $H$ belongs to the metric space $\\mathcal {C}(H)$ .",
"Moreover, let us recall that a closed operator $T$ is called Fredholm if its kernel and cokernel are of finite dimension.",
"In what follows, we denote the subset of $\\mathcal {C}(H)$ consisting of all $T$ which are selfadjoint and Fredholm by $\\mathcal {CF}^\\textup {sa}(H)$ .",
"It is well known that the spectrum $\\sigma (T)$ of every selfadjoint operator is real.",
"Moreover, if $T\\in \\mathcal {CF}^\\textup {sa}(H)$ then 0 is either in the resolvent set or an isolated eigenvalue of finite multiplicity (see e.g.",
"[21]).",
"It was shown in [3] that for every $T\\in \\mathcal {CF}^\\textup {sa}(H)$ there is $\\varepsilon >0$ and a neighbourhood $\\mathcal {N}_{T,\\varepsilon }\\subset \\mathcal {CF}^\\textup {sa}(H)$ of $T$ such that $\\pm \\varepsilon \\notin \\sigma (S)$ and the spectral projection $\\chi _{[-\\varepsilon ,\\varepsilon ]}(S)$ is of finite rank for all $S\\in \\mathcal {N}_{T,\\varepsilon }$ .",
"Let us now consider a path $\\mathcal {A}=\\lbrace \\mathcal {A}_\\lambda \\rbrace _{\\lambda \\in I}$ in $\\mathcal {CF}^\\textup {sa}(H)$ .",
"There are $0=\\lambda _0<\\lambda _1<\\ldots <\\lambda _N=1$ such that the restriction of the path $\\mathcal {A}$ to $[\\lambda _{i-1},\\lambda _i]$ is entirely contained in a neighbourhood $\\mathcal {N}_{T_i,\\varepsilon _i}$ as above for some $T_i\\in \\mathcal {CF}^\\textup {sa}(H)$ and some $\\varepsilon _i>0$ .",
"The spectral flow of the path $\\mathcal {A}$ is defined as $\\operatorname{sf}(\\mathcal {A})=\\sum ^N_{i=1}{\\left(\\dim (\\operatorname{im}(\\chi _{[0,\\varepsilon _i]}(\\mathcal {A}_{\\lambda _i}))-\\dim (\\operatorname{im}(\\chi _{[0,\\varepsilon _i]}(\\mathcal {A}_{\\lambda _{i-1}}))\\right)}.$ It follows by an argument of Phillips [15] that $\\operatorname{sf}(\\mathcal {A})$ only depends on the path $\\mathcal {A}$ , and that the following fundamental property holds (see also [3]).",
"Let $h:I\\times I\\rightarrow \\mathcal {CF}^\\textup {sa}(H)$ be a homotopy such that the dimensions of the kernels of $h(s,0)$ and $h(s,1)$ are constant for all $s\\in I$ .",
"Then $\\operatorname{sf}(h(0,\\cdot ))=\\operatorname{sf}(h(1,\\cdot )).$ Moreover, it is easily seen from the definition of the spectral flow that if the dimension of the kernel of $\\mathcal {A}_\\lambda $ is constant for all $\\lambda \\in I$ , then $\\operatorname{sf}(\\mathcal {A})=0$ ; if $\\mathcal {A}^1$ and $\\mathcal {A}^2$ are two paths in $\\mathcal {CF}^\\textup {sa}(H)$ such that $\\mathcal {A}^1_1=\\mathcal {A}^2_0$ , then $\\operatorname{sf}(\\mathcal {A}^1\\ast \\mathcal {A}^2)=\\operatorname{sf}(\\mathcal {A}^1)+\\operatorname{sf}(\\mathcal {A}^2).$ Let us finally note two further elementary properties of the spectral flow which play a crucial role in our proof of Theorem REF below.",
"The first of them has been used, e.g., in [14].",
"Lemma 2.1 Let $\\mathcal {A}:I\\rightarrow \\mathcal {CF}^\\textup {sa}(H)$ be gap-continuous and set $\\mathcal {A}^\\delta =\\mathcal {A}+\\delta I_H$ for $\\delta \\in \\mathbb {R}$ .",
"Then, for any sufficiently small $\\delta >0$ , $\\mathcal {A}^\\delta $ is a gap-continuous path in $\\mathcal {CF}^\\textup {sa}(H)$ and $\\operatorname{sf}(\\mathcal {A})=\\operatorname{sf}(\\mathcal {A}^\\delta ).$ We note at first that the operators $\\mathcal {A}^\\delta _\\lambda $ are selfadjoint and Fredholm for $\\delta $ sufficiently small, which follows from standard stability theory (see e.g.",
"[10]).",
"Moreover, the path $\\mathcal {A}^\\delta $ is gap-continuous by [10], and so $\\operatorname{sf}(\\mathcal {A}^\\delta )$ is well defined.",
"To show (REF ), let $0=\\lambda _0<\\ldots <\\lambda _N=1$ be a partition of the unit interval and $\\varepsilon _i>0$ , $i=1,\\ldots ,N$ , for $\\mathcal {A}$ as in (REF ).",
"Let $\\mathcal {N}_{T,\\varepsilon _i}$ be an open neighbourhood of some $T\\in \\mathcal {CF}^{sa}(H)$ as in the construction of the spectral flow such that $\\mathcal {A}_\\lambda \\in \\mathcal {N}_{T,\\varepsilon _i}$ for all $\\lambda \\in [\\lambda _{i-1},\\lambda _i]$ .",
"Now there is $\\delta _i>0$ such that $\\mathcal {A}^{s\\delta _i}_\\lambda \\in \\mathcal {N}_{T,\\varepsilon _i}$ for all $s\\in [0,1]$ and all $\\lambda \\in [\\lambda _{i-1},\\lambda _i]$ , i.e.",
"the spectral projections $\\chi _{[-\\varepsilon _i,\\varepsilon _i]}(\\mathcal {A}^{s\\delta _i}_{\\lambda })$ are of the same finite rank.",
"Moreover, by choosing $\\delta _i>0$ smaller, we can assume that $\\sigma (\\mathcal {A}_{\\lambda _i})\\cap [-\\delta _i,0)=\\sigma (\\mathcal {A}_{\\lambda _{i-1}})\\cap [-\\delta _i,0)=\\lbrace 0\\rbrace .$ Then, as $\\sigma (\\mathcal {A}^{\\delta _i}_\\lambda )=\\sigma (\\mathcal {A}_\\lambda )+\\delta _i$ , we see that $\\dim (\\operatorname{im}(\\chi _{[0,\\varepsilon _i]}(\\mathcal {A}_{\\lambda })))=\\dim (\\operatorname{im}(\\chi _{[0,\\varepsilon _i]}(\\mathcal {A}^{\\delta _i}_{\\lambda }))),\\quad \\lambda =\\lambda _{i-1},\\lambda _i.$ If we now set $\\delta =\\min \\lbrace \\delta _1,\\ldots ,\\delta _N\\rbrace >0$ , then $\\dim (\\operatorname{im}(\\chi _{[0,\\varepsilon _i]}(\\mathcal {A}_{\\lambda })))=\\dim (\\operatorname{im}(\\chi _{[0,\\varepsilon _i]}(\\mathcal {A}^{\\delta }_{\\lambda }))),\\quad \\lambda =\\lambda _{i-1},\\lambda _i$ holds simultaneously for this $\\delta $ and all $i=1,\\ldots ,N$ , and so the assertion follows from the definition (REF ).",
"Finally, let us note the following stability of the spectral flow under conjugation by invertible operators, where we denote by $M^T$ the adjoint of an operator in the real Hilbert space $H$ .",
"Lemma 2.2 Let $\\mathcal {A}:I\\rightarrow \\mathcal {CF}^\\textup {sa}(H)$ be a gap-continuous path and $M:I\\rightarrow GL(H)$ a continuous family of bounded invertible operators.",
"Then $\\lbrace M^T_\\lambda \\mathcal {A}_\\lambda M_\\lambda \\rbrace _{\\lambda \\in I}$ is gap-continuous and $\\operatorname{sf}(M^T\\mathcal {A}M)=\\operatorname{sf}(\\mathcal {A}).$ Note that $\\operatorname{graph}(M^T_\\lambda \\mathcal {A}_\\lambda M_\\lambda )&=\\lbrace (u,M^T_\\lambda \\mathcal {A}_\\lambda M_\\lambda u):\\, u\\in M^{-1}_\\lambda (\\mathcal {D}(\\mathcal {A}_\\lambda ))\\rbrace =\\lbrace (M^{-1}_\\lambda v,M^T_\\lambda \\mathcal {A}_\\lambda v):\\,v\\in \\mathcal {D}(\\mathcal {A}_\\lambda )\\rbrace \\\\&=\\begin{pmatrix}M^{-1}_\\lambda &0\\\\0&M^T_\\lambda \\end{pmatrix}\\,\\operatorname{graph}(\\mathcal {A}_\\lambda )=:N_\\lambda \\operatorname{graph}(\\mathcal {A}_\\lambda )\\subset H\\times H,$ and so $\\lbrace N_\\lambda P_{\\operatorname{graph}(\\mathcal {A}_\\lambda )}N^{-1}_\\lambda \\rbrace _{\\lambda \\in I}$ is a continuous family of oblique projections onto$\\lbrace \\operatorname{graph}(M^T_\\lambda \\mathcal {A}_\\lambda M_\\lambda )\\rbrace _{\\lambda \\in I}$ in $\\mathcal {L}(H\\times H)$ .",
"By [10], we have for the corresponding orthogonal projections $P_{\\operatorname{graph}(M^T_\\lambda \\mathcal {A}_\\lambda M_\\lambda )}$ onto $\\operatorname{graph}(M^T_\\lambda \\mathcal {A}_\\lambda M_\\lambda )$ the inequality $\\Vert P_{\\operatorname{graph}(M^T_\\mu \\mathcal {A}_\\mu M_\\mu )}-P_{\\operatorname{graph}(M^T_\\lambda \\mathcal {A}_\\lambda M_\\lambda )}\\Vert \\le \\Vert N_\\mu P_{\\operatorname{graph}(\\mathcal {A}_\\mu )}N^{-1}_\\mu -N_\\lambda P_{\\operatorname{graph}(\\mathcal {A}_\\lambda )}N^{-1}_\\lambda \\Vert ,\\quad \\mu ,\\lambda \\in I.$ Consequently, $\\lbrace P_{\\operatorname{graph}(M^T_\\lambda \\mathcal {A}_\\lambda M_\\lambda )}\\rbrace _{\\lambda \\in I}$ is continuous, which shows that $M^T\\mathcal {A}M$ is gap-continuous.",
"For the equality of the spectral flows, we just need to note that $M$ is homotopic inside $GL(H)$ to the constant path given by the identity $I_H$ .",
"Let us point out that this does not even require Kuiper's Theorem as we just need to shrink $M$ to a constant path and use that $GL(H)$ is connected.",
"As the conjugation preserves kernel dimensions, we obtain by the homotopy invariance (i) from above $\\operatorname{sf}(M^T\\mathcal {A}M)=\\operatorname{sf}(\\mathcal {A}).$"
],
[
"Proof of Theorem ",
"The proof of Theorem REF falls naturally into two parts.",
"In the first part we deal with the continuity of families of the type (REF ), where we actually consider a slightly more general setting.",
"In the second part we show the spectral flow formula in Theorem REF ."
],
[
"Continuity",
"To simplify notation, we set $E=L^2(I,\\mathbb {R}^{2n})$ and $H=H^1(I,\\mathbb {R}^{2n})$ .",
"The aim of this step is to prove the following proposition, which we will later apply in the cases $X=I$ and $X=I\\times I$ .",
"Proposition 3.1 Let $X$ be a metric space and $\\gamma _1,\\gamma _2:X\\rightarrow \\Lambda (n)$ two families of Lagrangian subspaces in $\\mathbb {R}^{2n}$ .",
"Then $\\mathcal {A}:X\\rightarrow \\mathcal {CF}^\\textup {sa}(E),\\quad (\\mathcal {A}_\\lambda u)(t)=Ju^{\\prime }(t),$ where $\\mathcal {D}(\\mathcal {A}_\\lambda )=\\lbrace u\\in H: u(0)\\in \\gamma _1(\\lambda ), u(1)\\in \\gamma _2(\\lambda )\\rbrace ,$ is continuous with respect to the gap-metric on $\\mathcal {CF}^\\textup {sa}(E)$ .",
"We want to use (REF ) and consider $\\delta (\\operatorname{graph}(\\mathcal {A}_\\lambda ),\\operatorname{graph}(\\mathcal {A}_{\\lambda _0})).$ Note at first that for $u\\in \\mathcal {D}(\\mathcal {A}_\\lambda )$ and $v\\in \\mathcal {D}(\\mathcal {A}_{\\lambda _0})$ $\\begin{split}\\Vert (u,\\mathcal {A}_\\lambda u)-(v,\\mathcal {A}_{\\lambda _0}v)\\Vert _{E\\oplus E}&=\\Vert (u-v,J(u^{\\prime }-v^{\\prime }))\\Vert _{E\\oplus E}\\\\&\\le \\left(\\Vert u-v\\Vert ^2_{E}+\\Vert J\\Vert \\Vert u^{\\prime }-v^{\\prime }\\Vert ^2_{E}\\right)^\\frac{1}{2}\\\\&=\\Vert u-v\\Vert _{H},\\end{split}$ where we have used that $\\Vert J\\Vert =1$ .",
"Let us recall that the topology of $G_n(\\mathbb {R}^{2n})$ is induced by the metric $d(L,M)=\\Vert P_L-P_M\\Vert $ , where $P_L, P_M\\in M(2n,\\mathbb {R})$ are the orthogonal projections onto $L$ and $M$ , respectively.",
"Hence, by the continuity of $\\gamma _1$ and $\\gamma _2$ , there are two families of orthogonal projections $\\hat{P},\\tilde{P}:X\\rightarrow M(2n,\\mathbb {R})$ such that $\\operatorname{im}(\\hat{P}_\\lambda )=\\gamma _1(\\lambda ),\\quad \\operatorname{im}(\\tilde{P}_\\lambda )=\\gamma _2(\\lambda ),\\quad \\lambda \\in X.$ We define for $w\\in H$ $(P_\\lambda w)(t)=w(t)-(1-t)(I_{2n}-\\hat{P}_\\lambda )w(0)-t(I_{2n}-\\tilde{P}_\\lambda )w(1).$ It is easily seen that $P^2_\\lambda w=P_\\lambda w$ , as well as $P_\\lambda w\\in \\mathcal {D}(\\mathcal {A}_\\lambda )$ for all $w\\in H$ and $\\lambda \\in X$ , which shows that $\\inf _{v\\in \\mathcal {D}(\\mathcal {A}_{\\lambda _0})}\\Vert u-v\\Vert _{H}\\le \\Vert u-P_{\\lambda _0}u\\Vert _{H}.$ As $u(t)-(P_{\\lambda _0}u)(t)=(1-t)(I_{2n}-\\hat{P}_{\\lambda _0})u(0)+t(I_{2n}-\\tilde{P}_{\\lambda _0})u(1),$ it follows for $u\\in \\mathcal {D}(\\mathcal {A}_\\lambda )$ that $\\begin{split}\\Vert u-P_{\\lambda _0}u\\Vert _{H}&\\le 2(\\Vert (I_{2n}-\\hat{P}_{\\lambda _0})u(0)\\Vert +\\Vert (I_{2n}-\\tilde{P}_{\\lambda _0})u(1)\\Vert )\\\\&= 2(\\Vert (I_{2n}-\\hat{P}_{\\lambda _0})\\hat{P}_\\lambda u(0)\\Vert +\\Vert (I_{2n}-\\tilde{P}_{\\lambda _0})\\tilde{P}_\\lambda u(1)\\Vert )\\\\&\\le 2(\\Vert (I_{2n}-\\hat{P}_{\\lambda _0})\\hat{P}_\\lambda \\Vert \\Vert u(0)\\Vert +\\Vert (I_{2n}-\\tilde{P}_{\\lambda _0})\\tilde{P}_\\lambda \\Vert \\Vert u(1)\\Vert ),\\end{split}$ where we have used that $u\\in \\mathcal {D}(\\mathcal {A}_\\lambda )$ and so $\\hat{P}_\\lambda u(0)=u(0)$ and $\\tilde{P}_\\lambda u(1)=u(1)$ .",
"Let us note that the factor 2 appears in the previous estimate as we are dealing with the norm on $H$ and so we also need to take into account the derivatives of $u-P_{\\lambda _0}u$ with respect to $t$ .",
"Since the point evaluation is continuous in $H$ , there is a constant $\\alpha >0$ such that for $t=0$ and $t=1$ $\\Vert u(t)\\Vert \\le \\alpha \\Vert u\\Vert _{H}=\\alpha \\left(\\Vert u\\Vert ^2_{E}+\\Vert u^{\\prime }\\Vert ^2_{E}\\right)^\\frac{1}{2}=\\alpha \\left(\\Vert u\\Vert ^2_{E}+\\Vert Ju^{\\prime }\\Vert ^2_{E}\\right)^\\frac{1}{2},$ where we use that $J$ is an isometry on $\\mathbb {R}^{2n}$ .",
"Hence, by (REF )–(REF ), $d((u,\\mathcal {A}_\\lambda u),\\operatorname{graph}(\\mathcal {A}_{\\lambda _0}))&=\\inf _{v\\in \\mathcal {D}(\\mathcal {A}_{\\lambda _0})}\\Vert (u,\\mathcal {A}_\\lambda u)-(v,\\mathcal {A}_{\\lambda _0}v)\\Vert _{E\\oplus E}\\\\&\\le \\inf _{v\\in \\mathcal {D}(\\mathcal {A}_{\\lambda _0})}\\Vert u-v\\Vert _{H}\\le \\Vert u-P_{\\lambda _0}u\\Vert _{H}\\\\&\\le 2(\\Vert (I_{2n}-\\hat{P}_{\\lambda _0})\\hat{P}_\\lambda \\Vert \\Vert u(0)\\Vert +\\Vert (I_{2n}-\\tilde{P}_{\\lambda _0})\\tilde{P}_\\lambda \\Vert \\Vert u(1)\\Vert )\\\\&\\le 2\\alpha (\\Vert (I_{2n}-\\hat{P}_{\\lambda _0})\\hat{P}_\\lambda \\Vert +\\Vert (I_{2n}-\\tilde{P}_{\\lambda _0})\\tilde{P}_\\lambda \\Vert )(\\Vert u\\Vert ^2_{E}+\\Vert Ju^{\\prime }\\Vert ^2_{E})^\\frac{1}{2}.$ As the unit sphere in $\\operatorname{graph}(\\mathcal {A}_\\lambda )$ is given by $\\lbrace (u,\\mathcal {A}_\\lambda u):\\, u\\in \\mathcal {D}(\\mathcal {A}_\\lambda ),\\, \\Vert u\\Vert ^2_{E}+\\Vert Ju^{\\prime }\\Vert ^2_{E}=1\\rbrace ,$ we finally get $\\begin{split}\\delta (\\operatorname{graph}(\\mathcal {A}_\\lambda ),\\operatorname{graph}(\\mathcal {A}_{\\lambda _0}))&=\\sup \\lbrace d((u,\\mathcal {A}_\\lambda u),\\operatorname{graph}(\\mathcal {A}_{\\lambda _0})):\\,u\\in \\mathcal {D}(\\mathcal {A}_\\lambda ),\\,\\Vert u\\Vert ^2+\\Vert Ju^{\\prime }\\Vert ^2=1\\rbrace \\\\&\\le 2\\alpha (\\Vert (I_{2n}-\\hat{P}_{\\lambda _0})\\hat{P}_\\lambda \\Vert +\\Vert (I_{2n}-\\tilde{P}_{\\lambda _0})\\tilde{P}_\\lambda \\Vert ).\\end{split}$ Note that if we swap $\\lambda $ and $\\lambda _0$ and repeat the above argument, we also have $\\delta (\\operatorname{graph}(\\mathcal {A}_{\\lambda _0}),\\operatorname{graph}(\\mathcal {A}_{\\lambda }))\\le 2\\alpha (\\Vert (I_{2n}-\\hat{P}_{\\lambda })\\hat{P}_{\\lambda _0}\\Vert +\\Vert (I_{2n}-\\tilde{P}_{\\lambda })\\tilde{P}_{\\lambda _0}\\Vert ).$ To finish the proof, we need the following well-known theorem that can be found, e.g., in [10].",
"Theorem 3.2 Let $E$ be a Hilbert space and $P,Q$ orthogonal projections in $E$ .",
"If $\\Vert (I_E-P)Q\\Vert <1\\,\\,\\text{and } \\Vert (I_E-Q)P\\Vert <1,$ then $\\Vert (I_E-P)Q\\Vert =\\Vert (I_E-Q)P\\Vert =\\Vert P-Q\\Vert .$ Now, as $(I_{2n}-\\hat{P}_{\\lambda })\\hat{P}_{\\lambda _0}=(I_{2n}-\\hat{P}_{\\lambda _0})\\hat{P}_{\\lambda }=0$ for $\\lambda =\\lambda _0$ , we have for all $\\lambda $ in a neighbourhood of $\\lambda _0$ $\\Vert (I_{2n}-\\hat{P}_{\\lambda })\\hat{P}_{\\lambda _0}\\Vert =\\Vert (I_{2n}-\\hat{P}_{\\lambda _0})\\hat{P}_{\\lambda }\\Vert =\\Vert \\hat{P}_{\\lambda }-\\hat{P}_{\\lambda _0}\\Vert $ and likewise $\\Vert (I_{2n}-\\tilde{P}_{\\lambda })\\tilde{P}_{\\lambda _0}\\Vert =\\Vert (I_{2n}-\\tilde{P}_{\\lambda _0})\\tilde{P}_{\\lambda }\\Vert =\\Vert \\tilde{P}_\\lambda -\\tilde{P}_{\\lambda _0}\\Vert .$ Consequently, we obtain from (REF ), (REF ) and (REF ) for all $\\lambda $ sufficiently close to $\\lambda _0$ $d_G(\\mathcal {A}_\\lambda ,\\mathcal {A}_{\\lambda _0})&=\\max \\lbrace \\delta (\\operatorname{graph}(\\mathcal {A}_\\lambda ),\\operatorname{graph}(\\mathcal {A}_{\\lambda _0})),\\delta (\\operatorname{graph}(\\mathcal {A}_{\\lambda _0}),\\operatorname{graph}(\\mathcal {A}_{\\lambda }))\\rbrace \\\\&\\le 2\\alpha (\\Vert \\hat{P}_\\lambda -\\hat{P}_{\\lambda _0}\\Vert +\\Vert \\tilde{P}_\\lambda -\\tilde{P}_{\\lambda _0}\\Vert ),$ which shows that $\\mathcal {A}=\\lbrace \\mathcal {A}_\\lambda \\rbrace _{\\lambda \\in X}$ is indeed continuous in $\\mathcal {CF}(E)$ .",
"Hence Proposition REF is shown."
],
[
"The Spectral Flow Formula",
"We now prove the spectral flow formula in Theorem REF in two steps."
],
[
"Step 1: Theorem ",
"We begin this first step of our proof with the following elementary observation.",
"Lemma 3.3 The set of all transversal pairs in $\\Lambda (n)$ , i.e.",
"$\\lbrace (L_1,L_2)\\in \\Lambda (n)\\times \\Lambda (n):\\, L_1\\cap L_2=\\lbrace 0\\rbrace \\rbrace \\subset \\Lambda (n)\\times \\Lambda (n),$ is path-connected.",
"Let us first recall the well-known fact that $\\Lambda _0(L_0)$ is contractible, and hence path-connected, for any $L_0\\in \\Lambda (n)$ (see [16]).",
"Now let $(L_1,L_2)$ and $(L_3,L_4)$ be two transversal pairs.",
"As in the construction of the Maslov index in Section REF , $L^{\\prime }_1=e^{\\Theta J}L_1$ is transversal to $L_2$ and $L_4$ for any sufficiently small $\\Theta >0$ .",
"In particular, we obtain a path connecting $(L_1,L_2)$ and $(L^{\\prime }_1,L_2)$ inside (REF ).",
"Also, as $\\Lambda _0(L^{\\prime }_1)$ is path-connected, there is a path connecting $(L^{\\prime }_1,L_2)$ and $(L^{\\prime }_1,L_4)$ inside (REF ).",
"Finally, there is a path from $(L^{\\prime }_1,L_4)$ to $(L_3,L_4)$ inside (REF ) as $\\Lambda _0(L_4)$ is path-connected.",
"This step of the proof is based on the following proposition in which we denote by $\\Omega ^2$ the set of all admissible pairs of paths in $\\Lambda (n)$ (see (REF )).",
"Let us note that by Section REF and (v') in Section REF , $\\mu _{Mas}(\\gamma _{nor},L_1)=1$ and $\\mu _{Mas}(L_0,\\gamma ^{\\prime }_{nor})=-1$ , where $L_0=\\mathbb {R}^n\\times \\lbrace 0\\rbrace $ and $L_1=\\lbrace 0\\rbrace \\times \\mathbb {R}^n$ .",
"Proposition 3.4 Let $\\mu :\\Omega ^2\\rightarrow \\mathbb {Z}$ be a map such that the same properties (i')-(iii') from Section REF are satisfied, as well as (N) $\\mu (\\gamma _{nor},L_1)=1$ and $\\mu (L_0,\\gamma ^{\\prime }_{nor})=-1$ , where $L_0=\\mathbb {R}^n\\times \\lbrace 0\\rbrace $ and $L_1=\\lbrace 0\\rbrace \\times \\mathbb {R}^n$ .",
"Then $\\mu =\\mu _{Mas}$ on $\\Omega ^2$ .",
"We note at first that we have by the properties (ii') and (iii') homomorphisms $\\mu , \\mu _{Mas}:\\pi _1(\\Lambda (n)\\times \\Lambda (n),(L_0,L_1))\\rightarrow \\mathbb {Z}$ and we now claim that they coincide.",
"We first note that $\\pi _1(\\Lambda (n)\\times \\Lambda (n),(L_0,L_1))\\cong \\pi _1(\\Lambda (n),L_0)\\times \\pi _1(\\Lambda (n),L_1)\\cong \\mathbb {Z}\\oplus \\mathbb {Z},$ where the first isomorphism is induced by the projections onto the components and the second one is given by the Maslov index.",
"As $\\gamma _{nor}(0)=L_0$ , $\\gamma ^{\\prime }_{nor}(0)=L_1$ and $\\mu _{Mas}(\\gamma _{nor})=\\mu _{Mas}(\\gamma ^{\\prime }_{nor})=1$ , we see that the pairs of paths $\\lbrace (\\gamma _{nor},L_1), (L_0,\\gamma ^{\\prime }_{nor})\\rbrace $ define a basis of $\\pi _1(\\Lambda (n)\\times \\Lambda (n),(L_0,L_1))$ .",
"Since the homomorphisms in (REF ) coincide on this basis by (N), it follows that $\\mu $ and $\\mu _{Mas}$ are indeed equal for closed paths based at $(L_0,L_1)$ .",
"Let us now assume that $(\\gamma _1,\\gamma _2)\\in \\Omega ^2$ is an arbitrary admissible pair of paths.",
"We connect $(L_0,L_1)$ to $(\\gamma _1(0),\\gamma _2(0))$ by a pair of paths $(\\gamma _3,\\gamma _4)$ and $(\\gamma _1(1),\\gamma _2(1))$ to $(L_0,L_1)$ by a pair of paths $(\\gamma _5,\\gamma _6)$ , where we can assume by Lemma REF that $\\gamma _3(\\lambda )\\cap \\gamma _4(\\lambda )=\\gamma _5(\\lambda )\\cap \\gamma _6(\\lambda )=\\lbrace 0\\rbrace $ for all $\\lambda $ .",
"Then by (i'), (ii') and the first step of our proof $\\mu (\\gamma _1,\\gamma _2)&=\\mu (\\gamma _3,\\gamma _4)+\\mu (\\gamma _1,\\gamma _2)+\\mu (\\gamma _5,\\gamma _6)\\\\&=\\mu ((\\gamma _3,\\gamma _4)\\ast (\\gamma _1,\\gamma _2)\\ast (\\gamma _5,\\gamma _6))=\\mu _{Mas}((\\gamma _3,\\gamma _4)\\ast (\\gamma _1,\\gamma _2)\\ast (\\gamma _5,\\gamma _6))\\\\&=\\mu _{Mas}(\\gamma _3,\\gamma _4)+\\mu _{Mas}(\\gamma _1,\\gamma _2)+\\mu _{Mas}(\\gamma _5,\\gamma _6)=\\mu _{Mas}(\\gamma _1,\\gamma _2),$ which proves the proposition.",
"Remark 3.5 Let $(\\gamma _1,\\gamma _2)$ be a pair of paths in $\\Lambda (n)$ as in (i'), i.e.",
"$\\gamma _1(\\lambda )\\cap \\gamma _2(\\lambda )=\\lbrace 0\\rbrace $ for all $\\lambda \\in I$ .",
"Then $(\\gamma _1,\\gamma _2)$ is homotopic to the constant pair of paths $(\\widetilde{\\gamma }_1(\\lambda ),\\widetilde{\\gamma }_2(\\lambda ))=(\\gamma _1(0),\\gamma _2(0))$ , $\\lambda \\in I$ , by a homotopy of admissible pairs.",
"Hence $\\mu (\\gamma _1,\\gamma _2)=\\mu (\\widetilde{\\gamma }_1,\\widetilde{\\gamma }_2)$ by (iii').",
"As $\\mu (\\widetilde{\\gamma }_1,\\widetilde{\\gamma }_2)=\\mu ((\\widetilde{\\gamma }_1,\\widetilde{\\gamma }_2)\\ast (\\widetilde{\\gamma }_1,\\widetilde{\\gamma }_2))=\\mu (\\widetilde{\\gamma }_1,\\widetilde{\\gamma }_2)+\\mu (\\widetilde{\\gamma }_1,\\widetilde{\\gamma }_2)$ by (ii'), we see that $\\mu (\\gamma _1,\\gamma _2)=\\mu (\\widetilde{\\gamma }_1,\\widetilde{\\gamma }_2)=0$ and so (i') follows from (ii') and (iii').",
"Consequently, Proposition REF actually characterises the Maslov index by the three axioms (ii'), (iii') and (N).",
"We now define $\\mu :\\Omega ^2\\rightarrow \\mathbb {Z},\\quad \\mu (\\gamma _1,\\gamma _2)=\\operatorname{sf}(\\mathcal {A}),$ where $\\mathcal {A}$ is the path of differential operators (REF ) for the pair $(\\gamma _1,\\gamma _2)$ .",
"We aim to use Proposition REF to show Theorem REF and so we need to check the properties (i'), (ii'), (iii') and (N).",
"Let us first note that (i') follows immediately from (ii) in Section REF and the fact that $\\ker (\\mathcal {A}_\\lambda )=\\gamma _1(\\lambda )\\cap \\gamma _2(\\lambda )$ .",
"Also, (ii') follows from (iii) in Section REF .",
"Finally, (ii') is an immediate consequence of the homotopy invariance (i) of the spectral flow and Proposition REF .",
"Hence it remains to show that $\\mu (\\gamma _{nor},L_1)=1$ and $\\mu (L_0,\\gamma ^{\\prime }_{nor})=-1$ , which will be a direct consequence of the following lemma.",
"Lemma 3.6 The spectra of the operators $\\mathcal {A}_\\lambda $ in (REF ) are (i) for $(\\gamma _1,\\gamma _2)=(\\gamma _{nor},L_1)$ $\\sigma (\\mathcal {A}_\\lambda )=\\left\\lbrace \\pi \\lambda -\\frac{\\pi }{2}+\\pi k:\\,k\\in \\mathbb {Z}\\right\\rbrace \\cup \\left\\lbrace \\frac{\\pi }{2}+k\\pi :\\,k\\in \\mathbb {Z} \\right\\rbrace ,$ (ii) for $(\\gamma _1,\\gamma _2)=(L_0,\\gamma ^{\\prime }_{nor})$ $\\sigma (\\mathcal {A}_\\lambda )=\\left\\lbrace -\\pi \\lambda +\\frac{\\pi }{2}+\\pi k:\\,k\\in \\mathbb {Z}\\right\\rbrace \\cup \\left\\lbrace \\frac{\\pi }{2}+k\\pi :\\,k\\in \\mathbb {Z}\\right\\rbrace .$ We consider $Ju^{\\prime }=\\mu u$ and note that the solutions of this equation are $u(t)=\\exp (-\\mu tJ)v,\\quad t\\in I, v\\in \\mathbb {R}^{2n}.$ Let us first consider the path in (i).",
"Then $u$ belongs to the domain of $\\mathcal {A}_\\lambda $ if and only if $u(0)=v\\in \\gamma _{nor}(\\lambda ),\\quad u(1)=\\exp (-\\mu J)v\\in L_1.$ As $\\exp (-\\mu J)v\\in L_1$ if and only if $v\\in \\exp (\\mu J)L_1$ , and $\\exp (\\mu J)=\\cos (\\mu )I_{2n}+\\sin (\\mu )J$ , we see that (REF ) is equivalent to $(\\cos (\\mu )I_{2n}+\\sin (\\mu )J)(\\lbrace 0\\rbrace \\times \\mathbb {R}^n)\\cap \\left(\\mathbb {R}(\\cos (\\pi \\lambda )e_1+\\sin (\\pi \\lambda )e_{n+1})+\\sum ^{n}_{j=2}{\\mathbb {R}e_j}\\right)\\ne \\lbrace 0\\rbrace .$ There are two different cases where these spaces intersect non-trivially.",
"Firstly, if $\\cos (\\mu )=0$ , i.e.",
"$\\mu =\\frac{\\pi }{2}+k\\pi $ for $k\\in \\mathbb {Z}$ .",
"Secondly, if there is an $\\alpha \\ne 0$ such that $\\sin (\\pi \\lambda )e_{n+1}=\\alpha \\cos (\\mu )e_{n+1}$ and $\\cos (\\pi \\lambda )e_1=-\\alpha \\sin (\\mu )e_1$ , where we use that $Je_{n+1}=-e_1$ .",
"Of course, the latter equations are equivalent to $\\sin (\\pi \\lambda )=\\alpha \\cos (\\mu )$ and $\\cos (\\pi \\lambda )=-\\alpha \\sin (\\mu )$ , which can be rewritten as $e^{i\\pi \\lambda }&=\\cos (\\pi \\lambda )+i\\sin (\\pi \\lambda )=\\alpha (-\\sin (\\mu )+i\\cos (\\mu ))=\\alpha i e^{i\\mu }=\\alpha e^{i(\\mu +\\frac{\\pi }{2})}.$ Hence $|\\alpha |=1$ , and this equation holds if and only if $\\pi \\lambda =\\mu +\\frac{\\pi }{2}+k\\pi $ , or equivalently $\\mu =\\pi \\lambda -\\frac{\\pi }{2}-k\\pi $ .",
"In (ii), $u$ belongs to the domain of $\\mathcal {A}_\\lambda $ if and only if $u(0)=v\\in L_0,\\quad u(1)=\\exp (-\\mu J)v\\in \\gamma ^{\\prime }_{nor},$ which is equivalent to $(\\cos (\\mu )I_{2n}-\\sin (\\mu )J)(\\mathbb {R}^n\\times \\lbrace 0\\rbrace )\\cap \\left(\\mathbb {R}(\\sin (\\pi \\lambda )e_1-\\cos (\\pi \\lambda )e_{n+1})+\\sum ^{2n}_{j=n+2}{\\mathbb {R}e_j}\\right)\\ne \\lbrace 0\\rbrace .$ Again, there are two cases in which this intersection is non-trivial.",
"Firstly, $\\mu =k\\pi +\\frac{\\pi }{2}$ where $\\cos (\\mu )=0$ .",
"Secondly, if there is some $\\alpha \\ne 0$ such that $\\alpha \\cos (\\mu )e_1=\\sin (\\pi \\lambda )e_1$ and $\\alpha \\sin (\\mu )e_{n+1}=\\cos (\\pi \\lambda )e_{n+1}$ , which is equivalent to $e^{i\\pi \\lambda }=\\cos (\\pi \\lambda )+i\\sin (\\pi \\lambda )=\\alpha (sin(\\mu )+i\\cos (\\mu ))=\\alpha i e^{-i\\mu }=\\alpha e^{i(\\frac{\\pi }{2}-\\mu )}.$ Hence $|\\alpha |=1$ , and the latter equation holds if and only if $\\pi \\lambda =\\frac{\\pi }{2}-\\mu +k\\pi $ which finally shows that $\\mu =-\\pi \\lambda +\\frac{\\pi }{2}+k\\pi $ .",
"We see from the previous lemma that in both cases there is only one eigenvalue of $\\mathcal {A}_\\lambda $ that crosses the axis whilst the parameter $\\lambda $ travels from 0 to 1.",
"It is now an immediate consequence of the definition of the spectral flow that $\\operatorname{sf}(\\mathcal {A})=1$ for $(\\gamma _1,\\gamma _2)=(\\gamma _{nor},L_1)$ and $\\operatorname{sf}(\\mathcal {A})=-1$ for $(\\gamma _1,\\gamma _2)=(L_0,\\gamma ^{\\prime }_{nor})$ .",
"Hence Theorem REF is shown in the admissible case.",
"Let $(\\gamma _1,\\gamma _2)$ be a pair of paths in $\\Lambda (n)$ which is not necessarily admissible, and let $\\mathcal {A}$ be the path (REF ).",
"Let $\\delta >0$ be as in Lemma REF such that $\\operatorname{sf}(\\mathcal {A})=\\operatorname{sf}(\\mathcal {A}^{\\delta _0})$ for all $0\\le \\delta _0\\le \\delta $ .",
"We consider the solution $\\Psi :I\\rightarrow \\operatorname{Sp}(2n,\\mathbb {R})$ of the differential equation ${\\left\\lbrace \\begin{array}{ll}J\\Psi ^{\\prime }(t)+\\delta _0\\Psi (t)=0,\\quad t\\in I\\\\\\Psi (0)=I_{2n},\\end{array}\\right.",
"}$ and the operator $M\\in GL(L^2(I,\\mathbb {R}^{2n}))$ given by $(Mu)(t)=\\Psi (t)u(t)$ , $t\\in I$ .",
"Then, as $\\mathcal {D}(\\mathcal {A}^{\\delta _0}_\\lambda )=\\mathcal {D}(\\mathcal {A}_\\lambda )$ , $M^T\\mathcal {A}^{\\delta _0}_\\lambda M$ is defined on the domain $\\mathcal {D}(M^T\\mathcal {A}^{\\delta _0}_\\lambda M)&=M^{-1}(\\mathcal {D}(\\mathcal {A}^{\\delta _0}_\\lambda ))=\\lbrace \\Psi (\\cdot )^{-1}u\\in H^1(I,\\mathbb {R}^{2n}):u(0)\\in \\gamma _1(\\lambda ),\\, u(1)\\in \\gamma _2(\\lambda )\\rbrace \\\\&=\\lbrace v\\in H^1(I,\\mathbb {R}^{2n}):\\, v(0)\\in \\gamma _1(\\lambda ), v(1)\\in \\Psi (1)^{-1}\\gamma _2(\\lambda )\\rbrace $ and given by $(M^T\\mathcal {A}^{\\delta _0}_\\lambda Mu)(t)&=M^T(J\\Psi ^{\\prime }(t)u(t)+J\\Psi (t)u^{\\prime }(t)+\\delta _0\\Psi (t)u(t))\\\\&=M^T(-\\delta _0\\Psi (t)u(t)+J\\Psi (t)u^{\\prime }(t)+\\delta _0\\Psi (t)u(t))=\\Psi (t)^TJ\\Psi (t)u^{\\prime }(t)=Ju^{\\prime }(t).$ As $\\Psi (t)=\\exp (\\delta _0 Jt)$ , $t\\in I$ , we see that $\\Psi (1)^{-1}=\\exp (-\\delta _0 J)$ .",
"Finally, if $\\delta _0>0$ is sufficiently small, we obtain by Step 1, Proposition REF and the definition of the Maslov index for non-admissible paths in Section REF , $\\operatorname{sf}(\\mathcal {A})=\\operatorname{sf}(\\mathcal {A}^{\\delta _0})=\\mu _{Mas}(\\gamma _1,e^{-\\delta _0 J}\\gamma _2)=\\mu _{Mas}(\\gamma _1,\\gamma _2),$ which proves Theorem REF in the general case."
],
[
"A Spectral Flow Formula for Hamiltonian Systems",
"Let $\\gamma _1,\\gamma _2:I\\rightarrow \\Lambda (n)$ be two paths of Lagrangian subspaces in $\\mathbb {R}^{2n}$ .",
"We note for later reference the following two standard properties of the Maslov index (see e.g.",
"[17]) If $\\Psi :I\\rightarrow \\operatorname{Sp}(2n,\\mathbb {R})$ is a path of symplectic matrices, then $\\mu _{Mas}(\\Psi \\gamma _1,\\Psi \\gamma _2)=\\mu _{Mas}(\\gamma _1,\\gamma _2).$ If $\\gamma ^{\\prime }_1,\\gamma ^{\\prime }_2:I\\rightarrow \\Lambda (n)$ denote the reverse paths defined by $\\gamma ^{\\prime }_1(\\lambda )=\\gamma _1(1-\\lambda )$ and $\\gamma ^{\\prime }_2(\\lambda )=\\gamma _2(1-\\lambda )$ , then $\\mu _{Mas}(\\gamma ^{\\prime }_1,\\gamma ^{\\prime }_2)=-\\mu _{Mas}(\\gamma _1,\\gamma _2).$ Moreover, we need below the following homotopy invariance property which is an immediate consequence of (iii') in Section REF and the definition of the Maslov index for non-admissible pairs of paths: $\\mu _{Mas}(\\gamma _1,\\gamma _2)=\\mu _{Mas}(\\gamma _3,\\gamma _4)$ if $\\gamma _1\\simeq \\gamma _3$ and $\\gamma _2\\simeq \\gamma _4$ are homotopic by homotopies with fixed endpoints.",
"Let now $S:I\\times I\\rightarrow M(2n,\\mathbb {R})$ be a two parameter family of symmetric matrices and let us consider $\\left\\lbrace \\begin{aligned}Ju^{\\prime }(t)+S_\\lambda (t)u(t)&=0,\\quad t\\in I\\\\(u(0),u(1))\\in \\gamma _1(\\lambda )&\\times \\gamma _2(\\lambda ),\\end{aligned}\\right.$ as well as the differential operators $\\mathcal {A}_\\lambda :\\mathcal {D}(\\mathcal {A}_\\lambda )\\subset L^2(I,\\mathbb {R}^{2n})\\rightarrow L^2(I,\\mathbb {R}^{2n}),\\quad (\\mathcal {A}_\\lambda u)(t)=Ju^{\\prime }(t)+S_\\lambda (t)u(t)$ on the domains $\\mathcal {D}(\\mathcal {A}_\\lambda )=\\lbrace u\\in H^1(I,\\mathbb {R}^{2n}):\\, u(0)\\in \\gamma _1(\\lambda ),\\, u(1)\\in \\gamma _2(\\lambda )\\rbrace $ .",
"We denote for $\\lambda \\in I$ by $\\Psi _\\lambda :I\\rightarrow \\operatorname{Sp}(2n,\\mathbb {R})$ the matrices defined by $\\left\\lbrace \\begin{aligned}J\\Psi ^{\\prime }_\\lambda (t)+S_\\lambda (t)\\Psi _\\lambda (t)&=0,\\quad t\\in I\\\\\\Psi _\\lambda (0)&=I_{2n},\\end{aligned}\\right.$ and we set $(\\Psi \\gamma _1)(\\lambda )=\\Psi _\\lambda (1) \\gamma _1(\\lambda )$ .",
"The aim of this final section is to obtain the following spectral flow formula from Theorem REF .",
"Theorem 4.1 Under the assumptions above, $\\mathcal {A}$ is a gap-continuous path of selfadjoint Fredholm operators on $L^2(I,\\mathbb {R}^{2n})$ and $\\operatorname{sf}(\\mathcal {A})=\\mu _{Mas}(\\Psi \\gamma _1,\\gamma _2).$ We define a continuous family of bounded invertible operators on $L^2(I,\\mathbb {R}^{2n})$ by $(M_\\lambda u)(t)=\\Psi _\\lambda (t) u(t)$ , $t\\in I$ .",
"Then $(M^T_\\lambda \\mathcal {A}_\\lambda M_\\lambda u)(t)&= \\Psi ^T_\\lambda (t)(J\\Psi ^{\\prime }_\\lambda (t)u(t)+J\\Psi _\\lambda (t)u^{\\prime }(t)+S_\\lambda (t)\\Psi _\\lambda (t)u(t))\\\\&=\\Psi ^T_\\lambda (t)(-S_\\lambda (t)\\Psi _\\lambda (t)u(t))+\\Psi ^T_\\lambda (t)J\\Psi _\\lambda (t)u^{\\prime }(t)+\\Psi ^T_\\lambda (t)S_\\lambda (t)\\Psi _\\lambda (t)u(t)\\\\&=Ju^{\\prime }(t)$ and $\\mathcal {D}(M^T_\\lambda \\mathcal {A}_\\lambda M_\\lambda )&= M^{-1}_\\lambda (\\mathcal {D}(\\mathcal {A}_\\lambda ))=\\lbrace u\\in H^1(I,\\mathbb {R}^{2n}):\\, u(0)\\in \\gamma _1(\\lambda ), u(1)\\in \\Psi _\\lambda (1)^{-1}\\gamma _2(\\lambda )\\rbrace .$ By Theorem REF , $M^T\\mathcal {A}M$ is gap-continuous, and it follows from Lemma REF that $\\mathcal {A}$ is gap-continuous as well.",
"Moreover, we obtain from Theorem REF and (REF ) that $\\operatorname{sf}(\\mathcal {A})=\\operatorname{sf}(M^T\\mathcal {A}M)=\\mu _{Mas}(\\gamma _1,\\Psi _{(\\cdot )}(1)^{-1}\\gamma _2)=\\mu _{Mas}(\\Psi \\gamma _1,\\gamma _2).$ Note that we obtain from $\\Psi $ and $\\gamma _1$ further pairs of paths in $\\Lambda (n)$ by $I\\ni t\\mapsto \\Psi _0(t)\\gamma _1(0)\\in \\Lambda (n),\\quad I\\ni t\\mapsto \\Psi _1(t)\\gamma _1(1)\\in \\Lambda (n).$ The following corollary is an easy reformulation of the previous theorem.",
"Corollary 4.2 Under the previous assumptions, $\\operatorname{sf}(\\mathcal {A})=\\mu _{Mas}(\\Psi _1(\\cdot )\\gamma _1(1),\\gamma _2(1))+\\mu _{Mas}(\\gamma _1,\\gamma _2)-\\mu _{Mas}(\\Psi _0(\\cdot )\\gamma _1(0),\\gamma _2(0)).$ We consider the family $\\Gamma :I\\times I\\rightarrow \\Lambda (n)\\times \\Lambda (n)$ defined by $\\Gamma (\\lambda ,t)=(\\Psi _\\lambda (t)\\gamma _1(\\lambda ),\\gamma _2(\\lambda ))$ .",
"We set $\\eta _1(t)=\\Gamma (0,t), \\eta _2(\\lambda )=\\Gamma (\\lambda ,1), \\eta _3(t)=\\Gamma (1,1-t), \\eta _4(\\lambda )=\\Gamma (1-\\lambda ,0).$ As $I\\times I$ is contractible, $\\eta _1\\ast \\eta _2\\ast \\eta _3\\ast \\eta _4$ is homotopic to a constant path by a homotopy with fixed endpoints.",
"Hence the Maslov index of $\\eta _1\\ast \\eta _2\\ast \\eta _3\\ast \\eta _4$ vanishes by (viii').",
"As $\\mu _{Mas}(\\eta _4)=-\\mu _{Mas}(\\gamma _1,\\gamma _2)$ , $\\mu _{Mas}(\\eta _3)=-\\mu _{Mas}(\\Psi _1(\\cdot )\\gamma _1(1),\\gamma _2(1))$ and $\\Psi _\\lambda (0)=I_{2n}$ for all $\\lambda \\in I$ , it follows that $\\mu _{Mas}(\\Psi \\gamma _1,\\gamma _2)=-\\mu _{Mas}(\\Psi _0(\\cdot )\\gamma _1(0),\\gamma _2(0))+\\mu _{Mas}(\\gamma _1,\\gamma _2)+\\mu _{Mas}(\\Psi _1(\\cdot )\\gamma _1(1),\\gamma _2(1)).$ The corollary is now an immediate consequence of the previous theorem.",
"Note that if $S_0(t)=S_1(t)$ for all $t\\in I$ , then $\\Psi _0(t)=\\Psi _1(t)$ , $t\\in I$ .",
"If, moreover, $\\gamma _1(0)=\\gamma _1(1)$ and $\\gamma _2(0)=\\gamma _2(1)$ , then we obtain from the previous corollary that $\\operatorname{sf}(\\mathcal {A})=\\mu _{Mas}(\\gamma _1,\\gamma _2).$ Consequently, under these assumptions the paths (REF ) and (REF ) have the same spectral flow and so the spectral flow of (REF ) does not depend on the family of matrices $S$ .",
"Note that each $S_\\lambda $ is $\\mathcal {A}_\\lambda $ -compact, i.e.",
"$S_\\lambda :\\mathcal {D}(\\mathcal {A}_\\lambda )\\rightarrow L^2(I,\\mathbb {R}^{2n})$ is compact with respect to the graph norm of $\\mathcal {A}_\\lambda $ on $\\mathcal {D}(\\mathcal {A}_\\lambda )$ .",
"Let us point out that for closed paths of bounded selfadjoint Fredholm operators, the spectral flow is invariant under perturbations by compact selfadjoint operators (see [7]).",
"Let us now consider again the general setting of Corollary REF , let $\\alpha , \\beta :[0,1]\\rightarrow [0,1]$ be two continuous functions such that $\\beta (\\lambda )=\\alpha (\\lambda )+\\lambda ,\\quad \\lambda \\in [0,1].$ Our final result generalises Theorem 2 of [9], where the following spectral flow formula was shown for a particular class of functions $\\alpha , \\beta $ that satisfy (REF ).",
"Corollary 4.3 Under the assumptions of Corollary REF , $\\operatorname{sf}(\\mathcal {A})&=\\mu _{Mas}(\\Psi _0(\\alpha (\\cdot ))\\gamma _1(0),\\Psi _0(\\beta (\\cdot ))\\Psi _0(1)^{-1}\\gamma _2(0))+\\mu _{Mas}(\\gamma _1,\\gamma _2)\\\\&-\\mu _{Mas}(\\Psi _1(\\alpha (\\cdot ))\\gamma _1(1),\\Psi _1(\\beta (\\cdot ))\\Psi _1(1)^{-1}\\gamma _2(1)).$ We define maps $h_1,h_2:I\\times I\\rightarrow I$ by $h_1(s,\\lambda )=(1-s)\\alpha (\\lambda )+s(1-\\lambda ),\\qquad h_2(s,\\lambda )=(1-s)\\beta (\\lambda )+s,\\qquad $ and consider for $i=1,2$ the homotopies $H_i:I\\times I\\rightarrow \\Lambda (n)\\times \\Lambda (n),\\quad H_i(s,\\lambda )=(\\Psi _i(h_1(s,\\lambda ))\\gamma _1(i),\\Psi _i(h_2(s,\\lambda ))\\Psi _i(1)^{-1}\\gamma _2(i)).$ As $\\alpha (0)=\\beta (0)$ , we see that $h_1(s,0)=h_2(s,0)$ and so $H_i(s,0)=(\\Psi _i(h_1(s,0))\\gamma _1(i),\\Psi _i(h_2(s,0))\\Psi _i(1)^{-1}\\gamma _2(i))=(\\gamma _1(i),\\Psi _i(1)^{-1}\\gamma _2(i))$ is independent of $s$ , where we have used (vi').",
"Moreover, since $\\alpha (1)=0$ , $\\beta (1)=1$ , and $\\Psi _i(0)=I_{2n}$ , $H_i(s,1)=(\\Psi _i(0)\\gamma _1(i),\\Psi _i(1)\\Psi _i(1)^{-1}\\gamma _2(i))=(\\gamma _1(i),\\gamma _2(i)),$ and so $H_i$ is a homotopy with fixed endpoints.",
"Hence $\\mu _{Mas}(H_i(0,\\cdot ))=\\mu _{Mas}(H_i(1,\\cdot ))$ by (viii') from above.",
"Finally, we note that $H_i(0,\\lambda )&=(\\Psi _i(\\alpha (\\lambda ))\\gamma _1(i),\\Psi _i(\\beta (\\lambda ))\\Psi _i(1)^{-1}\\gamma _2(i)),\\\\H_i(1,\\lambda )&=(\\Psi _i(1-\\lambda )\\gamma _1(i),\\Psi _i(1)\\Psi _i(1)^{-1}\\gamma _2(i))=(\\Psi _i(1-\\lambda )\\gamma _1(i),\\gamma _2(i))$ for all $\\lambda \\in I$ , and $\\mu _{Mas}(\\Psi _i(1-\\cdot )\\gamma _1(i),\\gamma _2(i))=-\\mu _{Mas}(\\Psi _i(\\cdot )\\gamma _1(i),\\gamma _2(i)),$ where we have used (vii').",
"Now the assertion of the corollary follows from Corollary REF .",
"Finally, let us briefly point out that a version of the Morse Index Theorem in semi-Riemannian geometry from [12] can easily be derived from Theorem REF as well.",
"We do not intend to explain the geometric content of the theorem, but just mention that it deals with non-trivial solutions of boundary value problems of the type $\\left\\lbrace \\begin{aligned}Ju^{\\prime }(t)+S_\\lambda (t)u(t)&=0,\\quad t\\in I\\\\u(0),u(1)\\in \\lbrace 0\\rbrace &\\times \\mathbb {R}^n\\end{aligned}\\right.$ where $J$ is as in (REF ) and $S_\\lambda $ is again a family of symmetric $2n\\times 2n$ matrices.",
"If we consider the operators $\\mathcal {A}_\\lambda $ in (REF ) for the equations (REF ), then $\\operatorname{sf}(\\mathcal {A})=\\mu _{Mas}(\\Psi (\\lbrace 0\\rbrace \\times \\mathbb {R}^n),\\lbrace 0\\rbrace \\times \\mathbb {R}^n)$ by Theorem REF , where $\\Psi =\\lbrace \\Psi _\\lambda (1)\\rbrace _{\\lambda \\in I}$ is the path in $\\operatorname{Sp}(2n,\\mathbb {R})$ obtained as in (REF ).",
"This is Proposition 6.1 in [12].",
"Note that in this setting the path $\\mathcal {A}=\\lbrace \\mathcal {A}_\\lambda \\rbrace _{\\lambda \\in I}$ has the constant domain $\\mathcal {D}(\\mathcal {A}_\\lambda )=\\lbrace u\\in H^1(I,\\mathbb {R}^{2n}):\\, u(0),u(1)\\in \\lbrace 0\\rbrace \\times \\mathbb {R}^n\\rbrace $ , which allows to compute its spectral flow by crossing forms (see [18] and [22]) and yields the different proof of (REF ) given in [12]."
]
] | 1808.08464 |
[
[
"On non-orientable surfaces in 4-manifolds"
],
[
"Abstract We find conditions under which a non-orientable closed surface S embedded into an orientable closed 4-manifold X can be represented by a connected sum of an embedded closed surface in X and an unknotted projective plane in a 4-sphere.",
"This allows us to extend the Gabai 4-dimensional light bulb theorem and the Auckly-Kim-Melvin-Ruberman-Schwartz \"one is enough\" theorem to the case of non-orientable surfaces."
],
[
"Introduction",
"The goal of the present note is to determine conditions under which a non-orientable closed surface $S$ embedded into a closed 4-manifold $X$ admits a splitting into the connected sum of an embedded surface $S^{\\prime }$ and an unknotted projective plane $P^2$ in a 4-sphere, i.e., there exists a diffeomorphism of pairs $(X, S) \\approx (X, S^{\\prime })\\# (S^4, P^2).$ The important ingredient for the existence of the splitting is the existence of an embedded transverse sphere for $S$ .",
"We say that a sphere $G$ embedded into $X$ is an embedded transverse sphere for $S$ if the Euler normal number of $G$ is trivial and $G$ intersects $S$ transversally at a unique point.",
"The surface $S$ is $G$ -inessential if the induced homomorphism $\\pi _1(S\\setminus G)\\rightarrow \\pi _1(X\\setminus G)$ is trivial.",
"When there is such a transverse sphere, any $P^2$ summand in $S$ can be split off.",
"Theorem 1 Let $S$ be a connected non-orientable closed surface in a closed orientable 4-manifold $X$ .",
"Suppose that $S$ is $G$ -inessential for a transverse sphere $G$ .",
"Let $P^2$ be a projective plane summand of $S$ .",
"Then the pair $(X, S)$ splits as in (REF ) with $P^2$ unknotted, and with the surface $S^{\\prime }$ still $G$ -inessential for the transverse sphere $G$ .",
"Remark 1 Let $M\\subset S^3\\subset S^4$ be the standard Möbius band.",
"The boundary of $M$ can be pushed radially into the upper hemisphere of $S^4$ where it bounds a unique disc $D^2$ up to isotopy.",
"The union of the Möbius band and the disc $D^2$ is an embedded, unknotted projective plane $P^2$ in $S^4$ .",
"Depending on the sign of the half-twist of the Möbius band, there are two non-isotopic unknotted projective planes, $P^2_+$ and $P^2_-$ .",
"These can be detected by one of two invariants: the normal Euler number, or the Brown invariant, see section .",
"Remark 2 If the Euler characteristics of $S$ is odd, then we may choose the projective plane $P^2$ in $S$ so that the surface $S^{\\prime }$ in the splitting (REF ) is orientable.",
"When the Euler characteristics of $S$ is even, we may split off two unknotted projective planes leaving an orientable surface.",
"It also follows that there is a diffeomorphism of pairs $(X, S)\\approx (X, S^2)\\,\\,\\#\\,\\, k(S^4, P^2_+)\\,\\,\\#\\,\\, \\ell (S^4, P^2_-)$ where $S^2$ is a 2-sphere embedded into $X$ , and $k+\\ell $ is the cross-cap number of $S$ .",
"As a consequence of the splitting theorem (Theorem REF ) we show that a version of the recent Gabai light bulb theorem (Theorem REF ) holds true for non-orientable surfaces as well.",
"Theorem 2 Let $X$ be an orientable 4-manifold such that $\\pi _1(X)$ has no 2-torsion.",
"Let $S_1$ and $S_2$ be two homotopic embedded $G$ -inessential closed surfaces with common transverse sphere $G$ .",
"Suppose that the normal Euler numbers of $S_1$ and $S_2$ agree.",
"Then the surfaces are ambiently isotopic via an isotopy that fixes the transverse sphere pointwise.",
"In the orientable case considered by Gabai the normal Euler number does not play a role.",
"However, this invariant is critical when the surfaces are non-orientable, see Remark REF .",
"Theorem REF has a number of applications.",
"One source of applications is to stabilization of smoothly knotted surfaces in 4-manifolds.",
"If $(X,S)$ is a pair consisting of a 4-manifold and an embedded surface, one has four types of stabilization: external stabilization $(X,S)\\# (S^2\\times S^2,\\emptyset )$ , pairwise stabilization $(X,S)\\# (S^2\\times S^2,\\lbrace \\text{pt}\\rbrace \\times S^2)$ , internal stabilization $(X,S)\\# (S^4,T^2)$ and non-orintable internal stabilization $(X,S)\\# (S^4,P^2)$ .",
"Baykur and Sunukjian proved that a sufficient number of internal stabilizations results in isotopic surfaces [2].",
"In [8], S. Kamada shows that two embedded surfaces become isotopic after enough internal non-orientable stabilizations.",
"The hypothesis that $S$ is $G$ -inessential for a transverse sphere $G$ is always satisfied after taking the connected sum of the pair $(X, S)$ with $(S^2\\times S^2, \\lbrace *\\rbrace \\times S^2)$ which immediately implies the following: Corollary 3 When $\\pi _1(X)$ has no 2-torsion, regularly homotopic embedded surfaces in $X$ become isotopic after just one pairwise stabilizatoin.",
"Quinn [16] and Perron [14], [15] show that topologically isotopic surfaces in simply-connected 4-manifolds become isotopic after sufficiently many external stabilizations.",
"Auckly, Kim, Melvin and Ruberman proved that just one external stabilization was enough for ordinary topologically isotopic orientable surfaces [1].",
"A second consequence of the splitting theorem is a non-orientable version of this “one is enough\" theorem.",
"Theorem 4 Let $S_1$ and $S_2$ be regularly homotopic (possibly non-orientable), embedded surfaces in a 4-manifold $X$ , each with simply-connected complement.",
"If the homology class $[S_1]=[S_2]$ in $H_2(X; \\mathbb {Z}_2)$ is ordinary, then $S_1$ is isotopic to $S_2$ in $X\\# (S^2\\times S^2)$ .",
"If the homology class is characteristic, then the surfaces are isotopic in $X\\# (S^2\\tilde{\\times }S^2)$ .",
"In § we review the notion of an unknotted projected plane as well as the definition of the normal Euler number.",
"In § we prove the splitting theorem (Theorem REF ).",
"The hypothesis in Theorem REF that there exists a transverse sphere $G$ is essential.",
"In § we give examples of surfaces with no transverse spheres that do not admit splittings.",
"In general, the isotopy class of the surface $S\\#S^{\\prime }$ in the pair $(X, S)\\#(X^{\\prime }, S^{\\prime })$ may change when $S$ and $S^{\\prime }$ are changed by isotopy.",
"In contrast, in § we show that the isotopy class of $S\\#S^{\\prime }$ is well defined when $X^{\\prime }$ is a sphere, see Lemma REF .",
"Lemma REF is essential for the proof of the Gabai theorem for non-orientable surfaces (Theorem REF ).",
"Another preliminary statement is proved in § where we show that homotopic surfaces with the same normal Euler numbers are regularly homotopic.",
"Theorem REF is proved in §.",
"Finally, Theorem REF is proved in §.",
"We are grateful to Victor Turchin for helpful suggestions and comments."
],
[
"Background",
"A pair of manifolds $(X, S)$ is a manifold $X$ together with an embedded submanifold $S$ .",
"The connected sum of pairs [10] is denoted by $(X_1, S_1)\\,\\#\\, (X_2, S_2) = (X_1\\,\\#\\, X_2, S_1\\,\\#\\, S_2).$ Given a possibly non-orientable surface $S$ in an oriented 4-manifold $X$ , one may define the self-intersection (or, normal Euler) number $e(S)$ .",
"Take a small isotopic displacement $\\tilde{S}$ of $S$ in the normal directions and count the algebraic number of intersection points in $S\\cap \\tilde{S}$ .",
"The sign of an intersection point $p$ is positive (respective, negative) if $(e_1, e_2, \\tilde{e}_1, \\tilde{e}_2)$ is positively (respectively, negatively) oriented, where $e_1, e_2$ is an arbitrary basis of the tangent space $T_pS$ and $\\tilde{e}_1$ and $\\tilde{e}_2$ the image of $e_1$ and $e_2$ in $T_p\\tilde{S}$ .",
"Remark 3 The normal Euler number is well defined up to regular homotopy, i.e., homotopy through immersions.",
"Figure: The normal Euler number of P + 2 P_+^2 is -2-2.",
"Not shown is the vector e ˜ 2 \\tilde{e}_2 directed into the interior of the lower half space ℝ - 4 \\mathbb {R}^4_-.The positive unknotted projective plane $P_+^2$ in $\\mathbb {R}^4\\subset S^4$ is obtained by capping off the gray right-handed Möbius band in Figure REF with a disc $D$ in the upper half space $\\mathbb {R}^4_+= [0,\\infty )\\times \\mathbb {R}^3$ .",
"There is a displacement $\\tilde{D}$ of $D$ in $\\mathbb {R}^4$ that has an empty intersection with $D$ .",
"It is bounded by the red curve in Figure REF .",
"This curve has zero linking number with the boundary of the Möbius band.",
"The red curve may be extended to the lower half space $\\mathbb {R}^4_-$ and then capped with a red Möbius band to obtain a displacement $\\tilde{P}_+^2$ .",
"The only points of intersection are the two green points in Figure REF and REF .",
"Orienting the tangent space of the grey Möbius band at a green point by vectors $e_1$ and $e_2$ and taking displaced vectors $\\tilde{e}_1$ and $\\tilde{e}_2$ in the tangent space of the red Möbius band, we can see that the two intersection points are counted negatively.",
"Therefore, the normal Euler number of $P_+^2$ is negative two.",
"The negative unknotted projective plane $P_-^2$ is obtained by capping off a left-handed Möbius band with a disc in the upper half space $\\mathbb {R}^4_+$ .",
"Its normal Euler number is 2.",
"Remark 4 Let $r$ be the linear transformation of $\\mathbb {R}^4$ given by $(t, x, y, z)\\mapsto (-t, -x, y, z)$ .",
"It takes the projective plane obtained by capping off the right-handed Möbius band with a disk in the upper half-space to the projective plane obtained by capping off a left-handed Möbius band with a disc in the lower half space $\\mathbb {R}^4_-$ .",
"Remark 5 There is a different invariant defined when $S$ is a characteristic (possibly non-orientable) closed surface in a closed 4-manifold $X$ .",
"It is called the Brown invariant [3].",
"The Brown invariant will not play a role in this paper."
],
[
"The splitting theorem",
"In this section we prove the main splitting theorem.",
"We begin with a simple observation that any splitting is determined by a special disk.",
"Lemma 5 Let $S$ be a closed surface embedded into a closed manifold $X$ .",
"Suppose that there is an open 4-disc $U\\subset X$ such that the intersection $U\\cap S$ is a Möbius band $M$ and $\\partial U$ is an embedded submanifold of $X$ intersecting $S$ transversally.",
"Suppose that $\\partial M$ is an unknot in $\\partial U\\approx S^3$ .",
"Then $(X, S)$ is diffeomorphic to the connected sum of pairs of manifolds $(X, S^{\\prime })$ and $(S^4, P^2_\\pm )$ where $S^{\\prime }$ is a surface obtained from $S\\setminus U$ by capping off its only boundary component with a 2-disc.",
"Since $\\partial M$ is an unknot in $\\partial U$ , the boundary of each of the pairs $(X\\setminus U, S\\setminus U)$ and $(\\bar{U}, \\bar{U}\\cap S)$ is diffeomorphic to the boundary of the pair $(D^4, D^2)$ of standard discs.",
"In other words, the boundary of each of the two pairs can be capped off by the pair of standard discs to produce pairs $(X, S^{\\prime })$ and $(S^4, P^2_\\pm )$ whose connected sum is diffeomorphic to $(X, S)$ .",
"In practice, the open set $U$ in Lemma REF is constructed by taking a neighborhood of a 2-disc $D$ such that $\\partial D$ is the central closed curve of the Möbius band $M$ , the interior of $D$ does not contain points of $S$ and $D$ is nowhere tangent to $S$ .",
"If such a disc $D$ exists, then we say that $D$ is the core of the splitting of Lemma REF .",
"Suppose a connected surface $S$ possesses a transverse sphere $G$ .",
"Given another surface $S^{\\prime }\\subset X$ , an intersection point $p\\in S\\cap S^{\\prime }$ can be tubed off using $G$ along a path $\\gamma $ in $S$ from the point $p$ to $G^{\\prime }\\cap S$ , where $G^{\\prime }$ is a parallel copy of $G$ , see Figure REF .",
"The result of this procedure is a new surface $S^{\\prime \\prime }$ obtained from $S^{\\prime }$ by taking the union of $S^{\\prime }$ and a copy $G^{\\prime }$ , removing a disc neighborhood $D_G$ of $G^{\\prime }\\cap S$ in $G^{\\prime }$ , removing a disc neighborhood $D_S$ of $p$ in $S^{\\prime }$ , attaching a tube $S^1\\times [0,1]$ along $\\gamma $ to the two new boundary components of $S^{\\prime }\\setminus D_S$ and $G^{\\prime }\\setminus D_G$ , and smoothing the corners.",
"Figure: Using a finger move to remove an intersection point.Let $\\alpha $ be a simple closed orientation reversing curve in $S$ that is disjoint from $G$ .",
"Since $S$ is $G$ -inessential, the curve $\\alpha $ bounds an immersed disc $D$ in $X\\setminus G$ .",
"We may join any self intersection point $p$ of $D$ with a point in $\\partial D$ by a curve and use a finger move to eliminate the self intersection point $p$ of $D$ , see Figure REF .",
"By repeating this procedure we obtain an embedded disc $D\\subset X\\setminus G$ that may intersect $S$ in interior points.",
"We may then use the transverse sphere $G$ to tube off the intersection points of $D$ with $S$ , see Figure REF .",
"Thus there exists an embedded 2-disc $D$ in $X\\setminus G$ such that the intersection $D\\cap S$ is the curve $\\alpha $ .",
"Furthermore, we may assume that $D$ approaches $S$ orthogonally (with respect to a Riemannian metric on $X$ ).",
"Figure: A twist of the 1-handleA neighborhood $S^1\\times D^3$ of $\\alpha $ in $X$ is diffeomorphic to the complement in $D^4$ of a neighborhood of $D^2$ .",
"This is depicted by a dotted circle representing the boundary of the disc $D^2$ , see Figure REF .",
"Such a neighborhood of $\\alpha $ already contains the Möbius band neighborhood of $\\alpha $ in $S$ .",
"With respect to a trivialization of $S^1\\times D^2$ , it twists $k+1/2$ times for some integer $k$ .",
"The rest of the neighborhood of $S$ in $X$ is obtained from the described neighborhood of $\\alpha $ by attaching 1-handles that correspond to thickenings of 1-handles of $S$ and one 2-handle $h_S$ which corresponds to the thickening of the 2-cell of $S$ of cell decomposition of $S$ corresponding to a perfect Morse function.",
"The neighborhood of a transverse sphere contributes a 2-handle attached along a meridian of $S$ with zero framing.",
"The neighborhood of the disc $D$ also contributes a 2-handle $h_D$ attached along a circle that passes over the 1-handle $h_1$ once and which, a priori, could be linked with the attaching circle of $h_S$ .",
"Using the transverse sphere $G$ , the attaching circle of $h_D$ can be unlinked from the attaching circle of $h_S$ .",
"We denote by $m$ and $n$ the framings of the attaching circles of the 2-handles $h_D$ and $h_S$ respectively.",
"Figure: Sliding the 2-handle h D h_D over the 1-handle links the attaching sphere h D h_D with the attaching sphere h S h_S and changes the framing by ±2\\pm 2.Giving one of the attaching discs of the 1-handle a full left rotation results in linking the attaching circles of $h_D$ with $h_S$ as well as decreasing simultaneously $m$ and $k$ by 1 and $n$ by 4, see Figure REF .",
"(A right rotation has the opposite effect.)",
"Thus, we may assume that $k$ is 0 or $-1$ , corresponding to $\\pm 1/2$ twists, and $m$ is even.",
"If $k=0$ (respectively, $k=-1$ ) and $m$ is odd, then a full $-1$ rotation (respectively, $+1$ rotation) of the 1-handle results in even $m$ and $k=-1$ (respectively, 0).",
"Sliding the attaching circle of $h_D$ along $h_1$ , links the attaching circles of $h_D$ and $h_S$ and changes the framing $m$ by $\\pm 2$ , see Figure REF .",
"In view of the transverse sphere, we may again unlink the attaching circles $h_D$ from $h_S$ .",
"To summarize, we may assume that in Figure REF , the twisting number $k$ is 0 or $-1$ , and the framing $m$ is 0.",
"Theorem REF now follows from Lemma REF ."
],
[
"Problems with splitting and sums",
"In the absence of a transverse sphere the splitting surgery along arbitrary 1-sided curves may not be possible, see Example REF .",
"Example 1 According to the Massey theorem, there exists an embedding of a closed non-orientable surface $S\\subset S^4$ of Euler characteristics $\\chi $ with normal Euler number $\\nu =2\\chi -4, 2\\chi , ..., 4-2\\chi $ .",
"We may choose an embedding so that $\\nu \\ne \\pm 2$ .",
"Suppose that there exists a splitting surgery representing $(S^4, S)$ by a connected sum of $(S^4, S^{\\prime })$ and $(S^4, P^2_\\pm )$ where $S^{\\prime }$ is a closed orientable surface.",
"Since the normal Euler number of $P^2_\\pm $ is $\\mp 2$ and the normal Euler number of a closed orientable surface in $S^4$ is trivial, we deduce that the normal Euler number of their connected sum $S$ is $\\pm 2$ , which contradicts the assumption that $\\nu \\ne \\pm 2$ .",
"Therefore, such a splitting surgery does not exist.",
"In fact it may be the case that no splitting is possible as in the following example.",
"Example 2 The rational elliptic surface has a cusp fiber $F$ .",
"In Kirby calculus, a neighborhood of this fiber is obtained by attaching a 0-framed 2-handle to $D^4$ along a right-handed trefoil.",
"The right handed trefoil bounds an obvious Möbius band.",
"Capping the Möbius band with the core of the 2-handle results in an embedded $P^2$ representing the fiber class $[F]$ .",
"Notice that this class is characteristic.",
"A splitting of the form $(E(1),P^2) \\cong (E(1),S^2)\\# (S^4,P^2)$ would imply the existence of a smoothly embedded sphere representing the fiber class $[F]$ in $E(1)$ contradicting the Kervaire-Milnor theorem.",
"When splitting is possible, it need not be unique.",
"Indeed changing the homotopy class of the core of the splitting disk can change the homology class of the summands.",
"Example 3 Suppose that $D$ is a core of splitting in a pair $(X, S)$ .",
"Suppose that the splitting results in the decomposition $(X, S) \\simeq (X, S_D^{\\prime }) \\# (S^4, P_\\pm ).$ Let $A$ be an embedded sphere in $X\\setminus (S\\cup D)$ with trivial normal bundle.",
"Consider the splitting with the core $D\\# A$ , $(X, S) \\simeq (X, S_{D\\# A}^{\\prime }) \\# (S^4, P_\\pm ).$ Then $[S_{D\\# A}^{\\prime }]=[S_D^{\\prime }]+2[A]$ , where the orientation of $A$ agrees with the orientation of $D\\# A$ .",
"In the last example, the $\\mathbb {Z}_2$ homology classes of the surfaces $S_{D\\# A}^{\\prime }$ and $S_D^{\\prime }$ in the decomposition agree.",
"However, the integral homology classes of the surfaces are not the same.",
"Thus, Example REF shows that the connected sum decomposition of pairs is not unique.",
"This should not be surprising as the connected sum decomposition of manifolds is not unique in 4-dimensions.",
"We now give one more example where by changing the core of the splitting disk we are able to change from splitting off a copy of $P_+$ to splitting off a copy of $P_-$ .",
"Example 4 The projective plane in $(X,S^2\\times \\lbrace 0\\rbrace )\\# (S^4,P_+)$ has transverse sphere $\\lbrace *\\rbrace \\times S^2$ in the connected factor $S^2\\times S^2$ , where $X=\\mathbb {C}P^2\\#(S^2\\times S^2)$ .",
"We claim that we can split off either $P_+$ or $P_-$ .",
"Indeed, the $P_+$ -splitting is obvious.",
"To describe the $P_-$ -splitting, let $D$ denote the core of the $P_+$ -splitting.",
"We note that this corresponds to $k=0$ and $m=0$ in Figure REF using the trivialization of $D$ .",
"Replacing $D$ with its connected sum with $\\mathbb {C}P^1\\subset \\mathbb {C}P^2$ , results in a model corresponding to $k=0$ and $m=1$ , see Figure REF where the 2-handle corresponding to $\\mathbb {C}P^1$ is denoted by $u$ .",
"To view the neighborhood in the trivialization of the new disk we apply a twist to the 1-handle.",
"This model corresponds to $k=-1$ and $m=0$ , see Figure REF .",
"Now we may slide the handle $D+u$ along the 2-handle $-G$ twice to obtain the core for a $P_-$ -summand.",
"In other words, $(X,S^2\\times \\lbrace 0\\rbrace )\\# (S^4,P_+) \\simeq (X,S^{\\prime })\\# (S^4,P_-).$ The normal Euler number of $P_-$ is 2, while the normal Euler number of $P_+$ is $-2$ .",
"Consequently, the normal Euler number of $S^{\\prime }$ is $-4$ .",
"In fact, $[S^{\\prime }] = [S^2\\times \\lbrace *\\rbrace ]+2[\\mathbb {C}P^1]-4[\\lbrace *\\rbrace \\times S^2]$ in the homology group $H_2(\\mathbb {C}P^2\\#S^2\\times S^2)$ .",
"Figure: A twist of the 1-handleThe connected sum of two manifolds $X$ and $Y$ is defined by removing coordinate open balls $D_X\\subset X$ and $D_Y\\subset Y$ , and then identifying the new boundaries in $X\\setminus D_X$ and $Y\\setminus D_Y$ appropriately.",
"The connected sum of pairs of manifolds is defined similarly by means of pairs of balls.",
"We say that $(D_X, D_S)\\subset (X, S)$ is a pair of coordinate open balls if $\\partial D_X$ intersects $S$ along $\\partial D_S$ and the pair $(D_X, D_S)$ is parametrized by a diffeomorphism from the standard pair of unit balls in $(\\mathbb {R}^4, \\mathbb {R}^2)$ .",
"The connected sum of pairs $(X, S)$ and $(Y, R)$ is defined by removing pairs of coordinate open discs $(D_X, D_S)\\subset (X, S)$ and $(D_Y, D_R)\\subset (Y, R)$ and then identifying the new boundaries appropriately.",
"In this paper we are interested in internal sums, which are defined by taking the connected sum with $(Y, R)$ where $R$ is a surface in $Y=S^4$ .",
"In this case, without loss of generality we may assume that $D_Y$ is the lower hemisphere, and identify $Y\\setminus D_Y$ with a closed coordinate ball.",
"Then $X\\#S^4$ is canonically diffeomorphic to the original manifold $X$ as the connected sum operation replaces the coordinate open ball $D_X$ with the interior of the coordinate ball $S^4\\setminus D_Y$ .",
"We will write $S\\#_i R$ for the resulted surface in $X=X\\# S^4$ when $i\\colon \\!D_X\\rightarrow X$ is a specified inclusion and $D_Y$ is an open lower hemisphere in $Y=S^4$ .",
"To motivate Lemma REF , we note that in the case of connected sums of pairs of manifolds there is an additional subtelety.",
"Namely, let $S_0$ and $S_1$ be two isotopic surfaces in $X$ that agree in a coordinate open ball $D_X$ .",
"Furthermore, suppose that $(X, S_0)$ and $(X, S_1)$ share the same pair of coordinate balls $(D_X, D_S)$ .",
"Then the ambient space $X\\#Y$ in the pair $(X, S_0)\\# (Y, R)$ coincides with the ambient space in the pair $(X, S_1)\\# (Y, R)$ .",
"However, in general, the surface $S_0\\# R$ may not be isotopic to $S_1\\# R$ in $X\\# Y$ , as the isotopy of $X\\setminus \\lbrace 0\\rbrace $ , where $\\lbrace 0\\rbrace $ is the center of the coordinate ball $D_X$ , may not admit an extension to an isotopy of $X\\#Y$ .",
"Given a pair $(X, S)$ , we say that a tuple of vectors $v_1,..., v_4$ at a point $x\\in S$ is an adapted frame if it is a basis for the tangent space $T_xX$ and if the vectors $v_1$ and $v_2$ form a basis for the tangent space $T_xS$ .",
"If $X$ and $S$ are oriented, then we additionally require that the basis $\\lbrace v_1,..., v_4\\rbrace $ for $T_xX$ and $\\lbrace v_1, v_2\\rbrace $ for $T_xS$ are positively oriented.",
"We note that up to isotopy the pair of the coordinate open balls $(D_X, D_S)$ in $(X, S)$ determines and is determined by the standard coordinate adopted frame $e_1,..., e_4$ in $T_{0}D_X\\subset TX$ .",
"Lemma 6 Let $S_0$ and $S_1$ be connected isotopic surfaces in an oriented connected closed 4-manifold $X$ , and $R$ be a surface in $S^4$ .",
"Let $i_0$ and $i_1$ be possibly different orientation preserving embeddings of an open coordinate 4-ball $D^4$ into $X$ such that $(i_kD^4, i_kD^2)$ is a pair of coordinate open discs in $(X, S_k)$ for $k=0,1$ .",
"In the case where $S_0$ and $S_1$ are oriented, suppose that the isotopy from $S_0$ to $S_1$ is orientation preserving, and the frames associated with the pairs $(i_kD^4, i_kD^2)$ of coordinate balls are adapted.",
"Then the surface $S_0\\#_{i_0} R$ is isotopic to the surface $S_1\\#_{i_1} R$ in $X$ .",
"The ambient isotopy taking $S_0$ to $S_1$ takes the connected sum $S_0\\#_{i_0}R$ to $S_1\\#_{j}R$ for some embedding $j\\colon \\!D^4\\rightarrow X$ .",
"Let $\\lbrace v_i\\rbrace $ and $\\lbrace w_i\\rbrace $ denote the frames over $(X, S_1)$ corresponding to the embeddings $j$ and $i_1$ respectively.",
"Since the frames are adapted, by applying an ambient isotopy of $X$ that fixes $S_1$ setwise, we may assume that $v_1=w_1$ and $v_2=w_2$ .",
"Since both $\\lbrace v_i\\rbrace $ and $\\lbrace w_i\\rbrace $ are positively oriented frames over $X$ , the coincidences $v_1=w_1$ and $v_2=w_2$ imply that there is an ambient isotopy of $X$ that fixes $S_1$ setwise and takes the frame $\\lbrace v_i\\rbrace $ to the frame $\\lbrace w_i\\rbrace $ .",
"Thus, the surface $S_0\\#_{i_0} R$ is isotopic to the surface $S_1\\# R$ .",
"This completes the proof of Lemma REF ."
],
[
"Regularly homotopic surfaces",
"At the first step in the Gabai's proof of the 4-dimensional light bulb theorem [5] one modifies a given homotopy between orientable surfaces into a regular homotopy.",
"In this section we will show that the hypotheses on the surfaces in the non-orientable version of the Gabai theorem (Theorem REF ) guarantee that the (possibly non-orientable) surfaces are still regularly homotopic.",
"By the Smale-Hirsch theorem, the space of immersions of a manifold $S$ into a manifold $X$ is weakly homotopy equivalent to the space ${\\mathop \\mathrm {Imm}}^F(S, X)$ of smooth injective bundle homomorphisms $TS\\rightarrow TX$ provided that $\\dim S\\le \\dim X$ or that $S$ is open.",
"There is a natural fibration: ${\\mathop \\mathrm {Imm}}^F(S, X) \\rightarrow C^\\infty (S,X),$ where $C^\\infty (S, X)$ is the space of smooth maps $f\\colon \\!S\\rightarrow X$ .",
"Its fiber over the path component of $f$ is the space $\\Gamma (TS,f^*TX))$ of sections of the bundle $V(TS,f^*TX)\\rightarrow S$ of injective bundle homomorphisms $TS\\rightarrow f^*TX$ .",
"When the dimension of $S$ and $X$ is 4, the fiber of the fiber bundle $V(TS,f^*TX)$ over $S$ is homotopy equivalent to $O(4)$ .",
"The following theorem is known.",
"The orientable case is used in [5].",
"Since we need the non-orientable case, we give a quick outline of its proof here.",
"Theorem 7 Suppose that $f$ and $g$ are two homotopic embeddings of a closed connected surface $S$ into $X$ .",
"If the surface $S$ is non-orientable, suppose, in addition, that the normal Euler numbers of $f$ and $g$ agree.",
"Then the embeddings $f$ and $g$ are regularly homotopic.",
"Choose a handle decomposition of $S$ with a unique 2-cell.",
"By a general position argument, we may assume that a homotopy of $f$ to $g$ restricts to an isotopy of a neighborhood of the 1-skeleton of $S$ .",
"Furthermore, the isotopy of this neighborhood extends to an isotopy of the ambient manifold $X$ .",
"Thus, we may assume that $f$ and $g$ agree in the neighborhood of the 1-skeleton of $S$ .",
"Consequently, the normal bundles $N(f)$ and $N(g)$ agree over the same neighborhood.",
"Since $f$ and $g$ are homotopic, their normal Euler numbers agree when $S$ is orientable.",
"In particular, in both cases, when $S$ is orientable or non-orientable, under the hypotheses of Theorem REF , the normal bundle of the immersion $f$ is isomorphic to that of $g$ .",
"Even more is true, the isomorphism already given over the 1-skeleton extends to an isomorphism $N(f)\\approx N(g)$ .",
"Let $\\hat{f}$ denote the inclusion of $N=N(f)$ into $X$ , and $\\hat{g}\\colon \\!N\\rightarrow X$ denote the composition of the isomorhism $N\\approx N(g)$ and the inclusion.",
"Since $\\pi _2(O_4)=0$ we can extend the constant path from $d\\hat{f}$ to $d\\hat{g}$ defined over the 1-skeleton to a path defined over all of $S$ .",
"It follows that $\\hat{f}$ is regularly homotopic to $\\hat{g}$ , hence $f$ is regularly homotopic to $g$ .",
"Remark 6 By a general position argument, one may assume that the regular homotopy in the conclusion of Theorem REF restricts to an isotopy away from any disk in the surface $S$ ."
],
[
"The Gabai light-bulb theorem for non-orientable surfaces",
"Recently Gabai proved the following theorem, see [5].",
"Theorem 8 (Gabai, [5]) Let $X$ be an orientable 4-manifold such that $\\pi _1(X)$ has no 2-torsion.",
"Two homotopic embedded $G$ -inessential orientable surfaces $S_0$ and $S_1$ with common transverse sphere $G$ are ambiently isotopic via an isotopy that fixes the transverse sphere pointwise.",
"We prove the Gabai result is still true for non-orientable surfaces as well, provided that the normal Euler numbers of the surfaces agree.",
"Remark 7 If a surface $S$ has a transverse sphere, then $S$ is ordinary.",
"Thus the Brown invairant does not play a role in this theorem.",
"The idea of the proof is to use the splitting theorem (Theorem REF ) to reduce the general case of possibly non-orientable surfaces $S_0$ and $S_1$ to the case of orientable surfaces by representing $S_0$ and $S_1$ as internal connected sums of orientable surfaces $S_0^{\\prime }$ and $S_1^{\\prime }$ with unknotted projective planes.",
"As examples in § show, in general the surfaces $S_0^{\\prime }$ and $S_1^{\\prime }$ may not even be homotopic.",
"Lemma REF below shows that we may assume that the surfaces agree away from an open ball.",
"For surfaces meeting the conclusion of Lemma REF , we prove Lemma REF ensuring that there is a splitting such that the surfaces $S_0^{\\prime }$ and $S_1^{\\prime }$ are homotopic.",
"Lemma 9 Let $S_0$ and $S_1$ be regularly homotopic surfaces in a 4-manifold $X$ .",
"Then there exist a surface $S_2$ in $X$ and a neighborhood $U$ of a point in $S_0$ such that $S_0$ agrees with $S_2$ on the complement of $U$ , and $S_2$ is regularly homotopic to $S_0$ by a homotopy that is constant on the complement of $U$ , and $S_1$ is isotopic to $S_2$ .",
"Let $R:I\\times S_0\\rightarrow X$ be a regular homotopy with $R_0(S_0)=S_0$ and $R_1(S_0)=S_1$ .",
"By Remark REF , we may assume that $R$ restricts to an isotopy in a neighborhood of a 1-skeleton of $S_0$ with complement an open disk $U$ .",
"By the isotopy extension theorem, there is an ambient isotopy $J:I\\times X\\rightarrow X$ that agrees with $R$ restricted to $S_0\\backslash U$ .",
"Clearly, the surface $S_2:=J_1^{-1}(S_1)$ is isotopic to $S_1$ , and $S_0$ agrees with $S_2$ in the complement to $U$ .",
"The required regular homotopy of $S_0$ to $S_2$ fixing the complement of $U$ is given by $\\hat{R}_t(x):=J_t^{-1}\\circ R_t(x)$ .",
"Lemma REF is the first step in the proof of Theorem REF .",
"It establishes an isotopy of $S_1$ to $S_2$ so that $S_0$ and $S_2$ agree away from a neighborhood of a point.",
"The following lemma establishes that there are core disks such that the surfaces $S_0^{\\prime }$ and $S_2^{\\prime }$ obtained by splitting off unknotted projective planes from $S_0$ and $S_2$ respectively are still homotopic.",
"Lemma 10 Let $S_0$ be a $G$ -inessential orientable surface with $D_0$ a core of a splitting.",
"Let $S_2$ be a surface that agrees with $S_0$ away from a neighborhood $U\\subset S_0$ of a point such that $\\partial D_0\\cap U=\\emptyset $ .",
"Suppose that $S_0$ is regularly homotopic to $S_2$ relative to the complement to $U$ .",
"Then there exists a core $D_2$ of a splitting for $S_2$ that agrees with $D_0$ in a neighborhood of $\\partial D_0$ and such that $D_0\\cup D_2$ is null-homotopic.",
"If the intersection of the interior of $D_0$ and $S_2$ is empty, then $D_1=D_0$ is a desired core of a splitting for $S_1$ .",
"Otherwise, if the intersection is not empty, then, as in the proof of the splitting theorem, we may tube the intersection points off to copies of $G$ to remove them.",
"A bit of care is necessary to insure that the resulting disc $D_1$ is homotopic to $D_0$ relative to the boundary.",
"To establish this homotopy, it will suffice to establish that a certain lift of $D_0\\cup D_1$ is zero homologous in the universal cover of $X\\setminus G$ .",
"By a slight perturbation of $D_0$ with support in the interior, we may assume that $S_2$ is transverse to the interior of $D_0$ .",
"By assumption a neighborhood of $G$ is diffeomorphic to $G\\times D^2$ .",
"Pick a collection $\\lbrace G_k\\rbrace $ of distinct parallel spheres; one sphere $G_k$ for each intersection point in $S_2\\cap D_0$ .",
"Since $S_2$ is $G$ -inessential and $G_k$ and $D_0$ are simply connected, all lift to copies in the universal cover $\\widetilde{X\\backslash G}$ .",
"Pick reference lifts $S_2^1$ , $G_k^1$ and $D_0^1$ so that $\\partial D_0^1\\subset S_2^1$ and $G_k^1\\cap S_2^1\\ne \\emptyset $ .",
"Let $S_2^\\tau $ , $G_k^\\tau $ and $D_0^\\tau $ denote the translates of these chosen lifts by an element $\\tau $ of the deck group.",
"The regular homotopy relative to the complement of $U$ lifts to homotopies of each $S_2^\\tau $ to a surface $S_0^\\tau $ .",
"There will now be two cases.",
"The first is when the complement of $\\partial D_0$ in $S_2$ is orientable, the second is when it is non-orientable.",
"Suppose the complement to $\\partial D_0$ is orientable.",
"By choosing an orientation on $S_2\\setminus \\partial D_0$ as well as on $X$ , all of the lifts inherit orientations and we may associate a sign to each intersection point.",
"We now wish to show that the algebraic count of the intersection points between $S_2^1$ and $D_0^\\tau $ is zero.",
"(By equivariance this will show that the algebraic count of intersections between $S_2^\\alpha $ and $D_0^\\beta $ is zero as well.)",
"There is a problem with the usual intersection theory argument because intersection points can run off the boundary of $D_0^\\tau $ .",
"We fix this by adding a correction term.",
"Consider what takes place in a regular homotopy between $S_0^1$ and $S_2^1$ .",
"Such a homotopy is comprised of isotopy and finger moves introducing oppositely oriented pairs of self intersections of the surface as well as Whitney moves removing pairs of intersections.",
"By general position these finger moves and Whitney moves may be assumed to take place away from $\\partial D_0^\\tau $ .",
"There will also be finger moves and Whiteny moves introducing and removing pairs of intersections between $S_2^1$ and translates $S_2^\\alpha $ as well as between the surface $S_2^1$ and the disk $D^\\tau _0$ .",
"Finally, an intersection point (self-intersection point if $\\tau =1$ ) in $S_2^1\\cap S_2^\\tau $ may slide past $\\partial D_0^\\tau $ .",
"We explore this posibility in Figure REF .",
"Here we use the vertical red line to represent a portion of $D_0^\\tau $ and the horizontal black line to represent a portion of the surface $S_2^\\tau $ .",
"Both of these extend forward and backward in time.",
"The thimble shape represents a portion of the surface $S_2^1$ existing in the present.",
"A positive intersection point (recall the complement of $\\partial D_0^\\tau $ in $S_2$ is oriented) becomes a negative intersection point and a new intersection point is formed between the surface and the disk.",
"We take our orientation convention for the disk so that this point will be counted positively.",
"Figure: Sliding a self intersection point past the boundary of the core diskLet $S_t^1$ be the surfaces in a regulary homotopy parametrized by $t\\in [0,2]$ .",
"For generic $t$ we define the following invariant obtaind as a combination of algebraic intersection numbers $\\psi (S_t^1,D_0^\\tau ):=S_t^1\\cdot D_0^\\tau +\\frac{1}{2} S_t^1\\cdot S_t^\\tau .$ Since $S_0^1\\cdot D_0^\\tau = S_0^1\\cdot S_0^\\tau =0$ , we see that $\\psi (S_0^1,D_0^\\tau )=0$ .",
"As $\\psi (S_t^1,D_0^\\tau )$ is invariant under the basic moves and is equal to zero at $t=0$ , we conclude that it is equal to zero at $t=2$ .",
"Since $S_2^1$ is embedded $S_2^1\\cdot S_2^\\tau =0$ and we conclude that $S_2^1\\cdot D_0^\\tau =0$ .",
"The same may be said for all translates, e.g., $S_2^\\tau \\cdot D_0^1=0$ Tube each intersection point $p_k\\in S_2^1\\cap D_0^\\tau $ to a $G_k^\\tau $ .",
"The arc that each tube follows projects to an embedded arc because $S_2$ is $G$ -inessential.",
"Thus we can tube in the universal cover and in the base at the same time.",
"Denote the resulting embedded disks $D_2^\\tau $ in the cover and $D_2$ in the base.",
"Since the algebraic intersection number of $D_0^1$ with each lift $S_2^\\tau $ is zero, an algebraically trivial number of copies of each $G_k^\\tau $ is added.",
"It follows that $D_2^1\\cup _\\partial D_0^1$ is null-homologous and $D_1\\cup _\\partial D_0$ is null-homotopic.",
"Now turn to the case where the complement of $\\partial D_0$ is non-orientable.",
"The function $\\psi (S_t^1,D_0^\\tau )$ is no longer well-defined since a self-intersection point may move around an orientation-reversing loop in the complement of the boundary of the core disk.",
"(Such an isotopy would change the sign of the intersection point.)",
"However the parity of $\\psi (S_t^1,D_0^\\tau )$ is well-defined.",
"We conclude that the parity of each intersection number $\\#(S_2^\\tau \\cap D_0^1)$ is even.",
"The result of tubing an intersection point $p_k$ to $G_k^\\tau $ along one path $\\gamma $ will change from $[D_0^1\\cup D_2^1]+[G_k^\\tau ]$ to $[D_0^1\\cup D_2^1]\\pm [G_k^\\tau ]$ in $H_2(\\widetilde{X\\backslash G});\\mathbb {Z})$ when using a second path $\\delta $ depending on whether $\\gamma \\cup \\delta $ is an orientation preserving or orientation reversing loop.",
"Thus by making the correct path choices we may assume that $D_0\\cup D_2$ is null-homotopic.",
"We first prove the result for surfaces of odd Euler characteristic.",
"By Lemma REF we may isotope $S_2$ to agree with $S_1$ away from a neighborhood of a point, and have a further regular homotopy relative to the complement of this neighborhood taking $S_1$ to $S_2$ .",
"Since the Euler characteristic of $S_1$ is odd there is a simple closed curve in $S_1$ having orientable complement.",
"By the splitting theorem (Theorem REF ) we know that there is an embedded disk $D_1$ with boundary this curve that forms the core of a splitting of $S_1$ .",
"By Lemma REF we also know that there is a core for $S_2$ so that $D_2^1\\cup _\\partial D_0^1$ is null-homologous and $D_1\\cup _\\partial D_0$ is null-homotopic.",
"Let $S_j^{\\prime }$ denote the result of splitting surgery of $S_j$ along $D_j$ .",
"One sees that $S_j^{\\prime }$ is homotopy equivalent to $S_j\\cup D_j$ and $S_1\\cap D_1$ is homotopic to $S_2\\cap D_2$ .",
"If follows that $S_1^{\\prime }$ and $S_2^{\\prime }$ are orientable surfaces in $X$ satisfying the hypothesis of the light-bulb theorem.",
"They are therefore isotopic.",
"Now $(X,S_1)\\cong (X,S_1^{\\prime })\\# (S^4,P_\\pm )$ and $(X,S_2)\\cong (X,S_2^{\\prime })\\# (S^4,P_\\pm )$ implies that $S_1$ is isotopic to $S_2$ as proved in Lemma REF .",
"Turn now to the case where $S_j$ is non-orientable with even Euler charasteric.",
"The proof is similar to the odd Euler characteristic case.",
"The difference is that the complement in $S_1$ of $\\partial D_1$ must be non-orientable.",
"Lemma REF still generates a disk $D_2$ forming the core of a splitting so that $D_1$ is homotopic rel boundary to $D_2$ .",
"Let $S_k^{\\prime }$ be the surfaces that result after the splitting.",
"They have odd Euler characteristic and are homotopic.",
"The normal Euler numbers of $S_1$ and $S_2$ agree since they are regularly homotopic.",
"It follows that the normal Euler numbers of $S_1^{\\prime }$ and $S_2^{\\prime }$ agree, so they are regularly homotopic.",
"Thus by the odd Euler characteristic case we know that $S_1^{\\prime }$ is isotopic to $S_2^{\\prime }$ which using Lemma REF completes the proof."
],
[
"Isotopy of surfaces in 4-manifolds.",
"In view of the Gabai's theorem for non-orientable surfaces (Theorem REF ), we are in position to extend a recent theorem of [1] to the case of non-orientable surfaces.",
"In the case where $S_1$ and $S_2$ are oriented surfaces, Theorem REF is established in [1].",
"In view of Theorem REF , the proof of Theorem REF in the non-orientable case follows in the same way as in the orientable case with one minor change.",
"We provide an abbreviated outline referring to [1] for longer exposition.",
"As $X\\backslash S_1$ is simply-connected, it contains an immersed disk bounded by a meridian of $S_1$ .",
"Capping the immersed disc with a fiber of the normal bundle results in an immersed dual $\\Sigma $ to $S_1$ .",
"If the homology class of $S_1$ is ordinary, then, after taking the sum with an immersed sphere disjoint form $S_1$ , we may assume that the self-intersection of $\\Sigma $ is even.",
"In contrast to the orientable case, in the present case only the parity of the intersection number $\\Sigma \\cdot S_1$ is well-defined up to regular homotopy.",
"It follows that $\\Sigma \\cdot S_2$ is odd.",
"We can reduce the geometric intersection number $\\#(\\Sigma \\cap S_2)$ down to one.",
"Indeed, if there is excess intersection, pick a pair of intersection points.",
"Given an arc in $S_2$ joining the two intersection points, one may associate a relative sign to the intersection points via an orientation of a neighborhood of the path.",
"Since $S_2$ is non-orientable, one may arrange that the relative sign is negative via a suitable choice of path.",
"The proof now continues as in the orientable case in[1].",
"Namely, as $X\\backslash S_2$ is simply-connected, the union of the chosen path in $S_2$ and a path in $\\Sigma $ bounds an immersed disk in the complement of $S_2$ .",
"Via finger moves intersections between the disk and $S_1$ may be removed.",
"The correct framing may be obtained by boundary twisting along the portion of the boundary of the disk meeting $\\Sigma $ .",
"The result is an immersed Whitney disk.",
"Sliding $\\Sigma $ across this disk removes a pair of intersection points.",
"If follows that one may assume that $\\Sigma $ intersects $S_1$ and $S_2$ , each at exactly one point.",
"In the ordinary case the self-intersection of $\\Sigma $ is even.",
"It follows that by taking the connected sum of pairs $(X,\\Sigma )\\# (S^2\\times S^2,\\lbrace \\text{pt}\\rbrace \\times S^2) = (X\\# (S^2\\times S^2),\\widetilde{\\Sigma })$ and tubing with copies of $S^2\\times \\lbrace \\text{pt}\\rbrace $ one may eliminate the self-intersections of $\\widetilde{\\Sigma }$ and adjust the square to zero.",
"The result now follows from the orientable version of the light bulb theorem.",
"In the characteristic case the self-intersection number of $\\Sigma $ is odd.",
"Here one takes the sum $(X,\\Sigma )\\# (S^2\\widetilde{\\times }S^2,\\text{zero section}) = (X\\# (S^2\\widetilde{\\times }S^2),\\widetilde{\\Sigma })$ to obtain an immersed dual with even square.",
"Tubing with copies of the fiber will remove the self-intersections and adjust the framing to zero."
]
] | 1808.08605 |
[
[
"Resonance spin transfer torque in ferromagnetic/normal/ferromagnetic\n spin-valve structure of topological insulators"
],
[
"Abstract We theoretically study the spin current and spin-transfer torque generation in a conventional spin- valve hybrid structure of type ferromagnetic/normal metal/ferromagnetic (FM/NM/FM) made of the topological insulator (TI), in which a gate voltage is attached to the normal layer.",
"We demonstrate the penetration of the spin-transfer torque into the right ferromagnetic layer and show that, unlike graphene spin-valve junction, the spin-transfer torque in TI is very sensitive to the chemical potential of the NM region.",
"As an important result, by changing the chemical potential of the NM spacer and magnetization directions, one can control all components of the STT.",
"Interestingly, both the resonance spin current and the resonance spin-transfer torque appear for energies determined from a resonance equation.",
"By increasing the chemical potential of the NM spacer, the amplitude of the STTs decreases while at large chemical potentials of $\\mu_N$ there are intervals of chemical potential in which both the spin current and the spin-transfer torque become zero.",
"These findings could open new perspectives for applications in spin-transfer torque magnetic random access memory (STT-MRAM) devices based on TI."
],
[
"Introduction",
"The electric current modulation of the magnetic properties of magnetic materials instead of externally applied magnetic fields has paved the way to integrate magnetic functionalities into electric-current-controlled spintronics devices with reduced dimensions and energy consumption compared with conventional magnetic field actuation.",
"The conservation of angular momentum between itinerant electrons and localized magnetization in magnetic heterostructures leads to the of particular interest concept of spin-transfer torque (STT) [1], [2], plays a major role in spintronic devices [3], [4], [5], [6].",
"In this phenomenon, the spin angular momentum of electrons in a spin-polarized current, generated by passing an electrical current through a ferromagnet layer, exerts a torque on the second magnetization, enabling magnetization switching or precession [7], [8], for sufficiently large currents without the need for an external field.",
"It is found to be important because of its potential for applications in spin-torque diode effect [9], microwave-assisted recording of hard-disk drives [10], [11], high-performance, and high-density magnetic storage devices [12], [13], [14].",
"Compared with current memory devices which use magnetic fields to reorient magnetization to store information ,Spin-transfer torque magnetic random access memory (STT-MRAM) devices, which store information in the magnetization of a nanoscale magnet, is a promising candidate for the last two decades [7], [8], [15], [16], [17].",
"Magnetic-nonmagnetic multilayers such as magnetic tunnel junctions, spin valves, point contacts, nanopillars, and nanowires [8] are common structures and device geometries that are applicable for STT proposal.",
"Among them, as originally proposed [18], [19], magnetic tunnel junctions were used as a high-performance, non-volatile magnetic memory cells in MRAMs [20].",
"As a large current is needed for current-induced magnetization dynamics, for creating the current densities required for the onset of magnetic instabilities ($10^{8} A/cm^{2}$ ) nanometre-scale devices should be used.",
"Despite the explosive growth of the field of STT in three-dimensional materials, only a few works have studied the spin-transfer torque of two-dimensional heterostructures.",
"STT generation in ferromagnetic-normal-ferromagnetic bulk graphene junctions has been studied theoretically in Ref.",
"[21], then possibility of current-induced STT in ferromagnetic-normal-ferromagnetic graphene nanoribbon junction studied by Ding et al. [22].",
"Very recently in a detailed study, we theoretically investigated the transport and STT in phosphorene-based multilayers with noncollinear magnetizations[23], [24].",
"In the present work, motivated by the recent measurements of the STT induced by a topological insulator [25], we theoretically study the generation of the spin currents and STT in F/N/F trilayer heterostructures of TI.",
"Within the scattering formalism, we find that the application of a local gate voltage to the N region of the FM/NM/FM structure leads to both the spin current and the spin-transfer torque resonance.",
"Depending on the chemical potential of the NM region ($\\mu _N$ ), and the configuration of the magnetization vectors one can has STTs.",
"Figure: (Color online) Schematic illustration of a FM-TI/NM-TI/FM-TI heterostructure, where the total charge current is flowing along the x axis through the left ferromagnetic layer (F 1 F_1) to the right ferromagnetic one (F 2 F_2).",
"The green arrows represent the local magnetic moments with overall magnetization directions 𝐦 1 ,𝐦 2 {\\bf m}_1,{\\bf m}_2.This paper is organized in the following way: In Sec.",
", we introduce the low-energy effective Hamiltonian of the ferromagnetic topological insulator and establish the theoretical framework which is used to calculate the spin current and spin-transfer torque generation in a conventional spin-valve hybrid structure of type ferromagnetic/normal metal/ferromagnetic (FM/NM/FM) made of the topological insulator (TI), in which a gate voltage is attached to the normal layer.",
"In Sec.",
", we discuss our numerical results for the proposed FM/NM/FM hetrostructure.",
"Finally, our conclusions are summarized in Sec.",
"."
],
[
"MODEL AND BASIC FORMALISM",
"As illustrated in Fig.",
"REF , we consider a conventional spin-valve hybrid structure of type ferromagnetic ($F_1$ )/normal metal (NM)/ferromagnetic ($F_2$ ) made of the topological insulator, with a normal spacer of width $L$ .",
"In general ${\\bf m}_1$ and ${\\bf m}_2$ in which ${\\bf m}=(m_x,m_y,m_z)=|{\\bf m}|(\\sin \\theta \\cos \\phi ,\\sin \\theta \\sin \\phi ,\\cos \\theta )$ are vectors along the magnetization of the left and right layers, respectively which are uniform and can point along any general direction.",
"$\\theta , \\phi $ denote polar and azimuthal angles in the spherical coordinate, respectively.",
"Flowing a current from layer $F_1$ into region $F_2$ induces a spin accumulation in $NM/F_2$ interface, exserted a spin-transfer torque on the magnetization $F_2$ [1], [2].",
"We suppose a normal metal spacer much thinner than the spin relaxation length in TI.",
"It is worth mentioning that in order to realize ferromagnetic TI, one may utilize either doping the TI with magnetic impurities [26], [27], [28] or using the proximity effect by coating it with a ferromagnetic insulator[29], [25].",
"The low-energy effective Hamiltonian of the ferromagnetic TI near the Dirac point, can be written as [30] $ \\hat{H}=\\hbar v\\hat{{\\sigma }}\\cdot ({\\bf k}\\times {\\bf z})+ \\hat{{\\sigma }}\\cdot {\\bf m}-\\varepsilon _{\\rm F},$ where $\\hat{{\\sigma }}$ and $v$ are the spin space Pauli matrices and the Fermi velocity, respectively and ${\\bf z}$ denotes the unit vector in the $z$ direction.",
"For simplicity, hereafter we set $\\hbar v=1$ The electron transport is confined in the $x$ -$y$ plane and ${\\bf k} =(k_x,k_y,0)=k(\\cos \\phi _k,\\sin \\phi _k,0)$ .",
"The second term in Eq.",
"(REF ) is the exchange coupling between itinerant and local spins and $\\varepsilon _{\\rm F}$ is the Fermi energy.",
"Figure REF shows the band structure of the pristine and ferromagnetic TI, (a) $m_x=m_y=m_z=0$ , (b) $m_x=m_y=0,m_z=0.016$ , (c) $m_y=m_z=0,m_x=0.15$ and (d) $m_x=m_z=0,m_y=0.15$ .",
"In the absence of the magnetization (the case of (a)), the band structure consists of a massless Dirac cone.",
"An energy gap can be opened in the spectrum of TI when the magnetization lies out of plane ($m_z\\ne 0$ (b)).",
"It is worth mentioning that the $x$ and $y$ components of the magnetization, have no effect on the gap modification of the TI band structure and only shift the Dirac cone along the $y$ and $x$ -momentum axis, respectively (c,d).",
"Figure: (Color online) The band structure of the topological insulator (a)in the absence of the magnetization (m x =m y =m z =0m_x=m_y=m_z=0), (b) m x =m y =0,m z =0.016m_x=m_y=0,m_z=0.016, (c) m y =m z =0,m x =0.15m_y=m_z=0,m_x=0.15 and (d) m x =m z =0,m y =0.15m_x=m_z=0,m_y=0.15, in the presence of the magnetization.The wave functions that diagonalize the unperturbed Dirac Hamiltonian $\\hat{H}$ (Eqn.REF ) are explicitly given as $\\mathinner {|{u_+^{F}}\\rangle }=\\begin{pmatrix}e^{i\\alpha _k}\\cos \\frac{\\beta _k}{2}\\\\ \\sin \\frac{\\beta _k}{2}\\end{pmatrix},\\;\\mathinner {|{u_{-}^{F}}\\rangle }=\\begin{pmatrix}-e^{i\\alpha _k}\\sin \\frac{\\beta _k}{2}\\\\\\cos \\frac{\\beta _k}{2}\\end{pmatrix},$ with $\\alpha _k=\\tan ^{-1}\\left[ \\frac{\\hbar vk\\cos \\phi _k-m\\sin \\phi \\sin \\theta }{\\hbar vk\\sin \\phi _k+m\\cos \\phi \\sin \\theta } \\right],\\;\\beta _k=\\cos ^{-1}\\left[\\frac{m}{|\\varepsilon _{\\bf k}|}\\cos \\theta \\right],$ and $\\varepsilon _{k}^{\\pm }=\\pm \\sqrt{\\hbar ^2v^2k^2+m^2+2\\hbar vkm\\sin \\theta \\sin (\\phi _k-\\phi )}$ .",
"For the normal region the spinors are as $\\mathinner {|{u_{\\pm }^{N}}\\rangle }=\\begin{pmatrix}\\pm e^{i\\alpha ^N_k}\\\\1 \\end{pmatrix},$ in which, $\\alpha ^N_k=\\tan ^{-1}\\left[ \\frac{k_x}{k_y} \\right]$ .",
"When a spin-polarized current interacts with a ferromagnetic layer due to the spin filtering, a spin transfer torque is applied to the magnetic layer.",
"Supposing that there is no spin-flipping processes, overall transmission and reflection amplitudes for spin-up electrons ($t_{\\uparrow }$ , $r_{\\uparrow }$ ) are different from those of spin-down electrons ($t_{\\downarrow }$ , $r_{\\downarrow }$ ).",
"Total wave functions in the two ferromagnetic regions are as $\\psi ^{F_1}_{\\rm in} = {e^{ik_x x} \\over \\sqrt{{\\Omega }}}\\Big ( e^{i\\alpha ^+_k}\\cos (\\beta _k/2) \\left|\\uparrow \\right\\rangle +\\sin (\\beta _k/2) \\left|\\downarrow \\right\\rangle \\Big ).$ $\\psi ^{F_1}_{\\rm ref} = {e^{-ik_x x} \\over \\sqrt{{\\Omega }}}\\Big ( r_{\\uparrow }e^{i\\alpha ^-_k}\\cos (\\beta _k/2) \\left|\\uparrow \\right\\rangle +r_{\\downarrow }\\sin (\\beta _k/2) \\left|\\downarrow \\right\\rangle \\Big ).$ $\\psi ^{F_2}_{\\rm tran} = {e^{ik_x x} \\over \\sqrt{{\\Omega }}}\\Big ( t_{\\uparrow }e^{i\\alpha ^+_k}\\cos (\\beta _k/2) \\left|\\uparrow \\right\\rangle +t_{\\downarrow }\\sin (\\beta _k/2) \\left|\\downarrow \\right\\rangle \\Big ).$ The corresponding eigenvectors in the normal region can be written as $\\psi ^{\\pm N} = {e^{\\pm ik^N_x x} \\over \\sqrt{{2\\Omega }}}\\Big ( a e^{i\\alpha _k^{\\pm N}} \\left|\\uparrow \\right\\rangle +b \\left|\\downarrow \\right\\rangle \\Big )$ Here, $\\alpha _k^{\\pm N}=\\alpha _k^N(\\pm \\phi _k)$ and ${\\Omega }$ is a normalization area.",
"The two propagation directions along the $x$ axis are denoted by $\\pm $ in $\\Psi ^{\\pm N}$ .",
"By matching the wave functions and their first derivatives at the interfaces $x = 0$ and $x = L$ , we obtain the coefficients in the wave functions as $r_{\\uparrow }&=& \\frac{e^{i(\\alpha ^+_{1k}-\\alpha ^-_{1k})}(e^{2ik_x^{N}L}(k_ {x}^{F2}-k_x^{N})(k_ {x}^{F1}+k_x^{N})-(k_ {x}^{F1}-k_x^{N})(k_ {x}^{F2}+k_x^{N})}{e^{2ik_x^{N}L}(k_x^{N}-k_ {x}^{F1})(k_x^{N}-k_ {x}^{F2})-(k_x^{N}+k_ {x}^{F1})(k_x^{N}+k_ {x}^{F2})}\\nonumber \\\\r_{\\downarrow }&=&- \\frac{e^{2ik_x^{N}L}(k_x^{N}+k_ {x}^{F1})(k_x^{N}-k_ {x}^{F2})+(k_ {x}^{F1}-k_x^{N})(k_ {x}^{F2}+k_x^{N})}{e^{2ik_x^{N}L}(k_x^{N}-k_ {x}^{F1})(k_x^{N}-k_ {x}^{F2})-(k_x^{N}+k_ {x}^{F1})(k_x^{N}+k_{x}^{F2})}$ $t_{\\uparrow }&=& \\frac{e^{i(\\alpha ^+_{1k}-\\alpha ^+_{2k}+(k_x^{N}-k_ {x}^{F_2})L)}k_ {x}^{F1}k_x^{N}\\cos (\\beta _{1k}/2)\\sec (\\beta _{2k}/2)}{e^{2ik_x^{N}L}(k_x^{N}-k_ {x}^{F1})(k_x^{N}-k_ {x}^{F2})-(k_x^{N}+k_ {x}^{F1})(k_x^{N}+k_ {x}^{F2})}\\nonumber \\\\t_{\\downarrow }&=& -\\frac{e^{i(k_x^{N}-k_ {x}^{F_2})L}k_ {x}^{F1}k_x^{N}\\sin (\\beta _{1k}/2)\\csc (\\beta _{2k}/2)}{e^{2ik_x^{N}L}(k_x^{N}-k_ {x}^{F1})(k_x^{N}-k_ {x}^{F2})-(k_x^{N}+k_ {x}^{F1})(k_x^{N}+k_ {x}^{F2})}$ In the steady state, the spin transfer torque acting on a volume $V$ of material (by conservation of angular momentum) can be computed simply by determining the net flux of non-equilibrium spin current ${\\bf J^S}$ through the surfaces of that volume as ${\\bf \\tau }_{\\rm stt} =- \\int _{V} dV \\nabla \\cdot {\\bf J^S},$ Note that since ${\\bf J^S}$ is a tensor, its dot product with a vector in real space leaves a vector in spin space.",
"For a single-electron wavefunction $\\psi $ , similar to the more-familiar probability current density $(\\hbar /m){\\rm Im}(\\psi ^*{\\bf \\nabla } \\psi )$ , the spin current density can be rewritten as ${\\bf J^S_{ij}} = {\\hbar \\over m} {\\rm Im}(\\psi ^* {\\bf S_i} \\otimes {\\partial _j}\\psi ),$ Here $i, j=x,y$ , with $i$ indicating the spin component and $j$ the transport direction.",
"$m$ is the electron mass, and ${\\bf S}$ represents the Pauli matrices $S_x$ , $S_y$ , and $S_z$ .",
"The three spin current density components can be determined substituting Eqs.",
"(REF -REF ) into Eq.REF as $J^S_{xx(y),trans} & = & {\\hbar ^2 k \\over 2m{\\Omega }} 2 {\\rm Re({\\rm Im})}[t_{\\uparrow }e^{i\\alpha ^+_{2k}}\\cos (\\beta _{2k}/2)t_{\\downarrow }\\sin (\\beta _{2k}/2)] \\nonumber \\\\J^S_{xz,trans} & = & {\\hbar ^2 k \\over 2m{\\Omega }} [|t_{\\uparrow }|^2\\cos ^2(\\beta _{2k}/2) - |t_{\\downarrow }|^2 \\sin ^2(\\beta _{2k}/2)].",
"\\nonumber $ $J^S_{xx(y),in} & = & {\\hbar ^2 k \\over 2m{\\Omega }} 2 {\\rm Re({\\rm Im})}[e^{i\\alpha ^+_{1k}}\\cos (\\beta _{1k}/2)\\sin (\\beta _{1k}/2)] \\nonumber \\\\J^S_{xz,in} & = & {\\hbar ^2 k \\over 2m{\\Omega }} \\cos (\\beta _{1k}).",
"\\nonumber $ $J^S_{xx(y),ref} & = & {\\hbar ^2 k \\over 2m{\\Omega }} 2 {\\rm Re({\\rm Im})}[r_{\\uparrow }e^{i\\alpha ^-_{1k}}\\cos (\\beta _{1k}/2)r_{\\downarrow }\\sin (\\beta _{1k}/2)] \\nonumber \\\\J^S_{xz,ref} & = & {\\hbar ^2 k \\over 2m{\\Omega }} [|r_{\\uparrow }|^2\\cos ^2(\\beta _{1k}/2) - |r_{\\downarrow }|^2 \\sin ^2(\\beta _{1k}/2)].",
"\\nonumber $ It is clear that the total spin current is not conserved during the filtering process because the spin current density flowing on the left of the magnet ${\\bf J^S_{\\rm in}} + {\\bf J^S_{\\rm refl}}$ is not equal to the spin current density on the right ${\\bf J^S_{\\rm trans}} $ .",
"Using Eq.",
"(REF ), the spin transfer torque ${\\bf \\tau }_{\\rm stt}$ on an area $A$ of the ferromagnet is equal to the net spin current transferred from the electron to the ferromagnet, and is given by ${\\bf \\tau }_{\\rm stt}= A {\\bf \\hat{x}} \\cdot ({\\bf J^S_{\\rm in}} + {\\bf J^S_{\\rm refl}}- {\\bf J^S_{\\rm trans}})$ .",
"Using the scattering theory as well as the incoherency of spin-up and -down states inside the ferromagnet, the STT can be formulated in terms of the spin dependence of the transmission and reflection coefficients as ${ \\tau ^{x(y)}_{\\rm st}}&=& {A \\over \\Omega }{\\hbar ^2 k \\over m} {\\rm Re({\\rm Im})}\\Big [k_1\\cos (\\beta _{1k}/2)\\sin (\\beta _{1k}/2)(e^{i\\alpha ^+_{1k}}\\nonumber \\\\&-&r_{\\uparrow }r_{\\downarrow }^*e^{i\\alpha ^-_{1k}}) -k_2 t_{\\uparrow }t_{\\downarrow }^*e^{i\\alpha ^+_{2k}}\\cos (\\beta _{2k}/2)\\sin (\\beta _{2k}/2) \\Big ]\\nonumber \\\\{ \\tau ^z_{\\rm st}}&=& {A \\over \\Omega }{\\hbar ^2 k \\over m} \\Big [|t_{\\uparrow }|^2(k_1\\cos ^2(\\beta _{1k}/2)-k_2\\cos ^2(\\beta _{2k}/2))\\nonumber \\\\&-&|t_{\\downarrow }|^2(k_1\\sin ^2(\\beta _{1k}/2)-k_2\\sin ^2(\\beta _{2k}/2)) \\Big ]$ We have used the fact that $|t_{\\uparrow }|^2 + |r_{\\uparrow }|^2 = 1$ and $|t_{\\downarrow }|^2 + |r_{\\downarrow }|^2 = 1$ .",
"It is worth mentioning that for a symmetric F/N/F junction there is no component of spin torque in the ${\\bf \\hat{z}}$ direction and the other two components are as follow ${ \\tau ^{x(y)}_{\\rm st}}&=& {A \\over \\Omega }{\\hbar ^2 k \\over 2m} \\sin (\\beta _k) {\\rm Re({\\rm Im})}\\Big [e^{i\\alpha ^+_k}(1-t_{\\uparrow }t_{\\downarrow }^*) - e^{i\\alpha ^-_k} r_{\\uparrow }r_{\\downarrow }^* \\Big ]\\nonumber \\\\$ By including all transverse modes, the total STT of the proposed structure at zero temperature is given by $\\tau _{tot}^i (E) =\\int _{0}^{k_{y}^{max}(E)} \\tau ^i(E,k_y)\\ dk_y,$ $k_{y}^{max}(E)$ is the maximum value of the transverse momentum."
],
[
"Nsumerical results",
"In this section, we present our numerical results.",
"As described in the introduction, a conventional spin-valve structure has the general ferromagnetic/normal/ferromagnetic structure.",
"We study the generation of the spin currents and the spin-transfer torque in a spin-valve hybrid structure of type topological insulator junctions.",
"As we are interested in both the metallic (where the chemical potential stands away from the charge neutrality point) and the zero energy regime, a gate voltage is attached to the normal layer.",
"We set $m_1=m_2=m=0.1$ eV in all figures and results presented in this section.",
"Figure: (Color online) The transmitted spin current density (in units of ℏ 2 /mΩ{\\hbar ^2}/{m\\Omega }) versus the chemical potential of the NM spacer (μ N \\mu _N), for when the first and second magnetizations are fixed along the zz (θ 1 =0)(\\theta _1=0) and -z-z (θ 2 =π)(\\theta _2=\\pi ) axes, respectively.",
"(a)J xx S {J}_{xx}^S (b) J xy S {J}_{xy}^S (c) J xz S {J}_{xz}^S.",
"The red curves are for L=10L=10 and the blue ones are for L=100L=100.",
"The other parameters are taken as m=0.1m=0.1 eV, μ F =0.2\\mu _F=0.2 eV.Figure REF shows the transmitted spin current densities (a)${J}_{xx}^S$ (b) ${J}_{xy}^S$ (c) ${J}_{xz}^S$ (in units of ${\\hbar ^2}/{m\\Omega }$ ) versus the chemical potential of the NM spacer ($\\mu _N$ ), when the first and second magnetizations are fixed along the $z$ $(\\theta _1=0)$ and $-z$ $(\\theta _2=\\pi )$ axes, respectively.",
"The red curves are for $L=10$ and the blue ones are for $L=100$ .",
"The other parameters are taken as $m=0.1$ eV, $\\mu _F=0.2$ eV.",
"As can be seen, the spin current density is an oscillatory function of the chemical potential of the NM spacer and amplitude of the oscillations drops with increasing the chemical potential of the NM region.",
"The spin current density of a junction with a thicker normal region exhibits faster oscillations.",
"Interestingly, resonant spin current peaks appear at the chemical potentials of the NM region that satisfy the equation $k_x^NL=2n\\pi $ with $n$ the positive integer where $k_x^N$ and $L$ are the wavevector and width of the NM region, respectively.",
"Figure: (Color online) The spin-transfer torques (in units of Aℏ 2 /mΩ{A\\hbar ^2}/{m\\Omega }) versus the polar angle of the second magnetization vector 𝐦 2 \\bf {m}_2 (θ 2 \\theta _2) for various chemical potential of the NM region (μ N \\mu _N).",
"(a,d) τ STT x {\\tau }_{STT}^x and (b,e) τ STT y {\\tau }_{STT}^y and (c,f) τ STT z {\\tau }_{STT}^z.",
"Top and bottom panels are for φ 1(2) =0(π)\\phi _{1(2)}=0(\\pi ) and φ 1(2) =π/2(3π/2)\\phi _{1(2)}=\\pi /2(3\\pi /2), respectively.",
"The other parameters are taken as θ 1 =0\\theta _1=0, μ F =0.2\\mu _F=0.2 eV, m=0.1m=0.1 eV and L=10L=10.The dependence of the STT components on the polar angle of the second magnetization ($\\theta _2$ ), for various chemical potentials of the NM region $\\mu _N$ is presented in Fig.REF , (a,d) ${\\tau }_{STT}^x$ and (b,e) ${\\tau }_{STT}^y$ and (c,f) ${\\tau }_{STT}^z$ .",
"Top and bottom panels are for $\\phi _{1(2)}=0(\\pi )$ and $\\phi _{1(2)}=\\pi /2(3\\pi /2)$ , respectively.",
"The other parameters in this figure are taken as $\\theta _1=0$ , $\\mu _F=0.2$ eV, $m=0.1$ eV and $L=10$ .",
"In both panels the first magnetization fixed along the $z$ -axis.",
"The second magnetization rotates from the $z$ -axis to the $-x$ -axis, (inside the $x$ -$z$ plane) in the top panel and from the $z$ -axis to the $-y$ -axis, (inside the $y-z$ plane) in the bottom panel.",
"As a whole, we see that the STT components decrease with increasing the chemical potential of the NM region.",
"Maximum STTs are related to the zero gate voltage.",
"For the configuration $(\\theta _{1(2)},\\phi _{1(2)})=(0(\\theta _2),0(\\pi ))$ , in a certain angle, STTs reached to the maximum value.",
"At the configuration $(\\theta _{1(2)},\\phi _{1(2)})=(0(\\theta _2),\\pi /2(3\\pi /2))$ , ${\\tau }_{STT}^x$ and ${\\tau }_{STT}^y$ are symmetric for the interval $[0,\\pi ]$ .",
"The $z$ component of the STT (${\\tau }_{STT}^z$ ) in the parallel configuration ($\\theta _1=\\theta _2=0$ ), for each chemical potential $\\mu _N$ becomes zero.",
"The $x, y$ components of the STT reach their maximum value at $\\theta _2=\\pi /2$ (parallel configuration), while the ${\\tau }_{STT}^z$ component obtains its maximum at $\\theta _2=\\pi $ (antiparallel configuration).",
"Figure: (Color online) The spin-transfer torques (in units of Aℏ 2 /mΩ{A\\hbar ^2}/{m\\Omega }) versus the azimuthal angle of the second magnetization vector 𝐦 2 \\bf {m}_2 (φ 2 \\phi _2), for various chemical potential of the NM spacer (μ N \\mu _N).",
"(a,b) τ STT x {\\tau }_{STT}^x (c,d) τ STT z {\\tau }_{STT}^z.",
"Left (a,c) and right (b,d) panels are for θ 1(2) =π/2(π/2)\\theta _{1(2)}=\\pi /2(\\pi /2) and θ 1(2) =π/2(3π/2)\\theta _{1(2)}=\\pi /2(3\\pi /2), respectively.",
"The other parameters are taken as φ 1 =0\\phi _1=0, μ F =0.2\\mu _F=0.2 eV, m=0.1m=0.1 eV and L=10L=10.",
"In both configurations, the yy-component of the STT (τ STT y {\\tau }_{STT}^y) becomes zero.The dependence of the STTs on the azimuthal angle, for different values of the chemical potential of the NM region $(\\mu _N)$ , is shown in Fig.REF .",
"As the sign tunability in the STTs devices is crucial, we see that one can simply control STTs both in terms of sign and magnitude.",
"As seen, regardless of the magnetization configuration, the dependence on the azimuthal angle essentially follows the usual sinusoidal behavior, in agreement with Ref. [25].",
"Except $\\mu _N=0$ , the oscillations amplitude of the STTs decreases with increasing chemical potential of the normal TI.",
"STTs of these two configurations have a phase difference of 180 degrees.",
"In both configurations, the case of $\\mu _N=0$ has a phase shift of 180 degrees in ${\\tau }_{STT}^x$ , relative to other chemical potentials of the NM region.",
"Figure: (Color online) The spin-transfer torques (a) τ STT x {\\tau }_{STT}^x and (b) τ STT y {\\tau }_{STT}^y and (c) τ STT z {\\tau }_{STT}^z (in units of Aℏ 2 /mΩ{A\\hbar ^2}/{m\\Omega }) versus the chemical potential of the NM spacer (μ N \\mu _N), for various configurations of the magnetizations.",
"The other parameters are taken as μ F =0.2\\mu _F=0.2 eV, m=0.1m=0.1 eV and L=10L=10.Figure: (Color online) The spin-transfer torques (a) τ STT x {\\tau }_{STT}^x and (b) τ STT y {\\tau }_{STT}^y and (c) τ STT z {\\tau }_{STT}^z (in units of Aℏ 2 /mΩ{A\\hbar ^2}/{m\\Omega }) versus the length of the NM region (LL), for various chemical potential of the NM spacer (μ N \\mu _N).",
"The left (𝐦 1 \\bf {m}_1) and right (𝐦 2 \\bf {m}_2) magnetizations are fixed along the zz and yy-axis, respectively.In Figure REF , we show the effect of chemical potential of the NM region ($\\mu _N$ ) on the spin-transfer torque components, (a,d) ${\\tau }_{STT}^x$ and (b,e) ${\\tau }_{STT}^y$ and (c,f) ${\\tau }_{STT}^z$ .",
"The results are shown for different configurations of the magnetizations $\\bf {m}_1$ and $\\bf {m}_2$ .",
"The other parameters are taken as $\\mu _F=0.2$ eV, $m=0.1$ eV and $L=10$ .",
"As an important result, we see that by changing $\\mu _N$ , one can control the STT that could be a useful consequence for the applications in TI-based nano-electronic devices.",
"Also, note that the amplitude of the STT oscillations decreases as the chemical potential of the NM region increases.",
"Furthermore, the formation of resonant-STT in the right ferromagnetic region is achievable by changing the $\\mu _N$ .",
"In excellent agreement with Ref.",
"[31], a clear oscillatory behavior with sharp peaks in STTs is observed .",
"It is also found that more peaks appear in a same Fermi energy region with enhancing the width of the NM spacer.",
"Increasing the chemical potential of the NM region leads to the STT resonance occurs at values of the energies determined from the equation $k_x^NL=2n\\pi $ .",
"Interestingly, all of the components of the STT are symmetric with respect to the sign reversal of the chemical potential.",
"It is further seen that at large chemical potentials $\\mu _N$ there are intervals of potential in which the spin-transfer torques become zero.",
"In Figure REF , we plot the spin-transfer torque versus the thickness of the central NM layer, when the first and second magnetizations fixed along the $y$ and $z$ -axis, respectively.",
"The STTs display rapid oscillations as a function of normal TI (spacer) width.",
"It is easily seen that for each configuration, the amplitude of the STT oscillations decays with the spacer width.",
"The magnitude of the STTs oscillations decreases as the chemical potential of the NM region increases."
],
[
"summary",
"In summary, we theoretically study the spin current and spin-transfer torque generation in a conventional spin-valve hybrid structure of type ferromagnetic/normal metal/ferromagnetic (FM/NM/FM) made of the topological insulator (TI), in which a gate voltage is attached to the normal layer.",
"We demonstrate the penetration of the spin current and the spin-transfer torque into the right ferromagnetic region and show that, unlike graphene spin-valve junction, the spin-transfer torque in TI is very sensitive to the chemical potential of the NM region.",
"As an important result, by changing the chemical potential of the NM spacer and magnetization directions, one can control all components of the STT.",
"It is interesting to note that both the resonance spin current and the resonance spin-transfer torque appear for the energies determined from the equation $k_x^NL=2n\\pi $ , where $k_x^N$ and $L$ are the wavevector and width of the NM region, respectively.",
"By increasing the chemical potential of the NM spacer, the amplitude of the STTs decreases while at large chemical potentials of $\\mu _N$ there are intervals of chemical potential in which both the spin current and the spin-transfer torque become zero.",
"Moreover, we find that the spin-transfer torques versus the thickness of the central NM layer, display rapid oscillations as a function of the normal TI width.",
"It is easily seen that for each configuration, the amplitude of the STT oscillations decays with the spacer width.",
"The magnitude of the STT oscillations decreases as the chemical potential of the NM region increases.",
"These findings could open new perspectives for applications in spin-transfer torque magnetic random access memory (STT-MRAM) devices based on TI."
]
] | 1808.08379 |
[
[
"Kummer rigidity for K3 surface automorphisms via Ricci-flat metrics"
],
[
"Abstract We give an alternative proof of a result of Cantat and Dupont, showing that any automorphism of a K3 surface with measure of maximal entropy in the Lebesgue class must be a Kummer example.",
"Our method exploits the existence of Ricci-flat metrics on K3s and also covers the non-projective case."
],
[
"Introduction",
"A basic result of Yomdin [39], known previously as the Shub Entropy Conjecture, says that the topological entropy of any smooth map of a compact manifold is bounded below by the spectral radius of the action on homology.",
"Gromov [27] showed that in fact for compact Kähler manifolds and holomorphic automorphisms, this lower bound is always achieved.",
"Thus entropy can be computed from linear-algebraic data, and when it is positive the measure of maximal entropy is unique on compact Kähler surfaces by [21].",
"For an introduction to complex dynamics in higher dimensions see [23], [37], or [15].",
"Kummer examples are tori with automorphisms that become affine on the universal cover, together with their modifications using basic operations of birational geometry.",
"A characteristic feature of Kummer examples is that the measure of maximal entropy is in the Lebesgue class.",
"In [16], Cantat-Dupont proved that automorphisms with positive topological entropy of projective surfaces with measure of maximal entropy in the Lebesgue class are Kummer examples.",
"The main goal of this article is to give an alternative proof in the case of K3 surfaces, which in addition covers the non-projective case."
],
[
"Main statements",
"Theorem 1.1.1 Let $X$ be a $K3$ surface, $T\\colon X\\rightarrow X$ an automorphism with positive topological entropy whose measure of maximal entropy is absolutely continuous with respect to the Lebesgue measure.",
"Then $X$ is a Kummer K3 and $T$ is induced by an (affine) automorphism of the corresponding torus.",
"This resolves a conjecture of Cantat [11] and McMullen [34], including the case of non-projective $K3$ surfaces.",
"Combining [16] with our main theorem we easily obtain the following generalization: Corollary Let $X$ be a compact complex surface, $T\\colon X\\rightarrow X$ an automorphism with positive topological entropy whose measure of maximal entropy is absolutely continuous with respect to the Lebesgue measure.",
"Then $(X,T)$ is a Kummer example.",
"As another application, our main result implies that the measure of maximal entropy in McMullen's construction of a Siegel disc [33] (which are never projective) cannot be in the Lebesgue class.",
"Whether the complement of the Siegel disc can have positive Lebesgue measure remains an interesting open question.",
"For rational maps of $\\mathbb {P}^1$ an analogous result was established by Zdunik [40], and for general endomorphisms of $\\mathbb {P}^n$ by Berteloot–Dupont [2] and Berteloot–Loeb [4], with the role of Kummer examples now played by Lattès maps.",
"For related results in the case of general endomorphisms of Kähler manifolds see [14].",
"For an introduction to K3 surfaces, including their Ricci-flat metrics, see [29]."
],
[
"Proof Outline",
"Let $\\mu $ be the measure of maximal entropy and $\\operatorname{dVol}$ the normalized volume form induced by the holomorphic 2-form on the K3 surface.",
"The assumption says that $\\mu =f\\operatorname{dVol}$ for some $f\\in L^1(\\operatorname{dVol})$ .",
"Because $\\operatorname{dVol}$ is invariant under any holomorphic automorphism and since $\\mu $ is ergodic (in fact mixing by [13]), it follows that $f$ is the normalized indicator function of a set of positive Lebesgue measure and hence we can assume $\\mu = \\frac{1}{\\operatorname{dVol}(S)}\\operatorname{dVol}\\vert _S$ for some $T$ -invariant set $S$ of positive Lebesgue measure.",
"The proof is then naturally divided into three separate steps: Step 1: We prove the result in the special case $\\mu = \\operatorname{dVol}$ (i.e.",
"$S=X$ above).",
"Step 2: We prove the result when $\\mu $ is uniformly hyperbolic and in the Lebesgue class, by reducing to the previous case.",
"Step 3: We prove that $\\mu $ is uniformly hyperbolic, assuming that it is in the Lebesgue class.",
"Step 1 is handled in sec:ergodiccase using the Ricci-flat Kähler metric provided by Yau's theorem [38].",
"This key input, combined with Jensen's inequality, allows for an elementary proof.",
"It already covers the case when the stable/unstable eigencurrents are smooth (since the indicator function of the set $S$ has to be smooth, hence $S=X$ ), which was conjectured by Cantat [11].",
"The two key properties that are used throughout are that the metric is Kähler (so certain integrals are computed cohomologically) and that the metric induces the same volume form as the holomorphic 2-form.",
"Each of these conditions can be easily ensured individually, but their simultaneous validity is Yau's theorem.",
"Step 2 is handled in sec:theuniformlyhyperboliccase and is quite general: it would apply to smooth volume-preserving complex surface diffeomorphisms which are uniformly hyperbolic on an ergodic component of Lebesgue measure.",
"Finally, Step 3 is handled in gen. We show that the expansion/contraction coefficients of the dynamics are in fact cohomologous to a constant (in the dynamical sense) and then show that the coboundaries are, in fact, uniformly bounded.",
"We use that $\\mu =\\eta _+\\wedge \\eta _-$ for two positive closed currents $\\eta _\\pm $ called the stable/unstable eigencurrents.",
"The coboundary structure follows from comparing the conditional measures of $\\mu $ on the stable/unstable foliations and the restriction of $\\eta _\\pm $ to these foliations.",
"To prove the $L^\\infty $ bound for the coboundaries, we first show that the restriction of the stable/unstable eigencurrent to a stable/unstable manifold is flat.",
"This is then used to derive a contradiction if the uniform bound does not hold by constructing a Brody curve (in an orbifold $K3$ surface which is obtained by contracting some $(-2)$ -curves in $X$ ) which intersects trivially both the stable and unstable eigencurrents.",
"Such an entire curve is known not to exist by work of Dinh–Sibony [20].",
"This idea is also a key input in the work of Cantat–Dupont [16].",
"sec:alternativeargumentsfortheexistenceofthecoboundaries contains an alternative derivation of the coboundary structure, based again on Ricci-flat metrics, some estimates of Birkhoff sums, and ideas along the lines of the Gottschalk–Hedlund theorem.",
"These arguments give a priori exponential integrability of the coboundaries, and it is possible that one could prove ergodicity of $\\operatorname{dVol}$ using this weaker property, rather than the $L^\\infty $ bound used in sec:theuniformlyhyperboliccase.",
"Orbifolds.",
"Because the automorphism potentially (in the Kummer case, always) has periodic curves, these must be contracted by a map $\\nu \\colon X\\rightarrow Y$ and many arguments happen on $Y$ instead of $X$ .",
"In the K3 case $Y$ has only orbifold singularities and the needed properties are explained in ssec:preliminariesoncurrentsandorbifolds.",
"On a first reading, one can assume that $X=Y$ and skip contract.",
"Comparison to the approach of Cantat–Dupont.",
"Our use of Ricci-flat metrics on K3 surfaces allows for several simplifications compared to the approach in [16].",
"A key step in showing that the K3 is Kummer is based on the existence of expanded/contracted foliations with the required smoothness properties.",
"In our case, this follows directly from the equality case in Jensen's inequality.",
"Moreover, the Ricci-flat metric already provides the flat metric on the torus (see ssec:analternativeargument) and leads to an alternative proof that we have a Kummer example.",
"To establish ergodicity (Step 2), we use a standard tool in dynamics, the Hopf argument (in a quantitative form).",
"[16] is based on a topological approach, showing that the set where the foliations are defined is both open and closed.",
"In our case, we construct the expansion/contraction factors $\\alpha _u,\\alpha _s$ and then establish their uniform boundedness.",
"We establish their existence using results of De Thélin and Dinh [21], whereas similar objects are constructed in [16] using the $\\lambda $ -lemma, holonomy maps, and renormalization along stable/unstable manifolds.",
"Finally, the use of Dinh–Sibony's entire and the construction of a Brody curve on which the stable/unstable currents vanish is common to both proofs.",
"We use it to establish uniform hyperbolicity of the measure of maximal entropy, while [16] use it to obtain a compactness property of the family of stable/unstable manifolds.",
"The affine structure on these manifolds, a standard fact in dynamics, also appears in both proofs.",
"The use of the contraction $\\nu \\colon X\\rightarrow Y$ is unavoidable in both proofs.",
"Acknowledgments.",
"We would like to thank Alex Eskin, Carlos Matheus, Federico Rodriguez-Hertz, and Amie Wilkinson for discussions, Mattias Jonsson for useful comments on an earlier draft, and Serge Cantat, Curt McMullen and the referee for extensive feedback that significantly improved the exposition.",
"This research was partially conducted during the period the first-named author served as a Clay Research Fellow, and during the second-named author's visits to the Center for Mathematical Sciences and Applications at Harvard University and to the Institut Henri Poincaré in Paris (supported by a Chaire Poincaré funded by the Clay Mathematics Institute) which he would like to thank for the hospitality and support.",
"The first-named author gratefully acknowledges support from the Institute for Advanced Study.",
"This material is based upon work supported by the National Science Foundation under Grants No.",
"DMS-1638352, DMS-1610278 and DMS-1903147."
],
[
"Preliminaries",
"In this section we collect some preliminaries discussions, mostly well-known, and fix notation for the rest of the paper.",
"On a first reading, the construction of the orbifold $Y$ in contract and its Ricci-flat metrics can be skipped and one can assume that $X=Y$ .",
"An introduction to dynamics on surfaces can be found in the survey [15]."
],
[
"The eigencurents",
"For a detailed discussion of the results quoted in this section, see [13].",
"Let $T$ be an automorphism with positive topological entropy $h>0$ of a complex $K3$ surface $X$ .",
"Then there exist two closed positive $(1,1)$ -currents $\\eta _\\pm $ on $X$ which satisfy $T^*\\eta _{\\pm }=e^{\\pm h}\\eta _{\\pm }$ and have continuous local potentials, in fact Hölder continuous by Dinh–Sibony [20].",
"The currents yield cohomology classes $[\\eta _\\pm ]$ and we normalize them to have $\\int _X [\\eta _+]\\wedge [\\eta _-]=1$ .",
"These classes are nef, i.e.",
"are limits of Kähler cohomology classes, and have vanishing self-intersection $\\int _X [\\eta _{\\pm }]^2=0$ .",
"We will be particularly interested in the cohomology $(1,1)$ -class $[\\eta _+]+[\\eta _-]$ which is also clearly nef and satisfies $\\int _X([\\eta _+]+[\\eta _-])^2=2\\int _X[\\eta _+]\\wedge [\\eta _-]=2$ ."
],
[
"The picture in $H^{1,1}$",
"The intersection pairing on $H^{1,1}(X,\\mathbb {R})$ has signature $(1,19)$ and the class $[\\eta _+]+[\\eta _-]$ sits on one of the two hyperboloids of classes with square 2.",
"The hyperboloid is naturally identified with hyperbolic 19-space and contains a geodesic determined by intersecting with the two-dimensional real space spanned by $[\\eta _+],[\\eta _-]$ .",
"Cup product has signature $(1,1)$ on this plane, and is negative definite on its orthogonal complement.",
"The pullback action of the automorphism $T$ acts as an isometry on the complement to the 2-plane, and as a translation by $h$ along the geodesic that the plane determines.",
"The Kähler classes in $H^{1,1}(X,\\mathbb {R})$ are those that pair positively against $(-2)$ curves (see [29]) and the Kähler cone contains the geodesic in its closure.",
"Typically the geodesic is strictly in the interior and the orbifold construction below is unnecessary (this can be assumed on a first reading).",
"When the geodesic lies in the boundary, one has to contract the $(-2)$ curves that pair to zero against both $[\\eta _+],[\\eta _-]$ as in contract below.",
"For more on constructing automorphisms of K3 surfaces using this point of view, see [35]."
],
[
"The associated orbifold",
"The following result is well-known, but for the reader's convenience we provide a proof.",
"Proposition Let $V\\subset X$ be the union of all irreducible compact holomorphic curves $C\\subset X$ which satisfy $\\int _C([\\eta _+]+[\\eta _-])=0$ .",
"There are finitely many compact holomorphic curves periodicA $T$ -periodic curve is defined to satisfy $T(C)=C$ set-theoretically, but not necessarily pointwise.",
"under $T$ , and $V$ is their union.",
"There exists an orbifold $Y$ and a holomorphic map $\\nu \\colon X\\rightarrow Y$ , which is an isomorphism away from $V$ and contracts each connected component of $V$ to an orbifold point of $Y$ .",
"There exists a holomorphic automorphism $T_Y$ of $Y$ with the same topological entropy as $T$ , such that $\\nu \\circ T=T_Y\\circ \\nu $ .",
"From this result it follows that $X=Y$ if and only if $T$ has no periodic curves, or equivalently the class $[\\eta _+]+[\\eta _-]$ is Kähler on $X$ .",
"For part (i), it is clear that $T$ -periodic curves are contained in $V$ , since $\\int _C \\eta _\\pm = \\int _{T^{{-N}}C}(T^N)^*\\eta _\\pm = e^{\\pm Nh}\\int _{T^{-N}C}\\eta _\\pm $ for any compact curve $C$ and integer $N$ .",
"A similar calculation show that $V$ is $T$ -invariant and the general result in [18] implies that $V$ is the union of finitely many curves, proving (i).",
"For part (ii), since the intersection form is negative definite on the complement of the span of $[\\eta _+],[\\eta _-]$ , it follows that if we write $V=\\cup _{i=1}^NC_i$ for the decomposition of $V$ into irreducible components, then the intersection matrix $(C_i\\cdot C_j)$ is negative definite.",
"By a theorem of Grauert [26] there is then a contraction map $\\nu \\colon X\\rightarrow Y$ onto an irreducible normal compact complex surface, which contracts each connected component of $V$ to a point.",
"Next we claim that each connected component of $V$ is in fact an $ADE$ curve (i.e.",
"its irreducible components are smooth rational curves with selfintersection equal to $-2$ [3]).",
"Indeed, the adjunction formula gives $p_a(C_i)=1+\\frac{(K_X\\cdot C_i)+(C_i^2)}{2}\\leqslant \\frac{1}{2},$ using of course that $K_X\\cong \\mathcal {O}_X,$ hence $p_a(C_i)=0$ and so each $C_i$ is a smooth rational curve with $(C_i^2)=-2$ , as claimed.",
"The fact that $V$ is composed of $ADE$ curves now implies that all singular points of $Y$ are rational double points, which in particular are orbifold points (locally isomorphic to the quotient $\\mathbb {C}^2/\\Gamma $ for certain finite subgroups $\\Gamma \\subset \\operatorname{{SU}}(2)$ acting freely on the unit sphere, see [22] for more).",
"For part (iii), following e.g.",
"[30] since $T$ maps $V$ onto itself, it descends to an automorphism $T_Y:Y\\rightarrow Y$ with $\\nu \\circ T=T_Y\\circ \\nu $ , and the topological entropy of $T_Y$ equals the one of $T$ , namely $h$ ."
],
[
"The invariant measures",
"Because the currents $\\eta _\\pm $ have continuous local potentials, their wedge product in the sense of Bedford-Taylor [9] is well-defined and gives a $T$ -invariant probability measure $\\mu :=\\eta _+\\wedge \\eta _-$ on $X$ .",
"It is the unique measure of maximal entropy.",
"On the other hand, the K3 surface $X$ carries a nowhere vanishing holomorphic 2-form $\\Omega $ , which we normalize to have $\\int _X\\Omega \\wedge \\overline{\\Omega }=1$ .",
"The probability measure $\\operatorname{dVol}:=\\Omega \\wedge \\overline{\\Omega }$ is automatically $T$ -invariant, and will be referred to as the Lebesgue measure of $X$ .",
"It will sometimes be denoted by $\\operatorname{dVol}_X$ .",
"Applying $\\nu _*$ gives a $T_Y$ -invariant volume form $\\operatorname{dVol}_Y:=\\nu _*(\\operatorname{dVol}_X)$ , which is nowhere vanishing in the orbifold sense."
],
[
"Orbifold Ricci-flat metrics",
"The orbifold $K3$ surface $Y$ constructed in contract admits orbifold Kähler metrics, as is well-known (see e.g.",
"[10]).",
"In fact, for every $t\\in \\mathbb {R}$ there is an orbifold Kähler class $[\\omega _{Y,t}]$ on $Y$ such that $e^{t}[\\eta _+]+e^{-t}[\\eta _-]=\\nu ^*[\\omega _{Y,t}]$ holds.",
"The (nontrivial) proof of this fact is given in [24].",
"In our setting, Ricci-flatness of a (normalized) Kähler metric $\\omega _Y$ is equivalent to $\\omega _Y^2 = \\operatorname{dVol}_Y$ .",
"Proposition With the notation as in eqn:pullbackfromorbifold, we have: The Kähler class $[\\omega _{Y,t}]$ on the orbifold $Y$ contains a unique orbifold Ricci-flat Kähler metric $\\omega _{Y,t}$ .",
"The metrics satisfy $\\omega _{Y,t+h}:=T_Y^*(\\omega _{Y,t})$ .",
"For (i), the proof of Yau's theorem [38] extends to orbifolds (see e.g.",
"[10]), and so we conclude that the class $[\\omega _{Y,t}]$ contains a unique orbifold Ricci-flat Kähler metric $\\omega _{Y,t}$ .",
"Part (ii) follows by uniqueness of the Ricci-flat metric in its cohomology class: both $\\omega _{Y,t+h}$ and $T_Y^*(\\omega _{Y,t})$ are Ricci-flat and we have $\\begin{split}\\nu ^*[T_Y^*(\\omega _{Y,t})]&=T^*\\nu ^*[\\omega _{Y,t}]=T^*(e^{t}[\\eta _+]+e^{-t}[\\eta _-])=e^{t+h}[\\eta _+]+e^{-t-h}[\\eta _-]\\\\&=\\nu ^*[\\omega _{Y,t+h}],\\end{split}$ and the map $\\nu ^*$ is injective in cohomology."
],
[
"General remarks about Lyapunov exponents",
"Let $Z$ denote either $X$ or $Y$ , and $T_Z$ the corresponding automorphism.",
"Let $m$ be any $T_Z$ -invariant ergodic probability measure.",
"Fix a smooth hermitian metric for computing all norms below and recall that the Lyapunov exponent of $m$ is defined by (see [31]): $\\lambda (m) := \\lim _{N\\rightarrow \\infty } \\frac{1}{N} I_N \\text{ with } I_N:=\\int _Z \\log \\left\\Vert DT_Z^N\\right\\Vert \\, dm$ The limit exists since $I_n$ satisfies the subadditivity property $ I_{k+\\ell }\\leqslant I_k + I_\\ell $ : $\\int _Z \\log \\left\\Vert D_xT^{k+\\ell }\\right\\Vert \\, dm(x) & \\leqslant \\int _Z \\log \\left( \\left\\Vert D_xT^k\\right\\Vert \\cdot \\left\\Vert D_{T^k x}T^\\ell \\right\\Vert \\right) dm(x)\\\\& = \\int _Z \\log \\left\\Vert D_xT^k\\right\\Vert dm(x) + \\int _Z \\log \\left\\Vert D_{T^kx}T^\\ell \\right\\Vert dm(x)\\\\& = I_k + \\int _Z \\log \\left\\Vert D_xT^\\ell \\right\\Vert dm(T^{-k}x) = I_k + I_\\ell $ where we have used the $T_Z$ -invariance of $m$ and the inequality $\\left\\Vert A\\cdot B\\right\\Vert \\leqslant \\left\\Vert A\\right\\Vert \\cdot \\left\\Vert B\\right\\Vert $ for linear maps $A,B$ .",
"In fact by Fekete's lemma $\\lambda (m)=\\inf _N \\frac{1}{N}I_N$ and in particular $\\frac{1}{N}I_N\\geqslant \\lambda (m) \\quad \\forall N \\geqslant 1.$ The exponent does not depend on the fixed ambient metric, since any two will be uniformly comparable."
],
[
"Stable/Unstable directions",
"If $m$ is a $T_Z$ -invariant ergodic probability measure with strictly positive Lyapunov exponent, then the Oseledets theorem (see e.g.",
"[31]) implies that there exist measurable $DT_Z$ -invariant complex line subbundles $W^\\pm (x)$ of the tangent bundle of $Y$ , defined for $m$ -a.e.",
"$x$ , such that $\\lim _{N\\rightarrow \\infty }\\frac{1}{N}\\log \\left\\Vert D_xT^N_Z\\vert _{W^\\pm (x)}\\right\\Vert = \\pm \\lambda (m)$ for $m$ -a.e.",
"$x$ (the positive and negative exponents have the same absolute value because $T_Z$ is volume-preserving).",
"We will use alternatively the notation $W^s$ for $W^-$ and call it the stable direction, and $W^u$ for $W^+$ for the unstable."
],
[
"Absolute continuity",
"Suppose now that the measure of maximal entropy $\\mu $ is absolutely continuous with respect to the Lebesgue measure.",
"Then Ledrappier–Young [32] implies that the Lyapunov exponent of $\\mu $ is $\\frac{h}{2}$ , since the real dimension of the unstable subspace is 2 and the entropy is $h$ ."
],
[
"The ergodic case",
"In this section we assume that in fact $\\mu =\\operatorname{dVol}$ and give an easy proof that $(X,T)$ is a Kummer example, using Ricci-flat metrics and the equality case of Jensen's inequality."
],
[
"Using Jensen's inequality",
"For the orbifold Ricci-flat metrics on $Y$ constructed in prop:kahler, let $\\omega _Y:=\\omega _{Y,0}$ be the fixed reference metric.",
"Definition (Expansion factor) For $x\\in Y$ let $\\lambda (x,N)$ denote the expansion factor (or pointwise Lyapunov exponent) after $N$ iterates of the map $T_Y$ .",
"Namely, at a given point $x$ the metrics $\\omega _Y$ and $\\omega _{Y,Nh}=(T_Y^N)^*\\omega _{Y}$ are of the form, in an appropriate basis: $\\begin{split}\\omega _Y(x) & = |dz_1|^2 + |dz_2|^2\\\\\\omega _{Y,Nh}(x) & = |e^{\\lambda (x,N)}dz_1|^2 + |e^{-\\lambda (x,N)}dz_2|^2\\end{split}$ where as is customary the Kähler forms are identified with their corresponding metrics.",
"More intrinsically, we can define $\\lambda (x,N)$ to be equal to $\\frac{1}{2}\\log $ of the largest eigenvalue of the hermitian form $\\omega _{Y,Nh}(x)$ with respect to $\\omega _Y(x)$ .",
"Remark The next observation regarding expansion factors will be useful in the proof of prop:exponentfoliationsergodiccase below (see also ssec:yauvsoseledetscurve).",
"Suppose $V\\cong \\mathbb {C}^2$ and is equipped with a volume form.",
"Assume that $h_1,h_2$ are two hermitian metrics on $V$ , inducing the background volume form.",
"Then the expansion factor can be alternatively defined as $\\operatorname{{dist}}(h_1,h_2):= \\max _{\\left\\Vert v\\right\\Vert _{h_1}=1}\\log \\left\\Vert v\\right\\Vert _{h_2}$ Observe that the quantity is in fact symmetric (because of volume compatibility), i.e.",
"$\\operatorname{{dist}}(h_1,h_2)=\\operatorname{{dist}}(h_2,h_1)$ , and nonzero if $h_1\\ne h_2$ .",
"It indeed defines a distance function, making the space of hermitian metrics isometric to hyperbolic 3-space (see sssec:hyperbolicdistances).",
"Observe now that if $h_1\\ne h_2$ then the maximizing vector in the definition is unique up to scaling by $S^1\\subset \\mathbb {C}$ , i.e.",
"it determines a unique complex line, call it the maximizing line between $h_1$ and $h_2$ .",
"Now suppose that $h_1,h_2,h_3$ are three distinct metrics that satisfy $\\operatorname{{dist}}(h_1,h_2) + \\operatorname{{dist}}(h_2,h_3) = \\operatorname{{dist}}(h_1,h_3)$ Then the maximizing line between the metrics is the same.",
"Indeed, let $v^{\\prime }$ be a maximizing vector for $\\operatorname{{dist}}(h_1,h_3)$ and observe that $\\operatorname{{dist}}(h_1,h_2) + \\operatorname{{dist}}(h_2,h_3) & \\geqslant \\log \\left(\\frac{\\left\\Vert v^{\\prime }\\right\\Vert _{h_2}}{\\left\\Vert v^{\\prime }\\right\\Vert _{h_1}}\\right)+\\log \\left(\\frac{\\left\\Vert v^{\\prime }\\right\\Vert _{h_3}}{\\left\\Vert v^{\\prime }\\right\\Vert _{h_2}}\\right)\\\\& = \\operatorname{{dist}}(h_1,h_3)$ The first inequality is an equality if and only if $v^{\\prime }$ is (up to scaling) maximizing for the pairs $h_1,h_2$ and $h_2,h_3$ so the conclusion follows.",
"The following simple observation is the key which yields the main theorem in the case when $\\mu =\\operatorname{dVol}$ .",
"Note that the assumption is on $X$ , but the conclusions are on $Y$ .",
"Proposition Assume that $\\mu =\\operatorname{dVol}$ on $X$ .",
"Then We have $\\lambda (x,N)=\\frac{Nh}{2}$ for every $x\\in Y$ and every $N$ .",
"There exist at every $x\\in Y$ two orthogonal (for $\\omega _Y$ ) tangent directions $W^\\pm (x)$ such that $\\log \\left\\Vert DT_Y\\vert _{W^{\\pm }(x)}\\right\\Vert = \\pm \\frac{h}{2}$ .",
"The directions $W^\\pm (x)$ vary real-analytically in $x$ , are $T_Y$ -invariant, and agree with the directions provided by the Oseledets theorem applied to $\\mu $ .",
"For part (i), the cohomological calculation $\\int _Y \\omega _Y\\wedge \\omega _{Y,Nh}=e^{Nh}+e^{-Nh}$ and Jensen's inequality give: $\\begin{split}\\log \\left( e^{Nh} + e^{-Nh} \\right) &= \\log \\left(\\int _Y \\omega _Y \\wedge \\omega _{Y,Nh}\\right)\\\\&\\geqslant \\int _Y\\log \\left(\\frac{\\omega _Y \\wedge \\omega _{Y,Nh}}{\\operatorname{dVol}_Y}\\right)\\operatorname{dVol}_Y\\\\&=\\int _Y \\log \\left( e^{2\\lambda (x,N) } + e^{-2\\lambda (x,N)} \\right) \\operatorname{dVol}_Y.\\end{split}$ We established in eqn:feketelowerbound that $\\int _Y \\lambda (x,N)\\operatorname{dVol}_Y\\geqslant \\frac{Nh}{2}.$ Indeed, in the case at hand $\\log \\left\\Vert D_xT_Y^N\\right\\Vert = \\lambda (x,N)$ by definition, and the Lyapunov exponent is $h/2$ by sssec:absolutecontinuity.",
"Note that the function $\\log (e^x+e^{-x})$ is convex and increasing so we can apply Jensen again.",
"Recall that $2I_N := \\int _Y 2\\lambda (x,N)\\operatorname{dVol}_Y\\geqslant Nh$ to find: $\\int _Y \\log \\left( e^{2\\lambda (x,N) } + e^{-2\\lambda (x,N)} \\right) \\operatorname{dVol}_Y \\geqslant \\log (e^{2I_N} + e^{-{2I_N}}) \\geqslant \\log (e^{Nh}+ e^{-Nh})$ So from jen1 and jen2 it follows that we must have equality pointwise a.e., that is $\\lambda (x,N)=Nh/2$ pointwise a.e., for all $N$ .",
"Since the function $\\lambda (-,N)$ is continuous, the result holds everywhere on $Y$ .",
"For part (ii), the equality case in Jensen plus the equality case in $I_k+I_\\ell \\leqslant I_{k+\\ell }$ (see ssec:generalremarksaboutlyapunovexponents) imply that at every point $x\\in Y$ , the directions $dz_1,dz_2$ appearing in eqn:omegaomegaN are independent of $N$ and determine the spaces $W^{\\pm }(x)$ .",
"Indeed, we can apply rmk:distancebetweenmetrics since we have equality in the triangle inequality and hence the direction of maximal expansion at each point is independent of the iterate of the dynamics.",
"The same applies to the inverse dynamics, giving the two directions.",
"The directions are orthogonal for $\\omega _Y$ by the spectral theorem for hermitian matrices.",
"Moreover, for any real-analytic Kähler metrics $\\alpha ,\\beta $ , the direction of maximal expansion of $\\alpha $ relative to $\\beta $ varies real-analytically, away from the locus where the direction is not unique (empty in our case).",
"We can now finish off the proof of main when $\\mu =\\operatorname{dVol}$ .",
"Theorem 3.1.4 Let $X$ be a $K3$ surface, $T\\colon X\\rightarrow X$ an automorphism with positive topological entropy whose measure of maximal entropy $\\mu $ equals Lebesgue measure $\\operatorname{dVol}$ .",
"Then $X$ is a Kummer K3 and $T$ is induced by an (affine) automorphism of the corresponding torus.",
"prop:exponentfoliationsergodiccase(ii) gives two line subbundles of the tangent bundle of $Y$ , invariant and uniformly expanded/contracted by the dynamics.",
"By Ghys [25] [16], these give two holomorphic foliations on the orbifold $Y$ .",
"These can be pulled back to holomorphic foliations on $X\\backslash V$ , which automatically extend to $X$ exactly as in [16]), which are preserved by $T$ (alternatively, as Cantat pointed out to us, in the present case the extension of the holomorphic foliations follows from the explicit description of the singular points of $Y$ as quotient $ADE$ singularities).",
"At this point we can apply either a result of Cantat [13], or a later result of Cantat-Favre [17] (which only needs one invariant foliation), to conclude that $X$ is a Kummer K3 and $T$ is induced by an automorphism of the corresponding torus.",
"Note that these results apply in the Kähler case (for [17] one needs to use results of Brunella [7], [8])."
],
[
"An alternative argument",
"At the end of the proof of main2 above, after obtaining two $T$ -invariant holomorphic foliations on $X$ , we appealed to the general results of Cantat [13] or Cantat-Favre [17] to conclude that $(X,T)$ is a Kummer example.",
"We now explain how to circumvent in our case some of the just cited arguments using the differential geometry of Ricci-flat metrics.",
"We suppose that on $X$ we have $\\mu =\\operatorname{dVol}$ , and we have applied prop:exponentfoliationsergodiccase, to obtain two $T_Y$ -invariant transverse holomorphic foliations on $Y$ , as above.",
"Proposition In this setting, the orbifold Ricci-flat Kähler metric $\\omega _Y$ on $Y$ is in fact flat.",
"At any point $x\\in Y$ we have the two orbifold Ricci-flat Kähler metrics $\\omega _Y$ and $\\omega _{Y,h}=T_Y^*\\omega _Y$ , which have the property that the eigenvalues of $\\omega _{Y,h}$ with respect to $\\omega _Y$ are $e^h$ and $e^{-h}$ , and the corresponding eigenvectors (these are eigenvectors of the endomorphism of the tangent space given by composing $\\omega _{Y,h}$ with $\\omega _Y^{-1}$ ) span the stable and unstable holomorphic foliations $\\mathcal {F}_{\\pm }$ respectively.",
"By construction, these eigenvectors are $\\omega _Y$ -orthogonal, hence so are $\\mathcal {F}_+$ and $\\mathcal {F}_-$ .",
"Near $x$ we can find local holomorphic functions $z_1,z_2$ such that $\\ker dz_1=\\mathcal {F}_+,\\ker dz_2=\\mathcal {F}_-$ , which implies that $z_1,z_2$ give local holomorphic coordinates near $x$ (on the local orbifold cover if $x$ is singular).",
"Therefore near $x$ we can write $\\omega _Y=a idz_1\\wedge d\\overline{z}_1+b idz_2\\wedge d\\overline{z}_2,$ where $a,b$ are local smooth positive functions.",
"But then $0=d\\omega _Y=\\left(\\frac{\\partial a}{\\partial z_2}dz_2+\\frac{\\partial a}{\\partial \\overline{z}_2}d\\overline{z}_2\\right)\\wedge idz_1\\wedge d\\overline{z}_1+\\left(\\frac{\\partial b}{\\partial z_1}dz_1+\\frac{\\partial b}{\\partial \\overline{z}_1}d\\overline{z}_1\\right)\\wedge idz_2\\wedge d\\overline{z}_2,$ which imply that $a$ is independent of the $z_2,\\overline{z}_2$ variables and $b$ is independent of the $z_1,\\overline{z}_1$ directions.",
"From the definition of curvature we have $R_{i\\overline{j}k\\overline{\\ell }}=-\\partial _i\\partial _{\\overline{j}}g_{k\\overline{\\ell }}+g^{p\\overline{q}}\\partial _i g_{k\\overline{q}}\\partial _{\\overline{j}}g_{p\\overline{\\ell }},$ where in our coordinates $g_{1\\overline{2}}=g_{2\\overline{1}}=g^{1\\overline{2}}=g^{2\\overline{1}}=0,\\quad g_{1\\overline{1}}=a, g_{2\\overline{2}}=b,\\quad g^{1\\overline{1}}=a^{-1},g^{2\\overline{2}}=b^{-1},$ $\\partial _2g_{1\\overline{1}}=\\partial _{\\overline{2}}g_{1\\overline{1}}=0,\\quad \\partial _1g_{2\\overline{2}}=\\partial _{\\overline{1}}g_{2\\overline{2}}=0.$ In particular $R_{2\\overline{2}1\\overline{1}}=-\\partial _2\\partial _{\\overline{2}}a+g^{p\\overline{q}}\\partial _2 g_{1\\overline{q}}\\partial _{\\overline{2}}g_{p\\overline{1}}=g^{1\\overline{1}}\\partial _2 g_{1\\overline{1}}\\partial _{\\overline{2}}g_{1\\overline{1}}=0,$ while Ricci-flatness gives $0=R_{1\\overline{1}}=g^{p\\overline{q}}R_{p\\overline{q}1\\overline{1}}=g^{1\\overline{1}}R_{1\\overline{1}1\\overline{1}}+g^{2\\overline{2}}R_{2\\overline{2}1\\overline{1}}=a^{-1}R_{1\\overline{1}1\\overline{1}},$ giving $R_{1\\overline{1}1\\overline{1}}=0$ , and $0=R_{2\\overline{2}}=g^{p\\overline{q}}R_{p\\overline{q}2\\overline{2}}=g^{1\\overline{1}}R_{1\\overline{1}2\\overline{2}}+g^{2\\overline{2}}R_{2\\overline{2}2\\overline{2}}=b^{-1}R_{2\\overline{2}2\\overline{2}},$ giving $R_{2\\overline{2}2\\overline{2}}=0$ (using the Kähler identities).",
"Next, $R_{1\\overline{1}1\\overline{2}}=-\\partial _1\\partial _{\\overline{1}}g_{1\\overline{2}}+g^{p\\overline{q}}\\partial _1 g_{1\\overline{q}}\\partial _{\\overline{1}}g_{p\\overline{2}}=0,$ $R_{1\\overline{2}2\\overline{2}}=-\\partial _1\\partial _{\\overline{2}}g_{2\\overline{2}}+g^{p\\overline{q}}\\partial _1 g_{2\\overline{q}}\\partial _{\\overline{2}}g_{p\\overline{2}}=g^{2\\overline{2}}\\partial _1 g_{2\\overline{2}}\\partial _{\\overline{2}}g_{2\\overline{2}}=0.$ Thanks to the Kähler identities, we thus obtain that $R_{i\\overline{j}k\\overline{\\ell }}=0$ for all $i,j,k,\\ell $ , hence $\\omega _Y$ is a flat orbifold Kähler metric.",
"Once we know that $\\omega _Y$ is flat, this in turn implies that there is a finite orbifold cover $\\pi \\colon Z\\rightarrow Y$ with $Z$ a compact complex 2-torus (cf.",
"the discussion in [19]).",
"The arguments in [13], [17] can then be used to show that $T_Y$ lifts to an automorphism of $Z$ , which is then affine linear, and that the map $\\pi :Z\\rightarrow Y$ is the quotient by an involution, and so $X$ is a Kummer K3 and $T$ is induced by an automorphism of the corresponding torus."
],
[
"The uniformly hyperbolic case",
"After recalling in ssec:recollectionsfrompesintheory the needed facts from Pesin theory, we prove in ssec:ergodicityintheuniformlyhyperboliccase that if $\\mu $ is uniformly hyperbolic and in the Lebesgue class, then it is in fact equal to $\\operatorname{dVol}$ ."
],
[
"Recollections from Pesin theory",
"This section collects some concepts and results from Pesin theory that will be used to prove ergodicity of $\\operatorname{dVol}$ , under an extra assumption of uniform hyperbolicity.",
"While Pesin theory is concerned with the non-uniformly hyperbolic setting, its conclusions apply to sets of almost full measure, on which hyperbolicity is uniform.",
"For the discussion in this subsection, $Z$ can be any compact complex surface, possibly an orbifold, $T_Z\\colon Z\\rightarrow Z$ a holomorphic automorphism, and $m$ a $T_Z$ -invariant ergodic probability measure with nonzero Lyapunov exponents in the sense of ssec:generalremarksaboutlyapunovexponents.",
"Definition (Uniform hyperbolicity) We say that $m$ is uniformly hyperbolic if there exists a constant $C>0$ such that for $m$ -a.e.",
"$x$ , we have for $N\\geqslant 1$ that $ \\log \\left\\Vert D_xT^{-N}_Z\\vert _{W^u(x)}\\right\\Vert & \\leqslant -\\frac{1}{C}N + C\\\\ \\log \\left\\Vert D_xT^N_Z\\vert _{W^s(x)}\\right\\Vert & \\leqslant -\\frac{1}{C}N + C$ and the angle between $W^u(x)$ and $W^s(x)$ is bounded below by $\\frac{1}{C}$ , where everything is measured relative to a fixed smooth Riemannian metric.",
"In [6] sets $\\Lambda ^\\ell _{\\lambda \\mu \\varepsilon j}$ are defined to which most considerations in loc.cit.",
"apply.",
"A $T_Z$ -invariant set of points $x$ which satisfy def:uniformexpansion is then contained in such a $\\Lambda ^\\ell _{\\lambda \\mu \\varepsilon j}$ with $j=1$ , $\\varepsilon =0$ and $\\ell ,\\lambda ,\\mu $ only depending on $C$ ."
],
[
"Stable and Unstable manifolds",
"The following discussion is expanded in [6].",
"Because the invariant measure $m$ has nonzero Lyapunov exponents, there exist for $m$ -a.e.",
"$x$ unique global immersed stable manifolds ${W}^s(x)$ which contain $x$ and with tangent space equal to $W^s(x)$ there.",
"For distinct points, stable manifolds either coincide or are disjoint.",
"Since in our case the stable manifolds are complex 1-dimensional they are parametrized by $\\mathbb {C}$ [5].",
"For convenience of notation, we will use the canonical parametrizations given by the complex line $W^u(x)$ , namely the holomorphic maps $\\xi ^s_x\\colon W^s(x) \\tilde{\\longrightarrow }{W}^s(x)$ normalized to have derivative the identity at the basepoints (where $0\\mapsto x$ ).",
"Because the only holomorphic automorphisms of $\\mathbb {C}$ are affine, we immediately deduce that For two points $x,y$ on the same stable manifold the composed map $(\\xi ^s_y)^{-1}\\circ \\xi ^s_x$ is an affine map from $W^s(x)$ to $W^s(y)$ .",
"The maps $\\xi ^s_{x}$ and $T^{-1}_Z\\circ \\xi _{T_Z(x)}^s \\circ D_{x}T_Z $ coincide, because they both induce parametrizations of the stable manifolds at $x$ and have the same derivative at the origin.",
"Equivalently $T_Z\\circ \\xi ^s_{x} = \\xi _{T_Z(x)}^s\\circ D_xT_Z$ .",
"Therefore the stable manifolds carry canonical affine structures and are parametrized equivariantly for the dynamics.",
"The same discussion applies to unstable manifolds when using $T_Z^{-1}$ instead of $T_Z$ , and we will denote their parametrizations by $\\xi ^u_x$ ."
],
[
"Charts and size",
"To discuss the geometry of stable manifolds, fix finitely many open charts covering $Z=\\cup _\\alpha U_\\alpha $ and view each chart as equipped with its flat Euclidean metric (in the orbifold sense when necessary).",
"There exists an $\\varepsilon >0$ which is a Lebesgue number of this covering, i.e.",
"for any $x\\in Z$ there exists a chart $U_\\alpha $ such that the ball of radius $\\varepsilon $ around $Z$ is contained in $U_\\alpha $ .",
"All considerations below will be in these charts and all objects will be considered only in balls of radius at most $\\varepsilon /2$ , so that the Euclidean and fixed background metric are comparable, up to uniform constants.",
"Implicit constants occurring below will be called uniform if they only depend on the automorphism $T_Z$ , the covering fixed above, and a fixed smooth ambient metric."
],
[
"Local stable manifolds",
"Recall that the global stable manifolds ${W}^s(x)$ are only immersed and are patched from local stable manifolds ${W}^s_{loc}(x)$ .",
"The geometry of the local stable manifolds is described in [6].",
"Most importantly, under the uniform hyperbolicity assumptions in def:uniformexpansion the constants (that appear in loc.cit.)",
"$r$ giving the radius, and $D$ giving the Hölder constant of the derivative, depend only on the constant describing the uniform hyperbolicity (by Thm.",
"7.5.1(5) of loc.cit.).",
"As a consequence, using the charts from sssec:chartsandsize to map a ball of radius $r$ in $W^{s/u}(x)$ to $Z$ , the stable resp.",
"unstable manifolds will be contained in disjoint cones around $W^{s}(x)$ resp.",
"$W^u(x)$ , with angle between the cones uniformly bounded below."
],
[
"Ergodicity in the uniformly hyperbolic case",
"Theorem 4.2.1 Suppose that the measure of maximal entropy $\\mu $ on $X$ is in the Lebesgue class and additionally under the contraction $\\nu \\colon X\\rightarrow Y$ from ssec:preliminariesoncurrentsandorbifolds, the measure $\\nu _*\\mu $ is uniformly hyperbolic on $Y$ in the sense of def:uniformexpansion.",
"Then $\\mu =\\operatorname{dVol}$ on $X$ , and so $(X,T)$ is a Kummer example by main2.",
"The statements $\\mu =\\operatorname{dVol}_X$ and $\\nu _*\\mu = \\operatorname{dVol}_Y$ are equivalent, since the set contracted by $\\nu $ has Lebesgue measure zero.",
"However, even when $(X,T)$ is a Kummer example, $\\operatorname{dVol}_X$ is not uniformly hyperbolic for a smooth metric on $X$ .",
"Let $S\\subset X$ be the set defined in eqn:muSdVol, so that $\\mu =\\frac{\\operatorname{dVol}}{|S|}\\bigg |_S$ .",
"All arguments below are on $Y$ so for simplicity of notation let $S$ denote the image of this set under $\\nu $ .",
"Assume that the uniform hyperbolicity condition holds for every point of $S$ (otherwise replace $S$ with the intersection of all iterates of the set on which uniform hyperbolicity holds, still a set of full $\\mu $ -measure).",
"It suffices to show that there exists a uniform $\\varepsilon >0$ such that for Lebesgue-a.e.",
"$x\\in S$ the ball $B(x,\\varepsilon )$ of radius $\\varepsilon $ around $x$ satisfies $|B(x,\\varepsilon )\\cap S|=|B(x,\\varepsilon )|$ , i.e.",
"the ball is essentially (up to Lebesgue measure 0) contained in $S$ (we'll use $|B|$ to denote the volume of $B$ for the canonical volume form).",
"This last property, in turn, follows from the Hopf argument and the uniformity of hyperbolicity.",
"Specifically, from Pesin theory there exist stable and unstable manifolds ${W}^s(x),{W}^u(x)$ through $\\mu $ -a.e.",
"point $x$ .",
"From our uniform hyperbolicity assumption, the sizes of the local stable/unstable manifolds in the sense of [6] are uniformly bounded below for every $x\\in S$ , and so are their angles (this is [6]).",
"Indeed there exists a single $\\ell >0$ such that the Oseledets-regular level sets $\\Lambda ^\\ell _{\\bullet }$ that appear in [6] are equal to $S$ .",
"Furthermore, the local stable and unstable manifolds depend continuously in the $C^1$ -topology for points in $S$ , by [6] (even better, Hölder continuity holds along the lines in [1]).",
"Now we apply the Hopf argument.",
"From [6], there exists a set $B_S$ of zero Lebesgue measure such that if $y\\in S$ and $z,w\\in {W}^u(y)\\setminus B_S$ then $z,w\\in S$ .",
"Moreover, by the absolute continuity of the unstable foliations [6], for $\\mu $ -a.e.",
"$y\\in S$ the set $B_S\\cap {W}^{u}(y)$ has zero Lebesgue measure in ${W}^u(y)$ .",
"Let $S^{\\prime }\\subset S$ be the set of $y$ with this property, so that $|S\\setminus S^{\\prime }|=0$ .",
"For $y\\in S^{\\prime }$ we have that Lebesgue-a.e.",
"$z\\in {W}^u(y)$ is in $S$ , hence admits stable manifolds ${W}^s(z)$ of size bounded below and which depend continuously on $z$ .",
"Hence we get a continuous injective map $\\Delta \\times \\Delta \\rightarrow Y$ , where $\\Delta =\\left\\lbrace |z|<1 \\colon z\\in \\mathbb {C}\\right\\rbrace $ , as follows.",
"The map is defined by $(z,w)\\mapsto \\xi ^s_z(w)$ where the first factor of $\\Delta \\times \\Delta $ is identified with a disc of size uniformly bounded below in ${W}^u(y)$ , and the second factor with a disc in $W^s(z)$ as $z$ varies in ${W}^u(y)$ (the disc $\\Delta $ of radius 1 is rescaled from the bounded below radius $r$ discussed in sssec:localstablemanifolds).",
"While the map is defined only for Lebesgue-a.e.",
"$z$ , since it is continuous as a map $z\\mapsto Hol(\\Delta ,Y)$ (by [6]), it extends to the desired continuous map $\\Delta \\times \\Delta \\rightarrow Y$ .",
"By the Invariance of Domain Theorem ([28]), this continuous injective map contains an open set around $y\\in Y$ .",
"It remains to check that this open set contains a ball of size uniformly bounded below.",
"For this, it suffices to check that there exists a uniform $\\varepsilon >0$ such that if $(z,w)\\in \\Delta \\times \\Delta $ , $y^{\\prime }=\\xi _z^s(w)$ and $|y-y^{\\prime }|\\leqslant \\varepsilon $ then $|z|+|w|\\leqslant \\frac{1}{10}$ .",
"Provided this last property, it is clear that every point within $\\varepsilon $ of $y$ is in the image, since the set in question is both open (by invariance of domain) and closed by the property that remains to be checked.",
"Suppose therefore that $y^{\\prime }=\\xi _z^s(w)$ .",
"Recall that in Euclidean triangles, by the law of sines if $a$ is the length of one side and $\\alpha $ is the opposite angle, then $\\frac{a}{\\sin \\alpha }$ controls any other side.",
"In the charts described in sssec:chartsandsize, the stable and unstable manifolds at $z$ are contained in cones around $W^s(z)$ and $W^u(z)$ with angle between them uniformly bounded below (see sssec:localstablemanifolds).",
"Therefore there exists a uniform constant $A$ such that $A|y-y^{\\prime }|\\geqslant |z|+|w|$ , which suffices for our purposes."
],
[
"The general case",
"We complete the proof of the main theorem by establishing Step 3 from the introduction (ssec:proofoutline)."
],
[
"Proving uniform hyperbolicity",
"Our goal is the following, which thanks to main2 (Step 1) and thm:uniformlyhyperboliccase (Step 2) completes the proof of main: Theorem 5.1.1 Suppose that the measure of maximal entropy $\\mu $ on $X$ is in the Lebesgue class.",
"Then under the contraction $\\nu \\colon X\\rightarrow Y$ , the measure $\\nu _*\\mu $ is uniformly hyperbolic on $Y$ .",
"Because sets contracted under $\\nu \\colon X\\rightarrow Y$ have zero Lebesgue measure, $\\mu $ -measurable functions on $X$ are naturally identified with $\\nu _*\\mu $ -measurable ones on $Y$ .",
"We will not distinguish in notation between functions identified in this manner."
],
[
"Expansion/Contraction factors",
"Recall (see sssec:stableunstabledirections and sssec:absolutecontinuity) that $\\nu _*\\mu $ -a.e.",
"$x\\in Y$ has a decomposition of the tangent space: $\\mathbf {T}_x Y = W^s(x) \\oplus W^u(x)$ which is $T_Y$ -invariant.",
"i.e.",
"$DT_Y(W^{u/s}(x))= W^{u/s}(T_Yx)$ .",
"Fix now the Calabi–Yau metric $\\omega _Y$ on $Y$ (see sssec:orbifoldricciflatmetrics) and define the $\\nu _*\\mu $ -measurable functions: $\\rho ^u(x):= \\log \\left\\Vert DT_Y\\vert _{W^u(x)}\\right\\Vert _{\\omega _Y}, \\quad \\rho ^s(x):= \\log \\left\\Vert DT_Y\\vert _{W^s(x)}\\right\\Vert _{\\omega _Y}$ where the norm of the operator is for the metrics $\\omega _Y(x)$ and $\\omega _Y(T_Yx)$ on the source and target tangent spaces.",
"The functions are $\\nu _*\\mu $ -measurable since the spaces $W^{u/s}$ depend measurably on $x$ ."
],
[
"Computation in a basis",
"Let now $e_1\\in W^u(x), e_2\\in W^s(x)$ be a unimodular basis, i.e.",
"using the dual basis $\\sqrt{-1}e_1^\\vee \\wedge \\overline{e_1}^\\vee \\wedge \\sqrt{-1}e_2^\\vee \\wedge \\overline{e_2}^\\vee $ is the fixed volume form $\\operatorname{dVol}_Y$ .",
"There are two metrics on $\\mathbf {T}_x Y$ , one is $\\omega _Y(x)$ and the other one is $\\omega _{Y,Nh}(x)$ , which is the pull-back by $T_Y^N$ of $\\omega _Y(T_Y^Nx)$ .",
"Suppose that in the fixed basis $\\lbrace e_1,e_2\\rbrace $ the metrics are represented by $\\omega _Y(x) =\\begin{bmatrix}a_0 & b_0 \\\\\\overline{b_0} & d_0 \\\\\\end{bmatrix},\\quad \\omega _{Y,Nh}(x) =\\begin{bmatrix}a_{Nh} & b_{Nh} \\\\\\overline{b_{Nh}} & d_{Nh} \\\\\\end{bmatrix}$ and these matrices have determinant equal to 1.",
"The relation to the functions defined in eqn:rhousdefinition is: $\\rho ^u(x) = \\frac{1}{2} \\log \\left( \\frac{a_h}{a_0} \\right),\\quad \\rho ^s(x) = \\frac{1}{2} \\log \\left( \\frac{d_h}{d_0} \\right)$"
],
[
"Defining $\\beta $",
"Consider now the quantity $\\beta (x):= \\frac{1}{2} \\log (a_0d_0)$ where $a_0,d_0$ are the entries in eqn:omega0Nhexplicit.",
"Note that $\\beta (x)\\geqslant 0$ and it is independent of the earlier choice of unimodular basis.",
"Indeed $\\log (a_0 d_0)$ can be expressed geometrically as follows.",
"The decomposition $\\mathbf {T}_xY = W^s(x)\\oplus W^u(x)$ and the metric $\\omega _Y$ determine another metric $\\widetilde{\\omega _Y}$ defined as the restriction of $\\omega _Y$ to $W^s$ and $W^u$ and declaring $W^s$ and $W^u$ to be orthogonal.",
"The log of the ratio of volume forms determined by $\\omega _Y$ and $\\widetilde{\\omega _Y}$ is exactly $2\\beta (x)=\\log (a_0 d_0)$ .",
"In particular $\\beta $ is also a measurable function, since it is defined using standard constructions on measurable objects.",
"Finally, we have: $T^*_Y\\beta - \\beta = \\rho ^s + \\rho ^u$ which says that $\\rho ^s+\\rho ^u$ is a coboundary (in the dynamical sense).",
"Indeed, in explicit coordinates as above: $\\rho ^s(x) + \\rho ^u(x) = \\frac{1}{2}\\log (a_h d_h) - \\frac{1}{2}\\log (a_0 d_0)$ and the formula follows."
],
[
"The cohomological equation",
"In this subsection we show that the expansion/contraction factors $\\rho ^u,\\rho ^s$ are cohomologous to constants via certain dynamical coboundary measurable functions $\\alpha _s,\\alpha _u$ .",
"Crucially, $\\alpha _s,\\alpha _u$ will be shown to be bounded in bounded."
],
[
"Conditional measures",
"To compare the restriction of the currents or Kähler metrics to stable/unstable manifolds, it is useful to go through the intermediate notion of conditional measure.",
"Specifically, there exists by [6] a partition $\\zeta ^s$ of $X$ , with atom containing $x$ denoted $\\zeta ^s(x)$ , with the following properties.",
"The partition is measurable, i.e.",
"the atoms can be identified with the fibers of a measurable map $X\\rightarrow X^{\\prime }$ to a standard measure space, and for $\\nu _*\\mu $ -almost every $x\\in X$ the atom $\\zeta ^s(x)$ is an open set in ${W}^s(x)$ ; such partitions are called subordinate to the stable foliation.",
"Additionally the partition is contracted by the dynamics, i.e.",
"$T_Y(\\zeta ^s(x))\\subset \\zeta ^s(T_Y(x))$ .",
"An analogous unstable partition $\\zeta ^u$ exists, but which is expanded by the dynamics.",
"Any measure on $X$ induces conditional (probability) measures on the atoms of $\\zeta ^{s/u}$ .",
"By [32] because $\\mu $ is in the Lebesgue class, the conditional measures are also in the Lebesgue class.",
"Proposition (Cohomologous to a constant) In the setting of main, there exist $\\nu _*\\mu $ -measurable functions $\\alpha _s,\\alpha _u$ such that $\\nu _*\\eta _+\\vert _{{W}^u(x)} &= e^{-2\\alpha _u}\\omega _Y\\vert _{{W}^u(x)}\\\\\\nu _*\\eta _-\\vert _{{W}^s(x)} &= e^{-2\\alpha _s}\\omega _Y\\vert _{{W}^s(x)}$ for $\\nu _*\\mu $ -a.e.",
"$x\\in Y$ , where ${W}^{s/u}$ denote the stable/unstable manifolds.",
"Moreover the functions satisfy: $\\rho ^u = T_Y^*\\alpha _u - \\alpha _u + \\frac{h}{2},\\quad \\rho ^s = T_Y^*\\alpha _s - \\alpha _s - \\frac{h}{2}$ To check the existence of functions satisfying 1, 2, it suffices to show that for $\\nu _*\\mu $ -a.e.",
"$x$ the measure $\\nu _*\\eta _+\\vert _{{W}^u(x)}$ is in the Lebesgue class, since then it must have a Radon-Nikodym derivative $e^{-2\\alpha _u}$ relative to the measure $\\omega _Y\\vert _{{W}^u(x)}$ .",
"But by [21] the measures $\\nu _*\\eta _+\\vert _{{W}^u(x)}$ are in the same measure class as the conditional measures of $\\mu $ along the unstable foliation (regardless of any assumptions on $\\mu $ ).",
"This property on $X$ pushes down to $Y$ .",
"By the discussion in sssec:conditionalmeasures, the conditional measures are in the Lebesgue class on the atoms of their partitions $\\nu _*\\zeta ^u$ .",
"The existence of $\\alpha _u,\\alpha _s$ follows in the interior of the atoms of the partition.",
"Note finally that is $\\zeta ^u$ is a partition subordinated to the unstable foliation, then so is $T^i(\\zeta ^u)$ for any $i\\geqslant 0$ .",
"Since $\\mu $ -a.e.",
"point has positive Lyapunov exponent, $\\mu $ -a.e.",
"point will be in the interior of some atom of $T^i(\\zeta ^u)$ along the unstable foliation for some $i\\geqslant 0$ .",
"The relations in eqn:rhocohomologoustoconstant follow from a computation: $e^{2\\rho ^u-2T_Y^*\\alpha _u}\\omega _Y\\vert _{{W}^u(x)}=e^{-2T_Y^*\\alpha _u}\\omega _{Y,h}\\vert _{{W}^u(x)}=\\\\= \\nu _*T^*\\eta _+\\vert _{{W}^u(x)}=e^h\\nu _*\\eta _+\\vert _{{W}^u(x)} = e^{h-2\\alpha _u}\\omega _Y\\vert _{{W}^u(x)}$ and similarly for $\\rho ^s$ .",
"Note also that combining eqn:thesumisacoboundary and prop:cohomologoustoaconstant we see that $T_Y^*(\\alpha _u+\\alpha _s-\\beta )-(\\alpha _u+\\alpha _s-\\beta )=\\rho ^u+\\rho ^s-(\\rho ^u+\\rho ^s)=0,$ and so by ergodicity of $\\mu $ : $\\alpha _u+\\alpha _s=\\beta +\\delta ,$ $\\mu $ -a.e., for some constant $\\delta \\in \\mathbb {R}$ .",
"The next proposition is analogous to [16], where it is proved by a different argument involving renormalization along the stable manifolds.",
"Proposition (The restricted current is flat) For $\\nu _*\\mu $ -a.e.",
"$x$ we have that $(\\xi _x^u)^*\\nu _*\\eta _+ = e^{-2\\alpha _u(x)}\\omega _Y(x)\\big \\vert _{W^u(x)}$ i.e.",
"the pulled back current to the unstable tangent space is a flat metric.",
"Consider the function $f(x,r)$ defined for $\\nu _*\\mu $ -a.e.",
"$x$ and $r>0$ : $f(x,r):= \\frac{\\int _{W^u(x,r)}(\\xi _x^u)^*\\nu _*\\eta _+}{\\pi r^2}$ where $W^u(x,r)$ denotes the ball of radius $r$ for the metric $e^{-2\\alpha _u(x)}\\omega _Y(x)$ on $W^u(x)$ .",
"Recall also that we normalized the derivative of $\\xi _x^{u}$ at 0 to be the identity.",
"Therefore, by the Lebesgue density theorem combined with 1, we have that $\\lim _{r\\rightarrow 0}f(x,r)=1$ for $\\nu _*\\mu $ -a.e.",
"$x$ .",
"Applying the automorphism and using the coboundary property, we have that $f(T_Yx,e^{h}r) = f(x,r) \\text{ or equivalently } f(T_Y^{-1}x,e^{-h}r)= f(x,r).$ Now for any $\\varepsilon >0$ there exists $r_\\varepsilon >0$ such that the set $Y_\\varepsilon :=\\left\\lbrace x\\colon |1-f(x,r)| \\leqslant \\varepsilon , \\forall r\\in (0,r_\\varepsilon )\\right\\rbrace $ has positive Lebesgue measure.",
"By ergodicity of $T_Y$ , $\\nu _*\\mu $ -a.e.",
"$x$ visits $Y_\\varepsilon $ for arbitrarily large negative times, so combined with the above equation it follows that $f(x,r) = 1$ for $\\nu _*\\mu $ -a.e.",
"$x$ and any $r>0$ .",
"Take now a point $y\\in {W}^u(x)$ with $v_y\\in W^u(x)$ such that $y=\\xi _x^u(v_y)$ .",
"We know that the composition $(\\xi _y^{u})^{-1}\\circ \\xi _x^u$ is an affine map and let $C_{x,y}$ be the derivative of this map for the metrics $e^{-2\\alpha _u(p)}\\omega _Y(p)$ with $p=x,y$ .",
"It suffices to show that $C_{x,y}=1$ , since then it follows that ${W}^u(p)$ carries a canonical flat metric independent of $p$ , and the Radon-Nikodym derivative of $(\\nu _*\\eta _+)\\vert _{{W}^u(p)}$ relative to this metric is identically 1 (because $f(x,r)=1$ ).",
"By symmetry of the next argument in the two variables, $C_{x,y}\\leqslant 1$ also suffices.",
"Since $f(y,r)=1$ for $y$ in a set of full Lebesgue measure on ${W}^u(x)$ (and we pick $y$ in this set) it follows that $\\frac{\\int _{W^u(y,r)}(\\xi _y^u)^*\\nu _*\\eta _+}{\\pi r^2} = 1 \\text{ for all }r>0.$ Transporting this identity back to $W^u(x)$ using $(\\xi _x^{u})^{-1}\\circ \\xi ^u_{y}$ and using that this map takes a ball of radius $r$ in $W^u(y)$ to a ball of radius $\\frac{r}{C_{xy}}$ in $W^u(x)$ , it follows that: $\\frac{\\int _{W^u(v_y,r)}(\\xi _x^u)^*\\nu _*\\eta _+}{\\pi r^2} = C^2_{xy}$ where $W^u(v_y,r)$ denotes the ball of radius $r$ at $v_y$ in the metric $e^{-2\\alpha _u(x)}\\omega _Y(x)\\vert _{W^u(x)}$ .",
"But $W^u(v_y,r)\\subset W^u(x,r+|v_y|))$ and $\\eta _+\\geqslant 0$ so we have: $1 = \\frac{\\int _{W^u(x,r+|v_y|)}(\\xi _x^u)^*\\nu _*\\eta _+}{\\pi (r+|v_y|)^2} \\geqslant \\frac{\\int _{W^u(v_y,r)}(\\xi _x^u)^*\\nu _*\\eta _+}{\\pi (r+|v_y|^2)} = C^2_{xy} \\frac{r^2}{(r+|v_y|)^2}$ Letting $r\\rightarrow \\infty $ the desired conclusion follows."
],
[
"Boundedness of the dynamical coboundaries",
"The next step, which was hinted to earlier, consists in showing that the dynamical coboundary functions $\\alpha _u$ and $\\alpha _s$ are in fact bounded: Theorem 5.3.1 The coboundary functions $\\alpha _u,\\alpha _s$ belong to $L^\\infty (\\nu _*\\mu )$ .",
"Remark Before the proof of bounded, note that for $\\mu $ -a.e.",
"$x$ we have that $(\\xi ^u_x)^*\\nu _*\\eta _-=0$ by say [13], or a combination of [21] in the non-projective case.",
"We will restrict to such $x$ and their image in $Y$ and will eventually construct an entire curve on $Y$ for which the pullback of both $\\nu _*\\eta _+$ and $\\nu _*\\eta _-$ vanish.",
"This idea is also a key point in the work of Cantat–Dupont [16].",
"Remark Recall that the map $\\nu \\colon X\\rightarrow Y$ contracts the analytic subset $V\\subset X$ to the singular points of $Y$ , and that by prop:kahler we can write $[\\eta _+]+[\\eta _-]=\\nu ^*[\\omega _Y]$ for a Ricci-flat orbifold Kähler metric $\\omega _Y$ on $Y$ .",
"For later use, the following observation will be useful: on $Y$ we can write $\\nu _*(\\eta _++\\eta _-)=\\omega _Y+i\\partial \\overline{\\partial }\\varphi ,$ where $\\varphi \\in C^0(Y)$ .",
"To see this, we fix smooth representatives $\\alpha _{\\pm }$ of $[\\eta _{\\pm }]$ on $X$ , and then we can then write $\\eta _{\\pm }=\\alpha _{\\pm }+i\\partial \\overline{\\partial }\\varphi _{\\pm }$ where $\\varphi _{\\pm }$ are Hölder continuous functions on $X$ , as recalled in sssec:theeigencurents.",
"On the other hand we can also write $\\nu ^*\\omega _Y=\\alpha _++\\alpha _-+i\\partial \\overline{\\partial }u$ for some continuous function $u$ on $X$ .",
"It follows that $\\eta _++\\eta _-=\\nu ^*\\omega _Y+i\\partial \\overline{\\partial }(\\varphi _++\\varphi _--u),$ and restricting this to any irreducible component $C$ of $V$ (which as we know is contracted to a point by $\\nu $ ) we have $0\\leqslant (\\eta _++\\eta _-)|_C=i\\partial \\overline{\\partial }(\\varphi _++\\varphi _--u)|_C,$ so $\\varphi _++\\varphi _--u$ is a plurisubharmonic function on the compact curve $C$ , hence constant.",
"It follows that we have $\\varphi _++\\varphi _--u=\\nu ^*\\varphi $ for some continuous function $\\varphi $ on $Y$ such that $\\omega _Y+i\\partial \\overline{\\partial }\\varphi $ is a closed positive current on $Y$ which satisfies $\\nu ^*(\\omega _Y+i\\partial \\overline{\\partial }\\varphi )=\\eta _++\\eta _-$ , and (REF ) follows from this.",
"By sumbeta together with the fact that $\\beta \\geqslant 0$ , it suffices to show that $\\alpha _u\\leqslant C, \\alpha _s\\leqslant C$ on a $\\nu _*\\mu $ -full measure set.",
"We give the argument for $\\alpha _u$ , the one for $\\alpha _s$ being identical.",
"Suppose that $\\alpha _u$ is not bounded on the full $\\nu _*\\mu $ -measure set of points which are not orbifold singularities, for which unstable manifolds exist, and on which prop:therestrictedcurrentisflat, prop:cohomologoustoaconstant, and rmk:vanishingpullback hold.",
"So there is a sequence of such points $x_i$ with $\\alpha _u(x_i)\\rightarrow +\\infty $ .",
"Therefore the unstable parametrizations $\\xi ^u_{x_i}\\colon W^u(x_i)\\rightarrow Y$ satisfy $\\xi ^u_{x_i}(0)=x_i, D_0\\xi _x^u=\\mathbf {1}$ and $(\\xi _{x_i}^u)^*\\nu _*\\eta _+ = e^{-2\\alpha _u(x_i)}\\omega _Y(x_i)\\vert _{W^u(x_i)}, $ which goes to zero.",
"Fix a sequence $R_i\\rightarrow \\infty $ .",
"Suppose first that there is $C$ such that $\\sup _i\\sup _{D_{R_i}(0)}|D\\xi ^u_{x_i}|\\leqslant C,$ relative to $\\omega _Y$ on $Y$ and the flat metric $\\omega _Y(x_i)$ on $W^u(x_i)$ .",
"Then by Ascoli–Arzelà up to passing to a subsequence, the maps $\\xi _{x_i}^u|_{D_{R_i}(0)}$ converge locally uniformly to a nonconstant entire curve $\\xi :\\mathbb {C}\\rightarrow Y$ .",
"By construction $\\alpha _u(x_i)\\rightarrow +\\infty $ , the current $\\nu _*(\\eta _++\\eta _-)$ has locally continuous potentials (see potentials), and the convergence $\\xi _i\\rightarrow \\xi $ is locally uniform, we can exchange limits to conclude that $\\xi ^*(\\nu _*(\\eta _++\\eta _-))=0$ (using rmk:vanishingpullback).",
"If there is no such $C$ , we can find points $z_i\\in \\overline{D_{R_i}(0)}$ such that $|D\\xi ^u_{x_i}|(z_i)\\rightarrow \\infty $ (up to subsequence).",
"Working in $D_1(z_i),$ we apply the standard Brody reparametrization argument.",
"Namely for each $i$ we pick a point $y_i\\in D_1(z_i)$ which maximizes $\\delta _i(z)=\\operatorname{{dist}}(z,\\partial D_1(z_i))|D\\xi ^u_{x_i}|(z).$ Call $r_i=\\operatorname{{dist}}(y_i,\\partial D_1(z_i)), a_i=|D\\xi ^u_{x_i}|(y_i),$ so that $a_i\\geqslant a_ir_i\\geqslant \\delta _i(z_i)=|D\\xi ^u_{x_i}|(z_i)\\rightarrow \\infty ,$ while for all $z\\in D_{r_i/2}(y_i)$ we have $a_ir_i\\geqslant \\operatorname{{dist}}(z,\\partial D_1(z_i))|D\\xi ^u_{x_i}|(z)\\geqslant \\frac{r_i}{2}|D\\xi ^u_{x_i}|(z),$ hence $|D\\xi ^u_{x_i}|\\leqslant 2a_i$ on $D_{r_i/2}(y_i)$ .",
"Let now $\\tilde{\\xi }_i\\colon D_{a_ir_i/2}(0)\\rightarrow Y,\\quad \\tilde{\\xi }_i(z)=\\xi ^u_{x_i}\\left(y_i+\\frac{z}{a_i}\\right), $ which are defined on bigger and bigger discs and satisfy $\\sup _{D_{a_ir_i/2}(0)}|D\\tilde{\\xi }_i|\\leqslant 2,\\quad |D\\tilde{\\xi }_i|(0)=1,$ $\\tilde{\\xi }_i^* \\nu _*(\\eta _+) = \\frac{e^{-2\\alpha _u(x_i)}}{a_i^2}\\omega _Y(x_i)\\vert _{W^u(x_i)},$ which goes to zero.",
"Again by Ascoli–Arzelà up to passing to a subsequence, the maps $\\tilde{\\xi }_i$ converge locally uniformly to a nonconstant entire curve $\\xi \\colon \\mathbb {C}\\rightarrow Y$ .",
"Since $a_i\\rightarrow +\\infty $ , reasoning as in the case above we again conclude that $\\xi ^*(\\nu _*(\\eta _++\\eta _-))=0$ .",
"In both cases, the existence of such an entire curve $\\xi $ is a contradiction to entire below.",
"In the proof above we used the following proposition, which is due to Dinh–Sibony [20] in general (see [16]).",
"In the $K3$ case we can give a very simple proof: Proposition There is no nonconstant entire curve $\\xi :\\mathbb {C}\\rightarrow Y$ such that $\\xi ^*(\\nu _*(\\eta _++\\eta _-))=0.$ Recall from potentials that on $Y$ we can write $\\nu _*(\\eta _++\\eta _-)=\\omega _Y+i\\partial \\overline{\\partial }\\varphi ,$ where $\\omega _Y$ is a Ricci-flat orbifold Kähler metric on $Y$ and $\\varphi \\in C^0(Y)$ .",
"For every $r>0$ choose $\\chi _r$ a nonnegative radial cutoff function which equals 1 on $D_r$ , is supported in $D_{2r}$ , and such that $|i\\partial \\overline{\\partial }\\chi _r|\\leqslant \\frac{C}{r^2},$ for a constant $C$ independent of $r$ .",
"This can be done by letting $\\chi _r(z)=\\eta \\left(\\frac{|z|}{r}\\right),$ where $\\eta $ is a nonnegative cutoff function on $\\mathbb {R}_{\\geqslant 0}$ which equals 1 on $[0,1]$ and vanishes on $[2,\\infty )$ .",
"We then compute $i\\partial \\overline{\\partial }\\chi _r=i\\partial \\left(\\eta ^{\\prime }\\cdot \\frac{z}{2|z|r}d\\overline{z}\\right)=\\left(\\frac{\\eta ^{\\prime \\prime }}{4r^2}+\\frac{\\eta ^{\\prime }}{4r|z|}\\right)idz\\wedge d\\overline{z},$ which is nonzero only for $r\\leqslant |z|\\leqslant 2r$ and so satisfies $|i\\partial \\overline{\\partial }\\chi _r|\\leqslant \\frac{C}{r^2}$ everywhere.",
"Then for all $r>0$ we have $0=\\int _{D_{2r}}\\chi _r\\xi ^*(\\nu _*(\\eta _++\\eta _-))=\\int _{D_{2r}}\\chi _r\\xi ^*(\\omega _Y+i\\partial \\overline{\\partial }\\varphi ),$ and so the area of $\\xi (D_r)$ is bounded above by $\\int _{D_{2r}}\\chi _r\\xi ^*\\omega _Y=\\int _{D_{2r}}\\chi _r \\xi ^*(-i\\partial \\overline{\\partial }\\varphi )=\\int _{D_{2r}}(-\\varphi \\circ \\xi )i\\partial \\overline{\\partial }\\chi _r\\leqslant \\frac{Cr^2}{r^2}\\leqslant C,$ and so the entire curve $\\xi $ has finite area, hence it extends to a holomorphic map $\\xi \\colon \\mathbb {P}^1\\rightarrow Y$ , see Moncet [36].",
"If $\\xi $ is nonconstant, then by construction the rational curve $C=\\xi (\\mathbb {P}^1)$ satisfies $C\\cdot ([\\nu _*(\\eta _++\\eta _-)])=0$ , which is impossible since $[\\nu _*(\\eta _++\\eta _-)]=[\\omega _Y]$ is Kähler."
],
[
"Completion of the proof of the main results",
"In this subsection we complete the proof of main3 (and therefore also of main), and we also derive coro.",
"From bounded we know that the coboundaries $\\alpha _u,\\alpha _s$ belong to $L^\\infty (\\nu _*\\mu )$ .",
"Since $\\rho ^s(x)=\\log \\Vert DT_Y|_{W^s(x)}\\Vert _{\\omega _Y}$ , it follows that for all $N\\geqslant 1$ we have $\\begin{split}\\log \\left\\Vert D_xT^{N}_Y\\vert _{W^s(x)}\\right\\Vert _{\\omega _Y}&=\\rho ^s(x)+\\cdots +\\rho ^s(T_Y^{N-1}x)\\\\&={\\alpha _s(T_Y^{N}x)-\\alpha _s(x)}-\\frac{Nh}{2}\\\\&\\leqslant -\\frac{Nh}{2}+C,\\end{split}$ using the coboundary relation (REF ) and the $L^\\infty $ bound for $\\alpha _s$ .",
"This proves (), and a similar argument shows (REF ).",
"The $L^\\infty $ bound for $\\alpha _u,\\alpha _s$ together with $\\beta =\\alpha _u+\\alpha _s-\\delta $ (by sumbeta) show that $\\beta $ is uniformly bounded.",
"From eqn:omega0Nhexplicit and the definition of $\\beta $ in eqn:definitionbeta, a uniform upper bound on $\\beta $ gives a uniform lower bound on the angle between $W^u$ and $W^s$ , measured relative to $\\omega _Y$ .",
"Lastly, we prove coro: Cantat has proved [12] that if a compact complex surface $X$ admits an automorphism $T$ with positive topological entropy then $X$ is either a torus, K3, Enriques, a blowup of these, or rational.",
"These are all projective, except for non-projective tori and $K3$ and their blowups, so thanks to Cantat-Dupont [16] (which assume projectivity) we may assume that $X$ is a torus, a $K3$ surface or a blowup of these.",
"First, if $X$ is a torus then $T$ is induced by an affine transformation of $\\mathbb {C}^2$ , and therefore $(X,T)$ is trivially a Kummer example.",
"Second, if $X$ is $K3$ then the result follows from our main main.",
"Lastly, if $X$ is a blowup of a torus or $K3$ surface, say $\\pi \\colon X\\rightarrow Y$ is the sequence of blowups, then $T$ induces a bimeromorphic map $T_Y$ of $Y$ , which must be a biholomorphism (see e.g.",
"[3]).",
"Furthermore the topological entropy of $T_Y$ equals the one of $T$ by the same argument as in [30].",
"From the relation $T_Y\\circ \\pi =\\pi \\circ T$ we deduce that the eigencurrents $\\eta _\\pm $ for $T$ on $X$ are equal to the pullbacks of the corresponding eigencurrents for $T_Y$ on $Y$ , hence the measures of maximal entropy satisfy $\\mu =\\pi ^*\\mu _Y$ .",
"Since by assumption $\\mu $ is absolutely continuous with respect to the Lebesgue measure, we conclude that $\\mu _Y$ is also absolutely continuous with respect to the Lebesgue measure.",
"By the previous cases, we see that $(Y,T_Y)$ is a Kummer example, and hence by definition so is $(X,T)$ ."
],
[
"Alternative arguments for the existence of the coboundaries",
"In this section we give an alternative argument for the existence of the expansion/contraction coboundaries in prop:cohomologoustoaconstant.",
"Instead of using the theories of Pesin and Ledrappier–Young and the result by De Thélin–Dinh, we rather exploit our specific geometric setup, and in particular the hyperbolic geometry of the space of hermitian metrics on a tangent space with given volume form.",
"The coboundaries thus constructed have better integrability properties than those given in prop:cohomologoustoaconstant, which are just measurable.",
"We hope that the ideas below may prove useful in related problems.",
"To simplify notation, in this section we work on $X$ as if it was the orbifold $Y$ , so that $[\\eta _+]+[\\eta _-]$ is a Kähler class and we will write $\\omega _{Nh}$ for the Ricci-flat metrics on $X$ that play the role of $\\omega _{Y,Nh}$ , so that $\\omega _0$ replaces $\\omega _Y$ .",
"In general one would apply the arguments below to the orbifold $Y$ ."
],
[
"A simple Lemma",
"The following simple lemma is reminescent of the Gottschalk–Hedlund theorem.",
"Lemma (Finding a coboundary) Let $T\\colon (X,\\mu )\\rightarrow (X,\\mu )$ be a mixing transformation of a probability measure space.",
"Suppose that for $f\\in L^2(\\mu )$ there exists $C\\geqslant 0$ such that $\\left\\Vert f + T^* f + \\cdots +(T^n)^*f\\right\\Vert _{L^2(\\mu )} \\leqslant C,$ for all $n\\geqslant 1.$ Then there exists $h\\in L^2(\\mu )$ such that $f = h - T^* h$ .",
"First note that cond, combined with the von Neumann ergodic theorem, implies that $\\int _X fd\\mu = 0$ .",
"By the mixing property of $T$ , this implies that the only possible weak limit of $(T^n)^*f$ in $L^2(\\mu )$ is 0.",
"Thanks again to the uniform $L^2$ boundedness of the Birkhoff sums of $f$ in cond, there is some weak limit $h\\in L^2(\\mu )$ of $f + T^* f + \\cdots + (T^{n_j})^* f$ along some subsequence $\\left\\lbrace n_j \\right\\rbrace $ .",
"Then using the above remark that the weak limit of $(T^{n_j+1})^* f$ is 0, it follows that $(T^* h) = \\lim _{n_j} \\left[ \\left( f + T^*f \\cdots + (T^{n_j+1})^*f \\right) -f \\right] = h- f$ which is the desired conclusion.",
"Note that $T^*$ is weakly continuous, since it is an isometry and hence has an adjoint (which can be used to obtain weak continuity).",
"Recall that for automorphisms of $K3$ surfaces with positive entropy, the measure of maximal entropy is mixing by [13]."
],
[
"Hyperbolic geometry",
"Let $V$ be a complex 2-dimensional vector space, equipped with a non-degenerate complex volume form $\\Omega \\in \\Lambda ^2(V^\\vee )$ .",
"Let $\\mathbb {H}^3(V)$ be the space of hermitian metrics which induce the same (real) volume on $V$ as $\\Omega $ ; this space is naturally isomorphic to real hyperbolic 3-space, since it can be described as the quotient $\\operatorname{{SL}}_2(\\mathbb {C})/\\operatorname{{SU}}(2)$ .",
"Consider now a decomposition $V= W_+ \\oplus W_-$ into two complex lines.",
"This determines a subset $\\gamma \\subset \\mathbb {H}^3(V)$ of hermitian metrics for which the decomposition of $V$ is orthogonal.",
"There is a natural nearest point projection $\\pi _\\gamma :\\mathbb {H}^3(V)\\rightarrow \\gamma $ , which we now make explicit.",
"Note that $\\gamma $ is a geodesic for the hyperbolic metric."
],
[
"Working in coordinates",
"Assume that we have $\\mathbb {C}^2 = \\mathbb {C}\\oplus \\mathbb {C}$ as our decomposition.",
"A hermitian metric, inducing the standard volume form, is given by a $2\\times 2$ matrix: $\\begin{bmatrix}a & b \\\\\\overline{b} & d \\\\\\end{bmatrix}\\text{ with }a,d \\in \\mathbb {R}, b\\in \\mathbb {C}\\text{ and } ad - |b|^2=1.$ The metrics on $\\gamma $ , for which the decomposition is orthogonal, have $b=0$ .",
"By symmetry considerations, we must have $\\pi _\\gamma \\left( \\begin{bmatrix}a & b \\\\\\overline{b} & a \\\\\\end{bmatrix} \\right) =\\begin{bmatrix}1 & 0 \\\\0 & 1 \\\\\\end{bmatrix}$ since the map $\\pi _\\gamma $ is equivariant for the action of the matrices in $\\operatorname{{GL}}_2(\\mathbb {C})$ which preserve the decomposition $\\mathbb {C}^2 = \\mathbb {C}\\oplus \\mathbb {C}$ , and the transformation exchanging the two axes is in there.",
"Finally, using equivariance under the action of scaling the coordinates by $e^{t/2}$ and $e^{-t/2}$ respectively, it follows that: $\\pi _\\gamma \\left(\\begin{bmatrix}e^{t/2} & 0 \\\\0 & e^{-t/2} \\\\\\end{bmatrix}\\begin{bmatrix}a & b \\\\\\overline{b} & a \\\\\\end{bmatrix}\\begin{bmatrix}e^{t/2} & 0 \\\\0 & e^{-t/2} \\\\\\end{bmatrix}\\right)= \\pi _\\gamma \\left(\\begin{bmatrix}e^t a & b \\\\\\overline{b} & e^{-t} a \\\\\\end{bmatrix}\\right)=\\begin{bmatrix}e^t & 0 \\\\0 & e^{-t} \\\\\\end{bmatrix}$ which can be rewritten as $\\pi _\\gamma \\left( \\begin{bmatrix}a & b \\\\\\overline{b} & d \\\\\\end{bmatrix}\\right)=\\begin{bmatrix}\\sqrt{a/d} & 0 \\\\0 & \\sqrt{d/a} \\\\\\end{bmatrix} $"
],
[
"Hyperbolic Distances",
"Given two Hermitian forms $\\omega _1,\\omega _2$ on $V$ (compatible with the volume) the distance in hyperbolic space between them is defined by $\\frac{\\omega _1\\wedge \\omega _2}{\\Omega \\wedge \\overline{\\Omega }} = \\frac{1}{2} \\left( e^{\\operatorname{{dist}}(\\omega _1,\\omega _2)} + e^{-\\operatorname{{dist}}(\\omega _1,\\omega _2)} \\right).$ Equivalently, one can pick a basis in which $\\omega _1$ is standard Euclidean and diagonalize $\\omega _2$ using the spectral theorem to define the distance as the logarithm of the largest (relative) singular value."
],
[
"Yau and Oseledets curves",
"We now apply the discussion above to $V=\\mathbf {T}_xX$ with $x$ in the $\\mu $ -full measure set where the Oseledets theorem gives us the decomposition $\\mathbf {T}_xX=W^s(x)\\oplus W^u(x)$ .",
"This determines the Oseledets curve $\\gamma \\subset \\mathbb {H}^3(V)$ , a hyperbolic geodesic.",
"On the other hand, by Yau's Theorem [38] we have Ricci-flat metrics $\\omega _t$ on $X$ in the class $e^t[\\eta _+]+e^{-t}[\\eta _-]$ , which together define the Yau curve in $\\mathbb {H}^3(V)$ (although we will be mostly interested only in the values $t=Nh$ , $N\\geqslant 1$ ).",
"Recall that the expansion/contraction factors defined in eqn:rhousdefinition are equal to $\\rho ^u(x) = \\frac{1}{2} \\log \\left( \\frac{a_h}{a_0} \\right),\\quad \\rho ^s(x) = \\frac{1}{2} \\log \\left( \\frac{d_h}{d_0} \\right)$ and so a telescoping sum gives $\\rho ^u(x) + \\rho ^u(Tx) + \\cdots + \\rho ^u(T^{N-1}x) = \\frac{1}{2} \\log \\left( \\frac{a_{Nh}}{a_0} \\right)$ and similarly for $\\rho ^s(x)$ .",
"We will later need the following: Proposition We have the identities: $\\begin{split}\\int _X \\rho ^u(x)\\, d\\mu (x) & = \\frac{h}{2}\\\\\\int _X \\rho ^s(x)\\, d\\mu (x) & = -\\frac{h}{2}\\end{split}$ Recall that the Lyapunov exponent of $\\mu $ is $h/2$ , see sssec:absolutecontinuity.",
"Once the stable and unstable bundles are given, and since in our case they are line bundles, the exponent can be computed from the formula: $\\lambda = \\int _X \\log \\left\\Vert DT|_{W^u(x)}\\right\\Vert d\\mu (x)$ where we compute the norm of $DT\\colon W^u(x)\\rightarrow W^u(Tx)$ for one fixed ambient metric.",
"Note that if the metric is changed, then the quantity $\\log \\left\\Vert DT|_{W^u(x)}\\right\\Vert $ changes by a coboundary, i.e.",
"$\\alpha (x)-\\alpha (Tx)$ where $e^{\\alpha (x)}$ is the constant of proportionality between the old and the new metric, when restricted to $W^u(x)$ .",
"In particular, the integral is independent of the metric.",
"Since by definition $\\rho ^u(x)$ is the pointwise norm of $DT$ on the unstable for the Ricci-flat metric, the claimed identity follows.",
"By the discussion in REF there are also the “Oseledets-projected” metrics $\\theta _0, \\theta _{Nh}$ which correspond to the projection of $\\omega _0,\\omega _{Nh}$ to the geodesic $\\gamma $ determined by the stable/unstable decomposition.",
"In the fixed basis $\\lbrace e_1,e_2\\rbrace $ as in REF , the metrics are: $\\theta _0 =\\begin{bmatrix}\\sqrt{\\frac{a_0}{d_0}} & 0 \\\\0 & \\sqrt{\\frac{d_0}{a_0}} \\\\\\end{bmatrix},\\quad \\theta _{Nh} =\\begin{bmatrix}\\sqrt{\\frac{a_{Nh}}{d_{Nh}}} & 0 \\\\0 & \\sqrt{\\frac{d_{Nh}}{a_{Nh}}} \\\\\\end{bmatrix}$ The distance-decreasing property of projections in the hyperbolic metric on $\\mathbb {H}^3(V)$ , together with the formula for the hyperbolic distance in sssec:hyperbolicdistances, gives $\\operatorname{{dist}}(\\theta _0,\\theta _{Nh}) \\leqslant \\operatorname{{dist}}(\\omega _0,\\omega _{Nh}) = 2 \\lambda (x,N).$ The distance $\\operatorname{{dist}}(\\theta _0, \\theta _{Nh})$ is computed explicitly as: $\\begin{split}\\operatorname{{dist}}(\\theta _0,\\theta _{Nh}) &= \\frac{1}{2} \\left| \\log \\left( \\frac{a_{Nh}d_0}{a_0 d_{Nh}} \\right) \\right| \\\\&=\\left| \\rho ^u(x) + \\cdots + \\rho ^u(T^{N-1}x) - \\rho ^s(x) - \\cdots - \\rho ^s(T^{N-1}x)\\right|\\\\& = |S_N \\rho ^u(x) - S_N \\rho ^s(x)|\\end{split}$ at $\\mu $ -a.e.",
"point, where $S_N f $ denotes the Birkhoff sum of the function $f$ , $S_Nf(x) := f(x) + \\cdots + f(T^{N-1}x).$ Combining this identity with the previous inequality gives $|S_N \\rho ^u(x) - S_N \\rho ^s(x)| \\leqslant 2 \\lambda (x,N)$ The following observation is going to be crucial: Proposition Suppose that $\\mu $ is absolutely continuous with respect to the Lebesgue measure, so that $\\mu =\\frac{\\operatorname{dVol}}{|S|}\\bigg |_S.$ Then the Birkhoff sums of $\\rho ^u-\\rho ^s-h$ satisfy $\\int _X e^{S_N (\\rho ^u-\\rho ^s - h)}d\\mu \\leqslant \\frac{2}{|S|},\\text{ for all }N\\geqslant 1.$ We have $\\begin{split}2e^{Nh}&\\geqslant e^{Nh} + e^{-Nh} = \\int _X \\omega _0 \\wedge \\omega _{Nh}\\\\&=\\int _X (e^{2\\lambda (x,N)}+e^{-2\\lambda (x,N)})\\operatorname{dVol}\\geqslant \\int _X e^{2\\lambda (x,N)}\\operatorname{dVol},\\end{split}$ i.e.",
"$\\int _X e^{2\\lambda (x,N)-Nh}\\operatorname{dVol}\\leqslant 2,$ but from eqn:Birkhoffsumbound we also have $S_N \\rho ^u(x) - S_N \\rho ^s(x) \\leqslant 2 \\lambda (x,N)$ , and so $\\int _X e^{S_N (\\rho ^u-\\rho ^s - h)}\\operatorname{dVol}=\\int _X e^{S_N \\rho ^u - S_N \\rho ^s-Nh}\\operatorname{dVol}\\leqslant 2.$ This finally gives: $\\int _X e^{S_N (\\rho ^u-\\rho ^s - h)}d\\mu \\leqslant \\frac{2}{|S|}.$ Next, observe that both $\\rho ^u$ and $\\rho ^s$ are in $L^\\infty $ , with a uniform bound which only depends on the transformation $T$ and the Ricci-flat metric $\\omega _0$ .",
"In particular the function $\\rho ^u-\\rho ^s-h$ is also in $L^\\infty $ , and thanks to prop:integralsofrhous it satisfies $\\int _X(\\rho ^u-\\rho ^s-h)d\\mu =0.$"
],
[
"Exponential integrability of Birkhoff sums",
"Our goal now is to use exp+ and exp0 to prove an exponential integrability bound for $|S_N(\\rho ^u-\\rho ^s-h)|$ .",
"As it turns out this is an essentially formal consequence of exp+ and exp0, as we now show.",
"Proposition (Exponential integrability of Birkhoff sums) Let $T\\colon (X,\\mu )\\rightarrow (X,\\mu )$ be an invertible transformation of a probability measure space.",
"Suppose that $f\\in L^1(X,\\mu )$ with $\\int _X f\\, d\\mu =0$ is such that $\\int _X e^{S_Nf}d\\mu \\leqslant C$ for a uniform constant $C$ and all $N\\geqslant 1$ .",
"Then for every $0<\\gamma <\\frac{1}{6}$ there is $C^{\\prime }=C^{\\prime }(C,\\gamma )$ , which does not depend on $f$ , such that $\\int _X e^{\\gamma |S_Nf|}d\\mu \\leqslant C^{\\prime }, \\text{ for all }N\\geqslant 1.$ Decompose $f=f^+ - f^-$ into its positive and negative parts, and for Birkhoff sums denote by $S_Nf = S_N^+f - S_N^-f$ the decomposition into positive and negative parts.",
"Suppose we show that there is a constant $C^{\\prime }$ that depends only on $C$ such that the negative part $f^-$ satisfies the bound $\\mu \\left( \\left\\lbrace x:f^-(x)\\geqslant L \\right\\rbrace \\right) \\leqslant C^{\\prime }e^{-\\frac{L}{6}}$ for all $L\\geqslant 0$ .",
"Then this can be applied to the function $S_Nf$ with the transformation $T^N$ , which would thus give us $\\mu \\left( \\left\\lbrace x: S_N^-f(x)\\geqslant L \\right\\rbrace \\right) \\leqslant C^{\\prime }e^{-\\frac{L}{6}}$ for all $L\\geqslant 0$ and all $N\\geqslant 1$ .",
"On the other hand assn directly implies $\\int _X e^{S^+_Nf}d\\mu \\leqslant C,$ which together with the Chebyshev-Markov inequality implies that $\\mu \\left( \\left\\lbrace x \\colon S_N^+f(x)\\geqslant L \\right\\rbrace \\right) \\leqslant Ce^{-L}$ for all $L\\geqslant 0$ and all $N\\geqslant 1$ , and so concl follows from these bounds together with the elementary formula $\\int _X e^{\\gamma |u|}d\\mu =\\gamma \\int _0^\\infty \\mu (\\lbrace |u|\\geqslant t\\rbrace )e^{\\gamma t} dt.$ So it suffices to prove topr.",
"Define the set of interest as $B_L := \\left\\lbrace x \\colon f^-(x)\\geqslant L \\right\\rbrace $ Consider now the set where the positive parts of the Birkhoff sums are large (but on a smaller scale): $P_{j,L} := \\left\\lbrace x \\colon S_j^+f(x)\\geqslant \\frac{1}{2}(L-1) \\right\\rbrace $ which by cheb satisfy $\\mu (P_{j,L})\\leqslant C^{\\prime }e^{-\\frac{L}{2}}$ .",
"Set $C_L := B_L \\setminus \\bigcup _{i=-e^{\\frac{L}{6}}}^{e^{\\frac{L}{6}}} \\bigcup _{j=1}^{e^{\\frac{L}{6}}} T^i (P_{j,L})$ which satisfies the size bound: $\\mu (C_L) \\geqslant \\mu (B_L) - 3e^{\\frac{L}{3}}\\cdot C\\cdot e^{-\\frac{L}{2}}=\\mu (B_L)-C^{\\prime }e^{-\\frac{L}{6}}.$ Any point $x\\in C_L$ has the property that $f^-(x)\\geqslant L$ and for any other point in its orbit $T^ix$ (with $i=-e^{\\frac{L}{6}}\\ldots e^{\\frac{L}{6}}$ ) any Birkhoff sum (with $j=1\\ldots e^{\\frac{L}{6}}$ ) satisfies $S_j^+f(T^ix)\\leqslant \\frac{L-1}{2}$ .",
"Consider now any point $y$ in the support of the function $g_L:=\\mathbf {1}_{C_L} + \\mathbf {1}_{T^{-1}C_L} + \\cdots + \\mathbf {1}_{T^{-e^{\\frac{L}{6}}}C_L}$ Then $g_L(y)$ is the number of visits of $y$ to $C_L$ in the times $1\\ldots e^{\\frac{L}{6}}$ .",
"The Birkhoff sum $S_{e^{\\frac{L}{6}}}f(y)$ can be divided into at most $g_L(y)+1$ intervals where the positive part is bounded above by $\\frac{L-1}{2}$ by the construction of $C_L$ , and the $g_L(y)$ points where $f^-\\geqslant L$ .",
"This implies that $S_{e^{\\frac{L}{6}}}^-f(y) \\geqslant g_L(y)\\left(L-2\\frac{L-1}{2}\\right)=g_L(y)$ for all $y$ in the support of $g_L$ .",
"Integrating over all $y$ gives: $\\int _{\\mathrm {Spt}(g_L)} S_{e^{\\frac{L}{6}}}^-f(y) d\\mu (y) \\geqslant e^{\\frac{L}{6}}\\cdot \\mu (C_L)$ On the other hand, the condition that $\\int _X f d\\mu = 0$ implies $\\int _X S_N^-f d\\mu =\\int _X S_N^+f d\\mu $ .",
"Using trivial, together with the trivial inequality $x\\leqslant e^x$ for $x\\geqslant 0$ , we immediately get $\\int _X S_N^+f d\\mu \\leqslant C$ for all $N\\geqslant 1$ .",
"Together with numbered, this then implies $ \\mu (C_L)\\leqslant C^{\\prime }e^{-\\frac{L}{6}}$ .",
"Using now the lower bound on $\\mu (C_L)$ in terms of $B_L$ gives the desired $\\mu (B_L)\\leqslant C^{\\prime \\prime }e^{-\\frac{L}{6}}$ .",
"Thanks to exp0 and exp+, prop:exponentialintegrabilityofbirkhoffsums applies to $f=\\rho ^u-\\rho ^s-h$ , and so we conclude that for every $0<\\gamma <\\frac{1}{6}$ there is $C^{\\prime }>0$ such that $\\int _X e^{\\gamma |S_N(\\rho ^u-\\rho ^s-h)|}d\\mu \\leqslant C^{\\prime },$ for all $N\\geqslant 1$ .",
"In particular, thanks to the elementary inequality $\\frac{\\gamma ^2}{2}|S_N(\\rho ^u-\\rho ^s-h)|^2\\leqslant e^{\\gamma |S_N(\\rho ^u-\\rho ^s-h)|},$ we see that the hypotheses of lem:findingacoboundary are satisfied by $f=\\rho ^u-\\rho ^s - h$ .",
"Indeed, the left hand side in lem:findingacoboundary is just $\\left\\Vert S_Nf\\right\\Vert _{L^2(\\mu )}^{1/2}$ .",
"We thus obtain Corollary There is a function $\\alpha \\in L^2(\\mu )$ such that $\\rho ^u-\\rho ^s-h=T^*\\alpha -\\alpha .$ Hence, combining this with eqn:thesumisacoboundary, the functions $\\alpha _u:=\\frac{\\alpha +\\beta }{2},\\quad \\alpha _s:=\\frac{\\beta -\\alpha }{2},$ satisfy eqn:rhocohomologoustoconstant.",
"Remark By working in a suitable Orlicz space instead of $L^2(\\mu )$ , and applying the analog of lem:findingacoboundary, it is not hard to see that the coboundary $\\alpha $ is in fact exponentially integrable, in the sense that $e^{\\gamma |\\alpha |}\\in L^1(\\mu )$ for some $\\gamma >0$ .",
"With more work, one can deduce the same integrability for $\\beta $ , and hence for $\\alpha _u$ and $\\alpha _s$ .",
"However, to show that $\\alpha _u$ and $\\alpha _s$ are in fact bounded, still requires the arguments that we used in gen."
]
] | 1808.08673 |
[
[
"A note On subgroups in a division ring that are left algebraic over a\n division subring"
],
[
"Abstract Let $D$ be a division ring with center $F$ and $K$ a division subring of $D$.",
"In this paper, we show that a non-central normal subgroup $N$ of the multiplicative group $D^*$ is left algebraic over $K$ if and only if so is $D$ provided $F$ is uncountable and contained in $K$.",
"Also, if $K$ is a field and the $n$-th derived subgroup $D^{(n)}$ of $D^{*}$ is left algebraic of bounded degree $d$ over $K$, then $\\dim_FD\\le d^2$."
],
[
"all"
]
] | 1808.08452 |
[
[
"$\\Delta\\phi$ and multi-jet correlations with CMS"
],
[
"Abstract We present angular correlations in multi-jet events at highest center-of-mass energies and compare the measurements to theoretical predictions including higher order parton radiation and coherence effects."
],
[
"Introduction",
"Particle jets with large transverse momenta, $\\rm {p}_T$ , are abundantly produced in highly energetic proton-proton collisions at the CERN LHC, when two partons interact with high momentum transfer under the strong interaction.",
"The process is described by Quantum Chromodynamics (QCD) using perturbative techniques (pQCD).",
"The two-final state partons, at leading order (LO) in pQCD, are produced back-to-back in the transverse plane and thus the azimuthal angular separation between the two highest $\\rm {p}_T$ jets in the transverse plane, $\\Delta \\phi _{1,2}=\\vert \\phi _\\mathrm {jet1}-\\phi _\\mathrm {jet2} \\vert $ , equals $\\pi $ .",
"The production of a third or more high-$\\rm {p}_T$ jets leads to a deviation from $\\pi $ in the azimuthal angle.",
"The measurement of the azimuthal angular correlation (or decorrelation from $\\pi $ ) in inclusive 2-jet topologies is proven to be an interesting tool to gain insight into multijet production processes.",
"Previous measurements of azimuthal correlation in inclusive 2-jet topologies were reported by the D0 Collaboration [1], [2], ATLAS Collaboration [3], and CMS Collaboration [4], [5].",
"Multijet correlations have been measured by the ATLAS collaboration at $\\sqrt{s}=7$ TeV and $\\sqrt{s}=8$ TeV [6], [7].",
"This paper reports measurements of the normalized inclusive 2-jet cross sections as a function of the azimuthal angular separation between the two leading $\\rm {p}_T$ jets for several intervals of the leading jet $\\rm {p}_T$ ($\\rm {p}_T^{max}$ ).",
"The measurements are done in the region $\\pi /2 < \\Delta \\phi _{1,2} \\le \\pi $ .",
"Measurement of inclusive 3-jet and 4-jet cross sections are also available in [8].",
"Experimental and theoretical uncertainties are reduced by normalizing the $\\Delta \\phi _{1,2}$ distribution to the total dijet cross section.",
"The measurement is performed using data collected during 2016 with the CMS experiment at the CERN LHC, corresponding to an integrated luminosity of 35.9$fb^{-1}$ of proton-proton collisions at $\\sqrt{s}=13 TeV$ .",
"Concerning the final event selection, in this analysis first we consider all jets with a minimum $\\rm {p}_T$ of 100 GeV and $\\vert y \\vert <5$ .",
"Then for inclusive 2-jets events we require at least 2 jets whith $\\vert y_1 \\vert <2.5$ and $\\vert y_2 \\vert <2.5$ .",
"For inclusive 3-jet (4-jet) all jets must have $\\vert y \\vert <2.5$ ."
],
[
"Results and Conclusions.",
"Predictions from different MC event generators are compared to data.",
"The HERWIG$++$ and the PYTHIA8 event generators are considered.",
"Both of them are based on LO $2\\rightarrow 2$ matrix element calculations.",
"For PYTHIA8, the CUETP8M1 tune [9], which is based on NNPDF2.3LO [10], [11], is considered, while HERWIG$++$ uses the CUETHppS1 tune [9], based on the CTEQ6L1 PDF set [12].",
"The MADGRAPH [13] event generator provides LO matrix element calculations with up to four outgoing partons.",
"The NNPDF2.3LO PDF set is used in the matrix element calculation.",
"It is interfaced to PYTHIA8 with tune CUETP8M1.",
"For the matching with PYTHIA8, the kt-MLM matching procedure [14] is used.",
"Predictions based on NLO pQCD are considered using the POWHEG package [15], [16], [17] and the HERWIG7 [18] event generator.",
"The events simulated with POWHEG are matched to PYTHIA8 or to HERWIG$++$ parton showers and MPI, while HERWIG7 uses similar parton shower and MPI models as HERWIG$++$ .",
"In this analysis, POWHEG provides an NLO dijet calculation [19], referred to as POWHEG 2jet, and an NLO three-jet calculation [20], referred to as POWHEG 3jet, both using the NNPDF30nlo PDF set [21].",
"The POWHEG 2jet is matched to PYTHIA8 with tune CUETP8M1 and HERWIG$++$ with tune CUETHppS1, while the POWHEG 3jet is matched only to PYTHIA8 with tune CUETP8M1.",
"Predictions from the HERWIG7 event generator make use of NLO dijet matrix elements calculated with the MMHT 2014 PDF set [22] and use the default tune H7-UE-MMHT [18] for the UE simulation.",
"Parton shower contributions are matched to the matrix element within the MC$@$ NLO procedure [23], [24] through angular-ordered emissions.",
"The unfolded, normalized inclusive 2-jet cross sections differential in $\\Delta \\phi _{1,2}$ are shown in Fig.",
"REF for the various $\\rm {p}_T^{max}$ regions considered.",
"The distributions are strongly peaked at $\\pi $ and become steeper with increasing $\\rm {p}_T^{max}$ .",
"Overlaid with the data are predictions from POWHEG 2jet $+$ PYTHIA8 event generator.",
"The error bars on the data points represent the total experimental uncertainty, which is the quadratic sum of the statistical and systematic uncertainties.",
"Figures REF (left) shows the ratios of the PYTHIA8, HERWIG$++$ , MADGRAPH $+$ PYTHIA8 event generators predictions to the normalized inclusive 2-jet cross section differential in $\\Delta \\phi _{1,2}$ , for all $\\rm {p}_T^{max}$ regions.",
"The solid band indicates the total experimental uncertainty and the error bars on the MC points represent their statistical uncertainties.",
"Among the LO dijet event generators HERWIG$++$ exhibits the largest deviations from the measurements.",
"PYTHIA8 behaves much better than HERWIG$++$ exhibiting some deviations particular around $\\Delta \\phi = 5\\pi /6$ .",
"The MADGRAPH $+$ PYTHIA8 event generator provides the best description of the measurements.",
"Figures REF (right) shows the ratios of the POWHEG 2jet matched to PYTHIA8 and HERWIG$++$ , POWHEG 3jet $+$ PYTHIA8, and HERWIG7 event generators predictions to the normalized inclusive 2-jet cross section differential in $\\Delta \\phi _{1,2}$ , for all $\\rm {p}_T^{max}$ regions.",
"The solid band indicates the total experimental uncertainty and the error bars on the MC points represent the statistical uncertainties in the simulated data.",
"The predictions of POWHEG 2jet or POWHEG 3jet exhibit large deviations from the measurements.",
"It has been checked that POWHEG 2jet predictions at parton level, i.e.",
"without the simulation of MPI, HAD and parton showers, give a reasonable description of the measurement for values of $\\Delta \\phi _{1,2}$ greater than $\\approx 2\\pi /3$ , while they completely fail for smaller values, where the parton shower has a crucial role.",
"Adding parton showers fills the phase space at low values of $\\Delta \\phi _{1,2}$ and brings the POWHEG 2jet predictions closer to data, however with the parameter setting used the agreement is not optimal.",
"Unfortunately, no big effect is observed when parton-shower is included.",
"Further investigation showed that the POWHEG 2jet calculation and the POWHEG three-jet calculation at LO are equivalent when initial- and final-state radiation is switched off.",
"The predictions from POWHEG 2jet matched to PYTHIA8 are describing the normalized cross sections better than those where POWHEG 2jet is matched to HERWIG$++$ .",
"Since the hard process calculation is the same, the difference between the two predictions is entirely due to different parton shower in PYTHIA8 and HERWIG$++$ , which also use different $\\alpha _S$ values for initial- and final-state emissions, in addition to a different upper scale used for the parton shower simulation, which is higher in PYTHIA8 than in HERWIG$++$ .",
"The dijet NLO event generator HERWIG7 provides the best description of the measurements, showing a very large improvement in comparison to HERWIG$++$ .",
"For this observable MC@NLO method of combining parton shower with the NLO parton level calculations has advantages compared to the POWHEG method.",
"All these observations emphasize the need to improve predictions for multijet production.",
"Similar observations, for the inclusive 2-jet cross sections differential in $\\Delta \\phi _{1,2}$ , were reported previously by CMS [5] at a different centre-of-mass energy.",
"Figure: Normalized inclusive 2-jet cross section differential in Δφ 1,2 \\Delta \\phi _{1,2}for nine p T max \\rm {p}_T^{max} regions, scaled by multiplicative factors forpresentation purposes .",
"The error bars on the data points includestatistical and systematic uncertainties.",
"Overlaid with the dataare predictions from the POWHEG 2jet + PYTHIA8 event generator.Figure: Ratios of PYTHIA8, HERWIG++++, MADGRAPH ++ PYTHIA8 (left), and POWHEG 2jet, POWHEG 3jet, Herwig7 (right) predictions, to the normalizedinclusive 2-jet cross section differential in Δφ 1,2 \\Delta \\phi _{1,2}, for all p T max \\rm {p}_T^{max} regions .The solid band indicates the total experimental uncertainty andthe error bars on the MC points represent the statisticaluncertainties of the simulated data."
]
] | 1808.08614 |
[
[
"Analytical Soft SUSY Spectrum in Mirage-Type Mediation Scenarios"
],
[
"Abstract We derive explicitly the soft SUSY breaking parameters at arbitrary low energy scale in the (deflected) mirage type mediation scenarios with possible gauge or Yukawa mediation contributions.",
"Based on the Wilsonian effective action after integrating out the messengers, we obtain analytically the boundary value (at the GUT scale) dependencies of the effective wavefunctions and gauge kinetic terms.",
"Note that the messenger scale dependencies of the effective wavefunctions and gauge kinetic terms had already been discussed in GMSB.",
"The RGE boundary value dependencies, which is a special feature in (deflected) mirage type mediation, is the key new ingredients in this study.",
"The appearance of $'mirage'$ unification scale in mirage mediation is proved rigorously with our analytical results.",
"We also discuss briefly the new features in deflected mirage mediation scenario in the case the deflection comes purely from the Kahler potential and the case with messenger-matter interactions."
],
[
"Introduction",
"After the discovery of the 125 GeV Higgs boson in 2012 at the CERN LHC[1], [2], the long missing particle content of the Standard Model(SM) has finally been verified.",
"In spite of the impressive triumph of SM, many physicists still believe that new physics may be revealed at LHC.",
"Among the many new physics models that can solve the fine-tuning problem, the most elegant and compelling resolution is low energy supersymmetry.",
"Augmented with weak scale soft SUSY breaking terms, the quadratic cutoff dependence is absent, leaving only relatively mild but intertwined logarthmic sensitivity to high scale physics.",
"As such soft SUSY breaking spectrum is determined by the SUSY breaking mechanism, it is interesting to survey the the phenomenology related to supersymmetry breaking mechanism.",
"In Type IIB string theory compactified on a Calabi-Yau (CY) orientifold, the presence of NS and RR 3-form background fluxes can fix the dilaton and the complex structure moduli, leaving only the Kahler moduli in the Wilsonian effective supergravity action after integrating out the superheavy complex structure moduli and dilaton.",
"The remaining Kahler moduli fields could be stabilized by non-perturbative effects, such as instanton or gaugino condensation.",
"In order to generates SUSY breaking in the observable sector and obtain a very tiny positive cosmological constant, Kachru-Kallosh-Linde- Trivedi (KKLT)[3] propose to add an anti-D3 brane at the tip of the Klebanov-Strassler throat (or adding F-term, D-term SUSY breaking contributions[4]) to explicitly break SUSY and lift the AdS universe to obtain a dS one.",
"In addition to the anomaly mediation contributions, SUSY breaking effects from the light Kahler moduli fields could also be mediated to the visible sector and result in a mixed modulus-anomaly mediation SUSY breaking scenario [5], [6].",
"It is interesting to note that the involved modulus mediated SUSY breaking contributions can be comparable to that of the anomaly mediation [7].",
"With certain assumptions on the Yukawa couplings and the modular weights, the SUSY breaking contributions from the renormalization group running and anomaly mediation could cancel each other at a $^{\\prime }mirage^{\\prime }$ unification scale, leading to a compressed low energy SUSY breaking spectrum [8].",
"Such a mixed modulus-anomaly mediation SUSY breaking mechanism is dubbed as $^{\\prime }{\\rm mirage~ mediation}^{\\prime }$ .",
"Anomaly mediation contribution is a crucial ingredient of such a mixed modulus-anomaly mediation.",
"It is well known that the pure anomaly mediation is bothered by the tachyonic slepton problem [9].",
"One of its non-trivial extensions with messenger sectors, namely the deflected anomaly mediated SUSY breaking (AMSB), can elegantly solve such a tachyonic slepton problem through the deflection of the renormalization group equation (RGE) trajectory [10], [11], [12].",
"Such a messenger sector can also be present in the mirage mediation so that additional gauge contributions by the messengers[13] can deflect the RGE trajectory and change the low energy soft SUSY predictions.",
"Additional deflection in mirage mediation can be advantageous in phenomenological aspect.",
"For example, apparent gaugino mass unification at TeV scale could still be realized with the simplest $^{\\prime }no~scale^{\\prime }$ Kahler potential, which, in ordinary mirage mediation, can only be possible with the not UV-preferable $\\alpha =2$ case.",
"Relevant discussions on mirage-type mediation scenarios can be seen, for example, in[14], [15], [16], [17].",
"In mirage type mediation scenarios, analytical expressions for the soft SUSY breaking parameters are no not given at the messenger scale $M$ (or scale below $M$ ), but given at the GUT scale instead.",
"One needs to numerically evolve the spectrum with GUT scale input to obtain the low energy SUSY spectrum.",
"This procedure obscures the appearance of $^{\\prime }mirage^{\\prime }$ unification scale from the input.",
"In mirage mediation scenarios with deflection from Kahler potential, analytical results of mirage mediation are necessary to predict the low energy SUSY spectrum.",
"So it is preferable to give the analytical expressions for the soft SUSY breaking parameters in mirage type mediation scenarios at arbitrary low energy scale.",
"Besides, possible new Yukawa-type interactions involving the messengers may give additional Yukawa mediation contributions to the low energy soft SUSY spectrum (See [18] for example).",
"Such a generalization of deflected mirage mediation scenario shows new features in phenomenological studies.",
"The inclusion of Yukawa mediation contributions at (or below) the messenger scale $M$ are non-trivial and again prefer analytical expressions near the messenger scale.",
"This paper is organized as follows.",
"We briefly review the mirage type mediation scenarios in Sec.. A general discussion on the analytical expressions for the soft SUSY parameters in the generalized deflected mirage mediation is given in Sec.. We discuss some applications of our analytical results in Sec., including the proof of the $^{\\prime }mirage^{\\prime }$ unification scale in mirage mediation with our analytical results and the discussions on deflection from Kahler potential.",
"Sec.",
"contains our conclusions."
],
[
"Brief Review of the Mirage Type Mediation Scenarios",
"Inspired by string-motivated KKLT approach to moduli stabilization within Type IIB string theory, mirage mediation supersymmetry breaking is proposed, in which the modulus mediated supersymmetry breaking terms are suppressed by numerically a loop factor so that the anomaly mediated terms can be competitive.",
"After fixing and integrating out the dilaton and the complex structure moduli, the four-dimensional Wilsonian effective supergravity action (defined at the boundary scale $\\Lambda $ ) in terms of compensator field and a single Kahler modulus parameterizing the overall size of the compact space[8] is given as $e^{-1}{\\cal L}=\\int d^4\\theta \\left[\\phi ^\\dagger \\phi \\left(-3 e^{-K/3}\\right)-(\\phi ^\\dagger \\phi )^2\\bar{\\theta }^2\\theta ^2 {\\cal P}_{lift}\\right]+\\int d^2\\theta \\phi ^3 W+ \\int d^2\\theta \\frac{f_i}{4} W_i^a W_i^a$ with a holomorphic gauge kinetic term $f_i=\\frac{1}{g_i^2}+i\\frac{\\theta }{8\\pi }.$ The Kahler potential takes the form $K&=&-3\\ln (T+T^\\dagger )+Z_X(T^\\dagger ,T) X^\\dagger X+Z_\\Phi (T^\\dagger ,T)\\Phi ^\\dagger \\Phi \\nonumber \\\\& & +\\sum \\limits _{i}Z_{P_i,\\bar{P}_i}(T^\\dagger ,T)\\left[ P_i^\\dagger P_i+\\bar{P}_i^\\dagger \\bar{P}_i\\right]~,$ with the $^{\\prime }no-scale^{\\prime }$ kinetic term for the Kahler modulus $T$ .",
"The gauge kinetic term $f_i$ , the messenger superfields $P_i$ , the MSSM superfields $\\Phi $ and the pseudo-moduli superfields are all assumed to depend non-trivially on the Kahler moduli $T$ as $Z_X(T^\\dagger ,T)&=&\\frac{1}{(T^\\dagger + T)^{n_X}}~,~~Z_\\Phi (T^\\dagger ,T)=\\frac{1}{(T^\\dagger + T)^{n_\\Phi }}~,~\\nonumber \\\\f_i(T)&=& T^{l_i}~,~~~~~~~~~~~~~Z_{P_i,\\bar{P}_i}(T^\\dagger ,T)=\\frac{1}{(T^\\dagger + T)^{n_P}}~.$ Choices of $n_X,n_\\Phi ,n_P,l_i$ depend on the location of the fields on the D3/D7 branes.",
"Besides, universal $l_i=1$ are adopted in our scenario to keep gauge coupling unification, so the gauge fields should reside on the D7 brane.",
"The superpotential takes the most general form involving the KKLT setup[3], the messenger sectors $W_M$ and visible sector $W_{\\overline{MSSM}}$ $W=\\left(\\omega _0-A e^{-aT}\\right)+W_{M}+W_{\\overline{MSSM}}~,$ where the first term is generated from the fluxes and the second term from non-perturbative effects, such as gaugino condensation or D3-instanton.",
"Within $W_M$ , interactions between messengers and MSSM fields can possibly arise which will be discussed subsequently.",
"The modulus $T$ , which is not fixed by the background flux, can be stabilized by non-perturbative gaugino condensation with its VEV satisfying $a~\\Re {\\langle T \\rangle }\\approx \\ln \\left(\\frac{A}{\\omega _0}\\right)\\approx \\ln \\left(\\frac{M_{Pl}}{m_{3/2}}\\right)\\approx 4\\pi ^2~$ up to ${\\cal O}(\\ln [{M_{Pl}}/{m_{3/2}}]^{-1})$ .",
"Boundary value of the soft SUSY breaking parameters at the GUT scale can be seen in [8]."
],
[
"Analytical Expressions of Soft SUSY Breaking Parameters",
"Mirage mediation can be seen as a typical mixed modulus-anomaly mediation SUSY breaking mechanism with each contribution of similar size.",
"Adding a messenger sector will add additional gauge mediation contributions.",
"Besides, upon the messenger thresholds, new Yukawa interactions involving the messengers could arise.",
"Such interactions may cause new contributions to trilinear couplings and sfermion masses (As an example, see our previous work [18]).",
"Additional deflection with Yukawa mediation can be advantageous in several aspects.",
"The value of trilinear coupling $|A_t|$ can be increased by additional contributions involving the new Yukawa interactions.",
"Larger value of $A_t$ is always welcome in MSSM and NMSSM not only to accommodate the 125 GeV Higgs but also to reduce[19] the EW fine tuning[20] involved.",
"As noted in [18], [21], [22], pure gauge mediation contributions are not viable to generate either trilinear couplings $A_\\kappa ,A_\\lambda $ or soft scalar masses $m_S^2$ for singlet superfields $S$ which are crucial to solve the mu-problem of NMSSM.",
"Deflection with Yukawa interactions will readily solve such difficulty.",
"To take into account such Yukawa mediation contributions in soft SUSY breaking parameters, it is better to derive the most general results involving the deflection.",
"There are two approaches to obtain the low energy SUSY spectrum in the (deflected) mirage type mediation scenario: In the first approach, the mixed modulus-anomaly mediation soft SUSY spectrum is given by their boundary values at the GUT scale[8].",
"Such a spectrum will receive additional contributions towards its RGE running to low energy scale, especially the threshold corrections related to the appearance of messengers[23], [24].",
"The soft SUSY breaking parameters are obtained by combing numerical RGE evolutions with threshold corrections.",
"In [23], following this approach, some analytical expressions of the soft SUSY spectrum, for example the gaugino masses, are given.",
"General expressions of the soft scalar masses and trilinear couplings are not given explicitly except for some simplified cases.",
"In the second approach which we will adopt, the soft SUSY spectrum at low energy scale is derived directly from the low energy effective action.",
"We know that the SUGRA description in eq.",
"(REF ) can be seen as a Wilsonian effective action after integrating out the complex structure moduli and dilaton field.",
"After the pseudo-modulus acquires a VEV and determines the messenger threshold, the messenger sector can be integrated out to obtain a low energy effective action below the messenger threshold.",
"So we anticipate the Kahler metric $Z_\\Phi $ and gauge kinetic $f_i$ will depend non-trivially on the messenger threshold $M_{mess}^2/\\phi ^\\dagger \\phi $ and $M_{mess}/\\phi $ , respectively.",
"The resulting soft SUSY spectrum below the messenger threshold can be derived from the wavefunction renormalization approach [25].",
"The main difficulty here is to find the boundary value dependencies of the wavefunction and gauge kinetic term.",
"In this approach, the most general expressions for soft SUSY breaking parameters in deflected modulus-anomaly (mirage) mediation SUSY breaking mechanism are derived below.",
"Ordinary mirage mediation results can be obtained by setting the deflection parameter $^{\\prime } d ^{\\prime }$ to zero.",
"The gaugino masses are given by $M_i&=& -g_i^2\\left(\\frac{F_T}{2}\\frac{\\partial }{\\partial T} -\\frac{F_\\phi }{2}\\frac{\\partial }{\\partial \\ln \\mu }+\\frac{d F_\\phi }{2}\\frac{\\partial }{\\partial \\ln |X|}\\right) f_a(T,\\frac{\\mu }{\\phi },\\sqrt{\\frac{X^\\dagger X}{\\phi ^\\dagger \\phi }})~,$ The trilinear terms are given by $&&A_{Y_{abc}}\\equiv A_{abc}/y_{abc}~\\\\&=&\\frac{1}{2}\\sum \\limits _{i=a,b,c}\\left({F^T}\\frac{\\partial }{\\partial T}-{F_\\phi }\\frac{\\partial }{\\partial \\ln \\mu }+{d F_\\phi }\\frac{\\partial }{\\partial \\ln |X|}\\right) \\ln \\left[e^{-K_0/3}Z_i(\\mu ,X,T)\\right]~.\\nonumber $ The soft sfermion masses are given by $-m^2_{soft}(\\mu )&=&\\left|\\frac{F_T}{2}\\frac{\\partial }{\\partial T}-\\frac{F_\\phi }{2}\\frac{\\partial }{\\partial \\ln \\mu }+\\frac{d}{2} F_\\phi \\frac{\\partial }{\\partial \\ln |X|}\\right|^2 \\ln \\left[e^{-K_0/3}Z_i(\\mu ,X,T)\\right]~\\\\&=&\\left(\\frac{|F_T|^2}{4}\\frac{\\partial ^2}{\\partial T\\partial T^*}+\\frac{F_\\phi ^2}{4}\\frac{\\partial ^2}{\\partial (\\ln \\mu )^2}+\\frac{d^2F^2_\\phi }{4}\\frac{\\partial ^2}{\\partial (\\ln |X|)^2}-\\frac{F_TF_\\phi }{2}\\frac{\\partial ^2}{\\partial T\\partial \\ln \\mu }\\right.~\\nonumber \\\\&&~~+\\left.\\frac{dF_TF_\\phi }{2}\\frac{\\partial ^2}{\\partial T\\partial \\ln |X|}-\\frac{d F^2_\\phi }{2}\\frac{\\partial ^2}{\\partial \\ln |X|\\partial \\ln \\mu }\\right) \\ln \\left[e^{-K_0/3}Z_i(\\mu ,X,T)\\right],\\nonumber $ From the previous general expressions, we can deduce the concrete analytical results for soft SUSY parameters.",
"In our notation, we define the modulus mediation part $M_0\\equiv \\frac{F_T}{T+T^*}~,~~~q_{Y_{ijk}}\\equiv 3-(n_i+n_j+n_k)~.$ The gauge and Yukawa couplings are used in the form $\\alpha _i=\\frac{g_i^2}{4\\pi }~,~~~\\alpha _{\\lambda _{ijk}}=\\frac{\\lambda ^2_{ijk}}{4\\pi }~.$"
],
[
"Gaugino Mass",
"The gaugino mass below the messenger scale can be obtained from Eqn.",
"(REF ).",
"At the GUT (compactification scale) $M_G$ , the gauge coupling unification requires $T^{l_a}= \\frac{1}{g^2(GUT)}~,$ The gauge coupling at scale $\\mu $ just below the messenger threshold $M$ is given as $\\frac{1}{g_i^2(\\mu )}&=&\\frac{1}{g_i^2(GUT)}+\\frac{b_i+\\Delta b_i}{8\\pi ^2}\\ln \\frac{M_G}{|X|}+\\frac{b_i }{8\\pi ^2}\\ln \\frac{|X|}{\\mu }~,\\nonumber \\\\&=& T^{l_a}+\\frac{b_i+\\Delta b_i}{8\\pi ^2}\\ln \\frac{M_G}{M}+\\frac{b_i}{8\\pi ^2}\\ln \\frac{M}{\\mu }~.$ The derivatives are given as $\\frac{\\partial }{\\partial \\ln \\mu }\\left(\\frac{1}{g_i^2(\\mu )}\\right)&=&-\\frac{b_i}{8\\pi ^2}~,~~~~~~\\frac{\\partial }{\\partial \\ln M}\\left(\\frac{1}{g_i^2(\\mu )}\\right)=-\\frac{\\Delta b_i}{8\\pi ^2}~,$ and $&&\\frac{\\partial }{\\partial T}\\left(\\frac{1}{g_a^2(\\mu )}\\right)=l_a T^{l_a-1}~\\Longrightarrow -2\\frac{1}{g_a^3}\\frac{\\partial g_a(\\mu )}{\\partial T}=l_a T^{l_a-1}~,$ So we can obtain the analytical results for gaugino mass $M_i(\\mu )=g_i^2(\\mu )\\left[l_a\\frac{F_T}{2T}\\frac{1}{g_a^2(GUT)}+\\frac{F_\\phi }{2}\\frac{b_i}{8\\pi ^2}-\\frac{d}{2} F_\\phi \\frac{\\Delta b_i}{8\\pi ^2}\\right]~.$ with $\\Delta b_i\\equiv b_i^\\prime -b_i$ and $b_i^\\prime ,~b_i$ the gauge beta function upon and below the messenger thresholds, respectively.",
"This results can coincide with the gaugino masses predicted from RGE running with threshold corrections at the messenger scale.",
"Following the approach in [13], the gaugino mass at the scale $\\mu $ slightly below the messenger scale $M$ will receive additional gauge mediation contributions $M_i(\\mu \\lesssim M)&=&\\frac{g_i^2( M)}{g_i^2(GUT)}M_i(GUT)-F_\\phi \\frac{g_i^2( M)}{16\\pi ^2} (d+1) \\Delta b_i~,\\nonumber \\\\&=&g_i^2( M)\\left[l_a\\frac{F_T}{2T}\\frac{1}{g_a(GUT)}+\\frac{F_\\phi }{2}\\frac{b_i+\\Delta b_i}{8\\pi ^2}\\right]-F_\\phi \\frac{g_i^2( M)}{16\\pi ^2} (d+1) \\Delta b_i,$ with $M_i(GUT)=g_i^2(GUT)\\left[l_a\\frac{F_T}{2T}\\frac{1}{g_a^2(GUT)}+\\frac{F_\\phi }{2}\\frac{b_i+\\Delta b_i}{8\\pi ^2}\\right].$ Then we can obtain the gaugino mass at scale $\\mu < M$ from one-loop RGE $M_i(\\mu )&=&\\frac{g_i^2(\\mu )}{g_i^2( M)} M_i(\\mu \\lesssim \\ln M)~,\\nonumber \\\\&=&g_i^2(\\mu )\\left[l_a\\frac{F_T}{2T}\\frac{1}{g_a^2(GUT)}+\\frac{F_\\phi }{2}\\frac{b_i}{8\\pi ^2}\\right]-F_\\phi \\frac{g_i^2(\\mu )}{16\\pi ^2} d \\Delta b_i~,$ So we can see that the two results agree with each other."
],
[
"Trilinear Terms",
"From the form of wavefunction $Z_i(\\mu )=Z_i(\\Lambda )\\prod \\limits _{l=y_t,y_b,y_\\tau } \\left(\\frac{y_l(\\mu )}{y_l(\\Lambda )}\\right)^{A_l}\\prod \\limits _{k=1,2,3}\\left(\\frac{g_k(\\mu )}{g_k(\\Lambda )}\\right)^{B_k}$ we can obtain the trilinear terms for scales below the messenger $M$ from Eqn.",
"(REF ).",
"The main challenge is the calculation of $\\partial Z_i/\\partial T$ .",
"Before we derive the final results involving all $y_t,y_b,y_\\tau $ and $g_3,g_2,g_1$ , we will study first the simplest case in which only the top Yukawa $\\alpha _t\\equiv y_t^2/4\\pi $ and $\\alpha _s\\equiv g_3^2/4\\pi $ are kept in the anomalous dimension.",
"The RGE equation for $\\alpha _t$ and $\\alpha _s$ takes the form $\\frac{d}{dt}\\ln \\alpha _t&=&\\frac{1}{\\pi }\\left(3\\alpha _t-\\frac{8}{3}\\alpha _s\\right)~,~~~~~~~~~~~~~\\frac{d}{dt}\\ln \\alpha _s=-\\frac{1}{2\\pi } b_3\\alpha _s~,$ Note the definition $b_3$ differs by a minus sign.",
"Define $A=\\ln \\left(\\alpha _t \\alpha _s^{-\\frac{16}{3b_3}}\\right)$ , the equation can be written as $\\frac{d}{dt} e^{-A}=-\\frac{3}{\\pi }\\alpha _s^{\\frac{16}{3b_3}}~,$ So we can exactly solve the differential equation to get $\\left[\\frac{\\alpha _t(\\mu )}{\\alpha _t(\\Lambda )}\\left( \\frac{\\alpha _s(\\mu )}{\\alpha _s(\\Lambda )}\\right)^{-\\frac{16}{3b_3}}\\right]^{-1}=1-\\frac{3\\alpha _t(\\Lambda )}{\\pi }\\frac{2\\pi }{\\frac{16}{3}-b_3}\\left[\\alpha _s(\\Lambda )^{-1}-\\left(\\frac{\\alpha _s(\\mu )}{\\alpha _s(\\Lambda )}\\right)^{\\frac{16}{3b_3}}\\alpha ^{-1}_s(\\mu )\\right].$ Expanding the expressions and neglect high order terms, we finally have $\\frac{\\partial }{\\partial T} \\left[\\ln {\\alpha _t(\\mu )}-\\ln {\\alpha _t(\\Lambda )}\\right]&\\approx &\\frac{\\partial }{\\partial T}\\left[-\\frac{8}{3\\pi }\\alpha _s(\\mu )+\\frac{3}{\\pi }\\alpha _t(\\mu )\\right]\\ln \\left(\\frac{\\Lambda }{\\mu }\\right).$ after calculations.",
"It can be observed that the expression within the square bracket is just the beta function of top Yukawa coupling.",
"Now we will calculate $\\partial Z_i/\\partial T$ with all $y_t,y_b,y_\\tau $ and $g_3,g_2,g_1$ taking into account in the expression.",
"Deduction of $\\partial Z_i/\\partial T$ without messenger deflections: From the form of wavefunction $Z_i(\\mu )=Z_i(M_G)\\prod \\limits _{l=y_t,y_b,y_\\tau } \\left(\\frac{y_l(\\mu )}{y_l(M_G)}\\right)^{A_l}\\prod \\limits _{k=1,2,3}\\left(\\frac{g_k(\\mu )}{g_k(M_G)}\\right)^{B_k}$ and renormalizatoin $Z=Z_0(1+\\delta Z)$ , we have $\\frac{\\partial \\ln e^{-K_0/3} Z_i}{\\partial T}&=&\\frac{\\partial }{\\partial T}\\ln e^{-K_0/3}Z_i(M_G)+\\frac{\\partial }{\\partial T} \\delta Z_i~,\\nonumber \\\\&=&\\frac{1-n_i}{T} +\\sum \\limits _{m=1,2}\\left[\\sum \\limits _{a}\\frac{\\partial g_{a;m}}{\\partial T}\\frac{\\partial \\delta Z_i}{\\partial g_{a;m}}+\\sum \\limits _{a,b,c} \\frac{\\partial \\ln y_{abc;m}}{\\partial T}\\frac{\\partial \\delta Z_i}{\\partial \\ln y_{abc;m}}\\right]~,\\nonumber $ with $m=1,2$ corresponding to the value at the scale $\\mu $ and the $GUT$ scale, respectively.",
"The derivative with respect to $g_m$ gives $\\frac{\\partial \\ln e^{-K_0/3} Z_i}{\\partial g_m(\\mu )}=\\frac{B_m}{g_m(\\mu )}~,~~~~~\\frac{\\partial \\ln e^{-K_0/3} Z_i}{\\partial g_m(M_G)}=-\\frac{B_m}{g_m(M_G)}~,$ and $\\frac{\\partial g_i(\\mu )}{\\partial T}=-\\frac{l_a T^{l_a-1}}{2}g^3_i(\\mu )~,~~~~~\\frac{\\partial g_i(M_G)}{\\partial T}=-\\frac{l_a T^{l_a-1}}{2}g^3_i(M_G)~.$ The derivative with respect to $y_l$ gives $\\frac{\\partial \\ln e^{-K_0/3} Z_i}{\\partial y_l(\\mu )}=\\frac{A_l}{y_l(\\mu )}~,~~~\\frac{\\partial \\ln e^{-K_0/3} Z_i}{\\partial y_l(M_G)}=-\\frac{A_l}{y_l(M_G)}~,$ and $\\frac{\\partial y_l(M_G)}{\\partial T}&=&-\\frac{y_l(M_G)}{2}\\left[\\frac{3-a_{ijk}}{T}\\right],~~~~~\\frac{\\partial \\ln \\alpha _{Y_{abc}}(\\mu )}{\\partial T}=-\\left[\\frac{3-a_{ijk}}{T}\\right]~~.$ From the beta function of the Yukawa couplings, we have $\\frac{\\partial \\ln \\alpha _{Y_{abc}}(\\mu )}{\\partial T}&=&\\frac{\\partial \\ln \\alpha _{Y_{abc}}(M_G)}{\\partial T}-\\frac{\\partial }{\\partial T}\\int \\limits _{\\mu }^{M_G}\\left( \\frac{d}{d\\ln \\mu ^\\prime } \\ln \\alpha _{Y_{abc}}\\right)~ d\\ln \\mu ^\\prime ~,\\nonumber \\\\&=&-\\frac{3-a_{abc}}{T}-\\frac{1}{2\\pi } \\int \\limits _{\\mu }^{M_G} d\\ln \\mu ^\\prime \\left(\\sum \\limits _{Y_{lmn}}c_{lmn}\\frac{\\partial }{\\partial T}\\alpha _{Y_{lmn}}(\\mu ^\\prime )+\\sum \\limits _{m}d_m\\frac{\\partial }{\\partial T}\\alpha _m(\\mu ^\\prime )\\right)~,\\nonumber \\\\&\\approx &-\\frac{3-a_{abc}}{T}+\\frac{1}{2\\pi }\\left[ \\sum \\limits _{Y_{lmn}}c_{lmn}\\frac{3-a_{lmn}}{T}\\alpha _{Y_{lmn}}(\\mu )+\\sum \\limits _{m}d_m\\frac{l_a}{T}\\frac{\\alpha _m^2(\\mu )}{\\alpha _m(M_G)}\\right]\\ln \\left(\\frac{{M_G}}{\\mu }\\right)~,\\nonumber $ with $a_{abc}=n_a+n_b+n_c$ .",
"So the derivative with respect to $T$ is given by $\\frac{\\partial }{\\partial T} \\delta Z_i&=&\\sum \\limits _{m=1,2}\\left[\\sum \\limits _{a}\\frac{\\partial \\ln \\alpha _{a;m}}{\\partial T}\\frac{\\partial \\delta Z_i}{\\partial \\ln \\alpha _{a;m}}+\\sum \\limits _{Y_{abc}} \\frac{\\partial \\ln \\alpha _{Y_{abc;m}}}{\\partial T}\\frac{\\partial \\delta Z_i}{\\partial \\ln \\alpha _{Y_{abc;m}}}\\right]~,\\nonumber \\\\&=&\\sum \\limits _{a}\\frac{B_a}{2}\\left[\\frac{\\partial }{\\partial T} \\ln \\left(\\frac{\\alpha _a(\\mu )}{\\alpha _a(\\Lambda )}\\right)\\right]+\\sum \\limits _{Y_{abc}}\\frac{A_{Y_{abc}}}{2}\\left[\\frac{\\partial }{\\partial T} \\ln \\left(\\frac{\\alpha _{Y_{abc}}(\\mu )}{\\alpha _{Y_{abc}}(\\Lambda )}\\right)\\right]~,\\nonumber \\\\&\\approx &\\sum \\limits _{a}\\frac{B_a}{2}\\left[\\frac{\\partial }{\\partial T}\\left(-\\frac{b_a}{2\\pi }\\alpha _a(\\mu )\\ln \\left(\\frac{\\Lambda }{\\mu }\\right)\\right) \\right]~\\nonumber \\\\&+&\\sum \\limits _{Y_{abc}}\\frac{A_{Y_{abc}}}{2}\\frac{1}{2\\pi }\\left[ \\sum \\limits _{Y_{lmn}}c_{lmn}\\frac{3-a_{lmn}}{T}\\alpha _{Y_{lmn}}(\\mu )-\\sum \\limits _{m}d_m\\frac{\\partial }{\\partial T}\\alpha _m(\\mu )\\right]\\ln \\left(\\frac{\\Lambda }{\\mu }\\right)~,\\nonumber \\\\$ We know from the expression of the wavefunction, the coefficients satisfy $\\sum \\limits _{Y_{abc}}\\frac{A_{Y_{abc}}}{2}d_m+ b_m \\frac{B_m}{2}=-\\frac{\\partial G_{Z_i}}{\\partial \\alpha _m},$ for coefficients of $\\alpha _m$ .",
"While the coefficients for Yukawa couplings $Y_{lmn}$ within $Z_i$ satisfy $\\sum \\limits _{Y_{abc}}\\frac{A_{Y_{abc}}}{2}c_{lmn}=-\\frac{\\partial G_{Z_i}}{\\partial \\alpha _{Y_{lmn}}},$ So the final results reduces to $\\frac{\\partial }{\\partial T} \\ln e^{-K_0/3}Z_i&\\approx &-\\frac{1}{2\\pi }\\left[ \\frac{d_{jk}^i}{2} \\frac{3-a_{Y_{ijk}}}{T}\\alpha _{Y_{ijk}}(\\mu )-2C_a(i)\\frac{l_a}{T}{\\alpha _a(\\mu )}\\right]\\ln \\left(\\frac{GUT}{\\mu }\\right)+\\frac{1-n_i}{T}~,\\nonumber \\\\&=&\\frac{1}{2\\pi }\\frac{\\partial }{\\partial T}\\left[\\frac{d_{jk}^i}{2}\\alpha _{Y_{ijk}}(\\mu )-2C_a(i)\\alpha _a(\\mu )\\right]\\ln \\left(\\frac{GUT}{\\mu }\\right)+\\frac{1-n_i}{T}$ with the expressions in the second square bracket being the anomalous dimension of $Z_i$ .",
"Deduction of $\\partial Z_i/\\partial T$ with messenger deflections: From the form of wavefunction $Z_i(\\mu )&=&Z_i({M_G})\\prod \\limits _{l=y_t,y_b,y_\\tau } \\left(\\frac{y_l(M)}{y_l({M_G})}\\right)^{A_l}\\prod \\limits _{k=1,2,3}\\left(\\frac{g_k(M)}{g_k({M_G})}\\right)^{B_k}\\prod \\limits _{k=y_U} \\left(\\frac{y_k(M)}{y_k({M_G})}\\right)^{C_k}\\nonumber \\\\&&~~~~~~~~\\prod \\limits _{l=y_t,y_b,y_\\tau } \\left(\\frac{y_l(\\mu )}{y_l(M)}\\right)^{A_l^\\prime }\\prod \\limits _{k=1,2,3}\\left(\\frac{g_k(\\mu )}{g_k(M)}\\right)^{B_k^\\prime }~,$ with $y_U$ the interactions involving the messengers which will be integrated below the messenger scale.",
"We have $\\frac{\\partial \\ln e^{-K_0/3} Z_i}{\\partial T}&=&\\frac{\\partial }{\\partial T}\\ln e^{-K_0/3}Z_i^0+\\frac{\\partial }{\\partial T} \\delta Z_i~,\\nonumber \\\\&=&\\left[\\sum \\limits _{g_a}\\frac{\\partial \\ln \\left(\\frac{g_{a}(\\mu )}{g_a(M)}\\right)}{\\partial T}\\frac{\\partial \\delta Z_i}{\\partial \\ln \\left(\\frac{g_{a}(\\mu )}{g_a(M)}\\right)}+\\sum \\limits _{y_{abc}} \\frac{\\partial \\ln \\left(\\frac{y_{abc}(\\mu )}{y_{abc}(M)} \\right)}{\\partial T}\\frac{\\partial \\delta Z_i}{\\partial \\ln \\left(\\frac{y_{abc}(\\mu )}{y_{abc}(M)} \\right)}\\right.~,\\nonumber \\\\&+&\\sum \\limits _{g_a}\\frac{\\partial \\ln \\left(\\frac{g_{a}(M)}{g_a({M_G})}\\right)}{\\partial T}\\frac{\\partial \\delta Z_i}{\\partial \\ln \\left(\\frac{g_{a}(M)}{g_a({M_G})}\\right)}+\\sum \\limits _{y_{abc}} \\frac{\\partial \\ln \\left(\\frac{y_{abc}(M)}{y_{abc}({M_G})} \\right)}{\\partial T}\\frac{\\partial \\delta Z_i}{\\partial \\ln \\left(\\frac{y_{abc}(M)}{y_{abc}({M_G})} \\right)},\\nonumber \\\\&+&\\left.\\sum \\limits _{y_U} \\frac{\\partial \\ln \\left(\\frac{y_{U}(M)}{y_{U}({M_G})} \\right)}{\\partial T}\\frac{\\partial \\delta Z_i}{\\partial \\ln \\left(\\frac{y_{U}(M)}{y_{U}({M_G})} \\right)}\\right]+\\frac{1-n_i}{T}~,\\nonumber \\\\$ with $m=1,2$ corresponding to the value at the scale $\\mu $ and the $GUT$ scale, respectively.",
"Using similar deductions for Yukawa couplings, we can obtain $\\frac{\\partial \\ln e^{-K_0/3} Z_i}{\\partial T}&=&\\frac{1-n_i}{T}-\\frac{1}{4\\pi }\\sum \\limits _{g_a} \\left( B_{a} b^\\prime _a \\frac{\\partial \\alpha _a(M)}{\\partial T}\\ln \\left(\\frac{M_G}{M}\\right)+B_{a}^\\prime b_a \\frac{\\partial \\alpha _a(M) }{\\partial T}\\ln \\left(\\frac{M }{\\mu }\\right)\\right)\\nonumber \\\\&+&\\sum \\limits _{y_{abc}\\in y_t,y_b,y_\\tau } A_{y_{abc}}\\ln \\left(\\frac{{M_G}}{M}\\right) \\frac{1}{4\\pi }\\left[ \\sum \\limits _{Y_{lmn}\\in y_t,y_b,y_\\tau }c_{lmn}\\frac{3-a_{lmn}}{T}\\alpha _{Y_{lmn}}(M)\\right.",
"~,\\nonumber \\\\&& \\left.~~~~~~~~~~~~~~~~~~~~~~~~~~~+\\sum \\limits _{ {Y}_{lmn}\\in y_U}\\tilde{c}_{lmn}\\frac{3-a_{U}}{T}\\alpha _{Y_U}(M)-\\sum \\limits _{g_m}d_m\\frac{\\partial }{\\partial T}\\alpha _m(M)\\right] ~.\\nonumber \\\\&+&\\sum \\limits _{y_{abc}\\in y_t,y_b,y_\\tau }A_{y_{abc}}^\\prime \\ln \\left(\\frac{\\mu }{M}\\right)\\frac{1}{4\\pi }\\left[ \\sum \\limits _{Y_{lmn}\\in y_t,y_b,y_\\tau }c_{lmn}\\frac{3-a_{lmn}}{T}\\alpha _{Y_{lmn}}(M)-\\sum \\limits _{g_m}d_m\\frac{\\partial }{\\partial T}\\alpha _m(M)\\right] ~.\\nonumber \\\\&+&\\sum \\limits _{y_U} C_{y_U} \\ln \\left(\\frac{M}{{M_G}}\\right)\\frac{1}{4\\pi }\\left[ \\sum \\limits _{\\tilde{Y}_{lmn}\\in y_t,y_b,y_\\tau }d_{lmn}\\frac{3-a_{lmn}}{T}\\alpha _{Y_{lmn}}(M)-\\sum \\limits _{g_m}f_m\\frac{\\partial }{\\partial T}\\alpha _m(M)~,\\right.\\nonumber \\\\&& \\left.~~~~~~~~~~~~~~~~~~~~~~~~~~~+\\sum \\limits _{\\tilde{Y}_{lmn}\\in y_U}\\tilde{d}_{lmn}\\frac{3-a_{U}}{T}\\alpha _{Y_U}(M)\\right]~.$ with the beta function for $y_t,y_b,y_\\tau $ Yukawa couplings $16\\pi ^2 \\beta _{Y_{abc}}(\\mu )&=&\\left\\lbrace \\begin{array}{c}\\sum \\limits _{Y_{lmn}\\in y_t,y_b,y_\\tau }c_{lmn}\\alpha _{Y_{lmn}}+\\sum \\limits _{Y_{lmn}\\in y_U}\\tilde{c}_{lmn}\\alpha _{Y_{U}}-\\sum \\limits _{g_m}d_m \\alpha _m~,~~~~~ \\mu \\gtrsim M,\\nonumber \\\\\\sum \\limits _{Y_{lmn}\\in y_t,y_b,y_\\tau }c_{lmn}\\alpha _{Y_{lmn}}-\\sum \\limits _{g_m}d_m \\alpha _m~,~~~~ \\mu \\lesssim M,\\nonumber \\\\ \\end{array}\\right.$ and the beta function for new messenger-matter $y_U$ Yukawa couplings $16\\pi ^2 \\beta _{Y_{U}}&=& \\sum \\limits _{\\tilde{Y}_{lmn}\\in y_t,y_b,y_\\tau }d_{lmn}\\alpha _{\\tilde{Y}_{lmn}}+\\sum \\limits _{\\tilde{Y}_{lmn}\\in y_U}\\tilde{d}_{lmn}\\alpha _{Y_{U}}-\\sum \\limits _{m}f_m \\alpha _m~.$ The coefficients satisfy $&&\\sum \\limits _{Y_{abc}}\\left(A_{y_{abc}}-A_{y_{abc}}^\\prime \\right) c_{lmn}+\\sum \\limits _{Y_U}C_{y_U}d_{lmn}=0~,~~~~~~({\\rm for} ~y_t,y_b,y_\\tau ~ {\\rm coefficients})\\nonumber \\\\&&\\sum \\limits _{Y_{abc}} A_{y_{abc}}\\tilde{c}_{lmn} + \\sum \\limits _{Y_U} C_{y_U} \\tilde{d}_{lmn}=0~,~~~~({\\rm for ~y_U~ coefficients})\\nonumber \\\\&& B_{m} b^\\prime _m+ \\sum \\limits _{Y_{abc}} A_{y_{abc}} d_m+\\sum \\limits _{Y_U} C_{Y_U} f_m=B^\\prime _{m} b_m+ \\sum \\limits _{Y_{abc}} A^\\prime _{y_{abc}} d_m$ and similarly for $g_m$ , the sum then reduces to the previous case.",
"So we have for $\\mu <M$ $&&\\frac{\\partial }{\\partial T} \\ln \\left(e^{-K_0/3} Z_i(\\mu )\\right)-\\frac{1-n_i}{T}~,\\\\& \\approx &-\\frac{1}{2\\pi }\\left[ \\frac{1}{2}d_{jk}^i \\frac{3-a_{Y_{ijk}}}{T}\\alpha _{Y_{ijk}}(\\mu )-2C_a(i)\\frac{l_a}{T}\\alpha _a(\\mu )\\right]\\ln \\left(\\frac{M_G}{\\mu }\\right)~.\\nonumber $ Note that the expressions within the square bracket agree with the anomalous dimension of $Z_i^-$ below the messenger threshold $M$ $G_i^-\\equiv \\frac{d Z_{i}^-}{d\\ln \\mu }\\equiv -\\frac{1}{8\\pi ^2}\\left( \\frac{1}{2} d_{kl}^i\\lambda ^2_{ikl}-2c_r^ig_r^2\\right).$ The $G_i^+$ , which is the anomalous dimension of $Z_i$ upon the messenger threshold $M$ , do not appear in the final expressions.",
"The dependence of $Z_i$ on messenger scale $M$ can be derived following the techniques [26], [27] developed in gauge mediated SUSY breaking (GMSB)[28] scenarios.",
"From the expressions of the wavefunction, we can obtain $\\frac{\\partial }{\\partial \\ln M}\\ln \\left[e^{-K_0/3}Z_i\\right]&=&\\frac{1}{4\\pi }\\sum \\limits _{g_k}\\left[(B_k-B_k^\\prime ) (b_k+N_F) \\alpha _k(M)+B_k^\\prime N_F \\alpha _k(\\mu )\\right]\\\\&+&\\sum \\limits _{Y_l}\\left[({A_l-A_l^\\prime })G^+_{Y_l}(\\ln M)+ {A_l^\\prime }\\frac{\\partial Y_l(\\ln \\mu ,M)}{\\partial \\ln M}\\right]+\\sum \\limits _{Y_U}\\left[{C_l} G^+_{Y_U}(\\ln M)\\right]~.\\nonumber $ So the main challenge is to calculate $\\partial \\ln Y_a(\\mu ,\\ln M)/\\partial \\ln M$ .",
"From the beta functions for Yukawa couplings upon and below the messenger thresholds, the Yukawa couplings at scale $\\mu <M$ is given as $\\ln Y_a(\\mu ,\\ln M)=\\ln Y_a({M_G})+\\int \\limits _{{M_G}}^{\\ln M} G^+_{Y_a}(t^\\prime ) dt^\\prime +\\int \\limits _{\\ln M}^{\\ln \\mu } G^-_{Y_a}(t^\\prime , \\ln M) dt^\\prime ~,$ with the Yukawa beta functions expressed as $\\beta _{Y_a}\\equiv G_{Y_a} &\\equiv &-\\frac{1}{2}\\sum \\limits _{i\\in a} G^i\\equiv \\frac{1}{4\\pi }\\left(\\frac{1}{2}\\tilde{d}_{kl}^i\\alpha _{\\lambda _{ikl}}-2\\tilde{c}_r\\alpha _r\\right)~,~\\nonumber \\\\G_i=\\frac{d \\ln Z_{i}}{d\\ln \\mu }&\\equiv &-\\frac{1}{2\\pi }\\left(\\frac{1}{2}d_{kl}^i\\alpha _{\\lambda _{ikl}}-2c_r^i\\alpha _r\\right).$ We can derive the Yukawa couplings dependence on $^{\\prime }\\ln M^{\\prime }$ at scale $\\mu <M$ $\\frac{\\partial }{\\partial \\ln M} \\ln Y_a(\\mu ,\\ln M)&=&\\left[ G^+_{Y_a}(\\ln M)-G_{Y_a}^-(\\ln M,\\ln M)\\right]+\\int \\limits _{\\ln M}^{\\ln \\mu } \\frac{\\partial }{\\partial \\ln M}G^-_{Y_a}(t^\\prime , \\ln M) dt^\\prime ~,\\nonumber \\\\&\\approx &\\Delta G_{Y_a}(\\ln M)-\\frac{1}{16\\pi ^2}\\left[\\tilde{d}_{kl}^i\\lambda _{ikl}(\\mu )\\Delta G_{\\lambda _{ikl}}-4\\tilde{c}_r \\frac{\\Delta b_r}{16\\pi ^2} g_r^4(\\mu )\\right]\\ln \\left(\\frac{M}{\\mu }\\right)~,\\nonumber $ In the case $\\Delta G=0$ in which no additional Yukawa couplings involving the messengers are present, we have $\\frac{\\partial }{\\partial \\ln M}\\ln Y_a(\\mu ,\\ln M)&\\approx & \\frac{\\tilde{c}_r}{4\\pi ^2} {\\Delta b_r} \\alpha _r^2(\\mu )\\ln \\left(\\frac{M}{\\mu }\\right)~.$ Note that at the messenger scale $\\frac{\\partial }{\\partial \\ln M}\\ln Y_a(\\ln M,\\ln M)=\\Delta G_a(\\ln M).$ The expressions takes a simple form at the scale $\\mu $ slightly below the messenger scale $M$ $&& A_{Y_{abc}}(\\mu \\lesssim M)-\\left(3-a_{abc}\\right)\\frac{F_T}{T+T^*}\\nonumber \\\\&=&\\sum \\limits _{l=a,b,c}\\left\\lbrace -\\frac{F_T}{T+T^*}\\frac{1}{2\\pi }\\left[ \\frac{1}{2}d_{jk}^i (3-a_{Y_{ijk}})\\alpha _{Y_{ijk}}(\\mu )-2C_a(i){l_a}\\alpha _a(\\mu )\\right]\\ln \\left(\\frac{GUT}{\\mu }\\right)\\right.\\nonumber \\\\&& \\left.+d F_\\phi \\frac{\\Delta G_i}{2}-\\frac{F_\\phi }{2} G_i^-\\frac{}{}\\right\\rbrace ~,\\nonumber $ with $\\Delta G_i\\equiv G_i^+-G_i^-$ [here $^{\\prime }G_i^+(G_i^-)^{\\prime }$ denotes respectively the anomalous dimension of $Z_i$ upon (below) the messenger threshold] the discontinuity of anomalous dimension across the messenger threshold."
],
[
"Soft Scalar Masses",
"The soft scalar masses are given as $-m^2_{soft}&=&\\left|\\frac{F_T}{2}\\frac{\\partial }{\\partial T}-\\frac{F_\\phi }{2}\\frac{\\partial }{\\partial \\ln \\mu }+d F_\\phi \\frac{\\partial }{\\partial \\ln X}\\right|^2 \\ln \\left[e^{-K_0/3}Z_i(\\mu ,X,T)\\right]~,\\nonumber \\\\&=&\\left(\\frac{|F_T|^2}{4}\\frac{\\partial ^2}{\\partial T\\partial T^*}+\\frac{F_\\phi ^2}{4}\\frac{\\partial ^2}{\\partial (\\ln \\mu )^2}+\\frac{d^2F^2_\\phi }{4}\\frac{\\partial }{\\partial (\\ln |X|)^2}-\\frac{F_TF_\\phi }{2}\\frac{\\partial ^2}{\\partial T\\partial \\ln \\mu }\\right.~\\nonumber \\\\&&~~+\\left.\\frac{dF_TF_\\phi }{2}\\frac{\\partial ^2}{\\partial T\\partial \\ln |X|}-\\frac{d F^2_\\phi }{2}\\frac{\\partial ^2}{\\partial \\ln |X|\\partial \\ln \\mu }\\right) \\ln \\left[e^{-K_0/3}Z_i(\\mu ,X,T)\\right],$ The new ingredients are the second derivative of $Z_i$ with respect to $T$ $&&\\frac{\\partial ^2}{\\partial T^2} \\ln \\left[e^{-3K_0} Z_i\\right]\\nonumber \\\\&=&-\\frac{1}{2\\pi }\\frac{\\partial }{\\partial T}\\left[\\frac{1}{2} d_{jk}^i \\frac{3-a_{Y_{ijk}}}{T}\\alpha _{Y_{ijk}}(\\mu )-2C_a(i)\\frac{l_a}{T}{\\alpha _a(\\mu )}\\right]\\ln \\left(\\frac{GUT}{\\mu }\\right)-\\frac{1-n_i}{T^2}~,\\nonumber \\\\&=&-\\frac{1}{2\\pi }\\left[\\frac{1}{2}d_{jk}^i \\frac{3-a_{Y_{ijk}}}{T}\\alpha _{Y_{ijk}}(\\mu )\\left[-\\frac{3-a_{Y_{ijk}}}{T}+\\frac{1}{2\\pi }\\left(\\frac{\\tilde{d}^p_{mn}}{2}\\frac{3-a_{Y_{ijk}}}{T}\\alpha _{Y_{mnp}}-2c_r\\frac{l_a}{T}{\\alpha _a}\\right)\\ln \\left(\\frac{GUT}{\\mu }\\right)\\right]\\right.\\nonumber \\\\&-& \\left.",
"\\frac{1}{2}d_{jk}^i \\frac{(3-a_{Y_{ijk}})}{T^2}\\alpha _{Y_{ijk}}(\\mu )- 2C_a(i)\\left(-\\frac{l_a}{T^2}{\\alpha _a(\\mu )}-\\frac{l_a^2}{T^2}\\frac{\\alpha _a^2(\\mu )}{\\alpha _a(GUT)}\\right)\\right]\\ln \\left(\\frac{GUT}{\\mu }\\right)-\\frac{1-n_i}{T^2}~.\\nonumber \\\\$ with the beta function of $Y_{ijk}$ given by $\\frac{d \\ln Y_{ijk}}{d\\ln \\mu }=\\frac{1}{16\\pi ^2}\\left[\\frac{\\tilde{d}^p_{mn}}{2}\\alpha _{Y_{mnp}}-2c_r^i{\\alpha _i}\\right].$ and $\\frac{\\alpha _a(\\mu )}{\\alpha _a(GUT)}=1-\\frac{b_a}{2\\pi }\\alpha _a(\\mu )\\ln \\left(\\frac{GUT}{\\mu }\\right).$ The other terms within Eqn.",
"(REF ) can be found in GMSB (not involving $\\partial T$ ) or calculated directly using Eqn.",
"(REF ) and Eqn.",
"(REF ) (involving $\\partial T$ ).",
"We list the analytical results of deflected mirage mediation in Appendix B."
],
[
"Analytical Results for Mirage Mediation",
"Equipped with the previous deduction, we can readily reproduce the ordinary mirage mediation results by setting $d\\rightarrow 0$ .",
"As the visible gauge fields originate from D7 branes and gauge coupling unification is always assumed, we adopt $l_a=1$ .",
"The following definitions are used $M_0\\equiv \\frac{F_T}{2T}\\equiv \\frac{F_\\phi }{\\alpha \\ln \\left(\\frac{M_{Pl}}{m_{3/2}}\\right)}\\approx \\frac{F_\\phi }{4\\pi ^2\\alpha }~.$ with the parameter $\\alpha $ defined as the ratio between the anomaly mediation and modulus mediation contributions and the approximation $\\ln ({M_{Pl}}/{m_{3/2}}) \\approx 4 \\pi ^2$ .",
"We have Gaugino mass: $M_i(\\mu )&=&l_a M_0\\frac{g_i^2(\\mu )}{g_a^2(GUT)}+\\frac{F_\\phi }{16\\pi ^2}b_i g_i^2(\\mu )~,\\nonumber \\\\&=&l_a M_0 \\left[ 1-\\frac{b_i}{8\\pi ^2}g_i^2(\\mu )\\ln \\frac{GUT}{\\mu }\\right]+\\frac{M_0}{4}\\alpha b_i g_i^2(\\mu )~.$ So we can see that at the scale $\\mu _{Mi}$ which satisfies $\\frac{1}{8\\pi ^2}\\ln \\left(\\frac{M_{GUT}}{\\mu _{Mi}}\\right)=\\frac{\\alpha }{4}.$ the gaugino masses unify at such $^{\\prime }mirage^{\\prime }$ unification scale $\\mu _{Mi}=M_{GUT} e^{-2\\alpha \\pi ^2}\\approx M_{GUT} \\left(\\frac{m_{3/2}}{M_{Pl}}\\right)^{\\frac{\\alpha }{2}}.$ Trilinear Term: $&& A_{Y_{abc}}(\\mu \\lesssim M)\\nonumber \\\\&=&\\sum \\limits _{l=a,b,c}\\left\\lbrace -M_0\\frac{1}{2\\pi }\\left[ \\frac{1}{2}d_{jk}^i (3-a_{Y_{ijk}})\\alpha _{Y_{ijk}}(\\mu )-2C_a(i){l_a}\\alpha _a(\\mu )\\right]\\ln \\left(\\frac{GUT}{\\mu }\\right)\\right.\\nonumber \\\\&& +\\frac{F_\\phi }{4\\pi }\\left[\\frac{1}{2}d_{jk}^i \\alpha _{Y_{ijk}}(\\mu )-2C_a(i)\\alpha _a(\\mu )\\right]+\\left(3-a_{abc}\\right)M_0~,\\nonumber $ In case the effect of Yukawa couplings are negligible or $a_{Y_{ijk}}=2$ , the trilinear term also $\"unify\"$ at a mirage scale at which the last two terms cancel $\\frac{1}{2\\pi }\\ln \\left(\\frac{M_{GUT}}{\\mu _{Mi}}\\right)=\\pi {\\alpha }.$ which is just the mirage scale for gaugino mass $\"unification\"$ .",
"Soft Scalar Masses: $-m_i^2&=&\\frac{M_0^2}{2\\pi }\\ln \\left(\\frac{M_{GUT}}{\\mu }\\right)\\left\\lbrace \\frac{d_{jk}^i}{2}\\left( q_{Y_{ijk}}^2+q_{Y_{ijk}}\\right)\\alpha _{Y_{ijk}}(\\mu )- 2C_a(i)\\left({l_a}+{l_a^2}\\right)\\alpha _a\\right.\\nonumber \\\\&+&\\left.\\frac{1}{2\\pi }\\left[\\frac{d_{jk}^i}{2}\\alpha _{Y_{ijk}}(\\mu )\\left(-\\frac{\\tilde{d}^p_{mn}}{2}q_{Y_{mnp}}\\alpha _{Y_{mnp}}+2c_r l_a{\\alpha _a}\\right)+2C_a(i) b_a \\alpha ^2_a\\right]\\ln \\left(\\frac{GUT}{\\mu }\\right) \\right\\rbrace \\nonumber \\\\&+&\\frac{M_0 F_\\phi }{2\\pi }\\left[\\frac{d_{jk}^i}{2}\\alpha _{Y_{ijk}}\\left(-q_{Y_{ikl}}+\\frac{1}{2\\pi }\\left[\\frac{\\tilde{d}^p_{mn}}{2} {q_{Y_{mnp}}}\\alpha _{Y_{mnp}}-2c_r {l_r} {\\alpha _r}\\right]\\ln \\left(\\frac{M_{GUT}}{\\mu }\\right)\\right) +2C_a(i) l_a \\frac{\\alpha _a^2}{\\alpha _a(GUT)}\\right]\\nonumber \\\\&-&\\frac{F_\\phi ^2}{8\\pi }\\left[\\frac{d_{jk}^i}{2}\\frac{1}{2\\pi }\\left(\\frac{\\tilde{d}^p_{mn}}{2}\\alpha _{Y_{mnp}}-2c_r{\\alpha _r}\\right) \\alpha _{Y_{ijk}}- 2C_a(i)\\frac{b_a}{2\\pi }\\alpha _a^2\\right]-(1-n_i)M_0^2.$ with $q_{Y_{ijk}}\\equiv 3-(n_i+n_j+n_k)=3-a_{ijk}$ .",
"Again, we can check that for $q_{Y_{ijk}}=1$ or negligible Yukawa couplings, the soft scalar masses apparent unify at $\\mu _{Mi}$ defined above $&&-m_i^2+ (1-n_i)M_0^2\\nonumber \\\\&=& \\pi \\alpha M_0^2 \\left\\lbrace 2\\frac{d_{jk}^i}{2}\\alpha _{Y_{ijk}}(\\mu )- 4C_a(i)\\alpha _a+\\frac{1}{2\\pi }\\left[\\frac{d_{jk}^i}{2}\\alpha _{Y_{ijk}}\\left(-\\frac{\\tilde{d}^p_{mn}}{2}\\alpha _{Y_{mnp}}+2c_r{\\alpha _a}\\right)+2C_a(i) b_a \\alpha ^2_a\\right]2\\pi ^2\\alpha \\right\\rbrace \\nonumber \\\\&+&2\\pi \\alpha M_0^2\\left[\\frac{d_{jk}^i}{2}\\alpha _{Y_{ijk}}\\left(-1+\\frac{1}{2\\pi }\\left[\\frac{\\tilde{d}^p_{mn}}{2}\\alpha _{Y_{mnp}}-2c_r^i {\\alpha _i}\\right]2\\pi ^2\\alpha \\right) +2C_a(i) \\left(\\alpha _a-\\frac{1}{2\\pi }b_a\\alpha _a^2 2\\pi ^2\\alpha \\right)\\right]\\nonumber \\\\&-&2\\pi ^3\\alpha ^2M_0^2\\left[\\frac{d_{jk}^i}{2}\\frac{1}{2\\pi }\\left(\\frac{\\tilde{d}^p_{mn}}{2}\\alpha _{Y_{mnp}}-2c_r^i{\\alpha _i}\\right) \\alpha _{Y_{ijk}}- 2C_a(i)\\frac{b_a}{2\\pi }\\alpha _a^2\\right]\\nonumber \\\\&=&0.$ The subleading terms within $\\partial ^2 Z_i/\\partial T^2$ are crucial for the exact cancelation of anomaly mediation and RGE effects.",
"So the numerical results of $^{\\prime }mirage^{\\prime }$ unification can be proved rigourously with our analytical expressions."
],
[
"Deflection in Mirage Mediation From The Kahler Potential",
"It is well known that AMSB is bothered by tachyonic slepton problems.",
"Such a problem in AMSB can be solved by the deflection of RGE trajectory with the introduction of messenger sector.",
"There are two possible ways to deflect the AMSB trajectory with the presence of messengers, either by pseudo-moduli field[10] or holomorphic terms (for messengers) in the Kahler potential[29].",
"Mirage mediation is a typical mixed modulus-anomaly mediation scenario.",
"So the messenger sector, which can give additional gauge or Yukawa mediation contributions, can also be added in the Kahler potential.",
"The Kahler potential involving the vector-like messengers $\\bar{P}_i, P_i$ contain the ordinary kinetic terms as well as new holomorphic terms $K\\supseteq \\phi ^\\dagger \\phi \\left[ Z_{P_i,\\bar{P}_i}(T^\\dagger ,T)\\left( P_i^\\dagger P_i+\\bar{P}_i^\\dagger \\bar{P}_i\\right)+\\left(\\tilde{Z}_{P_i,\\bar{P}_i}(T^\\dagger ,T)c_P \\bar{P}_i P_i+h.c.\\right)\\right]~,$ with $Z_{P_i,\\bar{P}_i}(T^\\dagger ,T)=\\frac{1}{(T+T^\\dagger )^{{n}_{P}}}~,~~~\\tilde{Z}_{P_i,\\bar{P}_i}(T^\\dagger ,T)=\\frac{1}{(T+T^\\dagger )^{\\tilde{n}_{P}}}.$ After normalizing and rescaling each superfield with the compensator field $\\Phi \\rightarrow \\phi \\Phi $ and substituting the F-term VEVs of the compensator field $\\phi =1+F_\\phi \\theta ^2$ , the relevant Kahler potential reduces to $W=\\int d^4 \\theta \\frac{\\phi ^\\dagger }{\\phi }\\frac{1}{(T+T^\\dagger )^{\\tilde{n}_P-n_P}} \\left( c_P R \\bar{P}{P}\\right),$ For simply, we define $\\tilde{n}_P-n_P\\equiv a_P$ .",
"Especially, $a_P=n_P$ for $\\tilde{n}_P=0$ .",
"The SUSY breaking effects can be taken into account by introducing a spurion superfields $R$ with with the spurion VEV as $R\\equiv M_R+\\theta ^2 F_R=\\frac{1}{(2T)^{a_P}}\\left(F_\\phi -\\frac{a_P}{2T}F_T\\right)+\\theta ^2 \\left[{a_P(a_P+1)}\\frac{ |F_T|^2}{4T^2}-|F_\\phi |^2\\right].$ with the value of the deflection parameter $d\\equiv \\frac{F_R}{M_R F_\\phi }-1~,$ depending on the choice of $a_P$ and $\\alpha $ which gives $d=-2$ for $a_P=0$ .",
"We can see that adding messenger sector in the Kahler potential within mirage mediation will display a new feature in contrast to the AMSB case which always predicts $d=-2$ .",
"The appearance of spurion messenger threshold will affect the AMSB RGE trajectory after integrating out the heavy messenger modes.",
"The soft SUSY breaking parameters can be obtained by substituting $^{\\prime }d^{\\prime }$ into the general formula given in the appendix.",
"Note that we can derive the final results directly with its low energy analytical expressions.",
"Besides, we can also add messenger-matter mixing to induce new Yukawa couplings between the messengers and the MSSM fields.",
"In this case, new Yukawa mediated contributions will also contribute to the low energy soft SUSY parameters (See Ref.",
"[31] for an example in AMSB)."
],
[
"Deflected Mirage Mediation With Messenger-Matter Interactions",
"In ordinary deflected mirage mediation SUSY breaking scenarios, additional messengers are introduced merely to amend the gauge beta functions which will subsequently feed into the low energy soft SUSY breaking parameters.",
"In general, it is possible that the messengers will share some new Yukawa-type interactions with the visible (N)MSSM superfields, which subsequently will appear in the anomalous dimension of the superfields and contribute to the low energy soft SUSY breaking parameters.",
"Such realizations have analogs in AMSB (see [31]) and can be readily extended to include the modulus mediation contributions.",
"Similar to the deflected mirage mediation scenarios, the superpotential include possible pseudo-modulus superfields $X$ , the relevant nearly flat superpotential $W(X)$ to determine the deflection and a new part that includes messenger-matter interactions $W_{mm}=\\lambda _{\\phi ij}X Q_iQ_j+y_{ijk}Q_i Q_j Q_k+W(X)~.$ with the Kahler potential $K_m=Z_U\\left(T+T^\\dagger ,\\frac{\\mu }{\\sqrt{\\phi ^\\dagger \\phi }}\\right)\\frac{1}{(T+T^\\dagger )^{n_{Q_i}}} Q_i^\\dagger Q_i~,$ Here $^{\\prime }\\phi ^{\\prime }$ denotes the compensator field with Weyl weight 1.",
"The indices $^{\\prime }i,j^{\\prime }$ run over all MSSM and messenger fields and the subscripts $^{\\prime }U,D^{\\prime }$ denote the case upon and below the messenger threshold, respectively.",
"After integrating out the heavy messenger fields, the visible sector superfields $Q_a$ will receive wavefunction normalization ${\\cal L}=\\int d^4\\theta Q_a^\\dagger Z_D^{ab}(T+T^\\dagger ,\\frac{\\mu }{\\sqrt{\\phi ^\\dagger \\phi }},\\sqrt{\\frac{X^\\dagger X}{\\phi ^\\dagger \\phi }}) Q_b+\\int d^2\\theta y_{abc}Q^aQ^bQ^c~,$ which can give additional contributions to soft supersymmetry breaking parameters.",
"Here the analytic continuing threshold superfield $^{\\prime }X^{\\prime }$ will trigger SUSY breaking mainly from the anomaly induced SUSY breaking effects with the form $<X>=M+\\theta ^2 F_X$ .",
"So we have $\\tilde{X}\\equiv \\frac{X}{\\phi }&=& \\frac{M+F_X\\theta ^2}{1+F_\\phi \\theta ^2}\\equiv M(1+d F_\\phi \\theta ^2),$ with the value of the deflection parameter $^{\\prime }d^{\\prime }$ determined by the form of superpotential $W(X)$ .",
"Integrating out the messengers, the messenger-matter interactions will cause the discontinuity of the anomalous dimension upon and below the threshold.",
"Such discontinuity will appear not only directly in the expressions for the trilinear couplings but also indirectly in the soft scalar masses.",
"For example, the trilinear couplings at the messenger scale receive additional contributions $\\left.\\Delta A_{ijk}\\right|_{\\mu =M}&=&\\sum \\limits _{a=i,j,k} \\frac{d}{2} F_\\phi \\frac{\\partial }{\\partial \\ln |X|} \\left.\\ln \\left[e^{-K_0/3}Z_a(\\mu ,X,T)\\right]\\right|_{\\mu =M}\\nonumber \\\\&=&\\frac{d}{2} F_\\phi \\sum \\limits _{a=i,j,k} \\left.\\Delta G_{i}\\right|_{\\mu =M}.$ We know that large trilinear couplings, especially $A_t$ , is welcome in low energy phenomenological studies to reduce fine tuning and increase the Higgs mass.",
"So the introduction of messenger-matter interactions can open new possibilities for mirage phenomenology."
],
[
"Conclusions",
"We derive explicitly the soft SUSY breaking parameters at arbitrary low energy scale in the (deflected) mirage type mediation scenarios with possible gauge or Yukawa mediation contributions.",
"Based on the Wilsonian effective action after integrating out the messengers, we obtain analytically the boundary value (at the GUT scale) dependencies of the effective wavefunctions and gauge kinetic terms.",
"Note that the messenger scale dependencies of the effective wavefunctions and gauge kinetic terms had already been discussed in GMSB.",
"The RGE boundary value dependencies, which is a special feature in (deflected) mirage type mediation, is the key new ingredients in this study.",
"The appearance of $^{\\prime }mirage^{\\prime }$ unification scale in mirage mediation is proved rigorously with our analytical results.",
"We also discuss briefly the new features in deflected mirage mediation scenario in the case the deflection comes purely from the Kahler potential and the case with messenger-matter interactions.",
"We should note that our approach is in principle different from that of Ref.",
"[23] in which the soft SUSY breaking parameters are obtained by numerical RGE evolution, matching and threshold corrections.",
"For example, mixed gauge-modulus mediation contributions, which will not appear in previous approach, will be necessarily present for the soft scalar masses in our approach."
],
[
"Acknowledgement",
"This work was supported by the Natural Science Foundation of China under grant numbers 11675147,11775012; by the Innovation Talent project of Henan Province under grant number 15HASTIT017.",
"We can construct the RGE invariants $\\frac{d}{dt} \\ln Z_i=\\sum \\limits _{l=y_t,y_b,y_\\tau } A_l \\frac{d \\ln y_{l}}{dt} +\\sum \\limits _{l=g_3,g_2,g_1} B_l \\frac{d \\ln g_{l}}{dt}~,$ by solving the equation in the basis of $(y_t^2,y_b^2,y_\\tau ^2,g_3^2,g_2^2,g_1^2)$ $\\left(\\begin{array}{cccccc}6&1&0&0&0&0\\\\1&6&3&0&0&0\\\\0&1&4&0&0&0\\\\-\\frac{16}{3}&-\\frac{16}{3}&0&b_3&0&0\\\\-3&-3&-3&0&b_2&0\\\\-\\frac{13}{15}&-\\frac{7}{15}&-\\frac{9}{5}&0&0&b_1\\end{array}\\right)\\left(\\begin{array}{c}A_t\\\\A_b\\\\A_\\tau \\\\B_3\\\\B_{2}\\\\B_{1}\\end{array}\\right)=\\left(\\begin{array}{c}-2c_1\\\\-2c_2\\\\-2c_3\\\\-2d_1\\\\-2d_2\\\\-2d_3\\end{array}\\right)~£¬$ with $c_1,c_2,c_3,d_1,d_2,d_3$ the relevant coefficients of $(y_t^2,y_b^2,y_\\tau ^2,g_3^2,g_2^2,g_1^2)$ within the anomalous dimension.",
"So from $\\frac{d}{dt}\\left[ Z_i(\\mu )\\prod \\limits _{l=y_t,y_b,y_\\tau } [y_l(\\mu )]^{-A_l}\\prod \\limits _{k=1,2,3}[g_k(\\mu )]^{-B_k}\\right]=0~,$ we have $Z_i(\\mu )=Z_i(\\Lambda )\\prod \\limits _{l=y_t,y_b,y_\\tau } \\left(\\frac{y_l(\\mu )}{y_l(\\Lambda )}\\right)^{A_l}\\prod \\limits _{k=1,2,3}\\left(\\frac{g_k(\\mu )}{g_k(\\Lambda )}\\right)^{B_k}$ The general expressions of wavefunction at ordinary scale $\\mu $ below the messenger scale $M$ are given as $Z_i(\\mu )&=&Z_i(\\Lambda )\\prod \\limits _{l=y_t,y_b,y_\\tau } \\left(\\frac{y_l(M)}{y_l(\\Lambda )}\\right)^{A_l}\\prod \\limits _{k=1,2,3}\\left(\\frac{g_k(M)}{g_k(\\Lambda )}\\right)^{B_k}\\prod \\limits _{k=y_U} \\left(\\frac{y_k(M)}{y_k(\\Lambda )}\\right)^{C_k}\\nonumber \\\\&&~~~~~~~~\\prod \\limits _{l=y_t,y_b,y_\\tau } \\left(\\frac{y_l(\\mu )}{y_l(M)}\\right)^{A_l^\\prime }\\prod \\limits _{k=1,2,3}\\left(\\frac{g_k(\\mu )}{g_k(M)}\\right)^{B_k^\\prime }~,$ with $y_U$ the interactions involving the messengers which will be integrated below the messenger scale.",
"The coefficients are listed in Table.REF and Table.REF .",
"Table: Relevant coefficients in wavefunction expansion with N F =0N_F=0 messengers.Table: The coefficients with N F =0,1,2,4N_F=0,1,2,4 messengers without new Yukawa couplings involving the messengers-matter interactions.The coefficients for y t ,y b ,y τ y_t,y_b,y_\\tau , namely A 1 (y t )A_1(y_t),A 2 (y b )A_2(y_b),A 3 (y τ )A_3(y_\\tau ), are the same as the case N F =1N_F=1.In order to show some essential features of our effective theory results, we list the predicted soft SUSY breaking parameters in deflected mirage mediation mechanism with $N_F$ messengers in ${\\bf 5}\\oplus \\bar{\\bf 5}$ representations of SU(5).",
"At energy $\\mu $ below the messenger thresholds, we have The gaugino masses: $M_i(\\mu )=l_a M_0\\frac{g_i^2(\\mu )}{g_a^2(GUT)}+\\frac{F_\\phi }{16\\pi ^2}b_i g_i^2(\\mu )-d \\frac{F_\\phi }{16\\pi ^2}N_F g_i^2(\\mu )~.$ with $(b_3~,b_2~,b_1)=(-3,~1,\\frac{33}{5}).$ The trilinear couplings $A_t,A_b$ and $A_\\tau $ : Note that at the messenger scale, the third contribution $\\partial _X Z_i$ vanishes.",
"The trilinear $A_t$ term is given at arbitrary low energy scale $\\mu < M$ $&& A_t(\\mu )-q_{y_t}M_0\\\\&=&\\frac{M_0}{2\\pi }\\left[6\\frac{q_{y_t}}{2}\\alpha _{y_t}(\\mu )+\\frac{q_{y_b}}{2}\\alpha _{y_b}(\\mu )-\\frac{16}{3}l_3\\alpha _3(\\mu )-3l_2\\alpha _2(\\mu )-\\frac{13}{15}l_1\\alpha _1(\\mu )\\right]\\ln \\left(\\frac{M_{GUT}}{\\mu }\\right)~\\nonumber \\\\&+&\\frac{F_\\phi }{4\\pi }\\left[6\\alpha _{y_t}(\\mu )+\\alpha _{y_b}(\\mu )-\\frac{16}{3}\\alpha _3(\\mu )-3\\alpha _2(\\mu )-\\frac{13}{15}\\alpha _1(\\mu ) \\right]+\\delta _G~,$ Note that additional GMSB-type contributions are $\\delta _G&=&d\\frac{F_\\phi }{8\\pi }\\sum \\limits _{k=1,2,3}\\sum \\limits _{F=Q_L^3,U_3,H_u}\\left[\\frac{}{}(B_k(F)-B_k^\\prime (F)) (b_k+N_F) \\alpha _k(M)+B_k^\\prime (F) N_F \\alpha _k(\\mu )\\right]~\\nonumber \\\\&+&d\\frac{F_\\phi }{8\\pi }\\sum \\limits _{y_l=y_t,y_b,y_\\tau }\\sum \\limits _{k=1,2,3}{A_l^\\prime }\\frac{1}{4\\pi ^2} N_F\\tilde{c}_r(y_l)\\alpha ^2_r(\\mu )\\ln \\left(\\frac{M}{\\mu }\\right) ~\\nonumber \\\\&=&d\\frac{F_\\phi }{8\\pi }\\left[-2\\frac{1}{8\\pi ^2} N_F\\left( \\frac{16}{3}\\alpha ^2_3(\\mu )+3\\alpha ^2_2(\\mu )+\\frac{13}{15}\\alpha ^2_1(\\mu )\\right)\\ln \\left(\\frac{M}{\\mu }\\right)\\right]~,$ with $2\\tilde{c}_r(y_l)$ the coefficients of $g_r^2$ within $-16\\pi ^2 \\beta _{y_l}$ and $\\sum \\limits _{F=Q_L^3,U_3,H_u}B_k(F)=\\sum \\limits _{F=Q_L^3,U_3,H_u}B_k^\\prime (F)=0.$ The trilinear $A_b$ term is $&& A_b(\\mu )-q_{y_b}M_0\\\\&=&\\frac{M_0}{2\\pi }\\left[\\frac{q_{y_t}}{2}\\alpha _{y_t}(\\mu )+6\\frac{q_{y_b}}{2}\\alpha _{y_b}(\\mu )+\\frac{q_{y_\\tau }}{2}\\alpha _{y_\\tau }(\\mu )-\\frac{16}{3}l_3\\alpha _3(\\mu )-3l_2\\alpha _2(\\mu )-\\frac{7}{15}l_1\\alpha _1(\\mu )\\right]\\ln \\left(\\frac{M_{GUT}}{\\mu }\\right)~\\nonumber \\\\&+&\\frac{F_\\phi }{4\\pi }\\left[\\alpha _{y_t}(\\mu )+6\\alpha _{y_b}(\\mu )+\\alpha _{y_\\tau }(\\mu )-\\frac{16}{3}\\alpha _3(\\mu )-3\\alpha _2(\\mu )-\\frac{7}{15}\\alpha _1(\\mu ) \\right]~\\nonumber \\\\&+&d\\frac{F_\\phi }{8\\pi }\\left[-2\\frac{1}{8\\pi ^2} N_F\\left( \\frac{16}{3}\\alpha ^2_3(\\mu )+3\\alpha ^2_2(\\mu )+\\frac{7}{15}\\alpha ^2_1(\\mu )\\right)\\ln \\left(\\frac{M}{\\mu }\\right)\\right]~.$ The trilinear $A_\\tau $ term is $&& A_\\tau (\\mu )-q_{y_\\tau }M_0\\\\&=&\\frac{M_0}{2\\pi }\\left[3\\frac{q_{y_b}}{2}\\alpha _{y_b}(\\mu )+4\\frac{q_{y_\\tau }}{2}\\alpha _{y_\\tau }(\\mu )-3l_2\\alpha _2(\\mu )-\\frac{9}{5}l_1\\alpha _1(\\mu )\\right]\\ln \\left(\\frac{M_{GUT}}{\\mu }\\right)~\\nonumber \\\\&+&\\frac{F_\\phi }{4\\pi }\\left[3\\alpha _{y_b}(\\mu )+4\\alpha _{y_\\tau }(\\mu )-3\\alpha _2(\\mu )-\\frac{9}{5}\\alpha _1(\\mu ) \\right]~\\nonumber \\\\&+&d\\frac{F_\\phi }{8\\pi }\\left[-2\\frac{1}{8\\pi ^2} N_F\\left( 3\\alpha ^2_2(\\mu )+\\frac{9}{5}\\alpha ^2_1(\\mu )\\right)\\ln \\left(\\frac{M}{\\mu }\\right)\\right]~.$ The soft SUSY breaking scalar masses are parameterized by several terms: $&& -m_0^2=-(1-n_i)M_0^2+\\delta _I+\\delta _{II}+\\delta _{III}+\\delta _{IV}+\\delta _{V}.~$ The anomalous dimension of $Z_i$ is supposed to take the form $G^i\\equiv \\frac{d \\ln Z_{i}}{d\\ln \\mu }=-\\frac{1}{2\\pi }\\left(\\frac{1}{2}d_{kl}^i\\alpha _{\\lambda _{ikl}} -2C_a(i)\\alpha _a\\right).$ with $\\alpha _{\\lambda _{ikl}}=\\lambda ^2_{ikl}/4\\pi $ and $\\alpha _a=g_a^2/4\\pi $ .",
"Pure modulus mediation contributions $\\delta _I&=&\\frac{M_0^2}{2\\pi }\\ln \\left(\\frac{M_{GUT}}{\\mu }\\right)\\left\\lbrace \\frac{d_{jk}^i}{2}\\left( q_{Y_{ijk}}^2+q_{Y_{ijk}}\\right)\\alpha _{Y_{ijk}}(\\mu )- 2C_a(i)\\left({l_a}+{l_a^2}\\right)\\alpha _a\\right.\\nonumber \\\\&+&\\left.\\frac{1}{2\\pi }\\left[\\frac{d_{jk}^i}{2}\\alpha _{Y_{ijk}}(\\mu )\\left(-\\frac{\\tilde{d}^p_{mn}}{2}q_{Y_{mnp}}\\alpha _{Y_{mnp}}+2c_r l_a{\\alpha _a}\\right)+2C_a(i) b_a \\alpha ^2_a\\right]\\ln \\left(\\frac{GUT}{\\mu }\\right) \\right\\rbrace \\nonumber \\\\$ Pure anomaly mediation contributions $\\delta _{II}&=&\\frac{F_\\phi ^2}{4}\\frac{\\partial ^2}{\\partial (\\ln \\mu )^2}\\ln \\left[e^{-K_0/3}Z_i\\right]\\\\&=&-\\frac{F_\\phi ^2}{8\\pi }\\frac{\\partial }{\\partial (\\ln \\mu )}\\left[\\frac{1}{2}d_{kl}^i\\alpha _{\\lambda _{ikl}} -2C_a(i)\\alpha _a\\right]~,\\nonumber \\\\&=&-\\frac{F_\\phi ^2}{8\\pi } \\left[\\frac{1}{2}d_{kl}^i \\alpha _{\\lambda _{ikl}}2 G^-_{\\lambda _{ikl}} -2C_a(i)\\frac{1}{2\\pi }b_a\\alpha ^2_a\\right].$ with the beta function for Yukawa coupling $\\lambda _{ikl}$ being $\\frac{d \\ln \\lambda _{ikl}}{d \\ln \\mu }=G_{\\lambda _{ikl}}=\\frac{1}{4\\pi }\\left[ \\frac{1}{2}{d^p_{mn}}\\alpha _{\\lambda _{mnp}}-2c_r \\alpha _r \\right]~.$ Pure gauge mediation contributions As no new interactions involving the messengers are present, we have $\\frac{\\partial }{\\partial \\ln M}\\ln \\left[e^{-K_0/3}Z_i\\right]&=&\\frac{1}{4\\pi }\\sum \\limits _{g_k}\\left[(B_k-B_k^\\prime ) (b_k+N_F) \\alpha _k(M)+B_k^\\prime N_F \\alpha _k(\\mu )\\right]\\nonumber \\\\&+&\\sum \\limits _{Y_l} {A_l^\\prime }\\frac{\\tilde{c}_r}{4\\pi ^2} {\\Delta b_r} \\alpha _r^2(\\mu )\\ln \\left(\\frac{M}{\\mu }\\right)~,$ So $\\delta _{III}&=&d^2\\frac{F_\\phi ^2}{4}\\frac{\\partial ^2}{\\partial (\\ln M)^2}\\ln \\left[e^{-K_0/3}Z_i\\right]\\\\&=&d^2\\frac{F_\\phi ^2}{32\\pi ^2}\\left\\lbrace \\sum \\limits _{g_k}\\left[(B_k-B_k^\\prime ) (b_k+N_F)^2 \\alpha ^2_k(M)+B_k^\\prime N_F^2 \\alpha ^2(\\mu )\\right]+\\sum \\limits _{Y_l} {A_l^\\prime } \\frac{\\tilde{c}_r}{4\\pi ^2} {\\Delta b_r} \\alpha _r^2(\\mu )\\right\\rbrace ~.\\nonumber $ Here $\\frac{\\partial }{\\partial \\ln M}\\alpha _k(M)&=&\\frac{b_k^+}{2\\pi }\\alpha _k(M)~,\\nonumber \\\\\\frac{\\partial }{\\partial \\ln M}\\alpha _k(\\mu , M)&=&\\frac{b_k^+-b_k^-}{2\\pi }\\alpha _k(\\mu ,M)\\equiv \\frac{\\Delta b_k }{2\\pi }\\alpha _k(\\mu ,M)~,$ The gauge-anomaly interference term $\\delta _{IV}&=&-\\frac{d F^2_\\phi }{2}\\frac{\\partial ^2}{\\partial \\ln M\\partial \\ln \\mu }\\ln \\left[e^{-K_0/3}Z_i(\\mu ,X,T)\\right]~,\\nonumber \\\\&=&-\\frac{d F_\\phi ^2}{2}\\frac{\\partial }{\\partial \\ln \\mu }\\left\\lbrace \\frac{1}{4\\pi }\\sum \\limits _{g_k}\\left[(B_k-B_k^\\prime ) (b_k+N_F) \\alpha _k(M)+B_k^\\prime N_F \\alpha _k(\\mu )\\right]\\right.\\nonumber \\\\&&~~~~~~~~~~~~~~~+\\left.\\sum \\limits _{Y_l} {A_l^\\prime }\\frac{\\tilde{c}_r}{4\\pi ^2} {\\Delta b_r} \\alpha _r^2(\\mu )\\ln \\left(\\frac{M}{\\mu }\\right)\\right\\rbrace ,~\\nonumber \\\\&=&-\\frac{d F_\\phi ^2}{16\\pi ^2} B_k^\\prime b_k N_F \\alpha ^2_k(\\mu )-d F_\\phi ^2\\frac{\\tilde{c}_r}{8\\pi ^2}{A_l^\\prime }{\\Delta b_r} \\alpha _r^2(\\mu ).$ The modulus-anomaly and modulus-gauge interference terms are given as $\\delta _{V}&=&-\\frac{F_T F_\\phi }{2}\\frac{\\partial ^2}{\\partial T\\partial \\ln \\mu }\\ln \\left[e^{-K_0/3}Z_i\\right]+\\frac{d F_T F_\\phi }{2}\\frac{\\partial ^2}{\\partial T\\partial \\ln |X|}\\ln \\left[e^{-K_0/3}Z_i\\right]~,\\nonumber \\\\&=&\\frac{F_T F_\\phi }{4\\pi }\\frac{\\partial }{\\partial T}\\left[\\frac{1}{2}d_{kl}^i\\alpha _{\\lambda _{ikl}}-2C_a(i)\\alpha _a\\right]\\nonumber \\\\&+&\\frac{d F_T F_\\phi }{2}\\frac{\\partial }{\\partial T}\\left\\lbrace \\frac{1}{4\\pi }\\sum \\limits _{g_k}\\left[(B_k-B_k^\\prime ) (b_k+N_F) \\alpha _k(M)+B_k^\\prime N_F \\alpha _k(\\mu )\\right]\\right.\\nonumber \\\\&&~~~~~~~~~~~~~~~~~+\\left.\\sum \\limits _{Y_l} {A_l^\\prime }\\frac{\\tilde{c}_r}{4\\pi ^2} {\\Delta b_r} \\alpha _r^2(\\mu )\\ln \\left(\\frac{M}{\\mu }\\right)~\\right\\rbrace ~,\\nonumber \\\\&=&\\frac{M_0 F_\\phi }{2\\pi }\\left[\\frac{d_{kl}^i}{2}\\alpha _{\\lambda _{ikl}} \\left(-{q_{y_{\\lambda _{ikl}}}}+\\frac{1}{2\\pi }\\left[\\frac{d^p_{mn}}{2} {q_{y_{\\lambda _{mnp}}}}\\alpha _{\\lambda _{mnp}}-2c_r {l_r} \\alpha _r\\right]\\ln \\left[\\frac{M_G}{\\mu }\\right]\\right)\\right.\\nonumber \\\\&&~~~~~~~~~\\left.+ 2C_a(i)\\frac{l_a}{T} \\frac{\\alpha _a^2}{\\alpha _a(GUT)} \\right]\\nonumber \\\\&-&\\frac{d F_\\phi M_0}{4\\pi }\\sum \\limits _{g_k}\\left[(B_k-B_k^\\prime ) (b_k+N_F) l_k \\frac{ \\alpha ^2_k(M)}{\\alpha _k(GUT)}+B_k^\\prime N_F l_k \\frac{\\alpha _k^2(\\mu )}{\\alpha _k(GUT)}\\right]\\nonumber \\\\&&~~~~~~~~~~~~~~~~~-\\left.\\sum \\limits _{Y_l}{A_l^\\prime }\\frac{d M_0 F_\\phi }{4\\pi ^2} \\tilde{c}_r {\\Delta b_r}l_r \\frac{2\\alpha _r^3(\\mu )}{\\alpha _r(GUT)}\\ln \\left(\\frac{M}{\\mu }\\right)~\\right\\rbrace ~,$ with $\\frac{\\partial }{\\partial T}\\alpha _k(\\mu )=-\\frac{l_k}{T}\\frac{\\alpha _k(\\mu )}{\\alpha _k(GUT)}\\alpha _k(\\mu )~,$"
]
] | 1808.08529 |
[
[
"Drawing Subcubic 1-Planar Graphs with Few Bends, Few Slopes, and Large\n Angles"
],
[
"Abstract We show that the 1-planar slope number of 3-connected cubic 1-planar graphs is at most 4 when edges are drawn as polygonal curves with at most 1 bend each.",
"This bound is obtained by drawings whose vertex and crossing resolution is at least $\\pi/4$.",
"On the other hand, if the embedding is fixed, then there is a 3-connected cubic 1-planar graph that needs 3 slopes when drawn with at most 1 bend per edge.",
"We also show that 2 slopes always suffice for 1-planar drawings of subcubic 1-planar graphs with at most 2 bends per edge.",
"This bound is obtained with vertex resolution $\\pi/2$ and the drawing is RAC (crossing resolution $\\pi/2$).",
"Finally, we prove lower bounds for the slope number of straight-line 1-planar drawings in terms of number of vertices and maximum degree."
],
[
"Introduction",
"A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once.",
"The notion of 1-planarity naturally extends planarity and received considerable attention since its first introduction by Ringel in 1965 [33], as witnessed by recent surveys [14], [27].",
"Despite the efforts made in the study of 1-planar graphs, only few results are known concerning their geometric representations (see, e.g., [1], [4], [7], [11]).",
"In this paper, we study the existence of 1-planar drawings that simultaneously satisfy the following properties: edges are polylines using few bends and few distinct slopes for their segments, edge crossings occur at large angles, and pairs of edges incident to the same vertex form large angles.",
"For example, Fig.",
"REF shows a 1-bend drawing of a 1-planar graph (i.e., a drawing in which each edge is a polyline with at most one bend) using 4 distinct slopes, such that edge crossings form angles at least $\\pi /4$ , and the angles formed by edges incident to the same vertex are at least $\\pi /4$ .",
"In what follows, we briefly recall known results concerning the problems of computing polyline drawings with few bends and few slopes or with few bends and large angles.",
"Related work.",
"The $k$ -bend (planar) slope number of a (planar) graph $G$ with maximum vertex degree $\\Delta $ is the minimum number of distinct edge slopes needed to compute a (planar) drawing of $G$ such that each edge is a polyline with at most $k$ bends.",
"When $k=0$ , this parameter is simply known as the (planar) slope number of $G$ .",
"Clearly, if $G$ has maximum vertex degree $\\Delta $ , at least $\\lceil \\Delta /2 \\rceil $ slopes are needed for any $k$ .",
"While there exist non-planar graphs with $\\Delta \\ge 5$ whose slope number is unbounded with respect to $\\Delta $ [3], [32], Keszegh et al.",
"[25] proved that the planar slope number is bounded by $2^{O(\\Delta )}$ .",
"Several authors improved this bound for subfamilies of planar graphs (see, e.g., [22], [26], [28]).",
"Concerning $k$ -bend drawings, Angelini et al.",
"[2] proved that the 1-bend planar slope number is at most $\\Delta -1$ , while Keszegh et al.",
"[25] proved that the 2-bend planar slope number is $\\lceil \\Delta /2 \\rceil $ (which is tight).",
"Special attention has been paid in the literature to the slope number of (sub)cubic graphs, i.e., graphs having vertex degree (at most) 3.",
"Mukkamala and Pálvölgyi showed that the four slopes $\\lbrace 0, \\frac{\\pi }{4}, \\frac{\\pi }{2}, \\frac{3\\pi }{4}\\rbrace $ suffice for every cubic graph [31].",
"For planar graphs, Kant and independently Dujmović et al.",
"proved that cubic 3-connected planar graphs have planar slope number 3 disregarding the slopes of three edges on the outer face [15], [23], while Di Giacomo et al.",
"[13] proved that the planar slope number of subcubic planar graphs is 4.",
"We also remark that the slope number problem is related to orthogonal drawings, which are planar and with slopes $\\lbrace 0, \\frac{\\pi }{2}\\rbrace $ [16], and with octilinear drawings, which are planar and with slopes $\\lbrace 0, \\frac{\\pi }{4}, \\frac{\\pi }{2}, \\frac{3\\pi }{4}\\rbrace $ [5].",
"All planar graphs with $\\Delta \\le 4$ (except the octahedron) admit 2-bend orthogonal drawings [6], [29], and planar graphs admit octilinear drawings without bends if $\\Delta \\le 3$ [13], [23], with 1 bend if $\\Delta \\le 5$ [5], and with 2 bends if $\\Delta \\le 8$ [25].",
"Of particular interest for us is the $k$ -bend 1-planar slope number of 1-planar graphs, i.e., the minimum number of distinct edge slopes needed to compute a 1-planar drawing of a 1-planar graph such that each edge is a polyline with at most $k \\ge 0$ bends.",
"Di Giacomo et al.",
"[12] proved an $O(\\Delta )$ upper bound for the 1-planar slope number ($k=0$ ) of outer 1-planar graphs, i.e., graphs that can be drawn 1-planar with all vertices on the external boundary.",
"Finally, the vertex resolution and the crossing resolution of a drawing are defined as the minimum angle between two consecutive segments incident to the same vertex or crossing, respectively (see, e.g., [17], [20], [30]).",
"A drawing is RAC (right-angle crossing) if its crossing resolution is $\\pi /2$ .",
"Eades and Liotta proved that 1-planar graphs may not have straight-line RAC drawings [18], while Chaplick et al.",
"[8] and Bekos et al.",
"[4] proved that every 1-planar graph has a 1-bend RAC drawing that preserves the embedding.",
"Our contribution.",
"We prove upper and lower bounds on the $k$ -bend 1-planar slope number of 1-planar graphs, when $k \\in \\lbrace 0,1,2\\rbrace $ .",
"Our results are based on techniques that lead to drawings with large vertex and crossing resolution.",
"In Section , we prove that every 3-connected cubic 1-planar graph admits a 1-bend 1-planar drawing that uses at most 4 distinct slopes and has both vertex and crossing resolution $\\pi /4$ .",
"In Section , we show that every subcubic 1-planar graph admits a 2-bend 1-planar drawing that uses at most 2 distinct slopes and has both vertex and crossing resolution $\\pi /2$ .",
"These bounds on the number of slopes and on the vertex/crossing resolution are clearly worst-case optimal.",
"In Section REF , we give a 3-connected cubic 1-plane graph for which any embedding-preserving 1-bend drawing uses at least 3 distinct slopes.",
"The lower bound holds even if we are allowed to change the outer face.",
"In Section REF , we present 2-connected subcubic 1-plane graphs with $n$ vertices such that any embedding-preserving straight-line drawing uses $\\Omega (n)$ distinct slopes, and 3-connected 1-plane graphs with maximum degree $\\Delta \\ge 3$ such that any embedding-preserving straight-line drawing uses at least $9(\\Delta -1)$ distinct slopes, which implies that at least 18 slopes are needed if $\\Delta =3$ .",
"Preliminaries can be found in Section , while open problems are in Section ."
],
[
"Preliminaries",
"We only consider simple graphs with neither self-loops nor multiple edges.",
"A drawing $\\Gamma $ of a graph $G$ maps each vertex of $G$ to a point of the plane and each edge to a simple open Jordan curve between its endpoints.",
"We always refer to simple drawings where two edges can share at most one point, which is either a common endpoint or a proper intersection.",
"A drawing divides the plane into topologically connected regions, called faces; the infinite region is called the outer face.",
"For a planar (i.e., crossing-free) drawing, the boundary of a face consists of vertices and edges, while for a non-planar drawing the boundary of a face may also contain crossings and parts of edges.",
"An embedding of a graph $G$ is an equivalence class of drawings of $G$ that define the same set of faces and the same outer face.",
"A (1-)plane graph is a graph with a fixed (1-)planar embedding.",
"Given a 1-plane graph $G$ , the planarization $G^*$ of $G$ is the plane graph obtained by replacing each crossing of $G$ with a dummy vertex.",
"To avoid confusion, the vertices of $G^*$ that are not dummy are called real.",
"Moreover, we call fragments the edges of $G^*$ that are incident to a dummy vertex.",
"The next lemma will be used in the following and can be of independent interest, as it extends a similar result by Fabrici and Madaras [19].",
"The proof is given in Appendix .",
"crossingMinimal Let $G=(V,E)$ be a 1-plane graph and let $G^*$ be its planarization.",
"We can re-embed $G$ such that each edge is still crossed at most once and [label=()] no cutvertex of $G^*$ is a dummy vertex, and if $G$ is 3-connected, then $G^*$ is 3-connected.",
"A drawing $\\Gamma $ is straight-line if all its edges are mapped to segments, or it is $k$ -bend if each edge is mapped to a chain of segments with at most $k>0$ bends.",
"The slope of an edge segment of $\\Gamma $ is the slope of the line containing this segment.",
"For convenience, we measure the slopes by their angle with respect to the $x$ -axis.",
"Let $\\mathcal {S}=\\lbrace \\alpha _1,\\dots ,\\alpha _t\\rbrace $ be a set of $t$ distinct slopes.",
"The slope number of a $k$ -bend drawing $\\Gamma $ is the number of distinct slopes used for the edge segments of $\\Gamma $ .",
"An edge segment of $\\Gamma $ uses the north (N) port (south (S) port) of a vertex $v$ if it has slope $\\pi /2$ and $v$ is its bottommost (topmost) endpoint.",
"We can define analogously the west (W) and east (E) ports with respect to the slope 0, the north-west (NW) and south-east (SE) ports with respect to slope $3\\pi /4$ , and the south-west (SW) and north-east (NE) ports with respect to slope $\\pi /4$ .",
"Any such port is free for $v$ if there is no edge that attaches to $v$ by using it.",
"We will use a decomposition technique called canonical ordering [24].",
"Let $G=(V,E)$ be a 3-connected plane graph.",
"Let $\\delta = \\lbrace 1,\\dots ,K\\rbrace $ be an ordered partition of $V$ , that is, $1 \\cup \\dots \\cup K = V$ and $i \\cap j = \\emptyset $ for $i \\ne j$ .",
"Let $G_i$ be the subgraph of $G$ induced by $1 \\cup \\dots \\cup i$ and denote by $C_i$ the outer face of $G_i$ .",
"The partition $\\delta $ is a canonical ordering of $G$ if: (i) $1=\\lbrace v_1,v_2\\rbrace $ , where $v_1$ and $v_2$ lie on the outer face of $G$ and $(v_1,v_2) \\in E$ .",
"(ii) $K = \\lbrace v_n\\rbrace $ , where $v_n$ lies on the outer face of $G$ , $(v_1,v_n) \\in E$ .",
"(iii) Each $C_i$ ($i > 1$ ) is a cycle containing $(v_1,v_2)$ .",
"(iv) Each $G_i$ is 2-connected and internally 3-connected, that is, removing any two interior vertices of $G_i$ does not disconnect it.",
"(v) For each $i \\in \\lbrace 2, \\dots , K-1\\rbrace $ , one of the following conditions holds: (a) $i$ is a singleton $v^i$ that lies on $C_i$ and has at least one neighbor in $G \\setminus G_i$ ; (b) $i$ is a chain $\\lbrace v^i_1,\\dots , v^i_l\\rbrace $ , both $v^i_1$ and $v^i_l$ have exactly one neighbor each in $C_{i-1}$ , and $v^i_2, \\ldots , v^i_{l-1}$ have no neighbor in $C_{i-1}$ .",
"Since $G$ is 3-connected, each $v^i_j$ has at least one neighbor in $G \\setminus G_i$ .",
"Let $v$ be a vertex in $i$ , then its neighbors in $G_{i-1}$ (if $G_{i-1}$ exists) are called the predecessors of $v$ , while its neighbors in $G\\setminus G_{i}$ (if $G_{i+1}$ exists) are called the successors of $v$ .",
"In particular, every singleton has at least two predecessors and at least one successor, while every vertex in a chain has either zero or one predecessor and at least one successor.",
"Kant [24] proved that a canonical ordering of $G$ always exists and can be computed in $O(n)$ time; the technique in [24] is such that one can arbitrarily choose two adjacent vertices $u$ and $w$ on the outer face so that $u=v_1$ and $w=v_2$ in the computed canonical ordering.",
"An $n$ -vertex planar $st$ -graph $G=(V,E)$ is a plane acyclic directed graph with a single source $s$ and a single sink $t$ , both on the outer face [10].",
"An $st$ -ordering of $G$ is a numbering $\\sigma : V\\rightarrow \\lbrace 1,2,\\dots ,n\\rbrace $ such that for each edge $(u,v) \\in E$ , it holds $\\sigma (u) < \\sigma (v)$ (thus $\\sigma (s)=1$ and $\\sigma (t)=n$ ).",
"For an $st$ -graph, an $st$ -ordering can be computed in $O(n)$ time (see, e.g., [9]) and every biconnected undirected graph can be oriented to become a planar $st$ -graph (also in linear time).",
"1-bend Drawings of 3-connected cubic 1-planar graphs Let $G$ be a 3-connected 1-plane cubic graph, and let $G^*$ be its planarization.",
"We can assume that $G^*$ is 3-connected (else we can re-embed $G$ by Lemma ).",
"We choose as outer face of $G$ a face containing an edge $(v_1,v_2)$ whose vertices are both real (see Fig.",
"REF ).",
"Such a face exists: If $G$ has $n$ vertices, then $G^*$ has fewer than $3n/4$ dummy vertices because $G$ is subcubic.",
"Hence we find a face in $G^*$ with more real than dummy vertices and hence with two consecutive real vertices.",
"Let $\\delta = \\lbrace 1,\\dots ,K\\rbrace $ be a canonical ordering of $G^*$ , let $G_i$ be the graph obtained by adding the first $i$ sets of $\\delta $ and let $C_i$ be the outer face of $G_i$ .",
"Note that a real vertex $v$ of $G_i$ can have at most one successor $w$ in some set $j$ with $j>i$ .",
"We call $w$ an L-successor (resp., R-successor) of $v$ if $v$ is the leftmost (resp., rightmost) neighbor of $j$ on $C_{i}$ .",
"Similarly, a dummy vertex $x$ of $G_i$ can have at most two successors in some sets $j$ and $l$ with $l \\ge j > i$ .",
"In both cases, a vertex $v$ of $G_i$ having a successor in some set $j$ with $j>i$ is called attachable.",
"We call $v$ L-attachable (resp., R-attachable) if $v$ is attachable and has no L-successor (resp., R-successor) in $G_i$ .",
"We will draw an upward edge at $u$ with slope $\\pi /4$ (resp., $3\\pi /4$ ) only if it is L-attachable (resp., R-attachable).",
"Figure: (a) A 3-connected 1-plane cubic graph GG;(b) a canonical ordering δ\\delta of the planarization G * G^* of GG—the real (dummy) vertices are black points (white squares);(c) the edges crossed by the dashed line are a uvuv-cut of G 5 G_5 with respect to (u,w)(u,w)—the two components have a yellow and a blue background, respectively;(d) a 1-bend 1-planar drawing with 4 slopes of GGLet $u$ and $v$ be two vertices of $C_i$ , for $i>1$ .",
"Denote by $P_i(u,v)$ the path of $C_i$ having $u$ and $v$ as endpoints and that does not contain $(v_1,v_2)$ .",
"Vertices $u$ and $v$ are consecutive if they are both attachable and if $P_i(u,v)$ does not contain any other attachable vertex.",
"Given two consecutive vertices $u$ and $v$ of $C_i$ and an edge $e$ of $C_i$ , a $uv$ -cut of $G_i$ with respect to $e$ is a set of edges of $G_i$ that contains both $e$ and $(v_1,v_2)$ and whose removal disconnects $G_i$ into two components, one containing $u$ and one containing $v$ (see Fig.",
"REF ).",
"We say that $u$ and $v$ are L-consecutive (resp., R-consecutive) if they are consecutive, $u$ lies to the left (resp., right) of $v$ on $C_i$ , and $u$ is L-attachable (resp., R-attachable).",
"We construct an embedding-preserving drawing $\\Gamma _i$ of $G_i$ , for $i=2,\\dots ,K$ , by adding one by one the sets of $\\delta $ .",
"A drawing $\\Gamma _i$ of $G_i$ is valid, if: It uses only slopes in the set $\\lbrace 0,\\frac{\\pi }{4},\\frac{\\pi }{2},\\frac{3\\pi }{4}\\rbrace $ ; It is a 1-bend drawing such that the union of any two edge fragments that correspond to the same edge in $G$ is drawn with (at most) one bend in total.",
"A valid drawing $\\Gamma _K$ of $G_K$ will coincide with the desired drawing of $G$ , after replacing dummy vertices with crossing points.",
"Construction of $\\Gamma _2$ .",
"We begin by showing how to draw $G_2$ .",
"We distinguish two cases, based on whether 2 is a singleton or a chain, as illustrated in Fig.",
"REF .",
"Construction of $\\Gamma _i$ , for $2 < i < K$ .",
"We now show how to compute a valid drawing of $G_i$ , for $i=3,\\dots ,K-1$ , by incrementally adding the sets of $\\delta $ .",
"We aim at constructing a valid drawing $\\Gamma _i$ that is also stretchable, i.e., that satisfies the following two more properties; see Fig.",
"REF .",
"These two properties will be useful to prove Lemma , which defines a standard way of stretching a drawing by lengthening horizontal segments.",
"The edge $(v_1,v_2)$ is drawn with two segments $s_1$ and $s_2$ that meet at a point $p$ .",
"Segment $s_1$ uses the SE port of $v_1$ and $s_2$ uses the SW port of $v_2$ .",
"Also, $p$ is the lowest point of $\\Gamma _i$ , and no other point of $\\Gamma _i$ is contained by the two lines that contain $s_1$ and $s_2$ .",
"Figure: Γ i \\Gamma _i is stretchable.",
"For every pair of consecutive vertices $u$ and $v$ of $C_i$ with $u$ left of $v$ on $C_i$ , it holds that [label=()] lebel=P0 If $u$ is L-attachable (resp., $v$ is R-attachable), then the path $P_i(u,v)$ is such that for each vertical segment $s$ on this path there is a horizontal segment in the subpath before $s$ if $s$ is traversed upwards when going from $u$ to $v$ (resp., from $v$ to $u$ ); lfbel=P0 if both $u$ and $v$ are real, then $P_i(u,v)$ contains at least one horizontal segment; and lgbel=P0 for every edge $e$ of $P_i(u,v)$ such that $e$ contains a horizontal segment, there exists a $uv$ -cut of $G_i$ with respect to $e$ whose edges all contain a horizontal segment in $\\Gamma _i$ except for $(v_1,v_2)$ , and such that there exists a $y$ -monotone curve that passes through all and only such horizontal segments and $(v_1,v_2)$ .",
"stretch Suppose that $\\Gamma _i$ is valid and stretchable, and let $u$ and $v$ be two consecutive vertices of $C_i$ .",
"If $u$ is L-attachable (resp., $v$ is R-attachable), then it is possible to modify $\\Gamma _i$ such that any half-line with slope $\\pi /4$ (resp., $3\\pi /4$ ) that originates at $u$ (resp., at $v$ ) and that intersects the outer face of $\\Gamma _i$ does not intersect any edge segment with slope $\\pi /2$ of $P_i(u,v)$ .",
"Also, the modified drawing is still valid and stretchable.",
"Crossings between such half-lines and vertical segments of $P_i(u,v)$ can be solved by finding suitable $uv$ -cuts and moving everything on the right/left side of the cut to the right/left.",
"The full proof is given in Appendix Let $P$ be a set of ports of a vertex $v$ ; the symmetric set of ports $P^{\\prime }$ of $v$ is the set of ports obtained by mirroring $P$ at a vertical line through $v$ .",
"We say that $\\Gamma _i$ is attachable if the following two properties also apply.",
"At any attachable real vertex $v$ of $\\Gamma _i$ , its N, NW, and NE ports are free.",
"Let $v$ be an attachable dummy vertex of $\\Gamma _i$ .",
"If $v$ has two successors, there are four possible cases for its two used ports, illustrated with two solid edges in Fig.",
"REF –fig:case-d.",
"If $v$ has only one successor not in $\\Gamma _i$ , there are eight possible cases for its three used ports, illustrated with two solid edges plus one dashed or one dotted edge in Fig.",
"REF –fig:case-e (see Fig.",
"in Appendix ).",
"Figure: Illustration for .",
"If vv has two successors not in Γ i \\Gamma _i,then the edges connecting vv to its two neighbors in Γ i \\Gamma _i are solid.",
"Ifvv has one successor in Γ i \\Gamma _i, then the edge between vv andthis successor is dashed or dotted.",
"Observe that $\\Gamma _2$ , besides being valid, is also stretchable and attachable by construction (see also Fig.",
"REF ).",
"Assume that $G_{i-1}$ admits a valid, stretchable, and attachable drawing $\\Gamma _{i-1}$ , for some $2 \\le i < K-1$ ; we show how to add the next set $i$ of $\\delta $ so to obtain a drawing $\\Gamma _i$ of $G_i$ that is valid, stretchable and attachable.",
"We distinguish between the following cases.",
"\\newlabelc:singleton111c:singleton Case 1.",
"$i$ is a singleton, i.e., $i=\\lbrace v^i\\rbrace $ .",
"Note that if $v^i$ is real, it has two neighbors on $C_{i-1}$ , while if it is dummy, it can have either two or three neighbors on $C_{i-1}$ .",
"Let $u_l$ and $u_r$ be the first and the last neighbor of $v^i$ , respectively, when walking along $C_{i-1}$ in clockwise direction from $v_1$ .",
"We will call $u_l$ (resp., $u_r$ ) the leftmost predecessor (resp., rightmost predecessor) of $v^i$ .",
"\\newlabelc:singleton-real1.111.1c:singleton-real Case 1.1.",
"Vertex $v^i$ is real.",
"Then, $u_l$ and $u_r$ are its only two neighbors in $C_{i-1}$ .",
"Each of $u_l$ and $u_r$ can be real or dummy.",
"If $u_l$ (resp., $u_r$ ) is real, we draw $(u_l,v^i)$ (resp., $(u_r,v^i)$ ) with a single segment using the NE port of $u_l$ and the SW port of $v^i$ (resp., the NW port of $u_r$ and the SE port of $v^i$ ).",
"If $u_l$ is dummy and has two successors not in $\\Gamma _{i-1}$ , we distinguish between the cases of Fig.",
"REF as shown in Fig.",
"REF .",
"The symmetric configuration of $C3$ is only used for connecting to $u_r$ .",
"Figure: A real singleton when u l u_l is dummy with two successors not in Γ i-1 \\Gamma _{i-1}If $u_l$ is dummy and has one successor not in $\\Gamma _{i-1}$ , we distinguish between the various cases of Fig.",
"REF as indicated in Fig.",
"REF (see Fig.",
"for all cases in Appendix ).",
"Observe that $C1$ requires a local reassignment of one port of $u_l$ .",
"The edge $(u_r,v^i)$ is drawn by following a similar case analysis(depicted in Fig.",
"of Appendix ).",
"Vertex $v^i$ is then placed at the intersection of the lines passing through the assigned ports, which always intersect by construction.",
"In particular, the S port is only used when $u_l$ has one successor, but the same situation cannot occur when drawing $(u_r,v^i)$ .",
"Otherwise, there is a path of $C_{i-1}$ from $u_l$ via its successor $x$ on $C_{i-1}$ to $u_r$ via its successor $y$ on $C_{i-1}$ .",
"Note that $x=y$ is possible but $x\\ne u_r$ .",
"Since the first edge on this path goes from a predecessor to a successor and the last edge goes from a successor to a predecessor, there has to be a vertex $z$ without a successor on the path; but then $u_l$ and $u_r$ are not consecutive.",
"To avoid crossings between $\\Gamma _{i-1}$ and the new edges $(u_l,v^i)$ and $(u_r,v^i)$ , we apply Lemma to suitably stretch the drawing.",
"In particular, possible crossings can occur only with vertical edge segments of $P_{i-1}(u_l,u_r)$ , because when walking along $P_{i-1}(u_l,u_r)$ from $u_l$ to $u_r$ we only encounter a (possibly empty) set of segments with slopes in the range $\\lbrace 3\\pi /4,\\pi /2,0\\rbrace $ , followed by a (possibly empty) set of segments with slopes in the range $\\lbrace \\pi /2,\\pi /4,0\\rbrace $ .",
"Figure: Some cases for the addition of a real singleton when u l u_l is dummy with one successor not in Γ i-1 \\Gamma _{i-1}Figure: Illustration for the addition of a dummy singleton\\newlabelc:singleton-dummy1.211.2c:singleton-dummy Case 1.2.",
"Vertex $v^i$ is dummy.",
"By 1-planarity, the two or three neighbors of $v^i$ on $C_{i-1}$ are all real.",
"If $v^i$ has two neighbors, we draw $(u_l,v^i)$ and $(u_r,v^i)$ as shown in Fig.",
"REF , while if $v^i$ has three neighbors, we draw $(u_l,v^i)$ and $(u_r,v^i)$ as shown in Fig.",
"REF .",
"Analogous to the previous case, vertex $v^i$ is placed at the intersection of the lines passing through the assigned ports, which always intersect by construction, and avoiding crossings between $\\Gamma _{i-1}$ and the new edges $(u_l,v^i)$ and $(u_r,v^i)$ by applying Lemma .",
"In particular, if $v^i$ has three neighbors on $C_{i-1}$ , say $u_l$ , $w$ , and $u_r$ , by REF there is a horizontal segment between $u_l$ and $w$ , as well as between $w$ and $u_r$ .",
"Thus, Lemma can be applied not only to resolve crossings, but also to find a suitable point where the two lines with slopes $\\pi /4$ and $3\\pi /4$ meet along the line with slope $\\pi /2$ that passes through $w$ .",
"\\newlabelc:chain212c:chain Case 2.",
"$i$ is a chain, i.e., $i=\\lbrace v^i_1,v^i_2,\\dots ,v^i_l\\rbrace $ .",
"We find a point as if we had to place a vertex $v$ whose leftmost predecessor is the leftmost predecessor of $v^i_1$ and whose rightmost predecessor is the rightmost predecessor of $v^i_l$ .",
"We then draw the chain slightly below this point by using the same technique used to draw 2.",
"Again, Lemma can be applied to resolve possible crossings.",
"We formally prove the correctness of our algorithm in Appendix .",
"gkvalid Drawing $\\Gamma _{K-1}$ is valid, stretchable, and attachable.",
"Figure: Illustration for the addition of kkConstruction of $\\Gamma _K$ .",
"We now show how to add $K=\\lbrace v_n\\rbrace $ to $\\Gamma _{K-1}$ so as to obtain a valid drawing of $G_K$ , and hence the desired drawing of $G$ after replacing dummy vertices with crossing points.",
"Recall that $(v_1,v_n)$ is an edge of $G$ by the definition of canonical ordering.",
"We distinguish whether $v_n$ is real or dummy; the two cases are shown in Fig.",
"REF .",
"Note that if $v_n$ is dummy, its four neighbors are all real and hence their N, NW, and NE ports are free by REF .",
"If $v_n$ is real, it has three neighbors in $\\Gamma _{K-1}$ , $v_1$ is real by construction, and the S port can be used to attach with a dummy vertex.",
"Finally, since $\\Gamma _{K-1}$ is attachable, we can use Lemma to avoid crossings and to find a suitable point to place $v_n$ .",
"A complete drawing is shown in Fig.",
"REF .",
"The theorem follows immediately by the choice of the slopes.",
"Every 3-connected cubic 1-planar graph admits a 1-bend 1-planar drawing with at most 4 distinct slopes and angular and crossing resolution $\\pi /4$ .",
"2-bend drawings Liu et al.",
"[29] presented an algorithm to compute orthogonal drawings for planar graphs of maximum degree 4 with at most 2 bends per edge (except the octahedron, which requires 3 bends on one edge).",
"We make use of their algorithm for biconnected graphs.",
"The algorithm chooses two vertices $s$ and $t$ and computes an $st$ -ordering of the input graph.",
"Let $V=\\lbrace v_1,\\ldots ,v_n\\rbrace $ with $\\sigma (v_i)=i$ , $1\\le i\\le n$ .",
"Liu et al.",
"now compute an embedding of $G$ such that $v_2$ lies on the outer face if $\\deg (s)=4$ and $v_{n-1}$ lies on the outer face if $\\deg (t)=4$ ; such an embedding exists for every graph with maximum degree 4 except the octahedron.",
"The edges around each vertex $v_i,1\\le i\\le n$ , are assigned to the four ports as follows.",
"If $v_i$ has only one outgoing edge, it uses the N port; if $v_i$ has two outgoing edges, they use the N and E port; if $v_i$ has three outgoing edges, they use the N, E, and W port; and if $v_i$ has four outgoing edges, they use all four ports.",
"Symmetrically, the incoming edges of $v_i$ use the S, W, E, and N port, in this order.",
"The edge $(s,t)$ (if it exists) is assigned to the W port of both $s$ and $t$ .",
"If $\\deg (s)=4$ , the edge $(s,v_2)$ is assigned to the S port of $s$ (otherwise the port remains free); if $\\deg (t)=4$ , the edge $(t,v_{n-1})$ is assigned to the N port of $t$ (otherwise the port remains free).",
"Note that every vertex except $s$ and $t$ has at least one incoming and one outgoing edge; hence, the given embedding of the graph provides a unique assignment of edges to ports.",
"Finally, they place the vertices bottom-up as prescribed by the $st$ -ordering.",
"The way an edge is drawn is determined completely by the port assignment, as depicted in Fig.",
"REF .",
"Let $G=(V,E)$ be a subcubic 1-plane graph.",
"We first re-embed $G$ according to Lemma .",
"Let $G^*$ be the planarization of $G$ after the re-embedding.",
"Then, all cutvertices of $G^*$ are real vertices, and since they have maximum degree 3, there is always a bridge connecting two 2-connected components.",
"Let $G_1,\\ldots ,G_k$ be the 2-connected components of $G$ , and let $G_i^*$ be the planarization of $G_i,1\\le i\\le k$ .",
"We define the bridge decomposition tree $\\mathcal {T}$ of $G$ as the graph having a node for each component $G_i$ of $G$ , and an edge $(G_i,G_j)$ , for every pair $G_i, G_j$ connected by a bridge in $G$ .",
"We root $\\mathcal {T}$ in $G_1$ .",
"For each component $G_i,2\\le i\\le k$ , let $u_i$ be the vertex of $G_i$ connected to the parent of $G_i$ in $\\mathcal {T}$ by a bridge and let $u_1$ be an arbitrary vertex of $G_1$ .",
"We will create a drawing $\\Gamma _i$ for each component $G_i$ with at most 2 slopes and 2 bends such that $u_i$ lies on the outer face.",
"To this end, we first create a drawing $\\Gamma _i^*$ of $G_i^*$ with the algorithm of Liu et al.",
"[29] and then modify the drawing.",
"Throughout the modifications, we will make sure that the following invariants hold for the drawing $\\Gamma _i^*$ .",
"$\\Gamma _i^*$ is a planar orthogonal drawing of $G_i^*$ and edges are drawn as in Fig.",
"REF ; $u_i$ lies on the outer face of $\\Gamma _i^*$ and its N port is free; every edge is $y$ -monotone from its source to its target; every edge with 2 bends is a C-shape, there are no edges with more bends; if a C-shape ends in a dummy vertex, it uses only E ports; and if a C-shape starts in a dummy vertex, it uses only W ports.",
"Figure: The shapes to draw edgesbendInvariants Every $G_i^*$ admits a drawing $\\Gamma _i^*$ that satisfies invariants (I1)–(I6).",
"We choose $t=u_i$ and some real vertex $s$ and use the algorithm by Liu et al.",
"to draw $G_i^°$ .",
"Since $s$ and $t$ are real, there are no U-shapes.",
"Since no real vertex can have an outgoing edge at its W port or incoming edge at its E port, the invariants follow.",
"The full proof is given in Appendix .",
"We now iteratively remove the C-shapes from the drawing while maintaining the invariants.",
"We make use of a technique similar to the stretching in Section .",
"We lay an orthogonal $y$ -monotone curve $S$ through our drawing that intersects no vertices.",
"Then we stretch the drawing by moving $S$ and all features that lie right of $S$ to the right, and stretching all points on $S$ to horizontal segments.",
"After this stretch, in the area between the old and the new position of $S$ , there are only horizontal segments of edges that are intersected by $S$ .",
"The same operation can be defined symmetrically for an $x$ -monotone curve that is moved upwards.",
"bendBicon Every $G_i$ admits an orthogonal 2-bend drawing such that $u_i$ lies on the outer face and its N port is free.",
"We start with a drawing $\\Gamma _i^*$ of $G_i^*$ that satisfies invariants (I1)–(I6), which exists by Lemma .",
"By (I2), $u_i$ lies on the outer face and its N port is free.",
"If no dummy vertex in $\\Gamma _i^*$ is incident to a C-shape, by (I4) all edges incident to dummy vertices are drawn with at most 1 bend, so the resulting drawing $\\Gamma _i$ of $G_i$ is an orthogonal 2-bend drawing.",
"Otherwise, there is a C-shape between a real vertex $u$ and a dummy vertex $v$ .",
"We show how to eliminate this C-shape without introducing new ones while maintaining all invariants.",
"We prove the case that $(u,v)$ is directed from $u$ to $v$ , so by (I5) it uses only E ports; the other case is symmetric.",
"We do a case analysis based on which ports at $u$ are free.",
"We show one case here and the rest in Appendix .",
"\\newlabelc:bottomleft111c:bottomleft Case 1.",
"The N port at $u$ is free; see Fig.",
"REF .",
"Create a curve $S$ as follows: Start at some point $p$ slightly to the top left of $u$ and extend it downward to infinity.",
"Extend it from $p$ to the right until it passes the vertical segment of $(u,v)$ and extend it upwards to infinity.",
"Place the curve close enough to $u$ and $(u,v)$ such that no vertex or bend point lies between $S$ and the edges of $u$ that lie right next to it.",
"Then, stretch the drawing by moving $S$ to the right such that $u$ is placed below the top-right bend point of $(u,v)$ .",
"Since $S$ intersected a vertical segment of $(u,v)$ , this changes the edge to be drawn with 4 bends.",
"However, now the region between $u$ and the second bend point of $(u,v)$ is empty and the N port of $u$ is free, so we can make an L-shape out of $(u,v)$ that uses the N port at $u$ .",
"This does not change the drawing style of any edge other than $(u,v)$ , so all the invariants are maintained and the number of C-shapes is reduced by one.",
"Figure: Proof of Lemma , Case Finally, we combine the drawings $\\Gamma _i$ to a drawing $\\Gamma $ of $G$ .",
"Recall that every cutvertex is real and two biconnected components are connected by a bridge.",
"Let $G_j$ be a child of $G_i$ in the bridge decomposition tree.",
"We have drawn $G_j$ with $u_j$ on the outer face and a free N port.",
"Let $v_i$ be the neighbor of $u_j$ in $G_i$ .",
"We choose one of its free ports, rotate and scale $\\Gamma _j$ such that it fits into the face of that port, and connect $u_j$ and $v_i$ with a vertical or horizontal segment.",
"Doing this for every biconnected component gives an orthogonal 2-bend drawing of $G$ .",
"Every subcubic 1-plane graph admits a 2-bend 1-planar drawing with at most 2 distinct slopes and both angular and crossing resolution $\\pi /2$ .",
"Lower bounds for 1-plane graphs 1-bend drawings of subcubic graphs There exists a subcubic 3-connected 1-plane graph such that any embedding-preserving 1-bend drawing uses at least 3 distinct slopes.",
"The lower bound holds even if we are allowed to change the outer face.",
"Let $G$ be the $K_4$ with a planar embedding.",
"The outer face is a 3-cycle, which has to be drawn as a polygon $\\Pi $ with at least four (nonreflex) corners.",
"Since we allow only one bend per edge, one of the corners of $\\Pi $ has to be a vertex of $G$ .",
"The vertex in the interior has to connect to this corner, however, all of its free ports lie on the outside.",
"Thus, no drawing of $G$ is possible.",
"the idea of making the example larger does not work .... need to think about it.",
"Straight-line drawings The full proofs for this section are given in Appendix .",
"lowerStraight There exist 2-regular 2-connected 1-plane graphs with $n$ vertices such that any embedding-preserving straight-line drawing uses $\\Omega (n)$ distinct slopes.",
"Figure: The constructions for the results of Section Let $G_k$ be the graph given by the cycle $a_1\\ldots ,a_{k+1},b_{k+1},\\ldots ,b_1,a_1$ and the embedding shown in Fig.",
"REF .",
"Walking along the path $a_1,\\ldots ,a_{k+1}$ , we find that the slope has to increase at every step.",
"There exist 3-regular 3-connected 1-plane graphs such that any embedding-preserving straight-line drawing uses at least 18 distinct slopes.",
"Consider the graph depicted in Fig.",
"REF .",
"We find that the slopes of the edges $(a_i,b_i),(a_i,c_i),(c_i,d_i),(c_i,e_i),(e_i,d_i),(e_i,a_{i+1})$ have to be increasing in this order for every $i=1,2,3$ .",
"There exist 3-connected 1-plane graphs such that any embedding-preserving straight-line drawing uses at least $9(\\Delta -1)$ distinct slopes.",
"Consider the graph depicted in Fig.",
"REF .",
"The degree of $a_i$ , $c_i$ , and $e_i$ is $\\Delta $ .",
"We repeat the proof of Lemma REF , but observe that the slopes of the $9(\\Delta -3)$ added edges lie between the slopes of $(a_i,b_i),(a_i,c_i),(c_i,e_i)$ , and $(e_i,a_{i+1})$ .",
"Open problems The research in this paper gives rise to interesting questions, among them: (1) Is it possible to extend Theorem to all subcubic 1-planar graphs?",
"(2) Can we drop the embedding-preserving condition from Theorem REF ?",
"(3) Is the 1-planar slope number of 1-planar graphs bounded by a function of the maximum degree?"
],
[
"1-bend Drawings of 3-connected cubic 1-planar graphs",
"Let $G$ be a 3-connected 1-plane cubic graph, and let $G^*$ be its planarization.",
"We can assume that $G^*$ is 3-connected (else we can re-embed $G$ by Lemma ).",
"We choose as outer face of $G$ a face containing an edge $(v_1,v_2)$ whose vertices are both real (see Fig.",
"REF ).",
"Such a face exists: If $G$ has $n$ vertices, then $G^*$ has fewer than $3n/4$ dummy vertices because $G$ is subcubic.",
"Hence we find a face in $G^*$ with more real than dummy vertices and hence with two consecutive real vertices.",
"Let $\\delta = \\lbrace 1,\\dots ,K\\rbrace $ be a canonical ordering of $G^*$ , let $G_i$ be the graph obtained by adding the first $i$ sets of $\\delta $ and let $C_i$ be the outer face of $G_i$ .",
"Note that a real vertex $v$ of $G_i$ can have at most one successor $w$ in some set $j$ with $j>i$ .",
"We call $w$ an L-successor (resp., R-successor) of $v$ if $v$ is the leftmost (resp., rightmost) neighbor of $j$ on $C_{i}$ .",
"Similarly, a dummy vertex $x$ of $G_i$ can have at most two successors in some sets $j$ and $l$ with $l \\ge j > i$ .",
"In both cases, a vertex $v$ of $G_i$ having a successor in some set $j$ with $j>i$ is called attachable.",
"We call $v$ L-attachable (resp., R-attachable) if $v$ is attachable and has no L-successor (resp., R-successor) in $G_i$ .",
"We will draw an upward edge at $u$ with slope $\\pi /4$ (resp., $3\\pi /4$ ) only if it is L-attachable (resp., R-attachable).",
"Figure: (a) A 3-connected 1-plane cubic graph GG;(b) a canonical ordering δ\\delta of the planarization G * G^* of GG—the real (dummy) vertices are black points (white squares);(c) the edges crossed by the dashed line are a uvuv-cut of G 5 G_5 with respect to (u,w)(u,w)—the two components have a yellow and a blue background, respectively;(d) a 1-bend 1-planar drawing with 4 slopes of GGLet $u$ and $v$ be two vertices of $C_i$ , for $i>1$ .",
"Denote by $P_i(u,v)$ the path of $C_i$ having $u$ and $v$ as endpoints and that does not contain $(v_1,v_2)$ .",
"Vertices $u$ and $v$ are consecutive if they are both attachable and if $P_i(u,v)$ does not contain any other attachable vertex.",
"Given two consecutive vertices $u$ and $v$ of $C_i$ and an edge $e$ of $C_i$ , a $uv$ -cut of $G_i$ with respect to $e$ is a set of edges of $G_i$ that contains both $e$ and $(v_1,v_2)$ and whose removal disconnects $G_i$ into two components, one containing $u$ and one containing $v$ (see Fig.",
"REF ).",
"We say that $u$ and $v$ are L-consecutive (resp., R-consecutive) if they are consecutive, $u$ lies to the left (resp., right) of $v$ on $C_i$ , and $u$ is L-attachable (resp., R-attachable).",
"We construct an embedding-preserving drawing $\\Gamma _i$ of $G_i$ , for $i=2,\\dots ,K$ , by adding one by one the sets of $\\delta $ .",
"A drawing $\\Gamma _i$ of $G_i$ is valid, if: It uses only slopes in the set $\\lbrace 0,\\frac{\\pi }{4},\\frac{\\pi }{2},\\frac{3\\pi }{4}\\rbrace $ ; It is a 1-bend drawing such that the union of any two edge fragments that correspond to the same edge in $G$ is drawn with (at most) one bend in total.",
"A valid drawing $\\Gamma _K$ of $G_K$ will coincide with the desired drawing of $G$ , after replacing dummy vertices with crossing points.",
"Construction of $\\Gamma _2$ .",
"We begin by showing how to draw $G_2$ .",
"We distinguish two cases, based on whether 2 is a singleton or a chain, as illustrated in Fig.",
"REF .",
"Construction of $\\Gamma _i$ , for $2 < i < K$ .",
"We now show how to compute a valid drawing of $G_i$ , for $i=3,\\dots ,K-1$ , by incrementally adding the sets of $\\delta $ .",
"We aim at constructing a valid drawing $\\Gamma _i$ that is also stretchable, i.e., that satisfies the following two more properties; see Fig.",
"REF .",
"These two properties will be useful to prove Lemma , which defines a standard way of stretching a drawing by lengthening horizontal segments.",
"The edge $(v_1,v_2)$ is drawn with two segments $s_1$ and $s_2$ that meet at a point $p$ .",
"Segment $s_1$ uses the SE port of $v_1$ and $s_2$ uses the SW port of $v_2$ .",
"Also, $p$ is the lowest point of $\\Gamma _i$ , and no other point of $\\Gamma _i$ is contained by the two lines that contain $s_1$ and $s_2$ .",
"Figure: Γ i \\Gamma _i is stretchable.",
"For every pair of consecutive vertices $u$ and $v$ of $C_i$ with $u$ left of $v$ on $C_i$ , it holds that [label=()] lebel=P0 If $u$ is L-attachable (resp., $v$ is R-attachable), then the path $P_i(u,v)$ is such that for each vertical segment $s$ on this path there is a horizontal segment in the subpath before $s$ if $s$ is traversed upwards when going from $u$ to $v$ (resp., from $v$ to $u$ ); lfbel=P0 if both $u$ and $v$ are real, then $P_i(u,v)$ contains at least one horizontal segment; and lgbel=P0 for every edge $e$ of $P_i(u,v)$ such that $e$ contains a horizontal segment, there exists a $uv$ -cut of $G_i$ with respect to $e$ whose edges all contain a horizontal segment in $\\Gamma _i$ except for $(v_1,v_2)$ , and such that there exists a $y$ -monotone curve that passes through all and only such horizontal segments and $(v_1,v_2)$ .",
"stretch Suppose that $\\Gamma _i$ is valid and stretchable, and let $u$ and $v$ be two consecutive vertices of $C_i$ .",
"If $u$ is L-attachable (resp., $v$ is R-attachable), then it is possible to modify $\\Gamma _i$ such that any half-line with slope $\\pi /4$ (resp., $3\\pi /4$ ) that originates at $u$ (resp., at $v$ ) and that intersects the outer face of $\\Gamma _i$ does not intersect any edge segment with slope $\\pi /2$ of $P_i(u,v)$ .",
"Also, the modified drawing is still valid and stretchable.",
"Crossings between such half-lines and vertical segments of $P_i(u,v)$ can be solved by finding suitable $uv$ -cuts and moving everything on the right/left side of the cut to the right/left.",
"The full proof is given in Appendix Let $P$ be a set of ports of a vertex $v$ ; the symmetric set of ports $P^{\\prime }$ of $v$ is the set of ports obtained by mirroring $P$ at a vertical line through $v$ .",
"We say that $\\Gamma _i$ is attachable if the following two properties also apply.",
"At any attachable real vertex $v$ of $\\Gamma _i$ , its N, NW, and NE ports are free.",
"Let $v$ be an attachable dummy vertex of $\\Gamma _i$ .",
"If $v$ has two successors, there are four possible cases for its two used ports, illustrated with two solid edges in Fig.",
"REF –fig:case-d.",
"If $v$ has only one successor not in $\\Gamma _i$ , there are eight possible cases for its three used ports, illustrated with two solid edges plus one dashed or one dotted edge in Fig.",
"REF –fig:case-e (see Fig.",
"in Appendix ).",
"Figure: Illustration for .",
"If vv has two successors not in Γ i \\Gamma _i,then the edges connecting vv to its two neighbors in Γ i \\Gamma _i are solid.",
"Ifvv has one successor in Γ i \\Gamma _i, then the edge between vv andthis successor is dashed or dotted.",
"Observe that $\\Gamma _2$ , besides being valid, is also stretchable and attachable by construction (see also Fig.",
"REF ).",
"Assume that $G_{i-1}$ admits a valid, stretchable, and attachable drawing $\\Gamma _{i-1}$ , for some $2 \\le i < K-1$ ; we show how to add the next set $i$ of $\\delta $ so to obtain a drawing $\\Gamma _i$ of $G_i$ that is valid, stretchable and attachable.",
"We distinguish between the following cases.",
"\\newlabelc:singleton111c:singleton Case 1.",
"$i$ is a singleton, i.e., $i=\\lbrace v^i\\rbrace $ .",
"Note that if $v^i$ is real, it has two neighbors on $C_{i-1}$ , while if it is dummy, it can have either two or three neighbors on $C_{i-1}$ .",
"Let $u_l$ and $u_r$ be the first and the last neighbor of $v^i$ , respectively, when walking along $C_{i-1}$ in clockwise direction from $v_1$ .",
"We will call $u_l$ (resp., $u_r$ ) the leftmost predecessor (resp., rightmost predecessor) of $v^i$ .",
"\\newlabelc:singleton-real1.111.1c:singleton-real Case 1.1.",
"Vertex $v^i$ is real.",
"Then, $u_l$ and $u_r$ are its only two neighbors in $C_{i-1}$ .",
"Each of $u_l$ and $u_r$ can be real or dummy.",
"If $u_l$ (resp., $u_r$ ) is real, we draw $(u_l,v^i)$ (resp., $(u_r,v^i)$ ) with a single segment using the NE port of $u_l$ and the SW port of $v^i$ (resp., the NW port of $u_r$ and the SE port of $v^i$ ).",
"If $u_l$ is dummy and has two successors not in $\\Gamma _{i-1}$ , we distinguish between the cases of Fig.",
"REF as shown in Fig.",
"REF .",
"The symmetric configuration of $C3$ is only used for connecting to $u_r$ .",
"Figure: A real singleton when u l u_l is dummy with two successors not in Γ i-1 \\Gamma _{i-1}If $u_l$ is dummy and has one successor not in $\\Gamma _{i-1}$ , we distinguish between the various cases of Fig.",
"REF as indicated in Fig.",
"REF (see Fig.",
"for all cases in Appendix ).",
"Observe that $C1$ requires a local reassignment of one port of $u_l$ .",
"The edge $(u_r,v^i)$ is drawn by following a similar case analysis(depicted in Fig.",
"of Appendix ).",
"Vertex $v^i$ is then placed at the intersection of the lines passing through the assigned ports, which always intersect by construction.",
"In particular, the S port is only used when $u_l$ has one successor, but the same situation cannot occur when drawing $(u_r,v^i)$ .",
"Otherwise, there is a path of $C_{i-1}$ from $u_l$ via its successor $x$ on $C_{i-1}$ to $u_r$ via its successor $y$ on $C_{i-1}$ .",
"Note that $x=y$ is possible but $x\\ne u_r$ .",
"Since the first edge on this path goes from a predecessor to a successor and the last edge goes from a successor to a predecessor, there has to be a vertex $z$ without a successor on the path; but then $u_l$ and $u_r$ are not consecutive.",
"To avoid crossings between $\\Gamma _{i-1}$ and the new edges $(u_l,v^i)$ and $(u_r,v^i)$ , we apply Lemma to suitably stretch the drawing.",
"In particular, possible crossings can occur only with vertical edge segments of $P_{i-1}(u_l,u_r)$ , because when walking along $P_{i-1}(u_l,u_r)$ from $u_l$ to $u_r$ we only encounter a (possibly empty) set of segments with slopes in the range $\\lbrace 3\\pi /4,\\pi /2,0\\rbrace $ , followed by a (possibly empty) set of segments with slopes in the range $\\lbrace \\pi /2,\\pi /4,0\\rbrace $ .",
"Figure: Some cases for the addition of a real singleton when u l u_l is dummy with one successor not in Γ i-1 \\Gamma _{i-1}Figure: Illustration for the addition of a dummy singleton\\newlabelc:singleton-dummy1.211.2c:singleton-dummy Case 1.2.",
"Vertex $v^i$ is dummy.",
"By 1-planarity, the two or three neighbors of $v^i$ on $C_{i-1}$ are all real.",
"If $v^i$ has two neighbors, we draw $(u_l,v^i)$ and $(u_r,v^i)$ as shown in Fig.",
"REF , while if $v^i$ has three neighbors, we draw $(u_l,v^i)$ and $(u_r,v^i)$ as shown in Fig.",
"REF .",
"Analogous to the previous case, vertex $v^i$ is placed at the intersection of the lines passing through the assigned ports, which always intersect by construction, and avoiding crossings between $\\Gamma _{i-1}$ and the new edges $(u_l,v^i)$ and $(u_r,v^i)$ by applying Lemma .",
"In particular, if $v^i$ has three neighbors on $C_{i-1}$ , say $u_l$ , $w$ , and $u_r$ , by REF there is a horizontal segment between $u_l$ and $w$ , as well as between $w$ and $u_r$ .",
"Thus, Lemma can be applied not only to resolve crossings, but also to find a suitable point where the two lines with slopes $\\pi /4$ and $3\\pi /4$ meet along the line with slope $\\pi /2$ that passes through $w$ .",
"\\newlabelc:chain212c:chain Case 2.",
"$i$ is a chain, i.e., $i=\\lbrace v^i_1,v^i_2,\\dots ,v^i_l\\rbrace $ .",
"We find a point as if we had to place a vertex $v$ whose leftmost predecessor is the leftmost predecessor of $v^i_1$ and whose rightmost predecessor is the rightmost predecessor of $v^i_l$ .",
"We then draw the chain slightly below this point by using the same technique used to draw 2.",
"Again, Lemma can be applied to resolve possible crossings.",
"We formally prove the correctness of our algorithm in Appendix .",
"gkvalid Drawing $\\Gamma _{K-1}$ is valid, stretchable, and attachable.",
"Figure: Illustration for the addition of kkConstruction of $\\Gamma _K$ .",
"We now show how to add $K=\\lbrace v_n\\rbrace $ to $\\Gamma _{K-1}$ so as to obtain a valid drawing of $G_K$ , and hence the desired drawing of $G$ after replacing dummy vertices with crossing points.",
"Recall that $(v_1,v_n)$ is an edge of $G$ by the definition of canonical ordering.",
"We distinguish whether $v_n$ is real or dummy; the two cases are shown in Fig.",
"REF .",
"Note that if $v_n$ is dummy, its four neighbors are all real and hence their N, NW, and NE ports are free by REF .",
"If $v_n$ is real, it has three neighbors in $\\Gamma _{K-1}$ , $v_1$ is real by construction, and the S port can be used to attach with a dummy vertex.",
"Finally, since $\\Gamma _{K-1}$ is attachable, we can use Lemma to avoid crossings and to find a suitable point to place $v_n$ .",
"A complete drawing is shown in Fig.",
"REF .",
"The theorem follows immediately by the choice of the slopes.",
"Every 3-connected cubic 1-planar graph admits a 1-bend 1-planar drawing with at most 4 distinct slopes and angular and crossing resolution $\\pi /4$ ."
],
[
"2-bend drawings",
"Liu et al.",
"[29] presented an algorithm to compute orthogonal drawings for planar graphs of maximum degree 4 with at most 2 bends per edge (except the octahedron, which requires 3 bends on one edge).",
"We make use of their algorithm for biconnected graphs.",
"The algorithm chooses two vertices $s$ and $t$ and computes an $st$ -ordering of the input graph.",
"Let $V=\\lbrace v_1,\\ldots ,v_n\\rbrace $ with $\\sigma (v_i)=i$ , $1\\le i\\le n$ .",
"Liu et al.",
"now compute an embedding of $G$ such that $v_2$ lies on the outer face if $\\deg (s)=4$ and $v_{n-1}$ lies on the outer face if $\\deg (t)=4$ ; such an embedding exists for every graph with maximum degree 4 except the octahedron.",
"The edges around each vertex $v_i,1\\le i\\le n$ , are assigned to the four ports as follows.",
"If $v_i$ has only one outgoing edge, it uses the N port; if $v_i$ has two outgoing edges, they use the N and E port; if $v_i$ has three outgoing edges, they use the N, E, and W port; and if $v_i$ has four outgoing edges, they use all four ports.",
"Symmetrically, the incoming edges of $v_i$ use the S, W, E, and N port, in this order.",
"The edge $(s,t)$ (if it exists) is assigned to the W port of both $s$ and $t$ .",
"If $\\deg (s)=4$ , the edge $(s,v_2)$ is assigned to the S port of $s$ (otherwise the port remains free); if $\\deg (t)=4$ , the edge $(t,v_{n-1})$ is assigned to the N port of $t$ (otherwise the port remains free).",
"Note that every vertex except $s$ and $t$ has at least one incoming and one outgoing edge; hence, the given embedding of the graph provides a unique assignment of edges to ports.",
"Finally, they place the vertices bottom-up as prescribed by the $st$ -ordering.",
"The way an edge is drawn is determined completely by the port assignment, as depicted in Fig.",
"REF .",
"Let $G=(V,E)$ be a subcubic 1-plane graph.",
"We first re-embed $G$ according to Lemma .",
"Let $G^*$ be the planarization of $G$ after the re-embedding.",
"Then, all cutvertices of $G^*$ are real vertices, and since they have maximum degree 3, there is always a bridge connecting two 2-connected components.",
"Let $G_1,\\ldots ,G_k$ be the 2-connected components of $G$ , and let $G_i^*$ be the planarization of $G_i,1\\le i\\le k$ .",
"We define the bridge decomposition tree $\\mathcal {T}$ of $G$ as the graph having a node for each component $G_i$ of $G$ , and an edge $(G_i,G_j)$ , for every pair $G_i, G_j$ connected by a bridge in $G$ .",
"We root $\\mathcal {T}$ in $G_1$ .",
"For each component $G_i,2\\le i\\le k$ , let $u_i$ be the vertex of $G_i$ connected to the parent of $G_i$ in $\\mathcal {T}$ by a bridge and let $u_1$ be an arbitrary vertex of $G_1$ .",
"We will create a drawing $\\Gamma _i$ for each component $G_i$ with at most 2 slopes and 2 bends such that $u_i$ lies on the outer face.",
"To this end, we first create a drawing $\\Gamma _i^*$ of $G_i^*$ with the algorithm of Liu et al.",
"[29] and then modify the drawing.",
"Throughout the modifications, we will make sure that the following invariants hold for the drawing $\\Gamma _i^*$ .",
"$\\Gamma _i^*$ is a planar orthogonal drawing of $G_i^*$ and edges are drawn as in Fig.",
"REF ; $u_i$ lies on the outer face of $\\Gamma _i^*$ and its N port is free; every edge is $y$ -monotone from its source to its target; every edge with 2 bends is a C-shape, there are no edges with more bends; if a C-shape ends in a dummy vertex, it uses only E ports; and if a C-shape starts in a dummy vertex, it uses only W ports.",
"Figure: The shapes to draw edgesbendInvariants Every $G_i^*$ admits a drawing $\\Gamma _i^*$ that satisfies invariants (I1)–(I6).",
"We choose $t=u_i$ and some real vertex $s$ and use the algorithm by Liu et al.",
"to draw $G_i^°$ .",
"Since $s$ and $t$ are real, there are no U-shapes.",
"Since no real vertex can have an outgoing edge at its W port or incoming edge at its E port, the invariants follow.",
"The full proof is given in Appendix .",
"We now iteratively remove the C-shapes from the drawing while maintaining the invariants.",
"We make use of a technique similar to the stretching in Section .",
"We lay an orthogonal $y$ -monotone curve $S$ through our drawing that intersects no vertices.",
"Then we stretch the drawing by moving $S$ and all features that lie right of $S$ to the right, and stretching all points on $S$ to horizontal segments.",
"After this stretch, in the area between the old and the new position of $S$ , there are only horizontal segments of edges that are intersected by $S$ .",
"The same operation can be defined symmetrically for an $x$ -monotone curve that is moved upwards.",
"bendBicon Every $G_i$ admits an orthogonal 2-bend drawing such that $u_i$ lies on the outer face and its N port is free.",
"We start with a drawing $\\Gamma _i^*$ of $G_i^*$ that satisfies invariants (I1)–(I6), which exists by Lemma .",
"By (I2), $u_i$ lies on the outer face and its N port is free.",
"If no dummy vertex in $\\Gamma _i^*$ is incident to a C-shape, by (I4) all edges incident to dummy vertices are drawn with at most 1 bend, so the resulting drawing $\\Gamma _i$ of $G_i$ is an orthogonal 2-bend drawing.",
"Otherwise, there is a C-shape between a real vertex $u$ and a dummy vertex $v$ .",
"We show how to eliminate this C-shape without introducing new ones while maintaining all invariants.",
"We prove the case that $(u,v)$ is directed from $u$ to $v$ , so by (I5) it uses only E ports; the other case is symmetric.",
"We do a case analysis based on which ports at $u$ are free.",
"We show one case here and the rest in Appendix .",
"\\newlabelc:bottomleft111c:bottomleft Case 1.",
"The N port at $u$ is free; see Fig.",
"REF .",
"Create a curve $S$ as follows: Start at some point $p$ slightly to the top left of $u$ and extend it downward to infinity.",
"Extend it from $p$ to the right until it passes the vertical segment of $(u,v)$ and extend it upwards to infinity.",
"Place the curve close enough to $u$ and $(u,v)$ such that no vertex or bend point lies between $S$ and the edges of $u$ that lie right next to it.",
"Then, stretch the drawing by moving $S$ to the right such that $u$ is placed below the top-right bend point of $(u,v)$ .",
"Since $S$ intersected a vertical segment of $(u,v)$ , this changes the edge to be drawn with 4 bends.",
"However, now the region between $u$ and the second bend point of $(u,v)$ is empty and the N port of $u$ is free, so we can make an L-shape out of $(u,v)$ that uses the N port at $u$ .",
"This does not change the drawing style of any edge other than $(u,v)$ , so all the invariants are maintained and the number of C-shapes is reduced by one.",
"Figure: Proof of Lemma , Case Finally, we combine the drawings $\\Gamma _i$ to a drawing $\\Gamma $ of $G$ .",
"Recall that every cutvertex is real and two biconnected components are connected by a bridge.",
"Let $G_j$ be a child of $G_i$ in the bridge decomposition tree.",
"We have drawn $G_j$ with $u_j$ on the outer face and a free N port.",
"Let $v_i$ be the neighbor of $u_j$ in $G_i$ .",
"We choose one of its free ports, rotate and scale $\\Gamma _j$ such that it fits into the face of that port, and connect $u_j$ and $v_i$ with a vertical or horizontal segment.",
"Doing this for every biconnected component gives an orthogonal 2-bend drawing of $G$ .",
"Every subcubic 1-plane graph admits a 2-bend 1-planar drawing with at most 2 distinct slopes and both angular and crossing resolution $\\pi /2$ ."
],
[
"1-bend drawings of subcubic graphs",
"There exists a subcubic 3-connected 1-plane graph such that any embedding-preserving 1-bend drawing uses at least 3 distinct slopes.",
"The lower bound holds even if we are allowed to change the outer face.",
"Let $G$ be the $K_4$ with a planar embedding.",
"The outer face is a 3-cycle, which has to be drawn as a polygon $\\Pi $ with at least four (nonreflex) corners.",
"Since we allow only one bend per edge, one of the corners of $\\Pi $ has to be a vertex of $G$ .",
"The vertex in the interior has to connect to this corner, however, all of its free ports lie on the outside.",
"Thus, no drawing of $G$ is possible.",
"the idea of making the example larger does not work .... need to think about it."
],
[
"Straight-line drawings",
"The full proofs for this section are given in Appendix .",
"lowerStraight There exist 2-regular 2-connected 1-plane graphs with $n$ vertices such that any embedding-preserving straight-line drawing uses $\\Omega (n)$ distinct slopes.",
"Figure: The constructions for the results of Section Let $G_k$ be the graph given by the cycle $a_1\\ldots ,a_{k+1},b_{k+1},\\ldots ,b_1,a_1$ and the embedding shown in Fig.",
"REF .",
"Walking along the path $a_1,\\ldots ,a_{k+1}$ , we find that the slope has to increase at every step.",
"There exist 3-regular 3-connected 1-plane graphs such that any embedding-preserving straight-line drawing uses at least 18 distinct slopes.",
"Consider the graph depicted in Fig.",
"REF .",
"We find that the slopes of the edges $(a_i,b_i),(a_i,c_i),(c_i,d_i),(c_i,e_i),(e_i,d_i),(e_i,a_{i+1})$ have to be increasing in this order for every $i=1,2,3$ .",
"There exist 3-connected 1-plane graphs such that any embedding-preserving straight-line drawing uses at least $9(\\Delta -1)$ distinct slopes.",
"Consider the graph depicted in Fig.",
"REF .",
"The degree of $a_i$ , $c_i$ , and $e_i$ is $\\Delta $ .",
"We repeat the proof of Lemma REF , but observe that the slopes of the $9(\\Delta -3)$ added edges lie between the slopes of $(a_i,b_i),(a_i,c_i),(c_i,e_i)$ , and $(e_i,a_{i+1})$ ."
],
[
"Open problems",
"The research in this paper gives rise to interesting questions, among them: (1) Is it possible to extend Theorem to all subcubic 1-planar graphs?",
"(2) Can we drop the embedding-preserving condition from Theorem REF ?",
"(3) Is the 1-planar slope number of 1-planar graphs bounded by a function of the maximum degree?"
]
] | 1808.08496 |
[
[
"The correlation of synthetic UV color vs Mg II index along the solar\n cycle"
],
[
"Abstract Modeling of planets' climate and habitability requires as fundamental input the UV emission of the hosting star.",
"\\citet{lovric2017} employed SORCE/SOLSTICE solar observations to introduce a UV color index which is a descriptor of the UV radiation that modulates the photochemistry of planets atmospheres.",
"After correcting the SOLSTICE data for residual instrumental effects that produced asymmetric signals during different phases of the cycles analyzed, the authors found that the UV color index is linearly correlated with the Mg II index.In this paper we employ an irradiance reconstruction technique to synthetize the UV color and Mg II index with the purpose of investigating whether the correction applied by \\citet{lovric2017} to SORCE/SOLSTICE data might have compensated for solar variations, and to investigate the physical mechanisms that produce such a strong correlation between the UV color index and the solar activity.",
"Reconstructed indices reproduce very well the observations and present the same strong linear dependence.",
"Moreover our reconstruction, which extends back to 1989, shows that the UV color - Mg II index relation can be described by the same linear relation for almost three cycles, thus ruling out an overcompensation of SORCE/SOLTICE data in the analysis of \\citet{lovric2017}.",
"We suggest that the strong correlation between the indices results from the fact that most of the Far- and Middle- UV radiation originates in the chromosphere, where atmosphere models of quiet and magnetic features present similar temperature and density gradients."
],
[
"Introduction",
"Variations of Total Solar Irradiance (TSI) and Spectral Solar Irradiance (SSI) measured at different temporal scales affect the Earth's chemistry and dynamics, thus affecting the Earth's climate [47], [32], [44], [21], [63], [62], [48].",
"It is well known that most of these variations are modulated by the magnetic activity, in particular by the appeareance on the solar surface of magnetic structures as plages and sunspots, which, in turn, modify the radiative properties of the solar atmosphere.",
"The principal temporal scale of variability is the 11-year cycle, observed since the 17th century, along which both, the number of active regions and the total and spectral intensity vary.",
"Of particular importance are variations measured at wavelengths shorter than approximately 400 nm, which is mostly absorbed by the high and medium layers of the Earth atmosphere.",
"In particular, solar radiation at wavelengths shorter than 120 nm (Extreme UV) plays an important role in creating the Earth ionosphere.",
"Irradiance variations at these wavelengths are responsible for variations of geomagnetic activity, may create disturbances in radio wave propagations and may modify satellite orbits due to the air-drug increase [23].",
"In the stratosphere, heating is caused by direct absorption of near-UV radiation in the Hartley (200-300 nm) and Huggins (320-360 nm) bands, whereas ozone photo-dissociation peaks in the Hartley continuum, at about 250 nm [33].",
"Variations of ozone abundance, in turn, cause further heating which, through the top-down mechanism, changes the atmospheric dynamics, thus affecting the tropospheric circulation patterns [60], [7], [31], [48].",
"Variations of solar irradiance in the UV have been continuously monitored with various instruments since the launch of NIMBUS-7 in 1978 [17], whereas continuous monitoring of the EUV started only in 2002 with the launch of the Thermosphere Ionosphere Mesosphere Energetic and Dynamics (TIMED).",
"Unfortunately, measurements of solar irradiance are known to be affected by long-term instrumental degradation effects, so that while variations obtained with different instruments typically agree when compared on temporal scales of the order of a few solar rotations, large differences are found at the solar cycle and longer temporal scales [17], [21], [81].",
"Such discrepancies hinder estimates of the impact of solar activity on the Earth atmosphere properties and in particular of the ozone production [50], [18], [48].",
"In order to provide the community with consistent spectra over long temporal scales (decades), and to extend them to times where measurements are not available, TSI and SSI measurements are complemented with independent estimates obtained with semi-empirical approaches , , , and/or through regression analyses with proper activity indices [19], [6], [70], [12].",
"In the recent past, studies of solar variability have been also motivated by the necessity of improving our understanding of stellar variability [22], with the aim of characterizing the habitable zones of stars and the atmospheres of their exo-planets.",
"Likewise the Earth's atmosphere, modeling of exoplanets requires as fundamental input the radiation emitted by the host star in the UV and shorter bands [71], [46], [65], [52].",
"For this reason, large efforts have been dedicated to measure stellar UV and XUV spectra [39], with recent particular interest in spectra of K and M dwarfs [30], [83], their planets being suitable for spectroscopic biomarker searches [61], [14], [34].",
"In the case of stellar fluxes, measurements in the FUV and shorter wavelengths are strongly hampered by interstellar medium absorption, which is significant even for relatively close stars.",
"Consequently, estimates of spectra at these spectral ranges have to rely on models [79], [83], [27] or proxies [43], [66], [67].",
"It is important to note that there is no space mission scheduled in the near future aimed at observing stellar spectra in the UV and EUV ranges, so that after the Hubble Space Telescope will cease operations, estimates of spectra at short wavelengths will necessarily rely on the use of models and/or proxies measured at longer spectral ranges.",
"Within this framework, [45] introduced a spectral color index in the solar UV that is linked to the ratio between the flux integrate over the Far-UV and Middle-UV spectral broad-bands.",
"Such a descriptor can be used to characterize UV stellar emission which modulates the photo-chemistry of molecular species, e.g., oxygen, in the atmospheres of planets [71].",
"By using solar irradiance measurements obtained with radiometers aboard the Solar Radiation and Climate Experiment satellite [49] for almost a solar cycle, the authors showed that the color index so defined is linearly correlated with the Bremen Magnesium II index, which is an excellent proxy of the magnetic activity [77].",
"Lovirc et a. showed that the correlation coefficient is slightly different for the descending phase of Cycle 23 and the ascending phase of the subsequent cycle.",
"Such difference was ascribed to residual instrumental effects, which, when compensated for, lead to a correlation coefficient constant with time.",
"On the other hand, several solar photospheric and chromospheric indices present a clear asymmetry during different phases of a cycle [5], [15], [59], whereas correlation coefficients between indices may vary from cycle to cycle [8], [69], so that the question arises whether, and to what extent, the data corrections applied by [45] might include a physical variation of solar emission.",
"The purposes of this paper are to answer this question and to investigate the physical mechanisms determining the high linear relation between the UV spectral color index and solar activity measured with the Mg II index.",
"To this aim we compare the results of [45] with synthetic indices obtained with an irradiance reconstruction technique based on the use of semi-empirical atmosphere models and full-disk observations.",
"The paper is organized as follows.",
"In Sec.",
"we describe the solar irradiance measurements analyzed, we describe the UV color index and briefly summarize the correction procedure applied to the data.",
"In Sec.",
"we describe in detail the irradiance reconstruction technique, and the input data and the radiative transfer utilized.",
"Results are presented in Sec.",
"and discussed in Sec.r̃efsec:disc.",
"Finally, our conclusions are drawn in Sec.",
"."
],
[
"The solar UV color index and solar data analysis",
"Our investigation is based on the analysis of two solar datasets.",
"The first consists of Far-UV (FUV, 115-180 nm) and Middle-UV (MUV, 180-310 nm) spectral irradiance observations acquired with the Solar Stellar Irradiance Comparison Experiment (SOLSTICE) available at http://lasp.colorado.edu/lisird/ aboard the SORCE satellite, for the period May 2003 - January 2015.",
"The second is the Bremen composite of the Magnesium II core-to-wing ratio, i.e., Mg II index [77] available at http://www.iup.uni-bremen.de/UVSAT/Datasets/mgii.",
"We provide here a basic description of the datasets, their treatment and a definition of the color index.",
"An extensive description of the datasets, data analysis and correlations with solar magnetic activity of the computed indices is provided in [45].",
"Both SOLSTICE FUV and MUV spectral irradiance observations were interpolated using a Piecewise Cubic Hermite Interpolating Polynomial scheme to take into account of missing data.",
"Fluxes in each band were then integrated using a Riemann integration algorithm, and finally the monthly averages of FUV and MUV spectral fluxes were computed.",
"The fluxes were used to derive the FUV and MUV magnitudes necessary to compute the [FUV-MUV] color index, defined as: $[FUV- MUV] = - 2.5 \\cdot log \\frac{F_{FUV}}{F_{MUV}}$ Where $F_{FUV}$ and $F_{MUV}$ are the fluxes integrated in the corresponding bands.",
"The zero points usually present in the color definition are here arbitrarily set to zero corresponding to a simple constant offset in the final magnitude difference.",
"Results presented in [45] show that the [FUV-MUV] color index is highly correlated with the Mg II index, indicating that the ratio of the fluxes in FUV and MUV bands is proportional to the solar activity on the time scale of 11 years.",
"Nevertheless, the correlation between the [FUV-MUV] color and the Mg II index shows slightly different coefficients for the descending phase of Cycle 23 and the ascending phase of Cycle 24 [45].",
"The observed [FUV-MUV] color vs Mg II index correlation can be completely reproduced by a simple model including appropriate color temperature values changing during the solar activity cycle and a suitable UV instrumental degradation.",
"Consequently, this model was only used to estimate the possible instrument degradation effect in the FUV SOLSTICE fluxes able to reproduce the measured slope difference during the descending and the ascending phase of solar cycle.",
"Assuming the same degradation process for all wavelengths in the FUV an average reduction of about 0.0002% efficiency per month is enough to remove the observed slope difference [45].",
"Under this hypothesis the [FUV-MUV] color index shows the same linear relation over the whole analyzed temporal range."
],
[
"Synthetic reconstruction of UV color and Mg II index",
"Irradiance variations of the FUV and MUV bands were reconstructed using a semi-empirical approach similar to the one described in [55], [56], which is based on the widely accepted assumption that irradiance variations observed at temporal scales from days to decades are modulated by variations of surface magnetism.",
"In summary, spectra synthetized in Local Thermodynamic Equilibrium (LTE) are generated using semi-empirical atmosphere models representing archetypal magnetic and quiet structures.",
"These are then weighed with area coverages of magnetic features as derived from full-disk CaIIK and red photospheric continua.",
"The variations of the Mg II index are estimated in the same way, with the main difference of using a Non Local Thermodynamic Equilibrium (NLTE) synthesis.",
"The spectral synthesis, data employed and procedure are describe in more detail below."
],
[
"Atmosphere models",
"Various atmosphere models have been presented in the literature to describe solar and stellar atmospheres [39].",
"For our reconstructions we employed the set of seven semi-empirical atmosphere models from the Solar Irradiance Physical Modeling (SRPM) system described in [26] (FAL2011 hereafter).",
"The different models correspond to different types of quiet and magnetic features, and consist of two quiet models (model 1000 and model 1001), two network models (model 1002 and model 1003), two facular models (model 1004 and model 1005), and an umbral (model 1006) and a penumbral (model 1007) model.",
"To avoid unrealistic chromospheric and coronal line emissions that would result from the LTE synthesis in the FUV and MUV spectral ranges, the atmosphere models were truncated at about 1500 km above the photosphere, which corresponds roughly to the base of the transition region [26].",
"This approximation is also motivated by previous studies showing that the majority of the radiation in these bands forms in the higher photosphere and chromosphere [70]; in particular, only continua at wavelengths shorter than approximately 160 nm have appreciable contribution above the temperature minimum [74], [29], [28].",
"The modification of atmosphere models for LTE synthesis in the UV is rather common, especially in irradiance reconstruction techniques based on LTE synthesis, although typically the modification is performed by linearly extrapolating the atmosphere beyond the temperature minimum, thus replacing the chromospheric temperature rise with a monotonic temperature decrease.",
"This is for instance the approach adopted in the Spectral And Total Irradiance REconstructions [75], [35] and in [55], [53].",
"As very well explained in [35], while this approximation allows to reproduce relatively well the observed solar spectrum in the MUV, it produces discrepancies at shorter wavelengths, especially in the FUV, where the spectrum is largely underestimated.",
"To improve the synthesis in this spectral range, in SATIRE the synthetic spectrum is multiplied by correction coefficients derived by the observed spectrum; old versions of SATIRE employed SUSIM data [35], while more recent versions employ the \"Solar Spectral Irradiance Reference Spectra for Whole Heliosphere Interval (WHI) 2008\" [80] and SORCE/SOLSTICE observations [82].",
"Because one of the objectives of our analysis is to verify the goodness of the instrumental degradation correction proposed in [45], and verify that this does not overcompensate for 'real' solar trends, we decided not to apply any correction factor derived from observations to our synthetic FUV spectrum.",
"The choice of truncating the atmosphere at 1500 km in height is in this respect a 'compromise' that allows to avoid unrealistic line emission, especially in the FUV, while reducing the discrepancies between the observed and synthetic spectrum."
],
[
"Synthesis of MUV and FUV spectrum",
"Numerical computation of stellar atmosphere opacities strongly relies on approximations and is affected by our limited knowledge of atomic and molecular parameters [2].",
"The discrepancies with observations are enhanced in NLTE computations, which produce over-ionization of metals and a consequent underestimate of line and continuum opacities.",
"The problem is particularly evident at wavelengths shorter than about 300 nm [9], [10], [3], [64], most likely due to uncertainties in estimates of atomic and molecular photolysis (photo-ionization and photo-dissociation) parameters.",
"As a consequence, LTE computations may perform better in reproducing the UV spectrum than NLTE ones [3], [64].",
"For our computations of the MUV and FUV spectral ranges we therefore decided to employ an LTE approach.",
"To this aim, we used the SPECTRUM code, written by R.O.",
"Gray.",
"It is a stellar spectral synthesis program, written in the C language and distributed with an atomic and molecular line list for the optical spectral region from 90 nm to 4000 nm.",
"It uses as input the run of the temperature structure and the electronic density, computing by itself the pressure and the density of the more important species.",
"Here we use the latest version, SPECTRUM 2.76 The code and the manual are available at http://www.appstate.edu/ grayro/spectrum/spectrum.html., but good results were also obtained with older versions [53], [54], [56].",
"As explained in Sec.",
"REF , to avoid unrealistic emission of chromospheric and coronal lines, especially in the FUV, which would result from the LTE treatment, the models were truncated at approximately 1500 km, corresponding to the end of the temperature plateau.",
"The LTE treatment in the UV with models with chromospheric rise is a critical point of our analysis and is further discussed in Sec. .",
"Here we note that, overall, our synthesis seems to reproduce reasonably well the observed spectrum, as shown in Fig.",
"REF .",
"In particular, the plot shows the synthetized spectrum using quiet sun model 1001 and the WHI observed spectrum.",
"The agreement is obviously more satisfactory in the MUV than in the FUV, where emission lines, especially the strong hydrogen Ly$\\alpha $ line, are not reproduced.",
"On the other hand, we note that that the slope of the continuum intensity is roughly reproduced, although the flux is underestimated at wavelengths shorter than approximately 170 nm.",
"Figure: Comparison of the spectral synthesis (black) in the FUV (left of the vertical dashed line) and MUV (right of the vertical dashed line) spectral ranges with the WHI Reference Spectrum (red).",
"The irradiance scale is logarithmic.Synthetic spectra were degraded to the spectral resolution of the observations."
],
[
"Synthesis of the Mg II line",
"The Mg II h&k lines were synthetized in NLTE [51], [4], [38], [68] with the RH code [73] together with the original (i.e.",
"not-truncated) FAL2011 models described above.",
"The adopted model atom, energy levels, lines, photo-ionization and collisional parameters are described in [36].",
"Lines in the spectral range 275-286 nm from the Kurucz database where included in the computations using a LTE approximation.",
"As describe in previous section, the continuum opacity is underestimated at these wavelength ranges.",
"To improve the agreement with the observed spectrum, the continuum opacity was therefore multiplied by correction factors as described in [9].",
"The comparison of the synthetic spectrum obtained with model 1001 at disk center with the Hawaii UV Atlas [1] measurements is shown in Fig.",
"REF .",
"The agreement is excellent in the core, while in the wings the agreement is less satisfactory.",
"The residual difference has to be ascribed to uncertainties in the temperature stratification of the model as well as 'missing' line opacity [10], [13], [25].",
"Figure: Comparison of Mg II line spectral synthesis obtained with model 1001 at disk center (black) and the Hawaii UV Atlas (red).",
"Synthetic spectrum was convolved with a 0.0014 nm width Gaussian function to matchthe resolution of the observations."
],
[
"Synthetic reconstruction of UV spectrum variability",
"Radiance variations were estimated by weighing the synthetic spectra obtained for the different atmosphere models with corresponding coverage factors [55], [56], [20], [26], [81] as: $F(\\lambda ) = \\sum _{j} \\alpha _{j} F_{j}(\\lambda ),$ where $\\alpha _j$ is the coverage factor corresponding to the j-th structure (quiet sun, network, facula or spot umbra and spot penumbra) and $F_{j}(\\lambda )$ is the corresponding flux.",
"In this model we suppose that the coverage factors are independent of disk position and use the disk integrated fluxes.",
"In [56] is shown that the error in neglecting the disk-position of magnetic features is negligible.",
"We are able to reconstruct the UV variations over the time range from 1988 to 2015 by utilizing magnetic features area coverages obtained from two different sets of full-disk observations.",
"From 1988 to 2004 we use area coverages of sunspot and faculae obtained at the San Fernando Observatory [78], [57] previously employed by [56] to reproduce TSI variations.",
"From 2005 to 2015, we used magnetic features area coverages obtained with the Precision Solar Photometric Telescope [58] located at the Mauna Loa Solar Observatory (MLSO), using the SRPM system [26], [24].",
"Both dataset were derived from full-disk CaIIK and red continuum images, but while the former distinguishes only between faculae and sunspots, the SRPM system provides area and position over the disk for features corresponding to the FAL atmosphere models utilized for the spectral synthesis.",
"In order to homogenize the datasets, we associated the sum of the areas of models 1004 and 1005 to faculae $(A_{fac})$ and the sum of the area of models 1002 and 1003 to network $(A_{net})$ , moreover we took into account that from 1988 to 2004 we have a spot coverage considered as inclusive of umbra and penumbra (i.e.",
"as sum of the areas of models 1006 and 1007), while better reconstructions are obtained by keeping separate their contributions.",
"We were able to separate the contribution of umbra $(A_{um})$ and penumbra $(A_{penum})$ in the overall spot coverage $(A_{spot}=A_{um}+A_{penum})$ , by using the relation $A_{penum}=1.6 \\cdot A_{um}$ , as derived from PSPT data.",
"The network area prior to 2004, which is not provided in the San Fernando database, was estimated by computing the correlation coefficient between the network area derived by the PSPT images and the Mg II index.",
"We synthetized in this way the daily fluxes in the spectral ranges of interest, for each of the days in which the coverage of magnetic features were available.",
"The values were then converted into irradiance and integrated in the FUV and MUV ranges.",
"In the case of the Mg II h&k lines, the spectra were convolved with a 1 nm Gaussian function and the Mg II index was then computed following the formula in [81].",
"In this section we compare the reconstructed spectral irradiance temporal variations with variations obtained from measurements.",
"In particular, we compare the relative variation $\\Delta {F}(t)= [{F}(t) -{F}(t_{min.})]/<{F}(t_{min.",
"})>$ , where ${F}(t)$ is the spectral irradiance or index at time $t$ and $<{F}(t_{min.",
"})>$ is the monthly average during a period of minimum of activity, namely April 2009.",
"Reconstructions between April 1988 and January 2015 of monthly relative variations of the Mg II index are shown in Fig.",
"REF together with the Bremen index composite variability.",
"The agreement between synthetic and measured Mg II index variability is very good over the whole period analyzed, for which the Pearson's correlation coefficient is 0.98.",
"The largest difference between measurements and reconstruction is found in 1992, during the maximum of Cycle 22, for which the Mg II index variability is overestimated up to approximately 4.%.",
"Reconstructions of monthly variations in the FUV and MUV spectral ranges are shown in Fig.",
"REF .",
"In this case, comparison of synthetic data with observations are limited to the time in which SOLSTICE data became available, that is after April 2004.",
"We note that overall the agreement between synthetic and observed variability is better in the FUV range than in the MUV.",
"Indeed, the Pearson's correlation coefficients between the observed and reconstructed variability is 0.815 and 0.94 for the MUV and FUV spectral ranges, respectively.",
"In the case of the FUV, the variability is consistently overestimated, with maximum difference between observed and synthetic data being of about 6% between 2012 and 2013.",
"In the case of the MUV, our synthesis consistently underestimates the variability, with a maximum difference of about 0.8% found in 2014 data.",
"Figure: Top: reconstruction (black) of the Mg II index monthly variations from 1988 to 2015 compared to the Bremen composite variations (red).",
"Bottom: difference between the synthetic and measured Mg II index variability.Figure: Top: monthly averages of variability obtained from synthesis (black) and SOLSTICE observations (red) in the integrated FUV (left) and MUV (right) spectral ranges.",
"Bottom: differences between synthetic and observed FUV(left) and MUV (right) variability.We finally show the dependence of relative variations of the [FUV-MUV] Color Index on relative variations of the Mg II index in Fig.",
"REF , as obtained by our reconstructions and SOLSTICE measurements.",
"Following [45] the scatter plot shows monthly averaged data and the differences are computed with respect to the average value over the period analyzed, that is $\\delta {F}(t)= [{F}(t) -<{F}(t)>)]/<{F}(t)>$ .",
"The different symbols in the plot show variations derived from our reconstruction and observations.",
"To investigate to what extent the comparison is affected by the fact that reconstructions and measurements were derived from time series of different length and sampling, we also show reconstructions computed interpolating the data at the same temporal interval and sampling of the SOLSTICE observations.",
"The agreement between reconstructions and observations is very good, especially for reconstructions interpolated at the SOLSTICE temporal sampling.",
"The plot confirms that indices obtained from both, the observations and synthetic reconstruction, are highly linearly correlated, the Spearman's correlation coefficient being in all three cases larger than 0.99.",
"These results support findings presented in [45] that the [FUV-MUV] color index strongly correlates with the Mg II index, with the [FUV-MUV] color index decreasing as the solar activity increases.",
"Moreover, the same assumptions made in the synthetic model of atmosphere lead to the exclusion of a dependence of the [FUV-MUV] color with the ascending and descending phase of the solar cycle and support the conclusion that UV instrumental degradation must be included in the analysis of the SOLSTICE FUV flux.",
"The comparison of results presented in Fig.",
"REF with those presented in Fig.",
"REF suggests that the small differences between the observed and reconstructed FUV and MUV fluxes are somewhat compensated for in the computation of the [FUV-MUV] color index.",
"On the other hand, the small deviation from linearity noticeable in Fig.",
"REF at high values of Mg II index variations has to be more likely imputed to overestimation of the reconstructed Mg II index variability, as shown in Fig.",
"REF .",
"Figure: Correlations between observed and reconstructed [FUV-MUV] color and Mg II index.",
"Black dots: indices reconstructed over the temporal range 1988-2015.",
"Red stars: indices computed from SORCE data over the temporalrange 2004-2015.",
"Black diamonds: indices reconstructed and interpolated over the same temporal range as SOLSTICE observations.",
"Note that variations have been computed with respectto the average value over the indicated temporal range (see text).",
"The dashed line represents the linear fit obtained from reconstructed data over the period 1988-2015."
],
[
"Discussion",
"This paper presents synthetic reconstructions of the [FUV-MUV] color and Mg II index that well reproduce observations obtained with SORCE/SOLSTICE over the period May 2003 - January 2015.",
"The presented reconstruction method combines spectra obtained with semi-empirical atmosphere models and area coverage of magnetic features derived from the analysis of full-disk images.",
"Our approach is similar to other semi-empirical approaches previously presented in the literature, such as SATIRE [76], [35], COSI [64], SRPM [26], [28] and OAR [55], [56], [20].",
"Figure: Variations of the MUV (left) and FUV (right) versus variations of the Mg II index as derived by FAL2011 models (plus) and SOLSTICE observations (red stars).",
"Only FAL models representing quiet,network and facular features (from model 1000 to model 1005) are shown in the plots.A very important indication, in our opinion, that our synthesis reproduces correctly the main properties of the FUV and MUV regions of the solar spectrum is supported by the results reported in Fig.",
"REF , which show the correlation between lambda-integrated MUV and FUV values and Mg II index computed by using different FAL2011 models (i.e.",
"by supposing a sun completely quiet, a sun completely covered with network, etc.)",
"together with the values measured with SOLSTICE along the solar cycle.",
"The plot highlights that the \"real\" Sun is located in the \"middle\" of the network (model 1003) and facula (model 1004) models.",
"This result is in agreement with the finding by [41] and [42] that the measured EUV and FUV continuous emission from stars of various level of activity well correlate with synthetic spectra obtained with FAL sets of models [26], [28] using an NLTE synthesis.",
"In other words, although it is established that the UV radiation is not reproducible in LTE, results presented in this paper suggest that the UV irradiance variability, at least in the FUV and MUV ranges, can be well reproduced considering a simple LTE approach.",
"This is not entirely surprising for the MUV, for which, as already discussed in detail in Sec.",
"REF , LTE synthesis have been already employed in other reconstruction techniques, but is rather unexpected for the FUV, which is flecked by a number of emission lines not reproduced by our synthesis, especially at wavelengths shorter than approximately 150 nm (see Fig.",
"REF ).",
"In this respect it is important to note that wavelengths longer than 150 nm contribute for more than 65% to the whole FUV even during periods of high activity.",
"The strong linearity between the two indices shown in Fig.",
"REF must be ascribed to their strong temperature dependence and to the proximity of the Mg II core and FUV formation heights on one side, and Mg II continuum and MUV on the other.",
"Most of the radiation contributing to the FUV and MUV originates in the higher photosphere and chromosphere [70], where atmosphere models of quiet and magnetic structures present similar (and rather small) temperature and density gradients [26].",
"This means that, at least at first order, an increase of the activity corresponds to an increase of equal amount of the temperature and density from which both MUV and FUV, as well as Mg II wings and core, originate.",
"Under the assumption that the lambda-integrated fluxes can be described by a Plank function, it easy to show that the color index must scale linearly with the temperature, at least for small temperature variations.",
"Therefore, since the spectral brightness variations over a magnetic cycle do not exceed few tens of K [26], the [FUV-MUV] color index must also scale linearly with the activity.",
"In LTE, a linear dependence of the core/wing ratio is expected [16], but such dependence is probably less obvious for the Mg II h&k line, whose cores form in NLTE conditions.",
"Nevertheless, collisions play an important role in the formation of the line [40], making the core of this doublet strongly sensitive to temperature [11].",
"Indeed, numerical magneto hydrodynamic simulations indicate that the brightness temperature derived from the Mg II h&k cores is a good approximation for the plasma temperature at the corresponding formation heights [37].",
"Moreover, the Bremen Mg II index is derived from data of relatively modest spectral resolution, so that the cores are not well resolved.",
"This means that the core intensities used to construct the Mg II index must have a non-negligible contribution from several layers of the atmosphere, from the lower photosphere, where the response function of the spectral region between the two h&k peaks reaches its maximum [72], to the base of the transition region [70], [37], [11], where the cores form.",
"Employing again a Plank function to describe both, the core and the wing intensities of the Mg II doublet, it is easy to show that the Mg II index must scale linearly with temperature, at least for small temperature perturbations."
],
[
"Conclusions",
"Results produced by the solar UV irradiance reconstructions presented in this paper reproduce with an excellent agreement the SORCE/SOLSTICE data trends compensated for instrumental degradation according to the method describe in [45].",
"This suggests that the proposed technique does not overcompensate irradiance variations and, at the moment, rules out possible temporal variations of the [FUV-MUV] color - Mg II index relation, at least during the time range analyzed, confirming that the [FUV-MUV] color index strongly correlates with the Mg II index, with the UV color index decreasing as the magnetic activity increases."
],
[
"Acknowledgments",
"The National Solar Observatory is operated by the Association of Universities for Research in Astronomy, Inc. (AURA) under cooperative agreement with the National Science Foundation.",
"This work was partially supported by the Joint Research PhD Program in “Astronomy, Astrophysics and Space Science” between the universities of Roma Tor Vergata, Roma Sapienza and INAF.",
"The authors are grateful to Dr. Jerry Harder for providing the PSPT masks, Dr. Dario del Moro for helping with the data format conversions and Dr. Han Uitenbroek for reading the paper and providing useful comments.",
"The following institutes are acknowledged for providing the data: Laboratory for Atmospheric and Space Physics (Boulder, CO) for SORCE SOLSTICE SSI data (http://lasp.colorado.edu/home/sorce/data/) and University of Bremen (Bremen, Germany) for Mg II index data (http://www.iup.uni-bremen.de/gome/gomemgii.html).",
"The authors are also thankful to the International Space Science Institute, Bern, for the support provided to the science team 335."
]
] | 1808.08439 |
[
[
"Constraining the p-mode--g-mode tidal instability with GW170817"
],
[
"Abstract We analyze the impact of a proposed tidal instability coupling $p$-modes and $g$-modes within neutron stars on GW170817.",
"This non-resonant instability transfers energy from the orbit of the binary to internal modes of the stars, accelerating the gravitational-wave driven inspiral.",
"We model the impact of this instability on the phasing of the gravitational wave signal using three parameters per star: an overall amplitude, a saturation frequency, and a spectral index.",
"Incorporating these additional parameters, we compute the Bayes Factor ($\\ln B^{pg}_{!pg}$) comparing our $p$-$g$ model to a standard one.",
"We find that the observed signal is consistent with waveform models that neglect $p$-$g$ effects, with $\\ln B^{pg}_{!pg} = 0.03^{+0.70}_{-0.58}$ (maximum a posteriori and 90% credible region).",
"By injecting simulated signals that do not include $p$-$g$ effects and recovering them with the $p$-$g$ model, we show that there is a $\\simeq 50\\%$ probability of obtaining similar $\\ln B^{pg}_{!pg}$ even when $p$-$g$ effects are absent.",
"We find that the $p$-$g$ amplitude for 1.4 $M_\\odot$ neutron stars is constrained to $\\lesssim \\text{few}\\times10^{-7}$, with maxima a posteriori near $\\sim 10^{-7}$ and $p$-$g$ saturation frequency $\\sim 70\\, \\mathrm{Hz}$.",
"This suggests that there are less than a few hundred excited modes, assuming they all saturate by wave breaking.",
"For comparison, theoretical upper bounds suggest a $p$-$g$ amplitude $\\lesssim 10^{-6}$ and $\\lesssim 10^{3}$ modes saturating by wave breaking.",
"Thus, the measured constraints only rule out extreme values of the $p$-$g$ parameters.",
"They also imply that the instability dissipates $\\lesssim 10^{51}\\, \\mathrm{ergs}$ over the entire inspiral, i.e., less than a few percent of the energy radiated as gravitational waves."
],
[
"Introduction",
"Detailed analysis of the gravitational-wave (GW) signal received from the first binary neutron star (NS) coalescence event (GW170817 [1]) constrains the tidal deformability of NSs and thus the equation of state (EOS) above nuclear saturation density [2], [3], [4].",
"Studies of NS tidal deformation typically focus on the linear, quasi-static tidal bulge induced in each NS by its companion.",
"Such deformations modify the system's binding energy and GW luminosity and thereby alter its orbital dynamics.",
"The degree of deformation is often expressed in terms of the tidal deformability $\\Lambda _i \\propto (R_i/m_i)^5$ of each component [5], or a particular mass-weighted average thereof ($\\tilde{\\Lambda }$ ) [2].",
"The strong dependence on compactness $R/m$ means that a stiffer EOS, which has larger $R$ for the same $m$ , imprints a larger tidal signals than a softer EOS.",
"Current analyses of GW data from the LIGO [6] and Virgo [7] detectors favor a soft EOS [3], [8].",
"Specifically, [2] finds $\\tilde{\\Lambda } \\lesssim \\leavevmode {\\color {black}730}$ at the 90% credible level for all waveform models considered, allowing for the components to spin rapidly.",
"The pressure at twice nuclear saturation density is also constrained to $P = \\leavevmode {\\color {black}3.5^{+2.7}_{-1.7}\\times 10^{34}\\ \\mathrm {dyn}/\\mathrm {cm}^2}$ (median and 90% credible region) [3] assuming small component spins.",
"In addition to GW phasing, the EOS-dependence of $\\tilde{\\Lambda }$ should correlate with post-merger signals [9], possible tidal disruptions, and kilonova observations [10].",
"Observed light-curves for the kilonova suggest a lower bound of $ \\tilde{\\Lambda }\\gtrsim \\leavevmode {\\color {black}200}$ [11], [12].",
"Although some dynamical tidal effects are incorporated in these analyses (see, e.g., [13], [2]), the impact of several types of dynamical tidal effects are neglected because they are assumed to be small or have large theoretical uncertainties.",
"These effects arise because tidal fields, in addition to raising a quasi-static bulge, excite stellar normal modes.",
"Three such excitation mechanisms are (i) resonant linear excitation, (ii) resonant nonlinear excitation, and (iii) non-resonant nonlinear excitation (see, e.g., [14]).",
"The first occurs when the GW frequency (the oscillation frequency of the tidal field) sweeps through a mode's natural frequency (see, e.g., [15], [16], [17], [18], [19], [20], [21], [22]).",
"However, since the GW frequency increases rapidly during the late inspiral, the time spent near resonance is too short to excite modes to large amplitudes.",
"As a result, for modes with natural frequencies within the sensitive bands of ground-based GW detectors, the change in orbital phasing is expected to be small ($\\Delta \\Psi \\lesssim 10^{-2}\\textrm { rad}$ ) unless the stars are rapidly rotating [17], [18], [19].",
"The impact of resonant nonlinear mode excitation (i.e., the parametric subharmonic instability) is likewise limited by the swiftness of the inspiral [23].",
"The proposed $p\\text{-}g$ tidal instability is a non-resonant, nonlinear instability in which the tidal bulge excites a low-frequency buoyancy-supported $g$ -mode and a high-frequency pressure-supported $p$ -mode [23], [24], [25], [26].",
"It occurs in the inner core of the NS, where the stratification is weak and the shear due to the tidal bulge is especially susceptible to instability.",
"Unlike resonantly excited modes, an unstable $p\\text{-}g$ pair continuously drains energy from the orbit once excited, even after the orbital frequency changes significantly.",
"There are many potentially unstable $p\\text{-}g$ pairs, each becoming unstable at a different frequency and growing at a different rate.",
"Although there is considerable uncertainty about the number of unstable pairs, their exact growth rates, and how they saturate, estimates suggest that the impact could be measurable with current detectors [27].",
"In this letter, we investigate the possible impact of the $p\\text{-}g$ instability on GW170817 using the phenomenological model developed in [27].",
"The model describes the energy dissipated by the instability within each NS, indexed by $i$ , in terms of three parameters: (i) an overall amplitude $A_i$ , which is related to the number of modes participating in the instability, their growth rates, and their saturation energies, (ii) a frequency $f_i$ corresponding to when the instability saturates, and (iii) a spectral index $n_i$ describing how the saturation energy evolves with frequency.",
"In Section , we describe our models in detail.",
"In Section , we compare the statistical evidence for models that include the $p\\text{-}g$ instability relative to those that do not.",
"In Section , we investigate the constraints on the $p\\text{-}g$ parameters from GW170817, and in Section we conclude."
],
[
"Phenomenological Model",
"Following [27], we extend a post-Newtonian (PN) waveform by including a parametrized model of the $p\\text{-}g$ instability.",
"For the initial PN model, we use the TaylorF2 frequency-domain approximant (see, e.g., [28]) terminated at the inner-most stable circular orbit, which includes the effects of linear tides ($\\tilde{\\Lambda }$ ) and spins aligned with the orbital angular momentum (the impact of mis-aligned spins on $p\\text{-}g$ effects is not known).",
"Waveform systematics between different existing approximants may be important for small $p\\text{-}g$ effects.",
"However, by comparing the waveform mismatches between several other models (TaylorF2, SEOBNRT, PhenomDNRT, and PhenomPNRT, see [2]), we find these systematic become important for $p\\text{-}g$ effects roughly an order of magnitude smaller than the upper limits set by our analysis (see Section ).",
"We expect TaylorF2 to be reasonably accurate and defer a complete analysis of waveform systematics to future work.",
"Assuming the $p\\text{-}g$ effects are a perturbation to TaylorF2, we find that they modify the phase in the frequency-domain by $\\Delta \\Psi (f) = - \\frac{2C_1}{3B^2(3-n_1)(4-n_1)}\\left[ \\Theta _1 \\left(\\frac{f}{f_{\\rm ref}}\\right)^{n_1-3} + (1-\\Theta _1) \\left(\\frac{f_1}{f_{\\rm ref}}\\right)^{n_1-3}\\left( \\left(4-n_1\\right) - \\left(3-n_1\\right)\\left(\\frac{f}{f_1}\\right)\\right)\\right] + (1 \\leftrightarrow 2),$ where $f_i$ is the saturation frequency, $f_\\mathrm {ref} \\equiv 100\\, \\mathrm {Hz}$ is a reference frequency with no intrinsic significance, $C_i = [2m_i/(m_1+m_2)]^{2/3} A_i$ , $B = (32/5)(G\\mathcal {M}\\pi f_\\mathrm {ref}/c^3)^{5/3}$ , $\\mathcal {M}=(m_1 m_2)^{3/5}/(m_1+m_2)^{1/5}$ , and $\\Theta _i = \\Theta (f-f_i)$ where $\\Theta $ is the Heaviside function.",
"This approximant is slightly different than that of [27] because they incorrectly applied the saddle-point approximation to obtain the frequency-domain waveform from time-domain phasing [29].",
"This correction renders the $p\\text{-}g$ instability slightly more difficult to measure than predicted in [27], although the observed behavior is qualitatively similar.",
"Specifically, we find that in order to achieve the same $|\\Delta \\Psi |$ , $A_i$ needs to be larger than [27] found by a factor of $\\sim (4-n_i)$ , although the precise factor also depends on the other $p\\text{-}g$ parameters.",
"The $\\Delta \\Psi $ expression contains three types of terms: a constant term, a linear term $\\propto (1-\\Theta _i)f$ , and a power-law term $\\propto \\Theta _i f^{n_i-3}$ .",
"The constant term corresponds to an overall phase offset and is degenerate with the orbital phase at coalescence.",
"The linear term corresponds to a change in the time of coalescence; because the $p\\text{-}g$ instability transfers energy from the orbit to stellar normal modes, the binary inspirals faster than it would if the effect was absent.",
"The power-law term accounts for the competition between the rate of $p\\text{-}g$ energy dissipation and the rate of inspiral, both of which increase as $f$ increases.",
"As argued in [27], we expect $n_i<3$ , which implies that the phase shift accumulates primarily at frequencies just above the “turn-on\" (saturation) frequency $f\\gtrsim f_i$ .",
"When $n_i < 3$ , $p\\text{-}g$ effects are most important at lower frequencies whereas linear tides ($\\tilde{\\Lambda }$ ) and spins ($\\chi _i = c S_i/G m_i^2$ , where $S_i$ is the spin-angular momentum of each component) have their largest impact at higher frequencies (see, e.g., [30]).",
"The priors placed on the latter quantities can, however, affect our inference of $p\\text{-}g$ parameters.",
"In order to account for a possible dependence on the component masses ($m_i$ ), we parametrize our model using a Taylor expansion in the $p\\text{-}g$ parameters around $m_i=1.4M_\\odot $ and sample from the posterior using the first two coefficients.",
"Our model computes $A_i$ as $A_i(m_i) = A_0 + \\left(\\left.\\frac{dA}{dm}\\right|_{1.4M_\\odot }\\right)\\left(m_i - 1.4M_\\odot \\right),$ and uses $A_0$ and $dA/dm$ instead of $A_1$ and $A_2$ .",
"The model uses similar representations for $f_i$ and $n_i$ in terms of the parameters $f_0$ , $df/dm$ , $n_0$ , and $dn/dm$ .",
"We assume a uniform prior on $\\log _{10} A_0$ within $10^{-10} \\le A_0 \\le 10^{-5.5}$ , a uniform prior in $f_0$ within $10\\,\\mathrm {Hz} \\le f_0 \\le 100\\,\\mathrm {Hz}$ , and a uniform prior in $n_0$ within $-1 \\le n_0 \\le 3$ .",
"The priors on the first-order terms ($dA/dm, df/dm, dn/dm$ ) are the same as those in [27]; when $m_1 \\sim m_2$ , they imply $A_1 \\sim A_2$ , etc.",
"We investigate GW170817 using data from several different frequency bands and with different spin priors, but unless otherwise noted we focus on results for data above 30 Hz with $|\\chi _i|\\le 0.89$ .",
"Throughout this letter, results from GW170817 were obtained using the same data conditioning as [2], including the removal of a short-duration noise artifact from the Livingston data ([31] and discussion in [1]) along with other independently measured noise sources (see, e.g., [32], [33], [34], [35]), calibration [36], [37], marginalization over calibration uncertainties, and whitening procedures [38], [39]."
],
[
"Model Selection",
"Using GW data from GW170817, we perform Bayesian model selection.",
"We compare a model that includes linear tides, spin components alinged with the orbital angular momentum, and PN phasing effects up to 3.5 PN phase terms ($\\mathcal {H}_{!pg}$ ) to an extension of this model that also includes $p\\text{-}g$ effects ($\\mathcal {H}_{pg}$ ).",
"Since we have nested models ($\\mathcal {H}_{!pg}$ is obtained from $\\mathcal {H}_{pg}$ as $A_i\\rightarrow 0$ )Since we use a uniform-in-$\\log _{10} A_0$ prior, $\\mathcal {H}_{pg}$ does not formally include $A_i=0$ .",
"Nonetheless, our lower limit on $A_i$ is sufficiently small that $\\mathcal {H}_{!pg}$ is effectively nested in $\\mathcal {H}_{pg}$ ., we use the Savage-Dickey Density Ratio (see, e.g., [40], [41], [42]) to estimate the Bayes Factor ($B^{pg}_{!pg}=p(D|\\mathcal {H}_{pg})/p(D|\\mathcal {H}_{!pg})$ , where $D$ refers to the observed data).",
"Specifically, we sample from the model's posterior distribution [43] and calculate $\\lim \\limits _{A_i\\rightarrow 0}\\left[\\frac{p(A_i|D, \\mathcal {H}_{pg})}{p(A_i|\\mathcal {H}_{pg})}\\right] & = \\lim \\limits _{A_i\\rightarrow 0}\\left[\\frac{1}{p(D|\\mathcal {H}_{pg})} \\int d \\theta df_i \\, dn_i \\, p(D|\\theta , A_i, f_i, n_i; \\mathcal {H}_{pg}) \\, p(\\theta |\\mathcal {H}_{pg}) \\, p(f_i, n_i|A_i, \\mathcal {H}_{pg})\\right] \\nonumber \\\\& = \\frac{1}{p(D|\\mathcal {H}_{pg})} \\int d \\theta \\, p(D|\\theta ; \\mathcal {H}_{!pg}) \\, p(\\theta |\\mathcal {H}_{!pg}) \\left[\\frac{p(\\theta |\\mathcal {H}_{pg})}{p(\\theta |\\mathcal {H}_{!pg})}\\right] \\int df_i \\, dn_i \\, p(f_i, n_i|A_i, \\mathcal {H}_{pg}) \\nonumber \\\\& = \\frac{p(D|\\mathcal {H}_{!pg})}{p(D|\\mathcal {H}_{pg})} \\left< \\frac{p(\\theta |\\mathcal {H}_{pg})}{p(\\theta |\\mathcal {H}_{!pg})} \\right>_{p(\\theta |D, \\mathcal {H}_{!pg})},$ where $\\theta $ refers to all parameters besides the $p\\text{-}g$ phenomenological parameters, we note that $\\int df dn\\, p(f_i,n_i|A_i, \\mathcal {H}_{pg})=1\\, \\forall \\, A_i$ , and $\\left<x\\right>_p$ denotes the average of $x$ with respect to the measure defined by $p$ .",
"Assuming that $p(\\theta |\\mathcal {H}_{pg})=p(\\theta |\\mathcal {H}_{!pg})$ , we determine $\\ln B^{pg}_{!pg}$ from the ratio, as $A_i \\rightarrow 0$ , of the marginal distribution of $A_i$ a priori to the distribution a posteriori $\\ln B^{pg}_{!pg}= \\lim \\limits _{A_i\\rightarrow 0} \\left[\\ln p(A_i|\\mathcal {H}_{pg}) - \\ln p(A_i|D, \\mathcal {H}_{pg})\\right].$ We confirmed that this estimate agrees with estimates from both nested sampling [44] and thermodynamic integration [45].",
"Figure: Distributions of lnB !pg pg \\ln B^{pg}_{!pg} due to sampling uncertainty for different values of f low f_\\mathrm {low}.The solid red curves assume high-spin priors (|χ i |≤0.89|\\chi _i|\\le 0.89) and the dashed blue curves assume low-spin priors (|χ i |≤0.05|\\chi _i|\\le 0.05).Figure REF shows $\\ln B^{pg}_{!pg}$ as a function of $f_\\mathrm {low}$ , the minimum GW frequency considered.",
"At a given $f_{\\rm low}$ , we show the distribution of $\\ln B^{pg}_{!pg}$ due to the sampling uncertainty from the finite length of our MCMC chains.",
"The solid and dashed curves correspond to the high-spin ($|\\chi _i| \\le 0.89$ ) and low-spin ($|\\chi _i|\\le 0.05$ ) priors discussed in [1], [2], [3].",
"For certain combinations of $f_\\mathrm {low}$ and $|\\chi _i|$ , we find $\\ln B^{pg}_{!pg}>0$ , suggesting $\\mathcal {H}_{pg}$ is more likely than $\\mathcal {H}_{!pg}$ .",
"In order to assess how likely such values are, we calculate $\\ln B^{pg}_{!pg}$ for a large number of simulated, high-spin signals with $A_i=0$ and distinct realizations of detector noise from times near GW170817.",
"We find that simulated signals without $p\\text{-}g$ effects can readily produce $\\ln B^{pg}_{!pg}$ at least as large as the ones we measured from GW170817.",
"In particular, $\\ln B^{pg}_{!pg}$ for the 30 Hz, high-spin data corresponds to a False Alarm Probability (FAP) $\\approx \\leavevmode {\\color {black}50\\%}$ .",
"We focus on the 30 Hz, high-spin data because it corresponds to the largest bandwidth investigated and the largest signal-to-noise ratio.",
"The high-spin prior is the most inclusive prior considered, and therefore allows the most model freedom when fitting $p\\text{-}g$ effects.",
"In our model of the instability, the phase shift $\\Delta \\Psi $ accumulates primarily at frequencies just above the saturation frequency $f\\gtrsim f_i$ .",
"Therefore, if it is present, its impact should become more apparent as we decrease the minimum GW frequency considered from $f_\\mathrm {low} \\gg f_i$ to $f_{\\rm low}\\lesssim f_i$ .",
"We do see some indication of this behavior in Fig.",
"REF .",
"However, we note that if our phenomenological model breaks down at $f<f_i$ due to poor modeling of the pre-saturation behavior (e.g., if our step-function turn-on at $f_i$ is not a good approximation to the instability's induced phase shift), we might expect $\\ln B^{pg}_{!pg}$ to decrease as we lower $f_\\mathrm {low}$ below $f_i$ ."
],
[
"Parameter Inference",
"We now investigate the constraints obtained from GW170817.",
"Figure REF shows the joint posterior distributions for both $\\mathcal {H}_{pg}$ and $\\mathcal {H}_{!pg}$ .",
"We find that $\\mathcal {H}_{pg}$ and $\\mathcal {H}_{!pg}$ yield similar posterior distributions for all non-$p\\text{-}g$ parameters, including both extrinsic and intrinsic parameters.",
"The constraints on the chirp mass ($\\mathcal {M}$ ), effective spin $\\chi _\\mathrm {eff} = (m_1\\chi _1 + m_2\\chi _2)/(m_1+m_2)$ , and $\\tilde{\\Lambda }$ are slightly weaker in $\\mathcal {H}_{pg}$ than $\\mathcal {H}_{!pg}$ .",
"This is because $\\mathcal {H}_{pg}$ provides extra freedom to the signal's duration in the time-domain.",
"Regarding the $p\\text{-}g$ parameters, we find a noticeable peak near $A_0\\sim 10^{-7}$ with a flat tail to small $A_0$ .",
"We find $A_0 \\le \\leavevmode {\\color {black}3.3\\times 10^{-7}}$ assuming a uniform-in-$\\log _{10} A_0$ prior and $A_0 \\le \\leavevmode {\\color {black}6.8\\times 10^{-7}}$ assuming a uniform-in-$A_0$ prior, both at 90% confidence.The upper limit with a uniform-in-$A_0$ prior is larger only because we weight larger values of $A_0$ more a priori than with a uniform-in-$\\log _{10} A_0$ prior.",
"We also find a peak at $f_0\\sim \\leavevmode {\\color {black}70\\, \\mathrm {Hz}}$ .",
"The peaks persist when we analyze the data from each interferometer separately, with reasonably consistent locations and shapes (Fig.",
"REF ).",
"However, we find that the simulated signals with $A_i=0$ can produce similar peaks, suggesting they may be due to noise alone.",
"Similar to [27], we find that $n_i$ is not strongly constrained and the gradient terms in the Taylor expansions are not measurable.",
"Theoretical arguments suggest an upper bound of $A_0 \\lesssim 10^{-6}$ [27].",
"Therefore, our $A_0$ constraint only rules out the most extreme values of the $p\\text{-}g$ parameters."
],
[
"Discussion",
"While GW170817 is consistent with models that neglect $p\\text{-}g$ effects, it is also consistent with a broad range of $p\\text{-}g$ parameters.",
"The constraints from GW170817 imply that there are $\\lesssim \\leavevmode {\\color {black}200}$ excited modes at $f=100\\,\\mathrm {Hz}$ , assuming all modes grow as rapidly as possible and saturate at their breaking amplitudes ($\\lambda =\\beta =1$ in Eq.",
"(7) of [27]) and that the frequency at which modes become unstable is well approximated by $f_0$ .",
"For comparison, theoretical arguments suggest an upper bound of $\\sim 10^{3}$ modes saturating by wave breaking [27].",
"More modes may be excited if they grow more slowly or saturate below their wave breaking energy.",
"We can also use the measured constraints to place upper limits on the amount of energy dissipated by the $p\\text{-}g$ instability.",
"As Fig.",
"REF shows, $p\\text{-}g$ effects dissipate $\\lesssim \\leavevmode {\\color {black}2.7\\times 10^{51}\\, \\mathrm {ergs}}$ throughout the entire inspiral at 90% confidence.",
"In comparison, GWs carry away $\\gtrsim \\leavevmode {\\color {black}10^{53}\\, \\mathrm {ergs}}$ .",
"This implies time-domain phase shifts $|\\Delta \\phi | \\lesssim \\leavevmode {\\color {black}7.6\\, \\mathrm {rad}}$ ($0.6$ orbits) at $100\\textrm { Hz}$ and $|\\Delta \\phi | \\lesssim \\leavevmode {\\color {black}32\\, \\mathrm {rad}}$ ($2.6$ orbits) at $1000\\textrm { Hz}$ after accounting for the joint uncertainty in component masses, spins, linear tides, and $p\\text{-}g$ effects.",
"A $g$ -mode with natural frequency $f_g$ is predicted to become unstable at a frequency $f_{\\rm crit}\\simeq 45\\, \\mathrm {Hz} (f_g/10^{-4}\\lambda f_{\\rm dyn})^{1/2}$ , where $f_{\\rm dyn}$ is the dynamical frequency of the NS and $\\lambda $ is a slowly varying function typically between $0.1-1$ [25], [27].",
"Since the modes grow quickly, the frequency at which the instability saturates is likely close to the frequency at which the modes become unstable ($f_0\\simeq f_{\\rm crit}$ ).",
"If we assume that the observed peak near $f_0\\sim \\leavevmode {\\color {black}70\\, \\mathrm {Hz}}$ is not due to noise alone, then the maximum a posteriori estimate for $f_0$ along with approximate values for the masses (1.4 $M_\\odot $ ) and radii (11 km) of the components [3] imply $f_g \\simeq \\leavevmode {\\color {black}0.5\\, \\mathrm {Hz}}$ .",
"Figure: Upper limits on the cumulative enegy dissipated by the p-gp\\text{-}g instability as a function of frequency.We note the relatively strong constraints at lower frequencies, where p-gp\\text{-}g effects are more pronounced.With several more signals comparable to GW170817, it should be possible to improve the amplitude constraint to $A_0 \\lesssim 10^{-7}$ .",
"Obtaining even tighter constraints will likely require many more detections, especially since most events will have smaller SNR.",
"Future measurements will also benefit from a better understanding of how the instability saturates.",
"To date, there have only been detailed theoretical studies of the instability's threshold and growth rate [23], [24], [25], [26], not its saturation.",
"As a result, we cannot be certain of the fidelity of our phenomenological model.",
"While this letter was in final internal review, related work was posted [46] with, in particular, the conclusion that the $\\mathcal {H}_{!pg}$ model is strongly favored over the $\\mathcal {H}_{pg}$ model by a factor of at least $10^4$ .",
"We are investigating possible reasons for the differences between our conclusions.",
"The authors gratefully acknowledge the support of the United States National Science Foundation (NSF) for the construction and operation of the LIGO Laboratory and Advanced LIGO as well as the Science and Technology Facilities Council (STFC) of the United Kingdom, the Max-Planck-Society (MPS), and the State of Niedersachsen/Germany for support of the construction of Advanced LIGO and construction and operation of the GEO600 detector.",
"Additional support for Advanced LIGO was provided by the Australian Research Council.",
"The authors gratefully acknowledge the Italian Istituto Nazionale di Fisica Nucleare (INFN), the French Centre National de la Recherche Scientifique (CNRS) and the Foundation for Fundamental Research on Matter supported by the Netherlands Organisation for Scientific Research, for the construction and operation of the Virgo detector and the creation and support of the EGO consortium.",
"The authors also gratefully acknowledge research support from these agencies as well as by the Council of Scientific and Industrial Research of India, the Department of Science and Technology, India, the Science & Engineering Research Board (SERB), India, the Ministry of Human Resource Development, India, the Spanish Agencia Estatal de Investigación, the Vicepresidència i Conselleria d'Innovació, Recerca i Turisme and the Conselleria d'Educació i Universitat del Govern de les Illes Balears, the Conselleria d'Educació, Investigació, Cultura i Esport de la Generalitat Valenciana, the National Science Centre of Poland, the Swiss National Science Foundation (SNSF), the Russian Foundation for Basic Research, the Russian Science Foundation, the European Commission, the European Regional Development Funds (ERDF), the Royal Society, the Scottish Funding Council, the Scottish Universities Physics Alliance, the Hungarian Scientific Research Fund (OTKA), the Lyon Institute of Origins (LIO), the Paris Île-de-France Region, the National Research, Development and Innovation Office Hungary (NKFI), the National Research Foundation of Korea, Industry Canada and the Province of Ontario through the Ministry of Economic Development and Innovation, the Natural Science and Engineering Research Council Canada, the Canadian Institute for Advanced Research, the Brazilian Ministry of Science, Technology, Innovations, and Communications, the International Center for Theoretical Physics South American Institute for Fundamental Research (ICTP-SAIFR), the Research Grants Council of Hong Kong, the National Natural Science Foundation of China (NSFC), the Leverhulme Trust, the Research Corporation, the Ministry of Science and Technology (MOST), Taiwan and the Kavli Foundation.",
"The authors gratefully acknowledge the support of the NSF, STFC, MPS, INFN, CNRS and the State of Niedersachsen/Germany for provision of computational resources.",
"N. Weinberg was supported in part by NASA grant NNX14AB40G."
]
] | 1808.08676 |
[
[
"An experimental investigation of the laminar horseshoe vortex around an\n emerging obstacle"
],
[
"Abstract An emerging long obstacle placed in a boundary layer developing under a free-surface generates a complex horseshoe vortex (HSV) system, which is composed of a set of vortices exhibiting a rich variety of dynamics.",
"The present experimental study examines such flow structure and characterizes precisely, using PIV measurements, the evolution of the HSV geometrical and dynamical properties over a wide range of dimensionless parameters (Reynolds number $Re_h \\in [750, 8300]$, boundary layer development ratio $h/\\delta \\in [1.25, 4.25]$ and obstacle aspect ratio $W/h \\in [0.67, 2.33]$).",
"The dynamical study of the HSV is based on the categorization of the HSV vortices motion into an enhanced specific bi-dimensional typology, separating a coherent (due to vortex-vortex interactions) and an irregular evolution (due to appearance of small-scale instabilities).",
"This precise categorization is made possible thanks to the use of vortex tracking methods applied on PIV measurements, A semi-empirical model for the HSV vortices motion is then proposed to highlight some important mechanisms of the HSV dynamics, as (i) the influence of the surrounding vortices on a vortex motion and (ii) the presence of a phase shift between the motion of all vortices.",
"The study of the HSV geometrical properties (vortex position and characteristic lengths and frequencies) evolution with the flow parameters shows that strong dependencies exist between the streamwise extension of the HSV and the obstacle width, and between the HSV vortex number and its elongation.",
"Comparison of these data with prior studies for immersed obstacles reveals that emerging obstacles lead to greater adverse pressure gradients and down-flows in front of the obstacle."
],
[
"Context",
"An obstacle placed in a developing boundary layer over a flat plate creates an adverse pressure gradient which, if sufficiently strong, makes the boundary layer detach.",
"The boundary layer separation creates a shear-layer, separating the main (upper) flow and the back (bottom) flow, which can contains a succession of vortices [25].",
"Those vortices do not appear in the upstream-most part of the shear layer, called the separation surface in figure REF [59], but can exhibit complex dynamics (oscillating motion, merging by pairs, diffusion, turbulent behavior) in the downstream part [38].",
"The resulting set of vortices wraps around the obstacle with a particular shape, explaining the name given to the whole structure: the horseshoe vortex (HSV, as illustrated on figure REF a).",
"Depending on the obstacle shape and on the flow velocity, recirculation zones can also appear at the sides of the obstacle and behind it [35].",
"The HSV has been extensively studied since 1962 [50] for its numerous applications: (i) The HSV influences the amount of turbulence released in the downstream boundary layer, impacting the aerodynamic properties.",
"(ii) The shear stress at the bottom wall and at the obstacle are affected by the HSV, modifying both the thermal exchanges [48] and the scouring process in hydraulic with mobile bed [21].",
"(iii) HSV appear in transverse jets interacting with boundary layers [30] in flow control issues.",
"(iv) Finally, the force applied by the flow on the obstacle, of interest in turbo-machinery [18] and hydraulics, is also affected by the HSV.",
"Flow configurations vary in those studies, with different obstacle shapes (cylinders, prisms, foils) and emerging, immersed or traversing obstacles.",
"The present work focuses on emerging long rectangular prisms (see figure REF ).",
"Figure: Schematic representation of a laminar horse-shoe vortex (HSV), created by the interaction of a free-surface flow and an emerging obstacle of width WW:(a) 3D illustration of the HSV system showing 3 vortices (A, B, C).",
"(b) Side mid-plane view of the HSV system showing the control parameters (boundary layer thickness δ\\delta , water level hh, bulk velocity u D u_D and obstacle width WW and length LL) along with the definition of the HSV main geometrical characteristics (streamwise elongation λ\\lambda and vortex center positions x i x_i, y i y_i)."
],
[
"State-of-the-art",
"In the case of a laminar boundary layer interacting with an (immersed or emerging) obstacle, different HSV dynamics typologies have been reported, based either on the number of vortices in the HSV and/or on their dynamics [50], [3], [25], [37].",
"The most complete and generally accepted typology is the one from [25], obtained through flow visualizations, who defined five HSV regimes based on the dynamics of the vortices, namely: (i) Steady vortex system (stationary HSV), (ii) Oscillating vortex system (periodical HSV, with streamwise vortex position oscillation), (iii) Amalgamating vortex system (vortex creation in the upstream part of the HSV and periodical vortex merging by pairs in its downstream part), (iv) Breakaway vortex system (periodical vortex shedding from the HSV, disappearing by diffusion near the obstacle foot), and (iv) Transitional vortex system (aperiodical vortex dynamics).",
"This typology was later confirmed, partially or completely, by [33], [39], [32], [38] for immersed obstacles and by [52], [20] for emerging obstacles, using either flow visualization, PIV, numerical simulations and/or pointwise velocity measurements.",
"However, studies devoted to understand how those regimes evolve with the flow and obstacle parameters are rare in this context: for an immersed obstacle configuration, [38] recently showed that the HSV regimes evolution mainly depends on the Reynolds number based on the obstacle width $_W$ and on the ratio of the boundary layer thickness over the obstacle width $\\delta /W$ .",
"Turbulent HSV are characterized by temporally non-coherent vortices and therefore do not allow a typology definition.",
"The HSV is rather characterized by a bi-modal phenomenon: the time alternation between the so-called “zero-flow” and “back-flow”, first described by [13] and later confirmed by [2], [35], [15], [20].",
"[44] showed, using numerical simulations, that this bi-modal phenomenon was actually three-dimensional and linked to the Görtler instability developing under the downstream-most vortex.",
"As the transition to turbulence of the HSV has not been extensively studied, it is not known if turbulent HSV can arise from laminar boundary layers.",
"Regarding immsersed obstacles, [54] summarized the state-of-the-art concerning flows around blunt or streamlined obstacles.",
"For laminar HSV, he compared the HSV dynamics regimes observed by different authors.",
"For turbulent HSV, he summarized the different descriptions of the bimodality phenomenon.",
"He also discussed the effect of the HSV on the scouring process and the “bluntness factor” that allows to take the obstacle geometry into account in the dimensional analysis.",
"[8] collected existing data on the evolution of the main HSV geometrical characteristics (such as the separation distance $\\lambda $ and the vortices position, see figure REF ) for both laminar and turbulent HSV, using, inter alia, the works of [3], [4], [5], [6], [7].",
"They indicated that the obstacle width $W$ is the main parameter for the HSV evolution, the obstacle height $\\xi $ being significant only with low ratios of $\\xi /W$ .",
"They also concluded that the HSV increases significantly the bottom wall shear stress, making the HSV an important structure for the scouring process and thermal transfers.",
"The literature dedicated to emerging obstacles is far less exhaustive than for immersed obstacles.",
"The specificity of the emerging obstacle configuration is that the flow cannot pass over the obstacle and is forced to skirt the obstacle by its sides.",
"When studying the effect of the obstacle submergence (with varying obstacle heights from immersed to emerging) on turbulent HSV, [49] indicated that the separation distance $\\lambda $ and the shear stress below the HSV are more important in the case of emerging obstacles.",
"In fact, most studies with emerging obstacles are dedicated to turbulent HSV [11], [23], [28], [47], [43], [49], and draw similar qualitative conclusions regarding the turbulent HSV dynamics than in the immersed obstacle configuration.",
"[52] examined the laminar HSV upstream of an emerging obstacles, but they focused only on the breakaway vortex system regime.",
"Indeed, to the authors knowledge, only a few studies in laminar HSV around an emerging obstacle provide a comprehensive description of the HSV dynamics and their geometrical properties as a function of the flow and obstacle parameters.",
"To summarize, the HSV in immersed obstacle configuration is well-documented thanks to numerous studies (see [8] survey).",
"This is, nonetheless, not the case for the emerging obstacle configuration subjected to a laminar boundary layer.",
"While some characteristics, such as the HSV regimes or some parametric dependencies, seem to be qualitatively similar for both emerging and immersed configurations, the evolution of the HSV properties with the dimensionless parameters of the flow remains poorly known for emerging obstacles.",
"In addition, it is not clear yet how the confinement of the free surface, that should strongly influence the HSV, affects its geometrical and dynamical properties.",
"In this context, this work aims at characterizing the evolution of the HSV with the dimensionless flow parameters, in the case of a long obstacle emerging from a laminar free-surface flow.",
"The studied HSV characteristics are separated in two main parts in the sequel: (i) the HSV vortices dynamics, whose study is based on a dynamics typology, and (ii) the HSV geometrical properties (size, number of vortex and vortex average position)."
],
[
"Dimensional analysis",
"Any property of the HSV that forms in a laminar boundary layer facing an emerging rectangular obstacle can be expressed as a function of the fluid, flow and geometrical parameters, as: $X = f(\\nu , \\rho , \\sigma , u_D, \\delta , H, W, L, h, k_s, g)$ with $\\nu $ the kinematic viscosity, $\\rho $ the fluid density, $\\sigma $ the surface tension, $u_D$ the bulk velocity, $\\delta $ and $H=\\delta ^*/\\theta $ respectively the boundary layer thickness and shape factor at the obstacle face location before introducing it (with $\\delta ^*$ the boundary layer displacement thickness and $\\theta $ the boundary layer momentum thickness), $W$ the obstacle width (along $z$ ), $L$ the obstacle length (along $x$ ), $h$ the water level at the obstacle location before introducing it, $k_s$ the bed and obstacle roughness, and $g$ the gravity acceleration (see figures REF and REF ).",
"Those 11 parameters involve 3 scales.",
"Using $h$ as length scale, $h/u_D$ as time scale and $\\rho h^3$ as mass scale, Vaschy-Buckingham $\\Pi $ -Theorem then allows to reduce the dependency to the following 8 dimensionless parameters: $X^* = f\\left(~_h=\\frac{4 u_D h}{\\nu },~\\frac{W}{h},~\\frac{h}{\\delta },~Fr=\\frac{u_D}{\\sqrt{gh}}, \\frac{W}{L},~H,~We=\\frac{\\rho u_D^2 h}{\\sigma },~\\frac{k_s}{W} \\right)$ with $X^*$ any flow property made dimensionless using the appropriate scale, $_h$ the Reynolds number based on the hydraulic diameter $D_h=4bh/(b+2h) \\approx 4h$ herein, so that $_h \\approx 4Q/b$ with $Q$ the total discharge and $b$ the channel width, $Fr$ the Froude number and $We$ the Weber number.",
"Previous works suggested that the obstacle length $L$ has no impact on the HSV when studied in the vertical upstream plane of symmetry [11], [8].",
"However, preliminary velocity measurements with increasing obstacle lengths showed that the boundary layer separation average position and its transverse oscillation (along $z$ ) are affected by this parameter.",
"In order to avoid the effect of the wake and to neglect the influence of the aspect ratio parameter $W/L$ , all obstacles considered in the sequel will be chosen sufficiently long with respect to the obstacle width ($W/L < 0.3$ ).",
"The Froude number remains small enough throughout the present work ($Fr < 0.3$ ) to neglect its influence.",
"Preliminary water level measurements showed that the flow remains in the hydraulic smooth regime for the studied flow parameter domain, allowing to neglect the influence of the roughness parameter $k_s/W$ .",
"The shape factor $H$ remains fairly constant around the value of $2.68$ in this study.",
"The measured free-surface deformations are small ($\\Delta h/h < 0.13$ ) throughout the experiment, and so, surface tension effects can be safely disregarded.",
"Therefore, in the present work, any HSV dimensionless characteristic $X^*$ should depend only on 3 dimensionless parameters as $X^* = f\\left(_h,~\\frac{h}{\\delta },~\\frac{W}{h}\\right)$"
],
[
"Experimental set-up",
"The water table schematized in figure REF is used to generate a horseshoe vortex at the foot of long, emerging, rectangular obstacles with varying dimensionless parameters $_h$ , $h/\\delta $ and $W/h$ .",
"The water tank is fed by a pumping loop which includes a valve for discharge control, an electromagnetic flowmeter (Promag W, of Endress+Hauser, uncertainty of $0.01$ L/s, i.e a precision of $0.5\\%$ to $4\\%$ depending on the discharge value), a homogenization tank composed of several grids and honeycombs (1) and a vertical convergent (2) to compress the boundary layer.",
"The water then flows on a horizontal smooth plate (width $b$ =0.97m and length 1.3m) made of glass to allow optical access from the side and bottom walls (3).",
"Water level can be controlled by an adjustable weir (4) and is measured by a mechanical water-depth probe with digital display (uncertainty of $0.15$ mm, according to [46]).",
"The obstacle (5) with adjustable width $W$ is placed at a distance of $0.6$ m downstream the convergent end.",
"Figure: Experimental set-up consists of a feeding loop and an adjustable weir to control the flow parameters (u D u_D and hh) and a transparent water tank to visualize and measure, in vertical and horizontal planes, HSV around long emerging obstacle (width WW and length LL)."
],
[
"Measurement techniques",
"HSV measurements are obtained using either particle image velocimetry (PIV) or trajectographies in the vertical plane of symmetry ($z=0$ ) and, for section REF , in a horizontal plane near the bed at the elevation $y/h=0.01$ (marker 6 on figure REF ).",
"For both techniques, a 532nm, 4W continuous laser with a Powell lens is used to illuminate 10$$ m hollow glass spheres included as tracers in the flow.",
"The displacement of these particles is recorded with a mono-chromatic, 12bit, $2048 \\times 1088$ pixels camera.",
"For trajectographies, frames with adapted time-exposure are taken at frequencies ranging from 1 to 25Hz.",
"For PIV measurements, double frames are taken at frequencies from 1 to 2Hz, depending on the typical flow velocities and the frames spatial resolution.",
"Image processing and PIV computations are realized under DaVis software (Lavision) and further velocity field analyzes are performed using Python.",
"Image processing includes ortho-rectification, background removal, intensity capping [53] and/or moving average.",
"PIV computations are realized using cross-correlations with $50\\%$ overlapping and adaptive interrogation windows size decreasing from 64 to 16 pixels, leading to a spatial resolution of approximately $0.01h$ .",
"Measurement quality is ensured by following recommendations from [1] and others: (i) a thin laser sheet (1mm thick), ensuring that particles displacements in the direction normal to the measurement plane do not influence the measured velocity, (ii) a small Stokes number for the seeding particles (maximum $St=0.012$ , leading to less than $1\\%$ error according to [57]), (iii) a low sedimentation ratio (ratio between typical velocity and sedimentation velocity) of $0.004$ ensuring that sedimentation velocity of the particles is negligible in regard to the advection velocity, (iv) ortho-rectification of the obtained frames to avoid velocity errors due to optical deformation, (v) suitable particle concentration (at least 10 particles per interrogation windows), (vi) sufficiently large particle displacement between two frames (at least 10 pixels), leading to a roughly approximated uncertainty on the velocity of $1\\%$ , (vii) filtering of the obtained vectors in regard to the cross-correlation peak-ratio (with a minimum of $1.5$ ) to remove possibly wrong vectors (generally around $5\\%$ of the vectors in the present case).",
"The measurement protocol for each case is the following: (i) values for the dimensionless parameters and associated dimensional parameters are selected and the experimental set-up is tuned accordingly, without obstacle, setting the desired bulk velocity $u_D$ along with the water level $h$ at the $x$ future position of the obstacle upstream face.",
"(ii) The boundary layer profile at the future position of the obstacle face is measured using PIV to access the boundary layer properties.",
"(iii) An obstacle of given width $W$ is placed so that its upstream face is located at $x=60$ cm from the convergent end, and its lateral faces are parallel to $x$ axis (figure REF ).",
"This creates the adverse pressure gradient responsible for the boundary layer separation and the HSV appearance.",
"(iv) PIV measurements or trajectographies are performed, once the flow is established, in the vertical plane ($x, y$ ) of symmetry upstream from the obstacle (or in a horizontal plane $x, z$ upstream from the obstacle for section REF )."
],
[
"Characteristics of the flow without the obstacle",
"The state of the laminar boundary layer as it interacts with the obstacle is a key parameter for the formation and the evolution of the HSV.",
"It is then essential to ensure that the boundary layer is not polluted by experimental set-up biases.",
"Without obstacle, the boundary layer freely develops from the end of the convergent, where the velocity profile is uniform along a vertical profile.",
"The vertical velocity profile at a given $x$ value can be fully characterized by: (i) the boundary layer thickness $\\delta $ , defined as the vertical position where the velocity reaches $99\\%$ of the maximum velocity, and (ii) the shape factor $H$ .",
"The boundary layer thickness at the future position of the obstacle face depends on the bulk velocity $u_D$ , on the distance from the end of the convergent, but also, on the vertical confinement imposed by the free-surface, i.e.",
"on the water depth $h$ .",
"The vertical profile of streamwise velocity in the boundary layer is expected to fit the laminar Blasius solution for high $h/\\delta $ (unconfined situation) and the half-parabolic Poiseuille profile for low $h/\\delta $ (highly confined situation), by analogy with closed channel flows.",
"To confirm this statement, figure REF shows the measured boundary layer thicknesses and shape factors for all boundary layers used in the sequel, compared to the corresponding values expected for Blasius ($\\delta _B$ and $H_B = 2.59$ ) and Poiseuille ($\\delta _P$ and $H_P = 2.5$ ) profiles.",
"This figure confirms that the boundary layers match with Poiseuille profiles for highly confined flows and approach Blasius-like profiles as the confinement decreases.",
"It should be noted that the shape factor $H$ remains fairly constant around an average value of $2.68$ , ensuring that the boundary layer does not undergo a turbulent transition in the present cases.",
"This is also confirmed by the measured turbulent intensity (not shown here), remaining lower than $<3\\%$ .",
"These results show that the experimental apparatus is able to produce laminar boundary layers (despite the high Reynolds number of up to $_h=8000$ ) that are affected by the vertical free-surface confinement for $h/\\delta _B < 2$ .",
"Figure: Characteristics of the boundary layer before introducing the obstacle.",
"Left axis: measured boundary layer thickness δ\\delta deviation from the Blasius solution δ B \\delta _B (blue filled circles) and analytic solution for a parabolic profile (dash line).",
"Right axis: evolution of the measured shape factor HH (red open square symbol), with dotted and dashed lines representing respectively the Blasius shape factor (H B =2.59H_B = 2.59) and the parabolic shape factor (H P =2.5H_P = 2.5).",
"uncertainties on (δ B -δ)/δ B (\\delta _B - \\delta )/\\delta _B and HH are calculated with an estimated uncertainty of 0.03h0.03h on the measured values of δ\\delta .As expected, the boundary layer fits a laminar Blasius profile and tends to a Poiseuille profile for important free-surface confinements."
],
[
"Experiment plan",
"In order to have a good insight on how the three dimensionless flow parameters affect the HSV structure and dynamics, the dimensionless parameter space ($_h$ , $h/\\delta $ , $W/h$ ) is mapped such as presented in figure REF .",
"$_h$ and $W/h$ are well-controlled (respectively by the discharge $Q$ and the obstacle width $W$ ) and allow homogeneous mapping, avoiding inter-dependencies.",
"Tuning $h/\\delta $ , however, is more difficult, as the boundary layer thickness $\\delta $ depends on the bulk velocity $u_D$ the water level $h$ and the distance between the convergent and the obstacle, which has a very limited variation range in the present experimental set-up.",
"Measurements duration always exceeds at least 20 periods (in case of periodic HSV behavior) for the 75 cases of the parametric study, and at least 200 periods for the 13 cases of the detailed transition study (square symbol in figure REF ).",
"Time resolutions of the measurements ensure at least 15 velocity fields per period.",
"Figure: Experiment plan used to study the HSV geometrical and dynamical properties evolution.Blue open symbols represent cases investigated by trajectography and red filled symbols cases investigated by PIV.",
"For each circle symbol (hollow or filled), 5 different values of W/hW/h are considered.For the red square symbol, 13 values of W/hW/h have been measured in order to study the HSV regime transitions in details.The left zone is inaccessible to measurement, as the experimental set-up weir creates a recirculation impacting the HSV."
],
[
"Horseshoe vortex properties",
"This section presents the methods used to measure the HSV geometrical and dynamical properties from the acquired PIV velocity fields and trajectographies."
],
[
"Horseshoe properties measurement",
"One first main characteristic of interest is the vortex center positions (see figure REF b).",
"The vortices sharing the boundary layer vorticity sign are designated as $V_i$ , with $i$ the number of the vortex, starting with the downstream-most one, and ($x_i,y_i$ ) their positions.",
"In the sequel, $V_1$ and $V_2$ will also be referred as “main vortex” and “secondary vortex”.",
"The counter-rotating vortices, located near the bed and just upstream from the previously defined vortices are designated as $V_{ci}$ with $i$ the number of the associated vortex $V_i$ .",
"A second characteristic of the HSV is the location of the boundary layer separation, defined as the upstream-most $x$ position where the shear-stress along $x$ equals zero.",
"Its distance to the obstacle is noted $\\lambda $ (see figure REF b).",
"As proposed by [8] in order to enhance the comparison between obstacles of different shapes, the reference points for streamwise distances ($\\lambda $ and $x_i$ ) is not the obstacle face but rather a point located at $W/2$ behind it (see figure REF b).",
"Uncertainties regarding those distances are estimated as about $0.05h$ for trajectographies and $0.02h$ for PIV measurements (as the measurements spatial resolution depends on $h$ ).",
"The vortex circulation, which is also an interesting property, is quite challenging to define in the vicinity of a strong shear layer (the boundary layer in this case).",
"The classical method to estimate the vortex radius, consisting of defining a vorticity threshold, can be rendered inoperable by the strong boundary layer vorticity due to the shear.",
"The residual vorticity [34], known to be the vorticity associated to rotation, is in place used herein to get the vortex region."
],
[
"HSV dynamics characterization",
"In order to have quantitative data on the HSV dynamics and to establish a clear typology, vortex positions need to be followed in time.",
"To do so, critical points of the velocity field in the vertical plane of symmetry are detected and tracked in time (for PIV measurements), summarizing efficiently the HSV structure evolution.",
"Critical points are Lagrangian dependant and, as such, are unable to detect vortices advected at high velocities.",
"In those cases, gradient-based criteria such as the $\\lambda _2$ -criterion [27] or the residual vorticity [34] are far more efficient in detecting and tracking vortices.",
"However, as the present vortex advection velocities (velocities of their centers) remain small compared to the velocities they induce, and regarding the valuable additional topological information brought by the critical points, they are used in the sequel to characterize the HSV dynamics.",
"The method of detection and tracking, inspired from [24], [12] and [19], consists of six steps: (i) Pre-filtering of the time-resolved velocity fields, using POD reconstruction on a truncated modal base.",
"POD modes are not filtered using the classical energy criterion [45], but according to the dispersion of their spatial spectra, which is representative of the presence of large-scale structures.",
"This step aims at reducing the measurements noise, removing the small-scale fluctuations to promote the large-scale structures and replacing the missing velocity vectors by spatially and timely interpolated ones.",
"(ii) Detection, on each instantaneous velocity field, of the measured vector grid cells susceptible of containing a critical point using a scan of the Poincarré-Bendixson index [26].",
"(iii) Detection of the sub-grid position of the critical points, using [19] method.",
"(iv) Determination of the critical points type (saddles points, stable or unstable nodes, stable rotating or counter-rotating vortex centers) based on the local Jacobian matrix eigenvalues.",
"(v) Optional topological simplification using [24] $\\Gamma $ criterion.",
"This step allows to select only large-scale bounded critical points.",
"(vi) Trajectory reconstruction using distance sum minimization.",
"For a more synthetic visualization of the HSV vortices motion for a particular configuration, the vortex centers trajectories are averaged by group according to their similarities: (i) For each couple of trajectories, the normalized integral of the squared difference is computed: $\\epsilon _{mn} = \\frac{1}{2} \\frac{\\int \\left[ x_m(t) - x_n(t)\\right]^2 dt}{\\frac{1}{2} \\int \\left[ x_m(t) + x_n(t)\\right]^2 dt}+ \\frac{1}{2} \\frac{\\int \\left[ y_m(t) - y_n(t)\\right]^2 dt}{\\frac{1}{2} \\int \\left[ y_m(t) + y_n(t)\\right]^2 dt}$ where $(x_m(t), y_m(t))$ is the position of the trajectory $m$ at time $t$ and $\\epsilon $ is representative of the difference between the two trajectories (small for close trajectories, large for different trajectories).",
"(ii) The maximum difference between two trajectories considered similar is arbitrarily defined (here to $\\epsilon _{crit} = 0.07$ , based on the obtained results).",
"(iii) Trajectories are distributed in groups, ensuring that no group includes couples of trajectories with $\\epsilon > \\epsilon _{crit}$ .",
"(iv) Groups of similar trajectories are averaged to obtain mean trajectories.",
"One can finally associate instantaneous velocity fields to each of those mean trajectories and so perform a conditional averaging of the velocity fields on the vortex center position."
],
[
"Horizontal view of the HSV",
"Main HSV geometrical and dynamical properties can be observed from measurements in the vertical plane of symmetry.",
"Nonetheless, the HSV does not remain necessarily symmetric with respect to this plane at all time, and [18] pointed out that the vortex filament maximal radii and intensities can be located outside of the symmetry plane.",
"In this context, measurements in a horizontal plane near the bed are mandatory to ensure that the HSV driven phenomena can be deducted from 2D PIV measurements in the vertical plane of symmetry.",
"Figure REF presents a velocity field in the horizontal plane near the bed ($y/h = 0.01$ ).",
"The evolution of the main and secondary vortex ($V_1$ and $V_2$ ) filaments location while bypassing the obstacle can be evaluated on this figure with the help of the critical points and their associated lines.",
"Vortices do not undergo strong modifications in size and velocity while wrapping around the obstacle, until they interact with the complex lateral separation bubble on the sides of the obstacle.",
"Time-resolved measurements show that the position of the horizontal separation line (see figure REF ) remains constant with time (not shown here), ensuring that shedding from the lateral recirculation bubbles are not strong enough to disrupt the HSV and the boundary layer separation.",
"Those measurements also reveal that the dynamics of the HSV vortices does not evolve significantly along the filaments (while rolling around the obstacle).",
"Conclusion can be made, on this configuration, that: (i) the instantaneous HSV symmetry plane coincides at each time with the obstacle symmetry plane, (ii) the vortex filaments remain approximately of the same size and keep the same dynamics while wrapping around the obstacle (contrary to the observations of [17]), and consequently, (iii) a measurement on the vertical symmetry plane of the HSV ($z=0$ ) is an adapted and sufficient approach to characterize the HSV behavior.",
"The same conclusions apply for all flows investigated with horizontal measurements, allowing the present work to be based solely on the study of measurements in the vertical plane of symmetry.",
"Figure: Instantaneous velocity field in the horizontal plane near the bed (y=0.01hy=0.01h) for a laminar flow around an obstacle for h =4272,h/δ=2.70,W/h=1.67_h=4272,~h/\\delta =2.70,~W/h=1.67.Streamlines are coloured with velocity magnitude.Circles represent detected critical points: saddle points (yellow), unstable nodes (grey) and vortex centers (blue).Plain thick lines are streamlines coming from and going to saddle points and delimit the vortices within the HSV.",
"V 1 V_1 and V 2 V_2 are the main and secondary vortices.The topology of the flow and the velocity distribution show that the HSV vortices properties do not evolve drastically when rolling around the obstacle in the upstream part of the HSV (before x=0x=0)."
],
[
"Comparison between various obstacle shape and submergence",
"Most previous works concerning the HSV, in both immersed and emerged configuration, considered cylindrical obstacles, making the comparison with the present results with rectangular obstacles challenging.",
"[7] proposed, in order to solve this problem, to estimate that HSV from obstacles of different shapes are comparable if they have the same separation distance (method that requires to known the separation distance) while [8] considered comparable HSV from square obstacles of width $W$ and HSV from cylindrical obstacles of diameter $W$ .",
"A more systematic method based on the estimated adverse pressure gradient is proposed herein.",
"Because the whole HSV and notably the boundary layer separation is governed by the streamwise adverse pressure gradient, one can assume that two obstacles of different shapes creating the same adverse pressure gradient in the upstream symmetry plane should generate similar HSV.",
"Based on this assumption, and using pressure profiles from potential flow computation, an equivalent obstacle width $W_{eq}$ can be computed for cylindrical obstacles of diameter $D$ , so that a quadrilateral obstacle of width $W_{eq}$ and infinite length and a cylindrical one with equivalent width $W_{eq}$ lead to nearly the same upstream pressure gradient and consequently, comparable HSV.",
"This method can further be applied to obstacles with other shapes (such as quadrilateral obstacles with non-infinite length, foil, bevelled quadrilateral, oblong obstacles, $\\ldots $ ).",
"As the separation distance is linked to the adverse pressure gradient, this method is very similar to the one of [7], but does not necessitate prior knowledge of the separation distance $\\lambda $ .",
"Figure REF shows that pressure coefficients $C_p$ (from 2D potential flow computation) upstream of cylindrical, squared-shaped and infinitely long rectangular obstacles can effectively be aggregated by adjusting their size (diameter $D$ or width $W$ ).",
"The optimal size coefficients, in a least squares sense, have been found to be: $W_{eq} = W$ $W_{eq} = \\frac{W_s}{1.093} = 0.915 W_s$ $W_{eq} = \\frac{D}{1.29} = 0.775 D$ with $W$ the width of an infinitely long rectangular obstacle and $W_s$ the width of a square-shaped obstacle.",
"Moreover, the fair agreement between the potential flow computation and measurements from [3] shows that potential flow is effectively able to predict the pressure distribution.",
"This equivalence will be used in the sequel to modify empirical correlations from the literature so that they use $W_{eq}$ instead of $D$ and $W_s$ (counting respectively $1.29W_{eq}$ and $1.093W_{eq}$ ).",
"Figure: Pressure coefficients C p C_p obtained using 2D (in the horizontal plane) potential flow theory computations upstream of different obstacles:(1) quadrilateral obstacle of width W eq W_{eq} and infinite length,(2) squared-shaped obstacles of widths W s =W eq W_s=W_{eq},(3) squared-shaped obstacles of width W s =1.093W eq W_s=1.093W_{eq},(4) cylindrical obstacles of diameters D=W eq D=W_{eq}, and(5) cylindrical obstacles of diameters D=1.29W eq D=1.29W_{eq}.",
"(6) measurements from at the bed in front of a cylindrical immersed obstacle.These results show that the pressure gradients for different obstacles can be aggregated by modifying their characteristic sizes.To compare immersed and emerging obstacle data, the obstacle height $\\xi $ for immersed obstacles will be said analogous to the water depth $h$ for emerging obstacles, as proposed by [8], thus keeping the same wet portion of obstacle shape ratio ($\\xi /W$ for immersed obstacles and $h/W$ for emerging ones)."
],
[
"Dynamic typology",
"Defining a typology categorizing the different observed behaviors is a convenient way of studying the HSV dynamics.",
"Authors such as [3] establish typologies using the number of vortices in the HSV.",
"Nevertheless, as in [25] and [38], the present typology is rather based on the vortex dynamics.",
"The number of vortices will be discussed further on, in the section dedicated to the HSV geometrical properties.",
"For a given flow configuration, the HSV shows either large-scale, coherent and well-defined vortices (like in figure REF ), and will be said in “coherent regime”, or a non-stationary and aperiodical behavior, with presence of small-scale non-coherent structures and will be said in “irregular regime”.",
"In coherent regime, the HSV vortices can either be steady or follow a “pseudo-periodic” motion, showing an alternation of elementary processes of same duration $T$ (HSV period) called “phases” in the sequel.",
"For one flow configuration, successive phases can be substantially different but always bring back the HSV to its initial state (and consequently, all phases share the same initial state).",
"The different observed phases are described in section REF and are the base of the definition of coherent “sub-regimes”, defined in section REF , and presented in details in section REF .",
"The evolution from coherent to irregular regimes is presented in section REF .",
"Finally, the evolution of all those regimes (coherent regimes and irregular regime) with the flow parameters is detailed in section REF ."
],
[
"Phases (elementary processes) definition",
"All observed phases can be classified in four categories: (i) The “oscillating phase”, where the displacement of each vortex is a horizontal ellipse that brings it back near its initial position at the end of the phase.",
"(ii) The “merging phase”, where the main vortex ($V_1$ ) follows the same pattern as in the oscillating phase, but merges with the secondary vortex ($V_2$ ) at the end of the phase.",
"The vortex merging is defined, from a “critical points” view, as a bifurcation from a saddle point and two vortex centers to a single vortex center.",
"As it disappears in the merging, $V_2$ is replaced by the third vortex $V_3$ while a new vortex is created near the boundary layer separation point.",
"(iii) The “diffusing phase”, analogue to the merging phase, but for which the main vortex circulation and size decrease along the phase, reaching the merging position with low circulation compared to the secondary vortex.",
"A phase will be considered to be “diffusing” if the main vortex circulation is less than $25\\%$ of the secondary vortex circulation (this ratio can reach $0\\%$ if the vortex fully diffuses before reaching the merging).",
"(iv) The “breaking phase”, where the main vortex breaks (escapes) from the HSV, is advected further downstream and finally diffuses near the obstacle.",
"The main vortex is considered to break from the HSV if the distance between $V_1$ and $V_2$ exceeds two times the diameter of the bigger vortex.",
"From this time on, the breaking vortex is considered outside of the HSV, $V_2$ replaces $V_1$ and a new vortex arises near the boundary layer separation point."
],
[
"Regimes definition",
"The definition of the four phases, observed in the experiments, allow to classify the flow configurations in seven coherent regimes, represented on figure REF , plus two transitional regimes: (i) The “no-vortex regime”, where no vortex appears on the shear layer.",
"This regime was not observed in the present study, but was reported by [50] for immersed obstacles, and is expected to exist in the present emerging obstacle configuration for $_h$ lower than those considered herein.",
"(ii) The “stable regime”, where the location and the circulation of the vortices remain constant with time.",
"In practice, a HSV is considered in stable regime if the mean amplitude of the vortices displacements does not exceed $0.03W$ .",
"(iii) The “oscillating regime”, (iv) the “merging regime” and (v) the “diffusing regime”, composed by a succession of similar associated phases (in practice at least $95\\%$ of all phases).",
"(vi) The “oscillating-merging transitional regime”, composed by non-regular alternations of oscillating and merging phases.",
"(vii) The “merging-diffusing transitional regime”, composed by non-regular alternation of merging and diffusing phases.",
"(viii) The “breaking regime”, composed by a succession of breaking phases.",
"This regime was not observed in this study, but was reported by [55] and [25] for the immersed obstacle configuration.",
"Where this regime should be placed on the 2D typology (figure REF ) remains unclear.",
"(ix) The “complex regime”, composed by an apparently chaotic succession of oscillating, merging, diffusing and/or breaking phases.",
"This regime is characterized by an important phase dispersion, i.e.",
"successive phases notably differ from each others even if they have the same phase type, contrary to the previously defined transitional regimes.",
"These regimes will be detailed in the next section.",
"The typology defined here is partially similar to the one presented by [25] for the immersed obstacle configuration: the present stable, oscillating, merging, breaking and irregular regimes can be, respectively, assimilated to Greco's “steady vortex system”, “oscillating vortex system”, “amalgamating vortex system”, “breakaway vortex system” and “transitional vortex system”.",
"However, as the typology of [25] is based on flow visualization and does not precisely define the regimes boundaries, this terminology is not used herein.",
"Complex regime was not reported (to the authors knowledge) in the immersed nor emerging obstacle literature.",
"Figure: HSV regimes organization for a laminar flow around a long rectangular obstacle.Each circle represents an observed dynamics, categorized along the coherent evolution (vertically) and the irregular evolution (horizontally).White arrows represent the coherent transitions and black arrows the irregular transitions.Dotted circles and arrows correspond to regimes and transitions not observed herein, but expected to exist."
],
[
"Stable regime",
"In stable regime, each vortex remains at the same spatial location at all time.",
"Figure REF a shows an example of a HSV in stable regime where critical points detection indicates the presence of two vortices, confirmed by the streamlines pattern.",
"The steadiness of the flow dynamics is visible through the small size of the critical points presence zones.",
"The classical stable HSV topology, as discussed for instance in [59], is well represented, with: (i) a boundary layer separation at $x/W=-1.9$ , (ii) a separation surface evolving from $x/W=-1.9$ to $x/W=-1.2$ , (iii) a succession of clockwise-rotating vortices (two in the present case), separated by saddle points, (iv) counter-clockwise rotating vortices between the clockwise rotating ones (one in the present case) and (v) the down-flow, visible through the streamlines curvature, that makes the flow re-attach near the obstacle foot at $x/W \\approx -0.1$ .",
"The three-dimensionality of the flow is clearly visible as the upper flow slows down while approaching the obstacle and as the HSV vortices streamlines are spiralling toward the vortex centers.",
"Figure: Mean velocity field (magnitude-colored streamlines) in the vertical plane of symmetry for (a): a stable flow case ( h =4271,h/δ=2.70,W/h=0.79_h=4271,~h/\\delta =2.70,~W/h=0.79) and (b): an oscillating flow case ( h =4271,h/δ=2.70,W/h=1.23,T=21.01s_h=4271,~h/\\delta =2.70,~W/h=1.23,~T=21.01s).Zones where 99% of the critical points are found are calculated using kernel density estimator and are shown as filled contour (blue for vortex centers, red for counter-rotating vortex centers and yellow for saddle points).Green lines are the separation surface section on the symmetry plane.",
"(c) Time evolution of the critical points position along x/Wx/W for the oscillating regime (b).Plain (blue) lines stand for vortices, dashed (red) lines for counter-rotating vortices and dotted (yellow) lines for saddles points.Critical points displacement exhibit a phase shift from the main vortex toward upstream at a celerity of 0.29u D 0.29 u_D.Aspect ratio is conserved on velocity fields despite the use of dimensionless axes."
],
[
"Oscillating regime",
"The oscillating regime is illustrated for one flow configuration in figures REF b-c.",
"The topology of the mean flow remains the same as for the stable regime (with an additional vortex in the present case).",
"The vortex centers presence zones are elongated but do not collide with each other: vortices are sustainable in time.",
"It is to be noted that vortex centers remain fairly on the shear layer originating from the separation point and ending at the main vortex $V_1$ position, and separating the upper flow (going towards downstream), and the backflow (going back upstream).",
"Saddle points presence zones present the same behavior as vortex centers, showing that the oscillating motion is shared by the whole HSV structure.",
"Figure REF c shows the evolution of each critical point streamwise location over the time during 4 consecutive periods.",
"The periodic, quasi-sinusoidal streamwise displacement behavior is shared by all critical points of the HSV.",
"A phase-shift is nevertheless present between those oscillations, and reveals that the oscillating dynamics source of the HSV is the main vortex $V_1$ , while the other vortices follow its motion."
],
[
"Merging regime",
"Figure REF illustrates the evolution of the so-called merging regime during 3 periods.",
"The global topology, already mentioned for the stable (figure REF a) and oscillating regimes (figure REF b) is also valid for the merging regime.",
"The main difference is that new vortices appear periodically at the end of the separation surface to replace those disappearing by merging.",
"Consequently, vortices have a life cycle: the highlighted vortex in figure REF appears at $x/W=-1.15$ (and $y/\\delta =0.22$ ) at $t/T=0$ , is advected toward downstream while gaining in radius and circulation until $t/T=2.6$ , then slows down as its radius decreases and ends up going back upstream and merging with the previous vortex ($V_2$ ) at $t/T=3.2$ .",
"Figure: Successive instantaneous velocity fields in the vertical plane of symmetry for a merging regime ( h =6397,h/δ=3.69,W/h=1.67,T=20.02s_h=6397,~h/\\delta =3.69,~W/h=1.67,~T=20.02s).Coloured symbols are detected critical points (same colours scheme as figure ).Plain line is the highlighted vortex trajectory and dashed lines help follow the vortex of interest through presented instantaneous fields.Aspect ratio is conserved despite the use of dimensionless axes."
],
[
"Diffusing regime",
"The diffusing regime is quite similar to the merging regime, but the main vortex radius and circulation drop before reaching the merging.",
"The end of the life cycle of a vortex in this flow regime is shown on figure REF , from the moment when the main vortex $V_1$ starts to lose its circulation ($t/T=3.0$ ) to its disappearance ($t/T=3.9$ ).",
"At the merging, the main vortex has a very low circulation and is absorbed by the secondary one without changing $V_2$ properties nor trajectory.",
"The evolution of the circulation and radius of the vortices is similar for all diffusing phases (see circulation evolution on figure REF ): the vortices increase in size in the upstream part of the shear layer and decrease in size in the downstream part.",
"This can be explained as vortices size and circulation evolution is the result of the balance between: (i) The vertical gradient of streamwise velocity due to the boundary layer separation, strong in the upstream part of the HSV, feeding the vortices.",
"(ii) The opposite-sign vorticity generated at the wall that rolls around the vortices and decreases their circulation by vorticity diffusion [51].",
"It is to be noted that the main vortex disappearance (by diffusion) and the vortex creation at the end of the separation surface are not necessarily simultaneous, resulting in a varying instantaneous number of vortices (for instance 3 at $t/T=3.0$ or 4 at $t/T=3.5$ for instance in figure REF ).",
"Figure: Successive instantaneous fields in the vertical plane of symmetry for a diffusing regime ( h =4250,h/δ=2.7,W/h=2.36,T=16.5s_h=4250 ,~h/\\delta =2.7,~W/h=2.36,~T=16.5s).t/T=0t/T=0 corresponds to the birth of the vortex of interest at the end of the separation surface.See legend from figure for the velocity fields.Right plot shows the evolution of the vortex of interest circulation with time."
],
[
"Note on the vortex merging",
"Vortex merging appears in the merging and diffusing phases and is thus an important mechanism for the HSV dynamics.",
"Co-rotating vortex merging in uniform flows was described by [16], [56], [41], [29].",
"Those studies showed that, for 2 identical vortices, vortex merging occurs when the ratio of the vortices radius over their separation distance exceeds a threshold value.",
"For non-identical vortices, a general merging criterion is not established, as the merging has been shown to be governed by different processes (some of them leading to the destruction of the smaller/weaker vortex without increasing the larger vortex circulation, see [16]).",
"The situation is more complicated in the present study, as the vortices are not two-dimensional, and are surrounded by a complex flow.",
"The vortex-vortex interaction match an elastic-like behavior: two vortices start to interact as the distance separating them $\\Delta x$ roughly equals the sum of their radii $\\Delta x \\approx R_1 + R_2$ .",
"When getting closer, they repel each other, as can be seen in the oscillating phases when $V_1$ travels upstream and pushes $V_2$ in figure REF c. If they succeed in getting even closer, the two vortices merge, resulting in a briefly (regarding the time-scale of the HSV oscillations) disrupted vortex.",
"However, for vortices with very different sizes, as in the diffusing phases (see figure REF ), the weaker vortex is simply absorbed by the larger one, without any elastic-like behavior.",
"The appearance of this elastic-like behavior can be attributed to the restriction of the vortex position along $y$ (due to the presence of the bed and the boundary layer) that prevents the vortices from rolling around each other, which is the typical behavior for vortices in uniform flows (see [16] for instance)."
],
[
"Breaking phase",
"As no breaking regimes have been observed in the present work, the breaking phase presented in figure REF is part of a complex regime configuration, explaining why the flow pattern differs between $t/T=2.2$ and $t/T=3.1$ .",
"This phase differs from a merging phase by the fact that the main vortex escapes from the HSV, travels toward downstream and diffuses near the obstacle.",
"Figure REF shows the end of a vortex life cycle in a breaking phase.",
"The highlighted vortex (main vortex at $t/T>2.2$ ) breaks from the HSV at $t/T \\approx 2.8$ and is advected downstream at high velocity while loosing in radius and circulation until disappearing at $t/T=3.7$ .",
"Unlike observed by [15], in this case no velocity eruptions is at the origin of the breaking.",
"Figure: Successive instantaneous velocity fields in the vertical plane of symmetry for a breaking phase in a complex regime ( h =6397,h/δ=3.69,W/h=1.67,T=17.22s_h=6397,~h/\\delta =3.69,~W/h=1.67,~T=17.22s).t/T=0t/T=0 (not shown here) corresponds to the birth of the vortex of interest at the end of the separation surface at x/W=-0.9x/W=-0.9.See legend from figure for the velocity fields."
],
[
"Oscillating - merging regime transition",
"While the transition from the stable to the oscillating regime is simply a continuous increase in vortex motion amplitude, the transition from the oscillating regime to the merging regime is more complex.",
"This transition is investigated more in details on figure REF for different cases with increasing $W/h$ but constant $_h$ and $h/\\delta $ values (corresponding to the square symbol on figure REF ).",
"For $W/h < 0.9$ , the main and secondary vortices are far from each other and remain steady, the HSV being in stable regime.",
"As $W/h$ increases (from $0.9$ to $1.2$ ), the main vortex starts to oscillate and the average distance between the main and secondary vortices decreases.",
"After a critical value of $W/h = 1.2$ , $V_1$ and $V_2$ begin to merge for some periods i.e.",
"merging phases appear.",
"Occurrences of merging phases increase with $W/h$ until reaching the merging regime ($>95\\%$ merging phases) at $W/h=1.6$ .",
"The transition from oscillating to merging regimes (from $W/h=1.2$ to $W/h=1.6$ ) is asymmetrical: The number of oscillation periods decreases rapidly for $W/h \\approx 1.2$ and more slowly for $W/h \\approx 1.6$ (see fitting in figure REF a).",
"Oscillations amplitudes can only be measured for oscillating phases, which explain the saturation of the vortex spatial amplitude (for $W/h = 1.2$ to $1.5$ on figure REF b): this saturation value $\\delta x/W = 0.16$ can be understood as the maximum possible oscillation amplitude before appearance of merging between $V_1$ and $V_2$ (Note that the critical value $\\delta x/W=0.16$ is expected to differ regarding $h/\\delta $ and $_h$ values).",
"A general parameter governing the vortex merging occurrences is, in this situation, quite challenging to define.",
"It is to be noted that the Strouhal number, presented in figure REF c, is directly proportional to the oscillation frequency $f$ ($\\delta $ and $u_D$ being kept constant as $W/h$ increase).",
"Consequently, figure REF c only indicates that the obstacle aspect ratio $W/h$ and the HSV regime have no influence on the HSV oscillation frequency.",
"Figure: Evolution of some HSV characteristics for h =4272,h/δ=2.7_h=4272,~h/\\delta =2.7 and increasing W/hW/h values, corresponding to the square symbol on figure .",
"(a) Percentage of merging or diffusing phase.",
"(b) blue circles: mean distance between the main and the second vortices Δx avg \\Delta x_{avg}, red triangles: mean spatial oscillation amplitude of the main vortex δx 1 \\delta x_1, yellow unversed triangles: mean spatial oscillation amplitude of the secondary vortex δx 2 \\delta x_2.Those values are only defined, and consequently computed, for stable regimes and oscillating phases.",
"(c) Strouhal number based on the boundary layer thickness δ\\delta .These plots illustrate well the continuous transition from the oscillating to the merging regimes."
],
[
"Merging to diffusing regimes transition",
"Contrary to the stable/oscillating/merging transition that is continuous, the merging to diffusing transition is more complex.",
"Figure REF shows the evolution of mean trajectories with increasing values of $W/h$ , while passing from a merging regime to a diffusing one.",
"The mean trajectory shown in figure REF a is characteristic of a merging regime (see figure REF ): the vortex appears at the end of the separation surface ($x/W \\approx -1$ ), is advected downstream while gaining in size and ends up going back upstream and merging with the secondary vortex.",
"This last part of the trajectory is not well reconstructed, because of the main vortex high velocity.",
"Oppositely, the trajectory shown in figure REF c is characteristic of a diffusing regime (see figure REF ): the vortex is advected downstream, but diffuses at $x/W \\approx -0.2$ instead of going back upstream and merging with the previous vortex.",
"For the transitional case in figure REF b, the majority ($77\\%$ , red trajectory) of the vortices appearing at the end of the separation surface merge with the previous vortices at $x/W \\approx -0.6$ (merging phases).",
"However, after this merging, these vortices are advected further downstream and diffuse at $x/W = 0.25$ (diffusing phases).",
"Therefore, the HSV dynamics is ($77\\%$ of the time) a regular alternation between merging and diffusing phases.",
"$10\\%$ of the vortex (blue trajectory) are simply diffusing-like.",
"The remaining trajectories ($13\\%$ ) differ from those mean trajectories.",
"Nevertheless, the observation of this bi-modal transition does not exclude the possibility of a more simple and continuous transition: the main vortex being simply more and more diffused during a phase, as the transition from merging to diffusing regimes occurs.",
"Circulation evolution for those three cases are presented on figure REF .",
"For the merging-diffusing case, the abrupt circulation increase at $x/W=-0.6$ corresponds to the merging of the main vortex with the secondary one, and separates the vortex trajectory in two parts: (i) The first part, quite similar to the merging case: the vortex gains in circulation from $x/W \\approx -1$ to $x/W\\approx -0.8$ , and looses in circulation further downstream.",
"(ii) The second part, quite similar to the diffusing case: the vortex circulation decreases while approaching the obstacle, reaching a very low circulation ($\\Gamma /(u_D \\delta ) \\approx 0.1$ ) at $x/W\\approx -0.2$ .",
"At the beginning of the second part, the circulation of the vortex is the same as at $x/W=-0.75$ , showing that the diffusing behavior occurrences are not only related to high circulations.",
"Figure: Main vortex centers mean trajectories on the x,yx, y plane of symmetry for three cases with h =4272,h/δ=2.7_h=4272,~h/\\delta =2.7 and increasing W/hW/h (a: 1.51, b: 2.01, c: 2.45), corresponding to the square symbol on figure .Dashed contours represent detected vortex centers envelopes.Each mean trajectory begins at the leftmost point.Percentages represent the percentage of trajectories used to compute each mean trajectories, the rest is not considered as it differs too much.Large circles are simultaneous vortex positions at a given arbitrary time.For visibility, aspect ratios are not conserved (trajectories representation is stretched along the y/δy/\\delta axis).Mean trajectories are computed on approximately 200 single trajectories.The discontinuous transition from merging to diffusing regimes is clearly visible on these plots with the increase of W/hW/h.Figure: Dimensionless vortex circulation evolution along the streamwise direction for the mean trajectories of figure .These evolutions show another aspect of the merging-diffusing transition linked to the increase of the vortex circulation."
],
[
"Complex regime",
"An example of vortex center trajectory for a complex regime is presented on figure REF , where oscillating, merging, diffusing and breaking phases alternate during 20 consecutive periods.",
"It has been verified by additional measurements upstream of the obstacle (not shown here) that the phase alternation is not provoked by perturbations from outside the HSV.",
"No particular order can be seen in the phases organizations, or in the main vortex maximum and/or minimal positions.",
"In the case of oscillating, merging or diffusing regimes, the initial and final states of each phase are very close, i.e.",
"each phase brings the HSV back in its initial state.",
"The phase behavior being governed by the initial state, successive phases have no particular reasons to differ from each other.",
"In the case of the complex regime, each phase final state slightly differs from its initial state, leading to an alternation of different phase types.",
"With this point of view, the HSV dynamics can be considered as a dynamic system linking the next initial state to the current one: $\\phi _{n+1} = F(\\phi _{n})$ with $\\phi _n$ the $n\\textsuperscript {th}$ initial state and $F$ the function representing the system dynamics.",
"This system is expected to admit stable equilibrium positions for stable, oscillating, merging, diffusing and breaking regimes, to be chaotic for the complex regime and to undergo dynamic bifurcations leading to complex transitional behaviors, as it is the case for the transitional regime between the fusion and the diffusing regimes (figure REF b).",
"Further, studying this dynamical system properties would require measuring at least 1000 consecutive phases, which represents approximately 20000 instantaneous velocity fields and is out of the scope of the present study.",
"Figure: Time evolution of the streamwise vortex center positions for a complex regime ( h =6397,W/h=1.67,h/δ=3.69,T=17.22s_h=6397,~W/h=1.67,~h/\\delta =3.69,~T=17.22s).Identified phase types are presented on the right panel.Dashed lines highlight the initials states, here the upstream most position of the main vortex V 1 V_1.The unordered phase succession illustrates the chaotic behavior of the complex regime."
],
[
"Coherent to irregular transition",
"From the previously defined coherent regimes, the HSV can evolve to an irregular, aperiodical, turbulent-like state, despite the boundary layer remaining in a laminar state.",
"This transition is characterized by the appearance of small-scale non-coherent structures.",
"Transitional cases (between coherent and irregular regimes) will be said in “semi-irregular regime”, and consist of time periods of coherent regime, punctually disturbed by eruptions of small-scale perturbations.",
"Such transitions have been observed for oscillating, merging and complex regimes, as shown in figure REF .",
"The fact that the irregular transition occurs for different coherent regimes make necessary the two-dimensional typology presented on figure REF , separating the coherent and the irregular evolutions.",
"This distinction was not included in any previous work [50], [3], [25], [38], where typologies remained strictly one-dimensional.",
"The eruptions of small scale non-coherent structures are of two types.",
"First type eruptions are linked to the appearance of positive vorticity above the main vortex $V_1$ .",
"Figure REF illustrates this phenomenon and shows small-scale structures appearing above and downstream of the main vortex at $t/T=0.06$ , and provoking the appearance of positive vorticity of the same order of magnitude (but opposite sign) as the main vortex vorticity.",
"This positive vorticity is then advected around the main vortex from $t/T=0.06$ to $0.08$ and ends up creating a new vortex downstream from $V_1$ at $t/T = 0.11$ .",
"Second type eruptions are linked to the merging between two vortices, resulting in a destabilized main vortex (not shown here).",
"The bi-modal behavior [13], previously reported for fully turbulent HSV, has not been clearly identified for irregular nor semi-irregular regimes, even if zero-flow and back-flow occurrences can be seen episodically.",
"The conclusion may be that the bi-modal behavior mainly occurs for higher Reynolds numbers, such as those used in studies devoted to this phenomenon ($_W=115000$ for [13] and [44], $_W=20000$ and 39000 for [20], while $_W < 4950$ in the present study).",
"Figure: Successive instantaneous streamlines and vorticity contours in the vertical plane of symmetry for a semi-irregular complex regime ( h =6395_h=6395, h/δ=2.81h/\\delta =2.81, W/h=1.36W/h=1.36, T=16.37sT=16.37s), illustrating the first type eruption.Circles represent detected critical points.This set of instantaneous flows illustrate well the first type eruption, characterized by the apparition of negative vorticity above and downstream of the main vortex."
],
[
"Instabilities origins",
"Several hypotheses can be put forward to explain the appearance of turbulent-like (small-scale and non-coherent) structures in the HSV and, consequently, its transition to an irregular regime.",
"The first possible origin can be the boundary layer transition to turbulence.",
"Turbulence bursts coming from the upstream boundary layer can certainly destabilize the HSV and break its periodicity.",
"Yet, no perturbation in the upstream boundary layer could be observed herein, thus excluding this hypothesis.",
"Second, the interaction between proximate vortices can lead to the appearance of an elliptical instability [31] in the core region of the vortices.",
"This interaction cannot be the origin of the first type eruption (figure REF ), where the instability originates from above the vortex, but is the best candidate to explain instabilities resulting from merging vortices (second type eruption).",
"Third, [20] show that eruptions of vorticity from the wall, responsible for the bi-modal behavior of fully turbulent HSV, can be understood as thee-dimensional Görtler instabilities [22].",
"In such scenario, the vorticity of the first counter-rotating vortex $V_{c1}$ punctually wraps around the main vortex $V_1$ and destabilizes it.",
"While the first type eruption in figure REF could agree with this definition, there is no evidence that the positive vorticity, appearing at $t/T=0.06$ , originates from the counter-rotating vortex.",
"A definitive conclusion on the first type eruption origin cannot, unfortunately, be drawn without proper 3D information on the flow, out of the scope of the present work."
],
[
"Regimes evolution with flow parameters",
"The evolution of the flow regimes with the dimensionless flow parameters is presented on figure REF , separately for the coherent (a) and the irregular (b) evolutions.",
"Figure REF a clearly shows that the coherent regimes depend on the three dimensionless parameters $h/\\delta $ , $W/h$ and $_h$ : an increase of any one of them leads to a destabilization of the HSV, potentially modifying the regime toward the complex one (toward the top in figure REF ).",
"The influence of $_h$ is the most obvious for the studied domain.",
"The interpretation of the parametric dependencies of the irregular evolution (figure REF b) is more challenging.",
"Influence of the Reynolds number $_h$ and the aspect ratio $W/h$ is clearly visible, but the influence of $h/\\delta $ remains unclear.",
"Figure REF compares the evolution of the HSV regimes between an immersed obstacle configuration from [38] and the present emerging obstacle configuration.",
"This figure shows an overall agreement with the regimes boundaries of [38], with a main dependency to $_W$ .",
"Nevertheless, the boundary between steady and periodic vortex system is not well reproduced, and the transition from periodic to irregular regimes appears at higher $_W$ in the emerging obstacle configuration.",
"As the periodic and irregular regimes overlap, the couple of dimensionless parameters used by [38] ($_W$ and $h/\\delta $ ) should not be the leading parameters of the regime evolution for the emerging obstacle configuration.",
"Figure: (a) Coherent regimes evolution as a function of the three dimensionless parameters.Each circle, representing a measured flow, is coloured according to the observed HSV regime.",
"(b) Irregular regimes evolution.As coherent regimes cannot be defined on irregular regimes configurations, irregular HSV flows are not represented on (a).These evolutions establish well the dependence of the HSV coherent regimes to the three dimensionless parameters, and the main dependence of the irregular regimes to the Reynolds number.Figure: HSV regimes evolution comparison with typology from .Each point represents a flow of the present study, classified according to typology: filled symbols for the steady vortex system (equivalent to the stable regime), cross-filled (yellow) symbols for periodic vortex system (equivalent to the oscillating, merging and diffusing regimes) and hollow symbols (red) for irregular vortex system (equivalent to the irregular regime).Lines represent regime boundaries reported by for immersed obstacles, and adapted with the equivalent diameter method (section ).The previous section presented the dependence of the HSV dynamics to the flow dimensionless parameters, but few is known concerning the physical mechanisms at the origin of these transitions.",
"This section proposes a model which purpose is to identify the main mechanisms behind the HSV periodic motion detailed in the previous section.",
"After a brief note on the HSV vortex appearance in section REF , a semi-empirical correlation for the main vortex velocity is proposed in section REF and on top of this, a model for the HSV dynamics is proposed in section REF .",
"Results obtained with this model are presented in section REF and allow to draw conclusions on the leading mechanisms of the HSV dynamics.",
"The reader is reminded that the presented model is not an attempt to obtain a predictive model for the HSV dynamics, but is rather to gain information on the HSV dynamics main mechanisms."
],
[
"Vortex creation",
"Because of its curvature and the strong three-dimensionality of the flow, the shear layer differs from the classical straight shear layer (see for instance [58]).",
"Assumption can be made that it still behaves qualitatively like a classical shear layer and that its stability is governed by the shear layer Reynolds number: $_{sh} = \\frac{\\Delta U \\delta _{sh}}{\\nu }$ with $\\Delta U$ the velocity difference between the outer flow on both sides of the shear layer and $\\delta _{sh}$ the shear layer thickness.",
"For the present shear layer (see figure REF a), one can estimate an upper bound for the associated Reynolds number as: $_{sh, \\max } = \\frac{2 u_D y_{sh}}{\\nu }$ where $y_{sh}$ is the elevation of the downstream end of the shear layer.",
"In the present study, $_{sh, \\max }$ ranges from 105 to 1363.",
"By analogy with classical shear layers, different dynamics behavior are expected depending on the $_{sh}$ values: (i) For low $_{sh}$ ($_{sh} < 55$ for classical shear layers), the shear layer should be stable and laminar [10], and the HSV should exhibit no vortex, leading to the no-vortex regime (see figure REF and figure REF a).",
"[50] observed this no-vortex regime for $_{sh, \\max } = 78$ .",
"(ii) For high Reynolds numbers ($_{sh} > 10^4$ for classical shear layers), the shear layer and the HSV should be fully turbulent [14].",
"This has not been observed in this study, as other instabilities appear before reaching such $_{sh}$ values.",
"(iii) For moderate Reynolds numbers ($55 < _{sh} < 10^4$ for classical shear layers), coherent, large-scale vortices should be generated periodically in the shear layer and advected downstream (see [40] and [42] for $_{sh}$ up to respectively 9700 and 5000).",
"This range is in fair agreement with the present measured configurations ($Re_{sh, max} \\in [105, 1363]$ ).",
"One main difference between the classical shear layer and this curved shear layer is the fact that HSV can show steady vortices (stable regime) instead of continuously advected downstream vortices.",
"Nevertheless, no correlation could be obtained between the estimated $_{sh, max}$ and the HSV typology, promoting the hypothesis that the vortex dynamics and periodical behavior are not directly linked to the shear layer vortex shedding.",
"Figure: (a) Schematic representation of the boundary layer separation in the vertical plane of symmetry and the resulting shear layer at low sh _{sh} (i.e.",
"in no-vortex regime).",
"(b) Diagram showing the vortex formation on the shear layer at moderate sh _{sh} (here in stable regime).",
"(c) Illustrative diagram for the model on the main vortex velocity v adv v_{adv}, detailed in section ."
],
[
"Vortex motion",
" The velocity of a vortex along the shear layer depends on the surrounding flow.",
"For the main vortex $V_1$ , which is supposed to govern the HSV dynamics, its advection velocity will depend on: (i) The wall influence, whose induced velocity on $V_1$ can be estimated using the vortex mirror concept ([15], see figure REF c) as: $v_{wall} = - \\frac{\\Gamma }{4\\overline{y_{1}} u_D}$ with $\\Gamma $ the main vortex circulation and $\\overline{y_1}$ the average location of the main vortex along $y$ .",
"Note that only the bed mirror vortex ($V_{M1}$ on figure REF c) is taken into account, as the two others ($V_{M2}$ and $V_{M3}$ ) are much further and have negligible influences on $V_1$ .",
"(ii) The secondary vortex $V_2$ influence, which depends on the relative position of the two vortices ($\\Delta x$ , $\\Delta y$ ).",
"(iii) The global state of the flow induced by the boundary layer separation, which should be constant for a given configuration and depend only of the dimensionless parameters $_h$ , $h/\\delta $ and $W/h$ .",
"An empirical correlation for the main vortex velocity $v_{adv}$ can finally be found by using measurements of all these parameters instantaneous values over stable and oscillating regimes (using mean trajectories and their associated velocity fields): $\\frac{v_{adv}}{u_D} =\\underbrace{- \\frac{\\Gamma }{4\\overline{y_{1}} u_D}}_{v_{wall}} +\\underbrace{\\left[1.19\\frac{\\Delta y}{\\overline{y_{1}}} - 1.15\\left(\\frac{\\Delta y}{\\overline{y_{1}}}\\right)^2\\right] \\left(\\frac{h}{\\delta }\\right)^{3.73} \\left(\\frac{\\overline{y_{1}}}{h}\\right)^2}_{v_{flow}}$ with $R^2=0.89$ and $v_{flow}$ the influence of the flow, aggregating points (ii) and (iii).",
"The effect of $\\overline{y_{1}}/h$ on the main vortex velocity can be explained by the upper flow contraction caused by the HSV, which provokes an increased $v_{adv}$ .",
"The effect of $h/\\delta $ accounts for the impact of the shear layer shape (governed by $\\delta $ and the down-flow).",
"Note that for a given flow configuration, the only time-varying parameters are $\\Delta y$ and $\\Gamma $ .",
"This correlation is illustrated in figure REF for the oscillating regime previously presented in section REF .",
"Figure REF a presents the strong correlation between the main vortex velocity $v_{conv}$ and the difference of altitude between the main and the secondary vortex $\\Delta y$ , confirming the necessity to take this parameter into account in relation REF .",
"Figure REF b presents the comparison between the measured main vortex velocity $v_{conv}$ and its prediction using the correlation REF , along with the two components of the predicted main vortex velocity: $v_{flow}$ and $v_{wall}$ .",
"The good agreement allows to pass to the next step.",
"Figure: Evolution of different components of the correlation on one selected period of the oscillating case presented in figure .",
"(a) Comparison between the measured main vortex velocity v adv v_{adv} and the measured difference of altitude between the main and the secondary vortex Δy\\Delta y, showing the strong correlation existing between these two parameters.",
"(b) Comparison between the two components of correlation (v flow v_{flow} and v wall v_{wall}) and the main vortex measured velocity v conv v_{conv}, showing the good agreement obtained for the main vortex velocity correlation.It is to be noted that for a given flow, the sole non-constant parameters in the correlation are Δy\\Delta y and Γ\\Gamma ."
],
[
"Vortex dynamics reproduction",
"To identify the physical phenomenon at the origin of the HSV dynamics, a numerical model is established based on the followings hypotheses: (i) The vortices can travel upstream and downstream but remain along the shear layer, which shape is taken from measurements.",
"(ii) The main vortex instantaneous velocity $v_{adv}$ is estimated by equation REF .",
"(iii) The secondary vortex velocity $v_{2,adv}$ is equal to the velocity of the main one $v_{adv}$ .",
"(iv) The initial vortex locations are their measured equilibrium or mean positions plus a perturbation.",
"(v) The vortices velocity are initially set to zero.",
"Numerical simulations using this model succeed in replicating the vortex equilibrium position for stable regime configurations but do not exhibit a periodic behavior for the oscillating regime configurations (not shown here).",
"One missing characteristic of the HSV dynamics that may explain the lack of periodicity is the delay $\\Delta t$ between $V_1$ and $V_2$ motion (previously discussed in section REF , and presented on figure REF ).",
"Figure REF shows a scenario that illustrates how the complexity added by this delay can lead to a periodic behavior: (a) Vortices are placed on the shear layer, the main vortex being placed upstream of its equilibrium position.",
"(b) The main vortex naturally goes towards its position of equilibrium, according to the equation REF while the secondary vortex remains in place, due to the delay (exaggerated for the sake of this demonstration).",
"(c) The secondary vortex, after a delay of $\\Delta t$ , moves downstream, decreasing the $\\Delta y$ value and thus moving upstream the main vortex equilibrium position.",
"(d) The main vortex moves towards its new equilibrium position while the secondary vortex stays still due to the delay.",
"(e) The secondary vortex, after a delay $\\Delta t$ , moves upstream, increasing the $\\Delta y$ value, pushing downstream the equilibrium position and bringing the vortex system back to its initial state (similar to (a)).",
"(f) If the main and the secondary vortices come close enough to each other in (d), they merge, the secondary vortex is replaced by a new one from upstream, and the vortex system recovers its initial state (similar to (a)).",
"Figure: Schematic illustration of how the motion delay between the main and the secondary vortices can lead to an oscillating behavior (explained step by step in section ).The fact that the delay is at the origin of the oscillation process if further ensured by the strong correlation between measured delays $\\Delta t$ and oscillation frequencies $f$ : $f = \\frac{0.154}{\\Delta t}$ with $R^2 = 0.92$ on the dimensional correlation (see on figure REF ).",
"Unfortunately, no correlation could be obtained between the delay $\\Delta t$ and the dimensionless parameters $_h$ , $h/\\delta $ and $W/h$ .",
"The delay is expected to result from vortex-vortex interactions, and so to depend on vortex properties (position, radius, circulation), themselves depending on the boundary layer shape, which depends on the dimensionless parameters.",
"This dependency chain explains the difficulty to build direct correlations between the delay $\\Delta t$ and the dimensionless parameters.",
"Figure: Evolution of the averaged measured delay Δt\\Delta t between the main and the secondary vortices motions according to the measured oscillation period TT for 15 configurations ( h ∈[2113,6406]_h \\in [2113, 6406], h/δ∈[1.9,3.8]h/\\delta \\in [1.9, 3.8], W/h∈[0.46,1.00]W/h \\in [0.46, 1.00], in oscillating, merging or complex regimes).Dotted line is the best linear correlation (equation ), with a R 2 =0.92R^2=0.92, indicating a strong link between the delay and the periodic behavior of the HSV.The following section aims at checking if taking this delay into account in the model is sufficient to retrieve a self-sustainable periodic behavior for the selected oscillating regime flow i.e.",
"if a small perturbation applied on the main vortex position while at its equilibrium position can lead to a stabilized oscillation amplitude.",
"It is to be noted that the delay does not have to be constant but, as a result of the interaction between the main and the secondary vortices, should depend on the distance between the main and the secondary vortex and on their circulations.",
"This delay is added in the model by considering that $V_2$ velocity equals $V_1$ velocity, with a measured constant average delay $\\Delta t$ : $v_{2, adv}(t) = v_{adv}\\left(t - \\Delta t\\right).$ As a lot of hypotheses have been made, an adjustment variable is necessary for the model to be able to reproduce the observed HSV behavior.",
"The addition of an empirical factor of 3 on the delay $\\Delta t$ fairly close the system: $v_{2, adv}(t) = v_{adv}\\left(t - 3 \\Delta t\\right).$"
],
[
"Numerical simulations results",
"Figure REF shows the results of simulations using the model presented in section REF for a stable, an oscillating and a merging regime configurations.",
"Firstly, regarding the simulation of the stable regime configuration, $V_1$ is initially introduced away from its equilibrium position ($x/W=-0.5$ instead of $x/W=-0.7$ ).",
"Vortices indeed appear to rapidly reach their equilibrium positions at $t/T\\approx 30$ (with $T$ the vortex position oscillation period), after a transitional damped oscillation.",
"Then, regarding the oscillating regime configuration (figure REF b), a small perturbation is applied on the position of the main vortex ($x/W=-0.62$ instead of $x/W=-0.61$ ).",
"The oscillation amplitude increases with time and reaches a stable value after $t/T \\approx 100$ .",
"Finally, regarding the merging regime configuration, the same small perturbation is applied on the position of the main vortex.",
"The oscillation amplitude increases but does not reach a stable position.",
"The main and secondary vortices end up close to each other, in which case the simulation is stopped, as no vortex-vortex interaction model was implemented.",
"Despite the delay adjustment, the Strouhal number based on the delay $St_{\\Delta t}$ , shown to be nearly constant on experiments ($St_{\\Delta t} = 0.154$ ), reaches a similar value of $St_{\\Delta t}=0.146$ in the simulation.",
"This shows that the relation between the delay $\\Delta t$ and the periodic behavior at frequency $f$ is well simulated by this model and that the proposed model is able to recreate the HSV dynamics for the stable, oscillating and merging regimes.",
"Conclusions can be made regarding the origin of the periodic motion of the HSV that: (i) the secondary vortex position has a strong influence on the velocity of the main one, represented by the parameter $\\Delta y$ in the correlation REF .",
"This effect may be explained by the feeding of the main vortex from the main flow (quantity of fluid ending in the main vortex), which is reduced when the secondary vortex arises.",
"[59] linked, for stable HSV, this feeding and the vortex size, but a relation between the feeding and the vortex velocity has not been found yet.",
"(ii) The delay $\\Delta t$ between the motion of the main and the secondary vortices is strongly linked to the periodic behavior of the HSV, and makes its complex dynamics possible.",
"Figure: Numerical simulation results for the model presented in section , for:(a) a stable regime configuration ( h =4271_h=4271, h/δ=2.70h/\\delta =2.70, W/h=0.79W/h=0.79),(b) an oscillating regime configuration ( h =4271_h=4271, h/δ=2.70h/\\delta =2.70, W/h=1.23W/h=1.23) and(c) a merging regime configuration ( h =4271_h=4271, h/δ=2.70h/\\delta =2.70, W/h=1.37W/h=1.37).Upper figures show the shear layer measured shapes (black lines), the average measured vortex positions (dashed lines) and the computed extreme positions for the main vortex (black circles) and the secondary vortex (white circles).Bottom figures show the vortices streamwise trajectories with time (black lines) and the average measured vortex positions (dashed lines, hidden by the black lines for the left case).Vignettes show the time evolution of the main vortex oscillation amplitude.Those results are in good agreement with the observations, indicating that the mechanisms taken into account in the model are sufficient to reproduce the HSV dynamics."
],
[
"Note on the vortex breaking",
"The breaking regime particular dynamics can also be approached using this model.",
"For high delays $\\Delta t$ and high $V_1$ advection velocities, the maximum distance $\\Delta x$ between $V_1$ and $V_2$ is expected to increase.",
"If $\\Delta x$ exceeds a certain value ($\\approx R_1 + R_2$ according to the present observations), the main flow passes between the two vortices, cancelling the interaction between $V_1$ and $V_2$ .",
"The general vortex motion then results in a competition between the wall influence $v_{wall}$ and the flow influence, reduced to a simple boundary layer (without effect of $V_2$ through $\\Delta y$ ).",
"The resulting new equilibrium position is observed to be stable and located in the vicinity of the obstacle (where the flow influence is small enough to be balanced by the wall influence).",
"The breaking vortex rapidly reaches this position and disappears due to the strong stretching in the near obstacle zone increasing its diffusion.",
"This last section presents the evolution of the HSV geometrical characteristics with the dimensionless parameters of the flow and an in-depth comparison of these results with the well-documented immersed configuration."
],
[
"Separation distance $\\lambda $",
"Separation distance The separation distance $\\lambda $ (see figure REF ) is a crucial parameter for the HSV, as it governs the shear layer shape and the HSV streamwise dimension.",
"Using all PIV and trajectography measurements of the HSV, the following correlation was obtained: $\\frac{\\lambda }{W} = 1.91$ with a $R^2 = 0.95$ computed on the dimensional correlation: $\\lambda =1.91W$ .",
"The boundary layer separation position greatly depends on the adverse pressure gradient [36], which depends on the obstacle width, according to 2D potential flow computations, and explains this result.",
"It is interesting to see that the boundary layer separation position does not depend on $\\delta $ , the boundary layer thickness measured before placing the obstacle.",
"[9] and [6] proposed two correlations for the separation distance $\\lambda $ for immersed cylindrical obstacles: $\\frac{\\lambda _{Belik}}{D} = 0.5 + 35.5Re_{D}^{-0.424}$ $\\frac{\\lambda _{Baker}}{D} = 0.5 + 0.338Re_{\\delta ^*}^{0.48} \\left(\\frac{\\delta ^*}{D}\\right)^{0.48} \\tanh \\left( \\frac{3h}{D}\\right)$ with $D$ the obstacle diameter.",
"Application of these three correlations (equations REF , REF and REF ) are compared in figure REF for present data and data from the literature.",
"Both literature correlations for immersed obstacles underestimate the separation distance $\\lambda $ , revealing that this distance is greater for emerging obstacles than for immersed ones.",
"This difference can be explained by the fact that flow cannot pass above emerging obstacles, resulting in a higher pressure gradient than for immersed obstacles, and then, a precocious boundary layer separation.",
"[49] already noticed that the obstacle emergence increases the separation distance.",
"Application of equation REF on data from literature with emerging obstacles and turbulent boundary layers (figure REF ) overestimates the separation distance $\\lambda $ .",
"This can be explained by the fact that turbulent boundary layers are known to separate for higher pressure gradient than laminar ones.",
"The apparent simplicity of correlation REF compared to equations REF and REF is discussed here: (i) The boundary layer thickness $\\delta $ is an important parameter for immersed obstacles, as it governs the position of the stagnation point on the face of the obstacle, and so, the down-flow and the adverse pressure gradient.",
"For emerging obstacles, the down-flow is stronger [49] and the stagnation point position always high on the obstacle face, resulting in a strong adverse pressure gradient that is not influenced by the boundary layer thickness $\\delta $ .",
"(ii) Correlations from literature are made more complex by the addition of $0.5D$ due to the method from [8] (see section REF ), and, for equation REF , by the last term ($\\tanh \\left(3h/2D\\right)$ ) added afterwards to take into account the obstacle height.",
"Figure: Correlations quality for separation distance (λ\\lambda ) prediction.black filled circle symbols stands for equation applied on present data, with dotted lines for 20%20\\% confidence interval,hollow (red) squares the correlation of (equation ) applied on the present data andhollow (blue) circles represent the correlation of (equation ) applied on the present data.Crossed-squares (yellow) represent equation applied on data from literature on emerging obstacles with turbulent boundary layer.Displayed maximum uncertainty (caption) only takes into account measurement uncertainties on PIV and trajectographies (see section )."
],
[
"Vortex position",
"The present section aims at proposing empirical correlations for the main vortex position along $x$ and $y$ i.e.",
"the mean vortex position for the stable and oscillating regime configurations and the position of the vortex maximum circulation for the merging, diffusing and breaking regimes configurations.",
"They read: $\\frac{x_{1}}{W} = 1.01$ $\\frac{y_{1}}{\\delta } = 0.2784 + 0.0229 \\left(\\frac{W}{h}\\right)^2$ with $R^2$ of respectively $0.96$ and $0.93$ on the dimensional forms of the correlations.",
"The main vortex position corresponds to the downstream end of the shear layer where the down-flow forces the reattachment of the boundary layer.",
"Consequently, the main vortex position should be linked to the boundary layer thickness $\\delta $ (governing the shear layer shape) and the obstacle width $W$ (governing the down-flow), which explains the correlations REF and REF .",
"[6] and [38] correlations for vortex position in the case of immersed cylindrical obstacles read: $\\frac{x_{1, Baker}}{D} = 0.5 + 0.013Re_{\\delta ^*}^{0.67} \\tanh \\left(\\frac{3h}{D}\\right)$ $\\frac{x_{1, Lin}}{\\delta } = 0.5\\frac{D}{\\delta } + 0.518\\left(\\frac{h}{\\delta }\\right)^{0.87}\\left(\\frac{h}{D}\\right)^{-0.34}$ $\\frac{y_{1, Lin}}{\\delta } = 0.207\\left(\\frac{h}{\\delta }\\right)^{0.6}\\left(\\frac{h}{D}\\right)^{-0.77}$ Application of these correlations on the present data is plotted in figure REF and REF .",
"Equations REF and REF fairly fit the present data, showing that, contrary to the separation distance $\\lambda $ , the streamwise position of the main vortex $x_1$ does not differ for immersed and emerging obstacles (and that it is not governed by the adverse pressure gradient).",
"Moreover, figure REF gives confidence to the equivalent diameter method described in section REF and used herein.",
"Results for $y_1$ in figure REF show more dispersion, due to the higher relative uncertainty in the measurement of the vortex vertical location.",
"Correlations from the literature overestimate $y_{1}$ .",
"This difference may be linked to the free-surface, that confines the shear layer vertically.",
"Figure: Application of correlations for streamwise location of the main vortex (x 1 x_1):black filled circles: equation with dashed line for 10% confidence interval,hollow (blue) circles: equation from andhollow (red) squares: equation from .Displayed maximum uncertainty (caption) only takes into account measurement uncertainties on PIV and trajectographies (see section ).Figure: Application of correlations for the vertical location of the vortex (y 1 y_{1}):black filled circles: equation with dashed line for 20% confidence interval,hollow (red) squares: equation from .Displayed maximum uncertainty only takes into account measurement uncertainties on PIV or trajectographies (see section )."
],
[
"Frequency",
"The frequency $f$ associated with the HSV vortex motion (for oscillating, diffusing and complex regimes) is of great importance to understand the HSV dynamics.",
"No correlation could be found for Strouhal numbers generally used in the literature ($St_\\delta $ and $St_W$ ) nor for the Strouhal number based on the separation distance ($St_\\lambda $ ) or on the distance between the main and the secondary vortices ( and $St_{\\Delta x}$ ).",
"The frequency $f$ was however found to be mainly dependent on the bulk velocity $u_D$ as: $f = 211u_D^{2.33} \\\\$ with $R^2=0.96$ .",
"The main conclusion of this correlation is that the obstacle width $W$ (see also figure REF c) and the boundary layer thickness $\\delta $ have no influence on the HSV frequency.",
"[55] measured the oscillation frequency for immersed obstacles in a wind tunnel with increasing obstacle width and observed a linear correlation (for Reynolds $_W=u_DW/\\nu $ ranging from 2320 to 10800): $St_W = 2.47\\times 10^{-5} _W$ $\\frac{fW}{u_D} = 2.47\\times 10^{-5} \\frac{u_D W}{\\nu }$ that is: $f = 1.8 u_D^2$ which confirms that the obstacle transverse dimension $W$ has no influence on the oscillating frequency.",
"As seen in section REF , a correlation can still be found with the average delay $\\Delta t$ between the motion of $V_1$ and $V_2$ (equation REF )."
],
[
"vortices number",
"At a given time, the number of vortices is detected using critical points on PIV measurements, and using particle trajectories on trajectography measurements.",
"The average number of vortices (not including counter-rotating vortices $V_{ci}$ ) varies from 1 to $3.5$ on the observed coherent regimes configurations.",
"The evolution of the vortices number should be strongly correlated to the shear layer shape.",
"Indeed, vortices follow each other along the shear layer at a distance from each other roughly equal to the sum of the vortices radii (which is correlated to their elevation $y$ ).",
"In such case, the vortices number should increase with the streamwise extension of the shear layer (equal to $\\lambda - x_1$ ) and decrease with the shear layer elevation (because of the subsequent decrease of the vortices radius).",
"Figure REF , showing the evolution of the measured vortex number with these parameters, confirms this statement: (i) for a constant $\\lambda - x_1$ (dashed line at $\\lambda - x_1 = 0.05$ for instance), the vortices number decreases as $y_1$ increases and (ii) for a constant $y_1$ (dotted line at $y_1=0.003$ for instance), the vortices number increases as $\\lambda - x_1$ increases.",
"Comparison of the present data with the observations of [3] for immersed obstacles is presented in figure REF and reveals that the vortices number (and so the shear layer shape evolution) differs greatly for the two emerging and immersed obstacles configurations.",
"Figure: Average vortices number evolution with the main vortex altitude y 1 y_{1} and the shear layer streamwise extension: λ-x 1 \\lambda - x_{1} for the coherent regime configurations.Non-integer values for the vortex number are associated to HSV with non-constant number of vortex.The visible continuous evolution (despite the non-homogeneous mapping), indicates that the vortex number is, as expected, governed by the shear-layer shape.Figure: Comparison of the HSV number of vortices with the observations of .Circles represent data from the present study and are coloured according to the observed vortices number.δ * \\delta ^* is the boundary layer momentum thickness.Black lines represent the domain of constant vortices number observed by (1 vortex, 2 vortices, 3 vortices and unsteady system).The bad agreement illustrates the differences between the immersed and emerging configurations in terms of dynamics."
],
[
"Conclusion on the impact of the emergence",
"The main difference between emerging and immersed obstacle configurations is the impossibility for the flow to pass over the obstacle in the latter case.",
"This has three main consequences: (i) the adverse pressure gradient is stronger in the case of an emerging obstacle, leading to a precocious boundary layer separation (longer $\\lambda $ , see figure REF ).",
"(ii) For an emerging obstacle, the whole flow facing the obstacle is deflected by the obstacle while for an immersed obstacle, the position of the flow stagnation point elevation along the upstream obstacle face (dependent on the boundary layer thickness) governs the quantity of fluid to be deflected.",
"This lead to a bigger dependency to the boundary layer thickness $\\delta $ in the case of immersed obstacles.",
"(iii) The difference of shear layer shape leads to different vortex number.",
"Nevertheless, the two configurations (immersed and emerging) share similarities in terms of : (i) main vortex distance to the obstacle, (ii) observed frequency dependency to the bulk velocity $u_D$ , and (iii) HSV vortices dynamics regimes."
],
[
"Conclusion",
"Trajectographies and PIV measurements were performed to investigate the HSV developing at the toe of a rectangular obstacle emerging from a laminar free-surface flow.",
"In this context, vortex tracking methods based on critical points detection was shown to be valuable tools to extract and summarize the HSV vortices motion patterns.",
"The HSV dynamics was categorized using a typology adapted from the existing ones for immersed obstacle, by adding the newly observed complex regime, separating the coherent and the irregular evolutions (leading to a two-dimensional typology) and giving clear definitions of the typology regimes.",
"The observations of the complex regime and the investigation of the transitions between the different regimes showed the importance of considering the HSV as a dynamical system built on quasi-similar phases and undergoing a chaotic transition.",
"The coherent regime evolution was shown to depend on the three dimensionless parameters ($_h$ , $h/\\delta $ and $W/h$ ), in a continuous way from stable to complex regimes.",
"The irregular evolution is more complex, as it is linked to local vortex instabilities, but showed a main dependence to the Reynolds number $_h$ .",
"A model for the main and secondary vortices motion was proposed and succeeded in reproducing the dynamics of the stable, oscillating and merging regimes.",
"This result allowed to identify two mechanisms of importance for the HSV dynamics.",
"Firstly the secondary vortex position has an influence on the main vortex velocity, certainly by modifying the quantity of fluid feeding it.",
"And secondly, the phase shift existing between the main and secondary vortices motion allows the apparition of a periodic motion in the present model.",
"Besides, the extraction of the HSV geometrical parameters allowed to highlight their correlations with the dimensionless parameters of the flow in the case of an emerging obstacle, showing that: (i) the separation distance $\\lambda $ is mainly linked to the adverse pressure gradient, (ii) the main vortex position depends on the shear layer shape and on the downflow strength, (iii) the number of vortices composing the HSV is governed by the shear layer shape, and (iv) the effect of the free-surface confinement on the HSV is mainly indirect, through the limitation of the boundary layer thickness $\\delta $ .",
"Finally, the comparison with existing results for immersed obstacles indicated that the emerging configuration exhibits a higher adverse pressure gradient, due to a more important blocking effect, and a stronger downflow, due to the impossibility for the flow to bypass the obstacle by the top.",
"This implies a higher separation distance $\\lambda $ , a lower main vortex altitude $y_1$ , and consequently, a higher number of vortices in the HSV in the emerging obstacle configuration.",
"The modification of these basic HSV properties is supposed to result in strong modifications of the HSV dynamics, making the comparison of the typology evolution for immersed and emerging configurations challenging.",
"In light of these results, some questions arise and should be the subject of future works investigation on this topic.",
"Firstly, the impact of the obstacle elongation ($L/W$ ), kept very high in this study, on the HSV should be investigated.",
"This parameter modifies the adverse pressure gradient, and, for sufficiently low $L/W$ , the wake should be able to influence the HSV dynamics.",
"Secondly, the model presented in section should be extended to reproduce the other coherent regimes (merging, diffusing, breaking and complex).",
"This will only be possible by including additional ingredients in the model, such as the vortices circulation variation during a phase or the vortex merging, which are both challenging to handle.",
"Thirdly, little is known on the effect of a free-surface confinement on a HSV taking birth from a turbulent boundary layer.",
"Notably, the bi-modal behavior (previously reported for non-confined flows) should be influenced by the vertical confinement, as the zero-flow mode will be strongly modified.",
"Fourthly, experiments or numerical simulations on flow around obstacles with rounded or streamlined upstream faces would allow to confirm the validity of the equivalent diameter method for more complex obstacle shapes.",
"Finally, in a more application-oriented point of view, the effect of the HSV regime on the thermal exchanges, obstacle drag coefficient, wall shear stress and the downstream boundary layer properties could be investigated."
]
] | 1808.08472 |
[
[
"Representing Social Media Users for Sarcasm Detection"
],
[
"Abstract We explore two methods for representing authors in the context of textual sarcasm detection: a Bayesian approach that directly represents authors' propensities to be sarcastic, and a dense embedding approach that can learn interactions between the author and the text.",
"Using the SARC dataset of Reddit comments, we show that augmenting a bidirectional RNN with these representations improves performance; the Bayesian approach suffices in homogeneous contexts, whereas the added power of the dense embeddings proves valuable in more diverse ones."
],
[
"Introduction",
"Irony and sarcasmWe use “sarcasm” to include both sarcasm and irony, as the two are generally conflated in the literature we review.",
"are extreme examples of context-dependence in language.",
"Given only the text Great idea!",
"or What a hardship!, we cannot resolve the speaker's intentions unless we have insight into the circumstances of utterance – who is speaking, and to whom, and how the content relates to the preceding discourse [4].",
"While certain texts are biased in favor of sarcastic uses [15], [24], the non-literal nature of this phenomenon ensures that there is an important role for pragmatic inference [5].",
"The current paper is an in-depth study of one important aspect of the context dependence of sarcasm: the author.",
"Our guiding hypotheses are that authors vary in their propensity for using sarcasm, that this propensity is influenced by more general facts about the context, and that authors have their own particular ways of indicating sarcasm.",
"These hypotheses are well supported by psycholinguistic research [6], [11], [7], but our ability to test them at scale has until recently been limited by available annotated corpora.",
"With the release of the Self-Annotated Reddit Corpus (SARC), [14] have helped to address this limitation.",
"SARC is large and diverse, and its distribution of users across comments and forums makes it particularly well suited to modeling authors and their relationship to sarcasm.",
"Figure: The model architecture.",
"Look-ups are indicated by arrows, dense connections by diamonds.",
"The author embedding can be null (a text-only baseline), a prior reflecting the author's propensity for sarcasm, or a learned embedding.",
"There are potentially multiple layers between the initial example embedding and the output sigmoid layer.Our core model of comment texts is a bidirectional RNN with GRU cells.",
"To model authors, we propose two strategies for augmenting these RNN representations: a simple Bayesian method that captures only an author's raw propensity for sarcasm, and a dense embedding method that allows for complex interactions between author and text (Figure REF ).",
"We find that, on SARC, the simple Bayesian approach does remarkably well, especially in smaller, more focused forums.",
"On the full SARC dataset, author embeddings are able to encode more kinds of variation and interaction with the text, and thus they achieve the highest predictive accuracy.",
"These findings extend and reinforce the prior work on user-level modeling for sarcasm (Section ), and they indicate that simple representation methods are effective here."
],
[
"Previous Work",
"A substantial literature exists around sarcasm detection.",
"Many of the prior studies focus on the analysis of Twitter posts, which lend themselves well to sarcasm detection with NLP methods because they are available in large quantities, they tend to correspond roughly to a single utterance, and users' hashtags in tweets (e.g., #sarcasm, #not) can provide imperfect but useful labels.",
"A central theme of this literature is that bringing in contextual features helps performance.",
"[12] trained classifiers using a combination of lexical and pragmatic features, including emoticons and whether the user was responding to another tweet (see also [8]).",
"[2] extend this kind of analysis with additional information about the context.",
"Of special interest here are their contextual features: the author's historical sentiment, topics, and terms; the addressee; and features drawn from historical interactions between the author and addressee.",
"The study finds most features to be useful, but a model trained on the tweet and author features alone achieved essentially the same performance (84.9% accuracy) as a model trained on all features (85.1%).",
"In a similar vein, [19] used a complex combination of features from users' Twitter histories, including sentiment, grammar, and word choice, as inputs into their model, and report a $\\approx $ 7% gain in classification accuracy upon adding these features to a baseline n-gram classifier.",
"Recent papers have also applied deep learning methods to detecting sarcastic tweets.",
"[18] use a combination convolutional–SVM architecture with auxiliary sentiment input features.",
"The architecture of [26] includes an RNN, and uses contextual features as well as tweet text for inputs.",
"[1] extend the work of [2] by generating author embeddings to reflect users' word-usage patterns (but not sarcasm history) in a manner similar to the paragraph vectors introduced by [16].",
"With the inclusion of these embeddings, their convolutional neural network (CNN) achieves a 2% gain in accuracy over that of [2].",
"[9] present a combination CNN/LSTM (long short-term memory RNN) architecture that takes as inputs user affect inferred from recent tweets as well as the text of the tweet and that of the parent tweet.",
"When a tweet was addressed to someone by name, the name of the addressee was included in the text representation of the tweet, providing a loose link between interlocutors [25] and a $\\approx $ 1% gain in performance for some data sets.",
"There has also been a small amount of previous work on Reddit data for sarcasm [22], [10].",
"[24] explore a hand-labeled dataset of $\\approx $ 3K Reddit comments from six subreddits.",
"They report that, when human graders attempted to mark comments as sarcastic or not sarcastic, they needed additional context like subreddit norms and author history roughly 30% of the time, and that the comments which graders found ambiguous were largely the same as those on which a baseline bag-of-words classifier tended to make mistakes.",
"In a follow-up study, [23] find that semantic cues for sarcasm differ by subreddit, and they show classifier accuracy gains when modeling subreddit-specific variation.",
"The work that is closest to our own is that of [13], who also experiment on the SARC dataset.",
"Their model learns author, forum, and text embeddings, and they show that all three kinds of representation contribute positively to the overall performance.",
"We take a much simpler approach to author embeddings and do not include forum embeddings, and we report comparable performance (Section ).",
"We take this as further indication of the value of author features for modeling sarcasm."
],
[
"The SARC Dataset",
"The Self-Annotated Reddit Corpus (SARC) was created by [14].http://nlp.cs.princeton.edu/SARC/2.0/ It includes an unprecedented 533M comments.",
"The corpus is self-annotated in the sense that a comment is considered sarcastic if its author marked it with the “/s” tag.",
"As a result, the positive examples are essentially those which the authors considered ambiguous enough to explicitly tag as sarcastic, meaning that the prediction problem is actually to identify which comments are not only sarcastic but both sarcastic and not obviously so.",
"The dataset is filtered in numerous ways, and has good precision (only $\\approx $ 1% false positive rate) but poor recall (2% false negatives relative to 0.25% true positives, or $\\approx $ 11% recall).",
"To alleviate the issues caused by low recall, the dataset also includes a balanced sample, where comments are supplied in pairs, both responding to the same parent comment and with exactly one of the two tagged as sarcastic.",
"All comments are accompanied with ancestor comments from the original conversation, author information, and a score as voted on by Reddit users.",
"This dataset presents numerous advantages for sarcasm detection.",
"For one, it is vastly larger than past sarcasm datasets, which enables the training of more sophisticated models.",
"In addition, most work in sarcasm detection has focused on tweets, which are very short and tend to use abbreviated and atypical language.",
"Reddit comments are not constrained by length and are therefore more representative of how people typically write.",
"Finally, Reddit is organized into topically-defined communities known as subreddits, each of which has its own community norms and linguistic patterns.",
"By making available large amounts of data from a number of subreddits, SARC facilitates the comparative analysis of subreddits, and more generally provides a view into the differences between communities.",
"Table REF provides basic statistics on the entire corpus as well as the subreddits that we focus on in our experiments.",
"Table: Basic statistics for SARC."
],
[
"Models",
"Our baseline model is a bidirectional RNN with GRU cells (BiGRU; [3]).",
"We tried variants with LSTM cells and did not observe a significant difference in performance.",
"We therefore chose to use GRU cells as the model with fewer parameters.Our models and associated experiment code are available at https://github.com/kolchinski/reddit-sarc The inputs to the BiGRU model are users' comments, which are split into words (and in the case of conjunctions, subwords) and punctuation marks and are converted to word vectors.",
"The final states of the two directions of the BiGRU are concatenated with each other and run through either a single fully-connected linear layer or two fully-connected linear layers with a rectified linear unit in between.",
"The output of the final linear layer is fed through a sigmoid function which outputs the estimated probability of sarcasm.",
"This baseline does not take author information into account: for each comment, only the words of the comment are considered as inputs.",
"The Bayesian prior model extends the BiGRU with the sarcastic and non-sarcastic comment counts for authors seen in the training data, which serves as a prior for sarcasm frequency.",
"This version of the model takes as inputs both a representation of the comment and the author representation $x_{\\text{author}}\\in \\mathbb {Z}^2_{\\ge 0}$ to estimate the probability of sarcasm.",
"The model can be interpreted as computing a posterior probability of sarcasm given both the comment and the prior of previous sarcastic and non-sarcastic comment counts – author modeling reduced to a Bernoulli prior.",
"For previously unseen authors, $x_{\\text{author}}$ is set to $(0,0)$ .",
"The author embedding approach extends the baseline BiGRU in a more sophisticated way.",
"Here, each author seen in the training data is associated with a randomly initialized embedding vector $x_{\\text{author}} \\in \\mathbb {R}^{15}$ , which is then provided as an input to the model along with a representation of the words of the comment.",
"A special randomly initialized vector $x_{\\textsc {unk}}$ is used for previously unseen authors.",
"The author embeddings are updated during training, with the goal of learning more sophisticated individualized patterns of sarcasm than the Bayesian prior allows.",
"We experimented with training the $x_{\\textsc {unk}}$ vector on infrequently-seen authors (fewer than 5 comments in the training set) instead of using a random vector, and found some suggestions of improved performance.",
"However, as the differences in performance were not substantial enough to change the relative performance of the different models, we report the results for the simpler random-$x_{\\textsc {unk}}$ model."
],
[
"Experiments",
"We conducted three sets of experiments, one for each model, to evaluate the effectiveness of the different approaches to author modeling.",
"Each set of experiments was conducted on five datasets: the balanced version of the entire corpus as well as the balanced and unbalanced versions of the r/politics and r/AskReddit subcorpora (Table REF ).",
"In all cases, the raw comment data was tokenized into words and punctuation marks, with components of contractions treated as individual words.",
"We mapped tokens to FastText embedding vectors which had been trained, using subword infomation, on Wikipedia 2017, the UMBC webbase corpus, and the statmt.org news dataset [17].",
"While vectors existed for nearly 100% of tokens generated, exceptions were mapped to a randomly initialized UNK vector.",
"All models were trained with early stopping on a randomly partitioned holdout set of either 5% of the data for balanced subreddit corpora or 1% for the others.",
"The performance of the model, as used for hyperparameter tuning, was evaluated against a second holdout set, generated in the same manner as the first holdout set but disjoint from both it and the portion of the data used for training.",
"Hyperparameters were tuned to maximize model performance as evaluated in this manner, starting with a randomized search process and fine-tuned manually.",
"The final evaluation was conducted against the test set, with a single randomly partitioned holdout set from the training data again used for early stopping.",
"We applied dropout [21] during training before and between all linear layers.",
"For additional regularization, we also applied an l2-norm penalty to the linear weights but not to the GRU weights.",
"We attempted other model variations, including multiple GRU layers and an attention mechanism for GRU outputs, but did not observe any gains in performance from the larger models."
],
[
"Quantitive assessment",
"Table REF reports the means of 10 runs to control for variation deriving from randomness in the optimization process [20].",
"Where there is overlap between our experiments and those of [13] (CASCADE), our model is highly competitive.",
"We slightly under-perform on the full balanced dataset but come out ahead on r/politics.",
"This is striking because our model makes use of much less information.",
"First, unlike CASCADE, we do not have forum embeddings.",
"Second, CASCADE author embeddings involve extensive feature engineering including “stylometric” and “personality” features.",
"Our author embeddings, on the other hand, are either simple empirical estimates (Bayesian priors) or learned embeddings with random initializations, in both cases allowing simpler model specification and training, and more flexibility on the task for which they are used.",
"There is also evidence that the BiGRU yields better representations of texts than does [13]'s CNN-based model.",
"Our `No embed' model is akin to their CASCADE with no contextual features, which achieves only 0.66 on the full balanced corpus and 0.70 on the r/politics balanced dataset.",
"Both numbers are well behind our `No embed'.",
"Unfortunately, we do not have space for a fuller study of the similarities and differences between our model and CASCADE.",
"Both of our methods for representing authors perform well.",
"This is perhaps especially striking for the unbalanced experiments, where the percentage of sarcastic comments is tiny (Table REF ).",
"The two methods perform differently on individual forums than on the full dataset.",
"For the r/politics and r/AskReddit communities, the Bayesian priors give the best results.",
"The situation is reversed for the full dataset, where the high-dimensional embeddings outperform the Bayesian priors.",
"This likely reflects two interacting factors.",
"First, with smaller, more focused forums, it is harder to learn good author embeddings, so the simple prior is more reliable.",
"Second, on the full dataset, there are more examples, and also more complex interactions between authors and their texts, so the added representational power of the embeddings proves justified."
],
[
"Qualitative comparisons",
"Table REF provides example predictions from the different models.",
"Each example is taken from the holdout set of a run in which all three models were trained on the same training set and evaluation was conducted on the same holdout set.",
"For both sarcastic and non-sarcastic comments, author modeling can be helpful for disambiguation.",
"For instance, in examples 1 and 2, omitting author modeling led to incorrect predictions, but including the frequency of the author's sarcasm use alone was enough to change the prediction from incorrect to correct.",
"In cases like examples 3 and 4, where the Bayesian prior was insufficient, including a model of the author's individualized patterns of sarcasm was much more powerful.",
"That said, the more complex embedding model can misfire, as in example 5, where the simpler models make a correct prediction but it does not.",
"This appeared to happen more for non-sarcastic examples, where the embedding model would occasionally strongly influence the predicted probability of sarcasm upward.",
"Evidently, authors have more individualized patterns of sarcasm than of non-sarcasm.",
"Judging by the relative performance of the Bayesian and multidimensional-embedding models (Table REF ), the multidimensional model wins more disagreements than it loses with the Bayesian model when there is more training data available.",
"However, when there is not, it overfits to such a degree that its predictions of authors' sarcasm patterns are less useful than the Bayesian approach.",
"This suggests a future direction of exploration: the most useful model of all may be one that expands in complexity for authors with more examples available, and shrinks for those who have fewer."
],
[
"Conclusion",
"This paper evaluated two data-driven methods for modeling the role of the author in sarcasm detection.",
"Both prove effective.",
"As shown by [13], similar techniques can be extended to other aspects of the context.",
"While our experiments did not support adding these representations, we think listeners rely on them as well, so additional computational modeling work here is likely to prove fruitful."
]
] | 1808.08470 |
[
[
"Network Inference from Temporal-Dependent Grouped Observations"
],
[
"Abstract In social network analysis, the observed data is usually some social behavior, such as the formation of groups, rather than an explicit network structure.",
"Zhao and Weko (2017) propose a model-based approach called the hub model to infer implicit networks from grouped observations.",
"The hub model assumes independence between groups, which sometimes is not valid in practice.",
"In this article, we generalize the idea of the hub model into the case of grouped observations with temporal dependence.",
"As in the hub model, we assume that the group at each time point is gathered by one leader.",
"Unlike in the hub model, the group leaders are not sampled independently but follow a Markov chain, and other members in adjacent groups can also be correlated.",
"An expectation-maximization (EM) algorithm is developed for this model and a polynomial-time algorithm is proposed for the E-step.",
"The performance of the new model is evaluated under different simulation settings.",
"We apply this model to a data set of the Kibale Chimpanzee Project."
],
[
"Introduction",
"A network is a data structure consisting of nodes (vertices) connected by links (edges).",
"A network with $n$ nodes can be represented by an $n \\times n$ adjacency matrix $A=[A_{ij}]$ , where $A_{ij}>0$ if there is an edge between nodes $i$ and $j$ and $A_{ij}=0$ otherwise.",
"A network $A$ can be weighted, where $A_{ij}$ measures the link strength between node $i$ and $j$ .",
"Network analysis has drawn increasing attention in a number of fields such as social sciences [35], [23], physics [2], [28], computer science [13], biology [33] and statistics [4], [18].",
"Traditionally, statistical network analysis deals with inferences concerning parameters of an observed network, i.e., an observed adjacency matrix $A$ (see [27], [15], [39] for reviews of models and techniques for analyzing observed networks).",
"In this article, we focus on the case that the network is unobserved and to be estimated.",
"What we do observe is a collection of subsets of nodes.",
"Each subset is called a group by [40] and a data set consisting of such groups is referred to as grouped data.",
"We continue to use the term grouped data in this article.",
"To better explain the structure of grouped data, we introduce some notations.",
"For a set of $n$ individuals, $V=\\lbrace v_1,..., v_n\\rbrace $ , we observe $T$ subsets $V^{1},...,V^{T}$ at times $1,...,T$ , called groups.",
"Each observed subset $V^{t}$ can be represented as an $n$ length row vector $G^{t}$ , where $ G_i^{t} = \\left\\lbrace \\begin{array}{l l}1 & \\quad \\textnormal {if v_i\\in V^{t},}\\\\0 & \\quad \\textnormal {otherwise.",
"}\\end{array} \\right.$ Let $G$ be a $T \\times n$ matrix with $G^{t}$ being its rows.",
"For simplicity, we will slightly abuse the notation: we will also refer to the indicator vector $G^{t}$ as a group from now on.",
"In this article, we analyze the grouped data from the so-called social network perspective [26].",
"The grouping behavior of the individuals is presumed to be governed by a latent social network $A$ .",
"The objective of this article is to infer the latent network from the groups being observed.",
"In other words, we aim to estimate the link strength between individuals using the information about their presence in the groups.",
"Remark Throughout this paper, we consider only the case that there exists one and only one group at a time $t$ .",
"The groups at different times can overlap.",
"In fact, it is only plausible to make meaningful inferences of $A$ from $G$ if groups overlap.",
"If all groups are disjoint, the best inference is to use a clique, i.e., a fully connected subgraph to estimate the relationships within each group.",
"[35] discuss grouped data (which they called affiliation networks) as well as some empirical methods and graphical representations for this type of data in Chapter 8 of Social Network Analysis: Methods and Applications.",
"The authors give an illustrative example of six children and three birthday parties (page 299), which is shown in Table REF .",
"By the use of the notation above, $G^2_2=1$ since Drew attended Party 2, but $G^3_2=0$ since Drew did not attend Party 3.",
"Table: Dataset for six children and three birthday parties.",
"Adapted from Fig.",
"8.1, page 299 of .Numerous researchers in social sciences have been interested in grouped data, in particular how to infer social structures from such data.",
"[35] provide a list of such data sets (pages 295-296).",
"For example, [12] collected CEO membership data that consisted of their participation in clubs, cultural boards and corporate boards of directors (see [35] page 755 for the data).",
"As another example (not in [35]), [29] collected a data set of 79 terrorists' presence in meetings, trainings and other events.",
"Grouped data are also popular in the study of social behaviors of animals.",
"Social network analysis (SNA) has become an important tool in this area [37], [8], but direct linkages between animals are often difficult or expensive to record [8] and certain methods used for collecting human network data, such as surveys, are clearly impossible.",
"On the contrary, grouped data such as groups of dolphins [3] or flocks of birds [11] are relatively easier to identify and record.",
"[40] use group data to study novels by treating the characters of a novel appearing in the same paragraph as a group and using the inferred network structure to interpret the relationships between characters.",
"Despite the popularity of group data, existing methods for network inference from grouped data are mainly ad-hoc approaches from the social sciences literature.",
"A simple technique is to count the number of times that a pair of nodes appears in the same group.",
"This measure has been called different names by different authors, e.g., the co-citation matrix in Section 6.4 of [27] or the sociomatrix in Section 8.4 of [35].",
"[40] refer to this measure as the co-occurrence matrix.",
"The half weight index [7] is an alternative approach that uses the conditional frequencies of co-occurrences as estimates.",
"A common difficulty of such methods is that they provide no statistical model to connect these descriptive statistics with the latent network.",
"[40] recently proposed a model-based approach for grouped observations.",
"In the so-called hub model, $G^t$ s are modeled as independently and identically distributed random vectors and there is a central node called hub or group leader in each group, who gathers other members into the group.",
"For example, the hub is the child who hosted the party in Table REF .",
"A crucial assumption made in [40] is that the groups are assumed to be independently generated by the hub model.",
"In some cases, this assumption is reasonable if each group forms spontaneously.",
"The assumption can also be approximately satisfied if researchers collect grouped data with sufficiently long time intervals between observations (see [3] for discussion).",
"The independence assumption however may not be valid in other situations.",
"In most practical situations, the grouped observations are temporal-dependent by default.",
"For example, in a study of animal behavior, researchers may observe the behavior of animals on an hourly or daily basis.",
"In Section , we analyze such a data set consisting of groups of wild chimpanzees studied by the Kibale Chimpanzee Project.",
"It is inappropriate to assume that every group is independent from the previous group.",
"A more plausible point of view is that the group at a particular time is a transformation of the previous group.",
"That is, some new members may join the group and some may leave, but the group maintains a certain level of stability.",
"We generalize the idea of the hub model in order to accommodate temporal dependence between groups.",
"We call the new model the temporal-dependent hub model, or the temporal-dependent model in short.",
"This new model allows for dependency between group leaders as well as between other group members.",
"We explain both dependency assumptions in the next two paragraphs.",
"As in the classical hub model, we assume there is one leader for each group.",
"Leaders however are not sampled independently in the temporal-dependent model, but follow a Markov chain.",
"That is, the probability of a certain node being the current leader depends on the leader in the previous group.",
"For other group members, we consider the following two cases to make the model flexible enough.",
"If the current leader is inside the previous group, then we treat this group as a transformation of the previous one.",
"If the new leader is from outside the previous group (e.g., some event occurs and completely breaks the previous group) then we treat this group as the start of a new segment.",
"In this case, the leader will select the group members as in the classical hub model, i.e., independently of whether or not they were members of the previous group.",
"As shown in Section , the temporal-dependent hub model can be viewed as a generalization of the hidden Markov model (HMM) when the group leaders are latent.",
"An efficient algorithm is thus developed for model fitting.",
"Furthermore, the temporal-dependent hub model provides estimates of the elements of the adjacency matrix with lower mean squared errors according to numerical studies in Section .",
"Finally, we discuss some related works.",
"First, the temporal-dependent hub model is fundamentally different from many existing models for dynamic networks, such as the preferential attachment model [2], discrete/continuous time Markov models [32], [17], etc.",
"In these works, the observed data are snapshots of the network at different time points.",
"In this article, the unknown parameters are a single latent network and the observations are groups with temporal-dependent structures.",
"Second, there are recent studies on estimating latent networks or related latent structures in dynamic settings, but from data structures that are different from groups.",
"[16] propose a Bayesian model to infer latent relationships between people from a special type of data – the evolution of people's language over time.",
"[30] propose a latent process model for dynamic relational network data.",
"Such a data set consists of binary interactions at different times.",
"[5] propose a nonparametric Bayesian approach for estimating latent communities from a similar data type.",
"The grouped data we consider in this article are more complicated than binary interactions in the sense that, unlike a linked pair, the links within a group consisting of more than two members are unknown.",
"Third, there are other interesting works on modeling latent social networks from survey data and such data only provide partial information of a latent network.",
"These survey data also have different structures from grouped data.",
"[24] propose a latent surface model for aggregated relational data collected by asking respondents the number of connections they have with members of a certain subpopulation.",
"In this work, the network structure for the population is latent.",
"[1] fit exponential-family random graph models (ERGMs) to latent heterosexual partnership networks, with degree distributions and mixing totals being sufficient statistics in the exponential family.",
"Those statistics for the underlying population are inferred from cross-sectional survey data.",
"[22] fit ERGMs to egocentrically sampled data, which provide information about respondents and anonymized information on their network neighbors.",
"Model The classical hub model We briefly state the generating mechanism of the classical hub model [40].",
"The hub model assumes one leader for each group.",
"The leader of $G^{t}$ is denoted by $z^t$ .",
"Under the hub model, each group $G^{t}$ is independently generated by the following two steps.",
"The group leader is sampled from a multinomial distribution with parameter $\\rho =(\\rho _1,...,\\rho _n)$ , i.e., $\\mathbb {P}(z^t=i)=\\rho _i$ , with $\\sum _i \\rho _i=1$ .",
"The group leader, $v_i$ , will choose to include $v_j$ in the group with probability $A_{ij}$ , i.e., $ \\mathbb {P}(G_j^{t}=1|z^t=i)=A_{ij}$ .",
"Generating mechanism of the temporal-dependent hub model The hub model assumes that all the groups are generated independently across time.",
"In practice, it is more natural to model the groups as temporal-dependent observations.",
"We first explain the idea of the generating mechanism of temporal-dependent groups and then give the formal definition.",
"We generalize the idea of the hub model into the temporal-dependent setting.",
"Specifically, we assume there is only one leader $z^t$ at each time who brought the group together, but the group at time $t$ depends on the previous group, which is different from the classical hub model.",
"At time $t=1$ , the group is generated from the classical hub model.",
"For $t=2,...,T$ , the group leader $z^t$ can remain the same as the previous leader or change to a new one.",
"We assume that the leader $z^t$ will remain as $z^{t-1}$ with a higher probability than the probability of changing to any other node.",
"If the new leader is outside the previous group, then the current group is considered the start of a new segment and is generated by the classical hub model.",
"It is worth noting that technically, the generation of the new group however still depends on the previous group.",
"This will become clearer after we introduce the likelihood function.",
"For the case that the new leader is inside the previous group – that is, if the leader remains unchanged, or the leader changes but is still a member of the previous group – we propose the following In-and-Out procedure: for any node $v_j$ being in the previous group, it will remain in $G^{t}$ with a probability higher than $A_{z^t,j}$ – the probability in the classical hub model.",
"On the contrary, for any node $v_k$ not being in the previous group, it will enter $G^{t}$ with a probability lower than $A_{z^t,k}$ .",
"Intuitively, this In-and-Out procedure assumes that when a group forms, it will maintain a certain level of stability.",
"We now give the formal definition of the generating mechanism as follows: Step 1: (Classical hub model).",
"When $t=1$ , $G^{t}$ is generated by the following two substeps.",
"1) The leader is sampled from a multinomial distribution with parameter $\\rho =(\\rho _1,...,\\rho _n)$ , i.e., $ \\mathbb {P}(z^{t}=i)=\\rho _i \\overset{\\Delta }{=} \\frac{\\exp (u_i)}{\\sum _{k=1}^n \\exp (u_k)}, $ where $u_i \\in \\mathbb {R}$ for $i=1,...,n$ .",
"2) The leader $v_i$ will choose to include $v_j$ in the group with probability $A_{ij}$ , i.e., $ \\mathbb {P}(G_j^{t}=1|z^{t}=i)=A_{ij}$ , where $A_{ii} \\equiv 1$ and $A_{ij}= A_{ji}\\overset{\\Delta }{=}\\frac{\\exp (\\theta _{ij})}{1+\\exp ( \\theta _{ij})}.",
"$ Here, $\\theta _{ii}=\\infty $ for $i=1,...,n$ and $\\theta _{ij} \\in \\overline{\\mathbb {R}}$ for $i \\ne j$ .",
"We allow some $\\theta _{ij}$ to be $\\pm \\infty $ so that the corresponding $A_{ij}$ can be 1 or 0.",
"Step 2: (Leader change).",
"For $t=2,...,T$ , $\\mathbb {P}(z^t=i| z^{t-1}) = \\frac{\\exp (u_i+\\alpha I(z^{t-1}=i))}{\\sum _{k=1}^n \\exp (u_k+\\alpha I(z^{t-1}=k))},$ where $\\alpha \\in \\mathbb {R}$ .",
"Step 3: (In-and-Out procedure).",
"For $t=2,...,T$ , given $v_i$ being the leader, $G^{t}$ is generated by the following mechanism: If $v_i$ is not within $G^{t-1}$ , then it will include each $v_j$ in the group with probability $A_{ij}$ ; otherwise, see below: 1) If $G^{t-1}_j=1$ , $v_i$ will include $v_j$ in the group with probability $B_{ij}=B_{ji}\\overset{\\Delta }{=} \\frac{\\exp ( \\theta _{ij}+\\beta )}{1+\\exp ( \\theta _{ij}+\\beta )},$ where $\\beta \\in \\mathbb {R}$ .",
"2) If $G^{t-1}_j=0$ , $v_i$ will include $v_j$ in the group with probability $C_{ij}=C_{ji}\\overset{\\Delta }{=} \\frac{\\exp ( \\theta _{ij}+\\gamma )}{1+\\exp ( \\theta _{ij}+\\gamma )},$ where $\\gamma \\in \\mathbb {R}$ .",
"For clarity of notation, we now give the vector/matrix form.",
"Define $z=(z^{1},..., z^{T})$ , $u=(u_1,...,u_n)$ and $\\rho =(\\rho _1,...,\\rho _n)$ .",
"Define $\\theta =[\\theta _{ij}]_{1\\le i \\le n, 1\\le j \\le n}$ , $A=[A_{ij}]_{1\\le i \\le n, 1\\le j \\le n}$ , $B=[B_{ij}]_{1\\le i \\le n, 1\\le j \\le n}$ and $C=[C_{ij}]_{1\\le i \\le n, 1\\le j \\le n}$ .",
"Furthermore, we assume $\\theta $ , $A$ , $B$ and $C$ to be symmetric in order to avoid any issue of identifiability (see the discussion in [40]).",
"Remark In the definition above, $u_i$ and $\\theta _{ij}$ are simply a reparameterization of $\\rho _i$ and $A_{ij}$ in exponential form.",
"This is to make optimization more convenient, since log-likelihood is convex under this parametrization.",
"The parameters $\\alpha $ , $\\beta $ and $\\gamma $ characterize the dependency between the groups.",
"$\\alpha $ is the adjustment factor, which controls the probability that a leader in the previous group remains as a leader.",
"$\\beta $ is the adjustment factor for nodes being inside the previous group.",
"And $\\gamma $ is the adjustment factor for nodes being from outside the previous group.",
"We do not enforce $\\alpha >0$ , $\\beta >0$ and $\\gamma <0$ in the model fitting.",
"Instead, we test these assumptions for the data example in Section .",
"The parameters $A$ , $\\beta $ and $\\gamma $ are identifiable.",
"The key observation is that the identifiability of $A$ can simply be obtained by $G^1$ since the first group only depends on $\\rho $ and $A$ .",
"This is essentially the identifiability of the classical hub model.",
"The proof is given by Theorem 1 in [40], under the condition of $A$ being symmetrical.",
"With the “baseline” $A$ being separately identified, the two adjustment factors $\\beta $ and $\\gamma $ are accordingly identifiable.",
"The parameters $(u_1,...,u_n)$ are non-identifiable under this parametrization, since $(u_1+\\delta ,...,u_n+\\delta )$ gives the same likelihood.",
"We will discuss the solution to this problem in Section after introducing the algorithm.",
"Likelihood For notational convenience in the likelihood, we indicate the leader in group $G^t$ by an $n$ length vector, $S^{t}$ , where $ S_i^{t} = \\left\\lbrace \\begin{array}{l l}1 & \\quad \\text{if $z^t=i$},\\\\0 & \\quad \\text{otherwise}.\\end{array} \\right.$ Only one element of $S^{t}$ is allowed to be 1.",
"$S^{t}$ is simply another representation of $z^{t}$ .",
"Let $S$ be a $T \\times n$ matrix, with $S^{t}$ being its rows.",
"Clearly, $ \\lbrace S^1,...,S^T \\rbrace $ is a Markov chain according to the generating mechanism.",
"Let $\\Phi _{ij}=\\mathbb {P}(z^t=i|z^{t-1}=j)$ be the transition probability and $\\Phi =[\\Phi _{ij}]_{n\\times n}$ .",
"We summarize all introduced notations in Table REF .",
"Table: Summary of NotationWe now give the joint log-likelihood of $S$ and $G$ for the model defined in the previous subsection: $& \\log \\mathbb {P} (S,G|\\alpha , \\beta , \\gamma , \\theta , u) \\nonumber \\\\= & \\sum _{i=1}^n S_i^{1} \\log \\rho _i +\\sum _{t=2}^T \\sum _{i=1}^n \\sum _{j=1}^n S_i^{t}S_j^{t-1} \\log \\Phi _{ij} \\nonumber \\\\&+ \\sum _{i=1}^n \\sum _{j=1}^n \\left\\lbrace S_i^{1} G_j^{1} \\log A_{ij}+S_i^{1} (1-G_j^{1}) \\log (1-A_{ij}) \\right\\rbrace \\nonumber \\\\&+ \\sum _{t=2}^T \\sum _{i=1}^n \\sum _{j=1}^n \\left\\lbrace S_i^{t} (1-G_i^{t-1}) G_j^{t} \\log A_{ij}+S_i^{t}(1-G_i^{t-1}) (1-G_j^{t}) \\log (1-A_{ij}) \\right\\rbrace \\nonumber \\\\& + \\sum _{t=2}^T \\sum _{i=1}^n \\sum _{j=1}^n \\left\\lbrace S_i^{t}G_i^{t-1} G_j^{t-1} G_j^{t} \\log B_{ij}+ S_i^{t}G_i^{t-1} G_j^{t-1} (1-G_j^{t}) \\log (1-B_{ij}) \\right\\rbrace \\nonumber \\\\& + \\sum _{t=2}^T \\sum _{i=1}^n \\sum _{j=1}^n \\left\\lbrace S_i^{t}G_i^{t-1} (1-G_j^{t-1}) G_j^{t} \\log C_{ij}+ S_i^{t}G_i^{t-1} (1-G_j^{t-1}) (1-G_j^{t}) \\log (1-C_{ij}) \\right\\rbrace .",
"$ Note that $\\alpha , \\beta , \\gamma , \\theta $ and $u$ are essentially the parameters of this model and $\\rho $ , $\\Phi $ , $A$ , $B$ and $C$ are their functions.",
"Despite its length, Equation (REF ) has a clear structure.",
"The 1st line gives the log-likelihood of $S$ .",
"The 2nd line gives the log-likelihood of $G^1$ given $S^1$ .",
"The 3rd line gives the log-likelihood of $G^t$ given that the current leader $z^t$ is outside the previous group $G^{t-1}$ .",
"The 4th and 5th lines give the log-likelihood of $G^t$ given that $z^t$ is inside $G^{t-1}$ , based on the In-and-Out procedure.",
"Equivalent to (REF ), we can write the likelihood as a product of conditional probabilities: $\\mathbb {P} (S,G) = \\mathbb {P} (S^1) \\mathbb {P}(G^1|S^1) \\prod _{t=2}^T \\mathbb {P}(S^t|S^{t-1}) \\prod _{t=2}^T \\mathbb {P}(G^t|S^{t},G^{t-1}).$ This factorization can be represented by a Bayesian network (Figure REF ), where a node represents a variable and a directed arc is drawn from node $X$ to node $Y$ if $Y$ is conditioned on $X$ in the factorization.",
"(Refer to [19] for a comprehensive introduction to Bayesian networks).",
"This Bayesian network should not be confused with the latent network $A$ – the former is a representation of the dependency structure between variables while the latter reflects the relationships between the group members.",
"Furthermore, the group leaders $z^1,...,z^T$ are assumed to be latent (as are $S^1,...,S^T$ ) since in many applications only the groups themselves are observable.",
"Figure: A Bayesian network representing the temporal-dependent hub model.",
"Nodes with dark colors indicate the observed variables.",
"Model fitting In this section, we propose an algorithm to find the maximum likelihood estimators (MLEs) for $\\alpha ,\\beta ,\\gamma ,u$ and $\\theta $ .",
"With $S$ being the latent variables, an expectation-maximization (EM) algorithm will be used for this problem.",
"The EM algorithm maximizes the marginal likelihood of the observed data, which in our case is $G$ , by iteratively applying an E-step and an M-step.",
"Let $\\alpha ^{\\textnormal {old}},\\beta ^{\\textnormal {old}},\\gamma ^{\\textnormal {old}},u^{\\textnormal {old}}$ and $\\theta ^{\\textnormal {old}}$ be the estimates in the current iteration.",
"In the E-step, we calculate the conditional expectation of the complete log-likelihood given $G$ under the current estimate.",
"That is, $Q \\overset{\\Delta }{=} Q(\\alpha ,\\beta ,\\gamma ,u,\\theta | \\alpha ^{\\textnormal {old}},\\beta ^{\\textnormal {old}},\\gamma ^{\\textnormal {old}},u^{\\textnormal {old}},\\theta ^{\\textnormal {old}})= \\mathbb {E}_{\\alpha ^{\\textnormal {old}},\\beta ^{\\textnormal {old}},\\gamma ^{\\textnormal {old}},u^{\\textnormal {old}},\\theta ^{\\textnormal {old}}} \\left[ \\log \\mathbb {P} (S,G) | G \\right].$ In the M-step, we maximize this conditional expectation with respect to the unknown parameters.",
"That is, $(\\alpha ^{\\textnormal {new}},\\beta ^{\\textnormal {new}},\\gamma ^{\\textnormal {new}},u^{\\textnormal {new}},\\theta ^{\\textnormal {new}} ) = \\operatornamewithlimits{arg\\,max}_{\\alpha ,\\beta ,\\gamma ,u,\\theta } \\,\\, Q(\\alpha ,\\beta ,\\gamma ,u,\\theta | \\alpha ^{\\textnormal {old}},\\beta ^{\\textnormal {old}},\\gamma ^{\\textnormal {old}},u^{\\textnormal {old}},\\theta ^{\\textnormal {old}}).$ It has been proved by [38] that the EM algorithm converges to a local maximizer of the marginal likelihood.",
"(Refer to [25] for a comprehensive introduction to this algorithm).",
"We now give details of the two steps in our context.",
"E-step Since the complete log-likelihood $\\log \\mathbb {P} (S,G)$ is a linear function of $S_i^t \\,\\, (t=1,...,T;i=1,...,n)$ and $S_i^{t}S_j^{t-1} \\,\\,(t=2,...,T;i=1,...,n;j=1,...,n) $ , the computation of its conditional expectation is equivalent to calculating $\\mathbb {P}(S_i^t=1|G)$ and $\\mathbb {P}(S_i^t=1,S_j^{t-1}=1|G)$ .",
"From now on, all conditional probabilities are defined under the current estimates.",
"A brute-force calculation of these probabilities, such as $\\mathbb {P}(S_i^t=1|G) = \\mathbb {P} (z^t=i|G)=\\frac{\\sum _{z^1}\\cdots \\sum _{z^{t-1}}\\sum _{z^{t+1}}\\cdots \\sum _{z^t} \\mathbb {P}(z^1,...,z^{t-1},z^t=i,z^{t+1},...,z^T,G)}{\\mathbb {P}(G)},$ is infeasible since the numerator involves a sum of $n^{T-1}$ terms.",
"This is because $G^1,...,G^T$ are not independent according to our model.",
"An efficient algorithm is needed for all practical purposes.",
"The temporal-dependent hub model is similar to the hidden Markov model (HMM) (Figure REF ).",
"A polynomial-time algorithm for this model, called the forward-backward algorithm, was developed for computing the conditional probabilities.",
"See [31], [14] for tutorials on HMMs and this algorithm.",
"In the HMM, the observed variable at time $t$ only depends on the corresponding hidden state.",
"But in our model, $G^t$ depends on both the current leader $z^t$ and the previous group $G^{t-1}$ .",
"We develop a new forward-backward algorithm for our model, which has more steps than the original algorithm but is also polynomial-time.",
"We describe the algorithm here (see the Appendix for detailed derivation and justification).",
"Define $a=[a_i^t],b=[b_i^t]$ and $c=[c_i^t]$ as $T\\times n$ matrices.",
"These matrices are computed by the following recursive procedures.",
"$a_i^1 & = \\mathbb {P}(z^1=i,G^1) \\quad (i=1,...,n).",
"\\\\a_i^t & =\\sum _{k=1}^n a^{t-1}_k \\Phi _{ik} \\mathbb {P}(G^t|z^t=i,G^{t-1}) \\quad (t=2,...,T;i=1,...,n).",
"\\\\b_i^T & = 1 \\quad (i=1,...,n).",
"\\\\b_i^t & =\\sum _{k=1}^n b^{t+1}_k \\Phi _{ki} \\mathbb {P}(G^{t+1}|z^{t+1}=k, G^t) \\quad (t=T-1,...,1;i=1,...,n).",
"\\\\c_i^T & = \\mathbb {P}(G^T|z^{T}=i,G^{T-1}) \\quad (i=1,...,n).",
"\\\\c_i^t & = \\sum _{k=1}^n c^{t+1}_k \\Phi _{ki} \\mathbb {P}(G^t|z^t=i,G^{t-1}) \\quad (t=T-1,...,2;i=1,...,n).$ The matrices $a$ , $b$ and $c$ should not be confused with the matrices $A$ , $B$ and $C$ introduced in Section .",
"The symbols are case-sensitive throughout the paper.",
"With these quantities, $\\mathbb {P}(S_i^t=1|G) & = \\frac{a_i^t b_i^t}{\\sum _k a_k^t b_k^t} \\quad (t=2,...,T;i=1,...,n).",
"\\\\\\mathbb {P}(S_i^t=1,S_j^{t-1}=1|G) & = \\frac{a_j^{t-1} \\Phi _{ij} c_i^t}{\\sum _{kl} a_l^{t-1} \\Phi _{kl} c_k^t} \\quad (t=2,...,T;i=1,...,n;j=1,...,n).$ The complexity of this algorithm is $O(Tn^2)$ .",
"Note that the first row of $c$ is undefined but also unused.",
"Also note that the elements of $a,b$ and $c$ will quickly vanish as the recursions progress.",
"Therefore, we renormalize each row to sum to one at each step.",
"It can easily be verified that this normalization does not affect the conditional probabilities.",
"Finally, we emphasize that this algorithm gives the exact values of the conditional probabilities in a fixed number of steps – i.e., it is not an approximate or iterative method.",
"M-step The M-step is somewhat routine compared to the E-step.",
"First, it is clear that $\\lbrace \\alpha ,u \\rbrace $ and $\\lbrace \\beta ,\\gamma ,\\theta \\rbrace $ can be handled separately.",
"We apply the coordinate ascent method (see [6] for a comprehensive introduction) to iteratively update $\\alpha $ and $u$ , as well as $\\beta ,\\gamma $ and $\\theta $ .",
"Since the complete log-likelihood is concave and so is $Q$ , coordinate ascent can guarantee a global maximizer.",
"At each step, we optimize the log-likelihood over parameter one by one with the other parameters being fixed.",
"The procedure is repeated until convergence.",
"At each step, we use the standard Newton-Raphson method to solve each individual optimization problem.",
"Specifically, for a parameter $\\phi $ (here $\\phi $ can represent $\\alpha $ , $\\beta $ , $\\gamma $ , $u_i$ or $\\theta _{ij} \\,\\, (i < j)$ ), the estimate at $(m+1)$ -th iteration is updated by the following formula given its estimate at $m$ -th iteration: $\\hat{\\phi }_{m+1}=\\hat{\\phi }_m-\\left( \\frac{\\partial ^2 Q}{\\partial \\phi ^2 } \\bigg |_{\\phi =\\hat{\\phi }_m} \\right)^{-1} \\left( \\frac{\\partial Q}{\\partial \\phi } \\bigg |_{\\phi =\\hat{\\phi }_m}\\right).",
"$ The calculation of these derivatives is straightforward but tedious, so we provide the details in the Appendix.",
"As shown in Section REF , the model is not identifiable with respect to $u$ .",
"A standard solution to this problem is to set some $u_i \\equiv 0$ .",
"But it does not work for our case.",
"This is because for small data sets, some $\\hat{\\rho }_i$ estimated by the EM algorithm may be zero, implying that $v_i$ never became the leader.",
"Furthermore, these zero $\\rho _i$ cannot be predetermined since the leaders are unobserved.",
"We observe that without constraint on $u_i$ , the algorithm converges to different $\\hat{u}$ with different initial values, but the corresponding $\\hat{\\rho }$ will be the same.",
"Therefore, identifiability is not an issue for model fitting.",
"Initial value As with many optimization algorithms, the EM algorithm is not guaranteed to find the global maximizer.",
"Ideally, one should use multiple random initial values and find the best solution by comparing the marginal likelihoods $\\mathbb {P}(G)$ under the corresponding estimates.",
"In principle, $\\mathbb {P}(G)$ can be computed by $\\sum _k a_k^t b_k^t$ , as shown in Section REF .",
"But the marginal likelihood vanishes quickly, even with a moderate $T$ .",
"Note that we cannot renormalize $a$ and $b$ for the purpose of computing $\\mathbb {P}(G)$ .",
"Therefore, we use the half weight index [9], [7] as the initial value of $A$ , which is defined by $H_{ij} = \\frac{2\\sum _t G_i^{t} G_j^{t}}{\\sum _t G_i^{t}+\\sum _t G_j^{t}}.$ This measure estimates the conditional probability that two nodes co-occur given that one of them is observed, which is a reasonable initial guess of the strength of links.",
"Furthermore, we use zero for the initial values of $\\alpha , \\beta $ and $\\gamma $ , and $\\sum _{t} G_i^t/T $ for the initial value of $\\rho _i$ .",
"Simulation studies In all simulation studies, we fix the size of the network to be $n=50$ and set $\\beta =3$ and $\\gamma =-1$ .",
"We generate $u_i$ as independently and identically distributed variables with $N(0,2)$ and $\\rho _i= u_i/\\sum _k u_k $ .",
"The parameters $\\theta _{ij}\\,\\, (i<j)$ are generated independently with $N(-2,1)$ .",
"We generate $\\theta _{ij}$ in this way to control the average link density of the network ($\\approx $ 0.12), which is more realistic than a symmetric setting, i.e., $\\theta _{ij} \\sim N(0,1)$ .",
"For clarification, we will not use the prior information on $u$ and $\\theta $ in our estimating procedure.",
"That is, we still treat $u$ and $\\theta $ as unknown fixed parameters in the algorithm.",
"We generate them as random variables for the whole purpose of adding more variations to the parameter setup in our study.",
"We consider three levels of $\\alpha =\\log ((n-1)/2),\\log (n-1),\\log (2(n-1))$ , which correspond to a leader from the previous group remaining unchanged in the current group with probabilities $1/3,1/2,2/3$ on average.",
"For each $\\alpha $ we try five different sample sizes, $T=1000,1500, 2000, 2500$ and 3000.",
"Figure: Average RMSEs (across 100 replicates) for the estimated AA from the classical hub model (HM) and the temporal-dependent hub model (TDHM).Table: Average RMSEs for the estimated AA from the classical hub model (HM) and the temporal-dependent hub model (TDHM).",
"Standard deviations ×10 3 \\times 10^3 in parentheses.Figure REF shows the average root of mean squared errors (RMSEs) for the estimated $A$ over 100 replicates.",
"For each simulation, we compare two methods, the classical hub model (HM) and the temporal-dependent hub model (TDHM).",
"We assume the leaders are unknown under both models.",
"Table REF provides the same information (with standard deviations) in numerical form.",
"From Figure REF and Table REF , our first observation is simply that the RMSEs decrease as sample sizes increase, which is consistent with common sense in statistics.",
"Second, the RMSEs for all the parameters increase as $\\alpha $ increases.",
"This phenomenon can be interpreted as follows: with a larger value of $\\alpha $ , the correlation between adjacent groups becomes stronger and hence the effective sample size becomes smaller.",
"The ratio of the sample size to the number of parameters decreases with $\\alpha $ , which makes inferences more difficult.",
"Third, the temporal-dependent hub model always outperforms the hub model.",
"Moreover, the discrepancy between the temporal-dependent hub model estimates and the corresponding hub model estimates becomes larger as $\\alpha $ increase.",
"This is because the behavior of the temporal-dependent model deviates more from the classical hub model as $\\alpha $ increases.",
"The standard deviations and means show a similar trend.",
"That is, the standard deviations decrease as $n$ increases and $\\alpha $ decreases.",
"The standard deviations for the temporal-dependent hub model estimates are comparable to or slightly smaller than those of the hub model estimates.",
"A data example of group dynamics in chimpanzees Behavioral ecologists become increasingly interested in using social network analysis to understand social organization and animal behavior [3], [37], [8], [11].",
"The social relationships are usually inferred by using certain association metrics (e.g., the half weight index) on grouped data.",
"As indicated in the Introduction however, it is unclear how the inferred network relates to the observed groups without specifying a model.",
"In this section, we study a data set of groups formed by chimpanzees by the temporal-dependent hub model.",
"This data set is compiled from the results of the Kibale Chimpanzee Project, which is a long-term field study of the behavior, ecology and physiology of wild chimpanzees in the Kanyawara region of Kibale National Park, southwestern Uganda (https://kibalechimpanzees.wordpress.com/).",
"Our analysis focuses on grouping behavior.",
"We analyze the grouped data collected from January 1, 2009 to June 30, 2009 [20].",
"The group identification was taken at 1 p.m. daily during this time period.",
"If there is no group observed at 1 p.m. for a given day, it is not included in the data.",
"Only one group is observed at 1 p.m. in 75.29% of the remaining days over this period of six months.",
"In the other days, multiple groups (usually two) are observed at 1 p.m. For these cases, we keep the group that has the most overlap with the previous group in our analysis.",
"We use the Jaccard index to measure the overlap between two groups $G^{t-1}$ and $G^{t}$ , $J(G^{t-1},G^{t})=\\frac{\\sum _{j=1}^n G^{t-1}_j G^{t}_j }{\\sum _{j=1}^n [ n-(1-G^{t-1}_j)(1-G^{t}_j) ]},$ where the numerator is the size of the intersection of two groups and the denominator is the size of their union.",
"One may refer to [23] for an introduction to this measure.",
"Moreover, five chimpanzees never appear in any group and thus are removed.",
"After the preprocessing, the data set contains 170 groups with 40 chimpanzees.",
"Figure REF illustrates the data set in grayscale with rows representing the groups over time and columns representing the chimpanzees.",
"Black indicates $G^{t}_i=1$ at location $(t,i)$ while white indicates $G^{t}_i=0$ .",
"The pattern in Figure REF clearly demonstrates the existence of dependency between groups.",
"Figure REF also shows the inferred grouped leaders indicated in red with the inferred segments separated by blue lines.",
"By the inferred grouped leader for $G^t$ , we mean that the chimpanzee with the highest posterior probability will be the leader given $G^t$ .",
"As shown in Figure REF , the leaders retain a certain level of stability, which is consistent with the estimates of $\\alpha $ ($=1.7291$ ).",
"Also, recall that by our definition, a new segment starts if the current leader is not within the previous group.",
"From Figure REF , the inferred segments are coincident with the visualization of the data set.",
"The estimated values of the adjustment factors, $\\hat{\\beta }=2.5703$ and $\\hat{\\gamma }=-0.1922$ .",
"The magnitude of $\\hat{\\beta }$ is larger than that of $\\hat{\\gamma }$ , which suggests individuals have a stronger tendency to join a group than leave a group.",
"In other words, the groups may start with small size and grow larger over time.",
"This phenomenon is shown (Figure REF ).",
"Figure: Visualization of the chimpanzee data.",
"The 170 rows represent the groups over time and the 40 columns represent chimpanzees.",
"Gray indicates the presence of the group membership.",
"Red indicates the inferred group leaders.",
"The blue lines separate the inferred segments.Figure: Grayscale plot for the estimated adjacency matrices from the chimpanzee data by the classical hub model and the temporal-dependent hub model.",
"The rows and the columns represent the 40 chimpanzees.",
"Darker colors indicate stronger relationships.",
"The red blocks indicate the biological clusters of chimpanzees.Figure REF shows the result of estimated adjacency matrices by the classical hub model and the temporal-dependent hub model.",
"As in the previous figure, the darker color indicates a higher value of $\\hat{A}_{ij}$ .",
"The red blocks indicate clusters of chimpanzees in a biological sense.",
"The first cluster consists of 12 adult males and each of the other nine clusters consists of an adult female and its children.",
"From the estimates by both the classical hub model and the temporal-dependent hub model, there are strong connections within these biological clusters.",
"Both estimates suggest that in this data set of chimpanzees, adult males usually do activities together but females usually stay with their children.",
"The two estimated adjacency matrices are different, however.",
"Generally speaking, without properly considering the temporal-dependence between groups, the estimates of the relationships between individuals by the classical hub model can be biased.",
"That is, an individual may choose to stay within or out of a group not solely based on its relationship with the group leader but also because of the inertia.",
"The overall graph density $(= 0.2286)$ of the estimated network by the classical hub model is larger than the corresponding value $(= 0.1973)$ of the temporal-dependent hub model.",
"This is consistent with the fact that the magnitude of $\\hat{\\beta }$ is larger than the magnitude of $\\hat{\\gamma }$ .",
"Since the classical hub model does not incorporate the adjustment factors, bias is introduced to certain $\\hat{A}_{ij}$ so the model can match the overall frequency of occurrences for the individuals.",
"The significance of $\\alpha $ , $\\beta $ and $\\gamma $ is tested by the parametric bootstrap method [10].",
"Specifically, we generate 5000 independent data sets from the fitted temporal-dependent hub model to the original data and compute the MLEs for each simulated data set.",
"The parametric bootstrap was applied to HMMs and showed a good performance [34].",
"Figure REF shows the histograms of the MLEs for $\\alpha $ , $\\beta $ and $\\gamma $ .",
"The 95% bootstrap confidence intervals for $\\alpha $ , $\\beta $ and $\\gamma $ are (1.2177, 1.9774), (2.0710, 2.8944) and (-0.5208, 0.0410), which shows that the effects of $\\alpha $ and $\\beta $ are significant while $\\gamma $ is not at the 0.05 significance level.",
"This further supports the observation in the previous paragraph – chimpanzees have a stronger tendency to join a group than to leave a group in this data set.",
"Figure: Histograms of estimates from parametric bootstrap samples.",
"Red lines indicate the estimated values from the original data set.",
"Summary and discussion In this article, we generalize the idea of the hub model and propose a novel model for temporal-dependent grouped data.",
"This new model allows for dependency between groups.",
"Specifically, the group leaders follow a Markov chain and a group is either a transformation of the previous group or a new start, depending on whether the current leader is within the previous group.",
"An EM algorithm is applied to this model with a polynomial-time algorithm being developed for the E-step.",
"The setup of our model is different from some work on estimating time-varying networks by graphical models, e.g., [21] for discrete data and [41] for continuous data.",
"These papers assume that the observations are independent and the latent network changes smoothly or is piecewise constant.",
"In this paper, we instead focus on the dependence between groups.",
"Ideally, both aspects – dependence between groups and changes in networks – need to be considered in modeling.",
"This should be plausible when the sample size, i.e., the number of observed groups, is large.",
"When the sample size is moderate (as in the chimpanzee data set), the length of the time interval between observation plays a key role in determining which aspect is more important.",
"If the time interval is short, then dependence between groups is significant but the changes in the latent network are likely to be minor since the overall time window is not long.",
"On the contrary, the dependence between groups becomes weak when the time interval is long.",
"For future work, we plan to study the time-varying effect on the latent network for the grouped data.",
"When changepoints exist, a single network cannot accurately represent the link strengths in different time windows.",
"A change-point analysis for temporal-dependent grouped data is an intriguing research topic.",
"Alternative approaches may be based on penalizing the difference between networks at adjacent time points, although careful investigation is required to determine how tractable these methods are for temporal-dependent groups.",
"In addition to time-varying networks, the temporal-dependent hub model can also be extended in the following directions: first, a group may contain zero or multiple hubs.",
"Second, multiple groups may exist at the same time (with some of these groups being unobserved).",
"These generalizations however will significantly increase model complexity.",
"Therefore, the total number of possible leaders needs to be limited.",
"A method by the author and a collaborator [36] was proposed to reduce this upper bound.",
"More test-based and penalization methods are under development.",
"Furthermore, we also plan to investigate the theoretical properties of the proposed model.",
"When the size of the network is fixed and the number of observed groups goes to infinity, the theoretical properties of the MLE may be studied via a standard theory of the Markov chain.",
"The case that the size of the network also goes to infinity is more intriguing but more complicated since the number of parameters diverges.",
"Acknowledgments This research was supported by NSF Grant DMS 1513004.",
"We thank Dr. Richard Wrangham for sharing the research results of the Kibale Chimpanzee Project.",
"We thank Dr. Charles Weko for compiling the results from the chimpanzee project and preparing the data set.",
"Forward-backward algorithm in the E-steps We derive the forward-backward algorithm for the temporal-dependent hub model introduced in Section REF .",
"Before proceeding, we state two propositions of Bayesian networks.",
"These results (or the equivalent forms) can be found in a standard textbook or tutorial on Bayesian networks, for example, [19].",
"Here we follow [14].",
"Proposition A.1 Each node is conditionally independent from its non-descendents given its parents.",
"Here node $X$ is a parent of another node $Y$ if there is a directed arc from $X$ to $Y$ and if so, $Y$ is a child of $X$ .",
"The descendents of a node are its children, children's children, etc.",
"Proposition A.2 Two disjoint sets of nodes $\\mathcal {A}$ and $\\mathcal {B}$ are conditionally independent given another set $\\mathcal {C}$ , if on every undirected path between a node in $\\mathcal {A}$ and a node in $\\mathcal {B}$ , there is a node $X$ in $\\mathcal {C}$ that is not a child of both the previous and following nodes in the path.",
"Define $G^{s:t}$ as a collection of groups from time $s$ to time $t$ .",
"Let $a_i^t=\\mathbb {P}(z^t=i,G^{1:t})$ .",
"Then, $a_i^t & = \\sum _{k=1}^n\\mathbb {P}(z^t=i,z^{t-1}=k,G^{1:t-1},G^t) \\\\& = \\sum _{k=1}^n\\mathbb {P}(z^{t-1}=k,G^{1:t-1})\\mathbb {P}(z^t=i|z^{t-1}=k,G^{1:t-1})\\mathbb {P}(G^t|z^t=i,z^{t-1}=k,G^{1:t-1}) \\\\& = \\sum _{k=1}^n a^{t-1}_k \\Phi _{ik} \\mathbb {P}(G^t|z^t=i,G^{t-1}).$ The last equation holds by Proposition REF .",
"Similarly, let $b_i^t=\\mathbb {P}(G^{t+1:T}|z^t=i, G^t)$ .",
"Then, $b_i^t & = \\sum _{k=1}^n \\mathbb {P}(z^{t+1}=k, G^{t+1},G^{t+2:T}|z^t=i, G^t) \\\\& = \\sum _{k=1}^n \\mathbb {P}(G^{t+2:T}|z^{t+1}=k,G^{t+1}, z^t=i, G^t)\\mathbb {P}(z^{t+1}=k|z^t=i, G^t)\\mathbb {P}(G^{t+1}|z^{t+1}=k,z^t=i, G^t) \\\\& = \\sum _{k=1}^n b^{t+1}_k \\Phi _{ki} \\mathbb {P}(G^{t+1}|z^{t+1}=k, G^t).$ In the last equation, $\\mathbb {P}(G^{t+2:T}|z^{t+1}=k,G^{t+1}, z^t=i, G^t)$ holds by Proposition REF .",
"This is because a path from $\\lbrace G^{t+2:T}\\rbrace $ to $\\lbrace z^t, G^t\\rbrace $ must pass $z^{t+1}$ or $G^{t+1}$ .",
"If it only passes one of these two variables, then we can take that variable as $X$ in Proposition REF .",
"If it passes both, then take $z^{t+1}$ as $X$ .",
"The rest of the last equation holds by REF .",
"The computation of $a$ and $b$ is essentially the same as in the classical forward-backward algorithm for the HMM with minor modifications.",
"Unlike the HMM, the dependence between the current and the previous groups requires another quantity $c$ .",
"Let $c_i^t=\\mathbb {P}(G^{t:T}|z^t=i, G^{t-1}).$ $c_i^t & = \\sum _{k=1}^n \\mathbb {P}(z^{t+1}=k,G^{t+1:T},G^t|z^t=i,G^{t-1}) \\\\& = \\sum _{k=1}^n \\mathbb {P}(G^{t+1:T}|z^{t+1}=k,G^t,z^t=i,G^{t-1}) \\mathbb {P}(z^{t+1}=k|z^t=i,G^{t-1}) \\mathbb {P}(G^t|z^{t+1}=k,z^t=i,G^{t-1}) \\\\& = \\sum _{k=1}^n c^{t+1}_k\\Phi _{ki} \\mathbb {P}(G^t|z^t=i,G^{t-1}).$ The last equation can be justified by a similar argument as before.",
"Since $& \\mathbb {P}(z^t=i,G^{1:T}) \\\\= & \\mathbb {P}(z^t=i,G^{1:t})\\mathbb {P}(G^{t+1:T}| z^t=i, G^{1:t}) \\\\= & \\mathbb {P}(z^t=i,G^{1:t})\\mathbb {P}(G^{t+1:T}|z^t=i, G^t),$ $\\mathbb {P}(S_i^t=1|G) & = \\frac{a_i^t b_i^t}{\\sum _k a_k^t b_k^t}.$ Similarly, $& \\mathbb {P}(z^t=i,z^{t-1}=j,G^{1:T}) \\\\= & \\mathbb {P}(z^{t-1}=j,G^{1:t-1})\\mathbb {P}(z^t=i|z^{t-1}=j,G^{1:t-1})\\mathbb {P}(G^{t:T}|z^t=i,z^{t-1}=j,G^{1:t-1}) \\\\= & a_j^{t-1} \\Phi _{ij} \\mathbb {P}(G^{t:T}|z^t=i,G^{t-1}) ,$ which implies $\\mathbb {P}(S_i^t=1,S_j^{t-1}=1|G) & = \\frac{a_j^{t-1} \\Phi _{ij} c_i^t}{\\sum _{kl} a_l^{t-1} \\Phi _{kl} c_k^t}.$ Derivatives of $Q$ We give the first and second derivatives of $Q$ with respect to $\\alpha , \\beta , \\gamma , u_i$ and $\\theta _{ij}$ , which are used in the coordinate ascent method introduced in Section REF .",
"Define, $R_{i}^t &=\\mathbb {P}(S_i^t=1|G), \\\\V_{ij} &=\\sum _{t=2}^T \\mathbb {P}(S_i^t=1,S_j^{t-1}=1|G), \\\\D^1_{ij} &= R_i^{1} G_j^{1} +\\sum _{t=2}^T R_i^{t} (1-G_i^{t-1}) G_j^{t}, \\\\D^2_{ij} & =R_i^{1} (1-G_j^{1})+\\sum _{t=2}^T R_i^{t}(1-G_i^{t-1}) (1-G_j^{t}), \\\\D^3_{ij} & = \\sum _{t=2}^T R_i^{t}G_i^{t-1} G_j^{t-1} G_j^{t}, \\\\D^4_{ij} & = \\sum _{t=2}^T R_i^{t}G_i^{t-1} G_j^{t-1} (1-G_j^{t}), \\\\D^5_{ij} & = \\sum _{t=2}^T R_i^{t}G_i^{t-1} (1-G_j^{t-1}) G_j^{t}, \\\\D^6_{ij} & = \\sum _{t=2}^T R_i^{t}G_i^{t-1} (1-G_j^{t-1}) (1-G_j^{t}).$ Therefore, $Q = & \\sum _{i=1}^n R_i^{1} \\left[ u_i-\\log \\left\\lbrace \\sum _{k=1}^{n} \\exp (u_k) \\right\\rbrace \\right] \\\\& + \\sum _{i=1}^n V_{ii} \\left[u_i+\\alpha - \\log \\left\\lbrace \\sum _{k=1}^{n} \\exp (u_k+\\alpha I(k=i)) \\right\\rbrace \\right] \\\\& + \\sum _{i=1}^n \\sum _{j \\ne i}V_{ij} \\left[u_i- \\log \\left\\lbrace \\sum _{k=1}^{n} \\exp (u_k+\\alpha I(k=j)) \\right\\rbrace \\right] \\\\& + \\sum _{ij} \\left[ D^1_{ij} \\log \\frac{e^{\\theta _{ij}}}{1+e^{\\theta _{ij}}} + D^2_{ij} \\log \\frac{1}{1+e^{\\theta _{ij}}} \\right.",
"\\\\& \\quad \\quad + D^3 _{ij} \\log \\frac{e^{\\theta _{ij}+\\beta }}{1+e^{\\theta _{ij}+\\beta }} +D^4 _{ij} \\log \\frac{1}{1+e^{\\theta _{ij}+\\beta }} \\\\& \\quad \\quad + \\left.",
"D^5 _{ij} \\log \\frac{e^{\\theta _{ij}+\\gamma }}{1+e^{\\theta _{ij}+\\gamma }} +D^6 _{ij} \\log \\frac{1}{1+e^{\\theta _{ij}+\\gamma }} \\right] .$ The first and second order derivatives are given as follows, $\\frac{\\partial Q }{\\partial \\alpha } = & \\sum _{i=1}^nV_{ii} + \\sum _{j=1}^n \\left[\\sum _{i=1}^n V_{ij} \\right]\\left[- \\frac{\\exp (u_j+\\alpha )}{\\sum _{k=1}^{n} \\exp (u_k+\\alpha I(k=j)) } \\right], \\\\\\frac{\\partial ^2 Q }{\\partial \\alpha ^2}=& \\sum _{j=1}^n \\left[\\sum _{i=1}^n V_{ij} \\right]\\left[- \\frac{\\exp (u_j+\\alpha )\\sum _{k=1}^{n} \\exp (u_k+\\alpha I(k=j))-\\exp (u_j+\\alpha )\\exp (u_j+\\alpha )}{(\\sum _{k=1}^{n} \\exp (u_k+\\alpha I(k=j)))^2 }\\right], \\\\\\frac{\\partial Q }{\\partial u_r } = & R_r^{1} + \\left[ \\sum _{i=1}^n R_i^{1} \\right] \\left[-\\frac{\\exp (u_r)}{\\sum _{k=1}^{n} \\exp (u_k) } \\right], \\\\\\frac{\\partial ^2 Q }{\\partial u_r^2 } = & \\left[\\sum _{i=1}^n R_i^{1} \\right]\\left[-\\frac{\\exp (u_r)\\sum _{k=1}^{n} \\exp (u_k)-\\exp (u_r)\\exp (u_r)}{(\\sum _{k=1}^{n} \\exp (u_k))^2} \\right] \\\\& +\\sum _{j=1}^n \\left[\\sum _{i=1}^n V_{ij} \\right] \\left[ -\\frac{\\exp (u_r+\\alpha I(r=j))\\sum _{k=1}^{n} \\exp (u_k+\\alpha I(k=j))-(\\exp (u_r+\\alpha I(r=j)))^2}{(\\sum _{k=1}^{n} \\exp (u_k+\\alpha I(k=j)))^2 } \\right] \\\\& +\\sum _{j=1}^n V_{rj} + \\sum _{j=1}^n \\left[\\sum _{i=1}^n B_{ij} \\right] \\left[-\\frac{\\exp (u_r+\\alpha I(r=j))}{\\sum _{k=1}^{n} \\exp (u_k+\\alpha I(k=j)) } \\right], \\\\\\frac{\\partial Q}{\\partial \\beta } = & \\sum _{i \\ne j} D^3 _{ij}-( D^3 _{ij}+D^4_{ij}) \\frac{e^{\\theta _{ij}+\\beta }}{1+e^{\\theta _{ij}+\\beta }}, \\\\\\frac{\\partial ^2 Q}{\\partial \\beta ^2 } = & -\\sum _{i \\ne j} ( D^3 _{ij}+D^4_{ij}) \\frac{e^{\\theta _{ij}+\\beta }}{(1+e^{\\theta _{ij}+\\beta })^2}, \\\\\\frac{\\partial Q}{\\partial \\gamma } = & \\sum _{i \\ne j} D^5 _{ij}-( D^5 _{ij}+D^6_{ij}) \\frac{e^{\\theta _{ij}+\\gamma }}{1+e^{\\theta _{ij}+\\gamma }} , \\\\\\frac{\\partial ^2 Q}{\\partial \\gamma ^2 } = & -\\sum _{i \\ne j} ( D^5 _{ij}+D^6_{ij}) \\frac{e^{\\theta _{ij}+\\gamma }}{(1+e^{\\theta _{ij}+\\gamma })^2}, \\\\\\frac{\\partial Q}{\\partial \\theta _{ij}} = & (D^1_{ij}+D^1_{ji})-(D^1_{ij}+D^1_{ji}+D^2_{ij}+D^2_{ji}) \\frac{e^{\\theta _{ij}}}{1+e^{\\theta _{ij}}} \\\\& + (D^3_{ij}+D^3_{ji})-(D^3_{ij}+D^3_{ji}+D^4_{ij}+D^4_{ji}) \\frac{e^{\\theta _{ij}+\\beta }}{1+e^{\\theta _{ij}+\\beta }} \\\\& + (D^5_{ij}+D^5_{ji})-(D^5_{ij}+D^5_{ji}+D^6_{ij}+D^6_{ji}) \\frac{e^{\\theta _{ij}+\\gamma }}{1+e^{\\theta _{ij}+\\gamma }}, \\\\\\frac{\\partial ^2 Q}{\\partial \\theta _{ij}^2} = & -(D^1_{ij}+D^1_{ji}+D^2_{ij}+D^2_{ji}) \\frac{e^{\\theta _{ij}}}{(1+e^{\\theta _{ij}})^2} \\\\& -(D^3_{ij}+D^3_{ji}+D^4_{ij}+D^4_{ji}) \\frac{e^{\\theta _{ij}+\\beta }}{(1+e^{\\theta _{ij}+\\beta })^2} \\\\& -(D^5_{ij}+D^5_{ji}+D^6_{ij}+D^6_{ji}) \\frac{e^{\\theta _{ij}+\\gamma }}{(1+e^{\\theta _{ij}+\\gamma })^2}.$"
],
[
"The classical hub model",
"We briefly state the generating mechanism of the classical hub model [40].",
"The hub model assumes one leader for each group.",
"The leader of $G^{t}$ is denoted by $z^t$ .",
"Under the hub model, each group $G^{t}$ is independently generated by the following two steps.",
"The group leader is sampled from a multinomial distribution with parameter $\\rho =(\\rho _1,...,\\rho _n)$ , i.e., $\\mathbb {P}(z^t=i)=\\rho _i$ , with $\\sum _i \\rho _i=1$ .",
"The group leader, $v_i$ , will choose to include $v_j$ in the group with probability $A_{ij}$ , i.e., $ \\mathbb {P}(G_j^{t}=1|z^t=i)=A_{ij}$ ."
],
[
"Generating mechanism of the temporal-dependent hub model",
"The hub model assumes that all the groups are generated independently across time.",
"In practice, it is more natural to model the groups as temporal-dependent observations.",
"We first explain the idea of the generating mechanism of temporal-dependent groups and then give the formal definition.",
"We generalize the idea of the hub model into the temporal-dependent setting.",
"Specifically, we assume there is only one leader $z^t$ at each time who brought the group together, but the group at time $t$ depends on the previous group, which is different from the classical hub model.",
"At time $t=1$ , the group is generated from the classical hub model.",
"For $t=2,...,T$ , the group leader $z^t$ can remain the same as the previous leader or change to a new one.",
"We assume that the leader $z^t$ will remain as $z^{t-1}$ with a higher probability than the probability of changing to any other node.",
"If the new leader is outside the previous group, then the current group is considered the start of a new segment and is generated by the classical hub model.",
"It is worth noting that technically, the generation of the new group however still depends on the previous group.",
"This will become clearer after we introduce the likelihood function.",
"For the case that the new leader is inside the previous group – that is, if the leader remains unchanged, or the leader changes but is still a member of the previous group – we propose the following In-and-Out procedure: for any node $v_j$ being in the previous group, it will remain in $G^{t}$ with a probability higher than $A_{z^t,j}$ – the probability in the classical hub model.",
"On the contrary, for any node $v_k$ not being in the previous group, it will enter $G^{t}$ with a probability lower than $A_{z^t,k}$ .",
"Intuitively, this In-and-Out procedure assumes that when a group forms, it will maintain a certain level of stability.",
"We now give the formal definition of the generating mechanism as follows: Step 1: (Classical hub model).",
"When $t=1$ , $G^{t}$ is generated by the following two substeps.",
"1) The leader is sampled from a multinomial distribution with parameter $\\rho =(\\rho _1,...,\\rho _n)$ , i.e., $ \\mathbb {P}(z^{t}=i)=\\rho _i \\overset{\\Delta }{=} \\frac{\\exp (u_i)}{\\sum _{k=1}^n \\exp (u_k)}, $ where $u_i \\in \\mathbb {R}$ for $i=1,...,n$ .",
"2) The leader $v_i$ will choose to include $v_j$ in the group with probability $A_{ij}$ , i.e., $ \\mathbb {P}(G_j^{t}=1|z^{t}=i)=A_{ij}$ , where $A_{ii} \\equiv 1$ and $A_{ij}= A_{ji}\\overset{\\Delta }{=}\\frac{\\exp (\\theta _{ij})}{1+\\exp ( \\theta _{ij})}.",
"$ Here, $\\theta _{ii}=\\infty $ for $i=1,...,n$ and $\\theta _{ij} \\in \\overline{\\mathbb {R}}$ for $i \\ne j$ .",
"We allow some $\\theta _{ij}$ to be $\\pm \\infty $ so that the corresponding $A_{ij}$ can be 1 or 0.",
"Step 2: (Leader change).",
"For $t=2,...,T$ , $\\mathbb {P}(z^t=i| z^{t-1}) = \\frac{\\exp (u_i+\\alpha I(z^{t-1}=i))}{\\sum _{k=1}^n \\exp (u_k+\\alpha I(z^{t-1}=k))},$ where $\\alpha \\in \\mathbb {R}$ .",
"Step 3: (In-and-Out procedure).",
"For $t=2,...,T$ , given $v_i$ being the leader, $G^{t}$ is generated by the following mechanism: If $v_i$ is not within $G^{t-1}$ , then it will include each $v_j$ in the group with probability $A_{ij}$ ; otherwise, see below: 1) If $G^{t-1}_j=1$ , $v_i$ will include $v_j$ in the group with probability $B_{ij}=B_{ji}\\overset{\\Delta }{=} \\frac{\\exp ( \\theta _{ij}+\\beta )}{1+\\exp ( \\theta _{ij}+\\beta )},$ where $\\beta \\in \\mathbb {R}$ .",
"2) If $G^{t-1}_j=0$ , $v_i$ will include $v_j$ in the group with probability $C_{ij}=C_{ji}\\overset{\\Delta }{=} \\frac{\\exp ( \\theta _{ij}+\\gamma )}{1+\\exp ( \\theta _{ij}+\\gamma )},$ where $\\gamma \\in \\mathbb {R}$ .",
"For clarity of notation, we now give the vector/matrix form.",
"Define $z=(z^{1},..., z^{T})$ , $u=(u_1,...,u_n)$ and $\\rho =(\\rho _1,...,\\rho _n)$ .",
"Define $\\theta =[\\theta _{ij}]_{1\\le i \\le n, 1\\le j \\le n}$ , $A=[A_{ij}]_{1\\le i \\le n, 1\\le j \\le n}$ , $B=[B_{ij}]_{1\\le i \\le n, 1\\le j \\le n}$ and $C=[C_{ij}]_{1\\le i \\le n, 1\\le j \\le n}$ .",
"Furthermore, we assume $\\theta $ , $A$ , $B$ and $C$ to be symmetric in order to avoid any issue of identifiability (see the discussion in [40]).",
"Remark In the definition above, $u_i$ and $\\theta _{ij}$ are simply a reparameterization of $\\rho _i$ and $A_{ij}$ in exponential form.",
"This is to make optimization more convenient, since log-likelihood is convex under this parametrization.",
"The parameters $\\alpha $ , $\\beta $ and $\\gamma $ characterize the dependency between the groups.",
"$\\alpha $ is the adjustment factor, which controls the probability that a leader in the previous group remains as a leader.",
"$\\beta $ is the adjustment factor for nodes being inside the previous group.",
"And $\\gamma $ is the adjustment factor for nodes being from outside the previous group.",
"We do not enforce $\\alpha >0$ , $\\beta >0$ and $\\gamma <0$ in the model fitting.",
"Instead, we test these assumptions for the data example in Section .",
"The parameters $A$ , $\\beta $ and $\\gamma $ are identifiable.",
"The key observation is that the identifiability of $A$ can simply be obtained by $G^1$ since the first group only depends on $\\rho $ and $A$ .",
"This is essentially the identifiability of the classical hub model.",
"The proof is given by Theorem 1 in [40], under the condition of $A$ being symmetrical.",
"With the “baseline” $A$ being separately identified, the two adjustment factors $\\beta $ and $\\gamma $ are accordingly identifiable.",
"The parameters $(u_1,...,u_n)$ are non-identifiable under this parametrization, since $(u_1+\\delta ,...,u_n+\\delta )$ gives the same likelihood.",
"We will discuss the solution to this problem in Section after introducing the algorithm.",
"Likelihood For notational convenience in the likelihood, we indicate the leader in group $G^t$ by an $n$ length vector, $S^{t}$ , where $ S_i^{t} = \\left\\lbrace \\begin{array}{l l}1 & \\quad \\text{if $z^t=i$},\\\\0 & \\quad \\text{otherwise}.\\end{array} \\right.$ Only one element of $S^{t}$ is allowed to be 1.",
"$S^{t}$ is simply another representation of $z^{t}$ .",
"Let $S$ be a $T \\times n$ matrix, with $S^{t}$ being its rows.",
"Clearly, $ \\lbrace S^1,...,S^T \\rbrace $ is a Markov chain according to the generating mechanism.",
"Let $\\Phi _{ij}=\\mathbb {P}(z^t=i|z^{t-1}=j)$ be the transition probability and $\\Phi =[\\Phi _{ij}]_{n\\times n}$ .",
"We summarize all introduced notations in Table REF .",
"Table: Summary of NotationWe now give the joint log-likelihood of $S$ and $G$ for the model defined in the previous subsection: $& \\log \\mathbb {P} (S,G|\\alpha , \\beta , \\gamma , \\theta , u) \\nonumber \\\\= & \\sum _{i=1}^n S_i^{1} \\log \\rho _i +\\sum _{t=2}^T \\sum _{i=1}^n \\sum _{j=1}^n S_i^{t}S_j^{t-1} \\log \\Phi _{ij} \\nonumber \\\\&+ \\sum _{i=1}^n \\sum _{j=1}^n \\left\\lbrace S_i^{1} G_j^{1} \\log A_{ij}+S_i^{1} (1-G_j^{1}) \\log (1-A_{ij}) \\right\\rbrace \\nonumber \\\\&+ \\sum _{t=2}^T \\sum _{i=1}^n \\sum _{j=1}^n \\left\\lbrace S_i^{t} (1-G_i^{t-1}) G_j^{t} \\log A_{ij}+S_i^{t}(1-G_i^{t-1}) (1-G_j^{t}) \\log (1-A_{ij}) \\right\\rbrace \\nonumber \\\\& + \\sum _{t=2}^T \\sum _{i=1}^n \\sum _{j=1}^n \\left\\lbrace S_i^{t}G_i^{t-1} G_j^{t-1} G_j^{t} \\log B_{ij}+ S_i^{t}G_i^{t-1} G_j^{t-1} (1-G_j^{t}) \\log (1-B_{ij}) \\right\\rbrace \\nonumber \\\\& + \\sum _{t=2}^T \\sum _{i=1}^n \\sum _{j=1}^n \\left\\lbrace S_i^{t}G_i^{t-1} (1-G_j^{t-1}) G_j^{t} \\log C_{ij}+ S_i^{t}G_i^{t-1} (1-G_j^{t-1}) (1-G_j^{t}) \\log (1-C_{ij}) \\right\\rbrace .",
"$ Note that $\\alpha , \\beta , \\gamma , \\theta $ and $u$ are essentially the parameters of this model and $\\rho $ , $\\Phi $ , $A$ , $B$ and $C$ are their functions.",
"Despite its length, Equation (REF ) has a clear structure.",
"The 1st line gives the log-likelihood of $S$ .",
"The 2nd line gives the log-likelihood of $G^1$ given $S^1$ .",
"The 3rd line gives the log-likelihood of $G^t$ given that the current leader $z^t$ is outside the previous group $G^{t-1}$ .",
"The 4th and 5th lines give the log-likelihood of $G^t$ given that $z^t$ is inside $G^{t-1}$ , based on the In-and-Out procedure.",
"Equivalent to (REF ), we can write the likelihood as a product of conditional probabilities: $\\mathbb {P} (S,G) = \\mathbb {P} (S^1) \\mathbb {P}(G^1|S^1) \\prod _{t=2}^T \\mathbb {P}(S^t|S^{t-1}) \\prod _{t=2}^T \\mathbb {P}(G^t|S^{t},G^{t-1}).$ This factorization can be represented by a Bayesian network (Figure REF ), where a node represents a variable and a directed arc is drawn from node $X$ to node $Y$ if $Y$ is conditioned on $X$ in the factorization.",
"(Refer to [19] for a comprehensive introduction to Bayesian networks).",
"This Bayesian network should not be confused with the latent network $A$ – the former is a representation of the dependency structure between variables while the latter reflects the relationships between the group members.",
"Furthermore, the group leaders $z^1,...,z^T$ are assumed to be latent (as are $S^1,...,S^T$ ) since in many applications only the groups themselves are observable.",
"Figure: A Bayesian network representing the temporal-dependent hub model.",
"Nodes with dark colors indicate the observed variables.",
"Model fitting In this section, we propose an algorithm to find the maximum likelihood estimators (MLEs) for $\\alpha ,\\beta ,\\gamma ,u$ and $\\theta $ .",
"With $S$ being the latent variables, an expectation-maximization (EM) algorithm will be used for this problem.",
"The EM algorithm maximizes the marginal likelihood of the observed data, which in our case is $G$ , by iteratively applying an E-step and an M-step.",
"Let $\\alpha ^{\\textnormal {old}},\\beta ^{\\textnormal {old}},\\gamma ^{\\textnormal {old}},u^{\\textnormal {old}}$ and $\\theta ^{\\textnormal {old}}$ be the estimates in the current iteration.",
"In the E-step, we calculate the conditional expectation of the complete log-likelihood given $G$ under the current estimate.",
"That is, $Q \\overset{\\Delta }{=} Q(\\alpha ,\\beta ,\\gamma ,u,\\theta | \\alpha ^{\\textnormal {old}},\\beta ^{\\textnormal {old}},\\gamma ^{\\textnormal {old}},u^{\\textnormal {old}},\\theta ^{\\textnormal {old}})= \\mathbb {E}_{\\alpha ^{\\textnormal {old}},\\beta ^{\\textnormal {old}},\\gamma ^{\\textnormal {old}},u^{\\textnormal {old}},\\theta ^{\\textnormal {old}}} \\left[ \\log \\mathbb {P} (S,G) | G \\right].$ In the M-step, we maximize this conditional expectation with respect to the unknown parameters.",
"That is, $(\\alpha ^{\\textnormal {new}},\\beta ^{\\textnormal {new}},\\gamma ^{\\textnormal {new}},u^{\\textnormal {new}},\\theta ^{\\textnormal {new}} ) = \\operatornamewithlimits{arg\\,max}_{\\alpha ,\\beta ,\\gamma ,u,\\theta } \\,\\, Q(\\alpha ,\\beta ,\\gamma ,u,\\theta | \\alpha ^{\\textnormal {old}},\\beta ^{\\textnormal {old}},\\gamma ^{\\textnormal {old}},u^{\\textnormal {old}},\\theta ^{\\textnormal {old}}).$ It has been proved by [38] that the EM algorithm converges to a local maximizer of the marginal likelihood.",
"(Refer to [25] for a comprehensive introduction to this algorithm).",
"We now give details of the two steps in our context.",
"E-step Since the complete log-likelihood $\\log \\mathbb {P} (S,G)$ is a linear function of $S_i^t \\,\\, (t=1,...,T;i=1,...,n)$ and $S_i^{t}S_j^{t-1} \\,\\,(t=2,...,T;i=1,...,n;j=1,...,n) $ , the computation of its conditional expectation is equivalent to calculating $\\mathbb {P}(S_i^t=1|G)$ and $\\mathbb {P}(S_i^t=1,S_j^{t-1}=1|G)$ .",
"From now on, all conditional probabilities are defined under the current estimates.",
"A brute-force calculation of these probabilities, such as $\\mathbb {P}(S_i^t=1|G) = \\mathbb {P} (z^t=i|G)=\\frac{\\sum _{z^1}\\cdots \\sum _{z^{t-1}}\\sum _{z^{t+1}}\\cdots \\sum _{z^t} \\mathbb {P}(z^1,...,z^{t-1},z^t=i,z^{t+1},...,z^T,G)}{\\mathbb {P}(G)},$ is infeasible since the numerator involves a sum of $n^{T-1}$ terms.",
"This is because $G^1,...,G^T$ are not independent according to our model.",
"An efficient algorithm is needed for all practical purposes.",
"The temporal-dependent hub model is similar to the hidden Markov model (HMM) (Figure REF ).",
"A polynomial-time algorithm for this model, called the forward-backward algorithm, was developed for computing the conditional probabilities.",
"See [31], [14] for tutorials on HMMs and this algorithm.",
"In the HMM, the observed variable at time $t$ only depends on the corresponding hidden state.",
"But in our model, $G^t$ depends on both the current leader $z^t$ and the previous group $G^{t-1}$ .",
"We develop a new forward-backward algorithm for our model, which has more steps than the original algorithm but is also polynomial-time.",
"We describe the algorithm here (see the Appendix for detailed derivation and justification).",
"Define $a=[a_i^t],b=[b_i^t]$ and $c=[c_i^t]$ as $T\\times n$ matrices.",
"These matrices are computed by the following recursive procedures.",
"$a_i^1 & = \\mathbb {P}(z^1=i,G^1) \\quad (i=1,...,n).",
"\\\\a_i^t & =\\sum _{k=1}^n a^{t-1}_k \\Phi _{ik} \\mathbb {P}(G^t|z^t=i,G^{t-1}) \\quad (t=2,...,T;i=1,...,n).",
"\\\\b_i^T & = 1 \\quad (i=1,...,n).",
"\\\\b_i^t & =\\sum _{k=1}^n b^{t+1}_k \\Phi _{ki} \\mathbb {P}(G^{t+1}|z^{t+1}=k, G^t) \\quad (t=T-1,...,1;i=1,...,n).",
"\\\\c_i^T & = \\mathbb {P}(G^T|z^{T}=i,G^{T-1}) \\quad (i=1,...,n).",
"\\\\c_i^t & = \\sum _{k=1}^n c^{t+1}_k \\Phi _{ki} \\mathbb {P}(G^t|z^t=i,G^{t-1}) \\quad (t=T-1,...,2;i=1,...,n).$ The matrices $a$ , $b$ and $c$ should not be confused with the matrices $A$ , $B$ and $C$ introduced in Section .",
"The symbols are case-sensitive throughout the paper.",
"With these quantities, $\\mathbb {P}(S_i^t=1|G) & = \\frac{a_i^t b_i^t}{\\sum _k a_k^t b_k^t} \\quad (t=2,...,T;i=1,...,n).",
"\\\\\\mathbb {P}(S_i^t=1,S_j^{t-1}=1|G) & = \\frac{a_j^{t-1} \\Phi _{ij} c_i^t}{\\sum _{kl} a_l^{t-1} \\Phi _{kl} c_k^t} \\quad (t=2,...,T;i=1,...,n;j=1,...,n).$ The complexity of this algorithm is $O(Tn^2)$ .",
"Note that the first row of $c$ is undefined but also unused.",
"Also note that the elements of $a,b$ and $c$ will quickly vanish as the recursions progress.",
"Therefore, we renormalize each row to sum to one at each step.",
"It can easily be verified that this normalization does not affect the conditional probabilities.",
"Finally, we emphasize that this algorithm gives the exact values of the conditional probabilities in a fixed number of steps – i.e., it is not an approximate or iterative method.",
"M-step The M-step is somewhat routine compared to the E-step.",
"First, it is clear that $\\lbrace \\alpha ,u \\rbrace $ and $\\lbrace \\beta ,\\gamma ,\\theta \\rbrace $ can be handled separately.",
"We apply the coordinate ascent method (see [6] for a comprehensive introduction) to iteratively update $\\alpha $ and $u$ , as well as $\\beta ,\\gamma $ and $\\theta $ .",
"Since the complete log-likelihood is concave and so is $Q$ , coordinate ascent can guarantee a global maximizer.",
"At each step, we optimize the log-likelihood over parameter one by one with the other parameters being fixed.",
"The procedure is repeated until convergence.",
"At each step, we use the standard Newton-Raphson method to solve each individual optimization problem.",
"Specifically, for a parameter $\\phi $ (here $\\phi $ can represent $\\alpha $ , $\\beta $ , $\\gamma $ , $u_i$ or $\\theta _{ij} \\,\\, (i < j)$ ), the estimate at $(m+1)$ -th iteration is updated by the following formula given its estimate at $m$ -th iteration: $\\hat{\\phi }_{m+1}=\\hat{\\phi }_m-\\left( \\frac{\\partial ^2 Q}{\\partial \\phi ^2 } \\bigg |_{\\phi =\\hat{\\phi }_m} \\right)^{-1} \\left( \\frac{\\partial Q}{\\partial \\phi } \\bigg |_{\\phi =\\hat{\\phi }_m}\\right).",
"$ The calculation of these derivatives is straightforward but tedious, so we provide the details in the Appendix.",
"As shown in Section REF , the model is not identifiable with respect to $u$ .",
"A standard solution to this problem is to set some $u_i \\equiv 0$ .",
"But it does not work for our case.",
"This is because for small data sets, some $\\hat{\\rho }_i$ estimated by the EM algorithm may be zero, implying that $v_i$ never became the leader.",
"Furthermore, these zero $\\rho _i$ cannot be predetermined since the leaders are unobserved.",
"We observe that without constraint on $u_i$ , the algorithm converges to different $\\hat{u}$ with different initial values, but the corresponding $\\hat{\\rho }$ will be the same.",
"Therefore, identifiability is not an issue for model fitting.",
"Initial value As with many optimization algorithms, the EM algorithm is not guaranteed to find the global maximizer.",
"Ideally, one should use multiple random initial values and find the best solution by comparing the marginal likelihoods $\\mathbb {P}(G)$ under the corresponding estimates.",
"In principle, $\\mathbb {P}(G)$ can be computed by $\\sum _k a_k^t b_k^t$ , as shown in Section REF .",
"But the marginal likelihood vanishes quickly, even with a moderate $T$ .",
"Note that we cannot renormalize $a$ and $b$ for the purpose of computing $\\mathbb {P}(G)$ .",
"Therefore, we use the half weight index [9], [7] as the initial value of $A$ , which is defined by $H_{ij} = \\frac{2\\sum _t G_i^{t} G_j^{t}}{\\sum _t G_i^{t}+\\sum _t G_j^{t}}.$ This measure estimates the conditional probability that two nodes co-occur given that one of them is observed, which is a reasonable initial guess of the strength of links.",
"Furthermore, we use zero for the initial values of $\\alpha , \\beta $ and $\\gamma $ , and $\\sum _{t} G_i^t/T $ for the initial value of $\\rho _i$ .",
"Simulation studies In all simulation studies, we fix the size of the network to be $n=50$ and set $\\beta =3$ and $\\gamma =-1$ .",
"We generate $u_i$ as independently and identically distributed variables with $N(0,2)$ and $\\rho _i= u_i/\\sum _k u_k $ .",
"The parameters $\\theta _{ij}\\,\\, (i<j)$ are generated independently with $N(-2,1)$ .",
"We generate $\\theta _{ij}$ in this way to control the average link density of the network ($\\approx $ 0.12), which is more realistic than a symmetric setting, i.e., $\\theta _{ij} \\sim N(0,1)$ .",
"For clarification, we will not use the prior information on $u$ and $\\theta $ in our estimating procedure.",
"That is, we still treat $u$ and $\\theta $ as unknown fixed parameters in the algorithm.",
"We generate them as random variables for the whole purpose of adding more variations to the parameter setup in our study.",
"We consider three levels of $\\alpha =\\log ((n-1)/2),\\log (n-1),\\log (2(n-1))$ , which correspond to a leader from the previous group remaining unchanged in the current group with probabilities $1/3,1/2,2/3$ on average.",
"For each $\\alpha $ we try five different sample sizes, $T=1000,1500, 2000, 2500$ and 3000.",
"Figure: Average RMSEs (across 100 replicates) for the estimated AA from the classical hub model (HM) and the temporal-dependent hub model (TDHM).Table: Average RMSEs for the estimated AA from the classical hub model (HM) and the temporal-dependent hub model (TDHM).",
"Standard deviations ×10 3 \\times 10^3 in parentheses.Figure REF shows the average root of mean squared errors (RMSEs) for the estimated $A$ over 100 replicates.",
"For each simulation, we compare two methods, the classical hub model (HM) and the temporal-dependent hub model (TDHM).",
"We assume the leaders are unknown under both models.",
"Table REF provides the same information (with standard deviations) in numerical form.",
"From Figure REF and Table REF , our first observation is simply that the RMSEs decrease as sample sizes increase, which is consistent with common sense in statistics.",
"Second, the RMSEs for all the parameters increase as $\\alpha $ increases.",
"This phenomenon can be interpreted as follows: with a larger value of $\\alpha $ , the correlation between adjacent groups becomes stronger and hence the effective sample size becomes smaller.",
"The ratio of the sample size to the number of parameters decreases with $\\alpha $ , which makes inferences more difficult.",
"Third, the temporal-dependent hub model always outperforms the hub model.",
"Moreover, the discrepancy between the temporal-dependent hub model estimates and the corresponding hub model estimates becomes larger as $\\alpha $ increase.",
"This is because the behavior of the temporal-dependent model deviates more from the classical hub model as $\\alpha $ increases.",
"The standard deviations and means show a similar trend.",
"That is, the standard deviations decrease as $n$ increases and $\\alpha $ decreases.",
"The standard deviations for the temporal-dependent hub model estimates are comparable to or slightly smaller than those of the hub model estimates.",
"A data example of group dynamics in chimpanzees Behavioral ecologists become increasingly interested in using social network analysis to understand social organization and animal behavior [3], [37], [8], [11].",
"The social relationships are usually inferred by using certain association metrics (e.g., the half weight index) on grouped data.",
"As indicated in the Introduction however, it is unclear how the inferred network relates to the observed groups without specifying a model.",
"In this section, we study a data set of groups formed by chimpanzees by the temporal-dependent hub model.",
"This data set is compiled from the results of the Kibale Chimpanzee Project, which is a long-term field study of the behavior, ecology and physiology of wild chimpanzees in the Kanyawara region of Kibale National Park, southwestern Uganda (https://kibalechimpanzees.wordpress.com/).",
"Our analysis focuses on grouping behavior.",
"We analyze the grouped data collected from January 1, 2009 to June 30, 2009 [20].",
"The group identification was taken at 1 p.m. daily during this time period.",
"If there is no group observed at 1 p.m. for a given day, it is not included in the data.",
"Only one group is observed at 1 p.m. in 75.29% of the remaining days over this period of six months.",
"In the other days, multiple groups (usually two) are observed at 1 p.m. For these cases, we keep the group that has the most overlap with the previous group in our analysis.",
"We use the Jaccard index to measure the overlap between two groups $G^{t-1}$ and $G^{t}$ , $J(G^{t-1},G^{t})=\\frac{\\sum _{j=1}^n G^{t-1}_j G^{t}_j }{\\sum _{j=1}^n [ n-(1-G^{t-1}_j)(1-G^{t}_j) ]},$ where the numerator is the size of the intersection of two groups and the denominator is the size of their union.",
"One may refer to [23] for an introduction to this measure.",
"Moreover, five chimpanzees never appear in any group and thus are removed.",
"After the preprocessing, the data set contains 170 groups with 40 chimpanzees.",
"Figure REF illustrates the data set in grayscale with rows representing the groups over time and columns representing the chimpanzees.",
"Black indicates $G^{t}_i=1$ at location $(t,i)$ while white indicates $G^{t}_i=0$ .",
"The pattern in Figure REF clearly demonstrates the existence of dependency between groups.",
"Figure REF also shows the inferred grouped leaders indicated in red with the inferred segments separated by blue lines.",
"By the inferred grouped leader for $G^t$ , we mean that the chimpanzee with the highest posterior probability will be the leader given $G^t$ .",
"As shown in Figure REF , the leaders retain a certain level of stability, which is consistent with the estimates of $\\alpha $ ($=1.7291$ ).",
"Also, recall that by our definition, a new segment starts if the current leader is not within the previous group.",
"From Figure REF , the inferred segments are coincident with the visualization of the data set.",
"The estimated values of the adjustment factors, $\\hat{\\beta }=2.5703$ and $\\hat{\\gamma }=-0.1922$ .",
"The magnitude of $\\hat{\\beta }$ is larger than that of $\\hat{\\gamma }$ , which suggests individuals have a stronger tendency to join a group than leave a group.",
"In other words, the groups may start with small size and grow larger over time.",
"This phenomenon is shown (Figure REF ).",
"Figure: Visualization of the chimpanzee data.",
"The 170 rows represent the groups over time and the 40 columns represent chimpanzees.",
"Gray indicates the presence of the group membership.",
"Red indicates the inferred group leaders.",
"The blue lines separate the inferred segments.Figure: Grayscale plot for the estimated adjacency matrices from the chimpanzee data by the classical hub model and the temporal-dependent hub model.",
"The rows and the columns represent the 40 chimpanzees.",
"Darker colors indicate stronger relationships.",
"The red blocks indicate the biological clusters of chimpanzees.Figure REF shows the result of estimated adjacency matrices by the classical hub model and the temporal-dependent hub model.",
"As in the previous figure, the darker color indicates a higher value of $\\hat{A}_{ij}$ .",
"The red blocks indicate clusters of chimpanzees in a biological sense.",
"The first cluster consists of 12 adult males and each of the other nine clusters consists of an adult female and its children.",
"From the estimates by both the classical hub model and the temporal-dependent hub model, there are strong connections within these biological clusters.",
"Both estimates suggest that in this data set of chimpanzees, adult males usually do activities together but females usually stay with their children.",
"The two estimated adjacency matrices are different, however.",
"Generally speaking, without properly considering the temporal-dependence between groups, the estimates of the relationships between individuals by the classical hub model can be biased.",
"That is, an individual may choose to stay within or out of a group not solely based on its relationship with the group leader but also because of the inertia.",
"The overall graph density $(= 0.2286)$ of the estimated network by the classical hub model is larger than the corresponding value $(= 0.1973)$ of the temporal-dependent hub model.",
"This is consistent with the fact that the magnitude of $\\hat{\\beta }$ is larger than the magnitude of $\\hat{\\gamma }$ .",
"Since the classical hub model does not incorporate the adjustment factors, bias is introduced to certain $\\hat{A}_{ij}$ so the model can match the overall frequency of occurrences for the individuals.",
"The significance of $\\alpha $ , $\\beta $ and $\\gamma $ is tested by the parametric bootstrap method [10].",
"Specifically, we generate 5000 independent data sets from the fitted temporal-dependent hub model to the original data and compute the MLEs for each simulated data set.",
"The parametric bootstrap was applied to HMMs and showed a good performance [34].",
"Figure REF shows the histograms of the MLEs for $\\alpha $ , $\\beta $ and $\\gamma $ .",
"The 95% bootstrap confidence intervals for $\\alpha $ , $\\beta $ and $\\gamma $ are (1.2177, 1.9774), (2.0710, 2.8944) and (-0.5208, 0.0410), which shows that the effects of $\\alpha $ and $\\beta $ are significant while $\\gamma $ is not at the 0.05 significance level.",
"This further supports the observation in the previous paragraph – chimpanzees have a stronger tendency to join a group than to leave a group in this data set.",
"Figure: Histograms of estimates from parametric bootstrap samples.",
"Red lines indicate the estimated values from the original data set.",
"Summary and discussion In this article, we generalize the idea of the hub model and propose a novel model for temporal-dependent grouped data.",
"This new model allows for dependency between groups.",
"Specifically, the group leaders follow a Markov chain and a group is either a transformation of the previous group or a new start, depending on whether the current leader is within the previous group.",
"An EM algorithm is applied to this model with a polynomial-time algorithm being developed for the E-step.",
"The setup of our model is different from some work on estimating time-varying networks by graphical models, e.g., [21] for discrete data and [41] for continuous data.",
"These papers assume that the observations are independent and the latent network changes smoothly or is piecewise constant.",
"In this paper, we instead focus on the dependence between groups.",
"Ideally, both aspects – dependence between groups and changes in networks – need to be considered in modeling.",
"This should be plausible when the sample size, i.e., the number of observed groups, is large.",
"When the sample size is moderate (as in the chimpanzee data set), the length of the time interval between observation plays a key role in determining which aspect is more important.",
"If the time interval is short, then dependence between groups is significant but the changes in the latent network are likely to be minor since the overall time window is not long.",
"On the contrary, the dependence between groups becomes weak when the time interval is long.",
"For future work, we plan to study the time-varying effect on the latent network for the grouped data.",
"When changepoints exist, a single network cannot accurately represent the link strengths in different time windows.",
"A change-point analysis for temporal-dependent grouped data is an intriguing research topic.",
"Alternative approaches may be based on penalizing the difference between networks at adjacent time points, although careful investigation is required to determine how tractable these methods are for temporal-dependent groups.",
"In addition to time-varying networks, the temporal-dependent hub model can also be extended in the following directions: first, a group may contain zero or multiple hubs.",
"Second, multiple groups may exist at the same time (with some of these groups being unobserved).",
"These generalizations however will significantly increase model complexity.",
"Therefore, the total number of possible leaders needs to be limited.",
"A method by the author and a collaborator [36] was proposed to reduce this upper bound.",
"More test-based and penalization methods are under development.",
"Furthermore, we also plan to investigate the theoretical properties of the proposed model.",
"When the size of the network is fixed and the number of observed groups goes to infinity, the theoretical properties of the MLE may be studied via a standard theory of the Markov chain.",
"The case that the size of the network also goes to infinity is more intriguing but more complicated since the number of parameters diverges.",
"Acknowledgments This research was supported by NSF Grant DMS 1513004.",
"We thank Dr. Richard Wrangham for sharing the research results of the Kibale Chimpanzee Project.",
"We thank Dr. Charles Weko for compiling the results from the chimpanzee project and preparing the data set.",
"Forward-backward algorithm in the E-steps We derive the forward-backward algorithm for the temporal-dependent hub model introduced in Section REF .",
"Before proceeding, we state two propositions of Bayesian networks.",
"These results (or the equivalent forms) can be found in a standard textbook or tutorial on Bayesian networks, for example, [19].",
"Here we follow [14].",
"Proposition A.1 Each node is conditionally independent from its non-descendents given its parents.",
"Here node $X$ is a parent of another node $Y$ if there is a directed arc from $X$ to $Y$ and if so, $Y$ is a child of $X$ .",
"The descendents of a node are its children, children's children, etc.",
"Proposition A.2 Two disjoint sets of nodes $\\mathcal {A}$ and $\\mathcal {B}$ are conditionally independent given another set $\\mathcal {C}$ , if on every undirected path between a node in $\\mathcal {A}$ and a node in $\\mathcal {B}$ , there is a node $X$ in $\\mathcal {C}$ that is not a child of both the previous and following nodes in the path.",
"Define $G^{s:t}$ as a collection of groups from time $s$ to time $t$ .",
"Let $a_i^t=\\mathbb {P}(z^t=i,G^{1:t})$ .",
"Then, $a_i^t & = \\sum _{k=1}^n\\mathbb {P}(z^t=i,z^{t-1}=k,G^{1:t-1},G^t) \\\\& = \\sum _{k=1}^n\\mathbb {P}(z^{t-1}=k,G^{1:t-1})\\mathbb {P}(z^t=i|z^{t-1}=k,G^{1:t-1})\\mathbb {P}(G^t|z^t=i,z^{t-1}=k,G^{1:t-1}) \\\\& = \\sum _{k=1}^n a^{t-1}_k \\Phi _{ik} \\mathbb {P}(G^t|z^t=i,G^{t-1}).$ The last equation holds by Proposition REF .",
"Similarly, let $b_i^t=\\mathbb {P}(G^{t+1:T}|z^t=i, G^t)$ .",
"Then, $b_i^t & = \\sum _{k=1}^n \\mathbb {P}(z^{t+1}=k, G^{t+1},G^{t+2:T}|z^t=i, G^t) \\\\& = \\sum _{k=1}^n \\mathbb {P}(G^{t+2:T}|z^{t+1}=k,G^{t+1}, z^t=i, G^t)\\mathbb {P}(z^{t+1}=k|z^t=i, G^t)\\mathbb {P}(G^{t+1}|z^{t+1}=k,z^t=i, G^t) \\\\& = \\sum _{k=1}^n b^{t+1}_k \\Phi _{ki} \\mathbb {P}(G^{t+1}|z^{t+1}=k, G^t).$ In the last equation, $\\mathbb {P}(G^{t+2:T}|z^{t+1}=k,G^{t+1}, z^t=i, G^t)$ holds by Proposition REF .",
"This is because a path from $\\lbrace G^{t+2:T}\\rbrace $ to $\\lbrace z^t, G^t\\rbrace $ must pass $z^{t+1}$ or $G^{t+1}$ .",
"If it only passes one of these two variables, then we can take that variable as $X$ in Proposition REF .",
"If it passes both, then take $z^{t+1}$ as $X$ .",
"The rest of the last equation holds by REF .",
"The computation of $a$ and $b$ is essentially the same as in the classical forward-backward algorithm for the HMM with minor modifications.",
"Unlike the HMM, the dependence between the current and the previous groups requires another quantity $c$ .",
"Let $c_i^t=\\mathbb {P}(G^{t:T}|z^t=i, G^{t-1}).$ $c_i^t & = \\sum _{k=1}^n \\mathbb {P}(z^{t+1}=k,G^{t+1:T},G^t|z^t=i,G^{t-1}) \\\\& = \\sum _{k=1}^n \\mathbb {P}(G^{t+1:T}|z^{t+1}=k,G^t,z^t=i,G^{t-1}) \\mathbb {P}(z^{t+1}=k|z^t=i,G^{t-1}) \\mathbb {P}(G^t|z^{t+1}=k,z^t=i,G^{t-1}) \\\\& = \\sum _{k=1}^n c^{t+1}_k\\Phi _{ki} \\mathbb {P}(G^t|z^t=i,G^{t-1}).$ The last equation can be justified by a similar argument as before.",
"Since $& \\mathbb {P}(z^t=i,G^{1:T}) \\\\= & \\mathbb {P}(z^t=i,G^{1:t})\\mathbb {P}(G^{t+1:T}| z^t=i, G^{1:t}) \\\\= & \\mathbb {P}(z^t=i,G^{1:t})\\mathbb {P}(G^{t+1:T}|z^t=i, G^t),$ $\\mathbb {P}(S_i^t=1|G) & = \\frac{a_i^t b_i^t}{\\sum _k a_k^t b_k^t}.$ Similarly, $& \\mathbb {P}(z^t=i,z^{t-1}=j,G^{1:T}) \\\\= & \\mathbb {P}(z^{t-1}=j,G^{1:t-1})\\mathbb {P}(z^t=i|z^{t-1}=j,G^{1:t-1})\\mathbb {P}(G^{t:T}|z^t=i,z^{t-1}=j,G^{1:t-1}) \\\\= & a_j^{t-1} \\Phi _{ij} \\mathbb {P}(G^{t:T}|z^t=i,G^{t-1}) ,$ which implies $\\mathbb {P}(S_i^t=1,S_j^{t-1}=1|G) & = \\frac{a_j^{t-1} \\Phi _{ij} c_i^t}{\\sum _{kl} a_l^{t-1} \\Phi _{kl} c_k^t}.$ Derivatives of $Q$ We give the first and second derivatives of $Q$ with respect to $\\alpha , \\beta , \\gamma , u_i$ and $\\theta _{ij}$ , which are used in the coordinate ascent method introduced in Section REF .",
"Define, $R_{i}^t &=\\mathbb {P}(S_i^t=1|G), \\\\V_{ij} &=\\sum _{t=2}^T \\mathbb {P}(S_i^t=1,S_j^{t-1}=1|G), \\\\D^1_{ij} &= R_i^{1} G_j^{1} +\\sum _{t=2}^T R_i^{t} (1-G_i^{t-1}) G_j^{t}, \\\\D^2_{ij} & =R_i^{1} (1-G_j^{1})+\\sum _{t=2}^T R_i^{t}(1-G_i^{t-1}) (1-G_j^{t}), \\\\D^3_{ij} & = \\sum _{t=2}^T R_i^{t}G_i^{t-1} G_j^{t-1} G_j^{t}, \\\\D^4_{ij} & = \\sum _{t=2}^T R_i^{t}G_i^{t-1} G_j^{t-1} (1-G_j^{t}), \\\\D^5_{ij} & = \\sum _{t=2}^T R_i^{t}G_i^{t-1} (1-G_j^{t-1}) G_j^{t}, \\\\D^6_{ij} & = \\sum _{t=2}^T R_i^{t}G_i^{t-1} (1-G_j^{t-1}) (1-G_j^{t}).$ Therefore, $Q = & \\sum _{i=1}^n R_i^{1} \\left[ u_i-\\log \\left\\lbrace \\sum _{k=1}^{n} \\exp (u_k) \\right\\rbrace \\right] \\\\& + \\sum _{i=1}^n V_{ii} \\left[u_i+\\alpha - \\log \\left\\lbrace \\sum _{k=1}^{n} \\exp (u_k+\\alpha I(k=i)) \\right\\rbrace \\right] \\\\& + \\sum _{i=1}^n \\sum _{j \\ne i}V_{ij} \\left[u_i- \\log \\left\\lbrace \\sum _{k=1}^{n} \\exp (u_k+\\alpha I(k=j)) \\right\\rbrace \\right] \\\\& + \\sum _{ij} \\left[ D^1_{ij} \\log \\frac{e^{\\theta _{ij}}}{1+e^{\\theta _{ij}}} + D^2_{ij} \\log \\frac{1}{1+e^{\\theta _{ij}}} \\right.",
"\\\\& \\quad \\quad + D^3 _{ij} \\log \\frac{e^{\\theta _{ij}+\\beta }}{1+e^{\\theta _{ij}+\\beta }} +D^4 _{ij} \\log \\frac{1}{1+e^{\\theta _{ij}+\\beta }} \\\\& \\quad \\quad + \\left.",
"D^5 _{ij} \\log \\frac{e^{\\theta _{ij}+\\gamma }}{1+e^{\\theta _{ij}+\\gamma }} +D^6 _{ij} \\log \\frac{1}{1+e^{\\theta _{ij}+\\gamma }} \\right] .$ The first and second order derivatives are given as follows, $\\frac{\\partial Q }{\\partial \\alpha } = & \\sum _{i=1}^nV_{ii} + \\sum _{j=1}^n \\left[\\sum _{i=1}^n V_{ij} \\right]\\left[- \\frac{\\exp (u_j+\\alpha )}{\\sum _{k=1}^{n} \\exp (u_k+\\alpha I(k=j)) } \\right], \\\\\\frac{\\partial ^2 Q }{\\partial \\alpha ^2}=& \\sum _{j=1}^n \\left[\\sum _{i=1}^n V_{ij} \\right]\\left[- \\frac{\\exp (u_j+\\alpha )\\sum _{k=1}^{n} \\exp (u_k+\\alpha I(k=j))-\\exp (u_j+\\alpha )\\exp (u_j+\\alpha )}{(\\sum _{k=1}^{n} \\exp (u_k+\\alpha I(k=j)))^2 }\\right], \\\\\\frac{\\partial Q }{\\partial u_r } = & R_r^{1} + \\left[ \\sum _{i=1}^n R_i^{1} \\right] \\left[-\\frac{\\exp (u_r)}{\\sum _{k=1}^{n} \\exp (u_k) } \\right], \\\\\\frac{\\partial ^2 Q }{\\partial u_r^2 } = & \\left[\\sum _{i=1}^n R_i^{1} \\right]\\left[-\\frac{\\exp (u_r)\\sum _{k=1}^{n} \\exp (u_k)-\\exp (u_r)\\exp (u_r)}{(\\sum _{k=1}^{n} \\exp (u_k))^2} \\right] \\\\& +\\sum _{j=1}^n \\left[\\sum _{i=1}^n V_{ij} \\right] \\left[ -\\frac{\\exp (u_r+\\alpha I(r=j))\\sum _{k=1}^{n} \\exp (u_k+\\alpha I(k=j))-(\\exp (u_r+\\alpha I(r=j)))^2}{(\\sum _{k=1}^{n} \\exp (u_k+\\alpha I(k=j)))^2 } \\right] \\\\& +\\sum _{j=1}^n V_{rj} + \\sum _{j=1}^n \\left[\\sum _{i=1}^n B_{ij} \\right] \\left[-\\frac{\\exp (u_r+\\alpha I(r=j))}{\\sum _{k=1}^{n} \\exp (u_k+\\alpha I(k=j)) } \\right], \\\\\\frac{\\partial Q}{\\partial \\beta } = & \\sum _{i \\ne j} D^3 _{ij}-( D^3 _{ij}+D^4_{ij}) \\frac{e^{\\theta _{ij}+\\beta }}{1+e^{\\theta _{ij}+\\beta }}, \\\\\\frac{\\partial ^2 Q}{\\partial \\beta ^2 } = & -\\sum _{i \\ne j} ( D^3 _{ij}+D^4_{ij}) \\frac{e^{\\theta _{ij}+\\beta }}{(1+e^{\\theta _{ij}+\\beta })^2}, \\\\\\frac{\\partial Q}{\\partial \\gamma } = & \\sum _{i \\ne j} D^5 _{ij}-( D^5 _{ij}+D^6_{ij}) \\frac{e^{\\theta _{ij}+\\gamma }}{1+e^{\\theta _{ij}+\\gamma }} , \\\\\\frac{\\partial ^2 Q}{\\partial \\gamma ^2 } = & -\\sum _{i \\ne j} ( D^5 _{ij}+D^6_{ij}) \\frac{e^{\\theta _{ij}+\\gamma }}{(1+e^{\\theta _{ij}+\\gamma })^2}, \\\\\\frac{\\partial Q}{\\partial \\theta _{ij}} = & (D^1_{ij}+D^1_{ji})-(D^1_{ij}+D^1_{ji}+D^2_{ij}+D^2_{ji}) \\frac{e^{\\theta _{ij}}}{1+e^{\\theta _{ij}}} \\\\& + (D^3_{ij}+D^3_{ji})-(D^3_{ij}+D^3_{ji}+D^4_{ij}+D^4_{ji}) \\frac{e^{\\theta _{ij}+\\beta }}{1+e^{\\theta _{ij}+\\beta }} \\\\& + (D^5_{ij}+D^5_{ji})-(D^5_{ij}+D^5_{ji}+D^6_{ij}+D^6_{ji}) \\frac{e^{\\theta _{ij}+\\gamma }}{1+e^{\\theta _{ij}+\\gamma }}, \\\\\\frac{\\partial ^2 Q}{\\partial \\theta _{ij}^2} = & -(D^1_{ij}+D^1_{ji}+D^2_{ij}+D^2_{ji}) \\frac{e^{\\theta _{ij}}}{(1+e^{\\theta _{ij}})^2} \\\\& -(D^3_{ij}+D^3_{ji}+D^4_{ij}+D^4_{ji}) \\frac{e^{\\theta _{ij}+\\beta }}{(1+e^{\\theta _{ij}+\\beta })^2} \\\\& -(D^5_{ij}+D^5_{ji}+D^6_{ij}+D^6_{ji}) \\frac{e^{\\theta _{ij}+\\gamma }}{(1+e^{\\theta _{ij}+\\gamma })^2}.$"
],
[
"Model fitting",
"In this section, we propose an algorithm to find the maximum likelihood estimators (MLEs) for $\\alpha ,\\beta ,\\gamma ,u$ and $\\theta $ .",
"With $S$ being the latent variables, an expectation-maximization (EM) algorithm will be used for this problem.",
"The EM algorithm maximizes the marginal likelihood of the observed data, which in our case is $G$ , by iteratively applying an E-step and an M-step.",
"Let $\\alpha ^{\\textnormal {old}},\\beta ^{\\textnormal {old}},\\gamma ^{\\textnormal {old}},u^{\\textnormal {old}}$ and $\\theta ^{\\textnormal {old}}$ be the estimates in the current iteration.",
"In the E-step, we calculate the conditional expectation of the complete log-likelihood given $G$ under the current estimate.",
"That is, $Q \\overset{\\Delta }{=} Q(\\alpha ,\\beta ,\\gamma ,u,\\theta | \\alpha ^{\\textnormal {old}},\\beta ^{\\textnormal {old}},\\gamma ^{\\textnormal {old}},u^{\\textnormal {old}},\\theta ^{\\textnormal {old}})= \\mathbb {E}_{\\alpha ^{\\textnormal {old}},\\beta ^{\\textnormal {old}},\\gamma ^{\\textnormal {old}},u^{\\textnormal {old}},\\theta ^{\\textnormal {old}}} \\left[ \\log \\mathbb {P} (S,G) | G \\right].$ In the M-step, we maximize this conditional expectation with respect to the unknown parameters.",
"That is, $(\\alpha ^{\\textnormal {new}},\\beta ^{\\textnormal {new}},\\gamma ^{\\textnormal {new}},u^{\\textnormal {new}},\\theta ^{\\textnormal {new}} ) = \\operatornamewithlimits{arg\\,max}_{\\alpha ,\\beta ,\\gamma ,u,\\theta } \\,\\, Q(\\alpha ,\\beta ,\\gamma ,u,\\theta | \\alpha ^{\\textnormal {old}},\\beta ^{\\textnormal {old}},\\gamma ^{\\textnormal {old}},u^{\\textnormal {old}},\\theta ^{\\textnormal {old}}).$ It has been proved by [38] that the EM algorithm converges to a local maximizer of the marginal likelihood.",
"(Refer to [25] for a comprehensive introduction to this algorithm).",
"We now give details of the two steps in our context."
],
[
"E-step",
"Since the complete log-likelihood $\\log \\mathbb {P} (S,G)$ is a linear function of $S_i^t \\,\\, (t=1,...,T;i=1,...,n)$ and $S_i^{t}S_j^{t-1} \\,\\,(t=2,...,T;i=1,...,n;j=1,...,n) $ , the computation of its conditional expectation is equivalent to calculating $\\mathbb {P}(S_i^t=1|G)$ and $\\mathbb {P}(S_i^t=1,S_j^{t-1}=1|G)$ .",
"From now on, all conditional probabilities are defined under the current estimates.",
"A brute-force calculation of these probabilities, such as $\\mathbb {P}(S_i^t=1|G) = \\mathbb {P} (z^t=i|G)=\\frac{\\sum _{z^1}\\cdots \\sum _{z^{t-1}}\\sum _{z^{t+1}}\\cdots \\sum _{z^t} \\mathbb {P}(z^1,...,z^{t-1},z^t=i,z^{t+1},...,z^T,G)}{\\mathbb {P}(G)},$ is infeasible since the numerator involves a sum of $n^{T-1}$ terms.",
"This is because $G^1,...,G^T$ are not independent according to our model.",
"An efficient algorithm is needed for all practical purposes.",
"The temporal-dependent hub model is similar to the hidden Markov model (HMM) (Figure REF ).",
"A polynomial-time algorithm for this model, called the forward-backward algorithm, was developed for computing the conditional probabilities.",
"See [31], [14] for tutorials on HMMs and this algorithm.",
"In the HMM, the observed variable at time $t$ only depends on the corresponding hidden state.",
"But in our model, $G^t$ depends on both the current leader $z^t$ and the previous group $G^{t-1}$ .",
"We develop a new forward-backward algorithm for our model, which has more steps than the original algorithm but is also polynomial-time.",
"We describe the algorithm here (see the Appendix for detailed derivation and justification).",
"Define $a=[a_i^t],b=[b_i^t]$ and $c=[c_i^t]$ as $T\\times n$ matrices.",
"These matrices are computed by the following recursive procedures.",
"$a_i^1 & = \\mathbb {P}(z^1=i,G^1) \\quad (i=1,...,n).",
"\\\\a_i^t & =\\sum _{k=1}^n a^{t-1}_k \\Phi _{ik} \\mathbb {P}(G^t|z^t=i,G^{t-1}) \\quad (t=2,...,T;i=1,...,n).",
"\\\\b_i^T & = 1 \\quad (i=1,...,n).",
"\\\\b_i^t & =\\sum _{k=1}^n b^{t+1}_k \\Phi _{ki} \\mathbb {P}(G^{t+1}|z^{t+1}=k, G^t) \\quad (t=T-1,...,1;i=1,...,n).",
"\\\\c_i^T & = \\mathbb {P}(G^T|z^{T}=i,G^{T-1}) \\quad (i=1,...,n).",
"\\\\c_i^t & = \\sum _{k=1}^n c^{t+1}_k \\Phi _{ki} \\mathbb {P}(G^t|z^t=i,G^{t-1}) \\quad (t=T-1,...,2;i=1,...,n).$ The matrices $a$ , $b$ and $c$ should not be confused with the matrices $A$ , $B$ and $C$ introduced in Section .",
"The symbols are case-sensitive throughout the paper.",
"With these quantities, $\\mathbb {P}(S_i^t=1|G) & = \\frac{a_i^t b_i^t}{\\sum _k a_k^t b_k^t} \\quad (t=2,...,T;i=1,...,n).",
"\\\\\\mathbb {P}(S_i^t=1,S_j^{t-1}=1|G) & = \\frac{a_j^{t-1} \\Phi _{ij} c_i^t}{\\sum _{kl} a_l^{t-1} \\Phi _{kl} c_k^t} \\quad (t=2,...,T;i=1,...,n;j=1,...,n).$ The complexity of this algorithm is $O(Tn^2)$ .",
"Note that the first row of $c$ is undefined but also unused.",
"Also note that the elements of $a,b$ and $c$ will quickly vanish as the recursions progress.",
"Therefore, we renormalize each row to sum to one at each step.",
"It can easily be verified that this normalization does not affect the conditional probabilities.",
"Finally, we emphasize that this algorithm gives the exact values of the conditional probabilities in a fixed number of steps – i.e., it is not an approximate or iterative method."
],
[
"M-step",
"The M-step is somewhat routine compared to the E-step.",
"First, it is clear that $\\lbrace \\alpha ,u \\rbrace $ and $\\lbrace \\beta ,\\gamma ,\\theta \\rbrace $ can be handled separately.",
"We apply the coordinate ascent method (see [6] for a comprehensive introduction) to iteratively update $\\alpha $ and $u$ , as well as $\\beta ,\\gamma $ and $\\theta $ .",
"Since the complete log-likelihood is concave and so is $Q$ , coordinate ascent can guarantee a global maximizer.",
"At each step, we optimize the log-likelihood over parameter one by one with the other parameters being fixed.",
"The procedure is repeated until convergence.",
"At each step, we use the standard Newton-Raphson method to solve each individual optimization problem.",
"Specifically, for a parameter $\\phi $ (here $\\phi $ can represent $\\alpha $ , $\\beta $ , $\\gamma $ , $u_i$ or $\\theta _{ij} \\,\\, (i < j)$ ), the estimate at $(m+1)$ -th iteration is updated by the following formula given its estimate at $m$ -th iteration: $\\hat{\\phi }_{m+1}=\\hat{\\phi }_m-\\left( \\frac{\\partial ^2 Q}{\\partial \\phi ^2 } \\bigg |_{\\phi =\\hat{\\phi }_m} \\right)^{-1} \\left( \\frac{\\partial Q}{\\partial \\phi } \\bigg |_{\\phi =\\hat{\\phi }_m}\\right).",
"$ The calculation of these derivatives is straightforward but tedious, so we provide the details in the Appendix.",
"As shown in Section REF , the model is not identifiable with respect to $u$ .",
"A standard solution to this problem is to set some $u_i \\equiv 0$ .",
"But it does not work for our case.",
"This is because for small data sets, some $\\hat{\\rho }_i$ estimated by the EM algorithm may be zero, implying that $v_i$ never became the leader.",
"Furthermore, these zero $\\rho _i$ cannot be predetermined since the leaders are unobserved.",
"We observe that without constraint on $u_i$ , the algorithm converges to different $\\hat{u}$ with different initial values, but the corresponding $\\hat{\\rho }$ will be the same.",
"Therefore, identifiability is not an issue for model fitting."
],
[
"Initial value",
"As with many optimization algorithms, the EM algorithm is not guaranteed to find the global maximizer.",
"Ideally, one should use multiple random initial values and find the best solution by comparing the marginal likelihoods $\\mathbb {P}(G)$ under the corresponding estimates.",
"In principle, $\\mathbb {P}(G)$ can be computed by $\\sum _k a_k^t b_k^t$ , as shown in Section REF .",
"But the marginal likelihood vanishes quickly, even with a moderate $T$ .",
"Note that we cannot renormalize $a$ and $b$ for the purpose of computing $\\mathbb {P}(G)$ .",
"Therefore, we use the half weight index [9], [7] as the initial value of $A$ , which is defined by $H_{ij} = \\frac{2\\sum _t G_i^{t} G_j^{t}}{\\sum _t G_i^{t}+\\sum _t G_j^{t}}.$ This measure estimates the conditional probability that two nodes co-occur given that one of them is observed, which is a reasonable initial guess of the strength of links.",
"Furthermore, we use zero for the initial values of $\\alpha , \\beta $ and $\\gamma $ , and $\\sum _{t} G_i^t/T $ for the initial value of $\\rho _i$ ."
],
[
"Simulation studies",
"In all simulation studies, we fix the size of the network to be $n=50$ and set $\\beta =3$ and $\\gamma =-1$ .",
"We generate $u_i$ as independently and identically distributed variables with $N(0,2)$ and $\\rho _i= u_i/\\sum _k u_k $ .",
"The parameters $\\theta _{ij}\\,\\, (i<j)$ are generated independently with $N(-2,1)$ .",
"We generate $\\theta _{ij}$ in this way to control the average link density of the network ($\\approx $ 0.12), which is more realistic than a symmetric setting, i.e., $\\theta _{ij} \\sim N(0,1)$ .",
"For clarification, we will not use the prior information on $u$ and $\\theta $ in our estimating procedure.",
"That is, we still treat $u$ and $\\theta $ as unknown fixed parameters in the algorithm.",
"We generate them as random variables for the whole purpose of adding more variations to the parameter setup in our study.",
"We consider three levels of $\\alpha =\\log ((n-1)/2),\\log (n-1),\\log (2(n-1))$ , which correspond to a leader from the previous group remaining unchanged in the current group with probabilities $1/3,1/2,2/3$ on average.",
"For each $\\alpha $ we try five different sample sizes, $T=1000,1500, 2000, 2500$ and 3000.",
"Figure: Average RMSEs (across 100 replicates) for the estimated AA from the classical hub model (HM) and the temporal-dependent hub model (TDHM).Table: Average RMSEs for the estimated AA from the classical hub model (HM) and the temporal-dependent hub model (TDHM).",
"Standard deviations ×10 3 \\times 10^3 in parentheses.Figure REF shows the average root of mean squared errors (RMSEs) for the estimated $A$ over 100 replicates.",
"For each simulation, we compare two methods, the classical hub model (HM) and the temporal-dependent hub model (TDHM).",
"We assume the leaders are unknown under both models.",
"Table REF provides the same information (with standard deviations) in numerical form.",
"From Figure REF and Table REF , our first observation is simply that the RMSEs decrease as sample sizes increase, which is consistent with common sense in statistics.",
"Second, the RMSEs for all the parameters increase as $\\alpha $ increases.",
"This phenomenon can be interpreted as follows: with a larger value of $\\alpha $ , the correlation between adjacent groups becomes stronger and hence the effective sample size becomes smaller.",
"The ratio of the sample size to the number of parameters decreases with $\\alpha $ , which makes inferences more difficult.",
"Third, the temporal-dependent hub model always outperforms the hub model.",
"Moreover, the discrepancy between the temporal-dependent hub model estimates and the corresponding hub model estimates becomes larger as $\\alpha $ increase.",
"This is because the behavior of the temporal-dependent model deviates more from the classical hub model as $\\alpha $ increases.",
"The standard deviations and means show a similar trend.",
"That is, the standard deviations decrease as $n$ increases and $\\alpha $ decreases.",
"The standard deviations for the temporal-dependent hub model estimates are comparable to or slightly smaller than those of the hub model estimates."
],
[
"A data example of group dynamics in chimpanzees",
"Behavioral ecologists become increasingly interested in using social network analysis to understand social organization and animal behavior [3], [37], [8], [11].",
"The social relationships are usually inferred by using certain association metrics (e.g., the half weight index) on grouped data.",
"As indicated in the Introduction however, it is unclear how the inferred network relates to the observed groups without specifying a model.",
"In this section, we study a data set of groups formed by chimpanzees by the temporal-dependent hub model.",
"This data set is compiled from the results of the Kibale Chimpanzee Project, which is a long-term field study of the behavior, ecology and physiology of wild chimpanzees in the Kanyawara region of Kibale National Park, southwestern Uganda (https://kibalechimpanzees.wordpress.com/).",
"Our analysis focuses on grouping behavior.",
"We analyze the grouped data collected from January 1, 2009 to June 30, 2009 [20].",
"The group identification was taken at 1 p.m. daily during this time period.",
"If there is no group observed at 1 p.m. for a given day, it is not included in the data.",
"Only one group is observed at 1 p.m. in 75.29% of the remaining days over this period of six months.",
"In the other days, multiple groups (usually two) are observed at 1 p.m. For these cases, we keep the group that has the most overlap with the previous group in our analysis.",
"We use the Jaccard index to measure the overlap between two groups $G^{t-1}$ and $G^{t}$ , $J(G^{t-1},G^{t})=\\frac{\\sum _{j=1}^n G^{t-1}_j G^{t}_j }{\\sum _{j=1}^n [ n-(1-G^{t-1}_j)(1-G^{t}_j) ]},$ where the numerator is the size of the intersection of two groups and the denominator is the size of their union.",
"One may refer to [23] for an introduction to this measure.",
"Moreover, five chimpanzees never appear in any group and thus are removed.",
"After the preprocessing, the data set contains 170 groups with 40 chimpanzees.",
"Figure REF illustrates the data set in grayscale with rows representing the groups over time and columns representing the chimpanzees.",
"Black indicates $G^{t}_i=1$ at location $(t,i)$ while white indicates $G^{t}_i=0$ .",
"The pattern in Figure REF clearly demonstrates the existence of dependency between groups.",
"Figure REF also shows the inferred grouped leaders indicated in red with the inferred segments separated by blue lines.",
"By the inferred grouped leader for $G^t$ , we mean that the chimpanzee with the highest posterior probability will be the leader given $G^t$ .",
"As shown in Figure REF , the leaders retain a certain level of stability, which is consistent with the estimates of $\\alpha $ ($=1.7291$ ).",
"Also, recall that by our definition, a new segment starts if the current leader is not within the previous group.",
"From Figure REF , the inferred segments are coincident with the visualization of the data set.",
"The estimated values of the adjustment factors, $\\hat{\\beta }=2.5703$ and $\\hat{\\gamma }=-0.1922$ .",
"The magnitude of $\\hat{\\beta }$ is larger than that of $\\hat{\\gamma }$ , which suggests individuals have a stronger tendency to join a group than leave a group.",
"In other words, the groups may start with small size and grow larger over time.",
"This phenomenon is shown (Figure REF ).",
"Figure: Visualization of the chimpanzee data.",
"The 170 rows represent the groups over time and the 40 columns represent chimpanzees.",
"Gray indicates the presence of the group membership.",
"Red indicates the inferred group leaders.",
"The blue lines separate the inferred segments.Figure: Grayscale plot for the estimated adjacency matrices from the chimpanzee data by the classical hub model and the temporal-dependent hub model.",
"The rows and the columns represent the 40 chimpanzees.",
"Darker colors indicate stronger relationships.",
"The red blocks indicate the biological clusters of chimpanzees.Figure REF shows the result of estimated adjacency matrices by the classical hub model and the temporal-dependent hub model.",
"As in the previous figure, the darker color indicates a higher value of $\\hat{A}_{ij}$ .",
"The red blocks indicate clusters of chimpanzees in a biological sense.",
"The first cluster consists of 12 adult males and each of the other nine clusters consists of an adult female and its children.",
"From the estimates by both the classical hub model and the temporal-dependent hub model, there are strong connections within these biological clusters.",
"Both estimates suggest that in this data set of chimpanzees, adult males usually do activities together but females usually stay with their children.",
"The two estimated adjacency matrices are different, however.",
"Generally speaking, without properly considering the temporal-dependence between groups, the estimates of the relationships between individuals by the classical hub model can be biased.",
"That is, an individual may choose to stay within or out of a group not solely based on its relationship with the group leader but also because of the inertia.",
"The overall graph density $(= 0.2286)$ of the estimated network by the classical hub model is larger than the corresponding value $(= 0.1973)$ of the temporal-dependent hub model.",
"This is consistent with the fact that the magnitude of $\\hat{\\beta }$ is larger than the magnitude of $\\hat{\\gamma }$ .",
"Since the classical hub model does not incorporate the adjustment factors, bias is introduced to certain $\\hat{A}_{ij}$ so the model can match the overall frequency of occurrences for the individuals.",
"The significance of $\\alpha $ , $\\beta $ and $\\gamma $ is tested by the parametric bootstrap method [10].",
"Specifically, we generate 5000 independent data sets from the fitted temporal-dependent hub model to the original data and compute the MLEs for each simulated data set.",
"The parametric bootstrap was applied to HMMs and showed a good performance [34].",
"Figure REF shows the histograms of the MLEs for $\\alpha $ , $\\beta $ and $\\gamma $ .",
"The 95% bootstrap confidence intervals for $\\alpha $ , $\\beta $ and $\\gamma $ are (1.2177, 1.9774), (2.0710, 2.8944) and (-0.5208, 0.0410), which shows that the effects of $\\alpha $ and $\\beta $ are significant while $\\gamma $ is not at the 0.05 significance level.",
"This further supports the observation in the previous paragraph – chimpanzees have a stronger tendency to join a group than to leave a group in this data set.",
"Figure: Histograms of estimates from parametric bootstrap samples.",
"Red lines indicate the estimated values from the original data set."
],
[
"Summary and discussion",
"In this article, we generalize the idea of the hub model and propose a novel model for temporal-dependent grouped data.",
"This new model allows for dependency between groups.",
"Specifically, the group leaders follow a Markov chain and a group is either a transformation of the previous group or a new start, depending on whether the current leader is within the previous group.",
"An EM algorithm is applied to this model with a polynomial-time algorithm being developed for the E-step.",
"The setup of our model is different from some work on estimating time-varying networks by graphical models, e.g., [21] for discrete data and [41] for continuous data.",
"These papers assume that the observations are independent and the latent network changes smoothly or is piecewise constant.",
"In this paper, we instead focus on the dependence between groups.",
"Ideally, both aspects – dependence between groups and changes in networks – need to be considered in modeling.",
"This should be plausible when the sample size, i.e., the number of observed groups, is large.",
"When the sample size is moderate (as in the chimpanzee data set), the length of the time interval between observation plays a key role in determining which aspect is more important.",
"If the time interval is short, then dependence between groups is significant but the changes in the latent network are likely to be minor since the overall time window is not long.",
"On the contrary, the dependence between groups becomes weak when the time interval is long.",
"For future work, we plan to study the time-varying effect on the latent network for the grouped data.",
"When changepoints exist, a single network cannot accurately represent the link strengths in different time windows.",
"A change-point analysis for temporal-dependent grouped data is an intriguing research topic.",
"Alternative approaches may be based on penalizing the difference between networks at adjacent time points, although careful investigation is required to determine how tractable these methods are for temporal-dependent groups.",
"In addition to time-varying networks, the temporal-dependent hub model can also be extended in the following directions: first, a group may contain zero or multiple hubs.",
"Second, multiple groups may exist at the same time (with some of these groups being unobserved).",
"These generalizations however will significantly increase model complexity.",
"Therefore, the total number of possible leaders needs to be limited.",
"A method by the author and a collaborator [36] was proposed to reduce this upper bound.",
"More test-based and penalization methods are under development.",
"Furthermore, we also plan to investigate the theoretical properties of the proposed model.",
"When the size of the network is fixed and the number of observed groups goes to infinity, the theoretical properties of the MLE may be studied via a standard theory of the Markov chain.",
"The case that the size of the network also goes to infinity is more intriguing but more complicated since the number of parameters diverges."
],
[
"Acknowledgments",
"This research was supported by NSF Grant DMS 1513004.",
"We thank Dr. Richard Wrangham for sharing the research results of the Kibale Chimpanzee Project.",
"We thank Dr. Charles Weko for compiling the results from the chimpanzee project and preparing the data set."
],
[
"Forward-backward algorithm in the E-steps",
"We derive the forward-backward algorithm for the temporal-dependent hub model introduced in Section REF .",
"Before proceeding, we state two propositions of Bayesian networks.",
"These results (or the equivalent forms) can be found in a standard textbook or tutorial on Bayesian networks, for example, [19].",
"Here we follow [14].",
"Proposition A.1 Each node is conditionally independent from its non-descendents given its parents.",
"Here node $X$ is a parent of another node $Y$ if there is a directed arc from $X$ to $Y$ and if so, $Y$ is a child of $X$ .",
"The descendents of a node are its children, children's children, etc.",
"Proposition A.2 Two disjoint sets of nodes $\\mathcal {A}$ and $\\mathcal {B}$ are conditionally independent given another set $\\mathcal {C}$ , if on every undirected path between a node in $\\mathcal {A}$ and a node in $\\mathcal {B}$ , there is a node $X$ in $\\mathcal {C}$ that is not a child of both the previous and following nodes in the path.",
"Define $G^{s:t}$ as a collection of groups from time $s$ to time $t$ .",
"Let $a_i^t=\\mathbb {P}(z^t=i,G^{1:t})$ .",
"Then, $a_i^t & = \\sum _{k=1}^n\\mathbb {P}(z^t=i,z^{t-1}=k,G^{1:t-1},G^t) \\\\& = \\sum _{k=1}^n\\mathbb {P}(z^{t-1}=k,G^{1:t-1})\\mathbb {P}(z^t=i|z^{t-1}=k,G^{1:t-1})\\mathbb {P}(G^t|z^t=i,z^{t-1}=k,G^{1:t-1}) \\\\& = \\sum _{k=1}^n a^{t-1}_k \\Phi _{ik} \\mathbb {P}(G^t|z^t=i,G^{t-1}).$ The last equation holds by Proposition REF .",
"Similarly, let $b_i^t=\\mathbb {P}(G^{t+1:T}|z^t=i, G^t)$ .",
"Then, $b_i^t & = \\sum _{k=1}^n \\mathbb {P}(z^{t+1}=k, G^{t+1},G^{t+2:T}|z^t=i, G^t) \\\\& = \\sum _{k=1}^n \\mathbb {P}(G^{t+2:T}|z^{t+1}=k,G^{t+1}, z^t=i, G^t)\\mathbb {P}(z^{t+1}=k|z^t=i, G^t)\\mathbb {P}(G^{t+1}|z^{t+1}=k,z^t=i, G^t) \\\\& = \\sum _{k=1}^n b^{t+1}_k \\Phi _{ki} \\mathbb {P}(G^{t+1}|z^{t+1}=k, G^t).$ In the last equation, $\\mathbb {P}(G^{t+2:T}|z^{t+1}=k,G^{t+1}, z^t=i, G^t)$ holds by Proposition REF .",
"This is because a path from $\\lbrace G^{t+2:T}\\rbrace $ to $\\lbrace z^t, G^t\\rbrace $ must pass $z^{t+1}$ or $G^{t+1}$ .",
"If it only passes one of these two variables, then we can take that variable as $X$ in Proposition REF .",
"If it passes both, then take $z^{t+1}$ as $X$ .",
"The rest of the last equation holds by REF .",
"The computation of $a$ and $b$ is essentially the same as in the classical forward-backward algorithm for the HMM with minor modifications.",
"Unlike the HMM, the dependence between the current and the previous groups requires another quantity $c$ .",
"Let $c_i^t=\\mathbb {P}(G^{t:T}|z^t=i, G^{t-1}).$ $c_i^t & = \\sum _{k=1}^n \\mathbb {P}(z^{t+1}=k,G^{t+1:T},G^t|z^t=i,G^{t-1}) \\\\& = \\sum _{k=1}^n \\mathbb {P}(G^{t+1:T}|z^{t+1}=k,G^t,z^t=i,G^{t-1}) \\mathbb {P}(z^{t+1}=k|z^t=i,G^{t-1}) \\mathbb {P}(G^t|z^{t+1}=k,z^t=i,G^{t-1}) \\\\& = \\sum _{k=1}^n c^{t+1}_k\\Phi _{ki} \\mathbb {P}(G^t|z^t=i,G^{t-1}).$ The last equation can be justified by a similar argument as before.",
"Since $& \\mathbb {P}(z^t=i,G^{1:T}) \\\\= & \\mathbb {P}(z^t=i,G^{1:t})\\mathbb {P}(G^{t+1:T}| z^t=i, G^{1:t}) \\\\= & \\mathbb {P}(z^t=i,G^{1:t})\\mathbb {P}(G^{t+1:T}|z^t=i, G^t),$ $\\mathbb {P}(S_i^t=1|G) & = \\frac{a_i^t b_i^t}{\\sum _k a_k^t b_k^t}.$ Similarly, $& \\mathbb {P}(z^t=i,z^{t-1}=j,G^{1:T}) \\\\= & \\mathbb {P}(z^{t-1}=j,G^{1:t-1})\\mathbb {P}(z^t=i|z^{t-1}=j,G^{1:t-1})\\mathbb {P}(G^{t:T}|z^t=i,z^{t-1}=j,G^{1:t-1}) \\\\= & a_j^{t-1} \\Phi _{ij} \\mathbb {P}(G^{t:T}|z^t=i,G^{t-1}) ,$ which implies $\\mathbb {P}(S_i^t=1,S_j^{t-1}=1|G) & = \\frac{a_j^{t-1} \\Phi _{ij} c_i^t}{\\sum _{kl} a_l^{t-1} \\Phi _{kl} c_k^t}.$"
],
[
"Derivatives of $Q$",
"We give the first and second derivatives of $Q$ with respect to $\\alpha , \\beta , \\gamma , u_i$ and $\\theta _{ij}$ , which are used in the coordinate ascent method introduced in Section REF .",
"Define, $R_{i}^t &=\\mathbb {P}(S_i^t=1|G), \\\\V_{ij} &=\\sum _{t=2}^T \\mathbb {P}(S_i^t=1,S_j^{t-1}=1|G), \\\\D^1_{ij} &= R_i^{1} G_j^{1} +\\sum _{t=2}^T R_i^{t} (1-G_i^{t-1}) G_j^{t}, \\\\D^2_{ij} & =R_i^{1} (1-G_j^{1})+\\sum _{t=2}^T R_i^{t}(1-G_i^{t-1}) (1-G_j^{t}), \\\\D^3_{ij} & = \\sum _{t=2}^T R_i^{t}G_i^{t-1} G_j^{t-1} G_j^{t}, \\\\D^4_{ij} & = \\sum _{t=2}^T R_i^{t}G_i^{t-1} G_j^{t-1} (1-G_j^{t}), \\\\D^5_{ij} & = \\sum _{t=2}^T R_i^{t}G_i^{t-1} (1-G_j^{t-1}) G_j^{t}, \\\\D^6_{ij} & = \\sum _{t=2}^T R_i^{t}G_i^{t-1} (1-G_j^{t-1}) (1-G_j^{t}).$ Therefore, $Q = & \\sum _{i=1}^n R_i^{1} \\left[ u_i-\\log \\left\\lbrace \\sum _{k=1}^{n} \\exp (u_k) \\right\\rbrace \\right] \\\\& + \\sum _{i=1}^n V_{ii} \\left[u_i+\\alpha - \\log \\left\\lbrace \\sum _{k=1}^{n} \\exp (u_k+\\alpha I(k=i)) \\right\\rbrace \\right] \\\\& + \\sum _{i=1}^n \\sum _{j \\ne i}V_{ij} \\left[u_i- \\log \\left\\lbrace \\sum _{k=1}^{n} \\exp (u_k+\\alpha I(k=j)) \\right\\rbrace \\right] \\\\& + \\sum _{ij} \\left[ D^1_{ij} \\log \\frac{e^{\\theta _{ij}}}{1+e^{\\theta _{ij}}} + D^2_{ij} \\log \\frac{1}{1+e^{\\theta _{ij}}} \\right.",
"\\\\& \\quad \\quad + D^3 _{ij} \\log \\frac{e^{\\theta _{ij}+\\beta }}{1+e^{\\theta _{ij}+\\beta }} +D^4 _{ij} \\log \\frac{1}{1+e^{\\theta _{ij}+\\beta }} \\\\& \\quad \\quad + \\left.",
"D^5 _{ij} \\log \\frac{e^{\\theta _{ij}+\\gamma }}{1+e^{\\theta _{ij}+\\gamma }} +D^6 _{ij} \\log \\frac{1}{1+e^{\\theta _{ij}+\\gamma }} \\right] .$ The first and second order derivatives are given as follows, $\\frac{\\partial Q }{\\partial \\alpha } = & \\sum _{i=1}^nV_{ii} + \\sum _{j=1}^n \\left[\\sum _{i=1}^n V_{ij} \\right]\\left[- \\frac{\\exp (u_j+\\alpha )}{\\sum _{k=1}^{n} \\exp (u_k+\\alpha I(k=j)) } \\right], \\\\\\frac{\\partial ^2 Q }{\\partial \\alpha ^2}=& \\sum _{j=1}^n \\left[\\sum _{i=1}^n V_{ij} \\right]\\left[- \\frac{\\exp (u_j+\\alpha )\\sum _{k=1}^{n} \\exp (u_k+\\alpha I(k=j))-\\exp (u_j+\\alpha )\\exp (u_j+\\alpha )}{(\\sum _{k=1}^{n} \\exp (u_k+\\alpha I(k=j)))^2 }\\right], \\\\\\frac{\\partial Q }{\\partial u_r } = & R_r^{1} + \\left[ \\sum _{i=1}^n R_i^{1} \\right] \\left[-\\frac{\\exp (u_r)}{\\sum _{k=1}^{n} \\exp (u_k) } \\right], \\\\\\frac{\\partial ^2 Q }{\\partial u_r^2 } = & \\left[\\sum _{i=1}^n R_i^{1} \\right]\\left[-\\frac{\\exp (u_r)\\sum _{k=1}^{n} \\exp (u_k)-\\exp (u_r)\\exp (u_r)}{(\\sum _{k=1}^{n} \\exp (u_k))^2} \\right] \\\\& +\\sum _{j=1}^n \\left[\\sum _{i=1}^n V_{ij} \\right] \\left[ -\\frac{\\exp (u_r+\\alpha I(r=j))\\sum _{k=1}^{n} \\exp (u_k+\\alpha I(k=j))-(\\exp (u_r+\\alpha I(r=j)))^2}{(\\sum _{k=1}^{n} \\exp (u_k+\\alpha I(k=j)))^2 } \\right] \\\\& +\\sum _{j=1}^n V_{rj} + \\sum _{j=1}^n \\left[\\sum _{i=1}^n B_{ij} \\right] \\left[-\\frac{\\exp (u_r+\\alpha I(r=j))}{\\sum _{k=1}^{n} \\exp (u_k+\\alpha I(k=j)) } \\right], \\\\\\frac{\\partial Q}{\\partial \\beta } = & \\sum _{i \\ne j} D^3 _{ij}-( D^3 _{ij}+D^4_{ij}) \\frac{e^{\\theta _{ij}+\\beta }}{1+e^{\\theta _{ij}+\\beta }}, \\\\\\frac{\\partial ^2 Q}{\\partial \\beta ^2 } = & -\\sum _{i \\ne j} ( D^3 _{ij}+D^4_{ij}) \\frac{e^{\\theta _{ij}+\\beta }}{(1+e^{\\theta _{ij}+\\beta })^2}, \\\\\\frac{\\partial Q}{\\partial \\gamma } = & \\sum _{i \\ne j} D^5 _{ij}-( D^5 _{ij}+D^6_{ij}) \\frac{e^{\\theta _{ij}+\\gamma }}{1+e^{\\theta _{ij}+\\gamma }} , \\\\\\frac{\\partial ^2 Q}{\\partial \\gamma ^2 } = & -\\sum _{i \\ne j} ( D^5 _{ij}+D^6_{ij}) \\frac{e^{\\theta _{ij}+\\gamma }}{(1+e^{\\theta _{ij}+\\gamma })^2}, \\\\\\frac{\\partial Q}{\\partial \\theta _{ij}} = & (D^1_{ij}+D^1_{ji})-(D^1_{ij}+D^1_{ji}+D^2_{ij}+D^2_{ji}) \\frac{e^{\\theta _{ij}}}{1+e^{\\theta _{ij}}} \\\\& + (D^3_{ij}+D^3_{ji})-(D^3_{ij}+D^3_{ji}+D^4_{ij}+D^4_{ji}) \\frac{e^{\\theta _{ij}+\\beta }}{1+e^{\\theta _{ij}+\\beta }} \\\\& + (D^5_{ij}+D^5_{ji})-(D^5_{ij}+D^5_{ji}+D^6_{ij}+D^6_{ji}) \\frac{e^{\\theta _{ij}+\\gamma }}{1+e^{\\theta _{ij}+\\gamma }}, \\\\\\frac{\\partial ^2 Q}{\\partial \\theta _{ij}^2} = & -(D^1_{ij}+D^1_{ji}+D^2_{ij}+D^2_{ji}) \\frac{e^{\\theta _{ij}}}{(1+e^{\\theta _{ij}})^2} \\\\& -(D^3_{ij}+D^3_{ji}+D^4_{ij}+D^4_{ji}) \\frac{e^{\\theta _{ij}+\\beta }}{(1+e^{\\theta _{ij}+\\beta })^2} \\\\& -(D^5_{ij}+D^5_{ji}+D^6_{ij}+D^6_{ji}) \\frac{e^{\\theta _{ij}+\\gamma }}{(1+e^{\\theta _{ij}+\\gamma })^2}.$"
]
] | 1808.08478 |
[
[
"Efficient Single Image Super Resolution using Enhanced Learned Group\n Convolutions"
],
[
"Abstract Convolutional Neural Networks (CNNs) have demonstrated great results for the single-image super-resolution (SISR) problem.",
"Currently, most CNN algorithms promote deep and computationally expensive models to solve SISR.",
"However, we propose a novel SISR method that uses relatively less number of computations.",
"On training, we get group convolutions that have unused connections removed.",
"We have refined this system specifically for the task at hand by removing unnecessary modules from original CondenseNet.",
"Further, a reconstruction network consisting of deconvolutional layers has been used in order to upscale to high resolution.",
"All these steps significantly reduce the number of computations required at testing time.",
"Along with this, bicubic upsampled input is added to the network output for easier learning.",
"Our model is named SRCondenseNet.",
"We evaluate the method using various benchmark datasets and show that it performs favourably against the state-of-the-art methods in terms of both accuracy and number of computations required."
],
[
"Introduction",
"Super Resolution (SR) problem is defined as recovering a high resolution image from a low resolution image.",
"This is a highly ill-posed problem with multiple solutions possible for a single input image.",
"This problem finds many applications such as medical imaging, security and surveillance among others.",
"In recent years, deep learning methods have performed better as compared to interpolation-based[2], reconstruction-based [6], [7] or other example-based methods [3], [4], [5], [8] that have been used in the past.",
"This is proved by the fact that the first effort in the direction of deep learning for solving the problem of single image super resolution [9] performed better than several previous models not using deep learning algorithms.",
"This lead to development of several other methods that used deep learning[11], [12], [13], [14], [15], [16], [17], [18], [19].",
"However, all these methods in order to get a slight performance improvement (in terms of PSNR) promoted use of deep, computationally heavy CNN models.",
"It would be objectively correct to say that such heavy resources are not available at all situations for such lengthy periods of time.",
"In order to solve this problem, it is required to build a model that uses less number of multiplication-addition operations (FLOPs) to come up with a high resolution image.",
"In this work, we present a novel super resolution model termed SRCondeseNet that uses the concept of removing unused connections in the network to form group convolutions.",
"Normal group convolutions also help in reducing the number of connections but the latter method comes with a huge loss in accuracy.",
"Once the features are extracted using this reduced model, reconstruction is done using deconvolutional layers with 1x1 kernels to produce a high resolution image.",
"Also applied is the concept of residual learning i.e.",
"the bicubic upsampled input is added to network output so that the model only has to learn the difference[11].",
"Our contributions through this work are: Our model incorporates the use of group convolutions and pruning in the field of super resolution thereby producing a lightweight CNN model for this problem.",
"Setting state-of-the-art in terms of performance metrics such as PSNR and SSIM along with using less number of FLOPs as compared to current light weight SISR methods."
],
[
"Related Work",
"Here we focus on various deep learning methods that have been used to solve the SISR problem.",
"Also we go through various methods that have been proposed to come up with efficient, lightweight CNNs."
],
[
"Single Image Super Resolution",
"Various deep learning methods have been applied in the past, to solve the SISR problem, many of which have been summarized in [23].",
"First, Dong et al.",
"proposed in [9] the replacement of all steps to produce a high resolution image - feature extraction then mapping then reconstruction - by a single neural network.",
"The deep learning model performed better than other example-based methods.",
"However, it was proposed in [9] that deeper networks may not be effective for SISR.",
"This was proved wrong by Kim et al.",
"in [11].",
"They used a very deep CNN model that performed better than [9].",
"Kim et al.",
"in [11] used residual learning proposed by He et al.",
"in [30] to combat the problem of vanishing gradients that arises in deep models.",
"Since then, the concept of residual learning has been used by many CNN models[13], [15], [17], [18], [19], [20].",
"Hence, we also include the feature of global residual learning in order to avoid the vanishing gradient problem that is bound to happen in a deep model like ours.",
"Moreover, some models[12], [17], [19], [20] advocate the use of recursive layers in the CNN.",
"This helps in reducing the number of effective parameters required.",
"However, these models require heavy computation power, significantly higher than our model.",
"To achieve real time performance, [21] proposed use of sub-pixel convolutions instead of bicubic upsampled image taken as the input.",
"Similarly, [10], [15], [18] start with low-resolution image as input to the network.",
"This way the model works on low resolution image thereby helping in reducing the number of computations.",
"We also use this idea for the aforementioned purpose.",
"Some methods[13], [14] have advocated the use of GAN (Generative Adverserial Networks) to produce visually-pleasing images along with promising results on quantitative metrics like PSNR and SSIM."
],
[
"Efficient convolutional neural networks",
"Many attempts have been made to build CNNs that use less computation power without compromising accuracy.",
"One such method is weight pruning.",
"Weight pruning is removing unwanted connections in a neural network.",
"CondenseNets[1] which are explained below use weight pruning."
],
[
"CondenseNet",
"Our model employs blocks that are modified version of CondenseNet blocks[1].",
"Original CondenseNet blocks use learned group convolutions.",
"In this method, the model goes through two kinds of stages: condensing stages and optimization stage.",
"In the former, using sparsity inducing regularization, unimportant filters are removed.",
"The convolutional layers used here have 1x1 kernels.",
"Thus, number of connections depend on number of input channels and number of output channels only.",
"The condensation is done by calculating $L_1$ -norm over every incoming feature and every filter group.",
"Then we remove those columns that have $L_1$ -norm value lesser than other columns.",
"The number of feature map connections that are left after every condensing stage depends on the condensing factor $C$ .",
"Once we get the lighter model, it goes through optimization stage where it is trained.",
"Every block contains several $denselayers$ and structure of each original $denselayer$ is described in Figure: REF (left).",
"Figure: Comparison between structure of denselayerdenselayer of CondenseNet (left) and SRCondenseNet (right).",
"BN layer is removed and Relu is replaced by LeakyRelu."
],
[
"Proposed Method",
"In this section, we describe our proposed method, SRCondenseNet, in detail.",
"First we take as input the low resolution(LR) image and pass it into an input convolutional layer.",
"The output of this layer is fed into modified CondenseNet that contains $denselayers$ that are stacked into four blocks.",
"The output of the last block is sent to what is called as the reconstruction network.",
"It comprises of a bottleneck layer, a set of deconvolution layer, whose number depends on the scaling factor.",
"Next comes the reconstruction layer with one output channel to get the final image."
],
[
"Modified CondenseNet blocks",
"In section 2.2, we explained original CondenseNet blocks.",
"However, CondenseNet has been designed for classification task.",
"Hence, several modifications have been done to suit it for the SISR problem.",
"SRCondenseNet contains $denselayer$ structure, which has been depicted in Figure: REF (right).",
"Every block contains many $denselayer$ named structures.",
"We have removed Batch-Normalization layers as suggested by Nah et al.",
"[22] and Lim et al.[16].",
"This removes unnecessary computations.",
"Also Relu activation has been replaced by LeakyRelu to combat the “dying ReLU\" problem.",
"We stack up four blocks each containing 7 $denselayers$ (blue) as shown in Figure: REF .",
"Only one out of four blocks (black dashed line) is shown in the figure to avoid clutter.",
"Number of input channels in every $denselayer$ depends on growth rate and increase in number linearly according to it.",
"Every block has its own growth rate.",
"After testing several values for trade-off between model size and accuracy, we set growth rate of all the blocks to 20.",
"Thus, every subsequent $denselayer$ has 20 more input channels than the previous one.",
"Moreover, original CondenseNet contains transition layers between blocks comprising of average pooling layers.",
"For SISR problem, there was no need of pooling layers.",
"We have skipped these transition layers in our model.",
"Thus, width and height of input into first block is equal to width and height of output from last block.",
"Figure: Our model (SRCondenseNet) structure.",
"We have four such blocks(black dashed line) each containing seven denselayerdenselayer (blue) structures.",
"Only one block is shown for clarity.",
"This is followed by the reconstruction network containing bottleneck, deconvolution and reconstruction CNN at the end."
],
[
"Reconstruction Network",
"The reconstruction network comes after the modified CondenseNet blocks as shown in Figure REF .",
"It starts with the bottleneck layer (green) which is a 1x1 layer to reduce the output feature maps to a very less number thereby reducing the number of computations in further layers.",
"1x1 kernel also helps in the purpose.",
"Number of output feature maps are set to 128.",
"Next, this is followed by a set of deconvolutional layers (pink).",
"Their number depends on the scaling factor$(r)$ .",
"With r equal to 2, we have a single deconvolutional layer with stride 2.",
"Deconvolutional layers help in reducing the number of parameters and computational complexity by a factor of $r^{2}$ throughout the model.",
"This is because, by using bicubic interpolated image as an input, instead of upscaling it at the last using deconvolutional layers, increases the size of input to all feature extraction layers by a factor of $r^{2}$ .",
"This method of upscaling also improves performance in reconstruction.",
"Again, number of feature maps are set to 128 for all deconvolutional layers.",
"Finally, we end up the model with a convolutional layer (yellow) with one channel as output to get the final YCbCr image."
],
[
"Global Residual Learning",
"Deep CNN models with high number of layers tend to suffer from vanishing gradient problem.",
"Hence, as proposed by He et al.",
"in [30], this problem is solved by adding a global residual connection.",
"In our model, we add a residual connection in which we add a bicubic interpolated image to the output received from the model.",
"This makes the learning easier and more and more layers can be stacked."
],
[
"Datasets",
"We have used 91 images from Yang et al.",
"[24] and 200 images from the Berkeley Segmentation Dataset(BSD)[25] for training.",
"We cut out several patches of the original images with a stride of 64, size of which depends on the scaling factor.",
"For every case, during training, input size of the image to the network is 32x32.",
"Hence, for scaling factor of 2, we cut out patches of 64x64.",
"Further, we have performed data augmentation on these patches.",
"Eventually, we get five patches for a single original patch.",
"These are converted to YCbCr image and only Y-channel is processed.",
"We test our model on standard datasets: SET5[26], SET14[27] and Urban100 [28]."
],
[
"Implementation Details",
"We set the initial learning rate to 0.0001 and keep the cosine learning rate method as used by Huang et al.",
"in [1].",
"We run the network for 180 epochs with both condensing factor and number of groups set to 4 to have every condensing stage with 30 epochs.",
"LeakyReLUs have negative slope set to $0.1$ .",
"We train our network on a Tesla P40 GPU.",
"All networks were optimized using Adam[29].",
"We have used a robust Charbonnier loss function instead of $L1$ or $L2$ function that is generally used[9], [11], [12], [16] to aid high-quality reconstruction performance[18]."
],
[
"Comparison on the basis of accuracy",
"Peak signal-to-noise Ratio (PSNR) and structures similarity (SSIM) are the two standard metrics for comparison.",
"Table REF shows comparisons for SISR results for various models(scale = x2).",
"Clearly, our method performs handsomely when compared to current state-of-the-art models using similar computation power.",
"Various standard testing datasets have been used.",
"Figure REF shows a qualitative comparison of images from various testing datasets.",
"Table: Average PSNR/SSIM values for x2 scale factor for various models on different models.",
"red indicates best value and blue indicates second best value.Figure: Qualitative comparison.",
"The first row shows low resolution input, bicubic interpolation of LR input, output from our model, original HR image(left to right) of image from set5.",
"Similarly second row are images for img_013 from set14.Figure: Graph showing average PSNR vs FLOPs(SET5 scale x2) trade-off."
],
[
"Comparison on the basis of FLOPs",
"SRCondenseNet uses the concept of learned group convolutions.",
"Thus, it requires relatively less computation power to produce better results.",
"Here, for comparison, we have used the definition of FLOPs(number of multiplications and additions) to compare computational complexity.",
"Similar method was used in the original CondenseNet [1] paper.",
"We have used the same method to calculate FLOPs for all models.",
"Scale is taken as 2 here as well.",
"SRCNN [9], VDSR [11] & DRRN [19] take bicubic interpolated input, hence we take input image size as 64x64 for these models.",
"Whereas, we take 32x32 input image size for LapSRN [18] & our model as these models use original low resolution image.",
"Table REF shows that our model is lighter than most models.",
"There is a trade-off between computational complexity and PSNR as can be seen in figure REF .",
"SRCNN [9] contains only three parameterised convolutional layers and thus is unable to learn good enough mapping between a low resolution image and its coressponding high resolution image.",
"Number of layers in SRCNN [9] is very less as compared to all other models mentioned in table REF and table REF which makes it computationally less expensive (without explicitly applying any technique to reduce number of parameters) than other models (including ours).",
"However, it should also be noted that it produces significantly poorer results than all other models.",
"On the other hand, rest of all the models are computationally heavier than our model.",
"Table: FLOPs count (x1e6) for various models with suitable input to produce a size 64x64 output image and scale factor of 2. red indicates best value and blue indicates second best value."
],
[
"Conclusion and Future Work",
"In this paper, we propose a single image super resolution method that uses pruned CNNs to solve the problem using less number of computations.",
"The proposed method outperforms state-of-the-art by a considerable margin in terms of PSNR and SSIM while maintaining less number of FLOPs than comparable methods.",
"Learned group convolutions after our modifications are found to be performing well for the SISR task.",
"This work promotes the use of efficient CNNs that have been used widely in high-level computer vision tasks into low-level vision tasks such as SISR.",
"In this work, Charbonnier loss has been used throughout the process.",
"We intend on integrating perceptual loss in the proposed method in order to produce visually pleasing images as claimed by Ledig et al.",
"[13] and Sajjadi et al.",
"[14] in future."
],
[
"Acknowledgements",
"The authors are grateful to HP Inc. for their support to the Innovations Incubator Program.",
"They are thankful to other stakeholders of this program including Leadership, and Faculty Mentors at IIT-BHU, Drstikona and Nalanda Foundation.",
"Authors are also grateful to Dr. Prasenjit Banerjee, Nalanda Foundation for his mentoring and support"
]
] | 1808.08509 |
[
[
"Adversarially Regularising Neural NLI Models to Integrate Logical\n Background Knowledge"
],
[
"Abstract Adversarial examples are inputs to machine learning models designed to cause the model to make a mistake.",
"They are useful for understanding the shortcomings of machine learning models, interpreting their results, and for regularisation.",
"In NLP, however, most example generation strategies produce input text by using known, pre-specified semantic transformations, requiring significant manual effort and in-depth understanding of the problem and domain.",
"In this paper, we investigate the problem of automatically generating adversarial examples that violate a set of given First-Order Logic constraints in Natural Language Inference (NLI).",
"We reduce the problem of identifying such adversarial examples to a combinatorial optimisation problem, by maximising a quantity measuring the degree of violation of such constraints and by using a language model for generating linguistically-plausible examples.",
"Furthermore, we propose a method for adversarially regularising neural NLI models for incorporating background knowledge.",
"Our results show that, while the proposed method does not always improve results on the SNLI and MultiNLI datasets, it significantly and consistently increases the predictive accuracy on adversarially-crafted datasets -- up to a 79.6% relative improvement -- while drastically reducing the number of background knowledge violations.",
"Furthermore, we show that adversarial examples transfer among model architectures, and that the proposed adversarial training procedure improves the robustness of NLI models to adversarial examples."
],
[
"Introduction",
"An open problem in Artificial Intelligence is quantifying the extent to which algorithms exhibit intelligent behaviour .",
"In Machine Learning, a standard procedure consists in estimating the generalisation error, i.e.",
"the prediction error over an independent test sample .",
"However, machine learning models can succeed simply by recognising patterns that happen to be predictive on instances in the test sample, while ignoring deeper phenomena , .",
"Adversarial examples are inputs to machine learning models designed to cause the model to make a mistake , .",
"In Natural Language Processing (NLP) and Machine Reading, generating adversarial examples can be really useful for understanding the shortcomings of NLP models , and for regularisation .",
"In this paper, we focus on the problem of generating adversarial examples for Natural Language Inference (NLI) models in order to gain insights about the inner workings of such systems, and regularising them.",
"NLI, also referred to as Recognising Textual Entailment , , , is a central problem in language understanding , , , , and thus it is especially well suited to serve as a benchmark task for research in machine reading.",
"In NLI, a model is presented with two sentences, a premise $p$ and a hypothesis $h$ , and the goal is to determine whether $p$ semantically entails $h$ .",
"The problem of acquiring large amounts of labelled data for NLI was addressed with the creation of the SNLI and MultiNLI datasets.",
"In these processes, annotators were presented with a premise $p$ drawn from a corpus, and were required to generate three new sentences (hypotheses) based on $p$ , according to the following criteria: [a)] Entailment – $h$ is definitely true given $p$ ($p$ entails $h$ ); Contradiction – $h$ is definitely not true given $p$ ($p$ contradicts $h$ ); and Neutral – $h$ might be true given $p$ .",
"Given a premise-hypothesis sentence pair $(p, h)$ , a NLI model is asked to classify the relationship between $p$ and $h$ – i.e.",
"either entailment, contradiction, or neutral.",
"Solving NLI requires to fully capture the sentence meaning by handling complex linguistic phenomena like lexical entailment, quantification, co-reference, tense, belief, modality, and lexical and syntactic ambiguities .",
"In this work, we use adversarial examples for: [a)] identifying cases where models violate existing background knowledge, expressed in the form of logic rules, and training models that are robust to such violations.",
"The underlying idea in our work is that NLI models should adhere to a set of structural constraints that are intrinsic to the human reasoning process.",
"For instance, contradiction is inherently symmetric: if a sentence $p$ contradicts a sentence $h$ , then $h$ contradicts $p$ as well.",
"Similarly, entailment is both reflexive and transitive.",
"It is reflexive since a sentence $a$ is always entailed by (i.e.",
"is true given) $a$ .",
"It is also transitive, since if $a$ is entailed by $b$ , and $b$ is entailed by $c$ , then $a$ is entailed by $c$ as well.",
"$\\Box $ Example 1 (Inconsistency) Consider three sentences $a$ , $b$ and $c$ each describing a situation, such as: [a)] “The girl plays”, “The girl plays with a ball”, and “The girl plays with a red ball”.",
"Note that if $a$ is entailed by $b$ , and $b$ is entailed by $c$ , then also $a$ is entailed by $c$ .",
"If a NLI model detects that $b$ entails $a$ , $c$ entails $b$ , but $c$ does not entail $a$ , we know that it is making an error (since its results are inconsistent), even though we may not be aware of the sentences $a$ , $b$ , and $c$ and the true semantic relationships holding between them.",
"Our adversarial examples are different from those used in other fields such as computer vision, where they typically consist in small, semantically invariant perturbations that result in drastic changes in the model predictions.",
"In this paper, we propose a method for generating adversarial examples that cause a model to violate pre-existing background knowledge (sec:generating), based on reducing the generation problem to a combinatorial optimisation problem.",
"Furthermore, we outline a method for incorporating such background knowledge into models by means of an adversarial training procedure (sec:regularisation).",
"Our results (sec:experiments) show that, even though the proposed adversarial training procedure does not sensibly improve accuracy on SNLI and MultiNLI, it yields significant relative improvement in accuracy (up to 79.6%) on adversarial datasets.",
"Furthermore, we show that adversarial examples transfer across models, and that the proposed method allows training significantly more robust NLI models.",
"Background Neural NLI Models.",
"In NLI, in particular on the Stanford Natural Language Inference (SNLI) and MultiNLI datasets, neural NLI models – end-to-end differentiable models that can be trained via gradient-based optimisation – proved to be very successful, achieving state-of-the-art results , , .",
"Let $\\mathcal {S}$ denote the set of all possible sentences, and let $a = (a_{1}, \\ldots , a_{\\ell _{a}}) \\in \\mathcal {S}$ and $b = (b_{1}, \\ldots , b_{\\ell _{b}}) \\in \\mathcal {S}$ denote two input sentences – representing the premise and the hypothesis – of length $\\ell _{a}$ and $\\ell _{b}$ , respectively.",
"In neural NLI models, all words $a_{i}$ and $b_{j}$ are typically represented by $k$ -dimensional embedding vectors $\\mathbf {a}_{i}, \\mathbf {b}_{j} \\in \\mathbb {R}^{k}$ .",
"As such, the sentences $a$ and $b$ can be encoded by the sentence embedding matrices $\\mathbf {a}\\in \\mathbb {R}^{k \\times \\ell _{a}}$ and $\\mathbf {b}\\in \\mathbb {R}^{k \\times \\ell _{b}}$ , where the columns $\\mathbf {a}_{i}$ and $\\mathbf {b}_{j}$ respectively denote the embeddings of words $a_{i}$ and $b_{j}$ .",
"Given two sentences $a, b \\in \\mathcal {S}$ , the goal of a NLI model is to identify the semantic relation between $a$ and $b$ , which can be either entailment, contradiction, or neutral.",
"For this reason, given an instance, neural NLI models compute the following conditional probability distribution over all three classes: $ \\begin{aligned}p_{\\Theta }({}\\cdot {} \\mid a, b) & = & & \\operatorname{softmax}(\\operatorname{score}_{\\Theta }(\\mathbf {a}, \\mathbf {b}))\\end{aligned}$ where $\\operatorname{score}_{\\Theta } : \\mathbb {R}^{k \\times \\ell _{a}} \\times \\mathbb {R}^{k \\times \\ell _{b}} \\rightarrow \\mathbb {R}^{3}$ is a model-dependent scoring function with parameters $\\Theta $ , and $\\operatorname{softmax}(\\mathbf {x})_{i} = \\exp \\lbrace x_i\\rbrace /\\sum _j \\exp \\lbrace x_j\\rbrace $ denotes the softmax function.",
"Several scoring functions have been proposed in the literature, such as the conditional Bidirectional LSTM (cBiLSTM) , the Decomposable Attention Model (DAM) , and the Enhanced LSTM model (ESIM) .",
"One desirable quality of the scoring function $\\operatorname{score}_{\\Theta }$ is that it should be differentiable with respect to the model parameters $\\Theta $ , which allows the neural NLI model to be trained from data via back-propagation.",
"Model Training.",
"Let $ \\mathcal {D} = $ $\\lbrace (x_{1}, y_{1}),$ $\\ldots ,$ $(x_{m}, y_{m})\\rbrace $ represent a NLI dataset, where $x_{i}$ denotes the $i$ -th premise-hypothesis sentence pair, and $y_{i} \\in \\lbrace 1, \\ldots , K \\rbrace $ their relationship, where $K \\in \\mathbb {N}$ is the number of possible relationships – in the case of NLI, $K = 3$ .",
"The model is trained by minimising a cross-entropy loss $ \\mathcal {J} _{\\mathcal {D}} $ on $ \\mathcal {D} $ : $ \\mathcal {J} _{\\mathcal {D}} ( \\mathcal {D} , \\Theta ) = - \\sum _{i = 1}^{m} \\sum _{k = 1}^{K} \\mathbb {1}\\lbrace y_{i} = k \\rbrace \\log (\\hat{y}_{i, k})$ where $\\hat{y}_{i, k} = p_{\\Theta }(y_{i} = k \\mid x_{i})$ denotes the probability of class $k$ on the instance $x_{i}$ inferred by the neural NLI model as in eq:nnli.",
"In the following, we analyse the behaviour of neural NLI models by means of adversarial examples – inputs to machine learning models designed to cause the model to commit mistakes.",
"In computer vision models, adversarial examples are created by adding a very small amount of noise to the input , : these perturbations do not change the semantics of the images, but they can drastically change the predictions of computer vision models.",
"In our setting, we define an adversary whose goal is finding sets of NLI instances where the model fails to be consistent with available background knowledge, encoded in the form of First-Order Logic (FOL) rules.",
"In the following sections, we define the corresponding optimisation problem, and propose an efficient solution.",
"Background Knowledge For analysing the behaviour of NLI models, we verify whether they agree with the provided background knowledge, encoded by a set of FOL rules.",
"Note that the three NLI classes – entailment, contradiction, and neutrality – can be seen as binary logic predicates, and we can define FOL formulas for describing the formal relationships that hold between them.",
"In the following, we denote the predicates associated with entailment, contradiction, and neutrality as ${\\mathrm {ent}}$ , ${\\mathrm {con}}$ , and ${\\mathrm {neu}}$ , respectively.",
"By doing so, we can represent semantic relationships between sentences via logic atoms.",
"For instance, given three sentences $s_{1}, s_{2}, s_{3} \\in \\mathcal {S}$ , we can represent the fact that $s_{1}$ entails $s_{2}$ and $s_{2}$ contradicts $s_{3}$ by using the logic atoms ${\\mathrm {ent}}(s_{1}, s_{2})$ and ${\\mathrm {con}}(s_{2}, s_{3})$ .",
"Let $X_{1}, \\ldots , X_{n}$ be a set of universally quantified variables.",
"We define our background knowledge as a set of FOL rules, each having the following ${\\mathrm {body}} \\Rightarrow {\\mathrm {head}}$ form: $ {\\mathrm {body}}(X_{1}, \\ldots , X_{n}) \\Rightarrow {\\mathrm {head}}(X_{1}, \\ldots , X_{n}),$ where ${\\mathrm {body}}$ and ${\\mathrm {head}}$ represent the premise and the conclusion of the rule – if ${\\mathrm {body}}$ holds, ${\\mathrm {head}}$ holds as well.",
"In the following, we consider the rules $\\mathbf {R_{1}}, \\ldots , \\mathbf {R_{5}}$ outlined in tab:rules.",
"Rule $\\mathbf {R_{1}}$ enforces the constraint that entailment is reflexive; rule $\\mathbf {R_{2}}$ that contradiction should always be symmetric (if $s_{1}$ contradicts $s_{2}$ , then $s_{2}$ contradicts $s_{1}$ as well); rule $\\mathbf {R_{5}}$ that entailment is transitive; while rules $\\mathbf {R_{3}}$ and $\\mathbf {R_{4}}$ describe the formal relationships between the entailment, neutral, and contradiction relations.",
"In sec:generating we propose a method to automatically generate sets of sentences that violate the rules outlined in tab:rules – effectively generating adversarial examples.",
"Then, in sec:regularisation we show how we can leverage such adversarial examples by generating them on-the-fly during training and using them for regularising the model parameters, in an adversarial training regime.",
"Generating Adversarial Examples Table: First-Order Logic rules defining desired properties of NLI models: X i X_{i} are universally quantified variables, and operators ∧\\wedge , ¬\\lnot , and ⊤\\top denote logic conjunction, negation, and tautology.In this section, we propose a method for efficiently generating adversarial examples for NLI models – i.e.",
"examples that make the model violate the background knowledge outlined in sec:background.",
"Inconsistency Loss We cast the problem of generating adversarial examples as an optimisation problem.",
"In particular, we propose a continuous inconsistency loss that measures the degree to which a set of sentences causes a model to violate a rule.",
"$\\Box $ Example 2 (Inconsistency Loss) Consider the rule $\\mathbf {R_{2}}$ in tab:rules, i.e.",
"${\\mathrm {con}}(X_{1}, X_{2}) \\Rightarrow {\\mathrm {con}}(X_{2}, X_{1})$ .",
"Let $s_{1}, s_{2} \\in \\mathcal {S}$ be two sentences: this rule is violated if, according to the model, a sentence $s_{1}$ contradicts $s_{2}$ , but $s_{2}$ does not contradict $s_{1}$ .",
"However, if we just use the final decision made by the neural NLI model, we can simply check whether the rule is violated by two given sentences, without any information on the degree of such a violation.",
"Intuitively, for the rule being maximally violated, the conditional probability associated to ${\\mathrm {con}}(s_{1}, s_{2})$ should be very high ($\\approx 1$ ), while the one associated to ${\\mathrm {con}}(s_{2}, s_{1})$ should be very low ($\\approx 0$ ).",
"We can measure the extent to which the rule is violated – which we refer to as inconsistency loss $ \\mathcal {J} _{\\mathcal {I}} $ – by checking whether the probability of the body of the rule is higher than the probability of its head: $\\begin{aligned} \\mathcal {J} _{\\mathcal {I}} (S= \\lbrace & X_{1} \\mapsto s_{1}, X_{2} \\mapsto s_{2} \\rbrace ) \\\\& = \\left[ p_{\\Theta }({\\mathrm {con}} \\mid s_{1}, s_{2}) - p_{\\Theta }({\\mathrm {con}} \\mid s_{2}, s_{1}) \\right]_{+}\\end{aligned}$ where $S$ is a substitution set that maps the variables $X_{1}$ and $X_{2}$ in $\\mathbf {R_{2}}$ to the sentences $s_{1}$ and $s_{2}$ , $[x]_{+} = \\max (0, x)$ , and $p_{\\Theta }({\\mathrm {con}} \\mid s_{i}, s_{j})$ is the (conditional) probability that $s_{i}$ contradicts $s_{j}$ according to the neural NLI model.",
"Note that, in accordance with the logic implication, the inconsistency loss reaches its global minimum when the probability of the body is close to zero – i.e.",
"the premise is false – and when the probabilities of both the body and the head are close to one – i.e.",
"the premise and the conclusion are both true.",
"We now generalise the intuition in Ex.",
"REF to any FOL rule.",
"Let $r = ({\\mathrm {body}} \\Rightarrow {\\mathrm {head}})$ denote an arbitrary FOL rule in the form described in eq:rule, and let ${\\mathrm {vars}}(r) = \\lbrace X_{1}, \\ldots , X_{n} \\rbrace $ denote the set of universally quantified variables in the rule $r$ .",
"Furthermore, let $S= \\lbrace X_{1} \\mapsto s_{1}, \\ldots , X_{n} \\mapsto s_{n} \\rbrace $ denote a substitution set, i.e.",
"a mapping from variables in ${\\mathrm {vars}}(r)$ to sentences $s_{1}, \\ldots , s_{n} \\in \\mathcal {S}$ .",
"The inconsistency loss associated with the rule $r$ on the substitution set $S$ can be defined as: $ \\begin{aligned} \\mathcal {J} _{\\mathcal {I}} (S) = \\left[ p(S; {\\mathrm {body}}) - p(S; {\\mathrm {head}}) \\right]_{+}\\end{aligned}$ where $p(S; {\\mathrm {body}})$ and $p(S; {\\mathrm {head}})$ denote the probability of body and head of the rule, after replacing the variables in $r$ with the corresponding sentences in $S$ .",
"The motivation for the loss in eq:iloss is that logic implications can be understood as “whenever the body is true, the head has to be true as well”.",
"In terms of NLI models, this translates as “the probability of the head should at least be as large as the probability of the body”.",
"For calculating the inconsistency loss in eq:iloss, we need to specify how to calculate the probability of $\\mathrm {head}$ and $\\mathrm {body}$ .",
"The probability of a single ground atom is given by querying the neural NLI model, as in eq:nnli.",
"The head contains a single atom, while the body can be a conjunction of multiple atoms.",
"Similarly to , we use the Gödel t-norm, a continuous generalisation of the conjunction operator in logic , for computing the probability of the body of a clause: $p_{\\Theta }(a_{1} \\wedge a_{2}) = \\min \\lbrace p_{\\Theta }(a_{1}), p_{\\Theta }(a_{2}) \\rbrace $ where $a_{1}$ and $a_{2}$ are two clause atoms.",
"In this work, we cast the problem of generating adversarial examples as an optimisation problem: we search for the substitution set $S= \\lbrace X_{1} \\mapsto s_{1}, \\ldots , X_{n} \\mapsto s_{n} \\rbrace $ that maximises the inconsistency loss in eq:iloss, thus (maximally) violating the available background knowledge.",
"Constraining via Language Modelling Maximising the inconsistency loss in eq:iloss may not be sufficient for generating meaningful adversarial examples: they can lead neural NLI models to violate available background knowledge, but they may not be well-formed and meaningful.",
"For such a reason, in addition to maximising the inconsistency loss, we also constrain the perplexity of generated sentences by using a neural language model .",
"In this work, we use a LSTM neural language model $p_{\\mathcal {L}}(w_{1}, \\ldots , w_{t})$ for generating low-perplexity adversarial examples.",
"Searching in a Discrete Space As mentioned earlier in this section, we cast the problem of automatically generating adversarial examples – i.e.",
"examples that cause NLI models to violate available background knowledge – as an optimisation problem.",
"Specifically, we look for substitutions sets $S= \\lbrace X_{1} \\mapsto s_{1}, \\ldots , X_{n} \\mapsto s_{n} \\rbrace $ that jointly: [a)] maximise the inconsistency loss described in eq:iloss, and are composed by sentences with a low perplexity, as defined by the neural language model in ssec:language.",
"The search objective can be formalised by the following optimisation problem: $ \\begin{aligned}& \\underset{S}{\\text{maximise}}& & \\mathcal {J} _{\\mathcal {I}} (S) \\\\& \\text{subject to}& & \\log p_{\\mathcal {L}}(S) \\le \\tau \\\\\\end{aligned}$ where $\\log p_{\\mathcal {L}}(S)$ denotes the log-probability of the sentences in the substitution set $S$ , and $\\tau $ is a threshold on the perplexity of generated sentences.",
"For generating low-perplexity adversarial examples, we take inspiration from and generate the sentences by editing prototypes extracted from a corpus.",
"Specifically, for searching substitution sets whose sentences jointly have a high probability and are highly adversarial, as measured the inconsistency loss in eq:iloss, we use the following procedure: [a)] we first sample sentences close to the data manifold (i.e.",
"with a low perplexity), by either sampling from the training set or from the language model; we then make small variations to the sentences – analogous to adversarial images, which consist in small perturbations of training examples – so to optimise the objective in eq:dsearch.",
"When editing prototypes, we consider the following perturbations: [a)] change one word in one of the input sentences; remove one parse sub-tree from one of the input sentences; insert one parse sub-tree from one sentence in the corpus in the parse tree of one of the input sentences.",
"Note that the generation process can easily lead to ungrammatical or implausible sentences; however, these will be likely to have a high perplexity according to the language model (ssec:language), and thus they will be ruled out by the search algorithm.",
"Adversarial Regularisation We now show one can use the adversarial examples to regularise the training process.",
"We propose training NLI models by jointly: [a)] minimising the data loss (eq:loss), and minimising the inconsistency loss (eq:iloss) on a set of generated adversarial examples (substitution sets).",
"More formally, for training, we jointly minimise the cross-entropy loss defined on the data $ \\mathcal {J} _{\\mathcal {D}} (\\Theta )$ and the inconsistency loss on a set of generated adversarial examples $\\max _{S} \\mathcal {J} _{\\mathcal {I}} (S; \\Theta )$ , resulting in the following optimisation problem: $ \\begin{aligned}& \\underset{\\Theta }{\\text{minimise}}& & \\mathcal {J} _{\\mathcal {D}} ( \\mathcal {D} , \\Theta ) + \\lambda \\max _{S} \\mathcal {J} _{\\mathcal {I}} (S; \\Theta ) \\\\& \\text{subject to}& & \\log p_{\\mathcal {L}}(S) \\le \\tau \\end{aligned}$ where $\\lambda \\in \\mathbb {R}_{+}$ is a hyperparameter specifying the trade-off between the data loss $ \\mathcal {J} _{\\mathcal {D}} $ (eq:loss), and the inconsistency loss $ \\mathcal {J} _{\\mathcal {I}} $ (eq:iloss), measured on the generated substitution set $S$ .",
"In eq:jloss, the regularisation term $\\max _{S} \\mathcal {J} _{\\mathcal {I}} (S; \\Theta )$ has the task of generating the adversarial substitution sets by maximising the inconsistency loss.",
"Furthermore, the constraint $\\log p_{\\mathcal {L}}(S) \\le \\tau $ ensures that the perplexity of generated sentences is lower than a threshold $\\tau $ .",
"For this work, we used the $\\max $ aggregation function.",
"However, other functions can be used as well, such as the sum or mean of multiple inconsistency losses.",
"[t] Solving the optimisation problem in eq:jloss via Mini-Batch Gradient Descent [1] Dataset $ \\mathcal {D} $ , weight $\\lambda \\in \\mathbb {R}_{+}$ No.",
"of epochs $\\tau \\in \\mathbb {N}_{+}$ No.",
"of adv.",
"substitution sets $n_{a} \\in \\mathbb {N}_{+}$ Initialise the model parameters $\\hat{\\Theta }$ $\\hat{\\Theta } \\leftarrow \\text{initialise}()$ $i \\in \\lbrace 1, \\ldots , \\tau \\rbrace $ $ \\mathcal {D} _{j} \\in \\text{batches}( \\mathcal {D} )$ Generate the adv.",
"substitution sets $S_{i}$ $\\lbrace S_{1}, \\ldots , S_{n_{a}} \\rbrace \\leftarrow \\text{generate}( \\mathcal {D} _{j})$ Compute the gradient of eq:jloss $\\mathcal {L} \\leftarrow \\mathcal {J} _{\\mathcal {D}} ( \\mathcal {D} _{j}, \\hat{\\Theta }) + \\lambda \\sum _{k = 1}^{n_{a}} \\mathcal {J} _{\\mathcal {I}} (S_{k}; \\hat{\\Theta })$ $g \\leftarrow \\nabla _{\\Theta } \\mathcal {L}$ Update the model parameters $\\hat{\\Theta } \\leftarrow \\hat{\\Theta } - \\eta g$ return $\\hat{\\Theta }$ For minimising the regularised loss in eq:jloss, we alternate between two optimisation processes – generating the adversarial examples (eq:dsearch) and minimising the regularised loss (eq:jloss).",
"The algorithm is outlined in alg:opt.",
"At each iteration, after generating a set of adversarial examples $S$ , it computes the gradient of the regularised loss in eq:jloss, and updates the model parameters via a gradient descent step.",
"On line REF , the algorithm generates a set of adversarial examples, each in the form of a substitution set $S$ .",
"On line REF , the algorithm computes the gradient of the adversarially regularised loss – a weighted combination of the data loss in eq:loss and the inconsistency loss in eq:iloss.",
"The model parameters are finally updated on line REF via a gradient descent step.",
"Creating Adversarial NLI Datasets Table: Sample sentences from an Adversarial NLI Dataset generated using the DAM model, by maximising the inconsistency loss 𝒥 ℐ \\mathcal {J} _{\\mathcal {I}} .We crafted a series of datasets for assessing the robustness of the proposed regularisation method to adversarial examples.",
"Starting from the SNLI test set, we proceeded as follows.",
"We selected the $k$ instances in the SNLI test set that maximise the inconsistency loss in eq:iloss with respect to the rules in $\\mathbf {R_{1}}$ , $\\mathbf {R_{2}}$ , $\\mathbf {R_{3}}$ , and $\\mathbf {R_{4}}$ in tab:rules.",
"We refer to the generated datasets as $ \\mathcal {A}_{\\text{m}}^{k} $ , where $m$ identifies the model used for selecting the sentence pairs, and $k$ denotes number of examples in the dataset.",
"For generating each of the $ \\mathcal {A}_{\\text{m}}^{k} $ datasets, we proceeded as follows.",
"Let $\\mathcal {D} = \\lbrace (x_{1}, y_{i}), \\ldots , (x_{n}, y_{n}) \\rbrace $ be a NLI dataset (such as SNLI), where each instance $x_{i} = (p_{i}, h_{i})$ is a premise-hypothesis sentence pair, and $y_{i}$ denotes the relationship holding between $p_{i}$ and $h_{i}$ .",
"For each instance $x_{i} = (p_{i}, h_{i})$ , we consider two substitution sets: $S_{i} = \\lbrace X_{1} \\mapsto p_{i}, X_{2} \\mapsto h_{i} \\rbrace $ and $S_{i}^{\\prime } = \\lbrace X_{1} \\mapsto h_{i}, X_{2} \\mapsto p_{i} \\rbrace $ , each corresponding to a mapping from variables to sentences.",
"We compute the inconsistency score associated to each instance $x_{i}$ in the dataset $\\mathcal {D}$ as $ \\mathcal {J} _{\\mathcal {I}} (S_{i}) + \\mathcal {J} _{\\mathcal {I}} (S_{i}^{\\prime })$ .",
"Note that the inconsistency score only depends on the premise $p_{i}$ and hypothesis $h_{i}$ in each instance $x_{i}$ , and it does not depend on its label $y_{i}$ .",
"After computing the inconsistency scores for all sentence pairs in $\\mathcal {D}$ using a model $m$ , we select the $k$ instances with the highest inconsistency score, we create two instances $x_{i} = (p_{i}, h_{i})$ and $\\hat{x_{i}} = (h_{i}, p_{i})$ , and add both $(x_{i}, y_{i})$ and $(\\hat{x}_{i}, \\hat{y}_{i})$ to the dataset $ \\mathcal {A}_{\\text{m}}^{k} $ .",
"Note that, while $y_{i}$ is already known from the dataset $\\mathcal {D}$ , $\\hat{y}_{i}$ is unknown.",
"For this reason, we find $\\hat{y}_{i}$ by manual annotation.",
"Related Work Adversarial examples are receiving a considerable attention in NLP; their usage, however, is considerably limited by the fact that semantically invariant input perturbations in NLP are difficult to identify .",
"analyse the robustness of extractive question answering models on examples obtained by adding adversarially generated distracting text to SQuAD dataset instances.",
"also notice that character-level Machine Translation are overly sensitive to random character manipulations, such as typos.",
"show that simple character-level modifications can drastically change the toxicity score of a text.",
"proposes using paraphrasing for generating adversarial examples.",
"Our model is fundamentally different in two ways: [a)] it does not need labelled data for generating adversarial examples – the inconsistency loss can be maximised by just making an NLI model produce inconsistent results, and it incorporates adversarial examples during the training process, with the aim of training more robust NLI models.",
"Table: Accuracy on the SNLI and MultiNLI datasets with different neural NLI models before (left) and after (right) adversarial regularisation.Adversarial examples are also used for assessing the robustness of computer vision models , , , where they are created by adding a small amount of noise to the inputs that does not change the semantics of the images, but drastically changes the model predictions.",
"Experiments Table: Violations (%) of rules 𝐑 1 ,𝐑 2 ,𝐑 3 ,𝐑 4 \\mathbf {R_{1}}, \\mathbf {R_{2}}, \\mathbf {R_{3}}, \\mathbf {R_{4}} from tab:rules on the SNLI training set, yield by cBiLSTM, DAM, and ESIM.We trained DAM, ESIM and cBiLSTM on the SNLI corpus using the hyperparameters provided in the respective papers.",
"The results provided by such models on the SNLI and MultiNLI validation and tests sets are provided in tab:models.",
"In the case of MultiNLI, the validation set was obtained by removing 10,000 instances from the training set (originally composed by 392,702 instances), and the test set consists in the matched validation set.",
"Table: Accuracy of unregularised and regularised neural NLI models DAM, cBiLSTM, and ESIM, and their adversarially regularised versions DAM 𝒜ℛ ^{\\mathcal {AR}}, cBiLSTM 𝒜ℛ ^{\\mathcal {AR}}, and ESIM 𝒜ℛ ^{\\mathcal {AR}}, on the datasets 𝒜 m k \\mathcal {A}_{\\text{m}}^{k} .Background Knowledge Violations.",
"As a first experiment, we count the how likely our model is to violate rules $\\mathbf {R_{1}}, \\mathbf {R_{2}}, \\mathbf {R_{3}}, \\mathbf {R_{4}}$ in tab:rules.",
"In tab:violations we report the number sentence pairs in the SNLI training set where DAM, ESIM and cBiLSTM violate $\\mathbf {R_{1}}, \\mathbf {R_{2}}, \\mathbf {R_{3}}, \\mathbf {R_{4}}$ .",
"In the $\\left|{\\mathbf {B}}\\right|$ column we report the number of times the body of the rule holds, according to the model.",
"In the $\\left|{\\mathbf {B} \\wedge \\lnot {\\mathbf {H}}}\\right|$ column we report the number of times where the body of the rule holds, but the head does not – which is clearly a violation of available rules.",
"We can see that, in the case of rule $\\mathbf {R_{1}}$ (reflexivity of entailment), DAM and ESIM make a relatively low number of violations – namely 0.09 and 1.00 %, respectively.",
"However, in the case of cBiLSTM, we can see that, each sentence $s \\in \\mathcal {S}$ in the SNLI training set, with a 23.76 % chance, $s$ does not entail itself – which violates our background knowledge.",
"With respect to $\\mathbf {R_{2}}$ (symmetry of contradiction), we see that none of the models is completely consistent with the available background knowledge.",
"Given a sentence pair $s_{1}, s_{2} \\in \\mathcal {S}$ from the SNLI training set, if – according to the model – $s_{1}$ contradicts $s_{2}$ , a significant number of times (between 9.84% and 46.17%) the same model also infers that $s_{2}$ does not contradict $s_{1}$ .",
"This phenomenon happens 16.70 % of times with DAM, 9.84 % of times with ESIM, and 46.17 % with cBiLSTM: this indicates that all considered models are prone to violating $\\mathbf {R_{2}}$ in their predictions, with ESIM being the more robust.",
"In sec:a:adversarial we report several examples of such violations in the SNLI training set.",
"We select those that maximise the inconsistency loss described in eq:iloss, violating rules $\\mathbf {R_{2}}$ and $\\mathbf {R_{3}}$ .",
"We can notice that the presence of inconsistencies is often correlated with the length of the sentences.",
"The model tends to detect entailment relationships between longer (i.e., possibly more specific) and shorter (i.e., possibly more general) sentences.",
"Generation of Adversarial Examples In the following, we analyse the automatic generation of sets of adversarial examples that make the model violate the existing background knowledge.",
"We search in the space of sentences by applying perturbations to sampled sentence pairs, using a language model for guiding the search process.",
"The generation procedure is described in sec:generating.",
"Table: Solving the optimisation problem in eq:jloss via Mini-Batch Gradient Descent"
],
[
"Neural NLI Models.",
"In NLI, in particular on the Stanford Natural Language Inference (SNLI) and MultiNLI datasets, neural NLI models – end-to-end differentiable models that can be trained via gradient-based optimisation – proved to be very successful, achieving state-of-the-art results , , .",
"Let $\\mathcal {S}$ denote the set of all possible sentences, and let $a = (a_{1}, \\ldots , a_{\\ell _{a}}) \\in \\mathcal {S}$ and $b = (b_{1}, \\ldots , b_{\\ell _{b}}) \\in \\mathcal {S}$ denote two input sentences – representing the premise and the hypothesis – of length $\\ell _{a}$ and $\\ell _{b}$ , respectively.",
"In neural NLI models, all words $a_{i}$ and $b_{j}$ are typically represented by $k$ -dimensional embedding vectors $\\mathbf {a}_{i}, \\mathbf {b}_{j} \\in \\mathbb {R}^{k}$ .",
"As such, the sentences $a$ and $b$ can be encoded by the sentence embedding matrices $\\mathbf {a}\\in \\mathbb {R}^{k \\times \\ell _{a}}$ and $\\mathbf {b}\\in \\mathbb {R}^{k \\times \\ell _{b}}$ , where the columns $\\mathbf {a}_{i}$ and $\\mathbf {b}_{j}$ respectively denote the embeddings of words $a_{i}$ and $b_{j}$ .",
"Given two sentences $a, b \\in \\mathcal {S}$ , the goal of a NLI model is to identify the semantic relation between $a$ and $b$ , which can be either entailment, contradiction, or neutral.",
"For this reason, given an instance, neural NLI models compute the following conditional probability distribution over all three classes: $ \\begin{aligned}p_{\\Theta }({}\\cdot {} \\mid a, b) & = & & \\operatorname{softmax}(\\operatorname{score}_{\\Theta }(\\mathbf {a}, \\mathbf {b}))\\end{aligned}$ where $\\operatorname{score}_{\\Theta } : \\mathbb {R}^{k \\times \\ell _{a}} \\times \\mathbb {R}^{k \\times \\ell _{b}} \\rightarrow \\mathbb {R}^{3}$ is a model-dependent scoring function with parameters $\\Theta $ , and $\\operatorname{softmax}(\\mathbf {x})_{i} = \\exp \\lbrace x_i\\rbrace /\\sum _j \\exp \\lbrace x_j\\rbrace $ denotes the softmax function.",
"Several scoring functions have been proposed in the literature, such as the conditional Bidirectional LSTM (cBiLSTM) , the Decomposable Attention Model (DAM) , and the Enhanced LSTM model (ESIM) .",
"One desirable quality of the scoring function $\\operatorname{score}_{\\Theta }$ is that it should be differentiable with respect to the model parameters $\\Theta $ , which allows the neural NLI model to be trained from data via back-propagation."
],
[
"Model Training.",
"Let $ \\mathcal {D} = $ $\\lbrace (x_{1}, y_{1}),$ $\\ldots ,$ $(x_{m}, y_{m})\\rbrace $ represent a NLI dataset, where $x_{i}$ denotes the $i$ -th premise-hypothesis sentence pair, and $y_{i} \\in \\lbrace 1, \\ldots , K \\rbrace $ their relationship, where $K \\in \\mathbb {N}$ is the number of possible relationships – in the case of NLI, $K = 3$ .",
"The model is trained by minimising a cross-entropy loss $ \\mathcal {J} _{\\mathcal {D}} $ on $ \\mathcal {D} $ : $ \\mathcal {J} _{\\mathcal {D}} ( \\mathcal {D} , \\Theta ) = - \\sum _{i = 1}^{m} \\sum _{k = 1}^{K} \\mathbb {1}\\lbrace y_{i} = k \\rbrace \\log (\\hat{y}_{i, k})$ where $\\hat{y}_{i, k} = p_{\\Theta }(y_{i} = k \\mid x_{i})$ denotes the probability of class $k$ on the instance $x_{i}$ inferred by the neural NLI model as in eq:nnli.",
"In the following, we analyse the behaviour of neural NLI models by means of adversarial examples – inputs to machine learning models designed to cause the model to commit mistakes.",
"In computer vision models, adversarial examples are created by adding a very small amount of noise to the input , : these perturbations do not change the semantics of the images, but they can drastically change the predictions of computer vision models.",
"In our setting, we define an adversary whose goal is finding sets of NLI instances where the model fails to be consistent with available background knowledge, encoded in the form of First-Order Logic (FOL) rules.",
"In the following sections, we define the corresponding optimisation problem, and propose an efficient solution."
],
[
"Background Knowledge",
"For analysing the behaviour of NLI models, we verify whether they agree with the provided background knowledge, encoded by a set of FOL rules.",
"Note that the three NLI classes – entailment, contradiction, and neutrality – can be seen as binary logic predicates, and we can define FOL formulas for describing the formal relationships that hold between them.",
"In the following, we denote the predicates associated with entailment, contradiction, and neutrality as ${\\mathrm {ent}}$ , ${\\mathrm {con}}$ , and ${\\mathrm {neu}}$ , respectively.",
"By doing so, we can represent semantic relationships between sentences via logic atoms.",
"For instance, given three sentences $s_{1}, s_{2}, s_{3} \\in \\mathcal {S}$ , we can represent the fact that $s_{1}$ entails $s_{2}$ and $s_{2}$ contradicts $s_{3}$ by using the logic atoms ${\\mathrm {ent}}(s_{1}, s_{2})$ and ${\\mathrm {con}}(s_{2}, s_{3})$ .",
"Let $X_{1}, \\ldots , X_{n}$ be a set of universally quantified variables.",
"We define our background knowledge as a set of FOL rules, each having the following ${\\mathrm {body}} \\Rightarrow {\\mathrm {head}}$ form: $ {\\mathrm {body}}(X_{1}, \\ldots , X_{n}) \\Rightarrow {\\mathrm {head}}(X_{1}, \\ldots , X_{n}),$ where ${\\mathrm {body}}$ and ${\\mathrm {head}}$ represent the premise and the conclusion of the rule – if ${\\mathrm {body}}$ holds, ${\\mathrm {head}}$ holds as well.",
"In the following, we consider the rules $\\mathbf {R_{1}}, \\ldots , \\mathbf {R_{5}}$ outlined in tab:rules.",
"Rule $\\mathbf {R_{1}}$ enforces the constraint that entailment is reflexive; rule $\\mathbf {R_{2}}$ that contradiction should always be symmetric (if $s_{1}$ contradicts $s_{2}$ , then $s_{2}$ contradicts $s_{1}$ as well); rule $\\mathbf {R_{5}}$ that entailment is transitive; while rules $\\mathbf {R_{3}}$ and $\\mathbf {R_{4}}$ describe the formal relationships between the entailment, neutral, and contradiction relations.",
"In sec:generating we propose a method to automatically generate sets of sentences that violate the rules outlined in tab:rules – effectively generating adversarial examples.",
"Then, in sec:regularisation we show how we can leverage such adversarial examples by generating them on-the-fly during training and using them for regularising the model parameters, in an adversarial training regime."
],
[
"Generating Adversarial Examples",
"In this section, we propose a method for efficiently generating adversarial examples for NLI models – i.e.",
"examples that make the model violate the background knowledge outlined in sec:background."
],
[
"Inconsistency Loss",
"We cast the problem of generating adversarial examples as an optimisation problem.",
"In particular, we propose a continuous inconsistency loss that measures the degree to which a set of sentences causes a model to violate a rule.",
"$\\Box $ Example 2 (Inconsistency Loss) Consider the rule $\\mathbf {R_{2}}$ in tab:rules, i.e.",
"${\\mathrm {con}}(X_{1}, X_{2}) \\Rightarrow {\\mathrm {con}}(X_{2}, X_{1})$ .",
"Let $s_{1}, s_{2} \\in \\mathcal {S}$ be two sentences: this rule is violated if, according to the model, a sentence $s_{1}$ contradicts $s_{2}$ , but $s_{2}$ does not contradict $s_{1}$ .",
"However, if we just use the final decision made by the neural NLI model, we can simply check whether the rule is violated by two given sentences, without any information on the degree of such a violation.",
"Intuitively, for the rule being maximally violated, the conditional probability associated to ${\\mathrm {con}}(s_{1}, s_{2})$ should be very high ($\\approx 1$ ), while the one associated to ${\\mathrm {con}}(s_{2}, s_{1})$ should be very low ($\\approx 0$ ).",
"We can measure the extent to which the rule is violated – which we refer to as inconsistency loss $ \\mathcal {J} _{\\mathcal {I}} $ – by checking whether the probability of the body of the rule is higher than the probability of its head: $\\begin{aligned} \\mathcal {J} _{\\mathcal {I}} (S= \\lbrace & X_{1} \\mapsto s_{1}, X_{2} \\mapsto s_{2} \\rbrace ) \\\\& = \\left[ p_{\\Theta }({\\mathrm {con}} \\mid s_{1}, s_{2}) - p_{\\Theta }({\\mathrm {con}} \\mid s_{2}, s_{1}) \\right]_{+}\\end{aligned}$ where $S$ is a substitution set that maps the variables $X_{1}$ and $X_{2}$ in $\\mathbf {R_{2}}$ to the sentences $s_{1}$ and $s_{2}$ , $[x]_{+} = \\max (0, x)$ , and $p_{\\Theta }({\\mathrm {con}} \\mid s_{i}, s_{j})$ is the (conditional) probability that $s_{i}$ contradicts $s_{j}$ according to the neural NLI model.",
"Note that, in accordance with the logic implication, the inconsistency loss reaches its global minimum when the probability of the body is close to zero – i.e.",
"the premise is false – and when the probabilities of both the body and the head are close to one – i.e.",
"the premise and the conclusion are both true.",
"We now generalise the intuition in Ex.",
"REF to any FOL rule.",
"Let $r = ({\\mathrm {body}} \\Rightarrow {\\mathrm {head}})$ denote an arbitrary FOL rule in the form described in eq:rule, and let ${\\mathrm {vars}}(r) = \\lbrace X_{1}, \\ldots , X_{n} \\rbrace $ denote the set of universally quantified variables in the rule $r$ .",
"Furthermore, let $S= \\lbrace X_{1} \\mapsto s_{1}, \\ldots , X_{n} \\mapsto s_{n} \\rbrace $ denote a substitution set, i.e.",
"a mapping from variables in ${\\mathrm {vars}}(r)$ to sentences $s_{1}, \\ldots , s_{n} \\in \\mathcal {S}$ .",
"The inconsistency loss associated with the rule $r$ on the substitution set $S$ can be defined as: $ \\begin{aligned} \\mathcal {J} _{\\mathcal {I}} (S) = \\left[ p(S; {\\mathrm {body}}) - p(S; {\\mathrm {head}}) \\right]_{+}\\end{aligned}$ where $p(S; {\\mathrm {body}})$ and $p(S; {\\mathrm {head}})$ denote the probability of body and head of the rule, after replacing the variables in $r$ with the corresponding sentences in $S$ .",
"The motivation for the loss in eq:iloss is that logic implications can be understood as “whenever the body is true, the head has to be true as well”.",
"In terms of NLI models, this translates as “the probability of the head should at least be as large as the probability of the body”.",
"For calculating the inconsistency loss in eq:iloss, we need to specify how to calculate the probability of $\\mathrm {head}$ and $\\mathrm {body}$ .",
"The probability of a single ground atom is given by querying the neural NLI model, as in eq:nnli.",
"The head contains a single atom, while the body can be a conjunction of multiple atoms.",
"Similarly to , we use the Gödel t-norm, a continuous generalisation of the conjunction operator in logic , for computing the probability of the body of a clause: $p_{\\Theta }(a_{1} \\wedge a_{2}) = \\min \\lbrace p_{\\Theta }(a_{1}), p_{\\Theta }(a_{2}) \\rbrace $ where $a_{1}$ and $a_{2}$ are two clause atoms.",
"In this work, we cast the problem of generating adversarial examples as an optimisation problem: we search for the substitution set $S= \\lbrace X_{1} \\mapsto s_{1}, \\ldots , X_{n} \\mapsto s_{n} \\rbrace $ that maximises the inconsistency loss in eq:iloss, thus (maximally) violating the available background knowledge."
],
[
"Constraining via Language Modelling",
"Maximising the inconsistency loss in eq:iloss may not be sufficient for generating meaningful adversarial examples: they can lead neural NLI models to violate available background knowledge, but they may not be well-formed and meaningful.",
"For such a reason, in addition to maximising the inconsistency loss, we also constrain the perplexity of generated sentences by using a neural language model .",
"In this work, we use a LSTM neural language model $p_{\\mathcal {L}}(w_{1}, \\ldots , w_{t})$ for generating low-perplexity adversarial examples."
],
[
"Searching in a Discrete Space",
"As mentioned earlier in this section, we cast the problem of automatically generating adversarial examples – i.e.",
"examples that cause NLI models to violate available background knowledge – as an optimisation problem.",
"Specifically, we look for substitutions sets $S= \\lbrace X_{1} \\mapsto s_{1}, \\ldots , X_{n} \\mapsto s_{n} \\rbrace $ that jointly: [a)] maximise the inconsistency loss described in eq:iloss, and are composed by sentences with a low perplexity, as defined by the neural language model in ssec:language.",
"The search objective can be formalised by the following optimisation problem: $ \\begin{aligned}& \\underset{S}{\\text{maximise}}& & \\mathcal {J} _{\\mathcal {I}} (S) \\\\& \\text{subject to}& & \\log p_{\\mathcal {L}}(S) \\le \\tau \\\\\\end{aligned}$ where $\\log p_{\\mathcal {L}}(S)$ denotes the log-probability of the sentences in the substitution set $S$ , and $\\tau $ is a threshold on the perplexity of generated sentences.",
"For generating low-perplexity adversarial examples, we take inspiration from and generate the sentences by editing prototypes extracted from a corpus.",
"Specifically, for searching substitution sets whose sentences jointly have a high probability and are highly adversarial, as measured the inconsistency loss in eq:iloss, we use the following procedure: [a)] we first sample sentences close to the data manifold (i.e.",
"with a low perplexity), by either sampling from the training set or from the language model; we then make small variations to the sentences – analogous to adversarial images, which consist in small perturbations of training examples – so to optimise the objective in eq:dsearch.",
"When editing prototypes, we consider the following perturbations: [a)] change one word in one of the input sentences; remove one parse sub-tree from one of the input sentences; insert one parse sub-tree from one sentence in the corpus in the parse tree of one of the input sentences.",
"Note that the generation process can easily lead to ungrammatical or implausible sentences; however, these will be likely to have a high perplexity according to the language model (ssec:language), and thus they will be ruled out by the search algorithm.",
"Adversarial Regularisation We now show one can use the adversarial examples to regularise the training process.",
"We propose training NLI models by jointly: [a)] minimising the data loss (eq:loss), and minimising the inconsistency loss (eq:iloss) on a set of generated adversarial examples (substitution sets).",
"More formally, for training, we jointly minimise the cross-entropy loss defined on the data $ \\mathcal {J} _{\\mathcal {D}} (\\Theta )$ and the inconsistency loss on a set of generated adversarial examples $\\max _{S} \\mathcal {J} _{\\mathcal {I}} (S; \\Theta )$ , resulting in the following optimisation problem: $ \\begin{aligned}& \\underset{\\Theta }{\\text{minimise}}& & \\mathcal {J} _{\\mathcal {D}} ( \\mathcal {D} , \\Theta ) + \\lambda \\max _{S} \\mathcal {J} _{\\mathcal {I}} (S; \\Theta ) \\\\& \\text{subject to}& & \\log p_{\\mathcal {L}}(S) \\le \\tau \\end{aligned}$ where $\\lambda \\in \\mathbb {R}_{+}$ is a hyperparameter specifying the trade-off between the data loss $ \\mathcal {J} _{\\mathcal {D}} $ (eq:loss), and the inconsistency loss $ \\mathcal {J} _{\\mathcal {I}} $ (eq:iloss), measured on the generated substitution set $S$ .",
"In eq:jloss, the regularisation term $\\max _{S} \\mathcal {J} _{\\mathcal {I}} (S; \\Theta )$ has the task of generating the adversarial substitution sets by maximising the inconsistency loss.",
"Furthermore, the constraint $\\log p_{\\mathcal {L}}(S) \\le \\tau $ ensures that the perplexity of generated sentences is lower than a threshold $\\tau $ .",
"For this work, we used the $\\max $ aggregation function.",
"However, other functions can be used as well, such as the sum or mean of multiple inconsistency losses.",
"[t] Solving the optimisation problem in eq:jloss via Mini-Batch Gradient Descent [1] Dataset $ \\mathcal {D} $ , weight $\\lambda \\in \\mathbb {R}_{+}$ No.",
"of epochs $\\tau \\in \\mathbb {N}_{+}$ No.",
"of adv.",
"substitution sets $n_{a} \\in \\mathbb {N}_{+}$ Initialise the model parameters $\\hat{\\Theta }$ $\\hat{\\Theta } \\leftarrow \\text{initialise}()$ $i \\in \\lbrace 1, \\ldots , \\tau \\rbrace $ $ \\mathcal {D} _{j} \\in \\text{batches}( \\mathcal {D} )$ Generate the adv.",
"substitution sets $S_{i}$ $\\lbrace S_{1}, \\ldots , S_{n_{a}} \\rbrace \\leftarrow \\text{generate}( \\mathcal {D} _{j})$ Compute the gradient of eq:jloss $\\mathcal {L} \\leftarrow \\mathcal {J} _{\\mathcal {D}} ( \\mathcal {D} _{j}, \\hat{\\Theta }) + \\lambda \\sum _{k = 1}^{n_{a}} \\mathcal {J} _{\\mathcal {I}} (S_{k}; \\hat{\\Theta })$ $g \\leftarrow \\nabla _{\\Theta } \\mathcal {L}$ Update the model parameters $\\hat{\\Theta } \\leftarrow \\hat{\\Theta } - \\eta g$ return $\\hat{\\Theta }$ For minimising the regularised loss in eq:jloss, we alternate between two optimisation processes – generating the adversarial examples (eq:dsearch) and minimising the regularised loss (eq:jloss).",
"The algorithm is outlined in alg:opt.",
"At each iteration, after generating a set of adversarial examples $S$ , it computes the gradient of the regularised loss in eq:jloss, and updates the model parameters via a gradient descent step.",
"On line REF , the algorithm generates a set of adversarial examples, each in the form of a substitution set $S$ .",
"On line REF , the algorithm computes the gradient of the adversarially regularised loss – a weighted combination of the data loss in eq:loss and the inconsistency loss in eq:iloss.",
"The model parameters are finally updated on line REF via a gradient descent step.",
"Creating Adversarial NLI Datasets Table: Sample sentences from an Adversarial NLI Dataset generated using the DAM model, by maximising the inconsistency loss 𝒥 ℐ \\mathcal {J} _{\\mathcal {I}} .We crafted a series of datasets for assessing the robustness of the proposed regularisation method to adversarial examples.",
"Starting from the SNLI test set, we proceeded as follows.",
"We selected the $k$ instances in the SNLI test set that maximise the inconsistency loss in eq:iloss with respect to the rules in $\\mathbf {R_{1}}$ , $\\mathbf {R_{2}}$ , $\\mathbf {R_{3}}$ , and $\\mathbf {R_{4}}$ in tab:rules.",
"We refer to the generated datasets as $ \\mathcal {A}_{\\text{m}}^{k} $ , where $m$ identifies the model used for selecting the sentence pairs, and $k$ denotes number of examples in the dataset.",
"For generating each of the $ \\mathcal {A}_{\\text{m}}^{k} $ datasets, we proceeded as follows.",
"Let $\\mathcal {D} = \\lbrace (x_{1}, y_{i}), \\ldots , (x_{n}, y_{n}) \\rbrace $ be a NLI dataset (such as SNLI), where each instance $x_{i} = (p_{i}, h_{i})$ is a premise-hypothesis sentence pair, and $y_{i}$ denotes the relationship holding between $p_{i}$ and $h_{i}$ .",
"For each instance $x_{i} = (p_{i}, h_{i})$ , we consider two substitution sets: $S_{i} = \\lbrace X_{1} \\mapsto p_{i}, X_{2} \\mapsto h_{i} \\rbrace $ and $S_{i}^{\\prime } = \\lbrace X_{1} \\mapsto h_{i}, X_{2} \\mapsto p_{i} \\rbrace $ , each corresponding to a mapping from variables to sentences.",
"We compute the inconsistency score associated to each instance $x_{i}$ in the dataset $\\mathcal {D}$ as $ \\mathcal {J} _{\\mathcal {I}} (S_{i}) + \\mathcal {J} _{\\mathcal {I}} (S_{i}^{\\prime })$ .",
"Note that the inconsistency score only depends on the premise $p_{i}$ and hypothesis $h_{i}$ in each instance $x_{i}$ , and it does not depend on its label $y_{i}$ .",
"After computing the inconsistency scores for all sentence pairs in $\\mathcal {D}$ using a model $m$ , we select the $k$ instances with the highest inconsistency score, we create two instances $x_{i} = (p_{i}, h_{i})$ and $\\hat{x_{i}} = (h_{i}, p_{i})$ , and add both $(x_{i}, y_{i})$ and $(\\hat{x}_{i}, \\hat{y}_{i})$ to the dataset $ \\mathcal {A}_{\\text{m}}^{k} $ .",
"Note that, while $y_{i}$ is already known from the dataset $\\mathcal {D}$ , $\\hat{y}_{i}$ is unknown.",
"For this reason, we find $\\hat{y}_{i}$ by manual annotation.",
"Related Work Adversarial examples are receiving a considerable attention in NLP; their usage, however, is considerably limited by the fact that semantically invariant input perturbations in NLP are difficult to identify .",
"analyse the robustness of extractive question answering models on examples obtained by adding adversarially generated distracting text to SQuAD dataset instances.",
"also notice that character-level Machine Translation are overly sensitive to random character manipulations, such as typos.",
"show that simple character-level modifications can drastically change the toxicity score of a text.",
"proposes using paraphrasing for generating adversarial examples.",
"Our model is fundamentally different in two ways: [a)] it does not need labelled data for generating adversarial examples – the inconsistency loss can be maximised by just making an NLI model produce inconsistent results, and it incorporates adversarial examples during the training process, with the aim of training more robust NLI models.",
"Table: Accuracy on the SNLI and MultiNLI datasets with different neural NLI models before (left) and after (right) adversarial regularisation.Adversarial examples are also used for assessing the robustness of computer vision models , , , where they are created by adding a small amount of noise to the inputs that does not change the semantics of the images, but drastically changes the model predictions.",
"Experiments Table: Violations (%) of rules 𝐑 1 ,𝐑 2 ,𝐑 3 ,𝐑 4 \\mathbf {R_{1}}, \\mathbf {R_{2}}, \\mathbf {R_{3}}, \\mathbf {R_{4}} from tab:rules on the SNLI training set, yield by cBiLSTM, DAM, and ESIM.We trained DAM, ESIM and cBiLSTM on the SNLI corpus using the hyperparameters provided in the respective papers.",
"The results provided by such models on the SNLI and MultiNLI validation and tests sets are provided in tab:models.",
"In the case of MultiNLI, the validation set was obtained by removing 10,000 instances from the training set (originally composed by 392,702 instances), and the test set consists in the matched validation set.",
"Table: Accuracy of unregularised and regularised neural NLI models DAM, cBiLSTM, and ESIM, and their adversarially regularised versions DAM 𝒜ℛ ^{\\mathcal {AR}}, cBiLSTM 𝒜ℛ ^{\\mathcal {AR}}, and ESIM 𝒜ℛ ^{\\mathcal {AR}}, on the datasets 𝒜 m k \\mathcal {A}_{\\text{m}}^{k} .Background Knowledge Violations.",
"As a first experiment, we count the how likely our model is to violate rules $\\mathbf {R_{1}}, \\mathbf {R_{2}}, \\mathbf {R_{3}}, \\mathbf {R_{4}}$ in tab:rules.",
"In tab:violations we report the number sentence pairs in the SNLI training set where DAM, ESIM and cBiLSTM violate $\\mathbf {R_{1}}, \\mathbf {R_{2}}, \\mathbf {R_{3}}, \\mathbf {R_{4}}$ .",
"In the $\\left|{\\mathbf {B}}\\right|$ column we report the number of times the body of the rule holds, according to the model.",
"In the $\\left|{\\mathbf {B} \\wedge \\lnot {\\mathbf {H}}}\\right|$ column we report the number of times where the body of the rule holds, but the head does not – which is clearly a violation of available rules.",
"We can see that, in the case of rule $\\mathbf {R_{1}}$ (reflexivity of entailment), DAM and ESIM make a relatively low number of violations – namely 0.09 and 1.00 %, respectively.",
"However, in the case of cBiLSTM, we can see that, each sentence $s \\in \\mathcal {S}$ in the SNLI training set, with a 23.76 % chance, $s$ does not entail itself – which violates our background knowledge.",
"With respect to $\\mathbf {R_{2}}$ (symmetry of contradiction), we see that none of the models is completely consistent with the available background knowledge.",
"Given a sentence pair $s_{1}, s_{2} \\in \\mathcal {S}$ from the SNLI training set, if – according to the model – $s_{1}$ contradicts $s_{2}$ , a significant number of times (between 9.84% and 46.17%) the same model also infers that $s_{2}$ does not contradict $s_{1}$ .",
"This phenomenon happens 16.70 % of times with DAM, 9.84 % of times with ESIM, and 46.17 % with cBiLSTM: this indicates that all considered models are prone to violating $\\mathbf {R_{2}}$ in their predictions, with ESIM being the more robust.",
"In sec:a:adversarial we report several examples of such violations in the SNLI training set.",
"We select those that maximise the inconsistency loss described in eq:iloss, violating rules $\\mathbf {R_{2}}$ and $\\mathbf {R_{3}}$ .",
"We can notice that the presence of inconsistencies is often correlated with the length of the sentences.",
"The model tends to detect entailment relationships between longer (i.e., possibly more specific) and shorter (i.e., possibly more general) sentences.",
"Generation of Adversarial Examples In the following, we analyse the automatic generation of sets of adversarial examples that make the model violate the existing background knowledge.",
"We search in the space of sentences by applying perturbations to sampled sentence pairs, using a language model for guiding the search process.",
"The generation procedure is described in sec:generating.",
"Table: Solving the optimisation problem in eq:jloss via Mini-Batch Gradient Descent"
],
[
"Adversarial Regularisation",
"We now show one can use the adversarial examples to regularise the training process.",
"We propose training NLI models by jointly: [a)] minimising the data loss (eq:loss), and minimising the inconsistency loss (eq:iloss) on a set of generated adversarial examples (substitution sets).",
"More formally, for training, we jointly minimise the cross-entropy loss defined on the data $ \\mathcal {J} _{\\mathcal {D}} (\\Theta )$ and the inconsistency loss on a set of generated adversarial examples $\\max _{S} \\mathcal {J} _{\\mathcal {I}} (S; \\Theta )$ , resulting in the following optimisation problem: $ \\begin{aligned}& \\underset{\\Theta }{\\text{minimise}}& & \\mathcal {J} _{\\mathcal {D}} ( \\mathcal {D} , \\Theta ) + \\lambda \\max _{S} \\mathcal {J} _{\\mathcal {I}} (S; \\Theta ) \\\\& \\text{subject to}& & \\log p_{\\mathcal {L}}(S) \\le \\tau \\end{aligned}$ where $\\lambda \\in \\mathbb {R}_{+}$ is a hyperparameter specifying the trade-off between the data loss $ \\mathcal {J} _{\\mathcal {D}} $ (eq:loss), and the inconsistency loss $ \\mathcal {J} _{\\mathcal {I}} $ (eq:iloss), measured on the generated substitution set $S$ .",
"In eq:jloss, the regularisation term $\\max _{S} \\mathcal {J} _{\\mathcal {I}} (S; \\Theta )$ has the task of generating the adversarial substitution sets by maximising the inconsistency loss.",
"Furthermore, the constraint $\\log p_{\\mathcal {L}}(S) \\le \\tau $ ensures that the perplexity of generated sentences is lower than a threshold $\\tau $ .",
"For this work, we used the $\\max $ aggregation function.",
"However, other functions can be used as well, such as the sum or mean of multiple inconsistency losses.",
"[t] Solving the optimisation problem in eq:jloss via Mini-Batch Gradient Descent [1] Dataset $ \\mathcal {D} $ , weight $\\lambda \\in \\mathbb {R}_{+}$ No.",
"of epochs $\\tau \\in \\mathbb {N}_{+}$ No.",
"of adv.",
"substitution sets $n_{a} \\in \\mathbb {N}_{+}$ Initialise the model parameters $\\hat{\\Theta }$ $\\hat{\\Theta } \\leftarrow \\text{initialise}()$ $i \\in \\lbrace 1, \\ldots , \\tau \\rbrace $ $ \\mathcal {D} _{j} \\in \\text{batches}( \\mathcal {D} )$ Generate the adv.",
"substitution sets $S_{i}$ $\\lbrace S_{1}, \\ldots , S_{n_{a}} \\rbrace \\leftarrow \\text{generate}( \\mathcal {D} _{j})$ Compute the gradient of eq:jloss $\\mathcal {L} \\leftarrow \\mathcal {J} _{\\mathcal {D}} ( \\mathcal {D} _{j}, \\hat{\\Theta }) + \\lambda \\sum _{k = 1}^{n_{a}} \\mathcal {J} _{\\mathcal {I}} (S_{k}; \\hat{\\Theta })$ $g \\leftarrow \\nabla _{\\Theta } \\mathcal {L}$ Update the model parameters $\\hat{\\Theta } \\leftarrow \\hat{\\Theta } - \\eta g$ return $\\hat{\\Theta }$ For minimising the regularised loss in eq:jloss, we alternate between two optimisation processes – generating the adversarial examples (eq:dsearch) and minimising the regularised loss (eq:jloss).",
"The algorithm is outlined in alg:opt.",
"At each iteration, after generating a set of adversarial examples $S$ , it computes the gradient of the regularised loss in eq:jloss, and updates the model parameters via a gradient descent step.",
"On line REF , the algorithm generates a set of adversarial examples, each in the form of a substitution set $S$ .",
"On line REF , the algorithm computes the gradient of the adversarially regularised loss – a weighted combination of the data loss in eq:loss and the inconsistency loss in eq:iloss.",
"The model parameters are finally updated on line REF via a gradient descent step.",
"Creating Adversarial NLI Datasets Table: Sample sentences from an Adversarial NLI Dataset generated using the DAM model, by maximising the inconsistency loss 𝒥 ℐ \\mathcal {J} _{\\mathcal {I}} .We crafted a series of datasets for assessing the robustness of the proposed regularisation method to adversarial examples.",
"Starting from the SNLI test set, we proceeded as follows.",
"We selected the $k$ instances in the SNLI test set that maximise the inconsistency loss in eq:iloss with respect to the rules in $\\mathbf {R_{1}}$ , $\\mathbf {R_{2}}$ , $\\mathbf {R_{3}}$ , and $\\mathbf {R_{4}}$ in tab:rules.",
"We refer to the generated datasets as $ \\mathcal {A}_{\\text{m}}^{k} $ , where $m$ identifies the model used for selecting the sentence pairs, and $k$ denotes number of examples in the dataset.",
"For generating each of the $ \\mathcal {A}_{\\text{m}}^{k} $ datasets, we proceeded as follows.",
"Let $\\mathcal {D} = \\lbrace (x_{1}, y_{i}), \\ldots , (x_{n}, y_{n}) \\rbrace $ be a NLI dataset (such as SNLI), where each instance $x_{i} = (p_{i}, h_{i})$ is a premise-hypothesis sentence pair, and $y_{i}$ denotes the relationship holding between $p_{i}$ and $h_{i}$ .",
"For each instance $x_{i} = (p_{i}, h_{i})$ , we consider two substitution sets: $S_{i} = \\lbrace X_{1} \\mapsto p_{i}, X_{2} \\mapsto h_{i} \\rbrace $ and $S_{i}^{\\prime } = \\lbrace X_{1} \\mapsto h_{i}, X_{2} \\mapsto p_{i} \\rbrace $ , each corresponding to a mapping from variables to sentences.",
"We compute the inconsistency score associated to each instance $x_{i}$ in the dataset $\\mathcal {D}$ as $ \\mathcal {J} _{\\mathcal {I}} (S_{i}) + \\mathcal {J} _{\\mathcal {I}} (S_{i}^{\\prime })$ .",
"Note that the inconsistency score only depends on the premise $p_{i}$ and hypothesis $h_{i}$ in each instance $x_{i}$ , and it does not depend on its label $y_{i}$ .",
"After computing the inconsistency scores for all sentence pairs in $\\mathcal {D}$ using a model $m$ , we select the $k$ instances with the highest inconsistency score, we create two instances $x_{i} = (p_{i}, h_{i})$ and $\\hat{x_{i}} = (h_{i}, p_{i})$ , and add both $(x_{i}, y_{i})$ and $(\\hat{x}_{i}, \\hat{y}_{i})$ to the dataset $ \\mathcal {A}_{\\text{m}}^{k} $ .",
"Note that, while $y_{i}$ is already known from the dataset $\\mathcal {D}$ , $\\hat{y}_{i}$ is unknown.",
"For this reason, we find $\\hat{y}_{i}$ by manual annotation.",
"Related Work Adversarial examples are receiving a considerable attention in NLP; their usage, however, is considerably limited by the fact that semantically invariant input perturbations in NLP are difficult to identify .",
"analyse the robustness of extractive question answering models on examples obtained by adding adversarially generated distracting text to SQuAD dataset instances.",
"also notice that character-level Machine Translation are overly sensitive to random character manipulations, such as typos.",
"show that simple character-level modifications can drastically change the toxicity score of a text.",
"proposes using paraphrasing for generating adversarial examples.",
"Our model is fundamentally different in two ways: [a)] it does not need labelled data for generating adversarial examples – the inconsistency loss can be maximised by just making an NLI model produce inconsistent results, and it incorporates adversarial examples during the training process, with the aim of training more robust NLI models.",
"Table: Accuracy on the SNLI and MultiNLI datasets with different neural NLI models before (left) and after (right) adversarial regularisation.Adversarial examples are also used for assessing the robustness of computer vision models , , , where they are created by adding a small amount of noise to the inputs that does not change the semantics of the images, but drastically changes the model predictions.",
"Experiments Table: Violations (%) of rules 𝐑 1 ,𝐑 2 ,𝐑 3 ,𝐑 4 \\mathbf {R_{1}}, \\mathbf {R_{2}}, \\mathbf {R_{3}}, \\mathbf {R_{4}} from tab:rules on the SNLI training set, yield by cBiLSTM, DAM, and ESIM.We trained DAM, ESIM and cBiLSTM on the SNLI corpus using the hyperparameters provided in the respective papers.",
"The results provided by such models on the SNLI and MultiNLI validation and tests sets are provided in tab:models.",
"In the case of MultiNLI, the validation set was obtained by removing 10,000 instances from the training set (originally composed by 392,702 instances), and the test set consists in the matched validation set.",
"Table: Accuracy of unregularised and regularised neural NLI models DAM, cBiLSTM, and ESIM, and their adversarially regularised versions DAM 𝒜ℛ ^{\\mathcal {AR}}, cBiLSTM 𝒜ℛ ^{\\mathcal {AR}}, and ESIM 𝒜ℛ ^{\\mathcal {AR}}, on the datasets 𝒜 m k \\mathcal {A}_{\\text{m}}^{k} .Background Knowledge Violations.",
"As a first experiment, we count the how likely our model is to violate rules $\\mathbf {R_{1}}, \\mathbf {R_{2}}, \\mathbf {R_{3}}, \\mathbf {R_{4}}$ in tab:rules.",
"In tab:violations we report the number sentence pairs in the SNLI training set where DAM, ESIM and cBiLSTM violate $\\mathbf {R_{1}}, \\mathbf {R_{2}}, \\mathbf {R_{3}}, \\mathbf {R_{4}}$ .",
"In the $\\left|{\\mathbf {B}}\\right|$ column we report the number of times the body of the rule holds, according to the model.",
"In the $\\left|{\\mathbf {B} \\wedge \\lnot {\\mathbf {H}}}\\right|$ column we report the number of times where the body of the rule holds, but the head does not – which is clearly a violation of available rules.",
"We can see that, in the case of rule $\\mathbf {R_{1}}$ (reflexivity of entailment), DAM and ESIM make a relatively low number of violations – namely 0.09 and 1.00 %, respectively.",
"However, in the case of cBiLSTM, we can see that, each sentence $s \\in \\mathcal {S}$ in the SNLI training set, with a 23.76 % chance, $s$ does not entail itself – which violates our background knowledge.",
"With respect to $\\mathbf {R_{2}}$ (symmetry of contradiction), we see that none of the models is completely consistent with the available background knowledge.",
"Given a sentence pair $s_{1}, s_{2} \\in \\mathcal {S}$ from the SNLI training set, if – according to the model – $s_{1}$ contradicts $s_{2}$ , a significant number of times (between 9.84% and 46.17%) the same model also infers that $s_{2}$ does not contradict $s_{1}$ .",
"This phenomenon happens 16.70 % of times with DAM, 9.84 % of times with ESIM, and 46.17 % with cBiLSTM: this indicates that all considered models are prone to violating $\\mathbf {R_{2}}$ in their predictions, with ESIM being the more robust.",
"In sec:a:adversarial we report several examples of such violations in the SNLI training set.",
"We select those that maximise the inconsistency loss described in eq:iloss, violating rules $\\mathbf {R_{2}}$ and $\\mathbf {R_{3}}$ .",
"We can notice that the presence of inconsistencies is often correlated with the length of the sentences.",
"The model tends to detect entailment relationships between longer (i.e., possibly more specific) and shorter (i.e., possibly more general) sentences.",
"Generation of Adversarial Examples In the following, we analyse the automatic generation of sets of adversarial examples that make the model violate the existing background knowledge.",
"We search in the space of sentences by applying perturbations to sampled sentence pairs, using a language model for guiding the search process.",
"The generation procedure is described in sec:generating.",
"Table: Solving the optimisation problem in eq:jloss via Mini-Batch Gradient Descent"
],
[
"Creating Adversarial NLI Datasets",
"We crafted a series of datasets for assessing the robustness of the proposed regularisation method to adversarial examples.",
"Starting from the SNLI test set, we proceeded as follows.",
"We selected the $k$ instances in the SNLI test set that maximise the inconsistency loss in eq:iloss with respect to the rules in $\\mathbf {R_{1}}$ , $\\mathbf {R_{2}}$ , $\\mathbf {R_{3}}$ , and $\\mathbf {R_{4}}$ in tab:rules.",
"We refer to the generated datasets as $ \\mathcal {A}_{\\text{m}}^{k} $ , where $m$ identifies the model used for selecting the sentence pairs, and $k$ denotes number of examples in the dataset.",
"For generating each of the $ \\mathcal {A}_{\\text{m}}^{k} $ datasets, we proceeded as follows.",
"Let $\\mathcal {D} = \\lbrace (x_{1}, y_{i}), \\ldots , (x_{n}, y_{n}) \\rbrace $ be a NLI dataset (such as SNLI), where each instance $x_{i} = (p_{i}, h_{i})$ is a premise-hypothesis sentence pair, and $y_{i}$ denotes the relationship holding between $p_{i}$ and $h_{i}$ .",
"For each instance $x_{i} = (p_{i}, h_{i})$ , we consider two substitution sets: $S_{i} = \\lbrace X_{1} \\mapsto p_{i}, X_{2} \\mapsto h_{i} \\rbrace $ and $S_{i}^{\\prime } = \\lbrace X_{1} \\mapsto h_{i}, X_{2} \\mapsto p_{i} \\rbrace $ , each corresponding to a mapping from variables to sentences.",
"We compute the inconsistency score associated to each instance $x_{i}$ in the dataset $\\mathcal {D}$ as $ \\mathcal {J} _{\\mathcal {I}} (S_{i}) + \\mathcal {J} _{\\mathcal {I}} (S_{i}^{\\prime })$ .",
"Note that the inconsistency score only depends on the premise $p_{i}$ and hypothesis $h_{i}$ in each instance $x_{i}$ , and it does not depend on its label $y_{i}$ .",
"After computing the inconsistency scores for all sentence pairs in $\\mathcal {D}$ using a model $m$ , we select the $k$ instances with the highest inconsistency score, we create two instances $x_{i} = (p_{i}, h_{i})$ and $\\hat{x_{i}} = (h_{i}, p_{i})$ , and add both $(x_{i}, y_{i})$ and $(\\hat{x}_{i}, \\hat{y}_{i})$ to the dataset $ \\mathcal {A}_{\\text{m}}^{k} $ .",
"Note that, while $y_{i}$ is already known from the dataset $\\mathcal {D}$ , $\\hat{y}_{i}$ is unknown.",
"For this reason, we find $\\hat{y}_{i}$ by manual annotation."
],
[
"Related Work",
"Adversarial examples are receiving a considerable attention in NLP; their usage, however, is considerably limited by the fact that semantically invariant input perturbations in NLP are difficult to identify .",
"analyse the robustness of extractive question answering models on examples obtained by adding adversarially generated distracting text to SQuAD dataset instances.",
"also notice that character-level Machine Translation are overly sensitive to random character manipulations, such as typos.",
"show that simple character-level modifications can drastically change the toxicity score of a text.",
"proposes using paraphrasing for generating adversarial examples.",
"Our model is fundamentally different in two ways: [a)] it does not need labelled data for generating adversarial examples – the inconsistency loss can be maximised by just making an NLI model produce inconsistent results, and it incorporates adversarial examples during the training process, with the aim of training more robust NLI models.",
"Table: Accuracy on the SNLI and MultiNLI datasets with different neural NLI models before (left) and after (right) adversarial regularisation.Adversarial examples are also used for assessing the robustness of computer vision models , , , where they are created by adding a small amount of noise to the inputs that does not change the semantics of the images, but drastically changes the model predictions.",
"Experiments Table: Violations (%) of rules 𝐑 1 ,𝐑 2 ,𝐑 3 ,𝐑 4 \\mathbf {R_{1}}, \\mathbf {R_{2}}, \\mathbf {R_{3}}, \\mathbf {R_{4}} from tab:rules on the SNLI training set, yield by cBiLSTM, DAM, and ESIM.We trained DAM, ESIM and cBiLSTM on the SNLI corpus using the hyperparameters provided in the respective papers.",
"The results provided by such models on the SNLI and MultiNLI validation and tests sets are provided in tab:models.",
"In the case of MultiNLI, the validation set was obtained by removing 10,000 instances from the training set (originally composed by 392,702 instances), and the test set consists in the matched validation set.",
"Table: Accuracy of unregularised and regularised neural NLI models DAM, cBiLSTM, and ESIM, and their adversarially regularised versions DAM 𝒜ℛ ^{\\mathcal {AR}}, cBiLSTM 𝒜ℛ ^{\\mathcal {AR}}, and ESIM 𝒜ℛ ^{\\mathcal {AR}}, on the datasets 𝒜 m k \\mathcal {A}_{\\text{m}}^{k} .Background Knowledge Violations.",
"As a first experiment, we count the how likely our model is to violate rules $\\mathbf {R_{1}}, \\mathbf {R_{2}}, \\mathbf {R_{3}}, \\mathbf {R_{4}}$ in tab:rules.",
"In tab:violations we report the number sentence pairs in the SNLI training set where DAM, ESIM and cBiLSTM violate $\\mathbf {R_{1}}, \\mathbf {R_{2}}, \\mathbf {R_{3}}, \\mathbf {R_{4}}$ .",
"In the $\\left|{\\mathbf {B}}\\right|$ column we report the number of times the body of the rule holds, according to the model.",
"In the $\\left|{\\mathbf {B} \\wedge \\lnot {\\mathbf {H}}}\\right|$ column we report the number of times where the body of the rule holds, but the head does not – which is clearly a violation of available rules.",
"We can see that, in the case of rule $\\mathbf {R_{1}}$ (reflexivity of entailment), DAM and ESIM make a relatively low number of violations – namely 0.09 and 1.00 %, respectively.",
"However, in the case of cBiLSTM, we can see that, each sentence $s \\in \\mathcal {S}$ in the SNLI training set, with a 23.76 % chance, $s$ does not entail itself – which violates our background knowledge.",
"With respect to $\\mathbf {R_{2}}$ (symmetry of contradiction), we see that none of the models is completely consistent with the available background knowledge.",
"Given a sentence pair $s_{1}, s_{2} \\in \\mathcal {S}$ from the SNLI training set, if – according to the model – $s_{1}$ contradicts $s_{2}$ , a significant number of times (between 9.84% and 46.17%) the same model also infers that $s_{2}$ does not contradict $s_{1}$ .",
"This phenomenon happens 16.70 % of times with DAM, 9.84 % of times with ESIM, and 46.17 % with cBiLSTM: this indicates that all considered models are prone to violating $\\mathbf {R_{2}}$ in their predictions, with ESIM being the more robust.",
"In sec:a:adversarial we report several examples of such violations in the SNLI training set.",
"We select those that maximise the inconsistency loss described in eq:iloss, violating rules $\\mathbf {R_{2}}$ and $\\mathbf {R_{3}}$ .",
"We can notice that the presence of inconsistencies is often correlated with the length of the sentences.",
"The model tends to detect entailment relationships between longer (i.e., possibly more specific) and shorter (i.e., possibly more general) sentences.",
"Generation of Adversarial Examples In the following, we analyse the automatic generation of sets of adversarial examples that make the model violate the existing background knowledge.",
"We search in the space of sentences by applying perturbations to sampled sentence pairs, using a language model for guiding the search process.",
"The generation procedure is described in sec:generating.",
"Table: Solving the optimisation problem in eq:jloss via Mini-Batch Gradient Descent"
],
[
"Experiments",
"We trained DAM, ESIM and cBiLSTM on the SNLI corpus using the hyperparameters provided in the respective papers.",
"The results provided by such models on the SNLI and MultiNLI validation and tests sets are provided in tab:models.",
"In the case of MultiNLI, the validation set was obtained by removing 10,000 instances from the training set (originally composed by 392,702 instances), and the test set consists in the matched validation set.",
"Table: Accuracy of unregularised and regularised neural NLI models DAM, cBiLSTM, and ESIM, and their adversarially regularised versions DAM 𝒜ℛ ^{\\mathcal {AR}}, cBiLSTM 𝒜ℛ ^{\\mathcal {AR}}, and ESIM 𝒜ℛ ^{\\mathcal {AR}}, on the datasets 𝒜 m k \\mathcal {A}_{\\text{m}}^{k} ."
],
[
"As a first experiment, we count the how likely our model is to violate rules $\\mathbf {R_{1}}, \\mathbf {R_{2}}, \\mathbf {R_{3}}, \\mathbf {R_{4}}$ in tab:rules.",
"In tab:violations we report the number sentence pairs in the SNLI training set where DAM, ESIM and cBiLSTM violate $\\mathbf {R_{1}}, \\mathbf {R_{2}}, \\mathbf {R_{3}}, \\mathbf {R_{4}}$ .",
"In the $\\left|{\\mathbf {B}}\\right|$ column we report the number of times the body of the rule holds, according to the model.",
"In the $\\left|{\\mathbf {B} \\wedge \\lnot {\\mathbf {H}}}\\right|$ column we report the number of times where the body of the rule holds, but the head does not – which is clearly a violation of available rules.",
"We can see that, in the case of rule $\\mathbf {R_{1}}$ (reflexivity of entailment), DAM and ESIM make a relatively low number of violations – namely 0.09 and 1.00 %, respectively.",
"However, in the case of cBiLSTM, we can see that, each sentence $s \\in \\mathcal {S}$ in the SNLI training set, with a 23.76 % chance, $s$ does not entail itself – which violates our background knowledge.",
"With respect to $\\mathbf {R_{2}}$ (symmetry of contradiction), we see that none of the models is completely consistent with the available background knowledge.",
"Given a sentence pair $s_{1}, s_{2} \\in \\mathcal {S}$ from the SNLI training set, if – according to the model – $s_{1}$ contradicts $s_{2}$ , a significant number of times (between 9.84% and 46.17%) the same model also infers that $s_{2}$ does not contradict $s_{1}$ .",
"This phenomenon happens 16.70 % of times with DAM, 9.84 % of times with ESIM, and 46.17 % with cBiLSTM: this indicates that all considered models are prone to violating $\\mathbf {R_{2}}$ in their predictions, with ESIM being the more robust.",
"In sec:a:adversarial we report several examples of such violations in the SNLI training set.",
"We select those that maximise the inconsistency loss described in eq:iloss, violating rules $\\mathbf {R_{2}}$ and $\\mathbf {R_{3}}$ .",
"We can notice that the presence of inconsistencies is often correlated with the length of the sentences.",
"The model tends to detect entailment relationships between longer (i.e., possibly more specific) and shorter (i.e., possibly more general) sentences."
],
[
"Generation of Adversarial Examples",
"In the following, we analyse the automatic generation of sets of adversarial examples that make the model violate the existing background knowledge.",
"We search in the space of sentences by applying perturbations to sampled sentence pairs, using a language model for guiding the search process.",
"The generation procedure is described in sec:generating.",
"Table: Solving the optimisation problem in eq:jloss via Mini-Batch Gradient Descent"
]
] | 1808.08609 |
[
[
"Multi-fold contour integrals of certain ratios of Euler gamma functions\n from Feynman diagrams: orthogonality of triangles"
],
[
"Abstract We observe a property of orthogonality of the Mellin-Barnes transformation of the triangle one-loop diagrams, which follows from our previous papers [JHEP {\\bf 0808} (2008) 106, JHEP {\\bf 1003} (2010) 051, JMP {\\bf 51} (2010) 052304].",
"In those papers it has been established that Usyukina-Davydychev functions are invariant with respect to Fourier transformation.",
"This has been proved at the level of graphs and also via the Mellin-Barnes transformation.",
"We partially apply to one-loop massless scalar diagram the same trick in which the Mellin-Barnes transformation was involved and obtain the property of orthogonality of the corresponding MB transforms under integration over contours in two complex planes with certain weight.",
"This property is valid in an arbitrary number of dimensions."
],
[
"Foreword of I.K.",
"I met Sasha Vasil'ev for the first time in 2001 in Valparaiso when I came as a postdoctoral fellow to the high energy physics group at the Physics Department of UTFSM.",
"At that time Sasha and Irina were professors at the Mathematical Department.",
"He was a wonderful person and a brilliant mathematician.",
"He worked in many fields of mathematics.",
"Complex analysis was one of them.",
"My talk is dedicated to an application of complex analysis to the tasks that may appear from comparing integrals corresponding to different Feynman diagrams.",
"My talk is along the lines of paper [1].",
"Sasha was my friend and his pass away is a loss for me.",
"He was a real human."
],
[
"Introduction",
"Ladder diagrams are important family of Feynman diagrams.",
"The scalar ladder diagrams have been calculated in Refs.",
"[2], [3], [4] at the beginning of nineties in $d=4$ space-time dimensions.",
"The result is written in terms of Usyukina-Davydychev (UD) functions [3], [4].",
"The calculation was based on the loop reduction technique proposed in Ref.",
"[2].",
"With help of this loop reduction technique any diagram of this family may be reduced to a one-loop massless triangle diagram with a bit modified indices of propagators.",
"The trick of loop reduction has been generalized to an arbitrary space-time dimension in Refs.",
"[5], [6], however with the index of the rungs different from 1.",
"Mellin-Barnes (MB) transforms of the momentum integrals corresponding to ladder diagrams at any loop order have been studied in Refs.",
"[7], [1], [8], [9] in four-dimensional case.",
"In Ref.",
"[9] it has been shown that the loop reduction in terms of Mellin-Barnes transforms means the application of the first and the second Barnes lemmas [10], [11].",
"The main equation of loop reduction technique in terms of MB transforms may be written explicitly in two lines We omit the factor $1/2\\pi i$ in front of each contour integral in complex plane $ \\oint _C~dz_2dz_3~D^{(u,v)}[1+\\varepsilon _1-z_3,1+\\varepsilon _2-z_2,1+\\varepsilon _3]D^{(z_2,z_3)}[1+\\varepsilon _2,1+\\varepsilon _1,1+\\varepsilon _3] = \\nonumber \\\\J\\left[ \\frac{D^{(u,v-\\varepsilon _2)}[1-\\varepsilon _1]}{\\varepsilon _2\\varepsilon _3}+ \\frac{D^{(u,v)}[1+\\varepsilon _3]}{\\varepsilon _1\\varepsilon _2} + \\frac{ D^{(u-\\varepsilon _1,v)}[1-\\varepsilon _2]}{\\varepsilon _1\\varepsilon _3} \\right]$ and is given for the first time in Ref.",
"[1], where $D^{(u,v)}[\\nu _1,\\nu _2,\\nu _3]$ is MB transform of the one-loop massless triangle scalar momentum integral, $\\varepsilon _1,\\varepsilon _2,\\varepsilon _3$ are three complex numbers which satisfy the condition $\\varepsilon _1+\\varepsilon _2+\\varepsilon _3 = 0,$ and $J$ is a factor that depends on $\\varepsilon _1,\\varepsilon _2,\\varepsilon _3$ only.",
"Eq.",
"(REF ) has a structure similar to decomposition of tensor product in terms of irreducible components.",
"In the present paper we find an analog of orthogonality condition.",
"The key step to this aim is to repeat the proof that has been applied in Ref.",
"[7] in order to show invariance of Usyukina-Davydychev functions with respect to Fourier transformation.",
"This proof is simple and may be used to show the invariance with respect to Fourier transformation of any three-point Green function [12].",
"The orthogonality and the decomposition taken together suggest that in quantum field theory there are internal integrable structures for the Green functions.",
"Integrable structures are usually known for amplitudes [13] but not for the multi-point Green functions.",
"The invariance of UD functions with respect to Fourier transformation is known already for several years [14], [15], [12].",
"The Fourier invariance in these papers has been found at the level of graphs.",
"Then, this invariance has been proved via Mellin-Barnes transformation in Ref.",
"[7].",
"In the next sections we collect all the necessary tools to prove the orthogonality of the triangle MB transforms $D^{(u,v)}[\\nu _1,\\nu _2,\\nu _3]$ and find a weight with which this orthogonality appears inside complex integrals over contours."
],
[
"MB representation of one-loop triangle diagram",
"One-loop massless triangle diagram is depicted in Figure 1.",
"It contains three scalar propagators.",
"The $d$ -dimensional momenta $p_1$ , $p_2$ , $p_3$ enter this diagram.",
"They are related by momentum conservation $ p_1 + p_2 + p_3 = 0.$ This momentum integral $J(\\nu _1,\\nu _2,\\nu _3) = \\int ~Dk~\\frac{1}{\\left[(k + q_1)^2\\right]^{\\nu _1} \\left[(k + q_2)^2 \\right]^{\\nu _2}\\left[(k + q_3)^2\\right]^{\\nu _3}}$ corresponds to the diagram in Figure 1.",
"The running momentum $k$ is the integration variable.",
"Notation is chosen in such a way that the index of propagator $\\nu _1$ stands on the line opposite to the vertex of triangle into which the momentum $p_1$ enters.",
"Figure: One-loop massless scalar triangle in momentum spaceThe notation $q_1,$ $q_2$ and $q_3$ are taken from Ref.[3].",
"It follows from the diagram in Figure REF and the momentum conservation law that $p_1 = q_3 - q_2, ~~~p_2 = q_1 - q_3, ~~~p_3 = q_2 - q_1.$ To define the integral measure in momentum space, we use the notation from Ref.",
"[16] $Dk \\equiv \\pi ^{-\\frac{d}{2}}d^d k. $ Such a definition of the integration measure in the momentum space helps to avoid powers of $\\pi $ in formulas for momentum integrals which will appear in the next sections.",
"Mellin-Barnes representation of integral $J(\\nu _1,\\nu _2,\\nu _3)$ may be obtained via Feynman parameters [3] and has the form $J(\\nu _1,\\nu _2,\\nu _3) = \\frac{1}{\\Pi _{i} \\Gamma (\\nu _i) \\Gamma (d-\\Sigma _i \\nu _i)} \\frac{1}{{(p^2_3)}^{ \\Sigma \\nu _i -d/2}}\\oint _C dz_2~dz_3 \\left(\\frac{p^2_1}{p^2_3}\\right)^{z_2} \\left(\\frac{p^2_2}{p^2_3}\\right)^{z_3}\\left\\lbrace \\Gamma \\left(-z_2 \\right)\\Gamma \\left(-z_3 \\right)\\right.\\nonumber \\\\\\left.",
"\\Gamma \\left(-z_2 -\\nu _2-\\nu _3 + d/2 \\right)\\Gamma \\left(-z_3-\\nu _1-\\nu _3 + d/2 \\right)\\Gamma \\left(z_2 + z_3 + \\nu _3 \\right)\\Gamma \\left(\\Sigma \\nu _i - d/2 + z_3 + z_2 \\right)\\right\\rbrace \\equiv \\\\\\equiv \\frac{1}{{(p^2_3)}^{ \\Sigma \\nu _i -d/2}}\\oint _C dz_2~dz_3 x^{z_2} y^{z_3}D^{(z_2,z_3)} [\\nu _1,\\nu _2,\\nu _3].",
"\\nonumber $ We have used here the definition of the Mellin-Barnes transform $D^{(z_2,z_3)}[\\nu _1,\\nu _2,\\nu _3]$ from our Ref.",
"[1], $D^{(z_2,z_3)}[\\nu _1,\\nu _2,\\nu _3] = \\frac{ \\Gamma \\left(-z_2 \\right)\\Gamma \\left(-z_3 \\right)\\Gamma \\left(-z_2 -\\nu _2-\\nu _3 + d/2 \\right)\\Gamma \\left(-z_3-\\nu _1-\\nu _3 + d/2 \\right)}{\\Pi _{i} \\Gamma (\\nu _i) } \\nonumber \\\\\\times \\frac{ \\Gamma \\left(z_2 + z_3 + \\nu _3 \\right)\\Gamma \\left(\\Sigma \\nu _i - d/2 + z_3 + z_2 \\right)}{\\Gamma (d-\\Sigma _i \\nu _i)}.",
"$ The function $D^{(z_2,z_3)}[\\nu _1,\\nu _2,\\nu _3]$ has already appeared in Introduction in Eq.",
"(REF )."
],
[
"Fourier transformation",
"The formulas for the Fourier transforms of the massless propagators may be found, for example, in Ref. [1].",
"The formula we need is $\\int ~d^d p ~e^{ip x} \\frac{1}{(p^2)^\\alpha } = \\pi ^{d/2}\\frac{\\Gamma (d/2-\\alpha )}{\\Gamma (\\alpha )}\\left(\\frac{4}{x^2}\\right)^{d/2-\\alpha },$ or, equivalently, $\\frac{1}{(x^2)^{\\alpha }} = \\pi ^{-d/2} 4^{-\\alpha } \\frac{\\Gamma (d/2-\\alpha )}{\\Gamma (\\alpha )}\\int ~d^d p ~e^{ip x} \\frac{1}{(p^2)^{d/2-\\alpha }}.$"
],
[
"Star-triangle relation",
"For any three vectors $x_1,$ $x_2$ , $x_3$ in $d$ -dimensional Euclidean space there is so-called “star-triangle” relation, $ \\int Dx \\frac{1}{\\left[(x_1-x)^2\\right]^{\\alpha _1} \\left[(x_2-x)^2\\right]^{\\alpha _2} \\left[(x_3-x)^2\\right]^{\\alpha _3} } = \\frac{\\Gamma (d/2-\\alpha _1)}{\\Gamma (\\alpha _1)}\\frac{\\Gamma (d/2-\\alpha _2)}{\\Gamma (\\alpha _2)} \\times \\nonumber \\\\\\times \\frac{\\Gamma (d/2-\\alpha _3)}{\\Gamma (\\alpha _3)}\\frac{1}{\\left[(x_1-x_2)^2\\right]^{d/2-\\alpha _3}\\left[(x_2-x_3)^2\\right]^{d/2-\\alpha _1} \\left[(x_1-x_3)^2\\right]^{d/2-\\alpha _2}}.$ under the condition of “uniqueness” $\\alpha _1 + \\alpha _2 + \\alpha _3 = d$ for the indices in denominator, $\\lbrace \\alpha _1,\\alpha _2,\\alpha _3 \\rbrace \\notin \\lbrace d/2+n\\rbrace , n \\in \\mathbb {Z}_{+} $ This relation may be used in the momentum space as well as in the position space.",
"In the position space we use the same $d$ -dimensional measure we used in Eq.",
"(REF ) for the integration in the momentum space, $Dx \\equiv \\pi ^{-\\frac{d}{2}}d^d x$ in order to avoid a power of $\\pi $ on the r.h.s.",
"of Eq.",
"(REF ).",
"This appeared to be very useful measure redefinition in the position space too in Refs.",
"[1], [6] in order to develop the loop reduction technique in the position space.",
"This star-triangle relation (REF ) was widely applied to the integrals corresponding to Feynman diagrams in massless field theories.",
"It has been published in Refs.",
"[17], [18], [19] (for the brief review, see Ref.[20]).",
"The relation has been further developed to “stars” and “triangles” with one-step deviation from unique “stars” and “triangles” in Refs.",
"[21], [22].",
"For example, in Ref.",
"[21] a loop reduction effect has been discovered due to uniqueness method in a special limit of the indices of diagrams.",
"Later, in Refs.",
"[2], [22] the loop reduction technique has been established without this special limit due to uniqueness method, too.",
"The key point of the method has been published in Ref.[22].",
"All the details of the loop reduction technique are given in Ref.",
"[1] for $d=4$ case and in Ref.",
"[5], [6] for an arbitrary space-time dimension.",
"The star-triangle relation (REF ) is a basement for the uniqueness method.",
"It may be proved in many ways.",
"We need in the present paper a proof via the Mellin-Barnes transformation We do not know explicit reference to this proof of star-triangle relation via the MB transformation.",
"I.K.",
"explained this proof via MB transformation in his lectures on QFT at UdeC, Chile in May of 2009.",
"We assume in the rest of the paper the concise notation of Ref.",
"[16], where $[Ny]= (x_N - y)^2$ and analogously for $[Nz],$ and $[yz] = (y-z)^2$ , that is, $N=1,2,3$ stands for $x_N=x_1,x_2,x_3$ , respectively, which are the external points of the triangle diagram in Figure REF in the position space.",
"According to Eq.",
"(REF ) we may write in our notation $\\int Dx \\frac{1}{\\left[(x_1-x)^2\\right]^{\\alpha _1} \\left[(x_2-x)^2\\right]^{\\alpha _2} \\left[(x_3-x)^2\\right]^{\\alpha _3} } \\equiv \\int Dx \\frac{1}{[x1]^{\\alpha _1} [x2]^{\\alpha _2} [x3]^{\\alpha _3} } \\\\= \\frac{1}{\\Pi _{i} \\Gamma (\\alpha _i) \\Gamma (d-\\Sigma _i \\alpha _i)} \\frac{1}{[12]^{ \\Sigma \\alpha _i -d/2}}\\oint _C dz_2~dz_3 \\left(\\frac{[23]}{[12]}\\right)^{z_2} \\left(\\frac{[31]}{[12]}\\right)^{z_3}\\left\\lbrace \\Gamma \\left(-z_2 \\right)\\Gamma \\left(-z_3 \\right)\\right.\\nonumber \\\\\\left.",
"\\Gamma \\left(-z_2 -\\alpha _2-\\alpha _3 + d/2 \\right)\\Gamma \\left(-z_3-\\alpha _1-\\alpha _3 + d/2 \\right)\\Gamma \\left(z_2 + z_3 + \\alpha _3 \\right)\\Gamma \\left(\\Sigma \\alpha _i - d/2 + z_3 + z_2 \\right)\\right\\rbrace = \\\\= \\frac{1}{\\Pi _{i} \\Gamma (\\alpha _i) \\Gamma (0)} \\frac{1}{[12]^{d/2}}\\oint _C dz_2~dz_3 \\left(\\frac{[23]}{[12]}\\right)^{z_2} \\left(\\frac{[31]}{[12]}\\right)^{z_3}\\left\\lbrace \\Gamma \\left(-z_2 \\right)\\Gamma \\left(-z_3 \\right)\\Gamma \\left(-z_2 + \\alpha _1 - d/2 \\right)\\right.\\nonumber \\\\\\left.",
"\\Gamma \\left(-z_3+\\alpha _2 - d/2 \\right)\\Gamma \\left(z_2 + z_3 + \\alpha _3 \\right)\\Gamma \\left(d/2 + z_3 + z_2 \\right)\\right\\rbrace .$ In the denominator we have $\\Gamma (0)$ and this means we need to have the same factor in the numerator.",
"Otherwise, the result would be zero.",
"This may happen only for the residues $z_2 = \\alpha _1-d/2$ and $z_3 = \\alpha _2-d/2,$ because only for these residues the factor $\\Gamma \\left(z_2 + z_3 + \\alpha _3\\right)$ is equal to $\\Gamma \\left(0\\right)$ and cancels $\\Gamma \\left(0\\right)$ in the denominator.",
"Any other residues produced by Gamma functions with negative sign of arguments will have a vanishing contribution due to $\\Gamma \\left(0\\right)$ in the denominator.",
"Thus, $\\frac{1}{\\Pi _{i} \\Gamma (\\alpha _i) \\Gamma (0)} \\frac{1}{[12]^{d/2}}\\oint _C dz_2~dz_3 \\left(\\frac{[23]}{[12]}\\right)^{z_2} \\left(\\frac{[31]}{[12]}\\right)^{z_3}\\left\\lbrace \\Gamma \\left(-z_2 \\right)\\Gamma \\left(-z_3 \\right)\\Gamma \\left(-z_2 + \\alpha _1 - d/2 \\right)\\right.\\nonumber \\\\\\left.",
"\\Gamma \\left(-z_3+\\alpha _2 - d/2 \\right)\\Gamma \\left(z_2 + z_3 + \\alpha _3 \\right)\\Gamma \\left(d/2 + z_3 + z_2 \\right)\\right\\rbrace =$ $\\frac{1}{\\Pi _{i} \\Gamma (\\alpha _i) } \\frac{1}{[12]^{d/2}} \\left(\\frac{[23]}{[12]}\\right)^{\\alpha _1-d/2} \\left(\\frac{[31]}{[12]}\\right)^{\\alpha _2-d/2}\\Gamma \\left(d/2 - \\alpha _1 \\right)\\Gamma \\left(d/2 -\\alpha _2 \\right)\\Gamma \\left(d/2 - \\alpha _3 \\right)= \\\\\\frac{1}{\\Pi _{i} \\Gamma (\\alpha _i) } \\frac{\\Gamma \\left(d/2 - \\alpha _1 \\right)\\Gamma \\left(d/2 -\\alpha _2 \\right)\\Gamma \\left(d/2 - \\alpha _3 \\right)}{[12]^{d/2-\\alpha _3}[23]^{d/2-\\alpha _1}[31]^{d/2-\\alpha _2}}.$ This proves the star-triangle relation of Eq.",
"(REF )."
],
[
"Fourier invariance",
"As we have mentioned in Introduction, the proof of Fourier invariance of UD functions has been given in Refs.",
"[14], [15], [12] at the level of graphs.",
"Later, it has been proved via Mellin-Barnes transformation in Ref.",
"[7].",
"The method of this proof of Ref.",
"[7] will be the key tool in the next section, in which the main result of the paper is obtained.",
"We will briefly repeat the proof of Ref.",
"[7] in this section.",
"The Usyukina-Davydychev functions $\\Phi ^{(n)}(x,y)$ are functions of two variables.",
"They are the result of calculation of the integrals corresponding to triangle ladder diagrams [3], [4].",
"The explicit form of the UD functions $\\Phi ^{(n)}$ is given in the same Refs.",
"[3], [4] $\\Phi ^{(n)}\\left(x,y\\right)= -\\frac{1}{n!\\lambda }\\sum _{j=n}^{2n}\\frac{(-1)^j j!\\ln ^{2n-j}{(y/x)}}{(j-n)!(2n-j)!",
"}\\left[{\\rm Li_j}\\left(-\\frac{1}{\\rho x} \\right)- {\\rm Li}_j(-\\rho y)\\right], $ $\\rho = \\frac{2}{1-x-y+\\lambda }, ~~~~ \\lambda = \\sqrt{(1-x-y)^2-4xy}.$ In Refs.",
"[14], [15] Fourier invariance property for the UD functions, that is, $ \\frac{1}{[31]^2} \\Phi ^{(n)}\\left(\\frac{[12]}{[31]},\\frac{[23]}{[31]}\\right)=\\frac{1}{(2\\pi )^4}\\int ~d^4p_1d^4p_2d^4p_3 ~ \\delta (p_1 + p_2 + p_3) \\times \\nonumber \\\\\\times e^{ip_2x_2} e^{ip_1x_1} e^{ip_3x_3} \\frac{1}{(p_2^2)^2} \\Phi ^{(n)}\\left(\\frac{p_1^2}{p_2^2},\\frac{p_3^2}{p_2^2}\\right)$ has been found at the level of graphs.",
"As it has been mentioned in Ref.",
"[14], a hint for such a kind of relation (REF ) has appeared from the explicit calculation of a Green function in Ref.",
"[23].",
"To prove formula (REF ) via MB transformation means to substitute in Eq.",
"(REF ) the UD function as $\\Phi ^{(n)}\\left(x,y\\right)= \\oint _C dz_2dz_3 x^{z_2} y^{z_3} {\\cal M}^{(n)}\\left(z_2,z_3\\right),$ where ${\\cal M}^{(n)}\\left(z_2,z_3\\right)$ is the MB transformation of the UD functions $\\Phi ^{(n)}\\left(x,y\\right).$ The explicit form of ${\\cal M}^{(n)}\\left(z_2,z_3\\right)$ can be found in Refs.",
"[1], [8].",
"We do not need any use of the explicit form of those MB transforms.",
"The proof of the Fourier invariance (REF ) is $\\frac{1}{(2\\pi )^4}\\int ~d^4p_1d^4p_2d^4p_3 ~ \\delta (p_1 + p_2 + p_3)e^{ip_2x_2} e^{ip_1x_1} e^{ip_3x_3} \\frac{1}{(p_2^2)^2} \\Phi ^{(n)}\\left(\\frac{p_1^2}{p_2^2},\\frac{p_3^2}{p_2^2}\\right)\\\\= \\frac{1}{(2\\pi )^8}\\int ~d^4p_1d^4p_2d^4p_3d^4x_5 e^{ip_2(x_2-x_5)} e^{ip_1(x_1-x_5)} e^{ip_3(x_3-x_5)}\\frac{1}{(p_2^2)^2} \\Phi ^{(n)}\\left(\\frac{p_1^2}{p_2^2},\\frac{p_3^2}{p_2^2}\\right)\\\\= \\frac{1}{(2\\pi )^8}\\int d^4p_1d^4p_2d^4p_3d^4x_5\\oint _C dz_2dz_3 ~\\frac{e^{ip_2(x_2-x_5)} e^{ip_1(x_1-x_5)} e^{ip_3(x_3-x_5)}}{(p_2^2)^{2+z_2+z_3} (p_1^2)^{-z_2} (p_3^2)^{-z_3}}{\\cal M}^{(n)}\\left(z_2,z_3\\right)\\\\= \\frac{(4\\pi )^6}{(2\\pi )^8}\\int d^4x_5 \\oint _C dz_2dz_3 ~\\frac{\\Gamma (-z_2-z_3)}{\\Gamma (2+z_2+z_3)} \\frac{\\Gamma (2+z_2)}{\\Gamma (-z_2)} \\frac{\\Gamma (2+z_3)}{\\Gamma (-z_3)}\\\\\\times \\frac{2^{2z_2+2z_3-2(2+z_2+z_3)}{\\cal M}^{(n)}\\left(z_2,z_3\\right)}{[25]^{-z_2-z_3}[15]^{2+z_2} [35]^{2+z_3} } = \\\\= \\oint _C~dz_2dz_3 ~ \\frac{{\\cal M}^{(n)}\\left(z_2,z_3\\right)}{[12]^{-z_3}[23]^{-z_2}[31]^{2+z_2+z_3} }= \\frac{1}{[31]^2} \\Phi ^{(n)}\\left(\\frac{[12]}{[31]},\\frac{[23]}{[31]}\\right)$ As it has been remarked in Ref.",
"[12] the same property is valid for any other three-point scalar Green function in the massless field theory because the explicit form of the MB transforms ${\\cal M}^{(n)}\\left(z_2,z_3\\right)$ does not play any role in this proof."
],
[
"Orthogonality of MB transforms of triangles",
"In this Section we establish an orthogonality condition for the MB transforms $D^{(u,v)}[\\nu _1,\\nu _2,\\nu _3]$ of the one-loop massless scalar diagram.",
"All the content of the previous sections will be used in the calculations of this section, however, the key step is to repeat the way used in the previous section in order to show the invariance of three-point functions with respect to Fourier transformation.",
"We will find a weight with which this orthogonality appears inside complex integrals over contours in two complex planes.",
"Figure: One-loop massless scalar triangle in position spaceOne-loop diagram in the position space is depicted in Figure REF .",
"It does not contain any integration.",
"Integration in the position space should be done over internal points only (See Ref.",
"[1]), however this diagram does not have internal points at all.",
"The point $x_1,$ $x_2,$ $x_3$ are external points.",
"In the momentum space representation given in Figure REF the momenta $p_1,$ $p_2$ and $p_3$ enter.",
"According to our notation, we write for the diagram in Figure REF $\\frac{1}{[12]^{\\alpha _3}[23]^{\\alpha _1}[31]^{\\alpha _2}} = \\frac{ \\pi ^{-3d/2} 4^{-\\Sigma _i \\alpha _i} \\Gamma (d/2-\\alpha _1)\\Gamma (d/2-\\alpha _2) \\Gamma (d/2-\\alpha _3)}{\\Gamma (\\alpha _1)\\Gamma (\\alpha _2)\\Gamma (\\alpha _3)} \\times \\\\\\int ¬dq_1dq_2dq_3 \\frac{e^{iq_3(x_1-x_2)} e^{iq_1(x_2-x_3)} e^{iq_2(x_3-x_1)}}{(q_3^2)^{d/2-\\alpha _3}(q_1^2)^{d/2-\\alpha _1} (q_2^2)^{d/2-\\alpha _2}} = \\\\\\frac{\\pi ^{-3d/2} 4^{-\\Sigma _i \\alpha _i}\\Gamma (d/2-\\alpha _1)\\Gamma (d/2-\\alpha _2) \\Gamma (d/2-\\alpha _3)}{\\Gamma (\\alpha _1)\\Gamma (\\alpha _2)\\Gamma (\\alpha _3)}\\int ¬dq_1dq_2dq_3 \\frac{e^{ix_1(q_3-q_2)} e^{ix_2(q_1-q_3)} e^{ix_3(q_2-q_1)}}{(q_3^2)^{d/2-\\alpha _3}(q_1^2)^{d/2-\\alpha _1} (q_2^2)^{d/2-\\alpha _2}} \\\\= \\frac{\\pi ^{-3d/2} 4^{-\\Sigma _i \\alpha _i}\\Gamma (d/2-\\alpha _1)\\Gamma (d/2-\\alpha _2) \\Gamma (d/2-\\alpha _3)}{\\Gamma (\\alpha _1)\\Gamma (\\alpha _2)\\Gamma (\\alpha _3)} \\times \\\\\\int ¬dp_1dp_2dp_3dq_3\\delta (p_1+p_2+p_3) \\frac{e^{ix_1p_1} e^{ix_2p_2} e^{ix_3p_3}}{[q_3^2]^{d/2-\\alpha _3}[(p_2+q_3)^2]^{d/2-\\alpha _1} [(p_1-q_3)^2]^{d/2-\\alpha _2}} \\\\= \\frac{\\pi ^{-3d/2} 4^{-\\Sigma _i \\alpha _i}\\Gamma (d/2-\\alpha _1)\\Gamma (d/2-\\alpha _2) \\Gamma (d/2-\\alpha _3)}{(2\\pi )^d\\Gamma (\\alpha _1)\\Gamma (\\alpha _2)\\Gamma (\\alpha _3)} \\times \\\\\\int ¬dp_1dp_2dp_3dq_3dx_5 \\frac{e^{i(x_1-x_5)p_1} e^{i(x_2-x_5)p_2} e^{i(x_3-x_5)p_3}}{[q_3^2]^{d/2-\\alpha _3}[(p_2+q_3)^2]^{d/2-\\alpha _1} [(p_1-q_3)^2]^{d/2-\\alpha _2}} \\\\$ $= \\frac{\\pi ^{-d} 4^{-\\Sigma _i \\alpha _i}\\Gamma (d/2-\\alpha _1)\\Gamma (d/2-\\alpha _2) \\Gamma (d/2-\\alpha _3)}{(2\\pi )^d\\Gamma (\\alpha _1)\\Gamma (\\alpha _2)\\Gamma (\\alpha _3)} \\times \\\\\\int ¬dp_1dp_2dp_3dx_5 e^{i(x_1-x_5)p_1} e^{i(x_2-x_5)p_2} e^{i(x_3-x_5)p_3} J(d/2-\\alpha _1,d/2-\\alpha _2,d/2-\\alpha _3) \\\\= \\frac{\\pi ^{-d} 4^{-\\Sigma _i \\alpha _i}\\Gamma (d/2-\\alpha _1)\\Gamma (d/2-\\alpha _2) \\Gamma (d/2-\\alpha _3)}{(2\\pi )^d\\Gamma (\\alpha _1)\\Gamma (\\alpha _2)\\Gamma (\\alpha _3)} \\times \\\\\\int ¬dp_1dp_2dp_3dx_5 \\oint _C~dz_2dz_3 \\frac{e^{i(x_1-x_5)p_1} e^{i(x_2-x_5)p_2} e^{i(x_3-x_5)p_3}}{(p_3^2)^{d-\\Sigma _i \\alpha _i +z_2+z_3}(p_1^2)^{-z_2} (p_2^2)^{-z_3} }D^{(z_2,z_3)}[d/2-\\alpha _1,d/2-\\alpha _2,d/2-\\alpha _3] \\\\= \\frac{\\pi ^{d/2} 4^{d/2}\\Gamma (d/2-\\alpha _1)\\Gamma (d/2-\\alpha _2) \\Gamma (d/2-\\alpha _3)}{(2\\pi )^d\\Gamma (\\alpha _1)\\Gamma (\\alpha _2)\\Gamma (\\alpha _3)} \\times \\\\\\int ¬dx_5 \\oint _C~dz_2dz_3 \\frac{D^{(z_2,z_3)}[d/2-\\alpha _1,d/2-\\alpha _2,d/2-\\alpha _3]}{[35]^{\\Sigma _i \\alpha _i -z_2-z_3-d/2}[15]^{d/2+z_2} [25]^{d/2+z_3} } \\times \\\\\\frac{\\Gamma (\\Sigma _i \\alpha _i -z_2-z_3-d/2 )\\Gamma (d/2 + z_2) \\Gamma (d/2+ z_3)}{\\Gamma (d-\\Sigma _i \\alpha _i +z_2 + z_3 )\\Gamma (-z_2 )\\Gamma (-z_3)} = \\\\= \\frac{\\Gamma (d/2-\\alpha _1)\\Gamma (d/2-\\alpha _2) \\Gamma (d/2-\\alpha _3)}{\\Gamma (\\alpha _1)\\Gamma (\\alpha _2)\\Gamma (\\alpha _3)} \\times \\\\\\frac{1}{[12]^{\\Sigma _i \\alpha _i}}\\oint _C~dz_2dz_3dudv \\left(\\frac{[23]}{[12]}\\right)^u \\left(\\frac{[31]}{[12]}\\right)^v D^{(z_2,z_3)}[d/2-\\alpha _1,d/2-\\alpha _2,d/2-\\alpha _3] \\times \\\\D^{(u,v)} [d/2+z_2, d/2+z_3, \\Sigma _i \\alpha _i -z_2-z_3-d/2] \\frac{\\Gamma (\\Sigma _i \\alpha _i -z_2-z_3-d/2 )\\Gamma (d/2 + z_2) \\Gamma (d/2+ z_3)}{\\Gamma (d-\\Sigma _i \\alpha _i +z_2 + z_3 )\\Gamma (-z_2 )\\Gamma (-z_3)}.$ The r.h.s.",
"of this equation may be represented in the form $\\frac{1}{[12]^{\\alpha _3}[23]^{\\alpha _1}[31]^{\\alpha _2}} =\\frac{1}{[12]^{\\Sigma _i \\alpha _i}}\\oint _C~dudv \\left(\\frac{[23]}{[12]}\\right)^u \\left(\\frac{[31]}{[12]}\\right)^v f(u,v),$ from which we may conclude that function $f(u,v)$ has residues only at points $u=-\\alpha _1$ and $v=-\\alpha _2$ in the complex planes $u$ and $v.$ The function $f(u,v)$ may be written as double integral $f(u,v) = \\frac{\\Gamma (d/2-\\alpha _1)\\Gamma (d/2-\\alpha _2) \\Gamma (d/2-\\alpha _3)}{\\Gamma (\\alpha _1)\\Gamma (\\alpha _2)\\Gamma (\\alpha _3)} \\times \\\\\\oint _C~dz_2dz_3 D^{(z_2,z_3)}[d/2-\\alpha _1,d/2-\\alpha _2,d/2-\\alpha _3] D^{(u,v)} [d/2+z_2, d/2+z_3, \\Sigma _i \\alpha _i -z_2-z_3-d/2] \\\\\\frac{\\Gamma (\\Sigma _i \\alpha _i -z_2-z_3-d/2 )\\Gamma (d/2 + z_2) \\Gamma (d/2+ z_3)}{\\Gamma (d-\\Sigma _i \\alpha _i +z_2 + z_3 )\\Gamma (-z_2 )\\Gamma (-z_3)}.$ Thus, we came to a kind of orthogonality condition for the MB transforms, $ \\oint _C~dudv x^u y^v f(u,v) = x^{-\\alpha _1}y^{-\\alpha _2}.$"
],
[
"Proof of orthogonality via Barnes lemma",
"The orthogonality condition has been found in the previous section in an implicit way.",
"In this section we will prove it explicitly via Barnes lemmas [10], [11] Our results obtained in the previous works [1], [9] show that any relation for Mellin-Barnes transforms that follow from Feynman diagrams may be proved via Barnes lemmas.",
"From formula (REF ) we may write $D^{(z_2,z_3)}[d/2-\\alpha _1,d/2-\\alpha _2,d/2-\\alpha _3] = \\Gamma \\left(d- \\Sigma \\alpha _i + z_3 + z_2 \\right)\\times \\\\\\times \\frac{ \\Gamma \\left(-z_2 \\right)\\Gamma \\left(-z_3 \\right)\\Gamma \\left(-z_2 +\\alpha _2+\\alpha _3 - d/2 \\right)\\Gamma \\left(-z_3+\\alpha _1+\\alpha _3 - d/2 \\right)\\Gamma \\left(z_2 + z_3 + d/2-\\alpha _3 \\right)}{\\Gamma (d/2-\\alpha _1) \\Gamma (d/2-\\alpha _2) \\Gamma (d/2-\\alpha _3) \\Gamma \\left(\\Sigma \\alpha _i - d/2 \\right)}$ and $D^{(u,v)}[d/2+z_2, d/2+z_3, \\Sigma _i \\alpha _i -z_2-z_3-d/2] = \\Gamma \\left(\\Sigma \\alpha _i + u + v \\right)\\times \\\\\\times \\frac{ \\Gamma \\left(-u \\right)\\Gamma \\left(-v \\right)\\Gamma \\left(-u + z_2 - \\Sigma \\alpha _i + d/2 \\right)\\Gamma \\left(-v + z_3 - \\Sigma \\alpha _i + d/2 \\right)\\Gamma \\left(u + v - z_2 - z_3 + \\Sigma \\alpha _i - d/2 \\right)}{\\Gamma (d/2+z_2) \\Gamma (d/2+z_3) \\Gamma ( \\Sigma _i \\alpha _i -z_2-z_3-d/2 ) \\Gamma \\left(d/2-\\Sigma \\alpha _i \\right)}.$ We may write explicitly, $f(u,v) = \\frac{\\Gamma (d/2-\\alpha _1)\\Gamma (d/2-\\alpha _2) \\Gamma (d/2-\\alpha _3)}{\\Gamma (\\alpha _1)\\Gamma (\\alpha _2)\\Gamma (\\alpha _3)} \\times \\\\\\oint _C~dz_2dz_3 D^{(z_2,z_3)}[d/2-\\alpha _1,d/2-\\alpha _2,d/2-\\alpha _3] D^{(u,v)} [d/2+z_2, d/2+z_3, \\Sigma _i \\alpha _i -z_2-z_3-d/2] \\\\\\frac{\\Gamma (\\Sigma _i \\alpha _i -z_2-z_3-d/2 )\\Gamma (d/2 + z_2) \\Gamma (d/2+ z_3)}{\\Gamma (d-\\Sigma _i \\alpha _i +z_2 + z_3 )\\Gamma (-z_2 )\\Gamma (-z_3)} = \\\\\\frac{\\Gamma (d/2-\\alpha _1)\\Gamma (d/2-\\alpha _2) \\Gamma (d/2-\\alpha _3)}{\\Gamma (\\alpha _1)\\Gamma (\\alpha _2)\\Gamma (\\alpha _3)} \\times \\\\\\oint _C~dz_2dz_3 \\Gamma \\left(d- \\Sigma \\alpha _i + z_3 + z_2 \\right)\\times \\\\\\times \\frac{ \\Gamma \\left(-z_2 \\right)\\Gamma \\left(-z_3 \\right)\\Gamma \\left(-z_2 +\\alpha _2+\\alpha _3 - d/2 \\right)\\Gamma \\left(-z_3+\\alpha _1+\\alpha _3 - d/2 \\right)\\Gamma \\left(z_2 + z_3 + d/2-\\alpha _3 \\right)}{\\Gamma (d/2-\\alpha _1) \\Gamma (d/2-\\alpha _2) \\Gamma (d/2-\\alpha _3) \\Gamma \\left(\\Sigma \\alpha _i - d/2 \\right)} \\\\\\times \\Gamma \\left(\\Sigma \\alpha _i + u + v \\right)\\times \\\\\\times \\frac{ \\Gamma \\left(-u \\right)\\Gamma \\left(-v \\right)\\Gamma \\left(-u + z_2 - \\Sigma \\alpha _i + d/2 \\right)\\Gamma \\left(-v + z_3 - \\Sigma \\alpha _i + d/2 \\right)\\Gamma \\left(u + v - z_2 - z_3 + \\Sigma \\alpha _i - d/2 \\right)}{\\Gamma (d/2+z_2) \\Gamma (d/2+z_3) \\Gamma ( \\Sigma _i \\alpha _i -z_2-z_3-d/2 ) \\Gamma \\left(d/2-\\Sigma \\alpha _i \\right)} \\times \\\\\\frac{\\Gamma (\\Sigma _i \\alpha _i -z_2-z_3-d/2 )\\Gamma (d/2 + z_2) \\Gamma (d/2+ z_3)}{\\Gamma (d-\\Sigma _i \\alpha _i +z_2 + z_3 )\\Gamma (-z_2 )\\Gamma (-z_3)} = \\\\\\frac{\\Gamma \\left(\\Sigma \\alpha _i + u + v \\right)\\Gamma \\left(-u \\right)\\Gamma \\left(-v \\right)}{\\Gamma (\\alpha _1)\\Gamma (\\alpha _2)\\Gamma (\\alpha _3) \\Gamma \\left(d/2-\\Sigma \\alpha _i \\right)\\Gamma \\left(\\Sigma \\alpha _i - d/2 \\right)} \\times \\\\\\oint _C~dz_2dz_3 \\Gamma \\left(-z_2 +\\alpha _2+\\alpha _3 - d/2 \\right)\\Gamma \\left(-z_3+\\alpha _1+\\alpha _3 - d/2 \\right)\\Gamma \\left(z_2 + z_3 + d/2-\\alpha _3 \\right)\\\\\\times \\Gamma \\left(-u + z_2 - \\Sigma \\alpha _i + d/2 \\right)\\Gamma \\left(-v + z_3 - \\Sigma \\alpha _i + d/2 \\right)\\Gamma \\left(u + v - z_2 - z_3 + \\Sigma \\alpha _i - d/2 \\right).$ To calculate this integral we need the first Barnes lemma of Ref.",
"[10], [24].",
"$ \\oint _C~dz ~\\Gamma \\left(\\lambda _1 + z \\right)\\Gamma \\left(\\lambda _2 + z \\right)\\Gamma \\left(\\lambda _3 - z \\right)\\Gamma \\left(\\lambda _4 - z \\right)=\\frac{\\Gamma \\left(\\lambda _1 + \\lambda _3 \\right)\\Gamma \\left(\\lambda _1 + \\lambda _4 \\right)\\Gamma \\left(\\lambda _2 + \\lambda _3 \\right)\\Gamma \\left(\\lambda _2 + \\lambda _4 \\right)}{\\Gamma \\left(\\lambda _1 + \\lambda _2 + \\lambda _3 + \\lambda _4 \\right)},$ in which $\\lambda _1, \\lambda _2, \\lambda _3, \\lambda _4$ are complex numbers.",
"They are chosen in a such a way that on the r.h.s.",
"of Eq.",
"(REF ) there are no singularities.",
"This lemma will be applied in the next two integrals.",
"We first integrate over the variable $z_2,$ $\\oint _C~dz_2 \\Gamma \\left(-z_2 +\\alpha _2+\\alpha _3 - d/2 \\right)\\Gamma \\left(z_2 + z_3 + d/2-\\alpha _3 \\right)\\Gamma \\left(-u + z_2 - \\Sigma \\alpha _i + d/2 \\right)\\times \\\\\\Gamma \\left(u + v - z_2 - z_3 + \\Sigma \\alpha _i - d/2 \\right)= \\\\\\frac{\\Gamma \\left(z_3 + \\alpha _2 \\right)\\Gamma \\left(-u - \\alpha _1 \\right)\\Gamma \\left(u + v + \\alpha _1 + \\alpha _2 \\right)\\Gamma \\left(v - z_3 \\right)}{\\Gamma \\left(v + \\alpha _2 \\right)}.$ Then, we have $f(u,v) = \\frac{\\Gamma \\left(\\Sigma \\alpha _i + u + v \\right)\\Gamma \\left(-u \\right)\\Gamma \\left(-v \\right)\\Gamma \\left(-u - \\alpha _1 \\right)\\Gamma \\left(u + v + \\alpha _1 + \\alpha _2 \\right)}{\\Gamma (\\alpha _1)\\Gamma (\\alpha _2)\\Gamma (\\alpha _3) \\Gamma \\left(d/2-\\Sigma \\alpha _i \\right)\\Gamma \\left(\\Sigma \\alpha _i - d/2 \\right)\\Gamma \\left(v + \\alpha _2 \\right)} \\times \\\\\\oint _C~dz_3 \\Gamma \\left(-z_3+\\alpha _1+\\alpha _3 - d/2 \\right)\\Gamma \\left(-v + z_3 - \\Sigma \\alpha _i + d/2 \\right)\\Gamma \\left(z_3 + \\alpha _2 \\right)\\Gamma \\left(v - z_3 \\right)= \\\\\\frac{\\Gamma \\left(\\Sigma \\alpha _i + u + v \\right)\\Gamma \\left(-u \\right)\\Gamma \\left(-v \\right)\\Gamma \\left(-u - \\alpha _1 \\right)\\Gamma \\left(u + v + \\alpha _1 + \\alpha _2 \\right)}{\\Gamma (\\alpha _1)\\Gamma (\\alpha _2)\\Gamma (\\alpha _3) \\Gamma \\left(d/2-\\Sigma \\alpha _i \\right)\\Gamma \\left(\\Sigma \\alpha _i - d/2 \\right)\\Gamma \\left(v + \\alpha _2 \\right)} \\times \\\\\\frac{\\Gamma \\left(-v - \\alpha _2 \\right)\\Gamma \\left(\\Sigma \\alpha _i - d/2 \\right)\\Gamma \\left(d/2 - \\Sigma \\alpha _i \\right)\\Gamma \\left(v + \\alpha _2 \\right)}{\\Gamma \\left(0 \\right)} = \\\\\\frac{\\Gamma \\left(\\Sigma \\alpha _i + u + v \\right)\\Gamma \\left(-u \\right)\\Gamma \\left(-v \\right)\\Gamma \\left(-u - \\alpha _1 \\right)\\Gamma \\left(-v - \\alpha _2 \\right)\\Gamma \\left(u + v + \\alpha _1 + \\alpha _2 \\right)}{\\Gamma (\\alpha _1)\\Gamma (\\alpha _2)\\Gamma (\\alpha _3)\\Gamma \\left(0 \\right)}.$ This result for $f(u,v)$ allows us to prove the orthogonality condition (REF ), $\\oint _C~dudv x^u y^v f(u,v) = \\\\\\oint _C~dudv x^u y^v \\frac{\\Gamma \\left(\\Sigma \\alpha _i + u + v \\right)\\Gamma \\left(-u \\right)\\Gamma \\left(-v \\right)\\Gamma \\left(-u - \\alpha _1 \\right)\\Gamma \\left(-v - \\alpha _2 \\right)\\Gamma \\left(u + v + \\alpha _1 + \\alpha _2 \\right)}{\\Gamma (\\alpha _1)\\Gamma (\\alpha _2)\\Gamma (\\alpha _3)\\Gamma \\left(0 \\right)}.$ The situation is completely the same as in Section 5 where we have proved the star-triangle relation via the Mellin-Barnes transformation.",
"In the denominator we have $\\Gamma (0)$ and this means we need to have the same factor in the numerator.",
"Otherwise, the result would be zero.",
"This may happen only for the residues $u = -\\alpha _1$ and $v = -\\alpha _2,$ because only for these residues the factor $\\Gamma \\left(u + v + \\alpha _1 + \\alpha _2 \\right)$ is equal to $\\Gamma \\left(0\\right)$ and cancels $\\Gamma \\left(0\\right)$ in the denominator.",
"Any other residues produce by the Gamma functions with negative sign of arguments will have a vanishing contribution due to $\\Gamma \\left(0\\right)$ in the denominator.",
"Thus, $\\oint _C~dudv x^u y^v \\frac{\\Gamma \\left(\\Sigma \\alpha _i + u + v \\right)\\Gamma \\left(-u \\right)\\Gamma \\left(-v \\right)\\Gamma \\left(-u - \\alpha _1 \\right)\\Gamma \\left(-v - \\alpha _2 \\right)\\Gamma \\left(u + v + \\alpha _1 + \\alpha _2 \\right)}{\\Gamma (\\alpha _1)\\Gamma (\\alpha _2)\\Gamma (\\alpha _3)\\Gamma \\left(0 \\right)} \\\\= x^{-\\alpha _1}y^{-\\alpha _2}.$"
],
[
"Conclusion",
"In this section we have found the explicit form for the integral of two MB transforms over contours in two complex planes, $ \\oint _C~dz_2dz_3 ~ \\Delta ^{(\\alpha _1,\\alpha _2,\\alpha _3)}(z_2,z_3) D^{(u,v)} [d/2+z_2, d/2+z_3, \\Sigma _i \\alpha _i -z_2-z_3-d/2]\\times \\\\D^{(z_2,z_3)}[d/2-\\alpha _1,d/2-\\alpha _2,d/2-\\alpha _3] = \\nonumber $ $\\frac{\\Gamma \\left(\\Sigma \\alpha _i + u + v \\right)\\Gamma \\left(-u \\right)\\Gamma \\left(-v \\right)\\Gamma \\left(-u - \\alpha _1 \\right)\\Gamma \\left(-v - \\alpha _2 \\right)\\Gamma \\left(u + v + \\alpha _1 + \\alpha _2 \\right)}{\\Gamma (d/2-\\alpha _1)\\Gamma (d/2-\\alpha _2) \\Gamma (d/2-\\alpha _3) \\Gamma \\left(0 \\right)},$ where the weight $\\Delta ^{(\\alpha _1,\\alpha _2,\\alpha _3)}(z_2,z_3)$ is defined as $\\Delta ^{(\\alpha _1,\\alpha _2,\\alpha _3)}(z_2,z_3) = \\frac{\\Gamma (\\Sigma _i \\alpha _i -z_2-z_3-d/2 )\\Gamma (d/2 + z_2) \\Gamma (d/2+ z_3)}{\\Gamma (d-\\Sigma _i \\alpha _i +z_2 + z_3 )\\Gamma (-z_2 )\\Gamma (-z_3)}.$ The right hand side should be understood in a generalized sense because the presence of $\\Gamma (0)$ in the denominator suggests the integration in two complex planes $u$ and $v.$ This formula is valid for any space-time dimensions $d.$ On the other side we have Eq.",
"(REF ) in which the MB transforms of the one-loop massless scalar integral are involved too, $\\oint _C~dz_2dz_3~D^{(u,v)}[1+\\varepsilon _1-z_3,1+\\varepsilon _2-z_2,1+\\varepsilon _3]D^{(z_2,z_3)}[1+\\varepsilon _2,1+\\varepsilon _1,1+\\varepsilon _3] = \\nonumber \\\\J\\left[ \\frac{D^{(u,v-\\varepsilon _2)}[1-\\varepsilon _1]}{\\varepsilon _2\\varepsilon _3}+ \\frac{D^{(u,v)}[1+\\varepsilon _3]}{\\varepsilon _1\\varepsilon _2} + \\frac{ D^{(u-\\varepsilon _1,v)}[1-\\varepsilon _2]}{\\varepsilon _1\\varepsilon _3} \\right].$ This equation has been proved in Ref.",
"[9] by using the first and the second Barnes lemmas.",
"This equation is valid for $d=4$ space-time dimensions, however, in an arbitrary space-time dimension the analog should exist due to the consideration done in Ref.",
"[6].",
"As we have mention in Introduction, the integral relation (REF ) has a structure similar to decomposition of tensor product in terms of irreducible components, and the integral relation (REF ) has a structure similar to orthogonality condition.",
"This observation suggests that behind MB transforms $D^{(u,v)}[\\nu _1,\\nu _2,\\nu _3]$ an integrable structure may exist [1], [8], [9].",
"It is remarkable that both the relations (REF ) and (REF ) are written for the Green functions, that is, the integrable structure should exist for the Green functions, too.",
"Usually, integrable structures are studied for amplitudes [13]."
],
[
"Acknowledgments",
"The work of I.K.",
"was supported in part by Fondecyt (Chile) Grants Nos.",
"1040368, 1050512 and 1121030, by DIUBB (Chile) Grant Nos.",
"125009, GI 153209/C and GI 152606/VC.",
"Also, the work of I.K.",
"is supported by Universidad del Bío-Bío and Ministerio de Educacion (Chile) within Project No.",
"MECESUP UBB0704-PD018.",
"He is grateful to the Physics Faculty of Bielefeld University for accepting him as a visiting scientist and for the kind hospitality and the excellent working conditions during his stay in Bielefeld.",
"E.A.N.C.",
"work was partially supported by project DIULS PR15151, Universidad de La Serena.",
"The work of I.P.F.",
"was supported in part by Fondecyt (Chile) Grant No.",
"1121030 and by Beca Conicyt (Chile) via Doctoral fellowship CONICYT-DAAD/BECAS Chile, 2016/91609937.",
"This paper is based on the talk of I. K. at ICAMI 2017, San Andres, Colombia, November 27 - December 1, 2017, and he is grateful to ICAMI organizers for inviting him.",
"The financial support of I.K.",
"participation in ICAMI 2017 has been provided by DIUBB via Fapei funding."
]
] | 1808.08337 |
[
[
"Field-theoretic approach to the universality of branching processes"
],
[
"Abstract Branching processes are widely used to model phenomena from networks to neuronal avalanching.",
"In a large class of continuous-time branching processes, we study the temporal scaling of the moments of the instant population size, the survival probability, expected avalanche duration, the so-called avalanche shape, the $n$-point correlation function and the probability density function of the total avalanche size.",
"Previous studies have shown universality in certain observables of branching processes using probabilistic arguments, however, a comprehensive description is lacking.",
"We derive the field theory that describes the process and demonstrate how to use it to calculate the relevant observables and their scaling to leading order in time, revealing the universality of the moments of the population size.",
"Our results explain why the first and second moment of the offspring distribution are sufficient to fully characterise the process in the vicinity of criticality, regardless of the underlying offspring distribution.",
"This finding implies that branching processes are universal.",
"We illustrate our analytical results with computer simulations."
],
[
"Introduction ",
"Branching processes [1] are widely used for modelling phenomena in many different subject areas, such as avalanches [2], [3], [4], networks [5], [4], [3], [6], earthquakes [7], [8], family names [9], populations of bacteria and cells [10], [11], nuclear reactions [12], [13], cultural evolution [14] and neuronal avalanches [15], [16].",
"Because of their mathematical simplicity they play an important role in statistical mechanics [17] and the theory of complex systems [8].",
"Branching processes are a paradigmatic example of a system displaying a second-order phase transition between extinction (absorbing state) with probability one and non-zero probability of survival (non-absorbing state) in the infinite time limit.",
"The critical point in the parameter region at which this transition occurs is where branching and extinction rates exactly balance, namely when the expected number of offspring per particle is exactly unity [1], [8].",
"In the present work we study the continuous-time version of the Galton-Watson branching process [1], which is a generalisation of the birth-death process [18], [19].",
"In the continuous-time branching process, particles go extinct or replicate into a number of identical offspring at random and with constant Poissonian rates.",
"Each of the new particles follows the same process.",
"The difference between the original Galton-Watson branching process and the continuous-time branching process we consider here, lies in the waiting times between events.",
"In the original Galton-Watson branching process, updates occur in discrete time steps, while in the continuous-time process we consider, waiting times follow a Poisson process [18], [19].",
"However, both processes share many asymptotics [1], [17], and therefore we regard the continuous-time branching process as the continuum limit of the Galton-Watson branching process.",
"By using field-theoretic methods, we provide a general framework to determine universal, finite-time scaling properties of a wide range of branching processes close to the critical point.",
"The main advantages of this versatile approach are, on the one hand, the ease with which observables are calculated and, on the other hand, the use of diagrammatic language, which allows us to manipulate the sometimes cumbersome expressions in a neat and compact way.",
"Other methods in the literature developed to study problems related to branching processes, in particular relating branching processes to different forms of motion, include the formalism based on the Pal-Bell equation [20], [21], [22].",
"Moreover, our framework allows us to determine systematically observables that are otherwise complicated to manipulate if possible at all.",
"To illustrate this point, we have calculated in closed form a number of observables that describe different aspects of the process in the vicinity of the critical point: we have calculated the moments of the population size as a function of time, the probability distribution of the population size as a function of time, the avalanche shape, the two-time and $n$ -time correlation functions, and the total avalanche size and its moments.",
"Our results show that branching processes are universal in the vicinity of the critical point [23], [24] in the sense that exactly three quantities (the Poissonian rate and the first and second moments of the offspring distribution) are sufficient to describe the asymptotics of the process regardless of the underlying offspring distribution.",
"The contents of this paper are organised as follows.",
"In Sec.",
"we derive the field theory of the continuous-time branching process.",
"In Sec.",
"we use our formalism to calculate a number of observables in closed form, and in Sec.",
"we discuss our results and our conclusions.",
"Further details of the calculations can be found in the appendices."
],
[
"Field Theory of the continuous-time branching process",
"The continuous-time branching process is defined as follows.",
"We consider a population of $N(t)$ identical particles at time $t\\ge 0$ with initial condition $N(0)=1$ .",
"Each particle is allowed to branch independently into $\\kappa $ offspring with Poissonian rate $s>0$ , where $\\kappa \\in \\lbrace 0\\rbrace \\cup \\mathbb {N}$ is a random variable with probability distribution $P(\\kappa =k)=p_k\\in [0,1]$ [18], Fig.",
"REF .",
"In the language of chemical reactions, this can be written as the reaction $A\\rightarrow \\kappa A$ .",
"To derive the field theory of this process following the methods by Doi and Peliti [25], [26], [27], [17], we first write the master equation of the probability $P(N,t)$ to find $N$ particles at time $t$ , $\\frac{\\mathrm {d}P(N,t)}{\\mathrm {d}t} = s\\sum _k p_k (N-k+1) P(N-k+1,t)- sNP(N,t) ,$ with initial condition $P(N,0)=\\delta _{N,1}$ .",
"Following work by Doi [25], we cast the master equation in a second quantised form.",
"A system with $N$ particles is represented by a Fock-space vector $\\left|N\\right\\rangle $ .",
"We use the ladder operators $a^\\dagger {}$ (creation) and $a$ (annihilation), which act on $\\left|N\\right\\rangle $ such that $a\\left|N\\right\\rangle =N\\left|N-1\\right\\rangle $ and $a^\\dagger {}\\left|N\\right\\rangle =\\left|N+1\\right\\rangle $ , and satisfy the commutation relation $[a,a^\\dagger {}]=aa^\\dagger {}-a^\\dagger {}a=1$ .",
"The probabilistic state of the system is given by $\\left|\\Psi (t)\\right\\rangle = \\sum _NP(N,t) \\left|N\\right\\rangle ,$ and its time evolution is determined by Eq.",
"(REF ), $\\frac{\\mathrm {d}\\left|\\Psi (t)\\right\\rangle }{\\mathrm {d}t} = s\\left(f\\!\\left(a^\\dagger {}\\right) - a^\\dagger {}\\right) a \\left|\\Psi (t)\\right\\rangle ,$ using the probability generating function of $\\kappa $ , $f(z) =\\sum _{k=0}^\\infty p_k z^k = \\left\\langle z^\\kappa \\right\\rangle ,$ where $\\left\\langle \\bullet \\right\\rangle $ denotes expectation.",
"We define the mass $r$ as the difference between the extinction and the net branching rates, $r = s p_0 - s \\sum _{k\\ge 2}(k-1)p_k = s\\left(1-{\\langle \\kappa \\rangle }\\right),$ and the rates $q_j$ as $q_j = s\\sum _{k} \\binom{k}{j} p_k = s\\left\\langle \\binom{\\kappa }{j} \\right\\rangle =\\frac{s}{j!",
"}f^{(j)}(1),$ where $f^{(j)}(1)$ denotes the $j$ th derivative of the probability generating function Eq.",
"(REF ) evaluated at $z=1$ .",
"We assume that the rates $q_j$ are finite.",
"In this notation, the time evolution in Eq.",
"(REF ) can be written as $\\tilde{\\mathcal {A}}\\left|\\Psi (t)\\right\\rangle = \\frac{\\mathrm {d}}{\\mathrm {d}{t}}\\left|\\Psi (t)\\right\\rangle \\text{ and thus }\\left|\\Psi (t)\\right\\rangle = \\mathchoice{e^{\\tilde{\\mathcal {A}} t}}{\\operatorname{exp}\\!\\left(\\tilde{\\mathcal {A}} t\\right)}{\\operatorname{exp}\\!\\left(\\tilde{\\mathcal {A}} t\\right)}{\\operatorname{exp}\\!\\left(\\tilde{\\mathcal {A}} t\\right)} \\left|\\Psi (0)\\right\\rangle ,$ where $\\tilde{\\mathcal {A}}$ is the operator $\\tilde{\\mathcal {A}} =\\sum _{j\\ge 2} q_j \\widetilde{a}^j a-r\\widetilde{a}a ,$ and $\\widetilde{a}$ denotes the Doi-shifted creation operator, $a^\\dagger {} = 1+\\widetilde{a}$ .",
"Figure: Typical avalanche profiles N(t)N(t) of a binary branching process (blue)and a branching process with geometric distributionof the number of offspring (orange), both at criticality r=0r=0,and with Poissonian rate s=1s=1.The sign of the mass $r$ , Eq.",
"(REF ), determines in which regime a particular branching process is in; if $r=0$ the process is at the critical point, if $r>0$ the process is in the subcritical regime and if $r<0$ the process is in the supercritical regime.",
"Subcritical and critical processes are bound to go extinct in finite time, whereas supercritical process have a positive probability of survival [1].",
"Following the work by Peliti [26], Eq.",
"(REF ) can be cast in path integral form.",
"Here, the creation and annihilation operators $a^\\dagger {}$ and $a$ are transformed to time-dependent creation and annihilation fields $\\phi ^\\dagger (t)$ and $\\phi (t)$ respectively.",
"Similarly, the Doi-shifted operator $\\widetilde{a}$ is transformed to the time-dependent Doi-shifted field $\\widetilde{\\phi }(t)=\\phi ^\\dagger (t)-1$ .",
"The action functional of the resulting field theory is given by $\\mathcal {A}\\!\\left[\\widetilde{\\phi },\\phi \\right]\\!=\\!\\!\\!\\int \\limits _{-\\infty }^{\\infty }\\!\\!\\!\\mathrm {d}t\\left\\lbrace \\sum \\limits _{j\\ge 2}q_j\\widetilde{\\phi }^j(t)\\phi (t)-\\widetilde{\\phi }(t)\\left(\\frac{d}{dt}\\!+\\!r\\right)\\phi (t)\\right\\rbrace .$ Using the Fourier transform (t)=d d d d () e-texp(-t)exp(-t)exp(-t) with d d d d = d d d d 2, ()=dt dt dt dt (t) etexp(t)exp(t)exp(t) , and identically for $\\widetilde{\\phi }(t)$ , the action Eq.",
"(REF ) becomes local in $\\omega $ and the bilinear, i.e.",
"the Gaussian part $\\mathcal {A}_0\\left[\\widetilde{\\phi },\\phi \\right]=-\\int \\mathchoice{\\!\\mathrm {d}\\hspace{-3.33328pt}\\omega \\,}{\\!\\mathrm {d}\\hspace{-3.33328pt}\\omega \\,}{\\!\\mathrm {d}\\hspace{-3.33328pt}\\omega \\,}{\\!\\mathrm {d}\\hspace{-3.33328pt}\\omega \\,}\\widetilde{\\phi }(-\\omega )(-\\mathring{\\imath }\\omega +r)\\phi (\\omega )$ of the path integral can be determined in closed form.",
"The Gaussian path integral is well-defined only when the mass is positive, $r>0$ .",
"The non-linear terms, $j\\ge 2$ in Eq.",
"(REF ) are then treated as a perturbation about the Gaussian part, as commonly done in field theory [17], [28].",
"Table: Summary of observables including their equation number and what cases and limitsare exact.",
"The expressions referred to are, in the limit r→0r\\rightarrow 0, either: exact,exact for binary branching processes (i.e.",
"for otherbranching process, our result is the leading order in q 2 /rq_2/r for small rr at fixed rtrt),or provide exact asymptotes (that is, our result is the leading order for any kind of branching process).",
"The regime of validity near criticality is either critical and subcritical (r≥0r\\ge 0) or all encompassing (r∈ℝr\\in \\mathbb {R})."
],
[
"Observables ",
"We use the field theory described above to calculate a number of observables that have received attention in the literature in various settings.",
"In Table REF we list all the observables that we have calculated in closed form and the degree of approximation of our analytical result.",
"Some results are exact for any kind of branching process and other results are only exact for binary branching processes.",
"Those results that are an approximation have the exact asymptotic behaviour.",
"All observables are constructed on the basis of the probability vector $\\left|\\Psi (t)\\right\\rangle $ which evolves according to Eq.",
"(REF ).",
"If the initial state, $t=0$ , consists of a single particle, then $\\left|\\Psi (0)\\right\\rangle =a^\\dagger {}\\left|0\\right\\rangle $ and $\\left|\\Psi (t)\\right\\rangle =\\mathchoice{e^{\\tilde{\\mathcal {A}} t}}{\\operatorname{exp}\\!\\left(\\tilde{\\mathcal {A}} t\\right)}{\\operatorname{exp}\\!\\left(\\tilde{\\mathcal {A}} t\\right)}{\\operatorname{exp}\\!\\left(\\tilde{\\mathcal {A}} t\\right)} a^\\dagger {}\\left|0\\right\\rangle $ .",
"Probing the particle number requires the action of the operator $a^\\dagger {}a{}$ , whose eigenvectors are the pure states $\\left|N\\right\\rangle $ , such that $a^\\dagger {}a{} \\left|N\\right\\rangle =N\\left|N\\right\\rangle $ .",
"The components of the vector $a^\\dagger {}a{} \\mathchoice{e^{\\tilde{\\mathcal {A}} t}}{\\operatorname{exp}\\!\\left(\\tilde{\\mathcal {A}} t\\right)}{\\operatorname{exp}\\!\\left(\\tilde{\\mathcal {A}} t\\right)}{\\operatorname{exp}\\!\\left(\\tilde{\\mathcal {A}} t\\right)} a^\\dagger {}\\left|0\\right\\rangle $ are thus the probability that the process has generated $N$ particles weighted by $N$ .",
"To sum over all states, we further need the projection state $\\left\\langle \\right|=\\sum \\limits _{N=0}^\\infty \\left\\langle N\\right|=\\sum _{N=0}^\\infty \\frac{1}{N!}",
"\\left\\langle 0\\right|a^N = \\left\\langle 0\\right|\\mathchoice{e^{a}}{\\operatorname{exp}\\!\\left(a\\right)}{\\operatorname{exp}\\!\\left(a\\right)}{\\operatorname{exp}\\!\\left(a\\right)} \\ .$ The expected particle number at time $t$ may thus be written as $\\left\\langle N(t) \\right\\rangle =\\left\\langle \\right|a^\\dagger {}a{}\\, \\mathchoice{e^{\\tilde{\\mathcal {A}} t}}{\\operatorname{exp}\\!\\left(\\tilde{\\mathcal {A}} t\\right)}{\\operatorname{exp}\\!\\left(\\tilde{\\mathcal {A}} t\\right)}{\\operatorname{exp}\\!\\left(\\tilde{\\mathcal {A}} t\\right)} a^\\dagger {}\\left|0\\right\\rangle \\ .$ More complicated observables and intermediate temporal evolution can be compiled following the same pattern [28].",
"To perform any calculations, the operators need to be normal ordered and then mapped to fields as suggested above, $a^\\dagger {}\\rightarrow \\phi ^\\dagger (t)=1+\\widetilde{\\phi }(t)$ and $a{}\\rightarrow \\phi (t)$ , where the time $t$ corresponds to the total time the system has evolved for, Eq.",
"(REF ).",
"The expectation in Eq.",
"(REF ) can thus be written as $\\left\\langle N(t) \\right\\rangle = \\left\\langle \\phi ^\\dagger (t)\\phi (t)\\,\\phi ^\\dagger (0) \\right\\rangle ,$ where $\\left\\langle \\mathcal {O} \\right\\rangle $ denotes the path integral $\\left\\langle \\mathcal {O} \\right\\rangle =\\int \\mathcal {D}\\left[\\widetilde{\\phi },\\phi \\right]\\mathcal {O}\\mathchoice{e^{\\mathcal {A}\\left[\\widetilde{\\phi },\\phi \\right]}}{\\operatorname{exp}\\!\\left(\\mathcal {A}\\left[\\widetilde{\\phi },\\phi \\right]\\right)}{\\operatorname{exp}\\!\\left(\\mathcal {A}\\left[\\widetilde{\\phi },\\phi \\right]\\right)}{\\operatorname{exp}\\!\\left(\\mathcal {A}\\left[\\widetilde{\\phi },\\phi \\right]\\right)}.$ The resulting expressions are most elegantly expressed in terms of Feynman diagrams [17].",
"The bare propagator of the field $\\phi $ is read off from the bilinear part of the action which, in Fourier space, is $\\left\\langle \\phi (\\omega )\\widetilde{\\phi }(\\omega ^{\\prime }) \\right\\rangle =\\frac{\\delta \\hspace{-4.44443pt}(\\omega +\\omega ^{\\prime })}{-\\mathring{\\imath }\\omega +r}\\hat{=}[baseline=-2.5pt]{ [Aactivity] (0.5,0) -- (-0.5,0) node[at end,above] { }; },$ where $\\delta \\hspace{-4.44443pt}(\\omega +\\omega ^{\\prime })=2\\pi \\delta (\\omega +\\omega ^{\\prime })$ denotes the scaled Dirac-$\\delta $ function.",
"Diagrammatically, the bare propagator is represented by a straight directed line.",
"The directedness of the propagator reflects the causality (see Eq.",
"(REF )) of the process in the time domain as a particle has to be created before it can be annihilated but not vice versa.",
"By convention, in our Feynman diagrams time proceeds from right to left.",
"Using the fact that the mass $r$ is strictly positive for the Gaussian path integral to converge, we write the propagator in real time by Fourier transforming, $\\left\\langle \\phi (t)\\widetilde{\\phi }(t^{\\prime }) \\right\\rangle =\\int _{-\\infty }^\\infty \\mathchoice{\\!\\mathrm {d}\\hspace{-3.33328pt}\\omega \\,}{\\!\\mathrm {d}\\hspace{-3.33328pt}\\omega \\,}{\\!\\mathrm {d}\\hspace{-3.33328pt}\\omega \\,}{\\!\\mathrm {d}\\hspace{-3.33328pt}\\omega \\,}\\frac{\\delta \\hspace{-4.44443pt}(\\omega +\\omega ^{\\prime })}{-\\mathring{\\imath }\\omega +r}e^{-\\mathring{\\imath }\\omega t}e^{-\\mathring{\\imath }\\omega ^{\\prime } t^{\\prime }}\\\\=\\Theta (t-t^{\\prime })e^{-r(t-t^{\\prime })},$ where $\\Theta $ is the Heaviside step function.",
"If $r<0$ , the integral in Eq.",
"(REF ) is only convergent for $t<t^{\\prime }$ , which violates causality and therefore yields an unphysical result.",
"For this reason, we will assume $r>0$ and we will take the limit $r\\rightarrow 0$ where possible.",
"Therefore, in this paper, the analytical results obtained through this field theory hold for the critical and subcritical regimes only ($r\\ge 0$ ).",
"However, in some cases we may be able to use probabilistic arguments that allow us to extend our results to the supercritical case ($r<0$ ), see Section REF .",
"Furthermore, we will drop the cumbersome Heaviside $\\Theta $ functions, assuming suitable choices for the times, such as $t>t^{\\prime }$ above.",
"Each of the interaction terms of the form $\\widetilde{\\phi }^j\\phi $ with $j\\ge 2$ in the non-linear part of the action Eq.",
"(REF ) come with individual couplings $q_j$ , Eq.",
"(REF ).",
"These are to be expanded perturbatively in.",
"Following the canonical field theoretic procedure [28], [17], [26], they are represented by (tree-like) amputated vertices such as $[baseline=-2.5pt]{[Aactivity] (0.5,0) -- (0,0) node[at end,above] {\\,q_2};[Aactivity] (130:0.5) -- (0,0);[Aactivity] (-130:0.5) -- (0,0);},\\quad [baseline=-2.5pt]{[Aactivity] (0.5,0) -- (0,0) node[at end,above] {\\,q_3};[Aactivity] (130:0.5) -- (0,0);[Aactivity] (180:0.5) -- (0,0);[Aactivity] (-130:0.5) -- (0,0);},\\quad [baseline=-2.5pt]{[Aactivity] (0.5,0) -- (0,0) node[at end,above] {\\,q_4};[Aactivity] (130:0.5) -- (0,0);[Aactivity] (163:0.5) -- (0,0);[Aactivity] (-163:0.5) -- (0,0);[Aactivity] (-130:0.5) -- (0,0);}.$ These vertices are not to be confused with the underlying branching process, because after the Doi-shift, lines are not representative of particles, but of their correlations.",
"For example, the vertex with coupling $q_2$ in Eq.",
"(REF ), accounts for density-density correlations due to any branching or extinction, just like the propagator Eq.",
"(REF ) accounts for all particle density due to any branching or extinction.",
"After Fourier transforming, these processes are accounted for regardless of when they take place.",
"The directionality of the diagrams allows us to define incoming legs and outgoing legs of a vertex [17].",
"In the present branching process, all vertices have one incoming leg and $j$ outgoing legs.",
"We will refer to diagrams that are constructed solely from $q_2$ vertices as binary tree diagrams.",
"The most basic such diagram is $[baseline=-2.5pt]{[Aactivity] (0.3,0) -- (0,0);[Aactivity] (166:0.3) -- (0,0);[Aactivity] (-166:0.3) -- (0,0);}$ , which in real time reads $\\left\\langle \\phi ^2(t)\\phi ^\\dagger (0) \\right\\rangle \\hat{=}\\,2[baseline=-2.5pt]{[Aactivity] (0.5,0) -- (0,0) node[at end,above] {\\,q_2};[Aactivity] (130:0.5) -- (0,0);[Aactivity] (-130:0.5) -- (0,0);}\\\\= 2q_2 \\!\\!\\int \\!\\mathchoice{\\!\\mathrm {d}\\hspace{-3.33328pt}\\omega _1\\,}{\\!\\mathrm {d}\\hspace{-3.33328pt}\\omega _1\\,}{\\!\\mathrm {d}\\hspace{-3.33328pt}\\omega _1\\,}{\\!\\mathrm {d}\\hspace{-3.33328pt}\\omega _1\\,}\\mathchoice{\\!\\mathrm {d}\\hspace{-3.33328pt}\\omega _1^{\\prime }\\,}{\\!\\mathrm {d}\\hspace{-3.33328pt}\\omega _1^{\\prime }\\,}{\\!\\mathrm {d}\\hspace{-3.33328pt}\\omega _1^{\\prime }\\,}{\\!\\mathrm {d}\\hspace{-3.33328pt}\\omega _1^{\\prime }\\,}\\mathchoice{\\!\\mathrm {d}\\hspace{-3.33328pt}\\omega _2\\,}{\\!\\mathrm {d}\\hspace{-3.33328pt}\\omega _2\\,}{\\!\\mathrm {d}\\hspace{-3.33328pt}\\omega _2\\,}{\\!\\mathrm {d}\\hspace{-3.33328pt}\\omega _2\\,}\\mathchoice{\\!\\mathrm {d}\\hspace{-3.33328pt}\\omega _2^{\\prime }\\,}{\\!\\mathrm {d}\\hspace{-3.33328pt}\\omega _2^{\\prime }\\,}{\\!\\mathrm {d}\\hspace{-3.33328pt}\\omega _2^{\\prime }\\,}{\\!\\mathrm {d}\\hspace{-3.33328pt}\\omega _2^{\\prime }\\,}\\mathchoice{\\!\\mathrm {d}\\hspace{-3.33328pt}\\omega _3\\,}{\\!\\mathrm {d}\\hspace{-3.33328pt}\\omega _3\\,}{\\!\\mathrm {d}\\hspace{-3.33328pt}\\omega _3\\,}{\\!\\mathrm {d}\\hspace{-3.33328pt}\\omega _3\\,}\\mathchoice{\\!\\mathrm {d}\\hspace{-3.33328pt}\\omega _3^{\\prime }\\,}{\\!\\mathrm {d}\\hspace{-3.33328pt}\\omega _3^{\\prime }\\,}{\\!\\mathrm {d}\\hspace{-3.33328pt}\\omega _3^{\\prime }\\,}{\\!\\mathrm {d}\\hspace{-3.33328pt}\\omega _3^{\\prime }\\,}\\,\\mathchoice{e^{-\\mathring{\\imath }\\omega _2t}}{\\operatorname{exp}\\!\\left(-\\mathring{\\imath }\\omega _2t\\right)}{\\operatorname{exp}\\!\\left(-\\mathring{\\imath }\\omega _2t\\right)}{\\operatorname{exp}\\!\\left(-\\mathring{\\imath }\\omega _2t\\right)}\\mathchoice{e^{-\\mathring{\\imath }\\omega _3t}}{\\operatorname{exp}\\!\\left(-\\mathring{\\imath }\\omega _3t\\right)}{\\operatorname{exp}\\!\\left(-\\mathring{\\imath }\\omega _3t\\right)}{\\operatorname{exp}\\!\\left(-\\mathring{\\imath }\\omega _3t\\right)}\\\\\\times \\delta \\hspace{-4.44443pt}\\left(\\omega _1\\!+\\!\\omega _2^{\\prime }\\!+\\!\\omega _3^{\\prime }\\right)\\frac{\\delta \\hspace{-4.44443pt}\\left(\\omega _1+\\omega _1^{\\prime }\\right)}{-\\mathring{\\imath }\\omega _1+r}\\frac{\\delta \\hspace{-4.44443pt}\\left(\\omega _2+\\omega _2^{\\prime }\\right)}{-\\mathring{\\imath }\\omega _2+r}\\frac{\\delta \\hspace{-4.44443pt}\\left(\\omega _3+\\omega _3^{\\prime }\\right)}{-\\mathring{\\imath }\\omega _3+r}\\\\= 2\\frac{q_2}{r} \\mathchoice{e^{-rt}}{\\operatorname{exp}\\!\\left(-rt\\right)}{\\operatorname{exp}\\!\\left(-rt\\right)}{\\operatorname{exp}\\!\\left(-rt\\right)}\\left(1-\\mathchoice{e^{-rt}}{\\operatorname{exp}\\!\\left(-rt\\right)}{\\operatorname{exp}\\!\\left(-rt\\right)}{\\operatorname{exp}\\!\\left(-rt\\right)}\\right),$ where the pre-factor 2 is the combinatorial factor of this diagram.",
"The various observables that we calculate in the following are illustrated by numerics for two different kinds of continuous-time branching processes.",
"Firstly, the binary branching process with probabilities $p_0,p_2\\ge 0$ such that $p_0+p_2=1$ , and secondly, the branching process with geometric offspring distribution $p_k=p(1-p)^k$ with $p\\in [0,1]$ .",
"The mass $r$ (REF ) and the rates of the couplings $q_j$ (REF ) are, in each case, (binary) rB=s(1-2p2), qB2=sp2=s-rB2, qBj=0 for j3 , (geometric) rG=s2p-1p, qGj=s(1-pp)j = s(1-rGs)j .",
"Fig.",
"REF shows typical trajectories for each case."
],
[
"Moments $\\left\\langle N^n(t) \\right\\rangle $ and their universality",
"In the following we will calculate the moments of the number of particles $N(t)$ , which can be determined using the particle number operator $a^\\dagger {}a{}$ , as introduced above.",
"The $n$ th moment of $N(t)$ can be expressed as $\\left\\langle N^n(t) \\right\\rangle &=& \\left\\langle |\\left(a^\\dagger {}a\\right)^n|\\Psi (t)\\right\\rangle \\nonumber \\\\&=&\\sum \\limits _{\\ell =0}^n\\begin{Bmatrix}{n}\\\\{\\ell }\\end{Bmatrix}\\left\\langle |a^\\ell |\\Psi (t)\\right\\rangle \\nonumber \\\\&=&\\sum \\limits _{\\ell =0}^{n}\\begin{Bmatrix}{n}\\\\{\\ell }\\end{Bmatrix} \\left\\langle \\phi ^{\\ell }(t)\\widetilde{\\phi }(0) \\right\\rangle ,$ where $\\begin{Bmatrix}{n}\\\\{\\ell }\\end{Bmatrix}$ denotes the Stirling number of the second kind for $\\ell $ out of $n$ The Stirling numbers of the second kind can be calculated using the expression $\\begin{Bmatrix}{n}\\\\{\\ell }\\end{Bmatrix} = \\frac{1}{\\ell !",
"}\\sum _{j=0}^\\ell (-1)^{\\ell -j}\\binom{\\ell }{j}j^n.$ .",
"We define the dimensionless function $\\hat{g}_n(t)$ as the expectation $\\hat{g}_n(t)=\\left\\langle \\phi ^n(t)\\widetilde{\\phi }(0) \\right\\rangle \\hat{=}\\,[baseline=-2.5pt]{\\begin{scope}[Aactivity] (-150:0.2) -- (-150:0.8);[postaction={decorate,decoration={raise=0ex,text along path, text align={center}, text={|\\large |....}}}] (170:0.7cm) arc (170:210:0.8cm);[Aactivity] (170:0.2) -- (170:0.8);[Aactivity] (150:0.2) -- (150:0.8);\\end{scope}[Aactivity] (0:0.2) -- (0:0.8);[thick,fill=white] (0,0) circle (0.2cm);\\node at (190:1) {n};} ,$ with $\\hat{g}_0(t)=\\left\\langle \\widetilde{\\phi }(0) \\right\\rangle =0$ and $\\hat{g}_1(t) = \\left\\langle \\phi (t)\\widetilde{\\phi }(0) \\right\\rangle = e^{-rt}$ .",
"The black circle in the diagram of Eq.",
"(REF ) represents the sum of all possible intermediate nodes, allowing for internal lines.",
"For instance, g1(t) = [baseline=-2.5pt] [Aactivity] (0.8,0) – (-0.8,0); [thick,fill=white] (0,0) circle (0.2cm); = [baseline=-2.5pt] [Aactivity] (0.5,0) – (-0.5,0) node[at end,above] ; , g2(t) = [baseline=-2.5pt] [Aactivity] (0.8,0) – (0,0); [Aactivity] (150:0.8) – (0,0); [Aactivity] (-150:0.8) – (0,0); [thick,fill=white] (0,0) circle (0.2cm); = 2 [baseline=-2.5pt] [Aactivity] (0.5,0) – (0,0) node[at end,above] ;[Aactivity] (130:0.5) – (0,0); [Aactivity] (-130:0.5) – (0,0); , g3(t) = [baseline=-2.5pt] [Aactivity] (0.8,0) – (0,0); [Aactivity] (150:0.8) – (0,0); [Aactivity] (180:0.8) – (0,0); [Aactivity] (-150:0.8) – (0,0); [thick,fill=white] (0,0) circle (0.2cm); = 6 [baseline=-2.5pt] [Aactivity] (0.5,0) – (0,0) node[at end,above] ;[Aactivity] (130:0.5) – (0,0); [Aactivity] (180:0.5) – (0,0); [Aactivity] (-130:0.5) – (0,0); + 12 [baseline=-2.5pt] [Aactivity] (1,0) – (0,0) node [at end,above] ;[Aactivity] (130:0.5) – (0,0); [Aactivity] (-130:0.5) – (0,0); a) at (0.5,0) ; [Aactivity] (a)+(-130:0.5) – (0.5,0) node [at end,above] ; , g4(t) = [baseline=-2.5pt] [Aactivity] (0.8,0) – (0,0); [Aactivity] (150:0.8) – (0,0); [Aactivity] (170:0.8) – (0,0); [Aactivity] (-170:0.8) – (0,0); [Aactivity] (-150:0.8) – (0,0); [thick,fill=white] (0,0) circle (0.2cm); = 24 [baseline=-2.5pt] [Aactivity] (0.5,0) – (0,0) node[at end,above] ;[Aactivity] (130:0.5) – (0,0); [Aactivity] (163:0.5) – (0,0); [Aactivity] (-163:0.5) – (0,0); [Aactivity] (-130:0.5) – (0,0); + 48 [baseline=-2.5pt] [Aactivity] (1,0) – (0,0) node [at end,above] ;[Aactivity] (130:0.5) – (0,0); [Aactivity] (180:0.5) – (0,0); [Aactivity] (-130:0.5) – (0,0); a) at (0.5,0) ;[Aactivity] (a)+(-130:0.5) – (0.5,0); + 72 [baseline=-2.5pt] [Aactivity] (1,0) – (0,0) node [at end,above] ;[Aactivity] (130:0.5) – (0,0); [Aactivity] (-130:0.5) – (0,0); a) at (0.5,0) ;[Aactivity] (a)+(-130:0.5) – (0.5,0); [Aactivity] (a)+(130:0.5) – (0.5,0);+ 24 [baseline=-2.5pt] [Aactivity] (0.5,0) – (0,0) node[at end,above] ;a) at (-130:0.4) ;b) at (130:0.4) ;[Aactivity] (130:0.8) – (0,0); [Aactivity] (-130:0.8) – (0,0); [Aactivity] (-130:0.4)+(150:0.3) – (-130:0.5); [Aactivity] (130:0.4)+(-150:0.3) – (130:0.5); + 96 [baseline=-2.5pt] [Aactivity] (1.5,0) – (0,0) node [at end,above] ;[Aactivity] (130:0.5) – (0,0); [Aactivity] (-130:0.5) – (0,0); a) at (0.5,0) ;[Aactivity] (a)+(-130:0.5) – (0.5,0); b) at (1,0) ;[Aactivity] (b)+(-130:0.5) – (1,0); , where the coefficient in front of each diagram is its symmetry factor, which is included in the representation involving the black circle, Eq.",
"(REF ).",
"The tree diagrams follow a pattern, whereby $\\hat{g}_n$ involves all $\\hat{g}_m$ with $m<n$ .",
"For $n\\ge 2$ this can be expressed as the recurrence relation ${ }\\hat{g}_n(t)\\hat{=}\\!\\!\\sum _{k=2}^n \\sum _{m_1,\\ldots ,m_k=1}\\!\\!\\binom{n}{m_1,\\ldots ,m_k}\\!\\!",
"[baseline=-2.5pt,scale=0.8]{[Aactivity] (0.5,0) -- (0,0);[postaction={decorate,decoration={raise=0ex,text along path, text align={center}, text={|\\large |......}}}] (170:0.7cm) arc (170:220:0.8cm);\\node at (190:0.9) {k};\\node (a) at (170:1.5) {};[postaction={decorate,decoration={raise=0ex,text along path, text align={center}, text={|\\large |....}}}] (a)+(170:0.7cm) arc (170:210:0.8cm);[Aactivity] (a)+(-150:0.2) -- ++(-150:0.8);[Aactivity] (a)+(170:0.2) -- ++(170:0.8);[Aactivity] (a)+(150:0.2) -- ++(150:0.8);[Aactivity] (a)+(0:0.2) -- (0,0);[thick,fill=white] (a)+(0,0) circle (0.2cm);\\node at (178:2.5) {m_2};\\begin{scope}\\node (b) at (140:1.75) {};[postaction={decorate,decoration={raise=0ex,text along path, text align={center}, text={|\\large |....}}}] (b)+(170:0.7cm) arc (170:210:0.8cm);[Aactivity] (b)+(-150:0.2) -- ++(-150:0.8);[Aactivity] (b)+(170:0.2) -- ++(170:0.8);[Aactivity] (b)+(150:0.2) -- ++(150:0.8);\\end{scope}[Aactivity] (b)+(0:0.2) -- (0,0);[thick,fill=white] (b)+(0,0) circle (0.2cm);\\node at (160:2.5) {m_1};\\begin{scope}\\node (c) at (-140:1.75) {};[postaction={decorate,decoration={raise=0ex,text along path, text align={center}, text={|\\large |....}}}] (c)+(170:0.7cm) arc (170:210:0.8cm);[Aactivity] (c)+(-150:0.2) -- ++(-150:0.8);[Aactivity] (c)+(170:0.2) -- ++(170:0.8);[Aactivity] (c)+(150:0.2) -- ++(150:0.8);\\end{scope}[Aactivity] (c)+(0:0.2) -- (0,0);[thick,fill=white] (c)+(0,0) circle (0.2cm);\\node at (207:2.6) {m_k};},$ where $\\binom{n}{m_1,\\ldots ,m_k}$ denotes the multinomial coefficient with the implicit constraint of $m_1+\\ldots +m_k=n$ .",
"Including $\\hat{g}_1(t)$ from Eqs.",
"(REF ) and (REF ), this may be written as $\\hat{g}_n(t) =\\delta _{n,1}\\mathchoice{e^{-rt}}{\\operatorname{exp}\\!\\left(-rt\\right)}{\\operatorname{exp}\\!\\left(-rt\\right)}{\\operatorname{exp}\\!\\left(-rt\\right)}+\\Biggl (\\sum _{k=2}^n q_k \\sum _{m_1,\\ldots ,m_k}\\binom{n}{m_1,\\ldots ,m_k}\\\\\\times \\int _0^t \\mathchoice{\\!\\mathrm {d}t^{\\prime }\\,}{\\!\\mathrm {d}t^{\\prime }\\,}{\\!\\mathrm {d}t^{\\prime }\\,}{\\!\\mathrm {d}t^{\\prime }\\,} \\mathchoice{e^{-r(t-t^{\\prime })}}{\\operatorname{exp}\\!\\left(-r(t-t^{\\prime })\\right)}{\\operatorname{exp}\\!\\left(-r(t-t^{\\prime })\\right)}{\\operatorname{exp}\\!\\left(-r(t-t^{\\prime })\\right)}\\hat{g}_{m_1}(t^{\\prime })\\hat{g}_{m_2}(t^{\\prime })\\cdots \\hat{g}_{m_k}(t^{\\prime })\\Biggr )\\ ,$ where the integral accounts for the propagation up until time $t-t^{\\prime }\\in [0,t]$ when a branching into (at least) $k$ particles takes place, each of which will branch into (at least) $m_k$ particles at some later time within $[t-t^{\\prime },t]$ .",
"We proceed by determining the leading order behaviour of $\\hat{g}_n$ in small $r$ , starting with a dimensional argument.",
"Since $\\left\\langle N^n(t) \\right\\rangle =\\sum \\limits _{\\ell =0}^{n}\\begin{Bmatrix}{n}\\\\{\\ell }\\end{Bmatrix} \\hat{g}_\\ell (t)$ from Eqs.",
"(REF ) and (REF ), $\\left\\langle N^n(t) \\right\\rangle $ being dimensionless implies the same for $\\hat{g}_n(t)$ .",
"Our notation for the latter obscures the fact that $\\hat{g}_n(t)$ is also a function of $r$ and all $q_j$ , which, by virtue of $s$ , are rates and thus have the inverse dimension of $t$ .",
"We may therefore write $\\hat{g}_n(t) = \\overline{g}_n(rt;\\overline{q}_2,\\overline{q}_3,\\ldots )$ where $\\overline{q}_j=q_j/r$ are dimensionless couplings.",
"Dividing $q_j$ by any rate renders the result dimensionless, but only the particular choice of dividing by $r$ ensures that all couplings only ever enter multiplicatively (and never as an inverse), thereby enabling us to extract the asymptote of $\\hat{g}_n(t)$ in the limit of small $r$ , as we will see in the following.",
"Writing Eq.",
"(REF ) as $&&\\overline{g}_n(y;\\overline{q}_2,\\overline{q}_3,\\ldots ) =\\delta _{n,1}\\mathchoice{e^{-y}}{\\operatorname{exp}\\!\\left(-y\\right)}{\\operatorname{exp}\\!\\left(-y\\right)}{\\operatorname{exp}\\!\\left(-y\\right)}\\\\&&\\quad +\\!\\Biggl (\\sum _{k=2}^n \\overline{q}_k\\!\\!\\!",
"\\sum _{m_1,\\ldots ,m_k}\\!\\!\\!\\binom{n}{m_1,\\ldots ,m_k}\\int _0^y\\!\\!",
"\\mathchoice{\\!\\mathrm {d}y^{\\prime }\\,}{\\!\\mathrm {d}y^{\\prime }\\,}{\\!\\mathrm {d}y^{\\prime }\\,}{\\!\\mathrm {d}y^{\\prime }\\,}\\mathchoice{e^{-(y-y^{\\prime })}}{\\operatorname{exp}\\!\\left(-(y-y^{\\prime })\\right)}{\\operatorname{exp}\\!\\left(-(y-y^{\\prime })\\right)}{\\operatorname{exp}\\!\\left(-(y-y^{\\prime })\\right)}\\nonumber \\\\&&\\qquad \\times \\overline{g}_{m_1}\\!",
"(y^{\\prime };\\overline{q}_2,\\ldots )\\overline{g}_{m_2}\\!",
"(y^{\\prime };\\ldots )\\cdots \\overline{g}_{m_k}\\!",
"(y^{\\prime };\\ldots )\\!\\!\\Biggr )\\ ,\\nonumber $ the dominant terms in small $r$ and fixed $y=rt$ are those that contain products involving the largest number of factors of $\\overline{q}_j\\propto r^{-1}$ .",
"Since each $\\overline{q}_j$ corresponds to a branching, diagrammatically these terms are those that contain the largest number of vertices, i.e.",
"those that are entirely made up of binary branching vertices $q_2$ .",
"This coupling, $q_2=\\left\\langle \\kappa (\\kappa -1) \\right\\rangle \\!/2$ , cannot possibly vanish if there is any branching taking place at all.",
"From Eqs.",
"(REF ) and (REF ) it follows that $\\hat{g}_n(t)\\propto (q_2/r)^{(n-1)}$ to leading order in small $r$ at fixed $y=rt$ .",
"Terms of that order are due to binary tree diagrams, whose contribution we denote by $g_n(t)$ in the following.",
"For instance, $g_1(t)=\\hat{g}_1(t)$ , $g_2(t)=\\hat{g}_2(t)$ , g3(t) = 12 [baseline=-2.5pt] [Aactivity] (1,0) – (0,0) node [at end,above] ;[Aactivity] (130:0.5) – (0,0); [Aactivity] (-130:0.5) – (0,0); a) at (0.5,0) ;[Aactivity] (a)+(-130:0.5) – (0.5,0);, g4(t) = 24 [baseline=-2.5pt] [Aactivity] (0.5,0) – (0,0) node[at end,above] ;a) at (-130:0.4) ;b) at (130:0.4) ;[Aactivity] (130:0.8) – (0,0); [Aactivity] (-130:0.8) – (0,0); [Aactivity] (-130:0.4)+(150:0.3) – (-130:0.5); [Aactivity] (130:0.4)+(-150:0.3) – (130:0.5); + 96 [baseline=-2.5pt] [Aactivity] (1.5,0) – (0,0) node [at end,above] ;[Aactivity] (130:0.5) – (0,0); [Aactivity] (-130:0.5) – (0,0); a) at (0.5,0) ;[Aactivity] (a)+(-130:0.5) – (0.5,0); b) at (1,0) ;[Aactivity] (b)+(-130:0.5) – (1,0);.",
"To summarise, $\\hat{g}_n(t)$ is dominated by those terms that correspond to binary tree diagrams, which are the trees $g_n$ that have the largest number of vertices for any fixed $n$ , i.e.",
"$\\left\\langle \\phi ^n(t)\\widetilde{\\phi }(0) \\right\\rangle = \\hat{g}_n(t) = g_n(t)+\\mathcal {O}\\left(\\left(1-{\\langle \\kappa \\rangle }\\right)^{-(n-2)}\\right),$ where the correction in fact vanishes for $n<3$ .",
"As far as the asymptote in small $r$ is concerned, we may thus replace $\\hat{g}_\\ell $ in Eq.",
"(REF ) by $g_\\ell $ .",
"Among the $\\hat{g}_\\ell \\sim r^{-(\\ell -1)}$ with $\\ell =0,1,,\\dots ,n$ , the dominating term is $g_n$ so that the $n$ th moment of the particle number $N$ is, to leading order, $\\left\\langle N^n(t) \\right\\rangle \\simeq g_n(t),$ although exact results, as shown in Eq.",
"(), are easily derived using Eqs.",
"(REF ), (REF ), (REF ), and (REF ).",
"On the basis of (REF ) the recurrence relation of $g_n$ is give by $g_n(t)=\\delta _{n,1}\\mathchoice{e^{-rt}}{\\operatorname{exp}\\!\\left(-rt\\right)}{\\operatorname{exp}\\!\\left(-rt\\right)}{\\operatorname{exp}\\!\\left(-rt\\right)}+ q_2 \\sum _{m=1}^{n-1}\\binom{n}{m}\\int _0^t \\mathrm {d}t^{\\prime }\\mathchoice{e^{-r(t-t^{\\prime })}}{\\operatorname{exp}\\!\\left(-r(t-t^{\\prime })\\right)}{\\operatorname{exp}\\!\\left(-r(t-t^{\\prime })\\right)}{\\operatorname{exp}\\!\\left(-r(t-t^{\\prime })\\right)}g_{m}(t^{\\prime })g_{n-m}(t^{\\prime }),$ whose exact solution is $g_n(t) = n!\\mathchoice{e^{-rt}}{\\operatorname{exp}\\!\\left(-rt\\right)}{\\operatorname{exp}\\!\\left(-rt\\right)}{\\operatorname{exp}\\!\\left(-rt\\right)}\\left(\\frac{q_2}{r}\\left(1-\\mathchoice{e^{-rt}}{\\operatorname{exp}\\!\\left(-rt\\right)}{\\operatorname{exp}\\!\\left(-rt\\right)}{\\operatorname{exp}\\!\\left(-rt\\right)}\\right)\\right)^{n-1}.$ We draw two main conclusions from our results.",
"Firstly, that near the critical point $r=0$ , the branching process is solely characterised by the two parameters $r$ and $q_2$ .",
"We therefore conclude that this process displays universality, in the sense that its asymptotia is exactly the same for any given $r$ and $q_2$ regardless of the underlying offspring distribution.",
"In particular, certain ratios of the moments of the particle number are universal constants (they do not depend on any parameters nor variables).",
"For $k,\\ell \\in \\mathbb {N}$ and $m\\in {0,\\dots ,k-1}$ , we find the constant ratios $\\frac{\\left\\langle N^k(t) \\right\\rangle \\left\\langle N^\\ell (t) \\right\\rangle }{\\left\\langle N^{k-m}(t) \\right\\rangle \\left\\langle N^{\\ell +m} \\right\\rangle }=\\frac{k!\\ell !",
"}{(k-m)!",
"(\\ell +m)!",
"}.$ Secondly, our results show that the scaling form of the moments is $\\left\\langle N^n(t) \\right\\rangle \\simeq ({q_2}{t})^{n-1}\\mathcal {G}_n(rt),$ where $\\mathcal {G}_n$ is the scaling function $\\mathcal {G}_n(y) = n!\\mathchoice{e^{-y}}{\\operatorname{exp}\\!\\left(-y\\right)}{\\operatorname{exp}\\!\\left(-y\\right)}{\\operatorname{exp}\\!\\left(-y\\right)}\\left(\\frac{1-e^{-y}}{y}\\right)^{n-1},$ and the argument $y=rt$ is the rescaled time, Fig.",
"REF .",
"The asymptotes of $\\mathcal {G}_n(y)$ characterise the behaviour of the branching process in each regime, $\\mathcal {G}_n(y) \\simeq \\left\\lbrace \\begin{array}{l l}n!",
"& \\text{ for }y=0,\\\\n!",
"\\,y^{-(n-1)}e^{-y} & \\text{ for }y\\rightarrow \\infty .\\end{array}\\right.$ Moreover, from Eq.",
"(REF ), we find that the moment generating function $\\mathcal {M}_{{N}}\\left({z}\\right) = \\left\\langle e^{Nz} \\right\\rangle $ is $\\mathcal {M}_{{N}}\\left({z}\\right) \\simeq 1+\\frac{ze^{-rt}}{1+z\\frac{q_2}{r}\\left(e^{-rt}-1\\right)}.$ Figure: Data collapse of the moments N(t)\\left\\langle N(t) \\right\\rangle , N 2 (t)\\left\\langle N^2(t) \\right\\rangle and N 3 (t)\\left\\langle N^3(t) \\right\\rangle ,as a function of rescaled time rtrt as of Eq. ().",
"Symbols show results for the binary branching process (blue)and the branching process with geometric distribution of offspring (orange),both with r∈{10 -3 ,10 -2 ,10 -1 }r\\in \\lbrace 10^{-3}, 10^{-2},10^{-1}\\rbrace and s=1s=1.Solid lines indicate the exact solution in Eq.",
"() and dashed linesindicate our approximation in Eq.",
"()."
],
[
"Probability distribution of $N(t)$ , probability of survival {{formula:c79e6676-779a-412d-b4bb-101efa448ae0}} and expected avalanche duration {{formula:78c47031-420a-41df-a8c4-e748fe8ef1c6}}",
"Using Eq.",
"(REF ) and the identity The Stirling numbers of the second kind satisfy the identity $ N^n = \\sum \\limits _{\\ell =0}^{n}\\begin{Bmatrix}{n}\\\\{\\ell }\\end{Bmatrix} N(N-1)\\ldots (N-\\ell +1) .",
"$ of Stirling numbers of the second kind, we deduce that the falling factorial moments of $N(t)$ are $\\left\\langle \\phi ^{\\ell }(t)\\widetilde{\\phi }(0) \\right\\rangle = \\left\\langle N(t)(N(t)-1)\\ldots (N(t)-\\ell +1) \\right\\rangle .$ Therefore, the probability generating function of $N(t)$ is $\\mathcal {P}_{{N(t)}}\\left({z}\\right) &=& \\sum _{\\ell =0}^\\infty \\left\\langle N(t)(N(t)-1)\\ldots (N(t)-\\ell +1) \\right\\rangle \\frac{(z-1)^\\ell }{\\ell !",
"}\\nonumber \\\\&=& \\sum _{\\ell =0}^\\infty \\left\\langle \\phi ^{\\ell }(t)\\widetilde{\\phi }(0) \\right\\rangle \\frac{(z-1)^\\ell }{\\ell !",
"},$ and the probability distribution of $N(t)$ is, using Eqs.",
"(REF ) and (REF ), P(N,t) = 1N!",
"dNdzN(PN(t)(z))|z=0 =NN(-1)-N!g(t) = { l l 1-e-rt1+q2r(1-e-rt) if N=0, e-rt(q2r(1-e-rt))N-1(1+q2r(1-e-rt))N+1 if N>0, .",
"which satisfies the initial condition $P(N,0)=\\delta _{N,1}$ and is an exact result for binary branching processes, consistent with [13].",
"It is straightforward to check that Eq.",
"(REF ) satisfies the master equation (REF ) and the initial condition in the binary branching case.",
"Due to the uniqueness of solutions of a system of coupled linear ordinary differential equations, the solution in Eq.",
"(REF ) is the only solution.",
"In particular, this solution holds in the supercritical case, $r<0$ .",
"Reconstructing back the path that has lead us here, we find that $g_\\ell (t)$ is the $\\ell $ th falling factorial moment of $N(t)$ , $\\left\\langle N(t)(N(t)-1)\\ldots (N(t)-\\ell +1) \\right\\rangle $ , for binary branching processes including the supercritical case and, therefore, most expressions derived from $g_\\ell (t)$ can be extended to $r<0$ .",
"In what follows, we will specify which expressions hold in the supercritical case.",
"The probability of survival $P_\\text{s}(t)$ is the probability that there is at least one particle at time $t$ , i.e.",
"$P_\\text{s}(t)=P(N(t)\\ge 1)$ .",
"Therefore, from Eq.",
"(REF ), $P_\\text{s}(t) = 1-P(0,t) = \\frac{e^{-rt}}{1+\\frac{q_2}{r}\\left(1-e^{-rt}\\right)},$ and at the critical point, $\\lim \\limits _{r\\rightarrow 0}P_\\text{s}(t)\\simeq \\frac{1}{1+q_2 t},$ which is consistent with [31], [32], [33], Fig.",
"REF .",
"Figure: Probability of survival as a function of rescaled time rtrt as of Eq.",
"().Symbols show numerical results for the binary branching process (blue)and the branching process with geometric distribution of offspring (orange),both with r∈{10 -3 ,10 -2 ,10 -1 }r\\in \\lbrace 10^{-3}, 10^{-2}, 10^{-1}\\rbrace and s=1s=1.Lines indicate the result in Eq.",
"(), which is exact for binarybranching (solid lines) and approximate otherwise (dashed lines).As rr gets closer to the critical value, r=0r=0, the curves P s (t)P_\\text{s}(t)flatten and resemble the power law in Eq.",
"(), whichhas exponent -1-1.We define the avalanche duration $T$ as the exact time where an avalanche dies, i.e.",
"the time $t$ when the process reaches the absorbing state, $T=\\text{min}\\lbrace t|N(t)=0\\rbrace $ .",
"The probability of survival $P_\\text{s}(t)$ gives the probability that $T>t$ .",
"Thus, $1-P_s(t)$ is the cumulative distribution function of the time of death and its probability density function is $\\mathcal {P}_{{T}}\\left({t}\\right)=-\\frac{dP_\\text{s}(t)}{dt}\\simeq \\frac{re^{rt}\\left(1+\\frac{q_2}{r}\\right)}{\\left(\\frac{q_2}{r}-e^{rt}\\left(1+\\frac{q_2}{r}\\right)\\right)^2},$ and at the critical point, $\\lim _{r\\rightarrow 0}\\mathcal {P}_{{T}}\\left({t}\\right)\\simeq \\frac{q_2}{(1+q_2t)^2},$ see Fig.",
"REF .",
"It follows from (REF ) that the expected avalanche duration is $\\left\\langle T \\right\\rangle \\simeq \\frac{1}{q_2}\\log \\left(1+\\frac{q_2}{r}\\right).$ Because the derivation of Eq.",
"(REF ) relies on a finite termination time, we cannot assume that it remains valid in the supercritical case, and similarly for (REF ).",
"Figure: Probability density function of the avalancheduration 𝒫 T t\\mathcal {P}_{{T}}\\left({t}\\right) for the binary branching processwith r∈{0,10 -3 ,10 -2 ,10 -1 }r\\in \\lbrace 0,10^{-3}, 10^{-2}, 10^{-1}\\rbrace and s=1s=1.",
"Solid linesrepresent our result in Eqs.",
"() and (),which is exact for binary branching.",
"Symbols show numerical results."
],
[
"Avalanche shape $V\\left(t,T\\right)$ ",
"The avalanche shape $V\\left(t,T\\right)$ is defined as the average of the temporal profiles $N(t)$ conditioned to extinction at time $T$ [4], [34], [35], [36], [37], [38], [39].",
"Closed form expressions of the avalanche shape have been calculated in other models such as avalanches in elastic interfaces [34], the Barkhausen noise [36], the discrete-time Ornstein-Uhlenbeck process [37].",
"An implicit expression of avalanche shape of branching processes is given in [4].",
"To produce an explicit expression we first calculate the expected number of particles at time $t$ of a branching process conditioned to being extinct by time $T$ , $\\Big \\langle N(t) \\Big | N(T)=0\\Big \\rangle $ .",
"In terms of ladder operators, $\\Big \\langle N(t) \\Big | N(T)=0\\Big \\rangle =\\left\\langle 0|\\mathchoice{e^{\\hat{\\mathcal {A}}(T-t)}}{\\operatorname{exp}\\!\\left(\\hat{\\mathcal {A}}(T-t)\\right)}{\\operatorname{exp}\\!\\left(\\hat{\\mathcal {A}}(T-t)\\right)}{\\operatorname{exp}\\!\\left(\\hat{\\mathcal {A}}(T-t)\\right)}a^\\dagger a\\mathchoice{e^{\\hat{\\mathcal {A}}t}}{\\operatorname{exp}\\!\\left(\\hat{\\mathcal {A}}t\\right)}{\\operatorname{exp}\\!\\left(\\hat{\\mathcal {A}}t\\right)}{\\operatorname{exp}\\!\\left(\\hat{\\mathcal {A}}t\\right)}a^\\dagger |0\\right\\rangle ,$ which means that a particle is created from the vacuum, the system is allowed to evolve for time $t$ , the number of particles is measured, and the system evolves further for time $T-t$ .",
"Finally, all possible trajectories are \"sieved\" so that only those avalanches are whose number of particles is 0 at time $T$ taken into account.",
"The path integral expression of Eq.",
"(REF ) is ${}&&\\Big \\langle N(t) \\Big | N(T)=0\\Big \\rangle =\\left\\langle \\mathchoice{e^{-\\phi (T)}}{\\operatorname{exp}\\!\\left(-\\phi (T)\\right)}{\\operatorname{exp}\\!\\left(-\\phi (T)\\right)}{\\operatorname{exp}\\!\\left(-\\phi (T)\\right)}\\phi ^\\dagger (t)\\phi (t)\\phi ^\\dagger (0) \\right\\rangle \\nonumber \\\\&&=\\left\\langle \\phi (t)\\widetilde{\\phi }(0) \\right\\rangle + \\sum _{n\\ge 1}\\frac{(-1)^n}{n!",
"}\\\\&&\\quad \\times \\left(\\left\\langle \\phi ^n(T)\\phi (t)\\widetilde{\\phi }(0) \\right\\rangle \\nonumber +\\left\\langle \\phi ^n(T)\\widetilde{\\phi }(t)\\phi (t)\\widetilde{\\phi }(0) \\right\\rangle \\right).",
"\\nonumber $ The two terms in the bracket have asymptotes ${}&&\\left\\langle \\phi ^n(T)\\phi (t)\\widetilde{\\phi }(0) \\right\\rangle \\simeq \\sum _{k=1}^n\\sum _{m_1,\\ldots ,m_k}\\binom{n}{m_1,\\ldots ,m_k}\\nonumber \\\\&&\\qquad \\times \\frac{1}{k!",
"}\\,g_{m_1}(T-t)\\cdots g_{m_k}(T-t)g_{k+1}(t),$ and ${}&&\\left\\langle \\phi ^n(T)\\widetilde{\\phi }(t)\\phi (t)\\widetilde{\\phi }(0) \\right\\rangle \\simeq \\sum _{k=1}^n\\sum _{m_1,\\ldots ,m_k}\\binom{n}{m_1,\\ldots ,m_k}\\nonumber \\\\&&\\qquad \\times \\frac{1}{(k-1)!",
"}\\,g_{m_1}(T-t)\\cdots g_{m_k}(T-t)g_{k}(t),$ with the constraint $m_1+\\ldots +m_k=n$ in both cases.",
"Both expressions are exact in case of binary branching.",
"Their diagrammatic representation and closed form expressions can be found in Appendix .",
"Using the expression of $g_n(t)$ in Eq.",
"(REF ) and the number of combinations of $n$ legs into $k$ groups, we have $&&\\Big \\langle N(t) \\Big | N(T)=0\\Big \\rangle \\\\&&\\quad = \\mathchoice{e^{-rt}}{\\operatorname{exp}\\!\\left(-rt\\right)}{\\operatorname{exp}\\!\\left(-rt\\right)}{\\operatorname{exp}\\!\\left(-rt\\right)} - P_\\text{s}(T) \\left[ 1+\\frac{q_2}{r}\\left(1-\\mathchoice{e^{-rt}}{\\operatorname{exp}\\!\\left(-rt\\right)}{\\operatorname{exp}\\!\\left(-rt\\right)}{\\operatorname{exp}\\!\\left(-rt\\right)}\\right)\\left(2-\\frac{P_\\text{s}(T)}{P_\\text{s}(t)}\\right)\\right],\\nonumber $ where $P_\\text{s}(t)$ is given in Eq.",
"(REF ).",
"In order to account solely for those instances that become extinct exactly at time $T$ , the expectation $\\Big \\langle N(t) \\Big | N(T)=0\\Big \\rangle $ is to be differentiated with respect to $T$ , and in order to account for the factor due to conditioning to extinction, we need to divide the result by $-\\frac{\\mathrm {d}}{\\mathrm {d}t}P_\\text{s}(t)$ , yielding, $V\\left(t,T\\right)&=& \\frac{\\frac{\\mathrm {d}}{\\mathrm {d}T} \\Big \\langle N(t) \\Big | N(T)=0\\Big \\rangle }{-\\frac{\\mathrm {d}}{\\mathrm {d}t}P_\\text{s}(t)}\\nonumber \\\\&\\simeq & 1 + 2\\frac{q_2}{r}\\left(1-\\mathchoice{e^{-rt}}{\\operatorname{exp}\\!\\left(-rt\\right)}{\\operatorname{exp}\\!\\left(-rt\\right)}{\\operatorname{exp}\\!\\left(-rt\\right)}\\right)\\left(1-\\frac{P_\\text{s}(T)}{P_\\text{s}(t)}\\right),$ Fig.",
"REF .",
"Since the observable $V(t,T)$ suitably incorporates the condition $N(T)=0$ , the result in Eq.",
"(REF ) holds for the supercritical case as well.",
"At criticality, the avalanche shape is the parabola [4], [36], [37], [38] $\\lim _{r\\rightarrow 0}V\\left(t,T\\right) \\simeq 1+2\\frac{(q_2T)^2}{1+q_2T}\\left(1-\\frac{t}{T}\\right)\\frac{t}{T}.$ The avalanche shape $V\\left(t,T\\right)$ in Eq.",
"(REF ) is a symmetric function with its maximum at $t=T/2$ , which is bounded [37] by $\\lim _{T\\rightarrow \\infty }V\\left(\\frac{T}{2},T\\right) \\simeq 1+ 2\\frac{q_2}{r}.$ Figure:"
],
[
"Connected correlation function $\\text{Cov}\\left(N(t_1),N(t_2)\\right)$ ",
"To calculate the expectation $\\left\\langle N(t_1)N(t_2) \\right\\rangle $ we assume $0<t_1<t_2$ without loss of generality, N(t1)N(t2) = |aae-A(t2-t1)aae-A t1a|0 =(t2)(t1)(t1)(0) =(t2)(t1)(t1)(0) +(t2)(t1)(0) = [baseline=-2.5pt] [Aactivity] (-1,0) – (-0.1,0) node[at end,above] ; [Aactivity] (0.1,0) – (1,0) node[at end,above] ; + 2 [baseline=-2.5pt] [Aactivity] (0.5,0) – (0,0) node[at end,above] ; [Aactivity] (130:0.5) – (0,0); [Aactivity] (-130:0.5) – (0,0);.",
"The diagram on the left consists of two separate components.",
"We refer to diagrams of that kind as disconnected diagrams, in contrast to connected diagrams that only consist of one component as the one appearing on the right.",
"The connected correlation function is ${}&&\\text{Cov}\\left(N(t_1),N(t_2)\\right)=\\left\\langle N(t_1)N(t_2) \\right\\rangle - \\left\\langle N(t_1) \\right\\rangle \\left\\langle N(t_2) \\right\\rangle \\nonumber \\\\&&\\quad = \\left(2\\frac{q_2}{r}+1\\right)\\mathchoice{e^{-r(t_1+t_2)}}{\\operatorname{exp}\\!\\left(-r(t_1+t_2)\\right)}{\\operatorname{exp}\\!\\left(-r(t_1+t_2)\\right)}{\\operatorname{exp}\\!\\left(-r(t_1+t_2)\\right)}\\left(\\mathchoice{e^{r t_1}}{\\operatorname{exp}\\!\\left(r t_1\\right)}{\\operatorname{exp}\\!\\left(r t_1\\right)}{\\operatorname{exp}\\!\\left(r t_1\\right)}-1\\right)$ which is an exact result independent of the type of branching process, (i.e.",
"irrespective of the offspring distribution), Fig.",
"REF .",
"In particular, the variance is $\\text{Var}\\left(N(t)\\right)=\\text{Cov}\\left(N(t),N(t)\\right)$ [17].",
"Figure: Two-point correlation functionCovN(t a ),N(t b )\\text{Cov}\\left(N(t_a),N(t_b)\\right) of the binary continuous-time branching processwith r=10 -1 r=10^{-1} and s=1s=1.",
"Our numerical results shown as symbols are in perfect agreement with theexact expression in Eq.",
"() with t 1 =min(t a ,t b )t_1=\\text{min}(t_a,t_b) andt 2 =max(t a ,t b )t_2=\\text{max}(t_a,t_b) (solid lines).",
"We also showVar(N(t))=CovN(t),N(t)\\text{Var}(N(t))=\\text{Cov}\\left(N(t),N(t)\\right), which is the envelope."
],
[
" $n$ -point correlation function",
"We call $\\zeta _n(t_1,\\dots , t_n)$ , with $0<t_1,\\dots , t_n$ (not necessarily in order), the contribution of all binary, and therefore connected, diagrams to the $n$ -point correlation function, where the error term is controlled as $\\left\\langle N(t_1)\\ldots N(t_n) \\right\\rangle = \\zeta _n(t_1,\\dots ,t_n)+ \\mathcal {O}\\left(\\left(1-{\\langle \\kappa \\rangle }\\right)^{-(n-2)}\\right).$ The leading order contribution $\\zeta _n$ satisfies the following recurrence relation, n(t1,...,tn) = m=1n-1 {t1,...,tn} ||=m { [baseline=-2.5pt] [Aactivity] (0.75,0) – (0,0); b) at (150:1.5) ; [postaction=decorate,decoration=raise=0ex,text along path, text align=center, text=||....] (b)+(170:0.7cm) arc (170:210:0.8cm); [Aactivity] (b)+(-150:0.2) – ++(-150:0.8); [Aactivity] (b)+(170:0.2) – ++(170:0.8); [Aactivity] (b)+(150:0.2) – ++(150:0.8); [Aactivity] (b)+(0:0.2) – (0,0); [thick,fill=white] (b)+(0,0) circle (0.2cm); t (160:2.75) $\\sigma $ ; c) at (-150:1.5) ; [postaction=decorate,decoration=raise=0ex,text along path, text align=center, text=||....] (c)+(170:0.7cm) arc (170:210:0.8cm); [Aactivity] (c)+(-150:0.2) – ++(-150:0.8); [Aactivity] (c)+(170:0.2) – ++(170:0.8); [Aactivity] (c)+(150:0.2) – ++(150:0.8); [Aactivity] (c)+(0:0.2) – (0,0); [thick,fill=white] (c)+(0,0) circle (0.2cm); t (-160:2.75) $\\sigma ^\\text{C}$ ; } =q2m=1n-1 {t1,...,tn} ||=m 0tmin m(t(1)-t',...,t(m)-t') n-m(tc(m+1)-t',...,tc(n)-t')e-rt'dt', with $\\zeta _0=0$ and $\\zeta _1(t)=e^{-rt}$ , and $t_\\text{min}=\\text{min}\\lbrace t_1,\\dots ,t_n\\rbrace $ .",
"Here, $\\sigma $ is a subset of the set of times $\\lbrace t_1,\\dots , t_n\\rbrace $ , whose size is $|\\sigma |$ , and $\\sigma (1),\\dots ,\\sigma (m)$ is a list of its distinct elements.",
"Its complementary set is $\\sigma ^c=\\lbrace t_1,\\dots ,t_n\\rbrace \\backslash \\sigma $ , which contains the elements $\\sigma ^c(m+1),\\dots ,\\sigma ^c(n)$ .",
"Eq.",
"(REF ) is symmetric under exchange of any permutation of the times $t_1,\\dots , t_n$ , see the 3-point correlation function in Appendix .",
"This approximation is two-fold.",
"First, it neglects higher order branching vertices proportional to $q_{j \\ge 3}$ , and secondly, it neglects contributions from disconnected diagrams, cf. Eq.",
"(REF ).",
"Latter contributions are dominant only when $t_\\text{max}=\\text{max}\\lbrace t_1,\\dots ,t_n\\rbrace $ is smaller than $s^{-1}$ .",
"For times in $\\left(0,s^{-1}\\right)$ , the branching process has typically not yet undergone a change in the particle number."
],
[
"Distribution of the total avalanche size $S$",
"We define the total avalanche size as the time-integrated activity $S = s\\int \\mathrm {d}t N(t) $ .",
"Using $\\left\\langle N(t) \\right\\rangle =e^{-rt}$ and Eq.",
"(REF ), the first and second moments of the total avalanche size [8], [40] read S =sdt N(t) =sr=11-, S2 =s2dt1dt2 N(t1)N(t2) =s2r2(q2r+1).",
"To calculate $\\left\\langle S^n \\right\\rangle $ close to criticality, we use the approximation to the $n$ -point correlation function defined in Eq.",
"(REF ) and find the following recurrence relation, Sn sndt1...dtn n(t1,...,tn) q2rm=1n-1nmSm Sn-m snq2n-1r2n-12n-1(2n-3)!",
"!, see Appendix for a proof by induction of Eq.",
"(REF ).",
"Similarly to Eq.",
"(REF ), we find the universal constant ratios of the moments of $S$ , $\\frac{\\left\\langle S^k \\right\\rangle \\left\\langle S^\\ell \\right\\rangle }{\\left\\langle S^{k-m} \\right\\rangle \\left\\langle S^{\\ell +m} \\right\\rangle }=\\frac{(2k-3)!!",
"(2\\ell -3)!!}{(2(k-m)-3)!!",
"(2(\\ell +m)-3)!!",
"}$ with $k,\\ell \\in \\mathbb {N}$ and $m\\in \\lbrace 0,\\dots ,k-1\\rbrace $ .",
"The moment generating function of $S$ is $\\mathcal {M}_S(z)\\simeq 1+\\frac{r-\\sqrt{r^2-4sq_2z}}{2q_2},$ and its probability density function $\\mathcal {P}_{{S}}\\left({x}\\right)$ is the inverse Laplace transform of $\\mathcal {M}_S(-z)$ , $\\mathcal {P}_{{S}}\\left({x}\\right) \\simeq \\frac{1}{2}\\sqrt{\\frac{s}{q_2\\pi }}x^{-\\frac{3}{2}}\\mathchoice{e^{-\\frac{r^2x}{4q_2s}}}{\\operatorname{exp}\\!\\left(-\\frac{r^2x}{4q_2s}\\right)}{\\operatorname{exp}\\!\\left(-\\frac{r^2x}{4q_2s}\\right)}{\\operatorname{exp}\\!\\left(-\\frac{r^2x}{4q_2s}\\right)},$ which is a power law with exponent $-3/2$ with exponential decay, Fig.",
"REF .",
"At criticality, this distribution is a pure power law.",
"The approximation used to derive these results, Eq.",
"(REF ), consists in neglecting contributions of disconnected diagrams to the $n$ -point correlation function.",
"This approximation is unjustified for total avalanche sizes corresponding those realisations of branching processes that underwent no branching but a single extinction event, and whose sizes are therefore typically smaller than 1, because their $n$ -point correlation functions $\\left\\langle N(t_1)\\ldots N(t_n) \\right\\rangle $ vanish for $t_{\\mathrm {max}} \\gtrsim s^{-1}$ .",
"Consequently, the $n$ -point correlation functions are dominated by purely disconnected diagrams (cf. Sec.",
"REF ).",
"We therefore expect a breakdown of our approximation around $x = 1$ .",
"All three features of the distribution of the total avalanche size, the power-law behaviour, the exponential cutoff, and the breakdown of the approximation for $x < 1$ , are in good agreement with numerical simulations as shown in Fig.",
"REF .",
"Figure: Probability density function of the total avalanche size𝒫 S x\\mathcal {P}_{{S}}\\left({x}\\right) for the binary branching process (blue) and the branching processwith geometric distribution of offspring (orange), withr∈0,10 -3 ,10 -2 ,10 -1 r\\in \\left\\lbrace 0,10^{-3},10^{-2},10^{-1}\\right\\rbrace and s=1s=1.",
"Dashed lines indicate ourapproximation in Eq. ().",
"This approximation is not valid for smalltimes, which explains the disagreement between the numerical results andthe dashed lines for small values of xx."
],
[
"Discussion and conclusions",
"In this paper we study the continuous-time branching process following a field-theoretic approach.",
"We build on the wealth of existing results in the literature obtained through other methods.",
"Here, we demonstrate that the Doi-Peliti field theory provides an elegant, intuitive, and seemingly natural language for continuous-time branching processes.",
"We illustrate how to use the field theory to calculate a number of relevant observables, listed in Table REF .",
"Our results are valid for any offspring distribution in the vicinity of the critical point and at large times.",
"However, many of the results are exact for the binary branching process and others are exact for any branching process.",
"In principle, many observables can be calculated systematically using the field theory for any offspring distribution, for any time and any parameter set.",
"In this paper, we extend the existing results in the literature by finding explicit scaling functions and universal moment ratios for any offspring distribution.",
"We find that all the scaling laws derived above depend on two parameters only, namely $r$ and $q_2$ .",
"Therefore, one may argue that the master equation of any branching process close to the critical point and asymptotically in large times is captured by the action Eq.",
"(REF ) with couplings $r$ and $q_2$ only.",
"Having established the field-theoretic ground work, in particular the basic formalism and range of relevant observables, we may now proceed by extending the basic branching process into more sophisticated models of natural phenomena.",
"We hope that the methods established in this paper will help reaching new boughs, branches, and twigs of the many offspring of branching processes."
],
[
"Exact expressions",
"The continuous-time branching process is exactly solvable, that is, in principle, all moments and correlation functions can be calculated in exact form if all the terms in the (possibly infinite) sums are taken into account.",
"Here we show some exact expressions.",
"The exact first three moments of $N(t)$ are N(t) = e-rtexp(-rt)exp(-rt)exp(-rt), N2(t) = e-rtexp(-rt)exp(-rt)exp(-rt) ( 1+2q2r(1 - e-rtexp(-rt)exp(-rt)exp(-rt))), N3(t) =e-3rt(6q22r2-3q3r)-e-2rt(12q22r2+6q2r)+ +e-rt(6q22r2+3q3r+6q2r+1), and therefore the variance is $\\text{Var}\\left( N(t) \\right) = \\left(1+2\\frac{q_2}{r}\\right) \\mathchoice{e^{-rt}}{\\operatorname{exp}\\!\\left(-rt\\right)}{\\operatorname{exp}\\!\\left(-rt\\right)}{\\operatorname{exp}\\!\\left(-rt\\right)} \\left(1- \\mathchoice{e^{-rt}}{\\operatorname{exp}\\!\\left(-rt\\right)}{\\operatorname{exp}\\!\\left(-rt\\right)}{\\operatorname{exp}\\!\\left(-rt\\right)} \\right),$ which is consistent with Eq.",
"(REF ) and [27], [17], [32], [1], [13].",
"The three-point correlation function is, assuming $0\\le t_1\\le t_2\\le t_3$ and using Eq.",
"(REF ), $&&\\left\\langle N(t_1)N(t_2)N(t_3) \\right\\rangle \\simeq \\zeta (t_1,t_2,t_3)\\nonumber \\\\&&=2\\left(\\frac{q_2}{r}\\right)^2e^{-r(t_1+t_2+t_3)}\\\\&&\\quad \\times \\left(\\left(e^{rt_1}-1\\right)\\left(2e^{rt_1}+e^{rt_2}\\right)-\\frac{3}{2}\\left(e^{2rt_1}-1\\right)\\right).\\nonumber $"
],
[
"Diagrammatic representation and closed form expressions of Eqs. (",
"Defining $a=\\frac{e^{-r(T-t)}-e^{-rT}}{1-e^{-r(T-t)}},$ we have, firstly (REF ), n(T)(t)(0) = k=1nm1,...,mknm1,...,mk gm1(T-t)gmk(T-t)gk+1(t)1k!",
"=k=1nm1,...,mknm1,...,mk { [baseline=-2.5pt,scale=0.9] [thick,fill=white] (0,0) circle (0.2cm); [Aactivity] (0.75,0) – (0.2,0); [postaction=decorate,decoration=raise=0ex,text along path, text align=center, text=||......] (145:0.7cm) arc (145:170:0.8cm); t (155:0.9) $k$ ; a) at (175:1.5) ; [postaction=decorate,decoration=raise=0ex,text along path, text align=center, text=||....] (a)+(170:0.7cm) arc (170:210:0.8cm); [Aactivity] (a)+(-150:0.2) – ++(-150:0.8); [Aactivity] (a)+(170:0.2) – ++(170:0.8); [Aactivity] (a)+(150:0.2) – ++(150:0.8); [Aactivity] (a)+(0:0.2) – (175:0.2); [thick,fill=white] (a)+(0,0) circle (0.2cm); t (180:2.5) $m_k$ ; b) at (140:1.75) ; [postaction=decorate,decoration=raise=0ex,text along path, text align=center, text=||....] (b)+(170:0.7cm) arc (170:210:0.8cm); [Aactivity] (b)+(-150:0.2) – ++(-150:0.8); [Aactivity] (b)+(170:0.2) – ++(170:0.8); [Aactivity] (b)+(150:0.2) – ++(150:0.8); [Aactivity] (b)+(0:0.2) – (140:0.2); [thick,fill=white] (b)+(0,0) circle (0.2cm); t (160:2.5) $m_1$ ; c) at (-150:1.1) ; [Aactivity] (c)+(0:0.2) – (-150:0.2); } =n!e-rt(q2r)n (1-e-rT)n [a2(n-1)(1+a)2+2a1+a], and secondly (REF ), n(T)(t)(t)(0) =k=1nm1,...,mknm1,...,mk gm1(T-t)gmk(T-t)gk(t)1(k-1)!",
"=k=1nm1,...,mknm1,...,mk { [baseline=-2.5pt,scale=0.9] [thick,fill=white] (0,0) circle (0.2cm); [Aactivity] (0.75,0) – (0.2,0); [postaction=decorate,decoration=raise=0ex,text along path, text align=center, text=||......] (170:0.7cm) arc (170:220:0.8cm); t (190:0.9) $k$ ; a) at (170:1.5) ; [postaction=decorate,decoration=raise=0ex,text along path, text align=center, text=||....] (a)+(170:0.7cm) arc (170:210:0.8cm); [Aactivity] (a)+(-150:0.2) – ++(-150:0.8); [Aactivity] (a)+(170:0.2) – ++(170:0.8); [Aactivity] (a)+(150:0.2) – ++(150:0.8); [Aactivity] (a)+(0:0.2) – (170:0.2); [thick,fill=white] (a)+(0,0) circle (0.2cm); t (180:2.5) $m_2$ ; b) at (140:1.75) ; [postaction=decorate,decoration=raise=0ex,text along path, text align=center, text=||....] (b)+(170:0.7cm) arc (170:210:0.8cm); [Aactivity] (b)+(-150:0.2) – ++(-150:0.8); [Aactivity] (b)+(170:0.2) – ++(170:0.8); [Aactivity] (b)+(150:0.2) – ++(150:0.8); [Aactivity] (b)+(0:0.2) – (140:0.2); [thick,fill=white] (b)+(0,0) circle (0.2cm); t (160:2.5) $m_1$ ; c) at (-140:1.75) ; [postaction=decorate,decoration=raise=0ex,text along path, text align=center, text=||....] (c)+(170:0.7cm) arc (170:210:0.8cm); [Aactivity] (c)+(-150:0.2) – ++(-150:0.8); [Aactivity] (c)+(170:0.2) – ++(170:0.8); [Aactivity] (c)+(150:0.2) – ++(150:0.8); [Aactivity] (-140:1.75) – (-140:1.05); [Aactivity] (-140:0.85) – (-140:0.2); [thick,fill=white] (c)+(0,0) circle (0.2cm); t (207:2.6) $m_k$ ; } =n!e-rt1-e-rt(q2r)n-1 (1-e-rT)n [a2(n-1)(1+a)2+a1+a]."
],
[
"Averaged avalanche shape",
"In Section REF , we derive analytically the expected avalanche shape $V(t,T)$ for a specific time of death $T$ .",
"However, direct comparison with numerics is computationally very expensive as specific large times of death occur rarely for subcritical branching processes.",
"Here we describe an observable that is accessible both analytically and numerically: the averaged avalanche shape $\\left\\langle V(\\tau ) \\right\\rangle _T$ .",
"For a fixed parameter set, we first rescale time $\\tau =t/T$ and then average all the avalanche profiles irrespectively of $T$ .",
"Finally, to achieve convergence to a shape comparable across parameter settings, we normalise the result by area [38], V() T=1NV0dTPT(T)V(T,T), NV=010ddTPT(T)V(T,T).",
"The result [41] can be expressed with the Gaussian hypergeometric function ${_2F}_1(a,b,c,z)$ , $\\left\\langle V(\\tau ) \\right\\rangle _T=\\frac{1}{N_V}+\\tau (\\tau -1)\\frac{q_2\\,F(\\tau ,q_2,r)}{(q_2+r)N_V},$ where $F(\\tau ,q_2,r)&=&\\frac{{_2F}_1\\left(1,2-\\tau ,3-\\tau ,\\frac{q_2}{q_2+r}\\right)}{\\tau -2}\\nonumber \\\\&&-\\frac{{_2F}_1\\left(1,1+\\tau ,2+\\tau ,\\frac{q_2}{q_2+r}\\right)}{\\tau +1}.$ Both $F$ and $N_V$ diverge at the critical point with the limit r0F(,q2,r)NV=6"
],
[
"Proof of Eq. (",
"Eq.",
"(REF ) can be proved by induction.",
"In Eq.",
"(REF ) we see that it applies to $\\left\\langle S \\right\\rangle $ .",
"The approximation of binary tree diagrams of $\\left\\langle N(t_1)N(t_2) \\right\\rangle $ gives $\\left\\langle S^2 \\right\\rangle =s^2q_2/r^3$ , which also satisfies Eq.",
"(REF ).",
"The induction step is verified by ${}\\left\\langle S^n \\right\\rangle &=&\\frac{q_2}{r}\\sum \\limits _{m=1}^{n-1}\\binom{n}{m}\\left(\\frac{s^mq_2^{m-1}2^{m-1}(2m-3)!!",
"}{r^{2m-1}}\\right)\\\\&&\\qquad \\times \\left(\\frac{s^{n-m}q_2^{n-m-1}2^{n-m-1}(2(n-m)-3)!!",
"}{r^{2(n-m)-1}}\\right)\\nonumber \\\\&=&\\frac{s^nq_2^{n-1}}{r^{2n-1}}2^{n-2}\\sum \\limits _{m=1}^{n-1}\\binom{n}{m}(2m-3)!!(2(n-m)-3)!!",
".\\nonumber $ This sum is equivalent to m=1n-1nm(2m-3)!!(2(n-m)-3)!!",
"= 1n-1m=1n-1nm(2m-3)!!(2(n-m)-1)!!",
"=1n-1k=0n-2nk+1(2k-1)!!(2(n-k)-3)!!",
"=2(2n-3)!",
"!, where we have used the identity [42], $\\sum \\limits _{k=0}^{n-1}\\binom{n}{k+1}(2k-1)!!(2(n-k)-3)!!=(2n-1)!",
"!.$ Using Eq.",
"() in Eq.",
"(REF ) reproduces Eq.",
"(REF ), thereby completing the proof.",
"We thank Nanxin Wei, Stephanie Miller, Kay Wiese, Ignacio Bordeu Weldt, Eric Smith, David Krakauer, and Nicholas Moloney, for fruitful discussions.",
"We extend our gratitude to Andy Thomas for invaluable computing support."
]
] | 1808.08418 |
[
[
"Optimal conditions for Bell test using spontaneous parametric\n down-conversion sources"
],
[
"Abstract We theoretically and experimentally investigate the optimal conditions for the Bell experiment using spontaneous parametric down conversion (SPDC) sources.",
"In theory, we show that relatively large average photon number (typically $\\sim$0.5) is desirable to observe the maximum violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality.",
"In experiment, we perform the Bell experiment without postselection using polarization entangled photon pairs at 1550 nm telecommunication wavelength generated from SPDC sources.",
"While the violation of the CHSH inequality is not directly observed due to the overall detection efficiencies of our system, the experimental values agree well with those obtained by the theory with experimental imperfections.",
"Furthermore, in the range of the small average photon numbers ($\\leq0.1$), we propose and demonstrate a method to estimate the ideal CHSH value intrinsically contained in the tested state from the lossy experimental data without assuming the input quantum state."
],
[
"Introduction",
"Quantum mechanically entangled photon pairs are essential tools for various optical quantum information and communication protocols [1], [2].",
"Such entangled photon pairs can be generated with spontaneous parametric down conversion (SPDC).",
"To generate perfectly correlated pairs via the SPDC process which is probabilistic, it is frequently driven by weak pumping regime, such that emitted light only contains biphotons (a pair of single photons) and higher-order multi-photon emissions are sufficiently low.",
"This feature is useful when one makes postselection of the coincidence photon counting events.",
"The weakly-pumped SPDC source has also been used in the experiment without postselection.",
"One important example is a loophole-free test of the Bell inequality [3], [4].",
"Violation of the Bell inequality rules out the possibility of describing the correlation between two parties by the local hidden variable model.",
"To observe the genuine quantum correlation directly, it is important that the Bell test is performed without any loopholes, e.g.",
"the detection loophole.",
"In addition, the loophole-free Bell test implies new quantum information applications such as device-independent quantum key distribution (DIQKD) [5], [6] and random number generation [7].",
"So far, in photonic systems, the violation of the Bell inequality closing the detection and locality loopholes [8], [9], [10], [11] have been demonstrated by combining the weakly-pumped SPDC sources and highly efficient detectors.",
"Though these experiments successfully violates the Clauser-Horne-Shimony-Holt (CHSH) inequality [12], the amount of violation was limited to be small ($\\sim 10^{-4}$ ) since the weakly-pumped SPDC source mainly emits vacuum and only a few biphotons.",
"The average photon number is typically in the order of $10^{-2}$ .",
"That is, the major component of the quantum state is vacuum, which does not contribute to yield the violation of the CHSH inequality.",
"In contrast, very recently, larger violation of the CHSH inequality have been reported by using strongly-pumped SPDC sources which produce a non-negligible amount of multiple pairs [13], [14].",
"Moreover, the theoretical analysis [15] considering multi-photon pair emissions of the SPDC sources indicates that the maximum violation of the CHSH inequality is $\\sim 0.35$ which is much larger than those obtained in the previous experiments [8], [9], [10], [11].",
"Thus, further study is required for clarifying the best quantum state which maximizes the CHSH inequality violation.",
"In this paper, both theoretically and experimentally, we elucidate the optimal conditions for SPDC sources to achieve the maximum violation of the CHSH inequality.",
"First we construct a realistic model based on the characteristic function approach, which can take into account higher-order multi-photon pair emissions [16], [17].",
"Then we show the optimal parameters for the system for a given detection efficiency ($\\eta $ ) in detail, including average photon numbers ($\\lambda $ ) of the two SPDC sources and their relative ratio, and optimal measurement angles.",
"It is revealed that the maximal violation is obtained at relatively high average photon number regime where the contribution of multi-photon pair emissions is not negligible: $\\lambda >0.1$ in most cases, and $\\lambda =0.99$ is optimal for $\\eta =1$ .",
"We also show that the measurement angle of the Bell test is almost independent of the detection efficiency.",
"It is noteworthy that this feature allows us to reduce the number of optimization parameters, and therefore is practically useful for saving computational resources.",
"Second, to test the theoretical predictions, we perform the Bell-test experiment without postselections using polarization entangled photon pairs generated by SPDC.",
"We collected all the events including no-detection (vacuum) events, and calculated the CHSH value for each average photon number.",
"While the overall detection efficiencies of our system are insufficient to directly observe the violation of the CHSH inequality, the CHSH values obtained by the experiment well agree with the theory in a wide range of parameters.",
"Furthermore, for the low average photon number regime of $\\lambda \\le 0.1$ , we propose and demonstrate a method to estimate the ideal probability distributions of the Bell test from the lossy experimental data without assuming the input quantum state.",
"The results agree with the theory and thus provide a useful estimation technique for quantum optics experiments with certain amount of losses.",
"The paper is organized as follows.",
"In Sec.",
", we briefly review the Bell test using SPDC sources and describe our theoretical model including higher order photon numbers and experimental imperfections.",
"In Sec.",
", we present our numerical results.",
"The experimental setup is described in Sec. .",
"In Sec.",
", we present our experimental results and introduce the method to compensate the loss of the system.",
"We conclude the paper in Sec.",
"."
],
[
"Bell test via the SPDC sources",
"The schematic diagram of the Bell test is shown in Fig.",
"REF (a).",
"A pair of particles is distributed from the source to two receivers, Alice and Bob.",
"They randomly choose the measurement settings $X_i\\in \\lbrace X_1, X_2\\rbrace $ and $Y_j\\in \\lbrace Y_1, Y_2\\rbrace $ , respectively.",
"All the observables produce binary outcomes $a_i, b_j\\in \\lbrace -1, +1\\rbrace $ .",
"Alice and Bob repeat the measurement, and calculate the CHSH value $S=\\langle a_1b_1 \\rangle +\\langle a_2b_1 \\rangle +\\langle a_1b_2 \\rangle -\\langle a_2b_2 \\rangle ,$ where $\\langle a_ib_j \\rangle =P(a=b|X_i,Y_j)-P(a\\ne b|X_i,Y_j)$ .",
"Here, $S>2$ indicates that the particles shared between Alice and Bob possess nonlocal quantum correlation which cannot be reproduced by any local hidden variables.",
"The maximum value of $S$ allowed by quantum mechanics is $2\\sqrt{2}$ , which is known as the Cirelson bound [18] and achieved by using a maximally entangled pair.",
"Next, the realistic model of the Bell test with SPDCs is shown in Fig.",
"REF (b).",
"The SPDCs emit entangled photon pairs, or more precisely, the two-mode squeezed vacuum (TMSV) whose Hamiltonian is represented by $\\hat{H}=i\\hbar (\\zeta _1\\hat{a}^\\dagger _{H_A}\\hat{a}^\\dagger _{V_B}+\\zeta _2\\hat{a}^\\dagger _{V_A}\\hat{a}^\\dagger _{H_B}-\\rm {h.c.})$ , where $\\hat{a}_{j}^\\dagger $ is the photon creation operator in mode $j$ , and $\\zeta _k=|\\zeta _k|e^{i\\phi _k}$ is the coupling constant of TMSV$k$ ($k=1,2$ ) which is proportional to the complex amplitude of each pump.",
"In the following, $\\phi _k$ is fixed as $\\phi _1=0$ and $\\phi _2=\\pi $ .",
"$H$ and $V$ denote the horizontal and vertical polarizations, respectively.",
"The generated quantum state is described by $| \\Psi _{\\mathrm {ent}} \\rangle &=&\\mathrm {exp}(-i\\hat{H}t/\\hbar )| 0 \\rangle \\\\&=&\\sum ^{\\infty }_{n=0}\\frac{1}{\\mathrm {cosh}r_1\\mathrm {cosh}r_2}\\sqrt{n+1}| \\Phi _n \\rangle ,$ where $| \\Phi _n \\rangle =\\frac{(-i)^n}{n!\\sqrt{n+1}}(\\mathrm {tanh}r_1\\hat{a}^\\dagger _{H_A}\\hat{a}^\\dagger _{V_B}-\\mathrm {tanh}r_2\\hat{a}^\\dagger _{V_A}\\hat{a}^\\dagger _{H_B})^n| 0 \\rangle .$ Here $| 0 \\rangle $ is the vacuum state, and $r_k=|\\zeta _k|t$ is the squeezing parameter of TMSV$k$ .",
"Note that the average photon number of TMSV$k$ is given by $\\lambda _k=\\mathrm {sinh}^2r_k$ .",
"The state clearly consists of an infinite series and the contribution from higher order photon numbers cannot be negligible even with finite $\\lambda _k$ .",
"The polarizer with angle $\\theta $ works as a polarization-domain beamsplitter mixing the $H$ and $V$ modes where its transmittance and reflectance are $\\mathrm {cos}^2\\theta $ and $\\mathrm {sin}^2\\theta $ , respectively.",
"The overall detection efficiencies including the system transmittance and the imperfect quantum efficiencies of the detectors are denoted by $\\eta _l$ for $l=1, 2, 3, 4$ .",
"This is modeled by inserting the losses in each arm before the detectors with unit efficiency (see Fig.",
"REF (b)).",
"We consider that the detectors D1, D2, D3 and D4 are on-off type, single photon detectors with dark counts which only distinguish between vacuum (off: no-click) and non-vacuum (on: click) with dark count probability of $\\nu $ ."
],
[
"Numerical results",
"To numerically calculate $S$ in Eq.",
"(REF ) with the above SPDC model, we use the approach based on the characteristic function [16], [17].",
"This approach is applicable when the system is composed of Gaussian states and operations, and on-off detectors.",
"The Gaussian state is defined by the state whose characteristic function (or equivalently Wigner function) has a Gaussian distribution, including TMSV states.",
"The Gaussian operation is also defined as an operation transforming a Gaussian state to another Gaussian state, which includes the operations by linear optics and second-order nonlinear processes.",
"The setup in Fig.",
"REF (b) includes only these means and thus meets the condition above.",
"See Appendix and Ref.",
"[16] for more details of this approach.",
"Note that a similar calculation with a different approach is reported in Ref. [15].",
"We calculate the probability of all the combinations of the photon detection (click) and no-detection (no-click) events for each polarizer angle, and obtain the probability distributions.",
"We denote, for example, the probability of observing clicks in D1 and D2, and no-clicks in D3 an D4 as $P(\\mathrm {c1,c2,nc3,nc4})$ .",
"Each of Alice and Bob determines her/his local rule, and assign +1 or $-1$ for each detection event.",
"Since there are four possible local events for each of Alice and Bob, i.e., (i) only the one detector clicks, (ii) only the other detector clicks, (iii) both of the two detectors simultaneously click and (iv) no detector clicks, there are 16 possible choices for each of Alice and Bob to assign $\\pm 1$ .",
"We introduce the following simple local assignment strategy for Alice (Bob): only D1(D2) clicks$\\rightarrow -1$ and otherwise $\\rightarrow +1$ , respectively.",
"Under the condition that Alice (Bob) chooses the angle $\\theta _{A1}~(\\theta _{B1})$ , respectively, the probability that both of Alice and Bob obtain the outcome $-1$ is calculated by $P(-1,-1|\\theta _{A1},\\theta _{B1})$ =$P(\\mathrm {c1,c2,nc3,nc4})$ .",
"Similarly, the other conditional probabilities $P(+1,-1|\\theta _{A1},\\theta _{B1})$ , $P(-1,+1|\\theta _{A1},\\theta _{B1})$ and $P(+1,+1|\\theta _{A1},\\theta _{B1})$ are also calculated by the detection probabilities, which enables us to calculate $S$ .",
"See Appendix for the details of the formulas.",
"Figure: The setup for the Bell experiment.",
"To generate entangled photon pairs by SPDC,we used counter propagating pump pulses to excite the PPKTP crystal in the Sagnac loop interferometer.Alice and Bob choose the measurement angles{θ A1 ,θ A2 }\\lbrace \\theta _{A1},\\theta _{A2}\\rbrace and {θ B1 ,θ B2 }\\lbrace \\theta _{B1},\\theta _{B2}\\rbrace , respectively,and assign +1 or -1 for the each detection event tocalculate SS value.",
"DFB: distributed feedback laser, EDFA: erbium-doped fiber amplifier,PPLN/W: periodically poled lithium niobate waveguide, PPKTP: periodically poled potassium titanyl phosphate,QWP: quarter waveplate, HWP: half waveplate, DM: dichroic mirror, PBS: polarization beamsplitter, FPBS:fiber-based PBS.Fig.",
"REF (a) shows the relation between the average photon number and $S$ in an ideal system, where all the detection efficiencies are unity and detectors have no dark counts (i.e.",
"$\\eta _1=\\eta _2=\\eta _3=\\eta _4=1$ and $\\nu =0$ ).",
"We define the larger of $\\lambda _1$ and $\\lambda _2$ as $\\lambda $ , and then for given $\\lambda $ , numerically optimize the other average photon number and $\\lbrace \\theta _{A1(B1)}, \\theta _{A2(B2)}\\rbrace $ via the Nelder-Mead method such that $S$ is maximized.",
"$S$ at the maximum violation is around 2.31, which coincides with the theoretical result by Vivoli et al. [15].",
"We found that the maximum violation is obtained at $\\lambda =0.99$ which is much larger than those used in the previous experiments [8], [9], [10], [11].",
"Note that the maximum $S$ obtained in Fig.",
"REF (a) is robust against the dark counts.",
"In fact, when we add a dark count probability of $\\nu =10^{-4}$ to each detector as a realistic value, degradation of $S$ was as small as $5.0\\times 10^{-4}$ .",
"Next, we show the loss tolerance of $S$ for the three different ranges of $\\lambda $ in Fig.",
"REF (b).",
"In the simulation, we assumed that $\\eta _1=\\eta _2=\\eta _3=\\eta _4:=\\eta $ and $\\nu =0$ .",
"$\\lambda _1$ , $\\lambda _2$ and the measurement angles are optimized for each $\\eta $ .",
"The red thick curve, blue dashed curve, and yellow thin curve represent $S$ optimized under the conditions of $0\\le \\lambda $ , $0\\le \\lambda \\le 0.1$ , and $0\\le \\lambda \\le 0.01$ , respectively.",
"The figure shows that the limited $\\lambda $ strongly restricts the maximum $S$ in any $\\eta $ .",
"The result indicates that the maximum $S$ obtainable in the previous Bell experiments using SPDC sources with small $\\lambda $ is intrinsically limited and thus suggests a use of higher pumping of the SPDC sources to obtain larger CHSH violation.",
"Finally, we show the optimal parameters for given $\\eta $ in Figs.",
"REF (a)-(c).",
"The optimal average photon numbers are shown in Fig.",
"REF (a).",
"Even when $\\eta =1$ , the two optimal average photon numbers are unbalanced.",
"The ratio between $\\lambda _1$ and $\\lambda _2$ is shown in Fig.",
"REF (b).",
"We found that the ratio of $\\lambda _1/\\lambda _2$ monotonically and continuously decreases as $\\eta $ decreases.",
"This result qualitatively agrees with the analysis based on qubit systems in Ref. [19].",
"The optimal angles of the polarizers are shown in Fig.",
"REF (c).",
"Interestingly, the optimal angles are almost constant regardless of $\\eta $ ."
],
[
"Experimental setup",
"Theoretical predictions in the above section are verified using the experimental setup illustrated in Fig.",
"REF .",
"We choose the measurement angles as $\\lbrace \\theta _{A1},\\theta _{A2}\\rbrace =\\lbrace 0, \\pi /5\\rbrace $ and $\\lbrace \\theta _{B1},\\theta _{B2}\\rbrace =\\lbrace 3\\pi /5, -3\\pi /5\\rbrace $ by which $S$ is expected to be $S=2.30$ with $\\lambda _1=\\lambda _2=0.62$ when the overall detection efficiency is unity and the dark count probabilities are zero.",
"These angles are slightly different from those shown in Fig.",
"REF (b) since we apply the condition $\\lambda _1=\\lambda _2$ for simplicity.",
"A distributed feedback (DFB) laser generates pulsed light at 1550 nm.",
"The DFB laser is directly modulated by electrical pulses with 100 kHz repetition and 300 ns duration.",
"The output laser pulse is amplified by an erbium-doped fiber amplifier (EDFA).",
"The output of EDFA is vertically polarized by a half-waveplate (HWP) and a polarizing beamsplitter (PBS), and then coupled to the 34 mm-long type-0 periodically poled lithium niobate waveguide (PPLN/W) for second harmonic generation (SHG).",
"Amplified spontaneous emission from the EDFA and unconverted fundamental light of the SHG are removed by the dichroic mirrors (DMs).",
"The polarization of the SHG pulses are adjusted by using a HWP and a pair of quarter waveplates (QWPs).",
"The maximum pulse energy (average power) of our SHG pulses is 0.2 $\\mu $ J (20 mW).",
"To generate polarization entangled photon pairs by SPDC process, SHG pulses are used to pump a 30 mm-long, type II, periodically poled potassium titanyl phosphate (PPKTP) crystal in a Sagnac loop interferometer with a PBS [20].",
"The two-qubit component of the generated state forms a maximally entangled state $| \\Psi ^- \\rangle =(| HV \\rangle -| VH \\rangle )/\\sqrt{2}$ , where $| H \\rangle $ and $| V \\rangle $ denote the $H$ and $V$ polarization state of a single photon, respectively.",
"One half of the photon pair passes through the DM and goes to Alice's side while the other photon goes to Bob's side.",
"Alice and Bob set measurement angles $\\lbrace \\theta _{A1},\\theta _{A2}\\rbrace $ and $\\lbrace \\theta _{B1},\\theta _{B2}\\rbrace $ , respectively, by means of the HWPs and fiber-based PBSs (FPBSs).",
"Finally, the photons are detected by four superconducting single photon detectors (SSPDs) D1 and D3 for Alice, and D2 and D4 for Bob, respectively.",
"The quantum efficiencies of these SSPDs are around $75~\\%$ [21].",
"The dark count probabilities of the SSPDs are $3.0\\times 10^{-4}$ per a detection window of 300 ns corresponding to the pulse duration.",
"The modulation signal for the DFB laser is also used as a start signal for a time-to-digital converter (TDC), and the detection signals from D1, D2, D3 and D4 are used as stop signals of the TDC.",
"All combination of click and no-click events are collected without postselection.",
"We assign events of D1 (D2) clicks on Alice's (Bob's) side as -1 and all the others as +1, then calculate $S$ ."
],
[
"Experimental results",
"Before performing the Bell-test experiment, we estimate the overall detection efficiencies $\\eta _l$ .",
"Suppose a TMSV is detected by two detectors, D1 and D2.",
"The overall detection efficiencies of the two modes ($\\eta _1$ and $\\eta _2$ ) are well estimated by following equation [22], $\\eta _{1(2)}=\\frac{C_{12}}{S_{2(1)}}.$ Here $C_{12}$ is the coincidence count between D1 and D2, and $S_{2(1)}$ is the single detection count at D2(1).",
"Note that the average photon number of the TMSV photons is small enough for this measurement.",
"In our theoretical model shown in Fig.",
"REF (b), we have assumed the same detection efficiencies for TMSV1 and TMSV2.",
"Thus, in the experiment, we carefully align the optical system such that the overall detection efficiencies for TMSV1 and TMSV2 are the same as each other.",
"We estimated them as $\\eta _1=10.48\\pm 0.69~\\%$ , $\\eta _2=12.76\\pm 0.97~\\%$ , $\\eta _3=12.72\\pm 0.53~\\%$ and $\\eta _4=11.86\\pm 0.24~\\%$ .",
"Once $\\eta _l$ is estimated, the average photon number ($\\lambda _k$ ) of TMSV$k$ is calculated by using following relation: $\\frac{S_{1(2)}}{N}=\\frac{\\lambda _k\\eta _{1(2)}}{1+\\lambda _k\\eta _{1(2)}},$ where $N$ is the number of the total events, which corresponds to the number of the start signals of the TDC.",
"In the Bell-test experiment, the difference between $\\lambda _1$ and $\\lambda _2$ is set to be less than 1 %.",
"Thus we denote $\\lambda _1=\\lambda _2:=\\lambda $ in the following.",
"The results of the Bell experiment is shown in Fig.",
"REF (a).",
"We perform the Bell experiment for various values of $\\lambda $ by changing the energy of the pump pulse.",
"Though the overall detection efficiencies of our system are not in the range of directly observing the CHSH violation, it is still possible to compare our experimental results and the theory calculated with experimentally observed parameters: the average photon numbers, measurement angles, detection efficiencies and dark counts.",
"The experimental results (blue triangles) and theoretical values with experimental parameters (yellow circles) are in good agreement for each $\\lambda $ , which indicates that the theoretical model well explain the experimental results.",
"Figure: (a) The SS values obtained bythe theory with unity detection efficiencies (red square),the theory with experimental parameters (yellow circle) andthe experimental results (blue triangle) forthe various values of λ\\lambda .The black diamonds represent the SS values obtained by compensating the losses of the system in the range ofλ≤0.1\\lambda \\le 0.1.",
"(b) The enlarged figure of the enclosed part.In the low average photon number regime ($\\lambda \\le 0.1$ ), it is possible to compensate the imperfection of the overall detection efficiencies without assuming the quantum states distributed to Alice and Bob.",
"In other words, one can estimate the intrinsic nonlocality that could be observed with the unity detection efficiencies.",
"Under the assumption that each detector detects at most one photon, the experimentally-obtained probability distribution $\\mathbf {P}_{\\mathrm {exp}}=(p_1, p_2,\\cdots , p_{16})^T$ composed of the 16 combinations of the detection probabilities and the ideal probability distribution with the unity detection efficiencies $\\mathbf {Q}_{\\mathrm {ideal}}=(q_1, q_2,\\cdots , q_{16})^T$ are connected by the linear transmission matrix $\\mathbf {T}$ as $\\mathbf {P}_{\\rm exp}=\\mathbf {T}\\mathbf {Q}_{\\rm ideal}$ for each measurement setting.",
"Here, $p_1=P$ (nc1,nc2,nc3,nc4), $p_2=P$ (c1,nc2,nc3,nc4), and so on.",
"$\\mathbf {T}$ is the upper triangular matrix whose matrix elements are composed of the products of $\\eta _l$ and $(1-\\eta _l)$ .",
"For example, the four-fold coincidence probabilities $p_{16}$ and $q_{16}$ are connected by $p_{16}=q_{16}\\prod _{i=1}^{4}\\eta _i$ .",
"One may think that $\\mathbf {Q}_{\\rm ideal}$ is estimated by simply calculating $\\mathbf {Q}_{\\rm ideal}=\\mathbf {T}^{-1}\\mathbf {P}_{\\rm \\mathrm {exp}}$ .",
"However, in this case, the elements of $\\mathbf {Q}_{\\rm ideal}$ could be negative since $\\mathbf {P}_{\\rm exp}$ contains experimental errors.",
"Thus we determine the most likely elements of $\\mathbf {Q}_{\\rm ideal}$ such that the $L^2$ distance between $\\mathbf {P}_{\\rm exp}$ and $\\mathbf {T}\\mathbf {Q}_{\\rm ideal}$ is minimum under the condition that $q_i\\ge 0$ and $\\sum _{i=1}^{16} q_i=1$ .",
"Namely, we estimate the probability distribution $\\mathbf {Q}_{\\rm ideal}$ which minimizes the function: $\\sum _{i=1}^{16}(p_i-\\sum _{j=1}^{16}T_{ij}q_j)^2.$ The $S$ values calculated by $\\mathbf {Q}_{\\rm ideal}$ is shown in Fig.",
"REF (a) and (b) by the black triangles.",
"The results agree with, but slightly below the theory plots for ideal state and detectors (red square), which reflects the deviation of the generated state from ideal TMSVs due to experimental imperfections.",
"In particular, these two plots start to deviate in $\\lambda >0.05$ where the probability of detecting multi-photon at each detector starts to be non-negligible."
],
[
"Conclusion",
"In conclusion, we theoretically and experimentally investigate the optimal conditions for the Bell test with the SPDC sources.",
"We construct a numerical model including multi-photon emissions from the SPDC sources and various imperfections, and see the maximum violation of the CHSH inequality as $S=2.31$ which agrees with the previous result in Ref. [15].",
"Then we show that the optimal experimental parameters to maximize the CHSH values for given average photon number of TMSV or the overall detection efficiency by numerical simulations.",
"In particular, we show the CHSH value takes its maximum when the average photon number is much larger than those utilized in the previous experiments [8], [9], [10].",
"Next, we perform the Bell-test experiment without postselection using polarization entangled photon pairs generated by SPDC to test these theoretical predictions.",
"The experimentally-obtained CHSH values agree well with those obtained by the theory.",
"Moreover, in the range of small average photon numbers, we also propose and demonstrate a method to estimate the CHSH value of the quantum state before undergoing losses, by compensating the detection losses without assuming the input quantum state.",
"The result shows good agreement with the theory model in the range of $\\lambda \\le 0.1$ .",
"This approach is useful to estimate the property of quantum states via imperfect detectors.",
"We thank Kaushik P. Seshadreesan for helpful discussions.",
"This work was supported by JST CREST Grant No.",
"JPMJCR1772, MEXT/JSPS KAKENHI Grant No.",
"JP18K13487 and No. JP17K14130.",
"*"
],
[
"Detailed calculations based on the characteristic function",
"We describe a procedure to calculate the probability distributions and the CHSH value $S$ using the theoretical model given in Sec.. First, we review the basic tools used in the characteristic function approach which is often used in Gaussian continuous-variable quantum systems.",
"This method allows us to deal with the quantum state generated by the SPDC process without the need for any approximations such as photon number truncation.",
"Next, we present the method to calculate the detection probabilities.",
"Finally we describe the procedure to calculate $S$ using the obtained probability distribution."
],
[
"${Characteristic}$ {{formula:9fdf9d86-73ec-4011-ae52-6d23fb8ef092}}",
"Let us consider $N$ bosonic modes associated with a tensor product Hilbert space ${\\mathcal {H}^{\\otimes N}}=\\bigotimes _{j=1}^N\\mathcal {H}_j$ , where $\\mathcal {H}_j$ is an infinite dimensional Hilbert space.",
"We define annihilation and creation operators corresponding each mode as $\\hat{a}_j$ and $\\hat{a}^\\dagger _j$ , respectively.",
"They satisfy the commutation relation given by $[\\hat{a}_j,\\hat{a}^\\dagger _k]=\\delta _{jk}.$ We also define the quadrature operators of a bosonic mode as $\\hat{x}_j=\\frac{1}{\\sqrt{2}}(\\hat{a}_j+\\hat{a}^\\dagger _j),\\\\\\hat{p}_j=\\frac{1}{\\sqrt{2}i}(\\hat{a}_j-\\hat{a}^\\dagger _j).$ Note that we choose as a convention $\\hbar =\\omega =1$ .",
"Their commutation relation is calculated as $[\\hat{x}_j,\\hat{p}_k]=i\\delta _{jk}.$ We define a density operator acting on $\\mathcal {H}^{\\otimes N}$ as $\\hat{\\rho }$ .",
"The characteristic function of $\\hat{\\rho }$ is defined by $\\chi (\\xi )=\\mathrm {Tr}[\\hat{\\rho }\\hat{\\mathcal {W}}(\\xi )],$ where $\\hat{\\mathcal {W}}(\\xi )=\\mathrm {exp}(-i\\xi ^T\\hat{R})$ is the Weyl operator.",
"Here, $\\hat{R}=(\\hat{x}_1,\\dots ,\\hat{x}_N,\\hat{p}_1,\\dots ,\\hat{p}_N)$ and $\\xi =(\\xi _1,\\dots ,\\xi _{2N})$ are a 2$N$ vector consisting of quadrature operators and a 2$N$ real vector, respectively.",
"A Gaussian state is a quantum state whose characteristic function has a Gaussian distribution: $\\chi (\\xi )=\\mathrm {exp}(-\\frac{1}{4}\\xi ^T\\gamma \\xi -id^T\\xi ),$ where $\\gamma $ is a 2$N$ $\\times $ 2$N$ matrix called the covariance matrix and $d$ is a 2$N$ -dimensional vector known as the displacement vector.",
"The covariance matrix of the TMSV state generated by a SPDC source is given by $\\gamma ^{\\mathrm {TMSV}}(\\lambda )=\\left[\\begin{array}{cc}\\gamma ^+(\\lambda )&{0}\\\\{0}&\\gamma ^-(\\lambda )\\end{array}\\right],$ where $\\gamma ^{\\mathrm {\\pm }}=\\left[\\begin{array}{cc}2\\lambda +1&\\pm 2\\sqrt{\\lambda (\\lambda +1)}\\\\\\pm 2\\sqrt{\\lambda (\\lambda +1)}&2\\lambda +1\\end{array}\\right],$ while $d=0$ .",
"As is described in Sec.",
", $\\lambda =\\mathrm {sinh}^2r$ corresponds to the average photon number per mode.",
"Gaussian unitary operation is defined as an unitary operation transforming a Gaussian state to another Gaussian state, which includes the operations by linear optics and the second-order nonlinear process.",
"Any Gaussian unitary operation acting on a Gaussian state is characterized by the following symplectic transformations: $\\gamma \\rightarrow S^T\\gamma S,~~d\\rightarrow S^Td,$ where $S$ is a symplectic matrix corresponding to the Gaussian unitary operation.",
"The symplectic matrix for a beamsplitter on mode A and mode B is given by $S_{AB}^t=\\left[\\begin{array}{cccc}\\sqrt{t}&\\sqrt{1-t}&0&0\\\\-\\sqrt{1-t}&\\sqrt{t}&0&0\\\\0&0&\\sqrt{t}&\\sqrt{1-t}\\\\0&0&-\\sqrt{1-t}&\\sqrt{t}\\end{array}\\right],$ where $t$ is the transmittance of the beamsplitter.",
"Hereafter, we simplify the description of a block diagonalized matrix like Eq.",
"(REF ) as $S_{AB}^t=\\left[\\begin{array}{cc}\\sqrt{t}&\\sqrt{1-t}\\\\-\\sqrt{1-t}&\\sqrt{t}\\end{array}\\right]^{\\oplus 2}.$ We consider that the detectors D1, D2, D3 and D4 in Fig.",
"REF (b) are on-off type, single photon detectors, namely, they only distinguish between vacuum and non-vacuum.",
"Denoting the dark count probability of the detectors by $\\nu $ , the POVM elements of the on-off detectors are described by $\\hat{\\Pi }^{\\rm {off}}(\\nu )=(1-\\nu )| 0 \\rangle \\langle 0 |$ and $\\hat{\\Pi }^{\\rm {on}}(\\nu )=\\hat{I}-\\hat{\\Pi }^{\\rm {off}}(\\nu ),$ where $\\hat{I}$ is the identity operator.",
"The detection probability is calculated by introducing the characteristic functions of the POVM elements.",
"Similar to the state, the characteristic function of the POVM element $\\hat{\\Pi }$ is given by $\\chi _{\\Pi }(\\xi )=\\mathrm {Tr}[\\hat{\\Pi }\\hat{\\mathcal {W}}(\\xi )]$ .",
"When a single-mode Gaussian state $\\hat{\\rho }$ with characteristic function $\\chi _\\rho (\\xi )=\\mathrm {exp}(-\\frac{1}{4}\\xi ^T\\gamma \\xi )$ is measured, the detection probability is given by $P_{\\mathrm {on}}&=&\\mathrm {Tr}[\\hat{\\rho }\\hat{\\Pi }^{\\mathrm {on}}]=1-(1-\\nu )\\mathrm {Tr}[\\hat{\\rho }| 0 \\rangle \\langle 0 |]\\nonumber \\\\&=&1-\\frac{2(1-\\nu )}{\\sqrt{\\mathrm {det}(\\gamma +I)}}.$ The linear photon losses in the system such as a coupling efficiency to the single-mode fiber and imperfect quantum efficiency of the detectors are modeled by performing a beamsplitter transformation of transmittance $t$ between the lossy mode and a vacuum mode, and tracing out the vacuum mode.",
"The transformation of the linear loss on the state with covariance matrix $\\gamma $ can be described as $\\mathcal {L}^t\\gamma =K^T\\gamma K+\\alpha ,$ where $K=\\sqrt{t}I$ and $\\alpha =(1-t)I$ ."
],
[
"Detection probabilities",
"Using the above basic tools, we present the procedure to calculate the detection probabilities.",
"As shown in Fig.",
"REF (b) the entangled photon pair source consists of two TMSV sources over polarization modes.",
"The covariance matrix of the output state is given by $\\gamma _{{H_AV_AH_BV_B}}^{\\mathrm {TMSV12}}=\\left[\\begin{array}{cccc}2\\lambda _1+1&2\\sqrt{\\lambda _1(\\lambda _1+1)}&0&0\\\\2\\sqrt{\\lambda _1(\\lambda _1+1)}&2\\lambda _1+1&0&0\\\\0&0&2\\lambda _2+1&-2\\sqrt{\\lambda _2(\\lambda _2+1)}\\\\0&0&-2\\sqrt{\\lambda _2(\\lambda _2+1)}&2\\lambda _2+1\\end{array}\\right]^{\\oplus 2}.$ Note that the relative phase between TMSV1 and TMSV2 is set to $\\pi $ as described in Eq.",
"(REF ).",
"In the experiment, the two TMSV sources are embedded in the Sagnac loop.",
"In this case, the covariance matrix of the output state is transformed into [16] $\\gamma _{{H_AV_AH_BV_B}}^{\\mathrm {SL}}=\\left[\\begin{array}{cccc}2\\lambda _1+1&0&0&2\\sqrt{\\lambda _1(\\lambda _1+1)}\\\\0&2\\lambda _2+1&-2\\sqrt{\\lambda _2(\\lambda _2+1)}&0\\\\0&-2\\sqrt{\\lambda _2(\\lambda _2+1)}&2\\lambda _2+1&0\\\\2\\sqrt{\\lambda _1(\\lambda _1+1)}&0&0&2\\lambda _1+1\\end{array}\\right]^{\\oplus 2}.$ The covariance matrix in Eq.",
"(REF ) is first transformed by the (polarization-domain) beamsplitter operations $S^{\\theta _A}_{H_AV_A}S^{\\theta _B}_{H_BV_B}$ .",
"The covariance matrix after the transformation is given by $\\gamma _{{H_AV_AH_BV_B}}^{\\mathrm {BS}}:=S^{\\theta _B~T}_{H_BV_B}S^{\\theta _A~T}_{H_AV_A}\\gamma _{{H_AV_AH_BV_B}}^{\\mathrm {SL}}S^{\\theta _A}_{H_AV_A}S^{\\theta _B}_{H_BV_B}.$ The overall system losses including imperfect quantum efficiencies of the detectors are considered by performing linear-loss operations $\\mathcal {L}^{\\eta _1}_{H_A}$ , $\\mathcal {L}^{\\eta _3}_{V_A}$ , $\\mathcal {L}^{\\eta _2}_{H_B}$ and $\\mathcal {L}^{\\eta _4}_{V_B}$ on corresponding modes.",
"The covariance matrix just before the detectors is given by $\\gamma _{{H_AV_AH_BV_B}}^{\\mathrm {final}}&=&\\mathcal {L}^{\\eta _1}_{H_A}\\mathcal {L}^{\\eta _3}_{V_A}\\mathcal {L}^{\\eta _2}_{H_B}\\mathcal {L}^{\\eta _4}_{V_B} \\gamma _{{H_AV_AH_BV_B}}^{\\mathrm {BS}}\\\\&=&K^{\\eta _1\\eta _3\\eta _2\\eta _4~T}_{H_AV_AH_BV_B} \\gamma _{{H_AV_AH_BV_B}}^{\\mathrm {BS}}K^{\\eta _1\\eta _3\\eta _2\\eta _4}_{H_AV_AH_BV_B}+\\alpha ^{\\eta _1\\eta _3\\eta _2\\eta _4}_{H_AV_AH_BV_B}$ where $K^{\\eta _1\\eta _3\\eta _2\\eta _4}_{H_AV_AH_BV_B}= \\left[\\begin{array}{cccc}\\sqrt{\\eta _1}&0&0&0\\\\0&\\sqrt{\\eta _{3}}&0&0\\\\0&0&\\sqrt{\\eta _2}&0\\\\0&0&0&\\sqrt{\\eta _4}\\end{array}\\right]^{\\oplus 2}.$ and $\\alpha ^{\\eta _1\\eta _3\\eta _2\\eta _4}_{H_AV_AH_BV_B}= \\left[\\begin{array}{cccc}1-\\eta _1&0&0&0\\\\0&1-\\eta _3&0&0\\\\0&0&1-\\eta _2&0\\\\0&0&0&1-\\eta _4\\end{array}\\right]^{\\oplus 2}.$ The detection probabilities are calculated by performing $\\hat{\\Pi }^{\\rm {on}/\\rm {off}}(\\nu )$ on corresponding modes.",
"For example, the probability of observing clicks in D1 and D2 and no-clicks in D3 and D4 is given by $&P(\\mathrm {c1,c2,nc3,nc4}|\\theta _{A},\\theta _{B})=\\mathrm {Tr}[\\rho ^{\\gamma _{{H_AV_AH_BV_B}}^{\\mathrm {final}}}\\hat{\\Pi }^{\\rm {on}}_{H_A}(\\nu )\\hat{\\Pi }^{\\rm {on}}_{H_B}(\\nu )\\hat{\\Pi }^{\\rm {off}}_{V_A}(\\nu )\\hat{\\Pi }^{\\rm {off}}_{V_B}(\\nu )]\\\\&=\\mathrm {Tr}[\\rho ^{\\gamma _{{H_AV_AH_BV_B}}^{\\mathrm {final}}}(\\hat{I}-(1-\\nu )| 0 \\rangle \\langle 0 |_{H_A})(\\hat{I}-(1-\\nu )| 0 \\rangle \\langle 0 |_{H_B})(1-\\nu )| 0 \\rangle \\langle 0 |_{V_A}(1-\\nu )| 0 \\rangle \\langle 0 |_{V_B}]\\\\&=\\frac{4(1-\\nu )^2}{\\sqrt{\\mathrm {det}(\\gamma _{V_AV_B}^{\\mathrm {final}}+I)}}-\\frac{8(1-\\nu )^3}{\\sqrt{\\mathrm {det}(\\gamma _{H_AV_AV_B}^{\\mathrm {final}}+I)}}\\nonumber \\\\&-\\frac{8(1-\\nu )^3}{\\sqrt{\\mathrm {det}(\\gamma _{H_BV_AV_B}^{\\mathrm {final}}+I)}}+\\frac{16(1-\\nu )^4}{\\sqrt{\\mathrm {det}(\\gamma _{{H_AV_AH_BV_B}}^{\\mathrm {final}}+I)}}.$"
],
[
"Calculation of $S$",
"As in Eq.",
"(REF ), $S$ is obtained by calculating $P(a=b|\\theta _{A_i},\\theta _{B_j})$ and $P(a\\ne b|\\theta _{A_i},\\theta _{B_j})$ for $i, j=\\lbrace 1,2\\rbrace $ .",
"For simplicity, omitting the conditions of the angles, these conditional probabilities are given by $P(a=b)=P(+1,+1)+P(-1,-1)$ and $P(a\\ne b)=P(+1,-1)+P(-1,+1).$ In our model, each probability in the right hand side of Eq.",
"(REF ) and Eq.",
"(REF ) is calculated as follows: $P(-1,-1)&=&P(\\mathrm {c1,c2,nc3,nc4}), \\\\P(+1,-1)&=&P(\\mathrm {c1,c2,c3,nc4})+P(\\mathrm {nc1,c2,c3,nc4})\\nonumber \\\\&&+P(\\mathrm {nc1,c2,nc3,nc4}),\\\\P(-1,+1)&=&P(\\mathrm {c1,c2,nc3,c4})+P(\\mathrm {c1,nc2,nc3,c4})\\nonumber \\\\&&+P(\\mathrm {c1,nc2,nc3,nc4}),\\\\P(+1,+1)&=&1-P(-1,-1)\\nonumber \\\\&&-P(+1,-1)-P(-1,+1).$"
]
] | 1808.08370 |
[
[
"Models of semiconductor quantum dots blinking based on spectral\n diffusion"
],
[
"Abstract Three models of single colloidal quantum dot emission fluctuations (blinking) based on spectral diffusion were considered analytically and numerically.",
"It was shown that the only one of them, namely the Frantsuzov and Marcus model reproduces the key properties of the phenomenon.",
"The other two models, the Diffusion-Controlled Electron Transfer (DCET) model and the Extended DCET model predict that after an initial blinking period, most of the QDs should become permanently bright or permanently dark which is significantly different from the experimentally observed behavior."
],
[
"Introduction",
"Two decades have passed since the first observation of long-term fluorescence intensity fluctuations (blinking) of single colloidal CdSe quantum dots (QDs) with a ZnS shell [1].",
"In further experimental studies it was found (see [2], [3], [4], [5], [6], [7], [8], [9] and references therein) that these fluctuations have a wide spectrum of characteristic timescales, from hundreds of microseconds to hours.",
"The intensity traces (binned photon counting data) of CdSe/ZnS core/shell dots show the following key properties: 1.",
"The intensity distribution usually has two maxima, so-called ON and OFF intensity levels 2.",
"The ON-time and OFF-time distributions obtained by the threshold procedure have the truncated power-law form $p(t)\\sim t^{-m} \\exp (-t/T)$ 3.",
"The power spectral density of the trace has a $1/f^r$ dependence, where $r$ value is around 1.",
"This dependence changes to $f^{-2}$ at large frequencies [10].",
"Figure: The schematic picture of the DCET model.The potential surfaces of the neutral (bright) and the charged (dark) electronic states are represented bythe red and blue lines, respectively.",
"Vertical dotted line corresponds to the crossing point.Another interesting phenomenon that manifests in the emission of single quantum dots is the spectral diffusion showing characteristic time scales in the order of hundreds of seconds [11], [12].",
"It is not surprising that there are a number of models proposed to explain the blinking that relate the fluctuations in the emission intensity with slow variations in the exciton energy.",
"The first model of that kind suggested by Shimizu et al.",
"[13] is based on the Efros/Rosen charging mechanism (CM) [14].",
"The CM attributes the ON and OFF periods to neutral and charged QDs, respectively.",
"The light-induced electronic excitation in the charged QD is supposed to be quenched by a fast Auger recombination process.",
"The model of Shimizu et al.",
"[13] assumes that the charging/discharging events happen when the energies of the neutral exciton and the charged state are in resonance.",
"A more advanced version of this idea was used by Tang and Marcus in the DCET model [15], [16].",
"In 2014 Zhu and Marcus [17] presented an extension of the DCET model by introducing an additional biexciton charging channel.",
"Simultaneously with Tang and Marcus [15], [16], another diffusion model based on the alternative fluctuating rate mechanism (FRM) of blinking was suggested by Frantsuzov and Marcus [18].",
"The FRM assumes that the non-radiative relaxation rate of the exciton is subject to long term fluctuations caused by the rearrangement of surface atoms.",
"A basic life cycle of the QD within this mechanism begins with a photon absorption.",
"A relaxation of the excited state can go in one of of two paths.",
"The first path is relaxation via a photon emission.",
"The second path is a hole trapping followed by a consequent non-radiative recombination with a remaining electron.",
"The photoluminescence quantum yield (PLQY) of the QD emission in this case can be expressed as $Y(t) = \\frac{k_r}{k_r+k_t(t)}\\equiv k_r \\tau _{av}$ where $k_r$ is the radiative recombination rate, and $\\tau _{av}$ is the averaged exciton lifetime.",
"Thus the variations of the $k_t$ generate fluctuations of the emission intensity on a long time scale.",
"The Frantsuzov and Marcus model [18] connects the recombination rate with the fluctuating energy difference between 1S$_e$ and 1P$_e$ states.",
"In this article we are going to discuss the advantages and disadvantages of these models of single QD blinking based on spectral diffusion as well as their perspectives of further development."
],
[
"Diffusion-controlled electron transfer model",
"After introducing the Marcus reaction coordinate $Q$ , DCET model equations describing the evolution of its probability distribution density in the neutral state $\\varrho _1(Q,t)$ and in the charged state $\\varrho _2(Q,t)$ can be written in the following form: $\\frac{\\partial }{\\partial t}\\varrho _1(Q,t)= D_1 \\frac{\\partial }{\\partial Q}\\left(\\frac{\\partial }{\\partial Q}+ \\frac{U_1^{\\prime }(Q)}{kT} \\right) \\varrho _1(Q,t)$ $-2\\pi \\frac{V^2}{\\hbar }\\delta (U_1(Q)-U_2(Q))\\left(\\varrho _{1}(Q,t)-\\varrho _{2}(Q,t)\\right)$ $\\frac{\\partial }{\\partial t}\\varrho _2(Q,t)= D_2 \\frac{\\partial }{\\partial Q}\\left(\\frac{\\partial }{\\partial Q}+ \\frac{U_2^{\\prime }(Q)}{kT}\\right) \\varrho _2(Q,t)$ $-2\\pi \\frac{V^2}{\\hbar }\\delta (U_1(Q)-U_2(Q))\\left(\\varrho _{2}(Q,t)-\\varrho _{1}(Q,t)\\right),$ where $D_1$ and $D_2$ are diffusion coefficients in the neutral electronic state and charged state respectively, $V$ is the electronic coupling matrix element between the neutral and charged states, and $T$ is the effective temperature.",
"The potential surfaces of the neutral $U_1(Q)$ and charged $U_2(Q)$ states are Marcus' parabolas (see Fig.",
"REF ): $U_1(Q)=\\frac{(Q+E_r)^2}{4 E_r} \\qquad U_2(X)=\\frac{(Q-E_r)^2}{4 E_r}+\\Delta G$ characterized by the reorganization energy $E_r$ and the free energy gap $\\Delta G$ .",
"Transitions between the neutral and charged states are determined by the delta-functional sink in the crossing point $Q_c$ (local Golden rule), where $U_1(Q_c)=U_2(Q_c)$ $Q_c=\\Delta G$ Equations (REF -REF ) were initially introduced in 1980 independently by Zusman [19] and Burshten and Yakobson [20] for describing solvent effects in electron transfer reactions.",
"In the literature they are usually called Zusman equations (see for example the review article [21] and references therein).The rigorous derivation of the Eqs.",
"(REF -REF ) from the basic quantum level (Spin-Boson Hamiltonian) was made in Ref.",
"[22].",
"The characteristic time scales of diffusion in the process of the electron transfer are of the order of picoseconds.",
"That is to say that the equations (REF -REF ) were originally designed to work for completely different time scales.",
"The statistics of the ON time blinking periods within the DCET model can be calculated using the function $\\rho _1(Q,t)$ which is a solution of the equation (REF ) where the term describing the transfer from the charged state to the neutral one is omitted: $\\frac{\\partial }{\\partial t}\\rho _1(Q,t)= D_1 \\frac{\\partial }{\\partial Q}\\left(\\frac{\\partial }{\\partial Q}+ \\frac{U_1^{\\prime }(Q)}{kT} \\right) \\rho _1(Q,t)$ $-2\\pi \\frac{V^2}{\\hbar }\\delta \\left(U_1(Q)-U_2(Q)\\right)\\rho _{1}(Q,t)$ with the initial condition describing the distribution function right after the transition from the charged state: $\\rho _1(Q,0)=\\delta (Q-Q_c)$ The probability of the ON state being longer than $t$ (survival probability) is defined by the integral of the function $\\rho _1(Q,t)$ $S_{\\mbox{\\tiny ON}}(t)=\\int \\limits _{-\\infty }^\\infty \\rho _1(Q,t)\\,dQ$ The ON time distribution function is expressed as a derivative $p_{\\mbox{\\tiny ON}}(t)=-\\frac{d}{dt} S_{\\mbox{\\tiny ON}}(t)$ The analytical expression for the Laplace image of the ON time distribution function $ \\tilde{p}_{\\mbox{\\tiny ON}}(s)=\\int \\limits _0^{\\infty } p_{\\mbox{\\tiny ON}}(t) e^{-st} \\,dt$ was found by Tang and Marcus [15], [16] (derivation details are given in Appendix A): $\\tilde{p}_{\\mbox{\\tiny ON}}(s)=\\frac{W g_1(s)}{1+W g_1(s)}$ where $W=\\frac{\\sqrt{2 \\pi } V^2}{\\hbar \\sqrt{E_rkT}}$ Function $g_1(s)$ can be expressed as an integral $g_1(s)=\\int \\limits _0^\\infty \\frac{\\exp \\left[-st-\\frac{x_c^2}{2}\\tanh \\left({\\frac{t}{2\\tau _1}}\\right)\\right] }{\\sqrt{2\\pi \\left(1-e^{-2t/\\tau _1}\\right)}}\\,dt$ where $\\tau _1$ is the relaxation time in the the neutral state $\\tau _1=\\frac{2E_rkT}{D_1}$ and $x_c$ is the dimensionless crossing point coordinate $x_c=\\frac{E_r+\\Delta G}{\\sqrt{2E_rkT}}$ At a short time limit $t\\ll \\tau _1$ Tang and Marcus [15], [16] presented the following approximation for the ON time distribution (see Appendix B): $p_{\\mbox{\\tiny ON}}(t)=\\frac{\\exp (-\\Gamma _1 t)}{\\sqrt{\\pi t_c t}} \\left[1-\\sqrt{\\frac{\\pi t}{t_c}} \\exp \\left(\\frac{t}{t_c} \\right)\\mbox{erfc}\\left(\\sqrt{\\frac{t}{t_c}}\\right) \\right]$ where $\\Gamma _1=\\frac{x_c^2}{4\\tau _1}$ and $t_c$ is the critical time $t_c=\\frac{4}{W^2 \\tau _1}$ When $t$ is much shorter than the critical time Eq.",
"(REF ) can be approximated as $p_{\\mbox{\\tiny ON}}(t)\\approx \\frac{1}{\\sqrt{\\pi t_c}} t^{-1/2} , \\quad t\\ll t_c$ when for longer times $p_{\\mbox{\\tiny ON}}(t)\\approx \\frac{1}{2} \\sqrt{\\frac{t_c}{\\pi }} t^{-3/2}\\exp (-\\Gamma _1 t), \\quad t_c\\ll t \\ll \\tau _1$ The equation (REF ) reproduces the experimentally observed truncated power-law dependence Eq.",
"(REF ).",
"This dependence has to correspond to the power spectral density of the emission intensity $S(f)\\sim f^{-3/2}$ .",
"The experimentally observed transition of the power spectral density dependence to $f^{-2}$ at large frequencies [10] was explained by the changing of the ON time distribution function behavior from (REF ) to (REF ) at times $t\\sim t_c$ .",
"Figure: The ON time distribution function within the DCET model (thick red line), the first interval power law(black dashed line), the second interval power law (black dashed-dotted line), the Tang-Marcus approximation Eq.",
"() (thin black line) and thelong-time asymptotic Eq.",
"() (red dashed line).",
"Vertical dotted lines represent borders between characteristic intervals at t c t_c, 1/Γ 1 1/\\Gamma _1 and τ\\tau .The parameters of the model are τ 1 =100s\\tau _1=100\\,s, Γ 1 =0.1s -1 \\Gamma _1=0.1\\,s^{-1}, t c =10 -3 st_c=10^{-3}\\,s.Figure: The coordinate probability distribution function ρ 1 (x,t=10 -4 s)×10 -4 \\rho _1(x,t=10^{-4}\\,s)\\times 10^{-4} (black line),ρ 1 (x,t=1s)×0.25\\rho _1(x,t=1\\,s)\\times 0.25 (red line), ρ 1 (x,t=10.21s)\\rho _1(x,t=10.21\\,s) (green line), ρ 1 (x,t=10 5 s)\\rho _1(x,t=10^{5}\\,s) (blue line).The parameters of the model are τ 1 =100s\\tau _1=100\\,s, Γ 1 =0.1s -1 \\Gamma _1=0.1\\,s^{-1}, t c =10 -3 st_c=10^{-3}\\,s.",
"xx is the dimensionless coordinatex=(Q+E r )/2E r kTx=(Q+E_r)/\\sqrt{2E_rkT}Figure: The time dependence of the probability of finding the QD in the neutral (bright) state for the initial condition ()(red line) and the initial condition () (blue line).The parameters of the model are τ 1 =100s\\tau _1=100\\,s, Γ 1 =0.1s -1 \\Gamma _1=0.1\\,s^{-1}, t c =10 -3 st_c=10^{-3}\\,s, τ 2 =10 4 s\\tau _2=10^4\\,s, Γ 2 =10 -3 s -1 \\Gamma _2=10^{-3}\\,s^{-1}The problem is that for longer times $t \\gg \\tau _1$ the approximate formula (REF ) is not applicable.",
"It can be shown (see Appendix C) that at a very long time scale the ON time distribution shows slow exponential decay [16]: $p_{\\mbox{\\tiny ON}}(t)\\approx p_1 \\exp (-k_1t) , \\quad \\tau _1 \\ll t$ were $k_1$ is the decay rate $k_1=\\frac{W}{\\sqrt{2\\pi }(1+WB)}\\exp \\left(-\\frac{x_c^2}{2}\\right)$ $p_1$ is the amplitude $p_1=\\frac{k_1}{1+WB}$ and $B=\\int \\limits _0^\\infty \\left[\\frac{\\exp \\left(-\\frac{x_c^2}{2} \\tanh \\left({\\frac{t}{2\\tau _1}}\\right)\\right) }{\\sqrt{2\\pi \\left(1-e^{-2t/\\tau _1}\\right)}}-\\frac{\\exp \\left(-\\frac{x_c^2}{2}\\right)}{\\sqrt{2\\pi }}\\right]\\,dt$ The last integral can be expressed in terms of a generalized hypergeometric function $_2F_2$ [23]: $B=\\frac{\\tau _1}{\\sqrt{2\\pi }} \\exp \\left(-\\frac{x_c^2}{2}\\right)\\left[\\ln 2+x_c^2\\,{_2F_2}\\left(\\left.\\begin{array}{cc}1&1\\\\\\frac{3}{2}&2\\end{array}\\right|\\frac{x_c^2}{2}\\right)\\right]$ The simpler analytical expressions of $B$ can be found in the limiting cases [23]: $B\\approx \\left\\lbrace \\begin{array} {ll}\\tau _1 {\\ln 2}/\\sqrt{2 \\pi },& \\quad |x_c|\\ll 1\\\\\\tau _1/{|x_c|} , & \\quad |x_c|\\gg 1\\end{array} \\right.$ Equation (REF ) can be rewritten as $k_1=\\frac{W}{\\sqrt{2\\pi }(1+WB)}\\exp \\left(-\\frac{(E_r+\\Delta G)^2}{4E_rkT}\\right)$ This formula is well-known in electron transfer theory [21].",
"It describes the quasi-stationary rate of the electron transfer in the absence of back transitions.",
"The argument in the exponent reproduces the famous Marcus' Free Energy Gap law.",
"For low coupling values the rate Eq.",
"(REF ) is proportional to $V^2$ (the Golden Rule result): $ k_1= \\frac{V^2}{\\hbar \\sqrt{E_rkT}}\\exp \\left(-\\frac{(E_r+\\Delta G)^2}{4E_rkT}\\right)$ At high coupling values the rate is limited by the diffusion transport to the crossing point and so becomes independent of $V$ .",
"For the activated process $(E_r+\\Delta G)^2\\gg 4E_rkT$ from Eqs.",
"(REF ) and (REF ) we get: $k_1=\\frac{|E_r+\\Delta G|}{\\tau _1 \\sqrt{4\\pi E_rkT} }\\exp \\left(-\\frac{(E_r+\\Delta G)^2}{4E_rkT}\\right)$ The maximum rate is reached in the activationless case $(E_r+\\Delta G)^2\\ll 4E_rkT$ $ k_1= \\frac{1}{\\tau _1 \\ln 2}$ As we can see the rate $k_1$ is always less than $1/\\tau _1$ .",
"The OFF time distribution shows a similar behaviour: $p_{\\mbox{\\tiny OFF}}(t)\\approx \\frac{1}{\\sqrt{\\pi t_2}} t^{-1/2}, \\quad t\\ll t_2 $ $p_{\\mbox{\\tiny OFF}}(t)\\approx \\frac{1}{2} \\sqrt{\\frac{t_2}{\\pi }} t^{-3/2}\\exp (-\\Gamma _2 t), \\quad t_2\\ll t \\ll \\tau $ $p_{\\mbox{\\tiny OFF}}(t)\\approx p_2 \\exp (-k_2t), \\quad \\tau _2 \\ll t$ where $\\Gamma _2=\\frac{x_2^2}{4\\tau _2},\\quad t_2=\\frac{4}{W^2 \\tau _2}$ $k_2=\\frac{W}{1+WB_2}\\exp \\left(-\\frac{x_2^2}{2}\\right), \\quad p_2=\\frac{k_2}{\\sqrt{2\\pi }(1+WB_2)}$ and $B_2=\\frac{\\tau _2}{\\sqrt{2\\pi }}\\exp \\left(-\\frac{x_2^2}{2}\\right)\\left[\\ln 2+x_2^2\\,{_2F_2}\\left(\\left.\\begin{array}{cc}1&1\\\\\\frac{3}{2}&2\\end{array}\\right|\\frac{x_2^2}{2}\\right)\\right]$ According to Eq.",
"(REF ) and Eq.",
"(REF ) there are four characteristic time intervals of the $p_{\\mbox{\\tiny ON}}(t)$ behavior: Interval I: Power-law with $1/2$ exponent at $t\\ll t_c$ ; Interval II: Power-law with $3/2$ exponent at $t_c\\ll t \\ll 1/\\Gamma _1$ ; Interval III: Exponential decay at $1/\\Gamma _1 \\ll t \\ll \\tau _1$ ; Interval IV: Long time exponential decay $\\tau _1 \\ll t$ .",
"Note that Interval III can only exist if $\\Gamma _1\\tau _1\\gg 1$ We performed numerical simulations of Eq.",
"(REF ) using the SSDP program [24].",
"The results of the simulations for the parameters $\\tau _1=100\\,s$ , $\\Gamma _1=0.1\\,s^{-1}$ , $t_c=10^{-3}\\,s$ are presented in Fig.",
"REF .",
"The parameters are very close to the ones used in Ref.",
"[15] for fitting the experimental data.",
"The model parameters can be restored using Eqs.",
"(REF ) and (REF ): $x_c=\\sqrt{4\\Gamma _1\\tau _1}\\approx 6.32,\\quad W=\\sqrt{\\frac{4}{\\tau _1 t_c}}\\approx 6.32\\, s^{-1}$ The condition $x_c\\gg 1$ following from (REF )is satisfied.",
"Using Eq.",
"(REF ) we get $BW=\\frac{1}{\\sqrt{\\Gamma _1 t_c}}=100$ An expression for $k_1$ follows from Eq.",
"(REF ) $k_1= \\sqrt{\\frac{2 \\Gamma _1}{\\pi \\tau _1}} \\frac{ \\exp (-2 \\Gamma _1 \\tau _1)}{1+\\sqrt{\\Gamma _1 t_c}}\\approx 5.15\\times 10^{-11}\\, s^{-1} $ All four characteristic intervals of the ON time distribution dynamics are clearly seen on Fig.",
"REF .",
"The value $ p_{\\mbox{\\tiny ON}} (t) $ is very small at $t \\gg \\tau _1$ (interval IV), however the probability for the ON state to survive after $\\tau _1$ time is quite significant.",
"$S_1= S_{\\mbox{\\tiny ON}}(t \\gg \\tau _1) \\approx \\int \\limits _{0}^\\infty p_1 \\exp (-k_1 t)\\,dt$ From (REF ) we get $S_1\\approx \\frac{1}{1+BW}=\\frac{\\sqrt{\\Gamma _1 t_c}}{1+ \\sqrt{\\Gamma _1 t_c}} \\approx 0.01$ That is why the averaged ON time is extremely long: $\\langle t_{\\mbox{\\tiny ON}}\\rangle =\\int \\limits _{0}^\\infty t p_{\\mbox{\\tiny ON}}(t)\\,dt \\approx \\int \\limits _{0}^\\infty t p_1\\exp (-k_1 t)\\,dt$ and after integration: $\\langle t_{\\mbox{\\tiny ON}} \\rangle \\approx \\frac{k_1^{-1}}{1+BW}=\\sqrt{\\frac{\\tau _1 t_c}{2}}\\exp (2\\Gamma _1 \\tau _1)\\approx 1.08\\times 10^8 s$ The coordinate probability distribution function $\\rho _1(Q,t)$ within each interval is shown on Fig.",
"REF .",
"At a short time (Interval I) the distribution has one narrow maximum, its width increases with time $\\Delta Q = \\sqrt{2 D_1t}$ .",
"The distribution function value at the crossing point $\\rho _1(Q_c,t)$ decays as $\\sim t^{-1/2}$ and it follows the same power law form of the ON time distribution.",
"At longer times (Interval II) the delta-functional sink burns a hole in the distribution function, and it shows two maxima.",
"The distribution starts shifting towards the potential minimum within Interval III.",
"That shifting generates an exponential decreasing of the $\\rho _1(Q_c,t)$ and as a result the exponential decay of the ON time distribution function.",
"At times longer than $\\tau _1$ (Interval IV) the function $\\rho _1(Q,t)$ reaches the quasistationary distribution at the bottom of the parabolic potential $\\rho _1(Q,t) \\approx \\frac{S_1}{\\sqrt{4\\pi E_rkT}} \\exp \\left(-\\frac{(Q+E_r)^2}{4E_rkT}\\right)\\exp (-k_1t)$ As such, the transition to the OFF state can only occur at the $Q_c$ crossing point, which requires thermal activation.",
"This explains why the decay of the ON time distribution is so gradual within Interval IV.",
"As seen from the analytical analysis and numerical simulations the DCET model predicts the appearance of extremely long ON time periods in a single QD emission trace.",
"As seen on Fig.",
"REF such a period could last years, which is much longer than the duration of a typical experiment.",
"The probability of such a long duration of a single ON time blinking event $S_1$ is found to be in order of 1%.",
"Thus the QD can become permanently bright after about one hundred blinking cycles with a high probability.",
"All the predictions made about the ON time distribution can be applied for the OFF distribution as well.",
"In most experiments the OFF time distribution truncation time of the single QD emission trace is too long to be detected.",
"The only exceptions are the observations made on similar nanoobjects, namely nanorods [25] where the value $1/\\Gamma _2\\sim 2500\\,s$ was found.",
"Let us set $\\Gamma _2= 10^{-3}\\,s^{-1}$ and $\\tau _2=10^4\\,s$ .",
"The corresponding rate for long time decay is $k_2\\approx 5.15\\times 10^{-13}\\, s^{-1}$ The probability of an extremely long OFF time period is $S_2= \\frac{\\sqrt{\\Gamma _2 t_2}}{1+ \\sqrt{\\Gamma _2 t_2}} \\approx 10^{-4}$ This means that after about ten thousand blinking cycles the QD should become permanently bright or permanently dark.",
"This prediction significantly differs from the behavior of single quantum dots observed in numerous experiments.",
"The fact that $S_2$ is much smaller than $S_1$ ($S_2/S_1 \\approx 10^{-2}$ ) suggests that the most of the QDs should became permanently bright.",
"In order to verify that statement we used the SSDP program [24] for numerical simulations of the Eqs.",
"(REF -REF ) with two types of initial conditions: at the beginning of the ON time period (delta-functional distribution in the neutral state) $\\varrho _1(Q,0)=\\delta (Q-Q_c);\\quad \\varrho _2(Q,0)=0$ and at the beginning of the OFF time period $\\varrho _1(Q,0)=0;\\quad \\varrho _2(Q,0)=\\delta (Q-Q_c)$ As shown in Fig.",
"REF , the probability of finding the system in the ON state $ P_1 (t) = \\int \\limits _{-\\infty }^\\infty \\varrho _1 (Q, t) \\, dQ $ becomes very close to unity at times greater than 100 seconds for both cases."
],
[
"Extended DCET model",
"The extended DCET model of Zhu and Marcus [17] includes the equations describing the evolution of the probability density of the ground state $\\varrho _g(Q,t)$ , the excited state $\\varrho _e(Q,t)$ , the biexciton state $\\varrho _b(Q,t)$ , the charged (dark) state $\\varrho _d(Q,t)$ , and the excited dark state $\\varrho _{d^\\ast }(Q,t)$ : $\\frac{\\partial }{\\partial t}\\varrho _g(Q,t)=k_{eg}\\varrho _e(Q,t)-I_{ge}\\varrho _g(Q,t)$ $\\frac{\\partial }{\\partial t}\\varrho _e(Q,t)=I_{ge}\\varrho _g(Q,t)+L_e\\varrho _e(Q,t)+k_{be}\\varrho _b(Q,t)$ $-(k_{eg}+I_{eb})\\varrho _e(Q,t)-k_{ed}\\delta (Q-Q_c)\\varrho _e(Q,t)$ $\\frac{\\partial }{\\partial t}\\varrho _b(Q,t)=I_{eb}\\varrho _e(Q,t)+L_b\\varrho _b(Q,t)-(k_{be}+k_{bd^{\\prime }})\\varrho _b(Q,t)$ $\\frac{\\partial }{\\partial t}\\varrho _d(Q,t)=k_{d^\\ast d}\\varrho _{d^\\ast }(Q,t)-I_{dd^\\ast }\\varrho _d(Q,t)$ $ \\frac{\\partial }{\\partial t}\\varrho _{d^\\ast }(Q,t)=L_{d^\\ast }\\varrho _{d^\\ast }(Q,t)+I_{dd^\\ast }\\varrho _d(Q,t)$ $-k_{d^\\ast d}\\varrho _{d^\\ast }(Q,t)-k_{d^\\ast e}\\delta (Q-Q_c)\\varrho _{d^\\ast }(Q,t)$ where $L_e$ , $L_b$ and $L_{d^\\ast }$ are diffusion operators $L_e= D_e \\frac{\\partial }{\\partial Q} \\left(\\frac{\\partial }{\\partial Q}+ \\frac{U_e^{\\prime }(Q)}{kT} \\right)$ $L_b= D_b \\frac{\\partial }{\\partial Q} \\left(\\frac{\\partial }{\\partial Q}+ \\frac{U_b^{\\prime }(Q)}{kT} \\right)$ $L_{d^\\ast }= D_{d^\\ast } \\frac{\\partial }{\\partial Q} \\left(\\frac{\\partial }{\\partial Q}+ \\frac{U_{d^\\ast }^{\\prime }(Q)}{kT} \\right)$ $D_e$ , $D_b$ $D_{d^\\ast }$ are the diffusion coefficients, $U_e(Q)$ , $U_b(Q)$ and $U_{d^\\ast }(Q)$ are the potential surfaces of the excited state, the biexciton state and the dark excited state, respectively.",
"$ I_{ge}$ , $I_{eb}$ , $I_{dd^\\ast }$ , $k_{eg}$ , $k_{be}$ , $k_{bd^{\\prime }}$ , $k_{d^\\ast d}$ , $k_{d^\\ast e}$ , and $k_{ed}$ are the rate constants.",
"The equation for the probability density of the higher energy dark state has to be added to the equation system (REF -REF ): $\\frac{\\partial }{\\partial t}\\varrho _{d^{\\prime }}(Q,t)=k_{bd^{\\prime }}\\varrho _{b}(Q,t)-k_{d^{\\prime }d}\\varrho _{d^\\ast }(Q,t)$ As stated by Zhu and Marcus [17] quasiequilibrium is established between the ground, the excited state and the biexciton state.",
"We can see from Eq.",
"(REF ) that a quasistationary distribution of the the higher energy dark state is also determined by $\\varrho _e (Q,t)$ and it can also can be considered a part of the quasiequilibrium.",
"As such we can introduce the population of the integrated ON state $\\varrho _1 (Q,t)=\\varrho _g (Q,t)+\\varrho _e (Q,t)+\\varrho _b (Q,t)+\\varrho _{d^{\\prime }} (Q,t)$ Similarly, there is a quasiequilibrium between the dark and the excited dark states and the OFF state population can also be introduced $\\varrho _2 (Q,t)=\\varrho _d (Q,t)+\\varrho _{d^\\ast } (Q,t)$ The following kinetic equations for the functions $\\varrho _1(Q,t)$ and $\\varrho _2 (Q,t)$ were obtained from Eqs.",
"(REF -REF ) (see Appendix D): $\\frac{\\partial }{\\partial t}\\varrho _1(Q,t)=L_1\\varrho _1(Q,t)-k_{L}\\varrho _1(Q,t)$ $-W_1\\delta (Q-Q_c)\\varrho _1(Q,t) +W_2\\delta (Q-Q_c)\\varrho _2(Q,t)$ $\\frac{\\partial }{\\partial t}\\varrho _2(Q,t)=L_2\\varrho _1(Q,t)-W_2\\delta (Q-Q_c)\\varrho _2(Q,t)$ $+W_1\\delta (Q-Q_c)\\varrho _1(Q,t) +k_{LD}\\varrho _1(Q,t)$ where $L_1$ and $L_2$ are effective diffusion operators: $ L_1=C_1\\left(L_e+\\frac{I_{eb}}{k_{be}}L_b\\right); \\quad L_2=C_2 L_d$ $W_1$ , $W_2$ and $k_{L}$ are effective rates: $W_1=C_1 k_{ed}; \\quad W_2= C_2 k_{d^\\ast e}; \\quad k_{L} =C_1 k_{bd^{\\prime }} \\frac{I_{eb}}{k_{be}}$ and $C_1$ and $C_2$ are the coefficients: $ C_1=\\left( 1+\\frac{k_{eg}}{I_{ge}} +\\frac{I_{eb}}{k_{be}} +\\frac{k_{bd^{\\prime }}}{k_{d^{\\prime }d}} \\frac{I_{eb}}{k_{be}}\\right)^{-1};\\quad C_2=\\left(1+\\frac{k_{d^\\ast d}}{I_{ge}}\\right)^{-1}$ We have to note that the equations derived by Zhu and Marcus ( Eqs.",
"(11-12) in Ref.",
"[17]) using the same procedure are different from Eqs.",
"(REF -REF ).",
"The last term in Eq.",
"(REF ) was omitted in Eq.",
"(11) in Ref.",
"[17] and the two last terms in Eq.",
"(REF ) were omitted in Eq.",
"(12) in Ref.[17].",
"It can be seen that because of the absence of these terms, Eqs.",
"(11-12) of Zhu and Marcus [17] do not preserve the total probability.",
"Figure: The schematic picture of the Frantsuzov and Marcus model.The potential surface is represented by a line colored red in the bright region (Q>δQ>\\delta ) and blue in the dark region (Q<0Q <0).The black line represents the PLQY dependence on the coordinate.The ON time and OFF time distribution functions in the Extended DCET model (REF -REF ) can be found by solving the following equations: $\\frac{\\partial }{\\partial t}\\rho _1(Q,t)=L_1\\rho _1(Q,t) -W_1\\delta (Q-Q_c)\\rho _1(Q,t) -k_{L}\\rho _1(Q,t)$ $\\frac{\\partial }{\\partial t}\\rho _2(Q,t)=L_2\\rho _2(Q,t) -W_2\\delta (Q-Q_c)\\rho _2(Q,t)$ Transitions from the dark state to the bright state occur only at the point $Q_c$ , thus the initial distribution for the Eq.",
"(REF ) is a delta-function: $\\rho _1(Q,t)=\\delta (Q-Q_c)$ in contrast transitions from a bright state to a dark state can occur not only at the crossing point and the initial condition for the Eq.",
"(REF ) has the following form: $\\rho _2(Q,t)=\\int \\limits _0^\\infty (W_1\\delta (Q-Q_c) +k_{L})\\rho _1(Q,t)\\,dt$ The Eq.",
"(REF ) has an additional term $-k_{L}\\rho _1(Q,t)$ in comparison to Eq.",
"(REF ) which leads to an exponential cutoff of the survival probability (REF ) time dependence $S_{\\mbox{\\tiny ON}}(t)=S^0_{\\mbox{\\tiny ON}}(t)\\exp (-k_{L}t)$ where $S^0_{\\mbox{\\tiny ON}}(t)$ is the survival probability obtained from Eq.",
"(REF ) at $k_{L}=0$ .",
"As a result the ON time distribution function in the Extended DCET model has an exponential cutoff.",
"$p_{\\mbox{\\tiny ON}}(t)\\sim \\exp (-k_{L}t), \\quad t \\gg 1/k_{L} $ The Eq.",
"(REF ) is equivalent to Eq.",
"(REF ).",
"The difference in the initial distributions leads to the deviation of the OFF time distribution in the Extended DCET model in comparison with the original DCET model at times smaller than $\\tau _2$ .",
"The long time exponential asymptotic behavior, however, is the same $p_{\\mbox{\\tiny OFF}}(t)\\sim \\exp (-k_{2}t), \\quad t \\gg \\tau _2 $ These theoretical predictions are confirmed by numerical simulations (see Fig.",
"REF ) performed for the case of the symmetric system $Q_c=0$ , $W_1=W_2$ .",
"The rest of the parameters are $\\tau _1=\\tau _2=10^4\\,s$ , $\\Gamma _1=\\Gamma _2=10^{-3}\\,s^{-1}$ , $t_c=t_2=0.1\\,s$ , $K_{L}=10^{-1}\\,s^{-1}$ .",
"It can be concluded that the presence of a second ionization channel resolves the problem with very long ON times, but not with very long OFF times.",
"As a result, most of the QDs in the Extended DCET model have to become permanently dark as confirmed by numerical simulations (see Fig.",
"REF ).",
"That prediction also significantly differs from the experimentally observed behavior of single quantum dots."
],
[
"Frantsuzov and Marcus model",
"The Frantsuzov and Marcus model [18] is based on the fluctuating rate mechanism, thus it does not consider transitions between neutral and charged states.",
"Fluctuations of the emission intensity in the model are caused by variations of the PLQY (REF ).",
"The nonradiative recombination rate $k_n$ depends on the reaction coordinate $Q$ which is performing diffusive motion.",
"Within the generalized formulation of the model the probability distribution function $\\rho (Q,t)$ satisfies the equation $\\frac{\\partial }{\\partial t}\\varrho (Q,t)= \\frac{\\partial }{\\partial Q} D(Q)\\left(\\frac{\\partial }{\\partial Q}+ Q\\right) \\varrho (Q,t)$ where $D(Q)$ is the coordinate dependent diffusion coefficient.",
"To generate fast transitions from high to low emission intensity and back, the function $Y(Q)$ must grow dramatically from a minimal value to a maximum one on a tiny interval of $\\delta $ close to the origin (see Fig.",
"REF ).",
"Thus, the QD is bright when $Q>\\delta $ , dark when $Q<0$ , and has some intermediate florescence intensity within the interval of $\\delta \\ll 1$ .",
"Taking into account that a molecular mechanism of the spectral diffusion is light induced [12], [26], the diffusion coefficient $D(Q)$ has to depend on the excitation intensity.",
"It also means that the diffusion could be much faster for a bright QD than for a dark one [18].",
"As such, we can choose: $D(Q)= \\left\\lbrace \\begin{array}{ccl} 1 /{T_{\\mbox{\\tiny OFF}}}, & & Q<0\\\\1/ {T_{\\mbox{\\tiny ON}}}, & \\delta \\le &Q\\end{array} \\right.$ It was shown by Frantsuzov and Marcus [18] that the normalized ON time and OFF time distributions obtained by the threshold procedure have the following dependence (see Appendix E for details): $p(t)=\\frac{\\sqrt{\\tau _{m}}}{2} t^{-3/2},\\quad \\tau _{m}\\le t\\ll T_0$ $p(t)=\\sqrt{\\frac{2 \\tau _{m}}{T_0^3}} \\exp (-t/T_0),\\quad T_0 \\ll t$ where $\\tau _m$ is the minimum time interval of observation (bin time) and $T_0$ is equal to $T_{\\mbox{\\tiny ON}}$ and $T_{\\mbox{\\tiny OFF}}$ for the ON time and OFF time distribution, respectively.",
"That prediction is confirmed by the numerical simulations made using the SSDP program [24] (see Fig.",
"REF ).",
"The power spectral density $S(f)$ of the single QD emission at frequencies $f$ larger than $1/\\tau _m$ could be obtained without binning procedure by measuring the autocorrelation function [27], [10].",
"In order to calculate $S(f)$ within the model one needs to specify the $Y(Q)$ function in the intermediate interval.",
"Let's choose the simplest linear dependence: $Y(Q)= \\left\\lbrace \\begin{array}{ccl} 0, \\quad & &Q<0\\\\Q/\\delta , \\quad & 0\\le &Q < \\delta \\\\1,\\quad & \\delta \\le &Q\\end{array} \\right.$ The results of numerical calculations of the $S(f)$ in that case are presented in Fig.",
"REF (see Appendix F for the detailed calculation procedure).",
"The Figure clearly shows the transition from the $f^{-3/2}$ dependence to $f^{-2}$ at large frequencies in accordance with the experiment of Pelton et al.",
"[10]."
],
[
"Discussion",
"As a result of the above analytical and numerical studies it was found that two models of single QD blinking based on spectral diffusion, namely the DCET model [15], [16] and the Extended DCET model [17] predict that after an initial blinking period, most of the QDs should become permanently bright or permanently dark.",
"That prediction significantly differs from the behavior of single quantum dots observed in numerous experiments.",
"Another drawback of these models is the charging mechanism on which they are based.",
"Despite the fact that most of the theoretical models proposed in the literature are based on that mechanism [28], [13], [29], [30], [16], [31], [17], there is a number of sufficient experimental evidence indicating that the charging mechanism fails in explaining the QD blinking phenomenon.",
"In several experiments, the emission intensity of a single QD was observed below the charged state (trion) emission intensity [32], [33], [34], [35].",
"Another very important set of experiments showed that the existence of the distinct ON and OFF states is an illusion; there is a nearly continuous set of emission intensities [36], [37], [38], [39], [40].",
"Furthermore, it was also shown [41], [42], [43] that the parameters $m$ and $T$ of the ON and OFF time distributions strongly depend on the threshold value.",
"The Frantsuzov and Marcus model [18], based on fluctuating rate mechanism, reproduces the key properties of the QD blinking phenomenon.",
"Nonetheless there are a number of the experimental observations which are not explained by the model: 1.",
"The exponent value $m$ of the ON and OFF time distribution functions is reported in the range from 1.2 to 2.0 [2], and it strongly depends on the threshold value [41], [42], [43].",
"Meanwhile, in the model, $m$ is always equal to $3/2$ regardless of the threshold.",
"2.",
"The exponent $r$ of the emission power spectral density is found to be in the range from 0.7 to 1.2 [27], [10], [44], when the model predicts the exponent value of 3/2.",
"3.",
"The long-term correlations between subsequent ON and OFF times [45], [46].",
"There are no such correlations in the model.",
"A possible reason for this discrepancy is that the description of the spectral diffusion in the model does not fully correspond to its real properties.",
"It was shown that the squared frequency displacement of the single QD emission has an anomalous (sublinear) time dependence [47].",
"Plakhotnik et al.",
"[47] suggested an explanation of this behavior by introducing a number of stochastic two-level systems (TLS) having a wide spectrum of flipping rates.",
"A similar idea was applied by Frantsuzov, Volkan-Kacso and Janko in the Multiple Recombination Center (MRC) model of single QD blinking [41].",
"The MRC model, based on the fluctuating rate mechanism, also reproduces the key properties the single QD blinking.",
"But in addition it explains the power spectral density dependence close to $1/f$ [44], the threshold dependence of the $m$ and $T$ values [41], and the long-term correlations between subsequent blinking times [46].",
"This suggests that the spectral diffusion and the fluctuations of the emission intensity of a single QD can be explained by an unified model, which could become a generalization of the Frantsuzov and Marcus model.",
"In conclusion, we analytically and numerically considered three models of the single QD emission fluctuations (blinking) based on spectral diffusion.",
"Only one of them, the Frantsuzov and Marcus model [18], reproduces the key properties of the phenomenon.",
"The DCET model [15], [16] and the Extended DCET model [17] predict that after an initial blinking period, most of the QDs should become permanently bright or permanently dark which is significantly different from the experimentally observed behavior."
],
[
"Acknowledgement",
"The authors are very grateful to Professor Rudolph Marcus for fruitful discussions.",
"The study was supported by the Russian Foundation for Basic Research, project 16-02-00713.",
"Introducing a dimensionless coordinate $x$ $x=\\frac{Q+E_r}{\\sqrt{2E_rkT}}$ we can rewrite Eq.",
"(REF ) as $\\frac{\\partial }{\\partial t}\\rho _1(x,t)=\\frac{1}{\\tau _1} \\frac{\\partial }{\\partial x}\\left(\\frac{\\partial }{\\partial x}+ x \\right) \\rho _1(x,t) $ $-W \\delta (x-x_c)\\rho _{1}(x,t)$ with the initial condition $\\rho _1(x,0)=\\delta (x-x_c)$ where the relaxation time $\\tau _1$ is given by Eq.",
"(REF ), $x_c$ is the dimensionless crossing point coordinate Eq.",
"(REF ) and $W$ is given by Eq.",
"(REF ) Applying Eq.",
"(REF ), the ON time distribution function (REF ) can be expressed as $p_{\\mbox{\\tiny ON}}(t)=-\\frac{d}{dt}\\int _{-\\infty }^{\\infty } \\rho _1(x,t)\\,dx=W\\rho _1(x_c,t)$ The Laplace image of the function $\\rho _1(x,t)$ $\\tilde{\\rho }_1(x,s)=\\int _0^\\infty \\rho _1(x,t)e^{-st}\\,dt$ obeys the following equation $s\\tilde{\\rho }_1(x,s)-\\delta (x-x_c)= $ $\\frac{1}{\\tau _1} \\frac{\\partial }{\\partial x}\\left(\\frac{\\partial }{\\partial x}+ x \\right)\\tilde{\\rho }_1(x,s)-W \\delta (x-x_c)\\tilde{\\rho }_{1}(x,s)$ The Green's function of the differential operator in Eq.",
"(REF ) satisfies the equation $\\frac{\\partial }{\\partial t}G(x,x^{\\prime },t)= \\frac{1}{\\tau _1} \\frac{\\partial }{\\partial x}\\left(\\frac{\\partial }{\\partial x}+ x \\right) G(x,x^{\\prime },t)$ with the initial condition $ G(x,x^{\\prime },0)=\\delta (x-x^{\\prime })$ The Green's function and the Laplace image satisfies the equation $s\\tilde{G}(x,x^{\\prime },s)-$ $\\frac{1}{\\tau _1} \\frac{\\partial }{\\partial x}\\left(\\frac{\\partial }{\\partial x}+ x \\right) \\tilde{G}(x,x^{\\prime },s)=\\delta (x-x^{\\prime })$ Using Eq.",
"(REF ), Eq.",
"(REF ) can be rewritten as $\\tilde{\\rho }_1(x,s)=\\tilde{G}(x,x_c,s)-W\\tilde{G}(x,x_c,s)\\tilde{\\rho }_1(x_c,s)$ From Eq.",
"(REF ) we can find $\\tilde{\\rho }_1(x_c,s)$ $\\tilde{\\rho }_1(x_c,s)=\\frac{\\tilde{G}(x_c,x_c,s)}{1+W\\tilde{G}(x_c,x_c,s)}$ The Laplace image of the ON time distribution Eq.",
"(REF ) is given by $\\tilde{p}_{\\mbox{\\tiny ON}}(s)=W\\tilde{\\rho }_1(x_c,s)$ Substituting Eq.",
"(REF ) we get $\\tilde{p}_{\\mbox{\\tiny ON}}(s)=\\frac{W \\tilde{G}(x_c,x_c,s)}{1+W \\tilde{G}(x_c,x_c,s)}$ Green's function (REF ) is well-known: $G(x,x^{\\prime },t)=\\frac{1}{\\sqrt{2\\pi \\left(1-e^{-2t/\\tau _1}\\right)}}\\exp \\left[-\\frac{\\left(x-x^{\\prime } e^{-t/\\tau _1}\\right)^2}{2\\left(1-e^{-2t/\\tau _1}\\right)}\\right]$ Introducing the function $g_1(s)$ $g_1(s)=\\tilde{G}(x_c,x_c,s)$ we can express Eq.",
"(REF ) in the form Eq.",
"(REF ).",
"At a short time limit $t\\ll \\tau _1$ one has to find the function $g_1(s)$ at $s \\gg 1/\\tau _1$ .",
"Expanding the exponent's argument in the Eq.",
"(REF ) we get $g_1(s)=\\int _0^\\infty \\frac{\\exp (-st-\\Gamma _1 t) }{\\sqrt{4\\pi t/\\tau _1}}\\,dt=\\frac{1}{2} \\sqrt{\\frac{\\tau _1}{s+\\Gamma _1}}$ where $\\Gamma _1$ is given by Eq.",
"(REF ) Substituting Eq.",
"(REF ) into Eq.",
"(REF ) gives $\\tilde{p}_{\\mbox{\\tiny ON}}(s)= \\frac{1}{1+ \\sqrt{(s+\\Gamma _1)t_c}}$ and after the inverse Laplace transformation we get Eq.",
"(REF ).",
"The approximate formula (REF ) works for short times only.",
"In order to see the behavior of the function $p_{\\mbox{\\tiny ON}}(t)$ at a long time limit $t \\gg \\tau _1$ one has to consider its Laplace image $\\tilde{p}_{\\mbox{\\tiny ON}}(s)$ (REF ) at $s\\rightarrow 0$ .",
"If we expand the function $g_1(s)$ (REF ) into a series on $s$ : $g_1(s)\\approx \\frac{1}{s} A + B$ where $A=\\lim _{t\\rightarrow \\infty } G(x_c,x_c,t)$ and $B=\\int _0^\\infty \\lbrace G(x_c,x_c,t)-A\\rbrace \\,dt$ The Green's function (REF ) approaches the stationary distribution at long times $\\lim _{t\\rightarrow \\infty } G(x,x^{\\prime },t) =\\frac{1}{\\sqrt{2\\pi }} \\exp \\left(-\\frac{x^2}{2}\\right)$ Thus the constants $A$ and $B$ are $A=\\frac{1}{\\sqrt{2\\pi }} \\exp \\left(-\\frac{x_c^2}{2}\\right)$ $B=\\int _0^\\infty \\left[\\frac{\\exp \\left(-\\frac{1}{2}{x_c^2}\\tanh \\left({\\frac{t}{2\\tau _1}}\\right)\\right) }{\\sqrt{2\\pi \\left(1-e^{-2t/\\tau _1}\\right)}}-A\\right]\\,dt$ Substituting Eq.",
"(REF ) into Eq.",
"(REF ) we get the following dependence of $\\tilde{p}_{\\mbox{\\tiny ON}}(s)$ at small $s$ $\\tilde{p}_{\\mbox{\\tiny ON}}(s)\\approx \\frac{WB}{1+WB}+ \\frac{p_l}{s+k}$ which corresponds to the exponential behavior Eq.",
"(REF )of the ON time distribution function at long times.",
"If $k_{eg}$ is much larger than all other rates in Eq.",
"(REF ) a quasiequlibrium value of exciton population is established $\\varrho _e (Q,t)\\approx \\frac{I_{ge}}{k_{eg}} \\varrho _d (Q,t)$ Similarly if $k_{be}$ is much larger than all other rates in Eq.REF : $\\varrho _b (Q,t)\\approx \\frac{I_{eb}}{k_{be}} \\varrho _e (Q,t)$ If $k_{d^{\\prime }d} \\gg k_{bd^{\\prime }}$ $\\varrho _{d^{\\prime }} (Q,t)\\approx \\frac{k_{bd^{\\prime }}}{k_{d^{\\prime }d}} \\varrho _b (Q,t)\\approx \\frac{k_{bd^{\\prime }}}{k_{d^{\\prime }d}} \\frac{I_{eb}}{k_{be}} \\varrho _e (Q,t)$ Substituting Eqs.",
"(REF -REF ) with the definition Eq.",
"(REF ) into Eqs.",
"(REF -REF ) we get Eq.",
"(REF ).",
"If $k_{d^\\ast d}$ is much larger than all other rates in Eq.",
"(REF ) a quasiequlibrium value of the dark exciton population is established $\\varrho _{d^\\ast } (Q,t)=\\frac{I_{ge}}{k_{d^\\ast d}} \\varrho _d (Q,t)$ Substituting Eq.",
"(REF ) with the definition Eq.",
"(REF ) into Eqs.",
"(REF -REF ) we obtain Eqs.",
"(REF ).",
"The survival probability of the ON time within the Frantsuzov and Marcus model can be found as an integral $S_{\\mbox{\\tiny ON}}(t)=\\int \\limits _0^\\infty \\rho (Q,t)\\,dQ$ where $\\rho (Q,t)$ is a solution of the following equation $\\frac{\\partial }{\\partial t}\\rho (Q,t)= \\frac{1}{T_{\\mbox{\\tiny ON}}} \\frac{\\partial }{\\partial Q}\\left(\\frac{\\partial }{\\partial Q}+ Q\\right) \\rho (Q,t)$ with an absorbing boundary condition at the border (the first passage time problem) $\\left.\\rho (Q,t)\\right|_{Q=0} = 0$ The question of what to take as the initial distribution for the equation is not easily answered.",
"There is the minimal time $\\tau _m$ (bin time) of the ON time period which can be observed.",
"In accordance with Eq.",
"(REF ), if the ON time period is longer than $\\tau _m$ then the coordinate $Q$ has reached values larger than $\\sqrt{\\tau _m/T_{\\mbox{\\tiny ON}}}$ .",
"We can take any distribution located at a distance less than $\\sqrt{\\tau _m/T_{\\mbox{\\tiny ON}}}$ from the origin as an initial one.",
"For the sake of simplicity, we can take the initial distribution in the form of a delta function $\\rho (Q,0)=\\delta (Q-\\Delta )$ where $\\delta \\ll \\Delta \\ll \\sqrt{\\tau _m/T_{\\mbox{\\tiny ON}}}$ The solution of Eqs.",
"(REF -REF ) is well known $\\rho (Q,t)=G(Q,\\Delta ,t)-G(-Q,\\Delta ,t)$ where $G(x,x^{\\prime },t)$ is the Green's function of the Eq.",
"(REF ) $G(Q,Q^{\\prime },t)=\\frac{\\exp \\left\\lbrace -\\frac{\\left[Q-Q^{\\prime } \\exp (-t/T_{\\mbox{\\tiny ON}})\\right]^2}{2\\left(1-\\exp (-2t/T_{\\mbox{\\tiny ON}})\\right)}\\right\\rbrace }{\\sqrt{2\\pi \\left(1-\\exp (-2t/{T_{\\mbox{\\tiny ON}}})\\right)}}$ Using Eq.",
"(REF ) the survival probability (REF ) can be expressed as $ S_{\\mbox{\\tiny ON}}(t)=\\frac{ \\int \\limits _{-b}^b\\exp \\left[-\\frac{Q^2}{2\\left(1-\\exp (-2t/T_{\\mbox{\\tiny ON}})\\right)}\\right]\\,dQ}{\\sqrt{2\\pi \\left(1-\\exp (-2t/T_{\\mbox{\\tiny ON}})\\right)}}$ where $b=\\Delta \\exp (-t/T_{\\mbox{\\tiny ON}})$ .",
"At times $t>\\tau _m$ the expression can be rewritten as $S_{\\mbox{\\tiny ON}}(t)=\\frac{2\\Delta \\exp (-t/T_{\\mbox{\\tiny ON}})}{\\sqrt{2\\pi \\left(1-\\exp (-2t/T_{\\mbox{\\tiny ON}})\\right)}}$ This expression has the following behavior in the limiting cases $S_{\\mbox{\\tiny ON}}(t)= \\Delta \\sqrt{\\frac{T_{\\mbox{\\tiny ON}}}{\\pi t}}, \\quad \\Delta ^2 T_{\\mbox{\\tiny ON}} \\ll t \\ll T_{\\mbox{\\tiny ON}}$ $S_{\\mbox{\\tiny ON}}(t)= \\Delta \\sqrt{\\frac{2}{\\pi }}\\exp (-t/T_{\\mbox{\\tiny ON}}), \\quad T_{\\mbox{\\tiny ON}} \\ll t$ In the experiment one can see that only the ON times are longer than $\\tau _m$ , which means that the ON time distribution should be normalized as follows $\\int \\limits _{\\tau _m}^\\infty p_{\\mbox{\\tiny ON}}(t)\\,dt=1$ The normalization procedure is equivalent to scaling of the function $S_{\\mbox{\\tiny ON}}$ so that the following equality for the normalized survival probability is satisfied $\\bar{S}_{\\mbox{\\tiny ON}}(\\tau _m)=1$ Applying this normalization to Eqs.",
"(REF -REF ) we get $\\bar{S}_{\\mbox{\\tiny ON}}(t)= \\sqrt{\\frac{\\tau _m}{t}}, \\quad \\tau _m \\le t \\ll T_{\\mbox{\\tiny ON}}$ $\\bar{S}_{\\mbox{\\tiny ON}}(t)= \\sqrt{\\frac{2}{T_{\\mbox{\\tiny ON}} }}\\exp (-t/T_{\\mbox{\\tiny ON}}), \\quad T_{\\mbox{\\tiny ON}} \\ll t $ From Eq.",
"(REF ) we obtain the ON time distribution function $p_{\\mbox{\\tiny ON}}(t)= \\frac{1}{2} \\sqrt{\\tau _m}t^{-3/2}, \\quad \\tau _m \\le t \\ll T_{\\mbox{\\tiny ON}}$ $p_{\\mbox{\\tiny ON}}(t)= \\sqrt{\\frac{2 \\tau _m}{T_{\\mbox{\\tiny ON}}^3}} \\exp (-t/T_{\\mbox{\\tiny ON}}), \\quad T_{\\mbox{\\tiny ON}} \\ll t$ Similarly the expression for the OFF time distribution function can be obtained $p_{\\mbox{\\tiny OFF}}(t)= \\frac{1}{2} \\sqrt{\\tau _m}t^{-3/2}, \\quad \\tau _m \\le t \\ll T_{\\mbox{\\tiny OFF}}$ $p_{\\mbox{\\tiny ON}}(t)= \\sqrt{\\frac{2 \\tau _m}{T_{\\mbox{\\tiny OFF}}^3}} \\exp (-t/T_{\\mbox{\\tiny OFF}}), \\quad T_{\\mbox{\\tiny OFF}} \\ll t$ These expression are equivalent to the Eqs.",
"(REF -REF ).",
"The autocorrelation function of the emission intensity within the FRM is $C(t)=\\left\\langle Y\\left(Q(t)\\right)Y\\left(Q(0)\\right) \\right\\rangle $ where averaging is performed over the ensemble of realizations of the random process $Q(t)$ .",
"For the Frantsuzov and Marcus model the function $C(t)$ can be written as $C(t)=\\int \\limits _{-\\infty }^\\infty \\int \\limits _{-\\infty }^\\infty Y(Q) G(Q,Q^{\\prime },t) Y(Q^{\\prime })\\varrho _0(Q^{\\prime })\\,dQdQ^{\\prime }$ where $G(Q,Q^{\\prime },t)$ is the Green's function of Eq.",
"(REF ) and the stationary distribution $\\varrho _0$ is $\\varrho _0(Q)=\\frac{1}{\\sqrt{2\\pi }} \\exp (-\\frac{1}{2} Q^2)$ Eq.",
"(REF ) can be rewritten as $C(t)=\\int \\limits _{-\\infty }^\\infty Y(Q) \\varrho (Q,t)\\,dQ$ where $\\varrho (Q,t)$ is the solution of Eq.",
"(REF ) with the initial condition $\\varrho (Q,0)=Y(Q)\\varrho _0(Q)$ A numerical solution was obtained using the SSDP program [24].",
"The power spectral density was calculated using a cosine transform $S(f)=4\\int \\limits _0^\\infty C(t)\\cos (2 \\pi f t)\\,dt $"
]
] | 1808.08463 |
[
[
"Airy and Painlev\\'e asymptotics for the mKdV equation"
],
[
"Abstract We consider the higher order asymptotics for the mKdV equation in the Painlev\\'e sector.",
"We first show that the solution admits a uniform expansion to all orders in powers of $t^{-1/3}$ with coefficients that are smooth functions of $x(3t)^{-1/3}$.",
"We then consider the special case when the reflection coefficient vanishes at the origin.",
"In this case, the leading coefficient which satisfies the Painlev\\'e II equation vanishes.",
"We show that the leading asymptotics is instead described by the derivative of the Airy function.",
"We are also able to express the subleading term explicitly in terms of the Airy function."
],
[
"Introduction",
"The initial value problem for a nonlinear integrable evolution equation can be analyzed via the inverse scattering transform, which expresses the solution of the equation in terms of the solution of a matrix Riemann–Hilbert (RH) problem.",
"One of the greatest advantages of this approach is that it can be used to derive detailed asymptotic formulas for the long-time behavior of the solution.",
"In the pioneering work [4], the asymptotic behavior of the solution $u(x,t)$ of the modified Korteweg-de Vries (mKdV) equation $u_t - 6u^2u_x + u_{xxx} = 0$ was rigorously established via the application of a nonlinear version of the steepest descent method.",
"Six asymptotic sectors, denoted by I-VI, were identified (see Figure REF ), and in each sector the leading asymptotic term was determined explicitly in terms of the reflection coefficient $r(k)$ .",
"Since $r(k)$ is defined in terms of the initial data $u_0(x) = u(x,0)$ alone, this provides an effective solution of the problem.",
"In Sector IV, defined by $|x| \\le M t^{1/3}$ for some constant $M > 0$ , it was shown in [4] that the solution $u(x,t)$ obeys the asymptotics $u(x,t) = \\frac{u_1(y)}{t^{1/3}} + O(t^{-2/3}), \\qquad t \\rightarrow \\infty ,$ where the leading coefficient $u_1(y)$ is given by $u_1(y) = 3^{-1/3}u_P(y;s,0,-s), \\qquad y := x(3t)^{-1/3},$ with $u_P(y; s,0,-s)$ being a solution of the Painlevé II equation $u_{P}^{\\prime \\prime }(y) = yu_P(y) + 2u_P(y)^3$ determined by the value of $s := ir(0)$ .",
"In Sector II, defined by $-M t\\le x < 0$ and $|x|t^{-1/3}\\rightarrow \\infty $ , the solution instead asymptotes, to leading order, to a slowly decaying modulated sine wave.",
"Later, in [5], the same authors derived an asymptotic expansion valid to all orders as $(x,t) \\rightarrow \\infty $ in the subsector of Sector II given by $-M_1t \\le x \\le -M_2 t$ and recursive formulas were derived for the higher order coefficients.",
"Despite this progress, it appears that the higher order asymptotics in other asymptotic regions remains to be studied in detail.",
"In this paper, we consider the higher order asymptotics of (REF ) in Sector IV.",
"If $r(0) \\ne 0$ , then equation (REF ) provides the leading asymptotics in this sector.",
"However, if $r(0) = 0$ , then $s = 0$ and the associated solution $u_P(y; s,0,-s)$ of Painlevé II vanishes identically.",
"Thus $u_1(y) \\equiv 0$ and (REF ) does not provide any information on the leading term in this case.",
"Our first result (Theorem REF ) states that $u(x,t)$ admits a uniform expansion to all orders in Sector IV of the form $u(x,t) \\sim \\sum _{j=1}^\\infty \\frac{u_j(y)}{t^{j/3}} \\quad \\text{as $t \\rightarrow \\infty $},$ where $\\lbrace u_j(y)\\rbrace _1^\\infty $ are smooth functions of $y \\in {R}$ with the leading coefficient $u_1(y)$ given by (REF ).",
"In general, the higher order coefficients $\\lbrace u_j(y)\\rbrace _2^\\infty $ cannot be computed explicitly, but they are solutions of a hierarchy of ordinary differential equations of which Painlevé II is the first member.",
"Our second result (Theorem REF ) deals with the special case when $r(0) = 0$ .",
"Interestingly, in this case it is possible not only to compute the leading coefficient $u_2(y)$ explicitly, but also the subleading term $u_3(y)$ .",
"The coefficients are expressed in terms of the Airy function $\\text{\\upshape Ai}(y)$ .",
"More precisely, if $r(0) = 0$ , we show that the solution $u(x,t)$ obeys the asymptotic expansion $u(x,t) = \\frac{u_2(y)}{t^{2/3}} + \\frac{u_3(y)}{t} + \\cdots \\quad \\text{as $t \\rightarrow \\infty $}$ uniformly for $|x| \\le M t^{1/3}$ , where $u_2(y) = \\frac{r^{\\prime }(0)}{2\\times 3^{2/3}}\\text{\\upshape Ai}^{\\prime }(y),\\qquad u_3(y) = -\\frac{ir^{\\prime \\prime }(0)}{24}y\\text{\\upshape Ai}(y).$ Although we only present our results for the mKdV equation, it is clear that similar considerations apply also to other integrable equations with Painlevé type asymptotic sectors.",
"Let us finally mention some other works that also use nonlinear steepest descent techniques for Riemann–Hilbert problems to derive asymptotic formulas in related situations.",
"Asymptotic results for the KdV equation can be found in [1], [3], [6], [8]; asymptotic results for the mKdV equation on the half-line are presented in [2], [10]; and higher order asymptotics in sectors analogous to Sector II but for other equations have been derived in [9], [11].",
"The two main theorems are stated in Section .",
"Following a brief review of the RH formalism relevant for (REF ) in Section , the proofs are presented in Sections and .",
"A substantial part of the work—namely the constructions of appropriate local models—is deferred to the two appendices.",
"Figure: NO_CAPTION"
],
[
"Main results",
"Given initial data $u_0(x)$ , $x \\in {R}$ , in the Schwartz class $\\mathcal {S}({R})$ , we let $r(k)$ , $k \\in {R}$ , be the reflection coefficient associated to $u_0$ according to the inverse scattering transform.We use the same conventions for the definition of $r(k)$ as in [4].",
"The following two theorems are the main results of the paper.",
"Theorem 1 Let $u_0 \\in \\mathcal {S}({R})$ be a function in the Schwartz class and let $r \\in \\mathcal {S}({R})$ be the associated reflection coefficient.",
"Then the solution $u(x,t)$ of (REF ) with initial data $u_0(x)$ satisfies the following asymptotic formula as $t \\rightarrow \\infty $ : $u(x,t) = \\sum _{j=1}^N \\frac{u_j(y)}{t^{j/3}} + O\\bigg (\\frac{1}{t^{\\frac{N+1}{3}}}\\bigg ), \\qquad |x| \\le M t^{1/3},$ where The formula holds uniformly with respect to $x$ in the given range for any fixed $M > 0$ and $N \\ge 1$ .",
"The variable $y$ is defined by $y = \\frac{x}{(3t)^{1/3}}.$ $\\lbrace u_j(y)\\rbrace _1^\\infty $ are smooth functions of $y \\in {R}$ .",
"The function $u_1(y)$ is given by (REF ) where $s = ir(0)$ and $u_P(y; s, 0, -s)$ denotes the smooth real-valued solution of the Painlevé II equation (REF ) corresponding to $(s,0,-s)$ via (REF ).",
"Theorem 2 If the reflection coefficient $r(k)$ satisfies $r(0) = 0$ , then the asymptotic formula (REF ) holds with $u_1(y) \\equiv 0$ and with $u_2(y)$ and $u_3(y)$ given explicitly by (REF ).",
"Remark 2.1 (Hierarchy of differential equations) Substituting the asymptotic expansion (REF ) into (REF ) and identifying coefficients of powers of $t^{-1/3}$ , we see that $\\lbrace u_j(y)\\rbrace _1^\\infty $ are smooth real-valued solutions of a hierarchy of linear ordinary differential equations.",
"The first four members of this hierarchy are given by $\\nonumber & u_1^{\\prime \\prime \\prime }- y u_1^{\\prime } - u_1 = 2 \\times 3^{2/3} (u_1^3)^{\\prime },\\\\\\nonumber & u_2^{\\prime \\prime \\prime } - y u_2^{\\prime } - 2u_2 = 6 \\times 3^{2/3}(u_1^2 u_2)^{\\prime },\\\\\\nonumber & u_3^{\\prime \\prime \\prime } - y u_3^{\\prime } - 3u_3 = 6 \\times 3^{2/3}(u_1 u_2^2 + u_1^2 u_3)^{\\prime },\\\\& u_4^{\\prime \\prime \\prime } - y u_4^{\\prime } -4u_4 = 2 \\times 3^{2/3}(u_2^3 + 6u_1 u_2 u_3 + 3u_1^2 u_4)^{\\prime }.$ As expected, the first of these equations reduces to the Painlevé II equation (REF ) when $u_1(y)$ and $u^P(y)$ are related by (REF ).",
"Moreover, in the special case when $r(0) = 0$ , we have $s = ir(0) = 0$ , so $u_1(y) \\equiv 0$ and the first equations in the hierarchy are $\\nonumber & u_2^{\\prime \\prime \\prime } - y u_2^{\\prime } - 2u_2 = 0,\\\\\\nonumber & u_3^{\\prime \\prime \\prime } - y u_3^{\\prime } - 3u_3 = 0,\\\\\\nonumber & u_4^{\\prime \\prime \\prime } - y u_4^{\\prime } -4u_4 = 2 \\times 3^{2/3}(u_2^3)^{\\prime }.$ The expressions (REF ) for $u_2(y)$ and $u_3(y)$ are consistent with these equations, because, for each integer $j \\ge 1$ , $& f(y) = \\frac{d^{j-1}}{dy^{j-1}}\\text{\\upshape Ai}(y),$ satisfies the homogeneous linear ODE $f^{\\prime \\prime \\prime } - y f^{\\prime } - j f = 0$ and $\\text{\\upshape Ai}^{\\prime \\prime }(y) = y\\text{\\upshape Ai}(y)$ ."
],
[
"Riemann–Hilbert problem",
"According to the inverse scattering transform formalism, the solution $u(x,t)$ of the Cauchy problem for (REF ) on ${R}$ with initial data $u_0(x)$ in the Schwartz class $\\mathcal {S}({R})$ is given by $u(x,t) = 2\\lim _{k \\rightarrow \\infty } k(m(x,t,k))_{21},$ where $m(x,t,k)$ denotes the unique solution of the RH problem ${\\left\\lbrace \\begin{array}{ll}\\text{$m(x, t, k)$ is analytic for $k \\in {R}$,} \\\\\\text{$m_+(x,t,k) = m_-(x, t, k) v(x, t, k)$ for $k \\in {R}$,} \\\\\\text{$m(x,t,k) = I + O(k^{-1})$ as $k \\rightarrow \\infty $,}\\end{array}\\right.",
"}$ with jump matrix $v(x,t,k)$ given by (cf.",
"[4]) $& v(x,t,k) = \\begin{pmatrix} 1 - |r(k)|^2 & -\\overline{r(k)}e^{-t\\Phi (\\zeta ,k)} \\\\ r(k)e^{t \\Phi (\\zeta ,k)} & 1 \\end{pmatrix},\\\\& \\Phi (\\zeta , k) := 2i(\\zeta k + 4k^3), \\qquad \\zeta := \\frac{x}{t},$ and $m_+$ and $m_-$ denote the boundary values of $m$ from the upper and lower half-planes, respectively.",
"If $u_0 \\in \\mathcal {S}({R})$ , then the reflection coefficient $r(k)$ belongs to the Schwartz class and obeys the bound $\\sup _{k \\in {R}} |r(k)| < 1$ and the symmetry $r(k) = -\\overline{r(-k)}, \\qquad k \\in {R}.$ It follows that $v$ obeys the symmetries $v(x,t,k) = \\sigma _1\\overline{v(x,t,\\bar{k})}^{-1}\\sigma _1 = \\sigma _1 \\sigma _3 v(x,t,-k)^{-1}\\sigma _3\\sigma _1,\\qquad k \\in {R},$ where $\\sigma _1 = \\begin{pmatrix} 0 & 1 \\\\1 & 0 \\end{pmatrix}, \\quad \\sigma _2 = \\begin{pmatrix} 0 & -i \\\\ i & 0 \\end{pmatrix}, \\quad \\sigma _3 = \\begin{pmatrix} 1 & 0 \\\\0 & -1 \\end{pmatrix}.$ Hence, by uniqueness of the solution of the RH problem (REF ), $m$ satisfies $m(x,t,k) = \\sigma _1\\overline{m(x,t,\\bar{k})}\\sigma _1 = \\sigma _1 \\sigma _3 m(x,t,-k)\\sigma _3\\sigma _1, \\qquad k \\in {R}.$ Figure: NO_CAPTIONThe proofs of our two theorems rely on a Deift-Zhou [4], [5] steepest descent analysis of the RH problem (REF ).",
"There are two critical points (i.e., solutions of the equation $\\partial _k\\Phi = 0$ ) which are relevant for this analysis; they are given by $\\pm k_0$ , where (see Figure REF ) $k_0 := {\\left\\lbrace \\begin{array}{ll} i\\sqrt{\\frac{\\zeta }{12}}, & \\zeta \\ge 0, \\\\\\sqrt{\\frac{|\\zeta |}{12}}, & \\zeta < 0.\\end{array}\\right.",
"}$"
],
[
"Notation",
"We will derive the asymptotics in the two halves of Sector IV corresponding to $x \\le 0$ and $x \\ge 0$ separately; we use the notation $\\text{\\upshape IV}_\\le := \\text{\\upshape IV}\\cap \\lbrace x \\le 0\\rbrace \\quad \\text{and} \\quad \\text{\\upshape IV}_\\ge := \\text{\\upshape IV}\\cap \\lbrace x \\ge 0\\rbrace .$ If $D$ is an open connected subset of $ bounded by a piecewise smooth curve $ D := {}$, then we write $ E2(D)$ for the space of all analytic functions $ f:D with the property that there exist curves $\\lbrace C_n\\rbrace _1^\\infty $ in $D$ tending to $\\partial D$ in the sense that $C_n$ eventually surrounds each compact subset of $D$ and such that $\\sup _{n \\ge 1} \\int _{C_n} |f(z)|^2 |dz| < \\infty .$ If $D = D_1 \\cup \\cdots \\cup D_n$ is a finite union of such open subsets, then $\\dot{E}^2(D)$ denotes the space of functions $f:D\\rightarrow such that $ f|Dj E2(Dj)$ for each $ j$.We can formulate the classical RH problem (\\ref {preRHm}) in this setting as{\\begin{@align}{1}{-1}{\\left\\lbrace \\begin{array}{ll}m(x, t, \\cdot ) \\in I + \\dot{E}^2({R}),\\\\m_+(x,t,k) = m_-(x, t, k) v(x, t, k) \\quad \\text{for a.e.}",
"\\ k \\in {R}.\\end{array}\\right.",
"}\\end{@align}}We write $ E(D)$ for the space of bounded analytic functions on $ D$.We use $ c>0$ and $ C > 0$ to denote generic constants which may change within a computation.$ Given a piecewise smooth contour $\\Gamma \\subset \\hat{, we denote the Cauchy operator \\mathcal {C} \\equiv \\mathcal {C}^\\Gamma associated with \\Gamma by{\\begin{@align}{1}{-1}(\\mathcal {C}h)(k) = \\frac{1}{2\\pi i} \\int _{\\Gamma } \\frac{h(k^{\\prime })dk^{\\prime }}{k^{\\prime } - k}, \\qquad k \\in \\Gamma .\\end{@align}}If h \\in L^2(\\Gamma ), then the left and right nontangential boundary values of \\mathcal {C} h, which we denote by \\mathcal {C}_+ h and \\mathcal {C}_- h respectively, exist a.e.",
"on \\Gamma and belong to L^2(\\Gamma ).",
"We let \\mathcal {B}(L^2(\\Gamma )) denote the space of bounded linear operators on L^2(\\Gamma ) and, for w \\in L^\\infty (\\Gamma ), we define \\mathcal {C}_{w} \\in \\mathcal {B}(L^2(\\Gamma )) by \\mathcal {C}_{w}h = \\mathcal {C}_-(hw).We write f^*(k) := \\overline{f(\\bar{k})} for the Schwartz conjugate of a function f(k).If A is an n \\times m matrix, we define |A|\\ge 0 by|A|^2 = \\sum _{i,j} |A_{ij}|^2; then |A + B| \\le |A| + |B| and |AB| \\le |A| |B|.",
"}$"
],
[
"Asymptotics in Sector $\\text{\\upshape IV}_\\ge $",
"We first consider the asymptotics in Sector $\\text{\\upshape IV}_\\ge $ given by $\\text{\\upshape IV}_\\ge = \\lbrace (x,t) \\, | \\, t\\ge 1 \\; \\text{and} \\; 0 \\le x \\le M t^{1/3}\\rbrace ,$ where $M > 0$ is a constant.",
"In this sector, the critical points $\\pm k_0 = \\pm i \\sqrt{\\frac{x}{12 t}}$ are pure imaginary and approach 0 at least as fast as $t^{-1/3}$ as $t \\rightarrow \\infty $ , i.e., $|k_0| \\le C t^{-1/3}$ ."
],
[
"The solution $m^{(1)}$",
"The first step is to transform the RH problem in such a way that the new jump matrix approaches the identity matrix as $t \\rightarrow \\infty $ .",
"Let $\\Gamma ^{(1)} \\subset denote the contour$$\\Gamma ^{(1)} := {R}\\cup e^{\\frac{\\pi i}{6}}{R}\\cup e^{-\\frac{\\pi i}{6}}{R}$$oriented to the right as in Figure \\ref {Gamma1geq.pdf} and let $ V$ and $ V*$ denote the open subsets shown in the same figure:{\\begin{@align*}{1}{-1}& V = \\lbrace \\arg k \\in (0, \\pi /6)\\rbrace \\cup \\lbrace \\arg k \\in (5\\pi /6, \\pi )\\rbrace ,\\\\& V^* = \\lbrace \\arg k \\in (-\\pi , -5\\pi /6)\\rbrace \\cup \\lbrace \\arg k \\in (-\\pi /6, 0)\\rbrace .\\end{@align*}}$ We first need to decompose $r$ into an analytic part $r_a$ and a small remainder $r_r$ .",
"Let $N \\ge 2$ be an integer.",
"Assume for definiteness that $N$ is even.",
"Figure: NO_CAPTIONLemma 4.1 (Analytic approximation for $\\zeta \\ge 0$ ) There exists a decomposition $& r(k) = r_{a}(t, k) + r_{r}(t, k), \\qquad t \\ge 1, \\ k \\in {R},$ where the functions $r_{a}$ and $r_{r}$ have the following properties: For each $t\\ge 1$ , $r_{a}(t, k)$ is defined and continuous for $k \\in \\bar{V}$ and analytic for $k \\in V$ , where $\\bar{V}$ denotes the closure of $V$ .",
"The function $r_{a}$ obeys the following estimates uniformly for $\\zeta \\ge 0$ and $t \\ge 1$ : $& |r_{a}(t, k)| \\le \\frac{C}{1 + |k|} e^{\\frac{t}{4}|\\text{\\upshape Re\\,}\\Phi (\\zeta ,k)|}, \\qquad k \\in \\bar{V},$ and $\\bigg |r_{a}(t, k) - \\sum _{j=0}^N \\frac{r^{(j)}(0)}{j!}",
"k^j\\bigg | \\le C |k|^{N+1} e^{\\frac{t}{4}|\\text{\\upshape Re\\,}\\Phi (\\zeta ,k)|}, \\qquad k \\in \\bar{V}.$ The $L^1$ and $L^\\infty $ norms of $r_{r}(t, \\cdot )$ on ${R}$ are $O(t^{-N})$ as $t \\rightarrow \\infty $ .",
"$r_{a}(t, k) = -r_{a}^*(t, -k)$ for $k \\in \\bar{V}$ and $r_{r}(t, k) = -r_{r}^*(t, -k)$ for $k \\in {R}$ .",
"The symmetry (REF ) implies that $\\text{\\upshape Re\\,}r(k) = -\\text{\\upshape Re\\,}r(-k)$ and $\\text{\\upshape Im\\,}r(k) = \\text{\\upshape Im\\,}r(-k)$ for $k \\in {R}$ .",
"It follows that $(k^2 + 1)^{3N+4}k^{-1} \\text{\\upshape Re\\,}r(k) \\quad \\text{and} \\quad (k^2 + 1)^{3N+4}\\text{\\upshape Im\\,}r(k)$ are smooth even functions of $k \\in {R}$ .",
"Considering their Taylor expansions at $k = 0$ , we infer the existence of real coefficients $\\lbrace a_j, b_j\\rbrace _{j=0}^{2N+1}$ such that ${\\left\\lbrace \\begin{array}{ll}(k^2 + 1)^{3N+4}k^{-1}\\text{\\upshape Re\\,}r(k) = \\sum _{j = 0}^{2N+1} a_j k^{2j} + O(k^{4N+4}), \\\\(k^2 + 1)^{3N+4}\\text{\\upshape Im\\,}r(k) = \\sum _{j = 0}^{2N+1} b_j k^{2j} + O(k^{4N+4}),\\end{array}\\right.}",
"\\ k \\rightarrow 0,$ i.e., $r(k) = r_0(k) + O\\big (k^{4N+4}\\big ), \\qquad k \\rightarrow 0,$ where the rational function $r_0(k)$ is defined by $r_0(k) = \\frac{1}{(k^2+1)^{3N+4}} \\sum _{j = 0}^{2N+1} (ka_j + ib_j) k^{2j}.$ Note that $r_0(k)$ is analytic away from $k = \\pm i$ and satisfies $r_0(k) = -r_0^*(-k)$ .",
"We conclude that $f(k) := r(k) - r_0(k)$ also obeys the symmetry $f(k) = -f^*(-k)$ , $k \\in {R}$ , and that $\\frac{d^n f}{dk^n} (k) ={\\left\\lbrace \\begin{array}{ll}O(k^{4N+4-n}), \\quad & k \\rightarrow 0,\\\\O(k^{-2N-5}), & k \\rightarrow \\pm \\infty ,\\end{array}\\right.",
"}\\ n = 0,1,\\dots , N+1.$ The decomposition of $r(k)$ can now be derived as follows.",
"The map $k \\mapsto \\phi \\equiv \\phi (k)$ defined by $\\phi = -i\\Phi (0, k) = 8k^3$ is an increasing bijection ${R}\\rightarrow {R}$ , so we may define a function $F(\\phi )$ by $F(\\phi ) = \\frac{(k^2+1)^{N+2}}{k^{N+1}} f(k), \\qquad \\phi \\in {R}.$ The function $F(\\phi )$ is smooth for $\\phi \\in {R}\\setminus \\lbrace 0\\rbrace $ and $\\frac{d^n F}{d \\phi ^n}(\\phi ) = \\bigg (\\frac{1}{24k^2} \\frac{d}{d k}\\bigg )^n\\bigg [\\frac{(k^2+1)^{N+2}}{k^{N+1}} f(k)\\bigg ], \\qquad \\phi \\in {R}\\setminus \\lbrace 0\\rbrace .$ In view of (REF ), we have $F \\in C^N({R})$ , and, for $n = 0, 1, \\dots , N+1$ , $\\bigg \\Vert \\frac{d^n F}{d \\phi ^n}\\bigg \\Vert _{L^2({R})}^2= \\int _{R}\\bigg |\\frac{d^n F}{d \\phi ^n}(\\phi )\\bigg |^2 d\\phi = \\int _{R}\\bigg |\\frac{d^n F}{d \\phi ^n}(\\phi )\\bigg |^2 \\frac{d \\phi }{d k} dk < \\infty .$ It follows that $F$ belongs to the Sobolev space $H^{N+1}({R})$ and the Fourier transform $\\hat{F}(s)$ defined by $\\hat{F}(s) = \\frac{1}{2\\pi } \\int _{{R}} F(\\phi ) e^{-i\\phi s} d\\phi ,$ satisfies $F(\\phi ) = \\int _{{R}} \\hat{F}(s) e^{i\\phi s} ds$ and, by the Plancherel theorem, $\\Vert s^{N+1} \\hat{F}\\Vert _{L^2({R})} < \\infty $ .",
"By (REF ) and (REF ), we have $\\frac{k^{N+1}}{(k^2+1)^{N+2}} \\int _{{R}} \\hat{F}(s) e^{s\\Phi (0,k)} ds= f(k), \\qquad k \\in {R}.$ Let us write $f(k) = f_{a}(t, k) + f_{r}(t, k), \\qquad t\\ge 1, \\ k \\in {R},$ where the functions $f_a$ and $f_r$ are defined by $& f_a(t,k) = \\frac{k^{N+1}}{(k^2+1)^{N+2}}\\int _{-\\frac{t}{4}}^{\\infty } \\hat{F}(s) e^{s\\Phi (0,k)} ds, \\qquad t\\ge 1, \\ k \\in \\bar{V},\\\\& f_r(t,k) = \\frac{k^{N+1}}{(k^2+1)^{N+2}}\\int _{-\\infty }^{-\\frac{t}{4}} \\hat{F}(s) e^{s\\Phi (0,k)} ds,\\qquad t\\ge 1, \\ k \\in {R}.$ Since $\\text{\\upshape Re\\,}\\Phi (\\zeta , k) \\le \\text{\\upshape Re\\,}\\Phi (0,k) \\le 0$ for $\\zeta \\ge 0$ and $k \\in \\bar{V}$ , the function $f_a(t, \\cdot )$ is continuous in $\\bar{V}$ and analytic in $V$ , and we find $\\nonumber |f_a(t, k)|&\\le \\frac{|k|^{N+1}}{|k^2+1|^{N+2}} \\Vert \\hat{F}\\Vert _{L^1({R})} \\sup _{s \\ge -\\frac{t}{4}} e^{s \\text{\\upshape Re\\,}\\Phi (0,k)}\\le \\frac{C|k|^{N+1}}{1 + |k|^{2N+4}} e^{\\frac{t}{4} |\\text{\\upshape Re\\,}\\Phi (0,k)|}\\\\&\\le \\frac{C|k|^{N+1}}{1 + |k|^{2N+4}} e^{\\frac{t}{4} |\\text{\\upshape Re\\,}\\Phi (\\zeta ,k)|}, \\qquad \\zeta \\ge 0, \\ t \\ge 1, \\ k \\in \\bar{V},$ and, by the Cauchy-Schwartz inequality, $\\nonumber |f_r(t, k)| & \\le \\frac{|k|^{N+1}}{|k^2+1|^{N+2}} \\int _{-\\infty }^{-\\frac{t}{4}} s^{N+1} |\\hat{F}(s)| s^{-N-1} ds\\\\\\nonumber & \\le \\frac{C}{1 + |k|^2} \\Vert s^{N+1} \\hat{F}(s)\\Vert _{L^2({R})} \\sqrt{\\int _{-\\infty }^{-\\frac{t}{4}} |s|^{-2N-2} ds}\\\\ \\nonumber & \\le \\frac{C}{1 + |k|^2} t^{-N - \\frac{1}{2}}, \\qquad \\zeta \\ge 0, \\ t \\ge 1, \\ k \\in {R}.$ In particulary, the $L^1$ and $L^\\infty $ norms of $f_r$ on ${R}$ are $O(t^{-N - \\frac{1}{2}})$ .",
"The symmetries $f(k) = -f^*(-k)$ and $\\phi (k) = -\\phi (-k)$ together with the fact that $N$ is even imply that $F(\\phi ) = \\overline{F(-\\phi )}$ .",
"Hence $\\hat{F}(s)$ is real valued for $s \\in {R}$ .",
"Since $\\Phi (\\zeta , k) = \\Phi ^*(\\zeta , -k)$ , this leads to the symmetries $f_a(t, k) = -f_a^*(t, -k)$ and $f_r(t, k) = -f_r^*(t, -k)$ .",
"Letting $& r_{a}(t, k) = r_0(k) + f_a(t, k), \\qquad t \\ge 1, \\ k \\in \\bar{V},\\\\& r_{r}(t, k) = f_r(t, k), \\qquad t \\ge 1, \\ k \\in {R},$ we obtain a decomposition of $r$ with the asserted properties.",
"Let $r = r_a + r_r$ be a decomposition of $r$ as in Lemma REF and define the sectionally analytic function $m^{(1)}$ by $m^{(1)}(x,t,k) = m(x,t,k)G(x,t,k),$ where $G(x,t,k) = {\\left\\lbrace \\begin{array}{ll}\\begin{pmatrix}1 & 0 \\\\-r_{a} e^{t\\Phi } & 1\\end{pmatrix}, & k \\in V,\\\\\\begin{pmatrix}1 & - r_{a}^* e^{-t\\Phi } \\\\0 & 1\\end{pmatrix}, & k \\in V^*,\\\\I, & \\text{elsewhere}.\\end{array}\\right.",
"}$ Lemma REF implies that $G(x,t,\\cdot ) \\in I + (\\dot{E}^2 \\cap E^\\infty )(V \\cup V^*).$ Thus $m$ satisfies the RH problem () if and only if $m^{(1)}$ satisfies the RH problem ${\\left\\lbrace \\begin{array}{ll}m^{(1)}(x, t, \\cdot ) \\in I + \\dot{E}^2(\\Gamma ^{(1)}),\\\\m^{(1)}_+(x,t,k) = m^{(1)}_-(x, t, k) v^{(1)}(x, t, k) \\quad \\text{for a.e.}",
"\\ k \\in \\Gamma ^{(1)},\\end{array}\\right.",
"}$ where $\\nonumber & v_1^{(1)} = \\begin{pmatrix}1 & 0 \\\\r_{a} e^{t\\Phi } & 1\\end{pmatrix}, \\qquad v_2^{(1)} = \\begin{pmatrix}1 & - r_{a}^* e^{-t\\Phi } \\\\0 & 1\\end{pmatrix},\\\\& v_3^{(1)} = \\begin{pmatrix}1 - |r_{r}|^2 & - r_{r}^* e^{-t\\Phi } \\\\r_{r} e^{t\\Phi } & 1\\end{pmatrix},$ and $v_j^{(1)}$ denotes the restriction of $v^{(1)}$ to the subcontour labeled by $j$ in Figure REF ."
],
[
"Local model",
"The RH problem for $m^{(1)}$ has the property that the matrix $v^{(1)} - I$ is uniformly small as $t \\rightarrow \\infty $ everywhere on $\\Gamma ^{(1)}$ except possibly near the origin.",
"Hence we only have to consider a neighborhood of the origin when computing the long-time asymptotics of $m^{(1)}$ .",
"In this subsection, we find a local solution $m_0$ which approximates $m^{(1)}$ near 0.",
"We introduce the variables $y$ and $z$ by $y := \\frac{x}{(3t)^{1/3}}, \\qquad z := (3t)^{1/3}k.$ Note that $0 \\le y \\le C$ in Sector $\\text{\\upshape IV}_\\ge $ .",
"The definitions of $y$ and $z$ are chosen such that $t\\Phi (\\zeta , k) = 2i\\bigg (y z + \\frac{4z^3}{3}\\bigg ).$ Fix $\\epsilon > 0$ .",
"Let $D_\\epsilon (0) = \\lbrace k \\in | \\, |k| < \\epsilon \\rbrace $ denote the open disk of radius $\\epsilon $ centered at the origin.",
"The map $k \\mapsto z$ maps $D_\\epsilon (0)$ onto the open disk $D_{(3t)^{1/3}\\epsilon }(0)$ of radius $(3t)^{1/3}\\epsilon $ in the complex $z$ -plane.",
"Let $\\mathcal {Y}^\\epsilon = (\\Gamma ^{(1)} \\cap D_\\epsilon (0))\\setminus {R}$ .",
"The map $k \\mapsto z$ takes $\\mathcal {Y}^\\epsilon $ onto $Y \\cap \\lbrace |z| < (3t)^{1/3}\\epsilon \\rbrace $ , where $Y$ is the contour defined in (REF ).",
"We write $\\mathcal {Y}^\\epsilon = \\cup _{j=1}^4\\mathcal {Y}_j^\\epsilon $ , where $\\mathcal {Y}_j^\\epsilon $ denotes the part of $\\mathcal {Y}^\\epsilon $ that maps into $Y_j$ , see Figure REF .",
"Figure: NO_CAPTIONThe long-time asymptotics in Sector $\\text{\\upshape IV}_\\ge $ is related to the solution $m^Y(y,t,z)$ of the RH problem (REF ) with the polynomial $p$ in (REF ) given by $p(t,z) = \\sum _{j=0}^N \\frac{r^{(j)}(0)}{j!}",
"k^j = \\sum _{j=0}^N \\frac{r^{(j)}(0)}{j!3^{j/3}} \\frac{z^j}{t^{j/3}}.$ In particular, the first few coefficients in (REF ) are $s = r(0) \\in i{R}, \\qquad p_1 = \\frac{r^{\\prime }(0)}{3^{1/3}} \\in {R}, \\qquad p_2 = \\frac{r^{\\prime \\prime }(0)}{2 \\times 3^{2/3}} \\in i{R}.$ Define $m_0(x,t,k)$ for $k \\in D_\\epsilon (0)$ by $m_0(x, t, k) = m^Y(y, t, z), \\qquad k \\in D_\\epsilon (0).$ By Lemma REF , we can choose $T \\ge 1$ such that $m_0$ is well-defined whenever $(x,t) \\in \\text{\\upshape IV}_\\ge ^T$ , where $\\text{\\upshape IV}_\\ge ^T$ denotes the part of $\\text{\\upshape IV}_\\ge $ where $t\\ge T$ , i.e., $\\text{\\upshape IV}_\\ge ^T := \\text{\\upshape IV}_\\ge \\cap \\lbrace t \\ge T\\rbrace .$ By (REF ), we have $p(t,z) = -\\overline{p(t,-\\bar{z})}$ .",
"Hence (REF ) implies that $m_0$ obeys the same symmetries (REF ) as $m$ .",
"Lemma 4.2 For each $(x,t) \\in \\text{\\upshape IV}_\\ge ^T$ , the function $m_0(x,t,k)$ defined in (REF ) is an analytic function of $k \\in D_\\epsilon (0) \\setminus \\mathcal {Y}^\\epsilon $ such that $|m_0(x,t,k) | \\le C, \\qquad (x,t) \\in \\text{\\upshape IV}_\\ge ^T, \\ k \\in D_\\epsilon (0) \\setminus \\mathcal {Y}^\\epsilon .$ Across $\\mathcal {Y}^\\epsilon $ , $m_0$ obeys the jump condition $m_{0+} = m_{0-} v_0$ , where the jump matrix $v_0$ satisfies, for each $1 \\le p \\le \\infty $ , $\\Vert v^{(1)} - v_0\\Vert _{L^p(\\mathcal {Y}^\\epsilon )} \\le Ct^{-\\frac{N+1}{3}}, \\qquad (x,t) \\in \\text{\\upshape IV}_\\ge ^T.$ Furthermore, $m_0(x,t,k)^{-1} = I + \\sum _{j=1}^N \\sum _{l=0}^N \\frac{m_{0,jl}(y)}{k^j t^{(j+l)/3}} + O\\big (t^{-\\frac{N+1}{3}}\\big )$ uniformly for $(x,t) \\in \\text{\\upshape IV}_\\ge ^T$ and $k \\in \\partial D_\\epsilon (0)$ , where the coefficients $m_{0,jl}(y)$ are smooth functions of $y \\in [0,\\infty )$ .",
"In particular, $\\Vert m_0(x,t,\\cdot )^{-1} - I\\Vert _{L^\\infty (\\partial D_\\epsilon (0))} = O\\big (t^{-1/3}\\big ),$ and $& \\frac{1}{2\\pi i}\\int _{\\partial D_\\epsilon (0)}(m_0(x,t,k)^{-1} - I) dk= \\sum _{l=1}^N \\frac{g_l(y)}{t^{l/3}} + O\\big (t^{-\\frac{N+1}{3}}\\big ),$ uniformly for $(x,t) \\in \\text{\\upshape IV}_\\ge ^T$ , where $g_{l+1}(y) = m_{0,1l}(y)$ for each $l \\ge 0$ ; the first coefficient is given by $g_1(y) = -\\frac{3^{-1/3}}{2} \\begin{pmatrix} -i\\int _y^\\infty u_P^2(y^{\\prime }; s, 0, -s)dy^{\\prime } & u_P(y; s, 0, -s) \\\\ u_P(y; s, 0, -s) & i\\int _y^\\infty u_P^2(y^{\\prime }; s, 0, -s)dy^{\\prime } \\end{pmatrix}$ with $s = r(0)$ .",
"If $r(0) = 0$ , then $\\nonumber g_1(y) = &\\; 0,\\\\ \\nonumber g_2(y) = &\\; -\\frac{r^{\\prime }(0)}{4\\times 3^{2/3}} \\text{\\upshape Ai}^{\\prime }(y)\\sigma _1\\\\ g_3(y) = &\\; \\frac{i r^{\\prime }(0)^2}{24} \\int _y^{\\infty } (\\text{\\upshape Ai}^{\\prime }(y^{\\prime }))^2dy^{\\prime } \\sigma _3+ \\frac{ir^{\\prime \\prime }(0)}{48} y\\text{\\upshape Ai}(y) \\sigma _1.$ The analyticity of $m_0$ and the bound (REF ) are a consequence of Lemma REF .",
"Moreover, $& v^{(1)} - v_0 = {\\left\\lbrace \\begin{array}{ll}\\begin{pmatrix}0 & 0 \\\\(r_a(t,k) - p(t,z)) e^{t\\Phi } & 0\\end{pmatrix}, \\qquad k \\in \\mathcal {Y}_1^\\epsilon \\cup \\mathcal {Y}_2^\\epsilon ,\\\\\\begin{pmatrix}0 & -(r_a^*(t,k) - p^*(t, z)) e^{-t\\Phi } \\\\0 & 0\\end{pmatrix}, \\qquad k \\in \\mathcal {Y}_3^\\epsilon \\cup \\mathcal {Y}_4^\\epsilon .\\end{array}\\right.",
"}$ Since $\\text{\\upshape Re\\,}\\Phi (\\zeta , re^{\\frac{\\pi i}{6}}) = -8 r^3 - \\zeta r \\le -8r^3$ for $\\zeta \\ge 0$ and $r \\ge 0$ , we obtain $\\text{\\upshape Re\\,}\\Phi (\\zeta , k) \\le -8|k|^3, \\qquad k \\in \\mathcal {Y}_1^\\epsilon , \\ (x,t) \\in \\text{\\upshape IV}_\\ge ^T.$ In particular, $e^{-\\frac{3t}{4} |\\text{\\upshape Re\\,}\\Phi |} \\le Ce^{-6t|k|^3} \\le C e^{-2|z|^3}, \\qquad k \\in \\mathcal {Y}_1^\\epsilon .$ Thus, by (REF ), (REF ), and (REF ), $\\nonumber |v^{(1)} - v_0|& \\le C|r_a(t,k) - p(t,z)| e^{t \\text{\\upshape Re\\,}\\Phi }\\le C|zt^{-1/3}|^{N+1} e^{-\\frac{3t}{4} |\\text{\\upshape Re\\,}\\Phi |}\\\\& \\le C|zt^{-1/3}|^{N+1} e^{-2|z|^3}, \\qquad k \\in \\mathcal {Y}_1^\\epsilon .$ Consequently, writing $r = |z|$ , $& \\Vert v^{(1)} - v_0\\Vert _{L^\\infty (\\mathcal {Y}_1^{\\epsilon })}\\le C\\sup _{0 \\le r < \\infty } (rt^{-1/3})^{N+1} e^{-2r^3}\\le C t^{-(N+1)/3}$ and $& \\Vert v^{(1)} - v_0\\Vert _{L^1(\\mathcal {Y}_1^{\\epsilon })}\\le C\\int _0^{\\infty } (rt^{-1/3})^{N+1} e^{-2r^3} \\frac{dr}{t^{1/3}}\\le C t^{-(N+2)/3}.$ Since similar estimates apply to $\\mathcal {Y}_j^{\\epsilon }$ with $j = 2,3,4$ , this proves (REF ).",
"We next apply Lemma REF to determine the asymptotics of the solution $m_0$ .",
"The variable $z = (3t)^{1/3}k$ satisfies $|z| = (3t)^{1/3}\\epsilon $ if $|k| = \\epsilon $ .",
"Thus equation (REF ) yields $m_0(x,t,k) = &\\; I + \\sum _{j=1}^N \\sum _{l=0}^N \\frac{m_{jl}^Y(y)}{((3t)^{1/3}k)^j t^{l/3}}+ O\\big (t^{-\\frac{N+1}{3}}\\big )$ uniformly for $(x,t) \\in \\text{\\upshape IV}_\\ge ^T$ and $k \\in \\partial D_\\epsilon (0)$ .",
"It follows that the expansion (REF ) exists with coefficients $m_{0,jl}(y)$ which can be expressed in terms of the $m_{jl}^Y(y)$ ; the first coefficient is given by $m_{0,10}(y) = - 3^{-1/3}m_{10}^Y(y)$ .",
"Equation (REF ) and Cauchy's formula yield $\\frac{1}{2\\pi i}\\int _{\\partial D_\\epsilon (0)}(m_0^{-1} - I) dk= &\\; \\sum _{l=0}^{N} \\frac{m_{0,1l}(y)}{t^{\\frac{l+1}{3}}}+ O\\big (t^{-\\frac{N+2}{3}}\\big ).$ This proves (REF ).",
"The expression (REF ) follows from (REF ) because $g_1(y) = m_{0,10}(y) = - 3^{-1/3}m_{10}^Y(y).$ The estimate (REF ) follows from (REF ).",
"If $r(0) = 0$ , then $m_{10}^Y(y) = 0$ and the explicit expressions in (REF ) then follow from (REF ), (REF ), (REF ), and (REF ) by straightforward computations."
],
[
"The solution $\\hat{m}$",
"Let $\\hat{\\Gamma } := \\Gamma ^{(1)} \\cup \\partial D_\\epsilon (0)$ and assume that the boundary of $D_\\epsilon (0)$ is oriented counterclockwise, see Figure REF .",
"The function $\\hat{m}(x,t,k)$ defined by $\\hat{m}(x,t,k) = {\\left\\lbrace \\begin{array}{ll}m^{(1)}(x, t, k)m_0(x,t,k)^{-1}, & k \\in D_\\epsilon (0),\\\\m^{(1)}(x, t, k), & k \\in D_\\epsilon (0),\\end{array}\\right.",
"}$ satisfies the RH problem ${\\left\\lbrace \\begin{array}{ll}\\hat{m}(x, t, \\cdot ) \\in I + \\dot{E}^2(\\hat{\\Gamma }),\\\\\\hat{m}_+(x,t,k) = \\hat{m}_-(x, t, k) \\hat{v}(x, t, k) \\quad \\text{for a.e.}",
"\\ k \\in \\hat{\\Gamma },\\end{array}\\right.",
"}$ where the jump matrix $\\hat{v}(x,t,k)$ is given by $\\hat{v}= {\\left\\lbrace \\begin{array}{ll}m_{0-} v^{(1)} m_{0+}^{-1}, & k \\in \\hat{\\Gamma } \\cap D_\\epsilon (0), \\\\m_0^{-1}, & k \\in \\partial D_\\epsilon (0), \\\\v^{(1)}, & k \\in \\hat{\\Gamma } \\setminus \\overline{D_\\epsilon (0)}.\\end{array}\\right.",
"}$ We write $\\hat{\\Gamma }$ as the union of four subcontours as follows: $\\hat{\\Gamma } = \\partial D_\\epsilon (0) \\cup \\mathcal {Y}^\\epsilon \\cup {R}\\cup \\hat{\\Gamma }^{\\prime },$ where $\\hat{\\Gamma }^{\\prime } := \\Gamma ^{(1)} \\setminus ({R}\\cup \\overline{D_\\epsilon (0)})$ .",
"Figure: NO_CAPTIONLemma 4.3 Let $\\hat{w} = \\hat{v} - I$ .",
"For each $1 \\le p \\le \\infty $ , the following estimates hold uniformly for $(x,t) \\in \\text{\\upshape IV}_\\ge ^T$ : $& \\Vert \\hat{w}\\Vert _{L^p(\\partial D_\\epsilon (0))} \\le Ct^{-1/3},\\\\& \\Vert \\hat{w}\\Vert _{L^p(\\mathcal {Y}^\\epsilon )} \\le Ct^{-(N+1)/3},\\\\ & \\Vert \\hat{w}\\Vert _{L^p({R})} \\le C t^{-N},\\\\& \\Vert \\hat{w}\\Vert _{L^p(\\hat{\\Gamma }^{\\prime })} \\le Ce^{-ct}.$ The estimate (REF ) follows from (REF ).",
"For $k \\in \\mathcal {Y}^\\epsilon $ , we have $\\hat{w} = m_{0-} (v^{(1)} - v_0) m_{0+}^{-1},$ so (REF ) and (REF ) yield ().",
"On ${R}$ , the jump matrix $v^{(1)}$ involves the small remainder $r_r$ (see the expression for $v_3^{(1)}$ in (REF )), so the estimates () hold as a consequence of Lemma REF and (for the part of ${R}$ that lies in $D_\\epsilon (0)$ ) the boundedness (REF ) of $m_0$ .",
"Finally, () follows because $e^{-t|\\text{\\upshape Re\\,}\\Phi |} \\le Ce^{-ct}$ uniformly on $\\hat{\\Gamma }^{\\prime }$ .",
"The estimates in Lemma REF show that $\\Vert \\hat{w}\\Vert _{(L^1 \\cap L^\\infty )(\\hat{\\Gamma })} \\le Ct^{-1/3}, \\qquad (x,t) \\in \\text{\\upshape IV}_\\ge ^T.$ In particular, $\\Vert \\hat{w}\\Vert _{L^\\infty (\\hat{\\Gamma })} \\rightarrow 0$ uniformly as $t \\rightarrow \\infty $ .",
"Thus, increasing $T$ if necessary, we may assume that $\\Vert \\hat{\\mathcal {C}}_{\\hat{w}}\\Vert _{\\mathcal {B}(L^2(\\hat{\\Gamma }))} \\le C \\Vert \\hat{w}\\Vert _{L^\\infty (\\hat{\\Gamma })} \\le 1/2$ for all $(x,t) \\in \\text{\\upshape IV}_\\ge ^T$ , where $\\hat{\\mathcal {C}}_{\\hat{w}}f := \\hat{\\mathcal {C}}_-(f\\hat{w})$ and $\\hat{\\mathcal {C}}$ is the Cauchy operator associated with $\\hat{\\Gamma }$ , see Section REF .",
"It follows from a standard argument that if $(x,t) \\in \\text{\\upshape IV}_\\ge ^T$ , then the RH problem (REF ) for $\\hat{m}$ has a unique solution given by $\\hat{m}(x, t, k) = I + \\hat{\\mathcal {C}}(\\hat{\\mu } \\hat{w}) = I + \\frac{1}{2\\pi i}\\int _{\\hat{\\Gamma }} (\\hat{\\mu } \\hat{w})(x, t, s) \\frac{ds}{s - k},$ where $\\hat{\\mu }(x, t, k) \\in I + L^2(\\hat{\\Gamma })$ is defined by $\\hat{\\mu } = I + (I - \\hat{\\mathcal {C}}_{\\hat{w}})^{-1}\\hat{\\mathcal {C}}_{\\hat{w}}I$ .",
"Also, by (REF ), $\\Vert \\hat{\\mu }(x,t,\\cdot ) - I\\Vert _{L^2(\\hat{\\Gamma })}\\le \\frac{C\\Vert \\hat{w}\\Vert _{L^2(\\hat{\\Gamma })}}{1 - \\Vert \\hat{\\mathcal {C}}_{\\hat{w}}\\Vert _{\\mathcal {B}(L^2(\\hat{\\Gamma }))}}\\le C t^{-1/3}, \\qquad (x,t) \\in \\text{\\upshape IV}_\\ge ^T.$ By (REF ) and (REF ), $\\hat{w}$ has an expansion to order $O(t^{-(N+1)/3})$ in $t$ as $t \\rightarrow \\infty $ , i.e., $\\hat{w}(x, t,k) = \\frac{\\hat{w}_1(y,k)}{t^{1/3}}+ \\frac{\\hat{w}_2(y,k)}{t^{2/3}} + \\cdots + \\frac{\\hat{w}_N(y,k)}{t^{N/3}} + \\frac{\\hat{w}_{err}(x,t,k)}{t^{(N+1)/3}}, \\quad (x,t) \\in \\text{\\upshape IV}_\\ge ^T, \\ k \\in \\hat{\\Gamma },$ where the coefficients $\\lbrace \\hat{w}_j\\rbrace _1^N$ are nonzero only for $k \\in \\partial D_\\epsilon (0)$ and, for $1 \\le p \\le \\infty $ and $ j = 1, \\dots , N$ , ${\\left\\lbrace \\begin{array}{ll}\\Vert \\hat{w}_j(y,\\cdot )\\Vert _{L^p(\\hat{\\Gamma })} \\le C,\\\\\\Vert \\hat{w}_{err}(x,t,\\cdot )\\Vert _{L^p(\\hat{\\Gamma })} \\le C,\\end{array}\\right.}",
"\\qquad (x,t) \\in \\text{\\upshape IV}_\\ge ^T.$ Since $\\hat{\\mu } = \\sum _{j=0}^N \\hat{\\mathcal {C}}_{\\hat{w}}^jI + (I-\\hat{\\mathcal {C}}_{\\hat{w}})^{-1}\\hat{\\mathcal {C}}_{\\hat{w}}^{N+1}I$ and $\\hat{\\mathcal {C}}_{\\hat{w}} = \\frac{\\hat{\\mathcal {C}}_{\\hat{w}_1}}{t^{1/3}} + \\frac{\\hat{\\mathcal {C}}_{\\hat{w}_2}}{t^{2/3}} + \\cdots + \\frac{\\hat{\\mathcal {C}}_{\\hat{w}_N}}{t^{N/3}} + \\frac{\\hat{\\mathcal {C}}_{\\hat{w}_{err}}}{t^{(N+1)/3}},$ it follows that (see the explanation of (REF ) for more details of a similar argument) $\\hat{\\mu }(x, t,k) = I + \\frac{\\hat{\\mu }_1(y,k)}{t^{1/3}} + \\cdots + \\frac{\\hat{\\mu }_N(y,k)}{t^{N/3}} + \\frac{\\hat{\\mu }_{err}(x,t,k)}{t^{(N+1)/3}}, \\qquad (x,t) \\in \\text{\\upshape IV}_\\ge ^T, \\ k \\in \\hat{\\Gamma },$ where the coefficients $\\lbrace \\hat{\\mu }_j(y,k)\\rbrace _1^N$ depend smoothly on $y \\in [0,\\infty )$ and ${\\left\\lbrace \\begin{array}{ll}\\Vert \\hat{\\mu }_j(y,\\cdot )\\Vert _{L^2(\\hat{\\Gamma })} \\le C, &\\\\\\Vert \\hat{\\mu }_{err}(x,t,\\cdot )\\Vert _{L^2(\\hat{\\Gamma })} \\le C, &\\end{array}\\right.}",
"\\ (x,t) \\in \\text{\\upshape IV}_\\ge ^T, \\ j = 1, \\dots , N.$"
],
[
"Asymptotics of $u$",
"By expanding (REF ) and inverting the transformations (REF ) and (REF ), we find the relation $\\lim _{k\\rightarrow \\infty } k(m(x,t,k) - I)= -\\frac{1}{2\\pi i}\\int _{\\hat{\\Gamma }} \\hat{\\mu }(x, t, k) \\hat{w}(x, t, k) dk.$ We will use (REF ) to compute the large $t$ asymptotics of $\\lim _{k\\rightarrow \\infty } k(m - I)$ ; then the asymptotics of $u(x,t)$ will follow from (REF ).",
"From now on until the end of Section , all equations involving error terms of the form $O(\\cdot )$ will be valid uniformly for all $(x,t) \\in \\text{\\upshape IV}_\\ge ^T$ .",
"By (REF ), (REF ), and (REF ), the contribution from $\\partial D_\\epsilon (0)$ to the right-hand side of (REF ) is $& -\\frac{1}{2\\pi i} \\int _{\\partial D_\\epsilon (0)} \\hat{w} dk - \\frac{1}{2\\pi i} \\int _{\\partial D_\\epsilon (0)} (\\hat{\\mu } - I) \\hat{w} dk= -\\frac{1}{2\\pi i}\\int _{\\partial D_\\epsilon (0)} (m_0^{-1} - I) dk\\\\& - \\frac{1}{2\\pi i} \\int _{\\partial D_\\epsilon (0)} \\bigg (\\sum _{j=1}^N \\frac{\\hat{\\mu }_j(y,k)}{t^{j/3}} + \\frac{\\hat{\\mu }_{err}(x,t,k)}{t^{(N+1)/3}}\\bigg )\\bigg (\\sum _{j=1}^N \\frac{\\hat{w}_j(y,k)}{t^{j/3}} + \\frac{\\hat{w}_{err}(x,t,k)}{t^{(N+1)/3}}\\bigg ) dk\\\\& = - \\sum _{j=1}^{N} \\frac{h_j(y)}{t^{j/3}} + O\\big (t^{-\\frac{N+1}{3}}\\big ),$ where $\\lbrace h_j(y)\\rbrace _1^N$ are smooth matrix-valued functions of $y \\in [0,\\infty )$ and $h_1(y) = g_1(y)$ .",
"By () and (REF ), the contribution from $\\mathcal {Y}^\\epsilon $ to the right-hand side of (REF ) is $-\\frac{1}{2\\pi i}\\int _{\\mathcal {Y}^\\epsilon } \\hat{\\mu } \\hat{w} dk= O\\big (\\Vert \\hat{w}\\Vert _{L^1(\\mathcal {Y}^\\epsilon )}& + \\Vert \\hat{\\mu } - I\\Vert _{L^2(\\mathcal {Y}^\\epsilon )}\\Vert \\hat{w}\\Vert _{L^2(\\mathcal {Y}^\\epsilon )}\\big )= O(t^{-(N+1)/3}).$ Similarly, by () and (REF ), the contribution from ${R}$ is $O(t^{-N})$ , and, by () and (REF ), the contribution from $\\hat{\\Gamma }^{\\prime }$ is $O(e^{- c t})$ .",
"In summary, we arrive at $\\lim _{k\\rightarrow \\infty } k(m(x,t,k) - I)= - \\sum _{j=1}^{N} \\frac{h_j(y)}{t^{j/3}} + O\\big (t^{-\\frac{N+1}{3}}\\big ).$ Recalling (REF ), this leads to the asymptotic formula $u(x,t) & = 2 \\lim _{k\\rightarrow \\infty }k(m(x,t,k))_{21}= \\sum _{j=1}^{N} \\frac{u_j(y)}{t^{j/3}}+ O(t^{-(N+1)/3}),$ where the $u_j(y)$ are smooth functions of $y \\in [0,\\infty )$ given by $u_j(y) = -2 (h_j(y))_{21}, \\qquad j = 1, \\dots , N.$ Using the expression (REF ) for $h_1(y) = g_1(y)$ , we find that $u_1(y)$ is given by (REF ).",
"This completes the proof of Theorem REF for Sector $\\text{\\upshape IV}_\\ge $ .",
"Let us finally assume that $r(0) = 0$ .",
"Then $\\hat{w}_1 = \\hat{\\mu }_1 = 0$ , and so $h_j(y) = g_j(y)$ for $j = 1,2,3$ .",
"Substituting the formulas (REF ) for $\\lbrace g_j\\rbrace _1^3$ into (REF ) gives $u_1 = 0$ and the expressions in (REF ) for $u_2(y)$ and $u_3(y)$ , thus completing the proof in Sector $\\text{\\upshape IV}_\\ge $ also for Theorem REF ."
],
[
"Asymptotics in Sector $\\text{\\upshape IV}_\\le $",
"We now consider the asymptotics in the sector $\\text{\\upshape IV}_\\le $ defined by $\\text{\\upshape IV}_\\le = \\lbrace (x,t) \\, | \\, t\\ge 1 \\; \\text{and} \\; -M t^{1/3} \\le x \\le 0\\rbrace ,$ where $M > 0$ is a constant.",
"In this sector, the critical points $\\pm k_0 = \\pm \\sqrt{\\frac{|x|}{12 t}}$ are real and approach 0 at least as fast as $t^{-1/3}$ as $t \\rightarrow \\infty $ , i.e., $|k_0| \\le C t^{-1/3}$ ."
],
[
"The solution $m^{(1)}$",
"As in Section , we begin by decomposing $r$ into an analytic part $r_a$ and a small remainder $r_r$ .",
"This time we define the contour $\\Gamma ^{(1)} \\equiv \\Gamma ^{(1)}(\\zeta )$ and the open subsets $V \\equiv V(\\zeta )$ and $V^* \\equiv V^*(\\zeta )$ as in Figure REF .",
"We still assume that $N \\ge 2$ is an integer.",
"We let $A > 0$ be a constant.",
"Figure: NO_CAPTIONLemma 5.1 (Analytic approximation for $-A \\le \\zeta \\le 0$ ) There exists a decomposition $& r(k) = r_{a}(x, t, k) + r_{r}(x, t, k), \\qquad k \\in (-\\infty , -k_0)\\cup (k_0, \\infty ),$ where the functions $r_{a}$ and $r_{r}$ have the following properties: For each $\\zeta \\in [-A, 0]$ and $t\\ge 1$ , $r_{a}(x, t, k)$ is defined and continuous for $k \\in \\bar{V}$ and analytic for $k \\in V$ .",
"The function $r_{a}$ obeys the following estimates uniformly for $\\zeta \\in [-A, 0]$ and $t \\ge 1$ : $& |r_{a}(x, t, k)| \\le \\frac{C}{1 + |k|} e^{\\frac{t}{4}|\\text{\\upshape Re\\,}\\Phi (\\zeta ,k)|}, \\qquad k \\in \\bar{V},$ and $\\bigg |r_{a}(x, t, k) - \\sum _{j=0}^N \\frac{r^{(j)}(k_0)}{j!}",
"(k-k_0)^j\\bigg | \\le C |k-k_0|^{N+1} e^{\\frac{t}{4}|\\text{\\upshape Re\\,}\\Phi (\\zeta ,k)|}, \\qquad k \\in \\bar{V}.$ The $L^1$ and $L^\\infty $ norms of $r_{r}(x, t, \\cdot )$ on $(-\\infty , -k_0)\\cup (k_0, \\infty )$ are $O(t^{-N})$ as $t \\rightarrow \\infty $ uniformly with respect to $\\zeta \\in [-A, 0]$ .",
"$r_{a}(x, t, k) = -r_{a}^*(x, t, -k)$ for $k \\in \\bar{V}$ and $r_{r}(x, t, k) = -r_{r}^*(x, t, -k)$ for $k \\in (-\\infty , -k_0)\\cup (k_0, \\infty )$ .",
"We write $V = V^{\\prime } \\cup V^{\\prime \\prime }$ , where $V^{\\prime }$ and $V^{\\prime \\prime }$ denote the parts of $V$ in the right and left half-planes, respectively.",
"We will derive a decomposition of $r$ in $V^{\\prime }$ and then use the symmetry (REF ) to extend it to $V^{\\prime \\prime }$ .",
"Taylor expansion of $\\rho (k) := (k-i)^{4N+7} r(k)$ around $k_0$ gives $(k-i)^{4N+7} r(k) = \\sum _{j=0}^{4N+3} \\frac{\\rho ^{(j)}(k_0)}{j!",
"}(k-k_0)^j + \\frac{1}{(4N+3)!}",
"\\int _{k_0}^k \\rho ^{(4N+4)}(u) (k-u)^{4N+3} du.$ Thus the function $f(\\zeta ,k) := r(k) - r_0(\\zeta ,k)= \\frac{1}{(4N+3)!",
"(k-i)^{4N+7}} \\int _{k_0}^k \\rho ^{(4N+4)}(u) (k-u)^{4N+3} du,$ where $r_0(\\zeta , k) := \\sum _{j=0}^{4N+3} \\frac{\\rho ^{(j)}(k_0)}{j!",
"}\\frac{(k-k_0)^j}{(k-i)^{4N+7}},$ satisfies the following estimates uniformly for $\\zeta \\in [-A, 0]$ : $\\frac{\\partial ^n f}{\\partial k^n} (\\zeta , k) ={\\left\\lbrace \\begin{array}{ll}O((k-k_0)^{4N+4-n}), \\quad & k \\rightarrow k_0,\\\\O(k^{-4}), & k \\rightarrow \\infty ,\\end{array}\\right.",
"}\\ n = 0,1,\\dots , N+1.$ The decomposition of $r(k)$ can now be derived as follows.",
"For each $\\zeta \\in [-A, 0]$ , the map $k \\mapsto \\phi \\equiv \\phi (\\zeta , k)$ where $\\phi = -i\\Phi (\\zeta , k) = 2\\zeta k + 8k^3$ is an increasing bijection $[k_0, \\infty ) \\rightarrow [\\phi (\\zeta , k_0), \\infty )$ .",
"Hence we may define a function $F(\\zeta , \\phi )$ by $F(\\zeta , \\phi ) = {\\left\\lbrace \\begin{array}{ll} \\frac{(k-i)^{N+3}}{(k-k_0)^{N+1}} f(\\zeta , k), & \\phi \\ge \\phi (\\zeta , k_0), \\\\0, & \\phi < \\phi (\\zeta , k_0),\\end{array}\\right.}",
"\\qquad \\zeta \\in [-A, 0], \\ \\phi \\in {R}.$ For each $\\zeta \\in [-A, 0]$ , the function $F(\\zeta , \\phi )$ is smooth for $\\phi \\ne \\phi (\\zeta ,k_0)$ and $\\frac{\\partial ^n F}{\\partial \\phi ^n}(\\zeta , \\phi ) = \\bigg (\\frac{1}{\\partial \\phi /\\partial k} \\frac{\\partial }{\\partial k}\\bigg )^n\\bigg [\\frac{(k-i)^{N+3}}{(k-k_0)^{N+1}} f(\\zeta , k)\\bigg ], \\qquad \\phi \\ge \\phi (\\zeta ,k_0),$ where $\\frac{\\partial \\phi }{\\partial k} = 24(k^2 -k_0^2).$ By (REF ) and (REF ), we have $F(\\zeta , \\cdot ) \\in C^N({R})$ for each $\\zeta $ and $\\bigg | \\frac{\\partial ^n F}{\\partial \\phi ^n}(\\zeta , \\phi )\\bigg | \\le \\frac{C}{1 + |\\phi |^{2/3}}, \\qquad \\phi \\in (\\phi (\\zeta , k_0), \\infty ), \\ \\zeta \\in [-A, 0], \\ n = 0,1, \\dots , N+1.$ Hence $\\sup _{\\zeta \\in [-A, 0]} \\big \\Vert \\partial _\\phi ^n F(\\zeta , \\cdot )\\big \\Vert _{L^2({R})} < \\infty , \\qquad n = 0,1, \\dots , N+1.$ In particular, $F(\\zeta , \\cdot )$ belongs to the Sobolev space $H^{N+1}({R})$ for each $\\zeta \\in [-A,0]$ .",
"We conclude that the Fourier transform $\\hat{F}(\\zeta , s)$ defined by (REF ) satisfies (REF ) and $\\sup _{\\zeta \\in [-A,0]} \\Vert s^{N+1} \\hat{F}(\\zeta , s)\\Vert _{L^2({R})} < \\infty .$ Equations (REF ) and (REF ) imply $ \\frac{(k-k_0)^{N+1}}{(k-i)^{N+3}}\\int _{{R}} \\hat{F}(\\zeta , s) e^{s\\Phi (\\zeta ,k)} ds= {\\left\\lbrace \\begin{array}{ll} f(\\zeta , k), \\quad & k \\ge k_0, \\\\0, & k < k_0,\\end{array}\\right.}",
"\\ \\zeta \\in [-A,0].$ We write $f(\\zeta , k) = f_a(x, t, k) + f_r(x, t, k), \\qquad \\zeta \\in [-A,0], \\ t \\ge 1, \\ k \\ge k_0,$ where the functions $f_a$ and $f_r$ are defined by $& f_a(x,t,k) = \\frac{(k-k_0)^{N+1}}{(k-i)^{N+3}}\\int _{-\\frac{t}{4}}^{\\infty } \\hat{F}(\\zeta ,s) e^{s\\Phi (\\zeta ,k)} ds, \\qquad k \\in \\bar{V}^{\\prime },\\\\& f_r(x,t,k) = \\frac{(k-k_0)^{N+1}}{(k-i)^{N+3}}\\int _{-\\infty }^{-\\frac{t}{4}} \\hat{F}(\\zeta ,s) e^{s\\Phi (\\zeta ,k)} ds,\\qquad k \\ge k_0.$ Since $\\text{\\upshape Re\\,}\\Phi (\\zeta , k) \\le 0$ for $k \\in \\bar{V}^{\\prime }$ , $f_a(x, t, \\cdot )$ is continuous in $\\bar{V}^{\\prime }$ and analytic in $V^{\\prime }$ .",
"Furthermore, $\\nonumber |f_a(x, t, k)|&\\le \\frac{|k-k_0|^{N+1}}{|k-i|^{N+3}} \\Vert \\hat{F}(\\zeta ,\\cdot )\\Vert _{L^1({R})} \\sup _{s \\ge -\\frac{t}{4}} e^{s \\text{\\upshape Re\\,}\\Phi (\\zeta ,k)}\\le \\frac{C|k-k_0|^{N+1}}{|k-i|^{N+3}} e^{\\frac{t}{4} |\\text{\\upshape Re\\,}\\Phi (\\zeta ,k)|}\\\\& \\hspace{142.26378pt} \\zeta \\in [-A,0], \\ t \\ge 1, \\ k \\in \\bar{V}^{\\prime },$ and $\\nonumber |f_r(x, t, k)| & \\le \\frac{|k-k_0|^{N+1}}{|k-i|^{N+3}} \\int _{-\\infty }^{-\\frac{t}{4}} s^{N+1} |\\hat{F}(\\zeta ,s)| s^{-N-1} ds\\\\\\nonumber & \\le \\frac{C}{1 + |k|^2} \\Vert s^{N+1} \\hat{F}(\\zeta ,s)\\Vert _{L^2({R})} \\sqrt{\\int _{-\\infty }^{-\\frac{t}{4}} |s|^{-2N-2} ds}\\\\& \\le \\frac{C}{1 + |k|^2} t^{-N-1/2}, \\qquad \\zeta \\in [-A,0], \\ t \\ge 1, \\ k \\ge k_0.$ Letting $& r_{a}(x, t, k) = r_0(\\zeta , k) + f_a(x, t, k), \\qquad k \\in \\bar{V}^{\\prime },\\\\& r_{r}(x, t, k) = f_r(x, t, k), \\qquad k \\ge k_0.$ we find a decomposition of $r$ for $k \\in (k_0,\\infty )$ with the asserted properties.",
"We use the symmetry (REF ) to extend this decomposition to $k \\in (-\\infty , -k_0)$ .",
"Using a decomposition of $r$ as provided by Lemma REF , we define $m^{(1)}$ by (REF ) with $G(x,t,k)$ given by (REF ).",
"By Lemma REF , $G(x,t,\\cdot ) \\in I + (\\dot{E}^2 \\cap E^\\infty )(V \\cup V^*),$ hence $m$ satisfies the RH problem () if and only if $m^{(1)}$ satisfies the RH problem (REF ), where the jump matrix $v^{(1)}$ is given by (REF ) and $v_4^{(1)} = \\begin{pmatrix}1 - |r|^2 & - r^* e^{-t\\Phi } \\\\r e^{t\\Phi } & 1\\end{pmatrix}$ with subscripts referring to Figure REF ."
],
[
"Local model",
"As in Section , we introduce the new variables $y$ and $z$ by (REF ).",
"We now have $-C \\le y \\le 0$ .",
"Let $\\mathcal {Z}^\\epsilon = (\\Gamma ^{(1)} \\cap D_\\epsilon (0))\\setminus ((-\\infty ,-k_0) \\cup (k_0, \\infty ))$ .",
"The map $k \\mapsto z$ takes $\\mathcal {Z}^\\epsilon $ onto $Z \\cap \\lbrace |z| < (3t)^{1/3}\\epsilon \\rbrace $ , where $Z$ is the contour defined in (REF ) with $z_0 := (3t)^{1/3}k_0 = \\sqrt{|y|}/2$ .",
"We write $\\mathcal {Z}^\\epsilon = \\cup _{j=1}^5\\mathcal {Z}_j^\\epsilon $ , where $\\mathcal {Z}_j^\\epsilon $ denotes the part of $\\mathcal {Z}^\\epsilon $ that maps into $Z_j$ , see Figure REF .",
"Figure: NO_CAPTIONThe long-time asymptotics in Sector $\\text{\\upshape IV}_\\le $ is related to the solution $m^Z(y,t,z)$ of the RH problem (REF ) with $p$ given by (REF ).",
"Let $\\text{\\upshape IV}_\\le ^T := \\text{\\upshape IV}_\\le \\cap \\lbrace t \\ge T\\rbrace $ .",
"The triple $(y,t,z_0)$ belongs to the parameter set $\\mathcal {P}_T$ in (REF ) whenever $(x,t) \\in \\text{\\upshape IV}_\\le ^T$ .",
"Thus, by Lemma REF , we can choose $T \\ge 1$ such that $m_0(x, t, k) := m^Z(y, t, z_0, z), \\qquad k \\in D_\\epsilon (0),$ is well-defined whenever $(x,t) \\in \\text{\\upshape IV}_\\le ^T$ .",
"By (REF ), $m_0$ obeys the same symmetries (REF ) as $m$ .",
"Lemma 5.2 For each $(x,t) \\in \\text{\\upshape IV}_\\le ^T$ , the function $m_0(x,t,k)$ defined in (REF ) is an analytic function of $k \\in D_\\epsilon (0) \\setminus \\mathcal {Z}^\\epsilon $ such that $|m_0(x,t,k)| \\le C, \\qquad (x,t) \\in \\text{\\upshape IV}_\\le ^T, \\ k \\in D_\\epsilon (0) \\setminus \\mathcal {Z}^\\epsilon .$ Across $\\mathcal {Z}^\\epsilon $ , $m_0$ obeys the jump condition $m_{0+} = m_{0-} v_0$ , where the jump matrix $v_0$ satisfies, for each $1 \\le p \\le \\infty $ , $\\Vert v^{(1)} - v_0\\Vert _{L^p(\\mathcal {Z}^\\epsilon )} \\le Ct^{-\\frac{N+1}{3}}, \\qquad (x,t) \\in \\text{\\upshape IV}_\\le ^T.$ Furthermore, $m_0(x,t,k)^{-1} = I + \\sum _{j=1}^N \\sum _{l=0}^N \\frac{m_{0,jl}(y)}{k^j t^{(j+l)/3}} + O(t^{-(N+1)/3})$ uniformly for $(x,t) \\in \\text{\\upshape IV}_\\le ^T$ and $k \\in \\partial D_\\epsilon (0)$ , where the coefficients $m_{0,jl}(y)$ are smooth extensions to all $y \\in {R}$ of the coefficients in (REF ).",
"The function $m_0(x,t,k)$ satisfies (REF )-(REF ) uniformly for $(x,t) \\in \\text{\\upshape IV}_\\le ^T$ , where the coefficients $\\lbrace g_j(y)\\rbrace _1^N$ are smooth functions of $y\\in {R}$ such that (REF )-(REF ) hold.",
"The analyticity of $m_0$ follows directly from the definition.",
"The bound (REF ) is a consequence of (REF ).",
"Moreover, $& v^{(1)} - v_0 = {\\left\\lbrace \\begin{array}{ll}\\begin{pmatrix}0 & 0 \\\\(r_a(x,t,k) - p(t,z)) e^{t\\Phi } & 0\\end{pmatrix}, & k \\in \\mathcal {Z}_1^\\epsilon \\cup \\mathcal {Z}_2^\\epsilon ,\\\\\\begin{pmatrix}0 & -(r_a^*(x,t,k) - p^*(t, z)) e^{-t\\Phi } \\\\0 & 0\\end{pmatrix}, & k \\in \\mathcal {Z}_3^\\epsilon \\cup \\mathcal {Z}_4^\\epsilon ,\\\\\\begin{pmatrix} - |r|^2 + |p|^2& -(r^*(k) - p^*(t, z)) e^{-t\\Phi } \\\\(r(k) - p(t,z)) e^{t\\Phi } & 0\\end{pmatrix}, & k \\in \\mathcal {Z}_5^\\epsilon .\\end{array}\\right.",
"}$ Since $\\text{\\upshape Re\\,}\\Phi (\\zeta , k_0 + re^{\\frac{\\pi i}{6}}) = -4 r^2(3^{3/2} k_0 + 2r) \\le -8r^3$ for $\\zeta \\le 0$ and $r \\ge 0$ , we obtain $\\text{\\upshape Re\\,}\\Phi (\\zeta , k) \\le -8|k-k_0|^3, \\qquad k \\in \\mathcal {Z}_1^\\epsilon , \\ (x,t) \\in \\text{\\upshape IV}_\\le ^T.$ Suppose $k \\in \\mathcal {Z}_1^\\epsilon $ .",
"If $|k-k_0| \\ge k_0$ , then $|k-k_0| \\ge |k|/3$ , and so $e^{-6t|k-k_0|^3} \\le e^{-2t|k|^3}, \\qquad |k-k_0| \\ge k_0, \\ k \\in \\mathcal {Z}_1^\\epsilon .$ If $|k-k_0| \\le k_0$ , then $|k| \\le Ct^{-1/3}$ , and so $e^{-6t|k-k_0|^3} \\le 1 \\le Ce^{-2t|k|^3}, \\qquad |k-k_0| \\le k_0, \\ k \\in \\mathcal {Z}_1^\\epsilon .$ We conclude that $e^{-\\frac{3t}{4}|\\text{\\upshape Re\\,}\\Phi |} \\le e^{-6t|k-k_0|^3} \\le Ce^{-2t|k|^3} \\le C e^{-\\frac{2}{3}|z|^3}, \\qquad k \\in \\mathcal {Z}_1^\\epsilon , \\ (x,t) \\in \\text{\\upshape IV}_\\le .$ On the other hand, $\\nonumber \\sum _{j=0}^N \\frac{r^{(j)}(k_0)}{j!}",
"(k-k_0)^j& = \\sum _{j=0}^N\\sum _{l=0}^N \\frac{r^{(j+l)}(0)}{j!",
"l!}",
"k_0^l(k-k_0)^j + O(k_0^{N+1})\\\\\\nonumber & = \\sum _{l=0}^N \\sum _{s=l}^{N+l} \\frac{r^{(s)}(0)}{(s-l)!",
"l!}",
"k_0^l(k-k_0)^{s-l} + O(k_0^{N+1})\\\\\\nonumber &= \\sum _{s=0}^{2N} r^{(s)}(0) \\sum _{l=0}^s \\frac{k_0^l(k-k_0)^{s-l}}{(s-l)!",
"l!}",
"+ O(|k-k_0|^{N+1})+ O(k_0^{N+1})\\\\\\nonumber &= \\sum _{s=0}^{2N} \\frac{r^{(s)}(0)}{s!",
"}k^s + O(|k-k_0|^{N+1})+ O(k_0^{N+1})\\\\&= \\sum _{s=0}^{N} \\frac{r^{(s)}(0)}{s!",
"}k^s + O(|k|^{N+1})$ uniformly for $0 \\le k_0 \\le C$ and $k \\in \\mathcal {Z}^\\epsilon $ .",
"Equations (REF ), (REF ), (REF ), (REF ), and (REF ) imply, for $k \\in \\mathcal {Z}_1^\\epsilon $ , $|v^{(1)} - v_0| \\le &\\; C|r_a(x,t,k) - p(t,z)| e^{t \\text{\\upshape Re\\,}\\Phi }\\le \\bigg |r_{a}(x, t, k) - \\sum _{j=0}^N \\frac{r^{(j)}(k_0)}{j!}",
"(k-k_0)^j\\bigg | e^{t \\text{\\upshape Re\\,}\\Phi }\\\\& + \\bigg |\\sum _{j=0}^N \\frac{r^{(j)}(k_0)}{j!}",
"(k-k_0)^j - \\sum _{j=0}^N \\frac{r^{(j)}(0)}{j!}",
"k^j\\bigg | e^{t \\text{\\upshape Re\\,}\\Phi }\\\\\\le &\\; C |k-k_0|^{N+1} e^{-\\frac{3}{4}t |\\text{\\upshape Re\\,}\\Phi |} + C|k|^{N+1} e^{- t |\\text{\\upshape Re\\,}\\Phi |} \\le C|zt^{-1/3}|^{N+1} e^{-\\frac{2}{3}|z|^3}.$ As in the proof of Lemma REF , this implies $\\Vert v^{(1)} - v_0\\Vert _{(L^1 \\cap L^\\infty )(\\mathcal {Z}_1^{\\epsilon })} \\le Ct^{-(N+1)/3}$ .",
"Similar estimates apply to $\\mathcal {Z}_j^{\\epsilon }$ with $j = 2,3,4$ .",
"For $k \\in \\mathcal {Z}_5^{\\epsilon }$ , we have $\\text{\\upshape Re\\,}\\Phi = 0$ , and so, by (REF ) and (REF ), $|v^{(1)} - v_0| \\le C |r(k) - p(t,z)| \\le C |k|^{N+1} \\le Ct^{-\\frac{N+1}{3}}, \\qquad k \\in \\mathcal {Z}_5^{\\epsilon }.$ This proves (REF ).",
"The rest of the lemma follows in the same way as Lemma REF ."
],
[
"The solution $\\hat{m}$",
"Let $\\hat{\\Gamma } := \\Gamma ^{(1)} \\cup \\partial D_\\epsilon (0)$ (see Figure REF ) and define $\\hat{m}(x,t,k)$ by (REF ).",
"Then $\\hat{m}$ satisfies (REF ) with jump matrix $\\hat{v}$ given by (REF ).",
"Write $\\hat{\\Gamma }$ as the union of four subcontours as follows: $\\hat{\\Gamma } = \\partial D_\\epsilon (0) \\cup \\mathcal {Z}^\\epsilon \\cup ({R}\\setminus [-k_0,k_0])\\cup \\hat{\\Gamma }^{\\prime },$ where $\\hat{\\Gamma }^{\\prime } := \\Gamma ^{(1)} \\setminus ({R}\\cup \\overline{D_\\epsilon (0)})$ .",
"Figure: NO_CAPTIONLemma 5.3 Let $\\hat{w} = \\hat{v} - I$ .",
"For each $1 \\le p \\le \\infty $ , the following estimates hold uniformly for $(x,t) \\in \\text{\\upshape IV}_\\le ^T$ : $& \\Vert \\hat{w}\\Vert _{L^p(\\partial D_\\epsilon (0))} \\le Ct^{-1/3},\\\\& \\Vert \\hat{w}\\Vert _{L^p(\\mathcal {Z}^\\epsilon )} \\le Ct^{-(N+1)/3},\\\\ & \\Vert \\hat{w}\\Vert _{L^p({R}\\setminus [-k_0,k_0])} \\le C t^{-N},\\\\& \\Vert \\hat{w}\\Vert _{L^p(\\hat{\\Gamma }^{\\prime })} \\le Ce^{-ct}.$ The proof is analogous to the proof of Lemma REF .",
"The remainder of the derivation in Sector $\\text{\\upshape IV}_\\le $ now proceeds as in Sector $\\text{\\upshape IV}_\\ge $ ."
],
[
"Model problem for Sector $IV_\\ge $",
"We first need to review the RH approach to the Painlevé II equation."
],
[
"Painlevé II Riemann–Hilbert problem",
"Let $P = \\cup _{n=1}^6 P_n$ denote the contour consisting of the six rays $P_n = \\biggl \\lbrace z \\in \\bigg |\\, \\arg z = \\frac{\\pi }{6} + \\frac{\\pi (n-1)}{3}\\biggr \\rbrace , \\qquad n = 1, \\dots , 6,$ oriented away from the origin, see Figure REF .",
"Figure: NO_CAPTIONProposition 1.1 (Painlevé II Riemann–Hilbert problem) Let $\\mathcal {S} = \\lbrace s_1, s_2, s_3\\rbrace $ be a set of complex constants such that $s_1 -s_2 +s_3 + s_1s_2s_3 = 0$ and define the matrices $\\lbrace S_n\\rbrace _1^6$ by $S_n = \\begin{pmatrix} 1 & 0 \\\\ s_n & 1 \\end{pmatrix}, \\quad n \\text{ odd};\\qquad S_n = \\begin{pmatrix} 1 & s_n \\\\ 0 & 1 \\end{pmatrix}, \\quad n \\text{ even},$ where $s_{n+3} = - s_n$ , $n = 1,2,3$ .",
"Then there exists a countable set $Y_{\\mathcal {S}} = \\lbrace y_j\\rbrace _{j=1}^\\infty \\subset with $ yj $ as $ j $, such that the classical RH problem{\\begin{@align}{1}{-1}{\\left\\lbrace \\begin{array}{ll} m_+^P(y, z) = m_-^P(y, z) e^{-i(yz + \\frac{4}{3}z^3)\\sigma _3} S_n e^{i(yz + \\frac{4}{3}z^3)\\sigma _3}, & z \\in P_n\\setminus \\lbrace 0\\rbrace , \\ n = 1, \\dots , 6, \\\\m^P(y, z) = I + O(z^{-1}), & z \\rightarrow \\infty , \\\\m^P(y,z) = O(1), & z \\rightarrow 0,\\end{array}\\right.",
"}\\end{@align}}has a unique solution $ mP(y, z)$ for each $ y YS$.For each $ n$, the restriction of $ mP$ to $ z ((2n-3)6, (2n-1)6)$ admits an analytic continuation to $ (YS) .",
"Moreover, there are smooth functions $\\lbrace m_j^P(y)\\rbrace _1^\\infty $ of $y \\in Y_{\\mathcal {S}}$ such that, for each integer $N \\ge 0$ , $m^P(y, z) = I + \\sum _{j=1}^N \\frac{m_j^P(y)}{z^j} + O(z^{-N-1}), \\qquad z \\rightarrow \\infty ,$ uniformly for $y$ in compact subsets of $Y_{\\mathcal {S}}$ and for $\\arg z \\in [0,2\\pi ]$ .",
"The off-diagonal elements of the leading coefficient $m_1^P$ are given by $(m_1^P(y))_{12} = (m_1^P(y))_{21} = \\frac{1}{2} u_P(y),$ where $u_P(y) \\equiv u_P(y; s_1, s_2, s_3)$ satisfies the Painlevé II equation (REF ).",
"The map $(s_1,s_2,s_3) \\in \\mathcal {S} \\mapsto u_P(\\cdot ; s_1, s_2, s_3)$ is a bijection $\\lbrace (s_1,s_2,s_3) \\in 3 \\,|\\, s_1 -s_2 +s_3 + s_1s_2s_3 = 0\\rbrace \\rightarrow \\lbrace \\text{solutions of }(\\ref {painleve2})\\rbrace $ and $Y_{\\mathcal {S}}$ is the set of poles of $u_P(\\cdot ; s_1, s_2, s_3)$ .",
"The solution $u_P$ obeys the reality condition $u_P(y; s_1, s_2, s_3) = \\overline{u_P(\\bar{y}; s_1, s_2, s_3)}$ if and only if $\\lbrace s_n\\rbrace _1^3$ satisfy $s_3 = \\bar{s}_1$ and $s_2 = \\bar{s}_2$ .",
"If $\\mathcal {S} = (s,0,-s)$ where $s \\in i{R}$ and $|s| < 1$ , then $Y_{\\mathcal {S}} \\cap {R}= \\emptyset $ , the leading coefficient $m_1^P$ is given by $m_1^P(y) = \\frac{1}{2} \\begin{pmatrix} -i\\int _y^\\infty u_P(y^{\\prime })^2dy^{\\prime } & u_P(y) \\\\ u_P(y) & i\\int _y^\\infty u_P(y^{\\prime })^2dy^{\\prime } \\end{pmatrix},$ and, for each $C_1 > 0$ , $\\sup _{y \\ge -C_1} \\sup _{z \\in P} |m^P(y,z)| < \\infty .$ The proof uses the fact that the Painlevé II equation (REF ) is the compatibility condition of the Lax pair ${\\left\\lbrace \\begin{array}{ll}\\partial _y\\Psi + i z \\sigma _3\\Psi = -u_P \\sigma _2 \\Psi ,\\\\\\partial _z\\Psi + i(y + 4 z^2)\\sigma _3\\Psi = (-2iu_P^2 \\sigma _3 - 4zu_P\\sigma _2 - 2(u_P)^{\\prime }\\sigma _1) \\Psi ,\\end{array}\\right.",
"}$ see Theorem 3.4, Theorem 4.2 and Corollary 4.4 in [7].",
"The reality condition (REF ) can be found on p. 158 of [7].",
"Employing the relation $\\Psi = m^{P}e^{-i(yz+4/3 z^{3} )\\sigma _{3}}$ , the expression for $m_1^P(y)$ can be obtained by substituting the expansion (REF ) of $m^P$ into the first equation in (REF ) and identifying powers of $z$ ."
],
[
"Model problem for Sector $\\text{\\upshape IV}_\\ge $",
"Let $Y$ denote the contour $Y = \\cup _{j=1}^4 Y_j$ oriented to the right as in Figure REF , where $ \\nonumber &Y_1 = \\bigl \\lbrace re^{\\frac{i\\pi }{6}}\\, \\big | \\, 0 \\le r < \\infty \\bigr \\rbrace , && Y_2 = \\bigl \\lbrace re^{\\frac{5i\\pi }{6}}\\, \\big | \\, 0 \\le r < \\infty \\bigr \\rbrace ,\\\\ &Y_3 = \\bigl \\lbrace re^{-\\frac{5i\\pi }{6}}\\, \\big | \\, 0 \\le r < \\infty \\bigr \\rbrace , && Y_4 = \\bigl \\lbrace re^{-\\frac{i\\pi }{6}}\\, \\big | \\, 0 \\le r < \\infty \\bigr \\rbrace .$ The long-time asymptotics in Sector $\\text{\\upshape IV}_\\ge $ is related to the solution $m^Y$ of the following family of RH problems parametrized by $(y,t)$ : ${\\left\\lbrace \\begin{array}{ll}m^Y(y, t, \\cdot ) \\in I + \\dot{E}^2(Y),\\\\m_+^Y(y, t, z) = m_-^Y(y, t, z) v^Y(y, t, z) \\quad \\text{for a.e.}",
"\\ z \\in Y,\\end{array}\\right.",
"}$ where the jump matrix $v^Y(y, t, z)$ has the form $v^Y(y, t, z) = {\\left\\lbrace \\begin{array}{ll}\\begin{pmatrix}1 & 0 \\\\p(t, z)e^{2i(y z + \\frac{4z^3}{3})} & 1\\end{pmatrix}, & z \\in Y_1 \\cup Y_2,\\\\\\begin{pmatrix} 1 & -p^*(t, z)e^{-2i(y z + \\frac{4z^3}{3})} \\\\0 & 1\\end{pmatrix}, & z \\in Y_3 \\cup Y_4,\\end{array}\\right.",
"}$ with the function $p$ specified below.",
"Figure: NO_CAPTIONLemma 1.2 (Model problem for Sector $\\text{\\upshape IV}_\\ge $ ) Let, for some integer $n \\ge 0$ , $p(t,z) = s + \\sum _{j=1}^n \\frac{p_{j}z^j}{t^{j/3}},$ be a polynomial in $z t^{-1/3}$ with coefficients $s \\in \\lbrace ir \\, | -1 < r < 1\\rbrace $ and $\\lbrace p_j\\rbrace _1^n \\subset .$ There is a $T \\ge 1$ such that the RH problem (REF ) with jump matrix $v^Y$ given by (REF ) has a unique solution $m^Y(y, t, z)$ whenever $y \\ge 0$ and $t \\ge T$ .",
"For each integer $N \\ge 1$ , there are smooth functions $\\lbrace m_{jl}^Y(y)\\rbrace $ of $y \\in [0,\\infty )$ such that $& m^Y(y, t, z) = I + \\sum _{j=1}^N \\sum _{l=0}^N \\frac{m_{jl}^Y(y)}{z^j t^{l/3}} + O\\biggl (\\frac{t^{-(N+1)/3}}{|z|} + \\frac{1}{|z|^{N+1}}\\biggr ), \\qquad z \\rightarrow \\infty ,$ uniformly with respect to $\\arg z \\in [0, 2\\pi ]$ , $y \\ge 0$ , and $t \\ge T$ .",
"$m^Y$ obeys the bound $\\sup _{y \\ge 0} \\sup _{t \\ge T} \\sup _{z \\in Y} |m^Y(y, t, z)| < \\infty .$ $m^Y$ obeys the symmetry $m^Y(y, t, z) = \\sigma _1\\overline{m^Y(y, t, \\bar{z})} \\sigma _1.$ If $p(t,z) = -\\overline{p(t,-\\bar{z})}$ , then it also obeys the symmetry $m^Y(y, t, z) = \\sigma _1\\sigma _3m^Y(y, t, -z) \\sigma _3\\sigma _1.$ The leading coefficient in (REF ) is given by $m_{10}^Y(y) =\\frac{1}{2} \\begin{pmatrix} -i\\int _y^\\infty u_P(y^{\\prime }; s, 0, -s)^2dy^{\\prime } & u_P(y; s, 0, -s) \\\\ u_P(y; s, 0, -s) & i\\int _y^\\infty u_P(y^{\\prime }; s, 0, -s)^2dy^{\\prime } \\end{pmatrix},$ where $u_P(\\cdot ; s, 0, -s)$ denotes the smooth real-valued solution of Painlevé II corresponding to $(s,0,-s)$ via (REF ).",
"If $s = 0$ , $p_1 \\in {R}$ , and $p_2 \\in i{R}$ , then the leading coefficients are given by $\\nonumber & m_{10}^Y(y) = 0,\\\\\\nonumber & m_{11}^Y(y) = \\frac{p_1}{4}\\text{\\upshape Ai}^{\\prime }(y)\\sigma _1,\\\\\\nonumber & m_{12}^Y(y) = \\frac{p_1^2}{8i} \\bigg (\\int _y^{\\infty } (\\text{\\upshape Ai}^{\\prime }(y^{\\prime }))^2dy^{\\prime }\\bigg ) \\sigma _3+ \\frac{p_2}{8i} \\text{\\upshape Ai}^{\\prime \\prime }(y) \\sigma _1,\\\\ & m_{21}^Y(y) = -\\frac{p_1}{8i}\\text{\\upshape Ai}^{\\prime \\prime }(y) \\sigma _3\\sigma _1.$ Let $u_P(y; s, 0, -s)$ denote the solution of the Painlevé II equation (REF ) corresponding to $(s,0,-s)$ according to (REF ).",
"Since $s \\in i{R}$ and $|s| < 1$ , $u_P(y; s, 0, -s)$ is a smooth real-valued function of $y \\in {R}$ (see Proposition REF ).",
"Let $m^P(y,z) \\equiv m^P(y,z;s,0,-s)$ be the corresponding solution of the RH problem ().",
"Then $m^P(y,z)$ solves the RH problem obtained from (REF ) by replacing the polynomial $p(t,z)$ on the right-hand side of (REF ) with its leading term $s$ .",
"The function $m^Y$ satisfies (REF ) iff $\\hat{m}^Y := m^Y (m^P)^{-1}$ satisfies ${\\left\\lbrace \\begin{array}{ll}\\hat{m}^Y(y, t, \\cdot ) \\in I + \\dot{E}^2(Y),\\\\\\hat{m}_+^Y(y, t, z) = \\hat{m}_-^Y(y, t, z) \\hat{v}^Y(y, t, z) \\quad \\text{for a.e.}",
"\\ z \\in Y,\\end{array}\\right.",
"}$ where the jump matrix obeys the relation $\\hat{v}^Y -I = m_{-}^P (v^Y - v^P)(m_+^P)^{-1}$ .",
"Letting $\\hat{w}^Y := \\hat{v}^Y - I$ , we can write $\\hat{w}^Y(y, t, z) = \\frac{\\hat{w}_1^Y(y,z)}{t^{1/3}} + \\cdots + \\frac{\\hat{w}_n^Y(y,z)}{t^{n/3}},$ where $\\hat{w}_j^Y(y, z) = {\\left\\lbrace \\begin{array}{ll}m_-^P \\begin{pmatrix}0 & 0 \\\\p_{j}z^je^{2i(y z + \\frac{4z^3}{3})} & 0\\end{pmatrix}(m_+^P)^{-1}, & z \\in Y_1 \\cup Y_2,\\\\m_-^P\\begin{pmatrix} 0 & - \\bar{p}_{j}z^je^{-2i(y z + \\frac{4z^3}{3})} \\\\0 & 0\\end{pmatrix}(m_+^P)^{-1}, & z \\in Y_3 \\cup Y_4.\\end{array}\\right.",
"}$ We next note that $|e^{\\pm 2i(y z + \\frac{4z^3}{3})}|\\le e^{-\\frac{8}{3}|z|^3}, \\qquad y \\ge 0, \\ z \\in Y,$ where the plus (minus) sign applies for $z \\in Y_1 \\cup Y_2$ ($z \\in Y_3 \\cup Y_4$ ).",
"The estimates (REF ) and (REF ) give, for any integer $m \\ge 0$ , $|z^m\\hat{w}_j^Y(y,z)| \\le &\\; C |z|^{m+j} e^{-\\frac{8}{3}|z|^3} \\le Ce^{-c|z|^3}, \\qquad y \\ge 0, \\ z \\in Y, \\ j = 1, \\dots , n,$ and hence, for any integer $m \\ge 0$ and any $1 \\le p \\le \\infty $ , ${\\left\\lbrace \\begin{array}{ll}\\Vert z^m\\hat{w}_j^Y(y,z)\\Vert _{L^p(Y)} \\le C, & j = 1, \\dots , n,\\\\\\Vert z^m\\hat{w}^Y(y,t,z)\\Vert _{L^p(Y)} \\le Ct^{-1/3}, \\quad & t \\ge 1,\\end{array}\\right.}",
"\\;\\; y \\ge 0.$ In particular, $\\Vert \\mathcal {C}_{\\hat{w}^Y(y,t,\\cdot )}^Y\\Vert _{\\mathcal {B}(L^2(Y))} \\le C \\Vert \\hat{w}^Y\\Vert _{L^\\infty (Y)} \\le Ct^{-1/3}, \\qquad y \\ge 0, \\ t \\ge 1.$ Hence there exists a $T \\ge 1$ such that the RH problem (REF ) has a unique solution $\\hat{m}^Y \\in I + \\dot{E}^2(\\hat{\\Gamma })$ whenever $y \\ge 0$ and $t \\ge T$ .",
"This solution is given by $\\hat{m}^Y(y, t, z) = I + \\mathcal {C}^Y(\\hat{\\mu }^Y \\hat{w}^Y) = I + \\frac{1}{2\\pi i}\\int _{Y} (\\hat{\\mu }^Y \\hat{w}^Y)(y, t, s) \\frac{ds}{s - z},$ where $\\hat{\\mu }^Y(x, t, \\cdot ) \\in I + L^2(Y)$ is defined by $\\hat{\\mu }^Y = I + (I - \\mathcal {C}^Y_{\\hat{w}^Y})^{-1}\\mathcal {C}^Y_{\\hat{w}^Y}I.$ Let $N \\ge 1$ be an integer.",
"Using that $\\mathcal {C}^Y_{\\hat{w}^Y} = \\frac{\\mathcal {C}^Y_{\\hat{w}_1^Y}}{t^{1/3}} + \\frac{\\mathcal {C}^Y_{\\hat{w}_2^Y}}{t^{2/3}} + \\cdots + \\frac{\\mathcal {C}^Y_{\\hat{w}_n^Y}}{t^{n/3}},$ it follows from (REF ) and (REF ) that $\\nonumber \\hat{\\mu }^Y(y, t,z) & = \\sum _{r=0}^{N} (\\mathcal {C}^Y_{\\hat{w}^Y})^rI+ (I - \\mathcal {C}^Y_{\\hat{w}^Y})^{-1}(\\mathcal {C}^Y_{\\hat{w}^Y})^{N+1}I \\\\& = I + \\sum _{j=1}^N \\frac{\\hat{\\mu }_j^Y(y,z)}{t^{j/3}} + \\frac{\\hat{\\mu }_{err}^Y(y,t,z)}{t^{(N+1)/3}},$ where the coefficients $\\hat{\\mu }_j^Y$ and $\\hat{\\mu }_{err}^Y$ satisfy ${\\left\\lbrace \\begin{array}{ll}\\Vert \\hat{\\mu }_j^Y(y,\\cdot )\\Vert _{L^2(Y)} \\le C, & j = 1, \\dots , N,\\\\\\Vert \\hat{\\mu }_{err}^Y(y,t,\\cdot )\\Vert _{L^2(Y)} \\le C, \\quad & t \\ge T,\\end{array}\\right.}",
"\\;\\; y \\ge 0.$ Indeed, to obtain (REF ) we note that $\\hat{\\mu }_{j}^Y$ is a sum of terms of the form $\\mathcal {C}^Y_{\\hat{w}_{j_1}^Y}\\cdots \\mathcal {C}^Y_{\\hat{w}_{j_r}^Y}I$ where $j_1 + \\cdots j_r = j$ , which, by (REF ), can be estimated as $\\nonumber \\Vert \\mathcal {C}^Y_{\\hat{w}_{j_1}^Y}\\cdots \\mathcal {C}^Y_{\\hat{w}_{j_{r}}^Y}I\\Vert _{L^2(Y)} & \\le C\\Vert \\mathcal {C}_{\\hat{w}_{j_1}^Y}^Y\\Vert _{\\mathcal {B}(L^2(Y))}\\cdots \\Vert \\mathcal {C}_{\\hat{w}_{j_{r-1}}^Y}^Y\\Vert _{\\mathcal {B}(L^2(Y))}\\Vert \\hat{w}_{j_r}^Y\\Vert _{L^2(Y)}\\\\& \\le C\\Vert \\hat{w}_{j_1}^Y\\Vert _{L^\\infty (Y)}\\cdots \\Vert \\hat{w}_{j_{r-1}}^Y\\Vert _{L^\\infty (Y)}\\Vert \\hat{w}_{j_r}^Y\\Vert _{L^2(Y)} \\le C;$ the coefficient $\\hat{\\mu }_{err}^Y$ involves terms of the same form (but with $j_1 + \\cdots j_r \\ge N+1$ ) which can be estimated in the same way, as well as terms of the form $(I - \\mathcal {C}^Y_{\\hat{w}^Y})^{-1}\\bigg (\\prod _{s=1}^{N+1}\\mathcal {C}^Y_{\\hat{w}_{j_s}^Y}\\bigg )I$ which, employing (REF ), can be estimated as $\\Big \\Vert (I - \\mathcal {C}^Y_{\\hat{w}^Y})^{-1}\\bigg (\\prod _{s=1}^{N+1}\\mathcal {C}^Y_{\\hat{w}_{j_s}^Y}\\bigg )I\\Big \\Vert _{L^2(Y)} \\le C\\Vert (I - \\mathcal {C}^Y_{\\hat{w}^Y})^{-1}\\Vert _{\\mathcal {B}(L^2(Y))} \\le C.$ Substituting the expansions (REF ) and (REF ) into the representation (REF ) for $\\hat{m}^Y$ , we infer that the following formula holds uniformly for $y \\ge 0$ and $t \\ge T$ as $z \\in Y$ goes to infinity in any nontangential sector: $\\nonumber \\hat{m}^Y&(y, t, z) = I - \\sum _{j=1}^{N} \\frac{1}{2\\pi i z^j}\\int _{Y} s^{j-1}\\hat{\\mu }^Y \\hat{w}^Y ds+ \\frac{1}{2\\pi i}\\int _{Y} \\frac{s^N\\hat{\\mu }^Y \\hat{w}^Y}{z^N(s-z)}ds\\\\\\nonumber = &\\; I - \\sum _{j=1}^{N} \\frac{1}{2\\pi i z^j}\\int _{Y} s^{j-1}\\bigg (I + \\sum _{a=1}^N \\frac{\\hat{\\mu }_a^Y(y,z)}{t^{a/3}} + \\frac{\\hat{\\mu }_{err}^Y(y,t,z)}{t^{(N+1)/3}}\\bigg )\\bigg (\\sum _{b=1}^n \\frac{\\hat{w}_b^Y(y,z)}{t^{b/3}}\\bigg )ds\\\\& + O(|z|^{-N-1} \\Vert \\hat{\\mu }^Y\\Vert _{L^2(Y)} \\Vert s^N\\hat{w}^Y\\Vert _{L^2(Y)}),$ i.e., utilizing (REF ) and (REF ) and setting $\\hat{w}_i^Y \\equiv 0$ for $i \\ge n+1$ if $N \\ge n + 1$ , $\\nonumber \\hat{m}^Y(y, t, z) = &\\; I - \\sum _{j=1}^{N} \\frac{1}{2\\pi i z^j} \\bigg \\lbrace \\sum _{l=1}^N t^{-l/3} \\int _{Y} s^{j-1} \\bigg (\\hat{w}_l^Y + \\sum _{i=1}^{l-1} \\hat{\\mu }_{l-i}^Y \\hat{w}_i^Y\\bigg )ds+ O\\big (t^{-\\frac{N+1}{3}}\\big )\\bigg \\rbrace \\\\& + O\\big (|z|^{-N-1}t^{-1/3}\\big ).$ Repeating the above steps with $Y$ replaced by a slightly deformed contour $\\tilde{Y}$ , we see that in fact the condition that $z$ lies in a nontangential sector can be dropped in (REF ).",
"We deduce that $& \\hat{m}^Y(y, t, z) = I + \\sum _{j=1}^N \\sum _{l=1}^N \\frac{\\hat{m}_{jl}^Y(y)}{z^j t^{l/3}} + O\\biggl (\\frac{t^{-(N+1)/3}}{|z|} + \\frac{t^{-1/3}}{|z|^{N+1}}\\biggr ), \\qquad z \\rightarrow \\infty ,$ uniformly with respect to $\\arg z \\in [0, 2\\pi ]$ , $y \\ge 0$ , and $t \\ge T$ , where $\\hat{m}_{jl}^Y(y) = - \\frac{1}{2\\pi i} \\int _{Y} s^{j-1} \\bigg (\\hat{w}_l^Y + \\sum _{i=1}^{l-1}\\hat{\\mu }_{l-i}^Y \\hat{w}_i^Y\\bigg )(y,s)ds, \\qquad 1 \\le j,l \\le N.$ The smoothness of $\\hat{m}_{jl}^Y(y)$ follows from the fact (see (REF ) and (REF )) that $y \\mapsto (\\cdot )^m\\hat{w}_j^Y(y, \\cdot )$ and $y \\mapsto \\hat{\\mu }_j^Y(y, \\cdot )$ are smooth maps $[0,\\infty ) \\rightarrow L^p(Y)$ , $1 \\le p \\le \\infty $ , and $[0,\\infty ) \\rightarrow L^2(Y)$ , respectively.",
"The expansion (REF ) of $m^Y = \\hat{m}^Y m^P$ follows from the expansions (REF ) and (REF ).",
"The bound (REF ) follows from (REF ) and (REF ) and the fact that the contour can be deformed.",
"The symmetries (REF ) follow from the analogous symmetries for $v^Y$ , i.e., $v^Y(y,t,z) = \\sigma _1 \\overline{v^Y(y,t,\\bar{z})}^{-1} \\sigma _1$ and, if $p(t,z) = -p^*(t,-z)$ , $v^Y(y,t,z) = \\sigma _1 \\sigma _3 v^Y(y,t, -z)^{-1} \\sigma _3 \\sigma _1$ .",
"Finally, to prove (REF ), assume that $s = 0$ , $p_1 \\in {R}$ , and $p_2 \\in i{R}$ .",
"In this case, $m^P \\equiv I$ and (REF ) gives the following expressions for $w_1^Y = \\hat{w}_1^Y$ and $w_2^Y=\\hat{w}_2^Y$ : $& w_1^Y = \\begin{pmatrix}0 & -p_1ze^{-2i(y z + \\frac{4z^3}{3})} 1_{Y_3 \\cup Y_4}(z) \\\\p_1z e^{2i(y z + \\frac{4z^3}{3})} 1_{Y_1 \\cup Y_2}(z) & 0\\end{pmatrix},\\\\ & w_2^Y = \\begin{pmatrix}0 & p_2z^2e^{-2i(y z + \\frac{4z^3}{3})} 1_{Y_3 \\cup Y_4}(z) \\\\p_2z^2e^{2i(y z + \\frac{4z^3}{3})} 1_{Y_1 \\cup Y_2}(z) & 0\\end{pmatrix},$ where $1_{A}(z)$ denotes the characteristic function of the set $A \\subset .Now{\\begin{@align*}{1}{-1}\\int _{Y_1 \\cup Y_2} e^{2i(yz + \\frac{4z^3}{3})} dz= \\int _{Y_3 \\cup Y_4} e^{-2i(yz + \\frac{4z^3}{3})} dz= \\pi \\text{\\upshape Ai}(y)\\end{@align*}}so that, differentiating $ j$ times with respect to $ y$,{\\begin{@align}{1}{-1}\\int _{Y_1 \\cup Y_2} z^j e^{2i(yz + \\frac{4z^3}{3})} dz= (-1)^j \\int _{Y_3 \\cup Y_4} z^j e^{-2i(yz + \\frac{4z^3}{3})} dz= \\frac{\\pi \\text{\\upshape Ai}^{(j)}(y)}{(2i)^j}\\end{@align}}for each integer $ j 0$.",
"The coefficients $ m11Y = m11Y$ and $ m21Y = m21Y$ can now be computed:{\\begin{@align*}{1}{-1}m_{11}^Y(y) = & -\\frac{1}{2\\pi i} \\int _{Y} w_1^Y dz\\\\= & - \\frac{p_1}{2\\pi i} \\begin{pmatrix} 0 & - \\int _{Y_3 \\cup Y_4} z e^{-2i(yz + \\frac{4z^3}{3})} dz \\\\\\int _{Y_1 \\cup Y_2} z e^{2i(yz + \\frac{4z^3}{3})} dz & 0\\end{pmatrix}\\\\= & -\\frac{p_1}{2\\pi i} \\frac{\\pi \\text{\\upshape Ai}^{\\prime }(y)}{2i}\\begin{pmatrix} 0 & 1 \\\\1 & 0 \\end{pmatrix},\\\\m_{21}^Y(y) = & -\\frac{1}{2\\pi i}\\int _{Y} z w_1^Y dz\\\\= & -\\frac{p_1}{2\\pi i} \\begin{pmatrix} 0 & -\\int _{Y_3 \\cup Y_4} z^2 e^{-2i(yz + \\frac{4z^3}{3})} dz \\\\\\int _{Y_1 \\cup Y_2} z^2 e^{2i(yz + \\frac{4z^3}{3})} dz & 0 \\end{pmatrix}\\\\= &\\; \\frac{p_1}{2\\pi i} \\frac{\\pi \\text{\\upshape Ai}^{\\prime \\prime }(y)}{4} \\begin{pmatrix} 0 & -1 \\\\1 & 0 \\end{pmatrix}.\\end{@align*}}$ It only remains to derive the expression for $m_{12}^Y$ .",
"Using that $\\mu _1^Y(y,z) & = \\mathcal {C}^Y_{w_1^Y}I= \\mathcal {C}^Y_-(w_1^Y)= \\frac{1}{2\\pi i}\\int _Y \\frac{w_1^Y(y,s)ds}{s-z_-}\\\\& = \\frac{p_1}{2\\pi i} \\begin{pmatrix}0 & -\\int _{Y_3 \\cup Y_4} \\frac{se^{-2i(ys + \\frac{4s^3}{3})} ds}{s-z_-} \\\\\\int _{Y_1 \\cup Y_2} \\frac{se^{2i(ys + \\frac{4s^3}{3})} ds}{s-z_-} & 0 \\end{pmatrix},$ we find that the matrix-valued function $F(y)$ defined by $F(y) = \\int _{Y} \\mu _1^Y w_1^Y dz$ satisfies $F(y)& = -\\frac{p_1^2}{2\\pi i} \\begin{pmatrix} F_{1}(y) & 0 \\\\0 & F_{2}(y)\\end{pmatrix},$ where the diagonal entries are given by $& F_1(y) = \\int _{Y_1 \\cup Y_2} \\bigg ( \\int _{Y_3 \\cup Y_4} \\frac{sze^{-2i(ys + \\frac{4s^3}{3})}e^{2i(yz + \\frac{4z^3}{3})}}{s-z} ds \\bigg )dz,\\\\& F_2(y) = \\int _{Y_3 \\cup Y_4} \\bigg ( \\int _{Y_1 \\cup Y_2} \\frac{sze^{2i(ys + \\frac{4s^3}{3})}e^{-2i(yz + \\frac{4z^3}{3})}}{s-z} ds \\bigg ) dz.$ Fubini's theorem implies that $F_2(y) = -F_1(y)$ and so $F(y) = -\\frac{p_1^2}{2\\pi i}F_{1}(y)\\sigma _3.$ Differentiation with respect to $y$ yields $F^{\\prime }(y) & = \\frac{p_1^2}{\\pi }\\bigg (\\int _{Y_1 \\cup Y_2} ze^{2i(yz + \\frac{4z^3}{3})} dz\\bigg )\\bigg ( \\int _{Y_3 \\cup Y_4} s e^{-2i(ys + \\frac{4s^3}{3})} ds\\bigg )\\sigma _3\\\\& = \\frac{p_1^2}{\\pi }\\bigg (\\frac{\\pi \\text{\\upshape Ai}^{\\prime }(y)}{2i}\\bigg )\\bigg (-\\frac{\\pi \\text{\\upshape Ai}^{\\prime }(y)}{2i}\\bigg )\\sigma _3,$ where we have used () with $j = 1$ .",
"Using that $F(y) \\rightarrow 0$ as $y\\rightarrow \\infty $ , we arrive at $\\int _{Y} \\mu _1^Y(y,z) w_1^Y(y,z)dz = \\int _{+\\infty }^y F^{\\prime }(y^{\\prime }) dy^{\\prime } = \\frac{\\pi p_1^2}{4} \\int _{+\\infty }^y \\text{\\upshape Ai}^{\\prime }(y^{\\prime })^2dy^{\\prime }\\sigma _3.$ Taking advantage of this formula, recalling (REF ), and using () again, the coefficient $m_{12}^Y$ is easily computed: $m_{12}^Y(y) = & -\\frac{1}{2\\pi i}\\int _{Y} (w_2^Y(y,z) + \\mu _1^Y(y,z) w_1^Y(y,z)) dz\\\\= &- \\frac{p_2}{2\\pi i} \\begin{pmatrix} 0 & \\int _{Y_3 \\cup Y_4} z^2 e^{-2i(yz + \\frac{4z^3}{3})} dz \\\\\\int _{Y_1 \\cup Y_2} z^2 e^{2i(yz + \\frac{4z^3}{3})} dz & 0 \\end{pmatrix}\\\\& -\\frac{1}{2\\pi i}\\frac{\\pi p_1^2}{4} \\int _{+\\infty }^y (\\text{\\upshape Ai}^{\\prime }(y^{\\prime }))^2dy^{\\prime } \\sigma _3\\\\ = &\\; \\frac{p_2}{2\\pi i} \\frac{\\pi }{4} \\text{\\upshape Ai}^{\\prime \\prime }(y) \\sigma _1-\\frac{p_1^2}{8i} \\int _{+\\infty }^y (\\text{\\upshape Ai}^{\\prime }(y^{\\prime }))^2dy^{\\prime } \\sigma _3.$ This completes the proof of the lemma."
],
[
"Model problem for Sector $IV_\\le $",
"For each $z_0 \\ge 0$ , let $Z \\equiv Z(z_0)$ denote the contour $Z = \\cup _{j=1}^5 Z_j$ oriented as in Figure REF , where $ \\nonumber &Z_1 = \\bigl \\lbrace z_0+ re^{\\frac{i\\pi }{6}}\\, \\big | \\, 0 \\le r < \\infty \\bigr \\rbrace , && Z_2 = \\bigl \\lbrace -z_0 + re^{\\frac{5i\\pi }{6}}\\, \\big | \\, 0 \\le r < \\infty \\bigr \\rbrace ,\\\\ \\nonumber &Z_3 = \\bigl \\lbrace -z_0 + re^{-\\frac{5i\\pi }{6}}\\, \\big | \\, 0 \\le r < \\infty \\bigr \\rbrace , && Z_4 = \\bigl \\lbrace z_0 + re^{-\\frac{i\\pi }{6}}\\, \\big | \\, 0 \\le r < \\infty \\bigr \\rbrace ,\\\\ & Z_5 = \\bigl \\lbrace r\\, \\big | -z_0 \\le r \\le z_0\\bigr \\rbrace .$ The long-time asymptotics in Sector $\\text{\\upshape IV}_\\le $ is related to the solution $m^Z$ of the following family of RH problems parametrized by $y \\le 0$ , $t \\ge 0$ , and $z_0 \\ge 0$ : ${\\left\\lbrace \\begin{array}{ll}m^Z(y, t, z_0, \\cdot ) \\in I + \\dot{E}^2(Z),\\\\m_+^Z(y, t, z_0, z) = m_-^Z(y, t, z_0, z) v^Z(y, t, z_0, z) \\quad \\text{for a.e.}",
"\\ z \\in Z,\\end{array}\\right.",
"}$ where the jump matrix $v^Z(y, t, z_0, z)$ is defined by $v^Z(y, t, z_0, z) = {\\left\\lbrace \\begin{array}{ll}\\begin{pmatrix}1 & 0 \\\\p(t,z) e^{2i(y z + \\frac{4z^3}{3})} & 1\\end{pmatrix}, & z \\in Z_1 \\cup Z_2,\\\\\\begin{pmatrix} 1 & -p^*(t, z)e^{-2i(y z + \\frac{4z^3}{3})} \\\\0 & 1\\end{pmatrix}, & z \\in Z_3 \\cup Z_4,\\\\\\begin{pmatrix} 1 - |p(t, z)|^2 & -p^*(t, z)e^{-2i(y z + \\frac{4z^3}{3})} \\\\p(t,z) e^{2i(y z + \\frac{4z^3}{3})} & 1 \\end{pmatrix}, & z \\in Z_5,\\end{array}\\right.",
"}$ with $p(t,z)$ given by (REF ).",
"Define the parameter subset $\\mathcal {P}_T$ of ${R}^3$ by $\\mathcal {P}_T = \\lbrace (y,t,z_0) \\in {R}^3 \\, | \\, -C_1 \\le y \\le 0, \\, t \\ge T, \\, \\sqrt{|y|}/2 \\le z_0 \\le C_2\\rbrace ,$ where $C_1,C_2 > 0$ are constants.",
"Figure: NO_CAPTIONLemma 2.1 (Model problem in Sector $\\text{\\upshape IV}_\\le $ ) Let $p(t,z)$ be of the form (REF ) with $s \\in \\lbrace ir \\, | -1 < r < 1\\rbrace $ and $\\lbrace p_{j}\\rbrace _1^n \\subset .$ There is a $T \\ge 1$ such that the RH problem (REF ) with jump matrix $v^Z$ given by (REF ) has a unique solution $m^Z(y, t, z_0, z)$ whenever $(y,t,z_0) \\in \\mathcal {P}_T$ .",
"For each integer $N \\ge 1$ , $& m^Z(y, t, z_0, z) = I + \\sum _{j=1}^N \\sum _{l=0}^N \\frac{m_{jl}^Y(y)}{z^j t^{l/3}} + O\\biggl (\\frac{t^{-(N+1)/3}}{|z|} + \\frac{1}{|z|^{N+1}}\\biggr )$ uniformly with respect to $\\arg z \\in [0, 2\\pi ]$ and $(y,t,z_0) \\in \\mathcal {P}_T$ as $z \\rightarrow \\infty $ , where $\\lbrace m_{jl}^Y(y)\\rbrace $ are smooth functions of $y \\in {R}$ which coincide with the functions in (REF ) for $y \\ge 0$ .",
"$m^Z$ obeys the bound $\\sup _{(y,t,z_0) \\in \\mathcal {P}_T} \\sup _{z \\in Z} |m^Z(y, t, z_0, z)| < \\infty .$ $m^Z$ obeys the symmetry $m^Z(y, t, z_0, z) = \\sigma _1\\overline{m^Z(y, t, z_0, \\bar{z})} \\sigma _1.$ If $p(t,z) = -\\overline{p(t,-\\bar{z})}$ , then it also obeys the symmetry $m^Z(y, t, z_0, z) = \\sigma _1\\sigma _3m^Z(y, t, z_0, -z)\\sigma _3 \\sigma _1.$ The leading coefficient $m_{10}^Y(y)$ in (REF ) is given by (REF ); if $s = 0$ , $p_1 \\in {R}$ , and $p_2 \\in i{R}$ , then (REF ) holds.",
"As in the proof of Lemma REF , we let $u_P(y; s, 0, -s)$ denote the smooth real-valued solution of (REF ) corresponding to $(s,0,-s)$ and we let $m^P(y,z) \\equiv m^P(y,z;s,0,-s)$ be the corresponding solution of the Painlevé II RH problem ().",
"Let $\\lbrace V_j\\rbrace _1^4$ denote the open subsets of $ displayed in Figure \\ref {Zfourrays.pdf}.Defining $ mP1(y,z0,z)$ by$$m^{P1}(y,z) = m^{P}(y,z) \\times {\\left\\lbrace \\begin{array}{ll}\\begin{pmatrix} 1 & 0 \\\\ s e^{2i(yz + \\frac{4 z^3}{3})} & 1 \\end{pmatrix}, & z \\in V_1 \\cup V_2,\\\\\\begin{pmatrix} 1 & \\bar{s} e^{-2i(yz + \\frac{4 z^3}{3})} \\\\ 0 & 1 \\end{pmatrix}, & z \\in V_3 \\cup V_4,\\end{array}\\right.",
"}$$we see that $ mP1$ satisfies the RH problem{\\begin{@align*}{1}{-1}{\\left\\lbrace \\begin{array}{ll}m^{P1}(y, z_0, \\cdot ) \\in I + \\dot{E}^2(Z),\\\\m_+^{P1}(y, z_0, z) = m_-^{P1}(y, z_0, z) v^{P1}(y, z_0, z) \\quad \\text{for a.e.}",
"\\ z \\in Z,\\end{array}\\right.",
"}\\end{@align*}}where the jump matrix $ vP1(y, z0, z)$ is defined by{\\begin{@align*}{1}{-1}v^{P1}(y, z_0, z) = {\\left\\lbrace \\begin{array}{ll}\\begin{pmatrix}1 & 0 \\\\s e^{2i(y z + \\frac{4z^3}{3})} & 1\\end{pmatrix}, & z \\in Z_1 \\cup Z_2,\\\\\\begin{pmatrix} 1 & -\\bar{s} e^{-2i(y z + \\frac{4z^3}{3})} \\\\0 & 1\\end{pmatrix}, & z \\in Z_3 \\cup Z_4,\\\\\\begin{pmatrix} 1 - |s|^2 & -\\bar{s} e^{-2i(y z + \\frac{4z^3}{3})} \\\\s e^{2i(y z + \\frac{4z^3}{3})} & 1 \\end{pmatrix}, & z \\in Z_5.\\end{array}\\right.",
"}\\end{@align*}}In other words, $ mP1(y,z0, z)$ solves the RH problem obtained from (\\ref {RHmZIVg}) by replacing the polynomial $ p(t,z)$ on the right-hand side of (\\ref {vZdefIVg}) with its leading term $ s$.",
"The bound (\\ref {mPbounded}) implies that, for any choice of $ C1, C2 > 0$,{\\begin{@align}{1}{-1}\\sup _{-C_1 \\le y \\le 0} \\sup _{0 \\le z_0 \\le C_2} \\sup _{z \\in Z} |m^{P1}(y,z_0, z)| < \\infty .\\end{@align}}\\begin{figure}\\begin{center}\\begin{overpic}[width=.55]{Zfourrays.pdf}\\put (77,34){\\small V_1}\\put (18,34){\\small V_2}\\put (18.5,8){\\small V_3}\\put (77,8){\\small V_4}\\put (62,18){\\small z_0}\\put (49,17.5){\\small 0}\\put (33.5,18){\\small -z_0}\\end{overpic}\\begin{figuretext}The regions V_{1},...,V_{4}\\end{figuretext}\\end{center}\\end{figure}The function $ mZ$ satisfies (\\ref {RHmZIVg}) iff $ mZ := mZ (mP1)-1$ satisfies{\\begin{@align}{1}{-1}{\\left\\lbrace \\begin{array}{ll}\\hat{m}^Z(y, t, z_0, \\cdot ) \\in I + \\dot{E}^2(Z),\\\\\\hat{m}_+^Z(y, t, z_0, z) = \\hat{m}_-^Z(y, t, z_0, z) \\hat{v}^Z(y, t, z_0, z) \\quad \\text{for a.e.}",
"\\ z \\in Z,\\end{array}\\right.",
"}\\end{@align}}where $ wZ := vZ -I = m-P1 (vZ - vP1)(m+P1)-1$.For $ z k=14 Zk$ we have{\\begin{@align}{1}{-1}\\hat{w}^Z(y, t, z_0, z) = \\frac{\\hat{w}_1^Z(y,z_0, z)}{t^{1/3}} + \\cdots + \\frac{\\hat{w}_{n}^Z(y,z_0, z)}{t^{n/3}},\\end{@align}}where{\\begin{@align}{1}{-1}\\hat{w}_j^Z(y, z_0, z) = {\\left\\lbrace \\begin{array}{ll}m_-^{P1} \\begin{pmatrix}0 & 0 \\\\p_{j}z^je^{2i(y z + \\frac{4z^3}{3})} & 0\\end{pmatrix}(m_+^{P1})^{-1}, & z \\in Z_1 \\cup Z_2,\\\\m_-^{P1}\\begin{pmatrix} 0 & - \\bar{p}_{j}z^je^{-2i(y z + \\frac{4z^3}{3})} \\\\0 & 0\\end{pmatrix}(m_+^{P1})^{-1}, & z \\in Z_3 \\cup Z_4.\\end{array}\\right.",
"}\\end{@align}}On the other hand, for $ z Z5$ we have{\\begin{@align}{1}{-1}\\hat{w}^Z(y, t, z_0, z) = \\frac{\\hat{w}_1^Z(y,z_0, z)}{t^{1/3}} + \\cdots + \\frac{\\hat{w}_{2n}^Z(y,z_0, z)}{t^{2n/3}},\\end{@align}}where (since $ |e2i(y z + 4z3/3)| = 1$ for $ z Z5$){\\begin{@align}{1}{-1}|\\hat{w}_j^Z(y, z_0, z)| \\le C, \\qquad j = 1, \\dots , 2n,\\end{@align}}uniformly for $ z Z5$, $ -C1 y 0$, and $ 0 z0 C2$.$ For $z = z_0 + re^{\\frac{\\pi i}{6}} \\in Z_1$ with $r \\ge 0$ , $z_0 \\ge 0$ , and $-4z_0^2 \\le y \\le 0$ , we have $\\text{\\upshape Re\\,}\\bigg (2i\\bigg (yz + \\frac{4z^3}{3}\\bigg )\\bigg )= -\\frac{8 r^3}{3}-4 \\sqrt{3} r^2 z_0-r \\left(y+4 z_0^2\\right)\\le -\\frac{8 r^3}{3}-4 \\sqrt{3} r^2 z_0.$ Hence the following estimate holds uniformly for $(y,t,z_0) \\in \\mathcal {P}_1$ , where $\\mathcal {P}_1$ denotes the parameter subset defined in (REF ) with $T = 1$ : $|e^{2i(y z + \\frac{4z^3}{3})}| \\le Ce^{-|z-z_0|^2(z_0 + |z-z_0|)}, \\qquad z \\in Z_1.$ Analogous estimates hold also for $z \\in Z_j$ , $j = 2,3,4$ : $& |e^{2i(y z + \\frac{4z^3}{3})}| \\le Ce^{-|z+z_0|^2(z_0 + |z+z_0|)}, \\qquad z \\in Z_2,\\\\& |e^{-2i(y z + \\frac{4z^3}{3})}| \\le Ce^{-|z+z_0|^2(z_0 + |z+z_0|)}, \\qquad z \\in Z_3,\\\\& |e^{-2i(y z + \\frac{4z^3}{3})}| \\le Ce^{-|z-z_0|^2(z_0 + |z-z_0|)}, \\qquad z \\in Z_4.$ Using () and (), equation () implies that, for any integer $m \\ge 0$ , $|z^m\\hat{w}_j^Z(y,z_0, z)| \\le &\\; C |z|^{m+j} e^{-|z-z_0|^3} \\le Ce^{-c|z-z_0|^3}, \\qquad z \\in Z_1, \\ j = 1, \\dots , n,$ uniformly for $(y,t,z_0) \\in \\mathcal {P}_1$ .",
"Similar estimates hold also on $Z_2 \\cup Z_3 \\cup Z_4$ .",
"Recalling also (), we conclude that, for any integer $m \\ge 0$ and any $1 \\le p \\le \\infty $ , ${\\left\\lbrace \\begin{array}{ll}\\Vert z^m\\hat{w}_j^Z(y,z_0,z)\\Vert _{L^p(Z)} \\le C, & j = 1, \\dots , 2n,\\\\\\Vert z^m\\hat{w}^Z(y,t,z_0,z)\\Vert _{L^p(Z)} \\le Ct^{-1/3},\\end{array}\\right.",
"}$ uniformly for $(y,t,z_0) \\in \\mathcal {P}_1$ .",
"The estimates (REF ) play the same role in the present proof as the estimates (REF ) did in the proof of Lemma REF and the rest of the proof is now analogous to the proof of that lemma.",
"Acknowledgement The authors are grateful to the anonymous referee for valuable remarks.",
"Support is acknowledged from the Göran Gustafsson Foundation, the European Research Council, Grant Agreement No.",
"682537, and the Swedish Research Council, Grant No.",
"2015-05430."
]
] | 1808.08408 |
[
[
"Nowcasting the Stance of Social Media Users in a Sudden Vote: The Case\n of the Greek Referendum"
],
[
"Abstract Modelling user voting intention in social media is an important research area, with applications in analysing electorate behaviour, online political campaigning and advertising.",
"Previous approaches mainly focus on predicting national general elections, which are regularly scheduled and where data of past results and opinion polls are available.",
"However, there is no evidence of how such models would perform during a sudden vote under time-constrained circumstances.",
"That poses a more challenging task compared to traditional elections, due to its spontaneous nature.",
"In this paper, we focus on the 2015 Greek bailout referendum, aiming to nowcast on a daily basis the voting intention of 2,197 Twitter users.",
"We propose a semi-supervised multiple convolution kernel learning approach, leveraging temporally sensitive text and network information.",
"Our evaluation under a real-time simulation framework demonstrates the effectiveness and robustness of our approach against competitive baselines, achieving a significant 20% increase in F-score compared to solely text-based models."
],
[
"Introduction",
"Predicting user voting stance and final results in elections using social media content is an important area of research in social media analysis [39], [21] with applications in online political campaigning and advertising [25], [11].",
"It also provides political scientists with tools for qualitative analysis of electoral behaviour on a large scale [2].",
"Previous approaches mainly focus on predicting national general elections, which are regularly scheduled and where data of past results and opinion polls are available [33], [60].",
"However, there is no evidence of how such models would work during a sudden and major political event under time-constrained circumstances.",
"That forms a more challenging task compared to general elections, due to its spontaneous nature [35].",
"Building robust methods for voting intention of social media users under such circumstances is important for political campaign strategists and decision makers.",
"Our work focuses on nowcasting the voting intention of Twitter users in the 2015 Greek bailout referendum that was announced in June, 27th 2015 and was held eight days later.",
"We define a time-sensitive binary classification task where the aim is to classify a user's voting intention (YES/NO) at different time points during the entire pre-electoral period.",
"For this purpose, we collect a large stream of tweets in Greek and manually annotate a set of users for testing.",
"We also collect a set of users for training via distant supervision.",
"We predict the voting intention of the test users during the eight-day period until the day of the referendum with a multiple convolution kernel learning model.",
"The latter allows us to leverage both temporally sensitive textual and network information.",
"Collecting all the available tweets written in GreekAs per Twitter Streaming API limitations: https://developer.twitter.com/en/docs/basics/rate-limiting, enables us to study user language use and network dynamics in a complete way.",
"We demonstrate the effectiveness and robustness of our approach, achieving a significant 20% increase in F-score against competitive text-based baselines.",
"We also show the importance of combining text and network information for inferring users' voting intention.",
"Our paper makes the following contributions: We present the first systematic study on nowcasting the voting intention of Twitter users during a sudden and major political event.",
"We demonstrate that network and language information are complementary, by combining them with multiple convolutional kernels.",
"We highlight the importance of the temporal modelling of text for capturing the voting intention of Twitter users.",
"We provide qualitative insights on the political discourse and user behaviour during this major political crisis."
],
[
"Related Work",
"Most previous work on predicting electoral results focuses on forecasting the final outcome.",
"Early approaches based on word counts [63] fail to generalize well [39], [21], [29].",
"[33] presented a bilinear model based on text and user information, using opinion polls as the target variable.",
"[60] similarly predicted the election results in different countries using Twitter and polls while others used sentiment analysis methods and past results [44], [53], [10], [8].",
"More recently, [56] presented a method to forecast the results of the latest US presidential election from user predictions on Twitter.",
"The key difference between our task and this strand of previous work lies in its spontaneous and time-sensitive nature.",
"Incorporating opinion polls or past results is not feasible, due to the time-constrained referendum period and the lack of previous referendum cases, respectively.",
"Previous work on predicting the outcomes of referendums [9], [22], [36] is also different to our task, since they do not attempt to predict a single user's voting intention but rather make use of aggregated data coming from multiple users to predict the voting share of only a few test instances.",
"On the user-level, most past work has focused on identifying the political leaning (left/right) of a user.",
"Early work by [50] explored the linguistic aspect of the task; follow-up work has also incorporated features based on the user's network [46], [1], [16], [65], leading to improvements in performance.",
"However, most of this work predicts the (static) political ideology of clearly separated groups of users who are either declaring their political affiliation in their profiles, or following specific accounts related to a political party.",
"This has been demonstrated to be problematic when applying such models on users that do not express political opinion [12].",
"[49] proposed instead a non-binary, seven-point scale for measuring the self-reported political orientation of Twitter users, showcasing that the task is more difficult for users who are not necessarily declaring their political ideology.",
"Our work goes beyond political ideology prediction, by simulating a real-world setting on a dynamically evolving situation for which there is no prior knowledge.",
"A smaller body of research has focused on tasks that go beyond the classic left/right political leaning prediction.",
"[19] predicted the stance of Twitter users in the 2014 Scottish Independence referendum by analysing topics in related online discussions.",
"In a related task, [66] classified user stance in three independence movements while [55] analysed user linguistic identity in the Catalan referendum.",
"Albeit relevant, none of these works have actually studied the problem under a real-time evaluation setting or during a sudden event where the time between announcement and voting day is extremely limited (e.g., less than two weeks).",
"Previous work on social media analysis during the Greek bailout referendum [40], [4] has not studied the task of inferring user voting intention, whereas most of the past work in opinion mining in social media in the Greek language has focused primarily on tasks related to sentiment analysis [30], [45], [61].",
"To the best of our knowledge, this is the first work to (a) infer user voting intention under sudden circumstances and a major political crisis; and (b) model user information over time under such settings."
],
[
"The Greek Bailout Referendum",
"The period of the Greek economic crisis before the bailout referendum (2009-2015) was characterized by extreme political turbulence, when Greece faced six straight years of economic recession and five consecutive years under two bailout programs [62].",
"Greek governments agreed to implement austerity measures, in order to secure loans and avoid bankruptcy – a fact that caused massive unrest and demonstrations.",
"During the same period, political parties regardless of their side on the left-right political spectrum were divided into pro-austerity and anti-austerity, while the traditional two-party system conceived a big blow [6], [58], [52].",
"The Greek bailout referendum was announced on June, 27th 2015 and was held eight days later.",
"The Greek citizens were asked to respond as to whether they agree or not (YES/NO) with the new bailout deal proposed by the TroikaA decision group formed by the European Commission, the European Central Bank and the International Monetary Fund to deal with the Greek economic crisis.",
"to the Greek Government in order to extend its credit line.",
"The final result was 61.3%-38.7% in favor of the NO vote.",
"For more details on the Greek crisis, refer to [62]."
],
[
"Task Description",
"Our aim is to classify a Twitter user either as a YES or a NO voter in the Greek Bailout referendum over the eight-day period starting right before its announcement (26/6, day 0) and ending on the last day before it took place (4/7, day 8).",
"We assume a training set of users $D_t=\\lbrace (x_{t}^{(1)},y^{(1)}),...,(x_{t}^{(n)},y^{(n)})\\rbrace $ , where $x_{t}^{(i)}$ is a representation of user $i$ up to time step $t \\in [0,...,8]$ and $y^{(i)} \\in \\lbrace \\texttt {YES},\\texttt {NO}\\rbrace $ .",
"Given $D_t$ , we want to learn a function $f_t$ that maps a user $j$ to her or his stance $\\hat{y}^{(j)}=f_t(x_{t}^{(j)})$ at time $t$ .",
"Then, we update our model with new information shared by the users in our training set up to $t$ +1, to predict the test users voting intention at $t$ +1.",
"Therefore, we mimic a real-time setup, where we nowcast user voting intention, starting from the moment before the announcement of the referendum, until the day of the referendum.",
"Sections and present how we develop the training dataset $D_t$ and the function $f_t$ respectively."
],
[
"Data",
"Using the Twitter Streaming API during the period 18/6–16/7, we collected 14.62M tweets in Greek (from 304K users) containing at least one of 283 common Greek stopwords, starting eight days before the announcement of the referendum and stopping 11 days after the referendum date (see Figure REF ).",
"This provides us with a rare opportunity to study the interaction patterns among the users in a rather complete and unbiased setting, as opposed to the vast majority of past works, which track event-related keywords only.",
"For example, [4] collected 0.3M tweets using popular referendum-related hashtags during 25/06–05/07 – we have collected 6.4M tweets during the same period.",
"In the rest of this section, we provide details on how we processed the data in order to generate our training set in a semi-supervised way (REF ) and how we annotated the users that were used as our test set in our experiments (REF ).",
"Figure: Number of tweets in Greek per hour.",
"The period highlighted in red indicates the nine evaluation time points (see Section )."
],
[
"Training Set",
"Manually creating a training set would have required annotating users based on their voting preference on an issue that they had not been aware of prior to the referendum announcement.",
"However, the same does not hold for certain accounts (e.g., major political parties) whose stance on austerity had been known a-priori given their manifestos and previous similar votes in parliament [52].",
"Such accounts can be used as seeds to form a semi-supervised task, under the hypothesis that users who are re-tweeting a political party more often than others, are likely to follow its stance in the referendum, once this is announced.",
"Hence, we compile a set of 267 seed accounts (148 YES, 119 NO) focusing on the pre-announcement period including: (1) political parties; (2) members of parliament (MPs); and (3) political party members.",
"Political Parties: We add as seeds the Twitter accounts of nine major and minor partiesWe excluded KKE (Greek Communist Party) since an active official Twitter account did not exist at the time.",
"with a known stance on austerity before the referendum (5 YES, 4 NO, see Table REF ).",
"We assume that the pro-austerity parties will back the bailout proposal (YES), while the anti-austerity parties will reject it (NO).",
"The pro-/anti- austerity stance of the parties was known before the referendum, since the pro-austerity parties had already backed previous bailout programs in parliament or had a clear favorable stance towards them, whereas the opposite holds for the anti-austerity parties [52].",
"MPs The accounts of the (300) MPs of these parties were manually extracted and added as seeds.",
"153 such accounts were identified (82 YES, 71 NO) labelled according to the austerity stance of their affiliated party.",
"Political Party Members We finally compiled a set of politically related keywords to look up in Twitter user account names and descriptions (names/abbreviations of the nine parties and keywords such as “candidate”).",
"We identified 257 accounts (133 YES, 124 NO), which were manually inspected by human experts to filter out irrelevant ones (e.g., the word “River” might not refer to the political party) and kept only those that had at least one tweet during the period preceding the announcement of the referendum (44 NO, 61 YES).",
"Table: Political position, austerity, referendum stance and national election result (January 2015) of the political parties that are used as seeds in our modelling.To expand the set of seed accounts, we calculate for every user $u$ in our dataset during the pre-announcement period his/her score as: $score(u) = PMI(u, \\texttt {YES}) - PMI (u, \\texttt {NO}) ,$ where $PMI(u, lbl)$ is the pointwise mutual information between a certain user and the respective seeding class (YES/NO).",
"A high (low) score implies that the user is endorsing often YES-related (NO-related) accounts, thus he/she is more likely to follow their stance after the referendum is announced.",
"This approach has been successfully applied to other related natural language processing tasks, such as building sentiment analysis lexical resources using a pre-defined list of seed words [43].",
"Assigning class labels to the users based on their scores, we set up a threshold $tr = n(\\max (|scores|))$ , with $n \\in [0,1]$ .",
"We assign the label YES to a user $u$ if $score(u)>tr$ or NO if $score(u)<-tr$ .",
"Setting $n=0$ , would imply that we are assigning the label YES if the user has re-tweeted more YES-supporting accounts (and inversely), which might result into a low quality training set, whereas higher values for $n$ would imply a smaller (but of higher quality) training set.",
"During development, we empirically set $n=0.5$ to keep users who are fairly closer to one class than the other.",
"From the final set of 5,430 users that have re-tweeted any seed account, 2,121 were kept (along with the seed accounts) as our training set (965 YES, 1,156 NO).",
"Table: Number of users (u) and tweets (t) used in our experiments per evaluation day."
],
[
"Test Set",
"For evaluation purposes, we generate a test set of active users that are likely to participate in political conversations on Twitter.",
"First, we identify all users having tweeted at least 10 times after the referendum announcement (86,000 users).",
"From the 500 most popular hashtags in their tweets, we selected those that were clearly related to the referendum (189) which were then manually annotated with respect to potentially conveying the user's voting intention (e.g., “yesgreece”, “no” as opposed to neutral ones, such as “referendum”).",
"Finally, we selected a random sample of 2,700 users (out of 22K) that had used more than three such hashtags, to be manually annotated – without considering any user from the training set.",
"This is standard practice in related work [20], [55] and enables us to evaluate our models on a high quality test set, as opposed to previous related work which rely on keyword-matching approaches to generate their test set [19], [66].",
"Two authors of the paper (Greek native speakers) annotated each of the users in the test set, using the tweets after the referendum announcement.",
"Each annotator was allowed to label an account as YES, NO, or N/A, if uncertain.",
"There was an agreement on 2,365 users (Cohen's $\\kappa =.75$ ) that is substantially higher if the N/A labels are not considered ($\\kappa =.98$ ), revealing high quality in the annotations, i.e., in the upper part of the `substantial' agreement band [5].",
"We discarded all accounts labelled as N/A by an annotator and used the remaining accounts where the annotators agreed for the final test set, resulting to 2,197 users – similar test set sizes are used in related tasks [18].",
"The resulting user distribution (NO 77%, YES 23%) is more imbalanced compared to the actual result of the referendum, due to the demographic bias on Twitter [42].",
"To mimic a real-time scenario, we refrained from balancing our train/test sets, since it would have been rather impossible to know the voting intention distribution of Twitter users a-priori.",
"Overall, we use 18.9% (1.64M/8.66M) of the tweets written in Greek during that period in our experiments (see Table REF )."
],
[
"Convolution Kernels",
"Convolution kernels are composed of sub-kernels operating on the item-level to build an overall kernel for the object-level [23], [14] and can be used with any kernel based model such as Support Vector Machines (SVMs) [27].",
"Such kernels have been applied in various NLP tasks [14], [31], [64], [37].",
"Here we build upon the approach of [37] by combining convolution kernels operating on available (1) text; and (2) network information.",
"Let $a$ , $b$ denote two objects (e.g., social network users), represented by two $M\\times N$ matrices $Z_a$ and $Z_b$ respectively, where $M$ denotes the number of items representing the object and $N$ the dimensionality of an item vector.",
"For example, an item can be a user's tweet or network information.",
"A kernel $K$ between the two objects (users) $a$ and $b$ over $Z_a$ and $Z_b$ is defined as: $K_z(a, b) = \\frac{1}{\\left|Z_a\\right|\\left|Z_b\\right|}\\sum _{i, j} k_z(z_{a}^{i}, z_{b}^{j}),$ where $k_z$ is any standard kernel function such as a linear or a radial basis function (RBF).",
"One can also normalise $K_z$ by dividing its entries $K_z(i, j)$ by $\\sqrt{K_z(i, i) K_z(j, j)}$ .",
"The resulting kernel has the ability to capture the similarities across objects on a per-item basis.",
"However, unless restricted to operate on consecutive items (time-wise), it ignores their temporal aspect.",
"Given a set of associated timestamps $T_o=\\lbrace t_{o}^{1},...,t_{o}^{N}\\rbrace $ for the items of each object $o$ , [37] proposed to combine the temporal and the item aspects as: $K_{zt}(a, b)=\\frac{1}{\\left|Z_a\\right|\\left|Z_b\\right|}\\sum _{i, j}k_z(z_{a}^{i}, z_{b}^{j})k_t(t_{a}^{i}, t_{b}^{j}),$ where $k_{t}$ is any valid kernel function operating on the timestamps of the items.",
"Here, $K_{zt}$ is a matrix capturing the similarities across users by leveraging both the information between pairs of items and their temporal interaction.",
"Let $a$ , $b$ denote two users in a social network, posting messages $W_a=\\lbrace w_{a}^{1},...,w_{a}^{N}\\rbrace $ and $W_b=\\lbrace w_{b}^{1}$ , ..., $w_{b}^{M}\\rbrace $ with associated timestamps $T_a=\\lbrace t_{a}^{1},...,t_{a}^{N}\\rbrace $ and $T_b=\\lbrace t_{b}^{1}$ , ..., $t_{b}^{M}\\rbrace $ respectively.",
"We assume that a message $w_{i}^{j}$ of user $i$ at time $j$ is represented by the mean $k$ -dimensional embedding [41] of its constituent terms.",
"This way, we can obtain text convolution kernels, $K_{w}$ and $K_{wt}$ by simply replacing $Z$ and $z$ with $W$ and $w$ respectively in Equations REF and REF .",
"Following [37], we opted for a linear kernel operating on text and an RBF on time."
],
[
"Network Kernels",
"Let assume a set of directed weighted graphs $\\textstyle G=\\lbrace G_1(N_1, E_1), ..., G_t(N_t, E_t)\\rbrace $ , where $G_i(N_i, E_i)$ represents the retweeting activity graph of the $N_i$ users at a time point $ i\\in T=\\lbrace 1, .., t\\rbrace $ .",
"Let $L_a \\in \\mathbf {R}^{N,k}$ , $L_b \\in \\mathbf {R}^{M,k}$ denote the resulting matrices of a k-dimensional, network-based user representation for two users $a$ and $b$ across time.",
"Contrary to the textual vector representation $w_{i}^{j}$ that is defined over a fixed space given a pre-defined vocabulary, user network vector representations (e.g., graph embeddings [57]), are computed at each time step on a different network structure.",
"Thus, a standard similarity score between two user representations at timepoints $t$ and $t$ +1 cannot be used, since the network vector spaces are different.",
"To accommodate this, at each time point $t$ we calculate the median $L_{\\texttt {YES}}^t$ and $L_{\\texttt {NO}}^t$ vectors for each class of our training examples and update the respective user vectors as: Lu*t = d(LYESt, Lut) - d(LNOt, Lut), using some distance metric $d$ (for simplicity, we opted for the linear distance).",
"If a user has not retweeted, his/her original network representation $l_u^{t}$ is calculated as the average across all user representations at $t$ .",
"Finally, the network convolution kernels, $K_n$ and $K_{nt}$ are computed using Equations REF and REF respectively by simply replacing $Z$ with $L^*$ and $z$ with $l^*$ .",
"Similarly to text kernels, we use a linear kernel $k_n$ for the network and an RBF kernel $k_t$ for time."
],
[
"Kernel Summation",
"We can combine the text and network convolution kernels by summing them up: $K_{sum} = K_w+K_{wt}+K_n+K_{nt}$ .",
"This implies a simplistic assumption that the contribution of the different information sources with respect to our target is equal.",
"While this might hold for a small number of carefully designed kernels, it lacks the ability to generalise over multiple kernels of potentially noisy representations."
],
[
"SVMs with Convolution Kernels",
"Convolution kernels can be used with any kernel based model.",
"Here, we use them with SVMs.",
"First, a SVM$_{s}$ operates on a single information source $s=\\lbrace w,n\\rbrace $ , i.e., SVM$_{w}$ for text and SVM$_{n}$ for network.",
"Second, a SVM$_{st}$ takes temporal information into account combined with text (SVM$_{wt}$ ) and network (SVM$_{nt}$ ) information respectively.",
"Finally, we combine the text and the network information using a linear kernel summation ($K_{sum}$ ) of their respective kernels (SVM$_{sum}$ )."
],
[
"Multiple Convolution Kernel Learning (MCKL)",
"Multiple kernel learning methods learn a weight for each kernel instead of assigning equal importance to all of them allowing more flexibility.",
"Such approaches have been extensively used in tasks where different data modalities exist [26], [59], [48].",
"We build upon the approach of [54] to build a model based on labelled instances $x_i \\in I$ , by combining the different convolution kernels $K_s$ with some weight $w_s>0$ s.t.",
"$\\sum _{s}{w_s}$ =1 and apply: f(x)=sign(iIis ws Ks(x,xi) + b).",
"The parameters $\\alpha _i$ , the bias term $b$ and the kernel weights are estimated by minimising the expression: $\\mbox{min} && \\gamma -\\sum _{i\\in {I}}\\alpha _i\\\\ \\mbox{w.r.t.}",
"&& \\gamma \\in R, \\alpha \\in R_+^{|I|} \\nonumber \\\\ \\mbox{s.t.}",
"&& 0\\le \\alpha _i\\le C \\;\\forall i,\\;\\;\\sum _{i\\in {I}} \\alpha _i y_i=0 \\nonumber \\\\ && \\frac{1}{2}\\sum _{i\\in {I}}\\sum _{j \\in {I}} \\alpha _i \\alpha _j y_i y_j K_s({\\bf x}_i,{\\bf x}_j)\\le \\gamma \\;\\; \\forall s.\\\\ $ This way, the four convolution kernels are calculated individually and subsequently combined in a weighted scheme accounting for their contribution in the prediction task.",
"This allows us to combine external and asynchronous information (e.g., news articles), while adding other kernels capturing different aspects of the users (e.g., images) is straight-forward."
],
[
"Textual Information (TEXT)",
"We obtain word embeddings by training word2vec [41] on a collection of 14.7 non-retweeted tweets obtained by [61], collected in the exact same way as our dataset, over a separate time period.",
"We performed standard pre-processing steps including lowercasing, tokenising, removal of non-alphabetic characters, replacement of URLs, mentions and all-upper-case words with identifiers.",
"We used the CBOW architecture, opting for a 5-token window around the target word, discarding all words appearing less than 5 times and using negative sampling with 5 “noisy” examples.",
"After training, each word is represented as a 50-dimensional vector.",
"Each tweet in our training and test set is represented by averaging each dimension of its constituent words."
],
[
"Network Information (NETWORK)",
"We trained LINE [57] embeddings at different timesteps, by training on the graphs $\\lbrace G_1(N_1, E_1), ..., G_T(N_T, E_T)\\rbrace $ , where $N_i$ is the set of users and $E_i$ is the (directed, weighted) set of retweets amongst $N_i$ up to time $i$ .",
"We choose the “retweet” rather than the “user mention” network due to its more polarised nature, as indicated by past work [15]The “following” network cannot be constructed based on the JSON objects returned by Twitter Streaming API; to achieve this requires a very large number of API calls and cannot be constructed accurately in a realistic scenario.. LINE was preferred over alternative models [47], [51] due to its ability to model directed weighted graphs.",
"We construct the network $G_t$ every 12 hours based on the retweets among all users up to time $t$ , and LINE is trained on $G_t$ to create 50-dimensional user representations.",
"We used the second-order proximity, since it performed better than the first-order in early experimentation.",
"We also refrained from concatenating them to keep the dimensionality relatively low."
],
[
"Convolution Kernel Models",
"Our MCKL and our SVM models are fed with the convolution kernels operating on the tweet-level (for TEXT) and each NETWORK representation (derived every 12 hours), based on the tweets and re-tweeting activity respectively of the users up to the current evaluation time point."
],
[
"Baselines",
"We compare our proposed methods against competitive baselines that are commonly used in social media mining tasks trained on feature aggregates [38], [66].",
"We obtain a TEXT representation of a user at each time step $t$ by averaging embedding values across all his/her tweets until $t$ .",
"Similarly, a user NETWORK representation is computed from the retweeting graph up until $t$ .",
"Finally, we train a regularised Logistic Regression (LR) with $L_2$ regularisation [34], a feed-forward neural network (FF) [24], a Random Forest (RF) [7] and a SVM."
],
[
"Model Parameters",
"Parameter selection of our models and the baselines is performed using a 5-fold cross-validation on the training set.",
"We experiment with different regularisation strength ($10^{-3}, 10^{-2}, $ $..., 10^{3}$ ) for LR, different number of trees (50, 100, ..., 500) for RF, and different kernels (linear, RBF) and parameters C and $\\gamma $ ($10^{-3}, 10^{-2}, ..., 10^{3}$ ) for SVMs.",
"For FF, we stack dense layers, each followed by a ReLU activation and a 20% dropout layer, and a final layer with a sigmoid activation function.",
"We train our network using the Adam optimiser [32] with the binary cross-entropy loss function and experiment with different number of hidden layers (1, 2), units per layer (10, 25, 50, 75, 100, 150, 200), batch size (10, 25, 50, 75, 100) and number of epochs (10, 25, 50, 100).",
"For MCKL, we experiment with the same C values as in SVM and apply an $L_2$ regulariser.",
"Figure: Macro-average F-score across all evaluation days using TEXT, NETWORK and BOTH user representations."
],
[
"Evaluation",
"We train and test our models based on the data collected on a daily basis (every midnight), starting from the day before the announcement of the referendum (day 0) until the day before its due date (day 8).",
"This way, we mimic a real-time setting and gain better evaluation insights.",
"To evaluate our models, we compute the macro-average F-score, which forms a more challenging metric compared to micro-averaging, given the imbalanced distribution of our test set.",
"At each evaluation time point $t$ , we use information about the users in our training set up to $t$ , to classify the test users that have tweeted at least once up to $t$ (note that all of the users in our training set have tweeted before the announcement of the referendum, thus the size of the training set in terms of number of users remains constant).",
"This results into a different number of test instances per day (see Table REF ).",
"However, we did not observe any major differences in our evaluation by excluding newly added users.",
"Parameter selection is performed on every evaluation day using a 5-fold cross-validation on the training set."
],
[
"Nowcasting Voting Intention",
"Figure REF presents the macro-average F-scores obtained by the methods compared in all days from the announcement to the day of the referendum.",
"As expected, the closer the evaluation is to the referendum date, the more accurate the models since more information becomes available for each user.",
"Table REF shows the average (across-all-days) F-score by each model.",
"Table: Average F-score and standard deviation across all evaluation days using TEXT, NETWORK and BOTH user representations.",
"SVM s _{s} and SVM st _{st} denote the SVM with convolution kernels (SVM w _{w}, SVM n _{n}) and (SVM wt _{wt}, SVM nt _{nt}), respectively.Temporal convolution kernels using TEXT (SVM$_{wt}$ ) significantly outperform the best text-based baseline ($p=.001$ , Kruskal-Wallis test against SVM), with an average of 11.8% and 17.2% absolute and relative improvement respectively.",
"This demonstrates the model's ability on capturing the similarities between different users on a per-tweet basis compared to simpler models using tweet aggregates.",
"Also, SVM$_w$ and SVM$_{wt}$ implicitly capture similarities in the retweeting activity of the users.",
"This is important, since network information might not be easily accesible (e.g., due to API limitations) while it is expensive to compute at each timestep.",
"Hence, one can use SVM$_{wt}$ to model user written content and partially capture network information.",
"Classification accuracy consistently improves when using the NETWORK representation (i.e., graph embeddings).",
"RF achieves 94% F-score on the day before the referendum, whereas the worst-performing baseline (FF) still achieves 80.66% F-score on average.",
"SVM$_{nt}$ provides a small boost (1.6% on average) compared to the vanilla SVM, which uses only the user representations derived at the current time point.",
"This implies that the current network structure is indicative of users' voting intention, probably because the referendum was the dominant topic of discussion at the time, e.g., most of the retweeting activity was relevant the referendum (see Section ).",
"This is also in line with recent findings of [3] on predicting occupation class and income where network information is more predictive than language.",
"When combining the user text and network representation (BOTH), the baselines fail to improve over using only NETWORK.",
"In contrast, our MCKL improves by 4.28% over the best performing single convolution kernel model (SVM$_{nt}$ ).",
"This demonstrates that MCKL can effectively combine information from both representations by weighting their importance, and further improve the accuracy of the best performing single representation model.",
"Overall, MCKL significantly outperforms the best performing text-based baseline by approximately 20% in F-score ($p<.001$ , Kruskal-Wallis test)."
],
[
"Robustness Analysis",
"Due to the semi-supervised nature of our task, it is impossible to judge whether the small difference between MCKL and RF stems from a better designed model.",
"Furthermore, it is difficult to assess MCKL's effectiveness with respect to its ability to generalise over multiple and potentially noisy feature sources.",
"To assess the robustness of the best performing models (MCKL, RF) operating on BOTH information sources, here we perform experiments by adding random noise in their input.",
"We assume that there is a noisy source generating an extra K-dimensional representation $X$ for every user that we add as extra input to the models.",
"We set $K=25$ , so that (a) we account for a smaller noisy input compared to our features (25 vs 50) and (b) 1/5 of our kernels in MCKL and 25/125 input features in RF are noisy.",
"We perform 100 runs, each time drawing random noise $X\\sim N(0, 1)$ .",
"Our results indicate that RF is more sensitive to the noisy input compared to MCKL (see Figure REF ).",
"On average, RF achieves a small boost (0.04%) in performance with the added noise.",
"That together with the higher standard deviation reveal the vulnerability of RF to potentially corruption and stochasticity introduced in the input.",
"On the contrary, MCKL is consistently robust, achieving only a tiny reduction in performance on average across all days (0.02%) while the respective average standard deviation is lower than the one achieved by RF (0.12 vs 0.41).",
"This robustness is highly desirable is cases of such sudden political events and it also indicates that we can add kernels capturing different properties of our task (e.g., user-related information, images, etc.",
"), without having to decide a-priori which of them are indeed predictive of the user's voting intention.",
"We plan to investigate this in future work."
],
[
"Qualitative Analysis",
"In this section, we provide insights into the temporal variation observed in the users' shared content and the network structure during this major political crisis.",
"Besides performing a qualitative analysis during this time period, we believe that this analysis will also provide insights on (a) the reasons that trigger the significant improvement in performance of convolution kernels methods operating on TEXT, and (b) the reason that our non-temporally-sensitive baselines are rather competitive to our convolution kernel models, when using NETWORK information.",
"In the current section we provide details on both of these aspects."
],
[
"Language",
"We are interested in investigating which are the political-related entities that voters from both sides most likely mention.",
"We expect that this will shed light on the main focus of discussion in the political debates between the YES/NO voters that occurred after the announcement of the referendum.",
"For this, two authors manually compiled two lists of n-grams containing different ways of referringNote that Greek is a fully inflected language.",
"We opted not to apply stemming because inflected word forms carry meaningful information.",
"to the (a) the six major political parties and (b) their leaders (see Table REF ).",
"We represent every YES/NO user in the test set as aggregated tf-idf values of the ngrams (1-3) appearing in his/her concatenated tweets; then, we compute an n-gram 's $n$ score as $PMI(\\textit {n}, \\texttt {YES}) - PMI(\\textit {n}, \\texttt {NO})$ .",
"A positive score implies that it is highly associated with users who support the YES vote, and vice versa.",
"Figure REF shows that the parties and leaders that supported one side, mostly appear in tweets of users supporting the opposite side.",
"This is more evident when we consider tweets shared by the users after the announcement of the referendum.",
"Examining the content of highly-retweeted tweets, revealed sarcasm and hostility for the opposite side in the majority of them (see Table REF ).",
"Hostility is a frequent phenomenon in public debates [28] and our findings corroborate previous work showing that the political discourse on Twitter is polarised [15], [20].",
"Figure: Scores of n-grams related to the political parties/leaders, pre (18/06-26/06) and post (27/06-05/07) the referendum announcement.Table: Most similar words to YES and NO (translated to English), when training word2vec on different time periods.Finally, we examine the temporal variation of language over the same two periods.",
"Table REF shows the most similar words (translated to English) to the yes and no words, measured by cosine similarity, when training word2vec using the tweets of each time period.",
"The difference of the cosine similarities $cos_{post} - cos_{pre}$ between the yes/no vectors and each of their corresponding most similar words over these two periods is shown in Figure REF .",
"After the announcement, the context of the two words shifts towards the political domain.",
"That might explain why text aggregates become noisy, as shown in our results.",
"Convolution kernels are able to filter-out this noise since they operate on the tweet level by also taking the time into account.",
"We plan to study the semantic variation in language [17] in a more fine-grained way in future work.",
"Table: Examples of highly re-tweeted tweets after the announcement of the referendum.Figure: Difference in cosine similarity (cos post (w no/yes ,w)-cos pre (w no/yes ,w)cos_{post} (w_{no/yes}, w) - cos_{pre} (w_{no/yes}, w)) between the no/yes (red/blue) word vectors w no/yes w_{no/yes} and each of their most similar words in the two periods."
],
[
"Network",
"We explore the differences in retweeting behaviour of users over the same periods ((a) before the announcement of the referendum and (b) after and until the day of the referendum), by training two different LINE embedding models using tweets from the each period respectively.",
"Figure REF shows the plots of the first two dimensions of the graph embeddings before and after the announcement using principal component analysis.",
"The results unveil the effects of the referendum announcement and provide insights on the effectiveness of NETWORK information for predicting vote intention, as demonstrated in our results.",
"Before, YES and NO users appear to have similar retweeting behaviour, which changes after the announcement.",
"This finding illustrates the political homophily of the social network [13] and highlights the extremely polarised pre-election period [62].",
"Figure: Network representations of YES/NO (blue/red) users, before (above) and after (below) the referendum announcement.Next, we question whether the distance between the two classes of users through time changes according to time points at which real-world events occur.",
"To answer this, we compute the network embeddings of the train and test users every 12 hours, as in our experiments, and represent every class (YES/NO) at a certain time point $t$ by the average representations ($avg_{Y}^t$ , $avg_{N}^t$ ) of the corresponding users in the training set at $t$ .",
"Then, for every user $u$ in the test set, we use the cosine similarity $cos$ to calculate: $network\\_score_u^t = cos(u^t, avg_{Y}^t)-cos(u^t, avg_{N}^t).$ Finally, we calculate the average score of the YES and the NO users in the test set ($network_{Y}^t$ , $network_{N}^t$ ) at every time point $t$ and normalise the corresponding time series s.t.",
"$network_{Y}(0)$ =$network_{N}(0)=0$ .",
"We also employ an alternative approach, by generating the network embeddings on a seven-day sliding-window fashion and following the same process.",
"The results are shown in Figure REF .",
"In both cases, the YES/NO users start to deviate from each other right after the announcement of the referendum, with an upward/downward YES/NO trend until the day of the referendum.",
"This is effectively captured in our modelling and might explain the reason for the high accuracy achieved even by our baseline models, which are trained using the network representation of the users in the last day only.",
"However, the YES/NO users start to again approach each other only in the sliding window approach after the referendum day, since in our modelling the representations are built based on re-tweets aggregates over the whole period.",
"While this does not seem to have affected our performance, exploring the temporal structure of the network formations through time is of vital importance for longer lasting electoral cases.",
"Figure: Normalised difference of similarity of YES/NO (blue/red) users in our modelling (left) and in a sliding window approach (right)."
],
[
"Limitations and Future Work",
"Despite working under a real-time simulation setting, we are aware that our results come with some caution, owed to the selection of the users in our test set.",
"The limitations stem from the fact that we have selected highly active users that have used at least three polarised hashtags in their tweets after the announcement of the referendum.",
"As previous work has shown [12], [49], we expect that the performance of any model is likely to drop, if tested in a random sample of Twitter users.",
"We plan to investigate this, by annotating a random sample of Twitter users and comparing the performance in the two test sets, in our future work.",
"We also plan to assess the ability of MCKL to generalise, through exploring different referendum cases and incorporating more sources of information in our modelling.",
"Finally, we plan to study the temporal variation of language and network in a more fine-grained way."
],
[
"Conclusion",
"We presented a distant-supervised multiple convolution kernel approach, leveraging temporally sensitive language and network information to nowcast the voting stance of Twitter users during the 2015 Greek bailout referendum.",
"Following a real-time evaluation setting, we demonstrated the effectiveness and robustness of our approach against competitive baselines, showcasing the importance of temporal modelling for our task.",
"In particular, we showed that temporal modelling of the content generated by social media users provides a significant boost in performance (11%-19% in F-score) compared to traditional feature aggregate approaches.",
"Also, in line with past work on inferring the political ideology of social media users [16], [1], we showed that the network structure (in our case, the re-tweet network) of the social media users is more predictive of their voting intention, compared to the content they share.",
"By combining those two temporally sensitive aspects (text, network) of our task via a multiple kernel learning approach, we further boost the performance, leading to an overall significant 20% increase in F-score against the best performing, solely text-based feature aggregate baseline.",
"Finally, we provided qualitative insights on aspects related to the shift in online discussions and polarisation phenomena that occurred during this time period, which are effectively captured through our temporal modelling approach."
],
[
"Acknowledgements",
"The current work was supported by the EPSRC through the University of Warwick's Centre for Doctoral Training in Urban Science and Progress (grant EP/L016400/1) and through The Alan Turing Institute (grant EP/N510129/1)."
]
] | 1808.08538 |
[
[
"When You Should Use Lists in Haskell (Mostly, You Should Not)"
],
[
"Abstract We comment on the over-use of lists in functional programming.",
"With this respect, we review history of Haskell and some of its libraries, and hint at current developments."
],
[
"Introduction",
"It seems that the designers of the programming language Haskell [11] were madly in love with singly linked lists.",
"They even granted notational privileges: e.g., the data constructor for a non-empty list (“cons”) uses just one letter (:).",
"Looking at the Haskell standard, it also seems that singly linked lists are more important than the concept of static typing.",
"When declaring the type of an identifier, we must use two letters ::, because the natural (that is, mathematical) notation : is taken.",
"One goal of language design is to make typical expected usage easy to write.",
"Once the design is cast in stone, this becomes self-fulfilling: unsuspecting users of the language will write programs in what they think is idiomatic style, as suggested by built-in syntactical convenience and sugar.",
"Haskell lists have plenty of both: convenience by short notation, and sugar in the form of list comprehensions.",
"So it seems that it is typical Haskell to not declare types for identifiers, and use lists all over the place.",
"I will explain on the following pages that the purported connection between functional programming and lists is largely historical, and is detrimental to teaching and programming.",
"Lists do indeed serve a purpose, but a rather limited one: iteration: you use each element of a collection in order (FIFO or LIFO), on demand, and at most once.",
"If you use lists for anything else, in particular storage: you use elements out of order, e.g., by known index, or you access (“search”) elements by a property, then you're doing it wrong, wrong, wrong.",
"I will explain why, and what you should do instead.",
"As an appetizer, here are some bumper-sticker slogans extracted from the sections that follow.",
"If your program accesses a list by index (with the !",
"!",
"operator), then your program is wrong.",
"(Section ) If your program uses the length function, then your program is wrong.",
"If your program sorts a list, then your program is wrong.",
"If you wrote this sort function yourself, then it is doubly wrong.",
"The ideal use of a list is such that will be removed by the compiler.",
"(Section ) The enlightened programmer writes list-free code with Foldable.",
"(Section ) I will not enter arguments about syntax.",
"(The notation of cons as “:”, and type declarations with “::”, came to Haskell from Miranda [19], and has recently been reversed in Agda [13].)",
"This paper is about the semantics (and consequently, performance) of singly linked lists, as we have them in Haskell.",
"Note that the name List has a different meaning in, e.g., the Java standard library [14], where it denotes the abstract data type of sequences (collections whose elements can be accessed via a numerical index).",
"They have linked lists as one possible realization, but there are others, for which length and !",
"!",
"(indexing) are not atrocious.",
"Of course, there are Haskell libraries for efficient sequences [10].",
"This text was originally published (in 2017) at https://www.imn.htwk-leipzig.de/~waldmann/etc/untutorial/list-or-not-list/ Joachim Breitner, Bertram Felgenhauer, Henning Thielemann and Serge Le Huitouze have sent helpful remarks on earlier versions of this text, and I thank WFLP reviewers for detailed comments.",
"Of course, all remaining technical errors are my own, and I do not claim the commenters endorse my views."
],
[
"Where Do All These Lists Come From?",
"Certainly Haskell builds on the legacy of LISP [12], [7], the very first functional programming language - that has lists in its name already.",
"Indeed, singly linked lists were the only way to structure nested data in LISP.",
"This is a huge advantage (it's soo uniform), but also a huge dis-advantage (you cannot distinguish data - they all look like nested lists).",
"For discriminating between different shapes of nested data, people then invented algebraic data types, pattern matching, and static typing.",
"The flip side of static typing is that it seems to hinder uniform data processing.",
"So, language designers invented polymorphism [8] and generic programming, to get back flexibility and uniformity, while keeping static safety.",
"This was all well known at the time of Haskell's design, so why are lists that prominent in the language?",
"The one major design goal for Haskell was: to define a functional programming language with static typing and lazy evaluation.",
"For functional programming with static typing and strict evaluation, the language ML was already well established [20].",
"Haskell has lazy evaluation for function calls, including constructor applications, and a prime showcase for this is that we can handle apparently infinite data, like streams: data Stream a = Cons a (Stream a) Such streams do never end, but we can still do reasonable things with them, by evaluating finite prefixes on demand, in other words, lazily.",
"Now a standard Haskell list is just such a stream that can end, because there is an extra constructor for the empty stream data Stream a = Nil | Cons a (Stream a) and the problem with Haskell lists is that people tend to confuse what the type [ a ] stands for: is it: a lazy (possibly infinite) stream (yes) or is it: a random-access sequence (no); or a set (hell no) If you have an object-oriented background, then this is the confusion between an iterator (that allows to traverse data) and its underlying collection (that holds the actual data).",
"A Haskell list is a very good iterator - we even might say, it is the essence of iteration, but it makes for a very bad implementation of a collection."
],
[
"And Where Are These Lists Today?",
"The (historic) first impression of “functional programming is inherently list processing” seems hard to erase.",
"Indeed it is perpetuated by some Haskell teaching texts, some of which are popular.",
"If a text(book) aimed at beginners presents Int, String and [Int] on pages 1, 2, 3; but keywords data and case (and assorted concepts of algebraic data type and pattern matching) appear only after the first two hundred pages, well, then something's wrong.",
"For example, the shiny new (in 2015) http://haskell.org/ gets this exactly wrong.",
"At the very top of the page, it presents example code that is supposed to highlight typical features of the language.",
"We see a version of Eratosthenes' sieve that generates an infinite list of primes.",
"Declarative it certainly is.",
"But see - Lists, numbers.",
"Must be typical Haskell.",
"Also, that example code features list comprehensions - that's more syntactic sugar to misguide beginners.",
"And it claims “statically typed code” in the previous line, but - there is no type annotation in the code.",
"Yes, I know, the types are still there, because they are inferred by the compiler, but what does a beginner think?",
"The icing on all this is that the one thing that this example code really exemplifies, namely: infinite streams with lazy evaluation, is not mentioned anywhere in the vicinity."
],
[
"What Lists Are (Not) Good At",
"The answer is simple: never use lists for data storage with out-of-order access.",
"Accessing an element of a singly linked list at some arbitrary position $i$ takes time proportional to $i$ , and you do not want that.",
"The only operations that are cheap with a singly linked list are accessing its first element element (the head of list), and prepending a fresh first element.",
"So can we access any of the other elements in any reasonable way?",
"Yes, we can access the second element, but only after we drop the first one.",
"The second element becomes the “new head”, and the “old head” is then gone for good.",
"We are singly linked, we cannot go back.",
"If we want to access any element in a list, we have to access, and then throw away, all the elements to the left of it.",
"In other words, lists in Haskell realize the iterator design pattern [3] (Iterator in Java, IEnumerator in C#).",
"So - they are a means of expressing control flow.",
"As Cale Gibbard (2006) puts it [4], “lists largely take the place of what one would use loops for in imperative languages.” Or, we can think of a list as a stack, and going to the next element is its “pop” operation.",
"Can we really never go back?",
"Well, if xs is an iterator for some collection, then x : xs is an iterator that first will provide x, and after that, all elements of xs.",
"So, thinking of a list as a stack, the list constructor (:) realizes the “push” operation.",
"On the other hand, the “merge” function (of “mergesort” fame, it merges two monotonically increasing sequences) is a good use case for Haskell lists, since both inputs are traversed in order (individually - globally, the two traversals are interleaved).",
"Its source code is merge :: Ord a => [a] -> [a] -> [a] merge as@(a:as') bs@(b:bs') | a `compare` b == GT = b:merge as bs' | otherwise = a:merge as' bs merge [] bs = bs merge as [] = as We never copy lists, and we only ever pattern match on their beginning.",
"Indeed you can easily (in Java or C#) write a merge function that takes two iterators for monotone sequences, and produces one.",
"Try it, it's a fun exercise.",
"Use it to learn about yield return in C#.",
"Back to Haskell: how should we implement merge-sort then, given merge?",
"The truthful answer is: we should not, see below.",
"But for the moment, let's try anyway, as an exercise.",
"A straightforward approach is to split long input lists at (or near) the middle: msort :: Ord a => [a] -> [a] msort [] = [] ; msort [x] = [x] msort xs = let (lo,hi) = splitAt (div (length xs) 2) xs in merge (msort lo) (msort hi) but this is already wrong.",
"Here, xs is not used in a linear way, but twice: first in length xs (to compute to position of the split), and secondly, in splitAt ... xs, to actually do the split.",
"This is inefficient, since the list is traversed twice, which costs time – and space, since the spine of the list will be held in memory fully.",
"Indeed, the Haskell standard library contains an implementation of mergesort that does something different.",
"The basic idea is msort :: Ord a => [a] -> [a] msort xs = mergeAll (map (\\x -> [x]) xs) where mergeAll [x] = x mergeAll xs = mergeAll (mergePairs xs) mergePairs (a:b:xs) = merge a b: mergePairs xs mergePairs xs = xs But the actual implementation [21] has an extra provision for handling monotonic segments of the input more efficiently.",
"Make sure to read the references mentioned in the comments atop the source code of Data.OldList.sort, including [1].",
"But - if you sort a list in your application at all, then you're probably doing something wrong already.",
"There's a good chance that your data should have never been stored in a list from the start.",
"How do we store data, then?",
"This is the topic of each and every “Data Structures and Algorithms” course, and there are corresponding Haskell libraries.",
"For an overview, see [24]."
],
[
"Is There No Use For Lists As Containers?",
"There are two exemptions from the rules stated above.",
"If you are positively certain that your collections have a bounded size throughout the lifetime of your application, then you can choose to implement it in any way you want (and even use lists), because all operations will run in constant time anyway.",
"But beware - “the constant depends on the bound”.",
"So if the bound is three, then it's probably fine.",
"Then, education.",
"With lists, you can exemplify pattern matching and recursion (with streams, you exemplify co-recursion).",
"Certainly each student should see and do recursion early on in their first term.",
"But the teacher must make very clear that this is in the same ballpark with exercises of computing the Fibonacci sequence, or implementing multiplication and exponentiation for Peano numbers: the actual output of the computation is largely irrelevant.",
"What is relevant are the techniques they teach (algebraic data types, pattern matching, recursion, induction) because these have broad applications.",
"So, indeed, when I teach [25] algebraic data types, I do also use singly linked lists, among other examples (Bool, Maybe, Either, binary trees), but I always write (and have students write) data List a = Nil | Cons a (List a) I do avoid Haskell's built-in lists as long as possible, that is, until students understand their use (representing iterators) in the application programmer interfaces of recommended libraries like Data.Set and Data.Map."
],
[
"Lists That Are Not Really There",
"We learned above that the one legitimate use case for lists is iteration.",
"This is an important one.",
"It's so important that the Glasgow Haskell compiler (GHC) applies clever program transformations [6] in order to generate efficient machine code for programs that combine iterators (that is, lists) in typical ways.",
"The idea is that the most efficient code is always the one that is not there at all.",
"Indeed, several program transformations in the compiler remove intermediate iterators (lists).",
"Consider the expression sum $ map (\\x -> x*x) [1 .. 1000] which contains two iterations: the list [1 .. 1000] (the producer), and another list (of squares, built by map, which is a transformer).",
"Finally, we have sum as a consumer (its output is not a list, but a number).",
"We note that a transformer is just a consumer (of the input list) coupled to a producer (of the output list).",
"Now the important observation is that we can remove the intermediate list between a producer and the following consumer (where each of them may actually be part of a transformer).",
"This removal is called fusion.",
"With fusion, lists are used prominently in the source code, but the compiler will transform this into machine code that does not mention lists at all!",
"In particular, the above program will run in constant space, as it allocates nothing on the heap, and it needs no garbage collector.",
"For historic reference and detail, see [22]."
],
[
"Lists That Are Really Not There",
"Now I will present a more recent development (in the style of writing libraries for Haskell) that aims to remove lists from the actual source code.",
"Recall that deforestation removes lists from compiled code.",
"We have, e.g., and [True,False,True]=False, and you would think that and has type [Bool] -> Bool, as it maps a list of Booleans to a Boolean.",
"But no, the type is and :: Foldable t => t Bool -> Bool, and indeed we can apply and to any object of a type t a where t implements Foldable.",
"The constraint Foldable t for a type constructor t of kind * -> * means that there is an associated function toList :: t a -> [a].",
"So, c of type t a with Foldable t can be thought of as some kind of collection, and we can get the stream of its elements by toList c. This is similar to c.iterator() for Iterable<A> c in java.util.",
"For example, Data.Set [9] does this, so we can make a set s = S.fromList [False, True] and then write and s. It will produce the same result as the expression and (toList s).",
"Why all this?",
"Actually, this is not about toList.",
"Quite the contrary - this Foldable class exists to make it easier to avoid lists.",
"Typically, we would not implement or call the toList function, but use foldMap or foldr instead — or functions (like and) that are defined via foldMap.",
"We give another example.",
"Ancient libraries used to have a function with name and type sum :: Num a => [a] -> a but now its actual type is different: sum :: (Num a, Foldable t) => t a -> a, and so we can take the sum of all elements in a set s :: Set Int simply via sum s, instead of writing sum (toList s).",
"See - the lists are gone from the source code.",
"Let us look at the type of foldMap, and the class Monoid that it uses.",
"class Foldable t where foldMap :: Monoid m => (a -> m) -> t a -> m class Monoid m where mempty :: m ; mappend :: m -> m -> m This is fancy syntax for the following (slightly re-arranged) foldMap :: m -> (a -> m) -> (m -> m -> m) -> t a -> m where the first and third argument are the dictionary for instance Monoid m. So when you call foldMap over some collection (of type t a), then you have to supply (via the Monoid instance) the result value (mempty :: m) for the empty collection, the mapping (of type a -> m) to be applied to individual elements, and a binary operation mappend :: m -> m -> m to be used to combine results for sub-collections.",
"Let us see how this is used to sum over Data.Set, the type mentioned above, internally defined as size-balanced search trees in Data.Set.Internal [9] and repeated here for convenience: data Set a = Bin Size a (Set a) (Set a) | Tip instance Foldable Set where foldMap f t = go t where go Tip = mempty go (Bin 1 k _ _) = f k go (Bin _ k l r) = go l `mappend` (f k `mappend` go r) This means that foldMap f s, for s :: Set a of size larger than one, does the following: each branch node n is replaced by two calls of mappend that combine results of recursive calls in subtrees l and r of n, and f k for the key k of n, and we make sure to traverse the left subtree, then the key at the root, then the right subtree.",
"The definition [15] of sum is class Foldable t where sum :: Num a => t a -> a sum = getSum #.",
"foldMap Sum with the following auxiliary types [5] newtype Sum a = Sum { getSum :: a } instance Num a => Monoid (Sum a) where mempty = Sum 0 ; mappend (Sum x) (Sum y) = Sum (x + y) So, when we take the sum of some Set a, we do a computation that adds up all elements of the set, but without an explicit iterator - there is no list in sight."
],
[
"Specifically for Section ",
": When the types for sum, and, and many more functions were generalized from lists to Foldable a while back, some Foldable instances were introduced that result in funny-looking semantics like maximum (2,1) == 1, which is only “natural” if you think of the curried type constructor application (,) a b, remove the last argument, and demand that (,) a “obviously” must be Functor, Foldable, and so on.",
"This might save some keystrokes, but is has been criticized [18] because it “moves Haskell into the Matlab league of languages where any code is accepted, with unpredictable results.”"
],
[
"In general",
": This text is based on my experience in teaching (advanced) programming language concepts, for two decades now, and using Haskell for real-world code, e.g., an E-Learning/Assessment system [16], [23] in use since 2001, and an administration and presentation layer for Termination Competitions [26].",
"Other researchers and practitioners have expressed similar opinion on using (or, avoiding) lists.",
"Breitner teaches a course [2] that is similar to mine, and Thielemann recommends [17] to “choose types properly”, in particular, to not confuse lists with arrays."
],
[
"Previous discussion of this text",
": see https://mail.haskell.org/pipermail/haskell-cafe/2017-March/thread.html#126457 and https://www.reddit.com/r/haskell/comments/5yiusn/when_you_should_use_lists_in_haskell_mostly_you/."
]
] | 1808.08329 |
[
[
"Human-centric Indoor Scene Synthesis Using Stochastic Grammar"
],
[
"Abstract We present a human-centric method to sample and synthesize 3D room layouts and 2D images thereof, to obtain large-scale 2D/3D image data with perfect per-pixel ground truth.",
"An attributed spatial And-Or graph (S-AOG) is proposed to represent indoor scenes.",
"The S-AOG is a probabilistic grammar model, in which the terminal nodes are object entities.",
"Human contexts as contextual relations are encoded by Markov Random Fields (MRF) on the terminal nodes.",
"We learn the distributions from an indoor scene dataset and sample new layouts using Monte Carlo Markov Chain.",
"Experiments demonstrate that our method can robustly sample a large variety of realistic room layouts based on three criteria: (i) visual realism comparing to a state-of-the-art room arrangement method, (ii) accuracy of the affordance maps with respect to groundtruth, and (ii) the functionality and naturalness of synthesized rooms evaluated by human subjects.",
"The code is available at https://github.com/SiyuanQi/human-centric-scene-synthesis."
],
[
"Introduction",
"Traditional methods of 2D/3D image data collection and ground-truth labeling have evident limitations.",
"i) High-quality ground truths are hard to obtain, as depth and surface normal obtained from sensors are always noisy.",
"ii) It is impossible to label certain ground truth information, , 3D objects sizes in 2D images.",
"iii) Manual labeling of massive ground-truth is tedious and error-prone even if possible.",
"To provide training data for modern machine learning algorithms, an approach to generate large-scale, high-quality data with the perfect per-pixel ground truth is in need.",
"Figure: An example of synthesized indoor scene (bedroom) with affordance heatmap.",
"The joint sampling of a scene is achieved by alternative sampling of humans and objects according to the joint probability distribution.In this paper, we propose an algorithm to automatically generate a large-scale 3D indoor scene dataset, from which we can render 2D images with pixel-wise ground-truth of the surface normal, depth, and segmentation, .",
"The proposed algorithm is useful for tasks including but not limited to: i) learning and inference for various computer vision tasks; ii) 3D content generation for 3D modeling and games; iii) 3D reconstruction and robot mappings problems; iv) benchmarking of both low-level and high-level task-planning problems in robotics.",
"Figure: Scene grammar as an attributed S-AOG.",
"A scene of different types is decomposed into a room, furniture, and supported objects.",
"Attributes of terminal nodes are internal attributes (sizes), external attributes (positions and orientations), and a human position that interacts with this entity.",
"Furniture and object nodes are combined by an address terminal node and a regular terminal node.",
"A furniture node (, a chair) is grouped with another furniture node (, a desk) pointed by its address terminal node.",
"An object (, a monitor) is supported by the furniture (, a desk) it is pointing to.",
"If the value of the address node is null, the furniture is not grouped with any furniture, or the object is put on the floor.",
"Contextual relations are defined between the room and furniture, between a supported object and supporting furniture, among different pieces of furniture, and among functional groups.Synthesizing indoor scenes is a non-trivial task.",
"It is often difficult to properly model either the relations between furniture of a functional group, or the relations between the supported objects and the supporting furniture.",
"Specifically, we argue there are four major difficulties.",
"(i) In a functional group such as a dining set, the number of pieces may vary.",
"(ii) Even if we only consider pair-wise relations, there is already a quadratic number of object-object relations.",
"(iii) What makes it worse is that most object-object relations are not obviously meaningful.",
"For example, it is unnecessary to model the relation between a pen and a monitor, even though they are both placed on a desk.",
"(iv) Due to the previous difficulties, an excessive number of constraints are generated.",
"Many of the constraints contain loops, making the final layout hard to sample and optimize.",
"To address these challenges, we propose a human-centric approach to model indoor scene layout.",
"It integrates human activities and functional grouping/supporting relations as illustrated in Figure REF .",
"This method not only captures the human context but also simplifies the scene structure.",
"Specifically, we use a probabilistic grammar model for images and scenes [49] – an attributed spatial And-Or graph (S-AOG), including vertical hierarchy and horizontal contextual relations.",
"The contextual relations encode functional grouping relations and supporting relations modeled by object affordances [8].",
"For each object, we learn the affordance distribution, , an object-human relation, so that a human can be sampled based on that object.",
"Besides static object affordance, we also consider dynamic human activities in a scene, constraining the layout by planning trajectories from one piece of furniture to another.",
"In Section , we define the grammar and its parse graph which represents an indoor scene.",
"We formulate the probability of a parse graph in Section .",
"The learning algorithm is described in Section .",
"Finally, sampling an indoor scene is achieved by sampling a parse tree (Section ) from the S-AOG according to the prior probability distribution.",
"This paper makes three major contributions.",
"(i) We jointly model objects, affordances, and activity planning for indoor scene configurations.",
"(ii) We provide a general learning and sampling framework for indoor scene modeling.",
"(iii) We demonstrate the effectiveness of this structured joint sampling by extensive comparative experiments."
],
[
"Related Work",
"3D content generation is one of the largest communities in the game industry and we refer readers to a recent survey [13] and book [31].",
"In this paper, we focus on approaches related to our work using probabilistic inference.",
"Yu [44] and Handa [10] optimize the layout of rooms given a set of furniture using MCMC, while Talton [39] and Yeh [43] consider an open world layout using RJMCMC.",
"These 3D room re-arrangement algorithms optimize room layouts based on constraints to generate new room layouts using a given set of objects.",
"In contrast, the proposed method is capable of adding or deleting objects without fixing the number of objects.",
"Some literature focused on fine-grained room arrangement for specific problems, , small objects arrangement using user-input examples [6] and procedural modeling of objects to encourage volumetric similarity to a target shape [29].",
"To achieve better realism, Merrell [22] introduced an interactive system providing suggestions following interior design guidelines.",
"Jiang [17] uses a mixture of conditional random field (CRF) to model the hidden human context and arrange new small objects based on existing furniture in a room.",
"However, it cannot direct sampling/synthesizing an indoor scene, since the CRF is intrinsically a discriminative model for structured classification instead of generation.",
"Synthetic data has been attracting an increasing interest to augment or even serve as training data for object detection and correspondence [5], [21], [24], [34], [38], [46], [48], single-view reconstruction [16], pose estimation [3], [32], [37], [41], depth prediction [36], semantic segmentation [28], scene understanding [9], [10], [45], autonomous pedestrians and crowd [23], [26], [33], VQA [18], training autonomous vehicles [2], [4], [30], human utility learning [42], [50] and benchmarks [11], [27].",
"Stochastic grammar model has been used for parsing the hierarchical structures from images of indoor [20], [47] and outdoor scenes [20], and images/videos involving humans [25], [40].",
"In this paper, instead of using stochastic grammar for parsing, we forward sample from a grammar model to generate large variations of indoor scenes."
],
[
"Representation of Indoor Scenes",
"We use an attributed S-AOG [49] to represent an indoor scene.",
"An attributed S-AOG is a probabilistic grammar model with attributes on the terminal nodes.",
"It combines i) a probabilistic context free grammar (PCFG), and ii) contextual relations defined on an Markov Random Field (MRF), , the horizontal links among the nodes.",
"The PCFG represents the hierarchical decomposition from scenes (top level) to objects (bottom level) by a set of terminal and non-terminal nodes, whereas contextual relations encode the spatial and functional relations through horizontal links.",
"The structure of S-AOG is shown in Figure REF .",
"Formally, an S-AOG is defined as a 5-tuple: $\\mathcal {G} = \\langle S, V, R, P, E \\rangle $ , where we use notations $S$ the root node of the scene grammar, $V$ the vertex set, $R$ the production rules, $P$ the probability model defined on the attributed S-AOG, and $E$ the contextual relations represented as horizontal links between nodes in the same layer.",
"We use the term “vertices\" instead of “symbols\" (in the traditional definition of PCFG) to be consistent with the notations in graphical models.",
"Vertex Set $V$ can be decomposed into a finite set of non-terminal and terminal nodes: $V = V_{NT} \\cup V_{T}$ .",
"[wide,leftmargin=0cm,noitemsep,nolistsep] $V_{NT} = V^{And} \\cup V^{Or} \\cup V^{Set}$ .",
"The non-terminal nodes consists of three subsets.",
"i) A set of And-nodes $V^{And}$ , in which each node represents a decomposition of a larger entity (, a bedroom) into smaller components (, walls, furniture and supported objects).",
"ii) A set of Or-nodes $V^{Or}$ , in which each node branches to alternative decompositions (, an indoor scene can be a bedroom or a living room), enabling the algorithm to reconfigure a scene.",
"iii) A set of Set nodes $V^{Set}$ , in which each node represents a nested And-Or relation: a set of Or-nodes serving as child branches are grouped by an And-node, and each child branch may include different numbers of objects.",
"$V_{T} = V_T^r \\cup V_T^a$ .",
"The terminal nodes consists of two subsets of nodes: regular nodes and address nodes.",
"i) A regular terminal node $v \\in V_T^r$ represents a spatial entity in a scene (, an office chair in a bedroom) with attributes.",
"In this paper, the attributes include internal attributes $A_{int}$ of object sizes $(w, l, h)$ , external attributes $A_{ext}$ of object position $(x, y, z)$ and orientation ($x-y$ plane) $\\theta $ , and sampled human positions $A_h$ .",
"ii) To avoid excessively dense graphs, an address terminal node $v \\in V_T^a$ is introduced to encode interactions that only occur in a certain context but are absent in all others [7].",
"It is a pointer to regular terminal nodes, taking values in the set $V_T^r \\cup \\lbrace \\mbox{nil}\\rbrace $ , representing supporting or grouping relations as shown in Figure REF .",
"Figure: (a) A simplified example of a parse graph of a bedroom.",
"The terminal nodes of the parse graph form an MRF in the terminal layer.",
"Cliques are formed by the contextual relations projected to the terminal layer.",
"Examples of the four types of cliques are shown in (b)-(e), representing four different types of contextual relations.Contextual Relations $E$ among nodes are represented by the horizontal links in S-AOG forming MRFs on the terminal nodes.",
"To encode the contextual relations, we define different types of potential functions for different cliques.",
"The contextual relations $E = E_f \\cup E_o \\cup E_g \\cup E_r$ are divided into four subsets: i) relations among furniture $E_f$ ; ii) relations between supported objects and their supporting objects $E_o$ (, a monitor on a desk); iii) relations between objects of a functional pair $E_g$ (, a chair and a desk); and iv) relations between furniture and the room $E_r$ .",
"Accordingly, the cliques formed in the terminal layer could also be divided into four subsets: $C = C_f \\cup C_o \\cup C_g \\cup C_r$ .",
"Instead of directly capturing the object-object relations, we compute the potentials using affordances as a bridge to characterize the object-human-object relations.",
"A hierarchical parse tree $pt$ is an instantiation of the S-AOG by selecting a child node for the Or-nodes as well as determining the state of each child node for the Set-nodes.",
"A parse graph $pg$ consists of a parse tree $pt$ and a number of contextual relations $E$ on the parse tree: $pg = (pt, E_{pt})$ .",
"Figure REF illustrates a simple example of a parse graph and four types of cliques formed in the terminal layer."
],
[
"Probabilistic Formulation of S-AOG",
"A scene configuration is represented by a parse graph $pg$ , including objects in the scene and associated attributes.",
"The prior probability of $pg$ generated by an S-AOG parameterized by $\\Theta $ is formulated as a Gibbs distribution: $p(pg | \\Theta ) & = \\frac{1}{Z} \\exp \\lbrace -\\mathcal {E}(pg | \\Theta ) \\rbrace \\\\& = \\frac{1}{Z} \\exp \\lbrace -\\mathcal {E}(pt | \\Theta ) - \\mathcal {E}(E_{pt} | \\Theta ) \\rbrace ,$ where $\\mathcal {E}(pg | \\Theta )$ is the energy function of a parse graph, $\\mathcal {E}(pt | \\Theta )$ is the energy function of a parse tree, and $\\mathcal {E}(E_{pt} | \\Theta )$ is the energy term of the contextual relations.",
"$\\mathcal {E}(pt | \\Theta )$ can be further decomposed into the energy functions of different types of non-terminal nodes, and the energy functions of internal attributes of both regular and address terminal nodes: $\\mathcal {E}(pt | \\Theta )& = \\underbrace{\\sum _{{v \\in V^{Or}}} \\mathcal {E}_{\\Theta }^{Or}(v)+ \\sum _{{v \\in V^{Set}}} \\mathcal {E}_{\\Theta }^{Set}(v) }_\\text{non-terminal nodes}+ \\underbrace{\\sum _{{v \\in V_{T}^{r}}} \\mathcal {E}_{\\Theta }^{A_{in}}(v) }_\\text{terminal nodes},$ where the choice of the child node of an Or-node $v \\in V^{Or}$ and the child branch of a Set-node $v \\in V^{Set}$ follow different multinomial distributions.",
"Since the And-nodes are deterministically expanded, we do not have an energy term for the And-nodes here.",
"The internal attributes $A_{in}$ (size) of terminal nodes follows a non-parametric probability distribution learned by kernel density estimation.",
"$\\mathcal {E}(E_{pt} | \\Theta )$ combines the potentials of the four types of cliques formed in the terminal layer, integrating human attributes and external attributes of regular terminal nodes: $p(E_{pt} | \\Theta ) & = \\frac{1}{Z} \\exp \\lbrace -\\mathcal {E}(E_{pt} | \\Theta ) \\rbrace \\\\& =\\prod _{{c \\in C_f}} \\phi _f(c)\\prod _{{c \\in C_o}} \\phi _o(c)\\prod _{{c \\in C_g}} \\phi _g(c)\\prod _{{c \\in C_r}} \\phi _r(c).$ Human Centric Potential Functions: [wide,leftmargin=0cm,noitemsep,nolistsep] Potential function $\\phi _f(c)$ is defined on relations between furniture (Figure REF (b)).",
"The clique $c = \\lbrace f_i\\rbrace \\in C_f$ includes all the terminal nodes representing furniture: $\\phi _f(c) = \\frac{1}{Z} \\exp \\lbrace - \\lambda _f \\cdot \\langle \\sum _{f_i \\ne f_j} l_{\\textup {col}}(f_i, f_j), l_{\\textup {ent}}(c) \\rangle \\rbrace ,$ where $\\lambda _f$ is a weight vector, $<\\cdot , \\cdot >$ denotes a vector, and the cost function $l_{\\textup {col}}(f_i, f_j)$ is the overlapping volume of the two pieces of furniture, serving as the penalty of collision.",
"The cost function $l_{\\textup {ent}}(c) = -H(\\Gamma ) = \\Sigma _i p(\\gamma _i)\\log p(\\gamma _i) $ yields better utility of the room space by sampling human trajectories, where $\\Gamma $ is the set of planned trajectories in the room, and $H(\\Gamma )$ is the entropy.",
"The trajectory probability map is first obtained by planning a trajectory $\\gamma _i$ from the center of every piece of furniture to another one using bi-directional rapidly-exploring random tree (RRT) [19], which forms a heatmap.",
"The entropy is computed from the heatmap as shown in Figure REF .",
"Potential function $\\phi _o(c)$ is defined on relations between a supported object and the supporting furniture (Figure REF (c)).",
"A clique $c = \\lbrace f, a, o\\rbrace \\in C_o$ includes a supported object terminal node $o$ , the address node $a$ connected to the object, and the furniture terminal node $f$ pointed by $a$ : $\\phi _o(c) = \\frac{1}{Z} \\exp \\lbrace - \\lambda _o \\cdot \\langle l_{\\textup {hum}}(f, o) , l_{\\textup {add}}(a) \\rangle \\rbrace ,$ where the cost function $l_{\\textup {hum}}(f, o)$ defines the human usability cost—a favorable human position should enable an agent to access or use both the furniture and the object.",
"To compute the usability cost, human positions $h_i^o$ are first sampled based on position, orientation, and the affordance map of the supported object.",
"Given a piece of furniture, the probability of the human positions is then computed by: $l_{\\textup {hum}}(f, o) = \\max _i p(h_i^o | f).$ The cost function $l_{\\textup {add}}(a)$ is the negative log probability of an address node $v \\in V_{T}^{a}$ , treated as a certain regular terminal node, following a multinomial distribution.",
"Potential function $\\phi _g(c)$ is defined on functional grouping relations between furniture (Figure REF (d)).",
"A clique $c = \\lbrace f_i, a, f_j\\rbrace \\in C_g$ consists of terminal nodes of a core functional furniture $f_i$ , pointed by the address node $a$ of an associated furniture $f_j$ .",
"The grouping relation potential is defined similarly to the supporting relation potential $\\phi _g(c) = \\frac{1}{Z} \\exp \\lbrace - \\lambda _c \\cdot \\langle l_{\\textup {hum}}(f_i, f_j), l_{\\textup {add}}(a) \\rangle \\rbrace .$ Figure: Given a scene configuration, we use bi-directional RRT to plan from every piece of furniture to another, generating a human activity probability map.Other Potential Functions: [wide,leftmargin=0cm,noitemsep,nolistsep] Potential function $\\phi _r(c)$ is defined on relations between the room and furniture (Figure REF (e)).",
"A clique $c = \\lbrace f, r \\rbrace \\in C_r$ includes a terminal node $f$ and $r$ representing a piece of furniture and a room, respectively.",
"The potential is defined as $\\phi _r(c) =\\frac{1}{Z} \\exp \\lbrace - \\lambda _r \\cdot \\langle l_{\\textup {dis}}(f, r), l_{\\textup {ori}}(f, r) \\rangle \\rbrace ,$ where the distance cost function is defined as $l_{\\textup {dis}}(f, r) = -\\log p(d | \\Theta )$ , in which $d \\sim \\ln \\mathcal {N}(\\mu , \\sigma ^2)$ is the distance between the furniture and the nearest wall modeled by a log normal distribution.",
"The orientation cost function is defined as $l_{\\textup {ori}}(f, r) = -\\log p(\\theta | \\Theta )$ , where $\\theta \\sim p(\\mu , \\kappa ) = \\frac{e^{\\kappa \\cos (x-\\mu )}}{2 \\pi I_0(\\kappa )}$ is the relative orientation between the model and the nearest wall modeled by a von Mises distribution."
],
[
"Learning S-AOG",
"We use the SUNCG dataset [35] as training data.",
"It contains over 45K different scenes with manually created realistic room and furniture layouts.",
"We collect the statistics of room types, room sizes, furniture occurrences, furniture sizes, relative distances, orientations between furniture and walls, furniture affordance, grouping occurrences, and supporting relations.",
"The parameters $\\Theta $ of the probability model $P$ can be learned in a supervised way by maximum likelihood estimation (MLE).",
"Weights of Loss Functions: Recall that the probability distribution of cliques formed in the terminal layer is $p(E_\\mathit {pt} | \\Theta ) & = \\frac{1}{Z} \\exp \\lbrace -\\mathcal {E}(E_\\mathit {pt} | \\Theta ) \\rbrace \\\\& = \\frac{1}{Z} \\exp \\lbrace - \\langle \\lambda , l(E_\\mathit {pt}) \\rangle \\rbrace ,$ where $\\lambda $ is the weight vector and $l(E_\\mathit {pt})$ is the loss vector given by four different types of potential functions.",
"To learn the weight vector, the standard MLE maximizes the average log-likelihood: $\\mathcal {L}(E_\\mathit {pt} | \\Theta ) & = - \\frac{1}{N} \\sum _{n=1}^{N} \\langle \\lambda , l(E_\\mathit {pt_n}) \\rangle - \\log Z.$ This is usually maximized by following the gradient: $& \\frac{\\partial \\mathcal {L}(E_\\mathit {pt} | \\Theta )}{\\partial \\lambda } = - \\frac{1}{N} \\sum _{n=1}^{N} l(E_\\mathit {pt_n}) -\\frac{\\partial \\log Z}{\\partial \\lambda } \\\\& = - \\frac{1}{N} \\sum _{n=1}^{N} l(E_\\mathit {pt_n}) -\\frac{\\partial \\log \\sum _{pt}{\\exp \\lbrace - \\langle \\lambda , l(E_\\mathit {pt}) \\rangle \\rbrace }}{\\partial \\lambda } \\\\& = - \\frac{1}{N} \\sum _{n=1}^{N} l(E_\\mathit {pt_n}) +\\sum _{pt} \\frac{1}{Z} \\exp \\lbrace - \\langle \\lambda , l(E_\\mathit {pt}) \\rangle \\rbrace l(E_\\mathit {pt}) \\\\& = - \\frac{1}{N} \\sum _{n=1}^{N} l(E_\\mathit {pt_n}) +\\frac{1}{\\widetilde{N}} \\sum _{\\widetilde{n}=1}^{\\widetilde{N}} l(E_\\mathit {pt_{\\widetilde{n}}}),$ where $\\lbrace E_\\mathit {pt_{\\widetilde{n}}}\\rbrace _{\\widetilde{n}=1, \\cdots , \\widetilde{N}}$ is the set of synthesized examples from the current model.",
"It is usually computationally infeasible to sample a Markov chain that burns into an equilibrium distribution at every iteration of gradient ascent.",
"Hence, instead of waiting for the Markov chain to converge, we adopt the contrastive divergence (CD) learning that follows the gradient of difference of two divergences [14] $\\mbox{CD}_{\\widetilde{N}} = \\mbox{KL}(p_0 || p_{\\infty }) - \\mbox{KL}(p_{\\widetilde{n}} || p_{\\infty }),$ where $\\mbox{KL}(p_0 || p_{\\infty })$ is the Kullback-Leibler divergence between the data distribution $p_0$ and the model distribution $p_{\\infty }$ , and $p_{\\widetilde{n}}$ is the distribution obtained by a Markov chain started at the data distribution and run for a small number $\\widetilde{n}$ of steps.",
"In this paper, we set $\\widetilde{n}=1$ .",
"Contrastive divergence learning has been applied effectively to addressing various problems; one of the most notable work is in the context of Restricted Boltzmann Machines [15].",
"Both theoretical and empirical evidences shows its efficiency while keeping bias typically very small [1].",
"The gradient of the contrastive divergence is given by $\\frac{\\partial \\mbox{CD}_{\\widetilde{N}}}{\\partial \\lambda } = & \\frac{1}{N} \\sum _{n=1}^{N} l(E_\\mathit {pt_n}) -\\frac{1}{\\widetilde{N}} \\sum _{\\widetilde{n}=1}^{\\widetilde{N}} l(E_\\mathit {pt_{\\widetilde{n}}}) \\\\& - \\frac{\\partial p_{\\widetilde{n}}}{\\partial \\lambda } \\frac{\\partial \\mbox{KL} (p_{\\widetilde{n}} || p_{\\infty }) }{\\partial p_{\\widetilde{n}}}.$ Extensive simulations [14] showed that the third term can be safely ignored since it is small and seldom opposes the resultant of the other two terms.",
"Finally, the weight vector is learned by gradient descent computed by generating a small number $\\widetilde{N}$ of examples from the Markov chain $\\lambda _{t+1} & = \\lambda _{t} - \\eta _{t} \\frac{\\partial \\mbox{CD}_{\\widetilde{N}}}{\\partial \\lambda } \\\\& = \\lambda _{t} + \\eta _{t} \\left(\\frac{1}{\\widetilde{N}} \\sum _{\\widetilde{n}=1}^{\\widetilde{N}} l(E_\\mathit {pt_{\\widetilde{n}}}) -\\frac{1}{N} \\sum _{n=1}^{N} l(E_\\mathit {pt_n}) \\right).$ Branching Probabilities: The MLE of the branch probabilities $\\rho _i$ of Or-nodes, Set-nodes and address terminal nodes is simply the frequency of each alternative choice [49]: $\\rho _i = \\#(v \\rightarrow u_i)/\\sum \\limits _{j=1}^{n(v)}\\#(v \\rightarrow u_j)$ .",
"Figure: Examples of the learned affordance maps.",
"Given the object positioned in the center facing upwards, , coordinate of (0,0)(0, 0) facing direction (0,1)(0, 1), the maps show the distributions of human positions.",
"The affordance maps accurately capture the subtle differences among desks, coffee tables, and dining tables.",
"Some objects are orientation sensitive, , books, laptops, and night stands, while some are orientation invariant, , fruit bowls and vases.Figure: MCMC sampling process (from left to right) of scene configurations with simulated annealing.Grouping Relations: The grouping relations are hand-defined (, nightstands are associated with beds, chairs are associated with desks and tables).",
"The probability of occurrence is learned as a multinomial distribution, and the supporting relations are automatically extracted from SUNCG.",
"Room Size and Object Sizes: The distribution of the room size and object size among all the furniture and supported objects is learned as a non-parametric distribution.",
"We first extract the size information from the 3D models inside SUNCG dataset, and then fit a non-parametric distribution using kernel density estimation.",
"The distances and relative orientations of the furniture and objects to the nearest wall are computed and fitted into a log normal and a mixture of von Mises distributions, respectively.",
"Affordances: We learn the affordance maps of all the furniture and supported objects by computing the heatmap of possible human positions.",
"These position include annotated humans, and we assume that the center of chairs, sofas, and beds are positions that humans often visit.",
"By accumulating the relative positions, we get reasonable affordance maps as non-parametric distributions as shown in Figure REF ."
],
[
"Synthesizing Scene Configurations",
"Synthesizing scene configurations is accomplished by sampling a parse graph $pg$ from the prior probability $p(pg | \\Theta )$ defined by the S-AOG.",
"The structure of a parse tree $pt$ (, the selection of Or-nodes and child branches of Set-nodes) and the internal attributes (sizes) of objects can be easily sampled from the closed-form distributions or non-parametric distributions.",
"However, the external attributes (positions and orientations) of objects are constrained by multiple potential functions, hence they are too complicated to be directly sampled from.",
"Here, we utilize a Markov chain Monte Carlo (MCMC) sampler to draw a typical state in the distribution.",
"The process of each sampling can be divided into two major steps: Directly sample the structure of $pt$ and internal attributes $A_{in}$ : (i) sample the child node for the Or-nodes; (ii) determine the state of each child branch of the Set-nodes; and (iii) for each regular terminal node, sample the sizes and human positions from learned distributions.",
"Use an MCMC scheme to sample the values of address nodes $V^a$ and external attributes $A_{ex}$ by making proposal moves.",
"A sample will be chosen after the Markov chain converges.",
"We design two simple types of Markov chain dynamics which are used at random with probabilities $q_i, i=1, 2$ to make proposal moves: [wide,leftmargin=0cm,noitemsep,nolistsep] Dynamics $q_1$ : translation of objects.",
"This dynamic chooses a regular terminal node, and samples a new position based on the current position $x$ : $x \\rightarrow x + \\delta x$ , where $\\delta x$ follows a bivariate normal distribution.",
"Dynamics $q_2$ : rotation of objects.",
"This dynamic chooses a regular terminal node, and samples a new orientation based on the current orientation of the object: $\\theta \\rightarrow \\theta + \\delta \\theta $ , where $\\delta \\theta $ follows a normal distribution.",
"Adopting the Metropolis-Hastings algorithm, the proposed new parse graph $pg^{\\prime }$ is accepted according to the following acceptance probability: $\\alpha (pg^{\\prime } | pg, \\Theta ) &= \\min (1, \\frac{p(pg^{\\prime }|\\Theta ) p(pg | pg^{\\prime })}{p(pg|\\Theta ) p(pg^{\\prime } | pg)}) \\\\&= \\min (1, \\exp (\\mathcal {E}(pg|\\Theta ) - \\mathcal {E}(pg^{\\prime }|\\Theta ))),$ where the proposal probability rate is canceled since the proposal moves are symmetric in probability.",
"A simulated annealing scheme is adopted to obtain samples with high probability as shown in Figure REF .",
"Figure: Examples of scenes in ten different categories.",
"Top: top-view.",
"Middle: a side-view.",
"Bottom: affordance heatmap.Table: Classification results on segmentation maps of synthesized scenes using different methods vs. SUNCG.Table: Comparison between affordance maps computed from our samples and real dataTable: Human subjects' ratings (1-5) of the sampled layouts based on functionality (top) and naturalness (bottom)"
],
[
"Experiments",
"We design three experiments based on different criteria: i) visual similarity to manually constructed scenes, ii) the accuracy of affordance maps for the synthesized scenes, and iii) functionalities and naturalness of the synthesized scenes.",
"The first experiment compares our method with a state-of-the-art room arrangement method; the second experiment measures the synthesized affordances; the third one is an ablation study.",
"Overall, the experiments show that our algorithm can robustly sample a large variety of realistic scenes that exhibits naturalness and functionality.",
"Figure: Top-view segmentation maps for classification.Layout Classification.",
"To quantitatively evaluate the visual realism, we trained a classifier on the top-view segmentation maps of synthesized scenes and SUNCG scenes.",
"Specifically, we train a ResNet-152 [12] to classify top view layout segmentation maps (synthesized vs. SUNCG).",
"Examples of top-view segmentation maps are shown in Figure REF .",
"The reason to use segmentation maps is that we want to evaluate the room layout excluding rendering factors such as object materials.",
"We use two methods for comparison: i) a state-of-the-art furniture arrangement optimization method proposed by Yu [44], and ii) slight perturbation of SUNCG scenes by adding small Gaussian noise ($\\mu = 0, \\sigma = 0.1$ ) to the layout.",
"The room arrangement algorithm proposed by [44] takes one pre-fixed input room and re-organizes the room.",
"1500 scenes are randomly selected for each method and SUNCG: 800 for training, 200 for validation, and 500 for testing.",
"As shown in Table REF , the classifier successfully distinguishes Yu vs. SUNCG with an accuracy of 87.49%.",
"Our method achieves a better performance of 76.18%, exhibiting a higher realism and larger variety.",
"This result indicates our method is much more visually similar to real scenes than the comparative scene optimization method.",
"Qualitative comparisons of Yu and our method are shown in Figure REF .",
"Figure: Top: previous methods only re-arranges a given input scene with a fixed room size and a predefined set of objects.",
"Bottom: our method samples a large variety of scenes.Affordance Maps Comparison.",
"We sample 500 rooms of 10 different scene categories summarized in Table REF .",
"For each type of room, we compute the affordance maps of the objects in the synthesized samples, and calculate both the total variation distances and Hellinger distances between the affordance maps computed from the synthesized samples and the SUNCG dataset.",
"The two distributions are similar if the distance is close to 0.",
"Most sampled scenes using the proposed method show similar affordance distributions to manually created ones from SUNCG.",
"Some scene types (Storage) show a larger distance since they do not exhibit clear affordances.",
"Overall, the results indicate that affordance maps computed from the synthesized scenes are reasonably close to the ones computed from manually constructed scenes by artists.",
"Functionality and naturalness.",
"Three methods are used for comparison: (i) direct sampling of rooms according to the statistics of furniture occurrence without adding contextual relation, (ii) an approach that only models object-wise relations by removing the human constraints in our model, and (iii) the algorithm proposed by Yu [44].",
"We showed the sampled layouts using three methods to 4 human subjects.",
"Subjects were told the room category in advance, and instructed to rate given scene layouts without knowing the method used to generate the layouts.",
"For each of the 10 room categories, 24 samples were randomly selected using our method and [44], and 8 samples were selected using both the object-wise modeling method and the random generation.",
"The subjects evaluated the layouts based on two criteria: (i) functionality of the rooms, , can the “bedroom\" satisfies a human's needs for daily life; and (ii) the naturalness and realism of the layout.",
"Scales of responses range from 1 to 5, with 5 indicating perfect functionalilty or perfect naturalness and realism.",
"The mean ratings and the standard deviations are summarized in Table REF .",
"Our approach outperforms the three methods in both criteria, demonstrating the ability to sample a functionally reasonable and realistic scene layout.",
"More qualitative results are shown in Figure REF .",
"Complexity of synthesis.",
"The time complexity is hard to measure since MCMC sampling is adopted.",
"Empirically, it takes about 20-40 minutes to sample an interior layout (20000 iterations of MCMC), and roughly 12-20 minutes to render a 640$\\times $ 480 image on a normal PC.",
"The rendering speed depends on settings related to illumination, environments, and the size of the scene, ."
],
[
"Conclusion",
"We propose a novel general framework for human-centric indoor scene synthesis by sampling from a spatial And-Or graph.",
"The experimental results demonstrate the effectiveness of our approach over a large variety of scenes based on different criteria.",
"In the future, to synthesize physically plausible scenes, a physics engine should be integrated.",
"We hope the synthesized data can contribute to the broad AI community."
],
[
"Acknowledgment",
"The authors thank Professor Ying Nian Wu from UCLA Statistics Department and Professor Demetri Terzopoulos from UCLA Computer Science Department for insightful discussions.",
"The work reported herein is supported by DARPA XAI N66001-17-2-4029 and ONR MURI N00014-16-1-2007.",
"Supplementary Material for Human-centric Indoor Scene Synthesis Using Stochastic Grammar Siyuan Qi$^{1}$ Yixin Zhu$^{1}$ Siyuan Huang$^{1}$ Chenfanfu Jiang$^{2}$ Song-Chun Zhu$^1$ 1 $^1$ UCLA Center for Vision, Cognition, Learning and Autonomy $^2$ UPenn Computer Graphics Group"
],
[
"Simulated Annealing",
"The simulated annealing schedule is important for synthesizing realistic scenes.",
"In our experiments, we set the total sampling iterations to 20000, and it takes around 20 minutes to sample an interior layout.",
"We use the following simulated schedule for sampling: $\\begin{aligned}T(t) = \\frac{T_0}{\\ln (1+t)}\\end{aligned}$ where $T(t)$ is the temperature at iteration $t$ .",
"Geman proved that $T(t) \\ge \\frac{T_0}{\\ln (1+t)}$ is a necessary and sufficient condition to ensure convergence to the global minimum with probability one.",
"Table: Depth estimation with different training protocols.Table: Normal estimation with different training protocols."
],
[
"Data Effectiveness",
"We further demonstrate that our data can be utilized to improve performance on two scene understanding tasks: depth estimation and surface normal estimation from single RGB images.",
"We show that the performance of state-of-art methods can be improved when trained with our synthesized data along with natural images."
],
[
"Depth estimation",
"Single-image depth estimation is a fundamental problem in computer vision, which has found broad applications in scene understanding, 3D modeling, and robotics.",
"The problem is challenging since no reliable depth cues are available.",
"In this task, the algorithms output a depth image based on a single RGB input image.",
"To demonstrate the efficacy of our synthetic data, we compare the depth estimation results provided by models trained following protocols similar to those we used in normal prediction with the network in .",
"To perform a quantitative evaluation, we used the metrics applied in previous work : [leftmargin=*,noitemsep,nolistsep] Abs relative error: $\\frac{1}{N}\\sum _p\\frac{\\left|d_p-d_p^{gt}\\right|}{d_p^{gt}}$ , Square relative difference: $\\frac{1}{N}\\sum _p\\frac{{\\left|d_p-d_p^{gt}\\right|}^2}{d_p^{gt}}$ , Average $\\log _{10}$ error: $\\frac{1}{N}\\sum _x{\\left|\\log _{10}(d_p)-\\log _{10}(d_p^{gt})\\right|}$ , RMSE :$\\sqrt{{\\frac{1}{N}\\sum _x{\\left|d_p-d_p^{gt}\\right|}}^2}$ , Log RMSE:$\\sqrt{{\\frac{1}{N}\\sum _x{\\left|\\log (d_p)-\\log (d_p^{gt})\\right|}}^2}$ , Threshold: % of $d_p \\mbox{~s.t.",
"}\\max {(\\frac{d_p}{d_p^{gt}},\\frac{d_p^{gt}}{d_p}}) <\\mbox{threshold}$ , where $d_p$ and $d_p^{gt}$ are the predicted depths and the ground truth depths at the pixel indexed by $p$ , respectively, and $N$ is the number of pixels in all the evaluated images.",
"The first five metrics capture the error calculated over all the pixels; lower values are better.",
"The threshold criteria capture the estimation accuracy; higher values are better.",
"Table REF summarizes the results.",
"We can see that the model pretrained on our dataset and fine-tuned on the NYU-Depth V2 dataset achieves the best performance, both in error and accuracy.",
"This demonstrates the usefulness of our dataset in improving algorithm performance in scene understanding tasks."
],
[
"Surface normal estimation",
"Predicting surface normals from a single RGB image is an essential task in scene understanding since it provides important information in recovering the 3D structure of the scenes.",
"We train a neural network with our synthetic data to demonstrate that the perfect per-pixel ground truth generated using our pipeline could be utilized to improve upon the state-of-the-art performance on a specific scene understanding task.",
"Using the fully convolutional network model described by Zhang [46], we compare the normal estimation results given by models trained under two different protocols: (i) the network is directly trained and tested on the NYU-Depth V2 dataset, and (ii) the network is first pre-trained using our synthetic data, then fine-tuned and tested on NYU-Depth V2.",
"Following the standard evaluation protocol , , we evaluate a per-pixel error over the entire dataset.",
"To evaluate the prediction error, we computed the mean, median, and RMSE of angular error between the predicted normals and ground truth normals.",
"Prediction accuracy is given by calculating the fraction of pixels that are correct within a threshold $t$ , where $t = 11.25^{\\circ }, 22.5^{\\circ },30^{\\circ }$ .",
"Our experimental results are summarized in Table REF .",
"By utilizing our synthetic data, the model achieves better performance.",
"The error mainly accrues in the area where the ground truth normal map is noisy.",
"We argue that part of the reason is due to the sensor's noise or sensing distance limit.",
"Such results in turn imply the importance to have perfect per-pixel ground truth for training and evaluation."
],
[
"More Qualitative Results",
"See page 13-17."
]
] | 1808.08473 |
[
[
"CGIntrinsics: Better Intrinsic Image Decomposition through\n Physically-Based Rendering"
],
[
"Abstract Intrinsic image decomposition is a challenging, long-standing computer vision problem for which ground truth data is very difficult to acquire.",
"We explore the use of synthetic data for training CNN-based intrinsic image decomposition models, then applying these learned models to real-world images.",
"To that end, we present \\ICG, a new, large-scale dataset of physically-based rendered images of scenes with full ground truth decompositions.",
"The rendering process we use is carefully designed to yield high-quality, realistic images, which we find to be crucial for this problem domain.",
"We also propose a new end-to-end training method that learns better decompositions by leveraging \\ICG, and optionally IIW and SAW, two recent datasets of sparse annotations on real-world images.",
"Surprisingly, we find that a decomposition network trained solely on our synthetic data outperforms the state-of-the-art on both IIW and SAW, and performance improves even further when IIW and SAW data is added during training.",
"Our work demonstrates the suprising effectiveness of carefully-rendered synthetic data for the intrinsic images task."
],
[
"Introduction",
"Intrinsic images is a classic vision problem involving decomposing an input image $I$ into a product of reflectance (albedo) and shading images $R\\cdot S$ .",
"Recent years have seen remarkable progress on this problem, but it remains challenging due to its ill-posedness.",
"An attractive proposition has been to replace traditional hand-crafted priors with learned, CNN-based models.",
"For such learning methods data is key, but collecting ground truth data for intrinsic images is extremely difficult, especially for images of real-world scenes.",
"One way to generate large amounts of training data for intrinsic images is to render synthetic scenes.",
"However, existing synthetic datasets are limited to images of single objects [1], [2] (e.g., via ShapeNet [3]) or images of CG animation that utilize simplified, unrealistic illumination (e.g., via Sintel [4]).",
"An alternative is to collect ground truth for real images using crowdsourcing, as in the Intrinsic Images in the Wild (IIW) and Shading Annotations in the Wild (SAW) datasets [5], [6].",
"However, the annotations in such datasets are sparse and difficult to collect accurately at scale.",
"Inspired by recent efforts to use synthetic images of scenes as training data for indoor and outdoor scene understanding [7], [8], [9], [10], we present the first large-scale scene-level intrinsic images dataset based on high-quality physically-based rendering, which we call CGIntrinsics (CGI).",
"CGI consists of over 20,000 images of indoor scenes, based on the SUNCG dataset [11].",
"Our aim with CGI is to help drive significant progress towards solving the intrinsic images problem for Internet photos of real-world scenes.",
"We find that high-quality physically-based rendering is essential for our task.",
"While SUNCG provides physically-based scene renderings [12], our experiments show that the details of how images are rendered are of critical importance, and certain choices can lead to massive improvements in how well CNNs trained for intrinsic images on synthetic data generalize to real data.",
"We also propose a new partially supervised learning method for training a CNN to directly predict reflectance and shading, by combining ground truth from CGI and sparse annotations from IIW/SAW.",
"Through evaluations on IIW and SAW, we find that, surprisingly, decomposition networks trained solely on CGI can achieve state-of-the-art performance on both datasets.",
"Combined training using both CGI and IIW/SAW leads to even better performance.",
"Finally, we find that CGI generalizes better than existing datasets by evaluating on MIT Intrinsic Images, a very different, object-centric, dataset.",
"Figure: Overview and network architecture.",
"Our workintegrates physically-based rendered images from our CGIntrinsics datasetand reflectance/shading annotations from IIW and SAW in order totrain a better intrinsic decompositionnetwork."
],
[
"Related work",
"Optimization-based methods.",
"The classical approach to intrinsic images is to integrate various priors (smoothness, reflectance sparseness, etc.)",
"into an optimization framework [13], [14], [15], [16], [17], [5].",
"However, for images of real-world scenes, such hand-crafted prior assumptions are difficult to craft and are often violated.",
"Several recent methods seek to improve decomposition quality by integrating surface normals or depths from RGB-D cameras [18], [19], [20] into the optimization process.",
"However, these methods assume depth maps are available during optimization, preventing them from being used for a wide range of consumer photos.",
"Learning-based methods.",
"Learning methods for intrinsic images have recently been explored as an alternative to models with hand-crafted priors, or a way to set the parameters of such models automatically.",
"Barron and Malik [21] learn parameters of a model that utilizes sophisticated priors on reflectance, shape and illumination.",
"This approach works on images of objects (such as in the MIT dataset), but does not generalize to real world scenes.",
"More recently, CNN-based methods have been deployed, including work that regresses directly to the output decomposition based on various training datasets, such as Sintel [22], [23], MIT intrinsics and ShapeNet [2], [1].",
"Shu et al.",
"[24] also propose a CNN-based method specifically for the domain of facial images, where ground truth geometry can be obtained through model fitting.",
"However, as we show in the evaluation section, the networks trained on such prior datasets perform poorly on images of real-world scenes.",
"Two recent datasets are based on images of real-world scenes.",
"Intrinsic Images in the Wild (IIW) [5] and Shading Annotations in the Wild (SAW) [6] consist of sparse, crowd-sourced reflectance and shading annotations on real indoor images.",
"Subsequently, several papers train CNN-based classifiers on these sparse annotations and use the classifier outputs as priors to guide decomposition [6], [25], [26], [27].",
"However, we find these annotations alone are insufficient to train a direct regression approach, likely because they are sparse and are derived from just a few thousand images.",
"Finally, very recent work has explored the use of time-lapse imagery as training data for intrinsic images [28], although this provides a very indirect source of supervision.",
"Synthetic datasets for real scenes.",
"Synthetic data has recently been utilized to improve predictions on real-world images across a range of problems.",
"For instance, [7], [10] created a large-scale dataset and benchmark based on video games for the purpose of autonomous driving, and [29], [30] use synthetic imagery to form small benchmarks for intrinsic images.",
"SUNCG [12] is a recent, large-scale synthetic dataset for indoor scene understanding.",
"However, many of the images in the PBRS database of physically-based renderings derived from SUNCG have low signal-to-noise ratio (SNR) and non-realistic sensor properties.",
"We show that higher quality renderings yield much better training data for intrinsic images."
],
[
"To create our CGIntrinsics (CGI) dataset, we started from the SUNCG dataset [11], which contains over 45,000 3D models of indoor scenes.",
"We first considered the PBRS dataset of physically-based renderings of scenes from SUNCG [12].",
"For each scene, PBRS samples cameras from good viewpoints, and uses the physically-based Mitsuba renderer [31] to generate realistic images under reasonably realistic lighting (including a mix of indoor and outdoor illumination sources), with global illumination.",
"Using such an approach, we can also generate ground truth data for intrinsic images by rendering a standard RGB image $I$ , then asking the renderer to produce a reflectance map $R$ from the same viewpoint, and finally dividing to get the shading image $S = I / R$ .",
"Examples of such ground truth decompositions are shown in Figure REF .",
"Note that we automatically mask out light sources (including illumination from windows looking outside) when creating the decomposition, and do not consider those pixels when training the network.",
"Figure: Visual comparisons between our CGI and theoriginal SUNCG dataset.",
"Top row: images from SUNCG/PBRS.",
"Bottomrow: images from our CGI dataset.",
"The images in our datasethave higher SNR and are more realistic.However, we found that the PBRS renderings are not ideal for use in training real-world intrinsic image decomposition networks.",
"In fact, certain details in how images are rendered have a dramatic impact on learning performance: Rendering quality.",
"Mitsuba and other high-quality renderers support a range of rendering algorithms, including various flavors of path tracing methods that sample many light paths for each output pixel.",
"In PBRS, the authors note that bidirectional path tracing works well but is very slow, and opt for Metropolis Light Transport (MLT) with a sample rate of 512 samples per pixel [12].",
"In contrast, for our purposes we found that bidirectional path tracing (BDPT) with very large numbers of samples per pixel was the only algorithm that gave consistently good results for rendering SUNCG images.",
"Comparisons between selected renderings from PBRS and our new CGI images are shown in Figure REF .",
"Note the significantly decreased noise in our renderings.",
"This extra quality comes at a cost.",
"We find that using BDPT with 8,192 samples per pixel yields acceptable quality for most images.",
"This increases the render time per image significantly, from a reported 31s [12], to approximately 30 minutes.While high, this is still a fair ways off of reported render times for animated films.",
"For instance, each frame of Pixar's Monsters University took a reported 29 hours to render [32].",
"One reason for the need for large numbers of samples is that SUNCG scenes are often challenging from a rendering perspective—the illumination is often indirect, coming from open doorways or constrained in other ways by geometry.",
"However, rendering is highly parallelizable, and over the course of about six months we rendered over ten thousand images on a cluster of about 10 machines.",
"Tone mapping from HDR to LDR.",
"We found that another critical factor in image generation is how rendered images are tone mapped.",
"Renderers like Mitsuba generally produce high dynamic range (HDR) outputs that encode raw, linear radiance estimates for each pixel.",
"In contrast, real photos are usually low dynamic range.",
"The process that takes an HDR input and produces an LDR output is called tone mapping, and in real cameras the analogous operations are the auto-exposure, gamma correction, etc., that yield a well-exposed, high-contrast photograph.",
"PBRS uses the tone mapping method of Reinhard et al.",
"[33], which is inspired by photographers such as Ansel Adams, but which can produce images that are very different in character from those of consumer cameras.",
"We find that a simpler tone mapping method produces more natural-looking results.",
"Again, Figure REF shows comparisons between PBRS renderings and our own.",
"Note how the color and illumination features, such as shadows, are better captured in our renderings (we noticed that shadows often disappear with the Reinhard tone mapper).",
"In particular, to tone map a linear HDR radiance image $I_{\\mathsf {HDR}}$ , we find the 90th percentile intensity value $r_{90}$ , then compute the image $I_{\\mathsf {LDR}}= \\alpha I_{\\mathsf {HDR}}^{\\gamma }$ , where $\\gamma =\\frac{1}{2.2}$ is a standard gamma correction factor, and $\\alpha $ is computed such that $r_{90}$ maps to the value 0.8.",
"The final image is then clipped to the range $[0,1]$ .",
"This mapping ensures that at most 10% of the image pixels (and usually many fewer) are saturated after tone mapping, and tends to result in natural-looking LDR images.",
"Table: Comparisons of existing intrinsic image datasets withour CGIntrinsics dataset.",
"PB indicates physically-based rendering andnon-PB indicates non-physically-based rendering.Using the above rendering approach, we re-rendered $\\sim $ 20,000 images from PBRS.",
"We also integrated 152 realistic renderings from [30] into our dataset.",
"Table REF compares our CGI dataset to prior intrinsic image datasets.",
"Sintel is a dataset created for an animated film, and does not utilize physical-based rendering.",
"Other datasets, such as ShapeNet and MIT, are object-centered, whereas CGI focuses on images of indoor scenes, which have more sophisticated structure and illumination (cast shadows, spatial-varying lighting, etc).",
"Compared to IIW and SAW, which include images of real scenes, CGI has full ground truth and and is much more easily collected at scale."
],
[
"Learning Cross-Dataset Intrinsics",
"In this section, we describe how we use CGIntrinsics to jointly train an intrinsic decomposition network end-to-end, incorporating additional sparse annotations from IIW and SAW.",
"Our full training loss considers training data from each dataset: $\\mathcal {L} = \\mathcal {L}_{\\mathsf {CGI}}+ \\lambda _{\\mathsf {IIW}}\\mathcal {L}_{\\mathsf {IIW}}+ \\lambda _{\\mathsf {SAW}}\\mathcal {L}_{\\mathsf {SAW}}.$ where $\\mathcal {L}_{\\mathsf {CGI}}$ , $\\mathcal {L}_{\\mathsf {IIW}}$ , and $\\mathcal {L}_{\\mathsf {SAW}}$ are the losses we use for training from the CGI, IIW, and SAW datasets respectively.",
"The most direct way to train would be to simply incorporate supervision from each dataset.",
"In the case of CGI, this supervision consists of full ground truth.",
"For IIW and SAW, this supervision takes the form of sparse annotations for each image, as illustrated in Figure REF .",
"However, in addition to supervision, we found that incorporating smoothness priors into the loss also improves performance.",
"Our full loss functions thus incorporate a number of terms: $\\mathcal {L}_{\\mathsf {CGI}}= & {\\mathcal {L}_{\\mathsf {sup}}}+ \\lambda _{\\mathsf {ord}}{\\mathcal {L}_{\\mathsf {ord}}}+ \\lambda _{\\mathsf {rec}}{\\mathcal {L}_{\\mathsf {reconstruct}}}\\\\\\mathcal {L}_{\\mathsf {IIW}}= & \\lambda _{\\mathsf {ord}}{\\mathcal {L}_{\\mathsf {ord}}}+ \\lambda _{\\mathsf {rs}}{\\mathcal {L}_{\\mathsf {rsmooth}}}+\\lambda _{\\mathsf {ss}}{\\mathcal {L}_{\\mathsf {ssmooth}}}+ {\\mathcal {L}_{\\mathsf {reconstruct}}}\\\\\\mathcal {L}_{\\mathsf {SAW}}= & \\lambda _{\\mathsf {S/NS}}{\\mathcal {L}_{\\mathsf {S/NS}}}+ \\lambda _{\\mathsf {rs}}{\\mathcal {L}_{\\mathsf {rsmooth}}}+ \\lambda _{\\mathsf {ss}}{\\mathcal {L}_{\\mathsf {ssmooth}}}+ {\\mathcal {L}_{\\mathsf {reconstruct}}}$ We now describe each term in detail."
],
[
"Supervised losses",
"CGIntrinsics-supervised loss.",
"Since the images in our CGI dataset are equipped with a full ground truth decomposition, the learning problem for this dataset can be formulated as a direct regression problem from input image $I$ to output images $R$ and $S$ .",
"However, because the decomposition is only up to an unknown scale factor, we use a scale-invariant supervised loss, $\\mathcal {L}_{\\mathsf {siMSE}}$ (for “scale-invariant mean-squared-error”).",
"In addition, we add a gradient domain multi-scale matching term $\\mathcal {L}_{\\mathsf {grad}}$ .",
"For each training image in CGI, our supervised loss is defined as $\\mathcal {L}_{\\mathsf {sup}}=\\mathcal {L}_{\\mathsf {siMSE}}+ \\mathcal {L}_{\\mathsf {grad}}$ , where $\\mathcal {L}_{\\mathsf {siMSE}}= \\frac{1}{N} \\sum _{i=1}^N\\left( R_i^{*} - c_r R_i \\right)^2 + \\left( S_i^{*} - c_s S_i\\right)^2\\\\\\mathcal {L}_{\\mathsf {grad}}= \\sum _{l=1}^{L} \\frac{1}{N_l} \\sum _{i=1}^{N_l}\\left\\Vert \\nabla R^{*}_{l,i} - c_r \\nabla R_{l,i}\\right\\Vert _1 + \\left\\Vert \\nabla S^{*}_{l,i} - c_s \\nabla S_{l,i}\\right\\Vert _1.$ $R_{l,i}$ ($R_{l,i}^*$ ) and $S_{l,i}$ ($S_{l,i}^*$ ) denote reflectance prediction (resp.",
"ground truth) and shading prediction (resp.",
"ground truth) respectively, at pixel $i$ and scale $l$ of an image pyramid.",
"$N_l$ is the number of valid pixels at scale $l$ and $N =N_1$ is the number of valid pixels at the original image scale.",
"The scale factors $c_r$ and $c_s$ are computed via least squares.",
"In addition to the scale-invariance of $\\mathcal {L}_{\\mathsf {siMSE}}$ , another important aspect is that we compute the MSE in the linear intensity domain, as opposed to the all-pairs pixel comparisons in the log domain used in [22].",
"In the log domain, pairs of pixels with large absolute log-difference tend to dominate the loss.",
"As we show in our evaluation, computing $\\mathcal {L}_{\\mathsf {siMSE}}$ in the linear domain significantly improves performance.",
"Finally, the multi-scale gradient matching term $\\mathcal {L}_{\\mathsf {grad}}$ encourages decompositions to be piecewise smooth with sharp discontinuities.",
"Ordinal reflectance loss.",
"IIW provides sparse ordinal reflectance judgments between pairs of points (e.g., “point $i$ has brighter reflectance than point $j$ ”).",
"We introduce a loss based on this ordinal supervision.",
"For a given IIW training image and predicted reflectance $R$ , we accumulate losses for each pair of annotated pixels $(i,j)$ in that image: $\\mathcal {L}_{\\mathsf {ord}}(R) = \\sum _{(i,j)}e_{i,j}(R)$ , where $e_{i,j}(R) ={\\left\\lbrace \\begin{array}{ll}w_{i,j} (\\log R_i - \\log R_j)^2, & r_{i,j} = 0 \\\\w_{i,j} \\left( \\max (0, m - \\log R_i + \\log R_j) \\right)^2, & r_{i,j} = +1 \\\\w_{i,j} \\left( \\max (0, m - \\log R_j + \\log R_i) \\right)^2, & r_{i,j} = -1 \\end{array}\\right.",
"}$ and $r_{i,j}$ is the ordinal relation from IIW, indicating whether point $i$ is darker (-1), $j$ is darker (+1), or they have equal reflectance (0).",
"$w_{i,j}$ is the confidence of the annotation, provided by IIW.",
"Example predictions with and without IIW data are shown in Fig.",
"REF .",
"We also found that adding a similar ordinal term derived from CGI data can improve reflectance predictions.",
"For each image in CGI, we over-segment it using superpixel segmentation [36].",
"Then in each training iteration, we randomly choose one pixel from every segmented region, and for each pair of chosen pixels, we evaluate $\\mathcal {L}_{\\mathsf {ord}}$ similar to Eq.",
"REF , with $w_{i,j} = 1$ and the ordinal relation derived from the ground truth reflectance.",
"Figure: Examples of predictions with and without IIWtraining data.",
"Adding real IIW data can qualitatively improvereflectance and shading predictions.",
"Note for instance how thequilt highlighted in first row has a more uniform reflectanceafter incorporating IIW data, and similarly for the floorhighlighted in the second row.SAW shading loss.",
"The SAW dataset provides images containing annotations of smooth (S) shading regions and non-smooth (NS) shading points, as depicted in Figure REF .",
"These annotations can be further divided into three types: regions of constant shading, shadow boundaries, and depth/normal discontinuities.",
"We integrate all three types of annotations into our supervised SAW loss $\\mathcal {L}_{\\mathsf {S/NS}}$ .",
"For each constant shading region (with $N_c$ pixels), we compute a loss $\\mathcal {L}_{\\mathsf {constant-shading}}$ encouraging the variance of the predicted shading in the region to be zero: $\\mathcal {L}_{\\mathsf {constant-shading}}= \\frac{1}{N_c} \\sum _{i=1}^{N_c} (\\log S_i)^2 - \\frac{1}{N_c^2} \\left( \\sum _{i=1}^{N_c} \\log S_i \\right)^2.",
"$ SAW also provides individual point annotations at cast shadow boundaries.",
"As noted in [6], these points are not localized precisely on shadow boundaries, and so we apply a morphological dilation with a radius of 5 pixels to the set of marked points before using them in training.",
"This results in shadow boundary regions.",
"We find that most shadow boundary annotations lie in regions of constant reflectance, which implies that for all pair of shading pixels within a small neighborhood, their log difference should be approximately equal to the log difference of the image intensity.",
"This is equivalent to encouraging the variance of $\\log S_i - \\log I_i$ within this small region to be 0 [37].",
"Hence, we define the loss for each shadow boundary region (with $N_{\\mathsf {sd}}$ ) pixels as: $\\mathcal {L}_{\\mathsf {shadow}}=\\frac{1}{N_{\\mathsf {sd}}} \\sum _{i=1}^{N_{\\mathsf {sd}}} (\\log S_i - \\log I_i )^2 - \\frac{1}{N_{\\mathsf {sd}}^2} \\left( \\sum _{i=1}^{N_{\\mathsf {sd}}} ( \\log S_i -\\log I_i )\\right)^2 $ Finally, SAW provides depth/normal discontinuities, which are also usually shading discontinuities.",
"However, since we cannot derive the actual shading change for such discontinuities, we simply mask out such regions in our shading smoothness term $\\mathcal {L}_{\\mathsf {ssmooth}}$ (Eq.",
"REF ), i.e., we do not penalize shading changes in such regions.",
"As above, we first dilate these annotated regions before use in training.",
"Examples predictions before/after adding SAW data into our training are shown in Fig.",
"REF .",
"Figure: Examples of predictions with and without SAWtraining data.",
"Adding SAW training data can qualitativelyimprove reflectance and shading predictions.",
"Note the pictures/TVhighlighted in the decompositions in the first row, and theimproved assignment of texture to the reflectance channel for thepaintings and sofa in the second row."
],
[
"Smoothness losses",
"To further constrain the decompositions for real images in IIW/SAW, following classical intrinsic image algorithms we add reflectance smoothness $\\mathcal {L}_{\\mathsf {rsmooth}}$ and shading smoothness $\\mathcal {L}_{\\mathsf {ssmooth}}$ terms.",
"For reflectance, we use a multi-scale $\\ell _1$ smoothness term to encourage reflectance predictions to be piecewise constant: $\\mathcal {L}_{\\mathsf {rsmooth}}= \\sum _{l=1}^L \\frac{1}{N_l l} \\sum _{i=1}^{N_l} \\sum _{j \\in \\mathcal {N}(l,i)} v_{l,i,j} \\left\\Vert \\log R_{l,i} - \\log R_{l,j} \\right\\Vert _1$ where $\\mathcal {N}(l,i)$ denotes the 8-connected neighborhood of the pixel at position $i$ and scale $l$ .",
"The reflectance weight $v_{l,i,j}= \\exp \\left( - \\frac{1}{2} (\\mathbf {f}_{l,i} - \\mathbf {f}_{l,j})^T\\Sigma ^{-1} (\\mathbf {f}_{l,i} - \\mathbf {f}_{l,j}) \\right)$ , and the feature vector $\\mathbf {f}_{l,i}$ is defined as $[\\ \\mathbf {p}_{l,i},I_{l,i} , c_{l,i}^1, c_{l,i}^2 \\ ]$ , where $\\mathbf {p}_{l,i}$ and $I_{l,i}$ are the spatial position and image intensity respectively, and $c_{l,i}^1$ and $c_{l,i}^2$ are the first two elements of chromaticity.",
"$\\Sigma $ is a covariance matrix defining the distance between two feature vectors.",
"We also include a densely-connected $\\ell _2$ shading smoothness term, which can be evaluated in linear time in the number of pixels $N$ using bilateral embeddings [38], [28]: $\\mathcal {L}_{\\mathsf {ssmooth}}= & \\frac{1}{2N} \\sum _{i}^N \\sum _{j}^N \\hat{W}_{i,j} \\left( \\log S_i - \\log S_j \\right)^2\\approx \\frac{1}{N} \\mathbf {s}^\\top (I - N_b S_b^\\top \\bar{B_b} S_b N_b ) \\mathbf {s} $ where $\\hat{W}$ is a bistochastic weight matrix derived from $W$ and $W_{i,j} = \\exp \\left( - \\frac{1}{2} || \\frac{ \\mathbf {p}_i-\\mathbf {p}_j }{\\sigma _p} ||_2^2 \\right)$ .",
"We refer readers to [38], [28] for a detailed derivation.",
"As shown in our experiments, adding such smoothness terms to real data can yield better generalization."
],
[
"Reconstruction loss",
"Finally, for each training image in each dataset, we add a loss expressing the constraint that the reflectance and shading should reconstruct the original image: $\\mathcal {L}_{\\mathsf {reconstruct}}= \\frac{1}{N} \\sum _{i=1}^N \\left( I_i - R_i S_i \\right)^2.$"
],
[
"Network architecture",
"Our network architecture is illustrated in Figure REF .",
"We use a variant of the “U-Net” architecture [28], [39].",
"Our network has one encoder and two decoders with skip connections.",
"The two decoders output log reflectance and log shading, respectively.",
"Each layer of the encoder mainly consists of a $4\\times 4$ stride-2 convolutional layer followed by batch normalization [40] and leaky ReLu [41].",
"For the two decoders, each layer is composed of a $4\\times 4$ deconvolutional layer followed by batch normalization and ReLu, and a $1\\times 1$ convolutional layer is appended to the final layer of each decoder."
],
[
"Evaluation",
"We conduct experiments on two datasets of real world scenes, IIW [5] and SAW [6] (using test data unseen during training) and compare our method with several state-of-the-art intrinsic images algorithms.",
"Additionally, we also evaluate the generalization of our CGI dataset by evaluating it on the MIT Intrinsic Images benchmark [35].",
"Network training details.",
"We implement our method in PyTorch [42].",
"For all three datasets, we perform data augmentation through random flips, resizing, and crops.",
"For all evaluations, we train our network from scratch using the Adam [43] optimizer, with initial learning rate $0.0005$ and mini-batch size 16.",
"We refer readers to the supplementary material for the detailed hyperparameter settings."
],
[
"Evaluation on IIW",
"We follow the train/test split for IIW provided by [27], also used in [25].",
"We also conduct several ablation studies using different loss configurations.",
"Quantitative comparisons of Weighted Human Disagreement Rate (WHDR) between our method and other optimization- and learning-based methods are shown in Table REF .",
"Comparing direct CNN predictions, our CGI-trained model is significantly better than the best learning-based method [45], and similar to [44], even though [45] was directly trained on IIW.",
"Additionally, running the post-processing from [45] on the results of the CGI-trained model achieves a further performance boost.",
"Table REF also shows that models trained on SUNCG (i.e., PBRS), Sintel, MIT Intrinsics, or ShapeNet generalize poorly to IIW likely due to the lower quality of training data (SUNCG/PBRS), or the larger domain gap with respect to images of real-world scenes, compared to CGI.",
"The comparison to SUNCG suggests the key importance of our rendering decisions.",
"We also evaluate networks trained jointly using CGI and real imagery from IIW.",
"As in [25], we augment the pairwise IIW judgments by globally exploiting their transitivity and symmetry.",
"The right part of Table REF demonstrates that including IIW training data leads to further improvements in performance, as does also including SAW training data.",
"Table REF also shows various ablations on variants of our method, such as evaluating losses in the log domain and removing terms from the loss functions.",
"Finally, we test a network trained on only IIW/SAW data (and not CGI), or trained on CGI and fine-tuned on IIW/SAW.",
"Although such a network achieves $\\sim $ 19% WHDR, we find that the decompositions are qualitatively unsatisfactory.",
"The sparsity of the training data causes these networks to produce degenerate decompositions, especially for shading images."
],
[
"Evaluation on SAW",
"To evaluate our shading predictions, we test our models on the SAW [6] test set, utilizing the error metric introduced in [28].",
"We also propose a new, more challenging error metric for SAW evaluation.",
"In particular, we found that many of the constant-shading regions annotated in SAW also have smooth image intensity (e.g., textureless walls), making their shading easy to predict.",
"Our proposed metric downweights such regions as follows.",
"For each annotated region of constant shading, we compute the average image gradient magnitude over the region.",
"During evaluation, when we add the pixels belonging to a region of constant shading into the confusion matrices, we multiply the number of pixels by this average gradient.",
"This proposed metric leads to more distinguishable performance differences between methods, because regions with rich textures will contribute more to the error compared to the unweighted metric.",
"Figure REF and Table REF show precision-recall (PR) curves and average precision (AP) on the SAW test set with both unweighted [28] and our proposed challenge error metrics.",
"As with IIW, networks trained solely on our CGI data can achieve state-of-the-art performance, even without using SAW training data.",
"Adding real IIW data improves the AP in term of both error metrics.",
"Finally, the last column of Table REF shows that integrating SAW training data can significantly improve the performance on shading predictions, suggesting the effectiveness of our proposed losses for SAW sparse annotations.",
"Note that the previous state-of-the-art algorithms on IIW (e.g., Zhou et al.",
"[25] and Nestmeyer et al.",
"[45]) tend to overfit to reflectance, hurting the accuracy of shading predictions.",
"This is especially evident in terms of our proposed challenge error metric.",
"In contrast, our method achieves state-of-the-art results on both reflectance and shading predictions, in terms of all error metrics.",
"Note that models trained on the original SUNCG, Sintel, MIT intrinsics or ShapeNet datasets perform poorly on the SAW test set, indicating the much improved generalization to real scenes of our CGI dataset.",
"Figure: Qualitative comparisons on the IIW/SAW test sets.Our predictions show significant improvements compared tostate-of-the-art algorithms (Bell et al.",
"and Zhou et al. ).",
"In particular, ourpredicted shading channels include significantly less surfacetexture in several challengingsettings.Qualitative results on IIW/SAW.",
"Figure REF shows qualitative comparisons between our network trained on all three datasets, and two other state-of-the-art intrinsic images algorithms (Bell et al.",
"[5] and Zhou et al.",
"[25]), on images from the IIW/SAW test sets.",
"In general, our decompositions show significant improvements.",
"In particular, our network is better at avoiding attributing surface texture to the shading channel (for instance, the checkerboard patterns evident in the first two rows, and the complex textures in the last four rows) while still predicting accurate reflectance (such as the mini-sofa in the images of third row).",
"In contrast, the other two methods often fail to handle such difficult settings.",
"In particular, [25] tends to overfit to reflectance predictions, and their shading estimates strongly resemble the original image intensity.",
"However, our method still makes mistakes, such as the non-uniform reflectance prediction for the chair in the fifth row, as well as residual textures and shadows in the shading and reflectance channels."
],
[
"Evaluation on MIT intrinsic images",
"For the sake of completeness, we also test the ability of our CGI-trained networks to generalize to the MIT Intrinsic Images dataset [35].",
"In contrast to IIW/SAW, the MIT dataset contains 20 real objects with 11 different illumination conditions.",
"We follow the same train/test split as Barron et al.",
"[21], and, as in the work of Shi et al.",
"[2], we directly apply our CGI trained networks to MIT testset, and additionally test fine-tuning them on the MIT training set.",
"We compare our models with several state-of-the-art learning-based methods using the same error metrics as [2].",
"Table REF shows quantitative comparisons and Figure REF shows qualitative results.",
"Both show that our CGI-trained model yields better performance compared to ShapeNet-trained networks both qualitatively and quantitatively, even though like MIT, ShapeNet consists of images of rendered objects, while our dataset contains images of scenes.",
"Moreover, our CGI-pretrained model also performs better than networks pretrained on ShapeNet and Sintel.",
"These results further demonstrate the improved generalization ability of our CGI dataset compared to existing datasets.",
"Note that SIRFS still achieves the best results, but as described in [22], [2], their methods are designed specifically for single objects and generalize poorly to real scenes."
],
[
"Conclusion",
"We presented a new synthetic dataset for learning intrinsic images, and an end-to-end learning approach that learns better intrinsic image decompositions by leveraging datasets with different types of labels.",
"Our evaluations illustrate the surprising effectiveness of our synthetic dataset on Internet photos of real-world scenes.",
"We find that the details of rendering matter, and hypothesize that improved physically-based rendering may benefit other vision tasks, such as normal prediction and semantic segmentation [12].",
"Acknowledgments.",
"We thank Jingguang Zhou for his help with data generation.",
"This work was funded by the National Science Foundation through grant IIS-1149393, and by a grant from Schmidt Sciences."
]
] | 1808.08601 |
[
[
"Fast and Accurate Recognition of Chinese Clinical Named Entities with\n Residual Dilated Convolutions"
],
[
"Abstract Clinical Named Entity Recognition (CNER) aims to identify and classify clinical terms such as diseases, symptoms, treatments, exams, and body parts in electronic health records, which is a fundamental and crucial task for clinical and translation research.",
"In recent years, deep learning methods have achieved significant success in CNER tasks.",
"However, these methods depend greatly on Recurrent Neural Networks (RNNs), which maintain a vector of hidden activations that are propagated through time, thus causing too much time to train models.",
"In this paper, we propose a Residual Dilated Convolutional Neural Network with Conditional Random Field (RD-CNN-CRF) to solve it.",
"Specifically, Chinese characters and dictionary features are first projected into dense vector representations, then they are fed into the residual dilated convolutional neural network to capture contextual features.",
"Finally, a conditional random field is employed to capture dependencies between neighboring tags.",
"Computational results on the CCKS-2017 Task 2 benchmark dataset show that our proposed RD-CNN-CRF method competes favorably with state-of-the-art RNN-based methods both in terms of computational performance and training time."
],
[
"Introduction",
"Clinical Named Entity Recognition (CNER) is a critical task for extracting patient information from Electronic Health Records (EHRs) in clinical and translational research.",
"CNER aims to identify and classify clinical terms in EHRs, such as diseases, symptoms, treatments, exams, and body parts.",
"It is important to extract named entities from clinical texts because the clinical texts usually contains abundant healthcare information, while biomedical systems that rely on structured data are unable to access directly such information locked in the clinical texts.",
"Identification of the clinical named entities is a non-trivial task.",
"There are two main reasons.",
"The one is the richness of EHRs, i.e., the same word or sentence can refer to more than one kind of named entities, and various forms can describe the same named entities .",
"The other one is that a huge number of entities that rarely or even do not occur in the training set because of the use of non-standard abbreviations or acronyms, and multiple variations of same entities .",
"Furthermore, CNER in Chinese texts is more difficult compared to those in Romance languages due to the lack of word boundaries in Chinese and the complexity of Chinese composition forms .",
"Traditionally, rule-based approaches , , dictionary-based approaches , and machine learning approaches , , are applied to address the CNER tasks.",
"Recently, along with the development of deep learning, some Recurrent Neural Network (RNN) based models, especially for the Bi-LSTM-CRF models , , , have been successfully used and achieved the state-of-the-art results.",
"However, RNN models are dedicated sequence models which maintain a vector of hidden activations that are propagated through time, thus requiring too much time for training.",
"To solve this problem, in this paper, we propose a Residual Dilated Convolutional Neural Network with Conditional Random Field (RD-CNN-CRF) for the Chinese CNER.",
"In our method, Chinese CNER task is regarded as a sequence labeling task in character level in order to avoid introducing noise caused by segmentation errors, and dictionary features are utilized to help recognize rare and unseen clinical named entities.",
"More specifically, Chinese characters and dictionary features are first projected into dense vector representations, then they are fed into the residual dilated convolutional neural network to capture contextual features.",
"Finally, a conditional random field is employed to capture dependencies between neighboring tags.",
"Computational studies on the CCKS-2017 Task 2 benchmark datasetIt is publicly available at http://www.ccks2017.com/en/index.php/sharedtask/ show that our proposed method achieves the highly competitive performance compared with state-of-the-art RNN-based methods, and is able to significantly save training time.",
"In addition, we also observe that the Chinese CNER task do not necessarily rely on long-distance contextual information.",
"The main contributions of this work can be summarized as follows.",
"We propose a Residual Dilated Convolutional Neural Network with Conditional Random Field (RD-CNN-CRF) for the Chinese CNER.",
"It is the first time to introduce the residual dilated convolutions for the CNER tasks especially for the Chinese CNER.",
"Experimental results on the CCKS-2017 Task 2 benchmark dataset demonstrate that our proposed RD-CNN-CRF method achieves a highly competitive performance compared with state-of-the-art RNN-based methods.",
"Moreover, our RD-CNN-CRF method is able to speed up the training process and save computational time.",
"The rest of the paper is organized as follows.",
"We briefly review the related work on CNER and introduce the Chinese CNER in Section and Section , respectively.",
"In Section , we present the proposed RD-CNN-CRF model.",
"We report the computational results in Section .",
"Section is dedicated to experimentally investigate several key issues of our proposed model.",
"Finally, conclusions are given in Section ."
],
[
"Related Work",
"Due to the practical significance, Clinical Named Entity Recognition (CNER) has attracted considerable attention, and a lot of solution approaches have been proposed in the literature.",
"All these existing approaches can be roughly divided into four categories: rule-based approaches, dictionary-based approaches, machine learning approaches and deep learning approaches.",
"Rule-based approaches rely on heuristics and handcrafted rules to identify entities , , .",
"They were the dominant approaches in the early CNER systems.",
"However, it is quite impossible to list all the rules to model the structure of clinical named entities, especially for various medical entities, and this kind of handcrafted approach always leads to a relatively high system engineering cost.",
"Dictionary-based approaches employ existing clinical vocabularies to identify entities , , .",
"They were widely used because of their simplicity and their performance.",
"A dictionary-based CNER system can extract all the matched entities defined in a dictionary from given clinical texts.",
"However, it's unable to deal with out-of-dictionary entities, and consequently this kind of approach typically causes low recalls.",
"Machine learning approaches consider CNER as a sequence labeling problem where the goal is to find the best label sequence for a given input sentence , .",
"Typical methods are Hidden Markov Models (HMMs) , , Maximum Entropy Markov Models (MEMMs) , , Conditional Random Fields (CRFs) , , and Support Vector Machines (SVMs) , .",
"However, these statistical methods rely on pre-defined features, which makes their development costly.",
"More specifically, feature engineering process will cost much to find the best set of features which help to discern entities of a specific type from others.",
"And it's more of an art than a science, incurring extensive trial-and-error experiments.",
"Deep learning approaches , especially the methods based on Bidirectional RNN with CRF layer as the output interface (Bi-RNN-CRF) , achieve state-of-the-art performance in CNER tasks and outperform the traditional statistical models , , .",
"RNNs with gated recurrent cells, such as Long-Short Term Memory (LSTM) and Gated Recurrent Units (GRU) , are capable of capturing long dependencies and retrieving rich global information.",
"The sequential CRF on top of the recurrent layers ensures that the optimal sequence of tags over the entire sentence is obtained.",
"Some scholars also tried to integrate other features like n-gram features to improve the performance.",
"However, RNNs are dedicated sequence models which maintain a vector of hidden activation that are propagated through time, so the RNN-based models often take long time for training."
],
[
"Chinese Clinical Named Entity Recognition",
"The Chinese Clinical Named Entity Recognition (Chinese CNER) task can be regarded as a sequence labeling task.",
"Due to the ambiguity in the boundary of Chinese words, following our previous work , we label the sequence in the character level to avoid introducing noise caused by segmentation errors.",
"Thus, given a clinical sentence $X=<x_1,...,x_n>$ , our goal is to label each character $x_i$ in the sentence $X$ with BIEOS (Begin, Inside, End, Outside, Single) tag scheme.",
"An example of the tag sequence for “UTF8gbsn腹平坦，未见腹壁静脉曲张。” (The abdomen is flat and no varicose veins can be seen on the abdominal wall) can be found in Table REF .",
"Table: Acknowledgment"
]
] | 1808.08669 |
[
[
"Mass spectra of heavy mesons with instanton effects"
],
[
"Abstract We investigate the mass spectra of ordinary heavy mesons, based on a nonrelativistic potential approach.",
"The heavy-light quark potential contains the Coulomb-type potential arising from one-gluon exchange, the confining potential, and the instanton-induced nonperturbative local heavy-light quark potential.",
"All parameters are theoretically constrained and fixed.",
"We carefully examine the effects from the instanton vacuum.",
"Within the present form of the local potential from the instanton vacuum, we conclude that the instanton effects are rather marginal on the charmed mesons."
],
[
"Introduction",
"The structure of hadrons containing a heavy quark is systematically understood when the mass of the heavy quark is taken to infinity.",
"This is valid, since the heavy-quark mass $m_Q$ is much larger than the $\\Lambda _{\\mathrm {QCD}}$ , i.e.",
"$m_Q\\gg \\Lambda _{\\mathrm {QCD}}$ .",
"Then a new type of symmetry arises: the physics is not changed by the exchange of the heavy-quark flavor.",
"This is called heavy-quark flavor symmetry.",
"In this limit, the spin of the heavy quark $\\mathbf {S}_Q$ is conserved, which brings about the spin conservation of the light degrees of freedom $\\mathbf {S}_L$ .",
"So, the spin of a heavy hadron is also conserved in this limit: $\\mathbf {S}=\\mathbf {S}_L+\\mathbf {S}_Q$ .",
"This is often called heavy-quark spin symmetry [1], [2], [3].",
"The heavy quark is entirely decoupled from the internal dynamics of a heavy hadron in the limit of $m_Q\\rightarrow \\infty $ and the interaction among light degrees of freedom becomes spin-independent.",
"The infinitely heavy-quark mass limit allows one to use the inverse of the heavy-quark mass, $1/m_Q$ , as an expansion parameter.",
"The spin-dependent part of the interaction appears as the next-to-leading order in the $1/m_Q$ expansion, which is proportional to $1/m_Q$ and stems from the chromomagnetic moment of the quark (see, for example, reviews [4], [5], [6], [7] and books [8], [9]).",
"In the limit of $m_Q\\rightarrow \\infty $ , the classification of conventional heavy meson states $Q\\bar{q}$ with a single heavy quark $Q$ is rather simple, where $\\bar{q}$ denotes the light anti-quark constituting the heavy meson.",
"Since the heavy quark is decoupled in the $m_Q\\rightarrow \\infty $ limit, the flavor structure is solely governed by the light quarks.",
"Thus the lowest-lying states of the heavy meson is classified as the antitriplet meson $\\overline{\\mathbf {3}}$ .",
"Moreover, the mesons with spin $s=0$ and those with $s=1$ are found to be degenerate, so that the pseudoscalar and vector heavy mesons consist of the doublets in the limit of $m_Q\\rightarrow \\infty $ .",
"This degeneracy is lifted by introducing the spin-dependent interactions coming from $1/m_Q$ order.",
"Based on this heavy-quark flavor-spin symmetry, there has been a great deal of theoretical works on properties of both the lowest-lying and excited heavy mesons: lattice QCD [10], [11], [12], [13], [14], [15], the nonrelativistic and relativistic quark models [16], [17], [18], [19], [20], potential models [21], [22], [23], [24], [25], [26], QCD sum rules [27], [28], [29] , holographic QCD [30], and so on.",
"The potential models for the heavy mesons are usually based on two important physics: the quark confinement and the perturbative one-gluon exchange.",
"While these two ingredients of the potentials describe successfully both properties of quarkonia and heavy mesons, certain nonperturbative effects need to be considered.",
"Diakonov et al.",
"derived the central part of the heavy-quark potential from the instanton vacuum, using the Wilson loop [31].",
"The spin-dependent part can be easily constructed by employing the Eichten-Feinberg formalism [32].",
"The effects of the heavy-quark potential from the instanton were examined only very recently by computing the quarkonium spectra [33].",
"The results showed that the effects of the instanton turn out to be rather small on the quarkonium spectra.",
"Chernyshev et al.",
"investigated the effects of a random gas of instantons and anti-instantons on mesons and baryons containing one or several heavy quarks [34].",
"They first derived the local effective interactions from the random instanton-gas model (RIGM) and then employed them to estimate the heavy-hadron mass spectra within a simple variational method, including the harmonic oscillator potential as a simple expression of the quark confinement .",
"They obtained results in qualitative agreement with the experimental data on the low-lying heavy mesons.",
"However, it is of great importance to examine cautiously such nonperturbative effects on the heavy hadron spectra in a quantitative manner.",
"In the present work, we aim at exploring carefully the heavy-light quark potentials, which were derived from the RIGM, examining their effects on the mass spectra of the heavy mesons.",
"For simplicity and convenience, we will use the nonrelativistic framework in dealing with the heavy-light quark interactions from the RIGM.",
"In any potential models for describing the quarkonia and heavy mesons, there are two essential components: the quark confinement and the one-gluon exchange contribution, which we want to introduce in addition to the interaction from the instantons.",
"Instead of a simple variational method used in Ref.",
"[34], we employ a more elaborated and sophisticated framework, i.e.",
"the Gaussian expansion method (GEM), which is well known for the successful description of two- and few-body systems [35], [36], [37], [38], so that we reduce numerical uncertainties arising from the simple variational method.",
"As will be shown in this work, the present form of the heavy-light quark interaction based on the RIGM has only marginal effects on the mass spectra of the heavy mesons.",
"The quark potentials of one-gluon exchange and the quark confinement already reproduce approximately the experimental data on the spectra of the low-lying heavy mesons.",
"However, since the heavy-mesons contain a light quark, we still expect that certain nonperturbative effects will come into play.",
"We will discuss them also in the present work.",
"This paper is organized as follows: In Section II, we define the heavy-light quark potentials arising from one-gluon exchange and the quark confinement.",
"We then introduce the effective potential coming from the nonerpturbative heavy-light quark interactions based on the RIGM.",
"In Section III, we show how to solve the nonrelativistic Schrödinger equation with the heavy-light quark potential within the framework of the GEM.",
"That will be the framework for numerical calculations in the present work.",
"In Section IV, we present the results and discuss them in comparison with the experimental data.",
"The final Section is devoted to summary and conclusion.",
"We also discuss a possible future outlook."
],
[
"Heavy-light quark potential",
"The general structure of the heavy-light quark potentials is expressed as $V(r)= V_c (r)+V_{SS}(r)(\\mathbf {S}_Q\\cdot \\mathbf {S}_q)+V_{LS}(r)(\\mathbf {L}\\cdot \\mathbf {S}) + V_{T}(r) [3(\\mathbf {S}_Q\\cdot \\hat{\\mathbf {n}})(\\mathbf {S}_q\\cdot \\hat{\\mathbf {n}})- \\mathbf {S}_1\\cdot \\mathbf {S}_2],$ where $V_c$ is the central part of the potential.",
"The $V_{SS}$ , $V_{LS}$ , and $V_T$ are called respectively the spin-spin term, the $LS$ term that shows the coupling between the orbital angular momentum and the spin angular momentum, and the tensor term.",
"Following Ref.",
"[32], the spin-dependent potential is derived from the central potential.",
"$\\mathbf {S}_Q$ and $\\mathbf {S}_q$ denote the spin operators for the heavy and light quarks, respectively.",
"$\\mathbf {L}$ and $\\mathbf {S}$ represent respectively the operator of the relative orbital angular momentum and the total spin operator defined as $\\mathbf {S}=\\mathbf {S}_Q+\\mathbf {S}_q$ .",
"In a nonrelativistic constitutent-quark potential model, the heavy-light quark potential consists of two different contributions: the confining linear potential $V_{\\mathrm {conf}}(r) = \\varkappa \\, r$ with the parameter of the string tension $\\varkappa $ and the Coulomb-like interaction arising from one-gluon exchange $V_{\\mathrm {Coul}}(r)=-\\frac{4\\alpha _{\\mathrm {s}}}{3r},$ where $\\alpha _{\\mathrm {s}}$ is the strong running coupling constant at the one-loop level $\\alpha _{\\mathrm {s}}(\\mu ) = \\frac{1}{\\beta _0}\\frac{1}{\\ln (\\mu ^2/\\Lambda _{\\mathrm {QCD}}^2)}.$ The one-loop $\\beta $ function is given as $\\beta _0=(33-2N_f)/(12\\pi )$ .",
"The dimensional transmutation parameter are taken from the Particle Data Group (PDG) [40], i.e.",
"$\\Lambda _{\\mathrm {QCD}}=0.217\\,\\mathrm {GeV}$ .",
"Since we include the charmed quark, the number of flavor is given by $N_f=4$ .",
"The scale parameter $\\mu $ will be set equal to the mass of the charmed quark.",
"$V_c(r) = V_{\\mathrm {conf}}(r) + V_{\\mathrm {Coul}}(r)$ and the spin-dependent parts are generated from this central potential and are expressed as $V_{SS}(r)&=\\frac{32\\pi \\alpha _{\\mathrm {s}}}{9M_Q M_q}\\delta (\\mathbf {r}), \\cr V_{LS}(r) &= \\frac{1}{2M_Q M_q} \\left(\\frac{4\\alpha _{\\mathrm {s}}}{r^3}- \\frac{\\varkappa }{r}\\right), \\cr V_{T}(r)&=\\frac{4\\alpha _{\\mathrm {s}}}{3 M_Q M_q}\\frac{1}{r^3},$ where $M_Q$ and $M_q$ are stand for the dynamical heavy and light quark masses, respectively, which will be discussed shortly.",
"In a practical calculation, the point-like spin-spin interaction is required to be smeared by using the exponential form $\\delta _{\\sigma }(r)=(\\frac{\\sigma }{\\sqrt{\\pi }})^3e^{-\\sigma ^2r^2},$ where $\\sigma $ stands for the smearing factor.",
"Thus, one has a given set of parameters $\\varkappa $ and $\\sigma $ which are fit to the spectra of mesons.",
"In order to reduce the number of free parameters in the present work, we fix the strong running coupling constant $\\alpha _s=0.4106$ defined in Eq.",
"(REF ) at the the scale of the charmed quark mass: $\\mu =M_Q=m_c^{\\mathrm {current}}+\\Delta M_Q$ with $m_c^{\\mathrm {current}}=1.275$ GeV and $\\Delta M_Q=0.086$ GeV.",
"Here $\\Delta M_Q$ is the shift of the heavy quark mass caused by the heavy-light quark interactions that arise from a random instanton gas of the QCD vacuum.",
"Its numerical value used here is determined in Ref.",
"[34] (see also discussions in Ref. [33]).",
"The dynamical mass of the light quark arises from the spontaneous breakdown of chiral symmetry (SB$\\chi $ S).",
"The QCD instanton vacuum explains quantitatively the mechanism of the $SB\\chi $ S [41] (see also reviews [42], [43]).",
"In the present work, we take the value of $M_{\\mathrm {u,\\,d}}=340\\,\\mathrm {MeV}$ .",
"The strange dynamical quark mass is taken to be $M_{\\mathrm {s}}=m_{\\mathrm {s}}+M_q=(150+340)\\,\\mathrm {MeV}=\\,490\\,\\mathrm {MeV}$ .",
"Since the main purpose of the present work is to consider the contribution of the nonperturbative heavy-light quark interaction from the instanton vacuum, we will introduce the effective instanton-induced heavy-light quark potential.",
"For simplicity, we follow Ref.",
"[34], where the local effective interactions between the heavy and light quarks due to instantons were derived in terms of the heavy and light quark operators $Q$ and $q$ $\\mathcal {L}_{qQ}=&-\\left(\\frac{M_q \\Delta M_Q}{2n N_c}\\right)\\left(\\overline{Q} \\frac{1+\\gamma ^0}{2} Q \\overline{q}q+\\frac{1}{4} \\overline{Q} \\frac{1+\\gamma ^0}{2}\\lambda ^a Q \\overline{q}\\lambda ^a q\\right),\\cr \\mathcal {L}_{qQ}^{\\mathrm {spin}} =&-\\left(\\frac{M_q \\Delta M_Q^{\\mathrm {spin}}}{2n N_c}\\right)\\frac{1}{4} \\overline{Q} \\frac{1+\\gamma ^0}{2} \\lambda ^a\\sigma ^{\\mu \\nu } Q \\overline{q} \\lambda ^a \\sigma _{\\mu \\nu } q.$ The density parameter $n$ of the random instanton gas is defined by $N/2V_4 N_c$ , where $N/V_4\\sim 1\\,\\mathrm {fm}^{-4}$ is the instanton density with the four-dimensional volume $V_4$ and $N_c$ denotes the number of colors.",
"$\\Delta M_Q$ is the mass shift of the heavy quark caused by the instantons.",
"$\\Delta M_Q^{\\mathrm {spin}}$ arises from the $M_Q^{-1}$ -order chromomagnetic interaction and, therefore, its value is different from that of $\\Delta M_Q$ .",
"In Ref.",
"[34], the numerical value of $\\Delta M_Q^{\\mathrm {spin}}$ is determined to be $3\\,\\mathrm {MeV}$ for the charmed quark.",
"Other standard quantities in the Lagrangian are the Gell-Mann matrices for color space and the combinations from the Dirac matrices.",
"Consequently, the relevant two-body instanton-induced central and spin-spin potentials are expressed as $V^c_{\\mathrm {I}}(\\mathbf {r}) &= \\left(\\frac{M_q\\Delta M_Q}{2nN_c}\\right)\\left( 1 + \\frac{1}{4} \\lambda _q^a \\lambda _Q^a \\right)\\delta ^3 (\\mathbf {r}),\\\\V^{\\mathrm {spin}}_{\\mathrm {I}} (\\mathbf {r}) &= -\\left(\\frac{M_q \\Delta M_Q^{\\mathrm {spin}}}{2n N_c}\\right)\\mathbf {S}_q \\cdot \\mathbf {S}_Q \\lambda _q^a\\lambda _Q^a \\delta ^3 (\\mathbf {r}),$ where $\\mathbf {r}$ designates the relative coordinates $\\mathbf {r}=\\mathbf {r}_q-\\mathbf {r}_Q$ .",
"Yet another spin-dependent potentials [32] are derived from the central potential from the instanton vacuum as follows: $V_{SS}^I(r)&=\\frac{1}{3M_QM_q}\\nabla ^2V_I(r), \\cr V_{LS}^I(r)&=\\frac{1}{2M_QM_q}\\frac{1}{r}\\frac{dV_I(r)}{dr},\\cr V_T^I(r)&=\\frac{1}{3M_QM_q}\\left(\\frac{1}{r}\\frac{dV_I(r)}{dr}-\\frac{d^2V_I(r)}{dr^2} \\right),$ Since the central and spin-spin potentials are given as the Dirac delta functions, we need to introduce here also a smearing function to remove any divergence that would be caused by them.",
"So, we introduce the Gaussian type of the smearing function $\\delta _{\\sigma _I}(r)=(\\frac{\\sigma _I}{\\sqrt{\\pi }})^3e^{-\\sigma _I^2r^2}$ in both central and spin-spin potentials.",
"Here $\\sigma _I$ stands for the another smearing factor, of which the numerical value will not be much changed from that of $\\sigma $ to avoid any additional uncertainty.",
"The explicit forms of the spin-dependent potentials are obtained as $V_{SS}^I(r)=& \\left(\\frac{\\Delta M_{Q}}{6nN_{c} M_Q}\\right)\\left(1+\\frac{1}{4}\\lambda _{Q}^{a}\\lambda _{\\bar{q}}^{a}\\right)\\left(-6\\sigma _I^2+4\\sigma _I^4r^2\\right)\\delta _{\\sigma _I}(r), \\cr V_{LS}^I(r)=& \\left(\\frac{\\Delta M_{Q}}{4nN_{c} M_Q} \\right)\\left(1+\\frac{1}{4}\\lambda _{Q}^{a}\\lambda _{\\bar{q}}^{a}\\right) (-2\\sigma _I^2)\\delta _{\\sigma _I}(r),\\cr V_T^I(r)=& \\left(\\frac{\\Delta M_{Q}}{6nN_{c} M_Q}\\right)\\left(1+\\frac{1}{4}\\lambda _{Q}^{a}\\lambda _{\\bar{q}}^{a}\\right) (-4\\sigma _I^4r^2)\\delta _{\\sigma _I}(r).$ The total potential can be constructed by combining the potentials from the instanton vacuum given in Eqs.",
"(REF ), (), and (REF ) with those from the confining and Coulomb-like potentials in Eqs.",
"(REF ) and (REF ) $V_{Q\\bar{q}}(r) = V(r) + V_I(r).$ where $V_I(r)$ is defined as $V_I(r) = V_{\\mathrm {I}}^c (r)+V_I^{\\mathrm {spin}}(r)+V_{SS}^I(r)(\\mathbf {S}_Q\\cdot \\mathbf {S}_q)+V_{LS}^I(r)(\\mathbf {L}\\cdot \\mathbf {S}) + V_{T}^I(r) [3(\\mathbf {S}_Q\\cdot \\hat{\\mathbf {n}})(\\mathbf {S}_q\\cdot \\hat{\\mathbf {n}})- \\mathbf {S}_1\\cdot \\mathbf {S}_2].$ The matrix element of the potential in the ${}^{2S+1}L_J$ basis is given by $\\langle {}^{2S+1}L_J|V_{Q\\bar{q}} (\\mathbf {r}) | {}^{2S+1}L_J\\rangle &=\\tilde{V}_{c}(r)+\\left[\\frac{1}{2}S(S+1)-\\frac{3}{4}\\right]\\tilde{V}_{SS}(r)+\\frac{1}{2}\\langle \\mathbf {L}\\cdot \\mathbf {S}\\rangle \\tilde{V}_{LS}(r) \\cr & +\\left[-\\frac{2\\langle \\mathbf {L} \\cdot \\mathbf {S} \\rangle (2\\langle \\mathbf {L}\\cdot \\mathbf {S}\\rangle +1)}{4(2L-1)(2L+3)}+\\frac{S(S+1)L(L+1)}{3(2L-1)(2L+3)}\\right] \\tilde{V}_T(r),$ where $\\langle \\mathbf {L}\\cdot \\mathbf {S}\\rangle =[J(J+1)-L(L+1)-S(S+1)]/2.$ Here we have taken the conventional spectroscopic notation ${}^{2S+1}L_J$ given in terms of the total spin $S$ , the orbital angular momentum $L$ , and the total angular momentum $J$ with the addition of the angular momenta, $\\mathbf {J}=\\mathbf {L}+\\mathbf {S}$ .",
"The corresponding terms $\\tilde{V}_{c}(r)$ , $\\tilde{V}_{SS}(r)$ , $\\tilde{V}_{LS}(r)$ and $\\tilde{V}_T(r)$ denote generically the central, spin-spin, spin-orbit, and tensor parts of the total potential."
],
[
"Calculations and Results",
"In Ref.",
"[34], the mass spectra of the heavy mesons were already studied within a simple variational method, the potential from the instanton vacuum and the potential of the simple harmonic oscillator being combined.",
"The results from Ref.",
"[34] were in qualitative agreement with the experimental data.",
"However, it is essential to consider more realistic contributions such as the confining potential and the Coulomb-like potential from one-gluon exchange in order to understand the effects of the instantons on the mass spectra of the heavy mesons in a quantitative manner.",
"In the present work, we will include all the potentials mentioned in the previous section.",
"A nonrelativistic potential approach for a heavy-light quark system is represented by the time-independent Schrödinger equation with the static potential $V_{Q\\bar{q}}(\\mathbf {r})$ $\\left[-\\frac{\\hbar ^2}{\\tilde{\\mu }}\\nabla ^2+V_{Q\\bar{q}}(\\mathbf {r})-E\\right]\\Psi _{JM}(\\mathbf {r})=0,$ where $\\tilde{\\mu }$ denotes the reduced mass of the heavy meson system and $\\Psi _{JM}$ stands for the wavefunction of the state with the total angular momentum $J$ and its third component $M$ .",
"To solve the Schrödinger equation numerically, we employ the GEM which was successfully applied to describe few-body systems such as light nuclei (see a review [36] and references therein).",
"In the GEM the wavefunction is expanded in terms of a set of $L^2$ -integrable basis functions $\\lbrace \\Phi _{JM,k}^{LS};\\,k=1-k_{\\mathrm {max}}\\rbrace $ $\\Psi _{JM}(\\mathbf {r})=\\sum _{k=1}^{k_{\\mathrm {max}}}C_{k,LS}^{(J)}\\Phi _{JM,k}^{LS}(\\mathbf {r})$ and the Rayleigh-Ritz variational method is used.",
"So, we are able to formulate a generalized eigenvalue problem given as $\\sum _{m=1}^{k_{\\mathrm {max}}}\\left\\langle \\Phi _{JM,k}^{LS}\\left|-\\frac{\\hbar ^2}{\\tilde{\\mu }}\\nabla ^2 +V_{Q\\bar{q}}(\\mathbf {r})-E\\right|\\Phi _{JM,m}^{LS}\\right\\rangle C_{m,LS}^{(J)}=0\\,.$ The angular part of the basis function $\\Phi _{JM,k}^{LS}$ is expressed in terms of standard spherical harmonics and the normalized radial part $\\phi _{k}^{L}(r)$ is written in terms of the Gaussian basis functions $\\phi _{k}^{L}(r) = \\left(\\frac{2^{2L+\\frac{7}{2}}r_{k}^{-2L-3}}{\\sqrt{\\pi }(2L+1)!!",
"}\\right)^{1/2}r^{L}e^{-(r/r_{k})^2},$ where $r_k,\\, k=1,2,...,k_{max}$ designate variational parameters.",
"When it comes to the case of a two-body problem, the total number of the variational parameters is reduced by using the geometric progression in the form of $r_k=r_1a^{k-1}$ , which provides a good convergence of the results.",
"Thus, in the two-body problem, we need only three variational parameters, i.e.",
"$r_1$ , $a$ and $k_{\\mathrm {max}}$ .For more details, see Refs.",
"[35], [36], [37], [38], [39].",
"Once the Schrödinger equation is solved, the energy eigenvalue $E_N$ is found and the mass of the heavy meson is determined by $M= M_Q + M_q + E_N +\\Delta E_{q},$ where $\\Delta E_{q}$ is the overall energy shift in the spectra depending on the light-quark content of the meson and plays a role of a simple tuning parameter.",
"As mentioned already, $M_Q$ and $M_q$ are the dynamical masses of the heavy and light quarks, respectively.",
"Note that $M_Q$ contains also the mass shift arising from the instanton vacuum.",
"In this work we will slightly vary the total mass of the strange quark mass $M_s$ and try to analyze the corresponding effects.",
"Since, some of remaining parameters cannot be determined theoretically, we construct several sets of the parameters and call them Model I$^{\\prime }$ , Model I, Model II, and Model III, respectively.",
"The numerical values of model parameters are listed in Table REF and we use them to calculate the spectra of the heavy mesons.The corresponding explanation of model parameters will be given hereafter in the text.",
"Table: Free parameters of the model: m s m_s denote thedynamical mass of the strange quark,ϰ\\varkappa stands for the string tension, σ\\sigma and σ I \\sigma _Idesignate the smearing parameters corresponding to point likeinteractions in Eqs.",
"() and (),ΔE u,d \\Delta E_{u,d} and ΔE s \\Delta E_s are the constant overallenergy shifts of mesons corresponding to the up (down) andstrange quark constituents, and nn is the density of instantonmedium.The results of the charmed meson masses corresponding to the different models are listed in Table REF in comparison with the experimental data taken from the Particle Data Group (PDG) [40].",
"In the second column, the results without instanton-induced quark-quark interactions are presented.",
"It is called Model $\\mathrm {I}^{\\prime }$ that is obtained by including only the confining and Coulomb-like type interactions.",
"One can assume that in this model the nonperturbative effects are only taken into account by means of dynamically generated masses of the corresponding light quarks.",
"It is seen that the results are relatively in good agreement with the experimental data.",
"It indicates that a nonrelativistic approach to the heavy-light quark system works even quantitatively at least for the mass spectra of the conventional heavy mesons.",
"Model I has the same parameter set as Model $\\mathrm {I}^{\\prime }$ except for the instanton-induced potentials, which means that the parameters are not tuned but the instanton-induced heavy-light quark interactions are taken into account.",
"By doing this, we can examine how the instanton-induced quark-quark interactions affect the mass of each charmed meson.",
"The effects of instanton-induced quark-quark interactions are clearly seen in the ground state $D^\\pm $ meson, while they are rather tiny on other charmed mesons.",
"In particular, the effects are almost negligible on the $P$ -wave charmed meson spectra.",
"One can conclude that in general instanton-induced interactions do not affect much the spectra of heavy mesons and play only a role in the fine-tuning level.",
"Thus, we present the results of Model II in which the free parameters are fitted to the experimental data.",
"One can see that the results slightly change in comparison with the Model I$^\\prime $ and shows that the instanton-induced quark-quark interactions are seem to be important in the fine-tuning level.",
"In Model III, we change also the density of the instanton medium is slightly changed, considering it as an input parameter.",
"This is allowed, as was already discussed in Ref.",
"[33] in detail.",
"All other parameters are fitted to the experimental data as in the case of Model II.",
"Table: The results of the charmed DD-meson masses in units ofMeV.",
"The second column lists the results without the instanton-inducedquark-quark interactions and is coined as Model I ' \\mathrm {I}^{\\prime }.",
"Thethird, fourth, and fifth columns list those of Models I, II, andIII.",
"The last column shows the corresponding experimental data takenfrom PDG .The results of Model III are slightly better than those of Model II.",
"As expected from the comparison of Model I with Model $\\mathrm {I}^{\\prime }$ , the prediction of Model III is not much different from that of Model I$^\\prime $ .",
"Thus the potential from the instanton vacuum in the present form change slightly the mass spectrum of the charmed mesons and does not affect quantitatively the results from the calculation without instanton-induced quark-quark interactions.",
"Table: The results of the charmed strange D s D_s-mesonmasses in units of MeV.Other notations are same as in the case of Table .Table REF lists the results of the charmed strange meson masses.",
"As done in Table REF , we first compute the masses of the charmed strange mesons without the instanton contributions, which are listed in the second column of Table REF .",
"Then we include the instanton-induced quark-quark interactions, of which the results are presented in the other columns.",
"The effects of the instantons are similar to the case of the charmed mesons, that is, the instanton effects are noticeable only on the ground state $D_s^\\pm $ meson whereas they are negligibly small on the $P$ -wave charmed strange mesons.",
"Though the results of Model III seem slightly better than those of Model I$^\\prime $ , for the quark-quark potential from the instanton vacuum, at least in the present form, the improvement is marginal in the charmed strange meson mass spectrum.",
"Moreover, the effects of the instanton-induced potential on the charmed strange mesons are even smaller than on the charmed nonstrange ones.",
"Finally, we would like to note that although we have changed the density of instanton medium $n$ in Model III in comparison with Model II the mass contribution $\\Delta M_Q$ is unchanged and kept in both cases equal to 0.086 GeV.",
"However, $\\Delta M_Q$ is proportional to $n$ and therefore it must be also modified if the value of $n$ changes.",
"As a result, eigenfunctions and eigenvalues of the Hamiltonian should be also altered.",
"Consequently, a better fine-fitting of the whole mass spectra can be achieved by means of changes of instanton parameters in a self-consistent manner.",
"Though these selfconsitent changes of parameters are expected to improve the present results further, we do not perform it because in the present work we aim at examining the effects of the existing nonperturbative heavy-light quark potentials from the instanton vacuum on the conventional heavy mesons.",
"Table: The results of the instaton effects on the low-lying charmedheavy mesons in units of MeV.",
"The values of the relevant parametersare taken from those for Model I.In Table REF , we list the results of the contributions from the instanton-induced potentials.",
"While they have visible effects on the masses of the $D^{\\pm }$ and $D_s^{\\pm }$ mesons, and marginal contributions to the radially excited $S$ -wave $D^*(2^1 S_0)$ and $D_s^*(2^1 S_0)$ mesons, they have almost no impact on other excited $D$ and $D_s$ mesons.",
"Thus, in conclusion, the present form of the instanton-induced potentials contributes to some of the $D$ and $D_s$ mesons as explicitly shown in Tables REF , REF , and REF , its overall effects turn out to be marginal.",
"Possible ways of improving the present results will be mentioned in the next Section."
],
[
"Summary and outlook",
"In the present work, we have investigate the effects of the heavy-light quark potential from the instanton vacuum on the mass spectra of the conventional charmed mesons.",
"First, we have considered the confining potential that is proportional to the relative distance between the heavy and light quarks.",
"The Coulomb-like potential, which arises from one-gluon exchange, has been included.",
"The spin-dependent potentials were generated from the central part.",
"Then we have computed the mass spectra of the charmed mesons, employing the Gaussian expansion method to solve the nonrelativistic Schrödinger equation.",
"The results are in good agreement with the experimental data even without the potential from the instanton vacuum included.",
"Then, we have introduced the central and spin-dependent potentials from the instanton vacuum.",
"The additional spin part of the potential was obtained from the central part of the instanton-induced potential.",
"While the instanton effects are noticeable on the $S$ -wave charmed and charmed strange heavy mesons, the contribution from the instanton-induced potential is rather tiny to their masses.",
"Though the present form of the instanton-induced potential does not give any significant contribution to the heavy meson masses, there are some possible ways of elaborating the present analysis: The present work is based on the nonrelativistic Schrödinger equation, since we aim mainly at investigating the effects of the instanton-induced potential.",
"However, once the light quark is involved, it is inevitable to include certain relativistic effects.",
"The instanton-induced potentials used in the present work was derived from the random instanton gas model and are given as local ones.",
"However, if one uses the instanton liquid model, the interaction between the heavy and light quarks turn out to be nonlocal [44].",
"This nonlocality will have certain effects on the mass spectra of the heavy mesons.",
"Recently, Ref.",
"[45] showed that rescattering of gluons with instantons generates dynamically the effective momentum-dependent gluon mass that will cause the screened heavy-quark potential.",
"It indicates that certain nonperturbative effects from the instanton vacuum will contribute also to the heavy-light quark system.",
"Thus, one needs to study systematically nonperturbative effects on both heavy mesons and heavy baryons, arising from the instanton vacuum.",
"The corresponding investigations are under way.",
"H.-Ch.K is grateful to P. Gubler, A. Hosaka, T. Maruyama and M. Oka for useful discussions.",
"He wants to express his gratitude to the members of the Advanced Science Research Center at Japan Atomic Energy Agency for the hospitality, where part of the present work was done.",
"The work of QW is supported by National Natural Science Foundation of China (11475085, 11535005,11690030) and National Major state Basic Research and Development of China (2016YFE0129300).",
"The works HChK and UY are supported by Basic Science Research Program through the National Research Foundation (NRF) of Korea funded by the Korean government (Ministry of Education, Science and Technology, MEST), Grant No.",
"NRF2018R1A2B2001752 (HChK) and Grant No.",
"2016R1D1A1B03935053 (UY)."
]
] | 1808.08359 |
[
[
"Trace and Testing Metrics on Nondeterministic Probabilistic Processes"
],
[
"Abstract The combination of nondeterminism and probability in concurrent systems lead to the development of several interpretations of process behavior.",
"If we restrict our attention to linear properties only, we can identify three main approaches to trace and testing semantics: the trace distributions, the trace-by-trace and the extremal probabilities approaches.",
"In this paper, we propose novel notions of behavioral metrics that are based on the three classic approaches above, and that can be used to measure the disparities in the linear behavior of processes wrt trace and testing semantics.",
"We study the properties of these metrics, like non-expansiveness, and we compare their expressive powers."
],
[
"Introduction",
"A major task in the development of complex systems is to verify that an implementation of a system meets its specification.",
"Typically, in the realm of process calculi, implementation and specification are processes formalized with the same language, and the verification task consists in comparing their behavior, which can be done at different levels of abstraction, depending on which aspects of the behavior can be ignored or must be captured.",
"If one focuses on linear properties only, processes are usually compared on the basis of the traces they can execute, or accordingly to their capacity to pass the same tests.",
"This was the main idea behind the study of trace equivalence [21] and testing equivalence [14].",
"If we consider also probabilistic aspects of system behavior, reasoning in terms of qualitative equivalences is only partially satisfactory.",
"Any tiny variation of the probabilistic behavior of a system, which may be also due to a measurement error, will break the equality between processes without any further information on the distance of their behaviors.",
"Actually, many implementations can only approximate the specification; thus, the verification task requires appropriate instruments to measure the quality of the approximation.",
"For this reason, we propose to use hemimetrics measuring the disparities in process behavior wrt.",
"linear semantics also to quantify process verification.",
"Informally, we may see a specification not as the precise desired behavior of the system, but as set of minimum requirements on system behavior, such as the lower bounds on the probabilities to execute given traces or pass given tests.",
"Then, given a hemimetric $\\mathbf {h}$ expressing trace (resp.",
"testing) semantics, we can set a certain tolerance $\\varepsilon $ , related to the application context, and transform the verification problem into a verification up-to-$\\varepsilon $ , or $\\varepsilon $ -robustness problem: we say that an implementation $I$ is $\\varepsilon $ -trace-robust (resp.",
"$\\varepsilon $ -testing-robust) wrt.",
"a specification $S$ if whenever $S$ can perform a trace (resp.",
"pass a test) with a given probability $p$ , then $I$ can do the same with probability at least $p-\\varepsilon $ , namely if $\\mathbf {h}(S,I) \\le \\varepsilon $ .",
"Dually, we may see $S$ as giving an upper bound to undesired system behavior, and demand that whenever $S$ can perform a trace (resp.",
"pass a test) with a given probability $p$ , then $I$ can do the same with probability at most $p+\\varepsilon $ , namely if $\\mathbf {h}(I,S) \\le \\varepsilon $ .",
"In this paper, we consider nondeterministic probabilistic labeled transition systems (PTS) [25], a very general model in which nondeterminism and probability coexist, and we discuss the definition of hemimetrics and pseudometrics suitable to measure the differences in process behavior wrt.",
"trace and testing semantics.",
"We will see that the interplay of probability and nondeterminism lead to some difficulties in defining notions of behavioral distance, as already experienced in the case of equivalences [7].",
"For instance, in trace semantics, it is questionable whether the choice of the trace should precede or follow the choice by the scheduler.",
"Several approaches to probabilistic trace equivalence are discussed in [7]: [(i)] The trace distribution [24] approach, comparing entire resolutions created by schedulers by checking if they assign the same probability to the same traces; The trace-by-trace [4] approach, in which firstly we take a trace and then we check if there are resolutions for processes assigning the same probability to it; The extremal probabilities [5] approach, considering for each trace only the infima and suprema of the probabilities assigned to it over all resolutions for the processes.",
"We will argue that considering only supremal probabilities instead of both extremal probabilities is more tailored to reason on the verification problem.",
"Then, we propose three trace hemimetrics and pseudometrics as quantitative variants of trace distribution, trace-by-trace and supremal probabilities trace preorders and equivalences.",
"All these distances are parametric wrt.",
"the type of scheduler.",
"We consider deterministic and randomized schedulers, however an extension to other types of schedulers seems feasible.",
"Our results can be summarized as follows: We prove that, under each hemimetric/pseudometric, the pairs of processes at distance zero are precisely those related by the corresponding preorder/equivalence.",
"We prove that the hemimetrics/pseudometrics for trace-by-trace and supremal probabilities semantics are suitable for compositional reasoning, by showing their non-expansiveness [16] wrt.",
"parallel composition.",
"We study the differences in the expressive powers of these distances, thus composing them in a simple spectrum.",
"In particular, we show that the supremal probabilities semantics defined either on deterministic or randomized schedulers has the same expressive power of the trace-by-trace semantics on randomized schedulers.",
"This is a very interesting result in the perspective of an application to quantitative process verification: the comparison of the suprema execution probabilities of linear properties has the same expressive power of a pairwise comparison of the probabilities in all possible randomized resolutions of nondeterminism.",
"Then, we consider three approaches to testing semantics: [(i)] the may/must [29], the trace-by-trace [7], the supremal probabilities approach.",
"Briefly, in (REF ) the extremal probabilities of passing a test are considered whereas (REF )–(REF ) base on a traced view of testing, in that we compare the probabilities of passing the test via the execution of a given trace.",
"Actually, (REF )–(REF ) can be considered as the adaptation to testing semantics of the trace-by-trace and suprema probability approaches to trace semantics.",
"For each of these approaches, we present a hemimetric and a pseudometric as the quantitative variant of the related preorder and equivalence.",
"To the best of our knowledge, ours is the first attempt in this direction.",
"In detail: We prove that, under each hemimetric/pseudometric, the pairs of processes at distance zero are precisely those equated by the related testing preorder/equivalence.",
"We prove that all hemimetrics and pseudometrics are non-expansive.",
"We compose these testing distances in a simple spectrum and we also compare them with trace distances.",
"Background PTSs [25] are a very general model combining LTSs [23] and discrete time Markov chains [19], to model reactive behavior, nondeterminism and probability.",
"In a PTS, the state space is given by a set $\\mathbf {S}$ of $\\emph {processes}$ , ranged over by $s,t,\\dots $ and transition steps take processes to probability distributions over processes.",
"Probability distributions over $\\mathbf {S}$ are mappings $\\pi \\colon \\mathbf {S}\\rightarrow [0,1]$ with $\\sum _{s \\in \\mathbf {S}} \\pi (s) = 1$ .",
"By $\\Delta (\\mathbf {S})$ we denote the set of all distributions over $\\mathbf {S}$ , ranged over by $\\pi ,\\pi ^{\\prime },\\dots $ For $\\pi \\in \\Delta (\\mathbf {S})$ , the support of $\\pi $ is the set $\\mathsf {supp}(\\pi ) = \\lbrace s \\in \\mathbf {S}\\mid \\pi (s) >0\\rbrace $ .",
"We consider only distributions with finite support.",
"For $s \\in \\mathbf {S}$ , we let $\\delta _s$ denote the Dirac distribution on $s$ defined by $\\delta _s(s)= 1$ and $\\delta _s(t)=0$ for $t\\ne s$ .",
"Definition 1 (PTS, [25]) A nondeterministic probabilistic labeled transition system (PTS) is a triple $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ where: [(i)] $\\mathbf {S}$ is a countable set of processes, $\\mathcal {A}$ is a countable set of actions, and $\\xrightarrow{} \\subseteq {\\mathbf {S}\\times \\mathcal {A}\\times \\Delta (\\mathbf {S})}$ is a transition relation.",
"We write $s\\xrightarrow{}\\pi $ for $(s,a,\\pi ) \\in \\xrightarrow{}$ , $s \\xrightarrow{} $ if there is a distribution $\\pi $ with $s \\xrightarrow{} \\pi $ , and $s \\mathrel {{\\xrightarrow{}}\\makebox{[}0em][r]{\\lnot \\hspace{8.5pt}}}{\\!",
"}$ otherwise.",
"A PTS is fully nondeterministic if every transition has the form $s\\xrightarrow{}\\delta _t$ for some $t \\in \\mathbf {S}$ .",
"A PTS is fully probabilistic if at most one transition is enabled for each process.",
"$s \\in \\mathbf {S}$ is image-finite [20] if for each $a \\in \\mathcal {A}$ the number of $a$ -labeled transitions enabled for $s$ is finite.",
"We consider only image-finite processes.",
"Definition 2 (Parallel composition) Let $P_1 = (S_1, \\mathcal {A}, \\xrightarrow{}_1)$ and $P_2 = (S_2, \\mathcal {A}, \\xrightarrow{}_2)$ be two PTSs.",
"The (CSP-like [21]) synchronous parallel composition of $P_1$ and $P_2$ is the PTS $P_1 \\parallel P_2 = (S_1 \\times S_2, \\mathcal {A}, \\xrightarrow{})$ , where $\\xrightarrow{} \\subseteq (S_1 \\times S_2) \\times \\mathcal {A}\\times \\Delta (S_1 \\times S_2)$ is such that $(s_1,s_2) \\xrightarrow{} \\pi $ if and only if $s_1 \\xrightarrow{}_1 \\pi _1$ , $s_2 \\xrightarrow{}_2 \\pi _2$ and $\\pi (s_1^{\\prime },s_2^{\\prime }) = \\pi _1(s^{\\prime }_1) \\cdot \\pi _2(s^{\\prime }_2)$ for all $(s^{\\prime }_1,s^{\\prime }_2) \\in S_1 \\times S_2$ .",
"We proceed to recall some notions, mostly from [5], [7], [6], necessary to reason on trace and testing semantics.",
"A computation is a weighted sequence of process-to-process transitions.",
"Definition 3 (Computation) A computation from $s_0$ to $s_n$ has the form $\\begin{array}{c}\\hspace{99.58464pt}c := s_0 \\stackrel{a_1}{{\\twoheadrightarrow }} s_1 \\stackrel{a_2}{{\\twoheadrightarrow }} s_2 \\dots s_{n-1} \\stackrel{a_n}{{\\twoheadrightarrow }} s_n\\end{array}$ where, for all $i = 1,\\dots ,n$ , there is a transition $s_{i-1} \\xrightarrow{} \\pi _i$ with $s_i \\in \\mathsf {supp}(\\pi _i)$ .",
"Note that $\\pi _i(s_i)$ is the execution probability of step $s_{i-1} \\stackrel{a_i}{{\\twoheadrightarrow }} s_i$ conditioned on the selection of the transition $s_{i-1} \\xrightarrow{} \\pi _i$ at $s_{i-1}$ .",
"We denote by $\\mathrm {Pr}(c) = \\prod _{i = 1}^{n} \\pi _i(s_i)$ the product of the execution probabilities of the steps in $c$ .",
"A computation $c$ from $s$ is maximal if it is not a proper prefix of any other computation from $s$ .",
"We denote by $s)$ (resp.",
"${\\max }(s)$ ) the set of computations (resp.",
"maximal computations) from $s$ .",
"For any $s)$ , we define $\\mathrm {Pr}( = \\sum _{c \\in \\mathrm {Pr}(c) whenever none of the computations in is a proper prefix of any of the others.",
"}We denote by $ A$ the set of \\emph {finite traces} in $ A$ and write $ e$ for the empty trace.We say that a computation is \\emph {compatible} with the trace $ A$ if{f} the sequence of actions labeling the computation steps is equal to $$.We denote by $ s,) s)$ the set of computations from $ s$ that are compatible with $$, and by $ (s,)$ the set $ (s,) = (s) s,)$.$ To express linear semantics we need to evaluate and compare the probability of particular sequences of events to occur.",
"As in PTSs this probability highly depends also on nondeterminism, schedulers [24], [28], [18] (or adversaries) resolving it become fundamental.",
"They can be classified into two main classes: deterministic and randomized schedulers [24].",
"For each process, a deterministic scheduler selects exactly one transition among the possible ones, or none of them, thus treating all internal nondeterministic choices as distinct.",
"Randomized schedulers allow for a convex combination of the equally labeled transitions.",
"The resolution given by a deterministic scheduler is a fully probabilistic process, whereas from randomized schedulers we get a fully probabilistic process with combined transitions [26].",
"Definition 4 (Resolutions) Let $P = (\\mathbf {S}, \\mathcal {A},\\xrightarrow{})$ be a PTS and $s \\in \\mathbf {S}$ .",
"We say that a PTS $\\mathcal {Z}= (Z,\\mathcal {A},\\xrightarrow{}_{\\mathcal {Z}})$ is a deterministic resolution for $s$ iff there exists a function $\\mathrm {corr}_{\\mathcal {Z}} \\colon Z \\rightarrow \\mathbf {S}$ such that $s = \\mathrm {corr}_{\\mathcal {Z}}(z_s)$ for some $z_s \\in Z$ and moreover: (i) If $z \\xrightarrow{}_{\\mathcal {Z}} \\pi $ , then $\\mathrm {corr}_{\\mathcal {Z}}(z) \\xrightarrow{} \\pi ^{\\prime }$ with $\\pi (z^{\\prime }) = \\pi ^{\\prime }(\\mathrm {corr}_{\\mathcal {Z}}(z^{\\prime }))$ for all $z^{\\prime } \\in Z$ .",
"(ii) If $z \\xrightarrow{}_{\\mathcal {Z}} \\pi _1$ and $z \\xrightarrow{}_{\\mathcal {Z}} \\pi _2$ then $a_1 = a_2$ and $\\pi _1 = \\pi _2$ .",
"Conversely, we say that $\\mathcal {Z}$ is a randomized resolution for $s$ if item (i) is replaced by (i)' If $z \\xrightarrow{}_{\\mathcal {Z}} \\pi $ , then there are $n \\in \\mathbb {N}$ , $\\lbrace p_i \\in (0,1] \\mid \\sum _{i = 1}^n p_i = 1\\rbrace $ and $\\lbrace \\mathrm {corr}_{\\mathcal {Z}}(z) \\xrightarrow{} \\pi _i \\mid 1 \\le i \\le n\\rbrace $ s.t.",
"$\\pi (z^{\\prime }) = \\sum _{i = 1}^n p_i \\cdot \\pi _i(\\mathrm {corr}_{\\mathcal {Z}}(z^{\\prime }))$ for all $z^{\\prime } \\in Z$ .",
"Then, $\\mathcal {Z}$ is maximal iff it cannot be further extended in accordance with the graph structure of $P$ and the constraints above.",
"For $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ , we denote by $\\mathrm {Res}^{\\mathrm {x}}(s)$ the set of resolutions for $s$ and by $\\mathrm {Res}^{\\mathrm {x}}_{\\max }(s)$ the subset of maximal resolutions for $s$ .",
"We conclude this section by recalling the mathematical notions of hemimetric and pseudometric.",
"A 1-bounded pseudometric on $\\mathbf {S}$ is a function $d \\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ s.t.",
": [(i)] $d(s,s) =0$ , $d(s,t) = d(t,s)$ , $d(s,t) \\le d(s,u) + d(u,t)$ , for $s,t,u \\in \\mathbf {S}$ .",
"Then, $d$ is a hemimetric if it satisfies (REF ) and (REF ).",
"The kernel of a (hemi,pseudo)metric $d$ on $\\mathbf {S}$ the set of pairs of elements in $\\mathbf {S}$ which are at distance 0, namely $ker(d) = \\lbrace (s,t) \\in \\mathbf {S}\\times \\mathbf {S}\\mid d(s,t) = 0\\rbrace $ .",
"Non-expansiveness [16] of a (hemi,pseudo)metric is the quantitative analogue to the (pre)congruence property.",
"Here we propose also a stronger notion, called strict non-expansiveness that gives tighter bounds on the distance of processes composed in parallel.",
"Definition 5 ((Strict) non-expansiveness) Let $d$ be a (hemi,pseudo)metric on $\\mathbf {S}$ .",
"Following [16], we say that $d$ is non-expansive wrt.",
"the parallel composition operator if and only if for all $s_1,s_2,t_1,t_2 \\in \\mathbf {S}$ we have $d(s_1 \\parallel s_2, t_1 \\parallel t_2) \\le d(s_1,t_1) + d(s_2,t_2)$ .",
"Moreover, we say that $d$ is strictly non-expansive if $d(s_1 \\parallel s_2, t_1 \\parallel t_2) \\le d(s_1,t_1) + d(s_2,t_2) - d(s_1,t_1) \\cdot d(s_2,t_2)$ .",
"Finally, we remark that, as elsewhere in the literature, throughout the paper we may use the term metric in place of pseudometric.",
"Metrics for traces In this Section, we define the metrics measuring the disparities in process behavior wrt.",
"trace semantics.",
"We consider three approaches to the combination of nondeterminism and probability: the trace distribution, the trace-by-trace and the supremal probabilities approach.",
"In defining the behavioral distances, we assume a discount factor $\\lambda \\in (0,1]$, which allows us to specify how much the behavioral distance of future transitions is taken into account [2], [16].",
"The discount factor $\\lambda =1$ expresses no discount, so that the differences in the behavior between $s,t \\in \\mathbf {S}$ are considered irrespective of after how many steps they can be observed.",
"The trace distribution approach In [24] the observable events characterizing the trace semantics are trace distributions, ie.",
"probability distributions over traces.",
"Processes $s,t \\in \\mathbf {S}$ are trace distribution equivalent if, for any resolution for $s$ there is a resolution for $t$ exhibiting the same trace distribution, ie.",
"the execution probability of each trace in the two resolutions is exactly the same, and vice versa.",
"Definition 6 (Trace distribution equivalence [24], [4]) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"Processes $s,t \\in \\mathbf {S}$ are in the trace distribution preorder, written $s \\sqsubseteq _{\\mathrm {Tr,dis}}^{\\mathrm {x}}t$ , if: $\\text{for each } \\mathcal {Z}_s \\in \\mathrm {Res}^{\\mathrm {x}}(s) \\text{ there is } \\mathcal {Z}_t \\in \\mathrm {Res}^{\\mathrm {x}}(t) \\text{ s.t.\\ for each } \\alpha \\in \\mathcal {A}^{\\star } \\colon \\mathrm {Pr}(z_s,\\alpha )) = \\mathrm {Pr}(z_t,\\alpha ))$ .",
"Then, $s,t$ are trace distribution equivalent, notation $s \\sim _{\\mathrm {Tr,dis}}^{\\mathrm {x}}t$ , iff $s \\sqsubseteq _{\\mathrm {Tr,dis}}^{\\mathrm {x}}t$ and $t \\sqsubseteq _{\\mathrm {Tr,dis}}^{\\mathrm {x}}s$ .",
"The quantitative analogue to trace distribution equivalence is based on the evaluation of the differences in the trace distributions of processes: the distance between processes $s,t$ is set to $\\varepsilon \\ge 0$ if, for any resolution for $s$ there is a resolution for $t$ exhibiting a trace distribution differing at most by $\\varepsilon $, meaning that the execution probability of each trace in the two resolutions differs by at most $\\varepsilon $, and vice versa.",
"Definition 7 (Trace distribution metric) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"The trace distribution hemimetric and the trace distribution metric are the functions $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {x}}, \\mathbf {m}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {x}}\\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {x}}(s,t) = \\sup _{\\mathcal {Z}_s \\in \\mathrm {Res}^{\\mathrm {x}}(s)}\\, \\inf _{\\mathcal {Z}_t \\in \\mathrm {Res}^{\\mathrm {x}}(t)}\\, \\sup _{\\alpha \\in \\mathcal {A}^{\\star }} \\lambda ^{|\\alpha |-1}|\\mathrm {Pr}(z_s,\\alpha )) - \\mathrm {Pr}(z_t,\\alpha ))|$ $\\mathbf {m}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {x}}(s,t) = \\max \\lbrace \\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {x}}(s,t),\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {x}}(t,s)\\rbrace $ .",
"We observe that the expression $\\sup _{\\alpha \\in \\mathcal {A}^{\\star }} \\lambda ^{|\\alpha |-1}|\\mathrm {Pr}(z_s,\\alpha )) - \\mathrm {Pr}(z_t,\\alpha ))|$ used in Definition REF corresponds to the (weighted) total variation distance between the trace distributions given by the two resolutions $\\mathcal {Z}_s$ and $\\mathcal {Z}_t$ .",
"An equivalent formulation is given in [27], [12] via the Kantorovich lifting of the discrete metric over traces.",
"We now state that trace distribution hemimetrics and metrics are well-defined and that their kernels are the trace distribution preorders and equivalences, respectively.",
"Theorem 1 Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"Then: The function $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {x}}$ is a 1-bounded hemimetric on $\\mathbf {S}$ , with $\\sqsubseteq _{\\mathrm {Tr,dis}}^{\\mathrm {x}}$ as kernel.",
"The function $\\mathbf {m}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {x}}$ is a 1-bounded pseudometric on $\\mathbf {S}$ , with $\\sim _{\\mathrm {Tr,dis}}^{\\mathrm {x}}$ as kernel.",
"Figure: We will evaluate the trace distances between s p s_p and tt wrt.",
"the different approaches, schedulers and parameter p∈[0,1]p \\in [0,1].In all upcoming examples we will investigate only the traces that are significant for the evaluation of the considered distance.Example 1 Consider processes $s_p$ and $t$ in Figure REF , with $p \\in [0,1]$ .",
"First we evaluate $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {det}}(t,s_p)$ .",
"We expand only the case for the resolution $\\mathcal {Z}_t$ for $t$ obtained from its central $a$ -branch.",
"It assigns probability $0.5$ to both $ab$ and $ac$ .",
"Under deterministic schedulers, any resolution $\\mathcal {Z}_{s_p}$ for $s_p$ can assign positive probability to only one of these traces.",
"Assume this trace is $ab$ , the case $ac$ is analogous.",
"We have either $\\mathrm {Pr}(z_{s_p},ab)) = p$ or $\\mathrm {Pr}(z_{s_p},ab)) = 1$ .",
"Then, $|\\mathrm {Pr}(z_{t},ab)) - \\mathrm {Pr}(z_{s_p},ab))| \\in \\lbrace 0.5,|0.5-p|\\rbrace $ and $|\\mathrm {Pr}(z_{t},ac)) - \\mathrm {Pr}(z_{s_p},ac))| = |0.5-0|=0.5$ .",
"Therefore, $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {det}}(t,s_p) = \\lambda \\cdot 0.5$ , for all $p \\in [0,1]$ .",
"Now we show that $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {det}}(s_p,t) = \\lambda \\cdot \\min \\lbrace p, |0.5 - p|,1-p\\rbrace $ .",
"For each resolution $\\mathcal {Z}_{s_p}$ for $s_p$ we need the resolution for $t$ whose trace distribution is closer to that of $\\mathcal {Z}_{s_p}$ .",
"We expand only the case of $\\mathcal {Z}_{s_p}$ corresponding to the leftmost $a$ -branch of $s_p$ and giving probability 1 to trace $a$ and $p$ to trace $ab$ .",
"We distinguish three subcases, related to the value of $p$ : [(i)] $p \\in [0,0.25]$ : The resolution for $t$ minimizing the distance from $\\mathcal {Z}_{s_p}$ is $\\mathcal {Z}_t^1$ that selects no action for $z^1_{t_1}$ .",
"The distance between $\\mathcal {Z}_{s_p}$ and $\\mathcal {Z}_t^1$ is $\\lambda ^{|ab|-1}|\\mathrm {Pr}(z_{s_p},ab)) - \\mathrm {Pr}(z_t^1,ab))| = \\lambda \\cdot p$ .",
"Notice that in this case $p \\le |0.5-p|,1-p$ .",
"$p \\in (0.25,0.75]$ : The resolution for $t$ that minimizes the distance from $\\mathcal {Z}_{s_p}$ is $\\mathcal {Z}_t^2$ that performs an $a$ -move and evolves to $0.5 \\delta _{z^2_{t_2}} + 0.5 \\delta _{z^2_{t_3}}$ , where $z^2_{t_3}$ that executes no action.",
"The distance between $\\mathcal {Z}_{s_p}$ and $\\mathcal {Z}^2_t$ is $\\lambda ^{|ab|-1}|\\mathrm {Pr}(z_{s_p},ab)) - \\mathrm {Pr}(z^2_t,ab))| = \\lambda \\cdot |0.5 - p|$ .",
"Notice that in this case we have $|0.5-p| \\le p,1-p$ .",
"$p \\in (0.75,1]$ : The resolution for $t$ that minimizes the distance from $\\mathcal {Z}_{s_p}$ is $\\mathcal {Z}_t^3$ that corresponds to the leftmost branch of $t$ .",
"The distance between $\\mathcal {Z}_s$ and $\\mathcal {Z}^3_t$ is $\\lambda ^{|ab|-1}|\\mathrm {Pr}(z_{s_p},ab)) - \\mathrm {Pr}(z^3_t,ab))| = \\lambda \\cdot (1-p)$ .",
"Notice that in this case $1-p \\le |0.5-p|,p$ .",
"In the case of randomized schedulers, one can prove that, since both $s_p,t$ can perform traces $ab$ and $ac$ with probability 1, for any $p \\in [0,1]$ we get $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {rand}}(s_p,t) = \\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {rand}}(t,s_p) =0$ .",
"$$ The trace-by-trace approach Trace distribution equivalences come with some desirable properties, as the full backward compatibility with the fully nondeterministic and fully probabilistic cases (cf.",
"[7]).",
"However, they are not congruences wrt.",
"parallel composition [24], and thus the related metrics cannot be non-expansive.",
"Moreover, due to the crucial rôle of the schedulers in the discrimination process, trace distribution distances are sometimes too demanding.",
"Take, for example, processes $s,t$ in Figure REF , with $\\varepsilon _1,\\varepsilon _2 \\in [0,0.5]$ .",
"We have $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {det}}(s,t) = \\lambda \\cdot 0.5$ and $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {det}}(t,s) = \\lambda \\cdot \\max _{i \\in \\lbrace 1,2\\rbrace } \\max \\lbrace 0.5- \\varepsilon _i, \\varepsilon _i\\rbrace $ , thus giving $\\mathbf {m}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {det}}(s,t) = \\lambda \\cdot 0.5$ for all $\\varepsilon _1,\\varepsilon _2 \\in [0,0.5]$ .",
"However, $s$ and $t$ can perform the same traces with probabilities that differ at most by $\\max (\\varepsilon _1,\\varepsilon _2)$ , which suggests that their trace distance should be $\\lambda \\cdot \\max (\\varepsilon _1,\\varepsilon _2)$ .",
"Specially, for $\\varepsilon _1,\\varepsilon _2=0$ , $s,t$ can perform the same traces with exactly the same probability.",
"Despite this, $s,t$ are still distinguished by trace distribution equivalences.",
"These situations arise since the focus of trace distribution approach is more on resolutions than on traces.",
"To move the focus on traces, the trace-by-trace approach was proposed [4].",
"The idea is to choose first the event that we want to observe, namely a single trace, and only as a second step we let the scheduler perform its selection: processes $s,t$ are equivalent wrt.",
"the trace-by-trace approach if for each trace $\\alpha $ , for each resolution for $s$ there is a resolution for $t$ that assigns to $\\alpha $ the same probability, and vice versa.",
"Figure: For ε 1 ,ε 2 ∈[0,0.5]\\varepsilon _1,\\varepsilon _2 \\in [0,0.5], we have 𝐦 Tr , tbt λ, det (s,t)=𝐦 Tr , tbt λ, rand (s,t)=λ·max(ε 1 ,ε 2 )\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {det}}(s,t)=\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {rand}}(s,t)= \\lambda \\cdot \\max (\\varepsilon _1,\\varepsilon _2), 𝐦 Tr , dis λ, det (s,t)=λ·0.5\\mathbf {m}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {det}}(s,t)= \\lambda \\cdot 0.5 and 𝐦 Tr , dis λ, rand (s,t)=λ·max{0.25+ε 1 ,0.25+ε 2 }\\mathbf {m}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {rand}}(s,t) = \\lambda \\cdot \\max \\lbrace 0.25 + \\varepsilon _1, 0.25 + \\varepsilon _2\\rbrace .Definition 8 (Tbt-trace equivalence [4]) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"We say that $s,t \\in \\mathbf {S}$ are in the tbt-trace preorder, written $s \\sqsubseteq _{\\mathrm {Tr,tbt}}^{x}t$ , if for each $\\alpha \\in \\mathcal {A}^{\\star }$ $\\text{for each } \\mathcal {Z}_s \\in \\mathrm {Res}^{\\mathrm {x}}(s) \\text{ there is } \\mathcal {Z}_t \\in \\mathrm {Res}^{\\mathrm {x}}(t) \\text{ such that } \\mathrm {Pr}(z_s,\\alpha )) = \\mathrm {Pr}(z_t,\\alpha ))$ .",
"Then, $s,t\\in \\mathbf {S}$ are tbt-trace equivalent, notation $s \\sim _{\\mathrm {Tr,tbt}}^{x}t$ , iff $s \\sqsubseteq _{\\mathrm {Tr,tbt}}^{x}t$ and $t \\sqsubseteq _{\\mathrm {Tr,tbt}}^{x}s$ .",
"In [4] it was proved that tbt-trace equivalences enjoy the congruence property and are full backward compatible with the fully nondeterministic and the fully probabilistic cases.",
"We introduce now the quantitative analogous to tbt-trace equivalences.",
"Processes $s,t$ are at distance $\\varepsilon \\ge 0$ if, for each trace $\\alpha $ , for each resolution for $s$ there is a resolution for $t$ such that the two resolutions assign to $\\alpha $ probabilities that differ at most by $\\varepsilon $, and vice versa.",
"Definition 9 (Tbt-trace metric) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"For each $\\alpha \\in \\mathcal {A}^{\\star }$ , the function $\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{\\alpha ,}\\lambda ,\\mathrm {x}} \\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ is defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{\\alpha ,}\\lambda ,\\mathrm {x}}(s,t) = \\lambda ^{|\\alpha |-1}\\; \\sup _{\\mathcal {Z}_s \\in \\mathrm {Res}^{\\mathrm {x}}(s)}\\, \\inf _{\\mathcal {Z}_t \\in \\mathrm {Res}^{\\mathrm {x}}(t)} \\, |\\mathrm {Pr}(z_s,\\alpha )) - \\mathrm {Pr}(z_t,\\alpha ))|$ The tbt-trace hemimetric and the tbt-trace metric are the functions $\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {x}}, \\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {x}}\\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {x}}(s,t) = \\sup _{\\alpha \\in \\mathcal {A}^{\\star }}\\; \\mathbf {h}_{\\mathrm {Tr,tbt}}^{{\\alpha ,}\\lambda ,\\mathrm {x}}(s,t)$ $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {x}}(s,t) = \\max \\lbrace \\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {x}}(s,t), \\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {x}}(t,s)\\rbrace $ .",
"It is not hard to see that for processes in Figure REF we have $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {x}}(s,t) = \\lambda \\cdot \\max (\\varepsilon _1,\\varepsilon _2)$ (and $s \\sim _{\\mathrm {Tr,tbt}}^{x}t$ if $\\varepsilon _1,\\varepsilon _2=0$ ).",
"Notice that, since we consider image finite processes, we are guaranteed that for each trace $\\alpha \\in \\mathcal {A}^{\\star }$ the supremum and infimum in the definition of $\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{\\alpha ,}\\lambda ,\\mathrm {x}}$ are actually achieved.",
"We show now that tbt-trace hemimetrics and metrics are well-defined and that their kernels are the tbt-trace preorders and equivalences, respectively.",
"Theorem 2 Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"Then: The function $\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {x}}$ is a 1-bounded hemimetric on $\\mathbf {S}$ , with $\\sqsubseteq _{\\mathrm {Tr,tbt}}^{x}$ as kernel.",
"The function $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {x}}$ is a 1-bounded pseudometric on $\\mathbf {S}$ , with $\\sim _{\\mathrm {Tr,tbt}}^{x}$ as kernel.",
"Example 2 Consider Figure REF .",
"We get $\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {det}}(s_p,t)$ = $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {det}}(s_p,t)$ = (see Example REF ) $\\lambda \\cdot \\min \\lbrace p,|0.5-p|,1-p\\rbrace $ .",
"The reason why in this particular case the two pseudometrics coincide is that each resolution for $s_p$ gives positive probability to at most one of the traces $ab$ and $ac$ , so that quantifying on traces before or after quantifying on resolutions is irrelevant.",
"Let us evaluate now $\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {det}}(t,s_p)$ .",
"To this aim, we focus on trace $ab$ and the resolution $\\mathcal {Z}_t$ obtained from the central $a$ -branch of $t$ , for which we have $\\mathrm {Pr}(z_t,ab)) = 0.5$ .",
"We need the resolution $\\mathcal {Z}_{s_p}$ for $s_p$ that minimizes $|0.5 - \\mathrm {Pr}(z_{s_p},ab))|$ .",
"Since for any resolutions $\\mathcal {Z}_{s_p}$ for $s_p$ we have $\\mathrm {Pr}(z_{s_p},ab)) \\in \\lbrace 0,p,1\\rbrace $ , we infer that the resolution $\\mathcal {Z}_{s_p}$ we are looking for satisfies $\\mathrm {Pr}(z_{s_p},ab)) = p$ and, therefore, $|0.5 - \\mathrm {Pr}(z_{s_p},ab))|$ = $|0.5 - p|$ .",
"By considering also the other resolutions for $ab$ and, then, the other traces, we can check that $\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {det}}(t,s_p) = \\lambda \\cdot |0.5-p|$ .",
"In Example REF we showed that $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {det}}(t,s_p) = \\lambda \\cdot 0.5$ for all $p \\in [0,1]$ .",
"Hence, we get $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {det}}(t,s_p) = \\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {det}}(t,s_p)$ for $p \\in \\lbrace 0,1\\rbrace $ , and $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {det}}(t,s_p) > \\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {det}}(t,s_p)$ for $p \\in (0,1)$ .",
"This disparity is due to the fact that the trace distributions approach forced us to match the resolution for $t$ assigning positive probability to both $ab$ and $ac$ , whereas in the trace-by-trace approach one never consider two traces at the same time.",
"$$ We conclude this section by stating that tbt-trace distances are strictly non-expansive, As a corollary, we re-obtain the (pre)congurence properties for their kernels (proved in [7]).",
"Theorem 3 All distances $\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {det}}$ , $\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {rand}}$ , $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {det}}$ , $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {rand}}$ are strictly non-expansive.",
"The supremal probabilities approach The trace-by-trace approach improves on trace distribution approach since it supports equivalences and metrics that are compositional.",
"Moreover, by focusing on traces instead of resolutions, the trace-by-trace approach puts processes in Figure REF in the expected relations.",
"However, we argue here that trace-by-trace approach on deterministic schedulers still gives some questionable results.",
"Take, for example, processes $s,t$ in Figure REF .",
"We believe that these processes should be equivalent in any semantics approach, since, after performing the action $a$ , they reach two distributions that should be identified, as they assign total probability 1 to states with an identical behavior.",
"But, if we consider the trace $ab$ , the resolution $\\mathcal {Z}_t \\in \\mathrm {Res}^{\\mathrm {det}}(t)$ in Figure REF is such that $\\mathrm {Pr}(z_t, ab)) = 0.5$ , whereas the unique resolution for $s$ assigning positive probability to $ab$ is $\\mathcal {Z}_s$ in Figure REF , for which $\\mathrm {Pr}(z_s,ab)) = 1$ .",
"Hence no resolution in $\\mathrm {Res}^{\\mathrm {det}}(s)$ matches $\\mathcal {Z}_t$ on trace $ab$ , thus giving $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {det}}(s.t) = \\lambda \\cdot 0.5$ and, consequently, $s \\lnot \\sim _{\\mathrm {Tr,tbt}}^{\\mathrm {det}}t$ .",
"This motivates to look for an alternative approach that allows us to equate processes in Figure REF and, at the same time, preserves all the desirable properties of the tbt-trace semantics.",
"We take inspiration from the extremal probabilities approach proposed in [5], which bases on the comparison, for each trace $\\alpha $ , of both suprema and infima execution probabilities, wrt.",
"resolutions, of $\\alpha $ : two processes are equated if they assign the same extremal probabilities to all traces.",
"However, reasoning on infima may cause some arguable results.",
"In particular, it is unclear whether such infima should be evaluated over the whole class of resolutions or over a restricted class, as for instance the resolutions in which the considered trace is actually executed.",
"Besides, desirable properties like the backward compatibility and compositionality are not guaranteed.",
"For all these reasons, we find it more reasonable to define a notion of trace equivalence, and a related metric, based on the comparison of supremal probabilities only.",
"Notice that, if we focus on verification, the comparison of supremal probabilities becomes natural.",
"To exemplify, we let the non-probabilistic case guide us.",
"To verify whether a process $t$ satisfies the specification $S$ , we check that whenever $S$ can execute a particular trace, then so does $t$ .",
"Actually, only positive information is considered: if there is a resolution for $S$ in which a given trace is executed, then this information is used to verify the equivalence.",
"Still, resolutions in which such a trace is not enabled are not considered.",
"The same principle should hold for PTSs: a process should perform all the traces enabled in $S$ and it should do it with at least the same probability, in the perspective that the quantitative behavior expressed in the specification expresses the minimal requirements on process behavior.",
"Focusing on supremal probabilities means relaxing the tbt-trace approach by simply requiring that, for each trace $\\alpha $ and resolution $\\mathcal {Z}_s$ for process $s$ there is a resolution for $t$ assigning to $\\alpha $ at least the same probability given by $\\mathcal {Z}_s$ , and vice versa.",
"Figure: Processes ss and tt are distinguished by ∼ Tr , tbt det \\sim _{\\mathrm {Tr,tbt}}^{\\mathrm {det}}, but related by ∼ Tr ,⊔ det \\sim _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {det}}.We remark that tt and uu are related by all the relations in the three approaches to trace semantics.Definition 10 ($\\bigsqcup $ -trace equivalence) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS and $\\mathrm {x} \\in \\lbrace \\mathrm {det}, \\mathrm {rand}\\rbrace $ .",
"We say that $s,t \\in \\mathbf {S}$ are in the $\\bigsqcup $ -trace preorder, written $s \\sqsubseteq _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {x}}t$ , if for each $\\alpha \\in \\mathcal {A}^{\\star }$ $\\sup _{\\mathcal {Z}_s \\in \\mathrm {Res}^{\\mathrm {x}}(s)} \\mathrm {Pr}(z_s,\\alpha )) \\le \\sup _{\\mathcal {Z}_t \\in \\mathrm {Res}^{\\mathrm {x}}(t)} \\mathrm {Pr}(z_t,\\alpha ))$ .",
"Then, $s,t \\in \\mathbf {S}$ are $\\bigsqcup $ -trace equivalent, notation $s \\sim _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {x}}t$ , iff $s \\sqsubseteq _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {x}}t$ and $t \\sqsubseteq _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {x}}s$ .",
"We stress that all good properties of trace-by-trace approach, as the backward compatibility with the fully nondeterministic and fully probabilistic cases and the non-expansiveness of the metric wrt.",
"parallel composition, are preserved by the supremal probabilities approach (Proposition REF and Theorem REF below).",
"Let $\\sim _{\\mathrm {Tr}}^{\\mathbf {N}}$ denote the trace equivalence on fully nondeterministic systems [8] and $\\sim _{\\mathrm {Tr}}^{\\mathbf {P}}$ denote the one on fully-probabilistic systems [22].",
"Proposition 1 Assume a PTS $P = (\\mathbf {S}, \\mathcal {A}, \\xrightarrow{})$ and processes $s,t \\in \\mathbf {S}$ .",
"Then: If $P$ is fully-nondeterministic, then $s \\sim _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {det}}t \\Leftrightarrow s \\sim _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {rand}}t \\Leftrightarrow s \\sim _{\\mathrm {Tr}}^{\\mathbf {N}}t$ .",
"If $P$ is fully-probabilistic, then $s \\sim _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {det}}t \\Leftrightarrow s \\sim _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {rand}}t \\Leftrightarrow s \\sim _{\\mathrm {Tr}}^{\\mathbf {P}}t$ .",
"The idea behind the quantitative analogue of $\\bigsqcup $ -trace equivalence is that two processes are at distance $\\varepsilon \\ge 0$ if, for each trace, the supremal execution probabilities wrt.",
"the resolutions of nondeterminism for the two processes differ at most by $\\varepsilon $.",
"Definition 11 ($\\bigsqcup $ -trace metric) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"For each $\\alpha \\in \\mathcal {A}^{\\star }$ , the function $\\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{\\alpha ,}\\lambda ,\\mathrm {x}} \\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ is defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{\\alpha ,}\\lambda ,\\mathrm {x}}(s,t) = \\max \\big \\lbrace 0,\\lambda ^{|\\alpha |-1} \\big (\\sup _{\\mathcal {Z}_s \\in \\mathrm {Res}^{\\mathrm {x}}(s)} \\mathrm {Pr}(z_s,\\alpha )) - \\sup _{\\mathcal {Z}_t \\in \\mathrm {Res}^{\\mathrm {x}}(t)} \\mathrm {Pr}(z_t,\\alpha ))\\big )\\big \\rbrace .$ The $\\bigsqcup $ -trace hemimetric and the $\\bigsqcup $ -trace metric are the functions $\\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}, \\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {x}}\\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}(s,t) = \\sup _{\\alpha \\in \\mathcal {A}^{\\star }}\\; \\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{\\alpha ,}\\lambda ,\\mathrm {x}}(s,t)$ and $\\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {x}}(s,t) = \\max \\lbrace \\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}(s,t), \\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}(t,s) \\rbrace $ .",
"We can show that $\\bigsqcup $ -trace hemimetrics and metrics are well-defined and that their kernels are the $\\bigsqcup $ -trace preorders and equivalences, respectively.",
"Theorem 4 Assume a PTS $(\\mathbf {S}, \\mathcal {A}, \\xrightarrow{})$ , $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"Then: The function $\\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}$ is a 1-bounded hemimetric on $\\mathbf {S}$ , with $\\sqsubseteq _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {x}}$ as kernel.",
"The function $\\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {x}}$ is a 1-bounded pseudometric on $\\mathbf {S}$ , with $\\sim _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {x}}$ as kernel.",
"We conclude this section by showing that $\\bigsqcup $ -trace distances are strictly non-expansive.",
"As a corollary, we infer the (pre)congruence property of their kernels.",
"Theorem 5 All distances $\\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{}\\lambda ,\\mathrm {det}}$ , $\\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{}\\lambda ,\\mathrm {rand}}$ , $\\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {det}}$ , $\\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {rand}}$ are strictly non-expansive.",
"Remark 1 We can show that the upper bounds to the distance of composed processes provided in Thms.",
"REF and REF are tight, namely for each distance $d$ considered in these theorems, there are processes $s_1,s_2,t_1,t_2$ with $d(s_1 \\parallel s_2 , t_1 \\parallel t_2) = d(s_1,t_1) + d(s_2,t_2) - d(s_1,t_1) \\cdot d(s_2,t_2)$ .",
"Indeed, for $z_s,z_t$ in Fig.",
"REF , with $\\lambda = 1$ , we have $d(z_s,t_t) = 0.5$ and $d(z_s \\parallel z_s, z_t \\parallel z_t) = 0.75 = 0.5 + 0.5 - 0.5 \\cdot 0.5$ .",
"Comparing the distinguishing power of trace metrics So far, we have discussed the properties of trace-based behavioral distances under different approaches.",
"Our aim is now to place these distances in a spectrum.",
"More precisely, we will order them wrt.",
"their distinguishing power: given the metrics $d,d^{\\prime }$ on $\\mathbf {S}$ , we write $d > d^{\\prime }$ if and only if $d(s,t) \\ge d^{\\prime }(s,t)$ for all $s,t \\in \\mathbf {S}$ and $d(u,v) > d^{\\prime }(u,v)$ for some $u,v \\in \\mathbf {S}$ .",
"Intuitively, for trace distributions and tbt-trace semantics, the distances evaluated on deterministic schedulers are more discriminating than their randomized analogues.",
"Theorem 6 Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\lambda \\in (0,1]$ and $\\mathrm {y} \\in \\lbrace \\mathrm {dis},\\mathrm {tbt}\\rbrace $ .",
"Then: [1.]",
"$\\mathbf {h}_{\\mathrm {Tr,y}}^{\\lambda ,\\mathrm {rand}} < \\mathbf {h}_{\\mathrm {Tr,y}}^{\\lambda ,\\mathrm {det}}$ .",
"$\\mathbf {m}_{\\mathrm {Tr,y}}^{\\lambda ,\\mathrm {rand}} < \\mathbf {m}_{\\mathrm {Tr,y}}^{\\lambda ,\\mathrm {det}}$ .",
"As a corollary of Theorem REF , by using the relations between distances and equivalences in Theorems REF and REF , we re-obtain the relations $\\sim _{\\mathrm {Tr,dis}}^{\\mathrm {det}}\\subset \\sim _{\\mathrm {Tr,dis}}^{\\mathrm {rand}}$ and $\\sim _{\\mathrm {Tr,tbt}}^{\\mathrm {det}}\\subset \\sim _{\\mathrm {Tr,tbt}}^{\\mathrm {rand}}$ proved in [7].",
"Moreover, also the analogous results for preorders follow.",
"As one can expect, the metrics on trace distributions are more discriminating than their corresponding ones in the trace-by-trace approach.",
"Theorem 7 Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"Then: [1.]",
"$\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {x}} < \\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {x}}$ .",
"$\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {x}}< \\mathbf {m}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {x}}$ .",
"As a corollary, by using the kernel relations given in Theorems REF and REF , we re-obtain the relation $\\sim _{\\mathrm {Tr,dis}}^{\\mathrm {x}}\\subset \\sim _{\\mathrm {Tr,tbt}}^{x}$ proved in [7] and we get $\\sqsubseteq _{\\mathrm {Tr,dis}}^{\\mathrm {x}}\\subset \\sqsubseteq _{\\mathrm {Tr,tbt}}^{x}$ .",
"Moreover, we remark that $\\mathbf {m}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {rand}}$ is not comparable with $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {det}}$ .",
"This is mainly due to the randomization process and it is witnessed by processes in Figure REF , where $\\mathbf {m}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {rand}}(s,t)=\\lambda \\cdot \\max \\lbrace 0.25+\\varepsilon _1,0.25+\\varepsilon _2\\rbrace $ and $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {det}}(s,t) = \\lambda \\cdot \\max (\\varepsilon _1,\\varepsilon _2)$ and Figure REF , where $\\mathbf {m}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {rand}}(s,t)=0$ and $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {det}}(s,t) = \\lambda \\cdot 0.5$ .",
"We focus now on supremal probabilities approach, that comes with a particularly interesting result: the $\\bigsqcup $ -trace metric on deterministic schedulers coincides with tbt-trace metrics on randomized schedulers.",
"Moreover, $\\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {det}}$ coincides also with its randomized version.",
"Theorem 8 Assume a PTS $P = (\\mathbf {S}, \\mathcal {A}, \\xrightarrow{})$ and $\\lambda \\in (0,1]$ .",
"Then: [1.]",
"$\\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{}\\lambda ,\\mathrm {det}} = \\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{}\\lambda ,\\mathrm {rand}} = \\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {rand}}$ .",
"$\\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {det}}= \\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {rand}}= \\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {rand}}$ .",
"This result is fundamental in the perspective of the application of our trace metrics to process verification: by comparing solely the suprema execution probabilities of the linear properties of interest we get same expressive power of a pairwise comparison of the probabilities in all possible randomized resolutions of nondeterminism.",
"Clearly, Theorem REF together with the kernel relations from Thms REF and REF imply that the relations for the supremal probabilities semantics coincide with those for the tbt-trace semantics wrt.",
"randomized schedulers, ie.",
"$\\sqsubseteq _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {det}}= \\sqsubseteq _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {rand}}= \\sqsubseteq _{\\mathrm {Tr,tbt}}^{\\mathrm {rand}}$ and $\\sim _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {det}}= \\sim _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {rand}}= \\sim _{\\mathrm {Tr,tbt}}^{\\mathrm {rand}}$ .",
"Metrics for testing Testing semantics [14] compares processes according to their capacity to pass a test.",
"The latter is a PTS equipped with a distinguished state indicating the success of the test.",
"Definition 12 (Test) A nondeterministic probabilistic test transition systems (NPT) is a finite PTS $(\\mathbf {O}, \\mathcal {A}, \\xrightarrow{})$ where $\\mathbf {O}$ is a set of processes, called tests, containing a distinguished success process $\\surd $ with no outgoing transitions.",
"We say that a computation from $o \\in \\mathbf {O}$ is successful iff its last state is $\\surd $ .",
"Given a process $s$ and a test $o$ , we can consider the interaction system among the two.",
"This models the response of the process to the application of the test, so that $s$ passes the test $o$ if there is a computation in the interaction system that reaches $\\surd $ .",
"Informally, the interaction system is the result of the parallel composition of the process with the test.",
"Definition 13 (Interaction system) The interaction system of a PTS $(\\mathbf {S},\\mathcal {A},$ $\\xrightarrow{})$ and an NPT $(\\mathbf {O},\\mathcal {A},\\xrightarrow{}_{\\mathbf {O}})$ is the PTS $(\\mathbf {S}\\times \\mathbf {O}, \\mathcal {A},\\xrightarrow{}^{\\prime })$ where: [(i)] $(s,o) \\in \\mathbf {S}\\times \\mathbf {O}$ is called a configuration and is successful iff $o = \\surd $ ; a computation from $(s,o) \\in \\mathbf {S}\\times \\mathbf {O}$ is successful iff its last configuration is successful.",
"For $(s, o)$ and $\\mathcal {Z}_{s,o} \\in \\mathrm {Res}^{\\mathrm {x}}(s,o)$ , we let $\\mathbf {SC}(z_{s,o})$ be the set of successful computations from $z_{s,o}$ .",
"For $\\alpha \\in \\mathcal {A}^{\\star }$ , $\\mathbf {SC}(z_{s,o},\\alpha )$ is the set of $\\alpha $ -compatible successful computations from $z_{s,o}$ .",
"Testing semantics should compare processes wrt.",
"their probability to pass a test.",
"In this Section we consider three approaches to it: [(i)] the may/must, the trace-by-trace, and the supremal probabilities.",
"For each approach, we present (hemi,pseudo)metrics that provide a quantitative variant of the considered testing equivalence.",
"To the best of our knowledge, ours is the first attempt in this direction.",
"The may/must approach In the original work on nondeterministic systems [14], testing equivalence was defined via the may and must preorders.",
"The former expresses the ability of processes to pass a test.",
"The latter expresses the impossibility to fail a test.",
"When also probability is considered, these two preorders are defined, resp., in terms of suprema and infima success probabilities [29].",
"Definition 14 (May/must testing equivalence, [29]) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $(\\mathbf {O},\\mathcal {A},\\xrightarrow{}_O)$ an NPT and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"We say that $s,t \\in \\mathbf {S}$ are in the may testing preorder, written $s \\sqsubseteq _{\\mathrm {Te,may}}^{x}t$ , if for each $o \\in \\mathbf {O}$ $\\sup _{\\mathcal {Z}_{s,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o)} \\mathrm {Pr}(\\mathbf {SC}(z_{s,o})) \\le \\sup _{\\mathcal {Z}_{t,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o)} \\mathrm {Pr}(\\mathbf {SC}(z_{t,o}))$ .",
"Then, $s,t\\in \\mathbf {S}$ are may testing equivalent, written $s \\sim _{\\mathrm {Te,may}}^{x}t$ , iff $s \\sqsubseteq _{\\mathrm {Te,may}}^{x}t$ and $t \\sqsubseteq _{\\mathrm {Te,may}}^{x}s$ .",
"The notions of must testing preorder, $\\sqsubseteq _{\\mathrm {Te,must}}^{x}$ , and must testing equivalence, $\\sim _{\\mathrm {Te,must}}^{x}$ , are obtained by replacing the suprema in $\\sqsubseteq _{\\mathrm {Te,may}}^{x}$ and $\\sim _{\\mathrm {Te,may}}^{x}$ , resp., with infima.",
"Finally, we say that $s,t \\in \\mathbf {S}$ are in the may/must testing preorder, written $s \\sqsubseteq _{\\mathrm {Te,mM}}^{x}t$ , if $s \\sqsubseteq _{\\mathrm {Te,may}}^{x}t$ and $s \\sqsubseteq _{\\mathrm {Te,must}}^{x}t$ .",
"They are may/must testing equivalent, written $s \\sim _{\\mathrm {Te,mM}}^{x}t$ , iff $s \\sqsubseteq _{\\mathrm {Te,mM}}^{x}t$ and $t \\sqsubseteq _{\\mathrm {Te,mM}}^{x}s$ .",
"The quantitative analogue to may/must testing equivalence bases on the evaluation of the differences in the extremal success probabilities.",
"The distance between $s,t \\in \\mathbf {S}$ is set to $\\varepsilon \\ge 0$ if the maximum between the difference in the suprema and infima success probabilities wrt.",
"all resolutions of nondeterminism for $s$ and $t$ is at most $\\varepsilon $ .",
"We introduce a function $\\omega :\\mathbf {O}\\rightarrow (0,1]$ that assigns to each test $o$ the proper discount.",
"In fact, as the success probabilities in the may/must semantics are not related to the execution of a particular trace, in general we cannot define a discount factor as we did for the trace distances.",
"However, a similar construction may be regained when only tests with finite depth are considered.",
"In that case, we could define $\\omega (o) = \\lambda ^{\\mathrm {depth}(o)}$ , for $\\lambda \\in (0,1]$ .",
"We will use $\\mathbf {1}$ to denote the 1 constant function.",
"Definition 15 (May/must testing metric) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $(\\mathbf {O},\\mathcal {A},\\xrightarrow{}_O)$ an NPT, $\\omega :\\mathbf {O}\\rightarrow (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"For each $o\\in \\mathbf {O}$ , the function $\\mathbf {h}_{\\mathrm {Te,may}}^{{o,}\\omega ,\\mathrm {x}}\\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ is defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Te,may}}^{{o,}\\omega ,\\mathrm {x}}(s,t)=\\max \\Big \\lbrace 0 , \\omega (o)\\Big (\\sup _{\\mathcal {Z}_{s,o}\\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o)} \\mathrm {Pr}(\\mathbf {SC}(z_{s,o})) - \\sup _{\\mathcal {Z}_{t,o}\\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o)} \\mathrm {Pr}(\\mathbf {SC}(z_{t,o}))\\Big ) \\Big \\rbrace $ Function $\\mathbf {h}_{\\mathrm {Te,must}}^{{o,}\\omega ,\\mathrm {x}}\\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ is obtained by replacing the suprema in $\\mathbf {h}_{\\mathrm {Te,may}}^{{o,}\\omega ,\\mathrm {x}}$ with infima.",
"Given $\\mathrm {y} \\in \\lbrace \\mathrm {may}, \\mathrm {must}\\rbrace $ , the $\\mathrm {y}$ testing hemimetric and the $\\mathrm {y}$ testing metric are the functions $\\mathbf {h}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}, \\mathbf {m}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}} \\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}(s,t) = \\sup _{o \\in \\mathbf {O}}\\; \\mathbf {h}_{\\mathrm {Te,y}}^{o,\\omega ,\\mathrm {x}}(s,t)$ and $\\mathbf {m}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}(s,t) = \\max \\lbrace \\mathbf {h}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}(s,t), \\mathbf {h}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}(t,s) \\rbrace $ .",
"The may/must testing hemimetric and the may/must testing metric are the functions $\\mathbf {h}_{\\mathrm {Te,mM}}^{{}\\omega ,\\mathrm {x}}, \\mathbf {m}_{\\mathrm {Te,mM}}^{\\omega ,\\mathrm {x}}\\colon $ $\\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Te,mM}}^{{}\\omega ,\\mathrm {x}}(s,t) = \\max \\lbrace \\mathbf {h}_{\\mathrm {Te,may}}^{{}\\omega ,\\mathrm {x}}(s,t), \\mathbf {h}_{\\mathrm {Te,must}}^{{}\\omega ,\\mathrm {x}}(s,t) \\rbrace $ .",
"$\\mathbf {m}_{\\mathrm {Te,mM}}^{\\omega ,\\mathrm {x}}(s,t) = \\max \\lbrace \\mathbf {m}_{\\mathrm {Te,may}}^{\\omega ,\\mathrm {x}}{}(s,t), \\mathbf {m}_{\\mathrm {Te,must}}^{\\omega ,\\mathrm {x}}{}(s,t) \\rbrace $ .",
"Theorem 9 Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\omega :\\mathbf {O}\\rightarrow (0,1]$ , $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ and $\\mathrm {y} \\in \\lbrace \\mathrm {may},\\mathrm {must}, \\mathrm {mM}\\rbrace $ : The function $\\mathbf {h}_{\\mathrm {Te,y}}^{\\lambda ,\\mathrm {x}}$ is a 1-bounded hemimetric on $\\mathbf {S}$ , with $\\sqsubseteq _{\\mathrm {Te,y}}^{\\mathrm {x}}$ as kernel.",
"The function $\\mathbf {m}_{\\mathrm {Te,y}}^{\\lambda ,\\mathrm {x}}$ is a 1-bounded pseudometric on $\\mathbf {S}$ , with $\\sim _{\\mathrm {Te,y}}^{\\mathrm {x}}$ as kernel.",
"Figure: We use the tests o 1 ,o 2 o_1,o_2 to evaluate the distance between processes s,t,us,t,u in Fig.",
"wrt.",
"testing semantics.•\\bullet represents a generic configuration in the interaction system.In all upcoming examples we will consider only the tests and traces that are significant for the evaluations of the testing metrics.Example 3 Consider $t,u$ in Fig REF and their interactions with test $o_1$ in Fig REF .",
"Clearly, $(t,o_1)$ and $(u,o_1)$ have the same suprema success probabilities.",
"In fact, they both have a maximal resolution assigning probability 1 to the trace $ab$ , ie.",
"the only successful trace in the considered case.",
"As the same holds for all tests we get $\\mathbf {m}_{\\mathrm {Te,may}}^{\\omega ,\\mathrm {x}}(t,u) = 0$ .",
"Conversely, if we compare the infima success probabilities, we get $\\inf _{\\mathcal {Z}_{t,o_1} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o_1)} = 1$ since $(t,o_1)$ has only one maximal resolution corresponding to $(t,o_1)$ itself and that with probability 1 reaches $\\surd $ .",
"Still, $\\inf _{\\mathcal {Z}_{u,o_1} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(u,o_1)} = 0$ , given by the maximal resolution corresponding to $(u,o_1) \\stackrel{a}{{\\twoheadrightarrow }} \\mathrm {nil}$ .",
"Hence, we can infer $\\mathbf {m}_{\\mathrm {Te,must}}^{\\omega ,\\mathrm {x}}(t,u) = \\omega (o_1) \\cdot |1-0| = \\omega (o_1)$ .",
"$$ We can finally observe that both $\\mathbf {h}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}$ and $\\mathbf {m}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}$ are non-expansive.",
"Theorem 10 Let $\\omega :\\mathbf {O}\\rightarrow (0,1]$ and $\\mathrm {y} \\in \\lbrace \\mathrm {may},\\mathrm {must}, \\mathrm {mM}\\rbrace $ .",
"$\\mathbf {h}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}$ and $\\mathbf {m}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}$ are non-expansive.",
"The trace-by-trace approach In [7] it was proved that the may/must is fully backward compatible with the restricted class of processes only if the same restriction is applied to the class of tests, ie.",
"if we consider resp.",
"fully nondeterministic and fully probabilistic tests only.",
"This is due to the duplication ability of nondeterministic probabilistic tests.",
"However, by applying the trace-by-trace approach to testing semantics, we regain the full backward compatibility wrt.",
"all tests (cf.",
"[7]).",
"Definition 16 (Tbt-testing equivalence) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $(\\mathbf {O},\\mathcal {A},\\xrightarrow{}_O)$ an NPT, $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"We say that $s,t \\in \\mathbf {S}$ are in the tbt-testing preorder, written $s \\sqsubseteq _{\\mathrm {Te,tbt}}^{x}t$ , if for each $o \\in \\mathbf {O}$ and $\\alpha \\in \\mathcal {A}^{\\star }$ $\\text{for each } \\mathcal {Z}_{s,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o) \\text{ there is } \\mathcal {Z}_{t,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o) \\text{ st. } \\mathrm {Pr}(\\mathbf {SC}(z_{s,o},\\alpha )) = \\mathrm {Pr}(\\mathbf {SC}(z_{t,o},\\alpha ))$ .",
"Then, $s,t\\in \\mathbf {S}$ are tbt-testing equivalent, notation $s \\sim _{\\mathrm {Te,tbt}}^{x}t$ , iff $s \\sqsubseteq _{\\mathrm {Te,tbt}}^{x}t$ and $t \\sqsubseteq _{\\mathrm {Te,tbt}}^{x}s$ .",
"The definition of the tbt-testing metric naturally follows from Def.",
"REF .",
"Definition 17 (Tbt-testing metric) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $(\\mathbf {O},\\mathcal {A},\\xrightarrow{}_O)$ an NPT, $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"For each $o \\in \\mathbf {O}$ and $\\alpha \\in \\mathcal {A}^{\\star }$ , function $\\mathbf {h}_{\\mathrm {Te,tbt}}^{{o,\\alpha ,}\\lambda ,\\mathrm {x}} \\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ is defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Te,tbt}}^{{o,\\alpha ,}\\lambda ,\\mathrm {x}}(s,t) = \\lambda ^{|\\alpha |-1}\\; \\sup _{\\mathcal {Z}_{s,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o)}\\, \\inf _{\\mathcal {Z}_{t,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o)} \\, |\\mathrm {Pr}(\\mathbf {SC}(z_{s,o},\\alpha )) - \\mathrm {Pr}(\\mathbf {SC}(z_{t,o},\\alpha ))|$ The tbt-testing hemimetric and the tbt-testing metric are the functions $\\mathbf {h}_{\\mathrm {Te,tbt}}^{{}\\lambda ,\\mathrm {x}}, \\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}}\\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Te,tbt}}^{{}\\lambda ,\\mathrm {x}}(s,t) = \\sup _{o \\in \\mathbf {O}} \\; \\sup _{\\alpha \\in \\mathcal {A}^{\\star }}\\; \\mathbf {h}_{\\mathrm {Te,tbt}}^{{o,\\alpha ,}\\lambda ,\\mathrm {x}}(s,t)$ $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}}(s,t) = \\max \\lbrace \\mathbf {h}_{\\mathrm {Te,tbt}}^{{}\\lambda ,\\mathrm {x}}(s,t), \\mathbf {h}_{\\mathrm {Te,tbt}}^{{}\\lambda ,\\mathrm {x}}(t,s) \\rbrace $ .",
"Theorem 11 Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"Then: The function $\\mathbf {h}_{\\mathrm {Te,tbt}}^{{}\\lambda ,\\mathrm {x}}$ is a 1-bounded hemimetric on $\\mathbf {S}$ , with $\\sqsubseteq _{\\mathrm {Te,tbt}}^{x}$ as kernel.",
"The function $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}}$ is a 1-bounded pseudometric on $\\mathbf {S}$ , with $\\sim _{\\mathrm {Te,tbt}}^{x}$ as kernel.",
"Example 4 Consider $s,t$ in Fig.",
"REF and their interactions with test $o_2$ in Fig.",
"REF .",
"By the same reasoning detailed in the first paragraph of Sect.",
"REF , we get $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {det}}(s,t)=\\lambda \\cdot 0.5$ and $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {rand}}(s,t) = 0$ .",
"$$ When the tbt-approach is used to define testing metrics, we get a refinement of the non-expansiveness property to strict non-expansiveness.",
"Theorem 12 All distances $\\mathbf {h}_{\\mathrm {Te,tbt}}^{{}\\lambda ,\\mathrm {det}}$ , $\\mathbf {h}_{\\mathrm {Te,tbt}}^{{}\\lambda ,\\mathrm {rand}}$ , $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {det}}$ , $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {rand}}$ are strictly non-expansive.",
"The supremal probabilities approach If we focus on verification, we can use the testing semantics to verify whether a process will behave as intended by its specification in all possible environments, as modeled by the interaction with the tests.",
"Informally, we could see each test as a set of requests of the environment to the system: the ones ending in the success state are those that must be answered.",
"The interaction of the specification with the test then tells us whether the system is able to provide those answers.",
"Thus, an implementation has to guarantee at least all the answers provided by the specification.",
"For this reason we decided to introduce also a supremal probabilities variant of testing semantics: for each test and for each trace we compare the suprema wrt.",
"all resolutions of nondeterminism of the probabilities of processes to reach success by performing the considered trace.",
"Definition 18 ($\\bigsqcup $ -testing equivalence) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $(\\mathbf {O},\\mathcal {A},\\xrightarrow{}_O)$ an NPT and $\\mathrm {x} \\in \\lbrace \\mathrm {det}, \\mathrm {rand}\\rbrace $ .",
"We say that $s,t \\in \\mathbf {S}$ are in the $\\bigsqcup $ -testing preorder, written $s \\sqsubseteq _{\\mathrm {Te,}\\sqcup }^{\\mathrm {x}}t$ , if for each $o \\in \\mathbf {O}$ and $\\alpha \\in \\mathcal {A}^{\\star }$ $\\sup _{\\mathcal {Z}_{s,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o)} \\mathrm {Pr}(\\mathbf {SC}(z_{s,o},\\alpha )) \\le \\sup _{\\mathcal {Z}_{t,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o)} \\mathrm {Pr}(\\mathbf {SC}(z_{t,o},\\alpha ))$ .",
"Then, $s,t \\in \\mathbf {S}$ are $\\bigsqcup $ -testing equivalent, notation $s \\sim _{\\mathrm {Te,}\\sqcup }^{\\mathrm {x}}t$ , iff $s \\sqsubseteq _{\\mathrm {Te,}\\sqcup }^{\\mathrm {x}}t$ and $t \\sqsubseteq _{\\mathrm {Te,}\\sqcup }^{\\mathrm {x}}s$ .",
"We obtain the $\\bigsqcup $ -testing metric as a direct adaptation to tests of Definition REF .",
"Definition 19 ($\\bigsqcup $ -testing metric) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $(\\mathbf {O},\\mathcal {A},\\xrightarrow{}_O)$ an NPT, $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"For each $o \\in \\mathbf {O}$ , $\\alpha \\in \\mathcal {A}^{\\star }$ , the function $\\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{o,\\alpha ,}\\lambda ,\\mathrm {x}} \\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ is defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{o,\\alpha ,}\\lambda ,\\mathrm {x}}(s,t) = \\max \\Big \\lbrace 0,\\lambda ^{|\\alpha |-1} \\Big (\\sup _{\\mathcal {Z}_{s,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o)} \\!\\!\\!\\!\\!\\mathrm {Pr}(\\mathbf {SC}(z_{s,o},\\alpha )) - \\sup _{\\mathcal {Z}_{t,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o)}\\!\\!\\!\\!\\!",
"\\mathrm {Pr}(\\mathbf {SC}(z_{t,o},\\alpha ))\\Big )\\Big \\rbrace .$ The $\\bigsqcup $ -testing hemimetric and the $\\bigsqcup $ -testing metric are the functions $\\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}, \\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}\\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}(s,t) = \\sup _{o \\in \\mathbf {O}} \\; \\sup _{\\alpha \\in \\mathcal {A}^{\\star }}\\; \\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{o,\\alpha ,}\\lambda ,\\mathrm {x}}(s,t)$ ; $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}(s,t) = \\max \\lbrace \\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}(s,t), \\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}(t,s) \\rbrace $ .",
"Theorem 13 Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS and $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"Then: The function $\\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}$ is a 1-bounded hemimetric on $\\mathbf {S}$ , with $\\sqsubseteq _{\\mathrm {Te,}\\sqcup }^{\\mathrm {x}}$ as kernel.",
"The function $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}$ is a 1-bounded pseudometric on $\\mathbf {S}$ , with $\\sim _{\\mathrm {Te,}\\sqcup }^{\\mathrm {x}}$ as kernel.",
"Finally, we can show that both $\\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}$ and $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}$ are strictly non-expansive.",
"Theorem 14 All distances $\\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{}\\lambda ,\\mathrm {det}}$ , $\\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{}\\lambda ,\\mathrm {rand}}$ , $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {det}}$ , $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {rand}}$ are strictly non-expansive.",
"Remark 2 For all distances $d$ considered in Thms.",
"REF , REF , REF and processes $z_s,z_t$ in Fig.",
"REF , with $\\lambda =1$ , we have $d(z_s , z_t) = 0.5$ and $d(z_s \\parallel z_s , z_t \\parallel z_t) = 0.75 = 0.5 + 0.5 - 0.5 \\cdot 0.5$ .",
"Hence, the upper bounds to the distance between composed processes provided in Thms.",
"REF and REF are tight.",
"We leave as a future work the analogous result for distances considered in Thm.",
"REF .",
"Comparing the distinguishing power of testing metrics Figure: The spectrum of trace and testing (hemi)metrics.d→d ' d \\rightarrow d^{\\prime } stands for d>d ' d > d^{\\prime }.We present only the general form with 𝐝∈{𝐡,𝐦}\\mathbf {d} \\in \\lbrace \\mathbf {h},\\mathbf {m}\\rbrace as the relations among the hemimetrics are the same wrt.",
"those among the metrics.The complete spectrum can be obtained by relating each metric with the respective hemimetric.We study the distinguishing power of the testing metrics presented in this section and the trace metrics defined in Sect.",
", thus obtaining the spectrum in Fig.",
"REF .",
"Firstly, we compare the expressiveness of the testing metrics wrt.",
"the chosen class of schedulers.",
"The distinguishing power of testing metrics based on may-must and supremal probabilities approaches is not influenced by this choice.",
"Differently, in the tbt approach, the distances evaluated on deterministic schedulers are more discriminating than their analogues on randomized schedulers.",
"Theorem 15 Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\lambda \\in (0,1]$ , $\\omega \\colon \\mathbf {O}\\rightarrow (0,1]$ $\\mathrm {y} \\in \\lbrace \\mathrm {may},\\mathrm {must}, \\mathrm {mM}\\rbrace $ and $\\mathbf {d} \\in \\lbrace \\mathbf {h},\\mathbf {m}\\rbrace $ : $\\begin{array}{llll}1.\\, \\mathbf {d}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {rand}} = \\mathbf {d}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {det}} &2.\\, \\mathbf {d}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {rand}} < \\mathbf {d}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {det}} &3.\\, \\mathbf {d}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {rand}} = \\mathbf {d}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {det}}\\end{array}$ From Thm.",
"REF , by using the kernel relations in Thms.",
"REF and REF , we regain relations $\\sim _{\\mathrm {Te,may}}^{\\mathrm {rand}}= \\sim _{\\mathrm {Te,may}}^{\\mathrm {det}}$ , $\\sim _{\\mathrm {Te,must}}^{\\mathrm {rand}}= \\sim _{\\mathrm {Te,must}}^{\\mathrm {det}}$ , $\\sim _{\\mathrm {Te,mM}}^{\\mathrm {rand}}= \\sim _{\\mathrm {Te,mM}}^{\\mathrm {det}}$ , $\\sim _{\\mathrm {Te,tbt}}^{\\mathrm {det}}\\subset \\sim _{\\mathrm {Te,tbt}}^{\\mathrm {rand}}$ , and their analogues on preorders, proved in [7].",
"From Thm.",
"REF we get $\\sqsubseteq _{\\mathrm {Te,}\\sqcup }^{\\mathrm {rand}}= \\sqsubseteq _{\\mathrm {Te,}\\sqcup }^{\\mathrm {det}}$ and $\\sim _{\\mathrm {Te,}\\sqcup }^{\\mathrm {rand}}= \\sim _{\\mathrm {Te,}\\sqcup }^{\\mathrm {det}}$ .",
"The strictness of the inequality in Thm.",
"REF .2, is witnessed by processes $s,t$ in Fig REF and their interactions with the test $o_2$ in Fig REF .",
"The same reasoning applied in the first paragraph of Sect.",
"REF to obtain $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {det}}(s,t) = \\lambda \\cdot 0.5$ and $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {rand}}(s,t) = 0$ , gives $\\mathbf {h}_{\\mathrm {Te,tbt}}^{{o_2,}\\lambda ,\\mathrm {det}}(t,s) = \\lambda \\cdot 0.5 = \\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {det}}{}(s,t)$ and $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {rand}}{}(s,t) = 0$ .",
"We proceed to compare the expressiveness of each metric wrt.",
"the other semantics.",
"Our results are fully compatible with the spectrum on probabilistic relations presented in [7].",
"Theorem 16 Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\lambda \\in (0,1]$ , $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ and $\\mathbf {d} \\in \\lbrace \\mathbf {h},\\mathbf {m}\\rbrace $ : $\\begin{array}{llll}1.\\, \\mathbf {d}_{\\mathrm {Te,may}}^{\\omega ,\\mathrm {x}} < \\mathbf {d}_{\\mathrm {Te,mM}}^{\\omega ,\\mathrm {x}} &2.\\, \\mathbf {d}_{\\mathrm {Te,must}}^{\\omega ,\\mathrm {x}} < \\mathbf {d}_{\\mathrm {Te,mM}}^{\\omega ,\\mathrm {x}} &3.\\, \\mathbf {d}_{\\mathrm {Te,}\\sqcup }^{1,\\mathrm {x}} < \\mathbf {d}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}} &4.\\, \\mathbf {d}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}} < \\mathbf {d}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}} \\\\5.\\, \\mathbf {d}_{\\mathrm {Tr,dis}}^{1,\\mathrm {rand}} < \\mathbf {d}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}} &6.\\, \\mathbf {d}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {x}} < \\mathbf {d}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}} &7.\\, \\mathbf {d}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {x}} < \\mathbf {d}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}\\end{array}$ The following Examples prove the strictness of the inequalities in Thm.",
"REF and the non comparability of the (hemi)metrics as shown in Fig.",
"REF .",
"For simplicity, we consider only the cases of the metrics.",
"Figure: Processes s,ts,t and their interaction systems with the test o 2 o_2 in Fig.",
".Example 5 Non comparability of $\\mathbf {m}_{\\mathrm {Te,may}}^{\\omega ,\\mathrm {x}}$ with $\\mathbf {m}_{\\mathrm {Te,must}}^{\\omega ,\\mathrm {x}}$.",
"In Ex.",
"REF we showed that for $t,u$ in Fig.",
"REF from their interaction with the test $o_1$ in Fig.",
"REF we obtain that $\\mathbf {m}_{\\mathrm {Te,must}}^{\\omega ,\\mathrm {x}}(t,u) = \\omega (o_1)$ , whereas one can easily check that $\\mathbf {m}_{\\mathrm {Te,may}}^{\\omega ,\\mathrm {x}}(t,u) = 0$ .",
"Consider now $s,t$ and their interactions in Fig.",
"REF with the test $o_2$ from Fig.",
"REF .",
"Clearly, we have $\\sup _{\\mathcal {Z}_{s,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o)}\\mathrm {Pr}(\\mathbf {SC}(z_{s,o})) = 1$ and $\\sup _{\\mathcal {Z}_{t,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o)}\\mathrm {Pr}(\\mathbf {SC}(z_{t,o})) = 0.3$ and thus $\\mathbf {m}_{\\mathrm {Te,may}}^{\\omega ,\\mathrm {x}}(s,t) = 0.7 \\cdot \\omega (o_2)$ .",
"Conversely, if we consider infima success probabilities, we have $\\inf _{\\mathcal {Z}_{s,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o)}\\mathrm {Pr}(\\mathbf {SC}(z_{s,o})) = 0$ and $\\sup _{\\mathcal {Z}_{t,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o)}\\mathrm {Pr}(\\mathbf {SC}(z_{t,o})) = 0.3$ .",
"Thus, $\\mathbf {m}_{\\mathrm {Te,must}}^{\\omega ,\\mathrm {x}}(s,t) = 0.3\\cdot \\omega (o_2)$ .",
"$$ Figure: Processes s,ts,t are such that 𝐝 Te , tbt 1,x (s,t)=0\\mathbf {d}_{\\mathrm {Te,tbt}}^{1,\\mathrm {x}}(s,t) = 0 and 𝐝 Te , must 1,x (s,t)=0.5\\mathbf {d}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0.5, as witnessed by the test o 1/2 o^{1/2}.Example 6 Non comparability of $\\mathbf {m}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}$ with $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{1,\\mathrm {x}}$ , $\\mathbf {m}_{\\mathrm {Te,tbt}}^{1,\\mathrm {x}}$ , $\\mathbf {m}_{\\mathrm {Tr,dis}}^{1,\\mathrm {x}}$ , $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{1,\\mathrm {x}}$ and $\\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{1,\\mathrm {x}}$.",
"We start with $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{1,\\mathrm {x}}$ .",
"Form Ex.",
"REF we know that for $t,u$ in Fig.",
"REF it holds $\\mathbf {m}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}(t,u) = \\mathbf {1}$ .",
"Since both $t$ and $u$ have maximal resolutions giving probability 1 to either $ab$ or $ac$ , we get $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{1,\\mathrm {x}}(t,u) =0$ .",
"Consider now $s,t$ in Fig.",
"REF .",
"In Ex.",
"REF we showed that $\\mathbf {m}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0.3$ .",
"From the interaction systems in Fig.",
"REF , by considering the superma success probabilities of trace $ac$ , we obtain that $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{1,\\mathrm {x}} = 0.4$ .",
"Next we deal with the tbt-testing metrics.",
"Consider $s,t$ in Fig.",
"REF and the family of tests $O = \\lbrace o^p \\mid p \\in (0,1)\\rbrace $ , each duplicating the actions $b$ in the interaction with $s$ and $t$ .",
"For each $o^p \\in O$ , $\\inf _{\\mathcal {Z}_{s,o^p} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o^p)} \\mathrm {Pr}(\\mathbf {SC}(z_{s,o^p})) = 0$ and $\\inf _{\\mathcal {Z}_{t,o^p} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o^p)} \\mathrm {Pr}(\\mathbf {SC}(z_{t,o^p})) = \\min \\lbrace p,1-p\\rbrace $ , thus giving $\\mathbf {h}_{\\mathrm {Te,must}}^{o^p,\\mathbf {1},\\mathrm {x}}(t,s) = \\min \\lbrace p,1-p\\rbrace $ .",
"One can then easily check that $\\mathbf {m}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}(s,t) = \\sup _{p \\in (0,1)}\\min \\lbrace p,1-p\\rbrace = 0.5$ .",
"Conversely, as the tbt-testing metric compares the success probabilities related to the execution of a single trace per time, we get $\\mathbf {m}_{\\mathrm {Te,tbt}}^{1,\\mathrm {x}}(s,t) = 0$ .",
"Notice that in the case of randomized schedulers, all the randomized resolutions for $t,o^p$ combining the two $a$ -moves can be matched by $s,o^p$ by combining the $b$ -moves and vice versa.",
"Consider now $s,t$ in Fig.",
"REF .",
"Even under randomized schedulers, the tbt-testing distance on them is given by the difference in the success probability of the trace $ac$ (or equivalently $ad$ ) and thus $\\mathbf {m}_{\\mathrm {Te,tbt}}^{1,\\mathrm {x}}(s,t) = 0.4$ .",
"However, we have already showed that $\\mathbf {m}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0.3$ .",
"Finally, we consider the case of trace distances.",
"Consider $t,u$ in Fig.",
"REF .",
"Clearly, $\\mathbf {m}_{\\mathrm {Tr,dis}}^{1,\\mathrm {x}}(t,u) = \\mathbf {m}_{\\mathrm {Tr,tbt}}^{1,\\mathrm {x}}(t,u) = \\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{1,\\mathrm {x}}(t,u) = 0$ .",
"However, in Ex.",
"REF we showed that $\\mathbf {m}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}(t,u) = 1$ .",
"Consider now $s,t$ in Fig.",
"REF .",
"We have that $\\mathbf {m}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0.3$ , but $\\mathbf {m}_{\\mathrm {Tr,dis}}^{1,\\mathrm {x}}(s,t) = 0.7$ and $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{1,\\mathrm {x}}(s,t) = \\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{1,\\mathrm {x}}(s,t) = 0.4$ .",
"$$ Example 7 Non comparability of $\\mathbf {m}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}}$ with $\\mathbf {m}_{\\mathrm {Te,tbt}}^{1,\\mathrm {x}}$ , $\\mathbf {m}_{\\mathrm {Tr,dis}}^{1,\\mathrm {det}}$ and $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{1,\\mathrm {det}}$.",
"For the tbt-testing metrics, consider $s,t$ in Fig.",
"REF .",
"In Ex.",
"REF we showed that $\\mathbf {m}_{\\mathrm {Te,tbt}}^{1,\\mathrm {x}}(s,t) = 0$ .",
"However, the same reasoning giving $\\mathbf {m}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0.5$ , can be applied on suprema success probabilities thus giving $\\mathbf {m}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0.5$ .",
"Consider now $t,u$ in Fig.",
"REF and their interactions with test $o_1$ in Fig.",
"REF .",
"As we consider maximal resolutions only, for both classes of schedulers, the success probability of trace $ab$ evaluates to 1 on $t,o_1$ , whereas on $u,o_1$ it evaluates to 0, due to the maximal resolution corresponding to the rightmost $a$ -branch.",
"Hence $\\mathbf {m}_{\\mathrm {Te,tbt}}^{1,\\mathrm {x}}(t,u) = \\lambda $ , whereas one can easily check that $\\mathbf {m}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}}(t,u) = 0$ .",
"We now proceed to the case of trace distances.",
"For $s,t$ in Fig.",
"REF , we showed that $\\mathbf {m}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0.5$ .",
"However, as both processes have a single resolution each allowing them to execute either trace $abc$ or $abd$ , we can infer that $\\mathbf {m}_{\\mathrm {Tr,dis}}^{1,\\mathrm {x}}(s,t) = \\mathbf {m}_{\\mathrm {Tr,tbt}}^{1,\\mathrm {x}}(s,t) = 0$ .",
"Notice, that this also shows the strictness of the relation $\\mathbf {m}_{\\mathrm {Tr,dis}}^{1,\\mathrm {rand}} < \\mathbf {m}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}}$ .",
"Consider now $s,t$ in Fig.",
"REF .",
"As discussed in Sect.",
"REF we have that $\\mathbf {m}_{\\mathrm {Tr,dis}}^{1,\\mathrm {det}} \\ge \\mathbf {m}_{\\mathrm {Tr,tbt}}^{1,\\mathrm {det}}(s,t) = 0.5$ .",
"However, one can easily check that $\\mathbf {m}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0$ .",
"$$ Example 8 Strictness of $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{1,\\mathrm {x}} < \\mathbf {m}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}}$.",
"Consider $s,t$ in Fig.",
"REF .",
"In Ex.",
"REF we have shown that $\\mathbf {m}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0.7$ .",
"However, since the supremal probability approach to testing proceeds in a trace-by-trace fashion, the $\\sqcup $ -testing distance is given by the difference in the success probability of the trace $ac$ (or $ad$ ) and thus $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{1,\\mathrm {x}}(s,t) = 0.4$ .",
"$$ Example 9 Strictness of $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}< \\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}}$.",
"We stress that this relation is due to the restriction to maximal resolutions, necessary to reason on testing semantics.",
"Consider now $t,u$ in Fig.",
"REF and their interactions with test $o_1$ in Fig.REF .",
"In Ex.REF we have shown that $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}}(t,u) = \\lambda $ .",
"However, one can easily check that $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}(t,u) = 0$ .",
"$$ Example 10 Strictness of $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {x}}< \\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}}$ and of $\\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {x}}< \\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}$.",
"For $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {x}} < \\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}}$ consider $t,u$ in Fig.",
"REF and the test $o_1$ in Fig.",
"REF , by which we get $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {x}}(t,u) = 0$ and $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}}(t,u) = \\lambda $ .",
"Similarly, for $\\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {x}} < \\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}$ consider $s,t$ in Fig.",
"REF with $\\varepsilon _1=\\varepsilon _2=0$ .",
"We have $\\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {x}}(s,t) = 0$ and $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}(s,t) = \\lambda \\cdot 0.5$ , given by the test $o$ corresponding to the leftmost branch of $s$ .",
"$$ Related and future work Trace metrics have been thoroughly studied on quantitative systems, as testified by the spectrum of distances, defined as the generalization of a chosen trace distance, in [17] and the one on Metric Transition Systems (MTSs) in [1].",
"The great variety in these models and the PTSs prevent us to compare the obtained results in detail.",
"Notably, in [1] the trace distance is based on a propositional distance defined over valuations of atomic propositions that characterize the MTS.",
"If on one side such valuation could play the role of the probability distributions in the PTS, it is unclear whether we could combine the ground distance on atomic propositions and the propositional distance, to obtain trace distances comparable to ours.",
"In [3], [13] trace metrics on Markov Chains (MCs) are defined as total variation distances on the cones generated by traces.",
"As in MCs probability depends only on the current state and not on nondeterminism, our quantification over resolutions would be trivial on MCs, giving a total variation distance.",
"Although ours is the first proposal of a metric expressing testing semantics, testing equivalences for probabilistic processes have been studied also in [15], [4], [5].",
"In detail, [15] proposed notions of probabilistic may/must testing for a Kleisli lifting of the PTS model, ie.",
"the transition relation is lifted to a relation $(\\rightarrow )^{\\dagger } \\subseteq (\\Delta (\\mathbf {S}) \\times \\mathcal {A}\\times \\Delta (\\mathbf {S}))$ taking distributions over processes to distributions over processes.",
"Again, the disparity in the two models prevents us from thoroughly comparing the proposed testing relations.",
"As future work, we aim to extend the spectrum of metrics to (bi)simulation metrics [16] and to metrics on different semantic models, and to study their logical characterizations and compositional properties on the same line of [9], [10], [11].",
"Further, we aim to provide efficient algorithms for the evaluation of the proposed metrics and to develop a tool for quantitative process verification: we will use the distance between a process and its specification to quantify how much that process satisfies a given property.",
"Acknowledgements I wish to thank Michele Loreti and Simone Tini for fruitful discussions, and the anonymous referees for their valuable comments and suggestions that helped to improve the paper."
],
[
"Background",
"PTSs [25] are a very general model combining LTSs [23] and discrete time Markov chains [19], to model reactive behavior, nondeterminism and probability.",
"In a PTS, the state space is given by a set $\\mathbf {S}$ of $\\emph {processes}$ , ranged over by $s,t,\\dots $ and transition steps take processes to probability distributions over processes.",
"Probability distributions over $\\mathbf {S}$ are mappings $\\pi \\colon \\mathbf {S}\\rightarrow [0,1]$ with $\\sum _{s \\in \\mathbf {S}} \\pi (s) = 1$ .",
"By $\\Delta (\\mathbf {S})$ we denote the set of all distributions over $\\mathbf {S}$ , ranged over by $\\pi ,\\pi ^{\\prime },\\dots $ For $\\pi \\in \\Delta (\\mathbf {S})$ , the support of $\\pi $ is the set $\\mathsf {supp}(\\pi ) = \\lbrace s \\in \\mathbf {S}\\mid \\pi (s) >0\\rbrace $ .",
"We consider only distributions with finite support.",
"For $s \\in \\mathbf {S}$ , we let $\\delta _s$ denote the Dirac distribution on $s$ defined by $\\delta _s(s)= 1$ and $\\delta _s(t)=0$ for $t\\ne s$ .",
"Definition 1 (PTS, [25]) A nondeterministic probabilistic labeled transition system (PTS) is a triple $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ where: [(i)] $\\mathbf {S}$ is a countable set of processes, $\\mathcal {A}$ is a countable set of actions, and $\\xrightarrow{} \\subseteq {\\mathbf {S}\\times \\mathcal {A}\\times \\Delta (\\mathbf {S})}$ is a transition relation.",
"We write $s\\xrightarrow{}\\pi $ for $(s,a,\\pi ) \\in \\xrightarrow{}$ , $s \\xrightarrow{} $ if there is a distribution $\\pi $ with $s \\xrightarrow{} \\pi $ , and $s \\mathrel {{\\xrightarrow{}}\\makebox{[}0em][r]{\\lnot \\hspace{8.5pt}}}{\\!",
"}$ otherwise.",
"A PTS is fully nondeterministic if every transition has the form $s\\xrightarrow{}\\delta _t$ for some $t \\in \\mathbf {S}$ .",
"A PTS is fully probabilistic if at most one transition is enabled for each process.",
"$s \\in \\mathbf {S}$ is image-finite [20] if for each $a \\in \\mathcal {A}$ the number of $a$ -labeled transitions enabled for $s$ is finite.",
"We consider only image-finite processes.",
"Definition 2 (Parallel composition) Let $P_1 = (S_1, \\mathcal {A}, \\xrightarrow{}_1)$ and $P_2 = (S_2, \\mathcal {A}, \\xrightarrow{}_2)$ be two PTSs.",
"The (CSP-like [21]) synchronous parallel composition of $P_1$ and $P_2$ is the PTS $P_1 \\parallel P_2 = (S_1 \\times S_2, \\mathcal {A}, \\xrightarrow{})$ , where $\\xrightarrow{} \\subseteq (S_1 \\times S_2) \\times \\mathcal {A}\\times \\Delta (S_1 \\times S_2)$ is such that $(s_1,s_2) \\xrightarrow{} \\pi $ if and only if $s_1 \\xrightarrow{}_1 \\pi _1$ , $s_2 \\xrightarrow{}_2 \\pi _2$ and $\\pi (s_1^{\\prime },s_2^{\\prime }) = \\pi _1(s^{\\prime }_1) \\cdot \\pi _2(s^{\\prime }_2)$ for all $(s^{\\prime }_1,s^{\\prime }_2) \\in S_1 \\times S_2$ .",
"We proceed to recall some notions, mostly from [5], [7], [6], necessary to reason on trace and testing semantics.",
"A computation is a weighted sequence of process-to-process transitions.",
"Definition 3 (Computation) A computation from $s_0$ to $s_n$ has the form $\\begin{array}{c}\\hspace{99.58464pt}c := s_0 \\stackrel{a_1}{{\\twoheadrightarrow }} s_1 \\stackrel{a_2}{{\\twoheadrightarrow }} s_2 \\dots s_{n-1} \\stackrel{a_n}{{\\twoheadrightarrow }} s_n\\end{array}$ where, for all $i = 1,\\dots ,n$ , there is a transition $s_{i-1} \\xrightarrow{} \\pi _i$ with $s_i \\in \\mathsf {supp}(\\pi _i)$ .",
"Note that $\\pi _i(s_i)$ is the execution probability of step $s_{i-1} \\stackrel{a_i}{{\\twoheadrightarrow }} s_i$ conditioned on the selection of the transition $s_{i-1} \\xrightarrow{} \\pi _i$ at $s_{i-1}$ .",
"We denote by $\\mathrm {Pr}(c) = \\prod _{i = 1}^{n} \\pi _i(s_i)$ the product of the execution probabilities of the steps in $c$ .",
"A computation $c$ from $s$ is maximal if it is not a proper prefix of any other computation from $s$ .",
"We denote by $s)$ (resp.",
"${\\max }(s)$ ) the set of computations (resp.",
"maximal computations) from $s$ .",
"For any $s)$ , we define $\\mathrm {Pr}( = \\sum _{c \\in \\mathrm {Pr}(c) whenever none of the computations in is a proper prefix of any of the others.",
"}We denote by $ A$ the set of \\emph {finite traces} in $ A$ and write $ e$ for the empty trace.We say that a computation is \\emph {compatible} with the trace $ A$ if{f} the sequence of actions labeling the computation steps is equal to $$.We denote by $ s,) s)$ the set of computations from $ s$ that are compatible with $$, and by $ (s,)$ the set $ (s,) = (s) s,)$.$ To express linear semantics we need to evaluate and compare the probability of particular sequences of events to occur.",
"As in PTSs this probability highly depends also on nondeterminism, schedulers [24], [28], [18] (or adversaries) resolving it become fundamental.",
"They can be classified into two main classes: deterministic and randomized schedulers [24].",
"For each process, a deterministic scheduler selects exactly one transition among the possible ones, or none of them, thus treating all internal nondeterministic choices as distinct.",
"Randomized schedulers allow for a convex combination of the equally labeled transitions.",
"The resolution given by a deterministic scheduler is a fully probabilistic process, whereas from randomized schedulers we get a fully probabilistic process with combined transitions [26].",
"Definition 4 (Resolutions) Let $P = (\\mathbf {S}, \\mathcal {A},\\xrightarrow{})$ be a PTS and $s \\in \\mathbf {S}$ .",
"We say that a PTS $\\mathcal {Z}= (Z,\\mathcal {A},\\xrightarrow{}_{\\mathcal {Z}})$ is a deterministic resolution for $s$ iff there exists a function $\\mathrm {corr}_{\\mathcal {Z}} \\colon Z \\rightarrow \\mathbf {S}$ such that $s = \\mathrm {corr}_{\\mathcal {Z}}(z_s)$ for some $z_s \\in Z$ and moreover: (i) If $z \\xrightarrow{}_{\\mathcal {Z}} \\pi $ , then $\\mathrm {corr}_{\\mathcal {Z}}(z) \\xrightarrow{} \\pi ^{\\prime }$ with $\\pi (z^{\\prime }) = \\pi ^{\\prime }(\\mathrm {corr}_{\\mathcal {Z}}(z^{\\prime }))$ for all $z^{\\prime } \\in Z$ .",
"(ii) If $z \\xrightarrow{}_{\\mathcal {Z}} \\pi _1$ and $z \\xrightarrow{}_{\\mathcal {Z}} \\pi _2$ then $a_1 = a_2$ and $\\pi _1 = \\pi _2$ .",
"Conversely, we say that $\\mathcal {Z}$ is a randomized resolution for $s$ if item (i) is replaced by (i)' If $z \\xrightarrow{}_{\\mathcal {Z}} \\pi $ , then there are $n \\in \\mathbb {N}$ , $\\lbrace p_i \\in (0,1] \\mid \\sum _{i = 1}^n p_i = 1\\rbrace $ and $\\lbrace \\mathrm {corr}_{\\mathcal {Z}}(z) \\xrightarrow{} \\pi _i \\mid 1 \\le i \\le n\\rbrace $ s.t.",
"$\\pi (z^{\\prime }) = \\sum _{i = 1}^n p_i \\cdot \\pi _i(\\mathrm {corr}_{\\mathcal {Z}}(z^{\\prime }))$ for all $z^{\\prime } \\in Z$ .",
"Then, $\\mathcal {Z}$ is maximal iff it cannot be further extended in accordance with the graph structure of $P$ and the constraints above.",
"For $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ , we denote by $\\mathrm {Res}^{\\mathrm {x}}(s)$ the set of resolutions for $s$ and by $\\mathrm {Res}^{\\mathrm {x}}_{\\max }(s)$ the subset of maximal resolutions for $s$ .",
"We conclude this section by recalling the mathematical notions of hemimetric and pseudometric.",
"A 1-bounded pseudometric on $\\mathbf {S}$ is a function $d \\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ s.t.",
": [(i)] $d(s,s) =0$ , $d(s,t) = d(t,s)$ , $d(s,t) \\le d(s,u) + d(u,t)$ , for $s,t,u \\in \\mathbf {S}$ .",
"Then, $d$ is a hemimetric if it satisfies (REF ) and (REF ).",
"The kernel of a (hemi,pseudo)metric $d$ on $\\mathbf {S}$ the set of pairs of elements in $\\mathbf {S}$ which are at distance 0, namely $ker(d) = \\lbrace (s,t) \\in \\mathbf {S}\\times \\mathbf {S}\\mid d(s,t) = 0\\rbrace $ .",
"Non-expansiveness [16] of a (hemi,pseudo)metric is the quantitative analogue to the (pre)congruence property.",
"Here we propose also a stronger notion, called strict non-expansiveness that gives tighter bounds on the distance of processes composed in parallel.",
"Definition 5 ((Strict) non-expansiveness) Let $d$ be a (hemi,pseudo)metric on $\\mathbf {S}$ .",
"Following [16], we say that $d$ is non-expansive wrt.",
"the parallel composition operator if and only if for all $s_1,s_2,t_1,t_2 \\in \\mathbf {S}$ we have $d(s_1 \\parallel s_2, t_1 \\parallel t_2) \\le d(s_1,t_1) + d(s_2,t_2)$ .",
"Moreover, we say that $d$ is strictly non-expansive if $d(s_1 \\parallel s_2, t_1 \\parallel t_2) \\le d(s_1,t_1) + d(s_2,t_2) - d(s_1,t_1) \\cdot d(s_2,t_2)$ .",
"Finally, we remark that, as elsewhere in the literature, throughout the paper we may use the term metric in place of pseudometric.",
"Metrics for traces In this Section, we define the metrics measuring the disparities in process behavior wrt.",
"trace semantics.",
"We consider three approaches to the combination of nondeterminism and probability: the trace distribution, the trace-by-trace and the supremal probabilities approach.",
"In defining the behavioral distances, we assume a discount factor $\\lambda \\in (0,1]$, which allows us to specify how much the behavioral distance of future transitions is taken into account [2], [16].",
"The discount factor $\\lambda =1$ expresses no discount, so that the differences in the behavior between $s,t \\in \\mathbf {S}$ are considered irrespective of after how many steps they can be observed.",
"The trace distribution approach In [24] the observable events characterizing the trace semantics are trace distributions, ie.",
"probability distributions over traces.",
"Processes $s,t \\in \\mathbf {S}$ are trace distribution equivalent if, for any resolution for $s$ there is a resolution for $t$ exhibiting the same trace distribution, ie.",
"the execution probability of each trace in the two resolutions is exactly the same, and vice versa.",
"Definition 6 (Trace distribution equivalence [24], [4]) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"Processes $s,t \\in \\mathbf {S}$ are in the trace distribution preorder, written $s \\sqsubseteq _{\\mathrm {Tr,dis}}^{\\mathrm {x}}t$ , if: $\\text{for each } \\mathcal {Z}_s \\in \\mathrm {Res}^{\\mathrm {x}}(s) \\text{ there is } \\mathcal {Z}_t \\in \\mathrm {Res}^{\\mathrm {x}}(t) \\text{ s.t.\\ for each } \\alpha \\in \\mathcal {A}^{\\star } \\colon \\mathrm {Pr}(z_s,\\alpha )) = \\mathrm {Pr}(z_t,\\alpha ))$ .",
"Then, $s,t$ are trace distribution equivalent, notation $s \\sim _{\\mathrm {Tr,dis}}^{\\mathrm {x}}t$ , iff $s \\sqsubseteq _{\\mathrm {Tr,dis}}^{\\mathrm {x}}t$ and $t \\sqsubseteq _{\\mathrm {Tr,dis}}^{\\mathrm {x}}s$ .",
"The quantitative analogue to trace distribution equivalence is based on the evaluation of the differences in the trace distributions of processes: the distance between processes $s,t$ is set to $\\varepsilon \\ge 0$ if, for any resolution for $s$ there is a resolution for $t$ exhibiting a trace distribution differing at most by $\\varepsilon $, meaning that the execution probability of each trace in the two resolutions differs by at most $\\varepsilon $, and vice versa.",
"Definition 7 (Trace distribution metric) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"The trace distribution hemimetric and the trace distribution metric are the functions $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {x}}, \\mathbf {m}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {x}}\\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {x}}(s,t) = \\sup _{\\mathcal {Z}_s \\in \\mathrm {Res}^{\\mathrm {x}}(s)}\\, \\inf _{\\mathcal {Z}_t \\in \\mathrm {Res}^{\\mathrm {x}}(t)}\\, \\sup _{\\alpha \\in \\mathcal {A}^{\\star }} \\lambda ^{|\\alpha |-1}|\\mathrm {Pr}(z_s,\\alpha )) - \\mathrm {Pr}(z_t,\\alpha ))|$ $\\mathbf {m}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {x}}(s,t) = \\max \\lbrace \\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {x}}(s,t),\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {x}}(t,s)\\rbrace $ .",
"We observe that the expression $\\sup _{\\alpha \\in \\mathcal {A}^{\\star }} \\lambda ^{|\\alpha |-1}|\\mathrm {Pr}(z_s,\\alpha )) - \\mathrm {Pr}(z_t,\\alpha ))|$ used in Definition REF corresponds to the (weighted) total variation distance between the trace distributions given by the two resolutions $\\mathcal {Z}_s$ and $\\mathcal {Z}_t$ .",
"An equivalent formulation is given in [27], [12] via the Kantorovich lifting of the discrete metric over traces.",
"We now state that trace distribution hemimetrics and metrics are well-defined and that their kernels are the trace distribution preorders and equivalences, respectively.",
"Theorem 1 Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"Then: The function $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {x}}$ is a 1-bounded hemimetric on $\\mathbf {S}$ , with $\\sqsubseteq _{\\mathrm {Tr,dis}}^{\\mathrm {x}}$ as kernel.",
"The function $\\mathbf {m}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {x}}$ is a 1-bounded pseudometric on $\\mathbf {S}$ , with $\\sim _{\\mathrm {Tr,dis}}^{\\mathrm {x}}$ as kernel.",
"Figure: We will evaluate the trace distances between s p s_p and tt wrt.",
"the different approaches, schedulers and parameter p∈[0,1]p \\in [0,1].In all upcoming examples we will investigate only the traces that are significant for the evaluation of the considered distance.Example 1 Consider processes $s_p$ and $t$ in Figure REF , with $p \\in [0,1]$ .",
"First we evaluate $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {det}}(t,s_p)$ .",
"We expand only the case for the resolution $\\mathcal {Z}_t$ for $t$ obtained from its central $a$ -branch.",
"It assigns probability $0.5$ to both $ab$ and $ac$ .",
"Under deterministic schedulers, any resolution $\\mathcal {Z}_{s_p}$ for $s_p$ can assign positive probability to only one of these traces.",
"Assume this trace is $ab$ , the case $ac$ is analogous.",
"We have either $\\mathrm {Pr}(z_{s_p},ab)) = p$ or $\\mathrm {Pr}(z_{s_p},ab)) = 1$ .",
"Then, $|\\mathrm {Pr}(z_{t},ab)) - \\mathrm {Pr}(z_{s_p},ab))| \\in \\lbrace 0.5,|0.5-p|\\rbrace $ and $|\\mathrm {Pr}(z_{t},ac)) - \\mathrm {Pr}(z_{s_p},ac))| = |0.5-0|=0.5$ .",
"Therefore, $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {det}}(t,s_p) = \\lambda \\cdot 0.5$ , for all $p \\in [0,1]$ .",
"Now we show that $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {det}}(s_p,t) = \\lambda \\cdot \\min \\lbrace p, |0.5 - p|,1-p\\rbrace $ .",
"For each resolution $\\mathcal {Z}_{s_p}$ for $s_p$ we need the resolution for $t$ whose trace distribution is closer to that of $\\mathcal {Z}_{s_p}$ .",
"We expand only the case of $\\mathcal {Z}_{s_p}$ corresponding to the leftmost $a$ -branch of $s_p$ and giving probability 1 to trace $a$ and $p$ to trace $ab$ .",
"We distinguish three subcases, related to the value of $p$ : [(i)] $p \\in [0,0.25]$ : The resolution for $t$ minimizing the distance from $\\mathcal {Z}_{s_p}$ is $\\mathcal {Z}_t^1$ that selects no action for $z^1_{t_1}$ .",
"The distance between $\\mathcal {Z}_{s_p}$ and $\\mathcal {Z}_t^1$ is $\\lambda ^{|ab|-1}|\\mathrm {Pr}(z_{s_p},ab)) - \\mathrm {Pr}(z_t^1,ab))| = \\lambda \\cdot p$ .",
"Notice that in this case $p \\le |0.5-p|,1-p$ .",
"$p \\in (0.25,0.75]$ : The resolution for $t$ that minimizes the distance from $\\mathcal {Z}_{s_p}$ is $\\mathcal {Z}_t^2$ that performs an $a$ -move and evolves to $0.5 \\delta _{z^2_{t_2}} + 0.5 \\delta _{z^2_{t_3}}$ , where $z^2_{t_3}$ that executes no action.",
"The distance between $\\mathcal {Z}_{s_p}$ and $\\mathcal {Z}^2_t$ is $\\lambda ^{|ab|-1}|\\mathrm {Pr}(z_{s_p},ab)) - \\mathrm {Pr}(z^2_t,ab))| = \\lambda \\cdot |0.5 - p|$ .",
"Notice that in this case we have $|0.5-p| \\le p,1-p$ .",
"$p \\in (0.75,1]$ : The resolution for $t$ that minimizes the distance from $\\mathcal {Z}_{s_p}$ is $\\mathcal {Z}_t^3$ that corresponds to the leftmost branch of $t$ .",
"The distance between $\\mathcal {Z}_s$ and $\\mathcal {Z}^3_t$ is $\\lambda ^{|ab|-1}|\\mathrm {Pr}(z_{s_p},ab)) - \\mathrm {Pr}(z^3_t,ab))| = \\lambda \\cdot (1-p)$ .",
"Notice that in this case $1-p \\le |0.5-p|,p$ .",
"In the case of randomized schedulers, one can prove that, since both $s_p,t$ can perform traces $ab$ and $ac$ with probability 1, for any $p \\in [0,1]$ we get $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {rand}}(s_p,t) = \\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {rand}}(t,s_p) =0$ .",
"$$ The trace-by-trace approach Trace distribution equivalences come with some desirable properties, as the full backward compatibility with the fully nondeterministic and fully probabilistic cases (cf.",
"[7]).",
"However, they are not congruences wrt.",
"parallel composition [24], and thus the related metrics cannot be non-expansive.",
"Moreover, due to the crucial rôle of the schedulers in the discrimination process, trace distribution distances are sometimes too demanding.",
"Take, for example, processes $s,t$ in Figure REF , with $\\varepsilon _1,\\varepsilon _2 \\in [0,0.5]$ .",
"We have $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {det}}(s,t) = \\lambda \\cdot 0.5$ and $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {det}}(t,s) = \\lambda \\cdot \\max _{i \\in \\lbrace 1,2\\rbrace } \\max \\lbrace 0.5- \\varepsilon _i, \\varepsilon _i\\rbrace $ , thus giving $\\mathbf {m}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {det}}(s,t) = \\lambda \\cdot 0.5$ for all $\\varepsilon _1,\\varepsilon _2 \\in [0,0.5]$ .",
"However, $s$ and $t$ can perform the same traces with probabilities that differ at most by $\\max (\\varepsilon _1,\\varepsilon _2)$ , which suggests that their trace distance should be $\\lambda \\cdot \\max (\\varepsilon _1,\\varepsilon _2)$ .",
"Specially, for $\\varepsilon _1,\\varepsilon _2=0$ , $s,t$ can perform the same traces with exactly the same probability.",
"Despite this, $s,t$ are still distinguished by trace distribution equivalences.",
"These situations arise since the focus of trace distribution approach is more on resolutions than on traces.",
"To move the focus on traces, the trace-by-trace approach was proposed [4].",
"The idea is to choose first the event that we want to observe, namely a single trace, and only as a second step we let the scheduler perform its selection: processes $s,t$ are equivalent wrt.",
"the trace-by-trace approach if for each trace $\\alpha $ , for each resolution for $s$ there is a resolution for $t$ that assigns to $\\alpha $ the same probability, and vice versa.",
"Figure: For ε 1 ,ε 2 ∈[0,0.5]\\varepsilon _1,\\varepsilon _2 \\in [0,0.5], we have 𝐦 Tr , tbt λ, det (s,t)=𝐦 Tr , tbt λ, rand (s,t)=λ·max(ε 1 ,ε 2 )\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {det}}(s,t)=\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {rand}}(s,t)= \\lambda \\cdot \\max (\\varepsilon _1,\\varepsilon _2), 𝐦 Tr , dis λ, det (s,t)=λ·0.5\\mathbf {m}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {det}}(s,t)= \\lambda \\cdot 0.5 and 𝐦 Tr , dis λ, rand (s,t)=λ·max{0.25+ε 1 ,0.25+ε 2 }\\mathbf {m}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {rand}}(s,t) = \\lambda \\cdot \\max \\lbrace 0.25 + \\varepsilon _1, 0.25 + \\varepsilon _2\\rbrace .Definition 8 (Tbt-trace equivalence [4]) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"We say that $s,t \\in \\mathbf {S}$ are in the tbt-trace preorder, written $s \\sqsubseteq _{\\mathrm {Tr,tbt}}^{x}t$ , if for each $\\alpha \\in \\mathcal {A}^{\\star }$ $\\text{for each } \\mathcal {Z}_s \\in \\mathrm {Res}^{\\mathrm {x}}(s) \\text{ there is } \\mathcal {Z}_t \\in \\mathrm {Res}^{\\mathrm {x}}(t) \\text{ such that } \\mathrm {Pr}(z_s,\\alpha )) = \\mathrm {Pr}(z_t,\\alpha ))$ .",
"Then, $s,t\\in \\mathbf {S}$ are tbt-trace equivalent, notation $s \\sim _{\\mathrm {Tr,tbt}}^{x}t$ , iff $s \\sqsubseteq _{\\mathrm {Tr,tbt}}^{x}t$ and $t \\sqsubseteq _{\\mathrm {Tr,tbt}}^{x}s$ .",
"In [4] it was proved that tbt-trace equivalences enjoy the congruence property and are full backward compatible with the fully nondeterministic and the fully probabilistic cases.",
"We introduce now the quantitative analogous to tbt-trace equivalences.",
"Processes $s,t$ are at distance $\\varepsilon \\ge 0$ if, for each trace $\\alpha $ , for each resolution for $s$ there is a resolution for $t$ such that the two resolutions assign to $\\alpha $ probabilities that differ at most by $\\varepsilon $, and vice versa.",
"Definition 9 (Tbt-trace metric) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"For each $\\alpha \\in \\mathcal {A}^{\\star }$ , the function $\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{\\alpha ,}\\lambda ,\\mathrm {x}} \\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ is defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{\\alpha ,}\\lambda ,\\mathrm {x}}(s,t) = \\lambda ^{|\\alpha |-1}\\; \\sup _{\\mathcal {Z}_s \\in \\mathrm {Res}^{\\mathrm {x}}(s)}\\, \\inf _{\\mathcal {Z}_t \\in \\mathrm {Res}^{\\mathrm {x}}(t)} \\, |\\mathrm {Pr}(z_s,\\alpha )) - \\mathrm {Pr}(z_t,\\alpha ))|$ The tbt-trace hemimetric and the tbt-trace metric are the functions $\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {x}}, \\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {x}}\\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {x}}(s,t) = \\sup _{\\alpha \\in \\mathcal {A}^{\\star }}\\; \\mathbf {h}_{\\mathrm {Tr,tbt}}^{{\\alpha ,}\\lambda ,\\mathrm {x}}(s,t)$ $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {x}}(s,t) = \\max \\lbrace \\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {x}}(s,t), \\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {x}}(t,s)\\rbrace $ .",
"It is not hard to see that for processes in Figure REF we have $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {x}}(s,t) = \\lambda \\cdot \\max (\\varepsilon _1,\\varepsilon _2)$ (and $s \\sim _{\\mathrm {Tr,tbt}}^{x}t$ if $\\varepsilon _1,\\varepsilon _2=0$ ).",
"Notice that, since we consider image finite processes, we are guaranteed that for each trace $\\alpha \\in \\mathcal {A}^{\\star }$ the supremum and infimum in the definition of $\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{\\alpha ,}\\lambda ,\\mathrm {x}}$ are actually achieved.",
"We show now that tbt-trace hemimetrics and metrics are well-defined and that their kernels are the tbt-trace preorders and equivalences, respectively.",
"Theorem 2 Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"Then: The function $\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {x}}$ is a 1-bounded hemimetric on $\\mathbf {S}$ , with $\\sqsubseteq _{\\mathrm {Tr,tbt}}^{x}$ as kernel.",
"The function $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {x}}$ is a 1-bounded pseudometric on $\\mathbf {S}$ , with $\\sim _{\\mathrm {Tr,tbt}}^{x}$ as kernel.",
"Example 2 Consider Figure REF .",
"We get $\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {det}}(s_p,t)$ = $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {det}}(s_p,t)$ = (see Example REF ) $\\lambda \\cdot \\min \\lbrace p,|0.5-p|,1-p\\rbrace $ .",
"The reason why in this particular case the two pseudometrics coincide is that each resolution for $s_p$ gives positive probability to at most one of the traces $ab$ and $ac$ , so that quantifying on traces before or after quantifying on resolutions is irrelevant.",
"Let us evaluate now $\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {det}}(t,s_p)$ .",
"To this aim, we focus on trace $ab$ and the resolution $\\mathcal {Z}_t$ obtained from the central $a$ -branch of $t$ , for which we have $\\mathrm {Pr}(z_t,ab)) = 0.5$ .",
"We need the resolution $\\mathcal {Z}_{s_p}$ for $s_p$ that minimizes $|0.5 - \\mathrm {Pr}(z_{s_p},ab))|$ .",
"Since for any resolutions $\\mathcal {Z}_{s_p}$ for $s_p$ we have $\\mathrm {Pr}(z_{s_p},ab)) \\in \\lbrace 0,p,1\\rbrace $ , we infer that the resolution $\\mathcal {Z}_{s_p}$ we are looking for satisfies $\\mathrm {Pr}(z_{s_p},ab)) = p$ and, therefore, $|0.5 - \\mathrm {Pr}(z_{s_p},ab))|$ = $|0.5 - p|$ .",
"By considering also the other resolutions for $ab$ and, then, the other traces, we can check that $\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {det}}(t,s_p) = \\lambda \\cdot |0.5-p|$ .",
"In Example REF we showed that $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {det}}(t,s_p) = \\lambda \\cdot 0.5$ for all $p \\in [0,1]$ .",
"Hence, we get $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {det}}(t,s_p) = \\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {det}}(t,s_p)$ for $p \\in \\lbrace 0,1\\rbrace $ , and $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {det}}(t,s_p) > \\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {det}}(t,s_p)$ for $p \\in (0,1)$ .",
"This disparity is due to the fact that the trace distributions approach forced us to match the resolution for $t$ assigning positive probability to both $ab$ and $ac$ , whereas in the trace-by-trace approach one never consider two traces at the same time.",
"$$ We conclude this section by stating that tbt-trace distances are strictly non-expansive, As a corollary, we re-obtain the (pre)congurence properties for their kernels (proved in [7]).",
"Theorem 3 All distances $\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {det}}$ , $\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {rand}}$ , $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {det}}$ , $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {rand}}$ are strictly non-expansive.",
"The supremal probabilities approach The trace-by-trace approach improves on trace distribution approach since it supports equivalences and metrics that are compositional.",
"Moreover, by focusing on traces instead of resolutions, the trace-by-trace approach puts processes in Figure REF in the expected relations.",
"However, we argue here that trace-by-trace approach on deterministic schedulers still gives some questionable results.",
"Take, for example, processes $s,t$ in Figure REF .",
"We believe that these processes should be equivalent in any semantics approach, since, after performing the action $a$ , they reach two distributions that should be identified, as they assign total probability 1 to states with an identical behavior.",
"But, if we consider the trace $ab$ , the resolution $\\mathcal {Z}_t \\in \\mathrm {Res}^{\\mathrm {det}}(t)$ in Figure REF is such that $\\mathrm {Pr}(z_t, ab)) = 0.5$ , whereas the unique resolution for $s$ assigning positive probability to $ab$ is $\\mathcal {Z}_s$ in Figure REF , for which $\\mathrm {Pr}(z_s,ab)) = 1$ .",
"Hence no resolution in $\\mathrm {Res}^{\\mathrm {det}}(s)$ matches $\\mathcal {Z}_t$ on trace $ab$ , thus giving $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {det}}(s.t) = \\lambda \\cdot 0.5$ and, consequently, $s \\lnot \\sim _{\\mathrm {Tr,tbt}}^{\\mathrm {det}}t$ .",
"This motivates to look for an alternative approach that allows us to equate processes in Figure REF and, at the same time, preserves all the desirable properties of the tbt-trace semantics.",
"We take inspiration from the extremal probabilities approach proposed in [5], which bases on the comparison, for each trace $\\alpha $ , of both suprema and infima execution probabilities, wrt.",
"resolutions, of $\\alpha $ : two processes are equated if they assign the same extremal probabilities to all traces.",
"However, reasoning on infima may cause some arguable results.",
"In particular, it is unclear whether such infima should be evaluated over the whole class of resolutions or over a restricted class, as for instance the resolutions in which the considered trace is actually executed.",
"Besides, desirable properties like the backward compatibility and compositionality are not guaranteed.",
"For all these reasons, we find it more reasonable to define a notion of trace equivalence, and a related metric, based on the comparison of supremal probabilities only.",
"Notice that, if we focus on verification, the comparison of supremal probabilities becomes natural.",
"To exemplify, we let the non-probabilistic case guide us.",
"To verify whether a process $t$ satisfies the specification $S$ , we check that whenever $S$ can execute a particular trace, then so does $t$ .",
"Actually, only positive information is considered: if there is a resolution for $S$ in which a given trace is executed, then this information is used to verify the equivalence.",
"Still, resolutions in which such a trace is not enabled are not considered.",
"The same principle should hold for PTSs: a process should perform all the traces enabled in $S$ and it should do it with at least the same probability, in the perspective that the quantitative behavior expressed in the specification expresses the minimal requirements on process behavior.",
"Focusing on supremal probabilities means relaxing the tbt-trace approach by simply requiring that, for each trace $\\alpha $ and resolution $\\mathcal {Z}_s$ for process $s$ there is a resolution for $t$ assigning to $\\alpha $ at least the same probability given by $\\mathcal {Z}_s$ , and vice versa.",
"Figure: Processes ss and tt are distinguished by ∼ Tr , tbt det \\sim _{\\mathrm {Tr,tbt}}^{\\mathrm {det}}, but related by ∼ Tr ,⊔ det \\sim _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {det}}.We remark that tt and uu are related by all the relations in the three approaches to trace semantics.Definition 10 ($\\bigsqcup $ -trace equivalence) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS and $\\mathrm {x} \\in \\lbrace \\mathrm {det}, \\mathrm {rand}\\rbrace $ .",
"We say that $s,t \\in \\mathbf {S}$ are in the $\\bigsqcup $ -trace preorder, written $s \\sqsubseteq _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {x}}t$ , if for each $\\alpha \\in \\mathcal {A}^{\\star }$ $\\sup _{\\mathcal {Z}_s \\in \\mathrm {Res}^{\\mathrm {x}}(s)} \\mathrm {Pr}(z_s,\\alpha )) \\le \\sup _{\\mathcal {Z}_t \\in \\mathrm {Res}^{\\mathrm {x}}(t)} \\mathrm {Pr}(z_t,\\alpha ))$ .",
"Then, $s,t \\in \\mathbf {S}$ are $\\bigsqcup $ -trace equivalent, notation $s \\sim _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {x}}t$ , iff $s \\sqsubseteq _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {x}}t$ and $t \\sqsubseteq _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {x}}s$ .",
"We stress that all good properties of trace-by-trace approach, as the backward compatibility with the fully nondeterministic and fully probabilistic cases and the non-expansiveness of the metric wrt.",
"parallel composition, are preserved by the supremal probabilities approach (Proposition REF and Theorem REF below).",
"Let $\\sim _{\\mathrm {Tr}}^{\\mathbf {N}}$ denote the trace equivalence on fully nondeterministic systems [8] and $\\sim _{\\mathrm {Tr}}^{\\mathbf {P}}$ denote the one on fully-probabilistic systems [22].",
"Proposition 1 Assume a PTS $P = (\\mathbf {S}, \\mathcal {A}, \\xrightarrow{})$ and processes $s,t \\in \\mathbf {S}$ .",
"Then: If $P$ is fully-nondeterministic, then $s \\sim _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {det}}t \\Leftrightarrow s \\sim _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {rand}}t \\Leftrightarrow s \\sim _{\\mathrm {Tr}}^{\\mathbf {N}}t$ .",
"If $P$ is fully-probabilistic, then $s \\sim _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {det}}t \\Leftrightarrow s \\sim _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {rand}}t \\Leftrightarrow s \\sim _{\\mathrm {Tr}}^{\\mathbf {P}}t$ .",
"The idea behind the quantitative analogue of $\\bigsqcup $ -trace equivalence is that two processes are at distance $\\varepsilon \\ge 0$ if, for each trace, the supremal execution probabilities wrt.",
"the resolutions of nondeterminism for the two processes differ at most by $\\varepsilon $.",
"Definition 11 ($\\bigsqcup $ -trace metric) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"For each $\\alpha \\in \\mathcal {A}^{\\star }$ , the function $\\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{\\alpha ,}\\lambda ,\\mathrm {x}} \\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ is defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{\\alpha ,}\\lambda ,\\mathrm {x}}(s,t) = \\max \\big \\lbrace 0,\\lambda ^{|\\alpha |-1} \\big (\\sup _{\\mathcal {Z}_s \\in \\mathrm {Res}^{\\mathrm {x}}(s)} \\mathrm {Pr}(z_s,\\alpha )) - \\sup _{\\mathcal {Z}_t \\in \\mathrm {Res}^{\\mathrm {x}}(t)} \\mathrm {Pr}(z_t,\\alpha ))\\big )\\big \\rbrace .$ The $\\bigsqcup $ -trace hemimetric and the $\\bigsqcup $ -trace metric are the functions $\\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}, \\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {x}}\\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}(s,t) = \\sup _{\\alpha \\in \\mathcal {A}^{\\star }}\\; \\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{\\alpha ,}\\lambda ,\\mathrm {x}}(s,t)$ and $\\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {x}}(s,t) = \\max \\lbrace \\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}(s,t), \\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}(t,s) \\rbrace $ .",
"We can show that $\\bigsqcup $ -trace hemimetrics and metrics are well-defined and that their kernels are the $\\bigsqcup $ -trace preorders and equivalences, respectively.",
"Theorem 4 Assume a PTS $(\\mathbf {S}, \\mathcal {A}, \\xrightarrow{})$ , $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"Then: The function $\\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}$ is a 1-bounded hemimetric on $\\mathbf {S}$ , with $\\sqsubseteq _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {x}}$ as kernel.",
"The function $\\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {x}}$ is a 1-bounded pseudometric on $\\mathbf {S}$ , with $\\sim _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {x}}$ as kernel.",
"We conclude this section by showing that $\\bigsqcup $ -trace distances are strictly non-expansive.",
"As a corollary, we infer the (pre)congruence property of their kernels.",
"Theorem 5 All distances $\\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{}\\lambda ,\\mathrm {det}}$ , $\\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{}\\lambda ,\\mathrm {rand}}$ , $\\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {det}}$ , $\\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {rand}}$ are strictly non-expansive.",
"Remark 1 We can show that the upper bounds to the distance of composed processes provided in Thms.",
"REF and REF are tight, namely for each distance $d$ considered in these theorems, there are processes $s_1,s_2,t_1,t_2$ with $d(s_1 \\parallel s_2 , t_1 \\parallel t_2) = d(s_1,t_1) + d(s_2,t_2) - d(s_1,t_1) \\cdot d(s_2,t_2)$ .",
"Indeed, for $z_s,z_t$ in Fig.",
"REF , with $\\lambda = 1$ , we have $d(z_s,t_t) = 0.5$ and $d(z_s \\parallel z_s, z_t \\parallel z_t) = 0.75 = 0.5 + 0.5 - 0.5 \\cdot 0.5$ .",
"Comparing the distinguishing power of trace metrics So far, we have discussed the properties of trace-based behavioral distances under different approaches.",
"Our aim is now to place these distances in a spectrum.",
"More precisely, we will order them wrt.",
"their distinguishing power: given the metrics $d,d^{\\prime }$ on $\\mathbf {S}$ , we write $d > d^{\\prime }$ if and only if $d(s,t) \\ge d^{\\prime }(s,t)$ for all $s,t \\in \\mathbf {S}$ and $d(u,v) > d^{\\prime }(u,v)$ for some $u,v \\in \\mathbf {S}$ .",
"Intuitively, for trace distributions and tbt-trace semantics, the distances evaluated on deterministic schedulers are more discriminating than their randomized analogues.",
"Theorem 6 Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\lambda \\in (0,1]$ and $\\mathrm {y} \\in \\lbrace \\mathrm {dis},\\mathrm {tbt}\\rbrace $ .",
"Then: [1.]",
"$\\mathbf {h}_{\\mathrm {Tr,y}}^{\\lambda ,\\mathrm {rand}} < \\mathbf {h}_{\\mathrm {Tr,y}}^{\\lambda ,\\mathrm {det}}$ .",
"$\\mathbf {m}_{\\mathrm {Tr,y}}^{\\lambda ,\\mathrm {rand}} < \\mathbf {m}_{\\mathrm {Tr,y}}^{\\lambda ,\\mathrm {det}}$ .",
"As a corollary of Theorem REF , by using the relations between distances and equivalences in Theorems REF and REF , we re-obtain the relations $\\sim _{\\mathrm {Tr,dis}}^{\\mathrm {det}}\\subset \\sim _{\\mathrm {Tr,dis}}^{\\mathrm {rand}}$ and $\\sim _{\\mathrm {Tr,tbt}}^{\\mathrm {det}}\\subset \\sim _{\\mathrm {Tr,tbt}}^{\\mathrm {rand}}$ proved in [7].",
"Moreover, also the analogous results for preorders follow.",
"As one can expect, the metrics on trace distributions are more discriminating than their corresponding ones in the trace-by-trace approach.",
"Theorem 7 Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"Then: [1.]",
"$\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {x}} < \\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {x}}$ .",
"$\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {x}}< \\mathbf {m}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {x}}$ .",
"As a corollary, by using the kernel relations given in Theorems REF and REF , we re-obtain the relation $\\sim _{\\mathrm {Tr,dis}}^{\\mathrm {x}}\\subset \\sim _{\\mathrm {Tr,tbt}}^{x}$ proved in [7] and we get $\\sqsubseteq _{\\mathrm {Tr,dis}}^{\\mathrm {x}}\\subset \\sqsubseteq _{\\mathrm {Tr,tbt}}^{x}$ .",
"Moreover, we remark that $\\mathbf {m}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {rand}}$ is not comparable with $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {det}}$ .",
"This is mainly due to the randomization process and it is witnessed by processes in Figure REF , where $\\mathbf {m}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {rand}}(s,t)=\\lambda \\cdot \\max \\lbrace 0.25+\\varepsilon _1,0.25+\\varepsilon _2\\rbrace $ and $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {det}}(s,t) = \\lambda \\cdot \\max (\\varepsilon _1,\\varepsilon _2)$ and Figure REF , where $\\mathbf {m}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {rand}}(s,t)=0$ and $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {det}}(s,t) = \\lambda \\cdot 0.5$ .",
"We focus now on supremal probabilities approach, that comes with a particularly interesting result: the $\\bigsqcup $ -trace metric on deterministic schedulers coincides with tbt-trace metrics on randomized schedulers.",
"Moreover, $\\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {det}}$ coincides also with its randomized version.",
"Theorem 8 Assume a PTS $P = (\\mathbf {S}, \\mathcal {A}, \\xrightarrow{})$ and $\\lambda \\in (0,1]$ .",
"Then: [1.]",
"$\\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{}\\lambda ,\\mathrm {det}} = \\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{}\\lambda ,\\mathrm {rand}} = \\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {rand}}$ .",
"$\\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {det}}= \\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {rand}}= \\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {rand}}$ .",
"This result is fundamental in the perspective of the application of our trace metrics to process verification: by comparing solely the suprema execution probabilities of the linear properties of interest we get same expressive power of a pairwise comparison of the probabilities in all possible randomized resolutions of nondeterminism.",
"Clearly, Theorem REF together with the kernel relations from Thms REF and REF imply that the relations for the supremal probabilities semantics coincide with those for the tbt-trace semantics wrt.",
"randomized schedulers, ie.",
"$\\sqsubseteq _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {det}}= \\sqsubseteq _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {rand}}= \\sqsubseteq _{\\mathrm {Tr,tbt}}^{\\mathrm {rand}}$ and $\\sim _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {det}}= \\sim _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {rand}}= \\sim _{\\mathrm {Tr,tbt}}^{\\mathrm {rand}}$ .",
"Metrics for testing Testing semantics [14] compares processes according to their capacity to pass a test.",
"The latter is a PTS equipped with a distinguished state indicating the success of the test.",
"Definition 12 (Test) A nondeterministic probabilistic test transition systems (NPT) is a finite PTS $(\\mathbf {O}, \\mathcal {A}, \\xrightarrow{})$ where $\\mathbf {O}$ is a set of processes, called tests, containing a distinguished success process $\\surd $ with no outgoing transitions.",
"We say that a computation from $o \\in \\mathbf {O}$ is successful iff its last state is $\\surd $ .",
"Given a process $s$ and a test $o$ , we can consider the interaction system among the two.",
"This models the response of the process to the application of the test, so that $s$ passes the test $o$ if there is a computation in the interaction system that reaches $\\surd $ .",
"Informally, the interaction system is the result of the parallel composition of the process with the test.",
"Definition 13 (Interaction system) The interaction system of a PTS $(\\mathbf {S},\\mathcal {A},$ $\\xrightarrow{})$ and an NPT $(\\mathbf {O},\\mathcal {A},\\xrightarrow{}_{\\mathbf {O}})$ is the PTS $(\\mathbf {S}\\times \\mathbf {O}, \\mathcal {A},\\xrightarrow{}^{\\prime })$ where: [(i)] $(s,o) \\in \\mathbf {S}\\times \\mathbf {O}$ is called a configuration and is successful iff $o = \\surd $ ; a computation from $(s,o) \\in \\mathbf {S}\\times \\mathbf {O}$ is successful iff its last configuration is successful.",
"For $(s, o)$ and $\\mathcal {Z}_{s,o} \\in \\mathrm {Res}^{\\mathrm {x}}(s,o)$ , we let $\\mathbf {SC}(z_{s,o})$ be the set of successful computations from $z_{s,o}$ .",
"For $\\alpha \\in \\mathcal {A}^{\\star }$ , $\\mathbf {SC}(z_{s,o},\\alpha )$ is the set of $\\alpha $ -compatible successful computations from $z_{s,o}$ .",
"Testing semantics should compare processes wrt.",
"their probability to pass a test.",
"In this Section we consider three approaches to it: [(i)] the may/must, the trace-by-trace, and the supremal probabilities.",
"For each approach, we present (hemi,pseudo)metrics that provide a quantitative variant of the considered testing equivalence.",
"To the best of our knowledge, ours is the first attempt in this direction.",
"The may/must approach In the original work on nondeterministic systems [14], testing equivalence was defined via the may and must preorders.",
"The former expresses the ability of processes to pass a test.",
"The latter expresses the impossibility to fail a test.",
"When also probability is considered, these two preorders are defined, resp., in terms of suprema and infima success probabilities [29].",
"Definition 14 (May/must testing equivalence, [29]) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $(\\mathbf {O},\\mathcal {A},\\xrightarrow{}_O)$ an NPT and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"We say that $s,t \\in \\mathbf {S}$ are in the may testing preorder, written $s \\sqsubseteq _{\\mathrm {Te,may}}^{x}t$ , if for each $o \\in \\mathbf {O}$ $\\sup _{\\mathcal {Z}_{s,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o)} \\mathrm {Pr}(\\mathbf {SC}(z_{s,o})) \\le \\sup _{\\mathcal {Z}_{t,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o)} \\mathrm {Pr}(\\mathbf {SC}(z_{t,o}))$ .",
"Then, $s,t\\in \\mathbf {S}$ are may testing equivalent, written $s \\sim _{\\mathrm {Te,may}}^{x}t$ , iff $s \\sqsubseteq _{\\mathrm {Te,may}}^{x}t$ and $t \\sqsubseteq _{\\mathrm {Te,may}}^{x}s$ .",
"The notions of must testing preorder, $\\sqsubseteq _{\\mathrm {Te,must}}^{x}$ , and must testing equivalence, $\\sim _{\\mathrm {Te,must}}^{x}$ , are obtained by replacing the suprema in $\\sqsubseteq _{\\mathrm {Te,may}}^{x}$ and $\\sim _{\\mathrm {Te,may}}^{x}$ , resp., with infima.",
"Finally, we say that $s,t \\in \\mathbf {S}$ are in the may/must testing preorder, written $s \\sqsubseteq _{\\mathrm {Te,mM}}^{x}t$ , if $s \\sqsubseteq _{\\mathrm {Te,may}}^{x}t$ and $s \\sqsubseteq _{\\mathrm {Te,must}}^{x}t$ .",
"They are may/must testing equivalent, written $s \\sim _{\\mathrm {Te,mM}}^{x}t$ , iff $s \\sqsubseteq _{\\mathrm {Te,mM}}^{x}t$ and $t \\sqsubseteq _{\\mathrm {Te,mM}}^{x}s$ .",
"The quantitative analogue to may/must testing equivalence bases on the evaluation of the differences in the extremal success probabilities.",
"The distance between $s,t \\in \\mathbf {S}$ is set to $\\varepsilon \\ge 0$ if the maximum between the difference in the suprema and infima success probabilities wrt.",
"all resolutions of nondeterminism for $s$ and $t$ is at most $\\varepsilon $ .",
"We introduce a function $\\omega :\\mathbf {O}\\rightarrow (0,1]$ that assigns to each test $o$ the proper discount.",
"In fact, as the success probabilities in the may/must semantics are not related to the execution of a particular trace, in general we cannot define a discount factor as we did for the trace distances.",
"However, a similar construction may be regained when only tests with finite depth are considered.",
"In that case, we could define $\\omega (o) = \\lambda ^{\\mathrm {depth}(o)}$ , for $\\lambda \\in (0,1]$ .",
"We will use $\\mathbf {1}$ to denote the 1 constant function.",
"Definition 15 (May/must testing metric) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $(\\mathbf {O},\\mathcal {A},\\xrightarrow{}_O)$ an NPT, $\\omega :\\mathbf {O}\\rightarrow (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"For each $o\\in \\mathbf {O}$ , the function $\\mathbf {h}_{\\mathrm {Te,may}}^{{o,}\\omega ,\\mathrm {x}}\\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ is defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Te,may}}^{{o,}\\omega ,\\mathrm {x}}(s,t)=\\max \\Big \\lbrace 0 , \\omega (o)\\Big (\\sup _{\\mathcal {Z}_{s,o}\\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o)} \\mathrm {Pr}(\\mathbf {SC}(z_{s,o})) - \\sup _{\\mathcal {Z}_{t,o}\\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o)} \\mathrm {Pr}(\\mathbf {SC}(z_{t,o}))\\Big ) \\Big \\rbrace $ Function $\\mathbf {h}_{\\mathrm {Te,must}}^{{o,}\\omega ,\\mathrm {x}}\\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ is obtained by replacing the suprema in $\\mathbf {h}_{\\mathrm {Te,may}}^{{o,}\\omega ,\\mathrm {x}}$ with infima.",
"Given $\\mathrm {y} \\in \\lbrace \\mathrm {may}, \\mathrm {must}\\rbrace $ , the $\\mathrm {y}$ testing hemimetric and the $\\mathrm {y}$ testing metric are the functions $\\mathbf {h}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}, \\mathbf {m}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}} \\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}(s,t) = \\sup _{o \\in \\mathbf {O}}\\; \\mathbf {h}_{\\mathrm {Te,y}}^{o,\\omega ,\\mathrm {x}}(s,t)$ and $\\mathbf {m}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}(s,t) = \\max \\lbrace \\mathbf {h}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}(s,t), \\mathbf {h}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}(t,s) \\rbrace $ .",
"The may/must testing hemimetric and the may/must testing metric are the functions $\\mathbf {h}_{\\mathrm {Te,mM}}^{{}\\omega ,\\mathrm {x}}, \\mathbf {m}_{\\mathrm {Te,mM}}^{\\omega ,\\mathrm {x}}\\colon $ $\\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Te,mM}}^{{}\\omega ,\\mathrm {x}}(s,t) = \\max \\lbrace \\mathbf {h}_{\\mathrm {Te,may}}^{{}\\omega ,\\mathrm {x}}(s,t), \\mathbf {h}_{\\mathrm {Te,must}}^{{}\\omega ,\\mathrm {x}}(s,t) \\rbrace $ .",
"$\\mathbf {m}_{\\mathrm {Te,mM}}^{\\omega ,\\mathrm {x}}(s,t) = \\max \\lbrace \\mathbf {m}_{\\mathrm {Te,may}}^{\\omega ,\\mathrm {x}}{}(s,t), \\mathbf {m}_{\\mathrm {Te,must}}^{\\omega ,\\mathrm {x}}{}(s,t) \\rbrace $ .",
"Theorem 9 Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\omega :\\mathbf {O}\\rightarrow (0,1]$ , $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ and $\\mathrm {y} \\in \\lbrace \\mathrm {may},\\mathrm {must}, \\mathrm {mM}\\rbrace $ : The function $\\mathbf {h}_{\\mathrm {Te,y}}^{\\lambda ,\\mathrm {x}}$ is a 1-bounded hemimetric on $\\mathbf {S}$ , with $\\sqsubseteq _{\\mathrm {Te,y}}^{\\mathrm {x}}$ as kernel.",
"The function $\\mathbf {m}_{\\mathrm {Te,y}}^{\\lambda ,\\mathrm {x}}$ is a 1-bounded pseudometric on $\\mathbf {S}$ , with $\\sim _{\\mathrm {Te,y}}^{\\mathrm {x}}$ as kernel.",
"Figure: We use the tests o 1 ,o 2 o_1,o_2 to evaluate the distance between processes s,t,us,t,u in Fig.",
"wrt.",
"testing semantics.•\\bullet represents a generic configuration in the interaction system.In all upcoming examples we will consider only the tests and traces that are significant for the evaluations of the testing metrics.Example 3 Consider $t,u$ in Fig REF and their interactions with test $o_1$ in Fig REF .",
"Clearly, $(t,o_1)$ and $(u,o_1)$ have the same suprema success probabilities.",
"In fact, they both have a maximal resolution assigning probability 1 to the trace $ab$ , ie.",
"the only successful trace in the considered case.",
"As the same holds for all tests we get $\\mathbf {m}_{\\mathrm {Te,may}}^{\\omega ,\\mathrm {x}}(t,u) = 0$ .",
"Conversely, if we compare the infima success probabilities, we get $\\inf _{\\mathcal {Z}_{t,o_1} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o_1)} = 1$ since $(t,o_1)$ has only one maximal resolution corresponding to $(t,o_1)$ itself and that with probability 1 reaches $\\surd $ .",
"Still, $\\inf _{\\mathcal {Z}_{u,o_1} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(u,o_1)} = 0$ , given by the maximal resolution corresponding to $(u,o_1) \\stackrel{a}{{\\twoheadrightarrow }} \\mathrm {nil}$ .",
"Hence, we can infer $\\mathbf {m}_{\\mathrm {Te,must}}^{\\omega ,\\mathrm {x}}(t,u) = \\omega (o_1) \\cdot |1-0| = \\omega (o_1)$ .",
"$$ We can finally observe that both $\\mathbf {h}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}$ and $\\mathbf {m}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}$ are non-expansive.",
"Theorem 10 Let $\\omega :\\mathbf {O}\\rightarrow (0,1]$ and $\\mathrm {y} \\in \\lbrace \\mathrm {may},\\mathrm {must}, \\mathrm {mM}\\rbrace $ .",
"$\\mathbf {h}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}$ and $\\mathbf {m}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}$ are non-expansive.",
"The trace-by-trace approach In [7] it was proved that the may/must is fully backward compatible with the restricted class of processes only if the same restriction is applied to the class of tests, ie.",
"if we consider resp.",
"fully nondeterministic and fully probabilistic tests only.",
"This is due to the duplication ability of nondeterministic probabilistic tests.",
"However, by applying the trace-by-trace approach to testing semantics, we regain the full backward compatibility wrt.",
"all tests (cf.",
"[7]).",
"Definition 16 (Tbt-testing equivalence) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $(\\mathbf {O},\\mathcal {A},\\xrightarrow{}_O)$ an NPT, $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"We say that $s,t \\in \\mathbf {S}$ are in the tbt-testing preorder, written $s \\sqsubseteq _{\\mathrm {Te,tbt}}^{x}t$ , if for each $o \\in \\mathbf {O}$ and $\\alpha \\in \\mathcal {A}^{\\star }$ $\\text{for each } \\mathcal {Z}_{s,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o) \\text{ there is } \\mathcal {Z}_{t,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o) \\text{ st. } \\mathrm {Pr}(\\mathbf {SC}(z_{s,o},\\alpha )) = \\mathrm {Pr}(\\mathbf {SC}(z_{t,o},\\alpha ))$ .",
"Then, $s,t\\in \\mathbf {S}$ are tbt-testing equivalent, notation $s \\sim _{\\mathrm {Te,tbt}}^{x}t$ , iff $s \\sqsubseteq _{\\mathrm {Te,tbt}}^{x}t$ and $t \\sqsubseteq _{\\mathrm {Te,tbt}}^{x}s$ .",
"The definition of the tbt-testing metric naturally follows from Def.",
"REF .",
"Definition 17 (Tbt-testing metric) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $(\\mathbf {O},\\mathcal {A},\\xrightarrow{}_O)$ an NPT, $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"For each $o \\in \\mathbf {O}$ and $\\alpha \\in \\mathcal {A}^{\\star }$ , function $\\mathbf {h}_{\\mathrm {Te,tbt}}^{{o,\\alpha ,}\\lambda ,\\mathrm {x}} \\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ is defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Te,tbt}}^{{o,\\alpha ,}\\lambda ,\\mathrm {x}}(s,t) = \\lambda ^{|\\alpha |-1}\\; \\sup _{\\mathcal {Z}_{s,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o)}\\, \\inf _{\\mathcal {Z}_{t,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o)} \\, |\\mathrm {Pr}(\\mathbf {SC}(z_{s,o},\\alpha )) - \\mathrm {Pr}(\\mathbf {SC}(z_{t,o},\\alpha ))|$ The tbt-testing hemimetric and the tbt-testing metric are the functions $\\mathbf {h}_{\\mathrm {Te,tbt}}^{{}\\lambda ,\\mathrm {x}}, \\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}}\\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Te,tbt}}^{{}\\lambda ,\\mathrm {x}}(s,t) = \\sup _{o \\in \\mathbf {O}} \\; \\sup _{\\alpha \\in \\mathcal {A}^{\\star }}\\; \\mathbf {h}_{\\mathrm {Te,tbt}}^{{o,\\alpha ,}\\lambda ,\\mathrm {x}}(s,t)$ $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}}(s,t) = \\max \\lbrace \\mathbf {h}_{\\mathrm {Te,tbt}}^{{}\\lambda ,\\mathrm {x}}(s,t), \\mathbf {h}_{\\mathrm {Te,tbt}}^{{}\\lambda ,\\mathrm {x}}(t,s) \\rbrace $ .",
"Theorem 11 Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"Then: The function $\\mathbf {h}_{\\mathrm {Te,tbt}}^{{}\\lambda ,\\mathrm {x}}$ is a 1-bounded hemimetric on $\\mathbf {S}$ , with $\\sqsubseteq _{\\mathrm {Te,tbt}}^{x}$ as kernel.",
"The function $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}}$ is a 1-bounded pseudometric on $\\mathbf {S}$ , with $\\sim _{\\mathrm {Te,tbt}}^{x}$ as kernel.",
"Example 4 Consider $s,t$ in Fig.",
"REF and their interactions with test $o_2$ in Fig.",
"REF .",
"By the same reasoning detailed in the first paragraph of Sect.",
"REF , we get $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {det}}(s,t)=\\lambda \\cdot 0.5$ and $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {rand}}(s,t) = 0$ .",
"$$ When the tbt-approach is used to define testing metrics, we get a refinement of the non-expansiveness property to strict non-expansiveness.",
"Theorem 12 All distances $\\mathbf {h}_{\\mathrm {Te,tbt}}^{{}\\lambda ,\\mathrm {det}}$ , $\\mathbf {h}_{\\mathrm {Te,tbt}}^{{}\\lambda ,\\mathrm {rand}}$ , $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {det}}$ , $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {rand}}$ are strictly non-expansive.",
"The supremal probabilities approach If we focus on verification, we can use the testing semantics to verify whether a process will behave as intended by its specification in all possible environments, as modeled by the interaction with the tests.",
"Informally, we could see each test as a set of requests of the environment to the system: the ones ending in the success state are those that must be answered.",
"The interaction of the specification with the test then tells us whether the system is able to provide those answers.",
"Thus, an implementation has to guarantee at least all the answers provided by the specification.",
"For this reason we decided to introduce also a supremal probabilities variant of testing semantics: for each test and for each trace we compare the suprema wrt.",
"all resolutions of nondeterminism of the probabilities of processes to reach success by performing the considered trace.",
"Definition 18 ($\\bigsqcup $ -testing equivalence) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $(\\mathbf {O},\\mathcal {A},\\xrightarrow{}_O)$ an NPT and $\\mathrm {x} \\in \\lbrace \\mathrm {det}, \\mathrm {rand}\\rbrace $ .",
"We say that $s,t \\in \\mathbf {S}$ are in the $\\bigsqcup $ -testing preorder, written $s \\sqsubseteq _{\\mathrm {Te,}\\sqcup }^{\\mathrm {x}}t$ , if for each $o \\in \\mathbf {O}$ and $\\alpha \\in \\mathcal {A}^{\\star }$ $\\sup _{\\mathcal {Z}_{s,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o)} \\mathrm {Pr}(\\mathbf {SC}(z_{s,o},\\alpha )) \\le \\sup _{\\mathcal {Z}_{t,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o)} \\mathrm {Pr}(\\mathbf {SC}(z_{t,o},\\alpha ))$ .",
"Then, $s,t \\in \\mathbf {S}$ are $\\bigsqcup $ -testing equivalent, notation $s \\sim _{\\mathrm {Te,}\\sqcup }^{\\mathrm {x}}t$ , iff $s \\sqsubseteq _{\\mathrm {Te,}\\sqcup }^{\\mathrm {x}}t$ and $t \\sqsubseteq _{\\mathrm {Te,}\\sqcup }^{\\mathrm {x}}s$ .",
"We obtain the $\\bigsqcup $ -testing metric as a direct adaptation to tests of Definition REF .",
"Definition 19 ($\\bigsqcup $ -testing metric) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $(\\mathbf {O},\\mathcal {A},\\xrightarrow{}_O)$ an NPT, $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"For each $o \\in \\mathbf {O}$ , $\\alpha \\in \\mathcal {A}^{\\star }$ , the function $\\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{o,\\alpha ,}\\lambda ,\\mathrm {x}} \\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ is defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{o,\\alpha ,}\\lambda ,\\mathrm {x}}(s,t) = \\max \\Big \\lbrace 0,\\lambda ^{|\\alpha |-1} \\Big (\\sup _{\\mathcal {Z}_{s,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o)} \\!\\!\\!\\!\\!\\mathrm {Pr}(\\mathbf {SC}(z_{s,o},\\alpha )) - \\sup _{\\mathcal {Z}_{t,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o)}\\!\\!\\!\\!\\!",
"\\mathrm {Pr}(\\mathbf {SC}(z_{t,o},\\alpha ))\\Big )\\Big \\rbrace .$ The $\\bigsqcup $ -testing hemimetric and the $\\bigsqcup $ -testing metric are the functions $\\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}, \\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}\\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}(s,t) = \\sup _{o \\in \\mathbf {O}} \\; \\sup _{\\alpha \\in \\mathcal {A}^{\\star }}\\; \\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{o,\\alpha ,}\\lambda ,\\mathrm {x}}(s,t)$ ; $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}(s,t) = \\max \\lbrace \\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}(s,t), \\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}(t,s) \\rbrace $ .",
"Theorem 13 Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS and $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"Then: The function $\\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}$ is a 1-bounded hemimetric on $\\mathbf {S}$ , with $\\sqsubseteq _{\\mathrm {Te,}\\sqcup }^{\\mathrm {x}}$ as kernel.",
"The function $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}$ is a 1-bounded pseudometric on $\\mathbf {S}$ , with $\\sim _{\\mathrm {Te,}\\sqcup }^{\\mathrm {x}}$ as kernel.",
"Finally, we can show that both $\\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}$ and $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}$ are strictly non-expansive.",
"Theorem 14 All distances $\\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{}\\lambda ,\\mathrm {det}}$ , $\\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{}\\lambda ,\\mathrm {rand}}$ , $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {det}}$ , $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {rand}}$ are strictly non-expansive.",
"Remark 2 For all distances $d$ considered in Thms.",
"REF , REF , REF and processes $z_s,z_t$ in Fig.",
"REF , with $\\lambda =1$ , we have $d(z_s , z_t) = 0.5$ and $d(z_s \\parallel z_s , z_t \\parallel z_t) = 0.75 = 0.5 + 0.5 - 0.5 \\cdot 0.5$ .",
"Hence, the upper bounds to the distance between composed processes provided in Thms.",
"REF and REF are tight.",
"We leave as a future work the analogous result for distances considered in Thm.",
"REF .",
"Comparing the distinguishing power of testing metrics Figure: The spectrum of trace and testing (hemi)metrics.d→d ' d \\rightarrow d^{\\prime } stands for d>d ' d > d^{\\prime }.We present only the general form with 𝐝∈{𝐡,𝐦}\\mathbf {d} \\in \\lbrace \\mathbf {h},\\mathbf {m}\\rbrace as the relations among the hemimetrics are the same wrt.",
"those among the metrics.The complete spectrum can be obtained by relating each metric with the respective hemimetric.We study the distinguishing power of the testing metrics presented in this section and the trace metrics defined in Sect.",
", thus obtaining the spectrum in Fig.",
"REF .",
"Firstly, we compare the expressiveness of the testing metrics wrt.",
"the chosen class of schedulers.",
"The distinguishing power of testing metrics based on may-must and supremal probabilities approaches is not influenced by this choice.",
"Differently, in the tbt approach, the distances evaluated on deterministic schedulers are more discriminating than their analogues on randomized schedulers.",
"Theorem 15 Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\lambda \\in (0,1]$ , $\\omega \\colon \\mathbf {O}\\rightarrow (0,1]$ $\\mathrm {y} \\in \\lbrace \\mathrm {may},\\mathrm {must}, \\mathrm {mM}\\rbrace $ and $\\mathbf {d} \\in \\lbrace \\mathbf {h},\\mathbf {m}\\rbrace $ : $\\begin{array}{llll}1.\\, \\mathbf {d}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {rand}} = \\mathbf {d}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {det}} &2.\\, \\mathbf {d}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {rand}} < \\mathbf {d}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {det}} &3.\\, \\mathbf {d}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {rand}} = \\mathbf {d}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {det}}\\end{array}$ From Thm.",
"REF , by using the kernel relations in Thms.",
"REF and REF , we regain relations $\\sim _{\\mathrm {Te,may}}^{\\mathrm {rand}}= \\sim _{\\mathrm {Te,may}}^{\\mathrm {det}}$ , $\\sim _{\\mathrm {Te,must}}^{\\mathrm {rand}}= \\sim _{\\mathrm {Te,must}}^{\\mathrm {det}}$ , $\\sim _{\\mathrm {Te,mM}}^{\\mathrm {rand}}= \\sim _{\\mathrm {Te,mM}}^{\\mathrm {det}}$ , $\\sim _{\\mathrm {Te,tbt}}^{\\mathrm {det}}\\subset \\sim _{\\mathrm {Te,tbt}}^{\\mathrm {rand}}$ , and their analogues on preorders, proved in [7].",
"From Thm.",
"REF we get $\\sqsubseteq _{\\mathrm {Te,}\\sqcup }^{\\mathrm {rand}}= \\sqsubseteq _{\\mathrm {Te,}\\sqcup }^{\\mathrm {det}}$ and $\\sim _{\\mathrm {Te,}\\sqcup }^{\\mathrm {rand}}= \\sim _{\\mathrm {Te,}\\sqcup }^{\\mathrm {det}}$ .",
"The strictness of the inequality in Thm.",
"REF .2, is witnessed by processes $s,t$ in Fig REF and their interactions with the test $o_2$ in Fig REF .",
"The same reasoning applied in the first paragraph of Sect.",
"REF to obtain $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {det}}(s,t) = \\lambda \\cdot 0.5$ and $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {rand}}(s,t) = 0$ , gives $\\mathbf {h}_{\\mathrm {Te,tbt}}^{{o_2,}\\lambda ,\\mathrm {det}}(t,s) = \\lambda \\cdot 0.5 = \\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {det}}{}(s,t)$ and $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {rand}}{}(s,t) = 0$ .",
"We proceed to compare the expressiveness of each metric wrt.",
"the other semantics.",
"Our results are fully compatible with the spectrum on probabilistic relations presented in [7].",
"Theorem 16 Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\lambda \\in (0,1]$ , $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ and $\\mathbf {d} \\in \\lbrace \\mathbf {h},\\mathbf {m}\\rbrace $ : $\\begin{array}{llll}1.\\, \\mathbf {d}_{\\mathrm {Te,may}}^{\\omega ,\\mathrm {x}} < \\mathbf {d}_{\\mathrm {Te,mM}}^{\\omega ,\\mathrm {x}} &2.\\, \\mathbf {d}_{\\mathrm {Te,must}}^{\\omega ,\\mathrm {x}} < \\mathbf {d}_{\\mathrm {Te,mM}}^{\\omega ,\\mathrm {x}} &3.\\, \\mathbf {d}_{\\mathrm {Te,}\\sqcup }^{1,\\mathrm {x}} < \\mathbf {d}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}} &4.\\, \\mathbf {d}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}} < \\mathbf {d}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}} \\\\5.\\, \\mathbf {d}_{\\mathrm {Tr,dis}}^{1,\\mathrm {rand}} < \\mathbf {d}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}} &6.\\, \\mathbf {d}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {x}} < \\mathbf {d}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}} &7.\\, \\mathbf {d}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {x}} < \\mathbf {d}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}\\end{array}$ The following Examples prove the strictness of the inequalities in Thm.",
"REF and the non comparability of the (hemi)metrics as shown in Fig.",
"REF .",
"For simplicity, we consider only the cases of the metrics.",
"Figure: Processes s,ts,t and their interaction systems with the test o 2 o_2 in Fig.",
".Example 5 Non comparability of $\\mathbf {m}_{\\mathrm {Te,may}}^{\\omega ,\\mathrm {x}}$ with $\\mathbf {m}_{\\mathrm {Te,must}}^{\\omega ,\\mathrm {x}}$.",
"In Ex.",
"REF we showed that for $t,u$ in Fig.",
"REF from their interaction with the test $o_1$ in Fig.",
"REF we obtain that $\\mathbf {m}_{\\mathrm {Te,must}}^{\\omega ,\\mathrm {x}}(t,u) = \\omega (o_1)$ , whereas one can easily check that $\\mathbf {m}_{\\mathrm {Te,may}}^{\\omega ,\\mathrm {x}}(t,u) = 0$ .",
"Consider now $s,t$ and their interactions in Fig.",
"REF with the test $o_2$ from Fig.",
"REF .",
"Clearly, we have $\\sup _{\\mathcal {Z}_{s,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o)}\\mathrm {Pr}(\\mathbf {SC}(z_{s,o})) = 1$ and $\\sup _{\\mathcal {Z}_{t,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o)}\\mathrm {Pr}(\\mathbf {SC}(z_{t,o})) = 0.3$ and thus $\\mathbf {m}_{\\mathrm {Te,may}}^{\\omega ,\\mathrm {x}}(s,t) = 0.7 \\cdot \\omega (o_2)$ .",
"Conversely, if we consider infima success probabilities, we have $\\inf _{\\mathcal {Z}_{s,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o)}\\mathrm {Pr}(\\mathbf {SC}(z_{s,o})) = 0$ and $\\sup _{\\mathcal {Z}_{t,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o)}\\mathrm {Pr}(\\mathbf {SC}(z_{t,o})) = 0.3$ .",
"Thus, $\\mathbf {m}_{\\mathrm {Te,must}}^{\\omega ,\\mathrm {x}}(s,t) = 0.3\\cdot \\omega (o_2)$ .",
"$$ Figure: Processes s,ts,t are such that 𝐝 Te , tbt 1,x (s,t)=0\\mathbf {d}_{\\mathrm {Te,tbt}}^{1,\\mathrm {x}}(s,t) = 0 and 𝐝 Te , must 1,x (s,t)=0.5\\mathbf {d}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0.5, as witnessed by the test o 1/2 o^{1/2}.Example 6 Non comparability of $\\mathbf {m}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}$ with $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{1,\\mathrm {x}}$ , $\\mathbf {m}_{\\mathrm {Te,tbt}}^{1,\\mathrm {x}}$ , $\\mathbf {m}_{\\mathrm {Tr,dis}}^{1,\\mathrm {x}}$ , $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{1,\\mathrm {x}}$ and $\\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{1,\\mathrm {x}}$.",
"We start with $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{1,\\mathrm {x}}$ .",
"Form Ex.",
"REF we know that for $t,u$ in Fig.",
"REF it holds $\\mathbf {m}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}(t,u) = \\mathbf {1}$ .",
"Since both $t$ and $u$ have maximal resolutions giving probability 1 to either $ab$ or $ac$ , we get $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{1,\\mathrm {x}}(t,u) =0$ .",
"Consider now $s,t$ in Fig.",
"REF .",
"In Ex.",
"REF we showed that $\\mathbf {m}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0.3$ .",
"From the interaction systems in Fig.",
"REF , by considering the superma success probabilities of trace $ac$ , we obtain that $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{1,\\mathrm {x}} = 0.4$ .",
"Next we deal with the tbt-testing metrics.",
"Consider $s,t$ in Fig.",
"REF and the family of tests $O = \\lbrace o^p \\mid p \\in (0,1)\\rbrace $ , each duplicating the actions $b$ in the interaction with $s$ and $t$ .",
"For each $o^p \\in O$ , $\\inf _{\\mathcal {Z}_{s,o^p} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o^p)} \\mathrm {Pr}(\\mathbf {SC}(z_{s,o^p})) = 0$ and $\\inf _{\\mathcal {Z}_{t,o^p} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o^p)} \\mathrm {Pr}(\\mathbf {SC}(z_{t,o^p})) = \\min \\lbrace p,1-p\\rbrace $ , thus giving $\\mathbf {h}_{\\mathrm {Te,must}}^{o^p,\\mathbf {1},\\mathrm {x}}(t,s) = \\min \\lbrace p,1-p\\rbrace $ .",
"One can then easily check that $\\mathbf {m}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}(s,t) = \\sup _{p \\in (0,1)}\\min \\lbrace p,1-p\\rbrace = 0.5$ .",
"Conversely, as the tbt-testing metric compares the success probabilities related to the execution of a single trace per time, we get $\\mathbf {m}_{\\mathrm {Te,tbt}}^{1,\\mathrm {x}}(s,t) = 0$ .",
"Notice that in the case of randomized schedulers, all the randomized resolutions for $t,o^p$ combining the two $a$ -moves can be matched by $s,o^p$ by combining the $b$ -moves and vice versa.",
"Consider now $s,t$ in Fig.",
"REF .",
"Even under randomized schedulers, the tbt-testing distance on them is given by the difference in the success probability of the trace $ac$ (or equivalently $ad$ ) and thus $\\mathbf {m}_{\\mathrm {Te,tbt}}^{1,\\mathrm {x}}(s,t) = 0.4$ .",
"However, we have already showed that $\\mathbf {m}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0.3$ .",
"Finally, we consider the case of trace distances.",
"Consider $t,u$ in Fig.",
"REF .",
"Clearly, $\\mathbf {m}_{\\mathrm {Tr,dis}}^{1,\\mathrm {x}}(t,u) = \\mathbf {m}_{\\mathrm {Tr,tbt}}^{1,\\mathrm {x}}(t,u) = \\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{1,\\mathrm {x}}(t,u) = 0$ .",
"However, in Ex.",
"REF we showed that $\\mathbf {m}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}(t,u) = 1$ .",
"Consider now $s,t$ in Fig.",
"REF .",
"We have that $\\mathbf {m}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0.3$ , but $\\mathbf {m}_{\\mathrm {Tr,dis}}^{1,\\mathrm {x}}(s,t) = 0.7$ and $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{1,\\mathrm {x}}(s,t) = \\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{1,\\mathrm {x}}(s,t) = 0.4$ .",
"$$ Example 7 Non comparability of $\\mathbf {m}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}}$ with $\\mathbf {m}_{\\mathrm {Te,tbt}}^{1,\\mathrm {x}}$ , $\\mathbf {m}_{\\mathrm {Tr,dis}}^{1,\\mathrm {det}}$ and $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{1,\\mathrm {det}}$.",
"For the tbt-testing metrics, consider $s,t$ in Fig.",
"REF .",
"In Ex.",
"REF we showed that $\\mathbf {m}_{\\mathrm {Te,tbt}}^{1,\\mathrm {x}}(s,t) = 0$ .",
"However, the same reasoning giving $\\mathbf {m}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0.5$ , can be applied on suprema success probabilities thus giving $\\mathbf {m}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0.5$ .",
"Consider now $t,u$ in Fig.",
"REF and their interactions with test $o_1$ in Fig.",
"REF .",
"As we consider maximal resolutions only, for both classes of schedulers, the success probability of trace $ab$ evaluates to 1 on $t,o_1$ , whereas on $u,o_1$ it evaluates to 0, due to the maximal resolution corresponding to the rightmost $a$ -branch.",
"Hence $\\mathbf {m}_{\\mathrm {Te,tbt}}^{1,\\mathrm {x}}(t,u) = \\lambda $ , whereas one can easily check that $\\mathbf {m}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}}(t,u) = 0$ .",
"We now proceed to the case of trace distances.",
"For $s,t$ in Fig.",
"REF , we showed that $\\mathbf {m}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0.5$ .",
"However, as both processes have a single resolution each allowing them to execute either trace $abc$ or $abd$ , we can infer that $\\mathbf {m}_{\\mathrm {Tr,dis}}^{1,\\mathrm {x}}(s,t) = \\mathbf {m}_{\\mathrm {Tr,tbt}}^{1,\\mathrm {x}}(s,t) = 0$ .",
"Notice, that this also shows the strictness of the relation $\\mathbf {m}_{\\mathrm {Tr,dis}}^{1,\\mathrm {rand}} < \\mathbf {m}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}}$ .",
"Consider now $s,t$ in Fig.",
"REF .",
"As discussed in Sect.",
"REF we have that $\\mathbf {m}_{\\mathrm {Tr,dis}}^{1,\\mathrm {det}} \\ge \\mathbf {m}_{\\mathrm {Tr,tbt}}^{1,\\mathrm {det}}(s,t) = 0.5$ .",
"However, one can easily check that $\\mathbf {m}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0$ .",
"$$ Example 8 Strictness of $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{1,\\mathrm {x}} < \\mathbf {m}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}}$.",
"Consider $s,t$ in Fig.",
"REF .",
"In Ex.",
"REF we have shown that $\\mathbf {m}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0.7$ .",
"However, since the supremal probability approach to testing proceeds in a trace-by-trace fashion, the $\\sqcup $ -testing distance is given by the difference in the success probability of the trace $ac$ (or $ad$ ) and thus $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{1,\\mathrm {x}}(s,t) = 0.4$ .",
"$$ Example 9 Strictness of $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}< \\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}}$.",
"We stress that this relation is due to the restriction to maximal resolutions, necessary to reason on testing semantics.",
"Consider now $t,u$ in Fig.",
"REF and their interactions with test $o_1$ in Fig.REF .",
"In Ex.REF we have shown that $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}}(t,u) = \\lambda $ .",
"However, one can easily check that $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}(t,u) = 0$ .",
"$$ Example 10 Strictness of $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {x}}< \\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}}$ and of $\\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {x}}< \\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}$.",
"For $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {x}} < \\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}}$ consider $t,u$ in Fig.",
"REF and the test $o_1$ in Fig.",
"REF , by which we get $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {x}}(t,u) = 0$ and $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}}(t,u) = \\lambda $ .",
"Similarly, for $\\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {x}} < \\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}$ consider $s,t$ in Fig.",
"REF with $\\varepsilon _1=\\varepsilon _2=0$ .",
"We have $\\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {x}}(s,t) = 0$ and $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}(s,t) = \\lambda \\cdot 0.5$ , given by the test $o$ corresponding to the leftmost branch of $s$ .",
"$$ Related and future work Trace metrics have been thoroughly studied on quantitative systems, as testified by the spectrum of distances, defined as the generalization of a chosen trace distance, in [17] and the one on Metric Transition Systems (MTSs) in [1].",
"The great variety in these models and the PTSs prevent us to compare the obtained results in detail.",
"Notably, in [1] the trace distance is based on a propositional distance defined over valuations of atomic propositions that characterize the MTS.",
"If on one side such valuation could play the role of the probability distributions in the PTS, it is unclear whether we could combine the ground distance on atomic propositions and the propositional distance, to obtain trace distances comparable to ours.",
"In [3], [13] trace metrics on Markov Chains (MCs) are defined as total variation distances on the cones generated by traces.",
"As in MCs probability depends only on the current state and not on nondeterminism, our quantification over resolutions would be trivial on MCs, giving a total variation distance.",
"Although ours is the first proposal of a metric expressing testing semantics, testing equivalences for probabilistic processes have been studied also in [15], [4], [5].",
"In detail, [15] proposed notions of probabilistic may/must testing for a Kleisli lifting of the PTS model, ie.",
"the transition relation is lifted to a relation $(\\rightarrow )^{\\dagger } \\subseteq (\\Delta (\\mathbf {S}) \\times \\mathcal {A}\\times \\Delta (\\mathbf {S}))$ taking distributions over processes to distributions over processes.",
"Again, the disparity in the two models prevents us from thoroughly comparing the proposed testing relations.",
"As future work, we aim to extend the spectrum of metrics to (bi)simulation metrics [16] and to metrics on different semantic models, and to study their logical characterizations and compositional properties on the same line of [9], [10], [11].",
"Further, we aim to provide efficient algorithms for the evaluation of the proposed metrics and to develop a tool for quantitative process verification: we will use the distance between a process and its specification to quantify how much that process satisfies a given property.",
"Acknowledgements I wish to thank Michele Loreti and Simone Tini for fruitful discussions, and the anonymous referees for their valuable comments and suggestions that helped to improve the paper."
],
[
"Metrics for traces",
"In this Section, we define the metrics measuring the disparities in process behavior wrt.",
"trace semantics.",
"We consider three approaches to the combination of nondeterminism and probability: the trace distribution, the trace-by-trace and the supremal probabilities approach.",
"In defining the behavioral distances, we assume a discount factor $\\lambda \\in (0,1]$, which allows us to specify how much the behavioral distance of future transitions is taken into account [2], [16].",
"The discount factor $\\lambda =1$ expresses no discount, so that the differences in the behavior between $s,t \\in \\mathbf {S}$ are considered irrespective of after how many steps they can be observed."
],
[
"The trace distribution approach",
"In [24] the observable events characterizing the trace semantics are trace distributions, ie.",
"probability distributions over traces.",
"Processes $s,t \\in \\mathbf {S}$ are trace distribution equivalent if, for any resolution for $s$ there is a resolution for $t$ exhibiting the same trace distribution, ie.",
"the execution probability of each trace in the two resolutions is exactly the same, and vice versa.",
"Definition 6 (Trace distribution equivalence [24], [4]) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"Processes $s,t \\in \\mathbf {S}$ are in the trace distribution preorder, written $s \\sqsubseteq _{\\mathrm {Tr,dis}}^{\\mathrm {x}}t$ , if: $\\text{for each } \\mathcal {Z}_s \\in \\mathrm {Res}^{\\mathrm {x}}(s) \\text{ there is } \\mathcal {Z}_t \\in \\mathrm {Res}^{\\mathrm {x}}(t) \\text{ s.t.\\ for each } \\alpha \\in \\mathcal {A}^{\\star } \\colon \\mathrm {Pr}(z_s,\\alpha )) = \\mathrm {Pr}(z_t,\\alpha ))$ .",
"Then, $s,t$ are trace distribution equivalent, notation $s \\sim _{\\mathrm {Tr,dis}}^{\\mathrm {x}}t$ , iff $s \\sqsubseteq _{\\mathrm {Tr,dis}}^{\\mathrm {x}}t$ and $t \\sqsubseteq _{\\mathrm {Tr,dis}}^{\\mathrm {x}}s$ .",
"The quantitative analogue to trace distribution equivalence is based on the evaluation of the differences in the trace distributions of processes: the distance between processes $s,t$ is set to $\\varepsilon \\ge 0$ if, for any resolution for $s$ there is a resolution for $t$ exhibiting a trace distribution differing at most by $\\varepsilon $, meaning that the execution probability of each trace in the two resolutions differs by at most $\\varepsilon $, and vice versa.",
"Definition 7 (Trace distribution metric) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"The trace distribution hemimetric and the trace distribution metric are the functions $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {x}}, \\mathbf {m}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {x}}\\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {x}}(s,t) = \\sup _{\\mathcal {Z}_s \\in \\mathrm {Res}^{\\mathrm {x}}(s)}\\, \\inf _{\\mathcal {Z}_t \\in \\mathrm {Res}^{\\mathrm {x}}(t)}\\, \\sup _{\\alpha \\in \\mathcal {A}^{\\star }} \\lambda ^{|\\alpha |-1}|\\mathrm {Pr}(z_s,\\alpha )) - \\mathrm {Pr}(z_t,\\alpha ))|$ $\\mathbf {m}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {x}}(s,t) = \\max \\lbrace \\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {x}}(s,t),\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {x}}(t,s)\\rbrace $ .",
"We observe that the expression $\\sup _{\\alpha \\in \\mathcal {A}^{\\star }} \\lambda ^{|\\alpha |-1}|\\mathrm {Pr}(z_s,\\alpha )) - \\mathrm {Pr}(z_t,\\alpha ))|$ used in Definition REF corresponds to the (weighted) total variation distance between the trace distributions given by the two resolutions $\\mathcal {Z}_s$ and $\\mathcal {Z}_t$ .",
"An equivalent formulation is given in [27], [12] via the Kantorovich lifting of the discrete metric over traces.",
"We now state that trace distribution hemimetrics and metrics are well-defined and that their kernels are the trace distribution preorders and equivalences, respectively.",
"Theorem 1 Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"Then: The function $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {x}}$ is a 1-bounded hemimetric on $\\mathbf {S}$ , with $\\sqsubseteq _{\\mathrm {Tr,dis}}^{\\mathrm {x}}$ as kernel.",
"The function $\\mathbf {m}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {x}}$ is a 1-bounded pseudometric on $\\mathbf {S}$ , with $\\sim _{\\mathrm {Tr,dis}}^{\\mathrm {x}}$ as kernel.",
"Figure: We will evaluate the trace distances between s p s_p and tt wrt.",
"the different approaches, schedulers and parameter p∈[0,1]p \\in [0,1].In all upcoming examples we will investigate only the traces that are significant for the evaluation of the considered distance.Example 1 Consider processes $s_p$ and $t$ in Figure REF , with $p \\in [0,1]$ .",
"First we evaluate $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {det}}(t,s_p)$ .",
"We expand only the case for the resolution $\\mathcal {Z}_t$ for $t$ obtained from its central $a$ -branch.",
"It assigns probability $0.5$ to both $ab$ and $ac$ .",
"Under deterministic schedulers, any resolution $\\mathcal {Z}_{s_p}$ for $s_p$ can assign positive probability to only one of these traces.",
"Assume this trace is $ab$ , the case $ac$ is analogous.",
"We have either $\\mathrm {Pr}(z_{s_p},ab)) = p$ or $\\mathrm {Pr}(z_{s_p},ab)) = 1$ .",
"Then, $|\\mathrm {Pr}(z_{t},ab)) - \\mathrm {Pr}(z_{s_p},ab))| \\in \\lbrace 0.5,|0.5-p|\\rbrace $ and $|\\mathrm {Pr}(z_{t},ac)) - \\mathrm {Pr}(z_{s_p},ac))| = |0.5-0|=0.5$ .",
"Therefore, $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {det}}(t,s_p) = \\lambda \\cdot 0.5$ , for all $p \\in [0,1]$ .",
"Now we show that $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {det}}(s_p,t) = \\lambda \\cdot \\min \\lbrace p, |0.5 - p|,1-p\\rbrace $ .",
"For each resolution $\\mathcal {Z}_{s_p}$ for $s_p$ we need the resolution for $t$ whose trace distribution is closer to that of $\\mathcal {Z}_{s_p}$ .",
"We expand only the case of $\\mathcal {Z}_{s_p}$ corresponding to the leftmost $a$ -branch of $s_p$ and giving probability 1 to trace $a$ and $p$ to trace $ab$ .",
"We distinguish three subcases, related to the value of $p$ : [(i)] $p \\in [0,0.25]$ : The resolution for $t$ minimizing the distance from $\\mathcal {Z}_{s_p}$ is $\\mathcal {Z}_t^1$ that selects no action for $z^1_{t_1}$ .",
"The distance between $\\mathcal {Z}_{s_p}$ and $\\mathcal {Z}_t^1$ is $\\lambda ^{|ab|-1}|\\mathrm {Pr}(z_{s_p},ab)) - \\mathrm {Pr}(z_t^1,ab))| = \\lambda \\cdot p$ .",
"Notice that in this case $p \\le |0.5-p|,1-p$ .",
"$p \\in (0.25,0.75]$ : The resolution for $t$ that minimizes the distance from $\\mathcal {Z}_{s_p}$ is $\\mathcal {Z}_t^2$ that performs an $a$ -move and evolves to $0.5 \\delta _{z^2_{t_2}} + 0.5 \\delta _{z^2_{t_3}}$ , where $z^2_{t_3}$ that executes no action.",
"The distance between $\\mathcal {Z}_{s_p}$ and $\\mathcal {Z}^2_t$ is $\\lambda ^{|ab|-1}|\\mathrm {Pr}(z_{s_p},ab)) - \\mathrm {Pr}(z^2_t,ab))| = \\lambda \\cdot |0.5 - p|$ .",
"Notice that in this case we have $|0.5-p| \\le p,1-p$ .",
"$p \\in (0.75,1]$ : The resolution for $t$ that minimizes the distance from $\\mathcal {Z}_{s_p}$ is $\\mathcal {Z}_t^3$ that corresponds to the leftmost branch of $t$ .",
"The distance between $\\mathcal {Z}_s$ and $\\mathcal {Z}^3_t$ is $\\lambda ^{|ab|-1}|\\mathrm {Pr}(z_{s_p},ab)) - \\mathrm {Pr}(z^3_t,ab))| = \\lambda \\cdot (1-p)$ .",
"Notice that in this case $1-p \\le |0.5-p|,p$ .",
"In the case of randomized schedulers, one can prove that, since both $s_p,t$ can perform traces $ab$ and $ac$ with probability 1, for any $p \\in [0,1]$ we get $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {rand}}(s_p,t) = \\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {rand}}(t,s_p) =0$ .",
"$$ The trace-by-trace approach Trace distribution equivalences come with some desirable properties, as the full backward compatibility with the fully nondeterministic and fully probabilistic cases (cf.",
"[7]).",
"However, they are not congruences wrt.",
"parallel composition [24], and thus the related metrics cannot be non-expansive.",
"Moreover, due to the crucial rôle of the schedulers in the discrimination process, trace distribution distances are sometimes too demanding.",
"Take, for example, processes $s,t$ in Figure REF , with $\\varepsilon _1,\\varepsilon _2 \\in [0,0.5]$ .",
"We have $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {det}}(s,t) = \\lambda \\cdot 0.5$ and $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {det}}(t,s) = \\lambda \\cdot \\max _{i \\in \\lbrace 1,2\\rbrace } \\max \\lbrace 0.5- \\varepsilon _i, \\varepsilon _i\\rbrace $ , thus giving $\\mathbf {m}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {det}}(s,t) = \\lambda \\cdot 0.5$ for all $\\varepsilon _1,\\varepsilon _2 \\in [0,0.5]$ .",
"However, $s$ and $t$ can perform the same traces with probabilities that differ at most by $\\max (\\varepsilon _1,\\varepsilon _2)$ , which suggests that their trace distance should be $\\lambda \\cdot \\max (\\varepsilon _1,\\varepsilon _2)$ .",
"Specially, for $\\varepsilon _1,\\varepsilon _2=0$ , $s,t$ can perform the same traces with exactly the same probability.",
"Despite this, $s,t$ are still distinguished by trace distribution equivalences.",
"These situations arise since the focus of trace distribution approach is more on resolutions than on traces.",
"To move the focus on traces, the trace-by-trace approach was proposed [4].",
"The idea is to choose first the event that we want to observe, namely a single trace, and only as a second step we let the scheduler perform its selection: processes $s,t$ are equivalent wrt.",
"the trace-by-trace approach if for each trace $\\alpha $ , for each resolution for $s$ there is a resolution for $t$ that assigns to $\\alpha $ the same probability, and vice versa.",
"Figure: For ε 1 ,ε 2 ∈[0,0.5]\\varepsilon _1,\\varepsilon _2 \\in [0,0.5], we have 𝐦 Tr , tbt λ, det (s,t)=𝐦 Tr , tbt λ, rand (s,t)=λ·max(ε 1 ,ε 2 )\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {det}}(s,t)=\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {rand}}(s,t)= \\lambda \\cdot \\max (\\varepsilon _1,\\varepsilon _2), 𝐦 Tr , dis λ, det (s,t)=λ·0.5\\mathbf {m}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {det}}(s,t)= \\lambda \\cdot 0.5 and 𝐦 Tr , dis λ, rand (s,t)=λ·max{0.25+ε 1 ,0.25+ε 2 }\\mathbf {m}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {rand}}(s,t) = \\lambda \\cdot \\max \\lbrace 0.25 + \\varepsilon _1, 0.25 + \\varepsilon _2\\rbrace .Definition 8 (Tbt-trace equivalence [4]) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"We say that $s,t \\in \\mathbf {S}$ are in the tbt-trace preorder, written $s \\sqsubseteq _{\\mathrm {Tr,tbt}}^{x}t$ , if for each $\\alpha \\in \\mathcal {A}^{\\star }$ $\\text{for each } \\mathcal {Z}_s \\in \\mathrm {Res}^{\\mathrm {x}}(s) \\text{ there is } \\mathcal {Z}_t \\in \\mathrm {Res}^{\\mathrm {x}}(t) \\text{ such that } \\mathrm {Pr}(z_s,\\alpha )) = \\mathrm {Pr}(z_t,\\alpha ))$ .",
"Then, $s,t\\in \\mathbf {S}$ are tbt-trace equivalent, notation $s \\sim _{\\mathrm {Tr,tbt}}^{x}t$ , iff $s \\sqsubseteq _{\\mathrm {Tr,tbt}}^{x}t$ and $t \\sqsubseteq _{\\mathrm {Tr,tbt}}^{x}s$ .",
"In [4] it was proved that tbt-trace equivalences enjoy the congruence property and are full backward compatible with the fully nondeterministic and the fully probabilistic cases.",
"We introduce now the quantitative analogous to tbt-trace equivalences.",
"Processes $s,t$ are at distance $\\varepsilon \\ge 0$ if, for each trace $\\alpha $ , for each resolution for $s$ there is a resolution for $t$ such that the two resolutions assign to $\\alpha $ probabilities that differ at most by $\\varepsilon $, and vice versa.",
"Definition 9 (Tbt-trace metric) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"For each $\\alpha \\in \\mathcal {A}^{\\star }$ , the function $\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{\\alpha ,}\\lambda ,\\mathrm {x}} \\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ is defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{\\alpha ,}\\lambda ,\\mathrm {x}}(s,t) = \\lambda ^{|\\alpha |-1}\\; \\sup _{\\mathcal {Z}_s \\in \\mathrm {Res}^{\\mathrm {x}}(s)}\\, \\inf _{\\mathcal {Z}_t \\in \\mathrm {Res}^{\\mathrm {x}}(t)} \\, |\\mathrm {Pr}(z_s,\\alpha )) - \\mathrm {Pr}(z_t,\\alpha ))|$ The tbt-trace hemimetric and the tbt-trace metric are the functions $\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {x}}, \\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {x}}\\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {x}}(s,t) = \\sup _{\\alpha \\in \\mathcal {A}^{\\star }}\\; \\mathbf {h}_{\\mathrm {Tr,tbt}}^{{\\alpha ,}\\lambda ,\\mathrm {x}}(s,t)$ $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {x}}(s,t) = \\max \\lbrace \\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {x}}(s,t), \\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {x}}(t,s)\\rbrace $ .",
"It is not hard to see that for processes in Figure REF we have $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {x}}(s,t) = \\lambda \\cdot \\max (\\varepsilon _1,\\varepsilon _2)$ (and $s \\sim _{\\mathrm {Tr,tbt}}^{x}t$ if $\\varepsilon _1,\\varepsilon _2=0$ ).",
"Notice that, since we consider image finite processes, we are guaranteed that for each trace $\\alpha \\in \\mathcal {A}^{\\star }$ the supremum and infimum in the definition of $\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{\\alpha ,}\\lambda ,\\mathrm {x}}$ are actually achieved.",
"We show now that tbt-trace hemimetrics and metrics are well-defined and that their kernels are the tbt-trace preorders and equivalences, respectively.",
"Theorem 2 Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"Then: The function $\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {x}}$ is a 1-bounded hemimetric on $\\mathbf {S}$ , with $\\sqsubseteq _{\\mathrm {Tr,tbt}}^{x}$ as kernel.",
"The function $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {x}}$ is a 1-bounded pseudometric on $\\mathbf {S}$ , with $\\sim _{\\mathrm {Tr,tbt}}^{x}$ as kernel.",
"Example 2 Consider Figure REF .",
"We get $\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {det}}(s_p,t)$ = $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {det}}(s_p,t)$ = (see Example REF ) $\\lambda \\cdot \\min \\lbrace p,|0.5-p|,1-p\\rbrace $ .",
"The reason why in this particular case the two pseudometrics coincide is that each resolution for $s_p$ gives positive probability to at most one of the traces $ab$ and $ac$ , so that quantifying on traces before or after quantifying on resolutions is irrelevant.",
"Let us evaluate now $\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {det}}(t,s_p)$ .",
"To this aim, we focus on trace $ab$ and the resolution $\\mathcal {Z}_t$ obtained from the central $a$ -branch of $t$ , for which we have $\\mathrm {Pr}(z_t,ab)) = 0.5$ .",
"We need the resolution $\\mathcal {Z}_{s_p}$ for $s_p$ that minimizes $|0.5 - \\mathrm {Pr}(z_{s_p},ab))|$ .",
"Since for any resolutions $\\mathcal {Z}_{s_p}$ for $s_p$ we have $\\mathrm {Pr}(z_{s_p},ab)) \\in \\lbrace 0,p,1\\rbrace $ , we infer that the resolution $\\mathcal {Z}_{s_p}$ we are looking for satisfies $\\mathrm {Pr}(z_{s_p},ab)) = p$ and, therefore, $|0.5 - \\mathrm {Pr}(z_{s_p},ab))|$ = $|0.5 - p|$ .",
"By considering also the other resolutions for $ab$ and, then, the other traces, we can check that $\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {det}}(t,s_p) = \\lambda \\cdot |0.5-p|$ .",
"In Example REF we showed that $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {det}}(t,s_p) = \\lambda \\cdot 0.5$ for all $p \\in [0,1]$ .",
"Hence, we get $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {det}}(t,s_p) = \\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {det}}(t,s_p)$ for $p \\in \\lbrace 0,1\\rbrace $ , and $\\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {det}}(t,s_p) > \\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {det}}(t,s_p)$ for $p \\in (0,1)$ .",
"This disparity is due to the fact that the trace distributions approach forced us to match the resolution for $t$ assigning positive probability to both $ab$ and $ac$ , whereas in the trace-by-trace approach one never consider two traces at the same time.",
"$$ We conclude this section by stating that tbt-trace distances are strictly non-expansive, As a corollary, we re-obtain the (pre)congurence properties for their kernels (proved in [7]).",
"Theorem 3 All distances $\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {det}}$ , $\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {rand}}$ , $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {det}}$ , $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {rand}}$ are strictly non-expansive.",
"The supremal probabilities approach The trace-by-trace approach improves on trace distribution approach since it supports equivalences and metrics that are compositional.",
"Moreover, by focusing on traces instead of resolutions, the trace-by-trace approach puts processes in Figure REF in the expected relations.",
"However, we argue here that trace-by-trace approach on deterministic schedulers still gives some questionable results.",
"Take, for example, processes $s,t$ in Figure REF .",
"We believe that these processes should be equivalent in any semantics approach, since, after performing the action $a$ , they reach two distributions that should be identified, as they assign total probability 1 to states with an identical behavior.",
"But, if we consider the trace $ab$ , the resolution $\\mathcal {Z}_t \\in \\mathrm {Res}^{\\mathrm {det}}(t)$ in Figure REF is such that $\\mathrm {Pr}(z_t, ab)) = 0.5$ , whereas the unique resolution for $s$ assigning positive probability to $ab$ is $\\mathcal {Z}_s$ in Figure REF , for which $\\mathrm {Pr}(z_s,ab)) = 1$ .",
"Hence no resolution in $\\mathrm {Res}^{\\mathrm {det}}(s)$ matches $\\mathcal {Z}_t$ on trace $ab$ , thus giving $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {det}}(s.t) = \\lambda \\cdot 0.5$ and, consequently, $s \\lnot \\sim _{\\mathrm {Tr,tbt}}^{\\mathrm {det}}t$ .",
"This motivates to look for an alternative approach that allows us to equate processes in Figure REF and, at the same time, preserves all the desirable properties of the tbt-trace semantics.",
"We take inspiration from the extremal probabilities approach proposed in [5], which bases on the comparison, for each trace $\\alpha $ , of both suprema and infima execution probabilities, wrt.",
"resolutions, of $\\alpha $ : two processes are equated if they assign the same extremal probabilities to all traces.",
"However, reasoning on infima may cause some arguable results.",
"In particular, it is unclear whether such infima should be evaluated over the whole class of resolutions or over a restricted class, as for instance the resolutions in which the considered trace is actually executed.",
"Besides, desirable properties like the backward compatibility and compositionality are not guaranteed.",
"For all these reasons, we find it more reasonable to define a notion of trace equivalence, and a related metric, based on the comparison of supremal probabilities only.",
"Notice that, if we focus on verification, the comparison of supremal probabilities becomes natural.",
"To exemplify, we let the non-probabilistic case guide us.",
"To verify whether a process $t$ satisfies the specification $S$ , we check that whenever $S$ can execute a particular trace, then so does $t$ .",
"Actually, only positive information is considered: if there is a resolution for $S$ in which a given trace is executed, then this information is used to verify the equivalence.",
"Still, resolutions in which such a trace is not enabled are not considered.",
"The same principle should hold for PTSs: a process should perform all the traces enabled in $S$ and it should do it with at least the same probability, in the perspective that the quantitative behavior expressed in the specification expresses the minimal requirements on process behavior.",
"Focusing on supremal probabilities means relaxing the tbt-trace approach by simply requiring that, for each trace $\\alpha $ and resolution $\\mathcal {Z}_s$ for process $s$ there is a resolution for $t$ assigning to $\\alpha $ at least the same probability given by $\\mathcal {Z}_s$ , and vice versa.",
"Figure: Processes ss and tt are distinguished by ∼ Tr , tbt det \\sim _{\\mathrm {Tr,tbt}}^{\\mathrm {det}}, but related by ∼ Tr ,⊔ det \\sim _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {det}}.We remark that tt and uu are related by all the relations in the three approaches to trace semantics.Definition 10 ($\\bigsqcup $ -trace equivalence) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS and $\\mathrm {x} \\in \\lbrace \\mathrm {det}, \\mathrm {rand}\\rbrace $ .",
"We say that $s,t \\in \\mathbf {S}$ are in the $\\bigsqcup $ -trace preorder, written $s \\sqsubseteq _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {x}}t$ , if for each $\\alpha \\in \\mathcal {A}^{\\star }$ $\\sup _{\\mathcal {Z}_s \\in \\mathrm {Res}^{\\mathrm {x}}(s)} \\mathrm {Pr}(z_s,\\alpha )) \\le \\sup _{\\mathcal {Z}_t \\in \\mathrm {Res}^{\\mathrm {x}}(t)} \\mathrm {Pr}(z_t,\\alpha ))$ .",
"Then, $s,t \\in \\mathbf {S}$ are $\\bigsqcup $ -trace equivalent, notation $s \\sim _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {x}}t$ , iff $s \\sqsubseteq _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {x}}t$ and $t \\sqsubseteq _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {x}}s$ .",
"We stress that all good properties of trace-by-trace approach, as the backward compatibility with the fully nondeterministic and fully probabilistic cases and the non-expansiveness of the metric wrt.",
"parallel composition, are preserved by the supremal probabilities approach (Proposition REF and Theorem REF below).",
"Let $\\sim _{\\mathrm {Tr}}^{\\mathbf {N}}$ denote the trace equivalence on fully nondeterministic systems [8] and $\\sim _{\\mathrm {Tr}}^{\\mathbf {P}}$ denote the one on fully-probabilistic systems [22].",
"Proposition 1 Assume a PTS $P = (\\mathbf {S}, \\mathcal {A}, \\xrightarrow{})$ and processes $s,t \\in \\mathbf {S}$ .",
"Then: If $P$ is fully-nondeterministic, then $s \\sim _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {det}}t \\Leftrightarrow s \\sim _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {rand}}t \\Leftrightarrow s \\sim _{\\mathrm {Tr}}^{\\mathbf {N}}t$ .",
"If $P$ is fully-probabilistic, then $s \\sim _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {det}}t \\Leftrightarrow s \\sim _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {rand}}t \\Leftrightarrow s \\sim _{\\mathrm {Tr}}^{\\mathbf {P}}t$ .",
"The idea behind the quantitative analogue of $\\bigsqcup $ -trace equivalence is that two processes are at distance $\\varepsilon \\ge 0$ if, for each trace, the supremal execution probabilities wrt.",
"the resolutions of nondeterminism for the two processes differ at most by $\\varepsilon $.",
"Definition 11 ($\\bigsqcup $ -trace metric) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"For each $\\alpha \\in \\mathcal {A}^{\\star }$ , the function $\\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{\\alpha ,}\\lambda ,\\mathrm {x}} \\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ is defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{\\alpha ,}\\lambda ,\\mathrm {x}}(s,t) = \\max \\big \\lbrace 0,\\lambda ^{|\\alpha |-1} \\big (\\sup _{\\mathcal {Z}_s \\in \\mathrm {Res}^{\\mathrm {x}}(s)} \\mathrm {Pr}(z_s,\\alpha )) - \\sup _{\\mathcal {Z}_t \\in \\mathrm {Res}^{\\mathrm {x}}(t)} \\mathrm {Pr}(z_t,\\alpha ))\\big )\\big \\rbrace .$ The $\\bigsqcup $ -trace hemimetric and the $\\bigsqcup $ -trace metric are the functions $\\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}, \\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {x}}\\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}(s,t) = \\sup _{\\alpha \\in \\mathcal {A}^{\\star }}\\; \\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{\\alpha ,}\\lambda ,\\mathrm {x}}(s,t)$ and $\\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {x}}(s,t) = \\max \\lbrace \\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}(s,t), \\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}(t,s) \\rbrace $ .",
"We can show that $\\bigsqcup $ -trace hemimetrics and metrics are well-defined and that their kernels are the $\\bigsqcup $ -trace preorders and equivalences, respectively.",
"Theorem 4 Assume a PTS $(\\mathbf {S}, \\mathcal {A}, \\xrightarrow{})$ , $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"Then: The function $\\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}$ is a 1-bounded hemimetric on $\\mathbf {S}$ , with $\\sqsubseteq _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {x}}$ as kernel.",
"The function $\\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {x}}$ is a 1-bounded pseudometric on $\\mathbf {S}$ , with $\\sim _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {x}}$ as kernel.",
"We conclude this section by showing that $\\bigsqcup $ -trace distances are strictly non-expansive.",
"As a corollary, we infer the (pre)congruence property of their kernels.",
"Theorem 5 All distances $\\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{}\\lambda ,\\mathrm {det}}$ , $\\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{}\\lambda ,\\mathrm {rand}}$ , $\\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {det}}$ , $\\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {rand}}$ are strictly non-expansive.",
"Remark 1 We can show that the upper bounds to the distance of composed processes provided in Thms.",
"REF and REF are tight, namely for each distance $d$ considered in these theorems, there are processes $s_1,s_2,t_1,t_2$ with $d(s_1 \\parallel s_2 , t_1 \\parallel t_2) = d(s_1,t_1) + d(s_2,t_2) - d(s_1,t_1) \\cdot d(s_2,t_2)$ .",
"Indeed, for $z_s,z_t$ in Fig.",
"REF , with $\\lambda = 1$ , we have $d(z_s,t_t) = 0.5$ and $d(z_s \\parallel z_s, z_t \\parallel z_t) = 0.75 = 0.5 + 0.5 - 0.5 \\cdot 0.5$ .",
"Comparing the distinguishing power of trace metrics So far, we have discussed the properties of trace-based behavioral distances under different approaches.",
"Our aim is now to place these distances in a spectrum.",
"More precisely, we will order them wrt.",
"their distinguishing power: given the metrics $d,d^{\\prime }$ on $\\mathbf {S}$ , we write $d > d^{\\prime }$ if and only if $d(s,t) \\ge d^{\\prime }(s,t)$ for all $s,t \\in \\mathbf {S}$ and $d(u,v) > d^{\\prime }(u,v)$ for some $u,v \\in \\mathbf {S}$ .",
"Intuitively, for trace distributions and tbt-trace semantics, the distances evaluated on deterministic schedulers are more discriminating than their randomized analogues.",
"Theorem 6 Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\lambda \\in (0,1]$ and $\\mathrm {y} \\in \\lbrace \\mathrm {dis},\\mathrm {tbt}\\rbrace $ .",
"Then: [1.]",
"$\\mathbf {h}_{\\mathrm {Tr,y}}^{\\lambda ,\\mathrm {rand}} < \\mathbf {h}_{\\mathrm {Tr,y}}^{\\lambda ,\\mathrm {det}}$ .",
"$\\mathbf {m}_{\\mathrm {Tr,y}}^{\\lambda ,\\mathrm {rand}} < \\mathbf {m}_{\\mathrm {Tr,y}}^{\\lambda ,\\mathrm {det}}$ .",
"As a corollary of Theorem REF , by using the relations between distances and equivalences in Theorems REF and REF , we re-obtain the relations $\\sim _{\\mathrm {Tr,dis}}^{\\mathrm {det}}\\subset \\sim _{\\mathrm {Tr,dis}}^{\\mathrm {rand}}$ and $\\sim _{\\mathrm {Tr,tbt}}^{\\mathrm {det}}\\subset \\sim _{\\mathrm {Tr,tbt}}^{\\mathrm {rand}}$ proved in [7].",
"Moreover, also the analogous results for preorders follow.",
"As one can expect, the metrics on trace distributions are more discriminating than their corresponding ones in the trace-by-trace approach.",
"Theorem 7 Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"Then: [1.]",
"$\\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {x}} < \\mathbf {h}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {x}}$ .",
"$\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {x}}< \\mathbf {m}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {x}}$ .",
"As a corollary, by using the kernel relations given in Theorems REF and REF , we re-obtain the relation $\\sim _{\\mathrm {Tr,dis}}^{\\mathrm {x}}\\subset \\sim _{\\mathrm {Tr,tbt}}^{x}$ proved in [7] and we get $\\sqsubseteq _{\\mathrm {Tr,dis}}^{\\mathrm {x}}\\subset \\sqsubseteq _{\\mathrm {Tr,tbt}}^{x}$ .",
"Moreover, we remark that $\\mathbf {m}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {rand}}$ is not comparable with $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {det}}$ .",
"This is mainly due to the randomization process and it is witnessed by processes in Figure REF , where $\\mathbf {m}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {rand}}(s,t)=\\lambda \\cdot \\max \\lbrace 0.25+\\varepsilon _1,0.25+\\varepsilon _2\\rbrace $ and $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {det}}(s,t) = \\lambda \\cdot \\max (\\varepsilon _1,\\varepsilon _2)$ and Figure REF , where $\\mathbf {m}_{\\mathrm {Tr,dis}}^{\\lambda ,\\mathrm {rand}}(s,t)=0$ and $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {det}}(s,t) = \\lambda \\cdot 0.5$ .",
"We focus now on supremal probabilities approach, that comes with a particularly interesting result: the $\\bigsqcup $ -trace metric on deterministic schedulers coincides with tbt-trace metrics on randomized schedulers.",
"Moreover, $\\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {det}}$ coincides also with its randomized version.",
"Theorem 8 Assume a PTS $P = (\\mathbf {S}, \\mathcal {A}, \\xrightarrow{})$ and $\\lambda \\in (0,1]$ .",
"Then: [1.]",
"$\\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{}\\lambda ,\\mathrm {det}} = \\mathbf {h}_{\\mathrm {Tr,}\\sqcup }^{{}\\lambda ,\\mathrm {rand}} = \\mathbf {h}_{\\mathrm {Tr,tbt}}^{{}\\lambda ,\\mathrm {rand}}$ .",
"$\\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {det}}= \\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {rand}}= \\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {rand}}$ .",
"This result is fundamental in the perspective of the application of our trace metrics to process verification: by comparing solely the suprema execution probabilities of the linear properties of interest we get same expressive power of a pairwise comparison of the probabilities in all possible randomized resolutions of nondeterminism.",
"Clearly, Theorem REF together with the kernel relations from Thms REF and REF imply that the relations for the supremal probabilities semantics coincide with those for the tbt-trace semantics wrt.",
"randomized schedulers, ie.",
"$\\sqsubseteq _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {det}}= \\sqsubseteq _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {rand}}= \\sqsubseteq _{\\mathrm {Tr,tbt}}^{\\mathrm {rand}}$ and $\\sim _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {det}}= \\sim _{\\mathrm {Tr,}\\sqcup }^{\\mathrm {rand}}= \\sim _{\\mathrm {Tr,tbt}}^{\\mathrm {rand}}$ .",
"Metrics for testing Testing semantics [14] compares processes according to their capacity to pass a test.",
"The latter is a PTS equipped with a distinguished state indicating the success of the test.",
"Definition 12 (Test) A nondeterministic probabilistic test transition systems (NPT) is a finite PTS $(\\mathbf {O}, \\mathcal {A}, \\xrightarrow{})$ where $\\mathbf {O}$ is a set of processes, called tests, containing a distinguished success process $\\surd $ with no outgoing transitions.",
"We say that a computation from $o \\in \\mathbf {O}$ is successful iff its last state is $\\surd $ .",
"Given a process $s$ and a test $o$ , we can consider the interaction system among the two.",
"This models the response of the process to the application of the test, so that $s$ passes the test $o$ if there is a computation in the interaction system that reaches $\\surd $ .",
"Informally, the interaction system is the result of the parallel composition of the process with the test.",
"Definition 13 (Interaction system) The interaction system of a PTS $(\\mathbf {S},\\mathcal {A},$ $\\xrightarrow{})$ and an NPT $(\\mathbf {O},\\mathcal {A},\\xrightarrow{}_{\\mathbf {O}})$ is the PTS $(\\mathbf {S}\\times \\mathbf {O}, \\mathcal {A},\\xrightarrow{}^{\\prime })$ where: [(i)] $(s,o) \\in \\mathbf {S}\\times \\mathbf {O}$ is called a configuration and is successful iff $o = \\surd $ ; a computation from $(s,o) \\in \\mathbf {S}\\times \\mathbf {O}$ is successful iff its last configuration is successful.",
"For $(s, o)$ and $\\mathcal {Z}_{s,o} \\in \\mathrm {Res}^{\\mathrm {x}}(s,o)$ , we let $\\mathbf {SC}(z_{s,o})$ be the set of successful computations from $z_{s,o}$ .",
"For $\\alpha \\in \\mathcal {A}^{\\star }$ , $\\mathbf {SC}(z_{s,o},\\alpha )$ is the set of $\\alpha $ -compatible successful computations from $z_{s,o}$ .",
"Testing semantics should compare processes wrt.",
"their probability to pass a test.",
"In this Section we consider three approaches to it: [(i)] the may/must, the trace-by-trace, and the supremal probabilities.",
"For each approach, we present (hemi,pseudo)metrics that provide a quantitative variant of the considered testing equivalence.",
"To the best of our knowledge, ours is the first attempt in this direction.",
"The may/must approach In the original work on nondeterministic systems [14], testing equivalence was defined via the may and must preorders.",
"The former expresses the ability of processes to pass a test.",
"The latter expresses the impossibility to fail a test.",
"When also probability is considered, these two preorders are defined, resp., in terms of suprema and infima success probabilities [29].",
"Definition 14 (May/must testing equivalence, [29]) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $(\\mathbf {O},\\mathcal {A},\\xrightarrow{}_O)$ an NPT and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"We say that $s,t \\in \\mathbf {S}$ are in the may testing preorder, written $s \\sqsubseteq _{\\mathrm {Te,may}}^{x}t$ , if for each $o \\in \\mathbf {O}$ $\\sup _{\\mathcal {Z}_{s,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o)} \\mathrm {Pr}(\\mathbf {SC}(z_{s,o})) \\le \\sup _{\\mathcal {Z}_{t,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o)} \\mathrm {Pr}(\\mathbf {SC}(z_{t,o}))$ .",
"Then, $s,t\\in \\mathbf {S}$ are may testing equivalent, written $s \\sim _{\\mathrm {Te,may}}^{x}t$ , iff $s \\sqsubseteq _{\\mathrm {Te,may}}^{x}t$ and $t \\sqsubseteq _{\\mathrm {Te,may}}^{x}s$ .",
"The notions of must testing preorder, $\\sqsubseteq _{\\mathrm {Te,must}}^{x}$ , and must testing equivalence, $\\sim _{\\mathrm {Te,must}}^{x}$ , are obtained by replacing the suprema in $\\sqsubseteq _{\\mathrm {Te,may}}^{x}$ and $\\sim _{\\mathrm {Te,may}}^{x}$ , resp., with infima.",
"Finally, we say that $s,t \\in \\mathbf {S}$ are in the may/must testing preorder, written $s \\sqsubseteq _{\\mathrm {Te,mM}}^{x}t$ , if $s \\sqsubseteq _{\\mathrm {Te,may}}^{x}t$ and $s \\sqsubseteq _{\\mathrm {Te,must}}^{x}t$ .",
"They are may/must testing equivalent, written $s \\sim _{\\mathrm {Te,mM}}^{x}t$ , iff $s \\sqsubseteq _{\\mathrm {Te,mM}}^{x}t$ and $t \\sqsubseteq _{\\mathrm {Te,mM}}^{x}s$ .",
"The quantitative analogue to may/must testing equivalence bases on the evaluation of the differences in the extremal success probabilities.",
"The distance between $s,t \\in \\mathbf {S}$ is set to $\\varepsilon \\ge 0$ if the maximum between the difference in the suprema and infima success probabilities wrt.",
"all resolutions of nondeterminism for $s$ and $t$ is at most $\\varepsilon $ .",
"We introduce a function $\\omega :\\mathbf {O}\\rightarrow (0,1]$ that assigns to each test $o$ the proper discount.",
"In fact, as the success probabilities in the may/must semantics are not related to the execution of a particular trace, in general we cannot define a discount factor as we did for the trace distances.",
"However, a similar construction may be regained when only tests with finite depth are considered.",
"In that case, we could define $\\omega (o) = \\lambda ^{\\mathrm {depth}(o)}$ , for $\\lambda \\in (0,1]$ .",
"We will use $\\mathbf {1}$ to denote the 1 constant function.",
"Definition 15 (May/must testing metric) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $(\\mathbf {O},\\mathcal {A},\\xrightarrow{}_O)$ an NPT, $\\omega :\\mathbf {O}\\rightarrow (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"For each $o\\in \\mathbf {O}$ , the function $\\mathbf {h}_{\\mathrm {Te,may}}^{{o,}\\omega ,\\mathrm {x}}\\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ is defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Te,may}}^{{o,}\\omega ,\\mathrm {x}}(s,t)=\\max \\Big \\lbrace 0 , \\omega (o)\\Big (\\sup _{\\mathcal {Z}_{s,o}\\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o)} \\mathrm {Pr}(\\mathbf {SC}(z_{s,o})) - \\sup _{\\mathcal {Z}_{t,o}\\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o)} \\mathrm {Pr}(\\mathbf {SC}(z_{t,o}))\\Big ) \\Big \\rbrace $ Function $\\mathbf {h}_{\\mathrm {Te,must}}^{{o,}\\omega ,\\mathrm {x}}\\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ is obtained by replacing the suprema in $\\mathbf {h}_{\\mathrm {Te,may}}^{{o,}\\omega ,\\mathrm {x}}$ with infima.",
"Given $\\mathrm {y} \\in \\lbrace \\mathrm {may}, \\mathrm {must}\\rbrace $ , the $\\mathrm {y}$ testing hemimetric and the $\\mathrm {y}$ testing metric are the functions $\\mathbf {h}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}, \\mathbf {m}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}} \\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}(s,t) = \\sup _{o \\in \\mathbf {O}}\\; \\mathbf {h}_{\\mathrm {Te,y}}^{o,\\omega ,\\mathrm {x}}(s,t)$ and $\\mathbf {m}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}(s,t) = \\max \\lbrace \\mathbf {h}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}(s,t), \\mathbf {h}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}(t,s) \\rbrace $ .",
"The may/must testing hemimetric and the may/must testing metric are the functions $\\mathbf {h}_{\\mathrm {Te,mM}}^{{}\\omega ,\\mathrm {x}}, \\mathbf {m}_{\\mathrm {Te,mM}}^{\\omega ,\\mathrm {x}}\\colon $ $\\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Te,mM}}^{{}\\omega ,\\mathrm {x}}(s,t) = \\max \\lbrace \\mathbf {h}_{\\mathrm {Te,may}}^{{}\\omega ,\\mathrm {x}}(s,t), \\mathbf {h}_{\\mathrm {Te,must}}^{{}\\omega ,\\mathrm {x}}(s,t) \\rbrace $ .",
"$\\mathbf {m}_{\\mathrm {Te,mM}}^{\\omega ,\\mathrm {x}}(s,t) = \\max \\lbrace \\mathbf {m}_{\\mathrm {Te,may}}^{\\omega ,\\mathrm {x}}{}(s,t), \\mathbf {m}_{\\mathrm {Te,must}}^{\\omega ,\\mathrm {x}}{}(s,t) \\rbrace $ .",
"Theorem 9 Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\omega :\\mathbf {O}\\rightarrow (0,1]$ , $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ and $\\mathrm {y} \\in \\lbrace \\mathrm {may},\\mathrm {must}, \\mathrm {mM}\\rbrace $ : The function $\\mathbf {h}_{\\mathrm {Te,y}}^{\\lambda ,\\mathrm {x}}$ is a 1-bounded hemimetric on $\\mathbf {S}$ , with $\\sqsubseteq _{\\mathrm {Te,y}}^{\\mathrm {x}}$ as kernel.",
"The function $\\mathbf {m}_{\\mathrm {Te,y}}^{\\lambda ,\\mathrm {x}}$ is a 1-bounded pseudometric on $\\mathbf {S}$ , with $\\sim _{\\mathrm {Te,y}}^{\\mathrm {x}}$ as kernel.",
"Figure: We use the tests o 1 ,o 2 o_1,o_2 to evaluate the distance between processes s,t,us,t,u in Fig.",
"wrt.",
"testing semantics.•\\bullet represents a generic configuration in the interaction system.In all upcoming examples we will consider only the tests and traces that are significant for the evaluations of the testing metrics.Example 3 Consider $t,u$ in Fig REF and their interactions with test $o_1$ in Fig REF .",
"Clearly, $(t,o_1)$ and $(u,o_1)$ have the same suprema success probabilities.",
"In fact, they both have a maximal resolution assigning probability 1 to the trace $ab$ , ie.",
"the only successful trace in the considered case.",
"As the same holds for all tests we get $\\mathbf {m}_{\\mathrm {Te,may}}^{\\omega ,\\mathrm {x}}(t,u) = 0$ .",
"Conversely, if we compare the infima success probabilities, we get $\\inf _{\\mathcal {Z}_{t,o_1} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o_1)} = 1$ since $(t,o_1)$ has only one maximal resolution corresponding to $(t,o_1)$ itself and that with probability 1 reaches $\\surd $ .",
"Still, $\\inf _{\\mathcal {Z}_{u,o_1} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(u,o_1)} = 0$ , given by the maximal resolution corresponding to $(u,o_1) \\stackrel{a}{{\\twoheadrightarrow }} \\mathrm {nil}$ .",
"Hence, we can infer $\\mathbf {m}_{\\mathrm {Te,must}}^{\\omega ,\\mathrm {x}}(t,u) = \\omega (o_1) \\cdot |1-0| = \\omega (o_1)$ .",
"$$ We can finally observe that both $\\mathbf {h}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}$ and $\\mathbf {m}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}$ are non-expansive.",
"Theorem 10 Let $\\omega :\\mathbf {O}\\rightarrow (0,1]$ and $\\mathrm {y} \\in \\lbrace \\mathrm {may},\\mathrm {must}, \\mathrm {mM}\\rbrace $ .",
"$\\mathbf {h}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}$ and $\\mathbf {m}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}$ are non-expansive.",
"The trace-by-trace approach In [7] it was proved that the may/must is fully backward compatible with the restricted class of processes only if the same restriction is applied to the class of tests, ie.",
"if we consider resp.",
"fully nondeterministic and fully probabilistic tests only.",
"This is due to the duplication ability of nondeterministic probabilistic tests.",
"However, by applying the trace-by-trace approach to testing semantics, we regain the full backward compatibility wrt.",
"all tests (cf.",
"[7]).",
"Definition 16 (Tbt-testing equivalence) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $(\\mathbf {O},\\mathcal {A},\\xrightarrow{}_O)$ an NPT, $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"We say that $s,t \\in \\mathbf {S}$ are in the tbt-testing preorder, written $s \\sqsubseteq _{\\mathrm {Te,tbt}}^{x}t$ , if for each $o \\in \\mathbf {O}$ and $\\alpha \\in \\mathcal {A}^{\\star }$ $\\text{for each } \\mathcal {Z}_{s,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o) \\text{ there is } \\mathcal {Z}_{t,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o) \\text{ st. } \\mathrm {Pr}(\\mathbf {SC}(z_{s,o},\\alpha )) = \\mathrm {Pr}(\\mathbf {SC}(z_{t,o},\\alpha ))$ .",
"Then, $s,t\\in \\mathbf {S}$ are tbt-testing equivalent, notation $s \\sim _{\\mathrm {Te,tbt}}^{x}t$ , iff $s \\sqsubseteq _{\\mathrm {Te,tbt}}^{x}t$ and $t \\sqsubseteq _{\\mathrm {Te,tbt}}^{x}s$ .",
"The definition of the tbt-testing metric naturally follows from Def.",
"REF .",
"Definition 17 (Tbt-testing metric) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $(\\mathbf {O},\\mathcal {A},\\xrightarrow{}_O)$ an NPT, $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"For each $o \\in \\mathbf {O}$ and $\\alpha \\in \\mathcal {A}^{\\star }$ , function $\\mathbf {h}_{\\mathrm {Te,tbt}}^{{o,\\alpha ,}\\lambda ,\\mathrm {x}} \\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ is defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Te,tbt}}^{{o,\\alpha ,}\\lambda ,\\mathrm {x}}(s,t) = \\lambda ^{|\\alpha |-1}\\; \\sup _{\\mathcal {Z}_{s,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o)}\\, \\inf _{\\mathcal {Z}_{t,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o)} \\, |\\mathrm {Pr}(\\mathbf {SC}(z_{s,o},\\alpha )) - \\mathrm {Pr}(\\mathbf {SC}(z_{t,o},\\alpha ))|$ The tbt-testing hemimetric and the tbt-testing metric are the functions $\\mathbf {h}_{\\mathrm {Te,tbt}}^{{}\\lambda ,\\mathrm {x}}, \\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}}\\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Te,tbt}}^{{}\\lambda ,\\mathrm {x}}(s,t) = \\sup _{o \\in \\mathbf {O}} \\; \\sup _{\\alpha \\in \\mathcal {A}^{\\star }}\\; \\mathbf {h}_{\\mathrm {Te,tbt}}^{{o,\\alpha ,}\\lambda ,\\mathrm {x}}(s,t)$ $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}}(s,t) = \\max \\lbrace \\mathbf {h}_{\\mathrm {Te,tbt}}^{{}\\lambda ,\\mathrm {x}}(s,t), \\mathbf {h}_{\\mathrm {Te,tbt}}^{{}\\lambda ,\\mathrm {x}}(t,s) \\rbrace $ .",
"Theorem 11 Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"Then: The function $\\mathbf {h}_{\\mathrm {Te,tbt}}^{{}\\lambda ,\\mathrm {x}}$ is a 1-bounded hemimetric on $\\mathbf {S}$ , with $\\sqsubseteq _{\\mathrm {Te,tbt}}^{x}$ as kernel.",
"The function $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}}$ is a 1-bounded pseudometric on $\\mathbf {S}$ , with $\\sim _{\\mathrm {Te,tbt}}^{x}$ as kernel.",
"Example 4 Consider $s,t$ in Fig.",
"REF and their interactions with test $o_2$ in Fig.",
"REF .",
"By the same reasoning detailed in the first paragraph of Sect.",
"REF , we get $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {det}}(s,t)=\\lambda \\cdot 0.5$ and $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {rand}}(s,t) = 0$ .",
"$$ When the tbt-approach is used to define testing metrics, we get a refinement of the non-expansiveness property to strict non-expansiveness.",
"Theorem 12 All distances $\\mathbf {h}_{\\mathrm {Te,tbt}}^{{}\\lambda ,\\mathrm {det}}$ , $\\mathbf {h}_{\\mathrm {Te,tbt}}^{{}\\lambda ,\\mathrm {rand}}$ , $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {det}}$ , $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {rand}}$ are strictly non-expansive.",
"The supremal probabilities approach If we focus on verification, we can use the testing semantics to verify whether a process will behave as intended by its specification in all possible environments, as modeled by the interaction with the tests.",
"Informally, we could see each test as a set of requests of the environment to the system: the ones ending in the success state are those that must be answered.",
"The interaction of the specification with the test then tells us whether the system is able to provide those answers.",
"Thus, an implementation has to guarantee at least all the answers provided by the specification.",
"For this reason we decided to introduce also a supremal probabilities variant of testing semantics: for each test and for each trace we compare the suprema wrt.",
"all resolutions of nondeterminism of the probabilities of processes to reach success by performing the considered trace.",
"Definition 18 ($\\bigsqcup $ -testing equivalence) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $(\\mathbf {O},\\mathcal {A},\\xrightarrow{}_O)$ an NPT and $\\mathrm {x} \\in \\lbrace \\mathrm {det}, \\mathrm {rand}\\rbrace $ .",
"We say that $s,t \\in \\mathbf {S}$ are in the $\\bigsqcup $ -testing preorder, written $s \\sqsubseteq _{\\mathrm {Te,}\\sqcup }^{\\mathrm {x}}t$ , if for each $o \\in \\mathbf {O}$ and $\\alpha \\in \\mathcal {A}^{\\star }$ $\\sup _{\\mathcal {Z}_{s,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o)} \\mathrm {Pr}(\\mathbf {SC}(z_{s,o},\\alpha )) \\le \\sup _{\\mathcal {Z}_{t,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o)} \\mathrm {Pr}(\\mathbf {SC}(z_{t,o},\\alpha ))$ .",
"Then, $s,t \\in \\mathbf {S}$ are $\\bigsqcup $ -testing equivalent, notation $s \\sim _{\\mathrm {Te,}\\sqcup }^{\\mathrm {x}}t$ , iff $s \\sqsubseteq _{\\mathrm {Te,}\\sqcup }^{\\mathrm {x}}t$ and $t \\sqsubseteq _{\\mathrm {Te,}\\sqcup }^{\\mathrm {x}}s$ .",
"We obtain the $\\bigsqcup $ -testing metric as a direct adaptation to tests of Definition REF .",
"Definition 19 ($\\bigsqcup $ -testing metric) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $(\\mathbf {O},\\mathcal {A},\\xrightarrow{}_O)$ an NPT, $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"For each $o \\in \\mathbf {O}$ , $\\alpha \\in \\mathcal {A}^{\\star }$ , the function $\\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{o,\\alpha ,}\\lambda ,\\mathrm {x}} \\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ is defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{o,\\alpha ,}\\lambda ,\\mathrm {x}}(s,t) = \\max \\Big \\lbrace 0,\\lambda ^{|\\alpha |-1} \\Big (\\sup _{\\mathcal {Z}_{s,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o)} \\!\\!\\!\\!\\!\\mathrm {Pr}(\\mathbf {SC}(z_{s,o},\\alpha )) - \\sup _{\\mathcal {Z}_{t,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o)}\\!\\!\\!\\!\\!",
"\\mathrm {Pr}(\\mathbf {SC}(z_{t,o},\\alpha ))\\Big )\\Big \\rbrace .$ The $\\bigsqcup $ -testing hemimetric and the $\\bigsqcup $ -testing metric are the functions $\\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}, \\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}\\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}(s,t) = \\sup _{o \\in \\mathbf {O}} \\; \\sup _{\\alpha \\in \\mathcal {A}^{\\star }}\\; \\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{o,\\alpha ,}\\lambda ,\\mathrm {x}}(s,t)$ ; $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}(s,t) = \\max \\lbrace \\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}(s,t), \\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}(t,s) \\rbrace $ .",
"Theorem 13 Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS and $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"Then: The function $\\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}$ is a 1-bounded hemimetric on $\\mathbf {S}$ , with $\\sqsubseteq _{\\mathrm {Te,}\\sqcup }^{\\mathrm {x}}$ as kernel.",
"The function $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}$ is a 1-bounded pseudometric on $\\mathbf {S}$ , with $\\sim _{\\mathrm {Te,}\\sqcup }^{\\mathrm {x}}$ as kernel.",
"Finally, we can show that both $\\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}$ and $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}$ are strictly non-expansive.",
"Theorem 14 All distances $\\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{}\\lambda ,\\mathrm {det}}$ , $\\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{}\\lambda ,\\mathrm {rand}}$ , $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {det}}$ , $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {rand}}$ are strictly non-expansive.",
"Remark 2 For all distances $d$ considered in Thms.",
"REF , REF , REF and processes $z_s,z_t$ in Fig.",
"REF , with $\\lambda =1$ , we have $d(z_s , z_t) = 0.5$ and $d(z_s \\parallel z_s , z_t \\parallel z_t) = 0.75 = 0.5 + 0.5 - 0.5 \\cdot 0.5$ .",
"Hence, the upper bounds to the distance between composed processes provided in Thms.",
"REF and REF are tight.",
"We leave as a future work the analogous result for distances considered in Thm.",
"REF .",
"Comparing the distinguishing power of testing metrics Figure: The spectrum of trace and testing (hemi)metrics.d→d ' d \\rightarrow d^{\\prime } stands for d>d ' d > d^{\\prime }.We present only the general form with 𝐝∈{𝐡,𝐦}\\mathbf {d} \\in \\lbrace \\mathbf {h},\\mathbf {m}\\rbrace as the relations among the hemimetrics are the same wrt.",
"those among the metrics.The complete spectrum can be obtained by relating each metric with the respective hemimetric.We study the distinguishing power of the testing metrics presented in this section and the trace metrics defined in Sect.",
", thus obtaining the spectrum in Fig.",
"REF .",
"Firstly, we compare the expressiveness of the testing metrics wrt.",
"the chosen class of schedulers.",
"The distinguishing power of testing metrics based on may-must and supremal probabilities approaches is not influenced by this choice.",
"Differently, in the tbt approach, the distances evaluated on deterministic schedulers are more discriminating than their analogues on randomized schedulers.",
"Theorem 15 Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\lambda \\in (0,1]$ , $\\omega \\colon \\mathbf {O}\\rightarrow (0,1]$ $\\mathrm {y} \\in \\lbrace \\mathrm {may},\\mathrm {must}, \\mathrm {mM}\\rbrace $ and $\\mathbf {d} \\in \\lbrace \\mathbf {h},\\mathbf {m}\\rbrace $ : $\\begin{array}{llll}1.\\, \\mathbf {d}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {rand}} = \\mathbf {d}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {det}} &2.\\, \\mathbf {d}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {rand}} < \\mathbf {d}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {det}} &3.\\, \\mathbf {d}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {rand}} = \\mathbf {d}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {det}}\\end{array}$ From Thm.",
"REF , by using the kernel relations in Thms.",
"REF and REF , we regain relations $\\sim _{\\mathrm {Te,may}}^{\\mathrm {rand}}= \\sim _{\\mathrm {Te,may}}^{\\mathrm {det}}$ , $\\sim _{\\mathrm {Te,must}}^{\\mathrm {rand}}= \\sim _{\\mathrm {Te,must}}^{\\mathrm {det}}$ , $\\sim _{\\mathrm {Te,mM}}^{\\mathrm {rand}}= \\sim _{\\mathrm {Te,mM}}^{\\mathrm {det}}$ , $\\sim _{\\mathrm {Te,tbt}}^{\\mathrm {det}}\\subset \\sim _{\\mathrm {Te,tbt}}^{\\mathrm {rand}}$ , and their analogues on preorders, proved in [7].",
"From Thm.",
"REF we get $\\sqsubseteq _{\\mathrm {Te,}\\sqcup }^{\\mathrm {rand}}= \\sqsubseteq _{\\mathrm {Te,}\\sqcup }^{\\mathrm {det}}$ and $\\sim _{\\mathrm {Te,}\\sqcup }^{\\mathrm {rand}}= \\sim _{\\mathrm {Te,}\\sqcup }^{\\mathrm {det}}$ .",
"The strictness of the inequality in Thm.",
"REF .2, is witnessed by processes $s,t$ in Fig REF and their interactions with the test $o_2$ in Fig REF .",
"The same reasoning applied in the first paragraph of Sect.",
"REF to obtain $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {det}}(s,t) = \\lambda \\cdot 0.5$ and $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {rand}}(s,t) = 0$ , gives $\\mathbf {h}_{\\mathrm {Te,tbt}}^{{o_2,}\\lambda ,\\mathrm {det}}(t,s) = \\lambda \\cdot 0.5 = \\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {det}}{}(s,t)$ and $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {rand}}{}(s,t) = 0$ .",
"We proceed to compare the expressiveness of each metric wrt.",
"the other semantics.",
"Our results are fully compatible with the spectrum on probabilistic relations presented in [7].",
"Theorem 16 Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\lambda \\in (0,1]$ , $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ and $\\mathbf {d} \\in \\lbrace \\mathbf {h},\\mathbf {m}\\rbrace $ : $\\begin{array}{llll}1.\\, \\mathbf {d}_{\\mathrm {Te,may}}^{\\omega ,\\mathrm {x}} < \\mathbf {d}_{\\mathrm {Te,mM}}^{\\omega ,\\mathrm {x}} &2.\\, \\mathbf {d}_{\\mathrm {Te,must}}^{\\omega ,\\mathrm {x}} < \\mathbf {d}_{\\mathrm {Te,mM}}^{\\omega ,\\mathrm {x}} &3.\\, \\mathbf {d}_{\\mathrm {Te,}\\sqcup }^{1,\\mathrm {x}} < \\mathbf {d}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}} &4.\\, \\mathbf {d}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}} < \\mathbf {d}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}} \\\\5.\\, \\mathbf {d}_{\\mathrm {Tr,dis}}^{1,\\mathrm {rand}} < \\mathbf {d}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}} &6.\\, \\mathbf {d}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {x}} < \\mathbf {d}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}} &7.\\, \\mathbf {d}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {x}} < \\mathbf {d}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}\\end{array}$ The following Examples prove the strictness of the inequalities in Thm.",
"REF and the non comparability of the (hemi)metrics as shown in Fig.",
"REF .",
"For simplicity, we consider only the cases of the metrics.",
"Figure: Processes s,ts,t and their interaction systems with the test o 2 o_2 in Fig.",
".Example 5 Non comparability of $\\mathbf {m}_{\\mathrm {Te,may}}^{\\omega ,\\mathrm {x}}$ with $\\mathbf {m}_{\\mathrm {Te,must}}^{\\omega ,\\mathrm {x}}$.",
"In Ex.",
"REF we showed that for $t,u$ in Fig.",
"REF from their interaction with the test $o_1$ in Fig.",
"REF we obtain that $\\mathbf {m}_{\\mathrm {Te,must}}^{\\omega ,\\mathrm {x}}(t,u) = \\omega (o_1)$ , whereas one can easily check that $\\mathbf {m}_{\\mathrm {Te,may}}^{\\omega ,\\mathrm {x}}(t,u) = 0$ .",
"Consider now $s,t$ and their interactions in Fig.",
"REF with the test $o_2$ from Fig.",
"REF .",
"Clearly, we have $\\sup _{\\mathcal {Z}_{s,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o)}\\mathrm {Pr}(\\mathbf {SC}(z_{s,o})) = 1$ and $\\sup _{\\mathcal {Z}_{t,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o)}\\mathrm {Pr}(\\mathbf {SC}(z_{t,o})) = 0.3$ and thus $\\mathbf {m}_{\\mathrm {Te,may}}^{\\omega ,\\mathrm {x}}(s,t) = 0.7 \\cdot \\omega (o_2)$ .",
"Conversely, if we consider infima success probabilities, we have $\\inf _{\\mathcal {Z}_{s,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o)}\\mathrm {Pr}(\\mathbf {SC}(z_{s,o})) = 0$ and $\\sup _{\\mathcal {Z}_{t,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o)}\\mathrm {Pr}(\\mathbf {SC}(z_{t,o})) = 0.3$ .",
"Thus, $\\mathbf {m}_{\\mathrm {Te,must}}^{\\omega ,\\mathrm {x}}(s,t) = 0.3\\cdot \\omega (o_2)$ .",
"$$ Figure: Processes s,ts,t are such that 𝐝 Te , tbt 1,x (s,t)=0\\mathbf {d}_{\\mathrm {Te,tbt}}^{1,\\mathrm {x}}(s,t) = 0 and 𝐝 Te , must 1,x (s,t)=0.5\\mathbf {d}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0.5, as witnessed by the test o 1/2 o^{1/2}.Example 6 Non comparability of $\\mathbf {m}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}$ with $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{1,\\mathrm {x}}$ , $\\mathbf {m}_{\\mathrm {Te,tbt}}^{1,\\mathrm {x}}$ , $\\mathbf {m}_{\\mathrm {Tr,dis}}^{1,\\mathrm {x}}$ , $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{1,\\mathrm {x}}$ and $\\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{1,\\mathrm {x}}$.",
"We start with $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{1,\\mathrm {x}}$ .",
"Form Ex.",
"REF we know that for $t,u$ in Fig.",
"REF it holds $\\mathbf {m}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}(t,u) = \\mathbf {1}$ .",
"Since both $t$ and $u$ have maximal resolutions giving probability 1 to either $ab$ or $ac$ , we get $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{1,\\mathrm {x}}(t,u) =0$ .",
"Consider now $s,t$ in Fig.",
"REF .",
"In Ex.",
"REF we showed that $\\mathbf {m}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0.3$ .",
"From the interaction systems in Fig.",
"REF , by considering the superma success probabilities of trace $ac$ , we obtain that $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{1,\\mathrm {x}} = 0.4$ .",
"Next we deal with the tbt-testing metrics.",
"Consider $s,t$ in Fig.",
"REF and the family of tests $O = \\lbrace o^p \\mid p \\in (0,1)\\rbrace $ , each duplicating the actions $b$ in the interaction with $s$ and $t$ .",
"For each $o^p \\in O$ , $\\inf _{\\mathcal {Z}_{s,o^p} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o^p)} \\mathrm {Pr}(\\mathbf {SC}(z_{s,o^p})) = 0$ and $\\inf _{\\mathcal {Z}_{t,o^p} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o^p)} \\mathrm {Pr}(\\mathbf {SC}(z_{t,o^p})) = \\min \\lbrace p,1-p\\rbrace $ , thus giving $\\mathbf {h}_{\\mathrm {Te,must}}^{o^p,\\mathbf {1},\\mathrm {x}}(t,s) = \\min \\lbrace p,1-p\\rbrace $ .",
"One can then easily check that $\\mathbf {m}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}(s,t) = \\sup _{p \\in (0,1)}\\min \\lbrace p,1-p\\rbrace = 0.5$ .",
"Conversely, as the tbt-testing metric compares the success probabilities related to the execution of a single trace per time, we get $\\mathbf {m}_{\\mathrm {Te,tbt}}^{1,\\mathrm {x}}(s,t) = 0$ .",
"Notice that in the case of randomized schedulers, all the randomized resolutions for $t,o^p$ combining the two $a$ -moves can be matched by $s,o^p$ by combining the $b$ -moves and vice versa.",
"Consider now $s,t$ in Fig.",
"REF .",
"Even under randomized schedulers, the tbt-testing distance on them is given by the difference in the success probability of the trace $ac$ (or equivalently $ad$ ) and thus $\\mathbf {m}_{\\mathrm {Te,tbt}}^{1,\\mathrm {x}}(s,t) = 0.4$ .",
"However, we have already showed that $\\mathbf {m}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0.3$ .",
"Finally, we consider the case of trace distances.",
"Consider $t,u$ in Fig.",
"REF .",
"Clearly, $\\mathbf {m}_{\\mathrm {Tr,dis}}^{1,\\mathrm {x}}(t,u) = \\mathbf {m}_{\\mathrm {Tr,tbt}}^{1,\\mathrm {x}}(t,u) = \\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{1,\\mathrm {x}}(t,u) = 0$ .",
"However, in Ex.",
"REF we showed that $\\mathbf {m}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}(t,u) = 1$ .",
"Consider now $s,t$ in Fig.",
"REF .",
"We have that $\\mathbf {m}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0.3$ , but $\\mathbf {m}_{\\mathrm {Tr,dis}}^{1,\\mathrm {x}}(s,t) = 0.7$ and $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{1,\\mathrm {x}}(s,t) = \\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{1,\\mathrm {x}}(s,t) = 0.4$ .",
"$$ Example 7 Non comparability of $\\mathbf {m}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}}$ with $\\mathbf {m}_{\\mathrm {Te,tbt}}^{1,\\mathrm {x}}$ , $\\mathbf {m}_{\\mathrm {Tr,dis}}^{1,\\mathrm {det}}$ and $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{1,\\mathrm {det}}$.",
"For the tbt-testing metrics, consider $s,t$ in Fig.",
"REF .",
"In Ex.",
"REF we showed that $\\mathbf {m}_{\\mathrm {Te,tbt}}^{1,\\mathrm {x}}(s,t) = 0$ .",
"However, the same reasoning giving $\\mathbf {m}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0.5$ , can be applied on suprema success probabilities thus giving $\\mathbf {m}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0.5$ .",
"Consider now $t,u$ in Fig.",
"REF and their interactions with test $o_1$ in Fig.",
"REF .",
"As we consider maximal resolutions only, for both classes of schedulers, the success probability of trace $ab$ evaluates to 1 on $t,o_1$ , whereas on $u,o_1$ it evaluates to 0, due to the maximal resolution corresponding to the rightmost $a$ -branch.",
"Hence $\\mathbf {m}_{\\mathrm {Te,tbt}}^{1,\\mathrm {x}}(t,u) = \\lambda $ , whereas one can easily check that $\\mathbf {m}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}}(t,u) = 0$ .",
"We now proceed to the case of trace distances.",
"For $s,t$ in Fig.",
"REF , we showed that $\\mathbf {m}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0.5$ .",
"However, as both processes have a single resolution each allowing them to execute either trace $abc$ or $abd$ , we can infer that $\\mathbf {m}_{\\mathrm {Tr,dis}}^{1,\\mathrm {x}}(s,t) = \\mathbf {m}_{\\mathrm {Tr,tbt}}^{1,\\mathrm {x}}(s,t) = 0$ .",
"Notice, that this also shows the strictness of the relation $\\mathbf {m}_{\\mathrm {Tr,dis}}^{1,\\mathrm {rand}} < \\mathbf {m}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}}$ .",
"Consider now $s,t$ in Fig.",
"REF .",
"As discussed in Sect.",
"REF we have that $\\mathbf {m}_{\\mathrm {Tr,dis}}^{1,\\mathrm {det}} \\ge \\mathbf {m}_{\\mathrm {Tr,tbt}}^{1,\\mathrm {det}}(s,t) = 0.5$ .",
"However, one can easily check that $\\mathbf {m}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0$ .",
"$$ Example 8 Strictness of $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{1,\\mathrm {x}} < \\mathbf {m}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}}$.",
"Consider $s,t$ in Fig.",
"REF .",
"In Ex.",
"REF we have shown that $\\mathbf {m}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0.7$ .",
"However, since the supremal probability approach to testing proceeds in a trace-by-trace fashion, the $\\sqcup $ -testing distance is given by the difference in the success probability of the trace $ac$ (or $ad$ ) and thus $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{1,\\mathrm {x}}(s,t) = 0.4$ .",
"$$ Example 9 Strictness of $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}< \\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}}$.",
"We stress that this relation is due to the restriction to maximal resolutions, necessary to reason on testing semantics.",
"Consider now $t,u$ in Fig.",
"REF and their interactions with test $o_1$ in Fig.REF .",
"In Ex.REF we have shown that $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}}(t,u) = \\lambda $ .",
"However, one can easily check that $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}(t,u) = 0$ .",
"$$ Example 10 Strictness of $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {x}}< \\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}}$ and of $\\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {x}}< \\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}$.",
"For $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {x}} < \\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}}$ consider $t,u$ in Fig.",
"REF and the test $o_1$ in Fig.",
"REF , by which we get $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {x}}(t,u) = 0$ and $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}}(t,u) = \\lambda $ .",
"Similarly, for $\\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {x}} < \\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}$ consider $s,t$ in Fig.",
"REF with $\\varepsilon _1=\\varepsilon _2=0$ .",
"We have $\\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {x}}(s,t) = 0$ and $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}(s,t) = \\lambda \\cdot 0.5$ , given by the test $o$ corresponding to the leftmost branch of $s$ .",
"$$ Related and future work Trace metrics have been thoroughly studied on quantitative systems, as testified by the spectrum of distances, defined as the generalization of a chosen trace distance, in [17] and the one on Metric Transition Systems (MTSs) in [1].",
"The great variety in these models and the PTSs prevent us to compare the obtained results in detail.",
"Notably, in [1] the trace distance is based on a propositional distance defined over valuations of atomic propositions that characterize the MTS.",
"If on one side such valuation could play the role of the probability distributions in the PTS, it is unclear whether we could combine the ground distance on atomic propositions and the propositional distance, to obtain trace distances comparable to ours.",
"In [3], [13] trace metrics on Markov Chains (MCs) are defined as total variation distances on the cones generated by traces.",
"As in MCs probability depends only on the current state and not on nondeterminism, our quantification over resolutions would be trivial on MCs, giving a total variation distance.",
"Although ours is the first proposal of a metric expressing testing semantics, testing equivalences for probabilistic processes have been studied also in [15], [4], [5].",
"In detail, [15] proposed notions of probabilistic may/must testing for a Kleisli lifting of the PTS model, ie.",
"the transition relation is lifted to a relation $(\\rightarrow )^{\\dagger } \\subseteq (\\Delta (\\mathbf {S}) \\times \\mathcal {A}\\times \\Delta (\\mathbf {S}))$ taking distributions over processes to distributions over processes.",
"Again, the disparity in the two models prevents us from thoroughly comparing the proposed testing relations.",
"As future work, we aim to extend the spectrum of metrics to (bi)simulation metrics [16] and to metrics on different semantic models, and to study their logical characterizations and compositional properties on the same line of [9], [10], [11].",
"Further, we aim to provide efficient algorithms for the evaluation of the proposed metrics and to develop a tool for quantitative process verification: we will use the distance between a process and its specification to quantify how much that process satisfies a given property.",
"Acknowledgements I wish to thank Michele Loreti and Simone Tini for fruitful discussions, and the anonymous referees for their valuable comments and suggestions that helped to improve the paper."
],
[
"Metrics for testing",
"Testing semantics [14] compares processes according to their capacity to pass a test.",
"The latter is a PTS equipped with a distinguished state indicating the success of the test.",
"Definition 12 (Test) A nondeterministic probabilistic test transition systems (NPT) is a finite PTS $(\\mathbf {O}, \\mathcal {A}, \\xrightarrow{})$ where $\\mathbf {O}$ is a set of processes, called tests, containing a distinguished success process $\\surd $ with no outgoing transitions.",
"We say that a computation from $o \\in \\mathbf {O}$ is successful iff its last state is $\\surd $ .",
"Given a process $s$ and a test $o$ , we can consider the interaction system among the two.",
"This models the response of the process to the application of the test, so that $s$ passes the test $o$ if there is a computation in the interaction system that reaches $\\surd $ .",
"Informally, the interaction system is the result of the parallel composition of the process with the test.",
"Definition 13 (Interaction system) The interaction system of a PTS $(\\mathbf {S},\\mathcal {A},$ $\\xrightarrow{})$ and an NPT $(\\mathbf {O},\\mathcal {A},\\xrightarrow{}_{\\mathbf {O}})$ is the PTS $(\\mathbf {S}\\times \\mathbf {O}, \\mathcal {A},\\xrightarrow{}^{\\prime })$ where: [(i)] $(s,o) \\in \\mathbf {S}\\times \\mathbf {O}$ is called a configuration and is successful iff $o = \\surd $ ; a computation from $(s,o) \\in \\mathbf {S}\\times \\mathbf {O}$ is successful iff its last configuration is successful.",
"For $(s, o)$ and $\\mathcal {Z}_{s,o} \\in \\mathrm {Res}^{\\mathrm {x}}(s,o)$ , we let $\\mathbf {SC}(z_{s,o})$ be the set of successful computations from $z_{s,o}$ .",
"For $\\alpha \\in \\mathcal {A}^{\\star }$ , $\\mathbf {SC}(z_{s,o},\\alpha )$ is the set of $\\alpha $ -compatible successful computations from $z_{s,o}$ .",
"Testing semantics should compare processes wrt.",
"their probability to pass a test.",
"In this Section we consider three approaches to it: [(i)] the may/must, the trace-by-trace, and the supremal probabilities.",
"For each approach, we present (hemi,pseudo)metrics that provide a quantitative variant of the considered testing equivalence.",
"To the best of our knowledge, ours is the first attempt in this direction.",
"The may/must approach In the original work on nondeterministic systems [14], testing equivalence was defined via the may and must preorders.",
"The former expresses the ability of processes to pass a test.",
"The latter expresses the impossibility to fail a test.",
"When also probability is considered, these two preorders are defined, resp., in terms of suprema and infima success probabilities [29].",
"Definition 14 (May/must testing equivalence, [29]) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $(\\mathbf {O},\\mathcal {A},\\xrightarrow{}_O)$ an NPT and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"We say that $s,t \\in \\mathbf {S}$ are in the may testing preorder, written $s \\sqsubseteq _{\\mathrm {Te,may}}^{x}t$ , if for each $o \\in \\mathbf {O}$ $\\sup _{\\mathcal {Z}_{s,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o)} \\mathrm {Pr}(\\mathbf {SC}(z_{s,o})) \\le \\sup _{\\mathcal {Z}_{t,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o)} \\mathrm {Pr}(\\mathbf {SC}(z_{t,o}))$ .",
"Then, $s,t\\in \\mathbf {S}$ are may testing equivalent, written $s \\sim _{\\mathrm {Te,may}}^{x}t$ , iff $s \\sqsubseteq _{\\mathrm {Te,may}}^{x}t$ and $t \\sqsubseteq _{\\mathrm {Te,may}}^{x}s$ .",
"The notions of must testing preorder, $\\sqsubseteq _{\\mathrm {Te,must}}^{x}$ , and must testing equivalence, $\\sim _{\\mathrm {Te,must}}^{x}$ , are obtained by replacing the suprema in $\\sqsubseteq _{\\mathrm {Te,may}}^{x}$ and $\\sim _{\\mathrm {Te,may}}^{x}$ , resp., with infima.",
"Finally, we say that $s,t \\in \\mathbf {S}$ are in the may/must testing preorder, written $s \\sqsubseteq _{\\mathrm {Te,mM}}^{x}t$ , if $s \\sqsubseteq _{\\mathrm {Te,may}}^{x}t$ and $s \\sqsubseteq _{\\mathrm {Te,must}}^{x}t$ .",
"They are may/must testing equivalent, written $s \\sim _{\\mathrm {Te,mM}}^{x}t$ , iff $s \\sqsubseteq _{\\mathrm {Te,mM}}^{x}t$ and $t \\sqsubseteq _{\\mathrm {Te,mM}}^{x}s$ .",
"The quantitative analogue to may/must testing equivalence bases on the evaluation of the differences in the extremal success probabilities.",
"The distance between $s,t \\in \\mathbf {S}$ is set to $\\varepsilon \\ge 0$ if the maximum between the difference in the suprema and infima success probabilities wrt.",
"all resolutions of nondeterminism for $s$ and $t$ is at most $\\varepsilon $ .",
"We introduce a function $\\omega :\\mathbf {O}\\rightarrow (0,1]$ that assigns to each test $o$ the proper discount.",
"In fact, as the success probabilities in the may/must semantics are not related to the execution of a particular trace, in general we cannot define a discount factor as we did for the trace distances.",
"However, a similar construction may be regained when only tests with finite depth are considered.",
"In that case, we could define $\\omega (o) = \\lambda ^{\\mathrm {depth}(o)}$ , for $\\lambda \\in (0,1]$ .",
"We will use $\\mathbf {1}$ to denote the 1 constant function.",
"Definition 15 (May/must testing metric) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $(\\mathbf {O},\\mathcal {A},\\xrightarrow{}_O)$ an NPT, $\\omega :\\mathbf {O}\\rightarrow (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"For each $o\\in \\mathbf {O}$ , the function $\\mathbf {h}_{\\mathrm {Te,may}}^{{o,}\\omega ,\\mathrm {x}}\\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ is defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Te,may}}^{{o,}\\omega ,\\mathrm {x}}(s,t)=\\max \\Big \\lbrace 0 , \\omega (o)\\Big (\\sup _{\\mathcal {Z}_{s,o}\\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o)} \\mathrm {Pr}(\\mathbf {SC}(z_{s,o})) - \\sup _{\\mathcal {Z}_{t,o}\\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o)} \\mathrm {Pr}(\\mathbf {SC}(z_{t,o}))\\Big ) \\Big \\rbrace $ Function $\\mathbf {h}_{\\mathrm {Te,must}}^{{o,}\\omega ,\\mathrm {x}}\\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ is obtained by replacing the suprema in $\\mathbf {h}_{\\mathrm {Te,may}}^{{o,}\\omega ,\\mathrm {x}}$ with infima.",
"Given $\\mathrm {y} \\in \\lbrace \\mathrm {may}, \\mathrm {must}\\rbrace $ , the $\\mathrm {y}$ testing hemimetric and the $\\mathrm {y}$ testing metric are the functions $\\mathbf {h}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}, \\mathbf {m}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}} \\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}(s,t) = \\sup _{o \\in \\mathbf {O}}\\; \\mathbf {h}_{\\mathrm {Te,y}}^{o,\\omega ,\\mathrm {x}}(s,t)$ and $\\mathbf {m}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}(s,t) = \\max \\lbrace \\mathbf {h}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}(s,t), \\mathbf {h}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}(t,s) \\rbrace $ .",
"The may/must testing hemimetric and the may/must testing metric are the functions $\\mathbf {h}_{\\mathrm {Te,mM}}^{{}\\omega ,\\mathrm {x}}, \\mathbf {m}_{\\mathrm {Te,mM}}^{\\omega ,\\mathrm {x}}\\colon $ $\\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Te,mM}}^{{}\\omega ,\\mathrm {x}}(s,t) = \\max \\lbrace \\mathbf {h}_{\\mathrm {Te,may}}^{{}\\omega ,\\mathrm {x}}(s,t), \\mathbf {h}_{\\mathrm {Te,must}}^{{}\\omega ,\\mathrm {x}}(s,t) \\rbrace $ .",
"$\\mathbf {m}_{\\mathrm {Te,mM}}^{\\omega ,\\mathrm {x}}(s,t) = \\max \\lbrace \\mathbf {m}_{\\mathrm {Te,may}}^{\\omega ,\\mathrm {x}}{}(s,t), \\mathbf {m}_{\\mathrm {Te,must}}^{\\omega ,\\mathrm {x}}{}(s,t) \\rbrace $ .",
"Theorem 9 Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\omega :\\mathbf {O}\\rightarrow (0,1]$ , $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ and $\\mathrm {y} \\in \\lbrace \\mathrm {may},\\mathrm {must}, \\mathrm {mM}\\rbrace $ : The function $\\mathbf {h}_{\\mathrm {Te,y}}^{\\lambda ,\\mathrm {x}}$ is a 1-bounded hemimetric on $\\mathbf {S}$ , with $\\sqsubseteq _{\\mathrm {Te,y}}^{\\mathrm {x}}$ as kernel.",
"The function $\\mathbf {m}_{\\mathrm {Te,y}}^{\\lambda ,\\mathrm {x}}$ is a 1-bounded pseudometric on $\\mathbf {S}$ , with $\\sim _{\\mathrm {Te,y}}^{\\mathrm {x}}$ as kernel.",
"Figure: We use the tests o 1 ,o 2 o_1,o_2 to evaluate the distance between processes s,t,us,t,u in Fig.",
"wrt.",
"testing semantics.•\\bullet represents a generic configuration in the interaction system.In all upcoming examples we will consider only the tests and traces that are significant for the evaluations of the testing metrics.Example 3 Consider $t,u$ in Fig REF and their interactions with test $o_1$ in Fig REF .",
"Clearly, $(t,o_1)$ and $(u,o_1)$ have the same suprema success probabilities.",
"In fact, they both have a maximal resolution assigning probability 1 to the trace $ab$ , ie.",
"the only successful trace in the considered case.",
"As the same holds for all tests we get $\\mathbf {m}_{\\mathrm {Te,may}}^{\\omega ,\\mathrm {x}}(t,u) = 0$ .",
"Conversely, if we compare the infima success probabilities, we get $\\inf _{\\mathcal {Z}_{t,o_1} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o_1)} = 1$ since $(t,o_1)$ has only one maximal resolution corresponding to $(t,o_1)$ itself and that with probability 1 reaches $\\surd $ .",
"Still, $\\inf _{\\mathcal {Z}_{u,o_1} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(u,o_1)} = 0$ , given by the maximal resolution corresponding to $(u,o_1) \\stackrel{a}{{\\twoheadrightarrow }} \\mathrm {nil}$ .",
"Hence, we can infer $\\mathbf {m}_{\\mathrm {Te,must}}^{\\omega ,\\mathrm {x}}(t,u) = \\omega (o_1) \\cdot |1-0| = \\omega (o_1)$ .",
"$$ We can finally observe that both $\\mathbf {h}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}$ and $\\mathbf {m}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}$ are non-expansive.",
"Theorem 10 Let $\\omega :\\mathbf {O}\\rightarrow (0,1]$ and $\\mathrm {y} \\in \\lbrace \\mathrm {may},\\mathrm {must}, \\mathrm {mM}\\rbrace $ .",
"$\\mathbf {h}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}$ and $\\mathbf {m}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {x}}$ are non-expansive.",
"The trace-by-trace approach In [7] it was proved that the may/must is fully backward compatible with the restricted class of processes only if the same restriction is applied to the class of tests, ie.",
"if we consider resp.",
"fully nondeterministic and fully probabilistic tests only.",
"This is due to the duplication ability of nondeterministic probabilistic tests.",
"However, by applying the trace-by-trace approach to testing semantics, we regain the full backward compatibility wrt.",
"all tests (cf.",
"[7]).",
"Definition 16 (Tbt-testing equivalence) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $(\\mathbf {O},\\mathcal {A},\\xrightarrow{}_O)$ an NPT, $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"We say that $s,t \\in \\mathbf {S}$ are in the tbt-testing preorder, written $s \\sqsubseteq _{\\mathrm {Te,tbt}}^{x}t$ , if for each $o \\in \\mathbf {O}$ and $\\alpha \\in \\mathcal {A}^{\\star }$ $\\text{for each } \\mathcal {Z}_{s,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o) \\text{ there is } \\mathcal {Z}_{t,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o) \\text{ st. } \\mathrm {Pr}(\\mathbf {SC}(z_{s,o},\\alpha )) = \\mathrm {Pr}(\\mathbf {SC}(z_{t,o},\\alpha ))$ .",
"Then, $s,t\\in \\mathbf {S}$ are tbt-testing equivalent, notation $s \\sim _{\\mathrm {Te,tbt}}^{x}t$ , iff $s \\sqsubseteq _{\\mathrm {Te,tbt}}^{x}t$ and $t \\sqsubseteq _{\\mathrm {Te,tbt}}^{x}s$ .",
"The definition of the tbt-testing metric naturally follows from Def.",
"REF .",
"Definition 17 (Tbt-testing metric) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $(\\mathbf {O},\\mathcal {A},\\xrightarrow{}_O)$ an NPT, $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"For each $o \\in \\mathbf {O}$ and $\\alpha \\in \\mathcal {A}^{\\star }$ , function $\\mathbf {h}_{\\mathrm {Te,tbt}}^{{o,\\alpha ,}\\lambda ,\\mathrm {x}} \\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ is defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Te,tbt}}^{{o,\\alpha ,}\\lambda ,\\mathrm {x}}(s,t) = \\lambda ^{|\\alpha |-1}\\; \\sup _{\\mathcal {Z}_{s,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o)}\\, \\inf _{\\mathcal {Z}_{t,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o)} \\, |\\mathrm {Pr}(\\mathbf {SC}(z_{s,o},\\alpha )) - \\mathrm {Pr}(\\mathbf {SC}(z_{t,o},\\alpha ))|$ The tbt-testing hemimetric and the tbt-testing metric are the functions $\\mathbf {h}_{\\mathrm {Te,tbt}}^{{}\\lambda ,\\mathrm {x}}, \\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}}\\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Te,tbt}}^{{}\\lambda ,\\mathrm {x}}(s,t) = \\sup _{o \\in \\mathbf {O}} \\; \\sup _{\\alpha \\in \\mathcal {A}^{\\star }}\\; \\mathbf {h}_{\\mathrm {Te,tbt}}^{{o,\\alpha ,}\\lambda ,\\mathrm {x}}(s,t)$ $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}}(s,t) = \\max \\lbrace \\mathbf {h}_{\\mathrm {Te,tbt}}^{{}\\lambda ,\\mathrm {x}}(s,t), \\mathbf {h}_{\\mathrm {Te,tbt}}^{{}\\lambda ,\\mathrm {x}}(t,s) \\rbrace $ .",
"Theorem 11 Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"Then: The function $\\mathbf {h}_{\\mathrm {Te,tbt}}^{{}\\lambda ,\\mathrm {x}}$ is a 1-bounded hemimetric on $\\mathbf {S}$ , with $\\sqsubseteq _{\\mathrm {Te,tbt}}^{x}$ as kernel.",
"The function $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}}$ is a 1-bounded pseudometric on $\\mathbf {S}$ , with $\\sim _{\\mathrm {Te,tbt}}^{x}$ as kernel.",
"Example 4 Consider $s,t$ in Fig.",
"REF and their interactions with test $o_2$ in Fig.",
"REF .",
"By the same reasoning detailed in the first paragraph of Sect.",
"REF , we get $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {det}}(s,t)=\\lambda \\cdot 0.5$ and $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {rand}}(s,t) = 0$ .",
"$$ When the tbt-approach is used to define testing metrics, we get a refinement of the non-expansiveness property to strict non-expansiveness.",
"Theorem 12 All distances $\\mathbf {h}_{\\mathrm {Te,tbt}}^{{}\\lambda ,\\mathrm {det}}$ , $\\mathbf {h}_{\\mathrm {Te,tbt}}^{{}\\lambda ,\\mathrm {rand}}$ , $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {det}}$ , $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {rand}}$ are strictly non-expansive.",
"The supremal probabilities approach If we focus on verification, we can use the testing semantics to verify whether a process will behave as intended by its specification in all possible environments, as modeled by the interaction with the tests.",
"Informally, we could see each test as a set of requests of the environment to the system: the ones ending in the success state are those that must be answered.",
"The interaction of the specification with the test then tells us whether the system is able to provide those answers.",
"Thus, an implementation has to guarantee at least all the answers provided by the specification.",
"For this reason we decided to introduce also a supremal probabilities variant of testing semantics: for each test and for each trace we compare the suprema wrt.",
"all resolutions of nondeterminism of the probabilities of processes to reach success by performing the considered trace.",
"Definition 18 ($\\bigsqcup $ -testing equivalence) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $(\\mathbf {O},\\mathcal {A},\\xrightarrow{}_O)$ an NPT and $\\mathrm {x} \\in \\lbrace \\mathrm {det}, \\mathrm {rand}\\rbrace $ .",
"We say that $s,t \\in \\mathbf {S}$ are in the $\\bigsqcup $ -testing preorder, written $s \\sqsubseteq _{\\mathrm {Te,}\\sqcup }^{\\mathrm {x}}t$ , if for each $o \\in \\mathbf {O}$ and $\\alpha \\in \\mathcal {A}^{\\star }$ $\\sup _{\\mathcal {Z}_{s,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o)} \\mathrm {Pr}(\\mathbf {SC}(z_{s,o},\\alpha )) \\le \\sup _{\\mathcal {Z}_{t,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o)} \\mathrm {Pr}(\\mathbf {SC}(z_{t,o},\\alpha ))$ .",
"Then, $s,t \\in \\mathbf {S}$ are $\\bigsqcup $ -testing equivalent, notation $s \\sim _{\\mathrm {Te,}\\sqcup }^{\\mathrm {x}}t$ , iff $s \\sqsubseteq _{\\mathrm {Te,}\\sqcup }^{\\mathrm {x}}t$ and $t \\sqsubseteq _{\\mathrm {Te,}\\sqcup }^{\\mathrm {x}}s$ .",
"We obtain the $\\bigsqcup $ -testing metric as a direct adaptation to tests of Definition REF .",
"Definition 19 ($\\bigsqcup $ -testing metric) Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $(\\mathbf {O},\\mathcal {A},\\xrightarrow{}_O)$ an NPT, $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"For each $o \\in \\mathbf {O}$ , $\\alpha \\in \\mathcal {A}^{\\star }$ , the function $\\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{o,\\alpha ,}\\lambda ,\\mathrm {x}} \\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ is defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{o,\\alpha ,}\\lambda ,\\mathrm {x}}(s,t) = \\max \\Big \\lbrace 0,\\lambda ^{|\\alpha |-1} \\Big (\\sup _{\\mathcal {Z}_{s,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o)} \\!\\!\\!\\!\\!\\mathrm {Pr}(\\mathbf {SC}(z_{s,o},\\alpha )) - \\sup _{\\mathcal {Z}_{t,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o)}\\!\\!\\!\\!\\!",
"\\mathrm {Pr}(\\mathbf {SC}(z_{t,o},\\alpha ))\\Big )\\Big \\rbrace .$ The $\\bigsqcup $ -testing hemimetric and the $\\bigsqcup $ -testing metric are the functions $\\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}, \\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}\\colon \\mathbf {S}\\times \\mathbf {S}\\rightarrow [0,1]$ defined for all $s,t \\in \\mathbf {S}$ by $\\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}(s,t) = \\sup _{o \\in \\mathbf {O}} \\; \\sup _{\\alpha \\in \\mathcal {A}^{\\star }}\\; \\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{o,\\alpha ,}\\lambda ,\\mathrm {x}}(s,t)$ ; $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}(s,t) = \\max \\lbrace \\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}(s,t), \\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}(t,s) \\rbrace $ .",
"Theorem 13 Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS and $\\lambda \\in (0,1]$ and $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ .",
"Then: The function $\\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}$ is a 1-bounded hemimetric on $\\mathbf {S}$ , with $\\sqsubseteq _{\\mathrm {Te,}\\sqcup }^{\\mathrm {x}}$ as kernel.",
"The function $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}$ is a 1-bounded pseudometric on $\\mathbf {S}$ , with $\\sim _{\\mathrm {Te,}\\sqcup }^{\\mathrm {x}}$ as kernel.",
"Finally, we can show that both $\\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{}\\lambda ,\\mathrm {x}}$ and $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}$ are strictly non-expansive.",
"Theorem 14 All distances $\\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{}\\lambda ,\\mathrm {det}}$ , $\\mathbf {h}_{\\mathrm {Te,}\\sqcup }^{{}\\lambda ,\\mathrm {rand}}$ , $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {det}}$ , $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {rand}}$ are strictly non-expansive.",
"Remark 2 For all distances $d$ considered in Thms.",
"REF , REF , REF and processes $z_s,z_t$ in Fig.",
"REF , with $\\lambda =1$ , we have $d(z_s , z_t) = 0.5$ and $d(z_s \\parallel z_s , z_t \\parallel z_t) = 0.75 = 0.5 + 0.5 - 0.5 \\cdot 0.5$ .",
"Hence, the upper bounds to the distance between composed processes provided in Thms.",
"REF and REF are tight.",
"We leave as a future work the analogous result for distances considered in Thm.",
"REF .",
"Comparing the distinguishing power of testing metrics Figure: The spectrum of trace and testing (hemi)metrics.d→d ' d \\rightarrow d^{\\prime } stands for d>d ' d > d^{\\prime }.We present only the general form with 𝐝∈{𝐡,𝐦}\\mathbf {d} \\in \\lbrace \\mathbf {h},\\mathbf {m}\\rbrace as the relations among the hemimetrics are the same wrt.",
"those among the metrics.The complete spectrum can be obtained by relating each metric with the respective hemimetric.We study the distinguishing power of the testing metrics presented in this section and the trace metrics defined in Sect.",
", thus obtaining the spectrum in Fig.",
"REF .",
"Firstly, we compare the expressiveness of the testing metrics wrt.",
"the chosen class of schedulers.",
"The distinguishing power of testing metrics based on may-must and supremal probabilities approaches is not influenced by this choice.",
"Differently, in the tbt approach, the distances evaluated on deterministic schedulers are more discriminating than their analogues on randomized schedulers.",
"Theorem 15 Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\lambda \\in (0,1]$ , $\\omega \\colon \\mathbf {O}\\rightarrow (0,1]$ $\\mathrm {y} \\in \\lbrace \\mathrm {may},\\mathrm {must}, \\mathrm {mM}\\rbrace $ and $\\mathbf {d} \\in \\lbrace \\mathbf {h},\\mathbf {m}\\rbrace $ : $\\begin{array}{llll}1.\\, \\mathbf {d}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {rand}} = \\mathbf {d}_{\\mathrm {Te,y}}^{\\omega ,\\mathrm {det}} &2.\\, \\mathbf {d}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {rand}} < \\mathbf {d}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {det}} &3.\\, \\mathbf {d}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {rand}} = \\mathbf {d}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {det}}\\end{array}$ From Thm.",
"REF , by using the kernel relations in Thms.",
"REF and REF , we regain relations $\\sim _{\\mathrm {Te,may}}^{\\mathrm {rand}}= \\sim _{\\mathrm {Te,may}}^{\\mathrm {det}}$ , $\\sim _{\\mathrm {Te,must}}^{\\mathrm {rand}}= \\sim _{\\mathrm {Te,must}}^{\\mathrm {det}}$ , $\\sim _{\\mathrm {Te,mM}}^{\\mathrm {rand}}= \\sim _{\\mathrm {Te,mM}}^{\\mathrm {det}}$ , $\\sim _{\\mathrm {Te,tbt}}^{\\mathrm {det}}\\subset \\sim _{\\mathrm {Te,tbt}}^{\\mathrm {rand}}$ , and their analogues on preorders, proved in [7].",
"From Thm.",
"REF we get $\\sqsubseteq _{\\mathrm {Te,}\\sqcup }^{\\mathrm {rand}}= \\sqsubseteq _{\\mathrm {Te,}\\sqcup }^{\\mathrm {det}}$ and $\\sim _{\\mathrm {Te,}\\sqcup }^{\\mathrm {rand}}= \\sim _{\\mathrm {Te,}\\sqcup }^{\\mathrm {det}}$ .",
"The strictness of the inequality in Thm.",
"REF .2, is witnessed by processes $s,t$ in Fig REF and their interactions with the test $o_2$ in Fig REF .",
"The same reasoning applied in the first paragraph of Sect.",
"REF to obtain $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {det}}(s,t) = \\lambda \\cdot 0.5$ and $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {rand}}(s,t) = 0$ , gives $\\mathbf {h}_{\\mathrm {Te,tbt}}^{{o_2,}\\lambda ,\\mathrm {det}}(t,s) = \\lambda \\cdot 0.5 = \\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {det}}{}(s,t)$ and $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {rand}}{}(s,t) = 0$ .",
"We proceed to compare the expressiveness of each metric wrt.",
"the other semantics.",
"Our results are fully compatible with the spectrum on probabilistic relations presented in [7].",
"Theorem 16 Let $(\\mathbf {S},\\mathcal {A},\\xrightarrow{})$ be a PTS, $\\lambda \\in (0,1]$ , $\\mathrm {x} \\in \\lbrace \\mathrm {det},\\mathrm {rand}\\rbrace $ and $\\mathbf {d} \\in \\lbrace \\mathbf {h},\\mathbf {m}\\rbrace $ : $\\begin{array}{llll}1.\\, \\mathbf {d}_{\\mathrm {Te,may}}^{\\omega ,\\mathrm {x}} < \\mathbf {d}_{\\mathrm {Te,mM}}^{\\omega ,\\mathrm {x}} &2.\\, \\mathbf {d}_{\\mathrm {Te,must}}^{\\omega ,\\mathrm {x}} < \\mathbf {d}_{\\mathrm {Te,mM}}^{\\omega ,\\mathrm {x}} &3.\\, \\mathbf {d}_{\\mathrm {Te,}\\sqcup }^{1,\\mathrm {x}} < \\mathbf {d}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}} &4.\\, \\mathbf {d}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}} < \\mathbf {d}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}} \\\\5.\\, \\mathbf {d}_{\\mathrm {Tr,dis}}^{1,\\mathrm {rand}} < \\mathbf {d}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}} &6.\\, \\mathbf {d}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {x}} < \\mathbf {d}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}} &7.\\, \\mathbf {d}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {x}} < \\mathbf {d}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}\\end{array}$ The following Examples prove the strictness of the inequalities in Thm.",
"REF and the non comparability of the (hemi)metrics as shown in Fig.",
"REF .",
"For simplicity, we consider only the cases of the metrics.",
"Figure: Processes s,ts,t and their interaction systems with the test o 2 o_2 in Fig.",
".Example 5 Non comparability of $\\mathbf {m}_{\\mathrm {Te,may}}^{\\omega ,\\mathrm {x}}$ with $\\mathbf {m}_{\\mathrm {Te,must}}^{\\omega ,\\mathrm {x}}$.",
"In Ex.",
"REF we showed that for $t,u$ in Fig.",
"REF from their interaction with the test $o_1$ in Fig.",
"REF we obtain that $\\mathbf {m}_{\\mathrm {Te,must}}^{\\omega ,\\mathrm {x}}(t,u) = \\omega (o_1)$ , whereas one can easily check that $\\mathbf {m}_{\\mathrm {Te,may}}^{\\omega ,\\mathrm {x}}(t,u) = 0$ .",
"Consider now $s,t$ and their interactions in Fig.",
"REF with the test $o_2$ from Fig.",
"REF .",
"Clearly, we have $\\sup _{\\mathcal {Z}_{s,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o)}\\mathrm {Pr}(\\mathbf {SC}(z_{s,o})) = 1$ and $\\sup _{\\mathcal {Z}_{t,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o)}\\mathrm {Pr}(\\mathbf {SC}(z_{t,o})) = 0.3$ and thus $\\mathbf {m}_{\\mathrm {Te,may}}^{\\omega ,\\mathrm {x}}(s,t) = 0.7 \\cdot \\omega (o_2)$ .",
"Conversely, if we consider infima success probabilities, we have $\\inf _{\\mathcal {Z}_{s,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o)}\\mathrm {Pr}(\\mathbf {SC}(z_{s,o})) = 0$ and $\\sup _{\\mathcal {Z}_{t,o} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o)}\\mathrm {Pr}(\\mathbf {SC}(z_{t,o})) = 0.3$ .",
"Thus, $\\mathbf {m}_{\\mathrm {Te,must}}^{\\omega ,\\mathrm {x}}(s,t) = 0.3\\cdot \\omega (o_2)$ .",
"$$ Figure: Processes s,ts,t are such that 𝐝 Te , tbt 1,x (s,t)=0\\mathbf {d}_{\\mathrm {Te,tbt}}^{1,\\mathrm {x}}(s,t) = 0 and 𝐝 Te , must 1,x (s,t)=0.5\\mathbf {d}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0.5, as witnessed by the test o 1/2 o^{1/2}.Example 6 Non comparability of $\\mathbf {m}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}$ with $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{1,\\mathrm {x}}$ , $\\mathbf {m}_{\\mathrm {Te,tbt}}^{1,\\mathrm {x}}$ , $\\mathbf {m}_{\\mathrm {Tr,dis}}^{1,\\mathrm {x}}$ , $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{1,\\mathrm {x}}$ and $\\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{1,\\mathrm {x}}$.",
"We start with $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{1,\\mathrm {x}}$ .",
"Form Ex.",
"REF we know that for $t,u$ in Fig.",
"REF it holds $\\mathbf {m}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}(t,u) = \\mathbf {1}$ .",
"Since both $t$ and $u$ have maximal resolutions giving probability 1 to either $ab$ or $ac$ , we get $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{1,\\mathrm {x}}(t,u) =0$ .",
"Consider now $s,t$ in Fig.",
"REF .",
"In Ex.",
"REF we showed that $\\mathbf {m}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0.3$ .",
"From the interaction systems in Fig.",
"REF , by considering the superma success probabilities of trace $ac$ , we obtain that $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{1,\\mathrm {x}} = 0.4$ .",
"Next we deal with the tbt-testing metrics.",
"Consider $s,t$ in Fig.",
"REF and the family of tests $O = \\lbrace o^p \\mid p \\in (0,1)\\rbrace $ , each duplicating the actions $b$ in the interaction with $s$ and $t$ .",
"For each $o^p \\in O$ , $\\inf _{\\mathcal {Z}_{s,o^p} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(s,o^p)} \\mathrm {Pr}(\\mathbf {SC}(z_{s,o^p})) = 0$ and $\\inf _{\\mathcal {Z}_{t,o^p} \\in \\mathrm {Res}^{\\mathrm {x}}_{\\max }(t,o^p)} \\mathrm {Pr}(\\mathbf {SC}(z_{t,o^p})) = \\min \\lbrace p,1-p\\rbrace $ , thus giving $\\mathbf {h}_{\\mathrm {Te,must}}^{o^p,\\mathbf {1},\\mathrm {x}}(t,s) = \\min \\lbrace p,1-p\\rbrace $ .",
"One can then easily check that $\\mathbf {m}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}(s,t) = \\sup _{p \\in (0,1)}\\min \\lbrace p,1-p\\rbrace = 0.5$ .",
"Conversely, as the tbt-testing metric compares the success probabilities related to the execution of a single trace per time, we get $\\mathbf {m}_{\\mathrm {Te,tbt}}^{1,\\mathrm {x}}(s,t) = 0$ .",
"Notice that in the case of randomized schedulers, all the randomized resolutions for $t,o^p$ combining the two $a$ -moves can be matched by $s,o^p$ by combining the $b$ -moves and vice versa.",
"Consider now $s,t$ in Fig.",
"REF .",
"Even under randomized schedulers, the tbt-testing distance on them is given by the difference in the success probability of the trace $ac$ (or equivalently $ad$ ) and thus $\\mathbf {m}_{\\mathrm {Te,tbt}}^{1,\\mathrm {x}}(s,t) = 0.4$ .",
"However, we have already showed that $\\mathbf {m}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0.3$ .",
"Finally, we consider the case of trace distances.",
"Consider $t,u$ in Fig.",
"REF .",
"Clearly, $\\mathbf {m}_{\\mathrm {Tr,dis}}^{1,\\mathrm {x}}(t,u) = \\mathbf {m}_{\\mathrm {Tr,tbt}}^{1,\\mathrm {x}}(t,u) = \\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{1,\\mathrm {x}}(t,u) = 0$ .",
"However, in Ex.",
"REF we showed that $\\mathbf {m}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}(t,u) = 1$ .",
"Consider now $s,t$ in Fig.",
"REF .",
"We have that $\\mathbf {m}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0.3$ , but $\\mathbf {m}_{\\mathrm {Tr,dis}}^{1,\\mathrm {x}}(s,t) = 0.7$ and $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{1,\\mathrm {x}}(s,t) = \\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{1,\\mathrm {x}}(s,t) = 0.4$ .",
"$$ Example 7 Non comparability of $\\mathbf {m}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}}$ with $\\mathbf {m}_{\\mathrm {Te,tbt}}^{1,\\mathrm {x}}$ , $\\mathbf {m}_{\\mathrm {Tr,dis}}^{1,\\mathrm {det}}$ and $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{1,\\mathrm {det}}$.",
"For the tbt-testing metrics, consider $s,t$ in Fig.",
"REF .",
"In Ex.",
"REF we showed that $\\mathbf {m}_{\\mathrm {Te,tbt}}^{1,\\mathrm {x}}(s,t) = 0$ .",
"However, the same reasoning giving $\\mathbf {m}_{\\mathrm {Te,must}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0.5$ , can be applied on suprema success probabilities thus giving $\\mathbf {m}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0.5$ .",
"Consider now $t,u$ in Fig.",
"REF and their interactions with test $o_1$ in Fig.",
"REF .",
"As we consider maximal resolutions only, for both classes of schedulers, the success probability of trace $ab$ evaluates to 1 on $t,o_1$ , whereas on $u,o_1$ it evaluates to 0, due to the maximal resolution corresponding to the rightmost $a$ -branch.",
"Hence $\\mathbf {m}_{\\mathrm {Te,tbt}}^{1,\\mathrm {x}}(t,u) = \\lambda $ , whereas one can easily check that $\\mathbf {m}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}}(t,u) = 0$ .",
"We now proceed to the case of trace distances.",
"For $s,t$ in Fig.",
"REF , we showed that $\\mathbf {m}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0.5$ .",
"However, as both processes have a single resolution each allowing them to execute either trace $abc$ or $abd$ , we can infer that $\\mathbf {m}_{\\mathrm {Tr,dis}}^{1,\\mathrm {x}}(s,t) = \\mathbf {m}_{\\mathrm {Tr,tbt}}^{1,\\mathrm {x}}(s,t) = 0$ .",
"Notice, that this also shows the strictness of the relation $\\mathbf {m}_{\\mathrm {Tr,dis}}^{1,\\mathrm {rand}} < \\mathbf {m}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}}$ .",
"Consider now $s,t$ in Fig.",
"REF .",
"As discussed in Sect.",
"REF we have that $\\mathbf {m}_{\\mathrm {Tr,dis}}^{1,\\mathrm {det}} \\ge \\mathbf {m}_{\\mathrm {Tr,tbt}}^{1,\\mathrm {det}}(s,t) = 0.5$ .",
"However, one can easily check that $\\mathbf {m}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0$ .",
"$$ Example 8 Strictness of $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{1,\\mathrm {x}} < \\mathbf {m}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}}$.",
"Consider $s,t$ in Fig.",
"REF .",
"In Ex.",
"REF we have shown that $\\mathbf {m}_{\\mathrm {Te,may}}^{\\mathbf {1},\\mathrm {x}}(s,t) = 0.7$ .",
"However, since the supremal probability approach to testing proceeds in a trace-by-trace fashion, the $\\sqcup $ -testing distance is given by the difference in the success probability of the trace $ac$ (or $ad$ ) and thus $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{1,\\mathrm {x}}(s,t) = 0.4$ .",
"$$ Example 9 Strictness of $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}< \\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}}$.",
"We stress that this relation is due to the restriction to maximal resolutions, necessary to reason on testing semantics.",
"Consider now $t,u$ in Fig.",
"REF and their interactions with test $o_1$ in Fig.REF .",
"In Ex.REF we have shown that $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}}(t,u) = \\lambda $ .",
"However, one can easily check that $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}(t,u) = 0$ .",
"$$ Example 10 Strictness of $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {x}}< \\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}}$ and of $\\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {x}}< \\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}$.",
"For $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {x}} < \\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}}$ consider $t,u$ in Fig.",
"REF and the test $o_1$ in Fig.",
"REF , by which we get $\\mathbf {m}_{\\mathrm {Tr,tbt}}^{\\lambda ,\\mathrm {x}}(t,u) = 0$ and $\\mathbf {m}_{\\mathrm {Te,tbt}}^{\\lambda ,\\mathrm {x}}(t,u) = \\lambda $ .",
"Similarly, for $\\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {x}} < \\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}$ consider $s,t$ in Fig.",
"REF with $\\varepsilon _1=\\varepsilon _2=0$ .",
"We have $\\mathbf {m}_{\\mathrm {Tr,}\\sqcup }^{\\lambda ,\\mathrm {x}}(s,t) = 0$ and $\\mathbf {m}_{\\mathrm {Te,}\\sqcup }^{\\lambda ,\\mathrm {x}}(s,t) = \\lambda \\cdot 0.5$ , given by the test $o$ corresponding to the leftmost branch of $s$ .",
"$$ Related and future work Trace metrics have been thoroughly studied on quantitative systems, as testified by the spectrum of distances, defined as the generalization of a chosen trace distance, in [17] and the one on Metric Transition Systems (MTSs) in [1].",
"The great variety in these models and the PTSs prevent us to compare the obtained results in detail.",
"Notably, in [1] the trace distance is based on a propositional distance defined over valuations of atomic propositions that characterize the MTS.",
"If on one side such valuation could play the role of the probability distributions in the PTS, it is unclear whether we could combine the ground distance on atomic propositions and the propositional distance, to obtain trace distances comparable to ours.",
"In [3], [13] trace metrics on Markov Chains (MCs) are defined as total variation distances on the cones generated by traces.",
"As in MCs probability depends only on the current state and not on nondeterminism, our quantification over resolutions would be trivial on MCs, giving a total variation distance.",
"Although ours is the first proposal of a metric expressing testing semantics, testing equivalences for probabilistic processes have been studied also in [15], [4], [5].",
"In detail, [15] proposed notions of probabilistic may/must testing for a Kleisli lifting of the PTS model, ie.",
"the transition relation is lifted to a relation $(\\rightarrow )^{\\dagger } \\subseteq (\\Delta (\\mathbf {S}) \\times \\mathcal {A}\\times \\Delta (\\mathbf {S}))$ taking distributions over processes to distributions over processes.",
"Again, the disparity in the two models prevents us from thoroughly comparing the proposed testing relations.",
"As future work, we aim to extend the spectrum of metrics to (bi)simulation metrics [16] and to metrics on different semantic models, and to study their logical characterizations and compositional properties on the same line of [9], [10], [11].",
"Further, we aim to provide efficient algorithms for the evaluation of the proposed metrics and to develop a tool for quantitative process verification: we will use the distance between a process and its specification to quantify how much that process satisfies a given property.",
"Acknowledgements I wish to thank Michele Loreti and Simone Tini for fruitful discussions, and the anonymous referees for their valuable comments and suggestions that helped to improve the paper."
],
[
"Related and future work",
"Trace metrics have been thoroughly studied on quantitative systems, as testified by the spectrum of distances, defined as the generalization of a chosen trace distance, in [17] and the one on Metric Transition Systems (MTSs) in [1].",
"The great variety in these models and the PTSs prevent us to compare the obtained results in detail.",
"Notably, in [1] the trace distance is based on a propositional distance defined over valuations of atomic propositions that characterize the MTS.",
"If on one side such valuation could play the role of the probability distributions in the PTS, it is unclear whether we could combine the ground distance on atomic propositions and the propositional distance, to obtain trace distances comparable to ours.",
"In [3], [13] trace metrics on Markov Chains (MCs) are defined as total variation distances on the cones generated by traces.",
"As in MCs probability depends only on the current state and not on nondeterminism, our quantification over resolutions would be trivial on MCs, giving a total variation distance.",
"Although ours is the first proposal of a metric expressing testing semantics, testing equivalences for probabilistic processes have been studied also in [15], [4], [5].",
"In detail, [15] proposed notions of probabilistic may/must testing for a Kleisli lifting of the PTS model, ie.",
"the transition relation is lifted to a relation $(\\rightarrow )^{\\dagger } \\subseteq (\\Delta (\\mathbf {S}) \\times \\mathcal {A}\\times \\Delta (\\mathbf {S}))$ taking distributions over processes to distributions over processes.",
"Again, the disparity in the two models prevents us from thoroughly comparing the proposed testing relations.",
"As future work, we aim to extend the spectrum of metrics to (bi)simulation metrics [16] and to metrics on different semantic models, and to study their logical characterizations and compositional properties on the same line of [9], [10], [11].",
"Further, we aim to provide efficient algorithms for the evaluation of the proposed metrics and to develop a tool for quantitative process verification: we will use the distance between a process and its specification to quantify how much that process satisfies a given property."
],
[
"Acknowledgements",
"I wish to thank Michele Loreti and Simone Tini for fruitful discussions, and the anonymous referees for their valuable comments and suggestions that helped to improve the paper."
]
] | 1808.08649 |
[
[
"Persistent Stochastic Non-Interference"
],
[
"Abstract In this paper we present an information flow security property for stochastic, cooperating, processes expressed as terms of the Performance Evaluation Process Algebra (PEPA).",
"We introduce the notion of Persistent Stochastic Non-Interference (PSNI) based on the idea that every state reachable by a process satisfies a basic Stochastic Non-Interference (SNI) property.",
"The structural operational semantics of PEPA allows us to give two characterizations of PSNI: the first involves a single bisimulation-like equivalence check, while the second is formulated in terms of unwinding conditions.",
"The observation equivalence at the base of our definition relies on the notion of lumpability and ensures that, for a secure process P, the steady state probability of observing the system being in a specific state P' is independent from its possible high level interactions."
],
[
"Introduction",
"Non-Interference is an information flow security property which aims at protecting sensitive data from undesired accesses.",
"In particular, it consists in protecting the confidentiality of information by guaranteeing that high level, sensitive, information never flows to low level, unauthorized, users.",
"It is well known that access control policies or cryptographic protocols are, in general, not sufficient to forbid unwanted flows which may arise from the so called covert channels or from some weakness in the cryptographic algorithms.",
"The notion of Non-Interference for deterministic systems has been introduced in [17] and it has been extended to non-deterministic systems.",
"Non-Interference has been then studied in different settings such as programming languages [16], [30], [31], trace models [22], [25], cryptographic protocols [1], [6], [13], process calculi [7], [8], [12], [19], [29], probabilistic models [2], [10], timed models [14], [18], and stochastic models [2].",
"In this paper we study a notion of Non-Interference for stochastic, cooperating, processes expressed as terms of the Performance Evaluation Process Algebra (PEPA) [20].",
"We introduce the notion of Persistent Stochastic Non-Interference (PSNI) based on the idea that every state reachable by a process satisfies a basic Stochastic Non-Interference (SNI) property.",
"By imposing that security persists during process execution, the system is guaranteed to be dynamically secure in the sense that every potential transition leads the process to a secure state.",
"Property SNI is inspired by the Bisimulation-based Non-Deducibility on Compositions (BNDC) property defined in [11] for non-deterministic CCS processes.",
"In our setting, the definition has the following form: a process $P$ is secure if a low level observer cannot distinguish the behavior of $P$ in isolation from the behavior of $P$ cooperating with any possible high level process $H$ .",
"The notion of observation that we consider is based on the concept of lumpability for the underlying Markov chain [21], [23], [24].",
"Formally, property SNI is defined as: for any high level process $H$ which may enable only high level activities, $P\\setminus {\\cal H}\\approx _l (P{{\\cal H}}H)/{\\cal H}$ where $P\\setminus {\\cal H}$ represents the low level view of $P$ in isolation, while $(P{{\\cal H}}H)/{\\cal H}$ denotes the low level view of $P$ interacting with the high process $H$ .",
"The observation equivalence $\\approx _l$ is the lumpable bisimilarity defined in [21] which is a characterization of a lumpable relation over the terms of the process algebra PEPA preserving contextuality and inducing a lumping in the underlying Markov processes.",
"Notice that this basic security property, that we call Stochastic Non-Interference (SNI) is not persistent in the sense that it is not preserved during system execution.",
"Thus, it might happen that a system satisfying SNI reaches a state which is not secure.",
"To overcome this problem we introduce the notion of Persistent Stochastic Non-Interference (PSNI) which requires that every state reachable by the system is secure, i.e., $P$ is secure if and only if $\\forall P^{\\prime } \\mbox{ reachable from } P, \\ \\ P^{\\prime } \\mbox{ satisfies } SNI\\,.$ Notice that this property contains two universal quantifications: one over all the reachable states and another one, inside the definition of SNI, over all the possible high level processes which may interact with the considered system.",
"The main contributions of this paper are: we provide a characterization of PSNI in terms of a single bisimulation-based check thus avoiding the universal quantification over all the high level contexts; based on the structural operational semantics of PEPA, we provide a characterization of PSNI expressed in terms of unwinding conditions; we prove that PSNI is compositional with respect to low prefix, cooperation over low actions and hiding; we prove that if $P$ is secure then the equivalence class $[P]$ with respect to lumpable bisimilarity $\\approx _l$ is closed under PSNI; we show through an example that if $P$ is secure then, from the low level point of view, the steady state probability of observing the system being in a specific state $P^{\\prime }$ is independent from the possible high level interactions of $P$ .",
"Structure of the paper.",
"The paper is organized as follows: in Section we introduce the process algebra PEPA, its structural operational semantics, and the observation equivalence named lumpable bisimilarity.",
"The notion of Persistent Stochastic Non-Interference (PSNI) and its characterizations are presented in Section .",
"In Section we prove some compositionality result and other properties of PSNI.",
"Comparisons with other SOS-based persistent security properties are discussed in Section .",
"Finally, Section concludes the paper.",
"Table: Operational semantics for PEPA components"
],
[
"The Calculus",
"PEPA (Performance Evaluation Process Algebra) [20] is an algebraic calculus enhanced with stochastic timing information which may be used to calculate performance measures as well as prove functional system properties.",
"The basic elements of PEPA are components and activities.",
"Each activity is represented by a pair $(\\alpha , r)$ where $\\alpha $ is a label, or action type, and $r$ is its activity rate, that is the parameter of a negative exponential distribution determining its duration.",
"We assume that there is a countable set, $$ , of possible action types, including a distinguished type, $\\tau $ , which can be regarded as the unknown type.",
"Activity rates may be any positive real number, or the distinguished symbol $\\top $ which should be read as unspecified.",
"The syntax for PEPA terms is defined by the grammar: $\\begin{array}{cclccl}P & ::= & P {L} P \\mid P/L \\mid S\\\\[1mm]S & ::= & (\\alpha , r).S \\mid S+S \\mid A\\end{array}$ where $S$ denotes a sequential component, while $P$ denotes a model component which executes in parallel.",
"We assume that there is a countable set of constants, $A$ .",
"We write $$ for the set of all possible components."
],
[
"Structural Operational Semantics",
"PEPA is given a structural operational semantics, as shown in Table REF .",
"The component $(\\alpha , r).P$ carries out the activity $(\\alpha , r)$ of type $\\alpha $ at rate $r$ and subsequently behaves as $P$ .",
"When $a=(\\alpha , r)$ , the component $(\\alpha , r).P$ may be written as $a.P$ .",
"The component $P+Q$ represents a system which may behave either as $P$ or as $Q$ .",
"$P+Q$ enables all the current activities of both $P$ and $Q$ .",
"The first activity to complete distinguishes one of the components, $P$ or $Q$ .",
"The other component of the choice is discarded.",
"The component $P/L$ behaves as $P$ except that any activity of type within the set $L$ are hidden, i.e., they are relabeled with the unobservable type $\\tau $ .",
"The meaning of a constant $A$ is given by a defining equation such as $AP$ which gives the constant $A$ the behavior of the component $P$ .",
"The cooperation combinator ${L}$ is in fact an indexed family of combinators, one for each possible set of action types, $L\\subseteq \\setminus \\lbrace \\tau \\rbrace $ .",
"The cooperation set $L$ defines the action types on which the components must synchronize or cooperate (the unknown action type, $\\tau $ , may not appear in any cooperation set).",
"It is assumed that each component proceeds independently with any activities whose types do not occur in the cooperation set $L$ (individual activities).",
"However, activities with action types in the set $L$ require the simultaneous involvement of both components (shared activities).",
"These shared activities will only be enabled in $P{L}Q$ when they are enabled in both $P$ and $Q$ .",
"The shared activity will have the same action type as the two contributing activities and a rate reflecting the rate of the slower participant [20].",
"If an activity has an unspecified rate in a component, the component is passive with respect to that action type.",
"In this case the rate of the shared activity will be completely determined by the other component.",
"For a given $P$ and action type $\\alpha $ , this is the apparent rate [21] of $\\alpha $ in $P$ , denoted $r_{\\alpha }(P)$ , that is the sum of the rates of the $\\alpha $ activities enabled in $P$ .",
"The semantics of each term in PEPA is given via a labeled multi-transition system where the multiplicities of arcs are significant.",
"In the transition system, a state or derivative corresponds to each syntactic term of the language and an arc represents the activity which causes one derivative to evolve into another.",
"The set of reachable states of a model $P$ is termed the derivative set of $P$ , denoted by $ds(P)$ , and constitutes the set of nodes of the derivation graph of $P$ ($(P)$ ) obtained by applying the semantic rules exhaustively.",
"We denote by $(P)$ the set of all the current action types of $P$ , i.e., the set of action types which the component $P$ may next engage in.",
"We denote by $(P)$ the multiset of all the current activities of $P$ .",
"Finally we denote by ${\\vec{\\cal A}(P)}$ the union of all ${{(P^{\\prime })}}$ with $P^{\\prime }\\in ds(P)$ , i.e., the set of all action types syntactically occurring in $P$ .",
"For any component $P$ , the exit rate from $P$ will be the sum of the activity rates of all the activities enabled in $P$ , i.e., $ q(P) = \\sum _{a \\in (P)}r_a$ , with $r_a$ being the rate of activity $a$ .",
"If $P$ enables more than one activity, $|(P)|>1$ , then the dynamic behavior of the model is determined by a race condition.",
"This has the effect of replacing the nondeterministic branching of the pure process algebra with probabilistic branching.",
"The probability that a particular activity completes is given by the ratio of the activity rate to the exit rate from $P$ ."
],
[
"Underlying Stochastic Process",
"In [20] it is proved that for any finite PEPA model $PP_0$ with $ds(P)=\\lbrace P_0,\\ldots ,P_n\\rbrace $, if we define the stochastic process $X(t)$ , such that $X(t)=P_i$ indicates that the system behaves as component $P_i$ at time t, then $X(t)$ is a continuous time Markov chain.",
"The transition rate between two components $P_i$ and $P_j$ , denoted $q(P_i,P_j)$ , is the rate at which the system changes from behaving as component $P_i$ to behaving as $P_j$ .",
"It is the sum of the activity rates labeling arcs which connect the node corresponding to $P_i$ to the node corresponding to $P_j$ in $(P)$ , i.e., $ q(P_i,P_j)= \\sum _{a\\in (P_i| P_j)} r_a$ where $P_i\\ne P_j$ and $(P_i| P_j) = a\\in (P_i) |\\ P_i {a} P_j$ .",
"Clearly if $P_j$ is not a one-step derivative of $P_i$ , $q(P_i,P_j)=0$ .",
"The $q(P_i,P_j)$ (also denoted $q_{ij}$ ), are the off-diagonal elements of the infinitesimal generator matrix of the Markov process, ${\\bf Q}$ .",
"Diagonal elements are formed as the negative sum of the non-diagonal elements of each row.",
"We use the following notation: $q(P_i)=\\sum _{j\\ne i}q(P_i,P_j)$ and $q_{ii}=-q(P_i)$ .",
"For any finite and irreducible PEPA model $P$ , the steady-state distribution $\\mathrm {\\Pi }(\\cdot )$ exists and it may be found by solving the normalization equation and the global balance equations: $\\sum _{P_i\\in ds(P)}\\mathrm {\\Pi }(P_i)=1$ and $\\mathrm {\\Pi } \\mathbf {Q} =\\mathbf {0}$ .",
"The conditional transition rate from $P_i$ to $P_j$ via an action type $\\alpha $ is denoted $q(P_i,P_j,\\alpha )$ .",
"This is the sum of the activity rates labeling arcs connecting the corresponding nodes in the derivation graph which are also labeled by the action type $\\alpha $ .",
"It is the rate at which a system behaving as component $P_i$ evolves to behaving as component $P_j$ as the result of completing a type $\\alpha $ activity.",
"The total conditional transition rate from $P$ to $S\\subseteq ds(P)$ , denoted $q[P,S,\\alpha ]$ , is defined as $q[P,S,\\alpha ]=\\sum _{P^{\\prime }\\in S} q(P,P^{\\prime },\\alpha )$ where $q(P,P^{\\prime },\\alpha )=\\sum _{P \\xrightarrow{} P^{\\prime }} r_{\\alpha }$ ."
],
[
"Observation Equivalence",
"In a process algebra, actions, rather than states, play the role of capturing the observable behavior of a system model.",
"This leads to a formally defined notion of equivalence in which components are regarded as equal if, under observation, they appear to perform exactly the same actions.",
"In this section we recall a bisimulation-like relation, named lumpable bisimilarity, for PEPA models [21].",
"Two PEPA components are lumpably bisimilar if there is an equivalence relation between them such that, for any action type $\\alpha $ different from $\\tau $ , the total conditional transition rates from those components to any equivalence class, via activities of this type, are the same.",
"Definition 1 (Lumpable bisimulation) An equivalence relation over PEPA components, $\\subseteq \\times $ , is a lumpable bisimulation if whenever $(P,Q)\\in $ then for all $\\alpha \\in $ and for all $S\\in /$ such that either $\\alpha \\ne \\tau $ , or $\\alpha =\\tau $ and $P,Q\\notin S$ , it holds $q[P,S,\\alpha ]=q[Q,S,\\alpha ]\\, .$ It is clear that the identity relation is a lumpable bisimulation.",
"We are interested in the relation which is the largest lumpable bisimulation, formed by the union of all lumpable bisimulations.",
"Definition 2 (Lumpable bisimilarity) Two PEPA components $P$ and $Q$ are lumpably bisimilar, written $P\\approx _lQ$ , if $(P,Q)\\in $ for some lumpable bisimulation $$ , i.e., $\\approx _l \\ =\\bigcup \\ \\lbrace \\ |\\ \\mbox{ is a lumpable bisimulation}\\rbrace .$ $\\approx _l$ is called lumpable bisimilarity and it is the largest symmetric lumpable bisimulation over PEPA components.",
"In [21] we proved that lumpable bisimilarity is a congruence for the so-called evaluation contexts, i.e., if $P_1 \\approx _l P_2$ then $a.P_1\\approx _l a.P_2$ ; $P_1{L}Q\\approx _l P_2{L}Q\\ \\ $ for all $L\\subseteq $ .",
"$P_1/L\\approx _lP_2/L$ .",
"Notice that the notion of strong equivalence defined in [20] is stricter than that of lumpable bisimilarity because the latter allows arbitrary activities with type $\\tau $ among components belonging to the same equivalence class.",
"In [3] a notion of weak bisimulation for CTMCs is introduced.",
"This is based on the idea that the time-abstract behavior of equivalent states is weakly bisimilar and that the relative speed of these states to move to a different equivalence class is equal.",
"To capture this intuition, the authors propose a definition of weak-bisimulation which resembles our notion of lumpable bisimulation if we ignore action types and labels.",
"This bisimulation is defined in the context of both discrete and continuous time Markov chains without any notion of compositionality, and hence of contextuality.",
"Compositionality is considered in [2], [5], [9], where definitions of weak bisimilarities for stochastic process algebra based on the classical concept of weak action are proposed.",
"Our approach shares with these bisimilarities the idea of ignoring the rates for non-synchronizing (labeled $\\tau $ ) transitions between a state and the others belonging to the same equivalence class.",
"The main difference between our definition and those presented in [2], [5], [9] is that we explicitly studied the relationships between our lumpable bisimilarity at the process algebra level and the induced lumping of the underlying Markov chains.",
"This led to a coinductive characterization of a notion of contextual lumpability as described in [21].",
"The security property named Persistent Stochastic Non-Interference (PSNI) tries to capture every possible information flow from a classified (high) level of confidentiality to an untrusted (low) one.",
"A strong requirement of this definition is that no information flow should be possible even in the presence of malicious processes that run at the classified level.",
"The main motivation is to protect a system also from internal attacks, which could be performed by the so-called Trojan Horse programs, i.e., programs that appear honest but hide some malicious code inside them.",
"More precisely, the notion of PSNI consists of checking all the states reachable by the system against all high level potential interactions.",
"In order to formally define our security property, we partition the set $\\setminus \\lbrace \\tau \\rbrace $ of visible action types, into two sets, ${\\cal H}$ and ${\\cal L}$ of high and low level action types.",
"A high level PEPA component $H$ is a PEPA term such that for all $H^{\\prime }\\in ds(H)$ , $(H^{\\prime })\\subseteq {\\cal H}$ , i.e., every derivative of $H$ may next engage in only high level actions.",
"We denote by ${\\cal C}_H$ the set of all high level PEPA components.",
"A system $P$ satisfies PSNI if for every state $P^{\\prime }$ reachable from $P$ and for every high level process $H$ a low level user cannot distinguish $P^{\\prime }$ from $P^{\\prime }{{\\cal H}}H$ .",
"In other words, a system $P$ satisfies PSNI if what a low level user sees of the system is not modified when it cooperates with any high level process $H$ .",
"In order to formally define the $\\emph {PSNI}$ property, we denote by $P\\setminus {\\cal H}$ the PEPA component $(P{{\\cal H}}\\bar{H})$ where $\\bar{H}$ is any high level process that does not cooperate with $P$ , i.e., for all $P^{\\prime }\\in ds(P)$ , ${\\cal A}(P^{\\prime })\\cap {\\cal A}(\\bar{H})=\\emptyset $ .",
"Intuitively $P\\setminus {\\cal H}$ denotes the component $P$ prevented from performing high level actions.",
"Notice that the definition is well formed in the sense that if $\\bar{H_1}$ and $\\bar{H_2}$ are two high level processes that do not cooperate with $P$ , then the derivation graphs of $(P{{\\cal H}}\\bar{H_1})$ and $(P{{\\cal H}}\\bar{H_2})$ are isomorphic.",
"Properties $\\emph {SNI}$ and $\\emph {PSNI}$ are formally defined as follows.",
"Definition 3 (Stochastic Non-Interference) Let $P$ be a PEPA component.",
"$P\\in SNI \\mbox{ iff }\\forall H\\in {\\cal C}_H,$ $ P\\setminus {\\cal H}\\approx _l (P{{\\cal H}}H)/{\\cal H}\\,.$ Definition 4 (Persistent Stochastic Non-Interference) Let $P$ be a PEPA component.",
"$P\\in PSNI \\mbox{ iff }\\forall P^{\\prime }\\in ds(P), \\, \\forall H\\in {\\cal C}_H,$ $ P^{\\prime }\\in SNI, \\mbox{ i.e., } P^{\\prime }\\setminus {\\cal H}\\approx _l (P^{\\prime }{{\\cal H}}H)/{\\cal H}\\,.$ We introduce a novel bisimulation-based equivalence relation over PEPA components, named $\\approx _l^{hc}$ , that allows us to give a first characterization of PSNI with no quantification over all the high level components $H$ .",
"In particular, we show that $P\\in \\mathit {PSNI}$ if and only if $P\\setminus {\\cal H}$ and $P$ are not distinguishable with respect to $\\approx _l^{hc}$ .",
"Intuitively, two processes are $\\approx _l^{hc}$ -equivalent if they can simulate each other in any possible high context, i.e., in every context $C[\\_]$ of the form $(\\_ \\ {{\\cal H}}H)/{\\cal H}$ where $H\\in {\\cal C}_H$ .",
"Observe that for any high context $C[\\_]$ and PEPA model $P$ , all the states reachable from $C[P]$ have the form $C^{\\prime }[P^{\\prime }]$ with $C^{\\prime }[\\_]$ being a high context too and $P^{\\prime }\\in ds(P)$ .",
"We now introduce the concept of lumpable bisimulation on high contexts: the idea is that, given two PEPA models $P$ and $Q$ , when a high level context $C[\\_]$ filled with $P$ executes a certain activity moving $P$ to $P^{\\prime }$ then the same context filled with $Q$ is able to simulate this step moving $Q$ to $Q^{\\prime }$ so that $P^{\\prime }$ and $Q^{\\prime }$ are again lumpable bisimilar on high contexts, and vice-versa.",
"This must be true for every possible high context $C[\\_]$ .",
"It is important to note that the quantification over all possible high contexts is re-iterated for $P^{\\prime }$ and $Q^{\\prime }$ .",
"For a PEPA model $P$ , $\\alpha \\in $ , $S\\subseteq ds(P)$ and a high context $C[\\_]$ we define: $q_C(P,P^{\\prime },\\alpha )=\\sum _{C[P] \\xrightarrow{} C^{\\prime }[P^{\\prime }]} r_{\\alpha }$ and $q_C[P,S,\\alpha ]=\\sum _{P^{\\prime }\\in S} q_C(P,P^{\\prime },\\alpha )\\,.$ The notion of lumpable bisimulation on high contexts is defined as follows: Definition 5 (Lumpable bisimilarity on high contexts) An equivalence relation over PEPA components, $\\subseteq \\times $ , is a lumpable bisimulation on high contexts if whenever $(P,Q)\\in $ then for all high context $C[\\_]$ , for all $\\alpha \\in $ and for all $S\\in /$ such that either $\\alpha \\ne \\tau $ , or $\\alpha =\\tau $ and $P,Q\\notin S$ , it holds $q_C[P,S,\\alpha ]=q_C[Q,S,\\alpha ]\\, .$ Two PEPA components $P$ and $Q$ are lumpably bisimilar on high contexts, written $P\\approx ^{hc}_lQ$ , if $(P,Q)\\in $ for some lumpable bisimulation on high contexts $$ , i.e., $\\approx ^{hc}_l \\ =\\bigcup \\ \\lbrace \\ |\\ \\mbox{ is a lumpable bisimulation on high contexts}\\rbrace .$ $\\approx ^{hc}_l$ is called lumpable bisimilarity on high contexts and it is the largest symmetric lumpable bisimulation on high contexts over PEPA components.",
"It is easy to prove that $\\approx ^{hc}_l$ is an equivalence relation.",
"The next theorem gives a characterization of PSNI in terms of $\\approx ^{hc}_l$ .",
"Theorem 1 Let $P$ be a PEPA component.",
"Then $P\\in PSNI \\mbox{ iff } P\\setminus {\\cal H}\\approx ^{hc}_l P\\,.$ We first show that $P\\setminus {\\cal H}\\approx ^{hc}_l P$ implies $P\\in PSNI$ .",
"In order to do it we prove that ${\\cal R}=\\lbrace (P_1\\setminus {\\cal H}, (P_2{{\\cal H}}H)/{\\cal H})\\,|\\, H\\in {\\cal C}_H \\mbox{ and } P_1\\setminus {\\cal H}\\approx ^{hc}_l P_2\\rbrace $ is a lumpable bisimulation.",
"This is sufficient to say that $P\\in PSNI$ .",
"First observe that, if $P\\setminus {\\cal H}\\approx ^{hc}_l P$ then for all $P^{\\prime }\\in ds(P)$ there exists $P^{\\prime \\prime }\\setminus {\\cal H}\\in ds(P\\setminus {\\cal H})$ such that $P^{\\prime \\prime }\\setminus {\\cal H}\\approx ^{hc}_l P^{\\prime }$ and, by definition of ${\\cal R}$ , for all $H\\in {\\cal C}_H$ , $(P^{\\prime \\prime }\\setminus {\\cal H}, (P^{\\prime }{{\\cal H}}H)/{\\cal H})\\in {\\cal R}$ .",
"Since ${\\cal R}$ is a lumpable bisimulation, we have that for all $H\\in {\\cal C}_H$ , $P^{\\prime \\prime }\\setminus {\\cal H}\\approx _l (P^{\\prime }{{\\cal H}}H)/{\\cal H}$ .",
"In particular, there exists $\\bar{H}\\in {\\cal C}_H$ such that $(P^{\\prime }{{\\cal H}}\\bar{H})/{\\cal H}$ coincides with $P^{\\prime }\\setminus {\\cal H}$ .",
"Since $\\approx _l $ is an equivalence relation, by symmetry and transitivity, we have that for every $P^{\\prime }\\in ds(P)$ and for every $H\\in {\\cal C}_H$ , $P^{\\prime \\prime }\\setminus {\\cal H}\\approx _l P^{\\prime }\\setminus {\\cal H} \\approx _l (P^{\\prime }{{\\cal H}}H)/{\\cal H}$ , i.e., $P\\in PSNI$ .",
"The fact that ${\\cal R}$ is a lumpable bisimulation follows from: if $P_1\\setminus {\\cal H}\\approx ^{hc}_l P_2$ then for all $\\alpha \\in $ with $\\alpha \\ne \\tau $ and for all $S\\in /\\!\\approx ^{hc}_l$ and for all high context $C[\\_]$ , we have $q_C[P_1\\setminus {\\cal H}, S,\\alpha ]=q_C[P_2, S,\\alpha ]$ .",
"Since a high context can only perform high level activities, we have that for all high level context $C[\\_]$ , it holds that $q[P_1\\setminus {\\cal H}, S,\\alpha ]=q_C[P_1\\setminus {\\cal H}, S,\\alpha ]$ and then $q[P_1\\setminus {\\cal H}, S,\\alpha ]=q_C[P_2, S,\\alpha ]$ , i.e., we have that for all $(P_1\\setminus {\\cal H}, (P_2{{\\cal H}}H)/{\\cal H})\\in {\\cal R}$ and for all $S^{\\prime }\\in /{\\cal R}$ it holds $q[P_1\\setminus {\\cal H}, S^{\\prime },\\alpha ]=q[(P_2{{\\cal H}}H)/{\\cal H}, S^{\\prime },\\alpha ]$ .",
"if $P_1\\setminus {\\cal H}\\approx ^{hc}_l P_2$ then for $\\alpha = \\tau $ and for all $S\\in /\\!\\approx ^{hc}_l$ with $P_1\\setminus {\\cal H}, P_2\\notin S$ and for all high context $C[\\_]$ , we have $q_C[P_1\\setminus {\\cal H}, S,\\alpha ]=q_C[P_2, S,\\alpha ]$ .",
"Since a high context can only perform high level activities, we have that for all high level context $C[\\_]$ , it holds that $q[P_1\\setminus {\\cal H}, S,\\alpha ]=q_C[P_1\\setminus {\\cal H}, S,\\alpha ]$ .",
"Hence for all $(P_1\\setminus {\\cal H}, (P_2{{\\cal H}}H)/{\\cal H})\\in {\\cal R}$ and for all $S^{\\prime }\\in /{\\cal R}$ with $P_1\\setminus {\\cal H},(P_2{{\\cal H}}H)/{\\cal H})\\notin S^{\\prime }$ it holds $q[P_1\\setminus {\\cal H}, S^{\\prime },\\alpha ]=q[(P_2{{\\cal H}}H)/{\\cal H}, S^{\\prime },\\alpha ]$ .",
"We now show that if $P\\in PSNI$ then $P\\setminus {\\cal H}\\approx ^{hc}_l P$ .",
"To this end it is sufficient to prove that ${\\cal R}=\\lbrace (P_1\\setminus {\\cal H}, P_2)\\,|\\, P_1\\setminus {\\cal H}\\approx _l P_2\\setminus {\\cal H} \\mbox{ and } P_2\\in PSNI\\rbrace $ is a lumpable bisimulation on high contexts.",
"Indeed, let $C[\\_]$ be a high context and $\\alpha \\in $ .",
"Assume $\\alpha \\ne \\tau $ .",
"From $P_1\\setminus {\\cal H}\\approx _l P_2\\setminus {\\cal H}$ , we have that for all $S\\in /\\approx _l$ , $q[P_1\\setminus {\\cal H}, S,\\alpha ]=q[P_2\\setminus {\\cal H}, S,\\alpha ]$ .",
"Since a high context can only perform high level activities, we have that for all high context $C[\\_]$ it holds $q[P_1\\setminus {\\cal H}, S,\\alpha ]=q_C[P_1\\setminus {\\cal H}, S,\\alpha ]$ .",
"Moreover, since $\\alpha \\ne \\tau $ , $q[P_2\\setminus {\\cal H}, S,\\alpha ]=q_C[P_2, S^{\\prime },\\alpha ]$ where $S^{\\prime }=\\lbrace P\\,|\\, P\\setminus {\\cal H}\\in S\\rbrace $ , i.e., for all high context $C[\\_]$ and $S\\in /{\\cal R}$ it holds $q_C[P_1\\setminus {\\cal H}, S,\\alpha ]=q_C[P_2, S,\\alpha ] $ .",
"Consider now $\\alpha =\\tau $ .",
"From $P_1\\setminus {\\cal H}\\approx _l P_2\\setminus {\\cal H}$ , we have that for all $S\\in /\\approx _l$ such that $P_1\\setminus {\\cal H}$ , $P_2\\setminus {\\cal H}\\notin S$ , $q[P_1\\setminus {\\cal H}, S,\\alpha ]=q[P_2\\setminus {\\cal H}, S,\\alpha ]$ .",
"Since a high context can only perform high level activities and both $P_1\\setminus {\\cal H}$ and $P_2\\setminus {\\cal H}$ do not perform high activities, we have that $q[P_i\\setminus {\\cal H}, S,\\alpha ]=q_C[P_i\\setminus {\\cal H}, S,\\alpha ]$ for all high level context $C[\\_]$ and for $i\\in \\lbrace 1,2\\rbrace $ .",
"From the fact that $P_2\\in PSNI$ , we have $P_2\\setminus {\\cal H}\\approx _l (P_2{{\\cal H}}H)/{\\cal H}$ for all $H\\in {\\cal C}_H$ , and then $q[P_2\\setminus {\\cal H}, S,\\alpha ]=q_C[P_2\\setminus {\\cal H}, S,\\alpha ]=q_C[P_2, S^{\\prime },\\alpha ]$ for all high context $C[\\_]$ , $S\\in /\\approx ^{hc}_l$ and $S^{\\prime }\\in /{\\cal R}$ such that $P_2\\setminus {\\cal H}\\notin S$ and $P_2\\notin S^{\\prime }$ , i.e., $q_C[P_1\\setminus {\\cal H}, S,\\alpha ]=q_C[P_2, S,\\alpha ]$ for all high context $C[\\_]$ and $S\\in /{\\cal R}$ such that $P_1\\setminus {\\cal H},P_2\\notin S$ .",
"Finally, we show how it is possible to give a characterization of PSNI avoiding both the universal quantification over all the possible high level components and the universal quantification over all the possible reachable states.",
"Before we have shown how the idea of “being secure in every state” can be directly moved inside the lumpable bisimulation on high contexts notion ($\\approx ^{hc}_l$ ).",
"However this bisimulation notion implicitly contains a quantification over all possible high contexts.",
"We now prove that $\\approx ^{hc}_l$ can be expressed in a rather simpler way by exploiting local information only.",
"This can be done by defining a novel equivalence relation which focuses only on observable actions that do not belong to ${\\cal H}$ .",
"More in detail, we define an observation equivalence where actions from ${\\cal H}$ may be ignored.",
"We first introduce the notion of lumpable bisimilarity up to ${\\cal H}$.",
"Definition 6 (Lumpable bisimilarity up to ${\\cal H}$ ) An equivalence relation over PEPA components, $\\subseteq \\times $ , is a lumpable bisimulation up to ${\\cal H}$ if whenever $(P,Q)\\in $ then for all $\\alpha \\in $ and for all $S\\in /$ if $\\alpha \\notin {\\cal H}\\cup \\lbrace \\tau \\rbrace $ then $q[P,S,\\alpha ]=q[Q,S,\\alpha ]\\, ,$ if $\\alpha \\in {\\cal H}\\cup \\lbrace \\tau \\rbrace $ and $P,Q\\notin S$ , then $q[P,S,\\alpha ]=q[Q,S,\\alpha ]\\,.$ Two PEPA components $P$ and $Q$ are lumpably bisimilar up to ${\\cal H}$, written $P\\approx ^{\\cal H}_lQ$ , if $(P,Q)\\in $ for some lumpable bisimulation up to ${\\cal H}$ , i.e., $\\approx ^{\\cal H}_l \\ =\\bigcup \\ \\lbrace \\ |\\ \\mbox{ is a lumpable bisimulation up to \\mbox{${\\cal H}$}}\\rbrace .$ $\\approx ^{\\cal H}_l$ is called lumpable bisimilarity up to ${\\cal H}$ and it is the largest symmetric lumpable bisimulation up to ${\\cal H}$ over PEPA components.",
"The next theorem shows that the binary relations $\\approx ^{hc}_l$ and $\\approx ^{{\\cal H}}_l$ are equivalent.",
"Theorem 2 Let $P$ and $Q$ be two PEPA components.",
"Then $P \\approx ^{hc}_l \\, Q \\mbox{ if and only if }\\,P\\approx ^{\\cal H}_l Q\\,.$ We first show that $P\\approx ^{hc}_l Q$ implies $P\\approx ^{\\cal H}_l Q$ .",
"In order to do it we prove that ${\\cal R}=\\lbrace (P,Q)\\,|\\, P\\approx ^{hc}_l Q\\rbrace $ is a lumpable bisimulation up to ${\\cal H}$ .",
"This follows from the following cases.",
"First observe that, by definition of ${\\cal R}$ , $S\\in /\\!\\approx ^{hc}_l$ if and only if $S\\in /\\!",
"{\\cal R}$ .",
"Let $\\alpha \\notin {\\cal H}\\cup \\lbrace \\tau \\rbrace $ .",
"From the fact that $P\\approx ^{hc}_l Q$ it holds that for all $S\\in /\\!\\approx ^{hc}_l$ and for all high context $C[\\_]$ , $q_C[P, S,\\alpha ]=q_C[Q, S,\\alpha ]$ .",
"Since $\\alpha \\notin {\\cal H}\\cup \\lbrace \\tau \\rbrace $ , we have that $q[P, S,\\alpha ]=q[Q, S,\\alpha ]$ .",
"Let $\\alpha \\in {\\cal H}\\cup \\lbrace \\tau \\rbrace $ .",
"From the fact that $P\\approx ^{hc}_l Q$ it holds that for all $S\\in /\\!\\approx ^{hc}_l$ such that $P,Q\\notin S$ and for all high context $C[\\_]$ , $q_C[P, S,\\tau ]=q_C[Q, S,\\tau ]$ .",
"If $C[\\_]$ does not synchronize with $P$ , we have that $q[P, S,\\tau ]=q[Q, S,\\tau ]$ .",
"On the other hand, consider a context $C[\\_]$ with only one current action type $h\\in {\\cal H}$ .",
"Then, from $q_C[P, S,\\tau ]=q_C[Q, S,\\tau ]$ and $q[P, S,\\tau ]=q[Q, S,\\tau ]$ , it follows that if $P$ cooperates over $h$ then also $Q$ cooperates over $h$ and $q[P, S,h]=q[Q, S,h]$ .",
"We now show that if $P\\approx ^{\\cal H}_l Q$ then $P\\approx ^{hc}_l Q$ .",
"To this end it is sufficient to prove that ${\\cal R}=\\lbrace (P,Q)\\,|\\, P\\approx ^{\\cal H}_l Q\\rbrace $ is a lumpable bisimulation on high contexts.",
"This follows from the following cases.",
"First observe that, by definition of ${\\cal R}$ , $S\\in /\\!\\approx ^{hc}_l$ if and only if $S\\in /\\!",
"{\\cal R}$ .",
"Let $\\alpha \\notin {\\cal H}\\cup \\lbrace \\tau \\rbrace $ .",
"From the fact that $P\\approx ^{\\cal H}_l Q$ it holds that for all $S\\in /\\!\\approx ^{\\cal H}_l$ , $q[P, S,\\alpha ]=q[Q, S,\\alpha ]$ .",
"Since a high context can only perform high level activities, we have that $q[P, S,\\alpha ]=q_C[P, S,\\alpha ]$ and $q[Q, S,\\alpha ]=q_C[Q, S,\\alpha ]$ for all high context $C[\\_]$ .",
"Hence, $q_C[P, S,\\alpha ]=q_C[Q, S,\\alpha ]$ .",
"Let $\\alpha =\\tau $ .",
"From the fact that $P\\approx ^{\\cal H}_l Q$ it holds that for all $S\\in /\\!\\approx ^{\\cal H}_l$ such that $P,Q\\notin S$ , $q[P, S,\\alpha ]=q[Q, S,\\alpha ]$ .",
"Hence for all high level context that do not synchronize with $P$ and $Q$ we have that $q[P, S,\\alpha ]=q_C[P, S,\\alpha ]$ and $q[Q, S,\\alpha ]=q_C[Q, S,\\alpha ]$ , i.e., $q_C[P, S,\\alpha ]=q_C[Q, S,\\alpha ]$ .",
"Let $h\\in {\\cal H}$ .",
"From the fact that $P\\approx ^{\\cal H}_l Q$ it holds that for all $S\\in /\\!\\approx ^{\\cal H}_l$ such that $P,Q\\notin S$ , $q[P, S,h]=q[Q, S,h]$ .",
"From this and the fact that $q[P, S,\\tau ]=q[Q, S,\\tau ]$ it follows that for all high level context $C[\\_]$ with only one current action type $h\\in {\\cal H}$ , $q_C[P, S,\\tau ]=q_C[Q, S,\\tau ]$ .",
"By induction on the number of current action types of a high level context $C[\\_]$ , we obtain that for $\\alpha =\\tau $ , for all $S\\in /{\\cal R}$ with $P,Q\\notin S$ it holds $q_C[P, S,\\alpha ]=q_C[Q, S,\\alpha ]$ .",
"Theorem REF allows us to identify a local property of processes (with no quantification on the states and on the high contexts) which is a necessary and sufficient condition for PSNI.",
"This is stated by the following corollary: Corollary 1 Let $P$ be a PEPA component.",
"Then $P\\in PSNI \\mbox{ iff } P\\setminus {\\cal H}\\approx ^{\\cal H}_l P\\,.$ Finally we provide a characterization of PSNI in terms of unwinding conditions.",
"In practice, whenever a state $P^{\\prime }$ of a PSNI PEPA model $P$ may execute a high level activity leading it to a state $P^{\\prime \\prime }$ , then $P^{\\prime }$ and $P^{\\prime \\prime }$ are indistinguishable for a low level observer.",
"Theorem 3 Let $P$ be a PEPA component.",
"$P\\in PSNI \\mbox{ iff }\\forall P^{\\prime }\\in ds(P), \\, $ $P^{\\prime } {(h,r)} P^{\\prime \\prime } \\mbox{ implies }P^{\\prime } \\setminus {\\cal H}\\approx _l P^{\\prime \\prime }\\setminus {\\cal H}$ We first prove that if $P\\in PSNI $ then for all $P^{\\prime }\\in ds(P)$ , $P^{\\prime } {(h,r)}~P^{\\prime \\prime }$ implies $P^{\\prime } \\setminus {\\cal H}\\approx _l P^{\\prime \\prime }\\setminus {\\cal H}$ .",
"Indeed, by Definition REF , $P^{\\prime }\\in PSNI$ and therefore, by Corollary REF , $ P^{\\prime }\\setminus {\\cal H}\\approx ^{\\cal H}_l P^{\\prime }$ .",
"By Definition REF of $\\approx ^{\\cal H}_l$ , for all $S\\in {\\cal C}/\\approx ^{\\cal H}_l$ such that $P^{\\prime }\\setminus {\\cal H}, P^{\\prime }\\notin S$ , both $q[P^{\\prime }\\setminus {\\cal H},S,\\tau ]=q[P^{\\prime },S,\\tau ]$ and $q[P^{\\prime }\\setminus {\\cal H},S,h]=q[P^{\\prime },S,h]$ .",
"Since $P^{\\prime }\\setminus {\\cal H}$ does not perform any high level action, $q[P^{\\prime }\\setminus {\\cal H},S,h]=0$ while, since $P^{\\prime } {(h,r)} P^{\\prime \\prime }$ , $q[P^{\\prime },S,\\hat{h}]\\ne 0$ .",
"Therefore, from $ P^{\\prime }\\setminus {\\cal H}\\approx ^{\\cal H}_l P^{\\prime }$ , either $h$ is not a current action type of $P^{\\prime }$ or $P^{\\prime }\\setminus {\\cal H}, P^{\\prime } \\in S$ , i.e., $P^{\\prime }\\setminus {\\cal H}\\approx ^{\\cal H}_l P^{\\prime \\prime }$ .",
"Since also $P^{\\prime \\prime }\\in PSNI$ , from $P^{\\prime \\prime }\\setminus {\\cal H}\\approx ^{\\cal H}_l P^{\\prime \\prime }$ it follows that $P^{\\prime }\\setminus {\\cal H}\\approx ^{\\cal H}_l P^{\\prime \\prime }\\setminus {\\cal H}$ .",
"Finally, since both $P^{\\prime }\\setminus {\\cal H}$ and $P^{\\prime \\prime }\\setminus {\\cal H}$ do not perform any high level activity, $P^{\\prime }\\setminus {\\cal H}\\approx ^{\\cal H}_l P^{\\prime \\prime }\\setminus {\\cal H}$ is equivalent to $P^{\\prime }\\setminus {\\cal H}\\approx _l P^{\\prime \\prime }\\setminus {\\cal H}$ .",
"We now prove that if for all $P^{\\prime }\\in ds(P)$ , $P^{\\prime } {(h,r)}~P^{\\prime \\prime }$ implies $P^{\\prime } \\setminus {\\cal H}\\approx _l P^{\\prime \\prime }\\setminus {\\cal H}$ then $P\\in PSNI$ .",
"Indeed observe that for all $\\alpha \\notin {\\cal H}\\cup {\\tau }$ , and for all $S\\in {\\cal C}/\\approx ^{\\cal H}_l$ , $q[P^{\\prime }\\setminus {\\cal H},S,\\alpha ]=q[P^{\\prime },S,\\alpha ]$ .",
"Moreover, if $P^{\\prime }\\setminus {\\cal H}, P^{\\prime }\\notin S$ then $q[P^{\\prime }\\setminus {\\cal H},S,\\tau ]=q[P^{\\prime },S,\\tau ]$ .",
"This is sufficient to prove that $P^{\\prime }\\setminus {\\cal H}\\approx ^{\\cal H}_l P^{\\prime }$ , i.e., by Corollary REF , $P\\in PSNI$ ."
],
[
"Properties of Persistent Stochastic Non-Interference",
"In this section we prove some interesting propertis of $PSNI$ .",
"First we prove that $PSNI$ is compositional with respect to low prefix, cooperation over low actions and hiding.",
"Proposition 1 Let $P$ and $Q$ be two PEPA components.",
"If $P,Q\\in PSNI$ , then $(\\alpha ,r).P\\in PSNI\\,$ for all $\\alpha \\in {\\cal L}\\cup \\lbrace \\tau \\rbrace $ $P/L\\in PSNI\\,$ for all $L\\subseteq {\\cal A}$ $P{L}Q\\in PSNI\\,$ for all $L\\subseteq {\\cal L}$ Assume that $P,Q\\in PSNI$ .",
"If $P\\in PSNI$ then for all $P^{\\prime }\\in ds(P)$ , $P^{\\prime } {(h,r)} P^{\\prime \\prime }$ implies $P^{\\prime } \\setminus {\\cal H}\\approx _l P^{\\prime \\prime }\\setminus {\\cal H}$ .",
"This property is clearly maintained for the PEPA component $(\\alpha ,r).P$ when $\\alpha \\in {\\cal L}\\cup \\lbrace \\tau \\rbrace $ .",
"If $P\\in PSNI$ then for all $P^{\\prime }\\in ds(P)$ , $P^{\\prime } {(h,r)} P^{\\prime \\prime }$ implies $P^{\\prime } \\setminus {\\cal H}\\approx _l P^{\\prime \\prime }\\setminus {\\cal H}$ .",
"Let $L\\subseteq {\\cal A}$ and $P^{\\prime }/L\\in ds(P)$ .",
"Assume that $P^{\\prime }/L {(h,r)} P^{\\prime \\prime }/L$ .",
"From the fact that $P^{\\prime } \\setminus {\\cal H}\\approx _l P^{\\prime \\prime }\\setminus {\\cal H}$ we have that $(P^{\\prime }{{\\cal H}}\\bar{H})\\approx _l(P^{\\prime \\prime }{{\\cal H}}\\bar{H})$ for any high level PEPA component $\\bar{H}$ that does not cooperate with $P$ .",
"From the fact that lumpable bisimilarity is a congruence for the evaluation contexts, we have that for all $L\\subseteq {\\cal A}$ , $(P^{\\prime }{{\\cal H}}\\bar{H})/L\\approx _l(P^{\\prime \\prime }{{\\cal H}}\\bar{H})/L$ .",
"We can assume that $\\vec{\\cal A}(\\bar{H})\\cap L=\\emptyset $ and hence, since also $\\vec{\\cal A}(\\bar{H})\\cap \\vec{\\cal A}(\\bar{P})=\\emptyset $ , $(P^{\\prime }/L{{\\cal H}}\\bar{H})/L\\approx _l(P^{\\prime \\prime }/L{{\\cal H}}\\bar{H})/L$ , i.e., $(P^{\\prime }/L) \\setminus {\\cal H}\\approx _l (P^{\\prime \\prime }/L)\\setminus {\\cal H}$ .",
"If $P,Q\\in PSNI$ then for all $P^{\\prime }\\in ds(P)$ , $P^{\\prime } {(h,r)} P^{\\prime \\prime }$ implies $P^{\\prime } \\setminus {\\cal H}\\approx _l P^{\\prime \\prime }\\setminus {\\cal H}$ and for all $Q^{\\prime }\\in ds(Q)$ , $Q^{\\prime } {(h,r)} Q^{\\prime \\prime }$ implies $Q^{\\prime } \\setminus {\\cal H}\\approx _l Q^{\\prime \\prime }\\setminus {\\cal H}$ .",
"Let $L\\subseteq {\\cal L}$ and $P^{\\prime }{L}Q^{\\prime }\\in ds(P{L}Q)$ .",
"Assume that $P^{\\prime }{L}Q^{\\prime }{(h,r)} P^{\\prime \\prime }{L}Q^{\\prime \\prime }$ .",
"In this case, either $P^{\\prime }{(h,r)} P^{\\prime \\prime }$ or $Q^{\\prime }{(h,r)} Q^{\\prime \\prime }$ .",
"Assume that $P^{\\prime }{(h,r)} P^{\\prime \\prime }$ and then $P^{\\prime }{L}Q^{\\prime }{(h,r)} P^{\\prime \\prime }{L}Q^{\\prime }$ .",
"From the hypothesis that $P\\in PSNI$ we have that $P^{\\prime } \\setminus {\\cal H}\\approx _l P^{\\prime \\prime }\\setminus {\\cal H}$ , i.e., $(P^{\\prime }{{\\cal H}}\\bar{H})\\approx _l(P^{\\prime \\prime }{{\\cal H}}\\bar{H})$ for any high level PEPA component $\\bar{H}$ that does not cooperate with $P$ and $Q$ .",
"From the fact that $\\approx _l$ is a congruence with respect to the cooperation operator we have $(P^{\\prime }{{\\cal H}}\\bar{H}){L}(Q^{\\prime }{{\\cal H}}\\bar{H})\\approx _l(P^{\\prime \\prime }{{\\cal H}}\\bar{H}){L}(Q^{\\prime }{{\\cal H}}\\bar{H})$ , moreover sice ${\\cal H}\\cap L=\\emptyset $ we obtain $(P^{\\prime }{L}Q^{\\prime }){{\\cal H}}\\bar{H} \\approx _l (P^{\\prime \\prime }{L}Q^{\\prime }){{\\cal H}}\\bar{H} $ , i.e., $(P^{\\prime }{L}Q^{\\prime })\\setminus {\\cal H} \\approx _l (P^{\\prime \\prime }{L}Q^{\\prime })\\setminus {\\cal H}$ .",
"In the case that $Q^{\\prime }{(h,r)} Q^{\\prime \\prime }$ the proof is analogous.",
"Notice that the fact that $PSNI$ is not preserved by the choice operatior is a consequence of the fact that lumpable bisimilarity is not a congruence for this operator.",
"We now prove that if $P\\in PSNI$ then the equivalence class $[P]$ with respect to lumpable bisimilarity $\\approx _l$ is closed under PSNI.",
"Proposition 2 Let $P$ and $Q$ be two PEPA components.",
"If $P\\in PSNI$ and $P\\approx _l Q$ then also $Q\\in PSNI$ .",
"Let $P\\in PSNI$ such that $P\\approx _l Q$ .",
"Let $ Q^{\\prime }\\in ds(Q)$ such that $Q^{\\prime } {(h,r)} Q^{\\prime \\prime }$ .",
"From the hypothesis that $P\\approx _l Q$ , there exist $P^{\\prime },P^{\\prime \\prime }\\in ds(P)$ such that $P^{\\prime }\\approx _l Q^{\\prime }$ and $P^{\\prime \\prime }\\approx _l Q^{\\prime \\prime }$ .",
"Hence there exists $r^{\\prime }$ such that $P^{\\prime } {(h,r^{\\prime })} P^{\\prime \\prime }$ and $P^{\\prime } \\setminus {\\cal H}\\approx _l P^{\\prime \\prime }\\setminus {\\cal H}$ .",
"From the fact that $\\approx _l$ is a congruence with respect to the cooperation operator we have $Q^{\\prime } \\setminus {\\cal H}\\approx _l Q^{\\prime \\prime }\\setminus {\\cal H}$ and then also $Q\\in PSNI$ ."
],
[
"Comparison with other SOS-based persistent security properties",
"The security property presented in this paper is persistent in the sense that if a model $P$ is secure then all the states reachable by $P$ during its execution are also secure.",
"Persistence is not a common feature of Non-Interference properties.",
"For example, many properties based on trace models, like generalized Non-Inference and separability [25], and the non local bisimulation based noninterference properties for the Markovian process calculus defined in [2] are not persistent.",
"Persistence is used in program verification techniques based on type-systems to provide sufficient conditions to Non-Interference properties, like, e.g., in [1], [19], [30], [31].",
"In this setting persistence provides sufficient static conditions which are invariant with respect to execution and imply the desired dynamic property.",
"In [15], a persistent property named P_BNDC has been proposed for non-deterministic CCS processes.",
"The aim of this definition is to capture a robust notion of security for processes which may move in the middle of a computation.",
"In this context persistence ensures that a secure process always migrates to a secure state.",
"Notice that if the system satisfies a non-persistent property then it might migrate when it is executing in an insecure state and then, from the point of view of the new host, the incoming process is insecure and, consequently, it should not be executed.",
"As our Persistent Stochastic Non-Interference property PSNI, property P_BNDC is provided with two sound and complete characterizations: one in terms of a behavioural equivalence between processes up to high level contexts and another one in terms of unwinding conditions.",
"Let us compare the expressivity of P_BNDC and PSNI by considering their SOS-based characterization in terms of unwinding conditions.",
"The formal unwinding characterization of P_BNDC for CCS processes is the following: Definition 7 Let $P$ be a CCS process and $H$ denote the set of all high level actions.",
"$P\\in P\\_BNDC \\mbox{ iff }\\, \\forall P^{\\prime }\\mbox{ reachable from P }, \\, $ $P^{\\prime } {h} P^{\\prime \\prime } \\mbox{ implies }P^{\\prime } {\\hat{\\tau }} P^{\\prime \\prime \\prime } \\mbox{ and }P^{\\prime \\prime } \\setminus {H}\\approx P^{\\prime \\prime \\prime }\\setminus {H}$ where ${\\hat{\\tau }}$ represents a possibly empty sequence of $\\tau $ transitions and $\\approx $ denotes Milner's weak bisimulation relation [27].",
"Both PEPA and CCS are provided with a structural operational semantics that allows us to compare the definitions of PSNI (for PEPA processes) and P_BNDC (for CCS processes) just by considering the processes label transition systems eventually removing information concerning the activity rates.",
"Consider for instance the simple process depicted in Figure REF .",
"If we discard the activity rates we can interpret the graph as the labeled transition system of a CCS process $P$ .",
"According to Definition REF we have that $P$ satisfies P_BNDC.",
"On the contrary, when we consider the activity rates we have the model of a PEPA process $P$ which, according to Theorem REF , does not satisfies PSNI.",
"Indeed we cannot find a lumpable bisimulation such that $P \\setminus {\\cal H}\\approx _l P^{\\prime }\\setminus {\\cal H}$ .",
"The unwinding definition of PSNI resembles the definition of Strong BNDC (SBNC) which has been introduced in [4] as a sufficient condition for verifying P_BNDC.",
"Figure: A simple two state model.Recently, in [2] Non-Interference properties for processes expressed as terms of a Markovian process calculus are introduced.",
"The calculus presented in the paper allows the authors to model three kinds of actions: exponentially timed actions, immediate actions and passive actions.",
"As a consequence, the proposed process algebra encompasses nondeterminism, probability, priority and stochastic time.",
"The behavioral observation defined by the authors extends the classical bisimulation relation of Milner [27].",
"The property named Bisimulation-based Strong Stochastic Local Non-Interference (BSSLNI) is defined in the style of our unwinding conditions but it is based on an observation equivalence named $\\approx _{EMB}$ which abstracts from internal $\\tau $ actions with zero duration.",
"In particular, the relation $\\approx _{EMB}$ is based on the idea that if a given class of processes is not reachable directly after executing a certain action, then one has to explore the possibility of reaching that class indirectly via a finite-length path $\\pi $ of internal actions with zero duration but with a specific probability of execution $prob(\\pi )$ .",
"As observed by the authors, in general the performance indices of a system satisfying BSSLNI are not independent from the presence or the absence of high level interactions.",
"Figure: A simple three state model.On the contrary, the observation equivalence at the base of our definition relies on the notion of lumpability and ensures that, for a secure process $P$ , the steady state probability of observing the system being in a specific state $P^{\\prime }$ is independent from its possible high level interactions.",
"In order to show it consider the simple three state system depicted in Figure REF .",
"In this case, following Theorem REF , we can prove that $P_1\\in PSNI$ .",
"Indeed, it is easy to prove that $P_1 \\setminus {\\cal H}\\approx _l P_2\\setminus {\\cal H}$ when $\\approx _l$ is the lumpable bisimilarity.",
"In particular, the probability for a low level user to observe, in steady state, the system being in state $P_3$ is independent from whether or not $P_1$ has performed the high level activity $(h,\\lambda )$ .",
"To prove this, suppose that $P_1$ synchronizes over $h$ .",
"Then, for a low level observer, the system behaves as $P_1/{\\cal H}$ depicted in Figure REF $(a)$ .",
"We can compute the steady state distribution of $P_1/{\\cal H}$ by solving the global balance equations together with the normalization condition, obtaining: $\\begin{array}{rcl}\\pi _1*2\\lambda & =& \\pi _3*\\rho \\\\\\pi _2*\\lambda & =& \\pi _1*\\lambda \\\\\\pi _3*\\rho & =& \\pi _1*\\lambda +\\pi _2*\\lambda \\\\\\pi _1+\\pi _2+\\pi _3& =& 1\\\\\\end{array}$ whose solution is $\\begin{array}{cccccc}\\pi _1=\\frac{\\rho }{2(\\lambda +\\rho )} & & \\pi _2=\\frac{\\rho }{2(\\lambda +\\rho )} & & \\pi _3=\\frac{\\lambda }{\\lambda +\\rho } \\\\\\end{array}$ where $\\pi _1,\\pi _2$ and $\\pi _3$ denote the steady state probabilities of states $P_1/{\\cal H}$ , $P_2/{\\cal H}$ and $P_3/{\\cal H}$ , respectively.",
"Consider now the case in which $P_1$ does not synchronize over $h$ .",
"Then the low level view of the system is represented by $P_1\\setminus {\\cal H}$ depicted in Figure REF $(b)$ .",
"Again we can compute the steady state distribution of $P_1\\setminus {\\cal H}$ by solving the global balance equations together with the normalization condition, obtaining: $\\begin{array}{rcl}\\pi _1*\\lambda & =& \\pi _3*\\rho \\\\\\pi _3*\\rho & =& \\pi _1*\\lambda \\\\\\pi _1+\\pi _3& =& 1\\\\\\end{array}$ whose solution is $\\begin{array}{cccccc}\\pi _1=\\frac{\\rho }{\\lambda +\\rho } & & \\pi _3=\\frac{\\lambda }{\\lambda +\\rho } \\\\\\end{array}$ where $\\pi _1$ and $\\pi _3$ are the steady state probabilities of states $P_1\\setminus {\\cal H}$ and $P_3\\setminus {\\cal H}$ , respectively.",
"This proves that, from the low level point of view, the steady state probability of $P_3$ is independent from the fact that $P$ has cooperated with a high level context or not.",
"Figure: The models of P 1 /ℋP_1/{\\cal H} and P 1 ∖ℋP_1\\setminus {\\cal H}."
],
[
"Conclusion",
"In this paper we presented a persistent information flow security property for stochastic processes expressed as terms of the PEPA process algebra.",
"Our property, named Persistent Stochastic Non-Interference (PSNI) is based on a structural operational semantics and a bisimulation based observation equivalence for the PEPA terms.",
"We provide two characterizations for PSNI: one in terms of a bisimulation-like equivalence relation and another one in terms of unwinding conditions.",
"The first characterization allows us to perform the verification of PSNI for finite state processes in polynomial time with respect to the number of states of the system [28].",
"The second characterization is based on unwinding conditions.",
"This kind of conditions for possibilistic security properties have been already explored in the literature, like, e.g., in [29], [26], [22].",
"Such unwinding conditions have been proposed for traces-based models and represent only sufficient conditions for their respective security properties.",
"Differently, our unwinding conditions provide both necessary and sufficient conditions for PSNI.",
"Finally, in this paper we also deal with compositionality issues.",
"Indeed, the development of large and complex systems strongly depends on the ability of dividing the task of the system into subtasks that are solved by system subcomponents.",
"Thus, it is useful to define properties which are compositional in the sense that if the properties are satisfied by the system subcomponents then the system as a whole will satisfy the desired property by construction.",
"We show that PSNI is compositional with respect to low prefix, cooperation over low actions and hiding.",
"Acknowledgments: The work described in this paper has been partially supported by the Università Ca' Foscari Venezia - DAIS within the IRIDE program.",
"It has been also partially supported by the PRID project ENCASE financed by Università degli Studi di Udine and by GNCS-INdAM project Metodi Formali per la Verifica e la Sintesi di Sistemi Discreti e Ibridi."
]
] | 1808.08650 |
[
[
"Simple Proofs for the Derivative Estimates of the Holomorphic Motion\n near Two Boundary Points of the Mandelbrot Set"
],
[
"Abstract For the complex quadratic family $q_c:z\\mapsto z^2+c$, it is known that every point in the Julia set $J(q_c)$ moves holomorphically on $c$ except at the boundary points of the Mandelbrot set.",
"In this note, we present short proofs of the following derivative estimates of the motions near the boundary points $1/4$ and $-2$: for each $z = z(c)$ in the Julia set, the derivative $dz(c)/dc$ is uniformly $O(1/\\sqrt{1/4-c})$ when real $c\\nearrow 1/4$; and is uniformly $O(1/\\sqrt{-2-c})$ when real $c\\nearrow -2$.",
"These estimates of the derivative imply Hausdorff convergence of the Julia set $J(q_c)$ when $c$ approaches these boundary points.",
"In particular, the Hausdorff distance between $J(q_c)$ with $0\\le c<1/4$ and $J(q_{1/4})$ is exactly $\\sqrt{1/4-c}$."
],
[
"Introduction",
"For the family of quadratic maps $q_{c}$ of $, $ zz2+c$, with $ c$ a complex number not locating on the boundary of the Mandelbrot set $ M$,it is well-known that every point in the Julia set $ J(qc)$ moves holomorphically with respect to $ c$, i.e.",
"the {\\it holomorphic motion} \\cite {L, MSS}.Note that $ J(qc)$ is a Cantor set when $ cM$ andis connected when $ cM$.A parameter $ c$ is called {\\it hyperbolic} if the orbit of the originaccumulates on an attracting cycle or diverges to infinity.A {\\it hyperbolic component} is a connected component of $ M$containing hyperbolic parameters.It is conjectured that the set $ M$consists of only the hyperbolic parameters.See \\cite [Exposé I]{DH} for example.$ Let $\\mathbb {D}$ denote the open disk of radius one centered at the origin.",
"There is a biholomorphic function $\\Phi $ from $\\overline{\\mathbb {C}}- \\mathbb {M}$ to $\\overline{\\mathbb {C}}- \\overline{\\mathbb {D}}$ with which the set $ \\mathcal {R}(\\theta ):=\\lbrace \\Phi ^{-1}(r e^{i2\\pi \\theta })|~ 1< r\\le \\infty \\rbrace $ is defined and called the parameter ray of angle $\\theta \\in \\mathbb {R}/\\mathbb {Z}$ of the Mandelbrot set $\\mathbb {M}$ .",
"Given $\\theta $ , if $\\lim _{r\\searrow 1}\\Phi ^{-1}(re^{i2\\pi \\theta })$ exists, then this limit is called the landing point of the parameter ray $\\mathcal {R}(\\theta )$ .",
"A parameter $\\hat{c}$ in $\\partial \\mathbb {M}$ is called semi-hyperbolic if the critical point is non-recurrent and belongs to the Julia set [1].",
"A typical example is a Misiurewicz parameter, that is, for which the critical point eventually lands on a repelling periodic point.",
"The set consisting of semi-hyperbolic parameters is dense with Hausdorff dimension 2 in $\\partial \\mathbb {M}$ [12].",
"For each semi-hyperbolic parameter ${\\hat{c}}\\in \\partial \\mathbb {M}$ , there exists at least one parameter ray ${\\mathcal {R}}(\\theta )$ landing at ${\\hat{c}}$ .",
"(See [4].)",
"In a recent paper [2], we proved the following result concerning the estimate for the derivative of the holomorphic motion.",
"Theorem 1.1 Let $\\hat{c} \\in \\partial \\mathbb {M}$ be a semi-hyperbolic parameter that is the landing point of $\\mathcal {R}(\\theta )$ .",
"Then there exists a constant $K>0$ that depends only on $\\hat{c}$ such that for any $c\\in \\mathcal {R}(\\theta )$ sufficiently close to ${\\hat{c}}$ and any $z = z(c)\\in J(q_c)$ , the point $z(c)$ moves holomorphically with ${\\left| \\frac{dz(c)}{dc} \\right|}\\le \\frac{K}{\\sqrt{{\\left| c-\\hat{c} \\right|}}}.",
"$ This result enables us to obtain one-sided Hölder continuity of the holomorphic motion along the parameter ray (i.e.",
"the holomorphic motion lands).",
"More precisely, let $\\hat{c}\\in \\partial \\mathbb {M}$ be a semi-hyperbolic parameter that is a landing point of $\\mathcal {R}(\\theta )$ , and let $c=c(r):=\\Phi ^{-1}(re^{i2\\pi \\theta })$ with $r \\in (1,2]$ .",
"Then for any $z(c(2))$ in $J(q_{c(2)})$ , the improper integral $z(\\hat{c}) := z(c(2))+ \\lim _{\\delta \\searrow 0} \\int _{2}^{1+\\delta } \\frac{dz(c)}{dc} ~\\frac{dc(r)}{dr}~dr$ exists in the Julia set $J(q_{{\\hat{c}}})$ .",
"In particular, $z(c)$ is uniformly one-sided Hölder continuous of exponent $1/2$ at $c = \\hat{c}$ along $\\mathcal {R}(\\theta )$ : There exists a constant $K^{\\prime }$ depending only on ${\\hat{c}}$ such that $|z(c)-z({\\hat{c}})| \\le K^{\\prime } \\sqrt{|c-{\\hat{c}}|} $ for any $c =c(r) \\in \\mathcal {R}(\\theta )$ with $1<r\\le 2$ .",
"The primary aims of this paper are two-folds.",
"The first is to show that the same estimate as (REF ) holds when $c$ approaches $1/4$ , which is a parabolic parameter, along the real line in the interior of the Mandelbrot set, namely Theorem 1.2 For any point $z = z(c)$ in the Julia set $J(q_c)$ , we have that $z(c)$ moves holomorphically with derivative ${\\left| \\frac{dz(c)}{dc} \\right|}\\le \\frac{1}{2\\sqrt{1/4-c}} $ as $c \\nearrow 1/4$ along the real line in the interior of $\\mathbb {M}$ .",
"Figure: Top: The Julia set J(q c )J(q_c) for c=k/20(k=0,1,⋯,5)c= k/20~ (k=0, 1, \\cdots , 5).Bottom: Real analytic motion of the preimages of the repelling fixed point for 0≤c<1/40 \\le c <1/4.The second is to present a simple proof of Theorem REF for the case ${\\hat{c}}=-2$ .",
"(See Remark REF .)",
"As a matter of fact, what we present is a proof of Theorem REF for the logistic map $f_\\mu :, $ zz(1-z)$, with the semi-hyperbolic (Misiurewicz) parameter $ =4$ case, stated as follows.$ Theorem 1.3 For $z = z(\\mu )$ in the Julia set $J(f_\\mu )$ , the point $z(\\mu )$ moves holomorphically with derivative ${\\left| \\frac{dz(\\mu )}{d\\mu } \\right|}= O{\\left( \\frac{1}{\\sqrt{\\mu -4}} \\right)}$ as $\\mu \\searrow 4$ along the real line.",
"Figure: Real analytic motion of the Julia set J(f μ )J(f_\\mu ) forμ↘4\\mu \\searrow 4.",
"See Figure 2 of for the corresponding motion for q c q_c for c↗-2c \\nearrow -2.The proof of Theorem REF in [2] relies on the hyperbolic metric, the hyperbolicity of the $\\omega $ -limit set of a semi-hyperbolic parameter ${\\hat{c}}$ , the John condition on $\\mathbb {C}-J(q_{\\hat{c}})$ , and the asymptotic similarity between $J(q_{\\hat{c}})$ and $\\mathbb {M}$ at $\\hat{c}$ .",
"To the best of our knowledge, the difficulty of the proof for parameter ${\\hat{c}}=-2$ is essentially the same as the general semi-hyperbolic parameter case.",
"Moreover, the difficulty of proof remains unchanged even if we consider quadratic maps in the form of logistic maps for general semi-hyperbolic parameter case.",
"To our surprise, however, we find that for the logistic map with $\\mu =4$ case, the difficulty of the proof can be substantially reduced.",
"We only need a singular metric, and the proof is very straightforward.",
"This is one of the motivations of this paper.",
"Remark 1.4 The logistic map $f_\\mu $ is affinely conjugate to $q_c$ via the conjugacy $G(\\cdot , \\mu ):z \\mapsto w=-\\mu z+\\frac{\\mu }{2} \\qquad (\\mbox{so that}~ G(\\cdot , \\mu )\\circ f_\\mu \\circ G(\\cdot , \\mu )^{-1}=q_c) $ with $c=\\frac{\\mu (2-\\mu )}{4} \\qquad (\\mu \\ne 0).",
"$ Fix a point $z_0\\in J(f_\\mu )$ , and let $w_0=G(z_0, \\mu )\\in J(q_c)$ .",
"When $\\mu $ varies, let $z(\\mu )$ be the holomorphic motion for $f_\\mu $ , and $w(c)$ be the corresponding holomorphic motion for $q_c$ via the above relation (REF ).",
"From the conjugacy (REF ), the derivative $dw(c)/dc$ can be obtained from $dz(\\mu )/d\\mu $ : $\\frac{dw(c)}{dc} &=& \\frac{\\partial G}{\\partial z}\\cdot \\frac{dz}{d\\mu }\\cdot \\frac{d\\mu }{dc}+\\frac{\\partial G}{\\partial \\mu }\\cdot \\frac{d\\mu }{dc} \\nonumber \\\\&=& \\left(-\\mu \\cdot \\frac{dz}{d\\mu }-z+\\frac{1}{2}\\right)\\cdot \\frac{2}{1-\\mu } \\qquad (\\mu \\ne 0~\\mbox{or}~1).",
"$ Also, we have $\\frac{dz(\\mu )}{d\\mu } = \\frac{\\mu -1}{2\\mu }\\cdot \\frac{dw}{dc}+\\frac{w}{\\mu ^2} \\qquad (\\mu \\ne 0~\\mbox{or}~1).",
"$ Theorem REF for ${\\hat{c}}=-2$ can be derived by using (REF ): The fact that $\\mu \\searrow 4$ means $c\\nearrow -2$ leads to $\\left|\\frac{dw(c)}{dc}\\right| &=& O\\left(\\frac{1}{\\sqrt{\\mu -4}}\\right) \\qquad \\mbox{(by Theorem \\ref {Main_thm})}\\\\&=& O\\left(\\frac{2}{\\sqrt{(\\mu +2)(\\mu -4)}}\\right) \\\\&=& O\\left(\\frac{1}{\\sqrt{-2-c}}\\right) \\qquad \\mbox{(by the identity (\\ref {cmu}))}$ as $c\\nearrow -2$ , in which we have used $-z+1/2=w/\\mu $ and the uniform boundedness of $J(q_c)$ .",
"(An estimate of the size of the Julia set is given in Lemma REF .)",
"Remark 1.5 The estimates in theorems REF and REF are optimal.",
"For example, let $z=z(c)$ be the repelling fixed point of $q_c$ for $0\\le c<1/4$ , then $\\frac{dz(c)}{dc}=- \\frac{1}{2\\sqrt{1/4-c}}.$ (This means that the equality in (REF ) is attained.)",
"Also, let $z=z(\\mu )$ be a pre-image point of $1\\in J(f_\\mu )$ , then $\\frac{dz(\\mu )}{d\\mu }=\\pm \\frac{1}{\\mu \\sqrt{\\mu }\\sqrt{\\mu -4}}.$"
],
[
"Hausdorff convergence and dynamical degeneration",
"In this section, we discuss the convergence of the Julia set $J(q_c)$ and the degeneration of dynamics of $q_c$ on $J(q_c)$ when $c$ approaches the boundary points $c=1/4$ or $-2$ of the Mandelbrot set.",
"We also discuss corresponding properties for the map $f_\\mu $ on its Julia set $J(f_\\mu )$ by using (REF ) and (REF ).",
"The equality (REF ) shows that the map from real $\\mu $ to real $c$ in $(-\\infty , 1/4]$ is two-fold, therefore $\\mu \\searrow 1$ or $\\mu \\nearrow 1$ when $c\\nearrow 1/4$ , and $\\mu \\searrow 4$ or $\\mu \\nearrow -2$ as $c\\nearrow -2$ (or more precisely, $\\mu =1\\pm \\sqrt{4\\epsilon }$ when $c=1/4-\\epsilon $ and $\\mu =1\\pm \\sqrt{9+4\\epsilon }$ when $c=-2-\\epsilon $ , with $\\epsilon >0$ ).",
"It is easy to see that the dynamics of $f_{1-\\sqrt{4\\epsilon }}$ is a trivial copy of that of $f_{1+\\sqrt{4\\epsilon }}$ , and the same triviality holds for $f_{1-\\sqrt{9+4\\epsilon }}$ and $f_{1+\\sqrt{9+4\\epsilon }}$ .",
"Hence, we shall restrict our discussion to the cases $\\mu \\searrow 1$ and $\\mu \\searrow 4$ only."
],
[
"$c\\nearrow 1/4$ or {{formula:b9405910-f19f-49b1-9ff7-ebfbc17aae66}}",
"Theorem REF imples one-sided Hölder continuity of the holomorphic motion as $c\\nearrow 1/4$ along the real line: The improper integral $z(1/4) := z(0)+ \\lim _{\\delta \\searrow 0} \\int _{0}^{1/4-\\delta } \\frac{dz(c)}{dc} ~dc$ exists in the Julia set $J(q_{1/4})$ .",
"In particular, $|z(c)-z(1/4)| \\le \\sqrt{1/4-c} $ for any real $c\\in [0,1/4)$ .",
"It is well-known that $q_c$ is hyperbolic for $c$ in the hyperbolic components of $\\mathbb {M}$ .",
"Therefore, $(J(q_c), q_c)$ , the restriction of $q_c$ to $J(q_c)$ , is topologically conjugate to $(J(q_0), q_0)$ via a conjugacy $h_c(\\cdot ;0)$ from $J(q_0)$ to $J(q_c)$ for $c$ in the main cardioid of $\\mathbb {M}$ .",
"Inequalities (REF ) and (REF ) lead to a result that the conjugacy $h_c(\\cdot ;0)$ converges uniformly to a semiconjugacy $h_{1/4}(\\cdot ;0):J(q_{0}) \\rightarrow J(q_{1/4})$ , $z(0)\\mapsto z(1/4)$ , as $c$ increases from 0 to $1/4$ .",
"As a matter of fact, $h_{1/4}(\\cdot ;0)$ is a conjugacy (see [7] for example).",
"So, $(J(q_{1/4}), q_{1/4})$ is topologically conjugate to $( \\mathcal {T})$ , where $\\mathcal {T}:~ t\\mapsto 2t ~(\\bmod ~ 1)$ is the angle-doubling map."
],
[
"Hausdorff distance.",
"The estimate (REF ) and the existence of the conjugacy above between $(J(q_c),q_c)$ and $(J(q_{1/4}),q_{1/4})$ imply that the Hausdorff distance between $J(q_c)$ and $J(q_{1/4})$ is at most $\\sqrt{1/4-c}$ .",
"The distance is also at least $\\sqrt{1/4-c}$ , because the distance between the parabolic fixed point $1/2 \\in J(q_{1/4})$ and the Julia set $J(q_c)$ is attained by the repelling fixed point $(1+\\sqrt{1-4c})/2=1/2+\\sqrt{1/4-c}$ of $q_c$ .",
"(Indeed, in the proof of Theorem REF we will show that $|z| \\ge (1+\\sqrt{1-4c})/2$ for any $z \\in J(q_c)$ .)",
"Hence we obtain the following: Corollary 2.1 For $0 \\le c <1/4$ , the Hausdorff distance between the Julia sets $J(q_c)$ and $J(q_{1/4})$ is exactly $\\sqrt{1/4-c}$ .",
"Remark 2.2 From (REF ) and Theorem REF , when $1<\\mu <2$ we get $\\left| \\frac{dz(\\mu )}{d\\mu }\\right| &\\le & \\frac{\\mu -1}{2\\mu }\\cdot \\frac{1}{\\sqrt{1-4c}}+\\frac{1+\\sqrt{1+4c}}{2}\\cdot \\frac{1}{\\mu ^2} \\qquad \\mbox{(by Lemma \\ref {uniform_bound_q})}\\\\&=& \\frac{1}{2\\mu } +\\frac{1+\\sqrt{1+2\\mu -\\mu ^2}}{2\\mu ^2} \\qquad \\mbox{(by the identity (\\ref {cmu}))} \\\\&\\le & \\frac{2+\\sqrt{2}}{2}.$ This unexpected result implies that the Hausdorff distance between $J(f_\\mu )$ and $J(f_1)$ is at most $(2+\\sqrt{2})(\\mu -1)/2$ as $\\mu \\searrow 1$ along the real axis."
],
[
"Notation.",
"When variables $X, Y \\ge 0$ satisfy $X/C \\le Y \\le CX$ with a uniform constant $C>1$ , we denote this by $X \\asymp Y$ .",
"Remark 2.3 Let $\\dim _H J(q_c)$ denote the Hausdorff dimension of $J(q_c)$ .",
"It has been known from [6] that $ 1<\\dim _H J(q_{\\frac{1}{4}})<\\frac{3}{2}$ and that there exists $c_0<1/4$ such that for all $c\\in [c_0,1/4)$ one has $\\frac{d}{dc}\\dim _H J(q_c)\\asymp \\left(\\frac{1}{4}-c \\right)^{\\dim _H J(q_{\\frac{1}{4}})-\\frac{3}{2}}.",
"$ Therefore, the derivative of the Hausdorff dimension of the Julia set $J(q_c)$ with respect to $c$ tends to infinity from the left of $1/4$ , and the graph of $\\dim _H J(q_c)$ versus $c$ has a vertical tangent on the left at $1/4$ .",
"The unexpected result in Remark REF suggests us to examine whether or not a similar vertical tangency holds for the graph of the Hausdorff dimension $\\dim _H J(f_\\mu )$ as $\\mu $ decreases to 1.",
"What we find is that the graph $\\dim _HJ(f_\\mu )$ versus $\\mu $ has a horizontal tangent on the right at 1.",
"(See the proposition below.)",
"Proposition 2.4 There exists $\\mu _0>1$ such that $-\\frac{d}{d\\mu }\\dim _H J(f_\\mu )\\asymp \\left( \\mu -1 \\right)^{2\\dim _H J(f_1)-2}$ for any $\\mu \\in (1,\\mu _0]$ ."
],
[
"Proof.",
"For $\\mu \\ne 0$ , the affine map $G(\\cdot , \\mu )$ in (REF ) sends $J(f_\\mu )$ to $J(q_{\\mu (2-\\mu )/4})$ , thus $ \\dim _H J(f_\\mu )=\\dim _H J(q_{\\frac{\\mu (2-\\mu )}{4}})$ for $\\mu \\ne 0$ .",
"In particular, $ \\dim _H J(f_1)=\\dim _H J(q_{\\frac{1}{4}}).$ Consequently, for $\\mu \\ne 0$ or 1, $-\\frac{d}{d\\mu }\\dim _H J(f_\\mu )&=&\\frac{d}{dc}\\dim _H J(q_c)\\cdot \\frac{\\mu -1}{2} \\\\&\\asymp & \\left( \\frac{1}{4}-\\frac{\\mu (2-\\mu )}{4}\\right)^{\\dim _HJ(f_1)-\\frac{3}{2}}\\cdot \\frac{\\mu -1}{2} \\qquad \\mbox{(by (\\ref {ddcdimH14}))}\\\\&=&\\left(\\frac{\\mu -1}{2}\\right)^{2\\dim _H J(f_1)-2}\\\\&\\asymp &\\left(\\mu -1\\right)^{2\\dim _H J(f_1)-2},$ as asserted.",
"The value of $\\mu _0$ can be obtained by solving $c_0=\\mu _0(2-\\mu _0)/4$ from (REF ) with the $c_0$ in Remark REF .",
"$\\blacksquare $"
],
[
"$c\\nearrow -2$ or {{formula:487dafc3-2079-4921-8b44-04febe9a7bae}}",
"For $c$ in the exterior of $\\mathbb {M}$ , the restriction of $q_c$ to $J(q_c)$ is topologically conjugate to the one-sided left shift with two symbols.",
"Therefore, there exists a conjugacy $h_c(\\cdot ;c_0)$ from $J(q_{c_0})$ to $J(q_c)$ for any fixed $c_0$ and $c$ not belonging to $\\mathbb {M}$ .",
"The results (REF ) and (REF ) give rise to a consequence that for any semi-hyperbolic parameter ${\\hat{c}}\\in \\partial \\mathbb {M}$ , any parameter ray $\\mathcal {R}(\\theta )$ landing at ${\\hat{c}}$ , and any $c_0\\in \\mathcal {R}(\\theta )$ , the conjugacy $h_c(\\cdot ;c_0)$ converges uniformly to a semiconjugacy $h_{{\\hat{c}}}(\\cdot ;c_0):J(q_{c_0}) \\rightarrow J(q_{\\hat{c}}), \\quad z(c)\\mapsto z({\\hat{c}}), $ as $c \\rightarrow {\\hat{c}}$ along $\\mathcal {R}(\\theta )$ .",
"This further implies that the Hausdorff distance between $J(q_c)$ and $J(q_{\\hat{c}})$ is $O(\\sqrt{|c-{\\hat{c}}|})$ as $c \\rightarrow {\\hat{c}}$ along ${\\mathcal {R}}(\\theta )$ .",
"Let $\\Sigma :=\\Big \\lbrace {\\bf s}=\\lbrace s_0,s_1,s_2,\\ldots \\rbrace |~ s_n=0\\ \\mbox{or}\\ 1~ \\mbox{for all}~ n\\ge 0\\Big \\rbrace $ be the space consisting of sequences of 0's and 1's with the product topology, and $\\sigma $ be the left shift in $\\Sigma $ , $\\sigma ( {\\bf s})={\\bf s}^{\\prime }=\\lbrace s^{\\prime }_0, s^{\\prime }_1,s^{\\prime }_2,\\cdots \\rbrace $ with $s^{\\prime }_i=s_{i+1}$ .",
"Fix $\\theta \\in \\lbrace 0\\rbrace $ , the two points $\\theta /2$ and $(\\theta +1)/2$ divide $ into two open semi-circles $ 0$ and $ 1$ with $ 0$.",
"Let $$ be such an angle that $ Tn(){2,+12}$ for all $ n0$.",
"Define the {\\it kneading sequence} of $$ under $ T$ as $ E()={ E()n }n0$ with$$ \\mathcal {E}(\\theta )_n={\\left\\lbrace \\begin{array}{ll}0 & \\mbox{for}~ \\mathcal {T}^n(\\theta )\\in 0^\\theta \\\\1 & \\mbox{for}~ \\mathcal {T}^n(\\theta )\\in 1^\\theta .\\end{array}\\right.",
"}$$Note that the kneading sequence of non-recurrent $$ is well-defined.$ A point ${\\bf e}\\in \\Sigma $ is said to be aperiodic if $\\sigma ^n({\\bf e})\\ne {\\bf e}$ for any $n\\ge 0$ .",
"Two points ${\\bf a}$ and ${\\bf s}$ in $\\Sigma $ are said to be equivalent with respect to aperiodic ${\\bf e}\\in \\Sigma $ , denoted by ${\\bf a}\\sim _{\\bf e}{\\bf s}$ , if there is $k\\ge 0$ such that $a_n=s_n$ for all $n\\ne k$ and $\\sigma ^{k+1}({\\bf a})=\\sigma ^{k+1}({\\bf s})={\\bf e}$ .",
"In [2], we proved that the semiconjugacy $h_{\\hat{c}}(\\cdot ; c_0)$ described in (REF ) leads to the following result.",
"Let ${\\hat{c}}$ be a semi-hyperbolic parameter with an external angle $\\theta $ and ${\\bf e}=\\mathcal {E}(\\theta )$ be the kneading sequence of $\\theta $ .",
"Then $(J(q_{\\hat{c}}), q_{\\hat{c}})$ is topologically conjugate to $(\\Sigma /{\\sim _{\\bf e}}, \\tilde{\\sigma })$ , where $\\tilde{\\sigma }$ is induced by the shift transformation $\\sigma $ .",
"The dynamical degeneration for $q_c$ as $c\\nearrow -2$ along real axis, namely, along $\\mathcal {R}(\\theta )$ with $\\theta =1/2$ is the same as the one for the logistic map $f_\\mu $ as $\\mu \\searrow 4$ along the real axis.",
"Since $\\mathcal {E}(1/2)=\\lbrace 0,1,1,\\ldots \\rbrace $ , the dynamical degeneration of $q_c$ at $c=-2$ or $f_\\mu $ at $\\mu =4$ is that both $(J(q_{-2}),q_{-2})$ and $(J(f_4),f_4)$ are topologically conjugate to $(\\Sigma /{\\sim _{\\lbrace 0,1,1,\\ldots \\rbrace }},\\tilde{\\sigma })$ .",
"Note that there is another way to interpret the kneading sequence: For the logistic map $f_\\mu $ , the Julia set $J(f_\\mu )$ is a Cantor set contained in the real interval $[0,1]$ when $\\mu $ is real and greater than 4.",
"When $\\mu = 4$ , $J(f_4)$ is the whole interval $[0,1]$ .",
"If the critical point $1/2$ belongs to the Julia set, one can define the kneading sequence $I(f_\\mu )=\\lbrace I(f_\\mu )_n\\rbrace _{n\\ge 0}$ for $f_\\mu $ by $I(f_\\mu )_n=1$ if $f_\\mu ^{1+n}(1/2)\\in [0,1/2] \\cap J(f_\\mu )$ and $I(f_\\mu )_n=0$ if $f_\\mu ^{1+n}(1/2)\\in [1/2, 1] \\cap J(f_\\mu )$ .",
"The sequence $I(f_\\mu )$ is well-defined if $f_\\mu ^{1+n}(1/2)\\ne 1/2$ for all $n\\ge 0$ .",
"Then, it is not difficult to see that $\\mathcal {E}(1/2)=I(f_4)$ ."
],
[
"Proof of Theorem ",
"Assume $0\\le c < 1/4$ .",
"Let $r=(1+\\sqrt{1-4c})/2$ , the distance of the repelling fixed point of $q_c$ from the critical point 0.",
"Suppose $z\\in J(q_c)$ and $|z|<r$ .",
"Then, $|q_c(z)|=|z^2+c|<r^2+c=r$ .",
"Thus, $|q_c^n(z)|<r$ for all $n\\ge 0$ .",
"Since this is an open condition, this means that $z$ belongs to the filled Julia set but not to the Julia set, a contradiction.",
"Therefore, $\\inf _{z\\in J(q_c)}|z|=r,$ and $|Dq_c(z)|=2|z|\\ge 1+\\sqrt{1-4c}$ for all $z\\in J(q_c)$ (where $D=d/dz$ ).",
"For any $c$ not belonging to $\\mathbb {M}$ or in a hyperbolic component of $\\mathbb {M}$ , and for any $z =z(c) \\in J(q_c)$ , in [2] we proved the following derivative formula $\\frac{dz(c)}{dc} =-\\sum _{n = 1}^\\infty \\frac{1}{Dq_c^{{n}}({z}(c))}.$ Hence, $\\left|\\frac{dz(c)}{dc}\\right| \\le \\sum _{n\\ge 1} \\frac{1}{|Dq_c^n(z)|} \\le \\sum _{n\\ge 1}\\frac{1}{\\left(1+\\sqrt{1-4c}\\right)^n}=\\frac{1}{2\\sqrt{1/4-c}}.$ $\\blacksquare $ Remark 3.1 (i) Compared with our proof of Theorem REF in [2] for $c$ approaching a semi-hyperbolic parameter, or even with that of Theorem REF , to come in the next section, we find that the proof of Theorem REF is surprisingly simple.",
"(ii) In fact, by combining with the lemma below, the Julia set $J(q_c)$ locates inside the annulus $\\lbrace z|~ (1+\\sqrt{1-4c})/2\\le |z|\\le (1+\\sqrt{1+4c})/2\\rbrace $ for $0\\le c<1/4$ .",
"Lemma 3.2 $|z|\\le (1+\\sqrt{1+4|c|})/2$ for any $z\\in J(q_c)$ and $c\\in \\mathbb {C}$ ."
],
[
"Proof.",
"Let $M=M(c)= (1+\\sqrt{1+4|c|})/2$ and notice that $M^2-M-|c|=0$ .",
"If $z$ is such a point that $|z|=M+R$ for some $R>0$ , then $|q_c(z)|-|z|=|z^2+c|-|z|\\ge |z|^2-|z|-|c|=(2M-1)R+R^2 $ .",
"This implies that the orbit of $z$ tends to infinity thus $z\\notin J(q_c)$ .",
"$\\blacksquare $"
],
[
"Proof of Theorem ",
"Note that $f_\\mu (1) = 0$ and the origin $z = 0$ is a repelling fixed point of multiplier $\\mu $ for any real $\\mu \\ge 4$ .",
"Hence there exists a linearizing coordinate $\\phi _\\mu :\\tilde{U}_0 \\rightarrow defined ona fixed neighborhood $ U0$ of $ 0$ such that\\begin{enumerate}\\item for any z \\in f_\\mu ^{-1}(\\tilde{U}_0)\\cap \\tilde{U}_0,\\phi _\\mu (f_\\mu (z)) = \\mu \\phi _\\mu (z).\\item |\\phi _\\mu (z)| \\asymp |z| when \\mu is sufficiently close to 4.\\end{enumerate}$ Fix a point $z = z(\\mu )$ in the Julia set $J(f_\\mu )$ for $\\mu >4$ .",
"Set $f := f_\\mu $ and $z_n =z_n(\\mu ): = f_\\mu ^n(z)~(n \\ge 0)$ .",
"As in [3], we can derive $\\frac{dz_{n + 1}}{d\\mu }=Df(z_n)\\frac{dz_{n}}{d\\mu }+ \\frac{1}{\\mu } z_{n + 1}$ (where $D = d/dz$ ) and thus we have a `formal' expansion $\\frac{dz}{d\\mu }=-\\frac{1}{\\mu } \\sum _{n \\ge 1}\\frac{z_{n}}{Df^n(z)}.",
"$ Now suppose that $\\mu $ is sufficiently close to 4.",
"We will show that the formal expansion above converges absolutely.",
"The main idea is to consider a fixed `singular' metric of the following explicit form $\\gamma (z) |dz|:= \\frac{|dz|}{\\sqrt{|z|~|z-1|}}$ on ${\\left\\lbrace 0,1 \\right\\rbrace }$ (inspired by [8]; see also [11]).",
"Let us fix a $z \\in J(f)\\cap (0,1)$ .",
"Then it is easy to see $\\gamma (z) \\ge 2$ .",
"Moreover, $\\frac{\\gamma (f(z))|Df(z)|}{\\gamma (z)} =\\frac{2 \\sqrt{\\mu }~|z-1/2|}{\\sqrt{1-f(z)}}.$ Note that $f(z) \\in J(f) \\subset [0,1]$ , hence $1-f(z) \\ge 0$ .",
"By $f(z) = \\mu /4-\\mu (z-1/2)^2$ , we have $\\frac{\\gamma (f(z))|Df(z)|}{\\gamma (z)} =\\frac{2 \\sqrt{\\mu /4 -f(z)}}{\\sqrt{1-f(z)}}.$ The right hand side takes its infimum as $f(z) \\searrow 0$ : $\\frac{\\gamma (f(z))|Df(z)|}{\\gamma (z)} \\ge 2 \\sqrt{\\frac{\\mu }{4}}= \\sqrt{\\mu }=:A ~(= 2 + O(\\mu -4)).$ Hence for any fixed $z= z_0 \\in J(f)$ whose forward orbit never lands on the fixed point 0, with the help of the identity $\\frac{\\gamma (z_n)}{\\gamma (z)}Df^n(z)=\\prod _{k=0}^{n-1} \\frac{\\gamma (z_{k+1})}{\\gamma (z_k)}Df(z_k),$ we obtain $\\frac{1}{|Df^n(z)|} \\le \\frac{\\gamma (z_n)}{A^n \\gamma (z)}\\le \\frac{\\gamma (z_n)}{2A^n}\\asymp \\frac{\\gamma (z_n)}{A^n}.",
"$ Now we have ${\\left| \\frac{dz}{d\\mu } \\right|}\\le \\mu ^{-1} \\sum _{n \\ge 1}\\frac{|z_{n}|}{|Df^n(z)|}\\le \\mu ^{-1} \\sum _{n \\ge 1}\\frac{|z_{n}|\\gamma (z_n)}{2A^n}\\asymp \\sum _{n \\ge 1}\\frac{|z_{n}|\\gamma (z_n)}{A^n}$ for such a $z$ .",
"This implies that if the forward orbit ${\\left\\lbrace z_n \\right\\rbrace }_{n \\ge 0}$ is a certain distance away from 0 (and 1), then $|z_n|\\gamma (z_n) \\asymp 1$ and the derivative $\\dfrac{dz}{d\\mu }$ is uniformly bounded by a constant.",
"More precisely, we have: Proposition 4.1 For any real $\\mu \\ge 4$ and any forward orbit ${\\left\\lbrace z_n \\right\\rbrace }_{n \\ge 0}$ , if there exists some $\\delta >0$ such that $\\delta \\le z_n \\le 1-\\delta $ for all $n$ , then ${\\left| \\dfrac{dz}{d\\mu } \\right|}\\le \\frac{1}{8\\delta }.$"
],
[
"Proof.",
"By assumption we have $|z_n| \\le 1$ and $\\gamma (z_n) \\le 1/\\delta $ .",
"Thus (REF ) implies ${\\left| \\frac{dz}{d\\mu } \\right|}\\le \\mu ^{-1} \\sum _{n \\ge 1}\\frac{|z_{n}|\\gamma (z_n)}{2A^n}\\le \\mu ^{-1}\\cdot \\frac{1}{2\\delta }\\cdot \\frac{A^{-1}}{1-A^{-1}}.$ Since $\\mu = A^2 \\ge 4$ we have the desired estimate.",
"$\\blacksquare $"
],
[
"Non-pre-fixed case.",
"Next we suppose that ${\\left\\lbrace z_n \\right\\rbrace }_{n \\ge 0}$ accumulates on 0 but never lands on 0.",
"We may assume that $f^{-1}(\\tilde{U}_0)$ is the union of disjoint neighborhoods $U_0$ of 0 and $U_1$ of 1.",
"Now there exist $N$ and $m \\ge 1$ such that $z_{N-1} \\in U_1 \\cap J(f)$ , $z_{N},z_{N + 1},\\, \\ldots , \\, z_{N + m-1} \\in U_0 \\cap J(f)$ , $z_{N + m} \\in (\\tilde{U}_0-U_0) \\cap J(f)$ .",
"By the linearizing coordinate $\\phi = \\phi _\\mu :\\tilde{U}_0 \\rightarrow ,we have$$\\phi (z_{N + m}) = \\phi (f^{m-i}(z_{N + i}))= \\mu ^{m-i}\\phi (z_{N + i})$$for $ 0 i m$.Since $ |(z)| |z|$, we have$ |(zN + m )| |zN + m | 1$ and thus$$|z_{N + i}| \\asymp \\mu ^{-m + i}.$$Since $ (z) 1/z$ when $ 0<z<1/2$, we have$$\\sum _{i = 0}^{m}\\frac{|z_{N + i}|\\gamma (z_{N + i})}{A^{N + i}}\\asymp \\sum _{i = 0}^{m}\\frac{\\mu ^{-m + i}\\cdot \\sqrt{\\mu ^{m - i}}}{A^{N + i}}= \\frac{m+1}{A^{N + m}}.$$Hence the sum for `near zero^{\\prime } orbit points$ zN, ..., zN + m U0$are bounded by $ (m + 1)/AN + m$.Since $ (m + 1)/Am 0$ as $ m $,there exits a constant $ K>0$ independent of $ m$ with$ (m + 1)/AN + m K/AN$.\\footnote {By calculating the function z \\mapsto (z + 1)/A^z,we will find K = A/(e\\log A) = 2/(e\\log 2) + O(\\mu -4).",
"}$ Let us give an estimate of $\\dfrac{|z_{N -1}|}{|Df^{N-1}(z)|}$ for $z_{N-1} \\in U_1$ .",
"When $N = 1$ this term is not counted in the formal sum expansion of $dz/d\\mu $ , so we may assume that $N \\ge 2$ .",
"One can show that $1-z_{N-1} \\asymp \\mu ^{-m-1}$ .",
"Since $|Df(z)| = \\mu ~ |1-2z| = 2\\sqrt{\\mu } \\sqrt{\\frac{\\mu }{4} -f(z)},$ we have $|Df(z_{N-2})|= 2\\sqrt{\\mu } \\sqrt{\\frac{\\mu -4}{4} + 1-z_{N-1}}\\ge \\sqrt{\\mu } \\sqrt{\\mu -4}.$ Moreover, by using (REF ) and $z_{N-2} \\approx 1/2$ , we have $\\frac{1}{|Df^{N-2}(z)|}= O{\\left( \\frac{\\gamma (z_{N-2})}{A^{N-2}} \\right)}= O{\\left( \\frac{1}{A^N} \\right)}.$ Hence $\\frac{|z_{N -1}|}{|Df^{N-1}(z)|}=\\frac{|z_{N -1}|}{|Df^{N-2}(z)\\cdot Df(z_{N-2})|}= O{\\left( \\frac{1}{A^{N}}\\cdot \\frac{1}{\\sqrt{\\mu -4}} \\right)}.$ By assumption, we have infinitely many `near singular' orbit points $z_{N_1-1}, \\ldots , z_{N_1 + m_1},z_{N_2-1}, \\ldots , z_{N_2 + m_2},\\ldots \\in \\tilde{U}_0 \\cup U_1$ with strictly increasing $N_j$ .",
"Hence the original sum (REF ) can be estimated as follows: ${\\left| \\frac{dz}{d\\mu } \\right|}&\\le \\sum _{z_n \\notin \\tilde{U}_0 \\cup U_1}\\frac{|z_{n}|\\gamma (z_n)}{A^n}+\\sum _{z_n \\in \\tilde{U}_0 }\\frac{|z_{n}|\\gamma (z_n)}{A^n}+ \\sum _{z_n \\in U_1}\\frac{|z_{n}|\\gamma (z_n)}{A^n}\\\\& \\asymp \\sum _{z_n \\notin \\tilde{U}_0 \\cup U_1}\\frac{1}{A^n}+\\sum _{j\\ge 1}\\frac{m_j + 1}{A^{N_j + m_j}}+\\sum _{j\\ge 1}O{\\left( \\frac{1}{A^{N_j}}\\cdot \\frac{1}{\\sqrt{\\mu -4}} \\right)}\\nonumber \\\\&=O(1)+\\sum _{j\\ge 1}\\frac{K}{A^{N_j}}+O{\\left( \\frac{1}{\\sqrt{\\mu -4}} \\right)}\\cdot \\sum _{j\\ge 1}\\frac{1}{A^{N_j}}\\nonumber \\\\& = O{\\left( \\frac{1}{\\sqrt{\\mu -4}} \\right)}~~(\\mu \\searrow 4).",
"\\nonumber $ Since the diameter of $\\tilde{U}_0$ is fixed for $\\mu \\approx 4$ , one can check that the estimate above is independent of both $\\mu $ and $z = z(\\mu ) \\in J(f_\\mu )$ ."
],
[
"Pre-fixed case.",
"Next we consider the case where the orbit of $z = z(\\mu )$ eventually lands on 0.",
"Assume that $z_{n} = 0$ if and only if $n \\ge N \\ge 0$ .",
"Then the derivative is written as a finite sum $\\frac{dz}{d\\mu } &= -\\frac{1}{\\mu } \\sum _{1 \\le n < N} \\frac{z_n}{Df^n(z)}$ and the sum is divided into $\\sum _{1 \\le n < N} =\\sum _{z_n \\notin \\tilde{U}_0 \\cup U_1}+\\sum _{z_n \\in \\tilde{U}_0 }+ \\sum _{z_n \\in U_1}$ as (REF ) in the previous case.",
"Hence we obtain the same estimate $|dz/d\\mu | = O(1/\\sqrt{\\mu -4})$ without extra effort.",
"$\\blacksquare $"
],
[
"Acknowledgments",
"Chen was partly supported by MOST 106-2115-M-001-007.",
"Kawahira was partly supported by JSPS KAKENHI Grant Number 16K05193.",
"They thank the hospitality of Academia Sinica, Nagoya University, RIMS in Kyoto University, and Tokyo Institute of Technology where parts of this research were carried out.",
"Yi-Chiuan Chen Institute of Mathematics Academia Sinica Taipei 10617, Taiwan YCChen@math.sinica.edu.tw Tomoki Kawahira Department of Mathematics Tokyo Institute of Technology Tokyo 152-8551, Japan kawahira@math.titech.ac.jp Mathematical Science Team RIKEN Center for Advanced Intelligence Project (AIP) 1-4-1 Nihonbashi, Chuo-ku Tokyo 103-0027, Japan"
]
] | 1808.08335 |
[
[
"$L_p$ and almost sure convergence of estimation on heavy tail index\n under random censoring"
],
[
"Abstract In this paper, we prove $L_p,\\ p\\geq 2$ and almost sure convergence of tail index estimator mentioned in \\cite{grama2008} under random censoring and several assumptions.",
"$p$th moment of the error of the estimator is proved to be of order $O\\left(\\frac{1}{\\log^{m\\kappa/2}n}\\right)$ with given assumptions.",
"We also perform several finite sample simulations to quantify performance of this estimator.",
"Finite sample results show that the proposed estimator is effective in finding underlying tail index even when censor rate is high."
],
[
"Introduction",
"Research on heavy tail data is relevant to numerous statistical application, such as actuarial science [2], economics [3] and etc.",
"Tail index is one of the most crucial factor for long tail data since it is related with extreme quantiles of the underlying distribution, see [4] and [5] for further discussion.",
"Hill [6] proposed an estimator for tail index and this estimator has been proved convergence under several assumptions and situations, we refer [7] and [8] as two examples.",
"Apart from traditional Hill's estimator, Grama and Spokoiny [1] applied Kullback-Leibler divergence to estimate heavy tail index and proposed an estimator based on maximization local log-likeliihood method, Kratz and Resnick [9] proposed a qq type estimator and Politis et.al.",
"[10] proposed a truncated ratio statistics and proved its $L^p$ convergence.",
"On the other hand, it is common for dealing with incomplete observations, especially right censor data, in practical researches.",
"Klein and Moeschberger [11] provide detail discussion and examples on this topic.",
"For heavy tail data, censoring is more likely to occur.",
"For example, in clinical trial, if the survival time obeys long tail distribution, it is more likely for the patient to survive after trial ends.",
"Therefore, how to estimate heavy tail index under censoring is worth discussion.",
"Beirlant and Guillou [12] proved consistency of a modified Hill's estimator under mild censoring, Einmahl et.al.",
"[13] applied moment estimator in this problem and proved the asymptotic normality of the proposed estimator.",
"Ndao et.al.",
"[14] and Stupfler[4] generalized the result to the conditional heavy tail index.",
"Instead of convergence in probability, we mainly focus on almost sure convergence and $L^p$ convergence of estimator proposed in [1] for censoring data.",
"Similar to [13] and [10], we apply a truncated version ratio type statistics for heavy tail index and use method proposed by Vasiliev [15] for proving convergence of the estimator.",
"We will give the basic assumption and statistics in REF .",
"In section and , we respectively discuss $L^p$ convergence and almost sure convergence of the proposed estimator, and numerical examples can be seen in section .",
"Finally, we make conclusions in section ."
],
[
"Basic assumptions and main results ",
"In this part, we introduce basic assumptions, frequently used notations and the main statistics in this paper.",
"The notations that are not listed below will be defined when being used.",
"Suppose $(X_1,Y_1),\\ (X_2,Y_2),...,(X_n,Y_n)$ are i.i.d data from underlying distribution whose tail functions are respectively $P_X(x)=L_X(x)x^{-\\alpha _X}$ , $P_Y(x)=L_Y(x)x^{-\\alpha _Y}$ , and we further assume that $X_i,Y_i,\\ i=1,2,...,n$ are mutually independent.",
"Suppose the observed data are $(Z_i,\\delta _i),\\ Z_i=X_i\\bigwedge Y_i=\\min (X_i,Y_i)$ and $\\delta _i=\\mathbf {1}_{X_i\\le Y_i}$ .",
"Under this assumption, it is obvious that tail function of $Z_i$ , $P_Z(x)=P_X(x)P_Y(x)=L_Z(x)x^{-\\alpha _Z}$ , here $L_Z(x)=L_X(x)L_Y(x),\\ \\alpha _Z=\\alpha _X+\\alpha _Y$ Moreover, we suppose $L_X,\\ L_Y$ are slow varying function (see [16]).",
"According to Karamata's theorem[16], the slow varying function $L_K(x),\\ K=X,Y$ satisfy $\\log (L_K(x))=c_K(x)+\\int _{a}^{x}\\frac{\\epsilon _K(y)}{y}dy,\\ c_K(x)\\rightarrow c,\\ \\epsilon _K\\rightarrow 0$ as $x\\rightarrow \\infty $ , and if we further assume that $L_K$ is differentiable, then formula REF intuitively implies that $c^{^{\\prime }}_K(x)\\rightarrow 0$ and $\\frac{xL_K^{^{\\prime }}(x)}{L_K(x)}=xc_K^{^{\\prime }}(x)+\\epsilon _K(x)$ Apart from the independent censor assumption, we hope the absolute value of derivative of $L_K,\\ K=X,Y$ to be small enough as $x$ being sufficiently large so that the influence of $L_K$ on estimating tail index can be controlled by taking logarithm.",
"This idea leads to assumption A2.",
"The third assumption comes form [10].",
"Assumption A1: Suppose $X_i$ being i.i.d data and $Y_i$ being i.i.d censor time, $X_i,Y_i$ are mutually independent and respectively have tail function(that is, 1-cumulative distribution) $P_X(x)=L_X(x)x^{-\\alpha _X}$ , $P_Y(x)=L_Y(x)x^{-\\alpha _Y}$ .",
"Thus the tail index of the data and censor time are $\\gamma _X=1/\\alpha _X,\\ \\gamma _Y=1/\\alpha _Y$ Assumption A2: Suppose $L_K,\\ K=X,Y$ are differentiable and there exists a number $\\kappa >0$ such that, for $K=X,Y$ , as $x\\rightarrow \\infty $ .",
"$\\vert \\frac{xL_K^{^{\\prime }}(x)}{L_K(x)}\\vert =O\\left(\\frac{1}{\\log ^{\\kappa } x}\\right)$ Assumption A3: There exists a known constant $\\gamma _0>0$ such that $\\gamma _X\\ge 2\\gamma _0,\\ \\gamma _Y\\ge 2\\gamma _0$ From REF this implies that $\\alpha _Z\\le 1/\\gamma _0$ and corresponding $\\gamma _Z\\ge \\gamma _0$ Table REF displays the frequently used notations and their meanings.",
"In order to illustrate the main estimator, we first introduce several intermediate statistics.",
"Table: Frequently used notationsDefinition 1 ($\\widehat{p}(x)$ and $\\widehat{q}(x)$ ) Suppose $x>0$ and sample size is $n$ , then we respectively define $\\widehat{p}(x)$ and $\\widehat{q}(x)$ as $\\widehat{p}(x)=\\frac{1}{n}\\Sigma _{i=1}^n \\mathbf {1}_{Z_i\\ge x}$ and $\\widehat{q}(x)=\\frac{1}{n}\\Sigma _{i=1}^n \\delta _i\\mathbf {1}_{Z_i\\ge x}$ It is not difficult to see that $\\widehat{p}(x)$ is estimator for $P_Z(x)$ and $\\widehat{q}(x)$ is an estimator for $Prob(Z\\le x\\cap X\\le Y)$ .",
"REF demonstrates the motivation for us to estimate this probability.",
"Similar with [13], the second estimator $\\widehat{\\rho }$ is applied to estimate $\\frac{\\gamma _Y}{\\gamma _X+\\gamma _Y}$ .",
"Definition 2 With the notation in table REF , we define estimator $\\widehat{\\rho }$ as $\\widehat{\\rho }=\\frac{\\widehat{q}(t(n))}{\\widehat{p}(t(n))}\\mathbf {1}_{\\widehat{p}\\ge s(n)}$ Here $t(n)\\rightarrow \\infty $ and $s(n)\\rightarrow 0$ satisfying ${\\left\\lbrace \\begin{array}{ll}t(n)=n^\\beta ,\\ s(n)=n^{-c},\\ \\beta <\\frac{\\gamma _0}{2},\\ \\frac{\\beta }{\\gamma _0}<c<1/2,\\ \\text{If A3 is satisfied}\\\\t(n)=\\log ^\\beta n,\\ s(n)=n^{-c},\\ \\beta >0,\\ 0<c<1/2\\ \\text{If A3 is not satisfied}\\end{array}\\right.",
"}$ We apply estimator defined in [1], $\\zeta $ , for estimating tail index of the censored data $\\gamma _Z$ .",
"According to REF , it is reasonable for expecting $\\zeta $ to converge to $\\frac{\\gamma _X\\gamma _Y}{\\gamma _X+\\gamma _Y}$ .",
"Definition 3 (Estimator for tail index $\\gamma _Z$ ) With the notation in table REF , we define the estimator for tail index $\\gamma _Z$ as $\\widehat{\\zeta }=\\frac{1}{\\widehat{p}(t(n))}\\int _{t(n)}^\\infty \\frac{\\widehat{p}(y)}{y}dy\\mathbf {1}_{\\widehat{p}(t(n))\\ge s(n)}=\\frac{\\mathbf {1}_{\\widehat{p}(t(n))\\ge s(n)}}{n\\widehat{p}(t(n))}\\left(\\Sigma _{i=1}^n\\log \\left(\\frac{Z_i}{t(n)}\\right)\\mathbf {1}_{Z_i\\ge t(n)}\\right)$ Here we use convention that $\\infty \\times 0=0$ .",
"Since $\\gamma _X=\\frac{\\gamma _X\\gamma _Y}{\\gamma _X+\\gamma _Y}/\\frac{\\gamma _Y}{\\gamma _X+\\gamma _y}$ , estimator $\\frac{\\widehat{\\zeta }}{\\widehat{\\rho }}$ is a candidate for estimating tail index of $X$ .",
"We will use a truncated version of this estimator.",
"The key results of this paper is presented in theorem REF and REF .",
"Theorem 1 Suppose A1, A2, A3 and choose $H_n=\\frac{1}{\\log \\log n}$ ,$m\\ge 4$ , then we have $\\vert \\vert \\frac{\\widehat{\\zeta }}{\\widehat{\\rho }}\\mathbf {1}_{\\widehat{\\rho }\\ge H_n}-\\gamma _X\\vert \\vert _{m/2}=O\\left(\\frac{1}{\\log ^{\\kappa }n}\\right)$ Theorem 2 Suppose condition A1-A3 and choose $H_n$ as in theorem REF , then we have $\\frac{\\widehat{\\zeta }}{\\widehat{\\rho }}\\mathbf {1}_{\\widehat{\\rho }\\ge H_n}\\rightarrow _{a.s.}\\gamma _X$"
],
[
"$L_p$ convergence of truncated statistics",
"We first provide several crucial lemma that will be frequently used in the following proof.",
"Lemma 1 Suppose $X$ and $Y$ satisfy A1 and A2, then $Z=min(X,Y)$ has tail index $\\gamma _Z=\\frac{\\gamma _X\\gamma _Y}{\\gamma _X+\\gamma _Y}$ , and $P_Z(x)=L_Z(x)x^{-1/\\gamma _Z}$ with $L_Z$ satisfies A2 Because of independents, we have, for arbitrary large $x$ , $P_Z(x)=L_X(x)L_Y(x)x^{-\\alpha _X-\\alpha _Y}$ Therefore, $\\gamma _Z=1/(\\alpha _X+\\alpha _Y)=\\frac{\\gamma _X\\gamma _Y}{\\gamma _X+\\gamma _Y}$ and the first part is proved.",
"For the second part, notice that $\\vert \\frac{xL_Z^{^{\\prime }}(x)}{L_Z(x)}\\vert \\le \\vert \\frac{xL_X^{^{\\prime }}}{L_X}\\vert +\\vert \\frac{xL_Y^{^{\\prime }}}{L_Y}\\vert =O\\left(\\frac{1}{\\log ^\\kappa n}\\right)$ and the result is proved.",
"The next one is introduced to provide a bound for the slow varying function.",
"Lemma 2 Suppose $L_K,\\ K=X,Y$ satisfies condition A2, then for $\\forall \\epsilon >0$ being given, for sufficiently large $x$ , we have $x^{-\\epsilon }\\le L_K(x)\\le x^\\epsilon ,\\ K=X,Y,Z$ This is equivalent as $\\vert \\log L_K(x)\\vert \\le \\epsilon \\log x$ for large $x$ .",
"According to lemma REF , $L_Z(x)$ also satisfies A2.",
"We suppose $\\kappa <1$ and if $\\kappa \\ge 1$ , the derivative is of $o\\left(\\frac{1}{\\log ^{0.5} x}\\right)$ for large $x$ .",
"Because of A2, there exists a constant $C>0$ and $x_0>0$ such that for arbitrary $x>x_0$ , $\\vert \\log L_K(x)\\vert \\le \\vert \\log L_K(x_0)\\vert +\\frac{C}{1-\\kappa }(\\log ^{1-\\kappa }x-\\log ^{1-\\kappa }x_0)\\le \\epsilon \\log x$ for large $x$ , and the result is proved.",
"The third one involves a frequently used inequality.",
"Lemma 3 Suppose $K_i,\\ i=1,2,...,n$ being i.i.d random variables and define $I_i=\\mathbf {1}_{K_i\\in A_n}$ , here suppose $A_n$ is a Borel set with positive measure as a function of sample size $n$ .",
"Define $\\widehat{r}=\\frac{\\Sigma _{i=1}^n I_i}{n}$ and $r=\\mathbf {E} I_i$ , and suppose $m\\ge 2$ being a constant, then there exists a constant $C_m$ such that $\\vert \\vert \\widehat{r}-r\\vert \\vert _m\\le \\frac{C_m (r(1-r))^{1/m}}{\\sqrt{n}}$ Define $f_k$ as $f_k={\\left\\lbrace \\begin{array}{ll}\\Sigma _{i=1}^k (I_i-r),\\ k\\le n\\\\\\Sigma _{i=1}^n (I_i-r),\\ k>n\\end{array}\\right.",
"}$ Then, $f_k$ is a martingale.",
"Since $m\\ge 2$ , from Minkowski inequality and Burkholder inequality [17], we have $\\begin{aligned}n\\vert \\vert \\widehat{r}-r\\vert \\vert _m\\le C_m\\vert \\vert \\sqrt{\\Sigma _{i=1}^n (I_i-r)^2}\\vert \\vert _m\\\\=C_m\\sqrt{\\vert \\vert \\Sigma _{i=1}^n (I_i-r)^2\\vert \\vert _{m/2}}\\\\\\le C_m\\sqrt{\\Sigma _{i=1}^n\\vert \\vert (I_i-r)^2\\vert \\vert _{m/2}}\\end{aligned}$ Since $m\\ge 2$ , we have $\\mathbf {E}\\vert I_i-r\\vert ^m\\le r(1-r)$ and the result is proved.",
"Now, we start proving the $L_p$ convergence of estimator $\\widehat{\\rho }$ .",
"Theorem 3 Suppose A1 and A2 and $m\\ge 2$ , then we have $\\vert \\vert \\frac{\\widehat{q}(t(n))}{\\widehat{p}(t(n))}\\mathbf {1}_{\\widehat{p}(t(n))\\ge s(n)}-\\frac{\\gamma _Y}{\\gamma _X+\\gamma _Y}\\vert \\vert _m=O\\left(\\frac{1}{s(n)\\sqrt{n}}\\right)+O\\left(\\frac{1}{P_Z(t(n))\\sqrt{n}}\\right)+O\\left(\\frac{1}{\\log ^\\kappa t(n)}\\right)$ According to [18], we have that $\\delta _i$ has the same distribution as $\\mathbf {1}_{U_i\\le \\lambda (Z_i)}$ , here $U_i$ is uniform $[0,1]$ random variable being independent with $Z_i$ and $\\lambda (x)=Prob(X\\le Y\\vert Z=x)=\\frac{f_X(x)P_Y(x)}{f_X(x)P_Y(x)+f_Y(x)P_X(x)}=\\frac{1/\\gamma _X-\\frac{xL_X^{^{\\prime }}(x)}{L_X(x)}}{1/\\gamma _X+1/\\gamma _Y-\\frac{xL_X^{^{\\prime }}(x)}{L_X(x)}-\\frac{xL_Y^{^{\\prime }}(x)}{L_Y(x)}}$ Because of condition A2, for sufficiently large $x$ , we have $\\vert \\lambda (x)-\\frac{\\gamma _Y}{\\gamma _X+\\gamma _Y}\\vert \\le \\frac{2}{(\\gamma _X+\\gamma _Y)^2}\\left(\\gamma _X\\gamma _Y^2\\vert \\frac{xL_Y^{^{\\prime }}}{L_Y}\\vert +\\gamma ^2_X\\gamma _Y\\vert \\frac{xL_X^{^{\\prime }}}{L_X}\\vert \\right)$ Thus, there exists a constant $C$ such that for sufficiently large $x$ , $\\vert \\lambda (x)-\\frac{\\gamma _Y}{\\gamma _X+\\gamma _Y}\\vert \\le \\frac{C}{\\log ^\\kappa (x)}$ and correspondingly, from mean value theorem, we have $\\vert \\mathbf {E}\\widehat{q}(t(n))-\\frac{\\gamma _Y P_Z(t(n))}{\\gamma _X+\\gamma _Y}\\vert \\le \\int _{t(n)}^\\infty \\vert \\lambda (x)-\\frac{\\gamma _X}{\\gamma _X+\\gamma _Y}\\vert f_Z(x)dx\\le \\frac{C}{\\log ^\\kappa t(n)}P_Z(t(n))$ From Minkowski inequality, we have $\\begin{aligned}\\vert \\vert \\frac{\\widehat{q}(t(n))}{\\widehat{p}(t(n))}\\mathbf {1}_{\\widehat{p}(t(n))\\ge s(n)}-\\frac{\\gamma _Y}{\\gamma _X+\\gamma _Y}\\vert \\vert _m\\\\\\le \\vert \\vert \\left(\\frac{\\widehat{q}(t(n))}{P_Z(t(n))}-\\frac{\\gamma _Y}{\\gamma _X+\\gamma _Y}\\right)\\frac{P_Z(t(n))}{\\widehat{p}(t(n))}\\mathbf {1}_{\\widehat{p}(t(n))\\ge s(n)}\\vert \\vert _m+\\frac{\\gamma _Y}{\\gamma _X+\\gamma _Y}\\vert \\vert \\frac{P_Z(t(n))}{\\widehat{p}(t(n))}\\mathbf {1}_{\\widehat{p}(t(n))\\ge s(n)}-1\\vert \\vert _m\\end{aligned}$ If A3 is satisfied, then for sufficiently large $n$ , from lemma REF and REF , for $\\forall \\epsilon >0$ $P_Z(t(n))\\ge L_Z(t(n))t^{-1/\\gamma _0}(n)\\ge n^{-\\beta (1/\\gamma _0+\\epsilon )}$ choose small $\\epsilon $ we have $c>\\epsilon +\\frac{\\beta }{\\gamma _0}$ and thus for large $n$ , we have $P_Z(t(n))/2>s(n)$ .",
"If A3 is not satisfied, from REF , similar with REF , we have $P_Z(t(n))\\ge \\log ^{-\\beta (\\epsilon +1/\\gamma _0)} n>2s(n)$ for large $n$ .",
"Thus from Chebyshev's inequality and lemma REF , we have $\\begin{aligned}\\vert \\vert \\frac{P_Z(t(n))}{\\widehat{p}(t(n))}\\mathbf {1}_{\\widehat{p}(t(n))\\ge s(n)}-1\\vert \\vert _m\\le \\vert \\vert \\left(\\frac{P_Z(t(n))}{\\widehat{p}(t(n))}-1\\right)\\mathbf {1}_{\\widehat{p}(t(n))\\ge s(n)}\\vert \\vert _m+Prob\\left(\\widehat{p}(t(n))<s(n)\\right)^{1/m}\\\\\\le \\frac{\\vert \\vert \\widehat{p}(t(n))-P_Z(t(n))\\vert \\vert _m}{s(n)}+\\frac{2\\vert \\vert \\widehat{p}(t(n))-P_Z(t(n))\\vert \\vert _m}{P_Z(t(n))}\\\\=O\\left(\\frac{1}{s(n)\\sqrt{n}}\\right)+O\\left(\\frac{1}{P_Z(t(n))\\sqrt{n}}\\right)\\end{aligned}$ This directly implies that $\\vert \\vert \\frac{P_Z(t(n))}{\\widehat{p}(t(n))}\\mathbf {1}_{\\widehat{p}(t(n))\\ge s(n)}\\vert \\vert _m=O\\left(1\\right)$ For the first term, from Cauchy inequality, there exists a constant $C$ such that we have $\\begin{aligned}\\vert \\vert \\left(\\frac{\\widehat{q}(t(n))}{P_Z(t(n))}-\\frac{\\gamma _Y}{\\gamma _X+\\gamma _Y}\\right)\\frac{P_Z(t(n))}{\\widehat{p}(t(n))}\\mathbf {1}_{\\widehat{p}(t(n))\\ge s(n)}\\vert \\vert _m \\\\ \\le \\vert \\vert \\left(\\frac{\\widehat{q}(t(n))}{P_Z(t(n))}-\\frac{\\gamma _Y}{\\gamma _X+\\gamma _Y}\\right)\\vert \\vert _{2m}\\vert \\vert \\frac{P_Z(t(n))}{\\widehat{p}(t(n))}\\mathbf {1}_{\\widehat{p}(t(n))\\ge s(n)}\\vert \\vert _{2m}\\\\\\le C \\left(\\frac{1}{P_Z(t(n))}\\vert \\vert \\widehat{q}(t(n))-\\mathbf {E}\\widehat{q}(t(n))\\vert \\vert _{2m}+O\\left(\\frac{1}{\\log ^\\kappa t(n)}\\right)\\right)\\end{aligned}$ From lemma REF , we get the result.",
"Notice that, if we assume A3, form REF , we know that the convergence rate is of $O\\left(\\frac{1}{\\log ^\\kappa n}\\right)$ , otherwise the convergence rate is of $O\\left(1/\\log ^\\kappa \\log n\\right)$ .",
"In the next part, we will concentrate on estimating $\\gamma _Z$ .",
"According to [18], since $\\gamma _Z=\\frac{\\gamma _X\\gamma _Y}{\\gamma _X+\\gamma _Y}$ , if we can find a suitable estimator $\\widehat{\\zeta }$ of $\\gamma _Z$ , since $\\gamma _X=\\gamma _Z/\\frac{\\gamma _Y}{\\gamma _X+\\gamma _Y}$ , it is reasonable to consider $\\widehat{\\zeta }/\\widehat{\\rho }$ .",
"We will prove $L_p$ convergence of its truncated version below.",
"First we give a lemma.",
"Lemma 4 Suppose A1, A2, then as $t\\rightarrow \\infty $ , we have $\\frac{1}{P_Z(t)}\\int _{t}^\\infty \\frac{P_Z(y)}{y}dy=\\gamma _Z+O\\left(\\frac{1}{\\log ^\\kappa t} \\right)$ Since $\\gamma _Z=\\frac{1}{P_Z(t)}\\int _{t}^\\infty \\frac{L_Z(t)}{y^{(1+\\alpha _Z)}}dy$ , from Fubini-Tonelli theorem and A2, for large $t$ , there exists constant $C$ such that $\\begin{aligned}\\vert \\frac{1}{P_Z(t)}\\int _{t}^\\infty \\frac{P_Z(y)}{y}dy-\\gamma _Z\\vert =\\frac{1}{P_Z(t)}\\vert \\int _{t}^\\infty \\frac{L_Z(y)-L_Z(t)}{y^{\\alpha _Z+1}}dy\\vert \\\\\\le \\frac{1}{P_Z(t)}\\int _{t}^\\infty \\frac{dy}{y^{\\alpha _Z+1}}\\int _{t}^y \\vert L_Z^{^{\\prime }}(z)\\vert dz\\\\=\\frac{\\gamma _Z}{P_Z(t)}\\int _{t}^\\infty \\vert L_Z^{^{\\prime }}(z)\\vert z^{-\\alpha _Z}dz\\\\\\le \\frac{\\gamma _Z C}{P_Z(t)}\\int _{t}^\\infty \\frac{L_Z(z)}{\\log ^\\kappa z}z^{-\\alpha _Z-1}dz\\end{aligned}$ From mean value theorem, suppose $\\alpha _Z=1/\\gamma _Z$ $\\begin{aligned}\\int _{t}^\\infty \\frac{L_Z(z)}{\\log ^\\kappa z}z^{-1/\\gamma _Z-1}dz=\\Sigma _{n=0}^\\infty \\int _{t\\log ^n t}^{t\\log ^{n+1}t}\\frac{L_Z(z)}{z^{\\alpha _Z+1}\\log ^\\kappa z}dz\\\\=\\Sigma _{n=0}^\\infty \\frac{\\gamma _Z L_Z(\\eta _n)}{\\log ^\\kappa \\eta _n}\\frac{1}{t^{\\alpha _Z}\\log ^{n\\alpha _Z}t}\\left(1-\\frac{1}{\\log ^{\\alpha _Z} t}\\right)\\\\\\le \\Sigma _{n=0}^\\infty \\frac{\\gamma _Z L_Z(\\eta _n)}{\\left(\\log t+n\\log \\log t\\right)^\\kappa }\\frac{1}{t^{\\alpha _Z}\\log ^{n\\alpha _Z}t}\\end{aligned}$ Notice that, from assumption A2, if $\\kappa \\ne 1$ , then $\\begin{aligned}\\vert \\log L_Z(\\eta _n)-\\log L_Z(t)\\vert \\le \\int _{t}^{\\eta _n} \\vert \\frac{L_Z^{^{\\prime }}(z)}{L_Z(z)}\\vert dz\\le \\frac{C}{1-\\kappa }(\\log ^{1-\\kappa }\\eta _n-\\log ^{1-\\kappa }t)\\\\\\le \\frac{C}{\\vert 1-\\kappa \\vert }\\vert (\\log t+(n+1)\\log \\log t)^{1-\\kappa }-\\log ^{1-\\kappa } t\\vert \\end{aligned}$ And if $\\kappa =1$ , then similarly we have $\\vert \\log L_Z(\\eta _n)-\\log L_Z(t)\\vert \\le $ , here $C$ is a constant.",
"We continue proof with 3 different cases.",
"Case 1: $0<\\kappa <1$ .",
"In this case, for a given constant $D$ and sufficiently large $t$ , equation REF is less than $\\frac{C}{1-\\kappa }\\left(\\frac{\\log t}{\\log ^\\kappa t}+\\frac{((n+1)\\log \\log t)}{((n+1)\\log \\log t)^\\kappa }-\\log ^{1-\\kappa } t\\right)\\le \\frac{(n+1)^{1-\\kappa }}{(1-\\kappa )D}\\log \\log t$ This implies that $\\frac{L_Z(\\eta _n)}{L_Z(t)}\\le (\\log t)^{\\frac{(n+1)^{1-\\kappa }}{D(1-\\kappa )}}$ , combine with REF and REF , we have $\\vert \\frac{1}{P_Z(t)}\\int _{t}^\\infty \\frac{P_Z(y)}{y}dy-\\gamma _Z\\vert \\le C\\gamma _z^2\\left(\\frac{L_Z(\\eta _0)}{L_Z(t)\\log ^\\kappa t}+\\Sigma _{n=1}^\\infty \\frac{1}{(\\log t+n\\log \\log t)^\\kappa (\\log t)^{n\\alpha _Z-\\frac{(n+1)^{1-\\kappa }}{D(1-\\kappa )}}}\\right)$ Since $\\log L_Z(\\eta _0)-\\log L_Z(t)\\le \\frac{1}{1-\\kappa }\\left(\\log ^{1-\\kappa } t\\left(1+\\frac{\\log \\log t}{\\log t}\\right)^{1-\\kappa }-\\log ^{1-\\kappa } t\\right)\\le \\frac{1}{1-\\kappa } \\frac{\\log \\log t}{\\log ^\\kappa t}=o(1)$ Thus, for large $t$ , $\\frac{L_Z(\\eta _0)}{L_Z(t)}<2$ .",
"Also, for $n\\ge 1$ , we have $n+1\\le 2n$ and $\\frac{n\\alpha _Z}{2}>\\frac{(n+1)^{1-\\kappa }}{D(1-\\kappa )}\\Leftarrow Dn^\\kappa \\alpha _Z>\\frac{2^{2-\\kappa }}{1-\\kappa }\\Leftarrow D\\alpha _Z>\\frac{2^{2-\\kappa }}{1-\\kappa }$ choose $D$ satisfies this condition then $\\Sigma _{n=1}^\\infty \\frac{1}{(\\log t)^{n\\alpha _Z-\\frac{(n+1)^{1-\\kappa }}{D(1-\\kappa )}}}\\le \\Sigma _{n=1}^\\infty \\frac{1}{4^{n\\alpha _Z-\\frac{(n+1)^{1-\\kappa }}{D(1-\\kappa )}}}\\le \\Sigma _{n=1}^\\infty \\frac{1}{2^{n\\alpha _Z}}<\\infty $ And we prove the result.",
"Case 2: $\\kappa =1$ .",
"If $\\kappa =1$ , from REF , we have $\\frac{L_Z(\\eta _n)}{L_Z(t)}\\le \\left(1+\\frac{(n+1)\\log \\log t}{\\log t}\\right)^C$ , correspondingly, combine with REF and REF , for large $t$ , we have $\\begin{aligned}\\vert \\frac{1}{P_Z(t)}\\int _{t}^\\infty \\frac{P_Z(y)}{y}dy-\\gamma _Z\\vert \\le \\frac{C\\gamma _Z^2}{\\log t}\\Sigma _{n=0}^\\infty \\frac{1}{\\log ^{n\\alpha _Z}t}\\left(1+\\frac{(n+1)\\log \\log t}{\\log t}\\right)^C\\\\\\le \\frac{C\\gamma _Z^2}{\\log t}\\left(2^C+\\Sigma _{n=1}^\\infty \\frac{(n+2)^C}{2^{n\\alpha _Z}}\\right)\\end{aligned}$ Since $\\Sigma _{n=1}^\\infty \\frac{(n+2)^C}{2^{n\\alpha _Z}}<\\infty $ , the result is proved.",
"Case 3: $\\kappa >1$ .",
"If $\\kappa >1$ , from REF , we have $\\frac{L_Z(\\eta _n)}{L_Z(t)}\\le \\exp (2C/(\\kappa -1))$ , combine with REF and we prove the result.",
"Now, we start to prove the $L_p$ convergence of statistics $\\widehat{\\zeta }$ .",
"Theorem 4 Suppose A1,A2 and $m\\ge 2$ then we have $\\vert \\vert \\widehat{\\zeta }-\\gamma _Z\\vert \\vert _m=O\\left(\\frac{1}{s(n)\\sqrt{n}}+\\frac{1}{P_Z(t(n))\\sqrt{n}}+\\frac{1}{\\log ^\\kappa t(n)}\\right)$ From definition of $\\widehat{\\zeta }$ (see REF ) and Minkowski inequality, $\\begin{aligned}\\vert \\vert \\widehat{\\zeta }-\\gamma _Z\\vert \\vert _m\\le \\vert \\vert \\left(\\frac{1}{P_Z(t(n))}\\int _{t(n)}^\\infty \\frac{\\widehat{p}(y)}{y}dy-\\gamma _Z\\right)\\frac{P_Z(t(n))}{\\widehat{p}(t(n))}\\mathbf {1}_{\\widehat{p}(t(n))\\ge s(n)}\\vert \\vert _m+\\\\\\gamma _Z\\vert \\vert \\left(\\frac{P_Z(t(n))}{\\widehat{p}(t(n))}-1\\right)\\mathbf {1}_{\\widehat{p}(t(n))\\ge s(n)}\\vert \\vert _m+\\gamma _Z Prob\\left(\\widehat{p}(t(n))<s(n)\\right)^{1/m}\\end{aligned}$ For the second and the third term, from REF we know that these term is of order $O\\left(\\frac{1}{s(n)\\sqrt{n}}+\\frac{1}{P_Z(t(n))\\sqrt{n}}\\right)=o(1)$ .",
"For the first term, from Cauchy inequality and Minkowski inequality, it is less than $\\vert \\vert \\frac{P_Z(t(n))}{\\widehat{p}(t(n))}\\mathbf {1}_{\\widehat{p}(t(n))\\ge s(n)}\\vert \\vert _{2m}\\left(\\frac{1}{P_Z(t(n))}\\vert \\vert \\int _{t(n)}^\\infty \\frac{\\widehat{p}(y)-P_Z(y)}{y}dy \\vert \\vert _{2m}+\\vert \\frac{1}{P_Z(t(n))}\\int _{t(n)}^\\infty \\frac{P_Z(y)}{y}dy-\\gamma _Z\\vert \\right)$ From REF , $\\vert \\vert \\frac{P_Z(t(n))}{\\widehat{p}(t(n))}\\mathbf {1}_{\\widehat{p}(t(n))\\ge s(n)}\\vert \\vert _{2m}=O\\left(1\\right)$ , from integral version Minkowski inequality and lemma REF , we have $\\begin{aligned}\\vert \\vert \\int _{t(n)}^\\infty \\frac{\\widehat{p}(y)-P_Z(y))}{y}dy \\vert \\vert _{2m}\\le \\int _{t(n)}^\\infty \\frac{\\vert \\vert \\widehat{p}(y)-P_Z(y)\\vert \\vert _{2m}}{y}dy\\\\\\le \\int _{t(n)}^\\infty \\frac{C_{2m}P_Z^{1/2m}(y)}{\\sqrt{n}y}dy\\\\=\\int _{t(n)}^\\infty \\frac{C_{2m}L_Z^{1/2m}(y)}{\\sqrt{n}y^{1+\\alpha _Z/2m}}dy\\end{aligned}$ From lemma REF , choose $\\epsilon =\\alpha _Z/2$ , for sufficiently large $y$ , $L_Z(y)^{1/2m}\\le y^{\\alpha _Z/4m}$ and thus the integration is less than $\\frac{4mC_{2m}}{\\alpha _Z\\sqrt{n}}t(n)^{-\\alpha _Z/4m}=o(1/\\sqrt{n})$ .",
"Thus, $\\frac{1}{P_Z(t(n))}\\vert \\vert \\int _{t(n)}^\\infty \\frac{\\widehat{p}(y)-P_Z(y)}{y}dy \\vert \\vert _{2m}=O\\left(\\frac{1}{P_Z(t(n))\\sqrt{n}}\\right)$ For the second term in REF , use lemma REF and we prove the result.",
"Similarly, if in addition we assume A3, from REF we know that the convergence rate is of $O\\left(\\frac{1}{\\log ^\\kappa n}\\right)$ and otherwise the convergence rate becomes $O\\left(\\frac{1}{\\log ^\\kappa \\log n}\\right)$ .",
"Finally, we apply discussions above to prove theorem REF .",
"[Proof for theorem REF ] We choose $\\mu =\\nu =m/2$ in theorem 1 of $\\cite {Vasiliev2014}$ , according to REF and REF , since exists constant $C$ such that for sufficiently large $n$ , $\\begin{aligned}\\mathbf {E}\\vert \\widehat{\\zeta }-\\gamma _Z\\vert ^{2\\nu }=\\vert \\vert \\widehat{\\zeta }-\\gamma _Z\\vert \\vert _m^m\\le \\frac{C^m}{\\log ^{m\\kappa } n}\\\\w_n=\\mathbf {E}\\vert \\widehat{\\rho }-\\frac{\\gamma _Y}{\\gamma _X+\\gamma _Y}\\vert ^{2\\mu }\\le \\frac{C^m}{\\log ^{m\\kappa } n}\\end{aligned}$ Choose $\\phi _n(m)=\\frac{2^{m-1}}{\\left(\\gamma _Y/(\\gamma _X+\\gamma _Y)\\right)^m}\\frac{C^m(\\gamma _Z^m+\\left(\\frac{\\gamma _X}{\\gamma _X+\\gamma _Y}\\right)^m)}{\\log ^{m\\kappa } n}=O\\left(\\frac{1}{\\log ^{m\\kappa }n}\\right)$ and choose $\\beta =m/4$ , $\\beta \\ge 1$ , then we have $\\mathbf {E}\\vert \\frac{\\widehat{\\zeta }}{\\widehat{\\rho }}\\mathbf {1}_{\\widehat{\\rho }\\ge H_n}-\\gamma _X\\vert ^{2\\beta }\\le V_n(\\beta )$ Here $V_n(\\beta )=O\\left(\\phi _n(2\\beta )+\\frac{\\phi _n^{1/2}(m)w_n^{1/4}}{H_n^\\beta }+\\frac{\\phi _n^{1/2}(m)w_n^{1/2}}{H_n^{2\\beta }}+w_n\\right)=O\\left(\\frac{1}{\\log ^{m\\kappa /2} n}\\right)$ Combine REF and REF , we prove the result.",
"In particular, this directly proves the $L_{m/2}$ convergence of the statistics."
],
[
"Almost sure convergence of tail index estimator",
"In this section, we try to prove the almost sure convergence of the tail index estimator under assumption A1-A3.",
"We first introduce two lemma.",
"Theorem 5 Suppose A1-A3, and $t(n), s(n)$ are chosen as in REF , then we have $\\frac{\\widehat{q}(t(n))}{\\widehat{p}(t(n))}\\mathbf {1}_{\\widehat{p}(t(n))\\ge s(n)}\\rightarrow _{a.s.}\\frac{\\gamma _Y}{\\gamma _X+\\gamma _Y}$ From Borel-Cantelli lemma [19], it suffices to show that, for $\\forall \\epsilon >0$ , $\\Sigma _{n=1}^\\infty Prob\\left(\\vert \\frac{\\widehat{q}(t(n))}{\\widehat{p}(t(n))}\\mathbf {1}_{\\widehat{p}(t(n))\\ge s(n)}-\\frac{\\gamma _Y}{\\gamma _X+\\gamma _Y}\\vert > 3\\epsilon \\right)<\\infty $ Since $\\begin{aligned}Prob\\left(\\vert \\frac{\\widehat{q}(t(n))}{\\widehat{p}(t(n))}\\mathbf {1}_{\\widehat{p}(t(n))\\ge s(n)}-\\frac{\\gamma _Y}{\\gamma _X+\\gamma _Y}\\vert >3\\epsilon \\right)\\\\\\le Prob\\left(\\vert \\left(\\frac{\\widehat{q}(t(n))}{P_Z(t(n))}-\\frac{\\mathbf {E}\\widehat{q}(t(n))}{P_Z(t(n))}\\right)\\left(\\frac{P_Z(t(n))}{\\widehat{p}(t(n))}-1\\right)\\mathbf {1}_{\\widehat{p}(t(n))\\ge s(n)}\\vert \\ge \\epsilon \\right)\\\\+Prob\\left(\\vert \\frac{\\mathbf {E}\\widehat{q}(t(n))}{P_Z(t(n))}\\frac{P_Z(t(n))}{\\widehat{p}(t(n))}\\mathbf {1}_{\\widehat{p}(t(n))\\ge s(n)}-\\frac{\\gamma _Y}{\\gamma _X+\\gamma _Y}\\vert \\ge \\epsilon \\right)\\\\+Prob\\left(\\vert \\frac{\\widehat{q}(t(n))-\\mathbf {E}\\widehat{q}(t(n))}{P_Z(t(n))}\\vert \\mathbf {1}_{\\widehat{p}(t(n))\\ge s(n)}\\ge \\epsilon \\right)\\end{aligned}$ We will separately discuss these 3 terms below.",
"For the first term, notice that for $\\forall k>1$ , from mean value inequality and Minkowski inequality, we have $\\begin{aligned}Prob\\left(\\vert \\left(\\frac{\\widehat{q}(t(n))-\\mathbf {E}\\widehat{q}(t(n))}{P_Z(t(n))}\\right)\\left(\\frac{P_Z(t(n))}{\\widehat{p}(t(n))}-1\\right)\\vert \\mathbf {1}_{\\widehat{p}(t(n))\\ge s(n)}\\ge \\epsilon \\right)\\\\\\le \\frac{1}{\\epsilon ^k}\\vert \\vert \\left(\\frac{\\widehat{q}(t(n))-\\mathbf {E} \\widehat{q}(t(n))}{P_Z(t(n))}\\right)\\left(\\frac{P_Z(t(n))}{\\widehat{p}(t(n))}-1\\right)\\mathbf {1}_{\\widehat{p}(t(n))\\ge s(n)}\\vert \\vert _k^k\\\\\\le \\frac{1}{2^k\\epsilon ^k}\\vert \\vert \\left(\\frac{\\widehat{q}(t(n))-\\mathbf {E}\\widehat{q}(t(n))}{P_Z(t(n))}\\right)^2+\\left(\\frac{\\widehat{p}(t(n))-\\mathbf {E} \\widehat{p}(t(n))}{\\widehat{p}(t(n))}\\right)^2\\mathbf {1}_{\\widehat{p}(t(n))\\ge s(n)}\\vert \\vert _k^k\\\\\\le \\frac{1}{2^k\\epsilon ^k}\\left(\\vert \\vert \\frac{\\widehat{q}(t(n))-\\mathbf {E}\\widehat{q}(t(n))}{P_Z(t(n))}\\vert \\vert _{2k}^2+\\frac{1}{s(n)^2}\\vert \\vert \\widehat{p}(t(n))-\\mathbf {E}\\widehat{p}(t(n))\\vert \\vert _{2k}^2\\right)^k\\end{aligned}$ From lemma REF , $\\vert \\vert \\widehat{q}(t(n))-\\mathbf {E} \\widehat{q}(t(n))\\vert \\vert _{2k}^2=O\\left(\\frac{P_Z^{1/k}(t(n))}{n}\\right)$ and $\\vert \\vert \\widehat{p}(t(n))-\\mathbf {E}\\widehat{p}(t(n))\\vert \\vert _{2k}^2=O\\left(\\frac{P_Z^{1/k}(t(n))}{n}\\right)$ , choose $k>\\max \\left(\\frac{1}{1/2-\\beta /\\gamma _0},\\frac{1}{1-2c}\\right)$ then the convergence of summation of first term is proved.",
"For the second term, notice that it is smaller than $Prob\\left(\\vert \\frac{\\mathbf {E}\\widehat{q}(t(n))}{P_Z(t(n))}\\frac{P_Z(t(n))}{\\widehat{p}(t(n))}\\mathbf {1}_{\\widehat{p}(t(n))\\ge s(n)}-\\frac{\\mathbf {E}\\widehat{q}(t(n))}{P_Z(t(n))}\\vert \\ge \\epsilon -\\vert \\frac{\\mathbf {E}\\widehat{q}(t(n))}{P_Z(t(n))}-\\frac{\\gamma _Y}{\\gamma _X+\\gamma _Y}\\vert \\right)$ From REF , for sufficiently large $n$ , $\\vert \\frac{\\mathbf {E}\\widehat{q}(t(n))}{P_Z(t(n))}-\\frac{\\gamma _Y}{\\gamma _X+\\gamma _Y}\\vert \\le \\epsilon /2$ , and $\\frac{\\mathbf {E}\\widehat{q}(t(n))}{P_Z(t(n))}\\le \\frac{2\\gamma _Y}{\\gamma _X+\\gamma _Y}$ .",
"Since for $\\forall k>1$ $\\begin{aligned}Prob\\left(\\vert \\frac{P_Z(t(n))}{\\widehat{p}(t(n))}\\mathbf {1}_{\\widehat{p}(t(n))\\ge s(n)}-1\\vert \\ge \\frac{(\\gamma _X+\\gamma _Y)\\epsilon }{4\\gamma _Y}\\right)\\\\\\le \\left(\\frac{4\\gamma _Y}{\\epsilon (\\gamma _X+\\gamma _Y)}\\right)^k\\left(\\frac{1}{s(n)}\\vert \\vert \\widehat{p}(t(n))-\\mathbf {E}\\widehat{p}(t(n))\\vert \\vert _k+Prob(\\widehat{p}(t(n))<s(n))\\right)^k\\end{aligned}$ From REF and similar to REF , choose sufficiently large $k$ and we know that summation of this term converges.",
"For the third term, similar with REF and we prove can prove the convergence of summation.",
"Since REF is true, almost sure convergence is proved as well.",
"Theorem 6 Suppose A1-A3 and suppose $s(n)$ and $t(n)$ are chosen as in REF , then we have $\\widehat{\\zeta }\\rightarrow _{a.s.}\\gamma _Z$ definition of $\\widehat{\\zeta }$ is in REF .",
"From Borel-Cantelli [19] lemma, it suffices to show that, for $\\forall \\epsilon >0$ , $\\Sigma _{n=1}^\\infty Prob\\left(\\vert \\widehat{\\zeta }-\\gamma _Z\\vert \\ge 4\\epsilon \\right)<\\infty $ Since the above term is less than $\\vert \\widehat{\\zeta }-\\frac{1}{P_Z(t(n))}\\int _{t(n)}^\\infty \\frac{\\widehat{p}(y)}{y}dy\\vert +\\vert \\frac{1}{P_Z(t(n))}\\int _{t(n)}^\\infty \\frac{\\widehat{p}(y)-P_Z(y)}{y}dy\\vert +\\vert \\frac{1}{P_Z(t(n))}\\int _{t(n)}^\\infty \\frac{P_Z(y)}{y}dy-\\gamma _Z\\vert $ According to lemma REF , for sufficiently large $n$ , $\\vert \\frac{1}{P_Z(t(n))}\\int _{t(n)}^\\infty \\frac{P_Z(y)}{y}dy-\\gamma _Z\\vert <2\\epsilon $ .",
"Thus, there exists a constant $n_0$ such that $\\begin{aligned}\\Sigma _{n=n_0}^\\infty Prob\\left(\\vert \\widehat{\\zeta }-\\gamma _Z\\vert \\ge 4\\epsilon \\right)\\\\\\le \\Sigma _{n=n_0}^\\infty Prob\\left(\\vert \\widehat{\\zeta }-\\frac{1}{P_Z(t(n))}\\int _{t(n)}^\\infty \\frac{\\widehat{p}(y)}{y}dy\\vert \\ge \\epsilon \\right)+Prob\\left(\\frac{1}{P_Z(t(n))}\\vert \\int _{t(n)}^\\infty \\frac{\\widehat{p}(y)-P_Z(y)}{y}dy\\vert \\ge \\epsilon \\right)\\end{aligned}$ For the second term, notice for arbitrary $k>1$ , from REF $\\begin{aligned}Prob\\left(\\frac{1}{P_Z(t(n))}\\vert \\int _{t(n)}^\\infty \\frac{\\widehat{p}(y)-P_Z(y)}{y}dy\\vert \\ge \\epsilon \\right)\\le \\frac{1}{\\epsilon ^k}\\left(\\frac{1}{P_Z(t(n))}\\vert \\vert \\int _{t(n)}^\\infty \\frac{\\widehat{p}-P_Z(y)}{y}dy\\vert \\vert _k\\right)^k\\\\=O\\left(\\frac{1}{P_Z^k(t(n))n^{k/2}}\\right)\\end{aligned}$ Choose $k>\\frac{2}{\\frac{1}{2}-\\frac{\\beta }{\\gamma _0}}$ we prove the convergence of summation for the second term.",
"The first term of REF is less than $\\begin{aligned}Prob\\left(\\frac{\\int _{t(n)}^\\infty \\frac{P_Z(y)}{y}dy}{P_Z(t(n))}\\vert \\frac{P_Z(t(n))\\mathbf {1}_{\\widehat{p}(t(n))\\ge s(n)}}{\\widehat{p}(t(n))}-1\\vert \\ge \\frac{\\epsilon }{2}\\right)\\\\+Prob\\left(\\frac{1}{P_Z(t(n))}\\vert \\left(\\frac{P_Z(t(n))\\mathbf {1}_{\\widehat{p}(t(n))\\ge s(n)}}{\\widehat{p}(t(n))}-1\\right)\\left(\\int _{t(n)}^\\infty \\frac{\\widehat{p}(y)-P_Z(y)}{y}dy\\right)\\vert \\ge \\frac{\\epsilon }{2}\\right)\\end{aligned}$ According to lemma REF , For large $n$ , $\\int _{t(n)}^\\infty \\frac{P_Z(y)}{y}dy/P_Z(t(n))\\le 2\\gamma _Z$ , so for sufficiently large $n$ , $\\begin{aligned}Prob\\left(\\frac{\\int _{t(n)}^\\infty \\frac{P_Z(y)}{y}dy}{P_Z(t(n))}\\vert \\frac{P_Z(t(n))\\mathbf {1}_{\\widehat{p}(t(n))\\ge s(n)}}{\\widehat{p}(t(n))}-1\\vert \\ge \\frac{\\epsilon }{2}\\right)\\\\\\le Prob\\left(\\frac{\\vert P_Z(t(n))-\\widehat{p}(t(n))\\vert }{s(n)}\\ge \\frac{\\epsilon }{4\\gamma _Z}\\right)+Prob\\left(\\widehat{p}(t(n))<s(n)\\right)\\end{aligned}$ From REF we know the convergence of summation on this term.",
"Also, from mean value inequality, $\\begin{aligned}Prob\\left(\\frac{1}{P_Z(t(n))}\\vert \\left(\\frac{P_Z(t(n))\\mathbf {1}_{\\widehat{p}(t(n))\\ge s(n)}}{\\widehat{p}(t(n))}-1\\right)\\int _{t(n)}^\\infty \\frac{\\widehat{p}(y)-P_Z(y)}{y}dy\\vert \\ge \\frac{\\epsilon }{2}\\right)\\\\\\le Prob\\left(\\frac{1}{P_Z^2(t(n))}\\vert \\int _{t(n)}^\\infty \\frac{\\widehat{p}(y)-P_Z(y)}{y}dy\\vert ^2\\ge \\frac{\\epsilon }{2}\\right)\\\\+ Prob\\left(\\left(\\frac{P_Z(t(n))\\mathbf {1}_{\\widehat{p}(t(n))\\ge s(n)}}{\\widehat{p}(t(n))}-1\\right)^2\\ge \\epsilon /2\\right)\\end{aligned}$ From REF and REF we know the convergence of summation.",
"Thus, REF is proved and the almost sure convergence is also proved.",
"Finally, we prove the almost sure convergence of $\\frac{\\widehat{\\zeta }}{\\widehat{\\rho }}\\mathbf {1}_{\\widehat{\\rho }\\ge H_n}$ .",
"[Proof for theorem REF ] According to theorem REF and REF , since $\\gamma _X,\\gamma _Y>0$ , we have $\\frac{\\widehat{\\zeta }}{\\widehat{\\rho }}\\rightarrow _{a.s.}\\gamma _X$ Since $H_n\\rightarrow 0$ as $n\\rightarrow \\infty $ , according to theorem REF , there exists a $n_0>0$ such that $\\begin{aligned}\\Sigma _{n=n_0}^\\infty \\mathbf {1}_{\\widehat{\\rho }<H_n}\\le \\Sigma _{n=n_0}^\\infty \\mathbf {1}_{\\vert \\widehat{\\rho }-\\frac{\\gamma _Y}{\\gamma _X+\\gamma _Y}\\vert >\\frac{\\gamma _Y}{2(\\gamma _X+\\gamma _Y)}}<\\infty \\end{aligned}$ which means that $\\mathbf {1}_{\\widehat{\\rho }\\ge H_n}\\rightarrow _{a.s.} 1$ .",
"Thus, the product of these two terms converges to $\\gamma _X$ ."
],
[
"Simulations and numerical examples",
"In this section, we suppose $X_i$ and $Y_i,\\ i=1,2,...n$ obey log gamma distribution, whose density is $f_K(x)=C_{f_K}x^{-\\alpha _K-1}\\log ^{\\beta _K-1} x,\\ K=X,Y,\\ x\\ge 1,\\ \\alpha _K>0$ We first prove that this distribution satisfies assumption A2.",
"Theorem 7 Distribution with density REF satisfies condition A2 Slow varying part of distribution REF is $L_K(x)=C_{f_K}x^{\\alpha _K}\\int _{x}^\\infty y^{-\\alpha _K-1}\\log ^{\\beta _K-1}y\\ dy$ Since $\\alpha _K\\int _x^\\infty y^{-\\alpha _K-1}\\log ^{\\beta _K-1}y\\ dy=x^{-\\alpha _K}\\log ^{\\beta _K-1}x+(\\beta _K-1)\\int _x^\\infty y^{-\\alpha _K-1}\\log ^{\\beta _K-2}y \\ dy$ We have $\\begin{aligned}\\vert \\frac{xL_K^{^{\\prime }}(x)}{L_K(x)}\\vert =\\frac{\\vert (\\alpha _K\\int _x^\\infty y^{-\\alpha _K-1}\\log ^{\\beta _K-1}y\\ dy)-x^{-\\alpha _K}\\log ^{\\beta _K-1}x \\vert }{\\int _x^\\infty y^{-\\alpha _K-1}\\log ^{\\beta _K-1}y\\ dy}\\\\=\\frac{\\vert \\beta _K-1\\vert \\int _x^\\infty y^{-\\alpha _K-1}\\log ^{\\beta _K-2}y\\ dy}{\\int _x^\\infty y^{-\\alpha _K-1}\\log ^{\\beta _K-1}y\\ dy}\\end{aligned}$ Suppose $h_K(x)=\\int _x^\\infty y^{-\\alpha _K-1}\\log ^{\\beta _K-1}y\\ dy$ , since $h_K$ is decreasing and $h_K(\\infty )=0$ , then $\\frac{\\int _x^\\infty \\frac{-h_K^{^{\\prime }}(y)}{\\log y}\\ dy}{h_K(x)}\\le \\frac{1}{\\log x}$ and thus assumption A2 is satisfied with $\\kappa =1$ .",
"Figure REF to REF demonstrates the performance of estimator $\\frac{\\widehat{\\zeta }}{\\widehat{\\rho }}\\mathbf {1}_{\\widehat{\\rho }\\ge H_n}$ under different conditions.",
"Parameters we choose for simulation is listed in table REF , sample size is assumed to be 10000 for case 1-5 and 50000 for case 6.",
"We use relative error $\\delta =\\frac{\\vert \\frac{\\widehat{\\zeta }}{\\widehat{\\rho }}\\mathbf {1}_{\\widehat{\\rho }\\ge H_n}-\\gamma _X\\vert }{\\gamma _X}$ to evaluate finite sample performance of our estimator.",
"We perform 50 times numerical experiments and the error bars in figure REF to REF show the maximum, minimum and average relative error under different $t(n)$ .",
"Following definition REF , $t(n)=n^\\beta $ , $\\beta $ coincides with notation Beta in figure REF -REF .",
"As we can see, 1.",
"If tail index of censor time is less than the underlying data, performance of tail index estimator will be inferior.",
"2.",
"Choosing suitable $t(n)$ is critical for making tail index estimator reliable.",
"Choosing too small or too big $t(n)$ leads to increase of relative error.",
"3.",
"For suitable $t(n)$ , tail index estimator has good performance even when censor rate is high.",
"Table: Parameter for simulation, definition of β K ,K=X,Y\\beta _K,K=X,Y see Figure: Numerical experiment case 1(left) and 2(right), sample size is 10000, parameters used in these cases coincide with table Figure: Numerical experiment case 3(left) and 4(right), sample size is 10000, parameters used in these cases coincide with table Figure: Numerical experiment case 5(left) and 6(right), sample size is 10000 for case 5 and 50000 for case 6, parameters used in these cases coincide with table"
],
[
"Conclusion",
"In this paper, we focus on proving almost sure convergence and $L_p$ convergence of estimator provided by Grama and Spokoiny [1] under random censoring and condition A1-A3.",
"We also perform numerical experiments with data satisfying log gamma distribution.",
"Numerical results demonstrate the usefulness of our tail index estimator when sample size is finite."
]
] | 1808.08320 |
[
[
"Linearized Reconstruction for Diffuse Optical Spectroscopic Imaging"
],
[
"Abstract In this paper, we present a novel reconstruction method for diffuse optical spectroscopic imaging with a commonly used tissue model of optical absorption and scattering.",
"It is based on linearization and group sparsity, which allows recovering the diffusion coefficient and absorption coefficient simultaneously, provided that their spectral profiles are incoherent and a sufficient number of wavelengths are judiciously taken for the measurements.",
"We also discuss the reconstruction for imperfectly known boundary and show that with the multi-wavelength data, the method can reduce the influence of modelling errors and still recover the absorption coefficient.",
"Extensive numerical experiments are presented to support our analysis."
],
[
"Introduction",
"Diffuse optical spectroscopy (DOS) is a noninvasive and quantitative medical imaging modality, for reconstructing absorption and scattering properties from optical measurements at multiple wavelengths excited by the near-infrared (NIR) light.",
"NIR light is particularly attractive for oncological applications because of its deep tissue penetrance and high sensitivity to haemoglobin concentration and oxygenation state [14].",
"DOS imaging effectively exploits the wavelength dependences of tissue optical properties (e.g.",
"absorption, scattering, anisotropy, reduced scattering and refractive index), and such dependences have been measured and tabulated for various tissues [10], [21].",
"It was reported that the spectral dependence of tissue scattering contains much useful information for functional imaging [13], [28], [29].",
"For example, dual-wavelength spectroscopy has been widely used to determine the absorption coefficient and hence the concentrations of reduced hemoglobin and oxygenated hemoglobin in tissue [28].",
"Thus, DOS imaging holds significant potentials for many biomedical applications, e.g., breast oncology functional brain imaging, stroke monitoring, neonatal hymodynamics and imaging of breast tumours [12], [13], [14], [24].",
"In many biomedical applications, it is realistic to assume that the absorption coefficient is linked to the concentrations and the spectra of chromophores through a linear map (cf.",
"(REF ) below), and the spectra of chromophores are known from experiments.",
"Then the goal in DOS is to recover the individual concentrations from the measurements taken at multiple wavelengths.",
"This task represents one of the most fundamental problems arising in accurate functional and molecular imaging.",
"Theoretically, very little is known about uniqueness and stability of DOS (see [5], [6], [7] for related work in electrical impedance tomography (EIT) and [1], [3], [4], [9] for quantitative photoacoustic imaging).",
"Note that despite the linear dependence of absorption on the concentrations, the dependence of imaging data on the absorption coefficient is highly nonlinear.",
"Further, it may suffer from severe ill-posedness and possible nonuniqueness; the latter is inherent to diffuse optical tomography of reconstructing simultaneously the absorption and diffusion coefficients [8].",
"Thus, the imaging problem is numerically very challenging.",
"Nonetheless, there have been many important efforts in developing effective reconstruction algorithms using multi-wavelength data to obtain images and estimates of spatially varying concentrations of chromophores inside an optically scattering medium such as biological tissues.",
"These include straightforward least-squares minimization [13], models-based minimization [23] and Bayesian approach [25].",
"Generally, there are two different ways to use the spectral measurements.",
"One is to recover the optical parameters at each different wavelength separately and then fit the spectral parameter model to these optical parameters [9], [16], [26]; and the other is to express the optical parameters as a function of the spectral parameter model and then estimate the spectral parameter directly [23], [25], [31].",
"We refer the interested readers to the survey [15] and the references therein for detailed discussions.",
"In this paper, we shall develop a simple and efficient linearized reconstruction method for DOS to recover the absorption and diffusion coefficients.",
"We employ the diffusion approximation to the radiative transfer equation for light transport, which has been used widely in biomedical optical imaging [8].",
"Our main contributions are as follows.",
"First, we show that within the linearized regime, incoherent spectral dependence allows recovering the concentrations and diffusion coefficient simultaneously, provided that a sufficient number of measurements are judiciously taken.",
"However, generally there is no explicit criterion on the number of wavelength, except in a few special cases (see Remark REF ).",
"Second, we demonstrate that with multi-wavelength measurements, the chromophore concentrations can be still be reasonably recovered even if the domain boundary is only imperfectly known.",
"Thus, DOS can partially alleviate the deleterious effect of modeling errors, in a manner similar to multifrequency EIT [2].",
"Third and last, these analytical findings are verified by extensive numerical experiments, where the reconstruction is performed via a group sparse type recovery technique developed in [2].",
"The rest of the paper is organized as follows.",
"In Section , we derive the linearized model, and discuss the conditions for simultaneous recovery (and also the one group sparse reconstruction technique).",
"Then in Section , we demonstrate the potential of the multi-wavelength data for handling modeling errors, especially imperfectly known boundaries.",
"We show that the chromophore concentrations can still be reasonably recovered from multi-wavelength data, but the diffusion coefficient is lost due to the corruption of domain deformation.",
"Extensive numerical experiments are carried out in Section to support the theoretical analysis.",
"Finally, some concluding remarks are provided in Section ."
],
[
"The linearized diffuse optical spectroscopy model",
"In this section, we mathematically formulate the linearized multi-wavelength method in DOS."
],
[
"Diffuse optical tomography",
"First, we introduce the diffuse optical tomography (DOT) model.",
"Let $\\Omega \\subset \\mathbb {R}^d$ $(d = 2, 3)$ be an open bounded domain with a smooth boundary $\\partial \\Omega $ .",
"The photon diffusion equation for the photon fluence rate $u$ in the frequency domain takes the following form: $\\left\\lbrace \\begin{aligned}-\\nabla \\cdot (D(x,\\lambda )\\nabla u) + \\mu _a(x,\\lambda )u &= 0 \\quad &\\mbox{in } \\Omega , \\\\D(x,\\lambda )\\frac{\\partial u}{\\partial \\nu } + \\alpha u&= S \\quad \\quad &\\mbox{on } \\partial \\Omega ,\\end{aligned} \\right.$ where $\\nu $ is the unit outward normal vector to the boundary $\\partial \\Omega $ , the nonnegative functions $D$ and $\\mu _a$ denote the photon diffusion coefficient and absorption coefficient, respectively.",
"In practice, the source function $S(x)$ is often taken to be a smooth approximation of the Dirac function $\\delta _y(x)$ located at $y\\in \\partial \\Omega $ [27].",
"The parameter $\\alpha $ in the boundary condition is formulated as $\\alpha =\\frac{1-R}{2(1+R)}$ in DOT model, where $R$ is a directionally varying refraction parameter [8].",
"Throughout, the parameter $\\alpha $ is assumed to be independent of the wavelength $\\lambda $ .",
"The weak formulation of problem (REF ) is to find $u\\in H^1(\\Omega ):=\\lbrace v \\in L^2(\\Omega ): \\nabla v \\in L^2(\\Omega )\\rbrace $ such that $\\int _{\\Omega }D(x,\\lambda )\\nabla u\\cdot \\nabla v +\\mu _a(x,\\lambda )uv \\mathrm {d}x+ \\int _{\\partial \\Omega }\\alpha uv\\mathrm {d}s=\\int _{\\partial \\Omega } Sv\\mathrm {d}s, \\quad \\forall v\\in H^1(\\Omega ).$ In practice, the coefficients $\\mu _a$ and $D$ are actually depending on the light wavelength $\\lambda $ .",
"The optical properties of the tissue can be expressed using their spectral representations.",
"Commonly used spectral models for optical properties can be written as [9], [23], [26], [31]: $\\mu _a(x,\\lambda ) & = \\sum _{k=1}^K \\mu _k(x)s_k(\\lambda ),\\\\\\mu _s(x,\\lambda ) & = \\mu _{s,\\rm ref}(\\lambda /\\lambda _{\\rm ref})^{-b}.",
"$ That is, the absorption coefficient $\\mu _a$ is expressed as a weighted sum of chromophore concentrations $\\mu _k$ and the corresponding absorption spectra $s_k$ of $K$ known chromophore.",
"The scattering coefficient $\\mu _s$ is given, according to Mie scattering theory, as being proportional to $\\mu _{s,\\rm ref}$ and (scattering) power $-b$ of a relative wavelength $\\lambda /\\lambda _{\\rm ref}$ [11], [18], [29].",
"The coefficient $\\mu _{s,\\rm ref}$ is known as the reduced scattering coefficient at a reference wavelength $\\lambda _{\\rm ref}$ , and it can be spatially dependent.",
"In many types of tissues (e.g., muscle and skin tissues), the wavelength dependence of $\\mu _s$ has been measured and can often be accurately approximated by $\\mu _s(x,\\lambda )=a(x)\\lambda ^{-b}$ , where the exponent $b$ is recovered from experiments [10], [21].",
"The optical diffusion coefficient $D(x,\\lambda )$ is given by $D(x,\\lambda )=[3(\\mu _a+\\mu _s)]^{-1}$ .",
"The condition $\\mu _s \\gg \\mu _a$ is usually considered valid in order to ensure the accuracy of the diffusion approximation to the radiative transfer equation [15], [16], [8].",
"Hence, we assume below that the diffusion coefficient $D(x,\\lambda )$ has the form: $D(x,\\lambda )=d(x)s_0(\\lambda ),$ where the wavelength dependence $s_0(\\lambda )$ is known from experiments.",
"In DOT experiments, the tissue under consideration is illuminated with $M$ sources, and measurements are taken at detectors.",
"In this work, we assume for the sake of simplicity that the positions $x_n$ of the sources and detectors are the same, and are distributed over the boundary $\\partial \\Omega $ .",
"The spectroscopic inverse problem is to recover the spatially-dependent coefficient $d(x)$ and the concentrations $\\mu _k(x)$ of the chromophores given the measured data $u$ (corresponding to the known sources $S$ ) on the detectors distributed on the boundary $\\partial \\Omega $ measured at several wavelengths $\\lambda _i$ .",
"It is well-known that the inverse problem of recovering both coefficients $d(x)$ and $\\mu _k(x)$ is quite ill-posed [16], since two different pairs of scattering and absorption coefficients can lead to identical measured data.",
"The multi-wavelength method is a promising approach to resolve this challenging nonuniqueness issue.",
"It is also reasonable to assume that the wavelength dependence $s_0(\\lambda )$ and the absorption spectra $s_k(\\lambda )$ are linearly independent so as to distinguish the diffusion coefficient and the chromophore concentrations by effectively using the information contained in the multi-wavelength data."
],
[
"The linearized diffuse optical spectroscopy model",
"Now we derive the linearized DOS model, which plays a crucial role in the reconstruction technique.",
"We discuss the cases of known and unknown diffusion coefficient separately."
],
[
"Unknown diffusion coefficient",
"First, we derive the linearized model with both diffusion coefficient $D(x,\\lambda )$ and concentrations $\\mu _k(x)$ being unknown.",
"For simplicity, we assume that the coefficient $d(x)$ is a small perturbation of the background, which is taken to be 1, i.e., $d(x) = 1+ \\delta d(x),$ where the unknown perturbation $\\delta d(x)$ has a compact support in the domain $\\Omega $ and is small (in suitable $L^p(\\Omega )$ norms).",
"For the inversion, smooth approximations $S_n$ of the Dirac masses at $\\lbrace \\delta _{x_n}\\rbrace _{n=1}^N$ are applied and the corresponding fluence rates $u_n$ are measured on the detectors located at all $x_n$ over the boundary $\\partial \\Omega $ to gain sufficient information about the diffusion coefficient $D(x,\\lambda )$ and absorption coefficient $\\mu _a(x,\\lambda )$ .",
"That is, let $\\lbrace u_n\\equiv u_n(x,\\lambda )\\rbrace _{n=1}^N\\subset H^1(\\Omega )$ be the corresponding solutions to (REF ), i.e., $\\int _{\\Omega }D(x,\\lambda )\\nabla u_n\\cdot \\nabla v +\\mu _a( x,\\lambda )u_nv \\mathrm {d}x+ \\int _{\\partial \\Omega }\\alpha u_nv\\mathrm {d}s=\\int _{\\partial \\Omega } S_nv \\mathrm {d}s, \\quad \\forall v\\in H^1(\\Omega ).$ Next we derive the linearized multi-wavelength model for the DOT problem based on an integral representation.",
"Let $v_m\\equiv v_m(\\lambda )\\in H^1(\\Omega )$ be the background solution corresponding to $D(x,\\lambda )\\equiv s_0(\\lambda )$ and $\\mu _a \\equiv 0$ with the excitation $S_m$ , i.e., $v_m$ fulfils $\\int _{\\Omega }s_0(\\lambda )\\nabla v_m\\cdot \\nabla v\\mathrm {d}x+ \\int _{\\partial \\Omega }\\alpha v_m v \\mathrm {d}s=\\int _{\\partial \\Omega } S_m v \\mathrm {d}s, \\quad \\forall v\\in H^1(\\Omega ).$ Note that unless $s_0(\\lambda )$ is independent of the wavelength $\\lambda $ , the dependence of the background solution $v_m$ on the wavelength $\\lambda $ cannot be factorized out.",
"Taking $v=v_m$ in (REF ) and $v=u_n$ in (REF ) and subtracting the two identities yield $s_0(\\lambda )\\int _\\Omega \\delta d(x) \\nabla u_n \\cdot \\nabla v_m\\mathrm {d}x+ \\sum _{k=1}^Ks_k(\\lambda ) \\int _\\Omega \\mu _k(x) u_n v_m\\mathrm {d}x= \\int _{\\partial \\Omega }( S_nv_m-S_mu_n)\\mathrm {d}s.$ Since $\\delta d$ and $\\mu _k$ s are assumed to be small, we can derive the approximations $\\nabla u_n(x,\\lambda )\\approx \\nabla v_n(x,\\lambda )$ and $u_n(x,\\lambda )\\approx v_n(x,\\lambda )$ in the domain $\\Omega $ (valid in the linear regime), and hence arrive at the following linearized model $ s_0(\\lambda )\\int _\\Omega \\delta d(x) \\nabla v_n \\cdot \\nabla v_m\\mathrm {d}x+ \\sum _{k=1}^Ks_k(\\lambda ) \\int _\\Omega \\mu _k(x) v_n v_m\\mathrm {d}x= \\int _{\\partial \\Omega }( S_nv_m-S_mu_n)\\mathrm {d}s.$ Note that since $\\int _{\\partial \\Omega } S_mu_n\\mathrm {d}s$ is the measured data on the detector located at $x_m$ and $\\int _{\\partial \\Omega } S_nv_m\\mathrm {d}s$ can be computed given the background spectra $s_0(\\lambda )$ , the right-hand side of the model (REF ) is completely known and can be readily computed.",
"The DOT imaging problem for the linearized model is to recover $\\delta d$ and the chromophore concentrations $\\lbrace \\mu _k\\rbrace _{k=1}^K$ from $\\lbrace u_n(x,\\lambda )\\rbrace _{n=1}^N$ on the boundary $\\partial \\Omega $ at several wavelengths $\\lbrace \\lambda _q\\rbrace _{q=1}^Q$ .",
"For the reconstruction, we divide the domain $\\Omega $ into a shape regular quasi-uniform mesh of elements $\\lbrace \\Omega _l\\rbrace _{l=1}^L$ such that $\\overline{\\Omega }=\\cup _{l=1}^L\\Omega _l$ , and consider a piecewise constant approximation of the coefficient $\\delta d(x)$ and the concentrations $\\lbrace \\mu _k\\rbrace _{k=1}^K$ of the chromophores as follows $\\begin{aligned}\\delta d(x) &\\approx \\sum _{l=1}^L (\\delta d)_l\\chi _{\\Omega _l}(x),\\\\\\mu _k(x) &\\approx \\sum _{l=1}^L (\\mu _k)_l\\chi _{\\Omega _l}(x),\\quad k=1,\\ldots ,K,\\end{aligned}$ where $\\chi _{\\Omega _l}$ is the characteristic function of the $l$ th element $\\Omega _l$ , and $(\\mu _k)_l$ denotes the value of the concentration $\\mu _k$ of the $k$ th chromophore in the $l$ th element $\\Omega _l$ , so is $(\\delta d)_l$ .",
"Upon substituting the approximation into (REF ), we have a finite-dimensional linear inverse problem $s_0(\\lambda )\\sum _{l=1}^L(\\delta d)_l\\int _{\\Omega _l} \\nabla v_n\\cdot \\nabla v_m\\mathrm {d}x+ \\sum _{k=1}^K s_k(\\lambda )\\sum _{l=1}^L(\\mu _k)_l\\int _{\\Omega _l} v_n v_m\\mathrm {d}x=\\int _{\\partial \\Omega }( S_nv_m-S_mu_n)\\mathrm {d}s.$ Finally, we introduce the sensitivity matrix $M^0(\\lambda )$ , $M^1(\\lambda )$ and the data vector $X$ .",
"We use a single index $j=1,\\ldots ,J$ with $J=N^2$ for the index pair $(m,n)$ with $j=N(m-1)+ n$ , and introduce the sensitivity matrix $M^0(\\lambda )=[M^0_{jl}]\\in \\mathbb {R}^{J\\times L}$ and $M^1(\\lambda )=[M^1_{jl}]\\in \\mathbb {R}^{J\\times L}$ with its entries $M^1_{jl}$ given by $\\begin{aligned}M^0_{jl}(\\lambda )=\\int _{\\Omega _l} \\nabla v_n\\cdot \\nabla v_m \\mathrm {d}x\\quad (j\\leftrightarrow (m,n)),\\\\M^1_{jl}(\\lambda )=\\int _{\\Omega _l} v_n v_m \\mathrm {d}x\\quad (j\\leftrightarrow (m,n)),\\end{aligned}$ which is independent of the wavelength $\\lambda $ .",
"Likewise, we introduce a data vector $X(\\lambda )\\in \\mathbb {R}^J$ with its $j$ th entry $X_j(\\lambda )$ given by $X_j(\\lambda ) = \\int _{\\partial \\Omega }( S_nv_m-S_mu_n)\\mathrm {d}s\\quad (j\\leftrightarrow (m,n)).$ By writing the vectors $A_0 = (\\delta d)_l\\in \\mathbb {R}^L$ and $A_k = (\\mu _k)_l\\in \\mathbb {R}^L$ , $k=0,\\ldots ,K$ , we obtain the following linear system (parameterized by the light wavelength $\\lambda $ ) $M^0(\\lambda )s_0(\\lambda )A_0 + M^1(\\lambda ) \\sum _{k=0}^Ks_k(\\lambda ) A_k = X(\\lambda ).$"
],
[
"Known diffusion coefficient",
"If the diffusion coefficient $D(x,\\lambda )$ is known, then the goal is to recover the concentrations $\\lbrace \\mu _k\\rbrace _{k=1}^K$ of the chromophores.",
"As before, we assume the unknowns $\\mu _k$ are small (in suitable $L^p(\\Omega )$ norms).",
"We can repeat the above procedure, except that the background solution $v_m\\in H^1(\\Omega )$ is now defined by $\\int _{\\Omega }D(x,\\lambda )\\nabla v_m \\cdot \\nabla v\\mathrm {d}x+ \\int _{\\partial \\Omega }\\alpha v_m v \\mathrm {d}s=\\int _{\\partial \\Omega } S_m v \\mathrm {d}s, \\quad \\forall v\\in H^1(\\Omega ).$ Taking $v=v_m$ in (REF ) and $v=u_n$ in (REF ) and subtracting the two identities gives $\\sum _{k=1}^Ks_k(\\lambda ) \\int _\\Omega \\mu _k(x) u_n v_m\\mathrm {d}x= \\int _{\\partial \\Omega }( S_nv_m-S_mu_n)\\mathrm {d}s.$ Using the approximation $u_n(x,\\lambda )\\approx v_n(x,\\lambda )$ in the domain $\\Omega $ (which is valid in the linear regime), we arrive at the following linearized model $\\sum _{k=1}^Ks_k(\\lambda ) \\int _\\Omega \\mu _k(x) v_n v_m\\mathrm {d}x= \\int _{\\partial \\Omega }( S_nv_m-S_mu_n)\\mathrm {d}s.$ As before, we divide the domain $\\Omega $ into a shape regular quasi-uniform mesh of elements $\\lbrace \\Omega _l\\rbrace _{l=1}^L$ such that $\\overline{\\Omega }=\\cup _{l=1}^L\\Omega _l$ , and consider piecewise constant approximations of the chromophore concentrations $\\mu _k$ : $\\begin{aligned}\\mu _k(x) \\approx \\sum _{l=1}^L (\\mu _k)_l\\chi _{\\Omega _l}(x),\\quad k=1,\\ldots ,K.\\end{aligned}$ Then we obtain the following finite-dimensional linear inverse problem $\\sum _{k=1}^K s_k(\\lambda )\\sum _{l=1}^L(\\mu _k)_l\\int _{\\Omega _l} v_n v_m\\mathrm {d}x= \\int _{\\partial \\Omega }( S_nv_m-S_mu_n)\\mathrm {d}s.$ Using the sensitivity matrix $M$ and the data vector $X$ in (REF ), we get the following parameterized linear system $M^1(\\lambda ) \\sum _{k=1}^Ks_k(\\lambda ) A_k = X(\\lambda ).$"
],
[
"The linearized DOT with multi-wavelength data",
"In the two linearized DOT inverse problems in Section REF , the vectors $A_0$ (if $d(x)$ is unknown) and $\\lbrace A_k\\rbrace ^K_{k=1}$ are the quantities of interest and are to be estimated from the wavelength dependent data $X(\\lambda )$ , given the spectra $s_0(\\lambda )$ and $s_k(\\lambda )$ .",
"These quantities directly contain the information of the locations and supports of $\\delta d(x)$ and all the chromophores $\\mu _k$ .",
"Now we describe a procedure for recovering the coefficient $d(x)$ (if unknown) and the concentrations $\\mu _k(x)$ of the chromophores simultaneously.",
"The diffusion wavelength dependence $s_0(\\lambda )$ is always known (e.g., $s_0(\\lambda )=c\\lambda ^{b}$ , where the parameter $b$ is known from experiments [10]).",
"We first formulate the inversion method for the case an unknown diffusion coefficient $D(x)$ .",
"We consider the case when all the absorption coefficient spectra $\\lbrace s_k(\\lambda )\\rbrace _{k=1}^K$ are known.",
"Suppose that we have collated the measured data at $Q$ distinct wavelengths $\\lbrace \\lambda _q\\rbrace _{q=1}^Q$ .",
"We write $S=(s_k(\\lambda _q))\\in \\mathbb {R}^{K\\times Q}$ , $S_0=(s_0(\\lambda ))\\in \\mathbb {R}^{1\\times Q}$ .",
"We also introduce the measured matrix $X=[X^t(\\lambda _1)\\ \\ldots \\ X^t(\\lambda _Q)]^t\\in \\mathbb {R}^{J\\times Q,1}$ and the vector of unknowns $A=[A^t_1\\ \\ldots \\ A^t_K, A^t_0]^t\\in \\mathbb {R}^{L\\times (K+1),1}$ .",
"Here, the superscript $t$ denotes the matrix/vector transpose.",
"Then we define a total sensitivity matrix $M$ constructed by $Q\\times (K+1)$ blocks.",
"The $(i,j)$ th block of $M$ is $M^1(\\lambda _i)s_j(\\lambda _i)$ for $1\\le i \\le Q$ , $1\\le j\\le K$ and the $(i,K+1)$ th block of $M$ is $M^1(\\lambda _i)s_0(\\lambda _i)$ for $1\\le i \\le Q$ .",
"Hence, (REF ) yields the following linear system: $MA=X.$ Similarly, when the diffusion coefficient $D(x,\\lambda )$ is known, we define another total sensitivity matrix $M$ constructed by $Q\\times K$ blocks.",
"The $(i,j)$ th block of $M$ is $M^1(\\lambda _i)s_j(\\lambda _i)$ for $1\\le i \\le Q$ , $1\\le j\\le K$ and the vector of unknowns $A=[A^t_1\\ \\ldots \\ A^t_K]^t\\in \\mathbb {R}^{L\\times K,1}$ .",
"We get from (REF ) the following linear system: $MA=X.$ Remark 1 Under the condition that the wavelength dependence of $M^0(\\lambda )$ and $M^1(\\lambda )$ can be factorized out (e.g., $s_0$ is independent of $\\lambda $ ) (and thus can be absorbed into the spectra $s_k(\\lambda )$ ), the linear systems can be decoupled to gain further insight.",
"To see this, we consider the case (REF ).",
"Since all the spectra are assumed to be linearly independent, when a sufficient number of wavelengths $\\lbrace \\lambda _q\\rbrace _{q=1}^Q$ are judiciously taken in the experiment, the corresponding spectral matrix $\\tilde{S}^t =\\left[S_0^t\\ \\ S^t \\right]$ is incoherent in the sense that $Q\\ge K+1$ and $\\mathrm {rank}(S)=K+1$ and $\\tilde{S}$ is also well-conditioned.",
"Then the matrix $\\tilde{S}$ has a right inverse $\\tilde{S}^{-1}$ .",
"By letting $\\tilde{Y}=X\\tilde{S}^{-1}$ , we obtain $[M^0A_0,M^1A] = \\tilde{Y}.$ These are $K+1$ decoupled linear systems.",
"By letting $\\tilde{Y}=[\\tilde{Y}_0\\ \\ldots \\ \\tilde{Y}_K]\\in \\mathbb {R}^{J\\times (K+1)}$ , we have $K+1$ independent (finite-dimensional) linear inverse problems $\\begin{aligned}M^0A_0&=\\tilde{Y}_0,\\\\M^1A_k&=\\tilde{Y}_k,\\quad k = 1,\\ldots ,K,\\end{aligned}$ where $A_0$ represents the diffusion coefficient $\\delta d(x)$ and $A_k$ (for $1\\le k \\le K$ ) represents the $k$ th chromophore $\\mu _k(x)$ .",
"Note that each linear system determines one and only one unknown concentration $A_k$ .",
"Similarly, for the case of a known diffusion coefficient, the matrix $S$ has a right inverse $S^{-1}$ , under the given incoherence assumption.",
"By letting $Y=XS^{-1}$ , we obtain $M^1A = Y.$ These are $K$ decoupled linear systems.",
"By letting $Y=[Y_1\\ \\ldots \\ Y_K]\\in \\mathbb {R}^{J\\times (K+1)}$ , we have $K$ independent (finite-dimensional) linear inverse problems $M^1A_k=Y_k,\\quad k = 1,\\ldots ,K,$ where $A_k$ (for $1\\le k \\le K$ ) represents the $k$ th chromophore concentrations $\\mu _k(x)$ .",
"Below, we describe a group sparse reconstruction method developed in [2] to solve the ill-conditioned linear systems (REF ) and (REF )."
],
[
"Group sparse reconstruction algorithm",
"Upon linearization and decoupling steps (see Remark REF ), one arrives at decoupled linear systems of the form: $Dx=b,$ where $D = M_0$ or $M\\in \\mathbb {R}^{J\\times L}$ is the sensitivity matrix, $x=A_k\\in \\mathbb {R}^{L}$ ($0\\le k \\le K$ ) is the unknown vector, and $b=Y_k\\in \\mathbb {R}^J$ ($0\\le k \\le K$ ) is a known measured data.",
"These linear systems are often under-determined, and severely ill-conditioned, due to the inherent ill-posed nature of the DOT inverse problem.",
"We adapt a numerical sparse method developed in [2] to solve (REF ).",
"The algorithm takes the following two aspects into consideration: Under the assumption that the unknowns $\\delta d$ and $\\mu _k$ are small, we may assume that $x$ is sparse.",
"This suggests to solve the minimization problem $\\min _{x\\in {\\Lambda }} \\Vert x\\Vert _1 \\quad \\mbox{subject to } \\Vert Dx-b\\Vert \\le \\epsilon ,$ where $\\Vert \\cdot \\Vert _1$ denotes the $\\ell ^1$ norm of a vector.",
"Here, ${\\Lambda }$ represents an admissible constraint on the unknowns $x$ , since they are bounded from below and above, and $\\epsilon >0$ is an estimate of the noise level of $b$ .",
"In DOT applications, it is also reasonable to assume that each concentration of chromophore $\\mu _k$ is clustered, and this refers to the concept of group sparsity.",
"The grouping effect is useful to remove the undesirable spikes typically observed for the $\\ell ^1$ penalty alone.",
"Now we describe the algorithm, i.e., group iterative soft thresholding, listed in Algorithm REF , adapted from iterative soft thresholding for $\\ell ^1$ optimization [17].",
"Here, $N$ is the maximum number of iterations, $w_{lk}$ are nonnegative weights controlling the strength of interaction, and $\\mathcal {N}_l$ denotes the neighborhood of the $l$ th element.",
"We take $w_{lk}=\\beta $ , for some $\\beta >0$ (default: $\\beta =0.5$ ), and $\\mathcal {N}_l$ consists of all elements in the triangulation sharing one edge with the $l$ th element.",
"Since the solution $x$ is expected to be sparse, a natural choice of the initial guess $x^0$ is the zero vector.",
"The regularization parameter $\\gamma $ plays a crucial role in the performance of the reconstruction quality: the larger the value $\\gamma $ is, the sparser the reconstructed solution becomes.",
"There are several possible strategies to determine its value, e.g., discrepancy principle and balancing principle, or a trial-and-error manner [20].",
"Group iterative soft thresholding.",
"[1] Input $D$ , $b$ , $W$ , $\\mathcal {N}$ , $\\gamma $ , $N$ and $x^0$ ; $j=1,\\ldots ,N$ Compute the proxy $g^j$ by $g^j=x^j-s^jD^t(Dx^j-b);$ Compute the generalized proxy $d^j$ by $d^j_l = |g^j_l|^2 + \\sum _{k\\in \\mathcal {N}_l}w_{lk}|g^j_k|^2;$ Compute the normalized proxy $\\bar{d}^j$ by $\\bar{d}^j= \\max (d^j)^{-1}d^j;$ Adapt the regularization parameter $\\bar{\\alpha }^j$ by $\\bar{\\alpha }_l^j=\\gamma /\\bar{d}^j_l,\\quad l=1,\\ldots ,L;$ Update $x^{j+1}$ by the group thresholding $x^{j+1} = P_{\\Lambda }(S_{s^j\\bar{\\alpha }^j} (g^j));$ Check the stopping criterion.",
"Below we briefly comment on the main steps of the algorithm and refer to [2] for details.",
"$g^j$ is a gradient descent update of $x^j$ , and $s^j>0$ is the step length, e.g., $s^j=1/\\Vert D\\Vert ^2$ .",
"This step takes into account the neighboring influence.",
"$\\bar{d}^j$ indicates a grouping effect: the larger $\\bar{d}^j_l$ is, the more likely the $l$ th element belongs to the group.",
"This step rescales $\\gamma $ to be inversely proportional to $\\bar{d}^j_l$ .",
"This step performs the projected thresholding with a spatially variable $\\bar{\\alpha }^j$ .",
"$P_\\Lambda $ denotes the pointwise projection onto the constraint set $\\Lambda $ and $S_\\lambda $ for $\\lambda >0$ is defined by $S_\\lambda (t) = \\max (|t|-\\lambda ,0)\\,\\mathrm {sign}(t)$ .",
"In our numerical experiments, we apply Algorithm REF to the coupled linear systems (REF ) and (REF ) directly.",
"This can be achieved by a simple change to Algorithm REF .",
"Specifically, at Step 3 of the algorithm, instead we compute the gradient of the least-squares functional $\\tfrac{1}{2}\\Vert MA-Y\\Vert ^2$ by $g^j=A^j-s^jM^t(Mx^j-Y).$ The remaining steps of the algorithm are applied to each component $A_i$ independently.",
"Note that one can easily incorporate separately a regularization parameter $\\gamma $ on each component, which is useful since the diffusion and scattering coefficients are likely to have different magnitudes."
],
[
"Imperfectly known boundary",
"Now, in order to show the potentials of DOS for handling modelling errors, we consider the case where the boundary of the domain of interest is not perfectly known.",
"This is one type of the modelling errors that occurs whenever the positions of the point sources and detectors or the domain of interest are not perfectly modelled.",
"We denote the true but unknown physical domain by $\\widetilde{\\Omega }$ , and the computational domain by $\\Omega $ , which approximates $\\widetilde{\\Omega }$ .",
"Next, we introduce a forward map $F:\\widetilde{\\Omega }\\rightarrow \\Omega $ , $\\widetilde{x}\\rightarrow x$ , which is assumed to be a smooth orientation-preserving map with a sufficiently smooth inverse map $F^{-1}: \\Omega \\rightarrow \\widetilde{\\Omega }$ .",
"We denote the Jacobian of the map $F$ by $J_F$ , and the Jacobian of $F$ with respect to the surface integral by $J^S_{F}$ .",
"Suppose now that the function $\\widetilde{u}_n(\\widetilde{x},\\lambda )$ satisfies problem (REF ) in the true domain $\\widetilde{\\Omega }$ with the diffusion coefficient $\\widetilde{D}(\\widetilde{x},\\lambda )$ , absorption coefficient $\\widetilde{\\mu }_a(\\widetilde{x},\\lambda )$ and source $\\widetilde{S}(\\widetilde{x})$ , namely $\\left\\lbrace \\begin{aligned}-\\nabla _{\\widetilde{x}}\\cdot (\\widetilde{D}(\\widetilde{x},\\lambda )\\nabla _{\\widetilde{x}}\\widetilde{u}_n(\\widetilde{x},\\lambda )) + \\widetilde{\\mu }_a(\\widetilde{x},\\lambda )\\widetilde{u}_n& = 0 \\quad \\mbox{ in }\\widetilde{\\Omega },\\\\\\widetilde{D}(\\widetilde{x},\\lambda )\\frac{\\partial \\widetilde{u}_n}{\\partial \\widetilde{\\nu }}(\\widetilde{x}) + \\alpha \\widetilde{u}_n(\\widetilde{x}) &= \\widetilde{S}_n(\\widetilde{x}) \\quad \\qquad \\mbox{on } \\partial \\widetilde{\\Omega }.\\end{aligned}\\right.$ Here, $\\widetilde{S}_n$ is a smooth approximation of the Dirac mass at the true position $\\widetilde{x}_n$ .",
"The wavelength-dependent absorption coefficient $\\widetilde{\\mu }(\\widetilde{x},\\lambda )$ also has a separable form related to the true concentrations $\\widetilde{\\mu }_k(\\widetilde{x})$ of the chromophres: $\\widetilde{\\mu }_a(\\widetilde{x},\\lambda )=\\sum _{k=1}^K \\widetilde{\\mu }_k(\\widetilde{x})s_k(\\lambda ),$ where $\\widetilde{\\mu }_k$ are assumed to be small.",
"Furthermore, the diffusion coefficient $D$ takes the linear form: $\\widetilde{D}(\\widetilde{x},\\lambda ) =s_0(\\lambda )(1+\\widetilde{\\delta d}(\\widetilde{x})).$ The weak formulation of problem (REF ) is given by: find $\\widetilde{u}_n(\\cdot ,\\lambda ) \\in H^1(\\widetilde{\\Omega })$ such that $\\int _{\\widetilde{\\Omega }}\\widetilde{D}( \\widetilde{x},\\lambda )\\nabla _{\\widetilde{x}} \\widetilde{u}_n\\cdot \\nabla _{\\widetilde{x}} \\widetilde{v} +\\widetilde{\\mu }_a( \\widetilde{x},\\lambda )\\widetilde{u}_n \\widetilde{v} \\mathrm {d}\\widetilde{x}+ \\int _{\\partial \\widetilde{\\Omega }}\\alpha \\widetilde{u}_n \\widetilde{v} \\mathrm {d}\\widetilde{s}=\\int _{\\widetilde{\\Omega }} \\widetilde{S}_n \\widetilde{v} \\mathrm {d}\\widetilde{s}, \\quad \\forall \\widetilde{v}\\in H^1(\\widetilde{\\Omega }).$ In the experimental settings, $\\widetilde{u}_n$ is assumed to be measured on the boundary $\\partial \\widetilde{\\Omega }$ .",
"However, because of the incorrect knowledge of $\\partial \\widetilde{\\Omega }$ , the measured quantity is in fact $u_n:=\\widetilde{u}_n \\circ F^{-1}$ restricted to the computational boundary $\\partial \\Omega $ .",
"Below we consider only the case that the domain $\\Omega $ is a small variation of the true physical one $\\widetilde{\\Omega }$ , so that the linearized regime is valid.",
"Specifically, the map $F:\\widetilde{\\Omega }\\rightarrow \\Omega $ is given by $F(\\widetilde{x})=\\widetilde{x} + \\epsilon \\widetilde{\\phi }(\\widetilde{x})$ , where $\\epsilon $ is a small scalar and the smooth function $\\widetilde{\\phi }(\\widetilde{x})$ characterizes the domain deformation.",
"Further, let $F^{-1}(x)=x+\\epsilon \\phi (x)$ be the inverse map, which is also smooth.",
"In order to analyze the influence of the domain deformation on the linearized DOT problem, we introduce the solution $v_m\\in H^1(\\Omega )$ corresponding to $\\bar{D}(\\lambda ,x)\\equiv s_0(\\lambda )$ and $\\mu _a \\equiv 0$ with $S_m$ being a smooth approximation of $\\delta _{x_m}$ , i.e., $v_m$ fulfils $\\int _{\\Omega }s_0(\\lambda )\\nabla v_m \\cdot \\nabla v\\mathrm {d}x+ \\int _{\\partial \\Omega }\\alpha v_m v \\mathrm {d}s= \\int _{\\partial \\Omega } S_m v \\mathrm {d}s, \\qquad v\\in H^1(\\Omega ).$ Now we can state the corresponding linearized DOT problem with an unknown boundary.",
"The result indicates that even for an isotropic diffusion coefficient $\\widetilde{D}$ in the true domain $\\widetilde{\\Omega }$ , in the computational domain $\\Omega $ the equivalent diffusion coefficient $D$ is generally anisotropic, and there is an additional perturbation factor on the boundary $\\partial \\Omega $ .",
"Proposition 3.1 Let $\\mu _a = \\widetilde{\\mu }_a \\circ F^{-1}$ .",
"The linearized inverse problem on the domain $\\Omega $ is given by $\\begin{split}s_0 (\\lambda )\\left(\\int _\\Omega (\\delta d(x)+\\epsilon \\Psi ) \\nabla v_n\\cdot \\nabla v_m\\mathrm {d}x\\right) + \\int _{\\partial \\Omega }\\alpha \\epsilon \\psi v_nv_m\\mathrm {d}s+ \\sum _{k=1}^Ks_k(\\lambda ) \\int _\\Omega \\mu _k(x) v_n v_m\\mathrm {d}x\\\\= \\int _{\\partial \\Omega } (S_n v_m - S_m u_n) \\mathrm {d}s,\\end{split}$ for some smooth functions $\\Psi :\\Omega \\rightarrow \\mathbb {R}^{d\\times d}$ and $\\psi :\\partial \\Omega \\rightarrow \\mathbb {R}$ , which are independent of the wavelength $\\lambda $ .",
"First, we derive the governing equation for the variable $u_n=\\widetilde{u}_n\\circ F^{-1}$ in the domain $\\Omega $ from (REF ).",
"Let $v=\\widetilde{v}\\circ F^{-1} \\in H^1(\\Omega )$ .",
"By the chain rule, we have $\\nabla _{\\widetilde{x}}\\widetilde{u}_n\\circ F^{-1}=(J_F^t\\circ F^{-1})\\nabla _xu_n$ , where the superscript $t$ denotes the matrix transpose.",
"Thus, we deduce that $\\begin{split}& \\quad \\int _{\\widetilde{\\Omega }}\\widetilde{D}(\\widetilde{x}, \\lambda )\\nabla _{\\widetilde{x}} \\widetilde{u}_n(\\widetilde{x})\\!\\cdot \\!\\nabla _{\\widetilde{x}} \\widetilde{v}(\\widetilde{x}){\\rm d}\\widetilde{x} \\\\& =\\!",
"\\int _\\Omega (\\widetilde{D}\\circ F^{-1})(x) (J_F^t\\circ F^{-1}) (x)\\nabla u_n(x)\\cdot (J_F^t\\circ F^{-1})(x) \\nabla v(x) |\\det J_{F}(x)|^{-1} \\mathrm {d}x\\\\& =\\!",
"\\int _\\Omega (J_F\\circ F^{-1})(x) (\\widetilde{D}\\circ F^{-1})(x) (J_F^t\\circ F^{-1}) (x)\\nabla u_n(x)\\cdot \\nabla v(x) |\\det J_{F}(x)|^{-1} \\mathrm {d}x\\\\& = \\int _\\Omega D(x,\\lambda ) \\nabla u_n(x)\\cdot \\nabla v(x) \\,\\mathrm {d}x,\\end{split}$ where the transformed diffusion coefficient $D(x,\\lambda )$ is given by [30], [22], [2] $D(x,\\lambda ) = \\left(\\frac{J_F(\\cdot )\\widetilde{D}(\\cdot ,\\lambda )J_F^t(\\cdot )}{|\\det J_F(\\cdot )|}\\circ F^{-1}\\right)(x).$ Similarly, we obtain $\\int _{\\widetilde{\\Omega }}\\widetilde{\\mu }_a(\\widetilde{x}, \\lambda ) \\widetilde{u}_n(\\widetilde{x})\\widetilde{v}(\\widetilde{x}){\\rm d}\\widetilde{x} & = \\int _\\Omega \\mu _a(x,\\lambda ) u_n(x) v(x) |{\\det J_{F}(x)}|^{-1} \\,\\mathrm {d}x,\\\\\\int _{\\partial \\Omega }\\alpha \\widetilde{u}_n \\widetilde{v} \\mathrm {d}\\widetilde{s} & = \\int _{\\partial \\Omega }\\alpha u_n v |{\\det J^S_{F}(x)}|^{-1} \\mathrm {d}s.$ Here we use the fact that $\\widetilde{D}\\equiv 1$ near the boundary in the second equation, since $\\delta d$ is compactly supported in the domain.",
"From (REF ), it follows that $u_n$ satisfies $\\int _{\\Omega }D(x,\\lambda )\\nabla u_n\\cdot \\nabla v &+\\mu _a(x,\\lambda )u_nv |\\det J_{F}(x)|^{-1}\\mathrm {d}x\\nonumber \\\\&+ \\int _{\\partial \\Omega }\\alpha u_n v |\\det J^S_{F}(x)|^{-1}\\mathrm {d}s= \\int _{\\partial \\Omega } S_n v \\mathrm {d}s, \\quad \\forall v\\in H^1(\\Omega ).$ Then by choosing $v=v_m$ in (REF ) and $v=u_n$ in (REF ), we arrive at $\\begin{split}\\int _\\Omega (D(x,\\lambda )- s_0(\\lambda )) \\nabla v_n\\cdot \\nabla v_m \\mathrm {d}x&+ \\int _\\Omega \\mu _a(x) v_n v_m|\\det J_{F}(x)|^{-1} \\mathrm {d}x\\\\& + \\int _{\\partial \\Omega }\\alpha (|\\det J^S_{F}(x)|^{-1}-1) u_nv\\mathrm {d}s= \\int _{\\partial \\Omega } (S_n v_m - S_m u_n) \\mathrm {d}s.\\end{split}$ Note that $J_F= I + \\epsilon J_{\\widetilde{\\phi }}$ , and $J_{F^{-1}}=I+\\epsilon J_\\phi = I-\\epsilon J_{\\widetilde{\\phi }}\\circ F^{-1}+O(\\epsilon ^2)$ , since $\\epsilon $ is small.",
"It is known that $|\\det J_F|= 1-\\epsilon \\mathrm {div}\\widetilde{\\phi }+O(\\epsilon ^2)$ [19], we similarly derive $|\\det J^S_F|= 1+\\epsilon \\psi +O(\\epsilon ^2)$ .",
"Then $D(x,\\lambda )$ is given by $\\begin{split}D(x,\\lambda ) & = \\widetilde{D}(\\cdot ,\\lambda )(1+\\epsilon \\mathrm {div}\\widetilde{\\phi }(\\cdot ))^{-1}(I+\\epsilon (J_{\\widetilde{\\phi }}(\\cdot )+J_{\\widetilde{\\phi }}^t(\\cdot )))\\circ F^{-1}(x)+O(\\epsilon ^2)\\\\&= \\widetilde{D}(\\cdot ,\\lambda )((1-\\epsilon \\mathrm {div}\\widetilde{\\phi }(\\cdot ))I + \\epsilon (J_{\\widetilde{\\phi }}(\\cdot )+J^t_{\\widetilde{\\phi }}(\\cdot )))\\circ F^{-1}(x) + O(\\epsilon ^2) \\\\&= \\widetilde{D}(\\cdot ,\\lambda )(1 + \\Psi \\epsilon )\\circ F^{-1}(x) + O(\\epsilon ^2),\\end{split}$ where $\\Psi =(J_{\\widetilde{\\phi }}+J_{\\widetilde{\\phi }}^t- \\mathrm {div}\\widetilde{\\phi }I)$ is smooth and independent of $\\lambda $ .",
"This and the linear form of $\\widetilde{\\mu }_a(\\widetilde{x},\\lambda )$ in (REF ) yield $D(x,\\lambda ) \\approx s_0(\\lambda )I +\\epsilon s_0(\\lambda )\\Psi (x)\\quad \\mbox{and}\\quad \\mu _a(x)|{\\det J_{F}(x)}|^{-1}=\\sum _{k=1}^K \\mu _k(x)s_k(\\lambda ) + o(\\epsilon ),$ where we have used the assumption that the $\\mu _k$ are small.",
"Upon substituting the above expressions into (REF ) and the approximations $\\nabla u_n\\approx \\nabla v_n$ , $u_n\\approx v_n$ in the domain, we obtain (REF ).",
"By Proposition REF , in the presence of an imperfectly known boundary with its magnitude $\\epsilon $ being comparable with the concentrations $\\lbrace \\mu _k\\rbrace _{k=1}^K$ and the perturbation $\\delta d$ , the perturbed sensitivity system contains significantly modeling errors resulting from the domain deformation.",
"Consequently, a direct inversion of the linearized model (REF ) is unsuitable.",
"This issue can be resolved using the multi-wavelength approach as follows.",
"Since (REF ) is completely analogous to (REF ), with the only difference lying in the additional terms in $s_0(\\lambda )$ (corresponding to the diffusion coefficient) and the edge perturbation $\\int _{\\partial \\Omega }\\alpha \\epsilon \\psi v_nv_m\\mathrm {d}s$ .",
"However, the edge perturbation on the boundary $\\partial \\Omega $ can be treated as unknowns corresponding to an additional spectral profile $s_*(\\lambda )\\equiv 1$ .",
"Thus, one may apply the multi-wavelength approach to recover the quantities of interest.",
"Specifically, assume that the spectral profiles $s_0, s_1,...,s_K,$ and $s_*$ are incoherent.",
"Then the method in Section REF may be applied straightforwardly, since the right-hand side is known.",
"However, the diffusion perturbation $\\delta d$ will never be properly reconstructed, due to the pollution of the error term $\\epsilon \\Psi $ (resulting from the domain perturbation).",
"The concentrations of chromophores $\\mu _k$ corresponding to the wavelength spectrum $s_k$ , $k=1,\\ldots ,K$ may be reconstructed, since they are affected by the deformation only through the transformation $\\mu _k = \\widetilde{\\mu }_k\\circ F^{-1}$ .",
"That is, the location and shape can be slightly deformed, provided that the deformation magnitude $\\epsilon $ is small.",
"Only the information of the diffusion coefficient is affected, and cannot be reconstructed.",
"In summary, multi-wavelength DOT is very effective to eliminate the modelling errors caused by the boundary uncertainty, at least in the linearized regime.",
"We have discussed that the influence of an uncertain boundary in the case where both diffusion and absorption coefficients are unknown.",
"We can also analyze for the case with a known diffusion coefficient similarly.",
"Specifically, one may assume the deformed diffusion coefficient on the domain $\\bar{D}(x,\\lambda )= \\widetilde{D}(\\widetilde{x},\\lambda )\\circ F^{-1}$ and repeat the procedure of Proposition REF .",
"We just give the conclusion: when the diffusion coefficient is known, the domain deformation contributes to a perturbation inside the spectrum $s_0$ , and the boundary deformation pollutes the known diffusion term, and the concentration of chromophores $\\mu _k$ corresponding to the wavelength spectrum $s_k$ , $k=1,\\ldots ,K$ could be reconstructed."
],
[
"Numerical experiments",
"Now we show some numerical results to illustrate our analytical findings.",
"The general setting for the numerical experiments is as follows.",
"The domain $\\Omega $ is taken to be the unit circle $\\Omega =\\lbrace (x_1,x_2):x_1^2+x_2^2<1\\rbrace $ .",
"There are 16 point sources uniformly distributed along the boundary $\\partial \\Omega $ ; see Figure REF for a schematic illustration of the domain $\\Omega $ , the point sources and the detectors.",
"Furthermore, we assume that the spectral profile $s_0(\\lambda )$ for the diffusion coefficient is $s_0(\\lambda )=0.2\\lambda ^b$ , where the parameter $b$ is known from experiments [10].",
"In all the examples below, we take $b=1.5$ .",
"We will also see that $\\mu _s \\gg \\mu _a$ is fulfilled in all the numerical examples.",
"We take a directionally varying refraction parameter $R=0.2$ , so that $\\alpha =\\frac{1-R}{2(1+R)}=1/3$ .",
"We use a piecewise linear finite element method on a shape regular quasi-uniform triangulation of the domain $\\Omega $ .",
"The unknowns are represented on a coarser finite element mesh using a piecewise constant finite element basis.",
"We measure the data $u_n(x_m,\\lambda ) (:= \\int _{\\partial \\Omega } S_m u_n \\mathrm {d}s)$ on the detectors located at $x_m$ .",
"The noisy data $u_n^\\delta (x_m,\\lambda )$ is generated by adding Gaussian noise to the exact data $u^\\dag _n(x_m,\\lambda )$ corresponding to the true diffusion coefficient $D(x,\\lambda )$ and absorption coefficient $\\mu _a(x,\\lambda )$ by $u_n^\\delta (x_m,\\lambda ) = u^\\dag _n(x_m,\\lambda ) + \\eta \\max _{l} |u^\\dag _n(x_m,\\lambda )-v_n(x_m,\\lambda )| \\xi _{n,m},$ where $\\eta $ is the noise level, and $\\xi _{n,m}$ follows the standard normal distribution.",
"Figure: The true boundary shape, the positions of sources and detectors.We present the numerical results for the cases with known boundary and with imperfectly known boundary separately, and for each case, we also present the examples with the diffusion coefficient being known and unknown.",
"In Algorithm REF , we take a constant step size to solve (REF )."
],
[
"Perfectly known boundary",
"First, we show numerical results for the case with a perfectly known boundary shape.",
"We shall test the robustness of the algorithm against the noise, and show that the multi-wavelength approach could reduce the deleterious effects of the noise in the measured data.",
"The regularization parameter $\\gamma $ was determined by a trial-and-error manner, and it was fixed at $\\gamma =5\\times 10^{-3}$ for the diffusion coefficient and $\\gamma =1\\times 10^{-4}$ for the absorption coefficient in all the numerical examples with perfectly known boundary.",
"This algorithm is always initialized with a zero vector.",
"Example 4.1 Consider a known diffusion coefficient $D(\\lambda ,x)=s_0(\\lambda )=0.2\\lambda ^{1.5}$ , and two chromophores inside the domain: the wavelength dependence of the chromophore on the top is $s_1(\\lambda )=0.2\\lambda $ , and the one on the bottom is $s_2(\\lambda )=0.2(\\lambda -1)^2$ ; See Figure REF for an illustration.",
"We take measurements at $Q=3$ wavelengths with $\\lambda _1=1$ , $\\lambda _2=1.5$ and $\\lambda _3=2$ .",
"The numerical results for Example REF are presented in Figure REF .",
"It is observed that the recovery is very localized within a clean background even with $10\\%$ noise in the data, and the supports of the recovered concentrations of the chromophores agree closely with the true ones and the magnitudes are well-retrieved.",
"Remarkably, the increase of the noise level from $1\\%$ to $10\\%$ does not influence much the shape of the recovered concentrations.",
"Therefore, if the given spectral profiles $s_k(\\lambda )$ are sufficiently incoherent, the corresponding unknowns can be fairly recovered.",
"This example also show that the proposed multi-wavelength approach is very robust to data noise, due to strong prior imposed by Algorithm REF .",
"Figure: Numerical results for Example : (a) exact μ 1 \\mu _1 and μ 2 \\mu _2 of two chromophores; (b)–(c) recovered results with η=1%\\eta =1\\% noise level; (d)–(e) recovered results with η=10%\\eta =10\\% noise level.The next example shows the approach for reconstructing three chromophores inside the domain.",
"Example 4.2 Consider the case with a known diffusion coefficient $D(\\lambda ,x)=s_0(\\lambda )=0.2\\lambda ^{1.5}$ , and 3 chromophores inside the domain: (i) The two chromophores on the top share the wavelength dependence $s_1(\\lambda )=0.2\\lambda $ , and the one on the bottom has a second spectral profile $s_2(\\lambda )=0.2(\\lambda -1)^2$ ; (ii) The wavelength dependence of the chromophore on the top right is $s_1(\\lambda )=0.2(\\lambda -1)^2$ , of the top left one is $s_2(\\lambda )=0.2\\lambda $ and of the bottom is $s_3(\\lambda )=0.2(\\lambda -1)^3$ .",
"We take measurements at $Q=3$ wavelengths with $\\lambda _1=1$ , $\\lambda _2=1.5$ and $\\lambda _3=2$ and the noise level is set to be $\\eta =1\\%$ .",
"The reconstruction results for Example REF are shown in Figure REF .",
"Figure REF indicates that the unknowns corresponding to two or three spectral profiles can be fairly recovered in terms of both the supports and magnitudes.",
"In case (i), the two chromophores on the top share the wavelength dependence, and they are recovered simultaneously; whereas in case (ii), the chromophores have three incoherent wavelength dependences, and they can be recovered separately.",
"The next example aims at recovering both diffusion and absorption coefficients, which is known to be very challenging in the absence of multi-wavelength data.",
"Figure: Numerical results for Example : (a) exact μ 1 \\mu _1 and μ 2 \\mu _2 of two chromophores; (b)-(c) recovered results for case (i) (noise level η=1%\\eta =1\\%); (d)-(f) recovered results for case (ii) (noise level η=1%\\eta =1\\%).Example 4.3 Consider the case of an unknown diffusion coefficient given by $D(\\lambda ,x)=s_0(\\lambda )(1+0.1\\delta d(x))=0.2\\lambda ^{1.5} (1+0.25\\delta d(x))$ .",
"Similar to Example REF , consider two chromophores inside the domain: the wavelength dependence of the chromophore on the top is $s_1(\\lambda )=0.2\\lambda $ , and that of the bottom is $s_2(\\lambda )=0.2(\\lambda -1)^2$ .",
"The measurements are taken at $Q=3$ wavelengths with $\\lambda _1=1$ , $\\lambda _2=1.5$ and $\\lambda _3=2$ , and the noise level $\\eta $ is fixed at $\\eta =1\\%$ .",
"Figure: Numerical results for Example : (a) exact μ 1 \\mu _1 and μ 2 \\mu _2 of two chromophores;(b)-(c) recovered results for two chromophores; (d)-(e) true and recovered δd(x)\\delta d(x).The numerical results for Example REF are shown in Figure REF .",
"Simultaneously reconstructing the diffusion and absorption coefficients is more sensitive to data noise, when compared with the case of a known diffusion coefficient.",
"Recall that the problem of recovering both coefficients is quite ill-posed [16]: two different pairs of scattering and absorption coefficients can give rise to identical measured data.",
"However, it is observed from Example REF that the multi-wavelength approach allows overcoming this nonuniqueness issue, provided that the spectra are indeed incoherent.",
"The next example shows that multi-wavelength data can mitigate the effects of the noise.",
"Example 4.4 Consider the setting of Example REF , but with a noise level $\\eta =30\\%$ .",
"We study two different numbers of wavelengths.",
"(i) The measurements are taken at $Q=3$ wavelengths with $\\lambda _i=1+(i-1)/2$ , $i=1,\\ldots ,3$ ; (ii) The measurements are taken at $Q=30$ wavelengths with $\\lambda _i=1+(i-1)/29$ , $i=1,\\ldots ,30$ .",
"Numerical results for Example REF are shown in Figure REF .",
"When using only data for 3 wavelengths, the recovered images are blurred by $ 30\\%$ noise.",
"However, using data for 30 wavelengths, both the diffusion coefficient $\\delta d$ and two chromophore concentrations $\\mu _k$ are much better resolved than using data with 3 wavelengths.",
"Hence, more wavelength observations can greatly mitigate the effects of data noise; which concurs with the observations from the experimental study [29].",
"Figure: Numerical results for Example : (a)-(c) the recovered results for case (i) (with 3 wavelengths); (d)-(f): results for case (ii) (with 30 wavelengths)."
],
[
"Imperfectly known boundary",
"Now we illustrate the approach in the case of an imperfectly known boundary.",
"The (unknown) true domain $\\widetilde{\\Omega }$ is an ellipse centered at the origin with semi-axes $a$ and $b$ , $\\mathcal {E}_{a,b} =\\lbrace (x_1,x_2): x_1^2/a^2+x_2^2/b^2<1\\rbrace $ , and the computational domain $\\Omega $ is the unit disk.",
"In this part, the regularization parameter $\\gamma $ was determined by a trial-and-error manner, and it was fixed at $\\gamma =5\\times 1^{-2}$ for the diffusion coefficient, $\\gamma =1\\times 10^{-4}$ for the absorption coefficient and $\\gamma =1\\times 10^{-4}$ for the edge perturbation in all the numerical examples with imperfectly known boundary.",
"This algorithm is always initialized with a zero vector.",
"Example 4.5 Consider the case of a known diffusion coefficient $D(\\lambda ,x)=s_0(\\lambda )=0.2\\lambda ^{1.5}$ , and two different shape deformations: $(i)$ $\\widetilde{\\Omega }$ is an ellipse with $a=1.1$ and $b=0.9$ and $(ii)$ $\\widetilde{\\Omega }$ is an ellipse with $a=1.2$ and $b=0.8$ .",
"Consider two chromophores inside $\\widetilde{\\Omega }$ : the wavelength dependence of the chromophore on the top is $s_1(\\lambda )=0.5(\\lambda -1)$ , and that of the bottom is $s_2(\\lambda )=0.5(\\lambda -1)^2$ .",
"The measurements are taken at $Q=3$ wavelengths with $\\lambda _1=1$ , $\\lambda _2=1.5$ and $\\lambda _3=2$ , and the noise level is fixed at $\\eta =1\\%$ .",
"The numerical results for Example REF are shown in Figure REF .",
"This example illustrates the influence of the deformation scale on the reconstruction.",
"The numerical results show clearly the potential of the multi-wavelength approach: Even using the wrong domain for the inversion step, we can still recover the concentrations of the chromophores (or more precisely the deformed concentrations $\\mu _k = \\widetilde{\\mu }_k \\circ F^{-1}$ ).",
"The numerical results also show even we assume known diffusion coefficient in case (i), we should also use all the spectra $s_0$ , $s_k$ , and $s_*$ to recover the right concentrations of the chromophores (Figures REF (d) and (e)), or the results will be ruined by the shape deformation (Figures REF (b) and (c)).",
"Figure: Numerical results for Example : (a),(f) exact μ 1 \\mu _1 and μ 2 \\mu _2 of two chromophores in Ω ˜\\widetilde{\\Omega };(b)-(c) recovered results for two chromophores in Ω\\Omega for case (i) only using the spectra s k s_k of the chromophores; (d)-(e)recovered results for two chromophores in Ω\\Omega for case (i) using the spectra s k s_k of the chromophores, the spectrum s 0 s_0 ofthe diffusion coefficient and the spectrum s * (λ)≡1s_*(\\lambda )\\equiv 1 of the edge perturbation; (g)-(h) the recovered results for case(ii) using all the spectra s 0 s_0, s k s_k, and s * s_*.Example 4.6 Consider the case of an unknown diffusion coefficient $D(\\lambda ,x)=s_0(\\lambda )(1+0.1\\delta d(x))=0.2\\lambda ^{1.5}(1+0.25\\delta d(x))$ , and the unknown true domain $\\widetilde{\\Omega }$ is an ellipse with $a=1.1$ and $b=0.9$ .",
"Consider two chromophores inside the domain $\\widetilde{\\Omega }$ : the wavelength dependence of the chromophore on the top is $s_1(\\lambda )=0.5(\\lambda -1)$ , and that of the bottom is $s_2(\\lambda )=0.5(\\lambda -1)^2$ .",
"The measurements are taken at $Q=3$ wavelengths with $\\lambda _1=1$ , $\\lambda _2=1.5$ and $\\lambda _3=2$ , and the noise level is fixed at $\\eta =1\\%$ .",
"The numerical results for Example REF are shown in Figure REF .",
"It is observed that the two chromophores are recovered well in spite of the imperfectly known boundary, while the diffusion coefficient is totally distorted by domain deformation, and thus it cannot be accurately recovered.",
"The empirical observations on Examples REF and REF concur with the theoretical predictions in Section : Proposition REF implies that the domain deformation will be added to the recovered results corresponding to the spectral profile $s_0(\\lambda )$ and the diffusion coefficient cannot be recovered due to the domain deformation, but the deformed chromophore concentrations $\\mu _k = \\widetilde{\\mu }_k\\circ F^{-1}$ can still be recovered.",
"Figure: Numerical results for Example : (a),(d): exact μ 1 \\mu _1 and μ 2 \\mu _2 of two chromophores in Ω ˜\\widetilde{\\Omega };(b)-(c): recovered results for two chromophores in Ω\\Omega ; (e) the recovered result corresponding to s 0 (λ)s_0(\\lambda )."
],
[
"Conclusion",
"In this work, we have introduced a novel reconstruction technique for diffuse optical imaging with multi-wavelength data.",
"The approach is based on a linearized model and a group sparsity approach.",
"We have shown that within the linear regime, our reconstruction technique allows recovering the concentration of individual chromophore and the diffusion coefficients, provided that their spectral profiles are known and incoherent.",
"Furthermore, we have demonstrated that the multi-wavelength data can significantly reduce modelling errors associated with an imperfectly known boundary.",
"In fact, it allows recovering well (deformed) concentrations of the chromophores.",
"These findings are fully supported by extensive numerical experiments."
]
] | 1808.08557 |
[
[
"Discrete Decreasing Minimization, Part II: Views from Discrete Convex\n Analysis"
],
[
"Abstract We continue to consider the discrete decreasing minimization problem on an integral base-polyhedron treated in Part I.",
"The problem is to find a lexicographically minimal integral vector in an integral base-polyhedron, where the components of a vector are arranged in a decreasing order.",
"This study can be regarded as a discrete counter-part of the work by Fujishige (1980) on the lexicographically optimal base and the principal partition of a base-polyhedron in continuous variables.",
"The objective of Part II is two-fold.",
"The first is to offer structural views from discrete convex analysis (DCA) on the results of Part I obtained by the constructive and algorithmic approach.",
"The second objective is to pave the way of DCA approach to discrete decreasing minimization on other discrete structures such as the intersection of M-convex sets, flows, and submodular flows.",
"We derive the structural results in Part I from fundamental facts on M-convex sets and M-convex functions in DCA.",
"A direct characterization is given to the canonical partition, which was constructed by an iterative procedure in Part I.",
"This reveals the precise relationship between the canonical partition for the discrete case and the principal partition for the continuous case.",
"Moreover, this result entails a proximity theorem, stating that every decreasingly minimal element is contained in the small box containing the (unique) fractional decreasingly minimal element (the minimum-norm point), leading further to a continuous relaxation algorithm for finding a decreasingly minimal element of an M-convex set.",
"Thus the relationship between the continuous and discrete cases is completely clarified.",
"Furthermore, we present DCA min-max formulas for network flows, the intersection of two M-convex sets, and submodular flows."
],
[
"Introduction",
"We continue to consider discrete decreasing minimization on an integral base-polyhedron studied in Part I.",
"The problem is to find a lexicographically minimal (dec-min) integral vector in an integral base-polyhedron, where the components of a vector are arranged in a decreasing order (see Section REF for precise description of the problem).",
"While our present study deals with the discrete case, the continuous case was investigated by Fujishige [11] around 1980 under the name of lexicographically optimal bases of a base-polyhedron, as a generalization of lexicographically optimal maximal flows considered by Megiddo [27].",
"Our study can be regarded as a discrete counter-part of the work by Fujishige [11], [12] on the lexicographically optimal base and the principal partition of a base-polyhedron.",
"The objective of Part II is two-fold.",
"The first is to offer structural views from discrete convex analysis (DCA) on the results of Part I obtained by the constructive and algorithmic approach.",
"The second objective is to pave the way of DCA approach to discrete decreasing minimization on other discrete structures such as the intersection of M-convex sets, flows, and submodular flows.",
"In Part I of this paper, we have shown the following: A characterization of decreasing minimality by 1-tightening steps (exchange operations), A (dual) characterization of decreasing minimality by the canonical chain, The structure of the dec-min elements as a matroidal M-convex set, A characterization of a dec-min element as a minimizer of square-sum of components, A min-max formula for the square-sum of components, A strongly polynomial algorithm for finding a dec-min element and the canonical chain, Applications.",
"In contrast to the constructive and algorithmic approach in Part I, Part II offers structural views from discrete convex analysis (DCA) as well as from majorization.",
"The concept of majorization ordering offers a useful general framework to discuss decreasing minimality.",
"The relevance of DCA to decreasing minimization is not surprising, since an M-convex set is nothing but the set of integral points of an integral base-polyhedron and a separable convex function on an M-convex set is an M-convex function.",
"In particular, the square-sum of components of a vector in an M-convex set is an M-convex function.",
"It will be shown that most of the important structural results obtained in Part I can be derived from the Fenchel-type discrete duality theorem, which is a main characteristic of DCA as compared with other theories of discrete functions such as [35].",
"In Section of this paper the basic facts about majorization are described.",
"In Section we derive the characterization of decreasing minimality in terms of 1-tightening steps (exchange operations) from the local characterization of global minimality for M-convex functions, known as M-optimality criterion in DCA.",
"In Section , the min-max formulas, including the one for the square-sum of components, are derived as special cases of the Fenchel-type discrete duality in DCA.",
"We also show a novel min-max formula, which reinforces the link between the present study and the theory of majorization.",
"In Section we use a general result on the Fenchel-type discrete duality in DCA for a short alternative proof to the statement that the decreasingly minimal elements of an M-convex set form a matroidal M-convex set.",
"The relationship between the continuous and discrete cases is clarified in Section .",
"We reveal the precise relation between the canonical partition and the principal partition by establishing an alternative direct characterization of the canonical partition, which was constructed by an iterative procedure in Part I.",
"The obtained result provides a proximity theorem, stating that every dec-min element is contained in the small box (unit box) containing the (unique) fractional dec-min element (the minimum-norm point), and hence a continuous relaxation algorithm for finding a decreasingly minimal element of an M-convex set.",
"In Section we present DCA results relevant discrete decreasing minimization for the set of integral feasible flows, the intersection of two M-convex sets, and the set of integral members of an integral submodular flow polyhedron.",
"In Appendix we offer a brief survey of early papers and books related to decreasing minimization on base-polyhedra."
],
[
"Definition and notation",
"We review some definitions and notations introduced in Part I [9]."
],
[
"Decreasing minimality",
"For a vector $x$ , let $x {\\downarrow }$ denote the vector obtained from $x$ by rearranging its components in a decreasing order.",
"For example, $x{\\downarrow }=(5,5,4,2,1)$ when $x = (2,5,5,1,4)$ .",
"We call two vectors $x$ and $y$ (of same dimension) value-equivalent if $x{\\downarrow }= y{\\downarrow }$ .",
"For example, $(2,5,5,1,4)$ and $(1,4,5,2,5)$ are value-equivalent while the vectors $(3,5,5,3,4)$ and $(3,4,5,4,4)$ are not.",
"A vector $x$ is decreasingly smaller than vector $y$ , in notation $x <_{\\rm dec} y$ , if $x{\\downarrow }$ is lexicographically smaller than $y{\\downarrow }$ in the sense that they are not value-equivalent and $x{\\downarrow }(j)<y{\\downarrow }(j)$ for the smallest subscript $j$ for which $x{\\downarrow }(j)$ and $y{\\downarrow }(j)$ differ.",
"For example, $x = (2,5,5,1,4)$ is decreasingly smaller than $y =(1,5,5,5,1)$ since $x{\\downarrow }=(5,5,4,2,1)$ is lexicographically smaller than $y{\\downarrow }=(5,5,5,1,1)$ .",
"We write $x\\le _{\\rm dec} y$ to mean that $x$ is decreasingly smaller than or value-equivalent to $y$ .",
"For a set $Q$ of vectors, $x\\in Q$ is decreasingly minimal (dec-min, for short) if $x \\le _{\\rm dec} y$ for every $y\\in Q$ .",
"Note that the dec-min elements of $Q$ are value-equivalent.",
"An element $m$ of $Q$ is dec-min if its largest component is as small as possible, within this, its second largest component (with the same or smaller value than the largest one) is as small as possible, and so on.",
"An element $x$ of $Q$ is said to be a max-minimized element (a max-minimizer, for short) if its largest component is as small as possible.",
"In an analogous way, for a vector $x$ , we let $x {\\uparrow }$ denote the vector obtained from $x$ by rearranging its components in an increasing order.",
"A vector $y$ is increasingly larger than vector $x$ , in notation $y >_{\\rm inc} x$ , if they are not value-equivalent and $y{\\uparrow }(j)> x{\\uparrow }(j)$ holds for the smallest subscript $j$ for which $y{\\uparrow }(j)$ and $x{\\uparrow }(j)$ differ.",
"We write $y \\ge _{\\rm inc} x$ if either $y >_{\\rm inc} x$ or $x$ and $y$ are value-equivalent.",
"Furthermore, we call an element $m$ of $Q$ increasingly maximal (inc-max for short) if its smallest component is as large as possible over the elements of $Q$ , within this its second smallest component is as large as possible, and so on.",
"The decreasing minimization problem is to find a dec-min element of a given set $Q$ of vectors.",
"When the set $Q$ consists of integral vectors, we speak of discrete decreasing minimization.",
"In Parts I and II of this series of papers, we deal with the case where the set $Q$ is an M-convex set, i.e., the set of integral members of an integral base-polyhedron.",
"In Part III, the set $Q$ will be the integral feasible flows.",
"The set $Q$ can be the intersection of two M-convex sets, or more generally, the set of integral members of an integral submodular flow polyhedron.",
"Throughout the paper, $S$ denotes a finite nonempty ground-set.",
"For a vector $m\\in {\\bf R}^{S}$ (or function $m:S\\rightarrow {\\bf R}$ ) and a subset $X \\subseteq S$ , we use the notation $\\widetilde{m}(X)=\\sum [m(v): v\\in X]$ .",
"The characteristic (or incidence) vector of a subset $Z \\subseteq S$ is denoted by $\\chi _{Z}$ , that is, $\\chi _{Z}(v)=1$ if $v\\in Z$ and $\\chi _{Z}(v)=0$ otherwise.",
"For a polyhedron $B$ , notation $\\overset{....}{B}$ (pronounced: dotted $B$ ) means the set of integral members (elements, vectors, points) of $B$ .",
"Let $b$ be a set-function for which $b(\\emptyset )=0$ and $b(X)=+\\infty $ is allowed but $b(X)=-\\infty $ is not.",
"The submodular inequality for subsets $X,Y\\subseteq S$ is defined by $ b(X) + b(Y) \\ge b(X\\cap Y) + b(X\\cup Y).$ We say that $b$ is submodular if the submodular inequality holds for every pair of subsets $X, Y\\subseteq S$ with finite $b$ -values.",
"A set-function $p$ is supermodular if $-p$ is submodular.",
"A (possibly unbounded) base-polyhedron $B$ in ${\\bf R}^{S}$ is defined by $ B=B(b)=\\lbrace x\\in {\\bf R}^{S}: \\widetilde{x}(S)=b(S), \\ \\widetilde{x}(Z)\\le b(Z) \\ \\hbox{ for every } \\ Z\\subset S\\rbrace .$ A nonempty base-polyhedron $B$ can also be defined by a supermodular function $p$ for which $p(\\emptyset )=0$ and $p(S)$ is finite as follows: $ B=B^{\\prime }(p)=\\lbrace x\\in {\\bf R}^{S}: \\widetilde{x}(S)=p(S), \\ \\widetilde{x}(Z)\\ge p(Z) \\ \\hbox{ for every } \\ Z\\subset S\\rbrace .$ We call the set $\\overset{....}{B}$ of integral elements of an integral base-polyhedron $B$ an M-convex set.",
"Originally, this basic notion of discrete convex analysis was defined as a set of integral points in ${\\bf R}^{S}$ satisfying certain exchange axioms, and it has been known that the two properties are equivalent ([33]).",
"For a function $\\varphi : {\\bf Z}\\rightarrow {\\bf R}\\cup \\lbrace -\\infty , +\\infty \\rbrace $ the effective domain of $\\varphi $ is denoted as ${\\rm dom\\,}\\varphi = \\lbrace k \\in {\\bf Z}: -\\infty < \\varphi (k) < +\\infty \\rbrace $ .",
"A function $\\varphi : {\\bf Z}\\rightarrow {\\bf R}\\cup \\lbrace +\\infty \\rbrace $ is called discrete convex (or simply convex) if $ \\varphi (k-1) + \\varphi (k+1) \\ge 2 \\varphi (k)$ for all $k \\in {\\rm dom\\,}\\varphi $ , and strictly convex if ${\\rm dom\\,}\\varphi = {\\bf Z}$ and $\\varphi (k-1) + \\varphi (k+1) > 2 \\varphi (k)$ for all $k \\in {\\bf Z}$ .",
"A function $\\Phi : {\\bf Z}^{S} \\rightarrow {\\bf R}\\cup \\lbrace +\\infty \\rbrace $ of the form $ \\Phi (x) = \\sum [\\varphi _{s}(x(s)): s\\in S]$ is called a separable (discrete) convex function if, for each $s \\in S$ , $\\varphi _{s}: {\\bf Z}\\rightarrow {\\bf R}\\cup \\lbrace +\\infty \\rbrace $ is a discrete convex function.",
"We call $\\Phi $ a symmetric separable convex function if $\\varphi _{s}$ does not depend on $s$ , that is, if $\\varphi _{s}=\\varphi $ for all $s \\in S$ for some discrete convex function $\\varphi $ .",
"We call $\\Phi $ a symmetric separable strictly convex function if $\\varphi $ is strictly convex."
],
[
"Connection to majorization",
"Majorization ordering (or dominance ordering) is a well-established notion studied in diverse contexts including statistics and economics, as described in Arnold–Sarabia [4] and Marshall–Olkin–Arnold [26].",
"In this section we describe the relevant results known in the literature of majorization, and indicate a close relationship to decreasing minimality investigated in our series of papers.",
"We have dual objectives in this section.",
"First, we intend to reinforce the connection between majorization and combinatorial optimization.",
"It is also hoped that this will lead to future applications of our results in areas like statistics and economics, in addition to those areas related to graphs, networks, and matroids mentioned in the introduction of Part I [9].",
"In economics, for example, egalitarian allocation for indivisible goods can possibly be formulated and analyzed by means of discrete decreasing minimization.",
"Second, we point out substantial technical connections between majorization and our results in Part I.",
"We argue that some of our results can be derived from the combination of the classical results about majorization and the results of Groenevelt [15] for the minimization of separable convex functions over the integer points in an integral base-polyhedron.",
"We also point out that some of the standard characterizations for least majorization are associated with min-max duality relations in the case where the underlying set is the integer points of an integral base-polyhedron or the intersection of two integral base-polyhedra."
],
[
"Majorization ordering",
"We review standard results known in the literature of majorization in a way suitable for our discussion.",
"Recall that $x{\\downarrow }$ denotes the vector obtained from a vector $x\\in {\\bf R}^{n}$ by rearranging its components in a decreasing order.",
"Let $\\overline{x}$ denote the vector whose $k$ -th component $\\overline{x}(k)$ is equal to the sum of the first $k$ components of $x{\\downarrow }$ .",
"A vector $x$ is said to be majorized by another vector $y$ , in notation $x \\prec y$ , if $\\overline{x} \\le \\overline{y}$ and $\\overline{x}(n) = \\overline{y}(n)$ .",
"It is easy to see [26] that $ x \\prec y \\iff -x \\prec -y.$ (At first glance, the equivalence in (REF ) may look strange, but observe that $x \\prec y$ means that $x$ is more uniform than $y$ , which is equivalent to saying that $-x$ is more uniform than $-y$ .)",
"Majorization is discussed more often for real vectors, but here we are primarily interested in integer vectors.",
"As an immediate adaptation of the standard results [26], the following proposition gives equivalent conditions for majorization for integer vectors.",
"A $T$ -transform (also called a Robin Hood operation) means a linear transformation of the form $T = (1 - \\lambda ) I + \\lambda Q$ , where $0 \\le \\lambda \\le 1$ and $Q$ is a permutation matrix that interchanges just two elements (transposition).",
"In other words, a $T$ -transform is a mapping of the form $x \\mapsto x + \\hat{\\lambda }(\\chi _{s}-\\chi _{t})$ with $0 \\le \\hat{\\lambda }\\le x(t)-x(s)$ .",
"It is noteworthy that this operation with $\\hat{\\lambda }=1$ corresponds to the basis exchange in an integral base-polyhedron.",
"Proposition 2.1 The following conditions are equivalent for $x, y \\in {\\bf Z}^{n}:$ (i) $x \\prec y$ ($x$ is majorized by $y$ ), that is, $ \\sum _{i=1}^{k} x{\\downarrow }(i) \\le \\sum _{i=1}^{k} y{\\downarrow }(i)\\quad (k=1,\\ldots ,n-1),\\qquad \\sum _{i=1}^{n} x{\\downarrow }(i) = \\sum _{i=1}^{n} y{\\downarrow }(i).$ (ii) $x = y P$ for some doubly stochastic matrix $P$ , where $x$ and $y$ are regarded as row vectors.",
"(iii) $x$ can be derived from y by successive applications of a finite number of $T$ -transforms.",
"(iv) $\\displaystyle \\sum _{i=1}^{n} \\varphi ( x(i) ) \\le \\sum _{i=1}^{n} \\varphi ( y(i) )$ for all discrete convex functions $\\varphi : {\\bf Z}\\rightarrow {\\bf R}$ .",
"(v) $\\displaystyle \\sum _{i=1}^{n} x(i)= \\sum _{i=1}^{n} y(i)$ and $\\displaystyle \\sum _{i=1}^{n} (x(i) - a )^{+} \\le \\sum _{i=1}^{n} (y(i) - a )^{+}$ for all $a \\in {\\bf Z}$ .",
"where $(z)^{+} = \\max \\lbrace 0, z \\rbrace $ for any $z \\in {\\bf Z}$ .",
"$\\rule {0.17cm}{0.17cm}$ Let $D$ be an arbitrary subset of ${\\bf Z}^{n}$ .",
"An element $x$ of $D$ is said to be least majorized in $D$ if $x$ is majorized by all $y \\in D$ .",
"A least majorized element may not exist in general, as the following example shows.",
"Example 2.1 Let $D= \\lbrace (2, 0, 0, 0), \\ (1, -1, 1, 1)\\rbrace $ .",
"For $x = (2, 0, 0, 0)$ and $y = (1, -1, 1, 1)$ we have $x{\\downarrow } =(2,0,0,0)$ and $y{\\downarrow } =(1,1,1,-1)$ .",
"Therefore, $x = (2, 0, 0, 0)$ is increasingly maximal in $D$ and $y = (1, -1, 1, 1)$ is decreasingly minimal in $D$ .",
"However, there exists no least majorized element in $D$ , since $\\overline{x} = (2,2,2,2)$ and $\\overline{y} = (1,2,3,2)$ , for which neither $\\overline{x} \\le \\overline{y}$ nor $\\overline{y} \\le \\overline{x}$ holds.",
"We note that $D$ here arises from the intersection of two integral base-polyhedra (see Section 3.4 of Part I [9]).",
"$\\rule {0.17cm}{0.17cm}$ Remark 2.1 In discussing the existence and properties of a least majorized element, we are primarily concerned with a subset $D$ of ${\\bf Z}^{n}$ whose elements have a constant component-sum.",
"If the component-sum is not constant on $D$ , we need to introduce a more general notion [38].",
"A vector $x$ is said to be weakly submajorized by another vector $y$ , denoted $x \\prec _{\\rm w} y$ , if $\\overline{x} \\le \\overline{y}$ .",
"An element $x$ of $D$ is said to be least weakly submajorized in $D$ if $x$ is weakly submajorized by all $y \\in D$ .",
"The distinction of “weakly submajorized” and “majorized” is not necessary for a base-polyhedron or the intersection of base-polyhedra, whereas we have to distinguish these concepts for a g-polymatroid and a submodular flow polyhedron.",
"$\\rule {0.17cm}{0.17cm}$ Remark 2.2 The characterization of a least majorized element in (iv) in Proposition REF can be associated with a min-max duality relation, which is given by (REF ) in Section REF when the underlying set $D$ is an M-convex set (= the integer points of an integral base-polyhedron), and by (REF ) in Section REF when $D$ is the intersection of two M-convex sets.",
"For an M-convex set, the min-max formula associated with (v) in Proposition REF is given by (REF ) in Theorem REF in Section REF .",
"$\\rule {0.17cm}{0.17cm}$"
],
[
"Majorization and decreasing-minimality",
"Majorization and decreasing-minimality are closely related, as is explicit in Tamir [38].",
"Proposition 2.2 If $x \\prec y$ , then $x \\le _{\\rm dec} y$ and $x \\ge _{\\rm inc} y$ .",
"Suppose that $x \\prec y$ .",
"If $\\overline{x} = \\overline{y}$ , then $x{\\downarrow } = y{\\downarrow }$ , and hence $x$ and $y$ are value-equivalent.",
"If $\\overline{x} < \\overline{y}$ , then there exists an index $k$ with $1 \\le k \\le n$ such that $x{\\downarrow }(i) = y{\\downarrow }(i)$ for $i=1,\\ldots , k-1$ and $x{\\downarrow }(k) < y{\\downarrow }(k)$ .",
"This shows that $x$ is decreasingly smaller than $y$ .",
"In either case, we have $x \\le _{\\rm dec} y$ .",
"Since $x \\prec y$ , we have $-x \\prec -y$ by (REF ).",
"By the above argument applied to $(-x,-y)$ , we obtain $-x \\le _{\\rm dec} -y $ , which is equivalent to $x \\ge _{\\rm inc} y$ .",
"Remark 2.3 The converse of Proposition REF is not true.",
"That is, $x \\prec y$ does not follow from $x \\le _{\\rm dec} y \\ \\ \\mbox{\\rm and} \\ \\ x \\ge _{\\rm inc} y$ .",
"For instance, for $x = (2, 2, -2, -2)$ and $y = (3, 0, 0, -3)$ we have $x \\le _{\\rm dec} y$ and $x \\ge _{\\rm inc} y$ , but $x \\lnot \\prec y$ since $\\overline{x}=(2,4,2,0)$ and $\\overline{y}=(3,3,3,0)$ .",
"$\\rule {0.17cm}{0.17cm}$ Proposition 2.3 Let $D$ be an arbitrary subset of ${\\bf Z}^{n}$ and assume that $D$ admits a least majorized element.",
"For any $x \\in D$ the following three conditions are equivalent.",
"(A) $x$ is least majorized in $D$ .",
"(B) $x$ is decreasingly minimal in $D$ .",
"(C) $x$ is increasingly maximal in $D$ .",
"(A)$\\rightarrow $ (B) By Proposition REF , a least majorized element is decreasingly minimal.",
"(B)$\\rightarrow $ (A) Take a least majorized element $y$ , which exists by the assumption.",
"By definition we have $\\overline{y} \\le \\overline{x}$ .",
"Since $x \\le _{\\rm dec} y$ , we have either $x{\\downarrow }= y{\\downarrow }$ or there exists an index $k$ with $1 \\le k \\le n$ such that $x{\\downarrow }(i) = y{\\downarrow }(i)$ for $i=1,\\ldots , k-1$ and $x{\\downarrow }(k) < y{\\downarrow }(k)$ .",
"In the latter case we have $\\overline{x}(k) < \\overline{y}(k)$ , which contradicts $\\overline{y} \\le \\overline{x}$ .",
"Therefore we have $x{\\downarrow }= y{\\downarrow }$ , which implies that $x$ is a least majorized element.",
"(A)$\\leftrightarrow $ (C) For any $y \\in D$ , we have $x \\prec y\\iff -x \\prec -y\\iff -x \\le _{\\rm dec} -y\\iff x \\ge _{\\rm inc} y$ by (REF ) and (A)$\\leftrightarrow $ (B) for $(-x,-y)$ ."
],
[
"Majorization in integral base-polyhedra",
"In this section we consider majorization ordering for integer points in an integral base-polyhedron.",
"In discrete convex analysis, the set of the integer points of an integral base-polyhedron is called an M-convex set.",
"The following fundamental fact has long been recognized by experts, though it was difficult for the present authors to identify its origin in the literature (see Remark REF ).",
"Theorem 2.4 The set of the integer points of an integral base-polyhedron admits a least majorized element.",
"$\\rule {0.17cm}{0.17cm}$ This fact can be regarded as a corollary of the following fundamental result of Groenevelt [15], which is already mentioned in Section 6 of Part I [9].",
"Proposition 2.5 (Groenevelt [15]; cf.",
"[12]) Let $B$ be an integral base-polyhedron, $\\overset{....}{B}$ be the set of its integral elements, and $\\Phi (x) = \\sum [\\varphi _{s}(x(s)): s\\in S]$ for $x\\in {\\bf Z}^{S}$ , where $\\varphi _{s}: {\\bf Z}\\rightarrow {\\bf R}\\cup \\lbrace +\\infty \\rbrace $ is a discrete convex function for each $s\\in S$ .",
"An element $m$ of $\\overset{....}{B}$ is a minimizer of $\\Phi (x)$ if and only if $\\varphi _{s}(m(s)+1) + \\varphi _{t}(m(t)-1) \\ge \\varphi _{s}(m(s)) + \\varphi _{t}(m(t))$ whenever $m+\\chi _{s}-\\chi _{t} \\in \\overset{....}{B}$ .",
"$\\rule {0.17cm}{0.17cm}$ Theorem REF can be derived from the combination of Proposition REF with Proposition REF .",
"Let $m\\in \\overset{....}{B}$ be a minimizer of the square-sum $\\sum [x(s)^{2}: s\\in S]$ over $\\overset{....}{B}$ ; note that such $m$ exists.",
"Then, by Proposition REF (only-if part), we have $(m(s)+1)^{2} + (m(t)-1)^{2} \\ge m(s)^{2} + m(t)^{2}$ whenever $m+\\chi _{s}-\\chi _{t} \\in \\overset{....}{B}$ .",
"Here the inequality $(m(s)+1)^{2} + (m(t)-1)^{2} \\ge m(s)^{2} + m(t)^{2}$ is equivalent to $m(s) - m(t) + 1 \\ge 0$ , which implies $\\varphi (m(s)+1) + \\varphi (m(t)-1) \\ge \\varphi (m(s)) + \\varphi (m(t))$ for any discrete convex function $\\varphi : {\\bf Z}\\rightarrow {\\bf R}$ .",
"Therefore, by Proposition REF (if part), $m$ is a minimizer of any symmetric separable convex function $\\sum [\\varphi (x(s)): s\\in S]$ over $\\overset{....}{B}$ .",
"By the equivalence of (i) and (iv) in Proposition REF , this element $m$ is a least majorized element of $\\overset{....}{B}$ .",
"The combination of Theorem REF and Proposition REF implies the following.",
"Theorem 2.6 Let $B$ be an integral base-polyhedron and $\\overset{....}{B}$ be the set of its integral elements.",
"An element $m$ of $\\overset{....}{B}$ is decreasingly minimal if and only if $m$ is least majorized in $\\overset{....}{B}$ .",
"$\\rule {0.17cm}{0.17cm}$ Remark 2.4 In Theorem 3.5 of Part I [9] we have shown that a dec-min element of $\\overset{....}{B}$ has the property (REF ), which is referred to as “min $k$ -largest-sum” in [9].",
"This implies that any dec-min element of $\\overset{....}{B}$ is a least majorized element of $\\overset{....}{B}$ .",
"Since a dec-min element always exists, this theorem also implies the existence of a least majorized element in $\\overset{....}{B}$ .",
"$\\rule {0.17cm}{0.17cm}$ Remark 2.5 A variant of majorization concept, “weak submajorization” (cf., Remark REF ), is investigated for integral g-polymatroids by Tamir [38] and for jump systems by Ando [2].",
"These results are a direct extension of Theorem REF .",
"Therefore, we may safely say that Theorem REF with the above proof was known to experts before 1995.",
"$\\rule {0.17cm}{0.17cm}$"
],
[
"Convex minimization and decreasing minimality",
"In this section we shed the light of discrete convex analysis on the following results obtained in Part I [9].",
"More specifically, we derive these results from the optimality criterion for M-convex functions, which is described in Section REF .",
"Theorem 3.1 ([9]) An element $m$ of $\\overset{....}{B}$ is a dec-min element of $\\overset{....}{B}$ if and only if there is no 1-tightening step for $m$ .",
"$\\rule {0.17cm}{0.17cm}$ Theorem 3.2 ([9]) Let $\\Phi (x) = \\sum [\\varphi (x(s)): s\\in S]$ be a symmetric separable convex function with $\\varphi : {\\bf Z}\\rightarrow {\\bf R}$ .",
"An element $m$ of $\\overset{....}{B}$ is a minimizer of $\\Phi $ if $m$ is a dec-min element of $\\overset{....}{B}$ , and the converse is also true if, in addition, $\\Phi $ is strictly convex.",
"$\\rule {0.17cm}{0.17cm}$ It should be clear in the above that $\\overset{....}{B}$ denotes an M-convex set (the set of integral points of an integral base-polyhedron), and a 1-tightening step for $m\\in \\overset{....}{B}$ means the operation of replacing $m$ to $m+\\chi _{s}-\\chi _{t}$ for some $s, t \\in S$ such that $m(t)\\ge m(s)+2$ and $m+\\chi _{s}-\\chi _{t} \\in \\overset{....}{B}$ ."
],
[
"Convex formulation of decreasing minimality",
"A dec-min element can be characterized as a minimizer of `rapidly increasing' convex function.",
"This characterization enables us to make use of discrete convex analysis in investigating decreasing minimality.",
"We say that a positive-valued function $\\varphi : {\\bf Z}\\rightarrow {\\bf R}$ is $N$ -increasing, where $N>0$ , if $ \\varphi (k+1) \\ge N \\ \\varphi (k) > 0\\qquad (k \\in {\\bf Z}) .$ With the choice of a sufficiently large $N$ , this concept formulates the intuitive notion that $\\varphi $ is “rapidly increasing.” An $N$ -increasing function $\\varphi $ with $N \\ge 2$ is strictly convex, since $\\varphi (k-1) + \\varphi (k+1) > \\varphi (k+1) \\ge N \\varphi (k) \\ge 2 \\varphi (k)$ .",
"As is easily expected, $x <_{\\rm dec} y$ is equivalent to $\\Phi (x) < \\Phi (y)$ defined by such $\\varphi $ , as follows.",
"Proposition 3.3 Assume $|S| \\ge 2$ and that $\\varphi $ is $|S|$ -increasing.",
"A vector $x \\in {\\bf Z}^{S}$ is decreasingly-smaller than a vector $y \\in {\\bf Z}^{S}$ if and only if $\\Phi (x) < \\Phi (y)$ .",
"For $x \\in {\\bf Z}^{S}$ and $k \\in {\\bf Z}$ , let $\\Theta (x,k)$ denote the number of elements $s$ of $S$ with $x(s)=k$ , i.e., $\\Theta (x,k) = |\\lbrace s \\in S : x(s)=k \\rbrace | $ .",
"Then we have $ \\Phi (x) = \\sum _{k} \\Theta (x,k) \\varphi (k) .$ Obviously, $\\Phi (x) = \\Phi (y)$ if $x$ and $y$ are value-equivalent.",
"Suppose that $x$ is not value-equivalent to $y$ , and let $\\hat{k}$ be the largest $k$ with $\\Theta (x,k) \\ne \\Theta (y,k)$ .",
"By definition, $x$ is decreasingly-smaller than $y$ if and only if $\\Theta (x,\\hat{k}) < \\Theta (y, \\hat{k})$ .",
"We show that $\\Theta (x,\\hat{k}) < \\Theta (y,\\hat{k})$ implies $\\Phi (x) < \\Phi (y)$ .",
"Then the converse also follows from this (by exchanging the roles of $x$ and $y$ ).",
"Let $T:= \\sum _{k > \\hat{k}} \\Theta (x,k) \\varphi (k)= \\sum _{k > \\hat{k}} \\Theta (y,k) \\varphi (k) $ .",
"It follows from $\\Phi (x)& = T + \\Theta (x,\\hat{k}) \\varphi (\\hat{k})+ \\sum _{k < \\hat{k}} \\Theta (x,k) \\varphi (k)\\\\& \\le T + \\Theta (x,\\hat{k}) \\varphi (\\hat{k})+ \\varphi (\\hat{k} - 1) \\sum _{k < \\hat{k}} \\Theta (x,k)\\\\& \\le T + \\Theta (x,\\hat{k}) \\varphi (\\hat{k})+ \\varphi (\\hat{k}) \\ \\frac{1}{|S|} \\sum _{k < \\hat{k}} \\Theta (x,k)\\\\& \\le T + ( \\Theta (x,\\hat{k}) + 1) \\varphi (\\hat{k}) ,\\\\\\Phi (y)& = T + \\Theta (y,\\hat{k}) \\varphi (\\hat{k})+ \\sum _{k < \\hat{k}} \\Theta (y,k) \\varphi (k)\\\\& \\ge T + \\Theta (y,\\hat{k}) \\varphi (\\hat{k})$ that $\\Phi (y) - \\Phi (x)& \\ge ( \\Theta (y,\\hat{k}) - \\Theta (x, \\hat{k}) -1 ) \\varphi (\\hat{k})\\ge 0.$ Here we can exclude the possibility of equality.",
"Suppose we have equalities in (REF ).",
"This implies that $\\Theta (y,\\hat{k}) = \\Theta (x, \\hat{k}) + 1$ and that we have equalities throughout (REF ) and (REF ).",
"From (REF ) we obtain $\\sum _{k < \\hat{k}} \\Theta (x,k) = |S|$ , from which $\\Theta (x,k) = 0$ for all $k \\ge \\hat{k}$ .",
"Therefore we have $\\Theta (y,k) = 0$ for all $k > \\hat{k}$ and $\\Theta (y,\\hat{k}) = 1$ .",
"From (), on the other hand, we obtain $\\Theta (y,k) = 0$ for all $k < \\hat{k}$ .",
"This contradicts the relation $\\sum _{k} \\Theta (y,k) = |S| \\ge 2$ .",
"By Proposition REF above, the problem of finding a dec-min element can be recast into a convex minimization problem.",
"It is emphasized that for this equivalence, the underlying set may be any subset of ${\\bf Z}^{S}$ (not necessarily an M-convex set).",
"Proposition 3.4 Let $D$ be an arbitrary subset of ${\\bf Z}^{S}$ , where $|S| \\ge 2$ , and assume that $\\varphi $ is $|S|$ -increasing.",
"An element $m$ of $D$ is decreasingly-minimal in $D$ if and only if it minimizes $\\Phi (x) = \\sum _{s \\in S} \\varphi ( x(s) )$ among all members of $D$ .",
"$\\rule {0.17cm}{0.17cm}$ Remark 3.1 The characterization of a decreasingly-minimal elements as a minimizer of a rapidly increasing convex function in Proposition REF is not particularly new.",
"Similar ideas are scattered in the literature of related topics such as majorization (Marshall–Olkin–Arnold [26]) and shifted optimization (Levin–Onn [25]).",
"$\\rule {0.17cm}{0.17cm}$ Remark 3.2 The relations of being majorized ($\\prec $ ), weakly submajorized ($\\prec _{\\rm w}$ ), and decreasingly-smaller ($\\le _{\\rm dec}$ ) are characterized with reference to different classes of symmetric separable convex functions as follows (Proposition REF , [26], and Proposition REF ): $x \\prec y$ $\\iff $ $\\displaystyle \\sum _{i=1}^{n} \\varphi ( x(i) ) \\le \\sum _{i=1}^{n} \\varphi ( y(i) )$ for all convex $\\varphi $ , $x \\prec _{\\rm w} y$ $\\iff $ $\\displaystyle \\sum _{i=1}^{n} \\varphi ( x(i) ) \\le \\sum _{i=1}^{n} \\varphi ( y(i) )$ for all increasing (nondecreasing) convex $\\varphi $ , $x \\le _{\\rm dec} y$ $\\iff $ $\\displaystyle \\sum _{i=1}^{n} \\varphi ( x(i) ) \\le \\sum _{i=1}^{n} \\varphi ( y(i) )$ for all rapidly increasing convex $\\varphi $ .",
"$\\rule {0.17cm}{0.17cm}$"
],
[
"M-convex function minimization in discrete convex analysis",
"In this section we introduce M-convex functions, a fundamental concept in discrete convex analysis [33], along with a local optimality condition for a minimizer of an M-convex function.",
"Since a separable convex function on an M-convex set is an M-convex function (cf.",
"Section REF ), this optimality criterion renders alternative proofs of Theorems REF and REF about the dec-min elements of an M-convex set (cf.",
"Section REF ).",
"For a vector $z \\in {\\bf R}^{S}$ in general, we define the positive and negative supports of $z$ as $ {\\rm supp}^{+}(z) = \\lbrace s \\in S : z(s) > 0 \\rbrace ,\\qquad {\\rm supp}^{-}(z) = \\lbrace t \\in S : z(t) < 0 \\rbrace .$ For a function $f: {\\bf Z}^{S} \\rightarrow {\\bf R}\\cup \\lbrace -\\infty , +\\infty \\rbrace $ , the effective domain is defined as ${\\rm dom\\,}f = \\lbrace x \\in {\\bf Z}^{S} : -\\infty < f(x) < +\\infty \\rbrace $ .",
"A function $f: {\\bf Z}^{S} \\rightarrow {\\bf R}\\cup \\lbrace +\\infty \\rbrace $ with ${\\rm dom\\,}f \\ne \\emptyset $ is called M-convex if, for any $x, y \\in {\\bf Z}^{S}$ and $s \\in {\\rm supp}^{+}(x-y)$ , there exists some $t \\in {\\rm supp}^{-}(x-y)$ such that $ f(x) + f(y) \\ge f(x-\\chi _{s}+\\chi _{t}) + f(y+\\chi _{s}-\\chi _{t}) .$ In the above statement we may change “for any $x, y \\in {\\bf Z}^{S}$ ” to “for any $x, y \\in {\\rm dom\\,}f$ ” since if $x \\notin {\\rm dom\\,}f$ or $y \\notin {\\rm dom\\,}f$ , (REF ) trivially holds with $f(x) + f(y) = +\\infty $ .",
"We often refer to this defining property as the exchange property of an M-convex function.",
"It follows from this definition that ${\\rm dom\\,}f$ consists of the integer points of an integral base-polyhedron (an M-convex set).",
"A function $f$ is called M-concave if $-f$ is M-convex.",
"We remark that the exchange property (REF ) of an M-convex function is a quantitative extension of the symmetric exchange property of matroid bases.",
"A function $f: {\\bf Z}^{S} \\rightarrow {\\bf R}\\cup \\lbrace +\\infty \\rbrace $ with ${\\rm dom\\,}f \\ne \\emptyset $ is called M$^{\\natural }$ -convex if, for any $x, y \\in {\\bf Z}^{S}$ and $s \\in {\\rm supp}^{+}(x-y)$ , we have (i) $ f(x) + f(y) \\ge f(x - \\chi _{s}) + f(y+\\chi _{s})$ or (ii) there exists some $t \\in {\\rm supp}^{-}(x-y)$ for which (REF ) holds.",
"It follows from this definition that the effective domain of an M$^{\\natural }$ -convex function consists of the integer points of an integral g-polymatroid [8]; such a set is called M$^{\\natural }$ -convex set in DCA.",
"An M-convex function is M$^{\\natural }$ -convex.",
"A function $f$ is called M$^{\\natural }$ -concave if $-f$ is M$^{\\natural }$ -convex.",
"The following is a local characterization of global minimality for M- or M$^{\\natural }$ -convex functions, called the M-optimality criterion.",
"Theorem 3.5 ([33]) Let $f: {\\bf Z}^{S} \\rightarrow {\\bf R}\\cup \\lbrace +\\infty \\rbrace $ be an M$^{\\natural }$ -convex function, and $x^{*} \\in {\\rm dom\\,}f$ .",
"Then $x^{*} $ is a minimizer of $f$ if and only if it is locally minimal in the sense that $& f(x^{*}) \\le f(x^{*} + \\chi _{s} - \\chi _{t}) \\quad \\mbox{\\rm for all } \\ s, t \\in S ,\\\\& f(x^{*}) \\le f(x^{*} + \\chi _{s}) \\quad \\quad \\quad \\mbox{\\rm for all } \\ s \\in S ,\\\\& f(x^{*}) \\le f(x^{*} - \\chi _{t}) \\quad \\quad \\quad \\mbox{\\rm for all } \\ t \\in S .$ If $f$ is M-convex, $x^{*} $ is a minimizer of $f$ if and only if (REF ) holds.",
"$\\rule {0.17cm}{0.17cm}$"
],
[
"Separable convex function minimization in discrete convex analysis",
"Minimization of a separable convex function over the set of integral points of an integral base-polyhedron can be treated successfully as a special case of M-convex function minimization presented in Section REF .",
"We consider a function $\\Phi : {\\bf Z}^{S} \\rightarrow {\\bf R}\\cup \\lbrace +\\infty \\rbrace $ of the form $ \\Phi (x) = \\sum [\\varphi _{s}(x(s)): s\\in S] ,$ where, for each $s \\in S$ , the function $\\varphi _{s}: {\\bf Z}\\rightarrow {\\bf R}\\cup \\lbrace +\\infty \\rbrace $ is discrete convex (i.e., $\\varphi _{s}(k-1) + \\varphi _{s}(k+1) \\ge 2 \\varphi _{s}(k)$ for all $k \\in {\\rm dom\\,}\\varphi _{s}$ ).",
"Such function $\\Phi $ is called a separable (discrete) convex function.",
"We call $\\Phi $ symmetric if $\\varphi _{s}=\\varphi $ for all $s \\in S$ .",
"Let $\\overset{....}{B}$ be the set of integral points of an integral base-polyhedron $B$ .",
"The problem we consider is: $ \\mbox{Minimize } \\ \\Phi (x) = \\sum [\\varphi _{s}(x(s)): s\\in S]\\ \\mbox{ subject to } \\ x \\in \\overset{....}{B}.$ Using the indicator function $\\delta : {\\bf Z}^{S} \\rightarrow {\\bf R}\\cup \\lbrace +\\infty \\rbrace $ of $\\overset{....}{B}$ defined as $ \\delta (x) =\\left\\lbrace \\begin{array}{ll}0 & (x \\in \\overset{....}{B}), \\\\+\\infty & (\\mbox{otherwise}), \\\\\\end{array} \\right.$ we can rewrite (REF ) as $ \\mbox{Minimize } \\ \\Phi (x) + \\delta (x).$ This problem is amenable to discrete convex analysis, since the separable convex function $\\Phi $ is M$^{\\natural }$ -convex, the indicator function $\\delta $ of an M-convex set is M-convex, and moreover, the function $\\Phi + \\delta $ is M-convex.",
"Indeed it is easy to verify that these functions satisfy the defining exchange property.",
"In this connection it is noted that the sum of an M-convex function and an M$^{\\natural }$ -convex function is not necessarily M$^{\\natural }$ -convex, but the sum of an M-convex function and a separable convex function is always M-convex (cf.",
"Remark REF in Section REF ).",
"An application of the M-optimality criterion (Theorem REF ) to our function $\\Phi + \\delta $ gives the important result due to Groenevelt [15] shown in Proposition REF .",
"In the special case of symmetric separable convex functions, with $\\varphi _{s} = \\varphi $ for all $s \\in S$ , we can relate the condition given in Proposition REF to 1-tightening steps.",
"Recall that a 1-tightening step for $m\\in \\overset{....}{B}$ means the operation of replacing $m$ to $m+\\chi _{s}-\\chi _{t}$ for some $s, t \\in S$ such that $m(t)\\ge m(s)+2$ and $m+\\chi _{s}-\\chi _{t} \\in \\overset{....}{B}$ .",
"Proposition 3.6 For any symmetric separable discrete convex function $\\Phi (x) = \\sum [\\varphi (x(s)): s\\in S]$ with $\\varphi : {\\bf Z}\\rightarrow {\\bf R}\\cup \\lbrace +\\infty \\rbrace $ , an element $m$ of $\\overset{....}{B}$ is a minimizer of $\\Phi $ over $\\overset{....}{B}$ if there is no 1-tightening step for $m$ .",
"The converse is also true if $\\varphi $ is strictly convex.",
"By Proposition REF , $m$ is a minimizer of $\\Phi $ if and only if $ \\varphi (m(s)+1) + \\varphi (m(t)-1) \\ge \\varphi (m(s)) + \\varphi (m(t))$ for all $s, t \\in S$ such that $m+\\chi _{s}-\\chi _{t} \\in \\overset{....}{B}$ .",
"By the convexity of $\\varphi $ , we have this inequality if $m(t)\\le m(s)+1$ , and the converse is also true when $\\varphi $ is strictly convex.",
"Finally we note that there is no 1-tightening step for $m$ if and only if $m(t)\\le m(s)+1$ for all $s, t \\in S$ such that $m+\\chi _{s}-\\chi _{t} \\in \\overset{....}{B}$ ."
],
[
"DCA-based proofs of the theorems",
"The combination of Proposition REF with Proposition REF provides alternative proofs of Theorems REF and REF ."
],
[
"Proof of Theorem ",
"Let $\\Phi $ be a symmetric separable convex function with rapidly increasing $\\varphi $ .",
"By Proposition REF , $m$ is dec-min if and only if $m$ is a minimizer of $\\Phi $ .",
"On the other hand, since $\\Phi $ is strictly convex, Proposition REF shows that $m$ is a minimizer of $\\Phi $ if and only if there is no 1-tightening step for $m$ .",
"Therefore, $m$ is a dec-min element of $\\overset{....}{B}$ if and only if there is no 1-tightening step for $m$ ."
],
[
"Proof of Theorem ",
"Let $\\Phi $ be a symmetric separable convex function.",
"By Proposition REF , $m$ is a minimizer of $\\Phi $ if there is no 1-tightening step for $m$ ; and the converse is also true for strictly convex $\\Phi $ .",
"Theorem REF , on the other hand, shows that there is no 1-tightening step for $m$ if and only if $m$ is a dec-min element.",
"Therefore, $m$ is a minimizer of $\\Phi $ if $m$ is a dec-min element of $\\overset{....}{B}$ ; and the converse is also true for strictly convex $\\Phi $ ."
],
[
"Extension to generalized polymatroids",
"In this section we shed a light of DCA on the majorization ordering and decreasing minimality in generalized polymatroids (g-polymatroids).",
"Let $Q$ be an integral g-polymatroid on the ground set $S$ and $\\overset{....}{Q}$ the set of its integral points; see [8] for the basic facts about g-polymatroids.",
"It is shown by Tamir [38] that $\\overset{....}{Q}$ admits a least weakly submajorized element (cf., Remark REF for this terminology).",
"By Remark REF this is equivalent to saying that there exists an element of $\\overset{....}{Q}$ that simultaneously minimizes all symmetric separable functions $\\sum _{s \\in S} \\varphi ( x(s) )$ defined by an increasing discrete convex function $\\varphi $ .",
"A least weakly submajorized element of $\\overset{....}{Q}$ is a decreasingly minimal element of $\\overset{....}{Q}$ (cf., Remark REF ).",
"G-polymatroids fit in the framework of DCA, because the set $\\overset{....}{Q}$ of integral points of an integral g-polymatroid $Q$ is nothing but an M$^{\\natural }$ -convex set, and accordingly, the indicator function of $\\overset{....}{Q}$ is an M$^{\\natural }$ -convex function.",
"An M$^{\\natural }$ -convex set is exactly the projection of an M-convex set, which is a classic result [8], [12] expressed in the language of DCA.",
"See [33] for more about M$^{\\natural }$ -convexity.",
"The M-optimality criterion (Theorem REF ) immediately implies the following generalization of Proposition REF .",
"Proposition 3.7 Let $Q$ be an integral g-polymatroid and $\\overset{....}{Q}$ be the set of its integral elements.",
"An element $m$ of $\\overset{....}{Q}$ is a minimizer of a separable convex function $\\Phi (x) = \\sum [\\varphi _{s}(x(s)): s\\in S]$ over $\\overset{....}{Q}$ if and only if each of the following three conditions holds: $\\varphi _{s}(m(s)+1) + \\varphi _{t}(m(t)-1) \\ge \\varphi _{s}(m(s)) + \\varphi _{t}(m(t))$ whenever $m+\\chi _{s}-\\chi _{t} \\in \\overset{....}{Q}$ , $\\varphi _{s}(m(s)+1) \\ge \\varphi _{s}(m(s))$ whenever $m+\\chi _{s} \\in \\overset{....}{Q}$ , and $\\varphi _{t}(m(t)-1) \\ge \\varphi _{t}(m(t))$ whenever $m-\\chi _{t} \\in \\overset{....}{Q}$ .",
"$\\rule {0.17cm}{0.17cm}$ Proposition REF for a symmetric separable convex function $\\Phi (x) = \\sum [\\varphi (x(s)): s\\in S]$ on base-polyhedra can be adapted to g-polymatroids under the additional assumption of monotonicity of $\\varphi $ .",
"Let $B$ denote the set of minimal elements of an integral g-polymatroid $Q$ , and $\\overset{....}{B}$ the set of integral members of $B$ .",
"When $Q$ is defined by a paramodular pair $(p,b)$ of an integer-valued supermodular function $p$ and an integer-valued submodular function $p$ , it has a minimal element precisely if $p(S)$ is finite [8].",
"That is, $B$ is nonempty if and only if $p(S)$ is finite.",
"If $B \\ne \\emptyset $ , $B$ is an integral base-polyhedron and $\\overset{....}{B}$ is an M-convex set.",
"Note that $\\overset{....}{B} \\ne \\emptyset $ if and only if $B \\ne \\emptyset $ , and $\\overset{....}{B}$ is the set of minimal elements of $\\overset{....}{Q}$ .",
"Proposition 3.8 Let $\\Phi $ be a symmetric separable convex function represented as $\\Phi (x) = \\sum [\\varphi (x(s)): s\\in S]$ with monotone nondecreasing discrete convex $\\varphi : {\\bf Z}\\rightarrow {\\bf R}\\cup \\lbrace +\\infty \\rbrace $ .",
"There exists a minimizer of $\\Phi $ in $\\overset{....}{Q}$ if and only if $\\overset{....}{B}$ is nonempty.",
"An element $m$ of $\\overset{....}{Q}$ is a minimizer of $\\Phi $ over $\\overset{....}{Q}$ if $m$ belongs to $\\overset{....}{B}$ and $m(t)\\le m(s)+1$ whenever $m+\\chi _{s}-\\chi _{t}$ is in $\\overset{....}{B}$ .",
"The converse is also true if $\\varphi $ is strictly convex and strictly monotone increasing.",
"$\\rule {0.17cm}{0.17cm}$ Let $m$ be an element of $\\overset{....}{Q}$ that minimizes $\\Phi (x) = \\sum [\\varphi (x(s)): s\\in S]$ for an arbitrarily chosen strictly convex and strictly monotone increasing $\\varphi $ .",
"Then Proposition REF implies that $m$ is a universal minimizer of all such $\\Phi (x)$ , since the condition $ m+\\chi _{s}-\\chi _{t} \\in \\overset{....}{B} \\ \\Rightarrow \\ m(t)\\le m(s)+1$ is independent of $\\varphi $ .",
"Therefore, $m$ is a least weakly submajorized element of $\\overset{....}{Q}$ .",
"By adapting the above results to decreasing minimality, we see that $\\overset{....}{Q}$ has a dec-min element if and only if $\\overset{....}{B}$ is nonempty, and that a member $m$ of $\\overset{....}{Q}$ is decreasingly minimal in $\\overset{....}{Q}$ if and only if $m \\in \\overset{....}{B}$ and (REF ) holds, which is equivalent, by Theorem 3.3 of Part I, to $m$ being a dec-min element of $\\overset{....}{B}$ ."
],
[
"Min-max formulas",
"Key min-max formulas on discrete decreasing minimization, established by constructive methods in Part I [9], are derived here from the Fenchel-type discrete duality in discrete convex analysis.",
"These formulas can in fact be derived from a special case of the Fenchel-type discrete duality where a separable convex function is minimized over an M-convex set.",
"This special case often provides interesting min-max relations in applications and deserves particular attention.",
"The (general) Fenchel-type discrete duality is described in Section REF and its special case for separable convex functions in Section REF ."
],
[
"Min-max formulas for decreasing minimization",
"In this section we treat the formulas (REF ), (REF ), (REF ), and (REF ) below.",
"Recall that $p$ is an integer-valued (fully) supermodular function on the ground-set $S$ describing a base-polyhedron $B$ and $\\hat{p}$ is the linear extension (Lovász extension) of $p$ , whose definition is given in (REF ) in Section REF .",
"[9] For the square-sum we have $\\min \\lbrace \\sum _{s \\in S} m(s)^{2} : m\\in \\overset{....}{B} \\rbrace = \\max \\lbrace \\hat{p}(\\pi ) - \\sum _{s\\in S}\\left\\lfloor {\\pi (s) \\over 2}\\right\\rfloor \\left\\lceil {\\pi (s) \\over 2}\\right\\rceil :\\pi \\in {\\bf Z}^{S} \\rbrace .$ [9] For the largest component $\\beta _{1}$ of a max-minimizer of $\\overset{....}{B}$ , we have $\\beta _{1}=\\max \\lbrace \\left\\lceil {p(X) \\over \\vert X\\vert } \\right\\rceil :\\emptyset \\ne X\\subseteq S\\rbrace .$ Recall that $\\beta _{1}$ is equal to the largest component of any dec-min element of $\\overset{....}{B}$ .",
"[9] For the minimum number $r_{1}$ of $\\beta _{1}$ -valued components of a $\\beta _{1}$ -covered member of $\\overset{....}{B}$ , we have $r_{1}= \\max \\lbrace p(X) - (\\beta _{1}-1)\\vert X\\vert : X\\subseteq S\\rbrace .$ Recall that $r_{1} = | \\lbrace s \\in S : m(s) = \\beta _{1} \\rbrace |$ for any dec-min element $m$ of $\\overset{....}{B}$ .",
"Moreover, the following min-max formula will be established in Section REF as a generalization of (REF ).",
"We refer to $\\sum _{s \\in S} (m(s) - a)^{+}$ in the minimization below as the total $a$ -excess of $m$ .",
"For each integer $a$ , we have $ \\min \\lbrace \\sum _{s \\in S} (m(s) - a)^{+} : m \\in \\overset{....}{B} \\rbrace = \\max \\lbrace p(X) - a \\vert X\\vert : X\\subseteq S\\rbrace .$ Note that this formula (REF ) for $a=\\beta _{1} - 1$ reduces to the formula (REF ) for $r_{1}$ .",
"It will be shown in Theorem REF that an element of $\\overset{....}{B}$ is decreasingly minimal if and only if it is a minimizer of the left-hand side of (REF ) universally for all $a \\in {\\bf Z}$ .",
"We remark that the minimization problem above is known to be most fundamental in the literature of majorization, whereas the function $p(X) - a \\vert X\\vert $ to be maximized plays the pivotal role in characterizing the canonical partition and the essential value-sequence (cf., Section REF ).",
"Thus the min-max formula (REF ) reinforces the link between the present study and the theory of majorization."
],
[
"Fenchel-type discrete duality in discrete convex analysis",
"In this section we describe an important result in DCA, the Fenchel-type discrete duality theorem, which we use to derive the min-max formulas related to dec-min elements.",
"The Fenchel-type discrete duality theorem in DCA originates in Murota [30] and is formulated for integer-valued functions in [31], [33].",
"For any integer-valued functions $f: {\\bf Z}^{S} \\rightarrow {\\bf Z}\\cup \\lbrace +\\infty \\rbrace $ and $h: {\\bf Z}^{S} \\rightarrow {\\bf Z}\\cup \\lbrace -\\infty \\rbrace $ , we define their (convex and concave) conjugate functions by $f^{\\bullet }(\\pi )&=& \\sup \\lbrace \\langle \\pi , x \\rangle - f(x) : x \\in {\\bf Z}^{S} \\rbrace \\qquad ( \\pi \\in {\\bf Z}^{S}), \\\\h^{\\circ }(\\pi )&=& \\inf \\lbrace \\langle \\pi , x \\rangle - h(x) : x \\in {\\bf Z}^{S} \\rbrace \\qquad ( \\pi \\in {\\bf Z}^{S}),$ where $\\langle \\pi , x \\rangle $ means the (standard) inner product of vectors $\\pi $ and $x$ .",
"Note that both $x$ and $\\pi $ are integer vectors.",
"Since the functions are integer-valued, the supremum in (REF ) is attained if it is finite-valued.",
"Similarly for the infimum in ().",
"Accordingly, we henceforth write “$\\max $ ” and “$\\min $ ” in place of “$\\sup $ ” in (REF ) and “$\\inf $ ” in (), respectively.",
"The Fenchel-type discrete duality is concerned with the relationship between the minimum of $f(x) - h(x)$ over $x \\in {\\bf Z}^{S}$ and the maximum of $h^{\\circ }(\\pi ) - f^{\\bullet }(\\pi )$ over $\\pi \\in {\\bf Z}^{S}$ .",
"By the definition of the conjugate functions in (REF ) and () we have inequalities (called the Fenchel–Young inequalities) $f(x) + f^{\\bullet }(\\pi ) &\\ge & \\langle \\pi , x \\rangle ,\\\\h(x) + h^{\\circ }(\\pi ) &\\le & \\langle \\pi , x \\rangle $ for any $x$ and $\\pi $ , and hence $ f(x) - h(x) \\ge h^{\\circ }(\\pi ) - f^{\\bullet }(\\pi )$ for any $x$ and $\\pi $ .",
"Therefore we have weak duality: $ \\min \\lbrace f(x) - h(x) : x \\in {\\bf Z}^{S} \\rbrace \\ge \\max \\lbrace h^{\\circ }(\\pi ) - f^{\\bullet }(\\pi ) : \\pi \\in {\\bf Z}^{S} \\rbrace .$ It is noted, however, that in this expression using “$\\min $ ” and “$\\max $ ” we do not exclude the possibility of the unbounded case where $\\min \\lbrace \\cdots \\rbrace $ and/or $\\max \\lbrace \\cdots \\rbrace $ are equal to $-\\infty $ or $+\\infty $ (we avoid using “$\\inf $ ” and “$\\sup $ ” for wider audience).",
"Here we note the following.",
"If ${\\rm dom\\,}f \\cap {\\rm dom\\,}h \\ne \\emptyset $ and ${\\rm dom\\,}f^{\\bullet } \\cap {\\rm dom\\,}h^{\\circ } \\ne \\emptyset $ , both $\\min \\lbrace f(x) - h(x) : x \\in {\\bf Z}^{S} \\rbrace $ and $\\max \\lbrace h^{\\circ }(\\pi ) - f^{\\bullet }(\\pi ) : \\pi \\in {\\bf Z}^{S} \\rbrace $ are finite integers and the minimum and the maximum are attained by some $x$ and $\\pi $ since the functions are integer-valued.",
"If ${\\rm dom\\,}f \\cap {\\rm dom\\,}h = \\emptyset $ , we understand (by convention) that the minimum of $f - h$ is equal to $+\\infty $ , that is, $\\min \\lbrace f(x) - h(x) : x \\in {\\bf Z}^{S} \\rbrace =+\\infty $ .",
"If ${\\rm dom\\,}f^{\\bullet } \\cap {\\rm dom\\,}h^{\\circ } = \\emptyset $ , we understand (by convention) that the maximum of $h^{\\circ } - f^{\\bullet }$ is equal to $-\\infty $ , that is, $\\max \\lbrace h^{\\circ }(\\pi ) - f^{\\bullet }(\\pi ) : \\pi \\in {\\bf Z}^{S} \\rbrace =-\\infty $ .",
"We say that strong duality holds if equality holds in (REF ).",
"The strong duality does hold for a pair of an M$^{\\natural }$ -convex function $f$ and an M$^{\\natural }$ -concave function $h$ , as the following theorem shows.",
"This is called the Fenchel-type discrete duality theorem [31], [33].",
"To be more precise, we need to assume that at lease one of the following two conditions is satisfied: (i) there exists $x$ for which both $f(x)$ and $h(x)$ are finite (primal feasibility, ${\\rm dom\\,}f \\cap {\\rm dom\\,}h \\ne \\emptyset $ ), (ii) there exists $\\pi $ for which both $f^{\\bullet }(\\pi )$ and $h^{\\circ }(\\pi )$ are finite (dual feasibility, ${\\rm dom\\,}f^{\\bullet } \\cap {\\rm dom\\,}h^{\\circ } \\ne \\emptyset $ ).",
"Note that these two feasibility conditions, (i) and (ii), are mutually independent, and there is an example for which both conditions fail simultaneously [33].",
"Theorem 4.1 (Fenchel-type discrete duality theorem [31], [33]) Let $f: {\\bf Z}^{S} \\rightarrow {\\bf Z}\\cup \\lbrace +\\infty \\rbrace $ be an integer-valued M$^{\\natural }$ -convex function and $h: {\\bf Z}^{S} \\rightarrow {\\bf Z}\\cup \\lbrace -\\infty \\rbrace $ be an integer-valued M$^{\\natural }$ -concave function such that ${\\rm dom\\,}f \\cap {\\rm dom\\,}h \\ne \\emptyset $ or ${\\rm dom\\,}f^{\\bullet } \\cap {\\rm dom\\,}h^{\\circ } \\ne \\emptyset $ .",
"Then we have $ \\min \\lbrace f(x) - h(x) : x \\in {\\bf Z}^{S} \\rbrace = \\max \\lbrace h^{\\circ }(\\pi ) - f^{\\bullet }(\\pi ) : \\pi \\in {\\bf Z}^{S} \\rbrace .$ This common value is finite if and only if ${\\rm dom\\,}f \\cap {\\rm dom\\,}h \\ne \\emptyset $ and ${\\rm dom\\,}f^{\\bullet } \\cap {\\rm dom\\,}h^{\\circ } \\ne \\emptyset $ , and then the minimum and the maximum are attained.",
"$\\rule {0.17cm}{0.17cm}$ The essential content of the above theorem may be expressed as follows: If $f(x) - h(x)$ is bounded from below, then $h^{\\circ }(\\pi ) - f^{\\bullet }(\\pi )$ is bounded from above, and the minimum of $f(x) - h(x)$ and the maximum of $h^{\\circ }(\\pi ) - f^{\\bullet }(\\pi )$ coincide.",
"Remark 4.1 The Fenchel-type duality theorem is the central duality theorem in discrete convex analysis.",
"The duality phenomenon captured by this theorem can be formulated in several different, mutually equivalent, forms including the M-separation theorem [33], the L-separation theorem [33], and the M-convex intersection theorem [33].",
"These duality theorems include a number of important results as special cases such as Edmonds' intersection theorem, Fujishige's Fenchel-type duality theorem [12] for submodular set functions, the discrete separation theorem [8] for submodular/supermodular functions, and the weight splitting theorem [8] for the weighted matroid intersection problem.",
"See [33] for this relationship.",
"$\\rule {0.17cm}{0.17cm}$ Remark 4.2 The Fenchel-type discrete duality theorem offers an optimality certificate for the minimization problem of $f(x) - h(x)$ .",
"Two cases are to be distinguished.",
"If the explicit forms of the conjugate functions $f^{\\bullet }(\\pi )$ and $h^{\\circ }(\\pi )$ are known, we can easily evaluate the value of $h^{\\circ }(\\pi ) - f^{\\bullet }(\\pi )$ for any integer vector $\\pi $ .",
"Given an integral vector $\\pi $ as a certificate of optimality for an allegedly optimal $x$ , we only have to compute the values of $f(x) - h(x)$ and $h^{\\circ }(\\pi ) - f^{\\bullet }(\\pi )$ and compare the two values (integers) for their equality.",
"Thus the availability of explicit forms of the conjugate functions is computationally convenient as well as intuitively appealing.",
"The min-max formula (REF ) for the square-sum minimization over an M-convex set falls into this case.",
"Even if explicit forms of the conjugate functions are not available, the Fenchel-type discrete duality theorem offers a computationally efficient (polynomial-time) method for verifying the optimality if it is combined with the M-optimality criterion (Theorem REF ).",
"We shall discuss this method in Section REF ; see Remark REF .",
"$\\rule {0.17cm}{0.17cm}$ The conjugate of an M$^{\\natural }$ -convex function is endowed with another kind of discrete convexity, called L$^{\\natural }$ -convexity.",
"A function $g: {\\bf Z}^{S} \\rightarrow {\\bf R}\\cup \\lbrace +\\infty \\rbrace $ with ${\\rm dom\\,}g \\ne \\emptyset $ is called L$^{\\natural }$ -convex if it satisfies the inequality $ g(\\pi ) + g(\\tau ) \\ge g \\left(\\left\\lceil \\frac{\\pi +\\tau }{2} \\right\\rceil \\right)+ g \\left(\\left\\lfloor \\frac{\\pi +\\tau }{2} \\right\\rfloor \\right)\\qquad (\\pi , \\tau \\in {\\bf Z}^{S}) ,$ where, for $z \\in {\\bf R}$ in general, $\\left\\lceil z \\right\\rceil $ denotes the smallest integer not smaller than $z$ (rounding-up to the nearest integer) and $\\left\\lfloor z \\right\\rfloor $ the largest integer not larger than $z$ (rounding-down to the nearest integer), and this operation is extended to a vector by componentwise applications.",
"The property (REF ) is referred to as discrete midpoint convexity.",
"A function $g$ is called L$^{\\natural }$ -concave if $-g$ is L$^{\\natural }$ -convex.",
"The following is a local characterization of global maximality for L$^{\\natural }$ -concave functions, called the L-optimality criterion (concave version).",
"Theorem 4.2 ([33]) Let $g: {\\bf Z}^{S} \\rightarrow {\\bf R}\\cup \\lbrace -\\infty \\rbrace $ be an L$^{\\natural }$ -concave function, and $\\pi ^{*} \\in {\\rm dom\\,}g$ .",
"Then $\\pi ^{*} $ is a maximizer of $g$ if and only if it is locally maximal in the sense that $& g(\\pi ^{*}) \\ge g(\\pi ^{*} - \\chi _{Y}) \\quad \\mbox{\\rm for all } \\ Y \\subseteq S ,\\\\& g(\\pi ^{*}) \\ge g(\\pi ^{*} + \\chi _{Y}) \\quad \\mbox{\\rm for all } \\ Y \\subseteq S .$ $\\rule {0.17cm}{0.17cm}$ The reader is referred to [33] for more properties of L$^{\\natural }$ -convex functions and [33] for the conjugacy between M$^{\\natural }$ -convexity and L$^{\\natural }$ -convexity.",
"In particular, [33] offers the whole picture of conjugacy relationship.",
"Remark 4.3 In Theorem REF the functions $f(x)$ and $-h(x)$ are both M$^{\\natural }$ -convex, but the function $f(x) - h(x)$ to be minimized on the left-hand side of (REF ) is not necessarily M$^{\\natural }$ -convex, since the sum of M$^{\\natural }$ -convex functions may not be M$^{\\natural }$ -convex.",
"To see this, consider two M-convex sets $\\overset{....}{B}_{1}$ and $\\overset{....}{B}_{2}$ associated with integral base-polyhedra $B_{1}$ and $B_{2}$ , respectively, and for $i=1,2$ , let $f_{i}$ be the indicator function of $\\overset{....}{B}_{i}$ (i.e., $f_{i}(x) = 0$ if $x \\in \\overset{....}{B}_{i}$ , and $f_{i}(x) = +\\infty $ if $x \\in {\\bf Z}^{S} \\setminus \\overset{....}{B}_{i}$ ).",
"The function $f_{1}+f_{2}$ is the indicator function of the set of integer points in the intersection $B_{1} \\cap B_{2}$ , which is not a base-polyhedron in general.",
"This argument also shows that the left-hand side of (REF ) is a nonlinear generalization of the weighted polymatroid intersection problem; see [33] for details.",
"$\\rule {0.17cm}{0.17cm}$ Remark 4.4 Functions $h^{\\circ }(\\pi )$ and $f^{\\bullet }(\\pi )$ in Theorem REF are L$^{\\natural }$ -concave and L$^{\\natural }$ -convex, respectively.",
"Since the sum of L$^{\\natural }$ -concave functions is L$^{\\natural }$ -concave, the function $h^{\\circ }(\\pi ) - f^{\\bullet }(\\pi )$ to be maximized on the right-hand side of (REF ) is an L$^{\\natural }$ -concave function.",
"In contrast, the function $f(x) - h(x)$ to be minimized on the left-hand side of (REF ) is not an M$^{\\natural }$ -convex function, as explained in Remark REF above.",
"In this sense, the left-hand side (minimization) and the right-hand side (maximization) are not symmetric.",
"$\\rule {0.17cm}{0.17cm}$"
],
[
"Min-max formula for separable convex functions on an M-convex set",
"In this section the Fenchel-type discrete duality theorem is tailored to the problem of minimizing a separable convex function over an M-convex set.",
"This special case deserves particular attention as it is suitable and sufficient for our use in decreasing minimization.",
"Consider the problem of minimizing an integer-valued separable convex function $ \\Phi (x) = \\sum [\\varphi _{s}(x(s)): s\\in S]$ over an M-convex set $\\overset{....}{B}$ , where each $\\varphi _{s}: {\\bf Z}\\rightarrow {\\bf Z}\\cup \\lbrace +\\infty \\rbrace $ is an integer-valued discrete convex function in a single integer variable.",
"This problem is equivalent to minimizing $\\Phi (x) + \\delta (x)$ , where $\\delta $ denotes the indicator function of $\\overset{....}{B}$ defined in (REF ).",
"In Section REF we have regarded the function $\\Phi + \\delta $ as an M-convex function and applied the M-optimality criterion to derive some results obtained in Part I [9].",
"In contrast, we are now going to apply the Fenchel-type discrete duality theorem to the minimization of the function $\\Phi + \\delta = \\Phi - (-\\delta )$ .",
"In so doing we can separate the roles of the constraining M-convex set and the objective function $\\Phi (x)$ itself.",
"With the choice of $f = \\Phi $ and $h= -\\delta $ in the min-max relation $\\min \\lbrace f(x) - h(x) \\rbrace = \\max \\lbrace h^{\\circ }(\\pi ) - f^{\\bullet }(\\pi ) \\rbrace $ in (REF ), the left-hand side represents minimization of $\\Phi $ over the M-convex set $\\overset{....}{B}$ .",
"We denote the conjugate function of $\\varphi _{s}$ by $\\psi _{s}$ , which is a function $\\psi _{s}: {\\bf Z}\\rightarrow {\\bf Z}\\cup \\lbrace +\\infty \\rbrace $ defined by $ \\psi _{s}(\\ell ) = \\max \\lbrace k \\ell - \\varphi _{s}(k) : k \\in {\\bf Z}\\rbrace \\qquad (\\ell \\in {\\bf Z}).$ Then the conjugate function of $f$ is given by $f^{\\bullet }(\\pi ) = \\sum [\\psi _{s}(\\pi (s)): s\\in S]\\qquad ( \\pi \\in {\\bf Z}^{S}) .$ On the other hand, the conjugate function $h^{\\circ }$ of $h$ is given by $h^{\\circ }(\\pi )= \\min \\lbrace \\langle \\pi , x \\rangle + \\delta (x) : x \\in {\\bf Z}^{S} \\rbrace =\\min \\lbrace \\langle \\pi , x \\rangle : x \\in \\overset{....}{B} \\rbrace =\\hat{p}(\\pi )\\quad ( \\pi \\in {\\bf Z}^{S})$ in terms of the linear extension (Lovász extension) $\\hat{p}$ of $p$ .",
"Recall that, for any set function $p$ , $\\hat{p}$ is defined [9] as $ \\hat{p}(\\pi ) = p(I_n)\\pi (s_n)+ \\sum _{j=1}^{n-1} p(I_j)[\\pi (s_j)-\\pi (s_{j+1})] ,$ where $n = |S|$ , the elements of $S$ are indexed in such a way that $\\pi (s_1)\\ge \\cdots \\ge \\pi (s_n)$ , and $I_j=\\lbrace s_1,\\dots ,s_j\\rbrace $ for $j=1,\\dots ,n$ .",
"If $p$ is supermodular, we have $ \\hat{p}(\\pi ) = \\min \\lbrace \\pi x : x \\in \\overset{....}{B} \\rbrace .$ Substituting (REF ) and (REF ) into (REF ) we obtain (REF ) below.",
"Theorem 4.3 Assume that (i) there exists $x \\in \\overset{....}{B}$ such that $\\varphi _{s}(x(s)) < +\\infty $ for all $s \\in S$ (primal feasibility) or (ii) there exists $\\pi \\in {\\bf Z}^{S}$ such that $\\hat{p}(\\pi ) > -\\infty $ and $\\psi _{s}(\\pi (s)) < +\\infty $ for all $s \\in S$ (dual feasibility).",
"Then we have the min-max relation: $ \\min \\lbrace \\sum _{s \\in S} \\varphi _{s} ( x(s) ) : x \\in \\overset{....}{B} \\rbrace = \\max \\lbrace \\hat{p}(\\pi ) - \\sum _{s \\in S} \\psi _{s}(\\pi (s)) : \\pi \\in {\\bf Z}^{S} \\rbrace .$ The unbounded case with both sides being equal to $-\\infty $ or $+\\infty $ is also a possibility.",
"$\\rule {0.17cm}{0.17cm}$ Since $\\hat{p}(\\pi )$ is an L$^{\\natural }$ -concave function and $\\sum [\\psi _{s}(\\pi (s)): s\\in S]$ is an L$^{\\natural }$ -convex function, the function $g(\\pi ) := \\hat{p}(\\pi ) - \\sum [\\psi _{s}(\\pi (s)): s\\in S]$ to be maximized on the right-hand side of (REF ) is an L$^{\\natural }$ -concave function (cf.",
"Remark REF ).",
"We state this as a proposition for later reference.",
"Proposition 4.4 The function $g(\\pi ) = \\hat{p}(\\pi ) - \\sum [\\psi _{s}(\\pi (s)): s\\in S]$ is L$^{\\natural }$ -concave.",
"$\\rule {0.17cm}{0.17cm}$ When specialized to a symmetric function $\\Phi $ , the min-max formula (REF ) is simplified to $ \\min \\lbrace \\sum _{s\\in S} \\varphi (x(s)) : x \\in \\overset{....}{B} \\rbrace = \\max \\lbrace \\hat{p}(\\pi ) - \\sum _{s\\in S} \\psi (\\pi (s)) : \\pi \\in {\\bf Z}^{S} \\rbrace ,$ where $\\varphi : {\\bf Z}\\rightarrow {\\bf Z}\\cup \\lbrace +\\infty \\rbrace $ is any integer-valued discrete convex function and $\\psi : {\\bf Z}\\rightarrow {\\bf Z}\\cup \\lbrace +\\infty \\rbrace $ is the conjugate of $\\varphi $ defined as $\\psi (\\ell ) = \\max \\lbrace k \\ell - \\varphi (k) : k \\in {\\bf Z}\\rbrace $ for $\\ell \\in {\\bf Z}$ .",
"With appropriate choices of $\\varphi $ in (REF ) we shall derive the formulas (REF ), (REF ), and (REF ).",
"In applications of (REF ) (resp., (REF )) with concrete functions $\\varphi _{s}$ (resp., $\\varphi $ ), it is often the case that the conjugate functions $\\psi _{s}$ (resp., $\\psi $ ) can be given explicitly.",
"This is illustrated in Section REF ."
],
[
"DCA-based proof of the min-max formula for the square-sum",
"The min-max formula (REF ) for the square-sum can be derived immediately from our duality formula (REF ).",
"For $\\varphi (k)=k^{2}$ , the conjugate function $\\psi (\\ell )$ for $\\ell \\in {\\bf Z}$ is given explicitly as $\\psi (\\ell )= \\max \\lbrace k \\ell - k^{2} : k \\in {\\bf Z}\\rbrace = \\max \\lbrace k \\ell - k^{2}: k \\in \\lbrace \\left\\lfloor {\\ell }/{2} \\right\\rfloor ,\\left\\lceil {\\ell }/{2} \\right\\rceil \\rbrace \\rbrace = \\left\\lfloor \\frac{\\ell }{2} \\right\\rfloor \\cdot \\left\\lceil \\frac{\\ell }{2} \\right\\rceil .$ The substitution of (REF ) into (REF ) yields (REF ).",
"Note that the primal feasibility is satisfied since $\\varphi (k)$ is finite for all $k$ .",
"Remark 4.5 In Part I [9] we provided a relatively simple algorithmic proof for the min-max formula (REF ), which did not use any tool from DCA.",
"However, to figure out the min-max formula itself without the DCA background seems rather difficult.",
"Indeed, the present authors first identified the formula (REF ) via DCA as above, and then came up with the algorithmic proof.",
"This example demonstrates the role and effectiveness of DCA.",
"$\\rule {0.17cm}{0.17cm}$ Remark 4.6 In Part I [9] we have characterized a dec-min element as a square-sum minimizer and also as a difference-sum minimizer.",
"Whereas a min-max formula can be obtained by DCA for the square-sum, this is not the case with the difference-sum.",
"This is because the difference-sum is not M$^{\\natural }$ -convex (though it is L-convex), and therefore difference-sum minimization over an M-convex set does not fit into the framework of the Fenchel-type discrete duality in DCA.",
"$\\rule {0.17cm}{0.17cm}$"
],
[
"DCA-based proof of the formula for $\\beta _{1}$",
"The formula (REF ) for the largest component $\\beta _{1}$ of a max-minimizer of $\\overset{....}{B}$ can also be derived from our duality formula (REF ).",
"With an integer parameter $\\alpha $ we choose $\\varphi (k) =\\left\\lbrace \\begin{array}{ll}0 & (k \\le \\alpha ) , \\\\+\\infty & (k \\ge \\alpha + 1) \\\\\\end{array} \\right.$ in (REF ).",
"By the definition of $\\beta _{1}$ , the left-hand side of (REF ) is equal to zero if $\\alpha \\ge \\beta _{1}$ , and equal to $+\\infty $ if $\\alpha \\le \\beta _{1} -1$ .",
"Hence $\\beta _{1}$ is equal to the minimum of $\\alpha $ for which the left-hand side is equal to zero.",
"The conjugate function $\\psi $ of $\\varphi $ is given by $\\psi (\\ell ) =\\max \\lbrace k \\ell : k \\le \\alpha \\rbrace =\\left\\lbrace \\begin{array}{ll}+\\infty & (\\ell \\le - 1) , \\\\0 & (\\ell = 0) , \\\\\\alpha \\ell & (\\ell \\ge 1) .",
"\\\\\\end{array} \\right.$ Both $\\hat{p}(\\pi )$ and $\\psi (\\ell )$ are positively homogeneous (i.e., $\\hat{p}(\\lambda \\pi ) = \\lambda \\hat{p}(\\pi )$ and $\\psi (\\lambda \\ell )= \\lambda \\psi (\\ell )$ for nonnegative integers $\\lambda $ ).",
"This implies, in particular, that the maximization problem on the right-hand side of (REF ) is feasible for all $\\alpha $ and hence the identity (REF ) holds, which reads either $0 = 0$ or $+\\infty = +\\infty $ .",
"Since $\\beta _{1}$ is the minimum of $\\alpha $ for which the left-hand side is equal to 0, we can say that $\\beta _{1}$ is the minimum of $\\alpha $ for which the right-hand side is equal to 0.",
"Finally, we consider the condition that ensures $\\pi ^{*} = {\\bf 0}$ to be a maximizer of the function $g(\\pi ):= \\hat{p} (\\pi ) - \\sum _{s \\in S} \\psi (\\pi (s))$ .",
"By the L$^{\\natural }$ -concavity of this function we can make use of Theorem REF (L-optimality criterion).",
"The first condition (REF ) in Theorem REF is satisfied trivially by (REF ), whereas the second condition () reads $g(\\pi ^{*} + \\chi _{Y}) = p(Y) -\\alpha |Y| \\le 0$ .",
"Therefore, the right-hand side of (REF ) is equal to zero if and only if $\\max \\lbrace p(Y) - \\alpha | Y | : Y \\subseteq S\\rbrace = 0$ , from which follows the formula (REF )."
],
[
"DCA-based proof of the formula for $r_{1}$",
"The formula (REF ) for the minimum number $r_{1}$ of $\\beta _{1}$ -valued components of a $\\beta _{1}$ -covered member of $\\overset{....}{B}$ can also be derived from our duality formula (REF ).",
"We choose $ \\varphi (k) =\\left\\lbrace \\begin{array}{ll}0 & (k \\le \\beta _{1} - 1) , \\\\1 & (k = \\beta _{1}) , \\\\+\\infty & (k \\ge \\beta _{1} + 1) , \\\\\\end{array} \\right.$ whose graph is given by the left of Fig.",
"REF .",
"By the definitions of $\\beta _{1}$ and $r_{1}$ , the minimum in (REF ) is equal to $r_{1}$ .",
"In particular, the primal problem is feasible, and hence the identity (REF ) holds.",
"Figure: Mutually conjugate discrete convex functions ϕ\\varphi and ψ\\psi in () and ()The conjugate function $\\psi $ of $\\varphi $ is given by $\\psi (\\ell ) &=&\\max \\big \\lbrace \\ \\max \\lbrace k \\ell : k \\le \\beta _{1} -1 \\rbrace , \\ \\ \\beta _{1} \\ell -1\\ \\big \\rbrace \\nonumber \\\\&=&\\left\\lbrace \\begin{array}{ll}+\\infty & (\\ell \\le - 1) , \\\\0 & (\\ell = 0) , \\\\\\beta _{1} \\ell -1 & (\\ell \\ge 1) , \\\\\\end{array} \\right.$ whose graph is given by the right of Fig.",
"REF .",
"In considering the maximum of $g(\\pi ):=\\hat{p} (\\pi ) - \\sum _{s \\in S} \\psi (\\pi (s))$ over $\\pi \\in {\\bf Z}^{S}$ , we may restrict $\\pi $ to $\\lbrace 0,1 \\rbrace $ -vectors, as shown in Lemma REF below.",
"For $\\pi = \\chi _{X} \\in \\lbrace 0, 1 \\rbrace ^{S}$ with $X \\subseteq S$ , we have $\\hat{p}(\\pi ) = \\hat{p}(\\chi _{X}) = p(X)$ and $\\sum _{s \\in S} \\psi (\\pi (s)) = \\sum _{s \\in S} \\psi (\\chi _{X}(s))= \\sum _{s \\in X} \\psi (1) = (\\beta _{1} -1) |X|$ , and therefore, the right-hand side of (REF ) is equal to $\\max \\lbrace p(X) - (\\beta _{1} -1)\\vert X\\vert : X\\subseteq S\\rbrace $ .",
"Thus the formula (REF ) is derived.",
"Lemma 4.5 There exists a $\\lbrace 0,1 \\rbrace $ -vector $\\pi $ that attains the maximum of $g(\\pi )$ over $\\pi \\in {\\bf Z}^{S}$ .",
"Note first that $g$ is an L$^{\\natural }$ -concave function, and define $a = \\beta _{1} - 1$ .",
"Let $A \\subseteq S$ be a maximizer of $p(X) - a \\vert X\\vert $ over all subsets of $S$ , and $\\pi ^{*} = \\chi _{A}$ .",
"Then $g(\\pi ^{*}) = p(A) - a \\vert A \\vert $ .",
"We will show that the conditions (REF ) and () in the L-optimality criterion (Theorem REF ) are satisfied.",
"Proof of $g(\\pi ^{*}) \\ge g(\\pi ^{*} - \\chi _{Y})$ in (REF ): We may assume $Y \\subseteq A$ , since, otherwise, $\\pi ^{*} - \\chi _{Y} \\notin {\\rm dom\\,}g$ by (REF ).",
"If $Y \\subseteq A$ , we have $\\pi ^{*} - \\chi _{Y} = \\chi _{A \\setminus Y} = \\chi _{Z}$ , where $Z = A \\setminus Y$ .",
"Hence, $g(\\pi ^{*} - \\chi _{Y}) = g(\\chi _{Z}) = p(Z) - a \\vert Z \\vert \\le p(A) - a \\vert A \\vert = g(\\pi ^{*}).$ Proof of $g(\\pi ^{*}) \\ge g(\\pi ^{*} + \\chi _{Y})$ in (): Since $(\\pi ^{*} + \\chi _{Y})(s) = (\\chi _{A} + \\chi _{Y})(s) =\\left\\lbrace \\begin{array}{ll}2 & (s \\in A \\cap Y) , \\\\1 & (s \\in (A \\cup Y) \\setminus (A \\cap Y)) , \\\\0 & (s \\in S \\setminus (A \\cup Y)) , \\\\\\end{array} \\right.$ we have $\\hat{p}(\\pi ^{*} + \\chi _{Y})&= p( A \\cap Y ) + p( A \\cup Y ),\\\\\\sum _{s\\in S} \\psi ((\\pi ^{*} + \\chi _{Y})(s))& =(2 \\beta _{1} -1) |A \\cap Y| + (\\beta _{1} -1) |(A \\cup Y) \\setminus (A \\cap Y)|=\\beta _{1} |A \\cap Y| + a |A \\cup Y|$ by the definition (REF ) of $\\hat{p}$ and the expression (REF ) of the conjugate function $\\psi $ .",
"Hence $g(\\pi ^{*} + \\chi _{Y})&=\\big ( p( A \\cap Y ) + p( A \\cup Y ) \\big )- \\big ( \\beta _{1} |A \\cap Y| + a |A \\cup Y| \\big )\\\\&=\\big ( p( A \\cap Y ) - \\beta _{1} |A \\cap Y| \\big )+ \\big ( p( A \\cup Y ) - a |A \\cup Y| \\big ).$ Here we have $& p( A \\cap Y )- \\beta _{1} |A \\cap Y| \\le 0,\\\\ &p( A \\cup Y ) - a |A \\cup Y| \\le p( A ) - a |A| = g(\\pi ^{*}) ,$ since $(\\beta _{1}, \\beta _{1}, \\ldots , \\beta _{1})$ belongs to the supermodular polyhedra defined by $p$ , $A$ is a maximizer of $p(X) - a \\vert X\\vert $ , and $p( A ) - a |A| = g(\\pi ^{*})$ .",
"Therefore, $g(\\pi ^{*} + \\chi _{Y}) \\le g(\\pi ^{*})$ ."
],
[
"Total $a$ -excess and decreasing minimality",
"In this section, we explore a link between decreasing minimality and the total $a$ -excess announced at the beginning of Section .",
"The minimization problem in (REF ) (or (REF ) below) is most fundamental in the literature of majorization.",
"Indeed, a least majorized element is characterized as a universal minimizer for all $a \\in {\\bf Z}$ (Proposition REF ).",
"On the other hand, the function $p(X) - a \\vert X\\vert $ to be maximized plays the pivotal role in characterizing the canonical partition and the essential value-sequence (cf., Section REF ).",
"As a preparation, we recall ([8], [37]) that, for a nonnegative and (fully) supermodular function $p_{0}$ , the polyhedron $C= \\lbrace x: \\widetilde{x}\\ge p_{0}\\rbrace $ is called a contra-polymatroid.",
"Note that the nonnegativity and supermodularity of $p_{0}$ imply that $p_{0}$ is monotone non-decreasing and that $C\\subseteq {\\bf R}_{+}^{S}$ , that is, every member of $C$ is a nonnegative vector.",
"When $p_{0}$ is integer-valued, $C$ is an integer polyhedron.",
"The corresponding version of Edmonds' greedy algorithm for polymatroids implies that $C$ uniquely determines $p_{0}$ , namely, $p_{0}(X)= \\min \\lbrace \\ \\widetilde{z}(X): z\\in C\\rbrace .$ It is known that, for a supermodular function $p_{1}$ with possibly negative values, the polyhedron $C(p_{1}) := \\lbrace x: x\\ge \\mathbf {0}, \\ \\widetilde{x}\\ge p_{1}\\rbrace $ is a contra-polymatroidIn the literature, (REF ) is used sometimes as the definition of a contra-polymatroid., for which the unique nonnegative supermodular bounding function $p_{0}$ is given by $p_{0}(X)= \\max \\lbrace p_{1}(Y) : Y\\subseteq X\\rbrace .$ It follows from (REF ), (REF ), and the integrality of the polyhedron $C(p_{1})$ that $\\min \\lbrace \\ \\widetilde{z}(S) : z \\in \\overset{....}{C}(p_{1})\\rbrace = \\max \\lbrace p_{1}(X) : X\\subseteq S \\rbrace ,$ where $\\overset{....}{C}(p_{1})$ denotes the set of the integral members of $C(p_{1})$ .",
"Lemma 4.6 Let $B=B^{\\prime }(p)$ be an (integral) base-polyhedron defined by an integer-valued supermodular function $p$ .",
"For a vector $g:S\\rightarrow {\\bf Z}$ , $\\min \\lbrace \\sum [ (m(s) - g(s))^+ : s\\in S] : m\\in \\overset{....}{B} \\rbrace = \\max \\lbrace p(X) - \\widetilde{g}(X): X\\subseteq S\\rbrace .$ It is known (for example, from the discrete separation theorem for submodular set functions or from a version of Edmonds' polymatroid intersection theorem) that, for a function $g^{\\prime }:S\\rightarrow {\\bf Z}$ , there is an element $m\\in \\overset{....}{B}$ for which $m\\le g^{\\prime }$ if and only if $p\\le \\widetilde{g}^{\\prime }$ .",
"Therefore the minimization problem on the left-hand side of (REF ) is equivalent to finding a lowest lifting $g^{\\prime }:=g+ z$ of $g$ with $z\\ge \\mathbf {0}$ such that $p \\le \\widetilde{g}^{\\prime }$ .",
"That is, the minimum on the left-hand side of (REF ) is equal to $\\min \\lbrace \\ \\widetilde{z}(S) :z\\ge \\mathbf {0}, \\ \\widetilde{z} \\ge p - \\widetilde{g} \\ \\rbrace $ .",
"By applying (REF ) to $p_{1}:= p - \\widetilde{g}$ , we obtain that this latter minimum is indeed equal to the right-hand side of (REF ).",
"The following theorem reinforces the link between the present study and the theory of majorization.",
"Theorem 4.7 Let $B$ be a base-polyhedron described by an integer-valued supermodular function $p$ and $\\overset{....}{B}$ the set of integral elements of $B$ .",
"For each integer $a$ , we have the following min-max relation for the minimum of the total $a$ -excess of the members of $\\overset{....}{B}$ : $ \\min \\lbrace \\sum _{s \\in S} (m(s) - a)^{+} : m \\in \\overset{....}{B} \\rbrace = \\max \\lbrace p(X) - a \\vert X\\vert : X\\subseteq S\\rbrace .$ Moreover, an element of $\\overset{....}{B}$ is a dec-min element of $\\overset{....}{B}$ if and only if it is a minimizer on the left-hand side for every $a \\in {\\bf Z}$ .",
"The min-max formula (REF ) follows from Lemma REF as it is a special case of (REF ) when $g=(a,a,\\dots ,a)$ .",
"Theorem REF shows that any dec-min element of $\\overset{....}{B}$ is a minimizer in (REF ) for every $a \\in {\\bf Z}$ .",
"The converse is also true, since $\\sum [(x(s) - a)^{+}: s \\in S] = \\sum [(y(s) - a)^{+}: s \\in S]$ for every $a \\in {\\bf Z}$ implies $x{\\downarrow }= y{\\downarrow }$ .",
"Therefore, an element of $\\overset{....}{B}$ is dec-min if and only if it is a universal minimizer for every $a \\in {\\bf Z}$ .",
"The established formula (REF ) generalizes the formula (REF ) for $r_{1}$ .",
"Indeed, the total $a$ -excess for $a=\\beta _{1} - 1$ is given as $\\sum _{s \\in S} (m(s) - a)^{+} = \\sum _{s \\in S} (m(s) - (\\beta _{1} - 1))^{+} =| \\lbrace s \\in S : m(s) = \\beta _{1} \\rbrace | = r_{1}$ for any dec-min element $m$ of $\\overset{....}{B}$ .",
"For any dec-min element $m$ of $\\overset{....}{B}$ and for $k=\\beta _{1}, \\beta _{1}-1, \\beta _{1}-2, \\ldots $ , let $\\Theta (m,k)$ denote the number of components of $m$ whose value are equal to $k$ , that is, $\\Theta (m,k)= | \\lbrace s \\in S : m(s) = k \\rbrace | .$ Note that $\\Theta (m,\\beta _{1}) = r_{1}$ and $\\Theta (m,k)$ does not depend on the choice of $m$ .",
"Since $\\sum _{s \\in S} (m(s) - (\\beta _{1} -i-1) )^{+}=\\sum _{j=0}^{i} (i+1-j) \\, \\Theta (m,\\beta _{1}-j)\\qquad (i=0,1,2,\\ldots ),$ the formula (REF ) implies $\\sum _{j=0}^{i} (i-j+1) \\, \\Theta (m,\\beta _{1}-j)= \\max \\lbrace p(X) - (\\beta _{1} -i-1)\\vert X\\vert : X\\subseteq S\\rbrace \\qquad (i=0,1,2,\\ldots ).$ This formula gives a recurrence formula for $\\Theta (m,\\beta _{1}), \\Theta (m,\\beta _{1}-1), \\Theta (m,\\beta _{1}-2), \\ldots $ as $\\Theta (m,\\beta _{1}) &= \\max \\lbrace p(X) - (\\beta _{1} -1)\\vert X\\vert : X\\subseteq S\\rbrace ,\\\\\\Theta (m,\\beta _{1} -1) &= \\max \\lbrace p(X) - (\\beta _{1} -2)\\vert X\\vert : X\\subseteq S\\rbrace - 2 \\, \\Theta (m,\\beta _{1}) ,\\\\\\Theta (m,\\beta _{1} -2) &= \\max \\lbrace p(X) - (\\beta _{1} -3)\\vert X\\vert : X\\subseteq S\\rbrace - 3 \\, \\Theta (m,\\beta _{1}) - 2 \\, \\Theta (m,\\beta _{1} -1),\\\\& \\quad \\cdots \\cdots \\cdots \\cdots \\cdots \\cdots $ Remark 4.7 A DCA-based proof of the formula (REF ) is as follows.",
"In (REF ) we choose $ \\varphi (k) = (k-a)^{+} =\\left\\lbrace \\begin{array}{ll}0 & (k \\le a) , \\\\k - a & (k \\ge a + 1) .",
"\\\\\\end{array} \\right.$ The left-hand side of (REF ) coincides with that of (REF ).",
"The conjugate function $\\psi $ is given by $ \\psi (\\ell ) =\\left\\lbrace \\begin{array}{ll}0 & (\\ell = 0) , \\\\a & (\\ell = 1) , \\\\+\\infty & (\\ell \\notin \\lbrace 0,1 \\rbrace ) .",
"\\\\\\end{array} \\right.$ Therefore, we may restrict $\\pi $ to $\\lbrace 0,1 \\rbrace $ -vectors in considering the maximum of $g(\\pi ):=\\hat{p} (\\pi ) - \\sum _{s \\in S} \\psi (\\pi (s))$ over $\\pi \\in {\\bf Z}^{S}$ .",
"For $\\pi = \\chi _{X} \\in \\lbrace 0, 1 \\rbrace ^{S}$ with $X \\subseteq S$ , we have $\\hat{p}(\\pi ) = \\hat{p}(\\chi _{X}) = p(X)$ and $\\sum _{s \\in S} \\psi (\\pi (s)) = \\sum _{s \\in S} \\psi (\\chi _{X}(s)) = \\sum _{s \\in X} \\psi (1) = a |X|$ , and therefore, the right-hand side of (REF ) is equal to $\\max \\lbrace p(X) - a \\vert X\\vert : X\\subseteq S\\rbrace $ .",
"Thus the formula (REF ) is derived.",
"$\\rule {0.17cm}{0.17cm}$"
],
[
"Structure of optimal solutions to square-sum minimization",
"In this section we offer the DCA view on the structure of optimal solutions of the min-max formula: $\\min \\lbrace \\sum [m(s)^{2}: s\\in S]: m\\in \\overset{....}{B} \\rbrace = \\max \\lbrace \\hat{p}(\\pi ) - \\sum _{s\\in S}\\left\\lfloor {\\pi (s) \\over 2}\\right\\rfloor \\left\\lceil {\\pi (s) \\over 2}\\right\\rceil :\\pi \\in {\\bf Z}^{S} \\rbrace ,$ to which a DCA-based proof has been given in Section REF .",
"Concerning the optimal solutions to (REF ) the following results were obtained in Part I [9].",
"Recall that $\\beta _{1}>\\beta _{2}>\\cdots >\\beta _{q}$ denotes the essential value-sequence, $C_{1} \\subset C_{2} \\subset \\cdots \\subset C_{q}$ is the canonical chain, $\\lbrace S_{1},S_{2},\\dots ,S_{q}\\rbrace $ is the canonical partition ($S_{i}= C_{i} - C_{i-1}$ and $C_{0}=\\emptyset $ ), $\\pi ^{*}$ and $\\Delta ^{*}$ are integral vectors defined by $\\pi ^{*}(s) =2\\beta _{i}-1, \\quad \\Delta ^{*}(s) =\\beta _{i}-1\\qquad (s\\in S_i; \\ i=1,2, \\dots ,q) ,$ and $M^{*}$ denotes the direct sum of matroids $M_{1}, M_{2}, \\ldots , M_{q}$ constructed in Section 5.3 of Part I [9].",
"Proposition 5.1 ([9]) The set $\\Pi $ of dual optimal integral vectors $\\pi $ in (REF ) is an L$^{\\natural }$ -convex set.",
"The unique smallest element of $\\Pi $ is $\\pi ^{*}$ .",
"$\\rule {0.17cm}{0.17cm}$ Theorem 5.2 ([9]) An integral vector $\\pi $ is a dual optimal solution in (REF ) if and only if the following three conditions hold for each $i=1,2,\\dots ,q:$ $&&\\hbox{\\rm $\\pi (s)=2\\beta _{i}-1$ \\ for every $s\\in S_i-F_i,$}\\ \\\\&&\\hbox{\\rm $2\\beta _{i}-1\\le \\pi (s) \\le 2\\beta _{i}+1$ \\ for every $s\\in F_i$,}\\ \\\\&&\\hbox{\\rm $\\pi (s)-\\pi (t) \\ge 0$ \\ whenever $s,t\\in F_i$ and $(s,t) \\in A_i$, }\\ $ where $F_{i}$ is the largest member of ${\\cal F}_{i} = \\lbrace X \\subseteq S_{i}: \\beta _{i} |X| = p(C_{i-1} \\cup X) - p(C_{i-1}) \\rbrace $ and $A_{i}$ is the set of pairs $(s,t)$ such that $s, t \\in F_{i}$ and there is no set in ${\\cal F}_{i}$ which contains $t$ and not $s$ .",
"$\\rule {0.17cm}{0.17cm}$ Theorem 5.3 ([9]) The set of dec-min elements of $\\overset{....}{B}$ is a matroidal M-convex set.In Part I, we have defined a matroidal M-convex set as the set of integral elements of a translated matroid base-polyhedron.",
"In other words, a matroidal M-convex set is an M-convex set in which the $\\ell _{\\infty }$ -distance of any two distinct members is equal to one.",
"More precisely, an element $m$ of $\\overset{....}{B}$ is decreasingly minimal if and only if $m$ can be obtained in the form $m=\\chi _L+ \\Delta ^{*}$ , where $L$ is a basis of the matroid $M^{*}$ .",
"$\\rule {0.17cm}{0.17cm}$ The objective of this section is to shed the light of DCA on these results.",
"It will turn out that the general results in DCA capture the structural essence of the above statements, but do not provide the full statements with specific details.",
"We first present a summary of the relevant results from DCA in Sections REF and REF ."
],
[
"General results on the optimal solutions in the Fenchel-type discrete duality",
"We summarize the fundamental facts about the optimal solutions in the Fenchel-type min-max relation $ \\min \\lbrace f(x) - h(x) : x \\in {\\bf Z}^{S} \\rbrace = \\max \\lbrace h^{\\circ }(\\pi ) - f^{\\bullet }(\\pi ) : \\pi \\in {\\bf Z}^{S} \\rbrace ,$ where $f$ is an integer-valued M$^{\\natural }$ -convex function and $h$ is an integer-valued M$^{\\natural }$ -concave function.",
"We assume that both ${\\rm dom\\,}f \\cap {\\rm dom\\,}h$ and ${\\rm dom\\,}f^{\\bullet } \\cap {\\rm dom\\,}h^{\\circ }$ are nonempty, in which case the common value in (REF ) is finite.",
"We denote the set of the minimizers by $\\mathcal {P}$ and the set of the maximizers by $\\mathcal {D}$ .",
"To derive the optimality criteria we recall the Fenchel–Young inequalities $f(x) + f^{\\bullet }(\\pi ) &\\ge \\langle \\pi , x \\rangle ,\\\\h(x) + h^{\\circ }(\\pi ) &\\le \\langle \\pi , x \\rangle ,$ which hold for any $x \\in {\\bf Z}^{S}$ and $\\pi \\in {\\bf Z}^{S}$ .",
"These inequalities immediately imply the weak duality $ f(x) - h(x) \\ge h^{\\circ }(\\pi ) - f^{\\bullet }(\\pi ) .$ The inequality in (REF ) turns into an equality if and only if the inequalities in (REF ) and () are satisfied in equalities.",
"The former condition is equivalent to saying that $x \\in \\mathcal {P}$ and $\\pi \\in \\mathcal {D}$ .",
"The equality condition for (REF ) can be rewritten as $f(x) - \\langle \\pi , x \\rangle = - f^{\\bullet }(\\pi )= - \\max \\lbrace \\langle \\pi , y \\rangle - f(y) : y \\in {\\bf Z}^{S} \\rbrace = \\min \\lbrace f(y) - \\langle \\pi , y \\rangle : y \\in {\\bf Z}^{S} \\rbrace .$ Similarly, the equality condition for () can be rewritten as $h(x) - \\langle \\pi , x \\rangle = - h^{\\circ }(\\pi )= - \\min \\lbrace \\langle \\pi , y \\rangle - h(y) : y \\in {\\bf Z}^{S} \\rbrace = \\max \\lbrace h(y) - \\langle \\pi , y \\rangle : y \\in {\\bf Z}^{S} \\rbrace .$ Therefore we have $ x \\in \\mathcal {P} \\ \\mbox{\\rm and} \\ \\pi \\in \\mathcal {D}\\iff x \\in \\arg \\min _{y} \\lbrace f(y) - \\langle \\pi , y \\rangle \\rbrace \\cap \\arg \\max _{y} \\lbrace h(y) - \\langle \\pi , y \\rangle \\rbrace .$ Furthermore, by the M-optimality criterion (Theorem REF ) applied to $f(y) - \\langle \\pi , y \\rangle $ , we have $x \\in \\arg \\min \\lbrace f(y) - \\langle \\pi , y \\rangle \\rbrace $ if and only if $f(x) - \\langle \\pi , x \\rangle & \\le f(x + \\chi _{s} - \\chi _{t}) - \\langle \\pi , x + \\chi _{s} - \\chi _{t} \\rangle \\qquad (\\forall s, t \\in S) ,\\nonumber \\\\f(x) - \\langle \\pi , x \\rangle & \\le f(x + \\chi _{s}) - \\langle \\pi , x + \\chi _{s} \\rangle \\qquad (\\forall s \\in S) ,\\nonumber \\\\f(x) - \\langle \\pi , x \\rangle & \\le f(x - \\chi _{t}) - \\langle \\pi , x - \\chi _{t} \\rangle \\qquad (\\forall t \\in S) ,$ that is, if and only if $\\pi (s) - \\pi (t) & \\le f(x + \\chi _{s} - \\chi _{t}) - f(x)\\qquad (\\forall s, t \\in S) ,\\\\f(x) - f(x - \\chi _{s}) & \\le \\pi (s) \\le f(x + \\chi _{s}) - f(x)\\qquad (\\forall s \\in S) .$ Similarly, we have $x \\in \\arg \\max \\lbrace h(y) - \\langle \\pi , y \\rangle \\rbrace $ if and only if $\\pi (s) - \\pi (t) & \\ge h(x + \\chi _{s} - \\chi _{t}) - h(x)\\qquad (\\forall s, t \\in S) ,\\\\h(x) - h(x - \\chi _{s}) & \\ge \\pi (s) \\ge h(x + \\chi _{s}) - h(x)\\qquad (\\forall s \\in S) .$ Therefore, $ x \\in \\mathcal {P} \\ \\mbox{\\rm and} \\ \\pi \\in \\mathcal {D}\\iff \\mbox{\\rm (\\ref {locoptfpi1}),(\\ref {locoptfpi2}),(\\ref {locopthpi1}),(\\ref {locopthpi2})hold}.$ Using the integer biconjugacy $f^{\\bullet \\bullet }=f$ and $h^{\\circ \\circ }=h$ for M$^{\\natural }$ -convex/concave functions with respect to the discrete conjugates in (REF ) and () (cf.",
"[33]), we can rewrite (REF ) and (REF ), respectively, as $f^{\\bullet }(\\pi ) - \\langle \\pi , x \\rangle & = -f(x) = - f^{\\bullet \\bullet }(x)= \\min \\lbrace f^{\\bullet }(\\tau ) - \\langle \\tau , x \\rangle : \\tau \\in {\\bf Z}^{S} \\rbrace ,\\\\h^{\\circ }(\\pi ) - \\langle \\pi , x \\rangle & = -h(x) = - h^{\\circ \\circ }(x)= \\max \\lbrace h^{\\circ }(\\tau ) - \\langle \\tau , x \\rangle : \\tau \\in {\\bf Z}^{S} \\rbrace .$ Hence the equivalence in (REF ) can be rephrased in terms of the conjugate functions as $ x \\in \\mathcal {P} \\ \\mbox{\\rm and} \\ \\pi \\in \\mathcal {D}\\iff \\pi \\in \\arg \\min _{\\tau } \\lbrace f^{\\bullet }(\\tau ) - \\langle \\tau , x \\rangle \\rbrace \\cap \\arg \\max _{\\tau } \\lbrace h^{\\circ }(\\tau ) - \\langle \\tau , x \\rangle \\rbrace .$ Furthermore, by the L-optimality criterion (Theorem REF ) applied to the L$^{\\natural }$ -convex function $f^{\\bullet }(\\tau ) - \\langle \\tau , x \\rangle $ , we have $\\pi \\in \\arg \\min \\lbrace f^{\\bullet }(\\tau ) - \\langle \\tau , x \\rangle \\rbrace $ if and only if $f^{\\bullet }(\\pi ) - \\langle \\pi , x \\rangle & \\le f^{\\bullet }(\\pi + \\chi _{Y}) - \\langle \\pi + \\chi _{Y}, x \\rangle \\qquad (\\forall Y \\subseteq S) ,\\nonumber \\\\f^{\\bullet }(\\pi ) - \\langle \\pi , x \\rangle & \\le f^{\\bullet }(\\pi - \\chi _{Y}) - \\langle \\pi - \\chi _{Y}, x \\rangle \\qquad (\\forall Y \\subseteq S) ,$ that is, if and only if $f^{\\bullet }(\\pi ) - f^{\\bullet }(\\pi - \\chi _{Y})\\le \\sum _{s \\in Y}x(s)\\le f^{\\bullet }(\\pi + \\chi _{Y}) - f^{\\bullet }(\\pi )\\qquad (\\forall Y \\subseteq S) .$ Similarly, we have $\\pi \\in \\arg \\max \\lbrace h^{\\circ }(\\tau ) - \\langle \\tau , x \\rangle \\rbrace $ if and only if $h^{\\circ }(\\pi ) - h^{\\circ }(\\pi - \\chi _{Y})\\ge \\sum _{s \\in Y}x(s)\\ge h^{\\circ }(\\pi + \\chi _{Y}) - h^{\\circ }(\\pi )\\qquad (\\forall Y \\subseteq S) .$ Therefore, $ x \\in \\mathcal {P} \\ \\mbox{\\rm and} \\ \\pi \\in \\mathcal {D}\\iff \\mbox{\\rm (\\ref {locoptfconjx}),(\\ref {locopthconjx})hold}.$ From the above argument we can obtain the following optimality criteria.",
"Theorem 5.4 Let $f$ be an integer-valued M$^{\\natural }$ -convex function and $h$ be an integer-valued M$^{\\natural }$ -concave function such that both $\\mathcal {P}_{0}:={\\rm dom\\,}f \\cap {\\rm dom\\,}h$ and $\\mathcal {D}_{0} := {\\rm dom\\,}f^{\\bullet } \\cap {\\rm dom\\,}h^{\\circ }$ are nonempty.",
"(1) Let $x \\in \\mathcal {P}_{0}$ and $\\pi \\in \\mathcal {D}_{0}$ .",
"Then the following three conditions are pairwise equivalent.",
"(a) $x$ and $\\pi $ are both optimal, that is, $x \\in \\mathcal {P}$ and $\\pi \\in \\mathcal {D}$ .",
"(b) The inequalities (REF ), (), (REF ), and () are satisfied by $x$ and $\\pi $ .",
"(c) The inequalities (REF ) and (REF ) are satisfied by $x$ and $\\pi $ .",
"(2) Let $\\hat{\\pi }\\in \\mathcal {D}$ be an arbitrary dual optimal solution.",
"Then $x^{*} \\in \\mathcal {P}_{0}$ is a minimizer of $f(x) - h(x)$ if and only if it is a minimizer of $f(x) - \\langle \\hat{\\pi }, x \\rangle $ and simultaneously a maximizer of $h(x) - \\langle \\hat{\\pi }, x \\rangle $ , or equivalently, $x^{*}$ satisfies (REF ) and (REF ) for $\\pi = \\hat{\\pi }$ .",
"Namely, $\\mathcal {P}& = \\arg \\min \\lbrace f(x) - \\langle \\hat{\\pi }, x \\rangle \\rbrace \\cap \\arg \\max \\lbrace h(x) - \\langle \\hat{\\pi }, x \\rangle \\rbrace \\\\ &= \\lbrace x \\in {\\bf Z}^{S} :\\mbox{\\rm (\\ref {locoptfpi1}),(\\ref {locoptfpi2}),(\\ref {locopthpi1}),(\\ref {locopthpi2})hold with $\\pi = \\hat{\\pi }$} \\rbrace \\\\ &= \\lbrace x \\in {\\bf Z}^{S} :\\mbox{\\rm (\\ref {locoptfconjx}) and (\\ref {locopthconjx})hold with $\\pi = \\hat{\\pi }$} \\rbrace .$ (3) Let $\\hat{x} \\in \\mathcal {P}$ be an arbitrary primal optimal solution.",
"Then $\\pi ^{*} \\in \\mathcal {D}_{0}$ is a maximizer of $h^{\\circ }(\\pi ) - f^{\\bullet }(\\pi )$ if and only if it is a minimizer of $f^{\\bullet }(\\pi ) - \\langle \\pi , \\hat{x} \\rangle $ and simultaneously a maximizer of $h^{\\circ }(\\pi ) - \\langle \\pi , \\hat{x} \\rangle $ , or equivalently, $\\pi ^{*}$ satisfies the inequalities (REF ), (), (REF ), and () for $x = \\hat{x}$ .",
"Namely, $\\mathcal {D}&= \\arg \\min \\lbrace f^{\\bullet }(\\pi ) - \\langle \\pi , \\hat{x} \\rangle \\rbrace \\cap \\arg \\max \\lbrace h^{\\circ }(\\pi ) - \\langle \\pi , \\hat{x} \\rangle \\rbrace \\\\ &= \\lbrace \\pi \\in {\\bf Z}^{S} :\\mbox{\\rm (\\ref {locoptfpi1}),(\\ref {locoptfpi2}),(\\ref {locopthpi1}),(\\ref {locopthpi2})hold with $x = \\hat{x}$} \\rbrace \\\\ &= \\lbrace \\pi \\in {\\bf Z}^{S} :\\mbox{\\rm (\\ref {locoptfconjx}) and (\\ref {locopthconjx})hold with $x = \\hat{x}$} \\rbrace .$ $\\rule {0.17cm}{0.17cm}$ It is emphasized that in the representation of $\\mathcal {P}$ , each of $\\arg \\min \\lbrace f(x) - \\langle \\hat{\\pi }, x \\rangle \\rbrace $ and $\\arg \\max \\lbrace h(x) - \\langle \\hat{\\pi }, x \\rangle \\rbrace $ depends on the choice of $\\hat{\\pi }$ , but their intersection is uniquely determined and equal to $\\mathcal {P}$ .",
"Similarly, in the representation of $\\mathcal {D}$ , each of $\\arg \\min \\lbrace f^{\\bullet }(\\pi ) - \\langle \\pi , \\hat{x} \\rangle \\rbrace $ and $\\arg \\max \\lbrace h^{\\circ }(\\pi ) - \\langle \\pi , \\hat{x} \\rangle \\rbrace $ depends on the choice of $\\hat{x}$ , but their intersection is uniquely determined and equal to $\\mathcal {D}$ .",
"The representation of $\\mathcal {P}$ in (REF ) (or ()) shows that $\\mathcal {P}$ is the intersection of two M$^{\\natural }$ -convex sets.",
"Such a set is called an M$_{2}^{\\natural }$ -convex set [33].",
"Note that the intersection of M$^{\\natural }$ -convex sets is not always M$^{\\natural }$ -convex.",
"The representation of $\\mathcal {D}$ in (REF ) (or ()) shows that $\\mathcal {D}$ is the intersection of two L$^{\\natural }$ -convex sets.",
"Since the intersection of two (or more) L$^{\\natural }$ -convex sets is again L$^{\\natural }$ -convex, $\\mathcal {D}$ is an L$^{\\natural }$ -convex set.",
"Proposition 5.5 In the Fenchel-type min-max relation (REF ) for M$^{\\natural }$ -convex/concave functions, the set $\\mathcal {P}$ of the minimizers is an M$_{2}^{\\natural }$ -convex set and the set $\\mathcal {D}$ of the maximizers is an L$^{\\natural }$ -convex set.",
"$\\rule {0.17cm}{0.17cm}$ Remark 5.1 In Remark REF we have discussed the role of the Fenchel-type discrete duality theorem for the certificate of optimality in minimizing $f(x) - h(x)$ .",
"We have distinguished two cases according to wheter the explicit forms of the conjugate functions $f^{\\bullet }(\\pi )$ and $h^{\\circ }(\\pi )$ are available or not.",
"If their explicit forms are known, we can verify the optimality of $x$ by simply computing the values of $f(x) - h(x)$ for $x$ and $h^{\\circ }(\\pi ) - f^{\\bullet }(\\pi )$ for a given dual optimal $\\pi $ .",
"Even if the explicit forms of the conjugate functions are not known, Theorem REF (2) above enables us to verify the optimality of $x$ by checking the inequalities (REF ), (), (REF ), and () for a given dual optimal $\\pi $ .",
"Note that we have $O(|S|^{2})$ inequalities in total.",
"We emphasize that Theorem REF (2) is derived from a combination of the Fenchel-type discrete duality theorem (Theorem REF ) with the M-optimality criterion (Theorem REF ).",
"$\\rule {0.17cm}{0.17cm}$ Remark 5.2 In convex analysis, as well as in discrete convex analysis, the optimality conditions such as those in Theorem REF are expressed usually in terms of subgradients and subdifferentials.",
"In this paper, however, we have intentionally avoided using these concepts for the sake of the audience from combinatorial optimization.",
"In this remark we will briefly indicate how the results in Theorem REF can be described and interpreted in terms of subgradients and subdifferentials.",
"Let $f: {\\bf Z}^{S} \\rightarrow {\\bf Z}\\cup \\lbrace +\\infty \\rbrace $ and $h: {\\bf Z}^{S} \\rightarrow {\\bf Z}\\cup \\lbrace -\\infty \\rbrace $ be integer-valued functions defined on ${\\bf Z}^{S}$ .",
"The integral subdifferential of $f$ at $x \\in {\\rm dom\\,}f$ and its concave version for $h$ at $x \\in {\\rm dom\\,}h$ are the sets of integer vectors defined as $\\partial f(x)& := &\\lbrace \\pi \\in {\\bf Z}^{S} :f(y) - f(x) \\ge \\langle \\pi , y - x \\rangle \\ \\ (\\forall y \\in {\\bf Z}^{S}) \\rbrace ,\\\\\\partial h(x)& := & \\lbrace \\pi \\in {\\bf Z}^{S} :h(y) - h(x) \\le \\langle \\pi , y - x \\rangle \\ \\ (\\forall y \\in {\\bf Z}^{S}) \\rbrace .$ A member of $\\partial f(x)$ is called a subgradient of $f$ at $x$ .",
"Accordingly, the integral subdifferentials of $f^{\\bullet }$ and $h^{\\circ }$ at $\\pi $ are defined as $\\partial f^{\\bullet }(\\pi )&:= & \\lbrace x \\in {\\bf Z}^{S} :f^{\\bullet }(\\tau ) - f^{\\bullet }(\\pi ) \\ge \\langle \\tau - \\pi , x \\rangle \\ (\\forall \\tau \\in {\\bf Z}^{S}) \\rbrace ,\\\\\\partial h^{\\circ }(\\pi )&:= & \\lbrace x \\in {\\bf Z}^{S} :h^{\\circ }(\\tau ) - h^{\\circ }(\\pi ) \\le \\langle \\tau - \\pi , x \\rangle \\ (\\forall \\tau \\in {\\bf Z}^{S}) \\rbrace ,$ where $\\partial f^{\\bullet }(\\pi )$ is defined for $\\pi \\in {\\rm dom\\,}f^{\\bullet }$ and $\\partial h^{\\circ }(\\pi )$ for $\\pi \\in {\\rm dom\\,}h^{\\circ }$ .",
"The following relations are straightforward translations of the corresponding results in (ordinary) convex analysis to the discrete setting (cf., [31], [33]): $\\pi \\in \\partial f(x) &\\iff \\mbox{equality holds in (\\ref {youngineqf})}\\iff x \\in \\partial f^{\\bullet }(\\pi ),\\\\\\pi \\in \\partial h(x) &\\iff \\mbox{equality holds in (\\ref {youngineqh})}\\iff x \\in \\partial h^{\\circ }(\\pi ),\\\\\\partial f(x) &= \\arg \\min _{\\pi } \\lbrace f^{\\bullet }(\\pi ) - \\langle \\pi , x \\rangle \\rbrace ,\\\\\\partial h(x) &= \\arg \\max _{\\pi } \\lbrace h^{\\circ }(\\pi ) - \\langle \\pi , x \\rangle \\rbrace ,\\\\\\partial f^{\\bullet }(\\pi )&= \\arg \\min _{x} \\lbrace f(x) - \\langle \\pi , x \\rangle \\rbrace ,\\\\\\partial h^{\\circ }(\\pi )&= \\arg \\max _{x} \\lbrace h(x) - \\langle \\pi , x \\rangle \\rbrace ,$ where the integer biconjugacy ($f^{\\bullet \\bullet }=f$ , $h^{\\circ \\circ }=h$ ) is assumed, which is true for M$^{\\natural }$ -convex/concave functions.",
"By using ()–() in (REF ), and ()–() in (REF ), respectively, we obtain the following representations of optimal solutions $\\mathcal {P}& = \\partial f^{\\bullet }(\\hat{\\pi }) \\cap \\partial h^{\\circ }(\\hat{\\pi }) ,\\\\\\mathcal {D}& = \\partial f(\\hat{x}) \\cap \\partial h(\\hat{x})$ for any $\\hat{\\pi }\\in \\mathcal {D}$ and $\\hat{x} \\in \\mathcal {P}$ .",
"We also have optimality criteria $x \\in \\mathcal {P}& \\iff \\partial f(x) \\cap \\partial h(x) \\ne \\emptyset ,\\\\\\pi \\in \\mathcal {D}& \\iff \\partial f^{\\bullet }(\\pi ) \\cap \\partial h^{\\circ }(\\pi ) \\ne \\emptyset .$ Finally it is worth mentioning that, by the M-L conjugacy [33], the subdifferential of an M$^{\\natural }$ -convex function $f$ (resp., an M$^{\\natural }$ -concave function $h$ ) is an L$^{\\natural }$ -convex set and the subdifferential of an L$^{\\natural }$ -convex function $f^{\\bullet }$ (resp., an L$^{\\natural }$ -concave function $h^{\\circ }$ ) is an M$^{\\natural }$ -convex set.",
"$\\rule {0.17cm}{0.17cm}$"
],
[
"Separable convex functions on an M-convex set",
"In Theorem REF we have shown a min-max formula $& \\min \\lbrace \\sum _{s \\in S} \\varphi _{s} ( x(s) ) :x \\in \\overset{....}{B} \\rbrace = \\max \\lbrace \\hat{p}(\\pi )- \\sum _{s \\in S} \\psi _{s} (\\pi (s) ) : \\pi \\in {\\bf Z}^{S} \\rbrace $ for an integer-valued separable convex function $ \\Phi (x) = \\sum [\\varphi _{s}(x(s)): s\\in S]$ on an M-convex set $\\overset{....}{B}$ .",
"Here we introduce notations for the set of feasible points: ${\\rm dom\\,}\\Phi &= \\lbrace x \\in {\\bf Z}^{S} : x(s) \\in {\\rm dom\\,}\\varphi _{s} \\mbox{ for each } s \\in S \\rbrace ,\\\\\\mathcal {P}_{0} &= \\overset{....}{B} \\cap {\\rm dom\\,}{\\Phi } =\\lbrace x \\in \\overset{....}{B}:x(s) \\in {\\rm dom\\,}\\varphi _{s} \\mbox{ for each } s \\in S \\rbrace ,\\\\\\mathcal {D}_{0} &= \\lbrace \\pi \\in {\\bf Z}^{S}:\\pi \\in {\\rm dom\\,}\\hat{p}, \\ \\pi (s) \\in {\\rm dom\\,}\\psi _{s} \\mbox{ for each } s \\in S \\rbrace .$ The min-max formula (REF ) holds under the assumption of primal feasibility ($\\mathcal {P}_{0} \\ne \\emptyset $ ) or dual feasibility ($\\mathcal {D}_{0} \\ne \\emptyset $ ).",
"The unbounded case with both sides of (REF ) being equal to $-\\infty $ or $+\\infty $ is also a possibility in general, but in this section we assume that the both sides are finite-valued and denote the set of the minimizers $x$ by $\\mathcal {P}$ and the set of the maximizers $\\pi $ by $\\mathcal {D}$ .",
"We can obtain the optimality conditions for (REF ) by applying Theorem REF with $& f(x) =\\sum [\\varphi _{s}(x(s)): s\\in S],\\qquad \\ h(x) = -\\delta (x),\\\\ &f^{\\bullet }(\\pi ) = \\sum [ \\psi _{s}(\\pi (s)): s\\in S],\\qquad h^{\\circ }(\\pi ) =\\hat{p}(\\pi ) ,$ where $\\delta $ is the indicator function of $\\overset{....}{B}$ defined in (REF ).",
"However, we present a direct derivation from (REF ) via weak duality ($\\min \\ge \\max $ ), as it should be more informative and convenient for readers.",
"For each conjugate pair $(\\varphi _{s}, \\psi _{s})$ , it follows from the definition (REF ) that $ \\varphi _{s}(k) + \\psi _{s} ( \\ell ) \\ge k \\ell \\qquad (k,\\ell \\in {\\bf Z}),$ which is known as the Fenchel–Young inequality, where the equality holds if and only if $ \\varphi _{s}(k) - \\varphi _{s}(k-1) \\le \\ell \\le \\varphi _{s}(k+1) - \\varphi _{s}(k).$ Let $x \\in \\mathcal {P}_{0}$ and $\\pi \\in \\mathcal {D}_{0}$ .",
"Then, using the Fenchel–Young inequality (REF ) as well as (REF ) for $p$ , we obtain the weak duality: $\\sum _{s \\in S} \\varphi _{s} ( x(s) )-\\left( \\hat{p}(\\pi )- \\sum _{s \\in S} \\psi _{s} (\\pi (s) )\\right)& = \\sum _{s \\in S} \\big [ \\varphi _{s} ( x(s) ) + \\psi _{s}(\\pi (s)) \\big ]\\ - \\hat{p}(\\pi )\\\\ &\\ge \\sum _{s \\in S} x(s) \\pi (s)\\ - \\hat{p}(\\pi )\\\\ &\\ge \\min \\lbrace \\pi z : z \\in \\overset{....}{B} \\rbrace - \\hat{p}(\\pi )\\ = 0.$ The optimality conditions can be obtained as the conditions for the inequalities in (REF ) and () to be equalities, as follows.",
"Proposition 5.6 Assume that both $\\mathcal {P}_{0}$ and $\\mathcal {D}_{0}$ in ()–() are nonempty.",
"(1) Let $x \\in \\mathcal {P}_{0}$ and $\\pi \\in \\mathcal {D}_{0}$ .",
"Then $x \\in \\mathcal {P}$ and $\\pi \\in \\mathcal {D}$ (that is, $x$ and $\\pi $ are both optimal) if and only if the following two conditions are satisfied: $&\\varphi _{s}(x(s)) - \\varphi _{s}(x(s)-1)\\le \\pi (s) \\le \\varphi _{s}(x(s)+1) - \\varphi _{s}(x(s))\\qquad (s \\in S),\\\\& \\mbox{ \\rm $\\pi (s) \\ge \\pi (t)$\\quad for every $(s,t)$ \\ with \\ $x + \\chi _{s} - \\chi _{t} \\in \\overset{....}{B}$}.$ (2) Let $\\hat{\\pi }\\in \\mathcal {D}$ be an arbitrary dual optimal solution.",
"Then $x^{*} \\in \\mathcal {P}_{0}$ is a minimizer of $\\Phi (x)$ over $\\overset{....}{B}$ if and only if it satisfies (REF ) and () for $\\pi = \\hat{\\pi }$ , or equivalently, it is a minimizer of $\\sum [\\varphi _{s}(x(s)) - \\hat{\\pi }(s) x(s) : s\\in S]$ and simultaneously a $\\hat{\\pi }$ -minimizer in $\\overset{....}{B}$ .",
"Namely, $\\mathcal {P}& = \\lbrace x \\in \\mathcal {P}_{0}:\\mbox{\\rm (\\ref {pisubgradBase}), (\\ref {piminzerBase}) hold with $\\pi = \\hat{\\pi }$} \\rbrace \\\\& =\\lbrace x \\in {\\rm dom\\,}\\Phi :\\mbox{\\rm (\\ref {pisubgradBase}) holds with $\\pi = \\hat{\\pi }$} \\rbrace \\cap \\lbrace x \\in \\overset{....}{B}:\\mbox{\\rm $x$ is a $\\hat{\\pi }$-minimizer in $\\overset{....}{B}$ } \\rbrace .$ (3) Let $\\hat{x} \\in \\mathcal {P}$ be an arbitrary primal optimal solution.",
"Then $\\pi ^{*} \\in \\mathcal {D}_{0}$ is a maximizer of $\\hat{p}(\\pi ) - \\sum _{s \\in S} \\psi _{s} (\\pi (s) )$ if and only if it satisfies the inequalities (REF ) and () for $x = \\hat{x}$ .",
"Namely, $ \\mathcal {D} = \\lbrace \\pi \\in \\mathcal {D}_{0} :\\mbox{\\rm (\\ref {pisubgradBase}), (\\ref {piminzerBase})hold with $x = \\hat{x}$} \\rbrace .$ The inequality (REF ) turns into an equality if and only if, for each $s \\in S$ , we have $\\varphi _{s} (k) + \\psi _{s} ( \\ell ) = k \\ell $ for $k= x(s)$ and $\\ell = \\pi (s)$ .",
"The latter condition is equivalent to (REF ) by (REF ).",
"The other inequality () turns into an equality if and only if $x$ is a $\\pi $ -minimizer in $\\overset{....}{B}$ , which is equivalent to ().",
"Finally, we see from (REF ) that the two inequalities in (REF ) and () simultaneously turn into equality if $x \\in \\mathcal {P}$ and $\\pi \\in \\mathcal {D}$ .",
"Proposition 5.7 In the min-max relation (REF ) for a separable convex function on an M-convex set, the set $\\mathcal {D}$ of the maximizers is an L$^{\\natural }$ -convex set and the set $\\mathcal {P}$ of the minimizers is an M-convex set.",
"The representation (REF ) shows that $\\mathcal {D}$ is described by the inequalities in (REF ) and ().",
"Hence $\\mathcal {D}$ is L$^{\\natural }$ -convex.",
"(The L$^{\\natural }$ -convexity of $\\mathcal {D}$ can also be obtained from Proposition REF .)",
"In the representation () of $\\mathcal {P}$ , the first set $\\lbrace x \\in {\\rm dom\\,}\\Phi : \\mbox{\\rm (\\ref {pisubgradBase}) holds with $\\pi = \\hat{\\pi }$} \\rbrace $ is a box of integers (the set of integers in an integral box), while the set of $\\hat{\\pi }$ -minimizers in $\\overset{....}{B}$ is an M-convex set.",
"Therefore, $\\mathcal {P}$ is an M-convex set."
],
[
"Dual optimal solutions to square-sum minimization",
"The min-max formula (REF ) for the square-sum minimization is a special case of the min-max formula (REF ) with $\\varphi _{s}(k)=\\varphi (k)=k^{2}$ and $\\psi _{s}(\\ell ) = \\psi (\\ell )= \\left\\lfloor {\\ell }/{2} \\right\\rfloor \\cdot \\left\\lceil {\\ell }/{2} \\right\\rceil $ for $k, \\ell \\in {\\bf Z}$ (cf., (REF )).",
"Accordingly, we can apply the general results (Proposition REF , in particular) for the analysis of the optimal solutions in the min-max formula (REF ).",
"In this section we consider the dual solutions, whereas the primal solutions are treated in Section REF .",
"The function $g(\\pi ) = \\hat{p}(\\pi ) - \\sum [\\psi (\\pi (s)): s\\in S]$ to be maximized in (REF ) is L$^{\\natural }$ -concave by Proposition REF , and the maximizers of an L$^{\\natural }$ -concave function form an L$^{\\natural }$ -convex set [33].",
"Therefore, the set $\\Pi $ of dual optimal solutions is an L$^{\\natural }$ -convex set, which is the first statement of Proposition REF .",
"The L$^{\\natural }$ -convexity of $\\Pi $ implies that there exists a unique smallest element of $\\Pi $ .",
"The second statement of Proposition REF shows that this smallest element is given by $\\pi ^{*}$ , but this fact is not easily shown by general arguments from discrete convex analysis.",
"Next we consider Theorem REF , which gives a representation of $\\Pi $ .",
"According to the general result stated in Proposition REF (3), we can obtain another representation of $\\Pi $ by choosing any dec-min element $\\hat{x}$ of $\\overset{....}{B}$ , which is a primal optimal solution for (REF ).",
"In this case the condition (REF ) reads $ 2 x(s) - 1 \\le \\pi (s) \\le 2 x(s) + 1\\qquad (s \\in S) ,$ since $\\varphi (k) - \\varphi (k-1) = k^{2} - (k-1)^{2} = 2k - 1$ and $\\varphi (k+1) - \\varphi (k) = (k+1)^{2} - k^{2} = 2k + 1$ .",
"Proposition 5.8 Let $m$ be any dec-min element of $\\overset{....}{B}$ .",
"The set $\\Pi $ of dual optimal solutions to (REF ) is represented as $\\Pi = I(m) \\cap P(m)$ , where $I(m) &= \\lbrace \\pi \\in {\\bf Z}^{S} :2 m(s) - 1 \\le \\pi (s) \\le 2 m(s) + 1 \\mbox{\\rm \\ for all $s \\in S$} \\rbrace ,\\\\P(m) &= \\lbrace \\pi \\in {\\bf Z}^{S} :\\mbox{\\rm $\\pi (s) \\ge \\pi (t)$\\ \\ for every $(s,t)$ \\ with \\ $x + \\chi _{s} - \\chi _{t} \\in \\overset{....}{B}$}\\rbrace .$ Hence $\\Pi $ is an L$^{\\natural }$ -convex set.",
"$\\rule {0.17cm}{0.17cm}$ Let us compare the representations of $\\Pi $ in Proposition REF and Theorem REF .",
"Roughly speaking, $I(m)$ corresponds to the first two conditions (REF ) and () in Theorem REF and $P(m)$ to the third condition ().",
"However, there is an essential difference between Proposition REF and Theorem REF .",
"Namely, each of $I(m)$ and $P(m)$ varies with the choice of $m$ , while their intersection is uniquely determined and equal to $\\Pi $ .",
"In this sense, the description of $\\Pi $ in Proposition REF is not canonical.",
"Theorem REF is a much stronger statement, giving a canonical description of $\\Pi $ without reference to a particular primal optimal solution.",
"Remark 5.3 Proposition REF above is equivalent to Proposition 6.11 of Part I [9], though in a slightly different form.",
"Recall the optimality criteria there:For a given vector $\\pi $ in ${\\bf R}^S$ , we call a nonempty set $X\\subseteq S$ a $\\pi $ -top set if $\\pi (u)\\ge \\pi (v)$ holds whenever $u\\in X$ and $v\\in S-X$ .",
"If $\\pi (u)>\\pi (v)$ holds whenever $u\\in X$ and $v\\in S-X$ , we speak of a strict $\\pi $ -top set.",
"We call a subset $X\\subseteq S$ $m$ -tight with respect to $p$ if $\\widetilde{m}(X)=p(X)$ .",
"${\\rm (O1)} & \\qquad m(s) \\in \\lbrace \\left\\lfloor \\pi (s)/2 \\right\\rfloor , \\left\\lceil \\pi (s)/2 \\right\\rceil \\rbrace \\mbox{\\ for each \\ } s \\in S,\\\\{\\rm (O2)} & \\qquad \\mbox{each strict $\\pi $-top-set is $m$-tight with respect to $p$.",
"}$ The set $I(m)$ corresponds to the first optimality criterion (O1), since $2 m(s) - 1 \\le \\pi (s) \\le 2 m(s) + 1$ if and only if $m(s) \\in \\lbrace \\left\\lfloor \\pi (s)/2 \\right\\rfloor , \\left\\lceil \\pi (s)/2 \\right\\rceil \\rbrace $ .",
"The equivalence of $P(m)$ to the second criterion (O2) is a well-known characterization of a minimum weight base.",
"$\\rule {0.17cm}{0.17cm}$"
],
[
"Primal optimal solutions to square-sum minimization",
"We now turn to the primal problem of (REF ), namely, the square-sum minimization.",
"Let ${\\rm dm}(\\overset{....}{B})$ denote the set of the dec-min elements of $\\overset{....}{B}$ .",
"By Theorem REF , ${\\rm dm}(\\overset{....}{B})$ coincides with the set of primal optimal solutions for (REF ).",
"According to the general result in Proposition REF (2), a representation of ${\\rm dm}(\\overset{....}{B})$ can be obtained by choosing any dual optimal solution $\\hat{\\pi }$ .",
"In this case the condition (REF ) is simplified to (REF ), which can be rewritten as $ x(s) \\in \\lbrace \\left\\lfloor \\pi (s)/2 \\right\\rfloor , \\left\\lceil \\pi (s)/2 \\right\\rceil \\rbrace \\qquad (s \\in S).$ Thus the following representation of the set of dec-min elements is obtained.",
"Proposition 5.9 Let $\\hat{\\pi }$ be any dual optimal solution to (REF ).",
"The set ${\\rm dm}(\\overset{....}{B})$ of dec-min elements of $\\overset{....}{B}$ is represented as ${\\rm dm}(\\overset{....}{B}) = T(\\hat{\\pi }) \\cap \\overset{....}{B^{\\circ }}(\\hat{\\pi })$ , where $T(\\hat{\\pi }) &= \\lbrace m \\in {\\bf Z}^{S} : m(s) \\in \\lbrace \\left\\lfloor \\hat{\\pi }(s)/2 \\right\\rfloor ,\\left\\lceil \\hat{\\pi }(s)/2 \\right\\rceil \\rbrace \\ (s \\in S) \\rbrace ,\\\\\\overset{....}{B^{\\circ }}(\\hat{\\pi }) &=\\lbrace m \\in \\overset{....}{B} : \\mbox{\\rm $m$ is a minimum $\\hat{\\pi }$-weight element of $\\overset{....}{B}$} \\rbrace .$ Hence ${\\rm dm}(\\overset{....}{B})$ is a matroidal M-convex set.",
"$\\rule {0.17cm}{0.17cm}$ Again, each of $T(\\hat{\\pi })$ and $\\overset{....}{B^{\\circ }}(\\hat{\\pi })$ varies with the choice of $\\hat{\\pi }$ , but their intersection is uniquely determined and is equal to ${\\rm dm}(\\overset{....}{B})$ .",
"Here, $\\overset{....}{B^{\\circ }}(\\hat{\\pi })$ is the integral elements of a face of $B$ , and is an M-convex set.",
"As for $T(\\hat{\\pi })$ , note that, for each $s \\in S$ , the two numbers $\\left\\lfloor \\hat{\\pi }(s)/2 \\right\\rfloor $ and $\\left\\lceil \\hat{\\pi }(s)/2 \\right\\rceil $ are the same integer or consecutive integers.",
"Therefore, ${\\rm dm}(\\overset{....}{B})$ is a matroidal M-convex set.",
"In other words, there exist a matroid $\\hat{M}$ and a translation vector $\\hat{\\Delta }\\in {\\bf Z}^{S}$ such that ${\\rm dm}(\\overset{....}{B}) = T(\\hat{\\pi }) \\cap \\overset{....}{B^{\\circ }}(\\hat{\\pi })= \\lbrace \\chi _L+ \\hat{\\Delta }: \\mbox{$L$ is a basis of $\\hat{M}$} \\rbrace .$ In this construction both $\\hat{M}$ and $\\hat{\\Delta }$ depend on the chosen $\\hat{\\pi }$ ; in particular, $\\hat{\\Delta }= \\left\\lfloor \\hat{\\pi }/2 \\right\\rfloor $ .",
"Theorem REF is significantly stronger than Proposition REF , in that it gives a concrete description of the matroid $\\hat{M}$ by referring to the canonical chain.",
"The translation vector $\\Delta ^{*}$ in Theorem REF corresponds to the choice of $\\hat{\\pi }= \\pi ^{*}$ ; note that we indeed have the relation $\\Delta ^{*} = \\left\\lfloor \\pi ^{*}/2 \\right\\rfloor $ .",
"Remark 5.4 Proposition REF implies, in particular, that the dec-min elements of an M-convex set is contained in a small box (unit box).",
"Note that such a property does not hold for an arbitrary integral polyhedron.",
"To see this, consider the line segment $P$ in ${\\bf R}^{3}$ connecting two points $(2,1,0)$ and $(1,0,2)$ .",
"This $P$ is an integral polyhedron, $\\overset{....}{P} = \\lbrace (2,1,0), (1,0,2) \\rbrace $ , and $\\overset{....}{P}$ is not an M-convex set.",
"Both $(2,1,0)$ and $(1,0,2)$ are dec-min in $\\overset{....}{P}$ , but there exists no small box (unit box) containing them, since their third components differ by 2.",
"In Part III prove that this small box (unit box) property also holds for network flows.",
"$\\rule {0.17cm}{0.17cm}$"
],
[
"Comparison of continuous and discrete cases",
"While our present study is focused on the discrete case for an M-convex set $\\overset{....}{B}$ , the continuous case for a base-polyhedron $B$ was investigated by Fujishige [11] around 1980 under the name of lexicographically optimal bases, as a generalization of lexicographically optimal maximal flows considered by Megiddo [27].",
"Lexicographically optimal bases are discussed in detail in [12].",
"Later in game theory Dutta–Ray [6] treated majorization ordering in the continuous case under the name of egalitarian allocation; see also Dutta [5].",
"See also the survey of related papers in Appendix .",
"Section REF offers comparisons of major ingredients in discrete and continuous cases.",
"These comparisons show that the discrete case is significantly different from the continuous case, being endowed with a number of intriguing combinatorial structures on top of the geometric structures known in the continuous case.",
"Section REF is devoted to a review of the principal partition (adapted to a supermodular function), Section REF gives an alternative characterization of the canonical partition, and Section REF clarifies their relationship.",
"Algorithmic implications are discussed in Section REF ."
],
[
"Summary of comparisons",
"The continuous case is referred to as Case ${\\bf R}$ and the discrete case as Case ${\\bf Z}$ .",
"We use notation $m_{{\\bf R}}$ and $m_{{\\bf Z}}$ for the dec-min element in Case ${\\bf R}$ and Case ${\\bf Z}$ , respectively."
],
[
"Underlying set",
"In Case ${\\bf R}$ we consider a base-polyhedron $B$ described by a real-valued supermodular function $p$ or a submodular function $b$ .",
"In Case ${\\bf Z}$ we consider the set $\\overset{....}{B}$ of integral members of an integral base-polyhedron $B$ described by an integer-valued $p$ or $b$ ."
],
[
"Terminology",
"In Case ${\\bf R}$ the terminology of “lexicographically optimal base” (or “lexico-optimal base”) is used in [11], [12].",
"A lexico-optimal base is the same as an inc-max element in our terminology, whereas a dec-min element is called a “co-lexicographically optimal base” in [12]."
],
[
"Weighting",
"In Case ${\\bf R}$ a weight vector is introduced to define and analyze lexico-optimality, while this is not the case in this paper for Case ${\\bf Z}$ .",
"In the following comparisons we always assume that no weighting is introduced in Cases ${\\bf R}$ and ${\\bf Z}$ .",
"In a forthcoming paper, we consider discrete decreasing minimality with respect to a weight vector."
],
[
"Decreasing minimality and increasing maximality",
"In Case ${\\bf Z}$ decreasing minimality in $\\overset{....}{B}$ is equivalent to increasing maximality.",
"This statement is also true in Case ${\\bf R}$ .",
"That is, an element of $B$ is dec-min in $B$ if and only if it is inc-max in $B$ .",
"Moreover, a least majorized element exists in $\\overset{....}{B}$ (in Case ${\\bf Z}$ ) and in $B$ (in Case ${\\bf R}$ )."
],
[
"Square-sum minimization",
"In both Cases ${\\bf Z}$ and ${\\bf R}$ , a dec-min element is characterized as a minimizer of square-sum of the components $W(x) = \\sum [x(s)^{2}: s\\in S]$ .",
"In Case ${\\bf R}$ , the minimizer is unique, and is often referred to as the minimum norm point."
],
[
"Uniqueness",
"The structures of dec-min elements have a striking difference in Cases ${\\bf R}$ and ${\\bf Z}$ .",
"In Case ${\\bf R}$ the dec-min element of $B$ is uniquely determined, and is given by the minimum norm point of $B$ .",
"In Case ${\\bf Z}$ the dec-min elements of $\\overset{....}{B}$ are endowed with the structure of basis family of a matroid, as formulated in Theorem REF .",
"The minimum norm point of $B$ can be expressed as a convex combination of the dec-min elements of $\\overset{....}{B}$ (cf., Theorem REF )."
],
[
"Proximity",
"Every dec-min element $m_{{\\bf Z}}$ of $\\overset{....}{B}$ is located near the minimum norm point $m_{{\\bf R}}$ of $B$ , satisfying $\\left\\lfloor m_{{\\bf R}} \\right\\rfloor \\le m_{{\\bf Z}} \\le \\left\\lceil m_{{\\bf R}} \\right\\rceil $ (cf., Theorem REF ).",
"However, not every integer vector $m_{{\\bf Z}}$ in $B$ satisfying $\\left\\lfloor m_{{\\bf R}} \\right\\rfloor \\le m_{{\\bf Z}} \\le \\left\\lceil m_{{\\bf R}} \\right\\rceil $ is a dec-min element of $\\overset{....}{B}$ , which is demonstrated by the following example.",
"Example 6.1 Let $\\overset{....}{B}$ be an M-convex set consisting of five vectors$\\overset{....}{B}$ is obtained from $\\lbrace (1,0,1,0), \\ (1,0,0,1), \\ (0,1,1,0), \\ (0,1,0,1), \\ (1,1,0,0) \\rbrace $ (basis family of rank 2 matroid) by a translation with $(1,1,0,0)$ .",
"$m_{1}=(2,1,1,0), \\quad m_{2}=(2,1,0,1), \\quad m_{3}=(1,2,1,0), \\quad m_{4}=(1,2,0,1), \\quad m_{5}=(2,2,0,0)$ and $B$ be its convex hull.",
"The dec-min elements of $\\overset{....}{B}$ are $m_{1}$ , $m_{2}$ , $m_{3}$ , and $m_{4}$ , whereas $ m_{5}=(2,2,0,0)$ is not dec-min.",
"The minimum norm point of the base-polyhedron $B$ is $m_{{\\bf R}} = (3/2, 3/2, 1/2, 1/2 )$ , for which $\\left\\lfloor m_{{\\bf R}} \\right\\rfloor = (1,1,0,0)$ and $\\left\\lceil m_{{\\bf R}} \\right\\rceil = (2,2,1,1)$ .",
"The point $m_{5}=(2,2,0,0)$ satisfies $\\left\\lfloor m_{{\\bf R}} \\right\\rfloor \\le m_{5} \\le \\left\\lceil m_{{\\bf R}} \\right\\rceil $ but it is not a dec-min element.",
"$\\rule {0.17cm}{0.17cm}$"
],
[
"Min-max formula",
"In Case ${\\bf Z}$ we have the min-max identity (REF ): $\\min \\lbrace \\sum [m(s)^{2}: s\\in S]: m\\in \\overset{....}{B} \\rbrace = \\max \\lbrace \\hat{p}(\\pi ) - \\sum _{s\\in S}\\left\\lfloor {\\pi (s) \\over 2}\\right\\rfloor \\left\\lceil {\\pi (s) \\over 2}\\right\\rceil :\\pi \\in {\\bf Z}^S \\rbrace .$ In Case ${\\bf R}$ the corresponding formula is $ \\min \\lbrace \\sum [x(s)^{2}: s\\in S]: x\\in B \\rbrace = \\max \\lbrace \\hat{p}(\\pi ) - \\sum _{s\\in S}\\left( {\\pi (s) \\over 2}\\right)^{2} : \\pi \\in {\\bf R}^S \\rbrace ,$ which may be regarded as an adaptation of the standard quadratic programming duality to the case where the feasible region is a base-polyhedron.",
"To the best knowledge of the authors, the formula (REF ) has never been shown in the literature."
],
[
"Principal partition vs canonical partition",
"The canonical partition for Case ${\\bf Z}$ is closely related to the principal partition for Case ${\\bf R}$ .",
"The principal partition (adapted to a supermodular function) is described in Section REF and the following relations are established in Sections REF and REF .",
"We denote the canonical partition by $\\lbrace S_{1}, S_{2}, \\ldots , S_{q} \\rbrace $ and the principal partition by $\\lbrace \\hat{S}_{1}, \\hat{S}_{2}, \\ldots , \\hat{S}_{r} \\rbrace $ .",
"They are constructed from the canonical chain $C_{1} \\subset C_{2} \\subset \\cdots \\subset C_{q}$ and the principal chain $\\hat{C}_{1} \\subset \\hat{C}_{2} \\subset \\cdots \\subset \\hat{C}_{r}$ , respectively, as the families of difference sets: $S_{j}=C_{j} - C_{j-1}$ for $j=1,2,\\ldots , q$ and $\\hat{S}_{i}=\\hat{C}_{i} - \\hat{C}_{i-1}$ for $i=1,2,\\ldots , r$ , where $C_{0} = \\hat{C}_{0} = \\emptyset $ .",
"We denote the essential values by $\\beta _{1} > \\beta _{2} > \\cdots > \\beta _{q}$ and the critical values by $\\lambda _{1} > \\lambda _{2} > \\cdots > \\lambda _{r}$ .",
"An integer $\\beta $ is an essential value for Case ${\\bf Z}$ if and only if there exists a critical value $\\lambda $ for Case ${\\bf R}$ satisfying $\\beta \\ge \\lambda > \\beta -1$ .",
"The essential values $\\beta _{1} > \\beta _{2} > \\cdots > \\beta _{q}$ are obtained from the critical values $\\lambda _{1} > \\lambda _{2} > \\cdots > \\lambda _{r}$ as the distinct members of the rounded-up integers $\\lceil \\lambda _{1} \\rceil \\ge \\lceil \\lambda _{2} \\rceil \\ge \\cdots \\ge \\lceil \\lambda _{r} \\rceil $ .",
"The canonical partition $\\lbrace S_{1}, S_{2}, \\ldots , S_{q} \\rbrace $ is obtained from the principal partition $\\lbrace \\hat{S}_{1}, \\hat{S}_{2}, \\ldots , \\hat{S}_{r} \\rbrace $ as an aggregation; we have $S_{j} = \\bigcup _{i \\in I(j)} \\hat{S}_{i}$ , where $I(j) = \\lbrace i : \\lceil \\lambda _{i} \\rceil = \\beta _{j} \\rbrace $ .",
"The canonical chain $\\lbrace C_{j} \\rbrace $ is a subchain of the principal chain $\\lbrace \\hat{C}_{i} \\rbrace $ ; we have $C_{j} = \\hat{C}_{i}$ for $i= \\max I(j)$ .",
"In Case ${\\bf R}$ , the dec-min element $m_{{\\bf R}}$ of $B$ is uniform on each member $\\hat{S}_{i}$ of the principal partition, i.e., $m_{{\\bf R}}(s) = \\lambda _{i}$ if $s \\in \\hat{S}_{i}$ , where $i=1,2,\\ldots , r$ (cf., Proposition REF ).",
"In Case ${\\bf Z}$ , the dec-min element $m_{{\\bf Z}}$ of $\\overset{....}{B}$ is near-uniform on each member $S_{j}$ of the canonical partition, i.e., $m_{{\\bf Z}}(s) \\in \\lbrace \\beta _{j}, \\beta _{j} -1 \\rbrace $ if $s \\in S_{j}$ , where $j=1,2,\\ldots ,q$ (cf., Theorem 5.1 of Part I [9])."
],
[
"Algorithm",
"In Case ${\\bf Z}$ we have developed a strongly polynomial algorithm for finding a dec-min element of $\\overset{....}{B}$ (Section 7 of Part I [9]).",
"In Case ${\\bf R}$ the decomposition algorithm of Fujishige [11] finds the minimum norm point $m_{{\\bf R}}$ in strongly polynomial time.",
"Our proximity result (Theorem REF ) leads to the following “continuous relaxation” approach.",
"Let $\\ell = \\left\\lfloor m_{{\\bf R}} \\right\\rfloor $ and $u= \\left\\lceil m_{{\\bf R}} \\right\\rceil $ , and let $\\overset{....}{B_{\\ell }^{u}}$ denote the intersection of $\\overset{....}{B}$ with the box (interval) $[\\ell , u]= [\\ell , u]_{{\\bf R}}$ (or $T(\\ell , u)$ in the notation of Part I).",
"The dec-min element of $\\overset{....}{B_{\\ell }^{u}}$ is also a dec-min element of $\\overset{....}{B}$ , since the box $[\\ell , u]$ contains all dec-min elements of $\\overset{....}{B}$ by Theorem REF .",
"Since $0 \\le u(s) - \\ell (s) \\le 1$ for all $s \\in S$ , $\\overset{....}{B_{\\ell }^{u}}$ can be regarded as a matroid translated by $\\ell $ , i.e., $\\overset{....}{B_{\\ell }^{u}} = \\lbrace \\ell + \\chi _{L} : L \\mbox{ is a base of $M$} \\rbrace $ , where $M$ is a matroid.",
"Therefore, the dec-min element of $\\overset{....}{B_{\\ell }^{u}}$ can be computed as the minimum weight base of matroid $M$ with respect to the weight vector $w$ defined by $w(s) = u(s)^{2} - \\ell (s)^{2}$ ($s \\in S$ ).",
"By the greedy algorithm we can find the minimum weight base of $M$ in strongly polynomial time.",
"Thus the total running time of this algorithm is bounded by strongly polynomial time.",
"Variants of such continuous relaxation algorithm are given in Section REF .",
"In the literature [12], [15], [18], [24] we can find continuous relaxation algorithms that are strongly polynomial for special classes of base-polyhedra; see Appendix for details."
],
[
"Review of the principal partition",
"As is pointed out by Fujishige [11], the dec-min element in the continuous case is closely related to the principal partition.",
"The principal partition is the central concept in a structural theory for submodular functions developed mainly in Japan; Iri gives an early survey in [22] and Fujishige provides a comprehensive historical and technical account in [13].",
"In this section we summarize the results that are relevant to the analysis of the dec-min element in the continuous case.",
"Originally [11], the results are stated for a real-valued submodular function, and the present version is a translation for a real-valued supermodular function $p: 2^{S} \\rightarrow {\\bf R}\\cup \\lbrace -\\infty \\rbrace $ .",
"For any real number $\\lambda $ , let $\\mathcal {L}(\\lambda )$ denote the family of all maximizers of $p(X) - \\lambda |X|$ .",
"Then $\\mathcal {L}(\\lambda )$ is a ring family (lattice), and we denote its smallest member by $L(\\lambda )$ .",
"That is, $L(\\lambda )$ denotes the smallest maximizer of $p(X) - \\lambda |X|$ .",
"The following is a well-known basic fact.",
"The proof is included for completeness.",
"Proposition 6.1 Let $\\lambda > \\lambda ^{\\prime }$ .",
"If $X \\in \\mathcal {L}(\\lambda )$ and $Y \\in \\mathcal {L}(\\lambda ^{\\prime })$ , then $X \\subseteq Y$ .",
"In particular, $L(\\lambda ) \\subseteq L(\\lambda ^{\\prime })$ .",
"Let $X \\in \\mathcal {L}(\\lambda )$ and $Y \\in \\mathcal {L}(\\lambda ^{\\prime })$ .",
"We have $p(X) + p(Y) & \\le p(X \\cap Y) + p(X \\cup Y),\\nonumber \\\\\\lambda |X| + \\lambda ^{\\prime } |Y|& = \\lambda |X \\cap Y| + \\lambda ^{\\prime } |X \\cup Y|+ (\\lambda - \\lambda ^{\\prime }) |X - Y|\\nonumber \\\\ &\\ge \\lambda |X \\cap Y| + \\lambda ^{\\prime } |X \\cup Y| .$ It follows from these inequalities that $( p(X) - \\lambda |X| ) + ( p(Y) - \\lambda ^{\\prime } |Y| ) \\le ( p(X \\cap Y) - \\lambda |X \\cap Y| ) + ( p(X \\cup Y) - \\lambda ^{\\prime } |X \\cup Y| ) .$ Here the reverse inequality $\\ge $ is also true by $X \\in \\mathcal {L}(\\lambda )$ and $Y \\in \\mathcal {L}(\\lambda ^{\\prime })$ .",
"Therefore, we have equality in (REF ), which implies $|X - Y|=0$ , i.e., $X \\subseteq Y$ .",
"There are finitely many numbers $\\lambda $ for which $| \\mathcal {L}(\\lambda )| \\ge 2$ .",
"We denote such numbers as $\\lambda _{1} > \\lambda _{2} > \\cdots > \\lambda _{r}$ , which are called the critical values.",
"It is easy to see that $\\lambda $ is a critical value if and only if $L(\\lambda ) \\ne L(\\lambda - \\varepsilon )$ for any $\\varepsilon > 0$ .",
"The principal partition $\\lbrace \\hat{S}_{1}, \\hat{S}_{2}, \\ldots , \\hat{S}_{r} \\rbrace $ is defined by $ \\hat{S}_{i} = \\max \\mathcal {L}(\\lambda _{i}) - \\min \\mathcal {L}(\\lambda _{i})\\qquad (i=1,2,\\ldots , r),$ which says that $\\hat{S}_{i}$ is the difference of the largest and the smallest element of $\\mathcal {L}(\\lambda _{i})$ .",
"Alternatively, $ \\hat{S}_{i} = L(\\lambda _{i} - \\varepsilon ) - L(\\lambda _{i})$ for a sufficiently small $\\varepsilon > 0$ .",
"By defining $\\hat{C}_{i} = \\hat{S}_{1} \\cup \\hat{S}_{2} \\cup \\cdots \\cup \\hat{S}_{i}$ for $i=1,2,\\ldots , r$ we obtain a chain: $\\hat{C}_{1} \\subset \\hat{C}_{2} \\subset \\cdots \\subset \\hat{C}_{r}$ , where $\\hat{C}_{1} \\ne \\emptyset $ and $\\hat{C}_{r} = S$ ; we also define $\\hat{C}_{0} = \\emptyset $ .",
"Then the chain $(\\emptyset =) \\hat{C}_{0} \\subset \\hat{C}_{1} \\subset \\hat{C}_{2}\\subset \\cdots \\subset \\hat{C}_{r} \\ (=S)$ is a maximal chain of the lattice $\\bigcup _{\\lambda \\in {\\bf R}} \\mathcal {L}(\\lambda )$ .",
"In this paper we call this chain the principal chain.",
"By slight abuse of terminology the principal chain sometime means the chain $\\hat{C}_{1} \\subset \\hat{C}_{2} \\subset \\cdots \\subset \\hat{C}_{r} \\ (=S)$ without $\\hat{C}_{0}$ ($=\\emptyset )$ .",
"Let $m_{{\\bf R}} \\in {\\bf R}^{S}$ be the minimum norm point of $B$ , which is the unique dec-min element of $B$ .",
"The critical values are exactly those numbers that appear as component values of $m_{{\\bf R}}$ .",
"Moreover, the vector $m_{{\\bf R}}$ is uniform on each member $\\hat{S}_{i}$ .",
"Proposition 6.2 (Fujishige [11]) $m_{{\\bf R}}(s) = \\lambda _{i}$ if $s \\in \\hat{S}_{i}$ , where $i=1,2,\\ldots , r$ .",
"$\\rule {0.17cm}{0.17cm}$"
],
[
"New characterization of the canonical partition",
"For the discrete case, the canonical partition describes the structure of dec-min elements.",
"In particular, a dec-min element is near-uniform on each member of the canonical partition.That is, $|m_{{\\bf Z}}(s) - m_{{\\bf Z}}(t)| \\le 1$ if $\\lbrace s, t \\rbrace \\subseteq S_{j}$ for some $S_{j}$ (cf., Theorem 5.1 of Part I [9]).",
"In Part I [9], the canonical partition has been defined iteratively using contractions.",
"In this section we give a non-iterative construction of this canonical partition, which reflects the underlying structure more directly.",
"This alternative construction enables us to reveal the precise relation between the discrete and continuous cases in Section REF .",
"We first recall the iterative construction from Section 5 of Part I [9].",
"Let $p: 2^{S} \\rightarrow {\\bf Z}\\cup \\lbrace -\\infty \\rbrace $ be an integer-valued supermodular function with $p(\\emptyset )=0$ and $p(S) > -\\infty $ , and $C_{0} = \\emptyset $ .",
"For $j=1,2,\\ldots , q$ , define $\\beta _{j} &= \\max \\left\\lbrace \\left\\lceil \\frac{p(X \\cup C_{j-1}) - p(C_{j-1})}{|X|} \\right\\rceil :\\emptyset \\ne X \\subseteq \\overline{C_{j-1}} \\right\\rbrace ,\\\\h_{j}(X) &= p(X \\cup C_{j-1}) - (\\beta _{j} - 1) |X| - p(C_{j-1})\\qquad (X \\subseteq \\overline{C_{j-1}}),\\\\S_{j} &= \\mbox{smallest subset of $\\overline{C_{j-1}}$ maximizing $h_{j}$},\\\\C_{j} &= C_{j-1} \\cup S_{j} ,$ where $\\overline{C_{j-1}} = S - C_{j-1}$ and the index $q$ is determined by the condition that $C_{q-1} \\ne S$ and $C_{q} = S$ .",
"According to the above definitions, we have that $ \\mbox{$C_{j}$ is the smallest maximizer of$p(X) - (\\beta _{j}-1) |X|$among all $Z \\supseteq C_{j-1}$}.$ We will show in Proposition REF below that $C_{j}$ is, in fact, the smallest maximizer of $p(X) - (\\beta _{j}-1) |X|$ among all subsets $X$ of $S$ .",
"For any integer $\\beta $ , let $\\mathcal {L}(\\beta )$ denote the family of all maximizers of $p(X) - \\beta |X|$ , and $L(\\beta )$ be the smallest element of $\\mathcal {L}(\\beta )$ , where the smallest element exists in $\\mathcal {L}(\\beta )$ since $\\mathcal {L}(\\beta )$ is a lattice (ring family).",
"(These notations are consistent with the ones introduced in Section REF .)",
"Proposition 6.3 (1) $\\beta _{1} > \\beta _{2} > \\cdots > \\beta _{q}$ .",
"(2) For each $j$ with $1 \\le j \\le q$ , $C_{j}$ is the smallest maximizer of $p(X) - (\\beta _{j}-1) |X|$ among all subsets $X$ of $S$ .",
"(1) The monotonicity of the $\\beta $ -values is already shown in Section 5 of Part I [9], but we give an alternative proof here.",
"Let $j \\ge 2$ .",
"By (REF ), we have $\\beta _{j-1} > \\beta _{j}$ if and only if $ \\beta _{j-1} > \\left\\lceil \\frac{p(X \\cup C_{j-1}) - p(C_{j-1})}{|X|} \\right\\rceil $ for every $X$ with $\\emptyset \\ne X \\subseteq \\overline{C_{j-1}}$ .",
"Furthermore, $(\\ref {prf1betaj1betaj})& \\iff \\beta _{j-1} -1 \\ge \\frac{p(X \\cup C_{j-1}) - p(C_{j-1})}{|X|}\\\\ & \\iff p(X \\cup C_{j-1}) - p(C_{j-1}) \\le (\\beta _{j-1} -1 ) |X|\\\\ & \\iff p(X \\cup C_{j-1}) - (\\beta _{j-1} -1 ) |X \\cup C_{j-1}| \\le p(C_{j-1}) - (\\beta _{j-1} -1 ) |C_{j-1}|.$ The last inequality holds, since the set $X \\cup C_{j-1}$ contains $C_{j-2}$ , whereas $C_{j-1}$ is the (smallest) maximizer of $p(X) - (\\beta _{j-1}-1) |X|$ among all $X$ containing $C_{j-2}$ .",
"We have thus shown $\\beta _{j-1} > \\beta _{j}$ .",
"(2) We prove $C_{j}= L(\\beta _{j} -1)$ for $j=1,2,\\ldots ,q$ by induction on $j$ .",
"This holds for $j=1$ by definition.",
"Let $j \\ge 2$ .",
"By Proposition REF for $\\lambda = \\beta _{j-1}-1$ and $\\lambda ^{\\prime } = \\beta _{j}-1$ , the smallest maximizer of $p(X) - (\\beta _{j}-1) |X|$ is a superset of $L(\\beta _{j-1} -1)$ , where $L(\\beta _{j-1} -1) =C_{j-1}$ by the induction hypothesis.",
"Combining this with (REF ), we obtain $C_{j}= L(\\beta _{j} -1)$ .",
"We now give an alternative characterization of the essential value-sequence $\\beta _{1} > \\beta _{2} > \\cdots > \\beta _{q}$ defined by (REF )–().",
"We consider the family $\\lbrace L(\\beta ) : \\beta \\in {\\bf Z}\\rbrace $ of the smallest maximizers of $p(X) - \\beta |X|$ for all integers $\\beta $ .",
"Each $C_{j}$ is a member of this family, since $C_{j} = L(\\beta _{j} - 1)$ ($j=1,2,\\ldots , q$ ) by Proposition REF (2).",
"Proposition 6.4 As $\\beta $ is decreased from $+\\infty $ to $-\\infty $ (or from $\\beta _{1}$ to $\\beta _{q}-1$ ), the smallest maximizer $L(\\beta )$ is monotone nondecreasing.",
"We have $L(\\beta ) \\ne L(\\beta -1)$ if and only if $\\beta $ is equal to an essential value.",
"Therefore, the essential value-sequence $\\beta _{1} > \\beta _{2} > \\cdots > \\beta _{q}$ is characterized by the propertyRecall that “$\\subset $ ” means “$\\subseteq $ and $\\ne $ .” $\\emptyset = L(\\beta _{1}) \\subset L(\\beta _{1}-1)= \\cdots = L(\\beta _{2}) \\subset L(\\beta _{2}-1)= \\cdots = L(\\beta _{q}) \\subset L(\\beta _{q}-1) = S.$ The monotonicity of $L(\\beta )$ follows from Proposition REF .",
"We will show (i) $L(\\beta _{1}) = \\emptyset $ , (ii) $L(\\beta _{j-1} -1) = L(\\beta _{j})$ for $j=2,\\ldots , q$ , and (iii) $L(\\beta _{j}) \\subset L(\\beta _{j}-1)$ for $j=1,2,\\ldots , q$ .",
"(i) Since $\\beta _{1} = \\max \\left\\lbrace \\left\\lceil p(X)/|X| \\right\\rceil : X \\ne \\emptyset \\right\\rbrace $ , we have $p(X) - \\beta _{1} |X| \\le 0$ for all $X \\ne \\emptyset $ , whereas $p(X) - \\beta _{1} |X| = 0$ for $X = \\emptyset $ .",
"Therefore, $L(\\beta _{1}) = \\emptyset $ .",
"(ii) Let $2 \\le j \\le q$ .",
"For short we write $C = C_{j-1}$ .",
"Define $h(Y) = p(Y) - \\beta _{j} |Y|$ for any subset $Y$ of $S$ , and let $A$ be the smallest maximizer of $h$ , which means $A = L(\\beta _{j})$ .",
"For any nonempty subset $X$ of $\\overline{C} \\ (= S - C)$ we have $& \\beta _{j}\\ge \\left\\lceil \\frac{p(X \\cup C) - p(C)}{|X|} \\right\\rceil \\ge \\frac{p(X \\cup C) - p(C)}{|X|} ,$ which implies $p(X \\cup C) - \\beta _{j} |X \\cup C| \\le p(C) - \\beta _{j} |C|$ , that is, $h(Y) \\le h(C)\\qquad \\mbox{for all \\ $Y \\supseteq C$}.$ By supermodularity of $p$ we have $ h(A) + h(C) \\le h(A \\cup C) + h(A \\cap C)$ , whereas $h(C) \\ge h(A \\cup C)$ by (REF ).",
"Therefore, $ h(A) \\le h(A \\cap C)$ .",
"Since $A$ is the smallest maximizer of $h$ , this implies that $A = A \\cap C$ , i.e., $A \\subseteq C$ .",
"Recalling $A = L(\\beta _{j})$ and $C = C_{j-1}= L(\\beta _{j-1} -1)$ , we obtain $L(\\beta _{j}) \\subseteq L(\\beta _{j-1} -1)$ .",
"We also have $L(\\beta _{j}) \\supseteq L(\\beta _{j-1} -1)$ by the monotonicity.",
"Therefore, $L(\\beta _{j}) = L(\\beta _{j-1} -1)$ .",
"(iii) Let $1 \\le j \\le q$ .",
"We continue to write $C = C_{j-1}$ .",
"Take a nonempty subset $Z$ of $\\overline{C}$ which gives the maximum in the definition of $\\beta _{j}$ , i.e., $& \\beta _{j}= \\max \\left\\lbrace \\left\\lceil \\frac{p(X \\cup C) - p(C)}{|X|} \\right\\rceil : \\emptyset \\ne X \\subseteq \\overline{C} \\right\\rbrace = \\left\\lceil \\frac{p(Z \\cup C) - p(C)}{|Z|} \\right\\rceil .$ Then we have $\\frac{p(Z \\cup C) - p(C)}{|Z|} > \\beta _{j} -1 ,$ which implies $p(Z \\cup C) - (\\beta _{j}-1) |Z \\cup C| > p(C) - (\\beta _{j}-1) |C|.$ This shows that $C = C_{j-1}= L(\\beta _{j-1} -1)$ is not a maximizer of $p(Y) - (\\beta _{j}-1) |Y|$ , and hence $L(\\beta _{j-1} -1) \\ne L(\\beta _{j}-1)$ .",
"On the other hand, we have $L(\\beta _{j-1} -1) = L(\\beta _{j})$ by (ii) and $L(\\beta _{j}) \\subseteq L(\\beta _{j}-1)$ by the monotonicity in Proposition REF .",
"Therefore, $L(\\beta _{j}) \\subset L(\\beta _{j}-1)$ .",
"Proposition REF justifies the following alternative definition of the essential value-sequence, the canonical chain, and the canonical partition: Consider the smallest maximizer $L(\\beta )$ of $p(X) - \\beta |X|$ for all integers $\\beta $ .",
"There are finitely many $\\beta $ for which $L(\\beta ) \\ne L(\\beta -1)$ .",
"Denote such integers as $\\beta _{1} > \\beta _{2} > \\cdots > \\beta _{q}$ and call them the essential value-sequence.",
"Furthermore, define $C_{j} = L(\\beta _{j}-1)$ for $j=1,2,\\ldots , q$ to obtain a chain: $C_{1} \\subset C_{2} \\subset \\cdots \\subset C_{q}$ .",
"Call this the canonical chain.",
"Finally define a partition $\\lbrace S_{1}, S_{2}, \\ldots , S_{q} \\rbrace $ of $S$ by $S_{j}=C_{j} - C_{j-1}$ for $j=1,2,\\ldots , q$ , where $C_{0} = \\emptyset $ , and call this the canonical partition.",
"This alternative construction clearly exhibits the parallelism between the canonical partition in Case ${\\bf Z}$ and the principal partition in Case ${\\bf R}$ .",
"In particular, the essential value-sequence is exactly the discrete counterpart of the critical values.",
"This is discussed in the next section."
],
[
"Canonical partition from the principal partition",
"The characterization of the canonical partition shown in Section REF enables us to obtain the canonical partition for Case ${\\bf Z}$ from the principal partition for Case ${\\bf R}$ as follows.",
"Theorem 6.5 (1) An integer $\\beta $ is an essential value if and only if there exists a critical value $\\lambda $ satisfying $\\beta \\ge \\lambda > \\beta -1$ .",
"(2) The essential values $\\beta _{1} > \\beta _{2} > \\cdots > \\beta _{q}$ are obtained from the critical values $\\lambda _{1} > \\lambda _{2} > \\cdots > \\lambda _{r}$ as the distinct members of the rounded-up integers $\\lceil \\lambda _{1} \\rceil \\ge \\lceil \\lambda _{2} \\rceil \\ge \\cdots \\ge \\lceil \\lambda _{r} \\rceil $ .",
"Let $I(j) = \\lbrace i : \\lceil \\lambda _{i} \\rceil = \\beta _{j} \\rbrace $ for $j=1,2,\\ldots , q$ .",
"(3) The canonical partition $\\lbrace S_{1}, S_{2}, \\ldots , S_{q} \\rbrace $ is obtained from the principal partition $\\lbrace \\hat{S}_{1}, \\hat{S}_{2}, \\ldots , \\hat{S}_{r} \\rbrace $ as an aggregation; it is given as $ S_{j} = \\bigcup _{i \\in I(j)} \\hat{S}_{i}\\qquad (j=1,2,\\ldots , q).$ (4) The canonical chain $\\lbrace C_{j} \\rbrace $ is a subchain of the principal chain $\\lbrace \\hat{C}_{i} \\rbrace $ ; it is given as $C_{j} = \\hat{C}_{i}$ for $i= \\max I(j)$ .",
"$\\rule {0.17cm}{0.17cm}$ In Case ${\\bf R}$ , the dec-min element $m_{{\\bf R}}$ of $B$ is uniform on each member $\\hat{S}_{i}$ of the principal partition, i.e., $m_{{\\bf R}}(s) = \\lambda _{i}$ if $s \\in \\hat{S}_{i}$ , where $i=1,2,\\ldots , r$ (cf., Proposition REF ).",
"In Case ${\\bf Z}$ , the dec-min element $m_{{\\bf Z}}$ of $\\overset{....}{B}$ is near-uniform on each member $S_{j}$ of the canonical partition, i.e., $m_{{\\bf Z}}(s) \\in \\lbrace \\beta _{j}, \\beta _{j} -1 \\rbrace $ if $s \\in S_{j}$ , where $j=1,2,\\ldots ,q$ (cf., Theorem 5.1 of Part I [9]).",
"Combining these results with Theorem REF above we can obtain a (strong) proximity theorem for dec-min elements.",
"Theorem 6.6 (Proximity) Let $m_{{\\bf R}}$ be the minimum norm point of $B$ .",
"Then every dec-min element $m_{{\\bf Z}}$ of $\\overset{....}{B}$ satisfies $\\left\\lfloor m_{{\\bf R}} \\right\\rfloor \\le m_{{\\bf Z}} \\le \\left\\lceil m_{{\\bf R}} \\right\\rceil $ .",
"For $s \\in S$ let $\\hat{S}_{i}$ denote the member of the principal partition containing $s$ , and $\\lambda _{i}$ be the associated critical value.",
"We have $m_{{\\bf R}}(s) = \\lambda _{i}$ by Proposition REF .",
"Let $\\beta _{j} = \\lceil \\lambda _{i} \\rceil $ .",
"This is an essential value, and the corresponding member $S_{j}$ of the canonical partition contains the element $s$ by Theorem REF .",
"We have $m_{{\\bf Z}}(s) \\in \\lbrace \\beta _{j}, \\beta _{j} -1 \\rbrace $ by Theorem 5.1 of Part I [9].",
"Therefore, $m_{{\\bf Z}} \\le \\left\\lceil m_{{\\bf R}} \\right\\rceil $ .",
"Next we apply the above argument to $-B$ , which is an integral base-polyhedron.",
"Since $-m_{{\\bf R}}$ is the minimum norm point of $-B$ and $-m_{{\\bf Z}}$ is a dec-min (=inc-max) element for $-\\overset{....}{B}$ , we obtain $-m_{{\\bf Z}} \\le \\left\\lceil -m_{{\\bf R}} \\right\\rceil $ , which is equivalent to $m_{{\\bf Z}} \\ge \\left\\lfloor m_{{\\bf R}} \\right\\rfloor $ .",
"Remark 6.1 Theorem REF implies a weaker statement that $ \\mbox{There exists a dec-min element $m_{{\\bf Z}}$ of $\\overset{....}{B}$ satisfying$\\left\\lfloor m_{{\\bf R}} \\right\\rfloor \\le m_{{\\bf Z}} \\le \\left\\lceil m_{{\\bf R}} \\right\\rceil $,}$ where $m_{{\\bf R}}$ is the minimum norm point of $B$ .",
"This statement (REF ) should not be confused with Proposition REF in Section REF , which is another proximity statement referring to a minimizer of the piecewise extension of the quadratic function, not to the minimum norm point (minimizer of the quadratic function itself).",
"$\\rule {0.17cm}{0.17cm}$ Theorem 6.7 The minimum norm point of $B$ can be represented as a convex combination of the dec-min elements of $\\overset{....}{B}$ .",
"On one hand, it was shown in Section 5.1 of Part I [9] that the dec-min elements of $\\overset{....}{B}$ lie on the face $B^{\\oplus }$ of $B$ defined by the canonical chain $C_{1} \\subset C_{2} \\subset \\cdots \\subset C_{q}$ .",
"This face is the intersection of $B$ with the hyperplanes $\\lbrace x\\in {\\bf R}^{S}: \\widetilde{x}(C_{j}) = p(C_{j}) \\rbrace $ $(j=1,2,\\ldots , q)$ .",
"On the other hand, it is known ([11], [12]) that the minimum norm point $m_{{\\bf R}}$ of $B$ lies on the face of $B$ defined by the principal chain $\\hat{C}_{1} \\subset \\hat{C}_{2} \\subset \\cdots \\subset \\hat{C}_{r}$ , which is the intersection of $B$ with the hyperplanes $\\lbrace x\\in {\\bf R}^{S}: \\widetilde{x}(\\hat{C}_{i}) = p(\\hat{C}_{i}) \\rbrace $ $(i=1,2,\\ldots , r)$ .",
"Since the principal chain is a refinement of the canonical chain (Theorem REF ), the latter face is a face of $B^{\\oplus }$ .",
"Therefore, $m_{{\\bf R}}$ belongs to $B^{\\oplus }$ .",
"The point $m_{{\\bf R}}$ also belongs to $T^{*}= \\lbrace x\\in {\\bf R}^{S}: \\ \\beta _{j}-1\\le x(s)\\le \\beta _{j}\\ \\ \\hbox{whenever}\\ s\\in S_{j} \\ (j=1,2,\\dots ,q)\\rbrace $ , since $m_{{\\bf R}}(s) = \\lambda _{i}$ for $s \\in \\hat{S}_{i}$ (Proposition REF ) and $ S_{j}= \\bigcup \\lbrace \\hat{S}_{i}: \\lceil \\lambda _{i} \\rceil = \\beta _{j} \\rbrace $ (Theorem REF ).",
"Therefore, $m_{{\\bf R}}$ is a member of $B^{\\bullet }= B^{\\oplus }\\cap T^{*}$ .",
"By recalling that $B^{\\bullet }$ is an integral base-polyhedron whose vertices are precisely the dec-min elements of $\\overset{....}{B}$ , we conclude that $m_{{\\bf R}}$ can be represented as a convex combination of the dec-min elements of $\\overset{....}{B}$ .",
"The following two examples illustrate Theorem REF .",
"Example 6.2 Let $S = \\lbrace s_1 , s_2 \\rbrace $ and $\\overset{....}{B} = \\lbrace (0,3), (1,2), (2,1) \\rbrace $ , where $B$ is the line segment connecting $(0,3)$ and $(2,1)$ .",
"For $\\overset{....}{B}$ there are two dec-min elements: $m_{{\\bf Z}}^{(1)}=(1,2)$ and $m_{{\\bf Z}}^{(2)}=(2,1)$ .",
"The minimum norm point (dec-min element) of $B$ is $m_{{\\bf R}}=(3/2,3/2)$ .",
"The supermodular function $p$ is given by $p(\\emptyset )= 0,\\quad p(\\lbrace s_{1} \\rbrace )= 0,\\quad p(\\lbrace s_{2} \\rbrace )= 1,\\quad p(\\lbrace s_{1},s_{2} \\rbrace ) = 3,$ and we have $ p(X) - \\lambda |X| =\\left\\lbrace \\begin{array}{ll}0 & (X = \\emptyset ), \\\\-\\lambda & (X = \\lbrace s_{1} \\rbrace ), \\\\1-\\lambda & (X = \\lbrace s_{2} \\rbrace ), \\\\3- 2\\lambda & (X = \\lbrace s_{1}, s_{2} \\rbrace ).",
"\\\\\\end{array} \\right.$ There is only one ($r=1$ ) critical value $\\lambda _{1} = 3/2$ and the associated sublattice is $\\mathcal {L}(\\lambda _{1}) = \\lbrace \\emptyset , S \\rbrace $ .",
"The principal partition is a trivial partition $\\lbrace S \\rbrace $ .",
"Since $\\lceil \\lambda _{1} \\rceil = 2$ , we have $\\beta _{1} = 2$ with $q=1$ , and the (only) member $S_{1}$ in the canonical partition is given by $S_{1} = L(\\beta _{1}-1)= L(1) = S$ .",
"Accordingly, the canonical chain consists of only one member ${C}_{1} = S$ .",
"$\\rule {0.17cm}{0.17cm}$ Example 6.3 We consider Example REF again.",
"We have $S = \\lbrace s_1, s_2, s_3, s_4 \\rbrace $ and $\\overset{....}{B}$ consists of five vectors: $ m_{1}=(2,1,1,0)$ , $m_{2}=(2,1,0,1)$ , $m_{3}=(1,2,1,0)$ , $m_{4}=(1,2,0,1)$ , and $m_{5}=(2,2,0,0)$ , of which the first four members, $m_{1}$ to $m_{4}$ , are the dec-min elements.",
"The supermodular function $p$ is given by $& p(\\emptyset )= 0, \\quad p(\\lbrace s_{1} \\rbrace )= p(\\lbrace s_{2} \\rbrace )= 1, \\quad p(\\lbrace s_{3} \\rbrace )= p(\\lbrace s_{4} \\rbrace )= 0,\\\\ &p(\\lbrace s_{1},s_{2} \\rbrace ) = 3, \\quad p(\\lbrace s_{3},s_{4} \\rbrace ) = 0, \\quad p(\\lbrace s_{1},s_{3} \\rbrace ) =p(\\lbrace s_{2},s_{3} \\rbrace ) =p(\\lbrace s_{1},s_{4} \\rbrace ) =p(\\lbrace s_{2},s_{4} \\rbrace ) = 1,\\\\ &p(\\lbrace s_{1},s_{2}, s_{3} \\rbrace ) =p(\\lbrace s_{1},s_{2}, s_{4} \\rbrace ) = 3, \\quad p(\\lbrace s_{1},s_{3}, s_{4} \\rbrace ) =p(\\lbrace s_{2},s_{3}, s_{4} \\rbrace ) = 2, \\quad \\\\ &p(\\lbrace s_{1},s_{2},s_{3}, s_{4} \\rbrace ) = 4.$ We have $ \\max \\lbrace p(X) - \\lambda |X| : X \\subseteq S \\rbrace =\\max \\lbrace 0, \\ 1- \\lambda , \\ 3- 2\\lambda , \\ 3- 3\\lambda , \\ 4- 4\\lambda \\rbrace .$ There are two ($r=2$ ) critical values $\\lambda _{1} = 3/2$ and $\\lambda _{2} = 1/2$ , with the associated sublattices $\\mathcal {L}(\\lambda _{1}) = \\lbrace \\emptyset , \\lbrace s_{1},s_{2} \\rbrace \\rbrace $ and $\\mathcal {L}(\\lambda _{2}) = \\lbrace \\lbrace s_{1},s_{2} \\rbrace , S \\rbrace $ .",
"The principal chain is given by $\\emptyset \\subset \\lbrace s_{1},s_{2} \\rbrace \\subset S$ , and the principal partition is a bipartition with $\\hat{S}_{1} = \\lbrace s_{1},s_{2} \\rbrace $ and $\\hat{S}_{2} = \\lbrace s_{3},s_{4} \\rbrace $ .",
"The minimum norm point of the base-polyhedron $B$ is given by $m_{{\\bf R}} = (3/2, 3/2, 1/2, 1/2 )$ by Proposition REF .",
"Since $\\lceil \\lambda _{1} \\rceil = 2$ and $\\lceil \\lambda _{2} \\rceil = 1$ , we have $\\beta _{1} = 2$ and $\\beta _{2} = 1$ with $q=2$ .",
"The canonical chain consists of two members $C_{1} = L(\\beta _{1}-1)= L(1) = \\lbrace s_{1},s_{2} \\rbrace $ and $C_{2} = L(\\beta _{2}-1)= L(0) = S$ .",
"Accordingly, the canonical partition is given by $S_{1} = \\lbrace s_{1},s_{2} \\rbrace $ and $S_{2} = \\lbrace s_{3},s_{4} \\rbrace $ .",
"$\\rule {0.17cm}{0.17cm}$"
],
[
"Continuous relaxation algorithms",
"In Section 7 of Part I [9], we have presented a strongly polynomial algorithm for finding a dec-min element of $\\overset{....}{B}$ as well as for finding the canonical partition.",
"This is based on an iterative approach to construct a dec-min element along the canonical chain.",
"By making use of the relation between Case ${\\bf R}$ and Case ${\\bf Z}$ , we can construct continuous relaxation algorithms, which first compute a real (fractional) vector that is guaranteed to be close to an integral dec-min element, and then find the integral dec-min element by solving a linearly weighted matroid optimization problem.",
"In our continuous relaxation algorithms, we first apply some algorithm for Case ${\\bf R}$ to find two integer vectors $\\ell $ and $u$ such that $\\mathbf {0} \\le u - \\ell \\le \\mathbf {1}$ , (i.e., $0 \\le u(s) - \\ell (s) \\le 1$ for all $s \\in S$ ) and the box $[\\ell , u]$ contains at least one dec-min element of $\\overset{....}{B}$ , i.e., $ \\ell \\le m_{{\\bf Z}} \\le u$ for some dec-min element $m_{{\\bf Z}}$ of $\\overset{....}{B}$ .",
"We denote the intersection of $\\overset{....}{B}$ and $[\\ell , u]$ by $\\overset{....}{B_{\\ell }^{u}}$ .",
"Then the dec-min element of $\\overset{....}{B_{\\ell }^{u}}$ is a dec-min element of $\\overset{....}{B}$ .",
"Since $\\mathbf {0} \\le u - \\ell \\le \\mathbf {1}$ , $\\overset{....}{B_{\\ell }^{u}}$ can be regarded as a matroid translated by $\\ell $ , i.e., $\\overset{....}{B_{\\ell }^{u}} = \\lbrace \\ell + \\chi _{L} : L \\mbox{ is a base of $M$} \\rbrace $ for some matroid $M$ .",
"Therefore, the dec-min element of $\\overset{....}{B_{\\ell }^{u}}$ can be computed as the minimum weight base of matroid $M$ with respect to the weight vector $w$ defined by $w(s) = u(s)^{2} - \\ell (s)^{2}$ ($s \\in S$ ).",
"By the greedy algorithm we can find the minimum weight base of $M$ in strongly polynomial time.",
"We can conceive two different algorithms for finding vectors $\\ell $ and $u$ ."
],
[
"(a) Using the minimum norm point",
"In Theorem REF we have shown that every dec-min element $m_{{\\bf Z}}$ of $\\overset{....}{B}$ satisfies $\\left\\lfloor m_{{\\bf R}} \\right\\rfloor \\le m_{{\\bf Z}} \\le \\left\\lceil m_{{\\bf R}} \\right\\rceil $ for the minimum norm point $m_{{\\bf R}}$ of $B$ .",
"Therefore, we can choose $\\ell = \\left\\lfloor m_{{\\bf R}} \\right\\rfloor $ and $u= \\left\\lceil m_{{\\bf R}} \\right\\rceil $ in (REF ).",
"With this choice of $(\\ell , u)$ , $\\overset{....}{B_{\\ell }^{u}}$ contains all dec-min elements of $\\overset{....}{B}$ .",
"The decomposition algorithm of Fujishige [11] (see also [12]) finds the minimum norm point $m_{{\\bf R}}$ in strongly polynomial time.",
"Therefore, the continuous relaxation algorithm using the minimum norm point is a strongly polynomial algorithm.",
"Example 6.4 We continue with Example REF , where $\\overset{....}{B}$ consists of five vectors: $ m_{1}=(2,1,1,0)$ , $m_{2}=(2,1,0,1)$ , $m_{3}=(1,2,1,0)$ , $m_{4}=(1,2,0,1)$ , and $m_{5}=(2,2,0,0)$ .",
"From the minimum norm point $m_{{\\bf R}} = (3/2, 3/2, 1/2, 1/2 )$ , we obtain $\\ell = (1,1,0,0)$ and $u=(2,2,1,1)$ , and hence $w = (3,3,1,1)$ .",
"Since $W(m_{i})=10$ for $i=1,\\ldots ,4$ and $W(m_{5})=12$ , the dec-min elements are given by $m_{1}$ to $m_{4}$ .",
"$\\rule {0.17cm}{0.17cm}$ The algorithm of Groenevelt [15] (see also [12]) employs a piecewise-linear extension of the objective function.",
"For the quadratic function $\\varphi (k)=k^{2}$ , the piecewise-linear extension $\\overline{\\varphi }: {\\bf R}\\rightarrow {\\bf R}$ is given by: $\\overline{\\varphi }(t)= (2k-1) t - k (k-1)$ if $k-1 \\le |t| \\le k$ for $k \\in {\\bf Z}$ .",
"The following proximity property is a special case of an observation of Groenevelt [15] (see also [12]).",
"Proposition 6.8 (Groenevelt [15]) For any minimizer $\\overline{m}_{{\\bf R}} \\in {\\bf R}^{S}$ of the function $\\overline{\\Phi }(x) = \\sum _{s \\in S} \\overline{\\varphi } ( x(s) )$ over $B$ , there exists a minimizer $m_{{\\bf Z}} \\in {\\bf Z}^{S}$ of $\\Phi (x) = \\sum _{s \\in S} x(s)^{2}$ over $\\overset{....}{B}$ satisfying $\\lfloor \\overline{m}_{{\\bf R}} \\rfloor \\le m_{{\\bf Z}} \\le \\lceil \\overline{m}_{{\\bf R}} \\rceil $ .",
"(We give a proof for completeness, though it is easy and standard.)",
"By the integrality of $B$ , we can express $\\overline{m}_{{\\bf R}}$ as a convex combination of integral member $z_{1}, z_{2}, \\ldots , z_{k}$ of $B$ satisfying $\\lfloor \\overline{m}_{{\\bf R}} \\rfloor \\le z_{i} \\le \\lceil \\, \\overline{m}_{{\\bf R}} \\rceil $ $(i=1,2,\\ldots ,k)$ , where $\\overline{m}_{{\\bf R}} = \\sum _{i=1}^{k} \\lambda _{i} z_{i}$ with $\\sum _{i=1}^{k} \\lambda _{i} =1$ and $\\lambda _{i} > 0 $ $(i=1,2,\\ldots ,k)$ .",
"Since $\\overline{\\Phi }$ is piecewise-linear, we have $\\overline{\\Phi }(\\overline{m}_{{\\bf R}}) =\\sum _{i=1}^{k} \\lambda _{i} \\Phi (z_{i})$ , in which $\\Phi (z_{i}) = \\overline{\\Phi }(z_{i}) \\ge \\overline{\\Phi }(\\overline{m}_{{\\bf R}})$ .",
"Therefore, $z_{1}, z_{2}, \\ldots , z_{k}$ are the minimizers of $\\Phi $ on $\\overset{....}{B}$ .",
"We can take any $z_{i}$ as $m_{{\\bf Z}}$ .",
"By Proposition REF we can take $\\ell = \\lfloor \\overline{m}_{{\\bf R}} \\rfloor $ and $u = \\lceil \\overline{m}_{{\\bf R}} \\rceil $ in (REF ).",
"In this case, however, $\\overset{....}{B_{\\ell }^{u}}$ may not contain all dec-min elements of $\\overset{....}{B}$ .",
"The complexity of computing $\\overline{m}_{{\\bf R}}$ is not fully analyzed in the literature [12], [15], [24].",
"See also Remark REF .",
"Remark 6.2 Minimization of a separable convex function on a base-polyhedron has been investigated in the literature of resource allocation under the name of “resource allocation problems under submodular constraints” (Hochbaum [18], Ibaraki–Katoh [20], Katoh–Ibaraki [23], Katoh–Shioura–Ibaraki [24]).",
"The continuous relaxation approach for the case of discrete variables is considered, e.g., by Hochbaum [17] and Hochbaum–Hong [19].",
"A more recent paper by Moriguchi–Shioura–Tsuchimura [29] discusses this approach in a more general context of M-convex function minimization in discrete convex analysis.",
"It is known ([19], [29], [24]) that a convex quadratic function $\\sum a_{i} x_{i}^{2}$ in discrete variables can be minimized over an integral base-polyhedron in strongly polynomial time if the base-polyhedron has a special structure like “Nested”, “Tree,” or “Network” in the terminology of [24].",
"$\\rule {0.17cm}{0.17cm}$"
],
[
"Min-max formulas for separable convex functions in DCA",
"The objective of this section is to pave the way of DCA approach to discrete decreasing minimization on other discrete structures such as the intersection of M-convex sets, network flows, submodular flows.",
"Min-max formulas for separable convex functions on the intersection of M-convex sets and ordinary/submodular flows are presented.",
"In Section REF we have considered the min-max formula $ \\min \\lbrace \\sum _{s \\in S} \\varphi _{s} ( x(s) ) : x \\in \\overset{....}{B} \\rbrace = \\max \\lbrace \\hat{p}(\\pi ) - \\sum _{s \\in S} \\psi _{s}(\\pi (s)) : \\pi \\in {\\bf Z}^{S} \\rbrace $ for a separable convex function on an M-convex set.",
"Here, $p$ is an integer-valued (fully) supermodular function on $S$ , $B$ is the base-polyhedron defined by $p$ , $\\overset{....}{B}$ is the set of integral points of $B$ , and $\\hat{p}$ is the linear extension (Lovász extension) of $p$ .",
"For each $s \\in S$ , $\\varphi _{s}: {\\bf Z}\\rightarrow {\\bf Z}\\cup \\lbrace +\\infty \\rbrace $ is an integer-valued (discrete) convex function and $\\psi _{s}$ is the conjugate function of $\\varphi _{s}$ .",
"Furthermore, the sets of primal and dual optimal solutions of (REF ) are described in Section REF .",
"These results have been used for the DCA-based proofs of some key results on decreasing minimization on an M-convex set in Sections REF , REF , and REF .",
"The min-max formula (REF ) for the square-sum has been obtained as a special case of (REF ) where the conjugate functions can be given explicitly.",
"To emphasize the role of explicit forms of conjugate functions, we offer in Section REF several examples of (discrete) convex functions that admit explicit expressions of conjugate functions.",
"These worked-out examples of conjugate functions and min-max formulas will hopefully trigger other applications of discrete convex analysis."
],
[
"Examples of explicit conjugate functions",
"In this section we offer several examples of (discrete) convex functions whose conjugate functions can be given explicitly.",
"An explicit representation of the conjugate function renders an easily checkable certificate of optimality in the min-max formulas such as (REF ).",
"For an integer-valued discrete convex function $\\varphi : {\\bf Z}\\rightarrow {\\bf Z}\\cup \\lbrace +\\infty \\rbrace $ , we denote its conjugate function $\\varphi ^{\\bullet }$ by $\\psi $ .",
"That is, function $\\psi : {\\bf Z}\\rightarrow {\\bf Z}\\cup \\lbrace +\\infty \\rbrace $ is defined by $ \\psi (\\ell ) = \\max \\lbrace k \\ell - \\varphi (k) : k \\in {\\bf Z}\\rbrace \\qquad (\\ell \\in {\\bf Z}).$ Obviously, we have $ \\varphi (k) + \\psi ( \\ell ) \\ge k \\ell \\qquad (k,\\ell \\in {\\bf Z}),$ which is known as the Fenchel–Young inequality, and the equality holds in (REF ) if and only if $ \\varphi (k) - \\varphi (k-1) \\le \\ell \\le \\varphi (k+1) - \\varphi (k).$ It is worth noting that, for $a, b, c \\in {\\bf Z}$ , the conjugate of the function $\\varphi _{a,b,c}(k) = \\varphi (k-a) + bk + c$ is given by $\\psi (\\ell - b) + a (\\ell -b) -c$ .",
"With abuse of notation we express this as $ \\big ( \\varphi (k-a) + bk + c \\big )^{\\bullet } = \\psi (\\ell - b) + a (\\ell -b) -c.$ In what follows we demonstrate how to calculate the conjugate functions for piecewise-linear functions, $\\ell _{1}$ -distances, quadratic functions, power products, and exponential functions."
],
[
"Piecewise-linear functions",
"Let $a$ be an integer.",
"For a piecewise-linear function $\\varphi $ defined by $ \\varphi (k) = (k-a)^{+} = \\max \\lbrace 0, k-a\\rbrace \\qquad (k \\in {\\bf Z}),$ the conjugate function $\\psi $ is given by $ \\psi (\\ell ) &=&\\left\\lbrace \\begin{array}{ll}0 & (\\ell = 0) , \\\\a & (\\ell = 1) , \\\\+\\infty & (\\ell \\notin \\lbrace 0,1 \\rbrace ) .",
"\\\\\\end{array} \\right.$ This explicit form can be used in the DCA-based proof of Theorem REF ; see Remark REF .",
"For another piecewise-linear function $\\varphi $ defined by $ \\varphi (k) =\\left\\lbrace \\begin{array}{ll}0 & (0 \\le k \\le a) , \\\\\\lambda (k - a) & (a \\le k \\le b) , \\\\+\\infty & (k \\le -1 \\mbox{ or } k \\ge b+1) \\\\\\end{array} \\right.$ for $a, b, \\lambda \\in {\\bf Z}$ with $0 \\le a \\le b$ and $\\lambda \\ge 0$ , the conjugate function $\\psi $ is given by $ \\psi (\\ell ) &=&\\left\\lbrace \\begin{array}{ll}0 & (\\ell \\le 0) , \\\\a \\ell & (0 \\le \\ell \\le \\lambda ) , \\\\b \\ell - (b - a) \\lambda & (\\ell \\ge \\lambda ) .",
"\\\\\\end{array} \\right.$"
],
[
"$\\ell _{1}$ -distances",
"Let $a$ be an integer.",
"For function $\\varphi $ defined by $ \\varphi (k) = | k - a |\\qquad (k \\in {\\bf Z}),$ the conjugate function $\\psi $ is given by $ \\psi ( \\ell ) =\\left\\lbrace \\begin{array}{ll}a \\ell & (\\ell =-1,0,+1) , \\\\+\\infty & (\\mbox{otherwise}) .\\end{array}\\right.$ A min-max relation for the minimum $\\ell _{1}$ -distance between an integer point of $B$ and a given integer point $c$ can be obtained from the min-max formula (REF ).",
"Recall that $p$ and $b$ are, respectively, the supermodular and submodular functions associated with $B$ , and our convention $\\widetilde{c}(X) = \\sum \\lbrace c(s): s \\in X \\rbrace $ .",
"Proposition 7.1 For $c \\in {\\bf Z}^{S}$ , $ & \\min \\lbrace \\sum _{s \\in S} |x(s) - c(s)| : x \\in \\overset{....}{B} \\rbrace \\nonumber \\\\&= \\max \\lbrace p(X) - b(Y)- \\widetilde{c}(X) + \\widetilde{c}(Y): X, Y \\subseteq S; \\ X \\cap Y = \\emptyset \\rbrace .$ We choose $\\varphi _{s}(k) = | k - c(s)|$ in (REF ).",
"By (REF ), we may assume $\\pi \\in \\lbrace -1,0,+1 \\rbrace ^{S}$ on the right-hand side of (REF ).",
"On representing $\\pi = \\chi _{X} - \\chi _{Y}$ with disjoint subsets $X$ and $Y$ , we obtain $\\hat{p}(\\pi ) = p(X) - b(Y)$ and $\\sum _{s \\in S} \\psi _{s} (\\pi (s)) = \\widetilde{c}(X) - \\widetilde{c}(Y)$ .",
"Therefore the right-hand side of (REF ) coincides with that of (REF ).",
"Let $a$ and $b$ be integers with $a \\le b$ , and define $\\varphi $ by $ \\varphi (k)= \\min \\lbrace | k - z | : a \\le z \\le b \\rbrace = \\max \\lbrace a - k, 0, k - b \\rbrace \\qquad (k \\in {\\bf Z}).$ This function represents the distance from an integer $k$ to the integer interval $[a,b]_{{\\bf Z}} := \\lbrace z \\in {\\bf Z}: a \\le z \\le b \\rbrace $ .",
"The conjugate function $\\psi $ is given by $ \\psi ( \\ell ) =\\left\\lbrace \\begin{array}{ll}- a & (\\ell =-1), \\\\0 & (\\ell =0) ,\\\\b & (\\ell =+1), \\\\+\\infty & (\\mbox{otherwise}).\\end{array}\\right.$ A min-max relation for the minimum $\\ell _{1}$ -distance between an integer point of $B$ and a given integer interval $[c,d]_{{\\bf Z}} := \\lbrace y \\in {\\bf Z}^{S} : c(s) \\le y(s) \\le d(s) \\ (s \\in S) \\rbrace $ can be obtained from the min-max formula (REF ), where $c, d \\in {\\bf Z}^{S}$ and $c \\le d$ .",
"Proposition 7.2 For $c, d \\in {\\bf Z}^{S}$ with $c \\le d$ , $ & \\min \\lbrace \\Vert x - y \\Vert _{1} : x \\in \\overset{....}{B}, \\ y \\in [c,d]_{{\\bf Z}} \\rbrace \\nonumber \\\\ &= \\max \\lbrace p(X) - b(Y)- \\widetilde{d}(X) + \\widetilde{c}(Y): X, Y \\subseteq S; \\ X \\cap Y = \\emptyset \\rbrace .$ With reference to (REF ), we define $\\varphi _{s}(k) = \\min \\lbrace | k - z | : c(d) \\le z \\le d(s) \\rbrace $ .",
"Then $& \\min \\lbrace \\ \\Vert x - y \\Vert _{1} : x \\in \\overset{....}{B}, \\ y \\in [c,d]_{{\\bf Z}} \\rbrace \\\\ & =\\min \\lbrace \\ \\sum _{s \\in S} | x(s) - y(s) | : x \\in \\overset{....}{B}, \\ y \\in [c,d]_{{\\bf Z}} \\rbrace \\\\ & =\\min \\lbrace \\ \\min \\lbrace \\sum _{s \\in S} | x(s) - y(s) | : c(d) \\le y(s) \\le d(s) \\ (s \\in S) \\rbrace : x \\in \\overset{....}{B} \\rbrace \\\\ & =\\min \\lbrace \\ \\sum _{s \\in S}\\min \\lbrace \\ | x(s) - y(s) | : c(d) \\le y(s) \\le d(s) \\rbrace : x \\in \\overset{....}{B} \\rbrace \\\\ & =\\min \\lbrace \\ \\sum _{s \\in S} \\varphi _{s}(x(s)) : x \\in \\overset{....}{B} \\rbrace .$ Thus the left-hand side of (REF ) is in the form of the left-hand side of the min-max formula (REF ).",
"By (REF ), we may assume $\\pi \\in \\lbrace -1,0,+1 \\rbrace ^{S}$ on the right-hand side of (REF ).",
"On representing $\\pi = \\chi _{X} - \\chi _{Y}$ with disjoint subsets $X$ and $Y$ , we obtain $\\hat{p}(\\pi ) = p(X) - b(Y)$ and $\\sum _{s \\in S} \\psi _{s} (\\pi (s))= \\widetilde{d}(X) - \\widetilde{c}(Y)$ .",
"Therefore the right-hand side of (REF ) coincides with that of (REF )."
],
[
"Quadratic functions",
"For a quadratic function $\\varphi $ defined by $ \\varphi (k)= a k^{2}\\qquad (k \\in {\\bf Z})$ with a positive integer $a$ , the conjugate function $\\psi $ is given (cf., Remark REF ) by $ \\psi (\\ell ) =\\left\\lfloor \\frac{1}{2} \\left( \\frac{\\ell }{a} + 1 \\right) \\right\\rfloor \\left(\\ell - a \\left\\lfloor \\frac{1}{2} \\left( \\frac{\\ell }{a} + 1 \\right) \\right\\rfloor \\right), \\ $ which admits the following alternative expressions: $\\psi (\\ell ) & =\\left\\lceil \\frac{1}{2} \\left( \\frac{\\ell }{a} - 1 \\right) \\right\\rceil \\left(\\ell - a \\left\\lceil \\frac{1}{2} \\left( \\frac{\\ell }{a} - 1 \\right) \\right\\rceil \\right), \\ \\\\\\psi (\\ell )&= \\max \\left\\lbrace \\ \\left\\lfloor \\frac{\\ell }{2 a} \\right\\rfloor \\left(\\ell - a \\left\\lfloor \\frac{\\ell }{2 a} \\right\\rfloor \\right), \\ \\ \\left\\lceil \\frac{\\ell }{2 a } \\right\\rceil \\left(\\ell - a \\left\\lceil \\frac{\\ell }{2 a} \\right\\rceil \\right) \\ \\right\\rbrace .$ If $a=1$ , these expressions reduce to $\\psi (\\ell ) = \\left\\lfloor {\\ell }/{2} \\right\\rfloor \\cdot \\left\\lceil {\\ell }/{2} \\right\\rceil $ in (REF ).",
"The min-max formula (REF ) for the square-sum can be extended for a nonsymmetric quadratic function $\\sum _{s \\in S} c(s) x(s)^{2}$ , where $c(s)$ is a positive integer for each $s \\in S$ .",
"Theorem 7.3 For an integer vector $c \\in {\\bf Z}^{S}$ with $c(s) \\ge 1$ for every $s \\in S$ , $& \\min \\lbrace \\sum _{s \\in S} c(s) x(s)^{2} : x \\in \\overset{....}{B} \\rbrace \\\\ &= \\max \\lbrace \\hat{p}(\\pi )- \\sum _{s \\in S}\\left\\lfloor \\frac{1}{2} \\left( \\frac{\\pi (s)}{c(s) } + 1 \\right) \\right\\rfloor \\left(\\pi (s) - c(s) \\left\\lfloor \\frac{1}{2} \\left( \\frac{\\pi (s)}{c(s) } + 1 \\right) \\right\\rfloor \\right) :\\pi \\in {\\bf Z}^{S} \\rbrace .$ $\\rule {0.17cm}{0.17cm}$ In the basic case where $c(s)=1$ for all $s \\in S$ , we had a combinatorial constructive proof in Part I [9].",
"Such a direct combinatorial proof, not relying on the Fenchel-type discrete duality in DCA, for the general case of (REF ) would be an interesting topic.",
"Remark 7.1 We derive (REF ), (REF ), and ().",
"Since $\\varphi $ is discrete convex, the maximum in the definition (REF ) of $\\psi (\\ell )$ is attained by $k$ satisfying $ \\varphi (k) - \\varphi (k-1) \\le \\ell \\le \\varphi (k+1) - \\varphi (k) .$ For $\\varphi (k)= a k^{2}$ this condition reads $a (2k-1) \\le \\ell \\le a (2k +1 )$ , or equivalently $\\frac{1}{2} \\left( \\frac{\\ell }{a} - 1 \\right) \\le k \\le \\frac{1}{2} \\left( \\frac{\\ell }{a} + 1 \\right) .$ Therefore, the maximum in (REF ) is attained by $k= \\left\\lfloor \\frac{1}{2} \\left( \\frac{\\ell }{a} + 1 \\right) \\right\\rfloor $ and also by $k= \\left\\lceil \\frac{1}{2} \\left( \\frac{\\ell }{a} - 1 \\right) \\right\\rceil $ .",
"This gives (REF ) and (REF ), respectively.",
"To derive () we consider $\\varphi (t)= a t^{2}$ in $t \\in {\\bf R}$ and its derivative $\\varphi ^{\\prime }(t)= 2a t$ .",
"Let $k_{\\ell }$ be the integer satisfying $ \\varphi ^{\\prime }(k_{\\ell }) \\le \\ell < \\varphi ^{\\prime }(k_{\\ell } +1) .$ Then the maximum in the definition (REF ) of $\\psi (\\ell )$ is attained by $k=k_{\\ell }$ if $\\varphi ^{\\prime }(k_{\\ell }) = \\ell $ , and otherwise by $k=k_{\\ell }$ or $k_{\\ell } +1$ .",
"For $\\varphi (k)= a k^{2}$ , we have $k_{\\ell }=\\left\\lfloor \\frac{\\ell }{2 a} \\right\\rfloor $ and the maximum is attained by $k = \\left\\lfloor \\frac{\\ell }{2 a} \\right\\rfloor $ or $k = \\left\\lceil \\frac{\\ell }{2 a } \\right\\rceil $ .",
"Hence we have ().",
"$\\rule {0.17cm}{0.17cm}$"
],
[
"Power products",
"For function $\\varphi $ defined by $ \\varphi (k)= a \\ k^{2b}\\qquad (k \\in {\\bf Z})$ with positive integers $a$ and $b$ , the conjugate function $\\psi $ is given (cf., Remark REF ) by $ \\psi (\\ell )&= \\max \\left\\lbrace \\ \\ell \\left\\lfloor K(\\ell ) \\right\\rfloor - a \\left\\lfloor K(\\ell ) \\right\\rfloor ^{2b}, \\ \\ \\ell \\left\\lceil K(\\ell ) \\right\\rceil - a \\left\\lceil K(\\ell ) \\right\\rceil ^{2b}\\right\\rbrace ,$ where $K(\\ell ) = \\left( \\frac{\\ell }{2 a b} \\right)^{1/(2b-1)}.$ By choosing $a=1$ and $b=2$ , for example, we obtain a min-max formula $& \\min \\lbrace \\sum _{s \\in S} x(s)^{4} : x \\in \\overset{....}{B} \\rbrace \\\\ &= \\max \\lbrace \\hat{p}(\\pi )- \\sum _{s \\in S}\\max \\left\\lbrace \\ \\pi (s)\\left\\lfloor ( \\pi (s)/4 )^{1/3} \\right\\rfloor - \\left\\lfloor ( \\pi (s)/4 )^{1/3} \\right\\rfloor ^{4}, \\ \\ \\right.\\\\ &\\phantom{AAAAAAAAAAAAAAA}\\left.\\pi (s)\\left\\lceil ( \\pi (s)/4 )^{1/3} \\right\\rceil - \\left\\lceil ( \\pi (s)/4 )^{1/3} \\right\\rceil ^{4}\\right\\rbrace : \\pi \\in {\\bf Z}^{S} \\rbrace .$ Remark 7.2 We derive (REF ) on the basis of (REF ) for $\\varphi (t)= a \\, t^{2b}$ and $\\varphi ^{\\prime }(t)= 2ab \\, t^{2b-1}$ .",
"We have $k_{\\ell }=\\left\\lfloor \\left( \\frac{\\ell }{2 a b} \\right)^{1/(2b-1)} \\right\\rfloor $ , and the maximum in (REF ) is attained by $k = \\left\\lfloor \\left( \\frac{\\ell }{2 a b} \\right)^{1/(2b-1)} \\right\\rfloor $ or $k = \\left\\lceil \\left( \\frac{\\ell }{2 a b} \\right)^{1/(2b-1)} \\right\\rceil $ .",
"Hence follows (REF ).",
"$\\rule {0.17cm}{0.17cm}$"
],
[
"Exponential functions",
"For an exponential function $\\varphi $ defined by $ \\varphi (k)=\\left\\lbrace \\begin{array}{ll}2^{k} & (k \\ge 0) , \\\\+\\infty & (\\mbox{otherwise}) ,\\end{array}\\right.$ the conjugate function $\\psi $ is given (cf., Remark REF ) by $ \\psi (\\ell ) = \\ell \\left\\lceil \\log _{2} \\ell \\right\\rceil - 2 ^{ \\lceil \\log _{2} \\ell \\rceil } .$ More generally, for function $\\varphi $ defined by $ \\varphi (k)=\\left\\lbrace \\begin{array}{ll}a\\, b^{k} & (k \\ge 0) , \\\\+\\infty & (\\mbox{otherwise})\\end{array}\\right.$ with integers $a \\ge 1$ and $b \\ge 2$ , the conjugate function $\\psi $ is given (cf., Remark REF ) by $ \\psi (\\ell ) =\\ell \\left\\lceil \\log _{b} \\left( \\frac{\\ell }{a(b-1)} \\right) \\right\\rceil - a\\, b^{ \\left\\lceil \\log _{b} \\left( \\frac{\\ell }{a(b-1)} \\right) \\right\\rceil } .$ Theorem 7.4 Assume that $B$ is contained in the nonnegative orthant ${\\bf Z}_{+}^{S}$ .",
"Then $& \\min \\lbrace \\sum _{s \\in S} 2^{x(s)} : x \\in \\overset{....}{B} \\rbrace \\\\ &= \\max \\lbrace \\hat{p}(\\pi )- \\sum _{s \\in S}\\left( \\pi (s) \\left\\lceil \\log _{2} \\pi (s) \\right\\rceil - 2 ^{ \\left\\lceil \\log _{2} \\pi (s) \\right\\rceil }\\right): \\pi \\in {\\bf Z}^{S} \\rbrace .$ More generally, for an integer vector $c, d \\in {\\bf Z}^{S}$ with $c(s) \\ge 1$ , $d(s) \\ge 2$ $(s \\in S)$ , $& \\min \\lbrace \\sum _{s \\in S} c(s) \\, d(s)^{x(s)} : x \\in \\overset{....}{B} \\rbrace \\\\ &= \\max \\lbrace \\hat{p}(\\pi )- \\sum _{s \\in S} \\left(\\pi (s) \\left\\lceil K(\\ell ) \\right\\rceil - c(s) \\, d(s)^{ \\left\\lceil K(\\ell ) \\right\\rceil }\\right): \\pi \\in {\\bf Z}^{S} \\rbrace ,$ where $K(\\ell ) = \\log _{d(s)} \\left( \\frac{\\pi (s)}{c(s)(d(s)-1)} \\right) .$ $\\rule {0.17cm}{0.17cm}$ Remark 7.3 We derive (REF ) on the basis of (REF ).",
"For $\\varphi (k)= a \\, b^{k}$ , the condition (REF ) reads $ a (b-1) b^{k-1} \\le \\ell \\le a (b-1) b^{k}$ , or equivalently $k-1 \\ \\le \\ \\log _{b} \\left( \\frac{\\ell }{a(b-1)} \\right)\\ \\le \\ k .$ Therefore, the maximum in (REF ) is attained by $k= \\left\\lceil \\log _{b} \\left( \\frac{\\ell }{a(b-1)} \\right) \\right\\rceil $ .",
"Hence follows (REF ).",
"By setting $a=1$ and $b=2$ in (REF ), we obtain (REF ).",
"$\\rule {0.17cm}{0.17cm}$"
],
[
"Separable convex functions on the intersection of M-convex sets",
"The duality formula (REF ) for separable convex functions on an M-convex set admits an extension to separable convex functions on the intersection of two M-convex sets.",
"This extension serves as a basis of the study of decreasing-minimality in the intersection of two M-convex sets (integral base-polyhedra).",
"Let $B_{1}$ and $B_{2}$ be two integral base-polyhedra, and $p_{1}$ and $p_{2}$ be the associated (integer-valued) supermodular functions.",
"For $i=1,2$ , the set of integer points of $B_{i}$ is denoted as $\\overset{....}{B_{i}}$ , and the linear extension (Lovász extension) of $p_{i}$ as $\\hat{p}_{i}$ .",
"For each $s \\in S$ , let $\\varphi _{s}: {\\bf Z}\\rightarrow {\\bf Z}\\cup \\lbrace +\\infty \\rbrace $ be an integer-valued discrete convex function.",
"As before we denote the conjugate function of $\\varphi _{s}$ by $\\psi _{s}: {\\bf Z}\\rightarrow {\\bf Z}\\cup \\lbrace +\\infty \\rbrace $ , which is defined by (REF ).",
"Recall notation ${\\rm dom\\,}\\Phi = \\lbrace x \\in {\\bf Z}^{S} : x(s) \\in {\\rm dom\\,}\\varphi _{s} \\mbox{ for each } s \\in S \\rbrace $ .",
"The following theorem gives a duality formula for separable discrete convex functions on the intersection of two M-convex sets.",
"We introduce notations for feasible vectors: $\\mathcal {P}_{0} &= \\lbrace x \\in \\overset{....}{B_{1}} \\cap \\overset{....}{B_{2}} :x(s) \\in {\\rm dom\\,}\\varphi _{s} \\mbox{ for each } s \\in S \\rbrace \\ = \\overset{....}{B_{1}} \\cap \\overset{....}{B_{2}} \\cap {\\rm dom\\,}\\Phi ,\\\\\\mathcal {D}_{0} &= \\lbrace (\\pi _{1}, \\pi _{2}) \\in {\\bf Z}^{S} \\times {\\bf Z}^{S}:\\pi _{i} \\in {\\rm dom\\,}\\hat{p}_{i} \\ (i=1,2), \\ \\pi _{1}(s) + \\pi _{2}(s) \\in {\\rm dom\\,}\\psi _{s} \\mbox{ for each } s \\in S \\rbrace .$ Theorem 7.5 Assume that $\\mathcal {P}_{0} \\ne \\emptyset $ (primal feasibility) or $\\mathcal {D}_{0} \\ne \\emptyset $ (dual feasibility) holds.",
"Then we have the min-max relation:The unbounded case with both sides of (REF ) being equal to $-\\infty $ or $+\\infty $ is also a possibility.",
"$& \\min \\lbrace \\sum _{s \\in S} \\varphi _{s} ( x(s) ) :x \\in \\overset{....}{B_{1}} \\cap \\overset{....}{B_{2}} \\rbrace \\\\ &= \\max \\lbrace \\hat{p}_{1}(\\pi _{1}) + \\hat{p}_{2}(\\pi _{2})- \\sum _{s \\in S} \\psi _{s} (\\pi _{1}(s) + \\pi _{2}(s) ) : \\pi _{1}, \\pi _{2} \\in {\\bf Z}^{S} \\rbrace .$ We give a proof based on an iterative application of the Fenchel duality theorem (Theorem REF ), while the weak duality ($\\min \\ge \\max $ ) is demonstrated in Remark REF .",
"We denote the indicator functions of $\\overset{....}{B}_{1}$ and $\\overset{....}{B}_{2}$ by $\\delta _{1}$ and $\\delta _{2}$ , respectively, and use the notation $\\Phi (x) = \\sum [\\varphi _{s}(x(s)): s\\in S]$ .",
"In the Fenchel-type discrete duality $ \\min \\lbrace f(x) - h(x) : x \\in {\\bf Z}^{S} \\rbrace = \\max \\lbrace h^{\\circ }(\\pi ) - f^{\\bullet }(\\pi ) : \\pi \\in {\\bf Z}^{S} \\rbrace $ in (REF ), we choose $f = \\delta _{2} + \\Phi $ and $h= -\\delta _{1}$ .",
"Since $f - h = \\Phi + \\delta _{1} + \\delta _{2}$ , the left-hand side of (REF ) coincides with the left-hand side of (REF ).",
"The conjugate function $f^{\\bullet }$ can be computed as follows.",
"For $\\pi \\in {\\bf Z}^{S}$ we define $\\varphi _{s}^{\\pi }(k) = \\varphi _{s}(k) - \\pi (s) k $ for $k \\in {\\bf Z}$ and $s \\in S$ .",
"Then the conjugate function $\\psi _{s}^{\\pi }$ of function $\\varphi _{s}^{\\pi }$ is given as $\\psi _{s}^{\\pi }(\\ell )&= \\max \\lbrace k \\ell - \\varphi _{s}^{\\pi }(k) : k \\in {\\bf Z}\\rbrace \\\\ &= \\max \\lbrace k ( \\ell + \\pi (s) ) - \\varphi _{s}(k) : k \\in {\\bf Z}\\rbrace \\\\ &= \\psi _{s}(\\ell + \\pi (s) )\\qquad (\\ell \\in {\\bf Z}).$ Using this expression and the min-max formula (REF ) for $B_{2}$ and $\\varphi _{s}^{\\pi }$ , we obtain $f^{\\bullet }(\\pi )&= \\max \\lbrace \\langle \\pi , x \\rangle - \\delta _{2}(x) - \\sum _{s \\in S} \\varphi _{s}(x(s)) : x \\in {\\bf Z}^{S} \\rbrace \\nonumber \\\\&= \\max \\lbrace \\sum _{s \\in S} \\big [ \\pi (s) x(s) - \\varphi _{s}(x(s)) \\big ] - \\delta _{2}(x): x \\in {\\bf Z}^{S} \\rbrace \\nonumber \\\\&= - \\min \\lbrace \\sum _{s \\in S} \\varphi _{s}^{\\pi }(x(s)) : x \\in \\overset{....}{B_{2}} \\rbrace \\nonumber \\\\&= - \\max \\lbrace \\hat{p}_{2}(\\pi ^{\\prime }) - \\sum _{s \\in S} \\psi _{s}^{\\pi }(\\pi ^{\\prime }(s)) : \\pi ^{\\prime } \\in {\\bf Z}^{S} \\rbrace \\nonumber \\\\&= - \\max \\lbrace \\hat{p}_{2}(\\pi ^{\\prime }) - \\sum _{s \\in S} \\psi _{s}(\\pi (s) + \\pi ^{\\prime }(s)) : \\pi ^{\\prime } \\in {\\bf Z}^{S} \\rbrace \\qquad ( \\pi \\in {\\bf Z}^{S}) .$ On the other hand, the conjugate function $h^{\\circ }$ of $h= -\\delta _{1}$ is equal to $\\hat{p}_{1}$ by (REF ), i.e., $h^{\\circ }(\\pi ) =\\hat{p}_{1}(\\pi )\\qquad ( \\pi \\in {\\bf Z}^{S}).$ The substitution of (REF ) and (REF ) into $h^{\\circ } - f^{\\bullet }$ shows that the right-hand side of (REF ) coincides with the right-hand side of (REF ).",
"Remark 7.4 The weak duality ($\\min \\ge \\max $ ) in (REF ) is shown here.",
"Let $x \\in \\mathcal {P}_{0}$ and $(\\pi _{1},\\pi _{2}) \\in \\mathcal {D}_{0}$ .",
"Then, using the Fenchel–Young inequality (REF ) for $(\\varphi _{s}, \\psi _{s})$ as well as (REF ) for $p=p_{i}$ $(i=1,2)$ , we obtain $& \\sum _{s \\in S} \\varphi _{s} ( x(s) )-\\left( \\hat{p}_{1}(\\pi _{1}) + \\hat{p}_{2}(\\pi _{2})- \\sum _{s \\in S} \\psi _{s} (\\pi _{1}(s) + \\pi _{2}(s) )\\right)\\\\ &= \\sum _{s \\in S} \\big [ \\varphi _{s} ( x(s) ) + \\psi _{s}(\\pi _{1}(s) + \\pi _{2}(s) ) \\big ]\\ - \\hat{p}_{1}(\\pi _{1}) - \\hat{p}_{2}(\\pi _{2})\\\\ &\\ge \\sum _{s \\in S} x(s) (\\pi _{1}(s) + \\pi _{2}(s) )\\ - \\hat{p}_{1}(\\pi _{1}) - \\hat{p}_{2}(\\pi _{2})\\\\ &=\\sum _{s \\in S} \\pi _{1}(s) x(s) + \\sum _{s \\in S} \\pi _{2}(s) x(s)\\ - \\hat{p}_{1}(\\pi _{1}) - \\hat{p}_{2}(\\pi _{2})\\\\ &\\ge \\min \\lbrace \\pi _{1} z : z \\in \\overset{....}{B_1} \\rbrace + \\min \\lbrace \\pi _{2} z : z \\in \\overset{....}{B_2} \\rbrace \\ - \\hat{p}_{1}(\\pi _{1}) - \\hat{p}_{2}(\\pi _{2})\\ = 0,$ showing the weak duality.",
"The optimality conditions can be obtained as the conditions for the inequalities in (REF ) and () to be equalities, as stated in Proposition REF below.",
"$\\rule {0.17cm}{0.17cm}$ In the min-max formula (REF ) we denote the set of the minimizers $x$ by $\\mathcal {P}$ and the set of the maximizers $(\\pi _{1},\\pi _{2})$ by $\\mathcal {D}$ .",
"The following proposition follows from the combination of Theorem REF and Remark REF .",
"We remark that this proposition is a special case of Theorem REF .",
"Proposition 7.6 Assume that both $\\mathcal {P}_{0}$ and $\\mathcal {D}_{0}$ in (REF )–() are nonempty.",
"(1) Let $x \\in \\mathcal {P}_{0}$ .",
"Then $x \\in \\mathcal {P}$ if and only if there exists $(\\pi _{1},\\pi _{2}) \\in \\mathcal {D}_{0}$ such that $&\\varphi _{s}(x(s)) - \\varphi _{s}(x(s)-1)\\le \\pi _{1}(s) + \\pi _{2}(s) \\le \\varphi _{s}(x(s)+1) - \\varphi _{s}(x(s))\\qquad (s \\in S),\\\\& \\mbox{\\rm $\\pi _{1}(s) \\ge \\pi _{1}(t)$\\quad for every $(s,t)$ \\ with \\ $x + \\chi _{s} - \\chi _{t} \\in \\overset{....}{B_{1}}$},\\\\& \\mbox{\\rm $\\pi _{2}(s) \\ge \\pi _{2}(t)$\\quad for every $(s,t)$ \\ with \\ $x + \\chi _{s} - \\chi _{t} \\in \\overset{....}{B_{2}}$}.$ (2) Let $(\\pi _{1},\\pi _{2}) \\in \\mathcal {D}_{0}$ .",
"Then $(\\pi _{1},\\pi _{2}) \\in \\mathcal {D}$ if and only if there exists $x \\in \\mathcal {P}_{0}$ that satisfies (REF ), (), and ().",
"(3) For any $(\\hat{\\pi }_{1},\\hat{\\pi }_{2}) \\in \\mathcal {D}$ , we have $\\mathcal {P} &= \\lbrace x \\in \\mathcal {P}_{0}:\\mbox{\\rm (\\ref {pisubgradInter}), (\\ref {pi1minzerInter}), (\\ref {pi2minzerInter})hold with $(\\pi _{1},\\pi _{2}) = (\\hat{\\pi }_{1},\\hat{\\pi }_{2})$} \\rbrace \\\\& = \\lbrace x \\in {\\rm dom\\,}\\Phi :\\mbox{\\rm (\\ref {pisubgradInter}) holds with $(\\pi _{1},\\pi _{2}) = (\\hat{\\pi }_{1},\\hat{\\pi }_{2})$} \\rbrace \\\\ & \\phantom{AA}\\cap \\lbrace x \\in \\overset{....}{B_{1}}:\\mbox{\\rm $x$ is a $\\hat{\\pi }_{1}$-minimizer in $\\overset{....}{B_{1}}$ } \\rbrace \\\\ & \\phantom{AA}\\cap \\lbrace x \\in \\overset{....}{B_{2}}:\\mbox{\\rm $x$ is a $\\hat{\\pi }_{2}$-minimizer in $\\overset{....}{B_{2}}$ } \\rbrace .$ (4) For any $\\hat{x} \\in \\mathcal {P}$ , we have $ \\mathcal {D} = \\lbrace (\\pi _{1},\\pi _{2}) \\in \\mathcal {D}_{0} :\\mbox{\\rm (\\ref {pisubgradInter}), (\\ref {pi1minzerInter}), (\\ref {pi2minzerInter})hold with $x = \\hat{x}$} \\rbrace .$ The inequality (REF ) turns into an equality if and only if, for each $s \\in S$ , we have $\\varphi _{s} (k) + \\psi _{s} ( \\ell ) = k \\ell $ for $k= x(s)$ and $\\ell = \\pi _{1}(s) + \\pi _{2}(s)$ .",
"The latter condition is equivalent to (REF ) by (REF ).",
"The other inequality () turns into an equality if and only if $x$ is a $\\pi _{i}$ -minimizer in $\\overset{....}{B_{i}}$ for $i=1,2$ , that is, () and () hold.",
"Finally, we see from Theorem REF that the two inequalities in (REF ) and () simultaneously turn into equality for some $x$ and $(\\pi _{1},\\pi _{2})$ .",
"Proposition 7.7 In the min-max relation (REF ) for a separable convex function on the intersection of two M-convex sets, the set $\\mathcal {D}^{\\prime } := \\lbrace (\\pi _{1},-\\pi _{2}) : (\\pi _{1},\\pi _{2}) \\in \\mathcal {D} \\rbrace $ corresponding to the maximizers is an L$^{\\natural }$ -convex set and the set $\\mathcal {P}$ of the minimizers is an M$_{2}^{\\natural }$ -convex set.",
"The representation (REF ) shows that $\\mathcal {D}$ is described by the inequalities in (REF ), (), and ().",
"Hence $\\mathcal {D}^{\\prime }$ is L$^{\\natural }$ -convex.",
"In the representation () of $\\mathcal {P}$ , the first set $\\lbrace x \\in {\\rm dom\\,}\\Phi : \\mbox{\\rm (\\ref {pisubgradInter}) holds with$(\\pi _{1},\\pi _{2}) = (\\hat{\\pi }_{1},\\hat{\\pi }_{2})$} \\rbrace $ is a box of integers (the set of integers in an integral box), while for each $i=1,2$ , the set of $\\hat{\\pi }_{i}$ -minimizers in $\\overset{....}{B_{i}}$ is an M-convex set.",
"Therefore, $\\mathcal {P}$ is an M$_{2}^{\\natural }$ -convex set.",
"When specialized to a symmetric function, the min-max formula (REF ) is simplified to $& \\min \\lbrace \\sum _{s \\in S} \\varphi ( x(s) ) :x \\in \\overset{....}{B_{1}} \\cap \\overset{....}{B_{2}} \\rbrace \\\\ &= \\max \\lbrace \\hat{p}_{1}(\\pi _{1}) + \\hat{p}_{2}(\\pi _{2})- \\sum _{s \\in S} \\psi (\\pi _{1}(s) + \\pi _{2}(s) ) : \\pi _{1}, \\pi _{2} \\in {\\bf Z}^{S} \\rbrace ,$ where $\\varphi : {\\bf Z}\\rightarrow {\\bf Z}\\cup \\lbrace +\\infty \\rbrace $ is any integer-valued discrete convex function and $\\psi : {\\bf Z}\\rightarrow {\\bf Z}\\cup \\lbrace +\\infty \\rbrace $ is the conjugate of $\\varphi $ .",
"The identity (REF ) will play a key role in the study of discrete decreasing minimization on the intersection of two M-convex sets, just as (REF ) did for an M-convex set.",
"As an example of (REF ) with explicit forms of $\\varphi $ and $\\psi $ , we mention a min-max formula for the minimum square-sum of components on the intersection of two M-convex sets, which is an extension of (REF ) for an M-convex set.",
"Theorem 7.8 $& \\min \\lbrace \\sum _{s \\in S} x(s)^{2} : x \\in \\overset{....}{B_{1}} \\cap \\overset{....}{B_{2}} \\rbrace \\\\ &= \\max \\lbrace \\hat{p}_{1}(\\pi _{1}) + \\hat{p}_{2}(\\pi _{2}) -\\sum _{s \\in S} \\left\\lfloor \\frac{\\pi _{1}(s) + \\pi _{2}(s)}{2} \\right\\rfloor \\cdot \\left\\lceil \\frac{\\pi _{1}(s) + \\pi _{2}(s)}{2} \\right\\rceil : \\pi _{1}, \\pi _{2} \\in {\\bf Z}^{S} \\rbrace .$ This is a special case of (REF ) with $\\varphi (k) = k ^{2}$ and $\\psi (\\ell ) = \\left\\lfloor {\\ell }/{2} \\right\\rfloor \\cdot \\left\\lceil {\\ell }/{2} \\right\\rceil $ (cf., (REF )).",
"If $\\overset{....}{B_{1}} \\cap \\overset{....}{B_{2}} \\ne \\emptyset $ , both sides of (REF ) are finite-valued, and the minimum and the maximum are attained.",
"If $\\overset{....}{B_{1}} \\cap \\overset{....}{B_{2}} = \\emptyset $ , the left-hand side of (REF ) is equal to $+\\infty $ by convention and the right-hand side is unbounded above (hence equal to $+\\infty $ ).",
"Note also that $\\overset{....}{B_{1}} \\cap \\overset{....}{B_{2}} \\ne \\emptyset $ if and only if $B_{1} \\cap B_{2} \\ne \\emptyset $ .",
"We can also formulate a min-max formula for a nonsymmetric quadratic function $\\sum _{s \\in S} c(s) x(s)^{2}$ , where $c(s)$ is a positive integer for each $s \\in S$ .",
"On recalling the conjugate function in (REF ), we obtain the following min-max formula.",
"Theorem 7.9 For an integer vector $c \\in {\\bf Z}^{S}$ with $c(s) \\ge 1$ for all $s \\in S$ , $& \\min \\lbrace \\sum _{s \\in S} c(s) x(s)^{2} : x \\in \\overset{....}{B_{1}} \\cap \\overset{....}{B_{2}} \\rbrace \\\\ &= \\max \\lbrace \\hat{p}_{1}(\\pi _{1}) + \\hat{p}_{2}(\\pi _{2})- \\sum _{s \\in S}\\left\\lfloor \\frac{1}{2} \\left( \\frac{\\pi (s)}{c(s) } + 1 \\right) \\right\\rfloor \\left(\\pi (s)- c(s) \\left\\lfloor \\frac{1}{2} \\left( \\frac{\\pi (s)}{c(s) } + 1 \\right) \\right\\rfloor \\right) :\\\\ &\\phantom{ \\max \\lbrace \\quad }\\pi = \\pi _{1}+ \\pi _{2}, \\ \\ \\pi _{1}, \\pi _{2} \\in {\\bf Z}^{S} \\rbrace .$ $\\rule {0.17cm}{0.17cm}$ We have obtained the min-max formulas (REF ) and (REF ) as special cases of the Fenchel-type discrete duality in DCA.",
"Direct algorithmic proofs, not relying on the DCA machinery, would be an interesting research topic."
],
[
"Separable convex functions on network flows",
"Let $D=(V,A)$ be a digraph, and suppose that we are given a finite integer-valued function $m$ on $V$ for which $\\widetilde{m}(V)=0$ .",
"A flow means simply a function on $A$ , and we are interested in flow $x$ that satisfies $ \\varrho _x(v)-\\delta _x(v)=m(v)\\qquad \\mbox{\\rm for each node $v\\in V$},$ where $\\varrho _x(v):= \\sum [x(uv):uv\\in A],\\qquad \\delta _x(v):= \\sum [x(vu): vu\\in A].$ A flow $x$ satisfying (REF ) will be referred to as an $m$ -flow.",
"We consider a convex cost integer flow problem.",
"For each edge $e \\in A$ , an integer-valued (discrete) convex function $\\varphi _{e}: {\\bf Z}\\rightarrow {\\bf Z}\\cup \\lbrace +\\infty \\rbrace $ is given, and we seek an integral flow $x$ that minimizes the sum of the edge costs $\\Phi (x) = \\sum _{e \\in A} \\varphi _{e} ( x(e) )$ subject to the constraint (REF ).",
"For the function value $\\Phi (x)$ to be finite, we must have $ x(e) \\in {\\rm dom\\,}\\varphi _{e} \\quad \\mbox{ for each edge $e \\in A$},$ and therefore, capacity constraints, if any, can be represented (implicitly) in terms of the cost function $\\varphi _{e}$ .",
"A flow $x$ is called feasible if it satisfies the conditions (REF ) and (REF ).",
"Convex cost flow problem (1): $\\mbox{Minimize \\ } & &\\Phi (x) = \\sum _{e \\in A} \\varphi _{e} ( x(e) ) \\\\\\mbox{subject to \\ }& & \\varrho _x(v)-\\delta _x(v)=m(v) \\qquad (v \\in V), \\\\& & x(e) \\in {\\bf Z}\\qquad (e \\in A) .$ The dual problem, in its integer version, is as follows (cf., e.g., [1], [21], [33], [36]).",
"For each $e \\in A$ , let $\\psi _{e}: {\\bf Z}\\rightarrow {\\bf Z}\\cup \\lbrace +\\infty \\rbrace $ denote the conjugate of $\\varphi _{e}$ , that is, $ \\psi _{e}(\\ell ) = \\max \\lbrace k \\ell - \\varphi _{e}(k) : k \\in {\\bf Z}\\rbrace \\qquad (\\ell \\in {\\bf Z}),$ which is also an integer-valued (discrete) convex function.",
"This function $\\psi _{e}$ represents the (dual) cost function associated with edge $e \\in A$ .",
"The decision variable in the dual problem is an integer-valued potential $\\pi : V \\rightarrow {\\bf Z}$ defined on the node-set $V$ .",
"Recall the notation $\\pi m = \\sum _{v \\in V} \\pi (v) m(v)$ .",
"Dual to the convex cost flow problem (1): $\\mbox{Maximize \\ } & &\\Psi (\\pi ) = \\pi m- \\sum _{e =uv \\in A} \\psi _{e} ( \\pi (v) - \\pi (u) ) \\\\\\mbox{subject to \\ }& & \\pi (v) \\in {\\bf Z}\\qquad (v \\in V) .$ We introduce notations for feasible flows and potentials: $\\mathcal {P}_{0} &= \\lbrace x \\in {\\bf Z}^{A} :\\mbox{\\rm $x$ satisfies (\\ref {modFdomphi1}) and (\\ref {modFflowdemand1})} \\rbrace ,\\\\\\mathcal {D}_{0} &= \\lbrace \\pi \\in {\\bf Z}^{V}:\\pi (v) - \\pi (u) \\in {\\rm dom\\,}\\psi _{e} \\ \\mbox{\\rm for each } \\ e=uv \\in A\\rbrace .$ Theorem 7.10 Assume primal feasibility ($\\mathcal {P}_{0} \\ne \\emptyset $ ) or dual feasibility ($\\mathcal {D}_{0} \\ne \\emptyset $ ).",
"Then we have the min-max relation: $ \\min \\lbrace \\Phi (x) :x \\in {\\bf Z}^{A} \\ \\mbox{\\rm \\ satisfies (\\ref {modFflowdemand1})} \\rbrace = \\max \\lbrace \\Psi (\\pi ) : \\pi \\in {\\bf Z}^{V} \\rbrace .$ The unbounded case with both sides being equal to $-\\infty $ or $+\\infty $ is also a possibility.",
"$\\rule {0.17cm}{0.17cm}$ As an example of (REF ) with explicit forms of $\\Phi $ and $\\Psi $ , we mention a min-max formula for the minimum square-sum of components of an integral $m$ -flow, where no capacity constrains are imposed.",
"Proposition 7.11 For a digraph $D=(V,A)$ and an integer vector $m$ on $V$ with $\\widetilde{m}(V)=0$ , we have $& \\min \\lbrace \\sum _{e \\in A} x(e)^{2} :\\mbox{\\rm $x$ is an integral $m$-flow} \\rbrace \\\\ &= \\max \\lbrace \\sum _{v \\in V} \\pi (v) m(v) -\\sum _{uv \\in A} \\left\\lfloor \\frac{ \\pi (v) - \\pi (u) }{2} \\right\\rfloor \\cdot \\left\\lceil \\frac{ \\pi (v) - \\pi (u) }{2} \\right\\rceil : \\pi \\in {\\bf Z}^{V} \\rbrace .$ This is a special case of (REF ) with $\\varphi _{e}(k) = k ^{2}$ and $\\psi _{e}(\\ell ) = \\left\\lfloor {\\ell }/{2} \\right\\rfloor \\cdot \\left\\lceil {\\ell }/{2} \\right\\rceil $ (cf., (REF )).",
"In using this min-max relation in Part III [10] it is convenient to introduce capacity constraints explicitly.",
"We denote the integer-valued lower and upper bound functions on $A$ by $f$ and $g$ , for which $f\\le g$ is assumed, and impose the capacity constraint $ f(e) \\le x(e) \\le g(e)\\qquad \\mbox{\\rm for each edge $e \\in A$} .$ With this explicit form of capacity constraints, a flow $x$ is called feasible if it satisfies the conditions (REF ), (REF ) and (REF ).",
"The primal problem reads as follows.",
"Convex cost flow problem (2): $\\mbox{Minimize \\ } & &\\Phi (x) = \\sum _{e \\in A} \\varphi _{e} ( x(e) ) \\\\\\mbox{subject to \\ }& & \\varrho _x(v)-\\delta _x(v)=m(v) \\qquad (v \\in V), \\\\& & f(e) \\le x(e) \\le g(e) \\qquad (e \\in A), \\\\& & x(e) \\in {\\bf Z}\\qquad (e \\in A) .$ The corresponding dual problem can be given as follows (cf., Remark REF ), where the decision variables consist of an integer-valued potential $\\pi : V \\rightarrow {\\bf Z}$ on $V$ and integer-valued functions $\\tau _{1}, \\tau _{2}: A \\rightarrow {\\bf Z}$ on $A$ .",
"The constraint () below says that the tension (potential difference) is split into two parts $\\tau _{1}$ and $\\tau _{2}$ .",
"Dual to the convex cost flow problem (2): $\\mbox{Maximize \\ } & &\\Psi (\\pi ,\\tau _{1}, \\tau _{2}) = \\pi m- \\sum _{e \\in A} \\bigg ( \\ \\psi _{e} ( \\tau _{1}(e) ) + \\max \\lbrace f(e) \\tau _{2}(e) , \\ g(e) \\tau _{2}(e) \\rbrace \\ \\bigg ) \\\\\\mbox{subject to \\ }& & \\pi (v) - \\pi (u) = \\tau _{1}(e) + \\tau _{2}(e)\\qquad (e = uv \\in A), \\\\& & \\pi (v) \\in {\\bf Z}\\qquad (v \\in V) , \\\\& & \\tau _{1}(e), \\tau _{2}(e) \\in {\\bf Z}\\qquad (e \\in A) .$ We introduce notations for feasible flows and potentials/tensions: $\\mathcal {P}_{0} &= \\lbrace x \\in {\\bf Z}^{A} :\\mbox{\\rm $x$ satisfies (\\ref {modFdomphi1}), (\\ref {modFflowdemand2}), (\\ref {modFflowcapconst2}) }\\rbrace ,\\\\\\mathcal {D}_{0} &= \\lbrace (\\pi , \\tau _{1}, \\tau _{2}) \\in {\\bf Z}^{V} \\times {\\bf Z}^{A} \\times {\\bf Z}^{A}:(\\ref {modFpotdifftension2}), \\ \\tau _{1}(e) \\in {\\rm dom\\,}\\psi _{e} \\ \\mbox{\\rm for each } \\ e \\in A\\rbrace .$ Theorem 7.12 Assume primal feasibility ($\\mathcal {P}_{0} \\ne \\emptyset $ ) or dual feasibility ($\\mathcal {D}_{0} \\ne \\emptyset $ ).",
"Then we have the min-max relation: $& \\min \\lbrace \\Phi (x) :x \\in {\\bf Z}^{A} \\ \\mbox{\\rm \\ satisfies(\\ref {modFflowdemand2}) and (\\ref {modFflowcapconst2})} \\rbrace \\\\&= \\max \\lbrace \\Psi (\\pi , \\tau _{1}, \\tau _{2}) :\\pi \\in {\\bf Z}^{V} \\ \\mbox{\\rm and } \\tau _{1}, \\tau _{2} \\in {\\bf Z}^{A} \\ \\mbox{\\rm \\ satisfy (\\ref {modFpotdifftension2})} \\rbrace .$ The unbounded case with both sides being equal to $-\\infty $ or $+\\infty $ is also a possibility.",
"$\\rule {0.17cm}{0.17cm}$ As an example of (REF ) with explicit forms of $\\Phi $ and $\\Psi $ , we can obtain the capacitated version of the min-max formula (REF ) in Proposition REF .",
"Proposition 7.13 For a digraph $D=(V,A)$ , an integer vector $m$ on $V$ with $\\widetilde{m}(V)=0$ , and integer-valued functions $f$ and $g$ on $A$ with $f\\le g$ , we have $& \\min \\lbrace \\sum _{e \\in A} x(e)^{2} :\\mbox{\\rm $x$ is an integral $m$-flow satisfying $f \\le x \\le g$} \\rbrace \\\\ &= \\max \\lbrace \\sum _{v \\in V} \\pi (v) m(v) -\\sum _{e \\in A} \\left( \\left\\lfloor \\frac{ \\tau _{1}(e) }{2} \\right\\rfloor \\cdot \\left\\lceil \\frac{ \\tau _{1}(e) }{2} \\right\\rceil + \\max \\lbrace f(e) \\tau _{2}(e) , \\ g(e) \\tau _{2}(e) \\rbrace \\right)\\\\ & \\phantom{=\\max \\ }: \\pi (v) - \\pi (u) = \\tau _{1}(e) + \\tau _{2}(e) \\ (e=uv), \\ \\pi \\in {\\bf Z}^{V}, \\ \\tau _{1}, \\tau _{2} \\in {\\bf Z}^{A}\\rbrace .$ For each $e \\in A$ , let $\\varphi _{e}(k) = k ^{2}$ , whose conjugate function is $\\psi _{e}(\\ell ) = \\left\\lfloor {\\ell }/{2} \\right\\rfloor \\cdot \\left\\lceil {\\ell }/{2} \\right\\rceil $ by (REF ).",
"Then (REF ) follows from (REF ).",
"As a special case of (REF ) we can obtain a min-max formula for non-negative flows.",
"Proposition 7.14 For a digraph $D=(V,A)$ and an integer vector $m$ on $V$ with $\\widetilde{m}(V)=0$ , we have $& \\min \\lbrace \\sum _{e \\in A} x(e)^{2} :\\mbox{\\rm $x$ is a non-negative integral $m$-flow} \\rbrace \\\\ &= \\max \\lbrace \\sum _{v \\in V} \\pi (v) m(v) -\\sum _{uv \\in A} \\left\\lfloor \\frac{ (\\pi (v) - \\pi (u))^{+} }{2} \\right\\rfloor \\cdot \\left\\lceil \\frac{ (\\pi (v) - \\pi (u))^{+} }{2} \\right\\rceil : \\pi \\in {\\bf Z}^{V} \\rbrace ,$ where $(\\pi (v) - \\pi (u))^{+} = \\max \\lbrace 0, \\pi (v) - \\pi (u) \\rbrace $ .",
"Let $f(e)=0$ and $g(e)=+\\infty $ for all $e \\in A$ .",
"In the maximization on the right-hand side of (REF ), we may assume $\\tau _{2}(e) \\le 0$ for all $e \\in A$ , since otherwise $\\max \\lbrace f(e) \\tau _{2}(e) , \\ g(e) \\tau _{2}(e) \\rbrace = +\\infty $ .",
"Then $\\tau _{1}(e) \\ge \\pi (v) - \\pi (u)$ .",
"Since $\\min \\lbrace \\left\\lfloor \\frac{ \\tau _{1}(e) }{2} \\right\\rfloor \\cdot \\left\\lceil \\frac{ \\tau _{1}(e) }{2} \\right\\rceil : \\tau _{1}(e) \\ge \\pi (v) - \\pi (u) \\rbrace = \\left\\lfloor \\frac{ (\\pi (v) - \\pi (u))^{+} }{2} \\right\\rfloor \\cdot \\left\\lceil \\frac{ (\\pi (v) - \\pi (u))^{+} }{2} \\right\\rceil ,$ the right-hand side of of (REF ) reduces to that of (REF ).",
"For the min-max formula (REF ) we can obtain the following optimality criterion, where we denote the set of the minimizers $x$ by $\\mathcal {P}$ and the set of the maximizers $(\\pi , \\tau _{1},\\tau _{2})$ by $\\mathcal {D}$ .",
"Proposition 7.15 Assume that both $\\mathcal {P}_{0}$ and $\\mathcal {D}_{0}$ in (REF )–() are nonempty.",
"(1) Let $x \\in \\mathcal {P}_{0}$ .",
"Then $x \\in \\mathcal {P}$ if and only if there exists $(\\pi , \\tau _{1}, \\tau _{2}) \\in \\mathcal {D}_{0}$ such that $&\\varphi _{e}(x(e)) - \\varphi _{e}(x(e)-1)\\le \\tau _{1}(e) \\le \\varphi _{e}(x(e)+1) - \\varphi _{e}(x(e))\\qquad (e \\in A),\\\\&\\tau _{2}(e)\\left\\lbrace \\begin{array}{ll}=0 & \\mbox{\\rm if \\ $f(e) +1 \\le x(e) \\le g(e) - 1$}, \\\\\\le 0 & \\mbox{\\rm if \\ $x(e) = f(e)$}, \\\\\\ge 0 & \\mbox{\\rm if \\ $x(e) = g(e)$} \\\\\\end{array} \\right.\\qquad (e \\in A).$ (2) Let $(\\pi , \\tau _{1}, \\tau _{2}) \\in \\mathcal {D}_{0}$ .",
"Then $(\\pi ,\\tau _{1},\\tau _{2}) \\in \\mathcal {D}$ if and only if there exists $x \\in \\mathcal {P}_{0}$ that satisfies (REF ) and ().",
"(3) For any $(\\hat{\\pi },\\hat{\\tau }_{1},\\hat{\\tau }_{2}) \\in \\mathcal {D}$ , we have $ \\mathcal {P} = \\lbrace x \\in \\mathcal {P}_{0}:\\mbox{\\rm (\\ref {pisubgradmodF2}) and (\\ref {capaslackmodF2})hold with $(\\pi ,\\tau _{1},\\tau _{2}) = (\\hat{\\pi },\\hat{\\tau }_{1},\\hat{\\tau }_{2})$} \\rbrace ,$ where the conditions in (REF ) and () can be rewritten as $&x(e) \\in \\arg \\min _{k} \\lbrace \\varphi _{e}(k) - \\tau _{1}(e) k \\rbrace \\qquad (e \\in A),\\\\&\\left\\lbrace \\begin{array}{ll}x(e) = f(e) & \\mbox{\\rm if} \\ \\ \\tau _{2}(e) < 0, \\\\f(e) \\le x(e) \\le g(e) & \\mbox{\\rm if} \\ \\ \\tau _{2}(e) =0, \\\\x(e) = g(e) & \\mbox{\\rm if} \\ \\ \\tau _{2}(e) > 0 \\\\\\end{array} \\right.\\qquad (e \\in A).$ (4) For any $\\hat{x} \\in \\mathcal {P}$ , we have $ \\mathcal {D} = \\lbrace (\\pi ,\\tau _{1},\\tau _{2}) \\in \\mathcal {D}_{0} :\\mbox{\\rm (\\ref {pisubgradmodF2}) and (\\ref {capaslackmodF2})hold with $x = \\hat{x}$} \\rbrace .$ This is a special case of Proposition REF in Section REF .",
"The condition (), or equivalently (), expresses the so-called kilter condition for flow $x(e)$ and tension $\\tau _{2}(e)$ , whereas the condition (REF ), or equivalently (REF ), can be regarded as a nonlinear version thereof for flow $x(e)$ and tension $\\tau _{1}(e)$ .",
"Remark 7.5 Here we derive the dual problem (REF )–() from the basic case in (REF )–().",
"For each $e \\in A$ , let $\\delta _{e}$ denote the indicator function of the integer interval $[f(e),g(e)]_{{\\bf Z}}$ , define $\\tilde{\\varphi }_{e} := \\varphi _{e} + \\delta _{e}$ , and let $\\tilde{\\psi }_{e}$ be the conjugate function of $\\tilde{\\varphi }_{e}$ .",
"By the claim below we obtain the following expression $\\tilde{\\psi }_{e}( \\pi (v) - \\pi (u) ) =\\min \\bigg \\lbrace \\psi _{e} ( \\ell _{1} ) + \\max \\lbrace f(e) \\ell _2 , g(e) \\ell _2 \\rbrace :\\ell _{1}, \\ell _{2} \\in {\\bf Z}, \\ \\ell _{1} + \\ell _{2} = \\pi (v) - \\pi (u)\\bigg \\rbrace .$ The substitution of this expression into (REF ) results in (REF )–().",
"Claim: Let $\\varphi : {\\bf Z}\\rightarrow {\\bf Z}\\cup \\lbrace +\\infty \\rbrace $ be a (discrete) convex function, $\\delta : {\\bf Z}\\rightarrow {\\bf Z}\\cup \\lbrace +\\infty \\rbrace $ the indicator function of an integer interval $[a,b]_{{\\bf Z}}$ with $a \\le b$ .",
"Then the conjugate function $( \\varphi + \\delta )^{\\bullet }$ of $\\varphi + \\delta $ is given by $ ( \\varphi + \\delta )^{\\bullet }(\\ell ) =\\min \\bigg \\lbrace \\varphi ^{\\bullet } ( \\ell _1 ) + \\max \\lbrace a \\ell _2 , b \\ell _2 \\rbrace :\\ell _{1}, \\ell _{2} \\in {\\bf Z}, \\ \\ell _{1} + \\ell _{2} = \\ell \\bigg \\rbrace .$ (Proof) By Theorem 8.36 of [33], $( \\varphi + \\delta )^{\\bullet }$ is equal to the infimum convolution of $ \\varphi ^{\\bullet }$ and $\\delta ^{\\bullet }$ , that is, $ ( \\varphi + \\delta )^{\\bullet }(\\ell ) =\\min \\bigg \\lbrace \\varphi ^{\\bullet } ( \\ell _1 ) + \\delta ^{\\bullet } (\\ell _{2}):\\ell _{1}, \\ell _{2} \\in {\\bf Z}, \\ \\ell _{1} + \\ell _{2} = \\ell \\bigg \\rbrace .$ Here we have $\\delta ^{\\bullet }(\\ell )= \\max \\lbrace k\\ell - \\delta (k) \\rbrace = \\max \\lbrace k\\ell : a \\le k \\le b \\rbrace = \\max \\lbrace a \\ell , b \\ell \\rbrace .$ Hence follows (REF ).",
"$\\rule {0.17cm}{0.17cm}$ Remark 7.6 The feasibility of the primal problems can be expressed by a variant of the Hoffman-condition.",
"Denote the integer interval of ${\\rm dom\\,}\\varphi _{e}$ by $[ f^{\\prime }(e), g^{\\prime }(e) ]_{{\\bf Z}}$ with $f^{\\prime }(e) \\in {\\bf Z}\\cup \\lbrace -\\infty \\rbrace $ and $g^{\\prime }(e) \\in {\\bf Z}\\cup \\lbrace +\\infty \\rbrace $ .",
"Then, by Hoffman's theorem, there exists a feasible flow for the basic problem (REF )–() if and only if $\\varrho _{g^{\\prime }}(Z)-\\delta _{f^{\\prime }}(Z) \\ge \\widetilde{m}(Z)\\qquad \\hbox{for all}\\quad Z\\subseteq V$ is satisfied.",
"For the problem (REF )–() with explicit capacity constraints, we replace $f^{\\prime }(e)$ and $g^{\\prime }(e)$ by $\\max \\lbrace f(e), f^{\\prime }(e) \\rbrace $ and $\\min \\lbrace g(e), g^{\\prime }(e) \\rbrace $ , respectively.",
"$\\rule {0.17cm}{0.17cm}$"
],
[
"Separable convex functions on submodular flows",
"Let $D=(V,A)$ be a digraph, and suppose that we are given an integral base-polyhedron $B$ with ground-set $V$ .",
"We assume that $B$ is described as $B = B^{\\prime }(p)$ in (REF ) by an integer-valued (fully) supermodular function $p: 2^{V} \\rightarrow {\\bf Z}\\cup \\lbrace -\\infty \\rbrace $ with $p(V)=0$ , which is equivalent to saying that $B$ is described as $B = B(b)$ in (REF ) by an integer-valued (fully) submodular function $b: 2^{V} \\rightarrow {\\bf Z}\\cup \\lbrace +\\infty \\rbrace $ with $b(V)=0$ , where $b$ is the complementary function of $p$ .",
"Here we are interested in an integral flow $x: A \\rightarrow {\\bf Z}$ such that the net-in-flow vector $(\\varrho _x(v)-\\delta _x(v) : v \\in V )$ belongs to $B$ , which we express as $ (\\varrho _x(v)-\\delta _x(v) : v \\in V ) \\in \\overset{....}{B} .$ Such a flow $x$ is called a submodular flow.",
"The constraint (REF ) for the ordinary flow problem in Section REF is a (very) special case of (REF ) where the bounding submodular function $b$ (or the supermodular function $p$ ) is a modular function $\\widetilde{m}$ defined by the vector $m$ .",
"We consider a convex cost integer submodular flow problem.",
"For each edge $e \\in A$ , an integer-valued (discrete) convex function $\\varphi _{e}: {\\bf Z}\\rightarrow {\\bf Z}\\cup \\lbrace +\\infty \\rbrace $ is given, and we seek an integral flow $x$ that minimizes the sum of the edge costs $ \\Phi (x) = \\sum _{e \\in A} \\varphi _{e} ( x(e) )$ subject to the submodular constraint (REF ).",
"For the function value $\\Phi (x)$ to be finite, we must have $ x(e) \\in {\\rm dom\\,}\\varphi _{e} \\quad \\mbox{ for each edge $e \\in A$},$ and therefore, capacity constraints, if any, can be represented (implicitly) in terms of the cost function $\\varphi _{e}$ .",
"A feasible submodular flow means a flow $x$ that satisfies the conditions (REF ) and (REF ).",
"Convex cost submodular flow problem (1): $\\mbox{Minimize \\ } & &\\Phi (x) = \\sum _{e \\in A} \\varphi _{e} ( x(e) ) \\\\\\mbox{subject to \\ }& & (\\varrho _x(v)-\\delta _x(v) : v \\in V ) \\in \\overset{....}{B}, \\\\& & x(e) \\in {\\bf Z}\\qquad (e \\in A) .$ In discrete convex analysis, a systematic study of convex-cost submodular flows has been conducted in a more general framework called the M-convex submodular flow problem, where particular emphasis is laid on duality theorems (Murota [32], [33]).",
"The decision variable in the dual problem is an integer-valued potential $\\pi : V \\rightarrow {\\bf Z}$ .",
"The objective function $\\Psi (\\pi )$ involves the linear extension (Lovász extension) $\\hat{p}(\\pi )$ of the supermodular function $p$ defining $B$ as well as the conjugate function $\\psi _{e}$ of $\\varphi _{e}$ for all $e \\in A$ .",
"It is worth noting that $\\pi m$ in (REF ) is replaced by $\\hat{p}(\\pi )$ in (REF ).",
"Dual to the convex cost submodular flow problem (1): $\\mbox{Maximize \\ } & &\\Psi (\\pi ) = \\hat{p}(\\pi )- \\sum _{e = uv \\in A} \\psi _{e} ( \\pi (v) - \\pi (u) ) \\\\\\mbox{subject to \\ }& & \\pi (v) \\in {\\bf Z}\\qquad (v \\in V) .$ The following min-max formula can be derived as a special case of a min-max formula [33] for M-convex submodular flows, while the weak duality ($\\min \\ge \\max $ ) is demonstrated in Remark REF below.",
"We also mention that the min-max formula (REF ) below can be regarded as being equivalent to the Fenchel-type discrete duality theorem (Theorem REF ); see [33] for the detail of this equivalence.",
"We introduce notations for feasible flows and potentials: $\\mathcal {P}_{0} &= \\lbrace x \\in {\\bf Z}^{A} :\\mbox{\\rm $x$ satisfies (\\ref {sbmFdomphi1}) and (\\ref {sbmFflowdemand1})} \\rbrace ,\\\\\\mathcal {D}_{0} &= \\lbrace \\pi \\in {\\bf Z}^{V}: \\pi \\in {\\rm dom\\,}\\hat{p}, \\ \\ \\pi (v) - \\pi (u) \\in {\\rm dom\\,}\\psi _{e} \\ \\mbox{\\rm for each } \\ e=uv \\in A\\rbrace .$ Theorem 7.16 Assume primal feasibility ($\\mathcal {P}_{0} \\ne \\emptyset $ ) or dual feasibility ($\\mathcal {D}_{0} \\ne \\emptyset $ ).",
"Then we have the min-max relation: $ \\min \\lbrace \\Phi (x) :x \\in {\\bf Z}^{A} \\ \\mbox{\\rm \\ satisfies (\\ref {sbmFflowdemand1})} \\rbrace = \\max \\lbrace \\Psi (\\pi ) : \\pi \\in {\\bf Z}^{V}\\rbrace .$ The unbounded case with both sides being equal to $-\\infty $ or $+\\infty $ is also a possibility.",
"$\\rule {0.17cm}{0.17cm}$ Remark 7.7 The weak duality $\\Phi (x) \\ge \\Psi (\\pi )$ is shown here.",
"Let $x$ and $\\pi $ be primal and dual feasible solutions.",
"Then, using the Fenchel–Young inequality (REF ) for $(\\varphi _{e}, \\psi _{e})$ and the feasibility condition () as well as the expression (REF ) for $\\hat{p}(\\pi )$ , we obtain $\\Phi (x) - \\Psi (\\pi )& = \\sum _{e = uv \\in A} [ \\varphi _{e} ( x(e) ) + \\psi _{e} ( \\pi (v) - \\pi (u) ) ]\\ -\\hat{p}(\\pi )\\\\ &\\ge \\sum _{e = uv \\in A} x(e) ( \\pi (v) - \\pi (u) ) \\ -\\hat{p}(\\pi )\\\\ &=\\sum _{v \\in V} \\pi (v) (\\varrho _x(v)-\\delta _x(v))\\ -\\hat{p}(\\pi )\\\\ &\\ge \\min \\lbrace \\pi z : z \\in \\overset{....}{B} \\rbrace \\ -\\hat{p}(\\pi )\\ = 0.$ This shows the weak duality.",
"The optimality conditions can be obtained as the conditions for the inequalities in (REF ) and () to be equalities.",
"See Proposition REF below.",
"$\\rule {0.17cm}{0.17cm}$ In the min-max formula (REF ) we denote the set of the minimizers $x$ by $\\mathcal {P}$ and the set of the maximizers $\\pi $ by $\\mathcal {D}$ .",
"Proposition 7.17 Assume that both $\\mathcal {P}_{0}$ and $\\mathcal {D}_{0}$ in (REF )–() are nonempty.",
"(1) Let $x \\in \\mathcal {P}_{0}$ .",
"Then $x \\in \\mathcal {P}$ if and only if there exists $\\pi \\in \\mathcal {D}_{0}$ such that $&\\varphi _{e}(x(e)) - \\varphi _{e}(x(e)-1)\\le \\pi (v) - \\pi (u) \\le \\varphi _{e}(x(e)+1) - \\varphi _{e}(x(e))\\qquad (e = uv \\in A),\\\\&\\mbox{\\rm Net-in-flow vector $(\\varrho _x(v)-\\delta _x(v) : v \\in V )$is a $\\pi $-minimizer in $\\overset{....}{B}$}.$ (2) Let $\\pi \\in \\mathcal {D}_{0}$ .",
"Then $\\pi \\in \\mathcal {D}$ if and only if there exists $x \\in \\mathcal {P}_{0}$ that satisfies (REF ) and ().",
"(3) For any $\\hat{\\pi }\\in \\mathcal {D}$ , we have $ \\mathcal {P} = \\lbrace x \\in \\mathcal {P}_{0}:\\mbox{\\rm (\\ref {pisubgradsbmF1}) and (\\ref {piminzersbmF1})hold with $\\pi = \\hat{\\pi }$} \\rbrace ,$ where the condition in (REF ) can be rewritten as $x(e) \\in \\arg \\min _{k} \\lbrace \\varphi _{e}(k) - ( \\pi (v) - \\pi (u) ) k \\rbrace \\qquad (e =(u,v) \\in A).$ (4) For any $\\hat{x} \\in \\mathcal {P}$ , we have $ \\mathcal {D} = \\lbrace \\pi \\in \\mathcal {D}_{0} :\\mbox{\\rm (\\ref {pisubgradsbmF1}) and (\\ref {piminzersbmF1})hold with $x = \\hat{x}$} \\rbrace .$ The inequality (REF ) turns into an equality if and only if, for each $e = uv \\in A$ , we have $\\varphi _{e} (k) + \\psi _{e} ( \\ell ) = k \\ell $ for $k= x(e)$ and $\\ell = \\pi (v) - \\pi (u)$ .",
"The latter condition is equivalent to (REF ) by (REF ).",
"The other inequality () is an equality if and only if () holds.",
"In applications it is often convenient to introduce capacity constraints explicitly as $ f(e) \\le x(e) \\le g(e)\\qquad \\mbox{\\rm for each edge $e \\in A$}.$ With this explicit form of capacity constraints, a flow $x$ is called a feasible submodular flow if it satisfies the conditions (REF ) and (REF ) as well as (REF ).",
"The primal problem reads as follows.",
"Convex cost submodular flow problem (2): $\\mbox{Minimize \\ } & &\\Phi (x) = \\sum _{e \\in A} \\varphi _{e} ( x(e) ) \\\\\\mbox{subject to \\ }& & (\\varrho _x(v)-\\delta _x(v) : v \\in V ) \\in \\overset{....}{B}, \\\\& & f(e) \\le x(e) \\le g(e) \\qquad (e \\in A), \\\\& & x(e) \\in {\\bf Z}\\qquad (e \\in A) .$ The corresponding dual problem can be derived from (REF )–() by the technique described in Remark REF .",
"The decision variables of the resulting dual problem consist of an integer-valued potential $\\pi : V \\rightarrow {\\bf Z}$ on $V$ and integer-valued functions $\\tau _{1}, \\tau _{2}: A \\rightarrow {\\bf Z}$ on $A$ .",
"The constraint () below says that the tension (potential difference) is split into two parts $\\tau _{1}$ and $\\tau _{2}$ .",
"Dual to the convex cost submodular flow problem (2): $\\mbox{Maximize \\ } & &\\Psi (\\pi ,\\tau _{1}, \\tau _{2}) = \\hat{p}(\\pi )- \\sum _{e \\in A} \\bigg ( \\ \\psi _{e} ( \\tau _{1}(e) ) + \\max \\lbrace f(e) \\tau _{2}(e) , \\ g(e) \\tau _{2}(e) \\rbrace \\ \\bigg ) \\\\\\mbox{subject to \\ }& & \\pi (v) - \\pi (u) = \\tau _{1}(e) + \\tau _{2}(e)\\qquad (e = uv \\in A), \\\\& & \\pi (v) \\in {\\bf Z}\\qquad (v \\in V) , \\\\& & \\tau _{1}(e), \\tau _{2}(e) \\in {\\bf Z}\\qquad (e \\in A) .$ We introduce notations for feasible flows and potentials/tensions: $\\mathcal {P}_{0} &= \\lbrace x \\in {\\bf Z}^{A} :\\mbox{\\rm $x$ satisfies (\\ref {sbmFdomphi1}), (\\ref {sbmFflowdemand2}), (\\ref {sbmFflowcapconst2}) }\\rbrace ,\\\\\\mathcal {D}_{0} &= \\lbrace (\\pi , \\tau _{1}, \\tau _{2}) \\in {\\bf Z}^{V} \\times {\\bf Z}^{A} \\times {\\bf Z}^{A}:(\\ref {sbmFpotdifftension2}), \\ \\pi \\in {\\rm dom\\,}\\hat{p}, \\ \\ \\tau _{1}(e) \\in {\\rm dom\\,}\\psi _{e} \\ \\mbox{\\rm for each } \\ e \\in A\\rbrace .$ Theorem 7.18 Assume primal feasibility ($\\mathcal {P}_{0} \\ne \\emptyset $ ) or dual feasibility ($\\mathcal {D}_{0} \\ne \\emptyset $ ).",
"Then we have the min-max relation: $& \\min \\lbrace \\Phi (x) :x \\in {\\bf Z}^{A} \\ \\mbox{\\rm \\ satisfies (\\ref {sbmFflowdemand2}) and (\\ref {sbmFflowcapconst2})} \\rbrace \\\\&= \\max \\lbrace \\Psi (\\pi , \\tau _{1}, \\tau _{2}) :\\pi \\in {\\bf Z}^{V} \\ \\mbox{\\rm and } \\tau _{1}, \\tau _{2} \\in {\\bf Z}^{A} \\ \\mbox{\\rm \\ satisfy (\\ref {sbmFpotdifftension2})} \\rbrace .$ The unbounded case with both sides being equal to $-\\infty $ or $+\\infty $ is also a possibility.",
"$\\rule {0.17cm}{0.17cm}$ In the min-max formula (REF ) we denote the set of the minimizers $x$ by $\\mathcal {P}$ and the set of the maximizers $(\\pi , \\tau _{1},\\tau _{2})$ by $\\mathcal {D}$ .",
"The optimality criterion in Proposition REF can be adapted for (REF ) as follows.",
"Proposition 7.19 Assume that both $\\mathcal {P}_{0}$ and $\\mathcal {D}_{0}$ in (REF )–() are nonempty.",
"(1) Let $x \\in \\mathcal {P}_{0}$ .",
"Then $x \\in \\mathcal {P}$ if and only if there exists $(\\pi , \\tau _{1}, \\tau _{2}) \\in \\mathcal {D}_{0}$ such that $&\\varphi _{e}(x(e)) - \\varphi _{e}(x(e)-1)\\le \\tau _{1}(e) \\le \\varphi _{e}(x(e)+1) - \\varphi _{e}(x(e))\\qquad (e \\in A),\\\\&\\tau _{2}(e)\\left\\lbrace \\begin{array}{ll}=0 & \\mbox{\\rm if \\ $f(e) +1 \\le x(e) \\le g(e) - 1$}, \\\\\\le 0 & \\mbox{\\rm if \\ $x(e) = f(e)$}, \\\\\\ge 0 & \\mbox{\\rm if \\ $x(e) = g(e)$} \\\\\\end{array} \\right.\\qquad (e \\in A),\\\\&\\mbox{\\rm Net-in-flow vector $(\\varrho _x(v)-\\delta _x(v) : v \\in V )$is a $\\pi $-minimizer in $\\overset{....}{B}$}.$ (2) Let $(\\pi , \\tau _{1}, \\tau _{2}) \\in \\mathcal {D}_{0}$ .",
"Then $(\\pi ,\\tau _{1},\\tau _{2}) \\in \\mathcal {D}$ if and only if there exists $x \\in \\mathcal {P}_{0}$ that satisfies (REF ), (), and ().",
"(3) For any $(\\hat{\\pi },\\hat{\\tau }_{1},\\hat{\\tau }_{2}) \\in \\mathcal {D}$ , we have $ \\mathcal {P} = \\lbrace x \\in \\mathcal {P}_{0}:\\mbox{\\rm (\\ref {pisubgradsbmF2}), (\\ref {capaslacksbmF2}), (\\ref {pi1minzersbmF2})hold with $(\\pi ,\\tau _{1},\\tau _{2}) = (\\hat{\\pi },\\hat{\\tau }_{1},\\hat{\\tau }_{2})$} \\rbrace ,$ where the conditions in (REF ) and () can be rewritten as $&x(e) \\in \\arg \\min _{k} \\lbrace \\varphi _{e}(k) - \\tau _{1}(e) k \\rbrace \\qquad (e \\in A),\\\\&\\left\\lbrace \\begin{array}{ll}x(e) = f(e) & \\mbox{\\rm if} \\ \\ \\tau _{2}(e) < 0, \\\\f(e) \\le x(e) \\le g(e) & \\mbox{\\rm if} \\ \\ \\tau _{2}(e)=0, \\\\x(e) = g(e) & \\mbox{\\rm if} \\ \\ \\tau _{2}(e) > 0 \\\\\\end{array} \\right.\\qquad (e \\in A).$ (4) For any $\\hat{x} \\in \\mathcal {P}$ , we have $ \\mathcal {D} = \\lbrace (\\pi ,\\tau _{1},\\tau _{2}) \\in \\mathcal {D}_{0} :\\mbox{\\rm (\\ref {pisubgradsbmF2}), (\\ref {capaslacksbmF2}), (\\ref {pi1minzersbmF2})hold with $x = \\hat{x}$} \\rbrace .$ Rather than translating the conditions in Proposition REF for the present case, we prove the claim by considering the weak duality $\\Phi (x) \\ge \\Psi (\\pi ,\\tau _{1},\\tau _{2})$ directly for this case.",
"Let $x$ and $(\\pi , \\tau _{1},\\tau _{2})$ be primal and dual feasible solutions.",
"Then, using the Fenchel–Young inequality (REF ) for $(\\varphi _{e}, \\psi _{e})$ and (), we obtain $& \\Phi (x) - \\Psi (\\pi ,\\tau _{1},\\tau _{2})\\\\ &=\\sum _{e \\in A} [ \\varphi _{e} ( x(e) ) + \\psi _{e} ( \\tau _{1}(e) ) ]+ \\sum _{e \\in A} \\max \\lbrace f(e) \\tau _{2}(e) , \\ g(e) \\tau _{2}(e) \\rbrace \\ -\\hat{p}(\\pi )\\\\ &\\ge \\sum _{e \\in A} x(e) \\tau _{1}(e)+ \\sum _{e \\in A} \\max \\lbrace f(e) \\tau _{2}(e) , \\ g(e) \\tau _{2}(e) \\rbrace \\ -\\hat{p}(\\pi )\\\\ &= \\sum _{e=uv \\in A} x(e) ( \\pi (v) - \\pi (u) - \\tau _{2}(e) )+ \\sum _{e \\in A} \\max \\lbrace f(e) \\tau _{2}(e) , \\ g(e) \\tau _{2}(e) \\rbrace \\ -\\hat{p}(\\pi )\\\\ &= \\sum _{e = uv \\in A} x(e) ( \\pi (v) - \\pi (u) )+ \\sum _{e \\in A} \\big [\\max \\lbrace f(e) \\tau _{2}(e) , \\ g(e) \\tau _{2}(e) \\rbrace - x(e) \\tau _{2}(e)\\big ]\\ -\\hat{p}(\\pi )\\\\ &= \\left( \\sum _{v \\in V} \\pi (v) (\\varrho _x(v)-\\delta _x(v))\\ -\\hat{p}(\\pi ) \\right)+ \\sum _{e \\in A} \\big [\\max \\lbrace f(e) \\tau _{2}(e) , \\ g(e) \\tau _{2}(e) \\rbrace - x(e) \\tau _{2}(e)\\big ] .$ For the former part of this expression we have $\\sum _{v \\in V} \\pi (v) (\\varrho _x(v)-\\delta _x(v))\\ -\\hat{p}(\\pi )\\ge \\min \\lbrace \\pi z : z \\in \\overset{....}{B} \\rbrace \\ -\\hat{p}(\\pi ) = 0$ by the feasibility condition () and the expression (REF ) for $\\hat{p}(\\pi )$ , whereas, for each summand in the latter part we have $\\max \\lbrace f(e) \\tau _{2}(e) , \\ g(e) \\tau _{2}(e) \\rbrace - x(e) \\tau _{2}(e) \\ge 0$ since $f(e) \\le x(e) \\le g(e)$ by the capacity constraint ().",
"Thus the weak duality is established.",
"The optimality conditions can be obtained as the conditions for the inequalities in (REF ), (REF ) and (REF ) to be equalities, as in the proof of Proposition REF .",
"Remark 7.8 The feasibility of the primal problems can be expressed in terms of the submodular function $b$ as follows, where we denote the integer interval of ${\\rm dom\\,}\\varphi _{e}$ by $[ f^{\\prime }(e), g^{\\prime }(e) ]_{{\\bf Z}}$ with $f^{\\prime }(e) \\in {\\bf Z}\\cup \\lbrace -\\infty \\rbrace $ and $g^{\\prime }(e) \\in {\\bf Z}\\cup \\lbrace +\\infty \\rbrace $ .",
"Then there exists a feasible flow for the basic problem (REF )–() if and only if $\\varrho _{f^{\\prime }}(Z)-\\delta _{g^{\\prime }}(Z) \\le b(Z)\\qquad \\hbox{for all}\\quad Z\\subseteq V$ is satisfied.",
"For the problem (REF )–() with explicit capacity constraints, we replace $f^{\\prime }(e)$ and $g^{\\prime }(e)$ by $\\max \\lbrace f(e), f^{\\prime }(e) \\rbrace $ and $\\min \\lbrace g(e), g^{\\prime }(e) \\rbrace $ , respectively.",
"$\\rule {0.17cm}{0.17cm}$"
],
[
"Survey of early papers",
"This appendix offers a brief survey of earlier papers and books that deal with topics closely related to decreasing minimization on base-polyhdera.",
"To be specific, we mention the following: Veinott [39] (1971), Megiddo [27] (1974), Fujishige [11] (1980), Groenevelt [15] (1985, 1991), Federgruen–Groenevelt [7] (1986), Ibaraki–Katoh [20] (1988), Dutta–Ray [6] (1989), Fujishige [12] (1991, 2005), Hochbaum [17] (1994), and Tamir [38] (1995).",
"Similar notions and terms are scattered in the literature such as “egalitarian,” “lexicographically optimal,” “least majorized,” “least weakly submajorized,” “decreasingly minimal (dec-min),” and “increasingly maximal (inc-max).” Unfortunately, these notions are discussed often independently in different context, without proper mutual recognition.",
"The term “least majorized” is used in Veinott [39] and “Least weakly submajorized” is used in Tamir [38].",
"These terms are not used in Marshall–Olkin–Arnold [26].",
"Dutta–Ray [6] uses “egalitarian” and does not use “majorization.” The term “lexicographically optimal” in Veinott [39], Megiddo [27], [28], and Fujishige [11], [12] means “increasingly maximal (inc-max).” Three notions “dec-min”, “inc-max”, and “least majorized” are different in general.",
"Generally, “least majorized” implies “dec-min” and “inc-max”, but the converse is not true (see Section REF ).",
"In base-polyhedron (in ${\\bf R}$ and ${\\bf Z}$ ), however, the three notions coincide (see Section REF ).",
"Another important aspect in majorization is minimization of symmetric separable convex functions.",
"An element is least majorized if and only if it simultaneously minimizes all symmetric separable convex functions (see Proposition REF ).",
"Therefore, if a least majorized is known to exist, then it can be computed as a minimizer of the square-sum."
],
[
"Veinott (1971) {{cite:cbd65d7783b60cc7fa5691c602b06b0b47a03c11}}",
"This paper deals with a network flow problem.",
"The ground set is a star of arcs, i.e., the set of arcs incident to a single node.",
"This amounts to considering a special case of a base-polyhedron.",
"The main result is the unique existence of a least majorized element in Case ${\\bf R}$ .",
"The computational aspect is also discussed.",
"The problem is reduced to separable quadratic network flow problem.",
"Then the paper describes an algorithm for nonlinear convex cost minimum flow problem.",
"It also defines the dual problem using the conjugate function.",
"Complexity of the algorithm is not discussed.",
"Case ${\\bf Z}$ is also treated.",
"Theorem 2 (1) shows the existence of an integral element that simultaneously minimizes all symmetric separable convex functions.",
"The proof is based on rounding argument (continuous relaxation).",
"That is, for a discrete convex function in integers, its piecewise-linear extension is considered and the integrality theorem is used to derive the existence of an integral minimizer.",
"Thus the existence of a least majorized element is shown for the network flow in Case ${\\bf Z}$ .",
"This paper deals with a network flow problem.",
"The ground set is the set of multi-terminals.",
"This is more general than a star considered in Veinott [39], but the difference is not really essential.",
"The paper defines the notions of “sink-optimality” and “source-optimality,” which are increasing-maximality for vectors on the sink and source terminals, respectively.",
"This paper considers Case ${\\bf R}$ only.",
"The main result is the characterization of an inc-max element using a chain of cuts in the network (Theorem 4.6).",
"The computational aspect is discussed in the companion paper [28], which gives an algorithm of complexity $O(n^{5}$ ).",
"This is the first paper that deals with base-polyhedra, beyond network flows.",
"It considers Case ${\\bf R}$ only.",
"Lexicographic optimality with respect to a weight vector is defined.",
"The lexicographically optimal base with respect to a uniform weight coincides with the inc-max element of the base-polyhedron.",
"The relation to weighted square-sum minimization is investigated in detail and the minimum norm point is highlighted.",
"The principal partition for base-polyhedra is introduced, as a generalization of the known construction for matroids.",
"The principal partition determines the lexico-optimal base.",
"The proposed decomposition algorithm finds the lexico-optimal base as well as the principal partition in strongly polynomial time.",
"While this paper covers various aspects of the lexico-optimal base, the majorization viewpoint is missing.",
"In particular, it is not stated that the minimum norm point is actually a minimizer of all symmetric separable convex functions.",
"The technical report appeared in 1985, and the journal version in 1991.",
"Already the technical report was influential, cited by [12], [17], and [20].",
"The main concern of this paper is separable convex minimization (not restricted to symmetric separable convex functions) on base-polyhedra.",
"Both continuous variables (Case ${\\bf R}$ ) and discrete variables (Case ${\\bf Z}$ ) are treated.",
"In particular, this is the first paper that addressed minimization of separable convex functions on base-polyhedra in discrete variables.",
"One of the results says that, in any integral base-polyhedron, there exists an integral element that is a (simultaneous) minimizer of all symmetric separable convex functions.",
"This paper does not discuss implications of this result to inc-maximality, dec-minimality, or majorization, though the result does imply the existence of a least majorized element by virtue of the well-known fact (Proposition REF ) about majorization.",
"The paper presents two kinds of algorithms, the marginal allocation algorithm (of incremental type) and the decomposition algorithm (DA).",
"Concerning complexity, the author argues that the algorithms are polynomial if the base-polyhedron are of some special types (tree-structured polymatroids, generalized symmetric polymatroids, network polymatroids).",
"We quote the following statements from [15], where $E$ denotes the ground set of a base-polyhedron and $N$ is the associated submodular function, which is integer-valued in Case ${\\bf Z}$ : The total complexity of DA is thus ${\\rm O}(|E| (\\tau _{1} + \\tau _{2} ))$ , where $\\tau _{1}=$ the number of operations needed to solve a single constraint problem, and $\\tau _{2}=$ the number of operations needed to perform one pass through Steps 2 and 3.",
"It is well-known that in the discrete case $\\tau _{1} = {\\rm O}(|E| \\log (N(E)/|E|))$ (see Frederickson and Johnson (1982)), and in the continuous case $\\tau _{1} = {\\rm O}(|E| \\log |E| +\\chi )$ , where $\\chi $ is the time needed to solve a certain type of non-linear equation (see Zipkin, 1980).",
"This paper was written in 1985 and at that time, no strongly polynomial algorithm for submodular function minimization was known; the strongly polynomial algorithm (using the ellipsoid method) first appeared in 1993 [16].",
"This paper deals with base-polyhedra in Case ${\\bf Z}$ .",
"Main concern of this paper is to offer a general framework in which a greedy procedure called the marginal allocation algorithm (MAA) works.",
"The concept of concave order is introduced as a class of admissible objective functions for which the greedy procedure works.",
"The main result (Corollary 1 in Sec.3) states, roughly, that the MAA gives an optimal solution for every weakly concave order on polymatroids.",
"This is the first comprehensive book for algorithmic aspects of the resource allocation problem and its extensions.",
"Chapter 9, entitled “Resource allocation problems under submodular constrains” presents the fundamental and up-to-date results at that time, including those by Fujishige [11], Groenevelt [15], and Federgruen–Groenevelt [7].",
"In particular, Theorem 9.2.2 [20] states that the decomposition algorithm runs in polynomial time in $|E|$ and $\\log M$ , where $E$ is the ground set and $M$ is an upper bound on $r(E)$ for the submodular function $r$ expressing the submodular constraint.",
"The contents of Chapter 9 of this book are updated in a handbook chapter by Ibaraki–Katoh [23] in 1998.",
"Its revised version by Katoh–Shioura–Ibaraki [24] in 2013 incorporates the views from discrete convex analysis.",
"This paper deals with base-polyhedra in the context of game theory.",
"Recall that the core of a convex game is nothing but the base-polyhedron.",
"Naturally this paper deals exclusively with Case ${\\bf R}$ .",
"According to Tamir [38], this is the first paper proving the existence of a least majorized element in a base-polyhedron.",
"Technically speaking, this result could be obtained from a combination of the results of Groenevelt [15] (which was written in 1985 and published in 1991) and a well-known fact “least majorized element $\\Leftrightarrow $ simultaneous minimizer of all symmetric separable convex functions” (see Proposition REF ).",
"However, Dutta–Ray [6] and Groenevelt [15] were unaware of each other; see Table REF at the end of Appendix.",
"We also note that Fujishige [11] deals with quadratic functions only, and hence the results of [11] do not imply the existence of a least majorized element.",
"This book offers a comprehensive exposition of the results of Fujishige [11] about the lexico-optimal (inc-max) element of a base-polyhedron in Case ${\\bf R}$ .",
"There is an explicit statement at the beginning of Section 9 that the argument is not applicable to Case ${\\bf Z}$ .",
"For separable convex minimization, both Cases ${\\bf R}$ and ${\\bf Z}$ are treated.",
"In particular, the results of Groenevelt [15] are described in a manner consistent with the other part of this book.",
"It is stated that the decomposition algorithm works for Cases ${\\bf R}$ and ${\\bf Z}$ , but complexity analysis is explicit only for Case ${\\bf R}$ .",
"It is shown that the decomposition algorithm is strong polynomial for Case ${\\bf R}$ .",
"As a natural consequence of the fact that lexico-optimal bases in Case ${\\bf Z}$ are not considered in this book, no connection is made between separable convex minimization and lexico-optimality (inc-max, dec-min) in Case ${\\bf Z}$ .",
"Majorization concept is not treated in the first edition, whereas in the second edition the definition is given in Section 2.3 (p. 44) and a reference to Dutta–Ray [6] is added in Section 9.2 (p. 264).",
"This paper shows that there exist no strongly polynomial time algorithms to solve the resource allocation problem with a separable convex cost function.",
"Subsequently, Hochbaum and her coworkers made significant contributions to resource allocation problems in discrete variables, dealing with important special cases and showing improved complexity bounds for the special cases (e.g., Hochbaum–Hong [19]).",
"The survey paper by Hochbaum [18] is informative and useful.",
"This papers deals with g-polymatroids in Case ${\\bf R}$ and Case ${\\bf Z}$ .",
"The relationship between majorization and decreasing-minimality is discussed explicitly.",
"The main result is the existence of a least weakly submajorized element in a g-polymatroid.",
"The following sentences concerning Case ${\\bf R}$ in pages 585–585 are informative: Fujishige (1980) extends the results of Megiddo to a general polymatroid and presents an algorithm to find a lexicographically optimal base of the polymatroid with respect to an arbitrary positive weight vector $d$ .",
"This weighted model is closely related to the concept of $d$ -majorization introduced by Veinott (1971).",
"Neither Megiddo nor Fujishige relate their results on lexicographically optimal bases to the stronger concept of majorization.",
"(From Proposition 2.1 we note that if an arbitrary set has a least majorized element it is clearly lexicographically optimal.",
"However, every convex and compact set $S$ has a unique lexicographically maximum element, but might not have a least majorized element.)",
"The fact that a polymatroid has a least majorized base is shown by Dutta and Ray (1989).",
"They consider the core of a convex game as defined by Shapley (1971), which corresponds to a polymatroid.",
"(Strictly speaking the former is defined as a contra-polymatroid; see next section.)",
"We will extend and unify the above results by proving that a bounded generalized polymatroid contains both least submajorized and least supermajorized elements.",
"For the complexity of finding the unique minimizer $x^{*} \\in {\\bf R}^{n}$ of the square-sum over a g-polymatroid (Case ${\\bf R}$ ), the following statement can be found in page 587: $x^{*}$ can be found in strongly polynomial time by modifying the procedure in Fujishige (1980) and Groenevelt (1991) which is applicable to polymatroids.",
"The latter procedure can now be implemented to solve any convex separable quadratic over a polymatroid in a strongly polynomial time since its complexity is dominated by the efforts to minimize a (strongly) polynomial number of submodular functions.",
"There is no statement about the complexity in Case ${\\bf Z}$ .",
"Table: Referencing relations between papers"
]
] | 1808.08477 |
[
[
"Bayesian inference for a single factor copula stochastic volatility\n model using Hamiltonian Monte Carlo"
],
[
"Abstract For modeling multivariate financial time series we propose a single factor copula model together with stochastic volatility margins.",
"This model generalizes single factor models relying on the multivariate normal distribution and allows for symmetric and asymmetric tail dependence.",
"We develop joint Bayesian inference using Hamiltonian Monte Carlo (HMC) within Gibbs sampling.",
"Thus we avoid information loss caused by the two-step approach for margins and dependence in copula models as followed by Schamberger et al(2017).",
"Further, the Bayesian approach allows for high dimensional parameter spaces as they are present here in addition to uncertainty quantification through credible intervals.",
"By allowing for indicators for different copula families the copula families are selected automatically in the Bayesian framework.",
"In a first simulation study the performance of HMC is compared to the Markov Chain Monte Carlo (MCMC) approach developed by Schamberger et al(2017) for the copula part.",
"It is shown that HMC considerably outperforms this approach in terms of effective sample size, MSE and observed coverage probabilities.",
"In a second simulation study satisfactory performance is seen for the full HMC within Gibbs procedure.",
"The approach is illustrated for a portfolio of financial assets with respect to one-day ahead value at risk forecasts.",
"We provide comparison to a two-step estimation procedure of the proposed model and to relevant benchmark models: a model with dynamic linear models for the margins and a single factor copula for the dependence proposed by Schamberger et al(2017) and a multivariate factor stochastic volatility model proposed by Kastner et al(2017).",
"Our proposed approach shows superior performance."
],
[
"Introduction",
"Multivariate time series models are employed to model the joint behaviour of stocks.",
"It is important to understand the dependence among these financial assets since it has high influence on the performance and the risk associated with a corresponding portfolio ([17], [14]).",
"Vine copulas ([3], [2]) have proven a useful tool to facilitate complex dependence structures ([39], [6], [1], [19], [37]).",
"A vine copula model consists of $\\frac{d(d-1)}{2}$ pair copulas, where $d$ is the number of assets.",
"So the number of parameters grows quadratically with $d$ .",
"[34] proposed the factor copula model, where the number of parameters grows only linearly in $d$ .",
"This model can be seen as a generalization of the Gaussian factor model.",
"The factor copula model provides much more flexibility, compared to the Gaussian one, as it is made up of different pair copulas that can be chosen arbitrarily.",
"Thus it covers a broad range of dependence structures that can accommodate symmetric as well as asymmetric tail dependence.",
"One way to construct multivariate time series models is to combine a univariate time series model for the margins with a dependence model such as the factor copula.",
"Univariate time series models for financial data need to account for typical characteristics like time varying volatility and volatility clustering.",
"Popular examples of such models include generalized autoregressive conditional heteroskedasticity (GARCH) models ([18], [5]), the more recently developed generalized autoregressive score (GAS) models ([12]) and stochastic volatility (SV) models ([32]).",
"Using the classification of [11] GARCH and GAS models are observation driven models, whereas the SV model is a parameter driven model.",
"In observation driven models volatility is modeled deterministically through the observed past and as such results cannot be transferred to other data sets following the same data generating process.",
"Inference for these observation driven models is often easier since evaluation of the likelihood is straightforward.",
"Inference for SV models is more involved since likelihood evaluation requires high dimensional integration.",
"But efficient MCMC algorithms have been developed ([30]).",
"In the SV model volatility is modeled as latent variables that follow an autoregressive process of order 1.",
"This representation has compared favorably to GARCH specifications in several data sets ([50], [8]).",
"We propose a copula based SV model.",
"The marginals follow a SV model and the dependence is modeled through a single factor copula.",
"In contrast to other factor SV models as proposed by [24] or [31] we only allow for one factor and dependence parameters remain constant.",
"But we do not assume that conditional on the volatilities the observed data is multivariate normal or Student t distributed.",
"Here we provide more flexibility through the choice of different pair copula families.",
"The single factor copula model has also been deployed by [43] who use dynamic linear models ([49]) as marginals and by [34] who use GARCH models as marginals.",
"As it is common in copula modeling, [43] and [34] both use a two-step approach for estimation.",
"They first estimate marginal parameters and based on these estimates they infer the dependence parameters.",
"[47] provide full Bayesian inference for a single factor copula based model, but their marginal models have only few parameters and the proposals for MCMC are built using independence among components.",
"However, for SV margins we need to estimate all $T$ log volatilities, where $T$ denotes the length of the time series.",
"Thus we have more than $T$ parameters to estimate per margin.",
"These more sophisticated marginal models for financial data are difficult to handle within a full Bayesian approach.",
"We are able to overcome the two-step approach commonly used in copula modeling and provide full Bayesian inference.",
"For this we develop and implement a Hamiltonian Monte Carlo (HMC) ([15], [38]) within Gibbs sampler.",
"In HMC information of the gradient of the log posterior density is used to propose new states which leads to an efficient sampling procedure.",
"The main novel contributions of this paper are: joint Bayesian inference of a single factor copula model with SV margins using HMC, automated selection of linking copula families and improved value at risk (VaR) forecasting over benchmark models in a financial application.",
"More precisely, we first demonstrate how HMC can be employed for the single factor copula model and compare the HMC approach for the copula part to the MCMC approach of [43] who use adaptive rejection Metropolis within Gibbs sampling ([22]).",
"HMC shows superior performance in terms of effective sample size, MSE and observed coverage probabilities.",
"Further, the HMC scheme is integrated within a Gibbs approach that allows for full Bayesian inference of the proposed single factor copula based SV model, including copula family selection.",
"Copula families are modeled with discrete indicator variables, which can be sampled directly from their full conditionals within our Gibbs approach.",
"Continuous parameters are updated with HMC.",
"Within the Bayesian procedure, marginal and dependence parameters are estimated jointly.",
"We stress that the joint estimation of marginal and dependence parameters is very demanding and is therefore most commonly avoided.",
"Instead a two-step approach is used where the marginal parameters are considered fixed when estimating dependence parameters, i.e.",
"uncertainty in the marginal parameters is ignored.",
"An advantage of the full Bayesian approach is that this uncertainty is not ignored and full uncertainty quantification is straightforward through credible intervals.",
"We further demonstrate the usefulness of the proposed single factor copula SV model with one-day ahead VaR prediction for financial data involving six stocks.",
"Within our full Bayesian approach a VaR forecast is obtained as an empirical quantile of simulations from the predictive distribution.",
"In addition we show that joint estimation leads to more accurate VaR forecasts than VaR forecasts obtained from a two-step approach.",
"The paper is organized as follows.",
"In Sections and we discuss the single factor copula model and the single factor copula SV model, respectively.",
"Both sections follow a similar structure.",
"We first specify a Bayesian model, propose a Bayesian inference approach and evaluate the performance of the approach with simulated data.",
"In Section the proposed single factor copula SV model is applied to financial returns data.",
"Section concludes."
],
[
"Bayesian inference for single factor copulas using the HMC approach",
"Hamiltonian dynamics describe the time evolution of a physical system through differential equations.",
"In Hamiltonian Monte Carlo (HMC) the posterior density is connected to the energy function of a physical system.",
"This makes it possible to propose states in the sampling process which are guided by appropriate differential equations.",
"New states are chosen utilizing information of the gradient of the log posterior density, which can lead to more efficient sampling procedures.",
"Therefore HMC has become popular.",
"For example [25] demonstrate how to estimate parameters of generalized extreme value distributions with HMC, while [41] use HMC to sample from truncated multivariate Gaussian distributions.",
"Especially with the development of the probabilistic programming language STAN by [7] its popularity is growing.",
"STAN allows easy model specification and deploys the No-U-Turn sampler of [26].",
"This extension of HMC automatically and adaptively selects the tuning parameters.",
"Instead of using STAN we provide our own implementation of HMC.",
"This allows us to use the HMC updates developed for single factor copulas within other samplers, as we will see in Section .",
"A short introduction to HMC based on [38] is provided in Appendix 6.1."
],
[
"Model specification",
"To illustrate the viability of HMC for factor copula models we start with the single factor copula model as a special case of the $p$ factor copula model according to [34].",
"We consider $d$ uniform(0,1) distributed variables $U_1, \\ldots U_d$ together with a uniform(0,1) distributed latent factor $V$ .",
"In the single factor copula model we assume that given $V$ , the variables $U_1, \\ldots , U_d$ are independent.",
"This implies that the joint density of $U_{1:d} = (U_1, \\ldots U_d)^\\top $ can be written as $\\begin{split}c_{U_{1:d}}(u_{1:d}) &= \\int _0^1 \\prod _{j=1}^d c_{j|V}(u_j|v) dv =\\int _0^1 \\prod _{j=1}^d c_{j}(u_j,v) dv,\\end{split}$ where $c_{j}$ is the density of $C_j$ , the copula of $(U_j,V)$ .",
"The copulas $C_{1}, \\ldots , C_{d}$ are called linking copulas as they link each of the observed copula variables $U_j$ to the latent factor $V$ .",
"For inference we use one parametric copula families, i.e.",
"we equip each linking copula density with a corresponding parameter $\\theta _j$ , and (REF ) becomes $c_{U_{1:d}}(u_{1:d}; \\theta _{1:d} ) = \\int _0^1 \\prod _{j=1}^d c_{j}(u_j,v;\\theta _j) dv.$ As it is common in Bayesian statistics we treat the latent variable $V$ as a parameter $v$ .",
"The joint density of $U_{1:d}$ , denoted by $c_{U_{1:d}}$ , given the parameters $(\\theta _{1:d} ,v)$ is obtained as $c_{U_{1:d}}(u_{1:d}; \\theta _{1:d} ,v) = \\prod _{j=1}^d c_{j}(u_j,v;\\theta _j) .$ Since the latent variable $V$ is random for each observation vector $(u_{t1}, \\ldots , u_{td})^\\top $ , we have $T$ latent parameters $v_{1:T} = (v_1, \\ldots , v_T)^\\top $ for $T$ time points.",
"The likelihood of the parameters $(\\theta _{1:d}, v_{1:T})$ given $T$ independent observations $U_{1:T,1:d}=(u_{tj})_{t = 1, \\ldots , T, j=1, \\ldots , d}$ is therefore $\\ell (\\theta _{1:d}, v_{1:T}|U_{1:T,1:d}) = \\prod _{t=1}^T\\prod _{j=1}^d c_{j}(u_{tj},v_t;\\theta _j).$"
],
[
"Bayesian inference",
"So far, Bayesian inference for the single factor copula model was addressed by [43] and [47].",
"Both approaches use Gibbs sampling where one can exploit the fact that the factors $v_1, \\ldots v_T$ are independent given the copula parameters $\\theta _1, \\ldots \\theta _d$ and vice versa.",
"We now show how HMC can be used for the single factor copula model.",
"Sampling with HMC is slower since it requires several evaluations of the gradient of the log posterior density.",
"However with HMC there is no blocking involved and we update the whole parameter vector, with well chosen proposals obtained from the Leapfrog approximation, at once.",
"We expect more accurate samples since this sampler suffers less from the dependence between factors and copula parameters.",
"To support this statement we compare HMC to adaptive rejection Metropolis sampling within Gibbs sampling (ARMGS) ([22]).",
"ARMGS is the sampler that worked best among several samplers that have been investigated by [43] for single factor copula models."
],
[
"Parameterization",
"Since HMC operates on unconstrained parameters we need to provide parameter transformations to remove the constraints present in our problem.",
"For many one parametric copula families there is a one-to-one correspondence between the copula parameter $\\theta _j$ and Kendall's $\\tau $ , i.e.",
"there is an invertible function $g_j$ such that $\\tau _j = g_j(\\theta _j)$ , see [27], Chapter 4.",
"For example $g_j(\\theta _j) = \\frac{2}{\\pi } \\arcsin \\left(\\theta _j\\right)$ is the corresponding Kendall's $\\tau $ for a Gaussian linking copula.",
"Furthermore we restrict the Kendall's $\\tau $ values to be in $(0,1)$ to avoid problems that might occur due to multimodal posterior distributions.",
"This is not a severe restriction for applications since $\\tau _{U_1,U_2} = -\\tau _{U_1,1-U_2}$$\\tau _{U_1,1-U_2} = P((U_1 - \\tilde{U}_1) ((1-U_2) - (1-\\tilde{U}_2))>0) - P((U_1 - \\tilde{U}_1) ((1-U_2) - (1-\\tilde{U}_2))<0) = P((U_1 - \\tilde{U}_1) (U_2 -\\tilde{U}_2)<0)-P((U_1 - \\tilde{U}_1) (U_2 -\\tilde{U}_2)>0) = - \\tau _{U_1,U_2}$ , where $(\\tilde{U}_1, \\tilde{U}_2)$ is an independent copy of $(U_1,U_2)$ ..",
"So we can replace $U_2$ by $1-U_2$ if we want to model negative dependence between $U_1$ and $U_2$ .",
"The components of the latent factors $v_{1:T}$ are also in $(0,1)$ .",
"To transform parameters on the $(0,1)$ scale to the unconstrained scale the logit function is a common choice.",
"Therefore we use the following transformations for the copula parameters $\\theta _{1:d}$ and the latent factors $v_{1:T}$ $\\delta _j = \\ln (\\frac{g_j(\\theta _j)}{1-g_j(\\theta _j)}),~~~w_t = \\ln (\\frac{v_t}{1-v_t}),$ and obtain unconstrained parameters $\\delta _j,v_t \\in \\mathbb {R}$ for $ j = 1,\\ldots d, t=1, \\ldots ,T$ .",
"We specify the prior distributions for $(\\delta _{1:d}, w_{1:T})$ such that the distributions implied for the corresponding Kendall's $\\tau $ and for $v_t$ are independently uniform on the interval $(0,1)$ .",
"Applying the density transformation law this implies that the factor copula ($FC$ ) prior density can be expressed as $\\begin{split}\\pi _{FC}(\\delta _{1:d}, w_{1:T}) &= \\prod _{t=1}^T \\pi _u(w_t) \\prod _{j=1}^d \\pi _u(\\delta _j), \\\\\\end{split}$ where $\\pi _{u}(x) = (1+\\exp (-x))^{-2} \\exp (-x), x \\in \\mathbb {R}.$ With these choices in (REF ) and (REF ) the posterior density is proportional to $\\begin{split}f(\\delta _{1:d}, w_{1:T}|U_{1:T,1:d}) \\propto l(\\theta _{1:d}, v_{1:T}|U_{1:T,1:d}) \\cdot \\pi _{FC}(\\delta _{1:d},w_{1:T}),\\\\\\end{split}$ where $\\theta _j$ and $v_t$ are functions of $\\delta _j$ and $w_t$ respectively.",
"Therefore the log posterior density is, up to an additive constant, given by $\\begin{split}\\mathcal {L}(\\delta _{1:d},w_{1:T}|U_{1:T,1:d}) \\propto & \\sum _{t=1}^T \\sum _{j=1}^d \\ln (c_j(u_{tj},v_t;\\theta _j))+ \\sum _{t=1}^T \\ln (\\pi _u(w_t))+\\sum _{j=1}^d \\ln (\\pi _u(\\delta _j)).\\end{split}$ Derivatives of the log posterior density with respect to all parameters are determined to perform Leapfrog approximations (see Appendix REF ).",
"With this at hand, HMC can be implemented as any Metropolis-Hastings sampler.",
"To run the algorithm we need to set values to the hyper parameters: the Leapfrog stepsize $\\epsilon $ , the number of Leapfrog steps $L$ and the mass matrix $M$ .",
"Choosing $\\epsilon $ and $L$ is not easy since good choices of these parameters can vary depending on different regions of the state space.",
"[38] suggest to randomly select $\\epsilon $ and $L$ from a set of values that may be appropriate for different regions.",
"This is the approach that we follow.",
"For our simulation study we have seen that choosing $\\epsilon $ uniformly between 0 and $0.2$ and choosing $L$ uniformly between 0 and 40 leads to reasonable mixing as measured by the effective sample size ([21], page 286).",
"The mass matrix $M$ is set equal to the identity matrix.",
"The MCMC procedure is implemented in $\\texttt {R}$ using the $\\texttt {R}$ package $\\texttt {Rcpp}$ by [16] which allows the integration of $\\texttt {C++}$ .",
"Effective sample sizes are calculated with the $\\texttt {R}$ package $\\texttt {coda}$ by [42]."
],
[
"Simulation study",
"To compare our approach we conduct the same simulation study as in [43].",
"For each of three scenarios, we simulate 100 data sets from the single factor copula model with $T=200$ and $d=5$ .",
"The three scenarios are characterized by the values of Kendall's $\\tau $ of the linking copulas and are denoted by the low $\\tau $, the high $\\tau $ and the mixed $\\tau $ scenario.",
"The Kendall's $\\tau $ values are shown in Table REF .",
"As linking copulas only Gumbel copulas are considered.",
"Based on these simulated data sets the samplers are run for 11000 iterations, whereas the first 1000 iterations are discarded for burn in.",
"Table: Kendall's τ\\tau values for the linking copulas C 1 ,...C 5 C_{1}, \\ldots C_{5} in the three scenarios.Table REF shows the results of the simulation study and compares them to the results obtained by [43] using adaptive rejection Metropolis sampling within Gibbs sampling (ARMGS).",
"The corresponding error statistics (e.g.",
"mean absolute deviation (MAD), mean squared error (MSE)) for each parameter is obtained from 100 replications.",
"Then, e.g.",
"the MSE for $\\tau $ in Table REF is computed as the average of MSE for $\\tau _1$ $,\\ldots ,$ MSE for $\\tau _5$ .",
"Since the objective is the comparison of our method to the method of [43] we follow their approach and calculate the error statistics from point estimates (marginal posterior mode estimates obtained as the estimated modes of univariate kernel density estimates).",
"Further we calculate the error statistics for $\\tau _{1:d}, v_{1:T}$ which are one-to-one transformations of $\\delta _{1:d}, w_{1:T}$ .",
"We see that a more accurate credible interval, a lower mean absolute deviation and a lower mean squared error is achieved in most cases by HMC compared to ARMGS.",
"Furthermore the effective sample size per minute is much higher for HMC.",
"Table REF shows the results of the simulation study in more detail, i.e.",
"we do not average over values of $\\tau _1, \\ldots , \\tau _5$ and $v_1, \\ldots v_{200}$ .",
"It is noticable that mixing is worse for higher values of Kendall's $\\tau $ in every scenario, whereas it is most extreme in the mixed $\\tau $ scenario.",
"This was also observed for ARMGS (see [43] Table 9 in the appendix).",
"Table: Comparison of the ARMGS and HMC method in terms of mean absolute deviation (MAD), mean squared error (MSE), effective sample size per minute (ESS/min) and observed coverage probability of the credible intervals (C.I.",
").Table: Detailed simulation results for the HMC method.",
"We show the estimated mean absolute deviation (MAD), mean squared error (MSE), effective sample size per minute (ESS/min) and observed coverage probability of the credible intervals (C.I.)",
"for τ 1 ,...,τ 5 \\tau _1, \\ldots , \\tau _5 and five selected latent variables v t ,t=10,50,100,150,190v_t, t = 10,50,100,150,190."
],
[
"The single factor copula stochastic volatility model",
"Now we combine the single factor copula with margins driven by a stochastic volatility model and develop a Bayesian approach to jointly estimate the parameters of the proposed model."
],
[
"The marginal model",
"We utilize the stochastic volatility model ([32]) as marginal model.",
"In this model the log variances $(s_1, \\ldots , s_T)^\\top $ of a conditionally normally distributed vector $(Z_1, \\ldots , Z_T)^\\top $ are modeled with a latent AR(1) process.",
"This AR(1) process has mean parameter $\\mu \\in \\mathbb {R}$ , persistence parameter $\\phi \\in (-1,1)$ and standard deviation parameter $\\sigma \\in (0,\\infty )$ .",
"More precisely, the stochastic volatility (SV) model is given by $\\begin{split}Z_t &= \\exp (\\frac{s_t}{2}) \\epsilon _t, ~~ t = 1, \\ldots T, \\\\s_t &= \\mu + \\phi (s_{t-1} - \\mu ) + \\sigma \\eta _t , ~~ t = 1, \\ldots T,\\end{split}$ where $s_0|\\mu , \\phi , \\sigma \\sim N\\left(\\mu , \\frac{\\sigma ^2}{1- \\phi ^2}\\right)$ and $\\epsilon _t, \\eta _t, \\sim N(0,1)$ independently, for $t =1, \\ldots , T$ .",
"[30] develop an MCMC algorithm for this model which uses the ancillarity-sufficiency interweaving strategy proposed by [51].",
"This strategy leads to an efficient MCMC sampling procedure which is implemented in the $\\texttt {R}$ package $\\texttt {stochvol}$ (see [28]).",
"We discuss the prior densities proposed by [28] since we also utilize them later.",
"The following priors for $\\mu , \\phi $ and $\\sigma $ are chosen $\\begin{split}&\\mu \\sim N(0,100), ~~ \\frac{\\phi + 1}{2} \\sim Beta(5,1.5), ~~\\sigma ^2 \\sim \\chi ^2_1.\\end{split}$ The prior for $\\mu $ is rather uninformative, whereas the prior for $\\phi $ puts more mass on higher values for the persistence parameter.",
"High persistence parameters are characteristic for financial time series.",
"The prior choice for $\\sigma ^2$ differs from the popular inverse Gamma prior.",
"In contrast to the inverse Gamma prior, the $\\chi ^2_1$ has more mass close to zero and thus allows for latent volatilities with less fluctuations.",
"Denoting by $s_{0:T} =(s_0, \\ldots , s_T)^\\top $ the vector of latent log variances, the prior density of $(\\mu , \\phi , \\sigma , s_{0:T}^\\top )^\\top $ is given by $\\begin{split}\\pi _{SV}(\\mu , \\phi , \\sigma , s_{0:T}) &= f( s_{0:T}|\\mu , \\phi , \\sigma ) f(\\mu , \\phi , \\sigma )\\\\&= \\varphi \\left(s_0 \\Big |\\mu , \\frac{\\sigma ^2}{1-\\phi ^2}\\right) \\prod _{t=1}^T \\varphi \\left(s_t|\\mu + \\phi (s_{t-1}-\\mu ),\\sigma ^2\\right) \\pi (\\mu ) \\pi (\\phi ) \\pi (\\sigma ),\\end{split}$ where $\\varphi \\left(\\cdot |\\mu _{normal},\\sigma _{normal}^2\\right)$ denotes a univariate normal density with mean $\\mu _{normal}$ and variance $\\sigma _{normal}^2$ and $\\pi (\\cdot )$ denotes the corresponding prior density as specified in (REF ).",
"We propose a multivariate dynamic model where each marginal follows a stochastic volatility model and the dependence between the marginals is captured by a single factor copula, the single factor copula stochastic volatility (factor copula SV) model.",
"In particular for $t=1,\\ldots , T, j=1, \\ldots ,d$ we assume that $\\begin{split}Z_{tj} &= \\exp (\\frac{s_{tj}}{2}) \\epsilon _{tj} \\\\s_{tj} &= \\mu _j + \\phi _j ( s_{t-1j} - \\mu _j) + \\sigma _j \\eta _{tj},\\end{split}$ where $\\mu _j \\in \\mathbb {R}, \\phi _j \\in (-1,1), \\sigma _j \\in (0, \\infty ), s_{0j}|\\mu _j, \\phi _j, \\sigma _j \\sim N\\left(\\mu _j, \\frac{\\sigma _j^2}{1- \\phi _j^2}\\right)$ and $\\eta _{tj} \\sim N(0,1)$ i.i.d.",
"holds.",
"The joint distribtion of the errors $\\epsilon _{tj}$ is now considered.",
"We model the dependence among the marginals by employing a factor copula model on the errors.",
"We further allow for Bayesian selection of the $d$ linking copula families of this factor copula instead of assuming that they were known as in Section and as in [43].",
"The families are chosen from a set $\\mathcal {M}$ of one parametric copula families, e.g.",
"$\\mathcal {M} = \\lbrace \\text{Gaussian, Gumbel, Clayton}\\rbrace $ .",
"[43] estimated one model for each specification of the linking copula families.",
"Since there are $|\\mathcal {M}|^d$ different specifications, they only considered factor copulas where all linking copulas belong to the same family.",
"With our Bayesian family selection, we can profit from the full flexibility of the factor copula model by allowing for all $|\\mathcal {M}|^d$ specifications.",
"In particular our modeling approach allows to combine different copula families.",
"Therefore we define $d$ family indicator variables $m_j \\in \\mathcal {M}, j=1,\\ldots , d $ .",
"Further, we introduce parameters $\\delta _j \\in \\mathbb {R}, j=1, \\ldots , d$ which are mapped to the corresponding Kendall's $\\tau $ with the sigmoid (inverse logit) transform.",
"This Kendall's $\\tau $ is then mapped to the corresponding copula parameter with the function $g^{-1}_{m_j}$ , i.e.",
"$\\theta _j^{m_j} = \\theta _j^{m_j}(\\delta _j) = g^{-1}_{m_j}\\left(\\frac{\\exp (\\delta _j)}{1+\\exp (\\delta _j)}\\right).$ Note that the parameter $\\delta _j$ has the same interpretation for different copula families: It is the logit transform of the associated Kendall's $\\tau $ value.",
"This allows to share this parameter among different copula families.",
"In the following, the with copula family $m_j$ associated copula parameter $\\theta _j^{m_j}$ is determined as a function of $\\delta _j$ and $m_j$ .",
"Since we model the dependence among the errors with a single factor copula, we assume that there exists a latent factor $v_t \\sim unif(0,1)$ for each $t$ such that the following holds for the error vector at time $t$ , $\\epsilon _{t\\cdot } = (\\epsilon _{t1}, \\ldots \\epsilon _{td})^\\top $ , $f(\\epsilon _{t\\cdot }|v_t, m_{1:d}, \\theta _1^{m_1}, \\ldots , \\theta _d^{m_d}) = \\prod _{j=1}^d \\Big [c^{m_j}_j(\\Phi (\\epsilon _{tj}),v_t;\\theta _j^{m_j}) \\varphi (\\epsilon _{tj})\\Big ],$ where $\\Phi $ denotes the standard normal distribution function.",
"In particular $\\epsilon _{tj} \\sim N(0,1)$ for any $t$ and $j$ .",
"Here $c^{m_j}_j(\\cdot ,\\cdot ;\\theta _j^{m_j})$ is the density of the bivariate copula family $m_j$ with parameter $\\theta _j^{m_j}$ .",
"Integrating out the factor $v_t$ in (REF ) yields $f(\\epsilon _{t\\cdot }|m_{1:d}, \\theta _1^{m_1}, \\ldots , \\theta _d^{m_d}) = \\Bigg [\\int _{(0,1)} \\prod _{j=1}^d c^{m_j}_j(\\Phi (\\epsilon _{tj}),v_t;\\theta _j^{m_j}) dv_t \\Bigg ] \\prod _{j=1}^d \\varphi (\\epsilon _{tj}).$ Furthermore we assume that the $T$ components of ($\\epsilon _{1\\cdot }$ , $\\ldots $ , $\\epsilon _{T\\cdot }$ ) are independent given the family indicators $m_{1:d}$ and the dependence parameters $\\delta _{1:d}, v_{1:T}$ .",
"To shorten notation we use the following abbreviations: $Z = (z_{tj})_{t=1,\\ldots , T, j=1,\\ldots d}$ the matrix of observations, $\\mathcal {E} = (\\epsilon _{tj})_{t=1,\\ldots , T, j=1,\\ldots d}$ the matrix of errors, $\\mu = (\\mu _j)_{j=1,\\ldots d}$ the vector of means of the marginal stochastic volatility models, $\\phi = (\\phi _j)_{j=1,\\ldots d}$ the vector of persistence parameters of the marginal stochastic volatility models, $\\sigma = (\\sigma _j)_{j=1,\\ldots d}$ the vector of standard deviations of the marginal stochastic volatility models, $S = (s_{tj})_{t=0,\\ldots , T, j=1,\\ldots d}$ the matrix of log variances, $s_{\\cdot j} = (s_{tj})_{t=0,\\ldots , T}$ the vector of log variances of the $j$ -th marginal, $v = (v_t)_{t=1,\\ldots , T}$ the vector of latent factors, $\\delta = (\\delta _j)_{j=1,\\ldots , d}$ the vector of transformed copula parameters, $m = (m_j)_{j=1,\\ldots , d}$ the vector of copula family indicators.",
"Utilizing these abbreviations, we can summarize the parameters of our model as $\\lbrace \\mu , \\phi , \\sigma , S, v, \\delta , m \\rbrace .$ For the special case where all linking copulas are Gaussian (i.e.",
"$m_j =$ Gaussian for $j = 1, \\ldots , d$ ), [34] show that the errors $\\epsilon _{tj}$ specified in (REF ) allow for the following stochastic representation $\\epsilon _{tj} = \\rho _j w_t + \\sqrt{1-\\rho _j^2} \\xi _{tj},$ where $w_t \\sim N(0,1)$ and $\\xi _{tj} \\sim N(0,1)$ independently and $\\rho _j = \\theta _j^{m_j}$ is the Gaussian copula parameter.",
"Therefore we obtain the following additive structure $Z_{tj} = \\rho _j \\exp (\\frac{s_{tj}}{2}) w_t + \\exp (\\frac{s_{tj}}{2}) \\sqrt{1-\\rho _j^2} \\xi _{tj}.$ This implies a time dynamic covariance matrix with elements $\\operatorname{cov}(Z_{tj},Z_{tk}) = \\rho _j \\rho _k \\exp (\\frac{s_{tj}}{2}) \\exp (\\frac{s_{tk}}{2}) \\text{ for } j \\ne k.$ The correlation matrix however remains constant as time evolves and its off-diagonal elements are given by $\\operatorname{cor}(Z_{tj},Z_{tk}) = \\rho _j \\rho _k \\text{ for } j \\ne k.$ The additive structure in (REF ) shows connections to other multivariate factor stochastic volatility models (see [9], [31]).",
"This can be seen by considering the following reparameterization $\\begin{split}s_{tj}^{\\prime } &{s_{tj}} + \\ln (1-\\rho _{j}^2), \\lambda _j \\frac{\\rho _j}{\\sqrt{1-\\rho _j^2}}, \\\\\\end{split}$ which implies the following representation of (REF ) $Z_{tj} = \\lambda _j \\exp (\\frac{s_{tj}^{\\prime }}{2}) w_t + \\exp (\\frac{s_{tj}^{\\prime }}{2}) \\xi _{tj}.$ Here $s_{tj}^{\\prime }$ is an AR(1) process with mean $\\mu _j + \\ln (1-\\rho _j^2)$ , persistence parameter $\\phi _j$ and standard deviation parameter $\\sigma _j$ .",
"For comparison, the model of [31] with one factor is given by $Z_{tj} = \\lambda _j \\exp (\\frac{s_{td+1}^{\\prime }}{2}) w_t + \\exp (\\frac{s_{tj}^{\\prime }}{2}) \\xi _{tj},$ with one additional latent AR(1) process $s_{td+1}^{\\prime }, t=1, \\ldots , T$ .",
"This implies time varying correlations given by $\\operatorname{cor}(Z_{tj},Z_{tk})=\\frac{\\lambda _j\\lambda _k \\exp (s_{td+1}^{\\prime })}{\\sqrt{\\lambda _j^2\\exp (s_{td+1}^{\\prime })+\\exp (s_{tj}^{\\prime })}\\sqrt{\\lambda _k^2\\exp (s_{td+1}^{\\prime })+\\exp (s_{tk}^{\\prime })}} \\text{ for } j \\ne k.$ Dividing $Z_{tj}$ by $\\exp (\\frac{s_{tj}^{\\prime }}{2})$ in (REF ) we recognize the structure of a standard factor model for $Z_{tj}^{\\prime } \\frac{Z_{tj}}{\\exp (\\frac{s_{tj}^{\\prime }}{2})}$ given by $Z_{tj}^{\\prime } = \\lambda _j w_t + \\xi _{tj},$ with factor loadings $\\lambda _1, \\ldots , \\lambda _d$ and factor $w_t$ .",
"In representation (REF ) the variance of $\\xi _{tj}$ is restricted to 1 whereas in the standard factor model (see e.g.",
"[36]) it is usually modeled through an additional variance parameter.",
"Since the variance of $\\epsilon _{tj}$ is already determined ($\\epsilon _{tj} \\sim N(0,1)$ ) we have this additional restriction compared to factor models with flexible marginal variance.",
"Note that $Z_{tj}$ still has flexible variance and the restriction for $\\epsilon _{tj}$ is necessary to ensure identifiability.",
"If all copula families are Gaussian other multivariate factor stochastic volatility models provide generalizations by allowing for more factors and for a time varying correlation.",
"We provide generalization with respect to the error distribution.",
"The choice of different pair copula families provides a flexible modeling approach and our model can accommodate features that can not be modeled with a multivariate normal distribution as e.g.",
"symmetric or asymmetric tail dependence.",
"[43] also use factor copulas to model dependence among financial assets.",
"Their approach differs to our approach in the choice of the marginal model.",
"They use dynamic linear models ([49]).",
"Secondly they assume the copula families to be known and they perform a two-step estimation approach, whereas we provide full Bayesian inference."
],
[
"Bayesian inference",
"In the following we develop a full Bayesian approach for the proposed model.",
"We use a block Gibbs sampler to sample from the posterior distribution.",
"We use $d$ blocks for the marginal parameters $(\\mu _j, \\phi _j , \\sigma _j, s_{\\cdot j})$ , $j=1, \\ldots ,d$ , one block for the dependence parameters $(\\delta , v)$ and $d$ blocks for the copula family indicators $m$ .",
"Sampling from the full conditionals is done with HMC for the first $d+1$ blocks.",
"Conditioning the dependence parameters on the marginal parameters and on the copula family indicators we are in the single factor copula framework of Section .",
"We have seen that HMC provides an efficient way to sample the dependence parameters.",
"Conditioned on the dependence parameters and on the family indicators, the marginal parameters corresponding to different dimensions are independent.",
"Each dimension can be considered as a generalized stochastic volatility model, where the distribution of the errors is determined by the corresponding linking copula.",
"Sampling from the posterior distribution is more involved than in the Gaussian case.",
"In the Gaussian case one can use an approximation of a mixture of normal distributions and rewrite the observation equation $Z_{tj} = \\exp (\\frac{s_{tj}}{2}) \\epsilon _{tj}$ as a linear, conditionally Gaussian state space model ([40], [30]).",
"This is not possible in our case and therefore HMC, which has already shown good performance for the copula part and only requires derivation of the derivatives, is our method of choice.",
"The family indicators $m$ are discrete variables which can be sampled directly from their full conditionals."
],
[
"Prior densities",
"For the copula family indicators we use independent discrete uniform priors, i.e $\\pi (m_j) = \\frac{1}{|\\mathcal {M}|}$ for $m_j \\in \\mathcal {M}, j=1, \\ldots , d$ independently.",
"The prior density of the other parameters is chosen as the product of the priors used for the marginal stochastic volatility model and for the single factor copula model, i.e.",
"$\\pi _J(\\mu , \\phi , \\sigma , S, \\delta ,v) = \\prod _{j=1}^d \\pi _{SV}(\\mu _j, \\phi _j,\\sigma _j, s_{\\cdot j}) \\pi _{u}(\\delta _j),$ where $\\pi _{SV}(\\cdot )$ and $\\pi _{u}(\\cdot )$ are specified in (REF ) and (REF ), respectively.",
"Further we assume that the family indicators are a priori independent of the parameters in (REF ).",
"The conditional independence of the $T$ components of ($\\epsilon _{1\\cdot }$ , $\\ldots $ , $\\epsilon _{T\\cdot }$ ) implies that the conditional distribution of the errors given the dependence parameters and the copula family indicators is $f(\\mathcal {E}|v, \\delta , m) = \\prod _{t=1}^T \\prod _{j=1}^d \\Big [c^{m_j}_j(\\Phi (\\epsilon _{tj}),v_t;\\theta _j^{m_j}) \\varphi (\\epsilon _{tj})\\Big ].$ Using the density transformation rule, the likelihood of parameters $(\\mu , \\phi , \\sigma , S, \\delta ,v, m)$ given the observation matrix $Z$ is obtained as $\\begin{split}\\ell (\\mu , \\phi , \\sigma , S, \\delta ,v, m|Z) &= \\prod _{t=1}^T\\prod _{j=1}^d \\left[ c_j^{m_j}(\\Phi \\left(\\frac{z_{tj}}{\\exp (\\frac{s_{tj}}{2})}\\right),v_t;\\theta _j^{m_j} ) \\varphi (\\frac{z_{tj}}{\\exp (\\frac{s_{tj}}{2})}) \\frac{1}{\\exp (\\frac{s_{tj}}{2})} \\right] .\\\\\\end{split}$ The conditional density we need to sample from is given by $\\begin{split}f(\\mu _j, \\phi _j, &\\sigma _j, s_{\\cdot j}| Z,\\mu _{-j},\\phi _{-j},\\sigma _{-j},S_{\\cdot -j},\\delta ,v, m)\\\\&\\propto \\ell (\\mu , \\phi , \\sigma , S, \\delta ,v, m|Z) \\pi _J(\\mu , \\phi , \\sigma , S, \\delta , v)\\\\&\\propto \\prod _{t=1}^T \\left[ c_j^{m_j}(\\Phi (\\frac{z_{tj}}{\\exp (\\frac{s_{tj}}{2})}),v_t;\\theta _j^{m_j} ) \\varphi (\\frac{z_{tj}}{\\exp (\\frac{s_{tj}}{2})}) \\frac{1}{\\exp (\\frac{s_{tj}}{2})} \\right] \\pi _{SV}(\\mu _j, \\phi _j,\\sigma _j, s_{\\cdot j}).",
"\\\\\\end{split}$ Here the abbreviation $x_{-j}$ refers to the vector $(x_1, \\ldots x_d)^\\top $ with the $j-$ th component removed and $X_{\\cdot -j}$ is the matrix $X$ with the $j$ -th column removed.",
"We sample from this density with HMC as will be outlined below.",
"Parameterization As in Section we need to provide parameterizations such that resulting parameters are unconstrained.",
"In particular we use the following transformations $\\begin{split}\\xi _j &= F_Z(\\phi _j), ~~~~~ \\psi _j =\\ln (\\sigma _j), \\\\\\end{split}$ where $F_Z(x) = \\frac{1}{2}\\ln (\\frac{1+x}{1-x})$ is Fisher's Z transformation.",
"Although the latent log variances are already unconstrained we make use of the following reparameterization $\\begin{split}\\tilde{s}_{0j} &= \\frac{(s_{0j}-\\mu _j)\\sqrt{1-\\phi _j^2}}{\\sigma _j} \\\\\\tilde{s}_{tj} &= \\frac{s_{tj}-\\mu _j-\\phi _j(s_{t-1j}-\\mu _j)}{\\sigma _j}, t=1, \\ldots , T. \\\\\\end{split}$ The transformation for $s_{\\cdot j}$ was proposed by the Stan [48] for the univariate stochastic volatility model and implies that $\\tilde{s}_{\\cdot j}|\\mu _j,\\phi _j,\\sigma _j \\sim N(0, I_{T+1})$ , where $I_{T+1}$ denotes the $(T+1)$ -dimensional identity matrix.",
"According to [51] the original parameterization in terms of $s_{tj}$ is a sufficient augmentation scheme, whereas the parameterization in terms of $\\tilde{s}_{tj}$ is an ancillary augmentation.",
"The performance of Markov Chain Monte Carlo methods can vary a lot for different parameterizations ([20], [46]).",
"[4] have seen better performance for the ancillary augmentation when sampling from the posterior distribution of hierarchical models with HMC.",
"Their explanation is that within the ancillary augmentation variables may be less correlated.",
"Here we also rely on the ancillary augmentation since we have seen much better performance for this parameterization in terms of effective sample size.",
"Prior densities We denote by $\\pi _{SV2}$ the joint prior density of the parameters $\\mu _j, \\xi _j, \\psi _j$ and $\\tilde{s}_{\\cdot j}$ .",
"The log of this joint prior density is, up to an additive constant, given by $\\ln (\\pi _{SV2}(\\mu _j, \\xi _j, \\psi _j,\\tilde{s}_{\\cdot j})) \\propto \\ln (\\pi (\\mu _j)) + \\ln (\\pi (\\xi _j)) + \\ln (\\pi (\\psi _j)) - \\frac{1}{2} \\sum _{t=0}^{T} \\tilde{s}_{tj}^2.$ where $\\pi (\\cdot )$ are the corresponding prior densities implied by (REF ) (see Appendix REF for details).",
"Posterior density The log posterior density we need to sample from is, up to an additive constant, given by $\\begin{split}\\mathcal {L} (\\mu _j, \\xi _j,& \\psi _j,\\tilde{s}_{\\cdot j}|Z,\\delta ,v, m) \\propto \\\\& \\sum _{t=1}^T \\left[ \\ln (c_j^{m_j}\\left(\\Phi \\left(\\frac{z_{tj}}{\\exp (\\frac{s_{tj}}{2})}\\right),v_t;\\theta _j^{m_j} \\right)) + \\ln (\\varphi \\left(\\frac{z_{tj}}{\\exp (\\frac{s_{tj}}{2})}\\right)) -\\frac{s_{tj}}{2} \\right] \\\\& +\\ln (\\pi _{SV2}(\\mu _j, \\xi _j, \\psi _j, \\tilde{s}_{\\cdot j})),\\end{split}$ where $s_{.j}$ is a function of $\\tilde{s}_{.j}$ (see (REF )).",
"The necessary derivatives of this log posterior are derived (see Appendix REF ) for the Leapfrog approximations and then sampling of the marginal parameters is straightforward.",
"The conditional density we need to sample from for the dependence parameters is proportional to $\\begin{split}f(\\delta ,v|Z,\\mu , \\phi , \\sigma , S, m) &\\propto \\ell (\\mu , \\phi , \\sigma , S, \\delta ,v, m|Z) \\pi _J(\\mu , \\phi , \\sigma , S, \\delta , v)\\\\& \\propto \\ell (\\mu , \\phi , \\sigma , S, \\delta ,v, m|Z) \\prod _{j=1}^d\\pi _{u}(\\delta _j) \\\\& \\propto \\prod _{t=1}^T \\prod _{j=1}^d c_j^{m_j}\\left(\\Phi \\left(\\frac{z_{tj}}{\\exp (\\frac{s_{tj}}{2})}\\right),v_t;\\theta _j^{m_j}\\right) \\prod _{j=1}^d\\pi _{u}(\\delta _j).\\end{split}$ To sample from this density we use the same HMC approach as in Section .",
"The full conditional of $m_j$ is obtained as $f(m_j|Z,\\mu ,\\phi ,\\sigma ,S,\\delta ,v, m_{-j}) = \\frac{ \\prod _{t=1}^T c_j^{m_j}\\left(\\Phi \\left(\\frac{z_{tj}}{\\exp (\\frac{s_{tj}}{2})}\\right),v_t;\\theta _j^{m_j}\\right)}{\\sum _{m_j^{\\prime } \\in \\mathcal {M}} \\prod _{t=1}^T c_j^{m_j^{\\prime }}\\left(\\Phi \\left(\\frac{z_{tj}}{\\exp (\\frac{s_{tj}}{2})}\\right),v_t;\\theta _j^{m_j^{\\prime }}\\right) }.$ We can sample directly from this discrete distribution and no MCMC updates are required here."
],
[
"Simulation study",
"We conduct a simulation study to evaluate the performance of the proposed joint HMC sampler.",
"We consider one scenario in five dimensions and one scenario in ten dimensions, as specified in Table REF .",
"We choose rather high values for the marginal persistence parameter $\\phi $ and moderate values for the dependence parameter Kendall's $\\tau $ .",
"These choices roughly correspond to what we expect to see in financial data.",
"For each scenario we simulate 100 data sets from the model introduced in Section REF .",
"The proposed MCMC sampler with HMC updates is then applied to the simulated data.",
"The sampler is run for 2500 iterations, whereas the first 500 iterations are discarded for burn in.",
"For family selection we consider the following set of one parametric copula families $\\lbrace $ Gaussian, Student t(df=4), Clayton, Gumbel$\\rbrace $ .",
"$\\begin{split}\\mu _{sim} &= (-6,-6,-7,-7,-8)\\\\\\phi _{sim} &= (0.7,0.8,0.85,0.9,0.95)\\\\\\sigma _{sim} &= (0.2,0.2,0.3,0.3,0.4)\\\\\\tau _{sim} &= (0.3,0.4,0.5,0.6,0.7)\\\\m_{sim} &= (\\text{Gaussian, Student t(df=4), Clayton, Gumbel, Gaussian})\\\\\\end{split}$ Table: Parameter specification for the two different scenarios in the simulation study.Table: MSE estimated using the posterior mode, observed coverage probability of the credible intervals (C.I.)",
"and effective samples size calculated from 2000 posterior draws for selected parameters (Scenario 1).The simulation results are summarized in Tables REF and REF for the five dimensional scenario and in Tables REF and REF in the appendix for the ten dimensional setup.",
"Comparing these two setups we see that the ESS of the Kendall's $\\tau $ parameters and of the latent states $v_t$ is better for the five dimensional scenario.",
"Besides that, the results of the five and ten dimensional setups are only slightly different and therefore we discuss only the five dimensional scenario.",
"Table: Proportion of how often the correct copula family was selected.",
"The selected copula family is the posterior mode estimate of m j m_j for j=1,...,5j=1, \\ldots , 5 (Scenario 1).Figure: Kernel density estimates of the posterior density of selected parameters obtained from a single MCMC run in Scenario 1.",
"The estimates are based on 2000 MCMC iterations after a burn in of 500.",
"The true parameter value is added in red.Comparing the simulation results for the factor copula parameters to the results of Section REF , we see that we perform worse in terms of observed coverage probabilities and MSE.",
"But this is not surprising because here we consider a much more complex model and also update the copula families and the marginal parameters.",
"Further we see that the ESS decreases from $\\tau _1$ up to $\\tau _5$ .",
"This is in line with our findings in Section REF where we have seen that mixing is worse for higher Kendall's $\\tau $ values.",
"We can also observe differences with respect to the observed coverage probability of credible intervals.",
"For a low marginal persistence parameter ($\\phi _1$ ) coverage probabilities are very high suggesting a broad posterior distribution.",
"For a high persistence parameter ($\\phi _5$ ) the observed coverage probabilities are lower.",
"Figures REF and REF show estimated posterior densities and trace plots of one MCMC run for the five dimensional setup.",
"These figures suggest that we achieve proper mixing.",
"Furthermore we see that the estimated posterior density of $\\phi _1$ is more dispersed compared to the estimated posterior density of $\\phi _5$ .",
"Table REF shows that the correct copula family was selected in at least 66 out of 100 cases.",
"This frequency is best for the first linking copula which has a low Kendall's $\\tau $ value and worst for the linking copula with the highest Kendall's $\\tau $ value.",
"Overall, the results suggest that the method performs well.",
"For all parameters we obtain reasonable MSE and ESS values and our method is able to select the correct copula family in most cases.",
"In particular our HMC schemes do a good job at jointly updating more than $T=1000$ parameters of one Gibbs block.",
"Figure: Trace plots of selected parameters obtained from a single MCMC run in Scenario 1.",
"The trace plots show 2000 MCMC iterations after a burn in of 500.",
"The true parameter value is added in red."
],
[
"Application",
"We illustrate our approach with one-day ahead value at risk (VaR) prediction for a portfolio consisting of several stocks.",
"These predictions can be obtained from simulations of the predictive distribution.",
"As before, $Z$ is the data matrix containing $T$ observations of the $d$ stocks.",
"We need to sample from the predictive distribution of the log returns at time $T+1$ , $Z_{T+1} = (Z_{T+1,1}, \\ldots Z_{T+1,d}) $ , given $Z$ .",
"We obtain simulations from the joint density $f(z_{T+1} ,s_{T+1\\cdot },S,\\mu ,\\phi ,\\sigma ,\\delta ,v_{T+1},v, m|Z),$ with the following steps: Simulate $S,\\mu ,\\phi ,\\sigma ,\\delta ,v, m$ from the corresponding posterior distribution given the data $Z$ with our sampler developed in Section .",
"We discard the first 500 samples for burnin and denote the remaining $R=2000$ samples by $S^r,\\mu ^r,\\phi ^r,\\sigma ^r,\\delta ^r,v^r, m^r$ , $r=1, \\ldots , R$ .",
"We proceed as follows for $r=1, \\ldots , R$ : Simulate $v^r_{T+1} \\sim unif(0,1)$ .",
"For $j = 1, \\ldots , d$ simulate $s^r_{T+1,j} \\sim N(\\mu _j^r + \\phi _j^r(s_{Tj}^r - \\mu _j^r), (\\sigma ^r_j)^2).$ To obtain the sample $z^r_{T+1}$ from $f(z_{T+1}| &s^r_{T+1\\cdot },S^r,\\mu ^r,\\phi ^r,\\sigma ^r,\\delta ^r,v^r_{T+1},v^r,m^r, Z) = \\\\ &\\prod _{j=1}^d \\Bigg [ c^{m^r_j}_j\\left(\\frac{z_{T+1j}}{\\exp (\\frac{s^r_{T+1j}}{2})},v^r_{T+1};\\theta _j^{m^r_j} \\right) \\varphi \\left(\\frac{z_{T+1j}}{\\exp (\\frac{s^r_{T+1j}}{2})}\\right)\\cdot \\frac{1}{\\exp (\\frac{s^r_{T+1j}}{2})} \\Bigg ]$ we simulate $u_j^{r}$ from $C_j^{m^r_j}\\left(\\cdot |v^r_{T+1};\\theta ^{m^r_j}_j\\right)$ and set $z^r_{T+1j} = \\Phi ^{-1}\\left(u_j^{r}|0,\\exp ({s^r_{T+1j}})\\right)$ for $j = 1, \\ldots , d$ .",
"Here $\\Phi (\\cdot |\\mu _{normal}, \\sigma _{normal}^2)$ is the distribution function of a normally distributed random variable with mean $\\mu _{normal}$ and variance $\\sigma _{normal}^2$ .",
"Figure: Observed daily log return of the portfolio and the estimated one-day ahead 90%90\\% VaR (red) and 95%95\\% VaR (blue) plotted against time in years.We consider an equally weighted portfolio consisting of 6 stocks from German companies (BASF, Fresenius Medical Care, Fresenius SE, Linde, Merck, K+S).",
"Since all companies are chosen from the chemical/pharmaceutical/medical industry we assume that a model with one factor is suitable to capture the dependence structure.",
"Our data, obtained from Yahoo Finance (https://finance.yahoo.com), contains daily log returns of these stocks from 2008 to 2017.",
"We use 1000 days as training period, which corresponds to data of approximately four years.",
"We set $T=1000$ and obtain simulations of the one-day ahead predictive distribution as described above for the first trading day in 2012.",
"Instead of refitting the model for each day we fix parameters that do not change over time ($\\mu , \\phi , \\sigma , \\delta , m$ ) at their posterior mode estimates and only update dynamic parameters ($S, v$ ) as in [33] for the remaining one-day ahead predicitve simulations.",
"For updating only the dynamic parameters we have seen that it is enough to use the last 100 time points and time needed for computation is reduced a lot.",
"We obtain 2000 simulations of the one-day ahead predictive distribution for each trading day in the period from January 2012 to December 2017.",
"From the simulations we calculate the portfolio value, and take the corresponding quantile to obtain the VaR prediction.",
"We consider the same VaR level of $90\\%$ as in [43] and additionally the $95\\%$ VaR.",
"The linking copulas are chosen from the following set of one parametric copula families: Gaussian, Student t with 4 degrees of freedom, (survival) Gumbel and (survival) Clayton.",
"For a copula with density $c(u_1,u_2)$ , the corresponding survival copula has density $c(1-u_1,1-u_2)$ .",
"More details about the considered bivariate copula families are given in [13], Chapter 3.",
"With these choices we cover a range of different tail dependence structures: no tail dependence (Gauss), symmetric tail dependence (Student t) and asymmetric tail dependence ((survival) Gumbel and (survival) Clayton).",
"As explained above, the copula family indicator was only updated for the first model we fitted and then kept fixed.",
"The linking copula families of BASF, Fresenius Medical Care, Fresenius SE, Linde, Merck and K+S with the highest posterior probabilities are Student t, survival Gumbel, survival Gumbel, Gaussian, survival Gumbel and Student t, respectively.",
"In particular, we obtain a model with asymmetric tail dependence structure.",
"Predicting the VaR for each trading day in six years results in 1521 VaR predictions.",
"The portfolio log returns and corresponding $90\\%$ and $95\\%$ VaR predictions are visualized in Figure REF .",
"We observe that the one-day ahead VaR forecast adapts to changes in the volatility.",
"To benchmark the proposed model (factor copula SV (fc SV)) we repeated the procedure for VaR prediction with two other models: marginal dynamic linear models combined with single factor copulas (fc dlm) estimated with a two-step procedure as proposed by [43] and a multivariate factor stochastic volatility model with dynamic factors (df Gauss SV) as proposed by [31].",
"The df Gauss SV model is here restricted to one factor.",
"To illustrate the necessity of copula family selection we further consider fc SV models with the restriction that all linking copula are chosen from the same family.",
"We consider the three copula families that were selected as linking copulas for the fc SV model and obtain the three restricted models fc SV (Ga), fc SV (t) and fc SV (sGu) which have only Gaussian, Student t(df=4) and survival Gumbel linking copulas, respectively.",
"Additionally we compare the proposed approach to a two-step estimation of the factor copula SV model (fc SV (ts)).",
"In this two-step approach we obtain simulations from the predictive distribution of the log returns at time $T+1$ , $Z_{T+1} $ , given $Z$ as follows: Estimate a SV model for each margin separately and obtain marginal posterior mode estimates for the latent log variances denoted by $\\hat{s}_{tj}$ for $t=1, \\ldots , T, j=1, \\ldots , d$ .",
"Use the probability integral transform to obtain data on the (0,1) scale, $u_{tj} \\Phi \\left(z_{tj} \\cdot \\exp (-\\frac{\\hat{s}_{tj}}{2})\\right)$ .",
"For the data $u_{tj}, t=1, \\ldots , T, j=1, \\ldots , d$ , we fit the single factor copula model with HMC as explained in Section , where we allow for Bayesian copula family selection and obtain posterior mode estimates of the corresponding parameters denoted by $\\hat{\\delta }_1, \\ldots \\hat{\\delta }_d, \\hat{m}_1, \\ldots \\hat{m}_d$ .",
"For each margin, we simulate from the predictive distribution of the log variances at time $T+1$ , i.e.",
"from $s_{T+1,j}|z_{1j}, \\ldots , z_{Tj}$ , and obtain marginal posterior mode estimates $\\hat{s}_{T+1,j}$ for $j=1, \\ldots , d$ .",
"For $r=1, \\ldots , R$ we proceed as follows: We simulate $u_{1}^r, \\ldots u_{d}^{r}$ from the single factor copula with parameters $\\hat{\\delta }_1, \\ldots \\hat{\\delta }_d, \\hat{m}_1, \\ldots \\hat{m}_d$ .",
"We set $z_{T+1,j}^r = \\Phi ^{-1}\\left(u_{j}^{r}|0,\\exp ({\\hat{s}_{T+1,j}})\\right)$ for $j=1, \\ldots , d$ .",
"Standard measures to compare the predictive accuracy between different models are the continuous ranked probability score ([23]) or log predictive scores as used in [29].",
"These scores evaluate the overall performance.",
"But we are interested in the VaR, a quantile of the predictive distribution, which is only one specific aspect.",
"Therefore we use the rate of VaR violations and Christoffersen's conditional coverage test ([10], Chapter 8), which are commonly used to compare VaR forecasts, as in [43] and [37].",
"From an optimal VaR measure at level $p$ we would expect that there are $(1-p) \\cdot 100 \\%$ VaR violations and that violations occur independently.",
"This constitutes the null hypotheses of Christoffersen's conditional coverage test.",
"The VaR violation rates for the different models are shown in Table REF .",
"For the $90\\%$ VaR, the violation rate of the df Gauss SV model is closest to the optimal rate of $10\\%$ , whereas for the $95\\%$ VaR the fc SV model performs best.",
"According to the p-values of Christoffersen's conditional coverage test in Table REF none of the considered models can be rejected at the $5\\%$ or $10\\%$ level with respect to $90\\%$ VaR prediction.",
"But with respect to $95\\%$ VaR prediction, every model except the fc SV model is rejected at the $5\\%$ or $10\\%$ level.",
"We conclude that the preferred model in this scenario is the fc SV model.",
"Table: The rate of 90%90\\% and 95%95\\% VaR violations for the seven models: fc SV, fc SV(Gauss), fc SV(t), fc SV(sGu), fc SV(ts), fc dlm, df Gauss SV.",
"The violation rate closest to the opimal value of 5%5\\% or 10%10 \\% is marked in bold.Table: The p-value of Christoffersen's conditional coverage test for the 90%90\\% and 95%95\\% VaR predictions of seven models: fc SV, fc SV(Gauss), fc SV(t), fc SV(sGu), fc SV(ts), fc dlm, df Gauss SV.",
"The highest p-value per row is marked in bold."
],
[
"Conclusion",
"We propose a single factor copula SV model, a combination of the SV model for the margins and factor copulas for the dependence.",
"Dependence and marginal parameters are estimated jointly within a Bayesian approach, avoiding a two-step estimation procedure which is commonly used for copula models.",
"The proposed model can be seen as one way to extend factor SV models that rely on Gaussian dependence to more complex dependence structures.",
"The necessity of such models was illustrated with one-day ahead value at risk prediction.",
"In the application our stocks were chosen such that one factor is suitable to describe dependencies.",
"However this might not be appropriate for different portfolios and the extension of the proposed model to multiple factors will be subject to future research.",
"This extension to multiple factors could exploit the partition of different stocks into sectors as in the structured factor copula model proposed by [35].",
"Another extension could allow for time varying dependence parameters or for copula families with two and more parameters."
],
[
"Acknowledgment",
"The first author acknowledges financial support by a research stipend of the Technical University of Munich.",
"The second author is supported by the German Research Foundation (DFG grant CZ 86/4-1).",
"Computations were performed on a Linux cluster supported by DFG grant INST 95/919-1 FUGG."
],
[
"Hamiltonian Monte Carlo",
"We provide a short introduction to HMC based on [38].",
"We start with the introduction of the Hamiltonian dynamics."
],
[
"Hamiltonian dynamics",
"We consider a position vector $q \\in \\mathbb {R}^d$ with associated momentum vector $p \\in \\mathbb {R}^d$ at time $t$ .",
"Their change over time is described through the function $H(p,q)$ , the Hamiltonian, which satisfies the following differential equations: $\\begin{split}\\frac{dq_i}{dt}&=\\frac{dH}{dp_i} \\\\\\frac{dp_i}{dt}&=-\\frac{dH}{dq_i}, i=1, \\ldots , d.\\end{split}$ Here $H$ represents the total energy of the system.",
"In HMC, it is assumed that $H$ can be expressed as $H(p,q) = U(q) + K(p) = U(q) + p^\\top M^{-1}p/2,$ where $U(q)$ is called the potential energy and $K(p)$ the kinetic energy.",
"Further $M$ is a symmetric positive definite mass matrix, which is usually assumed to be diagonal.",
"The Hamiltonian dynamics, specified in (REF ), can therefore be rewritten as $\\begin{split}\\frac{dq_i}{dt}&=(M^{-1}p)_i \\\\\\frac{dp_i}{dt}&=-\\frac{dU}{dq_i}, i=1, \\ldots , d.\\end{split}$ Since it is usually not possible to solve the system of differential equations given in (REF ) analytically we need to find iterative approximations.",
"Therefore we use the Leapfrog method, where the state one-step ahead of time $t$ with step size $\\epsilon $ , i.e.",
"the state at time $t+\\epsilon $ , is approximated by $\\begin{split}p_i(t+\\epsilon /2) &= p_i(t) - \\frac{\\epsilon }{2} \\frac{dU}{dq_i}(q(t)) \\\\q_i(t+\\epsilon ) &= q_i(t) + \\epsilon \\frac{p_i(t+\\epsilon /2)}{m_i} \\\\p_i(t+\\epsilon ) &= p_i(t+\\epsilon /2) - \\frac{\\epsilon }{2} \\frac{dU}{dq_i}(q(t+\\epsilon )), \\text{ for } i = 1, \\ldots d. \\\\\\end{split}$ To use Hamiltonian dynamics within MCMC sampling we need to relate the energy function to a probability distribution.",
"Therefore we utilize the canonical distribution $P(x)$ associated with a general energy function $E(x)$ with state $x$ defined through the density $p(x) \\frac{1}{Z} \\exp (-E(x)/T).$ Here $T$ is the temperature of the system and $Z$ the normalizing constant needed to satisfy the density constraint.",
"So the Hamiltonian $H(p,q)$ specified in (REF ) defines a probability density given by $p(q,p) = \\frac{1}{Z} \\exp (-H(p,q)/T) = \\frac{1}{Z} \\exp (-U(q)/T)\\exp (-K(p)/T),$ where $q$ and $p$ are independent.",
"In the following we assume $T=1$ .",
"In HMC we specify the corresponding energy function of $q$ and $p$ , i.e.",
"the Hamiltonian, and sample from the corresponding canonical distribution of $q$ and $p$ .",
"In a Bayesian setup we identify $q$ as our parameters of interest and $p$ are auxiliary variables.",
"Therefore we set $U(q) -\\ln (\\pi (q)\\ell (q|D)),$ where $\\pi (q)$ is the prior density and $\\ell (q|D)$ the likelihood function for the given data $D$ .",
"Therefore the canonical distribution of $q$ corresponds to the posterior distribution of $q$ , when $T=1$ .",
"Since $K(p) = p^\\top M^{-1}p/2$ , it holds that the auxiliary parameter vector $p$ is multivariate normal distributed with zero mean vector and covariance matrix $M$ .",
"Sampling is then done in the following way.",
"Sample $p$ from the normal distribution with zero mean vector and covariance matrix $M$ .",
"Metropolis update: Start with the current state $(q,p)$ and simulate $L$ steps of Hamiltonian dynamics with step size $\\epsilon $ using the Leapfrog method.",
"Obtain $(q^{\\prime },p^{\\prime })$ and accept this proposal with Metropolis acceptance probability $\\min (1,\\exp (-H(q^{\\prime },p^{\\prime })+H(q,p))) = \\min \\left(1,\\frac{\\pi (q^{\\prime })l(q^{\\prime }|D) \\exp (p^\\top M^{-1}p/2)}{\\pi (q)l(q|D) \\exp (p^{\\prime \\top }M^{-1}p^{\\prime }/2)}\\right).$"
],
[
"Derivatives for HMC for the single factor copula model",
"The derivatives of the log posterior density with respect to the parameters $\\delta _{j^{\\prime }}$ and $w_{t^{\\prime }}$ are given by $\\begin{split}{}{\\delta _{j^{\\prime }}}\\mathcal {L}(\\delta _{1:d}, w_{1:T}|U_{1:T,1:d}) =& \\sum _{j=1}^d \\sum _{t=1}^T {}{\\delta _{j^{\\prime }}} \\ln (c_j(u_{tj},v_t;\\theta _{j}))+{}{\\delta _{j^{\\prime }}}\\ln (\\pi _{FC}(\\delta _{1:d},w_{1:T})) \\\\=& \\sum _{t=1}^T {}{\\delta _{j^{\\prime }}} \\ln (c_j(u_{tj^{\\prime }},v_t;\\theta _{j^{\\prime }})) +{}{\\delta _{j^{\\prime }}}\\ln (\\pi _{FC}(\\delta _{1:d},w_{1:T})) \\\\=& \\sum _{t=1}^T {}{\\theta _{j^{\\prime }}} \\ln (c_j(u_{tj^{\\prime }},v_t;\\theta _{j^{\\prime }})) {\\theta _{j^{\\prime }}}{\\delta _{j^{\\prime }}} +{}{\\delta _{j^{\\prime }}}\\ln (\\pi _{FC}(\\delta _{1:d},w_{1:T})),\\\\\\end{split}$ and $\\begin{split}{}{w_{t^{\\prime }}}\\mathcal {L}(\\delta _{1:d},w_{1:T}|U_{1:T,1:d}) =& \\sum _{j=1}^d \\sum _{t=1}^T {}{w_{t^{\\prime }}} \\ln (c_j(u_{tj},v_t;\\theta _{j})) +{}{w_{t^{\\prime }}}\\ln (\\pi _{FC}(\\delta _{1:d},w_{1:T}))\\\\=& \\sum _{j=1}^d {}{w_{t^{\\prime }}} \\ln (c_j(u_{t^{\\prime }j},v_{t^{\\prime }};\\theta _{j})) +{}{w_{t^{\\prime }}}\\ln (\\pi _{FC}(\\delta _{1:d},w_{1:T}))\\\\=& \\sum _{j=1}^d {}{v_{t^{\\prime }}} \\ln (c_j(u_{t^{\\prime }j},v_{t^{\\prime }};\\theta _{j})) {v_{t^{\\prime }}}{w_{t^{\\prime }}} +{}{w_{t^{\\prime }}}\\ln (\\pi _{FC}(\\delta _{1:d},w_{1:T})) \\\\,\\end{split}$ The components of the derivative of the log posterior density are derived in the following."
],
[
"Derivatives of the log prior density",
"The derivative of the log prior density $\\pi _u$ is given by ${}{x}\\ln (\\pi _u(x)) = {}{x} -2 \\ln (1+\\exp (-x)) - x = 2 (1+\\exp (x))^{-1} - 1.$ We consider derivatives of the parameter transformation, i.e.",
"${\\theta _{j^{\\prime }}}{\\delta _{j^{\\prime }}}$ and ${v_{t^{\\prime }}}{w_{t^{\\prime }}}$ .",
"In this part we suppress the indices $j^{\\prime }$ and $t^{\\prime }$ .",
"We have that $v = (1 + \\exp (-w))^{-1},$ and the derivative is given by ${v}{w} = (1 + \\exp (-w))^{-2} \\exp (-w).$ Now we address the derivative ${\\theta }{\\delta }$ .",
"The parameter $\\delta $ was chosen to be the logit transform of the corresponding Kendall's $\\tau $ and so $\\tau $ can be written as $\\tau = (1 + \\exp (-\\delta ))^{-1},$ with corresponding derivative ${\\tau }{\\delta } = (1 + \\exp (-\\delta ))^{-2} \\exp (-\\delta ).$ The copula parameter $\\theta $ is a function of Kendall's $\\tau $ ($\\theta = g^{-1} (\\tau )$ ) and dependent on the copula family considered we obtain the following derivatives.",
"Gauss and Student t copula $\\begin{split}\\theta &=\\sin (\\frac{1}{2}\\pi \\tau )\\\\{\\theta }{\\delta } &= {\\theta }{\\tau } {\\tau }{\\delta }\\\\& =\\frac{1}{2}\\pi \\cos (\\frac{1}{2}\\pi \\tau ){\\tau }{\\delta }\\\\& = \\frac{1}{2}\\pi \\cos (\\frac{1}{2}\\pi (1 + \\exp (-\\delta ))^{-1}) (1 + \\exp (-\\delta ))^{-2} \\exp (-\\delta )\\end{split}$ Clayton copula $\\begin{split}\\theta &=\\frac{2\\tau }{1-\\tau }\\\\&=\\frac{2}{\\tau ^{-1}-1}\\\\&=\\frac{2}{1+\\exp (-\\delta )-1}\\\\&=2 \\exp (\\delta ) \\\\{\\theta }{\\delta }&= 2 \\exp (\\delta )\\end{split}$ Gumbel copula $\\begin{split}\\theta &=(1-\\tau )^{-1}\\\\{\\theta }{\\delta } &= {\\theta }{\\tau } {\\tau }{\\delta }\\\\& = (1-\\tau )^{-2} {\\tau }{\\delta }\\\\& = \\lbrace 1-[1+\\exp (-\\delta )]^{-1}\\rbrace ^{-2} [1+\\exp (-\\delta )]^{-2} \\exp (-\\delta )\\\\&=[1+\\exp (-\\delta )-1]^{-2} \\exp (-\\delta )\\\\&=\\exp (-\\delta )^{-2} \\exp (-\\delta )\\\\&=\\exp (\\delta )\\end{split}$ For all considered copula families [44] calculate the derivatives of the copula density with respect to the copula parameter $\\theta _j$ and with respect to the argument $v_t$ .",
"Based on their results the derivatives of the log copula density are easily derived.",
"The derivatives are also implemented in the $\\texttt {R}$ package $\\texttt {VineCopula}$ by [45]."
],
[
"Prior densities for transformed parameters",
"The prior densities for $\\mu _j,\\phi _j$ and $\\sigma ^2_j$ in (REF ) imply the following prior densities for $\\mu _j,\\xi _j$ and $\\psi _j$ .",
"We have that $\\mu _j \\sim N(0,\\sigma _{\\mu }^2) $ .",
"So the prior density for $\\mu _j$ is up to a constant given by $\\pi _{\\mu }(x) \\propto \\exp (-\\frac{x^2}{2\\sigma _{\\mu }^2}).$ We have that $\\frac{\\phi _j+1}{2} \\sim Beta(a,b)$ .",
"So the density of $\\phi _j$ is given by $f_{\\phi }(x)=f_{Beta}\\left(\\frac{x+1}{2}\\right) \\frac{1}{2}.$ This implies that the prior density of $\\xi _j$ is $\\begin{split}\\pi _{\\xi }(x)&=f_{\\phi }(F_Z^{-1}(x)) {{}{x}F_Z^{-1}(x)}\\\\&=\\frac{\\Gamma (a+b)}{\\Gamma (a)\\Gamma (b)} \\left(\\frac{F_Z^{-1}(x) + 1}{2}\\right)^{a-1} \\left(1-\\frac{F_Z^{-1}(x) + 1}{2}\\right)^{b-1}\\frac{1}{2}\\left(1-(F_Z^{-1}(x))^2 \\right).\\end{split}$ We have that $\\sigma _j^2 \\sim \\chi ^2_1$ , i.e.",
"$f_{\\sigma }(x) = 2 x f_{\\chi ^2_1} (x^2) .$ So the prior density for $\\psi _j$ is given by $\\begin{split}\\pi _{\\psi }(x) &= f_{\\sigma } (\\exp (x)) \\exp (x) \\\\&= 2 \\exp (x) \\exp (x) \\frac{1}{\\sqrt{2}\\Gamma (\\frac{1}{2})} \\exp (-x) \\exp (-\\frac{\\exp (2x)}{2}) \\\\&=2 \\frac{1}{\\sqrt{2}\\Gamma (\\frac{1}{2})} \\exp (x) \\exp (-\\frac{\\exp (2x)}{2}) .\\end{split}$"
],
[
"Derivatives for HMC for the stochastic volatility model",
"We need to calculate derivatives of the function $\\begin{split}\\mathcal {L}(\\mu _j, \\xi _j, \\psi _j,\\tilde{s}_{\\cdot j}|Z,\\delta ,v, m) \\propto & \\sum _{t=1}^T \\left[ \\ln (c_j^{m_j}\\left(\\Phi \\left(\\frac{Z_{tj}}{\\exp (\\frac{s_{tj}}{2})}\\right),v_t;\\theta _j^{m_j} \\right)) + \\ln (\\varphi \\left(\\frac{Z_{tj}}{\\exp (\\frac{s_{tj}}{2})}\\right)) -\\frac{s_{tj}}{2} \\right]\\\\& +\\ln (\\pi _{SV2}(\\mu _j, \\xi _j, \\psi _j,\\tilde{s}_{\\cdot j})),\\end{split}$ where $\\propto $ refers to proportionality up to an additive constant.",
"To shorten notation we omit the index $j$ in the following and consider the function $\\begin{split}\\mathcal {L}_2(\\mu , \\xi , \\psi ,\\tilde{s}_{0:T}|Z,\\delta , v, m) = & \\sum _{t=1}^T \\left[ \\ln (c\\left(\\Phi \\left(\\frac{Z_{t}}{\\exp (\\frac{s_{t}}{2})}\\right),v_t;\\theta \\right)) + \\ln (\\varphi \\left(\\frac{Z_{t}}{\\exp (\\frac{s_{t}}{2})}\\right)) -\\frac{s_{t}}{2} \\right]\\\\& +\\ln (\\pi _{SV2}(\\mu , \\xi , \\psi ,\\tilde{s}_{0:T})).\\end{split}$ We define $\\Omega (s_{1:T}) = \\sum _{t=1}^T \\left( \\ln (c\\left(\\Phi \\left(\\frac{Z_{t}}{\\exp (\\frac{s_{t}}{2})}\\right),v_t;\\theta \\right)) + \\ln (\\varphi \\left(\\frac{Z_{t}}{\\exp (\\frac{s_{t}}{2})}\\right)) -\\frac{s_{t}}{2} \\right),$ and the derivatives can be expressed as ${}{\\mu } \\mathcal {L}_2(\\mu , \\xi , \\psi ,\\tilde{s}_{0:T}) = {\\Omega (s_{1:T})}{s_{1:T}}{s_{1:T}}{\\mu }+{}{\\mu }\\ln (\\pi _{SV2}(\\mu , \\xi , \\psi ,\\tilde{s}_{0:T}))$ ${}{\\xi } \\mathcal {L}_2(\\mu , \\xi , \\psi ,\\tilde{s}_{0:T}) = {\\Omega (s_{1:T})}{s_{1:T}}{s_{1:T}}{\\phi }{\\phi }{\\xi } +{}{\\xi }\\ln (\\pi _{SV2}(\\mu , \\xi , \\psi ,\\tilde{s}_{0:T}))$ ${}{\\psi } \\mathcal {L}_2(\\mu , \\xi , \\psi ,\\tilde{s}_{0:T}) = {\\Omega (s_{1:T})}{s_{1:T}}{s_{1:T}}{\\sigma }{\\sigma }{\\psi }+{}{\\psi }\\ln (\\pi _{SV2}(\\mu , \\xi , \\psi ,\\tilde{s}_{0:T}))$ ${}{\\tilde{s}_{0:T}} \\mathcal {L}_2(\\mu , \\xi , \\psi ,\\tilde{s}_{0:T}) = {\\Omega (s_{1:T})}{s_{0:T}} J+{}{\\tilde{s}_{0:T}}\\ln (\\pi _{SV2}(\\mu , \\xi , \\psi ,\\tilde{s}_{0:T})),$ where $J \\in \\mathbb {R}^{(T+1) \\times (T+1)}$ denotes the corresponding Jacobian matrix, i.e.",
"$J_{tj} = {s_t}{\\tilde{s}_j}.$ The derivatives are calculated in the following.",
"${}{s_i} \\Omega (s_{1:T}) = {}{x} \\ln (c(x,v_i;\\theta ))\\Big |_{x=\\Phi \\left(\\frac{Z_{i}}{\\exp (\\frac{s_{i}}{2})}\\right)} \\varphi \\left(\\frac{Z_{i}}{\\exp (\\frac{s_{i}}{2})}\\right) \\frac{Z_{i}}{\\exp (\\frac{s_{i}}{2})} (-\\frac{1}{2}) + \\frac{Z_i^2}{2\\exp (s_i)} - \\frac{1}{2} \\text{ for }i=1, \\ldots , T$ We have that $s_0 = \\frac{\\tilde{s}_0 \\sigma }{\\sqrt{1-\\phi ^2}} + \\mu , ~~~ s_t = \\tilde{s}_t \\sigma + \\mu + \\phi (s_{t-1} - \\mu ), t=1, \\ldots , T\\\\$ and obtain Table: NO_CAPTION ${\\phi }{\\xi } = 1 - F^{-1}(\\xi )^2, ~~~{\\sigma }{\\psi } = \\exp (\\psi )$ ${}{\\mu } \\ln (\\pi _{SV2}(\\mu , \\xi , \\psi ,\\tilde{s}_{0:T})) = -\\frac{\\mu ^2}{\\sigma _{\\mu }^2}$ ${}{\\xi }\\ln (\\pi _{SV2}(\\mu , \\xi , \\psi ,\\tilde{s}_{0:T}))=(a-1)\\frac{(1-F_Z^{-1}(\\xi )^2)}{(F_Z^{-1}(\\xi )+1)} - (b-1)(1+F_Z^{-1}(\\xi )) - 2 F_Z^{-1}(\\xi )$ where $a=5$ and $b=1.5$ are the parameters of the beta distribution.",
"${}{\\psi }\\ln (\\pi _{SV2}(\\mu , \\xi , \\psi ,\\tilde{s}_{0:T})) = 1 - \\exp (2\\psi )$ ${}{\\tilde{s}_{0:T}}\\ln (\\pi _{SV2}(\\mu , \\xi , \\psi ,\\tilde{s}_{0:T})) = -\\tilde{s}_{0:T}$"
]
] | 1808.08624 |
[
[
"Event Detection with Neural Networks: A Rigorous Empirical Evaluation"
],
[
"Abstract Detecting events and classifying them into predefined types is an important step in knowledge extraction from natural language texts.",
"While the neural network models have generally led the state-of-the-art, the differences in performance between different architectures have not been rigorously studied.",
"In this paper we present a novel GRU-based model that combines syntactic information along with temporal structure through an attention mechanism.",
"We show that it is competitive with other neural network architectures through empirical evaluations under different random initializations and training-validation-test splits of ACE2005 dataset."
],
[
"Introduction",
"[1]Also associated with Oregon State University.",
"Events are the lingua franca of news stories and narratives and describe important changes of state in the world.",
"Identifying events and classifying them into different types is a challenging aspect of understanding text.",
"This paper focuses on the task of event detection, which includes identifying the “trigger\" words that indicate events and classifying the events into refined types.",
"Event detection is the necessary first step in inferring more semantic information about the events including extracting the arguments of events and recognizing temporal and causal relationships between different events.",
"Neural network models have been the most successful methods for event detection.",
"However, most current models ignore the syntactic relationships in the text.",
"One of the main contributions of our work is a new DAG-GRU architecture [3] that captures the context and syntactic information through a bidirectional reading of the text with dependency parse relationships.",
"This generalizes the GRU model to operate on a graph by novel use of an attention mechanism.",
"Following the long history of prior work on event detection, ACE2005 is used for the precise definition of the task and the data for the purposes of evaluation.",
"One of the challenges of the task is the size and sparsity of this dataset.",
"It consists of 599 documents, which are broken into a training, development, testing split of 529, 30, and 40 respectively.",
"This split has become a de-facto evaluation standard since [9].",
"Furthermore, the test set is small and consists only of newswire documents, when there are multiple domains within ACE2005.",
"These two factors lead to a significant difference between the training and testing event type distribution.",
"Though some work had been done comparing method across domains [15], variations in the training/test split including all the domains has not been studied.",
"We evaluate the sensitivity of model accuracy to changes in training and test set splits through a randomized study.",
"Given the limited amount of training data in comparison to other datasets used by neural network models, and the narrow margin between many high performance methods, the effect of the initialization of these methods needs to be considered.",
"In this paper, we conduct an empirical study of the sensitivity of the system performance to the model initialization.",
"Results show that our DAG-GRU method is competitive with other state-of-the-art methods.",
"However, the performance of all methods is more sensitive to the random model initialization than expected.",
"Importantly, the ranking of different methods based on the performance on the standard training-validation-test split is sometimes different from the ranking based on the average over multiple splits, suggesting that the community should move away from single split evaluations."
],
[
"Related Work",
"Event detection and extraction are well-studied tasks with a long history of research.",
"[15] used CNNs to represent windows around candidate triggers.",
"Each word is represented by a concatenation of its word and entity type embeddings with the distance to candidate trigger.",
"Global max-pooling summarizes the CNN filter and the result is passed to a linear classifier.",
"[16] followed up with a skip-gram based CNN model which allows the filter to skip non-salient or otherwise unnecessary words in the middle of word sequences.",
"[5] combined a CNN, similar to [15], with a bi-directional LSTM [7] to create a hybrid network.",
"The output of both networks was concatenated together and fed to a linear model for final predictions.",
"[14] uses a bidirectional gated recurrent units (GRUs) for sentence level encoding, and in conjunction with a memory network, to jointly predict events and their arguments.",
"[11] created a probablistic soft logic model incorporating the semantic frames from Framenet [1] in the form of extra training examples.",
"Based on the intuition that entity and argument information is important for event detection, [12] built an attention model over annotated arguments and the context surrounding event trigger candidates.",
"[10] created a cross language attention model for event detection and used it for event detection in both the English and Chinese portions of the ACE2005 data.",
"Recently, [17] used graph-CNN (GCCN) where the convolutional filters are applied to syntactically dependent words in addition to consecutive words.",
"The addition of the entity information into the network structure produced the state-of-the-art CNN model.",
"Another neural network model that includes syntactic dependency relationships is DAG-based LSTM [19].",
"It combines the syntactic hidden vectors by weighted average and adds them through a dependency gate to the output gate of the LSTM model.",
"To the best of our knowledge, none of the neural models combine syntactic information with attention, which motivates our research."
],
[
"DAG GRU Model",
"Event detection is often framed as a multi-class classification problem [2], [6].",
"The task is to predict the event label for each word in the test documents, and NIL if the word is not an event.",
"A sentence is a sequence of words $x_1 \\dots x_n$ , where each word is represented by a $k$ -length vector.",
"The standard GRU model works as follows: $r_t = \\sigma (W_r x_t + U_r h_{t -1} + b_r)$ $z_t = \\sigma (W_z x_t + U_z h_{t -1} + b_z)$ $\\tilde{h}_t = \\text{tanh}(W_h x_t + r_t \\odot U_h h_{t -1} + b_h)$ $h_t = (1 - z_t) \\odot h_{t-1} + z_t \\odot \\tilde{h}_t$ The GRU model produces a hidden vector $h_t$ for each word $x_t$ by combining its representation with the previous hidden vector.",
"Thus $h_t$ summarizes both the word and its prior context.",
"Our proposed DAG-GRU model incorporates syntactic information through dependency parse relationships and is similar in spirit to [17] and [19].",
"However, unlike those methods, DAG-GRU uses attention to combine syntactic and temporal information.",
"Rather than using an additional gate as in [19], DAG-GRU creates a single combined representation over previous hidden vectors and then applies the standard GRU model.",
"Each relationship is represented as an edge, $(t,t^\\prime , e)$ , between words at index $t$ and $t^\\prime $ with an edge type $e$ .",
"The standard GRU edges are included as $(t, t-1, temporal)$ .",
"Figure: The hidden state of “hacked” is a combination of previous output vectors.",
"In this case, three vectors are aggregated with DAG-GRU's attention model.",
"h t '' h_{t^{\\prime \\prime }}, is included in the input for the attention model since it is accessible through the “subj” dependency edge.",
"h t ' h_{t^\\prime } is included twice because it is connected through a narrative edge and a dependency edge with type “auxpass.” The input matrix is non-linearly transformed by U a U_a and tanh\\text{tanh}.",
"Next, w a w_a determines the importance of each vector in D t D_t.",
"Finally, the attention a t a_t is produced by tanh followed by softmax then applied to D t D_t.",
"The subject “members” would be distant under a standard RNN model, however the DAG-GRU model can focus on this important connection via dependency edges and attention.Each dependency relationship may be between any two words, which could produce a graph with cycles.",
"However, back-propagation through time [13] requires a directed acyclic graph (DAG), Hence the sentence graph, consisting of temporal and dependency edges $E$ , is split into two DAGs: a “forward” DAG $G_f$ that consists of only of edges $(t, t^\\prime , e)$ where $t^\\prime < t$ and a corresponding “backward” DAG $G_b$ where $t^\\prime > t$ .",
"The dependency relation between $t$ and $t^\\prime $ also includes the parent-child orientation, e.g., $nsubj\\text{-}parent$ or $nsubj\\text{-}child$ for a $nsubj$ (subject) relation.",
"An attention mechanism is used to combine the multiple hidden vectors.",
"The matrix $D_t$ is formed at each word $x_t$ by collecting and transforming all the previous hidden vectors coming into node $t$ , one per each edge type $e$ .",
"$\\alpha $ gives the attention, a distribution weighting importance over the edges.",
"Finally, the combined hidden vector $h_a$ is created by summing $D_t$ weighted by attention.",
"$&D_t^T =[ {\\text{tanh}}(U_e h_{t^\\prime })| (t, t^\\prime , e) \\in E] \\nonumber \\\\&\\alpha = \\text{softmax}(\\text{tanh}(D_t w_a)) \\nonumber \\\\&h_a = D_t^T \\alpha \\nonumber $ However, having a set of parameters $U_e$ for each edge type $e$ is over-specific for small datasets.",
"Instead a shared set of parameters $U_a$ is used in conjunction with an edge embedding $v_e$ .",
"$&D_t^T =[ {\\text{tanh}}(U_a h_{t^\\prime } \\circ v_e)| (t, t^\\prime , e) \\in E]$ The edge type embedding $v_e$ is concatenated with the hidden vector $h_{t^\\prime }$ and then transformed by the shared weights $U_a$ .",
"This limits the number of parameters while flexibly weighting the different edge types.",
"The new combined hidden vector $h_a$ is used instead of $h_{t-1}$ in the GRU equations.",
"The model is run forward and backward with the output concatenated, $h_{c,t} = h_{f,t} \\odot h_{b,t}$ , for a representation that includes the entire sentence's context and dependency relations.",
"After applying dropout [20] with $0.5$ rate to $h_{c,t}$ , a linear model with softmax is used to make predictions for each word at index $t$ ."
],
[
"Experiments",
"We use the ACE2005 dataset for evaluation.",
"Each word in each document is marked with one of the thirty-three event types or Nil for non-triggers.",
"Several high-performance models were reproduced for comparison.",
"Each is a good faith reproduction of the original with some adjustments to level the playing field.",
"For word embeddings, Elmo was used to generate a fixed representation for every word in ACE2005 [18].",
"The three vectors produced per word were concatenated together for a single representation.",
"We did not use entity type embeddings for any method.",
"The models were trained to minimize the cross entropy loss with Adam [8] with L2 regularization set at $0.0001$ .",
"The learning rate was halved every five epochs starting from $0.0005$ for a maximum of 30 epochs or until convergence as determined by F1 score on the development set.",
"The same training method and word embeddings were used across all the methods.",
"Based on preliminary experiments, these settings resulted in better performance than those originally specified.",
"However, notably both GRU [14] and DAG-LSTM [19] were not used as joint models.",
"Further, the GRU implementation did not use a memory network, instead we used the final vectors from the forward and backward pass concatenated to each timestep's output for additional context.",
"For CNNs [15] the number of filters was reduced to 50 per filter size.",
"The CNN+LSTM [5] model had no other modifications.",
"DAG-GRU used a hidden layer size of 128.",
"Variant A of the DAG-GRU model utilized the attention mechanism, while variant B used averaging, that is: $D_t = \\frac{1}{|E(t)|} \\underset{(t^\\prime , e) \\in E(t)}{\\sum } \\text{tanh}(U_a h_{t^\\prime }\\circ v_e)$"
],
[
"Effects of Random Initialization",
"Given that ACE2005 is small as far as neural network models are concerned, the effect of the random initialization of these models needs to be studied.",
"Although some methods include tests of significance, the type of statistical test is often not reported.",
"Simple statistical significance tests, such as the t-test, are not compatible with a single F1 score, instead the average of F1 scores should be tested [21].",
"We reproduced and evaluated five different systems with different initializations to empirically assess the effect of initialization.",
"The experiments were done on the standard ACE2005 split and the aggregated results over 20 random seeds were given in Table REF .",
"The random initializations of the models had a significant impact on their performance.",
"The variation was large enough that the observed range of the F1 scores overlapped across almost all the models.",
"However the differences in average performances of different methods, except for CNN and DAG-LSTM, were significant at $p < 0.05$ according to the t-test, not controlling for multiple hypotheses.",
"Both the GRU [14] and CNN [15] models perform well with their best scores being close to the reported values.",
"The CNN+LSTM model's results were significantly lower than the published values, though this method has the highest variation.",
"It is possible that there is some unknown factor such as the preprocessing of the data that significantly impacted the results or that the value is an outlier.",
"Likewise, the DAG-LSTM model underperformed.",
"However, the published results were based on a joint event and argument extraction model and probably benefited from the additional entity and argument information.",
"DAG-GRU A consistently and significantly outperforms the other methods in this comparison.",
"The best observed F1 score, 71.1%, for DAG-GRU is close to the published state-of-the-art scores of DAG-LSTM and GCNN at 71.9% and 71.4% respectively.",
"With additional entity information, GCNN achieves a score of 73.1%.",
"Also, the attention mechanism used in DAG-GRU A shows a significant improvement over the averaging method of DAG-GRU B.",
"This indicates that some syntactic links are more useful than others and that the weighting attention applies is necessary to utilize that syntactic information.",
"Another source of variation was the distributional differences between the development and testing sets.",
"Further, the testing set only include newswire articles whereas the training and dev.",
"sets contain informal writing such as web log (WL) documents.",
"The two sets have different proportions of event types and each model saw at least a 2% drop in performance between dev.",
"and test on average.",
"At worst, the DAG-LSTM model's drop was 5.26%.",
"This is a problem for model selection, since the dev.",
"score is used to choose the best model, hyperparameters, or random initialization.",
"The distributional differences mean that methods which outperform others on the dev.",
"set do not necessarily perform as well on the test set.",
"For example, DAG-GRU A performs worse that DAG-GRU B on the dev.",
"set, however it achieves a higher mean score on the testing set.",
"One method of model selection over random initializations is to train the model $k$ times and pick the best one based on the dev.",
"score.",
"Repeating this model selection procedure many times for each model is prohibitively expensive, so the experiment was approximated by bootstrapping the twenty samples per model [4].",
"For each model, 5 dev.",
"& test score pairs were sampled with replacement from the twenty available pairs.",
"The initialization with the best dev.",
"score was selected and the corresponding test score was taken.",
"This model selection process of picking the best of 5 random samples was repeated 1000 times and the results are shown in Table REF .",
"This process did not substantially increase the average performance beyond the results in Table REF , although it did reduce the variance, except for the CNN model.",
"It appears that using the dev.",
"score for model selection is only marginally helpful.",
"Table: The statistics of the 20 random initializations experiment.",
"†\\dagger denotes results are from a joint event and argument extraction model.Table: Bootstrap estimates on 1000 samples for each model after model selection based on dev set scores."
],
[
"Randomized Splits",
"In order to explore the effect of the training/testing split popularized by [9], a randomized cross validation experiment was conducted.",
"From the set of 599 documents in ACE2005, 10 random splits were created maintaining the same 529, 30, 40 document counts per split, training, development, testing, respectively.",
"This method was used to evaluate the effect of the standard split, since it maintains the same data proportions while only varying the split.",
"The results of the experiment are found in Table REF .",
"Table: Average results on 10 randomized splits.The effect of the split is substantial.",
"Almost all models' performance dropped except for DAG-LSTM, however the variance increased across all models.",
"In the worst case, the standard deviation increased threefold from 0.86% to 2.60% for the GRU model.",
"In fact, the increased variation of the splits means that the confidence intervals for all the models overlap.",
"This aligns with cross domain analysis, some domains such as WL are known to be much more difficult than the newswire domain which comprises all of the test data under the standard split [15].",
"Further, the effect of the difference in splits also negates the benefits of the attention mechanism of DAG-GRU A.",
"This is likely due to the test partitions' inclusion of WL and other kinds of informal writing.",
"The syntactic links are much more likely to be noisy for informal writing, reducing the syntactic information's usefulness and reliability.",
"All these sources of variation are greater than most advances in event detection, so quantifying and reporting this variation is essential when assessing model performance.",
"Further, understanding this variation is important for reproducibility and is necessary for making any valid claims about a model's relative effectiveness."
],
[
"Conclusions",
"We introduced and evaluated a DAG-GRU model along with four previous models in two different settings, the standard ACE2005 split with multiple random initializations and the same dataset with multiple random splits.",
"These experiments demonstrate that our model, which utilizes syntactic information through an attention mechanism, is competitive with the state-of-the-art.",
"Further, they show that there are several significant sources of variation which had not been previously studied and quantified.",
"Studying and mitigating this variation could be of significant value by itself.",
"At a minimum, it suggests that the community should move away from evaluations based on single random initializations and single training-test splits."
],
[
"Acknowledgments",
"This work was supported by grants from DARPA XAI (N66001-17-2-4030), DEFT (FA8750-13-2-0033), and NSF (IIS-1619433 and IIS-1406049)."
]
] | 1808.08504 |
[
[
"Spontaneous symmetry breaking induced by quantum monitoring"
],
[
"Abstract Spontaneous symmetry breaking (SSB) is responsible for structure formation in scenarios ranging from condensed matter to cosmology.",
"SSB is broadly understood in terms of perturbations to the Hamiltonian governing the dynamics or to the state of the system.",
"We study SSB due to quantum monitoring of a system via continuous quantum measurements.",
"The acquisition of information during the measurement process induces a measurement back-action that seeds SSB.",
"In this setting, by monitoring different observables, an observer can tailor the topology of the vacuum manifold, the pattern of symmetry breaking and the nature of the resulting domains and topological defects."
],
[
"Monitoring-induced symmetry breaking under a slow quench",
"We consider the dynamics of a continuously-monitored quantum phase transition from a paramagnetic to a ferromagnetic regime in a timescale $\\tau _Q$ , as opossed to a sudden quench.",
"The time-dependent Hamiltonian reads $H(t) = - \\Lambda \\left( 1- \\frac{t}{\\tau _Q} \\right) \\sum _{j=1}^{N} \\sigma _j^x - \\Lambda \\frac{t}{\\tau _Q} \\sum _{j=1}^{N} \\sigma _j^z\\sigma _{j+1}^z.$ For times $t~\\ge ~\\tau _Q$ we take $H(t) = H(\\tau _Q)$ .",
"By contrast to sudden-quenches, the finite-time crossing of the critical point reduces excitations and instead favors dynamics constrained to the lowest energy subspace [35].",
"Without access to measurement outcomes nothing in the dynamics breaks the symmetry between degenerate grounds states of the ferromagnetic Hamiltonian.",
"However, the monitoring of the system feeds in a seed of asymmetry.",
"Figures REF and REF illustrate the monitored dynamics for a variety of regimes.",
"Figure: Monitoring-induced symmetry breaking for a slow quench.A system undergoing a slow quench (1/τ Q ≪Λ1/\\tau _Q \\ll \\Lambda ) tends to remain close to the ground state, but in the absence of monitoring and other sources of symmetry breaking the system cannot select between degenerate ferromagnetic ground states |↑↑⋯↑↑〉\\mathinner {|{ \\uparrow \\uparrow \\dots \\uparrow \\uparrow }\\rangle } and |↓↓⋯↓↓〉\\mathinner {|{ \\downarrow \\downarrow \\dots \\downarrow \\downarrow }\\rangle }.",
"(a) A strong monitoring (1/τ m ∼1/τ Q ≪Λ1/\\tau _m \\sim 1/\\tau _Q \\ll \\Lambda ) of the individual spins generates a symmetry broken state where spins are uncorrelated.",
"(b) As measurement strength decreases, symmetry is broken with a stronger tendency to remain close to the ground state.",
"This favors formation of domains of spins collapsing to stable `up' or `down' configurations.",
"(c) A weak monitoring (1/τ m ≪{Λ,1/τ Q }1/\\tau _m \\ll \\lbrace \\Lambda ,1/\\tau _Q\\rbrace ) leads to an almost uniform domain, and a final randomly-selected state that is close to one of the possible symmetry-broken ground states of the ferromagnetic Hamiltonian.It is worth noting that the monitoring process affects the state of the system in timescales of order of τ m \\tau _m , so for this case the effect of the monitoring itself is small in the duration of the simulated experiment.Figure: Monitoring-induced symmetry breaking for a slow quench.For a different regime of the parameters {τ m ,τ Q ,Λ}\\lbrace \\tau _m,\\tau _Q, \\Lambda \\rbrace the same general behavior as in Fig.",
"is observed, with formation of larger homogeneous domains as the measurement strength is decreased, in which case the adiabatic dynamics dominates the properties of the resulting symmetry-broken state.There exists a competition between monitoring back-action, which attempts to single out individual spins, versus the tendency of the system to remain excitation-less due to adiabatic dynamics.",
"This competition can result in the formation of domains, that is, regions where spins locally align in a given direction, with different definite values of magnetization.",
"The size of such domains depends on the relative strength of the monitoring and of the quench time: while strong monitoring and fast quenches implies uncorrelated individual spins, weak monitoring and slow quenches favors formation of large domains of definite magnetization.",
"In the limit $1/\\tau _m \\ll \\lbrace \\Lambda ,1/\\tau _Q\\rbrace $ the system randomly selects one of the otherwise symmetric grounds states $\\mathinner {|{ \\uparrow \\uparrow \\dots \\uparrow \\uparrow }\\rangle }$ and $\\mathinner {|{ \\downarrow \\downarrow \\dots \\downarrow \\downarrow }\\rangle }$ ."
],
[
"Timescales of purity decay from monitoring-induced symmetry breaking",
"SSB in a standard setting, with no continuous measurements, can be induced by local perturbations, for example to the system Hamiltonian [31].",
"We consider a seed of SSB to be associated with a stochastic perturbation operator $V$ such that the total system Hamiltonian reads $H_{\\text{st}}(t) = H + \\lambda \\eta (t)V.$ The Hamiltonian $H$ is taken as the ferromagnetic one, and the setting considered is that of a sudden quench from paramagnetic to ferromagnetic.",
"Here $\\lambda $ is a dimensionless coupling constant characterizing the strength of the perturbation and $\\eta (t)$ corresponds to a real Gaussian random process.",
"For simplicity we assume $\\langle \\eta (t) \\rangle =0$ and $\\langle \\eta (t)\\eta (t^{\\prime }) \\rangle =\\delta (t-t^{\\prime })$ which corresponds to the white-noise limit.",
"The average over many-realizations of the noise makes the dynamics effectively open and described by the Lindblad operator [45], [46] $L[\\rho _t] &=-i \\left[H,\\rho _t \\right] - \\frac{\\lambda ^2}{2}[V,[V,\\rho _t ]].$ As a result, the evolution of an initial pure state decreases its purity.",
"The time scale in which this SSB seed acts can be identified from the short-time asymptotics of the purity decay and reads $\\frac{1}{\\tau _V}=\\frac{\\lambda ^2}{2}\\Delta V^2,$ where $\\Delta V^2$ is the variance of the SSB seed operator $V$ with respect to the initial state.",
"This provides the time-scale for SSB induced by the fluctuating operator $V$ .",
"Under continuous quantum measurements the dynamics averaged over many-realizations would be of the form $L[\\rho _t] &=-i \\left[H,\\rho _t \\right] - \\frac{\\lambda ^2}{2}[V,[V,\\rho _t ]] -\\sum _\\alpha \\frac{1}{8\\tau _m^\\alpha } \\left[A_\\alpha ,\\left[A_\\alpha ,\\rho _t \\right]\\right].$ The short-time purity decay is then set by $\\frac{1}{\\tau }= \\frac{\\lambda ^2}{2}\\Delta V^2+\\sum _\\alpha \\frac{1}{8\\tau _m^\\alpha } \\Delta A_\\alpha ^2,$ which illustrates the competition between the time scales associated with the stochastic perturbation and the continuous monitoring."
]
] | 1808.08343 |
[
[
"Formal Analysis of an E-Health Protocol"
],
[
"Abstract Given the sensitive nature of health data, security and privacy in e-health systems is of prime importance.",
"It is crucial that an e-health system must ensure that users remain private - even if they are bribed or coerced to reveal themselves, or others: a pharmaceutical company could, for example, bribe a pharmacist to reveal information which breaks a doctor's privacy.",
"In this paper, we first identify and formalise several new but important privacy properties on enforcing doctor privacy.",
"Then we analyse the security and privacy of a complicated and practical e-health protocol (DLV08).",
"Our analysis uncovers ambiguities in the protocol, and shows to what extent these new privacy properties as well as other security properties (such as secrecy and authentication) and privacy properties (such as anonymity and untraceability) are satisfied by the protocol.",
"Finally, we address the found ambiguities which result in both security and privacy flaws, and propose suggestions for fixing them."
],
[
"Introduction",
"The inefficiency of traditional paper-based health care and advances in information and communication technologies, in particular cloud computing, mobile, and satellite communications, constitute the ideal environment to facilitate the development of widespread electronic health care (e-health for short) systems.",
"E-health systems are distributed health care systems using devices and computers which communicate with each other, typically via the Internet.",
"E-health systems aim to support secure sharing of information and resources across different health care settings and workflows among different health care providers.",
"The services of such systems are intended to be more secure, effective, efficient and timely than the currently existing health care systems.",
"Given the sensitive nature of health data, handling this data must meet strict security and privacy requirements.",
"In traditional health care systems, this is normally implemented by controlling access to the physical documents that contain the health care data.",
"Security and privacy are then satisfied, assuming only legitimate access is possible and assuming that those with access do not violate security or privacy.",
"However, the introduction of e-health systems upends this approach.",
"The main benefit of e-health systems is that they facilitate digital exchange of information amongst the various parties involved.",
"This has two major consequences: first, the original health care data is shared digitally with more parties, such as pharmacists and insurance companies; and second, this data can be easily shared by any of those parties with an outsider.",
"Clearly, the assumption of a trusted network can no longer hold in such a setting.",
"Given that it is trivial for a malicious entity to intercept or even alter digital data in transit, access control approaches to security and privacy are no longer sufficient.",
"Therefore, we must consider security and privacy of the involved parties with respect to an outsider, the Dolev-Yao adversary [30], who controls the communication network (i.e., the adversary can observe, block, create and alter information).",
"Communication security against such an adversary is mainly achieved by employing cryptographic communication protocols.",
"Cryptography is also employed to preserve and enforce privacy, which prevents problems such as prescription bribery.",
"It is well known that designing such protocols is error-prone: time and again, flaws have been found in protocols that claimed to be secure (e.g., electronic voting systems [21], [42] have been broken [33], [43]).",
"Therefore, we must require that security and privacy claims of an e-health protocol are verified before the protocol is used in practice.",
"Without verifying that a protocol satisfies its security and privacy claims, subtle flaws may go undiscovered.",
"In order to objectively verify whether a protocol satisfies its claimed security and privacy requirements, each requirement must be formally defined as a property.",
"Various security and privacy properties have already been defined in the literature, such as secrecy, authentication, anonymity and untraceability.",
"We refer to these properties as regular security and privacy properties.",
"While they are necessary to ensure security and privacy, by themselves these regular properties are not sufficient.",
"Benaloh and Tuinstra pointed out the risk of subverting a voter [21] to sell her vote.",
"The idea of coercing or bribing a party into nullifying their privacy is hardly considered in the literature of e-health systems (notable exceptions include [47], [24]).",
"However, this concept impacts e-health privacy: for example, a pharmaceutical company could bribe doctors to prescribe only their medicine.",
"Therefore, we cannot only consider privacy with respect to the Dolev-Yao adversary.",
"To fully evaluate privacy of e-health systems, we must also consider this new aspect of privacy in the presence of an active coercer – someone who is bribing or threatening parties to reveal private information.",
"We refer to this new class of privacy properties as enforced privacy properties.",
"In particular, we identify the following regular and enforced privacy properties [27] to counter doctor bribery: prescription privacy: a doctor cannot be linked to his prescriptions; receipt-freeness: a doctor cannot prove his prescriptions to the adversary for preventing doctor bribes; independency of prescription privacy: third parties cannot help the adversary to link a doctor to the doctor's prescriptions for preventing others to reduce a doctor's prescription privacy; and independency of receipt-freeness: a doctor and third parties cannot prove the doctor's prescriptions to the adversary for preventing anyone from affecting a doctor's receipt-freeness.",
"Contributions.",
"We identify three enforced privacy properties in e-health systems and are the first to provide formal definitions for them.",
"In addition, we develop an in-depth applied pi model of the DLV08 e-health protocol [24].",
"As this protocol was designed for practical use in Belgium, it needed to integrate with the existing health care system.",
"As such, it has become a complicated system with many involved parties, that relies on complex cryptographic primitives to achieve a multitude of goals.",
"We formally analyse privacy and enforced privacy properties of the protocol, as well as regular security properties.",
"We identify ambiguities in the protocol description that cause both security and privacy flaws, and propose suggestions for fixing them.",
"The ProVerif code of modelling and full analysis of the DLV08 protocol can be found in [26].",
"Remark.",
"This article is a revised and extended version of [27] that appears in the proceedings of the 17th European Symposium on Research in Computer Security (ESORICS'12).",
"In this version we have added (1) the full formal modelling of the DLV08 protocol in the applied pi calculus (see Section ); (2) the detailed analysis of secrecy and authentication properties of the protocol (see Section ); and (3) details of the analysis of privacy properties of the protocol which are not described in the conference paper [27] (see Section ).",
"In addition, it contains an overview on privacy and enforced privacy in e-health systems (see Section ) and a brief but complete description of the applied pi calculus (see Section REF )."
],
[
"Privacy and enforced privacy in e-health",
"Ensuring privacy in e-health systems has been recognised as a necessary prerequisite for adoption such systems by the general public [50], [35].",
"However, due to the complexity of e-health settings, existing privacy control techniques, e.g., formal privacy methods, from domains such as e-voting (e.g., [29], [34]) and e-auctions (e.g., [25]) do not carry over directly.",
"In e-voting and e-auctions, there is a natural division into two types of roles: participants (voters, bidders) and authorities (who run the election/auction).",
"In contrast, e-health systems have to deal with a far more complex constellation of roles, including doctors, patients, pharmacists, insurance agencies, oversight bodies, etc.",
"These roles interact in various ways with each other, requiring private data of one another, which makes privacy even more complex.",
"Depending on the level of digitalisation, health care systems have different security requirements.",
"If electronic devices are only used to store patient records, then ensuring privacy mainly requires local access control.",
"On the other hand, if data is communicated over a network, then communication privacy becomes paramount.",
"Below, we sketch a typical situation of using a health care system, indicating what information is necessary where.",
"This will help to gain an understanding for the interactions and interdependencies between the various roles.",
"Typically, a patient is examined by a doctor, who then prescribes medicine.",
"The patient goes to a pharmacist to get the medicine.",
"The medicine is reimbursed by the patient's health insurance, and the symptoms and prescription of the patient may be logged with a medical research facility to help future research.",
"This overview hides many details.",
"The patient may possess medical devices enabling her to undergo the examination at home, after which the devices digitally communicate their findings to a remote doctor.",
"The findings of any examination (by doctor visit or by digital devices) need to be stored in the patients health record, either electronic or on paper, which may be stored at the doctor's office, on a server in the network, on a device carried by the patient, or any combination of these.",
"Next, the doctor returns a prescription, which also needs to be stored.",
"The pharmacist needs to know what medicine is required, which is privacy-sensitive information.",
"Moreover, to prevent abuse of medicine, the pharmacist must verify that the prescription came from an authorised doctor, is intended for this patient, and was not fulfilled before.",
"On top of that, the pharmacist may be allowed (or even required) to substitute medicine of one type for another (e.g., brand medicine for generic equivalents), which again must be recorded in the patient's health record.",
"For reimbursement, the pharmacist or the patient registers the transaction with the patient's health insurer.",
"In addition, regulations may require that such information is stored (in aggregated form or directly) for future research or logged with government agencies.",
"Some health care systems allow emergency access to health data, which complicates privacy matters even further.",
"Finally, although a role may need to have access to privacy-sensitive data of other roles, this does not mean that he is trusted to ensure the privacy of those other roles.",
"For instance, a pharmacist may sell his knowledge about prescription behaviour to a pharmaceutical company.",
"From the above overview on e-health systems, we can conclude that existing approaches to ensuring privacy from other domains deal with far simpler division of roles, and they are not properly equipped to handle the role diversity present in e-health systems.",
"Moreover, they do not address the influence of other roles on an individual's privacy.",
"Therefore, current privacy approaches cannot be lifted directly, but must be redesigned specifically for the e-health domain.",
"In the following discussions, we focus on the privacy of the main actors in health care: patient privacy and doctor privacy.",
"Privacy of roles such as pharmacists does not impact on the core process in health care, and is therefore relegated to future work.",
"We do not consider privacy of roles performed by public entities such as insurance companies, medical administrations, etc."
],
[
"Related work",
"The importance of patient privacy in e-health is traditionally seen as vital to establishing a good doctor-patient relationship.",
"This is even more pertinent with the emergence of the Electronic Patient Record [7].",
"A necessary early stage of e-health is to transform the paper-based health care process into a digital process.",
"The most important changes in this stage are made to patient information processing, mainly health care records.",
"Privacy policies are the de facto standard to properly express privacy requirements for such patient records.",
"There are three main approaches to implement these requirements: access control, architectural design, and the use of cryptography."
],
[
"Patient privacy by access control.",
"The most obvious way to preserve privacy of electronic health care records is to limit access to these records.",
"The need for access control is supported by several privacy threats to personal health information listed by Anderson [7].",
"Controlling access is not as straightforward as it sounds though: the need for access changes dynamically (e.g., a doctor only needs access to records of patients that he is currently treating).",
"Consequently, there exists a wide variety of access control approaches designed for patient privacy in the literature, from simple access rules (e.g., [7]), to consent-based access rules (e.g., [44]), role-based access control (RBAC) (e.g., [52]), organisation based access control (e.g., [36]), etc."
],
[
"Patient privacy by architectural design.",
"E-health systems cater to a number of different roles, including doctors, patients, pharmacists, insurers, etc.",
"Each such role has its own sub-systems or components.",
"As such, e-health systems can be considered as a large network of systems, including administrative system components, laboratory information systems, radiology information systems, pharmacy information systems, and financial management systems.",
"Diligent architectural design is an essential step to make such a complex system function correctly.",
"Since privacy is important in e-health systems, keeping privacy in mind when designing the architecture of such systems is a promising path towards ensuring privacy [55].",
"Examples of how to embed privacy constraints in the architecture are given by the architecture of wireless sensor networks in e-health [37], proxies that may learn location but not patient ID [49], an architecture for cross-institution image sharing in e-health [22], etc."
],
[
"Cryptographic approaches to patient privacy.",
"Cryptography is necessary to ensure private communication between system components over public channels (e.g., [10]).",
"For example, Van der Haak et al.",
"[58] use digital signatures and public-key authentication (for access control) to satisfy legal requirements for cross-institutional exchange of electronic patient records.",
"Ateniese et al.",
"[3] use pseudonyms to preserve patient anonymity, and enable a user to transform statements concerning one of his pseudonyms into statements concerning one of his other pseudonyms (e.g., transforming a prescription for the pseudonym used with his doctor to a prescription for the pseudonym used with the pharmacist).",
"Layouni et al.",
"[46] consider communication between health monitoring equipment at a patient's home and the health care centre.",
"They propose a protocol using wallet-based credentials (a cryptographic primitive) to let patients control when and how much identifying information is revealed by the monitoring equipment.",
"More recently, De Decker et al.",
"[24] propose a health care system for communication between insurance companies and administrative bodies as well as patients, doctors and pharmacists.",
"Their system relies on various cryptographic primitives to ensure privacy, including zero-knowledge proofs, signed proofs of knowledge, and bit-commitments.",
"We will explain this system in more detail in Section ."
],
[
"Doctor privacy.",
"A relatively understudied aspect is that of doctor privacy.",
"Matyáš [47] investigates the problem of enabling analysis of prescription information while ensuring doctor privacy.",
"His approach is to group doctors, and release the data per group, hiding who is in the group.",
"He does not motivate a need for doctor privacy, however.",
"Two primary reasons for doctor privacy have been identified in the literature: (1) (Ateniese et al.",
"[3]) to safeguard doctors against administrators setting specific efficiency metrics on their performance (e.g., requiring the cheapest medicine be used, irrespective of the patient's needs).",
"To address this, Ateniese et al.",
"[5], [3] propose an anonymous prescription system that uses group signatures to achieve privacy for doctors; (2) (De Decker et al.",
"[24]) to prevent a pharmaceutical company from bribing a doctor to prescribe their medicine.",
"A typical scenario can be described as follows.",
"A pharmaceutical company seeks to persuade a doctor to favour a certain kind of medicine by bribing or coercing.",
"To prevent this, a doctor should not be able to prove which medicine he is prescribing to this company (in general, to the adversary).",
"This implies that doctor privacy must be enforced by e-health systems.",
"De Decker et al.",
"also note that preserving doctor privacy is not sufficient to prevent bribery: pharmacists could act as intermediaries, revealing the doctor's identity to the briber, as pharmacists often have access to prescriptions, and thus know something about the prescription behaviour of a doctor.",
"This observation leads us to formulate a new but important requirement of independency of prescription privacy in this paper: no third party should be able to help the adversary link a doctor to his prescription."
],
[
"Observations",
"Current approaches to privacy in e-health, as witnessed from the literature study in Section REF , mostly focus on patient privacy as an access control or authentication problem.",
"Even though doctor privacy is also a necessity, research into ensuring doctor privacy is still in its infancy.",
"We believe that doctor privacy is as important as patient privacy and needs to be studied in more depth.",
"It is also clear from the analysis that privacy in e-health systems needs to be addressed at different layers: access control ensures privacy at the service layer; privacy by architecture design addresses privacy concerns at the system/architecture layer; use of cryptography guarantees privacy at the communication layer.",
"Since e-health systems are complex [56] and rely on correct communications between many sub-systems, we study privacy in e-health as a communication problem.",
"In fact, message exchanges in communication protocols may leak information which leads to a privacy breach [45], [23], [29].",
"Classical privacy properties, which are well-studied in the literature, attempt to ensure that privacy can be enabled.",
"However, merely enabling privacy is insufficient in many cases: for such cases, a system must enforce user privacy instead of allowing the user to pursue it.",
"One example is doctor bribery.",
"To avoid doctor bribery, we take into account enforced privacy for doctors.",
"In addition, we consider that one party's privacy may depend on another party (e.g., in the case of a pharmacist revealing prescription behaviour of a doctor).",
"In these cases, others can cause (some) loss of privacy.",
"Obviously, ensuring privacy in such a case requires more from the system than merely enabling privacy.",
"Consequently, we propose and study the following privacy properties for doctors in communication protocols in the e-health domain, in addition to regular security and privacy properties as we mentioned before in Section .",
"prescription privacy: A protocol preserves prescription privacy if the adversary cannot link a doctor to his prescriptions.",
"receipt-freeness: A protocol satisfies receipt-freeness if a doctor cannot prove his prescriptions to the adversary.",
"independency of prescription privacy: A protocol ensures independency of prescription privacy if third parties cannot help the adversary to link a doctor to the doctor's prescriptions.",
"independency of receipt-freeness: A protocol ensures independency of receipt-freeness if a doctor cannot prove his prescriptions to the adversary given that third parties sharing information with the adversary."
],
[
"Formalisation of privacy properties",
"In order to formally verify properties of a protocol, the protocol itself as well as the properties need to be formalised.",
"In this section, we focus on the formalisation of key privacy properties, while the formalisation of secrecy and authentication properties can be considered standard as studied in the literature [45], [15].",
"Thus secrecy and authentication properties are introduced later in the case study (Section REF ) and are omitted in this section.",
"We choose the formalism of the applied pi calculus, due to its capability in expressing equivalence based properties which is essential for privacy, and automatic verification supported by the tool ProVerif [15].",
"The applied pi calculus is introduced in Section REF .",
"Next, in Section REF , we show how to model e-health protocols in the applied pi calculus.",
"Then, from Section REF to Section REF , we formalise each of the privacy properties described in the end of Section REF .",
"Finally, in Sections REF and REF , we consider (strong) anonymity and (strong) untraceability in e-health, respectively.",
"These concepts have been formally studied in other domains (e.g., [54], [57], [13], [38], [4], [39]), and thus are only briefly introduced in this section."
],
[
"The applied pi calculus",
"The applied pi calculus is a language for modelling and analysing concurrent systems, in particular cryptographic protocols.",
"The following (mainly based on [6], [53]) briefly introduces its syntax, semantics and equivalence relations."
],
[
"Syntax",
"The calculus assumes an infinite set of names, which are used to model communication channels and other atomic data, an infinite set of variables, which are used to model received messages, and a signature $\\Sigma $ consisting of a finite set of function symbols, which are used to model cryptographic primitives.",
"Each function symbol has an arity.",
"A function symbol with arity zero is a constant.",
"Terms (which are used to model messages) are defined as names, variables, or function symbols applied to terms (see Figure REF ).",
"Figure: Terms in the applied pi calculus.Example 1 (function symbols and terms) Typical function symbols are enc with arity 2 for encryption, dec with arity 2 for decryption.",
"The term for encrypting $x$ with a key $k$ is ${\\sf {enc}}(x, k)$ .",
"The applied pi calculus assumes a sort system for terms.",
"Terms can be of a base type (e.g., Key or a universal base type Data) or type Channel$\\mathit {\\langle \\mathit {\\omega }\\rangle }$ where $\\mathit {\\omega }$ is a type.",
"A variable and a name can have any type.",
"A function symbol can only be applied to, and return, terms of base type.",
"Terms are assumed to be well-sorted and substitutions preserve types.",
"Terms are often equipped with an equational theory $E$ – a set of equations on terms.",
"The equational theory is normally used to capture features of cryptographic primitives.",
"The equivalence relation induced by $E$ is denoted as $=_E$ .",
"Example 2 (equational theory) The behaviour of symmetric encryption and decryption can be captured by the following equation: ${\\sf {dec}}({\\sf {enc}}(x, k), k)=_Ex,$ where $x$ and $k$ are variables.",
"Systems are described as processes: plain processes and extended processes (see Figure REF ).",
"Figure: Processes in the applied pi calculus.In Figure REF , $M, N$ are terms, ${\\tt {n}}$ is a name, $x$ is a variable and $v$ is a metavariable, standing either for a name or a variable.",
"The null process 0 does nothing.",
"The parallel composition $P \\mid Q$ represents the sub-process $P$ and the sub-process $Q$ running in parallel.",
"The replication $!P$ represents an infinite number of process $P$ running in parallel.",
"The name restriction $\\nu {\\tt {n}}.P$ binds the name ${\\tt {n}}$ in the process $P$ , which means the name ${\\tt {n}}$ is secret to the adversary.",
"The conditional evaluation $M =_EN$ represents equality over the equational theory rather than strict syntactic identity.",
"The message input ${\\sf {in}}(v, x).P$ reads a message from channel $v$ , and binds the message to the variable $x$ in the following process $P$ .",
"The message output ${\\sf {out}}(v, M).P$ sends the message $M$ on the channel $v$ , and then runs the process $P$ .",
"In both of these cases we may omit $P$ when it is 0.",
"Extended processes add variable restrictions and active substitutions.",
"The variable restriction $\\nu x.A $ binds the variable $x$ in the process $A$ .",
"The active substitution $\\lbrace M/x\\rbrace $ replaces variable $x$ with term $M$ in any process that it contacts with.",
"We say a process is sequential if it does not involve using the parallel composition $P\\mid Q$ , replication $!P$ , conditional, or active substitution.",
"That is, a sequential process is either null or constructed using name/variable restriction, message input/output.",
"In addition, applying syntactical substitution (i.e., “$\\mbox{let}\\ x=N \\ \\mbox{in} $ ” in ProVerif input language) to a sequential process still results in a sequential process.",
"For simplicity of presentation, we use $\\nu \\tilde{a}$ as an abbreviation for $\\nu a_1.\\nu a_2.\\cdots .\\nu a_n$ , where $\\tilde{a}$ is the sequence of names $a_1, a_2,\\cdots ,a_n$ .",
"We also use the abbreviation $P.Q$ to represent the process $\\mathit {action}_1.\\cdots .\\mathit {action}_n.Q$ , where $P:=\\mathit {action}_1.\\cdots .\\mathit {action}_n$ ; and an $\\mathit {action}_i$ is of the form $\\nu {\\tt {n}}$ , $\\nu x$ , ${\\sf {in}}(v, x)$ , ${\\sf {out}}(v, M)$ or $\\mbox{let}\\ x=N \\ \\mbox{in} \\ \\mathit {action}_j$ .",
"The intuition of $P.Q$ is that when a process consists of a sequential sub-process $P$ followed by a sub-process $Q$ , we write the process in an abbreviated manner as $P.Q$ .",
"In addition, we use the abbreviation $\\lbrace M_1/x_1,\\cdots , M_n/x_n\\rbrace $ to represent $\\lbrace M_1/x_1\\rbrace \\cdots \\lbrace M_n/x_n\\rbrace $ .",
"Names and variables have scopes.",
"A name is bound if it is under restriction.",
"A variable is bound by restrictions or inputs.",
"Names and variables are free if they are not delimited by restrictions or by inputs.",
"The sets of free names, free variables, bound names and bound variables of a process $A$ are denoted as ${\\sf {fn}}(A)$ , ${\\sf {fv}}(A)$ , ${\\sf {bn}}(A)$ and ${\\sf {bv}}(A)$ , respectively.",
"A term is ground when it does not contain variables.",
"A process is closed if it does not contain free variables.",
"Example 3 (processes) Consider a protocol in which $A$ generates a nonce ${\\tt {m}}$ , encrypts the nonce with a secret key ${\\tt {k}}$ , then sends the encrypted message to $B$ .",
"Denote with $\\mathit {Q}_A$ the process modelling the behaviour of $A$ , with $\\mathit {Q}_B$ the process modelling the behaviour of $B$ , and the whole protocol by $\\mathit {Q}$ : $\\begin{array}{ll}\\mathit {Q}_A & :=\\nu {\\tt {m}}.",
"{\\sf {out}}({\\tt {ch}}, {\\sf {enc}}({\\tt {m}}, {\\tt {k}})) \\\\\\mathit {Q}_B & :={\\sf {in}}({\\tt {ch}}, x) \\\\\\mathit {Q}& :=\\nu {\\tt {k}}.",
"(\\mathit {Q}_A \\mid \\mathit {Q}_B)\\end{array}$ Here, ${\\tt {ch}}$ is a free name representing a public channel.",
"Name ${\\tt {k}}$ is bound in process $\\mathit {Q}$ ; name ${\\tt {m}}$ is bound in process $\\mathit {Q}_A$ .",
"Variable $x$ is bound in process $\\mathit {Q}_B$ .",
"A frame is defined as an extended process built up from 0 and active substitutions by parallel composition and restrictions.",
"The active substitutions in extended processes allow us to map an extended process $A$ to its frame ${\\sf {frame}}(A)$ by replacing every plain process in $A$ with 0.",
"The domain of a frame $B$ , denoted as ${\\sf {domain}}(B)$ , is the set of variables for which the frame defines a substitution and which are not under a restriction.",
"Example 4 (frames) The frame of the process $\\nu {\\tt {m}}.",
"({\\sf {out}}({\\tt {ch}}, x)\\mid \\lbrace {\\tt {m}}/x\\rbrace )$ , denoted as ${\\sf {frame}}(\\nu {\\tt {m}}.\\linebreak ({\\sf {out}}({\\tt {ch}}, x)\\mid \\lbrace {\\tt {m}}/x\\rbrace ))$ is $\\nu {\\tt {m}}.",
"(0\\mid \\lbrace {\\tt {m}}/x\\rbrace )$ .",
"The domain of this frame, denoted as ${\\sf {domain}}(\\nu {\\tt {m}}.",
"(0\\mid \\lbrace {\\tt {m}}/x\\rbrace ))$ is $\\lbrace x\\rbrace $ .",
"A context ${\\mathcal {C}}[\\_]$ is defined as a process with a hole, which may be filled with any process.",
"An evaluation context is a context whose hole is not under a replication, a condition, an input or an output.",
"Example 5 (context) Process $\\nu {\\tt {k}}.",
"(\\mathit {Q}_A \\mid \\_)$ is an evaluation context.",
"When we fill the hole with process $\\mathit {Q}_B$ , we obtain the process $\\nu {\\tt {k}}.",
"(\\mathit {Q}_A \\mid \\mathit {Q}_B)$ , which is the process $Q$ ."
],
[
"Operational semantics",
"The operational semantics of the applied pi calculus is defined by: 1) structural equivalence ($\\equiv $ ), 2) internal reduction ($\\rightarrow $ ), and 3) labelled reduction ($\\xrightarrow{}$ ) of processes.",
"1) Intuitively, two processes are structurally equivalent if they model the same thing but differ in structure.",
"Formally, structural equivalence of processes is the smallest equivalence relation on extended process that is closed by $\\alpha $ -conversion on names and variables, by application of evaluation contexts as shown in Figure REF .",
"Figure: Structural equivalence in the applied pi calculus.2) Internal reduction is the smallest relation on extended processes closed under structural equivalence, application of evaluation of contexts as shown in Figure REF .",
"Figure: Internal reduction in the applied pi calculus.3) The labelled reduction models the environment interacting with the processes.",
"It defines a relation $A\\xrightarrow{} A^{\\prime }$ as in Figure REF .",
"The label $\\alpha $ is either reading a term from the process's environment, or sending a name or a variable of base type to the environment.",
"Figure: Labelled reduction in the applied pi calculus."
],
[
"Equivalences",
"The applied pi calculus defines observational equivalence and labelled bisimilarity to model the indistinguishability of two processes by the adversary.",
"It is proved that the two relations coincide, when active substitutions are of base type [6], [41].",
"We mainly use the labelled bisimilarity for the convenience of proofs.",
"Labelled bisimilarity is based on static equivalence: labelled bisimilarity compares the dynamic behaviour of processes, while static equivalence compares their static states (as represented by their frames).",
"Definition 1 (static equivalence) Two terms $M$ and $N$ are equal in the frame $B$ , written as $(M =_EN)B$ , iff there exists a set of restricted names $\\tilde{{\\tt {n}}}$ and a substitution $\\sigma $ such that $B \\equiv \\nu \\tilde{{\\tt {n}}}.",
"\\sigma $ , $M\\sigma =_EN\\sigma $ and $\\tilde{{\\tt {n}}} \\cap ({\\sf {fn}}(M) \\cup {\\sf {fn}}(N)) = \\emptyset $ .",
"Closed frames $B$ and $B^{\\prime }$ are statically equivalent, denoted as $B \\approx _{s}B^{\\prime }$ , if (1) ${\\sf {domain}}(B)={\\sf {domain}}(B^{\\prime })$ ; (2) $\\forall $ terms $M,N$ : $(M =_EN)B$ iff $(M =_EN)B^{\\prime }$ .",
"Extended processes $A$ , $A^{\\prime }$ are statically equivalent, denoted as $A \\approx _{s}A^{\\prime }$ , if their frames are statically equivalent: ${\\sf {frame}}(A) \\approx _{s}{\\sf {frame}}(A^{\\prime })$ .",
"Example 6 (equivalence of frames [6]) The frame $B$ and the frame $B^{\\prime }$ , are equivalent.",
"However, the two frames are not equivalent to frame $B^{\\prime \\prime }$ , because the adversary can discriminate $B^{\\prime \\prime }$ by testing $y=_Ex)$ .",
"$\\begin{array}{lcl}B&:=&\\nu M. \\lbrace M/x\\rbrace \\mid \\nu N. \\lbrace N/y\\rbrace \\\\B^{\\prime }&:=&\\nu M. (\\lbrace M)/x\\rbrace \\mid \\lbrace {\\sf g}(M)/y\\rbrace )\\\\B^{\\prime \\prime }&:=&\\nu M. (\\lbrace M/x\\rbrace \\mid \\lbrace M)/y\\rbrace )\\end{array}$ where $ and $g$ are two function symbols without equations.$ Example 7 (static equivalence) Process $\\lbrace M/x\\rbrace \\mid \\mathit {Q}_1$ is statically equivalent to process $\\lbrace M/x\\rbrace \\mid \\mathit {Q}_2$ where $\\mathit {Q}_1$ and $\\mathit {Q}_2$ are two closed plain process, because the frame of the two processes are statically equivalent, i.e., $\\lbrace M/x\\rbrace \\approx _{s}\\lbrace M/x\\rbrace $ .",
"Definition 2 (labelled bisimilarity) Labelled bisimilarity $(\\approx _{\\ell })$ is the largest symmetric relation $\\mathcal {R}$ on closed extended processes, such that $A\\, \\mathcal {R}\\, B$ implies: (1) $A \\approx _{s}B$ ; (2) if $A \\rightarrow A^{\\prime }$ then $B \\rightarrow ^* B^{\\prime }$ and $A^{\\prime }\\, \\mathcal {R}\\, B^{\\prime }$ for some $B^{\\prime }$ ; (3) if $A \\xrightarrow{} A^{\\prime }$ and ${\\sf {fv}}(\\alpha ) \\subseteq {\\sf {domain}}(A)$ and ${\\sf {bn}}(\\alpha ) \\cap {\\sf {fn}}(B)=\\emptyset $ ; then $B\\rightarrow ^*\\xrightarrow{}\\rightarrow ^* B^{\\prime }$ and $A^{\\prime }\\, \\mathcal {R}\\, B^{\\prime }$ for some $B^{\\prime }$ , where * denotes zero or more."
],
[
"E-health protocols",
"In the existing e-voting and (sealed bid) e-auction protocols, where bribery and coercion have been formally analysed using the applied pi calculus (see e.g., [29], [25]), the number of participants is determined a priori.",
"In contrast with these protocols, e-health systems should be able to handle newly introduced participants (e.g., patients).",
"To this end, we model user-types and each user-type can be instantiated infinite times."
],
[
"Roles.",
"An e-health protocol can be specified by a set of roles, each of which is modelled as a process, $\\mathit {P}_1, \\ldots , \\mathit {P}_n$ .",
"Each role specifies the behaviour of the user taking this role in an execution of the protocol.",
"By instantiating the free variables in a role process, we obtain the process of a specific user taking the role."
],
[
"Users.",
"Users taking a role can be modelled by adding settings (identity, pseudonym, encryption key, etc.)",
"to the process representing the role, that is $\\mathit {init}_i.\\mathit {P}_i$ , where $\\mathit {init}_i$ is a sequential process which generates names/terms modelling the data of the user (e.g., `$\\nu {\\tt {sk}}_{\\mathit {ph}}.\\mbox{let}\\ {\\tt {pk}}_\\mathit {ph}={\\sf pk}({\\tt {sk}}_{\\mathit {ph}})$ ' in Figure REF ), reads in setting data from channels (e.g., `${\\sf {in}}({\\tt {ch}}_{hp}, {{\\tt {Hii}}})$ ' in Figure REF ), or reveals data to the adversary (e.g., `${\\sf {out}}({\\tt {ch}}, {\\tt {pk}}_\\mathit {ph})$ ' in Figure REF ).",
"A user taking a role multiple times is captured by add replication to the role process, i.e., $\\mathit {init}_i.",
"!\\mathit {P}_i$ .",
"A user may also take multiple roles.",
"When the user uses two different settings in different roles, the user is treated as two separate users.",
"If the user uses shared setting in multiple roles, the user process is modelled as the user setting sub-process followed by the multiple role processes in parallel, e.g., $\\mathit {init}_{k}.!",
"(\\mathit {P}_i\\mid \\mathit {P}_j)$ when the user takes two roles $\\mathit {P}_i$ and $\\mathit {P}_j$ ,."
],
[
"User-types.",
"Users taking a specific role, potentially multiple times, belong to a user-type.",
"Hence, a user-type is modelled as $\\mathit {R}_i:=\\mathit {init}_i.",
"!\\mathit {P}_i$ .",
"The set of users of a type is captured by adding replication to the user-type process, i.e., $!\\mathit {R}_i$ .",
"A protocol with $n$ roles naturally forms $n$ user-types.",
"In protocols where users are allowed to take multiple roles with one setting, we consider these users form a new type.",
"For example, a challenge-response protocol, which specifies two roles – a role Initiator and a role Responder, has three user-types - the Initiator, the Responder and users taking both Initiator and Responder, assuming that a user taking both roles with the same setting is allowed.",
"A user-type with multiple roles is modelled as $\\mathit {init}_{k}.!",
"(\\mathit {P}_i\\mid \\ldots \\mid \\mathit {P}_j)$ , where $\\mathit {P}_i,\\ldots ,\\mathit {P}_j$ are the roles that a user of this type takes at the same time.",
"Since each user is an instance of a user-type, the formalisation of user-types allows us to model an unbounded number of users, by simply adding replication to the user-types.",
"In fact, in most cases, roles and user-types are identical, and the user-types that allow a user to take multiple roles can be considered as a new role as well.",
"Hence, we use roles and user-types interchangeably."
],
[
"Protocol instances.",
"Instances of an e-health protocol $\\mathit {\\mathit {P}_{\\!eh}}$ with $n$ roles/user-types are modelled in the following form: $\\mathit {\\mathit {P}_{\\!eh}}:=\\nu \\mathit {\\widetilde{mc}}.",
"\\mathit {init}.",
"(!\\mathit {R}_{1} \\mid \\ldots \\mid !\\mathit {R}_{n}),$ where process $\\nu \\mathit {\\widetilde{mc}}$ , which is the abbreviation for process $\\nu {\\tt {a}}_1.",
"\\ldots .",
"\\nu {\\tt {a}}_n.",
"\\nu {\\tt {c}}_1.\\ldots .\\nu {\\tt {c}}_n$ ($ {\\tt {a}}_i$ stands for private names, and ${\\tt {c}}_i$ stands for private channels), models the private names and channels in the protocol; $\\mathit {init}$ is a sequential process, representing settings of the protocol, such as generating/computing data and revealing information to the adversary (see Figure REF for example).",
"Essentially, $\\nu \\mathit {\\widetilde{mc}}.",
"\\mathit {init}$ models the global settings of an instance and auxiliary channels in the modelling of the protocol."
],
[
"Doctor role/user-type.",
"More specifically, we have a doctor role/user-type $\\mathit {R}_{\\mathit {dr}}$ of the form: $\\begin{array}{rcl}\\mathit {R}_{\\mathit {dr}}&:=&\\left\\lbrace \\begin{array}{ll}\\nu {{\\tt {Id}}_{\\mathit {dr}}}.",
"\\mathit {init}_{\\mathit {dr}}.",
"!\\mathit {P}_{\\mathit {dr}},& \\text{if doctor identity is not revealed by setting}\\\\\\nu {{\\tt {Id}}_{\\mathit {dr}}}.",
"{\\sf {out}}({\\tt {ch}}, {{\\tt {Id}}_{\\mathit {dr}}}).",
"\\mathit {init}_{\\mathit {dr}}.",
"!\\mathit {P}_{\\mathit {dr}},& \\text{if doctor identity is revealed by setting}\\end{array}\\right.\\\\\\mathit {P}_{\\mathit {dr}}&:=&\\nu {\\tt {presc}}.",
"\\mathit {\\mathit {main}_{\\mathit {dr}}}.\\end{array}$ In the following, we focus on the behaviour of a doctor, since our goal is to formalise privacy properties for doctors.",
"Each doctor is associated with an identity ($\\nu {{\\tt {Id}}_{\\mathit {dr}}}$ ) and can execute an infinite number of sessions (modelled by the exclamation mark `!'",
"in front of $\\mathit {P}_{\\mathit {dr}}$ ).",
"In case the doctor identity is revealed in the initialisation phase, we require that this unveiling does not appear in process $\\mathit {init}_{\\mathit {dr}}$ , for the sake of uniformed formalisations of the later defined privacy properties.",
"Instead, we model this case as identity generation ($\\nu {{\\tt {Id}}_{\\mathit {dr}}}$ ) immediately followed by unveiling the identity (${\\sf {out}}({\\tt {ch}}, {{\\tt {Id}}_{\\mathit {dr}}})$ ) on the public channel ${\\tt {ch}}$ .",
"Note that we reserve the name ${\\tt {ch}}$ for the adversary's public channel.",
"We require ${\\tt {ch}}$ to be free to model the public channel that is controlled by the adversary.",
"The adversary uses this channel by sending and receiving messages over ${\\tt {ch}}$ .",
"In fact, since the doctor identity ${{\\tt {Id}}_{\\mathit {dr}}}$ is defined outside of the process $\\mathit {init}_{\\mathit {dr}}.",
"!\\mathit {P}_{\\mathit {dr}}$ , the doctor identity appearing in the process is a free variable of the process.",
"Hence, in the case that the doctor identity is revealed, the doctor process can be simply modelled as $\\mathit {R}_{\\mathit {dr}}:=\\mathit {init}_{\\mathit {dr}}.",
"!\\mathit {P}_{\\mathit {dr}}$ , where doctor identity is a free variable.",
"To distinguish the free variable in process $\\mathit {init}_{\\mathit {dr}}.",
"!\\mathit {P}_{\\mathit {dr}}$ from the name ${{\\tt {Id}}_{\\mathit {dr}}}$ , we use the italic font to represent the free variable, i.e., ${\\mathit {Id_{\\mathit {dr}}}}$ .",
"Within each session, the doctor creates a prescription.",
"Since a prescription normally contains not only prescribed medicines but also the time/date that the prescription is generated as well as other identification information, we consider the prescriptions differ in sessions.",
"In the case that a prescription can be prescribed multiple times, one can add the replication mark $!$ in front of $\\mathit {\\mathit {main}_{\\mathit {dr}}}$ to model that the prescription ${\\tt {presc}}$ can be prescribed in infinite sessions, i.e., $\\mathit {P}_{\\mathit {dr}}:=\\nu {\\tt {presc}}.",
"!\\mathit {\\mathit {main}_{\\mathit {dr}}}$ .",
"Similarly, we use the italic font of the prescription, $\\mathit {presc}$ , to represent the free variable referring to the prescription in the process $\\mathit {\\mathit {main}_{\\mathit {dr}}}$ ."
],
[
"Well-formed.",
"We require that $\\mathit {\\mathit {P}_{\\!eh}}$ is well-formed, i.e., the process $\\mathit {\\mathit {P}_{\\!eh}}$ satisfies the following properties: $\\mathit {\\mathit {P}_{\\!eh}}$ is canonical: names and variables in the process never appear both bound and free, and each name and variable is bound at most once; data is typed, channels are ground, private channels are never sent on any channel; $\\nu \\mathit {\\widetilde{mc}}$ may be null; $\\mathit {init}$ and $\\mathit {init}_{\\mathit {dr}}$ are sequential processes; $\\mathit {init}$ , $\\mathit {init}_{\\mathit {dr}}$ and $\\mathit {\\mathit {main}_{\\mathit {dr}}}$ can be any process (possibly 0) such that $\\mathit {\\mathit {P}_{\\!eh}}$ is a closed plain process.",
"Furthermore, we use $\\mathcal {C}_\\mathit {eh}[\\_]$ to denote a context (a process with a hole) consisting of honest users, $\\mathcal {C}_\\mathit {eh}[\\_]:=\\nu \\mathit {\\widetilde{mc}}.",
"\\mathit {init}.",
"(!\\mathit {R}_{1} \\mid \\ldots \\mid !\\mathit {R}_{n}\\mid \\rule {0.3cm}{0.5pt}).$ Dishonest agents are captured by the adversary (Section REF ) with certain initial knowledge."
],
[
"The adversary",
"We consider security and privacy properties of e-health protocols with respect to the presence of active attackers – the Dolev-Yao adversary.",
"The adversary controls the network – the adversary can block, read and insert messages over the network; has computational power – the adversary can record messages and apply cryptographic functions to messages to obtain new messages; has a set of initial knowledge – the adversary knows the participants and public information of all participants, as well as a set of his own data; has the ability to initiate conversations – the adversary can take part in executions of protocols.",
"The adversary's behaviour models that of every dishonest agent (cf.",
"Section REF ), which is achieved by including the initial knowledge of each dishonest agent in the adversary's initial knowledge.",
"The behaviour of the adversary is modelled as a process running in parallel with the honest agents.",
"The adversary does whatever he can to break the security and privacy requirements.",
"We do not need to model the adversary explicitly, since he is embedded in the applied pi calculus as well as in the verification tool.",
"Modelling the honest users' behaviour is sufficient to verify whether the requirements hold."
],
[
"Limitations.",
"Note that the Dolev-Yao adversary model we use includes the “perfect cryptography” assumption.",
"This means that the adversary cannot infer any information from cryptographic messages for which he does not possess a key.",
"For instance, the attacker cannot decrypt a ciphertext without the correct key.",
"Moreover, the adversary does not have the ability to perform side-channel attacks.",
"For instance, fingerprinting a doctor based on his prescriptions is beyond the scope of this attacker model."
],
[
"Prescription privacy",
"Prescription privacy ensures unlinkability of a doctor and his prescriptions, i.e., the adversary cannot tell whether a prescription is prescribed by a doctor.",
"This requirement helps to prevent doctors from being influenced in the prescriptions they issue.",
"Normally, prescriptions are eventually revealed to the general public, for example, for research purposes.",
"In the DLV08 e-health protocol, prescriptions are revealed to the adversary observing the network.",
"Therefore, in the extreme situation where there is only one doctor, the doctor's prescriptions are obviously revealed to the adversary - all the observed prescriptions belong to the doctor.",
"To avoid such a case, prescription privacy requires at least one other doctor (referred to as the counter-balancing doctor).",
"This ensures that the adversary cannot tell whether the observed prescriptions belong to the targetted doctor or the counter-balancing doctor.",
"With this in mind, unlinkability of a doctor to a prescription is modelled as indistinguishability between two honest users that swap their prescriptions, analogously to the formalisation of vote-privacy [29].",
"By adopting the vote-privacy formalisation, prescription privacy is thus modelled as the equivalence of two doctor processes: in the first process, an honest doctor ${\\tt {d_A}}$ prescribes ${\\tt {p_A}}$ in one of his sessions and another honest doctor ${\\tt {d_B}}$ prescribes ${\\tt {p_B}}$ in one of his sessions; in the second one, ${\\tt {d_A}}$ prescribes ${\\tt {p_B}}$ and ${\\tt {d_B}}$ prescribes ${\\tt {p_A}}$ .",
"Definition 3 (prescription privacy) A well-formed e-health protocol $\\mathit {\\mathit {P}_{\\!eh}}$ with a doctor role $\\mathit {R}_{\\mathit {dr}}$ , satisfies prescription privacy if for all possible doctors ${\\tt {d_A}}$ and ${\\tt {d_B}}$ (${\\tt {d_A}}\\ne {\\tt {d_B}}$ ) we have $\\begin{array}{rl}&\\mathcal {C}_\\mathit {eh}[\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )\\mid \\\\&\\hspace{17.0pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )]\\\\\\approx _{\\ell }&\\mathcal {C}_\\mathit {eh}[\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )\\mid \\\\&\\hspace{17.0pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )],\\end{array}$ where ${\\tt {p_A}}$ and ${\\tt {p_B}}$ (${\\tt {p_A}}\\ne {\\tt {p_B}}$ ) are any two possible prescriptions, process $!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }$ and process $!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }$ can be 0.",
"Process $\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })$ models an instance of a doctor, with identity ${\\tt {d_A}}$ .",
"The sub-process $\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })$ models a prescribing session in which ${\\tt {d_A}}$ prescribes ${\\tt {p_A}}$ for a patient.",
"The sub-process $!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }$ models other prescribing sessions of ${\\tt {d_A}}$ .",
"Similarly, process $\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })$ models another doctor ${\\tt {d_B}}$ .",
"On the right-hand side of the equivalence, the two doctors, ${\\tt {d_A}}$ and ${\\tt {d_B}}$ , swap their prescriptions, ${\\tt {p_A}}$ and ${\\tt {p_B}}$ .",
"The labelled bisimilarity ($\\approx _{\\ell }$ ) captures that any dishonest third party (the adversary) cannot distinguish the two sides.",
"Doctor ${\\tt {d_B}}$ 's process is called the counter-balancing process.",
"We require the existence of the counter-balancing doctor ${\\tt {d_B}}$ and ${\\tt {p_A}}\\ne {\\tt {p_B}}$ to avoid the situation in which all patients prescribe the same prescription, and thus the prescription of all patients are simply revealed.",
"Note that ${\\mathit {Id_{\\mathit {dr}}}}$ and $\\mathit {presc}$ are free names in the processes in the definition.",
"${\\tt {d_A}}$ and ${\\tt {d_B}}$ are free names in the processes, when the doctor identities are initially public, and are private names in the processes, when the doctor identities are initially private.",
"Similarly, ${\\tt {p_A}}$ and ${\\tt {p_B}}$ are free names in the processes, when the prescriptions are revealed, and are private names in the processes, when the prescriptions are kept secret.",
"This holds for the following definitions as well."
],
[
"Receipt-freeness",
"Enforced privacy properties have been formally defined in e-voting and e-auctions.",
"Examples include receipt-freeness and coercion-resistance in e-voting [29], [34], and receipt-freeness for non-winning bidders in e-auctions [25].",
"De Decker et al.",
"[24] identify the need to prevent a pharmaceutical company from bribing a doctor to favour their medicine.",
"Hence, a doctor's prescription privacy must be enforced by the e-health system to prevent doctor bribery.",
"This means that intuitively, even if a doctor collaborates, the adversary cannot be certain that the doctor has followed his instructions.",
"Bribed users are not modelled as part of the adversary, as they may lie and are thus not trusted by the adversary.",
"Due to the domain differences – in e-voting and sealed-bid e-auctions, participants are fixed before the execution, whereas in e-health, participants may be infinitely involving; in e-voting and sealed-bid e-auctions, each participant executes the protocol exactly once, whereas in e-health, a participant may involve multiple/infinite times.",
"Thus, the formalisation in e-voting and sealed-bid e-auctions cannot be adopted.",
"Inspired by formalisations of receipt-freeness in e-voting [29] and e-auction [25], we define receipt-freeness to be satisfied if there exists a process where the bribed doctor does not follow the adversary's instruction (e.g., prescribing a particular medicine), which is indistinguishable from a process where she does.",
"Modelling this property necessitates modelling a doctor who genuinely reveals all her private information to the adversary.",
"This is achieved by the process transformation $P^{{\\tt {chc}}}$ by Delaune et al. [29].",
"This operation transforms a plain process $P$ into one which shares all private information over the channel ${\\tt {chc}}$ with the adversary.",
"The transformation $P^{{{\\tt {chc}}}}$ is defined as follows: Let $\\mathit {P}$ be a plain process and ${{\\tt {chc}}}$ a fresh channel name.",
"$P^{{{\\tt {chc}}}}$ , the process that shares all of $P$ 's secrets, is defined as: $0^{{{\\tt {chc}}}}$ $\\mathit {\\ \\hat{=}\\ }0$ , $(P\\mid Q)^{{{\\tt {chc}}}}$ $\\mathit {\\ \\hat{=}\\ }P^{{{\\tt {chc}}}}\\mid Q^{{{\\tt {chc}}}}$ , $(\\nu {\\tt {n}}.P)^{{{\\tt {chc}}}}$ $\\mathit {\\ \\hat{=}\\ }\\left\\lbrace \\begin{array}{lr}\\nu {\\tt {n}}.",
"{\\sf {out}}({{\\tt {chc}}}, {\\tt {n}}).P^{{{\\tt {chc}}}} &\\qquad \\qquad \\text{when ${\\tt {n}}$ is a name of base type,} \\\\\\nu {\\tt {n}}.P^{{{\\tt {chc}}}} &\\quad \\text{otherwise,}\\end{array}\\right.$ $({\\sf {in}}(v, x).P)^{{{\\tt {chc}}}}$ $\\mathit {\\ \\hat{=}\\ }\\left\\lbrace \\begin{array}{lr}{\\sf {in}}(v, x).",
"{\\sf {out}}({{\\tt {chc}}}, x).P^{{{\\tt {chc}}}} &\\qquad \\text{when $x$ is a variable of base type,} \\\\{\\sf {in}}(v, x).P^{{{\\tt {chc}}}} & \\text{otherwise,}\\end{array}\\right.$ $({\\sf {out}}(v, M).P)^{{{\\tt {chc}}}}$ $\\mathit {\\ \\hat{=}\\ }{\\sf {out}}(v, M).P^{{{\\tt {chc}}}}$ , $(!P)^{{{\\tt {chc}}}}$ $\\mathit {\\ \\hat{=}\\ }!P^{{{\\tt {chc}}}}$ , $(\\mbox{if}\\ \\ M=_EN \\ \\mbox{then} \\ P \\ \\mbox{else} \\ Q)^{{{\\tt {chc}}}}$ $\\mathit {\\ \\hat{=}\\ }\\mbox{if}\\ \\ M=_EN \\ \\mbox{then} \\ P^{{{\\tt {chc}}}} \\ \\mbox{else} \\ Q^{{{\\tt {chc}}}}$ .",
"In addition, we also use the transformation $P^{\\backslash {\\sf {out}}({\\tt {chc}}, \\cdot )}$ [29].",
"This models a process $P$ which hides all outputs on channel ${\\tt {chc}}$ .",
"Formally, $P^{\\backslash {\\sf {out}}({\\tt {chc}}, \\cdot )}:=\\nu {\\tt {chc}}.",
"(P \\mid \\, !",
"{\\sf {in}}({\\tt {chc}}, x))$ .",
"Definition 4 (receipt-freeness) A well-formed e-health protocol $\\mathit {\\mathit {P}_{\\!eh}}$ with a doctor role $\\mathit {R}_{\\mathit {dr}}$ , satisfies receipt-freeness if for any two doctors ${\\tt {d_A}}$ and ${\\tt {d_B}}$ (${\\tt {d_A}}\\ne {\\tt {d_B}}$ ) and any two possible prescriptions ${\\tt {p_A}}$ and ${\\tt {p_B}}$ (${\\tt {p_A}}\\ne {\\tt {p_B}}$ ), there exist processes $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ , such that: $\\begin{array}{lrl}1.\\ &&\\mathcal {C}_\\mathit {eh}[\\big (\\mathit {init}_{\\mathit {dr}}^{\\prime }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {P}_{\\mathit {dr}}^{\\prime })\\big )\\mid \\\\&&\\hspace{16.57497pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.(!",
"\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )]\\\\&\\approx _{\\ell }&\\mathcal {C}_\\mathit {eh}[\\big ((\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace })^{{\\tt {chc}}}.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid (\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })^{{\\tt {chc}}})\\big )\\mid \\\\&&\\hspace{16.57497pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )]; \\vspace{5.69054pt}\\\\2.\\ &&\\mathcal {C}_\\mathit {eh}[\\big ((\\mathit {init}_{\\mathit {dr}}^{\\prime }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {P}_{\\mathit {dr}}^{\\prime }))^{\\backslash {\\sf {out}}({\\tt {chc}}, \\cdot )}\\big )\\mid \\\\&&\\hspace{16.57497pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.(!",
"\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )]\\\\&\\approx _{\\ell }&\\mathcal {C}_\\mathit {eh}[\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )\\mid \\\\&&\\hspace{16.57497pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.(!",
"\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )],\\end{array}$ where $\\mathit {init}_{\\mathit {dr}}^{\\prime }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {P}_{\\mathit {dr}}^{\\prime })$ is a closed plain process, ${\\tt {chc}}$ is a free fresh channel name, process $!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }$ and $!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }$ can be 0.",
"In the definition, the sub-process $!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }$ models the sessions of ${\\tt {d_A}}$ that are not bribed, and the sub-processes $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ model the process in which the doctor ${\\tt {d_A}}$ lies to the adversary about one of his prescriptions.",
"The real prescription behaviour of ${\\tt {d_A}}$ is modelled by the second equivalence.",
"The first equivalence shows that the adversary cannot distinguish whether ${\\tt {d_A}}$ lied, given a counter-balancing doctor ${\\tt {d_B}}$ ."
],
[
"Remark",
"Receipt-freeness is stronger than prescription privacy (cf.",
"Figure REF ).",
"Intuitively, this is true since receipt-freeness is like prescription privacy except that the adversary may gain more knowledge.",
"Thus, if a protocol satisfies receipt-freeness (the adversary cannot break privacy with more knowledge), prescription privacy must also be satisfied (the adversary cannot break privacy with less knowledge).",
"We prove this formally, following the proof that receipt-freeness is stronger than vote-privacy in [4].",
"We prove that by applying an evaluation context that hides the channel ${\\tt {chc}}$ on both sides of the first equivalence in Definition REF , we can obtain Definition REF .",
"If a protocol satisfies receipt-freeness, there exists a closed plain process $\\mathit {init}_{\\mathit {dr}}^{\\prime }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {P}_{\\mathit {dr}}^{\\prime })$ such that the two equations in Definition REF are satisfied.",
"By applying the evaluation context $\\nu {\\tt {chc}}.",
"(\\_\\mid !",
"{\\sf {in}}({\\tt {chc}}, x))$ (defined as $P^{\\backslash {\\sf {out}}({\\tt {chc}}, \\cdot )}$ in Section REF ) on both sides of the first equation, we obtain $\\begin{array}{lrl}&&\\mathcal {C}_\\mathit {eh}[\\big (\\mathit {init}_{\\mathit {dr}}^{\\prime }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {P}_{\\mathit {dr}}^{\\prime })\\big )\\mid \\\\&&\\hspace{16.57497pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.(!",
"\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )]^{\\backslash {\\sf {out}}({\\tt {chc}}, \\cdot )}\\\\&\\approx _{\\ell }&\\mathcal {C}_\\mathit {eh}[\\big ((\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace })^{{\\tt {chc}}}.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid (\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })^{{\\tt {chc}}})\\big )\\mid \\\\&&\\hspace{16.57497pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )]^{\\backslash {\\sf {out}}({\\tt {chc}}, \\cdot )}\\end{array}$ Lemma 1 [4]: Let $\\mathit {\\mathcal {C}}_1=\\nu \\tilde{a_1}.",
"(\\_\\mid B_1)$ and $\\mathit {\\mathcal {C}}_2=\\nu \\tilde{a_2}.",
"(\\_\\mid B_2)$ be two evaluation contexts such that $\\tilde{a_1}\\cap ({\\sf {fv}}(B_2)\\cup {\\sf {fn}}(B_2))=\\emptyset $ and $\\tilde{a_2}\\cap ({\\sf {fv}}(B_1)\\cup {\\sf {fn}}(B_1))=\\emptyset $ .",
"We have that $\\mathit {\\mathcal {C}}_1[\\mathit {\\mathcal {C}}_2[A]]\\equiv \\mathit {\\mathcal {C}}_2[\\mathit {\\mathcal {C}}_1[A]]$ for any extended process $A$ .",
"Using Lemma 1, we can rewrite the left-hand side (1) and the right-hand side (2) of the equivalence as follows.",
"$\\begin{array}{lrl}(1) &&\\mathcal {C}_\\mathit {eh}[\\big (\\mathit {init}_{\\mathit {dr}}^{\\prime }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {P}_{\\mathit {dr}}^{\\prime })\\big )\\mid \\\\&&\\hspace{16.57497pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.(!",
"\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )]^{\\backslash {\\sf {out}}({\\tt {chc}}, \\cdot )}\\\\&\\equiv &\\mathcal {C}_\\mathit {eh}[\\big (\\mathit {init}_{\\mathit {dr}}^{\\prime }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {P}_{\\mathit {dr}}^{\\prime })\\big )^{\\backslash {\\sf {out}}({\\tt {chc}}, \\cdot )}\\mid \\\\&&\\hspace{16.57497pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.(!",
"\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )]\\\\~\\\\(2)&&\\mathcal {C}_\\mathit {eh}[\\big ((\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace })^{{\\tt {chc}}}.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid (\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })^{{\\tt {chc}}})\\big )\\mid \\\\&&\\hspace{16.57497pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )]^{\\backslash {\\sf {out}}({\\tt {chc}}, \\cdot )}\\\\&\\equiv &\\mathcal {C}_\\mathit {eh}[\\big ((\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace })^{{\\tt {chc}}}.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid (\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })^{{\\tt {chc}}})\\big )^{\\backslash {\\sf {out}}({\\tt {chc}}, \\cdot )}\\mid \\\\&&\\hspace{16.57497pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )]\\end{array}$ For equation $(1)$ , by the second equation in Definition REF , we have $\\begin{array}{lrl}&&\\mathcal {C}_\\mathit {eh}[\\big (\\mathit {init}_{\\mathit {dr}}^{\\prime }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {P}_{\\mathit {dr}}^{\\prime })\\big )\\mid \\\\&&\\hspace{16.57497pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.(!",
"\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )]^{\\backslash {\\sf {out}}({\\tt {chc}}, \\cdot )}\\\\&\\approx _{\\ell }&\\mathcal {C}_\\mathit {eh}[\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )\\mid \\\\&&\\hspace{16.57497pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.(!",
"\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )]\\end{array}$ Lemma 2 [4]: let $P$ be a closed plain process and ${\\tt {chc}}$ a channel name such that ${\\tt {chc}}\\notin {\\sf {fn}}(P)\\cup {\\sf {bn}}(P)$ .",
"We have $(P^{{\\tt {chc}}})^{\\backslash {\\sf {out}}({\\tt {chc}}, \\cdot )}\\approx _{\\ell }P$ .",
"For equation $(2)$ , using Lemma 2, we obtain that $\\begin{array}{lrl}&&\\mathcal {C}_\\mathit {eh}[\\big ((\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace })^{{\\tt {chc}}}.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid (\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })^{{\\tt {chc}}})\\big )^{\\backslash {\\sf {out}}({\\tt {chc}}, \\cdot )}\\mid \\\\&&\\hspace{16.57497pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )]\\\\&\\approx _{\\ell }&\\mathcal {C}_\\mathit {eh}[\\big ((\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }).",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid (\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace }))\\big )\\mid \\\\&&\\hspace{16.57497pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )]\\end{array}$ By transitivity, we have $\\begin{array}{lrl}&&\\mathcal {C}_\\mathit {eh}[\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )\\mid \\\\&&\\hspace{16.57497pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.(!",
"\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )]\\\\&\\approx _{\\ell }&\\mathcal {C}_\\mathit {eh}[\\big ((\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }).",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid (\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace }))\\big )\\mid \\\\&&\\hspace{16.57497pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )],\\end{array}$ which is exactly Definition REF .$\\Box $ The difference between this formalisation and receipt-freeness in e-voting [29] and in e-auctions [25] is that in this definition only a part of the doctor process (the initiation sub-process and a prescribing session) shares information with the adversary.",
"In e-voting, each voter only votes once.",
"In the contrast, a doctor prescribes multiple times for various patients.",
"As patients and situations of patients vary, a doctor cannot prescribe medicine from the bribing pharmaceutical company all the time.",
"Therefore, only part of the doctor process shares information with the adversary.",
"Note that we model only one bribed prescribing session, as it is the simplest scenario.",
"This definition can be extended to model multiple prescribing sessions being bribed, by replacing sub-process $(\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace })$ with the sub-process modelling multiple doctor sessions.",
"Note that the extended definition requires multiple sessions of the counter-balancing doctor or multiple counter-balancing doctors.",
"Assume $h$ sessions of ${\\tt {d_A}}$ are bribed, denoted as $\\mathit {\\mathit {main}^b_{\\mathit {dr}}}:=\\mathit {\\mathit {main}^1_{\\mathit {dr}}}\\mid \\ldots \\mid \\mathit {\\mathit {main}^h_{\\mathit {dr}}}$ , where $\\mathit {\\mathit {main}^i_{\\mathit {dr}}}:=(\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace })^{{\\tt {chc}}}$ .",
"An arbitrary instance of the bribed sessions is denoted as $\\mathit {\\mathit {main}^b_{\\mathit {dr}}}\\mathit {\\lbrace ({\\tt {p^1_A}}, \\ldots , {\\tt {p^h_A}})/\\mathit {presc}\\rbrace }:=\\mathit {\\mathit {main}^1_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {p^1_A}}/\\mathit {presc}\\rbrace }\\mid \\ldots \\mid \\mathit {\\mathit {main}^h_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {p^h_A}}/\\mathit {presc}\\rbrace }$ .",
"Assume there is a counter-balancing process $\\mathit {\\mathit {P}^c_{\\mathit {dr}}}$ of the bribed sessions.",
"The process $\\mathit {\\mathit {P}^c_{\\mathit {dr}}}$ has $h$ corresponding sessions from one or more honest doctors.",
"We use $\\mathit {\\mathit {P}^c_{\\mathit {dr}}}\\mathit {\\lbrace ({\\tt {p^1_A}}, \\ldots , {\\tt {p^h_A}})/\\mathit {presc}\\rbrace }$ to denote the counter-balancing process where the prescriptions of the $h$ sessions are ${\\tt {p^1_A}}, \\ldots , {\\tt {p^h_A}}$ , respectively.",
"Following definition REF , the multi-session receipt-freeness can be defined as follows.",
"Definition 5 (multi-session receipt-freeness) A well-formed e-health protocol $\\mathit {\\mathit {P}_{\\!eh}}$ with a doctor role $\\mathit {R}_{\\mathit {dr}}$ , satisfies multi-session receipt-freeness, if for any doctor ${\\tt {d_A}}$ with $h$ bribed sessions, denoted as $\\mathit {\\mathit {main}^b_{\\mathit {dr}}}:=\\mathit {\\mathit {main}^1_{\\mathit {dr}}}\\mid \\ldots \\mid \\mathit {\\mathit {main}^h_{\\mathit {dr}}}$ , where $\\mathit {\\mathit {main}^i_{\\mathit {dr}}}:=(\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace })^{{\\tt {chc}}}$ , for any instantiation of the prescriptions in the bribed sessions $\\mathit {\\mathit {main}^b_{\\mathit {dr}}}\\mathit {\\lbrace ({\\tt {p^1_A}}, \\ldots , {\\tt {p^h_A}})/\\mathit {presc}\\rbrace }$ , there exist processes $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ , such that $\\begin{array}{lrl}1.\\ &&\\mathcal {C}_\\mathit {eh}[\\big (\\mathit {init}_{\\mathit {dr}}^{\\prime }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {P}_{\\mathit {dr}}^{\\prime })\\big ) \\mid \\mathit {\\mathit {P}^c_{\\mathit {dr}}}\\mathit {\\lbrace ({\\tt {p^1_A}}, \\ldots , {\\tt {p^h_A}})/\\mathit {presc}\\rbrace }]\\\\&\\approx _{\\ell }&\\mathcal {C}_\\mathit {eh}[\\big ((\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace })^{{\\tt {chc}}}.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}^b_{\\mathit {dr}}}\\mathit {\\lbrace ({\\tt {p^1_A}}\\ldots , {\\tt {p^h_A}})/\\mathit {presc}\\rbrace }\\big )\\mid \\\\&&\\hspace{16.57497pt}\\mathit {\\mathit {P}^c_{\\mathit {dr}}}\\mathit {\\lbrace ({\\tt {p^1_B}}, \\ldots , {\\tt {p^h_B}})/\\mathit {presc}\\rbrace }]; \\vspace{5.69054pt}\\\\2.\\ &&\\mathcal {C}_\\mathit {eh}[\\big ((\\mathit {init}_{\\mathit {dr}}^{\\prime }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {P}_{\\mathit {dr}}^{\\prime }))^{\\backslash {\\sf {out}}({\\tt {chc}}, \\cdot )}\\big )\\mid \\mathit {\\mathit {P}^c_{\\mathit {dr}}}\\mathit {\\lbrace ({\\tt {p^1_A}}, \\ldots , {\\tt {p^h_A}})/\\mathit {presc}\\rbrace }]\\\\&\\approx _{\\ell }&\\mathcal {C}_\\mathit {eh}[\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p^1_B}}/\\mathit {presc}\\rbrace }\\mid \\ldots \\mid \\\\&&\\hspace{16.57497pt}\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p^h_B}}/\\mathit {presc}\\rbrace })\\big )\\mid \\mathit {\\mathit {P}^c_{\\mathit {dr}}}\\mathit {\\lbrace ({\\tt {p^1_A}}, \\ldots , {\\tt {p^h_A}})/\\mathit {presc}\\rbrace }],\\end{array}$ where $\\mathit {init}_{\\mathit {dr}}^{\\prime }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {P}_{\\mathit {dr}}^{\\prime })$ is a closed plain process, ${\\tt {chc}}$ is a free fresh channel name, process $!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }$ and $!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }$ can be 0, $\\mathit {\\mathit {P}^c_{\\mathit {dr}}}$ is less than $h$ doctor processes running in parallel and $\\mathit {\\mathit {P}^c_{\\mathit {dr}}}\\mathit {\\lbrace ({\\tt {p^1_A}}, \\ldots , {\\tt {p^h_A}})/\\mathit {presc}\\rbrace }$ denotes that in some sessions of the doctor processes, the prescriptions are instantiated with ${\\tt {p^1_A}}, \\ldots , {\\tt {p^h_A}}$ .",
"Definition REF (receipt-freeness) is a specific instance of this definition where only one session of the targeted doctor is bribed.",
"When multiple sessions are bribed, to ensure multi-session receipt-freeness, it requires the existence of more than one counter-balancing doctor sessions, and thus this extended definition is stronger than receipt-freeness (Definition REF ), meaning that if a protocol satisfied the multi-session receipt-freeness where multiple sessions are bribed, then the protocol satisfies receipt-freeness where only one session is bribed.",
"The intuition is that if there exists a lying process for multiple bribed sessions such that multi-session receipt-freeness is satisfied, by hiding the communications to the adversary in the lying process except one session, we can obtain the lying process such that receipt-freeness is satisfied.",
"More concretely, when receipt-freeness is satisfied, multi-session receipt-freeness may not be satisfied.",
"For example, when there are exactly two users and two nonces generated by each user, if the revealed information is only the two nonces, a bribed user can lie to the adversary about his nonce, since the link between the user and the nonce is private.",
"However, if the user is bribed on two sessions, i.e., the link between him and his two nonces, then at least one nonce has to be generated by the user, hence, multi-session receipt-freeness (two sessions in particular) is not satisfied.",
"Remark that we restrict the way a bribed user collaborates with the adversary: we only model forwarding information to the adversary.",
"The scenario that the adversary provides prescriptions for a bribed doctor, similar to in coercion in e-voting [29], is not modelled.",
"Although providing ready-made prescriptions is theoretically possible, we consider this to not be a practical attack: correctly prescribing requires professional (sometimes empirical) medical expertise and heavily depends on examination of the patient.",
"As such, no adversary can prepare an appropriate prescription without additional information.",
"Moreover, forwarding a non-appropriate description carries serious legal consequences for the forwarding doctor.",
"Therefore, we omit the case where the doctor merely forwards an adversary-prepared prescription.",
"The adversary could still prepare other information for the bribed doctor, for example, the randomness of a bit-commitment.",
"Such adversary-prepared information may lead to a stronger adversary than we are considering.",
"To model such a scenario, the verifier needs to specify exactly which information is prepared by the adversary.",
"Formalisation of such scenarios can follow the formal framework proposed in [29], [28]."
],
[
"Independency of prescription privacy",
"Usually, e-health systems have to deal with a complex constellation of roles: doctors, patients, pharmacists, insurance companies, medical administration, etc.",
"Each of these roles has access to different private information and has different privacy concerns.",
"An untrusted role may be bribed to reveal private information to the adversary such that the adversary can break the privacy of another role.",
"De Decker et al.",
"[24] note that pharmacists may have sensitive data which can be revealed to the adversary to break a doctor's prescription privacy.",
"To prevent a party from revealing sensitive data that affects a doctor's privacy, e-health protocols are required to satisfy independency of prescription privacy.",
"The DLV08 protocol, for example, requires prescription privacy independent of pharmacists [24].",
"Intuitively, independency of prescription privacy means that even if another party $\\mathit {R}_i$ reveals their information (i.e., $\\mathit {R}_i^{{\\tt {chc}}}$ ), the adversary is not able to break a doctor's prescription privacy.",
"Definition 6 (independency of prescription privacy) A well-formed e-health protocol $\\mathit {\\mathit {P}_{\\!eh}}$ with a doctor role $\\mathit {R}_{\\mathit {dr}}$ , satisfies prescription privacy independent of role $\\mathit {R}_i$, if for all possible doctors ${\\tt {d_A}}$ and ${\\tt {d_B}}$ (${\\tt {d_A}}\\ne {\\tt {d_B}}$ ) we have $\\begin{array}{lll}&\\mathcal {C}_\\mathit {eh}[!",
"{\\mathit {R}_{i}}^{{\\tt {chc}}}\\mid &\\hspace{-8.5pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )\\mid \\\\&&\\hspace{-8.5pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )]\\\\\\approx _{\\ell }&\\mathcal {C}_\\mathit {eh}[!",
"{\\mathit {R}_{i}}^{{\\tt {chc}}}\\mid &\\hspace{-8.5pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )\\mid \\\\&&\\hspace{-8.5pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )].\\end{array}$ where ${\\tt {p_A}}$ and ${\\tt {p_B}}$ (${\\tt {p_A}}\\ne {\\tt {p_B}}$ ) are any two possible prescriptions, $\\mathit {R}_i$ is a non-doctor role, process $!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }$ and $!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }$ can be 0.",
"Note that we assume a worst-case situation in which role $\\mathit {R}_i$ genuinely cooperates with the adversary.",
"For example, the pharmacist forwards all information obtained from channels hidden from the adversary.",
"The equivalence requires that no matter how role $\\mathit {R}_i$ cooperates with the adversary, the adversary cannot link a doctor to the doctor's prescriptions.",
"The cooperation between pharmacists and the adversary is modelled in the same way as the cooperation between bribed doctors and the adversary, i.e., $!",
"{\\mathit {R}_{i}}^{{\\tt {chc}}}$ .",
"We do not model the situation where the adversary prepares information for the pharmacists, as we focus on doctor privacy – information sent out by the pharmacist does not affect doctor privacy, so there is no reason to control this information.",
"Instead of modelling the pharmacists as compromised users, our modelling allows the definition to be easily extended to model new properties which capture situations where pharmacists lie to the adversary due to, for example, coalition between pharmacists and bribed doctors.",
"In addition, although we do not model delivery of medicine, pharmacists do need to adhere to regulations in providing medicine.",
"Thus, an adversary who only controls the network cannot impersonate a pharmacist.",
"Just as receipt-freeness is stronger than prescription privacy, independency of prescription privacy is stronger than prescription privacy (cf.",
"Figure REF ).",
"Intuitively, this holds since the adversary obtains at least as much information in independency of prescription privacy as in prescription privacy.",
"Formally, one can derive Definition REF from Definition REF by hiding channel ${\\tt {chc}}$ on the left-hand side as well as the right-hand side of the equivalence in Definition REF .",
"Consider a protocol that satisfies independency of prescription privacy.",
"This protocol thus satisfies definition REF .",
"By applying the evaluation context $\\nu {\\tt {chc}}.",
"(\\_\\mid !",
"{\\sf {in}}({\\tt {chc}}, x))$ to both the left-hand side and the right-hand side of Definition REF , we obtain $\\begin{array}{lll}&\\mathcal {C}_\\mathit {eh}[!",
"{\\mathit {R}_{i}}^{{\\tt {chc}}}\\mid &\\hspace{-8.5pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )\\mid \\\\&&\\hspace{-8.5pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )]^{\\backslash {\\sf {out}}({\\tt {chc}}, \\cdot )}\\\\\\approx _{\\ell }&\\mathcal {C}_\\mathit {eh}[!",
"{\\mathit {R}_{i}}^{{\\tt {chc}}}\\mid &\\hspace{-8.5pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )\\mid \\\\&&\\hspace{-8.5pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )]^{\\backslash {\\sf {out}}({\\tt {chc}}, \\cdot )}.\\end{array}$ According to Lemma 1, we have $\\begin{array}{lll}&\\mathcal {C}_\\mathit {eh}[!",
"{\\mathit {R}_{i}}^{{\\tt {chc}}}\\mid &\\hspace{-8.5pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )\\mid \\\\&&\\hspace{-8.5pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )]^{\\backslash {\\sf {out}}({\\tt {chc}}, \\cdot )}\\\\\\equiv &\\mathcal {C}_\\mathit {eh}[!",
"{{\\mathit {R}_{i}}^{{\\tt {chc}}}}^{\\backslash {\\sf {out}}({\\tt {chc}}, \\cdot )}\\mid &\\hspace{-8.5pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )\\mid \\\\&&\\hspace{-8.5pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )]\\end{array}$ $\\begin{array}{lll}&\\mathcal {C}_\\mathit {eh}[!",
"{\\mathit {R}_{i}}^{{\\tt {chc}}}\\mid &\\hspace{-8.5pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )\\mid \\\\&&\\hspace{-8.5pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )]^{\\backslash {\\sf {out}}({\\tt {chc}}, \\cdot )}\\\\\\equiv &\\mathcal {C}_\\mathit {eh}[!",
"{{\\mathit {R}_{i}}^{{\\tt {chc}}}}^{\\backslash {\\sf {out}}({\\tt {chc}}, \\cdot )}\\mid &\\hspace{-8.5pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )\\mid \\\\&&\\hspace{-8.5pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )]\\end{array}$ According to Lemma 2, we have $\\begin{array}{lll}& \\mathcal {C}_\\mathit {eh}[!",
"{{\\mathit {R}_{i}}^{{\\tt {chc}}}}^{\\backslash {\\sf {out}}({\\tt {chc}}, \\cdot )}\\mid &\\hspace{-8.5pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )\\mid \\\\&&\\hspace{-8.5pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )]\\\\\\approx _{\\ell }&\\mathcal {C}_\\mathit {eh}[!\\mathit {R}_{i}\\mid &\\hspace{-8.5pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )\\mid \\\\&&\\hspace{-8.5pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )]\\end{array}$ $\\begin{array}{lll}&\\mathcal {C}_\\mathit {eh}[!",
"{{\\mathit {R}_{i}}^{{\\tt {chc}}}}^{\\backslash {\\sf {out}}({\\tt {chc}}, \\cdot )}\\mid &\\hspace{-8.5pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )\\mid \\\\&&\\hspace{-8.5pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )]\\\\\\approx _{\\ell }&\\mathcal {C}_\\mathit {eh}[!\\mathit {R}_{i}\\mid &\\hspace{-8.5pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )\\mid \\\\&&\\hspace{-8.5pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )]\\end{array}$ Therefore, by transitivity, we have $\\begin{array}{lll}&\\mathcal {C}_\\mathit {eh}[!\\mathit {R}_{i}\\mid &\\hspace{-8.5pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )\\mid \\\\&&\\hspace{-8.5pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )]\\\\\\approx _{\\ell }&\\mathcal {C}_\\mathit {eh}[!\\mathit {R}_{i}\\mid &\\hspace{-8.5pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )\\mid \\\\&&\\hspace{-8.5pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )]\\end{array}$ which is exactly Definition REF .",
"$\\Box $ Note that the first step in the proof (application of an evaluation context) cannot be reversed.",
"Therefore, prescription privacy is weaker than independency of prescription privacy."
],
[
"Independency of receipt-freeness",
"We have discussed two situations where a doctor's prescription behaviour can be revealed when either the doctor or another different party cooperates with the adversary.",
"It is natural to consider the conjunction of these two, i.e., a situation in which the adversary coerces both a doctor and another party (not a doctor).",
"Since the adversary obtains more information, this constitutes a stronger attack on doctor's prescription privacy.",
"To address this problem, we define independency of receipt-freeness, which is satisfied when a doctor's prescription privacy is preserved even if both the doctor and another party reveal their private information to the adversary.",
"Definition 7 (independency of receipt-freeness) A well-formed e-health protocol $\\mathit {\\mathit {P}_{\\!eh}}$ with a doctor role $\\mathit {R}_{\\mathit {dr}}$ , satisfies receipt-freeness independent of role $\\mathit {R}_i$ if for any two doctors ${\\tt {d_A}}$ and ${\\tt {d_B}}$ (${\\tt {d_A}}\\ne {\\tt {d_B}}$ ) and any two possible prescriptions ${\\tt {p_A}}$ and ${\\tt {p_B}}$ (${\\tt {p_A}}\\ne {\\tt {p_B}}$ ), there exist processes $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ , such that: $\\begin{array}{lrl}1.\\ &&\\mathcal {C}_\\mathit {eh}[!\\mathit {R}_{i}^{{\\tt {chc}}}\\mid \\big (\\mathit {init}_{\\mathit {dr}}^{\\prime }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {P}_{\\mathit {dr}}^{\\prime })\\big )\\mid \\\\&&\\hspace{46.75pt} \\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )]\\\\&\\approx _{\\ell }&\\mathcal {C}_\\mathit {eh}[!\\mathit {R}_{i}^{{\\tt {chc}}}\\mid \\big ((\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace })^{{\\tt {chc}}}.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid (\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })^{{\\tt {chc}}})\\big )\\mid \\\\&&\\hspace{46.75pt} \\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )]; \\vspace{5.69054pt}\\\\2.\\ &&\\mathcal {C}_\\mathit {eh}[!\\mathit {R}_{i}^{{\\tt {chc}}}\\mid \\big ((\\mathit {init}_{\\mathit {dr}}^{\\prime }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {P}_{\\mathit {dr}}^{\\prime }))^{\\backslash {\\sf {out}}({\\tt {chc}}, \\cdot )}\\big )\\mid \\\\&&\\hspace{46.75pt} \\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )]\\\\& \\approx _{\\ell }&\\mathcal {C}_\\mathit {eh}[!\\mathit {R}_{i}^{{\\tt {chc}}}\\mid \\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )\\mid \\\\&&\\hspace{46.75pt} \\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )],\\end{array}$ where $\\mathit {init}_{\\mathit {dr}}^{\\prime }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {P}_{\\mathit {dr}}^{\\prime })$ is a closed plain process, $\\mathit {R}_i$ is a non-doctor role, ${\\tt {chc}}$ is a free fresh channel name, process $!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }$ and $!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }$ can be 0.",
"Independency of receipt-freeness implies receipt-freeness and independency of prescription privacy, each of which also implies prescription privacy (cf.",
"Figure REF ).",
"The proof follows the same reasoning as the proofs in [28].",
"Intuitively, the adversary obtains more information with independency of receipt-freeness (namely, from both doctor and pharmacist) than with either independency of prescription privacy (from pharmacist only) or receipt-freeness (from doctor only).",
"If the adversary is unable to break a doctor's privacy using this much information, the adversary will not be able to break doctor privacy using less information.",
"Therefore, if a protocol satisfies independency of receipt-freeness, then it must also satisfies independency of prescription privacy and receipt-freeness.",
"Similarly, since the adversary obtains more information in both independency of prescription privacy and in receipt-freeness than in prescription privacy, if a protocol satisfies either independency of prescription privacy or receipt-freeness, it must also satisfies prescription privacy."
],
[
"Anonymity and strong anonymity",
"Anonymity is a privacy property that protects users' identities.",
"We model anonymity as indistinguishability of processes initiated by two different users.",
"Definition 8 (doctor anonymity) A well-formed e-health protocol $\\mathit {\\mathit {P}_{\\!eh}}$ with a doctor role $\\mathit {R}_{\\mathit {dr}}$ satisfies doctor anonymity if for any doctor ${\\tt {d_A}}$ , there exists another doctor ${\\tt {d_B}}$ (${\\tt {d_B}}\\ne {\\tt {d_A}}$ ), such that $\\begin{array}{rl}\\mathcal {C}_\\mathit {eh}[\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }]\\approx _{\\ell }\\mathcal {C}_\\mathit {eh}[\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }].\\end{array}$ A stronger property of anonymity is defined in [4], capturing the situation that the adversary cannot even find out whether a user (with identity ${\\tt {d_A}}$ ) has participated in a session of the protocol or not.",
"Definition 9 (strong doctor anonymity [4]) A well-formed e-health protocol $\\mathit {\\mathit {P}_{\\!eh}}$ with a doctor role $\\mathit {R}_{\\mathit {dr}}$ satisfies strong doctor anonymity, if $\\mathit {\\mathit {P}_{\\!eh}}\\approx _{\\ell }\\nu \\mathit {\\widetilde{mc}}.",
"\\mathit {init}.\\big (!\\mathit {R}_{1}\\mid \\ldots \\mid !\\mathit {R}_{n}\\mid (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace })\\big ).$ Recall that the unveiling of a doctor's identity (when used) is performed outside the process $\\mathit {init}_{dr}$ (see Section REF ).",
"Therefore, the above two definitions do not include generation nor unveiling of doctor identities in the initialization phase.",
"Obviously, the concept of strong doctor anonymity is intended to be stronger than the concept of doctor anonymity.",
"We show that it is impossible to satisfy strong doctor anonymity without satisfying doctor anonymity (arrow $R_5$ in Figure REF ).",
"Assume that a protocol $\\mathit {\\mathit {P}_{\\!eh}}$ satisfies strong doctor anonymity but not doctor anonymity.",
"That is, $\\mathit {\\mathit {P}_{\\!eh}}$ satisfies Definition REF , i.e., $\\mathit {\\mathit {P}_{\\!eh}}\\approx _{\\ell }\\nu \\mathit {\\widetilde{mc}}.",
"\\mathit {init}.\\big (!\\mathit {R}_{1}\\mid \\ldots \\mid !\\mathit {R}_{n}\\mid (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace })\\big ), \\hspace{91.04872pt} (1)$ but there exists no ${\\tt {d_B}}$ such that the equation in Definition REF is satisfied.",
"That is, $\\nexists {\\tt {d_B}}$ s.t.",
"$\\begin{array}{rl}\\mathcal {C}_\\mathit {eh}[\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }]\\approx _{\\ell }\\mathcal {C}_\\mathit {eh}[\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }].",
"\\hspace{56.9055pt} (2)\\end{array}$ Since $\\mathcal {C}_\\mathit {eh}[\\_]:=\\nu \\mathit {\\widetilde{mc}}.",
"\\mathit {init}.",
"(!\\mathit {R}_{1} \\mid \\ldots \\mid !\\mathit {R}_{n}\\mid \\rule {0.3cm}{0.5pt}),$ we have $\\mathcal {C}_\\mathit {eh}[\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }] :=\\nu \\mathit {\\widetilde{mc}}.",
"\\mathit {init}.",
"(!\\mathit {R}_{1} \\mid \\ldots \\mid !\\mathit {R}_{n}\\mid (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace })).$ That is, the right-hand side of the equation $(1)$ is exactly the left-hand side of the equation $(2)$ .",
"Therefore, there exists no ${\\tt {d_B}}$ such that the following equation holds, $\\begin{array}{rl}\\mathit {\\mathit {P}_{\\!eh}}\\approx _{\\ell }\\mathcal {C}_\\mathit {eh}[\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }].",
"\\hspace{184.9429pt} (3)\\end{array}$ Since $\\mathit {\\mathit {P}_{\\!eh}}:=\\nu \\mathit {\\widetilde{mc}}.",
"\\mathit {init}.",
"(!\\mathit {R}_{1} \\mid \\ldots \\mid !\\mathit {R}_{n})$ , by letting ${\\tt {d_B}}$ be an identity of a doctor process in $\\mathit {\\mathit {P}_{\\!eh}}$ , the equation $(3)$ holds.",
"There obviously exists a ${\\tt {d_B}}$ such that the equation $(3)$ holds.",
"This contradicts the assumption.",
"$\\Box $ Anonymity and strong anonymity may be similarly defined for other roles.",
"We provide definitions for patient anonymity, anonymity for other roles is defined analogously.",
"Definition 10 (patient anonymity) A well-formed e-health protocol $\\mathit {\\mathit {P}_{\\!eh}}$ with a patient role $\\mathit {R}_{\\mathit {pt}}$ satisfies patient anonymity if for any patient ${\\tt {t_A}}$ , there exists another patient ${\\tt {t_B}}$ (${\\tt {t_B}}\\ne {\\tt {t_A}}$ ), such that $\\begin{array}{rl}\\mathcal {C}_\\mathit {eh}[\\mathit {init}_{\\mathit {pt}}\\mathit {\\lbrace {\\tt {t_A}}/{\\mathit {I}d_{\\mathit {pt}}}\\rbrace }.",
"!\\mathit {P}_{\\mathit {pt}}\\mathit {\\lbrace {\\tt {t_A}}/{\\mathit {I}d_{\\mathit {pt}}}\\rbrace }]\\approx _{\\ell }\\mathcal {C}_\\mathit {eh}[\\mathit {init}_{\\mathit {pt}}\\mathit {\\lbrace {\\tt {t_B}}/{\\mathit {I}d_{\\mathit {pt}}}\\rbrace }.",
"!\\mathit {P}_{\\mathit {pt}}\\mathit {\\lbrace {\\tt {t_B}}/{\\mathit {I}d_{\\mathit {pt}}}\\rbrace }].\\end{array}$ Definition 11 (strong patient anonymity [4]) A well-formed e-health protocol $\\mathit {\\mathit {P}_{\\!eh}}$ with a patient role $\\mathit {R}_{\\mathit {pt}}$ satisfies strong doctor anonymity, if $\\mathit {\\mathit {P}_{\\!eh}}\\approx _{\\ell }\\nu \\mathit {\\widetilde{mc}}.",
"\\mathit {init}.\\big (!\\mathit {R}_{1}\\mid \\ldots \\mid !\\mathit {R}_{n}\\mid (\\mathit {init}_{\\mathit {pt}}\\mathit {\\lbrace {\\tt {t_A}}/{\\mathit {I}d_{\\mathit {pt}}}\\rbrace }.",
"!\\mathit {P}_{\\mathit {pt}}\\mathit {\\lbrace {\\tt {t_A}}/{\\mathit {I}d_{\\mathit {pt}}}\\rbrace })\\big ).$ As with the case for doctor anonymity, strong patient anonymity is stronger than patient anonymity ($R_7$ in Figure REF ).",
"The proof is analoguous to the proof above."
],
[
"Untraceability and strong untraceability",
"Untraceability is a property preventing the adversary from tracing a user, meaning that he cannot tell whether two executions are initiated by the same user.",
"Definition 12 (doctor untraceability) A well-formed e-health protocol $\\mathit {\\mathit {P}_{\\!eh}}$ with a doctor role $\\mathit {R}_{\\mathit {dr}}$ satisfies doctor untraceability if, for any two doctors ${\\tt {d_A}}$ and ${\\tt {d_B}}\\ne {\\tt {d_A}}$ , $\\begin{array}{rl}&\\mathcal {C}_\\mathit {eh}[\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace })]\\\\\\approx _{\\ell }&\\mathcal {C}_\\mathit {eh}[(\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace })\\mid (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace })].\\end{array}$ A stronger version of untraceability, proposed in [4], captures the adversary's inability to distinguish the situation where one user executes the protocol multiple times from each user executing the protocol at most once.",
"Definition 13 (strong doctor untraceability [4]) A well-formed e-health protocol $\\mathit {\\mathit {P}_{\\!eh}}$ with a doctor role $\\mathit {R}_{\\mathit {dr}}$ being the $j^{\\mathit {th}}$ role, satisfies strong doctor untraceability, if $\\mathit {\\mathit {P}_{\\!eh}}\\approx _{\\ell }\\nu \\mathit {\\widetilde{mc}}.",
"\\mathit {init}.\\big (!\\mathit {R}_{1}\\mid \\ldots \\mid !\\mathit {R}_{j-1}\\mid !\\mathit {R}_{j+1} \\mid !\\mathit {R}_{n}\\mid !",
"(\\nu {{\\tt {Id}}_{\\mathit {dr}}}.",
"\\mathit {init}_{\\mathit {dr}}.\\mathit {P}_{\\mathit {dr}})\\big ).$ Similarly, we can define untraceability and strong untraceability for patient and other roles in a protocol, by replacing the doctor role with a different role.",
"Definition 14 (patient untraceability) A well-formed e-health protocol $\\mathit {\\mathit {P}_{\\!eh}}$ with a patient role $\\mathit {R}_{\\mathit {pt}}$ satisfies patient untraceability if, for any two patients ${\\tt {t_A}}$ and ${\\tt {t_B}}\\ne {\\tt {t_A}}$ , $\\begin{array}{rl}&\\mathcal {C}_\\mathit {eh}[\\mathit {init}_{\\mathit {pt}}\\mathit {\\lbrace {\\tt {t_A}}/{\\mathit {I}d_{\\mathit {pt}}}\\rbrace }.",
"(\\mathit {P}_{\\mathit {pt}}\\mathit {\\lbrace {\\tt {t_A}}/{\\mathit {I}d_{\\mathit {pt}}}\\rbrace }\\mid \\mathit {P}_{\\mathit {pt}}\\mathit {\\lbrace {\\tt {t_A}}/{\\mathit {I}d_{\\mathit {pt}}}\\rbrace })]\\\\\\approx _{\\ell }&\\mathcal {C}_\\mathit {eh}[(\\mathit {init}_{\\mathit {pt}}\\mathit {\\lbrace {\\tt {t_A}}/{\\mathit {I}d_{\\mathit {pt}}}\\rbrace }.\\mathit {P}_{\\mathit {pt}}\\mathit {\\lbrace {\\tt {t_A}}/{\\mathit {I}d_{\\mathit {pt}}}\\rbrace })\\mid (\\mathit {init}_{\\mathit {pt}}\\mathit {\\lbrace {\\tt {t_B}}/{\\mathit {I}d_{\\mathit {pt}}}\\rbrace }.\\mathit {P}_{\\mathit {pt}}\\mathit {\\lbrace {\\tt {t_B}}/{\\mathit {I}d_{\\mathit {pt}}}\\rbrace })].\\end{array}$ Definition 15 (strong patient untraceability [4]) A well-formed e-health protocol $\\mathit {\\mathit {P}_{\\!eh}}$ with a patient role $\\mathit {R}_{\\mathit {pt}}$ being the $j^{\\mathit {th}}$ role, satisfies strong doctor untraceability, if $\\mathit {\\mathit {P}_{\\!eh}}\\approx _{\\ell }\\nu \\mathit {\\widetilde{mc}}.",
"\\mathit {init}.\\big (!\\mathit {R}_{1}\\mid \\ldots \\mid !\\mathit {R}_{j-1}\\mid !\\mathit {R}_{j+1} \\mid !\\mathit {R}_{n}\\mid !",
"(\\nu {{\\tt {Id}}_{\\mathit {pt}}}.",
"\\mathit {init}_{\\mathit {pt}}.\\mathit {P}_{\\mathit {pt}})\\big ).$ Figure: Relations between privacy propertiesStrong (doctor/patient) anonymity is stronger than (doctor/patient) anonymity, as the two processes (left-hand side and right hand side of the equivalence) in (doctor/patient) anonymity are instances of the two processes in strong (doctor/patient) anonymity respectively.",
"Hence, if a protocol satisfies strong (doctor/patient) anonymity, it also satisfies (doctor/patient) anonymity.",
"It is evidenced by that the DLV08 protocol (see Section ) satisfies doctor anonymity (without doctor ID revealed), but does not satisfy strong doctor anonymity.",
"Once again, the strong notions of doctor/patient untraceability are stronger than the standard doctor/patient untraceability (see $R_6$ and $R_7$ , respectively, in Figure REF ).",
"The proof is again analoguous to the proof showing strong doctor anonymity is stronger than doctor anonymity.",
"Finally, note that the strong versions of anonymity are not comparable to the strong versions of untraceability (e.g.",
"strong patient anonymity is not comparable to strong patient untraceability).",
"Strong anonymity and strong untraceability capture different aspects of privacy – anonymity focuses on the link between participants and their identities, whereas untraceability focuses on the link between sessions of a participant.",
"This is supported by the case study in Section where the strong doctor anonymity and strong doctor untraceability fail due to different reasons – the model where strong doctor anonymity is satisfied, does not satisfy strong doctor untraceability, and vice versa.",
"Similarly, prescription privacy and doctor anonymity are not comparable either.",
"Doctor anonymity aims to protect the doctor identity whereas prescription privacy aims to protect the link between a doctor's identity and his prescriptions.",
"For instance, a system in which there is one doctor may satisfy doctor anonymity, if that the doctor's identity is perfectly protected.",
"However, the system would not satisfy prescription privacy, since there is no counter-balancing doctor.",
"Conversely, consider a system where two doctors send out their public keys over public channels and afterwards each doctor sends one prescription via a private channel to the trusted authority, who finally outputs both prescriptions.",
"Such a system may satisfy prescription privacy, due to the assumptions of private channel and trusted authority.",
"That is: on the left hand side of the equation in Definition REF , the adversary observes two public keys followed by two prescriptions; on the right hand side, the adversary observes exactly the same.",
"But this system does not satisfy anonymity since the adversary can block communication of participants in $C_{eh}$ and observe the public channel – on the left hand side of the equation in Definition REF , the adversary observes the public key of ${\\tt {d_A}}$ , while on the right hand side, the adversary observes the public key of ${\\tt {d_B}}$ ."
],
[
"Case study: the DLV08 protocol",
"In this section, we apply the above formal definitions for doctor privacy in a case study as a validation of the definitions.",
"We choose to analyse the DLV08 e-health protocol proposed by De Decker et al.",
"[24], as it claims enforced privacy for doctors.",
"However, our analysis is not restricted to doctor privacy.",
"We provide a rather complete analysis of the protocol including patient anonymity, patient untraceability, patient/doctor information secrecy and patient/doctor authentication as well.",
"The ProVerif code used to perform this analysis is available from [26].",
"The DLV08 protocol is a complex health care protocol for the Belgium situation.",
"It captures most aspects of the current Belgian health care practice and aims to provide a strong guarantee of privacy for patients and doctors.",
"Our analysis of this protocol focuses on the below properties.",
"For those that are explicitly claimed by DLV08, the corresponding claim identifier in that paper is given.",
"In addition to those, we analyse secrecy, prescription privacy, receipt-freeness, and independency of receipt-freeness, which are implicitly mentioned.",
"Secrecy of patient and doctor information: no other party should be able to know a patient or a doctor's information, unless the information is intended to be revealed in the protocol (for formal definitions, see Section REF and Section REF ).",
"Authentication ([24]: S1): all parties should properly authenticate each other (for formal definitions, see Section REF and Section REF ).",
"Patient anonymity ([24]: P3): no party should be able to determine a patient's identity.",
"Patient untraceability ([24]: P2): prescriptions issued to the same patient should not be linkable to each other.",
"Prescription privacy: the protocol protects a doctor's prescription behaviour.",
"Receipt-freeness: the protocol prevents bribery between doctors and pharmaceutical companies.",
"Independency of prescription privacy ([24]: P4): pharmacists should not be able to provide evidence to pharmaceutical companies about doctors' prescription.",
"Independency of receipt-freeness: pharmacists should not be able to provide evidence to pharmaceutical companies about doctors' prescription even if the doctor is bribed.",
"The rest of this section describes the DLV08 protocol in more detail."
],
[
"Roles",
"The protocol involves seven roles.",
"We focus on the five roles involved in the core process: doctor, patient, pharmacist, medicine prescription administrator (MPA) and health insurance institute (HII).",
"The other two roles, public safety organisation (PSO) and social security organisation (SSO), provide properties such as revocability and reimbursement.",
"As we do not focus on these properties, and as these roles are only tangentially involved in the core process, we omit these roles from our model.",
"The roles interact as follows: a doctor prescribes medicine to a patient; next the patient obtains medicine from a pharmacist according to the prescription; following that, the pharmacist forwards the prescription to his MPA, the MPA checks the prescription and refunds the pharmacist; finally, the MPA sends invoices to the patient's HII and is refunded."
],
[
"Cryptographic primitives",
"To ensure security and privacy properties, the DLV08 protocol employs several specific cryptographic primitives, besides the classical ones, like encryption.",
"We briefly introduce these cryptographic primitives."
],
[
"Bit-commitments.",
"The bit-commitments scheme consists of two phases, committing phase and opening phase.",
"On the committing phase, a message sender commits to a message.",
"This can be considered as putting the message into a box, and sending the box to the receiver.",
"Later in the opening phase, the sender sends the key of the box to the receiver.",
"The receiver opens the box and obtains the message."
],
[
"Zero-knowledge proofs.",
"A zero-knowledge proof is a cryptographic scheme which is used by one party (prover) to prove to another party (verifier) that a statement is true, without leaking secret information of the prover.",
"A zero-knowledge proof scheme may be either interactive or non-interactive.",
"We consider non-interactive zero-knowledge proofs in this protocol."
],
[
"Digital credentials.",
"A digital credential is a certificate, proving that the holder satisfies certain requirements.",
"Unlike paper certificates (such as passports) which give out the owner's identity, a digital credential can be used to authenticate the owner anonymously.",
"For example, a digital credential can be used to prove that a driver is old enough to drive without revealing the actual age of the driver."
],
[
"Anonymous authentication.",
"Anonymous authentication is a scheme for authenticating a user anonymously, e.g., [12].",
"The procedure of anonymous authentication is actually a zero-knowledge proof, with the digital credential being the public information of the prover.",
"In the scheme, a user's digital credential is used as the public key in a public key authentication structure.",
"Using this, a verifier can check whether a message is signed correctly by the prover (the person authenticating himself), while the verifier cannot identify the prover.",
"Thus, this ensures anonymous authentication."
],
[
"Verifiable encryptions.",
"Verifiable encryption is based on zero-knowledge proofs as well.",
"A prover encrypts a message, and uses zero-knowledge proofs to prove that the encrypted message satisfies specific properties without revealing the original message."
],
[
"Signed proofs of knowledge.",
"Signed proofs of knowledge provide a way of using proofs of knowledge as a digital signature scheme (cf. [20]).",
"Intuitively, a prover signs a message using secret information, which can be considered as a secret signing key.",
"The prover can convince the verifier using proofs of knowledge only if the prover has the right secret key.",
"Thus it proves the origination of the message."
],
[
"Setting",
"The initial information available to a participant is as follows.",
"A doctor has an identity (${{\\tt {Id}}_{\\mathit {dr}}}$ ), a pseudonym (${{\\tt {Pnym}}_{\\mathit {dr}}}$ ), and an anonymous doctor credential ($\\mathit {Cred_{\\mathit {dr}}}$ ) issued by trusted authorities.",
"A patient has an identity (${{\\tt {Id}}_{\\mathit {pt}}}$ ), a pseudonym (${{\\tt {Pnym}}_{\\mathit {pt}}}$ ), an HII (${{\\tt {Hii}}}$ ), a social security status (${{\\tt {Sss}}}$ ), a health expense account (${{\\tt {Acc}}}$ ) and an anonymous patient credential ($\\mathit {Cred_{\\mathit {pt}}}$ ) issued by trusted authorities.",
"Pharmacists, MPA, and HII are public entities, each of which has an identity (${\\tt {Id}}_{\\mathit {ph}}$ , ${{\\tt {Id}}_{\\mathit {mpa}}}$ , ${\\tt {Id}}_{\\mathit {hii}}$ ), a secret key (${\\tt {sk}}_{\\mathit {ph}}$ , ${\\tt {sk}}_{\\mathit {mpa}}$ , ${\\tt {sk}}_{\\mathit {hii}}$ ) and an authorised public key certificate (${\\tt {pk}}_\\mathit {ph}$ , ${\\tt {pk}}_\\mathit {mpa}$ , ${\\tt {pk}}_\\mathit {hii}$ ) issued by trusted authorities.",
"We assume that a user does not take two roles with the same identity.",
"Hence, one user taking two roles are considered as two individual users."
],
[
"Description of the protocol",
"The DLV08 protocol consists of four sub-protocols: doctor-patient sub-protocol, patient-pharmacist sub-protocol, pharmacist-MPA sub-protocol, and MPA-HII sub-protocol.",
"We describe the sub-protocols one by one."
],
[
"Doctor-patient sub-protocol",
"The doctor authenticates himself to a patient by anonymous authentication with the authorised doctor credential as public information.",
"The patient verifies the doctor credential.",
"If the verification passes, the patient anonymously authenticates himself to the doctor using the patient credential, sends the bit-commitments on his identity to the doctor, and proves to the doctor that the identity used in the credential is the same as in the bit-commitments.",
"After verifying the patient credential, the doctor generates a prescription, computes a prescription identity, computes the doctor bit-commitments.",
"Then the doctor combines these computed messages with the received patient bit-commitments; signs these messages using a signed proof of knowledge, which proves that the doctor's pseudonym used in the doctor credential is the same as in the doctor bit-commitments.",
"Together with the proof, the doctor sends the opening information, which is used to open the doctor bit-commitments.",
"The communication in the doctor-patient sub-protocol is shown as a message sequence chart (MSC, [48]) in Figure REF .",
"Figure: Doctor-Patient sub-protocol.Figure: Patient-Pharmacist sub-protocol.Figure: Pharmacist-MPA sub-protocol.Figure: MPA-HII sub-protocol."
],
[
"Patient-Pharmacist sub-protocol",
"The pharmacist authenticates himself to the patient using public key authentication.",
"The patient verifies the authentication and obtains, from the authentication, the pharmacist's identity and the pharmacist's MPA.",
"Then the patient anonymously authenticates himself to the pharmacist, and proves his social security status.",
"Next, the patient computes verifiable encryptions $\\mathit {vc}_1$ , $\\mathit {vc}_2$ , $\\mathit {vc}_3$ , $\\mathit {vc}^{\\prime }_3$ , $\\mathit {vc}_4$ , $\\mathit {vc}_5$ , where $\\mathit {vc}_1$ encrypts the patient's HII using the MPA's public key and proves that the HII encrypted in $\\mathit {vc}_1$ is the same as the one in the patient's credential.",
"$\\mathit {vc}_2$ encrypts the doctor's pseudonym using the MPA's public key and proves that the doctor's pseudonym encrypted in $\\mathit {vc}_2$ is the same as the one in the doctor commitment embedded in the prescription.",
"$\\mathit {vc}_3$ encrypts the patient's pseudonym using the public safety organisation's public key and proves that the pseudonym encrypted in $\\mathit {vc}_3$ is the same as the one in the patient's commitment.",
"$\\mathit {vc}^{\\prime }_3$ encrypts the patient's HII using the social security organisation's public key and proves that the content encrypted in $\\mathit {vc}^{\\prime }_3$ is the same as the HII in the patient's credential.",
"$\\mathit {vc}_4$ encrypts the patient's pseudonym using the MPA's public key and proves that the patient's pseudonym encrypted in $\\mathit {vc}_4$ is the same as the one in the patient's credential.",
"$\\mathit {vc}_5$ encrypts the patient's pseudonym using his HII's public key and proves that the patient's pseudonym encrypted in $\\mathit {vc}_5$ is the same as the one in the patient's credential.",
"5 encrypts $\\mathit {vc}_5$ using the MPA's public key.",
"The patient sends the received prescription to the pharmacist and proves to the pharmacist that the patient's identity in the prescription is the same as in the patient credential.",
"The patient sends $\\mathit {vc}_1, \\mathit {vc}_2, \\mathit {vc}_3, \\mathit {vc}^{\\prime }_3, \\mathit {vc}_4, 5$ as well.",
"The pharmacist verifies the correctness of all the received messages.",
"If every message is correctly formatted, the pharmacist charges the patient, and delivers the medicine.",
"Then the pharmacist generates an invoice and sends it to the patient.",
"The patient computes a receipt $\\mathit {ReceiptAck}$ : signing a message (consists of the prescription identity, the pharmacist's identity, $\\mathit {vc}_1$ , $\\mathit {vc}_2$ , $\\mathit {vc}_3$ , $\\mathit {vc}^{\\prime }_3$ , $\\mathit {vc}_4$ , $\\mathit {vc}_5$ ) using a signed proof of knowledge and proving that he knows the patient credential.",
"This receipt proves that the patient has received his medicine.",
"The pharmacist verifies the correctness of the receipt.",
"The communication in the patient-Pharmacist sub-protocol is shown in Figure REF .",
"Since the payment and medicine delivery procedures are out of the protocol scope, they are interpreted as dashed arrows in the figure."
],
[
"Pharmacist-MPA sub-protocol",
"The pharmacist and the MPA first authenticate each other using public key authentication.",
"Next, the pharmacist sends the received prescription and the receipt $\\mathit {ReceiptAck}$ , together with $\\mathit {vc}_1$ , $\\mathit {vc}_2$ , $\\mathit {vc}_3$ , $\\mathit {vc}^{\\prime }_3$ , $\\mathit {vc}_4$ , 5, to the MPA.",
"The MPA verifies correctness of the received information.",
"Then the MPA decrypts $\\mathit {vc}_1$ , $\\mathit {vc}_2$ , $\\mathit {vc}_4$ and 5, which provide the patient's HII, the doctor's pseudonym, the patient's pseudonym, and $\\mathit {vc}_5$ .",
"The communication in the pharmacist-MPA sub-protocol is shown in Figure REF .",
"Note that after authentication, two parties often establish a secure communication channel.",
"However, it is not mentioned in [24] that the pharmacist and the MPA agree on anything.",
"Nevertheless, this does not affect the properties that we verified, except authentication between pharmacist and MPA."
],
[
"MPA-HII sub-protocol",
"The MPA and the patient's HII first authenticate each other using public key authentication.",
"Then the MPA sends the receipt $\\mathit {ReceiptAck}$ to the patient's HII as well as the verifiable encryption $\\mathit {vc}_5$ which encrypts the patient's pseudonym with the patient's HII's public key.",
"The patient's HII checks the correctness of $\\mathit {ReceiptAck}$ , decrypts $\\mathit {vc}_5$ and obtains the patient's pseudonym.",
"From the patient pseudonym, the HII obtains the identity of the patient; then updates the patient's account and pays the MPA.",
"The MPA pays the pharmacist when he receives the payment.",
"The communication in the MPA-HII sub-protocol is shown in Figure REF .",
"Similar to the previous sub-protocol, there is nothing established during the authentication which can be used in the later message exchanges.",
"Note that in addition to authentications between MPA and HII, this affects the secrecy of a patient's pseudonym when the adversary controls dishonest patients (see Section REF )."
],
[
"Modelling DLV08",
"We model the DLV08 protocol in the applied pi calculus as introduced in Section REF .",
"For clarity, we also borrow some syntactic expressions from ProVerif, such as key words `$\\mathit {fun}$ ', `$\\mathit {private}$ $\\mathit {fun}$ ', `$\\mathit {reduc}$ ' and `$\\mathit {equation}$ ', and expression $`\\mbox{let}\\ x=N \\ \\mbox{in} \\ P^{\\prime }$ .",
"Particularly, `($\\mathit {private}$ ) $\\mathit {fun}$ ' denotes a constructor which uses terms to form a more complex term ('$\\mathit {private}$ ' means the adversary cannot use it).",
"`$\\mathit {reduc}$ ' and `$\\mathit {equation}$ ' are key words used to construct the equational theory $E$ .",
"`$\\mathit {reduc}$ ' denotes a destructor which retrieves sub-terms of a constructed term.",
"For the cryptographic primitives that cannot be captured by destructors, ProVerif provides `$\\mathit {equation}$ ' to capture the relationship between constructors.",
"The expression `$\\mbox{let}\\ x=N \\ \\mbox{in} \\ P$ ' is used as syntactical substitutions, i.e., $P\\lbrace N/x\\rbrace $ in the applied pi calculus.",
"It is an abbreviation of $`\\mbox{let}\\ x=N \\ \\mbox{in} \\ P \\ \\mbox{else} \\ Q^{\\prime }$ when $Q$ is the null process.",
"When $N$ is a destructor, there are two possible outcomes.",
"If the term $N$ does not fail, then $x$ is bound to $N$ and process $P$ is taken, otherwise $Q$ (in this case, the null process) is taken.",
"Since the description of the protocol in its original paper is not clear in some details, before modelling the protocol, several ambiguities need to be settled (Section REF ).",
"Next we explain the modelling of the cryptographic primitives (Section REF ), since security and privacy rely heavily on these cryptographic primitives in the protocol.",
"Then, we illustrate the modelling of the protocol (Section REF )."
],
[
"Underspecification of the DLV08 protocol",
"The DLV08 protocol leaves the following issues unspecified: Table: NO_CAPTIONTo be able to discover potential flaws on privacy, we make the following (weakest) assumptions in our modelling of the DLV08 protocol: Table: NO_CAPTIONNote that some assumptions may look weak to security experts, for example the assumption of deterministic encryption.",
"However, without explicit warning, deterministic encryption algorithms may be used, which will lead to security flaws.",
"With this in mind, we assume the weakest assumption when there is ambiguity.",
"By assuming weak assumptions and showing the security flaws with the assumptions, we provide security warnings for the implementation of the protocol."
],
[
"Modelling cryptographic primitives",
"The cryptographic primitives are modelled in the applied pi calculus using function symbols and equations.",
"All functions and equational theory are summarised in Figures REF , REF and REF .",
"Figure: Functions.Figure: Equational theory part i@: non-zero-knowledge part.Figure: Equational theory part ii@: zero-knowledge part."
],
[
"Bit-commitments.",
"The bit-commitments scheme is modelled as two functions: ${\\sf commit}$ , modelling the committing phase, and ${\\sf open}$ , modelling the opening phase.",
"The function ${\\sf commit}$ creates a commitment with two parameters: a message $m$ and a random number $r$ .",
"A commitment can only be opened with the correct opening information $r$ , in which case the message $m$ is revealed.",
"$\\begin{array}{rl}\\mathit {fun}\\ & {\\sf commit}/2.",
"\\\\\\mathit {reduc\\ }& {\\sf open}({\\sf commit}(m,r),r)=m.\\end{array}$"
],
[
"Zero-knowledge proofs.",
"Non-interactive zero-knowledge proofs can be modelled as function ${\\sf zk}(\\mathit {secrets}, \\mathit {pub\\_info})$ inspired by [19].",
"The public verification information $\\mathit {pub\\_info}$ and the secret information $\\mathit {secrets}$ satisfy a pre-specified relation.",
"Since the secret information is only known by the prover, only the prover can construct the zero-knowledge proof.",
"To verify a zero-knowledge proof is to check whether the relation between the secret information and the verification information is satisfied.",
"Verification of a zero-knowledge proof is modelled as function $\\mbox{{\\sf Vfy-{zk}}}({\\sf zk}(\\mathit {secrets},\\mathit {pub\\_info}), \\mathit {verif\\_info})$ , with a zero-knowledge proof to be verified ${\\sf zk}(\\mathit {secrets},\\linebreak \\mathit {pub\\_info})$ and the verification information $\\mathit {verif\\_info}$ .",
"Compared to the more generic definitions in [19], we define each zero-knowledge proof specifically, as only a limited number of zero-knowledge proofs are used in the protocol.",
"We specify each verification rule in Figure REF .",
"Since the $\\mathit {pub\\_info}$ and $\\mathit {verif\\_info}$ happen to be the same in all the zero-knowledge proofs verifications in this protocol, the generic structure of verification rule is given as $\\mbox{{\\sf Vfy-{zk}}}({\\sf zk}(\\mathit {secrets},\\mathit {pub\\_info}),\\mathit {pub\\_info})={\\sf true},$ where ${\\sf true}$ is a constant.",
"The specific function to check a zero-knowledge proof of type $z$ is denoted as $\\mbox{{\\sf Vfy-{zk}}}_z$ , e.g., verification of a patient's anonymous authentication modelled by function $\\mbox{\\sf Vfy-{zk}}_{\\sf Auth_{\\mathit {pt}}}$ ."
],
[
"Digital credentials.",
"A digital credential is issued by trusted authorities.",
"We assume the procedure of issuing a credential is perfect, which means that the adversary cannot forge a credential nor obtain one by impersonation.",
"We model digital credentials as a private function (declaimed by key word $\\mathit {private}\\ \\mathit {fun}\\ $ in ProVerif) which is only usable by honest users.",
"In the DLV08 protocol, a credential can have several attributes; we model these as parameters of the credential function.",
"$\\begin{array}{l@{\\hspace{28.45274pt}}l}\\mathit {private}\\ \\mathit {fun}\\ {\\sf drcred}/2.&\\mathit {private}\\ \\mathit {fun}\\ {\\sf ptcred}/5.\\end{array}$ There are two credentials in the DLV08 protocol: a doctor credential which is modelled as $\\mathit {Cred_{\\mathit {dr}}}:={\\sf drcred}({{\\tt {Pnym}}_{\\mathit {dr}}}, {{\\tt {Id}}_{\\mathit {dr}}})$ , and a patient credential which is modelled as $\\mathit {Cred_{\\mathit {pt}}}:=\\linebreak {\\sf ptcred}({{\\tt {Id}}_{\\mathit {pt}}}, {{\\tt {Pnym}}_{\\mathit {pt}}}, {{\\tt {Hii}}},{{\\tt {Sss}}}, {{\\tt {Acc}}})$ .",
"Unlike private data, the two private functions cannot be coerced, meaning that even by coercing, the adversary cannot apply the private functions.",
"Because a doctor having an anonymous credentials is a basic setting of the protocol, and thus the procedure of obtaining a credential is not assumed to be bribed or coerced.",
"However, the adversary can coerce patients or doctors for the credentials and parameters of the private functions."
],
[
"Anonymous authentication.",
"The procedure of anonymous authentication is a zero-knowledge proof using the digital credential as public information.",
"The anonymous authentication of a doctor is modelled as $\\mathit {Auth_{\\mathit {dr}}}:={\\sf zk}(({{\\tt {Pnym}}_{\\mathit {dr}}}, {{\\tt {Id}}_{\\mathit {dr}}}), {\\sf drcred}({{\\tt {Pnym}}_{\\mathit {dr}}},{{\\tt {Id}}_{\\mathit {dr}}})),$ and the verification of the authentication is modelled as $\\mbox{\\sf Vfy-{zk}}_{\\sf Auth_{\\mathit {dr}}}(\\mathit {Auth_{\\mathit {dr}}},{\\sf drcred}({{\\tt {Pnym}}_{\\mathit {dr}}},{{\\tt {Id}}_{\\mathit {dr}}})).$ The equational theory for the verification is $\\mathit {reduc\\ }\\mbox{\\sf Vfy-{zk}}_{\\sf Auth_{\\mathit {dr}}}({\\sf zk}(({{\\tt {Pnym}}_{\\mathit {dr}}},{{\\tt {Id}}_{\\mathit {dr}}}), {\\sf drcred}({{\\tt {Pnym}}_{\\mathit {dr}}},{{\\tt {Id}}_{\\mathit {dr}}})),{\\sf drcred}({{\\tt {Pnym}}_{\\mathit {dr}}},{{\\tt {Id}}_{\\mathit {dr}}}))={\\sf true}.$ The verification implies that the creator of the authentication is a doctor who has the credential ${\\sf drcred}({{\\tt {Pnym}}_{\\mathit {dr}}},{{\\tt {Id}}_{\\mathit {dr}}})$ .",
"Because only legitimate doctors can obtain a credential from authorities, i.e., use the function ${\\sf drcred}$ to create a credential; and the correspondence between the parameters of the anonymous authentication (the first parameter $({{\\tt {Pnym}}_{\\mathit {dr}}},{{\\tt {Id}}_{\\mathit {dr}}})$ in $\\mathit {Auth_{\\mathit {dr}}}$ ) and the parameters of the credential (parameters ${{\\tt {Pnym}}_{\\mathit {dr}}}$ and${{\\tt {Id}}_{\\mathit {dr}}}$ in ${\\sf drcred}({{\\tt {Pnym}}_{\\mathit {dr}}},{{\\tt {Id}}_{\\mathit {dr}}})$ ) ensures that the prover can only be the owner of the credential.",
"Other doctors may be able to use function ${\\sf drcred}$ but do not know ${{\\tt {Pnym}}_{\\mathit {dr}}}$ and ${{\\tt {Id}}_{\\mathit {dr}}}$ , and thus cannot create a valid proof.",
"The adversary can observe a credential ${\\sf drcred}({{\\tt {Pnym}}_{\\mathit {dr}}},{{\\tt {Id}}_{\\mathit {dr}}})$ , but does not know secrets ${{\\tt {Pnym}}_{\\mathit {dr}}}, {{\\tt {Id}}_{\\mathit {dr}}}$ , and thus cannot forge a valid zero-knowledge proof.",
"If the adversary forges a zero-knowledge proof with fake secret information ${{\\tt {Pnym}}_{\\mathit {dr}}}^{\\prime }$ and ${{\\tt {Id}}_{\\mathit {dr}}}^{\\prime }$ , the fake zero-knowledge proof will not pass verification.",
"For the same reason, a validated proof proves that the credential belongs to the creator of the zero-knowledge proof.",
"Similarly, an anonymous authentication of a patient is modelled as $\\begin{array}{rl}\\mathit {Auth_{\\mathit {pt}}}:={\\sf zk}(&\\hspace{-8.5pt}({{\\tt {Id}}_{\\mathit {pt}}}, {{\\tt {Pnym}}_{\\mathit {pt}}}, {{\\tt {Hii}}},{{\\tt {Sss}}}, {{\\tt {Acc}}}),{\\sf ptcred}({{\\tt {Id}}_{\\mathit {pt}}}, {{\\tt {Pnym}}_{\\mathit {pt}}}, {{\\tt {Hii}}},{{\\tt {Sss}}}, {{\\tt {Acc}}})),\\end{array}$ and the verification rule is modelled as $\\begin{array}{rl}\\mathit {reduc\\ }\\mbox{\\sf Vfy-{zk}}_{\\sf Auth_{\\mathit {pt}}}(&\\hspace{-8.5pt}{\\sf zk}(({{\\tt {Id}}_{\\mathit {pt}}}, {{\\tt {Pnym}}_{\\mathit {pt}}}, {{\\tt {Hii}}},{{\\tt {Sss}}}, {{\\tt {Acc}}}), \\\\& \\hspace{4.25pt}{\\sf ptcred}({{\\tt {Id}}_{\\mathit {pt}}}, {{\\tt {Pnym}}_{\\mathit {pt}}}, {{\\tt {Hii}}},{{\\tt {Sss}}}, {{\\tt {Acc}}})),\\\\& \\hspace{-8.5pt}{\\sf ptcred}({{\\tt {Id}}_{\\mathit {pt}}}, {{\\tt {Pnym}}_{\\mathit {pt}}}, {{\\tt {Hii}}},{{\\tt {Sss}}}, {{\\tt {Acc}}}))={\\sf true}.\\end{array}$"
],
[
"Verifiable encryptions.",
"A verifiable encryption is modelled as a zero-knowledge proof.",
"The encryption is embedded in the zero-knowledge proof as public information.",
"The receiver can obtain the cipher text from the proof.",
"For example, assume a patient wants to prove that he has encrypted a secret $s$ using a public key $k$ to a pharmacist, while the pharmacist does not know the corresponding secret key for $k$ .",
"The pharmacist cannot open the cipher text to test whether it uses the public key $k$ for encryption.",
"However, the zero-knowledge proof can prove that the cipher text is encrypted using $k$ , while not revealing the secret $s$ .",
"The general structure of the verification of a verifiable encryption is $\\mbox{{\\sf Vfy-{venc}}}({\\sf zk}(\\mathit {secrets}, (\\mathit {pub\\_info},\\mathit {cipher})),\\mathit {verif\\_info})={\\sf true},$ where $\\mathit {secrets}$ is private information, $\\mathit {pub\\_info}$ and $\\mathit {cipher}$ consist public information, $\\mathit {verif\\_info}$ is the verification information."
],
[
"Signed proofs of knowledge.",
"A signed proof of knowledge is a scheme which signs a message, and proves a property of the signer.",
"For the DLV08 protocol, this proof only concerns equality of attributes of credentials and commitments (e.g., the identity of this credential is the same as the identity of that commitment).",
"To verify a signed proof of knowledge, the verifier must know which credentials/commitments are considered.",
"Hence, this information must be obtainable from the proof, and thus is included in the model.",
"In general, a signed proof of knowledge is modelled as function ${\\sf spk}(\\mathit {secrets}, \\mathit {pub\\_info}, \\mathit {msg}),$ which models a signature using private value(s) secrets on the message msg, with public information $\\mathit {pub\\_info}$ as settings.",
"Similar to zero-knowledge proofs, $\\mathit {secrets}$ and $\\mathit {pub\\_info}$ satisfy a pre-specified relation.",
"$\\mathit {msg}$ can be any message.",
"What knowledge is proven, depends on the specific instance of the proof and is captured by the verification functions for the specific proofs.",
"For example, to prove that a user knows (a) all fields of a (simplified) credential, (b) all fields of a commitment to an identity, and (c) that the credential concerns the same identity as the commitment, he generates the following proof: $\\begin{array}{l@{\\hspace{-2.125pt}}l@{\\hspace{28.45274pt}}l}{\\sf spk}( & ({{\\tt {Id}}_{\\mathit {pt}}},{{\\tt {Pnym}}_{\\mathit {pt}}},{{\\tt {r}}_\\mathit {\\mathit {pt}}}),& (*\\mathit {secrets}*) \\\\\\qquad & ({\\sf ptcred}({{\\tt {Id}}_{\\mathit {pt}}},{{\\tt {Pnym}}_{\\mathit {pt}}}),{\\sf commit}({{\\tt {Id}}_{\\mathit {pt}}},{{\\tt {r}}_\\mathit {\\mathit {pt}}})), &(*\\mathit {public\\_info}*)\\\\& \\mathit {msg}).",
"&(*\\mathit {message}*)\\end{array}$ These proofs are verified by checking that the signature is correct, given the signed message and the verification information.",
"E.g., the above example proof can be verified as follows: $\\begin{array}{lll@{\\hspace{28.45274pt}}l}\\mathit {reduc\\ }\\mbox{{\\sf Vfy-{spk}}}( & \\hspace{-8.5pt}{\\sf spk}( &\\hspace{-8.5pt}({{\\tt {Id}}_{\\mathit {pt}}},{{\\tt {Pnym}}_{\\mathit {pt}}},{{\\tt {r}}_\\mathit {\\mathit {pt}}}), \\\\& & \\hspace{-8.5pt}({\\sf ptcred}({{\\tt {Id}}_{\\mathit {pt}}},{{\\tt {Pnym}}_{\\mathit {pt}}}),{\\sf commit}({{\\tt {Id}}_{\\mathit {pt}}},{{\\tt {r}}_\\mathit {\\mathit {pt}}})), \\\\& & \\hspace{-8.5pt}\\mathit {msg}\\ ), &(*\\mathit {signed\\_message}*)\\\\& \\multicolumn{2}{l}{\\hspace{-8.5pt}(\\ {\\sf ptcred}({{\\tt {Id}}_{\\mathit {pt}}},{{\\tt {Pnym}}_{\\mathit {pt}}}),{\\sf commit}({{\\tt {Id}}_{\\mathit {pt}}},{{\\tt {r}}_\\mathit {\\mathit {pt}}})\\ ),} &(*\\mathit {verify\\_info}*)\\\\\\end{array}&\\hspace{-8.5pt} \\mathit {msg} &&(*\\mathit {message}*)\\\\& \\multicolumn{2}{l}{\\hspace{-8.5pt}) = {\\sf true}.",
"}$ $$"
],
[
"Other cryptographic primitives.",
"Hash functions, encryptions and signing messages are modelled by functions ${\\sf hash}$ , $\\sf enc$ , and $\\sf sign$ , respectively (see Figure REF ).",
"Correspondingly, decryption, verifying a signature and retrieving the message from a signature are modelled as functions $\\sf dec$ , $\\mbox{\\sf Vfy-{sign}}$ and ${\\sf getsignmsg}$ (see Figure REF ).",
"Function ${\\sf pk}$ models the corresponding public key of a secret key, and function ${\\sf invoice}$ is used for a pharmacist to generate an invoice for a patient (see Figure REF ).",
"Functions ${\\sf getpublic}$ , ${\\sf getSpkVinfo}$ and ${\\sf getmsg}$ model retrieving public information from a zero-knowledge proof, from a signed proof of knowledge, and obtaining the message from a signed proof of knowledge, respectively (see Figure REF ).",
"Function ${\\sf key}$ models the public key of a user's identity and function ${\\sf host}$ retrieves the owner's identity from a public key (see Figure REF )."
],
[
"Modelling the DLV08 protocol",
"We first show how to model each of the sub-protocols and then how to compose them to form the full DLV08 protocol."
],
[
"Modelling the doctor-patient sub-protocol.",
"This sub-protocol is used for a doctor, whose steps are labelled d$i$ in Figure REF , to prescribe medicine for a patient, whose steps are labelled t$i$ in Figure REF .",
"Figure: The doctor process 𝑃 𝑑𝑟 \\mathit {P}_{\\mathit {dr}}.Figure: The patient process in the doctor-patient sub-protocol 𝑃 𝑝𝑡 ' \\mathit {P}^{\\prime }_{\\mathit {pt}}.First, the doctor anonymously authenticates to the patient using credential $\\mathit {Cred_{\\mathit {dr}}}$ (d1).",
"The patient reads in the doctor authentication (t1), obtains the doctor credential (t2), and verifies the authentication (t3).",
"If the verification in step (t3) succeeds, the patient anonymously authenticates himself to the doctor using his credential (t5, the first ${\\sf zk}$ function), generates a nonce ${{\\tt {r}}_\\mathit {\\mathit {pt}}}$ (t4), computes a commitment with the nonce as opening information, and proves that the patient identity used in the patient credential is the same as in the commitment, thus linking the patient commitment and the patient credential (t5, the second ${\\sf zk}$ ).",
"The doctor reads in the patient authentication as ${\\mathit {rcv\\_Auth_{\\mathit {pt}}}}$ and the patient proof as${\\mathit {rcv\\_\\mathit {PtProof}}}$ (d2), obtains the patient credential from the patient authentication (d3), obtains the patient commitment ${\\mathit {c\\_Comt_{\\mathit {pt}}}}$ and the patient credential from the patient proof, tests whether the credential matches the one embedded in the patient authentication (d4), then verifies the authentication (d5) and the patient proof (d6).",
"If the verification in the previous item succeeds, the doctor generates a prescription ${\\tt {presc}}$Note that a medical examination of the patient is not part of the DLV08 protocol.",
"(d7), generates a nonce ${\\tt {r}}_{\\mathit {dr}}$ (d8), computes a prescription identity $\\mathit {PrescriptID}$ (d9), and computes a commitment $\\mathit {Comt_{\\mathit {dr}}}$ using the nonce as opening information (d10).",
"Next, the doctor signs the message (${\\tt {presc}}$ , $\\mathit {PrescriptID}$ , $\\mathit {Comt_{\\mathit {dr}}}$ , ${\\mathit {c\\_Comt_{\\mathit {pt}}}}$ ) using a signed proof of knowledge.",
"This proves the pseudonym used in the credential $\\mathit {Cred_{\\mathit {dr}}}$ is the same as in the commitment $\\mathit {Comt_{\\mathit {dr}}}$ , thus linking the prescription to the credential.",
"The doctor sends the signed proof of knowledge together with the open information of the doctor commitment ${\\tt {r}}_{\\mathit {dr}}$ (d10).",
"The patient reads in the prescription as ${\\mathit {rcv\\_\\mathit {PrescProof}}}$ and the opening information of the doctor commitment (t6), obtains the prescription ${\\mathit {c\\_\\mathit {presc}}}$ , prescription identity ${\\mathit {c\\_\\mathit {PrescriptID}}}$ , doctor commitment ${\\mathit {c\\_Comt_{\\mathit {dr}}}}$ , and tests the patient commitment signed in the receiving message (t7).",
"Then the patient verifies the signed proof of prescription (t8).",
"If the verification succeeds, the patient obtains the doctor's pseudonym ${\\mathit {c\\_Pnym_{\\mathit {dr}}}}$ by opening the doctor commitment (t9) and continues the next sub-protocol behaving as in process $\\mathit {P}^{\\prime \\prime }_{\\mathit {pt}}$ .",
"Rationale for modelling of prescriptions.",
"In the description of DLV08 protocol [24], it is unclear precisely what information is included in a prescription.",
"Depending on the implementation, a prescription may contain various information, such as name of medicines prescribed, amount of medicine prescribed, the timestamp and organization that wrote the prescription, etc.",
"Some information in the prescription may reveal privacy of patients and doctors.",
"For instance, if the identities of patients and doctors are included in the prescription, then doctor and patient prescription privacy is trivially broken.",
"In addition, both (doctor/patient) anonymity and untraceability would also be trivially broken, if the prescriptions were revealed to the adversary.",
"In order to focus only on the logical flaws of the DLV08 protocol and exclude such dependencies, we assume that the prescriptions in the protocol are de-identified.",
"However, this may not be sufficient.",
"Doctors may e.g.",
"be identifiable by the way they prescribe, the order in which medicine appear on prescriptions, etc.",
"Such “fingerprinting” attacks would also trivially break prescription privacy.",
"For our analysis, we assume that a prescription cannot be linked to its doctor or patient by its content.",
"That is, the prescription shall not be modelled as a function of doctor or patient information.",
"To avoid any of the above concerns, we model prescriptions as abstract pieces of data: each prescription is represented by a single, unique name.",
"First, this modelling captures the assumption that prescriptions from different doctors are often different even for the same diagnose, due to different prescription styles.",
"Second, this allows us to capture an infinite number of prescriptions in infinite sessions, without introducing false attacks to the DLV08 protocol that are caused by the modelling of the prescriptions."
],
[
"Modelling the patient-pharmacist sub-protocol.",
"This sub-protocol is used for a patient, whose steps are labelled t$i$ in Figure REF , to obtain medicine from a pharmacist, whose steps are labelled h$i$ in Figure REF .",
"Figure: The patient process in the patient-pharmacist sub-protocol 𝑃 𝑝𝑡 '' \\mathit {P}^{\\prime \\prime }_{\\mathit {pt}}.Figure: The pharmacist process in the patient-pharmacist sub-protocol 𝑃 𝑝ℎ ' \\mathit {P}^{\\prime }_{\\mathit {ph}}.First, the pharmacist authenticates to the patient using a public key authentication (h1).",
"Note that the pharmacist does not authenticate anonymously, and that the pharmacists's MPA identity is embedded.",
"The patient reads in the pharmacist authentication ${\\mathit {rcv\\_Auth_{\\mathit {ph}}}}$ (t10) and verifies the authentication (t11).",
"If the verification succeeds, the pharmacist obtains the pharmacist's MPA identity from the authentication (t12), thus obtains the public key of MPA (t13).",
"Then the patient anonymously authenticates himself to the pharmacist, and proves his social security status using the proof $\\mathit {PtAuthSss}$ (t14).",
"The patient generates a nonce which will be used as a message in a signed proof of knowledge (t15), and computes verifiable encryptions $\\mathit {vc}_1$ , $\\mathit {vc}_2$ , $\\mathit {vc}_3$ , $\\mathit {vc}_3^{\\prime }$ , $\\mathit {vc}_4$ and $\\mathit {vc}_5$ (t16-t21).",
"These divulge the patient's HII, the doctor's pseudonym, and the patient's pseudonym to the MPA, the patient's pseudonym to the HII, and the patient pseudonym and HII to the social safety organisation, respectively.",
"The patient encrypts $\\mathit {vc}_5$ with MPA's public key as 5 (t22).",
"The patient computes a signed proof of knowledge $\\begin{array}{rl}\\mathit {PtSpk}=&{\\sf spk}(({{\\tt {Id}}_{\\mathit {pt}}}, {{\\tt {Pnym}}_{\\mathit {pt}}}, {{\\tt {Hii}}}, {{\\tt {Sss}}}, {{\\tt {Acc}}}),\\\\&\\quad \\ \\ ({\\sf ptcred}({{\\tt {Id}}_{\\mathit {pt}}}, {{\\tt {Pnym}}_{\\mathit {pt}}}, {{\\tt {Hii}}}, {{\\tt {Sss}}}, {{\\tt {Acc}}}),{\\sf commit}({{\\tt {Id}}_{\\mathit {pt}}}, {{\\tt {r}}_\\mathit {\\mathit {pt}}})),\\\\&\\qquad {\\tt {nonce}})\\end{array}$ which proves that the patient identity embedded in the prescription is the same as in his credential.",
"In the prescription, this identity is contained in a commitment.",
"For simplicity, we model the proof using the commitment instead of the prescription.",
"The link between commitment and prescription is ensured when the proof is verified (h10).",
"The patient sends the prescription ${\\mathit {rcv\\_\\mathit {PrescProof}}}$ , the signed proof $\\mathit {PtSpk}$ , and $\\mathit {vc}_1,\\mathit {vc}_2,\\mathit {vc}_3,\\linebreak \\mathit {vc}^{\\prime }_3,\\mathit {vc}_4,5$ to the pharmacist (t23).",
"The pharmacist reads in the authentication ${\\mathit {rcv\\_\\mathit {PtAuthSss}}}$ (h2), obtains the patient credential and his social security status (h3), verifies the authentication (h4).",
"If the verification succeeds, the pharmacist reads in the patient's prescription ${\\mathit {rcv_{\\mathit {ph}}\\_\\mathit {PrescProof}}}$ , the signed proof of knowledge ${\\mathit {r}cv_{\\mathit {ph}}\\_\\mathit {PtSpk}}$ , the verifiable encryptions ${\\mathit {rcv\\_vc}}_1$ , ${\\mathit {rcv\\_vc}}_2$ , ${\\mathit {rcv\\_vc}}_3$ , ${\\mathit {rcv\\_vc}}^{\\prime }_3$ , ${\\mathit {rcv\\_vc}}_4$ , and cipher text ${\\mathit {rcv\\_c}}_5$ (h5); and verifies ${\\mathit {rcv_{\\mathit {ph}}\\_\\mathit {PrescProof}}}$ (h6-h8), ${\\mathit {r}cv_{\\mathit {ph}}\\_\\mathit {PtSpk}}$ (h9-h10), and ${\\mathit {rcv\\_vc}}_1$ , ${\\mathit {rcv\\_vc}}_2$ , ${\\mathit {rcv\\_vc}}_3$ , ${\\mathit {rcv\\_vc}}^{\\prime }_3$ , ${\\mathit {rcv\\_vc}}_4$ (h11-h20).",
"If all the verifications succeed, the pharmacist charges the patient, and delivers the medicine (neither are modelled as they are out of DLV08's scope).",
"Then the pharmacist generates an invoice with the prescription identity embedded in it and sends the invoice to the patient (h21).",
"The patient reads in the invoice (t24), computes a receipt: a signed proof of knowledge $\\mathit {ReceiptAck}$ which proves that he receives the medicine (t25); and sends the signed proof of knowledge to the pharmacist (t26).",
"The pharmacist reads in the receipt ${\\mathit {rcv\\_ReceiptAck}}$ (h22), verifies its correctness (h23) and continues the next sub-protocol behaving as in $\\mathit {P}^{\\prime \\prime }_{\\mathit {ph}}$ ."
],
[
"Modelling the pharmacist-MPA sub-protocol.",
"The pharmacist-MPA sub-protocol is used for the pharmacist, whose steps are labelled h$i$ in Figure REF to report the received prescriptions to the MPA, whose steps are labelled m$i$ in Figure REF .",
"Figure: The pharmacist process in the pharmacist-MPA sub-protocol 𝑃 𝑝ℎ '' \\mathit {P}^{\\prime \\prime }_{\\mathit {ph}}.Figure: The MPA process in the pharmacist-MPA sub-protocol 𝑃 𝑚𝑝𝑎 \\mathit {P}_{\\mathit {mpa}}.As the pharmacist mostly forwards the information supplied by the patient, this protocol greatly resembles the patient-pharmacist protocol described above.",
"Each step is modelled in details as follows: The pharmacist authenticates himself to his MPA by sending his identity and the signed identities of the pharmacist and the MPA (h24).",
"The MPA stores this authentication in ${\\mathit {rcv_{\\mathit {mpa}}\\_Auth_{\\mathit {ph}}}}$ , and stores the pharmacist's identity in ${\\mathit {c_{\\mathit {mpa}}\\_Id_{\\mathit {ph}}}}$ (m1).",
"From the pharmacist's identity, the MPA obtains the pharmacist's public key (m2).",
"Then the MPA verifies the pharmacist's authentication against the pharmacist's public key (m3).",
"If the verification succeeds, according to the corresponding rule in the equational theory, and the MPA verifies that he is indeed the pharmacist's MPA (m4), the MPA then authenticates itself to the pharmacist by sending the signature of his identity (m5).",
"The pharmacist reads in the MPA's authentication in ${\\mathit {rcv\\_Auth_{\\mathit {mpa}}}}$ (h25), and verifies the authentication (h26).",
"If the verification succeeds, the pharmacist sends the following to the MPA: prescription ${\\mathit {rcv_{\\mathit {ph}}\\_\\mathit {PrescProof}}}$ , received receipt ${\\mathit {rcv\\_ReceiptAck}}$ , and verifiable encryptions ${\\mathit {rcv\\_vc}}_1$ , ${\\mathit {rcv\\_vc}}_2$ , ${\\mathit {rcv\\_vc}}_3$ , ${\\mathit {rcv\\_vc}}^{\\prime }_3$ , ${\\mathit {rcv\\_vc}}_4$ , ${\\mathit {rcv\\_c}}_5$ (h27).",
"The MPA reads in the information (m6) and verifies their correctness (m7-m24).",
"If the verifications succeed, the MPA decrypts the corresponding encryptions (${\\mathit {c_{\\mathit {mpa}}\\_Enc}}_1$ , ${\\mathit {c_{\\mathit {mpa}}\\_Enc}}_2$ , and ${\\mathit {c_{\\mathit {mpa}}\\_Enc}}_4$ ) embedded in ${\\mathit {rcv_{\\mathit {mpa}}\\_vc}}_1, {\\mathit {rcv_{\\mathit {mpa}}\\_vc}}_2, {\\mathit {rcv_{\\mathit {mpa}}\\_vc}}_4$ , and obtains the patient's HII (m12), the doctor pseudonym (m15), the patient pseudonym (m23).",
"Then the MPA continues the next sub-protocol behaving as in process $\\mathit {P}^{\\prime }_{\\mathit {mpa}}$ .",
"The storing information to database by the MPA is beyond our concern."
],
[
"Modelling the MPA-HII sub-protocol.",
"This protocol covers the exchange of information between the pharmacist's MPA, whose steps are labelled m$i$ in Figure REF and the patient's HII, whose steps are labelled i$i$ in Figure REF .",
"Figure: The MPA process in the MPA-HII sub-protocol 𝑃 𝑚𝑝𝑎 ' \\mathit {P}^{\\prime }_{\\mathit {mpa}}.Figure: The HII process 𝑃 ℎ𝑖𝑖 \\mathit {P}_{\\mathit {hii}}.The MPA sends his identity to the HII and authenticates to the HII using public key authentication (m25).",
"The HII stores the MPA's identity in $\\mathit {rcv_{\\mathit {hii}}\\_Id_{\\mathit {mpa}}}$ and stores the authentication in ${\\mathit {rcv_{\\mathit {hii}}\\_Auth_{\\mathit {mpa}}}}$ (i1).",
"From the MPA's identity, the HII obtains the MPA's public key (i2).",
"Then the HII verifies the MPA's authentication (i3).",
"If the verification succeeds, the HII authenticates to the MPA using public key authentication (i4).",
"The MPA stores the authentication in ${\\mathit {rcv_{\\mathit {mpa}}\\_Auth_{\\mathit {hii}}}}$ (m26).",
"Then the MPA obtains the HII's public key from the HII's identity (m27) and verifies the HII's authentication (m28).",
"If the verification succeeds, and the MPA verifies that the authentication is from the intended HII (m29), the MPA sends the receipt ${\\mathit {rcv_{\\mathit {mpa}}\\_\\mathit {PrescProof}}}$ and the patient pseudonym encrypted for the HII – verifiable encryption ${\\mathit {rcv_{\\mathit {mpa}}\\_vc}}_5={\\sf {dec}}({\\mathit {rcv_{\\mathit {mpa}}\\_c}}_5, {\\tt {sk}}_{\\mathit {mpa}})$ (m30).",
"The HII receives the receipt as ${\\mathit {rcv_{\\mathit {hii}}\\_ReceiptAck}}$ and the encrypted patient pseudonym for the HII as ${\\mathit {c_{\\mathit {hii}}\\_vc}}_5$ (i5).",
"The HII verifies the above two pieces of information (i6-i10).",
"If the verifications succeed, the HII decrypts the encryption ${\\mathit {c_{\\mathit {hii}}\\_Enc}}_5$ and obtains the patient's pseudonym (i11).",
"Finally, the HII sends an invoice of the prescription identity to the MPA (i12).",
"The MPA stores the invoice in ${\\mathit {rcv_{\\mathit {mpa}}\\_Invoice}}$ (m31).",
"Afterwards, the HII pays the MPA and updates the patient account.",
"As before, handling payment and storing information are beyond the scope of the DLV08 protocol and therefore, we do not model this stage."
],
[
"The full protocol.",
"In summary, the DLV08 protocol is composed as shown in Figure REF .",
"Figure: Overview of DLV08 protocol.The DLV08 protocol is modelled as the five roles $\\mathit {R}_{\\mathit {dr}}$ , $\\mathit {R}_{\\mathit {pt}}$ , $\\mathit {R}_{\\mathit {ph}}$ , $\\mathit {R}_{\\mathit {mpa}}$ , and $\\mathit {R}_{\\mathit {hii}}$ running in parallel (Figure REF ).",
"$\\begin{array}{rcl}\\mathit {\\mathit {P}_\\mathit {DLV08}}&:=& \\nu \\tilde{mc}.",
"\\mathit {init}.",
"(!\\mathit {R}_{\\mathit {pt}}\\mid !\\mathit {R}_{\\mathit {dr}}\\mid !\\mathit {R}_{\\mathit {ph}}\\mid !\\mathit {R}_{\\mathit {mpa}}\\mid !\\mathit {R}_{\\mathit {hii}})\\\\\\mathit {init}&:=&\\mbox{let}\\ {\\tt {pk}}_\\mathit {sso}={\\sf pk}({\\tt {sk}}_{\\mathit {sso}}) \\ \\mbox{in} \\ {\\sf {out}}({\\tt {ch}}, {\\tt {pk}}_\\mathit {sso})\\end{array}$ where $\\nu \\tilde{mc}$ represents global secrets ${\\tt {sk}}_{\\mathit {sso}}$ and private channels ${\\tt {ch}}_{hp}$ , ${\\tt {ch}}_{mp}$ , ${\\tt {ch}}_{phpt}$ ; process $\\mathit {init}$ initialises the settings of the protocol – publishing the public key ${\\tt {pk}}_\\mathit {sso}$ , so that the adversary knows it.",
"The roles $\\mathit {R}_{\\mathit {dr}}$ , $\\mathit {R}_{\\mathit {pt}}$ , $\\mathit {R}_{\\mathit {ph}}$ , $\\mathit {R}_{\\mathit {mpa}}$ and $\\mathit {R}_{\\mathit {hii}}$ are obtained by adding the settings of each role (see Section REF ) to the previously modelled corresponding process of the role as shown in Figure REF , Figure REF , Figure REF , Figure REF and Figure REF , respectivley.",
"Figure: The process for the DLV08 protocol.Each doctor has an identity ${{\\tt {Id}}_{\\mathit {dr}}}$ (rd1), a pseudonym ${{\\tt {Pnym}}_{\\mathit {dr}}}$ (rd2) and behaves like $\\mathit {P}_{\\mathit {dr}}$ (rd3) as shown in Figure REF .",
"The anonymous doctor credential is modelled by applying function ${\\sf drcred}$ on ${{\\tt {Id}}_{\\mathit {dr}}}$ and ${{\\tt {Pnym}}_{\\mathit {dr}}}$ .",
"Figure: The process for role doctor 𝑅 𝑑𝑟 \\mathit {R}_{\\mathit {dr}}.Each patient (as shown in Figure REF ) has an identity ${{\\tt {Id}}_{\\mathit {pt}}}$ (rt1), a pseudonym ${{\\tt {Pnym}}_{\\mathit {pt}}}$ , a social security status ${{\\tt {Sss}}}$ , a health expense account ${{\\tt {Acc}}}$ (rt2).",
"Unlike the identity and the pseudonym, which are attributes of a doctor, a doctor's ${{\\tt {Hii}}}$ is an association relation, and thus is modelled by reading in an HII identity to establish the relation (rt3).",
"In addition, a patient communicates with a pharmacist in each session.",
"Which pharmacist the patient communicates with is decided by reading in a pharmacist's public key (rt4).",
"From the pharmacist's public key, the patient can obtain the pharmacist's identity (rt5).",
"Finally the patient behaves as $\\mathit {P}^{\\prime }_{\\mathit {pt}}$ (rt6).",
"Figure: The process for role patient 𝑅 𝑝𝑡 \\mathit {R}_{\\mathit {pt}}.Each pharmacist has a secret key ${\\tt {sk}}_{\\mathit {ph}}$ (rh1), a public key ${\\tt {pk}}_\\mathit {ph}$ (rh2) and an identity ${\\tt {Id}}_{\\mathit {ph}}$ (rh3) as shown in Figure REF .",
"The public key of a pharmacist is published over channel ${\\tt {ch}}$ , so that the adversary knows it (rh4).",
"In addition, the public key is sent to the patients via private channel ${\\tt {ch}}_{phpt}$ (rh4), so that the patients can choose one to communicate with.",
"In each session, the pharmacist communicates with an MPA.",
"Which MPA the pharmacist communicates with is decided by reading in a public key of MPA (rh5).",
"From the public key, the pharmacist can obtain the identity of the MPA (rh6).",
"Finally, the pharmacist behave as $\\mathit {P}^{\\prime }_{\\mathit {ph}}$ .",
"Figure: The process for role pharmacist 𝑅 𝑝ℎ \\mathit {R}_{\\mathit {ph}}.Each MPA has a secret key ${\\tt {sk}}_{\\mathit {mpa}}$ (rm1), a public key ${\\tt {pk}}_\\mathit {mpa}$ (rm2) and an identity ${{\\tt {Id}}_{\\mathit {mpa}}}$ (rm3).",
"The MPA publishes his public key as well as sends his public key to pharmacists (rm4), and behaves as $\\mathit {P}_{\\mathit {mpa}}$ (rm5) as shown in Figure REF .",
"Figure: The process for role MPA 𝑅 𝑚𝑝𝑎 \\mathit {R}_{\\mathit {mpa}}.Similar to MPA, each HII (Figure REF ) has a secret key ${\\tt {sk}}_{\\mathit {hii}}$ (ri1), a public key ${\\tt {pk}}_\\mathit {hii}$ (ri2) and an identity ${\\tt {Id}}_{\\mathit {hii}}$ (ri3).",
"The public key is revealed to the adversary via channel ${\\tt {ch}}$ and sent to the patients via channel ${\\tt {ch}}_{hp}$ (ri4).",
"Then the HII behaves as $\\mathit {P}_{\\mathit {hii}}$ (ri5).",
"Figure: The process for role HII 𝑅 ℎ𝑖𝑖 \\mathit {R}_{\\mathit {hii}}."
],
[
"Analysis of DLV08",
"In this section, we analyse whether DLV08 satisfies the following properties: secrecy of patient and doctor information, authentication, (strong) patient and doctor anonymity, (strong) patient and doctor untraceability, (enforced) prescription privacy, and independence of (enforced) prescription privacy.",
"The properties doctor anonymity and untraceability are not required by the protocol but are still interesting to analyse.",
"The verification is supported by the automatic verification tool ProVerif [15], [16], [17].",
"The tool has been used to verify many secrecy, authentication and privacy properties, e.g., see [1], [2], [40], [11], [25].",
"The verification results for secrecy are summarised in Table REF , and those for authentication in Table REF .",
"As we are foremost interested in privacy properties, the verification results for privacy properties, and suggestions for improvements are discussed in Section .",
"Table REF summarises those results, causes of privacy weaknesses, suggested improvements, and the effect of the improvements.",
"In this section, we show the verification results of properties from basic to more complicated.",
"A flaw which fails a basic property is likely to fail a more complicated property as well.",
"Thus we first show flaws of basic properties and how to fix them, then we show new flaws of complicated properties based on the fixed model."
],
[
"ProVerif",
"ProVerif takes a protocol and a property modelled in the applied pi calculus as input (the input language (untyped version) differs slightly from applied pi, see [14]), and returns either a proof of correctness or potential attacks.",
"A protocol modelled in the applied pi calculus is translated to Horn clauses [32].",
"The adversary's capabilities are added as Horn clauses as well.",
"Using these clauses, verification of secrecy and authentication is equivalent to determining whether a certain clause is derivable from the set of initial clauses.",
"Secrecy of a term is defined as the adversary cannot obtain the term by communicating with the protocol and/or applying cryptography on the output of the protocol [1].",
"The secrecy property is modelled as a predicate in ProVerif: the query of secrecy of term $M$ is $``attacker: M\"$ [15].",
"ProVerif determines whether the term $M$ can be inferred from the Horn clauses representing the adversary knowledge.",
"Authentication is captured by correspondence properties of events in processes: if one event happens the other event must have happened before [2], [18].",
"Events are tags which mark important stages reached by the protocol.",
"Events have arguments, which allow us to express relationships between the arguments of events.",
"A correspondence property is a formula of the form: $\\mathit {ev}: \\bar{f}(M)==>\\mathit {ev}: \\bar{g}(N)$ .",
"That is, in any process, if event $\\bar{f}(M)$ has been executed, then the event $\\bar{g}(N)$ must have been previously executed, and any relationship between $M$ and $N$ must be satisfied.",
"To capture stronger authentication, where an injective relationship between executions of participants is required, an injective correspondence property $\\mathit {evinj}: \\bar{f}(M)==>\\mathit {evinj}: \\bar{g}(N)$ is defined: in any process, for each execution of event $\\bar{f}(M)$ , there is a distinct earlier execution of the event $\\bar{g}(N)$ , and the relationship between $M$ and $N$ is satisfied.",
"In addition, ProVerif provides automatic verification of labelled bisimilarity of two processes which differ only in the choice of some terms [9].",
"An operation $``choice[a, b]\"$ is introduced to model the different choices of a term in the two processes.",
"Using this operation, the two processes can be written as one process – a bi-process.",
"Example 8 To verify the equivalence $ \\nu {\\tt {a}}.",
"\\nu {\\tt {b}}.",
"{\\sf {out}}({\\tt {ch}}, {\\tt {a}}).",
"{\\sf {out}}({\\tt {ch}}, {\\tt {e}})\\approx _{\\ell }\\nu {\\tt {a}}.",
"\\nu {\\tt {b}}.",
"{\\sf {out}}({\\tt {ch}}, {\\tt {b}}).",
"{\\sf {out}}({\\tt {ch}}, {\\tt {d}})$ where ${\\tt {ch}}$ is a public channel, ${\\tt {e}}$ and ${\\tt {d}}$ are two free names, we can query the following bi-process in ProVerif: $\\mathit {P}:=\\nu {\\tt {a}}.",
"\\nu {\\tt {b}}.",
"{\\sf {out}}({\\tt {ch}}, {\\sf choice}[{\\tt {a}}, {\\tt {b}}]).",
"{\\sf {out}}({\\tt {ch}}, {\\sf choice}[{\\tt {e}}, {\\tt {d}}]).$ Using the first parameter of all $``choice\"$ operations in a bi-process $\\mathit {P}$ , we obtain one side of the equivalence (denoted as ${\\sf fst}(\\mathit {P})$ ); using the second parameters, we obtain the other side (denoted as ${\\sf snd}(\\mathit {P})$ ).",
"Example 9 For the bi-process in Example REF , using the first parameter to replace each $``choice\"$ operation, we obtain $\\nu {\\tt {a}}.",
"\\nu {\\tt {b}}.",
"{\\sf {out}}({\\tt {ch}}, {\\tt {a}}).",
"{\\sf {out}}({\\tt {ch}}, {\\tt {e}}),$ which is the left-hand side of the equivalence in Example REF ; using the second parameter to replace each $``choice\"$ operation, we obtain $\\nu {\\tt {a}}.",
"\\nu {\\tt {b}}.",
"{\\sf {out}}({\\tt {ch}}, {\\tt {b}}).",
"{\\sf {out}}({\\tt {ch}}, {\\tt {d}}),$ which is the right-hand side of the equivalence.",
"Given a bi-process $\\mathit {P}$ , ProVerif tries to prove that ${\\sf fst}(\\mathit {P})$ is labelled bisimilar to ${\\sf snd}(\\mathit {P})$ .",
"The fundamental idea is that ProVerif reasons on traces of the bi-process $\\mathit {P}$ : the bi-process $\\mathit {P}$ reduces when ${\\sf fst}(\\mathit {P})$ and ${\\sf snd}(\\mathit {P})$ reduce in the same way; when ${\\sf fst}(\\mathit {P})$ and ${\\sf snd}(\\mathit {P})$ do something that may differentiate them, the bi-process is stuck.",
"Formally, ProVerif shows that the bi-process $\\mathit {P}$ is uniform, that is, if ${\\sf fst}(\\mathit {P})$ can do a reduction to some $\\mathit {Q}_1$ , then the bi-process can do a reduction to some bi-process $\\mathit {Q}$ , such that ${\\sf fst}(\\mathit {Q})\\equiv \\mathit {Q}_1$ and symmetrically for ${\\sf snd}(\\mathit {P})$ taking a reduction to $\\mathit {Q}_2$ .",
"When the bi-process $\\mathit {P}$ always remains uniform after reduction and addition of an adversary, ${\\sf fst}(\\mathit {P})$ is labelled bisimilar to ${\\sf snd}(\\mathit {P})$ ."
],
[
"Secrecy of patient and doctor information",
"The DLV08 protocol claims to satisfy the following requirement: any party involved in the prescription processing workflow should not know the information of a patient and a doctor unless the information is intended to be revealed in the protocol.",
"In [24], this requirement is considered as an access control requirement.",
"We argue that ensuring the requirement with access control is not sufficient when considering a communication network.",
"A dishonest party could potentially act as an attacker from the network (observing the network and manipulating the protocol) and obtain information which he should not access.",
"It is not clearly stated which (if any) of the involved parties are honest.",
"We find that in such a way, some patient and doctor information may be revealed to parties who should not know the information.",
"We formalise the requirement as standard secrecy of patient and doctor information with respect to the Dolev-Yao adversary.",
"Standard secrecy of a term captures the idea that the adversary cannot access to that term (see Section REF ).",
"If a piece of information is known to the adversary, a dishonest party acting like the adversary can access to the information.",
"We do not consider strong secrecy, as it is unclear whether the information is guessable.",
"Recall that standard secrecy of a term $M$ is formally defined as a predicate $``attacker: M\"$ (see Section REF ).",
"By replacing $M$ with the listed private information, we obtain the formal definition of the secrecy of patient and doctor information.",
"The list of private information of patients and doctors that needs to be protected is: patient identity (${{\\tt {Id}}_{\\mathit {pt}}}$ ), doctor identity (${{\\tt {Id}}_{\\mathit {dr}}}$ ), patient pseudonym (${{\\tt {Pnym}}_{\\mathit {pt}}}$ ), doctor pseudonym (${{\\tt {Pnym}}_{\\mathit {dr}}}$ ), a patient's social security status (${{\\tt {Sss}}}$ ), and a patient's health insurance institute (${{\\tt {Hii}}}$ ).",
"Although DLV08 does not explicitly require it, we additionally analyse secrecy of the health expense account ${{\\tt {Acc}}}$ of a patient.",
"$\\begin{array}{lll}query\\ attacker: {{\\tt {Id}}_{\\mathit {pt}}}&query\\ attacker: {{\\tt {Id}}_{\\mathit {dr}}}&query\\ attacker: {{\\tt {Pnym}}_{\\mathit {pt}}}\\\\query\\ attacker: {{\\tt {Pnym}}_{\\mathit {dr}}}&query\\ attacker: {{\\tt {Sss}}}&query\\ attacker: {{\\tt {Hii}}}\\\\query\\ attacker: {{\\tt {Acc}}}\\end{array}$"
],
[
"Verification result.",
"We query the standard secrecy of the set of private information using ProVerif [15].",
"The verification results (see Table REF ) show that a patient's identity, pseudonym, health expense account, health insurance institute and identity of a doctor (${{\\tt {Id}}_{\\mathit {pt}}}$ , ${{\\tt {Pnym}}_{\\mathit {pt}}}$ , ${{\\tt {Hii}}}$ ${{\\tt {Acc}}}$ , ${{\\tt {Id}}_{\\mathit {dr}}}$ ) satisfy standard secrecy; a patient's social security status ${{\\tt {Sss}}}$ and a doctor's pseudonym ${{\\tt {Pnym}}_{\\mathit {dr}}}$ do not satisfy standard secrecy.",
"The ${{\\tt {Sss}}}$ is revealed by the proof of social security status from the patient to the pharmacist.",
"The ${{\\tt {Pnym}}_{\\mathit {dr}}}$ is revealed by the revealing of both the commitment of the patient's pseudonym and the open key to the commitment during the communication between the patient and the doctor.",
"Fixing secrecy of a patient's social security status requires that the proof of social security status only reveals the status to the pharmacist.",
"Since how a social security status is represented and what the pharmacist needs to verify are not clear, we cannot give explicit suggestions.",
"However, if the social security status is a number, and the pharmacist only needs to verify that the number is higher than a certain threshold, the patient can prove it using zero-knowledge proof without revealing the number; if the pharmacist needs to verify the exact value of the status, one way to fix its secrecy is that the pharmacist and the patient agree on a session key and the status is encrypted using the key.",
"Similarly, a way to fix the secrecy of ${{\\tt {Pnym}}_{\\mathit {dr}}}$ is to encrypt the opening information using the agreed session key.",
"Table: Verification results of secrecy for patients and doctors."
],
[
"Patient and doctor authentication",
"The protocol claims that all parties should be able to properly authenticate each other.",
"Compared to authentications between public entities, pharmacists, MPA and HII, we focus on authentications between patients and doctors, as patients and doctors use anonymous authentication.",
"Authentications between patients and pharmacists are sketched as well.",
"The DLV08 claims that no party should be able to succeed in claiming a false identity, or false information about his identity.",
"That is the adversary cannot pretend to be a patient or a doctor."
],
[
"Authentication from a patient to a doctor.",
"The authentication from a patient to a doctor is defined as when the doctor finishes his process and believes that he prescribed medicine for a patient, then the patient did ask the doctor for prescription.",
"To verify the authentication of a patient, we add an event ${\\sf EndDr}({\\mathit {c\\_Cred_{\\mathit {pt}}}},{\\mathit {c\\_Comt_{\\mathit {pt}}}})$ at the end of the doctor process (after line d10), meaning the doctor believes that he prescribed medicine for a patient who has a credential ${\\mathit {c\\_Cred_{\\mathit {pt}}}}$ and committed ${\\mathit {c\\_Comt_{\\mathit {pt}}}}$ ; and add an event ${\\sf StartPt}({\\sf ptcred}({{\\tt {Id}}_{\\mathit {pt}}},{{\\tt {Pnym}}_{\\mathit {pt}}},{{\\tt {Hii}}},{{\\tt {Sss}}}, {{\\tt {Acc}}}),{\\sf commit}({{\\tt {Id}}_{\\mathit {pt}}}, {{\\tt {r}}_\\mathit {\\mathit {pt}}}))$ in the patient process (between line t4 and line t5), meaning that the patient did ask for a prescription.",
"The definition is captured by the following correspondence property: $\\mathit {ev(inj)}: {\\sf EndDr}(x,y) ==> \\mathit {ev(inj)}: {\\sf StartPt}(x,y),$ meaning that when the event ${\\sf EndDr}$ is executed, there is a (unique) event ${\\sf StartPt}$ has been executed before."
],
[
"Authentication from a doctor to a patient.",
"Similarly, the authentication from a doctor to a patient is defined as when the patient believes that he visited a doctor, the doctor did prescribe medicine for the patient.",
"To authenticate a doctor, we add to the patient process an event ${\\sf EndPt}({\\mathit {c\\_Cred_{\\mathit {dr}}}},{\\mathit {c\\_Comt_{\\mathit {dr}}}},{\\mathit {c\\_\\mathit {presc}}},{\\mathit {c\\_\\mathit {PrescriptID}}})$ (after line t9), and add an event ${\\sf StartDr}({\\sf drcred}({{\\tt {Pnym}}_{\\mathit {dr}}},{{\\tt {Id}}_{\\mathit {dr}}}),{\\sf commit}({{\\tt {Pnym}}_{\\mathit {dr}}},{\\tt {r}}_{\\mathit {dr}}),{\\tt {presc}},\\mathit {PrescriptID})$ in the doctor process (between line d9 and line d10), then query $\\mathit {ev(inj)}: {\\sf EndPt}(x,y,z,t) ==> \\mathit {ev(inj)}: {\\sf StartDr}(x,y,z,t).$"
],
[
"Authentication from a patient to a pharmacist.",
"The authentication from a patient to a doctor is defined as when the pharmacist finishes a session and believes that he communicates with a patient, who is identified with the credential ${\\mathit {c_{\\mathit {ph}}\\_Cred_{\\mathit {pt}}}}$ , then the patient with the credential did communicate with the pharmacist.",
"to verify the authentication of a patient, we add to the pharmacist process the event ${\\sf EndPh}({\\mathit {c_{\\mathit {ph}}\\_Cred_{\\mathit {pt}}}})$ (after line h23), add to the patient process ${\\sf StartPtph}({\\sf ptcred}({{\\tt {Id}}_{\\mathit {pt}}},{{\\tt {Pnym}}_{\\mathit {pt}}},{{\\tt {Hii}}},{{\\tt {Sss}}}, {{\\tt {Acc}}}))$ (between line t13 and line t14), and query $\\mathit {ev(inj)}: {\\sf EndPh}(x)==>\\mathit {ev(inj)}:{\\sf StartPtph}(x)$ ."
],
[
"Authentication from a pharmacist to a patient.",
"The authentication from a pharmacist to a patient is defined as when the patient finishes a session and believes that he communicates with a pharmacist with the identity ${\\mathit {c_{\\mathit {pt}}\\_Id_{\\mathit {ph}}}}$ , then the pharmacist is indeed the one who communicated with the patient.",
"To verify this authentication, we add the event ${\\sf EndPtph}({\\mathit {c_{\\mathit {pt}}\\_Id_{\\mathit {ph}}}})$ into the patient process (after line t26), add the event ${\\sf StartPh}({\\tt {Id}}_{\\mathit {ph}})$ into the pharmacist process (between line h1 and line h2), and query $\\mathit {ev(inj)}:{\\sf EndPtph}(x) ==> \\mathit {ev(inj)}:{\\sf StartPh}(x)$ .",
"In addition, we add the conditional evaluation $\\mbox{if}\\ {\\mathit {rcv\\_Invoice}}={\\sf inv}({\\mathit {c\\_\\mathit {PrescriptID}}}) \\ \\mbox{then} $ before the end ${\\sf EndPtph}({\\mathit {c_{\\mathit {pt}}\\_Id_{\\mathit {ph}}}})$ in the patient process to capture that the patient checks the correctness of the invoice."
],
[
"Verification results.",
"The queries are verified using ProVerif.",
"The verification results show that doctor authentication, both injective and non-injective, succeed; non-injective patient authentication succeeds and injective patient authentication fails.",
"The failure is caused by a replay attack from the adversary.",
"That is, the adversary can impersonate a patient by replaying old messages from the patient.",
"This authentication flaw leads to termination of the successive procedure, the patient-pharmacist sub-process.",
"We verified authentication between patients and pharmacists as well.",
"Non-injective patient authentication succeeds, whereas injective patient authentication fails.",
"This means that the messages received by a pharmacist are from the correct patient, but not necessarily from this communication session.",
"Neither non-injective nor injective pharmacist authentication succeeds: the adversary can record and replay the first message which is sent from a pharmacist to a patient, and pass the authentication by pretending to be that pharmacist.",
"In addition, the adversary can prepare the second message sending from a pharmacist to a patient, and thus does not need to replay the second message.",
"Since the adversary alters messages, non-injective pharmacist authentication fails.",
"The verification results are summarised in Table REF .",
"The reason that injective patient authentication fails for both doctors and pharmacists is that they suffer from replay attacks.",
"One possible solution approach is to add a challenge sent from the doctor (respectively, the pharmacist) to the patient.",
"Then, when the patient authenticates to the doctor or pharmacist, the patient includes this challenge in the proofs.",
"This approach assures that the proof is freshly generated.",
"Therefore, this prevents the adversary replaying old messages.",
"The reason that (injective and non-injective) authentication from a pharmacist to a patient fails is that the adversary can generate an invoice to replace the one from the real pharmacist.",
"One solution is for the pharmacist to sign the invoice.",
"Table: Verification results of authentication of patients and doctors."
],
[
"Authentications between public entities.",
"The public entities – pharmacists, MPAs, HIIs, authenticate each other using public key authentication.",
"The authentication is often used to agree on a way for the later communication.",
"Since it is not mentioned in the original protocol that a key or a communication channel is established during authentication, we assume that the later message exchanges are over public channels, to model the worst case.",
"In this model, the authentications between public entities are obviously flawed, since the adversary can reuse messages from other sessions.",
"The flaws are confirmed by the verification results using ProVerif."
],
[
"(Strong) patient and doctor anonymity",
"The DLV08 protocol claims that no party should be able to determine the identity of a patient.",
"We define (strong) patient anonymity to capture the requirement.",
"Note that in the original paper of the DLV08 protocol, the terminology of the privacy property for capturing this requirement is patient untraceability.",
"Our definition of untraceability (Definition REF ) has different meaning from theirs (for details, see Section REF ).",
"Also note that the satisfaction of standard secrecy of patient identity does not fully capture this requirement, as the adversary can still guess about it."
],
[
"Patient and doctor anonymity.",
"Doctor anonymity is defined as in Definition REF .",
"Patient anonymity can be defined in a similar way by replacing the role of doctor with the role of patient.",
"$\\begin{array}{rl}\\mathcal {C}_\\mathit {eh}[\\mathit {init}_{\\mathit {pt}}\\mathit {\\lbrace {\\tt {t_A}}/{\\mathit {I}d_{\\mathit {pt}}}\\rbrace }.",
"!\\mathit {P}_{\\mathit {pt}}\\mathit {\\lbrace {\\tt {t_A}}/{\\mathit {I}d_{\\mathit {pt}}}\\rbrace }]\\approx _{\\ell }\\mathcal {C}_\\mathit {eh}[\\mathit {init}_{\\mathit {pt}}\\mathit {\\lbrace {\\tt {t_B}}/{\\mathit {I}d_{\\mathit {pt}}}\\rbrace }.",
"!\\mathit {P}_{\\mathit {pt}}\\mathit {\\lbrace {\\tt {t_B}}/{\\mathit {I}d_{\\mathit {pt}}}\\rbrace }].\\end{array}$ To verify doctor/patient anonymity, is to check the satisfiability of the corresponding equivalence between processes in the definition.",
"This is done by modelling the two processes on two sides of the equivalence as a bi-process, and verify the bi-process using ProVerif.",
"Recall that a bi-process models two processes sharing the same structure and differing only in terms or destructors.",
"The two processes are written as one process with choice-constructors which tells ProVerif the spots where the two processes differ.",
"The bi-process for verifying doctor anonymity is $\\nu \\tilde{mc}.",
"\\mathit {init}.",
"(!\\mathit {R}_{\\mathit {pt}}\\mid !\\mathit {R}_{\\mathit {dr}}\\mid !\\mathit {R}_{\\mathit {ph}}\\mid !\\mathit {R}_{\\mathit {mpa}}\\mid !\\mathit {R}_{\\mathit {hii}}\\mid (\\nu {{\\tt {Pnym}}_{\\mathit {dr}}}.\\mbox{let}\\ {{\\tt {Id}}_{\\mathit {dr}}}={\\sf choice}[{\\tt {d_A}}, {\\tt {d_B}}] \\ \\mbox{in} \\ !\\mathit {P}_{\\mathit {dr}}),$ and the bi-process for verifying patient anonymity is $\\begin{array}{l}\\nu \\tilde{mc}.",
"\\mathit {init}.",
"(!\\mathit {R}_{\\mathit {pt}}\\mid !\\mathit {R}_{\\mathit {dr}}\\mid !\\mathit {R}_{\\mathit {ph}}\\mid !\\mathit {R}_{\\mathit {mpa}}\\mid !\\mathit {R}_{\\mathit {hii}}\\mid (\\mbox{let}\\ {{\\tt {Id}}_{\\mathit {pt}}}={\\sf choice}[{\\tt {t_A}}, {\\tt {t_B}}] \\ \\mbox{in} \\\\\\hspace{38.25pt}\\nu {{\\tt {Pnym}}_{\\mathit {pt}}}.\\nu {{\\tt {Sss}}}.\\nu {{\\tt {Acc}}}.",
"{\\sf {in}}({\\tt {ch}}_{hp}, {{\\tt {Hii}}}).\\mbox{let}\\ \\mathit {c_{\\mathit {pt}}\\_pk_\\mathit {hii}}={\\sf key}({{\\tt {Hii}}}) \\ \\mbox{in} \\ !\\mathit {P}_{\\mathit {pt}}).\\end{array}$ Since the doctor identity is a secret information, we define ${\\tt {d_A}}$ and ${\\tt {d_B}}$ as private names $\\mathit {private}\\ \\mathit {free}\\ {\\tt {d_A}}.",
"$$\\mathit {private}\\ \\mathit {free}\\ {\\tt {d_B}}$ .",
"In addition, we consider a stronger version, in which the adversary knows the two doctor identities a priori, i.e., we verify whether the adversary can distinguish two known doctors as well.",
"This is modelled by defining the two doctor identities as free names, $\\mathit {free}\\ {\\tt {d_A}}.",
"\\mathit {free}\\ {\\tt {d_B}}$ .",
"Similarly, we verified two versions of patient anonymity - in one version, the adversary does not know the two patient identities, and in the other version, the adversary initially knows the two patient identities."
],
[
"Strong patient and doctor anonymity.",
"Strong doctor anonymity is defined as in Definition REF .",
"By replacing the role of doctor with the role of patient, we obtain the definition of strong patient anonymity.",
"The bi-process for verifying strong doctor anonymity is $\\begin{array}{l}\\mathit {free}\\ {\\tt {d_A}};\\\\\\nu \\tilde{mc}.",
"\\mathit {init}.",
"(!\\mathit {R}_{\\mathit {pt}}\\mid !\\mathit {R}_{\\mathit {dr}}\\mid !\\mathit {R}_{\\mathit {ph}}\\mid !\\mathit {R}_{\\mathit {mpa}}\\mid !\\mathit {R}_{\\mathit {hii}}\\mid \\\\\\hspace{38.25pt}(\\nu {\\tt {d_B}}.\\nu {\\tt {nPnym}}_{\\mathit {dr}}.\\mbox{let}\\ {{\\tt {Pnym}}_{\\mathit {dr}}}={\\tt {nPnym}}_{\\mathit {dr}}\\ \\mbox{in} \\ !",
"(\\mbox{let}\\ {{\\tt {Id}}_{\\mathit {dr}}}={\\sf choice}[{\\tt {d_B}}, {\\tt {d_A}}] \\ \\mbox{in} \\ \\mathit {P}_{\\mathit {dr}}))),\\end{array}$ and the bi-process for verifying strong patient anonymity is $\\begin{array}{l}\\mathit {free}\\ {\\tt {t_A}};\\\\\\nu \\tilde{mc}.",
"\\mathit {init}.",
"(!\\mathit {R}_{\\mathit {pt}}\\mid !\\mathit {R}_{\\mathit {dr}}\\mid !\\mathit {R}_{\\mathit {ph}}\\mid !\\mathit {R}_{\\mathit {mpa}}\\mid !\\mathit {R}_{\\mathit {hii}}\\mid (\\nu {\\tt {t_B}}.\\nu {{\\tt {Pnym}}_{\\mathit {pt}}}.\\nu {{\\tt {Sss}}}.\\nu {{\\tt {Acc}}}.\\\\\\hspace{38.25pt}{\\sf {in}}({\\tt {ch}}_{hp}, {{\\tt {Hii}}}).\\mbox{let}\\ \\mathit {c_{\\mathit {pt}}\\_pk_\\mathit {hii}}={\\sf key}({{\\tt {Hii}}}) \\ \\mbox{in} \\ !",
"(\\mbox{let}\\ {{\\tt {Id}}_{\\mathit {pt}}}={\\sf choice}[{\\tt {t_B}}, {\\tt {t_A}}] \\ \\mbox{in} \\ \\mathit {P}_{\\mathit {pt}}))).\\end{array}$ Note that by definition, the identities ${\\tt {d_A}}$ and ${\\tt {t_B}}$ is known by the adversary.",
"In the first bi-process, by choosing ${\\tt {d_B}}$ , we obtain $\\begin{array}{l}\\mathit {free}\\ {\\tt {d_A}};\\\\\\nu \\tilde{mc}.",
"\\mathit {init}.",
"(!\\mathit {R}_{\\mathit {pt}}\\mid !\\mathit {R}_{\\mathit {dr}}\\mid !\\mathit {R}_{\\mathit {ph}}\\mid !\\mathit {R}_{\\mathit {mpa}}\\mid !\\mathit {R}_{\\mathit {hii}}\\mid \\\\\\hspace{38.25pt}(\\nu {\\tt {d_B}}.\\nu {\\tt {nPnym}}_{\\mathit {dr}}.\\mbox{let}\\ {{\\tt {Pnym}}_{\\mathit {dr}}}={\\tt {nPnym}}_{\\mathit {dr}}\\ \\mbox{in} \\ !",
"(\\mbox{let}\\ {{\\tt {Id}}_{\\mathit {dr}}}={\\tt {d_B}}\\ \\mbox{in} \\ \\mathit {P}_{\\mathit {dr}}))).\\end{array}$ Since ${\\tt {d_A}}$ never appears in the remaining process, removing the declaration “$\\mathit {free}\\ {\\tt {d_A}};$ \" does not affect the process.",
"Since process “$\\nu {\\tt {d_B}}.\\nu {\\tt {nPnym}}_{\\mathit {dr}}.\\mbox{let}\\ {{\\tt {Pnym}}_{\\mathit {dr}}}={\\tt {nPnym}}_{\\mathit {dr}}\\ \\mbox{in} \\ !",
"(\\mbox{let}\\ {{\\tt {Id}}_{\\mathit {dr}}}={\\tt {d_B}}\\ \\mbox{in} \\ \\mathit {P}_{\\mathit {dr}})$ \" essentially renames the doctor role process “$\\mathit {R}_{\\mathit {dr}}$ \" (Figure REF ) - renaming ${{\\tt {Id}}_{\\mathit {dr}}}$ as ${\\tt {d_B}}$ and renaming ${{\\tt {Pnym}}_{\\mathit {dr}}}$ as ${\\tt {nPnym}}_{\\mathit {dr}}$ , we have that the above process is structurally equivalent to (using rule $\\textsc {REPL}$ ) $\\begin{array}{l}\\nu \\tilde{mc}.",
"\\mathit {init}.",
"(!\\mathit {R}_{\\mathit {pt}}\\mid !\\mathit {R}_{\\mathit {dr}}\\mid !\\mathit {R}_{\\mathit {ph}}\\mid !\\mathit {R}_{\\mathit {mpa}}\\mid !\\mathit {R}_{\\mathit {hii}}),\\end{array}$ which is the left-hand side of Definition REF - the $\\mathit {\\mathit {P}_\\mathit {DLV08}}$ in the case study.",
"On the other hand, by choosing ${\\tt {d_A}}$ , we obtain process $\\begin{array}{l}\\mathit {free}\\ {\\tt {d_A}};\\\\\\nu \\tilde{mc}.",
"\\mathit {init}.",
"(!\\mathit {R}_{\\mathit {pt}}\\mid !\\mathit {R}_{\\mathit {dr}}\\mid !\\mathit {R}_{\\mathit {ph}}\\mid !\\mathit {R}_{\\mathit {mpa}}\\mid !\\mathit {R}_{\\mathit {hii}}\\mid \\\\\\hspace{38.25pt}(\\nu {\\tt {d_B}}.\\nu {\\tt {nPnym}}_{\\mathit {dr}}.\\mbox{let}\\ {{\\tt {Pnym}}_{\\mathit {dr}}}={\\tt {nPnym}}_{\\mathit {dr}}\\ \\mbox{in} \\ !",
"(\\mbox{let}\\ {{\\tt {Id}}_{\\mathit {dr}}}={\\tt {d_A}}\\ \\mbox{in} \\ \\mathit {P}_{\\mathit {dr}}))).\\end{array}$ Since ${\\tt {d_B}}$ only appears in the sub-process “$\\nu {\\tt {d_B}}.$ \" which generates ${\\tt {d_B}}$ and never appears in the remaingin process, the process is structurally equivalent to (applying rule $\\textsc {NEW}-\\textsc {PAR}$ ) $\\begin{array}{l}\\mathit {free}\\ {\\tt {d_A}};\\\\{\\bf \\nu {\\tt {d_B}}.",
"}\\nu \\tilde{mc}.",
"\\mathit {init}.",
"(!\\mathit {R}_{\\mathit {pt}}\\mid !\\mathit {R}_{\\mathit {dr}}\\mid !\\mathit {R}_{\\mathit {ph}}\\mid !\\mathit {R}_{\\mathit {mpa}}\\mid !\\mathit {R}_{\\mathit {hii}}\\mid \\\\\\hspace{38.25pt}(\\nu {\\tt {nPnym}}_{\\mathit {dr}}.\\mbox{let}\\ {{\\tt {Pnym}}_{\\mathit {dr}}}={\\tt {nPnym}}_{\\mathit {dr}}\\ \\mbox{in} \\ !",
"(\\mbox{let}\\ {{\\tt {Id}}_{\\mathit {dr}}}={\\tt {d_A}}\\ \\mbox{in} \\ \\mathit {P}_{\\mathit {dr}}))).\\end{array}$ The above process is structurally equivalent to (proved later) $\\begin{array}{l}\\mathit {free}\\ {\\tt {d_A}};\\\\\\nu \\tilde{mc}.",
"\\mathit {init}.",
"(!\\mathit {R}_{\\mathit {pt}}\\mid !\\mathit {R}_{\\mathit {dr}}\\mid !\\mathit {R}_{\\mathit {ph}}\\mid !\\mathit {R}_{\\mathit {mpa}}\\mid !\\mathit {R}_{\\mathit {hii}}\\mid \\\\\\hspace{38.25pt}(\\nu {\\tt {nPnym}}_{\\mathit {dr}}.\\mbox{let}\\ {{\\tt {Pnym}}_{\\mathit {dr}}}={\\tt {nPnym}}_{\\mathit {dr}}\\ \\mbox{in} \\ !",
"(\\mbox{let}\\ {{\\tt {Id}}_{\\mathit {dr}}}={\\tt {d_A}}\\ \\mbox{in} \\ \\mathit {P}_{\\mathit {dr}}))),\\end{array}$ which is the right-hand side of Definition REF , where ${\\tt {d_A}}$ is a free name.",
"This structural equivalent relation is proved as follows.",
"Assuming the above process is $P$ (i.e., $P=\\nu \\tilde{mc}.",
"\\mathit {init}.",
"(!\\mathit {R}_{\\mathit {pt}}\\mid !\\mathit {R}_{\\mathit {dr}}\\mid !\\mathit {R}_{\\mathit {ph}}\\mid !\\mathit {R}_{\\mathit {mpa}}\\mid !\\mathit {R}_{\\mathit {hii}}\\mid (\\nu {\\tt {nPnym}}_{\\mathit {dr}}.\\mbox{let}\\ {{\\tt {Pnym}}_{\\mathit {dr}}}={\\tt {nPnym}}_{\\mathit {dr}}\\ \\mbox{in} \\ !",
"(\\mbox{let}\\ {{\\tt {Id}}_{\\mathit {dr}}}={\\tt {d_A}}\\ \\mbox{in} \\ \\mathit {P}_{\\mathit {dr}})))$ ), by applying rule $\\textsc {PAR}-0$ , we have $P\\equiv P\\mid 0$ .",
"By rule $\\textsc {NEW}-0$ , $\\nu {\\tt {d_B}}.0 \\equiv 0$ .",
"Thus, $P\\equiv P\\mid \\nu {\\tt {d_B}}.0$ .",
"Since ${\\tt {d_B}}$ never appears in the process $P$ , i.e., ${\\tt {d_B}}\\notin {\\sf {fn}}(P) \\cup {\\sf {fv}}(P)$ , by applying rule $\\textsc {NEW}-\\textsc {PAR}$ , we have $P\\mid \\nu {\\tt {d_B}}.0 \\equiv \\nu {\\tt {d_B}}.",
"(P\\mid 0) \\equiv \\nu {\\tt {d_B}}.",
"P$ .",
"Therefore, $P\\equiv \\nu {\\tt {d_B}}.",
"P$ ."
],
[
"Verification result.",
"The bi-processes are verified using ProVerif.",
"The verification results show that patient anonymity (with and without revealed patient identities a priori) and strong patient anonymity are satisfied; doctor anonymity is satisfied; neither doctor anonymity with revealed doctor identities nor strong doctor anonymity is satisfied.",
"For strong doctor anonymity, the adversary can distinguish a process initiated by an unknown doctor and a known doctor.",
"Given a doctor process, where the doctor has identity ${\\tt {d_A}}$ , pseudonym ${{\\tt {Pnym}}_{\\mathit {dr}}}$ , and credential ${\\sf drcred}({{\\tt {Pnym}}_{\\mathit {dr}}},{\\tt {d_A}})$ , the terms ${{\\tt {Pnym}}_{\\mathit {dr}}}$ and ${\\sf drcred}({{\\tt {Pnym}}_{\\mathit {dr}}},{\\tt {d_A}})$ are revealed.",
"We assume that the adversary knows another doctor identity ${\\tt {d_B}}$ .",
"The adversary can fake an anonymous authentication by faking the zero-knowledge proof as ${\\sf zk}(({{\\tt {Pnym}}_{\\mathit {dr}}},{\\tt {d_B}}),{\\sf drcred}({{\\tt {Pnym}}_{\\mathit {dr}}},{\\tt {d_A}}))$ .",
"If the zero-knowledge proof passes the corresponding verification $\\mbox{\\sf Vfy-{zk}}_{\\sf Auth_{\\mathit {dr}}}$ by the patient, then the adversary knows that the doctor process is executed by the doctor ${\\tt {d_B}}$ .",
"Otherwise, not.",
"For the same reason, doctor anonymity fails the verification.",
"Both flaws can be fixed by requiring a doctor to generate a new credential in each session (s4')."
],
[
"(Strong) patient and doctor untraceability",
"Even if a user's identity is not revealed, the adversary may be able to trace a user by telling whether two executions are done by the same user.",
"The DLV08 protocol claims that prescriptions issued to the same patient should not be linkable to each other.",
"In other words, the situation in which a patient executes the protocol twice should be indistinguishable from the situation in which two different patients execute the protocol individually.",
"To satisfy this requirement, patient untraceability is required.",
"(Remark that the original DLV08 paper calls this untraceability “patient unlinkability”.)"
],
[
"Patient and doctor untraceability.",
"Doctor untraceability has been defined in Definition REF , and patient untraceability can be defined in a similar style.",
"The bi-process for verifying doctor untraceability is $\\begin{array}{l}\\nu \\tilde{mc}.",
"\\mathit {init}.",
"(!\\mathit {R}_{\\mathit {pt}}\\mid !\\mathit {R}_{\\mathit {dr}}\\mid !\\mathit {R}_{\\mathit {ph}}\\mid !\\mathit {R}_{\\mathit {mpa}}\\mid !\\mathit {R}_{\\mathit {hii}}\\mid (\\nu {\\tt {nPnym}}_{\\mathit {dr}}.",
"\\nu {\\tt {wPnym}}_{\\mathit {dr}}.\\\\((\\mbox{let}\\ {{\\tt {Id}}_{\\mathit {dr}}}={\\tt {d_A}}\\ \\mbox{in} \\ \\mbox{let}\\ {{\\tt {Pnym}}_{\\mathit {dr}}}={\\tt {nPnym}}_{\\mathit {dr}}\\ \\mbox{in} \\ \\mathit {P}_{\\mathit {dr}})\\mid \\\\\\hspace{4.25pt}(\\mbox{let}\\ {{\\tt {Id}}_{\\mathit {dr}}}={\\sf choice}[{\\tt {d_A}}, {\\tt {d_B}}] \\ \\mbox{in} \\ \\mbox{let}\\ {{\\tt {Pnym}}_{\\mathit {dr}}}={\\sf choice}[{\\tt {nPnym}}_{\\mathit {dr}}, {\\tt {wPnym}}_{\\mathit {dr}}] \\ \\mbox{in} \\ \\mathit {P}_{\\mathit {dr}})))),\\end{array}$ and the bi-process for verifying patient untraceability is $\\begin{array}{l}\\nu \\tilde{mc}.",
"\\mathit {init}.",
"(!\\mathit {R}_{\\mathit {pt}}\\mid !\\mathit {R}_{\\mathit {dr}}\\mid !\\mathit {R}_{\\mathit {ph}}\\mid !\\mathit {R}_{\\mathit {mpa}}\\mid !\\mathit {R}_{\\mathit {hii}}\\mid \\\\(\\nu {\\tt {nPnym}}_{\\mathit {pt}}.\\nu {\\tt {nSss}}.\\nu {\\tt {nAcc}}.\\nu {\\tt {wPnym}}_{\\mathit {pt}}.\\nu {\\tt {wSss}}.\\nu {\\tt {wAcc}}.\\\\\\hspace{4.25pt}{\\sf {in}}({\\tt {ch}}_{hp}, \\mathit {nHii}).",
"{\\sf {in}}({\\tt {ch}}_{hp}, \\mathit {wHii}).\\\\\\hspace{4.25pt} \\mbox{let}\\ \\mathit {c_{\\mathit {pt}}\\_npk_\\mathit {hii}}={\\sf key}(\\mathit {nHii}) \\ \\mbox{in} \\ \\mbox{let}\\ \\mathit {c_{\\mathit {pt}}\\_wpk_\\mathit {hii}}={\\sf key}(\\mathit {wHii}) \\ \\mbox{in} \\\\(\\mbox{let}\\ {{\\tt {Hii}}}=\\mathit {nHii}\\ \\mbox{in} \\ \\mbox{let}\\ \\mathit {c_{\\mathit {pt}}\\_pk_\\mathit {hii}}=\\mathit {c_{\\mathit {pt}}\\_npk_\\mathit {hii}}\\ \\mbox{in} \\ \\mbox{let}\\ {{\\tt {Id}}_{\\mathit {pt}}}={\\tt {t_A}}\\ \\mbox{in} \\\\\\ \\mbox{let}\\ {{\\tt {Pnym}}_{\\mathit {pt}}}={\\tt {nPnym}}_{\\mathit {pt}}\\ \\mbox{in} \\ \\mbox{let}\\ {{\\tt {Sss}}}={\\tt {nSss}}\\ \\mbox{in} \\ \\mbox{let}\\ {{\\tt {Acc}}}={\\tt {nAcc}}\\ \\mbox{in} \\ \\mathit {P}_{\\mathit {pt}}) \\mid \\\\\\end{array}(\\mbox{let}\\ {{\\tt {Hii}}}={\\sf choice}[\\mathit {nHii}, \\mathit {wHii}] \\ \\mbox{in} \\ \\mbox{let}\\ \\mathit {c_{\\mathit {pt}}\\_pk_\\mathit {hii}}={\\sf choice}[\\mathit {c_{\\mathit {pt}}\\_npk_\\mathit {hii}}, \\mathit {c_{\\mathit {pt}}\\_wpk_\\mathit {hii}}] \\ \\mbox{in} \\\\\\ \\mbox{let}\\ {{\\tt {Id}}_{\\mathit {pt}}}={\\sf choice}[{\\tt {t_A}}, {\\tt {t_B}}] \\ \\mbox{in} \\ \\mbox{let}\\ {{\\tt {Pnym}}_{\\mathit {pt}}}={\\sf choice}[{\\tt {nPnym}}_{\\mathit {pt}}, {\\tt {wPnym}}_{\\mathit {pt}}] \\ \\mbox{in} \\\\\\ \\mbox{let}\\ {{\\tt {Sss}}}={\\sf choice}[{\\tt {nSss}}, {\\tt {wSss}}] \\ \\mbox{in} \\ \\mbox{let}\\ {{\\tt {Acc}}}={\\sf choice}[{\\tt {nAcc}}, {\\tt {wAcc}}] \\ \\mbox{in} \\ \\mathit {P}_{\\mathit {pt}}))).$ $We verified two versions of doctor and patient untraceability,- in one version, the adversary does not know the two doctor/patient identities, andin the other version, the adversary initially knows the two doctor/patient identities.\\paragraph {Strong patient and doctor untraceability.",
"}Strong untraceability is modelled as a patientexecuting the protocol repeatedly is indistinguishable from different patientsexecuting the protocol each once.",
"Strong doctor untraceabilityis defined as in Definition~\\ref {def:suntra} and strong patient untraceabilitycan be defined in the same manner.The bi-process for verifying strong doctor untraceability is$ l mc.",
"$\\mathit {init}$ .",
"(!$\\mathit {R}$$\\mathit {pt}$ !$\\mathit {R}$$\\mathit {ph}$ !$\\mathit {R}$$\\mathit {mpa}$ !$\\mathit {R}$$\\mathit {hii}$ !",
"(nId$\\mathit {dr}$ .",
"nPnym$\\mathit {dr}$ .",
"!",
"(wId$\\mathit {dr}$ .",
"wPnym$\\mathit {dr}$ .",
"let Id$\\mathit {dr}$ =choice[nId$\\mathit {dr}$ , wId$\\mathit {dr}$ ] in let Pnym$\\mathit {dr}$ =choice[nPnym$\\mathit {dr}$ , wPnym$\\mathit {dr}$ ] in $\\mathit {P}$$\\mathit {dr}$ ))), $and the bi-process for verifying strong patient untraceability is$ l mc.",
"$\\mathit {init}$ .",
"(!$\\mathit {R}$$\\mathit {dr}$ !$\\mathit {R}$$\\mathit {ph}$ !$\\mathit {R}$$\\mathit {mpa}$ !$\\mathit {R}$$\\mathit {hii}$ !",
"(nId$\\mathit {pt}$ .",
"nPnym$\\mathit {pt}$ .",
"nSss.",
"nAcc.",
"in(chhp, $\\mathit {nHii}$ ).",
"!",
"(wId$\\mathit {pt}$ .",
"wPnym$\\mathit {pt}$ .",
"wSss.",
"wAcc.",
"let Id$\\mathit {pt}$ =choice[nId$\\mathit {pt}$ , wId$\\mathit {pt}$ ] in let Pnym$\\mathit {pt}$ =choice[nPnym$\\mathit {pt}$ , wPnym$\\mathit {pt}$ ] in let Sss=choice[nSss, wSss] in let Acc=choice[nAcc, wSss] in in(chhp, $\\mathit {wHii}$ ).",
"let Hii=choice[$\\mathit {nHii}$ , $\\mathit {wHii}$ ] in let $\\mathit {c_{\\mathit {pt}}\\_pk_\\mathit {hii}}$ =key(Hii) in $\\mathit {P}$$\\mathit {pt}$ ))).",
"$This definition does not involve a specific doctor/patient, and thus needs not todistinguish whether the adversary knows the identities a priori.\\paragraph {Verification result.",
"}The bi-processes are verified using ProVerif.The verification results show that the DLV08 protocol does not satisfypatient/doctor untraceability (with/without revealed identities), nor strong untraceability.$ The strong doctor untraceability fail, because the adversary can distinguish sessions initiated by one doctor and by different doctors.",
"The doctor's pseudonym is revealed and a doctor uses the same pseudonym in all sessions.",
"Sessions with the same doctor pseudonyms are initiated by the same doctor.",
"For the same reasons, doctor untraceability without revealing doctor identities also fails.",
"Both of them can be fixed by requiring the representation of a doctor's pseudonym (${{\\tt {Sss}}}$ ) differ in each session (s3').",
"However, assuming s3' (doctor pseudonym is fresh in every sessions) is not sufficient for satisfying doctor anonymity with doctor identities revealed.",
"The adversary can still distinguish two sessions initiated by one doctor or by two different doctors, by comparing the anonymous authentications of the two sessions.",
"From the communication in the two sessions, the adversary is able to learn two doctor pseudonyms ${{\\tt {Pnym}}_{\\mathit {dr}}}^{\\prime }$ and ${{\\tt {Pnym}}_{\\mathit {dr}}}^{\\prime \\prime }$ , two doctor credentials $\\mathit {Cred_{\\mathit {dr}}}^{\\prime }$ and $\\mathit {Cred_{\\mathit {dr}}}^{\\prime \\prime }$ and two anonymous authentications $\\mathit {Auth_{\\mathit {dr}}}^{\\prime }$ and $\\mathit {Auth_{\\mathit {dr}}}^{\\prime \\prime }$ .",
"Since the adversary knows ${\\tt {d_A}}$ and ${\\tt {d_B}}$ in advance, he could construct the eight anonymous authentications by applying the zero-knowledge proof function, i.e., ${\\sf zk}(({\\mathit {Pnym}_{\\mathit {dr}}},{\\mathit {Id_{\\mathit {dr}}}}),\\mathit {Cred_{\\mathit {dr}}})$ , where ${\\mathit {Pnym}_{\\mathit {dr}}}={{\\tt {Pnym}}_{\\mathit {dr}}}^{\\prime }$ or ${\\mathit {Pnym}_{\\mathit {dr}}}={{\\tt {Pnym}}_{\\mathit {dr}}}^{\\prime \\prime }$ , ${\\mathit {Id_{\\mathit {dr}}}}={\\tt {d_A}}$ or ${\\mathit {Id_{\\mathit {dr}}}}={\\tt {d_B}}$ , $\\mathit {Cred_{\\mathit {dr}}}=\\mathit {Cred_{\\mathit {dr}}}^{\\prime }$ or $\\mathit {Cred_{\\mathit {dr}}}=\\mathit {Cred_{\\mathit {dr}}}^{\\prime }$ .",
"By comparing the constructed anonymous authentications with the observed ones, the adversary is able to tell who generated which anonymous authentication, and thus is able to tell whether the two sessions are initiated by the same doctor or different doctors.",
"This can be fixed by additionally requiring that the doctor anonymous authentication differs in every session (s4').",
"For strong patient untraceability, the adversary can distinguish sessions initiated by one patient (with identical social security statuses) and initiated by different patients (with different social security statuses).",
"Second, the adversary can distinguish sessions initiated by one patient (with identical cipher texts ${\\sf {enc}}({{\\tt {Pnym}}_{\\mathit {pt}}}, {\\tt {pk}}_\\mathit {sso})$ and identical cipher texts ${\\sf {enc}}({{\\tt {Hii}}}, {\\tt {pk}}_\\mathit {sso})$ ) and initiated by different patients (with different cipher texts ${\\sf {enc}}({{\\tt {Pnym}}_{\\mathit {pt}}}, {\\tt {pk}}_\\mathit {sso})$ and different cipher texts ${\\sf {enc}}({{\\tt {Hii}}}, {\\tt {pk}}_\\mathit {sso})$ ).",
"Third, since the patient credential is the same in all sessions and is revealed, the adversary can also trace a patient by the patient's credential.",
"Fourth, the adversary can distinguish sessions using the same HII and sessions using different HIIs.",
"For the same reasons, patient untraceability fails.",
"Both flaws can be fixed by requiring that the representation of a patient's social security status to be different in each session (s5'), the encryptions are probabilistic (s2'), a patient freshly generates a credential in each session (s4”), and patients who shall not be distinguishable share the same HII (s6')."
],
[
"Prescription privacy",
"Prescription privacy has been defined in Definition REF .",
"To verify the prescription privacy is to check the satisfaction of the equivalence in the definition.",
"The bi-process for verifying the equivalence is $\\begin{array}{ll}\\multicolumn{2}{l}{(\\mathit {private}) \\mathit {free}\\ {\\tt {d_A}}.",
"(\\mathit {private}) \\mathit {free}\\ {\\tt {d_B}}.\\mathit {free}\\ {\\tt {p_A}}.\\mathit {free}\\ {\\tt {p_B}}.",
"}\\\\\\nu \\tilde{mc}.",
"\\mathit {init}.",
"(!\\mathit {R}_{\\mathit {pt}}\\mid !\\mathit {R}_{\\mathit {dr}}\\mid !\\mathit {R}_{\\mathit {ph}}\\mid & (\\nu {\\tt {nPnym}}_{\\mathit {dr}}.",
"\\nu {\\tt {wPnym}}_{\\mathit {dr}}.\\\\&\\hspace{4.25pt}\\mbox{let}\\ {{\\tt {Id}}_{\\mathit {dr}}}={\\sf choice}[{\\tt {d_A}}, {\\tt {d_B}}] \\ \\mbox{in} \\\\&\\hspace{4.25pt}\\mbox{let}\\ {{\\tt {Pnym}}_{\\mathit {dr}}}={\\sf choice}[{\\tt {nPnym}}_{\\mathit {dr}}, {\\tt {wPnym}}_{\\mathit {dr}}] \\ \\mbox{in} \\\\&\\hspace{4.25pt}\\mbox{let}\\ {\\tt {presc}}={\\tt {p_A}}\\ \\mbox{in} \\ \\mathit {\\mathit {main}_{\\mathit {dr}}})\\mid \\\\& (\\nu {\\tt {nPnym}}_{\\mathit {dr}}.",
"\\nu {\\tt {wPnym}}_{\\mathit {dr}}.\\\\&\\hspace{4.25pt}\\mbox{let}\\ {{\\tt {Id}}_{\\mathit {dr}}}={\\sf choice}[{\\tt {d_B}}, {\\tt {d_A}}] \\ \\mbox{in} \\\\&\\hspace{4.25pt}\\mbox{let}\\ {{\\tt {Pnym}}_{\\mathit {dr}}}={\\sf choice}[{\\tt {nPnym}}_{\\mathit {dr}}, {\\tt {wPnym}}_{\\mathit {dr}}] \\ \\mbox{in} \\\\&\\hspace{4.25pt}\\mbox{let}\\ {\\tt {presc}}={\\tt {p_B}}\\ \\mbox{in} \\ \\mathit {\\mathit {main}_{\\mathit {dr}}})).\\end{array}$ Similarly, we verified two versions - in one version, the adversary does not know ${\\tt {d_A}}$ and ${\\tt {d_B}}$ , and in the other, the adversary knows ${\\tt {d_A}}$ and ${\\tt {d_B}}$ ."
],
[
"Verification result.",
"The verification, using ProVerif, shows that the DLV08 protocol satisfies prescription privacy when the adversary does not know the doctor identities a priori, and does not satisfy prescription privacy when the adversary knows the doctor identities a priori, i.e., the adversary can distinguish whether a prescription is prescribed by doctor ${\\tt {d_A}}$ or doctor ${\\tt {d_B}}$ , given the adversary knows ${\\tt {d_A}}$ and ${\\tt {d_B}}$ .",
"In the prescription proof, a prescription is linked to a doctor credential.",
"And a doctor credential is linked to a doctor identity.",
"Thus, the adversary can link a doctor to his prescription.",
"To break the link, one way is to make sure that the adversary cannot link a doctor credential to a doctor identity.",
"This can be achieved by adding randomness to the credential (s4')."
],
[
"Receipt-freeness",
"The definition of receipt-freeness is modelled as the existence of a process $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ , such that the two equivalences in Definition REF are satisfied.",
"Due to the existential quantification, we cannot verify the property directly using ProVerif.",
"Examining the DLV08 protocol, we find an attack on receipt-freeness, even with assumption s4' (after fixing prescription privacy with doctor ID revealed).",
"A bribed doctor is able to prove to the adversary of his prescription as follows: A doctor communicates with the adversary to agree on a bit-commitment that he will use, which links the doctor to the commitment.",
"The doctor uses the agreed bit-commitment in the communication with his patient.",
"This links the bit-commitment to a prescription.",
"Later, when the patient uses this prescription to get medicine from a pharmacist, the adversary can observe the prescription being used.",
"This proves that the doctor has really prescribed the medicine.",
"We formally confirm the attack using ProVerif, i.e., we show that in the protocol model, if a doctor reveals all his information to the adversary, the doctor's prescription privacy is broken.",
"The same attack exists for multi-session receipt-freeness as well – a bribed doctor is able to prove his prescriptions by agreeing with the adversary on the bit-commitments in each session.",
"Theorem 1 (receipt-freeness) The DLV08 protocol fails to satisfy receipt-freeness under both the standard assumption s4 (a doctor has the same credential in every session), and also under assumption s4' (a doctor generates a new credential for each session).",
"Formal proof of the theorem can be found in Appendix ."
],
[
"Independency of (enforced) prescription privacy",
"To determine whether the doctor's prescription privacy is independent of the pharmacist, we replace regular pharmacist role $\\mathit {R}_{i}$ with collaborating role $\\mathit {R}_{\\mathit {ph}}$ in Definition REF .",
"The bi-process for verifying the property is: $\\begin{array}{ll}\\multicolumn{2}{l}{(\\mathit {private}) \\mathit {free}\\ {\\tt {d_A}}.",
"(\\mathit {private}) \\mathit {free}\\ {\\tt {d_B}}.\\mathit {free}\\ {\\tt {p_A}}.\\mathit {free}\\ {\\tt {p_B}}.",
"}\\\\\\nu \\tilde{mc}.",
"\\mathit {init}.",
"(!\\mathit {R}_{\\mathit {pt}}\\mid !\\mathit {R}_{\\mathit {dr}}\\mid !",
"(\\mathit {R}_{\\mathit {ph}})^{{\\tt {chc}}}\\mid &(\\nu {\\tt {nPnym}}_{\\mathit {dr}}.",
"\\nu {\\tt {wPnym}}_{\\mathit {dr}}.\\\\&\\hspace{4.25pt}\\mbox{let}\\ {{\\tt {Id}}_{\\mathit {dr}}}={\\sf choice}[{\\tt {d_A}}, {\\tt {d_B}}] \\ \\mbox{in} \\\\&\\hspace{4.25pt}\\mbox{let}\\ {{\\tt {Pnym}}_{\\mathit {dr}}}={\\sf choice}[{\\tt {nPnym}}_{\\mathit {dr}}, {\\tt {wPnym}}_{\\mathit {dr}}] \\ \\mbox{in} \\\\&\\hspace{4.25pt}\\mbox{let}\\ {\\tt {presc}}={\\tt {p_A}}\\ \\mbox{in} \\ \\mathit {\\mathit {main}_{\\mathit {dr}}})\\mid \\\\&(\\nu {\\tt {nPnym}}_{\\mathit {dr}}.",
"\\nu {\\tt {wPnym}}_{\\mathit {dr}}.\\\\&\\hspace{4.25pt}\\mbox{let}\\ {{\\tt {Id}}_{\\mathit {dr}}}={\\sf choice}[{\\tt {d_B}}, {\\tt {d_A}}] \\ \\mbox{in} \\\\&\\hspace{4.25pt}\\mbox{let}\\ {{\\tt {Pnym}}_{\\mathit {dr}}}={\\sf choice}[{\\tt {nPnym}}_{\\mathit {dr}}, {\\tt {wPnym}}_{\\mathit {dr}}] \\ \\mbox{in} \\\\&\\hspace{4.25pt}\\mbox{let}\\ {\\tt {presc}}={\\tt {p_B}}\\ \\mbox{in} \\ \\mathit {\\mathit {main}_{\\mathit {dr}}})).\\end{array}$ Verification using ProVerif shows that the protocol (the original version where the adversary does not know ${\\tt {d_A}}$ and ${\\tt {d_B}}$ , and the version after fixing the flaw on prescription privacy with assumption s4' where the adversary knows ${\\tt {d_A}}$ and ${\\tt {d_B}}$ ) satisfies this property.",
"The case of pharmacist-independent receipt-freeness is treated analogously.",
"We replace regular pharmacist $\\mathit {R}_i$ with $\\mathit {R}_{\\mathit {ph}}$ in Definition REF .",
"The flaw described in Section REF also surfaces here.",
"This was expected: when a doctor can prove his prescription without the pharmacist sharing information with the adversary, the doctor can also prove this when the pharmacist genuinely cooperates with the adversary."
],
[
"Dishonest users",
"So far, we have considered security and privacy with respect to a Dolev-Yao style adversary (see Section REF ).",
"The initial knowledge of the adversary was modelled such, that the adversary could not take an active part in the execution of the protocol.",
"This constitutes the basic DY adversary, as shown in Table REF .",
"In more detail, for secrecy of private doctor and patient information (see Table REF ), [24] claims that no third party (including the basic DY adversary) shall be able to know a patient's or doctor's private information (refer to the beginning of Section ).",
"Similarly, the verification of authentication properties in Table REF is also with respect to the basic DY adversary.",
"This captures that no third party that does not participate in the execution shall be able to impersonate any party (involved in the execution).",
"The same basic DY adversary model is used to verify anonymity, untraceability and prescription privacy.",
"The exceptions are (1) for verifying receipt-freeness and independency of enforced prescription privacy, the basic DY adversary is extended with information from the targeted doctor; (2) for verifying independency of prescription privacy and independency of enforced prescription privacy, the basic DY adversary is extended with information from pharmacists.",
"Table: Summary of the respected adversaryIn this section, we consider dishonest users, that is, malicious users that collaborate with the adversary and are part of the execution, into consideration.",
"For each property previously verified, we analyse the result once again with respect to each dishonest role.",
"Dishonest users are modelled by providing the adversary certain initial knowledge such that the adversary can take part in the protocol.",
"To execute the protocol as a doctor, i.e., to instantiate the doctor process $\\mathit {P}_{\\mathit {dr}}$ , the adversary only needs to have an identity and a pseudonym.",
"Since the adversary is able to generate data, the adversary can create his own identity ${{\\tt {Id}}^{a}_{\\mathit {dr}}}$ and pseudonym ${{\\tt {Id}}^{a}_{\\mathit {dr}}}$ .",
"However, this is not sufficient, because an legitimate doctor has a credential issued by authorities.",
"The credential is captured by the private function ${\\sf drcred}$ .",
"The adversary cannot obtain this credential, since he cannot apply the function ${\\sf drcred}$ .",
"When the function ${\\sf drcred}$ is modelled as public, the adversary is able to obtain his credential ${\\sf drcred}({{\\tt {Id}}^{a}_{\\mathit {dr}}},{{\\tt {Id}}^{a}_{\\mathit {dr}}})$ , and thus has the ability to behave like a doctor.",
"Hence, by modelling the function ${\\sf drcred}$ as public, we allow the adversary to have the ability of dishonest doctors.",
"Note that an honest doctor's identity is secret (see Table REF ).",
"The attacker thus cannot forge credentials of honest doctors, as the doctor's identity must be known for this.",
"Thus, making the function ${\\sf drcred}$ public does not bestow extra power on the attacker.",
"Similarly, by allowing the adversary to have patient credentials, we strengthen the adversary with the ability to control dishonest patients.",
"This is modelled by changing the private functions ${\\sf ptcred}$ to be public.",
"Each public entity (pharmacist, MPA or HII) has a secret key as distinct identifier, i.e., its public key and identity can be derived from the secret key.",
"The adversary can create such a secret key by himself.",
"However, only the legitimate entities can participate in the protocol.",
"This is modelled using private channels – only the honest entities are allowed to publish their information to the channels, and participants only read in entities, which they are going to communicate with, from the private channels.",
"By changing the private channels to be public, the adversary is able to behave as dishonest public entities.",
"Note that when considering the adversary only controlling dishonest pharmacists among the public entities, for the sake of simplicity of modelling, a dishonest pharmacist is modelled as $\\mathit {R}_{\\mathit {ph}}^{{\\tt {chc}}}$ (same as in independency of prescription privacy) in which the pharmacist shares all his information with the adversary.",
"The above modelling of dishonest users captures the following scenarios.",
"For verifying secrecy of doctor/paitent information, the dishonest doctor/patient models other doctors/patients that may break secrecy of the target doctor/patient; the dishonest patient/doctor models the patients/doctors that may communicate with the target doctor/patient; the dishonest pharmacist, MPA and HII may participate in the same execution as the target doctor/patient.",
"Since secrecy is defined as no other party (including dishonest users) should be able to know a doctor's/patient's information, unless the information is intended to be revealed, we would not consider it as an attack if the dishonest user intends to receive the private information, for details, see Section REF .",
"For verifying authentication properties, for example, a doctor authenticates a patient, a dishonest doctor is not the doctor directly communicating with the patient, and a dishonest patient is not the one who directly communicates with the doctor, because it does not make sense to analyse a dishonest user authenticates or authenticates to another user.",
"Instead, the dishonest doctors and patients are observers who may participate in other execution sessions.",
"In general, if user $A$ authenticates to user $B$ , the dishonest users taking the same role of $A$ or $B$ are observers participating in different sessions, i.e., cannot be $A$ or $B$ .",
"For the dishonest pharmacists, MPAs and HIIs, since they are not the authentication parties, they can be users participating in the same session.",
"For verifying prescription privacy and receipt-freeness, the dishonest doctors are other doctors that aim to break the target doctor's privacy; the dishonest patients can be patients communicating with the target doctor; dishonest pharmacists, MPAs and HIIs can be users participating in the same session.",
"Note that the dishonest doctor differs from the bribed doctor, as the bribed doctor tries to break his own privacy, while dishonest doctor tries to break others' privacy.",
"For verifying independency of prescription privacy and independency of enforced prescription privacy, the dishonest doctors (not the target doctor) try to break the target doctor's privacy; the dishonest patients may directly communicate with the target doctor; the dishonest pharmacists are the same as the bribed pharmacists, since 1) the bribed pharmacists genuinely forward information to the adversary and 2) all the actions that a dishonest pharmacists can do can be simulated by the basic DY with the received information from the bribed pharmacists, i.e., there is no private functions or private channels that the dishonest pharmacists can use but the adversary with bribed pharmacists information cannot; the dishonest MPAs and HIIs can participate in the same session as the target doctor.",
"The verification with dishonest users shows similar results as the verification without dishonest users.",
"The reason is that if there is an attack with respect to the basic DY attacker when verifying a property, then the property is also broken when additionally considering dishonest users.",
"The exceptions (i.e., the additional identified attacks) are shown in Table REF , and the details of the additional attacks are shown in the remaining part of this section.",
"Table: Additional attacks when considering dishonest users"
],
[
"Secrecy",
"When considering dishonest doctors and pharmacists, secrecy results in Table REF do not change, since doctors do not receive any information that the adversary does not know (see Figure REF ).",
"When considering dishonest patients, an additional attack is found.",
"When a patient ${\\tt {t_A}}$ is dishonest, he can obtain another patient ${\\tt {t_B}}$ 's pseudonym by doing the following: ${\\tt {t_A}}$ observes ${\\tt {t_B}}$ 's communication and reads in $\\mathit {vc}_4$ (the verifiable encryption which encrypts ${\\tt {t_B}}$ 's pseudonym with the public key of an MPA).",
"Hence, from $\\mathit {vc}_4$ , ${\\tt {t_A}}$ can obtain the cipher-text ${\\sf {enc}}({{\\tt {Pnym}}^{B}_{\\mathit {pt}}}, \\mathit {c_{\\mathit {pt}}\\_pk_{\\mathit {mpa}}})$ , where ${{\\tt {Pnym}}^{B}_{\\mathit {pt}}}$ is ${\\tt {t_B}}$ 's pseudonym and $\\mathit {c_{\\mathit {pt}}\\_pk_{\\mathit {mpa}}}$ is the public key of the MPA.",
"${\\tt {t_A}}$ initiates the protocol with his own data.",
"In the communication with a pharmacist, ${\\tt {t_A}}$ replaces his 5 (which should be a verifiable encryption, containing a cipher-text from ${\\tt {t_A}}$ encrypted with the public key of the MPA) with $\\mathit {vc}_4$ .",
"On receiving $\\mathit {vc}_4$ (the fake 5), the pharmacist sends it to the MPA, and the MPA decrypts the cipher-text ${\\sf {enc}}({{\\tt {Pnym}}^{B}_{\\mathit {pt}}}, \\mathit {c_{\\mathit {pt}}\\_pk_{\\mathit {mpa}}})$ , embedded in $\\mathit {vc}_4$ and sends the decryption result to HII.",
"${\\tt {t_A}}$ observes the communication between the MPA and HII, and reads the decrypted text of the fake 5 (i.e., ${{\\tt {Pnym}}^{B}_{\\mathit {pt}}}$ ), which is ${\\tt {t_B}}$ 's pseudonym.",
"This attack does not exist when the attacker cannot participant as a patient.",
"Because the dishonest patient has to replace 5 in his own communication.",
"If an hones patient's 5 is replaced by the adversary, the pharmacist would detect it by verifying the receipt $\\mathit {ReceiptAck}$ , which should contain the correct 5.",
"This attack can be addressed by explicitly ask the MPA to verify the decrypted message of 5 to be a verifiable encryption before sending it out.",
"Alternatively, if the communication between MPA and the HII is secured after authentication, the adversary would not be able to observe ${\\tt {t_B}}$ 's pseudonym, and thus the attack would not happen.",
"When considering dishonest pharmacists, an additional attack may exist on a patient's pseudonym.",
"The dishonest pharmacist has/creates a secret key $y$ and obtains its corresponding public key ${\\sf pk}(y)$ .",
"The pharmacist creates a fake MPA identity using the secret key and public key, i.e., ${\\sf host}({\\sf pk}(y))$ .",
"The dishonest pharmacist provides the patient a fake MPA identity, from which the patient obtains the MPA public key ${\\sf pk}(y)$ .",
"Later, the patient encrypts his pseudonym using the fake MPA's public key ${\\sf {enc}}({{\\tt {Pnym}}_{\\mathit {pt}}}, {\\sf pk}(y))$ , and provides a verifiable encryption, $\\mathit {vc}_4$ in particular.",
"The verifiable encryption is sent to the pharmacist, from which he pharmacist can read the cipher-text ${\\sf {enc}}({{\\tt {Pnym}}_{\\mathit {pt}}}, {\\sf pk}(y))$ .",
"Since the pharmacist knows the secret key $y$ , he can decrypt the cipher-text and obtains the patient's pseudonym ${{\\tt {Pnym}}_{\\mathit {pt}}}$ .",
"Note that we model that the patient obtains the MPA of the pharmacist and the MPA's public key from the pharmacist, thus the attack may not happen if the patient initially knows the MPA of any pharmacist, or the patient has the ability to immediately check whether the MPA provided by a pharmacist is indeed an legitimate MPA.",
"When considering dishonest MPAs, the adversary additionally knows a patient's pseudonym.",
"However, this can hardly be an attack, as the patient pseudonym is intended to be known by the MPA.",
"Similarly, the HII is the intended receiver of a patient's pseudonym.",
"Other than a patient's pseudonym, the dishonest MPA and HII do not know any information that the adversary does not know without controlling dishonest agents.",
"Note that in reality, the MPA and HII may know more sensitive information, for example, from the pseudonym, the HII is able to obtain the patient's identity, and a dishonest MPA can claim that a prescription has medical issues and obtains the doctor identity in a procedure, which is beyond the scope of this protocol."
],
[
"Authentication",
"When considering dishonest users, the verification results of the authentication remain the same, except the authentication from the dishonest user to other parties.",
"For example, when doctors are dishonest, we do not need to consider the authentication from doctors to a patient, since the dishonest users are part of the adversary.",
"Similarly, when a patient is dishonest, the authentication from a patient to a doctor or a pharmacist is obviously unsatisfied, other authentication verification results remain the same.",
"When pharmacists, MPAs or HIIs are dishonest, the verification results remain unchanged."
],
[
"Privacy properties",
"For those privacy properties which are not satisfied with respect to the adversary controlling no dishonest agents, the properties are not satisfied when considering the adversary who controls dishonest users.",
"Thus, we only need to analyse the property that are satisfied with respect to the adversary controlling no dishonest agents, i.e.",
"(Strong) patient anonymity.",
"Obviously, two patients can be distinguished by the adversary who controls dishonest HIIs, when the two patients use different HIIs, because the patients use different HII public keys to encrypt his pseudonym.",
"When the two patients use different HIIs, and the HIIs are honest, the adversary, who controls dishonest MPAs, can still distinguish them, because a patient's HII is intended to be known by the MPA.",
"Finally, (strong) patient anonymity is satisfied with respect to the adversary controlling dishonest doctors and dishonest pharmacists."
],
[
"Addressing the flaws of the DLV08 protocol",
"To summarise, we present updates to the assumptions of Section REF to fix the flaws found in our analysis of the privacy properties.",
"s2' The encryptions are probabilistic.",
"s3' The value of ${{\\tt {Pnym}}_{\\mathit {dr}}}$ is freshly generated in every session.",
"s4' A doctor freshly generates an unpredictable credential in each session.",
"We model this with another parameter (a random number) of the credential.",
"Following this, anonymous authentication using these credentials proves knowledge of the used randomness.",
"s4” A patient freshly generates an unpredictable credential in each session.",
"Similar to s4', this can be achieved by add randomness in the credential.",
"The anonymous authentication using the credentials proves the knowledge of the used randomness.",
"s5' The values of ${{\\tt {Sss}}}$ differ in sessions.",
"s6' The value of ${{\\tt {Hii}}}$ shall be the same for all patients.",
"The proposed assumptions are provided on the model level.",
"Due to the ambiguities in the original protocol (e.g., it is not clear how a social security status is represented), it is difficult to propose detailed solutions.",
"To implement a proposed assumption, one only needs to capture its properties.",
"To capture s2', the encryption scheme can be ElGamal cryptosystem, or RSA cryptosystem with encryption padding, which are probabilistic.",
"In some systems, deterministic encryption, e.g., RSA without encryption padding, may be more useful than probabilistic encryption, for example for database searching of encrypted data.",
"In such systems, designers need to carefully distinguish which encryption scheme is used in which part of the system.",
"s3' can be achieved by directly requiring a doctor's pseudonym to be fresh in every session, for example, a doctor generates different pseudonyms in sessions and keeps the authorities, who maintain the relation between the doctor identity and pseudonyms, updated in a secure way; or before every session the doctor requests a pseudonym from the authorities.",
"Alternatively, it can be achieved by changing the value of ${{\\tt {Pnym}}_{\\mathit {dr}}}$ .",
"Assuming the authorities share a key with each doctor; instead of directly using pseudonym in a session, the doctor encrypts his pseudonym with the key using probabilistic encryption.",
"That is, the value of ${{\\tt {Pnym}}_{\\mathit {dr}}}$ is a cipher-text which differs in sessions.",
"When an MPA wants to find out the doctor of a prescription, he can contact the authorities to decrypt the ${{\\tt {Pnym}}_{\\mathit {dr}}}$ and finds out the pseudonym of the doctor or the identity of the doctor directly.",
"s4' and s4” together form the updates to s4.",
"We separate the update to the doctor credential in s4' and the update to the patient credential in s4” for the convenience of referring to them individually in other places.",
"Similar to s3', s5' can be achieved by directly requiring that a patient's social security status is different in each session, e.g., by embedding a timestamp in the status.",
"Alternatively, the value of ${{\\tt {Sss}}}$ can be a cipher-text which is a probabilistic encryption of a patient's social security status with the pharmacist's public key, since the social security status is used for the pharmacist to check the status of the patient.",
"s6' can be achieved by directly requiring that all patients share the same HII.",
"In the case of multiple HIIs, different HIIs should not be distinguishable, for example, HIIs may cooperate together and provide a uniformed reference (name and key).",
"In fact, if patients are satisfied with untraceability within a group of a certain amount of patients, patient untraceability can be satisfied as long as each HII has more patients than the expected size of the group.",
"If only untraceability is required (instead of strong untraceability), the use of a group key of all HIIs is sufficient.",
"The common key among HIIs can be established by using asymmetric group key agreement.",
"In this way, the HIIs cannot be distinguished by their keys.",
"In addition, the identities of HIIs are not revealed, and thus cannot be used to distinguish HIIS.",
"Hence, the common key ensures that two patients executing the protocol once and one patient executing the protocol twice cannot be distinguished by their HIIs.",
"The modified protocol was verified again using ProVerif.",
"The verification results show that the protocol with revised assumptions satisfies doctor anonymity, strong doctor anonymity, and prescription privacy, as well as untraceability and strong untraceability for both patient and doctor.",
"However, the modified protocol model does not satisfy receipt-freeness, to make the protocol satisfy receipt-freeness, we apply the following assumption on communication channels.",
"s8' Communication channels are untappable (i.e., the adversary does not observe anything from the channel), except those used for authentication, which remain public.",
"Table: Verification results of privacy properties and revised assumptions.Our model of the protocol is accordingly modified as follows: replacing channel ${\\tt {ch}}$ in lines d10, t6 with an untappable channel ${\\tt {ch}}_{dp}$ , replacing channel ${\\tt {ch}}$ in lines t23, t26, h5, h22 with an untappable channel ${\\tt {ch}}_{ptph}$ , and replacing channel ${\\tt {ch}}$ in lines t24, h21 with an untappable channel ${\\tt {ch}}_{phpt}$ .",
"The untappable channels are modelled as global private channels.",
"We prove that the protocol (with s4' and s8') satisfies receipt-freeness by showing the existence of a process $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ (as shown in Figure REF ) such that the equivalences in Definition REF are satisfied.",
"This was verified using ProVerif (for verification code, see [26]).",
"Figure: The doctor process 𝑃 𝑑𝑟 ' \\mathit {P}_{\\mathit {dr}}^{\\prime } (using untappable channels).Messages over untappable channels are assumed to be perfectly secret to the adversary (for example, the channels assumed in [29], [31]).",
"Thus, the security and classical privacy properties, which are satisfied in the model with public channels only, are also satisfied when replacing some public channels with untappable channels.",
"Similar to other proposed assumptions, the assumption of untappable channels is at the model level.",
"This is a strong assumption, as the implementation of an untappable channel is difficult [31].",
"However, as this assumption is often used in literature to achieve privacy in the face of bribery and coercion (e.g.",
"[51], [8], [31]), we feel that its use here is justifiable.",
"However, even with the above assumptions the DLV08 protocol does not satisfy independency of receipt-freeness.",
"The proof first shows that $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ is not sufficient for proving this with ProVerif.",
"Then we prove (analogous to the proof in Section REF ) that there is no alternative process $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ which satisfies Definition REF .",
"Intuitively, all information sent over untappable channels is received by pharmacists and can be genuinely revealed to the adversary (no lying assumption).",
"Hence, the links between a doctor, his nonces, his commitment, his credential and his prescription can still be revealed when the doctor is bribed/coerced to reveal those nonces.",
"Theorem 2 (independency of receipt-freeness) The DLV08 protocol fails to satisfy independency of receipt-freeness.",
"Formal proof the theorem can be found in Appendix .",
"Intuitively, a bribed doctor is linked to the nonces he sent to the adversary.",
"The nonces are linked to the doctor's prescription in a prescription proof.",
"A doctor's prescription proof is sent over untappable channels first to a patient and later from the patient to a pharmacist.",
"Malicious pharmacists reveal the prescription proof to the adversary.",
"If a bribed doctor lied about his prescription, the adversary can detect it by checking the doctor's corresponding prescription proof revealed by the pharmacist.",
"The untappable channel assumption enables the protocol to satisfy receipt-freeness but not independency of receipt-freeness because untappable channels enable a bribed doctor to hide his prescription proof and thus allow the doctor to lie about his prescription, however the pharmacist gives the prescription proof away, from which the adversary can detect whether the doctor lied about the prescription."
],
[
"Conclusions",
"In this paper, we have studied security and privacy properties, particular enforced privacy, in the e-health domain.",
"We identified the requirement that doctor privacy should be enforced to prevent doctor bribery by, for example, the pharmaceutical industry.",
"To capture this requirement, we first formalised the classical privacy property, i.e., prescription privacy, and its enforced privacy counterpart, i.e., receipt-freeness.",
"The cooperation between the bribed doctor and the adversary is formalised in the same way as in receipt-freeness in e-voting.",
"However, the formalisation of receipt-freeness differs from receipt-freeness, due to the domain requirement that only part of the doctor's process needs to share information with the adversary.",
"Next, we noted that e-health systems involve not necessarily trusted third parties, such as pharmacists.",
"Such parties should not be able to assist an adversary in breaking doctor privacy.",
"To capture this requirement, we formally defined independency of prescription privacy.",
"Moreover, this new requirement must hold, even if the doctor is forced to help the adversary.",
"To capture that, we formally defined independency of receipt-freeness.",
"These formalisations were validated in a case study of the DLV08 protocol.",
"The protocol was modelled in the applied pi calculus and verified with the help of the ProVerif tool.",
"In addition to the (enforced) doctor privacy properties, we also analysed secrecy, authentication, anonymity and untraceability for both patients and doctors.",
"Ambiguities in the original description of the protocol which may lead to flaws were found and addressed.",
"We notice that the property independency of receipt-freeness is not satisfied in the case study protocol, and we were not able to propose a reasonable fix for it.",
"Thus, it is interesting for us to design a new protocol to satisfy such strong property in the future.",
"Furthermore, when considering dishonest users, we did not consider one dishonest user taking multiple roles.",
"Thus, it would be interesting to analyse the security and privacy properties with respect to dishonest users taking various combination of roles."
],
[
"Proof of Theorem ",
"Theorem 1 (receipt-freeness).",
"The DLV08 protocol fails to satisfy receipt-freeness under both the standard assumption s4 (a doctor has the same credential in every session), and also under assumption s4' (a doctor generates a new credential for each session).",
"It is obvious that the DLV08 protocol fails to satisfy receipt-freeness under assumption s4 (a doctor has the same credential in every session), since DLV08 does not even satisfy prescription privacy with assumption s4.",
"The reasoning is as follows: since the adversary can link a prescription to a doctor without additional information from the bribed doctor, he can also link a prescription to a doctor when he has additional information from the bribed doctor.",
"Therefore, the adversary can always tell whether a bribed doctor lied.",
"Next we prove that the DLV08 protocol fails to satisfy receipt-freeness under assumption s4' (a doctor generates a new credential for each session).",
"That is to prove that there exists no indistinguishable process in which the doctor lies to the adversary.",
"To do so, we assume that there exists such a process $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ which satisfies the definition of receipt-freeness, and then derive some contradiction.",
"In generic terms, the proof runs as follows: a bribed doctor reveals the nonces used in the commitment and the credential to the adversary.",
"This allows the adversary to link a bribed doctor to his commitment and credential.",
"In the prescription proof, a prescription is linked to a doctor's commitment and credential.",
"Suppose there exists a process $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ in which the doctor lies to the adversary that he prescribed ${\\tt {p_A}}$ , while the adversary observes that the commitment or the credential is linked to ${\\tt {p_B}}$ .",
"The adversary can detect that the doctor has lied.",
"Assume there exist process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and $P_{\\mathit {dr}}^{\\prime }$ , so that the two equivalences in the definition of receipt-freeness are satisfied, i.e., $\\exists $ $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and $P_{\\mathit {dr}}^{\\prime }$ satisfying $\\begin{array}{lrl}1.\\ &&\\mathcal {C}_\\mathit {eh}[\\big (\\mathit {init}_{\\mathit {dr}}^{\\prime }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {P}_{\\mathit {dr}}^{\\prime })\\big )\\mid \\\\&&\\hspace{17.0pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.(!",
"\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )]\\\\&\\approx _{\\ell }&\\mathcal {C}_\\mathit {eh}[\\big ((\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace })^{{\\tt {chc}}}.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid (\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })^{{\\tt {chc}}})\\big )\\mid \\\\&&\\hspace{17.0pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )]; \\vspace{5.69054pt} \\text{and}\\\\2.\\ &&\\mathcal {C}_\\mathit {eh}[\\big ((\\mathit {init}_{\\mathit {dr}}^{\\prime }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {P}_{\\mathit {dr}}^{\\prime }))^{\\backslash {\\sf {out}}({\\tt {chc}}, \\cdot )}\\big )\\mid \\\\&&\\hspace{16.57497pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.(!",
"\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )]\\\\&\\approx _{\\ell }&\\mathcal {C}_\\mathit {eh}[\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )\\mid \\\\&&\\hspace{16.57497pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.(!",
"\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )].\\end{array}$ According to the definition of labelled bisimilarity (Definition REF ), if process $A$ can reach $A^{\\prime }$ ($A \\xrightarrow{} A^{\\prime }$ ) and $A\\approx _{\\ell }B$ , then $B$ can reach $B^{\\prime }$ ($B \\xrightarrow{} B^{\\prime }$ ) and $A^{\\prime }\\approx _{\\ell }B^{\\prime }$ .",
"Vice versa.",
"Note that we use $\\xrightarrow{}$ to denote one or more internal and/or labelled reductions.",
"According to Definition REF , if $A^{\\prime }\\approx _{\\ell }B^{\\prime }$ then $A^{\\prime } \\approx _{s}B^{\\prime }$ .",
"According to the definition of static equivalence (Definition REF ), if two processes are static equivalent $A^{\\prime } \\approx _{s}B^{\\prime }$ , then ${\\sf {frame}}(A^{\\prime })\\approx _{s}{\\sf {frame}}(B^{\\prime })$ .",
"Thus we have that $\\forall M, N$ , $(M =_E N){\\sf {frame}}(A^{\\prime })$ iff $(M =_E N){\\sf {frame}}(B^{\\prime })$ .",
"Let $A$ be the right-hand side of the first equivalence and $B$ be the left-hand side, i.e., $\\begin{array}{rcl}A &=& \\mathcal {C}_\\mathit {eh}[\\big ((\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace })^{{\\tt {chc}}}.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace } \\mid (\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })^{{\\tt {chc}}})\\big )\\mid \\\\&&\\hspace{17.0pt} \\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace } \\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )]\\\\\\end{array}B &=& \\mathcal {C}_\\mathit {eh}[\\big (\\mathit {init}_{\\mathit {dr}}^{\\prime }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace } \\mid \\mathit {P}_{\\mathit {dr}}^{\\prime })\\big ) \\mid \\\\&&\\hspace{17.0pt} \\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.(!",
"\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace } \\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )].$ $$ On the right-hand side of the first equivalence (process $A$ ), there exists an output of a prescription proof ${\\tt {\\mathit {PrescProof}^{\\mathit {r}}}}$ (together with the open information of the doctor commitment ${\\tt {r}}^{r}_{\\mathit {dr}}$ ), over public channels, from the process $(\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })^{{\\tt {chc}}}$ initiated by doctor ${\\tt {d_A}}$ .",
"Formally, $A \\xrightarrow{} A_i = {\\mathcal {C}}_i[{\\sf {out}}(ch, ({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},{\\tt {r}}^{r}_{\\mathit {dr}}))] \\equiv \\nu x.",
"({\\mathcal {C}}_i[{\\sf {out}}(ch, x)] \\mid \\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},{\\tt {r}}^{r}_{\\mathit {dr}})/x\\rbrace })\\\\\\xrightarrow{} {\\mathcal {C}}_i[0]\\mid \\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},{\\tt {r}}^{r}_{\\mathit {dr}})/x\\rbrace }$ .",
"Let $A^{\\prime \\prime }={\\mathcal {C}}_i[0]$ , we have $A\\xrightarrow{} \\xrightarrow{} A^{\\prime \\prime } \\mid \\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},{\\tt {r}}^{r}_{\\mathit {dr}})/x\\rbrace }$ .",
"Let $A^{\\prime }=A^{\\prime \\prime }\\mid \\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},{\\tt {r}}^{r}_{\\mathit {dr}})/x\\rbrace }$ , we have $A\\xrightarrow{} \\xrightarrow{} A^{\\prime }$ .",
"Since $A\\approx _{\\ell }B$ , we have that $B\\xrightarrow{} \\xrightarrow{} B^{\\prime }$ and $A^{\\prime }\\approx _{\\ell }B^{\\prime }$ .",
"Hence, $A^{\\prime }\\approx _{s}B^{\\prime }$ and thus ${\\sf {frame}}(A^{\\prime })\\approx _{s}{\\sf {frame}}(B^{\\prime })$ .",
"Since ${\\sf {frame}}(A^{\\prime })={\\sf {frame}}(A^{\\prime \\prime })\\mid \\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},{\\tt {r}}^{r}_{\\mathit {dr}})/x\\rbrace }$ , the adversary can obtain the prescription ${\\tt {p_A}}$ : ${\\tt {p_A}}={\\sf first}({\\sf getmsg}({\\sf first}(x)))$ , where function ${\\sf first}$ returns the first element of a tuple or a pair.",
"Since ${\\sf {frame}}(A^{\\prime })\\approx _{s}{\\sf {frame}}(B^{\\prime })$ , we should have the same relation ${\\tt {p_A}}={\\sf first}({\\sf getmsg}({\\sf first}(x)))$ in ${\\sf {frame}}(B^{\\prime })$ .",
"Intuitively, since on the right-hand side, the adversary can obtain the prescription ${\\tt {p_A}}$ from ${\\tt {\\mathit {PrescProof}^{\\mathit {r}}}}$ , due to $({\\tt {p_A}}, \\mathit {PrescriptID}^{r}, \\mathit {Comt^{r}_{\\mathit {dr}}}, {\\mathit {c\\_Comt^{r}_{\\mathit {pt}}}})={\\sf getmsg}({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}}),$ on the left-hand side of the first equivalence, there should also exist an output of a prescription proof $\\mathit {PrescProof}^{l}$ over public channels, from which the adversary can obtain a prescription ${\\tt {p_A}}$ , following the same relation: $({\\tt {p_A}}, \\mathit {PrescriptID}^{l}, \\mathit {Comt^{l}_{\\mathit {dr}}}, {\\mathit {c\\_Comt^{l}_{\\mathit {pt}}}})={\\sf getmsg}(\\mathit {PrescProof}^{l}).$ Next, we prove that the corresponding prescription proof $\\mathit {PrescProof}^{l}$ is indeed the prescription proof in the doctor sub-process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ in process $B$ , rather than other sub-processes.",
"Formally, the action $\\xrightarrow{}$ in process $B$ happens in sub-process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ .",
"On the right-hand side (process $A$ ), the doctor pseudonym ${\\tt {Pnym}}^{r}_{\\mathit {dr}}$ and the nonce for doctor commitment ${\\tt {r}}^{r}_{\\mathit {dr}}$ and the nonce for doctor credential ${\\tt {n}}^{r}_{\\mathit {dr}}$ (used for assumption s4') are revealed to the adversary on ${\\tt {chc}}$ channel.",
"Formally, $\\begin{array}{rll}A&\\xrightarrow{}\\xrightarrow{}&A_1\\mid \\mathit {\\lbrace {\\tt {Pnym}}^{r}_{\\mathit {dr}}/x_1\\rbrace }\\\\&\\xrightarrow{}\\xrightarrow{}&A_2\\mid \\mathit {\\lbrace {\\tt {Pnym}}^{r}_{\\mathit {dr}}/x_1\\rbrace }\\mid \\mathit {\\lbrace {\\tt {r}}^{r}_{\\mathit {dr}}/x_2\\rbrace }\\\\&\\xrightarrow{}\\xrightarrow{}&A_3\\mid \\mathit {\\lbrace {\\tt {Pnym}}^{r}_{\\mathit {dr}}/x_1\\rbrace }\\mid \\mathit {\\lbrace {\\tt {r}}^{r}_{\\mathit {dr}}/x_2\\rbrace }\\mid \\mathit {\\lbrace {\\tt {n}}^{r}_{\\mathit {dr}}/x_3\\rbrace }\\\\&\\xrightarrow{}\\xrightarrow{} & A_4\\mid \\mathit {\\lbrace {\\tt {Pnym}}^{r}_{\\mathit {dr}}/x_1\\rbrace }\\mid \\mathit {\\lbrace {\\tt {r}}^{r}_{\\mathit {dr}}/x_2\\rbrace }\\mid \\mathit {\\lbrace {\\tt {n}}^{r}_{\\mathit {dr}}/x_3\\rbrace }\\mid \\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},{\\tt {r}}^{r}_{\\mathit {dr}})/x\\rbrace }\\equiv A^{\\prime }.\\end{array}$ Since $A\\approx _{\\ell }B$ , we have that $\\begin{array}{rll}B&\\xrightarrow{}\\xrightarrow{}&B_1\\mid \\mathit {\\lbrace {\\tt {Pnym}}^{l}_{\\mathit {dr}}/x_1\\rbrace }\\\\&\\xrightarrow{}\\xrightarrow{}&B_2\\mid \\mathit {\\lbrace {\\tt {Pnym}}^{l}_{\\mathit {dr}}/x_1\\rbrace }\\mid \\mathit {\\lbrace {\\tt {r}}^{l}_{\\mathit {dr}}/x_2\\rbrace }\\\\&\\xrightarrow{}\\xrightarrow{}&B_3\\mid \\mathit {\\lbrace {\\tt {Pnym}}^{l}_{\\mathit {dr}}/x_1\\rbrace }\\mid \\mathit {\\lbrace {\\tt {r}}^{l}_{\\mathit {dr}}/x_2\\rbrace }\\mid \\mathit {\\lbrace {\\tt {n}}^{l}_{\\mathit {dr}}/x_3\\rbrace }\\\\&\\xrightarrow{}\\xrightarrow{} &B_4\\mid \\mathit {\\lbrace {\\tt {Pnym}}^{l}_{\\mathit {dr}}/x_1\\rbrace }\\mid \\mathit {\\lbrace {\\tt {r}}^{l}_{\\mathit {dr}}/x_2\\rbrace }\\mid \\mathit {\\lbrace {\\tt {n}}^{l}_{\\mathit {dr}}/x_3\\rbrace }\\mid \\mathit {\\lbrace (\\mathit {PrescProof}^{l},{\\tt {r}}^{l}_{\\mathit {dr}})/x\\rbrace } \\equiv B^{\\prime }.\\end{array}$ That is, on the left-hand side of the first equivalence, to be equivalent to the right-hand side, there also exist sub-processes which output messages on ${\\tt {chc}}$ channel.",
"Such sub-processes can only be $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ , because there is no output on ${\\tt {chc}}$ in other sub-processes in the left-hand side process (process $B$ ).",
"In ${\\sf {frame}}(A^{\\prime })={\\sf {frame}}(A_4)\\mid \\mathit {\\lbrace {\\tt {Pnym}}^{r}_{\\mathit {dr}}/x_1\\rbrace }\\mid \\mathit {\\lbrace {\\tt {r}}^{r}_{\\mathit {dr}}/x_2\\rbrace }\\mid \\mathit {\\lbrace {\\tt {n}}^{r}_{\\mathit {dr}}/x_3\\rbrace }\\mid \\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},{\\tt {r}}^{r}_{\\mathit {dr}})/x\\rbrace }$ , we have the following relation between two terms, $\\begin{array}{rcl}{\\sf first}(x)&=&{\\sf spk}((x_1,x_2,{\\tt {d_A}},x_3),\\\\&&\\hspace{17.0pt}({\\sf commit}(x_1,x_2),{\\sf drcred}(x_1,{\\tt {d_A}},x_3)),\\\\&&\\hspace{17.0pt}({\\tt {p_A}},\\mathit {PrescriptID}^{r}, {\\sf commit}(x_1,x_2),{\\mathit {c\\_Comt^{r}_{\\mathit {pt}}}})).\\end{array}$ Since $A^{\\prime }\\approx _{s}B^{\\prime }$ , we should have the same relation in ${\\sf {frame}}(B^{\\prime })$ .",
"$\\begin{array}{rcl}{\\sf first}(x)&=&{\\sf spk}((x_1,x_2,{\\tt {d_A}},x_3),\\\\&&\\hspace{17.0pt}({\\sf commit}(x_1,x_2),{\\sf drcred}(x_1,{\\tt {d_A}},x_3)),\\\\&&\\hspace{17.0pt}({\\tt {p_A}},\\mathit {PrescriptID}^{l}, {\\sf commit}(x_1,x_2),{\\mathit {c\\_Comt^{l}_{\\mathit {pt}}}})).\\end{array}$ On the left-hand side (process $B$ ), the terms sent by process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and/or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ over ${\\tt {chc}}$ – the doctor pseudonym and the nonces ${\\tt {Pnym}}^{l}_{\\mathit {dr}}$ , ${\\tt {r}}^{l}_{\\mathit {dr}}$ and ${\\tt {n}}^{l}_{\\mathit {dr}}$ (corresponding to ${\\tt {Pnym}}^{r}_{\\mathit {dr}}$ , ${\\tt {r}}^{r}_{\\mathit {dr}}$ and ${\\tt {n}}^{r}_{\\mathit {dr}}$ on the right-hand side), are essential to compute $\\mathit {PrescProof}^{l}$ .",
"Thus, process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and/or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ is able to compute and thus output the prescription proof $\\mathit {PrescProof}^{l}$ , given that the coerced doctor ${\\tt {d_A}}$ has the knowledge of ${\\tt {p_A}}$ , by applying the following function: $\\begin{array}{rcl}\\mathit {PrescProof}^{l}&=&{\\sf spk}(({\\tt {Pnym}}^{l}_{\\mathit {dr}}, {\\tt {r}}^{l}_{\\mathit {dr}}, {\\tt {d_A}}, {\\tt {n}}^{l}_{\\mathit {dr}}),\\\\&&\\hspace{17.0pt} ({\\sf commit}({\\tt {Pnym}}^{l}_{\\mathit {dr}},{\\tt {r}}^{l}_{\\mathit {dr}}),{\\sf drcred}({\\tt {Pnym}}^{l}_{\\mathit {dr}}, {\\tt {d_A}}, {\\tt {n}}^{l}_{\\mathit {dr}})),\\\\&&\\hspace{17.0pt} ({\\tt {p_A}},\\mathit {PrescriptID}^{l},{\\sf commit}({\\tt {Pnym}}^{l}_{\\mathit {dr}},{\\tt {r}}^{l}_{\\mathit {dr}}),{\\mathit {c\\_Comt^{l}_{\\mathit {pt}}}}).\\end{array}$ Now we have proved that the action of revealing $(\\mathit {PrescProof}^{l}, {\\tt {r}}^{l}_{\\mathit {dr}})$ ($\\xrightarrow{}$ ) can be taken in process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and/or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ in process $B$ .",
"Next we show that sub-processes except $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ in $B$ , cannot take the action of revealing $(\\mathit {PrescProof}^{l}, {\\tt {r}}^{l}_{\\mathit {dr}})$ , given the process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ does not replay the message $(\\mathit {PrescProof}^{l}, {\\tt {r}}^{l}_{\\mathit {dr}})$ in an honest doctor process.",
"By examining process $B$ , the sub-processes which send out a pair, the first element of which is a signed proof of knowledge, can only be doctor processes and the MPA processes, i.e., sub-processes that may send out a message $x$ potentially satisfying ${\\tt {p_A}}={\\sf first}({\\sf getmsg}({\\sf first}(x)))$ can only be doctor processes (at d10) or MPA processes (at m30).",
"Case 1: considering that message $x$ should also satisfy ${\\sf open}({\\sf third}({\\sf getmsg}({\\sf first}(x))),{\\sf snd}(x))={\\tt {Pnym}}^{l}_{\\mathit {dr}}$ , the processes revealing $x$ can only be doctor processes, because the second element in the message sent out at line m30 of an MPA process is a zero-knowledge proof, and thus cannot be used as a nonce to open a commitment third(getmsg(first(x))).",
"Case 2: considering that the message $x$ should satisfy ${\\sf first}(x)={\\sf spk}((x_1,x_2,{\\tt {d_A}},x_3),\\linebreak ({\\sf commit}(x_1,x_2),$${\\sf drcred}(x_1,{\\tt {d_A}},x_3)),$$({\\tt {p_A}},\\mathit {PrescriptID}^{l}, $${\\sf commit}(x_1,x_2),{\\mathit {c\\_Comt^{l}_{\\mathit {pt}}}}))$ , where $x_1$ is the doctor pseudonym, $x_2$ and $x_3$ are nonces, and the adversary receive $x_1$ , $x_2$ , $x_3$ from ${\\tt {chc}}$ channel, doctor sub-processes (except $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ ) cannot reveal the message $x$ .",
"Because these doctor sub-processes model honest doctor sessions, and thus use their own generated nonces to compute the signed proofs of knowledge (at line t23).",
"Such nonces are not sent to the adversary over ${\\tt {chc}}$ channel, since these doctor processes are not bribed or coerced.",
"Thus, the signed proofs of knowledge generated by these honest doctor prepossess cannot be the first element of the message $x$ , unless the process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and/or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ reuses one of the signed proofs of knowledge.",
"In the case that the process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and/or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ replay the message $(\\mathit {PrescProof}^{l}, {\\tt {r}}^{l}_{\\mathit {dr}})$ of an honest doctor process, $x_1$ needs to be the corresponding doctor pseudonym, and $x_2$ and $x_3$ need to be the corresponding nonces for the reused signed proof of knowledge the message.",
"Otherwise, $(\\mathit {PrescProof}^{l}, {\\tt {r}}^{l}_{\\mathit {dr}})$ will be detected as a fake message.",
"Thus, the message indeed represents the actual prescription in process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and/or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ .",
"Although the action of revealing message $(\\mathit {PrescProof}^{l}, {\\tt {r}}^{l}_{\\mathit {dr}})$ may be taken in an honest doctor process, the same action will be eventually taken in process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ .",
"Therefore, the process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and/or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ indeed outputs the prescription proof $\\mathit {PrescProof}^{l}$ on the left-hand side of the first equivalence, i.e., $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ /$\\mathit {P}_{\\mathit {dr}}^{\\prime }\\xrightarrow{}\\xrightarrow{} P\\mid \\mathit {\\lbrace (\\mathit {PrescProof}^{l},{\\tt {r}}^{l}_{\\mathit {dr}})/x\\rbrace }$ .",
"Let $C$ be the left-hand side of the second equivalence, and $D$ be the right-hand side.",
"$\\begin{array}{lrl}\\end{array}C&=&\\mathcal {C}_\\mathit {eh}[\\big ((\\mathit {init}_{\\mathit {dr}}^{\\prime }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {P}_{\\mathit {dr}}^{\\prime }))^{\\backslash {\\sf {out}}({\\tt {chc}}, \\cdot )}\\big )\\mid \\\\&&\\hspace{16.57497pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.(!",
"\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )]\\\\D&=&\\mathcal {C}_\\mathit {eh}[\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )\\mid \\\\&&\\hspace{16.57497pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.(!",
"\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )]$ $In process $ $, the sub-process $$\\mathit {init}$$\\mathit {dr}$ .",
"(!$\\mathit {P}$$\\mathit {dr}$ {dA/$\\mathit {Id_{\\mathit {dr}}}$ }$\\mathit {P}$$\\mathit {dr}$ ')out(chc, ):=chc.",
"($\\mathit {init}$$\\mathit {dr}$ '.",
"(!$\\mathit {P}$$\\mathit {dr}$ {dA/$\\mathit {Id_{\\mathit {dr}}}$ }$\\mathit {P}$$\\mathit {dr}$ ')!in(chc, y))$, according to the definition of $$\\mathit {P}$out(chc, )$.Since $$\\mathit {init}$$\\mathit {dr}$ '/$\\mathit {P}$$\\mathit {dr}$ '$ may take the action $ x. out(ch, x)$ where $ {($\\mathit {PrescProof}^{l}$ ,rl$\\mathit {dr}$ )/x}$, we have $$\\mathit {init}$$\\mathit {dr}$ '.",
"(!$\\mathit {P}$$\\mathit {dr}$ {dA/$\\mathit {Id_{\\mathit {dr}}}$ }$\\mathit {P}$$\\mathit {dr}$ ') *x. out(ch, x) P'{($\\mathit {PrescProof}^{l}$ ,rl$\\mathit {dr}$ )/x}$.",
"By filling it in the context $chc.",
"(_!in(chc, y))$, we have $chc.",
"(($\\mathit {init}$$\\mathit {dr}$ '.",
"(!$\\mathit {P}$$\\mathit {dr}$ {dA/$\\mathit {Id_{\\mathit {dr}}}$ }$\\mathit {P}$$\\mathit {dr}$ ')!in(chc, y))* x. out(ch, x)chc.",
"((P'{($\\mathit {PrescProof}^{l}$ ,rl$\\mathit {dr}$ )/x})!in(chc, y))$,and thus $ ($\\mathit {init}$$\\mathit {dr}$ '.",
"(!$\\mathit {P}$$\\mathit {dr}$ {dA/$\\mathit {Id_{\\mathit {dr}}}$ }$\\mathit {P}$$\\mathit {dr}$ ')out(chc, )*x. out(ch, x)chc.",
"(P'{($\\mathit {PrescProof}^{l}$ ,rl$\\mathit {dr}$ )/x}!in(chc, y))$.The sub-process $$\\mathit {\\mathit {main}_{\\mathit {dr}}}$$\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace }$ )$ also outputs a signed proof of knowledge from which the adversary obtains $pA$, i.e., $$\\mathit {\\mathit {main}_{\\mathit {dr}}}$$\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace }$ )* x. out(ch, x) C1{($\\mathit {PrescProof}$ ,r$\\mathit {dr}$ )/x1}$ and $pA=first(getmsg(first(x1)))$.Thus, in process $ C$, there are two outputs of a signed proof of knowledge, from which the adversary obtains $pA$.",
"Other signed proofs of knowledge will not lead to $pA$ or $pB$, as the prescription in process $ !",
"$\\mathit {P}$$\\mathit {dr}$ {dA/$\\mathit {Id_{\\mathit {dr}}}$ }$ and $ !",
"$\\mathit {P}$$\\mathit {dr}$ {dB/$\\mathit {Id_{\\mathit {dr}}}$ }$ are freshly generated.$ In process $D$ , the sub-process $\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace }$ outputs a prescription proof from which the adversary knows ${\\tt {p_B}}$ , sub-process $\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace }$ outputs a prescription proof from which the adversary knows ${\\tt {p_A}}$ , the prescriptions from the sub-process $!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }$ and $!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }$ are fresh names and thus cannot be ${\\tt {p_A}}$ or ${\\tt {p_B}}$ .",
"The adversary can detect that the process $C$ and $D$ are not equivalent: in process $C$ , the adversary obtains two ${\\tt {p_A}}$ , and in process $D$ , the adversary obtains one ${\\tt {p_A}}$ and one ${\\tt {p_B}}$ .",
"This contradicts the assumption that $C\\approx _{\\ell }D$ .",
"$\\Box $"
],
[
"Proof of Theorem ",
"Theorem 2 (independency of receipt-freeness).",
"The DLV08 protocol fails to satisfy independency of receipt-freeness.",
"Assume the DLV08 protocol satisfies independency of receipt-freeness.",
"That is, $\\exists $ $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and $P_{\\mathit {dr}}^{\\prime }$ satisfying the following two equivalences in the definition of independency of receipt-freeness (Definition REF ).",
"$\\begin{array}{lrl}1.\\ &&\\mathcal {C}_\\mathit {eh}[!\\mathit {R}_{\\mathit {ph}}^{{\\tt {chc}}}\\mid \\big (\\mathit {init}_{\\mathit {dr}}^{\\prime }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {P}_{\\mathit {dr}}^{\\prime })\\big )\\mid \\\\&&\\hspace{46.75pt} \\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )]\\\\&\\approx _{\\ell }&\\mathcal {C}_\\mathit {eh}[!\\mathit {R}_{\\mathit {ph}}^{{\\tt {chc}}}\\mid \\big ((\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace })^{{\\tt {chc}}}.\\\\&&\\hspace{53.125pt}(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid (\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })^{{\\tt {chc}}})\\big )\\mid \\\\&&\\hspace{46.75pt} \\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )]; \\vspace{5.69054pt} \\text{and}\\\\2.\\ &&\\mathcal {C}_\\mathit {eh}[!\\mathit {R}_{i}^{{\\tt {chc}}}\\mid \\big ((\\mathit {init}_{\\mathit {dr}}^{\\prime }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {P}_{\\mathit {dr}}^{\\prime }))^{\\backslash {\\sf {out}}({\\tt {chc}}, \\cdot )}\\big )\\mid \\\\&&\\hspace{46.75pt} \\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )]\\\\& \\approx _{\\ell }&\\mathcal {C}_\\mathit {eh}[!\\mathit {R}_{i}^{{\\tt {chc}}}\\mid \\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )\\mid \\\\&&\\hspace{46.75pt} \\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )].\\end{array}$ We prove that this assumption leads to contradictions.",
"Similar to the proof of Theorem REF , according to the definition of labelled bisimilarity (Definition REF ) and static equivalence (Definition REF ), Given $A\\approx _{\\ell }B$ , if $A\\xrightarrow{}A^{\\prime }$ and $M =_E N{\\sf {frame}}(A^{\\prime })$ , then $B\\xrightarrow{}B^{\\prime }$ and $M =_E N{\\sf {frame}}(B^{\\prime })$ .",
"Vice versa.",
"Let $A$ be the right-hand side of the first equivalence, $B$ be the left-hand side.",
"$\\begin{array}{lrl}A&=&\\mathcal {C}_\\mathit {eh}[!\\mathit {R}_{\\mathit {ph}}^{{\\tt {chc}}}\\mid \\big ((\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace })^{{\\tt {chc}}}.\\\\&&\\hspace{53.125pt}(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid (\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })^{{\\tt {chc}}})\\big )\\mid \\\\&&\\hspace{46.75pt} \\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )]\\\\B &=&\\mathcal {C}_\\mathit {eh}[!\\mathit {R}_{\\mathit {ph}}^{{\\tt {chc}}}\\mid \\big (\\mathit {init}_{\\mathit {dr}}^{\\prime }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {P}_{\\mathit {dr}}^{\\prime })\\big )\\mid \\\\&&\\hspace{46.75pt} \\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )]\\end{array}$ In process $A$ , the doctor ${\\tt {d_A}}$ computed a signed proof of knowledge ${\\tt {\\mathit {PrescProof}^{\\mathit {r}}}}$ in the sub-process $(\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })^{{\\tt {chc}}}$ .",
"The signed proof of knowledge is sent to a patient over private channel.",
"In addition, the signed proof of knowledge is also sent to the adversary over ${\\tt {chc}}$ together with a nonce (the message sent over ${\\tt {chc}}$ is $({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},{\\tt {r}}^{r}_{\\mathit {dr}})$ ).",
"On receiving the signed proof of knowledge, the patient sends it together with other information to a pharmacist over a private channel.",
"On receiving the message from the patient over private channel, the pharmacist forwards the message to the adversary over ${\\tt {chc}}$ (the message sent over ${\\tt {chc}}$ is $({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},\\mathit {PtSpk^r},\\mathit {vc}_1^r,\\mathit {vc}_2^r,\\mathit {vc}_3^r,\\mathit {vc}^{r^{\\prime }}_3,\\mathit {vc}_4^r,5^r)$ ).",
"Another sub-process $\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace }$ also generates a signed proof of knowledge ${\\tt {\\mathit {PrescProof}^{\\mathit {zr}}}}$ .",
"This signed proof of knowledge is sent to a patient in a message over private channel (but it is not sent to the adversary over ${\\tt {chc}}$ , as this sub-process is not bribed or coerced), and then sent to a pharmacist in another message via private channel.",
"Finally, the pharmacist, who receives the message containing ${\\tt {\\mathit {PrescProof}^{\\mathit {zr}}}}$ , sends the message to the adversary over channel ${\\tt {chc}}$ (the message sent to ${\\tt {chc}}$ is $({\\tt {\\mathit {PrescProof}^{\\mathit {zr}}}},\\mathit {PtSpk^{zr}},\\mathit {vc}_1^{zr},\\linebreak \\mathit {vc}_2^{zr},\\mathit {vc}_3^{zr},\\mathit {vc}^{zr^{\\prime }}_3,\\mathit {vc}_4^{zr},5^{zr})$ ).",
"Formally, there is a trace in process $A$ as follows.",
"$\\begin{array}{rll}A&\\xrightarrow{}\\xrightarrow{} &A_1\\mid \\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},{\\tt {r}}^{r}_{\\mathit {dr}})/x\\rbrace }\\\\&\\xrightarrow{}\\xrightarrow{}& A_2\\mid \\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},{\\tt {r}}^{r}_{\\mathit {dr}})/x\\rbrace }\\mid \\\\&&\\hspace{20.0pt}px\\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},\\mathit {PtSpk^r},\\mathit {vc}_1^r,\\mathit {vc}_2^r,\\mathit {vc}_3^r,\\mathit {vc}^{r^{\\prime }}_3,\\mathit {vc}_4^r,5^r)/y\\rbrace }\\\\&\\xrightarrow{}\\xrightarrow{}& A_3\\mid \\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},{\\tt {r}}^{r}_{\\mathit {dr}})/x\\rbrace }\\mid \\\\&&\\hspace{20.0pt}px\\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},\\mathit {PtSpk^r},\\mathit {vc}_1^r,\\mathit {vc}_2^r,\\mathit {vc}_3^r,\\mathit {vc}^{r^{\\prime }}_3,\\mathit {vc}_4^r,5^r)/y\\rbrace }\\mid \\\\&&\\hspace{20.0pt}px\\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {zr}}}},\\mathit {PtSpk^{zr}},\\mathit {vc}_1^{zr},\\mathit {vc}_2^{zr},\\mathit {vc}_3^{zr},\\mathit {vc}^{zr^{\\prime }}_3,\\mathit {vc}_4^{zr},5^{zr})/z\\rbrace }\\end{array}$ Let $A^{\\prime }=A_3\\mid \\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},{\\tt {r}}^{r}_{\\mathit {dr}})/x\\rbrace }\\mid \\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},\\mathit {PtSpk^r},\\mathit {vc}_1^r,\\mathit {vc}_2^r,\\mathit {vc}_3^r,\\mathit {vc}^{r^{\\prime }}_3,\\mathit {vc}_4^r,5^r)/y\\rbrace }\\mid \\linebreak \\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {zr}}}},\\mathit {PtSpk^{zr}},\\mathit {vc}_1^{zr},\\mathit {vc}_2^{zr},\\mathit {vc}_3^{zr},\\mathit {vc}^{zr^{\\prime }}_3,\\mathit {vc}_4^{zr},5^{zr})/z\\rbrace }$ .",
"We have ${\\tt {p_A}}={\\sf first}({\\sf getmsg}({\\sf first}(x)))={\\sf first}({\\sf getmsg}({\\sf first}(y)))$ and ${\\tt {p_B}}={\\sf first}({\\sf getmsg}({\\sf first}(z)))$ at frame ${\\sf {frame}}(A^{\\prime })$ .",
"Since $A\\approx _{\\ell }B$ , we should have that $\\begin{array}{rll}B&\\xrightarrow{}\\xrightarrow{} &B_1\\mid \\mathit {\\lbrace (\\mathit {PrescProof}^{l},{\\tt {r}}^{l}_{\\mathit {dr}})/x\\rbrace }\\\\&\\xrightarrow{}\\xrightarrow{}& B_2\\mid \\mathit {\\lbrace (\\mathit {PrescProof}^{l},{\\tt {r}}^{l}_{\\mathit {dr}})/x\\rbrace }\\mid \\\\&&\\hspace{20.0pt}px \\mathit {\\lbrace (\\mathit {PrescProof}^{l},\\mathit {PtSpk^l},\\mathit {vc}_1^l,\\mathit {vc}_2^l,\\mathit {vc}_3^l,\\mathit {vc}^{l^{\\prime }}_3,\\mathit {vc}_4^l,5^l)/y\\rbrace }\\mid \\\\&\\xrightarrow{}\\xrightarrow{}& B_3\\mid \\mathit {\\lbrace (\\mathit {PrescProof}^{l},{\\tt {r}}^{l}_{\\mathit {dr}})/x\\rbrace }\\mid \\\\&&\\hspace{20.0pt}px\\mathit {\\lbrace (\\mathit {PrescProof}^{l},\\mathit {PtSpk^l},\\mathit {vc}_1^l,\\mathit {vc}_2^l,\\mathit {vc}_3^l,\\mathit {vc}^{l^{\\prime }}_3,\\mathit {vc}_4^l,5^l)/y\\rbrace }\\mid \\\\&&\\hspace{20.0pt}px\\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {zl}}}},\\mathit {PtSpk^{zl}},\\mathit {vc}_1^{zl},\\mathit {vc}_2^{zl},\\mathit {vc}_3^{zl},\\mathit {vc}^{zl^{\\prime }}_3,\\mathit {vc}_4^{zl},5^{zl})/z\\rbrace }.\\end{array}$ Let $B^{\\prime }=B_3\\mid \\mathit {\\lbrace (\\mathit {PrescProof}^{l},{\\tt {r}}^{l}_{\\mathit {dr}})/x\\rbrace }\\mid \\mathit {\\lbrace (\\mathit {PrescProof}^{l},\\mathit {PtSpk^l},\\mathit {vc}_1^l,\\mathit {vc}_2^l,\\mathit {vc}_3^l,\\mathit {vc}^{l^{\\prime }}_3,\\mathit {vc}_4^l,5^l)/y\\rbrace }\\mid \\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {zl}}}},\\mathit {PtSpk^{zl}},\\mathit {vc}_1^{zl},\\mathit {vc}_2^{zl},\\mathit {vc}_3^{zl},\\mathit {vc}^{zl^{\\prime }}_3,\\mathit {vc}_4^{zl},5^{zl})/z\\rbrace }$ .",
"We should have $A^{\\prime }\\approx _{\\ell }B^{\\prime }$ , and thus, ${\\tt {p_A}}={\\sf first}({\\sf getmsg}({\\sf first}(x)))={\\sf first}({\\sf getmsg}({\\sf first}(y)))$ and ${\\tt {p_B}}={\\sf first}({\\sf getmsg}({\\sf first}(z)))$ at frame ${\\sf {frame}}(B^{\\prime })$ .",
"In process $B$ , the sub-process $\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace }$ generates a signed proof of knowledge $f$ and ${\\tt {p_A}}={\\sf first}({\\sf getmsg}(f))$ .",
"This signed proof of knowledge will be eventually revealed by a pharmacist, $B\\xrightarrow{}\\xrightarrow{}B_4\\mid \\mathit {\\lbrace (f,\\mathit {PtSpk^{f}},\\mathit {vc}_1^{f},\\mathit {vc}_2^{f},\\mathit {vc}_3^{f},\\mathit {vc}^{f^{\\prime }}_3,\\mathit {vc}_4^{f},5^{f})/h\\rbrace }$ and ${\\tt {p_A}}={\\sf first}({\\sf getmsg}({\\sf first}(h)))={\\sf first}({\\sf getmsg}(f))$ .",
"By examining process $B$ , we observe that $y=h$ .",
"The reason is as follows: since ${\\tt {p_A}}={\\sf first}({\\sf getmsg}({\\sf first}(y)))$ , sub-process $!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }$ generate fresh prescriptions and thus cannot be ${\\tt {p_A}}$ , therefore, $!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }$ does not generate a prescription which eventually leads to the action of sending $y$ .",
"Thus the possible sub-process which generates the prescription ${\\tt {p_A}}$ and potentially leads to sending $y$ can only be $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ , $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ or $\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace }$ .",
"Assume $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ generates prescription ${\\tt {p_A}}$ which leads to the action of sending $y$ and $y\\ne h$ , then, the adversary obtains three ${\\tt {p_A}}$ in process $B$ : one from ${\\tt {p_A}}={\\sf first}({\\sf getmsg}({\\sf first}(h)))$ , one from ${\\tt {p_A}}={\\sf first}({\\sf getmsg}({\\sf first}(y)))$ , and one from ${\\tt {p_A}}={\\sf first}({\\sf getmsg}({\\sf first}(x)))$ .",
"However, in process $A$ , the adversary can only observe two ${\\tt {p_A}}$ : one from ${\\tt {p_A}}={\\sf first}({\\sf getmsg}({\\sf first}(x)))$ and one from ${\\tt {p_A}}={\\sf first}({\\sf getmsg}({\\sf first}(y)))$ .",
"This contradicts the assumption that $A\\approx _{\\ell }B$ .",
"Therefore, the prescription ${\\tt {p_A}}$ which leads to the action of revealing $y$ is generated in sub-process $\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace }$ , and thus $y=h$ .",
"In addition, in process $B$ , we observe that the prescription ${\\tt {p_B}}$ is generated in process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ .",
"As sub-process $!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }$ generates fresh prescriptions and thus cannot be ${\\tt {p_B}}$ , and sub-process $\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace }$ generates ${\\tt {p_A}}$ (${\\tt {p_A}}\\ne {\\tt {p_B}}$ ), the only sub-process can generate ${\\tt {p_B}}$ is $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ .",
"The second equivalence also confirms this observation.",
"Let $C$ be the left-hand side of the second equivalence, and $D$ be the right-hand side.",
"$\\begin{array}{lrl}C &=&\\mathcal {C}_\\mathit {eh}[!\\mathit {R}_{i}^{{\\tt {chc}}}\\mid \\big ((\\mathit {init}_{\\mathit {dr}}^{\\prime }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {P}_{\\mathit {dr}}^{\\prime }))^{\\backslash {\\sf {out}}({\\tt {chc}}, \\cdot )}\\big )\\mid \\\\&&\\hspace{46.75pt} \\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )]\\\\D& = &\\mathcal {C}_\\mathit {eh}[!\\mathit {R}_{i}^{{\\tt {chc}}}\\mid \\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )\\mid \\\\&&\\hspace{46.75pt} \\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )].\\end{array}$ In process $D$ , the adversary can obtain one ${\\tt {p_A}}$ and one ${\\tt {p_B}}$ .",
"Since $C\\approx _{\\ell }D$ , in process $D$ , the adversary should also obtain one ${\\tt {p_A}}$ and one ${\\tt {p_B}}$ .",
"Since sub-process $\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace }$ generates ${\\tt {p_A}}$ and sub-process $!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }$ and $!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }$ cannot generates ${\\tt {p_B}}$ , it must be process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ who generates ${\\tt {p_B}}$ .",
"As the generated ${\\tt {p_B}}$ in process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ is first sent to patient, then sent to a pharmacist and thus leads to a message sending over ${\\tt {chc}}$ .",
"The message revealed by the pharmacist is $z$ , because ${\\tt {p_B}}={\\sf first}({\\sf getmsg}({\\sf first}(z)))$ , and on other process can generate ${\\tt {p_B}}$ in process $B$ .",
"By examining process $B$ , the only sub-process which can take the action of sending $x$ is process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ , as process $!\\mathit {R}_{\\mathit {ph}}^{{\\tt {chc}}}$ does not send a message $x$ which is a pair and thus satisfies $x={\\sf pair}({\\sf first}(x), {\\sf snd}(x))$ , and other processes does not involving using channel ${\\tt {chc}}$ .",
"Intuitively, in process $B$ , sub-process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and/or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ sends ${\\tt {p_B}}$ to the patient which leads to the action of sending $z$ ; meanwhile, the sub-process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and/or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ lies to the adversary that the singed proof of knowledge for prescription is $\\mathit {PrescProof}^{l}$ by sending $x$ .",
"In process $A$ , in addition to ${\\tt {\\mathit {PrescProof}^{\\mathit {r}}}}$ , process $(\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })^{{\\tt {chc}}}$ also sends other information over channel ${\\tt {chc}}$ .",
"Formally, $\\begin{array}{rll}A&\\xrightarrow{}\\xrightarrow{}&A^1\\mid \\mathit {\\lbrace {\\tt {Pnym}}^{r}_{\\mathit {dr}}/x_1\\rbrace }\\\\&\\xrightarrow{}\\xrightarrow{}&A^2\\mid \\mathit {\\lbrace {\\tt {Pnym}}^{r}_{\\mathit {dr}}/x_1\\rbrace }\\mid \\mathit {\\lbrace {\\tt {r}}^{r}_{\\mathit {dr}}/x_2\\rbrace }\\\\&\\xrightarrow{}\\xrightarrow{}&A^3\\mid \\mathit {\\lbrace {\\tt {Pnym}}^{r}_{\\mathit {dr}}/x_1\\rbrace }\\mid \\mathit {\\lbrace {\\tt {r}}^{r}_{\\mathit {dr}}/x_2\\rbrace }\\mid \\mathit {\\lbrace {\\tt {n}}^{r}_{\\mathit {dr}}/x_3\\rbrace }\\\\&\\xrightarrow{}\\xrightarrow{} & A_1\\mid \\mathit {\\lbrace {\\tt {Pnym}}^{r}_{\\mathit {dr}}/x_1\\rbrace }\\mid \\mathit {\\lbrace {\\tt {r}}^{r}_{\\mathit {dr}}/x_2\\rbrace }\\mid \\mathit {\\lbrace {\\tt {n}}^{r}_{\\mathit {dr}}/x_3\\rbrace }\\mid \\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},{\\tt {r}}^{r}_{\\mathit {dr}})/x\\rbrace },\\end{array}$ and $x_1$ , $x_2$ , $x_3$ and $x$ satisfy $\\begin{array}{rcl}{\\sf first}(x)&=&{\\sf spk}((x_1,x_2,{\\tt {d_A}},x_3),\\\\&&\\hspace{17.0pt}({\\sf commit}(x_1,x_2),{\\sf drcred}(x_1,{\\tt {d_A}},x_3)),\\\\&&\\hspace{17.0pt}({\\tt {p_A}},{\\sf snd}({\\sf getmsg}({\\sf first}(x))), {\\sf commit}(x_1,x_2),{\\sf fourth}({\\sf getmsg}({\\sf first}(x))))).\\end{array}$ Since $A\\approx _{\\ell }B$ , we should have that $\\begin{array}{rll}B&\\xrightarrow{}\\xrightarrow{}&B^1\\mid \\mathit {\\lbrace {\\tt {Pnym}}^{l}_{\\mathit {dr}}/x_1\\rbrace }\\\\&\\xrightarrow{}\\xrightarrow{}&B^2\\mid \\mathit {\\lbrace {\\tt {Pnym}}^{l}_{\\mathit {dr}}/x_1\\rbrace }\\mid \\mathit {\\lbrace {\\tt {r}}^{l}_{\\mathit {dr}}/x_2\\rbrace }\\\\&\\xrightarrow{}\\xrightarrow{}&B^3\\mid \\mathit {\\lbrace {\\tt {Pnym}}^{l}_{\\mathit {dr}}/x_1\\rbrace }\\mid \\mathit {\\lbrace {\\tt {r}}^{l}_{\\mathit {dr}}/x_2\\rbrace }\\mid \\mathit {\\lbrace {\\tt {n}}^{l}_{\\mathit {dr}}/x_3\\rbrace }\\\\&\\xrightarrow{}\\xrightarrow{} &B_1\\mid \\mathit {\\lbrace {\\tt {Pnym}}^{l}_{\\mathit {dr}}/x_1\\rbrace }\\mid \\mathit {\\lbrace {\\tt {r}}^{l}_{\\mathit {dr}}/x_2\\rbrace }\\mid \\mathit {\\lbrace {\\tt {n}}^{l}_{\\mathit {dr}}/x_3\\rbrace }\\mid \\mathit {\\lbrace (\\mathit {PrescProof}^{l},{\\tt {r}}^{l}_{\\mathit {dr}})/x\\rbrace },\\end{array}$ and the same relation holds between $x_1$ , $x_2$ , $x_3$ and $x$ , $\\begin{array}{rcl}{\\sf first}(x)&=&{\\sf spk}((x_1,x_2,{\\tt {d_A}},x_3),\\\\&&\\hspace{17.0pt}({\\sf commit}(x_1,x_2),{\\sf drcred}(x_1,{\\tt {d_A}},x_3)),\\\\&&\\hspace{17.0pt}({\\tt {p_A}},{\\sf snd}({\\sf getmsg}({\\sf first}(x))), {\\sf commit}(x_1,x_2),{\\sf fourth}({\\sf getmsg}({\\sf first}(x))))).",
"\\end{array}\\qquad \\mathrm {(eq1)}$ The sub-process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ do not know ${\\tt {Pnym}}^{l}_{\\mathit {dr}}, {\\tt {n}}^{l}_{\\mathit {dr}}$ since the two information satisfies secrecy.",
"Thus sub-process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and/or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ cannot send $x_1$ and $x_3$ over ${\\tt {chc}}$ channel.",
"Even assume the process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ know the private information ${\\tt {Pnym}}^{l}_{\\mathit {dr}}, {\\tt {r}}^{l}_{\\mathit {dr}}, {\\tt {n}}^{l}_{\\mathit {dr}}$ for constructing $\\mathit {PrescProof}^{l}$ , since $\\mathit {PrescProof}^{l}$ is actually generated by $\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace }$ , we have the following relation: $\\begin{array}{rcl}f=\\mathit {PrescProof}^{l}&=&{\\sf spk}(({\\tt {Pnym}}^{l}_{\\mathit {dr}}, {\\tt {r}}^{l}_{\\mathit {dr}}, {\\tt {d_B}}, {\\tt {n}}^{l}_{\\mathit {dr}}),\\\\&&\\hspace{17.0pt} ({\\sf commit}({\\tt {Pnym}}^{l}_{\\mathit {dr}},{\\tt {r}}^{l}_{\\mathit {dr}}),{\\sf drcred}({\\tt {Pnym}}^{l}_{\\mathit {dr}}, {\\tt {d_B}}, {\\tt {n}}^{l}_{\\mathit {dr}})),\\\\&&\\hspace{17.0pt} ({\\tt {p_A}},\\mathit {PrescriptID}^{l},{\\sf commit}({\\tt {Pnym}}^{l}_{\\mathit {dr}},{\\tt {r}}^{l}_{\\mathit {dr}}),{\\mathit {c\\_Comt^{l}_{\\mathit {pt}}}}).\\end{array}$ that is, $\\begin{array}{rcl}{\\sf first}(x)&=&{\\sf spk}((x_1,x_2,{\\tt {d_B}},x_3),\\\\&&\\hspace{17.0pt}({\\sf commit}(x_1,x_2),{\\sf drcred}(x_1,{\\tt {d_B}},x_3)),\\\\&&\\hspace{17.0pt}({\\tt {p_A}},{\\sf snd}({\\sf getmsg}({\\sf first}(x))), {\\sf commit}(x_1,x_2),{\\sf fourth}({\\sf getmsg}({\\sf first}(x))))).\\end{array}\\qquad \\mathrm {(eq2)}$ and thus, the adversary can detect that $\\mathit {PrescProof}^{l}$ is generated by ${\\tt {d_B}}$ by telling the difference between (eq1) and (eq2).",
"This contradicts the assumption that $A\\approx _{\\ell }B$ .",
"$\\Box $"
],
[
"Case study: the DLV08 protocol",
"In this section, we apply the above formal definitions for doctor privacy in a case study as a validation of the definitions.",
"We choose to analyse the DLV08 e-health protocol proposed by De Decker et al.",
"[24], as it claims enforced privacy for doctors.",
"However, our analysis is not restricted to doctor privacy.",
"We provide a rather complete analysis of the protocol including patient anonymity, patient untraceability, patient/doctor information secrecy and patient/doctor authentication as well.",
"The ProVerif code used to perform this analysis is available from [26].",
"The DLV08 protocol is a complex health care protocol for the Belgium situation.",
"It captures most aspects of the current Belgian health care practice and aims to provide a strong guarantee of privacy for patients and doctors.",
"Our analysis of this protocol focuses on the below properties.",
"For those that are explicitly claimed by DLV08, the corresponding claim identifier in that paper is given.",
"In addition to those, we analyse secrecy, prescription privacy, receipt-freeness, and independency of receipt-freeness, which are implicitly mentioned.",
"Secrecy of patient and doctor information: no other party should be able to know a patient or a doctor's information, unless the information is intended to be revealed in the protocol (for formal definitions, see Section REF and Section REF ).",
"Authentication ([24]: S1): all parties should properly authenticate each other (for formal definitions, see Section REF and Section REF ).",
"Patient anonymity ([24]: P3): no party should be able to determine a patient's identity.",
"Patient untraceability ([24]: P2): prescriptions issued to the same patient should not be linkable to each other.",
"Prescription privacy: the protocol protects a doctor's prescription behaviour.",
"Receipt-freeness: the protocol prevents bribery between doctors and pharmaceutical companies.",
"Independency of prescription privacy ([24]: P4): pharmacists should not be able to provide evidence to pharmaceutical companies about doctors' prescription.",
"Independency of receipt-freeness: pharmacists should not be able to provide evidence to pharmaceutical companies about doctors' prescription even if the doctor is bribed.",
"The rest of this section describes the DLV08 protocol in more detail."
],
[
"Roles",
"The protocol involves seven roles.",
"We focus on the five roles involved in the core process: doctor, patient, pharmacist, medicine prescription administrator (MPA) and health insurance institute (HII).",
"The other two roles, public safety organisation (PSO) and social security organisation (SSO), provide properties such as revocability and reimbursement.",
"As we do not focus on these properties, and as these roles are only tangentially involved in the core process, we omit these roles from our model.",
"The roles interact as follows: a doctor prescribes medicine to a patient; next the patient obtains medicine from a pharmacist according to the prescription; following that, the pharmacist forwards the prescription to his MPA, the MPA checks the prescription and refunds the pharmacist; finally, the MPA sends invoices to the patient's HII and is refunded."
],
[
"Cryptographic primitives",
"To ensure security and privacy properties, the DLV08 protocol employs several specific cryptographic primitives, besides the classical ones, like encryption.",
"We briefly introduce these cryptographic primitives."
],
[
"Bit-commitments.",
"The bit-commitments scheme consists of two phases, committing phase and opening phase.",
"On the committing phase, a message sender commits to a message.",
"This can be considered as putting the message into a box, and sending the box to the receiver.",
"Later in the opening phase, the sender sends the key of the box to the receiver.",
"The receiver opens the box and obtains the message."
],
[
"Zero-knowledge proofs.",
"A zero-knowledge proof is a cryptographic scheme which is used by one party (prover) to prove to another party (verifier) that a statement is true, without leaking secret information of the prover.",
"A zero-knowledge proof scheme may be either interactive or non-interactive.",
"We consider non-interactive zero-knowledge proofs in this protocol."
],
[
"Digital credentials.",
"A digital credential is a certificate, proving that the holder satisfies certain requirements.",
"Unlike paper certificates (such as passports) which give out the owner's identity, a digital credential can be used to authenticate the owner anonymously.",
"For example, a digital credential can be used to prove that a driver is old enough to drive without revealing the actual age of the driver."
],
[
"Anonymous authentication.",
"Anonymous authentication is a scheme for authenticating a user anonymously, e.g., [12].",
"The procedure of anonymous authentication is actually a zero-knowledge proof, with the digital credential being the public information of the prover.",
"In the scheme, a user's digital credential is used as the public key in a public key authentication structure.",
"Using this, a verifier can check whether a message is signed correctly by the prover (the person authenticating himself), while the verifier cannot identify the prover.",
"Thus, this ensures anonymous authentication."
],
[
"Verifiable encryptions.",
"Verifiable encryption is based on zero-knowledge proofs as well.",
"A prover encrypts a message, and uses zero-knowledge proofs to prove that the encrypted message satisfies specific properties without revealing the original message."
],
[
"Signed proofs of knowledge.",
"Signed proofs of knowledge provide a way of using proofs of knowledge as a digital signature scheme (cf. [20]).",
"Intuitively, a prover signs a message using secret information, which can be considered as a secret signing key.",
"The prover can convince the verifier using proofs of knowledge only if the prover has the right secret key.",
"Thus it proves the origination of the message."
],
[
"Setting",
"The initial information available to a participant is as follows.",
"A doctor has an identity (${{\\tt {Id}}_{\\mathit {dr}}}$ ), a pseudonym (${{\\tt {Pnym}}_{\\mathit {dr}}}$ ), and an anonymous doctor credential ($\\mathit {Cred_{\\mathit {dr}}}$ ) issued by trusted authorities.",
"A patient has an identity (${{\\tt {Id}}_{\\mathit {pt}}}$ ), a pseudonym (${{\\tt {Pnym}}_{\\mathit {pt}}}$ ), an HII (${{\\tt {Hii}}}$ ), a social security status (${{\\tt {Sss}}}$ ), a health expense account (${{\\tt {Acc}}}$ ) and an anonymous patient credential ($\\mathit {Cred_{\\mathit {pt}}}$ ) issued by trusted authorities.",
"Pharmacists, MPA, and HII are public entities, each of which has an identity (${\\tt {Id}}_{\\mathit {ph}}$ , ${{\\tt {Id}}_{\\mathit {mpa}}}$ , ${\\tt {Id}}_{\\mathit {hii}}$ ), a secret key (${\\tt {sk}}_{\\mathit {ph}}$ , ${\\tt {sk}}_{\\mathit {mpa}}$ , ${\\tt {sk}}_{\\mathit {hii}}$ ) and an authorised public key certificate (${\\tt {pk}}_\\mathit {ph}$ , ${\\tt {pk}}_\\mathit {mpa}$ , ${\\tt {pk}}_\\mathit {hii}$ ) issued by trusted authorities.",
"We assume that a user does not take two roles with the same identity.",
"Hence, one user taking two roles are considered as two individual users."
],
[
"Description of the protocol",
"The DLV08 protocol consists of four sub-protocols: doctor-patient sub-protocol, patient-pharmacist sub-protocol, pharmacist-MPA sub-protocol, and MPA-HII sub-protocol.",
"We describe the sub-protocols one by one."
],
[
"Doctor-patient sub-protocol",
"The doctor authenticates himself to a patient by anonymous authentication with the authorised doctor credential as public information.",
"The patient verifies the doctor credential.",
"If the verification passes, the patient anonymously authenticates himself to the doctor using the patient credential, sends the bit-commitments on his identity to the doctor, and proves to the doctor that the identity used in the credential is the same as in the bit-commitments.",
"After verifying the patient credential, the doctor generates a prescription, computes a prescription identity, computes the doctor bit-commitments.",
"Then the doctor combines these computed messages with the received patient bit-commitments; signs these messages using a signed proof of knowledge, which proves that the doctor's pseudonym used in the doctor credential is the same as in the doctor bit-commitments.",
"Together with the proof, the doctor sends the opening information, which is used to open the doctor bit-commitments.",
"The communication in the doctor-patient sub-protocol is shown as a message sequence chart (MSC, [48]) in Figure REF .",
"Figure: Doctor-Patient sub-protocol.Figure: Patient-Pharmacist sub-protocol.Figure: Pharmacist-MPA sub-protocol.Figure: MPA-HII sub-protocol."
],
[
"Patient-Pharmacist sub-protocol",
"The pharmacist authenticates himself to the patient using public key authentication.",
"The patient verifies the authentication and obtains, from the authentication, the pharmacist's identity and the pharmacist's MPA.",
"Then the patient anonymously authenticates himself to the pharmacist, and proves his social security status.",
"Next, the patient computes verifiable encryptions $\\mathit {vc}_1$ , $\\mathit {vc}_2$ , $\\mathit {vc}_3$ , $\\mathit {vc}^{\\prime }_3$ , $\\mathit {vc}_4$ , $\\mathit {vc}_5$ , where $\\mathit {vc}_1$ encrypts the patient's HII using the MPA's public key and proves that the HII encrypted in $\\mathit {vc}_1$ is the same as the one in the patient's credential.",
"$\\mathit {vc}_2$ encrypts the doctor's pseudonym using the MPA's public key and proves that the doctor's pseudonym encrypted in $\\mathit {vc}_2$ is the same as the one in the doctor commitment embedded in the prescription.",
"$\\mathit {vc}_3$ encrypts the patient's pseudonym using the public safety organisation's public key and proves that the pseudonym encrypted in $\\mathit {vc}_3$ is the same as the one in the patient's commitment.",
"$\\mathit {vc}^{\\prime }_3$ encrypts the patient's HII using the social security organisation's public key and proves that the content encrypted in $\\mathit {vc}^{\\prime }_3$ is the same as the HII in the patient's credential.",
"$\\mathit {vc}_4$ encrypts the patient's pseudonym using the MPA's public key and proves that the patient's pseudonym encrypted in $\\mathit {vc}_4$ is the same as the one in the patient's credential.",
"$\\mathit {vc}_5$ encrypts the patient's pseudonym using his HII's public key and proves that the patient's pseudonym encrypted in $\\mathit {vc}_5$ is the same as the one in the patient's credential.",
"5 encrypts $\\mathit {vc}_5$ using the MPA's public key.",
"The patient sends the received prescription to the pharmacist and proves to the pharmacist that the patient's identity in the prescription is the same as in the patient credential.",
"The patient sends $\\mathit {vc}_1, \\mathit {vc}_2, \\mathit {vc}_3, \\mathit {vc}^{\\prime }_3, \\mathit {vc}_4, 5$ as well.",
"The pharmacist verifies the correctness of all the received messages.",
"If every message is correctly formatted, the pharmacist charges the patient, and delivers the medicine.",
"Then the pharmacist generates an invoice and sends it to the patient.",
"The patient computes a receipt $\\mathit {ReceiptAck}$ : signing a message (consists of the prescription identity, the pharmacist's identity, $\\mathit {vc}_1$ , $\\mathit {vc}_2$ , $\\mathit {vc}_3$ , $\\mathit {vc}^{\\prime }_3$ , $\\mathit {vc}_4$ , $\\mathit {vc}_5$ ) using a signed proof of knowledge and proving that he knows the patient credential.",
"This receipt proves that the patient has received his medicine.",
"The pharmacist verifies the correctness of the receipt.",
"The communication in the patient-Pharmacist sub-protocol is shown in Figure REF .",
"Since the payment and medicine delivery procedures are out of the protocol scope, they are interpreted as dashed arrows in the figure."
],
[
"Pharmacist-MPA sub-protocol",
"The pharmacist and the MPA first authenticate each other using public key authentication.",
"Next, the pharmacist sends the received prescription and the receipt $\\mathit {ReceiptAck}$ , together with $\\mathit {vc}_1$ , $\\mathit {vc}_2$ , $\\mathit {vc}_3$ , $\\mathit {vc}^{\\prime }_3$ , $\\mathit {vc}_4$ , 5, to the MPA.",
"The MPA verifies correctness of the received information.",
"Then the MPA decrypts $\\mathit {vc}_1$ , $\\mathit {vc}_2$ , $\\mathit {vc}_4$ and 5, which provide the patient's HII, the doctor's pseudonym, the patient's pseudonym, and $\\mathit {vc}_5$ .",
"The communication in the pharmacist-MPA sub-protocol is shown in Figure REF .",
"Note that after authentication, two parties often establish a secure communication channel.",
"However, it is not mentioned in [24] that the pharmacist and the MPA agree on anything.",
"Nevertheless, this does not affect the properties that we verified, except authentication between pharmacist and MPA."
],
[
"MPA-HII sub-protocol",
"The MPA and the patient's HII first authenticate each other using public key authentication.",
"Then the MPA sends the receipt $\\mathit {ReceiptAck}$ to the patient's HII as well as the verifiable encryption $\\mathit {vc}_5$ which encrypts the patient's pseudonym with the patient's HII's public key.",
"The patient's HII checks the correctness of $\\mathit {ReceiptAck}$ , decrypts $\\mathit {vc}_5$ and obtains the patient's pseudonym.",
"From the patient pseudonym, the HII obtains the identity of the patient; then updates the patient's account and pays the MPA.",
"The MPA pays the pharmacist when he receives the payment.",
"The communication in the MPA-HII sub-protocol is shown in Figure REF .",
"Similar to the previous sub-protocol, there is nothing established during the authentication which can be used in the later message exchanges.",
"Note that in addition to authentications between MPA and HII, this affects the secrecy of a patient's pseudonym when the adversary controls dishonest patients (see Section REF )."
],
[
"Modelling DLV08",
"We model the DLV08 protocol in the applied pi calculus as introduced in Section REF .",
"For clarity, we also borrow some syntactic expressions from ProVerif, such as key words `$\\mathit {fun}$ ', `$\\mathit {private}$ $\\mathit {fun}$ ', `$\\mathit {reduc}$ ' and `$\\mathit {equation}$ ', and expression $`\\mbox{let}\\ x=N \\ \\mbox{in} \\ P^{\\prime }$ .",
"Particularly, `($\\mathit {private}$ ) $\\mathit {fun}$ ' denotes a constructor which uses terms to form a more complex term ('$\\mathit {private}$ ' means the adversary cannot use it).",
"`$\\mathit {reduc}$ ' and `$\\mathit {equation}$ ' are key words used to construct the equational theory $E$ .",
"`$\\mathit {reduc}$ ' denotes a destructor which retrieves sub-terms of a constructed term.",
"For the cryptographic primitives that cannot be captured by destructors, ProVerif provides `$\\mathit {equation}$ ' to capture the relationship between constructors.",
"The expression `$\\mbox{let}\\ x=N \\ \\mbox{in} \\ P$ ' is used as syntactical substitutions, i.e., $P\\lbrace N/x\\rbrace $ in the applied pi calculus.",
"It is an abbreviation of $`\\mbox{let}\\ x=N \\ \\mbox{in} \\ P \\ \\mbox{else} \\ Q^{\\prime }$ when $Q$ is the null process.",
"When $N$ is a destructor, there are two possible outcomes.",
"If the term $N$ does not fail, then $x$ is bound to $N$ and process $P$ is taken, otherwise $Q$ (in this case, the null process) is taken.",
"Since the description of the protocol in its original paper is not clear in some details, before modelling the protocol, several ambiguities need to be settled (Section REF ).",
"Next we explain the modelling of the cryptographic primitives (Section REF ), since security and privacy rely heavily on these cryptographic primitives in the protocol.",
"Then, we illustrate the modelling of the protocol (Section REF )."
],
[
"Underspecification of the DLV08 protocol",
"The DLV08 protocol leaves the following issues unspecified: Table: NO_CAPTIONTo be able to discover potential flaws on privacy, we make the following (weakest) assumptions in our modelling of the DLV08 protocol: Table: NO_CAPTIONNote that some assumptions may look weak to security experts, for example the assumption of deterministic encryption.",
"However, without explicit warning, deterministic encryption algorithms may be used, which will lead to security flaws.",
"With this in mind, we assume the weakest assumption when there is ambiguity.",
"By assuming weak assumptions and showing the security flaws with the assumptions, we provide security warnings for the implementation of the protocol."
],
[
"Modelling cryptographic primitives",
"The cryptographic primitives are modelled in the applied pi calculus using function symbols and equations.",
"All functions and equational theory are summarised in Figures REF , REF and REF .",
"Figure: Functions.Figure: Equational theory part i@: non-zero-knowledge part.Figure: Equational theory part ii@: zero-knowledge part."
],
[
"Bit-commitments.",
"The bit-commitments scheme is modelled as two functions: ${\\sf commit}$ , modelling the committing phase, and ${\\sf open}$ , modelling the opening phase.",
"The function ${\\sf commit}$ creates a commitment with two parameters: a message $m$ and a random number $r$ .",
"A commitment can only be opened with the correct opening information $r$ , in which case the message $m$ is revealed.",
"$\\begin{array}{rl}\\mathit {fun}\\ & {\\sf commit}/2.",
"\\\\\\mathit {reduc\\ }& {\\sf open}({\\sf commit}(m,r),r)=m.\\end{array}$"
],
[
"Zero-knowledge proofs.",
"Non-interactive zero-knowledge proofs can be modelled as function ${\\sf zk}(\\mathit {secrets}, \\mathit {pub\\_info})$ inspired by [19].",
"The public verification information $\\mathit {pub\\_info}$ and the secret information $\\mathit {secrets}$ satisfy a pre-specified relation.",
"Since the secret information is only known by the prover, only the prover can construct the zero-knowledge proof.",
"To verify a zero-knowledge proof is to check whether the relation between the secret information and the verification information is satisfied.",
"Verification of a zero-knowledge proof is modelled as function $\\mbox{{\\sf Vfy-{zk}}}({\\sf zk}(\\mathit {secrets},\\mathit {pub\\_info}), \\mathit {verif\\_info})$ , with a zero-knowledge proof to be verified ${\\sf zk}(\\mathit {secrets},\\linebreak \\mathit {pub\\_info})$ and the verification information $\\mathit {verif\\_info}$ .",
"Compared to the more generic definitions in [19], we define each zero-knowledge proof specifically, as only a limited number of zero-knowledge proofs are used in the protocol.",
"We specify each verification rule in Figure REF .",
"Since the $\\mathit {pub\\_info}$ and $\\mathit {verif\\_info}$ happen to be the same in all the zero-knowledge proofs verifications in this protocol, the generic structure of verification rule is given as $\\mbox{{\\sf Vfy-{zk}}}({\\sf zk}(\\mathit {secrets},\\mathit {pub\\_info}),\\mathit {pub\\_info})={\\sf true},$ where ${\\sf true}$ is a constant.",
"The specific function to check a zero-knowledge proof of type $z$ is denoted as $\\mbox{{\\sf Vfy-{zk}}}_z$ , e.g., verification of a patient's anonymous authentication modelled by function $\\mbox{\\sf Vfy-{zk}}_{\\sf Auth_{\\mathit {pt}}}$ ."
],
[
"Digital credentials.",
"A digital credential is issued by trusted authorities.",
"We assume the procedure of issuing a credential is perfect, which means that the adversary cannot forge a credential nor obtain one by impersonation.",
"We model digital credentials as a private function (declaimed by key word $\\mathit {private}\\ \\mathit {fun}\\ $ in ProVerif) which is only usable by honest users.",
"In the DLV08 protocol, a credential can have several attributes; we model these as parameters of the credential function.",
"$\\begin{array}{l@{\\hspace{28.45274pt}}l}\\mathit {private}\\ \\mathit {fun}\\ {\\sf drcred}/2.&\\mathit {private}\\ \\mathit {fun}\\ {\\sf ptcred}/5.\\end{array}$ There are two credentials in the DLV08 protocol: a doctor credential which is modelled as $\\mathit {Cred_{\\mathit {dr}}}:={\\sf drcred}({{\\tt {Pnym}}_{\\mathit {dr}}}, {{\\tt {Id}}_{\\mathit {dr}}})$ , and a patient credential which is modelled as $\\mathit {Cred_{\\mathit {pt}}}:=\\linebreak {\\sf ptcred}({{\\tt {Id}}_{\\mathit {pt}}}, {{\\tt {Pnym}}_{\\mathit {pt}}}, {{\\tt {Hii}}},{{\\tt {Sss}}}, {{\\tt {Acc}}})$ .",
"Unlike private data, the two private functions cannot be coerced, meaning that even by coercing, the adversary cannot apply the private functions.",
"Because a doctor having an anonymous credentials is a basic setting of the protocol, and thus the procedure of obtaining a credential is not assumed to be bribed or coerced.",
"However, the adversary can coerce patients or doctors for the credentials and parameters of the private functions."
],
[
"Anonymous authentication.",
"The procedure of anonymous authentication is a zero-knowledge proof using the digital credential as public information.",
"The anonymous authentication of a doctor is modelled as $\\mathit {Auth_{\\mathit {dr}}}:={\\sf zk}(({{\\tt {Pnym}}_{\\mathit {dr}}}, {{\\tt {Id}}_{\\mathit {dr}}}), {\\sf drcred}({{\\tt {Pnym}}_{\\mathit {dr}}},{{\\tt {Id}}_{\\mathit {dr}}})),$ and the verification of the authentication is modelled as $\\mbox{\\sf Vfy-{zk}}_{\\sf Auth_{\\mathit {dr}}}(\\mathit {Auth_{\\mathit {dr}}},{\\sf drcred}({{\\tt {Pnym}}_{\\mathit {dr}}},{{\\tt {Id}}_{\\mathit {dr}}})).$ The equational theory for the verification is $\\mathit {reduc\\ }\\mbox{\\sf Vfy-{zk}}_{\\sf Auth_{\\mathit {dr}}}({\\sf zk}(({{\\tt {Pnym}}_{\\mathit {dr}}},{{\\tt {Id}}_{\\mathit {dr}}}), {\\sf drcred}({{\\tt {Pnym}}_{\\mathit {dr}}},{{\\tt {Id}}_{\\mathit {dr}}})),{\\sf drcred}({{\\tt {Pnym}}_{\\mathit {dr}}},{{\\tt {Id}}_{\\mathit {dr}}}))={\\sf true}.$ The verification implies that the creator of the authentication is a doctor who has the credential ${\\sf drcred}({{\\tt {Pnym}}_{\\mathit {dr}}},{{\\tt {Id}}_{\\mathit {dr}}})$ .",
"Because only legitimate doctors can obtain a credential from authorities, i.e., use the function ${\\sf drcred}$ to create a credential; and the correspondence between the parameters of the anonymous authentication (the first parameter $({{\\tt {Pnym}}_{\\mathit {dr}}},{{\\tt {Id}}_{\\mathit {dr}}})$ in $\\mathit {Auth_{\\mathit {dr}}}$ ) and the parameters of the credential (parameters ${{\\tt {Pnym}}_{\\mathit {dr}}}$ and${{\\tt {Id}}_{\\mathit {dr}}}$ in ${\\sf drcred}({{\\tt {Pnym}}_{\\mathit {dr}}},{{\\tt {Id}}_{\\mathit {dr}}})$ ) ensures that the prover can only be the owner of the credential.",
"Other doctors may be able to use function ${\\sf drcred}$ but do not know ${{\\tt {Pnym}}_{\\mathit {dr}}}$ and ${{\\tt {Id}}_{\\mathit {dr}}}$ , and thus cannot create a valid proof.",
"The adversary can observe a credential ${\\sf drcred}({{\\tt {Pnym}}_{\\mathit {dr}}},{{\\tt {Id}}_{\\mathit {dr}}})$ , but does not know secrets ${{\\tt {Pnym}}_{\\mathit {dr}}}, {{\\tt {Id}}_{\\mathit {dr}}}$ , and thus cannot forge a valid zero-knowledge proof.",
"If the adversary forges a zero-knowledge proof with fake secret information ${{\\tt {Pnym}}_{\\mathit {dr}}}^{\\prime }$ and ${{\\tt {Id}}_{\\mathit {dr}}}^{\\prime }$ , the fake zero-knowledge proof will not pass verification.",
"For the same reason, a validated proof proves that the credential belongs to the creator of the zero-knowledge proof.",
"Similarly, an anonymous authentication of a patient is modelled as $\\begin{array}{rl}\\mathit {Auth_{\\mathit {pt}}}:={\\sf zk}(&\\hspace{-8.5pt}({{\\tt {Id}}_{\\mathit {pt}}}, {{\\tt {Pnym}}_{\\mathit {pt}}}, {{\\tt {Hii}}},{{\\tt {Sss}}}, {{\\tt {Acc}}}),{\\sf ptcred}({{\\tt {Id}}_{\\mathit {pt}}}, {{\\tt {Pnym}}_{\\mathit {pt}}}, {{\\tt {Hii}}},{{\\tt {Sss}}}, {{\\tt {Acc}}})),\\end{array}$ and the verification rule is modelled as $\\begin{array}{rl}\\mathit {reduc\\ }\\mbox{\\sf Vfy-{zk}}_{\\sf Auth_{\\mathit {pt}}}(&\\hspace{-8.5pt}{\\sf zk}(({{\\tt {Id}}_{\\mathit {pt}}}, {{\\tt {Pnym}}_{\\mathit {pt}}}, {{\\tt {Hii}}},{{\\tt {Sss}}}, {{\\tt {Acc}}}), \\\\& \\hspace{4.25pt}{\\sf ptcred}({{\\tt {Id}}_{\\mathit {pt}}}, {{\\tt {Pnym}}_{\\mathit {pt}}}, {{\\tt {Hii}}},{{\\tt {Sss}}}, {{\\tt {Acc}}})),\\\\& \\hspace{-8.5pt}{\\sf ptcred}({{\\tt {Id}}_{\\mathit {pt}}}, {{\\tt {Pnym}}_{\\mathit {pt}}}, {{\\tt {Hii}}},{{\\tt {Sss}}}, {{\\tt {Acc}}}))={\\sf true}.\\end{array}$"
],
[
"Verifiable encryptions.",
"A verifiable encryption is modelled as a zero-knowledge proof.",
"The encryption is embedded in the zero-knowledge proof as public information.",
"The receiver can obtain the cipher text from the proof.",
"For example, assume a patient wants to prove that he has encrypted a secret $s$ using a public key $k$ to a pharmacist, while the pharmacist does not know the corresponding secret key for $k$ .",
"The pharmacist cannot open the cipher text to test whether it uses the public key $k$ for encryption.",
"However, the zero-knowledge proof can prove that the cipher text is encrypted using $k$ , while not revealing the secret $s$ .",
"The general structure of the verification of a verifiable encryption is $\\mbox{{\\sf Vfy-{venc}}}({\\sf zk}(\\mathit {secrets}, (\\mathit {pub\\_info},\\mathit {cipher})),\\mathit {verif\\_info})={\\sf true},$ where $\\mathit {secrets}$ is private information, $\\mathit {pub\\_info}$ and $\\mathit {cipher}$ consist public information, $\\mathit {verif\\_info}$ is the verification information."
],
[
"Signed proofs of knowledge.",
"A signed proof of knowledge is a scheme which signs a message, and proves a property of the signer.",
"For the DLV08 protocol, this proof only concerns equality of attributes of credentials and commitments (e.g., the identity of this credential is the same as the identity of that commitment).",
"To verify a signed proof of knowledge, the verifier must know which credentials/commitments are considered.",
"Hence, this information must be obtainable from the proof, and thus is included in the model.",
"In general, a signed proof of knowledge is modelled as function ${\\sf spk}(\\mathit {secrets}, \\mathit {pub\\_info}, \\mathit {msg}),$ which models a signature using private value(s) secrets on the message msg, with public information $\\mathit {pub\\_info}$ as settings.",
"Similar to zero-knowledge proofs, $\\mathit {secrets}$ and $\\mathit {pub\\_info}$ satisfy a pre-specified relation.",
"$\\mathit {msg}$ can be any message.",
"What knowledge is proven, depends on the specific instance of the proof and is captured by the verification functions for the specific proofs.",
"For example, to prove that a user knows (a) all fields of a (simplified) credential, (b) all fields of a commitment to an identity, and (c) that the credential concerns the same identity as the commitment, he generates the following proof: $\\begin{array}{l@{\\hspace{-2.125pt}}l@{\\hspace{28.45274pt}}l}{\\sf spk}( & ({{\\tt {Id}}_{\\mathit {pt}}},{{\\tt {Pnym}}_{\\mathit {pt}}},{{\\tt {r}}_\\mathit {\\mathit {pt}}}),& (*\\mathit {secrets}*) \\\\\\qquad & ({\\sf ptcred}({{\\tt {Id}}_{\\mathit {pt}}},{{\\tt {Pnym}}_{\\mathit {pt}}}),{\\sf commit}({{\\tt {Id}}_{\\mathit {pt}}},{{\\tt {r}}_\\mathit {\\mathit {pt}}})), &(*\\mathit {public\\_info}*)\\\\& \\mathit {msg}).",
"&(*\\mathit {message}*)\\end{array}$ These proofs are verified by checking that the signature is correct, given the signed message and the verification information.",
"E.g., the above example proof can be verified as follows: $\\begin{array}{lll@{\\hspace{28.45274pt}}l}\\mathit {reduc\\ }\\mbox{{\\sf Vfy-{spk}}}( & \\hspace{-8.5pt}{\\sf spk}( &\\hspace{-8.5pt}({{\\tt {Id}}_{\\mathit {pt}}},{{\\tt {Pnym}}_{\\mathit {pt}}},{{\\tt {r}}_\\mathit {\\mathit {pt}}}), \\\\& & \\hspace{-8.5pt}({\\sf ptcred}({{\\tt {Id}}_{\\mathit {pt}}},{{\\tt {Pnym}}_{\\mathit {pt}}}),{\\sf commit}({{\\tt {Id}}_{\\mathit {pt}}},{{\\tt {r}}_\\mathit {\\mathit {pt}}})), \\\\& & \\hspace{-8.5pt}\\mathit {msg}\\ ), &(*\\mathit {signed\\_message}*)\\\\& \\multicolumn{2}{l}{\\hspace{-8.5pt}(\\ {\\sf ptcred}({{\\tt {Id}}_{\\mathit {pt}}},{{\\tt {Pnym}}_{\\mathit {pt}}}),{\\sf commit}({{\\tt {Id}}_{\\mathit {pt}}},{{\\tt {r}}_\\mathit {\\mathit {pt}}})\\ ),} &(*\\mathit {verify\\_info}*)\\\\\\end{array}&\\hspace{-8.5pt} \\mathit {msg} &&(*\\mathit {message}*)\\\\& \\multicolumn{2}{l}{\\hspace{-8.5pt}) = {\\sf true}.",
"}$ $$"
],
[
"Other cryptographic primitives.",
"Hash functions, encryptions and signing messages are modelled by functions ${\\sf hash}$ , $\\sf enc$ , and $\\sf sign$ , respectively (see Figure REF ).",
"Correspondingly, decryption, verifying a signature and retrieving the message from a signature are modelled as functions $\\sf dec$ , $\\mbox{\\sf Vfy-{sign}}$ and ${\\sf getsignmsg}$ (see Figure REF ).",
"Function ${\\sf pk}$ models the corresponding public key of a secret key, and function ${\\sf invoice}$ is used for a pharmacist to generate an invoice for a patient (see Figure REF ).",
"Functions ${\\sf getpublic}$ , ${\\sf getSpkVinfo}$ and ${\\sf getmsg}$ model retrieving public information from a zero-knowledge proof, from a signed proof of knowledge, and obtaining the message from a signed proof of knowledge, respectively (see Figure REF ).",
"Function ${\\sf key}$ models the public key of a user's identity and function ${\\sf host}$ retrieves the owner's identity from a public key (see Figure REF )."
],
[
"Modelling the DLV08 protocol",
"We first show how to model each of the sub-protocols and then how to compose them to form the full DLV08 protocol."
],
[
"Modelling the doctor-patient sub-protocol.",
"This sub-protocol is used for a doctor, whose steps are labelled d$i$ in Figure REF , to prescribe medicine for a patient, whose steps are labelled t$i$ in Figure REF .",
"Figure: The doctor process 𝑃 𝑑𝑟 \\mathit {P}_{\\mathit {dr}}.Figure: The patient process in the doctor-patient sub-protocol 𝑃 𝑝𝑡 ' \\mathit {P}^{\\prime }_{\\mathit {pt}}.First, the doctor anonymously authenticates to the patient using credential $\\mathit {Cred_{\\mathit {dr}}}$ (d1).",
"The patient reads in the doctor authentication (t1), obtains the doctor credential (t2), and verifies the authentication (t3).",
"If the verification in step (t3) succeeds, the patient anonymously authenticates himself to the doctor using his credential (t5, the first ${\\sf zk}$ function), generates a nonce ${{\\tt {r}}_\\mathit {\\mathit {pt}}}$ (t4), computes a commitment with the nonce as opening information, and proves that the patient identity used in the patient credential is the same as in the commitment, thus linking the patient commitment and the patient credential (t5, the second ${\\sf zk}$ ).",
"The doctor reads in the patient authentication as ${\\mathit {rcv\\_Auth_{\\mathit {pt}}}}$ and the patient proof as${\\mathit {rcv\\_\\mathit {PtProof}}}$ (d2), obtains the patient credential from the patient authentication (d3), obtains the patient commitment ${\\mathit {c\\_Comt_{\\mathit {pt}}}}$ and the patient credential from the patient proof, tests whether the credential matches the one embedded in the patient authentication (d4), then verifies the authentication (d5) and the patient proof (d6).",
"If the verification in the previous item succeeds, the doctor generates a prescription ${\\tt {presc}}$Note that a medical examination of the patient is not part of the DLV08 protocol.",
"(d7), generates a nonce ${\\tt {r}}_{\\mathit {dr}}$ (d8), computes a prescription identity $\\mathit {PrescriptID}$ (d9), and computes a commitment $\\mathit {Comt_{\\mathit {dr}}}$ using the nonce as opening information (d10).",
"Next, the doctor signs the message (${\\tt {presc}}$ , $\\mathit {PrescriptID}$ , $\\mathit {Comt_{\\mathit {dr}}}$ , ${\\mathit {c\\_Comt_{\\mathit {pt}}}}$ ) using a signed proof of knowledge.",
"This proves the pseudonym used in the credential $\\mathit {Cred_{\\mathit {dr}}}$ is the same as in the commitment $\\mathit {Comt_{\\mathit {dr}}}$ , thus linking the prescription to the credential.",
"The doctor sends the signed proof of knowledge together with the open information of the doctor commitment ${\\tt {r}}_{\\mathit {dr}}$ (d10).",
"The patient reads in the prescription as ${\\mathit {rcv\\_\\mathit {PrescProof}}}$ and the opening information of the doctor commitment (t6), obtains the prescription ${\\mathit {c\\_\\mathit {presc}}}$ , prescription identity ${\\mathit {c\\_\\mathit {PrescriptID}}}$ , doctor commitment ${\\mathit {c\\_Comt_{\\mathit {dr}}}}$ , and tests the patient commitment signed in the receiving message (t7).",
"Then the patient verifies the signed proof of prescription (t8).",
"If the verification succeeds, the patient obtains the doctor's pseudonym ${\\mathit {c\\_Pnym_{\\mathit {dr}}}}$ by opening the doctor commitment (t9) and continues the next sub-protocol behaving as in process $\\mathit {P}^{\\prime \\prime }_{\\mathit {pt}}$ .",
"Rationale for modelling of prescriptions.",
"In the description of DLV08 protocol [24], it is unclear precisely what information is included in a prescription.",
"Depending on the implementation, a prescription may contain various information, such as name of medicines prescribed, amount of medicine prescribed, the timestamp and organization that wrote the prescription, etc.",
"Some information in the prescription may reveal privacy of patients and doctors.",
"For instance, if the identities of patients and doctors are included in the prescription, then doctor and patient prescription privacy is trivially broken.",
"In addition, both (doctor/patient) anonymity and untraceability would also be trivially broken, if the prescriptions were revealed to the adversary.",
"In order to focus only on the logical flaws of the DLV08 protocol and exclude such dependencies, we assume that the prescriptions in the protocol are de-identified.",
"However, this may not be sufficient.",
"Doctors may e.g.",
"be identifiable by the way they prescribe, the order in which medicine appear on prescriptions, etc.",
"Such “fingerprinting” attacks would also trivially break prescription privacy.",
"For our analysis, we assume that a prescription cannot be linked to its doctor or patient by its content.",
"That is, the prescription shall not be modelled as a function of doctor or patient information.",
"To avoid any of the above concerns, we model prescriptions as abstract pieces of data: each prescription is represented by a single, unique name.",
"First, this modelling captures the assumption that prescriptions from different doctors are often different even for the same diagnose, due to different prescription styles.",
"Second, this allows us to capture an infinite number of prescriptions in infinite sessions, without introducing false attacks to the DLV08 protocol that are caused by the modelling of the prescriptions."
],
[
"Modelling the patient-pharmacist sub-protocol.",
"This sub-protocol is used for a patient, whose steps are labelled t$i$ in Figure REF , to obtain medicine from a pharmacist, whose steps are labelled h$i$ in Figure REF .",
"Figure: The patient process in the patient-pharmacist sub-protocol 𝑃 𝑝𝑡 '' \\mathit {P}^{\\prime \\prime }_{\\mathit {pt}}.Figure: The pharmacist process in the patient-pharmacist sub-protocol 𝑃 𝑝ℎ ' \\mathit {P}^{\\prime }_{\\mathit {ph}}.First, the pharmacist authenticates to the patient using a public key authentication (h1).",
"Note that the pharmacist does not authenticate anonymously, and that the pharmacists's MPA identity is embedded.",
"The patient reads in the pharmacist authentication ${\\mathit {rcv\\_Auth_{\\mathit {ph}}}}$ (t10) and verifies the authentication (t11).",
"If the verification succeeds, the pharmacist obtains the pharmacist's MPA identity from the authentication (t12), thus obtains the public key of MPA (t13).",
"Then the patient anonymously authenticates himself to the pharmacist, and proves his social security status using the proof $\\mathit {PtAuthSss}$ (t14).",
"The patient generates a nonce which will be used as a message in a signed proof of knowledge (t15), and computes verifiable encryptions $\\mathit {vc}_1$ , $\\mathit {vc}_2$ , $\\mathit {vc}_3$ , $\\mathit {vc}_3^{\\prime }$ , $\\mathit {vc}_4$ and $\\mathit {vc}_5$ (t16-t21).",
"These divulge the patient's HII, the doctor's pseudonym, and the patient's pseudonym to the MPA, the patient's pseudonym to the HII, and the patient pseudonym and HII to the social safety organisation, respectively.",
"The patient encrypts $\\mathit {vc}_5$ with MPA's public key as 5 (t22).",
"The patient computes a signed proof of knowledge $\\begin{array}{rl}\\mathit {PtSpk}=&{\\sf spk}(({{\\tt {Id}}_{\\mathit {pt}}}, {{\\tt {Pnym}}_{\\mathit {pt}}}, {{\\tt {Hii}}}, {{\\tt {Sss}}}, {{\\tt {Acc}}}),\\\\&\\quad \\ \\ ({\\sf ptcred}({{\\tt {Id}}_{\\mathit {pt}}}, {{\\tt {Pnym}}_{\\mathit {pt}}}, {{\\tt {Hii}}}, {{\\tt {Sss}}}, {{\\tt {Acc}}}),{\\sf commit}({{\\tt {Id}}_{\\mathit {pt}}}, {{\\tt {r}}_\\mathit {\\mathit {pt}}})),\\\\&\\qquad {\\tt {nonce}})\\end{array}$ which proves that the patient identity embedded in the prescription is the same as in his credential.",
"In the prescription, this identity is contained in a commitment.",
"For simplicity, we model the proof using the commitment instead of the prescription.",
"The link between commitment and prescription is ensured when the proof is verified (h10).",
"The patient sends the prescription ${\\mathit {rcv\\_\\mathit {PrescProof}}}$ , the signed proof $\\mathit {PtSpk}$ , and $\\mathit {vc}_1,\\mathit {vc}_2,\\mathit {vc}_3,\\linebreak \\mathit {vc}^{\\prime }_3,\\mathit {vc}_4,5$ to the pharmacist (t23).",
"The pharmacist reads in the authentication ${\\mathit {rcv\\_\\mathit {PtAuthSss}}}$ (h2), obtains the patient credential and his social security status (h3), verifies the authentication (h4).",
"If the verification succeeds, the pharmacist reads in the patient's prescription ${\\mathit {rcv_{\\mathit {ph}}\\_\\mathit {PrescProof}}}$ , the signed proof of knowledge ${\\mathit {r}cv_{\\mathit {ph}}\\_\\mathit {PtSpk}}$ , the verifiable encryptions ${\\mathit {rcv\\_vc}}_1$ , ${\\mathit {rcv\\_vc}}_2$ , ${\\mathit {rcv\\_vc}}_3$ , ${\\mathit {rcv\\_vc}}^{\\prime }_3$ , ${\\mathit {rcv\\_vc}}_4$ , and cipher text ${\\mathit {rcv\\_c}}_5$ (h5); and verifies ${\\mathit {rcv_{\\mathit {ph}}\\_\\mathit {PrescProof}}}$ (h6-h8), ${\\mathit {r}cv_{\\mathit {ph}}\\_\\mathit {PtSpk}}$ (h9-h10), and ${\\mathit {rcv\\_vc}}_1$ , ${\\mathit {rcv\\_vc}}_2$ , ${\\mathit {rcv\\_vc}}_3$ , ${\\mathit {rcv\\_vc}}^{\\prime }_3$ , ${\\mathit {rcv\\_vc}}_4$ (h11-h20).",
"If all the verifications succeed, the pharmacist charges the patient, and delivers the medicine (neither are modelled as they are out of DLV08's scope).",
"Then the pharmacist generates an invoice with the prescription identity embedded in it and sends the invoice to the patient (h21).",
"The patient reads in the invoice (t24), computes a receipt: a signed proof of knowledge $\\mathit {ReceiptAck}$ which proves that he receives the medicine (t25); and sends the signed proof of knowledge to the pharmacist (t26).",
"The pharmacist reads in the receipt ${\\mathit {rcv\\_ReceiptAck}}$ (h22), verifies its correctness (h23) and continues the next sub-protocol behaving as in $\\mathit {P}^{\\prime \\prime }_{\\mathit {ph}}$ ."
],
[
"Modelling the pharmacist-MPA sub-protocol.",
"The pharmacist-MPA sub-protocol is used for the pharmacist, whose steps are labelled h$i$ in Figure REF to report the received prescriptions to the MPA, whose steps are labelled m$i$ in Figure REF .",
"Figure: The pharmacist process in the pharmacist-MPA sub-protocol 𝑃 𝑝ℎ '' \\mathit {P}^{\\prime \\prime }_{\\mathit {ph}}.Figure: The MPA process in the pharmacist-MPA sub-protocol 𝑃 𝑚𝑝𝑎 \\mathit {P}_{\\mathit {mpa}}.As the pharmacist mostly forwards the information supplied by the patient, this protocol greatly resembles the patient-pharmacist protocol described above.",
"Each step is modelled in details as follows: The pharmacist authenticates himself to his MPA by sending his identity and the signed identities of the pharmacist and the MPA (h24).",
"The MPA stores this authentication in ${\\mathit {rcv_{\\mathit {mpa}}\\_Auth_{\\mathit {ph}}}}$ , and stores the pharmacist's identity in ${\\mathit {c_{\\mathit {mpa}}\\_Id_{\\mathit {ph}}}}$ (m1).",
"From the pharmacist's identity, the MPA obtains the pharmacist's public key (m2).",
"Then the MPA verifies the pharmacist's authentication against the pharmacist's public key (m3).",
"If the verification succeeds, according to the corresponding rule in the equational theory, and the MPA verifies that he is indeed the pharmacist's MPA (m4), the MPA then authenticates itself to the pharmacist by sending the signature of his identity (m5).",
"The pharmacist reads in the MPA's authentication in ${\\mathit {rcv\\_Auth_{\\mathit {mpa}}}}$ (h25), and verifies the authentication (h26).",
"If the verification succeeds, the pharmacist sends the following to the MPA: prescription ${\\mathit {rcv_{\\mathit {ph}}\\_\\mathit {PrescProof}}}$ , received receipt ${\\mathit {rcv\\_ReceiptAck}}$ , and verifiable encryptions ${\\mathit {rcv\\_vc}}_1$ , ${\\mathit {rcv\\_vc}}_2$ , ${\\mathit {rcv\\_vc}}_3$ , ${\\mathit {rcv\\_vc}}^{\\prime }_3$ , ${\\mathit {rcv\\_vc}}_4$ , ${\\mathit {rcv\\_c}}_5$ (h27).",
"The MPA reads in the information (m6) and verifies their correctness (m7-m24).",
"If the verifications succeed, the MPA decrypts the corresponding encryptions (${\\mathit {c_{\\mathit {mpa}}\\_Enc}}_1$ , ${\\mathit {c_{\\mathit {mpa}}\\_Enc}}_2$ , and ${\\mathit {c_{\\mathit {mpa}}\\_Enc}}_4$ ) embedded in ${\\mathit {rcv_{\\mathit {mpa}}\\_vc}}_1, {\\mathit {rcv_{\\mathit {mpa}}\\_vc}}_2, {\\mathit {rcv_{\\mathit {mpa}}\\_vc}}_4$ , and obtains the patient's HII (m12), the doctor pseudonym (m15), the patient pseudonym (m23).",
"Then the MPA continues the next sub-protocol behaving as in process $\\mathit {P}^{\\prime }_{\\mathit {mpa}}$ .",
"The storing information to database by the MPA is beyond our concern."
],
[
"Modelling the MPA-HII sub-protocol.",
"This protocol covers the exchange of information between the pharmacist's MPA, whose steps are labelled m$i$ in Figure REF and the patient's HII, whose steps are labelled i$i$ in Figure REF .",
"Figure: The MPA process in the MPA-HII sub-protocol 𝑃 𝑚𝑝𝑎 ' \\mathit {P}^{\\prime }_{\\mathit {mpa}}.Figure: The HII process 𝑃 ℎ𝑖𝑖 \\mathit {P}_{\\mathit {hii}}.The MPA sends his identity to the HII and authenticates to the HII using public key authentication (m25).",
"The HII stores the MPA's identity in $\\mathit {rcv_{\\mathit {hii}}\\_Id_{\\mathit {mpa}}}$ and stores the authentication in ${\\mathit {rcv_{\\mathit {hii}}\\_Auth_{\\mathit {mpa}}}}$ (i1).",
"From the MPA's identity, the HII obtains the MPA's public key (i2).",
"Then the HII verifies the MPA's authentication (i3).",
"If the verification succeeds, the HII authenticates to the MPA using public key authentication (i4).",
"The MPA stores the authentication in ${\\mathit {rcv_{\\mathit {mpa}}\\_Auth_{\\mathit {hii}}}}$ (m26).",
"Then the MPA obtains the HII's public key from the HII's identity (m27) and verifies the HII's authentication (m28).",
"If the verification succeeds, and the MPA verifies that the authentication is from the intended HII (m29), the MPA sends the receipt ${\\mathit {rcv_{\\mathit {mpa}}\\_\\mathit {PrescProof}}}$ and the patient pseudonym encrypted for the HII – verifiable encryption ${\\mathit {rcv_{\\mathit {mpa}}\\_vc}}_5={\\sf {dec}}({\\mathit {rcv_{\\mathit {mpa}}\\_c}}_5, {\\tt {sk}}_{\\mathit {mpa}})$ (m30).",
"The HII receives the receipt as ${\\mathit {rcv_{\\mathit {hii}}\\_ReceiptAck}}$ and the encrypted patient pseudonym for the HII as ${\\mathit {c_{\\mathit {hii}}\\_vc}}_5$ (i5).",
"The HII verifies the above two pieces of information (i6-i10).",
"If the verifications succeed, the HII decrypts the encryption ${\\mathit {c_{\\mathit {hii}}\\_Enc}}_5$ and obtains the patient's pseudonym (i11).",
"Finally, the HII sends an invoice of the prescription identity to the MPA (i12).",
"The MPA stores the invoice in ${\\mathit {rcv_{\\mathit {mpa}}\\_Invoice}}$ (m31).",
"Afterwards, the HII pays the MPA and updates the patient account.",
"As before, handling payment and storing information are beyond the scope of the DLV08 protocol and therefore, we do not model this stage."
],
[
"The full protocol.",
"In summary, the DLV08 protocol is composed as shown in Figure REF .",
"Figure: Overview of DLV08 protocol.The DLV08 protocol is modelled as the five roles $\\mathit {R}_{\\mathit {dr}}$ , $\\mathit {R}_{\\mathit {pt}}$ , $\\mathit {R}_{\\mathit {ph}}$ , $\\mathit {R}_{\\mathit {mpa}}$ , and $\\mathit {R}_{\\mathit {hii}}$ running in parallel (Figure REF ).",
"$\\begin{array}{rcl}\\mathit {\\mathit {P}_\\mathit {DLV08}}&:=& \\nu \\tilde{mc}.",
"\\mathit {init}.",
"(!\\mathit {R}_{\\mathit {pt}}\\mid !\\mathit {R}_{\\mathit {dr}}\\mid !\\mathit {R}_{\\mathit {ph}}\\mid !\\mathit {R}_{\\mathit {mpa}}\\mid !\\mathit {R}_{\\mathit {hii}})\\\\\\mathit {init}&:=&\\mbox{let}\\ {\\tt {pk}}_\\mathit {sso}={\\sf pk}({\\tt {sk}}_{\\mathit {sso}}) \\ \\mbox{in} \\ {\\sf {out}}({\\tt {ch}}, {\\tt {pk}}_\\mathit {sso})\\end{array}$ where $\\nu \\tilde{mc}$ represents global secrets ${\\tt {sk}}_{\\mathit {sso}}$ and private channels ${\\tt {ch}}_{hp}$ , ${\\tt {ch}}_{mp}$ , ${\\tt {ch}}_{phpt}$ ; process $\\mathit {init}$ initialises the settings of the protocol – publishing the public key ${\\tt {pk}}_\\mathit {sso}$ , so that the adversary knows it.",
"The roles $\\mathit {R}_{\\mathit {dr}}$ , $\\mathit {R}_{\\mathit {pt}}$ , $\\mathit {R}_{\\mathit {ph}}$ , $\\mathit {R}_{\\mathit {mpa}}$ and $\\mathit {R}_{\\mathit {hii}}$ are obtained by adding the settings of each role (see Section REF ) to the previously modelled corresponding process of the role as shown in Figure REF , Figure REF , Figure REF , Figure REF and Figure REF , respectivley.",
"Figure: The process for the DLV08 protocol.Each doctor has an identity ${{\\tt {Id}}_{\\mathit {dr}}}$ (rd1), a pseudonym ${{\\tt {Pnym}}_{\\mathit {dr}}}$ (rd2) and behaves like $\\mathit {P}_{\\mathit {dr}}$ (rd3) as shown in Figure REF .",
"The anonymous doctor credential is modelled by applying function ${\\sf drcred}$ on ${{\\tt {Id}}_{\\mathit {dr}}}$ and ${{\\tt {Pnym}}_{\\mathit {dr}}}$ .",
"Figure: The process for role doctor 𝑅 𝑑𝑟 \\mathit {R}_{\\mathit {dr}}.Each patient (as shown in Figure REF ) has an identity ${{\\tt {Id}}_{\\mathit {pt}}}$ (rt1), a pseudonym ${{\\tt {Pnym}}_{\\mathit {pt}}}$ , a social security status ${{\\tt {Sss}}}$ , a health expense account ${{\\tt {Acc}}}$ (rt2).",
"Unlike the identity and the pseudonym, which are attributes of a doctor, a doctor's ${{\\tt {Hii}}}$ is an association relation, and thus is modelled by reading in an HII identity to establish the relation (rt3).",
"In addition, a patient communicates with a pharmacist in each session.",
"Which pharmacist the patient communicates with is decided by reading in a pharmacist's public key (rt4).",
"From the pharmacist's public key, the patient can obtain the pharmacist's identity (rt5).",
"Finally the patient behaves as $\\mathit {P}^{\\prime }_{\\mathit {pt}}$ (rt6).",
"Figure: The process for role patient 𝑅 𝑝𝑡 \\mathit {R}_{\\mathit {pt}}.Each pharmacist has a secret key ${\\tt {sk}}_{\\mathit {ph}}$ (rh1), a public key ${\\tt {pk}}_\\mathit {ph}$ (rh2) and an identity ${\\tt {Id}}_{\\mathit {ph}}$ (rh3) as shown in Figure REF .",
"The public key of a pharmacist is published over channel ${\\tt {ch}}$ , so that the adversary knows it (rh4).",
"In addition, the public key is sent to the patients via private channel ${\\tt {ch}}_{phpt}$ (rh4), so that the patients can choose one to communicate with.",
"In each session, the pharmacist communicates with an MPA.",
"Which MPA the pharmacist communicates with is decided by reading in a public key of MPA (rh5).",
"From the public key, the pharmacist can obtain the identity of the MPA (rh6).",
"Finally, the pharmacist behave as $\\mathit {P}^{\\prime }_{\\mathit {ph}}$ .",
"Figure: The process for role pharmacist 𝑅 𝑝ℎ \\mathit {R}_{\\mathit {ph}}.Each MPA has a secret key ${\\tt {sk}}_{\\mathit {mpa}}$ (rm1), a public key ${\\tt {pk}}_\\mathit {mpa}$ (rm2) and an identity ${{\\tt {Id}}_{\\mathit {mpa}}}$ (rm3).",
"The MPA publishes his public key as well as sends his public key to pharmacists (rm4), and behaves as $\\mathit {P}_{\\mathit {mpa}}$ (rm5) as shown in Figure REF .",
"Figure: The process for role MPA 𝑅 𝑚𝑝𝑎 \\mathit {R}_{\\mathit {mpa}}.Similar to MPA, each HII (Figure REF ) has a secret key ${\\tt {sk}}_{\\mathit {hii}}$ (ri1), a public key ${\\tt {pk}}_\\mathit {hii}$ (ri2) and an identity ${\\tt {Id}}_{\\mathit {hii}}$ (ri3).",
"The public key is revealed to the adversary via channel ${\\tt {ch}}$ and sent to the patients via channel ${\\tt {ch}}_{hp}$ (ri4).",
"Then the HII behaves as $\\mathit {P}_{\\mathit {hii}}$ (ri5).",
"Figure: The process for role HII 𝑅 ℎ𝑖𝑖 \\mathit {R}_{\\mathit {hii}}."
],
[
"Analysis of DLV08",
"In this section, we analyse whether DLV08 satisfies the following properties: secrecy of patient and doctor information, authentication, (strong) patient and doctor anonymity, (strong) patient and doctor untraceability, (enforced) prescription privacy, and independence of (enforced) prescription privacy.",
"The properties doctor anonymity and untraceability are not required by the protocol but are still interesting to analyse.",
"The verification is supported by the automatic verification tool ProVerif [15], [16], [17].",
"The tool has been used to verify many secrecy, authentication and privacy properties, e.g., see [1], [2], [40], [11], [25].",
"The verification results for secrecy are summarised in Table REF , and those for authentication in Table REF .",
"As we are foremost interested in privacy properties, the verification results for privacy properties, and suggestions for improvements are discussed in Section .",
"Table REF summarises those results, causes of privacy weaknesses, suggested improvements, and the effect of the improvements.",
"In this section, we show the verification results of properties from basic to more complicated.",
"A flaw which fails a basic property is likely to fail a more complicated property as well.",
"Thus we first show flaws of basic properties and how to fix them, then we show new flaws of complicated properties based on the fixed model."
],
[
"ProVerif",
"ProVerif takes a protocol and a property modelled in the applied pi calculus as input (the input language (untyped version) differs slightly from applied pi, see [14]), and returns either a proof of correctness or potential attacks.",
"A protocol modelled in the applied pi calculus is translated to Horn clauses [32].",
"The adversary's capabilities are added as Horn clauses as well.",
"Using these clauses, verification of secrecy and authentication is equivalent to determining whether a certain clause is derivable from the set of initial clauses.",
"Secrecy of a term is defined as the adversary cannot obtain the term by communicating with the protocol and/or applying cryptography on the output of the protocol [1].",
"The secrecy property is modelled as a predicate in ProVerif: the query of secrecy of term $M$ is $``attacker: M\"$ [15].",
"ProVerif determines whether the term $M$ can be inferred from the Horn clauses representing the adversary knowledge.",
"Authentication is captured by correspondence properties of events in processes: if one event happens the other event must have happened before [2], [18].",
"Events are tags which mark important stages reached by the protocol.",
"Events have arguments, which allow us to express relationships between the arguments of events.",
"A correspondence property is a formula of the form: $\\mathit {ev}: \\bar{f}(M)==>\\mathit {ev}: \\bar{g}(N)$ .",
"That is, in any process, if event $\\bar{f}(M)$ has been executed, then the event $\\bar{g}(N)$ must have been previously executed, and any relationship between $M$ and $N$ must be satisfied.",
"To capture stronger authentication, where an injective relationship between executions of participants is required, an injective correspondence property $\\mathit {evinj}: \\bar{f}(M)==>\\mathit {evinj}: \\bar{g}(N)$ is defined: in any process, for each execution of event $\\bar{f}(M)$ , there is a distinct earlier execution of the event $\\bar{g}(N)$ , and the relationship between $M$ and $N$ is satisfied.",
"In addition, ProVerif provides automatic verification of labelled bisimilarity of two processes which differ only in the choice of some terms [9].",
"An operation $``choice[a, b]\"$ is introduced to model the different choices of a term in the two processes.",
"Using this operation, the two processes can be written as one process – a bi-process.",
"Example 8 To verify the equivalence $ \\nu {\\tt {a}}.",
"\\nu {\\tt {b}}.",
"{\\sf {out}}({\\tt {ch}}, {\\tt {a}}).",
"{\\sf {out}}({\\tt {ch}}, {\\tt {e}})\\approx _{\\ell }\\nu {\\tt {a}}.",
"\\nu {\\tt {b}}.",
"{\\sf {out}}({\\tt {ch}}, {\\tt {b}}).",
"{\\sf {out}}({\\tt {ch}}, {\\tt {d}})$ where ${\\tt {ch}}$ is a public channel, ${\\tt {e}}$ and ${\\tt {d}}$ are two free names, we can query the following bi-process in ProVerif: $\\mathit {P}:=\\nu {\\tt {a}}.",
"\\nu {\\tt {b}}.",
"{\\sf {out}}({\\tt {ch}}, {\\sf choice}[{\\tt {a}}, {\\tt {b}}]).",
"{\\sf {out}}({\\tt {ch}}, {\\sf choice}[{\\tt {e}}, {\\tt {d}}]).$ Using the first parameter of all $``choice\"$ operations in a bi-process $\\mathit {P}$ , we obtain one side of the equivalence (denoted as ${\\sf fst}(\\mathit {P})$ ); using the second parameters, we obtain the other side (denoted as ${\\sf snd}(\\mathit {P})$ ).",
"Example 9 For the bi-process in Example REF , using the first parameter to replace each $``choice\"$ operation, we obtain $\\nu {\\tt {a}}.",
"\\nu {\\tt {b}}.",
"{\\sf {out}}({\\tt {ch}}, {\\tt {a}}).",
"{\\sf {out}}({\\tt {ch}}, {\\tt {e}}),$ which is the left-hand side of the equivalence in Example REF ; using the second parameter to replace each $``choice\"$ operation, we obtain $\\nu {\\tt {a}}.",
"\\nu {\\tt {b}}.",
"{\\sf {out}}({\\tt {ch}}, {\\tt {b}}).",
"{\\sf {out}}({\\tt {ch}}, {\\tt {d}}),$ which is the right-hand side of the equivalence.",
"Given a bi-process $\\mathit {P}$ , ProVerif tries to prove that ${\\sf fst}(\\mathit {P})$ is labelled bisimilar to ${\\sf snd}(\\mathit {P})$ .",
"The fundamental idea is that ProVerif reasons on traces of the bi-process $\\mathit {P}$ : the bi-process $\\mathit {P}$ reduces when ${\\sf fst}(\\mathit {P})$ and ${\\sf snd}(\\mathit {P})$ reduce in the same way; when ${\\sf fst}(\\mathit {P})$ and ${\\sf snd}(\\mathit {P})$ do something that may differentiate them, the bi-process is stuck.",
"Formally, ProVerif shows that the bi-process $\\mathit {P}$ is uniform, that is, if ${\\sf fst}(\\mathit {P})$ can do a reduction to some $\\mathit {Q}_1$ , then the bi-process can do a reduction to some bi-process $\\mathit {Q}$ , such that ${\\sf fst}(\\mathit {Q})\\equiv \\mathit {Q}_1$ and symmetrically for ${\\sf snd}(\\mathit {P})$ taking a reduction to $\\mathit {Q}_2$ .",
"When the bi-process $\\mathit {P}$ always remains uniform after reduction and addition of an adversary, ${\\sf fst}(\\mathit {P})$ is labelled bisimilar to ${\\sf snd}(\\mathit {P})$ ."
],
[
"Secrecy of patient and doctor information",
"The DLV08 protocol claims to satisfy the following requirement: any party involved in the prescription processing workflow should not know the information of a patient and a doctor unless the information is intended to be revealed in the protocol.",
"In [24], this requirement is considered as an access control requirement.",
"We argue that ensuring the requirement with access control is not sufficient when considering a communication network.",
"A dishonest party could potentially act as an attacker from the network (observing the network and manipulating the protocol) and obtain information which he should not access.",
"It is not clearly stated which (if any) of the involved parties are honest.",
"We find that in such a way, some patient and doctor information may be revealed to parties who should not know the information.",
"We formalise the requirement as standard secrecy of patient and doctor information with respect to the Dolev-Yao adversary.",
"Standard secrecy of a term captures the idea that the adversary cannot access to that term (see Section REF ).",
"If a piece of information is known to the adversary, a dishonest party acting like the adversary can access to the information.",
"We do not consider strong secrecy, as it is unclear whether the information is guessable.",
"Recall that standard secrecy of a term $M$ is formally defined as a predicate $``attacker: M\"$ (see Section REF ).",
"By replacing $M$ with the listed private information, we obtain the formal definition of the secrecy of patient and doctor information.",
"The list of private information of patients and doctors that needs to be protected is: patient identity (${{\\tt {Id}}_{\\mathit {pt}}}$ ), doctor identity (${{\\tt {Id}}_{\\mathit {dr}}}$ ), patient pseudonym (${{\\tt {Pnym}}_{\\mathit {pt}}}$ ), doctor pseudonym (${{\\tt {Pnym}}_{\\mathit {dr}}}$ ), a patient's social security status (${{\\tt {Sss}}}$ ), and a patient's health insurance institute (${{\\tt {Hii}}}$ ).",
"Although DLV08 does not explicitly require it, we additionally analyse secrecy of the health expense account ${{\\tt {Acc}}}$ of a patient.",
"$\\begin{array}{lll}query\\ attacker: {{\\tt {Id}}_{\\mathit {pt}}}&query\\ attacker: {{\\tt {Id}}_{\\mathit {dr}}}&query\\ attacker: {{\\tt {Pnym}}_{\\mathit {pt}}}\\\\query\\ attacker: {{\\tt {Pnym}}_{\\mathit {dr}}}&query\\ attacker: {{\\tt {Sss}}}&query\\ attacker: {{\\tt {Hii}}}\\\\query\\ attacker: {{\\tt {Acc}}}\\end{array}$"
],
[
"Verification result.",
"We query the standard secrecy of the set of private information using ProVerif [15].",
"The verification results (see Table REF ) show that a patient's identity, pseudonym, health expense account, health insurance institute and identity of a doctor (${{\\tt {Id}}_{\\mathit {pt}}}$ , ${{\\tt {Pnym}}_{\\mathit {pt}}}$ , ${{\\tt {Hii}}}$ ${{\\tt {Acc}}}$ , ${{\\tt {Id}}_{\\mathit {dr}}}$ ) satisfy standard secrecy; a patient's social security status ${{\\tt {Sss}}}$ and a doctor's pseudonym ${{\\tt {Pnym}}_{\\mathit {dr}}}$ do not satisfy standard secrecy.",
"The ${{\\tt {Sss}}}$ is revealed by the proof of social security status from the patient to the pharmacist.",
"The ${{\\tt {Pnym}}_{\\mathit {dr}}}$ is revealed by the revealing of both the commitment of the patient's pseudonym and the open key to the commitment during the communication between the patient and the doctor.",
"Fixing secrecy of a patient's social security status requires that the proof of social security status only reveals the status to the pharmacist.",
"Since how a social security status is represented and what the pharmacist needs to verify are not clear, we cannot give explicit suggestions.",
"However, if the social security status is a number, and the pharmacist only needs to verify that the number is higher than a certain threshold, the patient can prove it using zero-knowledge proof without revealing the number; if the pharmacist needs to verify the exact value of the status, one way to fix its secrecy is that the pharmacist and the patient agree on a session key and the status is encrypted using the key.",
"Similarly, a way to fix the secrecy of ${{\\tt {Pnym}}_{\\mathit {dr}}}$ is to encrypt the opening information using the agreed session key.",
"Table: Verification results of secrecy for patients and doctors."
],
[
"Patient and doctor authentication",
"The protocol claims that all parties should be able to properly authenticate each other.",
"Compared to authentications between public entities, pharmacists, MPA and HII, we focus on authentications between patients and doctors, as patients and doctors use anonymous authentication.",
"Authentications between patients and pharmacists are sketched as well.",
"The DLV08 claims that no party should be able to succeed in claiming a false identity, or false information about his identity.",
"That is the adversary cannot pretend to be a patient or a doctor."
],
[
"Authentication from a patient to a doctor.",
"The authentication from a patient to a doctor is defined as when the doctor finishes his process and believes that he prescribed medicine for a patient, then the patient did ask the doctor for prescription.",
"To verify the authentication of a patient, we add an event ${\\sf EndDr}({\\mathit {c\\_Cred_{\\mathit {pt}}}},{\\mathit {c\\_Comt_{\\mathit {pt}}}})$ at the end of the doctor process (after line d10), meaning the doctor believes that he prescribed medicine for a patient who has a credential ${\\mathit {c\\_Cred_{\\mathit {pt}}}}$ and committed ${\\mathit {c\\_Comt_{\\mathit {pt}}}}$ ; and add an event ${\\sf StartPt}({\\sf ptcred}({{\\tt {Id}}_{\\mathit {pt}}},{{\\tt {Pnym}}_{\\mathit {pt}}},{{\\tt {Hii}}},{{\\tt {Sss}}}, {{\\tt {Acc}}}),{\\sf commit}({{\\tt {Id}}_{\\mathit {pt}}}, {{\\tt {r}}_\\mathit {\\mathit {pt}}}))$ in the patient process (between line t4 and line t5), meaning that the patient did ask for a prescription.",
"The definition is captured by the following correspondence property: $\\mathit {ev(inj)}: {\\sf EndDr}(x,y) ==> \\mathit {ev(inj)}: {\\sf StartPt}(x,y),$ meaning that when the event ${\\sf EndDr}$ is executed, there is a (unique) event ${\\sf StartPt}$ has been executed before."
],
[
"Authentication from a doctor to a patient.",
"Similarly, the authentication from a doctor to a patient is defined as when the patient believes that he visited a doctor, the doctor did prescribe medicine for the patient.",
"To authenticate a doctor, we add to the patient process an event ${\\sf EndPt}({\\mathit {c\\_Cred_{\\mathit {dr}}}},{\\mathit {c\\_Comt_{\\mathit {dr}}}},{\\mathit {c\\_\\mathit {presc}}},{\\mathit {c\\_\\mathit {PrescriptID}}})$ (after line t9), and add an event ${\\sf StartDr}({\\sf drcred}({{\\tt {Pnym}}_{\\mathit {dr}}},{{\\tt {Id}}_{\\mathit {dr}}}),{\\sf commit}({{\\tt {Pnym}}_{\\mathit {dr}}},{\\tt {r}}_{\\mathit {dr}}),{\\tt {presc}},\\mathit {PrescriptID})$ in the doctor process (between line d9 and line d10), then query $\\mathit {ev(inj)}: {\\sf EndPt}(x,y,z,t) ==> \\mathit {ev(inj)}: {\\sf StartDr}(x,y,z,t).$"
],
[
"Authentication from a patient to a pharmacist.",
"The authentication from a patient to a doctor is defined as when the pharmacist finishes a session and believes that he communicates with a patient, who is identified with the credential ${\\mathit {c_{\\mathit {ph}}\\_Cred_{\\mathit {pt}}}}$ , then the patient with the credential did communicate with the pharmacist.",
"to verify the authentication of a patient, we add to the pharmacist process the event ${\\sf EndPh}({\\mathit {c_{\\mathit {ph}}\\_Cred_{\\mathit {pt}}}})$ (after line h23), add to the patient process ${\\sf StartPtph}({\\sf ptcred}({{\\tt {Id}}_{\\mathit {pt}}},{{\\tt {Pnym}}_{\\mathit {pt}}},{{\\tt {Hii}}},{{\\tt {Sss}}}, {{\\tt {Acc}}}))$ (between line t13 and line t14), and query $\\mathit {ev(inj)}: {\\sf EndPh}(x)==>\\mathit {ev(inj)}:{\\sf StartPtph}(x)$ ."
],
[
"Authentication from a pharmacist to a patient.",
"The authentication from a pharmacist to a patient is defined as when the patient finishes a session and believes that he communicates with a pharmacist with the identity ${\\mathit {c_{\\mathit {pt}}\\_Id_{\\mathit {ph}}}}$ , then the pharmacist is indeed the one who communicated with the patient.",
"To verify this authentication, we add the event ${\\sf EndPtph}({\\mathit {c_{\\mathit {pt}}\\_Id_{\\mathit {ph}}}})$ into the patient process (after line t26), add the event ${\\sf StartPh}({\\tt {Id}}_{\\mathit {ph}})$ into the pharmacist process (between line h1 and line h2), and query $\\mathit {ev(inj)}:{\\sf EndPtph}(x) ==> \\mathit {ev(inj)}:{\\sf StartPh}(x)$ .",
"In addition, we add the conditional evaluation $\\mbox{if}\\ {\\mathit {rcv\\_Invoice}}={\\sf inv}({\\mathit {c\\_\\mathit {PrescriptID}}}) \\ \\mbox{then} $ before the end ${\\sf EndPtph}({\\mathit {c_{\\mathit {pt}}\\_Id_{\\mathit {ph}}}})$ in the patient process to capture that the patient checks the correctness of the invoice."
],
[
"Verification results.",
"The queries are verified using ProVerif.",
"The verification results show that doctor authentication, both injective and non-injective, succeed; non-injective patient authentication succeeds and injective patient authentication fails.",
"The failure is caused by a replay attack from the adversary.",
"That is, the adversary can impersonate a patient by replaying old messages from the patient.",
"This authentication flaw leads to termination of the successive procedure, the patient-pharmacist sub-process.",
"We verified authentication between patients and pharmacists as well.",
"Non-injective patient authentication succeeds, whereas injective patient authentication fails.",
"This means that the messages received by a pharmacist are from the correct patient, but not necessarily from this communication session.",
"Neither non-injective nor injective pharmacist authentication succeeds: the adversary can record and replay the first message which is sent from a pharmacist to a patient, and pass the authentication by pretending to be that pharmacist.",
"In addition, the adversary can prepare the second message sending from a pharmacist to a patient, and thus does not need to replay the second message.",
"Since the adversary alters messages, non-injective pharmacist authentication fails.",
"The verification results are summarised in Table REF .",
"The reason that injective patient authentication fails for both doctors and pharmacists is that they suffer from replay attacks.",
"One possible solution approach is to add a challenge sent from the doctor (respectively, the pharmacist) to the patient.",
"Then, when the patient authenticates to the doctor or pharmacist, the patient includes this challenge in the proofs.",
"This approach assures that the proof is freshly generated.",
"Therefore, this prevents the adversary replaying old messages.",
"The reason that (injective and non-injective) authentication from a pharmacist to a patient fails is that the adversary can generate an invoice to replace the one from the real pharmacist.",
"One solution is for the pharmacist to sign the invoice.",
"Table: Verification results of authentication of patients and doctors."
],
[
"Authentications between public entities.",
"The public entities – pharmacists, MPAs, HIIs, authenticate each other using public key authentication.",
"The authentication is often used to agree on a way for the later communication.",
"Since it is not mentioned in the original protocol that a key or a communication channel is established during authentication, we assume that the later message exchanges are over public channels, to model the worst case.",
"In this model, the authentications between public entities are obviously flawed, since the adversary can reuse messages from other sessions.",
"The flaws are confirmed by the verification results using ProVerif."
],
[
"(Strong) patient and doctor anonymity",
"The DLV08 protocol claims that no party should be able to determine the identity of a patient.",
"We define (strong) patient anonymity to capture the requirement.",
"Note that in the original paper of the DLV08 protocol, the terminology of the privacy property for capturing this requirement is patient untraceability.",
"Our definition of untraceability (Definition REF ) has different meaning from theirs (for details, see Section REF ).",
"Also note that the satisfaction of standard secrecy of patient identity does not fully capture this requirement, as the adversary can still guess about it."
],
[
"Patient and doctor anonymity.",
"Doctor anonymity is defined as in Definition REF .",
"Patient anonymity can be defined in a similar way by replacing the role of doctor with the role of patient.",
"$\\begin{array}{rl}\\mathcal {C}_\\mathit {eh}[\\mathit {init}_{\\mathit {pt}}\\mathit {\\lbrace {\\tt {t_A}}/{\\mathit {I}d_{\\mathit {pt}}}\\rbrace }.",
"!\\mathit {P}_{\\mathit {pt}}\\mathit {\\lbrace {\\tt {t_A}}/{\\mathit {I}d_{\\mathit {pt}}}\\rbrace }]\\approx _{\\ell }\\mathcal {C}_\\mathit {eh}[\\mathit {init}_{\\mathit {pt}}\\mathit {\\lbrace {\\tt {t_B}}/{\\mathit {I}d_{\\mathit {pt}}}\\rbrace }.",
"!\\mathit {P}_{\\mathit {pt}}\\mathit {\\lbrace {\\tt {t_B}}/{\\mathit {I}d_{\\mathit {pt}}}\\rbrace }].\\end{array}$ To verify doctor/patient anonymity, is to check the satisfiability of the corresponding equivalence between processes in the definition.",
"This is done by modelling the two processes on two sides of the equivalence as a bi-process, and verify the bi-process using ProVerif.",
"Recall that a bi-process models two processes sharing the same structure and differing only in terms or destructors.",
"The two processes are written as one process with choice-constructors which tells ProVerif the spots where the two processes differ.",
"The bi-process for verifying doctor anonymity is $\\nu \\tilde{mc}.",
"\\mathit {init}.",
"(!\\mathit {R}_{\\mathit {pt}}\\mid !\\mathit {R}_{\\mathit {dr}}\\mid !\\mathit {R}_{\\mathit {ph}}\\mid !\\mathit {R}_{\\mathit {mpa}}\\mid !\\mathit {R}_{\\mathit {hii}}\\mid (\\nu {{\\tt {Pnym}}_{\\mathit {dr}}}.\\mbox{let}\\ {{\\tt {Id}}_{\\mathit {dr}}}={\\sf choice}[{\\tt {d_A}}, {\\tt {d_B}}] \\ \\mbox{in} \\ !\\mathit {P}_{\\mathit {dr}}),$ and the bi-process for verifying patient anonymity is $\\begin{array}{l}\\nu \\tilde{mc}.",
"\\mathit {init}.",
"(!\\mathit {R}_{\\mathit {pt}}\\mid !\\mathit {R}_{\\mathit {dr}}\\mid !\\mathit {R}_{\\mathit {ph}}\\mid !\\mathit {R}_{\\mathit {mpa}}\\mid !\\mathit {R}_{\\mathit {hii}}\\mid (\\mbox{let}\\ {{\\tt {Id}}_{\\mathit {pt}}}={\\sf choice}[{\\tt {t_A}}, {\\tt {t_B}}] \\ \\mbox{in} \\\\\\hspace{38.25pt}\\nu {{\\tt {Pnym}}_{\\mathit {pt}}}.\\nu {{\\tt {Sss}}}.\\nu {{\\tt {Acc}}}.",
"{\\sf {in}}({\\tt {ch}}_{hp}, {{\\tt {Hii}}}).\\mbox{let}\\ \\mathit {c_{\\mathit {pt}}\\_pk_\\mathit {hii}}={\\sf key}({{\\tt {Hii}}}) \\ \\mbox{in} \\ !\\mathit {P}_{\\mathit {pt}}).\\end{array}$ Since the doctor identity is a secret information, we define ${\\tt {d_A}}$ and ${\\tt {d_B}}$ as private names $\\mathit {private}\\ \\mathit {free}\\ {\\tt {d_A}}.",
"$$\\mathit {private}\\ \\mathit {free}\\ {\\tt {d_B}}$ .",
"In addition, we consider a stronger version, in which the adversary knows the two doctor identities a priori, i.e., we verify whether the adversary can distinguish two known doctors as well.",
"This is modelled by defining the two doctor identities as free names, $\\mathit {free}\\ {\\tt {d_A}}.",
"\\mathit {free}\\ {\\tt {d_B}}$ .",
"Similarly, we verified two versions of patient anonymity - in one version, the adversary does not know the two patient identities, and in the other version, the adversary initially knows the two patient identities."
],
[
"Strong patient and doctor anonymity.",
"Strong doctor anonymity is defined as in Definition REF .",
"By replacing the role of doctor with the role of patient, we obtain the definition of strong patient anonymity.",
"The bi-process for verifying strong doctor anonymity is $\\begin{array}{l}\\mathit {free}\\ {\\tt {d_A}};\\\\\\nu \\tilde{mc}.",
"\\mathit {init}.",
"(!\\mathit {R}_{\\mathit {pt}}\\mid !\\mathit {R}_{\\mathit {dr}}\\mid !\\mathit {R}_{\\mathit {ph}}\\mid !\\mathit {R}_{\\mathit {mpa}}\\mid !\\mathit {R}_{\\mathit {hii}}\\mid \\\\\\hspace{38.25pt}(\\nu {\\tt {d_B}}.\\nu {\\tt {nPnym}}_{\\mathit {dr}}.\\mbox{let}\\ {{\\tt {Pnym}}_{\\mathit {dr}}}={\\tt {nPnym}}_{\\mathit {dr}}\\ \\mbox{in} \\ !",
"(\\mbox{let}\\ {{\\tt {Id}}_{\\mathit {dr}}}={\\sf choice}[{\\tt {d_B}}, {\\tt {d_A}}] \\ \\mbox{in} \\ \\mathit {P}_{\\mathit {dr}}))),\\end{array}$ and the bi-process for verifying strong patient anonymity is $\\begin{array}{l}\\mathit {free}\\ {\\tt {t_A}};\\\\\\nu \\tilde{mc}.",
"\\mathit {init}.",
"(!\\mathit {R}_{\\mathit {pt}}\\mid !\\mathit {R}_{\\mathit {dr}}\\mid !\\mathit {R}_{\\mathit {ph}}\\mid !\\mathit {R}_{\\mathit {mpa}}\\mid !\\mathit {R}_{\\mathit {hii}}\\mid (\\nu {\\tt {t_B}}.\\nu {{\\tt {Pnym}}_{\\mathit {pt}}}.\\nu {{\\tt {Sss}}}.\\nu {{\\tt {Acc}}}.\\\\\\hspace{38.25pt}{\\sf {in}}({\\tt {ch}}_{hp}, {{\\tt {Hii}}}).\\mbox{let}\\ \\mathit {c_{\\mathit {pt}}\\_pk_\\mathit {hii}}={\\sf key}({{\\tt {Hii}}}) \\ \\mbox{in} \\ !",
"(\\mbox{let}\\ {{\\tt {Id}}_{\\mathit {pt}}}={\\sf choice}[{\\tt {t_B}}, {\\tt {t_A}}] \\ \\mbox{in} \\ \\mathit {P}_{\\mathit {pt}}))).\\end{array}$ Note that by definition, the identities ${\\tt {d_A}}$ and ${\\tt {t_B}}$ is known by the adversary.",
"In the first bi-process, by choosing ${\\tt {d_B}}$ , we obtain $\\begin{array}{l}\\mathit {free}\\ {\\tt {d_A}};\\\\\\nu \\tilde{mc}.",
"\\mathit {init}.",
"(!\\mathit {R}_{\\mathit {pt}}\\mid !\\mathit {R}_{\\mathit {dr}}\\mid !\\mathit {R}_{\\mathit {ph}}\\mid !\\mathit {R}_{\\mathit {mpa}}\\mid !\\mathit {R}_{\\mathit {hii}}\\mid \\\\\\hspace{38.25pt}(\\nu {\\tt {d_B}}.\\nu {\\tt {nPnym}}_{\\mathit {dr}}.\\mbox{let}\\ {{\\tt {Pnym}}_{\\mathit {dr}}}={\\tt {nPnym}}_{\\mathit {dr}}\\ \\mbox{in} \\ !",
"(\\mbox{let}\\ {{\\tt {Id}}_{\\mathit {dr}}}={\\tt {d_B}}\\ \\mbox{in} \\ \\mathit {P}_{\\mathit {dr}}))).\\end{array}$ Since ${\\tt {d_A}}$ never appears in the remaining process, removing the declaration “$\\mathit {free}\\ {\\tt {d_A}};$ \" does not affect the process.",
"Since process “$\\nu {\\tt {d_B}}.\\nu {\\tt {nPnym}}_{\\mathit {dr}}.\\mbox{let}\\ {{\\tt {Pnym}}_{\\mathit {dr}}}={\\tt {nPnym}}_{\\mathit {dr}}\\ \\mbox{in} \\ !",
"(\\mbox{let}\\ {{\\tt {Id}}_{\\mathit {dr}}}={\\tt {d_B}}\\ \\mbox{in} \\ \\mathit {P}_{\\mathit {dr}})$ \" essentially renames the doctor role process “$\\mathit {R}_{\\mathit {dr}}$ \" (Figure REF ) - renaming ${{\\tt {Id}}_{\\mathit {dr}}}$ as ${\\tt {d_B}}$ and renaming ${{\\tt {Pnym}}_{\\mathit {dr}}}$ as ${\\tt {nPnym}}_{\\mathit {dr}}$ , we have that the above process is structurally equivalent to (using rule $\\textsc {REPL}$ ) $\\begin{array}{l}\\nu \\tilde{mc}.",
"\\mathit {init}.",
"(!\\mathit {R}_{\\mathit {pt}}\\mid !\\mathit {R}_{\\mathit {dr}}\\mid !\\mathit {R}_{\\mathit {ph}}\\mid !\\mathit {R}_{\\mathit {mpa}}\\mid !\\mathit {R}_{\\mathit {hii}}),\\end{array}$ which is the left-hand side of Definition REF - the $\\mathit {\\mathit {P}_\\mathit {DLV08}}$ in the case study.",
"On the other hand, by choosing ${\\tt {d_A}}$ , we obtain process $\\begin{array}{l}\\mathit {free}\\ {\\tt {d_A}};\\\\\\nu \\tilde{mc}.",
"\\mathit {init}.",
"(!\\mathit {R}_{\\mathit {pt}}\\mid !\\mathit {R}_{\\mathit {dr}}\\mid !\\mathit {R}_{\\mathit {ph}}\\mid !\\mathit {R}_{\\mathit {mpa}}\\mid !\\mathit {R}_{\\mathit {hii}}\\mid \\\\\\hspace{38.25pt}(\\nu {\\tt {d_B}}.\\nu {\\tt {nPnym}}_{\\mathit {dr}}.\\mbox{let}\\ {{\\tt {Pnym}}_{\\mathit {dr}}}={\\tt {nPnym}}_{\\mathit {dr}}\\ \\mbox{in} \\ !",
"(\\mbox{let}\\ {{\\tt {Id}}_{\\mathit {dr}}}={\\tt {d_A}}\\ \\mbox{in} \\ \\mathit {P}_{\\mathit {dr}}))).\\end{array}$ Since ${\\tt {d_B}}$ only appears in the sub-process “$\\nu {\\tt {d_B}}.$ \" which generates ${\\tt {d_B}}$ and never appears in the remaingin process, the process is structurally equivalent to (applying rule $\\textsc {NEW}-\\textsc {PAR}$ ) $\\begin{array}{l}\\mathit {free}\\ {\\tt {d_A}};\\\\{\\bf \\nu {\\tt {d_B}}.",
"}\\nu \\tilde{mc}.",
"\\mathit {init}.",
"(!\\mathit {R}_{\\mathit {pt}}\\mid !\\mathit {R}_{\\mathit {dr}}\\mid !\\mathit {R}_{\\mathit {ph}}\\mid !\\mathit {R}_{\\mathit {mpa}}\\mid !\\mathit {R}_{\\mathit {hii}}\\mid \\\\\\hspace{38.25pt}(\\nu {\\tt {nPnym}}_{\\mathit {dr}}.\\mbox{let}\\ {{\\tt {Pnym}}_{\\mathit {dr}}}={\\tt {nPnym}}_{\\mathit {dr}}\\ \\mbox{in} \\ !",
"(\\mbox{let}\\ {{\\tt {Id}}_{\\mathit {dr}}}={\\tt {d_A}}\\ \\mbox{in} \\ \\mathit {P}_{\\mathit {dr}}))).\\end{array}$ The above process is structurally equivalent to (proved later) $\\begin{array}{l}\\mathit {free}\\ {\\tt {d_A}};\\\\\\nu \\tilde{mc}.",
"\\mathit {init}.",
"(!\\mathit {R}_{\\mathit {pt}}\\mid !\\mathit {R}_{\\mathit {dr}}\\mid !\\mathit {R}_{\\mathit {ph}}\\mid !\\mathit {R}_{\\mathit {mpa}}\\mid !\\mathit {R}_{\\mathit {hii}}\\mid \\\\\\hspace{38.25pt}(\\nu {\\tt {nPnym}}_{\\mathit {dr}}.\\mbox{let}\\ {{\\tt {Pnym}}_{\\mathit {dr}}}={\\tt {nPnym}}_{\\mathit {dr}}\\ \\mbox{in} \\ !",
"(\\mbox{let}\\ {{\\tt {Id}}_{\\mathit {dr}}}={\\tt {d_A}}\\ \\mbox{in} \\ \\mathit {P}_{\\mathit {dr}}))),\\end{array}$ which is the right-hand side of Definition REF , where ${\\tt {d_A}}$ is a free name.",
"This structural equivalent relation is proved as follows.",
"Assuming the above process is $P$ (i.e., $P=\\nu \\tilde{mc}.",
"\\mathit {init}.",
"(!\\mathit {R}_{\\mathit {pt}}\\mid !\\mathit {R}_{\\mathit {dr}}\\mid !\\mathit {R}_{\\mathit {ph}}\\mid !\\mathit {R}_{\\mathit {mpa}}\\mid !\\mathit {R}_{\\mathit {hii}}\\mid (\\nu {\\tt {nPnym}}_{\\mathit {dr}}.\\mbox{let}\\ {{\\tt {Pnym}}_{\\mathit {dr}}}={\\tt {nPnym}}_{\\mathit {dr}}\\ \\mbox{in} \\ !",
"(\\mbox{let}\\ {{\\tt {Id}}_{\\mathit {dr}}}={\\tt {d_A}}\\ \\mbox{in} \\ \\mathit {P}_{\\mathit {dr}})))$ ), by applying rule $\\textsc {PAR}-0$ , we have $P\\equiv P\\mid 0$ .",
"By rule $\\textsc {NEW}-0$ , $\\nu {\\tt {d_B}}.0 \\equiv 0$ .",
"Thus, $P\\equiv P\\mid \\nu {\\tt {d_B}}.0$ .",
"Since ${\\tt {d_B}}$ never appears in the process $P$ , i.e., ${\\tt {d_B}}\\notin {\\sf {fn}}(P) \\cup {\\sf {fv}}(P)$ , by applying rule $\\textsc {NEW}-\\textsc {PAR}$ , we have $P\\mid \\nu {\\tt {d_B}}.0 \\equiv \\nu {\\tt {d_B}}.",
"(P\\mid 0) \\equiv \\nu {\\tt {d_B}}.",
"P$ .",
"Therefore, $P\\equiv \\nu {\\tt {d_B}}.",
"P$ ."
],
[
"Verification result.",
"The bi-processes are verified using ProVerif.",
"The verification results show that patient anonymity (with and without revealed patient identities a priori) and strong patient anonymity are satisfied; doctor anonymity is satisfied; neither doctor anonymity with revealed doctor identities nor strong doctor anonymity is satisfied.",
"For strong doctor anonymity, the adversary can distinguish a process initiated by an unknown doctor and a known doctor.",
"Given a doctor process, where the doctor has identity ${\\tt {d_A}}$ , pseudonym ${{\\tt {Pnym}}_{\\mathit {dr}}}$ , and credential ${\\sf drcred}({{\\tt {Pnym}}_{\\mathit {dr}}},{\\tt {d_A}})$ , the terms ${{\\tt {Pnym}}_{\\mathit {dr}}}$ and ${\\sf drcred}({{\\tt {Pnym}}_{\\mathit {dr}}},{\\tt {d_A}})$ are revealed.",
"We assume that the adversary knows another doctor identity ${\\tt {d_B}}$ .",
"The adversary can fake an anonymous authentication by faking the zero-knowledge proof as ${\\sf zk}(({{\\tt {Pnym}}_{\\mathit {dr}}},{\\tt {d_B}}),{\\sf drcred}({{\\tt {Pnym}}_{\\mathit {dr}}},{\\tt {d_A}}))$ .",
"If the zero-knowledge proof passes the corresponding verification $\\mbox{\\sf Vfy-{zk}}_{\\sf Auth_{\\mathit {dr}}}$ by the patient, then the adversary knows that the doctor process is executed by the doctor ${\\tt {d_B}}$ .",
"Otherwise, not.",
"For the same reason, doctor anonymity fails the verification.",
"Both flaws can be fixed by requiring a doctor to generate a new credential in each session (s4')."
],
[
"(Strong) patient and doctor untraceability",
"Even if a user's identity is not revealed, the adversary may be able to trace a user by telling whether two executions are done by the same user.",
"The DLV08 protocol claims that prescriptions issued to the same patient should not be linkable to each other.",
"In other words, the situation in which a patient executes the protocol twice should be indistinguishable from the situation in which two different patients execute the protocol individually.",
"To satisfy this requirement, patient untraceability is required.",
"(Remark that the original DLV08 paper calls this untraceability “patient unlinkability”.)"
],
[
"Patient and doctor untraceability.",
"Doctor untraceability has been defined in Definition REF , and patient untraceability can be defined in a similar style.",
"The bi-process for verifying doctor untraceability is $\\begin{array}{l}\\nu \\tilde{mc}.",
"\\mathit {init}.",
"(!\\mathit {R}_{\\mathit {pt}}\\mid !\\mathit {R}_{\\mathit {dr}}\\mid !\\mathit {R}_{\\mathit {ph}}\\mid !\\mathit {R}_{\\mathit {mpa}}\\mid !\\mathit {R}_{\\mathit {hii}}\\mid (\\nu {\\tt {nPnym}}_{\\mathit {dr}}.",
"\\nu {\\tt {wPnym}}_{\\mathit {dr}}.\\\\((\\mbox{let}\\ {{\\tt {Id}}_{\\mathit {dr}}}={\\tt {d_A}}\\ \\mbox{in} \\ \\mbox{let}\\ {{\\tt {Pnym}}_{\\mathit {dr}}}={\\tt {nPnym}}_{\\mathit {dr}}\\ \\mbox{in} \\ \\mathit {P}_{\\mathit {dr}})\\mid \\\\\\hspace{4.25pt}(\\mbox{let}\\ {{\\tt {Id}}_{\\mathit {dr}}}={\\sf choice}[{\\tt {d_A}}, {\\tt {d_B}}] \\ \\mbox{in} \\ \\mbox{let}\\ {{\\tt {Pnym}}_{\\mathit {dr}}}={\\sf choice}[{\\tt {nPnym}}_{\\mathit {dr}}, {\\tt {wPnym}}_{\\mathit {dr}}] \\ \\mbox{in} \\ \\mathit {P}_{\\mathit {dr}})))),\\end{array}$ and the bi-process for verifying patient untraceability is $\\begin{array}{l}\\nu \\tilde{mc}.",
"\\mathit {init}.",
"(!\\mathit {R}_{\\mathit {pt}}\\mid !\\mathit {R}_{\\mathit {dr}}\\mid !\\mathit {R}_{\\mathit {ph}}\\mid !\\mathit {R}_{\\mathit {mpa}}\\mid !\\mathit {R}_{\\mathit {hii}}\\mid \\\\(\\nu {\\tt {nPnym}}_{\\mathit {pt}}.\\nu {\\tt {nSss}}.\\nu {\\tt {nAcc}}.\\nu {\\tt {wPnym}}_{\\mathit {pt}}.\\nu {\\tt {wSss}}.\\nu {\\tt {wAcc}}.\\\\\\hspace{4.25pt}{\\sf {in}}({\\tt {ch}}_{hp}, \\mathit {nHii}).",
"{\\sf {in}}({\\tt {ch}}_{hp}, \\mathit {wHii}).\\\\\\hspace{4.25pt} \\mbox{let}\\ \\mathit {c_{\\mathit {pt}}\\_npk_\\mathit {hii}}={\\sf key}(\\mathit {nHii}) \\ \\mbox{in} \\ \\mbox{let}\\ \\mathit {c_{\\mathit {pt}}\\_wpk_\\mathit {hii}}={\\sf key}(\\mathit {wHii}) \\ \\mbox{in} \\\\(\\mbox{let}\\ {{\\tt {Hii}}}=\\mathit {nHii}\\ \\mbox{in} \\ \\mbox{let}\\ \\mathit {c_{\\mathit {pt}}\\_pk_\\mathit {hii}}=\\mathit {c_{\\mathit {pt}}\\_npk_\\mathit {hii}}\\ \\mbox{in} \\ \\mbox{let}\\ {{\\tt {Id}}_{\\mathit {pt}}}={\\tt {t_A}}\\ \\mbox{in} \\\\\\ \\mbox{let}\\ {{\\tt {Pnym}}_{\\mathit {pt}}}={\\tt {nPnym}}_{\\mathit {pt}}\\ \\mbox{in} \\ \\mbox{let}\\ {{\\tt {Sss}}}={\\tt {nSss}}\\ \\mbox{in} \\ \\mbox{let}\\ {{\\tt {Acc}}}={\\tt {nAcc}}\\ \\mbox{in} \\ \\mathit {P}_{\\mathit {pt}}) \\mid \\\\\\end{array}(\\mbox{let}\\ {{\\tt {Hii}}}={\\sf choice}[\\mathit {nHii}, \\mathit {wHii}] \\ \\mbox{in} \\ \\mbox{let}\\ \\mathit {c_{\\mathit {pt}}\\_pk_\\mathit {hii}}={\\sf choice}[\\mathit {c_{\\mathit {pt}}\\_npk_\\mathit {hii}}, \\mathit {c_{\\mathit {pt}}\\_wpk_\\mathit {hii}}] \\ \\mbox{in} \\\\\\ \\mbox{let}\\ {{\\tt {Id}}_{\\mathit {pt}}}={\\sf choice}[{\\tt {t_A}}, {\\tt {t_B}}] \\ \\mbox{in} \\ \\mbox{let}\\ {{\\tt {Pnym}}_{\\mathit {pt}}}={\\sf choice}[{\\tt {nPnym}}_{\\mathit {pt}}, {\\tt {wPnym}}_{\\mathit {pt}}] \\ \\mbox{in} \\\\\\ \\mbox{let}\\ {{\\tt {Sss}}}={\\sf choice}[{\\tt {nSss}}, {\\tt {wSss}}] \\ \\mbox{in} \\ \\mbox{let}\\ {{\\tt {Acc}}}={\\sf choice}[{\\tt {nAcc}}, {\\tt {wAcc}}] \\ \\mbox{in} \\ \\mathit {P}_{\\mathit {pt}}))).$ $We verified two versions of doctor and patient untraceability,- in one version, the adversary does not know the two doctor/patient identities, andin the other version, the adversary initially knows the two doctor/patient identities.\\paragraph {Strong patient and doctor untraceability.",
"}Strong untraceability is modelled as a patientexecuting the protocol repeatedly is indistinguishable from different patientsexecuting the protocol each once.",
"Strong doctor untraceabilityis defined as in Definition~\\ref {def:suntra} and strong patient untraceabilitycan be defined in the same manner.The bi-process for verifying strong doctor untraceability is$ l mc.",
"$\\mathit {init}$ .",
"(!$\\mathit {R}$$\\mathit {pt}$ !$\\mathit {R}$$\\mathit {ph}$ !$\\mathit {R}$$\\mathit {mpa}$ !$\\mathit {R}$$\\mathit {hii}$ !",
"(nId$\\mathit {dr}$ .",
"nPnym$\\mathit {dr}$ .",
"!",
"(wId$\\mathit {dr}$ .",
"wPnym$\\mathit {dr}$ .",
"let Id$\\mathit {dr}$ =choice[nId$\\mathit {dr}$ , wId$\\mathit {dr}$ ] in let Pnym$\\mathit {dr}$ =choice[nPnym$\\mathit {dr}$ , wPnym$\\mathit {dr}$ ] in $\\mathit {P}$$\\mathit {dr}$ ))), $and the bi-process for verifying strong patient untraceability is$ l mc.",
"$\\mathit {init}$ .",
"(!$\\mathit {R}$$\\mathit {dr}$ !$\\mathit {R}$$\\mathit {ph}$ !$\\mathit {R}$$\\mathit {mpa}$ !$\\mathit {R}$$\\mathit {hii}$ !",
"(nId$\\mathit {pt}$ .",
"nPnym$\\mathit {pt}$ .",
"nSss.",
"nAcc.",
"in(chhp, $\\mathit {nHii}$ ).",
"!",
"(wId$\\mathit {pt}$ .",
"wPnym$\\mathit {pt}$ .",
"wSss.",
"wAcc.",
"let Id$\\mathit {pt}$ =choice[nId$\\mathit {pt}$ , wId$\\mathit {pt}$ ] in let Pnym$\\mathit {pt}$ =choice[nPnym$\\mathit {pt}$ , wPnym$\\mathit {pt}$ ] in let Sss=choice[nSss, wSss] in let Acc=choice[nAcc, wSss] in in(chhp, $\\mathit {wHii}$ ).",
"let Hii=choice[$\\mathit {nHii}$ , $\\mathit {wHii}$ ] in let $\\mathit {c_{\\mathit {pt}}\\_pk_\\mathit {hii}}$ =key(Hii) in $\\mathit {P}$$\\mathit {pt}$ ))).",
"$This definition does not involve a specific doctor/patient, and thus needs not todistinguish whether the adversary knows the identities a priori.\\paragraph {Verification result.",
"}The bi-processes are verified using ProVerif.The verification results show that the DLV08 protocol does not satisfypatient/doctor untraceability (with/without revealed identities), nor strong untraceability.$ The strong doctor untraceability fail, because the adversary can distinguish sessions initiated by one doctor and by different doctors.",
"The doctor's pseudonym is revealed and a doctor uses the same pseudonym in all sessions.",
"Sessions with the same doctor pseudonyms are initiated by the same doctor.",
"For the same reasons, doctor untraceability without revealing doctor identities also fails.",
"Both of them can be fixed by requiring the representation of a doctor's pseudonym (${{\\tt {Sss}}}$ ) differ in each session (s3').",
"However, assuming s3' (doctor pseudonym is fresh in every sessions) is not sufficient for satisfying doctor anonymity with doctor identities revealed.",
"The adversary can still distinguish two sessions initiated by one doctor or by two different doctors, by comparing the anonymous authentications of the two sessions.",
"From the communication in the two sessions, the adversary is able to learn two doctor pseudonyms ${{\\tt {Pnym}}_{\\mathit {dr}}}^{\\prime }$ and ${{\\tt {Pnym}}_{\\mathit {dr}}}^{\\prime \\prime }$ , two doctor credentials $\\mathit {Cred_{\\mathit {dr}}}^{\\prime }$ and $\\mathit {Cred_{\\mathit {dr}}}^{\\prime \\prime }$ and two anonymous authentications $\\mathit {Auth_{\\mathit {dr}}}^{\\prime }$ and $\\mathit {Auth_{\\mathit {dr}}}^{\\prime \\prime }$ .",
"Since the adversary knows ${\\tt {d_A}}$ and ${\\tt {d_B}}$ in advance, he could construct the eight anonymous authentications by applying the zero-knowledge proof function, i.e., ${\\sf zk}(({\\mathit {Pnym}_{\\mathit {dr}}},{\\mathit {Id_{\\mathit {dr}}}}),\\mathit {Cred_{\\mathit {dr}}})$ , where ${\\mathit {Pnym}_{\\mathit {dr}}}={{\\tt {Pnym}}_{\\mathit {dr}}}^{\\prime }$ or ${\\mathit {Pnym}_{\\mathit {dr}}}={{\\tt {Pnym}}_{\\mathit {dr}}}^{\\prime \\prime }$ , ${\\mathit {Id_{\\mathit {dr}}}}={\\tt {d_A}}$ or ${\\mathit {Id_{\\mathit {dr}}}}={\\tt {d_B}}$ , $\\mathit {Cred_{\\mathit {dr}}}=\\mathit {Cred_{\\mathit {dr}}}^{\\prime }$ or $\\mathit {Cred_{\\mathit {dr}}}=\\mathit {Cred_{\\mathit {dr}}}^{\\prime }$ .",
"By comparing the constructed anonymous authentications with the observed ones, the adversary is able to tell who generated which anonymous authentication, and thus is able to tell whether the two sessions are initiated by the same doctor or different doctors.",
"This can be fixed by additionally requiring that the doctor anonymous authentication differs in every session (s4').",
"For strong patient untraceability, the adversary can distinguish sessions initiated by one patient (with identical social security statuses) and initiated by different patients (with different social security statuses).",
"Second, the adversary can distinguish sessions initiated by one patient (with identical cipher texts ${\\sf {enc}}({{\\tt {Pnym}}_{\\mathit {pt}}}, {\\tt {pk}}_\\mathit {sso})$ and identical cipher texts ${\\sf {enc}}({{\\tt {Hii}}}, {\\tt {pk}}_\\mathit {sso})$ ) and initiated by different patients (with different cipher texts ${\\sf {enc}}({{\\tt {Pnym}}_{\\mathit {pt}}}, {\\tt {pk}}_\\mathit {sso})$ and different cipher texts ${\\sf {enc}}({{\\tt {Hii}}}, {\\tt {pk}}_\\mathit {sso})$ ).",
"Third, since the patient credential is the same in all sessions and is revealed, the adversary can also trace a patient by the patient's credential.",
"Fourth, the adversary can distinguish sessions using the same HII and sessions using different HIIs.",
"For the same reasons, patient untraceability fails.",
"Both flaws can be fixed by requiring that the representation of a patient's social security status to be different in each session (s5'), the encryptions are probabilistic (s2'), a patient freshly generates a credential in each session (s4”), and patients who shall not be distinguishable share the same HII (s6')."
],
[
"Prescription privacy",
"Prescription privacy has been defined in Definition REF .",
"To verify the prescription privacy is to check the satisfaction of the equivalence in the definition.",
"The bi-process for verifying the equivalence is $\\begin{array}{ll}\\multicolumn{2}{l}{(\\mathit {private}) \\mathit {free}\\ {\\tt {d_A}}.",
"(\\mathit {private}) \\mathit {free}\\ {\\tt {d_B}}.\\mathit {free}\\ {\\tt {p_A}}.\\mathit {free}\\ {\\tt {p_B}}.",
"}\\\\\\nu \\tilde{mc}.",
"\\mathit {init}.",
"(!\\mathit {R}_{\\mathit {pt}}\\mid !\\mathit {R}_{\\mathit {dr}}\\mid !\\mathit {R}_{\\mathit {ph}}\\mid & (\\nu {\\tt {nPnym}}_{\\mathit {dr}}.",
"\\nu {\\tt {wPnym}}_{\\mathit {dr}}.\\\\&\\hspace{4.25pt}\\mbox{let}\\ {{\\tt {Id}}_{\\mathit {dr}}}={\\sf choice}[{\\tt {d_A}}, {\\tt {d_B}}] \\ \\mbox{in} \\\\&\\hspace{4.25pt}\\mbox{let}\\ {{\\tt {Pnym}}_{\\mathit {dr}}}={\\sf choice}[{\\tt {nPnym}}_{\\mathit {dr}}, {\\tt {wPnym}}_{\\mathit {dr}}] \\ \\mbox{in} \\\\&\\hspace{4.25pt}\\mbox{let}\\ {\\tt {presc}}={\\tt {p_A}}\\ \\mbox{in} \\ \\mathit {\\mathit {main}_{\\mathit {dr}}})\\mid \\\\& (\\nu {\\tt {nPnym}}_{\\mathit {dr}}.",
"\\nu {\\tt {wPnym}}_{\\mathit {dr}}.\\\\&\\hspace{4.25pt}\\mbox{let}\\ {{\\tt {Id}}_{\\mathit {dr}}}={\\sf choice}[{\\tt {d_B}}, {\\tt {d_A}}] \\ \\mbox{in} \\\\&\\hspace{4.25pt}\\mbox{let}\\ {{\\tt {Pnym}}_{\\mathit {dr}}}={\\sf choice}[{\\tt {nPnym}}_{\\mathit {dr}}, {\\tt {wPnym}}_{\\mathit {dr}}] \\ \\mbox{in} \\\\&\\hspace{4.25pt}\\mbox{let}\\ {\\tt {presc}}={\\tt {p_B}}\\ \\mbox{in} \\ \\mathit {\\mathit {main}_{\\mathit {dr}}})).\\end{array}$ Similarly, we verified two versions - in one version, the adversary does not know ${\\tt {d_A}}$ and ${\\tt {d_B}}$ , and in the other, the adversary knows ${\\tt {d_A}}$ and ${\\tt {d_B}}$ ."
],
[
"Verification result.",
"The verification, using ProVerif, shows that the DLV08 protocol satisfies prescription privacy when the adversary does not know the doctor identities a priori, and does not satisfy prescription privacy when the adversary knows the doctor identities a priori, i.e., the adversary can distinguish whether a prescription is prescribed by doctor ${\\tt {d_A}}$ or doctor ${\\tt {d_B}}$ , given the adversary knows ${\\tt {d_A}}$ and ${\\tt {d_B}}$ .",
"In the prescription proof, a prescription is linked to a doctor credential.",
"And a doctor credential is linked to a doctor identity.",
"Thus, the adversary can link a doctor to his prescription.",
"To break the link, one way is to make sure that the adversary cannot link a doctor credential to a doctor identity.",
"This can be achieved by adding randomness to the credential (s4')."
],
[
"Receipt-freeness",
"The definition of receipt-freeness is modelled as the existence of a process $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ , such that the two equivalences in Definition REF are satisfied.",
"Due to the existential quantification, we cannot verify the property directly using ProVerif.",
"Examining the DLV08 protocol, we find an attack on receipt-freeness, even with assumption s4' (after fixing prescription privacy with doctor ID revealed).",
"A bribed doctor is able to prove to the adversary of his prescription as follows: A doctor communicates with the adversary to agree on a bit-commitment that he will use, which links the doctor to the commitment.",
"The doctor uses the agreed bit-commitment in the communication with his patient.",
"This links the bit-commitment to a prescription.",
"Later, when the patient uses this prescription to get medicine from a pharmacist, the adversary can observe the prescription being used.",
"This proves that the doctor has really prescribed the medicine.",
"We formally confirm the attack using ProVerif, i.e., we show that in the protocol model, if a doctor reveals all his information to the adversary, the doctor's prescription privacy is broken.",
"The same attack exists for multi-session receipt-freeness as well – a bribed doctor is able to prove his prescriptions by agreeing with the adversary on the bit-commitments in each session.",
"Theorem 1 (receipt-freeness) The DLV08 protocol fails to satisfy receipt-freeness under both the standard assumption s4 (a doctor has the same credential in every session), and also under assumption s4' (a doctor generates a new credential for each session).",
"Formal proof of the theorem can be found in Appendix ."
],
[
"Independency of (enforced) prescription privacy",
"To determine whether the doctor's prescription privacy is independent of the pharmacist, we replace regular pharmacist role $\\mathit {R}_{i}$ with collaborating role $\\mathit {R}_{\\mathit {ph}}$ in Definition REF .",
"The bi-process for verifying the property is: $\\begin{array}{ll}\\multicolumn{2}{l}{(\\mathit {private}) \\mathit {free}\\ {\\tt {d_A}}.",
"(\\mathit {private}) \\mathit {free}\\ {\\tt {d_B}}.\\mathit {free}\\ {\\tt {p_A}}.\\mathit {free}\\ {\\tt {p_B}}.",
"}\\\\\\nu \\tilde{mc}.",
"\\mathit {init}.",
"(!\\mathit {R}_{\\mathit {pt}}\\mid !\\mathit {R}_{\\mathit {dr}}\\mid !",
"(\\mathit {R}_{\\mathit {ph}})^{{\\tt {chc}}}\\mid &(\\nu {\\tt {nPnym}}_{\\mathit {dr}}.",
"\\nu {\\tt {wPnym}}_{\\mathit {dr}}.\\\\&\\hspace{4.25pt}\\mbox{let}\\ {{\\tt {Id}}_{\\mathit {dr}}}={\\sf choice}[{\\tt {d_A}}, {\\tt {d_B}}] \\ \\mbox{in} \\\\&\\hspace{4.25pt}\\mbox{let}\\ {{\\tt {Pnym}}_{\\mathit {dr}}}={\\sf choice}[{\\tt {nPnym}}_{\\mathit {dr}}, {\\tt {wPnym}}_{\\mathit {dr}}] \\ \\mbox{in} \\\\&\\hspace{4.25pt}\\mbox{let}\\ {\\tt {presc}}={\\tt {p_A}}\\ \\mbox{in} \\ \\mathit {\\mathit {main}_{\\mathit {dr}}})\\mid \\\\&(\\nu {\\tt {nPnym}}_{\\mathit {dr}}.",
"\\nu {\\tt {wPnym}}_{\\mathit {dr}}.\\\\&\\hspace{4.25pt}\\mbox{let}\\ {{\\tt {Id}}_{\\mathit {dr}}}={\\sf choice}[{\\tt {d_B}}, {\\tt {d_A}}] \\ \\mbox{in} \\\\&\\hspace{4.25pt}\\mbox{let}\\ {{\\tt {Pnym}}_{\\mathit {dr}}}={\\sf choice}[{\\tt {nPnym}}_{\\mathit {dr}}, {\\tt {wPnym}}_{\\mathit {dr}}] \\ \\mbox{in} \\\\&\\hspace{4.25pt}\\mbox{let}\\ {\\tt {presc}}={\\tt {p_B}}\\ \\mbox{in} \\ \\mathit {\\mathit {main}_{\\mathit {dr}}})).\\end{array}$ Verification using ProVerif shows that the protocol (the original version where the adversary does not know ${\\tt {d_A}}$ and ${\\tt {d_B}}$ , and the version after fixing the flaw on prescription privacy with assumption s4' where the adversary knows ${\\tt {d_A}}$ and ${\\tt {d_B}}$ ) satisfies this property.",
"The case of pharmacist-independent receipt-freeness is treated analogously.",
"We replace regular pharmacist $\\mathit {R}_i$ with $\\mathit {R}_{\\mathit {ph}}$ in Definition REF .",
"The flaw described in Section REF also surfaces here.",
"This was expected: when a doctor can prove his prescription without the pharmacist sharing information with the adversary, the doctor can also prove this when the pharmacist genuinely cooperates with the adversary."
],
[
"Dishonest users",
"So far, we have considered security and privacy with respect to a Dolev-Yao style adversary (see Section REF ).",
"The initial knowledge of the adversary was modelled such, that the adversary could not take an active part in the execution of the protocol.",
"This constitutes the basic DY adversary, as shown in Table REF .",
"In more detail, for secrecy of private doctor and patient information (see Table REF ), [24] claims that no third party (including the basic DY adversary) shall be able to know a patient's or doctor's private information (refer to the beginning of Section ).",
"Similarly, the verification of authentication properties in Table REF is also with respect to the basic DY adversary.",
"This captures that no third party that does not participate in the execution shall be able to impersonate any party (involved in the execution).",
"The same basic DY adversary model is used to verify anonymity, untraceability and prescription privacy.",
"The exceptions are (1) for verifying receipt-freeness and independency of enforced prescription privacy, the basic DY adversary is extended with information from the targeted doctor; (2) for verifying independency of prescription privacy and independency of enforced prescription privacy, the basic DY adversary is extended with information from pharmacists.",
"Table: Summary of the respected adversaryIn this section, we consider dishonest users, that is, malicious users that collaborate with the adversary and are part of the execution, into consideration.",
"For each property previously verified, we analyse the result once again with respect to each dishonest role.",
"Dishonest users are modelled by providing the adversary certain initial knowledge such that the adversary can take part in the protocol.",
"To execute the protocol as a doctor, i.e., to instantiate the doctor process $\\mathit {P}_{\\mathit {dr}}$ , the adversary only needs to have an identity and a pseudonym.",
"Since the adversary is able to generate data, the adversary can create his own identity ${{\\tt {Id}}^{a}_{\\mathit {dr}}}$ and pseudonym ${{\\tt {Id}}^{a}_{\\mathit {dr}}}$ .",
"However, this is not sufficient, because an legitimate doctor has a credential issued by authorities.",
"The credential is captured by the private function ${\\sf drcred}$ .",
"The adversary cannot obtain this credential, since he cannot apply the function ${\\sf drcred}$ .",
"When the function ${\\sf drcred}$ is modelled as public, the adversary is able to obtain his credential ${\\sf drcred}({{\\tt {Id}}^{a}_{\\mathit {dr}}},{{\\tt {Id}}^{a}_{\\mathit {dr}}})$ , and thus has the ability to behave like a doctor.",
"Hence, by modelling the function ${\\sf drcred}$ as public, we allow the adversary to have the ability of dishonest doctors.",
"Note that an honest doctor's identity is secret (see Table REF ).",
"The attacker thus cannot forge credentials of honest doctors, as the doctor's identity must be known for this.",
"Thus, making the function ${\\sf drcred}$ public does not bestow extra power on the attacker.",
"Similarly, by allowing the adversary to have patient credentials, we strengthen the adversary with the ability to control dishonest patients.",
"This is modelled by changing the private functions ${\\sf ptcred}$ to be public.",
"Each public entity (pharmacist, MPA or HII) has a secret key as distinct identifier, i.e., its public key and identity can be derived from the secret key.",
"The adversary can create such a secret key by himself.",
"However, only the legitimate entities can participate in the protocol.",
"This is modelled using private channels – only the honest entities are allowed to publish their information to the channels, and participants only read in entities, which they are going to communicate with, from the private channels.",
"By changing the private channels to be public, the adversary is able to behave as dishonest public entities.",
"Note that when considering the adversary only controlling dishonest pharmacists among the public entities, for the sake of simplicity of modelling, a dishonest pharmacist is modelled as $\\mathit {R}_{\\mathit {ph}}^{{\\tt {chc}}}$ (same as in independency of prescription privacy) in which the pharmacist shares all his information with the adversary.",
"The above modelling of dishonest users captures the following scenarios.",
"For verifying secrecy of doctor/paitent information, the dishonest doctor/patient models other doctors/patients that may break secrecy of the target doctor/patient; the dishonest patient/doctor models the patients/doctors that may communicate with the target doctor/patient; the dishonest pharmacist, MPA and HII may participate in the same execution as the target doctor/patient.",
"Since secrecy is defined as no other party (including dishonest users) should be able to know a doctor's/patient's information, unless the information is intended to be revealed, we would not consider it as an attack if the dishonest user intends to receive the private information, for details, see Section REF .",
"For verifying authentication properties, for example, a doctor authenticates a patient, a dishonest doctor is not the doctor directly communicating with the patient, and a dishonest patient is not the one who directly communicates with the doctor, because it does not make sense to analyse a dishonest user authenticates or authenticates to another user.",
"Instead, the dishonest doctors and patients are observers who may participate in other execution sessions.",
"In general, if user $A$ authenticates to user $B$ , the dishonest users taking the same role of $A$ or $B$ are observers participating in different sessions, i.e., cannot be $A$ or $B$ .",
"For the dishonest pharmacists, MPAs and HIIs, since they are not the authentication parties, they can be users participating in the same session.",
"For verifying prescription privacy and receipt-freeness, the dishonest doctors are other doctors that aim to break the target doctor's privacy; the dishonest patients can be patients communicating with the target doctor; dishonest pharmacists, MPAs and HIIs can be users participating in the same session.",
"Note that the dishonest doctor differs from the bribed doctor, as the bribed doctor tries to break his own privacy, while dishonest doctor tries to break others' privacy.",
"For verifying independency of prescription privacy and independency of enforced prescription privacy, the dishonest doctors (not the target doctor) try to break the target doctor's privacy; the dishonest patients may directly communicate with the target doctor; the dishonest pharmacists are the same as the bribed pharmacists, since 1) the bribed pharmacists genuinely forward information to the adversary and 2) all the actions that a dishonest pharmacists can do can be simulated by the basic DY with the received information from the bribed pharmacists, i.e., there is no private functions or private channels that the dishonest pharmacists can use but the adversary with bribed pharmacists information cannot; the dishonest MPAs and HIIs can participate in the same session as the target doctor.",
"The verification with dishonest users shows similar results as the verification without dishonest users.",
"The reason is that if there is an attack with respect to the basic DY attacker when verifying a property, then the property is also broken when additionally considering dishonest users.",
"The exceptions (i.e., the additional identified attacks) are shown in Table REF , and the details of the additional attacks are shown in the remaining part of this section.",
"Table: Additional attacks when considering dishonest users"
],
[
"Secrecy",
"When considering dishonest doctors and pharmacists, secrecy results in Table REF do not change, since doctors do not receive any information that the adversary does not know (see Figure REF ).",
"When considering dishonest patients, an additional attack is found.",
"When a patient ${\\tt {t_A}}$ is dishonest, he can obtain another patient ${\\tt {t_B}}$ 's pseudonym by doing the following: ${\\tt {t_A}}$ observes ${\\tt {t_B}}$ 's communication and reads in $\\mathit {vc}_4$ (the verifiable encryption which encrypts ${\\tt {t_B}}$ 's pseudonym with the public key of an MPA).",
"Hence, from $\\mathit {vc}_4$ , ${\\tt {t_A}}$ can obtain the cipher-text ${\\sf {enc}}({{\\tt {Pnym}}^{B}_{\\mathit {pt}}}, \\mathit {c_{\\mathit {pt}}\\_pk_{\\mathit {mpa}}})$ , where ${{\\tt {Pnym}}^{B}_{\\mathit {pt}}}$ is ${\\tt {t_B}}$ 's pseudonym and $\\mathit {c_{\\mathit {pt}}\\_pk_{\\mathit {mpa}}}$ is the public key of the MPA.",
"${\\tt {t_A}}$ initiates the protocol with his own data.",
"In the communication with a pharmacist, ${\\tt {t_A}}$ replaces his 5 (which should be a verifiable encryption, containing a cipher-text from ${\\tt {t_A}}$ encrypted with the public key of the MPA) with $\\mathit {vc}_4$ .",
"On receiving $\\mathit {vc}_4$ (the fake 5), the pharmacist sends it to the MPA, and the MPA decrypts the cipher-text ${\\sf {enc}}({{\\tt {Pnym}}^{B}_{\\mathit {pt}}}, \\mathit {c_{\\mathit {pt}}\\_pk_{\\mathit {mpa}}})$ , embedded in $\\mathit {vc}_4$ and sends the decryption result to HII.",
"${\\tt {t_A}}$ observes the communication between the MPA and HII, and reads the decrypted text of the fake 5 (i.e., ${{\\tt {Pnym}}^{B}_{\\mathit {pt}}}$ ), which is ${\\tt {t_B}}$ 's pseudonym.",
"This attack does not exist when the attacker cannot participant as a patient.",
"Because the dishonest patient has to replace 5 in his own communication.",
"If an hones patient's 5 is replaced by the adversary, the pharmacist would detect it by verifying the receipt $\\mathit {ReceiptAck}$ , which should contain the correct 5.",
"This attack can be addressed by explicitly ask the MPA to verify the decrypted message of 5 to be a verifiable encryption before sending it out.",
"Alternatively, if the communication between MPA and the HII is secured after authentication, the adversary would not be able to observe ${\\tt {t_B}}$ 's pseudonym, and thus the attack would not happen.",
"When considering dishonest pharmacists, an additional attack may exist on a patient's pseudonym.",
"The dishonest pharmacist has/creates a secret key $y$ and obtains its corresponding public key ${\\sf pk}(y)$ .",
"The pharmacist creates a fake MPA identity using the secret key and public key, i.e., ${\\sf host}({\\sf pk}(y))$ .",
"The dishonest pharmacist provides the patient a fake MPA identity, from which the patient obtains the MPA public key ${\\sf pk}(y)$ .",
"Later, the patient encrypts his pseudonym using the fake MPA's public key ${\\sf {enc}}({{\\tt {Pnym}}_{\\mathit {pt}}}, {\\sf pk}(y))$ , and provides a verifiable encryption, $\\mathit {vc}_4$ in particular.",
"The verifiable encryption is sent to the pharmacist, from which he pharmacist can read the cipher-text ${\\sf {enc}}({{\\tt {Pnym}}_{\\mathit {pt}}}, {\\sf pk}(y))$ .",
"Since the pharmacist knows the secret key $y$ , he can decrypt the cipher-text and obtains the patient's pseudonym ${{\\tt {Pnym}}_{\\mathit {pt}}}$ .",
"Note that we model that the patient obtains the MPA of the pharmacist and the MPA's public key from the pharmacist, thus the attack may not happen if the patient initially knows the MPA of any pharmacist, or the patient has the ability to immediately check whether the MPA provided by a pharmacist is indeed an legitimate MPA.",
"When considering dishonest MPAs, the adversary additionally knows a patient's pseudonym.",
"However, this can hardly be an attack, as the patient pseudonym is intended to be known by the MPA.",
"Similarly, the HII is the intended receiver of a patient's pseudonym.",
"Other than a patient's pseudonym, the dishonest MPA and HII do not know any information that the adversary does not know without controlling dishonest agents.",
"Note that in reality, the MPA and HII may know more sensitive information, for example, from the pseudonym, the HII is able to obtain the patient's identity, and a dishonest MPA can claim that a prescription has medical issues and obtains the doctor identity in a procedure, which is beyond the scope of this protocol."
],
[
"Authentication",
"When considering dishonest users, the verification results of the authentication remain the same, except the authentication from the dishonest user to other parties.",
"For example, when doctors are dishonest, we do not need to consider the authentication from doctors to a patient, since the dishonest users are part of the adversary.",
"Similarly, when a patient is dishonest, the authentication from a patient to a doctor or a pharmacist is obviously unsatisfied, other authentication verification results remain the same.",
"When pharmacists, MPAs or HIIs are dishonest, the verification results remain unchanged."
],
[
"Privacy properties",
"For those privacy properties which are not satisfied with respect to the adversary controlling no dishonest agents, the properties are not satisfied when considering the adversary who controls dishonest users.",
"Thus, we only need to analyse the property that are satisfied with respect to the adversary controlling no dishonest agents, i.e.",
"(Strong) patient anonymity.",
"Obviously, two patients can be distinguished by the adversary who controls dishonest HIIs, when the two patients use different HIIs, because the patients use different HII public keys to encrypt his pseudonym.",
"When the two patients use different HIIs, and the HIIs are honest, the adversary, who controls dishonest MPAs, can still distinguish them, because a patient's HII is intended to be known by the MPA.",
"Finally, (strong) patient anonymity is satisfied with respect to the adversary controlling dishonest doctors and dishonest pharmacists."
],
[
"Addressing the flaws of the DLV08 protocol",
"To summarise, we present updates to the assumptions of Section REF to fix the flaws found in our analysis of the privacy properties.",
"s2' The encryptions are probabilistic.",
"s3' The value of ${{\\tt {Pnym}}_{\\mathit {dr}}}$ is freshly generated in every session.",
"s4' A doctor freshly generates an unpredictable credential in each session.",
"We model this with another parameter (a random number) of the credential.",
"Following this, anonymous authentication using these credentials proves knowledge of the used randomness.",
"s4” A patient freshly generates an unpredictable credential in each session.",
"Similar to s4', this can be achieved by add randomness in the credential.",
"The anonymous authentication using the credentials proves the knowledge of the used randomness.",
"s5' The values of ${{\\tt {Sss}}}$ differ in sessions.",
"s6' The value of ${{\\tt {Hii}}}$ shall be the same for all patients.",
"The proposed assumptions are provided on the model level.",
"Due to the ambiguities in the original protocol (e.g., it is not clear how a social security status is represented), it is difficult to propose detailed solutions.",
"To implement a proposed assumption, one only needs to capture its properties.",
"To capture s2', the encryption scheme can be ElGamal cryptosystem, or RSA cryptosystem with encryption padding, which are probabilistic.",
"In some systems, deterministic encryption, e.g., RSA without encryption padding, may be more useful than probabilistic encryption, for example for database searching of encrypted data.",
"In such systems, designers need to carefully distinguish which encryption scheme is used in which part of the system.",
"s3' can be achieved by directly requiring a doctor's pseudonym to be fresh in every session, for example, a doctor generates different pseudonyms in sessions and keeps the authorities, who maintain the relation between the doctor identity and pseudonyms, updated in a secure way; or before every session the doctor requests a pseudonym from the authorities.",
"Alternatively, it can be achieved by changing the value of ${{\\tt {Pnym}}_{\\mathit {dr}}}$ .",
"Assuming the authorities share a key with each doctor; instead of directly using pseudonym in a session, the doctor encrypts his pseudonym with the key using probabilistic encryption.",
"That is, the value of ${{\\tt {Pnym}}_{\\mathit {dr}}}$ is a cipher-text which differs in sessions.",
"When an MPA wants to find out the doctor of a prescription, he can contact the authorities to decrypt the ${{\\tt {Pnym}}_{\\mathit {dr}}}$ and finds out the pseudonym of the doctor or the identity of the doctor directly.",
"s4' and s4” together form the updates to s4.",
"We separate the update to the doctor credential in s4' and the update to the patient credential in s4” for the convenience of referring to them individually in other places.",
"Similar to s3', s5' can be achieved by directly requiring that a patient's social security status is different in each session, e.g., by embedding a timestamp in the status.",
"Alternatively, the value of ${{\\tt {Sss}}}$ can be a cipher-text which is a probabilistic encryption of a patient's social security status with the pharmacist's public key, since the social security status is used for the pharmacist to check the status of the patient.",
"s6' can be achieved by directly requiring that all patients share the same HII.",
"In the case of multiple HIIs, different HIIs should not be distinguishable, for example, HIIs may cooperate together and provide a uniformed reference (name and key).",
"In fact, if patients are satisfied with untraceability within a group of a certain amount of patients, patient untraceability can be satisfied as long as each HII has more patients than the expected size of the group.",
"If only untraceability is required (instead of strong untraceability), the use of a group key of all HIIs is sufficient.",
"The common key among HIIs can be established by using asymmetric group key agreement.",
"In this way, the HIIs cannot be distinguished by their keys.",
"In addition, the identities of HIIs are not revealed, and thus cannot be used to distinguish HIIS.",
"Hence, the common key ensures that two patients executing the protocol once and one patient executing the protocol twice cannot be distinguished by their HIIs.",
"The modified protocol was verified again using ProVerif.",
"The verification results show that the protocol with revised assumptions satisfies doctor anonymity, strong doctor anonymity, and prescription privacy, as well as untraceability and strong untraceability for both patient and doctor.",
"However, the modified protocol model does not satisfy receipt-freeness, to make the protocol satisfy receipt-freeness, we apply the following assumption on communication channels.",
"s8' Communication channels are untappable (i.e., the adversary does not observe anything from the channel), except those used for authentication, which remain public.",
"Table: Verification results of privacy properties and revised assumptions.Our model of the protocol is accordingly modified as follows: replacing channel ${\\tt {ch}}$ in lines d10, t6 with an untappable channel ${\\tt {ch}}_{dp}$ , replacing channel ${\\tt {ch}}$ in lines t23, t26, h5, h22 with an untappable channel ${\\tt {ch}}_{ptph}$ , and replacing channel ${\\tt {ch}}$ in lines t24, h21 with an untappable channel ${\\tt {ch}}_{phpt}$ .",
"The untappable channels are modelled as global private channels.",
"We prove that the protocol (with s4' and s8') satisfies receipt-freeness by showing the existence of a process $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ (as shown in Figure REF ) such that the equivalences in Definition REF are satisfied.",
"This was verified using ProVerif (for verification code, see [26]).",
"Figure: The doctor process 𝑃 𝑑𝑟 ' \\mathit {P}_{\\mathit {dr}}^{\\prime } (using untappable channels).Messages over untappable channels are assumed to be perfectly secret to the adversary (for example, the channels assumed in [29], [31]).",
"Thus, the security and classical privacy properties, which are satisfied in the model with public channels only, are also satisfied when replacing some public channels with untappable channels.",
"Similar to other proposed assumptions, the assumption of untappable channels is at the model level.",
"This is a strong assumption, as the implementation of an untappable channel is difficult [31].",
"However, as this assumption is often used in literature to achieve privacy in the face of bribery and coercion (e.g.",
"[51], [8], [31]), we feel that its use here is justifiable.",
"However, even with the above assumptions the DLV08 protocol does not satisfy independency of receipt-freeness.",
"The proof first shows that $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ is not sufficient for proving this with ProVerif.",
"Then we prove (analogous to the proof in Section REF ) that there is no alternative process $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ which satisfies Definition REF .",
"Intuitively, all information sent over untappable channels is received by pharmacists and can be genuinely revealed to the adversary (no lying assumption).",
"Hence, the links between a doctor, his nonces, his commitment, his credential and his prescription can still be revealed when the doctor is bribed/coerced to reveal those nonces.",
"Theorem 2 (independency of receipt-freeness) The DLV08 protocol fails to satisfy independency of receipt-freeness.",
"Formal proof the theorem can be found in Appendix .",
"Intuitively, a bribed doctor is linked to the nonces he sent to the adversary.",
"The nonces are linked to the doctor's prescription in a prescription proof.",
"A doctor's prescription proof is sent over untappable channels first to a patient and later from the patient to a pharmacist.",
"Malicious pharmacists reveal the prescription proof to the adversary.",
"If a bribed doctor lied about his prescription, the adversary can detect it by checking the doctor's corresponding prescription proof revealed by the pharmacist.",
"The untappable channel assumption enables the protocol to satisfy receipt-freeness but not independency of receipt-freeness because untappable channels enable a bribed doctor to hide his prescription proof and thus allow the doctor to lie about his prescription, however the pharmacist gives the prescription proof away, from which the adversary can detect whether the doctor lied about the prescription."
],
[
"Conclusions",
"In this paper, we have studied security and privacy properties, particular enforced privacy, in the e-health domain.",
"We identified the requirement that doctor privacy should be enforced to prevent doctor bribery by, for example, the pharmaceutical industry.",
"To capture this requirement, we first formalised the classical privacy property, i.e., prescription privacy, and its enforced privacy counterpart, i.e., receipt-freeness.",
"The cooperation between the bribed doctor and the adversary is formalised in the same way as in receipt-freeness in e-voting.",
"However, the formalisation of receipt-freeness differs from receipt-freeness, due to the domain requirement that only part of the doctor's process needs to share information with the adversary.",
"Next, we noted that e-health systems involve not necessarily trusted third parties, such as pharmacists.",
"Such parties should not be able to assist an adversary in breaking doctor privacy.",
"To capture this requirement, we formally defined independency of prescription privacy.",
"Moreover, this new requirement must hold, even if the doctor is forced to help the adversary.",
"To capture that, we formally defined independency of receipt-freeness.",
"These formalisations were validated in a case study of the DLV08 protocol.",
"The protocol was modelled in the applied pi calculus and verified with the help of the ProVerif tool.",
"In addition to the (enforced) doctor privacy properties, we also analysed secrecy, authentication, anonymity and untraceability for both patients and doctors.",
"Ambiguities in the original description of the protocol which may lead to flaws were found and addressed.",
"We notice that the property independency of receipt-freeness is not satisfied in the case study protocol, and we were not able to propose a reasonable fix for it.",
"Thus, it is interesting for us to design a new protocol to satisfy such strong property in the future.",
"Furthermore, when considering dishonest users, we did not consider one dishonest user taking multiple roles.",
"Thus, it would be interesting to analyse the security and privacy properties with respect to dishonest users taking various combination of roles."
],
[
"Proof of Theorem ",
"Theorem 1 (receipt-freeness).",
"The DLV08 protocol fails to satisfy receipt-freeness under both the standard assumption s4 (a doctor has the same credential in every session), and also under assumption s4' (a doctor generates a new credential for each session).",
"It is obvious that the DLV08 protocol fails to satisfy receipt-freeness under assumption s4 (a doctor has the same credential in every session), since DLV08 does not even satisfy prescription privacy with assumption s4.",
"The reasoning is as follows: since the adversary can link a prescription to a doctor without additional information from the bribed doctor, he can also link a prescription to a doctor when he has additional information from the bribed doctor.",
"Therefore, the adversary can always tell whether a bribed doctor lied.",
"Next we prove that the DLV08 protocol fails to satisfy receipt-freeness under assumption s4' (a doctor generates a new credential for each session).",
"That is to prove that there exists no indistinguishable process in which the doctor lies to the adversary.",
"To do so, we assume that there exists such a process $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ which satisfies the definition of receipt-freeness, and then derive some contradiction.",
"In generic terms, the proof runs as follows: a bribed doctor reveals the nonces used in the commitment and the credential to the adversary.",
"This allows the adversary to link a bribed doctor to his commitment and credential.",
"In the prescription proof, a prescription is linked to a doctor's commitment and credential.",
"Suppose there exists a process $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ in which the doctor lies to the adversary that he prescribed ${\\tt {p_A}}$ , while the adversary observes that the commitment or the credential is linked to ${\\tt {p_B}}$ .",
"The adversary can detect that the doctor has lied.",
"Assume there exist process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and $P_{\\mathit {dr}}^{\\prime }$ , so that the two equivalences in the definition of receipt-freeness are satisfied, i.e., $\\exists $ $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and $P_{\\mathit {dr}}^{\\prime }$ satisfying $\\begin{array}{lrl}1.\\ &&\\mathcal {C}_\\mathit {eh}[\\big (\\mathit {init}_{\\mathit {dr}}^{\\prime }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {P}_{\\mathit {dr}}^{\\prime })\\big )\\mid \\\\&&\\hspace{17.0pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.(!",
"\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )]\\\\&\\approx _{\\ell }&\\mathcal {C}_\\mathit {eh}[\\big ((\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace })^{{\\tt {chc}}}.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid (\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })^{{\\tt {chc}}})\\big )\\mid \\\\&&\\hspace{17.0pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )]; \\vspace{5.69054pt} \\text{and}\\\\2.\\ &&\\mathcal {C}_\\mathit {eh}[\\big ((\\mathit {init}_{\\mathit {dr}}^{\\prime }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {P}_{\\mathit {dr}}^{\\prime }))^{\\backslash {\\sf {out}}({\\tt {chc}}, \\cdot )}\\big )\\mid \\\\&&\\hspace{16.57497pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.(!",
"\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )]\\\\&\\approx _{\\ell }&\\mathcal {C}_\\mathit {eh}[\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )\\mid \\\\&&\\hspace{16.57497pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.(!",
"\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )].\\end{array}$ According to the definition of labelled bisimilarity (Definition REF ), if process $A$ can reach $A^{\\prime }$ ($A \\xrightarrow{} A^{\\prime }$ ) and $A\\approx _{\\ell }B$ , then $B$ can reach $B^{\\prime }$ ($B \\xrightarrow{} B^{\\prime }$ ) and $A^{\\prime }\\approx _{\\ell }B^{\\prime }$ .",
"Vice versa.",
"Note that we use $\\xrightarrow{}$ to denote one or more internal and/or labelled reductions.",
"According to Definition REF , if $A^{\\prime }\\approx _{\\ell }B^{\\prime }$ then $A^{\\prime } \\approx _{s}B^{\\prime }$ .",
"According to the definition of static equivalence (Definition REF ), if two processes are static equivalent $A^{\\prime } \\approx _{s}B^{\\prime }$ , then ${\\sf {frame}}(A^{\\prime })\\approx _{s}{\\sf {frame}}(B^{\\prime })$ .",
"Thus we have that $\\forall M, N$ , $(M =_E N){\\sf {frame}}(A^{\\prime })$ iff $(M =_E N){\\sf {frame}}(B^{\\prime })$ .",
"Let $A$ be the right-hand side of the first equivalence and $B$ be the left-hand side, i.e., $\\begin{array}{rcl}A &=& \\mathcal {C}_\\mathit {eh}[\\big ((\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace })^{{\\tt {chc}}}.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace } \\mid (\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })^{{\\tt {chc}}})\\big )\\mid \\\\&&\\hspace{17.0pt} \\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace } \\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )]\\\\\\end{array}B &=& \\mathcal {C}_\\mathit {eh}[\\big (\\mathit {init}_{\\mathit {dr}}^{\\prime }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace } \\mid \\mathit {P}_{\\mathit {dr}}^{\\prime })\\big ) \\mid \\\\&&\\hspace{17.0pt} \\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.(!",
"\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace } \\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )].$ $$ On the right-hand side of the first equivalence (process $A$ ), there exists an output of a prescription proof ${\\tt {\\mathit {PrescProof}^{\\mathit {r}}}}$ (together with the open information of the doctor commitment ${\\tt {r}}^{r}_{\\mathit {dr}}$ ), over public channels, from the process $(\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })^{{\\tt {chc}}}$ initiated by doctor ${\\tt {d_A}}$ .",
"Formally, $A \\xrightarrow{} A_i = {\\mathcal {C}}_i[{\\sf {out}}(ch, ({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},{\\tt {r}}^{r}_{\\mathit {dr}}))] \\equiv \\nu x.",
"({\\mathcal {C}}_i[{\\sf {out}}(ch, x)] \\mid \\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},{\\tt {r}}^{r}_{\\mathit {dr}})/x\\rbrace })\\\\\\xrightarrow{} {\\mathcal {C}}_i[0]\\mid \\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},{\\tt {r}}^{r}_{\\mathit {dr}})/x\\rbrace }$ .",
"Let $A^{\\prime \\prime }={\\mathcal {C}}_i[0]$ , we have $A\\xrightarrow{} \\xrightarrow{} A^{\\prime \\prime } \\mid \\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},{\\tt {r}}^{r}_{\\mathit {dr}})/x\\rbrace }$ .",
"Let $A^{\\prime }=A^{\\prime \\prime }\\mid \\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},{\\tt {r}}^{r}_{\\mathit {dr}})/x\\rbrace }$ , we have $A\\xrightarrow{} \\xrightarrow{} A^{\\prime }$ .",
"Since $A\\approx _{\\ell }B$ , we have that $B\\xrightarrow{} \\xrightarrow{} B^{\\prime }$ and $A^{\\prime }\\approx _{\\ell }B^{\\prime }$ .",
"Hence, $A^{\\prime }\\approx _{s}B^{\\prime }$ and thus ${\\sf {frame}}(A^{\\prime })\\approx _{s}{\\sf {frame}}(B^{\\prime })$ .",
"Since ${\\sf {frame}}(A^{\\prime })={\\sf {frame}}(A^{\\prime \\prime })\\mid \\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},{\\tt {r}}^{r}_{\\mathit {dr}})/x\\rbrace }$ , the adversary can obtain the prescription ${\\tt {p_A}}$ : ${\\tt {p_A}}={\\sf first}({\\sf getmsg}({\\sf first}(x)))$ , where function ${\\sf first}$ returns the first element of a tuple or a pair.",
"Since ${\\sf {frame}}(A^{\\prime })\\approx _{s}{\\sf {frame}}(B^{\\prime })$ , we should have the same relation ${\\tt {p_A}}={\\sf first}({\\sf getmsg}({\\sf first}(x)))$ in ${\\sf {frame}}(B^{\\prime })$ .",
"Intuitively, since on the right-hand side, the adversary can obtain the prescription ${\\tt {p_A}}$ from ${\\tt {\\mathit {PrescProof}^{\\mathit {r}}}}$ , due to $({\\tt {p_A}}, \\mathit {PrescriptID}^{r}, \\mathit {Comt^{r}_{\\mathit {dr}}}, {\\mathit {c\\_Comt^{r}_{\\mathit {pt}}}})={\\sf getmsg}({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}}),$ on the left-hand side of the first equivalence, there should also exist an output of a prescription proof $\\mathit {PrescProof}^{l}$ over public channels, from which the adversary can obtain a prescription ${\\tt {p_A}}$ , following the same relation: $({\\tt {p_A}}, \\mathit {PrescriptID}^{l}, \\mathit {Comt^{l}_{\\mathit {dr}}}, {\\mathit {c\\_Comt^{l}_{\\mathit {pt}}}})={\\sf getmsg}(\\mathit {PrescProof}^{l}).$ Next, we prove that the corresponding prescription proof $\\mathit {PrescProof}^{l}$ is indeed the prescription proof in the doctor sub-process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ in process $B$ , rather than other sub-processes.",
"Formally, the action $\\xrightarrow{}$ in process $B$ happens in sub-process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ .",
"On the right-hand side (process $A$ ), the doctor pseudonym ${\\tt {Pnym}}^{r}_{\\mathit {dr}}$ and the nonce for doctor commitment ${\\tt {r}}^{r}_{\\mathit {dr}}$ and the nonce for doctor credential ${\\tt {n}}^{r}_{\\mathit {dr}}$ (used for assumption s4') are revealed to the adversary on ${\\tt {chc}}$ channel.",
"Formally, $\\begin{array}{rll}A&\\xrightarrow{}\\xrightarrow{}&A_1\\mid \\mathit {\\lbrace {\\tt {Pnym}}^{r}_{\\mathit {dr}}/x_1\\rbrace }\\\\&\\xrightarrow{}\\xrightarrow{}&A_2\\mid \\mathit {\\lbrace {\\tt {Pnym}}^{r}_{\\mathit {dr}}/x_1\\rbrace }\\mid \\mathit {\\lbrace {\\tt {r}}^{r}_{\\mathit {dr}}/x_2\\rbrace }\\\\&\\xrightarrow{}\\xrightarrow{}&A_3\\mid \\mathit {\\lbrace {\\tt {Pnym}}^{r}_{\\mathit {dr}}/x_1\\rbrace }\\mid \\mathit {\\lbrace {\\tt {r}}^{r}_{\\mathit {dr}}/x_2\\rbrace }\\mid \\mathit {\\lbrace {\\tt {n}}^{r}_{\\mathit {dr}}/x_3\\rbrace }\\\\&\\xrightarrow{}\\xrightarrow{} & A_4\\mid \\mathit {\\lbrace {\\tt {Pnym}}^{r}_{\\mathit {dr}}/x_1\\rbrace }\\mid \\mathit {\\lbrace {\\tt {r}}^{r}_{\\mathit {dr}}/x_2\\rbrace }\\mid \\mathit {\\lbrace {\\tt {n}}^{r}_{\\mathit {dr}}/x_3\\rbrace }\\mid \\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},{\\tt {r}}^{r}_{\\mathit {dr}})/x\\rbrace }\\equiv A^{\\prime }.\\end{array}$ Since $A\\approx _{\\ell }B$ , we have that $\\begin{array}{rll}B&\\xrightarrow{}\\xrightarrow{}&B_1\\mid \\mathit {\\lbrace {\\tt {Pnym}}^{l}_{\\mathit {dr}}/x_1\\rbrace }\\\\&\\xrightarrow{}\\xrightarrow{}&B_2\\mid \\mathit {\\lbrace {\\tt {Pnym}}^{l}_{\\mathit {dr}}/x_1\\rbrace }\\mid \\mathit {\\lbrace {\\tt {r}}^{l}_{\\mathit {dr}}/x_2\\rbrace }\\\\&\\xrightarrow{}\\xrightarrow{}&B_3\\mid \\mathit {\\lbrace {\\tt {Pnym}}^{l}_{\\mathit {dr}}/x_1\\rbrace }\\mid \\mathit {\\lbrace {\\tt {r}}^{l}_{\\mathit {dr}}/x_2\\rbrace }\\mid \\mathit {\\lbrace {\\tt {n}}^{l}_{\\mathit {dr}}/x_3\\rbrace }\\\\&\\xrightarrow{}\\xrightarrow{} &B_4\\mid \\mathit {\\lbrace {\\tt {Pnym}}^{l}_{\\mathit {dr}}/x_1\\rbrace }\\mid \\mathit {\\lbrace {\\tt {r}}^{l}_{\\mathit {dr}}/x_2\\rbrace }\\mid \\mathit {\\lbrace {\\tt {n}}^{l}_{\\mathit {dr}}/x_3\\rbrace }\\mid \\mathit {\\lbrace (\\mathit {PrescProof}^{l},{\\tt {r}}^{l}_{\\mathit {dr}})/x\\rbrace } \\equiv B^{\\prime }.\\end{array}$ That is, on the left-hand side of the first equivalence, to be equivalent to the right-hand side, there also exist sub-processes which output messages on ${\\tt {chc}}$ channel.",
"Such sub-processes can only be $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ , because there is no output on ${\\tt {chc}}$ in other sub-processes in the left-hand side process (process $B$ ).",
"In ${\\sf {frame}}(A^{\\prime })={\\sf {frame}}(A_4)\\mid \\mathit {\\lbrace {\\tt {Pnym}}^{r}_{\\mathit {dr}}/x_1\\rbrace }\\mid \\mathit {\\lbrace {\\tt {r}}^{r}_{\\mathit {dr}}/x_2\\rbrace }\\mid \\mathit {\\lbrace {\\tt {n}}^{r}_{\\mathit {dr}}/x_3\\rbrace }\\mid \\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},{\\tt {r}}^{r}_{\\mathit {dr}})/x\\rbrace }$ , we have the following relation between two terms, $\\begin{array}{rcl}{\\sf first}(x)&=&{\\sf spk}((x_1,x_2,{\\tt {d_A}},x_3),\\\\&&\\hspace{17.0pt}({\\sf commit}(x_1,x_2),{\\sf drcred}(x_1,{\\tt {d_A}},x_3)),\\\\&&\\hspace{17.0pt}({\\tt {p_A}},\\mathit {PrescriptID}^{r}, {\\sf commit}(x_1,x_2),{\\mathit {c\\_Comt^{r}_{\\mathit {pt}}}})).\\end{array}$ Since $A^{\\prime }\\approx _{s}B^{\\prime }$ , we should have the same relation in ${\\sf {frame}}(B^{\\prime })$ .",
"$\\begin{array}{rcl}{\\sf first}(x)&=&{\\sf spk}((x_1,x_2,{\\tt {d_A}},x_3),\\\\&&\\hspace{17.0pt}({\\sf commit}(x_1,x_2),{\\sf drcred}(x_1,{\\tt {d_A}},x_3)),\\\\&&\\hspace{17.0pt}({\\tt {p_A}},\\mathit {PrescriptID}^{l}, {\\sf commit}(x_1,x_2),{\\mathit {c\\_Comt^{l}_{\\mathit {pt}}}})).\\end{array}$ On the left-hand side (process $B$ ), the terms sent by process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and/or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ over ${\\tt {chc}}$ – the doctor pseudonym and the nonces ${\\tt {Pnym}}^{l}_{\\mathit {dr}}$ , ${\\tt {r}}^{l}_{\\mathit {dr}}$ and ${\\tt {n}}^{l}_{\\mathit {dr}}$ (corresponding to ${\\tt {Pnym}}^{r}_{\\mathit {dr}}$ , ${\\tt {r}}^{r}_{\\mathit {dr}}$ and ${\\tt {n}}^{r}_{\\mathit {dr}}$ on the right-hand side), are essential to compute $\\mathit {PrescProof}^{l}$ .",
"Thus, process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and/or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ is able to compute and thus output the prescription proof $\\mathit {PrescProof}^{l}$ , given that the coerced doctor ${\\tt {d_A}}$ has the knowledge of ${\\tt {p_A}}$ , by applying the following function: $\\begin{array}{rcl}\\mathit {PrescProof}^{l}&=&{\\sf spk}(({\\tt {Pnym}}^{l}_{\\mathit {dr}}, {\\tt {r}}^{l}_{\\mathit {dr}}, {\\tt {d_A}}, {\\tt {n}}^{l}_{\\mathit {dr}}),\\\\&&\\hspace{17.0pt} ({\\sf commit}({\\tt {Pnym}}^{l}_{\\mathit {dr}},{\\tt {r}}^{l}_{\\mathit {dr}}),{\\sf drcred}({\\tt {Pnym}}^{l}_{\\mathit {dr}}, {\\tt {d_A}}, {\\tt {n}}^{l}_{\\mathit {dr}})),\\\\&&\\hspace{17.0pt} ({\\tt {p_A}},\\mathit {PrescriptID}^{l},{\\sf commit}({\\tt {Pnym}}^{l}_{\\mathit {dr}},{\\tt {r}}^{l}_{\\mathit {dr}}),{\\mathit {c\\_Comt^{l}_{\\mathit {pt}}}}).\\end{array}$ Now we have proved that the action of revealing $(\\mathit {PrescProof}^{l}, {\\tt {r}}^{l}_{\\mathit {dr}})$ ($\\xrightarrow{}$ ) can be taken in process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and/or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ in process $B$ .",
"Next we show that sub-processes except $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ in $B$ , cannot take the action of revealing $(\\mathit {PrescProof}^{l}, {\\tt {r}}^{l}_{\\mathit {dr}})$ , given the process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ does not replay the message $(\\mathit {PrescProof}^{l}, {\\tt {r}}^{l}_{\\mathit {dr}})$ in an honest doctor process.",
"By examining process $B$ , the sub-processes which send out a pair, the first element of which is a signed proof of knowledge, can only be doctor processes and the MPA processes, i.e., sub-processes that may send out a message $x$ potentially satisfying ${\\tt {p_A}}={\\sf first}({\\sf getmsg}({\\sf first}(x)))$ can only be doctor processes (at d10) or MPA processes (at m30).",
"Case 1: considering that message $x$ should also satisfy ${\\sf open}({\\sf third}({\\sf getmsg}({\\sf first}(x))),{\\sf snd}(x))={\\tt {Pnym}}^{l}_{\\mathit {dr}}$ , the processes revealing $x$ can only be doctor processes, because the second element in the message sent out at line m30 of an MPA process is a zero-knowledge proof, and thus cannot be used as a nonce to open a commitment third(getmsg(first(x))).",
"Case 2: considering that the message $x$ should satisfy ${\\sf first}(x)={\\sf spk}((x_1,x_2,{\\tt {d_A}},x_3),\\linebreak ({\\sf commit}(x_1,x_2),$${\\sf drcred}(x_1,{\\tt {d_A}},x_3)),$$({\\tt {p_A}},\\mathit {PrescriptID}^{l}, $${\\sf commit}(x_1,x_2),{\\mathit {c\\_Comt^{l}_{\\mathit {pt}}}}))$ , where $x_1$ is the doctor pseudonym, $x_2$ and $x_3$ are nonces, and the adversary receive $x_1$ , $x_2$ , $x_3$ from ${\\tt {chc}}$ channel, doctor sub-processes (except $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ ) cannot reveal the message $x$ .",
"Because these doctor sub-processes model honest doctor sessions, and thus use their own generated nonces to compute the signed proofs of knowledge (at line t23).",
"Such nonces are not sent to the adversary over ${\\tt {chc}}$ channel, since these doctor processes are not bribed or coerced.",
"Thus, the signed proofs of knowledge generated by these honest doctor prepossess cannot be the first element of the message $x$ , unless the process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and/or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ reuses one of the signed proofs of knowledge.",
"In the case that the process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and/or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ replay the message $(\\mathit {PrescProof}^{l}, {\\tt {r}}^{l}_{\\mathit {dr}})$ of an honest doctor process, $x_1$ needs to be the corresponding doctor pseudonym, and $x_2$ and $x_3$ need to be the corresponding nonces for the reused signed proof of knowledge the message.",
"Otherwise, $(\\mathit {PrescProof}^{l}, {\\tt {r}}^{l}_{\\mathit {dr}})$ will be detected as a fake message.",
"Thus, the message indeed represents the actual prescription in process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and/or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ .",
"Although the action of revealing message $(\\mathit {PrescProof}^{l}, {\\tt {r}}^{l}_{\\mathit {dr}})$ may be taken in an honest doctor process, the same action will be eventually taken in process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ .",
"Therefore, the process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and/or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ indeed outputs the prescription proof $\\mathit {PrescProof}^{l}$ on the left-hand side of the first equivalence, i.e., $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ /$\\mathit {P}_{\\mathit {dr}}^{\\prime }\\xrightarrow{}\\xrightarrow{} P\\mid \\mathit {\\lbrace (\\mathit {PrescProof}^{l},{\\tt {r}}^{l}_{\\mathit {dr}})/x\\rbrace }$ .",
"Let $C$ be the left-hand side of the second equivalence, and $D$ be the right-hand side.",
"$\\begin{array}{lrl}\\end{array}C&=&\\mathcal {C}_\\mathit {eh}[\\big ((\\mathit {init}_{\\mathit {dr}}^{\\prime }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {P}_{\\mathit {dr}}^{\\prime }))^{\\backslash {\\sf {out}}({\\tt {chc}}, \\cdot )}\\big )\\mid \\\\&&\\hspace{16.57497pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.(!",
"\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )]\\\\D&=&\\mathcal {C}_\\mathit {eh}[\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )\\mid \\\\&&\\hspace{16.57497pt}\\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.(!",
"\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )]$ $In process $ $, the sub-process $$\\mathit {init}$$\\mathit {dr}$ .",
"(!$\\mathit {P}$$\\mathit {dr}$ {dA/$\\mathit {Id_{\\mathit {dr}}}$ }$\\mathit {P}$$\\mathit {dr}$ ')out(chc, ):=chc.",
"($\\mathit {init}$$\\mathit {dr}$ '.",
"(!$\\mathit {P}$$\\mathit {dr}$ {dA/$\\mathit {Id_{\\mathit {dr}}}$ }$\\mathit {P}$$\\mathit {dr}$ ')!in(chc, y))$, according to the definition of $$\\mathit {P}$out(chc, )$.Since $$\\mathit {init}$$\\mathit {dr}$ '/$\\mathit {P}$$\\mathit {dr}$ '$ may take the action $ x. out(ch, x)$ where $ {($\\mathit {PrescProof}^{l}$ ,rl$\\mathit {dr}$ )/x}$, we have $$\\mathit {init}$$\\mathit {dr}$ '.",
"(!$\\mathit {P}$$\\mathit {dr}$ {dA/$\\mathit {Id_{\\mathit {dr}}}$ }$\\mathit {P}$$\\mathit {dr}$ ') *x. out(ch, x) P'{($\\mathit {PrescProof}^{l}$ ,rl$\\mathit {dr}$ )/x}$.",
"By filling it in the context $chc.",
"(_!in(chc, y))$, we have $chc.",
"(($\\mathit {init}$$\\mathit {dr}$ '.",
"(!$\\mathit {P}$$\\mathit {dr}$ {dA/$\\mathit {Id_{\\mathit {dr}}}$ }$\\mathit {P}$$\\mathit {dr}$ ')!in(chc, y))* x. out(ch, x)chc.",
"((P'{($\\mathit {PrescProof}^{l}$ ,rl$\\mathit {dr}$ )/x})!in(chc, y))$,and thus $ ($\\mathit {init}$$\\mathit {dr}$ '.",
"(!$\\mathit {P}$$\\mathit {dr}$ {dA/$\\mathit {Id_{\\mathit {dr}}}$ }$\\mathit {P}$$\\mathit {dr}$ ')out(chc, )*x. out(ch, x)chc.",
"(P'{($\\mathit {PrescProof}^{l}$ ,rl$\\mathit {dr}$ )/x}!in(chc, y))$.The sub-process $$\\mathit {\\mathit {main}_{\\mathit {dr}}}$$\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace }$ )$ also outputs a signed proof of knowledge from which the adversary obtains $pA$, i.e., $$\\mathit {\\mathit {main}_{\\mathit {dr}}}$$\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace }$ )* x. out(ch, x) C1{($\\mathit {PrescProof}$ ,r$\\mathit {dr}$ )/x1}$ and $pA=first(getmsg(first(x1)))$.Thus, in process $ C$, there are two outputs of a signed proof of knowledge, from which the adversary obtains $pA$.",
"Other signed proofs of knowledge will not lead to $pA$ or $pB$, as the prescription in process $ !",
"$\\mathit {P}$$\\mathit {dr}$ {dA/$\\mathit {Id_{\\mathit {dr}}}$ }$ and $ !",
"$\\mathit {P}$$\\mathit {dr}$ {dB/$\\mathit {Id_{\\mathit {dr}}}$ }$ are freshly generated.$ In process $D$ , the sub-process $\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace }$ outputs a prescription proof from which the adversary knows ${\\tt {p_B}}$ , sub-process $\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace }$ outputs a prescription proof from which the adversary knows ${\\tt {p_A}}$ , the prescriptions from the sub-process $!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }$ and $!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }$ are fresh names and thus cannot be ${\\tt {p_A}}$ or ${\\tt {p_B}}$ .",
"The adversary can detect that the process $C$ and $D$ are not equivalent: in process $C$ , the adversary obtains two ${\\tt {p_A}}$ , and in process $D$ , the adversary obtains one ${\\tt {p_A}}$ and one ${\\tt {p_B}}$ .",
"This contradicts the assumption that $C\\approx _{\\ell }D$ .",
"$\\Box $"
],
[
"Proof of Theorem ",
"Theorem 2 (independency of receipt-freeness).",
"The DLV08 protocol fails to satisfy independency of receipt-freeness.",
"Assume the DLV08 protocol satisfies independency of receipt-freeness.",
"That is, $\\exists $ $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and $P_{\\mathit {dr}}^{\\prime }$ satisfying the following two equivalences in the definition of independency of receipt-freeness (Definition REF ).",
"$\\begin{array}{lrl}1.\\ &&\\mathcal {C}_\\mathit {eh}[!\\mathit {R}_{\\mathit {ph}}^{{\\tt {chc}}}\\mid \\big (\\mathit {init}_{\\mathit {dr}}^{\\prime }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {P}_{\\mathit {dr}}^{\\prime })\\big )\\mid \\\\&&\\hspace{46.75pt} \\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )]\\\\&\\approx _{\\ell }&\\mathcal {C}_\\mathit {eh}[!\\mathit {R}_{\\mathit {ph}}^{{\\tt {chc}}}\\mid \\big ((\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace })^{{\\tt {chc}}}.\\\\&&\\hspace{53.125pt}(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid (\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })^{{\\tt {chc}}})\\big )\\mid \\\\&&\\hspace{46.75pt} \\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )]; \\vspace{5.69054pt} \\text{and}\\\\2.\\ &&\\mathcal {C}_\\mathit {eh}[!\\mathit {R}_{i}^{{\\tt {chc}}}\\mid \\big ((\\mathit {init}_{\\mathit {dr}}^{\\prime }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {P}_{\\mathit {dr}}^{\\prime }))^{\\backslash {\\sf {out}}({\\tt {chc}}, \\cdot )}\\big )\\mid \\\\&&\\hspace{46.75pt} \\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )]\\\\& \\approx _{\\ell }&\\mathcal {C}_\\mathit {eh}[!\\mathit {R}_{i}^{{\\tt {chc}}}\\mid \\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )\\mid \\\\&&\\hspace{46.75pt} \\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )].\\end{array}$ We prove that this assumption leads to contradictions.",
"Similar to the proof of Theorem REF , according to the definition of labelled bisimilarity (Definition REF ) and static equivalence (Definition REF ), Given $A\\approx _{\\ell }B$ , if $A\\xrightarrow{}A^{\\prime }$ and $M =_E N{\\sf {frame}}(A^{\\prime })$ , then $B\\xrightarrow{}B^{\\prime }$ and $M =_E N{\\sf {frame}}(B^{\\prime })$ .",
"Vice versa.",
"Let $A$ be the right-hand side of the first equivalence, $B$ be the left-hand side.",
"$\\begin{array}{lrl}A&=&\\mathcal {C}_\\mathit {eh}[!\\mathit {R}_{\\mathit {ph}}^{{\\tt {chc}}}\\mid \\big ((\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace })^{{\\tt {chc}}}.\\\\&&\\hspace{53.125pt}(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid (\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })^{{\\tt {chc}}})\\big )\\mid \\\\&&\\hspace{46.75pt} \\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )]\\\\B &=&\\mathcal {C}_\\mathit {eh}[!\\mathit {R}_{\\mathit {ph}}^{{\\tt {chc}}}\\mid \\big (\\mathit {init}_{\\mathit {dr}}^{\\prime }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {P}_{\\mathit {dr}}^{\\prime })\\big )\\mid \\\\&&\\hspace{46.75pt} \\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )]\\end{array}$ In process $A$ , the doctor ${\\tt {d_A}}$ computed a signed proof of knowledge ${\\tt {\\mathit {PrescProof}^{\\mathit {r}}}}$ in the sub-process $(\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })^{{\\tt {chc}}}$ .",
"The signed proof of knowledge is sent to a patient over private channel.",
"In addition, the signed proof of knowledge is also sent to the adversary over ${\\tt {chc}}$ together with a nonce (the message sent over ${\\tt {chc}}$ is $({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},{\\tt {r}}^{r}_{\\mathit {dr}})$ ).",
"On receiving the signed proof of knowledge, the patient sends it together with other information to a pharmacist over a private channel.",
"On receiving the message from the patient over private channel, the pharmacist forwards the message to the adversary over ${\\tt {chc}}$ (the message sent over ${\\tt {chc}}$ is $({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},\\mathit {PtSpk^r},\\mathit {vc}_1^r,\\mathit {vc}_2^r,\\mathit {vc}_3^r,\\mathit {vc}^{r^{\\prime }}_3,\\mathit {vc}_4^r,5^r)$ ).",
"Another sub-process $\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace }$ also generates a signed proof of knowledge ${\\tt {\\mathit {PrescProof}^{\\mathit {zr}}}}$ .",
"This signed proof of knowledge is sent to a patient in a message over private channel (but it is not sent to the adversary over ${\\tt {chc}}$ , as this sub-process is not bribed or coerced), and then sent to a pharmacist in another message via private channel.",
"Finally, the pharmacist, who receives the message containing ${\\tt {\\mathit {PrescProof}^{\\mathit {zr}}}}$ , sends the message to the adversary over channel ${\\tt {chc}}$ (the message sent to ${\\tt {chc}}$ is $({\\tt {\\mathit {PrescProof}^{\\mathit {zr}}}},\\mathit {PtSpk^{zr}},\\mathit {vc}_1^{zr},\\linebreak \\mathit {vc}_2^{zr},\\mathit {vc}_3^{zr},\\mathit {vc}^{zr^{\\prime }}_3,\\mathit {vc}_4^{zr},5^{zr})$ ).",
"Formally, there is a trace in process $A$ as follows.",
"$\\begin{array}{rll}A&\\xrightarrow{}\\xrightarrow{} &A_1\\mid \\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},{\\tt {r}}^{r}_{\\mathit {dr}})/x\\rbrace }\\\\&\\xrightarrow{}\\xrightarrow{}& A_2\\mid \\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},{\\tt {r}}^{r}_{\\mathit {dr}})/x\\rbrace }\\mid \\\\&&\\hspace{20.0pt}px\\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},\\mathit {PtSpk^r},\\mathit {vc}_1^r,\\mathit {vc}_2^r,\\mathit {vc}_3^r,\\mathit {vc}^{r^{\\prime }}_3,\\mathit {vc}_4^r,5^r)/y\\rbrace }\\\\&\\xrightarrow{}\\xrightarrow{}& A_3\\mid \\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},{\\tt {r}}^{r}_{\\mathit {dr}})/x\\rbrace }\\mid \\\\&&\\hspace{20.0pt}px\\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},\\mathit {PtSpk^r},\\mathit {vc}_1^r,\\mathit {vc}_2^r,\\mathit {vc}_3^r,\\mathit {vc}^{r^{\\prime }}_3,\\mathit {vc}_4^r,5^r)/y\\rbrace }\\mid \\\\&&\\hspace{20.0pt}px\\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {zr}}}},\\mathit {PtSpk^{zr}},\\mathit {vc}_1^{zr},\\mathit {vc}_2^{zr},\\mathit {vc}_3^{zr},\\mathit {vc}^{zr^{\\prime }}_3,\\mathit {vc}_4^{zr},5^{zr})/z\\rbrace }\\end{array}$ Let $A^{\\prime }=A_3\\mid \\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},{\\tt {r}}^{r}_{\\mathit {dr}})/x\\rbrace }\\mid \\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},\\mathit {PtSpk^r},\\mathit {vc}_1^r,\\mathit {vc}_2^r,\\mathit {vc}_3^r,\\mathit {vc}^{r^{\\prime }}_3,\\mathit {vc}_4^r,5^r)/y\\rbrace }\\mid \\linebreak \\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {zr}}}},\\mathit {PtSpk^{zr}},\\mathit {vc}_1^{zr},\\mathit {vc}_2^{zr},\\mathit {vc}_3^{zr},\\mathit {vc}^{zr^{\\prime }}_3,\\mathit {vc}_4^{zr},5^{zr})/z\\rbrace }$ .",
"We have ${\\tt {p_A}}={\\sf first}({\\sf getmsg}({\\sf first}(x)))={\\sf first}({\\sf getmsg}({\\sf first}(y)))$ and ${\\tt {p_B}}={\\sf first}({\\sf getmsg}({\\sf first}(z)))$ at frame ${\\sf {frame}}(A^{\\prime })$ .",
"Since $A\\approx _{\\ell }B$ , we should have that $\\begin{array}{rll}B&\\xrightarrow{}\\xrightarrow{} &B_1\\mid \\mathit {\\lbrace (\\mathit {PrescProof}^{l},{\\tt {r}}^{l}_{\\mathit {dr}})/x\\rbrace }\\\\&\\xrightarrow{}\\xrightarrow{}& B_2\\mid \\mathit {\\lbrace (\\mathit {PrescProof}^{l},{\\tt {r}}^{l}_{\\mathit {dr}})/x\\rbrace }\\mid \\\\&&\\hspace{20.0pt}px \\mathit {\\lbrace (\\mathit {PrescProof}^{l},\\mathit {PtSpk^l},\\mathit {vc}_1^l,\\mathit {vc}_2^l,\\mathit {vc}_3^l,\\mathit {vc}^{l^{\\prime }}_3,\\mathit {vc}_4^l,5^l)/y\\rbrace }\\mid \\\\&\\xrightarrow{}\\xrightarrow{}& B_3\\mid \\mathit {\\lbrace (\\mathit {PrescProof}^{l},{\\tt {r}}^{l}_{\\mathit {dr}})/x\\rbrace }\\mid \\\\&&\\hspace{20.0pt}px\\mathit {\\lbrace (\\mathit {PrescProof}^{l},\\mathit {PtSpk^l},\\mathit {vc}_1^l,\\mathit {vc}_2^l,\\mathit {vc}_3^l,\\mathit {vc}^{l^{\\prime }}_3,\\mathit {vc}_4^l,5^l)/y\\rbrace }\\mid \\\\&&\\hspace{20.0pt}px\\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {zl}}}},\\mathit {PtSpk^{zl}},\\mathit {vc}_1^{zl},\\mathit {vc}_2^{zl},\\mathit {vc}_3^{zl},\\mathit {vc}^{zl^{\\prime }}_3,\\mathit {vc}_4^{zl},5^{zl})/z\\rbrace }.\\end{array}$ Let $B^{\\prime }=B_3\\mid \\mathit {\\lbrace (\\mathit {PrescProof}^{l},{\\tt {r}}^{l}_{\\mathit {dr}})/x\\rbrace }\\mid \\mathit {\\lbrace (\\mathit {PrescProof}^{l},\\mathit {PtSpk^l},\\mathit {vc}_1^l,\\mathit {vc}_2^l,\\mathit {vc}_3^l,\\mathit {vc}^{l^{\\prime }}_3,\\mathit {vc}_4^l,5^l)/y\\rbrace }\\mid \\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {zl}}}},\\mathit {PtSpk^{zl}},\\mathit {vc}_1^{zl},\\mathit {vc}_2^{zl},\\mathit {vc}_3^{zl},\\mathit {vc}^{zl^{\\prime }}_3,\\mathit {vc}_4^{zl},5^{zl})/z\\rbrace }$ .",
"We should have $A^{\\prime }\\approx _{\\ell }B^{\\prime }$ , and thus, ${\\tt {p_A}}={\\sf first}({\\sf getmsg}({\\sf first}(x)))={\\sf first}({\\sf getmsg}({\\sf first}(y)))$ and ${\\tt {p_B}}={\\sf first}({\\sf getmsg}({\\sf first}(z)))$ at frame ${\\sf {frame}}(B^{\\prime })$ .",
"In process $B$ , the sub-process $\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace }$ generates a signed proof of knowledge $f$ and ${\\tt {p_A}}={\\sf first}({\\sf getmsg}(f))$ .",
"This signed proof of knowledge will be eventually revealed by a pharmacist, $B\\xrightarrow{}\\xrightarrow{}B_4\\mid \\mathit {\\lbrace (f,\\mathit {PtSpk^{f}},\\mathit {vc}_1^{f},\\mathit {vc}_2^{f},\\mathit {vc}_3^{f},\\mathit {vc}^{f^{\\prime }}_3,\\mathit {vc}_4^{f},5^{f})/h\\rbrace }$ and ${\\tt {p_A}}={\\sf first}({\\sf getmsg}({\\sf first}(h)))={\\sf first}({\\sf getmsg}(f))$ .",
"By examining process $B$ , we observe that $y=h$ .",
"The reason is as follows: since ${\\tt {p_A}}={\\sf first}({\\sf getmsg}({\\sf first}(y)))$ , sub-process $!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }$ generate fresh prescriptions and thus cannot be ${\\tt {p_A}}$ , therefore, $!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }$ does not generate a prescription which eventually leads to the action of sending $y$ .",
"Thus the possible sub-process which generates the prescription ${\\tt {p_A}}$ and potentially leads to sending $y$ can only be $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ , $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ or $\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace }$ .",
"Assume $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ generates prescription ${\\tt {p_A}}$ which leads to the action of sending $y$ and $y\\ne h$ , then, the adversary obtains three ${\\tt {p_A}}$ in process $B$ : one from ${\\tt {p_A}}={\\sf first}({\\sf getmsg}({\\sf first}(h)))$ , one from ${\\tt {p_A}}={\\sf first}({\\sf getmsg}({\\sf first}(y)))$ , and one from ${\\tt {p_A}}={\\sf first}({\\sf getmsg}({\\sf first}(x)))$ .",
"However, in process $A$ , the adversary can only observe two ${\\tt {p_A}}$ : one from ${\\tt {p_A}}={\\sf first}({\\sf getmsg}({\\sf first}(x)))$ and one from ${\\tt {p_A}}={\\sf first}({\\sf getmsg}({\\sf first}(y)))$ .",
"This contradicts the assumption that $A\\approx _{\\ell }B$ .",
"Therefore, the prescription ${\\tt {p_A}}$ which leads to the action of revealing $y$ is generated in sub-process $\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace }$ , and thus $y=h$ .",
"In addition, in process $B$ , we observe that the prescription ${\\tt {p_B}}$ is generated in process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ .",
"As sub-process $!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }$ generates fresh prescriptions and thus cannot be ${\\tt {p_B}}$ , and sub-process $\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace }$ generates ${\\tt {p_A}}$ (${\\tt {p_A}}\\ne {\\tt {p_B}}$ ), the only sub-process can generate ${\\tt {p_B}}$ is $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ .",
"The second equivalence also confirms this observation.",
"Let $C$ be the left-hand side of the second equivalence, and $D$ be the right-hand side.",
"$\\begin{array}{lrl}C &=&\\mathcal {C}_\\mathit {eh}[!\\mathit {R}_{i}^{{\\tt {chc}}}\\mid \\big ((\\mathit {init}_{\\mathit {dr}}^{\\prime }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {P}_{\\mathit {dr}}^{\\prime }))^{\\backslash {\\sf {out}}({\\tt {chc}}, \\cdot )}\\big )\\mid \\\\&&\\hspace{46.75pt} \\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )]\\\\D& = &\\mathcal {C}_\\mathit {eh}[!\\mathit {R}_{i}^{{\\tt {chc}}}\\mid \\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_B}}/\\mathit {presc}\\rbrace })\\big )\\mid \\\\&&\\hspace{46.75pt} \\big (\\mathit {init}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }.",
"(!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }\\mid \\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })\\big )].\\end{array}$ In process $D$ , the adversary can obtain one ${\\tt {p_A}}$ and one ${\\tt {p_B}}$ .",
"Since $C\\approx _{\\ell }D$ , in process $D$ , the adversary should also obtain one ${\\tt {p_A}}$ and one ${\\tt {p_B}}$ .",
"Since sub-process $\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace }$ generates ${\\tt {p_A}}$ and sub-process $!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }$ and $!\\mathit {P}_{\\mathit {dr}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}}\\rbrace }$ cannot generates ${\\tt {p_B}}$ , it must be process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ who generates ${\\tt {p_B}}$ .",
"As the generated ${\\tt {p_B}}$ in process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ is first sent to patient, then sent to a pharmacist and thus leads to a message sending over ${\\tt {chc}}$ .",
"The message revealed by the pharmacist is $z$ , because ${\\tt {p_B}}={\\sf first}({\\sf getmsg}({\\sf first}(z)))$ , and on other process can generate ${\\tt {p_B}}$ in process $B$ .",
"By examining process $B$ , the only sub-process which can take the action of sending $x$ is process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ , as process $!\\mathit {R}_{\\mathit {ph}}^{{\\tt {chc}}}$ does not send a message $x$ which is a pair and thus satisfies $x={\\sf pair}({\\sf first}(x), {\\sf snd}(x))$ , and other processes does not involving using channel ${\\tt {chc}}$ .",
"Intuitively, in process $B$ , sub-process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and/or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ sends ${\\tt {p_B}}$ to the patient which leads to the action of sending $z$ ; meanwhile, the sub-process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and/or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ lies to the adversary that the singed proof of knowledge for prescription is $\\mathit {PrescProof}^{l}$ by sending $x$ .",
"In process $A$ , in addition to ${\\tt {\\mathit {PrescProof}^{\\mathit {r}}}}$ , process $(\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_A}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace })^{{\\tt {chc}}}$ also sends other information over channel ${\\tt {chc}}$ .",
"Formally, $\\begin{array}{rll}A&\\xrightarrow{}\\xrightarrow{}&A^1\\mid \\mathit {\\lbrace {\\tt {Pnym}}^{r}_{\\mathit {dr}}/x_1\\rbrace }\\\\&\\xrightarrow{}\\xrightarrow{}&A^2\\mid \\mathit {\\lbrace {\\tt {Pnym}}^{r}_{\\mathit {dr}}/x_1\\rbrace }\\mid \\mathit {\\lbrace {\\tt {r}}^{r}_{\\mathit {dr}}/x_2\\rbrace }\\\\&\\xrightarrow{}\\xrightarrow{}&A^3\\mid \\mathit {\\lbrace {\\tt {Pnym}}^{r}_{\\mathit {dr}}/x_1\\rbrace }\\mid \\mathit {\\lbrace {\\tt {r}}^{r}_{\\mathit {dr}}/x_2\\rbrace }\\mid \\mathit {\\lbrace {\\tt {n}}^{r}_{\\mathit {dr}}/x_3\\rbrace }\\\\&\\xrightarrow{}\\xrightarrow{} & A_1\\mid \\mathit {\\lbrace {\\tt {Pnym}}^{r}_{\\mathit {dr}}/x_1\\rbrace }\\mid \\mathit {\\lbrace {\\tt {r}}^{r}_{\\mathit {dr}}/x_2\\rbrace }\\mid \\mathit {\\lbrace {\\tt {n}}^{r}_{\\mathit {dr}}/x_3\\rbrace }\\mid \\mathit {\\lbrace ({\\tt {\\mathit {PrescProof}^{\\mathit {r}}}},{\\tt {r}}^{r}_{\\mathit {dr}})/x\\rbrace },\\end{array}$ and $x_1$ , $x_2$ , $x_3$ and $x$ satisfy $\\begin{array}{rcl}{\\sf first}(x)&=&{\\sf spk}((x_1,x_2,{\\tt {d_A}},x_3),\\\\&&\\hspace{17.0pt}({\\sf commit}(x_1,x_2),{\\sf drcred}(x_1,{\\tt {d_A}},x_3)),\\\\&&\\hspace{17.0pt}({\\tt {p_A}},{\\sf snd}({\\sf getmsg}({\\sf first}(x))), {\\sf commit}(x_1,x_2),{\\sf fourth}({\\sf getmsg}({\\sf first}(x))))).\\end{array}$ Since $A\\approx _{\\ell }B$ , we should have that $\\begin{array}{rll}B&\\xrightarrow{}\\xrightarrow{}&B^1\\mid \\mathit {\\lbrace {\\tt {Pnym}}^{l}_{\\mathit {dr}}/x_1\\rbrace }\\\\&\\xrightarrow{}\\xrightarrow{}&B^2\\mid \\mathit {\\lbrace {\\tt {Pnym}}^{l}_{\\mathit {dr}}/x_1\\rbrace }\\mid \\mathit {\\lbrace {\\tt {r}}^{l}_{\\mathit {dr}}/x_2\\rbrace }\\\\&\\xrightarrow{}\\xrightarrow{}&B^3\\mid \\mathit {\\lbrace {\\tt {Pnym}}^{l}_{\\mathit {dr}}/x_1\\rbrace }\\mid \\mathit {\\lbrace {\\tt {r}}^{l}_{\\mathit {dr}}/x_2\\rbrace }\\mid \\mathit {\\lbrace {\\tt {n}}^{l}_{\\mathit {dr}}/x_3\\rbrace }\\\\&\\xrightarrow{}\\xrightarrow{} &B_1\\mid \\mathit {\\lbrace {\\tt {Pnym}}^{l}_{\\mathit {dr}}/x_1\\rbrace }\\mid \\mathit {\\lbrace {\\tt {r}}^{l}_{\\mathit {dr}}/x_2\\rbrace }\\mid \\mathit {\\lbrace {\\tt {n}}^{l}_{\\mathit {dr}}/x_3\\rbrace }\\mid \\mathit {\\lbrace (\\mathit {PrescProof}^{l},{\\tt {r}}^{l}_{\\mathit {dr}})/x\\rbrace },\\end{array}$ and the same relation holds between $x_1$ , $x_2$ , $x_3$ and $x$ , $\\begin{array}{rcl}{\\sf first}(x)&=&{\\sf spk}((x_1,x_2,{\\tt {d_A}},x_3),\\\\&&\\hspace{17.0pt}({\\sf commit}(x_1,x_2),{\\sf drcred}(x_1,{\\tt {d_A}},x_3)),\\\\&&\\hspace{17.0pt}({\\tt {p_A}},{\\sf snd}({\\sf getmsg}({\\sf first}(x))), {\\sf commit}(x_1,x_2),{\\sf fourth}({\\sf getmsg}({\\sf first}(x))))).",
"\\end{array}\\qquad \\mathrm {(eq1)}$ The sub-process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ do not know ${\\tt {Pnym}}^{l}_{\\mathit {dr}}, {\\tt {n}}^{l}_{\\mathit {dr}}$ since the two information satisfies secrecy.",
"Thus sub-process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and/or $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ cannot send $x_1$ and $x_3$ over ${\\tt {chc}}$ channel.",
"Even assume the process $\\mathit {init}_{\\mathit {dr}}^{\\prime }$ and $\\mathit {P}_{\\mathit {dr}}^{\\prime }$ know the private information ${\\tt {Pnym}}^{l}_{\\mathit {dr}}, {\\tt {r}}^{l}_{\\mathit {dr}}, {\\tt {n}}^{l}_{\\mathit {dr}}$ for constructing $\\mathit {PrescProof}^{l}$ , since $\\mathit {PrescProof}^{l}$ is actually generated by $\\mathit {\\mathit {main}_{\\mathit {dr}}}\\mathit {\\lbrace {\\tt {d_B}}/{\\mathit {Id_{\\mathit {dr}}}},{\\tt {p_A}}/\\mathit {presc}\\rbrace }$ , we have the following relation: $\\begin{array}{rcl}f=\\mathit {PrescProof}^{l}&=&{\\sf spk}(({\\tt {Pnym}}^{l}_{\\mathit {dr}}, {\\tt {r}}^{l}_{\\mathit {dr}}, {\\tt {d_B}}, {\\tt {n}}^{l}_{\\mathit {dr}}),\\\\&&\\hspace{17.0pt} ({\\sf commit}({\\tt {Pnym}}^{l}_{\\mathit {dr}},{\\tt {r}}^{l}_{\\mathit {dr}}),{\\sf drcred}({\\tt {Pnym}}^{l}_{\\mathit {dr}}, {\\tt {d_B}}, {\\tt {n}}^{l}_{\\mathit {dr}})),\\\\&&\\hspace{17.0pt} ({\\tt {p_A}},\\mathit {PrescriptID}^{l},{\\sf commit}({\\tt {Pnym}}^{l}_{\\mathit {dr}},{\\tt {r}}^{l}_{\\mathit {dr}}),{\\mathit {c\\_Comt^{l}_{\\mathit {pt}}}}).\\end{array}$ that is, $\\begin{array}{rcl}{\\sf first}(x)&=&{\\sf spk}((x_1,x_2,{\\tt {d_B}},x_3),\\\\&&\\hspace{17.0pt}({\\sf commit}(x_1,x_2),{\\sf drcred}(x_1,{\\tt {d_B}},x_3)),\\\\&&\\hspace{17.0pt}({\\tt {p_A}},{\\sf snd}({\\sf getmsg}({\\sf first}(x))), {\\sf commit}(x_1,x_2),{\\sf fourth}({\\sf getmsg}({\\sf first}(x))))).\\end{array}\\qquad \\mathrm {(eq2)}$ and thus, the adversary can detect that $\\mathit {PrescProof}^{l}$ is generated by ${\\tt {d_B}}$ by telling the difference between (eq1) and (eq2).",
"This contradicts the assumption that $A\\approx _{\\ell }B$ .",
"$\\Box $"
]
] | 1808.08403 |
[
[
"Bent Vectorial Functions, Codes and Designs"
],
[
"Abstract Bent functions, or equivalently, Hadamard difference sets in the elementary Abelian group $(\\gf(2^{2m}), +)$, have been employed to construct symmetric and quasi-symmetric designs having the symmetric difference property.",
"The main objective of this paper is to use bent vectorial functions for a construction of a two-parameter family of binary linear codes that do not satisfy the conditions of the Assmus-Mattson theorem, but nevertheless hold $2$-designs.",
"A new coding-theoretic characterization of bent vectorial functions is presented."
],
[
"11em Bent functions, or equivalently, Hadamard difference sets in the elementary Abelian group $({\\mathrm {GF}}(2^{2m}), +)$ , have been employed to construct symmetric and quasi-symmetric designs having the symmetric difference property [14], [8], [15], [11], [12].",
"The main objective of this paper is to use bent vectorial functions for a construction of a two-parameter family of binary linear codes that do not satisfy the conditions of the Assmus-Mattson theorem, but nevertheless hold 2-designs.",
"A new coding-theoretic characterization of bent vectorial functions is presented.",
"Keywords: bent function, bent vectorial function, linear code, 2-design.",
"MSC: 94B05, 94B15, 05B05."
],
[
"Introduction, motivations and objectives",
"We start with a brief review of combinatorial $t$ -designs (cf.",
"[1], [3], [22]).",
"Let ${\\mathcal {P}}$ be a set of $v \\ge 1$ elements, called points, and let ${\\mathcal {B}}$ be a collection of $k$ -subsets of ${\\mathcal {P}}$ , called blocks, where $k$ is a positive integer, $1 \\le k \\le v$ .",
"Let $t$ be a non-negative integer, $t \\le k$ .",
"The pair ${\\mathbb {D}}= ({\\mathcal {P}}, {\\mathcal {B}})$ is called a $t$ -$(v, k, \\lambda )$ design, or simply $t$ -design, if every $t$ -subset of ${\\mathcal {P}}$ is contained in exactly $\\lambda $ blocks of ${\\mathcal {B}}$ .",
"We usually use $b$ to denote the number of blocks in ${\\mathcal {B}}$ .",
"A $t$ -design is called simple if ${\\mathcal {B}}$ does not contain any repeated blocks.",
"In this paper, we consider only simple $t$ -designs.",
"Two designs are isomorphic if there is a bijection between their point sets that maps every block of the first design to a block of the second design.",
"An automorphism of a design is any isomorphism of the design to itself.",
"The set of all automorphisms of a design ${\\mathbb {D}}$ form the (full) automorphism group of ${\\mathbb {D}}$ .",
"It is clear that $t$ -designs with $k = t$ or $k = v$ always exist.",
"Such $t$ -designs are called trivial.",
"In this paper, we consider only $t$ -designs with $v > k > t$ .",
"The incidence matrix of a design ${\\mathbb {D}}$ is a $(0,1)$ -matrix $A=(a_{ij})$ with rows labeled by the blocks, columns labeled by the points, where $a_{i,j}=1$ if the $i$ th block contains the $j$ th point, and $a_{i,j}=0$ otherwise.",
"If the incidence matrix is viewed over ${\\mathrm {GF}}(q)$ , its rows span a linear code of length $v$ over ${\\mathrm {GF}}(q)$ , which is denoted by $q({\\mathbb {D}})$ and is called the code of the design.",
"Note that a $t$ -design can be employed to construct linear codes in different ways.",
"The supports of codewords of a given Hamming weight $k$ in a code $ may form a $ t$-design, which is referred toas a design supported by the code.$ A design is called symmetric if $v = b$ .",
"A 2-$(v, k, \\lambda )$ design is symmetric if and only if every two blocks share exactly $\\lambda $ points.",
"A 2-design is quasi-symmetric with intersection numbers $x$ and $y$ , ($x < y$ ) if any two blocks intersect in either $x$ or $y$ points.",
"Let ${\\mathbb {D}}=\\lbrace {\\mathcal {P}}, \\,{\\mathcal {B}}\\rbrace $ be a 2-$(v, k, \\lambda )$ symmetric design, where ${\\mathcal {B}}=\\lbrace B_1,\\, B_2,\\, \\cdots , \\, B_v\\rbrace $ and $v \\ge 2$ .",
"Then $(B_1, \\, \\lbrace B_2 \\cap B_1,\\, B_3 \\cap B_1,\\, \\cdots ,\\, B_v \\cap B_1 \\rbrace )$ is a 2-$(k,\\, \\lambda ,\\, \\lambda -1)$ design, and called the derived design of ${\\mathbb {D}}$ with respect to $B_1$ ; $(\\overline{B}_1,\\, \\lbrace B_2 \\cap \\overline{B}_1,\\, B_3 \\cap \\overline{B}_1,\\, \\cdots , B_v \\cap \\overline{B}_1 \\rbrace )$ is a 2-$(v-k,\\, k-\\lambda ,\\, \\lambda )$ design, called the residual design of ${\\mathbb {D}}$ with respect to $B_1$ , where $\\overline{B_1}={\\mathcal {P}}\\setminus B_1$ .",
"If a symmetric design ${\\mathbb {D}}$ has parameters $2-(2^{2m},\\, 2^{2m-1}-2^{m-1},\\, 2^{2m-2}-2^{m-1}),$ its derived designs have parameters $2-( 2^{2m-1}-2^{m-1},\\, 2^{2m-2}-2^{m-1},\\, 2^{2m-2}-2^{m-1}-1),$ and its residual designs have parameters $2-( 2^{2m-1}+2^{m-1},\\, 2^{2m-2},\\, 2^{2m-2}-2^{m-1}).$ A symmetric 2-design is said to have the symmetric difference property, or to be a symmetric SDP design, (Kantor [14], [15]), if the symmetric difference of any three blocks is either a block or the complement of a block.",
"Any derived or residual design of a symmetric SDP design is quasi-symmetric, and has the property that the symmetric difference of every two blocks is either a block or the complement of a block.",
"The derived and residual designs of a symmetric SDP design are called quasi-symmetric SDP designs [12].",
"The binary codes of quasi-symmetric SDP designs give rise to an exponentially growing number of inequivalent linear codes that meet the Grey-Rankin bound [11].",
"It was proved in [21] that any quasi-symmetric SDP design can be embedded as a derived or a residual design in exactly one (up to isomorphism) symmetric SDP design.",
"A coding-theoretical characterization of symmetric SDP designs was given by Dillon and Schatz [8], who proved that any symmetric SDP design with parameters (REF ) is supported by the codewords of minimum weight in a binary linear code $ of length $ 22m$, dimension$ 2m+2$ and weight enumerator given by\\begin{equation}1 + 2^{2m}z^{2^{2m-1}-2^{m-1}} + (2^{2m+1} - 2)z^{2^{2m-1}}+ 2^{2m}z^{2^{2m-1}+2^{m-1}} + z^{2m},\\end{equation}where $ is spanned by the first order Reed-Muller code ${\\mathrm {RM}}_2(1, 2m)$ and a vector $u$ being the truth table (introduced in Section ) of a bent function in $2m$ variables, or equivalently, $u$ is the incidence vector of a Hadamard difference set in the additive group of ${\\mathrm {GF}}(2)^{2m}$ with parameters $(2^{2m}, \\,2^{2m-1} \\pm 2^{m-1}, \\,2^{2m-2} \\pm 2^{m-1}).$ One of the objectives of this paper is to give a coding-theoretical characterization of bent vectorial functions (Theorem REF ), which generalizes the Dillon and Schatz characterization of single bent functions [8].",
"Another objective is to present in Theorem REF a two-parameter family of binary linear codes with parameters $ [2^{2m},2m+1 +\\ell ,2^{2m-1} - 2^{m-1}], \\ m \\ge 2, \\ 1\\le \\ell \\le m, $ that are based on bent vectorial functions and support 2-designs, despite that these codes do not satisfy the conditions of the Assmus-Mattson theorem (see Theorem REF ).",
"The subclass of codes with $\\ell =1$ consists of codes introduced by Dillon and Schatz [8] that are based on bent functions and support symmetric SDP designs.",
"Examples of codes with $\\ell =m$ are given that are optimal in the sence that they have the maximum possible minimum distance for the given length and dimension, or have the largest known minimum distance for the given length and dimension (see Note REF in Section , and the examples thereafter)."
],
[
"The classical constructions of $t$ -designs from codes",
"A simple sufficient condition for the supports of codewords of any given weight in a linear code to support a $t$ -design is that the code admits a $t$ -transitive or $t$ -homogeneous automorphism group.",
"All codes considered in this paper are of even length $n$ of the form $n=2^{2m}$ .",
"It is known that any 2-homogeneous group of even degree is necessarily 2-transitive (Kantor [13], [16]).",
"Another sufficient condition is given by the Assmus-Mattson theorem.",
"Let $ be a $ [v, , d]$ linear code over $ GF(q)$, and let $ Ai =Ai($be thenumber of codewords of Hamming weight $ i$ in $ ($0 \\le i \\le v$ ).",
"For each $k$ with $A_k \\ne 0$ , let ${\\mathcal {B}}_k$ denote the set of the supports of all codewords of Hamming weight $k$ in $,where the code coordinatesare indexed by $ 1,2, ..., v$.",
"Let $ P={ 1, 2, ..., v }$.The followingtheorem, proved by Assumus and Mattson, provides sufficient conditionsfor the pair $ (P, Bk)$ to be a $ t$-design.$ Theorem 1 (The Assmus-Mattson Theorem [2]) Let $ be a binary $ [v, , d]$ code, and let $ d$ be the minimumweight of the dual code $$.Suppose that $ Ai=Ai($ and $ Ai=Ai()$, $ 0 i v$, are theweight distributions of $ and $\\perp $ , respectively.",
"Fix a positive integer $t$ with $t < d$ , and let $s$ be the number of $i$ with $A_i^\\perp \\ne 0$ for $0 < i \\le v-t$ .",
"If $s \\le d -t$ , then the codewords of weight $i$ in $ hold a $ t$-design provided that $ Ai 0$ and$ d i v$, and\\item the codewords of weight $ i$ in the code $$ hold a $ t$-design provided that$ Ai0$ and $ di v-t$.$ The parameter $\\lambda $ of a $t$ -$(v,w,\\lambda )$ design supported by the codewords of weight $w$ in a binary code $ is determined by\\begin{equation*}A_w = \\lambda { v \\atopwithdelims ()t}/{w \\atopwithdelims ()t}.\\end{equation*}$ Bent functions and bent vectorial functions Let $f=f(x)$ be a Boolean function from ${\\mathrm {GF}}(2^{n})$ to ${\\mathrm {GF}}(2)$ .",
"The support $S_f$ of $f$ is defined as $S_f=\\lbrace x \\in {\\mathrm {GF}}(2^{n}) : f(x)=1\\rbrace \\subseteq {\\mathrm {GF}}(2^{n}).$ The $(0,1)$ incidence vector of $S_f$ , having its coordinates labeled by the elements of ${\\mathrm {GF}}(2^n)$ , is called the truth table of $f$ .",
"The Walsh transform of $f$ is defined by $\\hat{f}(w)=\\sum _{x \\in {\\mathrm {GF}}(2^{n})} (-1)^{f(x)+{\\mathrm {Tr}}_{n/1}(wx)}$ where $w \\in {\\mathrm {GF}}(2^{n})$ and ${\\mathrm {Tr}}_{n/n^{\\prime }}(x)$ denotes the trace function from ${\\mathrm {GF}}(2^n)$ to ${\\mathrm {GF}}(2^{n^{\\prime }})$ .",
"Two Boolean functions $f$ and $g$ from ${\\mathrm {GF}}(2^n)$ to ${\\mathrm {GF}}(2)$ are called weakly affinely equivalent or EA-equivalent if there are an automorphism $A$ of $({\\mathrm {GF}}(2^n), +)$ , a homomorphism $L$ from $({\\mathrm {GF}}(2^n),+)$ to $({\\mathrm {GF}}(2), +)$ , an element $a \\in {\\mathrm {GF}}(2^n)$ and an element $b \\in {\\mathrm {GF}}(2)$ such that $g(x)=f(A(x)+a)+ L(x) +b$ for all $x \\in {\\mathrm {GF}}(2^n)$ .",
"A Boolean function $f$ from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2)$ is called a bent function if $|\\hat{f}(w)|=2^{m}$ for every $w \\in {\\mathrm {GF}}(2^{2m})$ .",
"It is well known that a function $f$ from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2)$ is bent if and only if $S_f$ is a difference set in $({\\mathrm {GF}}(2^{2m}),\\,+)$ with parameters (REF ) [19].",
"A Boolean function $f$ from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2)$ is a bent function if and only if its truth table is at Hamming distance $2^{2m-1} \\pm 2^{m-1}$ from every codeword of the first order Read-Muller code ${\\mathrm {RM}}_2(1, 2m)$ [18].",
"It follows that $|S_f|=2^{2m-1} \\pm 2^{m-1}.$ There are many constructions of bent functions.",
"The reader is referred to [6] and [19] for detailed information about bent functions.",
"Let $\\ell $ be a positive integer, and let $f_1(x), \\cdots , f_\\ell (x)$ be Boolean functions from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2)$ .",
"The function $F(x)=(f_1(x), \\cdots , f_\\ell (x))$ from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2)^\\ell $ is called a $(2m,\\ell )$ vectorial Boolean function.",
"A $(2m,\\ell )$ vectorial Boolean function $F(x)=(f_1(x), \\cdots , f_\\ell (x))$ is called a bent vectorial function if $\\sum _{j=1}^\\ell a_j f_j(x)$ is a bent function for each nonzero $(a_1, \\cdots , a_\\ell )\\in {\\mathrm {GF}}(2)^\\ell $ .",
"For another equivalent definition of bent vectorial functions, see [7] or [19].",
"Bent vectorial functions exist only when $\\ell \\le m$ (cf.",
"[19]).",
"There are a number of known constructions of bent vectorial functions.",
"The reader is referred to [7] and [19] for detailed information.",
"Below we present a specific construction of bent vectorial functions from [7].",
"Example 2 [7].",
"Let $m \\ge 1$ be an odd integer, $\\beta _1, \\beta _2, \\cdots , \\beta _{m}$ be a basis of ${\\mathrm {GF}}(2^{m})$ over ${\\mathrm {GF}}(2)$ , and let $u \\in {\\mathrm {GF}}(2^{2m}) \\setminus {\\mathrm {GF}}(2^m)$ .",
"Let $i$ be a positive integer with $\\gcd (2m, i)=1$ .",
"Then $\\left({\\mathrm {Tr}}_{2m/1}(\\beta _1 u x^{2^i+1}), {\\mathrm {Tr}}_{2m/1}(\\beta _2 u x^{2^i+1}), \\cdots ,{\\mathrm {Tr}}_{2m/1}(\\beta _{m} u x^{2^i+1}) \\right)$ is a $(2m, m)$ bent vectorial function.",
"Under a basis of ${\\mathrm {GF}}(2^{\\ell })$ over ${\\mathrm {GF}}(2)$ , $({\\mathrm {GF}}(2^\\ell ), +)$ and $({\\mathrm {GF}}(2)^\\ell , +)$ are isomorphic.",
"Hence, any vectorial function $F(x)=(f_1(x), \\cdots , f_\\ell (x))$ from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2)^\\ell $ can be viewed as a function from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2^\\ell )$ .",
"It is well known that a function $F$ from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2^\\ell )$ is bent if and only if ${\\mathrm {Tr}}_{\\ell /1}(aF(x))$ is a bent Boolean function for all $a \\in {\\mathrm {GF}}(2^\\ell )^*$ .",
"Any such vectorial function $F$ can be expressed as ${\\mathrm {Tr}}_{2m/\\ell }(f(x))$ , where $f$ is a univariate polynomial.",
"This presentation of bent vectorial functions is more compact.",
"We give two examples of bent vectorial functions in this form.",
"Example 3 (cf.",
"[19]).",
"Let $m>1$ and $i \\ge 1$ be integers such that $2m/\\gcd (i, 2m)$ is even.",
"Then ${\\mathrm {Tr}}_{2m/m}(a x^{2^i+1})$ is bent if and only if $\\gcd (2^i+1, 2^m+1) \\ne 1$ and $a \\in {\\mathrm {GF}}(2^{2m})^* \\setminus \\langle \\alpha ^{\\gcd (2^i+1, 2^m+1)} \\rangle $ , where $\\alpha $ is a generator of ${\\mathrm {GF}}(2^{2m})^*$ .",
"Example 4 (cf.",
"[19]).",
"Let $m>1$ and $i \\ge 1$ be integers such that $\\gcd (i, 2m)=1$ .",
"Let $d=2^{2i}-2^i+1$ .",
"Let $m$ be odd.",
"Then ${\\mathrm {Tr}}_{2m/m}(a x^{d})$ is bent if and only if $a \\in {\\mathrm {GF}}(2^{2m})^* \\setminus \\langle \\alpha ^{3} \\rangle $ , where $\\alpha $ is a generator of ${\\mathrm {GF}}(2^{2m})^*$ .",
"A construction of codes from bent vectorial functions Let $q=2^{2m}$ , let ${\\mathrm {GF}}(q)=\\lbrace u_1, u_2, \\cdots , u_{q}\\rbrace $ , and let $w$ be a generator of ${\\mathrm {GF}}(q)^*$ .",
"For the purposes of what follows, it is convenient to use the following generator matrix of the binary $[2^{2m}, 2m+1,2^{2m-1}]$ first-order Reed-Muller code ${\\mathrm {RM}}_2(1,2m)$ : $G_0=\\left[\\begin{array}{cccc}1 & 1 & \\cdots & 1 \\\\{\\mathrm {Tr}}_{2m/1}(w^0u_1) & {\\mathrm {Tr}}_{2m/1}(w^0u_2) & \\cdots & {\\mathrm {Tr}}_{2m/1}(w^0u_q) \\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\{\\mathrm {Tr}}_{2m/1}(w^{2m-1}u_1) & {\\mathrm {Tr}}_{2m/1}(w^{2m-1}u_2) & \\cdots & {\\mathrm {Tr}}_{2m/1}(w^{2m-1}u_q)\\end{array}\\right].$ The weight enumerator of ${\\mathrm {RM}}_2(1,2m)$ is $1+(2^{2m+1}-2)z^{2^{2m-1}} + z^{2^{2m}}.$ Two binary linear codes are equivalent if there is a permutation of coordinates that sends the first code to the second.",
"Up to equivalence, ${\\mathrm {RM}}_2(1,2m)$ is the unique linear binary code with parameters $[2^{2m}, 2m+1,2^{2m-1}]$ [8].",
"Its dual code is the $[2^{2m}, 2^{2m}- 1 -2m,4]$ Reed-Muller code of order $2m-2$ .",
"Both codes hold 3-designs since they are invariant under a 3-transitive affine group.",
"Note that ${\\mathrm {RM}}_2(1,2m)^\\perp $ is the unique, up to equivalence, binary linear code for the given parameters, hence it is equivalent to the extended binary linear Hamming code.",
"Let $F(x)=(f_1(x), f_2(x), \\cdots , f_\\ell (x))$ be a $(2m, \\ell )$ vectorial function from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2)^\\ell $ .",
"For each $i$ , $1 \\le i \\le \\ell $ , we define a binary vector $F_i=(f_i(u_1), f_i(u_2), \\cdots , f_i(u_q)) \\in {\\mathrm {GF}}(2)^{2^{2m}},$ which is the truth table of the Boolean function $f_i(x)$ introduced in Section .",
"Let $\\ell $ be an integer in the range $1 \\le \\ell \\le m$ .",
"We now define a $(2m+1 + \\ell ) \\times 2^{2m}$ matrix $G=G(f_{1}, \\cdots , f_{\\ell })=\\left[\\begin{array}{c}G_0 \\\\F_{1} \\\\\\vdots \\\\F_{\\ell }\\end{array}\\right],$ where $G_0$ is the generator matrix of ${\\mathrm {RM}}_2(1,2m)$ .",
"Let $f_{1}, \\cdots , f_{\\ell })$ denote the binary code of length $2^{2m}$ with generator matrix $G(f_{1}, \\cdots , f_{\\ell })$ given by (REF ).",
"The dimension of the code has the following lower and upper bounds: $2m+1 \\le \\dim (f_{1}, \\cdots , f_{\\ell })) \\le 2m+1+\\ell .$ The following theorem gives a coding-theoretical characterization of bent vectorial functions.",
"Theorem 5 A $(2m, \\ell )$ vectorial function $F(x)=(f_1(x), f_2(x), \\cdots , f_\\ell (x))$ from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2)^\\ell $ is a bent vectorial function if and only if the code $f_1, \\cdots , f_\\ell )$ with generator matrix $G$ given by (REF ) has weight enumerator $1 + (2^\\ell -1)2^{2m} z^{2^{2m-1} - 2^{m-1}} + 2(2^{2m}-1)z^{2^{2m-1}}+ (2^\\ell -1)2^{2m} z^{2^{2m-1} + 2^{m-1}} + z^{2^{2m}}.$ By the definition of $G$ , the code $f_1, \\cdots , f_\\ell )$ contains the first-order Reed-Muller code ${\\mathrm {RM}}_2(1, 2m)$ as a subcode, having weight enumerator (REF ).",
"It follows from (REF ) that every codeword of $f_1, \\cdots , f_\\ell )$ must be the truth table of a Boolean function of the form $f_{(u, v, h)}(x)=\\sum _{i=1}^\\ell u_i f_i(x) + \\sum _{j=0}^{2m-1} v_j {\\mathrm {Tr}}_{2m/1}(w^jx) + h,$ where $u_i, v_j, h \\in {\\mathrm {GF}}(2)$ , $x \\in {\\mathrm {GF}}(2^{2m})$ .",
"Suppose that $F(x)=(f_1(x), f_2(x), \\cdots , f_\\ell (x))$ is a $(2m,\\ell )$ bent vectorial function.",
"When $(u_1, \\cdots , u_\\ell )=(0, \\cdots , 0)$ , $(v_0, v_1, \\cdots , v_{2m-1})$ runs over ${\\mathrm {GF}}(2)^{2m}$ and $h$ runs over ${\\mathrm {GF}}(2)$ , the truth tables of the functions $f_{(u, v, h)}(x)$ form the code ${\\mathrm {RM}}_2(1, 2m)$ .",
"Whenever $(u_1, \\cdots , u_\\ell ) \\ne (0, \\cdots , 0)$ , it follows from (REF ) that $f_{(u, v, h)}(x)$ is a bent function, and the corresponding codeword has Hamming weight $2^{2m-1} \\pm 2^{m-1}$ .",
"Since the all-one vector belongs to ${\\mathrm {RM}}_2(1, 2m)$ , the code $f_1, \\cdots , f_\\ell )$ is self-complementary, and the desired weight enumerator of $f_1, \\cdots , f_\\ell )$ follows.",
"Suppose that $f_1, \\cdots , f_\\ell )$ has weight enumerator given by (REF ).",
"Then $f_1, \\cdots , f_\\ell )$ has dimension $2m+1+\\ell $ .",
"Consequently, $\\sum _{i=1}^\\ell u_i f_i(x)$ is the zero function if and only if $(u_1, \\cdots , u_\\ell )=(0, \\cdots , 0)$ .",
"It then follows that the codewords corresponding to $f_{(u, v, h)}(x)$ must have Hamming weight $2^{2m-1} \\pm 2^{m-1}$ for all $u=(u_1, \\cdots , u_\\ell ) \\ne (0, \\cdots , 0)$ and all $(v_0, v_1, \\cdots , v_{2m-1}) \\in {\\mathrm {GF}}(2)^{2m}$ .",
"Notice that $\\sum _{j=0}^{2m-1} v_j {\\mathrm {Tr}}_{2m/1}(w^jx)$ ranges over all linear functions from ${\\mathrm {GF}}(2^m)$ to ${\\mathrm {GF}}(2)$ when $(v_0, v_1, \\cdots , v_{2m-1})$ runs over ${\\mathrm {GF}}(2)^{2m}$ .",
"Consequently, $F(x)$ is a bent vectorial function.",
"Note 6 Let $F(x)=(f_1(x), f_2(x), \\cdots , f_m(x))$ be a bent vectorial function from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2)^m$ .",
"Then the code $f_1, \\cdots , f_m)$ has parameters $ [2^{2m}, 3m+1, 2^{2m-1} - 2^{m-1}].",
"$ In particular, if $m=2$ , any code $f_1, f_2)$ based on a bent vectorial function from ${\\mathrm {GF}}(2^{4})$ to ${\\mathrm {GF}}(2)^2$ has parameters $[16, 7, 6]$ and is optimal (cf.",
"[10]).",
"An $[n,k,d]$ code is optimal if $d$ is the maximum possible minimum distance for the given $n$ and $k$ .",
"If $m=3$ , any code $f_1, f_2, f_3)$ based on a bent vectorial function from ${\\mathrm {GF}}(2^{6})$ to ${\\mathrm {GF}}(2)^3$ has parameters $[64, 10, 28]$ and is optimal [10].",
"If $m=4$ , any code $f_1, \\cdots , f_6)$ based on a bent vectorial function from ${\\mathrm {GF}}(2^{8})$ to ${\\mathrm {GF}}(2)^4$ has parameters $[256, 13, 120]$ and has the largest known minimum distance for the given code length and dimension [10].",
"Theorem 7 Up to equivalence, there is exactly one $[16,7,6]$ code that can be obtained from a $(4,2)$ bent vectorial function.",
"The weight enumerator of the second order Reed-Muller code ${\\mathrm {RM}}_{2}(2,4)$ is given by $1+140z^4 + 448z^6 + 870z^8 + 448z^{10} + 140z^{12} + z^{16}.$ The truth table of a bent function $f$ from ${\\mathrm {GF}}(2^4)$ to ${\\mathrm {GF}}(2)$ is a codeword $c_f$ of ${\\mathrm {RM}}_{2}(2,4)$ of weight 6.",
"The linear code $f)$ spanned by $c_f$ and ${\\mathrm {RM}}_{2}(1,4)$ is a subcode of ${\\mathrm {RM}}_{2}(2,4)$ of dimension 6, having weight enumerator $1 + 16z^6 + 30z^8 + 16z^{10} + z^{16}.$ The codewords of $f)$ of weight 6 form a symmetric 2-$(16,6,2)$ SDP design, whose blocks correspond to the supports of 16 bent functions.",
"Now, let $(f_1,f_2)$ be a $(4,2)$ bent vectorial function.",
"Then, the intersection of the codes $f_1)$ , $f_2)$ consists of the first order Reed-Muller code ${\\mathrm {RM}}_{2}(1,4)$ .",
"It follows that the set of 448 codewords of weight 6 in ${\\mathrm {RM}}_{2}(2,4)$ is a union $\\cal {U}$ of 28 pairwise disjoint subsets of size 16, corresponding to the incidence matrices of symmetric 2-$(16,6,2)$ SDP designs associated with 28 different $[16,6]$ codes defined by single bent functions.",
"If $f_1,f_2)$ is a $[16,7]$ code defined by a bent vectorial function $(f_1,f_2)$ , its weight enumerator is given by $1 + 48z^6 + 30z^8 + 48z^{10} +z^{16}.$ The set of 48 codewords of weight 6 of $f_1,f_2)$ is a union of the incidence matrices of three SDP designs from $\\cal {U}$ with pairwise disjoint sets of blocks.",
"A quick check shows that there are exactly 56 such collections of 48 codewords that generate a code having weight enumerator (REF ).",
"Therefore, the number of distinct $[16,7,6]$ subcodes of ${\\mathrm {RM}}_{2}(1,4)$ based on $(4,2)$ bent vectorial functions is 56.",
"The $7 \\times 16$ generator matrix $G$ of one such $[16,7,6]$ code is listed below: $\\left[\\begin{array}{cccccccccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 1& 1& 1& 1& 1& 1& 1& 1 \\\\0& 0& 0& 0& 1& 1& 1& 1& 0& 0& 0& 0& 1& 1& 1& 1 \\\\0& 0& 1& 1& 0& 0& 1& 1& 0& 0& 1& 1& 0& 0& 1& 1 \\\\0& 1& 0& 1& 0& 1& 0& 1& 0& 1& 0& 1& 0& 1& 0& 1 \\\\1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1 \\\\0& 0& 0& 1& 0& 1& 1& 1& 0& 1& 0& 0& 0& 0& 1& 0 \\\\0& 0& 0& 0& 0& 1& 0& 1& 0& 0& 1& 1& 0& 1& 1& 0\\end{array}\\right].$ The first five rows of $G$ form a generator matrix of ${\\mathrm {RM}}_{2}(1,4)$ , while the last two rows are codewords of weight 6 in ${\\mathrm {RM}}_{2}(2,4)$ .",
"The full automorphism group of the $[16,7,6]$ code generated by $G$ is of order 5760.",
"Since the order of the automorphism group of ${\\mathrm {RM}}_{2}(1,4)$ is 322560, and $ 322560/5760 = 56, $ it follows that all 56 $[16,7,6]$ codes based on $(4,2)$ bent vectorial functions are pairwise equivalent.",
"The next two examples illustrate that there are at least three inequivalent optimal $[64,10,28]$ codes that are obtainable from bent vectorial functions from ${\\mathrm {GF}}(2^{6})$ to ${\\mathrm {GF}}(2)^3$ .",
"The parameters $[64,10,28]$ correspond to $m=3$ in Note REF .",
"Example 8 The binary cyclic $[63,10]$ code $ with parity check polynomial$ h(x)=(x+1)(x3 + x2 +1)(x6 + x5 + x4 + x + 1)$has weight enumerator$$1 + 196z^{27} + 252z^{28} + 63z^{31} +63z^{32}+252z^{35} +196z^{36} +z^{63}.$$The $ [63,7]$ subcode $$ of $ having check polynomial $h^{\\prime }(x)=(x+1)(x^6 + x^5 + x^4 + x + 1)$ has weight enumerator $1 + 63z^{31} +63z^{32}+ z^{63}.$ The extended $[64,7]$ code $()^*$ of $$ has weight enumerator $ 1 + 126z^{32} + z^{64}, $ hence, $()^*$ is equivalent to the first order Reed-Muller code ${\\mathrm {RM}}_{2}(1,6)$ .",
"The extended $[64,10]$ code $*$ of $ has weight enumeratorgiven by\\begin{equation}1 + 448z^{28} + 126z^{32} + 448 z^{36} + z^{64}.\\end{equation}Since $ *$ contains a copy of the first order Reed-Muller code $ RM2(1,6)$ as a subcode, it follows from Theorem \\ref {thm-bentvectf}that $ *$ can be obtained from a $ (6,3)$ bent vectorial functionfrom $ GF(26)$ to $ GF(23)$.The full automorphism group of $ *$ is of order$$677,376 = 2^9 \\cdot 3^3 \\cdot 7^2.",
"$$Magma was used for these computations.$ Example 9 Let $M$ be the 7 by 64 $(0,1)$ -matrix with the following structure: the $i$ th column of the 6 by 64 submatrix $M^{\\prime }$ of $M$ consisting of its first six rows is the binary presentation of the number $i$ ($i =0, 1, \\ldots 63$ ), while the last row of $M$ is the all-one row.",
"Clearly, $M$ is a generator matrix of a binary linear $[64,7]$ code equivalent to the first order Reed-Muller code ${\\mathrm {RM}}_{2}(1,6)$ .",
"The first six rows of $M$ can be viewed as the truth tables of the single Boolean variables $x_1, x_2, \\dots x_6$ , while the seventh row of $M$ is the truth table of the constant ${\\bf 1}$ .",
"We consider the Boolean bent functions given by $f_{1}(x_1,\\ldots , x_6) & = & x_{1}x_6 + x_{2}x_{5} + x_{3}x_4, \\\\f_{2}(x_1,\\ldots , x_6) & = & x_{1}x_5 + x_{2}x_4 + x_{3}x_5 + x_{3}x_6,\\\\f_{3}(x_1,\\ldots , x_6) & = & x_{1}x_4 + x_{2}x_5 + x_{2}x_6 +x_{3}x_4 + x_{3}x_5 + x_{5}x_{6},\\\\f_{4}(x_1,\\ldots , x_6) & = & x_{1}x_4 +x_{2}x_3 + x_{3}x_6 + x_{5}x_6.$ The vectorial functions $F_{1}=(f_{1}, f_{2}, f_{3})$ , $F_{2}=(f_{1}, f_{2}, f_{4})$ give via Theorem REF binary linear codes $1, \\ 2$ with parameters $[64,10,28]$ , having weight enumerator given by ().",
"The automorphism groups of the codes $1, \\ 2$ were computed using the computer-algebra package Magma [5].",
"The code 1 has full automorphism group of order $ 10,752 = 2^{9}\\cdot 3\\cdot 7.",
"$ The code 2 has full automorphism group of order $ 4,032 = 2^{6}\\cdot 3^{2}\\cdot 7.",
"$ Thus, 1, 2 and the extended cyclic code $*$ from Example REF are pairwise inequivalent.",
"We note that the code 1 cannot be equivalent to any extended cyclic code because its group order is not divisible by 63.",
"Note 10 The full automorphism group of 1 from Example REF cannot be 2-transitive because its order is not divisible by 63.",
"Thus, the code 1 does not satisfy the classical sufficient condition to support 2-designs based on the 2-transitivity of its automorphism group (recall that according to [13], any 2-homogeneous group of degree 64 is necessarily 2-transitive).",
"In addition, the minimum distance of its dual code ${1}^\\perp $ is 4, thus the Assmus-Mattson theorem guarantees only 1-designs to be supported by 1.",
"We will prove in the next section that all codes obtained from bent vectorial functions support 2-designs.",
"A construction of 2-designs from bent vectorial functions The following theorem establishes that the binary codes based on bent vectorial functions support 2-designs, despite that these codes do not meet the conditions of the Assmus-Mattson theorem for 2-designs.",
"Theorem 11 Let $F(x)=(f_1(x), f_2(x), \\cdots , f_\\ell (x))$ be a bent vectorial function from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2)^\\ell $ , where $m\\ge 2$ and $1\\le \\ell \\le m$ .",
"Let $ f_1, \\cdots , f_\\ell )$ be the binary linear code with parameters $[2^{2m}, 2m+1 + \\ell , 2^{2m-1} - 2^{m-1}]$ defined in Theorem REF .",
"(a) The codewords of $ of minimum weight hold a 2-design $ D$with parameters\\begin{equation}2-(2^{2m}, 2^{2m-1} - 2^{m-1}, (2^\\ell -1)(2^{2m-2} - 2^{m-1})).\\end{equation}(b) The codewords of $ of weight $2^{2m-1} + 2^{m-1}$ hold a 2-design $\\overline{{\\mathbb {D}}}$ with parameters $2-(2^{2m}, 2^{2m-1} + 2^{m-1}, (2^\\ell -1)(2^{2m-2} + 2^{m-1})).$ Since $ contains $ RM2(1,2m)$, and the minimum distanceof $ RM2(1,2m)$ is 4,the minimum distance $ d$ of $$ is at least 4.",
"Applying theMacWilliams transform (see, for example \\cite [p. 41]{vanLint})to the weight enumerator (\\ref {eqn-wtenumerator111}) of $ shows that $d^{\\perp } =4$ .",
"It follows from the Assmus-Mattson theorem (Theorem REF ) that the codewords of any given nonzero weight $w< 2^{2m}$ in $ hold a 1-design.$ However, we will prove that $ actually holds 2-designs, despite thatthe Assmus-Mattson theorem guarantees only1-designs to be supported by $ .",
"Since the subcode ${\\mathrm {RM}}_2(1,2m)$ of $ containsall codewords of $ of weight $2^{2m-1}$ , the codewords of this weight hold a 3-design $\\cal {A}$ with parameters 3-$(2^{2m}, 2^{2m-1}, 2^{2m-2} -1)$ .",
"We note that $\\cal {A}$ is a 2-design with $\\lambda _2 = \\frac{2^{2m}-2}{2^{2m-1} -2}\\cdot (2^{2m-2} -1)=2^{2m-1}-1.$ Let ${\\mathbb {D}}$ be the 1-design supported by codewords of weight $2^{2m-1} - 2^{m-1}$ .",
"Since the number of codewords of weight $2^{2m-1} - 2^{m-1}$ is equal to $(2^\\ell -1)2^{2m}$ , ${\\mathbb {D}}$ is a 1-design with parameters 1-$(2^{2m}, 2^{2m-1} - 2^{m-1}, (2^\\ell -1)(2^{2m-1} - 2^{m-1}))$ .",
"Every codeword of $ of weight $ 22m-1 + 2m-1$ is the sum ofa codeword of weight $ 22m-1 - 2m-1$ and the all-one vector.Thus, the codewords of weight $ 22m-1 + 2m-1$ hold a 1-design $D$having parameters1-$ (22m, 22m-1 + 2m-1, (2-1)(22m-1 + 2m-1))$.Clearly, $D$is the complementary design of $ D$, that is, every block of $D$is the complement of some block of $ D$.$ Let $M$ be the $2^{2m+1+\\ell } \\times 2^{2m}$ $(0,1)$ -matrix having as rows the codewords of $.",
"Since $ d =4$, $ M$ is an orthogonal array of strength 3,that is, for every integer $ i$, $ 1 i 3$, and for every set of $ i$ distinct columnsof $ M$, every binary vector with $ i$ components appears exactly$ 22m+1+- i$ times among the rows of the $ 22m+1+ i$ submatrix of $ M$formed by the chosen $ i$ columns.In particular, any $ 22m+1+ 2$ submatrix consisting of two distinctcolumns of $ M$ contains the binary vector $ (1,1)$ exactly $ 22m+-1$ timesas a row.",
"Among these $ 22m+-1$ rows, one corresponds to the all-one codewordof $ , $2^{2m-1}-1$ rows correspond to codewords of weight $2^{2m-1}$ (by equation (REF )), and the remaining $2^{2m +\\ell -1} - 1 - (2^{2m-1}-1)=(2^{\\ell } -1)2^{2m-1}$ rows are labeled by codewords of weight $2^{2m-1} \\pm 2^{m-1}$ , corresponding to blocks of ${\\mathbb {D}}$ and $\\overline{{\\mathbb {D}}}$ .",
"Let now $1\\le c_1 < c_2 \\le 2^{2m}$ be two distinct columns of $M$ .",
"These two columns label two distinct points of ${\\mathbb {D}}$ (resp.",
"$\\overline{{\\mathbb {D}}}$ ).",
"Let $\\lambda $ denote the number of blocks of ${\\mathbb {D}}$ that are incident with $c_1$ and $c_2$ .",
"Then the pair $\\lbrace c_1, c_2 \\rbrace $ is incident with $(2^\\ell -1)2^{2m} -2(2^\\ell -1)(2^{2m-1}- 2^{m-1}) + \\lambda =(2^\\ell -1)2^m + \\lambda $ blocks of the complementary design $\\overline{{\\mathbb {D}}}$ .",
"It follows from (REF ) and (REF ) that $ (2^\\ell -1)2^m + 2\\lambda =(2^{\\ell } -1)2^{2m-1}, $ whence $ \\lambda = (2^\\ell -1)(2^{2m-2} - 2^{m-1}), $ and the statements (a) and (b) of the theorem follow.",
"The special case $\\ell =1$ in Theorem REF implies as a corollary the following result of Dillon and Schatz [8].",
"Theorem 12 Let $f(x)$ be a bent function from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2)$ .",
"Then the code $f)$ has parameters $[2^{2m}, 2m+2, 2^{2m-1} - 2^{m-1}]$ and weight enumerator ().",
"The minimum weight codewords form a symmetric SDP design with parameters (REF ).",
"The weight enumerator () is obtained by substitution $\\ell =1$ in (REF ).",
"Since the number of minimum weight vectors is equal to the code length $2^{2m}$ , the 2-design ${\\mathbb {D}}$ supported by the codewords of minimum weight is symmetric.",
"Since every two blocks $B_1, B_2$ of ${\\mathbb {D}}$ intersect in $\\lambda = 2^{2m-2} - 2^{m-1}$ points, the sum of the two codewords supporting $B_1$ , $B_2$ is a codeword $c_{1,2} $ of weight $2^{2m-1}$ that belongs to the subcode ${\\mathrm {RM}}_2(1,2m)$ .",
"Let $B_3$ be a block distinct from $B_1$ and $B_2$ , and let $c_3$ be the codeword associated with $B_3$ .",
"Since $c_3$ is the truth table of a bent function, the sum $c_{1,2}+c_3$ is a codeword of weight $2^{2m-1} \\pm 2^{m-1}$ , thus its support is either a block or the complement of a block of ${\\mathbb {D}}$ .",
"Therefore, ${\\mathbb {D}}$ is an SDP design.",
"Theorem 13 The code $ f_1, \\cdots , f_\\ell )$ from Theorem REF is spanned by the set of codewords of minimum weight.",
"All we need to prove is that the copy of ${\\mathrm {RM}}_{2}(1,2m)$ which is a subcode of $,is spanned by some minimum weight codewords of $ .",
"It is known that the 2-rank (that is, the rank over ${\\mathrm {GF}}(2)$ ) of the incidence matrix of any symmetric SDP design ${\\mathbb {D}}$ with $2^{2m}$ points is equal to $2m+2$ (for a proof, see [12]).",
"This implies that the binary code spanned by ${\\mathbb {D}}$ contains the first order Reed-Muller code ${\\mathrm {RM}}_{2}(1,2m)$ .",
"Consequently the minimum weight vectors of the subcode ${f_1} = f_1)$ of $f_1,\\ldots , f_{\\ell })$ span the subcode of $ beingequivalent to $ RM(1,2m)$.$ Corollary 14 Two codes $f =f_1, \\cdots , f_s)$ , $g =g_1, \\cdots , g_s)$ obtained from bent vectorial functions $F(f_1, \\cdots , f_s)$ , $F(g_1, \\cdots , g_s)$ are equivalent if and only if the designs supported by their minimum weight vectors are isomorphic.",
"Example 15 Let $m=5$ .",
"Let $w$ be a generator of ${\\mathrm {GF}}(2^{10})^*$ with $w^{10} + w^6 + w^5 +w^3 + w^2 + w + 1=0$ .",
"Let $\\beta =w^{2^5+1}$ .",
"Then $\\beta $ is a generator of ${\\mathrm {GF}}(2^5)^*$ .",
"Define $\\beta _j=\\beta ^j$ for $1 \\le j \\le 5$ .",
"Then $\\lbrace \\beta _1, \\beta _2, \\beta _3,\\beta _4, \\beta _5\\rbrace $ is a basis of ${\\mathrm {GF}}(2^5)$ over ${\\mathrm {GF}}(2)$ .",
"Now consider the bent vectorial function $(f_1, f_2, f_3, f_4, f_5)$ in Example REF and the code $f_1, f_2, f_3)$ .",
"When $i=1$ and $i=7$ , the two codes $f_1, f_2, f_3)$ have parameters $[1024, 14, 496]$ and weight enumerator $1 + 7168z^{496} + 2046z^{512} + 7168z^{528} +z^{1024}.$ The two codes are not equivalent according to Magma.",
"It follows from Corollary REF that the two designs with parameters 2-$(1024, 496, 1680)$ supported by these codes are not isomorphic.",
"Note 16 Examples REF and REF give three inequivalent $[64,10,28]$ codes, and Example REF lists two inequivalent codes with parameters $[1024, 14, 496]$ , obtained from bent vectorial functions.",
"As we pointed out in Note REF , the code 1 from Example REF , does not have a 2-transitive group.",
"These examples, as well as further evidence provided by Theorem REF below, suggest the following plausible statement that we formulate as a conjecture.",
"Conjecture 17 For any given $\\ell $ in the range $1\\le \\ell \\le m$ , the number of inequivalent codes with parameters $[2^{2m}, 2m+1 +\\ell , 2^{2m-1}-2^{m-1}]$ obtained from $(2m,\\ell )$ bent vectorial functions via Theorem REF , grows exponentially with linear growth of $m$ , and most of these codes do not admit a 2-transitive automorphism group.",
"As it is customary, by “most” we mean that the limit of the ratio of the number of 2-transitive codes divided by the total number of codes approaches zero when $m$ grows to infinity.",
"The next theorem proves Conjecture REF in the case $\\ell = 1$ .",
"Theorem 18 (i) The number of inequivalent $[2^{2m},2m+2,2^{2m-1}-2^{m-1}]$ codes obtained from single bent functions from $GF(2^{2m})$ to $GF(2)$ grows exponentially with linear growth of $m$ .",
"(ii) For every given $m\\ge 2$ , there is exactly one (up to equivalence) code with parameters $[2^{2m},2m+2,2^{2m-1}-2^{m-1}]$ obtained from a bent function from $GF(2^{2m})$ to $GF(2)$ , that admits a 2-transitive automorphism group.",
"(i) By the Dillon-Schatz Theorem REF , the minimum weight codewords of a code $f)$ with parameters $[2^{2m},2m+2,2^{2m-1}-2^{m-1}]$ obtained from a bent function $f$ form a symmetric SDP design ${\\mathbb {D}}(f)$ with parameters (REF ).",
"It follows from Theorem REF that two codes $f_1)$ , $f_2)$ obtained from bent functions $f_1$ , $f_2$ are equivalent if and only if the the corresponding designs ${\\mathbb {D}}(f_1)$ , ${\\mathbb {D}}(f_1)$ are isomorphic.",
"Since the number of nonisomorphic SDP designs with parameters (REF ) grows exponentially when $m$ grows to infinity (Kantor [15]), the proof of part (i) is complete.",
"(ii) It follows from Theorem REF that the automorphism group of a code $f)$ obtained from a bent function $f$ coincides with the automorphism group of the design ${\\mathbb {D}}(f)$ supported by the codewords of minimum weight.",
"The design ${\\mathbb {D}}(f)$ is a symmetric 2-design with parameters (REF ).",
"It was proved by Kantor [17] that for every $m\\ge 2$ , there is exactly one (up to isomorphism) symmetric design with parameters (REF ) that admits a 2-transitive automorphism group.",
"This completes the proof of part (ii).",
"By Theorem REF , the codes based on single bent functions support symmetric 2-designs.",
"The next theorem determines the block intersection numbers of the design ${\\mathbb {D}}(f_1, \\cdots , f_\\ell )$ supported by the minimum weight vectors in the code $f_1, \\cdots , f_\\ell )$ from Theorem REF .",
"Theorem 19 Let ${\\mathbb {D}}={\\mathbb {D}}(f_1,\\ldots ,f_\\ell )$ , ($1\\le \\ell \\le m$ ), be a 2-design with parameters $ 2-(2^{2m}, 2^{2m-1} - 2^{m-1}, (2^\\ell -1)(2^{2m-2} - 2^{m-1})) $ supported by the minimum weight codewords of a code $f_1,\\ldots ,f_\\ell )$ defined as in Theorem REF .",
"(a) If $\\ell =1$ , ${\\mathbb {D}}$ is a symmetric SDP design, with block intersection number $\\lambda = 2^{2m-2} - 2^{m-1}$ .",
"(b) If $2 \\le \\ell \\le m$ , ${\\mathbb {D}}$ has the following three block intersection numbers: $s_1 = 2^{2m -2} - 2^{m-2}, \\ s_2 = 2^{2m -2} - 2^{m-1}, \\ s_3 =2^{2m -2} - 3\\cdot 2^{m-2}.$ For every block of ${\\mathbb {D}}$ , these intersection numbers occur with multiplicities $n_1 =2^{m}(2^m +1)(2^{\\ell -1} -1), \\ n_2 = 2^{2m} -1, \\ n_3 = 2^{m}(2^m -1)(2^{\\ell -1} -1).$ Case (a) follows from Theorem REF .",
"(b) Assume that $2 \\le \\ell \\le m$ .",
"Let $w_1$ , $w_2$ be two distinct codewords of weight $2^{2m-1} - 2^{m-1}$ .",
"The Hamming distance $d(w_1,w_2)$ between $w_1$ and $w_2$ is equal to $ 2(2^{2m-1} - 2^{m-1}) - 2s, $ where $s$ is the size of the intersection of the supports of $w_1$ and $w_2$ .",
"Since the distance between $w_1$ and $w_2$ is either $2^{2m-1} - 2^{m-1}$ , or $2^{2m-1}$ , or $2^{2m-1} + 2^{m-1}$ , the size $s$ of the intersection of the two blocks of ${\\mathbb {D}}$ supported by $w_1$ , $w_2$ can take only the values $s_i$ , $1\\le i \\le 3$ , given by (REF ).",
"Let $B$ be a block of ${\\mathbb {D}}$ supported by a codeword of weight $2^{2m-1} - 2^{m-1}$ , and let $n_i$ , ($1\\le i \\le 3$ ), denote the number of blocks of ${\\mathbb {D}}$ that intersect $B$ in $s_i$ points.",
"Let ${\\bf r} = (2^\\ell -1)(2^{2m-1} - 2^{m-1})$ denote the number of blocks of ${\\mathbb {D}}$ containing a single point, and let $b = (2^\\ell -1)2^{2m}$ denote the total number of blocks of ${\\mathbb {D}}$ .",
"Finally, let $k = 2^{2m-1} - 2^{m-1}$ denote the size of a block, and let $\\lambda =(2^\\ell -1)(2^{2m-2} - 2^{m-1})$ denote the number of blocks containing two points.",
"We have $n_1 + n_2 + n_3 & = & b - 1,\\\\s_{1}n_1 + s_{2}n_2 + s_{3}n_3 & = & k({\\bf r} - 1), \\\\s_{1}(s_{1} - 1)n_1 + s_{2}(s_{2} -1)n_2 + s_{3}(s_{3} -1)n_3 & = &k(k-1)(\\lambda -1).$ The second and the third equation count in two ways the appearances of single points and ordered pairs of points of $B$ in other blocks of ${\\mathbb {D}}$ .",
"The unique solution of this system of equations for $n_1,\\ n_2, \\ n_3$ is given by (REF ).",
"Note 20 A bent set is a set $S$ of bent functions such that the sum of every two functions from $S$ is also a bent function [4].",
"Since every $(2m,\\ell )$ bent vectorial function gives rise to a bent set consisting of $2^\\ell $ functions [4], it follows from [4] that the set of blocks of the design ${\\mathbb {D}}$ is a union of $2^\\ell -1$ linked system of symmetric 2-$(2^{2m}, 2^{2m-1}-2^{m-1},2^{2m-2}-2^{m-1})$ designs.",
"This gives an alternative proof of Theorem REF and Theorem REF (b).",
"Note 21 For every integer $m \\ge 2$ , any code $f_1, f_2, \\ldots , f_m)$ based on a bent vectorial function $F(x)=(f_1(x), f_2(x), \\cdots , f_m(x))$ from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2)^m$ , contains $2^{m}-1$ subcodes $=(f_{j_1},\\ldots , f_{j_s})$ , $j_1 < \\cdots < j_s \\le m$ , such that ${\\mathrm {RM}}_2(1, 2m) \\subset \\subseteq f_1, \\ldots , f_m).$ Each subcode $$ holds 2-designs.",
"This may be the only known chain of linear codes, included in each other, other than the chain of the Reed-Muller codes, $ {\\mathrm {RM}}_2(1, 2m) \\subset {\\mathrm {RM}}_2(2, 2m) \\subset \\cdots \\subset {\\mathrm {RM}}_2(m-2, 2m).",
"$ such that all codes in the chain support nontrivial 2-designs.",
"Note 22 We would demonstrate that the characterization of bent vectorial functions in Theorem REF can be used to construct bent vectorial functions.",
"To this end, consider the extended binary narrow-sense primitive BCH code of length $2^{2m}-1$ and designed distance $2^{2m-1}-1-2^{m-1}$ , which is affine-invariant and holds 2-designs [9].",
"This code has the desired weight enumerator of (REF ) for $\\ell = m$ [9].",
"It can be proved with the Delsarte theorem that the trace representation of this code is equivalent to the following code: $\\left\\lbrace \\left(f_{a,b,h}(x)\\right)_{x\\in {\\mathrm {GF}}(2^{2m})}:a \\in {\\mathrm {GF}}(2^m), \\, b \\in {\\mathrm {GF}}(2^{2m}), \\, h \\in {\\mathrm {GF}}(2)\\right\\rbrace ,$ where $f_{a,b,h}(x)={\\mathrm {Tr}}_{m/1}\\left[a {\\mathrm {Tr}}_{2m/m}\\left(x^{1+2^{m-1}}\\right) \\right] + {\\mathrm {Tr}}_{2m/1}(bx) + h.$ It then follows from Theorem REF that ${\\mathrm {Tr}}_{2m/m}(x^{1+2^{m-1}})$ is a bent vectorial function from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2^m)$ .",
"Note that this bent vectorial function may not be new.",
"But our purpose here is to show that bent vectorial functions could be constructed from special linear codes.",
"Conversely, we could say that the extended narrow-sense BCH code of length $2^{2m}-1$ and designed distance $2^{2m-1}-1-2^{m-1}$ is in fact generated from the bent vectorial function ${\\mathrm {Tr}}_{2m/m}(x^{1+2^{m-1}})$ from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2^m)$ using the construction of Note REF .",
"Example REF gives a demonstration of that.",
"Thus, all known binary codes with the weight enumerator (REF ) for some $1\\le \\ell \\le m$ and arbitrary $m\\ge 2$ are obtained from the bent vectorial function construction.",
"As shown in Example REF , all $[16,7,6]$ codes obtained from $(4,2)$ bent vectorial functions are equivalent.",
"Example REF shows that there are at least three inequivalent $[64, 10, 28]$ binary codes from bent vectorial functions, one of these codes being an extended BCH code.",
"Note 23 It is known that two designs ${\\mathbb {D}}(f)$ and ${\\mathbb {D}}(g)$ from two single bent Boolean functions $f$ and $g$ on ${\\mathrm {GF}}(2^{2m})$ are isomorphic if and only if $f$ and $g$ are weakly affinely equivalent [8].",
"Although the classification of bent Boolean functions into weakly affinely equivalent classes is open, the results from [15] and [8] imply that the number of nonisomorphic SDP designs and inequivalent bent functions in $2m$ variables grows exponentially with linear growth of $m$ .",
"Note 24 Two $(n, \\ell )$ vectorial Boolean functions $(f_1(x), \\cdots , f_\\ell (x))$ and $(g_1(x), \\cdots , g_\\ell (x))$ from ${\\mathrm {GF}}(2^n)$ to ${\\mathrm {GF}}(2)^\\ell $ are said to be EA-equivalent if there are an automorphism of $({\\mathrm {GF}}(2^n), +)$ , a homomorphism $L$ from $({\\mathrm {GF}}(2^n),+)$ to $({\\mathrm {GF}}(2)^\\ell , +)$ , an $\\ell \\times \\ell $ invertible matrix $M$ over ${\\mathrm {GF}}(2)$ , an element $a \\in {\\mathrm {GF}}(2^n)$ , and an element $b \\in {\\mathrm {GF}}(2)^\\ell $ such that $(g_1(x), \\cdots , g_\\ell (x))=(f_1(A(x)+a), \\cdots , f_\\ell (A(x)+a))M +L(x) +b $ for all $x \\in {\\mathrm {GF}}(2^n)$ .",
"Let $(f_1(x), \\cdots , f_\\ell (x))$ and $(g_1(x), \\cdots , g_\\ell (x))$ be two bent vectorial functions from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2)^\\ell $ .",
"We conjecture that the designs ${\\mathbb {D}}(f_{1}, \\cdots , f_{\\ell })$ and ${\\mathbb {D}}(g_{1}, \\cdots , g_{\\ell })$ are isomorphic if and only if $(f_1(x), \\cdots , f_\\ell (x))$ and $(g_1(x), \\cdots , g_\\ell (x))$ are EA-equivalent.",
"The reader is invited to attack this open problem.",
"Suppose that ${\\mathbb {D}}$ is a 2-design with parameters () obtained from a bent vectorial function $F(x)=(f_1(x), f_2(x), \\cdots , f_\\ell (x))$ , ($1 \\le \\ell \\le m$ ), via the construction from Theorem REF .",
"Let $\\cal {B}$ be the block set of ${\\mathbb {D}}$ .",
"If $B$ is a block of ${\\mathbb {D}}$ , we consider the collection of new blocks ${\\mathcal {B}}^{de}$ consisting of intersections $B \\cap B^{\\prime }$ such that $B^{\\prime } \\in \\cal {B}$ and $| B \\cap B^{\\prime } |=2^{2m-2} - 2^{m-1}$ .",
"Theorem 25 For each $B \\in {\\mathbb {D}}$ , the incidence structure $(B, {\\mathcal {B}}^{de})$ is a quasi-symmetric design with parameters $2-( 2^{2m-1}-2^{m-1},\\, 2^{2m-2}-2^{m-1},\\, 2^{2m-2}-2^{m-1}-1)$ and intersection numbers $2^{2m-3} - 2^{m-2}$ and $2^{2m-3} - 2^{m-1}$ .",
"By Theorem REF , there are exactly $2^{2m} -1$ blocks that intersect $B$ in $2^{2m-2}-2^{m-1}$ points.",
"Together with $B$ , these blocks form a symmetric SDP design $D$ with parameters 2-$(2^{2m}, 2^{2m-1} - 2^{m-1}, 2^{2m-2} - 2^{m-1})$ .",
"The incidence structure $(B, {\\mathcal {B}})^{de}$ is a derived design of $D$ .",
"It was proved in [12] that each derived design of a symmetric SDP 2-$(2^{2m}, 2^{2m-1} - 2^{m-1}, 2^{2m-2} - 2^{m-1})$ design is quasi-symmetric design with intersection numbers $2^{2m-3} - 2^{m-2}$ and $2^{2m-3} - 2^{m-1}$ , and having the additional property that the symmetric difference of every two blocks is either a block or the complement of a block.",
"Note 26 Let $m >1$ be an integer.",
"Let $F$ be a bent vectorial function from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2^m)$ .",
"Let $A$ be a subgroup of order $2^s$ of $({\\mathrm {GF}}(2^m), +)$ .",
"Define a binary code by ${A}:=\\lbrace ({\\mathrm {Tr}}_{m/1}(aF(x))+{\\mathrm {Tr}}_{2m/1}(bx)+c)_{x \\in {\\mathrm {GF}}(2^{2m})}:a \\in A, b \\in {\\mathrm {GF}}(2^{2m}), c \\in {\\mathrm {GF}}(2)\\rbrace .$ It can be shown that ${A}$ can be viewed as a code $f_{i_1}, \\cdots , f_{i_s})$ obtained from a bent vectorial function $(f_{i_1}, \\cdots , f_{i_s})$ .",
"Summary and concluding remarks The contributions of this paper are the following.",
"A coding-theoretic characterization of bent vectorial functions (Theorem REF ).",
"A construction of a two-parameter family of four-weight binary linear codes with parameters $[2^{2m}, 2m+1+\\ell ,2^{2m-1}-2^{m-1}]$ for all $1 \\le \\ell \\le m$ and all $m\\ge 2$ , obtained from $(2m, \\ell )$ bent vectorial functions (Theorem REF ).",
"The parameters of these codes appear to be new when $2 \\le \\ell \\le m-1$ .",
"This family of codes includes some optimal codes, as well as codes meeting the BCH bound.",
"These codes do not satisfy the conditions of the Assmus-Mattson theorem, but nevertheless hold 2-designs.",
"It is plausible that most of these codes do not admit 2-transitive automorphism groups (Conjecture REF and Theorem REF ).",
"A new construction of a two-parameter family of 2-designs with parameters $2\\mbox{--}(2^{2m}, \\ 2^{2m-1}-2^{m-1}, \\ (2^\\ell -1)(2^{2m-2}-2^{m-1})),$ and having three block intersection numbers, where $2\\le \\ell \\le m$ , based on bent vectorial functions (Theorem REF and Theorem REF ).",
"This construction is a generalization of the construction of SDP designs from single bent functions given in [8].",
"The number of nonisomorphic designs with parameters (REF ) in the special case when $\\ell =1$ , grows exponentially with $m$ by a known theorem of Kantor [15].",
"It is an interesting open problem to prove that the number of nonisomorphic designs with parameters (REF ) grows exponentially for any fixed $\\ell >1$ .",
"Finally, we would like to mention that vectorial Boolean functions were employed in a different way to construct binary linear codes in [20].",
"The codes from [20] have different parameters from the codes described in this paper.",
"Acknowledgements Vladimir Tonchev acknowledges partial support by a Fulbright grant, and would like to thank the Hong Kong University of Science and Technology for the kind hospitality and support during his visit, when a large portion of this paper was written.",
"The research of Cunsheng Ding was supported by the Hong Kong Research Grants Council, under Grant No.",
"16300418.",
"The authors wish to thank the anonymous reviewers for their valuable comments and suggestions for improving the manuscript."
],
[
"Bent functions and bent vectorial functions",
"Let $f=f(x)$ be a Boolean function from ${\\mathrm {GF}}(2^{n})$ to ${\\mathrm {GF}}(2)$ .",
"The support $S_f$ of $f$ is defined as $S_f=\\lbrace x \\in {\\mathrm {GF}}(2^{n}) : f(x)=1\\rbrace \\subseteq {\\mathrm {GF}}(2^{n}).$ The $(0,1)$ incidence vector of $S_f$ , having its coordinates labeled by the elements of ${\\mathrm {GF}}(2^n)$ , is called the truth table of $f$ .",
"The Walsh transform of $f$ is defined by $\\hat{f}(w)=\\sum _{x \\in {\\mathrm {GF}}(2^{n})} (-1)^{f(x)+{\\mathrm {Tr}}_{n/1}(wx)}$ where $w \\in {\\mathrm {GF}}(2^{n})$ and ${\\mathrm {Tr}}_{n/n^{\\prime }}(x)$ denotes the trace function from ${\\mathrm {GF}}(2^n)$ to ${\\mathrm {GF}}(2^{n^{\\prime }})$ .",
"Two Boolean functions $f$ and $g$ from ${\\mathrm {GF}}(2^n)$ to ${\\mathrm {GF}}(2)$ are called weakly affinely equivalent or EA-equivalent if there are an automorphism $A$ of $({\\mathrm {GF}}(2^n), +)$ , a homomorphism $L$ from $({\\mathrm {GF}}(2^n),+)$ to $({\\mathrm {GF}}(2), +)$ , an element $a \\in {\\mathrm {GF}}(2^n)$ and an element $b \\in {\\mathrm {GF}}(2)$ such that $g(x)=f(A(x)+a)+ L(x) +b$ for all $x \\in {\\mathrm {GF}}(2^n)$ .",
"A Boolean function $f$ from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2)$ is called a bent function if $|\\hat{f}(w)|=2^{m}$ for every $w \\in {\\mathrm {GF}}(2^{2m})$ .",
"It is well known that a function $f$ from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2)$ is bent if and only if $S_f$ is a difference set in $({\\mathrm {GF}}(2^{2m}),\\,+)$ with parameters (REF ) [19].",
"A Boolean function $f$ from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2)$ is a bent function if and only if its truth table is at Hamming distance $2^{2m-1} \\pm 2^{m-1}$ from every codeword of the first order Read-Muller code ${\\mathrm {RM}}_2(1, 2m)$ [18].",
"It follows that $|S_f|=2^{2m-1} \\pm 2^{m-1}.$ There are many constructions of bent functions.",
"The reader is referred to [6] and [19] for detailed information about bent functions.",
"Let $\\ell $ be a positive integer, and let $f_1(x), \\cdots , f_\\ell (x)$ be Boolean functions from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2)$ .",
"The function $F(x)=(f_1(x), \\cdots , f_\\ell (x))$ from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2)^\\ell $ is called a $(2m,\\ell )$ vectorial Boolean function.",
"A $(2m,\\ell )$ vectorial Boolean function $F(x)=(f_1(x), \\cdots , f_\\ell (x))$ is called a bent vectorial function if $\\sum _{j=1}^\\ell a_j f_j(x)$ is a bent function for each nonzero $(a_1, \\cdots , a_\\ell )\\in {\\mathrm {GF}}(2)^\\ell $ .",
"For another equivalent definition of bent vectorial functions, see [7] or [19].",
"Bent vectorial functions exist only when $\\ell \\le m$ (cf.",
"[19]).",
"There are a number of known constructions of bent vectorial functions.",
"The reader is referred to [7] and [19] for detailed information.",
"Below we present a specific construction of bent vectorial functions from [7].",
"Example 2 [7].",
"Let $m \\ge 1$ be an odd integer, $\\beta _1, \\beta _2, \\cdots , \\beta _{m}$ be a basis of ${\\mathrm {GF}}(2^{m})$ over ${\\mathrm {GF}}(2)$ , and let $u \\in {\\mathrm {GF}}(2^{2m}) \\setminus {\\mathrm {GF}}(2^m)$ .",
"Let $i$ be a positive integer with $\\gcd (2m, i)=1$ .",
"Then $\\left({\\mathrm {Tr}}_{2m/1}(\\beta _1 u x^{2^i+1}), {\\mathrm {Tr}}_{2m/1}(\\beta _2 u x^{2^i+1}), \\cdots ,{\\mathrm {Tr}}_{2m/1}(\\beta _{m} u x^{2^i+1}) \\right)$ is a $(2m, m)$ bent vectorial function.",
"Under a basis of ${\\mathrm {GF}}(2^{\\ell })$ over ${\\mathrm {GF}}(2)$ , $({\\mathrm {GF}}(2^\\ell ), +)$ and $({\\mathrm {GF}}(2)^\\ell , +)$ are isomorphic.",
"Hence, any vectorial function $F(x)=(f_1(x), \\cdots , f_\\ell (x))$ from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2)^\\ell $ can be viewed as a function from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2^\\ell )$ .",
"It is well known that a function $F$ from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2^\\ell )$ is bent if and only if ${\\mathrm {Tr}}_{\\ell /1}(aF(x))$ is a bent Boolean function for all $a \\in {\\mathrm {GF}}(2^\\ell )^*$ .",
"Any such vectorial function $F$ can be expressed as ${\\mathrm {Tr}}_{2m/\\ell }(f(x))$ , where $f$ is a univariate polynomial.",
"This presentation of bent vectorial functions is more compact.",
"We give two examples of bent vectorial functions in this form.",
"Example 3 (cf.",
"[19]).",
"Let $m>1$ and $i \\ge 1$ be integers such that $2m/\\gcd (i, 2m)$ is even.",
"Then ${\\mathrm {Tr}}_{2m/m}(a x^{2^i+1})$ is bent if and only if $\\gcd (2^i+1, 2^m+1) \\ne 1$ and $a \\in {\\mathrm {GF}}(2^{2m})^* \\setminus \\langle \\alpha ^{\\gcd (2^i+1, 2^m+1)} \\rangle $ , where $\\alpha $ is a generator of ${\\mathrm {GF}}(2^{2m})^*$ .",
"Example 4 (cf.",
"[19]).",
"Let $m>1$ and $i \\ge 1$ be integers such that $\\gcd (i, 2m)=1$ .",
"Let $d=2^{2i}-2^i+1$ .",
"Let $m$ be odd.",
"Then ${\\mathrm {Tr}}_{2m/m}(a x^{d})$ is bent if and only if $a \\in {\\mathrm {GF}}(2^{2m})^* \\setminus \\langle \\alpha ^{3} \\rangle $ , where $\\alpha $ is a generator of ${\\mathrm {GF}}(2^{2m})^*$ ."
],
[
"A construction of codes from bent vectorial functions",
"Let $q=2^{2m}$ , let ${\\mathrm {GF}}(q)=\\lbrace u_1, u_2, \\cdots , u_{q}\\rbrace $ , and let $w$ be a generator of ${\\mathrm {GF}}(q)^*$ .",
"For the purposes of what follows, it is convenient to use the following generator matrix of the binary $[2^{2m}, 2m+1,2^{2m-1}]$ first-order Reed-Muller code ${\\mathrm {RM}}_2(1,2m)$ : $G_0=\\left[\\begin{array}{cccc}1 & 1 & \\cdots & 1 \\\\{\\mathrm {Tr}}_{2m/1}(w^0u_1) & {\\mathrm {Tr}}_{2m/1}(w^0u_2) & \\cdots & {\\mathrm {Tr}}_{2m/1}(w^0u_q) \\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\{\\mathrm {Tr}}_{2m/1}(w^{2m-1}u_1) & {\\mathrm {Tr}}_{2m/1}(w^{2m-1}u_2) & \\cdots & {\\mathrm {Tr}}_{2m/1}(w^{2m-1}u_q)\\end{array}\\right].$ The weight enumerator of ${\\mathrm {RM}}_2(1,2m)$ is $1+(2^{2m+1}-2)z^{2^{2m-1}} + z^{2^{2m}}.$ Two binary linear codes are equivalent if there is a permutation of coordinates that sends the first code to the second.",
"Up to equivalence, ${\\mathrm {RM}}_2(1,2m)$ is the unique linear binary code with parameters $[2^{2m}, 2m+1,2^{2m-1}]$ [8].",
"Its dual code is the $[2^{2m}, 2^{2m}- 1 -2m,4]$ Reed-Muller code of order $2m-2$ .",
"Both codes hold 3-designs since they are invariant under a 3-transitive affine group.",
"Note that ${\\mathrm {RM}}_2(1,2m)^\\perp $ is the unique, up to equivalence, binary linear code for the given parameters, hence it is equivalent to the extended binary linear Hamming code.",
"Let $F(x)=(f_1(x), f_2(x), \\cdots , f_\\ell (x))$ be a $(2m, \\ell )$ vectorial function from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2)^\\ell $ .",
"For each $i$ , $1 \\le i \\le \\ell $ , we define a binary vector $F_i=(f_i(u_1), f_i(u_2), \\cdots , f_i(u_q)) \\in {\\mathrm {GF}}(2)^{2^{2m}},$ which is the truth table of the Boolean function $f_i(x)$ introduced in Section .",
"Let $\\ell $ be an integer in the range $1 \\le \\ell \\le m$ .",
"We now define a $(2m+1 + \\ell ) \\times 2^{2m}$ matrix $G=G(f_{1}, \\cdots , f_{\\ell })=\\left[\\begin{array}{c}G_0 \\\\F_{1} \\\\\\vdots \\\\F_{\\ell }\\end{array}\\right],$ where $G_0$ is the generator matrix of ${\\mathrm {RM}}_2(1,2m)$ .",
"Let $f_{1}, \\cdots , f_{\\ell })$ denote the binary code of length $2^{2m}$ with generator matrix $G(f_{1}, \\cdots , f_{\\ell })$ given by (REF ).",
"The dimension of the code has the following lower and upper bounds: $2m+1 \\le \\dim (f_{1}, \\cdots , f_{\\ell })) \\le 2m+1+\\ell .$ The following theorem gives a coding-theoretical characterization of bent vectorial functions.",
"Theorem 5 A $(2m, \\ell )$ vectorial function $F(x)=(f_1(x), f_2(x), \\cdots , f_\\ell (x))$ from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2)^\\ell $ is a bent vectorial function if and only if the code $f_1, \\cdots , f_\\ell )$ with generator matrix $G$ given by (REF ) has weight enumerator $1 + (2^\\ell -1)2^{2m} z^{2^{2m-1} - 2^{m-1}} + 2(2^{2m}-1)z^{2^{2m-1}}+ (2^\\ell -1)2^{2m} z^{2^{2m-1} + 2^{m-1}} + z^{2^{2m}}.$ By the definition of $G$ , the code $f_1, \\cdots , f_\\ell )$ contains the first-order Reed-Muller code ${\\mathrm {RM}}_2(1, 2m)$ as a subcode, having weight enumerator (REF ).",
"It follows from (REF ) that every codeword of $f_1, \\cdots , f_\\ell )$ must be the truth table of a Boolean function of the form $f_{(u, v, h)}(x)=\\sum _{i=1}^\\ell u_i f_i(x) + \\sum _{j=0}^{2m-1} v_j {\\mathrm {Tr}}_{2m/1}(w^jx) + h,$ where $u_i, v_j, h \\in {\\mathrm {GF}}(2)$ , $x \\in {\\mathrm {GF}}(2^{2m})$ .",
"Suppose that $F(x)=(f_1(x), f_2(x), \\cdots , f_\\ell (x))$ is a $(2m,\\ell )$ bent vectorial function.",
"When $(u_1, \\cdots , u_\\ell )=(0, \\cdots , 0)$ , $(v_0, v_1, \\cdots , v_{2m-1})$ runs over ${\\mathrm {GF}}(2)^{2m}$ and $h$ runs over ${\\mathrm {GF}}(2)$ , the truth tables of the functions $f_{(u, v, h)}(x)$ form the code ${\\mathrm {RM}}_2(1, 2m)$ .",
"Whenever $(u_1, \\cdots , u_\\ell ) \\ne (0, \\cdots , 0)$ , it follows from (REF ) that $f_{(u, v, h)}(x)$ is a bent function, and the corresponding codeword has Hamming weight $2^{2m-1} \\pm 2^{m-1}$ .",
"Since the all-one vector belongs to ${\\mathrm {RM}}_2(1, 2m)$ , the code $f_1, \\cdots , f_\\ell )$ is self-complementary, and the desired weight enumerator of $f_1, \\cdots , f_\\ell )$ follows.",
"Suppose that $f_1, \\cdots , f_\\ell )$ has weight enumerator given by (REF ).",
"Then $f_1, \\cdots , f_\\ell )$ has dimension $2m+1+\\ell $ .",
"Consequently, $\\sum _{i=1}^\\ell u_i f_i(x)$ is the zero function if and only if $(u_1, \\cdots , u_\\ell )=(0, \\cdots , 0)$ .",
"It then follows that the codewords corresponding to $f_{(u, v, h)}(x)$ must have Hamming weight $2^{2m-1} \\pm 2^{m-1}$ for all $u=(u_1, \\cdots , u_\\ell ) \\ne (0, \\cdots , 0)$ and all $(v_0, v_1, \\cdots , v_{2m-1}) \\in {\\mathrm {GF}}(2)^{2m}$ .",
"Notice that $\\sum _{j=0}^{2m-1} v_j {\\mathrm {Tr}}_{2m/1}(w^jx)$ ranges over all linear functions from ${\\mathrm {GF}}(2^m)$ to ${\\mathrm {GF}}(2)$ when $(v_0, v_1, \\cdots , v_{2m-1})$ runs over ${\\mathrm {GF}}(2)^{2m}$ .",
"Consequently, $F(x)$ is a bent vectorial function.",
"Note 6 Let $F(x)=(f_1(x), f_2(x), \\cdots , f_m(x))$ be a bent vectorial function from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2)^m$ .",
"Then the code $f_1, \\cdots , f_m)$ has parameters $ [2^{2m}, 3m+1, 2^{2m-1} - 2^{m-1}].",
"$ In particular, if $m=2$ , any code $f_1, f_2)$ based on a bent vectorial function from ${\\mathrm {GF}}(2^{4})$ to ${\\mathrm {GF}}(2)^2$ has parameters $[16, 7, 6]$ and is optimal (cf.",
"[10]).",
"An $[n,k,d]$ code is optimal if $d$ is the maximum possible minimum distance for the given $n$ and $k$ .",
"If $m=3$ , any code $f_1, f_2, f_3)$ based on a bent vectorial function from ${\\mathrm {GF}}(2^{6})$ to ${\\mathrm {GF}}(2)^3$ has parameters $[64, 10, 28]$ and is optimal [10].",
"If $m=4$ , any code $f_1, \\cdots , f_6)$ based on a bent vectorial function from ${\\mathrm {GF}}(2^{8})$ to ${\\mathrm {GF}}(2)^4$ has parameters $[256, 13, 120]$ and has the largest known minimum distance for the given code length and dimension [10].",
"Theorem 7 Up to equivalence, there is exactly one $[16,7,6]$ code that can be obtained from a $(4,2)$ bent vectorial function.",
"The weight enumerator of the second order Reed-Muller code ${\\mathrm {RM}}_{2}(2,4)$ is given by $1+140z^4 + 448z^6 + 870z^8 + 448z^{10} + 140z^{12} + z^{16}.$ The truth table of a bent function $f$ from ${\\mathrm {GF}}(2^4)$ to ${\\mathrm {GF}}(2)$ is a codeword $c_f$ of ${\\mathrm {RM}}_{2}(2,4)$ of weight 6.",
"The linear code $f)$ spanned by $c_f$ and ${\\mathrm {RM}}_{2}(1,4)$ is a subcode of ${\\mathrm {RM}}_{2}(2,4)$ of dimension 6, having weight enumerator $1 + 16z^6 + 30z^8 + 16z^{10} + z^{16}.$ The codewords of $f)$ of weight 6 form a symmetric 2-$(16,6,2)$ SDP design, whose blocks correspond to the supports of 16 bent functions.",
"Now, let $(f_1,f_2)$ be a $(4,2)$ bent vectorial function.",
"Then, the intersection of the codes $f_1)$ , $f_2)$ consists of the first order Reed-Muller code ${\\mathrm {RM}}_{2}(1,4)$ .",
"It follows that the set of 448 codewords of weight 6 in ${\\mathrm {RM}}_{2}(2,4)$ is a union $\\cal {U}$ of 28 pairwise disjoint subsets of size 16, corresponding to the incidence matrices of symmetric 2-$(16,6,2)$ SDP designs associated with 28 different $[16,6]$ codes defined by single bent functions.",
"If $f_1,f_2)$ is a $[16,7]$ code defined by a bent vectorial function $(f_1,f_2)$ , its weight enumerator is given by $1 + 48z^6 + 30z^8 + 48z^{10} +z^{16}.$ The set of 48 codewords of weight 6 of $f_1,f_2)$ is a union of the incidence matrices of three SDP designs from $\\cal {U}$ with pairwise disjoint sets of blocks.",
"A quick check shows that there are exactly 56 such collections of 48 codewords that generate a code having weight enumerator (REF ).",
"Therefore, the number of distinct $[16,7,6]$ subcodes of ${\\mathrm {RM}}_{2}(1,4)$ based on $(4,2)$ bent vectorial functions is 56.",
"The $7 \\times 16$ generator matrix $G$ of one such $[16,7,6]$ code is listed below: $\\left[\\begin{array}{cccccccccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 1& 1& 1& 1& 1& 1& 1& 1 \\\\0& 0& 0& 0& 1& 1& 1& 1& 0& 0& 0& 0& 1& 1& 1& 1 \\\\0& 0& 1& 1& 0& 0& 1& 1& 0& 0& 1& 1& 0& 0& 1& 1 \\\\0& 1& 0& 1& 0& 1& 0& 1& 0& 1& 0& 1& 0& 1& 0& 1 \\\\1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1 \\\\0& 0& 0& 1& 0& 1& 1& 1& 0& 1& 0& 0& 0& 0& 1& 0 \\\\0& 0& 0& 0& 0& 1& 0& 1& 0& 0& 1& 1& 0& 1& 1& 0\\end{array}\\right].$ The first five rows of $G$ form a generator matrix of ${\\mathrm {RM}}_{2}(1,4)$ , while the last two rows are codewords of weight 6 in ${\\mathrm {RM}}_{2}(2,4)$ .",
"The full automorphism group of the $[16,7,6]$ code generated by $G$ is of order 5760.",
"Since the order of the automorphism group of ${\\mathrm {RM}}_{2}(1,4)$ is 322560, and $ 322560/5760 = 56, $ it follows that all 56 $[16,7,6]$ codes based on $(4,2)$ bent vectorial functions are pairwise equivalent.",
"The next two examples illustrate that there are at least three inequivalent optimal $[64,10,28]$ codes that are obtainable from bent vectorial functions from ${\\mathrm {GF}}(2^{6})$ to ${\\mathrm {GF}}(2)^3$ .",
"The parameters $[64,10,28]$ correspond to $m=3$ in Note REF .",
"Example 8 The binary cyclic $[63,10]$ code $ with parity check polynomial$ h(x)=(x+1)(x3 + x2 +1)(x6 + x5 + x4 + x + 1)$has weight enumerator$$1 + 196z^{27} + 252z^{28} + 63z^{31} +63z^{32}+252z^{35} +196z^{36} +z^{63}.$$The $ [63,7]$ subcode $$ of $ having check polynomial $h^{\\prime }(x)=(x+1)(x^6 + x^5 + x^4 + x + 1)$ has weight enumerator $1 + 63z^{31} +63z^{32}+ z^{63}.$ The extended $[64,7]$ code $()^*$ of $$ has weight enumerator $ 1 + 126z^{32} + z^{64}, $ hence, $()^*$ is equivalent to the first order Reed-Muller code ${\\mathrm {RM}}_{2}(1,6)$ .",
"The extended $[64,10]$ code $*$ of $ has weight enumeratorgiven by\\begin{equation}1 + 448z^{28} + 126z^{32} + 448 z^{36} + z^{64}.\\end{equation}Since $ *$ contains a copy of the first order Reed-Muller code $ RM2(1,6)$ as a subcode, it follows from Theorem \\ref {thm-bentvectf}that $ *$ can be obtained from a $ (6,3)$ bent vectorial functionfrom $ GF(26)$ to $ GF(23)$.The full automorphism group of $ *$ is of order$$677,376 = 2^9 \\cdot 3^3 \\cdot 7^2.",
"$$Magma was used for these computations.$ Example 9 Let $M$ be the 7 by 64 $(0,1)$ -matrix with the following structure: the $i$ th column of the 6 by 64 submatrix $M^{\\prime }$ of $M$ consisting of its first six rows is the binary presentation of the number $i$ ($i =0, 1, \\ldots 63$ ), while the last row of $M$ is the all-one row.",
"Clearly, $M$ is a generator matrix of a binary linear $[64,7]$ code equivalent to the first order Reed-Muller code ${\\mathrm {RM}}_{2}(1,6)$ .",
"The first six rows of $M$ can be viewed as the truth tables of the single Boolean variables $x_1, x_2, \\dots x_6$ , while the seventh row of $M$ is the truth table of the constant ${\\bf 1}$ .",
"We consider the Boolean bent functions given by $f_{1}(x_1,\\ldots , x_6) & = & x_{1}x_6 + x_{2}x_{5} + x_{3}x_4, \\\\f_{2}(x_1,\\ldots , x_6) & = & x_{1}x_5 + x_{2}x_4 + x_{3}x_5 + x_{3}x_6,\\\\f_{3}(x_1,\\ldots , x_6) & = & x_{1}x_4 + x_{2}x_5 + x_{2}x_6 +x_{3}x_4 + x_{3}x_5 + x_{5}x_{6},\\\\f_{4}(x_1,\\ldots , x_6) & = & x_{1}x_4 +x_{2}x_3 + x_{3}x_6 + x_{5}x_6.$ The vectorial functions $F_{1}=(f_{1}, f_{2}, f_{3})$ , $F_{2}=(f_{1}, f_{2}, f_{4})$ give via Theorem REF binary linear codes $1, \\ 2$ with parameters $[64,10,28]$ , having weight enumerator given by ().",
"The automorphism groups of the codes $1, \\ 2$ were computed using the computer-algebra package Magma [5].",
"The code 1 has full automorphism group of order $ 10,752 = 2^{9}\\cdot 3\\cdot 7.",
"$ The code 2 has full automorphism group of order $ 4,032 = 2^{6}\\cdot 3^{2}\\cdot 7.",
"$ Thus, 1, 2 and the extended cyclic code $*$ from Example REF are pairwise inequivalent.",
"We note that the code 1 cannot be equivalent to any extended cyclic code because its group order is not divisible by 63.",
"Note 10 The full automorphism group of 1 from Example REF cannot be 2-transitive because its order is not divisible by 63.",
"Thus, the code 1 does not satisfy the classical sufficient condition to support 2-designs based on the 2-transitivity of its automorphism group (recall that according to [13], any 2-homogeneous group of degree 64 is necessarily 2-transitive).",
"In addition, the minimum distance of its dual code ${1}^\\perp $ is 4, thus the Assmus-Mattson theorem guarantees only 1-designs to be supported by 1.",
"We will prove in the next section that all codes obtained from bent vectorial functions support 2-designs."
],
[
"A construction of 2-designs from bent vectorial functions",
"The following theorem establishes that the binary codes based on bent vectorial functions support 2-designs, despite that these codes do not meet the conditions of the Assmus-Mattson theorem for 2-designs.",
"Theorem 11 Let $F(x)=(f_1(x), f_2(x), \\cdots , f_\\ell (x))$ be a bent vectorial function from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2)^\\ell $ , where $m\\ge 2$ and $1\\le \\ell \\le m$ .",
"Let $ f_1, \\cdots , f_\\ell )$ be the binary linear code with parameters $[2^{2m}, 2m+1 + \\ell , 2^{2m-1} - 2^{m-1}]$ defined in Theorem REF .",
"(a) The codewords of $ of minimum weight hold a 2-design $ D$with parameters\\begin{equation}2-(2^{2m}, 2^{2m-1} - 2^{m-1}, (2^\\ell -1)(2^{2m-2} - 2^{m-1})).\\end{equation}(b) The codewords of $ of weight $2^{2m-1} + 2^{m-1}$ hold a 2-design $\\overline{{\\mathbb {D}}}$ with parameters $2-(2^{2m}, 2^{2m-1} + 2^{m-1}, (2^\\ell -1)(2^{2m-2} + 2^{m-1})).$ Since $ contains $ RM2(1,2m)$, and the minimum distanceof $ RM2(1,2m)$ is 4,the minimum distance $ d$ of $$ is at least 4.",
"Applying theMacWilliams transform (see, for example \\cite [p. 41]{vanLint})to the weight enumerator (\\ref {eqn-wtenumerator111}) of $ shows that $d^{\\perp } =4$ .",
"It follows from the Assmus-Mattson theorem (Theorem REF ) that the codewords of any given nonzero weight $w< 2^{2m}$ in $ hold a 1-design.$ However, we will prove that $ actually holds 2-designs, despite thatthe Assmus-Mattson theorem guarantees only1-designs to be supported by $ .",
"Since the subcode ${\\mathrm {RM}}_2(1,2m)$ of $ containsall codewords of $ of weight $2^{2m-1}$ , the codewords of this weight hold a 3-design $\\cal {A}$ with parameters 3-$(2^{2m}, 2^{2m-1}, 2^{2m-2} -1)$ .",
"We note that $\\cal {A}$ is a 2-design with $\\lambda _2 = \\frac{2^{2m}-2}{2^{2m-1} -2}\\cdot (2^{2m-2} -1)=2^{2m-1}-1.$ Let ${\\mathbb {D}}$ be the 1-design supported by codewords of weight $2^{2m-1} - 2^{m-1}$ .",
"Since the number of codewords of weight $2^{2m-1} - 2^{m-1}$ is equal to $(2^\\ell -1)2^{2m}$ , ${\\mathbb {D}}$ is a 1-design with parameters 1-$(2^{2m}, 2^{2m-1} - 2^{m-1}, (2^\\ell -1)(2^{2m-1} - 2^{m-1}))$ .",
"Every codeword of $ of weight $ 22m-1 + 2m-1$ is the sum ofa codeword of weight $ 22m-1 - 2m-1$ and the all-one vector.Thus, the codewords of weight $ 22m-1 + 2m-1$ hold a 1-design $D$having parameters1-$ (22m, 22m-1 + 2m-1, (2-1)(22m-1 + 2m-1))$.Clearly, $D$is the complementary design of $ D$, that is, every block of $D$is the complement of some block of $ D$.$ Let $M$ be the $2^{2m+1+\\ell } \\times 2^{2m}$ $(0,1)$ -matrix having as rows the codewords of $.",
"Since $ d =4$, $ M$ is an orthogonal array of strength 3,that is, for every integer $ i$, $ 1 i 3$, and for every set of $ i$ distinct columnsof $ M$, every binary vector with $ i$ components appears exactly$ 22m+1+- i$ times among the rows of the $ 22m+1+ i$ submatrix of $ M$formed by the chosen $ i$ columns.In particular, any $ 22m+1+ 2$ submatrix consisting of two distinctcolumns of $ M$ contains the binary vector $ (1,1)$ exactly $ 22m+-1$ timesas a row.",
"Among these $ 22m+-1$ rows, one corresponds to the all-one codewordof $ , $2^{2m-1}-1$ rows correspond to codewords of weight $2^{2m-1}$ (by equation (REF )), and the remaining $2^{2m +\\ell -1} - 1 - (2^{2m-1}-1)=(2^{\\ell } -1)2^{2m-1}$ rows are labeled by codewords of weight $2^{2m-1} \\pm 2^{m-1}$ , corresponding to blocks of ${\\mathbb {D}}$ and $\\overline{{\\mathbb {D}}}$ .",
"Let now $1\\le c_1 < c_2 \\le 2^{2m}$ be two distinct columns of $M$ .",
"These two columns label two distinct points of ${\\mathbb {D}}$ (resp.",
"$\\overline{{\\mathbb {D}}}$ ).",
"Let $\\lambda $ denote the number of blocks of ${\\mathbb {D}}$ that are incident with $c_1$ and $c_2$ .",
"Then the pair $\\lbrace c_1, c_2 \\rbrace $ is incident with $(2^\\ell -1)2^{2m} -2(2^\\ell -1)(2^{2m-1}- 2^{m-1}) + \\lambda =(2^\\ell -1)2^m + \\lambda $ blocks of the complementary design $\\overline{{\\mathbb {D}}}$ .",
"It follows from (REF ) and (REF ) that $ (2^\\ell -1)2^m + 2\\lambda =(2^{\\ell } -1)2^{2m-1}, $ whence $ \\lambda = (2^\\ell -1)(2^{2m-2} - 2^{m-1}), $ and the statements (a) and (b) of the theorem follow.",
"The special case $\\ell =1$ in Theorem REF implies as a corollary the following result of Dillon and Schatz [8].",
"Theorem 12 Let $f(x)$ be a bent function from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2)$ .",
"Then the code $f)$ has parameters $[2^{2m}, 2m+2, 2^{2m-1} - 2^{m-1}]$ and weight enumerator ().",
"The minimum weight codewords form a symmetric SDP design with parameters (REF ).",
"The weight enumerator () is obtained by substitution $\\ell =1$ in (REF ).",
"Since the number of minimum weight vectors is equal to the code length $2^{2m}$ , the 2-design ${\\mathbb {D}}$ supported by the codewords of minimum weight is symmetric.",
"Since every two blocks $B_1, B_2$ of ${\\mathbb {D}}$ intersect in $\\lambda = 2^{2m-2} - 2^{m-1}$ points, the sum of the two codewords supporting $B_1$ , $B_2$ is a codeword $c_{1,2} $ of weight $2^{2m-1}$ that belongs to the subcode ${\\mathrm {RM}}_2(1,2m)$ .",
"Let $B_3$ be a block distinct from $B_1$ and $B_2$ , and let $c_3$ be the codeword associated with $B_3$ .",
"Since $c_3$ is the truth table of a bent function, the sum $c_{1,2}+c_3$ is a codeword of weight $2^{2m-1} \\pm 2^{m-1}$ , thus its support is either a block or the complement of a block of ${\\mathbb {D}}$ .",
"Therefore, ${\\mathbb {D}}$ is an SDP design.",
"Theorem 13 The code $ f_1, \\cdots , f_\\ell )$ from Theorem REF is spanned by the set of codewords of minimum weight.",
"All we need to prove is that the copy of ${\\mathrm {RM}}_{2}(1,2m)$ which is a subcode of $,is spanned by some minimum weight codewords of $ .",
"It is known that the 2-rank (that is, the rank over ${\\mathrm {GF}}(2)$ ) of the incidence matrix of any symmetric SDP design ${\\mathbb {D}}$ with $2^{2m}$ points is equal to $2m+2$ (for a proof, see [12]).",
"This implies that the binary code spanned by ${\\mathbb {D}}$ contains the first order Reed-Muller code ${\\mathrm {RM}}_{2}(1,2m)$ .",
"Consequently the minimum weight vectors of the subcode ${f_1} = f_1)$ of $f_1,\\ldots , f_{\\ell })$ span the subcode of $ beingequivalent to $ RM(1,2m)$.$ Corollary 14 Two codes $f =f_1, \\cdots , f_s)$ , $g =g_1, \\cdots , g_s)$ obtained from bent vectorial functions $F(f_1, \\cdots , f_s)$ , $F(g_1, \\cdots , g_s)$ are equivalent if and only if the designs supported by their minimum weight vectors are isomorphic.",
"Example 15 Let $m=5$ .",
"Let $w$ be a generator of ${\\mathrm {GF}}(2^{10})^*$ with $w^{10} + w^6 + w^5 +w^3 + w^2 + w + 1=0$ .",
"Let $\\beta =w^{2^5+1}$ .",
"Then $\\beta $ is a generator of ${\\mathrm {GF}}(2^5)^*$ .",
"Define $\\beta _j=\\beta ^j$ for $1 \\le j \\le 5$ .",
"Then $\\lbrace \\beta _1, \\beta _2, \\beta _3,\\beta _4, \\beta _5\\rbrace $ is a basis of ${\\mathrm {GF}}(2^5)$ over ${\\mathrm {GF}}(2)$ .",
"Now consider the bent vectorial function $(f_1, f_2, f_3, f_4, f_5)$ in Example REF and the code $f_1, f_2, f_3)$ .",
"When $i=1$ and $i=7$ , the two codes $f_1, f_2, f_3)$ have parameters $[1024, 14, 496]$ and weight enumerator $1 + 7168z^{496} + 2046z^{512} + 7168z^{528} +z^{1024}.$ The two codes are not equivalent according to Magma.",
"It follows from Corollary REF that the two designs with parameters 2-$(1024, 496, 1680)$ supported by these codes are not isomorphic.",
"Note 16 Examples REF and REF give three inequivalent $[64,10,28]$ codes, and Example REF lists two inequivalent codes with parameters $[1024, 14, 496]$ , obtained from bent vectorial functions.",
"As we pointed out in Note REF , the code 1 from Example REF , does not have a 2-transitive group.",
"These examples, as well as further evidence provided by Theorem REF below, suggest the following plausible statement that we formulate as a conjecture.",
"Conjecture 17 For any given $\\ell $ in the range $1\\le \\ell \\le m$ , the number of inequivalent codes with parameters $[2^{2m}, 2m+1 +\\ell , 2^{2m-1}-2^{m-1}]$ obtained from $(2m,\\ell )$ bent vectorial functions via Theorem REF , grows exponentially with linear growth of $m$ , and most of these codes do not admit a 2-transitive automorphism group.",
"As it is customary, by “most” we mean that the limit of the ratio of the number of 2-transitive codes divided by the total number of codes approaches zero when $m$ grows to infinity.",
"The next theorem proves Conjecture REF in the case $\\ell = 1$ .",
"Theorem 18 (i) The number of inequivalent $[2^{2m},2m+2,2^{2m-1}-2^{m-1}]$ codes obtained from single bent functions from $GF(2^{2m})$ to $GF(2)$ grows exponentially with linear growth of $m$ .",
"(ii) For every given $m\\ge 2$ , there is exactly one (up to equivalence) code with parameters $[2^{2m},2m+2,2^{2m-1}-2^{m-1}]$ obtained from a bent function from $GF(2^{2m})$ to $GF(2)$ , that admits a 2-transitive automorphism group.",
"(i) By the Dillon-Schatz Theorem REF , the minimum weight codewords of a code $f)$ with parameters $[2^{2m},2m+2,2^{2m-1}-2^{m-1}]$ obtained from a bent function $f$ form a symmetric SDP design ${\\mathbb {D}}(f)$ with parameters (REF ).",
"It follows from Theorem REF that two codes $f_1)$ , $f_2)$ obtained from bent functions $f_1$ , $f_2$ are equivalent if and only if the the corresponding designs ${\\mathbb {D}}(f_1)$ , ${\\mathbb {D}}(f_1)$ are isomorphic.",
"Since the number of nonisomorphic SDP designs with parameters (REF ) grows exponentially when $m$ grows to infinity (Kantor [15]), the proof of part (i) is complete.",
"(ii) It follows from Theorem REF that the automorphism group of a code $f)$ obtained from a bent function $f$ coincides with the automorphism group of the design ${\\mathbb {D}}(f)$ supported by the codewords of minimum weight.",
"The design ${\\mathbb {D}}(f)$ is a symmetric 2-design with parameters (REF ).",
"It was proved by Kantor [17] that for every $m\\ge 2$ , there is exactly one (up to isomorphism) symmetric design with parameters (REF ) that admits a 2-transitive automorphism group.",
"This completes the proof of part (ii).",
"By Theorem REF , the codes based on single bent functions support symmetric 2-designs.",
"The next theorem determines the block intersection numbers of the design ${\\mathbb {D}}(f_1, \\cdots , f_\\ell )$ supported by the minimum weight vectors in the code $f_1, \\cdots , f_\\ell )$ from Theorem REF .",
"Theorem 19 Let ${\\mathbb {D}}={\\mathbb {D}}(f_1,\\ldots ,f_\\ell )$ , ($1\\le \\ell \\le m$ ), be a 2-design with parameters $ 2-(2^{2m}, 2^{2m-1} - 2^{m-1}, (2^\\ell -1)(2^{2m-2} - 2^{m-1})) $ supported by the minimum weight codewords of a code $f_1,\\ldots ,f_\\ell )$ defined as in Theorem REF .",
"(a) If $\\ell =1$ , ${\\mathbb {D}}$ is a symmetric SDP design, with block intersection number $\\lambda = 2^{2m-2} - 2^{m-1}$ .",
"(b) If $2 \\le \\ell \\le m$ , ${\\mathbb {D}}$ has the following three block intersection numbers: $s_1 = 2^{2m -2} - 2^{m-2}, \\ s_2 = 2^{2m -2} - 2^{m-1}, \\ s_3 =2^{2m -2} - 3\\cdot 2^{m-2}.$ For every block of ${\\mathbb {D}}$ , these intersection numbers occur with multiplicities $n_1 =2^{m}(2^m +1)(2^{\\ell -1} -1), \\ n_2 = 2^{2m} -1, \\ n_3 = 2^{m}(2^m -1)(2^{\\ell -1} -1).$ Case (a) follows from Theorem REF .",
"(b) Assume that $2 \\le \\ell \\le m$ .",
"Let $w_1$ , $w_2$ be two distinct codewords of weight $2^{2m-1} - 2^{m-1}$ .",
"The Hamming distance $d(w_1,w_2)$ between $w_1$ and $w_2$ is equal to $ 2(2^{2m-1} - 2^{m-1}) - 2s, $ where $s$ is the size of the intersection of the supports of $w_1$ and $w_2$ .",
"Since the distance between $w_1$ and $w_2$ is either $2^{2m-1} - 2^{m-1}$ , or $2^{2m-1}$ , or $2^{2m-1} + 2^{m-1}$ , the size $s$ of the intersection of the two blocks of ${\\mathbb {D}}$ supported by $w_1$ , $w_2$ can take only the values $s_i$ , $1\\le i \\le 3$ , given by (REF ).",
"Let $B$ be a block of ${\\mathbb {D}}$ supported by a codeword of weight $2^{2m-1} - 2^{m-1}$ , and let $n_i$ , ($1\\le i \\le 3$ ), denote the number of blocks of ${\\mathbb {D}}$ that intersect $B$ in $s_i$ points.",
"Let ${\\bf r} = (2^\\ell -1)(2^{2m-1} - 2^{m-1})$ denote the number of blocks of ${\\mathbb {D}}$ containing a single point, and let $b = (2^\\ell -1)2^{2m}$ denote the total number of blocks of ${\\mathbb {D}}$ .",
"Finally, let $k = 2^{2m-1} - 2^{m-1}$ denote the size of a block, and let $\\lambda =(2^\\ell -1)(2^{2m-2} - 2^{m-1})$ denote the number of blocks containing two points.",
"We have $n_1 + n_2 + n_3 & = & b - 1,\\\\s_{1}n_1 + s_{2}n_2 + s_{3}n_3 & = & k({\\bf r} - 1), \\\\s_{1}(s_{1} - 1)n_1 + s_{2}(s_{2} -1)n_2 + s_{3}(s_{3} -1)n_3 & = &k(k-1)(\\lambda -1).$ The second and the third equation count in two ways the appearances of single points and ordered pairs of points of $B$ in other blocks of ${\\mathbb {D}}$ .",
"The unique solution of this system of equations for $n_1,\\ n_2, \\ n_3$ is given by (REF ).",
"Note 20 A bent set is a set $S$ of bent functions such that the sum of every two functions from $S$ is also a bent function [4].",
"Since every $(2m,\\ell )$ bent vectorial function gives rise to a bent set consisting of $2^\\ell $ functions [4], it follows from [4] that the set of blocks of the design ${\\mathbb {D}}$ is a union of $2^\\ell -1$ linked system of symmetric 2-$(2^{2m}, 2^{2m-1}-2^{m-1},2^{2m-2}-2^{m-1})$ designs.",
"This gives an alternative proof of Theorem REF and Theorem REF (b).",
"Note 21 For every integer $m \\ge 2$ , any code $f_1, f_2, \\ldots , f_m)$ based on a bent vectorial function $F(x)=(f_1(x), f_2(x), \\cdots , f_m(x))$ from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2)^m$ , contains $2^{m}-1$ subcodes $=(f_{j_1},\\ldots , f_{j_s})$ , $j_1 < \\cdots < j_s \\le m$ , such that ${\\mathrm {RM}}_2(1, 2m) \\subset \\subseteq f_1, \\ldots , f_m).$ Each subcode $$ holds 2-designs.",
"This may be the only known chain of linear codes, included in each other, other than the chain of the Reed-Muller codes, $ {\\mathrm {RM}}_2(1, 2m) \\subset {\\mathrm {RM}}_2(2, 2m) \\subset \\cdots \\subset {\\mathrm {RM}}_2(m-2, 2m).",
"$ such that all codes in the chain support nontrivial 2-designs.",
"Note 22 We would demonstrate that the characterization of bent vectorial functions in Theorem REF can be used to construct bent vectorial functions.",
"To this end, consider the extended binary narrow-sense primitive BCH code of length $2^{2m}-1$ and designed distance $2^{2m-1}-1-2^{m-1}$ , which is affine-invariant and holds 2-designs [9].",
"This code has the desired weight enumerator of (REF ) for $\\ell = m$ [9].",
"It can be proved with the Delsarte theorem that the trace representation of this code is equivalent to the following code: $\\left\\lbrace \\left(f_{a,b,h}(x)\\right)_{x\\in {\\mathrm {GF}}(2^{2m})}:a \\in {\\mathrm {GF}}(2^m), \\, b \\in {\\mathrm {GF}}(2^{2m}), \\, h \\in {\\mathrm {GF}}(2)\\right\\rbrace ,$ where $f_{a,b,h}(x)={\\mathrm {Tr}}_{m/1}\\left[a {\\mathrm {Tr}}_{2m/m}\\left(x^{1+2^{m-1}}\\right) \\right] + {\\mathrm {Tr}}_{2m/1}(bx) + h.$ It then follows from Theorem REF that ${\\mathrm {Tr}}_{2m/m}(x^{1+2^{m-1}})$ is a bent vectorial function from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2^m)$ .",
"Note that this bent vectorial function may not be new.",
"But our purpose here is to show that bent vectorial functions could be constructed from special linear codes.",
"Conversely, we could say that the extended narrow-sense BCH code of length $2^{2m}-1$ and designed distance $2^{2m-1}-1-2^{m-1}$ is in fact generated from the bent vectorial function ${\\mathrm {Tr}}_{2m/m}(x^{1+2^{m-1}})$ from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2^m)$ using the construction of Note REF .",
"Example REF gives a demonstration of that.",
"Thus, all known binary codes with the weight enumerator (REF ) for some $1\\le \\ell \\le m$ and arbitrary $m\\ge 2$ are obtained from the bent vectorial function construction.",
"As shown in Example REF , all $[16,7,6]$ codes obtained from $(4,2)$ bent vectorial functions are equivalent.",
"Example REF shows that there are at least three inequivalent $[64, 10, 28]$ binary codes from bent vectorial functions, one of these codes being an extended BCH code.",
"Note 23 It is known that two designs ${\\mathbb {D}}(f)$ and ${\\mathbb {D}}(g)$ from two single bent Boolean functions $f$ and $g$ on ${\\mathrm {GF}}(2^{2m})$ are isomorphic if and only if $f$ and $g$ are weakly affinely equivalent [8].",
"Although the classification of bent Boolean functions into weakly affinely equivalent classes is open, the results from [15] and [8] imply that the number of nonisomorphic SDP designs and inequivalent bent functions in $2m$ variables grows exponentially with linear growth of $m$ .",
"Note 24 Two $(n, \\ell )$ vectorial Boolean functions $(f_1(x), \\cdots , f_\\ell (x))$ and $(g_1(x), \\cdots , g_\\ell (x))$ from ${\\mathrm {GF}}(2^n)$ to ${\\mathrm {GF}}(2)^\\ell $ are said to be EA-equivalent if there are an automorphism of $({\\mathrm {GF}}(2^n), +)$ , a homomorphism $L$ from $({\\mathrm {GF}}(2^n),+)$ to $({\\mathrm {GF}}(2)^\\ell , +)$ , an $\\ell \\times \\ell $ invertible matrix $M$ over ${\\mathrm {GF}}(2)$ , an element $a \\in {\\mathrm {GF}}(2^n)$ , and an element $b \\in {\\mathrm {GF}}(2)^\\ell $ such that $(g_1(x), \\cdots , g_\\ell (x))=(f_1(A(x)+a), \\cdots , f_\\ell (A(x)+a))M +L(x) +b $ for all $x \\in {\\mathrm {GF}}(2^n)$ .",
"Let $(f_1(x), \\cdots , f_\\ell (x))$ and $(g_1(x), \\cdots , g_\\ell (x))$ be two bent vectorial functions from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2)^\\ell $ .",
"We conjecture that the designs ${\\mathbb {D}}(f_{1}, \\cdots , f_{\\ell })$ and ${\\mathbb {D}}(g_{1}, \\cdots , g_{\\ell })$ are isomorphic if and only if $(f_1(x), \\cdots , f_\\ell (x))$ and $(g_1(x), \\cdots , g_\\ell (x))$ are EA-equivalent.",
"The reader is invited to attack this open problem.",
"Suppose that ${\\mathbb {D}}$ is a 2-design with parameters () obtained from a bent vectorial function $F(x)=(f_1(x), f_2(x), \\cdots , f_\\ell (x))$ , ($1 \\le \\ell \\le m$ ), via the construction from Theorem REF .",
"Let $\\cal {B}$ be the block set of ${\\mathbb {D}}$ .",
"If $B$ is a block of ${\\mathbb {D}}$ , we consider the collection of new blocks ${\\mathcal {B}}^{de}$ consisting of intersections $B \\cap B^{\\prime }$ such that $B^{\\prime } \\in \\cal {B}$ and $| B \\cap B^{\\prime } |=2^{2m-2} - 2^{m-1}$ .",
"Theorem 25 For each $B \\in {\\mathbb {D}}$ , the incidence structure $(B, {\\mathcal {B}}^{de})$ is a quasi-symmetric design with parameters $2-( 2^{2m-1}-2^{m-1},\\, 2^{2m-2}-2^{m-1},\\, 2^{2m-2}-2^{m-1}-1)$ and intersection numbers $2^{2m-3} - 2^{m-2}$ and $2^{2m-3} - 2^{m-1}$ .",
"By Theorem REF , there are exactly $2^{2m} -1$ blocks that intersect $B$ in $2^{2m-2}-2^{m-1}$ points.",
"Together with $B$ , these blocks form a symmetric SDP design $D$ with parameters 2-$(2^{2m}, 2^{2m-1} - 2^{m-1}, 2^{2m-2} - 2^{m-1})$ .",
"The incidence structure $(B, {\\mathcal {B}})^{de}$ is a derived design of $D$ .",
"It was proved in [12] that each derived design of a symmetric SDP 2-$(2^{2m}, 2^{2m-1} - 2^{m-1}, 2^{2m-2} - 2^{m-1})$ design is quasi-symmetric design with intersection numbers $2^{2m-3} - 2^{m-2}$ and $2^{2m-3} - 2^{m-1}$ , and having the additional property that the symmetric difference of every two blocks is either a block or the complement of a block.",
"Note 26 Let $m >1$ be an integer.",
"Let $F$ be a bent vectorial function from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2^m)$ .",
"Let $A$ be a subgroup of order $2^s$ of $({\\mathrm {GF}}(2^m), +)$ .",
"Define a binary code by ${A}:=\\lbrace ({\\mathrm {Tr}}_{m/1}(aF(x))+{\\mathrm {Tr}}_{2m/1}(bx)+c)_{x \\in {\\mathrm {GF}}(2^{2m})}:a \\in A, b \\in {\\mathrm {GF}}(2^{2m}), c \\in {\\mathrm {GF}}(2)\\rbrace .$ It can be shown that ${A}$ can be viewed as a code $f_{i_1}, \\cdots , f_{i_s})$ obtained from a bent vectorial function $(f_{i_1}, \\cdots , f_{i_s})$ ."
],
[
"Summary and concluding remarks",
"The contributions of this paper are the following.",
"A coding-theoretic characterization of bent vectorial functions (Theorem REF ).",
"A construction of a two-parameter family of four-weight binary linear codes with parameters $[2^{2m}, 2m+1+\\ell ,2^{2m-1}-2^{m-1}]$ for all $1 \\le \\ell \\le m$ and all $m\\ge 2$ , obtained from $(2m, \\ell )$ bent vectorial functions (Theorem REF ).",
"The parameters of these codes appear to be new when $2 \\le \\ell \\le m-1$ .",
"This family of codes includes some optimal codes, as well as codes meeting the BCH bound.",
"These codes do not satisfy the conditions of the Assmus-Mattson theorem, but nevertheless hold 2-designs.",
"It is plausible that most of these codes do not admit 2-transitive automorphism groups (Conjecture REF and Theorem REF ).",
"A new construction of a two-parameter family of 2-designs with parameters $2\\mbox{--}(2^{2m}, \\ 2^{2m-1}-2^{m-1}, \\ (2^\\ell -1)(2^{2m-2}-2^{m-1})),$ and having three block intersection numbers, where $2\\le \\ell \\le m$ , based on bent vectorial functions (Theorem REF and Theorem REF ).",
"This construction is a generalization of the construction of SDP designs from single bent functions given in [8].",
"The number of nonisomorphic designs with parameters (REF ) in the special case when $\\ell =1$ , grows exponentially with $m$ by a known theorem of Kantor [15].",
"It is an interesting open problem to prove that the number of nonisomorphic designs with parameters (REF ) grows exponentially for any fixed $\\ell >1$ .",
"Finally, we would like to mention that vectorial Boolean functions were employed in a different way to construct binary linear codes in [20].",
"The codes from [20] have different parameters from the codes described in this paper."
],
[
"Acknowledgements",
"Vladimir Tonchev acknowledges partial support by a Fulbright grant, and would like to thank the Hong Kong University of Science and Technology for the kind hospitality and support during his visit, when a large portion of this paper was written.",
"The research of Cunsheng Ding was supported by the Hong Kong Research Grants Council, under Grant No.",
"16300418.",
"The authors wish to thank the anonymous reviewers for their valuable comments and suggestions for improving the manuscript."
],
[
"A construction of 2-designs from bent vectorial functions",
"The following theorem establishes that the binary codes based on bent vectorial functions support 2-designs, despite that these codes do not meet the conditions of the Assmus-Mattson theorem for 2-designs.",
"Theorem 11 Let $F(x)=(f_1(x), f_2(x), \\cdots , f_\\ell (x))$ be a bent vectorial function from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2)^\\ell $ , where $m\\ge 2$ and $1\\le \\ell \\le m$ .",
"Let $ f_1, \\cdots , f_\\ell )$ be the binary linear code with parameters $[2^{2m}, 2m+1 + \\ell , 2^{2m-1} - 2^{m-1}]$ defined in Theorem REF .",
"(a) The codewords of $ of minimum weight hold a 2-design $ D$with parameters\\begin{equation}2-(2^{2m}, 2^{2m-1} - 2^{m-1}, (2^\\ell -1)(2^{2m-2} - 2^{m-1})).\\end{equation}(b) The codewords of $ of weight $2^{2m-1} + 2^{m-1}$ hold a 2-design $\\overline{{\\mathbb {D}}}$ with parameters $2-(2^{2m}, 2^{2m-1} + 2^{m-1}, (2^\\ell -1)(2^{2m-2} + 2^{m-1})).$ Since $ contains $ RM2(1,2m)$, and the minimum distanceof $ RM2(1,2m)$ is 4,the minimum distance $ d$ of $$ is at least 4.",
"Applying theMacWilliams transform (see, for example \\cite [p. 41]{vanLint})to the weight enumerator (\\ref {eqn-wtenumerator111}) of $ shows that $d^{\\perp } =4$ .",
"It follows from the Assmus-Mattson theorem (Theorem REF ) that the codewords of any given nonzero weight $w< 2^{2m}$ in $ hold a 1-design.$ However, we will prove that $ actually holds 2-designs, despite thatthe Assmus-Mattson theorem guarantees only1-designs to be supported by $ .",
"Since the subcode ${\\mathrm {RM}}_2(1,2m)$ of $ containsall codewords of $ of weight $2^{2m-1}$ , the codewords of this weight hold a 3-design $\\cal {A}$ with parameters 3-$(2^{2m}, 2^{2m-1}, 2^{2m-2} -1)$ .",
"We note that $\\cal {A}$ is a 2-design with $\\lambda _2 = \\frac{2^{2m}-2}{2^{2m-1} -2}\\cdot (2^{2m-2} -1)=2^{2m-1}-1.$ Let ${\\mathbb {D}}$ be the 1-design supported by codewords of weight $2^{2m-1} - 2^{m-1}$ .",
"Since the number of codewords of weight $2^{2m-1} - 2^{m-1}$ is equal to $(2^\\ell -1)2^{2m}$ , ${\\mathbb {D}}$ is a 1-design with parameters 1-$(2^{2m}, 2^{2m-1} - 2^{m-1}, (2^\\ell -1)(2^{2m-1} - 2^{m-1}))$ .",
"Every codeword of $ of weight $ 22m-1 + 2m-1$ is the sum ofa codeword of weight $ 22m-1 - 2m-1$ and the all-one vector.Thus, the codewords of weight $ 22m-1 + 2m-1$ hold a 1-design $D$having parameters1-$ (22m, 22m-1 + 2m-1, (2-1)(22m-1 + 2m-1))$.Clearly, $D$is the complementary design of $ D$, that is, every block of $D$is the complement of some block of $ D$.$ Let $M$ be the $2^{2m+1+\\ell } \\times 2^{2m}$ $(0,1)$ -matrix having as rows the codewords of $.",
"Since $ d =4$, $ M$ is an orthogonal array of strength 3,that is, for every integer $ i$, $ 1 i 3$, and for every set of $ i$ distinct columnsof $ M$, every binary vector with $ i$ components appears exactly$ 22m+1+- i$ times among the rows of the $ 22m+1+ i$ submatrix of $ M$formed by the chosen $ i$ columns.In particular, any $ 22m+1+ 2$ submatrix consisting of two distinctcolumns of $ M$ contains the binary vector $ (1,1)$ exactly $ 22m+-1$ timesas a row.",
"Among these $ 22m+-1$ rows, one corresponds to the all-one codewordof $ , $2^{2m-1}-1$ rows correspond to codewords of weight $2^{2m-1}$ (by equation (REF )), and the remaining $2^{2m +\\ell -1} - 1 - (2^{2m-1}-1)=(2^{\\ell } -1)2^{2m-1}$ rows are labeled by codewords of weight $2^{2m-1} \\pm 2^{m-1}$ , corresponding to blocks of ${\\mathbb {D}}$ and $\\overline{{\\mathbb {D}}}$ .",
"Let now $1\\le c_1 < c_2 \\le 2^{2m}$ be two distinct columns of $M$ .",
"These two columns label two distinct points of ${\\mathbb {D}}$ (resp.",
"$\\overline{{\\mathbb {D}}}$ ).",
"Let $\\lambda $ denote the number of blocks of ${\\mathbb {D}}$ that are incident with $c_1$ and $c_2$ .",
"Then the pair $\\lbrace c_1, c_2 \\rbrace $ is incident with $(2^\\ell -1)2^{2m} -2(2^\\ell -1)(2^{2m-1}- 2^{m-1}) + \\lambda =(2^\\ell -1)2^m + \\lambda $ blocks of the complementary design $\\overline{{\\mathbb {D}}}$ .",
"It follows from (REF ) and (REF ) that $ (2^\\ell -1)2^m + 2\\lambda =(2^{\\ell } -1)2^{2m-1}, $ whence $ \\lambda = (2^\\ell -1)(2^{2m-2} - 2^{m-1}), $ and the statements (a) and (b) of the theorem follow.",
"The special case $\\ell =1$ in Theorem REF implies as a corollary the following result of Dillon and Schatz [8].",
"Theorem 12 Let $f(x)$ be a bent function from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2)$ .",
"Then the code $f)$ has parameters $[2^{2m}, 2m+2, 2^{2m-1} - 2^{m-1}]$ and weight enumerator ().",
"The minimum weight codewords form a symmetric SDP design with parameters (REF ).",
"The weight enumerator () is obtained by substitution $\\ell =1$ in (REF ).",
"Since the number of minimum weight vectors is equal to the code length $2^{2m}$ , the 2-design ${\\mathbb {D}}$ supported by the codewords of minimum weight is symmetric.",
"Since every two blocks $B_1, B_2$ of ${\\mathbb {D}}$ intersect in $\\lambda = 2^{2m-2} - 2^{m-1}$ points, the sum of the two codewords supporting $B_1$ , $B_2$ is a codeword $c_{1,2} $ of weight $2^{2m-1}$ that belongs to the subcode ${\\mathrm {RM}}_2(1,2m)$ .",
"Let $B_3$ be a block distinct from $B_1$ and $B_2$ , and let $c_3$ be the codeword associated with $B_3$ .",
"Since $c_3$ is the truth table of a bent function, the sum $c_{1,2}+c_3$ is a codeword of weight $2^{2m-1} \\pm 2^{m-1}$ , thus its support is either a block or the complement of a block of ${\\mathbb {D}}$ .",
"Therefore, ${\\mathbb {D}}$ is an SDP design.",
"Theorem 13 The code $ f_1, \\cdots , f_\\ell )$ from Theorem REF is spanned by the set of codewords of minimum weight.",
"All we need to prove is that the copy of ${\\mathrm {RM}}_{2}(1,2m)$ which is a subcode of $,is spanned by some minimum weight codewords of $ .",
"It is known that the 2-rank (that is, the rank over ${\\mathrm {GF}}(2)$ ) of the incidence matrix of any symmetric SDP design ${\\mathbb {D}}$ with $2^{2m}$ points is equal to $2m+2$ (for a proof, see [12]).",
"This implies that the binary code spanned by ${\\mathbb {D}}$ contains the first order Reed-Muller code ${\\mathrm {RM}}_{2}(1,2m)$ .",
"Consequently the minimum weight vectors of the subcode ${f_1} = f_1)$ of $f_1,\\ldots , f_{\\ell })$ span the subcode of $ beingequivalent to $ RM(1,2m)$.$ Corollary 14 Two codes $f =f_1, \\cdots , f_s)$ , $g =g_1, \\cdots , g_s)$ obtained from bent vectorial functions $F(f_1, \\cdots , f_s)$ , $F(g_1, \\cdots , g_s)$ are equivalent if and only if the designs supported by their minimum weight vectors are isomorphic.",
"Example 15 Let $m=5$ .",
"Let $w$ be a generator of ${\\mathrm {GF}}(2^{10})^*$ with $w^{10} + w^6 + w^5 +w^3 + w^2 + w + 1=0$ .",
"Let $\\beta =w^{2^5+1}$ .",
"Then $\\beta $ is a generator of ${\\mathrm {GF}}(2^5)^*$ .",
"Define $\\beta _j=\\beta ^j$ for $1 \\le j \\le 5$ .",
"Then $\\lbrace \\beta _1, \\beta _2, \\beta _3,\\beta _4, \\beta _5\\rbrace $ is a basis of ${\\mathrm {GF}}(2^5)$ over ${\\mathrm {GF}}(2)$ .",
"Now consider the bent vectorial function $(f_1, f_2, f_3, f_4, f_5)$ in Example REF and the code $f_1, f_2, f_3)$ .",
"When $i=1$ and $i=7$ , the two codes $f_1, f_2, f_3)$ have parameters $[1024, 14, 496]$ and weight enumerator $1 + 7168z^{496} + 2046z^{512} + 7168z^{528} +z^{1024}.$ The two codes are not equivalent according to Magma.",
"It follows from Corollary REF that the two designs with parameters 2-$(1024, 496, 1680)$ supported by these codes are not isomorphic.",
"Note 16 Examples REF and REF give three inequivalent $[64,10,28]$ codes, and Example REF lists two inequivalent codes with parameters $[1024, 14, 496]$ , obtained from bent vectorial functions.",
"As we pointed out in Note REF , the code 1 from Example REF , does not have a 2-transitive group.",
"These examples, as well as further evidence provided by Theorem REF below, suggest the following plausible statement that we formulate as a conjecture.",
"Conjecture 17 For any given $\\ell $ in the range $1\\le \\ell \\le m$ , the number of inequivalent codes with parameters $[2^{2m}, 2m+1 +\\ell , 2^{2m-1}-2^{m-1}]$ obtained from $(2m,\\ell )$ bent vectorial functions via Theorem REF , grows exponentially with linear growth of $m$ , and most of these codes do not admit a 2-transitive automorphism group.",
"As it is customary, by “most” we mean that the limit of the ratio of the number of 2-transitive codes divided by the total number of codes approaches zero when $m$ grows to infinity.",
"The next theorem proves Conjecture REF in the case $\\ell = 1$ .",
"Theorem 18 (i) The number of inequivalent $[2^{2m},2m+2,2^{2m-1}-2^{m-1}]$ codes obtained from single bent functions from $GF(2^{2m})$ to $GF(2)$ grows exponentially with linear growth of $m$ .",
"(ii) For every given $m\\ge 2$ , there is exactly one (up to equivalence) code with parameters $[2^{2m},2m+2,2^{2m-1}-2^{m-1}]$ obtained from a bent function from $GF(2^{2m})$ to $GF(2)$ , that admits a 2-transitive automorphism group.",
"(i) By the Dillon-Schatz Theorem REF , the minimum weight codewords of a code $f)$ with parameters $[2^{2m},2m+2,2^{2m-1}-2^{m-1}]$ obtained from a bent function $f$ form a symmetric SDP design ${\\mathbb {D}}(f)$ with parameters (REF ).",
"It follows from Theorem REF that two codes $f_1)$ , $f_2)$ obtained from bent functions $f_1$ , $f_2$ are equivalent if and only if the the corresponding designs ${\\mathbb {D}}(f_1)$ , ${\\mathbb {D}}(f_1)$ are isomorphic.",
"Since the number of nonisomorphic SDP designs with parameters (REF ) grows exponentially when $m$ grows to infinity (Kantor [15]), the proof of part (i) is complete.",
"(ii) It follows from Theorem REF that the automorphism group of a code $f)$ obtained from a bent function $f$ coincides with the automorphism group of the design ${\\mathbb {D}}(f)$ supported by the codewords of minimum weight.",
"The design ${\\mathbb {D}}(f)$ is a symmetric 2-design with parameters (REF ).",
"It was proved by Kantor [17] that for every $m\\ge 2$ , there is exactly one (up to isomorphism) symmetric design with parameters (REF ) that admits a 2-transitive automorphism group.",
"This completes the proof of part (ii).",
"By Theorem REF , the codes based on single bent functions support symmetric 2-designs.",
"The next theorem determines the block intersection numbers of the design ${\\mathbb {D}}(f_1, \\cdots , f_\\ell )$ supported by the minimum weight vectors in the code $f_1, \\cdots , f_\\ell )$ from Theorem REF .",
"Theorem 19 Let ${\\mathbb {D}}={\\mathbb {D}}(f_1,\\ldots ,f_\\ell )$ , ($1\\le \\ell \\le m$ ), be a 2-design with parameters $ 2-(2^{2m}, 2^{2m-1} - 2^{m-1}, (2^\\ell -1)(2^{2m-2} - 2^{m-1})) $ supported by the minimum weight codewords of a code $f_1,\\ldots ,f_\\ell )$ defined as in Theorem REF .",
"(a) If $\\ell =1$ , ${\\mathbb {D}}$ is a symmetric SDP design, with block intersection number $\\lambda = 2^{2m-2} - 2^{m-1}$ .",
"(b) If $2 \\le \\ell \\le m$ , ${\\mathbb {D}}$ has the following three block intersection numbers: $s_1 = 2^{2m -2} - 2^{m-2}, \\ s_2 = 2^{2m -2} - 2^{m-1}, \\ s_3 =2^{2m -2} - 3\\cdot 2^{m-2}.$ For every block of ${\\mathbb {D}}$ , these intersection numbers occur with multiplicities $n_1 =2^{m}(2^m +1)(2^{\\ell -1} -1), \\ n_2 = 2^{2m} -1, \\ n_3 = 2^{m}(2^m -1)(2^{\\ell -1} -1).$ Case (a) follows from Theorem REF .",
"(b) Assume that $2 \\le \\ell \\le m$ .",
"Let $w_1$ , $w_2$ be two distinct codewords of weight $2^{2m-1} - 2^{m-1}$ .",
"The Hamming distance $d(w_1,w_2)$ between $w_1$ and $w_2$ is equal to $ 2(2^{2m-1} - 2^{m-1}) - 2s, $ where $s$ is the size of the intersection of the supports of $w_1$ and $w_2$ .",
"Since the distance between $w_1$ and $w_2$ is either $2^{2m-1} - 2^{m-1}$ , or $2^{2m-1}$ , or $2^{2m-1} + 2^{m-1}$ , the size $s$ of the intersection of the two blocks of ${\\mathbb {D}}$ supported by $w_1$ , $w_2$ can take only the values $s_i$ , $1\\le i \\le 3$ , given by (REF ).",
"Let $B$ be a block of ${\\mathbb {D}}$ supported by a codeword of weight $2^{2m-1} - 2^{m-1}$ , and let $n_i$ , ($1\\le i \\le 3$ ), denote the number of blocks of ${\\mathbb {D}}$ that intersect $B$ in $s_i$ points.",
"Let ${\\bf r} = (2^\\ell -1)(2^{2m-1} - 2^{m-1})$ denote the number of blocks of ${\\mathbb {D}}$ containing a single point, and let $b = (2^\\ell -1)2^{2m}$ denote the total number of blocks of ${\\mathbb {D}}$ .",
"Finally, let $k = 2^{2m-1} - 2^{m-1}$ denote the size of a block, and let $\\lambda =(2^\\ell -1)(2^{2m-2} - 2^{m-1})$ denote the number of blocks containing two points.",
"We have $n_1 + n_2 + n_3 & = & b - 1,\\\\s_{1}n_1 + s_{2}n_2 + s_{3}n_3 & = & k({\\bf r} - 1), \\\\s_{1}(s_{1} - 1)n_1 + s_{2}(s_{2} -1)n_2 + s_{3}(s_{3} -1)n_3 & = &k(k-1)(\\lambda -1).$ The second and the third equation count in two ways the appearances of single points and ordered pairs of points of $B$ in other blocks of ${\\mathbb {D}}$ .",
"The unique solution of this system of equations for $n_1,\\ n_2, \\ n_3$ is given by (REF ).",
"Note 20 A bent set is a set $S$ of bent functions such that the sum of every two functions from $S$ is also a bent function [4].",
"Since every $(2m,\\ell )$ bent vectorial function gives rise to a bent set consisting of $2^\\ell $ functions [4], it follows from [4] that the set of blocks of the design ${\\mathbb {D}}$ is a union of $2^\\ell -1$ linked system of symmetric 2-$(2^{2m}, 2^{2m-1}-2^{m-1},2^{2m-2}-2^{m-1})$ designs.",
"This gives an alternative proof of Theorem REF and Theorem REF (b).",
"Note 21 For every integer $m \\ge 2$ , any code $f_1, f_2, \\ldots , f_m)$ based on a bent vectorial function $F(x)=(f_1(x), f_2(x), \\cdots , f_m(x))$ from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2)^m$ , contains $2^{m}-1$ subcodes $=(f_{j_1},\\ldots , f_{j_s})$ , $j_1 < \\cdots < j_s \\le m$ , such that ${\\mathrm {RM}}_2(1, 2m) \\subset \\subseteq f_1, \\ldots , f_m).$ Each subcode $$ holds 2-designs.",
"This may be the only known chain of linear codes, included in each other, other than the chain of the Reed-Muller codes, $ {\\mathrm {RM}}_2(1, 2m) \\subset {\\mathrm {RM}}_2(2, 2m) \\subset \\cdots \\subset {\\mathrm {RM}}_2(m-2, 2m).",
"$ such that all codes in the chain support nontrivial 2-designs.",
"Note 22 We would demonstrate that the characterization of bent vectorial functions in Theorem REF can be used to construct bent vectorial functions.",
"To this end, consider the extended binary narrow-sense primitive BCH code of length $2^{2m}-1$ and designed distance $2^{2m-1}-1-2^{m-1}$ , which is affine-invariant and holds 2-designs [9].",
"This code has the desired weight enumerator of (REF ) for $\\ell = m$ [9].",
"It can be proved with the Delsarte theorem that the trace representation of this code is equivalent to the following code: $\\left\\lbrace \\left(f_{a,b,h}(x)\\right)_{x\\in {\\mathrm {GF}}(2^{2m})}:a \\in {\\mathrm {GF}}(2^m), \\, b \\in {\\mathrm {GF}}(2^{2m}), \\, h \\in {\\mathrm {GF}}(2)\\right\\rbrace ,$ where $f_{a,b,h}(x)={\\mathrm {Tr}}_{m/1}\\left[a {\\mathrm {Tr}}_{2m/m}\\left(x^{1+2^{m-1}}\\right) \\right] + {\\mathrm {Tr}}_{2m/1}(bx) + h.$ It then follows from Theorem REF that ${\\mathrm {Tr}}_{2m/m}(x^{1+2^{m-1}})$ is a bent vectorial function from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2^m)$ .",
"Note that this bent vectorial function may not be new.",
"But our purpose here is to show that bent vectorial functions could be constructed from special linear codes.",
"Conversely, we could say that the extended narrow-sense BCH code of length $2^{2m}-1$ and designed distance $2^{2m-1}-1-2^{m-1}$ is in fact generated from the bent vectorial function ${\\mathrm {Tr}}_{2m/m}(x^{1+2^{m-1}})$ from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2^m)$ using the construction of Note REF .",
"Example REF gives a demonstration of that.",
"Thus, all known binary codes with the weight enumerator (REF ) for some $1\\le \\ell \\le m$ and arbitrary $m\\ge 2$ are obtained from the bent vectorial function construction.",
"As shown in Example REF , all $[16,7,6]$ codes obtained from $(4,2)$ bent vectorial functions are equivalent.",
"Example REF shows that there are at least three inequivalent $[64, 10, 28]$ binary codes from bent vectorial functions, one of these codes being an extended BCH code.",
"Note 23 It is known that two designs ${\\mathbb {D}}(f)$ and ${\\mathbb {D}}(g)$ from two single bent Boolean functions $f$ and $g$ on ${\\mathrm {GF}}(2^{2m})$ are isomorphic if and only if $f$ and $g$ are weakly affinely equivalent [8].",
"Although the classification of bent Boolean functions into weakly affinely equivalent classes is open, the results from [15] and [8] imply that the number of nonisomorphic SDP designs and inequivalent bent functions in $2m$ variables grows exponentially with linear growth of $m$ .",
"Note 24 Two $(n, \\ell )$ vectorial Boolean functions $(f_1(x), \\cdots , f_\\ell (x))$ and $(g_1(x), \\cdots , g_\\ell (x))$ from ${\\mathrm {GF}}(2^n)$ to ${\\mathrm {GF}}(2)^\\ell $ are said to be EA-equivalent if there are an automorphism of $({\\mathrm {GF}}(2^n), +)$ , a homomorphism $L$ from $({\\mathrm {GF}}(2^n),+)$ to $({\\mathrm {GF}}(2)^\\ell , +)$ , an $\\ell \\times \\ell $ invertible matrix $M$ over ${\\mathrm {GF}}(2)$ , an element $a \\in {\\mathrm {GF}}(2^n)$ , and an element $b \\in {\\mathrm {GF}}(2)^\\ell $ such that $(g_1(x), \\cdots , g_\\ell (x))=(f_1(A(x)+a), \\cdots , f_\\ell (A(x)+a))M +L(x) +b $ for all $x \\in {\\mathrm {GF}}(2^n)$ .",
"Let $(f_1(x), \\cdots , f_\\ell (x))$ and $(g_1(x), \\cdots , g_\\ell (x))$ be two bent vectorial functions from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2)^\\ell $ .",
"We conjecture that the designs ${\\mathbb {D}}(f_{1}, \\cdots , f_{\\ell })$ and ${\\mathbb {D}}(g_{1}, \\cdots , g_{\\ell })$ are isomorphic if and only if $(f_1(x), \\cdots , f_\\ell (x))$ and $(g_1(x), \\cdots , g_\\ell (x))$ are EA-equivalent.",
"The reader is invited to attack this open problem.",
"Suppose that ${\\mathbb {D}}$ is a 2-design with parameters () obtained from a bent vectorial function $F(x)=(f_1(x), f_2(x), \\cdots , f_\\ell (x))$ , ($1 \\le \\ell \\le m$ ), via the construction from Theorem REF .",
"Let $\\cal {B}$ be the block set of ${\\mathbb {D}}$ .",
"If $B$ is a block of ${\\mathbb {D}}$ , we consider the collection of new blocks ${\\mathcal {B}}^{de}$ consisting of intersections $B \\cap B^{\\prime }$ such that $B^{\\prime } \\in \\cal {B}$ and $| B \\cap B^{\\prime } |=2^{2m-2} - 2^{m-1}$ .",
"Theorem 25 For each $B \\in {\\mathbb {D}}$ , the incidence structure $(B, {\\mathcal {B}}^{de})$ is a quasi-symmetric design with parameters $2-( 2^{2m-1}-2^{m-1},\\, 2^{2m-2}-2^{m-1},\\, 2^{2m-2}-2^{m-1}-1)$ and intersection numbers $2^{2m-3} - 2^{m-2}$ and $2^{2m-3} - 2^{m-1}$ .",
"By Theorem REF , there are exactly $2^{2m} -1$ blocks that intersect $B$ in $2^{2m-2}-2^{m-1}$ points.",
"Together with $B$ , these blocks form a symmetric SDP design $D$ with parameters 2-$(2^{2m}, 2^{2m-1} - 2^{m-1}, 2^{2m-2} - 2^{m-1})$ .",
"The incidence structure $(B, {\\mathcal {B}})^{de}$ is a derived design of $D$ .",
"It was proved in [12] that each derived design of a symmetric SDP 2-$(2^{2m}, 2^{2m-1} - 2^{m-1}, 2^{2m-2} - 2^{m-1})$ design is quasi-symmetric design with intersection numbers $2^{2m-3} - 2^{m-2}$ and $2^{2m-3} - 2^{m-1}$ , and having the additional property that the symmetric difference of every two blocks is either a block or the complement of a block.",
"Note 26 Let $m >1$ be an integer.",
"Let $F$ be a bent vectorial function from ${\\mathrm {GF}}(2^{2m})$ to ${\\mathrm {GF}}(2^m)$ .",
"Let $A$ be a subgroup of order $2^s$ of $({\\mathrm {GF}}(2^m), +)$ .",
"Define a binary code by ${A}:=\\lbrace ({\\mathrm {Tr}}_{m/1}(aF(x))+{\\mathrm {Tr}}_{2m/1}(bx)+c)_{x \\in {\\mathrm {GF}}(2^{2m})}:a \\in A, b \\in {\\mathrm {GF}}(2^{2m}), c \\in {\\mathrm {GF}}(2)\\rbrace .$ It can be shown that ${A}$ can be viewed as a code $f_{i_1}, \\cdots , f_{i_s})$ obtained from a bent vectorial function $(f_{i_1}, \\cdots , f_{i_s})$ ."
],
[
"Summary and concluding remarks",
"The contributions of this paper are the following.",
"A coding-theoretic characterization of bent vectorial functions (Theorem REF ).",
"A construction of a two-parameter family of four-weight binary linear codes with parameters $[2^{2m}, 2m+1+\\ell ,2^{2m-1}-2^{m-1}]$ for all $1 \\le \\ell \\le m$ and all $m\\ge 2$ , obtained from $(2m, \\ell )$ bent vectorial functions (Theorem REF ).",
"The parameters of these codes appear to be new when $2 \\le \\ell \\le m-1$ .",
"This family of codes includes some optimal codes, as well as codes meeting the BCH bound.",
"These codes do not satisfy the conditions of the Assmus-Mattson theorem, but nevertheless hold 2-designs.",
"It is plausible that most of these codes do not admit 2-transitive automorphism groups (Conjecture REF and Theorem REF ).",
"A new construction of a two-parameter family of 2-designs with parameters $2\\mbox{--}(2^{2m}, \\ 2^{2m-1}-2^{m-1}, \\ (2^\\ell -1)(2^{2m-2}-2^{m-1})),$ and having three block intersection numbers, where $2\\le \\ell \\le m$ , based on bent vectorial functions (Theorem REF and Theorem REF ).",
"This construction is a generalization of the construction of SDP designs from single bent functions given in [8].",
"The number of nonisomorphic designs with parameters (REF ) in the special case when $\\ell =1$ , grows exponentially with $m$ by a known theorem of Kantor [15].",
"It is an interesting open problem to prove that the number of nonisomorphic designs with parameters (REF ) grows exponentially for any fixed $\\ell >1$ .",
"Finally, we would like to mention that vectorial Boolean functions were employed in a different way to construct binary linear codes in [20].",
"The codes from [20] have different parameters from the codes described in this paper."
],
[
"Acknowledgements",
"Vladimir Tonchev acknowledges partial support by a Fulbright grant, and would like to thank the Hong Kong University of Science and Technology for the kind hospitality and support during his visit, when a large portion of this paper was written.",
"The research of Cunsheng Ding was supported by the Hong Kong Research Grants Council, under Grant No.",
"16300418.",
"The authors wish to thank the anonymous reviewers for their valuable comments and suggestions for improving the manuscript."
]
] | 1808.08487 |
[
[
"Longest increasing path within the critical strip"
],
[
"Abstract A Poisson point process of unit intensity is placed in the square $[0,n]^2$.",
"An increasing path is a curve connecting $(0,0)$ with $(n,n)$ which is non-decreasing in each coordinate.",
"Its length is the number of points of the Poisson process which it passes through.",
"Baik, Deift and Johansson proved that the maximal length of an increasing path has expectation $2n-n^{1/3}(c_1+o(1))$, variance $n^{2/3}(c_2+o(1))$ and that it converges to the Tracy-Widom distribution after suitable scaling.",
"Johansson further showed that all maximal paths have a displacement of $n^{\\frac23+o(1)}$ from the diagonal with probability tending to one as $n\\to \\infty$.",
"Here we prove that the maximal length of an increasing path restricted to lie within a strip of width $n^{\\gamma}, \\gamma<\\frac23$, around the diagonal has expectation $2n-n^{1-\\gamma+o(1)}$, variance $n^{1 - \\frac{\\gamma}{2}+o(1)}$ and that it converges to the Gaussian distribution after suitable scaling."
],
[
"Introduction",
"The problem of determining the distribution of the length $\\tilde{L}_n$ of the longest increasing subsequence in a random uniform permutation of $\\lbrace 1,2,\\cdots , n\\rbrace $ was first posed by Ulam and considered by Hammersley in his seminal paper [20].",
"It has since attracted a lot of attention in the mathematical community; see Romik's book [33] for a lively history of this problem and the remarkable progress that has been made since it was first introduced.",
"Using the subadditive ergodic theorem, Hammersley [20] was able to show the existence of a constant $c$ such that $ \\tilde{L}_n/ \\sqrt{n} \\rightarrow c$ in probability as $n \\rightarrow \\infty $ .",
"A few years later, Logan and Shepp [28], and Vershik and Kerov [39] were able to compute the exact value $c=2$ using an analysis of random Young tableaux.",
"A different proof was provided by Aldous and Diaconis [1] by studying of the hydrodynamical limit of an interacting particle system implicit in the work of Hammersley, which is now called the Hammersley process in his honor.",
"The above results can be considered as a law of large numbers for $\\tilde{L}_n$ .",
"The limiting distribution for $\\tilde{L}_n$ , under appropriate centering and scaling, was found to be the Tracy-Widom distribution in the landmark paper by Baik, Deift and Johansson [5].",
"The paper is a tour-de-force using the Robinson-Schensted-Knuth correspondence between permutations and Young tableaux, and elements of Riemann-Hilbert theory.",
"It is convenient to work with the following equivalent formulation of the problem, as done already by Hammersley [20] and as we will do in the rest of the paper.",
"Consider a Poisson point process of unit intensity in the square $[0,n]^2$ .",
"Let $P_n$ be the (random) number of points inside the square.",
"An increasing path is a curve connecting $(0,0)$ with $(n,n)$ which is non-decreasing in each coordinate.",
"Its length is the number of points of the Poisson process which it passes through.",
"Denote by $L_n$ the length of the longest increasing path.",
"The relation between $L_n$ and $\\tilde{L}_n$ is given by the fact, which is straightforward to check directly, that conditioned on $P_n = k$ , the distribution of $L_n$ equals the distribution of $\\tilde{L}_{k}$ .",
"One can therefore obtain the limiting distribution of $\\tilde{L}_{P_n}$ by studying the Poissonized problem, and obtain similar results for $\\tilde{L}_n$ via a de-Poissonization argument.",
"Among the key findings of [5] is that, as $n\\rightarrow \\infty $ , $&\\operatorname{{E}}L_n = 2n - n^{1/3}(c_1 + o(1)),\\\\&\\operatorname{Var}L_n = n^{2/3}(c_2 + o(1)),\\\\&\\frac{L_n - \\operatorname{{E}}L_n}{\\sqrt{\\operatorname{Var}L_n}}\\text{ converges in distribution to the Tracy-Widom distribution}$ with $c_1,c_2>0$ absolute constants.",
"In a subsequent development, Johansson [27] showed using a geometric argument that the transversal exponent of the longest increasing path is $\\frac{2}{3}$ .",
"More precisely, as the longest increasing path in $[0,n]^2$ need not be unique, it is shown there that with probability tending to 1 as $n\\rightarrow \\infty $ all the longest increasing paths have a displacement of order $n^{\\frac{2}{3} +o(1)}$ around the diagonal.",
"The result raises the question of studying the maximal length attainable for increasing paths restricted to lie closer to the diagonal.",
"Addressing this question is the main purpose of this paper.",
"Let $0< \\gamma <\\frac{2}{3}$ and set $L_n^{(\\gamma )}$ to be the maximal length of an increasing path restricted to lie in $[0,n]^2\\cap \\lbrace (x,y)\\colon |y-x|\\leqslant n^{\\gamma }\\rbrace $ .",
"As $\\gamma $ decreases, the length $L_n^{(\\gamma )}$ corresponds to a maximum over a smaller set.",
"It is then clear that $L_n^{(\\gamma )}$ decreases and one may further Theorem 1.1 Fix $0< \\gamma <\\frac{2}{3}$ .",
"We have the following asymptotic behavior as $n\\rightarrow \\infty $ : $\\begin{split} \\operatorname{{E}}L_n^{(\\gamma )} &= 2n - n^{1-\\gamma +o(1)}, \\\\\\operatorname{Var}L_n^{(\\gamma )} &= n^{1-\\frac{\\gamma }{2} +o(1)}\\end{split}$ and $\\frac{L_n^{(\\gamma )}-\\operatorname{{E}}L_n^{(\\gamma )}}{\\sqrt{\\operatorname{Var}{L_n^{(\\gamma )}}}} \\Rightarrow N(0,1).$ The Theorem is further refined by the results in Section REF below.",
"The Gaussian fluctuations of the length of the longest increasing path inside the strip $n^{\\frac{2}{3}-\\epsilon }$ are remarkably different from the Tracy-Widom fluctuations of the length of the longest path within the strip $n^{\\frac{2}{3}+\\epsilon }$ .",
"A tantalizing problem would be to study the transition when paths within the critical strip, having width of order $n^{\\frac{2}{3}}$ , are considered.",
"The heuristic for the proof is as follows.",
"Partition the strip $[0,n]^2\\cap \\lbrace (x,y)\\colon |y-x|\\leqslant n^{\\gamma }\\rbrace $ into $n^{1-\\frac{3\\gamma }{2}+o(1)}$ diagonally-aligned rectangles (which we shall call blocks) of length $n^{\\frac{3\\gamma }{2}+o(1)}$ (the strip is partitioned apart from initial and final portions which are small enough to be ignored; see Figure REF ).",
"The length of the longest increasing path within each block obeys the scaling given by (REF ), () as the height of each block is larger than the $2/3$ -power of its length, allowing for transversal fluctuations which are essentially unrestricted due to the result of [27].",
"If the length of the longest increasing path (LIP) in the entire strip was the sum of the lengths of the LIPs within each block, then we would have a sum of i.i.d.",
"random variables and we could apply the Lindeberg-Feller theorem to get our result.",
"However, this is not true – the restriction of a LIP (in the entire strip) to a block need not be maximal within this block.",
"A key ingredient in our proof is Theorem REF , as a consequence of which we are able to show the existence of many blocks with regeneration points (these are the points $\\mathbf {p}$ in (REF )).",
"A longest increasing path within the entire strip is obtained by concatenating the longest increasing paths between successive regeneration points.",
"The regeneration points provide the necessary independence structure required for a Gaussian limit.",
"A model related to the above is directed last passage percolation on the two-dimensional integer lattice.",
"One starts with a collection $\\lbrace Y_x\\rbrace _{x\\in \\mathbb {Z}_{\\geqslant 0}^2}$ of i.i.d.",
"random variables, and is interested in studying $G_{m,n}= \\max _{\\pi } \\sum _{x\\in \\pi } Y_x$ , where the maximum is over all up-right paths starting from the origin and terminating at $(m,n)$ .",
"A law of large numbers and a Tracy-Widom limit for $G_{n,n}$ have been obtained for the case of exponential and geometric weights $Y_x$ ([26], see the lecture notes [34]); the memory-less property of these distributions is used crucially.",
"The model with exponential weights is connected to the totally asymmetric simple exclusion process (TASEP) with step initial condition (a particle on every $x \\in \\mathbb {Z}_{<0}$ ).",
"The random variable $G_{m,n}$ has the same distribution as the time it takes for the $m$ th particle to move $n$ places to the right (see [34]).",
"One might ask if a result similar to Theorem REF holds for $G_{n,n}^{(\\gamma )}$ , where we change the definition of $G_{n,n}$ so that $\\pi $ lies within a strip of width $n^{\\gamma }$ around the diagonal.",
"While we have not attempted to do so, our arguments rely mainly on moderate deviation bounds of the type given in Lemma REF and Lemma REF and may adapt for other models where similar bounds are known.",
"Let us also mention the following equivalent formulation of $G_{m,n}^{(\\gamma )}$ .",
"Consider TASEP with particles to the left of the origin and a source with infinitely many particles at $-[n^{\\gamma }]$ .",
"Particles follow the same rules as ordinary TASEP except that now the particles at the source jump (in order) to the site $-[n^{\\gamma }]+1$ after it becomes vacant.",
"There is a sink at $[n^{\\gamma }]$ in which all particles eventually fall.",
"It can then be argued that the distribution of $G_{m,n}^{(\\gamma )}$ is the same as that of the time it takes for the $m$ th particle (which might now start from the source) to make $n$ steps to the right.",
"The literature on first/last passage percolation has several results of relevance to this work.",
"For (undirected) first passage percolation on $\\mathbb {Z}^d$ with weights $Y_x$ having all moments, [18] proved a Gaussian central limit theorem for the first passage time when the paths are restricted to thin rectangles of the form $[-h_n,h_n]\\times [-n,n]^{d-1}$ where $h_n =o(n^{1/(d+1)})$ .",
"Under natural but unproven assumptions they are able to extend their result up to $h_n=n^{\\xi ^{\\prime }}$ where $\\xi ^{\\prime }<\\xi $ , the transversal exponent (see [17], [2]).",
"For directed last passage percolation on $\\mathbb {Z}_{\\geqslant 0}^2$ and for $Y_x$ having all moments, one gets a Tracy-Widom limit for the last passage time for rectangles of the form $[0,n]\\times [0,n^a]$ where $a<\\frac{3}{7}$ ([7], [35], [16]).",
"This is remarkably different from the Gaussian behavior of first passage percolation.",
"In our present set-up, it is easy to see by a scaling argument that the behavior of the LIP in any rectangle with sides parallel to the axes is the same as the behavior of the LIP in a square of the same area, and we would have a Tracy-Widom limit.",
"With similar scaling, Theorem REF implies a Gaussian limit for the LIP in any inclined rectangle if its width is sufficiently smaller than its length.",
"Another result which deals with the length of the LIP in non-square domains is that of [6].",
"They consider the length of the LIP in a right-angled triangle with extra points distributed uniformly on the hypotenuse.",
"Using non-geometric arguments that link the model with random matrix theory, they are able to show different limiting behaviors depending on the number of points on the diagonal.",
"Lastly, we mention a recent result of [13] where they prove a law of large numbers for the length of the LIP in a square with points from a Poisson point process of intensity $\\lambda $ added to the diagonal.",
"Using geometric arguments of a similar flavor as ours, they are able to show that the limiting constant is strictly greater than 2 for any $\\lambda >0$ .",
"The analogue of this for the last passage percolation model is called the slow bond problem.",
"As an interesting aside, we note that the study of the longest increasing subsequence in a restricted geometry also arises in application areas; see [4], [3] for an application to airplane boarding times."
],
[
"Notation and Main Results",
"We consider a Poisson process of unit intensity on ${R}^2$ .",
"Instead of working with the usual $(x,y)$ coordinates in ${R}^2$ , we shall use diagonal coordinates $(t,s)$ .",
"Here $t$ measures the distance along the line $y=x$ and $s$ measures the distance along the line $y=-x$ .",
"The correspondence between the coordinate systems is $\\begin{split}&x = \\frac{t-s}{\\sqrt{2}},\\quad y = \\frac{t+s}{\\sqrt{2}}\\qquad \\text{ and } \\qquad t = \\frac{x+y}{\\sqrt{2}},\\quad s = \\frac{y-x}{\\sqrt{2}}.\\end{split}$ Thus a diagonal rectangle $R$ of length $t_0$ and width $s_0$ with lower-left corner at $(0,0)$ is the region $\\lbrace (t,s)\\colon 0\\leqslant t\\leqslant t_0,\\, 0\\leqslant s\\leqslant s_0\\rbrace $ (see Figure REF ).",
"From now on we will use always the $(t,s)$ coordinate system and thus an increasing path will refer to a path in the $t$ –$s$ plane whose slope at every point is in $[-1,1]$ (in the sense that $\\frac{s_2 - s_1}{t_2-t_1}\\in [-1,1]$ for pairs of distinct points $(t_1, s_1)$ , $(t_2, s_2)$ along the path).",
"Figure: Diagonal coordinate system with diagonal rectangle RRDenote by $\\mathcal {L}\\big ( (t_1,s_1), (t_2,s_2)\\big )$ the collection of LIPs between points $(t_1,s_1)$ and $(t_2,s_2)$ in the $t$ –$s$ plane.",
"The length of, that is the number of points on, any LIP in this collection is denoted by $L\\big ((t_1,s_1),(t_2,s_2)\\big )$ and its expectation by $E\\big ((t_1,s_1),(t_2,s_2)\\big )$ .",
"For a fixed domain $R$ (e.g.",
"a diagonal rectangle), $\\mathcal {L}^R\\big ((t_1,s_1),(t_2,s_2)\\big )$ will denote the collection of LIPs between $(t_1,s_1)$ and $(t_2,s_2)$ constrained to lie within the domain $R$ .",
"We will use the notation $L^R\\big ((t_1,s_1),(t_2,s_2)\\big )$ for the corresponding length and $E^R\\big ((t_1,s_1),(t_2,s_2)\\big )$ for the expectation of this length.",
"We write $L^R$ to denote the length of the LIP in $R$ when allowing arbitrary initial and end points in $R$ .",
"We shall use the shorthand $\\begin{split}L_t &:= L\\big ((0,0),(t,0)\\big ) \\qquad \\text{ and }\\qquad L_{t,s} := L \\big ((0,0),(t,s)\\big ).\\end{split}$ Note that $L_{t,s}=0$ if $|s|>t$ and $L_{t,s}$ equals $L_{\\sqrt{t^2-s^2}}$ in distribution when $|s|\\leqslant t$ , since the area of the rectangle determines the distribution of the length of the LIP (by applying a linear transformation to the Poisson process).",
"Correspondingly, we set $\\begin{split}E_t:= \\operatorname{{E}}L_t\\qquad & \\text{and} \\qquad E_{t,s} := \\operatorname{{E}}L_{t,s}\\,, \\\\\\sigma _t:= \\sqrt{\\operatorname{Var}(L_t)} \\qquad & \\text{and} \\qquad \\sigma _{t,s}:=\\sqrt{\\operatorname{Var}(L_{t,s})}\\,.\\end{split}$ Consider now a diagonal rectangle $R$ of length $\\ell $ and width $w$ .",
"The following theorem gives bounds on the expectation of $L^R$ .",
"The first assertion in (REF ) follows as a consequence.",
"Theorem 1.2 Fix $0<\\delta < 1/6$ .",
"Consider a diagonal rectangle $R$ of length $\\ell $ and width $w$ such that $w(\\log w)^{\\frac{5}{3}} \\leqslant \\ell ^{\\frac{2}{3}}$ .",
"There exist constants $c(\\delta ), \\, C(\\delta )$ depending only on $\\delta $ such that for $w$ large enough $\\sqrt{2} \\ell -C(\\delta )\\cdot (\\log w)^{\\frac{1}{3}+\\delta }\\cdot \\frac{\\ell }{w} \\leqslant \\operatorname{{E}}L^R \\leqslant \\sqrt{2} \\ell - \\frac{c(\\delta )}{ (\\log w)^{\\frac{5}{3}-\\delta } }\\cdot \\frac{\\ell }{w}$ The first term $\\sqrt{2} \\ell $ matches the leading term in the expectation of the length of the unrestricted LIP in square domains.",
"Our next result is a lower bound on the variance of $L^R$ .",
"Theorem 1.3 Consider a diagonal rectangle $R$ of length $\\ell $ and width $w$ with $\\frac{w}{\\log \\log w}\\leqslant \\ell ^{\\frac{2}{3}}$ .",
"There exists an $\\epsilon _0>0$ such that for any $0<\\epsilon <\\epsilon _0$ , we have $\\operatorname{Var}(L^R) \\geqslant \\left(\\frac{\\ell }{w^{\\frac{1}{2}}}\\right)^{1-\\varepsilon }$ for $w$ large enough.",
"Complementing this result, we provide upper bounds for all centered moments of $L^R$ .",
"These two theorems together prove the second assertion in (REF ).",
"Theorem 1.4 Consider a diagonal rectangle $R$ of length $\\ell $ and width $w$ such that $\\frac{w}{(\\log w)^{1/2}}\\leqslant \\ell ^{\\frac{2}{3}}$ .",
"For each fixed $k\\geqslant 1$ , there exists a constant $C(k)$ depending only on $k$ such that for $w$ large enough $\\left\\Vert L^R - \\operatorname{{E}}L^R\\right\\Vert _{2k} \\leqslant C(k)\\cdot \\frac{\\ell ^{\\frac{1}{2}}}{w^{\\frac{1}{4}}}\\cdot (\\log w)^{1+k},$ where we use the notation $\\Vert X\\Vert _p := (\\operatorname{{E}}X^p)^{1/p}$ .",
"The theorems above suggest that the variance decreases as the width increases for each fixed $\\ell $ .",
"It is worth noting that such a monotonicity property was a key (unproven) assumption made in the paper [18] for first passage percolation.",
"Our final theorem is a Gaussian limit theorem for $L^R$ when the width is less than the critical width.",
"Theorem 1.5 Fix $\\gamma <\\frac{2}{3}$ .",
"Consider a sequence of diagonal rectangles $R = R(\\ell )$ of length $\\ell $ and width $w(\\ell )$ such that $w(\\ell )\\leqslant \\ell ^\\gamma $ .",
"Then, as $\\ell \\rightarrow \\infty $ : $ \\frac{L^R-\\operatorname{{E}}L^R}{\\sqrt{\\operatorname{Var}\\, L^R}} \\Rightarrow N(0,1).$ Remark 1.1 In the above results we have considered $L^R$ , the length of the LIP within the rectangle $R$ .",
"One can also obtain similar results for $L^R(\\mathbf {a},\\mathbf {b})$ , the length of the LIP among paths starting at $\\mathbf {a}$ , ending at $\\mathbf {b}$ and restricted to lie within $R$ , where $\\mathbf {a},\\mathbf {b}$ are points on the left and right boundaries of the rectangle $R$ .",
"This can be done by arguing as in the proof of Theorem REF given in Section .",
"Remark 1.2 The main ingredients in our results are the tail estimates in Lemma REF and Lemma REF .",
"We believe that results of a similar flavor hold for other models where corresponding estimates are known.",
"We now describe the outline of the paper.",
"Section describes the moderate deviation results as well as the van-den-Berg-Kesten type inequality that our analysis relies upon, and collects several consequences.",
"Section gives a lower bound on the expectation of the length of the LIP in diagonal rectangles and Section provides error estimates for the length of the LIP in short blocks.",
"We next prove Theorem REF in Section .",
"Section contains the main argument of the paper in which we give a lower bound on the probability that LIPs (between arbitrary starting and ending points on the left and right boundaries respectively) in each block meet at a point.",
"We use this result to prove Theorem REF in Section .",
"After this we prove Theorem REF in Section and then Theorem REF in Section .",
"We end the paper with a few open questions for the reader.",
"Throughout the article, $c,C$ will be used to denote arbitrary positive constants which may change from line to line.",
"If the constants depend on a certain parameter this shall be indicated in parenthesis."
],
[
"A remark on timing",
"A preliminary version of this paper containing full statements and proofs of the above results is available since December 2015 on the website of the first author.",
"The second author presented these results at seminars in Bristol, Oxford and Sussex in 2015–16 while the third author discussed these at the June 2015 “Groups, Graphs and Stochastic Processes” Banff workshopA video is available at http://www.birs.ca/events/2015/5-day-workshops/15w5146/videos/watch/201506231638-Peled.html.",
"For various reasons, polishing of the draft and its final posting to the arXiv were greatly delayed, until its final appearance there on August 2018.",
"In the interim, many results on last passage percolation, in the continuum and on the lattice have appeared, including [12], [11], [10], [9], [21], [22], [23], [24], [25].",
"Among these, especially related to the present work is a recent result of Basu, Ganguly and Hammond [10] who consider the length of the longest increasing path between $(0,0)$ and $(n,n)$ , constrained to enclose an atypically large area of at least $(1/2+\\alpha )n^2$ .",
"Apart from finding the law of large numbers for the model (in fact, they also find a limiting curve), the key result of their work is that the fluctuations around the limiting curve are of order $n^{1/2}$ and the convex hull facet length is of order $n^{3/4}$ .",
"This should be contrasted with the orders $n^{1/3}$ and $n^{2/3}$ for the unconstrained model.",
"In a second related work which appeared very recently, Basu and Ganguly [9] find, among other results, upper and lower bounds on the variance of the last passage percolation time in thin cylinders which match up to a multiplicative constant.",
"The latter two works kindly acknowledged our preprint."
],
[
"Preliminaries",
"In this section we collect several results which we make use of in the sequel."
],
[
"Asymptotic Behavior",
"The following was proved in [5].",
"Lemma 2.1 There exist positive constants $c_0, c_1$ such that $\\sigma _t & = c_0 t^{1/3}\\big (1+o(1)\\big ) , \\\\E_t & = \\sqrt{2}\\, t - c_1 t^{1/3} \\big (1+o(1)\\big ) $ as $t\\rightarrow \\infty $ .",
"We also have for all $k \\geqslant 1$ , $ \\big \\Vert L_t- \\sqrt{2}\\, t \\big \\Vert _k \\leqslant C t^{1/3}.$ for some constant $C>0$ .",
"The next lemma gives tail bounds on the deviation of the length of the LIP from its mean.",
"This lemma is the black box for our arguments and shall be referred to several times during the course of the paper.",
"It collects results proved in [5], [29] and [30].",
"Lemma 2.2 The following hold.",
"(Upper tail bounds) There exist constants $c, C>0$ such that for sufficiently large $t$ , $c \\exp \\big (-CT^{3/2}\\big )\\leqslant \\operatorname{{P}}\\left(\\frac{L_t- E_t}{\\sigma _t} \\geqslant T\\right) \\leqslant C \\exp \\big (-c T^{3/2}\\big )$ for all $1\\leqslant T\\leqslant t^{2/3}$ .",
"(Lower tail bound) There exist constants $c, C>0$ such that for sufficiently large $t$ , $\\operatorname{{P}}\\left(\\frac{L_t- E_t}{\\sigma _t} \\leqslant -T\\right) \\leqslant C\\exp \\big (-c T^{3}\\big )$ for all $1\\leqslant T\\leqslant t^{2/3}$ .",
"The upper bounds in (REF ) and (REF ) have been proved in Lemma 7.1 of [5]; see also equations (1.4) and (1.5) in [27].",
"We now prove the lower bound in (REF ).",
"From Lemma REF and super-additivity of $E_{r}$ , we have that $0\\leqslant \\frac{\\sqrt{2}r-E_{r}}{\\sigma _{r}}\\leqslant a \\text{ and } b_1\\leqslant \\frac{\\sigma _{r}}{r^{1/3}}\\leqslant b_2 \\text{ for all } r\\geqslant 1$ for some universal constants $a,b_{1},b_{2}>0$ .",
"From the distributional convergence of $2^{1/6}t^{-1/3}(L_t-\\sqrt{2} t)$ to the Tracy-Widom disrtibution (see [5]) we note that there exists a number $\\lambda >0$ such that $\\operatorname{{P}}(L_q-E_q\\geqslant k \\sigma _q) \\geqslant e^{-\\lambda } \\quad \\mbox{ for all } q\\geqslant 1.$ Let $k=\\lceil a+{b_2}/{b_1}\\rceil $ and choose $r= t T^{-3/2}\\geqslant 1$ and now consider the event $A=\\bigcap _{j=0}^{\\lceil T^{3/2}\\rceil } \\Big \\lbrace L\\big ((jr,0), ((j+1)r,0)\\big ) -E_r \\geqslant k\\sigma _r\\Big \\rbrace $ The events in the braces are all independent and hence $\\operatorname{{P}}(A)\\geqslant e^{-\\lambda ( \\lceil T^{3/2}\\rceil +1)}$ .",
"Next note that on the event $A$ , we have $\\begin{split} L_t& \\geqslant \\sum _{j=0}^{\\lceil T^{3/2}\\rceil } L\\big ((jr,0), ((j+1)r,0)\\big ) \\\\& \\geqslant k T^{3/2} \\sigma _r + T^{3/2} E_r \\\\&\\geqslant \\Bigl (\\frac{\\sqrt{2}r-E_r}{\\sigma _r} + \\frac{b_2}{b_1}\\Bigr )T^{3/2} \\sigma _r+ T^{3/2} E_r \\\\&\\geqslant \\sqrt{2}rT^{3/2} + \\frac{\\sigma _t}{t^{1/3}}\\cdot \\frac{r^{1/3}}{\\sigma _r}\\cdot T^{3/2}\\sigma _r=\\sqrt{2}t +T\\sigma _t\\geqslant E_t+T\\sigma _t.\\end{split} $ This completes the proof of the lemma.",
"The following fundamental lemma is a consequence of Lemma REF .",
"Lemma 2.3 There exists a small $\\alpha _0>0$ such that the following hold.",
"There exist positive constants $c,C$ such that whenever $|s|/t\\leqslant \\alpha _0$ $\\frac{cs^2}{t}\\leqslant E_t - E_{t,s} \\leqslant \\frac{Cs^2}{t}$ There exist constants $c_2,c_3> 0$ such that whenever $|s|/t\\leqslant \\alpha _0$ $c_2 t^{1/3} \\leqslant \\sigma _{t,s} \\leqslant c_3 t^{1/3}$ when $t$ is large enough.",
"The upper bound holds for all $|s|\\leqslant t$ .",
"The tail bounds in Lemma REF hold (perhaps with different constants) with $L_{t}, E_t, \\sigma _t$ replaced by $L_{t,s}, E_{t,s}, \\sigma _{t,s}$ respectively.",
"The proof is a ready consequence of the distributional identity $L_{t,s}\\stackrel{d}{=} L_{\\sqrt{t^2-s^2}}$ and the fact that $\\sqrt{t^2-s^2} = t- \\frac{s^2}{2t}(1+O(\\frac{s^2}{t^2}))$ when $|s|\\ll t$ ."
],
[
"Transversal Fluctuations",
"In this section, we shall give a quantitative control on the transversal fluctuations of the LIP in square domains $[0,n]^2$ .",
"For this we shall discretize the space to a lattice of points of mesh size 1.",
"In order to show that the lengths of LIPs between points of the lattice are good approximations of the lengths of LIPs between general points we need a control over the number of Poisson points in small squares.",
"Thus we define the event $A_0:=\\left\\lbrace \\text{The number of points in each $1\\times 1$ squarecontained in $[0,n]^2$ is less than $(\\log n)^2$} \\right\\rbrace .$ Lemma 2.4 We have $\\operatorname{{P}}\\left(A_0^c\\right) \\leqslant C\\exp \\left(-c(\\log n)^2 \\log \\log n\\right).$ It suffices to cover $[0,n]^2$ by $n^2$ squares of side-length 1 and show that no such square contains more than $(\\log n)^2 / 4$ Poisson points with high probability.",
"This is because every square of side-length 1 intersects at most 4 of the $n^2$ squares.",
"The number of Poisson points in a square of side-length 1 is Poisson$(1)$ and hence the probability for it to be larger than $k$ is bounded by $\\exp (-k\\log k+k-1)$ , yielding the lemma.",
"Let $S$ denote the maximal absolute value of the $s$ coordinate of a point on a maximizing path connecting $(0,0)$ to $(n,0)$ (in the $t$ –$s$ plane).",
"More precisely, $S= \\max \\left\\lbrace |\\omega (t)|: 0\\leqslant t\\leqslant n,\\, t \\mapsto \\omega (t) \\text{ is a maximizing path in } \\mathcal {L}\\big ((0,0),(n,0)\\big )\\right\\rbrace $ Thus $S$ measures the transversal fluctuations of the path.",
"Lemma 2.5 (Transversal fluctuations upper bound) For every $0<\\delta <\\frac{1}{3}$ there exits a positive integer $n_0(\\delta )$ , such that for $n\\geqslant n_0(\\delta )$ we have $\\operatorname{{P}}\\left(S > n^{2/3}\\left(\\log n\\right)^{\\frac{1}{3}+\\delta }\\right)\\leqslant C\\exp \\big (-c(\\log n)^{1 + 3\\delta }\\big ).$ Assume the event $A_0$ occurs as the probability of $A_0^c$ is much smaller than the probability we are interested in.",
"Put a lattice of side length 1 inside the square with corner points $(0,0)$ and $(n,0)$ .",
"If $S > n^{2/3}\\left(\\log n\\right)^{\\frac{1}{3}+\\delta }$ there exists a point $(t,s)$ of the lattice with $\\big \\vert |s|-n^{2/3}\\left(\\log n\\right)^{\\frac{1}{3}+\\delta }\\big \\vert \\leqslant 1$ such that $L((0,0), (n,0)) \\leqslant L((0,0), (t,s)) + L((t,s), (n,0)) +(\\log n)^2.$ However, for each such $(t,s)$ $\\begin{split}E_n - E_{t,s} - E_{n - t, -s} & = (E_n-E_t-E_{n-t}) + (E_t-E_{t,s}) + (E_{n-t}-E_{n-t,-s})\\end{split}$ The first term in parenthesis on the right is of order $n^{1/3}$ .",
"The second and third terms in paranthesis are clearly nonnegative.",
"When $|s|/t\\leqslant \\alpha _0$ (the constant appearing in Lemma REF ) we have by (REF ) that $E_t - E_{t,s} \\geqslant \\frac{cs^2}{t}$ .",
"On the other hand when $|s|/t > \\alpha _0$ we have $|s|/(n-t) \\leqslant \\alpha _0$ and therefore $E_{n-t}-E_{n-t,-s} \\geqslant \\frac{cs^2}{n-t}$ .",
"In either case we obtain $E_n - E_{t,s} - E_{n - t, -s} \\geqslant cn^{1/3}(\\log n)^{2/3+2\\delta }$ since $\\big \\vert |s|-n^{2/3}\\left(\\log n\\right)^{\\frac{1}{3}+\\delta }\\big \\vert \\leqslant 1$ .",
"The standard deviations of $\\sigma _n, \\sigma _{t,s}$ and $\\sigma _{n-t,-s}$ are all of order at most $n^{1/3}$ .",
"Thus, by (REF ), (REF ) and similar bounds in Lemma REF it follows that $\\begin{split}& \\operatorname{{P}}\\left( L((0,0), (n,0)) \\leqslant L((0,0), (t,s)) + L((t,s), (n,0)) +(\\log n)^2\\right) \\\\& \\leqslant \\operatorname{{P}}\\left(| L_{t,s} - E_{t,s} | >c n^{1/3}(\\log n)^{2/3+2\\delta } \\right) + \\operatorname{{P}}\\left(| L_{n-t,-s} - E_{n-t,-s} | >c n^{1/3}(\\log n)^{2/3+2\\delta } \\right) \\\\& \\qquad \\qquad \\qquad + \\operatorname{{P}}\\left(| L_n- E_{n} | >c n^{1/3}(\\log n)^{2/3+2\\delta } \\right) \\\\&\\leqslant C\\exp \\big (-c(\\log n)^{1 + 3\\delta }\\big ).\\end{split}$ Therefore the probability that a maximal path from $(0,0)$ to $(n,0)$ passes at distance 1 from the lattice point $(t,s)$ is at most $C\\exp (-c(\\log n)^{1+3\\delta })$ .",
"As there are less than $Cn^2$ points in our lattice, the lemma follows.",
"The following definition will be useful for us.",
"It simplifies the exposition considerably.",
"Definition 2.1 We say an event $A$ depending on the points in $\\big [0,n]^2$ occurs with “overwhelming probability\" if there is a $\\theta >0$ such that $\\operatorname{{P}}(A^c) \\leqslant \\exp \\big (-c (\\log n)^{1+\\theta }\\big ).$ In particular the event $\\lbrace S\\leqslant n^{\\frac{2}{3}}(\\log n)^{\\frac{1}{3}+\\delta }\\rbrace $ occurs with overwhelming probability.",
"A consequence of the above definition which we shall use several times without explicitly referring to is Lemma 2.6 Let the event $A$ depending on the points in $\\big [0,n\\big ]^2$ occur with overwhelming probability.",
"Then for every $\\varepsilon >0$ and $k\\geqslant 1$ $\\big \\Vert L_{\\sqrt{2} n} \\cdot {1}_{A^c} \\big \\Vert _k \\leqslant n^\\varepsilon $ for $n$ large enough.",
"Let $K_n$ be the number of Poisson points in $[0,n]^2$ .",
"The following follows from a union bound by splitting the square into squares of side length 1.",
"$\\begin{split}\\operatorname{{P}}\\big (L_{\\sqrt{2}n} \\geqslant m\\big ) & \\leqslant \\operatorname{{P}}\\big ( K_n \\geqslant m\\big ) \\\\& \\leqslant n^2 \\operatorname{{P}}\\Big (\\text{Poisson}(1)> \\frac{m}{n^2}\\Big ) \\leqslant \\exp \\Big (-\\frac{m}{n^2}\\cdot \\log \\left(\\frac{m}{n^2}\\right)\\Big )\\end{split}$ Therefore $\\begin{split}\\operatorname{{E}}\\big (L_{\\sqrt{2}n}^k \\cdot {1}_{A^c}\\big ) & \\leqslant \\sum _{m \\leqslant n^3} n^{3k} \\operatorname{{E}}\\big ({1}_{A^c}\\cdot {1}\\lbrace L_{\\sqrt{2} n}\\leqslant n^3\\rbrace \\big ) \\\\&\\qquad + \\sum _{i=1}^{\\infty } n^{3(i+1)k} \\operatorname{{E}}\\big ({1}_{A^c}\\cdot {1}\\lbrace in^3\\leqslant L_{\\sqrt{2} n}\\leqslant (i+1)n^3\\rbrace \\big )\\end{split}$ which is smaller than $n^\\varepsilon $ when $n$ is large enough, since $A$ occurs with overwhelming probability and we have a fast decay in the tail probabilities of $L_{\\sqrt{2} n}$ from Lemma REF .",
"We will leave the details of this computation to the interested reader."
],
[
"BK inequality",
"We now state a generalization of the classical BK inequality in site percolation suitable for our case.",
"Denote by $\\Omega $ the collection of all realizations $\\omega $ of the Poisson point process on $\\mathbb {R}_+^2$ .",
"We denote $\\omega \\subseteq \\omega ^{\\prime }$ if all the Poisson points in $\\omega $ are also present in $\\omega ^{\\prime }$ and $\\omega ^{\\prime } \\backslash \\omega $ will then be the realization with all the points in $\\omega $ removed.",
"For a subset $K \\subseteq \\mathbb {R}_+^2$ we denote $\\omega _K$ to be the realization containing only the points of $\\omega $ in $K$ .",
"A set $A \\subseteq \\Omega $ is increasing if $\\omega \\in A$ implies $\\omega ^{\\prime } \\in A$ when $\\omega \\subseteq \\omega ^{\\prime }$ .",
"For increasing sets $A$ and $B$ define the disjoint occurence $\\begin{split}A \\Box B &= \\lbrace \\omega : \\text{there exist disjoint regions } K,\\, L\\subseteq \\mathbb {R}_+^2 \\text{ such that } \\omega _K \\in A \\text{ and } \\omega _L \\in B \\rbrace \\\\&= \\lbrace \\omega : \\text{there exists } \\omega ^{\\prime } \\subseteq \\omega \\text{ such that } \\omega ^{\\prime } \\in A \\text{ and } \\omega \\backslash \\omega ^{\\prime } \\in B\\rbrace .\\end{split}$ We then have Lemma 2.7 [38] For increasing events $A, B \\subseteq \\Omega $ , $\\operatorname{{P}}(A\\Box B) \\leqslant \\operatorname{{P}}(A)\\cdot \\operatorname{{P}}(B).$"
],
[
"Lower Bound of Expectation in Blocks",
"Consider the diagonal rectangle $R$ of length $\\ell $ and width $w$ .",
"Consider two points $(0,a)$ and $(\\ell ,b)$ on opposite sides of the rectangle.",
"Recall that $E^R\\big [(0,a),(\\ell ,b)\\big ]=\\operatorname{{E}}L^R\\big ((0,a),(\\ell ,b)\\big )$ is the expected value of the restricted LIP between the points $(0,a)$ and $(\\ell ,b)$ .",
"The following proposition shows that $E^R\\big [(0,a),(\\ell ,b)\\big ]$ is not very far from $E_{\\ell } $ .",
"Proposition 3.1 Fix $0<\\delta <\\frac{1}{3}$ and let $ \\ell ^{\\frac{2}{3}} (\\log \\ell )^{\\frac{1}{3}+\\delta } \\leqslant w \\leqslant \\alpha _0 \\ell $ with $\\ell $ large enough and $\\alpha _0$ as in Lemma REF .",
"There exists a constant $\\mathcal {C}=\\mathcal {C}(\\delta )$ such that for any $0\\leqslant a, b\\leqslant w$ , $E_{\\ell }- \\mathcal {C}\\frac{w^2}{\\ell } \\leqslant E^R\\big [(0,a),(\\ell ,b)\\big ] \\leqslant E_{\\ell }.$ The second inequality is immediate so let us consider the first inequality.",
"It is sufficient to show that $E^R\\big [(0,a), \\big (\\ell /2,w/2\\big )\\big ] \\geqslant E_{\\frac{\\ell }{2}} - C \\frac{w^2}{\\ell }.$ because by symmetry we have a similar relation for $E^R\\big [ \\big (\\ell /2,w/2\\big ), \\big (\\ell ,b\\big )\\big ]$ .",
"We shall prove this with $a=0$ and indicate at the end how to modify the argument for general $a$ .",
"We shall consider paths which enter the interior of $R$ via successively larger rectangles such that the paths in each sub rectangle behave just like unrestricted paths.",
"We make this precise as follows.",
"Let $t_0=s_0=\\ell ^{1/3}.$ We now define recursively two sequences $\\begin{split}s_{k+1}&= 2s_k, \\\\t_{k+1}&= t_k +\\frac{s_k^{3/2}\\ell }{2w^{3/2}}.\\end{split}$ Choose $k_0=O\\big (\\log _2 (w/\\ell ^{1/3})\\big )$ so that $s_{k_0}\\leqslant \\frac{w}{2} <s_{k_0+1}$ .",
"For this $k_0$ one can see that $\\frac{\\ell }{12} \\leqslant t_{k_0} \\leqslant \\frac{\\ell }{3}$ because $t_{k_0} = \\ell ^{1/3} +\\frac{\\ell ^{3/2}}{2 w^{3/2}}\\left[1+2^{3/2}+2^{2\\cdot 3/2}+\\cdots + 2^{(k_0-1)\\cdot 3/2}\\right]$ Figure: Construction of the restricted path.We now create an increasing (restricted to $R$ ) path from $(0,0)$ to $(\\ell /2,w/2)$ by first moving directly to $(t_0,s_0)$ (not passing through any Poisson points) and then for each $0\\leqslant k < k_0$ , taking a LIP in $R$ from $(t_k,s_k)$ to $(t_{k+1},s_{k+1})$ .",
"By using the transversal fluctuation Lemma REF , as $t_{k+1} - t_k$ is short compared with $s_k$ , we may treat each such LIP as an unrestricted LIP (not constrained to lie in $R$ ) as the unrestricted and restricted LIP coincide with overwhelming probability.",
"In the last step we take a LIP in $R$ from $(t_{k_0},s_{k_0})$ to $(\\ell /2,w/2)$ which again coincides with the unrestricted LIP with overwhelming probability.",
"In particular one can show that $\\big \\vert E^R\\big ((t_k,s_k), (t_{k+1},s_{k+1})\\big ) - E\\big ((t_k,s_k), (t_{k+1},s_{k+1})\\big ) \\big \\vert \\leqslant \\ell ^{1/6}$ and $\\Big \\vert E^R\\big ((t_{k_0},s_{k_0}),(\\ell /2,w/2)\\big ) - E\\big ((t_{k_0},s_{k_0}), (\\ell /2,w/2)\\big ) \\Big \\vert \\leqslant \\ell ^{1/6}.$ Let us show the first inequality (REF ) above and leave the second (REF ) to the reader.",
"$\\begin{split}&\\big \\vert E^R\\big ((t_k,s_k), (t_{k+1},s_{k+1})\\big ) - E\\big ((t_k,s_k), (t_{k+1},s_{k+1})\\big ) \\big \\vert \\\\& \\leqslant 2 \\operatorname{{E}}\\left[L\\big ((t_k, s_k),\\, (t_{k+1}, s_{k+1})\\big ) \\cdot {1}\\lbrace \\text{path goes out of the rectangle with endpoints } (t_k, s_k),\\, (t_{k+1}, s_{k+1})\\rbrace \\right] \\\\& \\leqslant 2\\ell \\cdot \\operatorname{{P}}\\left(\\lbrace \\text{path goes out of the rectangle with endpoints } (t_k, s_k),\\, (t_{k+1}, s_{k+1})\\rbrace \\right) \\\\&\\qquad +\\sum _{i=0}^\\infty (2\\ell + i \\ell ^{1/3})\\cdot \\operatorname{{P}}\\left(2\\ell +i\\ell ^{1/3}\\leqslant L\\big ((t_k, s_k),\\, (t_{k+1}, s_{k+1})\\big )\\leqslant 2\\ell +(i+1)\\ell ^{1/3}\\right).\\end{split}$ Thanks to the tail bounds in Lemma REF and the bound on the probability of large transversal fluctuations in Lemma REF , the inequality (REF ) follows for all large $\\ell $ .",
"Since the maximal restricted path in $R$ has larger length than the length of the path obtained from our construction we get, using Lemma REF $\\begin{split}& E_{\\frac{\\ell }{2}} - E^R\\big [(0,a), (\\ell /2,w/2)\\big ] \\\\&\\quad \\leqslant E_{\\frac{\\ell }{2}} -\\sum _{k=0}^{k_0-1}E\\big ((t_k,s_k), (t_{k+1},s_{k+1})\\big ) - E\\big ((t_{k_0},s_{k_0}), (\\ell /2,w/2)\\big ) + C\\ell ^{1/6} k_0\\\\&\\quad \\leqslant E_{\\frac{\\ell }{2}} -\\sum _{k=0}^{k_0-1}\\Big [E_{t_{k+1}-t_k} - C\\frac{(s_{k+1}-s_k)^2}{t_{k+1}-t_k}\\Big ] - \\Big [E_{\\frac{\\ell }{2}-t_{k_0}} - C\\frac{(w/2)^2}{\\frac{\\ell }{2}-t_{k_0}}\\Big ]+ C\\ell ^{1/6} k_0 \\\\&\\quad \\leqslant E_{\\frac{\\ell }{2}} -\\sum _{k=0}^{k_0-1} E_{t_{k+1}-t_k} -E_{\\frac{\\ell }{2}-t_{k_0}} + C\\Big [\\sum _{k=0}^{k_0-1}\\frac{(s_{k+1}-s_k)^2}{t_{k+1}-t_k} + \\frac{(w/2)^2}{\\frac{\\ell }{2}-t_{k_0}}\\Big ] + C\\ell ^{1/6} k_0 \\\\&\\quad \\leqslant C \\ell ^{1/3} + C \\sum _{k=0}^{k_0-1} \\Big \\lbrace \\frac{s_k^{3/2}\\ell }{2 w^{3/2}}\\Big \\rbrace ^{1/3} + C \\Big [\\sum _{k=0}^{k_0-1} \\frac{s_k^2}{s_k^{3/2}\\ell /2w^{3/2}}+ C\\frac{w^2}{\\ell }\\Big ] + C \\ell ^{1/6}k_0\\\\& \\quad \\leqslant C \\ell ^{1/3}+ \\frac{C\\ell ^{1/3}}{2^{1/3}w^{1/2}} \\sum _{k=0}^{k_0-1} 2^{k/2} \\ell ^{1/6} + C \\sum _{k=0}^{k_0-1} \\frac{2^{k/2}\\ell ^{1/6}w^{3/2}}{\\ell } + C \\frac{w^2}{\\ell } + C\\ell ^{1/6} k_0 \\\\& \\quad \\leqslant C \\frac{w^2}{\\ell }.\\end{split}$ We have used the bound to $2^{k_0/2} \\leqslant w^{1/2}/ \\ell ^{1/6}$ to arrive at the final step.",
"Now we explain how the argument can be extended for any $a$ .",
"For $a\\leqslant \\ell ^{1/3}$ , we follow the same rectangles.",
"For $\\ell ^{1/3}<a\\leqslant w/2$ and $s_i < a\\leqslant s_{i+1}$ , we take the maximal path to $(t_{i+1}, s_{i+1})$ (which will remain in $R$ with overwhelming probability) and then follow the remaining rectangles.",
"The argument for $a\\geqslant w/2$ follows along the same lines by symmetry."
],
[
"Upper bounds in blocks",
"Consider a diagonal rectangle $R$ of width $w$ and length $\\ell $ .",
"Let $\\Delta (R) := \\max _{\\mathbf {x}\\in B_l, \\, \\mathbf {y}\\in B_r} L^R(\\mathbf {x},\\mathbf {y}) - \\min _{\\mathbf {x}\\in B_l, \\, \\mathbf {y}\\in B_r} L^R(\\mathbf {x},\\mathbf {y}).$ Here $B_l,\\, B_r$ are the left and right boundaries of $R$ .",
"Proposition 4.1 Fix $k\\geqslant 1$ and $0<\\delta <\\frac{1}{3}$ and let $\\ell ^{\\frac{2}{3}}(\\log \\ell )^{\\frac{1}{3}+\\delta }\\leqslant w\\leqslant \\alpha _0\\ell $ with $\\alpha _0$ as in Lemma REF .",
"There exists a constant $C=C(\\delta ,k)$ such that $\\left\\Vert \\Delta (R) \\right\\Vert _k \\leqslant C\\frac{w^2}{\\ell }$ for large enough $\\ell $ .",
"We first bound $\\Delta (R)$ as follows.",
"$\\begin{split}\\Delta (R) &\\leqslant \\max _{\\mathbf {x}\\in B_l, \\, \\mathbf {y}\\in B_r} \\big [ L(\\mathbf {x},\\mathbf {y}) -\\operatorname{{E}}L(\\mathbf {x},\\mathbf {y})\\big ] + \\left[\\max _{\\mathbf {x}\\in B_l, \\, \\mathbf {y}\\in B_r} \\operatorname{{E}}L(\\mathbf {x},\\mathbf {y}) -\\min _{\\mathbf {x}\\in B_l, \\, \\mathbf {y}\\in B_r} \\operatorname{{E}}L(\\mathbf {x},\\mathbf {y}) \\right] \\\\&\\qquad - \\min _{\\mathbf {x}\\in B_l, \\, \\mathbf {y}\\in B_r} \\big [ L^R(\\mathbf {x},\\mathbf {y}) -\\operatorname{{E}}L(\\mathbf {x},\\mathbf {y})\\big ] \\\\&\\leqslant \\max _{\\mathbf {x}\\in B_l, \\, \\mathbf {y}\\in B_r} \\big [ L(\\mathbf {x},\\mathbf {y}) -\\operatorname{{E}}L(\\mathbf {x},\\mathbf {y})\\big ] + \\left[\\max _{\\mathbf {x}\\in B_l, \\, \\mathbf {y}\\in B_r} \\operatorname{{E}}L(\\mathbf {x},\\mathbf {y}) -\\min _{\\mathbf {x}\\in B_l, \\, \\mathbf {y}\\in B_r} \\operatorname{{E}}L(\\mathbf {x},\\mathbf {y}) \\right] \\\\&\\qquad - \\min _{\\mathbf {x}\\in B_l, \\, \\mathbf {y}\\in B_r} \\big [ \\tilde{L}^R(\\mathbf {x},\\mathbf {y}) -\\operatorname{{E}}L(\\mathbf {x},\\mathbf {y})\\big ]\\end{split}$ where $\\tilde{L}^R(\\mathbf {x},\\mathbf {y})$ are restricted paths from $\\mathbf {x}$ to $\\mathbf {y}$ passing through the sub-rectangles constructed as in Proposition REF .",
"By Lemma REF , $\\left[\\max _{\\mathbf {x}\\in B_l, \\, \\mathbf {y}\\in B_r} \\operatorname{{E}}L(\\mathbf {x},\\mathbf {y}) -\\min _{\\mathbf {x}\\in B_l, \\, \\mathbf {y}\\in B_r} \\operatorname{{E}}L(\\mathbf {x},\\mathbf {y}) \\right] \\leqslant C\\frac{ w^2}{\\ell }.$ Let us next look at the first term on the right hand side of (REF ).",
"We divide both $B_l$ and $B_r$ into intervals of length $\\ell ^{-1}$ .",
"Let $\\mathbf {a}_0,\\,\\mathbf {a}_1,\\cdots , \\mathbf {a}_{[w\\ell ]}$ denote the partition points on the left boundary and let $\\mathbf {b}_0,\\,\\mathbf {b}_1,\\cdots , \\mathbf {b}_{[w\\ell ]}$ denote the partition points on the right boundary.",
"We claim that $\\Big \\Vert \\max _{\\mathbf {x}\\in B_l, \\, \\mathbf {y}\\in B_r} \\big [ L(\\mathbf {x},\\mathbf {y}) -\\operatorname{{E}}L(\\mathbf {x},\\mathbf {y})\\big ]\\Big \\Vert _k \\leqslant \\Big \\Vert \\max _{i,j} \\big [ L(\\mathbf {a}_i,\\mathbf {b}_j) -\\operatorname{{E}}L(\\mathbf {a}_i,\\mathbf {b}_j)\\big ]\\Big \\Vert _k + C(k) \\frac{w^2}{\\ell }.$ This is because with overwhelming probability the number of points in each rectangle with vertices $\\mathbf {a}_i,\\mathbf {a}_{i+1},\\mathbf {b}_{j},\\mathbf {b}_{j+1}$ is at most $(\\log \\ell )^2$ .",
"An application of the tail bounds in Lemma REF along with a union bound gives $\\operatorname{{P}}\\left(\\left| L(\\mathbf {a}_i,\\mathbf {b}_j) -\\operatorname{{E}}L(\\mathbf {a}_i, \\mathbf {b}_j)\\right| > \\ell ^{\\frac{1}{3}} (\\log \\ell )^{\\frac{2}{3}+2\\delta } \\text{ for some } i,j \\right) \\leqslant \\exp \\big (-c(\\log \\ell )^{1+3\\delta }\\big ).$ From this it follows that $\\Big \\Vert \\max _{i,j} \\big [ L(\\mathbf {a}_i,\\mathbf {b}_j) -\\operatorname{{E}}L(\\mathbf {a}_i,\\mathbf {b}_j)\\big ]\\Big \\Vert _k \\leqslant C(\\delta ,k) \\cdot \\ell ^{\\frac{1}{3}}(\\log \\ell )^{\\frac{2}{3}+2\\delta } \\leqslant C(\\delta ,k)\\frac{w^2}{\\ell }$ which bounds the first term in (REF ).",
"Now we proceed to the last term in (REF ).",
"The $k$ th norm is bounded by $\\begin{split}& \\leqslant \\Big \\Vert \\max _{\\mathbf {x},\\mathbf {y}} \\big \\vert \\tilde{L}^R(\\mathbf {x},\\mathbf {y}) -\\operatorname{{E}}\\tilde{L}^R(\\mathbf {x},\\mathbf {y})\\big \\vert \\Big \\Vert _k +\\max _{\\mathbf {x},\\mathbf {y}} \\big \\vert \\operatorname{{E}}\\tilde{L}^R(\\mathbf {x},\\mathbf {y}) -\\operatorname{{E}}L(\\mathbf {x},\\mathbf {y})\\big \\vert \\\\&= \\Big \\Vert \\max _{i,j} \\big \\vert \\tilde{L}^R(\\mathbf {a}_i,\\mathbf {b}_j) -\\operatorname{{E}}\\tilde{L}^R(\\mathbf {a}_i,\\mathbf {b}_j)\\big \\vert \\Big \\Vert _k +\\max _{\\mathbf {x},\\mathbf {y}} \\big \\vert \\operatorname{{E}}\\tilde{L}^R(\\mathbf {x},\\mathbf {y}) -\\operatorname{{E}}L(\\mathbf {x},\\mathbf {y})\\big \\vert \\\\&\\leqslant \\Big \\Vert \\max _{i,j} \\big \\vert \\tilde{L}^R(\\mathbf {a}_i,\\mathbf {b}_j) -\\operatorname{{E}}\\tilde{L}^R(\\mathbf {a}_i,\\mathbf {b}_j)\\big \\vert \\Big \\Vert _k + C(\\delta ,k)\\frac{w^2}{\\ell }\\leqslant C(\\delta ,k)\\frac{w^2}{\\ell }.\\end{split}$ The second last inequality follows by an application of Proposition REF .",
"Since the number of rectangles in the construction of Proposition REF is $O(\\log (w/\\ell ^{1/3}))$ and the behavior of the path in each sub-rectangle is unrestricted, we can use a bound similar to (REF ) to conclude the last inequality.",
"We next give a bound on the $k$ th centered moments of $L^R := \\max _{\\mathbf {x}\\in B_l,\\, \\mathbf {y}\\in B_r} L^R(\\mathbf {x},\\mathbf {y}).$ Lemma 4.2 Fix $k\\geqslant 1$ and $0<\\delta <\\frac{1}{3}$ and let $\\ell ^{\\frac{2}{3}}(\\log \\ell )^{\\frac{1}{3}+\\delta }\\leqslant w\\leqslant \\alpha _0\\ell $ with $\\alpha _0$ as in Lemma REF .",
"There exists a constant $C=C(\\delta ,k)$ such that $\\big \\Vert L^R -\\operatorname{{E}}L^R\\big \\Vert _k \\leqslant C \\frac{w^2}{\\ell }.$ for large enough $\\ell $ .",
"We write $\\begin{split} L^R -\\operatorname{{E}}L^R & =\\big [ L^R - L^R\\big ( (0,w/2), (\\ell ,w/2)\\big ) \\big ] + \\big [ L^R\\big ( (0,w/2), (\\ell ,w/2)\\big ) - \\operatorname{{E}}L^R\\big ( (0,w/2), (\\ell ,w/2)\\big ) \\big ] \\\\&\\hspace{85.35826pt} +\\big [ \\operatorname{{E}}L^R\\big ( (0,w/2), (\\ell ,w/2)\\big )-\\operatorname{{E}}L^R \\big ]\\end{split} $ An application of Proposition REF gives us that $\\begin{split}\\big \\Vert L^R - L^R\\big ( (0,w/2), (\\ell ,w/2)\\big ) \\big \\Vert _k &\\leqslant \\Vert \\Delta (R) \\Vert _k \\leqslant C(\\delta ,k)\\frac{w^2}{\\ell } \\\\\\big |\\operatorname{{E}}L^R - \\operatorname{{E}}L^R\\big ( (0,w/2), (\\ell ,w/2)\\big ) \\big | & \\leqslant \\Vert \\Delta (R) \\Vert _k \\leqslant C(\\delta ,k)\\frac{w^2}{\\ell }\\end{split}$ As for the second term on the right hand side of (REF ) we observe by Lemma REF that with overwhelming probability the maximal path from $(0,w/2)$ to $(\\ell ,w/2)$ lies within the rectangle $R$ .",
"Hence $\\big \\Vert L^R\\big ( (0,w/2), (\\ell ,w/2)\\big ) - \\operatorname{{E}}L^R\\big ( (0,w/2), (\\ell ,w/2)\\big ) \\big \\Vert _k \\leqslant C(\\delta ,k)\\frac{w^2}{\\ell }.$ since the unrestricted LIP from $(0,w/2)$ to $(\\ell ,w/2)$ has moments of order $\\ell ^{1/3}$ by (REF ).",
"This completes the proof of the lemma."
],
[
"Proof of Theorem ",
"Fix $0<\\delta <\\frac{1}{6}$ .",
"Let us consider the lower bound first.",
"Choose $n$ so that $w=n^{\\frac{2}{3}}(\\log n)^{\\frac{1}{3}+\\delta }$ .",
"We divide $R$ into $m=\\lfloor \\ell /n \\rfloor $ blocks of length $n$ and the final block of length smaller than $n$ .",
"For each of the blocks $R_i$ , let $L^R_i$ denote the length of the restricted LIP from the midpoint of the left boundary to the midpoint of the right boundary.",
"Note that by Lemma REF the restricted LIP between the midpoints coincides with the unrestricted LIP with overwhelming probability.",
"Thus $\\operatorname{{E}}L^R_i = \\sqrt{2} n - c_1n^{1/3}\\big (1+o(1)\\big ), \\; 1\\leqslant i \\leqslant m$ and similarly for the final block.",
"One gets immediately $\\operatorname{{E}}L^R \\geqslant \\sum _{i=1}^{m+1} \\operatorname{{E}}L^R_i \\geqslant \\sqrt{2} \\ell - C \\cdot m n^{1/3} \\geqslant \\sqrt{2} \\ell - C(\\delta ) \\cdot (\\log w)^{\\frac{1}{3}+\\delta }\\cdot \\frac{\\ell }{w}.$ Let us now turn to the upper bound in (REF ).",
"For this we choose $n$ such that $n^{\\frac{2}{3}}(\\log n)^{\\frac{1}{3}+\\delta } \\leqslant w \\leqslant 2n^{\\frac{2}{3}}(\\log n)^{\\frac{1}{3}+\\delta }$ such that $\\frac{\\ell }{n(\\log n)^3}$ is a positive integer.",
"We provide some details on why we can choose such an $n$ .",
"It is easy to see that for any such $n$ we have that $n(\\log n)^3 \\leqslant c(\\delta ) w^{\\frac{3}{2}}(\\log w)^{\\frac{5}{2}-\\frac{3\\delta }{2}} \\ll l$ .",
"For the exact divisibility first choose $n_1$ so that $w= 2n_1^{\\frac{2}{3}}(\\log n_1)^{\\frac{1}{3}+\\delta }$ .",
"When dividing $\\ell $ by $n_1$ the remainder is at most $n_1(\\log n_1)^3$ .",
"It then follows that we can choose $n_2$ appropriately so that $n_1\\leqslant n_2\\leqslant 2n_1$ and $n_2(\\log n_2)^3$ exactly divides $\\ell $ .",
"It is clear that $w\\leqslant 2n_2^{\\frac{2}{3}}(\\log n_2)^{\\frac{1}{3}+\\delta }$ and $ w\\geqslant 2 (n_2/2)^{\\frac{2}{3}}\\big (\\log (n_2/2)\\big )^{\\frac{1}{3}+\\delta } \\geqslant n_2^{\\frac{2}{3}}(\\log n_2)^{\\frac{1}{3}+\\delta }$ when $w$ is large enough.",
"Let us therefore now assume that we have an $n$ such that $n^{\\frac{2}{3}}(\\log n)^{\\frac{1}{3}+\\delta } \\leqslant w \\leqslant 2n^{\\frac{2}{3}}(\\log n)^{\\frac{1}{3}+\\delta }$ and $n^{\\prime }=n(\\log n)^3$ exactly divides $\\ell $ .",
"We split $R$ into $m^{\\prime }=\\ell /n^{\\prime }$ blocks $R_i$ of length $n^{\\prime }$ .",
"For each $1\\leqslant i\\leqslant m^{\\prime }$ let $X_i$ denote the length of the restricted LIP in $R_i$ and let $L^R_i$ as before denote the length of the restricted (midpoint to midpoint) LIP.",
"Divide $R_i$ further into three sub-rectangles $S_i^{(1)}, \\, S_i^{(2)}, \\, S_i^{(3)}$ where the $S$ -rectangles on either side have length $n$ and $S_i^{(2)}$ has the remaining length.",
"Denote by $Y_i$ the length of restricted LIP in $S_i^{(2)}$ .",
"Figure: Division of block R i R_iIt is clear that $\\begin{split}X_i & \\leqslant \\max _{\\mathbf {x}_1, \\mathbf {y}_1} L^R(\\mathbf {x}_1,\\mathbf {y}_1) + Y_i + \\max _{\\mathbf {x}_3, \\mathbf {y}_3} L^R(\\mathbf {x}_3,\\mathbf {y}_3) \\\\L^R_i & \\geqslant \\min _{\\mathbf {x}_1, \\mathbf {y}_1} L^R(\\mathbf {x}_1,\\mathbf {y}_1) + Y_i + \\min _{\\mathbf {x}_3,\\mathbf {y}_3} L^R(\\mathbf {x}_3,\\mathbf {y}_3)\\end{split}$ where $\\mathbf {x}_1,\\mathbf {y}_1$ are generic points on the left and right boundaries of $S_i^{(1)}$ , and $\\mathbf {x}_3,\\mathbf {y}_3$ are generic points on the left and right boundaries of $S_i^{(3)}$ .",
"Therefore $\\operatorname{{E}}\\vert X_i - L^R_i\\vert \\leqslant 2 \\Vert \\Delta \\big (S_i^{(1)}\\big )\\Vert _1 \\leqslant C(\\delta )\\frac{w^2}{n}$ by Proposition REF .",
"This would then give $\\operatorname{{E}}L^R \\leqslant \\sum _{i=1}^{m^{\\prime }} \\operatorname{{E}}X_i \\leqslant \\sum _{i=1}^{m^{\\prime }} \\big (\\operatorname{{E}}L^R_i + C(\\delta )\\frac{w^2}{n}\\big ).$ We can bound $\\operatorname{{E}}L^R_i$ by the expected value of the length of the unrestricted (midpoint to midpoint) LIP to get an upper bound of $ \\sqrt{2} n^{\\prime }- c_1(n^{\\prime })^{1/3}\\big (1+o(1)\\big ) $ and thus $\\operatorname{{E}}L^R \\leqslant \\sqrt{2}\\ell - c m^{\\prime } (n^{\\prime })^{\\frac{1}{3}} + C(\\delta )\\cdot m^{\\prime }\\frac{w^2}{n} \\leqslant \\sqrt{2} \\ell - \\frac{C(\\delta )}{(\\log n)^{\\frac{5}{3}-\\delta }}\\cdot \\frac{\\ell }{w}.$ This gives the required upper bound because $\\log n$ is of the same order as $\\log w$ .",
"$\\blacksquare $"
],
[
"Meeting of Paths",
"Given $\\delta , \\delta ^{\\prime }>0$ and $n$ consider a diagonal rectangle $R$ of width $w^{(b)}=w^{(b)}(n)$ and length $\\ell ^{(b)}=\\ell ^{(b)}(n)$ with $ w^{(b)}= n^{\\frac{2}{3}}(\\log n)^{\\frac{1}{3} + \\delta }.$ and $\\ell ^{(b)}= n\\big [\\sqrt{\\log n}\\big ]+2\\tau \\quad \\text{where}\\quad \\tau :=\\frac{n}{(\\log n)^{\\delta ^{\\prime }}}.$ Rectangles of this type will form the basic blocks in Section .",
"In this section we give a lower bound on the probability that LIPs between arbitrary starting and ending points within the blocks meet.",
"This would help us get the necessary independence structure needed for getting a lower bound on the variance done in Section .",
"Denote by $\\mathbf {a}$ a generic point on the left boundary of $R$ and by $\\mathbf {b}$ a generic point on the right boundary of $R$ .",
"Define the event $\\Omega :=\\lbrace \\text{There exists $\\mathbf {p}\\in R$ such that for all $\\mathbf {a},\\, \\mathbf {b}$ there is a path in $\\mathcal {L}^R(\\mathbf {a},\\mathbf {b})$ passing through $\\mathbf {p}$}\\rbrace .$ By geometric considerations one observes that $\\Omega = \\lbrace \\text{A path in $\\mathcal {L}^R\\big ((0,0),(\\ell ^{(b)},0)\\big )$ intersects a path in$\\mathcal {L}^R\\big ((0,w^{(b)}),(\\ell ^{(b)}, w^{(b)})\\big )$}\\rbrace .$ In fact any point in the intersection of a path $\\tilde{p}$ in $\\mathcal {L}^R\\big ((0,0),(\\ell ^{(b)},0)\\big )$ and a path $\\tilde{q}$ in $\\mathcal {L}^R\\big ((0,w^{(b)}),(\\ell ^{(b)}, w^{(b)})\\big )$ would be such a point $\\mathbf {p}$ .",
"This is because from any path $\\tilde{r}$ in $\\mathcal {L}^R(\\mathbf {a},\\mathbf {b})$ which intersects either of these paths, one could construct another path in $\\mathcal {L}^R(\\mathbf {a},\\mathbf {b})$ passing through the point $\\mathbf {p}$ .",
"Note first that the path $\\tilde{r}$ would either intersect $\\tilde{p}$ or $\\tilde{q}$ at least two points.",
"Without loss of generality assume that it intersects $\\tilde{p}$ at points $p_1$ (the first intersecting point) and $p_2$ (the last intersecting point).",
"Construct another path following $\\tilde{r}$ till $p_1$ , then following $\\tilde{p}$ from $p_1$ to $p_2$ , and finally following $\\tilde{r}$ from $p_2$ to $\\mathbf {b}$ .",
"One can argue that the length of this new path between $p_1$ and $p_2$ is the same as the length of $\\tilde{r}$ between $p_1$ and $p_2$ .",
"Having a greater length would be a contradiction to the maximality of $\\tilde{r}$ while having a lower length would be a contradiction to the maximality of $\\tilde{p}$ .",
"Figure: Picture depicting Ωand𝐩\\Omega \\text{ and } \\mathbf {p}The following is the main result of the section.",
"Theorem 6.1 Fix $\\delta ,\\delta ^{\\prime }$ satisfying $\\quad 0<\\delta <\\frac{1}{12}\\quad \\text{and}\\quad 0<\\delta ^{\\prime }<\\frac{1}{6}-2\\delta .$ Let $R$ be a diagonal rectangle with length $\\ell ^{(b)}=\\ell ^{(b)}(n)$ and width $w^{(b)}=w^{(b)}(n)$ satisfying (REF ) and (REF ).",
"Then for any $0<\\epsilon <1$ we have $ \\operatorname{{P}}(\\Omega )\\geqslant n^{-\\epsilon }$ for all $n$ large enough.",
"Fix $0<\\epsilon <1$ and $\\delta , \\delta ^{\\prime }$ satisfying (REF ).",
"Define the following events.",
"$ \\begin{split}A_1&:=\\left\\lbrace L^R((0,0),(\\ell ^{(b)},0))\\geqslant E_{\\ell ^{(b)}} + 10 \\varepsilon ^{\\frac{2}{3}}n^{\\frac{1}{3}}( \\log n)^{\\frac{5}{6}}\\right\\rbrace ,\\\\A_1^{\\prime }&:=\\left\\lbrace L^R((0,w^{(b)}),(\\ell ^{(b)},w^{(b)}))\\geqslant E_{\\ell ^{(b)}} + \\varepsilon ^{\\frac{2}{3}}n^{\\frac{1}{3}}( \\log n)^{\\frac{5}{6}}\\right\\rbrace ,\\\\A_2&:=\\left\\lbrace A_1^{\\prime }\\text{ does not occur disjointly from $A_1$}\\right\\rbrace = \\left(A_1\\Box A_1^{\\prime }\\right)^c.\\end{split}$ Our approach to lower bound $\\operatorname{{P}}(\\Omega )$ proceeds by showing that $A_1$ has probability $n^{-c\\varepsilon }$ and $A_2$ is likely given $A_1$ .",
"Under the event $A_1$ , the LIP between the point $(0,0)$ and $(\\ell ^{(b)},0)$ is very long and the only potential way for $(0,w^{(b)})$ to be connected to $(\\ell ^{(b)},w^{(b)})$ via a path with similarly large length is by intersecting the first path.",
"To make this more precise, we need also control the loss incurred in connecting a corner point of $R$ to a point near the central line of $R$ (with $s$ coordinate roughly $w^{(b)}/2$ ).",
"To this end, define the events $ \\begin{split}A_3 & :=\\left\\lbrace \\forall \\, 0\\leqslant s\\leqslant w^{(b)},\\;\\,L^R\\big ((0,0),(\\tau , s)\\big )\\leqslant E_\\tau + n^{\\frac{1}{3}}(\\log n)^{\\frac{2}{3}}\\right\\rbrace ,\\\\A_{4,s^{\\prime }}& :=\\left\\lbrace L^R\\big ((0,0),(\\tau , s^{\\prime })\\big )\\geqslant E_\\tau -2\\mathcal {C} n^{\\frac{1}{3}}(\\log n)^{\\frac{2}{3}+2\\delta +\\delta ^{\\prime }}\\right\\rbrace ,\\qquad 0\\leqslant s^{\\prime }\\leqslant w^{(b)}\\end{split}$ and the symmetric events $ \\begin{split}A_3^{\\prime } &:=\\left\\lbrace \\forall \\, 0\\leqslant s\\leqslant w^{(b)},\\;\\,L^R\\big ((\\ell ^{(b)}-\\tau ,s),(\\ell ^{(b)},0)\\big )\\leqslant E_\\tau + n^{\\frac{1}{3}}(\\log n)^{\\frac{2}{3}}\\right\\rbrace \\\\A_{4,s^{\\prime }}^{\\prime } &:=\\left\\lbrace L^R\\big ((\\ell ^{(b)}-\\tau ,s^{\\prime }),(\\ell ^{(b)}, 0)\\big )\\geqslant E_\\tau -2\\mathcal {C} n^{\\frac{1}{3}}(\\log n)^{\\frac{2}{3} + 2\\delta +\\delta ^{\\prime }}\\right\\rbrace ,\\qquad 0\\leqslant s^{\\prime }\\leqslant w^{(b)}.\\end{split}$ where $\\mathcal {C}=\\mathcal {C}(\\delta )$ is the constant appearing in Proposition REF .",
"The following lemmas gather bounds which will be combined to prove Theorem REF .",
"Lemma 6.2 There exists a constant $c>0$ independent of $\\epsilon $ and $n_0= n_0(\\delta , \\delta ^{\\prime }, \\epsilon )$ such that for all $n\\geqslant n_0$ we have $\\operatorname{{P}}(A_2^c | A_1) \\leqslant n^{-c\\varepsilon }.$ The estimate for $\\operatorname{{P}}(A_2^c | A_1)$ follows by an application of Lemma REF and comparison with LIP which are not constrained to lie in $R$ .",
"$\\operatorname{{P}}(A_2^c | A_1)\\leqslant \\operatorname{{P}}(A_1^{\\prime })\\leqslant \\operatorname{{P}}\\big (L((0,w^{(b)}),(\\ell ^{(b)},w^{(b)}))\\geqslant E_{\\ell ^{(b)}} + \\varepsilon ^{\\frac{2}{3}}n^{\\frac{1}{3}}( \\log n)^{\\frac{5}{6}}\\big )\\leqslant Cn^{-c\\varepsilon },$ the last inequality following from (REF ) and (REF ).",
"Lemma 6.3 There exist constants $c, C>0$ and $n_0= n_0(\\delta , \\delta ^{\\prime })$ such that for all $n\\geqslant n_0$ we have $\\operatorname{{P}}(A_3^c)\\leqslant C\\exp \\big (-c(\\log n)^{1 + \\frac{\\delta ^{\\prime }}{2}}\\big ), $ A similar bound as (REF ) holds for $\\operatorname{{P}}\\big ((A^{\\prime }_3)^c\\big )$ .",
"As in Proposition REF we might as well discretize the possible values of $s$ to only about $\\tau w^{(b)}$ possible values at distance $\\tau ^{-1}$ apart; see the argument leading up to (REF ).",
"For each value of $s$ , we compare with the LIP which are not constrained to lie in $R$ to obtain $\\begin{split}\\operatorname{{P}}\\big [L^R\\big ((0,0),(\\tau , s)\\big )> E_\\tau + n^{\\frac{1}{3}}(\\log n)^{\\frac{2}{3}}\\big ]& \\leqslant \\operatorname{{P}}\\big [L\\big ((0,0),(\\tau , s)\\big )> E_\\tau + n^{\\frac{1}{3}}(\\log n)^{\\frac{2}{3}}\\big ] \\\\& = \\operatorname{{P}}\\big [L\\big ((0,0),(\\tau , s)\\big )> E_\\tau + \\tau ^{\\frac{1}{3}}(\\log n)^{\\frac{\\delta ^{\\prime }}{3}}(\\log n)^{\\frac{2}{3}}\\big ] \\\\& \\leqslant C\\exp \\big (-c(\\log n)^{1 + \\frac{\\delta ^{\\prime }}{2}}\\big ),\\end{split}$ the last inequality following from Lemma REF .",
"The lemma follows from a union bound.",
"Lemma 6.4 For any $D>0$ and $0\\leqslant s^{\\prime }\\leqslant w^{(b)}$ we have $\\operatorname{{P}}(A_{4,s^{\\prime }}^c)= o(n^{-D}) .",
"$ A similar bound as (REF ) holds for $\\operatorname{{P}}\\big ((A^{\\prime }_{4,s^{\\prime }})^c\\big )$ .",
"The argument is based on the proof of Proposition REF to which we shall refer.",
"Fix $D>0$ .",
"It is enough to prove $\\operatorname{{P}}\\left\\lbrace L^R\\big (\\mathbf {a},(\\tau /2, w^{(b)}/2)\\big )\\leqslant E_{\\frac{\\tau }{2}} -\\mathcal {C} n^{\\frac{1}{3}}(\\log n)^{\\frac{2}{3}+2\\delta +\\delta ^{\\prime }}\\right\\rbrace \\leqslant \\frac{1}{n^{D}}$ where $\\mathbf {a}$ is a point on the left boundary.",
"This is because we have a similar bound for $L^R\\big (\\big (\\frac{\\tau }{2}, \\frac{w^{(b)}}{2}\\big ), \\mathbf {b}\\big )$ where $\\mathbf {b}=(\\tau ,s^{\\prime })$ .",
"Let us assume without loss of generality that $\\mathbf {a}=0$ as in the proof of Proposition REF .",
"We have shown there that $\\sum _{k=0}^{k_0-1}E\\big ((t_k,s_k), (t_{k+1},s_{k+1})\\big ) + E\\big ((t_{k_0},s_{k_0}), (\\tau /2, w^{(b)}/2)\\big ) \\geqslant E_{\\frac{\\tau }{2}} -\\frac{\\mathcal {C}}{2}n^{\\frac{1}{3}} (\\log n)^{\\frac{2}{3} +2\\delta + \\delta ^{\\prime }}.$ By superadditivity it follows that $\\begin{split}L^R\\big ((0,0),(\\tau /2, w^{(b)}/2)\\big ) &\\geqslant \\sum _{k=0}^{k_0-1}L^R\\big ((t_k,s_k), (t_{k+1},s_{k+1})\\big ) + L^R\\big ((t_{k_0},s_{k_0}), (\\tau /2, w^{(b)}/2)\\big ) \\\\& = \\sum _{k=0}^{k_0-1}L\\big ((t_k,s_k), (t_{k+1},s_{k+1})\\big ) + L\\big ((t_{k_0},s_{k_0}), (\\tau /2, w^{(b)}/2)\\big ),\\end{split}$ the equality holding outside a set of probability less than $C\\exp \\big (-c(\\log n)^{1+3\\delta }\\big )$ by Lemma REF , and hence negligible compared to the bound we are trying to prove.",
"Thus it is enough to show that $\\begin{split} & \\sum _{k=0}^{k_0-1}\\Big \\vert L\\big ((t_k,s_k), (t_{k+1},s_{k+1})\\big ) -E\\big ((t_k,s_k), (t_{k+1},s_{k+1})\\big )\\Big \\vert \\\\&\\hspace{56.9055pt}+ \\Big \\vert L\\big ((t_{k_0},s_{k_0}), (\\tau /2, w^{(b)}/2)\\big ) - E\\big ((t_{k_0},s_{k_0}), (\\tau /2, w^{(b)}/2)\\big )\\Big \\vert \\leqslant \\frac{\\mathcal {C}}{2}n^{\\frac{1}{3}} (\\log n)^{\\frac{2}{3} +2\\delta + \\delta ^{\\prime }}\\end{split}$ with sufficiently large probability.",
"This follows from the tail bounds in Lemma REF .",
"Indeed for sufficiently large $B$ one gets $\\operatorname{{P}}\\Big (\\Big \\vert L\\big ((t_k,s_k), (t_{k+1},s_{k+1})\\big ) -E\\big ((t_k,s_k), (t_{k+1},s_{k+1})\\big )\\Big \\vert \\geqslant (t_{k+1}-t_k)^{\\frac{1}{3}} (B\\log n)^{\\frac{2}{3}}\\Big ) \\leqslant \\frac{1}{n^{ D}},$ the above holding for each $k$ .",
"A similar bound holds for the last difference in (REF ).",
"Since $k_0= O(\\log n)$ , $\\begin{split}&\\sum _{k=0}^{k_0-1}\\Big \\vert L\\big ((t_k,s_k), (t_{k+1},s_{k+1})\\big ) -E\\big ((t_k,s_k), (t_{k+1},s_{k+1})\\big )\\Big \\vert \\\\&\\hspace{56.9055pt}+ \\Big \\vert L\\big ((t_{k_0},s_{k_0}), (\\tau /2, w^{(b)}/2)\\big ) - E\\big ((t_{k_0},s_{k_0}), (\\tau /2, w^{(b)}/2)\\big )\\Big \\vert \\\\&= O\\Big (\\sum _{k=0}^{k_0-1} (t_{k+1}-t_k)^{1/3} (B\\log n)^{\\frac{2}{3}} + \\big (\\frac{\\tau }{2}-t_{k_0}\\big )^{\\frac{1}{3}} (B\\log n)^{\\frac{2}{3}} \\Big ) \\\\&= O\\Big (n^{\\frac{1}{3}} (\\log n)^{\\frac{2}{3} -\\frac{\\delta ^{\\prime }}{3}} \\Big )\\end{split}$ outside of a set of probability $\\frac{1}{n^{D}}$ .",
"This gives the bound (REF ) as required.",
"Lemma 6.5 There exist constants $C>0$ independent of $\\epsilon $ such that the following holds.",
"There exists $n_0= n_0(\\delta , \\delta ^{\\prime }, \\epsilon )$ such that for all $n\\geqslant n_0$ we have $\\operatorname{{P}}(A_1)\\geqslant n^{-C\\varepsilon }.",
"$ By superadditivity it follows that $L^R\\big ((0,0),(\\ell ^{(b)},0)\\big ) \\geqslant L^R\\big ((0,0),(\\tau , w^{(b)}/2)\\big )+ L^R\\big ((\\tau , w/2),(\\ell ^{(b)}-\\tau , w^{(b)}/2)\\big ) + L^R\\big ((\\ell ^{(b)}-\\tau ,w^{(b)}/2), (\\ell ,0)\\big ).$ Call $ A_4=A_{4,\\frac{w^{(b)}}{2}}$ and $ A_4^{\\prime }=A_{4,\\frac{w^{(b)}}{2}}^{\\prime }$ .",
"Note that on the event $A_{4}\\cap A_{4}^{\\prime } $ , $L^R\\big ((0,0),(\\ell ^{(b)},0)\\big ) \\geqslant 2E_\\tau - 4\\mathcal {C} n^{\\frac{1}{3}}(\\log n)^{\\frac{2}{3}+2\\delta +\\delta ^{\\prime }} + L^R\\big ((\\tau , w^{(b)}/2),(\\ell ^{(b)}-\\tau , w^{(b)}/2)\\big ).$ It is then easy to deduce that $\\begin{split}&\\operatorname{{P}}(A_1) \\geqslant \\operatorname{{P}}\\Big [L^R\\big ((\\tau , w^{(b)}/2),(\\ell ^{(b)}-\\tau ,w^{(b)}/2)\\big )\\geqslant E_{\\ell ^{(b)}} + 10\\varepsilon ^{\\frac{2}{3}}n^{\\frac{1}{3}}( \\log n)^{\\frac{5}{6}} - 2E_\\tau + 4\\mathcal {C}n^{\\frac{1}{3}}(\\log n)^{\\frac{2}{3}+2\\delta +\\delta ^{\\prime }}\\Big ] \\\\&\\hspace{113.81102pt}- 2\\operatorname{{P}}\\big (A_{4}^c\\big ).\\end{split}$ The term $\\operatorname{{P}}(A^c_{4})$ decays very fast in $n$ as we showed in (REF ) and we therefore focus on the first term on the right hand side.",
"We note that by () we have $|E_{\\ell ^{(b)}} - 2E_\\tau - E_{\\ell ^{(b)}-2\\tau }|\\leqslant Cn^{1/3}(\\log n)^{1/6}.$ Thus for large enough $n$ , $\\begin{split}& \\operatorname{{P}}\\left[L^R\\big ((\\tau , w^{(b)}/2),(\\ell ^{(b)}-\\tau , w^{(b)}/2)\\big )\\geqslant E_{\\ell ^{(b)}} + 10\\varepsilon ^{\\frac{2}{3}}n^{\\frac{1}{3}}( \\log n)^{\\frac{5}{6}} - 2E_\\tau + 4\\mathcal {C}n^{\\frac{1}{3}}(\\log n)^{\\frac{2}{3}+2\\delta +\\delta ^{\\prime }}\\right] \\\\&\\qquad \\geqslant \\operatorname{{P}}\\left[L^R\\big ((\\tau , w^{(b)}/2),(\\ell ^{(b)}-\\tau , w^{(b)}/2)\\big )\\geqslant E_{\\ell ^{(b)}-2\\tau } + 13 \\varepsilon ^{\\frac{2}{3}}n^{\\frac{1}{3}}( \\log n)^{\\frac{5}{6}}\\right]\\end{split} $ and we focus our attention on the right hand side.",
"We split the $t$ -interval $(\\tau ,\\ell ^{(b)}-\\tau )$ into $\\big [\\sqrt{\\log n}\\big ]$ intervals of length $n$ each and observe $L^R\\big ((\\tau , w^{(b)}/2), (\\ell ^{(b)}-\\tau , w^{(b)}/2)\\big ) \\geqslant \\sum _{i=0}^{[\\sqrt{\\log n}]-1}L^R\\big ((\\tau + in, w^{(b)}/2), (\\tau +(i+1)n, w^{(b)}/2)\\big ).$ Figure: Construction of a long path in A 1 A_1Using independence between different blocks, the right hand side of (REF ) is bounded below by $\\left(\\operatorname{{P}}\\left[L^R\\big ((\\tau , w^{(b)}/2), (\\tau +n, w^{(b)}/2)\\big ) \\geqslant \\frac{E_{\\ell ^{(b)}-2\\tau }}{\\big [\\sqrt{\\log n}\\big ]} + 14\\varepsilon ^{\\frac{2}{3}} n^{\\frac{1}{3}} (\\log n)^{\\frac{1}{3}}\\right]\\right)^{[\\sqrt{\\log n}]}.$ It is an easy computation to check that $E_{\\ell ^{(b)}-2\\tau } -\\big [\\sqrt{\\log n}\\big ] E_n= O\\big (n^{\\frac{1}{3}}\\sqrt{\\log n}\\big )$ and thus for large $n$ we have a lower bound of $\\left(\\operatorname{{P}}\\left[L^R\\big ((\\tau , w^{(b)}/2), (\\tau +n, w^{(b)}/2)\\big ) \\geqslant E_n + 15\\varepsilon ^{\\frac{2}{3}} n^{\\frac{1}{3}} (\\log n)^{\\frac{1}{3}}\\right]\\right)^{[\\sqrt{\\log n}]}.$ By the transversal fluctuations Lemma REF , we may lift the restriction of staying in $R$ from the last probability to get a lower bound of $ \\begin{split}&\\Big \\lbrace \\operatorname{{P}}\\left[L\\big ((\\tau , w^{(b)}/2), (\\tau +n, w^{(b)}/2)\\big ) \\geqslant E_n + 15\\varepsilon ^{\\frac{2}{3}} n^{\\frac{1}{3}} (\\log n)^{\\frac{1}{3}}\\right] -\\exp \\big (-c(\\log n)^{1+3\\delta }\\big )\\Big \\rbrace ^{[\\sqrt{\\log n}]} \\\\&\\geqslant \\left\\lbrace \\exp \\left(-C\\varepsilon (\\log n)^{\\frac{1}{2}}\\right)-\\exp \\big (-c(\\log n)^{1+3\\delta }\\big )\\right\\rbrace ^{[\\sqrt{\\log n}]} \\geqslant \\frac{1}{n^{C\\varepsilon }}\\end{split}$ for large $n$ , where we used (REF ) in the last line.",
"We can now combine all the lemmas above to prove Theorem REF .",
"Recall the events in (REF ), (REF ) and (REF ).",
"We have that $ \\Omega \\supseteq B_0\\cap A_1\\cap A_2\\cap \\tilde{A}_3\\cap \\tilde{A}_4.$ Here $\\tilde{A}_3$ is the event $A_3\\cap A_{3}^{\\prime }$ and the event $\\tilde{A}_4 =\\bigcap _{s^{\\prime }}\\big (A_{4,s^{\\prime }}^{\\prime } \\cap A_{4,s^{\\prime }}^{\\prime }\\big )$ , the intersection being over a discretization of $0\\leqslant s^{\\prime }\\leqslant w^{(b)}$ spaced distance $[\\ell ^{(b)}]^{-1}$ apart.",
"The event $B_0$ is that each of the strips of width $[\\ell ^{(b)}]^{-1}$ and length $\\ell ^{(b)}$ has at most $[\\log \\ell ^{(b)}]^2$ points.",
"One can bound the probability of this event just as in Lemma REF .",
"Using (REF ) one obtains $\\operatorname{{P}}\\big (\\tilde{A}_4^c\\big ) \\leqslant \\frac{1}{n^{D}}$ for some large $D$ since there are only $w^{(b)} \\ell ^{(b)}$ points in the discretization.",
"The reason for the containment (REF ) is that under $A_1$ there is a long (point to point, restricted) LIP from $(0,0)$ to $(\\ell ^{(b)},0)$ .",
"Note here that since $B_0$ occurs, one can assume that this path passes through one of the discretization points $s^{\\prime }$ , since the loss in length is comparitively small.",
"As $\\tilde{A}_3$ occurs, the path does not have a long subpath from $(0,0)$ to $(\\tau , s)$ for any $s$ or from $(\\ell ^{(b)}-\\tau , s)$ to $(\\ell ^{(b)},0)$ for any $s$ .",
"Then, as $A_2$ occurs, there is no similarly long LIP from $(0,w^{(b)})$ to $(\\ell ^{(b)},w^{(b)})$ which is disjoint from the first LIP.",
"However, as $\\tilde{A_4}$ occurs, it is possible to start from $(0,w^{(b)})$ , take an LIP to $(\\tau , s^{\\prime })$ for some suitable $s^{\\prime }$ , continue along the portion of the LIP from $(0,0)$ to $(\\ell ^{(b)},0)$ until some $(\\ell ^{(b)}-\\tau , s^{\\prime })$ for another suitable $s^{\\prime }$ , and then take an LIP to $(\\ell ^{(b)},w^{(b)})$ to altogether obtain a very long LIP from $(0,w)$ to $(\\ell ^{(b)},w^{(b)})$ .",
"Thus $\\Omega $ must occur.",
"Using the probability estimates (REF ), (REF ) and a similar bound as Lemma REF to bound $\\operatorname{{P}}(B_0)$ we get $\\operatorname{{P}}(B_0\\cap \\tilde{A}_3 \\cap \\tilde{A}_4) \\geqslant 1-n^{-D}.$ for all large enough $D$ .",
"On the other hand (REF ) and (REF ) give $\\operatorname{{P}}(A_2\\cap A_1)\\geqslant \\operatorname{{P}}(A_1)\\cdot \\left(1-n^{-c\\epsilon }\\right).$ for all large $n$ .",
"Thus $ \\begin{split}\\operatorname{{P}}(\\Omega ) &= \\operatorname{{P}}(A_1\\cap A_2) + \\operatorname{{P}}(B_0\\cap \\tilde{A}_3 \\cap \\tilde{A}_4) -\\operatorname{{P}}\\left((A_1\\cap A_2)\\cup (B_0\\cap \\tilde{A}_3 \\cap \\tilde{A}_4)\\right) \\\\&\\geqslant \\operatorname{{P}}(A_1\\cap A_2) +\\operatorname{{P}}(B_0\\cap \\tilde{A}_3 \\cap \\tilde{A}_4) -1\\\\&\\geqslant n^{-c\\epsilon }\\end{split}$ for some $c>0$ and all large $n$ , as we wanted to prove."
],
[
"Proof of Theorem ",
"Fix $\\delta , \\delta ^{\\prime }$ satisfying the conditions of Theorem REF and find $n$ so that $w=w^{(b)}= n^{\\frac{2}{3}}(\\log n)^{\\frac{1}{3}+\\delta }$ .",
"Now split the rectangle $R$ into basic blocks of length $\\ell ^{(b)}= n\\big [\\sqrt{\\log n}\\big ] +2\\tau $ where $\\tau = \\frac{n}{(\\log n)^{\\delta ^{\\prime }}}$ as defined in (REF ), with the last block of (possibly) smaller length.",
"We showed in Section that in each block with probability at least $n^{-c\\varepsilon }$ there exists a point $\\mathbf {p}$ (regeneration point) such that the following holds: For all points $\\mathbf {a}, \\mathbf {b}$ on the left and right boundaries of the block one can find a path in $\\mathcal {L}^R(\\mathbf {a},\\mathbf {b})$ connecting $\\mathbf {a}$ and $\\mathbf {b}$ and passing through $\\mathbf {p}$ .",
"The regeneration points $\\mathbf {p}$ give us the required independence in order to establish the lower bound on the variance.",
"It is clear that $L^R = \\text{Sum of restricted LIP between regeneration points}.$ We condition on $\\mathcal {G}$ , the $\\sigma $ -algebra generated by the Poisson points in alternate blocks $1,3,\\cdots $ .",
"Choose one regeneration point with respect to each odd block (odd regeneration point), if there are any present.",
"List the odd regeneration points as $\\mathbf {p}_1, \\mathbf {p}_2, \\cdots $ so that $\\mathbf {p}_1$ is closest to the left boundary of $R$ , $\\mathbf {p}_2$ is the second closest and so on.",
"By the conditional variance formula $\\begin{split}\\operatorname{Var}(L^R) & = \\operatorname{{E}}\\big (\\operatorname{Var}(L^R | \\mathcal {G}) \\big ) +\\operatorname{Var}\\big (\\operatorname{{E}}(L^R |\\mathcal {G})\\big ) \\\\&\\geqslant \\operatorname{{E}}\\Big [ \\sum _i \\operatorname{Var}( L^R(i)| \\mathcal {G}) \\Big ]\\end{split}$ where $L^R(i)$ is the length of LIP between the $\\mathbf {p}_i$ and $\\mathbf {p}_{i+1}$ .",
"This is further greater than $\\operatorname{{E}}\\Big [ \\sum _{i \\in \\mathcal {I}} \\operatorname{Var}( L^R(i)| \\mathcal {G}) \\Big ]$ where $\\mathcal {I}$ is the collection of all odd indices $i$ such that the $\\mathbf {p}_i$ and $\\mathbf {p}_{i+1}$ are in consecutive odd blocks.",
"Since the probability of having a regeneration point in a block is at least $n^{-\\varepsilon }$ , we get that $\\operatorname{Var}(L^R) \\geqslant \\frac{\\ell n^{-2\\varepsilon }}{5\\ell ^{(b)}} \\operatorname{{E}}\\big [\\operatorname{Var}( L^R(i)| \\mathcal {G})\\big ],$ where $i$ is any odd index such that $p_i$ and $p_{i+1}$ are in consecutive odd blocks.",
"The rest of the argument finds a lower bound for $\\operatorname{Var}( L^R(i)| \\mathcal {G})$ .",
"It is clear that $L^R(i) \\leqslant L^R_1(i)+L^R_2(i)+L^R_3(i)$ where $L^R_1(i)$ is the restricted LIP from $\\mathbf {p}_i $ to the right side of the block containing $\\mathbf {p}_i$ , $L^R_2(i)$ is the restricted LIP in the even block separating the odd blocks and $L^R_3(i)$ is the restricted LIP between the left side of the block containing $\\mathbf {p}_{i+1}$ and $\\mathbf {p}_{i+1}$ .",
"Figure: Picture illustrating L 1 R ,L 2 R ,L 3 R L^R_1, L^R_2, L^R_3We have from the upper bound in (REF ) (with the rectangle of length $\\ell ^{(b)}$ and width $w^{(b)}$ ) that $\\operatorname{{E}}L^R_2(i) \\leqslant \\sqrt{2} \\ell ^{(b)}-\\frac{\\ell ^{(b)}}{w^{(b)}(\\log w^{(b)})^{\\frac{5}{3}-\\delta }} \\leqslant E_{\\ell ^{(b)}}+ 2 c_1 (\\ell ^{(b)})^{\\frac{1}{3}} = E_{\\ell ^{(b)}}+ 3c_1 n^{\\frac{1}{3}}(\\log n)^{\\frac{1}{6}}$ from which it follows that $ \\begin{split}\\operatorname{{E}}\\big (L^R(i)\\vert \\mathcal {G}\\big ) &= L^R_1(i)+\\operatorname{{E}}\\big (L^R_2(i)\\big ) +L^R_3(i)\\\\&\\leqslant L^R_1(i)+E_{\\ell ^{(b)}}+ 3c_1 n^{\\frac{1}{3}}(\\log n)^{\\frac{1}{6}} +L^R_3(i).\\end{split}$ The argument used to prove (REF ) also shows that $\\operatorname{{P}}\\left(L^R(\\mathbf {c},\\mathbf {d})\\geqslant E_{\\ell ^{(b)}}+\\varepsilon ^{\\frac{2}{3}}n^{\\frac{1}{3}} (\\log n)^{\\frac{5}{6}} \\right) \\geqslant n^{-C\\varepsilon }$ where $\\mathbf {c}$ and $\\mathbf {d}$ are arbitrary points on the opposite sides of the even rectangle.",
"By taking $\\mathbf {c}$ (resp.",
"$\\mathbf {d}$ ) to be the points at which the LIP from $\\mathbf {p}_i$ (resp.",
"$\\mathbf {p}_{i+1}$ ) hit the left (resp.",
"right) sides of the even block, one gets $\\operatorname{{P}}\\left(L^R(i) \\geqslant L^R_1(i)+ E_{\\ell ^{(b)}}+\\varepsilon ^{\\frac{2}{3}}n^{\\frac{1}{3}} (\\log n)^{\\frac{5}{6}}+L^R_3(i) \\,\\big \\vert \\, \\mathcal {G} \\right) \\geqslant n^{-C\\varepsilon }$ Combining (REF ) and (REF ) we get $\\begin{split}\\operatorname{Var}(L^R) &\\geqslant \\frac{\\ell n^{-2\\varepsilon }}{5\\ell ^{(b)}} \\operatorname{{E}}\\big [\\operatorname{Var}( L^R(i)| \\mathcal {G})\\big ] \\\\& \\geqslant \\frac{\\ell n^{-2\\varepsilon }}{10\\ell ^{(b)}}\\cdot n^{-C\\varepsilon }\\cdot \\varepsilon ^{\\frac{4}{3}}n^{\\frac{2}{3}} (\\log n)^{\\frac{5}{3}} \\geqslant \\left(\\frac{\\ell }{w^{\\frac{1}{2}}}\\right)^{1-C\\varepsilon }.\\end{split}$ This completes the proof of the theorem.",
"$\\blacksquare $"
],
[
"Proof of Theorem ",
"Fix $0<\\delta <\\frac{1}{6}$ and find $n$ so that $w=n^{\\frac{2}{3}}(\\log n)^{\\frac{1}{3}+\\delta }$ .",
"The proof is based on a recursive splitting of the rectangle $R$ .",
"We start with the rectangle $R$ of length $\\ell _0=\\ell $ and divide $R$ into three sub-rectangles, the second of which has length $n$ , the first and last of length $\\ell _1$ where $\\ell _0=2\\ell _1+n$ .",
"Next split both of the rectangles of size $\\ell _1$ into three rectangles, the middle one of length $n$ and the ones on the sides of the same length.",
"Perform this process recursively so that the lengths of the side sub-rectangles are given by $\\ell _i = 2\\ell _{i+1}+n.",
"$ For a scale $\\ell _i$ , let $L^{R}(i)$ denote the length of the (restricted) maximal path in the left-most rectangle of length $\\ell _i$ , and let $L^{R}_1(i+1)$ and $L^{R}_2(i+1)$ denote the lengths of the (restricted) maximal paths of the sub-rectangles of size $\\ell _{i+1}$ inside it.",
"Let $S$ denote the middle rectangle of length $n$ and let $\\mathbf {x}$ (resp.",
"$\\mathbf {y}$ ) denote an arbitrary point on the left (resp.",
"right) boundary of $S$ .",
"It is clear that $L^R_1(i+1) +\\min _{\\mathbf {x},\\mathbf {y}} L^R(\\mathbf {x},\\mathbf {y}) + L^R_2(i+1) \\leqslant L^R(i) \\leqslant L^R_1(i+1) +\\max _{\\mathbf {x},\\mathbf {y}} L^R(\\mathbf {x},\\mathbf {y}) + L^R_2(i+1)$ Subtracting the means and moving terms we get from the above $\\begin{split}&\\min _{\\mathbf {x},\\mathbf {y}}L^R(\\mathbf {x},\\mathbf {y}) -\\operatorname{{E}}\\max _{\\mathbf {x},\\mathbf {y}} L^R(\\mathbf {x},\\mathbf {y}) \\\\&\\qquad \\leqslant L^R(i)-\\operatorname{{E}}L^R(i) -\\big \\lbrace \\big (L_1^R(i+1)- \\operatorname{{E}}L_1^R(i+1)\\big ) +\\big (L_2^R(i+1) - \\operatorname{{E}}L_2^R(i+1)\\big )\\big \\rbrace \\\\& \\qquad \\qquad \\leqslant \\max _{\\mathbf {x},\\mathbf {y}}L^R(\\mathbf {x},\\mathbf {y}) -\\operatorname{{E}}\\min _{\\mathbf {x},\\mathbf {y}} L^R(\\mathbf {x},\\mathbf {y}).\\end{split} $ An application of the triangle inequality then gives us $ \\begin{split}& \\big \\Vert L^R(i)-\\operatorname{{E}}L^R(i) \\big \\Vert _{2k^{\\prime }} \\\\& \\quad \\leqslant \\big \\Vert \\big (L_1^R(i+1) - \\operatorname{{E}}L_1^R(i+1)\\big ) +\\big (L_2^R(i+1) - \\operatorname{{E}}L_2^R(i+1)\\big ) \\big \\Vert _{2k^{\\prime }} \\\\&\\qquad \\quad + 2 \\big \\Vert \\max _{\\mathbf {x},\\mathbf {y}}L^R(\\mathbf {x},\\mathbf {y}) - \\min _{\\mathbf {x},\\mathbf {y}} L^R(\\mathbf {x},\\mathbf {y})\\big \\Vert _{2k^{\\prime }} + 2 \\big \\Vert \\max _{\\mathbf {x},\\mathbf {y}}L^R(\\mathbf {x},\\mathbf {y}) - \\operatorname{{E}}\\max _{\\mathbf {x},\\mathbf {y}} L^R(\\mathbf {x},\\mathbf {y})\\big \\Vert _{2k^{\\prime }} \\\\& \\quad \\leqslant \\big \\Vert \\big (L_1^R(i+1) - \\operatorname{{E}}L_1^R(i+1)\\big ) +\\big (L_2^R(i+1) - \\operatorname{{E}}L_2^R(i+1)\\big ) \\big \\Vert _{2k^{\\prime }} + C(\\delta , k^{\\prime }) \\cdot n^{\\frac{1}{3}} (\\log n)^{\\frac{2}{3}+2\\delta }\\end{split}$ where we used Proposition REF and Lemma REF in the last step.",
"We note now that $L_1^R(i+1)$ and $L_2^R(i+1)$ are independent and this will be useful for us in decomposing the first term.",
"The above inequality is the main ingredient in the proof of the theorem.",
"We shall use it recursively until the rectangles in the last step are of size close to $n$ .",
"The number of recursion steps will be $m = \\left\\lfloor \\log _2\\left(\\frac{\\ell }{n}\\right) \\right\\rfloor - 1$ so that $ \\ell _m = \\frac{\\ell }{2^m}- n \\left[\\frac{1}{2}+\\frac{1}{2^2}+\\cdots +\\frac{1}{2^m}\\right].$ It is easy to see that $n \\leqslant \\ell _m \\leqslant 4n$ .",
"We now claim Lemma 8.1 Fix $k\\geqslant 1$ .",
"For $n$ large enough $\\big \\Vert L^R(i)- \\operatorname{{E}}L^R(i)\\big \\Vert _{2k^{\\prime }} \\leqslant C(k) \\cdot 2^{\\frac{m-i}{2}} n^{\\frac{1}{3}} (\\log n)^{\\frac{2}{3}+2\\delta +k^{\\prime }}\\quad \\text{ for } 1\\leqslant k^{\\prime }\\leqslant k,\\, 0\\leqslant i\\leqslant m.$ Let us first consider the case $k^{\\prime }=1$ .",
"In this case (REF ) becomes $ \\big \\Vert L^R(i)-\\operatorname{{E}}L^R(i) \\big \\Vert _{2} \\leqslant \\sqrt{2} \\big \\Vert L^R(i+1)-\\operatorname{{E}}L^R(i+1) \\big \\Vert _{2}+ C(\\delta )\\cdot n^{\\frac{1}{3}} (\\log n)^{\\frac{2}{3}+2\\delta }.$ Proposition REF and Lemma REF continue to be valid with length of rectangle $\\ell _m$ and width $w=n^{\\frac{2}{3}}(\\log n)^{\\frac{1}{3}+\\delta }$ (we leave this as an exercise for the reader) and so $\\big \\Vert L^R(m)-\\operatorname{{E}}L^R(m) \\big \\Vert _{2} \\leqslant C(\\delta )\\cdot n^{\\frac{1}{3}} (\\log n)^{\\frac{2}{3}+2\\delta }.$ Using (REF ) recursively we get $\\big \\Vert L^R(i)-\\operatorname{{E}}L^R(i) \\big \\Vert _{2} \\leqslant C(\\delta )\\cdot n^{\\frac{1}{3}} (\\log n)^{\\frac{2}{3}+2\\delta } \\left[1+ \\sqrt{2}+ \\cdots + \\left(\\sqrt{2}\\right)^{m-i}\\right]$ which implies (REF ) for $k^{\\prime }=1$ .",
"We next consider $k^{\\prime }\\geqslant 2$ .",
"We prove this by backwards induction on $i$ .",
"As above the claim is true for $i=m$ by an application of Proposition REF and Lemma REF .",
"So now suppose it is true for $i+1$ .",
"Call $\\begin{split}X(i+1)&= L_1^R(i+1) - \\operatorname{{E}}L_1^R(i+1),\\\\Y(i+1) &= L_2^R(i+1) - \\operatorname{{E}}L_2^R(i+1).\\end{split}$ By the induction hypothesis and the independence of $X(i+1)$ and $Y(i+1)$ we have for large $n$ $\\begin{split}&\\operatorname{{E}}\\left[\\left\\lbrace X(i+1)+Y(i+1)\\right\\rbrace ^{2k^{\\prime }}\\right] \\\\&= \\operatorname{{E}}\\left[X(i+1)^{2k^{\\prime }}\\right] + \\frac{2k^{\\prime } (2k^{\\prime }-1)}{2} \\operatorname{{E}}\\left[X(i+1)^{2k^{\\prime }-2}\\right]\\cdot \\operatorname{{E}}\\left[Y(i+1)^{2}\\right] + \\cdots + \\operatorname{{E}}\\left[Y(i+1)^{2k^{\\prime }}\\right] \\\\& \\leqslant \\left\\lbrace 2^{\\frac{m-(i+1)}{2}} n^{\\frac{1}{3}} (\\log n)^{\\frac{2}{3}+2\\delta +k^{\\prime }} \\right\\rbrace ^{2k^{\\prime }} \\\\&\\qquad + \\frac{2k^{\\prime }(2k^{\\prime }-1)}{2}\\left\\lbrace 2^{\\frac{m-(i+1)}{2}} n^{\\frac{1}{3}} (\\log n)^{\\frac{2}{3}+2\\delta +k^{\\prime }-1} \\right\\rbrace ^{2k^{\\prime }-2} \\left\\lbrace 2^{\\frac{m-(i+1)}{2}} n^{\\frac{1}{3}} (\\log n)^{\\frac{2}{3}+2\\delta +1} \\right\\rbrace ^{2} \\\\&\\hspace{113.81102pt} + \\cdots + \\left\\lbrace 2^{\\frac{m-(i+1)}{2}} n^{\\frac{1}{3}} (\\log n)^{\\frac{2}{3}+2\\delta +k^{\\prime }} \\right\\rbrace ^{2k^{\\prime }} \\\\& \\leqslant 2^{\\frac{3}{2}} \\cdot \\left\\lbrace 2^{\\frac{m-(i+1)}{2}} n^{\\frac{1}{3}} (\\log n)^{\\frac{2}{3}+2\\delta +k^{\\prime }} \\right\\rbrace ^{2k^{\\prime }}.\\end{split}$ Plugging this in (REF ) completes the induction step, proving our claim (REF ).",
"The proof of the theorem follows by putting $i=0$ and substituting for $\\ell $ and $w$ in (REF ).",
"$\\blacksquare $"
],
[
"Proofs of Theorem ",
"Fix $0<\\delta < 1/12$ and choose $n$ so that $w = w(\\ell ) = n^{\\frac{2}{3}}(\\log n)^{\\frac{1}{3}+\\delta }$ .",
"We divide the rectangle into alternating long and short blocks, starting with a long block, where the long blocks have length $\\ell ^a$ and the short blocks have length $n$ .",
"The last block would have the remaining length.",
"Here $a$ is chosen so that $\\frac{2+3\\gamma }{4} < a< 1.$ The number of long and short blocks is $m=\\lfloor \\ell /(\\ell ^a+n)\\rfloor $ .",
"Since $w\\leqslant \\ell ^\\gamma $ we have $c\\ell ^{1-a} \\leqslant m \\leqslant C\\ell ^{1-a}$ for some positive constants $c, C$ .",
"Let $X_1, X_2,\\cdots , X_m$ denote the maximal lengths of the (restricted) paths in the consecutive long boxes.",
"Let $\\mathbf {x}_i, \\mathbf {y}_i$ be arbitrary left and right endpoints in the $i$ th short box.",
"By an argument similar to (REF ) one can conclude $\\sum _{i=1}^m \\min _{\\mathbf {x}_i, \\mathbf {y}_i} L^R(\\mathbf {x}_i, \\mathbf {y}_i) \\leqslant L^R - \\sum _{i=1}^m X_i \\leqslant \\sum _{i=1}^m \\max _{\\mathbf {x}_i,\\mathbf {y}_i} L^R(\\mathbf {x}_i, \\mathbf {y}_i).$ It follows from this that for any fixed $k\\geqslant 1$ , $\\begin{split}\\left\\Vert \\left[L^R- \\operatorname{{E}}L^R \\right] -\\sum _{i=1}^m \\left[ X_i - \\operatorname{{E}}X_i\\right]\\right\\Vert _k& \\leqslant \\sum _{i=1}^m \\left\\Vert \\max _{\\mathbf {x}_i,\\mathbf {y}_i} L^R(\\mathbf {x}_i, \\mathbf {y}_i) - \\operatorname{{E}}\\max _{\\mathbf {x}_i,\\mathbf {y}_i} L^R(\\mathbf {x}_i, \\mathbf {y}_i) \\right\\Vert _k \\\\& \\qquad + \\sum _{i=1}^m \\left\\Vert \\max _{\\mathbf {x}_i,\\mathbf {y}_i} L^R(\\mathbf {x}_i, \\mathbf {y}_i) - \\min _{\\mathbf {x}_i,\\mathbf {y}_i} L^R(\\mathbf {x}_i, \\mathbf {y}_i) \\right\\Vert _k \\\\& \\leqslant C(\\delta ,k)\\cdot m n^{\\frac{1}{3}} (\\log n)^{\\frac{2}{3}+2\\delta }\\end{split}$ for $n$ large enough, by an application of Proposition REF and Lemma REF .",
"It also follows by an application of the triangle inequality that $\\left| \\, \\sqrt{\\operatorname{Var}(L_N^R)} - \\sqrt{\\sum _{i=1}^m\\operatorname{Var}( X_i)} \\,\\right| \\leqslant C(\\delta )\\cdot m n^{\\frac{1}{3}} (\\log n)^{\\frac{2}{3}+2\\delta }.$ We need the following lemma to complete the proof of Theorem REF .",
"The lemma shows that the sum of the $X_i$ 's satisfy a Gaussian limit theorem.",
"Lemma 9.1 With the notation as above we have as $\\ell \\rightarrow \\infty $ : $\\frac{\\sum _{i=1}^m \\left[ X_i - \\operatorname{{E}}X_i \\right]}{\\sqrt{\\sum _{i=1}^m \\operatorname{Var}(X_i)}} \\Rightarrow N(0,1).$ We check Lindeberg's condition for proving the central limit theorem.",
"It is sufficient to check that $\\sum _{i=1}^m \\operatorname{{E}}\\left[T_i^4 \\right] \\rightarrow 0$ where $T_i = \\frac{X_i - \\operatorname{{E}}X_i }{\\sqrt{\\sum _{i=1}^m \\operatorname{Var}(X_i)}}.$ Using Theorem REF for the fourth moment and Theorem REF for the variance for $X_i$ one has $\\left\\Vert X_i -\\operatorname{{E}}X_i \\right\\Vert _4 \\leqslant \\frac{\\big (\\ell ^a\\big )^{\\frac{1}{2}}}{w^{\\frac{1}{4}}}(\\log w)^{4} \\quad \\text{ and }\\quad \\operatorname{Var}(X_i) \\geqslant \\left(\\frac{\\ell ^a}{w^{\\frac{1}{2}}}\\right)^{1-\\varepsilon }.$ Therefore $\\sum _{i=1}^m \\operatorname{{E}}\\left[T_i^4 \\right] = \\frac{1}{m} \\cdot \\frac{ \\Vert X_1-\\operatorname{{E}}X_1 \\Vert _4^4}{ \\operatorname{Var}(X_i)^2} \\leqslant \\frac{\\ell ^{2ac\\varepsilon }(\\log w)^{16}}{\\ell ^{1-a} w^{c\\varepsilon }}$ which tends to 0 because $a<1$ and $\\varepsilon $ is arbitrary.",
"It is now not difficult to prove (REF ).",
"Indeed (REF ) and Theorem REF gives $\\left\\Vert \\frac{\\left[L^R- \\operatorname{{E}}L^R \\right] -\\sum _{i=1}^m \\left[ X_i - \\operatorname{{E}}X_i\\right]}{\\sqrt{\\sum _{i=1}^m \\operatorname{Var}(X_i)}}\\right\\Vert _k\\leqslant \\frac{C(\\delta ,k) m n^{\\frac{1}{3}} (\\log n)^{\\frac{2}{3}+2\\delta }}{\\sqrt{m \\cdot \\big (\\ell ^a / w^{\\frac{1}{2}} \\big )^{1-\\varepsilon }}}\\longrightarrow 0$ since $a>\\frac{(2+3\\gamma )}{4}$ and $\\varepsilon >0$ is arbitrary.",
"Also by (REF ) and Theorem REF $\\sqrt{\\frac{\\operatorname{Var}(L^R)}{\\sum _{i=1}^m \\operatorname{Var}(X_i)} } \\rightarrow 1.$ This is more than enough to prove Theorem REF .",
"$\\blacksquare $ Finally we present the proof of Theorem REF stated in the introduction.",
"For this proof, we switch back to the usual $(x,y)$ coordinate system.",
"Let $0<\\varepsilon <\\frac{3\\gamma }{2}$ so that $(1-\\frac{\\gamma }{2}) \\cdot \\frac{1-\\epsilon }{2}> \\epsilon +\\frac{\\gamma }{2}$ .",
"Consider the square $[0,n]^2$ and the three regions $S_1$ , $R$ and $S_2$ in the extended strip as shown below.",
"The region $S_1$ is the rectangular region from the origin to the first anti-diagonal line at a distance $\\sqrt{2} n^{\\frac{3\\gamma }{2}-\\varepsilon }$ .",
"The region $S_2$ is the corresponding region on the top right.",
"Thus the length of the middle diagonal rectangle is $\\sqrt{2} n - 2\\sqrt{2} n^{\\frac{3\\gamma }{2}-\\varepsilon }$ .",
"Figure: The regions S 1 ,RS_1, R and S 2 S_2.Clearly we have $\\min _{\\mathbf {x}_1,\\mathbf {y}_1} L^{S_1} \\big (\\mathbf {x}_1, \\mathbf {y}_1\\big ) + L^R + \\min _{\\mathbf {x}_2, \\mathbf {y}_2} L^{S_2} \\big (\\mathbf {x}_2, \\mathbf {y}_2\\big ) \\leqslant L_n^{(\\gamma )} \\leqslant \\max _{\\mathbf {x}_1, \\mathbf {y}_1} L^{S_1} \\big ( \\mathbf {x}_1, \\mathbf {y}_1\\big ) + L^R + \\max _{\\mathbf {x}_2, \\mathbf {y}_2} L^{S_2} \\big (\\mathbf {x}_2, \\mathbf {y}_2\\big ).$ The points $\\mathbf {x}_1, \\mathbf {y}_1$ are arbitrary points on the left and right boundaries of $S_1$ and similarly $\\mathbf {x}_2,\\mathbf {y}_2$ are arbitrary boundary points for $S_2$ .",
"In particular this gives us by an application of Proposition REF $\\left|\\operatorname{{E}}L_n^{(\\gamma )} - \\operatorname{{E}}L^R - \\operatorname{{E}}L^{S_1} \\big (\\mathbf {0}, (n^{\\frac{3\\gamma }{2}-\\varepsilon },n^{\\frac{3\\gamma }{2}-\\varepsilon }) \\big )- \\operatorname{{E}}L^{S_2}\\big ( (n-n^{\\frac{3\\gamma }{2}-\\varepsilon }, n-n^{\\frac{3\\gamma }{2}-\\varepsilon }), (n,n)\\big )\\right|\\leqslant C n^{\\varepsilon +\\frac{\\gamma }{2}}.$ The first assertion in (REF ) follows from this, (REF ) and () .",
"Using (REF ) and applying Proposition REF and Lemma REF we get $\\left|\\sqrt{\\operatorname{Var}L_n^{(\\gamma )}} -\\sqrt{\\operatorname{Var}L^R}\\right|\\leqslant C n^{\\varepsilon +\\frac{\\gamma }{2}}.$ The second assertion in (REF ) follows from this.",
"This is because of Theorem REF and the bound $\\sqrt{\\operatorname{Var}L^R}\\gg n^{\\varepsilon +\\frac{\\gamma }{2}}$ , which follows from Theorem REF and our choice of $\\varepsilon $ .",
"For the final statement (REF ) note that $\\frac{L_n^{(\\gamma )} -\\operatorname{{E}}L_n^{(\\gamma )}}{\\sqrt{\\operatorname{Var}L_n^{(\\gamma )}}} = \\sqrt{\\frac{\\operatorname{Var}L^R}{\\operatorname{Var}L_n^{(\\gamma )}}} \\cdot \\left[\\frac{L^R-\\operatorname{{E}}L^R}{\\sqrt{\\operatorname{Var}L^R}} +\\frac{\\mathcal {E}}{\\sqrt{\\operatorname{Var}L^R}}\\right].$ Here the error term $\\mathcal {E}$ is of order $n^{\\varepsilon +\\frac{\\gamma }{2}}$ by (REF ), Proposition REF and Lemma REF , and hence small with respect to $\\sqrt{\\operatorname{Var}L^R}$ .",
"The above argument also shows that $\\operatorname{Var}L^R /\\operatorname{Var}L_n^{(\\gamma )} \\rightarrow 1$ as $n\\rightarrow \\infty $ .",
"This gives the central limit theorem for $L_n^{(\\gamma )}$ ."
],
[
"Open problems",
"In this section, we collect a few questions which are open.",
"Can one improve the results in Theorem REF and Theorem REF and get a more precise result for $\\operatorname{Var}(L^R)$ in rectangles $R$ of length $\\ell $ and width $1\\ll w\\ll \\ell ^{\\frac{2}{3}}$ ?",
"In particular is it true that $\\operatorname{Var}(L^R) = c \\frac{\\ell }{w^{\\frac{1}{2}}} \\big (1+o(1)\\big )$ as $\\ell \\rightarrow \\infty $ for an appropriate constant $c>0$ ?",
"Considering Theorem REF , can one get sharper results for $\\operatorname{{E}}L^R$ for a diagonal rectangle $R$ of length $\\ell $ and width $1\\ll w \\ll \\ell ^{\\frac{2}{3}}$ ?",
"Do there exist positive constants $c_1, c_2$ such that $\\operatorname{{E}}(L^R) = \\sqrt{2} \\ell - c_1 \\frac{\\ell }{w} + c_2 \\frac{\\ell ^{\\frac{1}{2}}}{w^{\\frac{1}{4}}} \\big ( 1+ o(1) \\big )$ as $\\ell \\rightarrow \\infty $ ?",
"Note that $\\ell ^{\\frac{1}{2}} / w^{\\frac{1}{4}}$ is our prediction in REF for the standard deviation.",
"Theorem REF gives a Gaussian limit for $L_n^{(\\gamma )}$ when $\\gamma <\\frac{2}{3}$ whereas the results of Baik, Deift and Johansson [5] imply a Tracy-Widom limit for $L_n^{(\\gamma )}$ when $\\gamma >\\frac{2}{3}$ .",
"What is the limiting distribution when we take the width of the strip to be $\\alpha \\cdot n^{\\frac{2}{3}}$ for fixed $\\alpha >0$ ?",
"As pointed out in the introduction, a generic $k$ -point configuration in the square $[0,n]^2$ corresponds naturally to a permutation $\\pi $ of $\\lbrace 1, 2, \\cdots , k\\rbrace $ .",
"Our results then fit within the framework of studying random permutations with a band structure, i.e.",
"permutations where $| \\pi (i)- i | $ is typically much smaller than $n$ .",
"Other models of this type include the interchange (or stirring) process on a finite segment (introduced in [36]), the Mallows model (introduced in [31]), the one-dimensional case of displacement-biased random permutations on the lattice [14], [19] and the model of $k$ -min permutations [37].",
"The longest increasing subsequence was analyzed in two of the above examples: the Mallows model [15], [8] (drawing on [32]) and the $k$ -min permutation [37].",
"The results of these studies, however, are not as detailed as the ones obtained here in the sense that the second-order correction to the expectation, the order of magnitude of the variance and the limit law have not been determined (for band widths growing with the size of the permutation).",
"It is natural to expect that the longest increasing subsequence in many random permutation models with a band structure exhibits similar behavior to the one obtained here and it is of great interest to obtain such results for a general class of models.",
"Acknowledgements: We thank Lucas Journel for a careful reading of an earlier draft and for suggesting several improvements.",
"We thank Eitan Bachmat for interesting discussions of related problems and application areas.",
"Most of this work was completed while M.J. was at the University of Sheffield, and he thanks the School of Mathematics and Statistics for a supportive environment.",
"The work of R.P.",
"was supported in part by Israel Science Foundation grant 861/15 and the European Research Council starting grant 678520 (LocalOrder)."
]
] | 1808.08407 |
[
[
"On Generalized Covering Groups of Topological Groups"
],
[
"Abstract It is well-known that a homomorphism p between topological groups K, G is a covering homomorphism if and only if p is an open epimorphism with discrete kernel.",
"In this paper we generalize this fact, in precisely, we show that for a connected locally path connected topological group G, a continuous map p is a generalized covering if and only if K is a topological group and p is an open epimorphism with prodiscrete (i.e, product of discrete groups) kernel.",
"To do this we first show that if G is a topological group and H is any generalized covering subgroup of fundamental group of G, then H is as intersection of all covering subgroups, which contain H. Finally, we show that every generalized covering of a connected locally path connected topological group is a fibration."
],
[
"Introduction",
"Chevalley [8] introduced a covering group theory for connected, locally path connected, and semi locally simply connected topological groups.",
"Rotman [12] proved that for every covering space $(\\widetilde{X}, p)$ of a connected, locally path connected, and semi locally simply connected topological group $G$ , $\\widetilde{X}$ is a topological group and $p$ is a homomorphism.",
"Recently, Torabi [15] developed this theory for connected locally path connected topological groups and gave a classification for covering groups of them.",
"He showed that the natural structure, which has been called the path space, and its relative endpoint projection map for a connected locally path connected topological group are a topological group and a group homomorphism, respectively.",
"Recall that for an arbitrary subgroup $ H $ of the fundamental group $ \\pi _1(X,x_0) $ , the path space is the set of all paths starting at $ x_0 $ , which is denoted by $ P(X,x_0) $ , with an equivalence relation $\\sim _H$ as follows: ${\\alpha }_1\\sim _H {\\alpha }_2$ if and only if ${\\alpha }_1(1)={\\alpha }_2(1)$ and $[{\\alpha }_1*{{\\alpha }_2}^{-1}]\\in H$ .",
"The equivalence class of $\\alpha $ is denoted by ${\\left\\langle \\alpha \\right\\rangle }_H$ .",
"One can consider the quotient space $\\widetilde{X}_H=P(X,x_0)/\\sim _H$ and the endpoint projection map $p_H: (\\widetilde{X}_H,e_H)\\rightarrow (X,x_0)$ defined by ${\\left\\langle \\alpha \\right\\rangle }_H \\mapsto \\alpha (1)$ , where $e_H$ is the class of the constant path at $x_0$ .",
"If $\\alpha \\in P(X,x_0)$ and $U$ is an open neighbourhood of $\\alpha (1)$ , then a continuation of $\\alpha $ in $U$ is a path $\\beta =\\alpha *\\gamma $ , where $\\gamma $ is a path in $U$ with $\\gamma (0)=\\alpha (1)$ .",
"Put $ N({\\left\\langle \\alpha \\right\\rangle }_H,U) =\\lbrace {\\left\\langle \\beta \\right\\rangle }_H\\in {\\widetilde{X}}_H \\ | \\ \\mathrm {\\beta \\ is\\ a\\ continuation\\ of\\ \\alpha \\ in\\ U}\\rbrace $ .",
"It is well known that the subsets $ N({\\left\\langle \\alpha \\right\\rangle }_H, U) $ form a basis for a topology on ${\\widetilde{X}}_H$ for which the function $p_H:{(\\widetilde{X}}_H,e_H)\\rightarrow (X,x_0)$ is continuous (see [13]).",
"Brodskiy et al.",
"[7] called this topology on ${\\widetilde{X}}_H$ the whisker topology.",
"Moreover, they were interested in studying the spaces whose local properties can extend to the entire space and introduced the notions of $ SLT $ and strong $ SLT $ spaces.",
"Definition 1.1 [7] A topological space $X$ is called strong small loop transfer space (strong $ SLT $ space for short) at $x_{0}$ if for each point $x \\in X$ and for every neighbourhood $U$ of $x_{0}$ , there is a neighbourhood $V$ of $x$ such that for every path $ \\alpha : I \\rightarrow X $ from $ x_0 $ to $ x $ and every loop $\\beta $ in $V$ based at $x$ there is a loop $\\gamma $ in $U$ based at $x_{0}$ which is homotopic to $ \\alpha \\ast \\beta \\ast \\alpha ^{-1}$ relative $\\dot{I}$ .",
"The space $X$ is called strong $ SLT $ space, if $X$ is strong $ SLT $ space at $x$ for every $x \\in {X}$ .",
"On the other hand, Torabi showed that every topological group is a strong $ SLT $ at the identity element [15].",
"In Section , we introduce the generalized covering group of a topological group and give some examples to clarify the difference between covering and generalized covering groups (Examples REF and REF ).",
"Moreover, we show that for a connected locally path connected strong $ SLT $ at $ x_0 $ space, every generalized covering subgroup $ H $ of the fundamental group can be written as the intersection of all covering subgroups, which contains $ H $ and vice versa.",
"In Section , we attempt to provide a method for classifying generalized covering groups of a topological group by studying the topology of the kernel of the relative generalized covering homomorphism.",
"Of cores, we extend the well-known result about covering groups (Remark REF ) for generalized covering groups and show that if $ G $ is a connected locally path connected topological group, $ (\\widetilde{G},p) $ is a generalized covering group of $ G $ if and only if $ p $ is an open epimorphism with prodiscrete (i.e, product of discrete groups) kernel (Corollary REF ).",
"In this regard, we first show that the central fibre of a generalized covering map of an arbitrary topological space is totally path disconnected ( Proposition REF ).",
"Counterexample REF show that it is not a sufficient condition for generalized covering subgroups, even in the case of topological groups.",
"After that we present our desirable definition of prodiscrete subgroups (Definition REF ) and show that for a connected locally path connected topological group $ G $ and a generalized covering subgroup $ H \\le \\pi _1(G,e) $ , the kernel of the endpoint projection homomorphism is a prodiscrete subgroup ( Theorem REF ).",
"Berestovskii et al.",
"in [4] provided a new definition by extending the concept of a cover of a topological group $ G $ such as the pair $ (\\widetilde{G},p) $ , where $ \\widetilde{G} $ is a topological group and the homomorphism $ p : \\widetilde{G} \\rightarrow G$ is an open epimorphism with prodiscrete kernel.",
"Note that the meaning of prodiscrete kernel in the sense of them was as the inverse limit of discrete groups.",
"In this paper we use Definition REF for a prodiscrete concept and show that if $ H $ is a prodiscrete normal subgroup of topological group $ G $ , then the pair $ (G,\\varphi _H) $ is a generalized covering group of $ \\frac{G}{H} $ (Theorem REF ).",
"Using this theorem, we conclude the main result in Corollary REF .",
"Finally, we show that in the case of topological groups the concepts of rigid covering fibrations (which was firstly introduced by Biss [5]) and generalized covering groups are coincide and conclude that every generalized covering group of a topological group is also a fibration."
],
[
"Generalized Coverings of Topological Groups",
"It is well-known that a continuous map $p:(\\widetilde{X},{\\tilde{x}}_0)\\rightarrow (X,x_0)$ has the unique lifting property, if for every connected, locally path connected space$\\ (Y,y_0)$ and every continuous map $f:(Y,y_0)\\rightarrow (X,x_0)$ with $f_*{\\pi }_1(Y,y_0)\\subseteq p_*{\\pi }_1(\\widetilde{X},{\\tilde{x}}_0)$ for ${\\tilde{x}}_0\\in p^{-1}(x_0)$ , there exists a unique continuous map $\\tilde{f}:(Y,y_0)\\rightarrow (\\widetilde{X},{\\tilde{x}}_0)$ with $p\\circ \\tilde{f}=f$ .",
"If $\\widetilde{X}$ is a connected, locally path connected space and $p:\\widetilde{X}\\rightarrow X$ has unique lifting property, then $p$ and $\\widetilde{X}$ are called a generalized covering map and a generalized covering space for $X$ , respectively.",
"Definition 2.1 Let $ G $ be a topological group.",
"By a generalized covering group of $ G $ , a pair $ (\\widetilde{G},p) $ is composed of a topological group $ \\widetilde{G} $ and a homomorphism $ p:(\\widetilde{G},\\tilde{e}) \\rightarrow (G,e) $ such that $ (\\widetilde{G},p) $ is a generalized covering space of $ G $ .",
"It is easy to check that for an arbitrary pointed topological space $ (X,x_0) $ if $ p:(\\widetilde{X},\\tilde{x}_0) \\rightarrow (X,x_0) $ is a generalized covering space, then the induced map $ p_* : \\pi _1 (\\widetilde{X},\\tilde{x}_0) \\rightarrow \\pi _1(X,x_0) $ is one to one.",
"Therefore, the image $ H=p_*\\pi _1 (\\widetilde{X},\\tilde{x}_0) $ is a subgroup of $ \\pi _1(X,x_0) $ , which is called generalized covering subgroup.",
"Remark 2.2 There is a well-known result about covering groups of topological groups which we will extend it for generalized covering groups.",
"A homomorphism $ p : \\widetilde{G} \\rightarrow G $ between topological groups is a covering homomorphism if and only if $ p $ is an open epimorphism with discrete kernel.",
"As mentioned in [2] and [9], the endpoint projection map $ p_H : \\widetilde{X}_H \\rightarrow X $ is surjective and open if $ G $ is path connected and locally path connected, respectively.",
"On the other hand, for an arbitrary pointed topological space $ (X,x_0) $ , Brazas [6] showed the relationship between the image of a generalized covering map and the space $ \\widetilde{X}_H $ as follows: Lemma 2.3 Suppose that $\\hat{p}:(\\hat{X},\\hat{x})\\rightarrow (X,x_0)$ has the unique lifting property with ${\\hat{p}}_*({\\pi }_1(\\hat{X},\\hat{x}))=H$ .",
"Then there is a homeomorphism $h:(\\hat{X},\\hat{x})\\rightarrow ({\\widetilde{X}}_H,e_H)$ such that $p_H\\circ h=\\hat{p}$ .",
"Remark 2.4 It has been indicated in [15] that for every subgroup $ H \\le \\pi _1(G,e_G) $ of the fundamental group of a topological group $ G $ , one can construct a multiplication on $ \\widetilde{G}_H $ , which makes $ \\widetilde{G}_H $ a topological group and the continuous map $ p_H: \\widetilde{G}_H \\rightarrow G $ a homomorphism.",
"The following proposition can be obtained by using the above remark and lemma.",
"Proposition 2.5 If $ p:(\\widetilde{G},\\tilde{e}) \\rightarrow (G,e) $ is a continuous map with unique lifting property, $ G$ and $ \\widetilde{G} $ are tow topological groups, and $ \\widetilde{G} $ is connected locally path connected, then $ (\\widetilde{G},p) $ is a generalized covering group.",
"If $ p:(\\widetilde{G},\\tilde{e}) \\rightarrow (G,e) $ has unique lifting property, then by the above lemma, there is a homeomorphism $ h: (\\widetilde{G},\\tilde{e}) \\rightarrow (\\widetilde{G}_H,e_{\\widetilde{G}_H }) $ such that $ p_H \\circ h = p $ .",
"As it was discussed in [15], $ \\widetilde{G}_H $ and so $ \\widetilde{G} $ are topological groups.",
"Moreover, Since the map $ p_H $ is a group homomorphism, then for $ g_1, g_2 \\in \\widetilde{G}$ , $ p(g_1g_2)=p_H \\circ h (g_1g_2) = p_H (h (g_1g_2)) = p_H(h(g_1)h(g_2)) = p_H(h(g_1))p_H(h(g_2)) = p_H \\circ h (g_1)p_H \\circ h (g_2)= p(g_1)p(g_2) $ .",
"Therefore, $ p $ is a homomorphism.",
"Corollary 2.6 Let $ G $ be a topological group, and let $ p:(\\widetilde{G},\\tilde{e}) \\rightarrow (G,e) $ .",
"Then $ (\\widetilde{G},p) $ is a generalized covering group if and only if it is a generalized covering space, or equivalently, $ p $ is a surjection with unique lifting property.",
"It is the immediate result of the definition.",
"Clearly, in the case of semilocally simply connected spaces the category of covering and generalized covering spaces are equivalent.",
"Out of semilocally simply connected spaces there may be a generalized covering, which is not a covering.",
"It seems interesting to find some examples in the case of topological groups.",
"Example 2.7 Let $ Y $ be the Tychonoff product of the infinite number of $ \\mathbb {S}^1 $ 's.",
"Clearly, $ Y $ is a topological group since the product of topological groups is also a topological group.",
"Moreover, every open neighbourhood of any $y \\in Y $ contains many $ \\mathbb {S}^1 $ 's except finite number and so $ Y$ is not a semilocally simply connected space at $ y \\in Y $ .",
"Since $ Y $ is locally path connected, hence $ Y $ does not have the classical universal covering space.",
"Then the universal path space $ p: {\\prod }_{i\\in I} \\mathbb {R} \\rightarrow Y $ is not a covering group of $ Y $ .",
"Although, it is a generalized covering group.",
"Since $ p_i: \\mathbb {R} \\rightarrow \\mathbb {S}^1 $ is a covering map for every $ i \\in I $ , then it is also a generalized covering map.",
"Therefore, from [6] the product of $ p_i $ 's, $ p $ , is also a generalized covering map.",
"In the above example, $ p_* \\pi _1(\\prod _{i \\in I} \\mathbb {R}, {0}) $ is a trivial subgroup of $ \\pi _1(Y,\\mathbf {0}) $ .",
"It seems interesting to introduce a nontrivial generalized covering subgroup of $ \\pi _1(Y,\\mathbf {0}) $ , which is not a covering subgroup.",
"Example 2.8 From the above example, let $ q: \\prod _{i \\in I}{{\\mathbb {R}} \\times \\mathbb {S}^1} \\rightarrow Y$ be as $ q|_{\\prod _{i\\in I}{\\mathbb {R}}} =p $ and identity on $ \\mathbb {S}^1 $ .",
"Clearly, $ p_*\\pi _1(\\prod _{i \\in I} {\\mathbb {R}} \\times \\mathbb {S}^1,({0},s)) = \\mathbb {Z} $ is a subgroup of the fundamental group of $ Y $ and $ q $ has unique path lifting property by the similar way of the above example.",
"Thus $ q $ is a generalized covering homomorphism.",
"We show that $ q $ is not a local homeomorphism, and so it is not a covering homomorphism.",
"Let $ ({x},s) \\in {\\prod _{i \\in I} \\mathbb {R}} \\times \\mathbb {S}^1 $ , and let $ U $ be an open subset containing $ ({x},s) $ .",
"By the definition of Tychonoff product topology, $ U $ and $ q(U) $ contain many $ \\mathbb {R} $ 's and $ \\mathbb {S}^1 $ 's except finite number, respectively.",
"Then for some $ i \\in I $ the restriction $ q|_{U_i}: \\mathbb {R} \\rightarrow \\mathbb {S}^1 $ is not injective.",
"This fact implies that $ q $ is not a local homeomorphism.",
"It was shown in [3] that if $G $ is a left topological group with continuous inverse operation and $ \\beta _e $ a local base of the space $ G $ at the identity element $ e $ , then, for every subset $ A $ of $ G $ $cl(A)=\\overline{A} = \\bigcap \\lbrace AU \\ | \\ U \\in \\beta _e \\rbrace .$ Recall from [7], [11] that for any path connected space $ X $ , any subgroup $ H \\le \\pi _1(X,x_0) $ and the endpoint projection map $p_H: \\widetilde{X}_H \\rightarrow X $ , the fibres $ p^{-1}_H(x) $ and $ p^{-1}_H(y) $ are homeomorphic, for every $ x,y \\in X $ , if and only if $ X $ is an $ SLT $ space.",
"Note that $X$ is called a small loop transfer space ($ SLT $ space for short) if for every $ x,y \\in X $ , for every path $ \\alpha : I \\rightarrow X $ from $ x $ to $ y $ and for every neighbourhood $U$ of $x$ there is a neighbourhood $V$ of $y$ such that for every loop $\\beta $ in $V$ based at $y$ there is a loop $\\gamma $ in $U$ based at $x$ , which is homotopic to $ \\alpha \\ast \\beta \\ast \\alpha ^{-1}$ relative $\\dot{I}$ .",
"Clearly, every strong $ SLT $ space is also an $ SLT $ space.",
"For an arbitrary pointed topological sapce $ (X,x_0) $ , Abdullahi et al.",
"in [2] showed that the fundamental group equipped with the whisker topology, $ \\pi _1^{wh}(X,x_0) $ , is a left topological group and the collection $ \\beta = \\lbrace i_*\\pi _1(U,x_0) \\ | \\ U \\ is \\ an \\ open \\ subset \\ of \\ X \\ contaning \\ x_0 \\rbrace $ forms a local basis at the identity element in $ \\pi _1^{wh}(X,x_0) $ .",
"Now the following proposition is obtained using these results.",
"Proposition 2.9 If $ (X,x_0) $ is $ SLT $ at $ x_0 $ , then every closed subset $ A $ of $ \\pi _1^{wh}(X,x_0) $ can be written as $ A = \\bigcap _{U} \\lbrace Ai_*\\pi _1(U,x_0) \\ | \\ U \\ is \\ an \\ open \\ subset \\ of \\ X \\ contaning \\ x_0 \\rbrace $ .",
"It is easily concluded from [11] that $ \\pi _1^{wh}(X,x_0) $ is a topological group, since $ X $ is $ SLT $ at $ x_0 $ .",
"Then the inverse operation is continuous.",
"Now the result comes from [2] and [3] .",
"If $ (X,x_0) $ is a strong $ SLT $ at $ x_0 $ space, then it is easily concluded from the definition that for any open neighbourhood $ U$ of $ x_0 $ in $ X $ , there is an open covering $ \\mathcal {U} $ of $ X $ such that $\\pi (\\mathcal {U}, x_0)\\le i_{*}\\pi _{1}(U,x_0) $ .",
"On the other hand, by [2] if $ H \\le \\pi _1(X,x_0) $ is a generalized covering subgroup, then $ H $ is closed under the whisker topology on the fundamental group.",
"The following theorem states a nice result of this fact.",
"Theorem 2.10 Let $ (X,x_0) $ be a connected locally path connected space which is strong $ SLT $ at $ x_0 $ , and let $ p:\\widetilde{X} \\rightarrow X $ be a map.",
"The pair $ (\\widetilde{X},p) $ is a generalized covering of $ X $ with $ p_*\\pi _1(\\widetilde{X},\\tilde{x}_0)=H \\le \\pi _1(X,x_0)$ if and only if $ H $ is as intersection of some covering subgroups in $ \\pi _1(X,x_0) $ .",
"It is easily concluded from [2] that the intersection of any collection of covering subgroups of the fundamental group is a generalized covering subgroup.",
"Conversely, Since $ X $ is strong $ SLT $ at $ x_0 $ , then for any open neighbourhood $ U$ of $ x_0 $ in $ X $ , there is an open covering $ \\mathcal {U} $ of $ X $ such that $\\pi (\\mathcal {U}, x_0)\\le i_{*}\\pi _{1}(U,x_0) $ .",
"It means that open subgroups of the whisker topology are covering subgroups.",
"Thus for any open neighbourhood $ U $ of $ X $ at $ x_0 $ and any subgroup $ H $ of $ \\pi _1(X,x_0) $ , the subgroup $ Hi_*\\pi _1(U,x_0) $ is a covering subgroup of $ \\pi _1(X,x_0) $ by [13].",
"Finally, if $ H $ is a generalized covering subgroup, then it is closed under the whisker topology on the fundamental group (see [2]) and so $ H = \\overline{H} = \\bigcap _{U} \\lbrace Hi_*\\pi _1(U,x_0) \\rbrace $ .",
"Corollary 2.11 Let $ (X,x_0) $ be a connected locally path connected space which is strong $ SLT $ at $ x_0 $ .",
"A subgroup $ H \\le \\pi _1(X,x_0) $ is a generalized covering subgroup if and only if $ H $ is an intersection of all covering subgroups which contain $ H $ .",
"Assume that $ \\Gamma _H $ is the collection of all covering subgroups of $ \\pi _1(X,x_0) $ containing $ H $ .",
"Clearly, $ \\bigcap _{K \\in \\Gamma _H} K $ is a generalized covering subgroup.",
"Conversely, by [1] and [2] for every $ K \\in \\Gamma _H $ , there is an open neighbourhood $ U_K $ of $ X $ at $ x_0 $ such that $ i_*\\pi _1(U_K,x_0) \\le K $ .",
"Thus $ Hi_*\\pi _1(U_K,x_0) \\le K $ .",
"As mentioned above for every open neighbourhood $ U $ of $ X $ at $ x_0 $ , there is an open covering $ \\mathcal {U} $ of $ X $ such that $\\pi (\\mathcal {U}, x_0)\\le i_{*}\\pi _{1}(U,x_0) \\le Hi_*\\pi _1(U,x_0) $ , which shows that $ Hi_*\\pi _1(U,x_0) $ is a covering subgroup.",
"Therefore, $ Hi_*\\pi _1(U,x_0) \\in \\Gamma _H $ , for every open neighbourhood $ U $ of $ X $ at $ x_0 $ .",
"It implies that $ \\bigcap _{K \\in \\Gamma _H} K = \\bigcap _{K \\in \\Gamma _H} Hi_*\\pi _1(U_{K},x_0)=\\overline{H}=H $ ."
],
[
"Kernel of Generalized Coverings of Topological Group",
"If $ p: \\widetilde{X} \\rightarrow X $ is a generalized covering map of a path connected $ SLT $ space $ X $ such that $ p_*\\pi _1(\\widetilde{X},\\tilde{x}_0)=H \\le \\pi _1(X,x_0) $ , since $\\widetilde{X} $ and $ \\widetilde{X}_H $ are homeomorphic, by Lemma REF , then for every $ x,y \\in X $ the fibres $ p^{-1}(x) $ and $ p^{-1}(y) $ are homeomorphic.",
"Corollary 3.1 If $ (\\widetilde{G},p) $ is a generalized covering group of a path connected topological group $ G $ , then the fibres of all elements of $ G $ are homeomorphic subspaces of $ \\widetilde{ G} $ .",
"By Theorem 2.11 in [15] every path connected topological group $ G $ is strong $ SLT $ and so $ SLT $ space.",
"Therefore, the result comes from the above assertion.",
"Proposition 3.2 Let $ (X,x_0) $ be an arbitrary topological space.",
"If $ (\\widetilde{X},p) $ is a generalized covering space, then the fibre $ p^{-1}(x_0) \\subseteq \\widetilde{X} $ is totally path disconnected.",
"Let $ \\alpha :I \\rightarrow p^{-1}(x_0) $ be a path with $ \\alpha (0)=\\tilde{x}_0 $ , and let $ C_{\\tilde{x}_0} $ be the constant path in $ \\tilde{x}_0 $ .",
"Clearly, $ p \\circ \\alpha = p \\circ C_{\\tilde{x}_0} = C_{x_0} $ and $ \\alpha (0)=C_{\\tilde{x}_0}(0) $ .",
"Since $ p $ has unique path lifting property, hence $ \\alpha =C_{\\tilde{x}_0} $ .",
"Therefore, $ p^{-1}(x_0) $ has no nonconstant path; that is, it is totally path disconnected space.",
"Corollary 3.3 If $ (\\widetilde{G},p) $ is a generalized covering group of connected locally path connected topological group $ G $ , then $ p $ is an open epimorphism with totally path disconnected kernel.",
"It is easy to check that $ p $ is epic and open, since $ G $ is connected and locally path connected, respectively.",
"The result comes from combination of Propositions REF and REF .",
"It seems interesting whether the converse statement of the above corollary can be correct.",
"There are some counterexamples even with extra conditions in very special case.",
"For instance, where $ p: G \\rightarrow G/H $ is an open epimorphism with totally path disconnected kernel and $ H $ is a normal subgroup of a topological group $ G $ .",
"In the next section we show that the necessary and sufficient condition to make $ p: G \\rightarrow G/H $ a generalized covering is the prodiscrete $ H $ .",
"Example 3.4 Let $ H= \\mathbb {S}^1 \\cap (\\mathbb {Q} \\times \\mathbb {Q}) $ be the subgroup of Abelian topological group $ \\mathbb {S}^1 $ (hence $ H \\unlhd \\mathbb {S}^1 $ ), and let $ p: \\mathbb {S}^1 \\rightarrow \\frac{\\mathbb {S}^1}{H} $ be the natural canonical map.",
"Theorem 4.14 from [14] showed that $ p $ is an onto, continuous, and open map.",
"It is easy to show that the kernel of $ p $ is equal to $ H $ .",
"Moreover, it is clear that every path in $ H $ is constant and thus it is a totally path disconnected subspace of $ \\mathbb {S}^1 $ .",
"Note that since $ H $ is a dense subgroup of $ \\mathbb {S}^1 $ , the quotient space $ \\frac{ \\mathbb {S}^1 }{H} $ and its relative fundamental group, $ \\pi _1(\\frac{ \\mathbb {S}^1 }{H},H) $ , are trivial space and group, respectively.",
"Therefore, if $ (\\mathbb {S}^1,p) $ is a generalized covering group of $ \\frac{ \\mathbb {S}^1 }{H} $ , then it is also a covering group, because the image of induced map $ p_* \\pi _1(\\mathbb {S}^1,0) $ is equal to $ \\pi _1(\\frac{ \\mathbb {S}^1 }{H},H) $ .",
"But it is impossible, since $ H $ is not discrete.",
"The concept of prodiscrete space has been defined in different ways in various sources.",
"In this paper, we define the concept of prodiscrete subgroup of a topological group as follows based on the need that we felt.",
"Definition 3.5 Let $ G $ be a topological group.",
"We call $ H \\le G $ a prodiscrete subgroup if there are some discrete groups $ H_i, \\ i \\in I $ , and an isomorphism homeomorphism $ \\psi : \\prod _{i \\in I} H_i \\rightarrow H $ such that $ \\psi (\\prod _{i \\in I, i \\ne j} H_i) \\trianglelefteq G $ for every $ j \\in I $ .",
"Recall from [6] that the pull-back construction helped to show that the intersection of any collection of generalized covering subgroup is also a generalized covering subgroup.",
"Although, there is another simple proof in [2]; using pull-backs of generalized coverings will be useful to study the fibres of generalized covering maps: Let $ p: \\widetilde{X} \\rightarrow X $ be a generalized covering of locally path connected space $ X $ , and let $ f: Y \\rightarrow X $ be a map.",
"The topological pull-back $ \\diamondsuit = \\lbrace (\\tilde{x},y) \\in \\widetilde{X} \\times Y \\ | \\ p(\\tilde{x})=f(y) \\rbrace $ is a subspace of the direct product $ \\widetilde{X} \\times Y $ .",
"Now for the base points $ y_0 \\in Y $ and $ x_0 = f(y_0) \\in X $ , pick $ \\tilde{x}_0 \\in p^{-1}(x_0) $ , and let $ f^\\# \\widetilde{X} $ be the path component of $ \\diamondsuit $ containing $ (\\tilde{x}_0,y_0) $ .",
"The projection $ f^\\#p : f^\\# \\widetilde{X} \\rightarrow Y $ with $ f^\\#p(\\tilde{x},y) = y $ is called the pull-back of $ \\widetilde{X} $ by $ f $ .",
"Brazas showed that for a generalized covering $ p: \\widetilde{X} \\rightarrow X $ and a map $ f: Y \\rightarrow X $ , the pull-back $ f^\\#p : f^\\# \\widetilde{X} \\rightarrow Y $ is also a generalized covering [6].",
"Theorem 3.6 If $ (G,e) $ is a connected locally path connected topological group and $ H $ is a generalized covering subgroup of $ \\pi _1(G,e) $ , then the kernel of the endpoint projection homomorphism $ p_H : (\\widetilde{G}_H, \\tilde{e}_H) \\rightarrow (G,e) $ is a prodiscrete subgroup.",
"Let $ \\lbrace K_{j} \\ | \\ j \\in J \\rbrace $ be the collection of all covering subgroups of $ \\pi _1(G,e) $ , which contain $ H $ .",
"By Corollary REF , $ H= \\bigcap _{j \\in J} K_j $ .",
"For every $ j \\in J $ , put $ p_{j} : (\\widetilde{G}_{j},\\tilde{e}_{j}) \\rightarrow (G,e) $ as the relative covering homomorphism of $ K_{j} $ ; that is, $ K_{j}=(p_{j})_*\\pi _1(\\widetilde{G}_{j},\\tilde{e}_{j}) $ .",
"Take the direct product $ \\widetilde{G}=\\prod _{j \\in J}\\widetilde{G}_{j} $ and $ p:(\\widetilde{G},\\tilde{e}) \\rightarrow (\\prod _{j \\in J}G,{e}) $ is the product homomorphism defined by $ p|_{\\widetilde{G}_{j}} = p_{j} $ , where $ {e}=(e,e,e,\\dots ) $ .",
"By Lemma 2.31 from [6], $ p $ is a generalized covering homomorphism.",
"Now consider the diagonal map $ \\Delta : (G,e) \\rightarrow (\\prod _{j \\in J}G,{e}) $ , and the pull-back of $ \\widetilde{G} $ by $ \\Delta $ is denoted by $ \\Delta ^{\\#}p: \\Delta ^{\\#}\\widetilde{G} \\rightarrow (G,e) $ where $ \\Delta ^{\\#}\\widetilde{G} = \\lbrace (\\tilde{t},g) \\in \\widetilde{G} \\times G \\ | \\ p(\\tilde{t}) = \\Delta (g) \\rbrace $ .",
"By Lemma 2.34 of [6], $ (\\Delta ^{\\#}\\widetilde{G},\\Delta ^{\\#}p) $ is also a generalized covering group of $ G $ .",
"Let $ x_0 = ((t_j),g_0) $ be the base point of $ \\Delta ^{\\#}\\widetilde{G} $ .",
"At first, we show that the image of $ (\\Delta ^{\\#}p)_* $ in $ \\pi _1(G,e) $ is $ H $ .",
"This implies that $ (\\widetilde{G}_H,p_H) $ is a generalized covering group by Lemma REF , since $ (\\Delta ^{\\#}\\widetilde{G},\\Delta ^{\\#}p) $ is a generalized covering group, and therefore $ (\\Delta ^{\\#}\\widetilde{G},\\Delta ^{\\#}p) $ and $ (\\widetilde{G}_H,p_H) $ are equivalent generalized covering groups.",
"After that by calculating the kernel of $ \\Delta ^{\\#}p $ , we find that the kernel of $ p_H $ is prodiscrete.",
"To do this, we claim that for a loop $ \\alpha \\in \\Omega (G,e) $ its unique lift $ \\beta \\in P(\\Delta ^{\\#}\\widetilde{G},x_0) $ (such that $ \\Delta ^{\\#}p \\circ \\beta = \\alpha $ ) is itself a loop if and only if $ [\\alpha ] \\in H $ .",
"Let $ q: \\Delta ^{\\#}\\widetilde{G} \\rightarrow \\widetilde{G} $ be the projection; so that $ p \\circ q = \\Delta \\circ \\Delta ^{\\#}p $ .",
"By the definition it is clear that $ \\beta = (\\gamma ,\\alpha )$ for a path $ \\gamma \\in P(\\widetilde{G},(t_j)) $ .",
"Let $ \\gamma _j \\in P(\\widetilde{G}_j,\\tilde{e}_j) $ be the $ j $ th component of $ \\gamma $ .",
"Now taking the $ j $ th component of the equation $p \\circ \\gamma = p \\circ q \\circ \\beta = \\Delta \\circ \\Delta ^{\\#}p \\circ \\beta = \\Delta \\circ \\alpha ,$ which concludes that $ p_j \\circ \\gamma _j = \\alpha $ for every $ j \\in J $ .",
"Therefore $\\beta \\ is \\ a \\ loop & \\Leftrightarrow \\ \\gamma \\ is \\ a \\ loop, & \\\\ &\\Leftrightarrow \\gamma _j \\ is \\ a \\ loop \\ for \\ every \\ j \\in J, & \\\\ & \\Leftrightarrow [\\alpha ] \\in (p_j)_*(\\pi _1(\\widetilde{G}_j,\\tilde{e}_j) = K_j \\ for \\ every \\ j \\in J, & \\\\ & \\Leftrightarrow [\\alpha ] \\in H.$ Clearly, $ ker( \\Delta ^{\\#}p) = \\lbrace (\\tilde{t},g) \\ | \\ \\Delta ^{\\#}p(\\tilde{t},g) = e \\rbrace $ .",
"This support that $ g=e $ and $ p(\\tilde{t})= \\Delta (e) = {e}=(e,e,\\dots )$ .",
"Hence $ p_j((\\tilde{t})_j) = e $ for every $ j \\in J $ .",
"It implies that $ \\tilde{t} \\in ker(p) = \\prod _{j \\in J} ker(p_j)$ .",
"Then $ ker( \\Delta ^{\\#}p) = \\lbrace (\\tilde{t},e) \\ | \\ \\tilde{t} \\in \\prod _{j \\in J} ker(p_j) \\rbrace $ .",
"Now since for every $ j \\in J $ , $ p_j : \\widetilde{G}_j \\rightarrow G $ is a covering homomorphism; thus its kernel is discrete.",
"Let $ h: \\Delta ^{\\#}\\widetilde{G} \\rightarrow \\widetilde{G}_H $ be the homeomorphism obtained from Lemma REF (see Diagram REF ).",
"It is easy to check that $ h $ is an isomorphism homeomorphism.",
"Therefore $ ker(p_H) $ is a prodiscrete subgroup of $ \\widetilde{G}_H $ , since $ker (\\Delta ^{\\#}p)$ is prodiscrete.",
"Figure: Diagram of G ˜ H \\widetilde{G}_H .Theorem 3.7 If $ G $ is a topological group and $ H $ is a prodiscrete normal subgroup of $ G $ such that $ \\frac{G}{H} $ is a connected locally path connected space, then the natural canonical homomorphism $ \\varphi : G \\rightarrow \\frac{G}{H} $ is a generalized covering homomorphism.",
"It is clear that $ \\frac{G}{H} $ is a topological group and by Theorem 4.14 from [14] $ \\varphi $ is an open epimorphism.",
"Let $ \\psi : \\prod _{j \\in J}H_j \\rightarrow H $ be the isomorphism homeomorphism of Definition REF .",
"For every $ j \\in J $ put $ K_j = \\psi (\\prod _{i \\in J, i \\ne j} H_i) $ .",
"Let $ p_j : \\frac{G}{K_j} \\rightarrow \\frac{G}{H}$ be the natural canonical homomorphism.",
"It is clear from the following diagram that $ p_j $ is continuous and open, because $ \\varphi $ and $ \\varphi ^{\\prime } $ are continuous and open.",
"Figure: NO_CAPTIONMoreover, for every $ j \\in J $ , $ p_j $ is also an epimorphism with the kernel $ \\frac{H}{K_j}$ .",
"Since $ \\psi ( H_j) $ is discrete, then by Remark REF the pair $ (\\frac{G}{K_j},p_j) $ is a covering group of $ \\frac{G}{H} $ for every $ j \\in J $ .",
"Take the direct product $ \\widetilde{G}=\\prod _{j \\in J} \\frac{G}{K_j} $ and $ p:(\\widetilde{G},\\tilde{e}) \\rightarrow (\\prod _{j \\in J}\\frac{G}{H},{e}) $ is the product homomorphism; that is, for every $ j \\in J $ , $ p|_{\\frac{G}{K_j}} = p_{j} $ and $ {e}=(H,H,\\dots ) $ .",
"It implies from Lemma 2.31 of [6] that $ p $ is a generalized covering homomorphism.",
"Now consider the diagonal map $ \\Delta : (\\frac{G}{H},H) \\rightarrow (\\prod _{j \\in J}\\frac{G}{H},{e}) $ , and the pull-back of $ \\widetilde{G} $ by $ \\Delta $ is denoted by $ \\Delta ^{\\#}p: \\Delta ^{\\#}\\widetilde{G} \\rightarrow (\\frac{G}{H},H) $ where $ \\Delta ^{\\#}\\widetilde{G} = \\lbrace ((g_jK_j)_{j \\in J},gH) \\in \\widetilde{G} \\times \\frac{G}{H} \\ | \\ p((g_jK_j)_{j \\in J}) = \\Delta (gH) \\rbrace $ .",
"By Lemma 2.34 of [6], $ (\\Delta ^{\\#}\\widetilde{G},\\Delta ^{\\#}p) $ is also a generalized covering group of $ \\frac{G}{H} $ .",
"To complete the proof, it is enough to show that $ (\\Delta ^{\\#}\\widetilde{G},\\Delta ^{\\#}p) $ , and $ (G,e) $ are equivalent generalized covering groups of $ \\frac{G}{H} $ .",
"Let $ x_0 = ((K_j)_{j \\in J},H) $ be the base point of $ \\Delta ^{\\#}\\widetilde{G} $ , and define homomorphism $ \\theta : (G,e) \\rightarrow (\\Delta ^{\\#}\\widetilde{G},x_0) $ with $ \\theta (g)= ((gK_j)_{j \\in J},gH) $ .",
"Since for every $ g \\in G $ , $ \\Delta ^{\\#}p \\circ \\theta (g) =\\Delta ^{\\#}p((gK_j)_{j \\in J},gH)= gH $ , hence the right triangle of Diagram REF is commutative.",
"Figure: Commutative diagramWe show that $ \\theta $ is an isomorphism homeomorphism.",
"Clearly, $ker(\\theta )=\\lbrace g \\in G \\ | \\ ((gK_j)_{j \\in J},gH) = x_0 = ((K_j)_{j \\in J},H) \\rbrace = \\bigcap _{j \\in J}K_j.$ Since $ \\bigcap _{j \\in J}K_j $ is the trivial subgroup, thus $ \\theta $ is one to one.",
"To show that $ \\theta $ is onto, let $ ((g_jK_j)_{j \\in J},gH) $ be an arbitrary element of $\\Delta ^{\\#}\\widetilde{G} $ , and let $ \\pi _j : H_j \\rightarrow \\prod _{j \\in J} H_j $ be the inclusion map.",
"By the definition of $\\Delta ^{\\#}\\widetilde{G} $ , for every $ j \\in J $ , there exists $ h_j \\in H_j $ such that $ g_j = g \\psi (\\pi _j(h_j)) $ .",
"If $ h = (h_j)_{j \\in J} \\in \\prod _{j \\in J} H_j $ and $ s \\in J $ , then $ h= \\pi _s(h_s).",
"(l_j)_{j \\in J} $ where $ l_j = h_j $ if $ j \\ne s $ and $ l_s=1 $ .",
"Hence, $\\psi (h)= \\psi (\\pi _s(h_s)(l_j)_{j \\in J}) = \\psi (\\pi _s(h_s)) \\psi ((l_j)_{j \\in J}).$ Then $ \\psi (h)K_s = \\psi (\\pi _s(h_s))K_s $ .",
"Therefore, $\\theta (g \\psi (h)) = ((g\\psi (h)K_j)_{j \\in J},ghH) = ((g\\psi (\\pi _j(h_j))K_j)_{j \\in J},gH) = ((g_jK_j)_{j \\in J},gH).$ It is clear from Diagram REF that $ \\theta = \\upsilon \\circ \\sigma $ , where $ \\sigma : G \\rightarrow \\widetilde{G} \\times \\frac{G}{H} $ is the product of natural canonical maps and $ \\upsilon : \\widetilde{G} \\times \\frac{G}{H} \\rightarrow \\Delta ^{\\#}\\widetilde{G} $ is the quotient map.",
"Hence $ \\theta $ is continuous because $ \\sigma $ and $ \\upsilon $ are continuous.",
"The continuity of $ \\theta ^{-1} $ follows from openness of $ \\varphi $ and continuity of $ \\Delta ^{\\#}p $ .",
"Corollary 3.8 Let $ G $ be a topological group, and let $ H \\trianglelefteq G$ be such that the quotient group $ \\frac{G}{H} $ is connected locally path connected.",
"The natural canonical homomorphism $ \\varphi : G \\rightarrow \\frac{G}{H} $ is a generalized covering homomorphism if and only if $ H $ is a prodiscrete subgroup of $ G $ .",
"It is clear that $ \\frac{G}{H} $ is a topological group.",
"If $ \\varphi $ is a generalized covering homomorphism, then by Theorem REF the kernel of $ \\varphi $ , $ H $ , is prodiscrete.",
"The converse statement is obtained from Theorem REF .",
"Corollary 3.9 Let $ G $ be connected locally path connected, and let $ p: \\widetilde{G} \\rightarrow G $ be a homomorphism on topological groups.",
"The pair $ (\\widetilde{G},p) $ is a generalized covering group of $ G $ if and only if $ p $ is an open epimorphism and the kernel of $ p $ is a prodiscrete subgroup of $ \\widetilde{G} $ .",
"Let $ p: \\widetilde{G} \\rightarrow G $ be a generalized covering homomorphism, and let $ H=p_* \\pi _1(\\widetilde{G},\\tilde{e}) $ .",
"By Lemma REF , there is a homeomorphism $ h: \\widetilde{G} \\rightarrow \\widetilde{G}_H$ such that $ p_H \\circ h = p $ .",
"Since $ p_H $ is open onto, then also $ p $ is.",
"Moreover, Theorem REF implies that the kernel of $ p_H $ and so $ p $ is prodiscrete.",
"Conversely, consider $ \\varphi : \\widetilde{G} \\rightarrow \\frac{\\widetilde{G}}{ker(p)}$ as the natural canonical homomorphism.",
"Since $ \\varphi $ is onto, then the homomorphism $ \\theta : \\frac{\\widetilde{G}}{ker(p)} \\rightarrow G $ is an isomorphism homeomorphism, where $ \\frac{\\widetilde{G}}{ker(p)} $ is equipped with the quotient topology.",
"Since $ ker(p) $ is a prodiscrete subgroup of $ \\widetilde{G} $ and $ \\frac{\\widetilde{G}}{ker(p)} $ is connected locally path connected (derived from $ G $ is connected locally path connected), it implies from Corollary REF that $ (\\widetilde{G},\\varphi ) $ is a generalized covering group of $ \\frac{\\widetilde{G}}{ker(p)} $ .",
"Therefore, $ (\\widetilde{G},p=\\varphi \\circ \\theta ) $ is a generalized covering group of $ G $ .",
"By Remark REF and Corollary REF one can easily conclude the following corollary which was promised in Remark REF .",
"Corollary 3.10 Let $ G $ be a connected locally path connected topological group, and let $ p: \\widetilde{G} \\rightarrow G $ be a continuous map.",
"The pair $ (\\widetilde{G},p) $ is a generalized covering group of $ G $ if and only if $ \\widetilde{G} $ is a topological group and $ p $ is an open epimorphism with prodiscrete kernel.",
"Example 3.11 It is easy to show that the kernel of generalized covering maps $ p:\\prod _{i \\in I} \\mathbb {R} \\rightarrow Y $ and $ q: \\prod _{i \\in I}{{\\mathbb {R}} \\times \\mathbb {S}^1} \\rightarrow Y$ in Examples REF and REF , respectively, both are $ \\prod _{i \\in I} \\mathbb {Z} $ which is the product of discrete groups.",
"For another example, let $ r_n :\\mathbb {S}^1 \\rightarrow \\mathbb {S}^1 $ with $ r_n (z)=z^n $ be the well-known covering homomorphism.",
"As mentioned before, the product homomorphism $ r = \\prod _{n \\in \\mathbb {N}} r_n: \\prod _{n \\in \\mathbb {N}} \\mathbb {S}^1 \\rightarrow \\prod _{n \\in \\mathbb {N}} \\mathbb {S}^1 $ is a generalized covering homomorphism and $ ker(r) = \\prod _{n \\in \\mathbb {N}} A_n $ where $ A_n = \\lbrace z \\in \\mathbb {S}^1 \\ | \\ z^n=1 \\rbrace $ , which is also a product of discrete groups.",
"Moreover, the kernel of $ p $ in Example REF is not prodiscrete.",
"In [5], Biss investigated on a kind of fibrations which is called rigid covering fibration with properties similar to covering spaces.",
"Nasri et al.",
"[10] simplified the definition of rigid covering fibration such as a fibration $ p: E \\rightarrow X $ with unique path lifting (unique lifting with respect to paths) property and concluded from [13] that a fibration $ p: E \\rightarrow X $ is a rigid covering fibration if and only if each fibre of $ p $ is totally path disconnected.",
"On the other hand it was shown in [13] that a rigid covering fibration has the unique lifting property and so it is a generalized covering spaces.",
"Although, the converse statement may not hold, in general, we show that it is right in the case of connected locally path connected topological groups.",
"Proposition 3.12 For a connected locally path connected topological group $ G $ , $ (\\widetilde{G},p) $ is a generalized covering group if and only if $ p: \\widetilde{G} \\rightarrow G$ is rigid covering fibration.",
"Let $ p: \\widetilde{G} \\rightarrow G $ be a generalized covering homomorphism, and let $ H= p_* \\pi _1(\\widetilde{G},\\tilde{e}) $ .",
"By Lemma REF , $ (\\widetilde{G},p) $ and $ (\\widetilde{G}_H,p_H) $ are equivalent generalized covering groups of $ G $ , and so the kernel of $ p_H $ is a prodiscrete subgroup, since the kernel of $ p $ is prodiscrete.",
"On the other hand, it was shown in [2] that the kernel of $ p_H $ and the left coset space $ \\frac{\\pi _1^{wh}(G,e)}{H} $ are homeomorphic, which implies that $ \\frac{\\pi _1^{wh}(G,e)}{H} $ is a prodiscrete space.",
"Then $ \\frac{\\pi _1^{wh}(G,e)}{H} $ is totally path disconnected; that is, $ \\frac{\\pi _1^{wh}(G,e)}{H} $ has no nonconstant paths.",
"As mentioned above, $ \\frac{\\pi _1^{qtop}(G,e)}{H} = \\frac{\\pi _1^{wh}(G,e)}{H} $ has no nonconstant paths.",
"Now use [5] which guarantees the existence of a rigid covering fibration $ q: E \\rightarrow G $ with $ p_*\\pi _1(E)=H $ .",
"Since every rigid covering fibration has unique lifting property (see [13]), there is a homeomorphism $ g $ between $ E $ and $ \\widetilde{G}_H $ (Lemma REF ).",
"Therefore, it is done as shown in Diagram REF .",
"Figure: DiagramThe converse statement obtains from [13].",
"The following corollary is the immediate result of the definition of rigid covering fibrations and Proposition REF .",
"Corollary 3.13 If $ G $ is a connected locally path connected topological group and $ (\\widetilde{G},p) $ is a generalized covering group of $ G $ , then $ p: \\widetilde{G} \\rightarrow G$ is a fibration.",
"Reference"
]
] | 1808.08514 |
[
[
"Gravitational backreaction on a cosmic string: Formalism"
],
[
"Abstract We develop a method for computing the linearized gravitational backreaction for Nambu-Goto strings using a fully covariant formalism.",
"We work with equations of motion expressed in terms of a higher dimensional analog of the geodesic equation subject to self-generated forcing terms.",
"The approach allows arbitrary spacetime and worldsheet gauge choices for the background and perturbation.",
"The perturbed spacetime metric may be expressed as an integral over a distributional stress-energy tensor supported on the string worldsheet.",
"By formally integrating out the distribution, this quantity may be re-expressed in terms of an integral over the retarded image of the string.",
"In doing so, one must pay particular attention to contributions that arise from the field point and from non-smooth regions of the string.",
"Then, the gradient of the perturbed metric decomposes into a sum of boundary and bulk terms.",
"The decomposition depends upon the worldsheet coordinates used to describe the string, but the total is independent of those considerations.",
"We illustrate the method with numerical calculations of the self-force at every point on the worldsheet for loops with kinks, cusps and self-intersections using a variety of different coordinate choices.",
"For field points on smooth parts of the worldsheet the self-force is finite.",
"As the field point approaches a kink or cusp the self-force diverges, but is integrable in the sense that the displacement of the worldsheet remains finite.",
"As a consistency check, we verify that the period-averaged flux of energy-momentum at infinity matches the direct work the self-force performs on the string.",
"The methodology can be applied to address many fundamental questions for string loop evolution."
],
[
"Cosmological superstrings",
"Cosmic superstrings are the strings of string theory stretched to macroscopic length scales by the universe's early phase of exponential, inflationary growth , , .",
"During subsequent epochs when the scale factor grows as a more leisurely power law of time a complicated network of various string elements forms , , .",
"Long, horizon-crossing strings stretch, short curved pieces accelerate and attempt to straighten, and, occasionally, individual segments intercommute (collide, break and reattach) chopping out loops and forming new, connected string pathways.",
"Analytic and numerical calculations demonstrate that these processes rapidly drive the network to a self-similar evolution with statistical properties largely determined by the string tension , , , , .",
"The energy densities in long strings, in loops, and in gravitational radiation divided by the critical energy density are all independent of time.",
"The distribution of loops of a given size relative to the horizon scale is also fixed.",
"An understanding of this evolution is informed by previous studies of one dimensional defects in the context of symmetry breaking in grand unified theories (GUTs; ; for a general review see ).",
"One important difference for superstrings is the expected value of the string tension.",
"In GUT theories the string tension $G \\mu /c^2 \\sim \\Lambda _{GUT}^2/M_p^2 \\sim 10^{-6}$ is fixed by the GUT energy scale $\\Lambda _{GUT}$ .",
"Observations of the microwave sky have ruled out GUT strings as the source of the cosmological perturbations , , and led to upper bounds on the tension.",
"Currently, broadly model-independent limits from lensing , , , , , , , , CMB studies , , , , , , , , , , , , and gravitational wave background and bursts , , , , , , , , , , , , , give $G \\mu /c^2 \\mathbin {\\unknown.",
"{<}}$ 10-7$.",
"More stringent but somewhat more model-dependent limits from pulsar timing\\cite {Bouchet:1989ck,Caldwell:1991jj,Kaspi:1994hp,Jenet:2006sv,DePies:2007bm}have regularly appeared.",
"Currently, thestrongest inferred limitis $ G /c2"
]
] | 1808.08631 |
[
[
"A local Bayesian optimizer for atomic structures"
],
[
"Abstract A local optimization method based on Bayesian Gaussian Processes is developed and applied to atomic structures.",
"The method is applied to a variety of systems including molecules, clusters, bulk materials, and molecules at surfaces.",
"The approach is seen to compare favorably to standard optimization algorithms like conjugate gradient or BFGS in all cases.",
"The method relies on prediction of surrogate potential energy surfaces, which are fast to optimize, and which are gradually improved as the calculation proceeds.",
"The method includes a few hyperparameters, the optimization of which may lead to further improvements of the computational speed."
],
[
"Introduction",
"One of the great successes of density functional theory (DFT) [1], [2] is its ability to predict ground state atomic structures.",
"By minimizing the total energy, the atomic positions in solids or molecules at low temperatures can be obtained.",
"However, the optimization of atomic structures with density functional theory or higher level quantum chemistry methods require substantial computer resources.",
"It is therefore important to develop new methods to perform the optimization efficiently.",
"It is of key interest here, that for a given atomic structure a DFT calculation provides not only the total electronic energy, but also, at almost no additional computational cost, the forces on the atoms, i.e.",
"the derivatives of the energy with respect to the atomic coordinates.",
"This means that for a system with $N$ atoms in a particular configuration only a single energy-value is obtained while $3N$ derivatives are also calculated.",
"It is therefore essential to include the gradient information in an efficient optimization.",
"A number of well-known function optimizers exploring gradient information exist [3] and several are implemented in standard libraries like the SciPy library [4] for use in Python.",
"Two much-used examples are the conjugate gradient (CG) method and the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm.",
"Both of these rely on line minimizations and perform particularly well for a nearly harmonic potential energy surface (PES).",
"In the CG method, a series of conjugated search directions are calculated, while the BFGS method gradually builds up information about the Hessian, i.e.",
"the second derivatives of the energy, to find appropriate search directions.",
"The Gaussian process (GP) method that we are going to present has the benefit that it produces smooth surrogate potential energy surfaces (SPES) even in regions of space where the potential is non-harmonic.",
"This leads to a generally improved convergence.",
"The number of algebraic operations that has to be carried out in order to move from one atomic structure to the next is much higher for the GP method than for the CG or BFGS methods, however, this is not of concern for optimizing atomic structures with DFT, because the electronic structure calculations themselves are so time consuming.",
"For more general optimization problems where the function evaluations are fast, the situation may be different.",
"Machine learning for PES modelling has recently attracted the attention of the materials modelling community [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18].",
"In particular, several methods have focused on fitting the energies of electronic structure calculations to expressions of the form $E(\\rho ) = \\sum _{i=1}^n \\alpha _i\\, k\\left(\\rho ^{(i)},\\; \\rho \\right).$ Here, $\\lbrace \\rho ^{(i)}\\rbrace _{i=1}^{n}$ are some descriptors of the $n$ atomic configurations sampled, $k\\left(\\rho ^{(i)},\\; \\rho \\right)$ is known as a kernel function and $\\lbrace \\alpha _i\\rbrace _{i=1}^n$ are the coefficients to be determined in the fit.",
"Since there are $n$ coefficients and $n$ free parameters, the SPES determined by this expression has the values of the calculations at the configurations on the training set.",
"Here we note that expression (REF ) can easily be extended to: $E(\\rho ) = \\sum _{i=1}^n \\alpha _i\\, k\\left(\\rho ^{(i)},\\; \\rho \\right) + \\sum _{i=1}^n \\sum _{j=1}^{3N} \\beta _{ij} \\; \\frac{\\partial k\\left(\\rho ^{(i)},\\; \\rho \\right)}{\\partial r_j^{(i)}} ,$ where $\\lbrace r_j^{(i)}\\rbrace _{j=1}^{3N}$ represent the coordinates of the N atoms in the $i-$ th configuration.",
"The new set of parameters $\\beta _{ij}$ together with $\\alpha _i$ can be adjusted so that not only the right energy of a given configuration $\\rho ^{(i)}$ is predicted, but also the right forces.",
"This approach has two advantages with respect to the previous one: (i) the information included in the model scales with the dimensionality, (ii) the new model is smooth and has the right gradients.",
"In the case of systems with many identical atoms or similar local atomic structures it becomes advantageous to construct SPESs based on descriptors or fingerprints characterizing the local environment [5], [6], [7], [8], [9], [10], [11].",
"The descriptors can then be constructed to obey basic principles as rotational and translational symmetries and invariance under exchange of identical atoms.",
"Here we shall develop an approach based on Gaussian processes which works directly with the atomic coordinates and effectively produces a surrogate PES of the type Eq.",
"(REF ) aimed at relaxing atomic structures.",
"We note, that Gaussian processes with derivatives for PES modeling is a field that is developing fast, with recent applications in local optimization [19] and path determination in elastic band calculations [13], [20], [21]."
],
[
"Gaussian process regression",
"We use Gaussian process regression with derivative information to produce a combined model for the energy $E$ and the forces $\\mathbf {f}$ of a configuration with atomic positions $\\mathbf {x} = ( \\mathbf {r}_1, \\mathbf {r}_2, \\dots , \\mathbf {r}_N)$ : $\\mathbf {U}(\\mathbf {x}) = \\left(E(\\mathbf {x}), -\\mathbf {f}(\\mathbf {x})\\right) \\sim \\mathcal {GP}\\left(\\mathbf {U}_p(\\mathbf {x}), K(\\mathbf {x},\\mathbf {x}^\\prime )\\right),$ where $\\mathbf {U}_p(\\mathbf {x}) = (E_p(\\mathbf {x}), \\nabla E_p(\\mathbf {x}))$ is a vector-valued function which constitutes the prior model for the PES and $K(\\mathbf {x},\\mathbf {x}^\\prime )$ is a matrix-valued kernel function that models the correlation between pairs of energy and force values as a function of the configuration space.",
"In this work, we choose the constant function $\\mathbf {U}_p(\\mathbf {x}) = (E_p, \\mathbf {0})$ as the prior function.",
"For the kernel, we use the squared-exponential covariance function to model the correlation between the energy of different configurations: $k(\\mathbf {x}, \\mathbf {x}^\\prime ) =\\sigma ^2_f e^{-\\Vert \\mathbf {x} -\\mathbf {x}^\\prime \\Vert ^2/2l^2},$ where $l$ is a typical scale of the problem and $\\sigma _f$ is a parameter describing the prior variance at any configuration $\\mathbf {x}$ .",
"The full kernel $K$ can be obtained by noting that [22], [23]: $\\mathrm {cov} \\left(E(\\mathbf {x}), E(\\mathbf {x}^\\prime )\\right) &= k(\\mathbf {x}, \\mathbf {x}^\\prime ) &\\\\\\mathrm {cov} \\left(E(\\mathbf {x}), \\frac{\\partial E(\\mathbf {x}^\\prime )}{\\partial x^\\prime _i} \\right) &= \\frac{\\partial k(\\mathbf {x}, \\mathbf {x}^\\prime )}{\\partial x^\\prime _i} & \\equiv J_i(\\mathbf {x},\\mathbf {x}^\\prime )\\\\\\mathrm {cov} \\left(\\frac{\\partial E(\\mathbf {x})}{\\partial x_i}, \\frac{\\partial E(\\mathbf {x}^\\prime )}{\\partial x^\\prime _j} \\right) &= \\frac{\\partial ^2 k(\\mathbf {x}, \\mathbf {x}^\\prime )}{\\partial x_i\\partial x^\\prime _j} & \\equiv H_{ij}(\\mathbf {x},\\mathbf {x}^\\prime ),$ and assembling these covariance functions in a matrix form: $K(\\mathbf {x}, \\mathbf {x}^\\prime ) = \\left( \\begin{array}{cc}k(\\mathbf {x} , \\mathbf {x}^\\prime ) & \\mathbf {J}(\\mathbf {x} , \\mathbf {x}^\\prime )\\\\\\mathbf {J}(\\mathbf {x}^\\prime , \\mathbf {x})^T & H(\\mathbf {x} , \\mathbf {x}^\\prime )\\end{array} \\right).$ The expressions for the mean and the variance for the posterior distribution follow the usual definitions incorporating the additional matrix structure.",
"Let $X = \\lbrace \\mathbf {x}^{(i)}\\rbrace _{i=1}^{n}$ denote the matrix containing $n$ training inputs and let $Y=\\lbrace \\mathbf {y}^{(i)}\\rbrace _{i=1}^n =\\lbrace \\left(E(\\mathbf {x}^{(i)}),-\\mathbf {f}(\\mathbf {x}^{(i)})\\right)\\rbrace _{i=1}^n $ be the matrix containing the corresponding training targets.",
"By defining $K(\\mathbf {x}, X) = \\left( K(\\mathbf {x}, \\mathbf {x}^{(1)}), K(\\mathbf {x}, \\mathbf {x}^{(2)}), \\dots , K(\\mathbf {x}, \\mathbf {x}^{(n)})\\right)$ and $\\left(K(X,X)\\right)_{ij} = K(\\mathbf {x}^{(i)}, \\mathbf {x}^{(j)}),$ we get the following expressions for the mean: $\\bar{\\mathbf {U}}(\\mathbf {x}) &=(\\bar{E}(\\mathbf {x}), -\\bar{\\mathbf {f}}(\\mathbf {x}))\\nonumber \\\\&= \\mathbf {U}_p(\\mathbf {x}) + K(\\mathbf {x}, X) \\mathbb {K}_X^{-1}(Y - \\mathbf {U}_p(X))$ and the variance: $\\mathbf {\\sigma }^2(\\mathbf {x}) = K(\\mathbf {x}, \\mathbf {x}) - K(\\mathbf {x}, X)\\mathbb {K}_X^{-1}K(X,\\mathbf {x})$ of the prediction, where $\\mathbb {K}_X = K(X,X)+\\Sigma _n^2$ .",
"Here, we have assumed an additive Gaussian noise term with covariance matrix $\\Sigma _n$ [22].",
"This term corrects only for the self covariance of the points in the training set, and thus, it is a diagonal matrix that models the self correlation of forces with a hyperparameter $\\sigma _n^2$ and the self correlation of energies with $\\sigma _n^2\\times l^2$ .",
"We note that even for computational frameworks where the energy and forces can be computed with very limited numerical noise, small non-zero values of $\\sigma _n$ are advantageous since they prevent the inversion of the covariance matrix $K(X,X)$ to be numerically ill-conditioned [13].",
"In the following, we will refer to $\\bar{E}(\\mathbf {x})$ as defined in equation (REF ) as the surrogate potential energy surface (SPES) and distinguish it from the first principles PES, $E(\\mathbf {x})$ ."
],
[
"Gaussian process minimizer: GPMin",
"The GP framework can be used to build an optimization algorithm.",
"In this section, we introduce the main ideas behind the proposed Gaussian process minimizer (denoted GPMin from hereon).",
"A more detailed description of the algorithm can be found in the Appendix in the form of pseudocode.",
"The GP regression provides a SPES that can be minimized using a gradient-based local optimizer.",
"For this purpose, we have used the L-BFGS-B algorithm as implemented in SciPy [24].",
"The prior value for the energy is initially set as the energy of the initial configuration and then the expression (REF ) is used to produce a SPES from that data point alone.",
"This model is then minimized, and the evaluation at the new local minimum generates new data that is then fed into the model to produce a new SPES that will have a different local minimum.",
"Before generating each new SPES the prior value for the energy is updated to the maximum value of the energies previously sampled.",
"This step is important because it makes the algorithm more stable.",
"If a high-energy configuration is sampled, the forces may be very large leading to a too large new step.",
"The increase of the prior value tends to dampen this by effectively reducing the step size.",
"The whole process is then iterated until convergence is reached.",
"It is illustrative to consider in more detail the first step of the algorithm.",
"It is straightforward to show using equations (REF ) - (REF ) that if only a single data point $\\mathbf {x}^{(1)}$ is known the SPES is given by $ \\bar{E}(\\mathbf {x}) = E^{(1)} -\\mathbf {f}^{(1)}\\cdot (\\mathbf {x} - \\mathbf {x}^{(1)}) \\,e^{-\\Vert \\mathbf {x} - \\mathbf {x}^{(1)}\\Vert ^2/2l^2},$ where $E^{(1)}$ and $\\mathbf {f}^{(1)}$ are the energy and forces of the SPES at the point $\\mathbf {x}^{(1)}$ , respectively.",
"We have here used that the prior energy is set to the energy of the first configuration $E^{(1)}$ .",
"One can confirm that this is the prior energy by noting that points far away from $\\mathbf {x}^{(1)}$ , where no information is available, take on this value for the energy.",
"It is seen that the initial force $\\mathbf {f}^{(1)}$ gives rise to a Gaussian depletion of the SPES.",
"The first step of the GPMin algorithm minimizes the SPES leading to a new configuration $ \\mathbf {x} = \\mathbf {x}^{(1)} + l \\frac{\\mathbf {f}^{(1)}}{\\Vert \\mathbf {f}^{(1)}\\Vert }.$ The first step is thus in the direction of the force with a step length of $l$ .",
"Considering the information available this is a very natural choice.",
"GPMin depends on a number of parameters: the length scale $l$ , the prior value of the energy $E_p$ , the energy width $\\sigma _f$ , and the noise or regularization parameter $\\sigma _n$ .",
"It can be seen from expressions (REF ) and (REF ) that the prediction of the SPES depends only on the ratio of $\\sigma _f$ and $\\sigma _n$ and not their individual values.",
"The prior energy $E_p$ is, as explained above, taken initially as the energy of the first configuration and then updated if larger energies are encountered.",
"It is important that the prior value is not too low to avoid large steps, since the prior energy is the value of the SPES for all configurations far away (on the scale of $l$ ) from previously investigated structures.",
"The scale $l$ is very important as it sets the distance over which the SPES relaxes back to the prior value $E_p$ when moving away from the region of already explored configurations.",
"It therefore also effectively determines a step length in the algorithm.",
"One interesting advantage of the Bayesian approach is that it allows for update of parameters (usually termed hyperparameters) based on existing data.",
"We investigate this option by allowing the value of the length scale $l$ to change.",
"Since the update procedure also depends on the width parameter $\\sigma _f$ , we update this as well.",
"The updated hyperparameters, $\\mathbf {\\theta } = (l, \\sigma _f)$ , are determined by maximizing the marginal likelihood: $\\mathbf {\\theta } = \\arg \\max _{\\mathbf {\\vartheta }} P\\left(Y\\vert X,\\mathbf {\\vartheta }\\right).$ The optimization may fail, for example if there is not enough evidence and the marginal likelihood is very flat, and if that happens, the previous scale is kept.",
"The update procedure allows the algorithm to find its own scale as it collects more information, producing a model that self-adapts to the problem at hand.",
"In section we shall consider in more depth the adequate choices for the values of the hyperparameters and the different strategies for the update of hyperparameters when the optimizers are applied to DFT calculations."
],
[
"Computational details",
"We illustrate and test the method on a variety of different systems using two different calculation methods: An interatomic effective medium theory potential (EMT) [25], [26] as implemented in ASE [27], [28] and DFT.",
"The DFT tests have been performed using GPAW [29] with the local density approximation (LDA) exchange-correlation functional and a plane wave basis set with an energy cutoff at 340 eV.",
"The Brillouin zone has been sampled using the Monkhorst-Pack scheme with a k-point density of 2.0/$(\\textup {Å}^{-1})$ in all three directions.",
"The PAW setup with one valence electron has been used for the sodium cluster for simplicity.",
"In addition to the default convergence criteria for GPAW, we specify that the maximum change in magnitude of the difference in force for each atom should be smaller than $10^{-4}\\textrm {eV}\\textup {Å}^{-1}$ for the self-consistent field iteration to terminate.",
"This improves the convergence of the forces.",
"All systems have been relaxed until the maximum force of the atoms was below 0.01 eV$\\textup {Å}^{-1}$ ."
],
[
"Example: Gold clusters described in effective medium theory",
"In the first example GPMin is used to find the structure of 10-atom gold clusters as described by the EMT potential, and the efficiency is compared with other common optimizers.",
"For this purpose, we generate 1000 random configurations of a 10-atom gold cluster.",
"The configurations are constructed by sequentially applying three uniform displacements for each atom in a cubic box with side length 4.8$\\textup {Å}$ and only keeping those that lie further than 1.7 times the atomic radius of gold away from any of the other atoms already present in the cluster.",
"Each configuration is then optimized with different choices of parameters for GPMin, and, for comparison, the same structures are optimized with the ASE implementations of FIRE [30] and BFGS Line Search, and the SciPy implementations of BFGS and the CG.",
"Figure: Statistics of the number of energy evaluationsfor 1000 relaxations of a 10-atom gold cluster.",
"The initial conditions have been randomlygenerated.",
"The left hand side of the plot shows the distribution of the number of energy evaluations for GPMin in its two variants for scales ranging from 0.3 to 0.8 Å: keeping the scale fixed or allowing it to be updated.",
"The right hand side shows the performance of other widely used optimizers, which have been sorted according to the average number of function evaluations.For the gold clusters, we have investigated the effect of updating $\\sigma _f$ and $l$ for six different initial scales between $0.3$ and $0.8\\,\\textup {Å}$ and initial $\\sigma _f=1.0$ eV.",
"Since the EMT potential has very small numerical noise, we choose a small value of $\\sigma _n/\\sigma _f = 5\\times 10^{-4} \\textrm {eV}\\textup {Å}^{-1}$ for the regularization.",
"In the update-version of the optimizer, we update the scale every 5th iteration.",
"The statistics of the number of energy evaluations are shown in Figure REF .",
"The GP optimizers are seen to be the fastest on average, with the appropriate choice of the hyperparameters.",
"For the initial scale of 0.5$\\textup {Å}$ , for example, the updated version of GPMin had relaxed the clusters after $42.1 \\pm 0.3$ energy evaluations and the non-updated one after $42.5 \\pm 0.3$ , as compared to $48.8 \\pm 0.3$ and $56.2 \\pm 0.5$ for the BFGS implementations in SciPy and ASE, respectively.",
"CG exhibits $79.7\\pm 0.7$ average number of steps and FIRE, $122.9\\pm 1.0$ .",
"Figure REF shows the trend in the performance as the scale is varied.",
"For this system, $l=0.5\\textup {Å}$ has the lowest average and variance for GPMin.",
"The performance depends rather sensitively on the scale parameter: reducing the scale results in a more conservative algorithm where more but smaller steps are needed.",
"Increasing the scale leads to a more explorative algorithm with longer steps that may fail to reduce the energy.",
"In the algorithm with updates, the scale is automatically modified to compensate for a non-optimal initial scale.",
"The update is particularly efficient for small scales where the local environment is sufficiently explored.",
"For larger scales the sampling is less informative and it takes longer for the algorithm to reduce the scale.",
"We note that under the appropriate choice of scale, both GPMin with and without update are among the fastest for the best case scenario, with 18 evaluations for the regular GPMin optimizer and 19 for the updated version with scale $l=0.5$ Å, compared to 19 for ASE BFGS, 27 and 34 for the SciPy implementations of BFGS and CG respectively and 70 for FIRE.",
"We further note that the updated version has by far the best worst-case performance.",
"Of the total of 18000 relaxations, only 17 failed to find a local minimum.",
"These 17 relaxations were all run with the GPMin optimizer with $l=0.8\\textup {Å}$ without the updates.",
"An optimizer with a too long scale fails to build a successful SPES: the minimum of the SPES often has a higher energy than the previously evaluated point.",
"Thus, we consider the optimization has failed if after 30 such catastrophic attempts, the optimizer has still not being able to identify a point that reduces the energy or if SciPy's BFGS cannot successfully optimize the predicted SPES."
],
[
"Determination of the hyperparameters",
"We now continue by considering the use of the GP optimizers more generally for systems with PESs described by DFT.",
"Default values of the hyperparameters should be chosen such that the algorithm performs well for a variety of atomic systems.",
"For this purpose, we have chosen a training set consisting of two different structures: (i) a 10-atom sodium cluster with random atomic positions and (ii) a carbon dioxide molecule on a (111) surface with two layers of gold and a $2\\times 2$ unit cell.",
"We have generated 10 slightly different initial configurations for each of the training systems by adding random numbers generated from a Gaussian distribution with standard deviation 0.1 $\\textup {Å}$ .",
"The training configurations are then relaxed using DFT energies and forces.",
"For each pair of the hyperparameters $(l,\\sigma _n/\\sigma _f)$ , we relax the training systems and average over the number of DFT evaluations the optimizer needs to find a local minimum.",
"The results are shown in Figure REF .",
"The plot shows that the metallic cluster benefits from relatively large scales, while the CO on gold system with tight CO bond requires a shorter scale.",
"A too long scale might even imply that the optimizer does not converge.",
"The set of hyperparameters $l=0.4 \\,\\textup {Å}$ , $\\sigma _n = 1\\ \\textrm {meV}\\textup {Å}^{-1}$ and $\\sigma _f=1\\ \\textrm {eV}$ seems to be a good compromise between the two cases and these are the default values we shall use in the following.",
"A similar procedure has been used to determine the default values of the hyperparameters and their initial values in the updated versions of GPMin.",
"Here, the hyperparameter $\\sigma _n/\\sigma _f$ is kept fixed during the optimization, whereas $l$ and $\\sigma _f$ are determined using expression (REF ).",
"The value of $\\sigma _n/\\sigma _f$ and the initial values of the other hyperparameters are then determined from the analysis of the performance of the optimizer on the two systems in the training set.",
"Figure: Evolution of the length scale ll with iteration for the three optimizers with update GPMin-5, GPMin-10%, and GPMin-20%.The upper panel shows the results for the sodium cluster, while the lower panel shows the evolution for the CO/Au system.",
"In allcases three different values l=0.2,0.3,0.4Ål = 0.2, 0.3, 0.4 \\textup {Å} for the initial scale has been considered.",
"For the sodium cluster the length scale is seen to increase significantly, while in the case of the CO/Au system, the length scale first decreases and then subsequently increases.",
"The final length scale various by about 30% dependent on the particular initial structure of the systems.The evolution of the hyperparameters depends on the details of the optimization of the marginal likelihood together with the frequency at which the hyperparameters are optimized.",
"Here, we explore three different strategies: Unconstrained maximization of the marginal log-likelihood every 5 energy evaluations (“GPMin-5\"), and two constrained optimization strategies, where the outcome of the optimization is constrained to vary in the range $\\pm 10\\%$ and $\\pm 20\\%$ of the value of the hyperparameter in the previous step (“GPMin-10%\" and “GPMin-20%\", respectively).",
"In the latter two cases we let the optimization take place whenever new information is added to the sample.",
"The algorithm used to maximize the marginal log-likelihood is L-BFGS-B [24] for all strategies.",
"We have relaxed the same 10 slightly different copies of the two training set systems described before using these three strategies for three different initial values of the scale (0.2, 0.3 and 0.4 $\\textup {Å}$ ), 8 different initial values of $\\sigma _f$ and 7 different values of the regularization parameter $\\sigma _n/\\sigma _f$ .",
"An overview of the full results can be found in the Supplementary Material [31].",
"The average numbers of energy evaluations needed to relax the training set for the different strategies and hyperparameters are shown in Figure REF .",
"The initial value of the scale is chosen as $0.3\\textup {Å}$ .",
"The plot shows the variation of the average number of energy evaluations with $\\sigma _n/\\sigma _f$ when the initial value of $\\sigma _f=1.8 \\textrm {eV}$ and the variation with $\\sigma _f$ when the value of $\\sigma _n/\\sigma _f=2 \\times 10^{-3} \\textup {Å}^{-1}$ .",
"The performance of the optimizers is seen to depend rather weakly on the parameter values in particular for the sodium cluster.",
"We shall therefore in the following use the values $\\sigma _f=1.8 \\textrm {eV}$ and $\\sigma _n/\\sigma _f=2 \\times 10^{-3} \\textup {Å}^{-1}$ .",
"From the figure it can also be seen that the versions of the optimizer with updates perform considerably better than GPMin without updates for the sodium cluster, while for the CO molecule on gold, the version without update works slightly better than the three optimizers with updates.",
"To understand this behavior further we consider in Figure REF the evolution of the length scale $l$ as it is being updated.",
"The scale is initially set at three different values $l = 0.2, 0.3, 0.4 \\textup {Å}$ .",
"For the sodium cluster the update procedure quickly leads to a much longer length scale around $1.5 \\textup {Å}$ .",
"For GPMin-5 the length scale is raised dramatically already at the first update after 5 energy evaluations, while for GPMin-10% and GPMin-20% the length scale increases gradually because of the constraint build into the methods.",
"The advantage of a longer length scale is in agreement with the results above for the gold cluster described with the EMT interatomic interactions, where a long length scale also led to faster convergence.",
"The situation is different for the CO/Au system, where the update leads first to a significant decrease in the scale and later to an increase saturating at a value around $0.3 \\textup {Å}$ .",
"This result was to be expected from the one shown in Figure REF for the performance of GPMin without hyperparameter update.",
"We interpret the variation of the scale for the CO/Au system as being due to the different length scales present in the system, where the CO bond is short and strong while the metallic bonds are much longer.",
"In the first part of the optimization the CO configuration is modified requiring a short scale, while the later stages involve the CO-metal and metal-metal distances.",
"Overall the update of the scale does not provide an advantage over the GPMin without updates where the scale is kept fixed at $l = 0.4 \\textup {Å}$ .",
"It can be seen that the final scales obtained, for example in the case of the sodium cluster optimized with GPMin-10%, varies by about 30%, where the variation depends on the particular system being optimized and not on the initial value for the length scale.",
"In the following we shall use $l = 0.3 \\textup {Å}$ as the initial scale for the optimizers with updates.",
"As shown in Figures S1, S2 and S3 in the supplementary material [31], the results do not depend very much on the initial scale in the range $0.2 - 0.4 \\textup {Å}$ .",
"Furthermore, the results for the EMT gold cluster indicate that long length scales should be avoided: it is easier for the algorithm to increase the length scale than to decrease it.",
"To summarize, we select the following default (initial) values of the hyperparameters for the updated versions of GPMin: $l=0.3\\,\\textup {Å}$ , $\\sigma _f = 2.0$ eV and $\\sigma _n = 0.004 \\,\\textrm {eV}\\textup {Å}^{-1}$ ($\\sigma _n/\\sigma _f = 0.002 \\textup {Å}^{-1}$ ).",
"These values are used in the rest of this paper."
],
[
"Results",
"To test the Bayesian optimizers we have investigated their performance for seven different systems with DFT: a CO molecule on a Ag(111) surface, a C adsorbate on a Cu(100) surface, a distorted Cu(111) surface, bulk copper with random displacements of the atoms with Gaussian distribution and width 0.1 $\\textup {Å}$ , an aluminum cluster with 13 atoms in a configuration close to fcc; the $\\rm {H}_2$ molecule, and the pentane molecule.",
"All surfaces are represented by 2 layer slabs with a $2\\times 2$ unit cell and periodic boundary conditions along the slab.",
"The bulk structure is represented by a $2\\times 2\\times 2$ supercell with periodic boundary conditions along the three unit cell vectors.",
"For each of the systems we have generated ten slightly different initial configurations by rattling the atoms by 0.1 Å.",
"The resulting configurations are then relaxed using the ASE and SciPy optimizers, together with the different GPMin optimizers.",
"It should be noted that in a few cases an optimizer fails to find a local minimum: an atomic configuration is suggested for which GPAW raises an error when it attempts to compute the energy, because two atoms are too close.",
"This happens for SciPy's BFGS for one of the CO/Ag configurations and for SciPy's conjugate gradient method for one of the hydrogen molecule configurations.",
"Figure: Number of DFT evaluations required to optimize a givenstructure.",
"For each structure 10 different initial configurationsare generated and optimized.",
"The vertical line represents theaverage number of steps of GPMin without parameter updates.",
"The error bar represents the error on the average.",
"A different color has been used to highlight the optimizers of the GPMin family.The results are collected in Figure REF .",
"For the sake of clarity, ASE FIRE has been excluded of the plot, since it takes about a factor of three more steps than the fastest optimizer for all systems.",
"The average number of DFT evaluations for the relaxation of the systems in the test set with the implementation of FIRE in ASE is $122 \\pm 4$ for CO/Ag, $91\\pm 5$ for the pentane molecule, $58\\pm 4$ for C/Cu, $85\\pm 3$ for the aluminum cluster, $62\\pm 2$ for the Cu slab, $53\\pm 1$ for Cu bulk and $30\\pm 3$ for H${}_2$ molecule.",
"The GP optimizers are seen to compare favorably or on par with the best one of the other optimizers in all cases.",
"GPMin without update is on average faster than the other optimizers for 6 of the 7 systems.",
"For the bulk Cu system, it is only slightly slower than the ASE-BFGS algorithm.",
"The updated GP optimizers perform even better with one exception: GPMin-5 is clearly worse than the other GP optimizers and ASE-BFGS for the copper bulk system.",
"Since the atomic displacements from the perfect crystal structure are quite small ($\\sim 0.1 \\,\\textup {Å}$ ), this system is probably within the harmonic regime and requires only few ($\\sim 10$ ) iterations to converge.",
"The ASE-BFGS can therefore be expected to perform well, which is also what is observed in Figure REF .",
"GPMin-5 does not update the scale for the first 5 iterations, and when it does so, the new scale does not lead to immediate convergence.",
"The plain GPMin and the two other optimizers with updates perform on par with ASE-BFGS.",
"Generally, the updated optimizers perform better than GPMin without updates, and both GPMin-10% and GPMin-20% with constrained update perform consistently very well.",
"The updated optimizers are clearly better than the plain GPMin for the Al cluster similar to the behavior for the Na cluster used in the determination of hyperparameters.",
"For the other training system, the CO/Au system, GPMin was seen to perform better than all the updated optimizers.",
"However, in Figure REF the scale was chosen to be $l=0.3 \\textup {Å}$ , which is superior for that particular system.",
"This behavior does not appear for any of the test systems including the CO/Ag system, which otherwise could be expected to be somewhat similar."
],
[
"Discussion",
"We ascribe the overall good performance of the GP optimizers to their ability to predict smooth potential energy surfaces covering both harmonic and anharmonic regions of the energy landscape.",
"Since the Gaussian functions applied in the construction of the SPES all have the scale $l$ , the SPES will be harmonic at scales much smaller than this around the minimum configuration.",
"If the initial configuration is in this regime the performance of the optimizer can be expected to be comparable to BFGS, which is optimal for a harmonic PES, and this is what is for example observed for the Cu bulk system.",
"We believe that the relatively worse performance of the SciPy implementation of BFGS can be attributed to an initial guess of the Hessian that is too far from the correct one.",
"Given the performance on both the training and test sets, GPMin-10% seems to be a good choice.",
"It should be noted that updating the hyperparameters require iteration over the the marginal log-likelihood leading to an increased computational cost.",
"However, this is not a problem at least for systems comparable in size to the ones considered here.",
"The current version of the algorithm still has room for improvement.",
"For example, different strategies for the update of hyperparameters may be introduced.",
"Another, maybe even more interesting possibility, is to use more advanced prior models of the PES than just a constant.",
"The prior model to the PES could for example be obtained from fast lower-quality methods.",
"Somewhat along these lines there have been recent attempts to use previously known semi-empirical potentials for preconditioning more traditional gradient-based optimizers [32], [33].",
"This approach might be combined with the GP framework suggested here.",
"We also note that the choice of the Gaussian kernel, even though encouraged by the characteristics of the resulting potential [22] and its previously reported success for similar problems [13], is to some extent arbitrary.",
"It would be worthwhile to test its performance against other kernel functions, for example the Matérn kernel, which has been reported to achieve better performance in different contexts [34], [35], [19].",
"The kernels used in the work here is also limited to considering only one length scale.",
"More flexible kernels allowing for different length scales for different types of bonds would be interesting to explore.",
"The probabilistic aspect, including the uncertainty as expressed in Eq.",
"(REF ), is presently used only in the update of the hyperparameters.",
"It could potentially lead to a further reduction of the number of function evaluations [13].",
"The uncertainty provides a measure of how much a region of configuration space has been explored and can thereby guide the search also in global optimization problems [34], [36], [16].",
"Finally, a note on the limitations of the present version of the optimizer.",
"The construction of the SPES involves the inversion of a matrix (Eq.",
"REF ) which is a square matrix, where the number of columns is equal to $n=N_c*(3*N+1)$ , where $N$ is the number of atoms in the system and $N_c$ the number of previously visited configurations.",
"This is not a problem for moderately sized systems, but for large systems, where the optimization also requires many steps, the matrix inversion can be very computationally time consuming, and the current version of the method will only be efficient if this time is still short compared to the time to perform the DFT calculations.",
"In addition, this can also result in a memory issue for large systems where the relaxation takes many steps.",
"These issues may be addressed by considering only a subset of the data points or other sparsification techniques.",
"Recently, Wang et al.",
"[37] showed that by using the Blackbox Matrix-Matrix multiplication algorithm it is possible to reduce the cost of training from $O(n^3)$ to $O(n^2)$ and then by using distributed memory and 8 GPUs they were able to train a Gaussian process of $n\\sim 4\\times 10^4$ (this would correspond to about 100 steps for 150 atoms with no constraints) in 50 seconds.",
"This time is negligible compared to the time for DFT calculations of systems of this size.",
"The GPMin optimizers are implemented in Python and available in ASE [27].",
"We appreciate fruitful conversations with Peter Bjørn Jørgensen.",
"This work was supported by Grant No.",
"9455 from VILLUM FONDEN."
],
[
"Appendix",
"The optimization algorithm can be represented in pseudocode as follows: Input: Initial structure: $\\mathbf {x}^{(0)} = (\\mathbf {r}_1, \\mathbf {r}_2, \\dots , \\mathbf {r}_N)$ Hyperparameters: $l$ , $\\sigma _n$ , Tolerance: $f_\\text{max}$ $E^{(0)}, \\mathbf {f}^{(0)} \\leftarrow $ Calculator$\\mathbf {x}^{(0)}$ $E_p \\leftarrow E^{(0)}$ $\\max _{i}\\vert \\mathbf {f}_i^{(0)}\\vert > f_\\text{max}$ $X, Y \\leftarrow $ Update$\\mathbf {x}^{(0)}, E^{(0)}, \\mathbf {f}^{(0)}$ $E_p \\leftarrow \\max Y_E$ $\\mathbf {x}^{(1)} \\leftarrow $ l-bfgs-bGP($X,Y$ ), start_from = $\\mathbf {x}^{(0)}$ $E^{(1)}, \\mathbf {f}^{(1)} \\leftarrow $ Calculator$\\mathbf {x}^{(1)}$ $E^{(1)} > E^{(0)}$ $X, Y \\leftarrow $ Update$ \\mathbf {x}^{(1)}, E^{(1)}, \\mathbf {f}^{(1)}$ $E_p \\leftarrow \\max Y_E$ $\\mathbf {x}^{(1)} \\leftarrow $ l-bfgs-bGP($X,Y$ ), start_from = $\\mathbf {x}^{(0)}$ $E^{(1)}, \\mathbf {f}^{(1)} \\leftarrow $ Calculator$\\mathbf {x}^{(1)}$ $\\max _{i}\\vert \\mathbf {f}_i^{(1)}\\vert > f_\\text{max}$ break $\\mathbf {x}^{(0)}, E^{(0)}, \\mathbf {f}^{(0)} \\leftarrow \\mathbf {x}^{(1)}, E^{(1)}, \\mathbf {f}^{(1)} $ Output: $\\mathbf {x}^{(0)}, E^{(0)}$"
]
] | 1808.08588 |
[
[
"Which part of a chain breaks"
],
[
"Abstract This work investigates the dynamics of a one-dimensional homogeneous harmonic chain on a horizontal table.",
"One end is anchored to a wall, the other (free) end is pulled by external force.",
"A Green's function is derived to calculate the response to a generic pulling force.",
"As an example, I assume that the magnitude of the pulling force increases with time at a uniform rate $\\beta$.",
"If the number of beads and springs used to model the chain is large, the extension of each spring takes a simple closed form, which is a piecewise-linear function of time.",
"Under an additional assumption that a spring breaks when its extension exceeds a certain threshold, results show that for large $\\beta$ the spring breaks near the pulling end, whereas the breaking point can be located close to the wall by choosing small $\\beta$.",
"More precisely, the breaking point moves back and forth along the chain as $\\beta$ decreases, which has been called \"anomalous\" breaking in the context of the pull-or-jerk experiment.",
"Although the experiment has been explained in terms of inertia, its meaning can be fully captured by discussing the competition between intrinsic and extrinsic time scales of forced oscillation."
],
[
"Introduction",
"The pull-or-jerk (or “inertia ball”) experiment is commonly used to demonstrate Newton's laws of motion: As depicted in Fig.",
"REF , a ball of mass $m$ is hung from a ceiling by a string, whose tension is denoted as $T_\\text{up}$ .",
"Another string is attached at the bottom of the ball, and its tension is denoted as $T_\\text{down}$ .",
"A downward “jerky” force $F$ is exerted at the end of the lower string, and the question is which string breaks.",
"We know from our experience that the answer depends on how quickly the magnitude of the force changes: If the force suddenly increases, the lower string breaks.",
"If, on the other hand, the force increases slowly, then the upper string breaks.",
"This phenomenon is sometimes explained as due to the inertia of the ball, described as a “tendency to resist changes in motion” (see, e.g., Refs.",
"hewitt2009conceptual and iop).",
"However, this does not clearly answer why the outcome varies with the jerkiness of the driving force, if all we know about mass is that it is a constant of motion independent of any specific dynamic process.",
"Moreover, this experiment has a counter-intuitive aspect, which may be completely baffling unless one makes good sense of inertia: Suppose that the force has constant jerkiness $\\alpha $ so that $F =\\alpha t$ for $t > 0$ .",
"By approximating each string as a spring with force constant $K$ , one can solve the equation of motion to obtain[3], [4] $\\left\\lbrace \\begin{array}{lcl}T_\\text{up} &=& \\alpha t - \\alpha \\omega _0^{-1}\\sin \\omega _0 t\\\\T_\\text{down} &=& \\alpha t,\\end{array} \\right.$ where $\\omega _0 \\equiv \\sqrt{K/m}$ and the gravitational force is neglected.",
"Interested readers are referred to Ref.",
"shima2014analytic for a thorough analysis of this system.",
"A simple assumption on the failure behavior is that the string breaks when $T$ exceeds a certain threshold, say, $T_c$ .",
"According to Eq.",
"(REF ), the lower string will break if $\\alpha > \\alpha _c \\equiv \\omega _0 \\pi ^{-1} T_c$ , which is consistent with our understanding.",
"At the same time, it also predicts the existence of “anomalous” breaking, which means that the force can break the lower string with even smaller jerkiness $\\alpha \\in \\left[ \\frac{1}{3} \\alpha _c, \\frac{1}{2} \\alpha _c \\right]$ .",
"In this sense, which string breaks is not really a matter of pull or jerk.",
"A natural question would be how this analysis generalizes to a chain of many beads and springs,[5] which I wish to address in this work.",
"A harmonic chain is a useful starting point to investigate properties of a macroscopic system near equilibrium.",
"It has been used to understand basic statistical properties of solids such as heat capacity[6] and thermal conductivity[7], [8], [9], [10], [11], [12] and the breaking strength of a polymer chain.",
"[13], [14] A ladder of resistively and capacitively shunted Josephson junctions can also be approximated by a harmonic chain through the mapping to a locally coupled one-dimensional Kuramoto model[15], [16], [17] when phase differences are small.",
"Specifically, I will consider a harmonic chain consisting of identical beads and springs on a horizontal table as depicted in Fig.",
"REF .",
"One end of the chain is fixed to the wall, and the other end is driven by a time-varying external force $F$ .",
"The primary goal of this paper is to give students a precise picture of this general many-body pull-or-jerk experiment.",
"Following Ref.",
"heald1996string, the spring is assumed to have a threshold of deformation above which it ceases to obey Hooke's law.",
"The question is which spring is the earliest that reaches the threshold under a given external force, and this spring will be regarded as a breaking point of the chain.",
"My finding is that the oscillatory motion in Eq.",
"(REF ) manifests itself as a wave traveling across this many-body system, implying that one should consider two competing time scales, one for external driving and the other for internal wave dynamics, to understand the failure behavior.",
"This study can also be thought of as an advanced exercise for physics majors because a harmonic chain is a representative mechanical example that is analytically soluble by means of undergraduate-level mathematics.",
"[18] This work is organized as follows: In Sec.",
", I calculate the Green's function for the model system by solving the full equation of motion.",
"It gives an approximate formula which holds in the continuum limit.",
"The case of a linearly increasing force is then investigated in Sec.",
"under the assumption that friction is negligibly small.",
"The analytic result is compared with numerical integration of the equations of motion.",
"I discuss implications of the observed behavior and conclude this work in Sec. .",
"A sample Python code is provided in the Appendix."
],
[
"Model",
"Consider longitudinal waves on a harmonic chain consisting of beads and springs.",
"Each bead has mass $m$ and every spring has the same spring constant $K$ , and the square root of their ratio is defined as $\\omega _0 \\equiv \\sqrt{K/m}$ .",
"The number of beads is $N$ , and their equilibrium positions in the absence of external force is denoted by $x_j = jl$ , where $l$ means the equilibrium length of the spring ($j=1,\\ldots ,N$ ).",
"The total length of the system in equilibrium therefore equals $x_N = Nl \\equiv L$ .",
"The displacement of the $j$ th bead from its equilibrium position is denoted by $y_j$ .",
"The number of springs is also $N$ , and the extension of the $j$ th spring is $z_j \\equiv y_j-y_{j-1}$ .",
"The $N$ th bead is pulled by a time-dependent external force $F(t)$ , where $t$ denotes time.",
"The equations of motion can thus be written as follows: $\\left\\lbrace \\begin{array}{ll}m \\frac{d^2}{dt^2} {y}_1 =K (- 2y_{1} + y_{2}) - \\Gamma \\frac{d}{dt} {y}_1 &\\\\m \\frac{d^2}{dt^2} {y}_j =K (y_{j-1} - 2y_{j} + y_{j+1}) - \\Gamma \\frac{d}{dt} {y}_j &\\mbox{for~}1 < j < N\\\\m \\frac{d^2}{dt^2} {y}_N =K (y_{N-1} - y_{N}) - \\Gamma \\frac{d}{dt} {y}_N + F(t) &\\end{array}\\right.$ where $\\Gamma $ is a friction coefficient.",
"With the Kronecker delta $\\delta _{k,l}$ and two auxiliary variables $y_0 \\equiv 0$ and $y_{N+1} \\equiv y_N$ , all the above cases can be covered by the following expression: $m \\frac{d^2}{dt^2} {y}_j =K (y_{j-1} - 2y_{j} + y_{j+1}) - \\Gamma \\frac{d}{dt} {y}_j + F(t) \\delta _{j,N},$ where $j = 1, \\ldots , N$ .",
"In this notation, $y_0 = 0$ and $y_{N+1} = y_N$ can be regarded as boundary conditions of Eq.",
"(REF ).",
"In particular, $y_0=0$ has direct physical meaning because the wall can be regarded as a fictitious bead with zero displacement (see Fig.",
"REF ).",
"In addition, the system is initially at rest with zero displacements, i.e., $y_j = 0$ and $d{y}_j/dt = 0$ for every $j$ at $t=0$ .",
"In a dimensionless form, the dynamics is now rewritten as $\\frac{d^2}{d\\tau ^2} \\psi _j = (\\psi _{j-1} - 2\\psi _{j} + \\psi _{j+1}) - 2\\gamma \\frac{d}{d\\tau } \\psi _j + f(\\tau ) \\delta _{j,N},$ where $\\tau \\equiv \\omega _0 t$ , $\\psi _j \\equiv y_j / l$ , $\\gamma \\equiv \\Gamma /(2m\\omega _0)$ , and $f \\equiv F / (K l)$ .",
"It is convenient to choose $f(\\tau ) =u(\\tau )$ and solve Eq.",
"(REF ) for $\\tau > 0$ , where $u$ means the Heaviside step function.",
"The unknown $\\psi _j$ is decomposed into homogeneous and particular parts, denoted by $\\psi _j^{\\text{(h)}}$ and $\\psi _j^{\\text{(p)}}$ , respectively, to have $\\psi _j =\\psi _j^{\\text{(h)}} + \\psi _j^{\\text{(p)}}$ .",
"It is easy to see that $\\psi _j^{\\text{(p)}} = j$ constitutes a particular solution for $j=1,\\ldots ,N$ with $\\psi _{0}^{\\text{(p)}} \\equiv 0$ and $\\psi _{N+1}^{\\text{(p)}} \\equiv \\psi _{N}^{\\text{(p)}}$ .",
"On the other hand, $\\psi _j^{\\text{(h)}}$ satisfies the following homogeneous equation: $\\frac{d^2}{d\\tau ^2} \\psi _j^{\\text{(h)}} = \\psi _{j-1}^{\\text{(h)}}- 2\\psi _{j}^{\\text{(h)}} + \\psi _{j+1}^{\\text{(h)}}- 2\\gamma \\frac{d}{d\\tau } \\psi _j^{\\text{(h)}}$ with $\\psi _0^{\\text{(h)}} \\equiv 0$ and $\\psi _{N+1}^{\\text{(h)}} \\equiv \\psi _{N}^{\\text{(h)}}$ .",
"The initial conditions are $\\psi _j^{\\text{(h)}} =-j$ and $d\\psi _j^{\\text{(h)}}/d\\tau = 0$ for $j=1,\\ldots ,N$ at $\\tau =0$ .",
"To construct a solution, one has to choose $\\psi _j^{\\text{(h)}}\\propto \\sin k j$ , considering $\\psi _0^{\\text{(h)}} = 0$ .",
"The other boundary condition $\\psi _{N+1}^{\\text{(h)}} = \\psi _{N}^{\\text{(h)}}$ is then rewritten as $\\sin k(N+1) = \\sin kN$ , which quantizes the wavenumber as $k_n = (n+\\frac{1}{2})\\pi /(N+\\frac{1}{2})$ with $n=0, 1, \\ldots , N-1$ .",
"Note that the resulting basis functions are orthogonal in the sense that $\\frac{4}{2N+1}\\sum _{j=1}^N \\sin k_n j \\sin k_m j = \\delta _{mn}$ .",
"With these basis functions, the homogeneous solution is represented as $\\psi _j^{\\text{(h)}} = \\sum _{n=0}^{N-1} a_n (\\tau )\\sin k_n j$ .",
"Substituting this into Eq.",
"(REF ), one sees that the coefficients have to satisfy $\\frac{d^2}{d\\tau ^2} a_n + 2\\gamma \\frac{d}{d\\tau } a_n = - \\Omega _n^2 a_n$ with $\\Omega _n \\equiv 2 \\sin (k_n/2)$ , which is just the dispersion relation for phonons.",
"The above differential equation can be solved by $a_n (\\tau ) = A e^{\\mu _n^+ \\tau } + B e^{\\mu _n^- \\tau }$ with $\\mu _n^\\pm \\equiv -\\gamma \\pm \\sqrt{\\gamma ^2 - \\Omega _n^2}$ and arbitrary constants $A$ and $B$ .",
"The constants are determined by applying the orthogonality relation to the initial conditions as follows: $a_n(0) &=& A+B = \\frac{4}{2N+1} \\sum _{j=1}^N (-j) \\sin k_n j =-\\frac{c_n}{2\\Omega _n^2}\\\\\\frac{d a_n}{d\\tau }(0) &=& A\\mu _n^+ + B\\mu _n^- = \\frac{4}{2N+1} \\sum _{j=1}^N 0\\times \\sin k_n j = 0,$ where $c_n \\equiv \\sin [(1+N)k_n] / \\left( N + \\frac{1}{2} \\right)$ .",
"After some algebra to compute $A$ and $B$ from the above set of equations, the solution is obtained as the following shifted discrete Fourier series: $\\psi _j^{\\text{(h)}}(\\tau ) = \\sum _{n=0}^{N-1}\\frac{c_n}{\\Omega _n^2\\sqrt{\\gamma ^2 - \\Omega _n^2}} \\left(\\mu _n^- e^{\\mu _n^+ \\tau } - \\mu _n^+ e^{\\mu _n^- \\tau } \\right) \\sin k_n j.$ Using the connection between the Heaviside step function and the Dirac delta function, i.e., $du/d\\tau = \\delta (\\tau )$ , one readily obtains the Green's function as follows: $G_j(\\tau ) = \\frac{d}{d \\tau } \\psi _j(\\tau )= \\sum _{n=0}^{N-1}\\frac{c_n}{\\sqrt{\\gamma ^2 - \\Omega _n^2}}\\left( e^{\\mu _n^+ \\tau } - e^{\\mu _n^- \\tau } \\right) \\sin k_n j,$ which is the response to $f(\\tau ) = \\delta (\\tau )$ .",
"Given any $f(\\tau )$ , the response can thus be calculated from the following convolution formula: $\\psi _j(\\tau ) = \\int _0^\\tau G_j(\\tau ^{\\prime }) f(\\tau -\\tau ^{\\prime }) d\\tau ^{\\prime }.$ It turns out that the case of $N \\gg 1$ greatly simplifies the analysis, making $k_n \\approx \\kappa _n \\equiv \\left( n + \\frac{1}{2} \\right) \\pi / N$ , $\\Omega _n \\approx \\kappa _n$ , and $\\mu _n^\\pm \\approx \\eta _n^\\pm \\equiv -\\gamma \\pm \\sqrt{\\gamma ^2 - \\kappa _n^2}$ for finite $n$ .",
"If a continuous variable $\\xi \\equiv x/l$ is introduced to replace the integer index $j$ , the Green's function becomes $G(\\xi ,\\tau )\\approx \\sum _{n=0}^{N-1} \\frac{(-1)^n}{N\\sqrt{\\gamma ^2 - \\kappa _n^2}}\\left( e^{\\eta _n^+ \\tau } - e^{\\eta _n^- \\tau } \\right) \\sin \\kappa _n \\xi ,$ with the displacement field, $\\psi (\\xi ,\\tau ) = \\int _0^\\tau G(\\xi ,\\tau ^{\\prime }) f(\\tau -\\tau ^{\\prime }) d\\tau ^{\\prime },$ as depicted in Fig.",
"REF (a).",
"The rescaled extension of the $j$ th spring, $\\phi _j \\equiv \\psi _j -\\psi _{j-1} = z_j/l$ , is approximated by $\\phi (\\xi , \\tau ) \\equiv \\frac{\\partial }{\\partial \\xi } \\psi (\\xi ,\\tau )$ evaluated at $\\xi =j$ .",
"In terms of Eq.",
"(REF ), it is written as $\\phi (\\xi , \\tau ) = \\int _0^\\tau \\frac{\\partial }{\\partial \\xi }G(\\xi , \\tau ^{\\prime })f(\\tau -\\tau ^{\\prime }) d\\tau ^{\\prime }.$ If $\\gamma = 0$ in Eq.",
"(REF ), the kernel function $\\partial G /\\partial \\xi $ takes the following form: $\\frac{\\partial G}{\\partial \\xi } (\\xi ,\\tau )&\\approx & \\sum _{n=0}^{N-1} \\frac{(-1)^n}{N} 2 \\sin \\kappa _n \\tau \\cos \\kappa _n \\xi \\\\&=& \\sum _{n=0}^{N-1} \\frac{(-1)^n}{N} \\left[ \\sin \\kappa _n (\\tau +\\xi ) + \\sin \\kappa _n (\\tau - \\xi ) \\right]$ inside the physical region, i.e., $\\tau > 0$ and $0 < \\xi < N$ .",
"Here, one can verify the following equality: $H_N(r) \\equiv \\sum _{n=0}^{N-1} \\frac{(-1)^n}{N} \\sin \\kappa _n r =\\frac{(-1)^{N-1} \\sin \\pi r}{2N\\cos \\frac{\\pi r}{2N}}$ by calculating geometric series.",
"When the argument $r$ is away from $(2p+1)N$ for any integer $p$ , the magnitude of $H_N(r)$ is small because of $N$ in the denominator.",
"If $r - (2p+1) N = \\epsilon \\ll 1$ , on the other hand, the cosine in the denominator behaves linearly as $\\epsilon $ varies.",
"It implies that $H_N$ is well approximated by the normalized sinc function, ${\\rm sinc}_\\pi (\\epsilon ) \\equiv \\sin \\pi \\epsilon / (\\pi \\epsilon )$ , and the sign depends on $p$ as follows: $H_N \\approx (-1)^p {\\rm sinc}_\\pi (\\epsilon ).$ To sum up, $H_N(r)$ can be regarded as a train of sinc-typed impulses at $r=(2p+1)N$ .",
"The factor of $(-1)^p$ means that two neighboring impulses have different signs, so the period of $H_N$ is $4N$ in total.",
"It would thus be useful to consider a convoluted function $W(r) \\equiv {\\rm sinc}_\\pi (r) * \\Sha _{4N}(r)$ , where $\\Sha _{4N}(r)$ is the Dirac comb with periodicity of $4N$ .",
"The alternating impulse train is then described as $H_N(r) \\approx W(r-N) - W(r-3N)$ to a good approximation.",
"Furthermore, one may simply take $W(r) \\approx \\Sha _{4N}(r)$ because the convoluted sinc function only modifies the peak shape without changing the essential physics.",
"This leads to a particularly handy formula, $H_N(r) \\approx \\Sha _{4N}(r-N) - \\Sha _{4N}(r-3N)$ .",
"Now, the kernel function simplifies to $\\frac{\\partial G}{\\partial \\xi } (\\xi ,\\tau ) &\\approx & H_N(\\tau +\\xi ) +H_N(\\tau -\\xi )\\\\&\\approx & \\Sha _{4N}(\\tau +\\xi -N) - \\Sha _{4N}(\\tau +\\xi -3N)\\nonumber \\\\&+& \\Sha _{4N}(\\tau -\\xi -N) - \\Sha _{4N}(\\tau -\\xi -3N),$ if $\\tau > 0$ and $0 < \\xi < N$ .",
"If the Dirac comb is written as an explicit sum of delta peaks, this can also be expressed as $\\frac{\\partial G}{\\partial \\xi } (\\xi ,\\tau )&\\approx & \\delta (\\tau +\\xi -N) - \\delta (\\tau +\\xi -3N) + \\delta (\\tau -\\xi -N) -\\delta (\\tau -\\xi -3N)\\nonumber \\\\&+& \\delta (\\tau +\\xi -5N) - \\delta (\\tau +\\xi -7N) + \\delta (\\tau -\\xi -5N) -\\delta (\\tau -\\xi -7N) + \\ldots \\\\&=& \\sum _{\\nu =0}^\\infty (-1)^\\nu \\delta [\\tau +\\xi -(2\\nu +1)N]+ \\sum _{\\nu =0}^\\infty (-1)^\\nu \\delta [\\tau -\\xi -(2\\nu +1)N].$ On the $(\\xi ,\\tau )$ plane, it is basically a pulse propagating back and forth between the two ends of the chain [Fig.",
"REF (b)].",
"The pulse undergoes a phase shift of $\\pi $ every time it hits the free end on the right.",
"Note that such soliton-like motion is due to the continuum approximation, in which the wave speed is given independent of the wave number when $\\gamma = 0$ .",
"In a finite-sized system, the pulse will eventually disperse.",
"Plugging Eq.",
"(REF ) into Eq.",
"(REF ), one approximately obtains the extension of the $j$ th spring as follows: $\\phi _j(\\tau ) \\approx \\sum _{\\nu =0}^{\\nu _{\\max }^+} (-1)^\\nu f[\\tau +j-(2\\nu +1)N]+ \\sum _{\\nu =0}^{\\nu _{\\max }^-} (-1)^\\nu f[\\tau -j-(2\\nu +1)N],$ where each of $\\nu _{\\max }^\\pm $ is defined as the greatest integer that makes positive the argument of every function in the summation.",
"An example is the periodic driving force $f(\\tau ) = \\sin (2\\pi \\tau /\\tau _0)$ with $\\tau _0 = 4N$ .",
"As expected, this induces resonant behavior [Fig.",
"REF (c)] because $\\partial G/\\partial \\xi $ has $4N$ -periodicity in time.",
"It is also clear that one can observe constructive or destructive interference at a specific spring by sending pulses with an appropriate time interval [Fig.",
"REF (d)]."
],
[
"Application",
"If $\\gamma =0$ and $f(\\tau ) = \\beta \\tau $ with a constant slope $\\beta $ , Eq.",
"(REF ) yields $\\psi (\\xi ,\\tau ) = \\beta \\tau \\xi -\\frac{16 \\beta N^2}{\\pi ^3} \\sum _{n=0}^\\infty \\frac{(-1)^n}{(2n+1)^3}\\sin \\left[ \\left(n + \\frac{1}{2} \\right) \\frac{\\pi \\xi }{N} \\right]\\sin \\left[ \\left(n + \\frac{1}{2} \\right) \\frac{\\pi \\tau }{N} \\right].$ A direct way to obtain Eq.",
"(REF ) is to note that $\\psi _j = \\beta \\tau j$ forms a particular solution for Eq.",
"(REF ) when $\\gamma = 0$ .",
"For general $\\gamma >0$ , however, one should employ the method of Green's functions.",
"The displacement field is obtained by differentiating Eq.",
"(REF ) with respect to $\\xi $ $\\phi (\\xi ,\\tau ) = \\beta \\tau -\\frac{8 \\beta N}{\\pi ^2} \\sum _{n=0}^\\infty \\frac{(-1)^n}{(2n+1)^2}\\sin \\left[ \\left(n + \\frac{1}{2} \\right) \\frac{\\pi \\tau }{N} \\right]\\cos \\left[ \\left(n + \\frac{1}{2} \\right) \\frac{\\pi \\xi }{N} \\right].$ At $\\xi =N$ , it reduces to $\\phi _N (\\tau ) \\approx \\phi (\\xi =N, \\tau ) = \\beta \\tau ,$ which means that the rightmost spring extends linearly in time.",
"At the other end of the chain, i.e., at $\\xi =0$ , the Fourier series on the right-hand side (RHS) of Eq.",
"(REF ) describes a triangle wave of period $4N$ and amplitude $\\beta N$ , which behaves as $-\\beta \\tau $ between $\\tau =0$ and $N$ .",
"Combining this result with the first term on the RHS of Eq.",
"(REF ), one can see that the extension of the leftmost spring fastened to the wall is described by the following piecewise linear function: $\\phi _1(\\tau ) \\approx \\left\\lbrace \\begin{array}{cl}0 & \\text{~~~if~~} 0 \\le \\tau < N\\\\2 \\beta \\left[ \\tau - (2\\nu +1) N \\right] & \\text{~~~if~~}(4\\nu +1)N \\le \\tau < (4\\nu +3)N\\\\4 \\beta (\\nu +1) N & \\text{~~~if~~} (4\\nu +3)N \\le \\tau < (4\\nu +5)N,\\end{array}\\right.$ where $\\nu = 0,1,2,\\ldots $ .",
"It is plausible that the spring remains at rest when $0 \\le \\tau < N$ because it takes time for the external perturbation to be transferred through $N$ intermediate springs.",
"However, the subsequent motion is not so self-evident: The spring suddenly begins to expand twice as fast as the rightmost one until the expansion stops abruptly at $\\tau =2N$ , and this pattern continues periodically.",
"Application of Eq.",
"(REF ) actually shows that every spring has such discontinuity in the time derivative of $\\phi _j$ except for $j=N$ : From the shape of $\\partial G/\\partial \\xi $ in Fig.",
"REF (b), it is easily seen that $\\phi _j(\\tau ) \\approx \\left\\lbrace \\begin{array}{cl}0 & \\text{~~~if~~} 0 \\le \\tau < N-j\\\\\\beta (\\tau +j-N) & \\text{~~~if~~} (4\\nu +1) N-j \\le \\tau < (4\\nu +1) N+j\\\\2\\beta [\\tau -(2\\nu +1) N] & \\text{~~~if~~} (4\\nu +1) N+j \\le \\tau < (4\\nu +3) N-j\\\\\\beta (\\tau -j+N) & \\text{~~~if~~} (4\\nu +3) N-j \\le \\tau < (4\\nu +3) N+j\\\\4 \\beta (\\nu +1) N & \\text{~~~if~~} (4\\nu +3) N+j \\le \\tau < (4\\nu +5) N-j,\\end{array}\\right.$ with $\\nu = 0,1,2,\\ldots $ [see, e.g., $\\phi _{N/2}$ in Fig.",
"REF (a)].",
"Note that Eq.",
"(REF ) is directly proportional to $\\beta $ .",
"It is because Eq.",
"(REF ) is linear and thus invariant under rescaling every $\\psi _j$ and $\\beta $ by a common factor $\\lambda > 0$ $\\lambda \\frac{d^2}{d\\tau ^2} \\psi _j = \\lambda (\\psi _{j-1} - 2\\psi _{j} +\\psi _{j+1}) -2\\gamma \\lambda \\frac{d}{d\\tau } \\psi _j+ \\lambda \\beta \\tau \\delta _{j,N}.$ In words, the sole effect of choosing a different value for $\\beta $ is to change the overall length scale.",
"No matter how slowly the end of the harmonic chain is pulled, the periodic discontinuity will not disappear.",
"Note also that Eqs.",
"(REF ) and (REF ) provide envelopes for every $\\phi _j$ in between [Eq.",
"(REF )], although the lines can sometimes coincide.",
"As demonstrated in Fig.",
"REF (a), the analytic predictions of Eq.",
"(REF ) are well substantiated by direct numerical integration of Eq.",
"(REF ).",
"One may also check the mechanical energy per particle $\\varepsilon = \\frac{1}{N} \\sum _{j=1}^N \\frac{1}{2} \\left( \\frac{1}{N}\\frac{d\\psi _j}{d\\tau } \\right)^2 + \\frac{1}{N} \\sum _{j=1}^N \\frac{1}{2} \\left(\\frac{\\psi _j - \\psi _{j-1}}{N} \\right)^2,$ where the first and second terms represent the kinetic and potential parts, respectively [Fig.",
"REF (b)].",
"The factor of $1/N$ inside each pair of parentheses is due to the fact that $\\psi _j \\sim O(N)$ [see Fig.",
"REF (a)].",
"The kinetic part of Eq.",
"(REF ) turns out to be a periodic function of $\\tau $ with a period of $4N$ .",
"The potential energy, on the other hand, keeps increasing as a quadratic function of $\\tau $ .",
"Recall the assumption that every spring obeys Hooke's law up to some threshold $\\phi _c >0$ , above which the spring breaks.",
"[3], [4], [5] The restoring force of the $j$ th spring is thus written as $f^{\\text{res}}_j = \\left\\lbrace \\begin{array}{cl}-K \\phi _j & \\text{if~~} |\\phi _j| < \\phi _c\\\\0 & \\text{otherwise (i.e., broken)}.\\end{array} \\right.$ Then, the above calculation implies that the magnitude of $\\beta $ is an important factor to determine which spring breaks.",
"It is related to the fact that a different value of $\\beta $ just rescales every $\\phi _j$ with exactly the same factor.",
"If $\\lambda = \\beta ^{-1}$ in Eq.",
"(REF ) and $\\zeta _j \\equiv \\beta ^{-1} \\phi _j$ , it is a harmonic chain defined by $\\frac{d^2}{d\\tau ^2} \\zeta _j = (\\zeta _{j-1} - 2\\zeta _{j} +\\zeta _{j+1}) -2\\gamma \\frac{d}{d\\tau } \\zeta _j + \\tau \\delta _{j,N},$ in which the threshold of a spring becomes $\\zeta _c \\equiv \\phi _c/\\beta $ .",
"Large jerkiness therefore maps to a low threshold in this derived system.",
"As illustrated in Fig.",
"REF (c), the $N$ th spring will break when $\\beta $ is large because it is the earliest one that extends to $\\zeta _c$ .",
"Conversely, small $\\beta $ can break a spring close to the wall.",
"Precisely speaking, one can only specify the range of springs to break in the latter case.",
"According to Eq.",
"(REF ), all the springs between $\\xi =0$ and $\\xi ^\\ast (\\le N)$ are the most extended ones in this chain when $\\tau = 3N-\\xi ^\\ast $ (mod $4N$ ).",
"In theory, therefore, it is possible to break every spring all at once by choosing a suitable value of $\\beta $ so that every $\\phi _j$ reaches the threshold $\\phi _c$ at the same time, which may happen at $\\tau = 2N$ .",
"In practice, however, this would mean that the breaking point becomes very sensitive to experimental noise and mechanical defects.",
"Another point of Fig.",
"REF (c) is that one can break the $N$ th spring by pulling the end even more slowly, which proves the existence of “anomalous” breaking in this system.",
"[3], [4], [5] If this anomaly is hardly observed, the reason could be that friction is not negligible in any experimental situation.",
"[3] If $\\gamma $ is positive yet so small that only the second summation contributes in Eq.",
"(REF ), the triangle wave in $\\phi _1$ will gradually decay as indicated by $e^{-\\gamma \\tau }$ in the summand.",
"On the other hand, it is reasonable to guess that $\\phi _N$ will not experience any notable change, considering that $\\partial G / \\partial \\xi $ containing the friction term identically vanishes at $\\xi = N$ due to the boundary condition.",
"Consequently, $\\phi _1$ is expected to lie below $\\phi _N$ in the long run [Fig.",
"REF (d)].",
"It implies that the anomaly can indeed be diminished by friction, but the price is that it also becomes hard to locate the breaking point close to the wall.",
"For sufficiently large $\\gamma $ , the one that breaks will always be the $N$ th spring where the force is acting."
],
[
"Discussion and Conclusion",
"To summarize, I have investigated the dynamics of a harmonic chain which is anchored to a wall at one end and subject to external force at the other end.",
"By using the method of Green's functions, one can calculate the response of the system to a general time-varying force.",
"A simple expression is obtained when the system becomes a continuous medium composed of a large number of beads and springs [Eq.",
"(REF )].",
"With the simple failure behavior assumed in Eq.",
"(REF ), anomalous breaking[3], [4], [5] is still a theoretical possibility in this many-body system when driven by a ramp force $F \\propto t$ .",
"A nontrivial difference from the common pull-or-jerk experiment is that every spring except the last one exhibits distinct stop-and-go behavior in its extension $\\phi _j$ [compare Eq.",
"(REF ) and Eq.",
"(REF )].",
"It implies that it roughly takes $\\Delta t \\sim O(\\omega _0^{-1})$ for the external perturbation to travel across a spring, and this is a fast process compared to system-wide dynamics when $N$ is large.",
"When one talks about the pull-or-jerk experiment in the context of Newton's law of inertia, the precise meaning is that $\\Delta t \\propto m^{1/2}$ .",
"If time is not enough to send an amount of energy across the chain, therefore, it will be the rightmost spring that breaks.",
"In other words, there are two competing time scales: One is the intrinsic time scale of the chain, and the other is that of the driving force.",
"The point is that the experiment should be understood in terms of these time scales of forced oscillation, in addition to the law of inertia.",
"From a technical point of view, the chain is described by a set of coupled, linear, ordinary differential equations.",
"Although it looks much more difficult than the one-body counterpart as in Fig.",
"REF , the problem can readily be handled by standard techniques such as separation of variables and Green's functions.",
"[18] It is instructive to check the validity of the analytic solution by performing numerical simulations, e.g., with the RK4 method as we have done throughout this work.",
"Figure REF (a) has already shown consistency between the analytic and numerical approaches, but the agreement is actually striking in every detail, as demonstrated in Fig.",
"REF .",
"For reference, a sample Python code is provided in the Appendix.",
"[19] Readers are also encouraged to extend the model to inhomogeneous or anharmonic cases to incorporate more realistic aspects of the chain dynamics.",
"[20] S.K.B.",
"gratefully acknowledges discussions with Julian Lee, Hang-Hyun Jo, and Hiroyuki Shima.",
"This work was supported by a research grant of Pukyong National University (2016)."
],
[
"Sample code",
"Here, I present a Python code to simulate the dynamics of $N=30$ beads with the RK4 method.",
"[19] Note that it includes $y_0 = 0$ explicitly because the index of an array begins from zero by default.",
"The array r contains both the displacement and velocity of every bead in such a way that its elements r[2*j] and r[2*j+1] correspond to $y_j$ and $\\frac{d}{dt} y_j$ , respectively.",
"from __future__ import print_function,division # for Python 2 from numpy import empty,array,zeros def drive(t): return beta*t def f(r,t): rdot = empty(2*N1, float) for i in range(0, 2*N1, 2): # time derivative of displacement rdot[i] = r[i+1] for i in range(1, 2*N1, 2): # time derivative of velocity if i==1: accel = 0. elif i>1 and i<2*N1-1: accel = r[i-3] - 2*r[i-1] + r[i+1] - gamma*r[i] elif i==2*N1-1: accel = r[i-3] - r[i-1] - gamma*r[i] + drive(t) rdot[i] = accel return rdot start, end = 0., 150.",
"# time domain max_step = 15000 # number of time steps h = (end - start) / max_step # time increment N1 = 31 # number of beads including the zeroth one (j=0) beta = 1.",
"# increasing rate of the driving force gamma = 0.",
"# friction coefficient r = zeros(2*N1, float) for step in range(max_step): t = h*step k1 = h*f(r,t) k2 = h*f(r+0.5*k1,t+0.5*h) k3 = h*f(r+0.5*k2,t+0.5*h) k4 = h*f(r+0.5*k3,t+h) r += (k1+2*k2+2*k3+k4)/6"
]
] | 1808.08668 |
[
[
"BOP: Benchmark for 6D Object Pose Estimation"
],
[
"Abstract We propose a benchmark for 6D pose estimation of a rigid object from a single RGB-D input image.",
"The training data consists of a texture-mapped 3D object model or images of the object in known 6D poses.",
"The benchmark comprises of: i) eight datasets in a unified format that cover different practical scenarios, including two new datasets focusing on varying lighting conditions, ii) an evaluation methodology with a pose-error function that deals with pose ambiguities, iii) a comprehensive evaluation of 15 diverse recent methods that captures the status quo of the field, and iv) an online evaluation system that is open for continuous submission of new results.",
"The evaluation shows that methods based on point-pair features currently perform best, outperforming template matching methods, learning-based methods and methods based on 3D local features.",
"The project website is available at bop.felk.cvut.cz."
],
[
"Introduction",
"Estimating the 6D pose, i.e.",
"3D translation and 3D rotation, of a rigid object has become an accessible task with the introduction of consumer-grade RGB-D sensors.",
"An accurate, fast and robust method that solves this task will have a big impact in application fields such as robotics or augmented reality.",
"Figure: A collection of benchmark datasets.",
"Top: Example test RGB-D images where the second row shows the images overlaid with 3D object models in the ground-truth 6D poses.",
"Bottom: Texture-mapped 3D object models.",
"At training time, a method is given an object model or a set of training images with ground-truth object poses.",
"At test time, the method is provided with one test image and an identifier of the target object.",
"The task is to estimate the 6D pose of an instance of this object.Many methods for 6D object pose estimation have been published recently, e.g.",
"[34], [24], [18], [2], [36], [21], [27], [25], but it is unclear which methods perform well and in which scenarios.",
"The most commonly used dataset for evaluation was created by Hinterstoisser et al.",
"[14], which was not intended as a general benchmark and has several limitations: the lighting conditions are constant and the objects are easy to distinguish, unoccluded and located around the image center.",
"Since then, some of the limitations have been addressed.",
"Brachmann et al.",
"[1] added ground-truth annotation for occluded objects in the dataset of [14].",
"Hodaň et al.",
"[16] created a dataset that features industry-relevant objects with symmetries and similarities, and Drost et al.",
"[8] introduced a dataset containing objects with reflective surfaces.",
"However, the datasets have different formats and no standard evaluation methodology has emerged.",
"New methods are usually compared with only a few competitors on a small subset of datasets.",
"This work makes the following contributions: Eight datasets in a unified format, including two new datasets focusing on varying lighting conditions, are made available (Fig.",
"REF ).",
"The datasets contain: i) texture-mapped 3D models of 89 objects with a wide range of sizes, shapes and reflectance properties, ii) 277K training RGB-D images showing isolated objects from different viewpoints, and iii) 62K test RGB-D images of scenes with graded complexity.",
"High-quality ground-truth 6D poses of the modeled objects are provided for all images.",
"An evaluation methodology based on [17] that includes the formulation of an industry-relevant task, and a pose-error function which deals well with pose ambiguity of symmetric or partially occluded objects, in contrast to the commonly used function by Hinterstoisser et al. [14].",
"A comprehensive evaluation of 15 methods on the benchmark datasets using the proposed evaluation methodology.",
"We provide an analysis of the results, report the state of the art, and identify open problems.",
"An online evaluation system at bop.felk.cvut.cz that allows for continuous submission of new results and provides up-to-date leaderboards.",
"The progress of research in computer vision has been strongly influenced by challenges and benchmarks, which enable to evaluate and compare methods and better understand their limitations.",
"The Middlebury benchmark [31], [32] for depth from stereo and optical flow estimation was one of the first that gained large attention.",
"The PASCAL VOC challenge [10], based on a photo collection from the internet, was the first to standardize the evaluation of object detection and image classification.",
"It was followed by the ImageNet challenge [29], which has been running for eight years, starting in 2010, and has pushed image classification methods to new levels of accuracy.",
"The key was a large-scale dataset that enabled training of deep neural networks, which then quickly became a game-changer for many other tasks [23].",
"With increasing maturity of computer vision methods, recent benchmarks moved to real-world scenarios.",
"A great example is the KITTI benchmark [11] focusing on problems related to autonomous driving.",
"It showed that methods ranking high on established benchmarks, such as the Middlebury, perform below average when moved outside the laboratory conditions.",
"Unlike the PASCAL VOC and ImageNet challenges, the task considered in this work requires a specific set of calibrated modalities that cannot be easily acquired from the internet.",
"In contrast to KITTY, it was not necessary to record large amounts of new data.",
"By combining existing datasets, we have covered many practical scenarios.",
"Additionally, we created two datasets with varying lighting conditions, which is an aspect not covered by the existing datasets."
],
[
"Evaluation Methodology",
"The proposed evaluation methodology formulates the 6D object pose estimation task and defines a pose-error function which is compared with the commonly used function by Hinterstoisser et al.",
"[13]."
],
[
"Formulation of the Task",
"Methods for 6D object pose estimation report their predictions on the basis of two sources of information.",
"Firstly, at training time, a method is given a training set $T = \\lbrace T_o\\rbrace _{o=1}^n$ , where $o$ is an object identifier.",
"Training data $T_o$ may have different forms, e.g.",
"a 3D mesh model of the object or a set of RGB-D images showing object instances in known 6D poses.",
"Secondly, at test time, the method is provided with a test target defined by a pair $(I, o)$ , where $I$ is an image showing at least one instance of object $o$ .",
"The goal is to estimate the 6D pose of one of the instances of object $o$ visible in image $I$ .",
"If multiple instances of the same object model are present, then the pose of an arbitrary instance may be reported.",
"If multiple object models are shown in a test image, and annotated with their ground truth poses, then each object model may define a different test target.",
"For example, if a test image shows three object models, each in two instances, then we define three test targets.",
"For each test target, the pose of one of the two object instances has to be estimated.",
"This task reflects the industry-relevant bin-picking scenario where a robot needs to grasp a single arbitrary instance of the required object, e.g.",
"a component such as a bolt or nut, and perform some operation with it.",
"It is the simplest variant of the 6D localization task [17] and a common denominator of its other variants, which deal with a single instance of multiple objects, multiple instances of a single object, or multiple instances of multiple objects.",
"It is also the core of the 6D detection task, where no prior information about the object presence in the test image is provided [17]."
],
[
"Measuring Error",
"A 3D object model is defined as a set of vertices in $\\mathbb {R}^3$ and a set of polygons that describe the object surface.",
"The object pose is represented by a $4\\times 4$ matrix $\\mathbf {P} = [\\mathbf {R} , \\mathbf {t}; \\mathbf {0}, 1]$ , where $\\mathbf {R}$ is a $3\\times 3$ rotation matrix and $\\mathbf {t}$ is a $3\\times 1$ translation vector.",
"The matrix $\\mathbf {P}$ transforms a 3D homogeneous point $\\mathbf {x}_m$ in the model coordinate system to a 3D point $\\mathbf {x}_c$ in the camera coordinate system: $\\mathbf {x}_c = \\mathbf {P}\\mathbf {x}_m$ ."
],
[
"Visible Surface Discrepancy.",
"To calculate the error of an estimated pose $\\hat{\\mathbf {P}}$ w.r.t.",
"the ground-truth pose $\\bar{\\mathbf {P}}$ in a test image $I$ , an object model $\\mathcal {M}$ is first rendered in the two poses.",
"The result of the rendering is two distance mapsA distance map stores at a pixel $p$ the distance from the camera center to a 3D point $\\mathbf {x}_p$ that projects to $p$ .",
"It can be readily computed from the depth map which stores at $p$ the $Z$ coordinate of $\\mathbf {x}_p$ and which can be obtained by a Kinect-like sensor.",
"$\\hat{S}$ and $\\bar{S}$ .",
"As in [17], the distance maps are compared with the distance map $S_I$ of the test image $I$ to obtain the visibility masks $\\hat{V}$ and $\\bar{V}$ , i.e.",
"the sets of pixels where the model $\\mathcal {M}$ is visible in the image $I$ (Fig.",
"REF ).",
"Given a misalignment tolerance $\\tau $ , the error is calculated as: $ e_\\mathrm {VSD}(\\hat{S}, \\bar{S}, S_I, \\hat{V}, \\bar{V}, \\tau ) =\\underset{p \\in \\hat{V} \\cup \\bar{V}}{\\mathrm {avg}}{\\left\\lbrace \\begin{array}{ll}0 & \\text{if $p \\in \\hat{V} \\cap \\bar{V} \\, \\wedge \\, |\\hat{S}(p) - \\bar{S}(p)| < \\tau $} \\\\1 & \\text{otherwise}.\\end{array}\\right.",
"}$ Figure: Comparison of e VSD e_\\mathrm {VSD} (bold, τ=20mm\\tau =20\\,{mm}) with e ADI /θ AD e_\\mathrm {ADI}/\\theta _\\mathrm {AD} (mm) on example pose estimates sorted by increasing e VSD e_\\mathrm {VSD}.Top: Cropped and brightened test images overlaid withrenderings of the model at i) the estimated pose 𝐏 ^\\hat{\\mathbf {P}} in blue, and ii) the ground-truth pose 𝐏 ¯\\bar{\\mathbf {P}} in green.",
"Only the part of the model surface that falls into the respective visibility mask is shown.",
"Bottom: Difference maps S Δ S_{\\Delta }.Case (b) is analyzed in Fig.",
"."
],
[
"Properties of $e_\\mathrm {VSD}$ .",
"The object pose can be ambiguous, i.e.",
"there can be multiple poses that are indistinguishable.",
"This is caused by the existence of multiple fits of the visible part of the object surface to the entire object surface.",
"The visible part is determined by self-occlusion and occlusion by other objects and the multiple surface fits are induced by global or partial object symmetries.",
"Pose error $e_\\mathrm {VSD}$ is calculated only over the visible part of the model surface and thus the indistinguishable poses are treated as equivalent.",
"This is a desirable property which is not provided by pose-error functions commonly used in the literature [17], including $e_\\mathrm {ADD}$ and $e_\\mathrm {ADI}$ discussed below.",
"As the commonly used pose-error functions, $e_\\mathrm {VSD}$ does not consider color information.",
"Definition (REF ) is different from the original definition in [17] where the pixel-wise cost linearly increases to 1 as $|\\hat{S}(p) - \\bar{S}(p)|$ increases to $\\tau $ .",
"The new definition is easier to interpret and does not penalize small distance differences that may be caused by imprecisions of the depth sensor or of the ground-truth pose."
],
[
"Criterion of Correctness.",
"An estimated pose $\\hat{\\mathbf {P}}$ is considered correct w.r.t.",
"the ground-truth pose $\\bar{\\mathbf {P}}$ if the error $e_\\mathrm {VSD} < \\theta $ .",
"If multiple instances of the target object are visible in the test image, the estimated pose is compared to the ground-truth instance that minimizes the error.",
"The choice of the misalignment tolerance $\\tau $ and the correctness threshold $\\theta $ depends on the target application.",
"For robotic manipulation, where a robotic arm operates in 3D space, both $\\tau $ and $\\theta $ need to be low, e.g.",
"$\\tau =20\\,{mm}$ , $\\theta =0.3$ , which is the default setting in the evaluation presented in Sec. .",
"The requirement is different for augmented reality applications.",
"Here the surface alignment in the $Z$ dimension, i.e.",
"the optical axis of the camera, is less important than the alignment in the $X$ and $Y$ dimension.",
"The tolerance $\\tau $ can be therefore relaxed, but $\\theta $ needs to stay low."
],
[
"Comparison to Hinterstoisser et al.",
"In [14], the error is calculated as the average distance from vertices of the model $\\mathcal {M}$ in the ground-truth pose $\\bar{\\mathbf {P}}$ to vertices of $\\mathcal {M}$ in the estimated pose $\\hat{\\mathbf {P}}$ .",
"The distance is measured to the position of the same vertex if the object has no indistinguishable views ($e_\\mathrm {ADD}$ ), otherwise to the position of the closest vertex ($e_\\mathrm {ADI}$ ).",
"The estimated pose $\\hat{\\mathbf {P}}$ is considered correct if $e \\le \\theta _\\mathrm {AD} = 0.1 d$ , where $e$ is $e_\\mathrm {ADD}$ or $e_\\mathrm {ADI}$ , and $d$ is the object diameter, i.e.",
"the largest distance between any pair of model vertices.",
"Error $e_\\mathrm {ADI}$ can be un-intuitively low because of many-to-one vertex matching established by the search for the closest vertex.",
"This is shown in Fig.",
"REF , which compares $e_\\mathrm {VSD}$ and $e_\\mathrm {ADI}$ on example pose estimates of objects that have indistinguishable views.",
"Overall, (f)-(n) yield low $e_\\mathrm {ADI}$ scores and satisfy the correctness criterion of Hinterstoisser et al.",
"These estimates are not considered correct by our criterion.",
"Estimates (a)-(e) are considered correct and (o)-(p) are considered wrong by both criteria."
],
[
"Datasets",
"We collected six publicly available datasets, some of which we reduced to remove redundanciesIdentifiers of the selected images are available on the project website.",
"and re-annotated to ensure a high quality of the ground truth.",
"Additionally, we created two new datasets focusing on varying lighting conditions, since this variation is not present in the existing datasets.",
"An overview of the datasets is in Fig.",
"REF and a detailed description follows."
],
[
"Training and Test Data",
"The datasets consist of texture-mapped 3D object models and training and test RGB-D images annotated with ground-truth 6D object poses.",
"The 3D object models were created using KinectFusion-like systems for 3D surface reconstruction [26], [33].",
"All images are of approximately VGA resolution.",
"For training, a method may use the 3D object models and/or the training images.",
"While 3D models are often available or can be generated at a low cost, capturing and annotating real training images requires a significant effort.",
"The benchmark is therefore focused primarily on the more practical scenario where only the object models, which can be used to render synthetic training images, are available at training time.",
"All datasets contain already synthesized training images.",
"Methods are allowed to synthesize additional training images, but this option was not utilized for the evaluation in this paper.",
"Only T-LESS and TUD-L include real training images of isolated, i.e.",
"non-occluded, objects.",
"To generate the synthetic training images, objects from the same dataset were rendered from the same range of azimuth/elevation covering the distribution of object poses in the test scenes.",
"The viewpoints were sampled from a sphere, as in [14], with the sphere radius set to the distance of the closest object instance in the test scenes.",
"The objects were rendered with fixed lighting conditions and a black background.",
"The test images are real images from a structured-light sensor – Microsoft Kinect v1 or Primesense Carmine 1.09.",
"The test images originate from indoor scenes with varying complexity, ranging from simple scenes with a single isolated object instance to very challenging scenes with multiple instances of several objects and a high amount of clutter and occlusion.",
"Poses of the modeled objects were annotated manually.",
"While LM, IC-MI and RU-APC provide annotation for instances of only one object per image, the other datasets provide ground-truth for all modeled objects.",
"Details of the datasets are in Tab.",
"REF .",
"Figure: NO_CAPTIONLM (a.k.a.",
"Linemod) has been the most commonly used dataset for 6D object pose estimation.",
"It contains 15 texture-less household objects with discriminative color, shape and size.",
"Each object is associated with a test image set showing one annotated object instance with significant clutter but only mild occlusion.",
"LM-O (a.k.a.",
"Linemod-Occluded) provides ground-truth annotation for all other instances of the modeled objects in one of the test sets.",
"This introduces challenging test cases with various levels of occlusion."
],
[
"IC-MI/IC-BIN {{cite:f4559c9c64d9ef1c60ace18913593ce8830a5246}}, {{cite:a5ae5b739821b8c021484298be2d911a9a9c6468}}.",
"IC-MI (a.k.a.",
"Tejani et al.)",
"contains models of two texture-less and four textured household objects.",
"The test images show multiple object instances with clutter and slight occlusion.",
"IC-BIN (a.k.a.",
"Doumanoglou et al., scenario 2) includes test images of two objects from IC-MI, which appear in multiple locations with heavy occlusion in a bin-picking scenario.",
"We have removed test images with low-quality ground-truth annotations from both datasets, and refined the annotations for the remaining images in IC-BIN."
],
[
"T-LESS {{cite:b0923850efaea5f5d06b486afb1a4f3b760b5181}}.",
"It features 30 industry-relevant objects with no significant texture or discriminative color.",
"The objects exhibit symmetries and mutual similarities in shape and/or size, and a few objects are a composition of other objects.",
"T-LESS includes images from three different sensors and two types of 3D object models.",
"For our evaluation, we only used RGB-D images from the Primesense sensor and the automatically reconstructed 3D object models."
],
[
"RU-APC {{cite:47775c7c26c12491c844a2c4efa6cfbe0bd00e9b}}.",
"This dataset (a.k.a.",
"Rutgers APC) includes 14 textured products from the Amazon Picking Challenge 2015 [6], each associated with test images of a cluttered warehouse shelf.",
"The camera was equipped with LED strips to ensure constant lighting.",
"From the original dataset, we omitted ten objects which are non-rigid or poorly captured by the depth sensor, and included only one from the four images captured from the same viewpoint."
],
[
"TUD-L/TYO-L.",
"Two new datasets with household objects captured under different settings of ambient and directional light.",
"TUD-L (TU Dresden Light) contains training and test image sequences that show three moving objects under eight lighting conditions.",
"The object poses were annotated by manually aligning the 3D object model with the first frame of the sequence and propagating the initial pose through the sequence using ICP.",
"TYO-L (Toyota Light) contains 21 objects, each captured in multiple poses on a table-top setup, with four different table cloths and five different lighting conditions.",
"To obtain the ground truth poses, manually chosen correspondences were utilized to estimate rough poses which were then refined by ICP.",
"The images in both datasets are labeled by categorized lighting conditions."
],
[
"Evaluated Methods",
"The evaluated methods cover the major research directions of the 6D object pose estimation field.",
"This section provides a review of the methods, together with a description of the setting of their key parameters.",
"If not stated otherwise, the image-based methods used the synthetic training images."
],
[
"Brachmann-14 {{cite:202a3f06c6e5aee3db0c1286b63d13abbcc78335}}.",
"For each pixel of an input image, a regression forest predicts the object identity and the location in the coordinate frame of the object model, a so called “object coordinate”.",
"Simple RGB and depth difference features are used for the prediction.",
"Each object coordinate prediction defines a 3D-3D correspondence between the image and the 3D object model.",
"A RANSAC-based optimization schema samples sets of three correspondences to create a pool of pose hypotheses.",
"The final hypothesis is chosen, and iteratively refined, to maximize the alignment of predicted correspondences, as well as the alignment of observed depth with the object model.",
"The main parameters of the method were set as follows: maximum feature offset: $20\\,\\textrm {px}$ , features per tree node: 1000, training patches per object: 1.5M, number of trees: 3, size of the hypothesis pool: 210, refined hypotheses: 25.",
"Real training images were used for TUD-L and T-LESS."
],
[
"Brachmann-16 {{cite:ac49d68646686a1390496b7d4ec5b39e49842c2d}}.",
"The method of [1] is extended in several ways.",
"Firstly, the random forest is improved using an auto-context algorithm to support pose estimation from RGB-only images.",
"Secondly, the RANSAC-based optimization hypothesizes not only with regard to the object pose but also with regard to the object identity in cases where it is unknown which objects are visible in the input image.",
"Both improvements were disabled for the evaluation since we deal with RGB-D input, and it is known which objects are visible in the image.",
"Thirdly, the random forest predicts for each pixel a full, three-dimensional distribution over object coordinates capturing uncertainty information.",
"The distributions are estimated using mean-shift in each forest leaf, and can therefore be heavily multi-modal.",
"The final hypothesis is chosen, and iteratively refined, to maximize the likelihood under the predicted distributions.",
"The 3D object model is not used for fitting the pose.",
"The parameters were set as: maximum feature offset: $10\\,\\textrm {px}$ , features per tree node: 100, number of trees: 3, number of sampled hypotheses: 256, pixels drawn in each RANSAC iteration: 10K, inlier threshold: $1\\,\\textrm {cm}$ ."
],
[
"Tejani-14 {{cite:f4559c9c64d9ef1c60ace18913593ce8830a5246}}.",
"Linemod [14] is adapted into a scale-invariant patch descriptor and integrated into a regression forest with a new template-based split function.",
"This split function is more discriminative than simple pixel tests and accelerated via binary bit-operations.",
"The method is trained on positive samples only, i.e.",
"rendered images of the 3D object model.",
"During the inference, the class distributions at the leaf nodes are iteratively updated, providing occlusion-aware segmentation masks.",
"The object pose is estimated by accumulating pose regression votes from the estimated foreground patches.",
"The baseline evaluated in this paper implements [34] but omits the iterative segmentation/refinement step and does not perform ICP.",
"The features and forest parameters were set as in [34]: number of trees: 10, maximum depth of each tree: 25, number of features in both the color gradient and the surface normal channel: 20, patch size: 1/2 the image, rendered images used to train each forest: 360."
],
[
"Kehl-16 {{cite:3abe12d24999673a7e04d669acf29f9934f94fc5}}.",
"Scale-invariant RGB-D patches are extracted from a regular grid attached to the input image, and described by features calculated using a convolutional auto-encoder.",
"At training time, a codebook is constructed from descriptors of patches from the training images, with each codebook entry holding information about the 6D pose.",
"For each patch descriptor from the test image, $k$ -nearest neighbors from the codebook are found, and a 6D vote is cast using neighbors whose distance is below a threshold $t$ .",
"After the voting stage, the 6D hypothesis space is filtered to remove spurious votes.",
"Modes are identified by mean-shift and refined by ICP.",
"The final hypothesis is verified in color, depth and surface normals to suppress false positives.",
"The main parameters of the method with the used values: patch size: $32\\times 32\\,\\textrm {px}$ , patch sampling step: $6\\,\\textrm {px}$ , $k$ -nearest neighbors: 3, threshold $t$ : 2, number of extracted modes from the pose space: 8.",
"Real training images were used for T-LESS."
],
[
"Hodaň-15 {{cite:284229fa7c78d72095a3f46fea633a2bcdd412b6}}.",
"A template matching method that applies an efficient cascade-style evaluation to each sliding window location.",
"A simple objectness filter is applied first, rapidly rejecting most locations.",
"For each remaining location, a set of candidate templates is identified by a voting procedure based on hashing, which makes the computational complexity largely unaffected by the total number of stored templates.",
"The candidate templates are then verified as in Linemod [14] by matching feature points in different modalities (surface normals, image gradients, depth, color).",
"Finally, object poses associated with the detected templates are refined by particle swarm optimization (PSO).",
"The templates were generated by applying the full circle of in-plane rotations with $10^\\circ $ step to a portion of the synthetic training images, resulting in 11–23K templates per object.",
"Other parameters were set as described in [18].",
"We present also results without the last refinement step (Hodaň-15-nr).",
"A method based on matching oriented point pairs between the point cloud of the test scene and the object model, and grouping the matches using a local voting scheme.",
"At training time, point pairs from the model are sampled and stored in a hash table.",
"At test time, reference points are fixed in the scene, and a low-dimensional parameter space for the voting scheme is created by restricting to those poses that align the reference point with the model.",
"Point pairs between the reference point and other scene points are created, similar model point pairs searched for using the hash table, and a vote is cast for each matching point pair.",
"Peaks in the accumulator space are extracted and used as pose candidates, which are refined by coarse-to-fine ICP and re-scored by the relative amount of visible model surface.",
"Note that color information is not used.",
"It was evaluated using function find_surface_model from HALCON 13.0.2 [12].",
"The sampling distances for model and scene were set to 3% of the object diameter, 10% of points were used as the reference points, and the normals were computed using the mls method.",
"Points further than 2 m were discarded."
],
[
"Drost-10-edge.",
"An extension of [9] which additionally detects 3D edges from the scene and favors poses in which the model contours are aligned with the edges.",
"A multi-modal refinement minimizes the surface distances and the distances of reprojected model contours to the detected edges.",
"The evaluation was performed using the same software and parameters as Drost-10, but with activated parameter train_3d_edges during the model creation."
],
[
"Vidal-18 {{cite:9443a4a0916aec46486d8c372b0e163d23b1c44f}}.",
"The point cloud is first sub-sampled by clustering points based on the surface normal orientation.",
"Inspired by improvements of [15], the matching strategy of [9] was improved by mitigating the effect of the feature discretization step.",
"Additionally, an improved non-maximum suppression of the pose candidates from different reference points removes spurious matches.",
"The most voted 500 pose candidates are sorted by a surface fitting score and the 200 best candidates are refined by projective ICP.",
"For the final 10 candidates, the consistency of the object surface and silhouette with the scene is evaluated.",
"The sampling distance for model, scene and features was set to 5% of the object diameter, and 20% of the scene points were used as the reference points."
],
[
"Buch-16 {{cite:bcd789c69a68c43f0ede730a37f274b47bba6e4b}}.",
"A RANSAC-based method that iteratively samples three feature correspondences between the object model and the scene.",
"The correspondences are obtained by matching 3D local shape descriptors and are used to generate a 6D pose candidate, whose quality is measured by the consensus set size.",
"The final pose is refined by ICP.",
"The method achieved the state-of-the-art results on earlier object recognition datasets captured by LIDAR, but suffers from a cubic complexity in the number of correspondences.",
"The number of RANSAC iterations was set to 10000, allowing only for a limited search in cluttered scenes.",
"The method was evaluated with several descriptors: 153d SI [19], 352d SHOT [30], 30d ECSAD [20], and 1536d PPFH [5].",
"None of the descriptors utilize color."
],
[
"Buch-17 {{cite:f50314ca6882156b56b51f3d8df713f0f55c55a2}}.",
"This method is based on the observation that a correspondence between two oriented points on the object surface is constrained to cast votes in a 1-DoF rotational subgroup of the full group of poses, SE(3).",
"The time complexity of the method is thus linear in the number of correspondences.",
"Kernel density estimation is used to efficiently combine the votes and generate a 6D pose estimate.",
"As Buch-16, the method relies on 3D local shape descriptors and refines the final pose estimate by ICP.",
"The parameters were set as in the paper: 60 angle tessellations were used for casting rotational votes, and the translation/rotation bandwidths were set to 10 mm/22.5$^\\circ $ ."
],
[
"Evaluation",
"The methods reviewed in Sec.",
"were evaluated by their original authors on the datasets described in Sec.",
", using the evaluation methodology from Sec.",
"."
],
[
"Fixed Parameters.",
"The parameters of each method were fixed for all objects and datasets.",
"The distribution of object poses in the test scenes was the only dataset-specific information used by the methods.",
"The distribution determined the range of viewpoints from which the object models were rendered to obtain synthetic training images."
],
[
"Pose Error.",
"The error of a 6D object pose estimate is measured with the pose-error function $e_\\mathrm {VSD}$ defined in Sec.",
"REF .",
"The visibility masks were calculated as in [17], with the occlusion tolerance $\\delta $ set to $15\\,{mm}$ .",
"Only the ground truth poses in which the object is visible from at least 10% were considered in the evaluation."
],
[
"Performance Score.",
"The performance is measured by the recall score, i.e.",
"the fraction of test targets for which a correct object pose was estimated.",
"Recall scores per dataset and per object are reported.",
"The overall performance is given by the average of per-dataset recall scores.",
"We thus treat each dataset as a separate challenge and avoid the overall score being dominated by larger datasets."
],
[
"Subsets Used for the Evaluation.",
"We reduced the number of test images to remove redundancies and to encourage participation of new, in particular slow, methods.",
"From the total of 62K test images, we sub-sampled 7K, reducing the number of test targets from 110K to 17K (Tab.",
"REF ).",
"Full datasets with identifiers of the selected test images are on the project website.",
"TYO-L was not used for the evaluation presented in this paper, but it is a part of the online evaluation.",
"Figure: NO_CAPTIONFigure: Left, middle: Average of the per-dataset recall scores for the misalignment tolerance τ\\tau fixed to 20mm20\\,{mm} and 80mm80\\,{mm}, and varying value of the correctness threshold θ\\theta .",
"The curves do not change much for τ>80mm\\tau > 80\\,{mm}.",
"Right: The recall scores w.r.t.",
"the visible fraction of the target object.",
"If more instances of the target object were present in the test image, the largest visible fraction was considered."
],
[
"Accuracy.",
"Tab.",
"REF and REF show the recall scores of the evaluated methods per dataset and per object respectively, for the misalignment tolerance $\\tau =20\\,{mm}$ and the correctness threshold $\\theta =0.3$ .",
"Ranking of the methods according to the recall score is mostly stable across the datasets.",
"Methods based on point-pair features perform best.",
"Vidal-18 is the top-performing method with the average recall of 74.6%, followed by Drost-10-edge, Drost-10, and the template matching method Hodaň-15, all with the average recall above 67%.",
"Brachmann-16 is the best learning-based method, with 55.4%, and Buch-17-ppfh is the best method based on 3D local features, with 54.0%.",
"Scores of Buch-16-si and Buch-16-shot are inferior to the other variants of this method and not presented.",
"Fig.",
"REF shows the average of the per-dataset recall scores for different values of $\\tau $ and $\\theta $ .",
"If the misalignment tolerance $\\tau $ is increased from $20\\,{mm}$ to $80\\,{mm}$ , the scores increase only slightly for most methods.",
"Similarly, the scores increase only slowly for $\\theta > 0.3$ .",
"This suggests that poses estimated by most methods are either of a high quality or totally off, i.e.",
"it is a hit or miss."
],
[
"Speed.",
"The average running times per test target are reported in Tab.",
"REF .",
"However, the methods were evaluated on different computersSpecifications of computers used for the evaluation are on the project website.",
"and thus the presented running times are not directly comparable.",
"Moreover, the methods were optimized primarily for the recall score, not for speed.",
"For example, we evaluated Drost-10 with several parameter settings and observed that the running time can be lowered by a factor of $$$$ 5$ to $ 0.5 s$ with only a relatively small drop of the average recall score from $ 68.1%$ to $ 65.8%$.",
"However, in Tab.~\\ref {tab:eval_per_dataset} we present the result with the highest score.",
"Brachmann-14 could be sped up by sub-sampling the 3D object models and Hodaň-15 by using less object templates.",
"A study of such speed/accuracy trade-offs is left for future work.$"
],
[
"Open Problems.",
"Occlusion is a big challenge for current methods, as shown by scores dropping swiftly already at low levels of occlusion (Fig.",
"REF , right).",
"The big gap between LM and LM-O scores provide further evidence.",
"All methods perform on LM by at least 30% better than on LM-O, which includes the same but partially occluded objects.",
"Inspection of estimated poses on T-LESS test images confirms the weak performance for occluded objects.",
"Scores on TUD-L show that varying lighting conditions present a serious challenge for methods that rely on synthetic training RGB images, which were generated with fixed lighting.",
"Methods relying only on depth information (e.g.",
"Vidal-18, Drost-10) are noticeably more robust under such conditions.",
"Note that Brachmann-16 achieved a high score on TUD-L despite relying on RGB images because it used real training images, which were captured under the same range of lighting conditions as the test images.",
"Methods based on 3D local features and learning-based methods have very low scores on T-LESS, which is likely caused by the object symmetries and similarities.",
"All methods perform poorly on RU-APC, which is likely because of a higher level of noise in the depth images."
],
[
"Conclusion",
"We have proposed a benchmark for 6D object pose estimation that includes eight datasets in a unified format, an evaluation methodology, a comprehensive evaluation of 15 recent methods, and an online evaluation system open for continuous submission of new results.",
"With this benchmark, we have captured the status quo in the field and will be able to systematically measure its progress in the future.",
"The evaluation showed that methods based on point-pair features perform best, outperforming template matching methods, learning-based methods and methods based on 3D local features.",
"As open problems, our analysis identified occlusion, varying lighting conditions, and object symmetries and similarities."
],
[
"Acknowledgements",
"We gratefully acknowledge Manolis Lourakis, Joachim Staib, Christoph Kick, Juil Sock and Pavel Haluza for their help.",
"This work was supported by CTU student grant SGS17/185/OHK3/3T/13, Technology Agency of the Czech Republic research program TE01020415 (V3C – Visual Computing Competence Center), and the project for GAČR, No.",
"16-072105: Complex network methods applied to ancient Egyptian data in the Old Kingdom (2700–2180 BC)."
]
] | 1808.08319 |
[
[
"Multi-Level Network Embedding with Boosted Low-Rank Matrix Approximation"
],
[
"Abstract As opposed to manual feature engineering which is tedious and difficult to scale, network representation learning has attracted a surge of research interests as it automates the process of feature learning on graphs.",
"The learned low-dimensional node vector representation is generalizable and eases the knowledge discovery process on graphs by enabling various off-the-shelf machine learning tools to be directly applied.",
"Recent research has shown that the past decade of network embedding approaches either explicitly factorize a carefully designed matrix to obtain the low-dimensional node vector representation or are closely related to implicit matrix factorization, with the fundamental assumption that the factorized node connectivity matrix is low-rank.",
"Nonetheless, the global low-rank assumption does not necessarily hold especially when the factorized matrix encodes complex node interactions, and the resultant single low-rank embedding matrix is insufficient to capture all the observed connectivity patterns.",
"In this regard, we propose a novel multi-level network embedding framework BoostNE, which can learn multiple network embedding representations of different granularity from coarse to fine without imposing the prevalent global low-rank assumption.",
"The proposed BoostNE method is also in line with the successful gradient boosting method in ensemble learning as multiple weak embeddings lead to a stronger and more effective one.",
"We assess the effectiveness of the proposed BoostNE framework by comparing it with existing state-of-the-art network embedding methods on various datasets, and the experimental results corroborate the superiority of the proposed BoostNE network embedding framework."
],
[
"Introduction",
"Learning meaningful and discriminative representations of nodes in a network is essential for various network analytical tasks as it avoids the laborious manual feature engineering process.",
"Additionally, as the node embedding representations are often learned in a task-agnostic fashion, they are generalizable to a number of downstream learning tasks such as node classification [33], community detection [44], link prediction [15], and visualization [37].",
"On top of that, it also has broader impacts in advancing many real-world applications, ranging from recommendation [45], polypharmacy side effects prediction [53] to name disambiguation [49].",
"The basic idea of network embedding is to represent each node by a low-dimensional vector in which the relativity information among nodes in the original network is maximally transcribed.",
"A vast majority of existing network embedding methods can be broadly divided into two different yet highly correlated categories.",
"Firstly, early network embedding methods are largely based on the matrix factorization approaches.",
"In particular, methods in this family represent the topological structure among nodes as a deterministic connectivity matrix and leverage low-rank matrix approximation strategies to obtain the node vector representations with the assumption that high-quality node embeddings are encoded in a small portion of latent factors.",
"Various methods differ in the way on how the connectivity matrix is built, typical examples include modularity matrix [40], Laplacian matrix [2], $k$ -step transition matrix [4] and higher-order adjacency matrix [47].",
"Secondly, the recent advances of network embedding research are largely influenced by the skip-gram model [31] in natural language processing.",
"The key idea behind these algorithms is that nodes tend to have similar embedding representations if they co-occur frequently on short random walks over the network [33], [15] or are directly connected with each other within certain contexts [39], [38].",
"As these methods often employ a flexible way to measure the node similarity, they have shown to achieve superior learning performance in many scenarios [16], [8].",
"Even though the two families of network embedding methods are distinct in nature, recent studies [34] found their inherent correlations and then cast the skip-gram inspired network embedding methods in the matrix factorization framework.",
"To this end, in this work, we investigate the network embedding problem within the framework of matrix factorization due to its broad generalizability.",
"As mentioned previously, after the closed-form connectivity matrix for each network embedding model is derived, the principled way to permit node embedding representation is to perform low-rank matrix approximation, such as with eigendecomposition, singular value decomposition (SVD) and nonnegative matrix factorization (NMF).",
"Here, the fundamental assumption is that the closed-form connectivity matrix is low-rank (a.k.a.",
"global low-rank), and the factorized matrix (which is also low-rank) is sufficient to provide a “one-size fits all\" representation to encode the connectivity information among nodes.",
"However, the widely perceived assumption is untenable in practice and may further hinder us learn effective node embedding representations due to the following reasons.",
"On one hand, the connectivity matrix in real-world scenarios is often not sparse and encodes diverse connectivity patterns among nodes, making the low-rank assumption improper [23], [22].",
"Hence, a global low-rank factorization cannot guarantee a good approximation of the closed-form connectivity matrix.",
"On the other hand, simply relying on the global low-rank property for a single representation for nodes cannot fully explain how the nodes are connected with each other in the network.",
"In a nutshell, it is desired to learn the embedding in a forward stagewise fashion to gradually shift the focus to the unexplained node connectivity behaviors as stages progress.",
"To address the above problems, we propose a multi-level network embedding framework BoostNE to obtain multiple granularity views (from coarse to fine) of the network over the full spectrum.",
"The proposed multi-level network embedding framework is motivated by the gradient boosting framework [14] in ensemble learning, which learns multiple weak learners sequentially and then combines them together to make the final prediction.",
"The main contributions of this paper are summarized as follows: Formulation We systematically examine the fundamental limitations of existing network embedding approaches, especially the methods that directly leverage or can be reduced to the framework of matrix factorization.",
"To alleviate the limitations, we propose to study a novel problem of multi-level network embedding by learning multiple node representations of different granularity.",
"Algorithm We propose a novel network embedding framework BoostNE to identify multiple embedding representations from coarse to fine in a forward stagewise manner.",
"The key idea is to leverage the principle of gradient boosting to successively factorize the residual of the connectivity matrix that is not well explained from the previous stage.",
"As we do not impose any assumption that the original connectivity matrix is global low-rank, the developed multi-level method yields better approximations and further leads to more effective embedding representation.",
"Evaluation We conduct experiments on multiple real-world networks from various domains to compare the proposed BoostNE with existing state-of-the-arts.",
"The results demonstrate the effectiveness of the proposed multi-level network embedding framework BoostNE.",
"Further studies are presented to understand why the proposed BoostNE work and how the number of levels impacts the final embedding representation.",
"The rest of this paper is structured as follows.",
"The problem statement of multi-level network embedding is introduced in Section II.",
"In Section III, we propose a novel framework BoostNE that is able to learn the multi-level network representations from coarse to fine with boosted low-rank matrix approximation.",
"Experimental evaluations on real-world datasets are presented in Section IV with discussions.",
"In Section V, we briefly review related work.",
"The conclusions and future work are presented in Section VI."
],
[
"Problem Definition",
"We first summarize the notations used in this paper.",
"We denote matrices using bold uppercase characters (e.g., ${\\bf X}$ ), vectors using bold lowercase characters (e.g., ${\\bf x}$ ), scalars using normal lowercase characters (e.g., $x$ ).",
"The $i$ -th element of vector ${\\bf x}$ is denoted by $x_{i}$ .",
"The $i$ -th row, the $j$ -th column and the $(i,j)$ -th entry of matrix ${\\bf X}$ are denoted by ${\\bf X}_{i*}$ , ${\\bf X}_{*j}$ , and ${\\bf X}_{ij}$ respectively.",
"The transpose of matrix ${\\bf X}$ is ${\\bf X}^{\\prime }$ , and its trace is $tr({\\bf X})$ if it is a square matrix.",
"The $\\ell _{2}$ -norm of the vector ${\\bf x}\\in \\mathbb {R}^{d}$ is denoted by $\\Vert {\\bf x}\\Vert _{2}=\\sqrt{\\sum _{i=1}^{d}x_{i}^{2}}$ .",
"The Frobenius norm of matrix ${\\bf X}\\in \\mathbb {R}^{d\\times k}$ is $\\Vert {\\bf X}\\Vert _{F}=\\sqrt{\\sum _{i=1}^{d}\\sum _{j=1}^{k}{\\bf X}_{ij}^{2}}$ .",
"The $r$ -th power of matrix ${\\bf X}$ is denoted as ${\\bf X}^{r}$ .",
"The main symbols used throughout this paper are summarized in Table REF .",
"Table: Symbols.Let $G=(\\mathcal {V},\\mathcal {E})$ be the given network, where $\\mathcal {V}=\\lbrace v_{1},...,v_{n}\\rbrace $ is the node set and $\\mathcal {E}=\\lbrace e_{1},...,e_{m}\\rbrace $ is the edge set.",
"We use the matrix ${\\bf A}\\in \\mathbb {R}^{n\\times n}$ to denote the adjacency matrix of the network, where ${\\bf A}_{ij}\\ge 0$ is a real number denoting the edge weight between node $v_{i}$ and node $v_{j}$ .",
"If there is no edge between $v_{i}$ and $v_{j}$ , then ${\\bf A}_{ij}=0$ .",
"In this work, we focus on undirected network such that ${\\bf A}_{ij}={\\bf A}_{ji}$ though our problem can be easily extended to directed networks.",
"Now, we formally define the studied problem of multi-level network embedding as follows.",
"Problem 1 (Multi-Level Network Embedding): Given a network $G=(\\mathcal {V},\\mathcal {E})$ with $n$ nodes, a predefined embedding dimensionality $d$ and the number of levels $k$ , the problem of multi-level network embedding is to learn a series of $k$ embeddings ${\\bf U}_{i}\\in \\mathbb {R}^{n\\times d_{s}}$ ($i=1,...,k$ ) for each node in the network ($d=kd_{s} \\ll n$ ).",
"The target is to ensure that the complex node connectivity information in the original network is gradually preserved from coarse to fine in a forward stagewise manner as the levels progress from 1 to $k$ ."
],
[
"The Proposed BoostNE Framework",
"In this section, we present the proposed multi-level network embedding framework with boosted low-rank matrix approximation - BoostNE in details.",
"The key idea is to first formulate the network embedding problem within the framework of matrix factorization by constructing a closed-form node connectivity matrix, then it successively factorizes the residual (unexplained part) obtained from the previous stage.",
"To this end, it enables us to generate a sequence of embedding representations of different granularity views from coarse to fine.",
"Afterwards, we ensemble all these weak embedding representations as the final embedding for downstream learning tasks.",
"An illustrative example of the workflow of the proposed BoostNE framework is shown in Fig.",
"REF .",
"Figure: An illustration of the proposed framework - BoostNE.",
"BoostNE first constructs the complex node connectivity matrix 𝐗{\\bf X} from the adjacency matrix 𝐀{\\bf A}.",
"Motivated by the principle of gradient boosting, the proposed BoostNE successively factorizes the residual of the connectivity matrix from the previous stage and obtains embedding representations from coarse (e.g., 𝐔 1 {\\bf U}_{1}) to fine (e.g., 𝐔 3 {\\bf U}_{3}).",
"The final embedding representation 𝐔{\\bf U} is obtained by ensembling all previous weak embedding representations."
],
[
"Network Embedding as Matrix Factorization",
"As mentioned previously, a large number of existing network embedding methods are fundamentally based on the technique of matrix factorization such as eigendecomposition, singular value decomposition (SVD) and nonnegative matrix factorization (NMF).",
"Concretely, they target at building a deterministic connectivity matrix of various forms to capture different types of connections among nodes in a network, including the first-order proximity, the higher-order proximity, the community structure, and etc.",
"The other category of methods are mostly inspired by the language model word2vec [31].",
"They either exploit random walks on the network to capture the node structural relations or directly perform edge modeling to learn the structure preserving node representations.",
"Even though their optimization target is distinct from matrix factorization based methods, recent studies [34], [46] examined and found that most of these algorithms can also be reduced to the matrix factorization framework, and the desired embedding representations can be derived by performing principled matrix factorization methods.",
"Next, we briefly show the connections between these sophisticated network embedding methods and matrix factorization.",
"Spectral Clustering [2], [41]: Given the adjacency matrix ${\\bf A}$ and the diagonal degree matrix ${\\bf D}$ , the node embedding representation ${\\bf U}$ can be obtained by concatenating the top-$d$ eigenvectors of the normalized adjacency matrix ${\\bf D}^{-1/2}{\\bf A}{\\bf D}^{-1/2}$ .",
"Graph Factorization [1]: The node embedding ${\\bf U}$ is obtained by performing symmetric matrix factorization on the adjacency matrix ${\\bf A}$ such that ${\\bf U}=\\arg \\!\\min _{{\\bf U}}\\Vert {\\bf A}-{\\bf U}{\\bf U}^{\\prime }\\Vert _{F}^{2}$ .",
"Social Dimension [40]: It first constructs the modularity matrix as ${\\bf A}-{\\bf d}{\\bf d}^{\\prime }/2m$ , where ${\\bf d}$ is the node degree vector.",
"Afterwards, the node embedding ${\\bf U}$ can be identified by taking the top-$d$ eigenvectors of the modularity matrix.",
"GraRep [4]: It concatenates the top left singular vectors of a series of transition matrices ${\\bf X}_{p}$ ($p=1,...,K$ ).",
"In particular, $K$ denotes the number of transition steps, ${\\bf X}_{p}=\\text{max}({\\bf Y}_{p},0)$ with $({\\bf Y}_{p})_{ij}=\\text{log}({\\bf S}^{p}_{ij}/\\sum _{t}{\\bf S}^{p}_{tj})-\\text{log}(b)$ , where $b$ denotes the log shift factor.",
"Hope [32]: It first derives the connectivity matrix as ${\\bf M}={\\bf M}_{g}^{-1}{\\bf M}_{l}$ where high-order proximity matrices ${\\bf M}_{g}$ and ${\\bf M}_{l}$ are built from ${\\bf A}$ .",
"Then it performs conventional low-rank matrix factorization on ${\\bf M}$ to get the node embeddings.",
"Deepwalk [33]: Deepwalk makes an analogy between truncated random walks on networks with sentences in a text corpus and learns the node embedding as the skip-gram model.",
"It has been shown [34] that Deepwalk is equivalent to factorize a transformation of network's normalized Laplacian matrix $\\text{log}(\\text{vol}(G)(\\frac{1}{T}\\sum _{r=1}^{\\prime }{\\bf S}^{r}){\\bf D}^{-1})-\\text{log}(b)$ , where $\\text{vol}(G)$ denotes the volume of the network $G$ , $T$ denotes the context window size and $b$ is the number of negative samples.",
"node2vec [15]: It actually factorizes a more complex matrix with strong connections to the stationary distribution and transition probability of second-order random walk.",
"It is also shown in [34] that the factorized matrix is in the following format $\\text{log}\\big (\\frac{\\frac{1}{2T}\\sum _{r=1}^{\\prime }(\\sum _{u}{\\bf Y}_{wu}{\\bf P}_{cwu}^{r}+\\sum _{u}{\\bf Y}_{cu}{\\bf P}_{wcu}^{r})}{(\\sum _{u}{\\bf Y}_{wu})(\\sum _{u}{\\bf Y}_{cu})}\\big )-\\text{log}(b)$ , where ${\\bf P}_{uvw}$ denotes the random walk probability to node $u$ given the current visited node $v$ and previously visited node $w$ .",
"${\\bf Y}$ encodes the second-order random walk stationary distribution.",
"LINE [39]: LINE is a special case of Deepwalk when the size of context is specified as one [34] and it is equivalent to factorize the following matrix $\\text{log}(\\text{vol}(G){\\bf S}{\\bf D}^{-1})-\\text{log}(b)$ .",
"In this following context, we focus on the node connectivity matrix of Deepwalk with the following reasons: (1) it is more general than other node connectivity matrices as it captures both local and global interactions among nodes in the network; (2) it often leads better learning performance as shown in [34]."
],
[
"Multi-Level Network Embedding with Boosted Low-Rank Matrix Approximation",
"As per the above summarization, after various node connectivity matrices are derived from the original network topology, a vast majority of existing network embedding methods discover the node embeddings with the prevalent assumption that the underlying connectivity matrix is of low-rank.",
"To this end, they target at learning the node embeddings with low-rank matrix approximation methods by conjecturing that the discriminative connectivity information should be well encoded within a single low-dimensional node representation.",
"As in the case of Deepwalk, the resultant node connectivity matrix is a nonnegative matrix, and to permit meaningful embedding representations, one possible solution is to perform nonnegative matrix factorization (NMF) on the connectivity matrix ${\\bf X}\\in \\mathbb {R}_{+}^{n\\times n}$ with a global low-rank assumption: $\\min _{{\\bf U},{\\bf V}\\ge 0}\\Vert {\\bf X}-{\\bf U}{\\bf V}\\Vert _{F}^{2},$ where $d$ denotes the dimension of the embedding representation.",
"The factorized matrices ${\\bf U}\\in \\mathbb {R}_{+}^{n\\times d}$ and ${\\bf V}\\in \\mathbb {R}_{+}^{d\\times n}$ are both low-rank ($d\\ll n$ ) and nonnegative.",
"In particular, we can interpret ${\\bf U}$ as the embedding of nodes that act like a “center\" node while ${\\bf V}^{\\prime }$ can be regarded as the embedding of nodes that play the role of “context\" node [26].",
"The above objective function of NMF can be solved by conventional optimization algorithms such as coordinate descent [21] and projected gradient descent [28].",
"In the above optimization problem, in order to perform NMF, the factorized matrix ${\\bf X}$ has to satisfy the property of global low-rank.",
"However, it is often argued that the low-rank assumption does not necessarily hold in real-world scenarios, especially when complex interactions among nodes are involved [23], [48], [51].",
"In light of this, performing a single NMF on the connectivity matrix ${\\bf X}$ may lead to suboptimal results and the obtained “one-size fits all\" embedding representation is insufficient to encode the connectivity patterns among nodes in the original network.",
"To find a better embedding representation that well approximates the node connectivity matrix ${\\bf X}$ , we relax the low-rank assumption of a single run of factorization.",
"Instead, we assume that the connectivity matrix can be well approximated by performing multiple levels of NMF, resulting in the following objective function: $\\min _{{\\bf U}_{i},{\\bf V}_{i}\\ge 0, i=1,...,k}\\Vert {\\bf X}-\\sum _{i=1}^{k}{\\bf U}_{i}{\\bf V}_{i}\\Vert _{F}^{2}.$ In the above objective function, $k$ denotes the number of levels, and ${\\bf U}_{i}\\in \\mathbb {R}_{+}^{n\\times d_{s}}$ and ${\\bf V}_{i}\\in \\mathbb {R}_{+}^{d_{s}\\times n}$ denotes the embedding representation of the center node and the context node in the $i$ -th level during the multi-level factorization process.",
"In this work, inspired by the well-known ensemble learning methods [10], [13], [36], we propose to learn multiple levels of node representations by using a forward stagewise strategy.",
"The essential idea is to perform a sequence of NMF operations with each single NMF operation focusing on the residual of the previously not well approximated part.",
"Hence, the initial embedding representations provide a coarse view of the node connectivity patterns while the latter embeddings provide finer-grained embedding representations.",
"In other words, different stages present various views of the embedding of different granularity.",
"More specifically, in the $i$ -th level, the not well explained residual matrix is defined as follows: ${\\bf R}_{i}=\\left\\lbrace \\begin{array}{ll}{\\bf X}& \\,\\,\\, \\text{if } i=1 \\\\\\text{max}({\\bf R}_{i-1}-{\\bf U}_{i-1}{\\bf V}_{i-1},0)& \\,\\,\\, \\text{if } i\\ge 2.\\\\\\end{array}\\right.$ The max operation implies that if there exists any negative elements after the approximation, we convert it to be zero.",
"With the above defined residual matrix, the embedding representation at the $i$ -the level is obtained by minimizing the following loss function: $\\min _{{\\bf U}_{i},{\\bf V}_{i}\\ge 0} \\Vert {\\bf R}_{i}-{\\bf U}_{i}{\\bf V}_{i}\\Vert _{F}^{2}.$ Compared with the objective function in Eq.",
"(REF ) which returns the node embedding representation in a single run with NMF, our proposed algorithm returns multiple weak representations from coarse to fine in a greedy fashion.",
"In other words, once the earlier level embeddings ${\\bf U}_{i}$ and ${\\bf V}_{i}^{\\prime }$ are obtained, they are fixed for the remaining operations.",
"Next, we briefly analyze the time complexity of the proposed BoostNE and NMF with a single run in Eq.",
"(REF ), here we specify $d=d_{s}k$ for a fair comparison.",
"The time complexity of optimizing Eq.",
"(REF ) is related to the number of nonzero elements in ${\\bf X}$ and the rank $d$ , which is $\\#iterations\\times O(\\text{nnz}({\\bf X})d)$ .",
"While for the proposed BoostNE, as can be observed from the illustrative example in Figure REF , the residual matrix (the unexplained part) becomes sparser and sparser as the level goes up, thus the computational cost of the proposed BoostNE is $\\#iterations\\times O(\\text{nnz}({\\bf R}_{1}+,...,+{\\bf R}_{k})d_{s})$ .",
"As $\\text{nnz}({\\bf R}_{1})+,...,+\\text{nnz}({\\bf R}_{k})<\\text{nnz}({\\bf X})k$ , the proposed BoostNE is more efficient than performing a single run NMF in obtaining a “one-size fits all\" embedding representation.",
"The detailed pseudo code of the proposed multi-level network embedding framework BoostNE is shown in Algorithm REF .",
"[!t] [1] A given network $G=(\\mathcal {V},\\mathcal {E})$ ; number of levels $k$ , embedding dimension in each level $d_{s}$ .",
"The node embedding representation ${\\bf U}_{1},...,{\\bf U}_{k}$ from coarse to fine and the final embedding representation ${\\bf U}$ .",
"Obtain the node connectivity matrix ${\\bf X}$ from the adjacency matrix ${\\bf A}$ of the network; $i = 1$ to $k$ Compute the residual matrix ${\\bf R}_{i}$ with Eq.",
"(REF ); Obtain the center node embedding representation ${\\bf U}_{i}$ and the context node embedding representation ${\\bf V}_{i}^{\\prime }$ by alternating optimization algorithms; Return the final embedding as ${\\bf U}=[{\\bf U}_{1},...,{\\bf U}_{k}]$ The proposed multi-level network embedding framework - BoostNE"
],
[
"Experiments",
"In this section, we conduct experiments to assess the effectiveness of the proposed BoostNE on the task of multi-label node classification.",
"Before introducing the detailed experimental results, we first introduce the used datasets, compared baseline methods, and experimental settings.",
"In this section, we also perform further analysis on why the multi-level embedding methods achieve better performance by studying the approximation error of the connectivity matrix.",
"At last, we investigate how the number of levels $k$ affects the final network embedding results."
],
[
"Dataset",
"We collect and use four real-world network datasets from different domains for experimental evaluation.",
"All these four datasets are publicly available and have been used in previous research [33], [46], [15].",
"The detailed descriptions of these datasets are as follow: Corahttps://linqs.soe.ucsc.edu/data: Cora is a citation network of scientific publications.",
"It consists of 2,708 papers from 7 classes representing different research areas.",
"There is a total number of 5,278 citation links in the dataset.",
"Wikihttps://github.com/thunlp/OpenNE/tree/master/data/wiki: Wiki is a collection of wikipedia documents that are inherently connected with each other via hyperlinks.",
"Each document is categorized into a number of predefined classes denoting their topics.",
"In total, we have 2,363 documents, 11,596 hyperlinks and 17 topics.",
"PPIhttps://snap.stanford.edu/node2vec/#datasets: It is a subgraph of the protein-protein interaction network of Homo Sapiens.",
"In the dataset, the labels of the protein are obtained from the hallmark gene sets and denote the biological states.",
"We have 3,860 proteins, 37,845 interaction links, and 50 states.",
"Blogcataloghttp://socialcomputing.asu.edu/datasets/BlogCatalog is a social blogging website in which users follow each other and post blogs under certain predefined categories.",
"The main categories of blogs by the users are regarded as the class labels of users.",
"In the dataset, we have 10,312 users, 333,983 user relations and 39 predefined categories.",
"For all the above mentioned datasets, we have removed all self-loop edges and have converted bi-directional edges to undirected ones for a fair comparison of various network embedding methods.",
"The detailed statistics of these datasets are listed in Table REF .",
"Table: Statistics of the used datasets."
],
[
"Compared Baseline Methods",
"In this subsection, we compare our proposed multi-level network embedding framework BoostNE with existing efforts from two main categories: matrix factorization based network embedding methods and skip-gram inspired network embedding methods.",
"Among them, Spectral Clustering, Social Dimension, GraRep belong to the former category while Deepwalk and LINE fall into the latter one.",
"In addition, we also compare with a recently proposed framework NetMF which provides a general framework to factorize a closed-form node connectivity matrix for embedding learning.",
"Spectral Clustering [41]: It is a typical matrix factorization based approach which takes the top-$d$ eigenvectors the normalized Laplacian matrix of network $G$ as the node embedding representation.",
"Modularity [40]: It is a kind of matrix factorization based method by taking the top-$d$ eigenvectors of the modularity matrix from the network $G$ why assuming that good embedding assignment can maximize the modularity of the node partition.",
"GraRep [4]: This method learns node embedding representation by capturing different $k$ -step relational information among nodes.",
"This method also generates multiple node embedding representations.",
"However, it is different from our method as they operate on multiple transitional matrices while our method focuses on a single node connectivity matrix.",
"Deepwalk [33]: Deepwalk is the first network embedding method which borrows the idea of word2vec in the NLP community.",
"Specifically, it performs truncated random walks on the network and the node embeddings are learned by capturing the node proximity information encoded in these short random walks.",
"LINE [39]: LINE carefully designs the objective function in preserving the first-order and the second-order node proximity for node embedding representation learning.",
"We concatenate the embedding representations from these two objective functions together.",
"NetMF [34]: It is a recently proposed network embedding method which bridges the gap between matrix factorization based approaches and skip-gram inspired methods.",
"In particular, it attempts to approximate the closed-form of the Deepwalk's implicit matrix.",
"In this work, to have a fair comparison with our method, we adapt it by performing NMF instead of SVD after the closed-form connectivity matrix is derived.",
"Table: Node classification performance comparison on the Cora dataset.Table: Node classification performance comparison on the Wiki dataset.Table: Node classification performance comparison on the PPI dataset.Table: Node classification performance comparison on the Blogcatalog dataset.The parameter settings of different embedding algorithms are as follows.",
"For all the compared network embedding methods, we follow [33], [15], [34] to specify the final embedding dimension as 128.",
"For GraRep, the log shit factor $b$ is set as $1/n$ , the transition step is specified as 8.",
"For Deepwalk, we set the number of walks as 10, the walk length as 80, and the window size as 10.",
"In terms of LINE, we concatenate both the first-order and the second-order embedding representations, the negative sample size is 5 and the number of training epoches is 100.",
"Lastly, for NetMF, we use the Deepwalk's implicit matrix by following the same setting as Deepwalk, and the parameter $h$ in NetMF is set as 256.",
"For the proposed BoostNE, the only additional parameter we need to specify is the number of levels $k$ , we set it as 8.",
"More discussions on the impact of $k$ will be presented later."
],
[
"Experimental Settings",
"To assess the effectiveness of the proposed multi-level network embedding framework BoostNE, we follow the commonly adopted setting to compare different embedding algorithms on the task of multi-label node classification.",
"In the task of multi-label node classification, each node is associated with multiple class labels, our target is to build a predictive learning model to predict the correct labels of nodes.",
"In particular, after the embedding representations of all $n$ nodes are obtained, we randomly sample a portion of nodes as the training data, use their embeddings and their class labels to build a predictive classification model, and then make the predictions with the embeddings of the remaining nodes.",
"In the experiments, we repeat the process 10 times and report the average classification performance in terms of Micro-F1 and Macro-F1, which are widely used multi-label classification evaluation metrics.",
"Specifically, Micro-F1 is a weighted average of F1-scores over different classes while Macro-F1 is an arithmetic average of F1-scores from different labels: $\\begin{split}\\text{Micro-F1}&=\\frac{\\sum _{i=1}^{c}\\text{2TP}^{i}}{\\sum _{i=1}^{c}(\\text{2TP}^{i}+\\text{FP}^{i}+\\text{FN}^{i})}\\\\\\text{Macro-F1}&=\\frac{1}{c}\\sum _{i=1}^{c}\\frac{\\text{2TP}^{i}}{(2\\text{TP}^{i}+\\text{FP}^{i}+\\text{FN}^{i})},\\end{split}$ where $\\text{TP}^{i}$ , $\\text{FP}^{i}$ , and $\\text{FN}^{i}$ denote the number of positives, false positives, and false negatives in the $i$ -th class, respectively.",
"Normally, higher values imply better classification performance, which further indicate better node embedding representations.",
"In the experiments, we vary the percentage of training data from 10% to 90%, and the logistic regression in Liblinear [12] is used as the discriminative classifier.",
"Figure: Approximation error of the node connectivity of NetMF and BoostNE with different number of levels kk."
],
[
"Embedding Results Comparison",
"We first compare the quality of embeddings from different methods in the task of multi-label node classification.",
"The comparison results on the Cora, Wiki, PPI and Blogcatalog datasets are shown in Table REF , Table REF , Table REF and Table REF , respectively.",
"We make the following observations from these tables: For all methods, the multi-label node classification performance w.r.t.",
"Micro-F1 and Macro-F1 gradually increases when the portion of training data is varied from 10% to 90%.",
"It implies that more training data can help us learn more effective embedding representations.",
"The proposed multi-level network embedding method BoostNE achieves the best classification performance in most of the cases.",
"To further validate the conclusion, we perform a pairwise Wilcoxon signed rank test between BoostNE and other embedding methods and the results indicate that BoostNE is significantly better when the $p$ -value threshold is set as 0.05.",
"Compared with NetMF which learns node embedding within a single run of low-rank matrix approximation, the proposed BoostNE learns multiple embedding representations from coarse to fine by gradually factorizing the residual matrix from previous stage.",
"The improvement of BoostNE over NetMF shows the effectiveness of the ensemble methods as multiple weak embedding representations lead to a more effective embedding.",
"Most of the time, the skip-gram inspired methods such as Deepwalk and LINE are better than conventional matrix factorization based methods.",
"Even though both NetMF and BoostNE are kind of matrix factorization based methods, their performance is much better than conventional matrix factorization methods as the connectivity matrix they factorize encode complex node interactions.",
"Figure: Blogcatalog Figure: Blogcatalog"
],
[
"Further Analysis w.r.t. Approximation Error",
"Until now, we have shown that the proposed multi-level network embedding method is superior to NetMF with a single run of NMF on the node connectivity matrix.",
"Despite its empirical success, the underlying reason why performing multiple stages of low-rank approximation leads better embedding representations still remain opaque.",
"In this section, we investigate the underlying mechanism from the approximation error perspective.",
"In particular, we vary the number of levels $k$ in the range of $\\lbrace 2, 4, 8, 16, 32, 64\\rbrace $ and compare the Frobenius norm of the residual matrix after multiple stages of low-rank approximation.",
"The baseline method NetMF can be regarded as a special case when $k=1$ .",
"The comparison results on these four datasets are shown in Fig.",
"REF .",
"As can be observed from the figure, the approximation of the node connectivity matrix from BoostNE is much smaller that of NetMF.",
"In particular, the approximation error gradually goes down when the value of $k$ is increased from 2 to 64.",
"This observation helps us explain why the proposed multi-level network embedding framework BoostNE learns better node representations than NetMF.",
"The main reason is that the low-rank assumption of the node connectivity matrix does not always hold in practice and through the multiple stages of low-rank approximation, the node connectivity patterns are better approximated in BoostNE, and further leads more discriminative node embeddings."
],
[
"Impact of the Number of Embedding Levels $k$",
"As shown in the previous subsection, larger embedding level $k$ often results in a smaller approximation error of the node connectivity matrix.",
"To further investigate how different values of $k$ impact the learned embeddings, we show the node classification performance w.r.t Micro-F1 and Macro-F1 of BoostNE with different values of $k$ and NetMF in Fig.",
"REF and Fig.",
"REF .",
"As can be observed from the figures, when we increase the value of $k$ , the classification performance first increases then reaches its peak and then gradually decreases.",
"The observations are consistent with different portions of training data (30%, 50%, and 70%) and the best classification performance is achieved when $k$ is set as 4 or 8, where the corresponding $d_{s}$ is 32 or 16.",
"In addition, we also observe that the performance of BoostNE is better than NetMF in a wide range (when $k$ is varied from 2 to 32), which further shows the validity of performing boosted low-rank approximation in learning node embeddings."
],
[
"Related Work",
"We briefly review related work on two aspects: (1) network embedding; and (2) ensemble-based matrix factorization.",
"Network Embedding The story of network embedding can be dated back to the early 2000s, when myriad of graph embedding algorithms [35], [2], [42] were developed, as a part of the general dimensionality reduction techniques.",
"Graph embedding first builds an affinity graph based on the feature representations of data instances, and then embeds the affinity graph into a low-dimensional feature space.",
"Even though these algorithms are not explicitly designed for networked data, we can trivially adapt them by feeding the adjacency matrix of networked data as the affinity graph.",
"Along this line, we witnessed a surge of factorization based network embedding methods in recent years, with the target to decompose a carefully designed affinity matrix in capturing the first-order [2], [1], higher-order [4], [32] node proximity or community structure [40], [44] of the underlying network.",
"Despite their empirical success, the factorization based network embedding methods have at least a quadratic time complexity w.r.t.",
"the number of nodes, prohibiting their practical usage on large-scale networks.",
"The recent advances of network representation learning is largely influenced by the word2vec [31] model in the NLP community.",
"The seminal work of Deepwalk [33] first makes an analogy between truncated random walks on a network and sentences in a corpus, and then learns the embedding representations of nodes with the same principle as word2vec.",
"Typical embedding methods along this line include LINE [39], node2vec [15] and PTE [38].",
"A recent work found that the embedding methods with negative sampling (e.g., Deepwalk, LINE, PTE, and node2vec) can be unified into a matrix factorization framework with closed-form solutions [34], which bridges the gap between these two families of network embedding methods.",
"Aforementioned methods mainly adopt a shallow model and the expressibility of the learned embedding representations are rather limited.",
"As a remedy, researchers also resort to deep learning techniques [6], [43], [5] to learn more complex and nonlinear mapping functions.",
"In addition to the raw network structure, real-world networks are often presented with different properties, thus there is a growing interest to learn the embedding representations of networks from different perspectives, such as attributed network [46], [18], [17], heterogeneous networks [7], [11], multi-dimensional networks [29], [50] and dynamic networks [27], [52].",
"Ensemble-based Matrix Factorization Ensemble methods have shown to be effective in improving the performance of single matrix factorization models, especially in the context of collaborative filtering.",
"DeCoste [9] made one of the first attempts to use ensemble methods to improve the prediction results of a single Maximum Margin Matrix Factorization (MMMF) model.",
"In the sequel, the effectiveness of ensemble models is further validated on the Nyström method [20] and Divide-and-Conquer matrix factorization [30].",
"The winner of Netflix Prize [3], [19] advanced the performance of recommendation by capitalizing the advantages of memory-based models and latent factor models.",
"Lee et al.",
"[25] proposed a feature induction algorithm that works in conjunction with stagewise least-squares, and the combination with induced features is superior to existing methods.",
"Most of the existing efforts on matrix factorization are fundamentally based on the prevalent assumption that the given matrix is low-rank, which is often too restrict in practice.",
"Lee et al.",
"developed a novel framework LLORMA model [23], [24] to combine the factorization results from multiple locally weighted submatrices, with the assumption that only the local submatrices are low-rank.",
"The success of LLORMA is further extended to other related problems, including social recommendation [51], topic discovery [36] and image processing [48]."
],
[
"Conclusions and Future Work",
"Network embedding is a fundamental task in graph mining.",
"Recent research efforts have shown that a vast majority of existing network embedding methods can be unified to the general framework of matrix factorization.",
"Specifically, these methods can be summarized with the following working mechanisms: first, construct the deterministic node connectivity matrix by capturing various types of node interactions, and then apply low-rank approximation techniques to obtain the final node embedding representations.",
"However, the fundamental low-rank assumption of the node connectivity matrix may not necessarily hold in practice, making the resultant low-dimensional node representation inadequate for downstream learning tasks.",
"To address this issue, we relax the global low-rank assumption and propose to learn multiple network representations of different granularity from coarse to fine in a forward stagewise fashion.",
"The superiority of the proposed multi-level network embedding framework over other popular network embedding methods is also in line with the success of gradient boosting framework, where the ensemble of multiple weak embedding representations (learners) leads to a better and more discriminative one (learner).",
"Future work can be focused on the following two aspects: first we will provide theoretical results to further understand the principles of the boosted NMF approach; second, we plan to investigate methods that can automatically learn the weights of different levels for more meaningful and discriminative embedding representations."
]
] | 1808.08627 |
[
[
"Parameter-wise co-clustering for high-dimensional data"
],
[
"Abstract In recent years, data dimensionality has increasingly become a concern, leading to many parameter and dimension reduction techniques being proposed in the literature.",
"A parameter-wise co-clustering model, for data modelled via continuous random variables, is presented.",
"The proposed model, although allowing more flexibility, still maintains the very high degree of parsimony achieved by traditional co-clustering.",
"A stochastic expectation-maximization (SEM) algorithm along with a Gibbs sampler is used for parameter estimation and an integrated complete log-likelihood criterion is used for model selection.",
"Simulated and real datasets are used for illustration and comparison with traditional co-clustering."
],
[
"Introduction",
"Clustering is the process of finding and analyzing underlying group structures in heterogenous data.",
"With the emergence of big data, the number of variables in a dataset is constantly increasing and in many areas of application it is not uncommon for the number of variables to exceed the number of observations.",
"In such situations where the dimension of the data is very high, traditional mixture modelling techniques for clustering oftentimes fail.",
"Co-clustering has become a very useful method for dealing with such scenarios.",
"Co-clustering aims to define a partition in the rows of the data matrix for clustering individuals, as well as a partition in the columns for clustering variables.",
"The result is partitioning the data matrix into homogenous blocks, or co-clusters, based on both individuals and variables.",
"A key assumption for maintaining parsimony is that observations within each block are independent and identically distributed.",
"Some of the earliest work in co-clustering was done by .",
"Since that time, model-based approaches have recently been shown to be effective for continuous data, , count data, and ordinal data, , to only name a few.",
"In traditional co-clustering, added flexibility is obtained by fitting more clusters in rows and columns; however, this is not generally advisable for parsimonious reasons.",
"Herein, we propose a co-clustering model for continuous data that separately clusters columns according to both means and variances using a Gaussian distribution.",
"This effectively breaks the identically distributed assumption of observations within each block while still maintaining the parsimony of traditional co-clustering that makes it attractive for really high dimensional data.",
"The remainder of this paper is laid out as follows.",
"Section 2 presents a detailed background on high dimensional clustering techniques as well as details on traditional co-clustering using the Gaussian distribution.",
"Section 3 presents the new model, parameter estimation, model selection criterion, and a non-exhaustive search algorithm for model selection.",
"In Sections 4 and 5, we look at the performance of the algorithms and parameter estimation as well as comparing the proposed model with traditional co-clustering on some synthetic datasets and on a real dataset respectively.",
"We end with a discussion of the results (Section 6)."
],
[
"Model-Based Clustering",
"Clustering is the process of finding underlying group structures in a dataset ${\\bf x}=({\\bf x}_1,{\\bf x}_2,\\ldots ,{\\bf x}_n)$ with $n$ individuals ${\\bf x}_i\\in \\mathbb {R}^p$ .",
"One common method for clustering is model-based clustering and this generally makes use of a finite mixture model.",
"A finite mixture model assumes that a real random vector ${\\bf X}_i$ of dimension $p$ has probability density function $f({\\bf x}_i|{{\\vartheta }})=\\sum _{g=1}^G\\pi _g f({\\bf x}_i|{\\Theta }_g),$ where $\\pi _g>0, \\forall g$ and $\\sum _{g=1}^G\\pi _g=1$ are the mixing proportions and $f(\\cdot |{\\Theta }_g)$ are the component density functions (among $G$ ) parameterized by ${\\Theta }_g$ .",
"${\\vartheta }$ represents all the mixture parameters and is given by ${\\vartheta }=(\\pi _1,\\ldots ,\\pi _G,{\\Theta }_1,\\ldots ,{\\Theta }_G)$ .",
"Because of its mathematical tractability, the multivariate Gaussian mixture model has been widely studied in the literature.",
"In this case each of the component densities is a multivariate Gaussian with density $f({\\bf x}_i|{\\Theta }_g)=\\frac{1}{(2\\pi )^{\\frac{p}{2}}|{\\Sigma }_g|^{\\frac{1}{2}}}\\exp \\left\\lbrace -\\frac{1}{2}({\\bf x}_i-{\\mu }_g)^{\\prime }{\\Sigma }_g^{-1}({\\bf x}_i-{\\mu }_g)\\right\\rbrace ,$ where ${\\Theta }_g=({\\mu }_g,{\\Sigma }_g)$ .",
"The number of parameters in a Gaussian mixture model is given by $\\text{\\#Params}_{\\text{GaussMix}}=(G-1)+Gp+Gp(p+1)/2.$ As is clearly evident, the number of parameters in this case is quadratic in the dimension of the data.",
"As a result, using this simple mixture of Gaussian distributions will usually fail when the dimension, $p$ , of the data reaches around 10.",
"In traditional model-based clustering, the group membership for observation ${\\bf x}_i$ is usually represented by the vector ${\\bf z}_i=(z_{i1},z_{i2},\\ldots ,z_{iG})$ , where $z_{ig}=1$ if observation ${\\bf x}_i$ belongs to group $g$ and 0 otherwise.",
"Moreover, ${\\bf z}_i\\sim \\text{multinomial}(1;{\\pi })$ where ${\\pi }=(\\pi _1,\\pi _2,\\ldots ,\\pi _G)$ .",
"In addition, all couples $({\\bf X}_i,{\\bf z}_i)$ are usually assumed to be independently drawed.",
"The earliest use of a Gaussian mixture model for clustering can be found in .",
"Other early work on Gaussian mixture models for clustering can be found in and .",
"A detailed review of model-based clustering and classification is given by , including related estimation and model selection procedures."
],
[
"High Dimensional Clustering Techniques",
"Although the Gaussian mixture model is widely used, problems arise when moving to high dimensional datasets.",
"The main impact of dimensionality is the number of parameters present in the component covariance matrices ${\\Sigma }_g$ which is quadratic in the dimension, see (REF ).",
"Therefore many methods try to first impose parsimonious constraints on ${\\Sigma }_g$ .",
"A detailed background on this can be found in .",
"One particular example to note is the mixture of factor analyzers model.",
"This model was first presented by and is a Gaussian mixture model with covariance structure ${\\Sigma }_g={\\Lambda }_g{\\Lambda }_g^{\\prime }+\\mathbf {\\Psi }$ , where ${\\Lambda }_g$ is a $p\\times q$ matrix of factors with $q<p$ .",
"A small extension was presented by , who utilize the more general structure ${\\Sigma }_g={\\Lambda }_g{\\Lambda }_g^{\\prime }+\\mathbf {\\Psi }_g$ .",
"introduce the closely-related mixture of PPCAs with ${\\Sigma }_g={\\Lambda }_g{\\Lambda }_g^{\\prime }+\\psi _g\\mathbf {I}$ .",
"constructed a family of eight parsimonious Gaussian models by considering the constraint $\\mathbf {\\Lambda }_g=\\mathbf {\\Lambda }$ in addition to $\\mathbf {\\Psi }_g=\\mathbf {\\Psi }$ and $\\mathbf {\\Psi }_g=\\psi _g\\mathbf {I}$ .",
"The main drawback to all these methods, is that although the number of parameters is reduced from quadratic to linear complexity in the dimension, it is still dependent on the dimension.",
"For example, for the fully constrained model in the number of parameters is $\\text{\\#Param}_{\\text{MFA}}=(G-1)+Gp+pq-q(q-1)/2+1.$ Therefore, these models will not perform well on very high dimensional datasets.",
"These methods also cannot be performed on datasets where $N>d$ which is common in applications such as gene expression data, word processing data, single nucleotide polymorphism (SNP) data, etc.",
"Alternatively, consider using the spectral decomposition of ${\\Sigma }_g$ ${\\Sigma }_g={\\bf D}_g{\\Delta }_g{\\bf D}^{\\prime }_g$ where ${\\bf D}_g$ is the orthogonal matrix of eigenvectors and ${\\Delta }_g$ is a diagonal matrix of corresponding eigenvalues for which they impose the structure ${\\Delta }_g=\\text{diag}(a_{1g},a_{2g},\\ldots ,a_{q_gg},b_g,b_g,\\ldots ,b_g)$ , where $a_{kg}$ are the $q_g$ largest eigenvalues and $b_g$ is average of the remaining $p-q_g$ eigenvalues.",
"This also greatly reduces the number of parameters, with the number of parameters given by ${\\text{\\#}Param}_{\\text{Bouveyron}}=(G-1)+Gp+\\sum _{g=1}^G q_g[p-(q_g+1)/2]+\\sum _{g=1}^G q_g+2G$ Finally, there are also variable selection procedures such as $\\ell _1$ penalization methods which take advantage of sparsity to perform variable selection and parameter estimation simultaneously.",
"The first such proposed method was presented by which considered equal, diagonal covariance matrices between groups and applied an $\\ell _1$ penalty to the mean vectors.",
"A lasso method is then used for parameter estimation.",
"This was extended by who considered unconstrained covariance matrices and applied an $\\ell _1$ penalty for both the mean and covariance parameters.",
"Although these methods are useful for dealing with the dimensionality problem, as discussed by , the $\\ell _1$ penalty shrinks the parameters, thus introducing bias.",
"Moreover, the Bayesian information criterion may not be suitable for high dimensional data.",
"A detailed review of each of these methods can be found in ."
],
[
"Co-Clustering and its Limits",
"Co-Clustering has become a very useful tool for high dimensional data.",
"This method essentially considers simultaneous clustering of rows and columns and then organizes the data into blocks.",
"As in clustering, in traditional co-clustering data is assumed to come in the form of an $n\\times p$ matrix ${\\bf x}$ with columns represented by ${\\bf x}_i$ .",
"Each individual element of ${\\bf x}_i$ will be denoted by $x_{ij}$ so that $x_{ij}$ is the observation in row $i$ and column $j$ .",
"Like in clustering, in co-clustering there is an unknown partition of the rows into $G$ clusters that can be represented by the indicator vector ${\\bf z}_i$ as defined previously.",
"We also, symmetrically and unlike clustering, have a partition of the columns into $L$ clusters that can be represented by the indicator vector ${\\bf w}_j=(w_{j1},w_{j2},\\ldots ,w_{jL})\\sim \\text{multinomial}(1;{\\rho })$ where $w_{jl}=1$ if column $j$ belongs to column cluster $l$ and 0 otherwise and ${\\rho }=(\\rho _1,\\rho _2,\\ldots ,\\rho _L)$ .",
"It is assumed that each data point $x_{ij}$ is independent once the ${\\bf z}_i$ and ${\\bf w}_j$ are fixed.",
"If in addition, all ${\\bf z}_i$ and ${\\bf w}_j$ are independent, then utilizing the latent block model for continuous data and using the Gaussian distribution, as discussed in , for continuous data the density of ${\\bf x}$ becomes $f({\\bf x};{\\vartheta })=\\sum _{{\\bf z}\\in \\mathcal {Z}}\\sum _{{\\bf w}\\in \\mathcal {W}}p({\\bf z};{\\pi })p({\\bf w};{\\rho })f({\\bf x}|{\\bf z},{\\bf w};{\\Theta })$ where $p({\\bf z};{\\pi })=\\prod _{i=1}^n\\prod _{g=1}^G\\pi _g^{z_{ig}},$ $p({\\bf w};{\\rho })=\\prod _{j=1}^p\\prod _{l=1}^{L}{\\rho _{l}}^{w_{jl}},$ and $f({\\bf x}|{\\bf z},{\\bf w}^{\\mu },{\\bf w}^{\\Sigma };{\\Theta })=\\prod _{i=1}^N\\prod _{g=1}^G\\prod _{j=1}^d\\prod _{l=1}^{L}\\left[\\frac{1}{\\sqrt{2\\pi }\\sigma _{gl}}\\exp \\left\\lbrace -\\frac{1}{2\\sigma ^2_{gl}}(x_{ij}-\\mu _{gl})^2\\right\\rbrace \\right]^{z_{ig}w_{jl}},$ where $\\mu _{gl}$ and $\\sigma _{gl}$ are the mean and standard deviation respectively for row cluster $g$ and column cluster $l$ and ${\\Theta }$ is the set of all $\\mu _{gl}$ and $\\sigma _{gl}$ and ${\\vartheta }=({\\pi },{\\rho },{\\Theta })$ The total number of free parameters in this co-clustering model is $G+L+2(GL-1)$ , which is not dependent on the dimension, making it a very parsimonious model with only the latent variables increasing with dimension.",
"Moreover, co-clustering will still work when $N>d$ .",
"Note that are two different ways that one can view co-clustering.",
"The first is that the main goal is the clustering of rows, and the clustering of columns is solely a way to solve the problem of dimensionality.",
"However, in certain applications, the clustering of the columns might also be of interest."
],
[
"Limitations of Co-Clustering",
"Although co-clustering has advantages over other high dimensional techniques (especially in the number of parameters), the model is fairly restrictive since each observation in a block is independent and identically distributed (iid) according to a Gaussian distribution with mean $\\mu _{gl}$ and variance $\\sigma ^2_{gl}$ .",
"More flexibility is obtained by fitting more clusters in columns and lines, which is not always possible or advisable.",
"What we propose in the present work is to relax the identically distributed assumption by clustering columns according to both mean and variance.",
"This is the reason why we adopt hereafter the denomination “parameter-wise\", that we present now in detail."
],
[
"A Model to Combine Two Latent Variables in Columns",
"Recall that traditional co-clustering aims to cluster data such that observations in the same block have the same distribution.",
"An extension of traditional co-clustering for Gaussian data is now considered.",
"Like in traditional co-clustering there is a partition in rows and columns, but now there are two partitions in the columns, specifically a partition with respect to means and a partition with respect to variances.",
"Recall also that the continuous data is represented as an $n\\times p$ matrix, ${\\bf x}=(x_{ij})_{1\\le i\\le n,1\\le j\\le p}$ .",
"The partition in rows is again represented by ${\\bf z}=({\\bf z}_1,{\\bf z}_2,\\ldots ,{\\bf z}_n)$ where ${\\bf z}_i$ is distributed the same as before."
],
[
"Two Partitions in Columns",
"The partition in columns by means is represented by ${\\bf w}^{\\mu }=({\\bf w}^{\\mu }_1,{\\bf w}^{\\mu }_2,\\ldots ,{\\bf w}^{\\mu }_p)$ where ${\\bf w}^{\\mu }_j=(w^{\\mu }_{j1},w^{\\mu }_{j2},\\ldots ,w^{\\mu }_{jL^{\\mu }})\\sim \\text{multinomial}(1;{\\rho }^{\\mu })$ with ${\\rho }^{\\mu }=(\\rho ^{\\mu }_1,\\rho ^{\\mu }_2,\\ldots ,\\rho ^{\\mu }_{L^{\\mu }})$ and also the partition in columns by variances is denoted by ${\\bf w}^{\\Sigma }=({\\bf w}^{\\Sigma }_1,{\\bf w}^{\\Sigma }_2,\\ldots ,{\\bf w}^{\\Sigma }_p)$ where ${\\bf w}^{\\Sigma }_j=(w^{\\Sigma }_{j1},w^{\\Sigma }_{j2},\\ldots ,w^{\\Sigma }_{jL^{\\Sigma }})\\sim \\text{multinomial}(1;{\\rho }^{\\Sigma }),$ with ${\\rho }^{\\Sigma }=(\\rho ^{\\Sigma }_1,\\rho ^{\\Sigma }_2,\\ldots ,\\rho ^{\\Sigma }_{L^{\\Sigma }})$ These two partitions in the columns is where the main novelty lies.",
"Note that $G,L^{\\mu }$ and $L^{\\Sigma }$ are the number of clusters in rows, columns by means, and columns by variances respectively."
],
[
"Log-Likelihood",
"Using a small extension of the latent block model the observed log-likelihood is then $f({\\bf x};{\\vartheta })=\\sum _{{\\bf z}\\in Z}\\sum _{{\\bf w}^{\\mu }\\in W^{\\mu }}\\sum _{{\\bf w}^{\\Sigma }\\in W^{\\Sigma }}p({\\bf z};{\\pi })p({\\bf w}^{\\mu };{\\rho }^{\\mu })p({\\bf w}^{\\Sigma };{\\rho }^{\\Sigma })f({\\bf x}|{\\bf z},{\\bf w}^{\\mu },{\\bf w}^{\\Sigma };{\\mu },{\\Sigma })$ where $p({\\bf z};{\\pi })=\\prod _{i=1}^n\\prod _{g=1}^G\\pi _g^{z_{ig}},$ $p({\\bf w}^{\\mu };{\\rho }^{\\mu })=\\prod _{j=1}^p\\prod _{l^{\\mu }=1}^{L^{\\mu }}{(\\rho ^{\\mu }_{l^{\\mu }})}^{w^{\\mu }_{jl^{\\mu }}},$ $p({\\bf w}^{\\Sigma };{\\rho }^{\\Sigma })=\\prod _{j=1}^p\\prod _{l^{\\Sigma }=1}^{L^{\\Sigma }}{(\\rho ^{\\Sigma }_{l^{\\Sigma }})}^{w^{\\Sigma }_{jl^{\\Sigma }}},$ and $f({\\bf x}|{\\bf z},{\\bf w}^{\\mu },{\\bf w}^{\\Sigma };{\\mu },{\\Sigma })=\\prod _{i=1}^n\\prod _{g=1}^G\\prod _{j=1}^p\\prod _{l^{\\mu }=1}^{L^{\\mu }}\\prod _{l^{\\Sigma }=1}^{L^{\\Sigma }}\\left[\\frac{1}{\\sqrt{2\\pi }\\sigma _{gl^{\\Sigma }}}\\exp \\left\\lbrace -\\frac{1}{2\\sigma ^2_{gl^{\\Sigma }}}(x_{ij}-\\mu _{gl^{\\mu }})^2\\right\\rbrace \\right]^{z_{ig}w^{\\mu }_{jl^{\\mu }}w^{\\Sigma }_{jl^{\\Sigma }}}.$ In terms of notation, ${\\mu }=({\\mu }_1,{\\mu }_2,\\ldots ,{\\mu }_G)$ where ${\\mu }_g=(\\mu _{g1},\\mu _{g2},\\ldots ,\\mu _{gL^{\\mu }})$ .",
"Note that $\\mu _{gl^{\\mu }}$ is the mean for row cluster $g$ and column cluster by means $l^{\\mu }$ .",
"Likewise, ${\\Sigma }=({\\Sigma }_1,{\\Sigma }_2,\\ldots ,{\\Sigma }_G)$ where ${\\Sigma }_{g}=(\\sigma _{g1}^2,\\sigma _{g2}^2,\\ldots ,\\sigma _{gL^{\\Sigma }}^2)$ and $\\sigma _{gl^{\\Sigma }}^2$ is the variance for row cluster $g$ and column cluster by variances $l^{\\Sigma }$ .",
"Finally, the complete-data log-likelihood is $\\begin{split}p({\\bf x},{\\bf z},{\\bf w}^{\\mu },{\\bf w}^{\\Sigma };{{\\vartheta }})=C+\\sum _{i=1}^N\\sum _{g=1}^G&z_{ig}\\log \\pi _g+\\sum _{j=1}^d\\sum _{l^{\\mu }=1}^{L^{\\mu }}w^{\\mu }_{jl^{\\mu }}\\log \\rho ^{\\mu }_{l^{\\mu }}+\\sum _{j=1}^d\\sum _{l^{\\Sigma }=1}^{L^{\\Sigma }}w^{\\Sigma }_{jl^{\\Sigma }}\\log \\rho ^{\\Sigma }_{l^{\\Sigma }}\\\\&-\\frac{1}{2}\\sum _{i=1}^N\\sum _{g=1}^G\\sum _{j=1}^d\\sum _{l^{\\mu }=1}^{L^{\\mu }}\\sum _{l^{\\Sigma }=1}^{L^{\\Sigma }}z_{ig}w^{\\mu }_{jl^{\\mu }}w^{\\Sigma }_{jl^{\\Sigma }}\\left[\\log \\sigma ^2_{gl^{\\Sigma }}+\\frac{(x_{ij}-\\mu _{gl^{\\mu }})^2}{\\sigma ^2_{gl^{\\Sigma }}}\\right]\\end{split}$ where $C$ is a constant with respect to the parameters and ${\\vartheta }=({\\pi },{\\rho }^{\\mu },{\\rho }^{\\Sigma },{\\mu },{\\Sigma })$ .",
"From this point on, we will refer to this model as non identically distributed (non-id) co-clustering."
],
[
"Number of Free Parameters",
"In the non-id co-clustering model the number of free parameters is $\\begin{split}{\\text{\\#}Param}_{\\text{new coclust}}&=G-1+L^{\\mu }-1+L^{\\Sigma }-1+GL^{\\mu }+GL^{\\Sigma }\\\\&=G+(L^{\\mu }+L^{\\Sigma })(G+1)-3.\\end{split}$ Therefore, like in traditional co-clustering, the number of free parameters is not dependent on the dimensionality of the data."
],
[
"Parameter Estimation Using the SEM Gibbs Algorithm",
"The SEM algorithm after initialization at iteration $q$ proceeds as follows.",
"SE Step: Generate the row partition ${\\bf z}^{(q+1)}$ according to $P(z_{ig}=1|{\\bf x},{{\\bf w}^{\\mu }}^{(q)},{{\\bf w}^{\\Sigma }}^{(q)};{\\mu }^{(q)},{\\Sigma }^{(q)},{\\pi }^{(q)})=\\frac{\\pi _g^{(q)}f({\\bf x}_i|{{\\bf w}^{\\mu }}^{(q)},{{\\bf w}^{\\Sigma }}^{(q)};{\\mu }_g^{(q)},{\\Sigma }_g^{(q)})}{\\sum _{g^{\\prime }}^G\\pi _{g^{\\prime }}^{(q)}f({\\bf x}_i|{{\\bf w}^{\\mu }}^{(q)},{{\\bf w}^{\\Sigma }}^{(q)};{\\mu }_{g^{\\prime }}^{(q)},{\\Sigma }_{g^{\\prime }}^{(q)})}$ where $f({\\bf x}_i|{{\\bf w}^{\\mu }}^{(q)},{{\\bf w}^{\\Sigma }}^{(q)};{\\mu }_g^{(q)},{\\Sigma }_g^{(q)})=\\prod _{j=1}^d\\prod _{l^{\\mu }=1}^{L^{\\mu }}\\prod _{l^{\\Sigma }=1}^{L^{\\Sigma }}\\left[\\frac{1}{\\sqrt{2\\pi }\\sigma ^{(q)}_{gl^{\\Sigma }}}\\exp \\left\\lbrace -\\frac{1}{2{\\sigma ^2}^{(q)}_{gl^{\\Sigma }}}(x_{ij}-\\mu ^{(q)}_{gl^{\\mu }})^2\\right\\rbrace \\right]^{{w^{\\mu }}^{(q)}_{jl^{\\mu }}{w^{\\Sigma }}^{(q)}_{jl^{\\Sigma }}}.$ Generate the column partition by means ${{\\bf w}^{\\mu }}^{(q+1)}$ according to $P(w^{\\mu }_{jl^{\\mu }}=1|{\\bf x},{{\\bf z}}^{(q+1)},{{\\bf w}^{\\Sigma }}^{(q)};{\\mu }^{(q)},{\\Sigma }^{(q)},{{\\rho }^{\\mu }}^{(q)})=\\frac{{\\rho ^{\\mu }}^{(q)}_{l^{\\mu }} f({\\bf x}_{\\cdot j}|{{\\bf z}}^{(q+1)},{{\\bf w}^{\\Sigma }}^{(q)};{\\mu }_{l^{\\mu }}^{(q)},{\\Sigma }^{(q)})}{\\sum _{{l^{\\mu }}^{\\prime }}^{L^{\\mu }}{\\rho ^{\\mu }}^{(q)}_{{l^{\\mu }}^{\\prime }} f({\\bf x}_{\\cdot j}|{{\\bf z}}^{(q+1)},{{\\bf w}^{\\Sigma }}^{(q)};{\\mu }_{{l^{\\mu }}^{\\prime }}^{(q)},{\\Sigma }^{(q)})}$ where ${\\bf x}_{\\cdot j}=(x_{1j},x_{2j},\\ldots ,x_{Nj})$ , ${\\mu }_{l^{\\mu }}^{(q)}=(\\mu _{1l^{\\mu }}^{(q)},\\mu _{2l^{\\mu }}^{(q)},\\ldots ,\\mu _{Gl^{\\mu }}^{(q)})$ and $f({\\bf x}_{\\cdot j}|{{\\bf z}}^{(q+1)},{{\\bf w}^{\\Sigma }}^{(q)};{\\mu }_{l^{\\mu }}^{(q)},{\\Sigma }^{(q)})=\\prod _{i=1}^N\\prod _{g=1}^G\\prod _{l^{\\Sigma }=1}^{L^{\\Sigma }}\\left[\\frac{1}{\\sqrt{2\\pi }\\sigma ^{(q)}_{gl^{\\Sigma }}}\\exp \\left\\lbrace -\\frac{1}{2{\\sigma ^2}^{(q)}_{gl^{\\Sigma }}}(x_{ij}-\\mu ^{(q)}_{gl^{\\mu }})^2\\right\\rbrace \\right]^{z_{ig}^{(q+1)}{w^{\\Sigma }}^{(q)}_{jl^{\\Sigma }}}.$ Generate the column partition by variances ${{\\bf w}^{\\Sigma }}^{(q+1)}$ according to $P(w^{\\Sigma }_{jl^{\\Sigma }}=1|{\\bf x},{{\\bf z}}^{(q+1)},{{\\bf w}^{\\mu }}^{(q+1)};{\\mu }^{(q)},{\\Sigma }^{(q)},{{\\rho }^{\\Sigma }}^{(q)})=\\frac{{\\rho ^{\\Sigma }}^{(q)}_{l^{\\Sigma }} f({\\bf x}_{\\cdot j}|{{\\bf z}}^{(q+1)},{{\\bf w}^{\\mu }}^{(q+1)};{\\mu }^{(q)},{\\Sigma }_{l^{\\Sigma }}^{(q)})}{\\sum _{{l^{\\Sigma }}^{\\prime }}^{L^{\\Sigma }}{\\rho ^{\\Sigma }}^{(q)}_{{l^{\\Sigma }}^{\\prime }} f({\\bf x}_{\\cdot j}|{{\\bf z}}^{(q+1)},{{\\bf w}^{\\mu }}^{(q+1)};{\\mu }^{(q)},{\\Sigma }_{{l^{\\Sigma }}^{\\prime }}^{(q)})}$ where ${\\Sigma }_{l^{\\Sigma }}^{(q)}=({\\sigma ^2}_{1l^{\\Sigma }}^{(q)},{\\sigma ^2}_{2l^{\\Sigma }}^{(q)},\\ldots ,{\\sigma ^2}_{Gl^{\\Sigma }}^{(q)})$ and $f({\\bf x}_{\\cdot j}|{{\\bf z}}^{(q+1)},{{\\bf w}^{\\mu }}^{(q+1)};{\\mu }^{(q)},{\\Sigma }_{l^{\\Sigma }}^{(q)})=\\prod _{i=1}^N\\prod _{g=1}^G\\prod _{l^{\\mu }=1}^{L^{\\mu }}\\left[\\frac{1}{\\sqrt{2\\pi }\\sigma ^{(q)}_{gl^{\\Sigma }}}\\exp \\left\\lbrace -\\frac{1}{2{\\sigma ^2}^{(q)}_{gl^{\\Sigma }}}(x_{ij}-\\mu ^{(q)}_{gl^{\\mu }})^2\\right\\rbrace \\right]^{z_{ig}^{(q+1)}{w^{\\mu }}^{(q+1)}_{jl^{\\mu }}}.$ M Step: Update the parameters according to $\\pi _{g}^{(q+1)}=\\frac{\\sum _{i=1}^nz_{ig}^{(q+1)}}{n}, \\qquad {\\rho ^{\\mu }_{l^{\\mu }}}^{(q+1)}=\\frac{\\sum _{j=1}^p{w^{\\mu }_{jl^{\\mu }}}^{(q+1)}}{p}, \\qquad {\\rho ^{\\Sigma }_{l^{\\Sigma }}}^{(q+1)}=\\frac{\\sum _{j=1}^p{w^{\\Sigma }_{jl^{\\Sigma }}}^{(q+1)}}{p},$ $\\begin{split}\\mu _{gl^{\\mu }}^{(q+1)}&=\\frac{\\sum _{i=1}^n\\sum _{j=1}^p\\sum _{l^{\\Sigma }=1}^{L^{\\Sigma }}z_{ig}^{(q+1)}{w^{\\mu }_{jl^{\\mu }}}^{(q+1)}{w^{\\Sigma }_{jl^{\\Sigma }}}^{(q+1)}x_{ij}}{\\sum _{i=1}^n\\sum _{j=1}^p\\sum _{l^{\\Sigma }=1}^{L^{\\Sigma }}z_{ig}^{(q+1)}{w^{\\mu }_{jl^{\\mu }}}^{(q+1)}{w^{\\Sigma }_{jl^{\\Sigma }}}^{(q+1)}}\\\\ \\\\&=\\frac{\\sum _{i=1}^n\\sum _{j=1}^pz_{ig}^{(q+1)}{w^{\\mu }_{jl^{\\mu }}}^{(q+1)}x_{ij}}{\\sum _{i=1}^N\\sum _{j=1}^pz_{ig}^{(q+1)}{w^{\\mu }_{jl^{\\mu }}}^{(q+1)}},\\end{split}$ ${\\sigma ^2_{gl^{\\Sigma }}}^{(q+1)}=\\frac{\\sum _{i=1}^n\\sum _{j=1}^p\\sum _{l^{\\mu }=1}^{L^{\\mu }}z_{ig}^{(q+1)}{w^{\\mu }_{jl^{\\mu }}}^{(q+1)}{w^{\\Sigma }_{jl^{\\Sigma }}}^{(q+1)}(x_{ij}-\\mu _{gl^{\\mu }}^{(q+1)})^2}{\\sum _{i=1}^n\\sum _{j=1}^p\\sum _{l^{\\mu }=1}^{L^{\\mu }}z_{ig}^{(q+1)}{w^{\\mu }_{jl^{\\mu }}}^{(q+1)}{w^{\\Sigma }_{jl^{\\Sigma }}}^{(q+1)}}.$ After a burn in period of the algorithm, the estimates of each of the parameters is just the mean of the runs of the SEM algorithm (the number of runs will be assessed experimentally in Section 4).",
"We denote these final estimates by $\\hat{{\\vartheta }}=(\\hat{{\\pi }},\\hat{{\\rho }^{\\mu }},\\hat{{\\rho }^{\\Sigma }},\\hat{{\\mu }},\\hat{{\\Sigma }})$ For the final partition of rows, columns by means, and columns by variances, we fix the parameters at their estimates, and run more iterations of the SE step.",
"The average of these partitions over these additional runs is calculated and the partition is then taken to be the maximum a posteriori estimates.",
"For our simulations and real data analysis, we took 20 such runs to obtain the final partitions $\\hat{{\\bf z}},\\hat{{\\bf w}^{\\mu }}$ , and $\\hat{{\\bf w}^{\\Sigma }}$ ."
],
[
"ICL-BIC",
"In the clustering scenario, the number of clusters in rows and columns by both means and variances is not known and therefore a model selection criterion is required.",
"Like in traditional co-clustering, the observed log-likelihood is intractable and therefore the Bayesian information criterion cannot be used.",
"Therefore, we propose using the integrated complete log-likelihood which relies on the complete data log-likelihood instead of the observed log-likelihood.",
"This criterion will be called ICL-BIC, similar to that used in and it is given by $\\text{ICL--BIC}=p({\\bf x},\\hat{{\\bf z}},\\hat{{\\bf w}^{\\mu }},\\hat{{\\bf w}^{\\Sigma }};\\hat{{\\vartheta }})-\\frac{G-1}{2}\\log N-\\frac{L^{\\mu }+L^{\\Sigma }-2}{2}\\log d-\\frac{G(L^{\\mu }+L^{\\Sigma })}{2}\\log Nd.$ From the property proven by , the BIC and ICL-BIC exhibit the same behaviour for large values of $n$ and/or $p$ , thus the number of blocks chosen by this criterion is consistent (under some conditions not mentioned here)."
],
[
"Search Algorithm",
"Because an extra layer of complexity, namely the two different partitions of the columns, is introduced with the non-id model, it may be take a very long time to perform an exhaustive search of all possible combinations of $G,L^{\\mu }$ and $L^{\\Sigma }$ in a pre-defined range.",
"This has been discussed in the literature, namely .",
"Specifically, the start values are set to $(G,L^{\\mu },L^{\\Sigma })=(G_1,L^{\\mu }_1,L^{\\Sigma }_1)$ .",
"We then fit three models with parameters $(G_1+1,L^{\\mu },L^{\\Sigma })$ , $(G_1,L^{\\mu }+1,L^{\\Sigma })$ and $(G_1,L^{\\mu },L^{\\Sigma }+1)$ .",
"The set with the highest ICL is retained and we get the set $(G_2,L^{\\mu }_2,L^{\\Sigma }_2)$ .",
"The procedure is then repeated until a maximum threshold is reached for these parameters or the ICL no longer increases."
],
[
"Algorithm and Parameter Estimation Evaluation",
"To evaluate the algorithm, we performed two simulations with different degrees of separation between the blocks.",
"The purpose of these simulations was to look at the number of iterations of the SEM algorithm that is needed for parameter estimation and classification performance."
],
[
"Simulation 1",
"The first simulation had good separation between the blocks.",
"We took $N=1000$ , $d=100$ , $G=3$ , $L^{\\mu }=2$ , $L^{\\Sigma }=3$ .",
"The means and variances were respectively taken to be ${\\mu }=\\left(\\begin{array}{ll}1 &-1\\\\2 & -2\\\\3 & -3\\\\\\end{array}\\right), \\qquad {\\Sigma }=\\left(\\begin{array}{ccc}1&0.5 &0.75\\\\2 & 1.75 & 0.25\\\\1.5 & 2.25 & 2.5\\\\\\end{array}\\right).$ Note that each cell of the matrices corresponds to the parameter for the corresponding cluster in lines and in columns.",
"So for example, the first row first column element of ${\\mu }$ represents the mean parameter for the first cluster in lines and the first cluster in columns according to the mean.",
"The mixing proportions were taken to be ${\\pi }=(0.3,0.3,0.4), \\qquad {\\rho }^{\\mu }=(0.4,0.6), \\qquad {\\rho }^{\\Sigma }=(0.3,0.3,0.4).$ 50 datasets were simulated according to these parameters.",
"A burn-in of 20 iterations of the SEM-Gibbs algorithm was used, followed by 100 iterations, followed by 20 iterations of the SE step for the final partitions.",
"In Table REF , we display the average errors of the estimates, up to a label switch.",
"We measure the mean estimate quality by $\\Delta {\\mu }=\\sum _{g,l^{\\mu }}|\\hat{\\mu }_{gl^{\\mu }}-\\mu _{gl^{\\mu }}|$ and similarly for the other parameters.",
"The errors are pretty small indicating that the parameter estimates from the algorithm are generally close to the true values.",
"Table REF displays the average ARI for the row, column by means and column by variance partitions.",
"Notice that the classification is perfect for both partitions by columns for all 50 datasets.",
"Moreover, the average ARI for the rows is also very high.",
"Table: Average error (and standard deviation) of the estimates over the 50 datasets for Simulation 1.Table: Average ARI (and standard deviation) for the rows (ARI r \\text{ARI}_r), column by means (ARI cμ \\text{ARI}_{c \\mu }), and column by variance (ARI cΣ \\text{ARI}_{c \\Sigma }) partitions over the 50 datasets for Simulation 1.In Figure REF , we show the progression of the parameter estimates over the course of the SEM-Gibbs algorithm for one of our datasets (all other datasets exhibit similar behaviour).",
"Notice that after a burn in period of 20, we obtain a very stable chain.",
"Figure: Simulation 1 SEM algorithm estimation progression.Finally, in Figure REF , we display the true clustering result (which in this case is equivalent to the estimate clustering result) for one of the 50 datasets.",
"In the co-clustering by means panel, the co-clustering results for the row partition and the column partition by means is shown.",
"The co-clustering by variances shows the co-clustering results for the row partition and the column partition by variances.",
"Finally, the combined co-clustering looks at all combinations of $l^{\\mu }$ and $l^{\\Sigma }$ and then organizes the columns so that each column in the same cluster have the same mean and variance as in regular co-clustering.",
"Specifically as seen here, the first cluster in columns would be those columns that were partitioned into cluster 1 for the means and cluster 1 for the variances, the second cluster would be those clustered into cluster 1 for the means and cluster 2 for the variances and so on.",
"In a general scenario, this would correspond to a maximum of $L^{\\mu }L^{\\Sigma }$ clusters in columns thus allowing more flexibility but not greatly increasing the number of parameters.",
"It is important to note, however, that there may be cases, as we will see with the real dataset, where no columns would be clustered into a particular pair of $l^{\\mu }$ and $l^{\\Sigma }$ and thus the combined co-clustering results might have fewer $L^{\\mu }L^{\\Sigma }$ clusters but never more.",
"From this figure it is clear that the blocks are well separated.",
"Figure: True co-clustering for one dataset from Simulation 1 which is the same as the predicted co-clustering results for this dataset.In Simulation 2, less separation between groups was considered.",
"This time we considered $n=200$ , $p=500$ , $G=3$ , $L^{\\mu }=3$ , $L^{\\Sigma }=2$ and took the means and variances to be ${\\mu }=\\left(\\begin{array}{lll}1 &1.25&0\\\\2 & 1.2&1\\\\1.5 & 1.9&0.5\\\\\\end{array}\\right), \\qquad {\\Sigma }=\\left(\\begin{array}{cc}1&0.5\\\\2 & 1.75\\\\1.5 & 2.25\\\\\\end{array}\\right),$ and the mixing proportions were ${\\pi }=(0.3,0.3,0.4), \\qquad {\\rho }^{\\mu }=(0.3,0.5,0.2), \\qquad {\\rho }^{\\Sigma }=(0.4,0.6).$ The algorithm was performed in the same way as before.",
"In Table REF we show the average error of the estimates with their standard deviations like before.",
"The ARI results are also shown in Table REF .",
"Figure REF shows the progression of the estimates.",
"We see this time that although it is not as flat as that seen in Simulation 1, we still obtain a fairly stable chain after a burn in of around 20 iterations of the SEM algorithm.",
"Finally, we display the true co-clustered data like before in Figure REF .",
"We see here that unlike in the first simulation, there is little separation between blocks.",
"Table: Average error (and standard deviation) of the estimates over the 50 datasets for Simulation 2.Table: Average ARI (and standard deviation) for the rows (ARI r \\text{ARI}_r), column by means (ARI cμ \\text{ARI}_{c \\mu }), and column by variance (ARI cΣ \\text{ARI}_{c \\Sigma }) partitions over the 50 datasets for Simulation 2.Figure: Simulation 2 SEM algorithm estimation progression.Figure: True co-clustering for one dataset from Simulation 2 which is the same as the predicted co-clustering results for this dataset."
],
[
"ICL-BIC Selection Criterion",
"In this simulation, we were interested in the performance of the ICL-BIC criterion selecting the correct number of partitions when using an exhaustive search.",
"Here we considered 2000 observations with $p=500$ .",
"We also choose 3 groups in lines, columns by means and columns by variances.",
"The parameters were ${\\mu }=\\left(\\begin{array}{lll}1 &1.25&0\\\\2 & 1.2&1\\\\1.5 & 1.9&0.5\\\\\\end{array}\\right), \\qquad {\\Sigma }=\\left(\\begin{array}{ccc}1&0.5&0.25\\\\2 & 1.75&0.5\\\\1.5 & 2.25&1\\\\\\end{array}\\right),$ and ${\\pi }=(0.3,0.3,0.4), \\qquad {\\rho }^{\\mu }=(0.3,0.4,0.3), \\qquad {\\rho }^{\\Sigma }=(0.4,0.3,0.3).$ An exhaustive search was performed considering each combination with $G,L^{\\mu },L^{\\Sigma }\\in \\lbrace 2,3,4\\rbrace $ .",
"In Table REF , we display the number of times each number of partitions was chosen by the ICL-BIC.",
"Notice that except for 1 dataset for clusters in lines, and 2 for clusters in columns for means and variances, the correct number of clusters was chosen for all 50 datasets.",
"Table: Frequency of the number of partitions chosen by the ICL-BIC over the 50 simulated datasets."
],
[
"Search Algorithm Evaluation",
"The last simulation looked at the non-exhaustive search algorithm described in Section 3.3.",
"There were 3 clusters in rows and columns by variances and 4 in columns by means.",
"The parameters were taken to be ${\\mu }=\\left(\\begin{array}{cccc}1& -0.25 & 0.3& -1\\\\1.25& 0& 0.1& -0.3\\\\0.5& -1& 0& 0.1\\\\\\end{array}\\right), \\qquad {\\Sigma }=\\left(\\begin{array}{ccc}1&0.5&0.25\\\\2 & 1.75&0.5\\\\1.5 & 2.25&1\\\\\\end{array}\\right),$ and ${\\pi }=(0.3,0.3,0.4), \\qquad {\\rho }^{\\mu }=(0.2,0.3,0.25,0.25), \\qquad {\\rho }^{\\Sigma }=(0.5,0.25,0.25).$ The procedure was performed for 25 datasets and with initial values of $(G_1,L^{\\mu }_1,L^{\\Sigma }_1)=(1,1,1)$ and the maximum values for all three were set to 5.",
"In Table REF we show the number of times each group was chosen using this non-exhaustive procedure.",
"We see that the procedure performs quite well with choosing the correct number of groups.",
"Table: Frequency of the number of clusters chosen by the ICL-BIC over the 25 simulated datasets when using the non-exhaustive search method.Figure: Traditional co-clustering results for the Jokes data.Figure: Non-Id co-clustering results for the Jokes data.Figure: ICL-BIC results for traditional and non-id co-clustering when using the exhaustive search algorithm for the Jokes data."
],
[
"Real Data Analysis",
"We consider the Jester dataset used by and use this to compare the non-id co-clustering method with traditional co-clustering.",
"Users gave 100 jokes a continuous rating from -10 to +10.",
"A total of 7200 users rated all 100 jokes.",
"We took a random sample of 2000 to make up our dataset.",
"The non-id model was fitted for 1 to 25 groups in lines and 1 to 7 groups for means and variances in columns.",
"Traditional co-clustering was performed for 1 to 25 groups in lines and 1 to 10 groups in columns.",
"When the non-exhaustive search algorithm was used the model selected using traditional co-clustering had 7 groups in rows and 3 groups in columns, leading to 50 estimated parameters.",
"The resulting ICL-BIC was -569487.",
"For non-id co-clustering, the final model had 17 groups in rows, 6 groups in columns by means and 4 groups in columns by variances with a total of 194 estimated parameters.",
"The resulting ICL-BIC was -561099 and the total number of groups in the combined co-clustering was 15.",
"In Figure REF and REF we show the visualization of the traditional co-clustering results and new method co-clustering results respectively.",
"Due to the ICL-BIC choosing more groups in lines for the new method we propose, we can see that there is more heterogeneity in the users than was seen when using traditional co-clustering.",
"For example, the 6th and 7th groups in lines from the top appear to be individuals who consistently rated all jokes high or all jokes low with very little variation.",
"Moreover, the group at the very bottom appears to be individuals who consistently ranked the jokes in the middle.",
"Although the traditional co-clustering seems to separate individuals in a similar fashion (groups 2, 3 and 5 from the top), there is clearly more variability.",
"It is also interesting to point out that for the non-id model a different number of groups in columns for means and variances were chosen again displaying the increased flexibility of the non-id model.",
"Finally, this very large increase in flexibility is obtained without a drastic increase in the number of estimated parameters (a difference of 144) and still obtaining a higher ICL-BIC than traditional co-clustering.",
"It is important to note that this example also illustrates the necessity of considering the visualization of the means and the variances separately as well as the combined co-clustering results when using the non-id method.",
"In the combined co-clustering visualization, it is a little difficult to see some of the clusters in columns because the number of columns in each cluster is very small; however, the two separate co-clusterings by means and variances in the second row provide a clearer visual.",
"In this particular application the interpretation for these two additional visuals is a little difficult since there isn't much information about the jokes, in an application such as gene expression data, Finally, in Figure REF , we display a plot of the highest ICL-BIC over all $L$ for traditional co-clustering and $L^{\\mu }$ and $L^{\\Sigma }$ for non-id co-clustering against $G$ (the number of groups in rows) when using the exhaustive search algorithm.",
"Although not displayed here, the values of $L$ for traditional and $L^{\\mu }$ and $L^{\\Sigma }$ for non-id co-clustering that resulted in the best ICL have a lot of variation.",
"Moreover, for traditional co-clustering we can see from the graph that there is a fair amount of variation in the ICL-BIC once $G$ approaches 10, and for non-iid co-clustering this occurs when $G$ is approximately 15.",
"Therefore, when using the exhaustive search algorithm, it can be difficult to choose the number of groups in lines and also in columns for both traditional and non-iid co-clustering.",
"Moreover, it is very computationally expensive to run the exhaustive search with the non-iid co-clustering taking a little over 24 hours using 25 1200MHz cores running continuously."
],
[
"Discussion",
"In this paper, we developed a co-clustering model that allowed more flexibility without greatly increasing the number of parameters by partitioning the columns of a continuous data matrix by both mean and variances.",
"This in effect relaxes the identically distributed assumption of traditional co-clustering.",
"Parameter estimation was carried out by the SEM-Gibbs algorithm and an ICL criterion was used to choose the number of blocks.",
"In addition, due to the increased number of possibilities when choosing the number of blocks, we proposed a non-exhaustive search method that was shown to perform well for both simulated and real data.",
"When analyzing the jokes dataset, the non-id method displayed three main advantages to traditional co-clustering.",
"The first is the increase in flexibility while still maintaining parsimony.",
"Specifically, we obtained a total of 15 groups in columns using the non-id method, compared to 3 when using traditional co-clustering resulting in an additional 144 estimated parameters.",
"This difference in parameters was also based on 17 groups in rows for non-id co-clustering verses 7 for traditional co-clustering.",
"It is worth noting that if $G$ was kept constant at 7 between the two methods with the same partitions in columns, there would only be an additional 34 parameters when using non-id co-clustering.",
"The non-id method also displayed easier interpretation.",
"Because the number of groups in columns according to means and variances are less numerous than the total number of groups in columns in the combined co-clustering, and traditional co-clustering when increasing $L$ , looking at the partitions according to means and variances offer a simplified visualization.",
"This was particularly evident in the jokes dataset as the two separate partitions offered a simplified visualization when compared to the combined co-clustering with 15 groups in columns.",
"Finally, a better ICL-BIC was achieved.",
"Although this method dealt with continuous data using the Gaussian distribution, it can be extended in various ways.",
"One example would be to use different distributions with more than one parameter.",
"For example, one could consider using distributions that account for tail weight, such as the $t$ -distribution, or skewness and tail weight like the skew-$t$ , variance gamma, or generalized hyperbolic distributions.",
"In these cases, one could take a partition in the columns according to location, scale, concentration and skewness.",
"This could also be applied to data that is not continuous like ordinal data where the columns could be partitioned according to mode and precision.",
"Again, the number of parameters in each of these cases will not depend on the dimensionality of the data thus preserving the parsimony that is inherent to co-clustering."
]
] | 1808.08366 |
[
[
"Analysis of the Morley element for the Cahn-Hilliard equation and the\n Hele-Shaw flow"
],
[
"Abstract The paper analyzes the Morley element method for the Cahn-Hilliard equation.",
"The objective is to derive the optimal error estimates and to prove the zero-level sets of the Cahn-Hilliard equation approximate the Hele-Shaw flow.",
"If the piecewise $L^{\\infty}(H^2)$ error bound is derived by choosing test function directly, we cannot obtain the optimal error order, and we cannot establish the error bound which depends on $\\frac{1}{\\epsilon}$ polynomially either.",
"To overcome this difficulty, this paper proves them by the following steps, and the result in each next step cannot be established without using the result in its previous one.",
"First, it proves some a priori estimates of the exact solution $u$, and these regularity results are minimal to get the main results; Second, it establishes ${L^{\\infty}(L^2)}$ and piecewise ${L^2(H^2)}$ error bounds which depend on $\\frac{1}{\\epsilon}$ polynomially based on the piecewise ${L^{\\infty}(H^{-1})}$ and ${L^2(H^1)}$ error bounds; Third, it establishes piecewise ${L^{\\infty}(H^2)}$ optimal error bound which depends on $\\frac{1}{\\epsilon}$ polynomially based on the piecewise ${L^{\\infty}(L^2)}$ and ${L^2(H^2)}$ error bounds; Finally, it proves the ${L^\\infty(L^\\infty)}$ error bound and the approximation to the Hele-Shaw flow based on the piecewise ${L^{\\infty}(H^2)}$ error bound.",
"The nonstandard techniques are used in these steps such as the generalized coercivity result, integration by part in space, summation by part in time, and special properties of the Morley elements.",
"If one of these techniques is lacked, either we can only obtain the sub-optimal piecewise ${L^{\\infty}(H^2)}$ error order, or we can merely obtain the error bounds which are exponentially dependent on $\\frac{1}{\\epsilon}$.",
"Numerical results are presented to validate the optimal $L^\\infty(H^2)$ error order and the asymptotic behavior of the solutions of the Cahn-Hilliard equation."
],
[
"Introduction",
"Consider the following Cahn-Hilliard equation with Neumann boundary conditions: $u_t +\\Delta (\\epsilon \\Delta u -\\frac{1}{\\epsilon }f(u)) &=0 &&\\quad \\mbox{in } \\Omega _T:=\\Omega \\times (0,T],\\\\\\frac{\\partial u}{\\partial n}=\\frac{\\partial }{\\partial n}(\\epsilon \\Delta u-\\frac{1}{\\epsilon }f(u))&=0 &&\\quad \\mbox{on } \\partial \\Omega _T:=\\partial \\Omega \\times (0,T],\\\\u &=u_0 &&\\quad \\mbox{in } \\Omega \\times \\lbrace t=0\\rbrace ,$ where $\\Omega \\subseteq \\mathbf {R}^2$ is a bounded domain, $f(u) = u^3 - u$ is the derivative of a double well potential $F(u)$ which is defined by $F(u)=\\frac{1}{4}(u^2-1)^2.$ The Allen-Cahn equation [3], [6], [12], [20], [17], [16], [19], [24] and the Cahn-Hilliard equation [2], [12], [25], [29] are two basic phase field models to describe the phase transition process.",
"They are also proved to be related to geometric flow.",
"For example, the zero-level sets of the Allen-Cahn equation approximate the mean curvature [15], [24] and the zero-level sets of the Cahn-Hilliard equation approximate the Hele-Shaw flow [28], [2].",
"The Cahn-Hilliard equation was introduced by J. Cahn and J. Hilliard in [11] to describe the process of phase separation, by which the two components of a binary fluid separate and form domains pure in each component.",
"It can be interpreted as the $H^{-1}$ gradient flow [2] of the Cahn-Hilliard energy functional $J_\\epsilon (v):= \\int _\\Omega \\Bigl ( \\frac{\\epsilon }{2} |\\nabla v|^2+\\frac{1}{\\epsilon } F(v) \\Bigr )\\, {\\rm d}x.$ There are a few papers [4], [30], [13], [14] discussing the error bounds, which depend on the exponential power of $\\frac{1}{\\epsilon }$ , of the numerical methods for Cahn-Hilliard equation.",
"Such an estimate is clearly not useful for small $\\epsilon $ , in particular, in addressing the issue whether the computed numerical interfaces converge to the original sharp interface of the Hele-Shaw problem.",
"Instead, the polynomial dependence in $\\frac{1}{\\epsilon }$ is proved in [21], [22] using the standard finite element method, and in [18], [26] using the discontinuous Galerkin method.",
"Due to the high efficiency of the Morley elements, compared with mixed finite element methods or $C^1$ -conforming finite element methods, the Morley finite element method is used to derive the error bound which depends on $\\frac{1}{\\epsilon }$ polynomially in this paper.",
"The highlights of this paper are fourfold.",
"First, it establishes the piecewise ${L^{\\infty }(L^2)}$ and ${L^2(H^2)}$ error bounds which depend on $\\frac{1}{\\epsilon }$ polynomially.",
"If the standard technique is used, we can only prove that the error bounds depend on $\\frac{1}{\\epsilon }$ exponentially, which can not be used to prove our main theorem.",
"To prove these bounds, special properties of the Morley elements are explored, i.e., Lemma 2.3 in [14], and piecewise ${L^{\\infty }(H^{-1})}$ and ${L^2(H^1)}$ error bounds [27] are required.",
"Second, by making use of the piecewise ${L^{\\infty }(L^2)}$ and ${L^2(H^2)}$ error bounds above, it establishes the piecewise ${L^{\\infty }(H^2)}$ error bound which depends on $\\frac{1}{\\epsilon }$ polynomially.",
"If the standard technique is used, we can only get the error bound in Remark REF , which does not have an optimal order.",
"The crux here is to employ the summation by part in time and integration by part in space techniques simultaneously to handle the nonlinear term, together with the special properties of the Morley elements.",
"Third, the minimal regularity of $u$ is used, i.e., $\\Vert u_{tt}\\Vert _{L^2(L^2)}$ regularity instead of $\\Vert u_{tt}\\Vert _{L^\\infty (L^2)}$ regularity is used, and the a priori estimate is derived in Theorem .",
"Fourth, the ${L^\\infty (L^\\infty )}$ error bound is established using the optimal piecewise ${L^{\\infty }(H^2)}$ error, by which the main result that the zero-level sets of the Cahn-Hilliard equation approximate the Hele-Shaw flow is proved in Section .",
"The organization of this paper is as follows.",
"In Section , the standard Sobolev space notation is introduced, some useful lemmas are stated, and a new a priori estimate of the exact solution $u$ is derived.",
"In Section , the fully discrete approximation based on the Morley finite element space is presented.",
"In Section , first the polynomially dependent piecewise ${L^{\\infty }(L^2)}$ and ${L^2(H^2)}$ error bounds are established based on piecewise ${L^{\\infty }(H^{-1})}$ and ${L^2(H^1)}$ error bounds, then the polynomially dependent piecewise ${L^{\\infty }(H^2)}$ error bound is established based on piecewise ${L^{\\infty }(L^2)}$ and ${L^2(H^2)}$ error bounds, by which the ${L^\\infty (L^\\infty )}$ error bound is proved.",
"In Section , the approximation of the zero-level sets of the Cahn-Hilliard equation of the Hele-Shaw flow is proved.",
"In Section , numerical tests are presented to validate our theoretical results, including the optimal error orders and the approximation of the Hele-Shaw flow."
],
[
"Preliminaries",
"In this section, we present some results which will be used in the following sections.",
"Throughout this paper, $C$ denotes a generic positive constant which is independent of interfacial length $\\epsilon $ , spacial size $h$ , and time step size $k$ , and it may have different values in different formulas.",
"The standard Sobolev space notation below is used in this paper.",
"$\\Vert v\\Vert _{0,p,A}&=\\bigg (\\int _{A}|v|^p\\,{\\rm d}x\\bigg )^{1\\slash p}\\qquad &&1\\le p<\\infty ,\\\\\\Vert v\\Vert _{0,\\infty ,A}&=\\underset{A}{\\mbox{\\rm ess sup }} |v|,\\\\|v|_{m,p,A}&=\\bigg (\\sum _{|\\alpha |=m}\\Vert D^{\\alpha }v\\Vert _{0,p,A}^p\\bigg )^{1\\slash p}\\qquad &&1\\le p<\\infty ,\\\\\\Vert v\\Vert _{m,p,A}&=\\bigg (\\sum _{j=0}^m|v|_{m,p,A}^p\\bigg )^{1\\slash p}.$ Here $A$ denotes some domain, i.e., a single mesh element $K$ or the whole domain $\\Omega $ .",
"When $A=\\Omega $ , $\\Vert \\cdot \\Vert _{H^k},\\Vert \\cdot \\Vert _{L^k}$ are used to denote $\\Vert \\cdot \\Vert _{H^k(\\Omega )},\\Vert \\cdot \\Vert _{L^k(\\Omega )}$ respectively, and $\\Vert \\cdot \\Vert _{0,2}$ is also used to denote $\\Vert \\cdot \\Vert _{L^2(\\Omega )}$ .",
"Let $\\mathcal {T}_h$ be a family of quasi-uniform triangulations of domain $\\Omega $ , and $\\mathcal {E}_h$ be a collection of edges, then the global mesh dependent semi-norm, norm and inner product are defined below $|v|_{j,p,h}&=\\bigg (\\sum _{K\\in \\mathcal {T}_h}|v|_{j,p,K}^p\\bigg )^{1\\slash p},\\\\\\Vert v\\Vert _{j,p,h}&=\\bigg (\\sum _{K\\in \\mathcal {T}_h}\\Vert v\\Vert _{j,p,K}^p\\bigg )^{1\\slash p},\\\\(w,v)_h&=\\sum _{K\\in \\mathcal {T}_h}\\int _Kw(x)v(x)\\,{\\rm d}x.$ Define $L^2_0(\\Omega )$ as the mean zero functions in $L^2(\\Omega )$ .",
"For $\\Phi \\in L_0^2(\\Omega )$ , let $u := -\\Delta ^{-1}\\Phi \\in H^1(\\Omega )\\cap L^2_0(\\Omega )$ such that $-\\Delta u &= \\Phi &&\\qquad \\mathrm {in}\\ \\Omega ,\\\\\\frac{\\partial u}{\\partial n}&= 0&&\\qquad \\mathrm {on}\\ \\partial \\Omega .$ Then we have $-(\\nabla \\Delta ^{-1}\\Phi ,\\nabla v) = (\\Phi ,v)\\quad \\mathrm {in}\\ \\Omega \\qquad \\forall v\\in H^1(\\Omega )\\cap L^2_0(\\Omega ).$ For $v\\in L^2_0(\\Omega )$ and $\\Phi \\in L^2_0(\\Omega )$ , define the continuous $H^{-1}$ inner product by $(\\Phi , v)_{H^{-1}} := (\\nabla \\Delta ^{-1}\\Phi ,\\nabla \\Delta ^{-1}v) =(\\Phi ,-\\Delta ^{-1}v) = (v,-\\Delta ^{-1}\\Phi ).$ As in [12], [18], [21], [22], [26], [27], we made the following assumptions on the initial condition.",
"These assumptions were used to derive the a priori estimates for the solution of problem (REF )–(REF ).",
"General Assumption (GA) (1) Assume that $m_0\\in (-1,1)$ where $m_0:=\\frac{1}{|\\Omega |}\\int _{\\Omega }u_0(x)\\,{\\rm d}x.$ (2) There exists a nonnegative constant $\\sigma _1$ such that $J_{\\epsilon }(u_0)\\le C\\epsilon ^{-2\\sigma _1}.$ (3) There exist nonnegative constants $\\sigma _2$ , $\\sigma _3$ and $\\sigma _4$ such that $\\big \\Vert -\\epsilon \\Delta u_0 +\\epsilon ^{-1} f(u_0)\\big \\Vert _{H^{\\ell }} \\le C\\epsilon ^{-\\sigma _{2+\\ell }}\\qquad \\ell =0,1,2.$ Under the above assumptions, the following a priori estimates of the solution were proved in [18], [21], [22], [26].",
"The solution $u$ of problem (REF )–(REF ) satisfies the following energy estimate: $&\\underset{t\\in [0,T]}{\\mbox{\\rm ess sup }}\\Bigl ( \\frac{\\epsilon }{2}\\Vert \\nabla u\\Vert _{L^2}^2+\\frac{1}{\\epsilon }\\Vert F(u)\\Vert _{L^1} \\Bigr )+\\int _{0}^{T}\\Vert u_t(s)\\Vert _{H^{-1}}^2\\, {\\rm d}s\\le J_{\\epsilon }(u_0).$ Moreover, suppose that GA (1)–(3) hold, $u_0\\in H^4(\\Omega )$ and $\\partial \\Omega \\in C^{2,1}$ , then $u$ satisfies the additional estimates: $&\\frac{1}{|\\Omega |}\\int _{\\Omega }u(x,t)\\, {\\rm d}x=m_0 \\quad \\forall t\\ge 0,\\\\&\\underset{t\\in [0,T]}{\\mbox{\\rm ess sup }}\\Vert \\Delta u\\Vert _{L^2}\\le C\\epsilon ^{-\\max \\lbrace \\sigma _1+\\frac{5}{2},\\sigma _3+1\\rbrace },\\\\&\\underset{t\\in [0,T]}{\\mbox{\\rm ess sup }}\\Vert \\nabla \\Delta u\\Vert _{L^2}\\le C\\epsilon ^{-\\max \\lbrace \\sigma _1+\\frac{5}{2},\\sigma _3+1\\rbrace },\\\\&\\epsilon \\int _0^{T}\\Vert \\Delta u_t\\Vert _{L^2}^2\\,{\\rm d}s+\\underset{t\\in [0,T]}{\\mbox{\\rm ess sup }}\\Vert u_t\\Vert _{L^2}^2\\le C\\epsilon ^{-\\max \\lbrace 2\\sigma _1+\\frac{13}{2},2\\sigma _3+\\frac{7}{2},2\\sigma _2+4,2\\sigma _4\\rbrace }.$ Furthermore, if there exists $\\sigma _5>0$ such that $\\mathop {\\rm {lim}}_{s\\rightarrow 0^{+}}\\limits \\Vert \\nabla u_t(s)\\Vert _{L^2}\\le C\\epsilon ^{-\\sigma _5},$ then there hold $&\\underset{t\\in [0,T]}{\\mbox{\\rm ess sup }}\\Vert \\nabla u_t\\Vert _{L^2}^2 + \\epsilon \\int _0^{T}\\Vert \\nabla \\Delta u_t\\Vert _{L^2}^2\\,{\\rm d}s\\le C\\rho _0(\\epsilon ),\\\\&\\int _0^{T}\\Vert u_{tt}\\Vert _{H^{-1}}^2\\,{\\rm d}s \\le C\\rho _1(\\epsilon ),$ where $\\rho _0(\\epsilon )&:=\\epsilon ^{-\\frac{1}{2}\\max \\lbrace 2\\sigma _1+5,2\\sigma _3+2\\rbrace -\\max \\lbrace 2\\sigma _1+\\frac{13}{2},2\\sigma _3+\\frac{7}{2},2\\sigma _2+4\\rbrace }+\\epsilon ^{-2\\sigma _5}\\\\&\\qquad +\\epsilon ^{-\\max \\lbrace 2\\sigma _1+7,2\\sigma _3+4\\rbrace },\\\\\\rho _1(\\epsilon ) &:=\\epsilon \\rho _0(\\epsilon ).$ Besides, an extra a priori estimates of solution $u$ is needed in this paper.",
"Under the assumptions of Theorem and if there exists $\\sigma _6>0$ such that $\\Vert \\Delta u_t(0)\\Vert _{L^2}\\le C\\epsilon ^{-\\sigma _6},$ then there hold $\\underset{t\\in [0,T]}{\\mbox{\\rm ess sup }}\\Vert \\Delta u_t\\Vert _{L^2}^2 + \\epsilon \\int _0^{T}\\Vert \\Delta ^2u_t\\Vert _{L^2}^2\\,{\\rm d}s &\\le C \\rho _2(\\epsilon ),\\\\\\underset{t\\in [0,T]}{\\mbox{\\rm ess sup }}\\epsilon \\Vert \\Delta u_t\\Vert _{L^2}^2+\\int _0^{T}\\Vert u_{tt}\\Vert _{L^2}^2\\,{\\rm d}s&\\le C \\rho _3(\\epsilon ),$ where $\\rho _2(\\epsilon )&:=\\epsilon ^{-\\max \\lbrace 2\\sigma _1+\\frac{13}{2},2\\sigma _3+\\frac{7}{2},2\\sigma _2+4,2\\sigma _4\\rbrace - \\max \\lbrace 2\\sigma _1+5, 2\\sigma _3+2\\rbrace - 3} \\\\& \\qquad + \\epsilon ^{-\\max \\lbrace \\sigma _1+\\frac{5}{2},\\sigma _3+1\\rbrace -3}\\rho _0(\\epsilon ) +\\epsilon ^{-2\\sigma _6},\\\\\\rho _3(\\epsilon )&:=\\epsilon \\rho _2(\\epsilon ).$ Using the Gagliardo-Nirenberg inequalities [1] in two-dimensional space, we have $\\Vert \\nabla u\\Vert _{L^{\\infty }}\\le C\\bigg (\\Vert \\nabla \\Delta u\\Vert _{L^2}^{\\frac{1}{2}}\\Vert u\\Vert _{L^{\\infty }}^{\\frac{1}{2}}+\\Vert u\\Vert _{L^{\\infty }}\\bigg )\\le C \\epsilon ^{-\\frac{1}{2} \\max \\lbrace \\sigma _1+\\frac{5}{2}, \\sigma _3 + 1\\rbrace }.$ Since $f^{\\prime }(u) = 3u^2 - 1$ , using Sobolev embedding theorem [1], (REF ), (), (), () and (REF ), we have $ &~\\quad \\int _0^T \\Vert \\Delta (f^{\\prime }(u)u_t)\\Vert _{L^2}^2 \\,{\\rm d}s \\\\&= \\int _0^T \\Vert 6uu_t \\Delta u + 12 u\\nabla u \\cdot \\nabla u_t +6u_t\\nabla u \\cdot \\nabla u + (3u^2 -1) \\Delta u_t\\Vert _{L^2}^2 \\,{\\rm d}s \\\\&\\le C\\int _0^T\\Vert \\Delta u\\Vert _{L^2}^2 \\Vert u_t\\Vert _{L^\\infty }^2\\, {\\rm d}s+ C\\int _0^T \\Vert \\nabla u\\Vert _{L^\\infty }^2 \\Vert \\nabla u_t\\Vert _{L^2}^2 \\,{\\rm d}s \\\\& ~\\quad + C\\int _0^T \\Vert \\nabla u\\Vert _{L^\\infty }^{4} \\Vert u_t\\Vert _{L^2}^2 \\,{\\rm d}s+ C\\int _0^T \\Vert \\Delta u_t\\Vert _{L^2}^2\\, {\\rm d}s \\\\&\\le C\\Vert \\Delta u\\Vert _{L^\\infty (L^2)}^2 \\int _0^T\\Vert u_t\\Vert _{H^2}^2\\, {\\rm d}s+ C\\Vert \\nabla u_t\\Vert _{L^\\infty (L^2)}^2 \\Vert \\nabla u\\Vert _{L^\\infty (L^\\infty )}^2 \\\\& ~\\quad + C\\Vert \\nabla u\\Vert _{L^\\infty (L^\\infty )}^{4}\\Vert u_t\\Vert _{L^\\infty (L^2)}^2+ C\\int _0^T \\Vert \\Delta u_t\\Vert _{L^2}^2\\,{\\rm d}s \\\\&\\le C \\epsilon ^{-\\max \\lbrace 2\\sigma _1+\\frac{13}{2},2\\sigma _3+\\frac{7}{2},2\\sigma _2+4,2\\sigma _4\\rbrace - \\max \\lbrace 2\\sigma _1+5, 2\\sigma _3+2\\rbrace -1} \\\\&~\\quad + C\\epsilon ^{-\\max \\lbrace \\sigma _1 + \\frac{5}{2},\\sigma _3+1\\rbrace }\\rho _0(\\epsilon )\\\\& ~\\quad + C \\epsilon ^{-\\max \\lbrace 2\\sigma _1+\\frac{13}{2},2\\sigma _3+\\frac{7}{2},2\\sigma _2+4,2\\sigma _4\\rbrace - \\max \\lbrace 2\\sigma _1+5, 2\\sigma _3 + 2\\rbrace } \\\\& ~\\quad + C \\epsilon ^{-\\max \\lbrace 2\\sigma _1+\\frac{13}{2},2\\sigma _3+\\frac{7}{2},2\\sigma _2+4,2\\sigma _4\\rbrace - 1} \\\\&\\le C \\epsilon ^{-\\max \\lbrace 2\\sigma _1+\\frac{13}{2},2\\sigma _3+\\frac{7}{2},2\\sigma _2+4,2\\sigma _4\\rbrace - \\max \\lbrace 2\\sigma _1+5,2\\sigma _3+2\\rbrace } \\\\&~\\quad + C\\epsilon ^{-\\max \\lbrace \\sigma _1 + \\frac{5}{2}, \\sigma _3+1\\rbrace }\\rho _0(\\epsilon ).$ Taking the derivative with respect to $t$ on both sides of (REF ), we get $u_{tt}+\\epsilon \\Delta ^2u_t-\\frac{1}{\\epsilon }\\Delta (f^{\\prime }(u)u_t)=0.$ Testing (REF ) with $\\Delta ^2u_t$ , and taking the integral over $(0,T)$ , we obtain $&~\\quad \\frac{1}{2}\\Vert \\Delta u_t(T)\\Vert _{L^2}^2+\\epsilon \\int _0^{T}\\Vert \\Delta ^2u_t\\Vert _{L^2}^2\\,{\\rm d}s \\\\& =\\frac{1}{\\epsilon }\\int _0^{T}(\\Delta (f^{\\prime }(u)u_t),\\Delta ^2 u_t)\\, {\\rm d}s +\\frac{1}{2}\\Vert \\Delta u_t(0)\\Vert _{L^2}^2 \\\\&\\le \\frac{C}{\\epsilon ^3}\\int _0^{T} \\Vert \\Delta (f^{\\prime }(u) u_t)\\Vert _{L^2}^2\\, {\\rm d}s+\\frac{\\epsilon }{2}\\int _0^{T}\\Vert \\Delta ^2u_t\\Vert _{L^2}^2\\,{\\rm d}s+C\\epsilon ^{-2\\sigma _6}.$ Then (REF ) is obtained by (REF ).",
"Next we bound ().",
"Testing (REF ) with $u_{tt}$ , taking the integral over $(0,T)$ , and using (REF ), we obtain $&~\\quad \\int _0^{T}\\Vert u_{tt}\\Vert _{L^2}^2\\,{\\rm d}s + \\frac{\\epsilon }{2}\\Vert \\Delta u_t(T)\\Vert _{L^2}^2 \\\\&\\le \\frac{\\epsilon }{2}\\Vert \\Delta u_t(0)\\Vert _{L^2}^2 +\\frac{C}{\\epsilon ^2}\\int _0^{T} \\Vert \\Delta (f^{\\prime }(u)u_t)\\Vert _{L^2}^2\\,{\\rm d}s+ \\frac{1}{2}\\int _0^{T}\\Vert u_{tt}\\Vert _{L^2}^2\\,{\\rm d}s .$ Then () is obtained by (REF ).",
"The next lemma gives an $\\epsilon $ -independent lower bound for the principal eigenvalue of the linearized Cahn-Hilliard operator $\\mathcal {L}_{CH}$ defined below.",
"The proof of this lemma can be found in [12].",
"Suppose that GA (1)–(3) hold.",
"Given a smooth initial curve/surface $\\Gamma _0$ , let $u_0$ be a smooth function satisfying $\\Gamma _0 =\\lbrace x\\in \\Omega ; u_0(x)=0\\rbrace $ and some profile described in [12].",
"Let $u$ be the solution to problem (REF )–(REF ).",
"Define $\\mathcal {L}_{CH}$ as $\\mathcal {L}_{CH} := \\Delta \\left(\\epsilon \\Delta -\\frac{1}{\\epsilon }f^{\\prime }(u)I\\right).$ Then there exists $0<\\epsilon _0\\ll 1$ and a positive constant $C_0$ such that the principle eigenvalue of the linearized Cahn-Hilliard operator $\\mathcal {L}_{CH}$ satisfies $\\lambda _{CH}:=\\mathop {\\inf }_{\\begin{array}{c}0\\ne \\psi \\in H^1(\\Omega )\\\\ \\Delta w=\\psi \\end{array}}\\limits \\frac{\\epsilon \\Vert \\nabla \\psi \\Vert _{L^2}^2+\\frac{1}{\\epsilon }(f^{\\prime }(u)\\psi ,\\psi )}{\\Vert \\nabla w\\Vert _{L^2}^2}\\ge -C_0$ for $t\\in [0,T]$ and $\\epsilon \\in (0,\\epsilon _0)$ ."
],
[
"Fully Discrete Approximation",
"In this section, the backward Euler is used for time stepping, and the Morley finite element discretization is used for space discretization."
],
[
"Morley finite element space",
"Define the Morley finite element spaces $S^h$ below [8], [10], [14]: $S^h := \\lbrace & v_h\\in L^{\\infty }(\\Omega ): v_h\\in P_2(K), v_h ~\\text{iscontinuous at the vertices of all triangles,} \\\\&\\frac{\\partial v_h}{\\partial n} \\text{ is continuous at the midpointsof interelement edges of triangles} \\rbrace .$ We use the following notation $H^j_E(\\Omega ):=\\lbrace v\\in H^j(\\Omega ): \\frac{\\partial v}{\\partial n}=0~\\text{on}~\\partial \\Omega \\rbrace \\qquad j=1, 2, 3.$ Corresponding to $H^j_E(\\Omega )$ , define $S^h_E$ as a subspace of $S^h$ below: $S^h_E := \\lbrace v_h\\in S^h: \\frac{\\partial v_h}{\\partial n}=0 \\text{ at themidpoints of the edges on } \\partial \\Omega \\rbrace .$ We also define $\\mathring{H}_E^j(\\Omega ) = H_E^j(\\Omega ) \\cap L_0^2(\\Omega ), j=1,2,3$ , and $\\mathring{S}^h_E = S^h_E \\cap L_0^2(\\Omega )$ , where $L_0^2(\\Omega )$ denotes the set of mean zero functions.",
"The enriching operator $\\widetilde{E}_h$ is restated [7], [8], [10].",
"Let $\\widetilde{S}_E^h$ be the Hsieh-Clough-Tocher macro element space, which is an enriched space of the Morley finite element space $S_E^h$ .",
"Let $p$ and $m$ be the internal vertices and midpoints of triangles $\\mathcal {T}_h$ .",
"Define $\\widetilde{E}_h: S_E^h\\rightarrow \\widetilde{S}_E^h$ by $(\\widetilde{E}_h v)(p) &= v(p),\\\\\\frac{\\partial (\\widetilde{E}_h v)}{\\partial n}(m) &= \\frac{\\partial v}{\\partial n}(m),\\\\(\\partial ^{\\beta }(\\widetilde{E}_h v))(p) &= \\text{average of } (\\partial ^{\\beta }v_i)(p)\\qquad |\\beta |=1,$ where $v_i=v|_{T_i}$ and triangle $T_i$ contains $p$ as a vertex.",
"Define the interpolation operator $I_h: H^2_E(\\Omega )\\rightarrow S_E^h$ such that $(I_h v)(p)&=v(p),\\\\\\frac{\\partial (I_h v)}{\\partial n}(m)&=\\frac{1}{|e|}\\int _e\\frac{\\partial v}{\\partial n}\\,{\\rm d}S,$ where $p$ ranges over the internal vertices of all the triangles $T$ , and $m$ ranges over the midpoints of all the edges $e$ .",
"It can be proved that [7], [8], [10], [14] $|v-I_hv|_{j,p,K}&\\le Ch^{3-j}|v|_{3,p,K}\\qquad &&\\forall K\\in \\mathcal {T}_h,\\quad \\forall v\\in H^3(K),\\quad j=0,1,2,\\\\\\Vert \\widetilde{E}_h v-v\\Vert _{j,2,h}&\\le Ch^{2-j}|v|_{2,2,h}\\quad &&\\forall v\\in S_E^h,\\quad j=0,1,2.$ Notice that $\\widetilde{E}_h$ and $I_h$ cannot preserve the mean zero functions.",
"Let $\\mathring{\\widetilde{S}_E^h}:= \\widetilde{S}_E^h \\cap L_0^2(\\Omega )$ .",
"Define $\\mathring{\\widetilde{E}_h}:\\mathring{S}_E^h \\mapsto \\mathring{\\widetilde{S}_E^h}$ such that $ \\mathring{\\widetilde{E}_h}v = \\widetilde{E}_h v - \\frac{1}{|\\Omega |}\\int _{\\Omega } \\widetilde{E}_h v \\,{\\rm d}x.$ Using (), we have $\\int _\\Omega \\widetilde{E}_h v \\,{\\rm d}x = (\\widetilde{E}_h v - v, 1) \\le |\\Omega |^{1/2}\\Vert \\widetilde{E}_h v -v\\Vert _{0,2} \\le Ch^2|v|_{2,2,h} \\qquad \\forall v \\in \\mathring{S}_E^h.$ Then $ \\Vert \\mathring{\\widetilde{E}_h} v-v\\Vert _{j,2,h}&\\le Ch^{2-j}|v|_{2,2,h}\\qquad \\forall v\\in \\mathring{S}_E^h,\\quad j=0,1,2.$ Finally the following spaces are needed $&H^{3,h}(\\Omega )=S^h\\oplus H^3(\\Omega ), &&\\qquad H_E^{3,h}(\\Omega )=S_E^h\\oplus H_E^3(\\Omega ),\\\\&H^{2,h}(\\Omega )=S^h\\oplus H^2(\\Omega ), &&\\qquad H_E^{2,h}(\\Omega )=S_E^h\\oplus H_E^2(\\Omega ),\\\\&H^{1,h}(\\Omega )=S^h\\oplus H^1(\\Omega ), &&\\qquad H_E^{1,h}(\\Omega )=S_E^h\\oplus H_E^1(\\Omega ),$ where, for instance, $S_E^h\\oplus H_E^3(\\Omega ):=\\lbrace u+v: u\\in S_E^h\\ \\ \\text{and}\\ \\ v\\in H_E^3(\\Omega )\\rbrace .$"
],
[
"Formulation",
"The weak form of (REF )–(REF ) is to seek $u(\\cdot ,t)\\in H^2_E(\\Omega )$ such that $(u_t,v)+\\epsilon a(u,v) +\\frac{1}{\\epsilon }(\\nabla f(u), \\nabla v)&= 0\\quad \\forall v\\in H_E^2(\\Omega ),\\\\u(\\cdot ,0)&=u_0\\in H_E^2(\\Omega ),$ where the bilinear form $a(\\cdot ,\\cdot )$ is defined as $a(u,v):=\\int _{\\Omega }\\Delta u\\Delta v+\\bigl (\\frac{\\partial ^2u}{\\partial x\\partial y}\\frac{\\partial ^2v}{\\partial x\\partial y}-\\frac{1}{2}\\frac{\\partial ^2u}{\\partial x^2}\\frac{\\partial ^2v}{\\partial y^2}-\\frac{1}{2}\\frac{\\partial ^2u}{\\partial y^2}\\frac{\\partial ^2v}{\\partial x^2}\\bigr )\\,{\\rm d}x {\\rm d}y$ with Poisson's ratio $\\frac{1}{2}$ .",
"Next define the discrete bilinear form $a_h(u,v)&:=\\sum _{K\\in \\mathcal {T}_h}\\int _K\\Delta u\\Delta v+\\bigl (\\frac{\\partial ^2u}{\\partial x\\partial y}\\frac{\\partial ^2v}{\\partial x\\partial y}-\\frac{1}{2}\\frac{\\partial ^2u}{\\partial x^2}\\frac{\\partial ^2v}{\\partial y^2}-\\frac{1}{2}\\frac{\\partial ^2u}{\\partial y^2}\\frac{\\partial ^2v}{\\partial x^2}\\bigr )\\,{\\rm d}x{\\rm d}y.$ Based on the bilinear form (REF ), a fully discrete Galerkin method is to seek $u_h^n\\in S^h_E$ such that $(d_tu_h^{n},v_h)+\\epsilon a_h(u_h^{n},v_h)+\\frac{1}{\\epsilon }(\\nabla f(u_h^{n}),\\nabla v_h)_h&=0\\quad \\forall v_h\\in S^h_E,\\\\u_h^0&=u_0^h\\in S^h_E,$ where the difference operator $d_tu_h^{n} :=\\frac{u_h^{n}-u_h^{n-1}}{k}$ and $u_0^h := P_hu(t_0)$ , where the operator $P_h$ is defined below."
],
[
"Elliptic operator $P_h$",
"We define $R:=\\bigl \\lbrace v\\in H_E^2(\\Omega ): \\Delta v\\in H_E^2(\\Omega )\\bigr \\rbrace .$ Then $\\forall v\\in R$ , define the elliptic operator $P_h$ (cf.",
"[14]) by seeking $P_hv\\in S_E^h$ such that $\\tilde{b}_h(P_hv,w):=(\\epsilon \\Delta ^2v-\\frac{1}{\\epsilon }\\nabla \\cdot (f^{\\prime }(u)\\nabla v)+\\alpha v,w)\\qquad \\forall w\\in S_E^h,$ where $\\tilde{b}_h(v,w):=\\epsilon a_h(v,w)+\\frac{1}{\\epsilon }(f^{\\prime }(u)\\nabla v,\\nabla w)_h+\\alpha (v,w),$ and $\\alpha $ should be chosen as $\\alpha = \\alpha _0 \\epsilon ^{-3}$ to guarantee the coercivity of $\\tilde{b}_h(\\cdot ,\\cdot )$ .",
"More precisely, first we cite some lemmas in [14], which will be used in this paper.",
"[Lemma 2.3 in [14]] Let $w,z \\in H_E^{2,h}(\\Omega )$ , then $\\left| \\sum _{K \\in \\mathcal {T}_h} \\int _{\\partial K} \\frac{\\partial w}{\\partial n}z \\,{\\rm d}S \\right|\\le Ch(h\\Vert w\\Vert _{2,2,h}\\Vert z\\Vert _{2,2,h} + \\Vert w\\Vert _{1,2,h}\\Vert z\\Vert _{2,2,h} +\\Vert w\\Vert _{2,2,h}\\Vert z\\Vert _{1,2,h}).$ [Lemma 2.5 in [14]] Let $z\\in H^{2,h}(\\Omega )$ and $w\\in H_E^2(\\Omega ) \\cap H^3(\\Omega )$ , and define $B_h(w,z)$ by $B_h(w,z) = \\sum _{K\\in \\mathcal {T}_h} \\int _{\\partial K} \\left(\\Delta w \\frac{\\partial z}{\\partial n} + \\frac{1}{2} \\frac{\\partial ^2w}{\\partial n \\partial s} - \\frac{1}{2}\\frac{\\partial ^2 w}{\\partial s^2}\\frac{\\partial z}{\\partial n}\\right)\\,{\\rm d}S,$ then we have $ |B_h(w,z)| \\le Ch |w|_{3,2,h}|z|_{2,2,h}.$ For any $w\\in S_E^h$ , using Lemma REF and the inverse inequality, we have $\\begin{aligned}|w|_{1,2,h}^2 & \\le |w|_{2,2,h}\\Vert w\\Vert _{0,2} + \\left| \\sum _{K \\in \\mathcal {T}_h} \\int _{\\partial K} \\frac{\\partial w}{\\partial n}z \\,{\\rm d}S\\right| \\le C \\Vert w\\Vert _{2,2,h}\\Vert w\\Vert _{0,2} \\\\&\\le C ( |w|_{2,2,h}\\Vert w\\Vert _{0,2} + |w|_{1,2,h}\\Vert w\\Vert _{0,2} +\\Vert w\\Vert _{0,2}^2 ).\\end{aligned}$ The kick-back argument gives $|w|_{1,2,h}^2 \\le C ( |w|_{2,2,h}\\Vert w\\Vert _{0,2} + \\Vert w\\Vert _{0,2}^2 ).$ Hence, $\\tilde{b}_h(w,w) &= \\epsilon a_h(w,w) + \\frac{1}{\\epsilon } (f^{\\prime }(u)\\nabla w, \\nabla w) + \\frac{\\alpha _0}{\\epsilon ^3}(w,w) \\\\& \\ge \\frac{1}{\\epsilon ^3} \\left(\\frac{\\epsilon ^4}{2}|w|_{2,2,h}^2 - C\\epsilon ^2|w|_{1,2,h}^2 +\\alpha _0\\Vert w\\Vert _{0,2}^2 \\right) \\\\& \\ge \\frac{1}{\\epsilon ^3} \\left(\\frac{\\epsilon ^4}{4}|w|_{2,2,h}^2 +(\\alpha _0 - C)\\Vert w\\Vert _{0,2}^2 \\right) ,$ which implies the coercivity of $\\tilde{b}_h(\\cdot ,\\cdot )$ when $\\alpha _0$ is large enough but independent of $\\epsilon $ .",
"Next we give the properties of $P_h$ .",
"Define $b_h(\\cdot ,\\cdot ) :=\\epsilon ^3 \\tilde{b}_h(\\cdot ,\\cdot )$ and a norm $\\left|\\hspace{-0.9pt}\\left|\\hspace{-0.9pt}\\left|v\\right|\\hspace{-0.9pt}\\right|\\hspace{-0.9pt}\\right|_{2,2,h}^2 := \\epsilon ^4 |v|_{2,2,h}^2 +\\epsilon ^2|v|_{1,2,h}^2 + \\Vert v\\Vert _{0,2}^2,\\qquad $ Consider the following problems: $b_h(v, \\eta ) &= F_h(\\eta ) \\quad \\forall \\eta \\in H_E^2(\\Omega ), \\\\b_h(v_h, \\chi ) &= \\widetilde{F}_h(\\chi ) \\quad \\forall \\chi \\in S_E^h.$ Then we have $& \\quad ~\\left|\\hspace{-0.9pt}\\left|\\hspace{-0.9pt}\\left|v - v_h\\right|\\hspace{-0.9pt}\\right|\\hspace{-0.9pt}\\right|_{2,2,h} \\\\& \\le Ch\\left\\lbrace (\\epsilon +h)^2|v|_{3,2} +|v|_{1,2} + \\sup _{\\chi \\in S_E^h} \\frac{F_h(\\widetilde{E}_h\\chi ) -\\widetilde{F}_h(\\chi ) + \\alpha _0(v, \\chi -\\widetilde{E}_h\\chi )}{\\left|\\hspace{-0.9pt}\\left|\\hspace{-0.9pt}\\left|\\chi \\right|\\hspace{-0.9pt}\\right|\\hspace{-0.9pt}\\right|_{2,2,h}} \\right\\rbrace .",
"$ Using (REF ) and the Strang Lemma, we have $\\begin{aligned}&\\quad ~ \\left|\\hspace{-0.9pt}\\left|\\hspace{-0.9pt}\\left|v - v_h\\right|\\hspace{-0.9pt}\\right|\\hspace{-0.9pt}\\right|_{2,2,h} \\\\& \\le C \\left( \\inf _{\\psi \\in S_E^h}\\left|\\hspace{-0.9pt}\\left|\\hspace{-0.9pt}\\left|v -\\psi \\right|\\hspace{-0.9pt}\\right|\\hspace{-0.9pt}\\right|_{2,2,h} + \\sup _{\\chi \\in S_E^h} \\frac{b_h(v, \\chi ) -\\widetilde{F}_h(\\chi )}{\\left|\\hspace{-0.9pt}\\left|\\hspace{-0.9pt}\\left|\\chi \\right|\\hspace{-0.9pt}\\right|\\hspace{-0.9pt}\\right|_{2,2,h}} \\right) \\\\& \\le C \\left( \\inf _{\\psi \\in S_E^h}\\left|\\hspace{-0.9pt}\\left|\\hspace{-0.9pt}\\left|v -\\psi \\right|\\hspace{-0.9pt}\\right|\\hspace{-0.9pt}\\right|_{2,2,h} + \\sup _{\\chi \\in S_E^h}\\frac{b_h(v, \\chi - \\widetilde{E}_h \\chi ) + b_h(v,\\widetilde{E}_h\\chi )-\\widetilde{F}_h(\\chi )}{\\left|\\hspace{-0.9pt}\\left|\\hspace{-0.9pt}\\left|\\chi \\right|\\hspace{-0.9pt}\\right|\\hspace{-0.9pt}\\right|_{2,2,h}} \\right) \\\\& \\le C \\left( \\inf _{\\psi \\in S_E^h}\\left|\\hspace{-0.9pt}\\left|\\hspace{-0.9pt}\\left|v -\\psi \\right|\\hspace{-0.9pt}\\right|\\hspace{-0.9pt}\\right|_{2,2,h} + \\sup _{\\chi \\in S_E^h}\\frac{b_h(v, \\chi - \\widetilde{E}_h \\chi ) + F_h(\\widetilde{E}_h\\chi )-\\widetilde{F}_h(\\chi )}{\\left|\\hspace{-0.9pt}\\left|\\hspace{-0.9pt}\\left|\\chi \\right|\\hspace{-0.9pt}\\right|\\hspace{-0.9pt}\\right|_{2,2,h}}\\right).\\end{aligned}$ Using Lemma REF and (), we have $\\begin{aligned}b_h(v, \\chi - \\widetilde{E}_h\\chi ) &= \\epsilon ^4 a_h(v, \\chi - \\widetilde{E}_h\\chi ) +\\epsilon ^2 (f^{\\prime }(u)\\nabla v, \\nabla (\\chi - \\widetilde{E}_h\\chi )) + (\\alpha _0v, \\chi - \\widetilde{E}_h\\chi ) \\\\& \\le Ch\\left(\\epsilon ^4 |v|_{3,2}|\\chi |_{2,2,h} + \\epsilon ^2|v|_{1,2}|\\chi |_{2,2,h}\\right) + (\\alpha _0 v, \\chi - \\widetilde{E}_h\\chi ) \\\\& \\le Ch\\left( \\epsilon ^2 |v|_{3,2} + |v|_{1,2} \\right)\\left|\\hspace{-0.9pt}\\left|\\hspace{-0.9pt}\\left|\\chi \\right|\\hspace{-0.9pt}\\right|\\hspace{-0.9pt}\\right|_{2,2,h}+ (\\alpha _0 v, \\chi - \\widetilde{E}_h\\chi ) \\\\\\end{aligned}$ Then we obtain the desired bound (REF ) by the approximation properties of Morley interpolation operator (REF ).",
"Suppose $u$ solves the Cahn-Hilliard equation (REF ) – (), then we have $& \\quad ~\\epsilon ^2|u - P_hu|_{2,2,h} + \\epsilon |u - P_h u|_{1,2,h} + \\Vert u -P_hu\\Vert _{0,2} \\\\& \\le Ch \\big ( (\\epsilon +h)^2|u|_{3,2} + |u|_{1,2} + \\epsilon h\\Vert u_t\\Vert _{0,2} \\big ), \\\\& \\quad ~\\epsilon ^2|u_t - (P_hu)_t|_{2,2,h} + \\epsilon |u_t - (P_h u)_t|_{1,2,h}+ \\Vert u_t - (P_hu)_t\\Vert _{0,2} \\\\& \\le Ch \\Big \\lbrace (\\epsilon +h)^2|u_t|_{3,2} + |u_t|_{1,2} +\\epsilon h \\Vert u_{tt}\\Vert _{0,2} + \\Vert u_t\\nabla u\\Vert _{0,2} \\\\&\\quad ~+ \\epsilon ^{-1}|\\ln h|^{1/2}\\Vert u_t\\Vert _{0,2} ((\\epsilon +h)^2|u|_{3,2} + |u|_{1,2} +\\epsilon h\\Vert u_t\\Vert _{0,2})\\Big \\rbrace .",
"$ Taking $v = u$ and $v_h = P_hu$ in Lemma REF , and noticing that $F_h(\\psi ) = \\tilde{F}_h(\\psi ) = (\\epsilon ^4 \\Delta ^2u -\\epsilon ^2\\Delta f(u) + \\alpha _0 u, \\psi ) = (\\epsilon ^3 u_t +\\alpha _0 u, \\psi ),$ we obtain the bound (REF ) from () and (REF ).",
"Taking $v = u_t$ and $v_h = (P_hu)_t$ , we have $\\begin{aligned}F_h(\\psi ) &= (\\epsilon ^4 \\Delta ^2 u_t - \\epsilon ^2 \\Delta f(u)_t +\\alpha _0 u_t, \\psi ) - (\\epsilon ^2 f^{\\prime \\prime }(u)u_t \\nabla u, \\nabla \\psi )_h, \\\\\\widetilde{F}_h(\\psi ) &= (\\epsilon ^4 \\Delta ^2 u_t - \\epsilon ^2 \\Delta f(u)_t + \\alpha _0 u_t, \\psi ) - (\\epsilon ^2 f^{\\prime \\prime }(u)u_t \\nabla P_hu,\\nabla \\psi )_h.\\end{aligned}$ Then we get $\\begin{aligned}& \\quad ~F_h(\\widetilde{E}_h \\chi ) - \\widetilde{F}(\\chi ) + \\alpha _0(u_t, \\chi -\\widetilde{E}_h \\chi ) \\\\&= (\\epsilon ^4 \\Delta ^2 u_t - \\epsilon ^2 \\Delta f(u)_t, \\widetilde{E}_h \\chi -\\chi ) \\\\& \\quad ~- (\\epsilon ^2 f^{\\prime \\prime }(u)u_t\\nabla u, \\nabla \\widetilde{E}_h\\chi - \\nabla \\chi ) -(\\epsilon ^2 f^{\\prime \\prime }(u)u_t\\nabla (u - P_hu), \\nabla \\chi )\\\\&\\le \\epsilon ^3h^2\\Vert u_{tt}\\Vert _{0,2}|\\chi |_{2,2,h} + C\\epsilon ^2 h\\Vert u_t\\nabla u\\Vert _{0,2} |\\chi |_{2,2,h} + C\\epsilon ^2\\Vert u_t\\Vert _{0,2}\\Vert \\nabla \\chi \\Vert _{0,\\infty } |u - P_hu|_{1,2,h} \\\\&\\le Ch\\Big \\lbrace \\epsilon h\\Vert u_{tt}\\Vert _{0,2} + \\Vert u_t\\nabla u\\Vert _{0,2} \\\\&\\quad ~+ \\epsilon ^{-1}|\\ln h|^{1/2}\\Vert u_t\\Vert _{0,2} ((\\epsilon +h)^2|u|_{3,2} + |u|_{1,2} +\\epsilon h\\Vert u_t\\Vert _{0,2})\\Big \\rbrace \\left|\\hspace{-0.9pt}\\left|\\hspace{-0.9pt}\\left|\\chi \\right|\\hspace{-0.9pt}\\right|\\hspace{-0.9pt}\\right|_{2,2,h},\\end{aligned}$ where we use the discrete Sobolev inequality and the fact that $\\nabla \\chi $ belongs to the Crouzeix-Raviar finite element space [9].",
"This implies the bound ().",
"Combining with the a priori estimates of the bounds given in Section , we have the following theorem.",
"Assume $h \\le C\\epsilon $ , then there hold $& \\epsilon ^4|u - P_hu|_{2,2,h}^2 + \\epsilon ^2|u - P_hu|_{1,2,h}^2 + \\Vert u - P_hu\\Vert _{0,2}^2\\le Ch^2\\rho _4(\\epsilon ), \\\\& \\quad ~ \\int _0^T \\epsilon ^4|u_t - (P_hu)_t|_{2,2,h}^2 +\\epsilon ^2|u_t - (P_hu)_t|_{1,2,h}^2 + \\Vert u_t - (P_hu)_t\\Vert _{0,2}^2 \\,{\\rm d}s \\\\&\\le Ch^2\\epsilon ^4\\rho _3(\\epsilon ) + Ch^2|\\ln h| \\rho _5(\\epsilon ),$ where $\\begin{aligned}\\rho _4(\\epsilon ) &:=\\epsilon ^{-\\max \\lbrace 2\\sigma _1+\\frac{13}{2},2\\sigma _3+\\frac{7}{2},2\\sigma _2+4,2\\sigma _4\\rbrace +4}, \\\\\\rho _5(\\epsilon ) &:=\\epsilon ^{-2\\max \\lbrace 2\\sigma _1+\\frac{13}{2},2\\sigma _3+\\frac{7}{2},2\\sigma _2+4,2\\sigma _4\\rbrace +2}.\\end{aligned}$ Using (REF ), () and (), we have $& \\quad ~(\\epsilon +h)^4 |u|_{3,2}^2 + |u|_{1,2}^2 +\\epsilon ^2 h^2 \\Vert u_t\\Vert _{0,2}^2 \\\\& \\le C\\epsilon ^{-\\max \\lbrace 2\\sigma _1+5, 2\\sigma _3+2\\rbrace +4} +C\\epsilon ^{-2\\sigma _1 - 1} +C\\epsilon ^{-\\max \\lbrace 2\\sigma _1+\\frac{13}{2},2\\sigma _3+\\frac{7}{2},2\\sigma _2+4,2\\sigma _4\\rbrace +4} \\\\& \\le C\\rho _4(\\epsilon ), $ which implies the bound (REF ) by (REF ).",
"Using (), (), (REF ) and (REF ), we obtain $\\begin{aligned}&\\quad ~ \\int _0^T (\\epsilon +h)^4|u_t|_{3,2}^2 +|u_t|_{1,2}^2 +\\epsilon ^2 h^2 \\Vert u_{tt}\\Vert _{0,2}^2 + \\Vert u_t\\nabla u\\Vert _{0,2}^2\\,{\\rm d}s \\\\& \\le C\\int _0^T \\epsilon ^4|u_t|_{3,2}^2+ |u_t|_{1,2}^2+ \\epsilon ^4\\Vert u_{tt}\\Vert _{0,2}^2+ \\Vert u_t\\Vert _{0,2}^2 \\Vert \\nabla u\\Vert _{0,\\infty }^2 \\,{\\rm d}s \\\\& \\le C\\epsilon ^3\\rho _0(\\epsilon ) + C\\rho _0(\\epsilon ) +C\\epsilon ^4\\rho _3(\\epsilon ) \\\\&~+ C\\epsilon ^{-\\max \\lbrace \\sigma _1+\\frac{5}{2},\\sigma _3+1\\rbrace - \\max \\lbrace 2\\sigma _1+\\frac{13}{2}, 2\\sigma _3+\\frac{7}{2},2\\sigma _2+4, 2\\sigma _4 \\rbrace } \\\\& \\le C\\epsilon ^4\\rho _3(\\epsilon ).\\end{aligned}$ Further, using () and (REF ), we obtain $\\begin{aligned}\\quad ~ \\int _0^T \\epsilon ^{-2} \\Vert u_t\\Vert _{0,2}^2((\\epsilon +h)^2|u|_{3,2} + |u|_{1,2} + \\epsilon h\\Vert u_t\\Vert _{0,2})^2\\,{\\rm d}s \\le C \\rho _5(\\epsilon ).\\end{aligned}$ This implies the bound ().",
"Under the condition that $ h \\le C\\epsilon ^2 \\rho _4^{-\\frac{1}{2}}(\\epsilon ), \\quad h \\le C \\rho _3^{-\\frac{1}{2}}(\\epsilon ), \\quad h|\\ln h|^{\\frac{1}{2}} \\le C \\epsilon ^2 \\rho _5^{-\\frac{1}{2}}(\\epsilon ),$ there hold $ |P_hu|_{j,2,h}^2 &\\le C(1+|u|_{j,2,h}^2) \\quad j=0,1,2, \\\\\\int _0^T |P_hu|_{j,2,h}^2 \\,{\\rm d}s&\\le C(1+\\int _0^T |u|_{j,2,h}^2) \\quad j=0,1,2, \\\\\\Vert P_h u\\Vert _{0,\\infty } &\\le C. $ By the Sobolev embedding and (REF ), we have $\\Vert P_h u\\Vert _{0,\\infty } \\le \\Vert u\\Vert _{0,\\infty } + \\Vert u - P_hu\\Vert _{2,2,h} \\le C + Ch\\epsilon ^{-2}\\rho _4^{1/2}(\\epsilon ) \\le C.$ The first two bounds are the direct consequences of Theorem REF ."
],
[
"Error Estimates",
"In this section, first we derive the piecewise ${L^{\\infty }(L^2)}$ and ${L^2(H^2)}$ error bounds which depend on $\\frac{1}{\\epsilon }$ polynomially based on the generalized coercivity result in Theorem REF , and piecewise ${L^{\\infty }(H^{-1})}$ and ${L^2(H^1)}$ error bounds.",
"Then we prove the piecewise ${L^{\\infty }(H^2)}$ error bound based on the piecewise ${L^{\\infty }(L^2)}$ and ${L^2(H^2)}$ error bounds.",
"Finally, the ${L^\\infty (L^\\infty )}$ error bound is established.",
"Decompose the error $u-u_h^n=(u-P_hu)+(P_hu-u_h^n):=\\rho ^n+\\theta ^n.$ The following two lemmas will be used in this section.",
"[Summation by parts] Suppose $\\lbrace a_n\\rbrace _{n=0}^\\ell $ and $\\lbrace b_n\\rbrace _{n=0}^\\ell $ are two sequences, then $\\sum _{n=1}^\\ell (a^n-a^{n-1},b^n) =(a^\\ell ,b^\\ell )-(a^0,b^0)-\\sum _{n=1}^\\ell (a^{n-1},b^n-b^{n-1}).$ Suppose $u(t_n)$ to be the solution of (REF )–(REF ), and $u_h^n$ to be the solution of (REF )–(), then $\\rho ^n \\in \\mathring{S}^h_E,\\quad \\theta ^n \\in \\mathring{S}^h_E.$ Testing (REF ) with constant 1, and then taking the integration over $(0,t)$ , we can obtain for any $t\\ge 0$ , $\\int _{\\Omega }u(t)dx=\\int _{\\Omega }u(0)dx.$ Then choosing $v=u(t), w=1$ in (REF ), we have for any $t\\ge 0$ , $\\int _{\\Omega }P_hu(t)~{\\rm d}x=\\int _{\\Omega }u(t)~{\\rm d}x.$ Choosing $v_h=1$ in (REF ), then $\\int _{\\Omega }u_h^n\\,{\\rm d}x=\\int _{\\Omega }u_h^{n-1}\\,{\\rm d}x =\\cdots =\\int _{\\Omega }u_h^{0}\\,{\\rm d}x.$ Therefore, if choosing $u_h^{0}=P_hu(0)$ , then $\\int _{\\Omega }u_h^n\\,{\\rm d}x &= \\int _{\\Omega }u_h^{0}\\,{\\rm d}x=\\int _{\\Omega }P_hu(0)\\,{\\rm d}x\\\\&=\\int _{\\Omega }u(0)\\,{\\rm d}x =\\int _{\\Omega }u(t_n)\\,{\\rm d}x=\\int _{\\Omega }P_hu(t_n)\\,{\\rm d}x.$ Hence, $P_hu(t_n)-u_h^n \\in \\mathring{S}^h_E$ ."
],
[
"Generalized coercivity result, piecewise $L^\\infty (H^{-1})$ and\n{{formula:193a3959-87e9-487b-b17c-20cd256b197a}} error estimates",
"We first cite the generalized coercivity result, piecewise $L^{\\infty }(H^{-1})$ and $L^2(H^1)$ error estimates established in [27].",
"[Generalized coercivity] Suppose there exists a positive number $\\gamma _3>0$ such that the solution $u$ of problem (REF )–(REF ) and elliptic operator $P_h$ satisfy $ \\Vert u-P_h u\\Vert _{L^{\\infty }((0,T);L^{\\infty })} \\le C_1 h\\epsilon ^{-\\gamma _3}.$ Then there exists an $\\epsilon $ -independent and $h$ -independent constant $C>0$ such that for $\\epsilon \\in (0,\\epsilon _0)$ , a.e.",
"$t\\in [0,T]$ , and for any $\\psi \\in \\mathring{S}_E^h$ , $(\\epsilon -\\epsilon ^4)(\\nabla \\psi ,\\nabla \\psi )_h+\\frac{1}{\\epsilon }(f^{\\prime }(P_hu(t))\\psi ,\\psi )_h\\ge -C\\Vert \\nabla \\Delta ^{-1}\\psi \\Vert _{L^2}^2-C\\epsilon ^{-2\\gamma _2-4}h^4,$ provided that $h$ satisfies the constraint $h &\\le (C_1C_2)^{-1}\\epsilon ^{\\gamma _3+3},$ where $\\gamma _2 = 2\\gamma _1 + \\sigma _1 + 6$ and $C_2$ is determined by $C_2:=\\max _{|\\xi |\\le \\Vert u\\Vert _{L^\\infty ((0,T); L^\\infty )}}|f{^{\\prime \\prime }}(\\xi )|.$ Remark 1 Thanks to the Sobolev embedding theorem and (REF ), we have $ \\Vert u - P_hu\\Vert _{0,\\infty } \\le \\Vert u - P_hu\\Vert _{2,2,h} \\le Ch \\epsilon ^{-2}\\rho _4^{\\frac{1}{2}}(\\epsilon ),$ which gives the explicit formulation of $\\gamma _3$ in (REF ).",
"[Piecewise $L^\\infty (H^{-1})$ and $L^2(H^1)$ error estimates] Assume $u$ is the solution of (REF )–(REF ), $u_h^n$ is the numerical solution of scheme (REF )–().",
"Under the mesh constraints in Theorem 3.15 in [27], we have the following error estimate $&\\frac{1}{4}\\Vert \\nabla \\widetilde{\\Delta }_h^{-1}\\theta ^{\\ell }\\Vert _{0,2,h}^2 +\\frac{k^2}{4}\\sum _{n=1}^\\ell \\Vert \\nabla \\widetilde{\\Delta }_h^{-1}d_t\\theta ^n\\Vert _{0,2,h}^2 +\\frac{\\epsilon ^4k}{16}\\sum _{n=1}^\\ell (\\nabla \\theta ^n,\\nabla \\theta ^n)_h\\\\& \\qquad + \\frac{k}{\\epsilon }\\sum _{n=1}^\\ell \\Vert \\theta ^n\\Vert _{0,4,h}^4\\le C(\\tilde{\\rho }_0(\\epsilon ) |\\ln h| h^2 +\\tilde{\\rho }_1(\\epsilon ) k^2),$ where $\\tilde{\\rho }_0(\\epsilon )$ and $\\tilde{\\rho }_1(\\epsilon )$ are polynomial $\\frac{1}{\\epsilon }$ -dependent functions and $\\widetilde{\\Delta }_h^{-1}$ is a discrete inverse Laplace operator defined in [27]."
],
[
"$L^\\infty (L^2)$ and piecewise {{formula:720af7bd-977b-4050-9a29-e9c5cddacfdb}} error estimates",
"Based on Theorem REF , the $L^\\infty (L^2)$ and piecewise $L^2(H^2)$ error estimates which depend on $\\frac{1}{\\epsilon }$ polynomially, instead of exponentially, are derived below.",
"Notice that the Theorem REF is used to circumvent the use of interpolation of $\\Vert \\cdot \\Vert _{1,2,h}$ between $\\Vert \\cdot \\Vert _{0,2,h}$ and $\\Vert \\cdot \\Vert _{2,2,h}$ , by which only the exponential dependence can be derived.",
"Assume $u$ is the solution of (REF )–(REF ), $u_h^n$ is the numerical solution of scheme (REF )–().",
"Under the mesh constraints in Theorem 3.15 in [27] and (REF ), the following $L^\\infty (L^2)$ and piecewise $L^2(H^2)$ error estimates hold $ &~\\quad \\Vert \\theta ^{\\ell }\\Vert _{0,2,\\Omega }^2+k\\sum _{n=1}^\\ell \\Vert d_t\\theta ^{n}\\Vert _{0,2,\\Omega }^2+ \\epsilon k\\sum _{n=1}^{\\ell }a_h(\\theta ^n,\\theta ^n)\\\\& \\le C\\tilde{\\rho }_2(\\epsilon ) |\\ln h|^2 h^2 +C\\tilde{\\rho }_3(\\epsilon ) |\\ln h| k^2,$ where $\\begin{aligned}\\tilde{\\rho }_2(\\epsilon ) &:= \\epsilon ^{4}\\rho _3(\\epsilon ) +\\epsilon ^{-2\\sigma _1-6}\\rho _4(\\epsilon ) + \\rho _5(\\epsilon ) +\\epsilon ^{-5}\\tilde{\\rho }_0(\\epsilon ) +\\epsilon ^{-2\\gamma _1 - 2\\gamma _2 - 2}\\tilde{\\rho }_0(\\epsilon ), \\\\\\tilde{\\rho }_3(\\epsilon ) &:= \\rho _3(\\epsilon ) +\\epsilon ^{-5}\\tilde{\\rho }_1(\\epsilon ) + \\epsilon ^{-2\\gamma _1 -2\\gamma _2 - 2}\\tilde{\\rho }_1(\\epsilon ).\\end{aligned}$ It follows from (REF ), (REF ), and (REF ) that for any $v_h\\in S^h_E$ , $&(d_t\\theta ^n,v_h)+\\epsilon a_h(\\theta ^n,v_h)\\\\=&~ [(d_tP_hu,v_h)+\\epsilon a_h(P_hu,v_h)]-[(d_tu_h^n,v_h)+\\epsilon a_h(u_h^n,v_h)]\\\\=& -(d_t\\rho ^n,v_h)+(u_t+\\epsilon \\Delta ^2u-\\frac{1}{\\epsilon }\\Delta f(u)+\\alpha u,v_h)+(R^n(u_{tt}),v_h)\\\\&-\\frac{1}{\\epsilon }(f^{\\prime }(u)\\nabla P_hu,\\nabla v_h)_h-\\alpha (P_hu,v_h)+\\frac{1}{\\epsilon }(\\nabla f(u_h^{n}),\\nabla v_h)_h\\\\=&~(-d_t\\rho ^n+\\alpha \\rho ^n,v_h)-\\frac{1}{\\epsilon }(f^{\\prime }(u)\\nabla P_hu-\\nabla f(u_h^n),\\nabla v_h)_h\\\\&+(R^n(u_{tt}),v_h), $ where the remainder $ R^n(u_{tt}):= \\frac{u(t_n) - u(t_{n-1})}{k} - u_t(t_n) =-\\frac{1}{k}\\int ^{t_n}_{t_{n-1}}(s-t_{n-1})u_{tt}(s)\\,{\\rm d}s.$ Choosing $v_h=\\theta ^n$ , taking summation over $n$ from 1 to $\\ell $ , multiplying $k$ on both sides of (REF ), we have $ & ~\\quad \\frac{1}{2}\\Vert \\theta ^\\ell \\Vert _{0,2}^2 +\\frac{k}{2}\\sum _{n=1}^\\ell \\Vert d_t \\theta ^n\\Vert _{0,2}^2 + \\epsilon k\\sum _{n=1}^\\ell a_h(\\theta ^n, \\theta ^n) \\\\&= k\\sum _{n=1}^\\ell (-d_t\\rho ^n+\\alpha \\rho ^n,\\theta ^n) -\\frac{k}{\\epsilon }\\sum _{n=1}^\\ell (f^{\\prime }(u)\\nabla P_hu-\\nabla f(u_h^n),\\nabla \\theta ^n)_h\\\\&~\\quad +k\\sum _{n=1}^\\ell (R^n(u_{tt}),\\theta ^n):= I_1 + I_2 + I_3.",
"$ Estimate of $I_1$ : The first term on the right hand side of (REF ) can be bounded by $I_1 &= k\\sum _{n=1}^\\ell (-d_t\\rho ^n+\\alpha \\rho ^n,\\theta ^n)\\\\&\\le Ck \\sum _{n=1}^\\ell \\Vert d_t\\rho ^n\\Vert _{0,2}^2 + Ck\\sum _{n=1}^\\ell \\alpha ^2 \\Vert \\rho ^n\\Vert _{0,2}^2 + Ck \\sum _{n=1}^\\ell \\Vert \\theta ^n\\Vert _{0,2}^2 \\\\&\\le C (\\epsilon ^4\\rho _3(\\epsilon ) + \\epsilon ^{-6}\\rho _4(\\epsilon )) h^2+ C\\rho _5(\\epsilon )|\\ln h|h^2 +Ck\\sum _{n=1}^\\ell \\Vert \\theta ^n\\Vert _{0,2}^2,$ where by (REF ) and () $k \\sum _{n=1}^\\ell \\Vert d_t \\rho ^n\\Vert _{0,2}^2 & = \\frac{1}{k}\\sum _{n=1}^\\ell \\Vert \\int _{t_{n-1}}^{t_n} \\rho _t \\,{\\rm d}s\\Vert _{0,2}^2\\le \\sum _{n=1}^\\ell \\int _{t_{n-1}}^{t_n} \\Vert \\rho _t \\Vert _{0,2}^2 \\,{\\rm d}s\\\\& \\le \\int _0^T \\Vert \\rho _t\\Vert _{0,2}^2 \\,{\\rm d}s \\le C\\epsilon ^4\\rho _3(\\epsilon )h^2 + C \\rho _5(\\epsilon )|\\ln h|h^2, \\\\k \\sum _{n=1}^\\ell \\alpha ^2 \\Vert \\rho ^n\\Vert _{0,2}^2 & \\le C \\epsilon ^{-6}\\underset{1\\le n \\le \\ell }{\\mbox{\\rm sup }}\\Vert \\rho ^n\\Vert _{0,2}^2 \\le C\\epsilon ^{-6}\\rho _4(\\epsilon )h^2.$ Estimate of $I_2$ : The second term on the right hand side of (REF ) can be written as $&-\\frac{k}{\\epsilon } \\sum _{n=1}^\\ell (f^{\\prime }(u)\\nabla P_hu-\\nabla f(u_h^n),\\nabla \\theta ^n)_h\\\\=&-\\frac{k}{\\epsilon }\\sum _{n=1}^\\ell (f^{\\prime }(u)\\nabla P_hu-f^{\\prime }(P_hu)\\nabla P_hu,\\nabla \\theta ^n)_h \\\\&- \\frac{k}{\\epsilon }\\sum _{n=1}^\\ell (\\nabla f(P_hu)-f^{\\prime }(P_hu)\\nabla u_h^n,\\nabla \\theta ^n)_h \\\\&-\\frac{k}{\\epsilon }\\sum _{n=1}^\\ell (f^{\\prime }(P_hu)\\nabla u_h^n-\\nabla f(u_h^n),\\nabla \\theta ^n)_h := J_1 + J_2 + J_3.",
"$ By (REF ), (REF ) and mesh condition (REF ), we have $\\Vert \\nabla P_hu\\Vert _{0,2}^2 \\le \\Vert \\nabla u\\Vert _{0,2}^2 + C \\le \\epsilon ^{-2\\sigma _1 - 1}.$ Then, using (REF ) and the piecewise $L^2(H^1)$ error estimate given in Theorem REF , the first term on the right-hand side of (REF ) can be bounded below $J_1 &= -\\frac{3k}{\\epsilon }\\sum _{n=1}^\\ell (\\rho ^n(u+P_hu)\\nabla P_hu,\\nabla \\theta ^n)_h \\\\& \\le \\frac{Ck}{\\epsilon }\\sum _{n=1}^\\ell \\Vert u+P_hu\\Vert _{0,\\infty }^2\\Vert \\rho ^n\\Vert _{0,\\infty }^2 \\Vert \\nabla P_h u\\Vert _{0,2}^2 +\\frac{Ck}{\\epsilon }\\sum _{n=1}^\\ell (\\nabla \\theta ^n, \\nabla \\theta ^n)_h \\\\& \\le C \\epsilon ^{-2\\sigma _1-6}\\rho _4(\\epsilon ) h^2 +C\\epsilon ^{-5}\\tilde{\\rho }_0(\\epsilon ){|\\ln h|}h^2 +C\\epsilon ^{-5}\\tilde{\\rho }_1(\\epsilon )k^2.",
"$ Again, thanks to the piecewise $L^2(H^1)$ error estimate given in Theorem REF , the second term on the right-hand side of (REF ) can be written as $J_2 & = -\\frac{k}{\\epsilon }\\sum _{n=1}^\\ell (f^{\\prime }(P_hu)\\nabla \\theta ^n,\\nabla \\theta ^n)_h \\le \\frac{Ck}{\\epsilon }\\sum _{n=1}^\\ell (\\nabla \\theta ^n, \\nabla \\theta ^n)_h \\\\&\\le C\\epsilon ^{-5}\\tilde{\\rho }_0(\\epsilon ){|\\ln h|}h^2 +C\\epsilon ^{-5}\\tilde{\\rho }_1(\\epsilon )k^2.",
"$ By the discrete Sobolev inequality and Theorem 3.14 in [27], we have for any $n$ , $\\Vert u_h^n\\Vert _{1,\\infty ,h} \\le C|\\ln h|^{\\frac{1}{2}}\\Vert u_h^n\\Vert _{2,2,h}\\le C\\epsilon ^{-\\gamma _2}|\\ln h|^{\\frac{1}{2}}.$ Then, the third term on the right-hand side of (REF ) can be bounded by $J_3 & = -\\frac{3k}{\\epsilon } \\sum _{n=1}^\\ell (\\theta ^n(P_hu + u_h)\\nabla u_h^n,\\nabla \\theta ^n) \\\\& \\le Ck \\sum _{n=1}^\\ell \\Vert \\theta ^n\\Vert _{0,2}^2 + \\frac{Ck}{\\epsilon ^2}\\sum _{n=1}^\\ell \\Vert P_hu + u_h^n\\Vert _{0,\\infty }^2 \\Vert u_h^n\\Vert _{1,\\infty ,h}^2\\Vert \\nabla \\theta ^n\\Vert _{0,2}^2 \\\\& \\le Ck \\sum _{n=1}^\\ell \\Vert \\theta ^n\\Vert _{0,2}^2 + C\\epsilon ^{-2\\gamma _1- 2\\gamma _2 - 2} |\\ln h| k\\sum _{n=1}^\\ell \\Vert \\nabla \\theta ^n\\Vert _{0,2}^2\\\\& \\le Ck \\sum _{n=1}^\\ell \\Vert \\theta ^n\\Vert _{0,2}^2 + C\\epsilon ^{-2\\gamma _1- 2\\gamma _2 - 2} (\\tilde{\\rho }_0(\\epsilon ){ |\\ln h|^2}h^2 +\\tilde{\\rho }_1(\\epsilon )|\\ln h|k^2 ).$ Estimate of $I_3$ : The third term on the right hand side of (REF ) can be bounded by $I_3 = k\\sum _{n=1}^\\ell (R^n(u_{tt}),\\theta ^n)& \\le Ck \\sum _{n=1}^\\ell \\Vert R^n(u_{tt})\\Vert _{0,2}^2 + Ck\\sum _{n=1}^\\ell \\Vert \\theta ^n\\Vert _{0,2}^2 \\\\& \\le C\\rho _3(\\epsilon )k^2 + Ck\\sum _{n=1}^\\ell \\Vert \\theta ^n\\Vert _{0,2}^2,$ where by () and (REF ), $k\\sum _{n=1}^{\\ell }\\Vert R^n(u_{tt})\\Vert _{0,2}^2&\\le \\frac{1}{k}\\sum _{n=1}^{\\ell } \\Bigl (\\int ^{t_n}_{t_{n-1}}(s-t_{n-1})^2\\,{\\rm d}s\\Bigr )\\Bigl (\\int ^{t_n}_{t_{n-1}}\\Vert u_{tt}(s)\\Vert _{0,2}^2\\,{\\rm d}s\\Bigr )\\\\&\\le C\\rho _3(\\epsilon )k^2.$ $L^\\infty (L^2)$ and piecewise $L^2(H^2)$ error estimates: Taking (REF ), (REF ), (REF ), (REF ), (REF ) into (REF ), we have $& \\quad ~ \\frac{1}{2} \\Vert \\theta ^{\\ell }\\Vert _{0,2}^2 + \\frac{k}{2}\\sum _{n=1}^\\ell \\Vert d_t\\theta ^{n}\\Vert _{0,2}^2 + \\epsilon k\\sum _{n=1}^{\\ell }a_h(\\theta ^n,\\theta ^n) \\\\&\\le Ck\\sum _{n=1}^{\\ell } \\Vert \\theta ^n\\Vert _{0,2}^2 \\\\&~~~ + C(\\epsilon ^{4}\\rho _3(\\epsilon ) +\\epsilon ^{-2\\sigma _1-6}\\rho _4(\\epsilon ) ) h^2\\\\&~~~ + C(\\rho _5(\\epsilon ) +\\epsilon ^{-5}\\tilde{\\rho }_0(\\epsilon ))|\\ln h| h^2+ \\epsilon ^{-2\\gamma _1 - 2\\gamma _2 -2}\\tilde{\\rho }_0(\\epsilon )|\\ln h|^2h^2 \\\\&~~~+ C(\\rho _3(\\epsilon ) + \\epsilon ^{-5}\\tilde{\\rho }_1(\\epsilon ))k^2+ C\\epsilon ^{-2\\gamma _1 - 2\\gamma _2 -2}\\tilde{\\rho }_1(\\epsilon )|\\ln h|k^2.",
"$ The desired result (REF ) is therefore obtained by the Gronwall's inequality."
],
[
"Piecewise $L^\\infty (H^2)$ and {{formula:571c5da4-666b-4326-b671-50a98cdc102b}} error estimates",
"In this subsection, we give the $\\Vert \\theta ^\\ell \\Vert _{2,2,h}^2$ estimate by taking the summation by parts in time and integration by parts in space, and using the special properties of the Morley element.",
"The $\\Vert \\theta ^\\ell \\Vert _{2,2,h}^2$ estimate below is “almost” optimal with respect to time and space.",
"Assume $u$ is the solution of (REF )–(REF ), $u_h^n$ is the numerical solution of scheme (REF )–().",
"Under the mesh constraints in Theorem 3.15 in [27] and (REF ), the following piecewise $L^\\infty (H^2)$ error estimate holds $&\\quad ~ k\\sum _{n=1}^{\\ell }\\Vert d_t\\theta ^n\\Vert _{L^2}^2+\\epsilon k^2 \\sum _{n=1}^{\\ell }a_h(d_t\\theta ^n, d_t\\theta ^n)+\\epsilon \\Vert \\theta ^\\ell \\Vert _{2,2,h}^2 \\\\&\\le C\\tilde{\\rho }_4(\\epsilon ) |\\ln h|^2 h^2+ C\\tilde{\\rho }_5(\\epsilon )|\\ln h| k^2, $ where $\\tilde{\\rho }_4(\\epsilon )& = \\epsilon ^{-2\\sigma _1 - 1}\\rho _3(\\epsilon )+ \\epsilon ^{-4}\\rho _0(\\epsilon )\\rho _4(\\epsilon ) +\\epsilon ^{-2\\sigma _1 - 5}\\rho _5(\\epsilon ) \\\\&~~~+ \\Big (\\epsilon ^{-4\\gamma _1-3} + \\epsilon ^{-4\\gamma _2-2}+ \\epsilon ^{-\\max \\lbrace 2\\sigma _1+5,2\\sigma _3+2\\rbrace -2} \\\\&\\qquad ~ + \\epsilon ^{2\\gamma _1-\\max \\lbrace 2\\sigma _1 + \\frac{13}{2}, 2\\sigma _{3} + \\frac{7}{2}, 2\\sigma _2+4,2\\sigma _4 \\rbrace - 1} \\Big ) \\tilde{\\rho }_2(\\epsilon ),\\\\\\tilde{\\rho }_5(\\epsilon )& = \\Big (\\epsilon ^{-4\\gamma _1-3} + \\epsilon ^{-4\\gamma _2-2}+ \\epsilon ^{-\\max \\lbrace 2\\sigma _1+5,2\\sigma _3+2\\rbrace -2} \\\\&\\qquad ~ + \\epsilon ^{2\\gamma _1-\\max \\lbrace 2\\sigma _1 + \\frac{13}{2}, 2\\sigma _{3} + \\frac{7}{2}, 2\\sigma _2+4,2\\sigma _4 \\rbrace - 1} \\Big ) \\tilde{\\rho }_3(\\epsilon ).$ Choosing $v_h = \\theta ^n - \\theta ^{n-1} = kd_t \\theta ^n$ in (REF ), taking summation over $n$ from 1 to $\\ell $ , we get $&\\quad ~ k \\sum _{n=1}^\\ell \\Vert d_t\\theta ^n\\Vert _{L^2}^2 +\\frac{\\epsilon }{2} a_h(\\theta ^\\ell , \\theta ^\\ell ) +\\frac{\\epsilon k^2}{2} \\sum _{n=1}^\\ell a_h(d_t\\theta ^n,d_t\\theta ^n)\\\\&= k\\sum _{n=1}^\\ell (-d_t\\rho ^n+\\alpha \\rho ^n,d_t\\theta ^n)-\\frac{k}{\\epsilon }\\sum _{n=1}^\\ell (f^{\\prime }(u)\\nabla P_hu-\\nabla f(u_h^n),\\nabla (d_t\\theta ^n))_h\\\\&\\quad + k \\sum _{n=1}^\\ell (R^n(u_{tt}),d_t\\theta ^n) := I_1 + I_2 +I_3.",
"$ Here we use the fact that $\\epsilon a_h(\\theta ^n,\\theta ^n-\\theta ^{n-1}) =\\frac{\\epsilon k^2}{2}a_h(d_t \\theta ^n, d_t \\theta ^n) +\\frac{\\epsilon }{2}a_h(\\theta ^n,\\theta ^n)-\\frac{\\epsilon }{2}a_h(\\theta ^{n-1},\\theta ^{n-1}).$ Estimates of $I_1$ and $I_3$: Similar to (REF ), using (REF ) and (), we have $I_1 &\\le Ck \\sum _{n=1}^\\ell \\Vert d_t\\rho ^n\\Vert _{L^2}^2 + Ck\\sum _{n=1}^\\ell \\alpha ^2 \\Vert \\rho ^n\\Vert _{L^2}^2 + \\frac{k}{8} \\sum _{n=1}^\\ell \\Vert d_t \\theta ^n\\Vert _{L^2}^2 \\\\&\\le C (\\epsilon ^4\\rho _3(\\epsilon ) + \\epsilon ^{-6}\\rho _4(\\epsilon )) h^2+ C\\rho _5(\\epsilon )|\\ln h|h^2 + \\frac{k}{8}\\sum _{n=1}^\\ell \\Vert d_t \\theta ^n\\Vert _{0,2,h}^2.$ From (REF ) and (REF ), we also obtain the estimate of $I_3$ below $I_3 = k\\sum _{n=1}^\\ell (R^n(u_{tt}),d_t\\theta ^n)& \\le Ck \\sum _{n=1}^\\ell \\Vert R^n(u_{tt})\\Vert _{L^2}^2 + \\frac{k}{8}\\sum _{n=1}^\\ell \\Vert d_t\\theta ^n\\Vert _{0,2}^2 \\\\& \\le C\\rho _3(\\epsilon )k^2 + \\frac{k}{8} \\sum _{n=1}^\\ell \\Vert d_t \\theta ^n\\Vert _{0,2}^2 .$ Estimate of $I_2$: Next we bound the more complicated term $I_2$ .",
"Using integration by parts, we have $ I_2 &= -\\frac{k}{\\epsilon }\\sum _{n=1}^\\ell (f^{\\prime }(u)\\nabla P_hu - \\nabla f(P_hu), d_t\\nabla \\theta ^n)_h - \\frac{k}{\\epsilon }\\sum _{n=1}^\\ell (\\nabla ( f(P_hu) - f(u_h^n)),d_t\\nabla \\theta ^n)_h \\\\&= -\\frac{k}{\\epsilon }\\sum _{n=1}^\\ell (f^{\\prime }(u)\\nabla P_hu - \\nabla f(P_hu), d_t\\nabla \\theta ^n)_h + \\frac{k}{\\epsilon }\\sum _{n=1}^\\ell (f(P_hu) - f(u_h^n), d_t\\Delta \\theta ^n)_h \\\\&~~~ - \\frac{k}{\\epsilon }\\sum _{n=1}^\\ell \\sum _{E \\in \\mathcal {E}_h}(\\lbrace f(P_hu) - f(u_h^n)\\rbrace , d_t \\llbracket \\nabla \\theta ^n\\rrbracket )_E\\\\&~~~ - \\frac{k}{\\epsilon }\\sum _{n=1}^\\ell \\sum _{E\\in \\mathcal {E}_h}(\\llbracket f(P_hu) - f(u_h^n)\\rrbracket , \\lbrace \\nabla d_t\\theta ^n\\rbrace )_E:= J_1 + J_2 + J_3 + J_4.$ Here we adopt the standard DG notation and the DG identity, see [5].",
"Next we bound $J_1$ to $J_4$ respectively."
],
[
"$\\bullet $ Estimate of {{formula:98454e9d-5e82-4d5c-8cf0-0696c992864a}}",
"Using summation by parts in Lemma , we have $ J_1 &= \\frac{k}{\\epsilon }\\sum _{n=1}^\\ell (d_t (\\rho (u+P_hu)\\nabla P_hu),\\nabla \\theta ^{n-1})_h - \\frac{1}{\\epsilon }(\\rho ^\\ell (u^\\ell +P_hu^\\ell )\\nabla P_hu^\\ell , \\nabla \\theta ^\\ell )_h.$ Thanks to (REF ), (), (REF ), (REF ), (), (REF ), and the piecewise $L^2(H^1)$ estimate in Theorem REF , the first term on the right hand side of (REF ) can be bounded by $ & \\quad ~ \\frac{k}{\\epsilon }\\sum _{n=1}^\\ell (d_t (\\rho (u+P_hu)\\nabla P_hu),\\nabla \\theta ^{n-1})_h \\\\& \\le \\frac{1}{k} \\sum _{n=1}^\\ell \\Vert \\int _{t_{n-1}}^{t_n} (\\rho (u+P_hu)\\nabla P_hu)_t \\,{\\rm d}s\\Vert _{0,2}^2 + C\\epsilon ^{-2} k \\sum _{n=1}^\\ell |\\theta ^{n-1}|_{1,2,h}^2 \\\\& \\le \\underset{t\\in [0,T]}{\\mbox{\\rm ess sup }}\\Vert \\nabla P_h u\\Vert _{0,2}^2\\int _0^T\\Vert \\rho _t\\Vert _{0,\\infty }^2 \\,{\\rm d}s +\\underset{t\\in [0,T]}{\\mbox{\\rm ess sup }}\\Vert \\rho \\Vert _{0,\\infty }^2 \\int _0^T \\Vert \\nabla (P_hu)_t\\Vert _{0,2}^2\\,{\\rm d}s\\\\& + \\underset{t\\in [0,T]}{\\mbox{\\rm ess sup }}\\Vert \\rho \\Vert _{0,\\infty }^2 \\Vert \\nabla P_hu \\Vert _{0,2}^2 \\int _0^T\\Vert u_t + (P_hu)_t\\Vert _{0,\\infty }^2 \\,{\\rm d}s + C\\epsilon ^{-2} k \\sum _{n=1}^\\ell |\\theta ^{n-1}|_{1,2,h}^2\\\\& \\le C\\epsilon ^{-2\\sigma _1 - 1}(\\rho _3(\\epsilon ) +\\epsilon ^{-4}\\rho _5(\\epsilon )|\\ln h|)h^2 + C\\epsilon ^{-4}\\rho _0(\\epsilon )\\rho _4(\\epsilon )h^2\\\\&~~~ + C\\epsilon ^{-2\\sigma _1 - 6 -\\max \\lbrace 2\\sigma _1 +\\frac{13}{2}, 2\\sigma _3+ \\frac{7}{2}, 2\\sigma _2 + 4,2\\sigma _4\\rbrace }\\rho _4(\\epsilon ) h^2 \\\\&~~~ + C\\epsilon ^{-6}\\tilde{\\rho }_0(\\epsilon )|\\ln h|h^2 +C\\epsilon ^{-6}\\tilde{\\rho }_1(\\epsilon )k^2.$ Thanks to (REF ), (REF ) and the $L^\\infty (L^2)$ estimate in Theorem REF , the second term on the right hand of (REF ) can be bounded by $ &\\quad ~ - \\frac{1}{\\epsilon }(\\rho ^\\ell (u^\\ell + P_hu^\\ell )\\nabla P_hu^\\ell , \\nabla \\theta ^\\ell )_h \\\\& \\le C \\epsilon ^{-2}\\Vert \\rho ^l\\Vert _{0,\\infty }^2 |P_hu^l|_{1,2,h}^2 +C\\epsilon ^{-1}\\Vert \\theta \\Vert _{0,2}^2 + \\frac{\\epsilon }{8}a_h(\\theta ^l,\\theta ^l) \\\\& \\le C\\epsilon ^{-2\\sigma _1 - 7}\\rho _4(\\epsilon ) h^2 +C\\epsilon ^{-1}\\tilde{\\rho }_2(\\epsilon )|\\ln h|^2 h^2 +C\\epsilon ^{-1}\\tilde{\\rho }_3(\\epsilon )|\\ln h|k^2 +\\frac{\\epsilon }{8}a_h(\\theta ^l, \\theta ^l).$ Combining (REF ) and (REF ), simplifying the coefficients according to the definition of $\\rho _i(\\epsilon )$ and $\\tilde{\\rho }_i(\\epsilon )$ , we obtain the bound for $J_1$ : $ J_1 &\\le C(\\epsilon ^{-2\\sigma _1 - 1}\\rho _3(\\epsilon ) +\\epsilon ^{-4}\\rho _0(\\epsilon )\\rho _4(\\epsilon ) +\\epsilon ^{-2\\sigma _1-5}\\rho _5(\\epsilon ) +\\epsilon ^{-1}\\tilde{\\rho }_2(\\epsilon ) ) |\\ln h|^2 h^2 \\\\&~~~ + C\\epsilon ^{-1}\\tilde{\\rho }_3(\\epsilon )|\\ln h|k^2 +\\frac{\\epsilon }{8}a_h(\\theta ^l, \\theta ^l).",
"$"
],
[
"$\\bullet $ Estimate of {{formula:0abceb82-f214-4383-8407-005c84057dee}}",
"Define $f(P_hu) - f(u_h^n) := M^n\\theta ^n$ , where $M^n$ is given as $M^n := (P_hu(t_n))^2 + P_hu(t_n)u_h^n + (u_h^n)^2 - 1.$ Using summation by parts in Lemma , we have $ J_2 &= -\\frac{k}{\\epsilon }\\sum _{n=1}^\\ell (d_t (M^n\\theta ^n), \\Delta \\theta ^{n-1})_h +\\frac{1}{\\epsilon }( M^l \\theta ^l, \\Delta \\theta ^l)_h \\\\& \\le \\frac{Ck}{\\epsilon } \\sum _{n=1}^\\ell \\Vert d_t(M^n\\theta ^n)\\Vert _{0,2}|\\theta |_{2,2,h} + \\frac{C}{\\epsilon }\\Vert M^l\\theta ^l\\Vert _{0,2}|\\theta ^l|_{2,2,h}.",
"$ Since $d_t u_h^n = d_t(P_hu^n) - d_t\\theta ^n$ , a direct calculation shows that $d_t(M^n \\theta ^n) &= \\theta ^n d_t M^n + M^{n-1}d_t \\theta ^n \\\\& = M^{n-1}d_t \\theta ^n + \\theta ^n(P_h u^{n} + P_h u^{n-1})d_t(P_hu^n) \\\\&~~~ + \\theta ^n u_h^n d_t(P_hu^n) + \\theta ^n P_h u^{n-1}d_t(P_hu^n)- \\theta ^n P_hu^{n-1} d_t \\theta ^n \\\\&~~~ + \\theta ^n(u_h^n+u_h^{n-1})d_t(P_hu^n) - \\theta ^n(u_h^n +u_h^{n-1})d_t\\theta ^n \\\\& = (M^{n-1} -\\theta ^nP_h u^{n-1} - \\theta ^n(u_h^n +u_h^{n-1}))d_t\\theta ^n \\\\&~~~ + (P_hu^n + 2P_hu^{n-1} + 2u_h^n + u_h^{n-1})\\theta ^nd_t(P_hu^n).$ Using the $L^2(H^2)$ error estimate (REF ) and the assumption on the $L^\\infty $ bound of $u_h^n$ , we get $ & \\quad ~\\frac{Ck}{\\epsilon }\\sum _{n=1}^\\ell \\Vert d_t(M^n\\theta ^n)\\Vert _{0,2}|\\theta ^n|_{2,2,h} \\\\& \\le C\\epsilon ^{-2\\gamma _1 - 1} k\\sum _{n=1}^\\ell \\Vert d_t\\theta ^n\\Vert _{0,2}|\\theta ^n|_{2,2,h}+ C\\epsilon ^{-\\gamma _1 - 1} k\\sum _{n=1}^\\ell \\Vert \\theta ^n d_t(P_hu)\\Vert _{0,2}|\\theta ^n|_{2,2,h} \\\\& \\le \\frac{k}{8}\\sum _{n=1}^\\ell \\Vert d_t \\theta ^n\\Vert _{0,2}^2 +C\\epsilon ^{-4\\gamma _1 - 2} k\\sum _{n=1}^\\ell |\\theta |_{2,2,h}^2 +C\\epsilon ^{2\\gamma _1}k\\sum _{n=1}^\\ell \\Vert \\theta d_t(P_h u)\\Vert _{0,2}^2\\\\& \\le \\frac{k}{8}\\sum _{n=1}^\\ell \\Vert d_t \\theta ^n\\Vert _{0,2}^2 +C\\epsilon ^{-4\\gamma _1 - 3}(\\tilde{\\rho }_2(\\epsilon )|\\ln h|^2h^2 +\\tilde{\\rho }_3(\\epsilon )|\\ln h|k^2) \\\\&~~~ +C \\epsilon ^{2\\gamma _1-\\max \\lbrace 2\\sigma _1 + \\frac{13}{2}, 2\\sigma _3+\\frac{7}{2},2\\sigma 2+4, 2\\sigma _4 \\rbrace -1}(\\tilde{\\rho }_2(\\epsilon )|\\ln h|^2 h^2 +\\tilde{\\rho }_3(\\epsilon )|\\ln h| k^2), $ where by () and the $L^\\infty (L^2)$ error estimate (REF ), $\\begin{aligned}&\\quad ~k\\sum _{n=1}^\\ell \\Vert \\theta d_t(P_h u)\\Vert _{0,2}^2 \\\\&\\le \\underset{1\\le n \\le \\ell }{\\mbox{\\rm sup }}\\Vert \\theta ^n\\Vert _{0,2}^2 \\frac{1}{k} \\Vert \\int _{t_{n-1}}^{t_n}(P_hu)_t \\,{\\rm d}s\\Vert _{0,\\infty }^2 \\\\& \\le \\underset{1\\le n \\le \\ell }{\\mbox{\\rm sup }}\\Vert \\theta ^n\\Vert _{0,2}^2 \\int _{0}^{T}\\Vert (P_hu)_t\\Vert _{0,\\infty }^2 \\,{\\rm d}s \\\\& \\le C \\epsilon ^{-\\max \\lbrace 2\\sigma _1 + \\frac{13}{2}, 2\\sigma _3+\\frac{7}{2}, 2\\sigma 2+4, 2\\sigma _4\\rbrace -1}(\\tilde{\\rho }_2(\\epsilon )|\\ln h|^2 h^2 +\\tilde{\\rho }_3(\\epsilon )|\\ln h| k^2).\\end{aligned}$ And the second term on the right hand side of (REF ) can be bounded by $ \\frac{C}{\\epsilon } \\Vert M^l\\theta ^l\\Vert _{0,2}|\\theta ^l|_{2,2,h}&\\le C^{-4\\gamma _1-3}\\Vert \\theta ^l\\Vert _{0,2}^2 +\\frac{\\epsilon }{8}a_h(\\theta ^l, \\theta ^l) \\\\&\\le C\\epsilon ^{-4\\gamma _1 - 3}(\\tilde{\\rho }_2(\\epsilon )|\\ln h|^2 h^2 +\\tilde{\\rho }_3(\\epsilon )|\\ln h|k^2)+ \\frac{\\epsilon }{8}a_h(\\theta ^l, \\theta ^l).",
"$ Combining (REF ) and (REF ), we obtain the bound for $J_2$ : $ J_2 &\\le \\frac{k}{8}\\sum _{n=1}^\\ell \\Vert d_t \\theta ^n\\Vert _{0,2}^2+ \\frac{\\epsilon }{8}a_h(\\theta ^l, \\theta ^l) +C\\epsilon ^{-4\\gamma _1 - 3}(\\tilde{\\rho }_2(\\epsilon )|\\ln h|^2 h^2 +\\tilde{\\rho }_3(\\epsilon )|\\ln h|k^2) \\\\&~~~ +C \\epsilon ^{2\\gamma _1-\\max \\lbrace 2\\sigma _1 + \\frac{13}{2}, 2\\sigma _3+\\frac{7}{2},2\\sigma _2+4, 2\\sigma _4 \\rbrace -1}(\\tilde{\\rho }_2(\\epsilon )|\\ln h|^2 h^2 +\\tilde{\\rho }_3(\\epsilon )|\\ln h| k^2).",
"$"
],
[
"$\\bullet $ Estimate of {{formula:ba3899ba-38c3-46c3-9f70-5e58537b8782}}",
"Notice that $\\theta ^n \\in S_E^h$ and $\\int _{E} \\llbracket \\nabla \\theta ^n\\rrbracket \\,{\\rm d}S = 0 \\qquad \\forall E \\in \\mathcal {E}_h.$ Using summation by parts in Lemma , Lemma 2.2 in [14] and inverse inequality, we have $J_3 &= \\frac{k}{\\epsilon }\\sum _{n=1}^\\ell \\sum _{E\\in \\mathcal {E}_h}(d_t \\lbrace M^n\\theta ^n\\rbrace , \\llbracket \\nabla \\theta ^{n-1}\\rrbracket )_E -\\frac{1}{\\epsilon }\\sum _{E\\in \\mathcal {E}_h}(\\lbrace M^\\ell \\theta ^{\\ell }\\rbrace , \\llbracket \\nabla \\theta ^{\\ell }\\rrbracket )_E \\\\& \\le \\frac{Ck}{\\epsilon } \\sum _{n=1}^\\ell \\Vert d_t(M^n\\theta ^n)\\Vert _{0,2}|\\theta |_{2,2,h} + \\frac{C}{\\epsilon }\\Vert M^\\ell \\theta ^\\ell \\Vert _{0,2}|\\theta ^l|_{2,2,h}.$ Hence, $J_3$ has the same bound as $J_2$ ."
],
[
"$\\bullet $ Estimate of {{formula:753fcd32-5f85-4a07-9d62-420dfe77cac4}}",
"Since $P_hu$ and $u_h$ are continuous at vertexes of $\\mathcal {T}_h$ , thanks to Lemma 2.6 in [14], we have $ J_4 & \\le \\frac{Ck}{\\epsilon } \\sum _{n=1}^\\ell h|M^n\\theta ^n|_{2,2,h}|d_t\\theta ^n|_{1,2,h}\\\\&\\le \\frac{Ck}{\\epsilon } \\sum _{n=1}^\\ell |M^n\\theta ^n|_{2,2,h}\\Vert d_t\\theta ^n\\Vert _{0,2} \\\\& \\le \\frac{Ck}{\\epsilon ^2} \\sum _{n=1}^\\ell |M^n\\theta ^n|_{2,2,h}^2 +\\frac{k}{8}\\sum _{n=1}^\\ell \\Vert d_t \\theta ^n\\Vert _{0,2}^2 .$ Using the piecewise $L^2(H^2)$ estimate given in Theorem REF , we have $ & \\quad ~\\frac{Ck}{\\epsilon ^2} \\sum _{n=1}^\\ell |M^n\\theta ^n|_{2,2,h}^2\\\\& \\le \\frac{Ck}{\\epsilon ^2} \\sum _{n=1}^\\ell \\left(\\Vert M^n\\Vert _{0,\\infty }^2 |\\theta ^n|_{2,2,h}^2 +|M^n|_{1,4,h}^2|\\theta ^n|_{1,4,h}^2 +|M^n|_{2,2,h}^2 \\Vert \\theta ^n\\Vert _{0,\\infty }^2\\right) \\\\& \\le \\frac{C}{\\epsilon ^2}\\underset{1\\le n \\le \\ell }{\\mbox{\\rm sup }}\\Vert M^n\\Vert _{2,2,h}^2 k\\sum _{n=1}^\\ell \\Vert \\theta ^n\\Vert _{2,2,h}^2 \\\\& \\le C(\\epsilon ^{-4\\gamma _2-2} + \\epsilon ^{-\\max \\lbrace 2\\sigma _1+5,2\\sigma _3+2\\rbrace -2})(\\tilde{\\rho }_2(\\epsilon )|\\ln h|^2 h^2 +\\tilde{\\rho }_3(\\epsilon )|\\ln h|k^2),$ where by () and the fact that $\\Vert u_h^n\\Vert _{2,2,h} \\le C\\epsilon ^{-\\gamma _2}$ (c.f.",
"[27]) $\\begin{aligned}\\Vert M^n\\Vert _{2,2,h} &\\le C (\\Vert (P_hu^n)^2\\Vert _{2,2,h} + \\Vert u_h^nP_hu^n\\Vert _{2,2,h} + \\Vert (u_h^n)^2\\Vert _{2,2,h}) \\\\& \\le C(\\Vert P_hu^n\\Vert _{2,2,h} + \\Vert P_hu^n\\Vert _{1,4,h}^2 +\\Vert u_h\\Vert _{0,\\infty }\\Vert u_h^n\\Vert _{2,2,h} + \\Vert u_h^n\\Vert _{1,4,h}^2\\\\&~~~ + \\Vert u_h^n\\Vert _{2,2,h} + \\Vert u_h^n\\Vert _{0,\\infty }\\Vert P_hu^n\\Vert _{2,2,h} +\\Vert u_h^n\\Vert _{1,4,h}\\Vert P_hu^n\\Vert _{1,4,h}) \\\\& \\le C(\\epsilon ^{-2\\gamma _2} + \\epsilon ^{-\\max \\lbrace 2\\sigma _1+5,2\\sigma _3 + 2\\rbrace }).\\end{aligned}$ Piecewise $L^\\infty (H^2)$ error estimate: Taking (REF ), (REF ), (REF ), (REF ) and (REF ) into (REF ), we obtain $&\\quad ~ \\frac{k}{8} \\sum _{n=1}^\\ell \\Vert d_t\\theta ^n\\Vert _{L^2}^2 +\\frac{\\epsilon }{8} a_h(\\theta ^\\ell , \\theta ^\\ell ) +\\frac{\\epsilon k^2}{2} \\sum _{n=1}^\\ell a_h(d_t\\theta ^n,d_t\\theta ^n) \\\\& \\le C (\\epsilon ^4\\rho _3(\\epsilon ) + \\epsilon ^{-6}\\rho _4(\\epsilon )) h^2+ C\\rho _5(\\epsilon )|\\ln h|^2 h^2 + C\\rho _3(\\epsilon )k^2\\\\&~~~ + C(\\epsilon ^{-2\\sigma _1 - 1}\\rho _3(\\epsilon ) +\\epsilon ^{-4}\\rho _0(\\epsilon )\\rho _4(\\epsilon ) +\\epsilon ^{-2\\sigma _1-5}\\rho _5(\\epsilon ) +\\epsilon ^{-1}\\tilde{\\rho }_2(\\epsilon ) ) |\\ln h|^2 h^2 \\\\&~~~ + C\\epsilon ^{-1}\\tilde{\\rho }_3(\\epsilon )|\\ln h|k^2 +C\\epsilon ^{-4\\gamma _1 - 3}(\\tilde{\\rho }_2(\\epsilon )|\\ln h|^2 h^2 +\\tilde{\\rho }_3(\\epsilon )|\\ln h|k^2) \\\\&~~~ + C \\epsilon ^{-\\max \\lbrace 2\\sigma _1 + \\frac{13}{2},2\\sigma _3+\\frac{7}{2}, 2\\sigma _2+4,2\\sigma _4\\rbrace -1}(\\tilde{\\rho }_2(\\epsilon )|\\ln h|^2 h^2 +\\tilde{\\rho }_3(\\epsilon )|\\ln h| k^2) \\\\&~~~ + C(\\epsilon ^{-4\\gamma _2-2} + \\epsilon ^{-\\max \\lbrace 2\\sigma _1+5,2\\sigma _3+2\\rbrace -2})(\\tilde{\\rho }_2(\\epsilon )|\\ln h|^2 h^2 +\\tilde{\\rho }_3(\\epsilon )|\\ln h|k^2).",
"$ Then the theorem can be proved by simplifying the coefficients according to the definitions of $\\rho _i(\\epsilon )$ and $\\tilde{\\rho }_i(\\epsilon )$ .",
"Remark 2 If the summation by part for time and integration by part for space techniques are not employed simultaneously, one can only obtain a coarse estimate $&\\quad ~\\Vert \\theta ^\\ell \\Vert _{2,2,h}^2 + k\\sum _{n=1}^{\\ell }\\Vert d_t\\theta ^n\\Vert _{L^2}^2+\\epsilon k^2 \\sum _{n=1}^{\\ell }a_h(d_t\\theta ^n, d_t \\theta ^n) \\\\&\\le Ck^{-\\frac{1}{2}}(\\epsilon ^{-\\gamma _4}|\\ln h|^2h^2+\\epsilon ^{-\\gamma _5}|\\ln h|k),$ where $\\gamma _4, \\gamma _5$ denote some positive constants.",
"Finally, using (REF ), Theorem REF and the Sobolev embedding theorem, we can prove the desired $L^\\infty (L^\\infty )$ error estimate.",
"Assume $u$ is the solution of (REF )–(REF ), $u_h^n$ is the numerical solution of scheme (REF )–().",
"Under the mesh constraints in Theorem 3.15 in [27] and (REF ), we have the $L^\\infty (L^\\infty )$ error estimate $ \\Vert u(t_n) - u_h^n\\Vert _{L^{\\infty }}\\le C|\\ln h|^{\\frac{1}{2}}((\\tilde{\\rho }_4(\\epsilon ))^{\\frac{1}{2}}|\\ln h|^{\\frac{1}{2}}h + (\\tilde{\\rho }_5(\\epsilon ))^{\\frac{1}{2}}k)\\quad \\forall 1 \\le n \\le \\ell .$ Remark 3 The mesh constraints in Theorem 3.15 in [27] and (REF ) can be achieved by $h = C\\epsilon ^{p_1}$ and $k = C\\epsilon ^{p_2}$ for certain positive $p_1, p_2$ .",
"Hence, the $|\\ln h| k^2$ decreases asymptoticly as $k^2$ when $\\epsilon $ goes to zero."
],
[
"Convergence of the Numerical Interface",
"In this section, we prove that the numerical interface defined as the zero level set of the Morley element interpolation of the solution $U^n$ converges to the moving interface of the Hele-Shaw problem under the assumption that the Hele-Shaw problem has a unique global (in time) classical solution.",
"We first cite the following convergence result established in [2].",
"Let $\\Omega $ be a given smooth domain and $\\Gamma _{00}$ be a smooth closed hypersurface in $\\Omega $ .",
"Suppose that the Hele-Shaw problem starting from $\\Gamma _{00}$ has a unique smooth solution $\\bigl (w,\\Gamma :=\\bigcup _{0\\le t\\le T}(\\Gamma _t\\times \\lbrace t\\rbrace ) \\bigr )$ in the time interval $[0,T]$ such that $\\Gamma _t\\subseteq \\Omega $ for all $t\\in [0,T]$ .",
"Then there exists a family of smooth functions $\\lbrace u_{0}^{\\epsilon }\\rbrace _{0<\\epsilon \\le 1}$ which are uniformly bounded in $\\epsilon \\in (0,1]$ and $(x,t)\\in \\overline{\\Omega }_T$ , such that if $u^{\\epsilon }$ solves the Cahn-Hilliard problem (REF )–(), then (i) $\\displaystyle {\\lim _{\\epsilon \\rightarrow 0}}u^{\\epsilon }(x,t)= {\\left\\lbrace \\begin{array}{ll}1 &\\qquad \\mbox{if}\\, (x,t)\\in \\mathcal {O}\\\\-1 &\\qquad \\mbox{if}\\, (x,t)\\in \\mathcal {I}\\end{array}\\right.",
"}\\,\\mbox{ uniformly on compact subsets}$ , where $\\mathcal {I}$ and $\\mathcal {O}$ stand for the “inside\" and “outside\" of $\\Gamma $ ; (ii) $\\displaystyle {\\lim _{\\epsilon \\rightarrow 0}}\\bigl ( \\epsilon ^{-1} f(u^{\\epsilon })-\\epsilon \\Delta u^{\\epsilon } \\bigr )(x,t)=-w(x,t)$ uniformly on $\\overline{\\Omega }_T$ .",
"We are now ready to state the first main theorem of this section.",
"Let $\\lbrace \\Gamma _t\\rbrace _{t\\ge 0}$ denote the zero level set of the Hele-Shaw problem and $U_{\\epsilon ,h,k}(x,t)$ denotes the piecewise linear interpolation in time of the numerical solution $u_h^n$ , namely, $U_{\\epsilon ,h,k}(x,t):=\\frac{t-t_{n-1}}{k}u_h^{n}(x)+\\frac{t_{n}-t}{k}u_h^{n-1}(x), $ for $t_{n-1}\\le t\\le t_{n}$ and $1\\le n\\le M$ .",
"Then, under the mesh and starting value constraints of Theorem REF and $k=O(h^q)$ with $0<q<1$ , we have (i) $U_{\\epsilon ,h,k}(x,t) \\stackrel{\\epsilon \\searrow 0}{\\longrightarrow } 1$ uniformly on compact subset of $\\mathcal {O}$ , (ii) $U_{\\epsilon ,h,k}(x,t) \\stackrel{\\epsilon \\searrow 0}{\\longrightarrow } -1$ uniformly on compact subset of $\\mathcal {I}$ .",
"For any compact set $A\\subset \\mathcal {O}$ and for any $(x,t)\\in A$ , we have $ |U_{\\epsilon ,h,k}-1|&\\le |U_{\\epsilon ,h,k}-u^{\\epsilon }(x,t)|+|u^{\\epsilon }(x,t)-1| \\\\&\\le |U_{\\epsilon ,h,k}-u^{\\epsilon }(x,t)|_{L^{\\infty }(\\Omega _T)}+|u^{\\epsilon }(x,t)-1|.\\nonumber $ Theorem REF infers that $|U_{\\epsilon ,h,k}-u^{\\epsilon }(x,t)|_{L^{\\infty }(\\Omega _T)}\\le C(\\tilde{\\rho }_6(\\epsilon ))^{\\frac{1}{2}}h^q|\\ln h|.$ where $\\tilde{\\rho }_6(\\epsilon )=\\max \\lbrace \\tilde{\\rho }_4(\\epsilon ),\\tilde{\\rho }_5(\\epsilon )\\rbrace .$ The first term on the right-hand side of (REF ) tends to 0 when $\\epsilon \\searrow 0$ (note that $h,k\\searrow 0$ , too).",
"The second term converges uniformly to 0 on the compact set $A$ , which is ensured by (i) of Theorem .",
"Hence, the assertion (i) holds.",
"To show (ii), we only need to replace $\\mathcal {O}$ by $\\mathcal {I}$ and 1 by $-1$ in the above proof.",
"The second main theorem addresses the convergence of numerical interfaces.",
"Let $\\Gamma _t^{\\epsilon ,h,k}:=\\lbrace x\\in \\Omega ;\\, U_{\\epsilon ,h,k}(x,t)=0\\rbrace $ be the zero level set of $U_{\\epsilon ,h,k}(x,t)$ , then under the assumptions of Theorem , we have $\\sup _{x\\in \\Gamma _t^{\\epsilon ,h,k}} \\mbox{\\rm dist}(x,\\Gamma _t)\\stackrel{\\epsilon \\searrow 0}{\\longrightarrow } 0 \\quad \\mbox{uniformly on $[0,T]$}.$ For any $\\eta \\in (0,1)$ , define the tabular neighborhood $\\mathcal {N}_{\\eta }$ of width $2\\eta $ of $\\Gamma _t$ $\\mathcal {N}_{\\eta }:=\\lbrace (x,t)\\in \\Omega _T;\\, \\mbox{\\rm dist}(x,\\Gamma _t)<\\eta \\rbrace .$ Let $A$ and $B$ denote the complements of the neighborhood $\\mathcal {N}_{\\eta }$ in $\\mathcal {O}$ and $\\mathcal {I}$ , respectively, $A=\\mathcal {O}\\setminus \\mathcal {N}_{\\eta } \\qquad \\mbox{and}\\qquad B=\\mathcal {I}\\setminus \\mathcal {N}_{\\eta }.$ Note that $A$ is a compact subset outside $\\Gamma _t$ and $B$ is a compact subset inside $\\Gamma _t$ .",
"By Theorem , there exists ${\\epsilon _1}>0$ , which only depends on $\\eta $ , such that for any $\\epsilon \\in (0,{\\epsilon _1})$ $&|U_{\\epsilon ,h,k}(x,t)-1|\\le \\eta \\quad \\forall (x,t)\\in A,\\\\&|U_{\\epsilon ,h,k}(x,t)+1|\\le \\eta \\quad \\forall (x,t)\\in B.$ Now for any $t\\in [0,T]$ and $x\\in \\Gamma _t^{\\epsilon ,h,k}$ , from $U_{\\epsilon ,h,k}(x,t)=0$ we have $&|U_{\\epsilon ,h,k}(x,t)-1|=1\\qquad \\forall (x,t)\\in A,\\\\&|U_{\\epsilon ,h,k}(x,t)+1|=1\\qquad \\forall (x,t)\\in B.$ (REF ) and (REF ) imply that $(x,t)$ is not in $A$ , and () and () imply that $(x,t)$ is not in $B$ , then $(x,t)$ must lie in the tubular neighborhood $\\mathcal {N}_{\\eta }$ .",
"Therefore, for any $\\epsilon \\in (0,\\epsilon _1)$ , $\\sup _{x\\in \\Gamma _t^{\\epsilon ,h,k}} \\mbox{\\rm dist}(x,\\Gamma _t) \\le \\eta \\qquad \\mbox{uniformly on $[0,T]$}.$ The proof is complete."
],
[
"Numerical experiments",
"In this section, we present two two-dimensional numerical tests to gauge the performance of the proposed fully discrete Morley finite element method for Cahn-Hilliard equation.",
"The square domain $\\Omega =[-1,1]^2$ is used in both tests."
],
[
"Test 1",
"Consider the Cahn-Hilliard problem with an ellipse initial interface determined by $\\Gamma _0: \\frac{x^2}{0.36} +\\frac{y^2}{0.04} = 0$ .",
"The initial condition is chosen to have the form $u_0(x,y) = \\tanh (\\frac{d_0(x, y)}{\\sqrt{2\\epsilon }})$ , where $d_0(x, y)$ denotes the signed distance from $(x,y)$ to the initial ellipse interface $\\Gamma _0$ and $\\tanh (t) = (e^t - e^{-t})/(e^t +e^{-t})$ .",
"Figure REF displays four snapshots at four fixed time points of the numerical interface with four different $\\epsilon $ 's.",
"Here time step size $k = 1\\times 10^{-4}$ and space size $h = 0.01$ are used.",
"They clearly indicate that at each time point the numerical interface converges to the sharp interface $\\Gamma _t$ of the Hele-Shaw flow as $\\epsilon $ tends to zero.",
"Note that this initial condition may not satisfy the General Assumption (GA) due to the singularity of the signed distance function.",
"We will adopt a smooth initial condition in the later test.",
"Figure: Test 1: Snapshots of the zero-level sets of u ϵ,k u^{\\epsilon , k}at t=0,0.005,0.015,0.03t=0,0.005,0.015,0.03 and ϵ=0.08,0.04,0.03,0.02\\epsilon = 0.08,0.04,0.03,0.02."
],
[
"Test 2",
"Consider the following initial condition, which is also adopted in [23], $u_0(x,y) = \\tanh \\Big ( ((x-0.3)^2 + y^2 - 0.25^2)/\\epsilon \\Big )\\tanh \\Big ( ((x+0.3)^2 + y^2 - 0.3^2)/\\epsilon \\Big ).$ Table REF and REF show the errors of spatial $L^2$ , $H^1$ and $H^2$ semi-norms and the rates of convergence at $T =0.0002$ and $T = 0.001$ .",
"$\\epsilon = 0.08$ is used to generate the table.",
"$k = 1\\times 10^{-5}$ is chosen so that the error in time is relatively small to the error in space.",
"The $L^\\infty (H^2)$ norm error is in agreement with the convergence theorem, but $L^\\infty (L^2)$ and $L^\\infty (H^1)$ norm errors are one order higher than our theoretical results.",
"We note that in [14], the second order convergence for both $L^\\infty (L^2)$ and $L^\\infty (H^1)$ norms are proved, whereas only $\\frac{1}{\\epsilon }$ -exponential dependence can be derived.",
"Table: Spatial errors and convergence rates of Test 2: ϵ=0.08\\epsilon =0.08, k=1×10 -5 k = 1\\times 10^{-5}, T=0.0002T = 0.0002.Table: Spatial errors and convergence rates of Test 2: ϵ=0.08\\epsilon =0.08, k=1×10 -5 k = 1\\times 10^{-5}, T=0.001T = 0.001.Figure REF displays six snapshots at six fixed time points of the numerical interface with four different $\\epsilon $ .",
"Again, they clearly indicate that at each time point the numerical interface converges to the sharp interface $\\Gamma _t$ of the Hele-€揝haw flow as $\\epsilon $ tends to zero.",
"Figure: Test 2: Snapshots of the zero-level sets of u ϵ,k u^{\\epsilon , k}at t=0,0.00005,0.0002,0.001,0.006,0.015t=0,0.00005,0.0002,0.001, 0.006, 0.015 and ϵ=0.08,0.04,0.03,0.02\\epsilon =0.08,0.04,0.03,0.02."
],
[
"Acknowledgements",
"The authors Shuonan Wu and Yukun Li highly thank Professor Xiaobing Feng in the University of Tennessee at Knoxville for his motivation for this paper."
]
] | 1808.08581 |
[
[
"Parameter estimation for Gaussian processes with application to the\n model with two independent fractional Brownian motions"
],
[
"Abstract The purpose of the article is twofold.",
"Firstly, we review some recent results on the maximum likelihood estimation in the regression model of the form $X_t = \\theta G(t) + B_t$, where $B$ is a Gaussian process, $G(t)$ is a known function, and $\\theta$ is an unknown drift parameter.",
"The estimation techniques for the cases of discrete-time and continuous-time observations are presented.",
"As examples, models with fractional Brownian motion, mixed fractional Brownian motion, and sub-fractional Brownian motion are considered.",
"Secondly, we study in detail the model with two independent fractional Brownian motions and apply the general results mentioned above to this model."
],
[
"Introduction",
"Gaussian processes with drift arise in many applied areas, in particular, in telecommunication and on financial markets.",
"An observed process often can be decomposed as the sum of a useful signal and a random noise, where the last one mentioned is usually modeled by a centered Gaussian process, see, e. g., [19].",
"The simplest example of such model is the process $Y_t=\\theta t+W_t,$ where $W$ is a Wiener process.",
"In this case the MLE of the drift parameter $\\theta $ by observations of $Y$ at points $0=t_0\\le t_1\\le \\ldots \\le t_{N}=T$ is given by $\\hat{\\theta }=\\frac{1}{t_{N}-t_0}\\sum _{i=0}^{N-1}\\left(Y_{t_{i+1}}-Y_{t_i}\\right)=\\frac{Y_{T}-Y_{0}}{T},$ and depends on the observations at two points, see e. g. [5].",
"Models of such type are widely used in finance.",
"For example, Samuelson's model [42] with constant drift parameter $\\mu $ and known volatility $\\sigma $ has the form $\\log S_t=\\left(\\mu -\\frac{\\sigma ^2}{2}\\right)t+\\sigma W_t,$ and the MLE of $\\mu $ equals $\\hat{\\mu }=\\frac{\\log S_T-\\log S_0}{T}+\\frac{\\sigma ^2}{2}.$ At the same time, the model with Wiener process is not suitable for many processes in natural sciences, computer networks, financial markets, etc., that have long- or short-term dependencies, i. e., the correlations of random noise in these processes decrease slowly with time (long-term dependence) or rapidly with time (short-term dependence).",
"In particular, the models of financial markets demonstrate various kinds of memory (short or long).",
"However, a Wiener process has independent increments, and, therefore, the random noise generated by it is “white”, i. e., uncorrelated.",
"The most simple way to overcome this limitation is to use fractional Brownian motion.",
"In some cases even more complicated models are needed.",
"For example, the noise can be modeled by mixed fractional Brownian motion [9], or by the sum of two fractional Brownian motions [28].",
"Moreover, recently Gaussian processes with non-stationary increments have become popular such as sub-fractional [6], bifractional [14] and multifractional [2], [36], [40] Brownian motions.",
"In this paper we study rather general model where the noise is represented by a centered Gaussian process $B=\\lbrace B_t, t\\ge 0\\rbrace $ with known covariance function, $B_0 = 0$ .",
"We assume that all finite-dimensional distributions of the process $\\lbrace B_t, \\; t>0\\rbrace $ are multivariate normal distributions with nonsingular covariance matrices.",
"We observe the process $X_t$ with a drift $\\theta G(t)$ , that is, $X_t = \\theta G(t) + B_t,$ where $G(t) = \\int _0^t g(s)\\, ds, $ and $g\\in L_1[0,t]$ for any $t>0$ .",
"The paper is devoted to the estimation of the parameter $\\theta $ by observations of the process $X$ .",
"We consider the MLEs for discrete and continuous schemes of observations.",
"The results presented are based on the recent papers [33], [32].",
"Note that in [32] the model (REF ) with $G(t)=t$ was considered, and the driving process $B$ was a process with stationary increments.",
"Then in [33] these results were extended to the case of non-linear drift and more general class of driving processes.",
"In the present paper we apply the theoretical results mentioned above to the models with fractional Brownian motion, mixed fractional Brownian motion and sub-fractional Brownian motion.",
"Similar problems for the model with linear drift driven by fractional Brownian motion were studied in [5], [16], [26], [35].",
"The mixed Brownian — fractional Brownian model was treated in [7].",
"In [4], [39] the nonparametric functional estimation of the drift of a Gaussian processes was considered (such estimators for fractional and subfractional Brownian motions were studied in [13] and [43] respectively).",
"In the present paper special attention is given to the model of the form$X_t=\\theta t + B^{H_1}_t+ B^{H_2}_t$ with two independent fractional Brownian motions $B^{H_1}$ and $B^{H_2}$ .",
"This model was first studied in [28], where a strongly consistent estimator for the unknown drift parameter $\\theta $ was constructed for $1/2<H_1<H_2<1$ and $H_2-H_1>1/4$ by continuous-time observations of $X$ .",
"Later, in [34], the strong consistency of this estimator was proved for arbitrary $1/2<H_1<H_2<1$ .",
"The details on this approach are given in Remark REF below.",
"However, the problem of drift parameter estimation by discrete observations in this model was still open.",
"Applying our technique, we obtain the discrete-time estimator of $\\theta $ and prove its strong consistency for any $H_1, H_2\\in (0,1)$ .",
"Moreover, we also construct the continuous-time estimator and prove the convergence of the discrete-time estimator to the continuous-time one in the case where $H_1\\in (1/2,3/4]$ and $H_2\\in (H_1,1)$ .",
"It is worth mentioning that the drift parameter estimation is developed for more general models involving fBm.",
"In particular, the fractional Ornstein–Uhlenbeck process is a popular and well-studied model with fBm.",
"The MLE of the drift parameter for this process was constructed in [20] and further investigated in [3], [44], [47].",
"Several non-standard estimators for the drift parameter of an ergodic fractional Ornstein–Uhlenbeck process were proposed in [15] and studied in [17].",
"The corresponding non-ergodic case was treated in [1], [10], [45].",
"In the papers [8], [11], [12], [18], [48], [49] drift parameter estimators were constructed via discrete observations.",
"More general fractional diffusion models were studied in [21], [27] for continuous-time estimators and in [23], [29], [31] for the case of discrete observations.",
"An estimator of the volatility parameter was constructed in [25].",
"For Hurst index estimators see, e. g., [22] and references cited therein.",
"Mixed diffusion model including fractional Brownian motion and Wiener process was investigated in [21].",
"We refer to the paper [30] for a survey of the results on parameter estimation in fractional and mixed diffusion models and to the books [24], [38] for a comprehensive study of this topic.",
"The paper is organized as follows.",
"In Section we construct the MLE by discrete-time observations and formulate the conditions for its strong consistency.",
"In Section we consider the estimator constructed by continuous-time observations and the relations between discrete-time and continuous-time estimators.",
"In Section these results are applied to various models mentioned above.",
"In particular, the new approach to parameter estimation in the model with two independent fractional Brownian motions is presented in Subsection REF .",
"Auxiliary results are proved in the appendices."
],
[
"Construction of drift parameter estimator for discrete-time observations",
"Let the process $X$ be observed at the points $0<t_1<t_2<\\ldots <t_N$ .",
"Then the vector of increments $\\Delta X^{(N)} = (X_{t_1}, \\: X_{t_2}-X_{t_1}, \\:\\ldots , \\: X_{t_N}-X_{t_{N-1}})^\\top $ is a one-to-one function of the observations.",
"We assume in this section that the inequality $G(t_k) \\ne 0$ holds at least for one $k$ .",
"Evidently, vector $\\Delta X^{(N)}$ has Gaussian distribution $\\mathcal {N}(\\theta \\Delta G^{(N)}, \\Gamma ^{(N)})$ , where $\\Delta G^{(N)} = \\bigl (G(t_1), \\: G(t_2)-G(t_1), \\:\\ldots , \\: G(t_N) - G(t_{N-1})\\bigr )^\\top .$ Let $\\Gamma ^{(N)}$ be the covariance matrix of the vector $\\Delta B^{(N)} = (B_{t_1}, \\: B_{t_2}-B_{t_1}, \\:\\ldots , \\: B_{t_N}-B_{t_{N-1}})^\\top .$ The density of the distribution of $\\Delta X^{(N)}$ w. r. t. the Lebesgue measure is $\\textstyle \\operatorname{pdf}_{\\Delta X^{(N)}}(x)=\\frac{(2 \\pi )^{-N/2}}{\\sqrt{\\det \\Gamma ^{(N)}}}\\exp \\left\\lbrace - \\frac{1}{2}\\left(x - \\theta \\Delta G^{(N)}\\right)^\\top \\left(\\Gamma ^{(N)}\\right)^{-1}\\left(x - \\theta \\Delta G^{(N)}\\right)\\right\\rbrace .$ Then one can take the density of the distribution of the vector $\\Delta X^{(N)}$ for a given $\\theta $ w. r. t. the density for $\\theta =0$ as a likelihood function: $L^{(N)}(\\theta ) =\\exp \\left\\lbrace \\theta (\\Delta G^{(N)})^\\top (\\Gamma ^{(N)})^{-1} \\Delta X^{(N)} -\\frac{\\theta ^2}{2}(\\Delta G^{(N)})^\\top (\\Gamma ^{(N)})^{-1} \\Delta G^{(N)}\\right\\rbrace .$ The corresponding MLE equals $\\hat{\\theta }^{(N)} = \\frac{\\left(\\Delta G^{(N)}\\right)^\\top \\left(\\Gamma ^{(N)}\\right)^{-1} \\Delta X^{(N)}}{\\left(\\Delta G^{(N)}\\right)^\\top \\left(\\Gamma ^{(N)}\\right)^{-1} \\Delta G^{(N)}}.$ [Properties of the discrete-time MLE [33]] 1.",
"The estimator $\\hat{\\theta }^{(N)}$ is unbiased and normally distributed: $\\hat{\\theta }^{(N)} - \\theta \\simeq \\mathcal {N}\\left(0,\\frac{1}{(\\Delta G^{(N)})^\\top (\\Gamma ^{(N)})^{-1} \\Delta G^{(N)}}\\right).$ 2.",
"Assume that $\\frac{\\operatorname{var}B_{t}}{G^2(t)} \\rightarrow 0, \\quad \\text{as } t\\rightarrow \\infty .$ If $t_{N} \\rightarrow \\infty $ , as $N\\rightarrow \\infty $ , then the discrete-time MLE $\\hat{\\theta }^{(N)}$ converges to $\\theta $ as $N\\rightarrow \\infty $ almost surely and in $L_2(\\Omega )$ ."
],
[
"Construction of drift parameter estimator for continuous-time observations",
"In this section we suppose that the process $X_t$ is observed on the whole interval $[0,T]$ .",
"We investigate MLE for the parameter $\\theta $ based on these observations.",
"Let $\\langle f,\\, g\\rangle = \\int _0^T f(t) g(t) \\, dt$ .",
"Assume that the function $G$ and the process $B$ satisfy the following conditions.",
"There exists a linear self-adjoint operator $\\Gamma =\\Gamma _T : L_2[0,T] \\rightarrow L_2[0,T]$ such that $\\operatorname{cov}(X_s, X_t) = \\operatorname{\\mathsf {E}}B_s B_t =\\int _0^t \\Gamma _T \\operatorname{\\mathsf {1}}\\nolimits _{[0,s]}(u) \\, du =\\langle \\Gamma _T \\operatorname{\\mathsf {1}}\\nolimits _{[0,s]}, \\, \\operatorname{\\mathsf {1}}\\nolimits _{[0,t]} \\rangle .$ The drift function $G$ is not identically zero, and in its representation $G(t) = \\int _0^t g(s)\\, ds$ the function $g \\in L_2[0,T]$ .",
"There exists a function $h_T \\in L_2[0,T]$ such that $g = \\Gamma h_T$ .",
"Note that under assumption $(A)$ the covariance between integrals of deterministic functions $f \\in L_2[0,T]$ and $g \\in L_2[0,T]$ w. r. t. the process $B $ equals $\\operatorname{\\mathsf {E}}\\int _0^T f(s)\\, dB_s \\, \\int _0^T g(t)\\, dB_t =\\langle \\Gamma _T f,\\, g\\rangle .$ [Likelihood function and continuous-time MLE [33]] Let $T$ be fixed, assumptions $(A)$ –$(C)$ hold.",
"Then one can choose $L(\\theta ) = \\exp \\left\\lbrace \\theta \\int _0^T h_T(s) \\, dX_s - \\frac{\\theta ^2}{2}\\int _0^T g(s) h_T(s) \\, ds\\right\\rbrace $ as a likelihood function.",
"The MLE equals $\\hat{\\theta }_T = \\frac{\\int _0^T h_T(s) \\, dX_s}{\\int _0^T g(s) h_T(s) \\, ds} .$ It is unbiased and normally distributed: $\\hat{\\theta }_T-\\theta \\simeq \\mathcal {N}\\left(0,\\frac{1}{\\int _0^T g(s) h_T(s) \\, ds} \\right).$ [Consistency of the continuous-time MLE [33]] Assume that assumptions $(A)$ –$(C)$ hold for all $T>0$ .",
"If, additionally, $\\liminf _{t\\rightarrow \\infty } \\frac{\\operatorname{var}B_t}{G(t)^2} = 0,$ then the estimator $\\hat{\\theta }_T$ converges to $\\theta $ as $T\\rightarrow \\infty $ almost surely and in mean square.",
"[Relations between discrete and continuous MLEs [33]] Let the assumptions of Theorem hold.",
"Construct the estimator $\\hat{\\theta }^{(N)}$ from (REF ) by observations $X_{Tk/N}$ , $k=1,\\ldots ,N$ .",
"Then the estimator $\\hat{\\theta }^{(N)}$ converges to $\\hat{\\theta }_T$ in mean square, as $N\\rightarrow \\infty $ , the estimator $\\hat{\\theta }^{(2^n)}$ converges to $\\hat{\\theta }_T$ almost surely, as $n\\rightarrow \\infty $ ."
],
[
"The model with fractional Brownian motion and linear drift",
"The fractional Brownian motion $B^H=\\left\\lbrace B^H_t,t\\ge 0\\right\\rbrace $ with Hurst index $H\\in (0,1)$ is a centered Gaussian process with $B_0=0$ and covariance function $\\operatorname{\\mathsf {E}}B^H_t B^H_s = \\tfrac{1}{2}\\left(t^{2H}+s^{2H}-\\left|t-s\\right|^{2H}\\right).$ Let $H\\in (0,1)$ be fixed.",
"Consider the model $X_t=\\theta t+B^H_t.$ where $X$ is an observed stochastic process, $B^H$ is an unobserved fractional Brownian motion with Hurst index $H$ , and $\\theta $ is a parameter of interest.",
"Any finite slice of the stochastic process $\\lbrace B^H_t, \\; t>0\\rbrace $ has a multivariate normal distribution with nonsingular covariance matrix.",
"Since $\\operatorname{var}\\left(B^H_t\\right) =t^{2H}$ , the random process $B^H$ satisfies Theorem .",
"Hence, we have the following result.",
"Under condition $t_N\\rightarrow +\\infty $ as $N\\rightarrow \\infty $ , the estimator $\\hat{\\theta }^{(N)}$ in the model (REF ) is $L_2$ -consistent and strongly consistent.",
"Bertin et al.",
"[5] considered the MLE in the model (REF ) in the discrete scheme of observations, where the trajectory of $X$ was observed at the points $t_k=\\frac{k}{N}$ , $k=1,2,\\ldots ,N^\\alpha $ , $\\alpha >1$ .",
"Hu et al.",
"[16] investigated the MLE by discrete observations at the points $tk=kh$ , $k=1,2,\\ldots ,N$ .",
"They considered even more general model of the form $X_t=\\theta t+\\sigma B^H_t$ with unknown $\\sigma $ .",
"In both papers $L_2$ -consistency and strongly consistency of the MLEs were proved.",
"Note that in Corollary REF both these schemes of observations are allowed, since the only condition $t_N\\rightarrow \\infty $ is required.",
"Now we consider the case of continuous-time observations and apply the results of Section to the model (REF ).",
"Let $H\\in (\\frac{1}{2},1)$ .",
"Denote by $\\Gamma _H$ the corresponding operator $\\Gamma $ for the model (REF ).",
"Then $(\\Gamma _H f)(t) = H(2H-1)\\int _0^T\\frac{f(s)}{\\left|t-s\\right|^{2-2H}}\\,ds.$ For the function $h_T(s) = C_Hs^{1/2-H} (T-s)^{1/2-H},$ $C_H=\\left(H (2H-1) \\mathrm {B} \\left(H-\\frac{1}{2}, \\frac{3}{2}-H\\right)\\right)^{-1}$ , we have that $\\Gamma _H h_T = \\operatorname{\\operatorname{\\mathsf {1}}\\nolimits _{[0,T]}},$ see [35].",
"The MLE is given by $\\hat{\\theta }_T = \\frac{T^{2H - 2}}{\\mathrm {B}(3/2-H,\\: 3/2-H)}\\int _0^T s^{1/2-H} (T-s)^{1/2-H} \\, dX_s.$ This estimator was studied in [26], [35], see also [32].",
"Let $H\\in (\\frac{1}{2},1)$ .",
"The conditions of Theorems , and , are satisfied.",
"The estimator $\\hat{\\theta }_T$ is $L_2$ -consistent and strongly consistent.",
"For fixed $T$ , it can be approximated by discrete-sample estimator in mean-square sense."
],
[
"Model with fractional Brownian motion and power drift",
"Now we generalize the model (REF ) for the case of the non-linear drift function $G(t)=t^{\\alpha + 1}$ .",
"Let $0 < H < 1$ and $\\alpha > -1$ .",
"Consider the process $X_t = \\theta t^{\\alpha + 1} + B_t^H$ This is a particular case of model (REF ), with $g(t) = (\\alpha + 1) t^\\alpha $ .",
"Now verify the conditions of the theorems.",
"The condition (REF ) holds true if and only if $\\alpha > H - 1$ .",
"If $\\alpha > H-1$ , the model (REF ) satisfies the conditions of Theorem .",
"The estimator $\\hat{\\theta }^{(N)}$ in the model (REF ) is $L_2$ -consistent and strongly consistent (provided that $\\lim _{N\\rightarrow \\infty } t_N = +\\infty $ ).",
"The condition $(B)$ : $g\\in L_2[0,T]$ holds true if and only if $\\alpha > -\\frac{1}{2}$ .",
"The integral equation $\\Gamma h = g$ is rewritten as $\\int _0^T \\frac{h(s)\\,ds}{|t-s|^{2H-2}} = \\frac{\\alpha +1}{(2H-1)H} t^\\alpha .$ If $\\alpha > 2H-2$ , then the solution is $h(t) = \\textrm {const} \\cdot \\left(\\frac{T^\\alpha }{t^{H-\\frac{1}{2}} (T-t)^{H-\\frac{1}{2}}} -\\alpha t^{\\alpha +1-2H} W\\left(\\textstyle \\frac{T}{t},\\; \\alpha ,\\; H-\\frac{1}{2}\\right)\\right),$ where $W\\left(\\textstyle \\frac{T}{t},\\: \\alpha ,\\: H-\\frac{1}{2}\\right)= \\int _0^{\\frac{T}{t}-1}(v+1)^{\\alpha -1} v^{\\frac{1}{2}-H} \\, dv$ .",
"The asymptotic behaviour of the function $W\\left(\\frac{T}{t},\\: \\alpha , \\: H-\\frac{1}{2}\\right)$ as $t \\rightarrow 0+$ is $W\\left(\\textstyle \\frac{T}{t},\\: \\alpha ,\\: H-\\frac{1}{2}\\right)\\sim {\\left\\lbrace \\begin{array}{ll}\\mathrm {B}\\left(\\frac{3}{2} - H, \\: H - \\frac{1}{2} - \\alpha \\right)& \\mbox{if $\\alpha < H - \\frac{1}{2}$},\\\\\\ln (T/t)& \\mbox{if $\\alpha = H - \\frac{1}{2}$},\\\\\\frac{2}{2\\alpha +1-2H}\\frac{T^{\\alpha -H+\\frac{1}{2}}}{t^{\\alpha -H+\\frac{1}{2}}}& \\mbox{if $\\alpha > H - \\frac{1}{2}$}.\\end{array}\\right.",
"}$ Therefore, the function $h(t)$ defined in (REF ) is square integrable if $\\alpha +1-2H - \\max \\left(0, \\: \\alpha - H + \\frac{1}{2}\\right) > -\\frac{1}{2}$ , which holds if $\\alpha > 2H - \\frac{3}{2}$ .",
"Note that if $\\alpha > 2H - \\frac{3}{2}$ , then the following inequalities hold true: $\\alpha > 2H - 2$ (whence $h$ defined in (REF ) is indeed a solution to the integral equation $\\Gamma h = g$ ), $\\alpha > H-1$ (whence conditions (REF ) and so (REF ) are satisfied), and $\\alpha > -\\frac{1}{2}$ (whence condition (B) is satisfied).",
"If $\\alpha > 2 H - \\frac{3}{2}$ , the conditions of Theorems , and , are satisfied.",
"The estimator $\\hat{\\theta }_T$ is $L_2$ -consistent and strongly consistent.",
"For fixed $T$ , it can be approximated by discrete-sample estimator in mean-square sense."
],
[
"The model with Brownian and fractional Brownian motion",
"Consider the following model: $X_t = \\theta t + W_t + B_t^{H},$ where $W$ is a standard Wiener process, $B^{H}$ is a fractional Brownian motion with Hurst index $H$ , and random processes $W$ and $B^{H}$ are independent.",
"The corresponding operator $\\Gamma $ is $\\Gamma = I + \\Gamma _H$ , where $\\Gamma _H$ is defined by (REF ).",
"The operator $\\Gamma _H$ is self-adjoint and positive semi-definite.",
"Hence, the operator $\\Gamma $ is invertible.",
"Thus Assumption (C) holds true.",
"In other words, the problem is reduced to the solving of the following Fredholm integral equation of the second kind $h_T(u) + H_2(2H_2-1)\\int _0^T h_T(s)\\left|s-u\\right|^{2H_2-2}\\,ds=1, \\quad u\\in [0,T].$ This approach to the drift parameter estimation in the model with mixed fractional Brownian motion was first developed in [7].",
"Note also that the function $h_T = \\Gamma _T^{-1} \\operatorname{\\operatorname{\\mathsf {1}}\\nolimits _{[0,T]}}$ can be evaluated iteratively $h_T = \\sum _{k=0}^\\infty \\frac{\\left( \\frac{1}{2} \\, \\left\\Vert \\Gamma ^H_T\\right\\Vert \\, I - \\Gamma ^H_T\\right)^k \\operatorname{\\operatorname{\\mathsf {1}}\\nolimits _{[0,T]}}}{\\left( 1 + \\frac{1}{2} \\, \\left\\Vert \\Gamma ^H_T\\right\\Vert \\right)^{k+1}} \\,.$"
],
[
"Model with subfractional Brownian motion",
"The subfractional Brownian motion $\\widetilde{B}^H=\\left\\lbrace \\widetilde{B}^H_t,t\\ge 0\\right\\rbrace $ with Hurst parameter $H\\in (0,1)$ is a centered Gaussian random process with covariance function $\\operatorname{cov}\\left(\\widetilde{B}^H_s,\\widetilde{B}^H_t \\right)= \\frac{2\\,|t|^{2H} + 2\\,|s|^{2H} - |t-s|^{2H} - |t+s|^{2H}}{2}.$ We refer [6], [46] for properties of this process.",
"Obviously, neither $\\widetilde{B}^H\\!$ , nor its increments are stationary.",
"If $\\lbrace B_t^H, \\; t\\in \\mathbb {R}\\rbrace $ is a fractional Brownian motion, then the random process $\\frac{B^H_t + B^H_{-t}}{\\sqrt{2}}$ is a subfractional Brownian motion.",
"Evidently, mixed derivative of the covariance function (REF ) equals $K_H(s,t) := \\frac{\\partial ^2 \\operatorname{cov}\\left(\\widetilde{B}^H_s, \\widetilde{B}^H_t\\right)}{\\partial t \\, \\partial s} =H \\, (2H-1) \\left(|t-s|^{2H-2} - |t+s|^{2H-2}\\right) .$ If $H\\in (\\frac{1}{2},1)$ , then the operator $\\Gamma = \\widetilde{\\Gamma }_H$ that satisfies (REF ) for $\\widetilde{B}^H$ equals $\\widetilde{\\Gamma }_H f(t) = \\int _0^T K_H(s,t) f(s) \\, ds .$ Consider the model (REF ) for $G(t) = t$ and $B=\\widetilde{B}^H$ : $X_t = \\theta t + \\widetilde{B}^H_t.$ Let us construct the estimators $\\hat{\\theta }^{(N)}$ and $\\hat{\\theta }_T$ from (REF ) and (REF ) respectively and establish their properties.",
"In particular, Proposition REF allows to define finite-sample estimator $\\hat{\\theta }^{(N)}$ .",
"The linear equation $\\widetilde{\\Gamma }_H f = 0$ has only trivial solution in $L_2[0,T]$ .",
"As a consequence, the finite slice $\\left(\\widetilde{B}^H_{t_1}, \\ldots , \\widetilde{B}^H_{t_N}\\right)$ with $0 < t_1 < \\ldots < t_N$ has a multivariate normal distribution with nonsingular covariance matrix.",
"Since $\\operatorname{var}\\left(\\widetilde{B}^H_t\\right) = \\left(2 - 2^{2 H -1}\\right) t^{2H}$ , the random process $\\widetilde{B}^H$ satisfies Theorem .",
"Hence, we have the following result.",
"Under condition $t_N\\rightarrow +\\infty $ as $N\\rightarrow \\infty $ , the estimator $\\hat{\\theta }^{(N)}$ in the model (REF ) is $L_2$ -consistent and strongly consistent.",
"In order to define the continuous-time MLE (REF ), we have to solve an integral equation.",
"The following statement guarantees the existence of the solution.",
"If $\\frac{1}{2} < H < \\frac{3}{4}$ , then the integral equation $\\widetilde{\\Gamma }_H h = \\operatorname{\\mathsf {1}}\\nolimits _{[0,T]}$ , that is $\\int _0^T K_H(s,t) h(s) \\, ds = 1 \\qquad \\mbox{for almost all $t \\in (0,T)$}$ has a unique solution $h \\in L_2[0,T]$ .",
"If $\\frac{1}{2} < H < \\frac{3}{4}$ , then the random process $\\widetilde{B}^H$ satisfies Theorems , , and .",
"As the result, $L(\\theta )$ defined in (REF ) is the likelihood function in the model (REF ), and $\\hat{\\theta }_T$ defined in (REF ) is the MLE.",
"The estimator is $L_2$ -consistent and strongly consistent.",
"For fixed $T$ , it can be approximated by discrete-sample estimator in mean-square sense."
],
[
"The model with two independent fractional Brownian motions",
"Consider the following model: $X_t = \\theta t + B^{H_1}_t + B^{H_2}_t,$ where $B^{H_1}$ and $B^{H_2}$ are two independent fractional Brownian motion with Hurst indices $H_1,H_2\\in (\\frac{1}{2},1)$ .",
"Obviously, the condition (REF ) is satisfied: $\\frac{\\operatorname{var}\\left(B^{H_1}_t + B^{H_2}_t\\right)}{t^2} =\\frac{t^{2H_1}+ t^{2H_2}}{t^2} \\rightarrow 0, \\quad t\\rightarrow \\infty .$ Under condition $t_N\\rightarrow +\\infty $ as $N\\rightarrow \\infty $ , the estimator $\\hat{\\theta }^{(N)}$ in the model (REF ) is $L_2$ -consistent and strongly consistent.",
"Evidently, the corresponding operator $\\Gamma $ for the model (REF ) equals $\\Gamma _{H_1}+\\Gamma _{H_2}$ , where $\\Gamma _H$ is defined by (REF ).",
"Therefore, in order to verify the assumptions of Theorem we need to show that there exists a function $h_T$ such that $\\left(\\Gamma _{H_1}+\\Gamma _{H_2}\\right) h_T = \\operatorname{\\mathsf {1}}\\nolimits _{[0,T]}$ .",
"This is equivalent to $\\left(I+\\Gamma _{H_1}^{-1}\\Gamma _{H_2}^{}\\right) h_T = \\Gamma _{H_1}^{-1}\\operatorname{\\mathsf {1}}\\nolimits _{[0,T]}$ , since the operator $\\Gamma _{H_1}$ is injective and its range contains $\\operatorname{\\mathsf {1}}\\nolimits _{[0,T]}$ , see (REF ) and Theorem .",
"Hence, it suffices to prove that the operator $I+\\Gamma _{H_1}^{-1}\\Gamma _{H_2}$ is invertible.",
"This is done in Theorem in Appendix for $H_1\\in (1/2,3/4]$ , $H_2\\in (H_1,1)$ .",
"Thus, in this case the assumptions of Theorem hold with $h_T = \\left(I+\\Gamma _{H_1}^{-1}\\Gamma _{H_2}\\right)^{-1}\\Gamma _{H_1}^{-1}\\operatorname{\\mathsf {1}}\\nolimits _{[0,T]}.$ Therefore, we have the following result for the estimator $\\hat{\\theta }_T = \\frac{\\int _0^T h_T(s) \\, dX_s}{\\int _0^T h_T(s) \\, ds} .$ If $H_1\\in (1/2,3/4]$ and $H_2\\in (H_1,1)$ , then the random process $B^{H_1} + B^{H_2}$ satisfies Theorems , , and .",
"As the result, $L(\\theta )$ defined in (REF ) is the likelihood function in the model (REF ), and $\\hat{\\theta }_T$ is the maximum likelihood estimator.",
"The estimator is $L_2$ -consistent and strongly consistent.",
"For fixed $T$ , it can be approximated by discrete-sample estimator in mean-square sense.",
"Another approach to the drift parameter estimation in the model with two fractional Brownian motions was proposed in [28] and developed in [34].",
"It is based on the solving of the following Fredholm integral equation of the second kind $(2-2H_1) \\tilde{h}_T(u)u^{1-2H_1}+\\int _0^T \\tilde{h}_T(s) k(s,u)\\,ds= (2-2H_1) u^{1-2H_1}, \\quad u\\in (0,T],$ where $k(s,u) =\\int _0^{s\\wedge u}\\partial _s K_{H_1,H_2}(s,v) \\partial _u K_{H_1,H_2}(u,v)\\,dv,\\\\K_{H_1,H_2}(t,s) = c_{H_1} \\beta _{H_2} s^{1/2-H_2}\\int _{s}^{t}(t-u)^{1/2-H_1}u^{H_2-H_1}(u-s)^{H_2-3/2}du,\\\\c_{H_1}=\\left(\\frac{\\Gamma (3-2H_1)}{2H_1\\Gamma (\\frac{3}{2}-H_1)^{3}\\Gamma (H_1+\\frac{1}{2})}\\right)^{\\frac{1}{2}}\\!,\\; \\beta _{H_2}= \\left(\\frac{2H_2\\left(H_2-\\frac{1}{2}\\right)^2\\Gamma (\\frac{3}{2}-H_2)}{\\Gamma (H_2+\\frac{1}{2})\\Gamma (2-2H_2)}\\right)^{\\frac{1}{2}}.$ Then for $1/2\\le H_1<H_2<1$ the estimator is defined as $\\hat{\\theta }(T)=\\frac{N(T)}{\\delta _{H_1}\\langle N\\rangle (T)},$ where $\\delta _{H_1}=c_{H_1}\\mathrm {B}\\left(\\frac{3}{2}-H_1,\\frac{3}{2}-H_1\\right)$ , $N(t)$ is a square integrable Gaussian martingale, $N(T)=\\int _0^{T}\\tilde{h}_{T}(t)\\,dX(t),$ $\\tilde{h}_{T}(t)$ is a unique solution to (REF ) and $\\langle N\\rangle (T)=(2-2H_1) \\int _0^{T} \\tilde{h}_{T}(t)t^{1-2H_1}\\,dt.$ This estimator is also unbiased, normal and strongly consistent.",
"The details of this method can be found also in [24].",
"Let $0< p < 1$ and $b>0$ .",
"If $y \\in L_1[0,b]$ is a solution to integral equation $\\int _0^b \\frac{y(s)\\, ds}{|t-s|^p} = f(t)\\quad \\mbox{for almost all $t\\in (0,b)$,}$ then $y(x)$ satisfies $y(x) = \\frac{\\Gamma (p) \\cos \\frac{\\pi p}{2}}{\\pi x^{(1-p)/2}}\\mathcal {D}^{(1-p)/2}_{b-}\\left(x^{1-p}\\mathcal {D}^{(1-p)/2}_{0+}\\left( \\frac{f(x)}{x^{(1-p)/2}} \\right)\\right)$ almost everywhere on $[0,b]$ , where $\\mathcal {D}^{\\alpha }_{a+}$ and $\\mathcal {D}^{\\alpha }_{b-}$ are the Riemann–Liouville fractional derivatives, that is $\\mathcal {D}^{\\alpha }_{a+} f(x) &=\\frac{1}{\\Gamma (1-\\alpha )} \\frac{d}{dx} \\biggl (\\int _a^x \\frac{f(t)}{(x-t)^\\alpha } \\, dt\\biggr ),\\\\\\mathcal {D}^{\\alpha }_{b-} f(x) &=\\frac{-1}{\\Gamma (1-\\alpha )} \\frac{d}{dx} \\biggl (\\int ^b_x \\frac{f(t)}{(t-x)^\\alpha } \\, dt\\biggr ).$ If $y_1 \\in L_1[0,b]$ and $y_2 \\in L_1[0,b]$ are two solutions to integral equation (REF ), then $y_1(x) = y_2(x)$ almost everywhere on $[0,b]$ .",
"If $y \\in L_1[0,b]$ satisfies (REF ) almost everywhere on $[0,b]$ and the fractional derivatives are solutions to respective Abel integral equations, that is $\\frac{1}{\\Gamma \\left(\\frac{1-p}{2}\\right)}\\int _0^t\\frac{\\mathcal {D}^{(1-p)/2}_{0+}(f(x) x^{(p-1)/2})}{(t-x)^{(p+1)/2}} \\, dx =\\frac{f(t)}{t^{(1-p)/2}},$ for almost all $t \\in (0,b)$ and $\\frac{1}{\\Gamma \\left(\\frac{1-p}{2}\\right)}\\int _x^b\\frac{\\pi y(s) s^{(1-p)/2}}{\\Gamma (p) \\cos \\frac{\\pi p}{2}}\\,\\frac{ds}{(s-x)^{(p+1)/2}}=x^{1-p}\\mathcal {D}^{(1-p)/2}_{0+}\\left( \\frac{f(x)}{x^{(1-p)/2}} \\right)$ for almost all $x \\in (0,b)$ , then $y(s)$ is a solution to integral equation (REF ).",
"Firstly, transform the left-hand side of (REF ).",
"By [35], for $0 < s < t$ $\\int _0^s \\frac{d\\tau }{(t-\\tau )^{(p+1)/2}(s-\\tau )^{(p+1)/2}\\tau ^{1-p} }= \\frac{\\mathrm {B}\\left(p, \\: \\frac{1-p}{2}\\right)}{s^{(1-p)/2}t^{(1-p)/2}(t-s)^p} .$ Hence, for $s>0$ , $t>0$ , $s\\ne t$ $\\int _0^{\\min (s,t)} \\frac{d\\tau }{(t-\\tau )^{(p+1)/2}(s-\\tau )^{(p+1)/2}\\tau ^{1-p} }= \\frac{\\mathrm {B}\\left(p, \\: \\frac{1-p}{2}\\right)}{s^{(1-p)/2}t^{(1-p)/2}|t-s|^p} .$ Hence $\\int _0^b \\frac{y(s)\\, ds}{|t-s|^p}= \\\\ =\\int _0^b \\frac{s^{(1-p)/2}t^{(1-p)/2}y(s)}{\\mathrm {B}\\left(p, \\: \\frac{1-p}{2}\\right)}\\int _0^{\\min (s,t)}\\frac{d\\tau }{(t-\\tau )^{(p+1)/2}(s-\\tau )^{(p+1)/2}\\tau ^{1-p} }\\, ds .$ Change the order of integration, noting that $\\lbrace (s, \\tau ) :0 < s < b,\\; 0 < \\tau < \\min (s, t)\\rbrace =\\lbrace (s, \\tau ) :0 < \\tau < t,\\; \\tau < s < b\\rbrace $ for $0<t<b$ : $\\int _0^b \\frac{y(s)\\, ds}{|t-s|^p}=\\frac{t^{(1-p)/2}}{\\mathrm {B}\\left(p, \\: \\frac{1-p}{2}\\right)}\\int _0^t \\frac{1}{(t-\\tau )^{(p+1)/2} \\tau ^{1-p}}\\int _{\\tau }^b\\frac{s^{(1-p)/2}y(s) \\, ds}{(s-\\tau )^{(p+1)/2} }\\, d\\tau .$ The right-hand side of (REF ) can be rewritten with fractional integration: $\\int _0^b \\frac{y(s)\\, ds}{|x-s|^p}=\\frac{\\Gamma \\left(\\frac{1-p}{2}\\right)^2}{\\mathrm {B}\\left(p, \\: \\frac{1-p}{2}\\right)}x^{(1-p)/2}I^{(1-p)/2}_{0+} \\left(\\frac{1}{x^{1-p}}I^{(1-p)/2}_{b-}( x^{(1-p)/2} y(x) )\\right)$ for $0 < x < b$ , where $I^\\alpha _{a+}$ and $I^\\alpha _{b-}$ are fractional integrals $I^{\\alpha }_{a+} f(x) =\\frac{1}{\\Gamma (\\alpha )} \\int _a^x \\frac{f(t)}{(x-t)^{(1-\\alpha )}} \\, dt$ and $I^{\\alpha }_{b-} f(x) =\\frac{1}{\\Gamma (\\alpha )} \\int ^b_x \\frac{f(t)}{(t-x)^{(1-\\alpha )}} \\, dt.$ The constant coefficient can be simplified: $\\frac{\\Gamma \\left(\\frac{1-p}{2}\\right)^2}{\\mathrm {B}\\left(p, \\: \\frac{1-p}{2}\\right)}=\\frac{\\Gamma \\left(\\frac{1-p}{2}\\right) \\Gamma \\left(\\frac{p+1}{2}\\right)}{\\Gamma (p)}=\\frac{\\pi }{\\Gamma (p) \\cos \\left(\\frac{\\pi p}{2}\\right)} .$ Thus integral equation (REF ) can be rewritten with use of fractional integrals: $\\frac{\\pi }{\\Gamma (p) \\cos \\left(\\frac{\\pi p}{2}\\right)}x^{(1-p)/2}I^{(1-p)/2}_{0+} \\left(\\frac{1}{x^{1-p}}I^{(1-p)/2}_{b-}( x^{(1-p)/2} y(x) )\\right)= f(x)$ for almost all $x \\in (0,b)$ .",
"Whenever $y \\in L_1[0,b]$ , the function $x^{(1-p)/2} y(x)$ is obviously integrable on $[0,b]$ .",
"Now prove that the function $x^{p-1} I^{(1-p)/2}_{b-}( x^{(1-p)/2} y(x) )$ is also integrable on $[0,b]$ .",
"Indeed, $\\left|I^{(1-p)/2}_{b-}( x^{(1-p)/2} y(x) )\\right| \\le \\frac{1}{\\Gamma \\left(\\frac{1-p}{2}\\right)}\\int ^b_x \\frac{t^{(1-p)/2} \\, |y(t)|}{(t-x)^{(p+1)/2}} \\, dt ,\\\\\\begin{aligned}&\\int _0^b \\left|\\frac{I^{(1-p)/2}_{b-}( x^{(1-p)/2} y(x) )}{x^{1-p}}\\right| dx\\le \\frac{1}{\\Gamma \\left(\\frac{1-p}{2}\\right)}\\int _0^b \\frac{1}{x^{1-p}}\\int ^b_x \\frac{t^{(1-p)/2} \\, |y(t)|\\, dt}{(t-x)^{(p+1)/2}}\\,dx\\\\&\\qquad =\\frac{1}{\\Gamma \\left(\\frac{1-p}{2}\\right)}\\int ^b_0 t^{(1-p)/2} \\, |y(t)|\\,\\int _0^t \\frac{dx}{x^{1-p} (t-x)^{(p+1)/2}}\\, dt\\\\&\\qquad =\\frac{\\mathrm {B}\\left(p,\\: \\frac{1-p}{2}\\right)}{\\Gamma \\left(\\frac{1-p}{2}\\right)}\\int ^b_0 |y(t)|\\, dt< \\infty ,\\end{aligned}$ and the integrability is proved.",
"Due to [41], the Abel integral equation $f(x) = I^\\alpha _{a+} \\phi (x)$ , $x \\in (a,b)$ , may have not more that one solution $\\phi (x)$ within $L_1[a,b]$ .",
"If the equation has such a solution, then the solution $\\phi (x)$ is equal to $\\mathcal {D}^\\alpha _{a+} f(x)$ .",
"Similarly, the Abel integral equation $f(x) = I^\\alpha _{b-} \\phi (x)$ may have not more that one solution $\\phi (x) \\in L_1[a,b]$ , and if it exists, $\\phi = \\mathcal {D}^\\alpha _{b-} f$ .",
"Therefore, if $y\\in L_1[0,b]$ is a solution to integral equation, then is also satisfies (REF ), so $I^{(1-p)/2}_{0+} \\left(\\frac{1}{x^{1-p}}I^{(1-p)/2}_{b-}\\left(\\frac{\\pi }{\\Gamma (p) \\cos \\left(\\frac{\\pi p}{2}\\right)}x^{(1-p)/2} y(x) \\right)\\right)= \\frac{f(x)}{x^{(1-p)/2}},\\\\\\frac{1}{x^{1-p}}I^{(1-p)/2}_{b-}\\left(\\frac{\\pi x^{(1-p)/2} y(x)}{\\Gamma (p) \\cos \\left(\\frac{\\pi p}{2}\\right)}\\right)=\\mathcal {D}^{(1-p)/2}_{0+}\\left(\\frac{f(x)}{x^{(1-p)/2}}\\right),\\nonumber \\\\\\frac{\\pi x^{(1-p)/2} y(x)}{\\Gamma (p) \\cos \\left(\\frac{\\pi p}{2}\\right)}=\\mathcal {D}^{(1-p)/2}_{b-} \\left(x^{1-p}\\mathcal {D}^{(1-p)/2}_{0+}\\left(\\frac{f(x)}{x^{(1-p)/2}}\\right)\\right)\\nonumber $ for almost all $x \\in (0, b)$ .",
"Thus $y(x)$ satisfies (REF ).",
"Statement 1 of Theorem is proved, and statement 2 follows from statement 1.",
"From equations (REF ) and (REF ), which can be rewritten with fractional integration operator, $I^{(1-p)/2}_{0+}\\mathcal {D}^{(1-p)/2}_{0+}(f(x) x^{(1-p)/2}) =\\frac{f(x)}{t^{(1-p)/2}},\\\\I^{(1-p)/2}_{b-}\\left(\\frac{\\pi y(x) x^{(1-p)/2}}{\\Gamma (p) \\cos \\frac{\\pi p}{2}}\\right)=x^{1-p}\\mathcal {D}^{(1-p)/2}_{0+}\\left( \\frac{f(x)}{x^{(1-p)/2}} \\right)$ (REF ) follows, and (REF ) is equivalent to (REF ).",
"Thus statement 3 of Theorem holds true.",
"$\\Box $ The integral equation (REF ) was solved explicitly in [26] under the assumption $f\\in C([0,b])$ .",
"Here we solve this equation in $L_1[0,b]$ and prove the uniqueness of a solution in this space.",
"Note also that the formula for solution in the handbook [37] is incorrect (it is derived from the incorrect formula 3.1.32 of the same book, where an operator of differentiation is missing; this error comes from the book [50])."
],
[
"Boundedness and invertibility of operators",
"This appendix is devoted to the proof of the following result, which plays the key role in the proof of the strong consistency of the MLE for the model with two independent fractional Brownian motions.",
"Let $H_1\\in \\left(\\frac{1}{2},\\frac{3}{4}\\right]$ , $H_2\\in (H_1,1)$ , and $\\Gamma _H$ be the operator defined by (REF ).",
"Then $\\Gamma ^{-1}_{H_1}\\Gamma _{H_2}\\colon L_2[0,T]\\rightarrow L_2[0,T]$ is a compact linear operator defined on the entire space $L_2[0,T]$ , and the operator $I+\\Gamma _{H_1}^{-1}\\Gamma _{H_2}$ is invertible.",
"The proof consists of several steps."
],
[
"Convolution operator",
"If $\\phi \\in L_1[-T,T]$ , then the following convolution operator $Lf(x) = \\int _0^T \\phi (t-s) f(s) \\, ds$ is a linear continuous operator $L_2[0,T] \\rightarrow L_2[0, T]$ , and $\\Vert L\\Vert \\le \\int _{-T}^T |\\phi (t)| \\, dt.$ Moreover, $L$ is a compact operator.",
"The adjoint operator of the operator (REF ) is $L^*f(x) = \\int _0^T \\phi (s-t) f(s) \\, ds .$ If the function $\\phi $ is even, then the linear operator $L$ is self-adjoint.",
"Let us consider the following convolution operators.",
"For $\\alpha >0$ , the Riemann–Liouville operators of fractional integration are defined as $I^\\alpha _{0+} f(t) = \\frac{1}{\\Gamma (\\alpha )} \\int _0^t \\frac{f(s)\\, ds}{(t-s)^{1-\\alpha }}, \\\\I^\\alpha _{T-} f(t) = \\frac{1}{\\Gamma (\\alpha )} \\int ^T_t \\frac{f(s)\\, ds}{(t-s)^{1-\\alpha }}.$ The operators $I^\\alpha _{0+}$ and $I^\\alpha _{T-}$ are mutually adjoint.",
"Their norm can be bounded as follows $\\Vert I^\\alpha _{T-}\\Vert = \\Vert I^\\alpha _{0+}\\Vert \\le \\frac{1}{\\Gamma (\\alpha )} \\int _0^T \\frac{ds}{s^{1-\\alpha }}=\\frac{T^\\alpha }{\\Gamma (\\alpha +1)} .$ Let $\\frac{1}{2} < H < 1$ and $\\Gamma _H$ be the operator defined by (REF ).",
"Then $\\Gamma _H = H\\Gamma (2H) \\left(I^{2H-1}_{0+} + I^{2H-1}_{T-}\\right).$ The linear operators $I^\\alpha _{0+}$ , $I^\\alpha _{T-}$ for $\\alpha > 0$ , and $\\Gamma _H$ for $\\frac{1}{2} < H < 1$ are injective."
],
[
"Semigroup property of the operator of fractional integration",
"For $\\alpha >0$ and $\\beta >0$ the following equalities hold $I^\\alpha _{0+} I^\\beta _{0+} = I^{\\alpha +\\beta }_{0+}, \\\\I^\\alpha _{T-} I^\\beta _{T-} = I^{\\alpha +\\beta }_{T-}.$ This theorem is a particular case of [41].",
"For $0 < \\alpha \\le \\frac{1}{2}$ and $f\\in L_2[0,T]$ , $\\langle I^\\alpha _{0+} f, \\: I^\\alpha _{T-}f \\rangle \\ge 0.$ Equality is achieved if and only if $f = 0$ almost everywhere on $[0, T]$ for $0 < \\alpha < \\frac{1}{2}$ ; $\\int _0^T f(t)\\, dt = 0$ for $\\alpha = \\frac{1}{2}$ .",
"Since the operators $I^\\alpha _{0+}$ and $I^\\alpha _{T-}$ are mutually adjoint, by semigroup property, we have that $\\langle I^\\alpha _{0+} f, \\: I^\\alpha _{T-}f \\rangle =\\langle I^\\alpha _{0+} I^\\alpha _{0+} f, f \\rangle =\\langle I^{2\\alpha }_{0+} f, f \\rangle , \\\\\\langle I^\\alpha _{0+} f, \\: I^\\alpha _{T-}f \\rangle =\\langle f, I^\\alpha _{T-} I^\\alpha _{T-}f \\rangle =\\langle f, I^{2\\alpha }_{T-}f \\rangle .$ Adding these equalities, we obtain $\\langle I^\\alpha _{0+} f, \\: I^\\alpha _{T-}f \\rangle =\\frac{1}{2}\\langle I^{2\\alpha }_{0+} f + I^{2\\alpha }_{T-}f, \\: f \\rangle .$ If $0 < \\alpha < \\frac{1}{2}$ , then $\\langle I^\\alpha _{0+} f, \\: I^\\alpha _{T-}f \\rangle &=\\frac{1}{2 H \\Gamma (2 H)}\\,\\langle \\Gamma _H f, \\: f \\rangle = \\\\ &=\\frac{1}{2 H \\Gamma (2 H)}\\operatorname{\\mathsf {E}}\\biggl (\\int _0^T f(t) \\, dB_t^H \\biggr )^2 \\ge 0.$ where $H = \\alpha + \\frac{1}{2}$ , $\\frac{1}{2} < H < 1$ , and $B_t^H$ is a fractional Brownian motion.",
"Let us consider the case $\\alpha = \\frac{1}{2}$ .",
"Since $I^1_{0+} f (t) + I^1_{T-}f (t)&= \\int _0^t f(s)\\, ds + \\int _t^T f(s)\\, ds =\\int _0^T f(s)\\, ds,\\\\I^1_{0+} f + I^1_{T-}f&= \\int _0^T f(s)\\, ds \\, \\operatorname{\\mathsf {1}}_{[0,T]},\\\\\\left\\langle I^1_{0+} f + I^1_{T-}f, \\: f \\right\\rangle &=\\int _0^T f(s)\\, ds \\: \\langle \\operatorname{\\mathsf {1}}_{[0,T]}, \\: f\\rangle =\\biggl ( \\int _0^T f(s)\\, ds \\biggr ) ^2,$ we see from (REF ) that $\\left\\langle I^{1/2}_{0+} f, \\: I^{1/2}_{T-}f \\right\\rangle =\\frac{1}{2} \\biggl ( \\int _0^T f(s)\\, ds \\biggr )^2 \\ge 0.$ Conditions for the equality $\\langle I^\\alpha _{0+} f, \\: I^\\alpha _{T-}f \\rangle = 0$ can be easily found by analyzing the proof.",
"Indeed, if $0 < \\alpha < \\frac{1}{2}$ and $H = \\alpha + \\frac{1}{2}$ , then $\\Gamma _H$ is a self-adjoint positive compact operator whose eigenvalues are all positive.",
"Then $2 H \\Gamma (2H) \\langle I^\\alpha _{0+} f, \\: I^\\alpha _{T-}f \\rangle = \\langle \\Gamma _H f, \\: f \\rangle = \\Vert \\Gamma _H^{1/2} f \\Vert ^2$ .",
"In this case, the equality $\\langle I^\\alpha _{0+} f, \\: I^\\alpha _{T-}f \\rangle = 0$ holds true if and only if $f = 0$ almost everywhere on $[0,T]$ .",
"If $\\alpha = \\frac{1}{2}$ , then the condition for the equality follows from (REF ).",
"$\\Box $ For $0 < \\alpha \\le \\frac{1}{2}$ and $f \\in L_2[0,T]$ , $\\Vert I^\\alpha _{0+} f \\Vert + \\Vert I^\\alpha _{T-}f \\Vert \\le \\sqrt{2}\\,\\Vert I^\\alpha _{0+} f + I^\\alpha _{T-}f \\Vert .$ Consequently, for $\\frac{1}{2} < H \\le \\frac{3}{4}$ $\\left\\Vert I^{2H-1}_{0+} f \\right\\Vert + \\left\\Vert I^{2H-1}_{T-}f \\right\\Vert \\le \\frac{\\sqrt{2}}{H \\Gamma (2H)}\\Vert \\Gamma _H f \\Vert .$ Taking into account Proposition REF , we get $(\\Vert I^\\alpha _{0+} f \\Vert + \\Vert I^\\alpha _{T-}f \\Vert )^2&\\le 2\\,\\Vert I^\\alpha _{0+} f \\Vert ^2 + 2\\,\\Vert I^\\alpha _{T-}f \\Vert ^2\\\\ &\\le 2\\,\\Vert I^\\alpha _{0+} f \\Vert ^2 + 4 \\, \\langle I^\\alpha _{0+} f, \\: I^\\alpha _{T-}f \\rangle + 2\\,\\Vert I^\\alpha _{T-}f \\Vert ^2\\\\ &=2 \\, \\Vert I^\\alpha _{0+} f + I^\\alpha _{T-}f \\Vert ^2 ,$ whence the inequality (REF ) follows.",
"The second statement is obtained by the representation (REF ).$\\Box $"
],
[
"Transposition of operators",
"Let $A$ be a linear continuous operator on $L_2[0,T]$ , and $B$ be an injective self-adjoint compact linear operator on $L_2[0,T]$ .",
"If the linear operator $A^* B^{-1}$ is bounded, that is $\\Vert A^* B^{-1}\\Vert = K < \\infty $ , then the linear operator $B^{-1} A$ is defined on the entire space $L_2[0,T]$ , bounded, and $\\Vert B^{-1} A\\Vert = K$ .",
"For the self-adjoint compact linear operator $B$ one can find an orthonormal eigenbasis $\\lbrace e_1, e_2, \\ldots \\rbrace $ such that $B\\biggl ( \\sum _{k=1}^\\infty x_k e_k \\biggr ) =\\sum _{k=1}^\\infty \\lambda _k x_k e_k\\quad \\text{for}\\quad \\sum _{k=1}^\\infty x_k^2 < +\\infty .$ Then $\\lim _{k \\rightarrow \\infty } \\lambda _k = 0$ (by compactness), but for all $k$ the inequality $\\lambda _k \\ne 0$ holds (by injectivity).",
"The inverse operator is a self-adjoint linear operator defined by the equation $B^{-1} \\biggl ( \\sum _{k=1}^\\infty x_k e_k \\biggr ) =\\sum _{k=1}^\\infty \\frac{x_k}{\\lambda _k} e_k\\quad \\text{for}\\quad \\sum _{k=1}^\\infty \\frac{x_k^2}{\\lambda _k^2} < +\\infty .$ The domain of the operator $B^{-1}$ is the subset $B(L_2[0,T]) = \\left\\lbrace \\sum _{k=1}^\\infty x_k e_k : \\sum _{k=1}^\\infty \\frac{x_k^2}{\\lambda _k^2} < \\infty \\right\\rbrace $ of the Hilbert space $L_2[0,T]$ .",
"Let us prove that the operator $B^{-1} A$ is defined on $L_2[0,T]$ .",
"Assume the opposite, i. e., $B^{-1} A$ is undefined at some point $f \\in L_2[0, T]$ .",
"This means that $A f \\notin B(L_2[0,T])$ .",
"Decompose $A f$ into a series by the eigenfunctions of the operator $B$ : $A f = \\sum _{k=1}^\\infty x_k e_k .$ Since $A f \\notin B(L_2[0,T])$ , we see that $\\sum _{k=1}^\\infty \\frac{x_k^2}{\\lambda _k^2} = +\\infty ,$ and for $s_n = \\sum _{k=1}^n \\frac{x_k^2}{\\lambda _k^2}$ it holds that $\\lim _{n\\rightarrow \\infty } s_n = +\\infty $ and $s_n \\ge 0$ for all $n\\in \\mathbb {N}$ .",
"Therefore, there exists $N\\in \\mathbb {N}$ such that $s_N > K^2 \\, \\Vert f\\Vert ^2 .$ Put $g = \\sum _{k=1}^N \\frac{x_k}{\\lambda _k} e_k .$ Then $\\Vert g\\Vert ^2 = \\sum _{k=1}^N \\frac{x_k^2}{\\lambda _k^2} = s_N, \\qquad \\Vert g\\Vert = \\sqrt{s_k}, \\qquad g \\in B(L_2[0, T]), \\\\B^{-1} g = \\sum _{k=1}^N \\frac{x_k}{\\lambda _k^2} e_k, \\qquad \\left\\langle A^* B^{-1} g, \\: f \\right\\rangle =\\left\\langle B^{-1} g, A f \\right\\rangle = \\sum _{k=1}^N \\frac{x_k}{\\lambda _k^2} \\, x_k = s_N .$ By the Cauchy–Schwarz inequality, $\\left| \\left\\langle A^* B^{-1} g, \\: f \\right\\rangle \\right| \\le \\left\\Vert A^* B^{-1} g \\right\\Vert \\, \\Vert f \\Vert \\le \\Vert A^* B^{-1} \\Vert \\,\\Vert g\\Vert \\, \\Vert f \\Vert = K \\, \\sqrt{s_N} \\, \\Vert f\\Vert .$ Hence, $s_N \\le K \\, \\sqrt{s_N} \\, \\Vert f\\Vert .$ The inequalities (REF ) and (REF ) contradict each other.",
"Thus, the operator $B^{-1} A$ is defined on the entire space $L_2[0,T]$ .",
"Now let us prove boundedness of the operator $B^{-1} A$ and the inequality $\\left\\Vert B^{-1} A \\right\\Vert \\le K$ .",
"Suppose that this is not so.",
"Then there exists an element $f$ of the space $L_2[0,T]$ such that $\\left\\Vert B^{-1} A f\\right\\Vert > K \\Vert f\\Vert .$ We use the same decomposition of the vector $Af$ into the eigenvectors of $B$ as above, see (REF ).",
"Then $B^{-1} A f = \\sum _{k=1}^\\infty \\frac{x_k}{\\lambda _k} e_k, \\qquad \\left\\Vert B^{-1} A f \\right\\Vert ^2 = \\sum _{k=1}^\\infty \\frac{x_k^2}{\\lambda _k^2}, \\\\\\lim _{n\\rightarrow \\infty } s_n = \\left\\Vert B^{-1} A f \\right\\Vert ^2 > K^2 \\Vert f\\Vert ^2,$ by (REF ).",
"Therefore, there exists $N\\in \\mathbb {N}$ such that the inequality (REF ) holds.",
"Arguing as above, we get a contradiction.",
"Hence, $\\Vert B^{-1} A\\Vert \\le K$ .",
"It remains to prove the opposite inequality $\\Vert B^{-1} A\\Vert \\ge K$ .",
"The operator $A^* B^{-1}$ is defined on the set $B(L_2[0,T])$ .",
"For all $f \\in B(L_2[0,T])$ from the domain of the operator $A^* B^{-1}$ , we have $\\Vert A^* B^{-1} f \\Vert ^2&= \\langle B^{-1} A A^* B^{-1} f, \\: f \\rangle \\le \\\\ &\\le \\Vert B^{-1} A A^* B^{-1} f \\Vert \\, \\Vert f\\Vert \\le \\Vert B^{-1} A \\Vert \\, \\Vert A^* B^{-1} f \\Vert \\, \\Vert f\\Vert ,$ whence $\\Vert A^* B^{-1} f \\Vert \\le \\Vert B^{-1} A \\Vert \\, \\Vert f\\Vert .$ Therefore $K = \\Vert A^* B^{-1} \\Vert \\le \\Vert B^{-1} A \\Vert $ .$\\Box $"
],
[
"The proof of boundedness and compactness",
"Let $\\frac{1}{2} < H_1 < H_2 < 1$ and $H_1 \\le \\frac{3}{4}$ .",
"Then $\\Gamma _{H_1}^{-1} \\Gamma _{H_2}^{}$ is a compact linear operator defined on the entire space $L_2[0,T]$ .",
"The operator $\\Gamma _{H_1}$ is an injective self-adjoint compact operator $L_2[0,T] \\rightarrow L_2[0,T]$ .",
"The inverse operator $\\Gamma _{H_1}^{-1}$ is densely defined on $L_2[0,T]$ .",
"By Proposition REF , the operators $I^{2H_1-1}_{0+} \\Gamma _{H_1}^{-1}$ and $I^{2H_1-1}_{T-} \\Gamma _{H_1}^{-1}$ are bounded.",
"Therefore, by Lemma REF , the operators $\\Gamma _{H_1}^{-1} I^{2H_1-1}_{T-}$ and $\\Gamma _{H_1}^{-1} I^{2H_1-1}_{0+}$ are also bounded and defined on the entire space $L_2[0,T]$ .",
"By (REF ) and the semigroup property (Theorem REF ), $\\Gamma _{H_1}^{-1} \\Gamma _{H_2}^{}&=H\\Gamma (2H) \\left( \\Gamma _{H_1}^{-1} I^{2H_2-1}_{0+} + \\Gamma _{H_1}^{-1} I^{2H_2-1}_{T-} \\right)= \\\\ &=H\\Gamma (2H) \\left( \\Gamma _{H_1}^{-1} I^{2H_1-1}_{0+} I^{2(H_2-H_1)}_{0+} + \\Gamma _{H_1}^{-1} I^{2H_1-1}_{T-} I^{2(H_2-H_1)}_{T-}\\right) .$ Since $I^{2(H_2-H_1)}_{0+}$ and $I^{2(H_2-H_1)}_{T-}$ are compact operators, the operator $\\Gamma _{H_1}^{-1} \\Gamma _{H_2}^{}$ is also compact.$\\Box $"
],
[
"The proof of invertibility",
"Now prove that $-1$ is not an eigenvalue of the linear operator $\\Gamma _{H_1}^{-1}\\Gamma _{H_2}^{}$ .",
"Indeed, if $\\Gamma _{H_1}^{-1}\\Gamma _{H_2}^{} f = -f$ for some function $f \\in L_2[0,T]$ , then $\\Gamma _{H_2} f + \\Gamma _{H_1} f = 0$ .",
"Since $\\Gamma _{H_2}$ and $\\Gamma _{H_1}$ are positive definite self-adjoint (and injective) operators, $\\Gamma _{H_2} + \\Gamma _{H_1}$ is also a positive definite self-adjoint and injective operator.",
"Hence $f = 0$ almost everywhere on $[0, T]$ .",
"Because $-1$ is not an eigenvalue of the compact linear operator $\\Gamma _{H_1}^{-1}\\Gamma _{H_2}^{}$ , $-1$ is a regular point, i.e., $-1 \\notin \\sigma (\\Gamma _{H_1}^{-1}\\Gamma _{H_2}^{})$ , and the linear operator $\\Gamma _{H_1}^{-1}\\Gamma _{H_2}^{} + I$ is invertible.",
"Acknowledgements The research of Yu.",
"Mishura was funded (partially) by the Australian Government through the Australian Research Council (project number DP150102758).",
"Yu.",
"Mishura and K. Ralchenko acknowledge that the present research is carried through within the frame and support of the ToppForsk project nr.",
"274410 of the Research Council of Norway with title STORM: Stochastics for Time-Space Risk Models."
]
] | 1808.08417 |
[
[
"Does Compton/Schwarzschild duality in higher dimensions exclude TeV\n quantum gravity?"
],
[
"Abstract In three spatial dimensions, the Compton wavelength $(R_C \\propto M^{-1}$) and Schwarzschild radius $(R_S \\propto M$) are dual under the transformation $M \\rightarrow M_{P}^2/M$, where $M_{P}$ is the Planck mass.",
"This suggests that there could be a fundamental link -- termed the Black Hole Uncertainty Principle or Compton-Schwarzschild correspondence -- between elementary particles with $M < M_{P}$ and black holes in the $M > M_{P}$ regime.",
"In the presence of $n$ extra dimensions, compactified on some scale $R_E$ exceeding the Planck length $R_P$, one expects $R_S \\propto M^{1/(1+n)}$ for $R_P < R < R_E$, which breaks this duality.",
"However, it may be restored in some circumstances because the {\\it effective} Compton wavelength of a particle depends on the form of the $(3+n)$-dimensional wavefunction.",
"If this is spherically symmetric, then one still has $R_C \\propto M^{-1}$, as in the $3$-dimensional case.",
"The effective Planck length is then increased and the Planck mass reduced, allowing the possibility of TeV quantum gravity and black hole production at the LHC.",
"However, if the wave function of a particle is asymmetric and has a scale $R_E$ in the extra dimensions, then $R_C \\propto M^{-1/(1+n)}$, so that the duality between $R_C$ and $R_S$ is preserved.",
"In this case, the effective Planck length is increased even more but the Planck mass is unchanged, so that TeV quantum gravity is precluded and black holes cannot be generated in collider experiments.",
"Nevertheless, the extra dimensions could still have consequences for the detectability of black hole evaporations and the enhancement of pair-production at accelerators on scales below $R_E$.",
"Though phenomenologically general for higher-dimensional theories, our results are shown to be consistent with string theory via the minimum positional uncertainty derived from $D$-particle scattering amplitudes."
],
[
"Introduction",
"A key feature of the microscopic domain is the (reduced) Compton wavelength for a particle of rest mass $M$ , which is $R_C = \\hbar /(Mc)$ .",
"In the $(M,R)$ diagram of Fig.",
"REF , the region corresponding to $R<R_C$ might be regarded as the `quantum domain' in the sense that the classical description breaks down there.",
"A key feature of the macroscopic domain is the Schwarzschild radius for a body of mass $M$ , which corresponds to the size of a black hole of this mass and is $R_S = 2GM/c^2$ .",
"The region $R<R_S$ might be regarded as the `relativistic domain' in the sense that there is no stable classical configuration in this part of Fig.",
"REF .",
"Figure: This shows the division of the (M,RM,R) diagram into different physical domains.Also shown are the Compton and Schwarzschild radii and the lines corresponding tothe Planck mass, length and density.",
"The broken line gives a smooth transition between the Compton and Schwarzschild lines, as postulated by the BHUP correspondence.Despite being essentially relativistic results, it is interesting that both these expressions can be derived from a semi-Newtonian treatment in which one invokes a maximum velocity $c$ but no other relativistic effects [1].",
"The Compton line can be derived from the Heisenberg Uncertainty Principle (HUP), which requires that the uncertainty in the position and momentum of a particle satisfy $\\Delta x \\gtrsim \\hbar /\\Delta p$ , by arguing that the momentum of a particle of mass $M$ is bounded by $Mc$ .",
"This implies that one cannot localize it on a scale less than $\\hbar /(Mc)$ and is equivalent to substituting $\\Delta x \\rightarrow R$ and $\\Delta p \\rightarrow Mc$ in the uncertainty relation.",
"Later, we discuss more rigorous ways of determining the Compton scale, in both relativisitc and non-relativistic quantum theory, though there is always some ambiguity in the precise numerical coefficient.",
"The expression for the Schwarzschild radius is derived rigorously from general relativity but exactly the same expression can be obtained by equating the escape velocity in Newtonian gravity to $c$ .",
"The Compton and Schwarzschild lines intersect at around the Planck scales, $ R_{P} = \\sqrt{ \\hbar G/c^3} \\simeq 10^{-33} \\mathrm {cm} \\, , \\quad M_{P} = \\sqrt{ \\hbar c/G} \\simeq 10^{-5} \\mathrm {g} \\, ,$ and naturally divide the $(M,R)$ diagram in Fig.",
"REF into three domains, which for convenience we label quantum, relativistic and classical.",
"There are several other interesting lines in Fig.",
"REF .",
"The vertical line $M=M_{P}$ marks the division between elementary particles ($M <M_{P}$ ) and black holes ($M > M_{P}$ ), since the event horizon of a black hole is usually required to be larger than the Compton wavelength associated with its mass.",
"The horizontal line $R=R_{P}$ is significant because quantum fluctuations in the metric should become important below this [2].",
"Quantum gravity effects should also be important whenever the density exceeds the Planck value, $\\rho _{P} = c^5/(G^2 \\hbar ) \\simeq 10^{94} \\mathrm {g \\, cm^{-3}}$ , corresponding to the sorts of curvature singularities associated with the big bang or the centres of black holes [3].",
"This implies $R < R_{P}(M/M_{P})^{1/3}$ , which is well above the $R = R_{P}$ line in Fig.",
"REF for $M \\gg M_P$ , so one might regard the shaded region as specifying the `quantum gravity' domain.",
"This point has been invoked to support the notion of Planck stars [4] and could have important implications for the detection of evaporating black holes [5].",
"The Compton and Schwarzschild lines transform into one another under the substitution $M \\rightarrow M_{P}^2/M$ , corresponding to a reflection in the line $M = M_{P}$ in Fig.",
"REF .",
"This interchanges sub-Planckian and super-Planckian mass scales and suggests some connection between elementary particles and black holes.",
"The lines also transform into each other under the transformation $R \\rightarrow R_{P}^2/R$ , corresponding to a reflection in the line $R = R_{P}$ .",
"This turns super-Planckian length scales into sub-Planckian ones, which might be regarded as unphysical.",
"However, we note that each line maps into itself under the combined T-duality transformation $ M \\rightarrow M_{P}^2/M, \\ \\ \\ R \\rightarrow R_{P}^2/R \\, .$ T-dualities arise naturally in string theory and are known to map momentum-carrying string states to winding states and vice-versa [6].",
"In addition, since they map sub-Planckian length scales to super-Planckian ones, this allows the description of physical systems in an otherwise inaccessible regime [7], [8].",
"Although the Compton and Schwarzschild boundaries correspond to straight lines in the logarithmic plot of Fig.",
"REF , this form presumably breaks down near the Planck point due to quantum gravity effects.",
"One might envisage two possibilities: either there is a smooth minimum, as indicated by the broken line in Fig.",
"REF , so the Compton and Schwarzschild lines in some sense merge, or there is some form of phase transition or critical point at the Planck scale, so that the separation between particles and black holes is maintained.",
"Which alternative applies has important implications for the relationship between elementary particles and black holes [9].",
"This may link to the issue of T-duality since this could also play a fundamental role in relating point particles and black holes.",
"One way of obtaining a smooth transition between the Compton and Schwarzschild lines is to invoke some connection between the uncertainty principle on microscopic scales and black holes on macroscopic scales.",
"This is termed the Black Hole Uncertainty Principle (BHUP) correspondence [10] and also the Compton-Schwarzschild correspondence when discussing an interpretation in terms of extended de Broglie relations [11].",
"It is manifested in a unified expression for the Compton wavelength and Schwarzschild radius.",
"The simplest expression of this kind would be $ R_{CS} = \\frac{\\beta \\hbar }{Mc} + \\frac{2GM}{c^2} \\, ,$ where $\\beta $ is the (somewhat arbitrary) constant appearing in the Compton wavelength.",
"In the sub-Planckian regime this can be interpreted as a modified Compton wavelength: $ R_C^{\\prime } = \\frac{\\beta \\hbar }{Mc} \\left[1 + \\frac{2}{\\beta } \\left( \\frac{M}{M_P} \\right)^2 \\right] \\quad (M \\ll M_P) \\, ,$ with the second term corresponding to a small correction of the kind invoked by the Generalised Uncertainty Principle [12].",
"In the super-Planckian regime, it can be interpreted as a modified Schwarzschild radius: $ R_S^{\\prime } = \\frac{2GM}{c^2} \\left[ 1 + \\frac{\\beta }{2} \\left(\\frac{M_P}{M} \\right)^2 \\right] \\quad (M \\gg M_P) \\, ,$ with the second term corresponding to a small correction to the Schwarzschild expression; this has been termed the Generalised Event Horizon [10].",
"More generally, the BHUP correspondence might allow any unified expression $R_{CS}(M)$ which has the asymptotic behaviour $\\beta \\hbar /(Mc)$ for $M \\ll M_{P}$ and $2GM/c^2$ for $M \\gg M_{P}$ .",
"One could envisage many such expressions but we are particularly interested in those which – like Eq.",
"(REF ) – are dual under under the transformation $M \\rightarrow M_{P}^2/M$ .",
"The considerations of this paper are not dependent on the validity of the BHUP correspondence itself but we mention this as an example of a particular context in which the duality arises.",
"The black hole boundary in Fig.",
"REF assumes there are three spatial dimensions but many theories suggest that dimensionality could increase on small scales.",
"In particular, superstring theory is consistent only in $(9+1)$ spacetime dimensions, even though our observable universe is $(3+1)$ -dimensional.",
"In current models, ordinary matter is described by open strings, whose end-points are confined to a $(p+1)$ -dimensional $D_p$ -brane, while gravity is described by closed strings that propagate in the bulk [6], [7], [8].",
"In the Randall-Sundrum picture [13], $p=3$ and one of the extra dimensions is large (i.e.",
"much larger than the Planck scale), so the universe corresponds to a $D_3$ brane in a 5-dimensional bulk.",
"The bulk dimension is usually viewed as being warped in an anti-de Sitter space, so that the $D_3$ -brane has some finite thickness, and this is equivalent to having a compactifed extra dimension.",
"One could also consider models with more than one large dimension and this might be compared to the model of Arkani-Hamed et al.",
"[14], in which there are $n$ extra spatial dimensions, all compactified on the same scale.",
"One could also consider models with a hierarchy of compacitifed dimensions, so that the dimensionality of the universe increases as one goes to smaller scales.",
"This motivates us to consider the behavior of black holes and quantum mechanical particles in spacetimes with extra directions.",
"For simplicity, we initially assume that all the extra dimensions in which matter is free to propagate are compactified on a single length scale $R_E$ , corresponding to a mass-scale $M_E \\equiv \\hbar /(c R_E)$ .",
"If there are $n$ extra dimensions, and black holes with $R_S < R_E$ are assumed to be approximately spherically symmetric with respect to the full $(3+n)$ -dimensional space, then the Schwarzschild radius scales as $M^{1/(1+n)}$ rather than $M$ for $M < c^2R_E/G = M_P^2/M_E$ [15], so the slope of the black hole boundary in Fig.",
"REF becomes shallower in this range of $M$ .",
"The question now arises of whether the $M$ dependence of $R_C$ is also affected by the extra dimensions.",
"The usual assumption is that it is not, so that one still has $R_C \\propto M^{-1}$ .",
"In this case, the intersect of the Schwarzschild and Compton lines becomes $ R_{P}^{\\prime } \\simeq (R_{P}^2R_E^n)^{1/(2+n)}, \\quad M_{P}^{\\prime } \\simeq (M_{P}^2M_E^n)^{1/(2+n)} \\,.$ This gives $M_{P}^{\\prime } \\simeq M_{P}$ and $R_{P}^{\\prime } \\simeq R_{P}$ for $R_E \\simeq R_{P}$ but $M_{P}^{\\prime } \\ll M_{P}$ and $R_{P}^{\\prime } \\gg R_{P}$ for $R_E \\gg R_{P}$ .",
"As is well known, the higher-dimensional Planck mass therefore decreases (allowing the possibility of TeV quantum gravity) and the higher-dimensional Planck length increases [14].",
"In principle, such effects would permit the production of small black holes at the Large Hadron Collider (LHC), with their evaporation leaving a distinctive signature [16], [17], [18].",
"However, there is still no evidence for this [19], which suggests that either the extra dimensions do not exist or they have a compactification scale $R_E$ which is so small that $M_P^{\\prime }$ exceeds the energy attainable by the LHC.",
"In this paper we point out another possible reason for the failure to produce black holes at accelerators.",
"We argue that in some circumstances one expects $R_C$ to scale as $M^{-1/(1+n)}$ rather than $M^{-1}$ .",
"This has the attraction that it preserves the T-duality between $R_C$ and $R_S$ ; later we present arguments for why one might expect this.",
"In this case, Eq.",
"(REF ) no longer applies.",
"Instead, the higher-dimensional Planck mass is unchanged but the Planck length is increased to $ R_{*} \\simeq (R_{P} R_E^n)^{1/(1+n)} \\, ,$ which is even larger than before.",
"While there is no TeV quantum gravity in this scenario, we will see that the preservation of duality has interesting physical implications.",
"The plan of the paper is as follows.",
"Sec.",
"considers the derivation of the standard expression for the 3D Compton wavelength.",
"Sec.",
"discusses the (well-known) expression for the Schwarzschild radius for a $(3 + n)$ -dimensional black hole.",
"Sec.",
"discusses the form of the Uncertainty Principle in higher dimensions, emphasizing that this depends crucially on the form assumed for the wave function in the higher-dimensional space.",
"Sec.",
"then derives the associated expressions for the effective Compton wavelength in higher dimensions.",
"Sec.",
"explores the consequences of our claim for the detectability of primordial black hole evaporations and recent $D$ -particle scattering results.",
"Sec.",
"draws some general conclusions and suggests future work."
],
[
"Derivations of the 3-dimensional Compton wavelength ",
"The Compton wavelength is defined as $R_C = h /(Mc)$ and first appeared historically in the expression for the Compton cross-section in the scattering of photons off electrons [20].",
"Subsequently, it has arisen in various other contexts.",
"For example, it is relevant to processes which involve turning photon energy ($hc/\\lambda $ ) into rest mass energy ($Mc^2$ ) and the reduced Compton wavelength $\\hbar /(Mc)$ appears naturally in the Klein-Gordon and Dirac equations.",
"One can also associate the Compton wavelength with the localisation of a particle and this is most relevant for the considerations of this paper.",
"There are both non-relativistic and relativistic arguments for this notion, so we will consider these in turn.",
"It is important to distinguish these diffferent contexts when discussing how the expression for the Compton wavelength is modified in higher-dimensional models but in this section we confine attention to the 3-dimensional case.",
"We first consider a non-relativistic argument which combines the de Broglie relations, $ E = \\hbar \\omega ,\\quad \\vec{p} = \\hbar \\vec{k} \\,,$ with the non-relativistic expression for the 3-momentum $\\vec{p} = M\\vec{v}$ and a maximum speed $|\\vec{v}| < c$ .",
"Then Eq.",
"(REF ) gives $ k = |\\vec{k}| < \\frac{Mc}{\\hbar } \\quad \\Rightarrow \\quad \\lambda = \\frac{2 \\pi }{k} > \\frac{2 \\pi \\hbar }{Mc} \\quad \\Rightarrow \\quad R_C = \\frac{h}{Mc} \\, .$ Though the numerical factors in this argument are imprecise, detailed calculations in quantum field theory and compelling observational evidence [21] suggest this result is at least qualitatively correct.",
"This argument can be related to the Uncertainty Principle if Eq.",
"(REF ) is viewed as giving an upper bound on the wave-number of the momentum operator eigenfunctions, or equivalently a lower bound on the de Broglie wavelength, such that: $ (\\Delta p_x)_{\\rm max} \\simeq M c \\, , \\ \\ \\ (\\Delta x)_{\\rm min} \\simeq R_C \\, .$ As discussed in Appendix A, one must distinguish between uncertainties in $x$ and $p_x$ associated with unavoidable noise in the measurement process and the standard deviation associated with repeated measurements which do not disturb the system prior to wave function collapse.",
"In the latter case, one often uses the notation $\\Delta _{\\psi }$ to stress the dependence on the wave vector $\\left|\\psi \\right\\rangle $ .",
"Throughout this paper, we refer to uncertainties in the latter sense but drop the subscript $\\psi $ for convenience.",
"We now present an alternative non-relativistic argument for identifying the maximum possible uncertainty in the momentum $(\\Delta p_x)_{\\rm max}$ with the rest mass of the particle in order to obtain a minimum value of the position uncertainty $(\\Delta x)_{\\rm min} \\simeq R_C$ .",
"Mathematically, this can be achieved by defining position and momentum operators, $\\hat{\\vec{r}}$ and $\\hat{\\vec{p}}$ , and their eigenfunctions in the position space representation, in the usual way, $ \\hat{\\vec{r}} = \\vec{r}, \\quad \\phi (\\vec{r}^{\\prime },\\vec{r}) = \\delta (\\vec{r}-\\vec{r}^{\\prime }); \\quad \\hat{\\vec{p}} = -i\\hbar \\vec{\\nabla }, \\quad \\phi (\\vec{k},\\vec{r}) = e^{i\\vec{k}.\\vec{r}},$ and then introducing an infrared cut-off in the expansion for $\\psi (\\hat{\\vec{r}})$ in terms of $\\phi ^{*}(\\vec{k},\\vec{r})$ or for $\\psi (\\vec{k})$ in terms of $\\phi (\\vec{k},\\vec{r})$ : $ \\psi (\\vec{r}) = \\int _{0}^{Mc/\\hbar } \\psi (\\vec{k}^{\\prime })e^{-i\\vec{k}^{\\prime }.\\vec{r}}d^3k^{\\prime }, \\ \\ \\ \\psi (\\vec{k}) = \\int _{h/(Mc)}^{\\infty } \\psi (\\vec{r}^{\\prime })e^{i\\vec{k}.\\vec{r}^{\\prime }}d^3r^{\\prime } \\, .$ While $\\psi $ is normalisable but not an eigenstate of $\\vec{r}$ and $\\vec{p}$ , $\\phi $ is an eigenstate but non-normalisable.",
"In the momentum space representation, $\\hat{\\vec{r}}$ and $\\hat{\\vec{p}}$ and their eigenfunctions take the form $ \\hat{\\vec{r}} = -i\\hbar \\vec{\\nabla }, \\quad \\phi (\\vec{k},\\vec{r}) = e^{i\\vec{k}.\\vec{r}}; \\quad \\hat{\\vec{p}} = \\vec{p},\\quad \\phi (\\vec{k}^{\\prime },\\vec{k}) = \\delta (\\vec{k}-\\vec{k}^{\\prime }),$ and consistency requires us to introduce an ultraviolet cut-off in $k$ at $k_{\\rm max} = Mc/\\hbar $ .",
"Ths implies an infrared cut-off, $r_{\\rm min} = h/(Mc)$ , so that the extension of $\\psi (\\vec{r})$ in position space is bounded from below by the Compton wavelength, the extension of $\\psi (\\vec{k})$ in $k$ -space is bounded from above by the corresponding wavenumber, and the extension of $\\psi (\\vec{p})$ in momentum-space is bounded by $Mc$ .",
"Since $\\Delta |\\vec{r}|$ and $\\Delta |\\vec{p}|$ are scalars, we may write these as $\\Delta R_{\\rm 3D}$ and $\\Delta P_{\\rm 3D}$ , respectively, where $R_{\\rm 3D} = |\\vec{r}|$ and $P_{\\rm 3D} = |\\vec{p}|$ .",
"For approximately spherically symmetric wave packets, we expect $\\langle {\\hat{\\vec{p}}}\\rangle \\simeq 0 \\, , \\ \\ \\ \\Delta P_{\\rm 3D} \\simeq \\sqrt{\\langle {\\hat{\\vec{p}}^2}\\rangle } \\lesssim Mc \\, , $ $\\langle {\\hat{\\vec{r}}}\\rangle \\simeq 0 \\, , \\ \\ \\ \\Delta R_{\\rm 3D} \\simeq \\sqrt{\\langle {\\hat{\\vec{r}}^2}\\rangle } \\gtrsim h/(Mc).",
"$ The commutator of $\\hat{\\vec{r}}$ and $\\hat{\\vec{p}}$ is $ [\\hat{\\vec{r}},\\hat{\\vec{p}}] = i\\hbar \\, ,$ which implies $ \\Delta R_{\\rm 3D} \\, \\Delta P_{\\rm 3D} \\ge \\hbar /2 \\, .$ From Eqs.",
"(REF )-(REF ), it is therefore reasonable to make the identifications $ \\left(\\Delta P_{\\rm 3D}\\right)_{\\rm max} \\simeq M c \\, , \\quad \\left(\\Delta R_{\\rm 3D}\\right)_{\\rm min} \\simeq R_C \\, ,$ where we will henceforth refer to the Compton wavelength as the Compton radius and restrict consideration to quasi-spherically symmetric distributions, the precise meaning of this term being explained in Sec. .",
"Under these conditions, the uncertainty relation for position and momentum allows us to recover the standard expression (REF ).",
"The advantage of the above non-relativistic arguments is that they can be readily extended to the higher-dimensional case with extra compactified dimensions.",
"The results obtained are phenomenologically robust, despite being derived in the approximate low-energy theory.",
"However, the problem is that the speed limit is put in by hand, without introducing additional relativistic effects, such as Lorentz invariance.",
"So how does one extend the above argument to the relativistic case?",
"Just as the non-relativistic relationship $E=p^2/(2M)$ corresponds to the Schrödinger equation, so the relativistic relationship $E^2= M^2 c^4 + P^2c^2$ corresponds to the Klein-Gordon equation, $- \\partial _t^2 \\psi /c^2 + \\nabla ^2 \\psi = (M c/ \\hbar )^2 \\psi \\, .$ Looking for a plane-wave solution $e^{i( \\vec{k} .",
"\\vec{r} - \\omega t)}$ leads to the dispersion relation $\\omega ^2 = c^2(k^2 - k_C^2) \\, ,$ where $ k_C =Mc/\\hbar $ is the reduced Compton wave-number.",
"In the time-independent case, one has a spherically symmetric solution $\\psi \\propto e^{- k_C r}/r$ , so the Compton wavelength can also be regarded as the scale on which the wave function decays or a correlation scale.",
"Another relativistic argument for the Compton wavelength is associated with pair-production.",
"This combines the relativistic energy-momentum relation with the de Broglie relations (REF ).",
"Since $E > Mc^2$ for $\\lambda < R_{C}$ , this shows that $R_C$ acts as a fundamental barrier beyond which pair-production occurs rather than further localization of the wave packet of the original particle.",
"While the de Broglie wavelength marks the scale at which non-relativistic quantum effects become important and the classical concept of a particle gives way to the idea of a wave packet, the Compton wavelength marks the point at which relativistic quantum effects become significant and the concept of a single wave packet as a state wih fixed particle number becomes invalid [21].",
"$R_C$ is an effective minimum width because, on smaller scales, the concept of a single quantum mechanical particle breaks down and we must switch to a field description in which particle creation and annihilation occur in place of further spatial localization.",
"However, as discussed in Appendix B, the minimum volume required for pair-production may be larger than $R_C^3$ when the wave packet is non-spherical.",
"This result is very relevant when we come to consider the higher dimensional case, especially in the context of particle production by black holes We have demonstrated that the existence of an effective cut-off for the maximum attainable energy/momentum in non-relativistic quantum mechanics implies the existence of a minimum attainable width for (almost) spherically symmetric wave functions, and this may be identified with the Compton radius for $P_{\\rm 3D} \\lesssim Mc$ .",
"For non-spherically symmetric systems we may still consider the upper bound on each momentum component, $p_i = \\hbar k_i < Mc$ , as giving rise to a lower bound for the spatial extent of the wave packet in $i^{th}$ spatial direction.",
"However, as demonstrated in Appendix B, the minimum volume required for pair-production may be much larger than $R_C^3$ .",
"In the presence of compact extra dimensions, in which asymmetry is the norm, the existence of a maximum spatial extent (the compactification scale) also gives rise to a minimum momentum uncertainty.",
"In particular, pair-production can occur for volumes exceeding $R_C^{3+n}$ .",
"As we shall see in Sec.",
", this has important implications for the physics of quantum particles in compactified spacetimes, which have hitherto not been considered in the literature."
],
[
"Higher-dimensional black holes and TeV quantum gravity",
"The black hole boundary in Fig.",
"REF assumes there are three spatial dimensions but many theories, including string theory, suggest that the dimensionality could increase on sufficiently small scales.",
"Although the extra dimensions are often assumed to be compactified on the Planck length, there are also models [13], [14], [15] in which they are either infinite or compactified on a scale much larger than $R_P$ , and these are the models of interest here.",
"In this section, we will assume that the standard expression for the Compton wavelength applies even in the higher-dimensional case and explain why the existence of large extra dimensions could then lead to TeV quantum gravity and the production of black holes at accelerators.",
"Although the argument is well-known, we present it in a way (cf.",
"[22]) which is useful when we come to consider non-standard models.",
"For simplicity, we first assume that the extra dimensions are associated with a single length scale $R_E$ .",
"If the number of extra dimensions is $n$ , then in the Newtonian approximation the gravitational potential generated by a mass $M$ is [23], [24] $ V_{\\mathrm {grav}} = \\frac{G_D M}{R^{1+n}} \\quad (R < R_E) \\, ,$ where $G_D$ is the higher-dimensional gravitational constant and $D = 3+n+1$ is the number of spacetime dimensions in the relativistic theory.",
"For $R>R_E$ , the factor $R^{1+n}$ is replaced by $R R_E^n$ , so one has $ V_{\\mathrm {grav}} = \\frac{G M}{R} \\quad \\mathrm {with} \\quad G = \\left(\\frac{G_D}{R_E^n}\\right) \\quad (R > R_E) \\, .$ Thus one recovers the usual form of the potential in this region.",
"The higher-dimensional nature of the gravitational force is only manifest for $R < R_E$ .",
"This follows directly from the fact that general relativity can be extended to an arbitrary number of dimensions, so we may take the weak field limit of Einstein's field equations in $3+n+1$ dimensions for $R < R_E$ .",
"In the Newtonian limit, the effective gravitational constants at large and small scales are different because of the dilution effect of the extra dimensions.",
"When considering scenarios with many extra dimensions, dimensional analysis may cease to be reliable for numerical estimates.",
"It works well with three dimensions because maximally symmetric volumes and areas scale as $V \\sim R^3$ and $A \\sim R^2$ , respectively, with the numerical coefficients being of order unity.",
"However, as pointed out by Barrow and Tipler [25], the volume of an $n$ -sphere of radius $R$ is $\\pi ^n (2R)^n \\Gamma (1+n/2)$ in Euclidean space, so there is an extra numerical factor $(2\\pi e/n)^{n/2} (n \\pi )^{-1/2}$ which decreases exponentially for large $n$ .",
"Similar deductions hold for (maximally symmetric) $n$ -dimensional surface areas.",
"Thus the dimensionless factors become important in higher dimensional spaces.",
"For present purposes, we may define an effective compactification scale $R_E \\equiv \\kappa (n)^{1/n}\\mathcal {R}_E$ , where $\\mathcal {R}_E$ is the true compactification scale of the extra dimensions and $\\kappa (n)$ is defined by the $n$ -dimensional volume being $V_{(n)} = \\kappa (n) \\mathcal {R}_E^{n}$ .",
"This yields $V_{(n)} \\sim R_E^{n}$ , as used in the estimate of $G$ in Eq.",
"(REF ), so the resulting expressions remain phenomenologically valid.",
"Similar arguments can be used to define effective length scales corresponding to highly asymmetric distributions, the simplest being $\\Delta R_{\\rm 3D} \\sim (\\Delta x \\Delta y \\Delta z)^{1/3}$ , for $\\Delta x \\ne \\Delta y \\ne \\Delta z$ , in three dimensions.",
"As we will show in Sec.",
", in higher dimensions such effective characteristic length scales quantify the asymmetry of a system and play a key role in determining its physics.",
"There are two interesting mass scales associated with the length scale $R_E$ : the mass whose Compton wavelength is $R_E$ , $ M_{E} \\equiv \\frac{\\hbar }{c R_E} \\simeq M_{P}\\frac{R_{P}}{R_E} \\, ,$ and the mass whose Schwarzschild radius is $R_E$ , $ M_E^{\\prime } \\equiv \\frac{c^2 R_E}{G} \\simeq M_{P} \\frac{R_E}{R_{P}} \\, .$ These mass scales are reflections of each other in the line $M=M_{P}$ in Fig 1, so that $M_{E}^{\\prime } = M_{P}^2/M_{E}$ .",
"An important implication of Eq.",
"(REF ) is that the usual expression for the Schwarzschild radius no longer applies for masses below $M_E^{\\prime }$ .",
"If the black hole is assumed to be (approximately) spherically symmetric in the higher-dimensional space on scales $R \\ll R_E$ , the expression for $R_S$ must be replaced with $ R_S \\simeq R_E \\left(\\frac{M}{M_E^{\\prime }} \\right)^{1/(n+1)} \\simeq R_{*}\\left(\\frac{M}{M_{P}}\\right)^{1/(1+n)} \\, ,$ where $R_{*}$ is defined by Eq.",
"(REF ).",
"Therefore, the slope of the black hole boundary in Fig.",
"REF becomes shallower for $M \\lesssim M_E^{\\prime }$ .",
"Strictly speaking, the metric associated with Eq.",
"(REF ) is only valid for infinite extra dimensions, since it assumes asymptotic flatness [26].",
"For black hole solutions with compact extra dimensions, one must ensure periodic boundary conditions with respect to the compact space.",
"However, Eq.",
"(REF ) should be accurate for black holes with $R_S \\ll R_E$ , so we adopt this for the entire range $R_{P} \\lesssim R \\lesssim R_E$ as a first approximation.",
"Similar problems arise, even in the Newtonian limit, since Eq.",
"(REF ) is also only valid for infinite extra dimensions and does not respect the periodicity of the internal space.",
"In practice, we expect corrections to smooth out the transition around $R_S \\simeq R_E$ , so that the true metric yields the asymptotic forms corresponding to the Schwarzschild radius of a $(3+1)$ -dimensional black hole on scales $R_S \\gg R_E$ and a $(3+n+1)$ -dimensional black hole on scales $R_S \\ll R_E$ .",
"This form of $R_S(M)$ for various values of $n$ is indicated in Fig. 2(a).",
"The intersect with the Compton boundary (assuming this is unchanged) is then given by Eq.",
"(REF ).",
"This implies $M_{P}^{\\prime } \\ll M_{P}$ and $R_{P}^{\\prime } \\gg R_{P}$ for $R_E \\gg R_{P}$ .",
"If the accessible energy is $E_{\\rm max}$ , then the extra dimensions can only be probed for $R_E > R_P \\left( \\frac{c^2M_P}{E_{\\rm max}} \\right)^{(2+n)/n} \\simeq 10^{(32/n) - 17} \\left(\\frac{E_{\\rm max}}{{\\rm 10 TeV}}\\right)^{-(2+n)/n}\\mathrm {cm} \\, ,$ where $E_{\\rm max}$ is normalised to 10 TeV, the order of magnitude energy associated with the Large Hadron Collider (LHC).",
"Thus black holes can be created at the LHC providing $ R_E > 10^{(30/n) - 18}\\, \\mathrm {cm}\\simeq {\\left\\lbrace \\begin{array}{ll}10^{12}\\, \\mathrm {cm}& (n=1) \\\\10^{-3} \\, \\mathrm {cm}& (n=2) \\\\10^{-14}\\, \\mathrm {cm}& (n=7) \\\\10^{-18} \\, \\mathrm {cm}& (n=\\infty ) \\, .\\end{array}\\right.",
"}$ Clearly, $n=1$ is excluded on empirical grounds but $n=2$ is possible.",
"One expects $n=7$ in M-theory [27], so it is interesting that $R_E$ must be of order a Fermi if all the dimensions are large.",
"$R_E \\rightarrow 10^{-18}$ cm as $n \\rightarrow \\infty $ since this is the smallest scale which can be probed by the LHC.",
"The above analysis assumes that all the extra dimensions have the same size.",
"One could also consider a hierarchy of compactification scales, $R_i = \\alpha _i R_{P}$ with $\\alpha _1 \\ge \\alpha _2 \\ge .... \\ge \\alpha _n \\ge 1$ , such that the dimensionality progressively increases as one goes to smaller distances [22].",
"In this case, the effective average length scale associated with the compact internal space is $\\langle R_E \\rangle = \\left( \\prod _{i=1}^{n} R_i \\right)^{1/n} = R_{P} \\left( \\prod _{i=1}^{n} \\alpha _i \\right)^{1/n} \\, .$ and the new effective Planck scales are $R_{P}^{\\prime } \\simeq \\left(R_{P}^2 \\prod _{i=1}^{n} R_i\\right)^{1/(2+n)} \\simeq (R_{P}^2 \\langle R_E \\rangle ^n)^{1/(2+n)}$ $M_{P}^{\\prime } \\simeq \\left(M_{P}^2 \\prod _{i=1}^{n}M_i^n\\right)^{1/(2+n)} \\simeq (M_{P}^2 \\langle M_E \\rangle ^{n})^{1/(2+n)} \\, ,$ where $M_i \\equiv \\hbar /(cR_i)$ and $\\langle M_E \\rangle \\simeq \\hbar /(c \\langle R_E \\rangle )$ .",
"For $R_{k+1} \\lesssim R \\lesssim R_{k}$ , the effective Schwarzschild radius is then given by $ R_S = R_{*(k)}\\left(\\frac{M}{M_{P}}\\right)^{1/(1+k)}, \\ \\ \\ R_{*(k)} = \\left(R_{P} \\prod _{i=1}^{k \\le n}R_{i}\\right)^{1/(1+k)}.$ This situation is represented in Fig. 2(b).",
"Clearly, for given $n$ , the Planck scales are not changed as much as in the scenario for which the extra dimensions all have the same scale.",
"Figure: Modification of the Schwarzschild line in the (M,R)(M,R) diagram in the presence of extra compact dimensions associated with a single length scale (a) or a hierarchy of length scales (b).",
"If the Compton scale preserves its usual form, the effective Planck scales are shifted as indicated.The relationship between the various key scales ($R_E, R_E^{\\prime },R_{P},R_{P}^{\\prime },M_{P},M_{P}^{\\prime },R_*$ ) in the above analysis is illustrated in Fig.",
"REF for the case of one extra spatial dimension ($n=1$ ).",
"This shows that the duality between the Compton and Schwarzschild length scales is lost if one introduces extra spatial dimensions.",
"However, this raises the issue of whether the expression for the Compton wavelength should also be modified in the higher-dimensional case.",
"We argue below that in this scenario a phenomenologically important length scale is the effective Compton wavelength, which may be identified with the minimum effective width (in 3-dimensional space) of the higher-dimensional wave packet $(\\Delta x)_{\\rm min}$ .",
"Figure: Key mass and length scales in the 3D case (solid lines) and 4D cases (dotted lines) if the extra dimension is compactified on a scale R E R_E.",
"The associated Compton and Schwarzschild masses are M E M_E and M E ' M_E^{\\prime }, respectively.",
"The revised Planck scales are M P ' M_{P}^{\\prime } and R P ' R_{P}^{\\prime } if duality is violated but M P M_{P} and R * R_* if it is preserved."
],
[
"Uncertainty Principle in higher dimensions",
"In this section, we consider whether the uncertainty in the momentum $\\Delta P$ in a ($3+n+1$ )-dimensional spacetime with $n$ compact dimensions scales inversely with the uncertainty in the position $\\Delta R$ , as in the 3-dimensional case, or according to a different law.",
"If we interpret $\\Delta R$ to mean the localisability of a particle, in the sense discussed in Sec.",
", we find that this depends crucially on the distribution of the wave packet in the extra dimensions, i.e.",
"on the degree of asymmetry between its size in the infinite and compact dimensions.i In 3-dimensional space with Cartesian coordinates $(x,y,z)$ , the uncertainty relations for position and momentum are $\\Delta x\\Delta p_{x} \\gtrsim \\hbar \\, , \\quad \\Delta y\\Delta p_{y} \\gtrsim \\hbar \\, , \\quad \\Delta z\\Delta p_{z} \\gtrsim \\hbar \\, .$ For spherically symmetric distributions, we have $ \\Delta x \\simeq \\Delta y \\simeq \\Delta z \\simeq \\Delta R_{\\rm 3D} \\, , \\quad \\Delta p_{x} \\simeq \\Delta p_{y} \\simeq \\Delta p_{z} \\simeq \\Delta P_{\\rm 3D} \\, ,$ where the axes are arbitrarily orientated, so that the relations (REF ) are each equivalent to $ \\Delta R_{\\rm 3D} \\, \\Delta P_{\\rm 3D} \\gtrsim c R_{P}M_{P} = \\hbar \\, .$ In $(3+n)$ spatial dimensions, we also have $ \\Delta x_i \\, \\Delta p_{i} \\gtrsim \\hbar \\quad (i = 1,2, .",
".",
".",
"n) \\, ,$ so for distributions that are spherically symmetric with respect to the three large dimensions we obtain $ \\Delta R_{\\rm 3D} \\, \\Delta P_{\\rm 3D} \\left(\\prod _{i=1}^{n} \\Delta \\, x_i\\Delta p_{i}\\right) \\gtrsim \\hbar ^{1+n} \\, ,$ where $\\Delta \\, x_i \\ne \\Delta R_{\\rm 3D}$ and $\\Delta p_{i} \\ne \\Delta P_{\\rm 3D}$ generally.",
"The exponent on the right is $1+n$ , rather than $3+n$ , because there is only one independent relation associated with the large spatial dimensions due to spherical symmetry.",
"Assuming, for simplicity, that the extra dimensions are compactified on a single length scale $R_E$ , then totally spherically symmetric wave functions are only possible on scales $\\Delta R < R_E$ in position space or $\\Delta P > c M_E$ in momentum space.",
"In this case, we may identify the standard deviations in the extra dimensions (i.e.",
"in both position and momentum space) with those in the infinite dimensions, $ \\Delta x_{i} \\simeq \\Delta R_{\\rm 3D} \\, , \\quad \\Delta p_{i} \\simeq \\Delta P_{\\rm 3D} \\, ,$ for all $i$ , so that Eq.",
"(REF ) reduces to (REF ).",
"Following the usual identifications, this gives the standard expression for the Compton wavelength in a higher-dimensional context.",
"However, this is not the only possibility.",
"The condition of spherical symmetry in the three large dimensions implies that the directly observable part of $\\psi $ is characterized by a single length scale, the 3-dimensional radius of the wave packet $\\Delta R_{\\rm 3D}$ .",
"One can therefore characterise the physical distribution of the wave packect by the ($1+n$ )-dimensional volume $ V_{(1+n)} \\simeq \\Delta R_{\\rm 3D} \\prod _{i=1}^{n}\\Delta x_i \\equiv (\\Delta \\mathcal {R})^{1+n} \\, ,$ where $\\Delta \\mathcal {R}$ corresponds to the effective ($1+n$ )-dimensional radius of the particle.",
"As demonstrated in Appendix B, for wave packets that are spherically symmetric in the large directions but irregular in the compact space, it is this length scale which controls pair-production rather than the geometric average over all $3+n$ dimensions, $[(\\Delta R_{\\rm 3D})^3\\Pi _{i=1}^{n} \\Delta x_i]^{1/(3+n)}$ .",
"Indeed, Appendix B suggests that $\\Delta \\mathcal {R}$ is the key length scale if only independent uncertainty relations contribute to the composite measurement.",
"This makes sense, since were we able to isolate our measurements of the 3-dimensional part of the wave packet, this would yield only a single length scale $\\Delta R_{\\rm 3D}$ ; any “smearing\" of this measurement due to the spread of the wave packet in the extra dimensions must be due to the $n$ additional independent widths, $\\Delta x_i \\ne \\Delta R_{\\rm 3D}$ .",
"We now consider scenarios with $\\Delta R_{\\rm 3D} \\ne \\Delta \\mathcal {R}$ and $\\Delta x_i \\ne \\Delta \\mathcal {R}$ , for at least some $i$ .",
"Such states may be considered “quasi-spherical\" in the sense that they are spherically symmetric with respect to the three large dimensions but possibly extremely irregular from the higher-dimensional perspective.",
"Note that $\\Delta R_{\\rm 3D} < R_{E}$ for $M>M_{E}$ and in this case Eq.",
"(REF ) becomes $ (\\Delta \\mathcal {R})^{1+n}\\Delta P_{\\rm 3D} \\left(\\prod _{i=1}^{n} \\Delta p_{i}\\right) \\gtrsim \\hbar ^{1+n} \\, .$ We restrict ourselves to states for which $ \\Delta p_{i} \\simeq \\kappa _{i}^{-1} cM_{P} \\, ,$ where the $\\kappa _i$ are dimensionless constants satisfying $ 1 \\le \\kappa _i \\le \\frac{R_E}{R_{P}} = \\frac{M_{P}}{M_E} \\, .$ This ensures that $ cM_{P} \\ge \\Delta p_i \\ge cM_E$ and restricts us to the higher-dimensional region of the $(M,R)$ diagram.",
"Conditions (REF ) then reduce to $ \\Delta x_i \\gtrsim \\kappa _i R_{P} \\, ,$ which together with Eq.",
"(REF ) ensures $R_{P} \\le (\\Delta x_i)_{\\rm min} \\le R_E \\, .$ Note that we are still considering the case in which all extra dimensions are compactified on a single length scale $R_E$ , so the $\\kappa _i$ have no intrinsic relationship with the constants $\\alpha _i$ used to characterize the hierarchy of length scales in Sec. .",
"They simply paramaterize the degree to which each extra dimension is “filled\" by the wave packet (e.g.",
"if $\\kappa _i = 1$ , the physical spread of the wave packet in the $i^{th}$ extra dimension is $R_{P}$ ).",
"Were we to consider a similar parameterization in the hierarchical case, it would follow immediately that $\\kappa _i \\le \\alpha _i$ .",
"Equation (REF ) now becomes $ \\Delta \\mathcal {R} \\gtrsim R_{P}\\left[\\frac{cM_{P}(\\prod _{i=1}^{n}\\kappa _{i})}{\\Delta P_{\\rm 3D}}\\right]^{1/(1+n)} \\, .$ The validity of this bound is subject to the quasi-spherical symmetry condition (REF ) but it is stronger than the equivalent condition (REF ) for fully spherically symmetric states (i.e.",
"the lower limit on $\\Delta \\mathcal {R}$ falls off more slowly with increasing $\\Delta P_{\\rm 3D}$ ).",
"By definition, such a wave packet is also quasi-spherically symmetric in momentum space, in the sense that it is spherically symmetric with respect to three infinite momentum dimensions, but not with respect to the full $(3+n)$ -dimensional momentum space.",
"For fully spherically symmetric states, we must put each $\\kappa _i$ equal to a single value $\\kappa $ , so that $\\Delta \\mathcal {R} \\equiv \\Delta R_{\\rm 3D}\\simeq \\kappa R_{P}$ and $\\Delta P_{\\rm 3D} \\simeq \\kappa ^{-1} cM_{P}$ , since the momentum space representation $\\psi (P)$ is given by the Fourier transform of $\\psi (R)$ .",
"Hence a wave function that is totally spherically symmetric in the $3+n$ dimensions of position space will also be totally spherically symmetric in the $3+n$ dimensions of momentum space.",
"For quasi-spherical states, the volume occupied by the particle in the $n$ extra dimensions of momentum space is $ V_{p(n)} \\simeq \\prod _{i=1}^{n}\\Delta p_i \\simeq (cM_{P})^{n}\\left(\\prod _{i=1}^{n}\\kappa _{i}\\right)^{-1}.$ In these states, we will assume that the extra-dimensional momentum volume remains fixed but the total momentum volume also depends on the 3-dimensional part $\\Delta P_{\\rm 3D}$ , which may take any value satisfying Eq.",
"(REF ).",
"We also fix the extra-dimensional physical volume.",
"The underlying physical assumption behind the mathematical requirement of fixed extra-dimensional volume is that the extra-dimensional space can only be probed indirectly – for example, via high-energy collisions between particles whose momenta in the compact directions cannot be directly controlled.",
"Therefore the net effect of any interaction is likely to leave the total extra-dimensional volume occupied by the wave packet unchanged, even if its 3-dimensional part can be successfully localized on scales below $R_E$ .",
"This is the mathematical expression of the fact that we have no control over the extra-dimensional part of any object - including that of the apparatus used to probe the higher-dimensional system.",
"As such, complete spherical symmetry in the higher-dimensional space is not expected and the most natural assumption is that asymmetry persists between the 3-dimensional and extra-dimensional parts of the wave function.",
"Indeed, the most natural assumption is $\\Delta x_i = R_i = R_E$ Since Eq.",
"(REF ) implies $ 1 \\le \\prod _{i=1}^{n}\\kappa _{i} \\le \\left(\\frac{R_{E}}{R_{P}}\\right)^{n},$ we have $ R_{P}\\left(\\frac{cM_{P}}{\\Delta P_{\\rm 3D}}\\right)^{1/(1+n)} \\lesssim (\\Delta \\mathcal {R})_{\\rm min} \\lesssim R_{*}\\left(\\frac{cM_{P}}{\\Delta P_{\\rm 3D}}\\right)^{1/(1+n)},$ where $R_{*}$ is defined by Eq.",
"(REF ) and we restrict ourselves to the higher-dimensional region of the $(M,R)$ diagram, $ cM_{P} \\ge \\Delta P_{\\rm 3D} \\ge cM_E \\, .$ In the extreme case $\\Delta P_{\\rm 3D} = cM_{P}$ , this gives $ R_{P} \\le (\\Delta \\mathcal {R})_{\\rm min} \\le R_*$ for any choice of the constants $\\kappa _i$ , with the extreme limits $(\\Delta \\mathcal {R})_{\\rm min} = R_{P}$ and $(\\Delta \\mathcal {R})_{\\rm min} = R_*$ corresponding to $\\kappa _i \\rightarrow 1$ and $\\kappa _i \\rightarrow R_E/R_{P}$ , respectively.",
"The first limit corresponds to the scenario $R_E \\rightarrow R_* \\rightarrow R_{P}$ , which recovers the standard Planck length bound on the minimum radius of a Planck mass particle.",
"In other words, if all the extra dimensions are compactified on the Planck scale, both the standard 3-dimensional Compton and Schwarzschild formulae hold all the way down to $R_{P}$ , giving the familiar intersect.",
"However, if $R_E > R_{P}$ , then $(\\Delta \\mathcal {R})_{\\rm min}$ for a Planck mass particle is larger than the Planck length and may be as large as the critical value $R_*$ .",
"This is the second limit and it occurs when the higher-dimensional part of the wave packet completely “fills\" the extra dimensions, each of these being compacified on the scale $R_E$ .",
"For $\\Delta P_{\\rm 3D} \\le cM_E$ , the same scenario gives $(\\Delta \\mathcal {R})_{\\rm min} \\ge R_E$ , which corresponds to the effectively 3-dimensional region of the $(M,R)$ plot.",
"In this region, the assumption of quasi-sphericity breaks down and the 3-dimensional and higher-dimensional parts of the wave packet decouple with respect to measurements which are unable to probe the length/mass scales associated with the extra dimensions.",
"We may therefore set $ \\Delta \\mathcal {R} \\gtrsim R_{*}\\left(\\frac{cM_{P}}{\\Delta P_{\\rm 3D}}\\right)^{1/(1+n)}$ as the strongest lower bound on $\\Delta \\mathcal {R}$ , since this is the upper bound on the value of $(\\Delta \\mathcal {R})_{\\rm min}$ .",
"To reiterate, this comes from combining two assumptions: (a) the wave function of the particle is quasi-spherically symmetric – in the sense of Eq.",
"(REF ) – with respect to the full higher-dimensional space on scales $\\Delta \\mathcal {R} \\lesssim R_E$ ; and (b) the wave packet is space-filling in the $n$ additional dimensions of position space.",
"Note that the unique 3-dimensional uncertainty relation and each of the $n$ independent uncertainty relations for the compact directions still hold individually.",
"However, the higher-dimensional uncertainty relations are satisfied for any choice of the constants $\\kappa _i$ in the range specified by Eq.",
"(REF ) and the remaining 3-dimensional relation $\\Delta R_{\\rm 3D} \\gtrsim \\hbar /\\Delta P_{\\rm 3D}$ is satisfied automatically for any $\\Delta P_{\\rm 3D}$ satisfying Eq.",
"(REF ).",
"In the limit $\\kappa _i \\rightarrow R_E/R_{P}$ for all $i$ , in which the wave packet completely fills the compact space, we have $(\\Delta R_{\\rm 3D})_{\\rm min} = (\\Delta \\mathcal {R})_{\\rm min} = R_E$ when $\\Delta P_{\\rm 3D} = cM_E$ , so that the 3-dimensional and $(3+n)$ -dimensional formulae match seamlessly.",
"Thus, for $\\Delta P_{\\rm 3D} \\simeq cM_{P}$ , we have $(\\Delta \\mathcal {R})_{\\rm min} \\simeq R_*$ but the genuine 3-dimensional radius of the wave packet is of order $(\\Delta R_{\\rm 3D})_{\\rm min} \\simeq R_{P}$ ."
],
[
"Compton wavelength and black holes in higher dimensions",
"In discussing the Compton wavelength of a particle in higher dimensions, the question of the experimental accessibility of the extra dimensions is crucial.",
"Unless the experimental set-up allows direct control over the size of the wave-packet in the compact space, we cannot assume that a probe energy $E \\sim \\hbar c/R_C$ implies the localisation of the (3+n)-dimensional wave-function within a volume $V \\sim R_C^{3+n}$ .",
"If there is only control over the three large dimensions, then the total volume of the wave-packet may be much larger than the minimum value.",
"In principle, this corresponds to a larger minimum width for the wave-packet (i.e.",
"a larger effective higher-dimensional Compton wavelength).",
"In this section we use the analysis of Sec.",
"4 and the identifications (REF ) to derive an expression for the effective higher-dimensional Compton wavelength: $ R_{C} \\simeq R_{E}\\left(\\frac{M_{E}}{M}\\right)^{1/(1+n)} \\simeq R_{*}\\left(\\frac{M_{P}}{M}\\right)^{1/(1+n)} \\, .$ This is equivalent to the identifications $ R_C \\simeq \\Delta \\mathcal {R} \\, , \\quad \\Delta P_{3D} \\simeq M c \\, ,$ where $ \\Delta \\mathcal {R}$ is given by Eq.",
"(REF ) with $\\Delta p_i \\simeq M_Ec$ for all $i$ .",
"Clearly, Eq.",
"(REF ) is consistent with the bound () since $M<M_P$ in the particle regime.",
"These arguments imply that, when extrapolating the usual arguments for the Compton wavelength in non-relativistic quantum theory to the case of compact extra dimensions, we should identify the geometric average of the spread of the wave packet in $1+n$ spatial dimensions with the effective particle `radius' $ \\Delta \\mathcal {R}$ but its spread in the large dimensions of momentum space with the rest mass.",
"The identifications (REF )-(REF ) are also consistent with the relativistic interpretation of the Compton wavelength as the minimum localization scale for the wave packet below which pair-production occurs (see Appendix B).",
"This is fortunate, as there is clearly a problem with identifying the standard deviation of the total higher-dimensional momentum, $P_T = \\sqrt{P_{\\rm 3D}^2 + P_{E}^2}$ where $P_{E}^2 = \\Sigma _{i=1}^{n}p_ip^i$ , with the rest mass of the particle.",
"Since the standard deviations of the individual extra-dimensional momenta are bound from below by $\\Delta p_i \\ge cM_E$ , we have $\\Delta P_T \\ge c M_E$ .",
"The identification $\\Delta P_T = M c$ would then imply $M > M_E$ .",
"Since $R_E$ must be very small to have avoided direct detection, $M_E$ must be large and the above requirement contradicts known physics as it requires all particles to have masses $M > M_E$ .",
"The manifest asymmetry of the wave packet in position space on scales less than $R_E$ (and on scales greater than $cM_E$ in momentum space) also requires identifications of the form (REF ) in order for the standard Compton formula to hold for $R \\ge R_E$ in a higher-dimensional setting.",
"What happens to the standard formula below this scale is unclear.",
"If the wave packet is able to adopt a genuinely spherically symmetric configuration in the full higher-dimensional space (including momentum space), then the above arguments suggest the identifications $(\\Delta R_{\\rm T})_{\\rm min} \\simeq R_C$ and $(\\Delta P_T)_{\\rm max} \\simeq Mc$ for $M_E \\le M \\le M_{P}$ , so that the usual Compton formula holds all the way down to $M \\simeq M_{P}$ .",
"The possible short-comings of this approach are that it would be valid only for spherically symmetric states and that it requires a change in the identification of the rest mass and particle radius at $M = M_E$ , i.e.",
"$R_C \\simeq (\\Delta R_{\\rm T})_{\\rm min} \\rightarrow R_C \\simeq (\\Delta R_{\\rm 3D})_{\\rm min}$ and $Mc \\simeq (\\Delta P)_{\\rm max} \\rightarrow Mc \\simeq (\\Delta P_{\\rm 3D})_{\\rm max}$ .",
"As wave packets will generally be asymmetric on scales $R \\ge R_E$ , it is reasonable to assume that the asymmetry will persist, even when we are able to (indirectly) probe scales associated with the extra dimensions.",
"For example, we may consider the following two gendanken experiments.",
"(i) We localize a particle in 3-dimensional space by constructing a spherically symmetric potential barrier.",
"We then gradually increase the steepness of the potential well, increasing the energy and localizing the particle on ever smaller length scales.",
"In principle, we may even shrink the 3-dimensional radius below the scale of the internal space.",
"But what about the width of the wave packet in the compact directions?",
"Since we did not design our initial potential to be spherically symmetric in $3+n$ spatial dimensions – having no direct manipulative control over its form in the extra dimensions – it is unlikely that one would suddenly obtain a fully spherically symmetric potential in higher-dimensional space simply by increasing the energy at which our “measuring device\" operates (i.e.",
"above $M_E c^2$ ).",
"(ii) We confine a particle within a spherical region of 3-dimensional space by bombarding it with photons from multiple angles.",
"Increasing the energy of the photons then reduces the radius of the sphere.",
"But how can we control the trajectories of the probing photons in the internal space?",
"Since, again, we do not have direct manipulative control in the compact space over the apparatus that creates the photons, it is impossible to ensure anything other than a random influx of photons (with random extra-dimensional momenta) in the $n$ compact directions.",
"In this case, we would expect to be able to measure the average photon energy and to relate this to a single average length scale, but we cannot ensure exact spherical symmetry with respect to all $3+n$ dimensions, or measure the spread of the wave packet in each individual extra dimension.",
"In the most extreme case, we may expect the combined effects of our (random) experimental probing of the extra dimensions to cancel each other out, leaving the total volume of the wave packet in the compact space unchanged.",
"This justifies Eq.",
"(REF ) but does not alter the reduction of the 3-dimensional and hence overall volume of the wave packet when the energy of the probe particles/potential barrier is increased.",
"Together, these considerations lead to the scaling predicted by Eq.",
"(REF ).",
"As this corresponds to the maximum possible asymmetry for which a single length scale can be associated with $\\psi $ , this should give the highest possible lower bound on the size of a quantum mechanical particle in a spacetime with $n$ compact extra dimensions.",
"Since the particle and black hole regimes are connected, it is also meaningful to ask if we can associate a wave function $\\psi $ with a black hole.",
"If so, should $\\psi $ be associated with the centre of mass of the black hole or with its event horizon at $R_S<R_E$ (cf.",
"Casadio [28])?",
"For classical non-extended bodies, i.e.",
"point-particles, such problems of quantization do not arise.",
"In the classical theory, with only infinite dimensions, a Schwarzschild black hole is the unique spherically symmetric vacuum solution [29].",
"However, in the quantum mechanical case, our previous analysis suggests that it may be possible to associate multiple quasi-spherically symmetric wave packets with the unique classical solution, just as we can for classical (spherically symmetric) point particles.",
"The investigation of both these points lies beyond the scope of this paper and is left for future work.",
"To summarize our results for higher-dimensional black holes and fundamental particles, we have $ R_{C} \\simeq {\\left\\lbrace \\begin{array}{ll}R_{P}\\frac{M_{P}}{M}& (R_{C} \\gtrsim R_E) \\\\R_{*}\\left(\\frac{M_{P}}{M}\\right)^{1/(1+n)}& (R_{C} \\lesssim R_E)\\end{array}\\right.",
"}$ $ R_{S} \\simeq {\\left\\lbrace \\begin{array}{ll}R_{P}\\frac{M}{M_{P}}& (R_{S} \\gtrsim R_E) \\\\R_{*}\\left(\\frac{M}{M_{P}}\\right)^{1/(1+n)}&(R_{S} \\lesssim R_E)\\end{array}\\right.",
"}$ for $n$ extra dimensions compactified on a single length scale $R_E$ , and these lines intersect at $(R_{*},M_{P})$ .",
"The crucial point is that there is no TeV quantum gravity in this scenario since the intersect of the Compton and Schwarzschild lines still occurs at $M \\simeq M_{P}$ .",
"The effective Planck length is increased to $R_*$ but this does not allow the production of higher-dimensional black holes at accelerators.",
"Thus, the constraint (REF ) on the scale $R_E$ in the conventional picture no longer applies.",
"Figure: Modifications of Fig.",
"2 for extra dimensions compactified on a single length scale (a) or a hierarchy of length scales (b) if one imposes quasi-spherical symmetry on the higher-dimensional wave packet, preserving the duality between the Compton and Schwarzschild expressions.This scenario is illustrated in Fig.",
"4(a) for extra dimensions compactified on a single length scale $R_E$ and in Fig.",
"4(b) for a hierarchy of length scales.",
"In the latter case, the expressions (REF )-(REF ) must be modified to $R_{C} \\simeq {\\left\\lbrace \\begin{array}{ll}R_{P}\\frac{M_{P}}{M}& (R_{C} \\gtrsim R_1) \\\\R_{*(k)}\\left(\\frac{M_{P}}{M}\\right)^{1/(1+k)}& (R_{k+1} \\lesssim R_{C} \\lesssim R_{k})\\end{array}\\right.",
"}$ $R_{S} \\simeq {\\left\\lbrace \\begin{array}{ll}R_{P}\\frac{M}{M_{P}}&(R_{C} \\gtrsim R_1) \\\\R_{*(k)}\\left(\\frac{M}{M_{P}}\\right)^{1/(1+k)}& (R_{k+1} \\lesssim R_{C} \\lesssim R_{k})\\end{array}\\right.",
"}$ where $R_1$ is the largest compact dimension and $R_{*(k)}$ is defined in Eq.",
"(REF )."
],
[
"Observational consequences",
"In this section we consider two possible observational consequences of retaining the duality between the Compton and Schwarzschild expressions.",
"The first relates to the detectability of exploding primordial black holes (PBHs).",
"There is still no unambiguous detection of such explosions but it has been claimed that some short-period gamma-ray bursts could be attributed to PBHs [35].",
"The second relates to high-energy scattering experments and the enhancement of pair-production at at accelerators on scales below $R_E$ ."
],
[
"Black hole evaporation",
"The Hawking temperature of a black hole of mass $M$ and radius $R_S$ in three dimensions is $ T_H \\simeq T_{P}\\frac{M_{P}}{M} \\simeq T_{P} \\frac{R_{P}}{R_S} \\, ,$ where $T_P = M_Pc^2/k_B$ is the Planck temperature.",
"This result can be obtained from the relations $\\Delta R_{\\rm 3D} \\simeq R_S \\propto M , \\quad \\Delta P_{\\rm 3D} \\propto 1/(\\Delta R_{\\rm 3D}), \\quad T_H \\propto \\Delta P_{\\rm 3D} \\propto M^{-1} \\, ,$ where the second relation is the standard Uncertainty Principle and the third relation assumes a black-body distribution for the emitted particles.",
"The temperature can also be obtained from the surface gravity: $T_H \\propto \\kappa \\propto M/R_S^{2} \\propto M^{-1} \\, ,$ this being equivalent to the relations $\\Delta R_{\\rm 3D} \\simeq R_S \\propto M , \\quad \\Delta P_{\\rm 3D} \\propto 1/(\\Delta R_{\\rm 3D}), \\quad T_H \\propto 1/ \\Delta R_{\\rm 3D} \\propto M^{-1} \\, .$ In this formulation the second expression is not needed to derive $T_H$ but is required by the HUP.",
"The only difference between (REF ) and (REF ) is that the first associates the temperature with a momentum and the second with a length but both sets of identifications yield Eq.",
"(REF ).",
"In the higher-dimensional case, if all the extra dimensions have the same compactification scale $R_E$ and one assumes the standard expression for the Compton wavelength, then the temperature is modified to [32], [33] $ T_H \\simeq T_P^{\\prime } \\left(\\frac{M_{P}^{\\prime }}{M}\\right)^{1/(1+n)} \\simeq T_*\\left(\\frac{M_{P}}{M}\\right)^{1/(1+n)} .$ Here $M_P^{\\prime }$ is given by Eq.",
"(REF ) and we have used the definitions $ T_P^{\\prime } \\equiv M_P^{\\prime } c^2/k_B, \\quad T_* \\equiv (T_PT_E^n)^{1/(1+n)}, \\quad T_E \\equiv M_Ec^2/k_B \\simeq T_{P} R_{P}/R_E \\, .$ The $M$ -dependence in Eq.",
"(REF ) can be derived from the relations $\\ \\Delta R_{\\rm T} \\simeq R_S \\propto M^{1/(1+n)} , \\quad \\Delta P_T \\propto 1/(\\Delta R_{\\rm T}) , \\quad T_H \\propto \\Delta P_T \\propto M^{-1/(1+n)} \\, ,$ where $\\Delta P_T$ and $\\Delta R_T$ appear because the wave function is assumed to be spherically symmetric in the full space.",
"The temperature can again be obtained from the surface gravity: $T_H \\propto \\kappa \\propto M/R_S^{2+n} \\propto M^{-1/(1+n)} \\, .$ Note that Eq.",
"(REF ) extends all way down to the reduced Planck scale $M_P^{\\prime }$ , where the temperature has the maximum possible value ($T_P^{\\prime } = M_P^{\\prime }$ ), while $T_*$ is the temperature of a black hole with the original Planck mass ($M_P$ ).",
"If the forms of the Uncertainty Principle and the Compton wavelength are modified to preserve duality in the higher-dimensional case, there are two ways to generalise the above result.",
"(i) The first way assumes that $\\Delta R_{3D}$ is replaced by $\\Delta \\mathcal {R}$ in Eq.",
"(REF ) but that $\\Delta P_{3D}$ is still the relevant momentum, this being associated with the emitted particle's rest mass.",
"One then has $\\Delta \\mathcal {R}\\simeq R_S \\propto M^{1/(1+n)} , \\quad \\Delta P_{3D} \\propto 1/( \\Delta \\mathcal {R})^{1+n} , \\quad T_H \\propto \\Delta P_{3D} \\propto M^{-1} \\, ,$ where the second relation comes from Eq.",
"(REF ) and the last one is consistent with the notion that a particle can only be emitted if $T_H$ exceeds its rest mass.",
"In this case, the black hole temperature reverts to the standard Hawking expression, without any dependence on $n$ , and the largest black hole temperature is just the maximum one allowed by the theory ($T_P$ ).",
"(ii) The second way identifies the temperature with the surface gravity (REF ), this not applying in the first case.",
"This corresponds to replacing Eq.",
"(REF ) with $\\Delta \\mathcal {R }\\simeq R_S \\propto M^{1/(1+n)} , \\quad \\Delta P_{3D} \\propto 1/( \\Delta R_{3D}) \\propto 1/M, \\quad T_H \\propto 1/ \\Delta \\mathcal {R} \\propto M^{-1/(1+n)} \\, ,$ where the second condition is required for consistency with Eq.",
"(REF ).",
"This is equivalent to the surface gravity argument, since $\\kappa \\propto M/R_S^{2+n} \\propto \\Delta \\mathcal {R}^{1+n}/ \\Delta \\mathcal {R}^{2+n} \\propto 1/\\Delta \\mathcal {R} \\, .$ so the black hole temperature is still given by Eq.",
"(REF ) and has a maximum value of $T_*$ .",
"Since both the above arguments are heuristic, we cannot be sure which one is correct, so we allow for both possibilities below.",
"The issue is whether one associates the black hole temperature with a length scale or a momentum scale in higher dmensions, these being inequivalent if the black hole is spherically symmetric but the particle wave-function is not.",
"The first argument has the attraction that black holes span the entire available temperature range; the second argument is more consistent with the standard higher-dimensional analysis.",
"We now consider the consequences of these results for PBH evaporation.",
"In the 3-dimensional model ($n=0$ ), PBHs complete their evaporation at the present epoch if they have an initial $M_0 \\simeq 10^{15}$ g and an initial radius $R_0 \\simeq 10^{-13}$ cm, comparable to the size of a proton [34].",
"For most of their lifetime these PBHs are producing photons with energy $E_0 \\simeq 100$ MeV, so the extragalactic $\\gamma $ -ray background at this energy places strong constraints on their number density and current explosion rate [35].",
"In principle, these PBHs could also contribute to cosmic-ray positrons and antiprotons, although there are other possible sources of these particles [34].",
"However, the black holes evaporating at the present epoch are necessarily higher dimensional if $R_E > 10^{-13}$ cm.",
"In the TeV quantum gravity scenario, for example, Eq.",
"(REF ) implies that this condition is always satisfied for $n< 7$ and this is expected in M-theory because the maximum number of compactified dimensions is 7.",
"Figure 5 shows the $(M,R)$ diagram for a hierarchical scenario with three extra dimensions, compactified on scales $R_1$ , $R_2$ and $R_3$ .",
"We therefore need to recalculate the critical mass and temperature of PBHs evaporating at the present epoch, distinguishing between the standard case in which duality is broken and the alternative case in which it is preserved.",
"If there are $n$ extra dimensions, each with compactification scale $R_E$ , and if the Compton wavelength has the standard form, then the density of black-body radiation of temperature $T$ is $\\rho _{BB} \\propto T/ R_C(T)^{n+3}\\propto T^{4+n} \\,$ and the black hole mass loss rate for $M<M_E$ is $dM/dt \\propto R_S^{2+n} T_H^{4+n} \\propto M^{-2/(1+n)} \\, ,$ where we assume that the emission is into the full ($3+n$ )-dimensional space.",
"This leads to a black hole lifetime $\\tau \\simeq \\frac{M}{dM/dt}\\simeq \\left( \\frac{M}{M_P} \\right)^{(3+n)/(1+n)} \\left( \\frac{R_E}{R_P} \\right) ^{2n/(1+n)} t_P \\, ,$ so the critical mass of the PBHs evaporating at the present epoch becomes $M_{crit} \\simeq 10^{15} {\\rm g} \\left( \\frac{t_0}{t_P} \\right) ^{2n/3(1+n)} \\left( \\frac{R_P}{R_E} \\right)^{2n/(3+n)}$ and the associated temperature is $T_{crit} \\simeq 100 \\, {\\rm MeV} \\left( \\frac{t_0}{t_P} \\right)^{n/3(3+n)} \\left( \\frac{R_P}{R_E} \\right) ^{n/(3+n)} \\, .$ Thus both $M_{crit}$ and $T_{crit}$ are modified compared to the 3-dimensional case ($n=0$ ).",
"This means that all the standard constraints on PBHs evaporating at the present epoch need to be recalculated, although we do not attempt this here.",
"If there is a hierarchy of extra dimensions, the value of $n$ in the above equations must be replaced by $k$ for $R_{k+1} < R_{crit} < R_k$ ($k \\le n$ ).",
"If T-duality is preserved, the situation is very different and there are several sources of uncertainty in modifying Eq.",
"(REF ).",
"The first concerns whether the back hole temperature is given by Eq.",
"(REF ) or (REF ).",
"The second concerns the power of $T$ in Eq.",
"(REF ), which depends on the density of black-body radiation with temperature $T$ in higher dimensions.",
"This also relates to whether the temperature is associated with a momentum or a length scale (i.e.",
"the first issue).",
"Perhaps the most natural assumption is that it is given by $\\rho _{BB} \\propto T/ R_C(T)^{n+3}\\propto T^{2(2+n)/(1+n)} \\,$ rather then Eq.",
"(REF ), where we have assumed $R_C \\propto T^{-1/(1+n)}$ in accordance with the expression for the modifed Compton wavelength.",
"However, if the particle wave-function is non-spherical, with $R_i =R_E$ in the extra dimensions, one might expect $\\rho _{BB} \\propto T/ [R_C(T)^3 R_E^n] \\propto T^4\\, .$ The third uncertainty concerns the power of $R_S$ in Eq.",
"(REF ).",
"This is $n+2$ in the totally spherically syymmetric case.",
"However, if black-body particles have a scale $R_i =R_E$ in the extra dimensions, one might expect them to be confined to that scale, in which case the effective black hole area scales as $R_S^2$ .",
"With so many uncertainities, we cannot advocate any expression for $dM/dt$ with confidence.",
"Only if one adopts the combination of Eqs.",
"(REF ), (REF ) and the last argument does one obtain the same scaling as in the standard 3-dimensional scenario, with the mass of PBHs evaporating today and the associated temperature preserving their standard Hawking values.",
"Figure: Showing the form of the Compton and Schwarzschild scalesfor a hierarchical model with three compactified dimensions in which the Compton-Schwarzschildduality is preserved.",
"In this case, the Planck length but not the Planck mass is modified and the collapsing matter may enter the quantum gravity regime at the modified Planck density."
],
[
"Consistency with $D$ -particle scattering results",
"We now consider the consistency of our phenomenologically general results with respect to the leading higher-dimensional theory of fundamental physics: string theory.",
"In particular, we focus on their consistency with minimum-radius results for higher-dimensional, non-relativistic and quantum mechanical particle-like objects, known as $D$ -particles.",
"The end points of open strings obey Neumann or Dirichlet boundary conditions (or a combination of both) and are restricted to $(p+1)$ -dimensional submanifolds, where $p \\le 3+n$ , called $D_p$ -branes.",
"Although these are composite rather than fundamental objects, they have dynamics in their own right and an intrinsic tension $\\mathcal {T}_p = (g_sl_s^{p+1})^{-1}$ , where $g_s$ denotes the string coupling and $l_s$ is the fundamental string length scale [36].",
"Thus, $D_0$ -branes, also referred to as $D$ -particles, are point-like, and possess internal structure only on scales $\\lesssim g_sl_s$ .",
"This may be seen as the analogue of the Compton wavelength in $D_0$ -brane models of fundamental particles.",
"At high energies, strings can convert kinetic into potential energy, thereby increasing their extension and counteracting attempts to probe smaller distances.",
"Therefore, the best way to probe $D_p$ -branes is by scattering them off each other, instead of using fundamental strings as probes [37].",
"$D$ -particle scattering has been studied in detail by Douglas et al [38], who showed that slow moving $D$ -particles can be used to probe distances down to $g_s^{1/3}l_s$ in $D=10$ spacetime dimensions.",
"This result may be obtained heuristically as follows [36].",
"Let us consider a perturbation of the metric component $g_{00} = 1+2V$ induced by the Newtonian potential $V$ of a higher-dimensional particle of mass $M$ .",
"In $D$ spacetime dimensions, this takes the form $ V \\simeq -\\frac{G_DM}{(\\Delta x)^{D-3}} \\, ,$ where $\\Delta x$ is the spatial extension of the particle and $G_D$ is the $D$ -dimensional Newton's constant, so that the horizon is located at $ \\Delta x \\simeq (G_DM)^{1/(D-3)}.$ (For convenience, we set $c=\\hbar =1$ throughout this section.)",
"In spacetimes with $n$ compact spatial dimensions, this is related to the $(3+1)$ -dimensional Newton's constant via $G \\simeq G_D/R_E^n$ , so that, for $D = 3+n+1$ , we simply recover the formula for the higher-dimensional Schwarzschild radius (REF ).",
"However, we may also use Eq.",
"(REF ) to derive the minimum length obtained from $D$ -particle scattering in [38] by first setting $M \\simeq 1/\\Delta t$ , where $\\Delta t$ is the time taken to test the geometry, and then using the higher-dimensional Newton's constant derived from string theory, $G_D \\simeq g_s^2 l_s^{D-2}$ [6].",
"This gives $ (\\Delta t)(\\Delta x)^{D-3} \\gtrsim g_s^2l_s^{D-2} \\, .$ Combining this with the spacetime uncertainty principle, which is thought to arise as a consequence of the conformal symmetry of the $D_p$ -brane world-volume [39], [40], [41], $ \\Delta x \\Delta t \\gtrsim l_s^2,$ we then have $ (\\Delta x)_{\\rm min} \\simeq g_s^{2/(D-4)}l_s, \\ \\ \\ (\\Delta t)_{\\rm min} \\simeq g_s^{-2/(D-4)}l_s.$ For $D=10$ , this gives $(\\Delta x)_{\\rm min} \\sim g_s^{1/3}l_s$ , as claimed.",
"Combining results from string theory and higher-dimensional general relativity by setting $G_D \\simeq g_s^2l_s^{D-2} \\simeq R_{P}^2R_E^{D-4}$ with $D=3+n+1$ , we obtain $ R_{P}^{\\prime } \\simeq (R_{P}^2R_E^n)^{1/(2+n)} \\simeq g_s^{2/(2+n)}l_s \\, ,$ which gives $R_{P}^{\\prime } \\simeq g_s^{1/4}l_s$ as the modified Planck length for $D = 10$ .",
"In fact, Eqs.",
"(REF )-(REF ) suggest that the minimum positional uncertainty for $D$ -particles cannot be identified with the modified Planck length obtained from the intersection of the higher-dimensional Schwarzschild line, $R_S \\sim M^{1/(1+n)}$ , and the standard Compton line, $R_C \\sim M^{-1}$ , in any number of dimensions.",
"Hence, the standard scenario is incompatible with $D$ -particle scattering results.",
"However, it is straightforward to verify that, if $R_{P} \\simeq g_s^{-2/n}l_s$ , we have $R_* \\simeq (R_{P}R_E^n)^{1/(1+n)} \\simeq (\\Delta x)_{\\rm min} \\simeq g_s^{2/n}l_s$ , so that the intersection of the higher-dimensional Schwarzschild and Compton lines is equal to the minimum length scale that can be probed by $D$ -particles.",
"In this scenario, $R_E \\simeq g_s^{2(1+n)/n^2}l_s$ , and we note that $R_E \\rightarrow R_* \\rightarrow R_{P}^{\\prime } \\rightarrow R_{P} \\rightarrow l_s$ for $g_s \\rightarrow 1$ , as required for consistency.",
"In general, $R_* > R_{P}^{\\prime }$ (or equivalently $R_E > R_{P}$ ) requires $g_s > 1$ ."
],
[
"Conclusions",
"We have addressed the question of how the effective Compton wavelength of a fundamental particle – defined as the minimum possible positional uncertainty over measurements in all independent spatial directions – scales with mass if there exist $n$ extra compact dimensions.",
"In $(3+1)$ -dimensional spacetime, the Compton wavelength scales as $R_C \\sim M^{-1}$ , whereas the Schwarzschild radius scales as $R_S \\sim M$ , so the two are related via $R_S \\sim R_{P}^2/R_C$ .",
"In higher-dimensional spacetimes with $n$ compact extra dimensions, $R_S \\sim M^{1/(1+n)}$ on scales smaller than the compactification radius $R_E$ , which breaks the symmetry between particles and black holes if the Compton scale remains unchanged.",
"However, we have argued that the effective Compton scale depends on the form of the wavefunction in the higher-dimensional space.",
"If this is spherically symmetric in the three large dimensions, but maximally asymmetric in the full $3+n$ spatial dimensions, then the effective radius scales as $R_C \\sim M^{-1/(1+n)}$ rather than $M^{-1}$ on scales less than $R_E$ and this preserves the symmetry about the $M \\simeq M_{P}$ line in ($M,R$ ) space.",
"In this scenario, the effective Planck length is increased but the Planck mass is unchanged, so quantum gravity and microscopic black hole production are associated with the standard Planck energy, as in the 3-dimensional scenario.",
"On the other hand, one has the interesting prediction that the Compton line – which marks the onset of pair-production – is “lifted\", relative to the 3-dimensional case, in the range $R_{P} < R < R_E$ , so that extra-dimensional effects may become visible via enhanced pair-production rates for particles with energies $E > M_Ec^2 = \\hbar c/R_E$ .",
"This prediction is consistent with minimum length uncertainty relations obtained from $D$ -particle scattering amplitudes in string theory.",
"Also, as indicated in Fig.",
"4, the existence of extra compact dimensions has crucial implications for the detectability of black holes evaporating at the present epoch, since they are necessarily higher-dimensional for $R_E > 10^{-13}$ cm.",
"In this paper, we have assumed that non-relativistic quantum mechanical particles obey the standard Heisenberg Uncertainty Principle (HUP) in each spatial direction.",
"The modified expression for the effective Compton line, which retains a simple power-law form until its intersection with the higher-dimensional Schwarzschild line in the ($M,R$ ) diagram, is seen to arise from the application of the HUP to maximally asymmetric wave functions.",
"These are spherically symmetric with respect to the three large dimensions but pancaked in the compact directions.",
"No allowance has been made for deviations from the HUP, as postulated by various forms of Generalised Uncertainty Principle (GUP) proposed in the quantum gravity literature, and no attempt has been made to smooth out the transition between particle and black hole states at the Planck point, as postulated by the Black Hole Uncertainty Principle (BHUP) correspondence [10].",
"These effects would entail different temperature predictions in the Planck regime even in the 3-dimensional case.",
"Our main intention here has been to examine the consequences of the existence of extra dimensions in the `standard' (i.e.",
"HUP-based) scenario.",
"Many other authors have studied the implication of the GUP for higher dimensional models [42] but without imposing (semi-)T-duality.",
"Finally, if we interpret the Compton wavelength as marking the boundary on the $(M,R$ ) diagram below which pair-production rates becomes significant, we expect the presence of compact extra dimensions to affect pair-production rates at high energies.",
"Specifically, we expect pair-production rates at energies above the mass scale associated with the compact space, $M_E \\equiv \\hbar /(cR_E)$ , to be enhanced relative to the 3-dimensional case.",
"This is equivalent to raising the Compton line, i.e.",
"increasing its (negative) gradient in the $(M,R)$ diagram.",
"A more detailed relativistic analysis would be needed to confirm whether this is a generic result for massive scalar fields (corresponding to uncharged matter).",
"There is tentative theoretical evidence that enhanced pair-production may be a generic feature of higher-dimensional theories in which some directions are compactified but the available literature on this is sparse (c.f.",
"[30], [31]).",
"Acknowledgments BC thanks the Research Center for the Early Universe (RESCEU), University of Tokyo, and ML thanks the Institute for Fundamental Study at Naresuan University for hospitality received during this work.",
"We thank John Barrow, Tiberiu Harko, Juan Maldacena, Shingo Takeuchi, Pichet Vanichchapongjaroen and Marek Miller for helpful comments and discussions."
],
[
"Intepretation of the uncertainty principle",
"In the form originally derived by Heisenberg, the uncertainty principle states that the product of the “uncertainties\" in the position and momentum of a quantum mechanical particle is of order of or greater than the reduced Planck's constant $\\hbar $ [43].",
"More generally, the rigorous definition of the uncertainty $\\Delta _\\psi O$ for an operator $\\hat{O}$ is the standard deviation for a large number $N$ of (absolutely precise) repeated measurements of an ensemble of identically prepared systems described by the wave vector $\\left|\\psi \\right\\rangle $ : $ \\Delta _\\psi O = \\sqrt{\\langle \\psi |\\hat{O}^2|\\psi \\rangle - \\langle \\psi |\\hat{O}|\\psi \\rangle ^2} \\, .$ Formally, this expression corresponds to the limit $N \\rightarrow \\infty $ and is generally $|\\psi \\rangle $ -dependent.",
"Thus the uncertainty $\\Delta _\\psi O$ does not correspond to incomplete knowledge about the value of the property $O$ for the system, since $\\left|\\psi \\right\\rangle $ need not possess a definite value of $O$ .",
"Consistency with the Hilbert space structure of quantum mechanics requires that the product of the uncertainties associated with arbitrary operators $\\hat{O}_1$ and $\\hat{O}_2$ satisfy the bound [20], [44] $ \\Delta _\\psi O_1 \\Delta _\\psi O_2 &\\ge & \\frac{1}{2}\\sqrt{|\\langle \\psi |[\\hat{O}_1,\\hat{O}_1]|\\psi \\rangle |^2 + |\\langle \\psi |[\\hat{A},\\hat{B}]_{+}|\\psi \\rangle |^2}\\ge \\frac{1}{2}|\\langle \\psi |[\\hat{O}_1,\\hat{O}_1]|\\psi \\rangle | \\, ,$ where $[\\hat{O}_1,\\hat{O}_2]$ is the commutator of $\\hat{O}_1$ and $\\hat{O}_2$ and $[\\hat{A},\\hat{B}]_{+}$ is the anticommutator of $\\hat{A} = \\hat{O}_1 - \\langle \\hat{O}_1\\rangle \\hat{\\mathbb {I}}$ and $\\hat{B} = \\hat{O}_2 - \\langle \\hat{O}_2\\rangle \\hat{\\mathbb {I}}$ .",
"This formulation, which was first presented in Refs.",
"[45], [46], can also be given a measurement-independent interpretation since, from a purely mathematical perspective, $\\Delta _\\psi O_1$ and $\\Delta _\\psi O_2$ represent the “widths\" of the wave function in the relevant physical space or phase space, regardless of whether a measurement is actually performed.",
"For the operators $\\hat{x}$ and $\\hat{p}_x$ , defined by $\\hat{x}\\psi (x)=x\\psi (x)$ and $\\hat{p}_x\\psi (p_x)=p_x\\psi (p_x)$ , the commutation relation $[\\hat{x},\\hat{p}_x] = i\\hbar $ gives $ \\Delta _\\psi x\\Delta _\\psi p_x \\ge \\hbar /2 \\, ,$ where $\\Delta _\\psi x$ and $\\Delta _\\psi p_x$ correspond to the standard deviations of $\\psi (x)$ in position space and $\\psi (p_x)$ in momentum space, respectively.",
"This formulation of the uncertainty principle for $\\hat{x}$ and $\\hat{p}_x$ was first given in Refs.",
"[47], [48] and, for this choice of operators, the $|\\psi \\rangle $ -dependent terms in Eq.",
"(REF ) are of subleading order, in accordance with Heisenberg's original result.",
"The underlying wave-vector in the Hilbert space of the theory is identical in either the physical or momentum space representations, which correspond to different choices for the basis vectors in the expansion of $|\\psi \\rangle $ [20], [44].",
"Although $\\Delta _\\psi x$ and $\\Delta _\\psi p_x$ do not refer to any unavoidable “noise\", “error\" or “disturbance\" introduced into the system by the measurement process, this was how Heisenberg interpreted his original result [43].",
"In order to distinguish between quantities representing such noise and the standard deviation of repeated measurements which do not disturb the state $\\mathinner {|{\\psi }\\rangle }$ prior to wave function collapse, within this Appendix (but not the main text) we use the notation $\\Delta O$ for the former and $\\Delta _\\psi O$ for the latter.",
"Strictly speaking, any disturbance to the state of the system caused by an act of measurement may also be $|\\psi \\rangle $ -dependent, but we adopt Heisenberg's original notation, in which the state-dependent nature of the disturbance is not explicit.",
"In this notation, Heisenberg's original formulation of the uncertainty principle may be written as $ \\Delta x\\Delta p_x \\gtrsim \\hbar \\ ,$ ignoring numerical factors.",
"It is well known that one can heuristically understand this result as reflecting the momentum transferred to the particle by a probing photon.",
"However, such a statement must be viewed as a postulate, with no rigorous foundation in the underlying mathematical structure of quantum theory.",
"Indeed, as a postulate, it has been shown to be manifestly false, both theoretically [49], [50] and experimentally [51], [52], [53], [54].",
"Despite this, the heuristic derivation of Eq.",
"(REF ) may be found in many older texts, alongside the more rigorous derivation of Eq.",
"(REF ) from basic mathematical principles (see, for example, [20]).",
"Unfortunately, it is not always made clear that the quantities involved in each expression are different, as clarified by the pioneering work of Ozawa [49], [50].",
"An excellent discussion of the various possible meanings and (often confused) interpretations of symbols like `$\\Delta x$ ' is given in [55].",
"We consider only uncertainties of the form $\\Delta _\\psi O$ , defined in Eq.",
"(REF ), and uncertainty relations derived from the general formula Eq.",
"(REF ).",
"However, for notational convenience we do not include the subscript $\\psi $ in the main text.",
"Unfortunately, Eq.",
"(REF ) is also sometimes referred to as the Generalized Uncertainty Principle or Generalized Uncertainty Relation (see, for example, [44]).",
"To avoid confusion, we use the term General Uncertainty Principle to refer to the most general uncertainty relation obtained from the Hilbert space structure of standard non-relativistic quantum mechanics (for arbitrary operators) and the term Generalized Uncertainty Principle to refer to the amended uncertainty relation for position and momentum in non-canonical theories."
],
[
"Pair-production in the non-spherical case",
"For collisions between pairs of non-relativistic free particles in momentum eigenstates with masses $M$ and $M^{\\prime }$ , pair-production of particles with rest mass $M$ is possible if the centre-of-mass frame energy satisfies $ E \\simeq \\frac{P_{\\rm 3D}^2}{2\\mu }\\gtrsim Mc^2 \\, ,$ where $\\mu = MM^{\\prime }/(M+M^{\\prime })$ is the reduced mass.",
"(Reversing the direction of the final equality is the condition for non-pair-production; all the inequalities below can be similarly negated but we will not state this explicitly.)",
"Here $P_{\\rm 3D}$ denotes the 3-momentum of each particle and their total 3-momentum in the centre-of-mass frame is zero by definition.",
"For identical particles, $M^{\\prime } = M$ , so $\\mu =M/2$ and Eq.",
"(REF ) reduces to $ P_{\\rm 3D}^2= \\hbar ^2k^2 = \\hbar ^2(k_x^2+k_y^2+k_z^2)= h^2\\left(\\frac{1}{\\lambda _x^2}+\\frac{1}{\\lambda _y^2}+\\frac{1}{\\lambda _z^2} \\right)\\gtrsim M^2c^2 \\,$ in three (infinite) spatial dimensions.",
"This may be written as $ \\lambda _{\\rm 3D} \\equiv \\frac{\\lambda _x\\lambda _y\\lambda _z}{\\sqrt{\\lambda _x^2\\lambda _y^2+\\lambda _x^2\\lambda _z^2+\\lambda _y^2\\lambda _z^2}} \\lesssim R_{\\rm C} =\\frac{h}{Mc} \\, ,$ where in this Appendix $R_C$ is always the standard Compton expression.",
"To within numerical factors of order unity, the final expression on the right-hand side of this equation remains valid when the particles have very different rest masses ($M \\ll M^{\\prime } \\Rightarrow \\mu \\approx M$ ) and when higher order relativistic effects are included in Eq.",
"(REF ).",
"For spherically symmetric states, $\\lambda _x = \\lambda _y = \\lambda _z = \\lambda _R $ , giving $\\lambda _{\\rm 3D} = \\lambda _{\\rm R}/\\sqrt{3}\\lesssim R_{\\rm C}$ and the volume required for pair-production is just $V_{\\rm min} \\simeq R_{\\rm C}^3$ .",
"However, for highly asymmetric states, the minimum volume required for pair-production may be much larger than this, so the effective “width\" of the wave-packet, averaged over all dimensions, may far exceed $R_{\\rm C}$ .",
"More specifically, if $\\lambda _y \\simeq \\lambda _z \\equiv \\lambda _{\\rm 2D}$ , we may have spindles with $\\lambda _x \\gg \\lambda _{\\rm 2D}$ or pancakes with $\\lambda _x \\ll \\lambda _{\\rm 2D}$ .",
"In these cases, we have $\\lambda _{\\rm 3D} = {\\rm min} (\\lambda _{2D},\\lambda _x) ={\\left\\lbrace \\begin{array}{ll}\\lambda _{\\rm 2D}&(\\lambda _x \\gg \\lambda _{\\rm 2D}) \\\\\\lambda _x&(\\lambda _x \\ll \\lambda _{\\rm 2D})\\end{array}\\right.",
"}$ and this must less than $R_C$ for pair-production.",
"The threshold volume for this is $V_{\\rm min} = \\lambda _x \\lambda _{\\rm 2D}^2 \\simeq {\\left\\lbrace \\begin{array}{ll}R_C^2 \\lambda _x&(\\lambda _x \\gg \\lambda _{\\rm 2D}) \\\\R_C \\lambda _{\\rm 2D}^2&(\\lambda _x \\ll \\lambda _{\\rm 2D})\\, ,\\end{array}\\right.",
"}$ with both expressions exceeding $R_C^3$ .",
"This may be contrasted with classical systems on the right-hand side of Fig.",
"1, for which $V \\lesssim R_{\\rm S}^3$ is required for gravitational collapse.",
"Similar considerations apply in the presence of extra dimensions.",
"In $3+n$ dimensions, the total ($3+n$ )-momentum may be decomposed into the 3-dimensional and extra-dimensional parts.",
"If the extra dimensions are large (or infinite), the condition for pair-production becomes $ P_T^2 = P_{\\rm 3D}^2 + P_{E}^2\\gtrsim M^2c^2 \\, ,$ where $ P_{\\rm 3D}^2 = p_x^2+p_y^2+p_z^2 \\, , \\quad P_{E}^2 \\equiv \\sum _{i=1}^{n} p_{i}^2 \\, .$ In this case, it is clear that the threshold for pair-production may be reached by increasing the momentum of the particle in either the 3-dimensional or extra-dimensional space or both.",
"The pair-production condition is changed in a non-trivial way if some of the extra dimensions are compact.",
"In the compact directions, the $i^{\\rm th}$ component of the de Broglie wavelength is bounded from above by the corresponding compactification scale $R_i$ , so $ \\lambda _{i} \\lesssim R_{i} \\, .$ This gives a lower bound on the $i^{\\rm th}$ extra-dimensional momentum component, $ p_{i} \\gtrsim M_{i}c \\equiv \\frac{\\hbar }{R_{i}} \\, ,$ which corresponds to the minimum-energy, space-filling ground state of the particle.",
"Since any newly created particle must also posses the minimum momentum in the compact space, the condition for pair-production becomes $ P_T^2 = P_{\\rm 3D}^2 + P_{E}^2\\gtrsim M^2c^2 + \\mathcal {M}_{E}^2c^2 \\, ,$ where $ \\mathcal {M}_{E}^2 \\equiv \\sum _{i=1}^{n} M_{i}^2 \\equiv \\frac{\\hbar ^2}{\\mathcal {R}_E^2} \\, .$ This condition can be written as $ \\lambda \\lesssim \\frac{h}{c \\sqrt{M^2 + \\mathcal {M}_E^2}} \\simeq {\\left\\lbrace \\begin{array}{ll}R_C&(M \\gtrsim \\mathcal {M}_E) \\\\\\mathcal {R}_E&(M \\lesssim \\mathcal {M}_E) \\, ,\\end{array}\\right.",
"}$ where $ \\lambda \\equiv \\lambda _{x}\\lambda _{y}\\lambda _{z} \\prod _{i=1}^{n}\\lambda _{i} \\times \\Bigg [\\left(\\lambda _{x}^2\\lambda _{y}^2 + \\lambda _{x}^2\\lambda _{z}^2 + \\lambda _{y}^2\\lambda _{z}^2\\right)\\prod _{i=1}^{n}\\lambda _{i}^2 + \\lambda _{x}^2 \\lambda _{y}^2 \\lambda _{z}^2 \\sum _{j=1}^{n}\\prod _{i \\ne j}\\lambda _{i}^2 \\Bigg ]^{-1/2} \\,$ is the higher-dimensional generalisation of the quantity $\\lambda _{\\rm 3D}$ defined by Eq.",
"(REF ).",
"For quasi-symmetric states, corresponding to $ \\lambda _{x} = \\lambda _{y} = \\lambda _{z} = \\lambda _{R} = \\sqrt{3} \\, \\lambda _{\\rm 3D}$ this becomes $ \\lambda \\equiv \\lambda _{\\rm 3D}\\prod _{i=1}^{n}\\lambda _{i} \\times \\Bigg [\\prod _{i=1}^{n}\\lambda _{i}^2 + \\lambda _{\\rm 3D}^2 \\sum _{j=1}^{n}\\prod _{i \\ne j}\\lambda _{i}^2 \\Bigg ]^{-1/2} \\, .$ For spindle configurations with $\\lambda _i \\gg \\lambda _{\\rm 3D}$ and pancake configurations with $\\lambda _i \\ll \\lambda _{\\rm 3D}$ , we have $\\lambda ={\\left\\lbrace \\begin{array}{ll}\\lambda _{\\rm 3D}&(\\lambda _i \\gg \\lambda _{\\rm 3D}) \\\\\\prod _{i=1}^{n}\\lambda _{i}/(\\sum _{j=1}^{n}\\prod _{i \\ne j}\\lambda _{i}^2)^{1/2}&(\\lambda _i \\ll \\lambda _{\\rm 3D}) \\, .\\end{array}\\right.",
"}$ The threshold volume for pair-production is $V_{\\rm min} = \\lambda _{\\rm 3D}^3 \\prod _{i=1}^{n}\\lambda _{i} \\simeq {\\left\\lbrace \\begin{array}{ll}R_C^3 \\prod _{i=1}^{n} \\lambda _{i}&(\\lambda _i \\gg \\lambda _{\\rm 3D}) \\\\\\lambda _{\\rm 3D}^3 R_C^{n}&(\\lambda _i \\ll \\lambda _{\\rm 3D})\\, ,\\end{array}\\right.",
"}$ both potentially exceeding $R_C^{n+3}$ .",
"However, this is not the key quantity controlling pair-production.",
"Rewriting the right-hand side of Eq.",
"(REF ) in terms of $R_C$ and $R_i$ , the pair-production condition for quasi-symmetric states can be written as $ \\lambda ^{1+n}\\equiv \\lambda _{\\rm 3D}\\prod _{i=1}^{n}\\lambda _i\\lesssim R_{\\rm C} \\prod _{i=1}^{n}R_{i}\\times \\Bigg [ \\frac{\\prod _{i=1}^{n}\\lambda _i^2 + \\lambda _{\\rm 3D}^2 \\sum _{j=1}^{n}\\prod _{i \\ne j}\\lambda _i^2}{\\prod _{i=1}^{n}R_{i}^2 + R_{C}^2\\sum _{j=1}^{n}\\prod _{i \\ne j}R_{i}^2 }\\Bigg ]^{1/2} \\, .$ For the pancake configurations corresponding to the experimental scenarios outlined in Sec.",
"5, one expects $\\lambda _{\\rm 3D} \\lesssim R_{C} \\lesssim \\lambda _i \\lesssim R_i$ for all $i$ , so the term in square brackets is less than 1, which implies $ \\lambda ^{1+n} \\equiv \\lambda _{\\rm 3D}\\prod _{i=1}^{n}\\lambda _i\\lesssim R_{C} \\prod _{i=1}^{n}R_{i} \\, .$ Since the last expression yields the threshold value of $\\lambda $ giving rise to pair-production, it represents a minimum width for the particle.",
"The critical limiting value on the right-hand side of Eq.",
"(REF ) is reached from below with respect to $\\lambda _i$ (i.e.",
"as $\\lambda _i \\rightarrow R_i^{-}$ ) but from above with respect to $\\lambda _{\\rm 3D}$ (i.e.",
"as $\\lambda _{\\rm 3D}\\rightarrow R_{C}^{+}$ ).",
"States with $\\lambda _i \\simeq R_i$ , where the volume of the wave function in the extra dimensions remains as large as possible, represent the maximal degree of asymmetry.",
"More generally, for particles that are not in momentum eigenstates, we may put $\\lambda _{\\rm 3D} \\rightarrow \\Delta R_{\\rm 3D}$ , $\\lambda _{i} \\rightarrow \\Delta x_{i}$ and $\\lambda ^n \\rightarrow (\\Delta \\mathcal {R})^{1+n}$ in Eq.",
"(REF ).",
"Analogous arguments to those given above then lead to $ (\\Delta \\mathcal {R})^{1+n} \\equiv \\Delta R_{\\rm 3D}\\prod _{i=1}^{n}\\Delta x_{i}\\lesssim R_{C} \\prod _{i=1}^{n}R_{i} \\, ,$ which is the converse of the (non-pair-production) condition (REF ).",
"The right-hand side of Eq.",
"(REF ) equals $R_*$ , given by Eq.",
"(REF ), when $R_i = R_{E}$ for all $i$ ."
]
] | 1808.08386 |
[
[
"Semi-continuous g-frames in Hilbert spaces"
],
[
"Abstract In this paper, we introduce the concept of semi-continuous $g$-frames in Hilbert spaces.",
"We first construct an example of semi-continuous $g$-frames using the Fourier transform of the Heisenberg group and study the structure of such frames.",
"Then, as an application we provide some fundamental identities and inequalities for semi-continuous $g$-frames.",
"Finally, we present a classical perturbation result and prove that semi-continuous $g$-frames are stable under small perturbations."
],
[
"Introduction",
"Discrete and continuous frames arise in many applications in mathematics and, in particular, they play important roles in scientific computations and digital signal processing.",
"The concept of a frame in Hilbert spaces has been introduced in 1952 by Duffin and Schaeffer [13], in the context of nonharmonic Fourier series (see [28]).",
"After the work of Daubechies et al.",
"[11] frame theory got considerable attention outside signal processing and began to be more broadly studied (see [8], [21]).",
"A frame for a Hilbert space is a redundant set of vectors in Hilbert space which provides non-unique representations of vectors in terms of frame elements.",
"The redundancy and flexibility offered by frames has spurred their application in several areas of mathematics, physics, and engineering such as wavelet theory, sampling theory, signal processing, image processing, coding theory and many other well known fields.",
"Applications of frames, especially in the last decade, motivated the researcher to find some generalization of frames like continuous frames [1], [22], $g$ -frames [26], Hilbert$-$ Schmidt frames [24], [25], $K$ -frames [19] and etc.",
"Our main purpose in this paper is to study a generalization of frames, name as semi-continuous $g$ -frames, which are natural generalizations of $g$ -frames and continuous $g$ -frames.",
"We investigate the structure of semi-continuous $g$ -frames and establish some identities and inequalities of these frames.",
"Also, we present a perturbation result and discuss the stability of the perturbation of a semi-continuous $g$ -frame.",
"Throughout this paper, ${\\mathcal {H}}$ and ${\\mathcal {K}}$ are two Hilbert spaces; $J$ is a countable index set; $({\\mathcal {X}}, \\mu )$ is a measure space with positive measure $\\mu $ ; $\\lbrace {\\mathcal {K}}_x\\rbrace _{x \\in {\\mathcal {X}}}$ is a sequence of closed subspaces of ${\\mathcal {K}}$ ; $\\mathcal {L}({\\mathcal {H}}, {\\mathcal {K}}_x)$ is the collection of all bounded linear operators from ${\\mathcal {H}}$ into ${\\mathcal {K}}_x$ ; if ${\\mathcal {K}}_x={\\mathcal {H}}$ for any $x \\in {\\mathcal {X}}$ , we denote $\\mathcal {L}({\\mathcal {H}}, {\\mathcal {K}}_x)$ by $\\mathcal {L}({\\mathcal {H}})$ .",
"We recall that a family $\\lbrace f_j\\rbrace _{j \\in J}$ in ${\\mathcal {H}}$ is called a (discrete) frame for ${\\mathcal {H}}$ , if there exist constants $0 < A \\le B < \\infty $ such that $A \\Vert f \\Vert ^2 \\le \\sum _{j \\in J } \\vert \\langle f,f_j \\rangle \\vert ^2 \\le B \\Vert f \\Vert ^2, \\quad \\forall f \\in {\\mathcal {H}}.$ The concept of the discrete frame was generalized to continuous frame by Kaiser [22] and independently by Ali et al.",
"[1].",
"A family of vectors $\\lbrace \\psi _x\\rbrace _{x \\in {\\mathcal {X}}} \\subseteq {\\mathcal {H}}$ is called a continuous frame for ${\\mathcal {H}}$ with respect to $({\\mathcal {X}}, \\mu )$ , if $\\lbrace \\psi _x\\rbrace _{x \\in {\\mathcal {X}}}$ is weakly-measurable, i.e., for any $f \\in {\\mathcal {H}}, \\; x \\rightarrow \\langle f, \\psi _x \\rangle $ is a measurable function on ${\\mathcal {X}}$ , and if there exist two constants $A,B > 0$ such that $ A \\Vert f \\Vert ^2 \\le \\int _{{\\mathcal {X}}} \\vert \\langle f,\\psi _x \\rangle \\vert ^2 d \\mu (x) \\le B \\Vert f \\Vert ^2, \\quad \\forall f \\in {\\mathcal {H}}.",
"$ Continuous frames have been widely applied in continuous wavelets transform [2] and the short-time Fourier transform [21].",
"We refer to [3], [16], [17] for more details on continuous frames.",
"The notion of a discrete frame extended to $g$ -frame by Sun [26], which generalized all the existing frames such as bounded quasi-projectors [15], frames of subspaces [7], pseudo-frames [23], oblique frames [9], etc.",
"$G$ -frames are natural generalizations of frames as members of a Hilbert space to bounded linear operators.",
"Let $\\lbrace {\\mathcal {K}}_j: j \\in J \\rbrace \\subset {\\mathcal {K}}$ be a sequence of Hilbert spaces.",
"A family $\\lbrace \\Lambda _j \\in \\mathcal {L}({\\mathcal {H}},{\\mathcal {K}}_j):j \\in J \\rbrace $ is called a $g$ -frame, for ${\\mathcal {H}}$ with respect to $\\lbrace {\\mathcal {K}}_j: j \\in J \\rbrace $ if there are two constants $A, B>0$ such that $A \\Vert f \\Vert ^2 \\le \\sum _{j \\in J } \\Vert \\Lambda _j(f) \\Vert ^2 \\le B \\Vert f \\Vert ^2, \\quad \\forall f \\in {\\mathcal {H}}.$ The continuous $g$ -frames were proposed by Dehghan and Hasankhani Fard in [12], which are an extension of $g$ -frames and continuous frames.",
"A family $\\lbrace \\Lambda _x \\in \\mathcal {L}({\\mathcal {H}},{\\mathcal {K}}_x):x \\in {\\mathcal {X}}\\rbrace $ is called a continuous $g$ -frame for ${\\mathcal {H}}$ with respect to $({\\mathcal {X}}, \\mu )$ , if $\\lbrace \\Lambda _x: x \\in {\\mathcal {X}}\\rbrace $ is weakly-measurable, i.e., for any $f \\in {\\mathcal {H}}, \\; x \\rightarrow \\Lambda _x(f)$ is a measurable function on ${\\mathcal {X}}$ , and if there exist two constants $A,B > 0$ such that $ A \\Vert f \\Vert ^2 \\le \\int _{{\\mathcal {X}}} \\Vert \\Lambda _x(f) \\Vert ^2 d \\mu (x) \\le B \\Vert f \\Vert ^2, \\quad \\forall f \\in {\\mathcal {H}}.",
"$ Notice that if ${\\mathcal {X}}$ is a countable set and $\\mu $ is a counting measure, then the continuous $g$ -frame is just the $g$ -frame.",
"By the Riesz representation theorem, for any $\\Lambda \\in \\mathcal {L}({\\mathcal {H}}, {\\mathbb {C}})$ , there exist a $h \\in {\\mathcal {H}}$ , such that $\\Lambda (f)=\\langle f,h \\rangle $ for all $f \\in {\\mathcal {H}}$ .",
"Hence, if ${\\mathcal {K}}_x={\\mathbb {C}}$ for any $x \\in {\\mathcal {X}}$ , then the continuous $g$ -frame is equivalent to the continuous frame.",
"This paper is organized as follows.",
"After the introduction, in Section , we introduce the semi-continuous $g$ -frames in Hilbert spaces and construct an example using the Fourier transform of the Heisenberg group.",
"Then we study the structure of semi-continuous $g$ -frames using shift-invariant spaces.",
"In Section , we first list some fundamental identities and inequalities of discrete frames just for the contrast to the main results of this section.",
"Then we derive some important identities and inequalities of semi-continuous $g$ -frames.",
"Finally, in Section , we present a classical perturbation result and prove that semi-continuous $g$ -frames are stable under small perturbations."
],
[
"Semi-continuous g-frames",
"Let $\\lbrace {\\mathcal {K}}_{x,j}:x \\in {\\mathcal {X}}, j \\in J \\rbrace \\subset {\\mathcal {K}}$ be a sequence of Hilbert spaces.",
"Definition 2.1 A family $\\lbrace \\Lambda _{x,j} \\in \\mathcal {L}({\\mathcal {H}},{\\mathcal {K}}_{x,j}):x \\in {\\mathcal {X}}, j \\in J \\rbrace $ is called a semi-continuous $g$ -frame for ${\\mathcal {H}}$ with respect to $({\\mathcal {X}}, \\mu )$ , if $\\lbrace \\Lambda _{x,j}: x \\in {\\mathcal {X}}, j \\in J\\rbrace $ is weakly-measurable, i.e., for any $f \\in {\\mathcal {H}}$ and any $j \\in J$ , the function $x \\rightarrow \\Lambda _{x,j}(f)$ is measurable on ${\\mathcal {X}}$ , and if there exist two constants $A,B > 0$ such that $A \\Vert f \\Vert ^2 \\le \\int _{{\\mathcal {X}}} \\sum _{j \\in J} \\Vert \\Lambda _{x,j}(f) \\Vert ^2 d \\mu (x) \\le B \\Vert f \\Vert ^2, \\quad \\forall f \\in {\\mathcal {H}}.$ If only the right-hand inequality of (REF ) is satisfied, we call $\\lbrace \\Lambda _{x,j}: x \\in {\\mathcal {X}}, j \\in J\\rbrace $ the semi-continuous $g$ -Bessel sequence for ${\\mathcal {H}}$ with respect to $({\\mathcal {X}}, \\mu )$ with Bessel bound $B$ .",
"Remark 2.2 If $0<\\mu ({\\mathcal {X}})< \\infty $ , and for any fixed $x \\in {\\mathcal {X}}$ , the sequence $\\lbrace \\Lambda _{x,j}:j \\in J\\rbrace $ is a $g$ -frame for ${\\mathcal {H}}$ with respect to $\\lbrace {\\mathcal {K}}_{x,j}:j \\in J\\rbrace $ , then $\\lbrace \\Lambda _{x,j}: x \\in {\\mathcal {X}}, j \\in J\\rbrace $ is a semi-continuous $g$ -frame for ${\\mathcal {H}}$ with respect to $({\\mathcal {X}}, \\mu )$ .",
"Moreover if $|J|< \\infty $ , and for any fixed $j \\in J$ , the sequence $\\lbrace \\Lambda _{x,j}:x \\in {\\mathcal {X}}\\rbrace $ is a continuous $g$ -frame for ${\\mathcal {H}}$ , then $\\lbrace \\Lambda _{x,j}: x \\in {\\mathcal {X}}, j \\in J\\rbrace $ is a semi-continuous $g$ -frame for ${\\mathcal {H}}$ with respect to $({\\mathcal {X}}, \\mu )$ .",
"In the following, we shall construct an example of such frames using the Fourier transform of the Heisenberg group."
],
[
"Heisenberg Group",
"The Heisenberg group ${\\mathbb {H}}$ is a Lie group whose underlying manifold is ${\\mathbb {R}}^3$ .",
"We denote points in ${\\mathbb {H}}$ by $(p, q, t)$ with $p, q, t \\in {\\mathbb {R}}$ , and define the group operation by $(p_1, q_1, t_1) (p_2, q_2, t_2)=(p_1 + p_2, q_1 + q_2, t_1 + t_2 + \\frac{1}{2} (p_1 q_2 - q_1 p_2)).$ It is easy to verify that this is a group operation, with the origin $0 = (0, 0, 0)$ as the identity element.",
"Notice that the inverse of $(p, q, t)$ is given by $(-p,-q,-t)$ .",
"The Haar measure on the group ${\\mathbb {H}}={\\mathbb {R}}^3$ is the usual Lebesgue measure.",
"The irreducible representations of the Heisenberg group has been identified by all non-zero elements in ${\\mathbb {R}}^*(={\\mathbb {R}}\\setminus \\lbrace 0 \\rbrace )$ (see [14]).",
"Indeed, for any $\\lambda \\in {\\mathbb {R}}^*$ , the associated irreducible representation $\\rho _{\\lambda }$ of ${\\mathbb {H}}$ is equivalent to Schrödinger representation into the class of unitary operators on $L^2({\\mathbb {R}})$ , such that for any $(p, q, t) \\in {\\mathbb {H}}$ and $f \\in L^2({\\mathbb {R}})$ , the operator $\\rho _{\\lambda }(p, q, t)$ is defined by $\\rho _{\\lambda }(p, q, t)f(x)=e^{i \\lambda t} e^{i \\lambda (px+\\frac{1}{2}(pq))} f(x+q).$ It is easy to see that $\\rho _{\\lambda }(p, q, t)$ is a unitary operator satisfying the homomorphism property: $ \\rho _{\\lambda }((p_1, q_1, t_1) (p_2, q_2, t_2))= \\rho _{\\lambda }(p_1, q_1, t_1) \\rho _{\\lambda }(p_2, q_2, t_2).",
"$ Thus each $\\rho _{\\lambda }$ is a strongly continuous unitary representation of ${\\mathbb {H}}$ .",
"By Stone and von Neumann theorem ([14]), $\\lbrace \\rho _{\\lambda }: \\lambda \\in {\\mathbb {R}}^* \\rbrace $ are all the infinite dimensional irreducible unitary representations of ${\\mathbb {H}}$ , whose set has non-zero Plancherel measure.",
"The measure $|\\lambda | d\\lambda $ is the Plancherel measure on the dual space $\\widehat{{\\mathbb {H}}}\\; (\\cong {\\mathbb {R}}^*)$ of ${\\mathbb {H}}$ , and $d \\lambda $ is the Lebesgue measure on ${\\mathbb {R}}^*$ .",
"For $\\varphi \\in L^2({\\mathbb {H}})$ and $\\lambda \\in {\\mathbb {R}}^*$ , we denote $\\widehat{\\varphi }(\\lambda )$ the operator-valued Fourier transform of $\\varphi $ at a given irreducible representation $\\rho _\\lambda $ , which is defined by $\\widehat{\\varphi }(\\lambda )=\\int _{{\\mathbb {H}}} \\varphi (x) \\rho _\\lambda (x) dx.$ The operator $\\widehat{\\varphi }(\\lambda )$ is a unitary map on $L^2({\\mathbb {R}})$ into $L^2({\\mathbb {R}})$ , such that for any $f \\in L^2({\\mathbb {R}})$ $ \\widehat{\\varphi }(\\lambda )f(y)=\\int _{{\\mathbb {H}}} \\varphi (x) \\rho _\\lambda (x) f(y) dx.",
"$ Therefore $\\widehat{\\varphi }(\\lambda )$ belongs to $L^2({\\mathbb {R}}) \\otimes L^2({\\mathbb {R}})$ .",
"If $\\varphi \\in L^2({\\mathbb {H}})$ , $\\widehat{\\varphi }(\\lambda )$ is actually a Hilbert$-$ Schmidt operator on $L^2({\\mathbb {R}})$ and from the Plancherel theorem we have $\\int _{{\\mathbb {H}}} |\\varphi (x)|^2 dx= \\int _{{\\mathbb {R}}^*} \\Vert \\widehat{\\varphi }(\\lambda ) \\Vert ^2_{H.S.}",
"\\; |\\lambda | d\\lambda ,$ the norm $\\Vert \\cdot \\Vert _{H.S.",
"}$ denotes the Hilbert$-$ Schmidt norm in $L^2({\\mathbb {R}}) \\otimes L^2({\\mathbb {R}})$ .",
"The proof of the Plancherel theorem for the Heisenberg group can be found in [20], and for more general groups, see [14].",
"To construct our example of semi-continuous $g$ -frames, we shall define another unitary operator as follows.",
"Let $\\Pi :=[0,1]$ and $\\mathfrak {L}:=\\ell ^2({\\mathbb {Z}}, L^2({\\mathbb {R}})\\otimes L^2({\\mathbb {R}}))$ be the Hilbert space of all sequences with values in the space $L^2({\\mathbb {R}})\\otimes L^2({\\mathbb {R}})$ , i.e., $ \\mathfrak {L}= \\Big \\lbrace \\lbrace a_n\\rbrace _{n \\in {\\mathbb {Z}}}: \\; a_n \\in L^2({\\mathbb {R}})\\otimes L^2({\\mathbb {R}}) \\mathrm {\\;and\\;} \\sum _{n \\in {\\mathbb {Z}}} \\Vert a_n \\Vert ^2_{H.S.",
"}< \\infty \\Big \\rbrace .",
"$ Lemma 2.3 For any $\\sigma \\in \\Pi $ , let $T_\\sigma : L^2({\\mathbb {H}}) \\rightarrow \\mathfrak {L}$ given by $T_\\sigma f(j)=|\\sigma +j|^{\\frac{1}{2}} \\widehat{f}(\\sigma +j)$ .",
"Then $T_\\sigma $ is well-defined and $\\sum _{j \\in {\\mathbb {Z}}}|\\sigma +j| \\; \\Vert \\widehat{f}(\\sigma +j)\\Vert ^2_{H.S.",
"}<\\infty .$ Let $f \\in L^2({\\mathbb {H}}).$ Using Plancherel theorem and an application of periodization method, we obtain $\\Vert f\\Vert ^2_{L^2({\\mathbb {H}})}=\\int _{{\\mathbb {R}}^*} \\Vert \\widehat{f}(\\lambda )\\Vert _{H.S.",
"}^2 |\\lambda | d\\lambda &=& \\int _{\\sigma \\in \\Pi } \\sum _{j \\in {\\mathbb {Z}}} |\\sigma +j| \\; \\Vert \\widehat{f}(\\sigma +j)\\Vert ^2_{H.S.}",
"d\\sigma \\\\&=& \\int _{\\sigma \\in \\Pi } \\sum _{j \\in {\\mathbb {Z}}} \\Vert T_\\sigma f(j) \\Vert _{H.S.",
"}^2 d\\sigma .$ Hence, the result follows from the fact that $f \\in L^2({\\mathbb {H}})$ .",
"Example 2.4 Consider ${\\mathcal {X}}=\\Pi $ and $J={\\mathbb {Z}}$ .",
"For any $\\sigma \\in \\Pi $ and $j \\in {\\mathbb {Z}}$ , define $\\Lambda _{\\sigma , j}: L^2({\\mathbb {H}}) \\rightarrow L^2({\\mathbb {R}}) \\otimes L^2({\\mathbb {R}})$ as $\\Lambda _{\\sigma , j}(f)=T_\\sigma f(j).$ Then for every $f \\in L^2({\\mathbb {H}})$ , using Lemma REF we get $\\int _{\\sigma \\in \\Pi } \\sum _{j \\in {\\mathbb {Z}}} \\Vert \\Lambda _{\\sigma , j}(f) \\Vert _{H.S.",
"}^2 d\\sigma &=&\\int _{\\sigma \\in \\Pi } \\sum _{j \\in {\\mathbb {Z}}} \\Vert T_\\sigma f(j) \\Vert _{H.S.",
"}^2 d\\sigma \\\\&=& \\int _{\\sigma \\in \\Pi } \\sum _{j \\in {\\mathbb {Z}}} \\left\\Vert |\\sigma +j|^{\\frac{1}{2}} \\widehat{f}(\\sigma +j) \\right\\Vert _{H.S.",
"}^2 d\\sigma \\\\&=& \\Vert f \\Vert ^2_{L^2({\\mathbb {H}})}.$ Therefor $\\lbrace \\Lambda _{\\sigma , j}: \\sigma \\in \\Pi , j \\in {\\mathbb {Z}}\\rbrace $ is a semi-continuous $g$ -frame with frame bounds $A=B=1$ .",
"Corollary 2.5 Let $0<\\mu ({\\mathcal {X}})<\\infty $ .",
"For any fixed $\\sigma \\in {\\mathcal {X}}$ , let $\\lbrace \\Lambda _{\\sigma , j}: j \\in J \\rbrace $ be a $g$ -frame for $L^2({\\mathbb {H}})$ .",
"Then $\\lbrace \\Lambda _{\\sigma , j}: \\sigma \\in {\\mathcal {X}}, j \\in J \\rbrace $ is a semi-continuous $g$ -frame for $L^2({\\mathbb {H}})$ with respect to $({\\mathcal {X}},\\mu )$ with unified frame bounds multiplied by $\\mu ({\\mathcal {X}})$ .",
"Since $\\lbrace \\Lambda _{\\sigma , j}: j \\in J \\rbrace $ be a $g$ -frame for $L^2({\\mathbb {H}})$ , there exist constants $A,B>0$ such that $ A\\Vert f \\Vert ^2_{L^2({\\mathbb {H}})} \\le \\sum _{j \\in J} \\Vert \\Lambda _{\\sigma , j}(f) \\Vert ^2 \\le B \\Vert f \\Vert ^2_{L^2({\\mathbb {H}})}, \\quad \\forall f \\in L^2({\\mathbb {H}}).$ Taking integral from all sides of the preceding inequality, we obtain $ A\\mu ({\\mathcal {X}}) \\Vert f \\Vert ^2_{L^2({\\mathbb {H}})} \\le \\int _{{\\mathcal {X}}} \\sum _{j \\in J} \\Vert \\Lambda _{\\sigma , j}(f) \\Vert ^2 d\\mu (\\sigma ) \\le B \\mu ({\\mathcal {X}}) \\Vert f \\Vert ^2_{L^2({\\mathbb {H}})}, \\quad \\forall f \\in L^2({\\mathbb {H}}).$ Hence, the result follows.",
"Now, we shall define shift-invariant spaces and give an example.",
"Definition 2.6 Let $\\Gamma $ be a countable subset of ${\\mathbb {H}}$ .",
"A subspace $\\mathcal {V} \\subset L^2({\\mathbb {H}})$ is called $\\Gamma $ -invariant if $L_{\\gamma }\\phi \\in \\mathcal {V}$ for all $\\gamma \\in \\Gamma $ and all $\\phi \\in \\mathcal {V}$ , where $L_{\\gamma }\\phi (w)=\\phi (\\gamma ^{-1}w), \\; w \\in {\\mathbb {H}}.$ If $\\Gamma $ is a discrete subset of ${\\mathbb {H}}$ , then $\\mathcal {V}$ is called shift-invariant.",
"Example 2.7 Let $\\phi \\in L^2({\\mathbb {H}})$ and $\\Gamma $ be a lattice.",
"Then the space $\\langle \\phi \\rangle _{\\Gamma }$ generated by $\\Gamma $ -shifts of $\\phi $ is a shift-invariant space.",
"Before we prove the main result of this section, we first need the following.",
"Let $T: L^2({\\mathbb {H}}) \\rightarrow L^2\\left(\\Pi , \\mathfrak {L} \\right)$ .",
"Then for any $\\sigma \\in \\Pi $ and $j \\in {\\mathbb {Z}}$ , $Tf(\\sigma )(j) \\in L^2({\\mathbb {R}})\\otimes L^2({\\mathbb {R}})$ .",
"By Lemma REF , it is clear that $Tf(\\sigma )=T_{\\sigma }f$ .",
"Let $ \\Gamma =\\Gamma _1\\Gamma _0=\\big \\lbrace xz \\in {\\mathbb {H}}: x \\in \\Gamma _1, z \\in \\Gamma _0 \\big \\rbrace ,$ where $\\Gamma _1$ be any discrete subset of ${\\mathbb {H}}$ and $\\Gamma _0$ be the lattice of integral points in ${\\mathbb {Z}}$ .",
"Then for $y \\in {\\mathbb {H}}$ and $\\sigma \\in \\Pi $ , define the unitary operator $\\tilde{\\rho }_{\\sigma }(y):\\mathfrak {L} \\rightarrow \\mathfrak {L}$ by $ (\\tilde{\\rho }_{\\sigma }(y)h)_j=\\rho _{\\sigma +j}(y)\\circ h_j, \\quad h \\in \\mathfrak {L},$ where $\\rho _{\\sigma +j}(y)\\circ h_j$ denotes function composition.",
"Also, define $\\tilde{\\rho }(y): L^2(\\Pi , \\mathfrak {L}) \\rightarrow L^2(\\Pi , \\mathfrak {L})$ by $ (\\tilde{\\rho }(y) a )(\\sigma )=\\tilde{\\rho }_{\\sigma }(y)a(\\sigma ), \\quad a \\in L^2(\\Pi , \\mathfrak {L}).$ Note that if $\\gamma \\in \\Gamma _0$ , then $(\\tilde{\\rho }(\\gamma ) a )(\\sigma )=e^{2 \\pi i \\langle \\sigma , \\gamma \\rangle } a(\\sigma )$ for all $a \\in L^2(\\Pi , \\mathfrak {L}).$ Further, the mapping $T$ is unitary, and for each $y \\in {\\mathbb {H}}$ , we have $ T(L_y \\phi )(\\sigma )=(\\tilde{\\rho }(y) T\\phi )(\\sigma ).",
"$ Proofs of these results and a more detailed study of these operators can be found in ([10], Section 3).",
"Fix a discrete subset $\\Gamma $ of ${\\mathbb {H}}$ of the form $\\Gamma _1 \\Gamma _0.$ Let $\\mathcal {V} \\subset L^2({\\mathbb {H}})$ be a countable set.",
"Define $E(\\mathcal {V})=\\lbrace L_{\\gamma }\\phi : \\gamma \\in \\Gamma , \\phi \\in \\mathcal {V}\\rbrace $ and put $\\mathcal {S}=\\overline{\\mathrm {span}}\\; E(\\mathcal {V})$ .",
"Let $R$ be the range function associated with $\\mathcal {S}$ .",
"Motivated by the results of Currey et al.",
"[10], we obtain the following.",
"Lemma 2.8 Let $f \\in L^2({\\mathbb {H}})$ , $\\Gamma \\subseteq {\\mathbb {H}}$ and $\\mathcal {V} \\subset L^2({\\mathbb {H}})$ .",
"Then $ \\sum _{\\phi \\in \\mathcal {V}, \\gamma \\in \\Gamma }|\\langle f, L_{\\gamma }\\phi \\rangle |^2=\\int _{\\Pi }\\sum _{\\phi \\in \\mathcal {V}, k \\in \\Gamma _1} |\\langle Tf(\\sigma ), T(L_k\\phi )(\\sigma ) \\rangle |^2 d\\sigma .",
"$ Let $ f \\in L^2({\\mathbb {H}})$ .",
"Since $\\Vert Tf\\Vert =\\Vert f\\Vert $ , we have $\\sum _{\\phi \\in \\mathcal {V}, \\gamma \\in \\Gamma }|\\langle f, L_{\\gamma }\\phi \\rangle |^2=\\sum _{\\phi \\in \\mathcal {V}, \\gamma \\in \\Gamma }|\\langle Tf, T(L_{\\gamma }\\phi ) \\rangle |^2&=& \\sum _{\\phi \\in \\mathcal {V}, \\gamma \\in \\Gamma }\\left| \\int _{\\Pi } \\langle Tf(\\sigma ), T(L_{\\gamma }\\phi )(\\sigma ) \\rangle \\; d \\sigma \\right|^2 \\\\&=& \\sum _{\\phi \\in \\mathcal {V}, \\gamma \\in \\Gamma }\\left| \\int _{\\Pi } \\langle Tf(\\sigma ), (\\tilde{\\rho }(\\gamma ) T\\phi )(\\sigma ) \\rangle \\; d \\sigma \\right|^2.$ Putting $\\gamma =kl$ , with $k \\in \\Gamma _1, l \\in \\Gamma _0$ , we get $ (\\tilde{\\rho }(kl) T\\phi )(\\sigma )=\\tilde{\\rho }_{\\sigma }(kl) T\\phi (\\sigma )=\\tilde{\\rho }_{\\sigma }(k)\\tilde{\\rho }_{\\sigma }(l) T\\phi (\\sigma )=e^{2 \\pi i \\langle \\sigma ,l \\rangle }\\tilde{\\rho }_{\\sigma }(k)T\\phi (\\sigma ).",
"$ Thus $\\sum _{\\phi \\in \\mathcal {V}, \\gamma \\in \\Gamma }\\left| \\int _{\\Pi } \\langle Tf(\\sigma ), (\\tilde{\\rho }(\\gamma ) T\\phi )(\\sigma ) \\rangle \\; d \\sigma \\right|^2= \\sum _{\\phi \\in \\mathcal {V}, (k,l) \\in \\Gamma }\\left| \\int _{\\Pi } \\langle Tf(\\sigma ), \\tilde{\\rho }_{\\sigma }(k)T\\phi (\\sigma ) \\rangle e^{-2 \\pi i \\langle \\sigma ,l \\rangle } d \\sigma \\right|^2.$ For each $k$ , define $F_k(\\sigma )=\\langle Tf(\\sigma ), \\tilde{\\rho }_{\\sigma }(k)T\\phi (\\sigma ) \\rangle .$ Then $F_k$ is integrable with square summable Fourier coefficients, hence $F_k \\in L^2(\\Pi )$ .",
"Using Fourier inversion formula we obtain $\\sum _{\\phi \\in \\mathcal {V}, (k,l) \\in \\Gamma }\\left| \\int _{\\Pi } \\langle Tf(\\sigma ), \\tilde{\\rho }_{\\sigma }(k)T\\phi (\\sigma ) \\rangle e^{-2 \\pi i \\langle \\sigma ,l \\rangle } d \\sigma \\right|^2&=&\\sum _{\\phi \\in \\mathcal {V}, (k,l) \\in \\Gamma } |\\hat{F_k}(l)|^2\\\\&=& \\sum _{\\phi \\in \\mathcal {V}, k \\in \\Gamma _1} \\Vert F_k\\Vert ^2 \\\\&=& \\sum _{\\phi \\in \\mathcal {V}, k \\in \\Gamma _1} \\int _{\\Pi } |F_k(\\sigma )|^2 d\\sigma .$ Again, by substituting $F_k(\\sigma )=\\langle Tf(\\sigma ), \\tilde{\\rho }_{\\sigma }(k)T\\phi (\\sigma ) \\rangle $ in the above we get $\\sum _{\\phi \\in \\mathcal {V}, \\gamma \\in \\Gamma }|\\langle f, L_{\\gamma }\\phi \\rangle |^2 =\\sum _{\\phi \\in \\mathcal {V}, k \\in \\Gamma _1} \\int _{\\Pi } |F_k(\\sigma )|^2 d\\sigma & =& \\sum _{\\phi \\in \\mathcal {V}, k \\in \\Gamma _1} \\int _{\\Pi } |\\langle Tf(\\sigma ), \\tilde{\\rho }_{\\sigma }(k)T\\phi (\\sigma ) \\rangle |^2 d\\sigma \\nonumber \\\\&= & \\int _{\\Pi } \\sum _{\\phi \\in \\mathcal {V}, k \\in \\Gamma _1} |\\langle Tf(\\sigma ), T(L_k\\phi )(\\sigma ) \\rangle |^2 d\\sigma .$ This completes the proof.",
"Now we are in a position to prove our main result of this section.",
"Theorem 2.9 If $\\lbrace T_{\\sigma }(L_k\\phi ): k \\in \\Gamma _1, \\phi \\in \\mathcal {V} \\rbrace $ is a frame for its spanned vector space for almost every $\\sigma \\in \\Pi $ .",
"Then $\\lbrace L_{\\gamma }\\phi : \\gamma \\in \\Gamma , \\phi \\in \\mathcal {V}\\rbrace $ is also a frame for its spanned vector space.",
"Suppose that $f \\in \\mathcal {S}$ , then $Tf(\\sigma ) \\in R(\\sigma )$ holds for a.e.",
"$\\sigma $ .",
"Since for a.e.",
"$\\sigma \\in \\Pi $ , $\\lbrace T_{\\sigma }(L_k\\phi ): k \\in \\Gamma _1, \\phi \\in \\mathcal {V} \\rbrace $ is a frame for its spanned vector space, there exist $0<A \\le B<\\infty $ such that $ A\\Vert Tf(\\sigma )\\Vert ^2 \\le \\int _{\\Pi }\\sum _{\\phi \\in \\mathcal {V}, k \\in \\Gamma _1}|\\langle Tf(\\sigma ), T(L_k\\phi )(\\sigma ) \\rangle |^2 d\\sigma \\le B\\Vert Tf(\\sigma )\\Vert ^2.",
"$ holds for a.e.",
"$\\sigma $ .",
"Integrating over $\\Pi $ yields $A\\Vert f\\Vert ^2=A\\Vert Tf\\Vert ^2=A \\int _{\\Pi }\\Vert Tf(\\sigma )\\Vert ^2 d\\sigma &\\le & \\int _{\\Pi }\\sum _{\\phi \\in \\mathcal {V}, k \\in \\Gamma _1}|\\langle Tf(\\sigma ), T(L_k\\phi )(\\sigma ) \\rangle |^2 d\\sigma \\\\&\\le & B \\int _{\\Pi }\\Vert Tf(\\sigma )\\Vert ^2 d\\sigma = B\\Vert f\\Vert ^2.$ Using (REF ) we obtain $ A\\Vert f\\Vert ^2 \\le \\sum _{\\phi \\in \\mathcal {V}, \\gamma \\in \\Gamma }|\\langle f, L_{\\gamma }\\phi \\rangle |^2 \\le B\\Vert f\\Vert ^2.",
"$ Hence, we have the desired result.",
"Remark 2.10 Notice that the family $\\lbrace T_{\\sigma }(L_k\\phi ): k \\in \\Gamma _1, \\phi \\in \\mathcal {V} \\rbrace $ constitutes a frame for the space which consists of all functions of the form $T_\\sigma f$ for every $f \\in L^2({\\mathbb {H}})$ .",
"Similarly, the above result can be extended for semi-continuous $g$ -frames using the Riesz representation theorem."
],
[
"Identities and inequalities for semi-continuous g-frames",
"Let $\\lbrace \\Lambda _{x,j} \\in \\mathcal {L}({\\mathcal {H}},{\\mathcal {K}}_{x,j}):x \\in {\\mathcal {X}}, j \\in J \\rbrace $ be a semi-continuous $g$ -frame for ${\\mathcal {H}}$ with respect to $({\\mathcal {X}}, \\mu )$ .",
"Then we define the semi-continuous $g$ -frame operator $S$ as follows: $S: {\\mathcal {H}}\\rightarrow {\\mathcal {H}}, \\quad Sf=\\int _{{\\mathcal {X}}}\\sum _{j \\in J} \\Lambda ^*_{x,j} \\Lambda _{x,j} f \\; d\\mu (x), $ where $\\Lambda ^*_{x,j}$ is the adjoint of $\\Lambda _{x,j}$ .",
"It is easy to show that $S$ is a bounded, invertible, self-adjoint and positive operator.",
"Therefore for any $f \\in {\\mathcal {H}}$ , we have the following reconstructions: $f=SS^{-1}f=\\int _{{\\mathcal {X}}}\\sum _{j \\in J} \\Lambda ^*_{x,j} \\Lambda _{x,j} S^{-1}f \\; d\\mu (x),$ $f=S^{-1}Sf=\\int _{{\\mathcal {X}}}\\sum _{j \\in J} S^{-1} \\Lambda ^*_{x,j} \\Lambda _{x,j} f \\; d\\mu (x).$ Denote $\\tilde{\\Lambda }_{x,j}=\\Lambda _{x,j} S^{-1}$ .",
"Then $\\lbrace \\tilde{\\Lambda }_{x,j} :x \\in {\\mathcal {X}}, j \\in J \\rbrace $ is also a semi-continuous $g$ -frame with frame bounds $\\frac{1}{B},\\frac{1}{A}$ , which we call the canonical dual frame of $\\lbrace \\Lambda _{x,j}:x \\in {\\mathcal {X}}, j \\in J \\rbrace $ .",
"A semi-continuous $g$ -frame $\\lbrace {\\mathcal {G}_{x,j}}\\in \\mathcal {L}({\\mathcal {H}},{\\mathcal {K}}_{x,j}):x \\in {\\mathcal {X}}, j \\in J \\rbrace $ is called an alternate dual frame of $\\lbrace \\Lambda _{x,j}:x \\in {\\mathcal {X}}, j \\in J \\rbrace $ if for all $f \\in {\\mathcal {H}}$ , the following identity holds: $f=\\int _{{\\mathcal {X}}}\\sum _{j \\in J} \\Lambda ^*_{x,j} {\\mathcal {G}_{x,j}}f \\; d\\mu (x)=\\int _{{\\mathcal {X}}}\\sum _{j \\in J} \\mathcal {G}^*_{x,j} \\Lambda _{x,j} f \\; d\\mu (x).$ A semi-continuous $g$ -frame $\\lbrace \\Lambda _{x,j}:x \\in {\\mathcal {X}}, j \\in J \\rbrace $ is called a Parseval semi-continuous $g$ -frame, if the frame bounds $A=B=1$ .",
"For any ${\\mathcal {X}}_1 \\subset {\\mathcal {X}}$ , we denote ${\\mathcal {X}}_1^c={\\mathcal {X}}\\setminus {\\mathcal {X}}_1$ , and define the following operator: $ S_{{\\mathcal {X}}_1}f=\\int _{{\\mathcal {X}}_1}\\sum _{j \\in J} \\Lambda ^*_{x,j} \\Lambda _{x,j} f \\; d\\mu (x).",
"$ In [4], the authors proved a longstanding conjecture of the signal processing community: a signal can be reconstructed without information about the phase.",
"While working on efficient algorithms for signal reconstruction, Balan et al.",
"[5] discovered a remarkable new identity for Parseval discrete frames, given in the following form.",
"Theorem 3.1 Let $\\lbrace f_j\\rbrace _{j \\in J }$ be a Parseval frame for ${\\mathcal {H}}$ , then for every $K\\subset J$ and every $f \\in {\\mathcal {H}}$ , we have $ \\sum _{j \\in K} \\vert \\langle f,f_j \\rangle \\vert ^2- \\bigg \\Vert \\sum _{j \\in K} \\langle f,f_j \\rangle f_j \\bigg \\Vert ^2=\\sum _{j \\in K^c} \\vert \\langle f,f_j \\rangle \\vert ^2- \\bigg \\Vert \\sum _{j \\in K^c} \\langle f,f_j \\rangle f_j \\bigg \\Vert ^2.",
"$ Theorem 3.2 If $\\lbrace f_j\\rbrace _{j \\in J}$ be a Parseval frame for ${\\mathcal {H}}$ , then for every $K\\subset J$ and every $f \\in {\\mathcal {H}}$ , we have $ \\sum _{j \\in K} \\vert \\langle f,f_j \\rangle \\vert ^2 + \\bigg \\Vert \\sum _{j \\in K^c} \\langle f,f_j \\rangle f_j \\bigg \\Vert ^2 \\ge \\frac{3}{4} \\Vert f \\Vert ^2.",
"$ In fact, the identity appears in Theorem REF was obtained in [5] as a particular case of the following result for general frames.",
"Theorem 3.3 Let $\\lbrace f_j\\rbrace _{j \\in J }$ be a frame for ${\\mathcal {H}}$ with canonical dual frame $\\lbrace \\tilde{f_j}\\rbrace _{ j \\in J }$ .",
"Then for every $K\\subset J$ and every $f \\in {\\mathcal {H}}$ , we have $ \\sum _{j \\in K} \\vert \\langle f,f_j \\rangle \\vert ^2- \\sum _{j \\in J} \\vert \\langle S_K f, \\tilde{f_j} \\rangle \\vert ^2 = \\sum _{j \\in K^c} \\vert \\langle f,f_j \\rangle \\vert ^2- \\sum _{j \\in J} \\vert \\langle S_{K^c} f, \\tilde{f_j} \\rangle \\vert ^2 .$ Motivated by these interesting results, the authors in [18], [29] generalized Theorems REF and REF to canonical and alternate dual frames.",
"In this section, we investigate the above mentioned results for semi-continuous $g$ -frames and derive some important identities and inequalities of these frames.",
"We first state a simple result on operators.",
"Lemma 3.4 [29] If $P,Q \\in \\mathcal {L}({\\mathcal {H}})$ satisfying $P+Q=I$ , then $P-P^{*}P=Q^*-Q^*Q.$ We compute $P-P^{*}P=(I-P^{*})P=Q^*(I-Q)=Q^*-Q^*Q.$ Theorem 3.5 Let $\\lbrace \\Lambda _{x,j}:x \\in {\\mathcal {X}}, j \\in J \\rbrace $ be a Parseval semi-continuous $g$ -frame for ${\\mathcal {H}}$ with respect to $({\\mathcal {X}}, \\mu )$ .",
"Then for every ${\\mathcal {X}}_1 \\subset {\\mathcal {X}}$ and every $f \\in {\\mathcal {H}}$ , we have $&& \\int _{{\\mathcal {X}}_1}\\sum _{j \\in J}\\Vert \\Lambda _{x,j} f \\Vert ^2 d\\mu (x) - \\bigg \\Vert \\int _{{\\mathcal {X}}_1}\\sum _{j \\in J} \\Lambda ^*_{x,j} \\Lambda _{x,j} f \\; d\\mu (x) \\bigg \\Vert ^2 \\nonumber \\\\&=& \\int _{{\\mathcal {X}}_1^c}\\sum _{j \\in J}\\Vert \\Lambda _{x,j} f \\Vert ^2 d\\mu (x) - \\bigg \\Vert \\int _{{\\mathcal {X}}_1^c}\\sum _{j \\in J} \\Lambda ^*_{x,j} \\Lambda _{x,j} f \\; d\\mu (x) \\bigg \\Vert ^2.$ Since $\\lbrace \\Lambda _{x,j}:x \\in {\\mathcal {X}}, j \\in J \\rbrace $ is a Parseval semi-continuous $g$ -frame, the corresponding frame operator $S=I$ , and hence $S_{{\\mathcal {X}}_1}+S_{{\\mathcal {X}}_1^c}=I$ .",
"Note that $S_{{\\mathcal {X}}_1^c}$ is a self-adjoint operator, and therefore $S_{{\\mathcal {X}}_1^c}^*=S_{{\\mathcal {X}}_1^c}$ .",
"Applying Lemma REF to the operators $S_{{\\mathcal {X}}_1}$ and $S_{{\\mathcal {X}}_1^c}$ , we obtain that for every $f \\in {\\mathcal {H}}$ $&& \\langle S_{{\\mathcal {X}}_1} f,f \\rangle - \\langle S_{{\\mathcal {X}}_1}^{*}S_{{\\mathcal {X}}_1} f,f \\rangle =\\langle S_{{\\mathcal {X}}_1^c}^* f, f \\rangle -\\langle S_{{\\mathcal {X}}_1^c}^* S_{{\\mathcal {X}}_1^c}f,f \\rangle \\nonumber \\\\\\Rightarrow && \\langle S_{{\\mathcal {X}}_1} f,f \\rangle - \\Vert S_{{\\mathcal {X}}_1} f \\Vert ^2 = \\langle S_{{\\mathcal {X}}_1^c} f,f \\rangle - \\Vert S_{{\\mathcal {X}}_1^c}f \\Vert ^2.$ We have $\\langle S_{{\\mathcal {X}}_1} f,f \\rangle =\\left\\langle \\int _{{\\mathcal {X}}_1}\\sum _{j \\in J} \\Lambda ^*_{x,j} \\Lambda _{x,j} f \\; d\\mu (x),f \\right\\rangle &=& \\int _{{\\mathcal {X}}_1}\\sum _{j \\in J}\\langle \\Lambda _{x,j} f, \\Lambda _{x,j}f \\rangle \\; d\\mu (x) \\nonumber \\\\&=& \\int _{{\\mathcal {X}}_1}\\sum _{j \\in J} \\Vert \\Lambda _{x,j} f \\Vert ^2 d\\mu (x).$ Similarly $\\langle S_{{\\mathcal {X}}_1^c} f,f \\rangle = \\int _{{\\mathcal {X}}_1^c}\\sum _{j \\in J} \\Vert \\Lambda _{x,j} f \\Vert ^2 d\\mu (x).$ Using equations (REF ) and (REF ) in (REF ), we obtain the desired result.",
"Now we generalize Theorem REF to dual semi-continuous $g$ -frames.",
"We first need the following lemma.",
"Lemma 3.6 [24] Let $P,Q \\in \\mathcal {L}({\\mathcal {H}})$ be two self-adjoint operators such that $P+Q=I$ .",
"Then for any $\\lambda \\in [0,1]$ and every $f \\in {\\mathcal {H}}$ we have $ \\Vert Pf \\Vert ^2+2 \\lambda \\langle Qf,f \\rangle = \\Vert Qf \\Vert ^2+2(1-\\lambda )\\langle Pf,f \\rangle + (2\\lambda -1)\\Vert f \\Vert ^2 \\ge (1-(\\lambda -1)^2)\\Vert f \\Vert ^2.",
"$ Theorem 3.7 Let $\\lbrace \\Lambda _{x,j}:x \\in {\\mathcal {X}}, j \\in J \\rbrace $ be a semi-continuous $g$ -frame for ${\\mathcal {H}}$ with respect to $({\\mathcal {X}}, \\mu )$ and $\\lbrace \\tilde{\\Lambda }_{x,j}:x \\in {\\mathcal {X}}, j \\in J \\rbrace $ be the canonical dual frame of $\\lbrace \\Lambda _{x,j}:x \\in {\\mathcal {X}}, j \\in J \\rbrace $ .",
"Then for any $\\lambda \\in [0,1]$ , for every ${\\mathcal {X}}_1 \\subset {\\mathcal {X}}$ and every $f \\in {\\mathcal {H}}$ , we have $&& \\int _{{\\mathcal {X}}}\\sum _{j \\in J} \\Vert \\tilde{\\Lambda }_{x,j} S_{{\\mathcal {X}}_1} f \\Vert ^2 d\\mu (x)+\\int _{{\\mathcal {X}}_1^c}\\sum _{j \\in J} \\Vert \\Lambda _{x,j} f \\Vert ^2 d\\mu (x) \\\\&&=\\int _{{\\mathcal {X}}}\\sum _{j \\in J} \\Vert \\tilde{\\Lambda }_{x,j} S_{{\\mathcal {X}}_1^c} f \\Vert ^2 d\\mu (x)+ \\int _{{\\mathcal {X}}_1}\\sum _{j \\in J} \\Vert \\Lambda _{x,j} f \\Vert ^2 d\\mu (x)\\\\&&\\ge (2 \\lambda - \\lambda ^2)\\int _{{\\mathcal {X}}_1}\\sum _{j \\in J} \\Vert \\Lambda _{x,j} f \\Vert ^2 d\\mu (x)+(1 - \\lambda ^2) \\int _{{\\mathcal {X}}_1^c}\\sum _{j \\in J} \\Vert \\Lambda _{x,j} f \\Vert ^2 d\\mu (x).$ Let $S$ be the frame operator for $\\lbrace \\Lambda _{x,j}:x \\in {\\mathcal {X}}, j \\in J \\rbrace $ .",
"Since $S_{{\\mathcal {X}}_1}+S_{{\\mathcal {X}}_1^c}=S$ , it follows that $S^{-1/2}S_{{\\mathcal {X}}_1} S^{-1/2}+S^{-1/2}S_{{\\mathcal {X}}_1^c}S^{-1/2}=I.$ Considering $P=S^{-1/2}S_{{\\mathcal {X}}_1}S^{-1/2}$ , $Q=S^{-1/2}S_{{\\mathcal {X}}_1^c}S^{-1/2}$ , and $S^{1/2}f$ instead of $f$ in Lemma REF , we obtain $\\nonumber && \\Vert S^{-1/2}S_{{\\mathcal {X}}_1}f \\Vert ^2+2 \\lambda \\langle S^{-1/2}S_{{\\mathcal {X}}_1^c}f,S^{1/2}f \\rangle \\\\&&= \\Vert S^{-1/2}S_{{\\mathcal {X}}_1^c}f \\Vert ^2+2(1-\\lambda )\\langle S^{-1/2}S_{{\\mathcal {X}}_1} f,S^{1/2}f \\rangle + (2\\lambda -1)\\Vert S^{1/2}f \\Vert ^2 \\nonumber \\\\ \\nonumber && \\ge (1-(\\lambda -1)^2)\\Vert S^{1/2}f \\Vert ^2 \\nonumber \\\\ \\nonumber && \\Rightarrow \\langle S^{-1}S_{{\\mathcal {X}}_1}f, S_{{\\mathcal {X}}_1}f \\rangle +\\langle S_{{\\mathcal {X}}_1^c}f,f \\rangle = \\langle S^{-1}S_{{\\mathcal {X}}_1^c}f, S_{{\\mathcal {X}}_1^c}f \\rangle +\\langle S_{{\\mathcal {X}}_1} f,f \\rangle \\\\&& \\ge (2 \\lambda - \\lambda ^2) \\langle S_{{\\mathcal {X}}_1}f,f \\rangle +(1 - \\lambda ^2) \\langle S_{{\\mathcal {X}}_1^c}f,f \\rangle .$ We have $\\langle S^{-1}S_{{\\mathcal {X}}_1}f, S_{{\\mathcal {X}}_1}f \\rangle &=& \\langle SS^{-1}S_{{\\mathcal {X}}_1}f, S^{-1}S_{{\\mathcal {X}}_1}f \\rangle \\nonumber \\\\&=& \\left\\langle \\int _{{\\mathcal {X}}}\\sum _{j \\in J} \\Lambda ^*_{x,j} \\Lambda _{x,j} S^{-1}S_{{\\mathcal {X}}_1}f d\\mu (x), S^{-1}S_{{\\mathcal {X}}_1}f \\right\\rangle \\nonumber \\\\&=& \\int _{{\\mathcal {X}}}\\sum _{j \\in J} \\langle \\Lambda _{x,j} S^{-1}S_{{\\mathcal {X}}_1}f, \\Lambda _{x,j} S^{-1}S_{{\\mathcal {X}}_1}f \\rangle \\; d\\mu (x) \\nonumber \\\\&=& \\int _{{\\mathcal {X}}}\\sum _{j \\in J} \\langle \\tilde{\\Lambda }_{x,j} S_{{\\mathcal {X}}_1}f, \\tilde{\\Lambda }_{x,j} S_{{\\mathcal {X}}_1}f \\rangle \\; d\\mu (x) \\nonumber \\\\&=& \\int _{{\\mathcal {X}}}\\sum _{j \\in J} \\Vert \\tilde{\\Lambda }_{x,j} S_{{\\mathcal {X}}_1}f\\Vert ^2 d\\mu (x).$ Similarly $ \\langle S^{-1}S_{{\\mathcal {X}}_1^c}f, S_{{\\mathcal {X}}_1^c}f \\rangle = \\int _{{\\mathcal {X}}}\\sum _{j \\in J} \\Vert \\tilde{\\Lambda }_{x,j} S_{{\\mathcal {X}}_1^c}f\\Vert ^2 d\\mu (x).$ $\\langle S_{{\\mathcal {X}}_1^c}f,f \\rangle =\\int _{{\\mathcal {X}}_1^c}\\sum _{j \\in J} \\Vert \\Lambda _{x,j} f \\Vert ^2 d\\mu (x).$ $\\langle S_{{\\mathcal {X}}_1}f,f \\rangle =\\int _{{\\mathcal {X}}_1}\\sum _{j \\in J} \\Vert \\Lambda _{x,j} f \\Vert ^2 d\\mu (x).$ Using equations (REF )–(REF ) in the inequality (REF ), we obtain the desired result.",
"Lemma 3.8 [24] If $P,Q \\in \\mathcal {L}({\\mathcal {H}})$ satisfy $P+Q=I$ , then for any $\\lambda \\in [0,1]$ and every $f \\in {\\mathcal {H}}$ we have $ P^{*}P+\\lambda (Q^{*}+Q)=Q^{*}Q+(1-\\lambda )(P^{*}+P)+(2\\lambda -1)I \\ge (1-(\\lambda -1)^2)I.",
"$ Theorem 3.9 Let $\\lbrace \\Lambda _{x,j}:x \\in {\\mathcal {X}}, j \\in J \\rbrace $ be a semi-continuous $g$ -frame for ${\\mathcal {H}}$ with respect to $({\\mathcal {X}}, \\mu )$ and $\\lbrace {\\mathcal {G}_{x,j}}:x \\in {\\mathcal {X}}, j \\in J \\rbrace $ be an alternate dual frame of $\\lbrace \\Lambda _{x,j}:x \\in {\\mathcal {X}}, j \\in J \\rbrace $ .",
"Then for any $\\lambda \\in [0,1]$ , for every ${\\mathcal {X}}_1 \\subset {\\mathcal {X}}$ and every $f \\in {\\mathcal {H}}$ , we have $&& Re \\bigg \\lbrace \\int _{{\\mathcal {X}}_1^c}\\sum _{j \\in J} \\langle {\\mathcal {G}_{x,j}}f, \\Lambda _{x,j} f \\rangle d\\mu (x) \\bigg \\rbrace + \\bigg \\Vert \\int _{{\\mathcal {X}}_1}\\sum _{j \\in J} \\Lambda ^*_{x,j} {\\mathcal {G}_{x,j}}f \\; d\\mu (x) \\bigg \\Vert ^2 \\\\&&= Re \\bigg \\lbrace \\int _{{\\mathcal {X}}_1}\\sum _{j \\in J} \\langle {\\mathcal {G}_{x,j}}f, \\Lambda _{x,j} f \\rangle d\\mu (x) \\bigg \\rbrace + \\bigg \\Vert \\int _{{\\mathcal {X}}_1^c}\\sum _{j \\in J} \\Lambda ^*_{x,j} {\\mathcal {G}_{x,j}}f \\; d\\mu (x) \\bigg \\Vert ^2 \\\\&& \\ge (2 \\lambda - \\lambda ^2) Re \\bigg \\lbrace \\int _{{\\mathcal {X}}_1}\\sum _{j \\in J} \\langle {\\mathcal {G}_{x,j}}f, \\Lambda _{x,j} f \\rangle d\\mu (x) \\bigg \\rbrace +(1-\\lambda ^2) Re \\bigg \\lbrace \\int _{{\\mathcal {X}}_1^c}\\sum _{j \\in J} \\langle {\\mathcal {G}_{x,j}}f, \\Lambda _{x,j} f \\rangle d\\mu (x) \\bigg \\rbrace .$ For ${\\mathcal {X}}_1 \\subset {\\mathcal {X}}$ and $f \\in {\\mathcal {H}}$ , define the operator $F_{{\\mathcal {X}}_1}$ by $F_{{\\mathcal {X}}_1}f=\\int _{{\\mathcal {X}}_1}\\sum _{j \\in J} \\Lambda ^*_{x,j} {\\mathcal {G}_{x,j}}f \\; d\\mu (x).$ Then $F_{{\\mathcal {X}}_1} \\in \\mathcal {L}({\\mathcal {H}})$ .",
"By (REF ), we have $F_{{\\mathcal {X}}_1}+F_{{\\mathcal {X}}_1^c}=I$ .",
"By Lemma REF , we get $&& (1-(\\lambda -1)^2)\\Vert f \\Vert ^2 \\le \\langle F_{{\\mathcal {X}}_1}^{*}F_{{\\mathcal {X}}_1} f,f \\rangle +\\lambda \\langle (F_{{\\mathcal {X}}_1^c}^{*}+F_{{\\mathcal {X}}_1^c})f,f \\rangle \\nonumber \\\\ &&= \\langle F_{{\\mathcal {X}}_1^c}^{*}F_{{\\mathcal {X}}_1^c}f,f \\rangle + (1-\\lambda ) \\langle (F_{{\\mathcal {X}}_1}^{*}+F_{{\\mathcal {X}}_1})f,f \\rangle +(2\\lambda -1)\\Vert f \\Vert ^2 \\nonumber \\\\&& \\Rightarrow (2 \\lambda - \\lambda ^2) Re(\\langle If,f \\rangle ) \\le \\Vert F_{{\\mathcal {X}}_1} f \\Vert ^2 +\\lambda ( \\overline{\\langle F_{{\\mathcal {X}}_1^c}f,f \\rangle } +\\langle F_{{\\mathcal {X}}_1^c}f,f \\rangle ) \\nonumber \\\\&&= \\Vert F_{{\\mathcal {X}}_1^c} f \\Vert ^2 +(1 - \\lambda ) ( \\overline{\\langle F_{{\\mathcal {X}}_1} f,f \\rangle } +\\langle F_{{\\mathcal {X}}_1} f,f \\rangle )+(2\\lambda -1)\\Vert f \\Vert ^2 \\nonumber \\\\&& \\Rightarrow (2 \\lambda - \\lambda ^2) Re(\\langle F_{{\\mathcal {X}}_1}f,f \\rangle )+(1 -\\lambda ^2) Re(\\langle F_{{\\mathcal {X}}_1^c}f,f \\rangle ) \\le \\Vert F_{{\\mathcal {X}}_1} f \\Vert ^2 + Re (\\langle F_{{\\mathcal {X}}_1^c}f,f \\rangle ) \\nonumber \\\\&&= \\Vert F_{{\\mathcal {X}}_1^c} f \\Vert ^2 + Re (\\langle F_{{\\mathcal {X}}_1} f,f \\rangle ) .$ We have $\\langle F_{{\\mathcal {X}}_1}f,f \\rangle = \\left\\langle \\int _{{\\mathcal {X}}_1}\\sum _{j \\in J} \\Lambda ^*_{x,j} {\\mathcal {G}_{x,j}}f \\; d\\mu (x),f \\right\\rangle = \\int _{{\\mathcal {X}}_1}\\sum _{j \\in J} \\langle {\\mathcal {G}_{x,j}}f , \\Lambda _{x,j}f \\rangle d\\mu (x).$ $\\langle F_{{\\mathcal {X}}_1^c}f,f \\rangle = \\int _{{\\mathcal {X}}_1^c}\\sum _{j \\in J} \\langle {\\mathcal {G}_{x,j}}f , \\Lambda _{x,j}f \\rangle d\\mu (x).$ Using equations (REF ), (REF ) and (REF ) in (REF ), we obtain the desired inequality.",
"Next we give a generalization of the above theorem to a more general form that does not involve the real parts of the complex numbers.",
"Theorem 3.10 Let $\\lbrace \\Lambda _{x,j}:x \\in {\\mathcal {X}}, j \\in J \\rbrace $ be a semi-continuous $g$ -frame for ${\\mathcal {H}}$ with respect to $({\\mathcal {X}}, \\mu )$ and $\\lbrace {\\mathcal {G}_{x,j}}:x \\in {\\mathcal {X}}, j \\in J \\rbrace $ be an alternate dual frame of $\\lbrace \\Lambda _{x,j}:x \\in {\\mathcal {X}}, j \\in J \\rbrace $ .",
"Then for every ${\\mathcal {X}}_1 \\subset {\\mathcal {X}}$ and every $f \\in {\\mathcal {H}}$ , we have $&& \\bigg (\\int _{{\\mathcal {X}}_1^c}\\sum _{j \\in J} \\langle {\\mathcal {G}_{x,j}}f, \\Lambda _{x,j} f \\rangle d\\mu (x) \\bigg )+ \\bigg \\Vert \\int _{{\\mathcal {X}}_1}\\sum _{j \\in J} \\Lambda ^*_{x,j} {\\mathcal {G}_{x,j}}f \\; d\\mu (x) \\bigg \\Vert ^2 \\\\&&= \\overline{\\bigg (\\int _{{\\mathcal {X}}_1}\\sum _{j \\in J} \\langle {\\mathcal {G}_{x,j}}f, \\Lambda _{x,j} f \\rangle d\\mu (x) \\bigg )}+ \\bigg \\Vert \\int _{{\\mathcal {X}}_1^c}\\sum _{j \\in J} \\Lambda ^*_{x,j} {\\mathcal {G}_{x,j}}f \\; d\\mu (x) \\bigg \\Vert ^2.$ For ${\\mathcal {X}}_1 \\subset {\\mathcal {X}}$ and $f \\in {\\mathcal {H}}$ , we define the operator $F_{{\\mathcal {X}}_1}$ as in Theorem REF .",
"Therefore, we have $F_{{\\mathcal {X}}_1}+F_{{\\mathcal {X}}_1^c}=I$ .",
"By Lemma REF , we have $&& \\bigg (\\int _{{\\mathcal {X}}_1^c}\\sum _{j \\in J} \\langle {\\mathcal {G}_{x,j}}f, \\Lambda _{x,j} f \\rangle d\\mu (x) \\bigg )+ \\bigg \\Vert \\int _{{\\mathcal {X}}_1}\\sum _{j \\in J} \\Lambda ^*_{x,j} {\\mathcal {G}_{x,j}}f \\; d\\mu (x) \\bigg \\Vert ^2 \\\\&& = \\langle F_{{\\mathcal {X}}_1^c}f,f \\rangle + \\langle F_{{\\mathcal {X}}_1}^* F_{{\\mathcal {X}}_1} f,f \\rangle = \\langle F_{{\\mathcal {X}}_1}^*f,f \\rangle +\\langle F_{{\\mathcal {X}}_1^c}^* F_{{\\mathcal {X}}_1^c} f,f \\rangle \\\\&& = \\overline{ \\langle F_{{\\mathcal {X}}_1} f,f \\rangle }+\\Vert F_{{\\mathcal {X}}_1^c} f \\Vert ^2 \\\\&& = \\overline{\\bigg (\\int _{{\\mathcal {X}}_1}\\sum _{j \\in J} \\langle {\\mathcal {G}_{x,j}}f, \\Lambda _{x,j} f \\rangle d\\mu (x) \\bigg )}+ \\bigg \\Vert \\int _{{\\mathcal {X}}_1^c}\\sum _{j \\in J} \\Lambda ^*_{x,j} {\\mathcal {G}_{x,j}}f \\; d\\mu (x) \\bigg \\Vert ^2.$ Hence the relation stated in the theorem holds."
],
[
"Stability of semi-continuous g-frames",
"The stability of frames is important in practice, so it has received much attentions and is, therefore, studied widely by many authors (see [8], [25], [27]).",
"In this section, we study the stability of semi-continuous $g$ -frames.",
"The following is a fundamental result in the study of the stability of frames.",
"Proposition 4.1 $($[6], $\\mathrm {Theorem \\;2})$ Let $\\lbrace f_i \\rbrace _{i=1}^\\infty $ be a frame for some Hilbert space ${\\mathcal {H}}$ with bounds $A, B$ .",
"Let $\\lbrace g_i \\rbrace _{i=1}^\\infty \\subseteq {\\mathcal {H}}$ and assume that there exist constants $\\lambda _1, \\lambda _2, \\mu \\ge 0$ such that $\\max (\\lambda _1+\\frac{\\mu }{\\sqrt{A}}, \\lambda _2)<1$ and $\\left\\Vert \\sum _{i=1}^n c_i (f_i-g_i) \\right\\Vert \\le \\lambda _1 \\left\\Vert \\sum _{i=1}^n c_i f_i \\right\\Vert +\\lambda _2 \\left\\Vert \\sum _{i=1}^n c_i g_i \\right\\Vert + \\mu \\left[ \\sum _{i=1}^n |c_i|^2 \\right]^{1/2}$ for all $c_1,...,c_n(n \\in {\\mathbb {N}}).$ Then $\\lbrace g_i \\rbrace _{i=1}^\\infty $ is a frame for ${\\mathcal {H}}$ with bounds $ A \\left( 1 - \\frac{\\lambda _1+\\lambda _2+\\frac{\\mu }{\\sqrt{A}}}{1+\\lambda _2} \\right)^2, \\;\\; B \\left( 1+ \\frac{\\lambda _1+\\lambda _2+\\frac{\\mu }{\\sqrt{B}}}{1-\\lambda _2} \\right)^2.$ Similar to discrete frames, semi-continuous $g$ -frames are stable under small perturbations.",
"The stability of semi-continuous $g$ -frames is discussed in the following theorem.",
"Theorem 4.2 Let $\\lbrace \\Lambda _{x,j}:x \\in {\\mathcal {X}}, j \\in J \\rbrace $ be a semi-continuous $g$ -frame for ${\\mathcal {H}}$ with respect to $({\\mathcal {X}}, \\mu )$ , with frame bounds $A$ and $B$ .",
"Suppose that ${\\Gamma _{x,j}}\\in \\mathcal {L}({\\mathcal {H}},{\\mathcal {K}}_{x,j})$ for any $x \\in {\\mathcal {X}}, \\; j \\in J$ and there exist constants $\\lambda _1, \\lambda _2, \\mu \\ge 0$ such that $\\max (\\lambda _1+\\frac{\\mu }{\\sqrt{A}}, \\lambda _2)<1$ and the following condition is satisfied $&& \\left( \\int _{{\\mathcal {X}}}\\sum _{j \\in J} \\Vert (\\Lambda _{x,j}-\\Gamma _{x,j})f \\Vert ^2 d\\mu (x) \\right)^{1/2} \\nonumber \\\\&& \\le \\lambda _1 \\left( \\int _{{\\mathcal {X}}}\\sum _{j \\in J} \\Vert \\Lambda _{x,j}(f) \\Vert ^2 d\\mu (x) \\right)^{1/2} + \\lambda _2 \\left( \\int _{{\\mathcal {X}}}\\sum _{j \\in J} \\Vert \\Gamma _{x,j}(f) \\Vert ^2 d\\mu (x) \\right)^{1/2} + \\mu \\Vert f \\Vert ,$ for all $f \\in {\\mathcal {H}}$ .",
"Then $\\lbrace {\\Gamma _{x,j}}: x \\in {\\mathcal {X}}, j \\in J\\rbrace $ is a semi-continuous $g$ -frame for ${\\mathcal {H}}$ with respect to $({\\mathcal {X}}, \\mu )$ , with frame bounds $A \\left( 1 - \\frac{\\lambda _1+\\lambda _2+\\frac{\\mu }{\\sqrt{A}}}{1+\\lambda _2} \\right)^2, \\;\\; B \\left( 1+ \\frac{\\lambda _1+\\lambda _2+\\frac{\\mu }{\\sqrt{B}}}{1-\\lambda _2} \\right)^2.$ Notice that $\\int _{{\\mathcal {X}}}\\sum _{j \\in J} \\Vert \\Lambda _{x,j}(f) \\Vert ^2 d\\mu (x) \\le B \\Vert f \\Vert ^2.",
"$ From (REF ) we see that $ \\left( \\int _{{\\mathcal {X}}}\\sum _{j \\in J} \\Vert (\\Lambda _{x,j}-\\Gamma _{x,j})f \\Vert ^2 d\\mu (x) \\right)^{1/2} \\le \\left( \\lambda _1 \\sqrt{B} + \\mu \\right)\\Vert f \\Vert + \\lambda _2 \\left( \\int _{{\\mathcal {X}}}\\sum _{j \\in J} \\Vert \\Gamma _{x,j}(f) \\Vert ^2 d\\mu (x) \\right)^{1/2}.",
"$ Using the triangle inequality, we get $&& \\left( \\int _{{\\mathcal {X}}}\\sum _{j \\in J} \\Vert (\\Lambda _{x,j}-\\Gamma _{x,j})f \\Vert ^2 d\\mu (x) \\right)^{1/2} \\\\&& \\ge \\left( \\int _{{\\mathcal {X}}}\\sum _{j \\in J} \\Vert \\Gamma _{x,j}(f) \\Vert ^2 d\\mu (x) \\right)^{1/2} - \\left( \\int _{{\\mathcal {X}}}\\sum _{j \\in J} \\Vert \\Lambda _{x,j}(f) \\Vert ^2 d\\mu (x) \\right)^{1/2}.$ Hence $&& (1-\\lambda _2) \\left( \\int _{{\\mathcal {X}}}\\sum _{j \\in J} \\Vert \\Gamma _{x,j}(f) \\Vert ^2 d\\mu (x) \\right)^{1/2} \\\\&& \\le \\left( \\lambda _1 \\sqrt{B} + \\mu \\right)\\Vert f \\Vert + \\left( \\int _{{\\mathcal {X}}}\\sum _{j \\in J} \\Vert \\Lambda _{x,j}(f) \\Vert ^2 d\\mu (x) \\right)^{1/2}\\le \\sqrt{B} \\bigg ( 1 + \\lambda _1 + \\frac{\\mu }{\\sqrt{B}} \\bigg )\\Vert f \\Vert .$ Therefore $\\int _{{\\mathcal {X}}}\\sum _{j \\in J} \\Vert \\Gamma _{x,j}(f) \\Vert ^2 d\\mu (x)\\le B \\left( 1+ \\frac{\\lambda _1+\\lambda _2+\\frac{\\mu }{\\sqrt{B}}}{1-\\lambda _2} \\right)^2 \\Vert f \\Vert ^2.",
"$ Similarly, we can prove that $\\int _{{\\mathcal {X}}}\\sum _{j \\in J} \\Vert \\Gamma _{x,j}(f) \\Vert ^2 d\\mu (x) \\ge A \\left( 1 - \\frac{\\lambda _1+\\lambda _2+\\frac{\\mu }{\\sqrt{A}}}{1+\\lambda _2} \\right)^2 \\Vert f \\Vert ^2.",
"$ This completes the proof.",
"Remark 4.3 In general, the inequality (REF ) does not imply that $\\lbrace {\\Gamma _{x,j}}: x \\in {\\mathcal {X}}, j \\in J\\rbrace $ is a semi-continuous $g$ -frame regardless how small the parameters $\\lambda _1, \\lambda _2, \\mu $ are.",
"A counterexample for $g$ -frames can be found in [27], and an example can be constructed similarly for semi-continuous $g$ -frames.",
"Corollary 4.4 Let $\\lbrace \\Lambda _{x,j}:x \\in {\\mathcal {X}}, j \\in J \\rbrace $ be a semi-continuous $g$ -frame for ${\\mathcal {H}}$ with respect to $({\\mathcal {X}}, \\mu )$ , with frame bounds $A, B$ , and let $\\lbrace {\\Gamma _{x,j}}: x \\in {\\mathcal {X}}, j \\in J\\rbrace $ be a sequence in $\\mathcal {L}({\\mathcal {H}},{\\mathcal {K}}_{x,j})$ for any $x \\in {\\mathcal {X}}, \\; j \\in J$ .",
"Assume that there exists a constant $0<M<A$ such that $\\int _{{\\mathcal {X}}}\\sum _{j \\in J} \\Vert (\\Lambda _{x,j}-\\Gamma _{x,j})f \\Vert ^2 d\\mu (x) \\le M \\Vert f \\Vert ^2, \\; \\forall f \\in {\\mathcal {H}}, $ then $\\lbrace {\\Gamma _{x,j}}: x \\in {\\mathcal {X}}, j \\in J\\rbrace $ is a semi-continuous $g$ -frame for ${\\mathcal {H}}$ with respect to $({\\mathcal {X}}, \\mu )$ , with bounds $A[1-(M/A)^{1/2}]^2$ and $B[1+(M/B)^{1/2}]^2$ .",
"Let $\\lambda _1=\\lambda _2=0$ and $\\mu =\\sqrt{M}.$ Since $M< A$ , $\\mu /\\sqrt{A}=\\sqrt{M/A}<1.$ So, by Theorem REF , $\\lbrace {\\Gamma _{x,j}}: x \\in {\\mathcal {X}}, j \\in J\\rbrace $ is a semi-continuous $g$ -frame for ${\\mathcal {H}}$ with respect to $({\\mathcal {X}}, \\mu )$ , with bounds $A[1-(M/A)^{1/2}]^2$ and $B[1+(M/B)^{1/2}]^2$ ."
],
[
"Acknowledgments",
"The author is deeply indebted to Dr. Azita Mayeli for several valuable comments and suggestions.",
"The author is grateful to the United States-India Educational Foundation for providing the Fulbright-Nehru Doctoral Research Fellowship, and Department of Mathematics and Computer Science, the Graduate Center, City University of New York, New York, USA for its kind hospitality during the period of this work.",
"He would also like to express his gratitude to the Norbert Wiener Center for Harmonic Analysis and Applications at the University of Maryland, College Park for its kind support."
]
] | 1808.08445 |
[
[
"Reversing Parallel Programs with Blocks and Procedures"
],
[
"Abstract We show how to reverse a while language extended with blocks, local variables, procedures and the interleaving parallel composition.",
"Annotation is defined along with a set of operational semantics capable of storing necessary reversal information, and identifiers are introduced to capture the interleaving order of an execution.",
"Inversion is defined with a set of operational semantics that use saved information to undo an execution.",
"We prove that annotation does not alter the behaviour of the original program, and that inversion correctly restores the initial program state."
],
[
"Introduction",
"Reverse execution of programs is the ability to take the final program state of an execution, undo all effects that were the result of that execution, and restore the exact initial program state.",
"This is a desirable capability as it has applications to many active research areas, including debugging [3] and Parallel Discrete Event Simulation [2].",
"When combined with parallelism, reverse execution removes issues relating to non-deterministic execution orders, allowing specific execution interleavings to be analysed easily.",
"In our previous work [10], we described a state-saving approach to reversible execution of an imperative while language.",
"Similarly to RCC [14], we generated two versions of a program, the augmented forwards version to save all necessary reversal information alongside its execution, and the inverted version that uses this saved data to undo all changes.",
"We proved that augmentation did not alter the behaviour of our program, and that inversion correctly restores the initial program state.",
"We then experimented with reversing a tiny language containing assignments and interleaving parallel composition.",
"In this paper, we extend the while language with blocks, local variables and procedures, as well as the parallel composition operator.",
"Local variables mean we must recognise scope, for example different versions of a shared name used in parallel.",
"Issues arise with the traditional approach, specifically with recursion, and calls to the same procedure executing in parallel.",
"Annotation and inversion are defined, allowing this extended language to be executed forwards with state-saving, as well as in reverse using this saved information.",
"The process of assigning identifiers to statements as we execute them is described, focusing on backtracking order, where statements are undone in the inverted order of the forwards execution.",
"We mention future work on causal-consistent reversibility [1], [13], [15] in the conclusion.",
"Consider the example shown in Figure REF , where w1.0 and $\\lambda $ can be ignored.",
"This is a simple model of a restaurant with two entrances.",
"One where a single person is continually allowed to enter, increasing the number of current single guests (c), until the total capacity (c + r) reaches the maximum (m).",
"The other allows a reserved group of two to enter, increasing the number of reserved guests (r).",
"Let the initial state be that m = 4, c = 0 and r = 0.",
"The execution begins with two full iterations of the while loop, allowing two people to enter meaning c = 2.",
"Next, the condition of the loop is evaluated, but the body is not yet executed.",
"Interleaving now occurs, setting r to 2.",
"Finally, the body of the loop is now executed, before the condition evaluates to false and the loop finishes.",
"The final state is m = 4, c = 3 and r = 2, which should be invalid as the total number of guests (c + r) $>$ m. This executed version of the annotated program is shown in Figure REF , where each statement now has a stack populated with identifiers in the order in which the statement occurred (starting at 1).",
"One solution to finding this bug is reverse execution.",
"The inverted version we generate (which, coincidentally is identical to Figure REF ) allows step-by-step reversal, using identifiers to remove non-determinism.",
"Backtracking through this execution removes the difficulties of cyclic debugging where different interleavings can occur.",
"Using this, we can see that we wrongly commit to allowing the third single person to enter, meaning the condition gave true when expected to give false.",
"Examining this further, we can see that the reserved guests are not considered until they have arrived, meaning this condition is not aware that the maximum capacity is actually m - 2.",
"This is an example of a race between the writing and reading of r. This is fixed using the condition ((m-c-2-1) >= 0).",
"Our main contributions are The definition of three sets of operational semantics for our language, namely for traditional forwards only execution, annotated forwards execution, and for reverse execution.",
"Annotation allows all necessary state-saving, and the use of identifiers to record the interleaving order of execution.",
"Inversion then uses the saved information to reverse via backtracking order.",
"Results showing that annotation does not alter the behaviour of the original program, and that inversion correctly restores to the initial program state.",
"We also have a prototype simulator under development.",
"This will be capable of implementing both the forwards and reverse execution, and used for both performance evaluation and validation of our results.",
"Figure: Executed annotated programProgram inversion has been the focus of many works for many years, including Jefferson [12], Gries [8] and Glück and Kawabe [4], [5].",
"The Reverse C Compiler (RCC) by Perumalla [14] describes a state-saving approach to reversibility of C programs.",
"The Backstroke framework [19] and extensions of it by Schordan et al [17] describe an approach to reversing C++ in the setting of Parallel Discrete Event Simulation [2].",
"The reversible programming language Janus, worked on in [20], [21], adds additional information into the source code, making all programs reversible.",
"More recent work on reversible imperative programs by Glück and Yokoyama [6], [7] introduce the languages R-WHILE and R-CORE.",
"Reversibility of algebraic process calculi is the focus of work by Phillips and Ulidowski [15], [16], where the notion of identifiers was introduced.",
"There has been work on reversible object oriented programming languages, including that of Schultz [18] and the language ROOPL [9].",
"The application of reverse computation to debugging of message passing concurrent programs is considered by Giachino et al [3]."
],
[
"Programming Language, Environments and Scope",
"Let P be the set of all programs and S be the set of all statements.",
"Each program P will be either a statement S, the sequential composition of programs P;Q or the parallel composition of programs P par Q (sometimes written as par {P}{Q}).",
"Each statement will either be a skip operation (empty statement), an assignment, a conditional, a loop, a block, a variable or procedure declaration, a variable or procedure removal or a call.",
"A block consists of the declaration of both local variables DV and procedures DP, a body that uses these, and then the removal of local procedures RP and variables RV.",
"Procedures do not have arguments, static scope is assumed and recursion is permitted.",
"The syntax of this language is shown below, including arithmetic and boolean expressions.",
"Note that the constructs runC and runB are reserved words that appear in our syntax, but not in original programs, and will be explained in Section .",
"These allow static operational semantics to be defined, needed to aid state-saving and in our results.",
"Many statements contain a path pa that is explained in Section REF .",
"Each conditional, loop, block, procedure and procedure call statement has a unique identifier named In, Wn, Bn, Pn and Cn respectively, each of which is an element of the sets In, Wn, Bn, Pn and Cn respectively.",
"The set union of these gives us the set of construct identifiers CI.",
"Note a procedure has a name from the set n (appears in code, and is potentially duplicated), as well as a unique identifier Pn.",
"$\\texttt {P} &::= \\texttt {\\varepsilon } ~|~ \\texttt {S} ~|~ \\texttt {P; P} ~|~ \\texttt {P par P} \\\\\\texttt {S} &::= \\texttt {skip} ~|~ \\texttt {X = E pa} ~|~ \\texttt {if In B then P else Q end pa} ~| \\\\ &\\phantom{\\texttt {::=}} \\texttt {while Wn B do P end pa} ~|~ \\texttt {begin Bn DV DP P RP RV end} ~| \\\\ &\\phantom{\\texttt {::=}}\\texttt {call Cn n pa} ~|~ \\texttt {runC Cn P end} ~|~ \\texttt {runB P end}$ $\\texttt {DV} &::= \\texttt {\\varepsilon } ~|~ \\texttt {var X = v pa; DV} &\\texttt {DP} &::= \\texttt {\\varepsilon } ~|~ \\texttt {proc Pn n is P pa; DP} \\\\\\texttt {RV} &::= \\texttt {\\varepsilon } ~|~ \\texttt {remove X = v pa; RV} &\\texttt {RP} &::= \\texttt {\\varepsilon } ~|~ \\texttt {remove Pn n is P pa; RP} \\\\\\texttt {E} &::= \\texttt {Var} ~|~ \\texttt {n} ~|~ \\texttt {(E)} ~|~ \\texttt {E Op E} &\\texttt {B} &::= \\texttt {T} ~|~ \\texttt {F} ~|~ \\lnot \\texttt {B}~|~ \\texttt {(B)} ~|~ \\texttt {E == E} ~|~ \\texttt {E > E} ~|~ \\texttt {B \\wedge B}$"
],
[
"Environments",
"We complete our setting with the definition of several environments.",
"Let V be the set of all program variables, Loc be the set of all memory locations, and Num be the set of integers.",
"As in [11], we first have a variable environment $\\gamma $ , responsible for mapping a variable name and the block to which it is local ($\\lambda $ in the case of global variables) to its bound memory location.",
"This is defined as $\\gamma : (\\textbf {V} \\times \\textbf {Bn}) \\mapsto \\textbf {Loc}$ .",
"The notation $\\gamma $ [(X,Bn) $\\Rightarrow $ l] indicates that the pair (X,Bn) maps to the memory location l, while $\\gamma $ [(X,Bn)] represents an update to $\\gamma $ with the mapping for the pair (X,Bn) removed.",
"We have a data store $\\sigma $ , responsible for mapping each memory location to the value it currently holds, defined as $\\sigma : (\\textbf {Loc} \\mapsto \\textbf {Num})$ .",
"The notion $\\sigma $ [l $\\mapsto $ v] indicates location l now holds the value v. The procedure environment $\\mu $ is responsible for mapping either a procedure or call identifier to both the actual procedure name (used in code) and (a copy of) the body.",
"This environment is defined as $\\mu : (\\textbf {Pn} \\cup \\textbf {Cn}) \\mapsto (\\textbf {n} \\times \\textbf {P})$ .",
"The notation $\\mu $ [Pn $\\Rightarrow $ (n,P)] represents that Pn maps to the pair (n,P), $\\mu $ [refC(Pn,P)] represents the updating of the mapping for Pn with changes retrieved from P, and $\\mu $ [Pn] indicates the removal of the mapping for Pn (in each case, Pn could also be Cn).",
"Finally, the while environment $\\beta $ is responsible for mapping a unique loop identifier to a copy of that loop.",
"This serves the purpose of storing both the original condition and program, allowing our semantics to be static.",
"This is defined as $\\beta : \\textbf {Wn} \\mapsto \\textbf {P}$ .",
"The notation $\\beta $ [Wn $\\Rightarrow $ P] indicates that Wn now maps to the program P, $\\beta $ [refW(Wn,P)] represents the updating of the mapping for Wn with changes retrieved from P, and the notation $\\beta $ [Wn] shows the removal of the mapping for Wn.",
"We now combine all environments, and use the notation $\\square $ to represent the set {$\\sigma $ ,$\\gamma $ ,$\\mu $ ,$\\beta $ }.",
"Each environment has a prime version, indicating a potential and arbitrary change."
],
[
"Scope",
"Local variables can share their name with a global variable, as well as local variables declared in different blocks.",
"The traditional method of handling this, as described for example in [11], is to implement a stack of environments, storing a copy for each scope.",
"This is not suitable when we use parallel composition as there can be several active scopes in an execution at once.",
"We therefore implement a single environment that will store all versions of variables.",
"Variables will either be global, or local to a specific block.",
"Using $\\lambda $ to represent the empty block name (a global variable), associating a variable with the identifier of the block in which it is declared is sufficient.",
"Now all versions of a variable name are stored distinctly.",
"We must be able to access the correct version of a given variable name.",
"Under the traditional approach, each environment will have only one mapping of any variable, something we do not have.",
"We must be able to determine the block identifier in which the variable was defined, which will not necessarily be the current block.",
"We achieve this by assigning a path to each statement.",
"Each path will be the sequence of the block identifiers Bn, for blocks in which this statement resides, separated using `*'.",
"Consider statement F = S (b2*b1,A) (from Figure REF ) that has a path b2*b1, meaning it occurs within a block b2, which is nested within b1.",
"Therefore we have the function evalV(), that takes a variable name and a path, traverses the sequence of block names until the first is found that has a local variable of this name, and uses this block name (or $\\lambda $ if no match) to return the desired memory location.",
"A similar reasoning, and function evalP(), exists to evaluate potentially shared procedure names, returning the correct unique procedure identifier.",
"One complication is code reuse, where the same program code is executed multiple times, with the two cases being procedure and loop bodies.",
"Consider two calls to the same procedure in parallel.",
"Both may create a local variable of a block, where on each side the block has the same name, meaning both will incorrectly use the same version of the local variable.",
"A similar case exists for recursive calls, as shown in Example 1.",
"Therefore, we rename any reused code prior to its execution.",
"This must make all constructs unique, a task achieved using the unique call identifier Cn.",
"All construct names are modified to now start with the unique call identifier, with paths also updated to reflect changes made to block identifiers.",
"Consider again the statement for F from Figure REF .",
"When block b2 is renamed to c1:c2:b2, the path becomes c1:c2:b2*b1 (Figure REF ).",
"This removes the issue described above, as each version of the variable will have a different block identifier.",
"This process must modify the call identifier of any recursive call statement similarly.",
"Therefore we have the function reP() that implements this renaming of procedure bodies, and as renaming occurs in reverse, we have IreP().",
"The reuse of loop bodies is different, as it is not possible for the same code (with same construct names) to be executed in parallel when not in a procedure body (handled above).",
"However, in order to keep all identifiers unique, and to aid future extensions to causal-consistent reversibility, we version each construct name, incrementing it by 1 for each loop iteration.",
"For example, a conditional statement i1.0 will become i1.1.",
"These versions are maintained via the function nextID(), used in $reL()$ that performs this renaming, and the function previousID(), used in the reverse renaming function $IreL()$ ."
],
[
"Forwards Only Operational Semantics",
"We now define the traditional, forwards only semantics of our language.",
"We give a set of transition rules for each construct of our language and a set of environments.",
"These rules specify how configurations (namely, pairs containing a program and a set of environments) compute by performing single transition steps.",
"The transition rules define our small step transition relation configuration $\\hookrightarrow $ configuration.",
"The transitive closure $\\hookrightarrow ^*$ represents executions of programs.",
"The forwards only semantics do not perform any state-saving, thus making them irreversible.",
"Transitions labelled with a or b are steps of arithmetic or boolean expression evaluation respectively, while $\\hookrightarrow ^*_{\\texttt {a}}$ and $\\hookrightarrow ^*_{\\texttt {b}}$ are the transitive closure of each.",
"Semantics of both are omitted as they are as expected, see [11].",
"By abuse of notation, we now use $\\square $ to represent all environments of the set {$\\sigma $ ,$\\gamma $ ,$\\mu $ ,$\\beta $ } that are not modified via the specific rule.",
"For example, if a rule only changes the procedure environment $\\mu $ , then $\\square $ will be the set {$\\sigma $ ,$\\gamma $ ,$\\beta $ }.",
"The semantics listed below are static, necessary for later sections including state-saving and our results.",
"Sequential and Parallel Composition Programs can be of the form S;P or P par Q.",
"As such, programs either execute sequentially, or allow each side of a parallel statement to interleave their execution.",
"$&\\text{[S1]} \\quad \\frac{(\\texttt {S} \\mid \\square ) \\hookrightarrow (\\texttt {S^{\\prime }} \\mid \\square ^{\\prime })}{(\\texttt {S; P} \\mid \\square ) \\hookrightarrow (\\texttt {S^{\\prime }; P} \\mid \\square ^{\\prime })} &\\quad &\\text{[S2]} \\quad \\frac{}{(\\texttt {skip; P} \\mid \\square ) \\hookrightarrow (\\texttt {P} \\mid \\square )} \\\\[6pt]&\\text{[P1]} \\quad \\frac{(\\texttt {P} \\mid \\square ) \\hookrightarrow (\\texttt {P^{\\prime }} \\mid \\square ^{\\prime })}{(\\texttt {P par Q} \\mid \\square ) \\hookrightarrow (\\texttt {P^{\\prime } par Q} \\mid \\square ^{\\prime })}&\\quad &\\text{[P2]} \\quad \\frac{(\\texttt {Q} \\mid \\square ) \\hookrightarrow (\\texttt {Q^{\\prime }} \\mid \\square ^{\\prime })}{(\\texttt {P par Q} \\mid \\square ) \\hookrightarrow (\\texttt {P par Q^{\\prime }} \\mid \\square ^{\\prime })} \\\\[6pt]&\\text{[P3]} \\quad \\frac{}{(\\texttt {P par skip} \\mid \\square ) \\hookrightarrow (\\texttt {P} \\mid \\square )} &\\quad &\\text{[P4]} \\quad \\frac{}{(\\texttt {skip par Q} \\mid \\square ) \\hookrightarrow (\\texttt {Q} \\mid \\square )}$ Assignment All assignments are considered destructive, with the overwritten value being lost.",
"A single atomic rule both evaluates the expression and assigns the new value to the appropriate memory location.",
"Arithmetic expressions do not contain side effects, meaning evaluation of these does not change the environments, as shown in rule [D1].",
"Similarly, this is also the case for boolean expressions, as shown in rule [I1] and there after.",
"$&\\text{[D1]} \\quad \\frac{(\\texttt {e pa} \\mid \\sigma ,\\gamma ,\\square ) \\hookrightarrow ^*_{\\texttt {a}} (\\texttt {v} \\mid \\sigma ,\\gamma ,\\square ) \\quad \\texttt {evalV(\\gamma ,pa,X) = l}}{(\\texttt {X = e pa} \\mid \\sigma ,\\gamma ,\\square ) \\hookrightarrow (\\texttt {skip} \\mid \\sigma [\\texttt {l \\mapsto v}],\\gamma ,\\square )}$ Conditional Condition evaluation is atomic via [I1], before the appropriate branch is executed completely to skip (potentially interleaved).",
"$&\\text{[I1]} \\quad \\frac{(\\texttt {b pa} \\mid \\square ) \\hookrightarrow ^*_{\\texttt {b}} (\\texttt {V} \\mid \\square )}{(\\texttt {if In b then P else Q end pa} \\mid \\square ) \\hookrightarrow (\\texttt {if In V then P else Q end pa} \\mid \\square )} \\\\[6pt]&\\text{[I2]} \\quad \\frac{(\\texttt {P} \\mid \\square ) \\hookrightarrow (\\texttt {P^{\\prime }} \\mid \\square ^{\\prime })}{(\\texttt {if In T then P else Q end pa} \\mid \\square ) \\hookrightarrow (\\texttt {if In T then P^{\\prime } else Q end pa} \\mid \\square ^{\\prime })} \\\\[6pt]&\\text{[I3]} \\quad \\frac{(\\texttt {Q} \\mid \\square ) \\hookrightarrow (\\texttt {Q^{\\prime }} \\mid \\square ^{\\prime })}{(\\texttt {if In F then P else Q end pa} \\mid \\square ) \\hookrightarrow (\\texttt {if In F then P else Q^{\\prime } end pa} \\mid \\square ^{\\prime })} \\\\[6pt]&\\text{[I4]} \\quad \\frac{}{(\\texttt {if In T then skip else Q end pa} \\mid \\square ) \\hookrightarrow (\\texttt {skip} \\mid \\square )} \\\\[6pt]&\\text{[I5]} \\quad \\frac{}{(\\texttt {if In F then P else skip end pa} \\mid \\square ) \\hookrightarrow (\\texttt {skip} \\mid \\square )}$ While Loop Evaluation of the condition is always atomic.",
"[W1] handles the first iteration of a loop, where no mapping for Wn is present in $\\beta $ .",
"The mapping Wn $\\Rightarrow $ R is inserted, and the condition is evaluated.",
"[W2] handles any other iteration, evaluating the condition and updating the mapping within $\\beta $ to Wn $\\Rightarrow $ R$^{\\prime }$.",
"Both rules rename the body, making constructs unique.",
"[W3] executes the body.",
"[W4] continues the loop using the program P, retrieved from $\\beta $ for Wn, until the condition is false, when [W5] will conclude the statement.",
"The premise $\\beta ($ Wn$)$ = R indicates an arbitrary mapping exists.",
"This is necessary as the rule [W5] requires the removal of some mapping via $\\beta [\\texttt {Wn}]$ .",
"Note that these semantics (and the semantics defined in Sections and ) are correct for all while loops with conditions that require evaluation.",
"In the case of b initially being T or F, there may be ambiguity in our rules.",
"$&\\text{[W1]} \\quad \\frac{\\texttt {\\beta (Wn) = }\\emph {und} \\quad (\\texttt {b pa} \\mid \\beta ,\\square ) \\hookrightarrow ^*_{\\texttt {b}} (\\texttt {V} \\mid \\beta ,\\square ) }{(\\texttt {while Wn b do P end pa} \\mid \\beta ,\\square ) \\hookrightarrow (\\texttt {while Wn V do reL(P) end pa} \\mid \\beta [\\texttt {Wn} \\Rightarrow \\texttt {R}],\\square )} \\\\[2pt] & \\phantom{\\text{[W1]} \\quad } \\text{where } \\texttt {R} = \\texttt {while Wn b do reL(P) end pa} \\\\[6pt]&\\text{[W2]} \\quad \\frac{\\texttt {\\beta (Wn) = while Wn b do P end pa} \\quad (\\texttt {b pa} \\mid \\beta ,\\square ) \\hookrightarrow ^*_{\\texttt {b}} (\\texttt {V} \\mid \\beta ,\\square ) }{(\\texttt {while Wn b do P end pa} \\mid \\beta ,\\square ) \\hookrightarrow (\\texttt {while Wn V do reL(P) end pa} \\mid \\beta [\\texttt {Wn} \\Rightarrow \\texttt {R^{\\prime }}],\\square )} \\\\[2pt] & \\phantom{\\text{[W2]} \\quad } \\text{where } \\texttt {R^{\\prime }} = \\texttt {while Wn b do reL(P) end pa} \\\\[6pt]&\\text{[W3]} \\quad \\frac{(\\texttt {R} \\mid \\square ) \\hookrightarrow (\\texttt {R^{\\prime }} \\mid \\square ^{\\prime })}{(\\texttt {while Wn T do R end pa} \\mid \\square ) \\hookrightarrow (\\texttt {while Wn T do R^{\\prime } end pa} \\mid \\square ^{\\prime })} \\\\[6pt]&\\text{[W4]} \\quad \\frac{\\texttt {\\beta (Wn) = P} }{(\\texttt {while Wn T do skip end pa} \\mid \\beta ,\\square ) \\hookrightarrow (\\texttt {P} \\mid \\beta ,\\square )} \\\\[6pt]&\\text{[W5]} \\quad \\frac{\\texttt {\\beta (Wn) = R}}{(\\texttt {while Wn F do P end pa} \\mid \\beta ,\\square ) \\hookrightarrow (\\texttt {skip} \\mid \\beta [\\texttt {Wn}],\\square )}$ Block Blocks begin with [B1] that creates the runB construct.",
"This then executes the block body via [B2], beginning with the declaration of local variables and procedures.",
"The program then executes using these local definitions, before all such information is removed.",
"Finally [B3] concludes the statement.",
"$&\\text{[B1]} \\quad \\frac{}{(\\texttt {begin Bn P end} \\mid \\square ) \\hookrightarrow (\\texttt {runB P end} \\mid \\square )} \\quad \\text{where } \\texttt {P } \\text{=} \\texttt { DV;DP;Q;RP;RV} \\\\[6pt]&\\text{[B2]} \\quad \\frac{(\\texttt {P} \\mid \\square ) \\hookrightarrow (\\texttt {P^{\\prime }} \\mid \\square ^{\\prime })}{(\\texttt {runB P end} \\mid \\square ) \\hookrightarrow (\\texttt {runB P^{\\prime } end} \\mid \\square ^{\\prime })} \\quad \\text{[B3]} \\quad \\frac{}{(\\texttt {runB skip end} \\mid \\square ) \\hookrightarrow (\\texttt {skip} \\mid \\square )}$ Variable and Procedure Declaration A variable declaration [L1] associates the given variable name and current block name Bn (first element of the sequence pa, written Bn*pa$^{\\prime }$) to the next available memory location l (via nextLoc()) in $\\gamma $ , while also mapping this location to the value v in $\\sigma $ (via the notation l $\\mapsto $ v).",
"A procedure declaration [L2] inserts the basis mapping between the unique procedure identifier Pn and a pair containing both the procedure name and the procedure body (n,P).",
"A call statement uses this mapping to create a renamed version.",
"$&\\text{[L1]} \\quad \\frac{\\texttt {nextLoc() = l} \\quad \\texttt {pa} = \\texttt {Bn*pa^{\\prime }}}{(\\texttt {var X = v pa} \\mid \\sigma ,\\gamma ,\\square ) \\hookrightarrow (\\texttt {skip} \\mid \\sigma [\\texttt {l} \\mapsto \\texttt {v}],\\gamma [(\\texttt {X},\\texttt {Bn}) \\Rightarrow \\texttt {l}],\\square )} \\\\[6pt]&\\text{[L2]} \\quad \\frac{}{(\\texttt {proc Pn n is P pa} \\mid \\mu ,\\square ) \\hookrightarrow (\\texttt {skip} \\mid \\mu [\\texttt {Pn} \\Rightarrow \\texttt {(n,P)}],\\square )}$ Procedure Call [G1] evaluates the procedure name to Pn, and retrieves the basis entry (n,P) from $\\mu $ .",
"The renamed version of P, written P$^{\\prime }$, is then inserted into $\\mu $ via the mapping Cn $\\Rightarrow $ (n,P$^{\\prime }$ ), and the runC construct is formed.",
"[G2] executes the body of the call statement, before [G3] concludes the statement by removing the mapping for Cn within $\\mu $ , written $\\mu $ [Cn].",
"$&\\text{[G1]} \\quad \\frac{\\texttt {evalP(n,pa) = Pn} \\quad \\texttt {\\mu (Pn) = (n,P)} \\quad \\texttt {reP(P,Cn) = P^{\\prime }}}{(\\texttt {call Cn n pa} \\mid \\mu ,\\square ) \\hookrightarrow (\\texttt {runC Cn P^{\\prime } end} \\mid \\mu [\\texttt {Cn} \\Rightarrow \\texttt {(n,P^{\\prime })}],\\square )} \\\\[6pt]&\\text{[G2]} \\quad \\frac{(\\texttt {P} \\mid \\square ) \\hookrightarrow (\\texttt {P^{\\prime }} \\mid \\square ^{\\prime })}{(\\texttt {runC Cn P end} \\mid \\square ) \\hookrightarrow (\\texttt {runC Cn P^{\\prime } end} \\mid \\square ^{\\prime })} \\\\[6pt]&\\text{[G3]} \\quad \\frac{}{(\\texttt {runC Cn skip end} \\mid \\mu ,\\square ) \\hookrightarrow (\\texttt {skip} \\mid \\mu [\\texttt {Cn}],\\square )}$ Variable and Procedure Removal A local variable X is local to the inner most block Bn of its path, written as Bn*pa$^{\\prime }$.",
"Removal of a variable [H1] removes the mapping of (X,Bn) from $\\gamma $ , written $\\gamma $ [(X,Bn)].",
"The location associated with this mapping is set to 0 (l $\\mapsto $ 0) and marked free for future use.",
"A procedure removal [H2] removes the mapping for the given procedure identifier Pn, written as $\\mu $ [Pn].",
"$&\\text{[H1]} \\quad \\frac{\\texttt {pa} = \\texttt {Bn*pa^{\\prime }} \\quad \\texttt {\\gamma (X,Bn) = l}}{(\\texttt {remove X = v pa} \\mid \\sigma ,\\gamma ,\\square ) \\hookrightarrow (\\texttt {skip} \\mid \\sigma [\\texttt {l} \\mapsto \\texttt {0}],\\gamma [(\\texttt {X},\\texttt {Bn})],\\square )} \\\\[6pt]&\\text{[H2]} \\quad \\frac{}{(\\texttt {remove Pn n is P pa} \\mid \\mu ,\\square ) \\hookrightarrow (\\texttt {skip} \\mid \\mu [\\texttt {Pn}],\\square )}$"
],
[
"Forwards Semantics of Annotated Programs",
"Annotation is the process of generating our annotated version of a program.",
"This will make small changes to the syntax of our program, adding the capability of storing necessary reversal information.",
"Before defining this in detail, we must have an environment for storing this information, keeping it separate from the program state.",
"To do this, we use an updated version of our auxiliary store $\\delta $ [10].",
"There is a stack for each program variable name, storing any overwritten values the variable holds throughout the execution, whether the variable is global, local or both.",
"Using one stack for all versions of a variable helps us to devise a technique to handle races on that variable.",
"There is a single stack B for all conditional statements.",
"After completion of the conditional, a pair containing an identifier and a boolean value indicating which branch was executed will be saved.",
"Identifiers allow us to resolve any races, meaning all pairs for all conditionals can be pushed to a single stack.",
"There is a single stack W for all while loops.",
"As explained and illustrated in our previous work [10], this stack will contain pairs of an identifier and a boolean value.",
"These pairs produce a sequence of boolean values necessary for inverse execution.",
"Stack WI stores all annotation information (order of identifiers) of a while loop, before it's removed from $\\beta $ .",
"Stack Pr performs a similar task for procedure bodies.",
"Let V be the set of all variables, $\\mathbb {S}(\\textbf {V})$ be a set of stacks, one for each element of $\\textbf {V}$ , $\\mathbb {B}$ be the set of boolean values, $\\mathbb {C}$ be the annotation information and K be the set of identifiers.",
"Then $ \\delta : (\\mathbb {S}(\\textbf {V}) \\mapsto (\\textbf {K} \\times \\textbf {Num})) \\cup (\\mathbb {S}(\\texttt {B}) \\mapsto (\\textbf {K} \\times \\mathbb {B})) \\cup (\\mathbb {S}(\\texttt {W}) \\mapsto (\\textbf {K} \\times \\mathbb {B})) \\cup (\\mathbb {S}(\\texttt {WI}) \\mapsto (\\textbf {K} \\times \\mathbb {C})) \\cup (\\mathbb {S}(\\texttt {Pr}) \\mapsto (\\textbf {K} \\times \\mathbb {C})) $ .",
"The notation $\\delta $ [el $\\rightharpoonup $ st] pushes el to the stack st, while $\\delta $ [st/st$^{\\prime }$] pops the top element of stack st, leaving the remaining stack st$^{\\prime }$.",
"We now define annotation.",
"Each statement, excluding blocks and parallel, receives a stack A for identifiers.",
"The association of a statement to its stack persists throughout the execution.",
"Each time a statement executes, the (global and atomic) function next() retrieves the next available identifier, which is pushed to that statements stack (and $\\delta $ if necessary).",
"The functions $ann()$ and $a()$ are defined below.",
"$ ann(\\texttt {$ $}) = \\texttt {$$} \\quad ann(\\texttt {S;P}) = a(\\texttt {S}); ann(\\texttt {P}) \\quad ann(\\texttt {P par Q}) = ann(\\texttt {P}) \\texttt { par } ann(\\texttt {Q}) $$ $ $a(\\texttt {skip}) &= \\texttt {skip I} \\\\a(\\texttt {X = e pa}) &= \\texttt {X = e (pa,A)} \\\\a(\\texttt {if In b then P else Q end pa}) &= \\texttt {if In b then ann(P) else ann(Q) end (pa,A)} \\\\a(\\texttt {while Wn b do P end pa}) &= \\texttt {while Wn b do ann(P) end (pa,A)} \\\\a(\\texttt {begin Bn DV DP P RP RV end}) &= \\texttt {begin Bn ann(DV) ann(DP) ann(P) ann(RP) ann(RV) end} \\\\a(\\texttt {var X = v pa}) &= \\texttt {var X = v (pa,A)} \\\\a(\\texttt {proc Pn n is P pa}) &= \\texttt {proc Pn n is ann(P) (pa,A)} \\\\a(\\texttt {call Cn n pa}) &= \\texttt {call Cn n (pa,A)} \\\\a(\\texttt {remove X = v pa}) &= \\texttt {remove X = v (pa,A)} \\\\a(\\texttt {remove Pn n is P pa}) &= \\texttt {remove Pn n is ann(P) (pa,A)}$ We use I to represent either nothing, a path, an identifier stack, or a pair consisting of both a path and an identifier stack.",
"After application of these functions, the resulting annotated version is of the following, modified syntax (with expressions omitted as they match Section ).",
"We note that $a(\\texttt {runB P end})$ is $\\texttt {runB $ ann($P$ )$ end}$ , and $a(\\texttt {runC Cn P end})$ is $\\texttt {runC Cn $ ann($P$ )$ end A}$ .",
"Execution of the annotated version produces the executed annotated version, an identical copy but with populated identifier stacks.",
"$\\texttt {AP} &::= \\texttt {\\varepsilon } ~|~ \\texttt {AS} ~|~ \\texttt {AP; AP} ~|~ \\texttt {AP par AP} \\\\\\texttt {AS} &::= \\texttt {skip I} ~|~ \\texttt {X = E (pa,A)} ~|~ \\texttt {if In B then AP else AQ end (pa,A)} ~| \\\\ &\\phantom{\\texttt {::=}} \\texttt {while Wn B do AP end (pa,A)} ~|~ \\texttt {begin Bn ADV ADP AP ARP ARV end} ~| \\\\ &\\phantom{\\texttt {::=}}\\texttt {call Cn n (pa,A)} ~|~ \\texttt {runC Cn AP end A} ~|~ \\texttt {runB AP end}$ $\\texttt {ADV} &::= \\texttt {\\varepsilon } ~|~ \\texttt {var X = v (pa,A); ADV} &\\texttt {ADP} &::= \\texttt {\\varepsilon } ~|~ \\texttt {proc Pn n is AP (pa,A); ADP} \\\\\\texttt {ARV} &::= \\texttt {\\varepsilon } ~|~ \\texttt {remove X = v (pa,A); ARV} &\\texttt {ARP} &::= \\texttt {\\varepsilon } ~|~ \\texttt {remove Pn n is AP (pa,A); ARP}$ As all statements within an annotated version will be of this syntax, this must be reflected in our environments, specifically $\\mu $ and $\\beta $ that store programs.",
"As a result, we now use $\\square $ to represent the set of annotated environments, unless explicitly stated otherwise.",
"Figure: Renamed and annotated programExample 1.",
"We now consider the example implementation of a Fibonacci sequence using our programming language above, shown in Figure REF .",
"Note that version numbers are omitted due to the absence of while loops, and all paths are initially just the most direct block name.",
"Let P denote the procedure body shown in the declaration statement, namely lines 3-13 of Figure REF .",
"The procedure removal statement on line 16 then uses this.",
"Assume global variables F=3, S=4 and N=4.",
"This program calculates the Nth element of the Fibonacci sequence beginning with the first and second elements F and S. After execution, the Nth element will be the value held by S. Before we execute this program forwards, it must first be both annotated and renamed, shown in Figure REF .",
"All paths have now been updated to include all block identifiers necessary for execution, and all appropriate statements now have a stack A for storing identifiers.",
"Line 7 of Figure REF has become line 7 of Figure REF , which now has the path b2*b1, meaning this statement appears directly within b2, and indirectly within b1.",
"Prior to defining the operational semantics of this, we must first introduce three functions.",
"The first, $getAI()$ , returns the order and application of identifiers to a given program.",
"This allows us to extract the annotation information that would otherwise be lost when a while or procedure environment mapping is removed.",
"The second, $\\emph {refW()}$ , reflects a given annotation change to the copy of the program mapped to the given while identifier.",
"The third, $\\emph {refC()}$ , is identical, but will reflect a change made to a procedure body using a given call identifier.",
"Recall functions $reL()$ and $reP()$ from Section REF .",
"We are now ready to give the operational semantics.",
"The transition rules are those in Section , but with $\\hookrightarrow $ replaced with $\\rightarrow $ , and with all state-saving performed.",
"We introduce m-rules, the name given to each transition rule that assigns an identifier.",
"All other rules that do not use identifiers are now named non m-rules.",
"Transitions $\\stackrel{m}{\\rightarrow }$ are also called identifier transitions.",
"Sequential and Parallel Composition These are identical to Section , but with the annotated syntax.",
"$&\\text{[S1a]} \\quad \\frac{(\\texttt {AS} \\mid \\square ) \\stackrel{\\circ }{\\rightarrow } (\\texttt {AS^{\\prime }} \\mid \\square ^{\\prime })}{(\\texttt {AS; AP} \\mid \\square ) \\stackrel{\\circ }{\\rightarrow } (\\texttt {AS^{\\prime }; AP} \\mid \\square ^{\\prime })} & \\quad &\\text{[S2a]} \\quad \\frac{}{(\\texttt {skip I; AP} \\mid \\square ) \\rightarrow (\\texttt {AP} \\mid \\square )} \\\\[6pt]&\\text{[P1a]} \\quad \\frac{(\\texttt {AP} \\mid \\square ) \\stackrel{\\circ }{\\rightarrow } (\\texttt {AP^{\\prime }} \\mid \\square ^{\\prime })}{(\\texttt {AP par AQ} \\mid \\square ) \\stackrel{\\circ }{\\rightarrow } (\\texttt {AP^{\\prime } par AQ} \\mid \\square ^{\\prime })} &\\quad &\\text{[P2a]} \\quad \\frac{(\\texttt {AQ} \\mid \\square ) \\stackrel{\\circ }{\\rightarrow } (\\texttt {AQ^{\\prime }} \\mid \\square ^{\\prime })}{(\\texttt {AP par AQ} \\mid \\square ) \\stackrel{\\circ }{\\rightarrow } (\\texttt {AP par AQ^{\\prime }} \\mid \\square ^{\\prime })} \\\\[6pt]&\\text{[P3a]} \\quad \\frac{}{(\\texttt {AP par skip I} \\mid \\square ) \\rightarrow (\\texttt {AP} \\mid \\square )} &\\quad &\\text{[P4a]} \\quad \\frac{}{(\\texttt {skip I par AQ} \\mid \\square ) \\rightarrow (\\texttt {AQ} \\mid \\square )}$ Assignment This is an m-rule, saving the old value and the next available identifier m, retrieved via the function next(), onto this variables stack on $\\delta $ .",
"$&\\text{[D1a]} \\quad \\frac{(\\texttt {e pa} \\mid \\delta ,\\sigma ,\\gamma ,\\square ) \\hookrightarrow ^*_{\\texttt {a}} (\\texttt {v} \\mid \\delta ,\\sigma ,\\gamma ,\\square ) \\quad \\texttt {m = next()} \\quad \\texttt {evalV(\\gamma ,pa,X) = l}}{(\\texttt {X = e (pa,A)} \\mid \\delta ,\\sigma ,\\gamma ,\\square ) \\stackrel{m}{\\rightarrow } (\\texttt {skip m:A} \\mid \\delta [\\texttt {(m,\\sigma (l)) \\rightharpoonup X}],\\sigma [\\texttt {l} \\mapsto \\texttt {v}],\\gamma ,\\square )}$ Conditional All rules follow as in Section , but with annotated programs, and [I4a] and [I5a] both being m-rules.",
"These save the next available identifier m (via next()) and a boolean value indicating which branch was executed (after execution of the branch) onto stack B on $\\delta $ .",
"$&\\text{[I1a]} \\quad \\frac{(\\texttt {b pa} \\mid \\square ) \\hookrightarrow ^*_{\\texttt {b}} (\\texttt {V} \\mid \\square )}{(\\texttt {if In b then AP else AQ end (pa,A)} \\mid \\square ) \\rightarrow (\\texttt {if In V then AP else AQ end (pa,A)} \\mid \\square )} \\\\[6pt]&\\text{[I2a]} \\quad \\frac{(\\texttt {AP} \\mid \\square ) \\stackrel{\\circ }{\\rightarrow } (\\texttt {AP^{\\prime }} \\mid \\square ^{\\prime })}{(\\texttt {if In T then AP else AQ end (pa,A)} \\mid \\square ) \\stackrel{\\circ }{\\rightarrow } (\\texttt {if In T then AP^{\\prime } else AQ end (pa,A)} \\mid \\square ^{\\prime })} \\\\[6pt]&\\text{[I3a]} \\quad \\frac{(\\texttt {AQ} \\mid \\square ) \\stackrel{\\circ }{\\rightarrow } (\\texttt {AQ^{\\prime }} \\mid \\square ^{\\prime })}{(\\texttt {if In F then AP else AQ end (pa,A)} \\mid \\square ) \\stackrel{\\circ }{\\rightarrow } (\\texttt {if In F then AP else AQ^{\\prime } end (pa,A)} \\mid \\square ^{\\prime })} \\\\[6pt]&\\text{[I4a]} \\quad \\frac{\\texttt {m = next()}}{(\\texttt {if In T then skip I else AQ end (pa,A)} \\mid \\delta ,\\square ) \\stackrel{m}{\\rightarrow } (\\texttt {skip m:A} \\mid \\delta [\\texttt {(m,T) \\rightharpoonup B}],\\square )} \\\\[6pt]&\\text{[I5a]} \\quad \\frac{\\texttt {m = next()}}{(\\texttt {if In F then AP else skip I end (pa,A)} \\mid \\delta ,\\square ) \\stackrel{m}{\\rightarrow } (\\texttt {skip m:A} \\mid \\delta [\\texttt {(m,F) \\rightharpoonup B}],\\square )}$ While Loop The first two rules are m-rules, saving the next available identifier m (via next()) and an element of the boolean sequence onto stack W on $\\delta $ .",
"[W1a] handles the first iteration of a loop, creating a mapping on $\\beta $ as in Section (but with annotated programs), and saving the first element F of the boolean sequence (see [10]).",
"[W2a] handles any other iteration, updating the current mapping as before and saving the next element T of the boolean sequence (see [10]).",
"Both rules rename the loop body.",
"The body executes via [W3a], now reflecting all annotation changes of AR$^{\\prime }$ into the stored copy, written using $\\beta ^{\\prime }$ [refW(Wn,AR$^{\\prime }$)].",
"Finally, a loop either continues via [W4a], or finishes via the m-rule [W5a].",
"This final rule stores the next available identifier m and all annotation information ($getAI()$ ) onto stack WI, before removing the mapping, written $\\beta $ [Wn].",
"$&\\text{[W1a]} \\quad \\frac{\\texttt {m = next()} \\quad \\texttt {\\beta (Wn) = }\\emph {und} \\quad (\\texttt {b pa} \\mid \\beta ,\\square ) \\hookrightarrow ^*_{\\texttt {b}} (\\texttt {V} \\mid \\beta ,\\square ) }{(\\texttt {S} \\mid \\delta ,\\beta ,\\square ) \\stackrel{m}{\\rightarrow } (\\texttt {while Wn V do reL(AP) end (pa,m:A)} \\mid \\delta [\\texttt {(m,F) \\rightharpoonup W}],\\beta [\\texttt {Wn} \\Rightarrow \\texttt {AR}],\\square )} \\\\[5pt] & \\phantom{\\text{[W1a]} \\quad } \\text{where } \\texttt {S} = \\texttt {while Wn b do AP end (pa,A)} \\text{ and } \\texttt {AR} = \\texttt {while Wn b do reL(AP) end (pa,m:A)} \\\\[6pt]&\\text{[W2a]} \\quad \\frac{\\texttt {m = next()} \\quad \\texttt {\\beta (Wn) = while Wn b do AQ end (pa,A)} \\quad (\\texttt {b pa} \\mid \\beta ,\\square ) \\hookrightarrow ^*_{\\texttt {b}} (\\texttt {V} \\mid \\beta ,\\square ) }{(\\texttt {S} \\mid \\delta ,\\beta ,\\square ) \\stackrel{m}{\\rightarrow } (\\texttt {while Wn V do reL(AQ) end (pa,m:A)} \\mid \\delta [\\texttt {(m,T) \\rightharpoonup W}],\\beta [\\texttt {Wn} \\Rightarrow \\texttt {AR^{\\prime }}],\\square )} \\\\[5pt] & \\phantom{\\text{[W2a]} \\quad } \\text{where } \\texttt {S} = \\texttt {while Wn b do AP end (pa,A)} \\text{ and } \\texttt {AR^{\\prime }} = \\texttt {while Wn b do reL(AQ) end (pa,m:A)} \\\\[6pt]&\\text{[W3a]} \\quad \\frac{(\\texttt {AR} \\mid \\beta ,\\square ) \\stackrel{\\circ }{\\rightarrow } (\\texttt {AR^{\\prime }} \\mid \\beta ^{\\prime },\\square ^{\\prime })}{(\\texttt {while Wn T do AR end (pa,A)} \\mid \\beta ,\\square ) \\stackrel{\\circ }{\\rightarrow } (\\texttt {while Wn T do AR^{\\prime } end (pa,A)} \\mid \\beta ^{\\prime \\prime },\\square ^{\\prime })} \\\\[4pt] & \\phantom{\\text{[W3a]} \\quad } \\text{where } \\texttt {\\beta ^{\\prime \\prime }} = \\texttt {\\beta ^{\\prime }}[\\emph {refW(}\\texttt {Wn,AR^{\\prime }}\\emph {)}] \\\\[6pt]&\\text{[W4a]} \\quad \\frac{\\texttt {\\beta (Wn) = AP} }{(\\texttt {while Wn T do skip I end (pa,A)} \\mid \\beta ,\\square ) \\rightarrow (\\texttt {AP} \\mid \\beta ,\\square )} \\\\[6pt]&\\text{[W5a]} \\quad \\frac{\\texttt {m = next()} \\quad \\texttt {\\beta (Wn) = AR}}{(\\texttt {while Wn F do AP end (pa,A)} \\mid \\delta ,\\beta ,\\square ) \\stackrel{m}{\\rightarrow } (\\texttt {skip m:A} \\mid \\delta [\\texttt {(m,getAI(\\beta (Wn))) \\rightharpoonup WI}],\\beta [\\texttt {Wn}],\\square )}$ Block These are identical to before, but using the annotated syntax.",
"$&\\text{[B1a]} \\quad \\frac{}{(\\texttt {begin Bn AP end} \\mid \\square ) \\rightarrow (\\texttt {runB AP end} \\mid \\square )} \\quad \\text{where } \\texttt {AP } \\text{=} \\texttt { ADV;ADP;AQ;ARP;ARV} \\\\[6pt]&\\text{[B2a]} \\quad \\frac{(\\texttt {AP} \\mid \\square ) \\stackrel{\\circ }{\\rightarrow } (\\texttt {AP^{\\prime }} \\mid \\square ^{\\prime })}{(\\texttt {runB AP end} \\mid \\square ) \\stackrel{\\circ }{\\rightarrow } (\\texttt {runB AP^{\\prime } end} \\mid \\square ^{\\prime })} \\quad \\text{[B3a]} \\quad \\frac{}{(\\texttt {runB skip I end} \\mid \\square ) \\rightarrow (\\texttt {skip} \\mid \\square )}$ Variable and Procedure Declaration Variable declarations [L1a] are as before, but are now m-rules without state-saving.",
"Procedure declarations [L2a] are also as before, but is also an m-rule without state-saving, and all programs are now annotated.",
"$&\\text{[L1a]} \\quad \\frac{\\texttt {m = next()} \\quad \\texttt {nextLoc() = l} \\quad \\texttt {pa} = \\texttt {Bn*pa^{\\prime }} }{(\\texttt {var X = v (pa,A)} \\mid \\sigma ,\\gamma ,\\square ) \\stackrel{m}{\\rightarrow } (\\texttt {skip m:A} \\mid \\sigma [\\texttt {l} \\mapsto \\texttt {v}],\\gamma [(\\texttt {X},\\texttt {Bn}) \\Rightarrow \\texttt {l}],\\square )} \\\\[6pt]&\\text{[L2a]} \\quad \\frac{\\texttt {m = next()}}{(\\texttt {proc Pn n is AP (pa,A)} \\mid \\mu ,\\square ) \\stackrel{m}{\\rightarrow } (\\texttt {skip m:A} \\mid \\mu [\\texttt {Pn} \\Rightarrow \\texttt {(n,AP)}],\\square )}$ Procedure Call [G1a] inserts a renamed copy of the basis entry onto $\\mu $ , exactly as in Section , but with an annotated program.",
"[G2a] uses the runC construct to execute this renamed copy, but now reflects any annotation changes to the stored copy, written $\\mu ^{\\prime }$ [refC(Cn,AP$^{\\prime }$)].",
"[G3a] is now an m-rule, saving all annotation changes from the copy to the stack Pr, alongside the next available identifier m (via next()).",
"The renamed copy is removed from $\\mu $ , written as $\\mu $ [Cn].",
"$&\\text{[G1a]} \\quad \\frac{\\texttt {evalP(n,pa) = Pn} \\quad \\texttt {\\mu (Pn) = (n,AP)} \\quad \\texttt {reP(AP,Cn) = AP^{\\prime }}}{(\\texttt {call Cn n (pa,A)} \\mid \\mu ,\\square ) \\rightarrow (\\texttt {runC Cn AP^{\\prime } end A} \\mid \\mu [\\texttt {Cn} \\Rightarrow \\texttt {(n,AP^{\\prime })}],\\square )} \\\\[6pt]&\\text{[G2a]} \\quad \\frac{(\\texttt {AP} \\mid \\mu ,\\square ) \\stackrel{\\circ }{\\rightarrow } (\\texttt {AP^{\\prime }} \\mid \\mu ^{\\prime },\\square ^{\\prime })}{(\\texttt {runC Cn AP end A} \\mid \\mu ,\\square ) \\stackrel{\\circ }{\\rightarrow } (\\texttt {runC Cn AP^{\\prime } end A} \\mid \\mu ^{\\prime }[\\emph {refC(}\\texttt {Cn},\\texttt {AP^{\\prime }}\\emph {)}],\\square ^{\\prime })} \\\\[6pt]&\\text{[G3a]} \\quad \\frac{\\texttt {m = next()} \\quad \\texttt {\\mu (Cn) = AP}}{(\\texttt {runC Cn skip I end A} \\mid \\delta ,\\mu ,\\square ) \\stackrel{m}{\\rightarrow } (\\texttt {skip m:A} \\mid \\delta [\\texttt {(m,getAI(\\mu (Cn))) \\rightharpoonup Pr}],\\mu [\\texttt {Cn}],\\square )}$ Variable and Procedure Removal Variable removal [H1a] is now an m-rule, saving the final value of that variable and the next available identifier m (via next()) onto that variables stack on $\\delta $ .",
"The mapping is removed as in Section .",
"Procedure removal [H2a] is as before, but an m-rule without state-saving.",
"$&\\text{[H1a]} \\quad \\frac{\\texttt {m = next()} \\quad \\texttt {pa} = \\texttt {Bn*pa^{\\prime }} \\quad \\texttt {\\gamma (X,Bn) = l}}{(\\texttt {remove X = v (pa,A)} \\mid \\delta ,\\sigma ,\\gamma ,\\square ) \\stackrel{m}{\\rightarrow } (\\texttt {skip m:A} \\mid \\delta [\\texttt {(m,\\sigma (l)) \\rightharpoonup X}],\\sigma [\\texttt {l} \\mapsto \\texttt {0}],\\gamma [(\\texttt {X},\\texttt {Bn})],\\square )} \\\\[6pt]&\\text{[H2a]} \\quad \\frac{\\texttt {m = next()} \\quad \\texttt {\\mu (Pn) = AQ}}{(\\texttt {remove Pn n is AP (pa,A)} \\mid \\mu ,\\square ) \\stackrel{m}{\\rightarrow } (\\texttt {skip m:A} \\mid \\mu [\\texttt {Pn}],\\square )}$"
],
[
"Results",
"We first define equivalence between traditional and annotated environments.",
"Definition 4.1 Let $\\sigma $ be a data store, $\\sigma _1$ be an annotated data store, $\\gamma $ be a variable environment and $\\gamma _1$ be an annotated variable environment.",
"We have $(\\sigma ,\\gamma )$ is equivalent to $(\\sigma _1,\\gamma _1)$ , written $(\\sigma ,\\gamma ) \\approx _S (\\sigma _1,\\gamma _1)$ , if and only if dom($\\gamma $) = dom($\\gamma _1$) and $\\sigma (\\gamma (\\textup {\\texttt {X}},\\textup {\\texttt {Bn}}))$ = $\\sigma _1(\\gamma _1(\\textup {\\texttt {X}},\\textup {\\texttt {Bn}}))$ for all X $\\in $ dom($\\gamma $) and block names Bn.",
"Definition 4.2 Let $\\mu $ be a procedure environment, and $\\mu _1$ be an annotated procedure environment.",
"We have that $\\mu $ is equivalent to $\\mu _1$ , written $\\mu \\approx _P \\mu _1$ , if and only if dom($\\mu $) = dom($\\mu _1$), $\\mu (\\textup {\\texttt {Pn}})$ = (n,P), $\\mu _1(\\textup {\\texttt {Pn}})$ = (n,AP) and $removeAnn($AP$)$ = P for all Pn $\\in $ dom($\\mu $).",
"(Note Pn could be Cn here).",
"Definition 4.3 Let $\\beta $ be a while environment, and $\\beta _1$ be an annotated while environment.",
"We have that $\\beta $ is equivalent to $\\beta _1$ , written $\\beta \\approx _W \\beta _1$ , if and only if dom($\\beta $) = dom($\\beta _1$), $\\beta (\\textup {\\texttt {Wn}})$ = P, $\\beta _1(\\textup {\\texttt {Wn}})$ = AP and $removeAnn($AP$)$ = P for all Wn $\\in $ dom($\\beta $).",
"Definition 4.4 Let $\\delta $ be an auxiliary store, and $\\delta _1$ be an annotated auxiliary store.",
"Firstly, we define the equivalence of stacks St and St$^{\\prime }$, written as St $\\approx _{ST}$ St$^{\\prime }$, as true if both stacks have matching elements.",
"We have that $\\delta $ is equivalent to $\\delta _1$ , written $\\delta \\approx _A \\delta _1$ , if and only if for each stack St $\\in $ $dom(\\delta )$, we have $\\delta ($St$) \\approx _{ST} \\delta _1(\\texttt {ST})$ .",
"Definition 4.5 Let $\\square $ represent the set of environments {$\\sigma $ ,$\\gamma $ ,$\\mu $ ,$\\beta $ } and $\\square _1$ represent the set of annotated environments {$\\sigma _1$ ,$\\gamma _1$ ,$\\mu _1$ ,$\\beta _1$ }.",
"We have that $\\square $ is equivalent to $\\square _1$ , written $\\square \\approx \\square _1$ , if and only if $(\\sigma ,\\gamma )~\\approx _S~(\\sigma _1,\\gamma _1)$ , $\\mu \\approx _P \\mu _1$ and $\\beta \\approx _W \\beta _1$ .",
"We now present our results.",
"Theorem REF states that identifiers are used in ascending order.",
"Theorem REF below states if an original program terminates, the annotated version will also, and annotation does not change the behaviour of the program w.r.t the stores $\\square $ , but does produce a populated auxiliary store $\\delta ^{\\prime }$ .",
"Theorem 1 Let AP and AQ be annotated programs, $\\square $ be the set of environments and $\\delta $ be an auxiliary store.",
"If $ (\\textup {\\texttt {AP} $$ $$, $$}) \\stackrel{\\circ }{\\rightarrow }^* (\\textup {\\texttt {AP$ '$} $$ $ '$, $ '$}) \\stackrel{\\text{n}}{\\rightarrow } (\\textup {\\texttt {AQ} $$ $ ”$, $ ”$}) \\rightarrow ^* (\\textup {\\texttt {AQ$ '$} $$ $ ”'$, $ ”'$}) \\stackrel{\\text{m}}{\\rightarrow } (\\textup {\\texttt {AQ$ ”$} $$ $ ””$, $ ””$}) $ , and the computation $ (\\textup {\\texttt {AQ} $$ $ ”$, $ ”$}) \\rightarrow ^* (\\textup {\\texttt {AQ$ '$} $$ $ ”'$, $ ”'$}) $ does not have any identifier transitions, then m = n + 1.",
"The order of identifiers used during execution is maintained using next().",
"The program AP will begin with any number of steps.",
"At some point, a transition occurs that will use the next available identifier n, while simultaneously incrementing next() by one to m. Any number of non m-rules can then apply, before the next m-action uses next() to get m. Hence, m = n + 1.",
"Lemma 1 Let P be a program, $\\square $ be the set {$\\sigma $ ,$\\gamma $ ,$\\mu $ ,$\\beta $ } of all environments, $\\square _1$ be the set {$\\sigma _1$ ,$\\gamma _1$ ,$\\mu _1$ ,$\\beta _1$ } of annotated environments such that $\\square \\approx \\square _1$ and $\\delta $ be the auxiliary store.",
"If $ (\\textup {\\texttt {P} $$ $$,$$}) \\hookrightarrow (\\textup {\\texttt {P$ '$} $$ $ '$,$$}) $ , for some $\\square ^{\\prime }$ , then there exists an execution $ (\\textup {\\texttt {$ ann($P$ )$} $$ $ 1$,$$}) \\stackrel{\\circ }{\\rightarrow } (\\textup {\\texttt {P$ ”$} $$ $ 1'$,$ '$}) $ , for some $\\square _1^{\\prime }$ , $\\delta ^{\\prime }$ and P$^{\\prime \\prime }$ = $ann($P$^{\\prime })$ such that $\\square ^{\\prime }$ $\\approx $ $\\square _1^{\\prime }$ .",
"Theorem 2 Let P be an original program, $\\square $ be the set {$\\sigma $ ,$\\gamma $ ,$\\mu $ ,$\\beta $ } of all environments, $\\square _1$ be the set {$\\sigma _1$ ,$\\gamma _1$ ,$\\mu _1$ ,$\\beta _1$ } of annotated environments such that $\\square \\approx \\square _1$ and $\\delta $ be the auxiliary store.",
"If $ (\\textup {\\texttt {P} $$ $$,$$}) \\hookrightarrow ^* (\\textup {\\texttt {skip} $$ $ '$,$$}) $ , for some $\\square ^{\\prime }$ , then there exists an execution $ (\\textup {\\texttt {$ ann($P$ )$} $$ $ 1$,$$}) \\stackrel{\\circ }{\\rightarrow }^* (\\textup {\\texttt {skip I} $$ $ 1'$,$ '$}) $ , for some I, $\\square _1^{\\prime }$ and $\\delta ^{\\prime }$ , such that $\\square ^{\\prime }$ $\\approx $ $\\square _1^{\\prime }$ .",
"The proof is by induction on the length of the sequence $(\\texttt {P} \\mid \\square ,\\delta ) \\hookrightarrow ^* (\\texttt {skip} \\mid \\square ^{\\prime },\\delta ^{\\prime })$ .",
"We consider P being either a sequential or parallel composition of programs.",
"Both cases hold using Lemma REF .",
"We note the implication holds in the opposite direction, but defer proof to future work."
],
[
"Reverse Semantics of Inverted Programs",
"Inversion is the process of generating the inverted version of a given program, produced from the executed annotated version, as the populated identifier stacks are necessary.",
"As the inverted version is a program that executes forwards, inversion inverts the overall statement order.",
"The functions that performs this, namely $inv()$ and the supplementary $i()$ , are defined below.",
"$ inv(\\texttt {$ $}) = \\texttt {$$} \\quad inv(\\texttt {AS;AP}) = inv(\\texttt {AP}); i(\\texttt {AS}) \\quad inv(\\texttt {AP par AQ}) = inv(\\texttt {AP}) \\texttt { par } inv(\\texttt {AQ}) $$ $ $i(\\texttt {skip I}) &= \\texttt {skip I} \\\\i(\\texttt {X = e (pa,A)}) &= \\texttt {X = e (pa,A)} \\\\i(\\texttt {if In b then AP else AQ end (pa,A)}) &= \\texttt {if In b then inv(AP) else inv(AQ) end (pa,A)} \\\\i(\\texttt {while Wn b do AP end (pa,A)}) &= \\texttt {while Wn b do inv(AP) end (pa,A)} \\\\i(\\texttt {begin Bn ADV ADP AP ARP ARV end}) &= \\texttt {begin Bn inv(ARV) inv(ARP) inv(AP) inv(ADP)} \\\\ &\\phantom{=== } \\texttt {inv(ADV) end} \\\\i(\\texttt {var X = v (pa,A)}) &= \\texttt {remove X = v (pa,A)} \\\\i(\\texttt {proc Pn n is AP (pa,A)}) &= \\texttt {remove Pn n is inv(AP) (pa,A)} \\\\i(\\texttt {call Cn n (pa,A)}) &= \\texttt {call Cn n (pa,A)} \\\\i(\\texttt {remove X = v (pa,A)}) &= \\texttt {var X = v (pa,A)} \\\\i(\\texttt {remove Pn n is AP (pa,A)}) &= \\texttt {proc Pn n is inv(AP) (pa,A)}$ All inverted programs are of the annotated syntax in Section , but with IP and IS used for inverted programs and statements respectively.",
"The inverse of runB and runC constructs simply invert the body.",
"Starting with the final state of all environments from the forwards execution, the inverted program will no longer perform any expression evaluation, offering potential time saving when compared to traditional cyclic debugging.",
"The result of any expression evaluation that happened during forwards execution is retrieved from the appropriate stack on $\\delta $ .",
"For example, a while loop will iterate until the top element of stack W on $\\delta $ is no longer true.",
"The non-determinism that possibly occurred during the forwards execution will also not feature in the inverted execution.",
"The identifiers assigned to statements ensure that any statement can only execute provided it has the highest unseen identifier.",
"Using the function previous(), and starting it with the final value of next(), all m-rules will be reversed in backtracking order.",
"There is freedom in the order of non m-rules, with examples being parallel skip operations, or parallel block closings.",
"Reversing these in any order produces no adverse effects.",
"We recognise that all environments will now store inverted programs wherever necessary.",
"Since an inverted program is of the same syntax as an annotated program, our current environments are sufficient.",
"We now return to our example discussed in Section .",
"Figure: Inverted versionExample 2.",
"We now execute the annotated version of our original program (Figure REF ), producing the final annotated version shown in Figure REF .",
"The initial value of next() is 1.",
"We enter block b1, and perform the procedure declaration, assigning the identifier 1 to its stack.",
"The call statement then happens, performing a renamed copy of the procedure body.",
"This will execute lines 3-9 using identifiers 2-6, before hitting the recursive call.",
"This call will then execute another renamed copy of the procedure body.",
"This renamed copy for the second call is shown in Figure REF , where the unique call name c1:c2 has been used to rename all constructs.",
"Lines 1-7 are executed using identifiers 7-11, before again hitting a recursive call.",
"This renamed version is then executed, with the conditional on line 5 evaluating to false, meaning the recursion is now finished.",
"This version uses the identifiers 12-14, before the recursive calls begin to close.",
"The second call (Figure REF ) then concludes, executing lines 8-11 using identifiers 15-17.",
"The first call then concludes using the identifiers 18-20.",
"Finally, the original program concludes, using identifier 21 to finish the call statement, and then the last identifier 22 (meaning next() = 23) to remove the procedure declaration.",
"This concludes our execution, producing the final state F=7, S=11 and N=2.",
"We now consider the inverted execution.",
"Application of the function $inv()$ to the executed annotated version (Figure REF ) produces the inverted version shown in Figure REF .",
"The initial value of previous() will be 22 (next() - 1).",
"Inverse execution begins by opening the outer block and performing the procedure declaration (the inverse of the procedure removal) using identifier 22.",
"The call statement is then entered using identifier 21.",
"A renamed copy of the procedure body is then executed, performing lines 3-6 using identifiers 20-18.",
"The recursive call is then hit, beginning the execution of another renamed copy, shown in Figure REF .",
"Lines 1-4 are executed using identifiers 17-15, before the recursive call is hit again.",
"The third renamed version now executes, and since the condition will be false (retrieved from the stack B on $\\delta $ ), recursion now stops.",
"This call concludes using identifiers 14-12, before the second call (Figure REF ) now concludes lines 5-11 using identifiers 11-7.",
"The first call then concludes using identifiers 6-2.",
"Finally, the original program concludes with the removal of the procedure (inverse of declaration) using identifier 1.",
"This execution order restores the initial program state.",
"Figure: Inverted version of 2nd callPrior to defining the inverse operational semantics, we must introduce the function setAI().",
"This takes the output of the function getAI() from Section and a program, and returns a copy of this program with the given annotation information inserted.",
"Recall the functions IreP and IreL from Section REF .",
"We now give the operational semantics of inverted programs.",
"We introduce reverse m-rules, the name given to all statements of the inverse execution that use an identifier.",
"Transitions $\\stackrel{m}{\\rightsquigarrow }$ are also called reverse identifier transitions.",
"All other rules remain non m-rules.",
"We note a correspondence between each m-rule and the matching reverse m-rule.",
"Sequential and Parallel Composition The inverted program is still executed forwards, meaning these are like those in Section , but with $\\rightsquigarrow $ replacing $\\rightarrow $ , and IP and IS replacing AP and AS respectively.",
"Each rule is named correspondingly to Section , but with the appended `a' replaced with `r'.",
"For example, rule [S1a] is now [S1r].",
"Assignment The inverse of an assignment will be a reverse m-rule, allowed to execute provided the top element of stack A is m (written A = m:A$^{\\prime }$), and m is the last used identifier (via previous()).",
"This retrieves the old value, with matching identifier m, from the stack on $\\delta $ for this variable, and assigns it to the corresponding location (evaluated as in Section ).",
"$&\\text{[D1r]} \\quad \\frac{\\texttt {A = m:A^{\\prime }} \\quad \\texttt {m = previous()} \\quad \\texttt {evalVar(\\gamma ,pa,X) = l} \\quad \\texttt {\\delta (X) = (m,v):X^{\\prime }}}{(\\texttt {X = e (pa,A)} \\mid \\delta ,\\sigma ,\\square ) \\stackrel{m}{\\rightsquigarrow } (\\texttt {skip A^{\\prime }} \\mid \\delta [\\texttt {X/X^{\\prime }}],\\sigma [\\texttt {l} \\mapsto \\texttt {v}],\\square )}$ Conditional This will begin with the reverse m-rule [I1r] that, provided this statement has the next identifier to invert (via previous() and A = m:A$^{\\prime }$) retrieves the boolean value from the stack B with matching identifier m on $\\delta $ .",
"Rules [I2r] and [I3r] follow Section , but with inverted programs.",
"Finally, [I4r] and [I5r] simply concludes the statement.",
"$&\\text{[I1r]} \\quad \\frac{\\texttt {A = m:A^{\\prime }} \\quad \\texttt {m = previous()} \\quad \\texttt {\\delta (B) = (m,V):B^{\\prime }}}{(\\texttt {S} \\mid \\delta ,\\square ) \\stackrel{m}{\\rightsquigarrow } (\\texttt {if In V then IP else IQ end (pa,A^{\\prime })} \\mid \\delta [\\texttt {B/B^{\\prime }}],\\square )} \\\\[4pt] & \\phantom{\\text{[I1r]} \\quad } \\text{where } \\texttt {S} = \\texttt {if In b then IP else IQ end (pa,A)} \\\\[6pt]&\\text{[I2r]} \\quad \\frac{(\\texttt {IP} \\mid \\square ) \\stackrel{\\circ }{\\rightsquigarrow } (\\texttt {IP^{\\prime }} \\mid \\square ^{\\prime })}{(\\texttt {if In T then IP else IQ end (pa,A)} \\mid \\square ) \\stackrel{\\circ }{\\rightsquigarrow } (\\texttt {if In T then IP^{\\prime } else IQ end (pa,A)} \\mid \\square ^{\\prime })} \\\\[6pt]&\\text{[I3r]} \\quad \\frac{(\\texttt {IQ} \\mid \\square ) \\stackrel{\\circ }{\\rightsquigarrow } (\\texttt {IQ^{\\prime }} \\mid \\square ^{\\prime })}{(\\texttt {if In F then IP else IQ end (pa,A)},\\square ) \\stackrel{\\circ }{\\rightsquigarrow } (\\texttt {if In F then IP else IQ^{\\prime } end (pa,A)} \\mid \\square ^{\\prime })} \\\\[6pt]&\\text{[I4r]} \\quad \\frac{}{(\\texttt {if In T then skip I else IQ end (pa,A)} \\mid \\square ) \\rightsquigarrow (\\texttt {skip} \\mid \\square )} \\\\[6pt]&\\text{[I5r]} \\quad \\frac{}{(\\texttt {if In F then IP else skip I end (pa,A)} \\mid \\square ) \\rightsquigarrow (\\texttt {skip} \\mid \\square )}$ While Loop The reverse m-rule [W1r] handles the first iteration of a loop, retrieving either the T or F from the stack B on $\\delta $ , and creating a mapping on $\\beta $ .",
"This mapping is similar to that of [W1a] but with an inverted copy of the loop IP$^{\\prime }$ that is both renamed and updated with annotated information C, retrieved from the stack WI (via $setAI()$ ).",
"The reverse m-rule [W2r] handles all iterations except the first, meaning the renaming is applied to the current mapping, written $\\beta $ [Wn $\\Rightarrow $ IR$^{\\prime }$].",
"Both rules only execute provided they have the last used identifier (via previous() and A = m:A$^{\\prime }$).",
"The body is then executed repeatedly with changes reflected to the stored copy, written $\\beta ^{\\prime }$ [Wn $\\Rightarrow $ IR$^{\\prime }$], via rule [W3r].",
"The loop continues through the rule [W4r], until the condition is false and thus the mapping removed by rule [W5r].",
"$&\\text{[W1r]} \\quad \\frac{\\texttt {m = previous()} \\quad \\texttt {A = m:A^{\\prime }} \\quad \\texttt {\\beta (Wn) = }\\emph {und} \\quad \\texttt {\\delta (WI)=(m,C):WI^{\\prime }} \\quad \\texttt {IP^{\\prime } = IreL(setAI(IP,C))}}{(\\texttt {S} \\mid \\delta ,\\beta ,\\square ) \\stackrel{m}{\\rightsquigarrow } (\\texttt {while Wn b do IP^{\\prime } end (pa,A^{\\prime })} \\mid \\delta [\\texttt {WI/WI^{\\prime }}],\\beta [\\texttt {Wn} \\Rightarrow \\texttt {IR}],\\square )} \\\\[4pt] & \\phantom{\\text{[W1r]} \\quad } \\text{where } \\texttt {S} = \\texttt {while Wn b do IP end (pa,A)} \\text{and } \\texttt {IR} = \\texttt {while Wn b do IP^{\\prime } end (pa,A^{\\prime })} \\\\[6pt]&\\text{[W2r]} \\quad \\frac{\\texttt {m = previous()} \\quad \\texttt {A = m:A^{\\prime }} \\quad \\texttt {\\beta (Wn) = while Wn b do IQ end (pa,A)} \\quad \\texttt {\\delta (W) = (m,V):W^{\\prime }}}{(\\texttt {S} \\mid \\delta ,\\beta ,\\square ) \\stackrel{m}{\\rightsquigarrow } (\\texttt {while Wn V do IreL(IQ) end (pa,A^{\\prime })} \\mid \\delta [\\texttt {W/W^{\\prime }}],\\beta [\\texttt {Wn} \\Rightarrow \\texttt {IR^{\\prime }}],\\square )} \\\\[4pt] & \\phantom{\\text{[W2r]} \\quad } \\text{where } \\texttt {S} = \\texttt {while Wn b do IP end (pa,A)} \\text{ and } \\texttt {IR^{\\prime }} = \\texttt {while Wn b do IreL(IQ) end (pa,A^{\\prime })} \\\\[6pt]&\\text{[W3r]} \\quad \\frac{(\\texttt {IR} \\mid \\delta ,\\beta ,\\square ) \\stackrel{\\circ }{\\rightsquigarrow } (\\texttt {IR^{\\prime }} \\mid \\delta ^{\\prime },\\beta ^{\\prime },\\square ^{\\prime })}{(\\texttt {while Wn T do IR end (pa,A)} \\mid \\delta ,\\beta ,\\square ) \\stackrel{\\circ }{\\rightsquigarrow } (\\texttt {while Wn T do IR^{\\prime } end (pa,A)} \\mid \\delta ^{\\prime },\\beta ^{\\prime \\prime },\\square ^{\\prime })} \\\\[4pt] & \\phantom{\\text{[W3r]} \\quad } \\text{where } \\texttt {\\beta ^{\\prime \\prime }} = \\texttt {\\beta ^{\\prime }}[\\emph {refW(}\\texttt {Wn,IR^{\\prime }}\\emph {)}] \\\\[6pt]&\\text{[W4r]} \\quad \\frac{\\texttt {\\beta (Wn) = IP} }{(\\texttt {while Wn T do skip I end (pa,A)} \\mid \\beta ,\\square ) \\rightsquigarrow (\\texttt {IP} \\mid \\beta ,\\square )} \\\\[6pt]&\\text{[W5r]} \\quad \\frac{\\texttt {\\beta (Wn) = IR}}{(\\texttt {while Wn F do IP end (pa,A)} \\mid \\beta ,\\square ) \\rightsquigarrow (\\texttt {skip} \\mid \\beta [\\texttt {Wn}],\\square )}$ Block The inversion of a block is very similar to that of Section , but with the inverted syntax.",
"Recall that a variable declaration in the inverted syntax is the inverse of a variable removal, and similarly for procedures.",
"This means an inverted block has a body of the form shown below as IP in [B1r].",
"$&\\text{[B1r]} \\quad \\frac{}{(\\texttt {begin Bn IP end} \\mid \\square ) \\rightsquigarrow (\\texttt {runB IP end} \\mid \\square )} \\quad \\text{where } \\texttt {IP } \\text{=} \\texttt { IDV;IDP;IQ;IRP;IRV} \\\\[6pt]&\\text{[B2r]} \\quad \\frac{(\\texttt {IP} \\mid \\square ) \\stackrel{\\circ }{\\rightsquigarrow } (\\texttt {IP^{\\prime }} \\mid \\square ^{\\prime })}{(\\texttt {runB IP end} \\mid \\square ) \\stackrel{\\circ }{\\rightsquigarrow } (\\texttt {runB IP^{\\prime } end} \\mid \\square ^{\\prime })} \\quad \\text{[B3r]} \\quad \\frac{}{(\\texttt {runB skip I end} \\mid \\square ) \\rightsquigarrow (\\texttt {skip} \\mid \\square )}$ Variable and Procedure Declaration Reverse variable declaration [L1r] is a reverse m-rule, allowed to execute provided it has the last used identifier (via previous() and A = m:A$^{\\prime }$).",
"The given value is ignored, and the variable is instead set to its final value retrieved from the corresponding stack on $\\delta $ (written $\\delta ($ X$)$ = (m,v$^{\\prime }$ ):X$^{\\prime }$).",
"The location l is used as in Section .",
"A procedure declaration [L2r] creates the basis mapping as in Section , but with inverted programs, provided its identifiers allow this.",
"$&\\text{[L1r]} \\quad \\frac{\\texttt {A = m:A^{\\prime }} \\quad \\texttt {m = previous()} \\quad \\texttt {\\delta (X) = (m,v^{\\prime }):X^{\\prime }} \\quad \\texttt {nextLoc() = l} \\quad \\texttt {pa} = \\texttt {Bn*pa^{\\prime }} }{(\\texttt {var X = v (pa,A)} \\mid \\delta ,\\sigma ,\\gamma ,\\square ) \\stackrel{m}{\\rightsquigarrow } (\\texttt {skip A^{\\prime }} \\mid \\delta [\\texttt {X/X^{\\prime }}],\\sigma [\\texttt {l} \\mapsto \\texttt {v^{\\prime }}],\\gamma [(\\texttt {X},\\texttt {Bn}) \\Rightarrow \\texttt {l}],\\square )} \\\\[6pt]&\\text{[L2r]} \\quad \\frac{\\texttt {A = m:A^{\\prime }} \\quad \\texttt {m = previous()}}{(\\texttt {proc Pn n is IP (pa,A)} \\mid \\mu ,\\square ) \\stackrel{m}{\\rightsquigarrow } (\\texttt {skip A^{\\prime }} \\mid \\mu [\\texttt {Pn} \\Rightarrow \\texttt {(n,IP)}],\\square )}$ Procedure Call The reverse m-rule [G1r] creates a renamed copy IP$^{\\prime }$ of the basis procedure body IP, that has annotated changes C from stack WI inserted.",
"This is inserted into $\\mu $ via $\\mu $ [Cn $\\Rightarrow $ (n,IP$^{\\prime }$ )].",
"Rule [G2r] follows Section but uses inverted programs, while [G3r] removes the mapping.",
"$&\\text{[G1r]} \\quad \\frac{\\texttt {m = previous()} \\quad \\texttt {A = m:A^{\\prime }} \\quad \\texttt {\\mu (evalP(n,pa)) = (n,IP)} \\quad \\texttt {\\delta (Pr) = (m,C):Pr^{\\prime }} }{(\\texttt {call Cn n (pa,A)} \\mid \\mu ,\\square ) \\stackrel{m}{\\rightsquigarrow } (\\texttt {runC Cn IP^{\\prime } end A^{\\prime }} \\mid \\mu [\\texttt {Cn} \\Rightarrow \\texttt {(n,IP^{\\prime })}],\\square )} \\\\&\\phantom{[RG1] \\quad } \\text{where } \\texttt {IP^{\\prime }} = \\texttt {IreP(setAI(IP,C),Cn)} \\\\[6pt]&\\text{[G2r]} \\quad \\frac{(\\texttt {IP} \\mid \\mu ,\\square ) \\stackrel{\\circ }{\\rightsquigarrow } (\\texttt {IP^{\\prime }} \\mid \\mu ^{\\prime },\\square ^{\\prime })}{(\\texttt {runC Cn IP end A} \\mid \\mu ,\\square ) \\stackrel{\\circ }{\\rightsquigarrow } (\\texttt {runC Cn IP^{\\prime } end A} \\mid \\mu ^{\\prime }[\\emph {refC(}\\texttt {Cn},\\texttt {IP^{\\prime }}\\emph {)}],\\square ^{\\prime })} \\\\[6pt]&\\text{[G3r]} \\quad \\frac{\\texttt {\\mu (Cn) = IP}}{(\\texttt {runC Cn skip I end A} \\mid \\mu ,\\square ) \\rightsquigarrow (\\texttt {skip A} \\mid \\mu [\\texttt {Cn}],\\square )}$ Variable and Procedure Removal Variable removal [H1r] is similar to Section , as no state-saving is required.",
"But it is a reverse m-rule and can execute provided A = m:A$^{\\prime }$ and m = previous().",
"Procedure removal [H2r] is a reverse m-rule with no state-saving, removing the mapping.",
"$&\\text{[H1r]} \\quad \\frac{\\texttt {A = m:A^{\\prime }} \\quad \\texttt {m = previous()} \\quad \\texttt {pa} = \\texttt {Bn*pa^{\\prime }} \\quad \\texttt {\\gamma (X,Bn) = l}}{(\\texttt {remove X = v (pa,A)} \\mid \\delta ,\\sigma ,\\gamma ,\\square ) \\stackrel{m}{\\rightsquigarrow } (\\texttt {skip A^{\\prime }} \\mid \\delta ,\\sigma [\\texttt {l} \\mapsto \\texttt {0}],\\gamma [(\\texttt {X},\\texttt {Bn})],\\square )} \\\\[2pt]&\\text{[H2r]} \\quad \\frac{\\texttt {A = m:A^{\\prime }} \\quad \\texttt {m = previous()} \\quad \\texttt {\\mu (Pn) = IQ}}{(\\texttt {remove Pn n is IP (pa,A)} \\mid \\mu ,\\square ) \\stackrel{m}{\\rightsquigarrow } (\\texttt {skip A^{\\prime }} \\mid \\mu [\\texttt {Pn}],\\square )}$"
],
[
"Results",
"Prior to describing our inversion results, we first note that the definitions in Section REF are now used to relate annotated environments with inverted environments, instead of traditional.",
"As an example, Definition REF would no longer have that P = $removeAnn($AP$)$ , but instead that IP = $inv($P$)$ .",
"Theorem REF states that identifiers are used in descending order throughout a reverse execution (opposite of Theorem REF ).",
"Theorem REF shows that if an original sequential program and its annotated execution terminate, then the reverse execution will also, and that the reverse execution beginning in the final state can restore the initial state.",
"Theorem 3 Let P and Q be original programs, AP and AQ be the annotated versions producing the executed versions AP$^{\\prime }$ and AQ$^{\\prime }$ respectively, and IP and IQ be the inverted versions $inv($ AP$^{\\prime })$ and $inv($ AQ$^{\\prime })$ respectively.",
"Further let $\\square $ be the set of all environments and $\\delta $ be the auxiliary store.",
"If $ (\\textup {\\texttt {IP} $$ $$, $$}) \\stackrel{\\circ }{\\rightsquigarrow }^* (\\textup {\\texttt {IP$ '$} $$ $ '$, $ '$}) \\stackrel{\\textup {\\text{n}}}{\\rightsquigarrow } (\\textup {\\texttt {IQ} $$ $ ”$, $ ”$}) \\rightsquigarrow ^* (\\textup {\\texttt {IQ$ '$} $$ $ ”'$, $ ”'$}) \\stackrel{\\textup {\\text{m}}}{\\rightsquigarrow } (\\textup {\\texttt {IQ$ ”$} $$ $ ””$, $ ””$}) $ , and provided that the computation $ (\\textup {\\texttt {IQ} $$ $ ”$, $ ”$}) \\rightsquigarrow ^* (\\textup {\\texttt {IQ$ '$} $$ $ ”'$, $ ”'$}) $ does not include any identifier transitions, then m = n - 1.",
"The order of identifiers is maintained using the function previous().",
"This proof follows closely to the argument within the proof of Theorem REF , but uses $\\stackrel{\\circ }{\\rightsquigarrow }$ in place of $\\stackrel{\\circ }{\\rightarrow }$ .",
"Theorem 4 Let P be a sequential program (does not contain par) and AP be $ann($ P$)$.",
"Further let $\\square $ be the set {$\\sigma $ ,$\\gamma $ ,$\\mu $ ,$\\beta $ } of all environments, $\\square _1$ be the set {$\\sigma _1$ ,$\\gamma _1$ ,$\\mu _1$ ,$\\beta _1$ } of annotated environments such that $\\square \\approx \\square _1$ , $\\square _1^{\\prime }$ be the set {$\\sigma _1^{\\prime }$ ,$\\gamma _1^{\\prime }$ ,$\\mu _1^{\\prime }$ ,$\\beta _1^{\\prime }$ } of final annotated environments, $\\square _2$ be the set {$\\sigma _2$ ,$\\gamma _2$ ,$\\mu _2$ ,$\\beta _2$ } of inverted environments such that $\\square _2 \\approx \\square _1^{\\prime }$ , $\\delta $ be the auxiliary store, $\\delta ^{\\prime }$ be the final auxiliary store and $\\delta _2$ be the inverted auxiliary store such that $\\delta _2 \\approx _A \\delta ^{\\prime }$ .",
"If $ (\\textup {\\texttt {P} $$ $$,$$}) \\hookrightarrow ^* (\\textup {\\texttt {skip} $$ $ '$,$$}) $ , for some $\\square ^{\\prime }$ , and there exists an annotated execution $ (\\textup {\\texttt {AP} $$ $ 1$,$$}) \\stackrel{\\circ }{\\rightarrow }^* (\\textup {\\texttt {skip I} $$ $ 1'$,$ '$}) $ , for some I, $\\square _1^{\\prime }$ and $\\delta ^{\\prime }$ such that the executed annotated version of AP produced by its execution is AP$^{\\prime }$, then there also exists $ (\\textup {\\texttt {IP} $$ $ 2$,$ 2$}) \\stackrel{\\circ }{\\rightsquigarrow }^* (\\textup {\\texttt {skip I$ '$} $$ $ 2'$,$ 2'$}) $ , for IP = $inv($ AP$^{\\prime })$ and some I$^{\\prime }$, $\\square _2^{\\prime }$ and $\\delta _2^{\\prime }$ .",
"(Termination) If $ (\\textup {\\texttt {P} $$ $$,$$}) \\hookrightarrow ^* (\\textup {\\texttt {skip} $$ $ '$,$$}) $ , for some $\\square ^{\\prime }$ , and there exists an annotated execution $ (\\textup {\\texttt {AP} $$ $ 1$,$$}) \\stackrel{\\circ }{\\rightarrow }^* (\\textup {\\texttt {skip I} $$ $ 1'$,$ '$}) $ , for some I, $\\square _1^{\\prime }$ and $\\delta ^{\\prime }$ , such that $\\square ^{\\prime } \\approx \\square _1^{\\prime }$ and that the executed annotated version of AP produced by its execution is AP$^{\\prime }$, then there also exists $ (\\textup {\\texttt {IP} $$ $ 2$,$ 2$}) \\stackrel{\\circ }{\\rightsquigarrow }^* (\\textup {\\texttt {skip I$ '$} $$ $ 2'$,$ 2'$}) $ , for IP = $inv($ AP$^{\\prime })$ and some I$^{\\prime }$, $\\square _2^{\\prime }$ and $\\delta _2^{\\prime }$ , such that $\\square _2^{\\prime } \\approx \\square $ and $\\delta _2^{\\prime } \\approx _A \\delta $ .",
"This proof is by induction on the length of the sequence $(\\texttt {P} \\mid \\square ,\\delta ) \\hookrightarrow ^* (\\texttt {skip} \\mid \\square ^{\\prime },\\delta ^{\\prime })$ .",
"The full version of Theorem REF , where programs can contain parallel composition, is currently being considered.",
"We note the implication in the other direction would be valid, but defer proof to future work."
],
[
"Conclusion",
"We have presented an approach to reversing a language containing blocks, local variables, procedures and the interleaving parallel composition.",
"We defined annotation, the process of creating a state-saving annotated version capable of assigning identifiers to capture the interleaving order.",
"This was proved to not alter the behaviour of the original program and to populate the auxiliary store.",
"Inversion creates an inverted version that uses this saved information to restore the initial program state, with this being proved to hold.",
"The auxiliary store is also restored, meaning it is garbage free.",
"We are also currently developing a simulator capable of implementing this approach, with one application being to aid the proof of our results.",
"The current prototype is capable of simulating the annotated forwards execution, with the simulation of the inverted execution currently being worked on.",
"This will be used to evaluate the performance overhead and costs associated with our approach to reversibility.",
"Our future work will continue the development of this simulator, as well as modify the approach allowing for causal-consistent reversibility."
],
[
"Acknowledgements",
"We are grateful to the referees for their detailed and helpful comments and suggestions.",
"The authors acknowledge partial support of COST Action IC1405 on Reversible Computation - extending horizons of computing.",
"The third author is supported by JSPS KAKENHI grant numbers 17H01722 and 17K19969."
]
] | 1808.08651 |
[
[
"A Taxonomy on Big Data: Survey"
],
[
"Abstract The Big Data is the most popular paradigm nowadays and it has almost no untouched area.",
"For instance, science, engineering, economics, business, social science, and government.",
"The Big Data are used to boost up the organization performance using massive amount of dataset.",
"The Data are assets of the organization, and these data gives revenue to the organizations.",
"Therefore, the Big Data is spawning everywhere to enhance the organizations' revenue.",
"Thus, many new technologies emerging based on Big Data.",
"In this paper, we present the taxonomy of Big Data.",
"Besides, we present in-depth insight on the Big Data paradigm."
],
[
"Introduction",
"BigData Big Data growing rapidly day-by-day due to producing a huge data from various sources.",
"The new emerging technologies act as catalyst in growing data where the growth of data get an exponential pace.",
"For instance, IoT.",
"Moreover, smart devices engender enormous data.",
"The smart devices are core component of smart cities (includes smart healthcare, smart home, smart irrigation, smart schools and smart grid), smart agriculture, smart pollution control, smart car and smart transportation [1], [2].",
"Data are generated not only by IoT devices, but also sciences, business, economics and government.",
"The science generates humongous dataset and these data are handled by Big Data.",
"For example, Large Hadron Collider in Geneva.",
"Moreover, the web of things also a big factor in engendering the huge size of data.",
"In addition, the particle analysis requires a huge data to be analyzed.",
"Moreover, the seismology also generates large dataset, and thus, the Big Data tools are deployed to analyze and predict.",
"Interestingly, the Big Data tools are deployed in diverse fields to handle very large scale dataset.",
"There are hundreds of application field of the Big Data which makes the Big Data paradigm glorious in this high competitive era.",
"The paper present following key point- Provides rich taxonomy of Big Data.",
"Presents Big Data in every aspect precisely.",
"Highlights on each technology.",
"The Big Data is categorized into seven key categories, namely, semantic, compute infrastructure, storage system, Big Data Management, Big Data Mining, Big Machine Learning, and Security & Privacy as shown in the figure REF .",
"The paper discusses on semantic on the Big Data and explore $V_3^{11}+C$ [3] in the section .",
"The paper also discusses on compute infrastructure and classifies the Big Data into three categories, namely, MapReduce, Bulk Synchronous Parallel, and Streaming in the section .",
"Besides, the Big Data storage system is classified into four categories, namely, storage architecture, storage implementation, storage structure and storage devices in section .",
"Figure: Taxonomy of Big Data Technology"
],
[
"Semantic of Big Data",
"There are many V's coming up to define the characteristics of Big Data.",
"Doug Laney defines Big Data using 3V's, namely, volume, velocity and variety.",
"Now, the $V_3^{11}+C$ is used to define the characteristics of Big Data [3].",
"The different kind of V's are shown in the figure REF .",
"The volatility and visibility is not the family of V [3].",
"The table REF defines the meaning of Vs precisely.",
"Table: Individual meaning the V family"
],
[
"MapReduce",
"The MapReduce [4] programming paradigm is the best parallel programming concept even if the MapReduce is purely based on the only Map and Reduce task.",
"However, the DryadLINQ [5] is also emerging based on Microsoft Dryad [6].",
"But, MapReduce can solve almost every problem of distributed and parallel computing, and large-scale data-intensive computing.",
"It has been widely accepted and demonstrated that the MapReduce is the most powerful programming for large scale cluster computing.",
"The conventional DBMS is designed to work with structured data and it can scale with scaling of expensive hardware, but not low-cost commodity hardware.",
"The MapReduce programming works on low-cost unreliable commodity hardware, and it is an extremely scalable RAIN cluster, fault tolerant yet easy to administer, and highly parallel yet abstracted [7].",
"The MapReduce is key/value pair computing wherein the input and output are in the form of key and value.",
"Map.",
"The Map function transforms the input key/value pair to intermediate key-value pair [4].",
"For instance, the key is the file name and value is its content.",
"The output of Map function is transferred to reducer function by shuffling and the sorting.",
"Sorting is done by some internal sorting techniques (Timsort for internal sorting or quick sort).",
"The input is fetched from the file system, namely, HDFS [8], and GFS [9].",
"The input size varies from 64 MB to 1GB and the performance can vary in a specific range, which is evaluated in the paper [10].",
"Reduce.",
"The Reduce function consists of three different phases, namely, copy phase, shuffle phase and reduce phase.",
"The copy phase fetches the output from the Map function [11].",
"The Map function spills the output when it reaches a specific size (for example, 10MB).",
"This spilling cause early starting of Reduce task, otherwise the reduce task has to wait until the Map task has not finished [12].",
"The shuffle phase sort the data using an external sorting algorithm [11].",
"The shuffle phase and copy phase takes a longer time to execute as compared other phases.",
"The reduce phase computes as the user defines and produces final output [11].",
"The final output is written to the file system.",
"The limitation of MapReduce is very limited and these are listed below: The computation depends on the previously computed value is not suitable for MapReduce.",
"If the computation cannot be converted into the Map and Reduce form.",
"MapReduce is not suitable for small-scale computing, such like RDBMS jobs.",
"No high-level language [13]: The MapReduce does not support SQL.",
"No schema and no index [13]: The MapReduce does not support schema and index, since, MapReduce is a programming language.",
"A Single fixed data flow [13]: The MapReduce strictly follows key/value computation style, and Map and Reduce function.",
"Withal, the SQL can be converted to MapReduce programming and can easily be made using these two roles (map and reduce).",
"Almost all distributed system's problems can be solved by MapReduce, albeit, folk can claim that the conversion of their task to the Map and Reduce functions is difficult and performing low.",
"Antithetically, the paper [14] reveals MapReduce as the query processing engine.",
"Moreover, the MapReduce can be used for indexing, schema support, structured, unstructured, and semi-structured information.",
"MapReduce is also used to process Big Graph [15].",
"It depends on the programming way and fine tune of the programmer.",
"Hence, the MapReduce is the most powerful engine to work out any sort of distributed system.",
"The MapReduce consists of Map and Reduce task spawned in several machines for maximizing the parallelism.",
"The tasks are split into several workers and assigned them to process.",
"Suppose, one of the workers become straggler [16], [17], then the entire process becomes Achilles' Heels.",
"That is, one task can complete a job $J$ in 10 minutes, then 10 tasks should complete the job $J$ in one Minutes.",
"Unfortunately, one task is taking 10 minutes, then the time to complete the job is 10 minutes, even though the task are running in parallel and works are divided equally likely.",
"Almost every problem is solvable in MapReduce, but, the solution may not be suitable or may not perform well.",
"A deliberate design and optimization are required for MapReduce to do comfortably.",
"For instance, thousands of Map with one Reduce task is obviously slower than many reduce tasks with thousands of Map tasks.",
"Some investigation on the unsuitability of conventional MapReduce is given below: The Reducer Task takes more time to finish.",
"If the reducer task becomes slow or straggler, then there is no guarantee that the job will be completed same time period with straggler and without straggler.",
"The copy of a straggler task can be scheduled in another suitable node which increases network traffic.",
"Making the decision of whether reduce task has to be rescheduled or not is a challenge itself.",
"For instance, reduce task is about to complete and straggling.",
"In this situation, the scheduler must examine whether scheduling a copy of that straggler task is profitable or not.",
"The SkewTune [18] schedules a partial copy of straggler task, where the partial copy is the remaining task to be finished.",
"The MapReduce does redundant computing in some situation [19].",
"For example, MapReduce has completed a job, word count problem.",
"Now, there is one or few word changes in the entire file system.",
"The MapReduce recompute the entire word count job and it does not reuse the previous result of a computation of the same job.",
"This problem has been addressed by ReStore [20], and Early Accurate Result Library (EARL) [21].",
"The MapReduce shows certain disadvantage in reading input several times redundantly, while the same job is run several times on the same data.",
"A careful implementation of MapReduce has to be done, while the data are coming continuously and writing to the file system.",
"In this continuous online data degrades the performance of MapReduce.",
"The sharing among multiple jobs can improve the performances [19].",
"This challenge is addressed in MRShare [22].",
"Giraph [23] is an open source system which is used for graph processing on big data.",
"It uses MapReduce implementation for graph processing.",
"In general, it follows a master/workers model.",
"Also it support multithreading by assigning each worker multiple graph partitioning.",
"During each superstep, an available worker pairs compute threads with uncomputable partitions.",
"And between supersteps, workers perform serial tasks (e.g.",
"blocking on global barriers, resolving mutations) by executing with a single thread.",
"Moreover Giraph implement BSP by maintaining two message stores for holding messages from previous and current supersteps respectively.",
"It reduces memory usage and computation time by using receiver-side message combining.",
"Also for global coordination or counters blocking, aggregators are used.",
"And master.compute() is used for serial computations at the master."
],
[
"Pregel",
"Pregel [24] is a system that provides a graph processing API along with BSP [25] with a vertex-centric, programming model.",
"Its programs are inspired by Valiant’s Bulk Synchronous Parallel model [26].",
"Directed graph is input to a Pregel computation where each vertex have a string vertex identifier for unique identification.",
"A typical Pregel computation have input (graph is initialized), then it have sequence of supersteps separated by global synchronization points, finally it give the output and then algorithm terminates.",
"In each superstep the vertices compute in parallel while executing the same user-defined function expressing the logic of the given algorithm.",
"However, algorithm terminates when every vertex vote to halt.",
"Pregel also have aggregators.",
"Aggregator is a mechanism for global data, monitoring, and communication.",
"And it can be used for global coordination To improve usability and performance Pregel keeps vertices and edges on the machine doing computation.",
"And it uses network only for messages.",
"Also Pregel programs are deadlock free.",
"Moreover algorithms developed by Pregel can be used to solve real problems such as Shortest Paths, Page Rank, Bipartite Matching, Semi-Clustering algorithm and so on."
],
[
"BSP ML",
"Bulk Synchronous Parallel ML language (BSML) [27] is a library for parallel programming along with functional language Objective Caml.",
"It is based on an extension of the λ-calculus by parallel operations on parallel vector.",
"Parallel vector is a parallel data structure.",
"Moreover, the BSML library provide a safe environment for declarative parallel programming.",
"And programs are similar to functional programs (in Objective Caml) but using few additional functions.",
"Furthermore, BSML have provided functions.",
"These functions are used for accessing the parameters of the parallel machine for creating and performing operations on parallel vectors.",
"BSML is based on a confluent extension of the λ-calculus, making it deterministic and deadlock free.",
"In BSML, programs can be easily composed, written and reused.",
"Also it has simpler semantics and better readability.",
"And it is implemented using Objective Caml in a modular approach.This approach helps to communicate with various communication libraries, makes it portable, and efficient, on a wide range of architectures."
],
[
"Hama",
"Hama [28] is a pure bulk synchronous parallel model which can do vast scientific computations, e.g.",
"matrix, graph, and network algorithms.",
"It’s internal architecture is different from other known computational frameworks because of its underlying BSP based communication and synchronization mechanisms.",
"Again it is based on Master-Slave model consisting of three major components, BSP master, Groom Server, and Zookeeper.",
"Some functions of BSP master are scheduling jobs, task assignment to a Groom Server, maintainance of the Groom Server status and job progress information.",
"Groom Server acts as a slave and it executes tasks assigned by the BSP Master.",
"And Zookeeper gives efficient barrier synchronization to the BSP tasks.",
"The robust BSP model helps in avoiding deadlines and conflicts during communication in Hama.",
"Also it is flexible so it can be used with any distributed file system.",
"However BSP Master is a single point of failure and the application will stop if it dies.",
"Additionaly the graph partitioning algorithm have to be customized, to avoid communication overhead between nodes."
],
[
"BSPLib",
"BSPLib [29] is a small communication library for BSP programming.",
"It is done in a Single Program Multiple Data (SPMD) manner.",
"It consist of only 20 basic operations.",
"It has two modes of communication, direct remote memory access (DRMA) and bulk synchronous message passing (BSMP) approach.",
"BSPLib provides the infrastructure required for the user for data distribution, and communication required for changing parts of the data structure present in a remote process.",
"Additionally, it provide a higher-level libraries or programming tools which is architecture independent and automatically distribute the problem domain among the processes."
],
[
"Infosphere",
"InfoSphere [30] is a component-based distributed stream processing platform.",
"The stream processing applications can be graphs of modular, reusable software components interconnected by data streams.",
"Likewise, Component-based programming model allows composition and reconfiguration of individual components to create different applications.",
"These applications can perform different types of analysis or answer different types of queries.",
"Also it helps in creation and deployment of new applications without disturbing existing ones.",
"It is used in sense-and-respond application domains.",
"And it provides both language and runtime to these applications to improve efficiency in processing data from high rate streams."
],
[
"Spark",
"Spark [31] is a new cluster computing framework.",
"It supports applications with working sets while providing scalability and fault tolerance properties to MapReduce.",
"Spark provides three data abstractions, resilient distributed datasets (RDDs), and two restricted types of shared variables: broadcast variables and accumulators.",
"RDD represents a read-only collection of objects.",
"These objects are partitioned across a set of systems and they can be rebuilt if a partition is lost.",
"Suppose a large read-only piece of data (e.g., a lookup table) is used in multiple parallel operations, so it should be distributed to the workers only once.",
"Similarly, Broadcast variable wraps the value and copy it once to each worker.",
"Likewise, Accumulators are variables that workers use an associative operation to “add”, and these variables can only be read by driver."
],
[
"Storm",
"Storm [32] software is a framework for building, processing applications that use the computing resources of all systems in a cluster.",
"Based on varying processing needs of such applications, the platform should automatically grow and shrink as per requirement.",
"Storm efficiently processes unbounded streams of data.",
"It give the ability to users, to transform an existing stream into a new stream using two primitives,a spout and a bolt.",
"A spout is a source of streams.",
"Usually, user have to provide code that reads data from source (such as a queue, a database or a website).",
"This data is then given to one or more bolts for processing.",
"LIkewise, a bolt takes input streams, does processing, and then gives new streams of data.",
"In addition, Storm cluster have two types of nodes, the master node and the processing nodes.",
"The master node runs a daemon called Nimbus.",
"Numbus is responsible for distribution of code around the cluster, task assignment, and monitoring their progression and failures.",
"Whereas, Processing nodes run a daemon called Supervisor.",
"Supervisor listens for work assignment to it's system.",
"It starts and stops processes based on the work assigned by the master node."
],
[
"Storage Architecture",
"The storage system architecture is broadly categorized into three categories, namely, Direct-Attached Storage (DAS), Network Attached Storage (NAS), and Storage Area Network (SAN) [33].",
"The three architectures have its own pros and cons, shown in table REF .",
"Table: Storage Architecture comparison of featuresDAS is digital storage, which attaches storage directly to the computer that accessing it [33], [34].",
"These storages are from USB drive, and by Bus, i.e., every server has its own storage space directly attached to it without using network accessing."
],
[
"Network Attached Storage (NAS)",
"The storage is attached through an Ethernet switch to scale the storage system.",
"The NAS uses TCP/IP protocol to access the storage [33], [34].",
"The application server is detached from the file system and data storage.",
"The advantages of detaching application server, and file system & storage system is incremental scalability.",
"It is really easy to design a disaster recovery system using NAS.",
"The performance is the major issue in the NAS."
],
[
"Storage Area Network (SAN)",
"The storage devices are attached with fiber channel and storage are networked together [33], [34].",
"Thus SAN strides the speed accessing of storage devices.",
"Storage is connected through fiber switch so that the accessing the data become faster.",
"The performance is the major advantage and scalability is the major issue in the SAN.",
"The SAN supports fast accessing of data through a fibre channel.",
"The SAN outperform NAS and DAS in performance, but NAS outperform SAN and DAS in scalability."
],
[
"Storage System Implementation",
"A billion dollar question is how do we store the Exabytes of data?",
"How do we process them efficiently?",
"The answer is partially given by Apache Hadoop [8] and GFS [9].",
"Another good example is Google Spanner [35] and Microsoft Dryad [6].",
"The assumed environment must not be Infiniband with a high-end server.",
"No doubt, the production version is deployed in the high-end server configuration, but our assumption is a low-end server.",
"To span the probable failure, the solution must assume, how to deploy thousands of low-cost commodity hardware.",
"The low-cost commodity hardware is more vulnerable to failure, and therefore, the replication technique is used to overcome the failure rate and achieves maximum parallel processing and reliability of the system.",
"Figure: The Hadoop StackFigure: Architecture of Block, File and Object store"
],
[
"File System for Big Data",
"The most popular file system like GFS [9] and HDFS [8] need to enhance their scalability, fine-tune their performance in bigger dimension.",
"Conventionally, the file system and block storage are different, but both are combined to form a modern distributed file system as shown in figure REF .",
"The file system holds some properties of a conventional file system and blocks storage system.",
"The data are stored in the form of a file within a specified range of file size and split into block otherwise.",
"Moreover, the files are kept same as originals if the file size falls within a specified range.",
"A file size varies from MB to TBs, that involves some low sized files are mixed in concert to restrict from generating more number of Metadata and split high sized file to maximize parallelism.",
"File system stores the data in a hard disk or SSD devices and store data information in RAM, known as Metadata [36], to enhance the access time.",
"A dedicated Metadata server (MDS) serves client queries for data.",
"The MDS become a bottleneck when smaller sized file.",
"The billions of small sized file create no more storage space, but produce the huge size of Metadata, results performance degradation [10], [37], [38], [39].",
"The paper [40] address the problem of small files.",
"On the other hand, the standalone MDS cannot serve billions of client request and that is why most of the modern file system uses distributed MDS (dMDS) for better scalability.",
"The most modern dMDS are Dr. Hadooop [41], CalvinFS [42], DROP [43], IndexFS and ShardFS [44], [45], and CephFS [46].",
"The issues of designing dMDS are Small file problem [40], scalability [41], consistency, latency [47], [48], [49], Hot-standby [37], partition tolerance, network traffic, hotspot problem, and disaster recovery and management [38].",
"The table REF shows the some important modern file system with respect to scalability, disaster recovery and management, hot-standby, single point of failure (SPoF), and types of metadata server (MDS).",
"Table: Modern File System comparison, * denotes limited, ×\\times denotes no and denotes yes"
],
[
"Object Storage for Big Data",
"Object storage is a basic storage unit for applications which stores data as objects and as a logical collection of bytes on a storage device along with the methods for accessing and describing the characteristics of data.",
"The object holds data and metadata [56].",
"The metadata is used to store the information about the content and context of the data.",
"To access data, traditionally, the methods for input and output are specified explicitly in the application or use other external ways, and then the object storage system map these files into objects.",
"The object is accessed using the globally unique identifier.",
"We need to sacrifice the hierarchical layout of file and directory as a conventional file system has, since, there is no silver bullet.",
"Object storage is designed to manage the heavy bit of unstructured files that need to be laid in.",
"As well, it is best for archiving the information when it is not frequently accessed."
],
[
"Block Storage System",
"The block storage system depends on the storage area network, and thus scaling is an issue this storage system.",
"The block storage uses FCoE or iSCSI protocol to access the stored data and it is stored in the block of storage media.",
"The block does not contain any information about the data, it contains only the raw data."
],
[
"Cloud Storage",
"NIST defines Cloud computing as a model for enabling ubiquitous, convenient, on-demand network access to a shared pool of configurable computing resources that can be rapidly provisioned and released with minimal management effort or service provider interaction [57].",
"In the other word, Cloud computing is Internet computing that shares resources seamlessly.",
"The cloud storage is a virtualized storage space where the actual data is stored in several servers.",
"The cloud storage space can use object storage, file system, and hybrid storage system.",
"The cloud storage provides high scalability, availability, security, fault tolerance, and cost-effective data services for those applications [58], [59].",
"In that respect are three layers of Cloud Storage Architecture: The user application layer is the interface between the user and the virtual storage media.",
"The Storage management layer is virtualization of the storage space.",
"The virtualization manages the data, and create an illusion of simplicity and single storage space, while the storage space is rather complicated and span on several servers or geographical area.",
"The Storage resources layer deals with the actual data to store.",
"Ordinarily, this layer uses file system or object storage system.",
"The most advanced system uses a hybrid storage system which combines both storage systems."
],
[
"Storage System Structure- NoSQL",
"NoSQL [60], is Not Only SQL, emerging due to its urgent necessity in the industries.",
"The NoSQL is the bigger dimension of SQL and non-SQL which is implemented for distributed or parallel computing.",
"The alternative to RDBMS is NoSQL which provides high availability, scalability, fault-tolerance, and Reliability [61], [62].",
"However, the NoSQL database does not perform well in OLTP due to ACID property requirement in OLTP process [63].",
"The comparison of the NoSQL architecture is presented in the table REF ."
],
[
"Key-value store",
"The key-value store is the most elementary sort of storage where replication, versioning, and distributed locking are supported.",
"The key-value store is very much useful in MapReduce environment."
],
[
"Column-Oriented Store",
"The main issue in RDBMS is one column is exceeding RAM size, then the processing performance becomes very poor.",
"Moreover, if it reaches to petabytes, then the conventional system does not work at all.",
"To overcome this limitation, Google implements BigTable, which is scalable, flexible, reliable and fault-tolerance.",
"In the current marketplace, there are a large amount of columnar database available, namely, HBase, HyperTable, Cassandra, Flink, eXtremeDB, and HPCC.",
"This columnar family can ameliorate the processing speed in unstructured data as well as structured data."
],
[
"Document-oriented store",
"The document store is similar to key-value store, except, the document store relies on the inner structure of the document to extract their metadata.",
"XML database, for instance.",
"The document store is a semi-structured database which provides fault-tolerance, and scalability in large-scale computation."
],
[
"Graph Database",
"Graph database[64] is more appropriate to deal with complex, densely connected, and semi-structured data.",
"Graph databases are extremely helpful in the industries, for example, online business solution, healthcare, online media, financial, social network, communication, retail, etc.",
"Graph database gives the response of complex queries in a few milliseconds because it stores the data in RAM.",
"The shared-memory Big Graph processing engine is centralized whereas the other one is decentralized [65].",
"The decentralized graph processing engine is easy to scale.",
"The issues and challenges of Big Graph are high-degree vertexes, sparseness, unstructured problems, in-memory challenge, poor locality, communication overhead, and load balancing [15].",
"Table: Category comparison.",
"Source It is the time to do research on latency and it is the hot potato in the research field.",
"The RAMCloud [66], Memcached, and Spark [31] deal with the latency issues.",
"The performance of a system depends on latency and the latency is the great impact factor of a performance of a particular system.",
"The latency of RAM is lower than the SSD and HDD.",
"The most prominent field of research is latency to reduce, and eventually, SSD has turned up.",
"Even though, the SSD cannot be as fast as RAM, but still reducing the latency.",
"The research challenge of HPC requires the highest performance with the lowest cost in $.",
"In-memory system requires SSD or HDD support for durability, otherwise, system shutdown causes data lost and it is not tolerated at any cost.",
"The race is now among Cache memory, RAM and SSD/PCIe PCM.",
"The main component of performance is RAM, but unfortunately, cost of RAM does not decrease satisfactory as in SSD or HDD as shown in the figure REF .",
"As the figure REF shown, the RAM cost is falling very slowly, and its size is increasing exponentially.",
"On the other hand, The HDD and SSD cost is really low and continuously falling.",
"Furthermore, the size of SSD and HDD is slowly rising.",
"Figure: RAM, SSD, and HDD cost per MB.",
"Source , , , respectively"
],
[
"Cache Memory",
"No doubt, cache memory is the fastest memory devices.",
"The challenges of designing algorithms for distributed system are to increase the cache hit ratio.",
"There are numerous researcher working on cache-aware algorithm, namely, scheduling, MDS designing."
],
[
"Primary Memory",
"Designing in-memory Big Data system is an open challenge.",
"There are many such systems has been unleashed, such like, RamCloud, H-Store, Big Graph Analytic software."
],
[
"SSD",
"The Solid-State Device (SSD) is the most advanced storage device for Big Data technology.",
"The SSD is the hybrid version of RAM and secondary memory.",
"The designing of the system must utilize recent the technology, such that the outcome should gain highest possible benefit from them.",
"The design decision must ensure that the new recent technology should not degrade the performance of the system."
],
[
"HDD",
"The most common form of storage device is Hard Disk (HDD) now-a-days.",
"The HDD has a latency problem due to finding the track and sector."
],
[
"Tape",
"The tape is the most inexpensive and bulk storage device, but the read/write performance is really inadequate.",
"This type of storage device has significant advantage in implementing a storage system for backup purpose."
],
[
"Data Acquisition",
"Data acquisition [71] is the process of collecting, filtering and removing any noise from data before they can be stored in any data warehouse or any storage system.",
"It adopts adaptive and time efficient algorithms for processing of high value data.",
"For data acquisition, frameworks are available that are based on predefined protocols.",
"However, many organizations that depend on big data processing have developed their own enterprise-specific protocols.",
"Most commonly used open protocol is Advanced Message Queuing Protocol (AMQP).",
"It satisfies a series of requirements compiled by 23 companies.",
"And it became an OASIS standard in October 2012.",
"It has characteristics such as ubiquity, safety, fidelity, applicability, interoperability, and manageability.",
"In addition, lots of software tools are also available for data acquisition (e.g.",
"Storm, S4)."
],
[
"Data Preprocessing",
"Data preprocessing [72] is the set of techniques used before the application of data processing techniques.",
"It removes data redundancies and inconsistencies, and make it suitable for application of data processing algorithms.",
"Some data preprocessing approaches are Dimensionality reduction and Instance reduction.",
"Dimensionality of data refers to the instances of the data.",
"And Dimensionality reduction include Feature selection and Space transformations.",
"Whereas, Instance reduction refers to reduction of size of data set.",
"It include Instance selection and Instance generation.",
"Additionaly, In Big Data, MLlib [73] is used for data preprocessing in Spark.",
"MLlib is a powerful machine learning library that helps in use of Spark in data analysis."
],
[
"Shared-nothing Architecture",
"The shared-nothing architecture does not share its resources to others.",
"The resources are HDD, SSD, and RAM.",
"The significant advantage of shared-nothing is fine-grained fault tolerance, scalability, and maximize the parallelism.",
"The table REF shows the technology that uses Shared-nothing architecture and shared-everything architecture."
],
[
"Shared-everything Architecture",
"Sharing always cause synchronization, whatever the implementation is.",
"Shared Memory- Even though the fastest inter-process communication using shared memory, but there arises some synchronization issues.",
"Careful design can lead to the best performance of the resources.",
"Shared Disc- The sharing storage devices implemented very widely.",
"Shared Both- The hybrid system always exists."
],
[
"Data Analytics",
"The Big Data Analytics is an logical analysis on large set of data for specific purposes.",
"It requires Data Mining algorithms which is classified into four key categories, namely, Machine Learning, Statistic, Artificial Intelligence, and data warehouse.",
"The Big Data analytics requires Statistics and Machine Learning.",
"The Machine Learning (ML) algorithms are applied to analyze and predict on very large dataset.",
"Moreover, the ML algorithms take huge time in execution.",
"The ML algorithms require new technologies to execute a mammoth sized data.",
"The Big Data tools efficiently handle ML algorithms in very large scale data-intensive computation.",
"Figure: Taxonomy of Big Data Analytics"
],
[
"Data Visualization",
"Spatial Layout visualization techniques refers to formulas that maps an input data uniquely to a specific point on the coordinate space.",
"Popular visualization techniques under it can be classified into chart and plots, and trees and graphs.",
"Example of some techniques coming under chart and plots are line and bar chart, and scatter plots.",
"Again latter have tree maps, arc diagram, and forced graph drawing.",
"Abstract/Summary Visualization techniques does abstraction or summarization before representation of data.",
"For that scaling is one of the technique.",
"It is done for easy understanding of data (e.g.",
"1cm=1km in map) which helps in finding meaningful correlations among them.",
"Common technique for data abstraction is binning it into histograms or data cubes.",
"Its advantage is providing a compressed, reduced dimensional representation of data.",
"Likewise, its techniques can also be classified into another group, clustering (e.g Hierarchical Aggregation).",
"Interactive/Real-Time Visualization techniques are recent techniques that have ability to adapt to user interactions in real-time.",
"Such techniques have capability to take less than a second for a real-time navigation of data by a user.",
"Such techniques are powerful because they can quickly discover important details in the data that helps to verify different data science theories.",
"For example Microsoft Pivot Table and Tableau, enables to pivot the data in Microsoft Excel, text file, .pdf, and Google Sheets data sources from crosstab format into columnar format for easy analysis."
],
[
"Big Data Mining",
"Too big data, too frequent data incoming, too frequent data changes, and too complex data.",
"These are the data to be envisioned.",
"Visualization of few MB data is not a big deal, but the huge data is a big sight.",
"The jewels have to be mined from the massive amount of data, analyzed for business purposes- such like business prediction, and use for growth of business.",
"Big data is the most hyped word in recent times which depicts prodigious volume of data consisting of various data formats and it is very difficult to process them using conventional mode of database, software methodologies and also it is successful enough to attract the attention of technology dwellers whereas data mining is the method or technique to mine useful information in the form of patterns that enlighten our vision about data and its utilities in every sphere of life starting from life science, business, etc.",
"Although big data and data mining are the two different perspectives of modern technology but they have a relationship of dealing with massive quantity of data, for example, Twitter termed their data mining experience as Big Data Mining which was treated as data mining [75].",
"Simply storing huge volume of data from time to time without using it for organizational benefit is merely wastage of resources.",
"So we should use this data for acquiring knowledge which can be useful for future work.",
"But it is very difficult to handle such enormous volume of data and analyze it, under such scenario comes the concept of data mining for getting specific information from the big data.",
"Many algorithms and techniques have been applied to mine useful information from the deep ocean of data.",
"Big data is an ocean of enormous volume of data but mining precious and useful information out of this can be done with the help of efficient use of data mining techniques such as classification, clustering, outlier detection, association rule, etc.",
"We encounter different levels of difficulty in mining useful information from large data sets as we need to handle our data with the changing requirement of technology.",
"But we get our required set of analyzed results with advent use of data mining techniques, for instance, sentimental analysis [76].",
"The data mining techniques applied on the Big data should be applicable for any kind of data.",
"The various data mining algorithms such as C4.5, K-Means, Support Vector Mechanism (SVM), Apriori, KNN, CART, Naive Bayes, Page Rank, etc.",
"are widely applied in the field of Big Data Analytics.",
"Likewise, the OLAP over Big Data is evolving [77]."
],
[
"Machine Learning",
"Machine learning is a field of computer science that try to enable computers with the ability to learn without explicitly programming them.",
"It$’$ s algorithms can be classified as following (Fig.14).",
"It is categorized into eleven categories depending on the characteristics of the algorithm as shown in the figure REF , albeit, the ML algorithms are categorized in three key categories, namely, supervised, unsupervised and semi-supervised ML algorithm.",
"The Big Data Analytics or Big Data Security analytics fully depends on Data Mining, where the Machine Learning techniques are subset of Data Mining.",
"The supervised learning algorithms are trained with a complete set of data and thus, the supervised learning algorithms are used to predict/forecast.",
"The unsupervised learning algorithms starts learning from scratch, and therefore, the unsupervised learning algorithms are used for clustering.",
"However, the semi-supervised learning combines both supervised and unsupervised learning algorithms.",
"The semi-supervised algorithms are trained, and the algorithms also include non-trained learning.",
"Figure: Taxonomy of Machine Learning Algorithms.",
"Source Regression algorithm: It predict output values using input features of the data provided to the system.",
"Most popular algorithms under it are Linear Regression, Logistic Regression Multivariate Adaptive Regression Splines (MARS) and Locally Estimated Scatter plot Smoothing (LOESS).",
"Instance based algorithm: It compares new problem instances with instances in training, that are stored in memory.",
"Common algorithms can be k-Nearest Neighbor (kNN), Learning Vector Quantization (LVQ), Self-Organizing Map (SOM) and Locally Weighted Learning (LWL).",
"Regularization algorithm: It is a process of providing additional information to prevent overfitting or solve an ill-posed problem.",
"Algorithms which are common under it are Least Absolute Shrinkage and Selection Operator (LASSO), Least-Angle Regression (LARS), Elastic Net and Ridge Regression.",
"Decision tree algorithm: It solves the problem using tree representation.",
"Some popular algorithms are Classification and Regression Tree (CART), C4.5 and C5.0, M5, and Conditional Decision Trees.",
"Bayesian algorithm: It is based on Bayesian method.",
"Popular algorithms are Naive Bayes, Gaussian Naive Bayes, Bayesian Network (BN), and Bayesian Belief Network (BBN).",
"Clustering algorithm: Grouping of data points based on similar features.",
"Some popular algorithms are K-Means, K-Medians, and Hierarchical Clustering.",
"Association Rule Learning Algorithms: It is used to discover relationship between data points.",
"Some common algorithms under it are Apriori algorithm, and Eclat algorithm.",
"Artificial Neural Network Algorithms: It is based on working of biological neural networks.",
"Example of some popular algorithms are Back-Propagation Neural Network(BPNN), Perceptron, and Radial Basis Function Network (RBFN).",
"Deep Learning Algorithms: It uses unsupervised learning to set each level of hierarchy of features using features discovered at previous level.",
"It has some popular algorithms such as Deep Boltzmann Machine (DBM), Deep Belief Networks (DBN), and Convolution Neural Network (CNN).",
"Dimensionality Reduction Algorithms: They reduces the number of feature by obtaining a set of principal variables.",
"Some algorithms commonly under it are Principal Component Analysis (PCA) and Principal Component Regression (PCR), Partial Least Squares Regression (PLSR) and Linear Discriminant Analysis (LDA), Mixture Discriminant Analysis (MDA) and Quadratic Discriminant Analysis (QDA) and Flexible Discriminant Analysis (FDA).",
"Ensemble Algorithms: It combines multiple learning algorithms to obtain better predictive performance.",
"Some well know algorithms are Random Forest, Gradient Boosted Regression Trees (GBRT), and Gradient Boosting Machines (GBM)."
],
[
"Security and Privacy",
"Analysis of Big data provides a large volume of knowledge which can be used many ways for the betterment of individual, society, nation, and even the world.",
"Nowadays people have become open with their thoughts to the world.",
"So it is the responsibility of the people who are using these data, to protect the data and prevent others from misusing it.",
"Figure: Security and privacy.",
"Source The Security and Privacy challenges for big data can be classified into four groups (Fig.15) such as Infrastructure Security, Data Privacy, Data Management, and Integrity and Reactive Security."
],
[
"Infrastructure Security",
"Infrastructure Security[79] in big data systems includes securing distributed computations and non-relational data stores.",
"As Hadoop framework is most commonly used for distributed system so its security is mostly researched to make it robust to threats.",
"Example of such a security model is G-Hadoop that implements users’ authentication and some security mechanisms in a simplified way to protect the system from traditional attacks.",
"Additionally, Distributed system uses parallelism for computation and storage of high volume of data.",
"Common example is Mapreduce framework so its security at mapper and reducer phase is important.",
"For that two main attack preventive measures are securing the mappers and securing the data during the presence of untrusted mapper.",
"Also big data use non-relational data stores so focus on its security is also important.",
"For such data stores NoSQL is used which don't have security provision so security in middleware is used.",
"However using clustering in NoSQL impose some security threats to it."
],
[
"Data Privacy",
"Organisations want to protect the privacy of data and also want to make a profit out of it.",
"With that aim several techniques and mechanisms were developed such as Privacy preserving data mining and analytics, Cryptography, Access Control.",
"Now, Privacy Preserving data mining and analytics means efficiently finding valuable data which are prone to misuse.",
"And Cryptography is the most commonly used mechanism for data security.",
"Some examples are Homomorphic Encryption (HE), secure Multi-Party Computation (MPC), and Verifiable Computation (VC).",
"Whereas Access control is to stop undesirable users from accessing the system."
],
[
"Data Management",
"Data Management[80] comes into picture after data is stored in big data environment.",
"From security point of view it include Secure data storage and transaction logs, Granular audits, and Data provenance.",
"Security to data storage and transaction logs are necessary.",
"As multi-tiered storage media are used to store data and transaction logs.",
"So, Auto-tiering is used to move data in data storage because of huge size of data.",
"But this disable the system to keep track of where data is stored.",
"So new mechanisms were developed for security and round the clock availability of data.",
"Similarly transaction logs security are essential as they contain all the changes made in system.",
"Similarly, Granular audits are crucial as it provide information related to attacks.",
"It have information about what happened and what preventive measures can be taken.",
"It also can be used for agreement, regulation and forensic investigation.",
"Likewise, Data provenance provide information about data and its origin that include input data, entities, systems and processes influencing that particular data.",
"But its complexity increases as provenance enabled programming environments in Big Data applications produces complex provenance graphs.",
"Analysis of such complex information is difficulty but required for security purposes."
],
[
"Integrity",
"Big data collects data from various sources and stores them in various formats.",
"So there comes the importance of integrity of data which is the accuracy, consistency, and trustworthiness of data.",
"As shown in the figure integrity can be classified into Endpoint validation and filtering, and Real-time security monitoring.",
"Before the data processing it is essential to validate the authenticity of input data otherwise system may be processing bad data.",
"For this purpose Endpoint validation and filtering is essential for producing good and trustworthy results.",
"Real-time security monitoring is used to monitor big data infrastructure.",
"However the security devices generates lots of security alarms and false positives.",
"And as it is big data these may increases further more in volume [81]."
],
[
"Conclusion",
"The Big Data is game changer paradigm in data-intensive field and it is very wide area to study.",
"The Big Data is a data silos.",
"It is very difficult to process, manage and store the data silos.",
"The data silos are formed not only in core computing area, but also science, engineering, economy, government, environment, society, etc.",
"There are a variety of processing engine to process mammoth sized data efficiently and effectively.",
"These processing engines are developed based on their requirements and characteristics.",
"Moreover, different kind of storage engines are also emerging, for instance, file system and object storage system.",
"Besides, machine learning algorithms are integrated with Big Data.",
"As a consequence, the Big Data Analytics is born."
]
] | 1808.08474 |
[
[
"New solitary wave and Multiple soliton solutions of (3 + 1)-dimensional\n KdV type equation by using Lie symmetry approach"
],
[
"Abstract Solitary waves are localized gravity waves that preserve their consistency and henceforth their visibility through properties of nonlinear hydrodynamics.",
"Solitary waves have finite amplitude and spread with constant speed and constant shape.",
"In this paper, we have used Lie group of transformation method to solve (3 + 1)-dimensional KdV type equation.",
"We have obtained the infinitesimal generators, commutator table of Lie algebra for the KdV type equation.",
"We have achieved a number of exact solutions of KdV type equation in the explicit form through similarity reduction.",
"All the reported results are expressed in analytic (closed form) and figured out graphically through their evolution solution profiles.",
"We characterized the physical explanation of the obtained solutions with the free choice of the particular parameters by plotting some 3D and 2D illustrations.",
"The geometrical analysis explains that the nature of solutions is travelling wave, kink wave, single solitons, doubly solitons and curve-shaped multisolitons."
],
[
"Introduction",
"Solitons and nonlinear evolution equations (NLEEs) are broadly used to explain nonlinear phenomena in many mathematical physics and emerging engineering areas, such as nonlinear optics, condensed matter, plasma physics, fluid dynamics, convictive fluids, solid-state physics, acoustics, and quantum field theory [1], [2], [3], [4], [5], [6], [7].",
"Owing to the significant role of solitons and nonlinear equations play in these scientific fields, constructing exact solutions for the NLEEs is of great value.",
"In the mid 19th century, John Scott Russell first investigated shallow water solitary waves experimentally and noted their importance through nonlinear interactions[36], [37].",
"Both Boussinesq and Rayleigh established mathematically the existence of steady solitary waves on shallow water, before Korteweg and de Vries (KdV) published their famous PDE, which was first originally derived by Boussinesq [8], [9], [10].",
"After this work, the theory of solitary waves remained almost untouched for 70 years until the mid 1960s when numerical studies by Zabusky and Kruskal [11] discovered the robust nature of soliton interactions, prompting an explosion of refined mathematical analysis on nonlinear PDEs.",
"The history of solitary waves has been analyzed by Miles[12], and that of water waves, more generally by Darrigol[13].",
"The Korteweg–de Vries (KdV) equation is given in the form of $u_t-6 u u_x+u_{xxx}=0.$ The KdV type equation is characterized by the special waves which is called solitons on shallow water surfaces [11].",
"Solitons are localized wave disturbances that proliferate without changing shape or spreading out [14].",
"The KdV equation has a number of connections with physical problems such as shallow water waves with weakly non-linear restoring forces, ion acoustic waves in plasma, long internal waves in a density-stratified ocean, acoustic waves on a crystal lattice [15].",
"The (3+1)-dimensional KdV type equation has many extensive applications in the field of condensed matter physics, fluid dynamics, plasma physics and optics [42].",
"After the inverse scattering method, which is established for solving the KdV equation many important approaches were contributed by many researchers.",
"In this view, we have observed many methods developed in the literature.",
"For getting analytical solutions of considered system, homogeneous balance method, Hirota bilinear method, ansatz method, exp function method, Kudryashov simplest equation method and Lie group of transformation method can be cited [16], [17], [18], [19], [20], [21], [23], [25], [22], [24], [26], [27], [28], [29], [32], [35], [38], [39], [40], [41].",
"In this work, we study a (3+1)-dimensional KdV type equation of the form $\\Delta := u_t+6 u_x u_y+u_{xxy}+u_{xxxxz}+60 u_x^2 u_z+10 u_{xxx} u_z+20 u_x u_{xxz}= 0$ which was introduced by Lou [30] and obtained the five types of multidromion solutions for its potentials form.",
"Further, Wazwaz [34] investigated one and two solitons solutions only.",
"Besides this, the same problem was tackled by Ünsal [31] and obtained complexiton and interaction solutions by Hirota Method.",
"This paper is a continuation to above mentioned earlier studies, in the sense that the (3+1)-dimensional KdV type equation is being analyzed using the Lie symmetry approach.",
"The main objective of this paper is to obtain the symmetry reductions and exact solutions by using the Lie group of transformation method.",
"We study various analytical (closed form) solutions of the equation via the symbolic calculations, including solitary waves solitons, single solitons, doubly solitons and multisolitons and travelling wave solitons.",
"Furthermore, the exact solutions of the equation are graphically analyzed though their evolution profiles.",
"The skeleton of this paper is organized as follows: In Sec.",
"2, we perform the Lie group analysis on the (3+1)-dimensional KdV type equation and present all the geometric vector fields.",
"Then the complete symmetry classification of the (3+1)-dimensional KdV type Eq.",
"(REF ) is performed.",
"In Sec.",
"3, we discuss the Lie symmetry group of Eq.",
"(REF ).",
"In Sec.",
"4, the symmetry reductions and exact solutions to the (3+1)-dimensional KdV type equation are investigated.",
"In Sec.",
"5, discussion on graphical illustration of solutions are presented.",
"Finally, the conclusion and some remarks are given in Sec.",
"6."
],
[
"Lie symmetry analysis for the (3+1) KdV type equation",
"Lie group analysis is a powerful mathematical tool to study the properties of NLEEs and for obtaining the invariant solutions.",
"If Eq (REF ) is invariant under a one-parameter Lie group of point transformations: $\\tilde{x} &=x+\\epsilon \\, \\xi ^1 (x,y,z,t,u) +O(\\epsilon ^2), \\nonumber \\\\\\tilde{y} &=y+\\epsilon \\, \\xi ^2 (x,y,z,t,u) +O(\\epsilon ^2), \\nonumber \\\\\\tilde{z} &=z+\\epsilon \\, \\xi ^3 (x,y,z,t,u) +O(\\epsilon ^2), \\nonumber \\\\\\tilde{t} &=t+\\epsilon \\, \\xi ^4 (x,y,z,t,u) +O(\\epsilon ^2), \\nonumber \\\\\\tilde{u} &=u+\\epsilon \\, \\eta (x,y,z,t,u) +O(\\epsilon ^2), \\nonumber $ where $\\epsilon $ is a small expansion parameter with infinitesimal generator ${\\bf V}=\\xi ^1(x,y,z,t,u) \\frac{\\partial }{\\partial x}+ \\xi ^2(x,y,z,t,u) \\frac{\\partial }{\\partial y}+\\xi ^3(x,y,z,t,u) \\frac{\\partial }{\\partial z}+\\xi ^4(x,y,z,t,u) \\frac{\\partial }{\\partial t}+\\eta (x,y,z,t,u)\\frac{\\partial }{\\partial u},$ then the vector field (REF ) generates a symmetry of Eq.",
"(REF ), and $V$ must satisfy Lie symmetry conditions $pr^{(5)}V(\\Delta )|_{\\Delta =0} = 0,$ where $pr^{(5)}V$ is the fifth prolongation of $V$ .",
"Applying the fifth prolongation $pr^{(5)}V$ to Eq.",
"(REF ), the invariant conditions given by $\\eta ^t+6 \\eta ^x u_y + 6 u_x \\eta ^y +\\eta ^{xxy}+\\eta ^{xxxxz}+&120 \\eta ^x u_x u_z+ 60 u_x^2 \\eta ^z+10 \\eta ^{xxx}u_z\\\\+&10 u_{xxx} \\eta ^z+20 \\eta ^x u_{xxz}+20 u_x \\eta ^{xxz} = 0,$ where $\\eta ^t, \\eta ^x, \\eta ^y, \\eta ^z, \\eta ^{xxy}, \\eta ^{xxz}, \\eta ^{xxx}$ and $\\eta ^{xxxxz}$ are the coefficients of $pr^{(5)}V(\\Delta )$ .",
"Moreover, we have $\\eta ^t &= D_t(\\eta )-u_x D_t (\\xi ^1)-u_y D_t (\\xi ^2)-u_z D_t (\\xi ^3)-u_t D_t (\\xi ^4),\\\\\\eta ^x &= D_x(\\eta )-u_x D_x (\\xi ^1)-u_y D_x (\\xi ^2)-u_z D_x (\\xi ^3)-u_t D_x (\\xi ^4),\\\\\\eta ^y &= D_y(\\eta )-u_x D_y (\\xi ^1)-u_y D_y (\\xi ^2)-u_z D_y (\\xi ^3)-u_t D_y (\\xi ^4),\\\\\\eta ^z &= D_z(\\eta )-u_x D_z (\\xi ^1)-u_y D_z (\\xi ^2)-u_z D_z (\\xi ^3)-u_t D_z (\\xi ^4),\\\\\\eta ^{xxx} &= D_x (\\eta _{xx})-u_{xxx} D_x (\\xi ^1)- u_{xxy} D_x (\\xi ^2)-u_{xxz} D_x (\\xi ^3)-u_{xxt} D_x (\\xi ^4),\\\\\\eta ^{xxy} &= D_y (\\eta _{xx})-u_{xxx} D_y (\\xi ^1)- u_{xxy} D_y (\\xi ^2)-u_{xxz} D_y (\\xi ^3)-u_{xxt} D_y (\\xi ^4),\\\\\\eta ^{xxz} &= D_z (\\eta _{xx})-u_{xxx} D_z (\\xi ^1)- u_{xxy} D_z (\\xi ^2)-u_{xxz} D_z (\\xi ^3)-u_{xxt} D_z (\\xi ^4),\\\\\\eta ^{xxxxz} &= D_z (\\eta _{xxxx})-u_{xxxxx} D_z (\\xi ^1)- u_{xxxxy} D_z (\\xi ^2)-u_{xxxxz} D_z (\\xi ^3)-u_{xxxxt} D_z (\\xi ^4),$ where $D_x$ , $D_y$ and $D_t$ represent the total derivative operators for $x$ , $y$ and $t$ , respectively.",
"For example, one of them can be given as $D_x = \\frac{\\partial }{\\partial x}+ u_x\\frac{\\partial }{\\partial u}+u_{xx}\\frac{\\partial }{\\partial u_x}+u_{xy}\\frac{\\partial }{\\partial u_y}+u_{xt}\\frac{\\partial }{\\partial u_t}+\\dots .$ In similar manner, we can use the total derivative operators for other variables.",
"Incorporating all the expressions of Eq.",
"(REF ) into Eq.",
"(REF ) and then comparing the various differential coefficients of $u$ to zero, we get the following determining system of equations: $\\eta _t=0,\\,\\,\\, 2 \\eta _u = -\\xi ^4_t+\\xi ^2_y, \\,\\,\\,6 \\eta _x= \\xi ^2_t,\\,\\,\\, 6 \\eta _y = \\xi ^1_t,\\\\10 \\eta _z =\\xi ^1_y,\\,\\,\\, \\xi ^4_u=\\xi ^4_x=\\xi ^4_y=\\xi ^4_z=\\xi ^4_{tt}=0,\\\\\\xi ^1_u=0,\\,\\,\\, 2 \\xi ^1_x =\\xi ^4_t-\\xi ^2_y,\\,\\,\\, \\xi ^1_z=\\xi ^1_{tt}=\\xi ^1_{ty}=\\xi ^1_{yy}=0,\\\\\\xi ^2_u=\\xi ^2_x= \\xi ^2_z=\\xi ^2_{tt}=\\xi ^2_{ty}=\\xi ^2_{yy}=0,\\\\\\xi ^3_t=\\xi ^3_u=\\xi ^3_x=0,\\,\\,\\, 3 \\xi ^3_y= 10 \\xi ^2_t,\\,\\,\\, \\xi ^3_z=-\\xi ^4_t+2\\xi ^2_y,$ where $\\eta _t=\\frac{\\partial \\eta }{\\partial t}, \\eta _x=\\frac{\\partial \\eta }{\\partial x},\\eta _u=\\frac{\\partial \\eta }{\\partial u},\\xi ^1_x=\\frac{\\partial \\xi ^1}{\\partial x}, \\xi ^4_{tt}=\\frac{\\partial ^2 \\xi ^4}{\\partial t^2},\\xi ^1_{ty}=\\frac{\\partial ^2 \\xi ^1}{\\partial t \\partial y}, etc$ .",
"Solutions of Eq.",
"(REF ) resulted in the following infinitesimal generators: $\\xi ^1 &= \\frac{1}{4} (a_1-a_5)x + a_8 t+ a_9 y +a_{10}, \\,\\, \\\\\\xi ^2 &= \\frac{1}{2} (a_1 +a_5)y+\\frac{3t}{10} a_4+a_7,\\,\\,\\\\\\xi ^3 &= a_4 y+ a_5 z + a_6,\\\\\\xi ^4&=a_1 t+a_3,\\\\\\eta &=\\frac{1}{4}(a_5-a_1)u+\\frac{x}{20} a_4+\\frac{y}{6}a_8 +\\frac{z}{10}a_9 +a_2,$ where $a_i, i=1,\\dots , 10$ are all integral constants.",
"For the above cumbersome calculations, computer software Maple is used.",
"Therefore, Lie algebra of infinitesimal symmetries of Eq.",
"(REF ) is spanned by the following ten vector fields: $v_1=&\\frac{x}{4}\\frac{\\partial }{\\partial x}+\\frac{y}{2}\\frac{\\partial }{\\partial y}+t\\frac{\\partial }{\\partial t}- \\frac{u}{4} \\frac{\\partial }{\\partial u}, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,v_2= \\frac{\\partial }{\\partial u}, \\,\\,\\,\\,\\,\\,v_3=\\frac{\\partial }{\\partial t}, \\,\\,\\,\\,\\,\\,v_4=\\frac{3t}{10}\\frac{\\partial }{\\partial y}+y \\frac{\\partial }{\\partial z}+\\frac{x}{20} \\frac{\\partial }{\\partial u}, \\\\ \\,\\,\\, \\,\\,\\,v_5=&\\frac{-x}{4}\\frac{\\partial }{\\partial x}+\\frac{y}{2} \\frac{\\partial }{\\partial y}+z \\frac{\\partial }{\\partial z}+\\frac{u}{4} \\frac{\\partial }{\\partial u}, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,v_6= \\frac{\\partial }{\\partial z}, \\,\\,\\,\\,\\,\\,v_7= \\frac{\\partial }{\\partial y},\\\\ \\,\\,\\,\\,\\,\\,v_8=& t\\frac{\\partial }{\\partial x}+ \\frac{y}{6}\\frac{\\partial }{\\partial u},\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,v_9= y\\frac{\\partial }{\\partial x}+\\frac{z}{10}\\frac{\\partial }{\\partial u},\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,v_{10}= \\frac{\\partial }{\\partial x}.$ Table: Commutation table of Lie algebraIt shows that the symmetry generators found in Eqs.",
"(REF ) form the 10 - dimensional Lie algebra.",
"Also, it is easy to check that the symmetry generators found in (REF ) form a closed Lie algebra whose commutation relations are given in Table REF .",
"Then, all of the infinitesimal of Eq.",
"(REF ) can be expressed as a linear combination of $v_i$ given as ${\\bf V}=a_1 v_1+a_2 v_2+a_3 v_3+a_4 v_4+a_5 v_5+a_6 v_6+ a_7 v_7+ a_8 v_8+a_9v_9+a_{10} v_{10}.$ The $(i, j)$ th entry of the Table REF is the Lie bracket $[v_i \\,\\, v_j] = v_i \\cdot v_j-v_j \\cdot v_i$ .",
"We observe that Table REF is skew-symmetric with zero diagonal elements.",
"Also, Table REF shows that the generators $v_i,\\,\\, 1 \\le i \\le 10$ are linearly independent."
],
[
"Symmetry group of (3+1)- dimensional KdV type equation",
"In this section, we obtain the group transformations $g_i:(x,y,z,t,u) \\rightarrow (\\tilde{x}, \\tilde{y}, \\tilde{z}, \\tilde{t}, \\tilde{u}),$ which is generated by the generators of infinitesimal transformations $v_i$ for $1 \\le i \\le 10$ .",
"In order to get some exact solutions from known ones, we should find the Lie symmetry groups from the related symmetries.",
"To get the Lie symmetry group, we should solve the following problems For this purpose, we need to solve, following system of ODE's $\\frac{d}{d \\epsilon } (\\tilde{x}, \\tilde{y}, \\tilde{z}, \\tilde{t}, \\tilde{u})&={\\bf \\sigma }(\\tilde{x}, \\tilde{y}, \\tilde{z}, \\tilde{t}, \\tilde{u}), \\\\(\\tilde{x}, \\tilde{y}, \\tilde{z}, \\tilde{t}, \\tilde{u})|_{\\epsilon =0} &= (x,y,z,t,u),$ where $\\epsilon $ is an arbitrary real parameter and ${\\bf \\sigma }=\\xi ^1 u_x+\\xi ^2 u_y+\\xi ^3 u_z+\\xi ^4 u_t+\\eta u.$ So, we can obtain the Lie symmetry group $g:(x,y,z,t,u) \\rightarrow (\\tilde{x}, \\tilde{y}, \\tilde{z}, \\tilde{t}, \\tilde{u}).$ According to different $\\xi ^1, \\xi ^2, \\xi ^3, \\xi ^4$ , and $\\eta $ , we have the following groups $g_1:&(x,y,z,t,u) \\rightarrow (x e^{\\epsilon }, y e^{\\epsilon }, z, t e^{4 \\epsilon }, u e^{-\\epsilon }),\\\\g_2:&(x,y,z,t,u) \\rightarrow (x,y,z,t,u+\\epsilon ),\\\\g_3:&(x,y,z,t,u) \\rightarrow (x,y,z,t+\\epsilon ,u),\\\\g_4:&(x,y,z,t,u) \\rightarrow ( x, y + 6 t \\epsilon , z + 60 t \\epsilon ^2+20 y \\epsilon , t, u+\\epsilon x ),\\\\g_5:&(x,y,z,t,u) \\rightarrow (x e^{-\\epsilon }, y e^{2 \\epsilon }, z e^{4 \\epsilon }, t, u e^{\\epsilon }),\\\\g_6:&(x,y,z,t,u) \\rightarrow (x,y,z+\\epsilon ,t,u),\\\\g_7:&(x,y,z,t,u) \\rightarrow (x,y+\\epsilon ,z,t,u),\\\\g_8:&(x,y,z,t,u) \\rightarrow (x+6 \\epsilon t ,y,z,t,u+ \\epsilon y),\\\\g_9:&(x,y,z,t,u) \\rightarrow (x+10 \\epsilon y,y,z,t,u+\\epsilon z),\\\\g_{10}:&(x,y,z,t,u) \\rightarrow (x+\\epsilon ,y,z,t,u).$ The entries on the right side give the transformed point $\\exp (x,y,z,t,u)=(\\tilde{x}, \\tilde{y}, \\tilde{z}, \\tilde{t}, \\tilde{u})$ .",
"The symmetry groups $g_2, g_3, g_6, g_7$ and $g_{10}$ demonstrate the space and time invariance of the equation.",
"The well known scaling symmetry turns up in $g_1, g_4, g_5, g_8$ and $g_9$ .",
"We can obtain the corresponding new solutions by applying above groups $g_i, 1 \\le i \\le 10$ .",
"If $u=f(x,y,z,t)$ is a known solution of Eq.",
"(REF ), then by using above groups $g_i, 1 \\le i \\le 10$ corresponding new solutions $u_i, 1 \\le i \\le 10$ can be obtained as follows $u_1 &= e^{\\epsilon } f_1(x e^{-\\epsilon }, y e^{-\\epsilon }, z, t e^{-4 \\epsilon }), \\\\u_2 &= f_2(x,y,z,t)-\\epsilon , \\\\u_3 &= f_3(x,y,z,t-\\epsilon ), \\\\u_4 &= f_4( x, y - 6 t \\epsilon , z - 60 t \\epsilon ^2 - 20 y \\epsilon , t)- \\epsilon x, \\\\u_5 &= e^{-\\epsilon }f_5(x e^{\\epsilon }, y e^{-2 \\epsilon }, z e^{-4 \\epsilon }, t), \\\\u_6 &= f_6(x,y,z-\\epsilon ,t), \\\\u_7 &= f_7(x,y-\\epsilon ,z,t), \\\\u_8 &= f_8(x-6 t \\epsilon ,y,z,t)-\\epsilon y, \\\\u_9 &= f_9(x-10 \\epsilon y,y,z,t)-\\epsilon z, \\\\u_{10} &= f_{10}(x-\\epsilon ,y,z,t),$ By selecting the arbitrary constants, one can obtain many new solutions.",
"Thus, to obtain the invariant solutions of Eq.",
"(REF ), the corresponding Lagrange system is $\\frac{dx}{\\xi ^1(x,y,z,t)}=\\frac{dy}{\\xi ^2(x,y,z,t)}=\\frac{dz}{\\xi ^3(x,y,z,t)}=\\frac{dt}{\\xi ^4(x,y,z,t)}=\\frac{du}{\\eta (x,y,z,t)}.$ The different forms of the invariant solutions of the equation are obtained by assigning the specific values to $a_i,\\,\\, 1 \\le i \\le 10$ .",
"Therefore, the Lie symmetry method predicts the following vector fields to generate the different forms of the invariant solutions."
],
[
"Symmetry reduction and closed-form solutions of (3+1) KdV type equation",
"In this section, we systematically derive the Lie point symmetries of the (3+1)-dimensional KdV type equation.",
"For this purpose, we reduce the characteristic equations of vector fields obtained in the previous section for getting reduction equations."
],
[
"Graphical illustration of the solutions",
"In this section, the numerical simulation of (3+1)-dimensional KdV type equation is demonstrated.",
"The proposed method provides more general and plentiful new kink wave solutions with some free parameters.",
"Kink waves are the type of solitary waves that preserve its shape when travelling down and do not change their shape through the propagation.",
"Kink waves are travelling waves which rise or go down from one asymptotic state to another.",
"Graphical representations of the solutions $u_3, u_4, u_5, u_{20}$ and $u_{21}$ are illustrated in Figures 1-8 for free choices of parameters.",
"The other exact solutions could be achieved from the remaining set of solutions."
],
[
"Conclusion",
"This research depicts that the Lie symmetry analysis method is quite well-organized for extracting the exact travelling wave solutions of (3+1)-dimensional KdV type equation.",
"In this paper, some general exact solutions in the form of kink wave, travelling wave, single soliton, doubly soliton, curved shaped multisoliton, explicit WeierstrassZeta and WeierstrassP function are constructed by Lie group of transformation method.",
"The results in this article are excogitated and continued version of previously reported results.",
"Many new exact solutions are derived, which have fruitful applications in different areas of mathematical physics, engineering, other fields of applied sciences and might provide a valuable help for researchers and physicists to study more complex nonlinear phenomena."
]
] | 1808.08364 |
[
[
"Scalable star-shape architecture for universal spin-based nonadiabatic\n holonomic quantum computation"
],
[
"Abstract Nonadiabatic holonomic quantum computation as one of the key steps to achieve fault tolerant quantum information processing has so far been realized in a number of physical settings.",
"However, in some physical systems particularly in spin qubit systems, which are actively considered for realization of quantum computers, experimental challenges are undeniable and the lack of a practically feasible and scalable scheme that supports universal holonomic quantum computation all in a single well defined setup is still an issue.",
"Here, we propose and discuss a scalable star-shape architecture with promising feasibility, which may open up for realization of universal (electron-)spin-based nonadiabatic holonomic quantum computation."
],
[
"Introduction",
"Holonomic quantum computation [1], [2], [3], [4] is recognised among key approaches to fault resistant quantum computation.",
"Nonadiabatic holonomic quantum computation [2], [3], [4] compared to its adiabatic counterpart [1] is more compatible with the short coherence time of quantum bits (qubits).",
"To achieve a feasible platform, nonadiabatic holonomic quantum computation has been adapted and developed for different physical settings [2], [3], [5], [6], [4], [7], [8], [9], [10].",
"Nonadiabatic holonomic quantum computation has also been combined with decoherence free subspaces [21], [22], [23], [24], [25], [26], [27], [28], noiseless subsystems [29], and dynamical decoupling [30] to further improve its robustness.",
"Experimental realizations of nonadiabatic holonomic quantum computation in various physical systems, such as NMR [11], [12], superconducting transmon [13], [14], [15], and NV centers in diamond [16], [17], [18], [19], [20] have been carried out.",
"Nevertheless, the implementation of nonadiabatic holonomic quantum computation in some physical systems particularly in spin qubit system, which is one of the natural and promising candidates to built quantum computers upon, has been remained at the level of single-qubit gates.",
"In fact, from practical perspectives, establishing a scalable multipartite scheme, which possesses full holonomic computational power for quantum processing, in these physical systems is still a challenge.",
"In this paper, we aim to address this issue by proposing a scalable architecture for universal spin-based nonadiabatic holonomic quantum computation, which enjoys a reasonable capability of being implemented with current technologies.",
"We consider a star-shape system, where in principal an arbitrary number of register spin qubits are arranged about and all coupled to an auxiliary spin qubit in the middle of architecture.",
"Universal holonomic single-qubit gates are achieved by controlling the coupling between two computational basis states of register qubits through local transverse magnetic fields.",
"The middle auxiliary spin qubit introduces an indirect bridge coupling between each pair of register qubits bringing about a double $\\Lambda $ structure, which permits to implement holonomic entangling gates between selected pair."
],
[
"Scalable architecture",
"The model system that we have in mind is a scalable $n$ register spin qubits coupled in a star-shape architecture through an auxiliary spin qubit as depicted in Fig.",
"REF .",
"Figure: (Color online) Scalable star-shape architecture for universal spin-based nonadiabatic holonomic quantum computation.",
"An arbitrary nn number of register spin qubits arranged about and all coupled to an auxiliary spin qubit deployed in the middle of architecture.",
"Each register qubit is allowed to interact with a controllable local magnetic field.Since only universal single-qubit and two-qubit gates are needed to achieve a universal quantum information processing, the Hamiltonian adapted here is a collective single-qubit and two-qubit Hamiltonians given by $H=\\sum _{k=1}^{n}H_{k}+\\sum _{k, l=1}^{n}H_{kl}.$ For single-qubit Hamiltonians we consider $H_{k}=B_{k}^{\\bot }\\cdot S^{(k)},$ which describes the interaction of the $k$ th spin qubit, $S^{(k)}=(S^{(k)}_{x} ,S^{(k)}_{y} ,S^{(k)}_{z})$ , with a local transverse magnetic field $B_{k}^{\\bot }=(B_{k}^{x}, B_{k}^{y}, 0)$ .",
"Two-qubit Hamiltonians read $H_{kl}=J_{k} H^{(k)}_{XY}+J_{l} H^{(l)}_{XY},$ where $H^{(\\bullet )}_{XY}=S^{(\\bullet )}_{x}S^{(a)}_{x}+S^{(\\bullet )}_{y}S^{(a)}_{y}$ , bullet stands for the corresponding superscript $k$ or $l$ , and $a$ represents the auxiliary qubit.",
"The $H_{kl}$ describes a three-body anisotropic interaction between the corresponding register qubits $k, l$ , and the auxiliary qubit.",
"The $J_{k}$ and $J_{l}$ are the exchange coupling strengths to the auxiliary qubit.",
"In fact, the two-qubit Hamiltonian in Eq.",
"(REF ) introduces an indirect coupling between the selected two register qubits $k$ and $l$ through the auxiliary qubit.",
"This can be seen in Fig.",
"REF as well.",
"In the following, we discuss the realization of a universal family of single-qubit and two-qubit gates in this setup."
],
[
"Singel-Qubit Gates",
"For a single-qubit gate on the given qubit $k$ , we only turn on the single-qubit Hamiltonian, $H_{k}$ , in the collective Hamiltonian given in Eq.",
"(REF ) by exposing the qubit $k$ to a local transverse magnetic field $B_{k}^{\\bot }$ .",
"During this implementation, we assume that the other terms in Eq.",
"(REF ) are kept off.",
"Thus, our effective Hamiltonian in this case is $H_{k}=B_{k}^{\\bot }\\cdot S^{(k)}=\\frac{B}{2}\\vec{n}\\cdot \\vec{\\sigma },$ where $B$ and $\\vec{n}=(\\cos \\beta , \\sin \\beta , 0)$ , respectively, describe the strength and the direction of the local transverse magnetic field $B_{k}^{\\bot }$ in the $xy$ plane.",
"The $\\vec{\\sigma }=(\\sigma _{x}, \\sigma _{y}, \\sigma _{z})$ is the standard Pauli operators and $\\hbar =1$ from now on.",
"To achieve holonomic single-qubit gates, we consider cyclic evolutions of an arbitrary qubit state $\\left| \\psi \\right\\rangle =\\cos \\frac{\\theta }{2}\\left| 0 \\right\\rangle +e^{i\\phi }\\sin \\frac{\\theta }{2}\\left| 1 \\right\\rangle ,$ in which only geometric phases are relevant.",
"Explicitly, we are interested in evolutions $\\mathcal {U}(\\tau _{0}, \\tau )\\left| \\psi \\right\\rangle =\\exp [-i\\int _{\\tau _{0}}^{\\tau }H_{k}(s)ds]\\left| \\psi \\right\\rangle $ , along which no dynamical phases occur, for instance the condition $\\left\\langle \\psi \\right|\\mathcal {U}^{\\dagger }(\\tau _{0}, t)H_{k}(t) \\mathcal {U}(\\tau _{0}, t)\\left| \\psi \\right\\rangle =0$ is satisfied at any time $t\\in [\\tau _{0}, \\tau ]$ [34].",
"Considering a local transverse magnetic field $B_{k}^{\\bot }=B\\vec{n}$ with constant phase $\\beta $ , reduces the condition in Eq.",
"(REF ) to one of the following simplified conditions: $(i)\\ && \\phi -\\beta =(2m+1)\\frac{\\pi }{2},\\ \\ \\ \\ \\ \\ \\ m=0, \\pm 1, \\pm 2, ... \\nonumber \\\\(ii) \\ &&\\theta =0\\ \\text{or}\\ \\pi $ This follows from the fact that $[H_{k}(t), \\mathcal {U}(\\tau _{0}, t)]=0$ at any time $t$ , when the phase $\\beta $ is fixed constant.",
"In the light of the above simplified conditions, we carry out our cyclic evolution in the following three steps: Step 1: We first evolve the general initial state $\\left| \\psi \\right\\rangle $ to the computational basis state $\\left| 0 \\right\\rangle $ by turning on the local transverse magnetic field $B_{k}^{\\bot }=B\\vec{n}$ for a time interval $[0, \\tau _{1}]$ with constant phase $\\beta =\\phi -\\frac{\\pi }{2}$ and (time-dependent) strength $B$ such that $\\int _{0}^{\\tau _{1}}Bdt=\\theta $ .",
"Hence, we have $\\mathcal {U}(0, \\tau _{1})\\left| \\psi \\right\\rangle =\\left| 0 \\right\\rangle .$ Step 2: Next, we evolve the state $\\left| 0 \\right\\rangle $ , all the way along the meridian of the Bloch sphere corresponding to the fixed azimuthal angle $\\tilde{\\phi }$ , to the state $e^{i\\tilde{\\phi }}\\left| 1 \\right\\rangle $ by employing the constant magnetic phase $\\beta =\\tilde{\\phi }+\\pi /2$ and (time-dependent) magnetic strength $B$ for a time interval $[\\tau _{1}, \\tau _{2}]$ such that $\\int _{\\tau _{1}}^{\\tau _{2}}Bdt=\\pi $ .",
"Namely, $\\mathcal {U}(\\tau _{1}, \\tau _{2})\\left| 0 \\right\\rangle =e^{i\\tilde{\\phi }}\\left| 1 \\right\\rangle .$ Step 3: Finally, we run the Hamiltonian $H_{k}$ for another time interval $[\\tau _{2}, \\tau _{3}]$ with fixed magnetic phase $\\beta =\\phi -\\pi /2$ and (time-dependent) magnetic strength $B$ such that $\\int _{\\tau _{2}}^{\\tau _{3}}Bdt=\\pi -\\theta $ .",
"This would evolve the final state of step 2, i.e., the state $e^{i\\tilde{\\phi }}\\left| 1 \\right\\rangle $ , into the final state $e^{i\\Delta \\phi }\\left| \\psi \\right\\rangle $ , where $\\Delta \\phi =\\tilde{\\phi }-\\phi $ .",
"In other words $\\\\\\mathcal {U}(\\tau _{2}, \\tau _{3})e^{i\\tilde{\\phi }}\\left| 1 \\right\\rangle =e^{i\\Delta \\phi }\\left| \\psi \\right\\rangle .$ Figure: (Color online) Cyclic evolution of a general qubit state ψ\\left| \\psi \\right\\rangle on Bloch sphere is carried out in three steps: step 1, which is illustrated in blue, evolves the state ψ\\left| \\psi \\right\\rangle to the state 0\\left| 0 \\right\\rangle along the meridian of the Bloch sphere corresponding to the fixed azimuthal angle φ\\phi ; step 2, which is shown in red, moves the north pole state 0\\left| 0 \\right\\rangle all the way down to the south pole state e iφ ˜ 1e^{i\\tilde{\\phi }}\\left| 1 \\right\\rangle along the meridian of the Bloch sphere corresponding to the fixed azimuthal angle φ ˜\\tilde{\\phi }; finally, step 3, which is depicted in black, evolves the state e iφ ˜ 1e^{i\\tilde{\\phi }}\\left| 1 \\right\\rangle back into the initial state with overall accumulated phase Δφ=φ ˜-φ\\Delta \\phi =\\tilde{\\phi }-\\phi , i.e., e iΔφ ψe^{i\\Delta \\phi }\\left| \\psi \\right\\rangle , along the meridian of the Bloch sphere corresponding to the fixed azimuthal angle φ\\phi .",
"The overall cyclic evolution introduces a parallel transport of the state ψ\\left| \\psi \\right\\rangle along an orange slice shaped path.",
"The solid angle Δφ\\Delta \\phi subtended by the orange slice shaped path, is the associated non-adiabatic Abelian geometric phase.We illustrate the above three steps evolution on the Bloch sphere in Fig.",
"REF .",
"As shown in Fig.",
"REF , the first and third steps evolve the qubit state along the meridian of the Bloch sphere corresponding to the fixed azimuthal angle $\\phi $ .",
"It is important to note that the parameters $\\theta $ and $\\phi $ are constant during the evolution and the magnetic strength $B$ is the only allowed time-dependent control variable.",
"At the completion of the three steps, we have a cyclic evolution $\\mathcal {U}(0, \\tau _{3})\\left| \\psi \\right\\rangle =\\mathcal {U}(\\tau _{2}, \\tau _{3})\\mathcal {U}(\\tau _{1}, \\tau _{2})\\mathcal {U}(0, \\tau _{1})\\left| \\psi \\right\\rangle =e^{i\\Delta \\phi }\\left| \\psi \\right\\rangle .\\nonumber \\\\$ In fact, this three-step evolution introduces a cyclic evolution of the general qubit state $\\left| \\psi \\right\\rangle $ about an orange slice shaped path on the Bloch sphere, where the two geodesic edges of the path differ as $\\Delta \\phi =\\tilde{\\phi }-\\phi $ in their azimuthal angles.",
"Note that the evolution in step 1 satisfies the condition $(i)$ of Eq.",
"(REF ) and the evolutions in step 2 and 3 satisfy the condition $(ii)$ in Eq.",
"(REF ), which indicate that no dynamical phases occur along the three step evolutions.",
"Therefore, the dynamical phase vanishs in the cyclic evolution of the general state $\\left| \\psi \\right\\rangle $ and the overall phase accumulated in this evolution, i.e., $\\Delta \\phi $ , is all geometric.",
"Strictly speaking, the phase $\\Delta \\phi $ , which is the solid angle subtended by the orange slice shaped path, is the non-adiabatic Abelian geometric phase accompanying the parallel transport of the state $\\left| \\psi \\right\\rangle $ along this orange slice shaped path [34].",
"One may further note that the orthogonal counterpart state of $\\left| \\psi \\right\\rangle $ , i.e., $\\left| \\psi ^{\\bot } \\right\\rangle =\\sin \\frac{\\theta }{2}\\left| 0 \\right\\rangle -e^{i\\phi }\\cos \\frac{\\theta }{2}\\left| 1 \\right\\rangle ,$ would accordingly evolve in a cyclic fashion giving rise to $\\mathcal {U}(0, \\tau _{3})\\left| \\psi ^{\\bot } \\right\\rangle =e^{-i\\Delta \\phi }\\left| \\psi ^{\\bot } \\right\\rangle .$ with geometric phase $-\\Delta \\phi $ .",
"Eqs.",
"(REF ) and (REF ) imply that the final time evolution operator $\\mathcal {U}(0, \\tau _{3})$ has actually a geometric structure given by $\\mathcal {U}(0, \\tau _{3})&=&\\mathcal {U}(\\tau _{2}, \\tau _{3})\\mathcal {U}(\\tau _{1}, \\tau _{2})\\mathcal {U}(0, \\tau _{1})\\nonumber \\\\&=&e^{i\\Delta \\phi }\\left| \\psi \\right\\rangle \\left\\langle \\psi \\right|+e^{-i\\Delta \\phi }\\left| \\psi ^{\\bot } \\right\\rangle \\left\\langle \\psi ^{\\bot } \\right|,\\nonumber \\\\$ which takes the following form in the qubit computational $\\lbrace \\left| 0 \\right\\rangle , \\left| 1 \\right\\rangle \\rbrace $ basis $\\mathcal {U}(0, \\tau _{3})=\\mathcal {R}_{\\vec{m}}(\\Delta \\phi )=\\cos \\Delta \\phi +i\\sin \\Delta \\phi [\\vec{m}\\cdot \\vec{\\sigma }]$ with $\\vec{m}=(\\sin \\theta \\cos \\phi , \\sin \\theta \\sin \\phi , \\cos \\theta )$ .",
"The Eq.",
"(REF ) indicates that the time evolution operator $\\mathcal {U}(0, \\tau _{3})$ is actually a general $SU(2)$ rotation about the rotation axis $\\vec{m}$ with rotation angle given by non-adiabatic Abelian geometric phase $\\Delta \\phi $ .",
"Thus, the proposed evolution $\\mathcal {U}(0, \\tau _{3})$ introduces a practical route to realize universal non-adiabatic geometric single-qubit gates."
],
[
"Two-Qubit Gates",
"A two-qubit gate on given register qubits $k$ and $l$ in the system is achieved by the two-qubit Hamiltonian $H_{kl}$ identified in Eq.",
"(REF ).",
"Hamiltonian $H_{kl}$ embeds the computation system of two register qubits $k$ and $l$ into a host three-qubit system via, as shown in Fig.",
"REF , indirect coupling of our register qubits $k$ and $l$ through a third auxiliary qubit $a$ .",
"The $H_{kl}$ in the computational basis takes the following double $\\Lambda $ structure $H_{kl}&=&\\frac{J_{l}}{2}\\left| 010 \\right\\rangle \\left\\langle 001 \\right|+\\frac{J_{k}}{2}\\left| 010 \\right\\rangle \\left\\langle 100 \\right|+\\nonumber \\\\&&\\frac{J_{k}}{2}\\left| 101 \\right\\rangle \\left\\langle 011 \\right|+\\frac{J_{l}}{2}\\left| 101 \\right\\rangle \\left\\langle 110 \\right|+h.c.,$ where 0 and 1 at each site in the basis states from left to right, respectively, represent the states of qubits $k$ , $a$ , $l$ .",
"Assume $(J_{k}, J_{l})=\\Omega (\\cos \\frac{\\theta }{2}, \\sin \\frac{\\theta }{2}),$ where $\\Omega =\\sqrt{J_{k}^{2}+J_{l}^{2}}$ .",
"If we fix the angle $\\theta $ and turn on the Hamiltonian $H_{kl}$ for a time interval $[0, \\tau ]$ such that $\\frac{1}{2}\\int _{0}^{\\tau }\\Omega dt=\\pi $ then the double $\\Lambda $ structure of $H_{kl}$ leads to the final time evolution operator $\\mathcal {U}(0,\\tau )=e^{-i\\int _{0}^{\\tau }H_{kl}dt}=\\mathcal {U}_{0}(0,\\tau )\\oplus \\mathcal {U}_{1}(0,\\tau )$ in the ordered basis $\\lbrace \\left| 000 \\right\\rangle , \\left| 001 \\right\\rangle , \\left| 100 \\right\\rangle , \\left| 101 \\right\\rangle , \\left| 010 \\right\\rangle , \\left| 011 \\right\\rangle ,$ $\\left| 110 \\right\\rangle , \\left| 111 \\right\\rangle \\rbrace $ , where $\\mathcal {U}_{0}(0,\\tau )&=&\\left(\\begin{array}{cccc}1 & 0 & 0 & 0 \\\\0 & \\cos \\theta & -\\sin \\theta & 0 \\\\0 & -\\sin \\theta & -\\cos \\theta & 0 \\\\0 & 0 & 0 & -1\\end{array}\\right)\\nonumber \\\\\\mathcal {U}_{1}(0,\\tau )&=&\\left(\\begin{array}{cccc}-1 & 0 & 0 & 0 \\\\0 & -\\cos \\theta & -\\sin \\theta & 0 \\\\0 & -\\sin \\theta & \\cos \\theta & 0 \\\\0 & 0 & 0 & 1\\end{array}\\right).$ We pursue with some remarks and properties of the system in Eq.",
"(REF ) and its time evolution operator described in Eq.",
"(REF ): Let us denote $\\mathcal {H}_{q}=\\text{span}\\lbrace \\left| 0q0 \\right\\rangle , \\left| 0q1 \\right\\rangle , \\left| 1q0 \\right\\rangle , \\left| 1q1 \\right\\rangle \\rbrace $ $q=0, 1$ .",
"Each of the subspaces $\\mathcal {H}_{q}$ , $q=0, 1$ , indeed corresponds to the four dimensional computational subspace of the two register qubits $k$ and $l$ , when the auxiliary qubit is fixed to the state $\\left| q \\right\\rangle $ .",
"Eq.",
"(REF ) indicates that the subspaces $\\mathcal {H}_{q}$ , $q=0, 1$ , evolve in cyclic manners during the time interval $[0, \\tau ]$ .",
"Therefore the time evolution operator components $\\mathcal {U}_{q}(0,\\tau )$ , $q=0,1$ , introduce two-qubit gates on register qubits $k$ and $l$ , if the auxiliary qubit is initialized and measured in the same basis state $\\left| q \\right\\rangle $ .",
"By evaluating the entangling powers [31], [35] $e_{p}[\\mathcal {U}_{0}(0,\\tau )]=e_{p}[\\mathcal {U}_{1}(0,\\tau )]=\\frac{2}{9}[1-\\cos ^{4}\\theta ],$ we obtain that both gates provide the same and full entangling power controlled by the parameter $\\theta $ .",
"For each $\\theta $ satisfying $|\\cos \\theta |<1$ , the gates $\\mathcal {U}_{q}(0,\\tau )$ , $q=0, 1$ , are entangling two-qubit gates and thus they allow for universal quantum information processing when accompanied with universal single-qubit gates given in Eq.",
"(REF ).",
"Moreover, we notice that the gates $\\mathcal {U}_{q}(0, \\tau )$ , $q=0,1$ , have remarkable holonomic natures, which can be verified from two points of views.",
"First: As mentioned in the first remark above, from Eq.",
"(REF ) we have that the subspaces $\\mathcal {H}_{q}$ evolve in cyclic fashions during the time interval $[0, \\tau ]$ .",
"These cyclic evolutions actually take place in the Grassmanian $G(8,4)$ , the space of all four dimensional subspaces of the eight dimensional Hilbert space of the three qubits $k$ , $l$ , and $a$ .",
"We may call $\\mathcal {C}_{q}$ the corresponding loops in the Grassmanian $G(8,4)$ .",
"In addition, for each $q=0, 1$ , one may observe that $\\mathcal {U}(0, t)\\mathbf {P}_{q}\\mathcal {U}^{\\dagger }(0, t)H_{kl}\\mathcal {U}(0, t)\\mathbf {P}_{q}\\mathcal {U}^{\\dagger }(0, t)=0,$ at each time $t\\in [0, \\tau ]$ , where $\\mathbf {P}_{q}$ is the projection operator on the subspace $\\mathcal {H}_{q}$ and $\\mathcal {U}(0, t)=\\exp [-i\\int _{0}^{t}H_{kl}ds]$ is the evolution operator at time $t$ .",
"The Eq.",
"(REF ) follows from $[H_{kl}, \\mathcal {U}(0, t)]=0$ at each time $t$ and that $\\mathbf {P}_{q}H_{kl}\\mathbf {P}_{q}=0$ .",
"Therefore, the Eqs.",
"(REF ) and (REF ) imply that for each $q=0,1$ , the subspace $\\mathcal {H}_{q}$ evolves cyclicly about the corresponding closed path $\\mathcal {C}_{q}$ in the Grassmanian $G(8,4)$ , along which no dynamical phases occur [36].",
"Mathematically speaking, the gate operator $\\mathcal {U}_{q}(0, \\tau )$ , which is actually the projection of the final time evolution operator $\\mathcal {U}(0, \\tau )$ into the subspace $\\mathcal {H}_{q}$ , i.e., $\\mathcal {U}_{q}(0, \\tau )=\\mathbf {P}_{q}\\mathcal {U}(0, \\tau )\\mathbf {P}_{q}$ , is the non-Abelian nonadiabatic quantum holonomy associated with the parallel transport of $\\mathcal {H}_{q}$ about the loop $\\mathcal {C}_{q}$ in the Grassmanian $G(8,4)$ [36].",
"Second: Looking more carefully into the final time evolution operator given by Eqs.",
"(REF , REF ) and the double $\\Lambda $ coupling structure of Eq.",
"(REF ), we see that the two-qubit entangling gates $\\mathcal {U}_{q}(0, \\tau )$ possess even richer holonomic structures.",
"For the sake of simplicity, in the following we restrict ourselves to explain the further holonomic structure of the gate $\\mathcal {U}_{0}(0, \\tau )$ , however the same type of explanation would exists for the gate $\\mathcal {U}_{1}(0, \\tau )$ .",
"We shall rewrite the two-qubit computational space $\\mathcal {H}_{0}$ given in Eq.",
"(REF ) in the following directsum form $\\mathcal {H}_{0}=\\mathcal {H}_{0}^{0}\\oplus \\mathcal {H}_{0}^{2}\\oplus \\mathcal {H}_{0}^{1},$ where $\\mathcal {H}_{0}^{0}=\\text{span}\\lbrace \\left| 000 \\right\\rangle \\rbrace $ , $\\mathcal {H}_{0}^{2}=\\text{span}\\lbrace \\left| 001 \\right\\rangle , \\left| 100 \\right\\rangle \\rbrace $ and $\\mathcal {H}_{0}^{1}=\\text{span}\\lbrace \\left| 101 \\right\\rangle \\rbrace $ .",
"Accordingly, we may put the gate operator $\\mathcal {U}_{0}(0, \\tau )$ in a directsum form as $\\mathcal {U}_{0}(0, \\tau )=(1)\\oplus \\left(\\begin{array}{cc}\\cos \\theta & -\\sin \\theta \\\\-\\sin \\theta & -\\cos \\theta \\end{array}\\right).\\oplus (-1).$ The state $\\left| 000 \\right\\rangle $ does not contribute into the Hamiltonian given in Eq.",
"(REF ) and thus it is kept unchanged during the time evolution of the system.",
"In other words, the evolution of the subspace $\\mathcal {H}_{0}^{0}$ would be stationary with associated trivial phase during any time interval.",
"This explains the first element, $(1)$ , in the right hand side directsum of Eq.",
"(REF ).",
"However, the double $\\Lambda $ structure of Eq.",
"(REF ) implies that the evolutions of $\\mathcal {H}_{0}^{1}$ and $\\mathcal {H}_{0}^{2}$ are non-stationary and, respectively, take place in the three dimensional invariant subspaces $\\text{span}\\lbrace \\left| 101 \\right\\rangle , \\left| 011 \\right\\rangle , \\left| 110 \\right\\rangle \\rbrace $ and $\\text{span}\\lbrace \\left| 001 \\right\\rangle , \\left| 010 \\right\\rangle , \\left| 100 \\right\\rangle \\rbrace $ .",
"Explicitly speaking, for each $d=1, 2$ , the evolution of $\\mathcal {H}_{0}^{d}$ specifies a non-trivial path in the Grassmanian $G(3,d)$ , which we may here call $\\mathcal {C}_{0}^{d}$ .",
"Moreover, the block diagonal form of the final time evolution operator, $\\mathcal {U}(0, \\tau )$ , in the corresponding ordered basis given below Eq.",
"(REF ) further implies that each of the subspaces $\\mathcal {H}_{0}^{d}$ , $d=1, 2$ , undergoes a cyclic evolution during the time interval $[0, \\tau ]$ and thus the corresponding path $\\mathcal {C}_{0}^{d}$ is a closed path in $G(3,d)$ .",
"If we assume $\\mathbf {P}_{0}^{d}$ to be the projection operator on the subspace $\\mathcal {H}_{0}^{d}$ then from $\\mathbf {P}_{0}^{d}H_{kl}\\mathbf {P}_{0}^{d}=0$ and $[H_{kl}, \\mathcal {U}(0, t)]=0$ we obtain $\\mathcal {U}(0, t)\\mathbf {P}_{0}^{d}\\mathcal {U}^{\\dagger }(0, t)H_{kl}\\mathcal {U}(0, t)\\mathbf {P}_{0}^{d}\\mathcal {U}^{\\dagger }(0, t)=0$ at each time $t$ , for $d=1,2$ , which indicates no dynamical phases occur along the cyclic evolutions $\\mathcal {C}_{0}^{d}$ , $d=1, 2$ [36].",
"All these verify that the subspaces $\\mathcal {H}_{0}^{d}$ , $d=1,2$ , are actually parallel transported about the loops $\\mathcal {C}_{0}^{d}$ giving rise to the following nonadiabatic quantum holonomies [36] $U(\\mathcal {C}_{0}^{1})&=&\\mathbf {P}_{0}^{1}\\mathcal {U}(0, t)\\mathbf {P}_{0}^{1}=(-1)\\nonumber \\\\U(\\mathcal {C}_{0}^{2})&=&\\mathbf {P}_{0}^{2}\\mathcal {U}(0, t)\\mathbf {P}_{0}^{2}=\\left(\\begin{array}{cc}\\cos \\theta & -\\sin \\theta \\\\-\\sin \\theta & -\\cos \\theta \\end{array}\\right).$ As a result, we see in Eq.",
"(REF ) that the two-qubit entangling gate $\\mathcal {U}_{0}(0, \\tau )$ is indeed composed of the holonomies in Eq.",
"(REF ), namely $\\mathcal {U}_{0}(0, \\tau )=(1)\\oplus U(\\mathcal {C}_{0}^{2})\\oplus U(\\mathcal {C}_{0}^{1}).$ In conclusion, our analysis above reveals the rich holonomic nature of the two-qubit entangling gate $\\mathcal {U}_{0}(0, \\tau )$ by demonstrating that the gate $\\mathcal {U}_{0}(0, \\tau )$ not only as a whole is a nonadiabatic holonomy but also each of its nonzero block constitutes is a nonadiabatic holonomy."
],
[
"Discussion and summary",
"As the nonadiabatic holonomies became an important approach for implementation of fast fault-tolerant quantum gates, experimental implementation of nonadiabatic holonomic quantum computation with spin qubits, as a natural and suitable platform for realization of quantum computers, caught increasing interests in recent years [16], [17], [18], [19], [20].",
"Despite a number of significant efforts in this area, only single-qubit gates have been addressed.",
"Therefore, still a lack of feasible scheme, which supports scalability as well as universal single-qubit and two-qubit entangling gates all in the same configuration is felt.",
"Compared to the existing works, our scheme above is consist of arbitrary $n$ register spin qubits arranged in a star-shape architecture about a shared single auxiliary spin qubit in the middle (see Fig.",
"REF ).",
"In addition to the scalability, the proposed star-shape configuration permits for universal nonadiabatic holonomic quantum computation, where an arbitrary holonomic single-qubit gate on each register qubit is achieved by a local transverse magnetic field and a two-qubit entangling gate between a given pair of register qubits is performed in a double $\\Lambda $ structure as demonstrated by indirect bridge coupling between the selected register qubits through the auxiliary qubit.",
"While single-qubit gates are realized through Abelian nonadiabatic holonomies [34], the proposed entangling two-qubit gates obey a rich holonomic description associated with non-Abelian as well as Abelian nonadiabatic holonomies [36].",
"All holonomic universal computations take place in the subspace of the system, where the state of auxiliary qubit is fixed to one of its computational basis states (say for instance the basis state $\\left| 0 \\right\\rangle $ ).",
"A universal circuit corresponding to our nonadiabatic holonomic scheme is depicted in Fig.",
"REF .",
"Figure: (Color online) Schematic digram of a universal holonomic quantum computation.",
"Holonomic single-qubit gates are implemented by local transverse magnetic fields, which introduce transverse coupling between two computational basis states of qubits.",
"The auxiliary qubit, which is illustrated in blue, only contributes in two-qubit gates.",
"Two register qubits are coupled through the auxiliary qubit in a double Λ\\Lambda structure allowing for implementation of two-qubit entangling gates.",
"The auxiliary qubit is initialized, and measured at the end of computation in the same computational basis state, which here we selected to be the state 0\\left| 0 \\right\\rangle .In summary, we have proposed a scalable spin-based setup for universal nonadiabatic holonomic quantum computation.",
"We hope the present scheme helps to overcome practical challenges and establish a feasible platform for realization of scalable universal nonadiabatic holonomic quantum computation particularly with spin qubits.",
"The discussion for the holonomic nature of the gates would further improve our understanding of the concept of quantum holonomy in solid state systems and its relation to quantum computation."
],
[
"Acknowledgment ",
"This work was supported by Department of Mathematics at University of Isfahan (Iran).",
"The author acknowledges financial support from the Iran National Science Foundation (INSF) through Grant No.",
"96008297."
]
] | 1808.08547 |
[
[
"No lattice tiling of $\\mathbb{Z}^n$ by Lee Sphere of radius 2"
],
[
"Abstract We prove the nonexistence of lattice tilings of $\\mathbb{Z}^n$ by Lee spheres of radius $2$ for all dimensions $n\\geq 3$.",
"This implies that the Golomb-Welch conjecture is true when the common radius of the Lee spheres equals $2$ and $2n^2+2n+1$ is a prime.",
"As a direct consequence, we also answer an open question in the degree-diameter problem of graph theory: the order of any abelian Cayley graph of diameter $2$ and degree larger than $5$ cannot meet the abelian Cayley Moore bound."
],
[
"Introduction",
"The Lee distance (also known as $\\ell _l$ -norm, taxicab metric, rectilinear distance or Manhattan distance) between two vectors $x=(x_1,x_2,\\cdots , x_n)$ and $y=(y_1,y_2,\\cdots , y_n)\\in \\mathbb {Z}^n$ is defined by $ d_L(x,y)=\\sum _{i=1}^n|x_i-y_i|.",
"$ Let $S(n,r)$ denote the Lee sphere of radius $r$ centered at the origin in $\\mathbb {Z}^n$ , i.e.",
"$S(n,r)=\\left\\lbrace (x_1,\\cdots , x_n)\\in \\mathbb {Z}^n: \\sum _{i=1}^{n}|x_i|\\le r\\right\\rbrace .",
"$ If there exists a subset $C\\in \\mathbb {Z}^n$ such that $\\mathcal {T}=\\lbrace S(n,r)+c: c\\in C \\rbrace $ forms a partition of $\\mathbb {Z}^n$ , then we say that $\\mathcal {T}$ is a tiling of $\\mathbb {Z}^n$ by $S(n,r)$ .",
"If $C$ is further a lattice, then we call $\\mathcal {T}$ a lattice tiling.",
"One may get a geometric interpretation of tilings of $\\mathbb {Z}^n$ by Lee spheres in the following way.",
"Let $\\mathbb {R}$ denote the set of real numbers and $C(x_1, \\cdots , x_n)=\\lbrace (y_1,\\cdots , y_n): |y_i-x_i|\\le 1/2 \\rbrace $ which is the $n$ -cube centered at $(x_1,\\cdots , x_n)\\in \\mathbb {R}^n$ .",
"Let $L(n,r)$ be the union of $n$ -cubes centered at each point in $S(n,r)$ .",
"Figure REF depicts $L(n,r)$ for $n=2,3$ and $r=1,2$ .",
"Figure: Figures of L(2,1)L(2,1), L(2,2)L(2,2), L(3,1)L(3,1) and L(3,2)L(3,2)It is easy to see that a tiling of $\\mathbb {Z}^n$ by $S(n,r)$ exits if and only if a tiling of $\\mathbb {R}^n$ by $L(n,r)$ exists.",
"Figure REF shows a (lattice) tiling of $\\mathbb {R}^2$ by $L(2,2)$ .",
"Actually lattice tilings of $\\mathbb {R}^n$ by $L(n,r)$ for $n=1,2$ and any radius $r$ always exist and lattice tilings of $\\mathbb {R}^n$ by $L(n,1)$ also exist for any $n$ ; see [6].",
"Figure: Tiling of ℝ 2 \\mathbb {R}^2 by L(2,2)L(2,2)This geometric interpretation in $\\mathbb {R}^n$ is quite important, because $L(n,r)$ is close to a cross-polytope when $r$ is large enough.",
"It follows that a tiling of $\\mathbb {R}^n$ with $L(n,r)$ induces a dense packing of $\\mathbb {R}^n$ by cross-polytopes.",
"One can use the cross-polytope packing density or the linear programming method which is originally applied on the Euclidean sphere packing density in [3] to show the following type of results.",
"Result 1.1 For any $n\\ge 3$ , there exists $r_n$ such that for $r>r_n$ , $\\mathbb {R}^n$ cannot be tiled by $L(n,r)$ .",
"Result REF was first obtained by Golomb and Welch who showed in [6] only the existence of $r_n$ .",
"However, the value of $r_n$ three is unspecified.",
"Later, several lower bounds on $r_n$ for the periodic case were obtained by Post [15] and Lepistö [13].",
"In [10] the very first lower bound on $r_n$ is stated.",
"In the same seminal paper [6], Golomb and Welch proposed the following conjecture originally given in the language of perfect Lee codes.",
"Conjecture 1 For $n\\ge 3$ and $r\\ge 2$ , there is no perfect $r$ -error-correcting Lee code in $\\mathbb {Z}^n$ , i.e.",
"$\\mathbb {Z}^n$ cannot be tiled by Lee spheres of radius $r$ .",
"Conjecture REF is still far from being solved, though various approaches have been applied on it.",
"We refer the reader to the recent survey [10] and the references therein.",
"In [10] Horak and Kim suggest that $r=2$ appears to be the most difficult case of Conjecture REF for two reasons.",
"First it is the threshold case, because $\\mathbb {Z}^n$ can always be tiled by $S(n,1)$ .",
"Second the proof of Conjecture REF for $3 \\le n \\le 5$ and all $r \\ge 2$ in [8] is based on the nonexistence of tilings of $\\mathbb {Z}^n$ by $S(n,2)$ for the given $n$ .",
"In this direction, there are several recent advances.",
"In [9], Conjecture REF is proved for $n\\le 12$ and $r=2$ .",
"In [12], Kim presents a method based on symmetric polynomials to show that Conjecture REF is true for $r=2$ and a certain class of $n$ satisfying that $|S(n,2)|$ is a prime.",
"This approach has been further applied to the lattice tilings of $\\mathbb {Z}^n$ by $S(n,r)$ with larger $r$ in [16] and [18].",
"In [19], Zhang and the second author translated the lattice tilings of $\\mathbb {Z}^n$ by $S(n,2)$ or $S(n,3)$ into group ring equations.",
"By applying group characters and algebraic number theory, they have obtained more nonexistence results for infinitely many $n$ with $r=2$ and 3.",
"In this paper, we completely solve the lattice tiling cases of Conjecture REF for $r=2$ and any $n$ .",
"Theorem 1.1 For any integer $n\\ge 3$ , there is no lattice tiling of $\\mathbb {Z}^n$ by $S(n,2)$ .",
"It is worth noting that, in contrast to Result REF which is proved for fixed dimension $n$ , Theorem REF is for fixed radius $r$ and arbitrary $n$ .",
"It is straightforward to show that $|S(n,2)|=2n^2+2n+1$ .",
"According to [17] (see [11] for an alternative proof), when $2n^2+2n+1$ is a prime, a tiling of $\\mathbb {Z}^n$ by $S(n,2)$ must be a lattice tiling.",
"Thus Theorem REF implies the following result.",
"Corollary 1.2 For $r=2$ and $n\\ge 3$ satisfying that $2n^2+2n+1$ is prime, the Golomb-Welch conjecture is true.",
"Our result also answers an important question in graph theory.",
"The degree-diameter problem is to determine the largest graph of given maximum degree $d$ and diameter $k$ .",
"For the general case, the famous Moore bound is an upper bound for the orders of such graphs.",
"Except for $k = 1$ or $d \\le 2$ , graphs achieving the Moore bound are only possible for $d = 3, 7, 57$ and $k = 2$ ; see [1] [4] and [7].",
"Let $G$ be a multiplicative group with the identity element $e$ and $S\\subseteq G$ such that $S^{-1}=S$ and $e\\notin S$ .",
"Here $S^{-1}=\\lbrace s^{-1}: s\\in S\\rbrace $ .",
"The (undirected) Cayley graph $\\Gamma (G,S)$ has a vertex set $G$ , and two distinct vertices $g,h$ are adjacent if and only if $g^{-1}h\\in S$ .",
"In particular, when $G$ is abelian, we call $\\Gamma (G,S)$ an abelian Cayley graph.",
"Let $AC(d,k)$ denote the largest order of abelian Cayley graphs of degree $d$ and diameter $k$ .",
"In [5], an upper bound for $AC(2n,r)$ is obtained which actually equals $|S(n,r)|=\\sum _{i=0}^{\\text{min}\\lbrace n,r\\rbrace }2^{i}\\binom{n}{i}\\binom{r}{i}.$ This value is often called the abelian Cayley Moore bound.",
"An important open question in graph theory is whether there exists an abelian graph whose order meets this bound.",
"For more details about the degree-diameter problems, we refer to the survey [14].",
"By checking the proof of the upper bound for $AC(2n,r)$ in [5], it is not difficult to see that an abelian Cayley graph of degree $2n$ and diameter $r$ achieves this upper bound if and only if there is a lattice tiling of $\\mathbb {Z}^n$ by $S(n,r)$ ; see [19] for the detail.",
"This link is also pointed out in [2].",
"Hence, Theorem REF is equivalent to the following statement.",
"Corollary 1.3 The number of vertices in any abelian Cayley graph of diameter 2 and even degree $d\\ge 6$ is strictly less than the abelian Cayley Moore bound.",
"The rest of this paper is organized as follows: In Section , we introduce the group ring conditions for the existence of a lattice tiling of $\\mathbb {Z}^n$ by $S(n,2)$ .",
"In Section , we prove Theorem REF ."
],
[
"Preliminaries",
"Let $\\mathbb {Z}[G]$ denote the set of formal sums $\\sum _{g\\in G} a_g g$ , where $a_g\\in \\mathbb {Z}$ and $G$ is any (not necessarily abelian) group which we write here multiplicatively.",
"The addition of elements in $\\mathbb {Z}[G]$ is defined componentwise, i.e.",
"$\\sum _{g\\in G} a_g g +\\sum _{g\\in G} b_g g :=\\sum _{g\\in G} (a_g+b_g) g.$ The multiplication is defined by $(\\sum _{g\\in G} a_g g )\\cdot (\\sum _{g\\in G} b_g g) :=\\sum _{g\\in G} (\\sum _{h\\in G} a_hb_{h^{-1}g})\\cdot g.$ Moreover, $\\lambda \\cdot (\\sum _{g\\in G} a_g g ):= \\sum _{g\\in G} (\\lambda a_g) g $ for $\\lambda \\in \\mathbb {Z}$ .",
"For $A=\\sum _{g\\in G} a_g g$ and $t\\in \\mathbb {Z}$ , we define $A^{(t)}:=\\sum _{g\\in G} a_g g^t.$ For any set $A$ whose elements belong to $G$ ($A$ may be a multiset), we can identify $A$ with the group ring element $\\sum _{g\\in G} a_g g$ where $a_g$ is the multiplicity of $g$ appearing in $A$ .",
"Moreover, we use $|A|$ to denote the number of distinct elements in $A$ , rather than the counting of elements with multiplicity.",
"The existence of a lattice tiling of $\\mathbb {Z}^n$ by $S(n,2)$ can be equivalently given by a collection of group ring equations.",
"Lemma 2.1 ([19]) Let $n\\ge 2$ .",
"There exists a lattice tiling of $\\mathbb {Z}^n$ by $S(n,2)$ if and only if there exists a finite abelian group $G$ of order $2n^{2}+2n+1$ and a subset $T$ of size $2n+1$ viewed as an element in $\\mathbb {Z}[G]$ satisfying the identity element $e$ belongs to $T$ , $T=T^{(-1)}$ , $T^{2}= 2G-T^{(2)} +2n$ .",
"We also need the following nonexistence results summarized in [19].",
"Lemma 2.2 For $3\\le n\\le 100$ , there is no lattice tiling of $\\mathbb {Z}^n$ by $S(n,2)$ except possibly for $n=16, 21, 36, 55, 64, 66, 78, 92$ ."
],
[
"Proof of the main result",
"Our objective is to show the nonexistence of $T$ satisfying Conditions REF –REF in Lemma REF .",
"To do so, we do assume such $T$ exists and try to deduce some necessary consequences.",
"The outline of our proof of Theorem REF is as follows: we first investigate $T^{(2)}T \\pmod {3}$ , which provides us some strong restrictions on the multiplicities of elements in $T^{(2)}T$ .",
"In particular, it leads to a proof of Theorem REF when $n\\equiv 0 \\pmod {3}$ ; see Proposition REF .",
"Then we further look at $T^{(4)}T\\pmod {5}$ .",
"For each of the rest 10 possible value of $n$ modulo 15, we can get a contradiction.",
"First, by using Condition REF , we immediately obtain the following: Lemma 3.1 For any $g\\in G\\setminus \\lbrace e\\rbrace $ , $ |\\lbrace (t_1,t_2)\\in T\\times T: t_1t_2=g \\rbrace |={\\left\\lbrace \\begin{array}{ll}1, & \\mbox{ if } g\\in T^{(2)};\\\\2, & \\mbox{ if } g\\in G\\backslash T^{(2)}\\end{array}\\right.",
"}.$ In particular, $T\\cap T^{(2)}=\\lbrace e\\rbrace $ .",
"Moreover, if $t, t_1,t_2\\in T$ , then $t_1t_2=t^2$ if and only if $t_1=t_2=t$ .",
"Observe that by REF , $ T^3 = 2(2n+1)G-T^{(2)}T+2nT$ which means $T^{(2)}T = 2(2n+1)G-T^3+2nT.$ Our strategy is to exploit the above equation.",
"For convenience, we keep the following notation through this section.",
"We write $ T^{(2)}T = \\sum _{i=0}^{N} iX_i$ where $\\lbrace X_i: i=0,1,\\dots , N\\rbrace $ forms a partition of $G$ .",
"It is easy to deduce the following: $2n^2+2n+1 = \\sum _{i=0}^{N} |X_i|,$ and $(2n+1)^2 = \\sum _{i=1}^N i|X_i|.$ Note that $|G|$ is odd and $|T^{(2)}|=2n+1$ .",
"Moreover, as $T^{(2)}\\cap T=\\lbrace e\\rbrace $ , it follows that $e\\in X_1$ .",
"Besides the above two equations on $|X_i|$ 's, we derive another equation based on the inclusion-exclusion principle as follows: Lemma 3.2 $\\sum _{i=1}^N|X_i| = 4n+1 + \\sum _{s=3}^{N}\\frac{(s-1)(s-2)}{2}|X_s|.$ By REF and REF , we can write $T^{(2)}= \\sum _{i=0}^{2n}a_i$ with $a_0=e$ and $a_{i}^{-1}=a_{2n+1-i}$ for $i=1,2,\\dots , 2n$ .",
"Clearly all the $a_i$ 's are distinct from each other and $T^{(2)}T=\\sum _{i=0}^{2n}a_iT.$ First, we prove the following claim.",
"Claim 1.",
"For $0\\le i<j\\le 2n$ , $|a_iT\\cap a_jT|={\\left\\lbrace \\begin{array}{ll}1, & 0=i<j;\\\\2, & 0<i<j.\\end{array}\\right.",
"}$ Observe that $a_it=a_jt^{\\prime }\\in a_iT\\cap a_jT$ if and only if $a_i a_j^{-1}=t^{-1}t^{\\prime }$ for some $t,t^{\\prime }\\in T$ .",
"Recall that $T^{(2)}$ also satisfies Condition REF in Lemma REF .",
"Hence, $a_i a_j^{-1}\\notin T^{(2)}$ if and only if $i\\ne 0$ .",
"If $a_ia_j^{-1} \\in G\\setminus T^{(2)}$ , then by Lemma REF , there exist two distinct $t_1, t_2\\in T$ such that $(t^{-1}, t^{\\prime })=(t_1,t_2)$ or $(t_2,t_1)$ .",
"Hence, $t=t_1^{-1}$ or $t_2^{-1}$ .",
"Consequently, $|a_iT\\cap a_jT|=2$ .",
"On the other hand, if $i=0$ , $a_ia_j^{-1}=e a_j^{-1}=s^2\\in T^{(2)}$ for some $s\\in T$ .",
"By Lemma REF , $t^{-1}=s=t^{\\prime }$ .",
"Hence, $|a_iT\\cap a_jT|=1$ .",
"By the inclusion–exclusion principle and (REF ), we count the distinct elements in $T^{(2)}T$ , $|T^{(2)}T| = \\sum _{i=0}^{2n} |a_iT| - \\sum _{i<j} |a_iT \\cap a_jT| + \\sum _{r\\ge 3} (-1)^{r-1} |a_{i_1}T\\cap a_{i_2}T \\cap \\cdots \\cap a_{i_r}T|,$ where $i_1<i_2<\\cdots <i_r$ cover all the possible values.",
"By definition of $X_i$ 's, its left-hand side equals $\\sum _{i=1}^N|X_i|$ .",
"It is clear that $\\sum _{i=0}^{2n} |a_iT|=(2n+1)^2.$ By Claim 1, $\\sum _{i<j} |a_iT \\cap a_jT| = 2n + 2\\binom{2n}{2}=4n^2.$ Suppose that $g\\in a_{i_1}T\\cap a_{i_2}T \\cap \\cdots \\cap a_{i_r}T$ with $r\\ge 3$ .",
"It means that $g\\in X_s$ for some $s\\ge 3$ .",
"Then the contribution for $g$ in the sum $\\sum _{r\\ge 3} (-1)^{r-1} |a_{i_1}T\\cap a_{i_2}T \\cap \\cdots \\cap a_{i_r}T|$ is $ (-1)^{3-1} \\binom{s}{3}+(-1)^{4-1} \\binom{s}{4} +\\cdots =\\binom{s}{2} -\\binom{s}{1} + \\binom{s}{0} = \\frac{(s-1)(s-2)}{2}.$ Therefore, $\\sum _{r\\ge 3} (-1)^{r-1} |a_{i_1}T\\cap a_{i_2}T \\cap \\cdots \\cap a_{i_r}T| = \\sum _{s\\ge 3} |X_s|\\frac{(s-1)(s-2)}{2}.$ Plugging the above equation, (REF ) and (REF ) into (REF ), we obtain (REF ).",
"Our strategy is to derive a contradiction using Equation (2), (3) and (4).",
"We need to further exploit (REF ).",
"It is natural to consider (REF ) modulus 3 as $T^3\\equiv T^{(3)} \\pmod {3}$ .",
"We then have $T^{(2)}T = 2(2n+1)G-T^{(3)}+2nT \\pmod {3}.$ Note that $|G|=2n^2+2n+1$ is not divisible by 3.",
"Therefore, $|T^{(3)}|=2n+1$ .",
"We first investigate the case when $n\\equiv 0 \\pmod {3}$ .",
"Proposition 3.3 Theorem REF is true for $n\\equiv 0\\pmod {3}$ .",
"Now (REF ) becomes $T^{(2)}T \\equiv 2G-T^{(3)}\\pmod {3}.$ Since all coefficients are non-negative, $|T^{(3)}|=2n+1$ and all coefficients of $T^{(3)}$ is 1, we conclude that $\\sum _{i=0}|X_{3i+1}| = 2n+1, \\quad \\sum _{i=0}|X_{3i+2}| = 2n^2 \\quad \\text{and}\\quad \\sum _{i=0}|X_{3i}| = 0.$ By (REF ) and (REF ), $2n= \\sum _{i=1} 3i (|X_{3i+1}|+|X_{3i+2}|).$ We recall that $N$ is the largest integer with $|X_N|\\ne 0$ .",
"By (REF ) and (REF ), we have $2n^2-2n=\\sum _{s=4}^N\\frac{(s-1)(s-2)}{2}|X_s| \\le \\frac{N-1}{2} \\sum _{s=4}^N (s-2) |X_s|.$ As $T^{(-1)}=T$ and $T^{(-2)}=T^{(2)}$ , it is clear that $X_N=X_N^{(-1)}$ .",
"Recall that $e\\in X_1$ .",
"It then follows that $|X_N|\\ge 2$ .",
"From (REF ), we derive that $2(N-2)\\le 2n$ .",
"By (REF ), $N\\equiv 1, 2\\pmod {3}$ and that $\\sum _{i=0}|X_{3i}|=0$ , we conclude $\\sum _{s=4}^N (s-2) |X_s|\\le 2n$ .",
"Therefore, we obtain from (REF ) that $ 2n^2-2n\\le (n+1)n$ .",
"This is possible only if $n\\le 3$ .",
"As for $n=3$ , it is already known in Lemma REF that Theorem REF is true.",
"Unfortunately, using the above argument in case $n\\equiv \\pm 1 \\pmod {3}$ does not rule the existence out.",
"But we are still able to obtain some essential informations in those cases.",
"Lemma 3.4 Suppose $n\\equiv 1\\pmod {3}$ .",
"Then, $|X_i|=0$ for all $i\\ge 4$ .",
"Moreover, we have $|X_3| = \\frac{4n(n-1)}{3}, \\quad |X_0|=\\frac{2n(n-1)}{3}, \\quad |X_2|=4n \\text{ and } |X_1|=1.$ Now (REF ) becomes $T^{(2)}T \\equiv 2T-T^{(3)} \\pmod {3}.$ We first show that $T\\cap T^{(3)}=\\lbrace e\\rbrace $ .",
"Suppose that $t$ and $t_0\\in T$ satisfy that $t=t_0^3\\in T$ .",
"Then $t t_0^{-1} = t_0^2\\in T^{(2)}$ which means $t=t_0^{-1}=t_0$ by REF .",
"Hence $t=t_0=e$ because $|G|$ is odd.",
"By comparing the coefficients in (REF ), we see that except for those elements in $T\\cup T^{(3)}$ , all are congruent to $0 \\bmod 3$ .",
"Since $T\\cap T^{(3)}=\\lbrace e\\rbrace $ and $e\\in X_1$ , the coefficients of all the elements in $T\\cup T^{(3)}\\backslash \\lbrace e\\rbrace $ are congruent to $2 \\bmod 3$ .",
"Therefore, we get $ |X_1|=1, \\ |X_{3i+1}|=0 \\mbox{ for } i\\ge 1 \\mbox{ and } \\sum _{i=0}|X_{3i+2}|=4n.$ Plugging them into (REF ), we get $ 1+|X_2| +\\sum _{i=3}^N|X_i| = \\sum _{i=0}|X_{3i+2}|+1 + \\sum _{s=3}^{N}\\frac{(s-1)(s-2)}{2}|X_s|,$ from which we conclude that $ 2|X_4| + 4|X_5| +9|X_6| +\\cdots \\le 0.",
"$ This implies that $|X_i|=0$ for $i\\ge 4$ and $|X_2|=4n$ .",
"Hence, by (REF ) and (REF ), $\\left\\lbrace \\begin{array}{l}2n^2+2n+1 = |X_0|+1+4n+|X_3|,\\\\4n^2+4n+1 = 1+2\\cdot 4n +3|X_3|.\\end{array}\\right.$ Solving the above equations, we get the desired result.",
"Next, we consider the case when $n \\equiv 2 \\pmod {3}$ .",
"Lemma 3.5 Suppose $n\\equiv 2\\pmod {3}$ .",
"Then, $|X_i|=0$ for all $i\\ge 5$ .",
"Moreover, we have $|X_1| = \\frac{4n^2-2n+3}{3},\\quad |X_2|= |X_3|=2n,\\quad |X_4|=\\frac{2n^2-4n}{3}\\text{ and }|X_0|=0.$ Now (REF ) becomes $T^{(2)}T \\equiv G-T^{(3)}+T \\pmod {3}.$ As shown before, we have $T\\cap T^{(3)} =\\lbrace e\\rbrace $ .",
"Thus, $X_0=\\emptyset $ and $ T^{(3)}\\backslash \\lbrace e\\rbrace =\\bigcup _{i=0} X_{3i},\\ T\\backslash \\lbrace e\\rbrace =\\bigcup _{i=0} X_{3i+2} \\mbox{ and } G\\backslash (T\\cup T^{(3)}) \\cup \\lbrace e\\rbrace =\\bigcup _{i=0} X_{3i+1}.$ Hence $\\sum _{i=0}|X_{3i+1}| = 2n^2-2n+1,$ $\\sum _{i=0}|X_{3i+2}| = 2n \\quad \\text{and}\\quad \\sum _{i=0}|X_{3i}| = 2n.$ By (REF ), we have $2n^2+2n+1 \\ge 4n+1 + \\sum _{s=3}^{N}\\frac{(s-1)(s-2)}{2}|X_s|.$ Summing up the above equation and (REF ), we have $4n^2-4n+1 \\ge |X_1| + |X_3| + 4|X_4| +6|X_5| + 10|X_6|+16|X_7|+\\cdots .$ On the other hand, plugging (REF ), (REF ) and $X_0=\\emptyset $ into $\\sum _{i=1}i|X_i|-2\\sum _{i=0} |X_{3i}|-2\\sum _{i=0} |X_{3i+2}|$ , we have $ 4n^2-4n+1=|X_1|+|X_3| + 4|X_4| + 3|X_5| + 4|X_6| + 7|X_7| +\\cdots .",
"$ Thus $|X_5|=|X_6|=\\cdots =0$ .",
"It follows from (REF ) that $|X_2|=2n$ .",
"By solving $\\left\\lbrace \\begin{array}{ll}4n^2-4n+1&=|X_1|+|X_3| + 4|X_4|,\\\\4n^2+4n+1&=|X_1|+2|X_2| + 3|X_3| + 4|X_4|,\\\\2n^2-2n+1 &= |X_1|+|X_4|.\\end{array}\\right.$ we get our desired result.",
"In view of the above results, we see that by just considering modulus 3, it doesn't rule out the case for $n\\equiv 1,2 \\pmod {3}$ .",
"It is natural to consider a similar equation modulus 5.",
"By REF , $ T^5 = (8n^2+8n+2)(2n+1)G - T^{(4)}T -4n T^{(2)}T + (4n^2+2n)T$ which implies $T^{(4)}T = (8n^2+8n+2)(2n+1)G - T^5 -4n T^{(2)}T + (4n^2+2n)T.$ As before, we write $ T^{(4)}T = \\sum _{i=0}^{M} iY_i $ where $\\lbrace Y_i: i=0,1,\\dots , M\\rbrace $ forms a partition of $G$ .",
"Since $|T^{(4)}|=2n+1$ , we have : $2n^2+2n+1 = \\sum _{i=0}^{M} |Y_i|,$ and $(2n+1)^2 = \\sum _{i=1}^M i|Y_i|.$ However, the situation is slightly different now.",
"Lemma 3.6 There exists an integer $\\Delta \\in [-2n,0]$ such that $\\sum _{i=1}^M|Y_i| = 4n+1 +\\Delta + \\sum _{s=3}^{M}\\frac{(s-1)(s-2)}{2}|Y_s|.$ Moreover, we have $2|Y_1| + 3|Y_2| +3|Y_3| +2 |Y_4| \\ge 4n^2+6n+2.$ The proof is quite similar to the one for Lemma REF .",
"The only different part is Claim 1.",
"By REF , we may write $T^{(4)}= \\sum _{i=0}^{2n}a^2_i$ with $a_0=e$ and $a_{i}^{-1}=a_{2n+1-i}$ for $i=1,2,\\dots , 2n$ .",
"Hence $T^{(4)}T=\\sum _{i=0}^{2n}a^2_iT.$ Claim 1.",
"$|T\\cap a^2_iT|=2$ for $i>0$ and $ 4n^2- 6n\\le \\sum _{0<i<j\\le 2n+1} |a_i^2T\\cap a_j^2T|\\le 4n^2-4n.$ Suppose that $t\\in T\\cap a^2_iT$ .",
"Then there exists $t_0$ such that $tt_0 = a_i^2\\in T^{(4)}\\setminus \\lbrace e\\rbrace $ .",
"Note that $a_i^2\\in G\\setminus T^{(2)}$ as $T^{(4)}\\cap T^{(2)}=\\lbrace e\\rbrace $ by REF .",
"Hence, there are two choices for $t$ and hence $|T\\cap a^2_iT|=2$ .",
"For $j>i>0$ , as shown before, $ |a_i^2T\\cap a_j^2T| = {\\left\\lbrace \\begin{array}{ll}1, &a_i^{-2}a_j^2\\in T^{(2)},\\\\2, & \\text{ otherwise.}\\end{array}\\right.}",
"$ To find the number of pairs $(i,j)$ with $0<i<j$ when $|a_i^2T\\cap a_j^2T| =1$ , we need to find for each $s\\in T^{(2)}$ , the number of pairs of $(i,j)$ with $i<j$ and $a_i^{-2}a_j^2=s$ .",
"Since $a_i^{-2}, a_j^2\\in T^{(4)}$ and $T^{(4)}$ also satisfies Condition REF , it follows that the number of pairs is at most 1.",
"Therefore, $0\\le \\delta =|\\lbrace (i,j): 0<i<j,~~a_i^{-2}a_j^2\\in T^{(2)}\\rbrace |\\le 2n.",
"$ Thus $\\sum _{i<j} |a^2_iT \\cap a^2_jT| = &\\sum _{i\\ne 0} |T \\cap a^2_iT|+\\sum _{0<i<j} |a^2_iT \\cap a^2_jT|\\\\=&2\\cdot 2n+ 2\\cdot (\\binom{2n}{2}-\\delta )+\\delta \\\\=& 4n^2+2n-\\delta .$ By applying a similar argument to $T^{(4)}T$ as in the proof of in Lemma REF , we obtain $\\sum _{i=1}^M|Y_i| = (2n+1)^2 -(4n^2+2n-\\delta ) + \\sum _{s=3}^{M}\\frac{(s-1)(s-2)}{2}|Y_s|.$ Setting $\\Delta = \\delta -2n$ , we obtain (REF ).",
"By adding up (REF ) and (REF ), we obtain $2|Y_1| + 3|Y_2| +3|Y_3| +2 |Y_4| = 4n^2+8n+2+\\Delta + \\sum _{s=5}(\\frac{(s-1)(s-2)}{2}-s-1)|Y_s|.$ Finally, as the last sum is always non-negative and $\\Delta \\ge -2n$ , we obtain (REF ).",
"Now, we are ready to resume the proof of Theorem REF .",
"First we consider a special case.",
"Proposition 3.7 Theorem REF is true for $n\\equiv 0\\pmod {5}$ .",
"By (REF ), we obtain $ T^{(4)}T \\equiv 2G-T^{(5)} \\pmod {5}.$ In this case, 5 doesn't divide $|G|$ .",
"Therefore, $|T^{(5)}|=2n+1$ .",
"Consequently, $ \\bigcup _{i=0} Y_{5i+1}=T^{(5)}, \\ \\bigcup _{i=1} Y_{5i+2}=G\\backslash T^{(5)}.$ Therefore, $|Y_i|=0$ for $i\\lnot \\equiv 1,2 \\pmod {5}$ and $\\sum _{i=0}|Y_{5i+1}| = 2n+1,\\quad \\sum _{i=0}|Y_{5i+2}| = 2n^2.$ Hence $\\sum _{i=0}|Y_{5i+1}|+ \\sum _{i=0}2|Y_{5i+2}| = 4n^2 + 2n +1.$ On the other hand, $|Y_1| + 2|Y_2| + 6|Y_6| + 7 |Y_7| + \\cdots =\\sum _{i=1}^M i|Y_i|=4n^2+4n+1.",
"$ Together with (REF ), we get $5\\sum _{i=1}i(|Y_{5i+1}|+|Y_{5i+2}|)=2n.$ Recall that $M=\\max \\lbrace i: Y_i\\ne \\emptyset \\rbrace $ .",
"By (REF ) and (REF ), $2n^2-2n-\\Delta = \\sum _{s=3}^M \\frac{(s-1)(s-2)}{2} |Y_s| \\le \\frac{M-1}{2} \\sum _{s=3}^M (s-2) |Y_s|.$ As $|Y_i|=0$ for $i\\lnot \\equiv 1,2 \\pmod {5}$ , it follows from (REF ) $\\sum _{s=3}^M (s-2)|Y_s|\\le 2n.$ Case (i) If $|Y_M|\\ge 2$ , then as in the proof of Proposition REF , we obtain $2(M-2)\\le 2n$ and $M-1\\le n+1$ .",
"Plugging them into (REF ) and (REF ), we get $2n^2-2n-\\Delta \\le (n+1)n$ and $n\\le 3$ .",
"This is impossible.",
"Case (ii) $|Y_M|=1$ .",
"Note that $Y_M=Y_M^{-1}$ .",
"Hence, $Y_M=\\lbrace e\\rbrace $ .",
"If $M<2n+1$ , then $M-1<2n$ .",
"Then, in view of (REF ), there exists $j\\ne M$ such that $|Y_j|\\ge 1$ .",
"Suppose $j=5i+c$ where $i\\ge 1$ and $c=1$ or 2.",
"Again, $Y_j=Y_j^{-1}$ implies, $M-2\\le 2n-10$ .",
"Consequently, $2n^2-2n-\\Delta \\le \\frac{(M-1)}{2}\\sum _{s=3}^M (s-2)|Y_s|\\le (2n-9)n.$ This is impossible as $\\Delta \\le 0$ .",
"Lastly, we assume $M=2n+1$ and $Y_M=\\lbrace e\\rbrace $ .",
"This is possible only when $T=T^{(4)}$ .",
"In that case, $T^{(4)}T=T^2=2G-T^{(2)}+2n$ .",
"It follows from (REF ) that $ T^{(5)}=T^{(2)}$ .",
"For any $t\\in T$ , there exists $s\\in T$ such that $t^5=s^2$ .",
"As $T^{(4)}=T$ , $t^4\\in T$ .",
"Hence, $t^4 t=s^2$ and $s^2\\in T^{(2)}$ .",
"By Lemma REF , this is possible only when $t=t^4=s$ .",
"But it then follows that $t^3=e$ .",
"Hence $|T|\\le 3$ which is impossible.",
"Proposition 3.8 Theorem REF is true if $n\\equiv 1 \\pmod {3}$ .",
"By Proposition REF , we only have to consider the 4 cases when $n\\lnot \\equiv 0 \\pmod {5}$ .",
"(i) $n\\equiv 1\\pmod {5}$ : By (REF ) and Lemma REF , $T^{(4)}T &\\equiv 4G+T^{(2)}T+T-T^{(5)} \\pmod {5}\\\\&\\equiv 4G+X_1 + 2 X_2 + 3X_3+T-T^{(5)} \\pmod {5}.$ As $e\\in T, T^{(5)}, X_1, G$ and $e\\notin X_2, X_3$ , the identity element $e$ appears in $Y_{5i}$ for some $i\\ge 1$ .",
"In view of the above equation, we deduce that $ e\\notin X_3\\setminus ( T\\cup T^{(5)})\\subset \\bigcup _{i=1} Y_{5i+2}, $ $ e\\notin X_0\\setminus ( T\\cup T^{(5)})\\subset \\bigcup _{i=1} Y_{5i+4}, \\mbox{ and }e\\notin X_2\\setminus ( T\\cup T^{(5)})\\subset \\bigcup _{i=1} Y_{5i+1}.$ In view of (REF ), we get $(2n+1)^2 &\\ge \\sum _{i=0}5i|Y_{5i}| +\\sum _{i=0}2|Y_{5i+2}| + \\sum _{i=0}4|Y_{5i+4}|+\\sum _{i=0}|Y_{5i+1}|\\\\&\\ge 5 + \\sum _{i=0}2|Y_{5i+2}| + \\sum _{i=0}4|Y_{5i+4}|+\\sum _{i=0}|Y_{5i+1}|\\\\&\\ge 5 + 2|X_3\\setminus ( T\\cup T^{(5)})|+4|X_0\\setminus ( T\\cup T^{(5)})| + |X_2\\setminus ( T\\cup T^{(5)})| \\\\&\\ge 5 + 2|X_3| + 4|X_0| + |X_2|-4|(T\\cup T^{(5)})\\setminus \\lbrace e\\rbrace |\\\\&\\ge 5+ \\frac{16n^2-52n}{3} \\qquad \\text{(by Lemma \\ref {lm:n=1mod3}),}$ where the second last inequality comes from the fact that the number of elements of $T\\cup T^{(5)}$ in the disjoint union of $X_0, X_2$ and $X_3$ is at most the size of $(T\\cup T^{(5)})\\setminus \\lbrace e\\rbrace $ .",
"This means $4n^2-64n+12\\le 0$ whence $n\\le 15$ .",
"However, according to Lemma REF , this is impossible.",
"(ii) $n\\equiv 2\\pmod {5}$ : $T^{(4)}T &\\equiv 2T^{(2)}T-T^{(5)} \\pmod {5}\\\\&\\equiv 0X_0+2X_1 + 4 X_2 + X_3-T^{(5)} \\pmod {5}.$ Recall that $X_0$ , $X_1$ , $X_2$ and $X_3$ form a partition of $G$ and all nonzero coefficients in $T^{(5)}$ are 1.",
"Therefore, $ Y_1 \\setminus T^{(5)}\\subset X_3 , Y_2\\setminus T^{(5)}\\subset X_1, Y_3\\setminus T^{(5)}=\\emptyset \\mbox{ and } Y_4\\setminus T^{(5)}\\subset X_2.$ It follows that $|Y_1| \\le |X_3|+x_1$ , $|Y_2|\\le |X_1|+x_2$ , $|Y_3|\\le x_3$ and $|Y_4|\\le |X_2|+x_3$ where $x_1+x_2+x_3+x_4\\le |T^{(5)}|=2n+1$ .",
"Hence, from (REF ) we can derive $2|Y_1| + 3|Y_2| +3|Y_3| +2 |Y_4| \\le \\frac{8n(n-1)}{3}+3+8n+3(2n+1)=\\frac{8}{3}n^2+\\frac{34}{3}n+6.$ On the other hand, it follows from (REF ) and (REF ), we have $\\frac{8}{3}n^2+\\frac{34}{3}n+6 \\ge 4n^2+6n+2,$ which means $n\\le 6$ .",
"Hence $n=2$ .",
"However, it contradicts the assumption that $n\\equiv 1 \\pmod {3}$ .",
"(iii) $n\\equiv 3\\pmod {5}$ : By (REF ) and Lemma REF , $T^{(4)}T &\\equiv G - 2T^{(2)}T+2T-T^{(5)} \\pmod {5}\\\\&\\equiv G+ 0X_0+3X_1 + X_2 + 4X_3 + 2T-T^{(5)} \\pmod {5}\\\\&\\equiv 1X_0+4X_1 + 2X_2 + 0X_3 + 2T-T^{(5)} \\pmod {5}.$ In this case, 5 divides $|G|$ and it is not necessarily true that all nonzero coefficients in $T^{(5)}$ are 1.",
"One may write $T^{(5)}=\\sum _{i=1}^{k} i Z_i $ where $\\bigcup _{i=1}^{k} Z_i=T^{(5)}$ and $\\sum _{i=1}^{k} i|Z_i|=2n+1$ .",
"However, as we will see below, we can still get some contradiction by checking the bounds for $|Y_1|, |Y_2|, |Y_3|$ and $|Y_4|$ as before.",
"Observe that $ Y_1\\backslash (T\\cup T^{(5)})\\subset X_0, Y_2\\backslash (T\\cup T^{(5)})\\subset X_2, $ $ Y_3\\backslash (T\\cup T^{(5)})=\\emptyset \\mbox{ and } Y_4\\backslash (T\\cup T^{(5)})=\\emptyset .$ Note that the last equation is true because $X_1=\\lbrace e\\rbrace $ .",
"$&2|Y_1| + 3|Y_2| +3|Y_3| +2 |Y_4|\\\\\\le & 2|X_0|+3|X_2|+3|T\\cup T^{(5)}|\\\\\\le & \\frac{4n(n-1)}{3} +12n+12n +3 \\qquad \\text{(by Lemma \\ref {lm:n=1mod3})}\\\\=& \\frac{4n^2}{3} +\\frac{68n}{3}+3.$ By (REF ), we have $\\frac{4n^2}{3} +\\frac{68n}{3} +3 \\ge 4n^2+6n+2,$ which implies $n\\le 6$ .",
"Taking account of the values of $n$ modulo 3 and 5, we see that $n=13$ .",
"However, according to Lemma REF , there is no lattice tiling of $\\mathbb {Z}^{13}$ by $S(13,2)$ .",
"(iv) $n\\equiv 4\\pmod {5}$ : By (REF ) and Lemma REF , $T^{(4)}T &\\equiv 3G - T^{(2)}T+2T-T^{(5)} \\pmod {5}\\\\&\\equiv 3G + 0X_0+4X_1 + 3X_2 + 2X_3 + 2T-T^{(5)} \\pmod {5}\\\\&\\equiv 3X_0+2X_1 + X_2 + 0X_3 + 2T-T^{(5)} \\pmod {5}.$ As before, we obtain $ Y_1\\backslash (T\\cup T^{(5)})\\subset X_2, Y_2\\backslash (T\\cup T^{(5)})=\\emptyset , $ $ Y_3\\backslash (T\\cup T^{(5)})\\subset X_0 \\mbox{ and } Y_4\\backslash (T\\cup T^{(5)})=\\emptyset .$ Together with (REF ) we get $&2|Y_1| + 3|Y_2| +3|Y_3| +2 |Y_4| \\\\\\le &2|X_2|+3|X_1|+3|X_0|+3|T\\cup T^{(5)}|\\\\\\le &8n+ 3\\frac{2n(n-1)}{3} +12n+3\\\\=&2n^2+18n+3.",
"$ By (REF ), $ 4n^2+6n+2\\le 2n^2+18n+3.$ Hence $n\\le 6$ which means $n=4$ .",
"However, this value has been already excluded by Lemma REF .",
"Proposition 3.9 Theorem REF is true for $n\\equiv 2 \\pmod {3}$ .",
"By Proposition REF , we only have to investigate the 4 cases when $n\\lnot \\equiv 0 \\pmod {5}$ .",
"(i) $n\\equiv 1\\pmod {5}$ : By (REF ) and Lemma REF , $T^{(4)}T &\\equiv 4G+T^{(2)}T+T-T^{(5)} \\pmod {5}\\\\&\\equiv 4G+X_1 + 2X_2 + 3X_3 + 4X_4+T-T^{(5)} \\pmod {5}.$ Thus, $ Y_1\\backslash (T\\cup T^{5)})\\subset X_2, Y_2\\backslash (T\\cup T^{5)})\\subset X_3,Y_3\\backslash (T\\cup T^{5)})\\subset X_4, Y_4\\backslash (T\\cup T^{5)})=\\emptyset .$ Hence, $&2|Y_1| + 3|Y_2| +3|Y_3| +2 |Y_4|\\\\\\le & 2|X_2|+3|X_2|+3|X_4|+3|T\\cup T^{(5)}|\\\\\\le & 10n+(2n^2-4n)+12n+3\\qquad (\\text{By Lemma \\ref {lm:n=2mod3}})\\\\=& 2n^2+18n+3.$ Therefore, $4n^2+6n+2\\le 2n^2+18n+3$ which means $n\\le 6$ .",
"By Lemma REF , this is impossible.",
"Alternatively, as $5\\mid 2n^2+2n+1$ , $8n-3\\ne 5k^2$ for some $k\\in \\mathbb {Z}$ , and $8n+1$ is not a square, one may also use [19] to prove this case.",
"(ii) $n\\equiv 2\\pmod {5}$ : By (REF ) and Lemma REF , $T^{(4)}T &\\equiv 2T^{(2)}T-T^{(5)} \\pmod {5}\\\\&\\equiv 2X_1 + 4 X_2 + X_3 + 3X_4-T^{(5)} \\pmod {5}.$ It is easy to see that $ X_1\\backslash T^{(5)}\\subset \\bigcup _{i=0} Y_{5i+2} \\mbox{ and }X_4\\backslash T^{(5)}\\subset \\bigcup _{i=0} Y_{5i+3}.$ It follows that $\\sum _{i=0}(5i+2)|Y_{5i+2}| \\ge 2|X_1|-2x \\mbox{ and } \\sum _{i=0}(5i+3)|Y_{5i+3}| \\ge 3 |X_4|-3y$ with $x+y\\le 2n+1$ .",
"Thus, by (REF ) $(2n+1)^2=\\sum _{i=1}^Mi|Y_i|\\ge 2|X_1|+3|X_4|-6n-3=\\frac{14n^2-34n}{3}-1.$ By calculation, we get $2n^2-46n-6\\le 0$ which implies that $n\\le 23$ .",
"As $n$ is congruent to $2 \\mod {5}$ , $n\\ne 16, 21$ .",
"Therefore by Lemma REF , we have a contradiction.",
"(iii) $n\\equiv 3\\pmod {5}$ : By (REF ) and Lemma REF , $T^{(4)}T &\\equiv G - 2T^{(2)}T+2T-T^{(5)} \\pmod {5}\\\\&\\equiv G+ 3X_1 + X_2 + 4X_3 + 2X_4 + 2T-T^{(5)} \\pmod {5}\\\\&\\equiv 4X_1 + 2X_2 + 0X_3 +3X_4 + 2T-T^{(5)} \\pmod {5}.$ This implies that $ X_1\\backslash (T\\cup T^{(5)})\\subset \\bigcup _{i=0} Y_{5i+4} \\mbox{ and }X_4\\backslash (T\\cup T^{(5)})\\subset \\bigcup _{i=0} Y_{5i+3}.$ Therefore, $\\sum _{i=0}|Y_{5i+4}| \\ge |X_1|-x,$ and $\\sum _{i=0}|Y_{5i+3}| \\ge |X_4|-y$ where $0\\le x+y\\le 4n+1$ .",
"Hence, by (REF ) $(2n+1)^2=\\sum _{i=1}^Mi|Y_i|\\ge 4|X_1|+3|X_4|-16n -4=\\frac{22n^2-68n}{3},$ which implies that $10n^2-74n-3\\le 0$ whence $n\\le 7$ .",
"According to Lemma REF , there is no such a lattice tiling of $\\mathbb {Z}^n$ by $S(n,2)$ .",
"(iv) $n\\equiv 4\\pmod {5}$ : By (REF ) and Lemma REF , $T^{(4)}T &\\equiv 3G - T^{(2)}T+2T-T^{(5)} \\pmod {5}\\\\&\\equiv 3G + 4X_1 + 3X_2 + 2X_3 + X_4 + 2T-T^{(5)} \\pmod {5}\\\\&\\equiv 2X_1 + X_2 + 0X_3 + 4X_4 + 2T-T^{(5)} \\pmod {5}.$ As before, we obtain $\\sum _{i=0}|Y_{5i+2}| \\ge |X_1|-x,$ and $\\sum _{i=0}|Y_{5i+4}| \\ge |X_4|-y$ where $0\\le x+y\\le 4n+1$ .",
"Hence, by (REF ) $(2n+1)^2=\\sum _{i=1}^Mi|Y_i|\\ge 2|X_1|+4|X_4|-16n-4=\\frac{16n^2-68n}{3}-2,$ which implies that $4n^2-74n-3\\le 0$ whence $n\\le 18$ .",
"As $n$ is congruent to $4 \\mod {5}$ , $n\\ne 16$ .",
"Thus by Lemma REF , this is impossible.",
"Propositions REF , REF and REF together form a complete proof of Theorem REF .",
"Remark 1 Although our main result shows that there is no lattice tiling of $\\mathbb {Z}^n$ by $S(n, 2)$ for all $n\\ge 3$ ; we still do not know whether Golomb-Welch conjecture has been proved in the case of $r = 2$ for infinitely many values of $n$ .",
"The reason is that we do not know whether $f(n) = 2n^2 +2n+1$ is a prime for infinitely many values of $n$ .",
"A positive answer to this question would solve a special case of the famous conjecture of Bunyakovsky (1857) that asks whether there exists an irreducible quadratic polynomial attaining a prime number value infinitely many times."
],
[
"Acknowledgment",
"The authors thank the referees for their helpful comments and suggestions, which improve the presentation of this paper.",
"The first author is supported by grantR-146-000-158-112, Ministry of Education, Singapore.",
"The second author is supported by the National Natural Science Foundation of China (No.",
"11771451) and Natural Science Foundation of Hunan Province (No.",
"2019RS2031)."
]
] | 1808.08520 |
[
[
"Improving the results of string kernels in sentiment analysis and Arabic\n dialect identification by adapting them to your test set"
],
[
"Abstract Recently, string kernels have obtained state-of-the-art results in various text classification tasks such as Arabic dialect identification or native language identification.",
"In this paper, we apply two simple yet effective transductive learning approaches to further improve the results of string kernels.",
"The first approach is based on interpreting the pairwise string kernel similarities between samples in the training set and samples in the test set as features.",
"Our second approach is a simple self-training method based on two learning iterations.",
"In the first iteration, a classifier is trained on the training set and tested on the test set, as usual.",
"In the second iteration, a number of test samples (to which the classifier associated higher confidence scores) are added to the training set for another round of training.",
"However, the ground-truth labels of the added test samples are not necessary.",
"Instead, we use the labels predicted by the classifier in the first training iteration.",
"By adapting string kernels to the test set, we report significantly better accuracy rates in English polarity classification and Arabic dialect identification."
],
[
"Introduction",
"In recent years, methods based on string kernels have demonstrated remarkable performance in various text classification tasks ranging from authorship identification [24] and sentiment analysis [9], [25] to native language identification [26], [18], [19], [17], dialect identification [16], [14], [4] and automatic essay scoring [6].",
"As long as a labeled training set is available, string kernels can reach state-of-the-art results in various languages including English [18], [9], [6], Arabic [13], [19], [14], [4], Chinese [25] and Norwegian [19].",
"Different from all these recent approaches, we use unlabeled data from the test set to significantly increase the performance of string kernels.",
"More precisely, we propose two transductive learning approaches combined into a unified framework.",
"We show that the proposed framework improves the results of string kernels in two different tasks (cross-domain sentiment classification and Arabic dialect identification) and two different languages (English and Arabic).",
"To the best of our knowledge, transductive learning frameworks based on string kernels have not been studied in previous works."
],
[
"Transductive String Kernels",
" Figure: The standard kernel learning pipeline based on the linear kernel.",
"Kernel normalization is not illustrated for simplicity.",
"Best viewed in color.String kernels.",
"Kernel functions [27] capture the intuitive notion of similarity between objects in a specific domain.",
"For example, in text mining, string kernels can be used to measure the pairwise similarity between text samples, simply based on character n-grams.",
"Various string kernel functions have been proposed to date [20], [27], [18].",
"Perhaps one of the most recently introduced string kernels is the histogram intersection string kernel [18].",
"For two strings over an alphabet $\\Sigma $ , $x,y \\in \\Sigma ^*$ , the intersection string kernel is formally defined as follows: $\\begin{split}k^{\\cap }(x,y)=\\sum \\limits _{v \\in \\Sigma ^p} \\min \\lbrace \\mbox{num}_v(x), \\mbox{num}_v(y) \\rbrace ,\\end{split}$ where $\\mbox{num}_v(x)$ is the number of occurrences of n-gram $v$ as a substring in $x$ , and $p$ is the length of $v$ .",
"The spectrum string kernel or the presence bits string kernel can be defined in a similar fashion [18].",
"The standard kernel learning pipeline is presented in Figure REF .",
"String kernels help to efficiently [25] compute the dual representation directly, thus skipping the first step in the pipeline illustrated in Figure REF .",
"Figure: The transductive kernel learning pipeline based on the linear kernel.",
"Kernel normalization and RBF kernel transformation are not illustrated for simplicity.",
"Best viewed in color.Transductive string kernels.",
"We propose a simple and straightforward approach to produce a transductive similarity measure suitable for strings, as illustrated in Figure REF .",
"We take the following steps to derive transductive string kernels.",
"For a given kernel (similarity) function $k$ , we first build the full kernel matrix $K$ , by including the pairwise similarities of samples from both the train and the test sets (step $S1$ in Figure REF ) .",
"For a training set $X = \\lbrace x_1, x_2, ..., x_m\\rbrace $ of $m$ samples and a test set $Y = \\lbrace y_1, y_2, ..., y_n\\rbrace $ of $n$ samples, such that $X \\cap Y = \\emptyset $ , each component in the full kernel matrix is defined as follows (step $S2$ in Figure REF ): $\\begin{split}K_{ij}= k(z_i, z_j),\\end{split}$ where $z_i$ and $z_j$ are samples from the set $Z = X \\cup Y = \\lbrace x_1, x_2, ..., x_m, y_1, y_2, ..., y_n\\rbrace $ , for all $1 \\le i,j \\le m + n$ .",
"We then normalize the kernel matrix by dividing each component by the square root of the product of the two corresponding diagonal components: $\\begin{split}\\hat{K}_{ij} = \\frac{K_{ij}}{\\sqrt{K_{ii} \\cdot K_{jj}}}.\\end{split}$ We transform the normalized kernel matrix into a radial basis function (RBF) kernel matrix as follows: $\\begin{split}\\tilde{K}_{ij} = exp \\left( - \\frac{\\displaystyle 1 - \\hat{K}_{ij}}{\\displaystyle 2 \\sigma ^2} \\right).\\end{split}$ As the kernel matrix is already normalized, we can choose $\\sigma ^2 = 0.5$ for simplicity.",
"Therefore, Equation (REF ) becomes: $\\begin{split}\\tilde{K}_{ij} = exp \\left(-1 + \\hat{K}_{ij}\\right).\\end{split}$ Each row in the RBF kernel matrix $\\tilde{K}$ is now interpreted as a feature vector, going from step $S2$ to step $S3$ in Figure REF .",
"In other words, each sample $z_i$ is represented by a feature vector that contains the similarity between the respective sample $z_i$ and all the samples in $Z$ (step $S3$ in Figure REF ).",
"Since $Z$ includes the test samples as well, the feature vector is inherently adapted to the test set.",
"Indeed, it is easy to see that the features will be different if we choose to apply the string kernel approach on a set of test samples $Y^{\\prime }$ , such that $Y^{\\prime } \\ne Y$ .",
"It is important to note that through the features, the subsequent classifier will have some information about the test samples at training time.",
"More specifically, the feature vector conveys information about how similar is every test sample to every training sample.",
"We next consider the linear kernel, which is given by the scalar product between the new feature vectors.",
"To obtain the final linear kernel matrix, we simply need to compute the product between the RBF kernel matrix and its transpose (step $S4$ in Figure REF ): $\\ddot{K} = \\tilde{K} \\cdot \\tilde{K}^{\\prime }.$ In this way, the samples from the test set, which are included in $Z$ , are used to obtain new (transductive) string kernels that are adapted to the test set at hand.",
"Transductive kernel classifier.",
"After obtaining the transductive string kernels, we use a simple transductive learning approach that falls in the category of self-training methods [22], [5].",
"The transductive approach is divided into two learning iterations.",
"In the first iteration, a kernel classifier is trained on the training data and applied on the test data, just as usual.",
"Next, the test samples are sorted by the classifier's confidence score to maximize the probability of correctly predicted labels in the top of the sorted list.",
"In the second iteration, a fixed number of samples (1000 in the experiments) from the top of the list are added to the training set for another round of training.",
"Even though a small percent (less than $8\\%$ in all experiments) of the predicted labels corresponding to the newly included samples are wrong, the classifier has the chance to learn some useful patterns (from the correctly predicted labels) only visible in the test data.",
"The transductive kernel classifier (TKC) is based on the intuition that the added test samples bring more useful information than noise, since the majority of added test samples have correct labels.",
"Finally, we would like to stress out that the ground-truth test labels are never used in our transductive algorithm.",
"The proposed transductive learning approaches are used together in a unified framework.",
"As any other transductive learning method, the main disadvantage of the proposed framework is that the unlabeled test samples from the target domain need to be used in the training stage.",
"Nevertheless, we present empirical results indicating that our approach can obtain significantly better accuracy rates in cross-domain polarity classification and Arabic dialect identification compared to state-of-the-art methods based on string kernels [9], [14].",
"We also report better results than other domain adaptation methods [23], [3], [8], [28], [11]."
],
[
"Polarity Classification",
"Data set.",
"For the cross-domain polarity classification experiments, we use the second version of Multi-Domain Sentiment Dataset [2].",
"The data set contains Amazon product reviews of four different domains: Books (B), DVDs (D), Electronics (E) and Kitchen appliances (K).",
"Reviews contain star ratings (from 1 to 5) which are converted into binary labels as follows: reviews rated with more than 3 stars are labeled as positive, and those with less than 3 stars as negative.",
"In each domain, there are 1000 positive and 1000 negative reviews.",
"Baselines.",
"We compare our approach with several methods [23], [3], [8], [28], [9], [11] in two cross-domain settings.",
"Using string kernels, franco-EACL-2017 reported better performance than SST [3] and KE-Meta [8] in the multi-source domain setting.",
"In addition, we compare our approach with SFA [23], KMM [10], CORAL [28] and TR-TrAdaBoost [11] in the single-source setting.",
"Table: Multi-source cross-domain polarity classification accuracy rates (in %\\%) of our transductive approaches versus a state-of-the-art (sota) baseline based on string kernels , as well as SST and KE-Meta .",
"The best accuracy rates are highlighted in bold.",
"The marker * indicates that the performance is significantly better than the best baseline string kernel according to a paired McNemar's test performed at a significance level of 0.010.01.Evaluation procedure and parameters.",
"We follow the same evaluation methodology of franco-EACL-2017, to ensure a fair comparison.",
"Furthermore, we use the same kernels, namely the presence bits string kernel ($K_{0/1}$ ) and the intersection string kernel ($K_{\\cap }$ ), and the same range of character n-grams (5-8).",
"To compute the string kernels, we used the open-source code provided by radu-marius-book-chap6-2016.",
"For the transductive kernel classifier, we select $r=1000$ unlabeled test samples to be included in the training set for the second round of training.",
"We choose Kernel Ridge Regression [27] as classifier and set its regularization parameter to $10^{-5}$ in all our experiments.",
"Although franco-EACL-2017 used a different classifier, namely Kernel Discriminant Analysis, we observed that Kernel Ridge Regression produces similar results ($\\pm 0.1\\%$ ) when we employ the same string kernels.",
"As franco-EACL-2017, we evaluate our approach in two cross-domain settings.",
"In the multi-source setting, we train the models on all domains, except the one used for testing.",
"In the single-source setting, we train the models on one of the four domains and we independently test the models on the remaining three domains.",
"Table: Single-source cross-domain polarity classification accuracy rates (in %\\%) of our transductive approaches versus a state-of-the-art (sota) baseline based on string kernels , as well as SFA , KMM , CORAL and TR-TrAdaBoost .",
"The best accuracy rates are highlighted in bold.",
"The marker * indicates that the performance is significantly better than the best baseline string kernel according to a paired McNemar's test performed at a significance level of 0.010.01.Results in multi-source setting.",
"The results for the multi-source cross-domain polarity classification setting are presented in Table REF .",
"Both the transductive presence bits string kernel ($\\ddot{K}_{0/1}$ ) and the transductive intersection kernel ($\\ddot{K}_{\\cap }$ ) obtain better results than their original counterparts.",
"Moreover, according to the McNemar's test [7], the results on the DVDs, the Electronics and the Kitchen target domains are significantly better than the best baseline string kernel, with a confidence level of $0.01$ .",
"When we employ the transductive kernel classifier (TKC), we obtain even better results.",
"On all domains, the accuracy rates yielded by the transductive classifier are more than $1.5\\%$ better than the best baseline.",
"For example, on the Books domain the accuracy of the transductive classifier based on the presence bits kernel ($84.1\\%$ ) is $2.1\\%$ above the best baseline ($82.0\\%$ ) represented by the intersection string kernel.",
"Remarkably, the improvements brought by our transductive string kernel approach are statistically significant in all domains.",
"Results in single-source setting.",
"The results for the single-source cross-domain polarity classification setting are presented in Table REF .",
"We considered all possible combinations of source and target domains in this experiment, and we improve the results in each and every case.",
"Without exception, the accuracy rates reached by the transductive string kernels are significantly better than the best baseline string kernel [9], according to the McNemar's test performed at a confidence level of $0.01$ .",
"The highest improvements (above $2.7\\%$ ) are obtained when the source domain contains Books reviews and the target domain contains Kitchen reviews.",
"As in the multi-source setting, we obtain much better results when the transductive classifier is employed for the learning task.",
"In all cases, the accuracy rates of the transductive classifier are more than $2\\%$ better than the best baseline string kernel.",
"Remarkably, in four cases (E$\\rightarrow $ B, E$\\rightarrow $ D, B$\\rightarrow $ K and D$\\rightarrow $ K) our improvements are greater than $4\\%$ .",
"The improvements brought by our transductive classifier based on string kernels are statistically significant in each and every case.",
"In comparison with SFA [23], we obtain better results in all but one case (K$\\rightarrow $ D).",
"With respect to KMM [10], we also obtain better results in all but one case (B$\\rightarrow $ E).",
"Remarkably, we surpass the other state-of-the-art approaches [28], [11] in all cases."
],
[
"Arabic Dialect Identification",
"Data set.",
"The Arabic Dialect Identification (ADI) data set [1] contains audio recordings and Automatic Speech Recognition (ASR) transcripts of Arabic speech collected from the Broadcast News domain.",
"The classification task is to discriminate between Modern Standard Arabic and four Arabic dialects, namely Egyptian, Gulf, Levantine, and Maghrebi.",
"The training set contains 14000 samples, the development set contains 1524 samples, and the test contains another 1492 samples.",
"The data set was used in the ADI Shared Task of the 2017 VarDial Evaluation Campaign [29].",
"Baseline.",
"We choose as baseline the approach of Radu-Andrei-ADI-2017, which is based on string kernels and multiple kernel learning.",
"The approach that we consider as baseline is the winner of the 2017 ADI Shared Task [29].",
"In addition, we also compare with the second-best approach (Meta-classifier) [21].",
"Evaluation procedure and parameters.",
"Radu-Andrei-ADI-2017 combined four kernels into a sum, and used Kernel Ridge Regression for training.",
"Three of the kernels are based on character n-grams extracted from ASR transcripts.",
"These are the presence bits string kernel ($K_{0/1}$ ), the intersection string kernel ($K_{\\cap }$ ), and a kernel based on Local Rank Distance ($K_{LRD}$ ) [12].",
"The fourth kernel is an RBF kernel ($K_{ivec}$ ) based on the i-vectors provided with the ADI data set [1].",
"In our experiments, we employ the exact same kernels as Radu-Andrei-ADI-2017 to ensure an unbiased comparison with their approach.",
"As in the polarity classification experiments, we select $r=1000$ unlabeled test samples to be included in the training set for the second round of training the transductive classifier, and we use Kernel Ridge Regression with a regularization of $10^{-5}$ in all our ADI experiments.",
"Table: Arabic dialect identification accuracy rates (in %\\%) of our adapted string kernels versus the 2017 ADI Shared Task winner (sota) and the first runner up .",
"The best accuracy rates are highlighted in bold.",
"The marker * indicates that the performance is significantly better than according to a paired McNemar's test performed at a significance level of 0.010.01.Results.",
"The results for the cross-domain Arabic dialect identification experiments on both the development and the test sets are presented in Table REF .",
"The domain-adapted sum of kernels obtains improvements above $0.8\\%$ over the state-of-the-art sum of kernels [14].",
"The improvement on the development set (from $64.17\\%$ to $65.42\\%$ ) is statistically significant.",
"Nevertheless, we obtain higher and significant improvements when we employ the transductive classifier.",
"Our best accuracy is $66.73\\%$ ($2.56\\%$ above the baseline) on the development set and $78.35\\%$ ($2.08\\%$ above the baseline) on the test set.",
"The results show that our domain adaptation framework based on string kernels attains the best performance on the ADI Shared Task data set, and the improvements over the state-of-the-art are statistically significant, according to the McNemar's test."
]
] | 1808.08409 |
[
[
"Multi-scale CNN stereo and pattern removal technique for underwater\n active stereo system"
],
[
"Abstract Demands on capturing dynamic scenes of underwater environments are rapidly growing.",
"Passive stereo is applicable to capture dynamic scenes, however the shape with textureless surfaces or irregular reflections cannot be recovered by the technique.",
"In our system, we add a pattern projector to the stereo camera pair so that artificial textures are augmented on the objects.",
"To use the system at underwater environments, several problems should be compensated, i.e., refraction, disturbance by fluctuation and bubbles.",
"Further, since surface of the objects are interfered by the bubbles, projected patterns, etc., those noises and patterns should be removed from captured images to recover original texture.",
"To solve these problems, we propose three approaches; a depth-dependent calibration, Convolutional Neural Network(CNN)-stereo method and CNN-based texture recovery method.",
"A depth-dependent calibration is our analysis to find the acceptable depth range for approximation by center projection to find the certain target depth for calibration.",
"In terms of CNN stereo, unlike common CNNbased stereo methods which do not consider strong disturbances like refraction or bubbles, we designed a novel CNN architecture for stereo matching using multi-scale information, which is intended to be robust against such disturbances.",
"Finally, we propose a multi-scale method for bubble and a projected-pattern removal method using CNNs to recover original textures.",
"Experimental results are shown to prove the effectiveness of our method compared with the state of the art techniques.",
"Furthermore, reconstruction of a live swimming fish is demonstrated to confirm the feasibility of our techniques."
],
[
"Introduction",
"There are strong demands on capturing dynamic scenes of underwater environments, , measurement of seabeds, capturing dynamic shape deformations of swimming fish or humans, inspection of water-filled nuclear tanks by autonomous robots, etc.",
"Passive stereo is a common solution for capturing 3D shapes because of its great advantage of simplicity; , it only requires two cameras in theory.",
"In addition, since the shapes are recovered only from a pair of stereo images, it can capture moving or deforming objects.",
"One severe problem on passive stereo is instability, , it fails to capture objects with textureless surfaces or irregular reflection.",
"To overcome the problem, using a pattern projector to add an artificial texture onto the objects has been proposed [15].",
"In the system, we also take the same approach to achieve robust and dense reconstruction.",
"Considering underwater environments, there are additional problems for shape reconstruction by stereo, such as refraction and disturbances by fluctuation and bubbles.",
"Further, since original textures of objects are interfered by projected patterns if active illumination is projected, they should be removed for obtaining both 3D shapes and textures.",
"In this paper, we propose three approaches to solve aforementioned problems.",
"For the refraction issue, a depth-dependent calibration where refractions are approximated by lens distortion of a center projection model is proposed [13].",
"In the paper, we analyze to find the acceptable depth range for the approximation and find the best depth for calibration.",
"For the problems of disturbances by obstacles, we propose Convolutional Neural Network(CNN)-based stereo as a solution.",
"Since captured images of underwater scenes are affected by mixtures of light attenuation caused by strong absorption of light intensity in water medium and strong disturbances such as bubbles, shadows of water surface or fluctuation, it is impossible to decompose them analytically.",
"To handle such difficult problems, learning-based approaches, especially CNN techniques, are proposed.",
"Our shape reconstruction consists of two techniques, such as CNN-based object segmentation and CNN-based stereo matching.",
"The CNN-based target object segmentation method efficiently segment a target object, , fish in our experiment, from background, which is not only useful for reducing calculation times, but also effective to achieve robust reconstruction by narrowing the search ranges of stereo disparities.",
"CNN-based stereo effectively works under common variations [31], however, there are strong disturbances at underwater environment.",
"In case of such strong disturbances, we propose a novel architecture of CNN, which uses multi-scale information of captured images.",
"For the texture recovery, we also propose a CNN-based method for projected-pattern removal and bubble cancellation.",
"Main contributions of the proposed technique are as follows: A practical technique is proposed to achieve dense and robust shape reconstruction based on passive stereo using active pattern projection.",
"A valid depth range for depth-dependent approximation by radial distortion is analysed.",
"A target-region detection method by CNN for robust stereo matching is proposed.",
"A multi-scale CNN-based stereo technique specialized for underwater environment is proposed.",
"A multi-scale CNN-based bubble and projected pattern removal method specialized for underwater environment is proposed.",
"Experimental results are shown to prove the effectiveness of our method by comparing the results with the previous method.",
"We also conduct demonstration to show the reconstructed sequence of a swimming fish."
],
[
"Related works",
"To recover shape and texture of underwater environment, many researches have been done.",
"Main issue for underwater environment is refraction and generally two types of solution are proposed; one is geometric approach and the other is approximation-based approach.",
"Geometric approach is based on physical models such as refractive index, distance to refraction interface, and normal of the interface.",
"Agrawal et al.",
"introduced polynomial formulation for the model [1].",
"Sedlazeck and Koch proposed structure from motion for underwater environment [11].",
"Kawahara et al.",
"proposed pixel-wise varifocal camera model [12].",
"In this model, appropriate focal lengths are assigned to each pixel.",
"Those techniques can calculate genuine light rays if parameters are correctly estimated and interface is completely planar, however, they are usually impractical.",
"On the other hand, approximation approach converts captured images into central projection images by lens distortion and focal length adjustment [7].",
"They assumed focal point moved backward to adjust light paths as linear as possible, then remaining error was treated as lens distortion.",
"Kawasaki et al.",
".",
"also proposed a simple method to approximate the refraction by radial distortion [13].",
"Since the parameter cannot be fixed for all the depth range, they proposed a depth dependent technique.",
"It works well in most cases, however in specific case it fails because refractive distortion depends on depth and effective range of depth is not thoroughly analyzed yet.",
"Another problem for underwater environment is disturbances by bubbles, water fluctuation and other effects.",
"Recently, convolutional neural network (CNN) based stereo matching becomes popular, which is robust to irregular distortion on image set.",
"Z̆bontar and LeCun proposed a CNN-based method to train network as a cost function of image patches [31].",
"Those techniques rather concentrate on textureless region recovery, but not noise compensation, which is a main problem for underwater stereo.",
"Since patch based technique is known to be slow, Luo et al.",
"proposed a speeding-up technique by substituting FCN to inner product at final stage [20].",
"Shaked and Wolf achieved high accuracy as well as fast calculation time by combining both FCN to inner product [26].",
"To fundamentally solve the calculation time, end-to-end approach called DispNet is proposed, but accuracy is not so high [21].",
"Another aspect for underwater environment is that range of the scale of obstacles is large.",
"Recently, to solve such scaling problem, multi-scale CNN technique is proposed.",
"Nah et al.",
"proposed a method for deblurring [22], Zhaowei et al.",
"proposed a method for dehaze [3] and Li et al.",
"proposed a method for object recognition [16], Yadati et al.",
", Lu et al.",
", and Chen et al.",
"[29], [19], [4] used multi-scale features for CNN-based stereo matching.",
"We also use multi-scale features for CNN-based stereo matching, but novel network architecture to recognize multi-scale information is proposed.",
"Collection of huge data for learning is another open problem for CNN-based stereo techniques.",
"For solution, Zhou et al.",
"proposed a technique without using ground truth depth data, but LR consistency as a loss function [33].",
"Tonioni et al.",
"proposed a unsupervised method by using existing stereo technique as an instruction [27].",
"Tulyakov and Ivanov proposed a multi-instance learning (MIL) method by using several constraints and cost functions [28].",
"We also take a similar approach to [28] and use several cost functions.",
"CNNs are also popular in the field of image restoration and segmentation.",
"In underwater environment, there are several noises, such as bubbles or shadows of water surfaces.",
"In addition, projected pattern onto the target object is also a severe noise.",
"To remove such a large noise, inpainting method based on a GAN is promising [10], [30].",
"However, since resolution of generative approaches are basically low, noise removal approach is better fit to our purpose.",
"For efficient noise removal, shallow CNN-based approach using residual is proposed [8].",
"The technique is also extended to remove reflection [6].",
"Liu and Fang propose an end-to-end architecture using the WIN5RB network [18] which outperform others.",
"We also use this technique, but data collection and multi-scale extension is novel.",
"Liao et al.",
"[17] denoised depth images both using depth image and RGB image.",
"Zhang et al.",
"[32] denoised images with CNNs with different noise levels taken into account.",
"Choi et al.",
"[5] proposed denoising with multi-scales with light-weight computation.",
"Nakamura et al.",
"[23] removed texts in natural scene images using multi-layers of convolutions and deconvolutions.",
"Image segmentation is also important for our system, since usually only the regions of the target object are enough for 3D shape reconstruction.",
"Badrinarayanan et al.",
".",
"proposed a network architecture for semantic segmentation called SegNet [2].",
"Ronneberger et al.",
"also proposed a network architecture called U-Net which is useful for biomedical image segmentation [24].",
"Since captured images do not look similar to scenery image, but rather close to biomedical image, we use U-Net for our segmentation."
],
[
"System Configuration",
"Our system consists of stereo camera pair and one laser projector as shown in Fig.",
"REF .",
"We prepare two systems for our experiments.",
"One is for evaluation purpose where two cameras and a projector are set outside a water tank.",
"The other is a practical system where devices are installed into a specially built waterproof housing in order to make distance between interface glass and camera lens to be relatively short.",
"For the both systems, the optical axes of the cameras are set orthogonal to glass surface so that error by refraction approximation is minimized.",
"The two cameras are synchronized by GPIO cable to capture dynamic scenes.",
"In terms of the pattern projector to add textures onto the objects, no synchronization is required since the pattern is static.",
"In our implementation, we use a laser projector where diffractive optic element (DOE) is used to configure wave pattern proposed in [25] without losing light power.",
"Figure: Left: Minimum system configuration of the proposed algorithm.",
"Right: Ourexperimental system for evaluation where two cameras and a projector are set outside a water tank.Figure: Overview of the algorithm.Figure: Depth-dependent error of approximation estimated by simulation."
],
[
"Algorithm",
"The algorithm of our underwater shape reconstruction will be explained by using Fig.",
"REF .",
"First, the camera pair is calibrated.",
"The refractions in the captured images are modeled and canceled by center projection approximation in our technique using depth-dependent intrinsic and extrinsic parameters which are acquired in advance.",
"In the measurement process, the targets are captured with stereo cameras.",
"Pattern illumination is projected onto the scene for adding features on it.",
"From captured images, target regions are detected by a CNN-based segmentation technique, where only fish regions are extracted.",
"Then, a stereo-matching method is applied to the target regions.",
"In our technique, a CNN-based stereo is applied to increase stability under the condition of dimmed patterns, disturbances by bubbles, and flickering shadows.",
"Then, 3D points are reconstructed from the disparity maps estimated by the stereo algorithm.",
"Outliers are removed from the point cloud and meshes are recovered by Poisson equation method [14].",
"Since textures are degraded by bubbles and projected patterns, they are efficiently recovered by CNN-based bubble canceling and pattern removal techniques.",
"Using the recovered 3D shapes and textures, we can render the dynamic and textured 3D scene."
],
[
"Depth-dependent calibration",
"Because of refractions, captured images of underwater scene are severely distorted.",
"In this paper, we undistort captured images by a lens distortion model [13].",
"The technique is only an approximation, because refraction effect is not strictly represented as the lens-distortion model, but it can be used for stereo matching for limited working distances [13].",
"For the actual process, a calibration tool, , planar board with checker pattern, is submerged to a water tank to retrieve intrinsic and extrinsic (camera-to-camera transformation) parameters, and thus, it is preferable if the best depth for approximation is known in advance.",
"In the paper, we simulate error using actual parameter of our system as shown in Fig.",
"REF , showing that a maximum error is below 0.8% if depth range is less than 1m.",
"Thus, we set all the devices as close as possible to the water interface so that the error becomes small enough to be ignored."
],
[
"CNN based stereo technique with pattern projection",
"In the technique, we first apply CNN-based target region extraction technique (Sec.",
"REF ) to increase robustness as well as decrease calculation time Then, multi-scale CNN stereo (Sec.",
"REF ) is applied to reconstruct 3D points."
],
[
"CNN-based target-region extraction",
"For many applications, reconstruction targets are recognizable, such as swimming fishes in the water.",
"In general, the wider the range of disparities considered in stereo-matching processes, the more ambiguities exist, leading to wrong correspondences.",
"Thus, by extracting the target regions from the input images and reducing possibilities of matching within the detected target regions, 3D reconstruction process becomes more robust.",
"To this purpose, we implemented an U-Net [24], an FCN with multi-scale feature extraction, and trained it.",
"We made training dataset from underwater image sequence contains live fish (since one of our applications is live fish measurement) where scenes are illumination by the pattern projector.",
"Since both the target and background regions are projected with the same pattern, segmentation between those regions was difficult.",
"From image sequences, 100 images were sampled and the target regions were masked with manual operations.",
"These training images were augmented by scalings, rotations, and translations.",
"As a result, we provide 980 pairs of source images and target-region masks for training U-Net.",
"We used softmax entropy for loss function.",
"The trained U-Net was tested for large number of images, we obtained qualitatively successful results in most examples (Fig.",
"REF ).",
"In the evaluation process, we have found that the numbers of resolution levels of the U-Net architecture is important.",
"By using only two or three levels of resolutions, we could not get sufficient results.",
"We finally reached the conclusion that the U-Net with five levels of resolutions works effectively with our dataset by qualitative evaluation increasing number of levels.",
"Regarding the number of training data pairs, 300 augmented image pairs from around 30 annotated data did not work sufficiently for a living fish, but at least 100 pairs were required.",
"Using the obtained results, rectified images are masked so that only measurement target is on the images.",
"We also use this mask image to limit the output disparity of stereo matching, which can drastically decrease calculation time as well as improve accuracy.",
"Figure: An example of CNN segmentation.",
"Left: Successful example.",
"Right: Minor failure example.",
"Patternless region was difficult to detect."
],
[
"CNN stereo matching by transfer learning",
"In general, normal stereo-matching methods such as SGBM are not robust against strong noises since they do not classify pixels into right intensity and wrong intensity [9].",
"Because CNN-based stereo proposed in [31] learns from real images, it is possible to cope with the noises.",
"In the technique, small image patches from stereo image pairs are processed by CNNs and their feature vectors are calculated.",
"Similarity measures of the feature vectors are used to find the best-matching disparities for every patches of the input images.",
"In the method, we propose an effective training method for CNN-based stereo specialized for bubble-disturbed images by applying a transfer-learning technique.",
"First, we made a training dataset disturbed by bubbles from Middlebury 2005 and 2006 dataset.",
"Middlebury dataset contains 1890 images in total, and we used 540 images of them.",
"To create images with bubbles, we set a display monitor behind a water tank and put a bubble generator inside a tank (Fig.",
"REF (left)).",
"The Middlebury images were presented on the monitor and captured by the camera in front of the water tank.",
"The captured images were warped both by the perspective projection and the refraction by the air-water interfaces.",
"To compensate for this, gray code was presented on the display screen and captured by the camera.",
"Then, lens distortion parameters are estimated, which approximate the refraction, by using the gray code.",
"The captured images were undistorted by the lens distortion parameters and rectified by homography transformation.",
"Examples of a source image and their bubble-disturbed images are shown in Fig.",
"REF (right).",
"Since Middlebury dataset is annotated with ground-truth disparities, we can get positive and negative pairs of image patches for stereo-matching training data.",
"The positive pairs of patches are sampled from stereo images with corresponding positions, whereas the negative pairs are sampled randomly.",
"Using these matching pair datasets, we additionally trained the CNN-based similarity measure pipeline with the captured dataset.",
"Figure: Capturing images through bubbles to create real learning dataset."
],
[
"Multi-scale CNN stereo",
"CNN-based stereo techniques usually take fixed-size image patches because a large number of patches with wide variation are trained.",
"However, it sometimes makes wrong correspondences unless wider regions are considered; repetitive pattern of windows are well known example.",
"Similarly, we assume bubbles whose shapes and sizes vary by large scale, the ambiguity can increase and cause serious failures.",
"Therefore, we propose a novel network architecture for stereo matching called multi-scale CNN Stereo, which can cope with such ambiguities (Fig.",
"REF (Left)).",
"The network takes two image patches as input, and outputs similarity score between the patches.",
"One input patch is processed by two CNN-layer pipelines, one is for low-resolution, wide-range process, and the other is high-resolution, narrow-range process.",
"The input patch is scaled to half through MaxPooling operation for low-res process, and the center sub-image of the input is considered for high-res process.",
"Each of the convolutional layers is composed of 3$\\times $ 3 convolution, batch normalization, and ReLU operation.",
"As a result, two processed patches (high and low-res) have the same sizes with half the original patches with 64 channels.",
"The high and low-res results are concatenated, and used as a feature vector to measure similarities.",
"The neural network parameters are optimized to minimize a hinge loss expressed as $loss = max(0, s_- - s_+ + m),$ , high similarity score is marked to positive patch pair, while low similarity score is marked to negative patch pair, where $s_-$ is output score of negative patch pair, $s_+$ is that of positive patch pair and $m$ is margin which means positive score must exceed negative score at this value.",
"In our training, we used $m = 0.2$ as the margin.",
"Using both high and low-res information helps recognizing wide area and narrow area similarities at the same time, and it leads to robustness against underwater disturbances.",
"The ability of Multi-scale CNN Stereo is shown in Fig.",
"REF .",
"We trained the multi-scale CNN with training dataset created from modified (, with bubble) Middlebury dataset similarly with section REF with data augmentation of random rotations, scalings, and brightness changes.",
"Note that input patches were explicitly extracted from the same epipolar lines of input images in training phase, but whole image can be inputted in estimation phase.",
"Figure: Left: Network architecture of multi-scale CNN Stereo.Right: Network architecture of multi-scale CNN pattern removal.Numbers of the data description (round-cornered rectangles) are data dimensions."
],
[
"Texture recovery from noise, bubble and projected pattern",
"For real situations, the captured images are often severely degraded by underwater environments, such as bubble and other noises, as well as projected pattern on the object surface.",
"In order to remove such undesirable effects, we propose a CNN-based texture recovery technique.",
"In our technique, we focus on two major problems, such as bubbles and projected patterns.",
"Although those two phenomena are totally different and have different optical attributes, it is common in the sense that appearances for both effects have a wide variation in scale.",
"Note that such wide variation depends on the distance between a target object, bubble and a projector.",
"Such a large variation of scale makes it difficult for removal by simple noise removal method.",
"Since multi-scale CNN is suitable to learn such a variation, we also use a multi-scale CNN for our bubble and pattern removal purpose.",
"The network for such obstacle removal is shown in Fig.",
"REF (Right).",
"In the figure, it is shown that an original image is converted to three different resolutions and trained by independent CNN.",
"Each output is up-sampled and concatenated to higher resolution.",
"This network is advantageous because it can handle a large structure of projected pattern, as well as it can be trained in a relatively short time.",
"We prepared two datasets to train the network for bubble removal and pattern removal.",
"For training bubble removal network, we also used Middlebury dataset containing bubbles mentioned in Sec.",
"REF .",
"For training pattern removal network, we captured several real targets with/without pattern projection to create training data.",
"However, the number of data is not sufficient to train the network, we synthesize training data by using CG.",
"We use Middlebury dataset and reconstruct 3D shape with texture map, and then, use virtual pattern projector to add pattern onto the object surface.",
"Then, images were translated, rotated, and scaled randomly for data augmentation.",
"The pattern removal ability of this network is shown in the experiment."
],
[
"Experiments",
"To evaluate proposed method, we conducted 4 experiments.",
"In Sec.",
"REF , we describe how our method is accurate and dense under depth-dependent calibration.",
"In Sec.",
"REF , it is examined that how our multi-scale CNN stereo is robust against underwater disturbances.",
"In Sec.",
"REF , qualitative evaluation results of texture recovery are shown.",
"Finally in Sec.",
"REF , we captured and reconstructed real swimming fish to confirm the feasibility of our method."
],
[
"Validation of shape reconstruction by depth-dependent calibration",
"For the experiments, we used Point Grey Grasshopper3 cameras and Canon LV-HD420 lamp projector.",
"To reproduce underwater environments, we used a water tank with a size of 90$\\times $ 45$\\times $ 45cm.",
"Target objects were a calibration board, a vinyl model of fish, and a silicon model of a human head as shown in Fig.",
"REF .",
"They are captured in the air and reconstructed with a structured-light technique to acquire the ground-truth.",
"The cameras and the projector were calibrated at a distance of 60cm by our depth-dependent calibration technique and captured images were converted to center projection image.",
"Each target object is placed at different distances, ranging from 40 to 80cm by 10cm intervals and captured with/without pattern projection, , in total 180 images were captured.",
"Then, all the objects were reconstructed by the proposed method and the numbers of the reconstructed points and measured ICP residual errors from the ground-truth were calculated.",
"The results are shown in Fig.",
"REF .",
"It is proved that all shapes are successfully recovered with our depth-dependent calibration technique.",
"Further, it can be confirmed that, in most cases, a larger number of points were reconstructed with pattern projection than without projection.",
"The accuracies were also better than without pattern projection in most cases.",
"Figure: Upper row shows target objects and bottom row shows reconstruction results.",
"Left to right:a calibration board, a vinyl fish and a mannequin head.Figure: Live fish experiment.",
"Top: Captured images.",
"Middle: Segmentation results.",
"Bottom: Reconstruction results.Figure: Graph of accuracy and density experiment.",
"Horizontal axis represents number of reconstructed points, vertical axis represents RMSE from the GT shape, and lower right point is better result.Our pattern projection based passive stereo method drasticallyimproves the RMSE as well as point density."
],
[
"Evaluation of various CNN stereo techniques",
"Next, we tested CNN-based stereo for underwater scene with bubbles.",
"For evaluation purpose, we prepared four implementations, such as CNN-based stereo of [31], multi-scale CNN stereo with linear combination (ms-cnn-lin), multi-scale CNN stereo with FCN (ms-cnn-fcn), and transfer learned ms-cnn-lin with bubble erased images (ms-cnn-lin(trans)).",
"The target objects were placed at a distance of 50, 60, 70cm and the depth-dependent calibration was applied as same as the previous experiment.",
"We intentionally made bubbles to interfere image capturing process.",
"We reproduced four bubble environments, , far little bubble, far much bubble, near little bubble, and near much bubble.",
"In addition, no bubble scenes as reference were prepared.",
"We captured three pairs of images for each target with five environments.",
"In total, 90 images were captured.",
"Then, we removed bubbles on the images with multi-scale bubble removal architecture, and reconstructed all the scenes and targets.",
"We calculated average RMSE from the GT shape of each target.",
"The results are shown in Fig.",
"REF .",
"From the graph, we can confirm that the accuracy of proposed CNN architecture is better than previous method, supporting the effectiveness of our method.",
"Fig.",
"REF shows examples of the reconstructed disparity maps (masked with segmentation results) for each technique confirming that shapes are recovered by our technique even if captured images are severely degraded by bubbles.",
"Figure: Comparison on proposed method and previous method.",
"Our methods(ms-cnn-fcn) performed best in most cases.",
"(erased) means result from bubble removed images.Figure: Difference of disparity maps between stereo methods in bubble scene.Bubble is so severe and almost any method can produce quite poor results, whereas our methodproduced much better results."
],
[
"Experiments of texture acquisition",
"We also tested the bubble-removal and the pattern-removal techniques.",
"The results are shown in Fig.",
"REF .",
"It is shown that bubbles in the source images were successfully removed as shown in the top row of the figure.",
"In the bottom row, we can also confirm that projected patterns are robustly removed by multi-scale CNN technique.",
"Figure: Result of texture acquisition experiment.",
"Left pane is results of bubble removal and Right pane is results of pattern removal.",
"Left: Input images.",
"Middle-left: Close-up view of input.",
"Middle-right: Output images.",
"Right: Close-up view of output.",
"(a, b): Middlebury dataset.",
"(c): Fish model.",
"(d, e): Pattern removal dataset we created.",
"(f): Live fish."
],
[
"Demonstration with a live fish",
"Finally, we captured a live swimming fish (filefish) at an aquarium.",
"We used a special experimental system with an aluminum housing.",
"Cameras are same as the above experiment, but a projector is substituted by laser pattern projector.",
"We captured and reconstructed 360 frames.",
"Five frames from the results are shown in Fig.",
"REF for example.",
"As shown in the figure, we can confirm that the target object is mostly successfully segmented by our CNN-based object segmentation method.",
"In addition, dense shapes of the swimming fish are successfully reconstructed, which proves the effectiveness and practicality of our method.",
"Texture are also partially recovered with our method."
],
[
"Conclusion",
"The paper presents a practical underwater dense shape reconstruction technique as well as texture refinement method using stereo cameras with a static-pattern projector.",
"Since underwater environments have severe conditions, such as refraction, light attenuation and disturbances by bubbles, we propose a CNN-based solutions, such as a target-object segmentation, robust stereo matching with a multi-scale CNN and CNN based texture-recovery method.",
"By comparing 3D shape reconstruction with various methods, since other methods are severely affected by bubbles and other degradation of underwater environment, our method achieved best among them.",
"Further, bubbles and projected patterns on the objects are successfully removed by our method.",
"We also conducted experiments to show that our approximation of refraction by radial distortion is feasible.",
"Our future plan is to apply the technique to a swimming human for sports analysis."
],
[
"Acknowledgment",
"This work was part supported by grant JSPS/KAKENHI 16H02849, 16KK0151, 18H04119, 18K19824 in Japan, and MSRA CORE14."
]
] | 1808.08348 |
[
[
"The TauSpinner approach for electroweak corrections in LHC Z to ll\n observables"
],
[
"Abstract The LHC enters era of the Standard Model Z-boson couplings precise measurements, to match precision of LEP.",
"The calculations of electroweak (EW) corrections in the Monte Carlo generators become of relevance.",
"Precise predictions of Z-boson production and decay require classes of QED/EW/QCD corrections, preferably in the manner which allows for separation from the QCD dynamics of the production.",
"At LEP, calculations, genuine weak and lineshape corrections were introduced into electroweak form-factors and Improved Born Approximation.",
"This was well suited for so-called doubly-deconvoluted observables around the Z-pole; observables for which the initial- and final-state QED real and virtual emissions are treated separately or integrated over.",
"This approach to EW corrections is followed for LHC pp collisions.",
"We focus on the EW corrections to doubly-deconvoluted observables of Z to ll process, in a form of per-event weight and on numerical results.",
"The reweighting technique of TauSpinner package is revisited and the program is enriched with the EW sector.",
"The Dizet library, as interfaced to KKMC Monte Carlo of the LEP era, is used to calculate O(alpha) weak loop corrections, supplemented by some higher-order terms.",
"They are used in the form of look-up tables by the TauSpinner package.",
"The size of the corrections is evaluated for the following observables: the Z-boson resonance line-shape, the outgoing leptons forward-backward asymmetry, effective leptonic weak mixing angles and the lepton distribution spherical harmonic expansion coefficients.",
"Evaluation of the EW corrections for observables with simplified calculations based on Effective Born of modified EW couplings, is also presented and compared with the predictions of Improved Born Approximation where complete set of EW form-factors is used."
],
[
"Introduction",
"A theoretically sound separation of QED/EW effects between the QED emissions and genuine weak effects was essential for the phenomenology of LEP precision physics [1].",
"It was motivated by the structure of the amplitudes for single $Z$ or (to a lesser degree) $WW$ pairs production in $e^+e^-$ collisions, and by the fact that QED bremsstrahlung occurs at a different energy scale than the electroweak processes.",
"Even more importantly, with this approach multi-loop calculations for complete electroweak sector could be avoided.",
"The QED terms could be resumed in an exclusive exponentiation scheme implemented in Monte Carlo [2].",
"Note that QED corrections modify the cross-section at the peak by as much as 40%.",
"The details of this paradigm are explained in [3].",
"It was obtained as a consequence of massive efforts, we will not recall them here.",
"For the present study, the observation that spin amplitudes semi-factorize into a Born-like terms and functional factors responsible for bremsstrahlung [4] was very important.",
"A similar separation can be also achieved for dynamics of production process in $pp$ collisions, which can be isolated from QED/EW corrections.",
"It was explored recently in the case of configurations with high-$p_T$ jets associated with the Drell-Yan production of $Z$ [5] or $W$ bosons [6] at LHC.",
"The potentially large electroweak Sudakov logarithmic corrections discussed in [7] (absent in our work) represent yet another class of weak effects, separable from those discussed throughout this paper.",
"They are very small for lepton pairs with a virtuality close to the $Z$ -boson pole mass and, if accompanied by the jet when virtuality of $\\ell \\ell j$ system is not much larger than 2 $M_W$ .",
"Otherwise the Sudakov corrections have to be revisited and calculation of electroweak corrections extended, even if invariant mass of the lepton pair is close to the $Z$ mass.",
"To assess precisely the size and impact of genuine weak corrections to the Born-like cross section for lepton pair production with a virtuality below threshold for $WW$ pair production, the precision calculations and programs prepared for the LEP era: KKMC Monte Carlo [8] and Dizet electroweak (EW) library, were adapted to provide pre-tabulated EW corrections to be used by LHC specific event reweighting programs like TauSpinner package [9].",
"Even at present KKMC Monte Carlo use Dizet version 6.21 [10], [11].",
"We restrict ourselves to that reference version.",
"The TauSpinner package was initially created as a tool to correct with per-event weight longitudinal spin effects in the generated event samples including $\\tau $ decays.",
"Algorithms implemented there turned out to be of more general usage.",
"The possibility to introduce one-loop electroweak corrections from SANC library [12] in case of Drell-Yan production of the $Z$ -boson became available in TauSpinner since [13].",
"Pre-tabulation prepared for EW corrections of SANC library, was useful to introduce weights for complete spin effects at each individual event level.",
"However no higher loop contributions were available.",
"TauSpinner provides a reweighting technique to modify hard process matrix elements (also matrix elements for $\\tau $ decays) which were used for Monte Carlo generation.",
"For each event no changes of any details for event kinematic configurations are introduced.",
"The reweighting algorithm can be used for events where final state QED bremsstrahlung photons and/or high $p_T$ jets are present.",
"For matrix element calculation used for re-weighting, some contributions such as of QED bremsstrahlung or of jet emissions have to be removed.",
"For that purpose factorization and detailed inspection of fixed order perturbation expansion amplitudes is necessary.",
"The most recent summary on algorithms and their applications is given in [14].",
"The reference explains in detail how kinematical configurations are reduced to Born-level configurations used for the correcting weights, also for electroweak correctionsIn Ref.",
"[15] (on Tauola Universal Interface) other than TauSpinner solution was prepared.",
"Then parton level history entries for generated event record were used.",
"For TauSpinner use of history event record entries was abandoned, because of too many variants how corresponding information was required to be interpreted.",
"Instead, contributions from all possible parton level processes, weighted with parton distribution functions are averaged.",
"This could also be used for configurations generated with multi jet matrix elements, when Born level marix element configurations can not be identified.. Used for both Tauola Univesal Interface and TauSpinner , SANC library [12] of year 2008 calculates one loop i.e.",
"NLO electroweak corrections in two $\\alpha (0)$ and $G\\mu $ ($G_F$ ) schemes.",
"It was found numerically insufficient for practical applications.",
"For example, it was missing sizable $\\alpha _s$ corrections to the calulated $Z$ boson width.",
"Two aspects of EW corrections implementation [13] had to be enhanced.",
"First, in [5], [6] we have studied separation of QCD higher order corrections and the Born-level spin amplitudes calculated in the adapted Mustraal lepton pair rest frameOver the paper we use several variants of coordinate system orientation for the lepton pair rest-frame.",
"The Mustraal frame resulted from careful analysis of the cross section for the initial and final state bremsstrahlung that is $e^+e^- \\rightarrow \\mu ^+\\mu ^- \\gamma $ .",
"It was found that it can be represented, without any approximation as sum of four incoherently added distributions with well defined probabilities (two for initial and two for final state emission), each factorized into Born cross section calculated in reference frame oriented as required by the form of matrix element and the factor dependent on kinematical variables for the $\\gamma $ .",
"One should keep in mind that the spin carried by the photon cancels out with its orbital momentum.",
"That property of the matrix element originates from the properties of the Lorentz group representations, their combinations for the ultra-relativistic states.",
"That is why it generalizes unchanged to the $q \\bar{q} \\rightarrow l^+l^- g$ and approximately also to other processes of single or even double jet emissions in a bulk of parton emissions in $pp$ collisions.",
"It was checked numerically in Refs.",
"[5], [6]..",
"It is defined like for QED bremsstrahlung of Ref.",
"[4].",
"The separation holds to a good approximation for the Drell-Yan processes where one or even two high $p_T$ jets are present.",
"This frame is now used as option for EW weight calculation.",
"Second, the TauSpinner package and algorithms are now adapted to EW corrections from the Dizet libraryThis legacy library of EW corrections, features numerically important, corrections beyond NLO, n particular to $Z$ and $\\gamma ^*$ propagators.",
"Contributions corresponding to QED are carefully removed and left for the independent treatment., more accurate than SANC.",
"The EW corrections are introduced with form-factor corrections of Standard Model couplings and propagators which enter spin amplitudes of the Improved Born Approximation, used for EW weights calculation.",
"They represent complete $\\cal O (\\alpha ) $ electroweak corrections with QED contributions removed but augmented with carefully selected dominant higher order terms.",
"This was very successful in analyses of LEP I precision physics.",
"We attempt a similar strategy for the $Z$ -boson pole LHC precision physics; the approach to EW corrections already attracted attention.",
"It was used in the preliminary measurement of effective leptonic weak mixing angle recently published by ATLAS Collaboration [16].",
"This paper is organized as follows.",
"In Section we collect the main formulae of the formalism, in particular we recall the definition of the Improved Born Approximation.",
"In Section we present numerical results for the electroweak form-factors.",
"Some details on commonly used EW schemes are discussed in Section , which also recall the definition of the Effective Born.",
"In Section we comment on the issues of using the Born approximation in $pp$ collisions and in Section we give more explanation why the Born approximation of the EW sector is still valid in the presence of NLO QCD matrix elements.",
"In Section we define the concept of EW weight which can be applied to introduce EW corrections into already existing samples, generated with Monte Carlo programs with EW LO hard process matrix elements only.",
"In Section we discuss, in numerical detail, EW corrections to different observables of interest for precision measurements: $Z$ -boson line-shape, lepton forward-backward asymmetry and for coefficients of lepton spherical harmonic expansion.",
"In this Section we include also a discussion of the effective weak mixing angle in case of $pp$ collision.",
"For results presented in Section we use QCD NLO Powheg+MiNLO [17] $Z+j$ Monte Carlo sample, generated for $pp$ collision with $\\sqrt{s}$ = 8 TeV and EW LO implementation in matrix elements.",
"Section summarizes the paper.",
"In Appendix details on the technical implementation of EW weight and how it can be calculated with help of the TauSpinner framework are given.",
"In Appendix formulae which have been implemented to allow variation of the weak mixing angle parameter of the Born spin amplitudes are discussed.",
"In Appendix initialization details, and options valuable for future discussions, for the Dizet library are collected."
],
[
"Improved Born Approximation",
"At LEP times, to match higher order QED effects with the loop corrections of electroweak sector, the concept of electroweak form-factors was introduced [3].",
"This arrangement was very beneficial and enabled common treatment of one loop electroweak effects with not only higher order QED corrections including bremsstrahlung, but also to incorporate higher order loops into $Z$ and photon propagators, see e.g.",
"documentation of KKMC Monte Carlo [2] or Dizet [11].",
"Such description has its limitations for the LHC applications, but for the processes of the Drell-Yan type with a moderate virtuality of produced lepton pairs is expected to be useful, even in the case when high $p_T$ jets are present.",
"For the LEP applications [1], the EW form-factors were used together with multi-photon bremsstrahlung amplitudes, but for the purpose of this paper we discuss their use with parton level Born processes only (no QED ISR/FSRPresence in reweighted events of QED initial and final state bremsstrahlung, does not lead to complications of principle, but would obscure presentation.",
"Necessary extensions [14] are technically simple, thanks to properties of QED matrix elements, presented for the first time in [4].).",
"The terminology double-deconvoluted observable was widely used since LEP time and is explained e.g.",
"in [18].",
"The so called Improved Born Approximation (IBA) [11] is employed.",
"It absorbs some of the higher order EW corrections into a redefinition of couplings and propagators of the Born spin amplitude.",
"This allows for straightforward calculation of doubly-deconvoluted observables like various cross-sections and asymmetries.",
"QED effects are then removed or integrated over.",
"It is possible, because the excluded initial/final QCD and QED corrections form separately gauge invariant subsets of diagrams [11].",
"The QED subset consists of QED-vertices, $\\gamma \\gamma $ and $\\gamma Z$ boxes and bremsstrahlung diagrams.",
"The subset corresponding to the initial/final QCD corrections can be constructed as well.",
"All the remaining corrections contribute to the IBA: purely EW loops, boxes and internal QCD corrections for loops (line-shape corrections).",
"They can be split into two more gauge-invariant subsets, giving rise to two improved (or dressed) amplitudes: (i) improved $\\gamma $ exchange amplitude with running QED coupling where fermion loops of low $Q^2$ contribute dominantly and (ii) improved $Z$ -boson exchange amplitude with four complex EW form-factors: $\\rho _{\\ell f}$ , ${K}_{\\ell }$ , ${K}_{f}$ , ${K}_{\\ell f}$ .",
"Components of those corrections are as follows: Corrections to photon propagator, where fermion loops contribute dominantly the so called vacuum-polarization corrections.",
"Corrections to $Z$ -boson propagator and couplings, called EW form-factors.",
"Contribution from the purely weak $WW$ and $ZZ$ box diagrams.",
"They are negligible at the $Z$ -peak (suppressed by the factor $(s-M^2_Z)/s$ ), but very important at higher energies.",
"They enter as corrections to form-factors and introduce non-polynomial dependence on the $\\cos $ of the scattering angle.",
"Mixed $O(\\alpha \\alpha _s, \\alpha \\alpha _s^2, ...)$ corrections which originate from gluon insertions to the fermionic components of bosonic self-energies.",
"They enter as corrections to all form-factors.",
"Below, to define notation we present the formula of the Born spin amplitude ${A}^{Born}$ .",
"We recall conventions from [11].",
"Let us start with defining the lowest order coupling constants (without EW corrections) of the $Z$ boson to fermions: $ s^2_W = 1- M_W^2/M_Z^2=\\sin \\theta ^2_W $ defines weak Weinberg angle in the on-mass-shell scheme and $T_3^{\\ell , f}$ third component of the isospin.",
"The vector $v_{\\ell }, v_f$ and axial $a_{\\ell }, a_f$ couplings for leptons and quarks are defined with the formulae belowWe will use “$\\ell $ ” for lepton, and “$f$ ” for quarks.",
"$v_{\\ell } && = (2 \\cdot T_3^{\\ell } - 4 \\cdot q_{\\ell } \\cdot s^2_W)/\\Delta , \\nonumber \\\\v_f && = (2 \\cdot T_3^f - 4 \\cdot q_f \\cdot s^2_W)/\\Delta , \\\\a_{\\ell } && = (2 \\cdot T_3^{\\ell } )/\\Delta , \\nonumber \\\\a_f && = (2 \\cdot T_3^f )/\\Delta .",
"\\nonumber $ where $\\Delta = \\sqrt{ 16 \\cdot s^2_W \\cdot (1 - s^2_W)} ,$ and $ q_f$ , $q_l$ denote charge of incoming fermion (quark) and outgoing lepton.",
"With this notation, the ${A}^{Born}$ spin amplitude for the $q \\bar{q} \\rightarrow Z/\\gamma ^* \\rightarrow \\ell ^+ \\ell ^- $ can be written as: $&& {A}^{Born} = \\frac{\\alpha }{s}\\ \\ \\lbrace \\nonumber \\\\&& [\\bar{u} \\gamma ^{\\mu } v g_{\\mu \\nu } \\bar{v} \\gamma ^{\\nu } u] \\cdot ( q_{\\ell } \\cdot q_f) \\cdot \\chi _{\\gamma }(s)+ [\\bar{u} \\gamma ^{\\mu } v g_{\\mu \\nu } \\bar{\\nu }\\gamma ^{\\nu } u \\cdot ( v_{\\ell } \\cdot v_f ) \\nonumber \\\\&& + \\bar{u} \\gamma ^{\\mu } v g_{\\mu \\nu } \\bar{\\nu }\\gamma ^{\\nu } \\gamma ^5 u \\cdot (v_{\\ell } \\cdot a_f) + \\bar{u} \\gamma ^{\\mu } \\gamma ^5 v g_{\\mu \\nu } \\bar{\\nu }\\gamma ^{\\nu } u \\cdot (a_{\\ell } \\cdot v_f)\\nonumber \\\\&&+ \\bar{u} \\gamma ^{\\mu } \\gamma ^5 v g_{\\mu \\nu }\\bar{\\nu }\\gamma ^{\\nu } \\gamma ^5 u \\cdot (a_{\\ell } \\cdot a_f) ] \\cdot \\chi _Z (s)\\ \\ \\ \\rbrace ,$ where $u, v$ denote fermion spinors and, $\\alpha $ stands for QED coupling constant.",
"The $Z$ -boson and photon propagators are defined respectively as: $\\chi _{\\gamma }(s) = 1, \\\\$ $\\chi _Z(s) = \\frac{G_{\\mu } \\cdot M_{z}^2 \\cdot \\Delta ^2 }{\\sqrt{2} \\cdot 8 \\pi \\cdot \\alpha }\\cdot \\frac{s}{s - M_Z^2 + i \\cdot \\Gamma _Z \\cdot s/M_Z}.",
"\\\\$ For the IBA, we redefine vector and axial couplings and introduce EW form-factors $\\rho _{\\ell f}(s,t), {K}_{\\ell }(s,t)$ , ${K}_f(s,t)$ , ${K}_{\\ell f} (s,t)$ as follows: $v_{\\ell } && = (2 \\cdot T_3^{\\ell } - 4 \\cdot q_{\\ell } \\cdot s^2_W \\cdot {K}_{\\ell }(s,t))/\\Delta , \\nonumber \\\\v_f && = (2 \\cdot T_3^f - 4 \\cdot q_f \\cdot s^2_W \\cdot {K}_f(s,t))/\\Delta , \\\\a_{\\ell } && = (2 \\cdot T_3^{\\ell } )/\\Delta , \\nonumber \\\\a_f && = (2 \\cdot T_3^f )/\\Delta .",
"\\nonumber $ Normalization correction $ Z_{V_{\\Pi }} $ to the $Z$ -boson propagator is defined as $Z_{V_{\\Pi }} = \\rho _{\\ell f}(s,t) \\ .$ Re-summed vacuum polarization corrections $ \\Gamma _{V_{\\Pi }}$ to the $\\gamma ^*$ propagator are expressed as $\\Gamma _{V_{\\Pi }} = \\frac{1}{ 2 - (1 + \\Pi _{\\gamma \\gamma }(s))},$ where $\\Pi _{\\gamma \\gamma }(s)$ denotes vacuum polarization loop corrections of virtual photon exchange.",
"Both $\\Gamma _{V_{\\Pi }}$ and $Z_{V_{\\Pi }}$ are multiplicative correction factors.",
"The $\\rho _{\\ell f}(s,t)$ could be also absorbed as multiplicative factor into the definition of vector and axial couplings.",
"The EW form-factors $\\rho _{\\ell f}(s,t), {K}_{\\ell }(s,t)$ , ${K}_f(s,t)$ , ${K}_{\\ell f}(s,t) $ depend on two Mandelstam invariants $(s,t)$ due to contributions of the $WW$ and $ZZ$ boxes.",
"The Mandelstam variables satisfy the identity $s+t+u = 0 \\ \\ \\ where \\ \\ \\ \\ t = -\\frac{s}{2}(1 - \\cos \\theta )$ and $ \\cos \\theta $ is the cosine of the scattering angle, i.e.",
"the angle between incoming and outgoing fermion directions.",
"Note, that in this approach the mixed EW and QCD loop corrections, originating from gluon insertions to fermionic components of bosonic self-energies, are included in $\\Gamma _{V_{\\Pi }}$ and $Z_{V_{\\Pi }}$ .",
"One has to pay special attention to the angle dependent product of the vector couplings.",
"The corrections break factorization, formula (REF ), of the couplings into ones associated with either $Z$ boson production or decay.",
"The mixed term has to be added: $vv_{\\ell f} =&& \\frac{1}{v_{\\ell } \\cdot v_f} [( 2 \\cdot T_3^{\\ell }) (2 \\cdot T_3^f) - 4 \\cdot q_{\\ell } \\cdot s^2_W \\cdot {K}_f(s,t)( 2 \\cdot T_3^{\\ell })\\nonumber \\\\&&- 4 \\cdot q_f \\cdot s^2_W \\cdot {K}_{\\ell }(s,t) (2 \\cdot T_3^f) \\\\&& + (4 \\cdot q_{\\ell } \\cdot s^2_W) (4 \\cdot q_f \\cdot s^2_W) {K}_{\\ell f}(s,t)] \\frac{1}{\\Delta ^2}.",
"\\nonumber $ Finally, we can write the spin amplitude for Born with EW corrections, ${A}^{Born+EW} $ , as: ${\\Huge A}^{Born+EW} &=& \\frac{\\alpha }{s} \\lbrace [\\bar{u} \\gamma ^{\\mu } v g_{\\mu \\nu } \\bar{v} \\gamma ^{\\nu } u] \\cdot ( q_{\\ell } \\cdot q_f)] \\cdot \\Gamma _{V_{\\Pi }} \\cdot \\chi _{\\gamma }(s) \\nonumber \\\\ &&+ [\\bar{u} \\gamma ^{\\mu } v g_{\\mu \\nu } \\bar{\\nu }\\gamma ^{\\nu } u \\cdot ( v_{\\ell } \\cdot v_f \\cdot vv_{\\ell f})\\\\&&+ \\bar{u} \\gamma ^{\\mu } v g_{\\mu \\nu } \\bar{\\nu }\\gamma ^{\\nu } \\gamma ^5 u \\cdot (v_{\\ell } \\cdot a_f) \\nonumber \\\\ &&+ \\bar{u} \\gamma ^{\\mu } \\gamma ^5 v g_{\\mu \\nu } \\bar{\\nu }\\gamma ^{\\nu } u \\cdot (a_{\\ell } \\cdot v_f)\\nonumber \\\\ &&+ \\bar{u} \\gamma ^{\\mu } \\gamma ^5 v g_{\\mu \\nu }\\bar{\\nu }\\gamma ^{\\nu } \\gamma ^5 u \\cdot (a_{\\ell } \\cdot a_f) ] \\cdot Z_{V_{\\Pi }}\\cdot \\chi _Z (s)\\rbrace .",
"\\nonumber $ The EW form-factor corrections: $\\rho _{\\ell f}, {K}_{\\ell }, {K}_f, {K}_{\\ell f}$ can be calculated using the Dizet library.",
"This library invokes also calculation of vacuum polarization corrections to the photon propagator $\\Pi _{\\gamma \\gamma }$ .",
"For the case of $pp$ collisions we do not introduce QCD corrections to vector and axial couplings of incoming fermions.",
"They are assumed to be included elsewhere as a part of the QCD NLO calculations for the initial parton state, including convolution with proton structure functions.",
"The Improved Born Approximation uses the spin amplitude ${A}^{Born+EW}$ of Eq.",
"(REF ) and $2 \\rightarrow 2$ body kinematics to define the differential cross-section with EW corrections for $q \\bar{q} \\rightarrow Z/\\gamma ^* \\rightarrow l l$ process.",
"The formulae presented above very closely follow the approach taken for implementationCompatibility with this program is also part of the motivation why we leave updates for the Dizet library to the forthcoming work.",
"Dizet 6.21 is also well documented.",
"of EW corrections to KKMC Monte Carlo [2]."
],
[
"Electroweak form-factors",
"For the calculation of EW corrections, we use the Dizet library, as of the 2010 KKMC Monte Carlo [2] version.",
"For this and related projects, massive theoretical effort was necessary.",
"Simultaneous study of several processes, like of $\\mu ^+\\mu ^-$ , $u \\bar{u}$ , $d \\bar{d}$ , $\\nu \\bar{\\nu }$ production in $e^+e^-$ collisions and also in $p \\bar{p}$ initiated parton processes, like at Tevatron, was performed.",
"Groups of diagrams for the $Z/\\gamma ^*$ propagators, production and decay vertices could be identified and incorporated into form-factors.",
"The core of the Dizet library relies on such separation.",
"It also opened the possibility that for one group of diagrams, such as vacuum polarizations, higher order contributions could be included while for others were not.",
"That was particularly important for quark contributions to vacuum polarizations.",
"Otherwise, the required precision would not be achieved.",
"The above short explanation only indicates fundamental importance of the topic, we delegate the reader to Refs.",
"[2], [19], [20] and experimental papers of LEP and Tevatron experiments quoting these papers.",
"The interface in KKMC prepares look-up tables with EW form-factors and vacuum polarization corrections.",
"The tabulation grid granularity and ranges of the centre-of-mass energy of outgoing leptons and lepton scattering angle are adapted to variation of the tabulated functions.",
"Theoretical uncertainties on the predictions for EW form-factors have been estimated in times of LEP precision measurements, in the context of either benchmark results like [18] or specific analyses [3].",
"The predictions are now updated with the known Higgs boson and top-quark masses.",
"In the existing code of the Dizet library, certain types of the corrections or options of the calculations of different corrections can be switched off/on.",
"In Appendix , we show in Table REF an almost complete list of options useful for discussions.",
"We do not attempt to estimate the size of theoretical uncertainties, delegating it to the follow up work in the context of LHC EW Precision WG studies.",
"The other versions of electroweak calculations, like of [12], [21], can and should be studied then as well.",
"Already now the precision requirements of LHC experiments [16] are comparable to those of individual LEP measurements, but phenomenology aspects are more involved."
],
[
"Input parameters to Dizet",
"The Dizet package relies on the so called on-mass-shell (OMS) normalization scheme [19], [20] but modifications are present.",
"The OMS uses the masses of all fundamental particles, both fermions and bosons, the electromagnetic coupling constant $\\alpha (0)$ and the strong coupling $\\alpha _s(M_Z^2)$ .",
"The dependence on the ill-defined masses of the light quarks u, d,c, s and b is solved by dispersion relations, for details see [11].",
"Another exception is the $W$ -boson mass $M_W$ , which still can be predicted with better theoretical accuracy than experimentally measured.",
"The Fermi constant $G_{\\mu }$ is precisely known from $\\mu $ -decay.",
"For this reason, $M_W$ was usually, in time of LEP analyses, replaced by $G_{\\mu }$ as an input.",
"The knowledge about the hadronic vacuum polarization is contained in $\\Delta \\alpha _h^{(5)}(s)$ , which is used as external, easy to change, parametrization.",
"It can be either computed from quark masses or, preferably, fitted to experimental low energy $e^+ e^- \\rightarrow hadrons$ data.",
"The $M_W$ is calculated iteratively from the equation $M_W = \\frac{M_Z}{\\sqrt{2}} \\sqrt{ 1 + \\sqrt{ 1 - \\frac{ 4 A^2_0}{M^2_Z ( 1 - \\Delta r)}}},$ where $A_0 = \\sqrt{ \\frac{\\pi \\alpha (0)}{\\sqrt{2}G_{\\mu }}}.$ The Sirlin's parameter $\\Delta r$ [22] $\\Delta r = \\Delta \\alpha (M_Z^2) + \\Delta r_{EW}$ is also calculated iteratively, and the definition of $ \\Delta r_{EW}$ involves re-summation and higher order corrections.",
"This term implicitly depends on $M_W$ and $M_Z$ , and the iterative procedure is needed.",
"The re-summation term in formula (REF ) is not formally justified by renormalisation group arguments, the correct generalization is to compute higher order corrections, see discussion in [11].",
"Note that once the $M_W$ is recalculated from formula (REF ), the lowest order Standard Model relationship between the weak and electromagnetic couplings $G_{\\mu } = \\frac{\\pi \\alpha }{\\sqrt{2} M^2_W \\sin ^2\\theta _W}$ is not fulfilled anymore, unless the $G_{\\mu }$ is redefined away from the measured value.",
"This is an approach of some EW LO schemes, but not the one used by Dizet.",
"It requires therefore the complete expression for $\\chi _Z(s)$ propagator in spin amplitude of Eq.",
"(REF ), as defined by formula (REF ).",
"In the OMS renormalisation scheme the weak mixing angle is defined uniquely through the gauge-boson masses: $\\sin ^2\\theta _W = s^2_W = 1 - \\frac{M^2_W}{M^2_Z}.$ With this scheme, measuring $\\sin ^2\\theta _W$ will be equivalent to indirect measurement of $M^2_W$ through the relation (REF ).",
"In Table REF we collect numerical values for all parameters used in the presented below evaluations.",
"Note that formally they are not representing EW LO scheme, as the relation (REF ) is not obeyed.",
"The $M_W$ in (REF ) is recalculated with (REF ) but $G_{\\mu }$ , $M_Z$ remain unchanged.",
"Table: The Dizet initialization: masses and couplings.",
"Thecalculated M W M_W and s W 2 s^2_Ware shown also."
],
[
"The EW form-factors",
"Real parts of the $\\rho _{\\ell f}(s,t)$ , ${K}_f(s,t)$ , ${K}_{\\ell }(s,t)$ , ${K}_{\\ell f}(s,t)$ EW form-factors are shown in Figure REF for a few values of $\\cos \\theta $ , the angle between directions of the incoming quark and the outgoing lepton, calculated in the outgoing lepton pair centre-of-mass frame.",
"Eq.",
"(REF ) relates Mandelstam variables $(s,t)$ to the invariant mass and $\\cos \\theta $ .",
"The $\\cos \\theta $ dependence of the box correction is more sizable for the up-quarks.",
"Note, that at the $Z$ -boson peak, Born-like couplings are only weakly modified; form-factors are close to 1 and of no numerically significant angular dependence.",
"At lower virtualities corrections are relatively larger because the $Z$ -boson contributions are non resonant and thus smaller.",
"In this phase-space region the $Z$ -boson is itself dominated by the virtual photon contribution.",
"Above the peak, the $WW$ and later also $ZZ$ boxes contributions become sizable, the dependence on the $\\cos \\theta $ appears; contributions become gradually doubly resonant and sizable.",
"Figure: Real parts of theρ e,up \\rho _{e, up}, K e {K}_{e}, K up {K}_{up} and K e,up {K}_{e, up}EW form-factors of qq ¯→Z→eeq \\bar{q} \\rightarrow Z \\rightarrow ee process,as a function of s\\sqrt{s} and for the few values of cosθ\\cos \\theta .For the up-type quark flavour, left sideplots are collected and for the down-type the right side plots.Note, that K e {K_e} depends on the flavour of incoming quarks."
],
[
"Running $\\alpha (s)$",
"Fermionic loop insertion of the photon propagator, i.e.",
"vacuum polarization corrections, are summed together as a multiplicative factor $\\Gamma _{V_{\\Pi }}$ , Eq.",
"(REF ), for the photon exchange in Eq.",
"(REF ).",
"But it can be interpreted as the running QED coupling: $\\alpha (s) = \\frac{\\alpha (0)}{1 - \\Delta \\alpha _h^{(5)}(s) - \\Delta \\alpha _{\\ell }(s) - \\Delta \\alpha _t(s) - \\Delta \\alpha ^{\\alpha \\alpha _s}(s)}.$ The hadronic contribution at $M_Z$ is a significant [11] correction: $ \\Delta \\alpha _h^{(5)}(M_Z^2)$ = 0.0280398.",
"It is calculated in the five flavour scheme with use of dispersion relation and input from low energy experiments.",
"We will continue to use LEP times parametrization, while the most recent measured $ \\Delta \\alpha _h^{(5)}(M_Z^2)$ = 0.02753 $\\pm $ 0.00009 [23].",
"The changed value modifies predicted form-factors, in particular the effective leptonic mixing angle $\\sin ^2\\theta _{eff}^{lep}(M_Z^2)=Re{({K}_l(M_Z^2))} s^2_W$ is shifted by almost $20 \\cdot 10^{-5}$ closer to the measured LEP value.",
"This is not included in the numerical results presented as we consistently remain with the defaults used in KKMC.",
"The leptonic loop contribution $\\Delta \\alpha _{\\ell }(s)$ is calculated analytically up to the 3-loops, and is a comparably significant correction, $ \\Delta \\alpha _{\\ell }(M_Z^2)$ = 0.0314976.",
"The other contributions are very small.",
"Fig.",
"REF shows the vacuum polarization corrections to the $\\chi _{\\gamma }(s)$ propagator, directly representing the ratio $\\alpha (s)/\\alpha (0)$ of Eq.",
"(REF ).",
"Figure: The vacuum polarization (α(s)/α(0)\\alpha (s)/\\alpha (0)) correction of γ\\gamma propagator,Eq.",
"()."
],
[
"EW input schemes and Effective Born",
"Formally, at the lowest EW order, only three independent parameters can be set, other are calculated following the structure of $ SU(2) \\times U(1)$ group from Standard Model constraints.",
"Formula (REF ) represents one of such constraints.",
"Following report [24], the most common choices at hadron colliders are: $G_{\\mu }$ scheme $(G_{\\mu }, M_Z, M_W)$ and $\\alpha (0)$ scheme $(\\alpha (0)$ , $M_Z, M_W)$ .",
"There exists by now a family of different modifications of the $G_{\\mu }$ scheme, see discussion in [24], and they are considered as preferred schemes for hadron collider physicsThe Monte Carlo generators usually allow user to define set of input parameters $(\\alpha , M_Z, M_W)$ , $(\\alpha , M_Z, G_{\\mu })$ or $(\\alpha , M_Z, s_W^2)$ .",
"However, within this flexibility, formally multiplicative factor $\\chi _Z(s)$ in the $Z$ -boson propagator, see formula (REF ), is always kept to be equal to 1: $\\frac{G_{\\mu } \\cdot M_{z}^2 \\cdot \\Delta ^2 }{\\sqrt{2} \\cdot 8 \\pi \\cdot \\alpha } = 1,$ where $\\Delta $ is given by Eq.",
"(REF ).",
"The multiplicative factor of (REF ) in the definition of $\\chi _Z(s)$ is quite often absent in the programs code.",
"With the choice of primary parameters, the others are adjusted to match the constraint Eq.",
"(REF ), regardless if they fall outside their measurement uncertainty window or not.",
".",
"Let us recall, that the calculations of EW corrections available in Dizet work with a variant of the $\\alpha (0)$ scheme.",
"It is defined by the input parameters $(\\alpha (0), G_{\\mu }, M_Z)$ .",
"Then $M_W$ is calculated iteratively from formula (REF ) and $s^2_W$ of Eq.",
"(REF ) uses that value of $M_W$ .",
"This formally brings it beyond EW LO scheme.",
"The numerical value of $s_W^2$ calculated from (REF ) does not fulfill the EW LO relation (REF ) anymore.",
"At this point we introduce two options for the Effective Born spin amplitudes parametrization, which works well for parametrizing EW corrections near the $Z$ -pole and denote them respectively as LEP and LEP with improved norm.",
": The LEP parametrization uses formula (REF ) for spin amplitude but with $\\alpha (s) = \\alpha (M_Z^2) = 1./128.8667$ , $s^2_W = \\sin ^2\\theta _W^{eff} (M_Z^2) = 0.23152$ , i.e.",
"as measured at the $Z$ -pole and reported in [25].",
"All form-factors are set to 1.0.",
"The LEP with improved norm.",
"parametrization also uses formula (REF ) for spin amplitude with parameters set as for LEP parametrization.",
"All form-factors are set to 1, but $\\rho _{\\ell f} = 1.005$ .",
"This corresponds to the measured $\\rho (M_Z^2)$ = 1.005, as reported in [25].",
"Table REF collects initialization constants of EW schemes relevant for our discussion.",
"We specify parameters which enter formula (REF ) for Born spin amplitudes used for: (i) actual MC events generation The EW LO initialization is consistent with PDG $\\sin ^2\\theta _{eff}^{lep}$ = 0.23113, but commonly used $G_{\\mu }$ scheme, ($G_{\\mu }=1.1663787 \\cdot 10^{-5}$ GeV$^{-2}$ , $M_Z$ = 91.1876 GeV, $M_W$ =80.385 GeV) correspond to $s^2_W$ = 0.2228972., (ii) the EW LO $\\alpha (0)$ scheme, (iii) effective Born (LEP) parametrization and (iv) effective Born (LEP with improved norm.).",
"In each case parameters are chosen such that the SM relation, formula (REF ), is obeyed.",
"In the Improved Born Approximation complete $O(\\alpha )$ EW corrections, supplemented by selected higher order terms, are handled thanks to s-, t-dependent form-factors, which multiply couplings and propagators of the usual Born expressions.",
"Instead, the Effective Born absorbs the bulk of EW corrections into a redefinition of a few fixed parameters (i.e.",
"couplings).",
"In the following, we will systematically compare predictions obtained with the EW corrections and those calculated with LEP or LEP with improved norm.",
"approximations.",
"As we will see, effective Born with LEP with improved norm.",
"works very well around $Z$ -pole both for the line-shape and forward-backward asymmetry.",
"Table: The EW parameters used for: (i) MC events generation, (ii) the EW LO α(0)\\alpha (0) scheme,(iii) effective Born spin amplitude around the ZZ-pole and (iv) effective Born with improved normalization.In each case parameters are chosen such that the SM relation, formula (), is obeyed.The G μ G_{\\mu } = 1.166389·10 -5 1.166389 \\cdot 10^{-5} GeV -2 ^{-2}, M Z M_Z = 91.1876 GeV and K f ,K e ,K ℓf {K}_f, {K}_e, {K}_{\\ell f} = 1 are taken."
],
[
"Born kinematic approximation and $p p$ scattering",
"The solution to define Born-like parton level kinematics for $p p$ scattering process is encoded in the TauSpinner package [14].",
"It does not exploit hard-process, so-called history entries which only sometimes are stored for the generated events.",
"In particular, the flavour and momenta of the incoming partons have to be emulated from the kinematics of final states and incoming protons momenta.",
"Probabilities calculated from parton level cross-sections and PDFs weight all possible contributions.",
"Let us now recall briefly principles and choices for optimization."
],
[
"Average over incoming partons flavour",
"The parton level Born cross-section $\\sigma ^{q \\bar{q}}_{Born}(\\hat{s}, \\cos \\theta )$ has to be convoluted with the structure functions, and summed over all possible flavours of incoming partons and all possible helicity states of outgoing leptons.",
"The lowest order formulaValid for the ultra-relativistic leptons.",
"is given below $d\\sigma _{Born}&&( x_1, x_2, \\hat{s}, \\cos \\theta ) =\\sum _{q_f, \\bar{q}_f} \\nonumber \\\\&[& f^{q_f}(x_1,...)f^{\\bar{q}_f}(x_2,...)d\\sigma ^{q_f \\bar{q}_f}_{Born}( \\hat{s}, \\cos \\theta ) \\\\&+& \\ f^{\\bar{q}_f}(x_1,...)f^{ q_f}(x_2,...)d\\sigma ^{ q_f \\bar{q}_f}_{Born}( \\hat{s}, -\\cos \\theta ) ] \\nonumber ,$ where $x_1$ , $x_2$ denote fractions of incoming protons momenta carried by the corresponding parton, $\\hat{s} = x_1\\ x_2\\ s $ and $f/\\bar{f}$ denotes parton (quark-/anti-quark) density functions.",
"We assume that kinematics is reconstructed from four-momenta of the outgoing leptons.",
"The incoming quark and anti-quark may come respectively either from the first and second proton or reversely from the second and first.",
"Both possibilities are taken into accountOne should mention photon induced contributions.",
"They are of the same coupling order as electroweak corrections.",
"For production of the lepton pairs in $pp$ collisions, contributions were evaluated e.g.",
"in [26].",
"In general, for the calculation of TauSpinner weights, sum over partons is not restricted as in eq.",
"(REF ) to the quarks and anti-quarks only.",
"Gluon PDF's are used when weight calculation with matrix elements for lepton pair with two jets in final state is used [27].",
"The $\\gamma \\gamma \\rightarrow l^+ l^-$ contributions can be then taken into account as a part of the $2 \\rightarrow 4$ matrix elements.",
"Photon induced processes are however usually generated and stored separately.",
"That is why our reweighting algorithm for EW corrections does not need to take such (rather small) contributions into account in eq.",
"(REF ).",
"by the two terms of (REF ).",
"The sign in front of $\\cos \\theta $ , the cosine of the scattering angle, is negative for the second term.",
"Then the parton of the first incoming proton which carries $x_1$ and follows the direction of the $z$ -axis is an anti-quark, not a quark.",
"The formula is used for calculating the differential cross-section $d\\sigma _{Born}( x_1, x_2, \\hat{s}, \\cos \\theta )$ of each analyzed event, regardless if its kinematics and flavours of incoming partons may be available from the event history entries or not.",
"The formula can be used to a good approximation in case of NLO QCD spin amplitudes.",
"The momenta of outgoing leptons are used to construct effective kinematics of the Drell-Yan production process and decay, without the need of information on parton-level hard-process itself.",
"Born-like kinematics can be constructed, as we will see later, even for events of quark-gluon or gluon-gluon parton level collisions (as inspected for test in the event history entries) too."
],
[
"Effective beams kinematics",
"The $x_1, x_2$ are calculated from the kinematics of outgoing leptons, following formulae of [15] $x_{1,2} = \\frac{1}{2}\\ {\\Big (}\\ \\pm \\frac{p_z^{ll}}{ E} + \\sqrt{ (\\frac{p_z^{ll}}{ E})^2 + \\frac{m^2_{ll}}{ E^2}} \\; \\; {\\Big )} ,$ where $E$ denotes energy of the proton beam and $p_z^{\\ell \\ell }$ denotes $z$ -axis momentum of outgoing lepton pair in the laboratory frame and $m_{ll}$ lepton pair virtuality.",
"Note that this formula can be used, as approximation, for the events with hard jets too."
],
[
"Definition of the polar angle",
"For the polar angle $\\cos \\theta $ , of factorized Born level $q \\bar{q} \\rightarrow Z \\rightarrow \\ell \\ell $ process, weighted average of the outgoing leptons angles with respect to the beams' directions, denoted as $\\cos \\theta ^*$ , was used.",
"In [28] it was found helpful to compensate the effect of initial state hard bremsstrahlung photons of $e^+e^- \\rightarrow Z n\\gamma $ , $Z \\rightarrow \\ell \\ell m\\gamma $ , where $m,\\; n$ denote the number of accompanying photons.",
"Extension to $pp$ collisions required to take both options in Eq.",
"(REF ) into account; when the $z$ -axis is parallel- and anti-parallel to the incoming quark.",
"For the further calculation, boost of all four-momenta (also of incoming beams) into the rest frame of the lepton pair need to be performed.",
"The $\\cos \\theta ^{*}$ is then calculated from $\\cos \\theta _1 = \\frac{\\tau _x^{(1)} b_x^{(1)} + \\tau _y^{(1)} b_y^{(1)} + \\tau _z^{(1)} b_z^{(1)}}{ | \\vec{\\tau }^{(1)}| |\\vec{b}^{(1)}|},\\nonumber \\\\\\cos \\theta _2 = \\frac{\\tau _x^{(2)} b_x^{(2)} + \\tau _y^{(2)} b_y^{(2)} + \\tau _z^{(2)} b_z^{(2)}}{ | \\vec{\\tau }^{(2)}| |\\vec{b}^{(2)}|},$ as follows: $\\cos \\theta ^* = \\frac{\\cos \\theta _1 \\sin \\theta _2 + \\cos \\theta _2 \\sin \\theta _1}{\\sin \\theta _1 + \\sin \\theta _2}$ where $\\vec{\\tau }^{(1)}, \\vec{\\tau }^{(2)}$ denote 3-vectors of outgoing leptons and $\\vec{b}^{(1)}, \\vec{b}^{(2)}$ denote 3-vectors of incoming beams' four-momenta.",
"The polar angle definition, Eq.",
"(REF ), is at present the TauSpinner default.",
"For tests we have used variants; Mustraal [4] and Collins-Soper [29] frames, which differ when high $p_T$ jets are present.",
"We will return later to the frame choice, best suitable when NLO QCD corrections are included in the production process of generated events."
],
[
"QCD corrections and angular coefficients",
"For the Drell-Yan production [30] one can separate QCD and EW components of the fully differential cross-section and describe the $Z/\\gamma ^* \\rightarrow \\ell \\ell $ sub-process with lepton angular ($\\theta , \\phi $ ) dependence $\\frac{ d\\sigma }{dp_T^2 dY d\\Omega } =\\Sigma _{ \\alpha =1}^{9} g_{ \\alpha }( \\theta , \\phi )\\frac{3}{ 16 \\pi } \\frac{d \\sigma ^{\\alpha }}{ dp_T^2 dY}, $ where the $ g_{ \\alpha }( \\theta , \\phi )$ denotes second order spherical harmonics, multiplied by normalization constants and $d \\sigma ^{\\alpha }$ denotes helicity cross-sections, for each of nine helicity configurations of $q \\bar{q} \\rightarrow Z/\\gamma ^* \\rightarrow \\ell \\ell $ .",
"The polar and azimuthal ($\\theta $ and $\\phi $ ) angles of $d\\Omega = d \\cos \\theta d\\phi $ are defined in the $Z$ -boson rest-frame.",
"The $p_T$ , $Y$ denote laboratory frame transverse momenta and rapidity of the intermediate $Z/\\gamma ^*$ -boson.",
"Thanks to th effort [31], [32], [33] from the early 90's one expects such factorization to break with non-logarithmic ${\\cal O}(\\alpha _s^2) \\sim 0.01$ QCD correctionsAlso the impact of final state QED bremsstrahlung can be overcome with a proper definition of frames.",
"The solution is available thanks to Ref. [4].",
"We use it with the definition of frames $A$ and $A^{\\prime }$ ; Section 3.1 of [14].",
"only.",
"There is some flexibility for the $Z$ -boson rest frame $z$ -axis choice.",
"The most common, so called helicity frame, is to take the $Z$ -boson laboratory frame momentum.",
"For the Collins-Soper frame it is defined from directions of the two beams in the $Z$ -boson rest frame and is signed with the $Z$ -boson $p_z$ laboratory frame sign.",
"Eq.",
"(REF ) with explicit spherical harmonics and coefficients reads $\\frac{ d\\sigma }{dp_T^2 dY d \\cos \\theta d\\phi } & = & \\frac{3}{ 16 \\pi } \\frac{d \\sigma ^{ U+ L }}{ dp_T^2 dY} [ (1 + \\cos ^2\\theta ) \\nonumber \\\\+ 1/2\\ A_0 (1 - 3 \\cos ^2\\theta ) &+& A_1 \\sin {2\\theta }\\cos \\phi \\\\+ 1/2\\ A_2 \\sin ^2\\theta \\cos ( 2 \\phi ) &+&A_3 \\sin \\theta \\cos \\phi + A_4 \\cos \\theta \\nonumber \\\\+ A_5 \\sin ^2 \\theta \\sin ( 2 \\phi ) + &A_6& \\sin {2\\theta } \\sin \\phi + A_7 \\sin \\theta \\ \\sin \\phi ], \\nonumber $ where $d \\sigma ^{ U+ L }$ denotes the unpolarised differential cross-section (notation used in several papers of the 80's).",
"The coefficients $A_i(p_T, Y)$ are related to ratios of definite intermediate state helicity contributions to the $d \\sigma ^{ U+ L }$ cross-sections.",
"The first term of the polynomial expansion is $ (1 + \\cos ^2\\theta )$ because intermediate boson is of the spin 1.",
"The dynamics of the production process is hidden in the angular coefficients $A_i (p_T, Y)$ .",
"In particular, all the hadronic physics is described implicitly by the angular coefficients and it decouples from the well understood leptonic and intermediate boson physics.",
"For the present paper, of particular interest are coupling constants present in coefficients $A_i$ of Eq.",
"(REF ) representing ratios of the so-called helicity cross sections [31], [32], [33]: $\\sigma ^{U+L } & \\sim & (v_{\\ell }^2 + a_{\\ell }^2)(v_{q}^2 + a_{q}^2), \\nonumber \\\\A_0, A_1, A_2 & \\sim & 1 , \\nonumber \\\\A_3, A_4 & \\sim & \\frac{ v_{\\ell } a_{\\ell } v_q a_q}{(v_{\\ell }^2 + a_{\\ell }^2)(v_{q}^2 + a_{q}^2)} , \\\\A_5, A_6 & \\sim & \\frac{(v_{\\ell }^2 + a_{\\ell }^2) ( v_q a_q)}{(v_{\\ell }^2 + a_{\\ell }^2)(v_{q}^2 + a_{q}^2)}, \\nonumber \\\\A_7 & \\sim & \\frac{ v_{\\ell } a_{\\ell } ( v_q^2 + a_q^2)}{(v_{\\ell }^2 + a_{\\ell }^2)(v_{q}^2 + a_{q}^2)}.",
"\\nonumber $ IntegrationOne can easily check that $A_{FB}$ of Eq.",
"(REF ) equals to $\\frac{3}{8} A_4$ .",
"over the azimuthal angle $\\phi $ reduces Eq.",
"(REF ) to $\\frac{ d\\sigma }{dp_T^2 dY d \\cos \\theta } = \\frac{3}{ 8 \\pi } \\frac{d \\sigma ^{ U+ L }}{ dp_T^2 dY}[ (1 + \\cos ^2\\theta )\\nonumber \\\\+ 1/2\\ A_0 (1 - 3 \\cos ^2\\theta ) + A_4 \\cos \\theta ].$ Both Eqs.",
"(REF ) and (REF ) are valid in any rest frame of the outgoing lepton pairs, however the $A_i(p_T, Y)$ are frame dependent.",
"The Collins-Soper frame is the most convenient and usual choice for the analyses dedicated to QCD dynamics.",
"In this frame, in the low $p_T$ limit, $A_4$ is the only non-zero coefficient.",
"It carries direct information on the EW couplings, as can be concluded from formulae (REF ).",
"All other coefficients depart from zero with increasing $p_T$ while at the same time $A_4$ gradually decreases.",
"Due to different transfer dependence of the $Z$ and $\\gamma ^*$ propagators, the $A_i$ vary with $m_{ll}$ .",
"The $A_i$ dependence on $(p_T,Y)$ , expressing production dynamics, differ with the frame definition variants of distinct coordinate system orientations.",
"For the studies of EW couplings, it is convenient when the lepton-pair rest-frame definition absorbs effects of production dynamics partly into the $z$ -axis choice.",
"Then, those $A_i$ coefficients which are proportional to the product of EW vector and axial couplings remain non-zero over the full range of $p_T$ .",
"Promising for that purpose frame was developed at LEP times for the Mustraal Monte Carlo program [4].",
"Recently, an extension of this Mustraal frame, for the case of hadron-hadron collisions, was introduced and discussed in [5].",
"As shown in that paper, both Collins-Soper and Mustraal frames are equivalent in the $p_T = 0$ limit.",
"Then $A_4$ is the only non-zero coefficient for both frames and is also numerically very close.",
"With increasing $p_T$ , in the Mustraal frame $A_4$ remains as the only sizably non-zero coefficient, while several $A_i$ coefficients depart from zero with the Collins-Soper frame.",
"In the collision of the same-charge protons the careful choice for the $z$ -axis orientation is necessary for the $A_4$ coefficient to remain non-zero.",
"For the Collins-Soper frame, the $z$ -axis follows the direction of the intermediate $Z$ -boson in the laboratory frame.",
"In case of the Mustraal frame the choice of the sign is made stochastically using information of the system of leptons and outgoing accompanying visible jets.",
"For details see [5], alternatively the same sign choice for the $z$ -axis as in the Collins-Soper case, can be used.",
"The shape of $A_i$ coefficients as a function of laboratory frame $Z$ -boson transverse momenta $p_T$ depends on the choice of lepton pairs rest-frame.",
"In Fig.",
"REF , $A_i$ coefficients of the Collins-Soper and Mustraal frames are shown.",
"As intended, even for large $p_T$ , with this frame, only $A_4$ coefficient is sizably non-zero.",
"Figure: The A i A_i coefficients for Z→e + e - Z\\rightarrow e^+e^- in lepton pair invariant mass range80<m ee <10080 < m_{ee} < 100 GeV.The Z+jZ+j production process in pppp collisionsat 8 TeV centre-of-mass energy, was used for the sample generation with Powheg+MiNLO Monte Carlo.The A i A_i coefficients are calculated in the Collins-Soper and Mustraal frames with momentsmethod ."
],
[
"Concept of the EW weight",
"The EW corrections enter the $\\sigma _{Born}( \\hat{s}, \\cos \\theta )$ through the definition of the vector and axial couplings, also photon and $Z$ -boson propagators.",
"They modify normalization of the cross-sections, the line-shape of the $Z$ -boson peak, polarization of the outgoing leptons and asymmetries.",
"Given that, we were able to factorize QCD and EW components of the cross-section to a good approximation and define per-event weights which specifically correct for EW effects.",
"Such a weight may modify events generated with EW LO to the ones including the EW corrections.",
"This is very much the same idea as already implemented in TauSpinner for introducing corrections for other effects: spin correlations, production process, etc.",
"The per-event $ wt^{EW} $ is defined as ratio of the Born-level cross-sections with and without EW corrections $wt^{EW} = \\frac{ d\\sigma _{Born+EW}( s, \\cos \\theta )}{d\\sigma _{Born}( s, \\cos \\theta )},$ where $\\cos \\theta $ can be taken according to $\\cos \\theta ^*$ , $\\cos \\theta ^{Mustraal}$ (Mustraal frame) or $\\cos \\theta ^{CS}$ (Collins-Soper frame) prescription.",
"For most events, the three choices will lead to numerically very close values for $\\cos \\theta $ and thus resulting $wt^{EW}$ .",
"The difference originates from distinct $\\cos \\theta $ dependence of $Z$ and $\\gamma ^*$ exchange amplitudes and not only from electroweak boxes.",
"The $wt^{EW}$ allows for flexible implementation of the EW corrections using TauSpinner framework and form-factors calculated e.g.",
"with Dizet.",
"The formula for $wt^{EW}$ can be used to re-weight from one EW LO scheme to another too.",
"In that case, both the numerator and denominator of Eq.",
"(REF ) will use lowest order $d\\sigma _{Born}$ , calculated in different EW schemesIn this way, in particular, the fixed width description for the $Z$ -boson propagator can be replaced with the $s$ dependent one.",
"though."
],
[
"EW corrections to doubly-deconvoluted observables",
"Now that all components needed for calculation of $wt^{EW}$ are explained, we can present results for selected examples of doubly-deconvoluted observables around the $Z$ -pole.",
"The Powheg+MiNLO Monte Carlo, with NLO QCD and LO EW matrix elements, was used to generate $Z+j$ events with $Z \\rightarrow e^+ e^-$ decays in $pp$ collisions at 8 TeV.",
"No selection was applied to generated events, except for an outgoing electron pair invariant mass range of $70 < m_{ee} < 150$ GeV.",
"For events generation, the EW parameters as shown in left-most column of Table REF were used.",
"It is often used as a default for phenomenological studies at LHC.",
"The $\\alpha $ and $s^2_W$ close to the ones of $\\overline{\\mathrm {M}S}$ scheme discussed in [25] were taken.",
"Note that they do not coincide accurately with the precise LEP experiments measurements at the $Z$ -pole [1].",
"To quantify the effect of the EW corrections, we re-weight events generated, to EW LO with the scheme used by the Dizet: Table REF second column.",
"Only then we gradually introduce EW corrections and form-factors calculated with that library.",
"For each step, the appropriate numerator of the $wt^{EW}$ is calculated, while for the denominator the EW LO ${A}^{Born}$ matrix element Eq.",
"(REF ) is used; parameters as in the left-most column of Table REF .",
"The sequential steps, in which we illustrate effects of EW corrections are given below: Re-weight with $wt^{EW}$ , from EW LO scheme used for MC events generation to EW LO scheme with $s^2_W$ = 0.21215, Table REF second column.",
"The ${A}^{Born}$ matrix element, Eq.",
"(REF ), is usedThe MC sample is generated with fixed width propagator.",
"We remain with this convention.",
"This could also be changed with the help of $wt^{EW}$ .",
"for calculating numerator of $wt^{EW}$ .",
"As in step (1), but include EW corrections to $M_W$ , effectively changing to $s^2_W$ = 0.22352 in calculation of $wt^{EW}$ .",
"Relation, formula (REF ), is not obeyed anymore.",
"As in step (2), but include EW loop corrections to the normalization of $Z$ -boson and $\\gamma ^*$ propagators, i.e.",
"QCD/EW corrections to $\\alpha (0)$ and $\\rho _{\\ell f}(s)$ form-factor calculated without box corrections.",
"The ${A}^{Born+EW}$ , Eq.",
"(REF ), is used for calculating numerator of $wt^{EW}$ .",
"As in step (3), but include EW corrections to $Z$ -boson vector couplings: ${K}_f, {K}_l, {K}_{\\ell f}$ , calculated without box corrections.",
"The ${A}^{Born+EW}$ is used for calculating numerator of $wt^{EW}$ .",
"As in step (4), but $\\rho _{\\ell f}, {K}_f, {K}_l, {K}_{\\ell f}$ form-factors include box corrections.",
"The ${A}^{Born+EW}$ is used for calculating numerator of $wt^{EW}$ .",
"After step (1) the sample is EW LO and QCD NLO, but with different EW scheme than used originally for events generation.",
"Then steps (2)-(5) introduce EW corrections.",
"Step (3) effectively changes $\\alpha $ back to be close to $\\alpha (M_Z^2)$ , while steps (4)-(5) effectively shift back $v_f, v_l$ close to the values used in generation.",
"Parameters for EW LO scheme used for event generation are already close to measured at the $Z$ -pole.",
"That is why we expect the total EW corrections to the generated sample to be roughly at the percent level only.",
"In the following, we will estimate how precise it would be to use effective Born approximation with LEP or LEP with improved norm.",
"parametrisations instead of complete EW corrections.",
"To obtain those predictions, re-weighting similar to step (1) listed above is needed, but in the numerator of $wt^{EW}$ the ${A}^{Born}$ parametrisations as specified in the right two columns of Table REF are used.",
"For LEP with improved norm.",
"the $\\rho _{\\ell ,f} = 1.005$ has to be included as well.",
"The important flexibility of the proposed approach is that $wt^{EW}$ can be calculated using $d \\sigma _{Born}$ in different frames: $\\cos \\theta ^*$ , Mustraal or Collins-Soper.",
"For some observables, frame choice used for $wt^{EW}$ calculation is not numerically relevant at all; the simplest $\\cos \\theta ^*$ frame can be used.",
"We show later an example, where only the Mustraal frame for the $wt^{EW}$ calculation leads to correct results."
],
[
"The $Z$ -boson line-shape",
"In the EW LO, the $Z$ -boson line-shape, assuming that the constraint (REF ) holds, depends predominantly on $M_Z$ and $\\Gamma _z$ .",
"The effects on the line-shape from EW loop corrections are due to corrections to the propagators: vacuum polarization corrections (running $\\alpha $ ) and $\\rho $ form-factor, which change relative contributions of the $Z$ to $\\gamma ^*$ and, the $Z$ -boson vector to axial coupling ratio ($\\sin ^2\\theta _{eff}$ ).",
"The above affects not only shape but also normalization of the cross-section.",
"In the formulae (REF ) we do not use running $Z$ -boson width, which remains fixed.",
"In Fig.",
"REF (top-left) distributions of generated and EW corrected line-shapes are shown.",
"With the logarithmic scale, a difference is barely visible.",
"With the following plots of the same Figure we study details.",
"The ratios of the line-shape distributions with gradually introduced EW corrections are shown.",
"For the reference distributions (ratio-histograms denominators) for the following three plots: (i) EW LO $\\alpha (0)$ scheme, (ii) effective Born (LEP) and (iii) effective Born (LEP with improved norm.)",
"are used.",
"At the $Z$ -pole, complete EW corrections contribute about 0.1% with respect to the one of effective Born (LEP with improved norm.).",
"A use of events generated with EW LO matrix element but of different parametrisations significantly reduce the numerical size of missing EW corrections.",
"Table REF details numerically EW corrections to the normalization (ratio of the cross-sections) integrated in the range $ 80 < m_{ee} <100$ GeV and $89 < m_{ee} < 93$ GeV.",
"Results from EW weight with the $\\cos \\theta ^*$ definition of the scattering angle are shown.",
"The total EW correction factor is about 0.965 for cross-section normalization and EW LO $\\alpha (0)$ , while the total correction for the effective Born (LEP with improved norm.)",
"is of about 1.001.",
"In Table REF results with $wt^{EW}$ calculated with different frames are compared.",
"If Mustraal or Collins-Soper frames are used instead of $\\cos \\theta ^*$ for weight calculations, the differences are at most at the 5-th significant digit.",
"Table: EW corrections for cross-sections integrated over the mass window around ZZ-pole;89<m ee <89 < m_{ee} < 93 GeV.",
"The EW weight is calculatedwith cosθ * \\cos \\theta ^*, cosθ Mustraal \\cos \\theta ^{Mustraal} or cosθ CS \\cos \\theta ^{CS} .Figure: Top-left: line-shape distribution as generated with Powheg+MiNLO (blue triangles)and after reweighting introducing all EW corrections (red triangles).",
"The two choices arebarely distinguishable.",
"Ratios of the line-shapes with gradually introduced EW correctionsare shown in consecutive plots, where as a reference (black dashed line) respectively:(i) EW LO α(0)\\alpha (0) scheme (top-right),(ii) effective Born (LEP) (bottom-left) and, (iii)effective Born (LEP with improved norm.)",
"(bottom-right), was used."
],
[
"The $A_{FB}$ distribution",
"The forward-backward asymmetry for $pp$ collisions reads $ A_{FB} = \\frac{\\sigma (\\cos \\theta > 0) - \\sigma (\\cos \\theta < 0)}{\\sigma (\\cos \\theta > 0) + \\sigma (\\cos \\theta < 0)},$ where $\\cos \\theta $ of the Collins-Soper frame is used.",
"The EW corrections change $A_{FB}$ , particularly around the $Z$ -pole.",
"In Fig.",
"REF (top-left), the $A_{FB}$ as generated (EW LO) and EW corrected is shown as a function of $ m_{ee}$ .",
"In the following plots of this Figure, we study details.",
"The $\\Delta A_{FB} = A_{FB} - A_{FB}^{ref}$ , with gradually introduced EW corrections to $A_{FB}$ is shown and compared with the following reference choices for $A_{FB}^{ref}$ : (i) EW LO $\\alpha (0)$ scheme, (ii) effective Born (LEP) and (iii) effective Born (LEP with improved norm.).",
"Complete EW corrections to predictions of EW LO $\\alpha (0)$ scheme for $A_{FB}$ integrated around $Z$ -pole give $\\Delta A_{FB}$ = -0.03534.",
"The EW correction $\\Delta A_{FB}$ to predicition of effective Born (LEP with improved norm.",
"), is only -0.00005.",
"We observe that effective Born (LEP improved norm.)",
"reproduces EW loop corrections precision better and $\\Delta A_{FB}$ = -0.0001 in the full presented mass range.",
"The remaining box corrections contribute around $ m_{ee}=150$ GeV about -0.002 to $\\Delta A_{FB}$ .",
"Table REF details numerically EW corrections, for $A_{FB}$ integrated over the $80 < m_{ee} < 100$ GeV and $ 89 < m_{ee} < 93$ GeV ranges.",
"For calculating EW weight, the $\\cos \\theta ^*$ definition of the scattering angle was used.",
"In Table REF results obtained with $wt^{EW}$ calculated in different frames are compared.",
"When the Mustraal or Collins-Soper frame is used instead of $\\cos \\theta ^*$ , the differences are at most at the 5-th significant digit, similar as for the line-shape.",
"Table: The difference ΔA FB \\Delta A_{FB} in forward-backward asymmetry around ZZ-pole,m ee m_{ee} = 89 - 93 GeV.The cosθ CS \\cos \\theta ^{CS} is used to define forward and backward hemispheres.The EW weight is calculated respectively from cosθ * \\cos \\theta ^*, cosθ Mustraal \\cos \\theta ^{Mustraal} or cos CS \\cos ^{CS}.Figure: Top-left: the A FB A_{FB} as generated with Powheg+MiNLO (blue triangles)and after reweighting introducing all EW corrections (red triangles).The two choices are barely distinguishable.",
"The differencesΔA FB =A FB -A FB ref \\Delta A_{FB} = A_{FB} - A_{FB}^{ref}, due to gradually introduced EW corrections are shownin consecutive plots, where as a reference (black dashed line) respectively:(i) EW LO α(0)\\alpha (0) scheme (top-right),(ii) effective Born (LEP) (bottom-left) and, (iii)effective Born (LEP with improved norm.)",
"(bottom-right), was used."
],
[
"Effective weak mixing angles",
"The forward-backward asymmetry $A_{FB}$ at the $Z$ -pole can be used as an observable for effective weak mixing Weinberg angles, dependent on the invariant mass of lepton pairs.",
"We extend standard LEP definition of effective weak mixing angles to $\\sin ^2\\theta ^f_{eff}(s,t) = Re({K}^f(s,t)) s^2_W + I^2_f(s,t),$ which is more suitable for LHC and for the off $Z$ -pole regions.",
"The flavour dependent effective weak mixing angles, calculated using: Eq.",
"(REF ), EW form-factors of Dizet library, and $s^2_W=0.22352$ are shown on Fig.",
"REF as a function of the invariant mass of outgoing lepton pair and for $\\cos \\theta = 0.5$ .",
"The imaginary part of $I^2_f(s,t)$ is about $10^{-4}$ only.",
"In Table REF we display effective weak mixing angles averaged over specified mass windows.",
"The effective $\\sin \\theta _{eff}^f$ on the $Z$ -pole, printed by Dizet is shown in Table REF .",
"It is numerically slightly different than of Table REF , which is an average over mass window close to $Z$ -pole.",
"Note, that the observed very good agreement at the $Z$ -pole between $A_{FB}$ predictions of effective Born with (LEP) or (LEP with improved norm.)",
"parametrisations and fully EW corrected is not reflected for predictions of flavour dependent effective weak Weinberg angles.",
"Effective Born (LEP) and (LEP with improved norm.)",
"are parametrised with $s^2_W = 0.23152$ , while Dizet library predicts leptonic effective weak mixing angle $\\sin ^2\\theta _{eff}^{\\ell }(M_Z^2)$ = 0.23176 which is about $20 \\cdot 10^{-5}$ different.",
"Why then such a good agreement on $\\Delta A_{FB}$ as seen on Fig.",
"REF bottom plots?",
"Certainly this requires further attention.",
"Table: The effective weak mixing angles sin 2 θ eff f \\sin ^2 \\theta _{eff}^{f}, for different mass windowswith/without box corrections.",
"The form-factor corrections are averaged with realistic line-shapeand cosθ\\cos \\theta distribution.Figure: Effective weak mixing angles sin 2 θ eff f (s,t)\\sin ^2\\theta _{eff}^f(s,t)as a function of m ee m_{ee} and cosθ\\cos \\theta = 0,without (left-hand plot) and with (right-hand plot) box corrections.The K f (s,t){K}^f(s,t) form-factor calculated using Dizetlibrary and on-mass-shell s W 2 =0.22352s^2_W=0.22352 were used.Only the real part is shown, imaginary part of I f 2 (s,t)I^2_f(s,t) is only about 10 -4 10^{-4}."
],
[
"The $A_{4}$ , {{formula:22b69db4-bf8a-415b-9564-50cc6132c011}} angular coefficients",
"To complete the discussion on doubly-deconvoluted observables, we turn our attention to angular coefficients $A_{4}$ and $A_{3}$ (proportional to product of vector and axial couplings) and to EW corrections.",
"The coefficients are calculated from the event sample with the moments methods [32] and in the Collins-Soper frame.",
"The EW weight $wt^{EW}$ is used to introduce EW corrections and is calculated with the help of $\\cos \\theta ^*$ , $\\cos \\theta ^{Mustraal}$ or $\\cos \\theta ^{CS}$ angles.",
"Similarly as for $A_{FB}$ , the EW corrections change overall size and the shape of $A_4$ as a function of $m_{ee}$ ; particularly around the $Z$ -pole.",
"In Fig.",
"REF (top-right), the $A_4$ for generated sample (EW LO) and EW corrected is shown as a function of $m_{ee}$ .",
"In the following plots of the figure details are studied.",
"The $\\Delta A_4 = A_4 - A_4^{ref}$ with gradually introduced EW corrections is shown and compared with the following reference choices for $A_{4}^{ref}$ : (i) EW LO $\\alpha (0)$ scheme, (ii) effective Born (LEP) and (iii) effective Born (LEP with improved norm.).",
"Conclusions are very similar as for previous $\\Delta A_{FB}$ discussion.",
"Note that $\\Delta A_4$ and $\\Delta A_{FB}$ scale approximately with the relation $A_4 = 8/3 A_{FB}$ .",
"The analogous set of plots, Fig.",
"REF , is prepared for $A_3$ .",
"In this case, only the Mustraal frame turned out to be adequate for $wt^{EW}$ calculation.",
"Both the $\\cos \\theta ^*$ and $\\cos \\theta ^{CS}$ were unable to fully capture the effects of EW corrections.",
"The results for $\\Delta A_{3}$ are collected in Table REF .",
"The mass window $80 < m_{ee} <100$ GeV and $p_T^{ee} < 30$ GeV are chosen.",
"The estimation for $\\Delta A_4$ differ little if $\\cos \\theta ^*$ , $\\cos \\theta ^{CS}$ or $\\cos \\theta ^{Mustraal}$ is used for calculations of EW corrections.",
"The $\\Delta A_3$ is non-zero, as it should be, only if the $\\cos \\theta ^{Mustraal}$ is used in $wt^{EW}$ calculation.",
"For $A_{4}$ , multiplied by $\\frac{8}{3}$ entries of Table REF are good enough.",
"Figure: Top-left: the A 4 A_4 as function of m ee m_{ee}.",
"Overlayed are generatedand EW corrected A 4 A_4 predictions.",
"These results are barely distinguishable.The differences ΔA 4 =A 4 -A 4 ref \\Delta A_{4} = A_4 - A_4^{ref} due to gradually introduced EW corrections areshown in consecutive plots, where as a reference A 4 ref A_4^{ref} (black dashed line) respectively(i) EW LO α(0)\\alpha (0) scheme (top-right),(ii) effective Born (LEP) (bottom-left) and (iii)effective Born (LEP with improved norm.)",
"(bottom-right) was used.Figure: Top-left: the A 3 A_3 as function of m ee m_{ee}.",
"Overlayed are generated and EW corrected A 3 A_3 predictions.",
"These results are barely distinguishable.The differences ΔA 3 =A 3 -A 3 ref \\Delta A_3 = A_3 - A_3^{ref} due to gradually introduced EW corrections areshown in consecutive plots, where as a reference A 3 ref A_3^{ref} (black dashed line) respectively(i) EW LO α(0)\\alpha (0) scheme (top-right),(ii) effective Born (LEP) (bottom-left) and (iii)effective Born (LEP with improved norm.)",
"(bottom-right) was used.",
"In this case,the EW weight is calculated with cosθ Mustraal \\cos \\theta ^{Mustraal}.Table: The ΔA 3 \\Delta A_3 shift of the A 3 A_3, due to EW corrections, averaged overp T ee <p_T^{ee} < 30 GeV and 80<m ee <10080 < m_{ee} < 100 GeV ranges.The cosθ CS \\cos \\theta ^{CS} is used for angular polynomials but for the EW weight calculationcosθ * \\cos \\theta ^*, cosθ Mustraal \\cos \\theta ^{Mustraal} or cosθ CS \\cos \\theta ^{CS} areused respectively."
],
[
"Summary",
"In this paper we have shown how the EW corrections for double-deconvoluted observables at LHC can be evaluated using Improved Born Approximation.",
"We have exploited a wealth of the LEP era results encapsulated in the Dizet library developed at that time.",
"We have used that formalism to calculate and present numerically EW corrections for doubly-deconvoluted observables, such as $Z$ -boson line-shape, forward-backward asymmetry $A_{FB}$ , effective weak mixing angles or lepton direction angular coefficients.",
"We have followed largely discussions available in Dizet documentation.",
"We have introduced the notion of the effective Born and explained how Monte Carlo events generated at NLO QCD can be transformed to reduced kinematics, of strong interaction lowest order, for the calculation of spin amplitudes $q \\bar{q} \\rightarrow Z/\\gamma ^* \\rightarrow \\ell \\ell $ .",
"This could be achieved thanks to properties of spin amplitudes discussed in [5], [6].",
"We explained how per-event weight $wt^{EW}$ , can be build and used to attribute EW corrections to already generated events.",
"We have re-visited the notion of Effective Born with LEP (or with LEP of improved norm.)",
"parametrisations where dominant parts of EW corrections are taken into accout with a redefinition of coupling constants.",
"We have evaluated how well it works for observables of the paper.",
"The discussed approach for treating EW corrections for Drell-Yan process in pp collisions has been implemented in the Tauola/TauSpinner package [15], [9] to be available starting from the forthcoming release.",
"Once the formalism was explained, numerical results of EW corrections to the $Z$ -boson line-shape, forward-backward asymmetries, lepton angular coefficients were presented.",
"Results were obtained using Dizet for calculating EW form-factors and Tauola/TauSpinner for calculating respective EW weights of Improved Born Approximation or Effective Born with LEP (or with LEP improved norm.)",
"parametrisations.",
"The choice of the version of EW library was dictated by the compatibility with the KKMC Monte Carlo [2], the program widely used at the LEP times.",
"It relies on a published version of Dizet, thus suits the purposes of a reference point well.",
"Also, omitted effects are rather small.",
"In the future, the algorithm of TauSpinner can be useful to quantify the differences among distinct implementations of the electroweak sector.",
"The numerical studies with the updates to Dizet version 6.42 [12], [21] and with other, sometimes unpublished electroweak codes are left for the future work.",
"One should stress the necessity of such future numerical discussion and updates, in particular due to the photonic vacuum polarization, e.g.",
"as provided in refs.",
"[34], [35] but absent in the last published (or presently public) version of Dizet 6.42.",
"This update is required already at LHC precision of $Z$ -boson couplings measurements.",
"In many applications focused on challenges of strong interactions, electroweak corrections are receiving rather minimal attention and in particular $Z$ boson fixed value width, or running only in proportion to the energy transfer, is used.",
"This may be inappropriate for large $s$ as found e.g.",
"in [36].",
"TauSpinner can be used to evaluate numerical consequences of such approximation.",
"Finally let us mention that presented implementation of EW corrections as per-even weight, was already found useful for experimental measurements [16] at LHC and for discussions during recent workshops, see e.g.",
"Ref. [37].",
"Acknowledgements E.R-W. would like to thank Daniel Froidevaux, Aaron Ambruster and colleagues from ATLAS Collaboration Standard Model Working Group for numerous inspiring discussions on the applications of presented here implementation of EW corrections to the $\\sin ^2\\theta _{eff}^{lep}$ measurement at LHC.",
"This project was supported in part from funds of Polish National Science Centre under decision UMO-2014/15/B/ST2/00049.",
"Majority of the numerical calculations were performed at the PLGrid Infrastructure of the Academic Computer Centre CYFRONET AGH in Krakow, Poland.",
"Comment on technical details of TauSpinner EW effects implementation Although the framework of Tauola/TauSpinner package [15], [9] has been used for numerical results presented in this paper, the code is not yet available with the public release but only in the private distribution and only partly in development version [38] which updates itself daily from our work repository.",
"Tests and some of the code developments need to be completed.",
"Once we achieve confidence the official stable version of the code will become public at [38] .",
"Let us nonetheless list main points of the implementation which was already used to obtain numerical results: Pre-tabulated EW corrections: form-factors, vacuum polarization corrections in form of 2D root histograms or alternatively ASCII files of the KMMC project [2] were used to assure modularity and to enable graphic tests.",
"Functions to calculate $\\cos \\theta ^*$ , $\\cos \\theta ^{Mustraal}$ , $\\cos \\theta ^{CS}$ from kinematics of outgoing final state (leptons and partons/jets) used for numerical results are already in part available in TAUOLA/TauSpinner/examples/ Dizet-example directory.",
"The README file of that directory is gradually filled with technical details.",
"Routine to initialize parameters of the Born function is provided.",
"The SUBROUTINE INITWK of TAUOLA/ src/tauolaFortranInterfaces/tauola_extras.f has been copied and extended.",
"It is available under the name INITWKSWDELT , with the following input: $G_{\\mu }$ , $\\alpha $ , $M_Z$ , $s$ , EW form-factors and vacuum polarization corrections, $s^2_W$ and parameters for couplings variations $\\delta _{s2W}$ , $\\delta _{V}$ , see Section for details.",
"To calculate $d\\sigma _{Born}$ and the $wt^{EW}$ the t_bornew function with flexible options for EW scheme and $\\delta _{s2W}$ , $\\delta _{V}$ , is prepared.",
"It is used by TauSpinner library function default_nonSM_born(ID, S, cost, H1, H2, key) now.",
"It is premature for complete documentation, but comments on the software used to obtain numerical results are in place.",
"How to vary $s^2_W$ beyond the EW LO schemes.",
"In the discussed EW scheme $(\\alpha (0), G_\\mu , M_Z)$ , the $s^2_W$ is not directly available for fits.",
"It is calculated from relation (REF ) of the Standard Model.",
"One possibility to vary $s^2_W$ , but stay within Standard Model framework is to vary some other constants which impact $s^2_W$ .",
"The candidates within Standard Model, which are also inputs to the Dizet library, are $G_{\\mu }$ or $m_{t}$ .",
"From the simple estimates, to allow $\\pm 100 \\cdot 10^{-5}$ variation of $s^2_W$ , those parameter will have to be varied far beyond their experimental ambiguitiesRange would be $\\pm 10$ GeV for $m_t$ or $\\pm 4\\cdot 10^{-8} GeV^{-2}$ for $G_\\mu $ .. One can extend formulae for $A^{Born+EW}$ (REF ) beyond the Standard Model too.",
"Additional v-like contribution to $Z$ -boson $v_{\\ell }, v_f$ couplings can be introduced with $\\delta _{S2W}$ or $\\delta _{V}$ as presented later.",
"Below few details and options on implementation into $A^{Born+EW}$ amplitudes are given: optME = 1: introduce unspecified heavy particle coupling to the $Z$ -boson, to modify fermions vector couplings $v_{\\ell } = && (2 \\cdot T_3^{\\ell } - 4 \\cdot q_{\\ell } \\cdot (s^2_W + \\delta _{S2W}) \\cdot {K}_{\\ell }(s,t))/\\Delta , \\nonumber \\\\v_f = && (2 \\cdot T_3^f - 4 \\cdot q_f \\cdot (s^2_W + \\delta _{S2W}) \\cdot {K}_f(s,t))/\\Delta , \\nonumber \\\\vv_{\\ell f} = && \\frac{1}{v_{\\ell } \\cdot v_f} [( 2 \\cdot T_3^{\\ell }) (2 \\cdot T_3^f) \\nonumber \\\\&& - 4 \\cdot q_{\\ell } \\cdot (s^2_W+ \\delta _{S2W}) \\cdot {K}_f(s,t)( 2 \\cdot T_3^{\\ell }) \\\\&& - 4 \\cdot q_f \\cdot (s^2_W+ \\delta _{S2W}) \\cdot {K}_{\\ell }(s,t) (2 \\cdot T_3^f) \\nonumber \\\\&& + (4 \\cdot q_{\\ell } \\cdot s^2_W) (4 \\cdot q_f \\cdot s^2_W) {K}_{\\ell f}(s,t) \\nonumber \\\\&& + 2 \\cdot (4 \\cdot q_{\\ell })) (4 \\cdot q_f \\cdot ) \\cdot s^2_W \\cdot \\delta _{S2W} ) {K}_{\\ell f}(s,t) ]\\ \\frac{1}{\\Delta ^2} \\nonumber $ but do not alter $\\Delta = \\sqrt{ 16 \\cdot s^2_W \\cdot (1 - s^2_W)}$ or any other $A^{Born+EW}$ (REF ) couplings or calculations of the EW form-factors.",
"optME = 2: recalculate $M_W$ for numerically modified $m_t$ or $G_\\mu $ and modify accordingly Standard Model $s^2_W = 1 -M_W^2/M_Z^2$ , for $s^2_W $ present in $A^{Born+EW}$ .",
"The form-factors are (are not) recalculatedThe optME = 1, 2, if form-factors are not recalculated, formally differ by the term proportional to $\\delta ^2_{S2W}$ and only in the expression for $vv_{\\ell f}$ .",
"Change of input parameters $G_{\\mu }$ or $m_{t}$ as a source for $s^2_W$ variations in optME = 2, implies changes of the couplings and thus for consistency, recalculation of form-factors.",
"All these options can be realized with the Tauola/TauSpinner package, of the development version..",
"In total, 3 variants of this option were used for Fig.",
"REF .",
"optME = 3: similar as optME = 1 but redefine directly fermions vector couplings with $\\delta _{V}$ .",
"We keep relative normalization (charge structure) of $\\delta _{V}$ similar to $\\delta _{S2W}$ , to facilitate comparisons.",
"Then $v_{\\ell } = && (2 \\cdot T_3^{\\ell } - 4 \\cdot q_{\\ell } \\cdot (s^2_W \\cdot {K}_{\\ell }(s,t) + \\delta _{V} ))/\\Delta , \\nonumber \\\\v_f = && (2 \\cdot T_3^f - 4 \\cdot q_f \\cdot (s^2_W \\cdot {K}_f(s,t) + \\delta _{V} ))/\\Delta , \\nonumber \\\\vv_{\\ell f} = && \\frac{1}{v_{\\ell } \\cdot v_f} [( 2 \\cdot T_3^{\\ell }) (2 \\cdot T_3^f) \\nonumber \\\\&& - 4 \\cdot q_{\\ell } \\cdot (s^2_W \\cdot {K}_f(s,t) + \\delta _{V} ) ( 2 \\cdot T_3^{\\ell }) \\\\&& - 4 \\cdot q_f \\cdot (s^2_W \\cdot {K}_{\\ell }(s,t) + \\delta _{V} ) (2 \\cdot T_3^f) \\nonumber \\\\&& + (4 \\cdot q_{\\ell } \\cdot s^2_W) (4 \\cdot q_f \\cdot s^2_W) {K}_{\\ell f}(s,t) \\nonumber \\\\&& + 2 \\cdot (4 \\cdot q_{\\ell })) (4 \\cdot q_f \\cdot ) \\cdot s^2_W \\cdot {K}_{\\ell f}(s,t) \\cdot \\delta _{V} ]\\ \\frac{1}{\\Delta ^2}.",
"\\nonumber $ The $\\delta _{V}$ shift is almost equivalent to $\\delta _{S2W}$ shift, but affects couplings in a $(s,t)$ independent manner.",
"Even though discussion of $s^2_W$ variation necessary for fits, is generally out of scope of the present paper and it can not be now exhausted, let us provide some numerical results to illustrate stability of the methodFor optME=2, the $m_t$ (or $G_{\\mu }$ ) have been shifted to move $s^2_W$ by $\\pm 100 \\cdot 10^{-5}$ .",
"Then the form-factors were recalculated, or optionally kept at nominal values..",
"The variations for $A_4(M_Z)$ are presented in Figure REF : as a function of $s^2_W$ on the left-hand side plot and as a function of $sin^2\\theta _{eff}^l$ on the right-hand side plot.",
"It is very reassuring, that all presented optME methods lead to the same slope of the $A_4(M_Z)$ as a function of $sin^2\\theta _{eff}$ .",
"Very similar curve could be presented for $A_{FB}$ , which would be scaled by $\\frac{3}{8}$ with respect to $A_4$ only.",
"Figure: The A 4 A_4 variation due to shifts induced with the presentedin Appendix options;as a functionof s W 2 s^2_W (left-hand side) and as a function of sin 2 θ eff l sin^2\\theta _{eff}^l (right-hand side).The “FF G μ G_\\mu varied”, FF m t m_t varied” correspond to the case whenform-factors were recalculated.",
"Otherwise they were kept at nominal values.",
"Initialization of the Dizet library There is a wealth of initialization constants and options available for Dizet library.",
"The documentation of that program and of its interface for KKMC, explains options available for the TauSpinner users as well.",
"Tables REF and REF recall available Dizet initialization, Table REF lists calculated by Dizet quantities for the use in TauSpinner library.",
"Table: Dizet initialization parameters: masses and couplings.In the present work, we have relied on the Dizet library version as installed in the KKMC Monte Carlo [8] and used at a time of LEP 1 in detector simulations.",
"Already for the data analysis and in particular for final fits [1], further effects of minor, but non-negligible numerical impact were taken into account.",
"Gradually, effects such as improved top contributions [39] or better photonic vacuum polarization [23], were taken into account.",
"This has to be updated for Dizet library too.",
"Such update is of importance also for the KKMC project itself because of forthcoming applications for the Future Circular Collider or for LHC [40].",
"Table: Dizet recalculated quantities available for the TauSpinner use.",
"For details of the ZPAR table see Refs.",
","
],
[
"Comment on technical details of ",
"Although the framework of Tauola/TauSpinner package [15], [9] has been used for numerical results presented in this paper, the code is not yet available with the public release but only in the private distribution and only partly in development version [38] which updates itself daily from our work repository.",
"Tests and some of the code developments need to be completed.",
"Once we achieve confidence the official stable version of the code will become public at [38] .",
"Let us nonetheless list main points of the implementation which was already used to obtain numerical results: Pre-tabulated EW corrections: form-factors, vacuum polarization corrections in form of 2D root histograms or alternatively ASCII files of the KMMC project [2] were used to assure modularity and to enable graphic tests.",
"Functions to calculate $\\cos \\theta ^*$ , $\\cos \\theta ^{Mustraal}$ , $\\cos \\theta ^{CS}$ from kinematics of outgoing final state (leptons and partons/jets) used for numerical results are already in part available in TAUOLA/TauSpinner/examples/ Dizet-example directory.",
"The README file of that directory is gradually filled with technical details.",
"Routine to initialize parameters of the Born function is provided.",
"The SUBROUTINE INITWK of TAUOLA/ src/tauolaFortranInterfaces/tauola_extras.f has been copied and extended.",
"It is available under the name INITWKSWDELT , with the following input: $G_{\\mu }$ , $\\alpha $ , $M_Z$ , $s$ , EW form-factors and vacuum polarization corrections, $s^2_W$ and parameters for couplings variations $\\delta _{s2W}$ , $\\delta _{V}$ , see Section for details.",
"To calculate $d\\sigma _{Born}$ and the $wt^{EW}$ the t_bornew function with flexible options for EW scheme and $\\delta _{s2W}$ , $\\delta _{V}$ , is prepared.",
"It is used by TauSpinner library function default_nonSM_born(ID, S, cost, H1, H2, key) now.",
"It is premature for complete documentation, but comments on the software used to obtain numerical results are in place."
],
[
"How to vary $s^2_W$ beyond the EW LO schemes.",
"In the discussed EW scheme $(\\alpha (0), G_\\mu , M_Z)$ , the $s^2_W$ is not directly available for fits.",
"It is calculated from relation (REF ) of the Standard Model.",
"One possibility to vary $s^2_W$ , but stay within Standard Model framework is to vary some other constants which impact $s^2_W$ .",
"The candidates within Standard Model, which are also inputs to the Dizet library, are $G_{\\mu }$ or $m_{t}$ .",
"From the simple estimates, to allow $\\pm 100 \\cdot 10^{-5}$ variation of $s^2_W$ , those parameter will have to be varied far beyond their experimental ambiguitiesRange would be $\\pm 10$ GeV for $m_t$ or $\\pm 4\\cdot 10^{-8} GeV^{-2}$ for $G_\\mu $ .. One can extend formulae for $A^{Born+EW}$ (REF ) beyond the Standard Model too.",
"Additional v-like contribution to $Z$ -boson $v_{\\ell }, v_f$ couplings can be introduced with $\\delta _{S2W}$ or $\\delta _{V}$ as presented later.",
"Below few details and options on implementation into $A^{Born+EW}$ amplitudes are given: optME = 1: introduce unspecified heavy particle coupling to the $Z$ -boson, to modify fermions vector couplings $v_{\\ell } = && (2 \\cdot T_3^{\\ell } - 4 \\cdot q_{\\ell } \\cdot (s^2_W + \\delta _{S2W}) \\cdot {K}_{\\ell }(s,t))/\\Delta , \\nonumber \\\\v_f = && (2 \\cdot T_3^f - 4 \\cdot q_f \\cdot (s^2_W + \\delta _{S2W}) \\cdot {K}_f(s,t))/\\Delta , \\nonumber \\\\vv_{\\ell f} = && \\frac{1}{v_{\\ell } \\cdot v_f} [( 2 \\cdot T_3^{\\ell }) (2 \\cdot T_3^f) \\nonumber \\\\&& - 4 \\cdot q_{\\ell } \\cdot (s^2_W+ \\delta _{S2W}) \\cdot {K}_f(s,t)( 2 \\cdot T_3^{\\ell }) \\\\&& - 4 \\cdot q_f \\cdot (s^2_W+ \\delta _{S2W}) \\cdot {K}_{\\ell }(s,t) (2 \\cdot T_3^f) \\nonumber \\\\&& + (4 \\cdot q_{\\ell } \\cdot s^2_W) (4 \\cdot q_f \\cdot s^2_W) {K}_{\\ell f}(s,t) \\nonumber \\\\&& + 2 \\cdot (4 \\cdot q_{\\ell })) (4 \\cdot q_f \\cdot ) \\cdot s^2_W \\cdot \\delta _{S2W} ) {K}_{\\ell f}(s,t) ]\\ \\frac{1}{\\Delta ^2} \\nonumber $ but do not alter $\\Delta = \\sqrt{ 16 \\cdot s^2_W \\cdot (1 - s^2_W)}$ or any other $A^{Born+EW}$ (REF ) couplings or calculations of the EW form-factors.",
"optME = 2: recalculate $M_W$ for numerically modified $m_t$ or $G_\\mu $ and modify accordingly Standard Model $s^2_W = 1 -M_W^2/M_Z^2$ , for $s^2_W $ present in $A^{Born+EW}$ .",
"The form-factors are (are not) recalculatedThe optME = 1, 2, if form-factors are not recalculated, formally differ by the term proportional to $\\delta ^2_{S2W}$ and only in the expression for $vv_{\\ell f}$ .",
"Change of input parameters $G_{\\mu }$ or $m_{t}$ as a source for $s^2_W$ variations in optME = 2, implies changes of the couplings and thus for consistency, recalculation of form-factors.",
"All these options can be realized with the Tauola/TauSpinner package, of the development version..",
"In total, 3 variants of this option were used for Fig.",
"REF .",
"optME = 3: similar as optME = 1 but redefine directly fermions vector couplings with $\\delta _{V}$ .",
"We keep relative normalization (charge structure) of $\\delta _{V}$ similar to $\\delta _{S2W}$ , to facilitate comparisons.",
"Then $v_{\\ell } = && (2 \\cdot T_3^{\\ell } - 4 \\cdot q_{\\ell } \\cdot (s^2_W \\cdot {K}_{\\ell }(s,t) + \\delta _{V} ))/\\Delta , \\nonumber \\\\v_f = && (2 \\cdot T_3^f - 4 \\cdot q_f \\cdot (s^2_W \\cdot {K}_f(s,t) + \\delta _{V} ))/\\Delta , \\nonumber \\\\vv_{\\ell f} = && \\frac{1}{v_{\\ell } \\cdot v_f} [( 2 \\cdot T_3^{\\ell }) (2 \\cdot T_3^f) \\nonumber \\\\&& - 4 \\cdot q_{\\ell } \\cdot (s^2_W \\cdot {K}_f(s,t) + \\delta _{V} ) ( 2 \\cdot T_3^{\\ell }) \\\\&& - 4 \\cdot q_f \\cdot (s^2_W \\cdot {K}_{\\ell }(s,t) + \\delta _{V} ) (2 \\cdot T_3^f) \\nonumber \\\\&& + (4 \\cdot q_{\\ell } \\cdot s^2_W) (4 \\cdot q_f \\cdot s^2_W) {K}_{\\ell f}(s,t) \\nonumber \\\\&& + 2 \\cdot (4 \\cdot q_{\\ell })) (4 \\cdot q_f \\cdot ) \\cdot s^2_W \\cdot {K}_{\\ell f}(s,t) \\cdot \\delta _{V} ]\\ \\frac{1}{\\Delta ^2}.",
"\\nonumber $ The $\\delta _{V}$ shift is almost equivalent to $\\delta _{S2W}$ shift, but affects couplings in a $(s,t)$ independent manner.",
"Even though discussion of $s^2_W$ variation necessary for fits, is generally out of scope of the present paper and it can not be now exhausted, let us provide some numerical results to illustrate stability of the methodFor optME=2, the $m_t$ (or $G_{\\mu }$ ) have been shifted to move $s^2_W$ by $\\pm 100 \\cdot 10^{-5}$ .",
"Then the form-factors were recalculated, or optionally kept at nominal values..",
"The variations for $A_4(M_Z)$ are presented in Figure REF : as a function of $s^2_W$ on the left-hand side plot and as a function of $sin^2\\theta _{eff}^l$ on the right-hand side plot.",
"It is very reassuring, that all presented optME methods lead to the same slope of the $A_4(M_Z)$ as a function of $sin^2\\theta _{eff}$ .",
"Very similar curve could be presented for $A_{FB}$ , which would be scaled by $\\frac{3}{8}$ with respect to $A_4$ only.",
"Figure: The A 4 A_4 variation due to shifts induced with the presentedin Appendix options;as a functionof s W 2 s^2_W (left-hand side) and as a function of sin 2 θ eff l sin^2\\theta _{eff}^l (right-hand side).The “FF G μ G_\\mu varied”, FF m t m_t varied” correspond to the case whenform-factors were recalculated.",
"Otherwise they were kept at nominal values."
],
[
"Initialization of the ",
"There is a wealth of initialization constants and options available for Dizet library.",
"The documentation of that program and of its interface for KKMC, explains options available for the TauSpinner users as well.",
"Tables REF and REF recall available Dizet initialization, Table REF lists calculated by Dizet quantities for the use in TauSpinner library.",
"Table: Dizet initialization parameters: masses and couplings.In the present work, we have relied on the Dizet library version as installed in the KKMC Monte Carlo [8] and used at a time of LEP 1 in detector simulations.",
"Already for the data analysis and in particular for final fits [1], further effects of minor, but non-negligible numerical impact were taken into account.",
"Gradually, effects such as improved top contributions [39] or better photonic vacuum polarization [23], were taken into account.",
"This has to be updated for Dizet library too.",
"Such update is of importance also for the KKMC project itself because of forthcoming applications for the Future Circular Collider or for LHC [40].",
"Table: Dizet recalculated quantities available for the TauSpinner use.",
"For details of the ZPAR table see Refs.",
","
]
] | 1808.08616 |
[
[
"Doubly Robust Sure Screening for Elliptical Copula Regression Model"
],
[
"Abstract Regression analysis has always been a hot research topic in statistics.",
"We propose a very flexible semi-parametric regression model called Elliptical Copula Regression (ECR) model, which covers a large class of linear and nonlinear regression models such as additive regression model,single index model.",
"Besides, ECR model can capture the heavy-tail characteristic and tail dependence between variables, thus it could be widely applied in many areas such as econometrics and finance.",
"In this paper we mainly focus on the feature screening problem for ECR model in ultra-high dimensional setting.",
"We propose a doubly robust sure screening procedure for ECR model, in which two types of correlation coefficient are involved: Kendall tau correlation and Canonical correlation.",
"Theoretical analysis shows that the procedure enjoys sure screening property, i.e., with probability tending to 1, the screening procedure selects out all important variables and substantially reduces the dimensionality to a moderate size against the sample size.",
"Thorough numerical studies are conducted to illustrate its advantage over existing sure independence screening methods and thus it can be used as a safe replacement of the existing procedures in practice.",
"At last, the proposed procedure is applied on a gene-expression real data set to show its empirical usefulness."
],
[
"Introduction",
"In the last decades, data sets with large dimensionality have arisen in various areas such as finance, chemistry and so on due to the great development of the computer storage capacity and processing power and feature selection with these big data is of fundamental importance to many contemporary applications.",
"The sparsity assumption is common in high dimensional feature selection literatures, i.e., only a few variables are critical in for in-sample fitting and out-sample forecasting of certain response of interest.",
"In specific, for the linear regression setting, statisticians care about how to select out the important variables from thousands or even millions of variables.",
"In fact, a huge amount of literature springs up since the appearance of the Lasso estimator [21].",
"To name a few, there exist SCAD by [9], Adaptive Lasso by [25], MCP by [23], the Dantzig selector by [2], group Lasso by [22].",
"This research area is very active, and as a result, this list of references here is illustrative rather than comprehensive.",
"The aforementioned feature selection methods perform well when the dimensionality is not “too” large, theoretically in the sense that it is of polynomial order of the sample size.",
"However, in the ultrahigh dimensional setting where the dimensionality is of exponential order of the sample size, the aforementioned methods may encounter both theoretical and computational issue.",
"Take the Dantzig selector for example, the Uniform Uncertainty Principle (UUP) condition to guarantee the oracle property may be difficult to satisfy, and the computational cost would increase dramatically by implementing linear programs in ultra-high dimension.",
"[10] first proposed Sure Independence Screening (SIS) and its further improvement, Iterative Sure Independence Screening (ISIS), to alleviate the computational burden in ultra-high dimensional setting.",
"The basic idea goes as follows.",
"In the first step, reduce the dimensionality to a moderate size against the sample size by sorting the marginal pearson correlation between covariates and the response and removing those covariates whose marginal correlation with response are lower than a certain threshold.",
"In the second stage perform Lasso, SCAD etc.",
"to the variables survived in the first step.",
"The SIS (ISIS) turns out to enjoy sure screening property under certain conditions, that is, with probability tending to 1, the screening procedure selects out all important variables.",
"The last decade has witnessed plenty of variants of SIS to handle the ultra-high dimensionality for more general regression models.",
"[7] proposed a sure screening procedure for ultra-high dimensional additive models.",
"[6] proposed a sure screening procedure for ultra-high dimensional varying coefficient models.",
"[19] proposed censored rank independence screening of high-dimensional survival data which is robust to predictors that contain outliers and works well for a general class of survival models.",
"[24] and [4] proposed model-free feature screening.",
"[14] proposed to screen Kendall's tau correlation while [15] proposed to screen distance correlation which both show robustness to heavy tailed data.",
"[13] proposed to screen the canonical correlation between the response and all possible sets of $k$ variables, which performs well particularly for selecting out variables that are pairwise jointly important with other variables but marginally insignificant.",
"This list of references for screening methods is also illustrative rather than comprehensive.",
"For the development of the screening methods in the last decade, we refer to the review paper of [16] and [5].",
"The main contribution of the paper is two-fold.",
"On the one hand, we innovatively propose a very flexible semi-parametric regression model called Elliptical Copula Regression (ECR) model, which can capture the thick-tail property of variables and the tail dependence between variables.",
"In specific, the ECR model has the following representation: $f_{0}(Y)=\\beta _0+\\sum _{j=1}^p\\beta _jf_{j}(X_j)+\\epsilon ,$ where $Y$ is response variable, $X_1\\ldots ,X_p$ are predictors, $f_{j}(\\cdot )$ are univariate monotonic functions.",
"We say $(Y,^\\top )^\\top =(Y,X_1,\\ldots ,X_p)^\\top $ satisfies a Elliptical copula regression model if the marginally transformed random vectors $\\tilde{}=(\\tilde{Y},\\tilde{}^\\top )^\\top \\overset{\\bigtriangleup }{=}(f_0(Y),f_1(X_1),\\ldots ,f_p(X_p))^\\top $ follows elliptical distribution.",
"From the representation of ECR model in (REF ), it can be seen that the ECR model covers a large class of linear and nonlinear regression models such as additive regression model, single index model which makes it more applicable in many areas such as econometrics, finance and bioinformatics.",
"On the other hand, we propose a doubly robust dimension reduction procedure for the ECR model in the ultrahigh dimensional setting.",
"The doubly robustness is achieved by combining two types of correlation, which are Kendall' tau correlation and canonical correlation.",
"The canonical correlation is employed to capture the joint information of a set of covariates and the joint relationship between the response and this set of covariates.",
"Note that for ECR model in (REF ), only $(Y,^\\top )^\\top $ is observable rather than the transformed variables.",
"Thus the Kendall's tau correlation is exploited to estimate the canonical correlations due to its invariance under strictly monotone marginal transformations.",
"The dimension reduction procedure for ECR model is achieved by sorting the estimated canonical correlations and leaving the variable that attributes a relatively high canonical correlation at least once into the active set.",
"The proposed screening procedure enjoys the sure screening property and reduces the dimensionality substantially to a moderate size under mild conditions.",
"Numerical results shows that the proposed approach enjoys great advantage over state-of-the-art procedures and thus it can be used as a safe replacement.",
"We introduce some notations adopted in the paper.",
"For any vector $=(\\mu _1,\\ldots ,\\mu _d) \\in ^d$ , let $_{-i}$ denote the $(d-1)\\times 1$ vector by removing the $i$ -th entry from $$ .",
"$||_0=\\sum _{i=1}^d I\\lbrace \\mu _i\\ne 0\\rbrace $ , $||_1=\\sum _{i=1}^d |\\mu _i|$ , $||_2=\\sqrt{\\sum _{i=1}^d\\mu _i^2}$ and $||_\\infty =\\max _i|\\mu _i|$ .",
"Let $=[a_{ij}]\\in \\mathbb {R}^{d\\times d}$ .",
"$\\Vert \\Vert _{L_1}=\\mathrm {max}_{1\\le j\\le d}\\sum _{i=1}^d|a_{ij}|$ , $\\Vert \\Vert _\\infty =\\mathrm {max}_{i,j}|a_{ij}|$ and $\\Vert \\Vert _1=\\sum _{i=1}^d\\sum _{j=1}^d|a_{ij}|$ .",
"We use $\\lambda _{\\mathrm {min}}()$ and $\\lambda _{\\mathrm {max}}()$ to denote the smallest and largest eigenvalues of $$ respectively.",
"For a set $\\mathcal {H}$ , denote by $|\\mathcal {H}|$ the cardinality of $\\mathcal {H}$ .",
"For a real number $x$ , denote by $\\lfloor x \\rfloor $ the largest integer smaller than or equal to $x$ .",
"For two sequences of real numbers $\\lbrace a_n\\rbrace $ and $\\lbrace b_n\\rbrace $ , we write $a_n=O(b_n)$ if there exists a constant $C$ such that $|a_n|\\le C|b_n|$ holds for all $n$ , write $a_n=o(b_n)$ if $\\lim _{n\\rightarrow \\infty } a_n/b_n=0$ , and write $a_n \\asymp b_n$ if there exist constants $c$ and $C$ such that $c\\le a_n/b_n \\le C$ for all $n$ .",
"The rest of the paper is organized as follows: in Section 2, we introduce the Elliptical copula regression model and present the proposed dimension reduction procedure by ranking the estimated canonical correlations.",
"In Section 3, we present the theoretical properties of the proposed procedure,with more detailed proofs collected in the Appendix.",
"In Section 4, we conduct thorough numerical simulations to investigate the empirical performance of the procedure.",
"In section 5, a real gene-expression data example is given to illustrate its empirical usefulness.",
"At last, we give a brief discussion on possible future directions in the last section."
],
[
"Elliptical Copula Regression Model",
"To present the Elliptical Copula Regression Model, we first need to introduce the elliptical distribution.",
"The elliptical distribution generalizes the multivariate normal distribution, which includes symmetric distributions with heavy tails, like the multivariate $t$ -distribution.",
"Elliptical distributions are commonly used in robust statistics to evaluate proposed multivariate-statistical procedures.",
"In specific, the definition of elliptical distribution is given as follows: (Elliptical distribution) Let $\\in ^p$ and $\\in ^{p\\times p}$ with $\\text{rank}()=q\\le p$ .",
"A $p$ -dimensional random vector $$ is elliptically distributed, denoted by $\\sim ED_p(,,\\zeta )$ , if it has a stochastic representation $\\overset{d}{=}+\\zeta .$ where $$ is a random vector uniformly distributed on the unit sphere $S^{q-1}$ in $^q$ , $\\zeta \\ge 0$ is a scalar random variable independent of $$ , $\\in ^{p\\times q}$ is a deterministic matrix satisfying $^\\top =$ with $$ called scatter matrix.",
"The representation $\\overset{d}{=}+\\zeta .$ is not identifiable since we can rescale $\\zeta $ and $$ .",
"We require $\\zeta ^2=q$ to make the model identifiable, which makes the covariance matrix of $$ to be $$ .",
"In addition, we assume $$ is non-singular, i.e., $q=p$ .",
"In this paper, we only consider continuous elliptical distributions with $\\text{Pr}(\\zeta =0)=0$ .",
"Another equivalent definition of the elliptical distribution is by its characteristic function, which has the form $\\exp (i^\\top ) \\psi (^\\top )$ , where $\\psi (\\cdot )$ is a properly defined characteristic function and $i=\\sqrt{-1}$ .",
"$\\zeta $ and $\\psi $ are mutually determined by each other.",
"Given the definition of Elliptical distribution, we are ready for introducing the Elliptical Copula Regression (ECR) model.",
"(Elliptical copula regression model) Let $f=\\lbrace f_0,f_1,\\ldots ,f_p\\rbrace $ be a set of monotone univariate functions and $$ be a positive-definite correlation matrix with diag($$ )=$$ .",
"We say a $d$ -dimensional random variable $=(Y,X_1,\\ldots ,X_p)^\\top $ satisfies the Elliptical Copula Regression model if and only if $\\tilde{}=(\\tilde{Y},\\tilde{}^\\top )^\\top =(f_0(Y),f_1(X_1),\\ldots ,f_p(X_p))^\\top \\sim ED_d(,,\\zeta )$ with $\\zeta ^2=d$ and $\\tilde{Y}=\\tilde{}^\\top +\\epsilon , \\ \\ \\text{or} \\ \\ \\text{equivalently}, \\ \\ f_0(Y)=\\sum _{j=1}^p\\beta _jf_{j}(X_j)+\\epsilon ,$ where $Y$ is the response and $=(X_1,\\ldots ,X_p)^\\top $ are covariates, $d=p+1$ .",
"We require diag($$ )=$$ in Definition REF for identifiability because the shifting and scaling are absorbed into the marginal functions $f$ .",
"For ease of presentation, we denote $Z=(Y,X_1,\\ldots ,X_p)^\\top \\sim \\text{ECR}(,\\zeta ,f)$ in the following sections.",
"The ECR model allows the data to come from heavy-tailed distribution and is thus more flexible and more useful in modelling many modern data sets, including financial data, genomics data and fMRI data.",
"Notice that the transformed variable $\\tilde{Y}$ and the transformed covariates $\\tilde{}$ obeys the linear regression model, however, the transformed variables are unobservable, only $Y,$ are observable.",
"In the following, by virtue of canonical correlation and Kendall's tau correlation, we will present an adaptive screening procedure without estimating the marginal transformation functions $f$ while capturing the joint information of a set of covariates and the joint relationship between the response and this set of covariates."
],
[
"Adaptive Doubly Robust Screening for ECR Model",
"In this section we will present the adaptive doubly robust screening for ECR model.",
"We first introduce the Kendall's tau-based estimator of correlation matrix in subsection REF , then we introduce the Kendall's tau-based estimator of canonical correlation in subsection REF , which are both of fundamental importance for the detailed doubly robust screening procedure introduced in subsection REF ."
],
[
"Kendall's tau Based Estimator of Correlation Matrix",
"In this section we present the estimator of the correlation matrix based on Kendall's tau.",
"Let $_{1},\\ldots ,_n$ be $n$ independent observations where $_i=(Y_i,X_{i1},\\ldots ,X_{ip})^\\top $ .",
"The sample Kendall's tau correlation of $Z_j$ and $Z_k$ is defined by $\\hat{\\tau }_{j,k}=\\frac{2}{n(n-1)}\\sum _{1\\le i< i^{\\prime }\\le n}\\text{sign}\\lbrace (Z_{ij}-Z_{i^\\prime j})(Z_{ik}-Z_{i^\\prime k})\\rbrace .$ Let $\\tilde{}_i=(\\tilde{Y_i},\\tilde{}_i)^\\top =(f_0(Y_i),f_1(X_{i1}),\\ldots ,f_p(X_{ip}))^\\top $ for $i=1,\\ldots ,n$ , then $\\tilde{}_i$ can be viewed as the latent observations from Elliptical distribution $ED(,,\\zeta )$ .",
"We can estimate $\\Sigma _{j,k}$ (the $(j,k)$ -th element of $$ ) by $\\hat{S}_{j,k}$ where $\\hat{S}_{j,k}=\\sin (\\frac{\\pi }{2}\\hat{\\tau }_{j,k}).$ This is because the Kendall's tau correlation is invariant under strictly monotone marginal transformations and the fact that $\\Sigma _{j,k}=\\sin (\\frac{\\pi }{2}\\tau _{j,k})$ holds for Elliptical distribution.",
"Define by $\\hat{}=[\\hat{S}_{j,k}]_{d\\times d}$ with $\\hat{S}_{j,k}$ defined in Equation (REF ).",
"We call $\\hat{}$ the rank-based estimator of correlation matrix."
],
[
"Kendall's tau Based Estimator of Canonical Correlation",
"Canonical Correlation (CC) could capture the pairwise correlations within a subset of covariates and the joint regression relationship between the response and the subset of covariates.",
"In this section, we present the Kendall's tau based estimator of CC between the transformed response $f_0(Y)$ and $k$ (a fixed number) transformed covariates $\\lbrace f_{m_1}({X}_{m_1}),\\ldots ,f_{m_k}({X}_{m_k})\\rbrace $ , which bypasses estimating the marginal transformation functions.",
"Recall that for ECR model, $(f_0(Y),f_1(X_{1}),\\ldots ,f_p(X_{p}))^\\top \\sim ED(,,\\zeta )$ and its corresponding correlation matrix is exactly $=(\\Sigma _{s,t})$ .",
"Denote $=\\lbrace 1\\rbrace $ and $=\\lbrace {m_1,\\ldots , m_k}\\rbrace $ , the CC between $\\tilde{Y}$ and $\\lbrace \\tilde{X}_{m_1},\\ldots ,\\tilde{X}_{m_k}\\rbrace $ is defined as $\\rho ^c=\\mathop {\\mathrm {sup}}_{,}\\frac{^\\top _{\\times }}{\\sqrt{^\\top _{\\times }}\\sqrt{^\\top _{\\times }}},$ where we define $_{\\times }=(\\Sigma _{s,t})_{s\\in {},t\\in {}}$ and suppress its dependence on parameter $k$ .",
"It can be shown that $(\\rho ^c)^2=_{\\times }_{\\times }^{-1}_{\\times }^\\top .$ In Section REF we present the Kendall's tau based estimator of $$ and denote it by $\\hat{}$ .",
"Thus the Canonical Correlation $\\rho ^c$ can be naturally estimated by: $\\hat{\\rho }^c=\\sqrt{\\hat{}_{\\times }\\hat{}_{\\times }^{-1}\\hat{}_{\\times }^\\top }.$ If $\\hat{}_{\\times }$ is not positive definite (not invserible), we first project $\\hat{}_{\\times }$ into the cone of positive semidefinite matrices.",
"In particular, we propose to solve the following convex optimization problem: $\\tilde{}_{\\times }=\\mathop {\\mathrm {arg\\ min}}_{}\\Vert \\hat{}_{\\times }-\\Vert _{\\infty }.$ The matrix element-wise infinity norm $\\Vert \\cdot \\Vert _\\infty $ is adopted for the sake of further technical developments.",
"Empirically, we can use a surrogate projection procedure that computes a singular value decomposition of $\\hat{}_{\\times }$ and truncates all of the negative singular values to be zero.",
"Numerical study shows that this procedure works well."
],
[
"Screening procedure",
"In this section, we present the screening procedure by sorting canonical correlation estimated by Kendall' tau.",
"The Screening procedure goes as follows: first collect all sets of $k$ transformed variables and total adds up to $_p^k$ , the combinatorial number, i.e., $\\lbrace \\tilde{X}_{l,m_1},\\ldots ,\\tilde{X}_{l,m_k}\\rbrace $ with $l=1,\\ldots ,_p^k$ .",
"For each $k$ -variable set $\\lbrace \\tilde{X}_{l,m_1},\\ldots ,\\tilde{X}_{l,m_k}\\rbrace $ , we denote its canonical correlation with $f_0(Y)$ by ${\\rho }_l^{c}$ and estimate it by $\\hat{\\rho }_l^{c}=\\sqrt{\\hat{}_{\\times }\\hat{}_{\\times }^{-1}\\hat{}_{\\times }^\\top }.$ where $\\hat{}$ is the rank-based estimator of correlation matrix introduced in Section REF .",
"Then we sort these canonical correlations $\\lbrace \\hat{\\rho }_l^{c},l=1,\\ldots ,_p^k\\rbrace $ and select the variables that attributes a relatively large canonical correlation at least once into the active set.",
"Specifically, let $_*=\\lbrace 1\\le i\\le p, \\beta _i\\ne 0\\rbrace $ be the true model with size $s=|_*|$ and define sets $_i^n=\\Big \\lbrace l; (X_i,X_{i_1},\\ldots ,X_{i_{k-1}}) \\ \\text{with}\\ \\mathop {\\mathrm {max}}_{1\\le m\\le k-1}|i_m-i|\\le k_n \\ \\text{is used in calculating} \\ \\hat{\\rho }_l^{c} \\Big \\rbrace , i=1,\\ldots ,p,$ where $k_n$ is a parameter determining a neighborhood set in which variables jointly with $X_i$ are included to calculate the canonical correlation with the response.",
"Finally we estimate the active set as follow: $\\hat{}_{t_n}=\\Big \\lbrace 1\\le i\\le p:\\mathop {\\mathrm {max}}_{l\\in _i^n}\\hat{\\rho }_l^c>t_n\\Big \\rbrace $ where $t_n$ is a threshold parameter which controls the the size of the estimated active set.",
"If we set $k_n=p$ , then all $k$ -variable sets including $X_i$ are considered in $_i^n$ .",
"However, if there is a natural index for all the covariates such that only the neighboring covariates are related, which is often the case in portfolio tracking in finance, it is more appropriate to consider a $k_n$ much smaller than $p$ .",
"As for the parameter $k$ , a relatively large $k$ may bring more accurate results, but will increase the computational burden.",
"Empirical simulation results show that the performance by by taking $k=2$ is already good enough and substantially better than taking $k=1$ which is equivalent to sorting marginal correlation."
],
[
"Theoretical properties",
"In this section, we present the theoretical properties of the proposed approach.",
"In the screening problem, what we care about most is whether the true non-zero index set $_*$ is contained in the estimated active set $\\hat{}_{t_n}$ with high probability for properly chosen threshold $t_n$ , i.e., whether the procedure has sure screening property.",
"To this end, we assume the following three assumptions hold.",
"Assumption 1 Assume $p>n$ and $\\log p=O(n^\\xi )$ for some $\\xi \\in (0,1-2\\kappa )$ .",
"Assumption 2 For all $l=1,\\ldots ,_p^k$ , $\\lambda _{\\max }((_{^l\\times ^l})^{-1})\\le c_0$ for some constant $c_0$ , where ${}^l=\\lbrace m_1^l,\\ldots ,m_k^l\\rbrace $ is the index set of variables in the $l$ -th $k$ -variable sets.",
"Assumption 3 For some $0\\le \\kappa \\le 1/2$ , $\\mathop {\\mathrm {min}}_{i\\in _*}\\mathop {\\mathrm {max}}_{l\\in _i^n}\\rho _l^c\\ge c_1 n^{-\\kappa }$ .",
"Assumption 1 specifies the scaling between the dimensionality $p$ and the sample size $n$ .",
"Assumption 2 requires that the minimum eigenvalue of the covariance matrix of any $k$ covariates is lower bounded.",
"Assumption 3 is the fundamental basis for guaranteeing the sure screening property, which means that any important variable is correlated to the response jointly with some other variables.",
"Technically, the Assumption 3 entails that an important variable would not be veiled by the statistical approximation error resulting from the estimated canonical correlation.",
"Assume that Assumptions 1-3 hold, then for some positive constants $c_1^*$ and $C$ , as $n$ goes to infinity, we have $\\left(\\mathop {\\mathrm {min}}_{i\\in _*}\\mathop {\\mathrm {max}}_{l\\in _i^n}\\hat{\\rho }_l^c\\ge c_1^* n^{-\\kappa }\\right)\\ge 1-O\\left(\\exp \\left(-{Cn^{1-2\\kappa }}\\right)\\right),$ and $\\left(_*\\subset \\hat{}_{c_1^* n^{-\\kappa }}\\right)\\ge 1-O\\left(\\exp \\left(-{Cn^{1-2\\kappa }}\\right)\\right).$ Theorem shows that, by setting the threshold of order $c_1^*n^{-\\kappa }$ , all important variables can be selected out with probability tending to 1.",
"However, the constant $c_1^*$ remains unknown.",
"To refine the theoretical result, we assume the following assumption holds.",
"Assumption 4 For some $0\\le \\kappa \\le 1/2$ , $\\mathop {\\mathrm {max}}_{i\\notin _*}\\mathop {\\mathrm {max}}_{l\\in _i^n}\\rho _l^c< c_1^* n^{-\\kappa }$ .",
"The Assumption 4 requires that if a variable $X_i$ is not important, then the canonical correlations between the response and all $k$ variables sets containing $X_i$ are all upper bounded by $c_1^* n^{-\\kappa }$ , and it uniformly holds for all unimportant variables.",
"Assume that Assumptions 1-4 hold, then for some constants $c_1^*$ and $C$ , we have $\\left(_*=\\hat{}_{c_1^* n^{-\\kappa }}\\right)\\ge 1-O\\left(\\exp \\left(-{Cn^{1-2\\kappa }}\\right)\\right)$ and in particular $\\left(|\\hat{}_{c_1^* n^{-\\kappa }}|=s\\right)\\ge 1-O\\left(\\exp \\left(-{Cn^{1-2\\kappa }}\\right)\\right)$ where $s$ is the size of $_*$ .",
"Theorem guarantees the exact sure screening property without any condition on $k_n$ .",
"Besides, the theorem guarantees the existence of $c_1^*$ and $C$ , however, it still remains unknown how to select the constant $c_1^*$ .",
"If we know that $s<n\\log n$ in advance, one can select a constant $c^*$ such that the size of $\\hat{}_{c^* n^{-\\kappa }}$ is approximately $n$ .",
"Obviously, we have $\\hat{}_{c_1^* n^{-\\kappa }}\\subset \\hat{}_{c^* n^{-\\kappa }}$ with probability tending to 1.",
"The following theorem is particularly useful in practice summarizing the above discussion.",
"Assume that Assumptions 1-4 hold, if $s=|_*|\\le n/\\log n$ , we have for any constant $\\gamma >0$ , $\\left(_*\\subset ^\\gamma \\right)1-O\\left(\\exp \\left(-{Cn^{1-2\\kappa }}\\right)\\right),$ where $^\\gamma =\\lbrace 1\\le i\\le p; \\mathop {\\mathrm {max}}_{l\\in _i^n} \\hat{\\rho }_l^c \\ \\text{is among the largest} \\ \\lfloor \\gamma n\\rfloor \\ \\text{of} \\ \\mathop {\\mathrm {max}}_{l\\in _1^n} \\hat{\\rho }_l^c,\\cdots , \\mathop {\\mathrm {max}}_{l\\in _p^n} \\hat{\\rho }_l^c\\rbrace $ The above theorem guarantees that one can reduce dimensionality to a moderate size against $n$ while ensuring the sure screening property, which further guarantees the validity of a more sophisticated and computationally efficient variable selection methods.",
"Theorem heavily relies on the Assumption 4.",
"If there is natural order of the variables, and any important variable together with only the adjacent variables contributes to the response variable, then Assumption 4 can be totally removed while exserting an constraint on the parameter $k_n$ .",
"The following theorem summarizes the above discussion.",
"Assume Assumptions 1-3 hold, $\\lambda _{\\max }()\\le c_2n^{\\tau }$ for some $\\tau \\ge 0$ and $c_2>0$ , and further assume $k_n=c_3n^{\\tau ^*}$ for some constants $c_3>0$ and $\\tau ^*\\ge 0$ .",
"If $2\\kappa +\\tau +\\tau ^*<1$ , then there exists some $\\theta \\in [0,1-2\\kappa -\\tau -\\tau ^*)$ such that for $\\gamma =c_4n^{-\\theta }$ with $c_4>0$ , we have for some constant $C>0$ , $\\left(_*\\subset ^\\gamma \\right)\\ge 1-O\\left(\\exp \\left(-{Cn^{1-2\\kappa }}\\right)\\right).$ The assumption $k_n=c_3n^{\\tau ^*}$ is reasonable in many fields such as Biology and Finance.",
"An intuitive example is in genomic association study, millions of genes tend to cluster together and functions together with adjacent genes.",
"The procedure by ranking the estimated canonical correlation and reducing the dimension in one step from a large $p$ to $\\lfloor n/\\log n\\rfloor $ is a crude and greedy algorithm and may result in many fake covariates due to the strong correlations among them.",
"Motivated by the ISIS method in [10], we propose a similar iterative procedure which achieve sure screening in multiple steps.",
"The iterative procedure works as follows.",
"Let the shrinking factor $\\delta \\rightarrow 0$ be properly chosen such that $\\delta n^{1-2\\kappa -\\tau -\\tau ^*}\\rightarrow \\infty $ as $n\\rightarrow \\infty $ and we successively perform dimensionality reduction until the number of remaining variables drops to below sample size $n$ .",
"In specific, define a subset $^1({\\delta })=\\left\\lbrace 1\\le i\\le p:\\mathop {\\mathrm {max}}_{l\\in _i^n}\\hat{\\rho }_l^c\\ \\text{is among the largest} [\\delta p] \\ \\text{of all}\\right\\rbrace .$ In the first step we select a subset of $\\lfloor \\delta p\\rfloor $ variables, $^1(\\delta )$ by Equation (REF ).",
"In the next step, we start from the variables indexed in $^1(\\delta )$ , and apply a similar procedure as (REF ), and again obtain a sub-model $^2(\\delta )\\subset ^1(\\delta )$ with size $\\lfloor \\delta ^2p\\rfloor $ .",
"Iterate the steps above and finally obtain a sub-model $^k(\\delta )$ , with size $[\\delta ^k p]<n$ .",
"Assume that the conditions in Theorem hold, let $\\delta \\rightarrow 0$ satisfying $\\delta n^{1-2\\kappa -\\tau -\\tau ^*}\\rightarrow \\infty $ as $n\\rightarrow \\infty $ , then we have $\\left(_*\\subset ^k(\\delta )\\right)\\ge 1-O\\left(\\exp \\left(-{Cn^{1-2\\kappa }}\\right)\\right).$ The above theorem guarantees the sure screening property of the iterative procedure and the step size $\\delta $ can be chosen in the same way as ISIS in [10]."
],
[
"Simulation Study",
"In this section we conduct thorough numerical simulation to illustrate the empirical performance of the proposed doubly robust screening procedure (denoted as CCH).",
"Besides, we compare the proposed procedure with three methods, the method proposed by [13] (denoted as CCK), the rank correlation screening approach proposed by [14] (denoted as RRCS) and the initially proposed SIS procedure by [10].",
"To illustrate the doubly robustness of the proposed procedure, we consider the following five models which includes linear regression with thick-tail covariates and error term, single-index model with thick-tail error term, additive model and more general regression model.",
"Model 1 Linear model setting adapted from [13]: $Y_i=0.9+\\beta _1X_{i1}+\\cdots +\\beta _p X_{ip}+\\epsilon _i$ , where $=(1,-0.5,0,0,\\ldots ,0)^\\top $ with the last $p-2$ components being 0.",
"The covariates $$ is sampled from multivariate normal $N(, )$ or multivariate $t$ with degree 1, noncentrality parameter $$ and scale matrix $$ .",
"The diagonal entries of $$ are 1 and the off-diagonal entries are $\\rho $ , the error term $\\epsilon $ is independent of $$ and generated from the standard normal distribution or the standard $t$ -distribution with degree 1.",
"Model 2 Linear model setting adapted from [14]: $Y_i=\\beta _1X_{i1}+\\cdots +\\beta _p X_{ip}+\\epsilon _i$ , where $=(5,5,5,0,\\ldots ,0)^\\top $ with the last $p-3$ components being 0.",
"The covariates $$ is sampled from multivariate normal $N(, )$ or multivariate $t$ with degree 1, noncentrality parameter $$ and scale matrix $$ .",
"The diagonal entries of $$ are 1 and the off-diagonal entries are $\\rho $ , the error term $\\epsilon $ is independent of $$ and generated from the standard normal distribution or the standard $t$ -distribution with degree 1.",
"Model 3 Single-index model setting: $H(Y)=^\\top +\\epsilon .$ We set $H(Y)=\\log (Y)$ which corresponds to a special case of BOX-COX transformation $({|Y|^\\lambda \\text{sgn}(Y)-1})/{\\lambda }$ with $\\lambda =1$ .",
"The error term $\\epsilon $ is independent of $$ and generated from the standard normal distribution or the standard $t$ -distribution with degree 3.",
"The regression coefficients $=(3,1.5,2,0,\\ldots ,0)^\\top $ with the last $p-3$ components being 0.",
"The covariates $$ is sampled from multivariate normal $N(, )$ or multivariate $t$ with degree 3, where the diagonal entries of $$ are 1 and the off-diagonal entries are $\\rho $ .",
"Model 4 Additive model from [17]: $Y_i=5f_1(X_{1i})+3f_2(X_{2i})+4f_3(X_{3i})+6f_4(X_{4i})+\\epsilon _i,$ with $f_1(x)=x, \\ \\ f_2(x)=(2x-1)^2, \\ \\ f_3(x)=\\frac{\\sin (2\\pi x)}{2-\\sin (2\\pi x)}$ and $\\begin{split}f_4(x)=&0.1\\sin (2\\pi x)+0.2\\cos (2\\pi x)+0.3\\sin ^2(2\\pi x)\\\\&+0.4\\cos ^3(2\\pi x)+0.5\\sin ^3(2\\pi x)\\end{split}$ The covariates $=(X_1,\\ldots ,X_p)^\\top $ are generated by $X_j=\\frac{W_j+tU}{1+t}, \\ \\ j=1,\\ldots ,p,$ where $W_1,\\ldots ,W_p$ and $U$ are i.i.d.",
"Uniform[0,1].",
"For $t=0$ , this is the independent uniform case while $t=1$ corresponds to a design with correlation 0.5 between all covariates.",
"The error term $\\epsilon $ are sampled from $N(0,1.74)$ .",
"The regression coefficients $$ in the model setting is obviously $(3,1.5,2,0,\\ldots ,0)^\\top $ with the last $p-4$ components being 0.",
"Model 5 A model generated by combining Model 3 and Model 4: $H(Y_i)=5f_1(X_{1i})+3f_2(X_{2i})+4f_3(X_{3i})+6f_4(X_{4i})+\\epsilon _i,$ where $H(Y)$ is the same with Model 3 and the functions $\\lbrace f_1,f_2,f_3,f_4\\rbrace $ are the same with Model 4.",
"The covariates $=(X_1,\\ldots ,X_p)^\\top $ are generated in the same way as in Model 4.",
"The error term $\\epsilon $ are sampled from $N(0,1.74)$ .",
"Table: The proportions of containing Model 4 and Model 5 in the active setFor models in which $\\rho $ involved, we take $\\rho =0,0.1,0.5,0.9$ .",
"For all the models, we consider four combinations of $(n,p)$ : (20,100), (50,100), (20,500), (50, 500).",
"All simulations results are based on the 500 replications.",
"We evaluate the performance of different screening procedures by the proportion that the true model is included in the selected active set in 500 replications.",
"To guarantee a fair comparison, for all the screening procedures, we choose the variables whose coefficients rank in the first largest $\\lfloor n/\\log n\\rfloor $ values.",
"For our method CCH and the method CCK in [13], two parameters $k_n$ and $k$ are involved.",
"The simulation study shows that when $k_n$ is small, the performance for different combination of $(k_n,k)$ are quite similar.",
"Thus we only presents the results of $(k_n,k)=(2,2), (2,3)$ for illustration, which are denoted as CCH1, CCH2 for our method and CCK1 and CCK2 for the method by [13].",
"From the simulation results, we can see that the proposed CCH methods detects the true model much more accurately than SIS, RRCS and CCK meothods in almost all cases.",
"In specific, for the motivating Model 1 in [13], from Table , we can see that when the correlations among covariates become large, all the SIS, RRCS and CCK meothods perform worse (the proportion of containing the true model drops sharply), but the proposed CCH procedure shows robustness against the correlations among covariates and detects the true model for each replication.",
"Besides, for the heavy tailed error term following $t(1)$ , we can see that all the SIS, RRCS and CCK meothods perform very bad while the CCH method still works very well.",
"For Model 2, from Table , we can see that when the covariates are multivariate normal and the error term is normal, then all the methods works well when the sample size is relatively large while CCK and CCH requires less sample size compared with RRCS and SIS.",
"If the error term is from $t(1)$ , then SIS, RRCS and CCK meothods perform bad especially when the ratio $p/n$ is large.",
"In contrast, the CCH approach still performs very well.",
"We should notice that the RRCS also shows certain robustness and CCK2 is slightly better than CCK1 because the important covariates are indexed by three consecutive integers.",
"The CCH's advantage over the CCK is mainly illustrated by the results of Model 3 to Model 5.",
"In fact, Model 3 is an example of single index model, Model 4 is an example of additive model and Model 5 is an example of more complex nonlinear regression model.",
"CCK approach relies heavily on the linear regression assumption while CCH is more applicable.",
"For the single index regression model, from Table , we can see that CCK performs badly especially when the ratio $p/n$ is large.",
"The approach RRCS ranks the Kenall' tau correlation which is invariant to monotone transformations, thus it exhibits robustness for Model 3, but it still performs much worse than CCH.",
"For the additive regression model and Model 5, by Table REF , similar conclusions can be drawn as discussed for Model 3.",
"It is worth mentioning that although we require the marginal transformation functions are monotone in theory, but simulation study shows that the proposed screening procedure is not sensitive to the requirement, and performs pretty well even the transformation functions are not monotone.",
"In fact, the marginal transformation functions $f_2,f_3,f_4$ in Model 4 and Model 5 are all not monotone.",
"In one word, the proposed CCH procedure performs very well not only for heavy tailed error terms, but also for various unknown transformation functions, which shows doubly robustness.",
"Thus in practice, CCH can be used as a safe replacement of the CCK, RRCS or SIS.",
"Figure: Boxplot of the ranks of the first 20 genes ordered by r ^ j U \\hat{r}_j^U.Figure: 3-dimensional plots of variables with Genralized Additive Model (GLM) fits."
],
[
"Real Example",
"In this section we apply the variable selection method to a gene expression data set for an eQTL experiment in rat eye reported in [18].",
"The data set has ever been analyzed by [11], [20] and [8] and can be downloaded from the Gene Expression Omnibus at accession number GSE5680.",
"For this data set, 120 12-week-old male rats were selected for harvesting of tissue from the eyes and subsequent microarray analysis.",
"The microarrays used to analyze the RNA from the eyes of the rats contain over 31,042 different probe sets (Affymetric GeneChip Rat Genome 230 2.0 Array).",
"The intensity values were normalized using the robust multi-chip averaging method [1], [12] to obtain summary expression values for each probe set.",
"Gene expression levels were analyzed on a logarithmic scale.",
"Similar to the work of [11] and [8], we are still interested in finding the genes correlated with gene TRIM32, which was found to cause Bardet¨CBiedl syndrome [3].",
"Bardet¨CBiedl syndrome is a genetically heterogeneous disease of multiple organ systems, including the retina.",
"Of more than 31,000 gene probes including $>$ 28,000 rat genes represented on the Affymetrix expression microarray, only 18,976 exhibited sufficient signal for reliable analysis and at least 2-fold variation in expression among 120 male rats generated from an SR/JrHsd $\\times $ SHRSP intercross.",
"The probe from TRIM32 is 1389163$\\_$ at, which is one of the 18, 976 probes that are sufficiently expressed and variable.",
"The sample size is $n=120$ and the number of probes is 18,975.",
"It's expected that only a few genes are related to TRIM32 such that this is a sparse high dimensional regression problem.",
"[h] The proportions of containing Model 1 in the active set Table: NO_CAPTION[!ht] The proportions of containing Model 2 in the active set Table: NO_CAPTION[!ht] The proportions of containing Model 3 in the active set Table: NO_CAPTIONDirect application of the proposed approach on the whole dataset is slow, thus we select 500 probes with the largest variances of the whole 18,975 probes.",
"[11] proposed nonparametric additive model to capture the relationship between expression of TRIM32 and candidates genes and find most of the plots of the estimated additive components are highly nonlinear, thus confirming the necessity of taking into account nonlinearity.",
"The Elliptical Copula Regression (ECR) model can also capture the nonlinear relationship and thus it is reasonable to apply the proposed doubly robust dimension reduction procedure on this data set.",
"For the real data example, we compare the selected genes by procedures introduced in the simulation study, which are the SIS ([10]), the RRCS procedure ([14]), CCK procedure ([13]) and the proposed CCH procedure.",
"To detect influential genes, we adopt the bootstrap procedure similar to [14], [13].",
"We denote the respective correlation coefficients calculated using the SIS, RRCS, CCK, CCH by $\\tilde{\\rho }_{sis},\\tilde{\\rho }_{rrcs},\\tilde{\\rho }_{cck}$ and $\\tilde{\\rho }_{cch}$ .",
"The detailed algorithm is presented in Algorithm .",
"[!ht] A bootstrap procedure to obtain influential genes Input: $=\\lbrace (_i,Y_i),i=1,\\ldots ,n\\rbrace $ Output: Index of influential genes [1] By the data set $\\lbrace (_i,Y_i),i=1,\\ldots ,n\\rbrace $ , calculate the correlations coefficients $\\tilde{\\rho }_{sis}^i,\\tilde{\\rho }_{rrcs}^i,\\tilde{\\rho }_{cck}^i$ and $\\tilde{\\rho }_{cch}^i$ and then order them as $\\tilde{\\rho }^{(\\hat{j}_1)}\\ge \\tilde{\\rho }^{(\\hat{j}_2)}\\ge \\cdots \\ge \\tilde{\\rho }^{(\\hat{j}_p)}$ , where $\\tilde{\\rho }$ can be $\\tilde{\\rho }_{sis},\\tilde{\\rho }_{rrcs},\\tilde{\\rho }_{cck}$ and $\\tilde{\\rho }_{cch}$ , thus the set $\\lbrace \\hat{j}_1,\\cdots ,\\hat{j}_p\\rbrace $ varies with different screening procedure.",
"We denote by $\\hat{j}_1\\succeq \\cdots \\succeq \\hat{j}_p$ to represent an empirical ranking of the component indices of $$ based on the contributions to the response, i.e., $s\\succeq t$ indicates $\\tilde{\\rho }^{(s)}\\ge \\tilde{\\rho }^{(t)}$ and we informally interpret as “ the $s$ th component of $$ has at least as much influence on the response as the $t$ th component.",
"The ranking $\\hat{r}_j$ of the $j$ th component is defined to be the value of $r$ such that $\\hat{j}_r=j$ .",
"For each $1\\le i\\le p$ , employ the SIS, RRCS, CCK and CCH procedures to calculate the $b$ th bootstrap version of $\\tilde{\\rho }^{i}$ , denotes as $\\tilde{\\rho }^{i}_b, b=1,\\ldots ,200$ .",
"Denote the ranks of $\\tilde{\\rho }_b^{1},\\ldots ,\\tilde{\\rho }_b^{p}$ by $\\hat{j}^1_b\\succeq \\cdots \\hat{j}^p_b$ and calculate the corresponding rank $\\hat{r}_j^b$ for the $j$ th component of $$ .",
"Given a value $\\alpha =0.05$ , compute the $(1-\\alpha )$ level, two-sides and equally tailed interval for the rank of the $j$ th component, i.e., an interval $[\\hat{r}_j^L,\\hat{r}_j^U]$ where $(\\hat{r}_j^b\\le \\hat{r}_j^L|)\\approx (\\hat{r}_j^b\\ge \\hat{r}_j^U|)\\approx \\frac{\\alpha }{2}.$ Treat a variable as influential if $\\hat{r}_j^U$ ranks in the top 20 positions.",
"The box-plot of the ranks of influential genes is illustrated in Figure REF , from which we can see that the proposed CCH procedure selects out three very influential genes $1373349\\_{at}$ , $1368887\\_{at}$ and $1382291\\_{at}$ (emphasized in Figure REF by blue color), which were not detected as influential by the other screening methods.",
"The reason we selects out the three influential genes is that there exists strong nonlinearity relationship between the response and the combination of the three covariates genes.",
"Figure REF illustrate the above findings.",
"Besides, gene $1398594\\_at$ is detected as influential by CCH and RRCS procedure, which is also emphasized by red colour in Figure REF .",
"By scatter plot, we find the nonlinearity between gene $1398594\\_at$ and TRIM 32 gene is obvious and CCH and RRCS procedure can both capture the nonlinear relationship.",
"The above findings are just based on statistical analysis, which need to be further validated by experiments in labs.",
"The screening procedure is particularly helpful by narrowing down the number of research targets to a few top ranked genes from the 500 candidates."
],
[
"Discussion",
"We propose a very flexible semi-parametric ECR model and consider the variable selection problem for ECR model in the ultra-high dimensional setting.",
"We propose a doubly robust sure screening procedure for ECR model Theoretical analysis shows that the procedure enjoys sure screening property, i.e., with probability tending to 1, the screening procedure selects out all important variables and substantially reduces the dimensionality to a moderate size against the sample size.",
"We set $k_n$ to be a small value and it performs well as long as there is a natural index for all the covariates such that the neighboring covariates are correlated.",
"If there is no natural index group in prior, we can do statistical clustering for the variables before screening.",
"The performance of the screening procedure then would rely heavily on the clustering performance, which we leave as a future research topic."
],
[
"Appendix: Proof of Main Theorems",
"First we introduce a useful lemma which is critical for the proof of the main results.",
"For any $c>0$ and some positive constant $C>0$ , we have $\\left(\\left|\\hat{S}_{s,t}-\\Sigma _{s,t}\\right|\\ge cn^{-\\kappa }\\right)=O\\left(\\exp (-Cn^{1-2\\kappa })\\right).$ Recall that $\\hat{S}_{s,t}=\\sin (\\frac{\\pi }{2}\\hat{\\tau }_{s,t})$ , then we have $\\begin{array}{lll}\\left(|\\hat{S}_{s,t}-\\Sigma _{s,t}|>t\\right)&=&\\left(\\left|\\sin (\\frac{\\pi }{2}\\hat{\\tau }_{s,t})-\\sin (\\frac{\\pi }{2}{\\tau }_{s,t})\\right|\\ge t\\right)\\\\&\\le & \\left(\\left|\\hat{\\tau }_{s,t}-{\\tau }_{s,t}\\right|\\ge \\frac{2}{\\pi }t\\right)\\\\&\\le & \\left(\\left|\\hat{\\tau }_{s,t}-{\\tau }_{s,t}\\right|\\ge \\frac{2}{\\pi }t\\right)\\end{array}$ Since $\\hat{\\tau }_{s,t}$ can be written in the form of U-statistic with a kernel bounded between -1 and 1, by Hoeffding's inequality, we have that $\\left(\\left|\\hat{\\tau }_{s,t}-{\\tau }_{s,t}\\right|\\ge \\frac{2}{\\pi }t\\right)\\le 2\\exp \\left(-\\frac{2}{\\pi ^2}nt^2\\right).$ By taking $t=cn^{-\\kappa }$ , we then have $\\left(\\left|\\hat{S}_{s,t}-\\Sigma _{s,t}\\right|\\ge cn^{-\\kappa }\\right)\\le 2\\exp \\left(-\\frac{2c^2}{\\pi ^2}n^{1-2\\kappa }\\right),$ which concludes the Lemma.",
"By the definition of CC, $(\\rho ^c_l)^2=_{\\times ^l}_{^l\\times ^l}^{-1}_{\\times ^l}^\\top .$ By Assumption 1, $\\lambda _{\\max }({_{^l\\times ^l}})\\le c_0$ , and note that $_{\\times ^l}=(\\Sigma _{1,m_1^l},\\ldots ,\\Sigma _{1,m_k^l})$ is a row vector, we have that $(\\rho _{l}^c)^2\\le c_0\\sum _{t=1}^k(\\Sigma _{1,m^{l}_t})^2.$ By Assumption 2, if $i\\in _*$ , then there exists $l_i\\in _i^n$ such that $\\rho _{l_i}^c\\ge c_1n^{-\\kappa }$ .",
"Without loss of generality, we assume that $|\\Sigma _{1,m_1^{l_i}}|=\\mathop {\\mathrm {max}}_{1\\le t\\le k}\\large |\\Sigma _{1,m_t^{l_i}}\\large |.$ By Equation (REF ), we have that $|\\Sigma _{1,m_1^{l_i}}|\\ge c_1^*n^{-\\kappa }$ for some $c_1^*>0$ .",
"For $\\Sigma _{1,m_1^{l_i}}$ , we denote the corresponding Kendall' tau estimator as $\\hat{S}_{1,m_1^{l_i}}\\in \\hat{}_{\\times ^{l_i}}$ , then we have the following result: $\\left(\\mathop {\\mathrm {min}}_{i\\in _*}|\\hat{S}_{1,m_1^{l_i}}|\\ge \\frac{c_1^*}{2}n^{-\\kappa }\\right)\\ge \\left(\\mathop {\\mathrm {min}}_{i\\in _*}\\left|\\hat{S}_{1,m_1^{l_i}}-\\Sigma _{1,m_1^{l_i}}\\right|\\le \\frac{c_1^*}{2}n^{-\\kappa }\\right).$ Furthermore, we have $\\begin{array}{llll}& \\left(\\mathop {\\mathrm {min}}_{i\\in _*}\\left|\\hat{S}_{1,m_1^{l_i}}-\\Sigma _{1,m_1^{l_i}}\\right|\\ge \\frac{c_1^*}{2}n^{-\\kappa }\\right) \\\\\\le & s*\\left(\\left|\\hat{S}_{1,m_1^{l_i}}-\\Sigma _{1,m_1^{l_i}}\\right|\\ge \\frac{c_1^*}{2}n^{-\\kappa }\\right)\\\\=& s*\\left(\\left|\\sin (\\frac{\\pi }{2}\\hat{\\tau }_{1,m_1^{l_i}})-\\sin (\\frac{\\pi }{2}{\\tau }_{1,m_1^{l_i}})\\right|\\ge \\frac{c_1^*}{2}n^{-\\kappa }\\right)\\\\\\le & s*\\left(\\left|\\hat{\\tau }_{1,m_1^{l_i}}-{\\tau }_{1,m_1^{l_i}}\\right|\\ge \\frac{c_1^*}{\\pi }n^{-\\kappa }\\right)\\\\\\le & p*\\left(\\left|\\hat{\\tau }_{1,m_1^{l_i}}-{\\tau }_{1,m_1^{l_i}}\\right|\\ge \\frac{c_1^*}{\\pi }n^{-\\kappa }\\right)\\end{array}$ Since $\\hat{\\tau }_{1,m_1^{l_i}}$ can be written in the form of U-statistic with a kernel bounded between -1 and 1, by Hoeffding's inequality, we have that $\\left(\\left|\\hat{\\tau }_{1,m_1^{l_i}}-{\\tau }_{1,m_1^{l_i}}\\right|\\ge \\frac{c_1^*}{\\pi }n^{-\\kappa }\\right)\\le 2\\exp \\left(-\\frac{c_1^{*2}}{2\\pi ^2}n^{1-2\\kappa }\\right).$ By Assumption 3, $\\log (p)=o(n^{1-2\\kappa })$ , we further have that for some constant $C$ , $\\left(\\mathop {\\mathrm {min}}_{i\\in _*}\\left|\\hat{S}_{1,m_1^{l_i}}-\\Sigma _{1,m_1^{l_i}}\\right|\\ge {c_1^*}n^{-\\kappa }\\right)\\le 2\\exp \\left(-Cn^{1-2\\kappa }\\right).$ Combining with Equation (REF ), we have $\\left(\\mathop {\\mathrm {min}}_{i\\in _*}|\\hat{S}_{1,m_1^{l_i}}|\\ge {c_1^*}n^{-\\kappa }\\right)\\ge 1-2\\exp \\left(-Cn^{1-2\\kappa }\\right).$ Besides, it is easy to show that $(\\hat{\\rho ^c_l})^2\\ge \\mathop {\\mathrm {max}}_{1\\le t\\le k}(\\hat{S}_{1,m_t^l})^2$ , and hence, $\\left(\\mathop {\\mathrm {min}}_{i\\in _*}\\mathop {\\mathrm {max}}_{l\\in _i^n}\\hat{\\rho }_l^c\\ge c_1^* n^\\kappa \\right)\\ge 1-2\\exp \\left(-Cn^{1-2\\kappa }\\right),$ which concludes $\\left(\\mathop {\\mathrm {min}}_{i\\in _*}\\mathop {\\mathrm {max}}_{l\\in _i^n}\\hat{\\rho }_l^c\\ge c_1^* n^\\kappa \\right)\\ge 1-O\\left(\\exp \\left(-{Cn^{1-2\\kappa }}\\right)\\right).$ The above result further implies that $\\left(_*\\subset \\hat{}_{c_1^* n^\\kappa }\\right)\\ge 1-O\\left(\\exp \\left(-{Cn^{1-2\\kappa }}\\right)\\right).$ The proof of Theorem is split into the following 2 steps.",
"(Step I) In this step we aim to prove the following result holds: $\\left(\\max _{1\\le l\\le _p^k} \\left|(\\hat{\\rho }_l^c)^2-(\\tilde{\\rho }_l^c)^2\\right|>cn^{-\\kappa }\\right)=O\\Big (\\exp (-Cn^{1-2\\kappa })\\Big ),$ where $(\\tilde{\\rho }_l^c)^2=\\hat{}_{\\times ^l}{}_{^l\\times ^l}^{-1}\\hat{}_{\\times ^l}^\\top $ .",
"Note that the determinants of matrices ${}_{^l\\times ^l}$ and $\\hat{}_{^l\\times ^l}$ are polynomials of finite order in their entries, thus we have the following inequality holds, $\\begin{array}{lll}\\left(\\left||\\hat{}_{^l\\times ^l}|-|{}_{^l\\times ^l}|\\right|>cn^{-\\kappa }\\right)&\\le & \\left(\\max _{1\\le s, t\\le k}|\\hat{S}_{s,t}-\\Sigma _{s,t}|>cn^{-\\kappa }\\right),\\\\&\\le & k^2*\\left(|\\hat{S}_{s,t}-\\Sigma _{s,t}|>cn^{-\\kappa }\\right)\\\\&=& k^2*\\left(\\left|\\sin (\\frac{\\pi }{2}\\hat{\\tau }_{s,t})-\\sin (\\frac{\\pi }{2}{\\tau }_{s,t})\\right|\\ge {c}n^{-\\kappa }\\right)\\\\&\\le & k^2*\\left(\\left|\\hat{\\tau }_{s,t}-{\\tau }_{s,t}\\right|\\ge \\frac{2c}{\\pi }n^{-\\kappa }\\right)\\\\&\\le & k^2*\\left(\\left|\\hat{\\tau }_{s,t}-{\\tau }_{s,t}\\right|\\ge \\frac{2c}{\\pi }n^{-\\kappa }\\right)\\end{array}$ Since $\\hat{\\tau }_{s,t}$ can be written in the form of U-statistic with a kernel bounded between -1 and 1, by Hoeffding's inequality, we have that $\\left(\\left|\\hat{\\tau }_{s,t}-{\\tau }_{s,t}\\right|\\ge \\frac{2c}{\\pi }n^{-\\kappa }\\right)\\le 2\\exp \\left(-\\frac{2c^{2}}{\\pi ^2}n^{1-2\\kappa }\\right).$ Thus we have for some positive constant $C^{*}$ , the following inequality holds: $\\left(\\left||\\hat{}_{^l\\times ^l}|-|{}_{^l\\times ^l}|\\right|>cn^{-\\kappa }\\right)\\le \\exp \\left(-C^*n^{1-2\\kappa }\\right)$ By Assumption 3, $\\log p=O(n^\\xi )$ with $\\xi \\in (0,1-2\\kappa )$ , we further have for some positive constant $C$ , $\\left(\\max _{1\\le l\\le _p^k}\\left||\\hat{}_{^l\\times ^l}|-|{}_{^l\\times ^l}|\\right|>cn^{-\\kappa }\\right)\\le \\exp \\left(-Cn^{1-2\\kappa }\\right).$ Note that $k$ is finite and by the adjoint matrix expansion of an inverse matrix, similar to the above analysis, we have for any positive $c$ , $\\left(\\max _{1\\le l\\le _p^k}\\left\\Vert (\\hat{}_{^l\\times ^l})^{-1}-({}_{^l\\times ^l})^{-1}\\right\\Vert _\\infty >cn^{-\\kappa }\\right)\\le \\exp \\left(-Cn^{1-2\\kappa }\\right).$ Notice that $\\begin{array}{lll}\\left|(\\hat{\\rho }_l^c)^2-(\\tilde{\\rho }_l^c)^2\\right|&\\le & \\Vert _{\\times }\\Vert _\\infty ^2\\left\\Vert (\\hat{}_{^l\\times ^l})^{-1}-({}_{^l\\times ^l})^{-1}\\right\\Vert _\\infty \\\\&\\le &\\left\\Vert (\\hat{}_{^l\\times ^l})^{-1}-({}_{^l\\times ^l})^{-1}\\right\\Vert _\\infty \\end{array}$ Thus $\\left(\\max _{1\\le l\\le _p^k} \\left|(\\hat{\\rho }_l^c)^2-(\\tilde{\\rho }_l^c)^2\\right|>cn^{-\\kappa }\\right)=O\\Big (\\exp (-Cn^{1-2\\kappa })\\Big )$ (Step II) In this step, we will first prove that for any $c>0$ , $\\left(\\mathop {\\mathrm {max}}_{1\\le l\\le _p^k}|\\tilde{\\rho }_l^c-\\rho _l^c|\\ge cn^{-\\kappa }\\right)=O\\left(\\exp (-Cn^{1-2\\kappa })\\right).$ By Lemma REF , we have that $\\left(\\left|\\hat{S}_{s,t}-\\Sigma _{s,t}\\right|\\ge cn^{-\\kappa }\\right)=O\\left(\\exp (-Cn^{1-2\\kappa })\\right).$ By Assumption 3, $\\log p=O(n^\\xi )$ with $\\xi \\in (0,1-2\\kappa )$ , thus we have $\\left(\\mathop {\\mathrm {max}}_{1\\le i\\le p}\\mathop {\\mathrm {max}}_{l\\in _i^n}\\mathop {\\mathrm {max}}_{s,t}|\\hat{S}_{s,t}-\\Sigma _{s,t}|\\right)=O\\left(\\exp (-Cn^{1-2\\kappa })\\right).$ Recall that $(\\tilde{\\rho }_l^c)^2=\\hat{}_{\\times ^l}{}_{^l\\times ^l}^{-1}\\hat{}_{\\times ^l}^\\top $ , by the property of ${}_{^l\\times ^l}$ , we have for any $c>0$ , $\\left(\\mathop {\\mathrm {max}}_{1\\le l\\le _p^k}|\\tilde{\\rho }_l^c-\\rho _l^c|\\ge cn^{-\\kappa }\\right)=O\\left(\\exp (-Cn^{1-2\\kappa })\\right).$ Further by Assumption 4, $\\mathop {\\mathrm {min}}_{i\\notin _*}\\mathop {\\mathrm {max}}_{l\\in _i^n}\\rho _l^c< c_1^* n^{-\\kappa }$ and the last equation, we have that $\\left(\\mathop {\\mathrm {max}}_{i\\notin _*}\\mathop {\\mathrm {max}}_{l\\in _i^n}\\tilde{\\rho }_l^c<c_1^*n^{-\\kappa }\\right)\\ge 1-O\\left(\\exp (-Cn^{1-2\\kappa })\\right).$ By the result in Step I, we have that $\\left(\\max _{1\\le l\\le _p^k} \\left|(\\hat{\\rho }_l^c)^2-(\\tilde{\\rho }_l^c)^2\\right|>cn^{-\\kappa }\\right)=O\\Big (\\exp (-Cn^{1-2\\kappa })\\Big )$ Thus we further have $\\left(\\mathop {\\mathrm {max}}_{i\\notin _*}\\mathop {\\mathrm {max}}_{l\\in _i^n}\\hat{\\rho }_l^c<c_1^*n^{-\\kappa }\\right)\\ge 1-O\\left(\\exp (-Cn^{1-2\\kappa })\\right).$ By Theorem , the following inequality holds: $\\left(\\mathop {\\mathrm {min}}_{i\\in _*}\\mathop {\\mathrm {max}}_{l\\in _i^n}\\hat{\\rho }_l^c\\ge c_1^* n^{-\\kappa }\\right)\\ge 1-O\\left(\\exp \\left(-{Cn^{1-2\\kappa }}\\right)\\right),$ then combining the result in Equation (REF ), we have $\\left(_*=\\hat{}_{c_1^* n^{-\\kappa }}\\right)\\ge 1-O\\left(\\exp \\left(-{Cn^{1-2\\kappa }}\\right)\\right),$ which concludes the theorem.",
"Let $\\delta \\rightarrow 0$ satisfying $\\delta n^{1-2\\kappa -\\tau -\\tau ^*}\\rightarrow \\infty $ as $n\\rightarrow \\infty $ and define $\\begin{split}(\\delta )=\\left\\lbrace 1\\le i\\le p:\\mathop {\\mathrm {max}}_{l\\in _i^n}\\hat{\\rho }_l^c \\ \\text{is among the largest} \\ \\lfloor \\delta p\\rfloor \\ \\text{of all}\\right\\rbrace \\\\\\tilde{}(\\delta )=\\left\\lbrace 1\\le i\\le p:\\mathop {\\mathrm {max}}_{l\\in _i^n}\\tilde{\\rho }_l^c \\ \\text{is among the largest} \\ \\lfloor \\delta p\\rfloor \\ \\text{of all}\\right\\rbrace \\end{split}$ where $(\\tilde{\\rho }_l^c)^2=\\hat{}_{\\times ^l}{}_{^l\\times ^l}^{-1}\\hat{}_{\\times ^l}^\\top $ and where $(\\hat{\\rho }_l^c)^2=\\hat{}_{\\times ^l}{\\hat{}}_{^l\\times ^l}^{-1}\\hat{}_{\\times ^l}^\\top $ .",
"We will first show that $\\Big (_*\\subset (\\delta )\\Big )\\ge 1-O\\left(\\exp \\left(-{Cn^{1-2\\kappa }}\\right)\\right)$ By Theorem , it is equivalent to show that $\\Big (_*\\subset (\\delta )\\cap \\hat{}_{c_1^*n^{-\\kappa }}\\Big )\\ge 1-O\\left(\\exp \\left(-{Cn^{1-2\\kappa }}\\right)\\right)$ By Step I in the proof of Theorem , it is also equivalent to show that $\\Big (_*\\subset \\tilde{}(\\delta )\\cap \\hat{}_{c_1^*n^{-\\kappa }}\\Big )\\ge 1-O\\left(\\exp \\left(-{Cn^{1-2\\kappa }}\\right)\\right).$ Finally, by Theorem again, to prove Equation (REF ) is equivalent to prove that $\\Big (_*\\subset \\tilde{}(\\delta )\\Big )\\ge 1-O\\left(\\exp \\left(-{Cn^{1-2\\kappa }}\\right)\\right).$ Recall that in the proof of Theorem , we obtained the following result: $\\left(\\mathop {\\mathrm {min}}_{i\\in _*}\\mathop {\\mathrm {max}}_{l\\in _i^n}\\tilde{\\rho }_l^c\\ge c_1^* n^\\kappa \\right)\\ge 1-O\\left(\\exp \\left(-{Cn^{1-2\\kappa }}\\right)\\right).$ If we can prove that $\\left(\\sum _{i=1}^p(\\mathop {\\mathrm {max}}_{l\\in _i^n}\\tilde{\\rho }_l^c)^2\\le cn^{-1+\\tau ^*+\\tau }p\\right)\\ge 1-O\\left(\\exp \\left(-{Cn^{1-2\\kappa }}\\right)\\right).$ Then we have, with probability larger than $1-O\\left(\\exp \\left(-{Cn^{1-2\\kappa }}\\right)\\right)$ , $\\text{Card}\\left\\lbrace 1\\le i\\le p; \\mathop {\\mathrm {max}}_{l\\in _i^n}\\tilde{\\rho }_l^c\\ge \\min _{i\\in _*}\\mathop {\\mathrm {max}}_{l\\in _i^n}\\tilde{\\rho }_l^c\\right\\rbrace \\le \\frac{cp}{n^{1-2\\kappa -\\tau -\\tau ^*}},$ which further indicate that the result in (REF ) holds due to $\\delta n^{1-2\\kappa -\\tau -\\tau ^*}\\rightarrow \\infty $ .",
"So to end the whole proof, we just need to show that the result in (REF ) holds.",
"For each $1\\le i\\le p$ , let $\\tilde{\\rho }^c_{i_0}=\\max _{l\\in _i^n}\\tilde{\\rho }^c_{l}$ .",
"Note that $(\\tilde{\\rho }_{i_0}^c)^2=\\hat{}_{\\times ^{i_0}}{}_{^{i_0}\\times ^{i_0}}^{-1}\\hat{}_{\\times ^{i_0}}^\\top $ , with $\\hat{}_{\\times ^{i_0}}=(\\hat{S}_{1,m_1^{i_0}},\\ldots ,\\hat{S}_{1,m_k^{i_0}})$ .",
"By Assumption 1, we have $(\\tilde{\\rho }_{i_0}^c)^2\\le c_0\\sum _{t=1}^k(\\hat{S}_{1,m_t^{i_0}})^2=c_0\\Vert \\hat{}_{\\times ^{i_0}}\\Vert _2^2,$ which further indicates $\\sum _{i=1}^p(\\tilde{\\rho }_{i_0}^c)^2\\le c_0k k_n \\Vert \\hat{}_{\\times }\\Vert _2^2, \\ \\text{with} \\ =\\lbrace 2,\\ldots ,d\\rbrace .$ Notice that $\\left(\\left|\\hat{S}_{s,t}-\\Sigma _{s,t}\\right|\\ge cn^{-\\kappa }\\right)\\le 2\\exp \\left(-\\frac{2c^2}{\\pi ^2}n^{1-2\\kappa }\\right),$ thus similar to the argument in [13], we can easily get that $\\left(\\sum _{i=1}^p(\\mathop {\\mathrm {max}}_{l\\in _i^n}\\tilde{\\rho }_l^c)^2\\le cn^{-1+\\tau ^*+\\tau }p\\right)\\ge 1-O\\left(\\exp \\left(-{Cn^{1-2\\kappa }}\\right)\\right).$ Finally, following the same idea of iterative screening as in the proof of Theorem 1 of [10], we finish the proof of the theorem.",
"Yong He's research is partially supported by the grant of the National Science Foundation of China (NSFC 11801316).",
"Xinsheng Zhang's research is partially supported by the grant of the National Science Foundation of China (NSFC 11571080).",
"Jiadong Ji's work is supported by the grant from the the grant of the National Science Foundation of China (NSFC 81803336) and Natural Science Foundation of Shandong Province (ZR2018BH033)."
]
] | 1808.08551 |
[
[
"On the motive of intersections of two Grassmannians in ${\\mathbb{P}}^9$"
],
[
"Abstract Using intersections of two Grassmannians in ${\\mathbb{P}}^9$, Ottem-Rennemo and Borisov-C\\u{a}ld\\u{a}raru-Perry have exhibited pairs of Calabi-Yau threefolds $X$ and $Y$ that are deformation equivalent, L-equivalent and derived equivalent, but not birational.",
"To complete the picture, we show that $X$ and $Y$ have isomorphic Chow motives."
],
[
"Introduction",
"Let $\\hbox{Var}($ denote the category of algebraic varieties over the complex numbers $, and let$ K0(Var()$ denote the Grothendieck ring.",
"This ring is a rather mysterious object.",
"Its intricacyis highlighted by Borisov \\cite {Bor}, who showed that the class of the affine line $ L$ is a zero--divisor in $ K0(Var()$.",
"Following on Borisov^{\\prime }s pioneering result, recent years have seen a flurry of constructions of pairs of Calabi--Yau varieties$ X, Y$ that are {\\em not} birational (and so $ [X]=[Y]$ in the Grothendieck ring), but$$ ([X] -[Y]) \\mathbb {L}^r=0\\ \\ \\ \\hbox{in}\\ K_0(\\hbox{Var}() \\ ,$$i.e., $ X$ and $ Y$ are ``L--equivalent^{\\prime \\prime } in the sense of \\cite {KS}.In most cases, the constructed varieties $ X$ and $ Y$ are also derived equivalent \\cite {IMOU}, \\cite {IMOU2}, \\cite {Mar}, \\cite {Kuz}, \\cite {OR}, \\cite {BCP}, \\cite {KS}, \\cite {HL}, \\cite {Man}, \\cite {KR}, \\cite {KKM}.$ According to a conjecture made by Orlov [40], derived equivalent smooth projective varieties should have isomorphic Chow motives.",
"This conjecture is true for $K3$ surfaces [17], but is still widely open for Calabi–Yau varieties of dimension $\\ge 3$ .",
"In [33], [34], I verified Orlov's conjecture for the Calabi–Yau threefolds of Ito–Miura–Okawa–Ueda [19], resp.",
"the Calabi–Yau threefolds of Kapustka–Rampazzo [27].",
"The aim of the present note is to check that Orlov's conjecture is also true for the Calabi–Yau threefolds studied recently by Borisov–Căldăraru–Perry [10], and independently by Ottem–Rennemo [41].",
"The threefolds of [41], [10] are called GPK$^3$ threefolds.",
"The shorthand “GPK$^3$ ” stands for Gross–Popescu–Kanazawa–Kapustka–Kapustka, the authors of the papers [13], [23], [24], [22] where they first appeared (the shorthand “GPK$^3$ ” is coined in [10]).",
"These threefolds are constructed as follows.",
"Given $W$ a 10–dimensional vector space over $, let $ P:=P(W)$.",
"Let $ V$ be a $ 5$--dimensional vector space over $ , and choose isomorphisms $\\phi _i\\colon \\wedge ^2 V\\rightarrow W\\ , \\ \\ i=1,2\\ .$ Composing the Plücker embedding with the induced isomorphisms $\\phi _i\\colon \\mathbb {P}(\\wedge ^2 V)\\cong \\mathbb {P}$ , one obtains two embeddings of the Grassmannian $Gr(2,V)$ in $\\mathbb {P}$ , whose images are denoted $Gr_i$ , $i=1,2$ .",
"For $\\phi _i$ generic, the intersection $ X:= Gr_1\\cap Gr_2\\ \\ \\ \\subset \\ \\mathbb {P}$ is a smooth Calabi–Yau threefold, called a GPK$^3$ threefold.",
"Let $Gr_i^\\vee $ be the projective dual of $Gr_i$ .",
"The intersection $ Y:=Gr_1^\\vee \\cap Gr_2^\\vee \\ \\ \\ \\subset \\ \\mathbb {P}^\\vee $ is again a smooth Calabi–Yau threefold, and it is deformation equivalent to $X$ .",
"The pair $X,Y$ are called GPK$^3$ double mirrors, and $X,Y$ are known to be Hodge equivalent, derived equivalent, L–equivalent, and in general not birational [10], [41].",
"In this note, we prove the following: -1mmTheorem (=theorem REF ) Let $X, Y$ be two GPK$^3$ double mirrors.",
"Then there is an isomorphism of Chow motives $ h(X)\\cong h(Y)\\ \\ \\ \\hbox{in}\\ \\mathcal {M}_{{\\rm rat}}\\ .$ The proof of theorem REF is an elementary exercice in manipulating Chow groups and correspondences, based on a nice geometric relation between $X$ and $Y$ established in [10] (cf.",
"proposition REF below).",
"The only ingredient in the proof that may perhaps not be completely standard is the use of Bloch's higher Chow groups ([5], cf.",
"also section below), and some results on higher Chow groups of piecewise trivial fibrations (section below).",
"-1mmConventions In this note, the word variety will refer to a reduced irreducible scheme of finite type over the field of complex numbers $.",
"{\\bf All Chow groups will be with \\mathbb {Q}--coefficients, unless indicated otherwise:} For a variety $ X$, we will write $ Aj(X):=CHj(X)Q$ for the Chow group of dimension $ j$ cycles on $ X$ with rational coefficients.For $ X$ smooth of dimension $ n$, the notations $ Aj(X)$ and $ An-j(X)$ will be used interchangeably.$ The notations $A^j_{hom}(X)$ (and $A^j_{AJ}(X)$ ) will be used to indicate the subgroups of homologically trivial (resp.",
"Abel–Jacobi trivial) cycles.",
"For a morphism between smooth varieties $f\\colon X\\rightarrow Y$ , we will write $\\Gamma _f\\in A^\\ast (X\\times Y)$ for the graph of $f$ , and ${}^t \\Gamma _f\\in A^\\ast (Y\\times X)$ for the transpose correspondence.",
"We will write $\\mathcal {M}_{{\\rm rat}}$ for the contravariant category of Chow motives (i.e., pure motives as in [43], [39], with Hom–groups defined using $A^\\ast (X\\times Y)_{}$ ).",
"We will write $H^j(X)=H^j(X,\\mathbb {Q})$ for singular cohomology, and $H_j(X)=H_j^{BM}(X,\\mathbb {Q})$ for Borel–Moore homology."
],
[
"The Calabi–Yau threefolds",
"In this section we consider GPK$^3$ threefolds, as defined in the introduction.",
"Proposition 2.1 (Ottem–Rennemo, Kanazawa [41], [22]) The family of GPK$^3$ threefolds is locally complete.",
"A GPK$^3$ threefold $X$ has Hodge numbers $ h^{1,1}(X)=1\\ ,\\ \\ \\ h^{2,1}(X)=51\\ .$ The first statement is [41].",
"The Hodge numbers are computed in [22].",
"Theorem 2.2 (Ottem–Rennemo, Borisov–Căldăraru-Perry [41], [10]) Let $X,Y$ be a general pair of GPK$^3$ double mirrors.",
"Then $X$ and $Y$ are not birational, and so $ [X]\\ne [Y]\\ \\ \\ \\hbox{in}\\ K_0(\\hbox{Var}() \\ .$ However, one has $ ([X] -[Y]) \\mathbb {L}^4=0\\ \\ \\ \\hbox{in}\\ K_0(\\hbox{Var}() \\ .$ Moreover, $X$ and $Y$ are derived equivalent, i.e.",
"there is an isomorphism of bounded derived categories $ D^b(X)\\cong D^b(Y)\\ .$ In particular, there is an isomorphism of polarized Hodge structures $ H^3(X,\\mathbb {Z})\\ \\cong \\ H^3(Y,\\mathbb {Z})\\ .$ Non–birationality is [10], and independently [41].",
"Thanks to the birational invariance of the MRC–fibration, $X$ and $Y$ are not stably birational (cf.",
"[9]).",
"The celebrated Larsen–Lunts result [31] implies that $[X]\\ne [Y]$ in the Grothendieck ring.",
"The L–equivalence is [10]; it is a corollary of the geometric relation of proposition REF below.",
"Derived equivalence is proven in [41], and also in [29].",
"The isomorphism of Hodge structures is a corollary of the derived equivalence, in view of [41].",
"The argument of this note crucially relies on the following (for the notion of piecewise trivial fibration, cf.",
"definition REF below).",
"Proposition 2.3 (Borisov–Căldăraru–Perry [10]) Let $X,Y$ be a pair of GPK$^3$ double mirrors.",
"There is a diagram $ \\begin{array}[c]{ccccccccc} && F & \\xrightarrow{} & Q & \\xleftarrow{} & G && \\\\&&&&&&&&\\\\&{}^{\\scriptstyle p_X} \\swarrow \\ \\ && {}^{\\scriptstyle p} \\swarrow \\ \\ \\ & & \\ \\ \\ \\searrow {}^{\\scriptstyle q} & & \\ \\ \\searrow {}^{\\scriptstyle q_Y} & \\\\&&&&&&&&\\\\X & \\xrightarrow{} & Gr_1 & & & & Gr_2^\\vee & \\leftarrow & Y\\\\\\end{array}$ Here, $ Q:= \\sigma \\times _{\\mathbb {P}\\times \\mathbb {P}^\\vee } (Gr_1\\times Gr_2^\\vee ) $ is the intersection of the natural incidence divisor $\\sigma \\subset \\mathbb {P}\\times \\mathbb {P}^\\vee $ with the product $Gr_1\\times Gr_2^\\vee \\subset \\mathbb {P}\\times \\mathbb {P}^\\vee $ , the morphisms $p$ and $q$ are induced by the natural projections, the closed subvarieties $F,G$ are defined as $p^{-1}(X)$ resp.",
"$q^{-1}(Y)$ , and $p_X, q_Y$ are defined as the restrictions $p\\vert _F$ resp.",
"$q\\vert _G$ .",
"The morphisms $p_X,q_Y$ are piecewise trivial fibrations with fibres $F_x$ resp.",
"$G_y$ verifying $ \\begin{split} &A_i(F_x)_=A_i(G_y)_={\\left\\lbrace \\begin{array}{ll} \\mathbb {Q}&\\hbox{if\\ }i=0,1,5\\ ,\\\\\\mathbb {Q}^2 &\\hbox{if\\ } i=2,3,4\\ ,\\\\\\end{array}\\right.}",
"\\\\&H_j(F_x)=H_j(G_y)=0\\ \\ \\ \\hbox{for\\ all\\ $j$\\ odd}\\ ,\\end{split} $ for all $x\\in X, y\\in Y$ .",
"However, over the open complements $ U:=Gr_1\\setminus X\\ ,\\ \\ \\ V:=Gr_2^\\vee \\setminus Y \\ ,$ the restrictions $p_U:= p\\vert _{p^{-1}(U)}, q_V:=q\\vert _{q^{-1}(V)}$ are piecewise trivial fibrations with fibres $Q_u:=p^{-1}(u)$ resp.",
"$Q_v:=q^{-1}(v)$ verifying $ A_i(Q_u)_=A_i(Q_v)_={\\left\\lbrace \\begin{array}{ll} \\mathbb {Q}&\\hbox{if\\ }i=0,1,4,5\\ ,\\\\\\mathbb {Q}^2 &\\hbox{if\\ } i=2,3\\ ,\\\\\\end{array}\\right.}",
"$ for all $u\\in U, v\\in V$ .",
"The diagram is constructed in [10].",
"The computation of homology and Chow groups of the fibres of $p$ easily follows from the explicit description of the fibres as hyperplane sections of the Grassmannian $Gr(2,V)$ [10].",
"Precisely, as explained in [10], there exists a closed subvariety $Z\\subset F_x$ such that $Z\\cong \\mathbb {P}^2$ , and the complement $C:=F_x\\setminus Z$ is a Zariski locally trivial fibration over $\\mathbb {P}^2$ , with fibres isomorphic to $\\mathbb {P}^3\\setminus \\mathbb {P}^1$ .",
"Since neither $C$ nor $Z$ have odd–degree Borel–Moore homology, the same holds for $F_x$ .",
"As for even–degree homology, there is a commutative diagram with exact rows $ \\begin{array}[c]{cccccc}\\rightarrow A_i(Z) &\\rightarrow & A_i(F_x) &\\rightarrow & A_i(C)&\\rightarrow 0\\\\&&&&&\\\\\\ \\ \\downarrow {\\scriptstyle \\cong }&&\\downarrow &&\\ \\ \\downarrow {\\scriptstyle \\cong }\\\\&&&&&\\\\0 \\rightarrow H_{2i}(Z) &\\rightarrow & H_{2i}(F_x) &\\rightarrow & H_{2i}(C)&\\ \\ \\rightarrow 0\\ ,\\\\\\end{array}$ where vertical arrows are cycle class maps.",
"The left and right vertical arrow are isomorphisms, because of the above explicit description of $Z$ and $C$ .",
"This implies that the cycle class map induces isomorphisms $A_i(F_x)\\cong H_{2i}(F_x)$ for all $i$ .",
"The bottom exact sequence of this diagram shows that $ H_{2i}(F_x) = {\\left\\lbrace \\begin{array}{ll} H_{2i}(C) &\\hbox{if\\ }i=3,4,5\\ ,\\\\H_{2i}(C)\\oplus \\mathbb {Q}&\\hbox{if\\ }i=0,1,2\\ .\\\\\\end{array}\\right.",
"}$ Next, one remarks that the open $C$ (being a fibration over $\\mathbb {P}^2$ with fibre $T\\cong \\mathbb {P}^3\\setminus \\mathbb {P}^1$ ) has $ H_{2i}(C)=\\bigoplus _{\\ell +m=2i} H_\\ell (\\mathbb {P}^2)\\otimes H_m(T)={\\left\\lbrace \\begin{array}{ll} 0 &\\hbox{if\\ }i=0,1\\ ,\\\\\\mathbb {Q}&\\hbox{if\\ }i=2,5\\ ,\\\\\\mathbb {Q}^2 &\\hbox{if\\ }i=3,4\\ .\\\\\\end{array}\\right.",
"}$ Putting things together, this shows the statement for $H_{2i}(F_x)$ .",
"(A more efficient, if less self–contained, way of determining the Betti numbers of $F_x$ is as follows.",
"One has equality in the Grothendieck ring [10] $ [ F_x]= (\\mathbb {L}^2 +\\mathbb {L}+1)(\\mathbb {L}^3 +\\mathbb {L}^2+1)\\ \\ \\ \\hbox{in}\\ K_0(\\hbox{Var})\\ .$ Let $W^\\ast $ denote Deligne's weight filtration on Borel–Moore homology [42].",
"The “virtual Betti number” $ P_{2i}():= \\sum _j (-1)^j \\dim \\hbox{Gr}_W^{-2i} H_j() $ is a functor on $K_0(\\hbox{Var})$ , and so $ P_{2i}(F_x) = P_{2i}\\Bigl ( (\\mathbb {L}^2 +\\mathbb {L}+1)(\\mathbb {L}^3 +\\mathbb {L}^2+1)\\Bigr ) = {\\left\\lbrace \\begin{array}{ll} 1 &\\hbox{if\\ }i=0,1,5\\ ,\\\\2 &\\hbox{if\\ } i=2,3,4\\ .\\\\\\end{array}\\right.}",
"$ On the other hand, $F_x$ has no odd–degree homology, and the fact that $H_{2i}(F_x)$ is algebraic implies that $H_{2i}(F_x)$ is pure of weight $-2i$ .",
"It follows that $P_{2i}(F_x)=\\dim H_{2i}(F_x)$ .)",
"The homology groups and Chow groups of the fibres $Q_u$ over $u\\in U$ are determined similarly: according to loc.",
"cit., there is a closed subvariety $Z\\subset Q_u$ such that $Z$ is isomorphic to a smooth quadric in $\\mathbb {P}^4$ , and the complement $Q_u\\setminus W$ is a Zariski locally trivial fibration over $\\mathbb {P}^3$ , with fibres isomorphic to $\\mathbb {P}^2\\setminus \\mathbb {P}^1$ .",
"We record a lemma for later use: Lemma 2.4 The open $U$ (and the open $V$ ) of proposition REF has trivial Chow groups, i.e.",
"cycle class maps $ A_i(U)\\ \\rightarrow \\ H_{2i}(U)\\ $ are injective.",
"This is a standard argument.",
"One has a commutative diagram with exact rows $ \\begin{array}[c]{cccccc}A_i(X)& \\rightarrow & A_i(Gr_1)& \\rightarrow & A_i(U) & \\rightarrow 0\\\\&&&&&\\\\\\downarrow &&\\ \\ \\downarrow {\\cong }&& \\downarrow {}&\\\\&&&&&\\\\H_{2i}(X)& \\rightarrow & H_{2i}(Gr_1)& \\rightarrow & H_{2i}(U) & \\rightarrow 0\\\\\\end{array}$ (the middle vertical arrow is an isomorphism, as $Gr_1$ is a Grassmannian).",
"Given $a\\in A_i(U)$ homologically trivial, there exists $\\bar{a}\\in A_i(Gr_1)$ such that the homology class of $\\bar{a}$ is supported on $X$ .",
"Using semisimplicity of polarized Hodge structures, the homology class of $\\bar{a}$ is represented by a Hodge class in $H_{2i}(X)$ .",
"But $X$ being three–dimensional, the Hodge conjecture is known for $X$ , and so $\\bar{a}\\in H_{2i}(Gr_1)$ is represented by a cycle $d\\in A_i(X)$ .",
"The cycle $\\bar{a}-d\\in A_i(Gr_1)$ thus restricts to $a$ and is homologically trivial, hence rationally trivial.",
"Remark 2.5 We observe in passing that the subvarieties $F,G$ in proposition REF must be singular.",
"Indeed, the fibres $F_x, G_y$ have Picard number 1, but the group of Weil divisors has dimension 2, and so the fibres $F_x,G_y$ are not $\\mathbb {Q}$ –factorial.",
"By generic smoothness [14], it follows that $F,G$ cannot be smooth.",
"Remark 2.6 As explained in [41], the 51–dimensional family of GPK$^3$ threefolds degenerates to the 50–dimensional family of Calabi–Yau threefolds first studied in [25], [18].",
"Generalized mirror pairs in this 50–dimensional family are also derived equivalent and L–equivalent [27], and have isomorphic Chow motives [34]."
],
[
"Higher Chow groups",
"Definition 3.1 (Bloch [5], [6]) Let $\\Delta ^j\\cong \\mathbb {A}^j($ denote the standard $j$ –simplex.",
"For any quasi–projective variety $M$ and any $i\\in \\mathbb {Z}$ , let $z_i^{simp}(M,\\ast )$ denote the simplicial complex where $z_i(X,j)$ is the group of $(i+j)$ –dimensional algebraic cycles in $M\\times \\Delta ^j$ that meet the faces properly.",
"Let $z_i^{}(M,\\ast )$ denote the single complex associated to $z_i^{simp}(M,\\ast )$ .",
"The higher Chow groups of $M$ are defined as $ A_i(M,j):= H^j( z_i^{}(M,\\ast )\\otimes \\mathbb {Q})\\ .$ Remark 3.2 Clearly one has $A_i(M,0)\\cong A_i(M)$ .",
"For a closed immersion, there is a long exact sequence of higher Chow groups [6], [35], extending the usual “localization exact sequence” of Chow groups.",
"Higher Chow groups are related to higher algebraic $K$ –theory: there are isomorphisms $ \\hbox{Gr}_\\gamma ^{n-i} G_j(M)_\\mathbb {Q}\\cong A_i(M,j)_{} \\ \\ \\ \\hbox{for\\ all\\ }i,j \\ ,$ where $G_j(M)$ is Quillen's higher $K$ –theory group associated to the category of coherent sheaves on $M$ , and $\\hbox{Gr}^\\ast _\\gamma $ is the graded for the $\\gamma $ –filtration [5].",
"Higher Chow groups are also related to Voevodsky's motivic cohomology (defined as hypercohomology of a certain complex of Zariski sheaves) [11], [38]."
],
[
"Operational Chow cohomology",
"In what follows, we will rely on the existence of operational Chow cohomology, as constructed by Fulton–MacPherson.",
"The precise definition does not matter here; we merely use the existence of a theory with good formal properties: Theorem 4.1 (Fulton [12]) There exists a contravariant functor $ A^\\ast ()\\colon \\ \\ \\ \\hbox{Var}_\\rightarrow \\ \\hbox{Rings}\\ $ (from the category of varieties with arbitrary morphisms to that of graded commutative rings), with the following properties: for any $X$ , and $b\\in A^j(X)$ there is a cap–product $ b \\cap ()\\colon \\ \\ A_i(X)\\ \\rightarrow \\ A_{i-j}(X)\\ ,$ making $A_\\ast (X)$ a graded $A^\\ast (X)$ –module; for $X$ smooth of dimension $n$ , the map $A^j(X)\\rightarrow A_{n-j}(X)$ given by $ b\\ \\mapsto \\ b \\cap [X]\\ \\in A_{n-j}(X) $ is an isomorphism for all $j$ ; for any proper morphism $f\\colon X\\rightarrow Y$ , there is a projection formula: $ f_\\ast ( f^\\ast (b)\\cap a)= b\\cap f_\\ast (a)\\ \\ \\ \\hbox{in}\\ A_{i-j}(Y)\\ \\ \\ \\hbox{for\\ any\\ } b\\in A^j(Y)\\ ,\\ a\\in A_i(X)\\ .",
"$ This is contained in [12].",
"The projection formula is [12].",
"Remark 4.2 For quasi–projective varieties, there is another cohomology theory to pair with Chow groups: the assignment $ CH^j(X):=\\varinjlim A^j(Y)\\ ,$ where the limit is over all smooth quasi–projective varieties $Y$ with a morphism to $X$ .",
"As shown in [3], [12], this theory satisfies the formal properties of theorem REF .",
"Since in this note, we are only interested in quasi–projective varieties, we might as well work with this theory rather than operational Chow cohomology."
],
[
"Piecewise trivial fibrations",
"This section contains two auxiliary results, propositions REF and REF .",
"The first is about Chow groups of the open complement $R:=Q\\setminus F$ of proposition REF ; the second concerns the Chow groups of the singular variety $F$ .",
"Definition 5.1 (Section 4.2 in [44]) Let $p\\colon M\\rightarrow N$ be a projective surjective morphism between quasi–projective varieties.",
"We say that $p$ is a piecewise trivial fibration with fibre $F$ if there is a finite partition $N=\\cup _j N_j$ , where $N_j\\subset N$ is locally closed and there is an isomorphism of $N_j$ –schemes $p^{-1}(N_j)\\cong N_j\\times F$ for all $j$ .",
"Proposition 5.2 Let $U:=Gr_1\\setminus X$ and $R:= Q\\setminus F$ and $p_U\\colon R\\rightarrow U$ be as in proposition REF .",
"(i) Let $h\\in A^1(R)$ be a hyperplane section, and let $h^j\\colon A_i(R)\\rightarrow A_{i-j}(R)$ denote the map induced by intersecting with $h^j$ .",
"There are isomorphisms $ \\begin{split} &\\Phi _0\\colon \\ \\ \\ A_{0}(U)\\ \\xrightarrow{}\\ A_0(R)\\ ,\\\\&\\Phi _1 \\colon \\ \\ \\ A_1(U)\\oplus A_{0}(U)\\ \\xrightarrow{}\\ A_1(R)\\ ,\\\\& \\Phi _2\\colon \\ \\ \\ A_2(U)\\oplus A_1(U) \\oplus A_0(U)^{\\oplus 2} \\ \\xrightarrow{}\\ A_2(R)\\ \\ ,\\\\&\\Phi _3\\colon \\ \\ \\ A_3(U)^{}\\oplus A_2(U)^{} \\oplus A_1(U)^{\\oplus 2} \\oplus A_0(U)^{\\oplus 2} \\ \\xrightarrow{}\\ A_3(R)\\ ,\\\\&\\Phi _4\\colon \\ \\ \\ A_3(U)\\oplus A_2(U)^{\\oplus 2}\\oplus A_1(U)^{\\oplus 2}\\oplus A_0(U)\\ \\xrightarrow{}\\ A_4(R)\\ ,\\\\&\\Phi _5\\colon \\ \\ \\ A_3(U)^{\\oplus 2}\\oplus A_2(U)^{\\oplus 2}\\oplus A_1(U)^{}\\oplus A_0(U)\\ \\xrightarrow{}\\ A_5(R)\\ .\\\\\\end{split}$ The maps are defined as $ \\begin{split}&\\Phi _0:= h^5\\circ (p_U)^\\ast \\ ,\\\\&\\Phi _1 := \\Bigl ( h^5\\circ (p_U)^\\ast , h^4\\circ (p_U)^\\ast \\Bigr ) \\ ,\\\\& \\Phi _2:=\\Bigl ( h^5\\circ (p_U)^\\ast , h^4\\circ (p_U)^\\ast , h^3\\circ (p_U)^\\ast , (b\\cdot h)\\circ (p_U)^\\ast \\Bigr )\\ ,\\\\& \\Phi _3:=\\Bigl ( h^5\\circ (p_U)^\\ast , h^4\\circ (p_U)^\\ast , h^3\\circ (p_U)^\\ast , (b\\cdot h)\\circ (p_U)^\\ast , h^2\\circ (p_U)^\\ast ,b\\circ (p_U)^\\ast \\Bigr )\\ ,\\\\& \\Phi _4:=\\Bigl ( h^4\\circ (p_U)^\\ast , h^3\\circ (p_U)^\\ast , (b\\cdot h)\\circ (p_U)^\\ast , h^2\\circ (p_U)^\\ast ,b\\circ (p_U)^\\ast , h\\circ (p_U)^\\ast \\Bigr )\\ ,\\\\& \\Phi _5:=\\Bigl ( h^3\\circ (p_U)^\\ast , (b\\cdot h)\\circ (p_U)^\\ast , h^2\\circ (p_U)^\\ast ,b\\circ (p_U)^\\ast , h\\circ (p_U)^\\ast , (p_U)^\\ast \\Bigr )\\ ,\\\\\\end{split} $ where $b\\in A^2(R)$ is a class made explicit in the proof (and $b\\colon A_i(R)\\rightarrow A_{i-2}(R)$ denotes the operation of intersecting with $b$ , and similarly for $b\\cdot h$ ).",
"(ii) $ A_i^{hom}(R)=0\\ \\ \\ \\forall i\\ .$ (i) As we have seen in proposition REF , the morphism $p_U\\colon R\\rightarrow U$ is a piecewise trivial fibration, with fibre $R_u$ .",
"Let $T$ denote the tautological bundle on the Grassmannian $Gr_2$ , and define $ b:= \\bigl ( Gr_1\\times c_2(T)\\bigr )\\vert _R\\ \\ \\ \\in \\ A^2(R)\\ .$ The (5–dimensional) fibres $R_u$ of the fibration $p_U\\colon R\\rightarrow U$ verify $ A^i(R_u)= {\\left\\lbrace \\begin{array}{ll} \\mathbb {Q}\\cdot h^i\\vert _{R_u} &\\ \\hbox{if}\\ i=0,1,4,5\\ ,\\\\\\mathbb {Q}\\cdot h^2\\vert _{R_u} \\oplus \\mathbb {Q}\\cdot b\\vert _{R_u} &\\ \\hbox{if}\\ i=2\\ ,\\\\\\mathbb {Q}\\cdot h^3\\vert _{R_u} \\oplus \\mathbb {Q}\\cdot (b\\cdot h)\\vert _{R_u} &\\ \\hbox{if}\\ i=3\\ .\\\\\\end{array}\\right.",
"}$ To prove the isomorphisms of Chow groups of (i), it is more convenient to prove a more general statement for higher Chow groups.",
"That is, we consider maps $ \\Phi _i^j\\colon \\ \\ \\ \\bigoplus A_{i-k}^{}(U,j)\\ \\xrightarrow{}\\ A_i^{}(R,j)\\ $ such that $\\Phi _i^0=\\Phi _i$ (the maps $\\Phi _i^j$ are defined just as the $\\Phi _i$ , using $(p_U)^\\ast $ and intersecting with $h$ and $b$ ).",
"We now claim that the maps $\\Phi _i^j$ are isomorphisms for all $i=0,\\ldots ,5$ and all $j$ .",
"The $j=0$ case of this claim proves (i).",
"To prove the claim, we exploit the piecewise triviality of the fibration $p_U$ .",
"Up to subdividing some more, we may suppose the strata $U_k$ (and hence also the strata $R_k$ ) are smooth.",
"We will use the notation $ U_{\\le s}:= \\bigcup _{i\\le s} U_i\\ ,\\ \\ \\ R_{\\le s}:=\\bigcup _{i\\le k} R_i\\ .$ For any $s$ , the morphism $p_{U_{\\le s}}\\colon R_{\\le s}\\rightarrow U_{\\le s}$ is a piecewise trivial fibration (with fibre $R_u$ ).",
"For $s$ large enough, $R_{\\le s}=R$ .",
"Since $U$ and $R$ are smooth, we may suppose the $U_s$ are ordered in such a way that the $U_{\\le s}$ (and hence the $R_{\\le s}$ ) are smooth.",
"The morphism $p_U$ is flat of relative dimension 5, and so there is a commutative diagram of complexes (where rows are exact triangles) $ \\begin{array}[c]{cccccc}z_{i+5}(R_{\\le s-1},\\ast ) &\\rightarrow & z_{i+5}(R_{\\le s},\\ast ) &\\rightarrow & z_{i+5}(R_s,\\ast ) &\\rightarrow \\\\&&&&&\\\\\\uparrow {\\scriptstyle (p_{U_{\\le s-1}})^\\ast }&& \\uparrow {\\scriptstyle p_{U_{\\le s}}^\\ast } && \\uparrow {\\scriptstyle (p_{U_s})^\\ast } \\\\&&&&&\\\\z_{i}(U_{\\le s-1},\\ast ) &\\rightarrow & z_{i}(U_{\\le s},\\ast ) &\\rightarrow & z_{i}(U_s,\\ast ) &\\rightarrow \\\\\\end{array}$ Also, given a codimension $\\ell $ subvariety $M\\subset R$ , let $z_i^M(R_{\\le k},\\ast )\\subset z_i(R_{\\le k},\\ast )$ denote the subcomplex formed by cycles in general position with respect to $M$ .",
"The inclusion $z_i^M(R_{\\le k},\\ast )\\subset z_i(R_{\\le k},\\ast )$ is a quasi–isomorphism [5].",
"The projection formula for higher Chow groups [5] gives a commutative diagram up to homotopy $ \\begin{array}[c]{ccccc}z_{i+5-\\ell }(R_{\\le s-1},\\ast ) & \\xrightarrow{} & z_{i+5-\\ell }(R_{\\le s},\\ast ) & \\rightarrow & z_{i+5-\\ell }(R_s,\\ast ) \\rightarrow \\\\&&&&\\\\\\uparrow {\\scriptstyle \\cdot {M\\vert _{R_{\\le s-1}}}}&&\\uparrow {\\scriptstyle \\cdot M\\vert _{R_{\\le s}}} &&\\uparrow {\\scriptstyle \\cdot M\\vert _{R_s}} \\\\&&&&\\\\z_{i+5}^{{M}}(R_{\\le s-1},\\ast ) & \\xrightarrow{} & \\ \\ z_{i+5}^M(R_{\\le s},\\ast ) & \\rightarrow & z_{i+5}^M(R_s,\\ast ) \\rightarrow \\ .\\\\\\end{array}$ In particular, these diagrams exist for $M$ being (a representative of) the classes $h^r, b\\in A^\\ast (R)$ that make up the definition of the map $\\Phi ^j_i$ .",
"The result of the above remarks is a commutative diagram with long exact rows $ \\begin{array}[c]{ccccc}\\rightarrow A_{i}(R_{s},j+1)& \\xrightarrow{}&\\ A_{i}(R_{\\le s-1},j )& \\xrightarrow{}& A_{i}(R_{\\le s},j)\\ \\rightarrow \\\\&&&&\\\\\\ \\ \\ \\ \\ \\ \\ \\ \\uparrow {\\scriptstyle \\Phi _i^{j+1}\\vert _{R_s}}&&\\ \\ \\ \\ \\ \\ \\ \\ \\uparrow {\\scriptstyle \\Phi _i^j\\vert _{R_{\\le s-1}}} &&\\ \\ \\ \\uparrow {\\scriptstyle \\Phi _i^j\\vert _{R_{\\le s}}} \\\\&&&&\\\\\\rightarrow \\bigoplus A_{i-k}^{}(U_s,j+1) &\\rightarrow & \\bigoplus A_{i-k}^{}(U_{\\le s-1},j) &\\xrightarrow{} & \\ \\bigoplus A_{i-k}(U_{\\le s},j)\\ \\rightarrow \\ .\\\\\\end{array}$ Applying noetherian induction and the five–lemma, one is reduced to proving the claim for $R_s\\rightarrow U_s$ .",
"But $R_s$ is isomorphic to the product $U_s\\times R_u$ and the fibre $R_u$ is a linear variety (i.e., $R_u$ can be written as a finite disjoint union of affine spaces $\\mathbb {A}^r$ ).",
"Cutting up the fibre $R_u$ and using another commutative diagram with long exact rows, one is reduced to proving that $A_i(U_s,j)\\cong A_{i+r}(U_s\\times \\mathbb {A}^r,j)$ , which is the homotopy property for higher Chow groups [5].",
"This proves the claim, and hence (i).",
"(ii) The point is that there is also a version in homology of (i).",
"That is, for any $j\\in \\mathbb {N}$ there are isomorphisms $\\Phi ^h_j\\colon \\ \\ \\ \\bigoplus H_{j-2k}(U)\\ \\xrightarrow{}\\ H_j(R)\\ ,$ where $\\Phi ^h_j$ is defined as $ \\Phi ^h_j:= \\Bigl ( h^5\\circ (p_U)^\\ast , h^4\\circ (p_U)^\\ast , h^3\\circ (p_U)^\\ast , (b\\cdot h)\\circ (p_U)^\\ast , h^2\\circ (p_U)^\\ast ,b\\circ (p_U)^\\ast , h\\circ (p_U)^\\ast , (p_U)^\\ast \\Bigr )\\ .$ This is proven just as (i), using homology instead of higher Chow groups.",
"Cycle class maps fit into a commutative diagram $ \\begin{array}[c]{ccc}\\bigoplus A_{i-k}(U)& \\xrightarrow{}& A_i(R)\\\\&&\\\\\\downarrow &&\\downarrow \\\\&&\\\\\\bigoplus H_{2i-2k}(U)& \\xrightarrow{}&\\ \\ H_{2i}(R)\\ .\\\\\\end{array}$ As the horizontal arrows are isomorphisms, and the left vertical arrow is injective (lemma REF ), the right vertical arrow is injective as well.",
"This proves statement (ii).",
"For later use, we record the following result: Corollary 5.3 One has $ H_j(R)=0\\ \\ \\ \\hbox{for\\ $j$\\ odd}\\ .$ The threefold $X$ has $H_{j-1}(X)=\\mathbb {Q}$ for any $j-1$ even, and the Grassmannian $Gr_1$ has $H_j(Gr_1)=0$ for $j$ odd.",
"The exact sequence $ H_j(Gr_1)\\ \\rightarrow \\ H_j(U)\\ \\rightarrow H_{j-1}(X)\\ \\rightarrow \\ H_{j-1}(Gr_1)\\ \\rightarrow \\ $ implies that the open $U:=Gr_1\\setminus X$ has no odd–degree cohomology.",
"In view of the isomorphism (REF ), $R$ has no odd–degree homology either.",
"Let us now turn to the fibration $F\\rightarrow X$ , where $F$ (but not $X$ ) is singular.",
"Proposition 5.4 Let $p_X\\colon F\\rightarrow X$ be as in proposition REF .",
"Let $h^k\\colon A_i(F)\\rightarrow A_{i-k}(F)$ denote the operation of intersecting with a hyperplane section.",
"(i) There are isomorphisms $ \\begin{split} &\\Phi _0\\colon \\ \\ \\ A_{0}(X)\\ \\xrightarrow{}\\ A_0(F)\\ ,\\\\&\\Phi _1 \\colon \\ \\ \\ A_1(X)\\oplus A_{0}(X)\\ \\xrightarrow{}\\ A_1(F)\\ ,\\\\& \\Phi _2\\colon \\ \\ \\ A_2(X)\\oplus A_1(X) \\oplus A_0(X)^{\\oplus 2} \\ \\xrightarrow{}\\ A_2(F)\\ \\ ,\\\\&\\Phi _3\\colon \\ \\ \\ A_3(X)^{}\\oplus A_2(X)^{}\\oplus A_1(X)^{\\oplus 2}\\oplus A_0(X)^{\\oplus 2} \\ \\xrightarrow{}\\ A_3(F)\\ ,\\\\&\\Phi _4\\colon \\ \\ \\ A_3(X)\\oplus A_2(X)^{\\oplus 2}\\oplus A_1(X)^{\\oplus 2}\\oplus A_0(X)^{\\oplus 2}\\ \\xrightarrow{}\\ A_4(F)\\ .\\\\\\end{split}$ The maps $\\Phi _j$ are defined as $ \\begin{split} &\\Phi _0:= h^5\\circ (p_X)^\\ast \\ ,\\\\&\\Phi _1 := \\Bigl ( h^5\\circ (p_X)^\\ast , h^4\\circ (p_X)^\\ast \\Bigr )\\ ,\\\\& \\Phi _2:=\\Bigl ( h^5\\circ (p_X)^\\ast , h^4\\circ (p_X)^\\ast , h^3\\circ (p_X)^\\ast , (b\\cdot h)\\circ (p_X)^\\ast \\Bigr )\\ ,\\\\& \\Phi _3:=\\Bigl ( h^5\\circ (p_X)^\\ast , h^4\\circ (p_X)^\\ast , h^3\\circ (p_X)^\\ast , (b\\cdot h)\\circ (p_X)^\\ast , h^2\\circ (p_X)^\\ast ,b\\circ (p_X)^\\ast \\Bigr )\\ ,\\\\& \\Phi _4:=\\Bigl ( h^4\\circ (p_X)^\\ast , h^3\\circ (p_X)^\\ast , (b\\cdot h)\\circ (p_X)^\\ast , h^2\\circ (p_X)^\\ast ,b\\circ (p_X)^\\ast , h\\circ (p_X)^\\ast , (p_X)^\\ast (-)\\cap e \\Bigr )\\ ,\\\\\\end{split} $ where $b\\in A^2(F)$ is a class made explicit in the proof (and $b\\colon A_i(R)\\rightarrow A_{i-2}(R)$ denotes the operation of intersecting with $b$ , and similarly for $b\\cdot h$ ), and $e\\in A_7(F)$ is a class made explicit in the proof (and the last $(p_X)^\\ast (-)$ means pullback of operational Chow cohomology).",
"(ii) The maps $\\Phi _i$ induce isomorphisms of homologically trivial cycles $ \\Phi _i\\colon \\ \\ \\ \\bigoplus A_{i-k}^{hom}(X)\\ \\xrightarrow{}\\ A_i^{hom}(F)\\ .$ (i) The element $b$ is defined just as in proposition REF : $ b:= (\\iota _F)^\\ast \\Bigl ( \\bigl ( Gr_1\\times c_2(T)\\bigr )\\vert _Q\\Bigr )\\ \\ \\ \\in \\ A^2(F)\\ ,$ where $\\iota _F\\colon F\\hookrightarrow Q$ denotes the inclusion morphism, and $A^\\ast (F)$ is operational Chow cohomology of the singular variety $F$ .",
"To define the element $e\\in A_7(F)$ , we return to the description of the fibres of $p_X\\colon F\\rightarrow X$ given in [10].",
"By definition of the variety $F$ , we have $ F=\\bigl \\lbrace (x,y) \\in Gr_1\\times Gr_2^\\vee \\ \\vert \\ x\\in X\\ ,\\ (x,y)\\in \\sigma \\bigr \\rbrace \\ ,$ where $\\sigma \\subset \\mathbb {P}\\times \\mathbb {P}^\\vee $ is the incidence divisor.",
"As in [10], given $x\\in \\mathbb {P}$ (or $y\\in \\mathbb {P}^\\vee $ ) let $ x_i:=\\phi _i^{-1}(x)\\in \\mathbb {P}(\\wedge ^2 V)$ (resp.",
"$y_i:= \\phi _i^\\vee (y)\\in \\mathbb {P}(\\wedge ^2 V^\\vee )$ ).",
"For a point $\\omega \\in \\mathbb {P}(\\wedge ^2 V)$ (or in $\\mathbb {P}(\\wedge ^2 V^\\vee )$ ), let $rk(\\omega )$ denote the rank of $\\omega $ considered as a skew–form; the rank of $\\omega $ is either 2 or 4.",
"As explained in loc.",
"cit., the expression for $F$ can be rewritten as $ \\begin{split} F&=\\bigl \\lbrace (x,y)\\in \\mathbb {P}\\times \\mathbb {P}^\\vee \\ \\vert \\ rk(x_i)=2 ,\\ rk(y_2)=2 ,\\ y\\in \\ H_{x_2}\\bigr \\rbrace \\ \\\\&= \\bigl \\lbrace (x,y) \\in \\ \\mathbb {P}\\times \\mathbb {P}^\\vee \\ \\vert \\ rk(x_i)=2 ,\\ rk(y_2)=2 ,\\ A_y\\cap \\ker (x_2)\\ne \\emptyset \\bigr \\rbrace \\ \\subset \\ \\mathbb {P}(\\wedge ^2 V)\\times \\mathbb {P}(\\wedge ^2 V^\\vee ) .",
"\\\\\\end{split} $ (Here, $A_y\\subset V^\\vee $ denotes the 2–dimensional subspace corresponding to $y$ , and $\\ker (x_2)\\subset V^\\vee $ denotes the kernel of the skew–form $x_2\\in \\mathbb {P}(\\wedge ^2 V)$ .)",
"The stratification of the fibres $F_x$ as given in loc.",
"cit.",
"can be done relatively over $X$ .",
"That is, we define $ \\mathcal {Z}:= \\bigl \\lbrace (x,y)\\ \\in \\mathbb {P}\\times \\mathbb {P}^\\vee \\ \\vert \\ (x,y)\\in F\\ ,\\ A_y\\subset \\ker (x_2)\\bigr \\rbrace \\ \\ \\ \\subset \\ F\\ .$ The intersection of $\\mathcal {Z}$ with a fibre $F_x$ is the variety $Z\\cong \\mathbb {P}^2$ of [10], and so $\\mathcal {Z}\\rightarrow X$ is a $\\mathbb {P}^2$ –fibration.",
"The complement $F^0:=F\\setminus \\mathcal {Z}$ can be described as $ F^0= \\bigl \\lbrace (x,y) \\in \\ \\mathbb {P}\\times \\mathbb {P}^\\vee \\ \\vert \\ rk(x_i)=2 ,\\ rk(y_2)=2 ,\\ \\dim (A_y\\cap \\ker (x_2))=1\\bigr \\rbrace \\ .$ The natural morphism $ F^0\\rightarrow X$ factors as $ F^0\\ \\rightarrow \\ \\mathcal {W}\\ \\rightarrow \\ X\\ ,$ where $ \\mathcal {W}:= \\bigl \\lbrace (x,s) \\in X\\times \\mathbb {P}(V^\\vee )\\ \\vert \\ s\\in \\mathbb {P}(\\ker (x_2))\\bigr \\rbrace \\ ,$ and $\\mathcal {W}\\rightarrow X$ is a $\\mathbb {P}^2$ –fibration.",
"As explained in loc.",
"cit., over each $x\\in X$ the morphism from $(F^0)_x$ to $\\mathcal {W}_x$ is a fibration with fibres isomorphic to $\\mathbb {P}^3\\setminus \\mathbb {P}^1$ .",
"Let $\\mathcal {W}^\\prime \\subset \\mathcal {W}$ be the divisor $ \\mathcal {W}^\\prime := \\bigl \\lbrace (x,s) \\in X\\times \\mathbb {P}(V^\\vee )\\ \\vert \\ s\\in \\mathbb {P}(\\ker (x_2))\\cap h\\bigr \\rbrace \\ ,$ where $h\\subset \\mathbb {P}(V^\\vee )$ is a hyperplane section.",
"The morphism $\\mathcal {W}^\\prime \\rightarrow X$ is a $\\mathbb {P}^1$ –fibration.",
"The class $e\\in A_7(F)$ is now defined as $ e:= (p_X\\vert _{F^0})^{-1}(\\mathcal {W}^\\prime )\\ \\ \\ \\in \\ A_7(F^0)\\cong A_7(F)\\ ,$ where $A_7(F^0)\\cong A_7(F)$ for dimension reasons.",
"We observe that $e\\in A_7(F)$ is not proportional to the class of a hyperplane section $h\\in A_7(F)$ .",
"(Indeed, let $x\\in X$ and $w\\in (\\mathcal {W}\\setminus \\mathcal {W}^\\prime )_x$ and let $\\nu \\in A^1((F^0)_x)$ be the tautological class with respect to the projective bundle structure of $(F^0)_x\\rightarrow \\mathcal {W}_x$ .",
"Then $C:=((F^0)_x)_w\\cdot \\nu ^2\\cong \\mathbb {P}^1$ is an effective curve disjoint from $e$ , whereas $h\\cap C$ has strictly positive degree.)",
"Since we know that the fibres $F_x$ have $A_4(F_x)\\cong \\mathbb {Q}^2$ (proposition REF ), it follows that $ h\\vert _{F_x}\\ ,\\ e\\vert _{F_x}\\ \\ \\ \\in \\ A_4(F_x) $ generate $A_4(F_x)$ .",
"(Here, $e\\vert _{F_x}\\in A_4(F_x)$ is defined as $\\tau ^\\ast (e)\\in A_4(F_x)$ where $\\tau ^\\ast $ is the refined Gysin homomorphism [12] associated to the regular morphism $\\tau \\colon x\\hookrightarrow X$ .)",
"The (5–dimensional) fibres $F_x$ of the fibration $p_X\\colon F\\rightarrow X$ thus verify $ A_i(F_x)= {\\left\\lbrace \\begin{array}{ll} \\mathbb {Q}\\cdot h^{5-i}\\vert _{F_x} &\\ \\hbox{if}\\ i=0,1,5\\ ,\\\\\\mathbb {Q}\\cdot h^3\\vert _{F_x} \\oplus \\mathbb {Q}\\cdot (b\\cdot h)\\vert _{F_x} &\\ \\hbox{if}\\ i=2\\ ,\\\\\\mathbb {Q}\\cdot h^2\\vert _{F_x} \\oplus \\mathbb {Q}\\cdot b\\vert _{F_x} &\\ \\hbox{if}\\ i=3\\ ,\\\\\\mathbb {Q}\\cdot h\\vert _{F_x}\\oplus \\mathbb {Q}\\cdot e\\vert _{F_x} &\\ \\hbox{if}\\ i=4\\ .\\\\\\end{array}\\right.",
"}$ We would like to prove proposition REF following the strategy of proposition REF , i.e.",
"invoking higher Chow groups.",
"The only delicate point is that $F$ is singular, and we need to make sense of the operation of “capping with $h^k$ (or $b$ )” on higher Chow groups.",
"Since this seems difficultIt is not clear whether operational Chow cohomology operates on higher Chow groups of a singular variety, which is a nuisance., we will prove proposition REF without using higher Chow groups.",
"The piecewise triviality of the fibration $p_X$ means that there exist opens $ F_0=F\\setminus F_{\\ge 1}\\ ,\\ \\ \\ X_0=X\\setminus X_{\\ge 1}\\ $ such that $F_0$ is isomorphic to the product $X_0\\times F_x$ , and $F_{\\ge 1}\\rightarrow X_{\\ge 1}$ is a piecewise trivial fibration (with fibre $F_x$ ).",
"There is a commutative diagram with long exact rows $ \\begin{array}[c]{ccccc}\\rightarrow A_{i}(F_{\\ge 1})& \\xrightarrow{}&\\ A_{i}(F )& \\xrightarrow{}& A_{i}(F_{0})\\ \\rightarrow 0 \\\\&&&&\\\\\\ \\ \\ \\ \\ \\ \\ \\ \\uparrow {\\scriptstyle \\Phi _i^{}\\vert _{F_{\\ge 1}}}&&\\ \\ \\ \\ \\ \\ \\ \\ \\uparrow {\\scriptstyle \\Phi _i{{}}} &&\\ \\ \\ \\uparrow {\\scriptstyle \\Phi _i\\vert _{F_0}} \\\\&&&&\\\\\\rightarrow \\bigoplus A_{i-k}^{}(X_{\\ge 1}) &\\rightarrow & \\bigoplus A_{i-k}^{}(X_{}) &\\xrightarrow{} & \\ \\bigoplus A_{i-k}(X_{0})\\ \\rightarrow \\ 0\\ .\\\\\\end{array}$ The arrow $\\Phi _i\\vert _{F_0}$ is an isomorphism, because the fibre $F_x$ is a linear variety in the sense of [45], which implies (by [45], cf.",
"also [46]) that the natural map $ \\bigoplus _{k+\\ell =i} A_k(M)\\otimes A_\\ell (F_x)\\ \\rightarrow \\ A_i(M\\times F_x) $ is an isomorphism for any variety $M$ , and thus in particular for $M=X_0$ .",
"By noetherian induction, we may assume that $\\Phi _i\\vert _{F_{\\ge 1}}$ is surjective.",
"Contemplating the diagram, we find that the middle arrow $\\Phi _i$ is also surjective.",
"It remains to prove injectivity.",
"To this end, let us define a map $ \\begin{split} \\Psi ^\\prime _i\\colon \\ \\ \\ A_i(F)\\ &\\rightarrow \\ \\bigoplus A_{i-k}(X)\\ ,\\\\a\\ &\\mapsto \\ \\Bigl ( (p_X)_\\ast (a), (p_X)_\\ast (h\\cap a), (p_X)_\\ast (h^2\\cap a), (p_X)_\\ast (h^2\\cap a),\\ldots , (p_X)_\\ast (h^5\\cap a)\\Bigr )\\ .\\\\\\end{split}$ Using the above–mentioned isomorphism $ \\bigoplus _{k+\\ell =i} A_k(X_0)\\otimes A_\\ell (F_x)\\ \\xrightarrow{}\\ A_i(X_0\\times F_x) \\ ,$ one finds that $ \\Psi ^\\prime _i\\vert _{F_0}\\circ \\Phi _i\\vert _{F_0}$ is given by an invertible diagonal matrix.",
"Dividing by some appropriate numbers, one can find $\\Psi _i$ such that $\\Psi _i\\vert _{F_0}\\circ \\Phi _i\\vert _{F_0}$ is the identity.",
"Using the projection formula, we see that there is a commutative diagram $ \\begin{array}[c]{ccccccc}\\rightarrow & \\bigoplus A_{i-k}^{}(X_{\\ge 1}) &\\rightarrow & \\bigoplus A_{i-k}^{}(X_{}) &\\xrightarrow{} & \\ \\bigoplus A_{i-k}(X_{0})& \\rightarrow \\ 0\\ \\\\&&&&&&\\\\& \\ \\ \\ \\ \\ \\ \\ \\ \\uparrow {\\scriptstyle \\Psi _i^{}\\vert _{F_{\\ge 1}}}&&\\ \\ \\ \\ \\ \\ \\ \\ \\uparrow {\\scriptstyle \\Psi _i{{}}} &&\\ \\ \\ \\uparrow {\\scriptstyle \\Psi _i\\vert _{F_0}} & \\\\&&&&&&\\\\\\rightarrow & A_{i}(F_{\\ge 1})& \\xrightarrow{}&\\ A_{i}(F )& \\xrightarrow{}& A_{i}(F_{0}) & \\rightarrow 0 \\\\&&&&&&\\\\& \\ \\ \\ \\ \\ \\ \\ \\ \\uparrow {\\scriptstyle \\Phi _i^{}\\vert _{F_{\\ge 1}}}&&\\ \\ \\ \\ \\ \\ \\ \\ \\uparrow {\\scriptstyle \\Phi _i{{}}} &&\\ \\ \\ \\uparrow {\\scriptstyle \\Phi _i\\vert _{F_0}} & \\\\&&&&&&\\\\\\rightarrow & \\bigoplus A_{i-k}^{}(X_{\\ge 1}) &\\rightarrow & \\bigoplus A_{i-k}^{}(X_{}) &\\xrightarrow{} & \\ \\bigoplus A_{i-k}(X_{0})& \\rightarrow \\ 0\\ .\\\\\\end{array}$ We now make the following claim: Claim 5.5 For any given $i$ , there exists a polynomial $p_i(x)\\in \\mathbb {Q}[x]$ such that $ a=p_i(\\Psi _i\\circ \\Phi _i)(a) \\ \\ \\ \\forall a\\in \\bigoplus A_{i-k}^{}(X_{}) \\ .$ Clearly, the claim implies injectivity of $\\Phi _i$ .",
"To prove the claim, we apply noetherian induction.",
"Given $a\\in \\bigoplus A_{i-k}^{}(X_{}) $ , we know that $ \\Psi _i\\vert _{F_0}\\circ \\Phi _i\\vert _{F_0}$ acts as the identity on the restriction $a\\vert _{X_0}\\in \\bigoplus A_{i-k}^{}(X_{0})$ .",
"It follows that we can write $ a - (\\Psi _i\\circ \\Phi _i)(a)= b\\ \\ \\ \\hbox{in}\\ \\bigoplus A_{i-k}^{}(X_{})\\ ,$ where $b$ is in the image of the pushforward map $ \\bigoplus A_{i-k}^{}(X_{\\ge 1}) \\rightarrow \\bigoplus A_{i-k}^{}(X_{}) $ .",
"By noetherian induction, we may assume the claim is true for the piecewise trivial fibration $F_{\\ge 1}\\rightarrow X_{\\ge 1}$ , and so there is a polynomial $q_i$ such that $ b=q_i (\\Psi _i\\circ \\Phi _i)(b)\\ \\ \\ \\hbox{in}\\ \\bigoplus A_{i-k}(X)\\ .$ Plugging this in (REF ), we find that $ a - (\\Psi _i\\circ \\Phi _i)(a)= q_i (\\Psi _i\\circ \\Phi _i) \\Bigl (a - (\\Psi _i\\circ \\Phi _i)(a)\\Bigr ) \\ \\ \\ \\hbox{in}\\ \\bigoplus A_{i-k}^{}(X_{}) \\ .$ It follows that $ a=p_i(\\Psi _i\\circ \\Phi _i)(a) \\ \\ \\ \\hbox{in}\\ \\bigoplus A_{i-k}^{}(X_{}) \\ ,$ where the polynomial $p_i$ is defined as $ p_i(x):= q_i(x) - x q_i(x) +x\\ \\ \\in \\mathbb {Q}[x]\\ .$ (ii) As in proposition REF , one can also prove a homology version of (i).",
"That is, for any $j\\in \\mathbb {N}$ there are isomorphisms $\\Phi ^h_j\\colon \\ \\ \\ \\bigoplus H_{j-2k}(X)\\ \\xrightarrow{}\\ H_j(F)\\ ,$ where $\\Phi ^h_j$ is now defined as $ \\begin{split} \\Phi ^h_j:= \\Bigl ( h^5\\circ (p_X)^\\ast , h^4\\circ (p_X)^\\ast , h^3\\circ (p_X)^\\ast , (b\\cdot h)\\circ (p_X)^\\ast ,h^2\\circ (p_X)^\\ast ,b\\circ (p_X)^\\ast ,&\\\\h\\circ (p_X)^\\ast , (p_X)^\\ast (-)\\cap e, (p_X)^\\ast & \\Bigr )\\ .",
"\\\\\\end{split} $ This is proven just as (i), using homology instead of higher Chow groups.",
"Cycle class maps fit into a commutative diagram $ \\begin{array}[c]{ccc}\\bigoplus A_{i-k}(X)& \\xrightarrow{}& A_i(F)\\\\&&\\\\\\downarrow &&\\downarrow \\\\&&\\\\\\bigoplus H_{2i-2k}(X)& \\xrightarrow{}&\\ \\ H_{2i}(F)\\ .\\\\\\end{array}$ Horizontal arrows being isomorphisms, this proves (ii).",
"Remark 5.6 Comparing propositions REF and REF , we observe that the only difference is the class $e\\in A_7(F)$ appearing in proposition REF but not in REF .",
"This “extra class” $e$ appears because of the singularities: the fibres of $p\\colon Q\\rightarrow G$ are smooth over the open $U\\subset Gr_1$ , but degenerate to singular fibres over $X\\subset Gr_1$ (cf.",
"remark REF ), and this causes an extra Weil divisor class $e$ to appear in the singular fibres.",
"This observation will be key to the proof of theorem REF ."
],
[
"Main result",
"Theorem 6.1 Let $X,Y$ be a pair of GPK$^3$ double mirrors.",
"Then there is an isomorphism $ h(X)\\cong h(Y)\\ \\ \\ \\hbox{in}\\ \\mathcal {M}_{{\\rm rat}}\\ .$ The proof is a four–step argument, which exploits that the threefolds $ \\begin{split} X&:= Gr_1\\cap Gr_2\\ \\ \\subset \\ \\mathbb {P}\\ ,\\\\Y&:=Gr_1^\\vee \\cap Gr_2^\\vee \\ \\ \\subset \\ \\mathbb {P}^\\vee \\ \\\\\\end{split}$ are geometrically related as in proposition REF .",
"In essence, the argument is similar to the proof that $X,Y$ are L–equivalent [10], by applying “cut and paste” to the diagram of proposition REF .",
"Here is an overview of the proof.",
"Let $ Q:= \\sigma \\times _{\\mathbb {P}\\times \\mathbb {P}^\\vee } (Gr_1\\times Gr_2^\\vee ) $ be the 11–dimensional intersection as in proposition REF .",
"Assuming $Q$ is non–singular, we prove there exist isomorphisms of Chow groups $ A^i_{hom}(X)\\ \\xrightarrow{}\\ A^{i+4}_{hom}(Q)\\ \\ \\ \\hbox{for\\ all\\ }i\\ .$ This is done in step 1 (for $i=2,3$ ) and step 2 (for $i=0,1$ ), and relies on the isomorphisms for the piecewise trivial fibrations established in the prior section.",
"In step 3, the isomorphism (REF ) is upgraded to an isomorphism of Chow motives $ h^3(X) \\cong h^{11}(Q)(-4)\\ \\ \\ \\hbox{in}\\ \\mathcal {M}_{\\rm rat}\\ .$ As $X$ and $Y$ are symmetric, this implies an isomorphism of Chow motives $h^3(X)\\cong h^3(Y)$ , and hence also $h(X)\\cong h(Y)$ .",
"Finally, in step 4 we show that we may “spread out” this isomorphism to all GPK$^3$ double mirrors.",
"Step 1: an isomorphism of Chow groups.",
"In this first step, we assume the $Gr_i$ are sufficiently general, so that $Q$ is non–singular (there is no loss in generality; the degenerate case where $Q$ may be singular will be taken care of in step 4 below).",
"The goal of this first step will be to construct an isomorphism between certain Chow groups of $X$ and $Y$ : Proposition 6.2 There exist correspondences $\\Gamma \\in A^{7}(X\\times Q)$ , $\\Psi \\in A^{7}(Y\\times Q)$ inducing isomorphisms $ \\begin{split} \\Gamma _\\ast \\colon \\ \\ \\ A^i_{hom}(X)\\ \\xrightarrow{}\\ A^{i+4}_{hom}(Q)\\ \\ \\ \\hbox{for}\\ i=2,3 ,\\\\\\Psi _\\ast \\colon \\ \\ \\ A^i_{hom}(Y)\\ \\xrightarrow{}\\ A^{i+4}_{hom}(Q)\\ \\ \\ \\hbox{for}\\ i=2,3 .\\\\\\end{split}$ Before proving this proposition, let us first establish two lemmas (in these lemmas, we continue to assume $X,Y$ are sufficiently general, so that $Q$ is smooth): Lemma 6.3 The pushforward map $ (\\iota _F)_\\ast \\colon \\ \\ \\ A_i^{hom}(F)\\ \\rightarrow \\ A_i^{hom}(Q) $ is surjective, for all $i$ .",
"As before, let $R$ denote the open complement $R:=Q\\setminus F$ .",
"There is a commutative diagram with exact rows $ \\begin{array}[c]{cccccccc}&\\rightarrow & A_i(F) &\\rightarrow & A_i(Q) &\\rightarrow & A_i(R)&\\rightarrow 0\\\\&&&&&&&\\\\&&\\downarrow &&\\downarrow &&\\downarrow &\\\\&&&&&&&\\\\0&\\rightarrow & H_{2i}(F,\\mathbb {Q}) &\\rightarrow & H_{2i}(Q,\\mathbb {Q}) &\\rightarrow &\\ \\ H_{2i}(R,\\mathbb {Q})\\ ,&\\\\\\end{array}$ where vertical arrows are cycle class maps.",
"Here, the lower left entry is 0 because $R$ has no odd–degree homology (corollary REF ).",
"The lemma follows from the fact that the right vertical arrow is injective, which is proposition REF (ii).",
"Lemma 6.4 Let $e\\in A_7(F)$ be as in proposition REF .",
"The composition $ A^i_{hom}(X)\\ \\xrightarrow{}\\ A^{i}_{hom}(F)\\ \\xrightarrow{}\\ A_{7-i}^{hom}(F)\\ \\xrightarrow{}\\ A_{7-i}^{hom}(Q)=A^{i+4}_{hom}(Q) $ is an isomorphism for $i=2,3$ .",
"(As before, $A^\\ast (F)$ denotes operational Chow cohomology of the singular variety $F$ .)",
"Let us treat the case $i=3$ in detail.",
"Proposition REF gives us an isomorphism $ \\Phi \\colon \\ \\ \\ A_3(X)\\oplus A_2(X)^{\\oplus 2}\\oplus A_1(X)^{\\oplus 2}\\oplus A_0(X)^{\\oplus 2}\\ \\xrightarrow{}\\ A_4(F)\\ ,\\\\ $ where $\\Phi :=\\Phi _4$ is defined as $ \\Phi = \\Bigl ( h^4\\circ (p_X)^\\ast , h^3\\circ (p_X)^\\ast , (b\\cdot h)\\circ (p_X)^\\ast , h^2\\circ (p_X)^\\ast ,b\\circ (p_X)^\\ast , h\\circ (p_X)^\\ast , (p_X)^\\ast (-)\\cap e \\Bigr )\\ .\\\\ $ We want to single out the part in $A_4(F)$ coming from the “extra class” $e\\in A_7(F)$ .",
"That is, we write the isomorphism $\\Phi $ as a decomposition $ A_4^{}(F) = A^\\perp \\oplus A \\ ,$ where $ \\begin{split}A&:= (p_X)^\\ast A_0{}(X)\\cap e\\ ,\\\\A^\\perp &:= \\hbox{Im}\\Bigl ( A_3(X)\\oplus A_2(X)^{\\oplus 2}\\oplus A_1(X)^{\\oplus 2}\\oplus A_0(X) \\ \\xrightarrow{}\\ A_4(F)\\Bigr )\\ ,\\\\\\Phi ^\\perp &:= \\Bigl ( h^4\\circ (p_X)^\\ast , h^3\\circ (p_X)^\\ast , (b\\cdot h)\\circ (p_X)^\\ast , h^2\\circ (p_X)^\\ast ,b\\circ (p_X)^\\ast , h\\circ (p_X)^\\ast \\Bigr )\\ .\\\\ \\end{split}$ The decomposition (REF ) also exists in cohomology, and so there is an induced decomposition $ A_4^{hom}(F) = A^\\perp _{hom} \\oplus A_{hom} \\ ,$ where we put $ A_{hom}:= A\\cap A_4^{hom}(F)\\ ,\\ \\ \\ A_{hom}^\\perp := A^\\perp \\cap A_4^{hom}(F)\\ .$ We now claim that there is a commutative diagram with exact rows $ \\begin{array}[c]{ccccc}A_{4}(R,1)& \\xrightarrow{}&\\ A_{4}(F)& \\xrightarrow{}& A_{4}(Q)\\ \\rightarrow \\\\&&&&\\\\\\uparrow {\\scriptstyle \\Phi _4^1}&&\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\uparrow {\\scriptstyle \\Phi ^\\perp } &&\\uparrow \\\\&&&&\\\\\\bigoplus A_{k}^{}(U,1) &\\xrightarrow{} & \\bigoplus A_{k}^{}(X) &\\xrightarrow{} & A_{0}(G)\\oplus A_1(G)^{\\oplus 2}\\ \\rightarrow \\ ,\\\\\\end{array}$ where $\\Phi _4^1$ is the isomorphism of proposition REF (and we use the shorthand $G:=Gr_1$ ).",
"Granting this claim, let us prove the lemma for $i=3$ .",
"The kernel of $(\\iota _F)_\\ast $ equals the image of the arrow $\\delta $ .",
"Since $\\Phi _4^1$ is an isomorphism, the image $\\hbox{Im}\\delta $ is contained in $\\hbox{Im}\\Phi ^\\perp =:A^\\perp $ .",
"In view of the decomposition (REF ), this implies injectivity $ (\\iota _F)_\\ast \\colon \\ \\ A\\ \\hookrightarrow \\ A_4(Q)\\ ,$ i.e.",
"the composition of lemma REF is injective for $i=3$ .",
"To prove surjectivity, let us consider a class $ b \\in A^\\perp _{hom}$ .",
"We know (from proposition REF (ii)) that $ b= \\Phi ^\\perp (\\beta )\\ , \\ \\ \\ \\beta \\in \\ A_1^{hom}(X)^{\\oplus 2}\\oplus A_0^{hom}(X)\\ .$ Referring to diagram (REF ), we see that $\\iota _\\ast (\\beta )$ must be 0 (for the Grassmannian $G$ has trivial Chow groups).",
"It follows that $\\beta $ is in the image of $\\delta _U$ , and so $b\\in \\hbox{Im}\\delta $ .",
"This shows that $ (\\iota _F)_\\ast ( A^\\perp _{hom})=0\\ ,$ and hence, in view of the decomposition (REF ), that $ (\\iota _F)_\\ast ( A_{hom})= (\\iota _F)_\\ast ( A_4^{hom}(F))\\ .",
"$ On the other hand, we know that $ (\\iota _F)_\\ast ( A_4^{hom}(F))= A_4^{hom}(Q)$ (lemma REF ), and so we get a surjection $ (\\iota _F)_\\ast \\colon \\ \\ A_{hom}\\ \\twoheadrightarrow \\ A_4^{hom}(Q)\\ .$ Combining (REF ) and (REF ), we see that the composition of lemma REF is an isomorphism for $i=3$ .",
"It remains to establish the claimed commutativity of diagram (REF ).",
"The morphism $p\\colon Q\\rightarrow G$ is equidimensional of relative dimension 5, and so there is a commutative diagram of complexes (where rows are exact triangles) $ \\begin{array}[c]{cccccc}z_{i+5}(F,\\ast ) &\\rightarrow & z_{i+5}(Q,\\ast ) &\\rightarrow & z_{i+5}(R,\\ast ) &\\rightarrow \\\\&&&&&\\\\\\uparrow {\\scriptstyle (p_X)^\\ast }&& \\uparrow {\\scriptstyle p^\\ast } && \\uparrow {\\scriptstyle (p_U)^\\ast } \\\\&&&&&\\\\z_{i}(X,\\ast ) &\\rightarrow & z_{i}(G,\\ast ) &\\rightarrow & z_{i+5}(U,\\ast ) &\\rightarrow \\\\\\end{array}$ Also, given a codimension $k$ subvariety $M\\subset Q$ , let $z_i^M(Q,\\ast )\\subset z_i(Q,\\ast )$ denote the subcomplex formed by cycles in general position with respect to $M$ .",
"The inclusion $z_i^M(Q,\\ast )\\subset z_i(Q,\\ast )$ is a quasi–isomorphism [5].",
"The diagram $ \\begin{array}[c]{cccccc}z_{i+5-k}(F,\\ast ) &\\rightarrow & z_{i+5-k}(Q,\\ast ) &\\rightarrow & z_{i+5-k}(R,\\ast ) &\\rightarrow \\\\ &&&&&\\\\&&\\uparrow {\\scriptstyle \\cdot M}&&\\uparrow {\\scriptstyle \\cdot M\\vert _R}\\\\&&&&&\\\\&& z_{i+5}^M(Q,\\ast ) &\\rightarrow & z_{i+5}^M(R,\\ast ) &\\rightarrow \\\\&&&&&\\\\&&\\downarrow {\\scriptstyle \\simeq }&&\\downarrow {\\scriptstyle \\simeq }\\\\&&&&&\\\\z_{i+5}(F,\\ast ) &\\rightarrow & z_{i+5}(Q,\\ast ) &\\rightarrow & z_{i+5}(R,\\ast ) &\\rightarrow \\\\\\end{array}$ (where $\\simeq $ indicates quasi–isomorphisms) defines an arrow in the homotopy category $ f_M\\colon \\ \\ z_{i+5}(F,\\ast )\\ \\rightarrow \\ z_{i+5-k}(F,\\ast )\\ .$ (The arrow $f_M$ represents “intersecting with $M$ ”.)",
"On the other hand, let $g\\colon \\widetilde{F}\\rightarrow F$ be a resolution of singularities, and let $\\widetilde{M}:=(\\iota _F\\circ g)^\\ast (M)\\in A^k(\\widetilde{F})$ .",
"The projection formula for higher Chow groups [5] gives a commutative diagram up to homotopy $ \\begin{array}[c]{ccc}z_{i+5-k}(\\widetilde{F},\\ast ) & \\xrightarrow{} & z_{i+5-k}(Q,\\ast )\\\\&&\\\\\\uparrow {\\scriptstyle \\cdot \\widetilde{M}}&&\\uparrow {\\scriptstyle \\cdot M}\\\\&&\\\\z_{i+5}^{\\vert \\widetilde{M}\\vert }(\\widetilde{F},\\ast ) & \\xrightarrow{} & \\ \\ z_{i+5}^M(Q,\\ast )\\ ,\\\\\\end{array}$ and so there is also a commutative diagram up to homotopy $ \\begin{array}[c]{ccc}z_{i+5-k}(\\widetilde{F},\\ast ) & \\xrightarrow{} & z_{i+5-k}(F,\\ast )\\\\&&\\\\\\uparrow {\\scriptstyle \\cdot \\widetilde{M}}&&\\uparrow {\\scriptstyle f_M}\\\\&&\\\\z_{i+5}^{}(\\widetilde{F},\\ast ) & \\xrightarrow{} & \\ \\ z_{i+5}(F,\\ast )\\ .\\\\\\end{array}$ In particular, this shows that $ f_M = (\\iota _F)^\\ast (M)\\cap (-)\\colon \\ \\ \\ A_{i+5}(F)\\ \\rightarrow \\ A_{i+5-k}(F) \\ ,$ where we consider $ (\\iota _F)^\\ast (M)\\in A^k(F)$ as an element in operational Chow cohomology.",
"Combining the above remarks, one obtains a commutative diagram with long exact rows $ \\begin{array}[c]{ccccc}A_{i+5-k}(R,1)& \\xrightarrow{}&\\ A_{i+5-k}(F)& \\xrightarrow{}& A_{i+5-k}(Q)\\ \\rightarrow \\\\&&&&\\\\\\ \\ \\ \\uparrow {\\scriptstyle \\cdot M\\vert _R}&&\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\uparrow {\\scriptstyle (\\iota _F)^\\ast (M)\\cap (-)} &&\\uparrow \\\\&&&&\\\\A_{i+5}(R,1)& \\xrightarrow{}&\\ A_{i+5}(F)& \\xrightarrow{}& A_{i+5}(Q)\\ \\rightarrow \\\\&&&&\\\\\\ \\ \\ \\uparrow {\\scriptstyle (p_U)^\\ast }&&\\ \\ \\uparrow {\\scriptstyle (p_X)^\\ast } &&\\ \\ \\ \\uparrow {\\scriptstyle p^\\ast }\\\\&&&&\\\\A_{i}^{}(U,1) &\\rightarrow & A_{i}^{}(X) &\\xrightarrow{} & A_{i}(G)\\ \\rightarrow \\ .\\\\\\end{array}$ In particular, these diagrams exist for $M$ being (a representative of) the classes $h^j, b\\cdot h, b \\in A^\\ast (Q)$ that make up the definition of the map $\\Phi ^\\perp $ .",
"It follows there is a commutative diagram with long exact rows $ \\begin{array}[c]{ccccc}A_{4}(R,1)& \\xrightarrow{}&\\ A_{4}(F)& \\xrightarrow{}& A_{4}(Q)\\ \\rightarrow \\\\&&&&\\\\\\uparrow {}&&\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\uparrow {\\scriptstyle \\Phi ^\\perp } &&\\uparrow \\\\&&&&\\\\\\bigoplus A_{k}^{}(U,1) &\\rightarrow & \\bigoplus A_{k}^{}(X) &\\xrightarrow{} & \\bigoplus A_{k}(G)\\ \\rightarrow \\ ,\\\\\\end{array}$ The $i=2$ case of the lemma is proven similarly: using proposition REF , we can write $ A_5^{hom}(F) = A_5^\\perp \\oplus (p_X)^\\ast A^2_{hom}(X)\\cap e \\ .$ Here $A_5^\\perp $ is $ A_5^\\perp = \\hbox{Im}\\Bigl ( A^3_{hom}(X)^{}\\oplus A^2_{hom}(X)^{} \\ \\xrightarrow{}\\ A_5(F)\\Bigr )\\ ,$ where $\\Phi ^\\perp $ is defined as $ \\Phi ^\\perp := \\Bigl ( (p_X)^\\ast (-), h\\cap (p_X)^\\ast (-)\\Bigr )\\ .$ As above, there is a commutative diagram with long exact rows $ \\begin{array}[c]{ccccc}A_{5}(R,1)& \\xrightarrow{}&\\ A_{5}(F)& \\xrightarrow{}& A_{5}(Q)\\ \\rightarrow \\\\&&&&\\\\\\uparrow {}&&\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\uparrow {\\scriptstyle \\Phi ^\\perp } &&\\uparrow \\\\&&&&\\\\A_{0}^{}(U,1)\\oplus A_1(U,2)^{} &\\rightarrow & A_{0}^{}(X)\\oplus A_1(X)^{} &\\xrightarrow{} & A_{0}(G)\\oplus A_1(G)^{}\\ \\rightarrow \\ .\\\\\\end{array}$ As above, chasing this diagram we conclude that $ (\\iota _F)_\\ast (A_5^\\perp )= (\\iota _F)_\\ast \\Phi ^\\perp \\Bigl ( A_{0}^{hom}(X)\\oplus A_1^{hom}(X)^{} \\Bigr ) =0\\ \\ \\ \\hbox{in}\\ A_5(Q)\\ .$ It follows that the restriction of $(\\iota _F)_\\ast $ to the second term of the decomposition (REF ) induces an isomorphism $ (\\iota _F)_\\ast \\colon \\ \\ (p_X)^\\ast A^2_{hom}(X)\\cap e \\ \\xrightarrow{}\\ A_5^{hom}(Q)\\ .",
"$ Let us now proceed to prove proposition REF .",
"We will construct the correspondence $\\Gamma $ (the construction of $\\Psi $ is only notationally different, the roles of $X$ and $Y$ being symmetric).",
"The variety $X$ is smooth, and the variety $Q$ of proposition REF is also smooth, by our generality assumptions.",
"The variety $F$ , however, is definitely singular (remark REF ), and so we need to desingularize.",
"Let $g\\colon \\widetilde{F}\\rightarrow F$ be a resolution of singularities.",
"We let $\\bar{e}\\subset \\widetilde{F}$ denote the strict transform of $e\\subset F$ , and $\\widetilde{e}\\rightarrow \\bar{e}$ a resolution of singularities, and we write $\\tau \\colon \\widetilde{e}\\rightarrow \\widetilde{F}$ for the composition of the resolution and the inclusion morphism.",
"The correspondence $\\Gamma $ will be defined as $ \\Gamma := \\Gamma _{\\iota _{{F}}}\\circ \\Gamma _g \\circ (\\Gamma _{{\\tau }} \\circ {}^t \\Gamma _{{\\tau }}) \\circ {}^t \\Gamma _g \\circ {}^t \\Gamma _{{p_X}}\\ \\ \\ \\in \\ A^7(X\\times Q)\\ .",
"$ By definition, the action of $\\Gamma $ decomposes as $ \\begin{split} \\Gamma _\\ast \\colon \\ \\ A^i(X)\\ \\xrightarrow{} \\ A^i(F) \\ \\xrightarrow{}\\ A^i(\\widetilde{F})\\ \\xrightarrow{}\\ A^{i+1}(\\widetilde{F})=A_{7-i}(\\widetilde{F})\\ \\xrightarrow{} &\\\\ A_{7-i}(F)\\ \\xrightarrow{}\\ A_{7-i}(Q)= A^{i+4}(Q)\\ .",
"&\\\\\\end{split}$ (Here $A^\\ast (F)$ refers to Fulton–MacPherson's operational Chow cohomology [12].)",
"The projection formula for operational Chow cohomology (theorem REF ) ensures that for any $b\\in A^i(F)$ one has $ g_\\ast ( g^\\ast (b)\\cdot \\bar{e})= g_\\ast ( g^\\ast (b)\\cap \\bar{e})=b\\cap e\\ \\ \\ \\hbox{in}\\ A_{7-i}(F)\\ ,$ and so the action of $\\Gamma $ simplifies to $ \\Gamma _\\ast \\colon \\ \\ A^i(X)\\ \\xrightarrow{} \\ A^i(F)\\ \\xrightarrow{}\\ A_{7-i}(F)\\ \\xrightarrow{}\\ A_{7-i}(Q)= A^{i+4}(Q)\\ .$ Lemma REF ensures that $\\Gamma _\\ast $ induces isomorphisms $ \\Gamma _\\ast \\colon \\ \\ A^i_{hom}(X)\\ \\xrightarrow{}\\ A^{i+4}_{hom}(Q)\\ \\ \\ (i=2,3)\\ ,$ and so we have proven proposition REF .",
"Step 2: Trivial Chow groups.",
"In this step, we study the Chow groups of the incidence variety $Q$ .",
"We continue to assume that $Q$ is smooth, just as in step 1.",
"The goal of step 2 will be to show that many Chow groups of $Q$ are trivial: Proposition 6.5 We have $ A^i_{hom}(Q)=0\\ \\ \\ \\hbox{for\\ all\\ }i\\notin \\lbrace 6,7\\rbrace \\ .$ (This means that $\\hbox{Niveau}(A^\\ast (Q))\\le 3$ in the language of [32], i.e.",
"the 11–dimensional variety $Q$ motivically looks like a variety of dimension 3.)",
"Suppose we can prove that $ A_j^{hom}(Q)=0\\ \\ \\ \\hbox{for}\\ j< 4\\ .$ Applying the Bloch–Srinivas argument [8], [32] to the smooth projective variety $Q$ , this implies that $ A^i_{AJ}(Q)=0\\ \\ \\ \\hbox{for\\ all\\ }i\\notin \\lbrace 6,7\\rbrace \\ .$ But $Q$ has no odd–degree cohomology except for degree 11 (proposition REF below), and so there is equality $A^i_{AJ}(Q)=A^i_{hom}(Q)$ for all $i\\ne 6$ .",
"That is, to prove proposition REF one is reduced to proving (REF ).",
"There is an exact sequence $ A_j(R,1)\\ \\xrightarrow{}\\ A_j(F)\\ \\xrightarrow{}\\ A_j(Q)\\ \\rightarrow \\ \\ \\ ,$ and we have seen (lemma REF ) that $(\\iota _F)_\\ast A_j^{hom}(F)=A_j^{hom}(Q)$ .",
"Thus, to prove the vanishing (REF ) it only remains to show that $ A_j^{hom}(F)\\ \\ \\subset \\ \\hbox{Im}\\, \\delta \\ \\ \\ \\hbox{for\\ }j< 4\\ .$ The inclusion (REF ) is proven by the same argument as that of lemma REF .",
"That is, we observe that propositions REF and REF give us isomorphisms $ \\begin{split} \\Phi _j\\colon \\ \\ \\bigoplus A_{j-k}^{}(X) \\ &\\xrightarrow{} \\ A_j^{}(F)\\ ,\\\\\\Phi ^1_j\\colon \\ \\ \\bigoplus A_{j-k}^{}(U,1) \\ &\\xrightarrow{} \\ A_j^{}(R,1)\\ ,\\\\\\end{split} $ where $\\Phi _j, \\Phi _j^1$ are defined as $ \\begin{split} \\Phi _j&= {\\left\\lbrace \\begin{array}{ll} h^5 \\circ (p_X)^\\ast &\\ \\ \\hbox{if}\\ j=0\\ ,\\\\\\sum _{k=0}^1 h^{5-k}\\circ (p_X)^\\ast &\\ \\ \\hbox{if}\\ j=1\\ ,\\\\\\sum _{k=0}^2 h^{5-k}\\circ (p_X)^\\ast + b \\circ (p_X)^\\ast &\\ \\ \\hbox{if}\\ j=2\\ ,\\\\\\sum _{k=0}^3 h^{5-k}\\circ (p_X)^\\ast + b \\circ (p_X)^\\ast + (b\\cdot h)\\circ (p_X)^\\ast &\\ \\ \\hbox{if}\\ j=3\\ ,\\\\\\end{array}\\right.}",
"\\\\\\Phi ^1_j&= {\\left\\lbrace \\begin{array}{ll} h^5 \\circ (p_U)^\\ast &\\ \\ \\hbox{if}\\ j=0\\ ,\\\\\\sum _{k=0}^1 h^{5-k}\\circ (p_U)^\\ast &\\ \\ \\hbox{if}\\ j=1\\ ,\\\\\\sum _{k=0}^2 h^{5-k}\\circ (p_U)^\\ast + b \\circ (p_U)^\\ast &\\ \\ \\hbox{if}\\ j=2\\ ,\\\\\\sum _{k=0}^3 h^{5-k}\\circ (p_X)^\\ast + b \\circ (p_U)^\\ast + (b\\cdot h)\\circ (p_U)^\\ast &\\ \\ \\hbox{if}\\ j=3\\ .\\\\\\end{array}\\right.}",
"\\\\\\end{split}$ In particular, we observe that for each $j\\le 3$ , the isomorphisms $\\Phi _j, \\Phi ^1_j$ of (REF ) have the same number of direct summands on the left–hand side (i.e., the “extra class” $e\\in A_7(F)$ does not appear).",
"For each $j\\le 3$ , we can construct a commutative diagram with exact rows $ \\begin{array}[c]{ccccc}A_j(R,1)& \\xrightarrow{}&\\ A_j(F)& \\xrightarrow{}& A_j(Q)\\ \\rightarrow \\\\&&&&\\\\\\ \\ \\uparrow {\\scriptstyle \\Phi _j^1}&&\\ \\ \\ \\uparrow {\\scriptstyle \\Phi _j} &&\\uparrow \\\\&&&&\\\\\\bigoplus A_{j-k}^{}(U,1) &\\xrightarrow{} & \\bigoplus A_{j-k}^{}(X) &\\xrightarrow{} & \\bigoplus A_{j-k}(Gr_1)\\\\\\end{array}$ (the commutativity of this diagram is checked as in the proof of lemma REF ).",
"Let $a\\in A_j^{hom}(F)$ , for $j\\le 3$ .",
"Then $a=\\Phi _j(\\alpha )$ for some $\\alpha \\in \\bigoplus A_{j-k}^{hom}(X)$ (proposition REF (ii)).",
"But then $\\iota _\\ast (\\alpha )=0$ , since the Grassmannian $Gr_1$ has trivial Chow groups.",
"Using the above diagram, it follows that $\\alpha $ is in the image of $\\delta _U$ , and hence $a\\in \\hbox{Im}\\delta $ .",
"This proves (REF ) and hence proposition REF .",
"(NB: in fact, the above argument does not need that $\\Phi ^1_j$ is an isomorphism; we merely need the fact that a map $\\Phi ^1_j$ fitting into the above commutative diagram exists.)",
"To close step 2, it only remains to prove the following proposition: Proposition 6.6 Assume $j$ is odd and $j\\ne 11$ .",
"Then $ H_j(Q)=0\\ .$ For $j$ odd and different from 11, there is a commutative diagram with exact row $ \\begin{array}[c]{cccccc}H_{j+1}(R)& \\xrightarrow{}&\\ H_j(F)& \\xrightarrow{}& H_j(Q) & \\rightarrow 0 \\\\&&&&&\\\\\\uparrow {\\scriptstyle \\Phi _{j+1}^h}&& \\uparrow {\\scriptstyle \\Phi ^h_j} &&&\\\\&&&&&\\\\\\bigoplus H_{j+1-2k}^{}(U) &\\rightarrow & \\bigoplus H_{j-2k}^{}(X) \\ .",
"& & \\ \\ & \\\\\\end{array}$ Here $\\Phi ^h_j$ and $\\Phi ^h_{j+1}$ are the isomorphisms of (REF ) resp.",
"(REF ).",
"The righthand 0 is because $H_j(R)=0$ for $j$ odd (corollary REF ).",
"We observe that $j\\ne 11$ implies that $k\\ne 4$ (the only odd homology of $X$ is $H_3(X)$ ), which means that the “extra class” $e\\in A_7(F)$ does not intervene in the map $\\Phi ^h_j$ .",
"It follows that there are the same number of direct summands in the isomorphisms $\\Phi ^h_j, \\Phi ^h_{j+1}$ (they are both defined in terms of $h^k$ and $b$ ).",
"Observing that a Grassmannian does not have odd–degree cohomology, we thus see that the lower horizontal arrow is surjective.",
"The diagram now shows that $\\delta $ is surjective, and hence $(\\iota _F)_\\ast $ is the zero–map.",
"The proposition is proven.",
"This ends the proof of proposition REF .",
"Step 3: an isomorphism of motives.",
"We continue to assume (as in steps 1 and 2) that $X,Y$ are general, so that $Q$ is smooth.",
"The assignment $\\pi ^3_X:=\\Delta _X-\\pi ^0_X-\\pi ^2_X-\\pi ^4_X-\\pi ^6_X$ (where the Künneth components $\\pi ^j_X$ , $j\\ne 3$ are defined using hyperplane sections) defines a motive $h^3(X)$ such that there is a splitting $ h(X)= {1} \\oplus {1}(1)\\oplus h^3(X) \\oplus {1}(2) \\oplus {1}(3) \\ \\ \\ \\hbox{in}\\ \\mathcal {M}_{\\rm rat}\\ ,$ where 1 is the motive of a point.",
"It follows that $ A^i(h^3(X))= A^i_{hom}(X)\\ \\ \\ \\forall i\\ .$ The variety $Q$ is a smooth ample divisor in the product $P:=Gr_1\\times Gr_2$ .",
"The product $P$ has trivial Chow groups, and hence in particular verifies the standard conjectures.",
"It follows that $P$ admits a (unique) Chow–Künneth decomposition $\\lbrace \\pi ^i_P\\rbrace $ , and that there exist correspondences $C^j\\in A^{12+j}(P\\times P)$ such that $ (C^j)_\\ast \\colon \\ \\ H^{12+j}(P)\\ \\rightarrow \\ H^{12-j}(P) $ is inverse to $ \\cup Q^j\\colon \\ \\ H^{12-j}(P)\\ \\rightarrow \\ H^{12+j}(P)\\ .$ (The correspondences $C^j\\in A^{12+j}(P\\times P)$ are well–defined, as rational and homological equivalence coincide on $P\\times P$ .)",
"Let $\\tau \\colon Q\\rightarrow P$ denote the inclusion morphism.",
"One can construct a Chow–Künneth decomposition for $Q$ , by setting $ \\pi ^i_Q := {\\left\\lbrace \\begin{array}{ll} {}^t \\Gamma _\\tau \\circ (\\Gamma _\\tau \\circ {}^t \\Gamma _\\tau )^{\\circ 11-i} \\circ C^{12-i} \\circ \\pi _P^{i} \\circ \\Gamma _\\tau & \\ \\ \\ \\hbox{if\\ } i<11\\ ,\\\\{}^t \\pi ^{22-i}_Q & \\ \\ \\ \\hbox{if\\ } i>11\\ ,\\\\\\Delta _Q -\\sum _{j\\ne 11} \\pi ^j_Q \\ \\ \\ \\ \\in \\ A^{11}(Q\\times Q)& \\ \\ \\ \\hbox{if\\ }i=11\\ .\\end{array}\\right.",
"}$ (To check this is indeed a Chow–Künneth decomposition, one remarks that $ (\\Gamma _\\tau \\circ {}^t \\Gamma _\\tau )^{\\circ 12-i} \\circ C^{12-i} \\circ \\pi _P^{i} = \\pi _P^i\\ \\ \\ \\hbox{in}\\ H^{24}(P\\times P)\\ ,$ and because $P\\times P$ has trivial Chow groups one has the same equality modulo rational equivalence: $ (\\Gamma _\\tau \\circ {}^t \\Gamma _\\tau )^{\\circ 12-i} \\circ C^{12-i} \\circ \\pi _P^{i} = \\pi _P^i\\ \\ \\ \\hbox{in}\\ A^{12}(P\\times P)\\ .$ It is now readily checked that $\\lbrace \\pi ^i_Q\\rbrace $ verifies $\\pi ^i_Q\\circ \\pi ^j_Q=\\delta _{ij} \\pi ^i_Q$ in $A^{11}(Q\\times Q)$ , where $\\delta _{ij}$ is the Kronecker symbol.)",
"Setting $h^i(Q):=(Q,\\pi ^i_Q,0)$ , this induces a decomposition of the motive of $Q$ as $ h(Q) =\\bigoplus {1}(\\ast )\\oplus h^{11}(Q)\\ \\ \\ \\hbox{in}\\ \\mathcal {M}_{\\rm rat}\\ ,$ and hence one has $ A^i(h^{11}(Q))= A^i_{hom}(Q)\\ \\ \\ \\forall i\\ .$ We now consider the homomorphism of motives $ \\Gamma \\colon \\ \\ \\ h^3(X)\\ \\rightarrow \\ h^{11}(Q)(-4)\\ \\ \\ \\hbox{in}\\ \\mathcal {M}_{\\rm rat}\\ ,$ where $\\Gamma \\in A^7(X\\times Q)$ is as in step 1.",
"We have seen in steps 1 and 2 that there are isomorphisms $ \\Gamma _\\ast \\colon \\ \\ \\ A^i_{hom}(X)\\ \\xrightarrow{}\\ A^{i+4}_{hom}(Q)\\ \\ \\ \\forall i\\ .$ In view of (REF ) and (REF ), this translates into $ \\Gamma _\\ast \\colon \\ \\ \\ A^i_{}(h^3(X))\\ \\xrightarrow{}\\ A^{i+4}_{}(h^{11}(Q))=A^i(h^{11}(Q)(-4))\\ \\ \\ \\forall i\\ .$ Using that the field $ is a universal domain, this implies (cf.",
"\\cite [Lemma 1.1]{Huy}) there is an isomorphism of motives$$ \\Gamma \\colon \\ \\ \\ h^3(X)\\ \\xrightarrow{}\\ h^{11}(Q)(-4)\\ \\ \\ \\hbox{in}\\ \\mathcal {M}_{\\rm rat}\\ .$$The roles of $ X$ and $ Y$ being symmetric, the same argument also furnishes an isomorphism$$ \\Psi \\colon \\ \\ \\ h^3(Y)\\ \\xrightarrow{}\\ h^{11}(Q)(-4)\\ \\ \\ \\hbox{in}\\ \\mathcal {M}_{\\rm rat}\\ .$$The result is an isomorphism$$ \\Psi ^{-1}\\circ \\Gamma \\colon \\ \\ \\ h^3(X)\\ \\xrightarrow{}\\ h^3(Y)\\ \\ \\ \\hbox{in}\\ \\mathcal {M}_{\\rm rat}\\ .$$Since the difference $ h(X)-h3(X)$ is just $ 1 1(1)1(2) 1(3)$, which is the same as $ h(Y)-h3(Y)$, there is also an isomorphism$$ h(X)\\ \\xrightarrow{}\\ h(Y)\\ \\ \\ \\hbox{in}\\ \\mathcal {M}_{\\rm rat}\\ .$$$ Step 4: Spreading out.",
"We have now proven that there is an isomorphism of Chow motives $h^3(X)\\cong h^3(Y)$ for a general pair of GPK$^3$ double mirrors.",
"To extend this to all pairs of double mirrors, we reason as follows.",
"Let $ \\pi _X\\colon \\ \\ \\ \\mathcal {X}\\ \\rightarrow \\ B $ denote the universal family of all GPK$^3$ threefolds (so $B$ is an open in $PGL(\\wedge ^2 V)$ ).",
"The double mirror construction corresponds to an involution $\\sigma $ on $B$ , such that $X_b$ and $Y_b:=X_{\\sigma (b)}$ are double mirrors.",
"We define the family $\\mathcal {Y}$ as the composition $ \\pi _Y:=\\sigma \\circ \\pi _X\\colon \\ \\ \\ \\mathcal {Y}:=\\mathcal {X}\\ \\rightarrow \\ B\\ .$ The above construction of the correspondences $\\Gamma ,\\Psi $ can be done relatively: over the open $B^0\\subset B$ where the incidence variety $Q$ is smooth, one obtains relative correspondences $ \\Gamma \\ \\ \\in A^3(\\mathcal {X}\\times _{B^0}\\mathcal {Y})\\ ,\\ \\ \\Psi \\ \\ \\in \\ A^3(\\mathcal {Y}\\times _{B^0}\\mathcal {X})\\ ,$ such that for each $b\\in B^0$ the restrictions $\\Gamma _b\\in A^3(X_b\\times Y_b)$ , $\\Psi _b\\in A^3(Y_b\\times X_b)$ verify $ \\begin{split} &\\pi ^3_{X_b}= \\Psi _b\\circ \\Gamma _b\\ \\ \\ \\hbox{in}\\ A^3(X_b\\times X_b)\\ ,\\\\&\\pi ^3_{Y_b}= \\Gamma _b\\circ \\Psi _b\\ \\ \\ \\hbox{in}\\ A^3(Y_b\\times Y_b)\\ .\\\\\\end{split}$ Taking the closure, one obtains extensions $\\bar{\\Gamma }\\in A^3(\\mathcal {X}\\times _B \\mathcal {Y})$ , $\\bar{\\Psi }\\in A^3(\\mathcal {Y}\\times _B \\mathcal {X})$ to the larger base $B$ , that restrict to $\\Gamma $ resp.",
"$\\Psi $ .",
"The correspondences $\\pi ^3_{X_b},\\pi ^3_{Y_b}$ also exist relatively (note that any GPK$^3$ threefold has Picard number 1, and so $\\pi ^2,\\pi ^4$ exist as relative correspondences).",
"The relative correspondences $ \\pi ^3_\\mathcal {X}- \\bar{\\Psi }\\circ \\bar{\\Gamma }\\ \\ \\in \\ A^3(\\mathcal {X}\\times _B \\mathcal {X})\\ ,\\ \\ \\ \\pi ^3_\\mathcal {Y}- \\bar{\\Gamma }\\circ \\bar{\\Psi }\\ \\ \\in \\ A^3(\\mathcal {Y}\\times _B \\mathcal {Y}) $ have the property that their restriction to a general fibre is rationally trivial.",
"But this implies (cf.",
"[50]) that the restriction to every fibre is rationally trivial, and hence we obtain an isomorphism of motives for all $b\\in B$ , i.e.",
"for all pairs $(X_b,Y_b)$ of double mirrors.",
"Remark 6.7 In the proof of theorem REF , we have not explicitly determined the inverse to the isomorphism $\\Gamma $ .",
"With some more work, it is actually possible to show that there exists $m\\in \\mathbb {Q}$ such that $ {1\\over m}\\, {}^t \\Gamma \\colon \\ \\ h^{11}(Q)\\ \\rightarrow \\ h^3(X)\\ \\ \\ \\hbox{in}\\ \\mathcal {M}_{\\rm rat} $ is inverse to $\\Gamma $ .",
"Remark 6.8 It would be interesting to extend theorem REF to the category $\\mathcal {M}_{\\mathbb {Z}rat}$ of Chow motives with integral coefficients.",
"Let $X,Y$ be as in theorem REF .",
"Is it true that $h(X)$ and $h(Y)$ are isomorphic in $\\mathcal {M}_{\\mathbb {Z}{\\rm rat}}$ ?",
"Steps 1 and 2 in the above proof probably still work for Chow groups with $\\mathbb {Z}$ –coefficients (one just needs to upgrade the fibration results of section to $\\mathbb {Z}$ –coefficients); steps 3 and 4, however, certainly need $\\mathbb {Q}$ –coefficients.",
"Remark 6.9 In all likelihood, an argument similar to that of theorem REF could also be applied to establish an isomorphism of Chow motives for the Grassmannian–Pfaffian Calabi–Yau varieties of [9], [37], as well as for the Calabi–Yau fivefolds of [36]."
],
[
"Some corollaries",
"Corollary 7.1 Let $X,Y$ be two GPK$^3$ double mirrors.",
"Let $M$ be any smooth projective variety.",
"Then there are isomorphisms $ N^j H^i(X\\times M,\\mathbb {Q})\\cong N^j H^i(Y\\times M,\\mathbb {Q})\\ \\ \\ \\hbox{for\\ all\\ }i,j\\ .$ (Here, $N^\\ast $ denotes the coniveau filtration [7].)",
"Theorem REF implies there is an isomorphism of Chow motives $h(X\\times M)\\cong h(Y\\times M)$ .",
"As the cohomology and the coniveau filtration only depend on the motive [2], [47], this proves the corollary.",
"Remark 7.2 It is worth noting that for any derived equivalent threefolds $X,Y$ , there are isomorphisms $ N^j H^i(X,\\mathbb {Q})\\cong N^j H^i(Y,\\mathbb {Q})\\ \\ \\ \\hbox{for\\ all\\ }i,j\\ ;$ this is proven in [1].",
"Corollary 7.3 Let $X,Y$ be two GPK$^3$ double mirrors.",
"Then there are (correspondence–induced) isomorphisms between higher Chow groups $ A^i(X,j)_{}\\ \\xrightarrow{}\\ A^i(Y,j)_{} \\ \\ \\ \\hbox{for\\ all\\ }i,j\\ .$ There are also (correspondence–induced) isomorphisms in higher algebraic K–theory $ G_j(X)_{\\mathbb {Q}}\\ \\xrightarrow{}\\ G_j(Y)_{\\mathbb {Q}}\\ \\ \\ \\hbox{for\\ all\\ }j\\ .$ This is immediate from the isomorphism of Chow motives $h(X)\\cong h(Y)$ .",
"-1mmAcknowledgements Thanks to Len, for telling me wonderful stories of Baba Yaga and vatrouchka."
]
] | 1808.08339 |
[
[
"Signatures of nodeless multiband superconductivity and particle-hole\n crossover in the vortex cores of FeTe$_{0.55}$Se$_{0.45}$"
],
[
"Abstract Scanning tunneling experiments on single crystals of superconducting FeTe$_{0.55}$Se$_{0.45}$ have recently provided evidence for discrete energy levels inside vortices.",
"Although predicted long ago, such levels are seldom resolved due to extrinsic (temperature, instrumentation) and intrinsic (quasiparticle scattering) limitations.",
"We study a microscopic multiband model with parameters appropriate for FeTe$_{0.55}$Se$_{0.45}$.",
"We confirm the existence of well-separated bound states and show that the chemical disorder due to random occupation of the chalcogen site does not affect significantly the vortex-core electronic structure.",
"We further analyze the vortex bound states by projecting the local density of states on angular-momentum eigenstates.",
"A rather complex pattern of bound states emerges from the multiband and mixed electron-hole nature of the normal-state carriers.",
"The character of the vortex states changes from hole-like with negative angular momentum at low energy to electron-like with positive angular momentum at higher energy within the superconducting gap.",
"We show that disorder in the arrangement of vortices most likely explains the differences found experimentally when comparing different vortices."
],
[
"Introduction",
"As a local and space-resolved probe of the electronic density of states, the scanning tunneling microscope (STM) [1] has enabled detailed testing of the Bardeen–Cooper–Schrieffer (BCS) theory of superconductivity [2]—generalized to encompass inhomogeneous order parameters by Bogoliubov and de Gennes [3]—especially in the interior of Abrikosov vortices [4], [5], [6], [7].",
"The predicted spectrum of one-electron excitations inside vortices is qualitatively different in superconductors with and without nodes in the order parameter, for in the first case the low-energy states attached to the cores are resonant with extended states outside the core and the resulting energy spectrum is continuous, while in the second case, where the superconducting state is fully gapped, the subgap vortex-core states are truly bound and the energy spectrum is discrete (assuming the vortex is isolated).",
"Hence an experimental detection of discrete levels in vortices is sufficient to rule out nodal order parameters.",
"Such an observation is possible only in the quantum regime, when the energy separation between the levels is larger than the energy broadening related to temperature and quasiparticle interactions.",
"The typical energy separation is $\\Delta ^2/E_{\\mathrm {F}}$ , where $\\Delta $ is the order-parameter scale and $E_{\\mathrm {F}}$ is the Fermi energy [8].",
"In the cuprates, the quantum regime is realized but the order parameter has nodes and the vortex-core spectrum is therefore continuous [7].",
"On the other hand, most nodeless superconductors are not in the quantum regime, such that the vortex states are dense and measurements by STM show broad features inside vortices [9].",
"Very recently, discrete vortex levels were discovered in FeTe$_{0.55}$ Se$_{0.45}$ [10].",
"In the quantum regime, the bound states appear as resolution-limited peaks in the local tunneling conductance.",
"Exactly at the core center, the tunneling spectrum predicted by the Bogoliubov–de Gennes theory breaks particle-hole symmetry [8], [11], [12], [6].",
"For an electron band (positive mass), the lowest state has energy $+\\frac{1}{2}\\Delta ^2/E_{\\mathrm {F}}$ and the electronic component of its wave function is maximal at the core center, while the hole component vanishes at the core center, such that there is no peak at energy $-\\frac{1}{2}\\Delta ^2/E_{\\mathrm {F}}$ .",
"The situation is reverted for a hole band (negative mass) [13], where the peak at the core center has energy $-\\frac{1}{2}\\Delta ^2/E_{\\mathrm {F}}$ and there is nothing at $+\\frac{1}{2}\\Delta ^2/E_{\\mathrm {F}}$ .",
"The second and higher-energy states have no weight at the core center.",
"These expectations contrast with the observations made in FeTe$_{0.55}$ Se$_{0.45}$ , where indeed some of the vortices show a single peak at positive energy at the core center, as may be expected for an electronic band, while another vortex shows an almost particle-hole symmetric pair of peaks and yet other vortices show an asymmetric spectrum with a tall peak at negative energy and a smaller peak at positive energy [10].",
"Moreover, some vortices show several additional peaks at the core center, where the theory predicts only one.",
"A characteristic of FeTe$_{0.55}$ Se$_{0.45}$ , shared with most iron chalcogenides and pnictides [14], is a disconnected Fermi surface presenting hole pockets around the $\\Gamma $ point of the Brillouin zone and electron pockets around the M points.",
"The minimal model for FeTe$_{0.55}$ Se$_{0.45}$ has one electron pocket and one hole pocket of nearly equal volumes.",
"Because of this, a tempting interpretation of the experimental data would be that vortices show a mixture of electron-like and hole-like bound states.",
"Motivated by measurements on pnictides, a number of authors have studied vortices in models featuring electron and hole bands [13], [15], [16], [17], [18], [19], [20], [21].",
"A comparison with FeTe$_{0.55}$ Se$_{0.45}$ is difficult, as these works either choose parameters not in the quantum regime, or use exact diagonalization on small systems and fail to achieve the high energy resolution required in order to address discrete vortex levels.",
"Thus, the theoretical vortex spectrum for a nearly compensated two-band metal in the quantum regime has not been computed so far.",
"Provided that the two-band scenario gives the correct interpretation, another mechanism must still be identified in order to explain the significant variations observed from one vortex to another.",
"As FeTe$_{0.55}$ Se$_{0.45}$ is microscopically disordered [22], one candidate explanation rests on the variations in the local potential landscape due to random occupation of the Te/Se sites.",
"Another possibility would be density inhomogeneities, which would locally change the balance between the electron and hole pockets.",
"One could also adduce the irregular distribution of vortices observed in the experiments, as simulations have shown that disorder in the vortex positions impacts the local density of states (LDOS) in the cores [23], [7].",
"A measure of the proximity to the quantum limit is provided by the product $k_{\\mathrm {F}}\\xi \\sim E_{\\mathrm {F}}/\\Delta $ of the Fermi wave vector and the coherence length.",
"The typical size of the Fermi pockets in FeTe$_{0.55}$ Se$_{0.45}$ is [24] $k_{\\mathrm {F}}\\approx 0.1$ Å$^{-1}$ and the coherence length is [25] $\\xi \\approx 20$ Å.",
"The value $k_{\\mathrm {F}}\\xi \\approx 2$ locates FeTe$_{0.55}$ Se$_{0.45}$ deeper in the quantum regime compared with the only other compound where discrete vortex levels have been experimentally observed [6], i.e., YNi$_2$ B$_2$ C. The theoretical calculations in Ref.",
"Kaneko-2012 ($k_{\\mathrm {F}}\\xi =10$ ) as well as the earlier LDOS calculations of Ref.",
"Hayashi-1998 ($k_{\\mathrm {F}}\\xi =8$ ) must be extended to lower values of $k_{\\mathrm {F}}\\xi $ , before a comparison with FeTe$_{0.55}$ Se$_{0.45}$ can be attempted.",
"These calculations have considered an isolated vortex in a clean superconductor with free-electron-like dispersion.",
"Further generalizations are needed in order to incorporate the electron-hole band structure and the chemical disorder present in FeTe$_{0.55}$ Se$_{0.45}$ and for studying the effects of intervortex interactions within a disordered vortex configuration like the one observed in Ref. Chen-2018.",
"Our aim is to extend and complement the pioneering calculations, coming closer to the actual experimental situation of FeTe$_{0.55}$ Se$_{0.45}$ .",
"We present our two-dimensional disordered tight-binding model and review its basic properties as well as our calculation methods in Sec. .",
"In Sec.",
", we study an isolated vortex and characterize the discrete bound states in terms of their approximate angular momentum and electron-like or hole-like character.",
"Section illustrates the changes that chemical disorder, possible density inhomogeneities, and disorder in the vortex positions can bring to vortex-core spectra.",
"We summarize our results and present discussions and perspectives in Sec.",
", concluding in Sec. .",
"Appendices to collect additional material."
],
[
"Model and method",
"The early photoemission studies have revealed a significant band renormalization; furthermore, the DFT calculations give quite different band structures close to the Fermi energy for FeSe and FeTe [26], [27].",
"These observations suggest that an effective model is preferable, in order to describe the low-energy region and the superconducting state of FeTe$_{0.55}$ Se$_{0.45}$ , to a model based on bare DFT bands.",
"The qualitative shape of the Fermi surface plays an important role in the quantum regime, because inhomogeneities of the condensate, in particular vortices, have dimensions comparable with the Fermi wavelength and scatter the Bogoliubov excitations of the superconductor at large angles, thus sensing the whole Fermi surface.",
"In order to keep the approach simple while preserving the relevant ingredients for a successful low-energy theory of FeTe$_{0.55}$ Se$_{0.45}$ , we build on the $S_4$ -symmetric microscopic model introduced in Ref.",
"Hu-2012 for iron pnictides and chalcogenides.",
"This is a four-orbital model that captures the low-energy features of the more familiar five-orbital model [29], [30] and explains the robustness of the $s$ -wave superconducting state in this class of materials.",
"In the simplest variant, the model has one twofold-degenerate electron-like band and one twofold-degenerate hole-like band.",
"The electron and hole bands follow from a tight-binding Hamiltonian with the structure shown in Fig.",
"REF (a).",
"Each Fe site is coupled to its first and third neighbors by isotropic hopping amplitudes $t_1$ and $t_3$ , while the coupling to second neighbors is anisotropic and rotated by 90 degrees at the two inequivalent Fe sites.",
"The dispersion relation measured from the chemical potential $\\mu $ for these electron and hole bands is $\\xi ^{\\pm }_{\\mathbf {k}}=4t_{2s}\\cos (k_xa)\\cos (k_ya)\\\\\\pm 2\\sqrt{t_1^2[\\cos (k_xa)+\\cos (k_ya)]^2+[2t_{2d}\\sin (k_xa)\\sin (k_ya)]^2}\\\\+2t_3[\\cos (2k_xa)+\\cos (2k_ya)]-\\mu ,$ where $t_{2s}=(t_2+t^{\\prime }_2)/2$ and $t_{2d}=(t_2-t^{\\prime }_2)/2$ are the $s$ - and $d_{x^2-y^2}$ -symmetric second-neighbor hopping amplitudes, respectively, and $a=2.69$ Å is the lattice parameter of the 1-Fe unit cell.",
"The second pair of bands has the roles of $t_2$ and $t_2^{\\prime }$ interchanged.",
"The difference between $t_2$ and $t_2^{\\prime }$ stems from the fact that these hopping amplitudes are generated through hybridization with the out-of-plane $p$ orbitals of the pnictogen or chalcogen atom.",
"The robust $d_{x^2-y^2}$ -symmetric component $t_{2d}$ stabilizes the $s$ -wave pairing on the second neighbors, very much like, in the cuprates, the $s$ -symmetric first-neighbor hopping stabilizes the $d_{x^2-y^2}$ pairing symmetry [28].",
"For the choice of the five parameters, we require that the dispersion satisfies the following conditions: (1) the Fermi surface has a hole pocket at $\\Gamma $ and an electron pocket at M; (2) the Fermi wave vectors on the $\\Gamma $ and M pockets are $k_{\\mathrm {F},\\Gamma }=0.15\\pi /a$ and $k_{\\mathrm {F},\\mathrm {M}}=0.12\\pi /a$ , respectively; (3) the hole band at $\\Gamma $ has its maximum at 13 meV; (4) the electron pocket has minimal anisotropy; (5) the self-consistent vortex-core size is 16 Å.",
"Condition (2) uses an average of the measured wave vectors at the $\\alpha _2$ and $\\alpha _3$ sheets in FeTe$_{0.42}$ Se$_{0.58}$ [27].",
"Condition (3) is based on the band parametrization used in Ref. Sarkar-2017.",
"Condition (4) is imposed for simplicity and used to fix the parameter $t_{2d}$ .",
"At low energy, the main effect of $t_{2d}$ is to change the anisotropy of the Fermi surface at the M point.",
"One can see this by expanding the dispersion around the $\\Gamma $ and M points.",
"At leading order, the dispersion around $\\Gamma $ is isotropic, while around M there is a term $t_1[(2t_{2d}/t_1)^2-1](k_x^2-k_y^2)$ .",
"The choice $t_{2d}=t_1/2$ cancels this term and minimizes the anisotropy of the M pocket.",
"Finally, condition (5) sets the overall bandwidth, or rather the average Fermi velocity, such that the low-energy theory properly reproduces the emerging length scale that controls the vortices [25], as developed further below For the determination of the bandwidth via the vortex-core size, I have mistakenly used the lattice parameter of the 2-Fe unit cell, as was realized after submission of this work.",
"I am grateful to Lingyuan Kong for pointing this out.",
"For this reason, the value 16 Å is a factor $\\sqrt{2}$ smaller than the value 22 Å of the coherence length reported in Ref. Shruti-2015.",
"This has only minor impact on the results, because the main parameter $k_{\\mathrm {F}}\\xi $ , being dimensionless, is not affected.",
"The resulting average value $k_{\\mathrm {F}}\\xi =2.5$ realizes the strong quantum regime to which FeTe$_{0.55}$ Se$_{0.45}$ belongs.",
"The model parameters satisfying conditions (1)–(5) are $(t_1,t_2,t_2^{\\prime },t_3,\\mu )=(53,55.6,2.6,14.4,-51)$ meV.",
"The Fermi surface and the dispersion in the low-energy sector probed by STM and angle-resolved photoemission (ARPES) experiments are depicted in Fig.",
"REF (b).",
"To describe the superconducting state, we adopt the functional gap dependence reported in Ref. Miao-2012.",
"This $s^{\\pm }$ state is realized in real space by means of isotropic pairing amplitudes $\\Delta _2/4>0$ and $\\Delta _3/4<0$ on all second- and third-neighbor bonds, respectively.",
"The corresponding momentum-space gap structure is $\\Delta _{\\mathbf {k}}=\\Delta _2\\cos (k_xa)\\cos (k_ya)\\\\+\\frac{\\Delta _3}{2}[\\cos (2k_xa)+\\cos (2k_ya)].$ The sign reversal of the order parameter between the two pockets is not an essential feature for the results presented in the present study.",
"What is essential, though, is that the order parameter is nodeless on the Fermi surface.",
"The ARPES values $\\Delta _2=3.55$ meV and $\\Delta _3=-0.95$ meV [24] produce a gap in the DOS that is approximately twice as wide as the gap seen in high-resolution STM experiments [see Fig.",
"REF (b) in Appendix ].",
"The optical conductivity also appears to be consistent with gaps larger than those seen by STM [33], [34].",
"A bump at $\\sim 4$ meV observed by STM [35] was interpreted as the signature of a large gap [24].",
"However, no systematic structure at $\\pm 4$ meV is seen in Refs Wang-2018, Chen-2018, despite the large number of spectra reported.",
"This discrepancy was ascribed to momentum selectivity of the tunneling matrix element [36], which would hide the large gap at the M points.",
"Note that, in the cuprates, the largest gap at the M point is consistently seen with the same amplitude in photoemission and tunneling [37], [38].",
"Since our focus here is on tunneling, we rescale the reported values by a factor of two and use $\\Delta _2=1.775$ meV and $\\Delta _3=-0.475$ meV.",
"These values yield LDOS spectra in semiquantitative agreement with the STM data if the resolution of the calculation is set to 0.64 meV [compare Fig.",
"REF (f) with Fig.",
"1c of Ref. Chen-2018].",
"In particular, the robust and sharp gap edge at $\\sim 1$ meV, which is seen in all STM measurements [35], [39], [36], [10], is well reproduced.",
"We also calculate the spectral function and obtain features much sharper than the ones observed in photoemission [Fig.",
"REF (e)], despite averaging over a disordered region as described below.",
"This raises the question of the origin of a broad signal in photoemission and how it may affect the determination of gap values that are similar to the resolution of the experiment (2 meV) [24].",
"Because the second-neighbor hopping is anisotropic, an isotropic order parameter between second neighbors can only be generated self-consistently by the Bogoliubov–de Gennes equations if the pairing interaction has a small $d_{x^2-y^2}$ component.",
"We find that an interaction $V_2=-28.4$ meV, $V_2^{\\prime }=-27.0$ meV, corresponding to $V_{2s}=-27.7$ meV and $V_{2d}=-0.7$ meV, produces the desired gap structure on the second neighbors, while $V_3=-27.2$ meV generates the order parameter on the third-neighbor bonds.",
"Figure: Low-energy tight-binding model for FeTe 0.55 _{0.55}Se 0.45 _{0.45}.",
"(a) Square lattice with two inequivalent Fe sites.",
"The black bonds indicate a homogeneous and isotropic first-neighbor hopping amplitude t 1 t_1; the third-neighbor hopping t 3 t_3, only one of which is drawn, is also uniform and isotropic.",
"The second-neighbor hopping amplitudes take the values t 2 t_2 and t 2 ' t_2^{\\prime } as indicated.",
"There is a second pair of states with the roles of t 2 t_2 and t 2 ' t_2^{\\prime } exchanged.",
"(b) Fermi surface in the 1-Fe Brillouin zone.",
"The inset shows the dispersion with hole and electron bands at Γ\\Gamma and M, respectively.",
"(c) 35a×35a35a\\times 35a region showing Te (bright) and Se (dark) unit cells; this represents 0.06% of the system size used in the simulations.",
"(d) Simulated STM tunneling topography of the same area.",
"(e) Momentum-resolved spectral function (left part of graphs) and same quantity at lower resolution and multiplied by a Fermi function at T=6T=6 K (right part of graphs) along the two cuts indicated in (b).",
"(f) Local density of states along the path shown in (d).",
"The energy resolution is set to 0.64 meV (N=7000N=7000) in (d) and (f), and in (e) to 0.25 meV (N=18000N=18\\,000) for the left parts and 2 meV (N=2250N=2250) for the right parts; see Appendix for a definition of the resolution.Se and Te are isoelectronic, the former being slightly more electronegative than the latter.",
"One may therefore expect the unit cells containing Se to be attractive compared with those containing Te.",
"This disorder is weak and leads to local displacements of spectral weight and contrast in the topography measured by STM.",
"In order to incorporate the chemical disorder, we introduce two on-site energies $V_{\\mathrm {Te}}$ and $V_{\\mathrm {Se}}$ and distribute them randomly with the required proportion on the Fe sites of the model [Fig.",
"REF (c)].",
"To keep the electron density fixed, the chemical potential must be shifted by the average on-site energy $xV_{\\mathrm {Te}}+(1-x)V_{\\mathrm {Se}}$ , where $x$ is the concentration of Te.",
"Relative to the disorder-free chemical potential $\\mu $ , the on-site energies are therefore $(1-x)(V_{\\mathrm {Te}}-V_{\\mathrm {Se}})\\equiv V$ in the unit cells containing Te and $-x/(1-x)V$ in those containing Se.",
"This model of disorder is certainly very crude, but has the advantage of reducing the number of unknown parameters to just one.",
"The resulting potential is random and bimodal without spatial correlations.",
"Because, in reality, the potential on Fe atoms originates from out-of-plane disorder, a better model would attach a screened Coulomb potential to each Se and Te atom, leading to a more smooth landscape with spatial correlations at the Fe sites.",
"Disorder models of this kind have been studied in relation to cuprate superconductors [40], [41], [42], [43].",
"This sort of refinement seems not essential for the vortex-core spectroscopy, which is the main focus of this study, and is therefore left for a future investigation.",
"We tentatively set the strength of the potential to $V=1$ meV.",
"In order to fix this value, we have studied the disorder-induced fluctuations of the LDOS.",
"On each given Fe site, the surrounding disorder configuration induces in the LDOS specific structures whose amplitudes scale with $V$ (see Fig.",
"REF in Appendix ).",
"Simultaneously, the disorder reduces the height of the superconducting coherence peaks.",
"With the value $V=1$ meV, the fluctuations of the calculated LDOS are comparable with the variations of the tunneling spectrum measured in zero field [10].",
"Figure REF (f) shows a series of 25 spectra taken along the path indicated in Fig.",
"REF (d).",
"The calculations reported in this study are based on the Bogoliubov–de Gennes theory at zero temperature, solved by means of the Chebyshev-expansion technique introduced in Ref.",
"Covaci-2010, and using the asymmetric singular gauge of Ref.",
"Berthod-2016 to describe disordered distributions of vortices.",
"The Bogoliubov–de Gennes equations we consider are $\\sum _{\\mathbf {r}^{\\prime }}\\begin{pmatrix}h_{\\mathbf {r}\\mathbf {r}^{\\prime }}^{\\alpha }&\\Delta _{\\mathbf {r}\\mathbf {r}^{\\prime }}^{\\alpha }\\\\[0.5em](\\Delta _{\\mathbf {r}^{\\prime }\\mathbf {r}}^{\\alpha })^* & -(h_{\\mathbf {r}\\mathbf {r}^{\\prime }}^{\\alpha })^*\\end{pmatrix}\\begin{pmatrix} u_{\\mathbf {r}^{\\prime }}^{\\alpha } \\\\[0.5em] v_{\\mathbf {r}^{\\prime }}^{\\alpha }\\end{pmatrix}=\\varepsilon \\begin{pmatrix} u_{\\mathbf {r}}^{\\alpha } \\\\[0.5em] v_{\\mathbf {r}}^{\\alpha }\\end{pmatrix},$ where $\\varepsilon $ denote the energy eigenvalue for the wave function $(u_{\\mathbf {r}}^{\\alpha },v_{\\mathbf {r}}^{\\alpha })$ , $\\mathbf {r}$ , $\\mathbf {r}^{\\prime }$ denote the lattice sites (Fe atoms), and $\\alpha $ labels the two orbitals on each Fe.",
"$h_{\\mathbf {r}\\mathbf {r}^{\\prime }}^{\\alpha }=\\delta _{\\mathbf {r}\\mathbf {r}^{\\prime }}(V_{\\mathbf {r}}-\\mu )+t_{\\mathbf {r}\\mathbf {r}^{\\prime }}^{\\alpha }$ contains the on-site energies $V_{\\mathbf {r}}$ representing chemical disorder and the hopping amplitudes $t_{\\mathbf {r}\\mathbf {r}^{\\prime }}^{\\alpha }$ , which in zero field are as described in Fig.",
"REF (a), and in a finite magnetic field carry a Peierls phase (see, e.g., Ref. Berthod-2016).",
"The order parameter $\\Delta _{\\mathbf {r}\\mathbf {r}^{\\prime }}^{\\alpha }$ is determined self-consistently from the pairing interaction $V_{\\mathbf {r}\\mathbf {r}^{\\prime }}$ according to $\\Delta _{\\mathbf {r}\\mathbf {r}^{\\prime }}^{\\alpha }=V_{\\mathbf {r}\\mathbf {r}^{\\prime }}\\sum _{\\varepsilon >0}\\left[u^{\\alpha }_{\\mathbf {r}}(v_{\\mathbf {r}^{\\prime }}^{\\alpha })^*f(\\varepsilon )-(v_{\\mathbf {r}}^{\\alpha })^*u^{\\alpha }_{\\mathbf {r}^{\\prime }}f(-\\varepsilon )\\right],$ where the sum runs over all eigenstates of positive energy and the Fermi factors reduce to $f(\\varepsilon )=0$ and $f(-\\varepsilon )=1$ at zero temperature.",
"In order to reach our target resolution (typical separation between the energies $\\varepsilon $ much smaller that the gap), we must consider square lattices of order $1500\\times 1500$ sites, for which the Hamiltonian in Eq.",
"(REF ) has dimensions of order $8\\cdot 10^6\\times 8\\cdot 10^6$ , preventing a direct solution by diagonalization.",
"The Chebyshev expansion allows one to compute physical quantities like the LDOS $N(\\mathbf {r},E)$ and the order parameter $\\Delta _{\\mathbf {r}\\mathbf {r}^{\\prime }}^{\\alpha }$ for each lattice point $\\mathbf {r}$ or each bond $(\\mathbf {r},\\mathbf {r}^{\\prime })$ independently, without diagonalizing the Hamiltonian.",
"The LDOS is computed as $N(\\mathbf {r},E)&=-\\frac{2}{\\pi }\\,\\mathrm {Im}\\sum _{\\alpha }G_{\\alpha \\alpha }(\\mathbf {r},\\mathbf {r},E)\\\\\\nonumber &\\approx \\frac{2}{\\pi \\mathfrak {a}}\\mathrm {Re}\\,\\left\\lbrace \\frac{1}{\\sqrt{1-\\tilde{E}^2}}\\left[2+2\\sum _{n=1}^Ne^{-in\\arccos (\\tilde{E})}c_nK_n\\right]\\right\\rbrace .$ The first line relates the LDOS to the retarded single-particle Green's function, while the second line expresses the Chebyshev expansion, which is approximate due to truncation at finite order $N$ .",
"In the time domain, the Green's function is defined as $G_{\\alpha ^{\\prime }\\alpha }(\\mathbf {r}^{\\prime },\\mathbf {r},t)=(-i/\\hbar )\\theta (t)\\langle [\\psi ^{\\phantom{\\dagger }}_{\\alpha ^{\\prime }}(\\mathbf {r}^{\\prime },t),\\psi ^{\\dagger }_{\\alpha }(\\mathbf {r},0)]_+\\rangle $ , where $\\theta (t)$ is the Heaviside function, $\\psi ^{\\dagger }_{\\alpha }(\\mathbf {r})$ creates an electron at site $\\mathbf {r}$ in orbital $\\alpha $ , $[\\cdot ,\\cdot ]_+$ is the anti-commutator, and the average $\\langle \\cdots \\hspace{-0.29999pt}\\rangle $ is taken with respect to the Bogoliubov–de Gennes Hamiltonian $H$ .",
"The Chebyshev coefficients are $c_n=\\sum _{\\alpha }\\langle \\mathbf {r}\\alpha |T_n(\\tilde{H})|\\mathbf {r}\\alpha \\rangle ,$ where $|\\mathbf {r}\\alpha \\rangle $ is the state representing an electron localized at point $\\mathbf {r}$ in orbital $\\alpha $ and $T_n(\\tilde{H})$ is the Chebyshev polynomial of order $n$ evaluated at the dimensionless rescaled Hamiltonian $\\tilde{H}=(H-\\mathfrak {b})/\\mathfrak {a}$ .",
"Likewise, $\\tilde{E}=(E-\\mathfrak {b})/\\mathfrak {a}$ with, in our case, $\\mathfrak {a}=605$ meV and $\\mathfrak {b}=162$ meV.",
"The coefficients $K_n=\\lbrace (N-n+1)\\cos [\\pi n/(N+1)]+\\sin [\\pi n/(N+1)]\\cot [\\pi /(N+1)]\\rbrace /(N+1)$ remove unphysical oscillations due to the truncation at order $N$ [45].",
"All calculations are performed on a lattice comprising 2 002 001 sites centered on the site where the LDOS is being calculated [23].",
"The calculation scales with the order $N$ , the energy resolution improving like $1/N$ , as illustrated in Appendix .",
"The simulated topography of Fig.",
"REF (d) was constructed by calculating the tunneling current at each of the $35\\times 35$ sites shown in Fig.",
"REF (c) as $I(\\mathbf {r})=\\int _0^{10~\\mathrm {meV}}dE\\,N(\\mathbf {r},E)$ , consistently with the experimental protocol [10], and attaching to each pixel a function with the shape of a rounded square colored by $I(\\mathbf {r})$ .",
"The spatial contrast results from LDOS fluctuations due to disorder.",
"Notice that the spatial tunneling current distribution is not bimodal and fails to show one-to-one correspondence with the distribution of Te and Se potentials, despite clear correlations between Figs.",
"REF (c) and REF (d).",
"We find a weak positive correlation coefficient $R=0.4$ (see Appendix ).",
"The simulated topography shows dark patches (lower current) in Se-rich regions and bright patches in Te-rich regions, in agreement with the experimental observations [46], [27], [35], [22], [47], [39], [36], [10].",
"A trace of 25 LDOS spectra is displayed in Fig.",
"REF (f).",
"The smaller and larger gaps, residing on the $\\Gamma $ and M pockets, respectively, are visible despite the relatively low resolution used (see Appendix for high-resolution LDOS curves).",
"The disorder induces fluctuations in $N(\\mathbf {r},E)$ , in particular fluctuations of the coherence-peak height.",
"One also notices small fluctuations of the gap width as reported in Ref. Lin-2013.",
"Those are mostly due to the disorder, but also reflect the self-consistent adjustment of the order parameter in the disordered landscape.",
"A few remarks are in order regarding self-consistency.",
"The gap equation (REF ) can be rewritten in a form similar to Eq.",
"(REF ) involving the anomalous Green's function and the corresponding Chebyshev coefficients [23].",
"In principle, Eq.",
"(REF ) must be solved self-consistently on each bond ($\\mathbf {r},\\mathbf {r}^{\\prime }$ ).",
"In a system of two millions sites, this is presently beyond reach.",
"For calculating the LDOS at a given site, however, precise self-consistent values of the order parameter at distant sites are not required and can be replaced by approximate values.",
"For the data shown in Figs.",
"REF (d)–REF (f), we limited the search for a self-consistent solution to the region shown in Fig.",
"REF (c), including a few sites around it, while keeping the order parameter fixed to its unperturbed value in the rest of the system.",
"The self-consistency brings only small quantitative changes to the various properties.",
"In particular, the spectral gap defined by half the energy separation between the two main coherence peaks differs by less than 0.2% on average (0.5% at maximum) between the self-consistent and non-self-consistent calculations (see Fig.",
"REF in Appendix ).",
"The real-space Green's function also gives access to the momentum-resolved spectral function measured by ARPES.",
"For a translation-invariant system, the Green's function depends on the relative coordinate $\\mathbf {r}^{\\prime }-\\mathbf {r}$ and the spectral function $A(\\mathbf {k},E)$ is proportional to the imaginary part of its Fourier transform with respect to $\\mathbf {r}^{\\prime }-\\mathbf {r}$ .",
"In a disordered system like FeTe$_{0.55}$ Se$_{0.45}$ , the spectral function must be spatially averaged over the center-of-mass coordinate $(\\mathbf {r}^{\\prime }+\\mathbf {r})/2$ , leading us to the expression $A(\\mathbf {k},E)&=\\frac{1}{M}\\sum _{\\mathbf {r}}\\left({-\\frac{2}{\\pi }}\\right)\\mathrm {Im}\\sum _{\\mathbf {r}^{\\prime }}e^{-i\\mathbf {k}\\cdot (\\mathbf {r}^{\\prime }-\\mathbf {r})}\\sum _{\\alpha \\alpha ^{\\prime }}G_{\\alpha ^{\\prime }\\alpha }(\\mathbf {r}^{\\prime },\\mathbf {r},E)\\\\\\nonumber &\\approx \\frac{2}{\\pi \\mathfrak {a}}\\mathrm {Re}\\,\\left\\lbrace \\frac{1}{\\sqrt{1-\\tilde{E}^2}}\\left[2+2\\sum _{n=1}^Ne^{-in\\arccos (\\tilde{E})}c^{\\prime }_nK_n^{\\phantom{^{\\prime }}}\\right]\\right\\rbrace .$ The spatial average is performed on $M$ sites labeled $\\mathbf {r}$ , while the $\\mathbf {r}^{\\prime }$ sum runs over the whole lattice.",
"The Chebyshev expansion has the same form as for the LDOS, except that the coefficients now depend on nonlocal matrix elements according to $c^{\\prime }_n=\\frac{1}{M}\\sum _{\\mathbf {r}}\\sum _{\\alpha \\alpha ^{\\prime }}\\sum _{\\mathbf {r}^{\\prime }}e^{-i\\mathbf {k}\\cdot (\\mathbf {r}^{\\prime }-\\mathbf {r})}\\langle \\mathbf {r}^{\\prime }\\alpha ^{\\prime }|T_n(\\tilde{H})|\\mathbf {r}\\alpha \\rangle .$ Figure REF (e) shows $A(\\mathbf {k},E)$ evaluated along the two cuts drawn in Fig.",
"REF (b), as well as $A(\\mathbf {k},E)f(E)$ calculated with a lower resolution for a temperature $T=6$ K [this temperature applies only to $f(E)$ ; $A(\\mathbf {k},E)$ remains the zero-temperature result].",
"Since the calculation of the spectral function is relatively time-consuming, the spatial average was restricted to the region shown in Figs.",
"REF (c) and REF (d).",
"Notice that the spatial average is not a disorder average in the usual meaning of statistical average over the disorder: here, a single configuration of the disorder was generated and kept for the whole study.",
"It is seen that the disorder has little effect on $A(\\mathbf {k},E)$ .",
"In particular, no broadening is observed: $A(\\mathbf {k},E)$ has the resolution implied by the Chebyshev expansion.",
"Note that the broadening of order $\\pi V^2N(0)$ that would be expected from a conventional disorder average, where $N(0)$ is the Fermi-level DOS, is here very small ($\\sim 0.04$ meV).",
"The gaps on the hole and electron pockets are visible, even if the resolution is lowered to 2 meV.",
"References Tamai-2010, Miao-2012 report different responses of the hole bands around $\\Gamma $ to light polarization, attributed to their different orbital characters.",
"These are optical selection rules associated with the photoemission matrix element.",
"In the present model, the spectral function itself has structure in momentum space, independent of any matrix-element effect, and, interestingly, the hole band has exactly zero spectral weight along $\\Gamma $ –M (see Appendix ).",
"For this reason, the hole band was imaged around $(\\pi ,\\pi )$ in Fig.",
"REF (e)."
],
[
"Isolated vortex without chemical disorder",
"We turn now to the vortex-core states, starting with a single isolated vortex without chemical disorder.",
"The effects of disorder and nonideal vortex lattice are the topics of the next section.",
"The order parameter $\\Delta _{\\mathbf {r}\\mathbf {r}^{\\prime }}^{\\alpha }$ for a vortex has a modulus that decreases from its asymptotic bulk value to zero when $\\mathbf {r}$ and $\\mathbf {r}^{\\prime }$ approach the vortex center, and a phase that winds by $2\\pi $ along any path encircling the vortex.",
"Solving Eq.",
"(REF ) in the neighborhood of the vortex, we find that the self-consistent order parameter modulus is very nearly isotropic around the vortex center and accurately parametrized by the form [23] $\\Delta (r)=\\Delta _0/[1+(\\xi _0/r)\\exp (-r/\\xi _1)]$ , where $r$ is the distance from the vortex center to the center of the bond where the order parameter $\\Delta $ is located, $\\Delta _0$ is the asymptotic bulk value (either $\\Delta _2$ or $\\Delta _3$ ), and $\\xi _0$ , $\\xi _1$ are two parameters with the unit of length.",
"This analytical expression is compared with the self-consistent profile in Fig.",
"REF (a).",
"The core sizes $\\xi _c$ , defined by $\\Delta (\\xi _c)=\\Delta _0/2$ , equal $6.4a$ for the large gap and $3.85a$ for the small gap, which, once weighted by the bulk gap amplitudes and averaged, give the overall core size of 16 Å.",
"Recall that the bandwidth was adjusted such as to achieve this value for the core size [32].",
"Note also that no unique definition of the core size exists, and that the value obtained with our definition based solely on the self-consistent order parameter may differ from values deduced from, e.g., spectroscopic data.",
"Regarding the phase of the vortex order parameter, we find that, within numerical accuracy, it is given by the angle defined by the center of the bond and the center of the vortex.",
"Figure: (a) Self-consistent vortex order parameter.",
"The red and blue points show the order-parameter modulus (multiplied by four) on bonds connected to lattice sites along the (10) and (11) direction, respectively; for instance, the two red points closest to the vortex center correspond to lattice coordinates (2,0), the center of the bond being at (2.5,0.5) for the large gap and (3,0) for the small gap.",
"The solid lines show the interpolation formula given in the text.",
"Note the absence of anisotropy between the (10) and (11) directions.",
"(b) LDOS at the vortex center (red) and without vortex (blue) calculated with a resolution of 0.25 meV (N=18000N=18\\,000).",
"The insets show the spatial distribution of the LDOS in a 51a×51a51a\\times 51a region around the vortex at the energies of the low-lying bound states.",
"The color scale is logarithmic, going from blue (minimal intensity) to white (maximal intensity).The LDOS at the vortex center shown in Fig.",
"REF (b) presents two peaks near $-0.11$ meV and $+0.53$ meV, reminiscent of the Caroli–de Gennes–Matricon bound states [8].",
"The width and shape of the peak at $+0.53$ meV are identical to those of the resolution function (see Appendix ), indicating that it corresponds to a single bound state.",
"This calculation was performed with $N=18\\,000$ , achieving an energy resolution of 0.25 meV.",
"For the taller peak at $-0.11$ meV, the width is slightly larger than the resolution: a spectral analysis reveals that this peak gets contributions from two bound states.",
"As our calculation method delivers the LDOS in terms of the Green's function, Eq.",
"(REF ), we miss a direct access to the energies and wave functions of the Bogoliubov–de Gennes Hamiltonian.",
"Nonetheless, in the subgap range where the spectrum is discrete, it is possible to extract the energies and wave functions by fitting the expression $N(\\mathbf {r},E)=\\sum _{\\varepsilon }\\left[|u_{\\varepsilon }(\\mathbf {r})|^2\\delta _N(E-\\varepsilon )+|v_{\\varepsilon }(\\mathbf {r})|^2\\delta _N(E+\\varepsilon )\\right]$ to the calculated LDOS.",
"$u_{\\varepsilon }(\\mathbf {r})$ and $v_{\\varepsilon }(\\mathbf {r})$ are the electron and hole amplitudes, respectively, and $\\varepsilon >0$ are the energies.",
"Equation (REF ) gives the exact LDOS if the energies $\\varepsilon $ are allowed to run over all states, both in the continuous and discrete parts of the spectrum, and if $\\delta _N(E)$ is the Dirac delta function.",
"In our case, we restrict ourselves to a small set of discrete energies in the subgap region and we replace the Dirac delta function by the resolution function of the Chebyshev expansion.",
"We consider a series of 26 LDOS spectra taken along the (10) direction at the positions $(x,y)=(ia,0)$ , $i=0,\\ldots ,25$ , and we fit Eq.",
"(REF ) to this whole data set at once.",
"The outcome of this fitting is displayed in Fig.",
"REF .",
"We expect the result to be independent of the direction because, as can be seen in the insets of Fig.",
"REF (b), the LDOS is almost perfectly isotropic.",
"Figure REF (a) shows the spectral decomposition of the LDOS as a sum of resolution-limited peaks.",
"Only the LDOS curves up to $x=10a$ are shown for clarity; the complete data set is displayed in Fig.",
"REF , Appendix .",
"The quality of the fit is very good, although not perfect.",
"Given the large number of adjustable parameters, finding the absolute minimum is somewhat chancy.",
"Nevertheless, the results make sense and are sufficient for our purposes.",
"Figure: (a) Spectral decomposition of the vortex LDOS.",
"The black curves show the LDOS at sites (ia,0)(ia,0) for i=0i=0 (left) to i=10i=10 (right), the shaded curves show the pairs of peaks |u ε (𝐫)| 2 δ N (E-ε)|u_{\\varepsilon }(\\mathbf {r})|^2\\delta _N(E-\\varepsilon ) and |v ε (𝐫)| 2 δ N (E+ε)|v_{\\varepsilon }(\\mathbf {r})|^2\\delta _N(E+\\varepsilon ) (one color per pair), and the dashed gray curves show the sum of all these peaks.",
"(b) Amplitudes |u ε (𝐫)| 2 |u_{\\varepsilon }(\\mathbf {r})|^2 (full symbols) and |v ε (𝐫)| 2 |v_{\\varepsilon }(\\mathbf {r})|^2 (empty symbols) for the four lowest-energy states and their interpretation in terms of angular-momentum eigenstates (solid lines).In the continuum model, an isolated vortex has cylindrical symmetry and the eigenstates can be chosen such that the electron and hole components $u$ and $v$ have well-defined angular momenta [8], [11], [48], [12].",
"The electron component has angular momentum $\\mu -1/2$ and behaves in the core like $J_{\\mu -1/2}(k_+r)$ , where $\\mu $ is half an odd integer and $k_+$ is a wave vector slightly above $k_{\\mathrm {F}}$ , while the hole component has angular momentum $-\\mu -1/2$ and behaves like $J_{\\mu +1/2}(k_-r)$ .",
"We conform to the traditional notation for the angular momentum $\\mu $ , since confusion with the chemical potential is unlikely.",
"All Bessel functions $J_n$ except $J_0$ vanish at the origin, such that the only angular-momentum eigenstates with a finite amplitude at the vortex center are $\\mu =\\pm 1/2$ .",
"For an electron band (i.e., a positive mass), the lowest Bogoliubov excitation is electron-like with momentum $+1/2$ and finite value of $u$ at the vortex center.",
"For a hole band, the lowest excitation is hole-like with momentum $-1/2$ and finite value of $v$ in the core [13].",
"The angular momentum is not a good quantum number on the lattice.",
"Nevertheless, because the Fermi energy is small in our model, the dispersion is nearly parabolic [see Fig.",
"REF (b)] and the effective low-energy Hamiltonian resembles the continuum, as exemplified by the approximate cylindrical symmetry of the LDOS in Fig.",
"REF (b).",
"One sees in Fig.",
"REF (b) that the lowest bound state ($\\varepsilon =0.09$ meV) has its hole component finite in the core ($\\propto J_0$ ), while the electron component vanishes ($\\propto J_1$ ) [see also the top panel in Fig.",
"REF (b)].",
"This suggests that the low-energy carriers are holes.",
"At first sight, one expects that the next bound state at $\\varepsilon =0.2$ meV has angular momentum $-3/2$ with its hole (electron) component vanishing in the core like $J_1$ ($J_2$ ), an expectation contradicted by the data in Figs.",
"REF (b) and REF (b).",
"Here we must remember that the model has four bands grouped in two degenerate pairs.",
"The correct expectation is therefore to observe two doubly-degenerate states at each angular momentum.",
"The second state at $\\varepsilon =0.2$ meV indeed has angular momentum $-1/2$ and does not vanish in the core.",
"In order to determine the angular momenta of the various bound states, we fit the form $\\hat{\\psi }_{\\mu }(\\mathbf {r})=\\begin{pmatrix}u^{(1)}J_{\\mu -\\frac{1}{2}}\\big (k_+^{(1)}r\\big )+u^{(2)}J_{\\mu -\\frac{1}{2}}\\big (k_+^{(2)}r\\big )\\\\[1em]v^{(1)}J_{\\mu +\\frac{1}{2}}\\big (k_-^{(1)}r\\big )+v^{(2)}J_{\\mu +\\frac{1}{2}}\\big (k_-^{(2)}r\\big )\\end{pmatrix}$ to the electron and hole components extracted from the LDOS via Eq.",
"(REF ).",
"Equation (REF ) represents an angular-momentum eigenstate with radial components at two different wave vectors, as appropriate in a two-band setup.",
"The state at $\\varepsilon =0.09$ meV is very well approximated by $\\hat{\\psi }_{-1/2}$ , confirming that this is a hole-like state with momentum $-1/2$ [see Fig.",
"REF (b)].",
"At $\\varepsilon =0.2$ meV, the fit yields two independent components with angular momenta $-1/2$ and $-3/2$ .",
"We believe that two bound states live there, too close in energy to be resolved.",
"One is the second hole-like state with momentum $-1/2$ , the other is the first state with momentum $-3/2$ .",
"The near degeneracy of two states here will be confirmed below (Fig.",
"REF ).",
"Something quite interesting happens as the energy increases: we find that the best fit to the state at $\\varepsilon =0.39$ meV is $\\hat{\\psi }_{+5/2}$ with a small admixture of $\\hat{\\psi }_{+3/2}$ .",
"Hence this is an electron-like state and the vortex electronic structure changes from hole- to electron-like between 0.2 and 0.4 meV.",
"This statement is confirmed by the state at $\\varepsilon =0.53$ meV, as well as two other states not shown in Fig.",
"REF (b) at $0.69$ and $0.84$ meV.",
"The superposition of two angular momenta signals the broken rotational symmetry, and/or the interband mixing.",
"At $\\varepsilon =0.53$ meV, we find a $+3/2$ state with a significant mixing of $+1/2$ , which explains why this state has amplitude at the vortex center despite its relatively high energy.",
"In Ref.",
"Kaneko-2012, a similar mixing was advocated to explain a second peak in YNi$_2$ B$_2$ C. Finally, states at 0.69 and 0.84 meV are found to have angular momenta $+7/2$ and $+5/2$ with admixture of $+5/2$ and $+3/2$ , respectively.",
"The data are summarized in Fig.",
"REF .",
"The two series of bound states obey approximately the scaling $\\varepsilon =|\\mu |\\Delta ^2/E_{\\mathrm {F}}$ if we take for $\\Delta $ the two spectral gaps of 1.2 and 2 meV, and for $E_{\\mathrm {F}}$ the value 10.4 meV, which is the average of the hole-band maximum at 13 meV and the electron-band minimum at $-7.8$ meV [see Fig.",
"REF (b)].",
"Figure: Spectrum of subgap bound states for an isolated vortex, classified according to their approximate angular momentum μ\\mu .",
"Filled (empty) symbols correspond to hole-like (electron-like) states with negative (positive) μ\\mu .",
"Dashed lines indicate mixing of angular momenta.The blue and red straight lines show the behavior |μ|Δ 2 /E F |\\mu |\\Delta ^2/E_{\\mathrm {F}} with Δ=1.2\\Delta =1.2 and 2 meV, respectively, and E F =10.4E_{\\mathrm {F}}=10.4 meV."
],
[
"Effects of chemical disorder, density inhomogeneity, and neighboring vortices on the vortex-core spectrum",
"Strong local scattering centers induce low-energy states in superconductors [49].",
"The effects of weak extended disorder like the chemical disorder in FeTe$_{0.55}$ Se$_{0.45}$ have been much less studied (for a recent perspective, see Ref.",
"Sulangi-2018 and references therein).",
"As the experiments [10] report variations in the spectroscopy from one vortex to the next, we first ask whether these variations could be attributed to the chemical disorder.",
"For this purpose, we consider an isolated vortex and move it across the disordered landscape of Fig.",
"REF (c).",
"For the order parameter, we use the approximate form deduced from the self-consistent calculation and shown in Fig.",
"REF (a).",
"As the self-consistent adjustment of the order parameter in the disorder has almost no effect on the LDOS (Appendix ), we neglect it here.",
"For each of the $35\\times 35$ possible positions, we compute the LDOS at the vortex center.",
"Overall, we find that the changes are tiny relative to the spectrum in the clean case [Fig.",
"REF (b)]: all LDOS curves have the same general structure with one tall peak at small negative energy and one weaker peak at higher positive energy.",
"We have previously attributed the tall peak to the superposition of two hole-like bound states of angular momentum $-1/2$ and energies 0.09 and 0.2 meV.",
"The weaker peak belongs to a single electron-like state at energy 0.53 meV with principal angular momentum $+3/2$ and an admixture of $+1/2$ .",
"We fit Eq.",
"(REF ) to each LDOS curve in order to determine the correlation between bound states and disorder.",
"The energies of the bound states vary as the vortex is moved over the disorder as shown in Fig.",
"REF .",
"The distribution of energies for each bound state is much narrower (typical variance 0.02 meV) than the corrugation of the potential (1 meV).",
"The lowest state $\\varepsilon _{-1/2}^{(1)}$ , being the most localized [see Fig.",
"REF (b)], varies on the same scale as the disorder and shows a weak anti-correlation with it: there is a tendency for the state to have higher energy in attractive Se-rich regions and lower energy in repulsive Te-rich regions.",
"These tendencies do not compensate, though, and on average the energy $\\varepsilon _{-1/2}^{(1)}$ is pushed down ($0.06$ meV) compared with the clean case ($0.09$ meV).",
"This average shift occurs in spite of the fact that, on average, the potential due to disorder is zero.",
"The behavior of the state $\\varepsilon _{-1/2}^{(2)}$ is similar (not shown in the figure), except that it displays almost no shift on average.",
"This may be due to the fact that this state is closer to the particle-hole crossover energy.",
"The less localized state $\\varepsilon _{+3/2}^{(2)}$ varies on a coarse-grained scale of the order of the coherence length [Fig.",
"REF (b)].",
"This state has a weak positive correlation with the disorder and is pushed on average up.",
"Hence electron-like and hole like vortex states behave differently in the presence of extended disorder.",
"Figure: Energy of the vortex bound state ε -1/2 (1) \\varepsilon _{-1/2}^{(1)} (a) and ε +3/2 (2) \\varepsilon _{+3/2}^{(2)} (b) as the vortex is moved over the disorder.",
"Each pixel represents a different calculation with the vortex centered at that pixel.",
"The correlation coefficients RR indicate weak negative and positive correlation with the disorder.",
"The color scales, together with the histograms of energies, are shown on top of each graph.",
"The white bars indicate the bound-state energy in the absence of disorder.",
"The scale bar in (b) shows the coherence length (average vortex-core size).Figure: Evolution of the LDOS at the vortex center with varying the chemical potential, all other parameters held fixed, in the absence of disorder.",
"(a) and (b) show data for increasing charge accumulation and charge depletion, respectively, from bottom to top.",
"The curves are shifted vertically for clarity.",
"The resolution is 0.25 meV (N=18000N=18\\,000).Within our model, the chemical disorder can not explain the variety of vortex-core spectra observed in FeTe$_{0.55}$ Se$_{0.45}$ , with sometimes a single peak, sometimes two peaks of equal heights, and sometimes a taller peak at negative energy, like in Fig.",
"REF (b).",
"As a second possibility, we study how the vortex spectrum changes when varying the chemical potential, as would occur in the presence of fluctuations in the electronic density on length scales comparable with or larger than the coherence length.",
"It is not obvious how such nanoscale variations could be stabilized.",
"The vortex charging effect [50], [51] is small and may explain a depletion of order $10^{-5}$ electrons per cell in the vortex, corresponding to a tiny change of chemical potential in the $\\mu $ eV range.",
"Much larger changes are needed in order to produce appreciable variations in the vortex spectrum.",
"Figure REF shows the evolution of the spectrum at the vortex center upon charge accumulation [Fig.",
"REF (a)] and charge depletion [Fig.",
"REF (b)] stemming from variations of $\\mu $ by $\\pm 15$ meV, which correspond roughly to $\\pm 0.2$ electrons per cell.",
"The vortex order parameter was held fixed in this calculation and only $\\mu $ was varied.",
"It is seen that the charge accumulation has little effect, until $\\Delta \\mu =13$ meV, which is the point where the hole pocket at the $\\Gamma $ point disappears [see Fig.",
"REF (b)].",
"From this point on, one of the two hole-like states forming the tall peak at negative energy moves towards the gap edge, leaving a single pair of peaks corresponding to the states $\\varepsilon _{-1/2}^{(1)}$ and $\\varepsilon _{+3/2}^{(2)}$ in the vortex core.",
"The charge depletion alters the spectrum more dramatically.",
"As the chemical potential is lowered and the electron band at the M point is progressively emptied, the electron-like state $\\varepsilon _{+3/2}^{(2)}$ moves to the gap edge and another hole-like state comes in from the negative-energy gap edge.",
"The two hole-like states $\\varepsilon _{-1/2}^{(1)}$ and $\\varepsilon _{-1/2}^{(2)}$ show a very intriguing dependence on $\\mu $ : they initially begin to split for low values of $\\Delta \\mu $ , before merging again into a single peak, splitting across zero energy, and merging back a second time.",
"Most remarkably, the first merging leads to a peak at exactly zero energy: this occurs for $\\Delta \\mu =-7.8$ meV, which puts the chemical potential at the quadratic touching point where the electron pocket at M disappears and a hole pocket appears instead [Fig.",
"REF (b)].",
"This accidental zero-energy peak is unrelated to Majorana-type physics [52], [36].",
"Although Fig.",
"REF presents some spectral variation, we believe that the source of the variability in the experimental spectra must be searched for elsewhere.",
"In addition to being unlikely for electrostatic reasons, the large variations of chemical potential do not produce the kind of spectra that are seen by STM.",
"Figure: Comparison of the vortex-core LDOS in an isolated vortex (red) and in three vortex lattices for a 5 T field.",
"The square vortex lattice has the same orientation as the Fe lattice (0 ∘ 0^{\\circ }), or is rotated by 45 degrees.",
"The triangular lattice has its main axis along the Fe lattice.",
"The resolution is 0.25 meV (N=18000N=18\\,000).Finally, we investigate the influence of nearby vortices.",
"In a perfect vortex lattice, the localized states of the isolated vortices hybridize to form bands.",
"In a 5 T field, the hybridization is weaker than the typical energy separation between core states and the latter remain visible in the LDOS as well-defined peaks.",
"Apart from an overall broadening, no significant qualitative change is expected in a field if the vortices are ordered.",
"Figure REF shows the results for square and triangular vortex lattices of various orientations.",
"The tiny dependence on vortex-lattice structure and orientation reflects the absence of anisotropy in the isolated vortex (Fig.",
"REF ), owing to low electron density and exponential localization of the bound states in the cores.",
"This contrasts with the case of $d$ -wave superconductors, where the isolated vortex defines preferred directions relative to the microscopic lattice [53], [54], [23], [7].",
"Note that the vortex-lattice order parameter was constructed by superimposing the modulus and winding phase of isolated vortices using the method described in Ref.",
"Berthod-2016, and a small self-consistent adjustment was neglected.",
"We expect this to play no role for the LDOS inside the cores.",
"The calculation is made in the limit of a large penetration depth, assuming a constant field throughout space.",
"The measurements of Ref.",
"Chen-2018 show a vortex distribution that is largely random at 5 T. The positional disorder in the vortices may explain the variety of spectra observed, as we show now.",
"We have extracted vortex positions from Fig.",
"1d of Ref. Chen-2018.",
"As the simulation uses a system $\\sim 4$ times larger than the field of view of that figure, we have generated random vortex positions outside the field of view with a distribution similar to that seen inside (Appendix ).",
"The disordered vortices are surrounded by an ordered square lattice at the same field in order to ensure the correct boundary condition for an infinite distribution of vortices [23].",
"The unknown vortex positions outside the field of view do influence the LDOS calculated inside [7].",
"Hence the simulation should not be seen as an attempt to precisely model the experimental data, but as a way of obtaining typical spectra that may occur in a disordered vortex configuration.",
"Figure: LDOS at the center (phase-singularity point) of 50 vortices sorted from bottom to top and from left to right by order of increasing distance from the center of the STM field of view (Fig.",
"1d of Ref.",
"Chen-2018; see Fig.",
"in Appendix ).",
"The curves are shifted vertically for clarity.",
"The resolution is 0.25 meV (N=18000N=18\\,000).Figure REF shows a collection of LDOS curves calculated at the core of the 50 vortices closest to the center of the field of view.",
"A remarkable diversity of behaviors is seen.",
"The red curves show vortices where the LDOS has one tall peak at negative energy and one weaker peak at positive energy, similar to what is found in the isolated vortex and in the ideal vortex lattices.",
"A shoulder is often seen at a small positive energy between the two peaks, like in Fig.",
"REF .",
"In the curves highlighted in orange, the shoulder develops into a peak and two additional features appear on the high-energy tails of the main peaks.",
"This further evolves into the yellow vortices, which show five peaks of roughly equal heights.",
"The blue-dotted lines define another class, characterized by a prominent peak at positive energy.",
"Inspection of Fig.",
"REF in Appendix reveals that these vortices are often (but not always) in the neighborhood of an area with lower vortex density.",
"A Lorentz force therefore pushes these vortices into the vortex-deficient area, leading to a polarization of the bound states in that direction [55].",
"Since a prominent peak at positive energy is characteristic of the LDOS four to seven lattice parameters away from the vortex center (see Figs.",
"REF and REF ), a possible interpretation of the blue curves is that, in these vortices, the singularity point where the LDOS was calculated is four to seven lattice parameters away from the “electronic” center of the vortex, where the low-lying bound states have their maximum.",
"If that is the case, there must be another point inside the vortex where the LDOS resembles the red curves.",
"Similarly, the orange and yellow curves are somewhat alike the LDOS two to three lattice parameters away from the center of an isolated vortex.",
"Still, there are spectral shapes, like the yellow curves with peaks of almost equal heights and especially the green curve with a peak at zero energy, that are not seen close to an isolated vortex.",
"Sorting out what stems from LDOS polarization—i.e., more or less rigid displacement of the LDOS relative to the phase-singularity point—from what would be genuinely new spectral shapes induced by positional disorder, requires a detailed analysis of the LDOS in the neighborhood of each vortex.",
"This is left for a future study, and will hopefully also explain the gray curves, which seem to have very mixed up spectral shapes."
],
[
"Discussion",
"Let us summarize the results first.",
"A four-band tight-binding model (two inequivalent sites with two orbitals per site) with a bimodal uncorrelated random disorder and $s^{\\pm }$ pairing symmetry explains well the STM topography recorded on cleaved FeTe$_{0.55}$ Se$_{0.45}$ crystals.",
"In order to explain the STM spectroscopy as well within this model, we need to assume gaps a factor of two smaller that those observed in ARPES and optical spectroscopy.",
"While the disorder model captures well the spatial fluctuations of the STM spectra, it gives a spectral function much sharper than seen in ARPES experiments.",
"The calculated topography shows some degree of correlation with the disorder, but the spectral gap (separation between coherence peaks) is uncorrelated with the disorder, even if the superconducting gap (order parameter) is allowed to adjust self-consistently in the disorder.",
"The self-consistent order parameter of an isolated vortex, solved without disorder, is almost perfectly isotropic (cylindrical symmetry).",
"The LDOS in the vortex shows resolution-limited peaks consistent with discrete levels.",
"A spectral decomposition allows us to extract energies and wave functions from the calculated LDOS and to project the wave functions on angular-momentum eigenstates.",
"We find that the bound states come in pairs at each angular momentum and can mix several angular momenta.",
"The bound states change from being hole-like (negative angular momentum with the hole part of the wave function finite in the core) to being electron-like (positive angular momentum with the electron part of the wave function finite in the core) at a crossover energy.",
"The hole-like states have higher energy in attractive Se-rich regions and lower energy in repulsive Te-rich regions, while electron-like states behave the opposite way.",
"These variations are minute, though.",
"Relatively large changes of the chemical potential are needed in order to substantially modify the vortex spectra, in ways that do not resemble the variations seen in the experiments.",
"Finally, the LDOS at the phase-singularity point of a given vortex can present a variety of shapes, depending on the positions of disordered nearby vortices.",
"The electron-hole crossover in vortex bound states raises the question as to where the “gender” of these states comes from.",
"It is tempting to bind each series of bound states in Fig.",
"REF to one of the Fermi-surface pockets.",
"This fails, because both series start with a hole-like state.",
"Figure REF contains further insight.",
"It shows two different transitions: (i) at $\\Delta \\mu =13$ meV, when the hole pocket at $\\Gamma $ disappears, and (ii) at $\\Delta \\mu =-7.8$ meV, when the electron pocket at M transforms into a hole pocket.",
"At the transition (i), one hole-like state leaves the core by moving into the continuum: this suggests that one series of hole-like core states is indeed bound to the hole band at $\\Gamma $ .",
"At the transition (ii), the number of core states is conserved but their gender changes: one electron-like state leaves the core to the continuum while one hole-like state leaves the continuum to the core, and at the same time one hole-like state crosses zero energy and becomes electron-like.",
"This makes sense if the band at M carries two core states, one being hole-like and one being electron-like, irrespective of whether the Fermi surface is electron- or hole-like.",
"A scenario is proposed in Fig.",
"REF .",
"The hole band at $\\Gamma $ behaves as usual: it carries one series of states with angular momenta $\\mu =-1/2, -3/2, \\ldots $ , only the first of which has weight in the core.",
"These levels scale with the small gap and progressively fill the core as the chemical potential enters the band.",
"The band at M carries two series of states with opposite genders that scale with the large gap.",
"When the band is electron-like [Figs.",
"REF (a)–REF (c)], the lowest hole-like state has momentum $-1/2$ and the lowest electron-like state has momentum $+3/2$ .",
"The latter hybridizes with $+1/2$ and thus shows up in the core.",
"As the band gets emptied, the $+3/2$ state moves to the gap edge, while the $-1/2$ state has a nonmonotonic dependence, first moving toward the gap edge, then toward zero energy (see Fig.",
"REF ).",
"When the chemical potential hits the quadratic touching point [Fig.",
"REF (d)], the $-1/2$ state crosses zero energy and has mixed gender, while the state $+3/2$ reaches the gap edge and a state $-3/2$ appears at the opposite edge.",
"Finally, when the band is hole-like [Fig.",
"REF (e)], the lowest electron-like bound state has momentum $+1/2$ and the lowest hole-like state has momentum $-3/2$ , hybridizes with $-1/2$ , and becomes visible in the core [topmost curves in Fig.",
"REF (b)].",
"This interpretation implies that, in the situation of Fig.",
"REF (a), two bound states of opposite genders are present in the core even if $\\Delta \\mu $ is increased to a point that the hole band is out of the game.",
"We have checked that this is actually true, and it takes another topological transition at $\\Delta \\mu =+99.5$ meV, where the M pockets touch and transform into $\\Gamma $ hole pockets, to recover a unique gender in the cores, hole-like in this case.",
"Likewise, a topological transition at $\\Delta \\mu =-55.4$ meV transforms the situation of Fig.",
"REF (e) into an electron-like system with pockets at $(\\pi /2,\\pi /2)$ .",
"The vortex-core spectrum of the model [Fig.",
"REF (b)] with one tall peak at negative energy and one smaller peak at positive energy bears some resemblance with the observations made in part of the FeTe$_{0.55}$ Se$_{0.45}$ vortices [10], especially considering that nearby vortices, temperature, and a finite experimental resolution smoothen the theoretical spectrum.",
"The energies do not match, though.",
"The experiment reports peaks at $-0.6$ , $+0.45$ , $+1.2$ , and $+1.9$ meV.",
"The model has peaks at $-0.09$ , $-0.2$ , $+0.39$ , $+0.53$ , $+0.69$ , and $+0.84$ meV.",
"In qualitative agreement, both series show more electron-like than hole-like states.",
"In qualitative disagreement, the measurement shows a hole-like state that is not the lowest, while the model has all hole-like states lying below the electron-like states.",
"We would like to point out that the experiment faces a difficulty in locating the precise vortex center.",
"The same difficulty arises in the theory when disordered nearby vortices polarize the LDOS (Sec. ).",
"For example, the series of spectra in Supplementary Fig.",
"5e of Ref.",
"Chen-2018 shows a tall peak at $-0.88$ meV and a small peak at $+0.35$ meV in the putative vortex center, as well as a peak at $-0.28$ meV that develops far from the core.",
"Such a configuration—with the lowest-lying state localized farther away from the core than higher-energy states—is quite unexpected and impossible for a clean isolated vortex.",
"We speculate that the actual vortex center may be closer to where the $-0.28$ meV state is localized.",
"The calculations show that the chemical disorder may shift the vortex levels to some extent; however these shifts are way too small to resolve the discrepancy.",
"The seemingly smaller bound-state energies in the model compared with the experiment brings us back to the disagreement between the ARPES and STM spectral gaps.",
"Scaling the gaps by a factor of two in order to match ARPES would scale the core levels by a factor of four, giving numbers somewhat more alike the experimental ones.",
"For reconciling such a scaled model with the zero-field STM conductance, a third band is required with a gap of $\\sim 1$ meV.",
"This gap is unresolved in photoemission, but needed to explain the sharp edge at 1 meV seen in tunneling.",
"Additionally, one would need to invoke a nonlocal tunneling process that hides the larger gap near the M point.",
"How the conductance delivered by this specific tunneling path relates to the LDOS inside vortices is an open question.",
"In any case, the model as it stands confirms that the peaks observed by STM correspond to discrete vortex levels.",
"This establishes a sharp contrast with the parent compound FeSe, which is claimed to host nodal [56], [57] or at least very anisotropic [58], [59], [60] gaps and shows a broad signature in vortices, with no sign of discrete states [61].",
"The ability to observe and identify the “conventional” vortex states is an asset for demonstrating the unconventional nature of the Majorana zero mode in vortices where both types of quasiparticles coexist [36], [62].",
"The intricate mixing of electron- and hole-like bound states in vortices is worth investigating further.",
"Our results indicate that working at low field to approach as much as possible the isolated-vortex limit may be a good strategy in order to address the genuine vortex signature.",
"It is expected that the vortices appear less distorted under the STM at low field than at 5 T and that the maximum of the zero-bias conductance does coincide with the electronic vortex center, while at higher fields they can be separated.",
"Very much like the Chebyshev expansion, the STM lacks a direct access to the energies and wave functions of the bound states.",
"A spectral decomposition of the tunneling conductance, analogous to the LDOS decomposition used in the present work—if it turns out to be feasible—would be of great value for distinguishing nearly degenerate states and determining their quantum numbers.",
"The reverse behaviors of electron- and hole-like states relative to the chemical disorder may also be used as an identification tool, by studying the statistics of a given bound state in many vortices and correlating it with the topography.",
"On the theory side, many questions remain.",
"Beside the ARPES/STM gap problem already discussed, a more realistic model of disorder may be needed: as the cartoon used here has almost no effect on the vortex spectroscopy, it does not provide a clue as to why the vortices are pinned at disordered positions.",
"Another intriguing issue is the connection between the symmetries of the vortex and those of the Fermi surface.",
"In the quantum regime, it is expected that the breaking of cylindrical symmetry is led by the Fermi surface rather than by the gap anisotropy [23], [21], [7].",
"In this work, we have deliberately made the Fermi pockets maximally symmetric by a choice of the parameter $t_{2d}$ (Sec. ).",
"It would be interesting to monitor the evolution of the insets in Fig.",
"REF (b) as the parameter $t_{2d}$ is varied and the electron pocket gains a fourfold harmonics oriented either along $(\\pi ,0)$ or along $(\\pi ,\\pi )$ , depending on $t_{2d}$ .",
"A further extension of our model should also consider the Zeeman splitting.",
"In a 5 T field, this amounts to an energy scale of 0.58 meV that is not small compared with the gap size, unlike for the cuprates at similar fields.",
"Going beyond the two-dimensional idealization, or at least modeling the three-dimensional effects and understanding how they change the spectra, is also a direction to explore.",
"Finally, it is necessary to clarify whether there exists a link between the electron- or hole-like nature of the vortex bound states and the charge of elementary carriers as measured by the Hall coefficient, which was sometimes found to change sign as a function of temperature [63], [64]."
],
[
"Conclusion",
"The direct observation of individual Caroli–de Gennes–Matricon bound states has been a rare event since their prediction in 1964.",
"It appears that the low-density nearly compensated metal FeTe$_{0.55}$ Se$_{0.45}$ has just the right set of band-structure and pairing parameters to enable this observation with high-resolution and low-temperature STM.",
"Yet, the multiband nature of the compound with mixed electron-hole character together with nonlocal pairing, not to mention the intrinsic chemical disorder, challenges the well-established understanding for a clean single-band metal with local pairing.",
"The model studied here is a first step in trying to understand how the vortex bound states behave in this more complex setup and deep in the quantum regime.",
"The energy scales involved and the requirement that the interlevel spacing due to finite-size effects be much smaller than the superconducting gap immediately prompts unusually large system sizes that defeat ordinary methods based on straight diagonalization.",
"The Chebyshev expansion allows one to get through this, with the advantage of being a real-space method well suited for disordered systems.",
"Our calculations give a number of original insights without providing a completely satisfactory description of the observations made in FeTe$_{0.55}$ Se$_{0.45}$ .",
"We hope that this will motivate follow-up studies to refine and clarify the experimental data and develop the theory further.",
"This research was supported by the Swiss National Science Foundation under Division II.",
"The calculations were performed at the University of Geneva with the clusters Mafalda and Baobab."
],
[
"Energy resolution of the Chebyshev expansion",
"One of the main advantages of the Chebyshev expansion (REF ) is that it provides for the LDOS a formula that is an analytical function of energy.",
"On the contrary, calculations of the Green's function performed by direct inversion of the Hamiltonian matrix must be repeated independently at each energy of interest.",
"The exact LDOS is a sum of Dirac delta functions centered at the energy eigenstates, a quantity that is discrete for any finite-size system, and which Eq.",
"(REF ) approximates by a continuous function where each delta function is replaced by a function $\\delta _N(E)$ of finite $N$ -dependent width.",
"The function $\\delta _N(E)$ equals the DOS calculated for a one-orbital Hamiltonian $H=0$ , that is, with a single eigenvalue at zero energy.",
"Since $T_n(0)=\\cos (n\\pi /2)$ , the resolution function is $\\delta _N(E)=\\frac{1}{\\pi \\mathfrak {a}}\\mathrm {Re}\\,\\left\\lbrace \\frac{1}{\\sqrt{1-(E/\\mathfrak {a})^2}}\\right.",
"\\\\ \\left.",
"\\times \\left[1+2\\sum _{n=1}^Ne^{-in\\arccos (E/\\mathfrak {a})}\\cos \\left(\\frac{n\\pi }{2}\\right)K_n\\right]\\right\\rbrace .$ This function is similar to a Gaussian, however with tiny oscillations in the tails.",
"In order to evaluate its width, we calculate $\\nonumber \\Delta E&=2\\sqrt{2\\ln (2)\\int _{-\\infty }^{\\infty }dE\\,E^2\\delta _N(E)}\\\\&=2\\mathfrak {a}\\sqrt{\\ln (2)(1-K_2)}\\approx \\frac{2\\sqrt{\\ln 4}\\pi \\mathfrak {a}}{N},$ where the approximate result is valid at large $N$ .",
"For a Gaussian, this definition yields the full width at half maximum.",
"The resolution functions calculated for $N=9000$ and $N=18\\,000$ are plotted in Fig.",
"REF (b) in Appendix .",
"The corresponding energy resolutions are $\\Delta E=0.5$ meV and 0.25 meV, respectively."
],
[
"Comparison of the DOS for full and halved ARPES gaps",
"Figure REF (b) shows the DOS of the clean system (without the disorder associated with Te and Se potentials), calculated using the superconducting gap parameters deduced from ARPES measurements.",
"Two gaps are clearly resolved, one of amplitude $\\Delta _2\\cos (k_{\\mathrm {F},\\Gamma }a)+\\Delta _3\\cos ^2(k_{\\mathrm {F},\\Gamma }a)=2.4$ meV located on the $\\Gamma $ pocket and one of amplitude $-\\Delta _2\\cos (k_{\\mathrm {F},\\Gamma }a)+\\Delta _3\\cos ^2(k_{\\mathrm {F},\\Gamma }a)=-4.1$ meV located on the M pocket.",
"The gap edges are square-root singularities as expected for an order parameter of $s$ symmetry, but appear rounded due to the finite resolution of the calculation truncated at order $N$ .",
"The amount of rounding can be compared with the resolution function, also displayed in Fig.",
"REF (b).",
"A structure near $-9$ meV marks the bottom of the electron band at $-7.8$ meV [see Fig.",
"REF (b)], shifted by the gap opening.",
"Figure REF (a) shows the DOS calculated with halved values of $\\Delta _2$ and $\\Delta _3$ .",
"The two gaps at 1.2 and 2 meV match the structures observed experimentally by STM [10], [36].",
"With a resolution of 0.5 meV ($N=9000$ ), the smaller gap still produces a small peak, while experimentally it appears more like a shoulder inside the main gap.",
"For this reason, we have used $N=7000$ ($\\Delta E=0.64$ meV) in our calculations of the LDOS in Fig.",
"REF ."
],
[
"Choice of the disorder strength",
"Figure REF shows how the LDOS depends on the disorder strength measured by the parameter $V$ .",
"We have picked three random sites on the lattice, two of which turning out to be Te sites and one a Se site.",
"We recall that relative to the unperturbed chemical potential, the disorder potential is $+V$ at the Te sites and $-(11/9)V$ at the Se sites, in such a way that the spatially averaged potential is zero.",
"One thus expects a shift of spectral weight towards positive energies at Te sites and towards negative energies at Se sites.",
"This is clearly seen at $E>0$ , where the LDOS is mostly increased by increasing disorder strength at the Te sites, and mostly decreased at the Se site.",
"As the tunneling current is given by the integral of these LDOS curves from 0 to 10 meV, it is smaller at Se sites than at Te sites, which provides the contrast in Fig.",
"REF (d).",
"The disorder induces structures in the LDOS at energies that depend on the disorder configuration, but not on its strength.",
"The amplitude of these structures scales with the strength of the disorder.",
"We estimate that the typical amplitude of LDOS fluctuations for $V=1$ meV is similar to what is observed experimentally [10].",
"Figure REF also shows that the small gap on the $\\Gamma $ pocket is more sensitive to disorder: there can be significant fluctuations of the coherence peak height at the edges of the small gap, while the fluctuations are weaker for the large gap.",
"We note that for the calculations shown in Fig.",
"REF , unlike for those shown in Figs.",
"REF (d) and REF (f), we neglected the self-consistent adjustment of the superconducting order parameter.",
"Nevertheless, the tendency for the small gap to be more affected by disorder is confirmed by a statistical analysis of the peak height in the fully self-consistent map of Fig.",
"REF (d).",
"We find that (i) relative to the unperturbed DOS at the coherence-peak maximum, which marks the large gap, and at the energy of the shoulder, which marks the small gap, the average LDOS for $V=1$ meV ($V=2$ meV) goes down for the large gap by 1% (5%), while for the small gap it goes up by 1% (2%); (ii) more importantly, the standard deviation of the LDOS distribution at these energies, relative to its average, is only 3% (6%) for the large gap for $V=1$ meV ($V=2$ meV), but as large as 8% (15%) for the small gap."
],
[
"Correlation of topography and gap map with disorder",
"In order to measure the correlation between the local tunneling current $I(\\mathbf {r})$ shown in Fig.",
"REF (d) and the disorder landscape shown in Fig.",
"REF (c), we compute the correlation coefficient $R=\\frac{\\sum _{\\mathbf {r}}[I(\\mathbf {r})-\\langle I\\rangle ][V_{\\mathbf {r}}-\\langle V\\rangle ]}{\\sqrt{\\sum _{\\mathbf {r}}[I(\\mathbf {r})-\\langle I\\rangle ]^2\\sum _{\\mathbf {r}}[V_{\\mathbf {r}}-\\langle V\\rangle ]^2}},$ where $V_{\\mathbf {r}}$ is the local value of the potential, while $\\langle I\\rangle $ and $\\langle V\\rangle $ are the average current and potential, respectively.",
"We find a value $R=0.4$ , which quantifies the positive—although relatively weak—correlation between disorder and tunneling current that can be perceived by the eye.",
"In Fig.",
"REF , we compare this weak correlation with the absence of correlation between the potential and the spatial distribution of gap values.",
"The spectral gap, defined as half the energy separation between the main coherence peaks [inset of Fig.",
"REF (c)] is sensitive to disorder even if the order parameter is uniform.",
"Figure REF (c) shows that the spectral gap, which takes the value 2.27 meV in the absence of disorder, varies between 2.25 and 2.3 meV when disorder is included without letting the order parameter adjust self-consistently.",
"These variations show almost zero correlation with the disorder.",
"With the self-consistent order parameter, the correlation is only marginally higher [Fig.",
"REF (d)]."
],
[
"ARPES structure factor",
"In a one-orbital system of noninteracting electrons, the spectral function is simply $A(\\mathbf {k},E)=\\delta (E-\\xi _{\\mathbf {k}})$ , where $\\xi _{\\mathbf {k}}$ is the dispersion.",
"Consequently, for any energy, the spectral weight is unity all along the constant-energy surface $\\xi _{\\mathbf {k}}=E$ .",
"The situation is different for multiorbital systems, where the spectral weight is a function of the wave vector $\\mathbf {k}$ .",
"To obtain this function for our model, we start from the expression (REF ) and note that, in the translation-invariant case, the Green's function can be expressed in terms of momentum eigenstates as $G_{\\alpha ^{\\prime }\\alpha }(\\mathbf {r}^{\\prime },\\mathbf {r},E)=\\langle \\mathbf {r}^{\\prime }\\alpha ^{\\prime }|\\sum _{\\mathbf {k}\\gamma =\\pm }\\frac{|\\psi _{\\mathbf {k}\\gamma }\\rangle \\langle \\psi _{\\mathbf {k}\\gamma }|}{E+i0-\\xi _{\\mathbf {k}}^{\\gamma }}|\\mathbf {r}\\alpha \\rangle .$ In this expression, $\\xi _{\\mathbf {k}}^{\\gamma }$ is the dispersion given by Eq.",
"(REF ) and the momentum eigenfunctions are $\\langle \\mathbf {r}\\alpha |\\psi _{\\mathbf {k}\\gamma }\\rangle =\\delta _{\\alpha \\gamma }\\times {\\left\\lbrace \\begin{array}{ll}\\psi _{\\mathbf {k}\\gamma }^{(1)}e^{i\\mathbf {k}\\cdot \\mathbf {r}} & \\mathbf {r}\\in \\text{first sublattice}\\\\[1em]\\psi _{\\mathbf {k}\\gamma }^{(2)}e^{i\\mathbf {k}\\cdot (\\mathbf {r}-\\mathbf {\\tau })} & \\mathbf {r}\\in \\text{second sublattice}\\end{array}\\right.",
"}$ with $\\mathbf {\\tau }$ the vector joining the two sublattices—i.e., $\\mathbf {\\tau }=(a,0)$ —and $\\psi _{\\mathbf {k}\\pm }^{(1)}$ , $\\psi _{\\mathbf {k}\\pm }^{(2)}$ the values of the wave function on the first and second sublattice, respectively, in the unit cell sitting at the origin, for the eigenstate of energy $\\xi _{\\mathbf {k}}^{\\pm }$ .",
"We insert these expressions into Eq.",
"(REF ), where $M$ is to be understood as the total number of Fe sites, $M/2$ in each sublattice, and we split the expression into four terms, depending on whether $\\mathbf {r}$ and $\\mathbf {r}^{\\prime }$ belong to the first or second sublattice.",
"If both $\\mathbf {r}$ and $\\mathbf {r}^{\\prime }$ belong to the first sublattice, we get $\\frac{M}{2}\\sum _{\\alpha }\\big |\\psi _{\\mathbf {k}\\alpha }^{(1)}\\big |^2\\delta (E-\\xi _{\\mathbf {k}}^{\\alpha })$ .",
"If both belong to the second sublattice, we get the same expression with $\\psi _{\\mathbf {k}\\alpha }^{(2)}$ instead of $\\psi _{\\mathbf {k}\\alpha }^{(1)}$ .",
"The normalization of the wave function is $\\big |\\psi _{\\mathbf {k}\\alpha }^{(1)}\\big |^2+\\big |\\psi _{\\mathbf {k}\\alpha }^{(2)}\\big |^2=\\frac{2}{M}$ , such that these two contributions together simply give $\\sum _{\\alpha }\\delta (E-\\xi _{\\mathbf {k}}^{\\alpha })$ .",
"Adding to this the contributions coming from $\\mathbf {r}$ and $\\mathbf {r}^{\\prime }$ being in different sublattices, we get $A(\\mathbf {k},E)=\\sum _{\\alpha =\\pm }\\left[1+M\\mathrm {Re}\\,\\psi _{\\mathbf {k}\\alpha }^{(1)}(\\psi _{\\mathbf {k}\\alpha }^{(2)})^*e^{i\\mathbf {k}\\cdot \\mathbf {\\tau }}\\right]\\delta (E-\\xi ^{\\alpha }_{\\mathbf {k}}).$ The second term in the square brackets modulates the spectral weight along the constant-energy surfaces.",
"After inserting the expressions of $\\psi _{\\mathbf {k}\\pm }^{(1,2)}$ , we are led to the final expression: $A(\\mathbf {k},E)=\\sum _{\\alpha =\\pm }\\left\\lbrace 1+\\alpha \\frac{t_1[\\cos (k_xa)+\\cos (k_ya)]}{\\sqrt{t_1^2[\\cos (k_xa)+\\cos (k_ya)]^2+[2t_{2d}\\sin (k_xa)\\sin (k_ya)]^2}}\\right\\rbrace \\delta (E-\\xi ^{\\alpha }_{\\mathbf {k}}).$ The function within curly braces vanishes for $k_y=0$ and $\\alpha =-1$ ; hence the hole band has no spectral weight along $\\Gamma $ –M."
],
[
"Spectral decomposition of the vortex LDOS",
"Figure REF displays a series of LDOS spectra (black lines) taken on a path going from the vortex center (bottom curve) to a distance of $25a$ along the (10) direction (top curve).",
"All spectra are normalized to the peak height for clarity.",
"The shaded curves show the spectral decomposition according to Eq.",
"(REF ); each shaded curve has the shape of the spectral-resolution function and pairs of peaks with the same color represent the electron and hole amplitudes of a given Bogoliubov excitation.",
"The dashed lines show the sum of all peaks.",
"Focusing one's attention on the black curves, one gets the impression that two peaks disperse in space as one moves away from the vortex center and end around $\\pm 0.8$ meV at $x=25a$ .",
"The spectral analysis shows how this apparent dispersion results from the superposition of non-dispersing peaks.",
"This effect is well known in cases where the spectrum of bound states is dense (see, e.g., Ref.",
"Gygi-1991)."
],
[
"Disordered configuration of vortices",
"Figure REF shows the distribution of vortices used in the simulations.",
"The gray square represents the 200 nm $\\times $ 200 nm STM field of view of Fig.",
"1d in Ref. Chen-2018.",
"In this area, the vortex positions are located at maxima of the measured STM conductance (white and colors).",
"There are 97 vortices, corresponding to a field of 5 T. The colored circles indicate the 50 vortices whose core LDOS is displayed in Fig.",
"REF with the same color code.",
"Outside the area, the vortex positions were generated randomly with the constraint of being at least 16 nm apart (cyan).",
"The disordered vortices are surrounded by a regular square vortex lattice extending to infinity (magenta), which fixes the boundary condition for the phase of the order parameter [23]."
]
] | 1808.08390 |
[
[
"Markov Chain Monte Carlo for Dummies"
],
[
"Abstract This is an introductory article about Markov Chain Monte Carlo (MCMC) simulation for pedestrians.",
"Actual simulation codes are provided, and necessary practical details, which are skipped in most textbooks, are shown.",
"The second half is written for hep-th and hep-lat audience.",
"It explains specific methods needed for simulations with dynamical fermions, especially supersymmetric Yang-Mills.",
"The examples include QCD and matrix integral, in addition to SYM."
],
[
"Introduction",
" Markov Chain Monte Carlo (MCMC) simulation is a very powerful tool for studying the dynamics of quantum field theory (QFT).",
"But in hep-th community people tend to think it is a very complicated thing which is beyond their imagination [1].",
"They tend to think that a simulation code requires a very complicated and long computer program, they need to hire special postdocs with mysterious skill sets, very expensive supercomputers which they will never have access are needed, etc.",
"It is a pity, because MCMC is actually (at least conceptually) very simple,Because I don't like black boxes, I usually code everything by myself from scratch.",
"Still it is extremely rare to use anything more than $+,-,\\times ,\\div ,\\sin ,\\cos ,\\exp ,\\log ,\\sqrt{\\ \\ }$ , “if\" and loop.",
"Sometimes a few linear algebra routines from LAPACK [5] are needed, but you can copy and paste them.",
"For the Matrix Model of M-theory [6], [7] you don't even need LAPACK.",
"In short: nothing more than high school math is needed.",
"We just have to remove bugs patiently.",
"and a lot of nontrivial simulations can be done by using a laptop.",
"Indeed I have several papers for which the coding took at most an hour and crucial parts of simulations were done on a laptop, e.g.",
"[2], [3], [4].",
"You can quickly write a simple code, say a simple integral with the Metropolis algorithm, and it teaches you all important concepts.",
"Of course for certain theories we have to invest a lot of computational resources.",
"If you wanted to compete with lattice QCD experts, a lot of sophisticated optimizations, sometimes at the level of hardware, would be needed.",
"However there are many other subjects — including many problems in hep-th field — which are not yet at that stage.",
"There are many sophisticated techniques which enables us to perform large scale simulations with realistic computational resources.",
"They are sometimes technically very complicated but almost always conceptually very simple.",
"It is not easy to invent new techniques by ourselves, but it is not hard to learn and use something experts have already invented.",
"In case you have to do a serious simulation, you may not be able to code everything by yourself.",
"But you can use open-source simulation codes,See e.g.",
"[8], [9], [10], [11] for supersymmetric theories.",
"or you can work with somebody who can write a code.",
"And running the code and getting results are rather straightforward, once you understand the very basics.",
"In this introductory article, I will present basic knowledge needed for the Monte Carlo study of SYM.",
"I have two kinds of audience in mind: string theorists who have no idea what is lattice Monte Carlo, and lattice QCD practitioners who know QCD but not SYM.",
"For the former, I provide plenty of examples, including sample codes, which are sufficient for running actual simulation codes for SYM.",
"(Sec.",
", Sec.",
"and a part of Sec.",
"would be useful for much broader audience including non-physicists.)",
"These materials can also be useful for students and postdocs already working with MCMC; the materials presented here are something all senior people working in MCMC expect their students/postdocs to know, but many students/postdocs do not have chance to learn.",
"I also explain the technical differences between lattice QCD and SYM simulations, which are useful for both string and lattice people."
],
[
"Note",
"This version is (probably) not final; more examples and sample codes will be added.",
"I have decided to post it to arXiv because lately I am too busy and do not have much time to work on this.",
"Sample codes can be downloaded from GitHub, https://github.com/MCSMC/MCMC_sample_codes.",
"The latest version of this review will be uploaded there as well.",
"Comments, requests and bug/typo reports will be appreciated."
],
[
"Markov Chain Monte Carlo (MCMC)",
"Suppose we want to perform a Euclidean path-integral with a partition function $Z=\\int [d\\phi ] e^{-S[\\phi ]},$ where the action $S[\\phi ]$ depends only on bosonic field(s) $\\phi $ .",
"(In later sections I will explain how to include fermions.)",
"Usually we are interested in the expectation values of operators, $\\langle {\\cal O}\\rangle =\\frac{1}{Z}\\int [d\\phi ] e^{-S[\\phi ]}{\\cal O}(\\phi ).$ In order to make sense of this expression, typically we regularize the theory on a lattice, so that the path-integral reduces to a usual integral with respect to finitely many variables.",
"Let's call these variables $x_1,x_2,\\cdots ,x_p$ .",
"Typically, the lattice action is so complicated that it is impossible to estimate the integral analytically.",
"Because we have to send $p$ to infinity in order to take the continuum limit or the large-volume limit, a naive numerical integral does not work either.",
"If we approximate the integral by a sum by dividing the integral region of each $x_i$ to $n$ intervals like in Fig.",
"REF , the calculation cost is proportional to $n^p$ , simply because we have to take a sum of $n^p$ numbers.",
"This is hopelessly hard for realistic values of $n$ and $p$ .",
"Suppose $n=100$ and $p=10$ .",
"Then we have to take a sum of $10^{20}$ numbers.",
"Let us convince my collaborators in Livermore Laboratory that this integral is extremely important, and use their supercomputer Sequoia, which was the fastest in the world from 2012 to 2013.",
"Its performance is 20PFLOPS, namely it can process $2\\times 10^{16}$ double-precision floating-point arithmetics every second.",
"Let's ignore the cost for calculating the value of the function at each point, and consider only a sum of given numbers.",
"(This is an unrealistic assumption, of course.)",
"But already it takes $10^{20}/(2\\cdot 10^{16})=5000$ seconds.",
"Well, it may be acceptable... but if you take $p=15$ , it takes $10^{30}/(2\\cdot 10^{16})=5\\times 10^{13}$ seconds, which is about 634,000 years.",
"For a 4d pure SU$(3)$ Yang-Mills on lattice with $10^4$ points, $p$ is $4\\times (3^2-1)\\times 10^4$ .",
"We cannot even take $n=2$ .",
"This is the notorious curse of dimensionality; when the dimension $p$ is large it is practically impossible to scan the phase space.",
"Perhaps when you are reading this article you have access to much better machines, but it will not give you much gain.",
"Figure: Approximate the integral by a sum of the area of rectangles.Markov Chain Monte Carlo (MCMC) circumvents the curse of dimensionality based on the idea of importance sampling.",
"In most cases of our interest, the majority of the phase space is irrelevant because the action $S$ is large and the weight $e^{-S}$ is very small.",
"If we can find important regions in the phase space and invest our resources there, we can avoid the curse of dimensionality.",
"MCMC enables us to actually do it.",
"We assume $S[x_1,x_2,\\cdots ,x_p]$ is real and the partition function $Z=\\int dx_1\\cdots dx_p e^{-S[x_1,x_2,\\cdots ,x_p]}$ is finite.",
"In MCMC simulations, we construct a chain of sets of variables $\\lbrace x^{(0)}\\rbrace \\rightarrow \\lbrace x^{(1)}\\rbrace \\rightarrow \\lbrace x^{(2)}\\rbrace \\rightarrow \\cdots \\lbrace x^{(k)}\\rbrace \\rightarrow \\lbrace x^{(k+1)}\\rbrace \\rightarrow \\cdots $ satisfying the following conditions: Markov Chain.",
"The probability of obtaining $\\lbrace x^{(k+1)}\\rbrace $ from $\\lbrace x^{(k)}\\rbrace $ does not depend on the previous configurations $\\lbrace x^{(0)}\\rbrace ,\\lbrace x^{(1)}\\rbrace ,\\cdots , \\lbrace x^{(k-1)}\\rbrace $ .",
"We denote this transition probability by $T[\\lbrace x^{(k)}\\rbrace \\rightarrow \\lbrace x^{(k+1)}\\rbrace ]$ .",
"Irreducibility.",
"Any two configurations are connected by finite steps.",
"Aperiodicity.",
"The period of a configuration $\\lbrace x\\rbrace $ is given by the greatest common divisor of possible numbers of steps to come back to itself.",
"When the period is 1 for all configurations, the Markov chain is called aperiodic.",
"Detailed balance condition.",
"The transition probability $T$ satisfies $e^{-S[\\lbrace x\\rbrace ]}T[\\lbrace x\\rbrace \\rightarrow \\lbrace x^{\\prime }\\rbrace ]=e^{-S[\\lbrace x^{\\prime }\\rbrace ]}T[\\lbrace x^{\\prime }\\rbrace \\rightarrow \\lbrace x\\rbrace ]$ .",
"Then, the probability distribution of $\\lbrace x^{(k)}\\rbrace (k=1,2,\\cdots )$ converges to $P(x_1,x_2,\\cdots ,x_p)=e^{-S(x_1,x_2,\\cdots ,x_p)}/Z$ as the chain becomes longer.",
"The expectation values are obtained by taking the average over the configurations, $\\langle \\hat{O}\\rangle =\\int dx_1\\cdots dx_p O(x_1,\\cdots ,x_p)P(x_1,x_2,\\cdots ,x_p)=\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\sum _{k=1}^n O(x_1^{(k)},\\cdots ,x_p^{(k)}).$ Note that this is not an approximation; this is exact.",
"Practically we can have only finitely many configurations, so we can only approximate the right hand side by a finite sum.",
"However there is a systematic way to improve it to arbitrary precision: just make the chain longer.",
"Although a proof is rather involved, the importance of each condition can easily be understood.",
"Probably the most nontrivial condition for most readers is the detailed balance.",
"Suppose the chain converged to a certain distribution $P[\\lbrace x\\rbrace ]$ .",
"Then it has to be `stationary', or equivalently, it should be invariant when shifted one step: $\\sum _{\\lbrace x\\rbrace }P[\\lbrace x\\rbrace ]T[\\lbrace x\\rbrace \\rightarrow \\lbrace x^{\\prime }\\rbrace ]=P[\\lbrace x^{\\prime }\\rbrace ]$ If $P[\\lbrace x\\rbrace ]\\propto e^{-S[\\lbrace x\\rbrace ]}$ , it follows from the detailed balance condition as $\\sum _{\\lbrace x\\rbrace }P[\\lbrace x\\rbrace ]T[\\lbrace x\\rbrace \\rightarrow \\lbrace x^{\\prime }\\rbrace ]&=&\\sum _{\\lbrace x\\rbrace }P[\\lbrace x^{\\prime }\\rbrace ]T[\\lbrace x^{\\prime }\\rbrace \\rightarrow \\lbrace x\\rbrace ]\\nonumber \\\\&=&P[\\lbrace x^{\\prime }\\rbrace ]\\sum _{\\lbrace x\\rbrace }T[\\lbrace x^{\\prime }\\rbrace \\rightarrow \\lbrace x\\rbrace ]\\nonumber \\\\&=&P[\\lbrace x^{\\prime }\\rbrace ].$ I recommend you to follow a complete proof once by looking at an appropriate textbook, but you don't have to keep it in your brain.",
"You will need it only when you try to invent something better than MCMC.",
"Note that, even if you use exactly the same simulation code, if you take different initial condition or use different sequence of random numbers, you get different chain.",
"Still, the chain always converges to the same statistical distribution."
],
[
"Off-topic: Bayesian analysis",
" MCMC is powerful outside physics as well.",
"To see a little bit of flavor, let us consider the Bayes's theorem, $P(B_i|A)=\\frac{P(A|B_i)P(B_i)}{\\sum _jP(A|B_j)P(B_j)}.$ Here $P(A|B)$ is the conditional probability: Probability that $A$ is true when the condition $B$ is satisfied.",
"For example $B_1,B_2,B_3\\cdots $ are physicists, high tech engineers, MLB players etc, and $A$ is millionaires.",
"Suppose $P(A|B_i)$ and $P(B_i)$ are given (e. g. $P(A|B_1)=10^{-4}, P(A|B_2)=0.05, P(A|B_3)=0.8,\\cdots $ ), and we want to derive $P(B_i|A)$ .",
"We can identify $B_j$ and $P(A|B_j)P(B_j)$ with the value of the field $\\phi $ , the path integral weight $e^{-S[\\phi ]}[d\\phi ]$ .",
"The denominator $\\sum _jP(A|B_j)P(B_j)=P(A)$ is regarded as the partition function $Z$ .",
"Then we can use MCMC to obtain $P(B_i|A)\\sim \\frac{e^{-S[\\phi ]}[d\\phi ]}{Z}$ via the Bayes's theorem; namely we can collect many samples and see the distribution.",
"Also if we know $f(B_i)\\equiv P(C|B_i)$ you can calculate $P(C|A)$ as $P(C|A)=\\langle f\\rangle .$ For example $C$ is nice muscle and $P(C|B_1)=P(C|B_2)=0.01, P(C|B_3)=0.99,\\cdots $ ."
],
[
"Integration of one-variable functions with Metropolis algorithm",
" Let us start with the integration of a one-variable function with the Metropolis algorithm [12].",
"In particular, we will consider the simplest example we can imagine: the Gaussian integral, $S(x)=x^2/2$ .",
"This very basic example contains essentially all important ingredients; all other cases are, ultimately, just technical improvements of this example.",
"Of course we can handle the Gaussian integral analytically.",
"Also there is a much better algorithm for generating Gaussian random numbers (see Appendix ).",
"We use it just for an educational purpose."
],
[
"Metropolis Algorithm",
" Let us consider the weight $e^{-S(x)}$ , where $S(x)$ is a continuous function of $x\\in {\\mathbb {R}}$ bounded from below.",
"We further assume that $\\int e^{-S(x)}dx$ is finite.",
"The Metropolis algorithm gives us a chain of configurations (or just `values' in the case of single variable) $x^{(0)}\\rightarrow x^{(1)}\\rightarrow x^{(2)}\\rightarrow \\cdots $ which satisfies the conditions listed above: Randomly choose $\\Delta x\\in {\\mathbb {R}}$ , and shift $x^{(k)}$ as $x^{(k)}\\rightarrow x^{\\prime }\\equiv x^{(k)}+\\Delta x$ .",
"($\\Delta x$ and $-\\Delta x$ must appear with the same probability, so that the detailed balance condition is satisfied.",
"Here we use the uniform random number between $\\pm c$ , where $c>0$ is the `step size'.)",
"Metropolis test: Generate a uniform random number $r$ between 0 and 1.",
"If $r<e^{-\\Delta S}$ , where $\\Delta S = S(x^{\\prime })-S(x^{(k)})$ , then $x^{(k+1)}=x^{\\prime }$ , i.e.",
"the new value is `accepted.'",
"Otherwise $x^{(k+1)}=x^{(k)}$ , i.e.",
"the new value is `rejected.'",
"Repeat the same for $k+1, k+2, \\cdots $ .",
"It is an easy exercise to see that all conditions explained above are satisfied: It is a Markov Chain, because the past history is not referred either for the selection of $\\Delta x$ or the Metropolis test.",
"It is irreducible; for example, any $x$ and $x^{\\prime }$ , by taking $n$ large we can make $\\frac{x-x^{\\prime }}{n}$ to be in $[-c,c]$ , and there is a nonzero probability that $\\Delta x=\\frac{x-x^{\\prime }}{n}$ appears $n$ times in a row and passes the Metropolis test every time.",
"For any $x$ , there is a nonzero probability of $\\Delta x=0$ .",
"Hence the period is one for any $x$ .",
"If $|x-x^{\\prime }|>c$ , $T[x\\rightarrow x^{\\prime }]=T[x^{\\prime }\\rightarrow x]=0$ .",
"When $|x-x^{\\prime }|\\le c$ , both $\\Delta x = x^{\\prime }-x$ and $\\Delta x^{\\prime } = x-x^{\\prime }$ are chosen with probability $\\frac{1}{2c}$ .",
"Let us assume $\\Delta S=S[x^{\\prime }]-S[x]>0$ , without a loss of generality.",
"Then the change $x\\rightarrow x^{\\prime }$ passes the Metropolis test with probability $e^{-\\Delta S}$ , while $x^{\\prime }\\rightarrow x$ is always accepted.",
"Hence $T[x\\rightarrow x^{\\prime }]=\\frac{e^{-\\Delta S}}{2c}$ and $T[x^{\\prime }\\rightarrow x]=\\frac{1}{2c}$ , and $e^{-S[x]}T[x\\rightarrow x^{\\prime }]=e^{-S[x^{\\prime }]}T[x^{\\prime }\\rightarrow x]=\\frac{e^{-S[x^{\\prime }]}}{2c}$ ."
],
[
"How it works",
"Let me show a sample code written in C: #include <stdio.h> #include <stdlib.h> #include <math.h> #include <time.h> int main(void){ int iter,niter=100; int naccept; double step_size=0.5e0; double x,backup_x,dx; double action_init, action_fin; double metropolis; srand((unsigned)time(NULL)); /*********************************/ /* Set the initial configuration */ /*********************************/ x=0e0; naccept=0; /*************/ /* Main loop */ /*************/ for(iter=1;iter<niter+1;iter++){ backup_x=x; action_init=0.5e0*x*x; dx = (double)rand()/RAND_MAX; dx=(dx-0.5e0)*step_size*2e0; x=x+dx; action_fin=0.5e0*x*x; /*******************/ /* Metropolis test */ /*******************/ metropolis = (double)rand()/RAND_MAX; if(exp(action_init-action_fin) > metropolis) /* accept */ naccept=naccept+1; else /* reject */ x=backup_x; /***************/ /* data output */ /***************/\t printf(\"} Let me explain the code line by line.",
"Firstly, by srand((unsigned)time(NULL)); the seed for the random number generator is set.",
"A default random number generator is used, by using the system clock time to set the seed randomly.",
"For more serious simulations, it is better to use a good generator, say the Mersenne twister.",
"Then we specify an initial configuration; here we took $x^{(0)}=0$ .",
"naccept counts how many times the new values are accepted.",
"x=0e0; naccept=0; Then we move on to the main part of the simulation, which is inside the following loop: for(iter=1;iter<niter+1;iter++){ .... } Here, iter corresponds to $k$ , and niter is the number of configurations we will collect during the simulation.",
"Inside the loop, the first thing we have to do is to save the value of x$=x^{(k)}$ , because it may or may not be updated: backup_x=x; Then action${\\ }$ init$=S(x^{(k)})$ is calculated.",
"Now we have to generate a random variation dx$=\\Delta x$ with an appropriate step size, and shift $x$ to $x^{\\prime }=x^{(k)}+\\Delta x$ .",
"We can generate a uniform random number in $[0,1]$ by rand()/RAND${\\ }$MAX.",
"From this we can easily get $-c<\\Delta x<c$ .",
"dx = (double)rand()/RAND_MAX; dx=(dx-0.5e0)*step_size*2e0; x=x+dx; By using $x^{\\prime }$ , action${\\ }$ fin$=S(x^{\\prime })$ is calculated.",
"Note that x and backup${\\ }$ x in the code correspond to $x^{\\prime }$ and $x^{(k)}$ .",
"Finally we perform the Metropolis test: metropolis = (double)rand()/RAND_MAX; if(exp(action_init-action_fin) > metropolis) /* accept */ naccept=naccept+1; else /* reject */ x=backup_x; metropolis is a uniform random number in $[0,1]$ , which corresponds to $r$ .",
"Depending on the result of the test, we accept or reject $x^{\\prime }$ .",
"We emphasize again that all MCMC simulations have exactly the same structure; there are many fancy algorithms, but essentially, they are all about improving the step $x\\rightarrow x+\\Delta x$ .",
"We take $x^{(0)}=0$ , and $\\Delta x$ to be a uniform random number between $-0.5$ and $0.5$ .",
"(As we will see later, this parameter choice is not optimal.)",
"In Fig.",
"REF , we show the distribution of $x^{(1)},x^{(2)},\\cdots ,x^{(n)}$ , for $n=10^3, 10^5$ and $10^7$ .",
"We can see that the distribution converges to $e^{-x^2/2}/\\sqrt{2\\pi }$ .",
"The expectation values $\\langle x\\rangle = \\frac{1}{n}\\sum _{k=1}^n x^{(k)}$ and $\\langle x^2\\rangle = \\frac{1}{n}\\sum _{k=1}^n \\left(x^{(k)}\\right)^2$ are plotted in Fig.",
"REF .",
"As $n$ becomes large, they converge to the right values, 0 and 1.",
"Note that the step size $c$ should be chosen so that the acceptance rate is not too high, not too low.",
"If $c$ is too large, the acceptance rate becomes extremely low, then the value is rarely updated.",
"If $c$ is too small, the acceptance rate is almost 1, but the change of the value at each step is extremely small.",
"In both cases, huge amount of configurations are needed in order to approximate the integration measure accurately.",
"The readers can confirm it by changing the step size in the sample code.",
"(We will demonstrate it in Sec.",
"REF .)",
"Typically the acceptance rate 30% – 80% is good.",
"But it can heavily depend on the detail of the system and algorithm; See Sec.",
"REF and Sec.",
"REF for details.",
"Figure: The distribution of x (1) ,x (2) ,⋯,x (n) x^{(1)},x^{(2)},\\cdots ,x^{(n)}, for n=10 3 ,10 5 n=10^3, 10^5 and 10 7 10^7,and e -x 2 /2 2π\\frac{e^{-x^2/2}}{\\sqrt{2\\pi }}.Figure: 〈x〉=1 n∑ k=1 n x (k) \\langle x\\rangle = \\frac{1}{n}\\sum _{k=1}^n x^{(k)} and 〈x 2 〉=1 n∑ k=1 n x (k) 2 \\langle x^2\\rangle = \\frac{1}{n}\\sum _{k=1}^n \\left(x^{(k)}\\right)^2.As nn becomes large, they converge to the right values, 0 and 1."
],
[
"A bad example",
" It is instructive to see wrong examples.",
"Let us take $\\Delta x$ from $\\left[-\\frac{1}{2}, 1\\right]$ , so that the detailed balance condition is violated; for example $0\\rightarrow 1$ has a finite probability while $1\\rightarrow 0$ is impossible.",
"Then, as we can see from Fig REF , the chain does not converge to the right probability distribution.",
"Figure: The distribution of x (1) ,x (2) ,⋯,x (n) x^{(1)},x^{(2)},\\cdots ,x^{(n)} for n=10 7 n=10^7, with wrong algorithm with Δx∈-1 2,1\\Delta x\\in \\left[-\\frac{1}{2}, 1\\right].The dotted line is the right Gaussian distribution e -x 2 /2 2π\\frac{e^{-x^2/2}}{\\sqrt{2\\pi }}."
],
[
"Autocorrelation and Thermalization",
" In MCMC, $x^{(k+1)}$ is obtained from $x^{(k)}$ .",
"In general, they are correlated.",
"This correlation is called autocorrelation.",
"The autocorrelation can exist over many steps.",
"The autocorrelation length depends on the detail of the theory, algorithm and the parameter choice.",
"Because of the autocorrelation, some cares are needed.",
"In the above, we have set $x^{(0)}=0$ , because we knew it is `the most important configuration'.",
"What happens if we start with an atypical value, say $x^{(0)}=100$ ?",
"It takes some time for typical values to appear, due to the autocorrelation.",
"The history of the Monte Carlo simulation with this initial condition is shown in Fig.",
"REF .",
"The value of $x$ eventually reaches to `typical values' $|x|\\lesssim 1$ — we often say `the configurations are thermalized' (note that the same term `thermalized' has another meaning as well, as we will see shortly) —, but a lot of steps are needed.",
"If we include `unthermalized' configurations when we estimate the expectation values, we will suffer from huge error unless the number of configurations are extremely large.",
"We should discard unthermalized configurations, say $n\\lesssim 1000$ .Number of steps needed for the thermalization is sometimes called `burn-in time' or `mixing time'.",
"In generic, more complicated situation, we don't a priori know what the typical configurations look like.",
"Still, whether the configurations are thermalized or not can be seen by looking at several observables.",
"As long as they are changing monotonically, it is plausible that the configuration is moving toward a typical one.",
"When they start to oscillate around certain values (see Fig.",
"REF , we can see a fluctuation around $x=0$ ), it is reasonable to think the configuration has been thermalized.For more careful analysis, we can vary the number of configurations removed and take it large enough so that the average values do not change any more.",
"Fig.",
"REF is a zoom-up of Fig.",
"REF , from $n=1000$ to $n=2000$ .",
"We can see that the values of $x$ can be strongly correlated unless they are 20 or 30 step separated.",
"(We will give a quantitative analysis in Sec.",
"REF .)",
"When we estimate the statistical error, we should not treat all configuration as independent; rather we have only 1 independent configuration every 20 or 30 steps.",
"Note also that we need sufficiently many independent configurations in order to estimate the expectation values reliably.",
"We often say the simulation has been thermalized when we have sufficiently many independent configurations so that the expectation values are stabilized.",
"Figure: Monte Calro history of the Gaussian integral with the Metropolis algorithm, Δx∈[-0.5,0.5]\\Delta x\\in [-0.5,0.5].We took the initial value to be a very atypical value, x=100x=100.",
"It takes a lot of steps to reach typical values |x|∼1|x|\\sim 1.Figure: A zoom-up of Fig.",
", from n=1000n=1000 to n=2000n=2000.The values of xx can be strongly correlated unless they are at least 20 or 30 steps separated."
],
[
"Jackknife method",
" The Jackknife method provides us with a simple way to estimate the autocorrelation length.",
"Here we assume the quantity of interest can be calculated for each sample.The correlation function is in this class.",
"The mass of particle excitation is not; we need to calculate two-point function by using many samples and then extract the mass from its exponential decay.",
"For more generic cases, see Appendix .",
"In the Jackknife method, we first divide the configurations to bins with width $w$ ; the first bin consists of $\\lbrace x^{(1)}\\rbrace , \\lbrace x^{(2)}\\rbrace ,\\cdots ,\\lbrace x^{(w)}\\rbrace $ , the second bin is $\\lbrace x^{(w+1)}\\rbrace , \\lbrace x^{(w+2)}\\rbrace ,\\cdots ,\\lbrace x^{(2w)}\\rbrace $ , and so on.",
"Suppose we have $n$ bins.",
"Then we define the average of an observable $f(x)$ with $k$ -th bin removed, $\\overline{f}^{(k,w)}\\equiv \\frac{1}{(n-1)w}\\sum _{j\\ \\notin \\ k{\\rm th\\ bin}}f(x^{(j)}).$ The average value $\\overline{f}\\equiv \\frac{1}{n}\\sum _{k}\\overline{f}^{(k,w)}$ is the same as the average of all samples, $\\frac{1}{nw}\\sum _j f(x^{(j)})$ , for the class of quantities we are discussing.",
"The Jackknife error is defined by $\\Delta _w\\equiv \\sqrt{\\frac{n-1}{n}\\sum _{k}\\left(\\overline{f}^{(k,w)}-\\overline{f}\\right)^2}.$ By using $\\tilde{f}^{(k,w)}\\equiv \\frac{1}{w}\\sum _{j\\ \\in \\ k{\\rm th\\ bin}}f(x^{(j)}),$ we can easily see $\\overline{f}^{(k,w)}-\\overline{f}=\\frac{\\overline{f}-\\tilde{f}^{(k,w)}}{n-1}.$ Hence $\\Delta _w\\equiv \\sqrt{\\frac{1}{n(n-1)}\\sum _{k}\\left(\\tilde{f}^{(k,w)}-\\overline{f}\\right)^2}.$ Namely $\\Delta _w$ is the standard error obtained by treating $\\tilde{f}^{(k,w)}$ to be independent samples.",
"Typically, as $w$ becomes large, $\\Delta _w$ increases and then becomes almost constant at certain value of $w$ , which we denote by $w_c$ .",
"This $w_c$ and $\\Delta _{w_c}$ give good estimates of the autocorrelation length and the error bar.",
"It can be understood as follows.",
"Let us consider two bin sizes $w$ and $2w$ .",
"Then $\\tilde{f}^{(k,2w)}=\\frac{\\tilde{f}^{(2k-1,w)}+\\tilde{f}^{(2k,w)}}{2},$ $\\Delta _{2w}&=&\\sqrt{\\frac{1}{\\frac{n}{2}\\left(\\frac{n}{2}-1\\right)}\\sum _{k=1}^{n/2}\\left(\\tilde{f}^{(k,2w)}-\\overline{f}\\right)^2}\\nonumber \\\\&=&\\sqrt{\\frac{4}{n(n-2)}\\sum _{k=1}^{n/2}\\left(\\frac{\\left(\\tilde{f}^{(2k-1,w)}-\\overline{f}\\right)}{2}+\\frac{\\left(\\tilde{f}^{(2k,w)}-\\overline{f}\\right)}{2}\\right)^2}.$ If $w$ is sufficiently large, $\\tilde{f}^{(2k-1,w)}-\\overline{f}$ and $\\tilde{f}^{(2k,w)}-\\overline{f}$ should be independent, and the cross-term $\\left(\\tilde{f}^{(2k-1,w)}-\\overline{f}\\right)\\cdot \\left(\\tilde{f}^{(2k,w)}-\\overline{f}\\right)$ should average to zero after summing up with respect to sufficiently many $k$ .",
"Then $\\Delta _{2w}\\sim \\sqrt{\\frac{1}{n^2}\\sum _{k=1}^{n}\\left(\\tilde{f}^{(k,w)}-\\overline{f}\\right)^2}\\sim \\Delta _{w}.$ In this way, $\\Delta _{w}$ becomes approximately constant when $w$ is large enough so that $\\tilde{f}^{(k,w)}$ can be treated as independent samples.",
"(Note that $n$ must also be large for the above estimate to hold.)",
"$\\Delta _{w}$ is the standard error of these `independent samples'.",
"In Fig.",
"REF , $\\langle x^2\\rangle $ and Jackknife error $\\Delta _{w}$ are shown by using first 50000 samples.",
"We can see that $w_c=50$ is a reasonably safe choice; $w_c=20$ is already in the right ballpark.",
"In Fig.",
"REF , bin-averaged values with $w=50$ are plotted.",
"They do look independent.",
"We obtained $\\langle x^2\\rangle =0.982\\pm 0.012$ , which agree reasonably well with the analytic answer, $\\langle x^2\\rangle =1$ .",
"Figure: 〈x 2 〉\\langle x^2\\rangle and Jackknife error Δ w \\Delta _{w} with 50000 samples.Figure: Bin-averaged version of Fig.",
", with bigger window for nn, with w=50w=50."
],
[
"Tuning the simulation parameters",
" In order to run the simulation efficiently, we should tune parameters so that we can obtain more independent samples with less cost.In parallelized simulations, the notion of the cost is more nontrivial because time is money.",
"Sometimes you may want to invest more electricity and machine resources to obtain the same result with shorter time.",
"In the current example (Gaussian integral with uniform random number), when the step size $c$ is too large, unless $\\Delta x\\lesssim 1$ the configuration is rarely updated; the acceptance rate and the autocorrelation length scale as $1/c$ and $c$ , respectively.",
"On the other hand, when $c$ is too small, the configurations are almost always updated, but only tiny amount.",
"This is just a random walk with a step size $c$ , and hence the average change after $n$ steps is $c\\sqrt{n}$ .",
"Therefore the autocorrelation length should scale as $n\\sim 1/c^2$ .",
"We expect the autocorrelation is minimized between these two regions.",
"In Table REF , we have listed the acceptance rate for several values of $c$ .",
"We can see that the large-$c$ scaling ($c\\times {\\rm acceptance}\\sim {\\rm const}$ ) sets in at around $c=2$ $\\sim $ $c=4$ .",
"In Fig.",
"REF we have shown how $\\langle x^2\\rangle $ converges to 1 as the number of configurations increases.",
"We can actually see that $c=2$ and $c=4$ show faster convergence compared to too small or too large $c$ .",
"Table: Step size vs acceptance rate, total 10000 samples.Figure: 〈x 2 〉=1 n∑ k=1 n x (k) 2 \\langle x^2\\rangle = \\frac{1}{n}\\sum _{k=1}^n \\left(x^{(k)}\\right)^2 for several different step sizes cc."
],
[
"How to calculate partition function",
" In MCMC, we cannot directly calculate the partition function $Z$ ; we can only see the expectation values.",
"Usually the partition function is merely a normalization factor of the path integral measure which does not affect the path integral, so we do not care.",
"But sometimes it has interesting physical meanings; for example it can be used to test the conjectured dualities between supersymmetric theories.",
"Suppose you want to calculate $Z=\\int dx e^{-S(x)}$ , where $S(x)$ is much more complicated than $S_0(x)=x^2/2$ .",
"By using MCMC, we can calculate the ratio between $Z$ and $Z_0=\\int dx e^{-S_0(x)}=\\sqrt{2\\pi }$ , $\\frac{Z}{Z_0}=\\frac{1}{Z_0}\\int dx e^{-S_0}\\cdot e^{S_0-S}=\\left\\langle e^{S_0-S}\\right\\rangle _0,$ where $\\langle \\ \\cdot \\ \\rangle _0$ stands for the expectation value with respect to the action $S_0$ .",
"Because we know $Z_0$ analytically, we can determine $Z$ ."
],
[
"Overlapping problem and its cure",
" The method described above can always work in principle.",
"In practice, however, it fails when the probability distributions $\\rho (x)=\\frac{e^{-S(x)}}{Z}$ and $\\rho _0(x)=\\frac{e^{-S_0(x)}}{Z_0}$ do not have sufficiently large overlap.",
"As a simple example, let us consider $S=(x-c)^2/2$ (though you can analytically handle it!).",
"Then $\\rho (x)$ and $\\rho _0(x)$ have peaks around $x=c$ and $x=0$ , respectively.",
"When $c$ is very large, say $c=100$ , the value of $e^{S_0-S}$ appearing in the simulation is almost always an extremely small number $\\sim e^{-5000}$ , and once every $e^{+5000}$ steps or so we get an extremely large number $\\sim e^{+5000}$ .",
"And they average to $\\frac{Z}{Z_0}=1$ .",
"Clearly, we cannot get an accurate number if we truncate the sum at a realistic number of configurations.",
"It happens because of the absence of the overlap of $\\rho (x)$ and $\\rho _0(x)$ , or equivalently, because important configurations in two different theories are different; hence the `operator' $e^{S_0-S}$ behaves badly at the tail of $\\rho _0(x)$ .",
"This is so-called overlapping problem.In SYM, the overlapping problem can appear combined with the sign problem; we will revisit this point in Sec.",
"REF .",
"In the current situation, the overlapping problem can easily be solved as follows.",
"Let us introduce a series of actions $S_0$ , $S_1$ , $S_2$ , ..., $S_k=S$ .",
"We choose them so that $S_i$ and $S_{i+1}$ are sufficiently close and the ratio $\\frac{Z_{i+1}}{Z_i}$ , where $Z_i=\\int dx e^{-S_i(x)}$ , can be calculated without the overlapping problem.",
"For example we can take $S_i=\\frac{1}{2}\\left(x-\\frac{i}{k}c\\right)^2$ with $\\frac{c}{k}\\sim 1$ .",
"Then we can obtain $Z=Z_k$ by calculating $\\frac{Z_1}{Z_0}$ , $\\frac{Z_2}{Z_1}$ , $\\cdots $ , $\\frac{Z_k}{Z_{k-1}}$ .",
"The same method can be applied to any complicated $S(x)$ , as long as $e^{-S(x)}$ is real and positive.",
"This rather primitive method is actually powerful; for example the partition function of ABJM theory [13] at finite coupling and finite $N$ has been calculated accurately by using this method [4]."
],
[
"Common mistakes",
" Let us see some common mistakes below."
],
[
"Don't change step size during the run",
" Imagine the probability distribution you want to study has a bottleneck like in Fig.",
"REF .",
"For example if $S(x)=-\\log \\left(e^{-\\frac{x^2}{2}}+e^{-\\frac{(x-100)^2}{2}}\\right)$ then $e^{-S(x)}$ is strongly suppressed between two peaks at $x=0$ and $x=100$ .",
"By using a small step size $c\\sim 1$ you can sample one of the peaks efficiently, but then the other peak cannot be sampled.",
"Then in order to go across the bottleneck you would be tempted to change the step size $c$ when you come close to the bottleneck.",
"You would want to make the step size larger so that you can jump over the bottle neck, or you would want to make the step size smaller so that you can slowly penetrate into the bottle neck.",
"But if you do so, you obtain a wrong result, because the transition probability can depend on the past history.",
"You must not change the step size during the simulation.",
"But it does not mean that you cannot use multiple fixed step sizes; it is allowed to change the step size if the conditions listed in Sec.",
"are not violated.",
"For example we can take $c=1$ for odd steps and $c=100$ for even steps; see Fig.",
"REF .",
"Or we can throw a dice, namely randomly choose step size $c=1,2,3,4,5,6$ with probability $1/6$ .",
"As long as the conditions listed in Sec.",
", in particular the detailed balance, are not violated, you can do whatever you want.",
"Figure: If the probability distribution has a bottleneck, the acceptance rate goes down there.Figure: A histogram for S(x)=-loge -x 2 2 +e -(x-100) 2 2 S(x)=-\\log \\left(e^{-\\frac{x^2}{2}}+e^{-\\frac{(x-100)^2}{2}}\\right)with Metropolis, step size c=1c=1 for odd steps and c=100c=100 for even steps, 10 7 10^7 samples.The solid curve (which is actually invisible because it agrees with the histogram too precisely...) is the exact answer, e -x 2 2 +e -(x-100) 2 2 22\\frac{e^{-\\frac{x^2}{2}}+e^{-\\frac{(x-100)^2}{2}}}{2\\sqrt{2}}.Similar temptation of evil is common in muilti-variable case.",
"For example in the lattice gauge theory simulation it often happens that the acceptance is extremely low until the system thermalizes.",
"Then we can use smaller step size just to make the system thermalize, and then start actual data-taking with a larger step size.Another common strategy to reach the thermalization is to turn off the Metropolis test.",
"Or it occasionally happens that the simulation is trapped at a rare configuration so that the acceptance rate becomes almost zero.",
"In such case, it would be useful to use multiple step sizes (the ordinary and very small)."
],
[
"Don't mix independent simulations with different step sizes",
" This is similar to Sec.",
"REF : when you have several independent runs with different step sizes, you must not mix them to evaluate the expectation value, unless you pay extra cares for the error analysis.",
"Although each stream are guaranteed to converge to the same distribution, if you truncate them at finite number of configurations each stream contains different uncontrollable systematic error.",
"Note however that, if you can estimate the autocorrelation time of each stream reliably (for that each stream has to be sufficiently long), you can mix different streams with proper weights, with a careful error analysis."
],
[
"Make sure that random numbers are really random",
" In actual simulations, random numbers are not really random, they are just pseudo-random.",
"But we have to make sure that they are sufficiently random.",
"In Fig.REF , we used the same sequence of random numbers repeatedly every 1000 steps.",
"The answer is clearly wrong.",
"This mistake is very common;Yes, I did.",
"when one simulates a large system, it will take days or months, so one has to split the simulation to small number of steps.",
"Then when one submits a new job by mistake one would use the same `random numbers' again, for example by reseting the seed of random numbers to the same number.",
"Figure: Gaussian integral with Metropolis, c=1c=1, with non-random numbers;we chose the same random number sequence every 1000 steps.〈x 2 〉=1 n∑ k=1 n x (k) 2 \\langle x^2\\rangle = \\frac{1}{n}\\sum _{k=1}^n \\left(x^{(k)}\\right)^2 converges to a wrong number,which is different from 1."
],
[
"Remark on the use of Mathematica for larger scale simulations",
" I saw several people tried to use Mathematica for MCMC of systems of moderate size (gauge theories consisting of $O(10^3)$ — $O(10^4)$ variables) and failed to run the code with acceptable speed.",
"Probably the problem was that unless one understands Mathematica well one can unintentionally use nice features such as symbolic calculations which are not needed in MCMC.",
"The same remark could apply to other advanced softwares as well.",
"The simplest solution to this problem is to keep advanced softwares for advanced tasks, and avoid using them for such simple things like MCMC.",
"Unless you use a rare special function or something, it is unlikely that you need anything more than C or Fortran.",
"But by using those softwares you may be able to save the time for coding.",
"As long as the system size you want to study is small, you don't have to to worry; 10 seconds and 10 minutes are not that different and you will spend more time for the coding anyways.",
"When you want to do heavier calculations, make sure to understand the software well and avoid using unnecessary features.Wolfram research can easily solve this issue, I suppose.",
"Or perhaps it can easily be avoided by using existing features.",
"If anybody knows how to solve this problem, please let me know.",
"Given that Mathematica is extremely popular among physicists, if Mathematica can handle MCMC in physics it will certainly lower the entrance threshold.",
"Mathematica can be a useful tool to generate a C/Fortran code, especially with the HMC algorithm Sec.",
"REF , by utilizing the symbolic calculations.",
"I personally think such direction is the right use of Mathematica in the context of MCMC."
],
[
"Sign problem",
" So far we have assumed $e^{-S(x)}\\ge 0$ .",
"This was necessary because we interpreted $e^{-S(x)}$ as a `probability'.",
"But in physics we often encounter $e^{-S(x)}< 0$ , or sometimes $e^{-S(x)}$ can be complex.",
"Then a naive MCMC approach does not work.",
"This is infamous sign problem (or phase problem, when $e^{-S(x)}$ is complex).",
"Although no generic solution of the sign problem is known, there are various theory-specific solutions.",
"We will come back to this point in Sec.",
"REF ."
],
[
"What else do we need for lattice gauge theory simulations?",
"The advantage of the Metropolis algorithm is clear: it is simple.",
"It is extremely simple and applicable to any theory, as long as the `sign problem' does not exist.",
"When it works, just use it.",
"For example, for simple matrix model calculations like the one in [2],In that paper, we studied the symmetry breaking in Twisted Eguchi-Kawai model at large $N$ .",
"In 1980's people used the best computers available and studied $N\\lesssim 16$ .",
"They did not observe a symmetry breaking.",
"In 2006, I wrote a Metropolis code spending an hour or so, and studied $25\\lesssim N\\lesssim 100$ with my laptop.",
"Within a few hours I could see a clear signature of the symmetry breaking.",
"In order to understand the detail of the symmetry breaking pattern we had to study many parameters, so we ran the same code on a cluster machine.",
"One of my collaborators was serious enough to write a more sophisticated code to go to much larger $N$ , with which we could study $N\\gtrsim 100$ .",
"Note that it is a story from 2006 to 2007; now you can do much better job with Metropolis and your laptop.",
"you don't need anything more than Metropolis and your laptop.",
"But our budget is limited and we cannot live forever.",
"So sometimes we have to reduce the cost and make simulations faster.",
"We should use better algorithms, better lattice actions and better observables, which are `better' in the following sense: The autocorrelation length is shorter.",
"Easier to parallelize.",
"In lattice gauge theory, it typically means that we should utilize the sparseness of the Dirac operator.",
"Find good observables and good measurement methods which are easier to calculate, have less statistical fluctuations, and/or show faster convergence to the continuum limit.",
"For quantum field theories, especially when the fermions are involved, HMC is effective.",
"RHMC is a variant of HMC which is applicable to SYM."
],
[
"Integration of multiple variables and bosonic QFT",
" Once a regularization is given, the path-integral is merely an integral with multiple variables.",
"Hence let us start with a simple case of a matrix integral, then proceed to QFT."
],
[
"Metropolis for multiple variables",
" Generalization of the Metropolis algorithm (Sec.",
"REF ) to multiple variables $(x_1,x_2,\\cdots ,x_p)$ is straightforward.",
"For example, we can do as follows: For all $i=1,2,\\cdots ,p$ , randomly choose $\\Delta x_i\\in [-c_i,+c_i]$ , and shift $x_i^{(k)}$ as $x_i^{(k)}\\rightarrow x^{\\prime }_i\\equiv x_i^{(k)}+\\Delta x_i$ .",
"Note that the step size $c_i$ can be different for different $x_i$ .",
"Metropolis test: Generate a uniform random number $r$ between 0 and 1.",
"If $r<e^{S[x^{(k)}]-S[x^{\\prime }]}$ , $\\lbrace x^{(k+1)}\\rbrace =\\lbrace x^{\\prime }\\rbrace $ , i.e.",
"the new configuration is `accepted.'",
"Otherwise $\\lbrace x^{(k+1)}\\rbrace =\\lbrace x^{(k)}\\rbrace $ , i.e.",
"the new configuration is `rejected.'",
"Repeat the same for $k+1,k+2,\\cdots $ .",
"One can also do as follows: Randomly choose $\\Delta x_1\\in [-c_1,+c_1]$ , and shift $x_1^{(k)}$ as $x_1^{(k)}\\rightarrow x^{\\prime }_1\\equiv x_1^{(k)}+\\Delta x_1$ .",
"Metropolis test: Generate a uniform random number $r$ between 0 and 1.",
"If $r<e^{S[x^{(k)}]-S[x^{\\prime }]}$ , $x_1^{(k+1)}=x^{\\prime }_1$ , i.e.",
"the new value is `accepted.'",
"Otherwise $x_1^{(k+1)}=x_1^{(k)}$ , i.e.",
"the new configuration is `rejected.'",
"For other values of $i$ we don't do anything, namely $x_{i}^{(k+1)}=x_{i}^{(k)}$ for $i=2,3,\\cdots ,p$ .",
"Repeat the same for $i=2,3,\\cdots ,p$ .",
"Repeat the same for $k+1,k+2,\\cdots $ ."
],
[
"How it works",
"Let us consider a one matrix model, $S[\\phi ]=N{\\rm Tr}\\left(\\frac{1}{2}\\phi ^2+V(\\phi )\\right), $ where $\\phi $ is an $N\\times N$ Hermitian matrix, $\\phi _{ji}=\\phi _{ij}^\\ast $ .",
"The potential $V(\\phi )$ can be anything as long as the partition function is convergent; say $V(\\phi )=\\phi ^4$ .",
"The code has exactly the same structure as the sample code in Sec.",
"REF ; we should calculate $S[\\phi ]$ instead of the Gaussian weight, and instead of $x$ we can shift $\\phi $ by using $N^2$ real random numbers.",
"As $N$ gets larger, more and more portion of the integral region becomes unimportant.",
"Therefore, if we vary all the components simultaneously, $\\Delta S$ is typically large and the acceptance rate is very small, unless we take the step size to be small.",
"To avoid it, we can vary one component at each time.",
"Note that, when only $\\phi _{ij}$ and $\\phi _{ji}=\\phi _{ij}^\\ast $ are varied, one should save the computational cost by calculating $\\phi _{ij}$ -dependent part instead of $S[\\phi ]$ itself; the latter costs $O(N^3)$ , though the former costs only $O(N^2)$ ."
],
[
"Hybrid Monte Carlo (HMC) Algorithm ",
" The important configurations are like bottom of a valley; the altitude is the value of the action.",
"This is a valley in the phase space, whose dimension is very large.",
"So if the configuration is literally randomly varied, like in the Metropolis algorithm, the action almost always increases a lot.",
"Hence with the Metropolis algorithm the acceptance rate is small unless the step size is extremely small, and it causes rather long autocorrelation length.",
"The Hybrid Monte Carlo (HMC) algorithm [14] avoids the problem of a long autocorrelation by effectively crawling along the bottom of the valley; this is a `hybrid' of molecular dynamical method and Metropolis algorithm.",
"In HMC algorithm, sets of configurations $\\lbrace x^{(k)}\\rbrace $ $(k=0,1,2,\\cdots )$ are generated in the following manner.",
"Firstly, $\\lbrace x^{(0)}\\rbrace $ can be arbitrary.",
"Once $\\lbrace x^{(k)}\\rbrace $ is obtained, $\\lbrace x^{(k+1)}\\rbrace $ is obtained as follows.",
"Randomly generate auxiliary momenta $P_i^{(k)}$ , which are `conjugate' to $x_i^{(k)}$ , with probabilities $\\frac{1}{\\sqrt{2\\pi }}e^{-(P_i^{(k)})^2/2}$ .",
"To generate Gaussian random numbers, the Box-Muller algorithm is convenient; see Appendix .",
"Calculate the `Hamiltonian' $H_i=S[x^{(k)}]+\\frac{1}{2}\\sum _i (P_i^{(k)})^2$ .",
"Then we consider `time evolution' along an auxiliary time $\\tau $ (which is not the Euclidean time!).",
"We set the initial condition to be $x^{(k)}(\\tau =0)=x^{(k)}$ and $P^{(k)}(\\tau =0)=P^{(k)}$ , and use the leap frog method (see below) to calculate $x^{(k)}(\\tau _{f})$ and $P^{(k)}(\\tau _{f})$ , where $\\tau _{f}$ is related to the input parameters $\\Delta \\tau $ and $N_\\tau $ by $\\tau _{f}=N_\\tau \\Delta \\tau $ .",
"This process is called `molecular evolution.'",
"Calculate $H_f=S[x^{(k)}(\\tau _{f})]+\\frac{1}{2}\\sum _i (P_i^{(k)}(\\tau _{f}))^2$ .",
"Metropolis test: Generate a uniform random number $r$ between 0 and 1.",
"If $r<e^{H_i-H_f}$ , $x^{(k+1)}=x^{(k)}(\\tau _{f})$ , i.e.",
"the new configuration is `accepted.'",
"Otherwise $x^{(k+1)}=x^{(k)}$ , i.e.",
"the new configuration is `rejected.'",
"At first sight it is a rather complicated algorithm.",
"Why do we introduce such auxiliary dynamical system?",
"The key is the `energy conservation'.",
"Just like Metropolis, the change of the configuration is random due to the randomness of the choice of auxiliary momenta $P_i$ .",
"If we just compared the initial and final values of the action, the change would be equally large.",
"However in HMC the change of the auxiliary Hamiltonian matters in the Metropolis test.",
"(Please accept this fact for the moment, in the next paragraph we will explain the reason.)",
"If we keep $N_\\tau \\Delta \\tau $ fixed and send $N_\\tau $ to infinity, then the Hamiltonian is exactly conserved and new configurations are always accepted.",
"By taking $N_\\tau \\Delta \\tau $ to be large, new configurations can be substantially different from the old ones.",
"(Of course, calculation cost increase with $N_\\tau $ .",
"So we have to find a sweet spot, with moderately large $N_\\tau $ and moderately small $\\Delta \\tau $ .)",
"To check the detailed balance condition $e^{-S[x]}T[\\lbrace x\\rbrace \\rightarrow \\lbrace x^{\\prime }\\rbrace ]=e^{-S[x^{\\prime }]}T[\\lbrace x^{\\prime }\\rbrace \\rightarrow \\lbrace x\\rbrace ]$ , note that the leap-frog method is designed so that the molecular evolution is reversible; namely, if we start with $x^{(k)}(\\tau _{f})$ and $-P^{(k)}(\\tau _{f})$ , the final configuration is $x^{(k)}(\\tau =0)$ and $-P^{(k)}(\\tau =0)$ .",
"Hence, if $\\lbrace x,p\\rbrace $ evolves to $\\lbrace x^{\\prime },p^{\\prime }\\rbrace $ , then (by assuming $H-H^{\\prime }<0$ without loss of generality) $e^{-S[x]}T[\\lbrace x\\rbrace \\rightarrow \\lbrace x^{\\prime }\\rbrace ]\\propto e^{-S[x]}e^{-p^2/2}$ , while $e^{-S[x^{\\prime }]}T[\\lbrace x^{\\prime }\\rbrace \\rightarrow \\lbrace x\\rbrace ]\\propto e^{-S[x^{\\prime }]}e^{-p^{\\prime 2}/2}e^{-S[x]-p^2/2+S[x^{\\prime }]+p^{\\prime 2}/2}=e^{-S[x]}e^{-p^2/2}$ , with the same proportionality factor.",
"In fact the HMC algorithm works even when we take different $\\tau $ for each $x_i$ ; see Sec.",
"REF .",
"We can use different $\\tau $ on depending the field, momentum of the mode, etc.",
"The HMC algorithm is powerful especially when we have to deal with fermions, as we will explain later."
],
[
"Leap frog method",
"The leap frog method is a clever way to discretize the continuum Hamiltonian equation keeping the reversibility, which is crucial for assuring the detailed balance condition.",
"The continuum Hamiltonian equation is given by $\\frac{dp_i}{d\\tau }=-\\frac{\\partial H}{\\partial x_i}=-\\frac{\\partial S}{\\partial x_i},\\qquad \\frac{d x_i}{d\\tau }=\\frac{\\partial H}{\\partial p_i}=p_i.$ The leap frog method goes as followsBe careful about a factor of $1/2$ and ordering of the operations.",
"These are crucial for the reversibility of the molecular evolution and the detailed balance condition.",
": $x_i(\\Delta \\tau /2)=x_i(0)+p_i(0)\\cdot \\frac{\\Delta \\tau }{2}$ (Step 1 in Fig.",
"REF ).",
"For $n=1,2,N_\\tau -1$ , repeat it: $p_i(n\\Delta \\tau )=p_i((n-1)\\Delta \\tau )-\\frac{\\partial S}{\\partial x_i}((n-1/2)\\Delta \\tau )\\cdot \\Delta \\tau $ (Step $2,4,\\cdots ,2N_\\tau -2$ in Fig.",
"REF ), then $x_i((n+1/2)\\Delta \\tau )=x_i((n-1/2)\\Delta \\tau )+p_i(n\\Delta \\tau )\\cdot \\Delta \\tau $ (Step $3,\\cdots ,2N_\\tau -1$ in Fig.",
"REF ).",
"Finally, $p_i(N_\\tau \\Delta \\tau )=p_i((N_\\tau -1)\\Delta \\tau )-\\frac{\\partial S}{\\partial x_i}((N_\\tau -1/2)\\Delta \\tau )\\cdot \\Delta \\tau $ (Step $2N_\\tau $ in Fig.",
"REF ), then $x_i(N_\\tau \\Delta \\tau )=x_i((N_\\tau -1/2)\\Delta \\tau )+p_i(N_\\tau \\Delta \\tau )\\cdot \\frac{\\Delta \\tau }{2}$ (Step $2N_\\tau +1$ in Fig.",
"REF ).",
"Figure: Leap-frog method."
],
[
"How it works, 1 — Gaussian Integral",
" As the simplest example, let us go back to the one-variable case and start with the Gaussian integral again.This is an extremely stupid example, given that we need the Gaussian random number for the HMC algorithm!",
"But I believe it is still instructive.",
"Here is a sample code: #include <iostream> #include <cmath> #include<fstream> const int niter=10000; const int ntau=40; const double dtau=1e0; /******************************************************************/ /*** Gaussian Random Number Generator with Box Muller Algorithm ***/ /******************************************************************/ int BoxMuller(double& p, double& q){ double pi; double r,s; pi=2e0*asin(1e0); //uniform random numbers between 0 and 1 r = (double)rand()/RAND_MAX; s = (double)rand()/RAND_MAX; //Gaussian random numbers, //with weights proportional to e^{-p^2/2} and e^{-q^2/2} p=sqrt(-2e0*log(r))*sin(2e0*pi*s); q=sqrt(-2e0*log(r))*cos(2e0*pi*s); return 0; } /*********************************/ /*** Calculation of the action ***/ /*********************************/ // When you change the action, you should also change dH/dx, // specified in \"calc_delh\".",
"double calc_action(const double x){ double action=0.5e0*x*x; return action; } /**************************************/ /*** Calculation of the Hamiltonian ***/ /**************************************/ double calc_hamiltonian(const double x,const double p){ double ham; ham=calc_action(x); ham=ham+0.5e0*p*p; return ham; } /****************************/ /*** Calculation of dH/Dx ***/ /****************************/ // Derivative of the Hamiltonian with respect to x, // which is equivalent to the derivative of the action.",
"// When you change \"calc_action\", you have to change this part as well.",
"double calc_delh(const double x){ double delh=x; return delh; } /***************************/ /*** Molecular evolution ***/ /***************************/ int Molecular_Dynamics(double& x,double& ham_init,double& ham_fin){ double p; double delh; double r1,r2; BoxMuller(r1,r2); p=r1; //*** calculate Hamiltonian *** ham_init=calc_hamiltonian(x,p); //*** first step of leap frog *** x=x+p*0.5e0*dtau; //*** 2nd, ..., Ntau-th steps *** for(int step=1; step!=ntau; step++){ delh=calc_delh(x); p=p-delh*dtau; x=x+p*dtau; } //*** last step of leap frog *** delh=calc_delh(x); p=p-delh*dtau; x=x+p*0.5e0*dtau; //*** calculate Hamiltonian again *** ham_fin=calc_hamiltonian(x,p); return 0; } int main() { double x; double backup_x; double ham_init,ham_fin,metropolis,sum_xx; srand((unsigned)time(NULL)); /*********************************/ /* Set the initial configuration */ /*********************************/ x=0e0; /*****************/ /*** Main part ***/ /*****************/ std::ofstream outputfile(\"output.txt\"); int naccept=0;//counter for the number of acceptance sum_xx=0e0;//sum of x^2, useed for <x^2> for(int iter=0; iter!=niter; iter++){ backup_x=x; Molecular_Dynamics(x,ham_init,ham_fin); metropolis = (double)rand()/RAND_MAX; if(exp(ham_init-ham_fin) > metropolis){ //accept naccept=naccept+1; }else{ //reject x=backup_x; } /*******************/ /*** data output ***/ /*******************/ sum_xx=sum_xx+x*x; // output x, <x^2>, acceptance std::cout << x << ' ' << sum_xx/((double)(iter+1)) << ' ' << ((double)naccept)/((double)iter+1) << std::endl; outputfile << x << ' ' << sum_xx/((double)(iter+1)) << ' ' << ((double)naccept)/((double)iter+1) << std::endl; } outputfile.close(); return 0; } At the beginning of the code, a few parameters are set.",
"niter is the number of samples we will collect; ntau is $N_\\tau $ ; and dtau is $\\Delta \\tau $ .",
"Then several routines/functions are defined: BoxMuller generates Gaussian random numbers by using the Box-Muller algorithm.",
"We have to be careful about the normalization of the Gaussian here.",
"It will be a kind of confusing when you go to complex variables; see the case of matrix integral in Sec.",
"REF .",
"calc${\\ }$action calculates the action $S[x]$ .",
"In this case it is just $S[x]=\\frac{x^2}{2}$ .",
"It is called in calc${\\ }$hamiltonian.",
"calc${\\ }$hamiltonian adds $\\frac{p^2}{2}$ to the action and returns the Hamiltonian.",
"It is called in Molecular${\\ }$Dynamics.",
"calc${\\ }$delh returns $\\frac{dH}{dx}=\\frac{dS}{dx}=x$ .",
"It is called in Molecular${\\ }$Dynamics.",
"Molecular${\\ }$Dynamics performs one molecular evolution and returns the value of $x$ after the evolution and the values of the Hamiltonian before and after the evolution.",
"When the action $S[x]$ is changed to more complicated functions, you have to rewrite calc${\\ }$action and calc${\\ }$delh accordingly.",
"In main, the only difference from Metropolis is that Molecular${\\ }$Dynamics is used instead of a naive random change ($x\\rightarrow x+\\Delta x$ with random $\\Delta x$ ), and the Metropolis test is performed by using $\\Delta H$ instead of $\\Delta S$ ."
],
[
"How it works, 2 — Matrix Integral",
" Next let us consider the same example as before, $S[\\phi ]=N{\\rm Tr}\\left(\\frac{1}{2}\\phi ^2+\\frac{1}{4}\\phi ^4\\right),$ where $\\phi $ is $N\\times N$ Hermitian, and use the convention explained above.",
"Then the force terms are $\\frac{dP_{ij}}{d\\tau }=-\\frac{\\partial S}{\\partial \\phi _{ji}}=-\\phi _{ij}-\\left(\\phi ^3\\right)_{ij},\\qquad \\frac{d \\phi _{ij}}{d\\tau }=P_{ij}.$ The simulation code is very simple.",
"Here is a one in Fortran 90:I realize that people grew up in the 21st century prefer C++.",
"Still I personally love Fortran.",
"program phi4 implicit none !--------------------------------- integer nmat parameter(nmat=100) integer ninit parameter(ninit=0)!ninit=1 -> new config; ninit=0 -> old config integer iter,niter parameter(niter=10000) integer ntau parameter(ntau=20) double precision dtau parameter(dtau=0.005d0) integer naccept double complex phi(1:NMAT,1:NMAT),backup_phi(1:NMAT,1:NMAT) double precision ham_init,ham_fin,action,sum_action double precision tr_phi,tr_phi2 double precision metropolis open(unit=10,status='REPLACE',file='matrix-HMC.txt',action='WRITE') !",
"************************************* !",
"*** Set the initial configuration *** !",
"************************************* call pre_random if(ninit.EQ.1)then phi=(0d0,0d0) else if(ninit.EQ.0)then open(UNIT=22, File ='config.dat', STATUS = \"OLD\", ACTION = \"READ\") read(22,*) phi close(22) end if sum_action=0d0 !",
"***************** !",
"*** Main part *** !",
"***************** naccept=0 !counter for the number of acceptance do iter=1,niter backup_phi=phi call Molecular_Dynamics(nmat,phi,dtau,ntau,ham_init,ham_fin) !",
"*********************** !",
"*** Metropolis test *** !",
"*********************** call random_number(metropolis) if(dexp(ham_init-ham_fin) > metropolis)then !accept naccept=naccept+1 else !reject phi=backup_phi end if !",
"******************* !",
"*** data output *** !",
"******************* call calc_action(nmat,phi,action) sum_action=sum_action+action write(10,*)iter,action/dble(nmat*nmat),sum_action/dble(iter)/dble(nmat*nmat),& \t&dble(naccept)/dble(iter) end do close(10) open(UNIT = 22, File = 'config.dat', STATUS = \"REPLACE\", ACTION = \"WRITE\") write(22,*) phi close(22) end program Phi4 Again, it is very similar to a Metropolis code; randomly change the configuration, perform the Metropolis test, randomly change the configuration, perform the Metropolis test,....",
"In Molecular${\\ }$Dynamics, random momentum is generated with the normalization explained below (REF ), the molecular evolution performed, and $H_i$ and $H_f$ are calculated.",
"Subroutines calc${\\ }$hamiltonian and calc${\\ }$force (which corresponds to calc${\\ }$delh in the previous example) return the Hamiltonian and the force term $\\frac{\\partial H}{\\partial \\phi _{ji}}=\\frac{\\partial S}{\\partial \\phi _{ji}}$ ; it literally calculates products of matrices.",
"Another subroutine calc${\\ }$action is also simple.",
"Let's see them one by one.For routines which are not explained below, please look at the sample code at https://github.com/MCSMC/MCMC_sample_codes."
],
[
"subroutine Molecular_Dynamics(nmat,phi,dtau,ntau,ham_init,ham_fin) implicit none integer nmat integer ntau double precision dtau double precision r1,r2 double precision ham_init,ham_fin double complex phi(1:NMAT,1:NMAT) double complex P_phi(1:NMAT,1:NMAT) double complex delh(1:NMAT,1:NMAT) integer imat,jmat,step !",
"*** randomly generate auxiliary momenta *** do imat=1,nmat-1 do jmat=imat+1,nmat call BoxMuller(r1,r2) P_phi(imat,jmat)=dcmplx(r1/dsqrt(2d0))+dcmplx(r2/dsqrt(2d0))*(0D0,1D0) P_phi(jmat,imat)=dcmplx(r1/dsqrt(2d0))-dcmplx(r2/dsqrt(2d0))*(0D0,1D0) end do end do do imat=1,nmat call BoxMuller(r1,r2) P_phi(imat,imat)=dcmplx(r1) end do !",
"*** calculate Hamiltonian *** call calc_hamiltonian(nmat,phi,P_phi,ham_init) !",
"*** first step of leap frog *** phi=phi+P_phi*dcmplx(0.5d0*dtau) !",
"*** 2nd, ..., Ntau-th steps *** step=1 do while (step.LT.ntau) step=step+1 call calc_force(delh,phi,nmat) P_phi=P_phi-delh*dtau phi=phi+P_phi*dcmplx(dtau) end do !",
"*** last step of leap frog *** call calc_force(delh,phi,nmat) P_phi=P_phi-delh*dtau phi=phi+P_phi*dcmplx(0.5d0*dtau) !",
"*** calculate Hamiltonian *** call calc_hamiltonian(nmat,phi,P_phi,ham_fin) return END subroutine Molecular_Dynamics The inputs are the matrix size nmat$=N$ , the matrix phi$=\\phi ^{(k)}$ , the step size and number of steps for the molecular evolution, dtau$=\\Delta \\tau $ and ntau$=N_\\tau $ .",
"The output is phi$=\\phi ^{\\prime }$ and ham${\\ }$init$=H_i$ , ham${\\ }$fin$=H_f$ .",
"Note that the auxiliary momentum is neither input nor output; it is randomly generated every time in this subroutine.",
"Firstly random momentum $P_\\phi $ is generated.",
"BoxMuller($r_1$ ,$r_2$ ) generates random numbers $r_1$ and $r_2$ with the Gaussian weight $\\frac{e^{-r_1^2/2}}{\\sqrt{2\\pi }}$ , $\\frac{e^{-r_2^2/2}}{\\sqrt{2\\pi }}$ .",
"Note that $P_\\phi $ is Hermitian, $P_\\phi =P_\\phi ^\\dagger $ .",
"Hence we take $P_{\\phi ,ii}$ to be real, $P_{\\phi ,ii}=r_1$ , and $P_{\\phi ,ij}=P_{\\phi ,ji}^\\ast =(r_1+ir_2)/\\sqrt{2}$ for $i<j$ .",
"A factor $1/\\sqrt{2}$ is necessary in order to adjust the normalization.",
"!",
"*** randomly generate auxiliary momenta *** do imat=1,nmat-1 do jmat=imat+1,nmat call BoxMuller(r1,r2) P_phi(imat,jmat)=dcmplx(r1/dsqrt(2d0))+dcmplx(r2/dsqrt(2d0))*(0D0,1D0) P_phi(jmat,imat)=dcmplx(r1/dsqrt(2d0))-dcmplx(r2/dsqrt(2d0))*(0D0,1D0) end do end do do imat=1,nmat call BoxMuller(r1,r2) P_phi(imat,imat)=dcmplx(r1) end do Then we calculate the initial value of the Hamiltonian: !",
"*** calculate Hamiltonian *** call calc_hamiltonian(nmat,phi,P_phi,ham_init) Because we have already taken a backup of $\\phi $ before using this subroutine, we do not take a backup here.",
"Then we perform the molecular evolution by using the leap frog method.",
"!",
"*** first step of leap frog *** phi=phi+P_phi*dcmplx(0.5d0*dtau) Note that we need a factor $1/2$ here!",
"Then we just repeat the leap-frog steps, !",
"*** 2nd, ..., Ntau-th steps *** step=1 do while (step.LT.ntau) step=step+1 call calc_force(delh,phi,nmat) P_phi=P_phi-delh*dtau phi=phi+P_phi*dcmplx(dtau) end do and we need a factor $1/2$ again at the end: !",
"*** last step of leap frog *** call calc_force(delh,phi,nmat) P_phi=P_phi-delh*dtau phi=phi+P_phi*dcmplx(0.5d0*dtau) Now the molecular evolution has been done.",
"In order to perform the Metropolis test, we need to calculate $H_f$ : !",
"*** calculate Hamiltonian *** call calc_hamiltonian(nmat,phi,P_phi,ham_fin) Next we need to understand other subroutines called in this subroutine.",
"We will skip BoxMuller because it is exactly the same as before.",
"The other three will be explained below; they are almost trivial as well though.",
"This subroutine calculates the force term $\\frac{\\partial H}{\\partial \\phi _{ji}}=\\frac{\\partial S}{\\partial \\phi _{ji}}=N\\left(\\phi +\\phi ^3\\right)_{ij}.$ We just do it without using thinking too much, in the following manner: subroutine calc_force(delh,phi,nmat) implicit none integer nmat double complex phi(1:NMAT,1:NMAT),phi2(1:NMAT,1:NMAT),phi3(1:NMAT,1:NMAT) double complex delh(1:NMAT,1:NMAT) integer imat,jmat,kmat !",
"*** phi2=phi*phi, phi3=phi*phi*phi *** phi2=(0d0,0d0) phi3=(0d0,0d0) do imat=1,nmat do jmat=1,nmat do kmat=1,nmat phi2(imat,jmat)=phi2(imat,jmat)+phi(imat,kmat)*phi(kmat,jmat) end do end do end do do imat=1,nmat do jmat=1,nmat do kmat=1,nmat phi3(imat,jmat)=phi3(imat,jmat)+phi2(imat,kmat)*phi(kmat,jmat) end do end do end do !",
"*** delh=dH/dphi *** delh=phi+phi3 delh=delh*dcmplx(nmat) return END subroutine Calc_Force This subroutine just returns $H=\\frac{1}{2}{\\rm Tr}P^2+S[\\phi ]$ .",
"Firstly another subroutine calc${\\ }$action, which calculates the action, is called.",
"Then $\\frac{1}{2}{\\rm Tr}P^2$ is added.",
"It is too simple and looks almost stupid, but most things needed for MCMC codes are like this.",
"SUBROUTINE calc_hamiltonian(nmat,phi,P_phi,ham) implicit none integer nmat double precision action,ham double complex phi(1:NMAT,1:NMAT) double complex P_phi(1:NMAT,1:NMAT) integer imat,jmat call calc_action(nmat,phi,action) ham=action do imat=1,nmat do jmat=1,nmat ham=ham+0.5d0*dble(P_phi(imat,jmat)*P_phi(jmat,imat)) end do end do return END SUBROUTINE calc_hamiltonian This subroutine just returns $S[\\phi ]$ .",
"We honestly write down everything.",
"It is tedious but straightforward.",
"You just have to be patient.",
"SUBROUTINE calc_action(nmat,phi,action) implicit none integer nmat double precision action double complex phi(1:NMAT,1:NMAT) double complex phi2(1:NMAT,1:NMAT) integer imat,jmat,kmat !",
"*** phi2=phi*phi *** phi2=(0d0,0d0) do imat=1,nmat do jmat=1,nmat do kmat=1,nmat phi2(imat,jmat)=phi2(imat,jmat)+phi(imat,kmat)*phi(kmat,jmat) end do end do end do action=0d0 !",
"*** Tr phi^2 term *** do imat=1,nmat action=action+0.5d0*dble(phi2(imat,imat)) end do !",
"*** Tr phi^4 term *** do imat=1,nmat do jmat=1,nmat action=action+0.25d0*dble(phi2(imat,jmat)*phi2(jmat,imat)) end do end do !",
"*** overall normalization *** action=action*dble(nmat) return END SUBROUTINE calc_action In order to see how we can adjust the simulation parameters, let us vary $N_\\tau $ and $\\Delta \\tau $ keeping the product $N_\\tau \\Delta \\tau $ to be $0.1$ .",
"Then the acceptance rate changes as follows shown in Table REF .",
"Table: Acceptance rate for several choices of N τ N_\\tau , with N τ Δτ=0.1N_\\tau \\Delta \\tau =0.1, matrix size N=100N=100.We started the measurement runs with well thermalized configurations and collected 10000 samples for each parameter choice.Roughly speaking, the simulation cost is proportional to $N_\\tau $ .",
"When $N_\\tau \\Delta \\tau $ is fixed, the matrix $\\phi $ changes more or less the same amount by the molecular evolution, regardless of $N_\\tau $ .",
"Therefore, the rate of change is proportional to the acceptance rate.",
"Hence the change per cost is $({\\rm acceptance})/N_\\tau $ .",
"We should maximize it.",
"So we should use $N_\\tau =8$ or 10.",
"In Fig.",
"REF we have plotted $\\langle S/N^2\\rangle = \\frac{1}{n}\\sum _{k=1}^n S[\\phi ^{(k)}]$ for several different values of $N_\\tau $ by taking the horizontal axis to be `cost'$=n\\times N_\\tau $ .",
"We can see that $N_\\tau =8,10$ are actually cost effective.",
"Ideally we should do similar cost analysis varying $N_\\tau \\Delta \\tau $ , and estimate the autocorrelation length as well, to achieve more independent configurations with less cost.",
"Figure: 〈S〉/N 2 =1 n∑ k=1 n S[φ (k) ]\\langle S\\rangle /N^2= \\frac{1}{n}\\sum _{k=1}^n S[\\phi ^{(k)}] with N=100N=100, for several different values of N τ N_\\tau .The horizontal axis is n×N τ n\\times N_\\tau , which is proportional to the cost (time and electricity needed for the simulation).We can see that N τ =8,10N_\\tau =8,10 are more cost effective; i.e.",
"better convergence with less cost.Note that we started the measurement runs with well thermalized configurations.Before closing this section, let us demonstrate the importance of the leap-frog method.",
"Let us try a wrong algorithm: we omit a factor $1/2$ in the final step in Fig.",
"REF .This was the bug in my first HMC code.",
"It took several days to find it.",
"The outcome is a disaster; as shown in Fig.",
"REF , different $N_\\tau $ and $\\Delta \\tau $ give different values.",
"Correct expectation value (which agree with the value in Fig.",
"REF up to a very small $1/N$ correction) is obtained only at $N_\\tau =\\infty $ with $N_\\tau \\Delta \\tau $ fixed.",
"It is also easy to see the importance of the normalization of the auxiliary momentum; you can try it by yourself.",
"Figure: 〈S〉/N 2 =1 n∑ k=1 n S[φ (k) ]\\langle S\\rangle /N^2= \\frac{1}{n}\\sum _{k=1}^n S[\\phi ^{(k)}] with N=10N=10, for several different values of N τ N_\\tau and Δτ\\Delta \\tau ,without a factor 1/21/2 in the final step of the leap frog.",
"(Left) N τ ΔτN_\\tau \\Delta \\tau is fixed to 0.10.1; (Right) N τ N_\\tau is fixed to 10.Correct expectation value is obtained only at N τ =∞N_\\tau =\\infty with N τ ΔτN_\\tau \\Delta \\tau fixed.",
"Above we have assumed that the variables $x_1,x_2,\\cdots $ are real.",
"When you have to deal with complex variables, Hermitian matrices etc, you can always rewrite everything by using real variables; for example one can write a Hermitian matrix $M$ as $M=\\sum _a M_a T^a$ , where $T^a$ are generators and $M_a$ are real-valued coefficients.",
"But it is tedious and we have seen so many lattice QCD practitioners, who almost always work on SU$(3)$ , waste time struggling with SU$(N)$ , just to fix the normalization.",
"So let us summarize the cautions regarding the normalization.",
"Let us consider the simplest case again: the Gaussian integral, $S[x]=\\frac{x^2}{2}$ .",
"The Hamiltonian is $H[x,p]=\\frac{x^2}{2}+\\frac{p^2}{2}$ , and the equations of motion are $\\frac{dp}{d\\tau }=-\\frac{\\partial H}{\\partial x}=-\\frac{\\partial S}{\\partial x}=-x,\\qquad \\frac{d x}{d\\tau }=\\frac{\\partial H}{\\partial p}=p.$ This $p$ should be generated with the weight $\\frac{1}{\\sqrt{2\\pi }}e^{-p^2/2}$ .",
"Now let $x$ be complex and $S[x]=|x|^2=\\bar{x}x$ .",
"Let $p$ be the conjugate of $\\bar{x}$ , then the Hamiltonian is $H[x,p]=\\bar{x}x+\\bar{p}p$ , and $\\frac{dp}{d\\tau }=-\\frac{\\partial H}{\\partial \\bar{x}}=-\\frac{\\partial S}{\\partial \\bar{x}}=-x,\\qquad \\frac{d x}{d\\tau }=\\frac{\\partial H}{\\partial \\bar{p}}=p.$ To rewrite it to real variables with the right normalization, we do as follows: $x=\\frac{x_R+ix_I}{\\sqrt{2}},\\qquad p=\\frac{p_R+ip_I}{\\sqrt{2}}.$ Then $H[x,p]=\\frac{x_R^2+x_I^2+p_R^2+p_I^2}{2}$ , and $(x_R, p_R)$ and $(x_I, p_I)$ are conjugate pairs.",
"We should generate $p_R$ and $p_I$ with weight $\\frac{1}{\\sqrt{2\\pi }}e^{-(p_R)^2/2}$ and $\\frac{1}{\\sqrt{2\\pi }}e^{-(p_I)^2/2}$ .",
"Now let $X_{ij}$ be a Hermitian matrix, $X_{ji}=X_{ij}^\\ast $ .",
"The conjugate $P$ is also a Hermitian matrix, and we can take $P_{ji}=P_{ij}^\\ast $ to be the conjugate of $X_{ij}$ .",
"A simple Hamiltonian $H=\\frac{1}{2}{\\rm Tr}X^2 + \\frac{1}{2}{\\rm Tr}P^2$ becomes $H=\\frac{1}{2}\\sum _i\\left(X_{ii}^2+P_{ii}^2\\right)+\\sum _{i<j}\\left(X_{ij}X_{ij}^\\ast +P_{ij}P_{ij}^\\ast \\right).$ Hence $P_{ii}$ should be generated with the weight $\\frac{1}{\\sqrt{2\\pi }}e^{-(P_{ii})^2/2}$ , while $P_{ij}$ can be obtained by rewriting it as $P_{ij}=\\frac{P_{ij,R}+iP_{ij,I}}{\\sqrt{2}}$ and generating $P_{ij,R}$ , $P_{ij,I}$ with the weight $\\frac{1}{\\sqrt{2\\pi }}e^{-(P_{ij,R})^2/2}$ , $\\frac{1}{\\sqrt{2\\pi }}e^{-(P_{ij,I})^2/2}$ .",
"The force terms are as follows: $\\frac{dP_{ij}}{d\\tau }=-\\frac{\\partial H}{\\partial X_{ji}}=-\\frac{\\partial S}{\\partial X_{ji}}=-X_{ij},\\qquad \\frac{d X_{ij}}{d\\tau }=\\frac{\\partial H}{\\partial P_{ji}}=P_{ij}.$ As the final example, let $X_{ij}$ be a complex matrix.",
"The conjugate $P$ is also a Hermitian matrix, and we can take $P_{ij}^\\ast $ to be the conjugate of $X_{ij}$ .",
"A simple Hamiltonian $H={\\rm Tr}X^\\dagger X + {\\rm Tr}P^\\dagger P$ becomes $H=\\sum _{i,j}\\left(X_{ij}X_{ij}^\\ast +P_{ij}P_{ij}^\\ast \\right).$ Hence $P_{ij}$ can be obtained by rewriting it as $P_{ij}=\\frac{P_{ij,R}+iP_{ij,I}}{\\sqrt{2}}$ and generating $P_{ij,R}$ , $P_{ij,I}$ with the weight $\\frac{1}{\\sqrt{2\\pi }}e^{-(P_{ij,R})^2/2}$ , $\\frac{1}{\\sqrt{2\\pi }}e^{-(P_{ij,I})^2/2}$ .",
"The force terms are as follows: $\\frac{dP_{ij}}{d\\tau }=-\\frac{\\partial S}{\\partial X_{ij}^\\ast }=-X_{ij},\\qquad \\frac{d X_{ij}}{d\\tau }=\\frac{\\partial S}{\\partial P_{ij}^\\ast }=P_{ij}.$ The case with generic actions $S$ should be apparent.",
"Note that, when you change the normalization of the $p^2$ term in the Hamiltonian, you have to change the width of random Gaussian appropriately.",
"Otherwise you will end up in getting wrong answers.",
"One extra bonus associated with HMC is that we can use the conservation of the Hamiltonian for debugging.",
"It is very rare to make bugs in the calculations of the action and force term consistently; so, practically, unless we code both correctly we cannot see the conservation of the Hamiltonian in the `continuum limit' $N_\\tau \\rightarrow \\infty $ , $\\Delta \\tau \\rightarrow 0$ with $N_\\tau \\Delta \\tau $ fixed.Note that the normalization of the auxiliary momentum and the factor $1/2$ at the first and last steps of the leap-frog evolution cannot be tested by the conservation of the Hamiltonian.",
"This is a very good check of the code; see Fig.",
"REF .",
"Figure: ΔH=H f -H i \\Delta H = H_f-H_i vs N τ N_\\tau , with N τ Δτ=0.1N_\\tau \\Delta \\tau =0.1 (fixed),in log-log scale.We picked up a thermalized configuration {φ}\\lbrace \\phi \\rbrace anda randomly generated auxiliary momentum {P φ }\\lbrace P_\\phi \\rbrace ,and used the same {φ,P φ }\\lbrace \\phi ,P_\\phi \\rbrace for all (N τ ,Δτ)(N_\\tau ,\\Delta \\tau ).The solid line is 18.5N τ -2 18.5N_\\tau ^{-2}.The action is () with N=100N=100.When you confirm the conservation of the Hamiltonian, debugging is more or less done."
],
[
"Multiple step sizes",
" Let us conside a two-matrix model $S[\\phi _1,\\phi _2]=N{\\rm Tr}\\left(V_1(\\phi _1)+V_2(\\phi _2)+\\phi _1\\phi _2\\right),$ where $V_1(\\phi _1)=\\frac{m_1^2}{2}\\phi _1^2+\\frac{1}{4}\\phi _1^4,\\qquad V_2(\\phi _2)=\\frac{m_2^2}{2}\\phi _2^2+\\frac{1}{4}\\phi _2^4.$ Imagine an extreme situation, $m_1=1$ and $m_2=1000000000$ , i.e.",
"$\\phi _2$ is much heavier.",
"Then typical value of $\\phi _1$ is much larger than that of $\\phi _2$ .",
"If we use the same step size for both, then in order to raise the acceptance rate for $\\phi _2$ we have to take the step size very small, which leads to a very long autocorrelation for $\\phi _1$ .",
"As we have emphasized in Sec.",
"REF , the choice of the step size has to be consistent with the requirements listed in Sec.",
", but other than that it is completely arbitrary; see Sec.",
"REF .",
"Hence we can simply use different step sizes for $\\phi _1$ and $\\phi _2$ .",
"This simple fact is very important in the simulation of QFT: we should use larger step size for lighter particle.",
"Also, in the momentum space, high-frequency modes are `heavier' in that $m^2+p^2$ behaves like a mass.",
"Hence we should use smaller step size for ultraviolet modes, larger step size for infrared modes.",
"This method is called `Fourier acceleration'."
],
[
"Different algorithms for different fields",
" We can even use different update algorithms for different variables, and as we will see, this is important when we study systems with fermions.",
"Let us consider $S(x,y)=y^2f(x)+g(x)$ where $f(x)$ and $g(x)$ are complicated functions.",
"Then we can repeat the following two steps, Update $y$ for fixed $x$ , Update $x$ for fixed $y$ .",
"In Sec.",
"REF we introduced essentially the same example, namely we adopted the Metropolis algorithm and varied $x_1,x_2,\\cdots $ one by one.",
"If $f(x)>0$ , then $z\\equiv y\\sqrt{f(x)}$ has a Gaussian weight for each fixed $x$ .",
"In this case, we can use the Box-Muller algorithm (Appendix ) to generate $z$ ; then there is no autocorrelation between $y$ 's.",
"Hence we can use the following method: Update $y$ for fixed $x$ , by generating Gaussian random $z$ and setting $y=z/\\sqrt{f(x)}$ .",
"Update $x$ for fixed $y$ , by using Metropolis or HMC.",
"Note that the use of the Box-Muller algorithm does not violate the conditions listed in Sec. .",
"It just gives us a very special Markov chain without autocorrelation."
],
[
"QFT example 1: 4d scalar theory",
" Let us consider 4d scalar theory, $S[\\phi ]=N\\int d^4x{\\rm Tr}\\left(\\frac{1}{2}\\left(\\partial _\\mu \\phi \\right)^2+\\frac{m^2}{2}\\phi ^2+V(\\phi )\\right), $ where the $N\\times N$ Hermitian matrices $\\phi (x)$ now depends on the coordinate $x$ .",
"For simplicity we assume the spacetime is compactified to a square four-torus with circumference $\\ell $ and volume $V=\\ell ^4$ .",
"It can be regularized by using an $n^4$ lattice with the lattice spacing $a=L/n$ as $S_{\\rm lattice}[\\phi ]=Na^4\\sum _{\\vec{x}}{\\rm Tr}\\left(\\frac{1}{2}\\sum _\\mu \\left(\\frac{\\phi _{\\vec{x}+\\hat{\\mu }}-\\phi _{\\vec{x}}}{a}\\right)^2+\\frac{m^2}{2}\\phi ^2_{\\vec{x}}+V(\\phi _{\\vec{x}})\\right), $ where $\\hat{\\mu }$ stands for a shift of one lattice unit along the $\\mu $ direction.",
"The path-integral is just a multi-variable integral with Hermitian matrices, the methods we have already explained can directly be applied."
],
[
"QFT example 2: Wilson's plaquette action (SU(N) pure Yang-Mills)",
" Let's move on to 4d SU($N$ ) Yang-Mills.",
"The continuum action we consider is $S_{\\rm continuum}=\\frac{1}{4g_{YM}^2}\\int d^4x{\\rm Tr}F_{\\mu \\nu }^2,$ where the field strength $F_{\\mu \\nu }$ is defined by $F_{\\mu \\nu }=\\partial _\\mu A_\\nu -\\partial _\\nu A_\\mu + i[A_\\mu ,A_\\nu ]$ .",
"Typically we take the 't Hooft coupling $\\lambda =g_{YM}^2N$ fixed when we take large $N$ .",
"As a lattice regularization we use Wilson's plaquette action,Often the overall factor $N$ is included in $\\beta $ and $\\beta ^{\\prime }=\\beta N$ is used as the lattice coupling.",
"It is simply a bad convention when we consider generic values of $N$ , because the coupling to be fixed as $N$ is varied is not $\\beta ^{\\prime }$ but $\\beta $ .",
"$S_{\\rm lattice}=-\\beta N\\sum _{\\mu \\ne \\nu }\\sum _{\\vec{x}}{\\rm Tr}\\ U_{\\mu ,\\vec{x}}U_{\\nu ,\\vec{x}+\\hat{\\mu }}U^\\dagger _{\\mu ,\\vec{x}+\\hat{\\nu }}U^\\dagger _{\\nu ,\\vec{x}}.$ Here $U_{\\mu ,\\vec{x}}$ is a unitary variable living on a link connecting to lattice points $\\vec{x}$ and $\\vec{x}+a\\hat{\\mu }$ , where $a$ is the lattice spacing and $\\hat{\\mu }$ is a unit vector along the $\\mu $ -th direction.",
"It is related to the gauge field $A_\\mu (\\vec{x})$ by $U_{\\mu ,\\vec{x}}=e^{iaA_\\mu (\\vec{x})}$ .",
"The lattice coupling constant $\\beta $ is the inverse of the 't Hooft coupling, $\\beta =1/\\lambda $ , and it should be scaled appropriately with the lattice spacing $a$ in order to achieve the right continuum limit."
],
[
"Metropolis for unitary variables",
" The only difference is that, instead of adding random numbers, we should multiply random unitary matrices.",
"A random unitary matrix can be generated as follows.",
"Firstly, we generate random Hermitian matrix $H$ by using random numbers.",
"For example we can generate it with Gaussian weight $\\sim e^{-{\\rm Tr}H^2/2\\sigma ^2}$ .",
"Then $V=e^{iH}$ is random unitary centered around $V=1$ .",
"When $\\sigma $ is small, it is more likely to be close to 1.",
"Hence $\\sigma $ is `step size'.",
"Of course, you can use the uniform random number to generate $H$ if you want.",
"Regardless, the Metropolis goes as follows: Generate $V$ randomly, change $U_{1,\\vec{x}}$ to $U_{1,\\vec{x}}^{\\prime }=U_{1,\\vec{x}}V$ , and perform the Metropolis test.",
"Do the same for $U_{2,\\vec{x}}$ , $U_{3,\\vec{x}}$ and $U_{4,\\vec{x}}$ .",
"Do the same for other lattice sites.",
"Repeat the same procedure many many times."
],
[
"HMC for Wilson's plaquette action",
"HMC for unitary variables goes as follows.",
"Let us define the momentum $p_\\mu ^{ij}$ conjugate to the gauge field $A_\\mu ^{ji}$ by $\\frac{dU}{d\\tau }=ipU,\\qquad \\frac{dU^\\dagger }{d\\tau }=-iUp.$ It generates $U\\rightarrow e^{i\\delta A}U,\\qquad U^\\dagger \\rightarrow U^\\dagger e^{-i\\delta A}.$ Therefore, $\\frac{dp_{ij}}{d\\tau }=-\\frac{\\partial S}{d A^{ji}}=-i\\left(U\\frac{\\partial S}{\\partial U}\\right)_{ij}+i\\left(U\\frac{\\partial S}{\\partial U}\\right)_{ji}^\\ast $ where the second term comes from the derivative w.r.t.",
"$U^\\dagger $ .",
"The discrete molecular evolution can be defined as follows: $& &U(\\Delta \\tau /2)=\\exp \\left(i\\frac{\\Delta \\tau }{2} p(0)\\right)\\cdot U(0),\\nonumber \\\\& &p(\\Delta \\tau )=p(0)+\\Delta \\tau \\cdot \\frac{dp}{d\\tau }(\\Delta \\tau /2).\\nonumber $ Repeat the following for $\\tau =\\Delta \\tau ,2\\Delta \\tau ,\\cdots ,(N_\\tau -1)\\Delta \\tau $ : $& &U(\\tau +\\Delta \\tau /2)=\\exp \\left(i\\Delta \\tau p(\\tau )\\right)\\cdot U(\\tau -\\Delta \\tau /2),\\nonumber \\\\& &p(\\tau +\\Delta \\tau )=p(\\tau )+\\Delta \\tau \\cdot \\frac{dp}{d\\tau }(\\tau +\\Delta \\tau /2).\\nonumber $ $& &U(N_{\\tau }\\Delta \\tau )=\\exp \\left(i\\frac{\\Delta \\tau }{2} p(N_{\\tau }\\Delta \\tau )\\right)\\cdot U\\Big ((N_{\\tau }-1/2)\\Delta \\tau \\Big ).\\nonumber $ In order to calculate $e^{i\\Delta \\tau p}$ , it is necessary to diagonalize $p$ .",
"As long as one considers SU$(N)$ theory with not very large $N$ , the diagonalization is not that costly.",
"(Note also that, when the fermions are introduced, this part cannot be a bottle-neck, so we do not have to care; we should spend our effort for improving other parts.)",
"In case we need to cut the cost as much as possible, we can approximate it by truncating the Taylor expansion of $e^{i\\Delta \\tau \\cdot p}$ at some finite order."
],
[
"Including fermions with HMC and RHMC",
" Let us consider the simplest example, $S[x,\\psi ,\\bar{\\psi }]=\\frac{x^2}{2}+\\bar{\\psi }D(x)\\psi ,$ where $\\psi $ is a complex Grassmann number and a function $D(x)$ is a `Dirac operator'.",
"We can integrate out $\\psi $ by hand, so that $Z=\\int dx d\\psi d\\bar{\\psi } e^{-S[x,\\psi ,\\bar{\\psi }]}=\\int dx D(x) e^{-x^2/2}=\\int dx e^{-x^2/2+\\log D(x)}.$ Hence we need to deal with the effective action in terms of $x$ , $S_{\\rm eff}=\\frac{x^2}{2}-\\log D(x)$ .",
"This is simple enough so that we don't need anything more sophisticated than the Metropolis algorithm.",
"However our life becomes a bit more complicated when there are multiple variables, $x_{1,2,\\cdots , n_x}$ and $\\psi _{1,2,\\cdots ,n_\\psi }$ .",
"Let us consider the action of the following form, $S[x,\\psi ,\\bar{\\psi }]=\\sum _{i=1}^{n_x}\\frac{x_i^2}{2}+\\sum _{a,b=1}^{n_\\psi }\\bar{\\psi }_aD_{ab}(x)\\psi _b.$ Now the Dirac operator $D_{ab}(x)$ is an $n_\\psi \\times n_\\psi $ matrix, and the partition function becomes $Z=\\int [dx] [d\\psi ] [d\\bar{\\psi }] e^{-S[x,\\psi ,\\bar{\\psi }]}=\\int [dx] \\det D(x)\\cdot e^{-\\sum _i x_i^2/2}.$ We can still use the Metropolis algorithm in principle,Here we assumed $\\det D(x)>0$ for any $x$ , i.e.",
"the sign problem is absent.",
"but the calculation of the determinant is very costly (cost$\\sim n_\\psi ^3$ ).",
"Even worse, though $D_{ab}(x)$ is typically sparse, it is not easy to utilize the sparseness when one calculates the determinant.",
"HMC and RHMC avoid the evaluation of the determinant and enable us to utilize the sparseness of $D_{ab}(x)$ ."
],
[
"2-flavor QCD with HMC",
" Let us consider 2-flavor QCD, whose action in the continuum is $S[A_\\mu ,\\psi _f,\\bar{\\psi }_f]=\\int d^4x\\left(\\frac{1}{4}Tr F_{\\mu \\nu }^2+\\sum _{f=1}^2\\bar{\\psi }^{(f)} D^{(f)}\\psi ^{(f)}\\right),$ where $A_\\mu (\\mu =1,2,3,4)$ is the SU(3) gauge field and $D_f=\\gamma ^\\mu D_\\mu + m_f$ is the Dirac operator with fermion mass $m_f$ .",
"For a lattice regularization, we use link variables as in Sec.REF , and put fermions on sites.",
"The latice action we consider is $S_{\\rm Lattice}[U_\\mu ,\\psi _f,\\bar{\\psi }_f]=S_B[U_\\mu ]+\\sum _{\\vec{x},\\vec{y}}\\sum _{f=1}^2\\bar{\\psi }^{(f)}_{\\vec{x}\\alpha } D^{(f)}_{\\vec{x}\\alpha ,\\vec{y}\\beta }\\psi ^{(f)}_{\\vec{y}\\beta },$ where $S_B[U_\\mu ]$ is the plaquette action (REF ).",
"There are various choices for a lattice Dirac operator $D^{(f)}_{\\vec{x}\\alpha ,\\vec{y}\\beta }$ ; we do not specify it here.",
"Below we assume $m_1=m_2=m$ (i.e.",
"neglect the difference of the mass of up and down quarks), then the partition function is $Z_{\\rm Lattice}=\\int [dU] \\left(\\det D\\right)^2 e^{-S_B}.$ Usually the determinant is real, and hence $\\left(\\det D\\right)^2=\\det (D^\\dagger D)$ .",
"Now we introduce pseudo-fermion $F$ , on which the Dirac operator acts just in the same way as on $\\psi $ .",
"But $F$ is a complex bosonic field.",
"Then $Z_{\\rm Lattice}=\\int [dU] [dF] e^{-S_B - F^\\dagger (D^\\dagger D)^{-1} F}.$ At first sight it may look like a stupid way of writing an easy thing in a complicated form.",
"However it enables us to avoid the calculation of $\\det D$ ; rather we have to calculate a linear equation $(D^\\dagger D)\\chi =F,$ for which the sparseness of $D$ can be fully utilized.",
"In order to obtain the solution $\\chi $ , we can use the conjugate gradient method; see Appendix .",
"The strategy is simply applying the HMC algorithm to $S[U_\\mu ,F]=S_B[U_\\mu ]+F^\\dagger (D^\\dagger D)^{-1} F,$ by using $A_\\mu $ and $F$ as dynamical variables.",
"The only nontrivial parts are the calculation of the force and Hamiltonian.",
"For that, we only need the solution $\\chi $ of $(D^\\dagger D)\\chi =F$ .",
"Indeed, the force is calculated as $-\\frac{\\partial S}{\\partial A_\\mu }=-\\frac{\\partial S_B}{\\partial A_\\mu }+\\chi ^\\dagger \\frac{\\partial (D^\\dagger D)}{\\partial A_\\mu }\\chi $ and $-\\frac{\\partial S}{\\partial F^\\dagger }=\\chi ,$ and the action is simply $S[U_\\mu ,F]=S_B[U_\\mu ]+F^\\dagger \\chi .$"
],
[
"A better way of treating $F$",
" It is possible to update the pseudo-fermion $F$ more efficiently, by using the idea explained in Sec.REF .",
"This is based on a simple observation that $\\Phi \\equiv \\left(D^\\dagger \\right)^{-1}F$ has the Gaussian weight and hence can easily be generated randomly by using the Gaussian random number generator.",
"Hence we can do as follows: Randomly generate $\\Phi ^{(k+1)}$ with the Gaussian weightBy using the Box-Muller algorithm explained in Appendix , real Gaussian random numbers with variance 1 is generated, i.e.",
"the weight of $x,y\\in {\\mathbb {R}}$ is $e^{-x^2/2}$ and $e^{-y^2/2}$ .",
"By dividing $\\tilde{x}=x/\\sqrt{2}$ and $\\tilde{y}=y/\\sqrt{2}$ have the weights $e^{-\\tilde{x}^2}$ and $e^{-\\tilde{y}^2}$ .",
"To reproduce the weight is $e^{-\\Phi ^\\dagger \\Phi }=e^{-({\\rm Re}\\Phi )^2-({\\rm Im}\\Phi )^2}$ , we should take $\\Phi =(\\tilde{x}+\\sqrt{-1}\\tilde{y})=(x+\\sqrt{-1}y)/\\sqrt{2}$ .",
"If you use a wrong normalization, you end up in a wrong result.",
"$e^{-\\Phi ^\\dagger \\Phi }$ .",
"Calculate the pseudo-fermion $F^{(k+1)}=D^\\dagger (U_\\mu ^{(k)})\\cdot \\Phi ^{(k+1)}$ .",
"Randomly generate auxiliary momenta $P_\\mu ^{(k)}$ , which are conjugate to $A_\\mu ^{(k)}$ , with probabilities proportional to $e^{-{\\rm Tr}(P_\\mu ^{(k)})^2/2}$ .",
"Calculate $H_i=S_B[U^{(k)}]+\\sum _\\mu {\\rm Tr}(P_\\mu ^{(k)})^2/2+(F^{(k+1)})^\\dagger (D^\\dagger (U^{(k)})\\cdot D(U^{(k)}))^{-1} F^{(k+1)}.$ Note that $(F^{(k+1)})^\\dagger (D^\\dagger (U^{(k)})\\cdot D(U^{(k)}))^{-1} F^{(k+1)}=(\\Phi ^{(k+1)})^\\dagger \\Phi ^{(k+1)}$ .",
"Then perform the Molecular evolution for $U_\\mu $ , fixing $F$ to be $F^{(k+1)}$ .",
"The force is $-\\frac{\\partial S_B}{\\partial A_\\mu }+\\chi ^\\dagger \\frac{\\partial (D^\\dagger D)}{\\partial A_\\mu }\\chi ,$ where $(D^\\dagger D)\\chi =F$ as before.",
"Then we obtain $U_\\mu ^{(k+1)}$ and $P_\\mu ^{(k+1)}$ .",
"Calculate $H_f=S_B[U^{(k+1)}]+\\sum _\\mu {\\rm Tr}(P_\\mu ^{(k+1)})^2/2+(F^{(k+1)})^\\dagger (D^\\dagger (U^{(k+1)})\\cdot D(U^{(k+1)}))^{-1} F^{(k+1)}.\\nonumber \\\\$ Metropolis test: Generate a uniform random number $r$ between 0 and 1.",
"If $r<e^{H_i-H_f}$ , $U_\\mu ^{(k+1)}=U_\\mu ^{(k)}(\\tau _{f})$ , otherwise $U_\\mu ^{(k+1)}=U_\\mu ^{(k)}$ .",
"Repeat 1–7."
],
[
"(2+1)-flavor QCD with RHMC",
" Strange quark has much bigger mass than up and down quarks.",
"So let us take $m_1=m_2\\ll m_3=m_s$ .",
"Let us denote the Dirac operators for up and down by $D$ , and the one for strange by $D_s$ .",
"The partition function is $Z_{\\rm Lattice}=\\int [dU] \\left(\\det D_s\\right) \\left(\\det D\\right)^2 e^{-S_B}.$ Hence we need to introduce two pseudo fermions $F$ and $F_s$ as$D_s^\\dagger D_s$ is used so that the action becomes positive definite.",
"$S[U_\\mu ,F]=S_B[U_\\mu ]+F^\\dagger (D^\\dagger D)^{-1} F+F_s^\\dagger (D_s^\\dagger D_s)^{-1/2} F_s.$ The last term in the right hand side is problematic because it is difficult (or costly) to solve $(D^\\dagger D)^{1/2}\\chi _s=F_s$ .",
"The Rational Hybrid Monte Carlo (RHMC) algorithm [15] evades this problem by utilizing a rational approximation, $(D^\\dagger D)^{-1/2}\\simeq a_0+\\sum _{i=1}^Q\\frac{a_i}{D^\\dagger D + b_i},$ where $a$ 's and $b$ 's are positive constants, and $Q$ is typically 10 or 20.",
"These numbers are adjusted so that the rational approximation is good for all samples appearing in the actual simulation.",
"A good code to find such numbers can be found at [17].",
"By using this rational approximation, we replace the last term by $S_{\\rm strange}=a_0 F_s^\\dagger F_s+\\sum _{i=1}^Qa_i F_s^\\dagger \\frac{1}{D_s^\\dagger D_s+b_i}F_s.$ To determine the force, we need to solve $(D_s^\\dagger D_s+\\beta _i)\\chi _{s,i}=F_s$ .",
"Rather surprisingly, we do not have to solve these equations $Q$ times; the multi-mass CG solver [16] solves all $Q$ equations simultaneously, essentially without any additional cost.",
"See Appendix .",
"Randomly generate $\\Phi ^{(k+1)}$ and $\\Phi _s^{(k+1)}$ with the Gaussian weights $e^{-\\Phi ^\\dagger \\Phi }$ and $e^{-\\Phi _s^\\dagger \\Phi _s}$ .",
"Calculate the pseudo-fermion $F^{(k+1)}=D^\\dagger (U_\\mu ^{(k)})\\cdot \\Phi ^{(k+1)}$ .",
"Calculate the pseudo-fermion $F_s^{(k+1)}=\\left(D^\\dagger (U_\\mu ^{(k)})\\cdot D(U_\\mu ^{(k)})\\right)^{1/4}\\Phi _s^{(k+1)}$ .",
"See the end of this section for an efficient way to do it.",
"Randomly generate auxiliary momenta $P_\\mu ^{(k)}$ , which are conjugate to $A_\\mu ^{(k)}$ , with probabilities proportional to $e^{-{\\rm Tr}(P_\\mu ^{(k)})^2/2}$ .",
"Calculate $H_i=H[U^{(k)},P^{(k)},F^{(k+1)},F_s^{(k+1)}]$ where $H[U,P,F,F_s]&=&S_B[U]+\\sum _\\mu {\\rm Tr}(P_\\mu )^2/2\\nonumber \\\\& &+F^\\dagger (D^\\dagger (U)\\cdot D(U))^{-1} F\\nonumber \\\\& &+a_0 F_s^\\dagger F_s+\\sum _{i=1}^Qa_i F_s^\\dagger \\frac{1}{D_s(U)^\\dagger D_s(U)+b_i}F_s.$ Note that, although the last line formally seems to agree with $(\\Phi _s^{(k+1)})^\\dagger \\Phi _s^{(k+1)}$ at $\\tau =0$ , it is not the case due to the approximations used for $(D^\\dagger D)^{-1/2}$ and $(D^\\dagger D)^{+1/4}$ are not exactly the `inverse'.",
"Then perform the Molecular evolution for $U_\\mu $ , fixing $F$ to be $F^{(k+1)}$ .",
"The force is $-\\frac{\\partial S_B}{\\partial A_\\mu }+\\chi ^\\dagger \\frac{\\partial (D^\\dagger D)}{\\partial A_\\mu }\\chi +\\sum _{i=1}^Q a_i\\chi _{s,i}^\\dagger \\frac{\\partial (D_s^\\dagger D_s)}{\\partial A_\\mu }\\chi _{s,i},$ where $(D^\\dagger D)\\chi =F$ and $(D_s^\\dagger D_s+b_i)\\chi _{s,i}=F_s$ .",
"Then we obtain $U_\\mu ^{(k+1)}$ and $P_\\mu ^{(k+1)}$ .",
"Calculate $H_f=H[U^{(k+1)},P^{(k+1)},F^{(k+1)},F_s^{(k+1)}]$ Metropolis test: Generate a uniform random number $r$ between 0 and 1.",
"If $r<e^{H_i-H_f}$ , $U_\\mu ^{(k+1)}=U_\\mu ^{(k)}(\\tau _{f})$ , i.e.",
"the new configuration is `accepted.'",
"Otherwise $U_\\mu ^{(k+1)}=U_\\mu ^{(k)}$ , i.e.",
"the new configuration is `rejected.'",
"Repeat 1 – 8."
],
[
"How to calculate $F=\\left(D^\\dagger D\\right)^{1/4}\\Phi $",
" We can use the rational approximation $x^{1/4} \\simeq a^{\\prime }_0 + \\sum _{k=1}^{Q^{\\prime }}\\frac{a^{\\prime }_k}{x+b^{\\prime }_k}.$ Then $F\\simeq a^{\\prime }_0\\Phi +a^{\\prime }_k\\chi ^{\\prime }_k,$ where $\\frac{1}{D_s^\\dagger D_s+b^{\\prime }_k}\\Phi =\\chi ^{\\prime }_k$ is obtained by solving $(D_s^\\dagger D_s+b^{\\prime }_k)\\chi ^{\\prime }_k=\\Phi .$ We can use the multi-mass solver; see Appendix ."
],
[
"Maximal SYM with RHMC",
"Sec.",
"REF is for lattice people who knows almost nothing about SYM.",
"String theorists, formal QFT people and lattice theorists who already know SYM can skip it."
],
[
"The theories",
" Supersymmetry relates bosons and fermions.",
"Because it changes the spin of the fields, if there are too many supercharges (i.e.",
"generators of supersymmetry transformation) higher spin fields have to be involved.",
"In order to construct a gauge theory without gravity, we cannot have massless fields with spin larger than one.",
"It restricts the possible number of supercharges: the maximal number is sixteen.",
"The maximally supersymmetric Yang-Mills theories (or maximal SYM) are the supersymmetric generalizations of Yang-Mills theory with sixteen supercharges.",
"They can be obtained from SYM in $(9+1)$ dimensions as follows.",
"Firstly, why do we care about $(9+1)$ dimensions?",
"It is related to sixteen.",
"In $(9+1)$ -dimensional Minkowski space, there is a sixteen-dimensional spinor representation, which is Majorana and Weyl.",
"Hence the minimal supersymmery transformation has sixteen components; and as we have seen above, it is maximal as well.",
"The field content is simple: gauge field $A_M (M=0,1,\\cdots ,9)$ and its superpartner (gaugino) $\\psi _\\alpha (\\alpha =1,2,\\cdots ,16)$ .",
"Both of them are $N\\times N$ Hermitian matrices.",
"The action isWe use $(-,+,\\cdots ,+)$ signature.",
"$S_{(9+1){\\rm d}}=\\frac{1}{g_{YM}^2}\\int d^{10}x{\\rm Tr}\\left(-\\frac{1}{4}F_{MM^{\\prime }}^2+\\frac{i}{2}\\bar{\\psi }\\gamma ^MD_M\\psi \\right),$ where $F_{MM^{\\prime }}=\\partial _MA_{M^{\\prime }}-\\partial _{M^{\\prime }}A_M+i[A_M,A_{M^{\\prime }}]$ and $D_M\\psi =\\partial _M\\psi +i[A_M,\\psi ]$ .",
"$\\gamma _M$ is the left-handed part of the 10d gamma matrices, which can be chosen as $\\gamma ^0&=&\\textbf {1}\\otimes \\textbf {1}\\otimes \\textbf {1}\\otimes \\textbf {1},\\nonumber \\\\\\gamma ^1&=&\\sigma _3\\otimes \\textbf {1}\\otimes \\textbf {1}\\otimes \\textbf {1},\\nonumber \\\\\\gamma ^2&=&\\sigma _2\\otimes \\sigma _2\\otimes \\sigma _2\\otimes \\sigma _2,\\nonumber \\\\\\gamma ^3&=&\\sigma _2\\otimes \\sigma _2\\otimes \\textbf {1}\\otimes \\sigma _1,\\nonumber \\\\\\gamma ^4&=&\\sigma _2\\otimes \\sigma _2\\otimes \\textbf {1}\\otimes \\sigma _3,\\nonumber \\\\\\gamma ^5&=&\\sigma _2\\otimes \\sigma _1\\otimes \\sigma _2\\otimes \\textbf {1},\\nonumber \\\\\\gamma ^6&=&\\sigma _2\\otimes \\sigma _3\\otimes \\sigma _2\\otimes \\textbf {1},\\nonumber \\\\\\gamma ^7&=&\\sigma _2\\otimes \\textbf {1}\\otimes \\sigma _1\\otimes \\sigma _2,\\nonumber \\\\\\gamma ^8&=&\\sigma _2\\otimes \\textbf {1}\\otimes \\sigma _3\\otimes \\sigma _2,\\nonumber \\\\\\gamma ^9&=&\\sigma _1\\otimes \\textbf {1}\\otimes \\textbf {1}\\otimes \\textbf {1}.$ They are all real and symmetric.",
"The fermion $\\psi $ is Majorana-Weyl, namely $\\psi _\\alpha ^\\dagger =\\psi _\\alpha $ .",
"Theories in lower spacetime dimensions can be obtained by the dimensional reduction.",
"Let us restrict spacetime to be $(p+1)$ -dimensional.",
"Namely, we do not allow the fields to depend on $x_{p+1},\\cdots ,x_9$ : $A_M(x_0,x_1,\\cdots ,x_{p},\\cdots ,x_9)&=&A_M(x_0,x_1,\\cdots ,x_{p}),\\nonumber \\\\\\psi _\\alpha (x_0,x_1,\\cdots ,x_{p},\\cdots ,x_9)&=&\\psi _\\alpha (x_0,x_1,\\cdots ,x_{p}).$ Let us use $\\mu ,\\nu ,\\cdots $ and $I,J\\cdots $ to denote $0,1,\\cdots ,p$ and $p+1,\\cdots ,9$ .",
"Also let us use $X_I$ to denote $A_I$ .",
"Then $F_{\\mu \\nu }&=&\\partial _\\mu A_\\nu -\\partial _\\nu A_\\mu +i[A_\\mu ,A_\\nu ],\\nonumber \\\\F_{\\mu I}&=&\\partial _\\mu X_I-\\partial _I A_\\mu +i[A_\\mu ,X_I]=\\partial _\\mu X_I+i[A_\\mu ,X_I]=D_\\mu X_I,\\nonumber \\\\F_{IJ}&=&\\partial _I X_J-\\partial _J X_I+i[X_I,X_J]=i[X_I,X_J].$ Here $D_\\mu X_I$ is the covariant derivative; $X_I$ behaves as an adjoint scalar after the dimensional reduction.",
"Hence the dimensionally reduced theory has the following action: $S_{(p+1){\\rm d}}=S_B + S_F,$ where $S_B=\\frac{1}{g_{YM}^2}\\int d^{p+1}x{\\rm Tr}\\left(-\\frac{1}{4}F_{\\mu \\nu }^2-\\frac{1}{2}(D_\\mu X_I)^2+\\frac{1}{4}[X_I,X_J]^2\\right)$ and $S_F=\\frac{1}{g_{YM}^2}\\int d^{p+1}x{\\rm Tr}\\left(\\frac{i}{2}\\bar{\\psi }\\Gamma ^\\mu D_\\mu \\psi -\\frac{1}{2}\\bar{\\psi }\\Gamma ^I [X_I,\\psi ]\\right).$ The Euclidean theory is obtained by performing the Wick rotation: $S_{\\rm Euclidean}&=&\\frac{1}{g_{YM}^2}\\int d^{p+1}x{\\rm Tr}\\left(\\frac{1}{4}F_{\\mu \\nu }^2+\\frac{1}{2}(D_\\mu X_I)^2-\\frac{1}{4}[X_I,X_J]^2\\right.\\nonumber \\\\& &\\hspace{99.58464pt}\\left.+\\frac{1}{2}\\bar{\\psi }D_t\\psi +\\frac{i}{2}\\sum _{\\mu =1}^3\\bar{\\psi }\\gamma ^\\mu D_\\mu \\psi -\\frac{1}{2}\\bar{\\psi }\\Gamma ^I [X_I,\\psi ]\\right).$"
],
[
"RHMC for SYM",
" By integrating out the fermions by hand, we obtain the PfaffianPfaffian is defined for $2n\\times 2n$ antisymmetric matrices $M$ , where $n=1,2,\\cdots $ .",
"Roughly speaking, the Pfaffian is the square root of the determinant: $({\\rm Pf} M)^2 = \\det M$ .",
"More precisely, ${\\rm Pf}M=\\frac{1}{2^nn!",
"}\\sum _{\\sigma \\in S_{2n}}{\\rm sgn}(\\sigma )\\prod _{i=1}^n M_{\\sigma (2i-1)\\sigma (2i)}$ .",
"of the Dirac operator: $Z=\\int [dA][dX] ({\\rm Pf} D[A,X])\\cdot e^{-S_B[A,X]}.$ Unfortunately, ${\\rm Pf}D$ is not positive definite.",
"Hence we use the absolute value of the pfaffian instead: $Z_{\\rm phase\\ quench}=\\int [dA][dX] |{\\rm Pf} D[A,X]|\\cdot e^{-S_B[A,X]}.$ For the justification of this phase quenching, see Sec.",
"REF .",
"In order to evaluate this integral, we use the RHMC algorithm.RHMC has been widely used in lattice QCD.",
"The first application to SYM can be found in [18].",
"The starting point is to rewrite $|{\\rm Pf}{\\cal D}|=\\left(\\det (D^\\dagger D)\\right)^{1/4}$ as $|{\\rm Pf}D|= \\int dF dF^* \\exp \\left(-((D^\\dagger D)^{-1/8} F)^\\dagger ((D^\\dagger D)^{-1/8} F)\\right).$ If we define $\\Phi $ by $\\Phi =(D^\\dagger D)^{-1/8} F$ , then $\\Phi $ can be generated by the Gaussian weight $e^{-\\Phi ^\\dagger \\Phi }$ , and $F$ can be obtained by $F={\\cal D}^{1/8} \\Phi $ .",
"We further rewrite this expression by using the rational approximation $x^{-1/4} \\simeq a_0 + \\sum _{k=1}^{Q}\\frac{a_k}{x+b_k}.$ Then we can replace the Pfaffian with $|{\\rm Pf}D|= \\int dF dF^* \\exp \\left(-S_{\\rm PF}\\right),$ where $S_{\\rm PF} =a_0 F^\\dagger F + \\sum _{k=1}^{Q}a_k F^\\dagger (D^\\dagger D+b_k)^{-1} F.$ The parameters $a_k$ , $b_k$ , $a^{\\prime }_k$ and $b^{\\prime }_k$ are real and positive, and can be chosen so that the approximation is sufficiently good within the range of $D^\\dagger D$ during the simulation.",
"Now it is clear that everything is the same as RHMC for strange quark except that the powers are different.",
"Therefore, the molecular evolution goes as follows: Generate $\\Phi $ by the Gaussian weight.",
"Calculate $F=(D^\\dagger D)^{1/8}\\Phi $ .",
"Calculate $H_i$ .",
"Fix $F$ and update $A_\\mu $ and $X_I$ .",
"Calculate $H_f$ .",
"Do the Metropois test.",
"Computationally most demanding part is solving the linear equations $({\\cal D}+b_k) \\chi _k =F \\quad (k=1, \\cdots , Q ),$ which appears in the derivative of $S_{PF}$ , $\\frac{\\partial S_{PF}}{\\partial A}=-\\sum _{k=1}^{Q}a_k \\chi _k^\\dagger \\frac{\\partial \\cal {D}}{\\partial A} \\chi _k,\\qquad \\frac{\\partial S_{PF}}{\\partial X}=-\\sum _{k=1}^{Q}a_k \\chi _k^\\dagger \\frac{\\partial \\cal {D}}{\\partial X} \\chi _k.$ This is much easier than evaluating the Pfaffian."
],
[
"Difference between SYM and QCD",
" Let us explain important differences between SYM and QCD simulations.This section has large overlap with other review articles I have written in the past, but I include it here in order to make this article self-contained."
],
[
"Parameter fine tuning problem and its cure",
" Symmetry plays important roles in physics.",
"For example, why is the pion so light?",
"Because QCD has approximate chiral symmetry and pion is the Nambu-Goldstone boson associated with this symmetry.",
"Finite quark mass breaks chiral symmetry softly, so that a light mass of pion is generated.",
"The Wilson fermion breaks chiral symmetry explicitly.",
"If we use it as a lattice regularization, the radiative corrections generate pion mass, and hence we have to fine-tune the bare quark mass to keep the pion light.",
"This is a well known example of the parameter fine tuning problem.",
"This problem does not exist if we use overlap fermion [19] or domain-wall fermion [20].",
"As another simple example, let us consider the plaquette action (REF ).",
"It has many exact symmetries at regularized level: Gauge symmetry $U_{\\mu ,\\vec{x}}\\rightarrow \\Omega _{\\vec{x}}U_{\\mu ,\\vec{x}}\\Omega _{\\vec{x}+\\hat{\\mu }}^\\dagger $ , where $\\Omega _{\\vec{x}}$ are unitary matrices defined on sites.",
"Discrete translation $\\vec{x}\\rightarrow \\vec{x}+\\hat{\\mu }$ 90-degree rotations, e. g. $\\hat{x}\\rightarrow \\hat{y}, \\hat{y}\\rightarrow -\\hat{x}$ Parity $\\hat{x},\\hat{y},\\hat{z}\\rightarrow -\\hat{x},-\\hat{y},-\\hat{z}$ Charge conjugation $U\\rightarrow U^\\dagger $ The integral measure is taken to be the Haar measure, which respects these symmetries.",
"Therefore the radiative corrections cannot break them: these are the symmetries of the quantum theory.In this case, discrete translation and rotation symmetries guarantee the invariance under continuous transformations in the continuum limit.",
"What happens if we use a regulator which breaks these symmetries, say a momentum cutoff which spoils gauge symmetry?",
"Then we need counter terms, whose coefficients are fine tuned, to restore the gauge invariance in the continuum.",
"So now we know the basic strategy for the lattice simulation of supersymmetric theories — let's keep supersymmetry exactly on a lattice!",
"But there is a one-sentence proof of a `no-go theorem': because supersymmetry algebra contains infinitesimal translation $\\lbrace Q_\\alpha ,\\bar{Q}_\\beta \\rbrace \\sim \\gamma ^\\mu P_\\mu $ , which is explicitly broken on any lattice by construction, it is impossible to keep entire algebra exactly.",
"Still, it is not the end of the game.",
"Exact symmetry at the regularized level is sufficient, but it is not always necessary.",
"Sometimes the radiative corrections can be controlled by using some other symmetries or a part of supersymmetry algebra.",
"The first breakthrough emerged for 4d ${\\cal N}=1$ pure SYM.",
"For this theory, the supersymmetric continuum limit is realized if the gaugino mass is set to zero.",
"This can be achieved if the chiral symmetry is realized on lattice [21], [22].",
"Of course it was not really a `solution' at that time, due to the Nielsen-Ninomiya no-go theorem [23].",
"Obviously, that the author of [21] proposed a way to circumvent the no-go theorem [20] was not a coincidence.",
"(Note that the one-parameter fine tuning can be tractable in this case.",
"See [24] for recent developments.)",
"The second breakthrough was directly motivated by the gauge/gravity duality [25].D. B.",
"Kaplan and M. Ünsal told me that their biggest motivation was in the gauge/gravity duality.",
"For several years this motivation has not been widely shared among the followers, probably because they did not mention it explicitly in their first paper [26].",
"The duality relates supersymmetric theories in various spacetime dimensions, including the ones with less than four dimensions, to string/M-theory [27].",
"Some prior proposals [6], [7], [29], [30] related lower dimensional theories to quantum gravity as well.",
"Because Yang-Mills in less than four dimensions are super-renormalizable, possible radiative corrections are rather constrained.",
"In particular, in two dimensions, supersymmetric continuum limit can be guaranteed to all order in perturbation theory by keeping some supersymmetries in a clever manner [26].",
"Yet another breakthrough came from superstring theory.",
"Maximally supersymmetric Yang-Mills in $(p+1)$ dimensions describe the low-energy dynamics of D$p$ -branes [31].",
"Furthermore higher dimensional D-branes can be made by collecting lower dimensional D-branes [32].",
"In terms of gauge theory, higher dimensional theories can be obtained as specific vacua of lower dimensional large-$N$ theories.",
"Or in other words, spacetime emerges from matrix degrees of freedom [33].",
"Based on these ideas, various regularization methods have been invented.",
"A list of regularization schemes for 4-, 8- and 16-SUSY theories without additional matter (i.e.",
"dimensional reductions of 4d, 6d and 10d ${\\cal N}=1$ pure super Yang-Mills) is shown below.",
"For more details see [34] and references therein.",
"Table: NO_CAPTION"
],
[
"Sign problem and its cure",
" A crucial assumption of MCMC is that the path-integral weight $e^{-S}$ is real and positive; otherwise it is not `probability'.",
"This condition is often broken.",
"The simplest example is the path-integral in Minkowski space — the weight $e^{iS}$ , where $S$ is real, is complex.",
"One may think this problem is gone in Euclidean space; the life is not that easy, the action $S$ can be complex after the Wick rotation; it only has to satisfy the reflection positivity, which is the counterpart of the reality in Minkowski space.",
"Well-known evils which cause the sign problem include the $\\theta $ -term and the Chern-Simons term in Yang-Mills theory, and QCD with finite baryon chemical potential.",
"Note that, although it is often called `fermion sign problem', the source of the sign is not necessary the fermionic part.",
"Maximal super Yang-Mills have the sign problem, because the pfaffian can take complex values.",
"There is no known generic solution of the sign problem.",
"It has been argued that the sign problem is NP-hard [35], and hence, unless you have a reason to think P$=$ NP, it is a waste of time to invest your time in searching for a generic solution.",
"However, theory-specific solutions are not excluded.",
"Probably the best case scenario is that somebody comes up with a clever change of variables so that the sign problem disappears.",
"I actually had such a lucky experience when he studied the ABJM matrix model in Ref.",
"[4], thanks to smart students who knew important analytic formulas developed in [36].",
"Certain supersymmetric theories seem to allow yet another beautiful solution, as we will explain below."
],
[
"Phase reweighting",
"Let us consider SYM as a concrete example.",
"The expectation value of an observable ${\\cal O}$ , which is written in terms of $A_\\mu $ and $X_I$ , with the `full' partition function (REF ) is $\\langle {\\cal O}(A,X)\\rangle _{\\rm full}=\\frac{\\int [dA][dX]({\\rm Pf} D[A,X])\\cdot e^{-S_B[A,X]}\\cdot {\\cal O}(A,X)}{\\int [dA][dX]({\\rm Pf}D[A,X])\\cdot e^{-S_B[A,X]}}.$ By writing ${\\rm Pf} D=|{\\rm Pf} D|e^{i\\theta }$ , we can easily see $\\langle {\\cal O}(A,X)\\rangle _{\\rm full}=\\frac{\\langle {\\cal O}(A,X)\\cdot e^{i\\theta [A,X]}\\rangle _{\\rm phase\\ quench}}{\\langle e^{i\\theta [A,X]}\\rangle _{\\rm phase\\ quench}},$ where $\\langle \\ \\cdot \\ \\rangle _{\\rm phase\\ quench}$ is the expectation value with the phase quenched path-integral (REF ).",
"By using this trivial identity, in principle we can calculate the expectation value in the full theory.",
"This method is called phase reweighting.",
"However there are several reasons we do not want to use the phase reweighting method.",
"Firstly, calculation of the pfaffian is numerically very demanding, and although we can avoid explicit evaluation of the Pfaffian in the configuration generation with the phase quenched ensemble, in order to calculate (REF ) we must calculate the Pfaffian explicitly.",
"Secondly, if the phase factor $e^{i\\theta }$ fluctuates rapidly, both the denominator and numerator becomes very small, and numerically it is difficult to distinguish them from zero.",
"Then the expression is practically $0/0$ , whose error bar is infinitely large.",
"The fluctuation is particularly violent when the full theory and phase quenched theory does not have substantial overlap (see Sec.",
"REF ), and it actually happens in finite density QCD.",
"As we will argue shortly, the full theory and phase quenched theory are rather close in the case of SYM.",
"Still, the phase fluctuation is very large and it is impossible to calculate the average phase at large $N$ and/or large volume.",
"Roughly speaking, $\\theta $ is the imaginary part of the action, which scales as $N^2$ times volume!",
"Although the phase reweighting is an unrealistic approach to SYM, it can be a practical tool when the simulation cost is cheaper.",
"For example it is a common strategy in certain area of condensed matter physics."
],
[
"Phase quench",
"Let us consider the phase quench approximation.",
"Namely, we calculate $\\langle {\\cal O}(A,X)\\rangle _{\\rm phase\\ quench}$ , which does not take into account the phase at all, instead of $\\langle {\\cal O}(A,X)\\rangle _{\\rm full}$ .",
"Such an `approximation' can make sense when ${\\cal O}$ and $e^{i\\theta }$ factorize, $\\langle {\\cal O}e^{i\\theta }\\rangle _{\\rm phase\\ quench}\\simeq \\langle {\\cal O}\\rangle _{\\rm phase\\ quench}\\times \\langle e^{i\\theta }\\rangle _{\\rm phase\\ quench}$ .",
"The phase quench approximation almost always fails.",
"Very fortunately, rare exceptions include SYM.",
"Firstly, at high temperature and not so large $N$ , $\\theta $ is close to zero and hence the phase can safely be ignored.",
"Rather surprisingly, $\\theta $ remains small down to rather low temperature and moderately large $N$ , which is relevant for testing the gauge/gravity duality; see e.g.",
"[37], [38], [39], [40], [41] for explicit checks for $(0+1)$ - and $(1+1)$ -dimensional theories.",
"At low temperature or with supersymmetric boundary condition, the phase factor does oscillate.",
"Still, as long as the fluctuation is not very rapid, we can defeat the sign by brute force.",
"Then we can confirm $\\langle {\\cal O}\\rangle _{\\rm phase\\ quench}= \\langle {\\cal O}\\rangle _{\\rm full}$ within numerical error, for some observables [40], [42].",
"When the phase factor fluctuates rapidly, or the dimension of the Dirac operator is too big, we cannot evaluate the effect of the phase.",
"Still we can make an indirect argument based on numerical evidence that the phase quench approximation is good for certain observables [43].",
"Note that the phase quench leads to a wrong result for the type IIB matrix model [28], which is the dimensional reduction of maximal SYM to zero dimension."
],
[
"Flat direction and its cure",
" Many supersymmetric Yang-Mills theories (practically all theories relevant in the context of the gauge/gravity duality) have adjoint scalar fields $X_I$ .",
"There are flat directions along which scalar matrices commute, $[X_I,X_J]=0$ , and the eigenvalues roll to infinity.",
"With 2 or 3 noncompact spatial dimensions, we can fix the eigenvalues by hand; it is a choice of moduli, due to the superselection.",
"However with compact spaceMore precisely, the torus compactification.",
"For the compactification to a curved manifold, the flat directions are lifted by a mass term associated with the curvature.",
"or lower dimensions the eigenvalue distribution should be determined dynamically.",
"There is nothing wrong with this: the flat direction is a feature of the theory, it is not a bug.",
"However it causes headache when we perform MCMC for two reasons: firstly, the partition function is not convergent, and hence the simulation never thermalizes [37].",
"Secondly, when the eigenvalues roll too far, the lattice simulation runs into an unphysical lattice artifact [44].",
"In order to tame the flat directions, we have to understand its physical meaning.",
"$(p+1)$ -d U($N$ ) SYM describe a system of $N$ D$p$ -branes sitting parallel to each other, and the eigenvalues $(X_1^{ii},X_2^{ii},\\cdots ,X_{9-p}^{ii})$ describe the location of the $i$ -th D$p$ -brane.",
"When eigenvalues form a bound state, it can be regarded as a black $p$ -brane.",
"When one of the eigenvalues is separated far from others, the interaction is very weak due to supersymmetry; the potential is proportional to $-f(T)/r^{8-p}$ [45], [46], where $r$ is the distance, $T$ is temperature of the black brane and $f(T)$ is a monotonically increasing function which vanishes at $T=0$ .",
"This potential is not strong enough to trap a D$p$ -brane once it is emitted from the black brane.",
"In the same manner, phases with multiple bunches of eigenvalues can exist; they describe multiple black branes.",
"This is the reason that the flat directions exist; the partition function is not convergent because there are too many different classes of configurations.",
"Now our task is apparent: we should cut out configurations describing certain physical situation we are interested.",
"The first thing we should study is a single black hole or black brane, namely a bound state of all eigenvalues.",
"This bound state is only metastable, but as $N$ increases it becomes stable enough so that we can collect sufficiently many configurations without seeing the emission of the eigenvalues [37].",
"Large-$N$ behavior of the $(0+1)$ -dimensional theory has been studied in this manner in [37], [38], [43].",
"When $N$ is small, we need to introduce a cutoff for the eigenvalues.",
"In [47] a cutoff is introduced for $(1/N)\\sum _M {\\rm Tr}X_M^2$ , and by carefully changing the cutoff the property of the single bunch phase has been extracted.",
"It is also possible to introduce a mass term $Nm^2\\int d^{p+1}x {\\rm Tr}X_M^2$ and take $m\\rightarrow 0$ [48], [44].",
"Note that one has to make sure that the flat direction is under control when $m^2$ is small, as demonstrated in [48], [44], because simulations often pick up the flat direction and end up in the U$(1)^N$ vacuum.",
"See also [49], [50][51], [52], [53], [41][54] for the details on how to control the flat directions in actual simulations.",
"It is also possible to add slightly more complicated deformation which preserves (at least a part of) supersymmetry, which is so-called `plane wave deformation' [55].",
"See [55], [56] for various $(0+1)$ -d theories and [57], [58], [59] for $(1+1)$ -d lattice construction.",
"See also [60] for other cases."
],
[
"Conclusion",
" In this article I have tried to explain the essence of Markov Chain Monte Carlo.",
"Simple algorithms like the Metropolis algorithm is good enough for solving nontrivial problems in hep-th literature on a laptop.",
"Serious simulations of super Yang-Mills require more efforts, but the basic idea is the same.",
"We just have to update the configurations more efficiently.",
"For that, we can use efficient techniques developed in lattice QCD community: HMC, RHMC and many more which I haven't explained in this article.",
"They can be implemented to our simulation code by using $+,-,\\times ,\\div ,\\sin ,\\cos ,\\exp ,\\log ,\\sqrt{\\ \\ }$ , “if\" and loop.",
"There are some challenging issues specific to supersymmetric theories, most notably the problem associated with flat directions, but we already know basic strategies to handle them.",
"In this article I did not explain the complete details of lattice SYM simulation.",
"The detail of the simulations of BFSS matrix model can be found in the article available at [8]."
],
[
"Acknowledgement",
" I thank Tatsuo Azeyanagi, Valentina Forini, Anosh Joseph, Michael Kroyter, So Matsuura, R. Loganayagam, Joao Penedones, Enrico Rinaldi, David Schaich, Masaki Tezuka and Toby Wiseman.",
"I was in part supported by JSPS KAKENHI Grants 17K14285.",
"Many materials in this article were prepared for “Nonperturbative and Numerical Approaches to Quantum Gravity, String Theory and Holography\", which was held from 27 January 2018 to 03 February 2018 at International Center for Theoretical Science.",
"I thank Anosh Joseph and R. Loganayagam for giving me that wonderful opportunity.",
"This work was also partially supported by the Department of Energy, award number DE-SC0017905."
],
[
"(Single-mass) CG method ",
"Let us first introduce the ordinary (or `single-mass') conjugate gradient method.",
"The notation is that of [61].",
"We want to solve the linear equation $A\\vec{x}=\\vec{b},$ where $A=D^\\dagger D$ .",
"We construct a sequence of approximate solutions $\\vec{x}_1,\\vec{x}_2,\\cdots $ which (almost always) converges to the solution.",
"We start with an initial trial solution $\\vec{x}_1$ , which is arbitrary.",
"From this we define $\\vec{r}_1,\\vec{\\bar{r}}_1,\\vec{p}_1,\\vec{\\bar{p}}_1$ as $\\vec{r}_1=\\vec{\\bar{r}}_1=\\vec{p}_1=\\vec{\\bar{p}}_1=\\vec{b}-A\\vec{x}_1.$ Then, we construct $\\vec{x}_k$ as follows: $\\alpha _k=\\frac{\\vec{r}^\\dagger _k\\cdot \\vec{r}_k}{\\vec{p}^\\dagger _k\\cdot A\\vec{p}_k}$ .",
"$\\vec{x}_{k+1}=\\vec{x}_k+\\alpha _k\\vec{p}_k.$ $\\vec{r}_{k+1}=\\vec{r}_k-\\alpha _k A\\vec{p}_k$ .",
"$\\beta _k=\\frac{\\vec{r}^\\dagger _{k+1}\\cdot \\vec{r}_{k+1}}{\\vec{r}^\\dagger _k\\cdot \\vec{r}_k}$ .",
"$\\vec{p}_{k+1}=\\vec{r}_{k+1}+\\beta _k\\vec{p}_{k}$ .",
"Note that $\\vec{r}_{k}=\\vec{b}-A\\vec{x}_{k}$ .",
"The norm of $\\vec{r}_k$ converges to zero, or equivalently, $\\vec{x}_{k}$ converges to the solution."
],
[
"Multi-mass CG method ",
"Let $A_\\sigma $ be `shifted' version of $A$ : $A_\\sigma =A+\\sigma \\cdot \\textbf {1}.$ If $A$ were the Laplacian, $\\sigma $ would be interpreted as a mass term.",
"The multi-mass CG solver [16] enables us to solve $A_\\sigma \\vec{x}=\\vec{b}$ for many different values of $\\sigma $ simultaneously, with only negligibly small additional cost.",
"The key idea is that, from the iterative series for $A$ , $\\vec{r}_{k+1}&=&\\vec{r}_k-\\alpha _k A\\vec{p}_k,\\nonumber \\\\\\vec{p}_{k+1}&=&\\vec{r}_{k+1}+\\beta _k\\vec{p}_{k},$ it is possible to construct a similar series for the shifted operator, $\\vec{r}_{k+1}^\\sigma &=&\\vec{r}_k^\\sigma -\\alpha _k^\\sigma A_\\sigma \\vec{p}_k^\\sigma ,\\nonumber \\\\\\vec{p}_{k+1}^\\sigma &=&\\vec{r}_{k+1}^\\sigma +\\beta _k^\\sigma \\vec{p}_{k}^\\sigma ,$ where $\\vec{r}_k^\\sigma =\\zeta _k^\\sigma \\vec{r}_k.$ Indeed, (REF ) can be satisfied by takingFrom (REF ) we have $\\vec{r}_{k+1}=\\left(1+\\frac{\\alpha _k\\beta _{k-1}}{\\alpha _{k-1}}\\right)\\vec{r}_k-\\alpha _k A\\vec{r}_k-\\frac{\\alpha _k\\beta _{k-1}}{\\alpha _{k-1}}\\vec{r}_{k-1}.$ Comparing the coefficients with those in a similar equation obtained from (REF ), we obtain (REF ).",
"$\\alpha _k^\\sigma &=&\\alpha _k\\cdot \\frac{\\zeta _{k+1}^\\sigma }{\\zeta _{k}^\\sigma },\\nonumber \\\\\\beta _k^\\sigma &=&\\beta _k\\cdot \\left(\\frac{\\zeta _{k+1}^\\sigma }{\\zeta _{k}^\\sigma }\\right)^2,\\nonumber \\\\\\zeta _{k+1}^\\sigma &=&\\frac{\\zeta _{k}^\\sigma \\zeta _{k-1}^\\sigma \\alpha _{k-1}}{\\alpha _{k-1}\\zeta _{k-1}^\\sigma (1+\\alpha _k\\sigma )+\\alpha _k\\beta _{k-1}(\\zeta _{k-1}^\\sigma -\\zeta _{k}^\\sigma )}.$ Therefore, we can generalize the usual BiCG solver in the following manner: $\\alpha _k=\\frac{\\vec{r}^\\dagger _k\\cdot \\vec{r}_k}{\\vec{p}^\\dagger _k\\cdot A\\vec{p}_k}$ .",
"Calculate $\\zeta _{k+1}^\\sigma $ and $\\alpha _k^\\sigma $ using (REF ).",
"$\\vec{x}_{k+1}^\\sigma =\\vec{x}_k^\\sigma +\\alpha _k^\\sigma \\vec{p}_k^\\sigma .$ $\\vec{r}_{k+1}=\\vec{r}_k-\\alpha _k A\\vec{p}_k$ .",
"$\\beta _k=\\frac{\\vec{r}^\\dagger _{k+1}\\cdot \\vec{r}_{k+1}}{\\vec{r}^\\dagger _k\\cdot \\vec{r}_k}$ .",
"Calculate $\\beta _k^\\sigma $ using (REF ).",
"$\\vec{p}_{k+1}=\\vec{r}_{k+1}+\\beta _k\\vec{p}_{k}$ , $\\vec{p}_{k+1}^\\sigma =\\vec{r}_{k+1}^\\sigma +\\beta _k^\\sigma \\vec{p}_{k}^\\sigma $ .",
"Then, $\\vec{x}^\\sigma _k$ is an approximate solution with the residual vector $\\vec{r}_k^\\sigma $ , $\\vec{r}_k^\\sigma =\\vec{b}-A_\\sigma \\vec{x}_k^\\sigma .$ Note that we cannot start with an arbitrary initial condition, because (REF ) is not satisfied then.",
"We choose the following special initial condition in order to satisfy (REF ): $& &\\vec{x}_1=\\vec{x}_1^\\sigma =\\vec{p}_0=\\vec{p}_0^\\sigma =\\vec{0},\\nonumber \\\\& &\\vec{r}_1=\\vec{r}_1^\\sigma =\\vec{r}_0=\\vec{r}_0^\\sigma =\\vec{p}_1=\\vec{p}_1^\\sigma =\\vec{b},\\nonumber \\\\& &\\zeta _0^\\sigma =\\zeta _1^\\sigma =\\alpha _0=\\alpha _0^\\sigma =\\beta _0=\\beta _0^\\sigma =1.$ As long as we stick to this initial condition, it is hard to implement any preconditioning.",
"If you know any preconditioning which can be used with multi-mass CG solver, please let us know."
],
[
"Box-Muller method",
"Let $p$ and $q$ be uniform random numbers in $[0,1]$ .",
"There are many random number generators which give you such $p$ and $q$ .",
"Then, $x = \\sqrt{-2\\log p}\\sin (2\\pi q)$ and $y = \\sqrt{-2\\log p}\\cos (2\\pi q)$ are random numbers with weights $\\frac{e^{-x^2/2}}{\\sqrt{2\\pi }}$ and $\\frac{e^{-y^2/2}}{\\sqrt{2\\pi }}$ ."
],
[
"Jackknife method: generic case",
"In Sec.",
"REF , we assumed that a quantity of interest can be calculated for each sample.",
"Let us consider more generic cases; for example in order to determine the mass of particle excitation we need to calculate two-point function by using many samples and then extract the mass from that, and hence the mass cannot be calculated sample by sample.",
"In the Jackknife method, we first divide the configurations to bins with width $w$ ; the first bin is $\\lbrace x^{(1)}\\rbrace , \\lbrace x^{(2)}\\rbrace ,\\cdots ,\\lbrace x^{(w)}\\rbrace $ , the second bin is $\\lbrace x^{(w+1)}\\rbrace , \\lbrace x^{(w+2)}\\rbrace ,\\cdots ,\\lbrace x^{(2w)}\\rbrace $ , etc.",
"Suppose we have $n$ bins.",
"Then we define the average of an observable $f(x)$ with $k$ -th bin removed, $\\overline{f}^{(k,w)}\\equiv \\left({\\rm the\\ value\\ calculated\\ after\\ removing} k{\\rm th\\ bin}\\right).$ The average value is defined by $\\overline{f}\\equiv \\frac{1}{n}\\sum _{k}\\overline{f}^{(k,w)}.$ The Jackknife error is defined by $\\Delta _w\\equiv \\sqrt{\\frac{n-1}{n}\\sum _{k}\\left(\\overline{f}^{(k,w)}-\\overline{f}\\right)^2}.$"
]
] | 1808.08490 |
[
[
"Spectral Efficiency Analysis of Multi-Cell Massive MIMO Systems with\n Ricean Fading"
],
[
"Abstract This paper investigates the spectral efficiency of multi-cell massive multiple-input multiple-output systems with Ricean fading that utilize the linear maximal-ratio combining detector.",
"We firstly present closed-form expressions for the effective signal-to-interference-plus-noise ratio (SINR) with the least squares and minimum mean squared error (MMSE) estimation methods, respectively, which apply for any number of base-station antennas $M$ and any Ricean $K$-factor.",
"Also, the obtained results can be particularized in Rayleigh fading conditions when the Ricean $K$-factor is equal to zero.",
"In the following, novel exact asymptotic expressions of the effective SINR are derived in the high $M$ and high Ricean $K$-factor regimes.",
"The corresponding analysis shows that pilot contamination is removed by the MMSE estimator when we consider both infinite $M$ and infinite Ricean $K$-factor, while the pilot contamination phenomenon persists for the rest of cases.",
"All the theoretical results are verified via Monte-Carlo simulations."
],
[
"Introduction",
"In the design of future communication systems, spectral efficiency (SE) becomes one of the dominant targets [1].",
"Massive multiple-input multiple-output (MIMO) [2], where the base station (BS) is equipped with hundreds of antennas to serve tens of users in the same time-frequency resource block, has attracted substantial attention from academia and industry thanks to the the high SE gains provided by the massive array [3], [4], [5].",
"The fundamental problem of placing a massive number of antennas in a confined space, can be addressed by pushing the operating frequency in the milimeter wave band, where the wavelengths become inherently small.",
"We recall that mm-wave channels are typically modeled via the Ricean fading distribution due to the presence of line-of-sight (LOS) or specular components [6].",
"Hence, the SE analysis in massive MIMO systems with Ricean fading becomes a problem of practical relevance.",
"We will now review the relevant state-of-the-art in the massive MIMO literature.",
"We firstly note the work of [3], which derived useful lower bounds on the uplink ergodic rate for different classical linear receivers and analyzed the SE performance over a single-cell massive MIMO system operating in Rayleigh fading for perfect channel state information (CSI) and imperfect CSI, respectively.",
"The same authors extended part of the above results into multi-cell systems and also obtained the corresponding lower bounds in [7].",
"The authors in [8] provided a closed-form expression of the SE, explored the SE maximizing problem in multi-cell systems, and proposed the optimal system parameters.",
"Recently, [4] considered the achievable uplink rates based on both the least squares (LS) and minimum mean squared error (MMSE) estimation methods in multi-cell massive MIMO systems and investigated the pilot power allocation scheme to maximize the minimum SE.",
"As a general comment, most issues pertaining to the achievable SE of massive MIMO, massive MIMO systems in Rayleigh fading have been largely and extensively characterized.",
"Apart from that, some recent investigations have analyzed the SE performance in massive MIMO systems based on Ricean fading.",
"For example, for a single-cell environment, [6] examined the scaling law and obtained the approximate expressions of uplink ergodic rate by considering Ricean fading and both perfect CSI and imperfect CSI.",
"Moreover, for multi-cell massive MIMO systems, [9] obtained a closed-form approximation and [10] provided a lower bound of the achievable uplink rate with imperfect CSI, respectively.",
"Also, [11] considered a similar scenario as [9] and pursued an asymptotic analysis of the achievable rate.",
"Recently, [12], [13] systematically investigated the SE performance with spatially correlated Ricean fading channels and obtained closed-form expressions of uplink/downlink SE for different channel estimation methods.",
"However, it appears that those expressions are untraceable to gain a very clear insight.",
"Also, the corresponding asymptotic analysis in massive antenna regime is obtained at the cost of the tougher conditions of the spatial covariance matrix and LOS component.",
"From the above discussion, it becomes apparent that a straightforward and exact theoretical analysis of massive MIMO systems with Ricean fading is missing from the open literature.",
"In this paper, we firstly introduce a general analytic framework for investigating the achievable uplink SE with a linear maximal-ratio combining (MRC) detector.",
"Then, we derive two exact closed-form expressions of the effective signal-to-interference-plus-noise ratio (SINR) based on the LS and MMSE estimation methods, respectively, which apply for any number of antennas $M$ and any Ricean $K$ -factor.",
"These expressions are particularly tractable for admitting fast and effective computation.",
"When the Ricean $K$ -factor is equal to zero, i.e., Rayleigh fading, our results are substantially simplified.",
"Also, based on the proposed expressions, we investigate in detail the performance of the effective SINR precisely by making either the $M$ or the Ricean $K$ -factor becomes infinite.",
"In both cases, it is shown that the effect of pilot contamination cannot be removed for both LS and MMSE estimation methods; however, if both the $M$ and Ricean $K$ -factor continue to grow unbound, the pilot contamination issue is eliminated only for the MMSE estimation method.",
"Finally, a set of Monte-Carlo simulations is conducted to validate the the above mentioned analytical results.",
"Notation: Lower-case and upper-case boldface letters denote vectors and matrices, respectively; ${\\mathbb {C}}^{M\\times N}$ denotes the $M\\times N$ complex space; ${\\bf A}^{\\text{T}}$ , ${\\bf A}^{\\dag }$ , and ${\\bf A}^{-1}$ denote the transpose, the Hermitian transpose, and the inverse of the matrix ${\\bf A}$ , respectively; ${\\bf I}_{M}$ denotes an $M\\times M$ identity matrix and ${\\bf 0}_{M\\times N}$ denotes an $M\\times N$ zero matrix.",
"The expectation operation is $\\mathbb {E}\\lbrace \\cdot \\rbrace $ .",
"A complex Gaussian random vector ${\\bf x}$ is denoted as ${\\bf x}\\sim \\mathcal {C}\\mathcal {N}(\\bar{{\\bf x}},{\\bf {\\Sigma }})$ , where the mean vector is $\\bar{{\\bf x}}$ and the covariance matrix is ${\\bf {\\Sigma }}$ , while $\\Vert \\cdot \\Vert _{2}$ denotes the 2-norm of a vector.",
"Finally, ${\\rm {diag}}\\left({\\bf a}\\right)$ denotes a diagonal matrix where the main diagonal entries are the elements of vector ${\\bf a}$ ."
],
[
"System Model",
"In this paper, we consider a typical uplink cellular communication system with $L$ hexagonal cells.",
"Each cell contains a BS and $N$ single-antenna users.",
"Each BS has a uniform linear array with $M$ ($M\\gg N$ ) antennas.",
"In the following, the time-division duplex (TDD) mode is adopted and we assume that the BSs and users in this system are perfectly synchronized.",
"For each channel use, the $M\\times 1$ received signal vector of the BS in cell $j$ is given by ${\\bf y}_{j}=\\sqrt{\\rho _{u}}\\sum \\limits _{l=1}^{L}{\\bf H}_{jl}{\\bf x}_{l}+{\\bf n}_{j},$ where ${\\bf x}_{l}$ denotes the $N\\times 1$ vector containing the transmitted signals from all the users in cell $l$ , which satisfies $\\mathbb {E}\\left\\lbrace {\\bf x}_{l}\\right\\rbrace ={\\bf 0}_{N\\times 1}$ and $\\mathbb {E}\\left\\lbrace {\\bf x}_{l}{\\bf x}_{l}^{\\dag }\\right\\rbrace ={\\bf I}_{N}$ , $\\rho _u$ is the average transmitted power of each user, and ${\\bf n}_{j}\\in \\mathbb {C}^{M\\times 1}\\sim \\mathcal {C}\\mathcal {N}({\\bf 0}_{M\\times 1},{\\bf I}_{M})$ is the additive white Gaussian noise (AWGN) vector.",
"Also, ${\\bf H}_{jl}\\in \\mathbb {C}^{M \\times N}$ represents the channel matrix between the users in cell $l$ and the BS in cell $j$ , whose its $n$ th column, ${\\bf h}_{jln}\\in \\mathbb {C}^{M\\times 1}$ , is the channel vector between the user $n$ in cell $l$ and the BS in cell $j$ .",
"Here, the block-fading model [3] is utilized where the large-scale fading coefficients are kept fixed over lots of coherence time intervals and the small-scale fading coefficients remain fixed within a coherence time interval and change between any two adjacent coherence time intervals.",
"Moreover, the large-scale fading coefficients are assumed to be perfectly known at the BS side due to their slow-varying nature.",
"We herein consider the Ricean fading model in [9] where both a LOS path and a non LOS (NLOS) component exist in the channels between the users and BS in the same cell, while only a NLOS component exists in the channels between the users and BS in different cells.",
"The above mentioned model is reasonable since a LOS component is more likely to kick in when the users and the BS are in the same cell.",
"Hence, ${\\bf h}_{jln}$ can be rewritten as $\\begin{split}{\\bf h}_{jln}\\!=\\!\\left\\lbrace \\begin{array}{*{20}l}\\!\\!\\!\\!\\!\\sqrt{\\!\\frac{K_{jn}}{K_{jn}+1}}{\\bf h}_{jjn, {\\rm {LOS}}}+\\!\\!\\sqrt{\\!\\frac{1}{K_{jn}+1}}{\\bf h}_{jjn, {\\rm {NLOS}}}, &l=j,\\\\\\\\{\\bf h}_{jln, {\\rm {NLOS}}}, &l\\ne j,\\\\\\end{array}\\right.\\end{split}$ where $K_{jn}$ denotes the Ricean $K$ -factor for the user $n$ in cell $j$ , ${\\bf h}_{jln, {\\rm {NLOS}}}\\sim \\mathcal {C}\\mathcal {N}({\\bf 0}_{M\\times 1},\\beta _{jln}{\\bf I}_{M})$ ($\\forall l,n$ ) denotes the NLOS component between the user $n$ in cell $l$ and the BS in cell $j$ , while $\\beta _{jln}$ is the corresponding large-scale fading coefficient.",
"Also, ${\\bf h}_{jjn, {\\rm {LOS}}}\\in \\mathbb {C}^{M \\times 1}$ denotes the LOS part between the user $n$ and the BS in cell $j$ , whose $m$ th entry $[{\\bf h}_{jjn, {\\rm {LOS}}}]_{m}$ is given by $[{\\bf h}_{jjn, {\\rm {LOS}}}]_{m}=\\beta _{jjn}^{\\frac{1}{2}}e^{-i(m-1)\\frac{2\\pi d}{\\lambda }\\sin (\\theta _{jn})},$ where $\\lambda $ is the wavelength, $d$ is the antenna spacing, $\\theta _{jn}\\in [0, 2\\pi )$ is the angle of arrival of the $n$ th user in cell $j$ , and $i$ denotes imaginary unit.",
"Based on the fact that $K_{jn}$ and $\\theta _{jn}$ can be obtained through a feedback link [14], we can also assume that the Ricean $K$ -factor and LOS path can be perfectly obtained by the BS and users.",
"In practical communication systems, the BS side does not have perfect CSI and the channel needs to be estimated at the BS.",
"In TDD mode, uplink pilot training is adopted for obtaining the estimated channel information before data transmission.",
"The worst-case of pilot sequence allocation is adopted here, where each cell's users utilize the same set of pilot sequences [4], which we denote as ${\\mathbf {\\Phi }}\\in \\mathbb {C}^{\\tau \\times N}$ .",
"Also, $\\tau $ is the length of the pilot sequence, which is larger than or equal to $N$ , and ${\\mathbf {\\Phi }}$ satisfies ${\\mathbf {\\Phi }}^{\\dag }{\\mathbf {\\Phi }}={\\bf I}_{N}$ .",
"Then, the $M\\times \\tau $ received pilot sequence signal matrix at the BS in cell $j$ is given by ${\\bf Y}_{j,\\text{train}}=\\sum \\limits _{l=1}^{L}{\\bf H}_{jl}({\\mathbf {\\Omega }}_{l}+{\\bf I}_{N})^{\\frac{1}{2}}{\\bf P}_{l}^{\\frac{1}{2}}{\\mathbf {\\Phi }}^{\\dag }+{\\bf N}_{j},$ where ${\\bf P}_l$ is the $N\\times N$ diagonal matrix with the power of the pilot sequence sent by user $n$ in cell $l$ is $[{\\bf P}_{l}]_{nn}=\\rho _{ln}$ , ${\\bf N}_{j}\\in \\mathbb {C}^{M\\times \\tau }$ is the AWGN matrix with the independent and identically distributed zero-mean and unit-variance elements, and ${\\mathbf {\\Omega }}_{l}={\\rm {diag}}([K_{l1},\\ldots , K_{ln},\\ldots ,K_{lN}]^{\\text{T}})\\in \\mathbb {C}^{N\\times N}$ .",
"Particularly, the reason for the pilot matrix ${\\bf P}_{l}^{\\frac{1}{2}}{\\mathbf {\\Phi }}^{\\dag }$ multiplied by $({\\mathbf {\\Omega }}_{l}+{\\bf I}_{N})^{\\frac{1}{2}}$ , which is sent by all users in cell $j$ , is to estimate the NLOS component in a simplified manner; note that a similar method followed in [6], [9].",
"By removing the known LOS part in (4) and utilizing the LS and MMSE estimation methods [4], [15], ${\\bf h}_{jjn, {\\rm {NLOS}}}$ is estimated as ${\\bf {\\hat{h}}}_{jjn, {\\rm {NLOS}}}^{\\text{LS}}&={\\bf {h}}_{jjn, {\\rm {NLOS}}}\\!+\\!\\!\\!\\sum \\limits _{l\\ne j}^{L}\\!\\!\\sqrt{\\frac{\\rho _{ln}\\!(\\!K_{ln}\\!\\!+\\!\\!1\\!",
")}{\\rho _{jn}}}{\\bf {h}}_{jln, {\\rm {NLOS}}}\\!+\\!\\frac{\\tilde{\\bf n}_{jn}}{\\sqrt{\\rho _{jn}}},\\\\{\\bf {\\hat{h}}}_{jjn, {\\rm {NLOS}}}^{\\text{MMSE}}&=\\chi _{jn}{\\bf {\\hat{h}}}_{jjn, {\\rm {NLOS}}}^{\\text{LS}},$ respectively, where ${\\bf {\\hat{h}}}_{jjn, {\\rm {NLOS}}}^{\\text{LS}}$ and ${\\bf {\\hat{h}}}_{jjn, {\\rm {NLOS}}}^{\\text{MMSE}}$ denote the estimators of ${\\bf h}_{jjn, {\\rm {NLOS}}}$ based on the LS and MMSE estimation methods, respectively, as well as, $\\tilde{\\bf n}_{jn}\\triangleq {\\bf N}_{j}{\\mathbf {\\phi }}_{n}$ .",
"Also, ${\\mathbf {\\phi }}_{n}$ is the $n$ th column of ${\\mathbf {\\Phi }}$ and $\\chi _{jn}\\triangleq \\frac{\\rho _{jn}\\beta _{jjn}}{\\rho _{jn}\\beta _{jjn}+\\sum \\limits _{l\\ne j}^{L}{\\rho _{ln}}(K_{ln}+1)\\beta _{jln}+1}.$ For convenience, we denote the estimator of ${\\bf {h}}_{jjn, {\\rm {NLOS}}}$ as ${\\bf {\\hat{h}}}_{jjn, {\\rm {NLOS}}}$ and the estimator of ${\\bf {h}}_{jjn}$ as ${\\bf {\\hat{h}}}_{jjn}=\\sqrt{\\frac{K_{jn}}{K_{jn}+1}}{\\bf h}_{jjn, {\\rm {LOS}}}+\\sqrt{\\frac{1}{K_{jn}+1}}{\\bf {\\hat{h}}}_{jjn, {\\rm {NLOS}}},$ respectively, for both the LS and MMSE estimation methods.",
"After channel estimation, we use the standard linear detector MRC [4] to detect the received data signal in (1).",
"Hence, for the BS in cell $j$ , (1) is separated into $N$ streams by multiplying it with the MRC detector, that is, ${\\bf r}_{j}=\\hat{\\bf H}_{jj}^{\\dag }{\\bf y}_{j}\\in \\mathbb {C}^{N \\times 1},$ where $\\hat{\\bf H}_{jj}\\triangleq [\\hat{\\bf h}_{jj1},\\ldots ,\\hat{\\bf h}_{jjn} ,\\ldots ,\\hat{\\bf h}_{jjN}]\\in \\mathbb {C}^{M \\times N}$ .",
"Then, for the $n$ th user in cell $j$ , we have ${r}_{jn}&=\\sqrt{\\rho _{u}}\\hat{\\bf h}_{jjn}^{\\dag }{\\bf h}_{jjn}x_{jn}+\\sqrt{\\rho _u}\\!\\!\\!\\!\\sum \\limits _{(l,t)\\ne (j,n)}\\!\\!\\!\\!\\hat{\\bf h}_{jjn}^{\\dag }{\\bf h}_{jlt}x_{lt}+\\hat{\\bf h}_{jjn}^{\\dag }{\\bf n}_{j},$ where ${r}_{jn}$ is the $n$ th entry of ${\\bf r}_{j}$ ."
],
[
"Spectral Efficiency",
"In this section, we obtain the closed-form expressions of the effective SINR for both LS and MMSE estimation methods and, thereafter, aim to analyze the SINR performance with the respect of the number of BS antennas $M$ and the Ricean $K$ -factor, respectively."
],
[
"Closed-Form of ${\\text{SINR}}_{jn}$",
"Since we want to obtain a computable expression of the achievable uplink SE and investigate it by a simple way, it is convenient to follow the methodology of [4] that assumes that the term $\\mathbb {E}\\lbrace {\\bf \\hat{h}}_{jjn}^{\\dag }{\\bf h}_{jjn}\\rbrace $ is known at the BS in cell $j$ perfectly.",
"Hence, (10) can be rewritten as ${r}_{jn}&=\\!\\!\\sqrt{\\rho _{u}}\\mathbb {E}\\left\\lbrace \\hat{\\bf h}_{jjn}^{\\dag }{\\bf h}_{jjn}\\right\\rbrace x_{jn}\\!+\\!\\sqrt{\\rho _u}\\!\\!\\!\\!\\!\\!\\sum \\limits _{(l,t)\\ne (j,n)}\\!\\!\\!\\!\\!\\!\\hat{\\bf h}_{jjn}^{\\dag }{\\bf h}_{jlt}x_{lt}+\\sqrt{\\rho _{u}}\\left\\lbrace \\hat{\\bf h}_{jjn}^{\\dag }{\\bf h}_{jjn}\\!\\!-\\!\\!\\mathbb {E}\\left\\lbrace \\hat{\\bf h}_{jjn}^{\\dag }{\\bf h}_{jjn}\\!\\right\\rbrace \\right\\rbrace \\!x_{jn}\\!+\\!\\hat{\\bf h}_{jjn}^{\\dag }{\\bf n}_{j}.$ Then, by using the definition of the effective SINR in multi-cell massive MIMO systems as in [4], the achievable uplink SE of the $n$ th user in cell $j$ , in units of bit/s/Hz, is given by ${R}_{jn}=\\frac{T-\\tau }{T}\\log _{2}\\left(1+{\\text{SINR}}_{jn}\\right),$ where $T$ denotes the channel coherence time interval, in terms of the number of symbols, while $\\tau $ symbols are utilized for channel estimation, and the ${\\text{SINR}}_{jn}$ is defined as ${\\text{SINR}}_{jn}\\triangleq \\frac{\\rho _{u}\\left|\\mathbb {E}\\left\\lbrace \\hat{\\bf h}_{jjn}^{\\dag }{\\bf h}_{jjn}\\right\\rbrace \\right|^{2}}{\\rho _{u} \\sum \\limits _{l=1}^{L}\\sum \\limits _{t=1}^{N} \\mathbb {E} \\left\\lbrace \\left|\\hat{\\bf h}_{jjn}^{\\dag }{\\bf h}_{jlt}\\right|^{2}\\right\\rbrace -\\rho _{u}\\left|\\mathbb {E}\\left\\lbrace \\hat{\\bf h}_{jjn}^{\\dag }{\\bf h}_{jjn}\\right\\rbrace \\right|^{2}+\\mathbb {E}\\left\\lbrace \\left\\Vert \\hat{\\bf h}_{jjn}\\right\\Vert _{2}^{2}\\right\\rbrace }.$ The following theorem presents a new general framework for the closed-form expression of ${\\text{SINR}}_{jn}$ , which applies for both the LS and MMSE estimation methods.",
"This constitutes a key contribution of this paper.",
"Theorem 1: The exact ${\\text{SINR}}_{jn}$ , for both the LS and MMSE estimation methods, can be analytically evaluated as $\\begin{split}{\\text{SINR}}_{jn}=\\left\\lbrace \\begin{array}{*{20}l}{\\text{SINR}}_{jn}^{\\rm {LS}}, &{\\text{LS}},\\\\\\\\{\\text{SINR}}_{jn}^{\\rm {MMSE}}, &{\\text{MMSE}},\\\\\\end{array}\\right.\\end{split}$ where ${\\text{SINR}}_{jn}^{\\rm {LS}}&\\triangleq \\frac{M\\rho _{jn}(K_{jn}+1)\\beta _{jjn}^2}{M\\psi _{jn}+\\zeta _{jn}\\vartheta _{j}+\\rho _{jn}K_{jn}\\beta _{jjn}\\varsigma _{jn}},\\\\{\\text{SINR}}_{jn}^{\\rm {MMSE}}&\\triangleq \\frac{M{\\rho _{jn}}\\left(K_{jn}\\!+\\!\\chi _{jn}\\right)^{2}\\beta _{jjn}^{2}}{M\\chi _{jn}^{2}\\!\\left(\\!K_{jn}\\!+\\!1\\right)\\psi _{jn}\\!+\\!\\rho _{jn}\\left(K_{jn}\\!+\\!\\chi _{jn}\\right)\\left(K_{jn}\\!+\\!1\\right)\\beta _{jjn}\\vartheta _{j}\\!+\\!\\rho _{jn}K_{jn}\\left(K_{jn}\\!+\\!1\\right)\\beta _{jjn}\\varsigma _{jn}}.$ Also, $\\psi _{jn}$ , $\\zeta _{jn}$ , $\\vartheta _{j}$ , and $\\varsigma _{jn}$ are denoted as $\\psi _{jn}&\\triangleq \\sum \\limits _{l\\ne j}^{L}\\rho _{ln}\\left(K_{ln}+1\\right)\\beta _{jln}^2,\\\\\\zeta _{jn}&\\triangleq \\sum \\limits _{c=1}^{L}\\rho _{cn}\\left(K_{cn}+1\\right)\\beta _{jcn}+1,\\\\\\vartheta _{j}&\\triangleq \\sum \\limits _{l=1}^{L}\\sum \\limits _{t=1}^{N}\\beta _{jlt}+\\frac{1}{\\rho _u},\\\\\\varsigma _{jn}&\\triangleq \\sum \\limits _{t\\ne n}^{N}\\frac{K_{jt}}{K_{jt}+1}\\frac{\\phi _{nt}^2}{M}\\beta _{jjt}-\\sum \\limits _{t=1}^{N}\\frac{K_{jt}}{K_{jt}+1}\\beta _{jjt},$ respectively, where $\\phi _{nt}&\\triangleq \\frac{\\sin \\left(\\frac{M\\pi }{2}\\left(\\sin (\\theta _{jn})-\\sin (\\theta _{jt})\\right)\\right)}{\\sin \\left(\\frac{\\pi }{2}\\left(\\sin (\\theta _{jn})-\\sin (\\theta _{jt})\\right)\\right)}.$ Proof: See Appendix A.",
"It is important to note that the expressions in Theorem 1 can be easily evaluated since they involve only the pilot sequence power, uplink data power, Ricean $K$ -factor, and large-scale fading coefficients, for all cases of interest.",
"Moreover, from Theorem 1, we see that the obtained effective SINRs based on the LS and MMSE channel estimation methods with MRC detector are different.",
"Note that when $K_{ln}=0\\ (\\forall l,n)$ , ${\\text{SINR}}_{jn}$ reduces to the special case of Rayleigh fading channel.",
"After performing some simplifications, for both LS and MMSE estimation methods, we have $&{\\text{SINR}}_{{\\rm {Rayleigh}},jn}=\\frac{M\\rho _{jn}\\beta _{jjn}^{2}}{M\\sum \\limits _{l\\ne j}^{L}\\rho _{ln}\\beta _{jln}^{2}\\!+\\!\\left(\\sum \\limits _{c=1}^{L}\\rho _{cn}\\beta _{jcn}+1\\!\\right)\\vartheta _{j}}.$ Interestingly, (22) is the effective SINR in Rayleigh fading channels given by [4].",
"Note that Theorem 1 gives a universal formula for the ${\\text{SINR}}_{jn}$ when Ricean fading is considered."
],
[
"Analysis of ${\\text{SINR}}_{jn}$",
"Now, we consider the ${\\text{SINR}}_{jn}$ limit when $M$ grows without bound.",
"To the best of our knowledge, this result is also new.",
"Corollary 1: If $M\\rightarrow \\infty $ , the exact analytical expression of the ${\\text{SINR}}_{jn}$ in (14) approaches to $\\begin{split}\\lim \\limits _{M\\rightarrow \\infty }{\\text{SINR}}_{jn}=\\left\\lbrace \\begin{array}{*{20}l}\\overline{{{\\text{SINR}}}}_{jn}^{\\rm {LS}}, &{\\text{LS}},\\\\\\\\\\overline{{\\text{SINR}}}_{jn}^{\\rm {MMSE}}, &{\\text{MMSE}},\\\\\\end{array}\\right.\\end{split}$ where $\\overline{{\\text{SINR}}}_{jn}^{\\rm {LS}}&\\triangleq \\frac{\\rho _{jn}\\left(K_{jn}+1\\right)\\beta _{jjn}^2}{\\psi _{jn}},\\\\\\overline{{\\text{SINR}}}_{jn}^{\\rm {MMSE}}&\\triangleq \\frac{\\rho _{jn}\\left(K_{jn}+\\chi _{jn}\\right)^{2}\\beta _{jjn}^2}{\\chi _{jn}^2\\left(K_{jn}+1\\right)\\psi _{jn}}.$ Proof: The proof is completed by calculating the limit of (14) when $M\\rightarrow \\infty $ .",
"Corollary 1 indicates that if the number of users is kept fixed and the number of receive antennas at the BS side is increased, then, the asymptotic ${\\text{SINR}}_{jn}$ is saturated.",
"Intuitively, this is due to the pilot contamination since the other cells' users adopt the same pilot sequence as the user in the target cell $j$ .",
"It is also worth noting that, based on (24) and (25), if $K_{ln}=0,\\ \\forall l,n$ , the limit of ${\\text{SINR}}_{jn}$ as $M\\rightarrow \\infty $ for both LS and MMSE estimation methods is given by $\\overline{{\\text{SINR}}}_{{\\rm {Rayleigh}}, jn}=\\frac{\\rho _{jn}\\beta _{jjn}^{2}}{\\sum \\limits _{l\\ne j}^{L}\\rho _{ln}\\beta _{jln}^2}.$ Again, it is important to note that (26) is the limit of effective SINR with imperfect CSI in Rayleigh fading channels given by [4].",
"Hence, (26) is a special case of (23) if the power of the LOS part of the Ricean fading channel is equal to zero.",
"To gain more insights, the exact ${\\text{SINR}}_{jn}$ admits further simplifications in the large Ricean $K$ -factor regime for both LS and MMSE estimation methods.",
"Corollary 2: If for any $l$ and $n$ , $K_{ln}=K\\rightarrow \\infty $ , (14) converges to $\\begin{split}\\lim \\limits _{K\\rightarrow \\infty }{\\text{SINR}}_{jn}=\\left\\lbrace \\begin{array}{*{20}l}\\widetilde{{\\text{SINR}}}_{jn}^{\\rm {LS}}, &{\\text{LS}},\\\\\\\\\\widetilde{{\\text{SINR}}}_{jn}^{\\rm {MMSE}}, &{\\text{MMSE}},\\\\\\end{array}\\right.\\end{split}$ where $\\widetilde{{\\text{SINR}}}_{jn}^{\\rm {LS}}$ and $\\widetilde{{\\text{SINR}}}_{jn}^{\\rm {MMSE}}$ are defined as $\\widetilde{{\\text{SINR}}}_{jn}^{\\rm {LS}}\\!&\\triangleq \\!\\frac{M\\!\\rho _{jn}\\beta _{jjn}^2}{M\\!\\!\\sum \\limits _{l\\ne j}^{L}\\!\\rho _{ln}\\beta _{jln}^2\\!\\!+\\!\\!\\sum \\limits _{c=1}^{L}\\!\\rho _{cn}\\beta _{jcn}\\!\\vartheta _{j}+\\rho _{jn}\\beta _{jjn}\\varrho _{jn}},\\\\\\widetilde{{\\text{SINR}}}_{jn}^{\\rm {MMSE}}&\\triangleq \\frac{M\\beta _{jjn}}{\\sum \\limits _{l\\ne j}^{L}\\sum \\limits _{t=1}^{N}\\beta _{jlt}+\\frac{1}{\\rho _{u}}+\\sum \\limits _{t\\ne n}^{N}\\frac{\\phi _{nt}^{2}}{M}},$ respectively.",
"Also, $\\varrho _{jn}$ is denoted as $\\varrho _{jn}\\triangleq \\sum \\limits _{t\\ne n}^{N}\\frac{\\phi _{nt}^{2}}{M}\\beta _{jjt}-\\sum \\limits _{t=1}^{N}\\beta _{jjt}.$ Proof: The proof is completed by calculating the limit of (14) when $K\\rightarrow \\infty $ .",
"It is interesting to note from Corollary 2 that as $K$ increases, the SINR based on LS and MMSE will approach two constant values, respectively.",
"Note that (29) is unbounded as $M\\rightarrow \\infty $ , whereas (28) is bounded for the same conditions.",
"In other words, in the high Ricean $K$ -factor regime with infinite $M$ , the pilot contamination effect will be completely eliminated for MMSE estimation, since this scheme accounts for the presence of a LOS component in (6).",
"On the other hand, pilot contamination effect cannot be be removed when using the LS estimation method since the LS estimation method regards the co-channel interference in (5) as just noise.",
"Note that, ${\\emph {Corollary 2}}$ reflects that massive MIMO systems have unlimited achievable uplink SE via different estimation methods, the number of BS antennas, and Ricean $K$ -factor, though in a slightly different way than in [16]."
],
[
"Numerical Results",
"In this section, we consider a hexagonal cellular network with a set of $L$ cells and radius (from center to vertex) $\\varpi $ meters where users are distributed uniformly in each cell.",
"Also, leveraging the large-scale fading model of[17], the large-scale fading coefficient between the user $n$ in cell $l$ and the BS in cell $j$ , $\\beta _{jln}$ , is given by $\\beta _{jln}=\\frac{v_{jln}}{1+\\left(\\frac{\\eta _{jln}}{\\eta _{\\rm {min}}}\\right)^{\\alpha }},$ where $v_{jln}$ is a log-normal random variable with standard deviation $\\xi $ , $\\alpha $ is the path loss exponent, $\\eta _{jln}$ is the distance between the user $n$ in cell $l$ and the BS in cell $j$ , and $\\eta _{\\rm {min}}$ is the reference distance.",
"In our simulations, we choose $L=7$ , $N=10$ , $\\varpi =500$ m, $\\xi =8$ dB, $\\alpha =3.8$ , $d=\\frac{\\lambda }{2}$ , and $\\eta _{\\rm {min}}=200$ m, which follow the methodology of [17], [6].",
"Here, the observed cell is the center cell and call it cell 1, i.e., $j=1$ .",
"Also, the pilot sequence symbol and the data symbol are assumed to be modulated based on orthogonal frequency-division multiplexing (OFDM).",
"By considering the long term evolution standard, the channel coherence time interval is equal to 196 OFDM symbols, i.e., $T=196$ [3], [6].",
"Since we assume that the noise variance is 1, we consider that each user has the same pilot sequence power denoted by $\\rho _p$ which is equal to 30dB, and the uplink data power $\\rho _{u}=20$ dB.",
"For convenience, we assume all the channels between the BS and the users in same cell have the same Ricean $K$ -factor, denoted by $K$ , and $\\tau =N=10$ OFDM symbols.",
"Finally, all the simulation results are obtained by averaging 100 realizations of all users' large-scale fading coefficients in all cells over 100 independent small-scale fading channels for each realization of users' large-scale fading coefficients.",
"In the following, we assess the accuracy of the achievable uplink SE given by (12) for both LS and MMSE estimation methods, the closed-form expression given in Theorem 1, and the results in Corollary 1 and Corollary 2.",
"For comparison, we define the metric called “Sum achievable uplink SE\" in target cell 1, which is given as $R_{\\rm {sum}}\\triangleq \\sum \\limits _{n=1}^{N}R_{1n}.$ Figure: Whole MM regimeFig.",
"1(a) gives the analytical and Monte-Carlo simulated sum achievable uplink SE $R_{\\rm {sum}}$ with the LS and MMSE estimation methods, respectively, in moderate-to-high $M$ regime.",
"Results are shown for different Ricean $K$ -factor, and pilot sequence power $\\rho _p=30$ dB with uplink data power $\\rho _u=20$ dB.",
"We see that in all cases the analytical curves (based on (14)) match precisely with the simulated curves (based on (13)), which proves the validity of Theorem 1.",
"Moreover, for all cases, when $M$ increases, $R_{\\rm {sum}}$ increases.",
"Also, when the power of the LOS path becomes zero, $R_{\\rm {sum}}$ for both the LS and MMSE estimation methods are identical for both the analytical and simulated curves, respectively, which not only shows the validity of the results (based on (22)) in previous literature [4], but also does prove that Theorem 1 can be applied into the Rayleigh fading environment.",
"Moreover, in Ricean fading conditions, the results in this figure show that the MMSE performance is always better than the LS.",
"Fig.",
"1(b) investigates the analytical results of sum achievable uplink SE $R_{\\rm {sum}}$ in the whole $M$ regime under the same parameter setting in Fig.",
"1(a).Since the match of the analytical results and simulation results has been examined in Fig.",
"1(a), for convenience, we only need to examine the analytical results in Fig.",
"1(b).",
"When $M\\rightarrow \\infty $ , we see that all $R_{\\rm {sum}}$ results tend to different constants, which match the asymptotic expression (based on (23)) for different Ricean $K$ -factor, respectively.",
"In other words, it justifies the effectiveness of Corollary 1.",
"Also, we note that, for the LS case with $K=0(-\\infty {\\text{dB}}), 3{\\text{dB}}, 6{\\text{dB}}, {\\text{and}}\\ 10{\\text{dB}}$ , as well as, the MMSE case with $K=0(-\\infty {\\text{dB}})$ , the asymptotic results are identical.",
"This phenomenon is caused by the following two reasons.",
"First, if all users' Ricean $K$ -factors are identical, (24) is uncorrelated with the Ricean $K$ -factor.",
"In other words, for the LS case, different Ricean $K$ -factor means the same asymptotic SE.",
"Second, when Ricean $K$ -factor is equal to zero, both the LS and MMSE cases have the same asymptotic SE since the current channel becomes the Rayleigh fading channel.",
"Hence, for both the LS and MMSE estimation methods, the pilot contamination exists such that the $R_{\\rm {sum}}$ saturates even when $M\\rightarrow \\infty $ .",
"Figure: The sum achievable uplink SE R sum R_{\\rm {sum}} as the Ricean KK-factor increases with ρ p =30\\rho _p=30dB, ρ u =20\\rho _u=20dB, as well as, M=125,250,500,and1000M=125, 250, 500, {\\text{and}}\\ 1000, for the LS and MMSE estimation methods, respectively.In Fig.",
"2, we investigate the impact of the Ricean $K$ -factor on the sum achievable uplink SE performance for $M=125, 250, 500, {\\text{and}}\\ 1000$ , with the LS and MMSE estimation methods, respectively.",
"In this figure, the pilot sequence power is 30dB and the uplink data power is 20dB.",
"It shows that the analytical values and simulation values are almost indistinguishable for both the LS and MMSE estimation methods, regardless of the number of BS antennas and Ricean $K$ -factors.",
"Moreover, across the entire Ricean $K$ -factor regime, for both LS and MMSE estimation methods, a larger $M$ means larger $R_{\\rm {sum}}$ .",
"Given the number of BS antennas $M$ , the sum achievable uplink SE performance difference between the LS and and MMSE is distinguishable expect in the low Ricean $K$ -factor regime since in this situation the Ricean fading channel tends to become a Rayleigh fading channel.In this figure, the asymptotic case when Ricean $K$ -factor approaches to $0(-\\infty {\\text{dB}})$ has not been shown since the Rayleigh fading case has been examined in Fig.",
"1.",
"When the Ricean $K$ -factor becomes infinite, the sum achievable uplink SE $R_{\\rm {sum}}$ approaches to different constant values, which match the asymptotic expressions (based on (27) in Corollary 2) well, respectively.",
"If also $M\\rightarrow \\infty $ , it can be shown that the pilot contamination is completely removed for the MMSE case."
],
[
"Conclusion",
"In this paper, a detailed statistical characterization of the SE for the muti-cell massive MIMO system with Ricean fading was presented.",
"In order to evaluate the SE performance, we first proposed two exact closed-form expressions for the effective SINR based on LS and MMSE estimation methods, respectively, which also can be adopted in Rayleigh fading.",
"Then, we analyzed the asymptotic properties of the effective SINR when the number of BS antennas $M$ and the Ricean $K$ -factor became infinite.",
"It was shown that, when the Ricean $K$ -factor became infinite or $M\\rightarrow \\infty $ , the SINR performance for the LS and MMSE estimation methods was saturated, which underlines the pilot contamination phenomenon.",
"However, if both the Ricean $K$ -factor and $M$ grow asymptotically large, the pilot contamination phenomenon disappeared for the MMSE estimation method, but persists for the LS estimation method."
],
[
"Proof of Theorem 1",
"To evaluate the ${\\text{SINR}}_{jk}$ in (13), we define six terms $\\mathfrak {A}&\\triangleq \\left|\\mathbb {E}\\left\\lbrace \\hat{\\bf h}_{jjn}^{\\dag }{\\bf h}_{jjn}\\right\\rbrace \\right|^2,\\\\\\mathfrak {B}&\\triangleq \\mathbb {E}\\left\\lbrace \\left|\\hat{\\bf h}_{jjn}^{\\dag }{\\bf h}_{jjn}\\right|^2\\right\\rbrace ,\\\\\\mathfrak {C}&\\triangleq \\mathbb {E}\\left\\lbrace \\left|\\hat{\\bf h}_{jjn}^{\\dag }{\\bf h}_{jjt}\\right|^2\\right\\rbrace , (t\\ne n), \\\\\\mathfrak {D}&\\triangleq \\mathbb {E}\\left\\lbrace \\left|\\hat{\\bf h}_{jjn}^{\\dag }{\\bf h}_{jln}\\right|^2\\right\\rbrace , (l\\ne j),\\\\\\mathfrak {E}&\\triangleq \\mathbb {E}\\left\\lbrace \\left|\\hat{\\bf h}_{jjn}^{\\dag }{\\bf h}_{jlt}\\right|^2\\right\\rbrace , (l\\ne j\\ \\&\\ t\\ne n),\\\\\\mathfrak {F}&\\triangleq \\mathbb {E}\\left\\lbrace \\left\\Vert \\hat{\\bf h}_{jjn}\\right\\Vert _2^2\\right\\rbrace .$ Although $\\hat{\\bf h}_{jjn}$ has different expressions for the LS and MMSE estimation methods, the corresponding proofs for ${\\text{SINR}}_{jn}$ are similar.",
"Hence, it is convenient to only study the case of the LS estimation.",
"Calculate $\\mathfrak {A}$ : Substituting (3), (5) and (8) into (33) , after much algebraic manipulation, it can be shown that $\\mathfrak {A}$ reduces to $\\mathfrak {A}&=M^2\\beta _{jjn}^2.",
"$ Calculate $\\mathfrak {B}$ : Substituting (2), (5), and (8) into (34), after some manipulations, it is easy to obtain $\\mathfrak {B}\\!&=\\underbrace{\\!\\mathbb {E}\\!\\left\\lbrace \\!\\left|{\\bf h}_{jjn}^{\\dag }\\!\\!\\left(\\!\\!\\sum \\limits _{l\\ne j}^{L}\\!\\sqrt{\\frac{\\rho _{ln}\\!\\left(K_{ln}\\!+\\!1\\right)}{\\rho _{jn}}}{\\bf h}_{jln,\\rm {NLOS}}\\!+\\!\\frac{\\tilde{\\bf n}_{jn}}{\\sqrt{\\rho _{jn}}}\\!\\!\\right)\\!\\right|^2\\!\\!\\right\\rbrace }_{B_1}\\frac{1}{K_{jn}+1}+\\underbrace{\\mathbb {E}\\left\\lbrace \\Vert {\\bf h}_{jjn}\\Vert _2^4\\right\\rbrace }_{B_2}, $ where the closed-form expression of $B_1$ can be obtained based on ${\\bf h}_{jjn}$ is uncorrelated with the rest of terms in $B_1$ and the distribution of ${\\bf h}_{jjn}$ .",
"Also, $B_2$ can be obtained based on the properties of non-central Wishart matrices [6].",
"After some algebraic manipulations, we write $\\mathfrak {B}$ as follows $\\mathfrak {B}&=M^2\\beta _{jjn}^2+M\\beta _{jjn}^2\\frac{K_{jn}}{\\left(K_{jn}+1\\right)^2}+M\\beta _{jjn}\\frac{\\sum \\limits _{l\\ne j}^{L}\\rho _{ln}\\left(K_{ln}+1\\right)\\beta _{jln}+1}{\\rho _{jn}\\left(K_{jn}+1\\right)}.$ Calculate $\\mathfrak {C}$ : Since the $\\hat{\\bf h}_{jjn}$ is uncorrelated with ${\\bf h}_{jjt}$ when $t\\ne n$ , we substitute (2), (5), and (8) into $\\mathfrak {C}$ , as well as, utilize [6].",
"Then, clearly $\\mathfrak {C}&=\\beta _{jjn}\\beta _{jjt}\\frac{K_{jn}K_{jt}\\phi _{nt}^2+M(K_{jn}+K_{jt})+M}{(K_{jn}+1)(K_{jt}+1)}+M\\beta _{jjt}\\frac{\\sum \\limits _{l\\ne j}^{L}\\!\\rho _{ln}\\!\\left(K_{ln}+1\\right)\\beta _{jln}+1}{\\rho _{jn}\\left(K_{jn}+1\\right)},$ where $\\phi _{nt}$ has been defined as in (21).",
"Calculate $\\mathfrak {D}$ : Substituting (2), (5), and (8) into (36), and performing some basic simplifications to obtain $\\mathfrak {D}&=\\mathbb {E}\\left\\lbrace \\left|D_1\\right|^2\\right\\rbrace +\\mathbb {E}\\left\\lbrace \\left|D_2\\right|^2\\right\\rbrace ,$ where $D_1$ and $D_2$ is denoted as $D_1&\\triangleq \\!\\left(\\!",
"{\\bf h}_{jjn,\\rm {LOS}}\\sqrt{\\frac{K_{jn}}{K_{jn}\\!+\\!1}}\\!+\\!\\left(\\!",
"{\\bf h}_{jjn,\\rm {NLOS}}\\!+\\!\\sum \\limits _{c\\ne j,l}^{L}\\sqrt{\\frac{\\rho _{cn}\\!\\left(K_{cn}\\!\\!+\\!\\!1\\right)}{\\rho _{jn}}}{\\bf h}_{jcn,\\rm {NLOS}}\\!\\!+\\!\\!\\frac{\\tilde{\\bf n}_{jn}}{\\sqrt{\\rho _{jn}}}\\!\\right)\\!\\!\\sqrt{\\frac{1}{K_{jn}\\!\\!+\\!\\!1}}\\right)^{\\!\\dag }\\\\&\\ \\ \\ \\times {\\bf h}_{jln,\\rm {NLOS}},\\\\D_2&\\triangleq \\sqrt{\\frac{\\rho _{ln}\\left(K_{ln}+1\\right)}{\\rho _{jn}\\left(K_{jn}+1\\right)}}\\left\\Vert {\\bf h}_{jln,\\rm {NLOS}}\\right\\Vert _2^2,$ respectively.",
"Based on the similar way for obtaining $\\mathfrak {C}$ , $\\mathbb {E}\\left\\lbrace \\left|D_1\\right|^2\\right\\rbrace $ reduces to $\\mathbb {E}\\left\\lbrace \\left|D_1\\right|^2\\right\\rbrace &=M\\beta _{jln}\\frac{\\sum \\limits _{c\\ne j,l}^{L}\\rho _{cn}\\left(K_{cn}+1\\right)\\beta _{jcn}+1}{\\rho _{jn}\\left(K_{jn}+1\\right)}+M\\beta _{jjn}\\beta _{jln}.$ By using the properties of Wishart matrices [18], we obtain $\\mathbb {E}\\left\\lbrace \\left|D_2\\right|^2\\right\\rbrace &=\\frac{\\rho _{ln}\\left(K_{ln}+1\\right)}{\\rho _{jn}\\left(K_{jn}+1\\right)}M\\left(M+1\\right)\\beta _{jln}^2.$ Therefore, substituting (46) and (47) into (43) and simplifying, we get $\\mathfrak {D}&=M^2\\beta _{jln}^2\\frac{\\rho _{ln}\\left(K_{ln}+1\\right)}{\\rho _{jn}\\left(K_{jn}+1\\right)}+M\\beta _{jjn}\\beta _{jln}+M\\beta _{jln}\\frac{\\sum \\limits _{c\\ne j}^{L}\\rho _{cn}\\left(K_{cn}+1\\right)\\beta _{jcn}+1}{\\rho _{jn}\\left(K_{jn}+1\\right)}.$ Calculate $\\mathfrak {E}$ : Based on the similar way for obtaining $B_1$ and $\\mathfrak {C}$ , $\\mathfrak {E}$ is given by $\\mathfrak {E}&=M\\beta _{jjn}\\beta _{jlt}+M\\beta _{jlt}\\frac{\\sum \\limits _{c\\ne j}^{L}\\rho _{cn}\\left(K_{cn}+1\\right)\\beta _{jcn}+1}{\\rho _{jn}\\left(K_{jn}+1\\right)}.$ Calculate $\\mathfrak {F}$ : With the help of (3), (5), and (8), we get $\\mathfrak {F}&=M\\beta _{jjn}+M\\frac{\\sum \\limits _{l\\ne j}^{L}\\rho _{ln}\\left(K_{ln}+1\\right)\\beta _{jln}+1}{\\rho _{jn}\\left(K_{jn}+1\\right)}.$ Finally, substituting (39), (41), (42), and (48)-(50) into (13) and simplifying, the closed-form expression for ${\\text{SINR}}_{jn}^{\\rm {LS}}$ is obtained."
]
] | 1808.08519 |
[
[
"Compactifications of $M_{0,n}$ associated with Alexander self-dual\n complexes: Chow ring, $\\psi$-classes and intersection numbers"
],
[
"Abstract An Alexander self-dual complex gives rise to a compactification of $M_{0,n}$, called ASD compactification, which is a smooth algebraic variety.",
"ASD compactifications include (but are not exhausted by) the polygon spaces, or the moduli spaces of flexible polygons.",
"We present an explicit description of the Chow rings of ASD compactifications.",
"We study the analogues of Kontsevich tautological bundles, compute their Chern classes, compute top intersections of the Chern classes, and derive a recursion for the intersection numbers."
],
[
"Introduction",
"The moduli space of $n$ -punctured rational curves ${\\mathcal {M}}_{0,n}$ and its various compactifications is a classical object, bringing together algebraic geometry, combinatorics, and topological robotics.",
"Recently, D.I.Smyth [1] classified all modular compactifications of ${\\mathcal {M}}_{0,n}$ .",
"We make use of an interplay between different compactifications, and: describe the classification in terms of (what we call) preASD simplicial complexes; describe the Chow rings of the compactifications arising from Alexander self-dual complexes (ASD compactifications); compute for ASD compactifications the associated Kontsevich's $\\psi $ -classes, their top monomials, and give a recurrence relation for the top monomials.",
"Oversimplifying, the main approach is as follows.",
"Some (but not all) compactifications are the well-studied polygon spaces, that is, moduli spaces of flexible polygons.",
"A polygon space corresponds to a threshold Alexander self-dual complex.",
"Its cohomology ring (which equals the Chow ring) is known due to J.-C. Hausmann and A. Knutson [2], and A. Klyachko[10].",
"The paper [3] gives a computation-friendly presentation of the ring.",
"Due to Smyth [1], all the modular compactifications correspond to preASD complexes, that is, to those complexes that are contained in an ASD complex.",
"A removal of a facet of a preASD complex amounts to a blow up of the associated compactification.",
"Each ASD compactification is achievable from a threshold ASD compactification by a sequence of blow ups and blow downs.",
"Since the changes in the Chow ring are controllable, one can start with a polygon space, and then (by elementary steps) reach any of the ASD compactifications and describe its Chow ring (Theorem REF ).",
"M. Kontsevich's $\\psi $ -classes [4] arise here in a standard way.",
"Their computation of is a mere modification of the Chern number count for the tangent bundle over $\\mathbb {S}^2$ (a classical exercise in a topology course).",
"The recursion (Theorem REF ) and the top monomial counts (Theorem REF ) follow.",
"It is worthy mentioning that a disguised compactification by simple games, i.e., ASD complexes, is discussed from a combinatorial viewpoint in [5].",
"Now let us give a very brief overview of moduli compactifications of ${\\mathcal {M}}_{0,n}$ .",
"A compactification by a smooth variety is very desirable since it makes intersection theory applicable.",
"We also expect that (1) a compactification is modular, that is, itself is the moduli space of some curves and marked points lying on it, and (2) the complement of ${\\mathcal {M}}_{0,n}$ (the “boundary”) is a divisor.",
"The space ${\\mathcal {M}}_{0,n}$ is viewed as the configuration space of $n$ distinct marked points (“particles”) living in the complex projective plane.",
"The space ${\\mathcal {M}}_{0,n}$ is non-compact due to forbidden collisions of the marked points.",
"Therefore, each compactification should suggest an answer to the question: what happens when two (or more) marked points tend to each other?",
"There exist two possible choices: either one allows some (not too many!)",
"points to coincide, either one applies a blow up.",
"It is important that the blow ups amount to adding points that correspond to $n$ -punctured nodal curves of arithmetic genus zero.",
"A compactification obtained by blow ups only is the celebrated Deligne–Mumford compactification.",
"If one avoids blow ups and allows (some carefully chosen) collections of points to coincide, one gets an ASD-compactification; among them are the polygon spaces.",
"Diverse combinations of these two options (in certain cases one allows points to collide, in other cases one applies a blow up) are also possible; the complete classification is due to [1].",
"Now let us be more precise and look at the compactifications in more detail."
],
[
"Deligne–Mumford compactification",
"Definition 1 [6] Let $B$ be a scheme.",
"A family of rational nodal curves with $n$ marked points over $B$ is a flat proper morphism $\\pi : C \\rightarrow B$ whose geometric fibers $E_{\\bullet }$ are nodal connected curves of arithmetic genus zero, and a set of sections $(s_{1}, \\dots , s_{n})$ that do not intersect nodal points of geometric fibers.",
"In this language, the sections correspond to marked points.",
"The above condition means that a nodal point of a curve may not be marked.",
"A family $(\\pi :C \\rightarrow B; s_{1}, \\dots , s_{n})$ is stable if the divisor $K_{\\pi }+s_{1}+\\dots +s_{n}$ is $\\pi $ -relatively ample.",
"Let us rephrase this condition: a family $(\\pi :C \\rightarrow B; s_{1}, \\dots , s_{n})$ is stable if each irreducible component of each geometric fiber has at least three special points (nodal points and points of the sections $s_{i}$ ).",
"Theorem 2 [6] (1) There exists a smooth and proper over $\\mathbb {Z}$ stack $\\overline{\\mathcal {M}}_{0,n}$ , representing the moduli functor of stable rational curves.",
"Corresponding moduli scheme is a smooth projective variety over $\\mathbb {Z}$ .",
"(2) The compactification equals the moduli space for $n$ -punctured stable curves of arithmetic genus zero with $n$ marked points.",
"A stable curve is a curve of arithmetic genus zero with at worst nodal singularities and finite automorphism group.",
"This means that (i) every irreducible component has at least three marked or nodal points, and (ii) no marked point is a nodal point.",
"The Deligne-Mumford compactification has a natural stratification by stable trees with $n$ leaves.",
"A stable tree with $n$ leaves is a tree with exactly $n$ leaves enumerated by elements of $[n]=\\lbrace 1,...,n\\rbrace $ such that each vertice is at least trivalent.",
"Here and in the sequel, we use the following notation: by vertices of a tree we mean all the vertices (in the usual graph-theoretical sense) excluding the leaves.",
"A bold edge is an edge connecting two vertices (see Figure REF ).",
"Figure: Stable nodal curves (left) and the corresponding trees (right)The initial space ${\\mathcal {M}}_{0,n}$ is a stratum corresponding to the one-vertex tree.",
"Two-vertex trees (Fig.REF (b)) are in a bijection with bipartitions of the set $[n]$ : $T\\sqcup T^{c} = [n]$ s.t.",
"$|T|, |T^{c}| > 1$ .",
"We denote the closure of the corresponding stratum by ${T}$ .",
"The latter are important since the (Poincaré duals of) closures of the strata generate the Chow ring $\\textbf {A}^{*}(\\overline{\\mathcal {M}}_{0,n})$ : Theorem 3 [7] The Chow ring $\\bf {A}^{*}$$(\\overline{\\mathcal {M}}_{0,n})$ is isomorphic to the polynomial ring $\\mathbb {Z}\\left[{T}:\\; T \\subset [n]; |T|>1, |T^{c}| >1\\right]$ factorized by the relations: ${T} = {T^{c}}$ ; ${T}{S} =0$ unless $S \\subset T$ or $T \\subset S$ or $S \\subset T^{c}$ or $T \\subset S^{c}$ ; For any distinct elements $i,j,k,l \\in [n]$ : $\\sum _{i,j \\in T; k,l \\notin T} {T} = \\sum _{i,k \\in T; j,l \\notin T} {T} = \\sum _{i,l \\in T; j,k \\notin T} {T}$"
],
[
"Weighted compactifications",
"The next breakthrough step was done by B. Hassett in [8].",
"Define a weight data as an element $\\mathcal {A} = (a_{1}, \\dots , a_{n}) \\in \\mathbb {R}^{n}$ such that $0 < a_{i} \\le 1$ for any $i \\in [n]$ , $a_{1}+ \\dots +a_{n} > 2$ .",
"Definition 4 Let $B$ be a scheme.",
"A family of nodal curves with $n$ marked points $(\\pi : C \\rightarrow B; s_{1}, \\dots , s_{n})$ is $\\mathcal {A}$ –stable if $K_{\\pi }+ a_{1}s_{1}+\\dots + a_{n}s_{n}$ is $\\pi $ -relatively ample, whenever the sections $\\lbrace s_{i}\\rbrace _{i \\in I}$ intersect for some $I \\subset [n]$ , one has $\\sum _{i \\in I} a_{i} < 1$ .",
"The first condition can be rephrased as: each irreducible component of any geometric fiber has at least three distinct special points.",
"Theorem 5 [8] For any weight data $\\mathcal {A}$ there exist a connected Deligne–Mumford stack $\\overline{\\mathcal {M}}_{0, \\mathcal {A}}$ smooth and proper over $\\mathbb {Z}$ , representing the moduli functor of $\\mathcal {A}$ –stable rational curves.",
"The corresponding moduli scheme is a smooth projective variety over $\\mathbb {Z}$ .",
"The Deligne–Mumford compactification arises as a special case for the weight data $ (1,\\dots , 1)$ .",
"It is natural to ask: how much does a weighted compactification $\\overline{\\mathcal {M}}_{0, \\mathcal {A}}$ depend on $\\mathcal {A}$ ?",
"Pursuing this question, let us consider the space of parameters: ${ \\mathcal {A}}_{n} = \\left\\lbrace \\mathcal {A} \\in \\mathbb {R}^{n}:\\; 0< a_{i} \\le 1, \\; \\sum _{i} a_{i} >2 \\right\\rbrace \\subset \\mathbb {R}^{n}.$ The hyperplanes $\\sum _{i \\in I} a_{i} = 1$ , $I \\subset [n], |I| \\ge 2$ , (called walls) cut the polytope ${ \\mathcal {A}}_{n}$ into chambers.",
"The Hassett compactification depends on a chamber only [8].",
"Combinatorial stratification of the space $\\overline{\\mathcal {M}}_{0, \\mathcal {A}}$ looks similarly to that of the Deligne–Mumford's with the only difference — some of the marked points now can coincide [9].",
"More precisely, a weighted tree $(\\gamma , I)$ is an ordered $k$ -partition $I_{1}\\sqcup \\dots \\sqcup I_{k} = [n]$ and a tree $\\gamma $ with $k$ ordered leaves marked by elements of the partition such that (1) $\\sum _{j \\in I_{m}} a_{j} \\le 1$ for any $m$ , and (2) for each vertex, the number of emanating bold edges plus the total weight is greater than 2.",
"Open strata are enumerated by weighted trees: the stratum of the space $\\overline{\\mathcal {M}}_{0,\\mathcal {A}}$ corresponding to a weighted tree $({\\gamma }, I)$ consists of curves whose irreducible components form the tree $\\gamma $ and collisions of sections form the partition $I$ .",
"Closure of this stratum is denoted by ${({\\gamma }, I)}$ ."
],
[
"Polygon spaces as compactifications of ${\\mathcal {M}}_{0,n}$",
"Assume that an $n$ -tuple of positive real numbers $\\mathcal {L} = (l_{1},...,l_{n})$ is fixed.",
"We associate with it a flexible polygon, that is, $n$ rigid bars of lengths $l_{i}$ connected in a cyclic chain by revolving joints.",
"A configuration of $\\mathcal {L}$ is an $n$ -tuple of points $(q_{1},...,q_{n}), \\; q_i \\in \\mathbb {R}^3,$ with $|q_i q_{i+1}|=l_{i}, \\; \\; |q_{n} q_{1}|=l_{n}$ .",
"The following two definitions for the polygon space, or the moduli space of the flexible polygon are equivalent: Definition 6 [2] I.",
"The moduli space $M_{\\mathcal {L}}$ is a set of all configurations of $\\mathcal {L}$ modulo orientation preserving isometries of $\\mathbb {R}^3$ .",
"II.",
"Alternatively, the space $M_{ \\mathcal {L}}$ equals the quotient of the space $\\left\\lbrace (u_1,...,u_n) \\in (\\mathbb {S}^2)^n : \\sum _{i=1}^n l_iu_i=0\\right\\rbrace $ by the diagonal action of the group $\\operatorname{SO}_3(\\mathbb {R})$ .",
"The second definition shows that the space $M_{\\mathcal {L}}$ does not depend on the ordering of $\\lbrace l_1,...,l_n\\rbrace $ ; however, it does depend on the values of $l_i$ .",
"Let us consider the parameter space $\\left\\lbrace (l_{1}, \\dots , l_{n}) \\in \\mathbb {R}^{n}_{>0}:\\ l_{i}< \\sum _{j\\ne i} l_{j} \\text{ for }i=1, \\dots , n\\right\\rbrace .$ This space is cut into open chambers by walls.",
"The latter are hyperplanes with defining equations $\\sum _{i\\in I} l_i = \\sum _{j\\notin I} l_j.$ The diffeomorphic type of $M_{ \\mathcal {L}}$ depends only on the chamber containing $\\mathcal {L}$ .",
"For a point $\\mathcal {L}$ lying strictly inside some chamber, the space $M_{\\mathcal {L}}$ is a smooth $(2n-6)$ -dimensional algebraic variety [10].",
"In this case we say that the length vector is generic.",
"Definition 7 For a generic length vector $\\mathcal {L}$ , we call a subset $J\\subset [n]$ long if $\\sum _{i \\in J} l_i > \\sum _{i\\notin J} l_i.$ Otherwise, $J$ is called short.",
"The set of all short sets we denote by $SHORT(\\mathcal {L})$ .",
"Each subset of a short set is also short, therefore $SHORT(\\mathcal {L})$ is a (threshold Alexander self-dual) simplicial complex.",
"Rephrasing the above, the diffeomorphic type of $M_{ \\mathcal {L}}$ is defined by the simplicial complex $SHORT(\\mathcal {L})$ ."
],
[
"ASD and preASD simplicial complexes",
"Simplicial complexes provide a necessary combinatorial framework for the description of the category of smooth modular compactifications of ${\\mathcal {M}}_{0,n}$ .",
"A simplicial complex (a complex, for short) $K$ is a subset of $2^{[n]}$ with the hereditary property: $A \\subset B\\in K$ implies $A \\in K$ .",
"Elements of $K$ are called faces of the complex.",
"Elements of $2^{[n]} \\setminus K$ are called non-faces.",
"The maximal (by inclusion) faces are called facets.",
"We assume that the set of 0-faces (the set of vertices) of a complex is $[n]$ .",
"The complex $2^{[n]}$ is denoted by $\\Delta _{n-1}$ .",
"Its $k$ -skeleton is denoted by $\\Delta _{n-1}^{k}$ .",
"In particular, $\\Delta _{n-1}^{n-2}$ is the boundary complex of the simplex $\\Delta _{n-1}$ .",
"Definition 8 For a complex $K \\subset 2^{[n]}$ , its Alexander dual is the simplicial complex $K^{\\circ }:= \\lbrace A \\subset [n]:\\; A^{c} \\notin K\\rbrace = \\lbrace A^{c}:\\; A \\in 2^{[n]}\\backslash K\\rbrace .$ Here and in the sequel, $A^c=[n]\\setminus A$ is the complement of $A$ .",
"A complex $K$ is Alexander self-dual (an ASD complex) if $K^{\\circ }=K$ .",
"A pre Alexander self-dual (a pre ASD) complex is a complex contained in some ASD complex.",
"In other words, ASD complexes (pre ASD complexes, respectively) are characterized by the condition: for any partition $[n]=A\\sqcup B$ , exactly one (at most one, respectively) of $A$ , $B$ is a face.",
"Some ASD complexes are threshold complexes: they equal $SHORT(\\mathcal {L})$ for some generic weight vectors $\\mathcal {L}$ (Section REF ).",
"It is known that threshold ASD complexes exhaust all ASD complexes for $n \\le 5$ .",
"However, for bigger $n$ this is no longer true.",
"Moreover, for $n \\rightarrow \\infty $ the percentage of threshold ASD complexes tends to zero.",
"To produce new examples of ASD complexes, we use flips: Definition 9 [5] For an ASD complex $K$ and a facet $A \\in K$ we build a new ASD complex $\\operatorname{flip}_{A}(K):= (K\\backslash A) \\cup A^{c}.$ It is easy to see that Proposition 10 (1) [5] Inverse of a flip is also some flip.",
"(2) [5] Any two ASD complexes are connected by a sequence of flips.",
"(3) For any ASD complex $K$ there exists a threshold ASD complex $K^{\\prime }$ that can be obtained from $K$ by a sequence of flips with some $A_{i}\\subset [n]$ such that $|A_{i}| > 2, |A_{i}^c| > 2$ .",
"We prove (3).",
"It is sufficient to show that for any ASD complex, there exists a threshold ASD complex with the same collection of 2-element non-faces.",
"For this, let us observe that any two non-faces of an ASD complex necessarily intersect.",
"Therefore, all possible collections of 2-element non-faces of an ASD complex (up to renumbering) are: empty set; $(12),\\ (23),\\ (31)$ ; $(12),(13),\\dots ,(1k)$ .",
"It is easy to find appropriate threshold ASD complexes for all these cases.",
"ASD complexes appear in the game theory literature as “simple games with constant sum” (see [11]).",
"One imagines $n$ players and all possible ways of partitioning them into two teams.",
"The teams compete, and a team looses if it belongs to $K$ .",
"In the language of flexible polygons, a short set is a loosing team.",
"Contraction, or freezing operation.",
"Given an ASD complex $K$ , let us build a new ASD complex $K_{(ij)}$ with $n-1$ vertices $\\lbrace 1,...,\\widehat{{i}},...,\\widehat{{j}},...,n,(i,j)\\rbrace $ by contracting the edge $\\lbrace i,j\\rbrace \\in K$ , or freezing $i$ and $j$ together.",
"The formal definition is: for $A\\subset \\lbrace 1,...,\\widehat{{i}},...,\\widehat{{j}},...,n\\rbrace $ , $A\\in K_{(ij)}$ iff $A\\in K$ , and $A\\cup \\lbrace (ij)\\rbrace \\in K_{(ij)}$ iff $A \\cup \\lbrace i,j\\rbrace \\in K$ .",
"Contraction $K_I$ of any other face $I\\in K$ is defined analogously.",
"Informally, in the language of simple game, contraction of an edge means making one player out of two.",
"In the language of flexible polygons, “freezing” means producing one new edge out of two old ones (the lengths sum up).",
"Figure: Contraction of {1,2}\\lbrace 1,2\\rbrace in a simplicial complex"
],
[
"Smooth extremal assignment compactifications",
"Now we review the results of [1] and [12], and indicate a relation with preASD complexes.",
"For a scheme $B$ , consider the space $\\mathcal {U}_{\\,0, n}(B)$ of all flat, proper, finitely-presented morphisms $\\pi : \\;\\mathcal {C} \\rightarrow B$ with $n$ sections $\\lbrace s_{i}\\rbrace _{i \\in [n]}$ , and connected, reduced, one-dimensional geometric fibers of genus zero.",
"Denote by $\\mathcal {V}_{0, n}$ the irreducible component of $\\mathcal {U}_{\\,0,n}$ that contains ${\\mathcal {M}}_{0,n}$ .",
"Definition 11 A modular compactification of ${\\mathcal {M}}_{0,n}$ is an open substack $\\mathcal {X} \\subset \\mathcal {V}_{0,n}$ that is proper over $\\mathbb {Z}$ .",
"A modular compactification is stable if every geometric point $(\\pi :\\;\\mathcal {C} \\rightarrow B; s_{1}, \\dots , s_{n})$ is stable.",
"We call a modular compactification smooth if it is a smooth algebraic variety.",
"Definition 12 A smooth extremal assignment $\\mathcal {Z}$ over $\\overline{\\mathcal {M}}_{0,n}$ is an assignment to each stable tree with $n$ leaves a subset of its vertices $\\gamma \\mapsto \\mathcal {Z}(\\gamma ) \\subset Vert(\\gamma )$ such that: for any tree $\\gamma $ , the assignment is a proper subset of vertices: $\\mathcal {Z}(\\gamma ) \\subsetneqq Vert(\\gamma )$ , for any contraction $\\gamma \\rightsquigarrow \\tau $ with $\\lbrace v_{i}\\rbrace _{i \\in I} \\subset Vert(\\gamma )$ contracted to $v \\in Vert(\\tau )$ , we have $v_{i}\\in \\mathcal {Z}(\\gamma )$ for all $i \\in I$ if and only if $v \\in \\mathcal {Z}(\\tau )$ .",
"for any tree $\\gamma $ and $v \\in \\mathcal {Z}(\\gamma )$ there exists a two-vertex tree $\\gamma ^{\\prime }$ and $v^{\\prime }\\in \\mathcal {Z}(\\gamma ^{\\prime })$ such that $\\gamma ^{\\prime } \\rightsquigarrow \\gamma \\text{ and }v^{\\prime } \\rightsquigarrow v.$ Definition 13 Assume that $\\mathcal {Z}$ is a smooth extremal assignment.",
"A curve $(\\pi :\\;\\mathcal {C} \\rightarrow B; s_{1}, \\dots , s_{n})$ is $\\mathcal {Z}$ –stable if it can be obtained from some Deligne–Mumford stable curve $(\\pi ^{\\prime }:\\;\\mathcal {C}^{\\prime } \\rightarrow B^{\\prime }; s_{1}^{\\prime }, \\dots , s_{n}^{\\prime })$ by (maximal) blowing down irreducible components of the curve $\\mathcal {C}^{\\prime }$ corresponding to the vertices from the set $\\mathcal {Z}(\\gamma (\\mathcal {C}^{\\prime }))$ .",
"A smooth assignment is completely defined by its value on two-vertex stable trees with $n$ leaves.",
"The latter bijectively correspond to unordered partitions $A\\sqcup A^{c} = [n]$ with $|A|, |A^c| > 1$ : sets $A$ and $A^{c}$ are affixed to two vertices of the tree.",
"The first condition of Definition REF implies that no more than one of $A$ and $A^{c}$ is “assigned”.",
"One concludes that preASD complexes are in bijection with smooth assignments.",
"All possible modular compactifications of ${\\mathcal {M}}_{0, n}$ are parametrized by smooth extremal assignments: Theorem 14 [1] and [12] For any smooth extremal assignment $\\mathcal {Z}$ of ${\\mathcal {M}}_{0,n}$ , or equivalently, for any preASD complex $K$ , there exists a stack $\\overline{\\mathcal {M}}_{0,\\mathcal {Z}} = \\overline{\\mathcal {M}}_{0,K} \\subset \\mathcal {V}_{0,n}$ parameterizing all $\\mathcal {Z}$ –stable rational curves.",
"For any smooth modular compactification $\\mathcal {X} \\subset \\mathcal {V}_{0,n}$ , there exist a smooth extremal assignment $\\mathcal {Z}$ (a preASD complex $K$ ) such that $\\mathcal {X} = \\overline{\\mathcal {M}}_{0,\\mathcal {Z}} = \\overline{\\mathcal {M}}_{0,K}$ .",
"There are two different ways to look at a moduli spaces.",
"In the present paper we look at the moduli space as at a smooth algebraic variety equipped with $n$ sections (fine moduli space).",
"The other way is to look at it as at a smooth algebraic variety (coarse moduli space).",
"Different preASD complexes give rise to different fine moduli spaces.",
"However, two different complexes can yield isomorphic coarse moduli spaces.",
"Indeed, consider two preASD complexes $K$ and $K\\cup \\lbrace ij\\rbrace $ (we abbreviate the latter as $K+(ij)$ ), assuming that $\\lbrace ij\\rbrace \\notin K$ .",
"The corresponding algebraic varieties $\\overline{\\mathcal {M}}_{0, K}$ and $\\overline{\\mathcal {M}}_{0, K + (ij)}$ are isomorphic.",
"A vivid explanation is: to let a couple of marked points to collide is the same as to add a nodal curve with these two points sitting alone on an irreducible component.",
"Indeed, this irreducible component would have exactly three special points, and $\\operatorname{PSL}_{2}$ acts transitively on triples.",
"Theorem 15 [12] The set of smooth modular compactifications of ${\\mathcal {M}}_{0,n}$ is in a bijection with objects of the ${\\bf preASD}_{n}/ \\sim $ , where $K\\sim L$ whenever $K\\setminus L$ and $L\\setminus K$ consist of two-element sets only.",
"Example 16 PreASD complexes and corresponding compactifications.",
"the 0-skeleton $\\Delta _{n-1}^{0} = [n]$ of the simplex $\\Delta _{n-1}$ corresponds to the Deligne–Mumford compactification; the complex $\\mathcal {P}_{n} : = {\\bf pt} \\sqcup \\Delta _{n-2}^{n-3}$ (disjoint union of a point and the boundary of a simplex $\\Delta _{n-2}$ ) is ASD.",
"It corresponds to the Hassett weights $(1, \\varepsilon , \\dots , \\varepsilon )$ ; this compactification is isomorphic to $\\mathbb {P}^{n-3}$ ; the Losev–Manin compactification $\\overline{\\mathcal {M}}_{0, n}^{LM}$ [13] corresponds to the weights $(1,1,\\varepsilon , \\dots , \\varepsilon )$ and to the complex ${\\bf pt}_{1} \\sqcup {\\bf pt}_{2} \\sqcup \\Delta _{n-3}$ ; the space $(\\mathbb {P}^{1})^{n-3}$ corresponds to weights $(1,1,1,\\varepsilon , \\dots , \\varepsilon )$ , and to the complex ${\\bf pt}_{1} \\sqcup {\\bf pt}_{2} \\sqcup {\\bf pt}_{3} \\sqcup \\Delta _{n-4}$ ."
],
[
"ASD compactifications\nvia stable point configurations",
"ASD compactifications can be explained in a self-contained way, without referring to [1].",
"Fix an ASD complex $K$ and consider configurations of $n$ (not necessarily all distinct) points $p_1,...,p_n$ in the projective line.",
"A configuration is called stable if the index set of each collection of coinciding points belongs to $K$ .",
"That is, whenever $p_{i_1}=...=p_{i_k}$ , we have $\\lbrace i_1,...,i_k\\rbrace \\in K$ .",
"Denote by $STABLE(K)$ the space of stable configurations in the complex projective line.",
"The group $\\mathrm {PSL}_{2}(\\mathbb {C})$ acts naturally on this space.",
"Set $\\overline{\\mathcal {M}}_{0, K}:=STABLE(K)/\\mathrm {PSL}_{2}(\\mathbb {C}).$ If $K$ is a threshold complex, that is, arises from some flexible polygon $\\mathcal {L}$ , then the space $\\overline{\\mathcal {M}}_{0, K}$ is isomorphic to the polygon space $M_{\\mathcal {L}}$ [10].",
"Although the next theorem fits in a broader context of [1], we give here its elementary proof.",
"Theorem 17 The space $\\overline{\\mathcal {M}}_{0, K}$ is a compact smooth variety with a natural complex structure.",
"$\\;$ Smoothness.",
"For a distinct triple of indices $i,j,k \\in [n]$ , denote by $U_{i,j,k}$ the subset of $\\overline{\\mathcal {M}}_{0, K}$ defined by $p_i\\ne p_j,$ $p_j \\ne p_k$ , and $p_i \\ne p_k$ .",
"For each of $U_{i,j,k}$ , we get rid of the action of the group $\\mathrm {PSL}_{2}(\\mathbb {C})$ , setting $U_{i,j,k}=\\big \\lbrace (p_1,...,p_n)\\in \\overline{\\mathcal {M}}_{0, K}:\\; p_i=0,p_j=1,\\text{ and }p_k=\\infty \\big \\rbrace .$ Clearly, each of the charts $U_{i,j,k}$ is an open smooth manifold.",
"Since all the $U_{i,j,k}$ cover $\\overline{\\mathcal {M}}_{0, K}$ , smoothness is proven.",
"Compactness.",
"Let us show that each sequence of $n$ -tuples has a converging subsequence.",
"Assume the contrary.",
"Without loss of generality, we may assume that the sequence $(p_1^i=0,p_2^i=1,p_3^i=\\infty ,p_4^i,...,p_n^i)_{i=1}^{\\infty }$ has no converging subsequence.",
"We may assume that for some set $I \\notin K$ , all $p^i_j$ with $j\\in I$ converge to a common point.",
"We say that we have a collapsing long set $I$ .",
"This notion depends on the choice of a chart.",
"We may assume that our collapsing long set has the minimal cardinality among all long sets that can collapse without a limit (that is, violate compactness) for this complex $K$ .",
"We may assume that $I=\\lbrace 3,4,5,...,k\\rbrace $ .",
"This long set can contain at most one of the points $p_1,p_2,p_3$ .",
"We consider the case when it contains $p_3$ ; other cases are treated similarly.",
"That is, all the points $p^i_4,...,p^i_k$ tend to $\\infty $ .",
"Denote by $C_i$ the circle with the minimal radius embracing the points $p^i_3=\\infty ,p^i_4,p^i_5,...,p^i_k$ .",
"The circle contains at least two points of $p^i_4,...,p^i_k, p_3=\\infty $ .",
"Apply a transform $\\phi _i \\in \\mathrm {PSL}_{2}(\\mathbb {C})$ which turns the radius of $C_i$ to 1, and keeps at least two of the points $p^i_4,...,p^i_k, p_3=\\infty $ away from each other.",
"In this new chart the cardinality of the collapsing long set gets smaller.",
"A contradiction to the minimality assumption.",
"A natural question is: what if one takes a simplicial complex (not a self-dual one), and cooks the analogous quotient space.",
"Some heuristics are: if the complex contains simultaneously some set $A$ and its complement $[n]\\setminus A$ , we have a stable tuple with a non-trivial stabilizer in $\\mathrm {PSL}_{2}(\\mathbb {C})$ , so the factor has a natural nontrivial orbifold structure.",
"If a simplicial complex is smaller than some ASD complex $K^{\\prime }$ , and therefore, we get a proper open subset of $\\overline{\\mathcal {M}}_{0, K^{\\prime }}$ , that is, we lose compactness."
],
[
"Perfect cycles",
"Assume that we have an ASD complex $K$ and the associated compactification $\\overline{\\mathcal {M}}_{0, K}$ .",
"Let $K_I$ be the contraction of some face $I\\in K$ .",
"Since the variety $\\overline{\\mathcal {M}}_{0, K_I}$ naturally embeds in $ \\overline{\\mathcal {M}}_{0, K}$ , the contraction procedure gives rise to a number of subvarieties of $\\overline{\\mathcal {M}}_{0, K}$ .",
"These varieties (1) “lie on the boundary” That is do not intersect the initial space ${\\mathcal {M}}_{0, n}$ .",
"and (2) generate the Chow ring (Theorem REF ).",
"Let us look at them in more detail.",
"An elementary perfect cycle $(ij)=(ij)_{K}\\subset \\overline{\\mathcal {M}}_{0, K}$ is defined as $(ij)=(ij)_K=\\lbrace (p_1,...,p_n)\\in \\overline{\\mathcal {M}}_{0, K}: p_i=p_j\\rbrace .$ Let $[n]=A_1\\sqcup ...\\sqcup A_k$ be an unordered partition of $[n]$ .",
"A perfect cycle associated to the partition $(A_1)\\cdot ...\\cdot (A_k)&=(A_1)_{K}\\cdot ...\\cdot (A_k)_{K} = \\\\&= \\lbrace (p_1,...,p_n)\\in \\overline{\\mathcal {M}}_{0, K}: i,j \\in A_m \\Rightarrow p_i=p_j\\rbrace .$ Each perfect cycle is isomorphic to $\\overline{\\mathcal {M}}_{0, K^{\\prime }}$ for some complex $K^{\\prime }$ obtained from $K$ by a series of contractions.",
"Singletons play no role, so we omit all one-element sets $A_i$ from our notation.",
"Consequently, all the perfect cycles are labeled by partitions of some subset of $[n]$ such that all the $A_i$ have at least two elements.",
"Note that for arbitrary $A \\in K$ , the complex $K_A$ might be ill-defined.",
"This happens if $A \\notin K$ .",
"In this case the associated perfect cycle $(A)$ is empty.",
"For each perfect cycle there is an associated Poincaré dual element in the cohomology ring.",
"These dual elements we denote by the same symbols as the perfect cycles.",
"The following rules allow to compute the cup-product of perfect cycles: Proposition 18 $\\;$ Let $A$ and $B$ be disjoint subsets of $[n]$ .",
"Then $(A)\\smile (B)=(A)\\cdot (B).$ $(Ai)\\smile (Bi)=(ABi).$ For $A\\notin K$ , we have $(A)=0$ .",
"If one of $A_k$ is a non-face of $K$ , then $(A_1)\\cdot ...\\cdot (A_k)=0$ .",
"The four-term relation: $(ij)+(kl)=(jk)+(il)$ holds for any distinct $i,j,k,l$ .",
"In the cases (1) and (2) we have a transversal intersection of holomorphically embedded complex varieties.",
"The item (3) will be proven in Theorem REF .",
"Examples: $(123)\\cdot (345)=(12345);\\ \\ (12)\\cdot (34) \\cdot (23)=(1234).$ A more sophisticated computation: $(12)\\cdot (12) = (12)\\cdot \\big ( (13) + (24) - (34) \\big ) = (123) + (124) -(12)\\cdot (34).$ Proposition 19 A cup product of perfect cycles is a perfect cycle.",
"Proof.",
"Clearly, each perfect cycle is a product of elementary ones.",
"Let us prove that the product of two perfect cycles is an integer linear combination of perfect cycles.",
"We may assume that the second factor is an elementary perfect cycle, say, $(12)$ .",
"Let the first factor be $(A_1)\\cdot (A_2)\\cdot (A_3)\\cdot \\ldots \\cdot (A_k)$ .",
"We need the following case analysis: If at least one of $1,2$ does not belong to $\\bigcup A_i$ , the product is a perfect cycle by Proposition REF , (1).",
"If 1 and 2 belong to different $A_i$ , we use the following: for any perfect cycle $(A_1)\\cdot (A_2)$ with $ i\\in A_1, j\\in A_2$ , we have $(A_1)\\cdot (A_2)\\smile (ij)=(A_1)\\smile (ij)\\smile (A_2)= (A_1j)\\smile (A_2)= (A_1\\cup A_2).$ Finally, assume that $1,2 \\in A_1$ .",
"Choose $i\\notin A_1,\\ \\ j\\notin A_1$ such that $i$ and $j$ do not belong to one and the same $A_k$ .",
"By Proposition REF , (3), $(A_1)\\cdot (A_2)\\cdot (A_3)\\cdot \\cdots \\cdot (A_k)\\smile (12)=(A_1)\\cdot (A_2)\\cdot (A_3)\\cdot \\ldots \\cdot (A_k)\\smile \\big ((1i)+(2j)-(ij)\\big ).$ After expanding the brackets, one reduces this to the above cases.$\\Box $ Lemma 20 For an ASD complex $K$ , let $A\\sqcup B\\sqcup C=[n]$ be a partition of $[n]$ into three faces.",
"Then $(A)\\cdot (B)\\cdot (C)=1$ in the graded component ${\\bf A}^{n-3}_K$ of the Chow ring, which is canonically identified with $\\mathbb {Z}$ .",
"Indeed, the cycles $(A)$ , $(B)$ , and $(C)$ intersect transversally at a unique point.",
"Now we see that the set of perfect cycles is closed under cup-product.",
"In the next section we show that the Chow ring equals the ring of perfect cycles."
],
[
"Flips and blow ups. ",
"Let $K$ be an ASD complex, and let $A \\subset [n]$ be its facet.",
"Lemma 21 The perfect cycle $(A)$ is isomorphic to $\\overline{\\mathcal {M}}_{0, \\mathcal {P}_{|A^c|+1}} \\cong \\mathbb {P}^{|A^c|-2}$ .",
"Contraction of $A$ gives the complex ${\\bf pt}\\;\\sqcup \\;\\Delta _{|A^c|}^{} = \\mathcal {P}_{|A^c|+1}$ from the Example REF , (2).",
"Lemma 22 For an ASD complex $K$ and its facet $A$ , there are two blow up morphisms $\\pi _{ A }: \\overline{\\mathcal {M}}_{0,K\\backslash A} \\rightarrow \\overline{\\mathcal {M}}_{0, K} \\text{ and }\\pi _{A^c}: \\overline{\\mathcal {M}}_{0,K\\backslash A} \\rightarrow \\overline{\\mathcal {M}}_{0, \\operatorname{flip}_{A} (K)}.$ The centers of these blow ups are the perfect cycles $(A)$ and $(A^c)$ respectively.",
"The exceptional divisors are equal: ${A} = {A^c}$ .",
"Both are isomorphic to $\\overline{\\mathcal {M}}_{0, \\mathcal {P}_{|A|+1}} \\times \\overline{\\mathcal {M}}_{0, \\mathcal {P}_{|A^c|+1}} \\cong \\mathbb {P}^{|A|} \\times \\mathbb {P}^{|A^c|}$ .",
"The maps $\\pi _{A}|_{{A}}$ and $\\pi _{A^c}|_{{A^c}}$ are projections to the first and the second components respectively.",
"The proof literally repeats [8]: $K$ –stable but not $K_{A}$ ($K_{A^c}$ )–stable curves have two connected components.",
"The marked points with indices from the set $A$ lie on one of the irreducible components, and marked points with indices from the set $A^c$ lie on the other.",
"$\\Box $ Corollary 23 For an ASD complex $K$ and its facet $A$ , the algebraic varieties $\\overline{\\mathcal {M}}_{0, K}$ and $\\overline{\\mathcal {M}}_{0, K \\backslash A}$ are HI–schemes, i.e., the canonical map from the Chow ring to the cohomology ring is an isomorphism.",
"Proof.",
"This follows from Lemma REF and Theorem REF .$\\Box $"
],
[
"Chow rings of ASD compactifications",
"As it was already mentioned, many examples of ASD compactifications are polygon spaces, that is, come from a threshold ASD complex.",
"Their Chow rings were computed in [2].",
"A more relevant to the present paper presentation of the ring is given in [3].",
"We recall it below.",
"Definition 24 Let ${\\bf A}^{*}_{univ} = {\\bf A}^{*}_{univ, n}$ be the ring $\\mathbb {Z}\\big [(I):\\; I \\subset [n], 2 \\le |I| \\le n-2 \\big ]$ factorized by relations: “The four-term relations”: $(ij)+(kl)-(ik)-(jl)=0$ for any $i,j,k,l \\in [n]$ .",
"“The multiplication rule”: $(Ik)\\cdot (Jk) = (IJk)$ for any disjoint $I, J \\subset [n]$ not containing element $k$ .",
"There is a natural graded ring homomorphism from ${\\bf A}^*_{univ}$ to the Chow ring of an ASD-compactification that sends each of the generators $(I)$ to the corresponding perfect cycle.",
"Theorem 25 [3] The Chow ring (it equals the cohomology ring) of a polygon space equals the ring ${\\bf A}^*_{univ}$ factorized by $(I)=0 \\ \\ \\ \\hbox{whenever $I$ is a long set.",
"}$ The following generalization of Theorem REF is the first main result of the paper: Theorem 26 For an ASD complex $L$ , the Chow ring ${\\bf A}^{*}_{L}:= {\\bf A}^{*}(\\overline{\\mathcal {M}}_{0, L})$ of the moduli space $\\overline{\\mathcal {M}}_{0, L}$ is isomorphic to the quotient ${\\bf A}^{*}_{univ}$ by the ideal $\\mathcal {I}_{L}:= \\big \\langle (I):\\;I\\notin L \\big \\rangle $ .",
"The idea of the proof is: the claim is true for threshold ASD complexes (i.e., for polygon spaces), and each ASD complex is achievable from a threshold ASD complex by a sequence of flips.",
"Therefore it is sufficient to look at a unique flip.",
"Let us consider an ASD complex $K + B$ where $B\\notin K$ is a facet in $K + B$ .",
"Set $A:=[n]\\setminus B$ , and consider the ASD complex $K + A= \\operatorname{flip}_{B}(K+B)$ .",
"[column sep=small] K [dl, hook] [dr, hook] K+ B [rr, dashrightarrow, \"flipB\"] K+A We are going to prove that if the claim of the theorem holds true for $K+B$ , then it also holds for $K+A$ .",
"By Lemma REF , the space $\\overline{\\mathcal {M}}_{0, K}$ is the blow up of $\\overline{\\mathcal {M}}_{0, K+B}$ along the subvariety $(B)$ and the blow up of $\\overline{\\mathcal {M}}_{0, K+A}$ along the subvariety $(A)$ .",
"The diagram of the blow ups looks as follows: (B)[d, hook, \"iB\"] [hook]djA = jB rgAlgB (A) [hook]diA M0, K+B M0, K [twoheadrightarrow]rA[l, two heads, \"B\"] M0, K+A The induced diagram of Chow rings is: A*(B) = A*P|A|+1 rgB* A*P|A|+1A*P|B|+1 A*(A) = A*P|B|+1 lgA* A*K + B[u, \"iB*\"][r, hookrightarrow, \"B*\" description] A*K [hook]ujA* = jB* A*K + A [l, hookrightarrow , \"A*\" description][u, \"iA*\" description] Let ${\\bf A}^{*}_{K+ A, comb}$ be the quotient of ${\\bf A}^{*}_{univ}$ by the ideal $\\mathcal {I}_{K+ A}$ .",
"We have a natural graded ring homomorphism $\\alpha = \\alpha _{K+A}:{\\bf A}^{*}_{K+ A, comb} \\rightarrow {\\bf A}^{*}_{K+ A}=:{\\bf A}^{*}_{K+ A, alg},$ where the map $\\alpha $ sends each symbol $(I)$ to the associated perfect cycle.",
"A remark on notation: as a general rule, all objects related to ${\\bf A}^{*}_{K+ A, comb}$ we mark with a subscript “comb”, and objects related to ${\\bf A}^{*}_{K+ A, alg}$ we mark with “alg”.",
"We shall show that $\\alpha $ is an isomorphism.",
"The outline of the proof is: The ring ${\\bf A}^{*}_{K+A,alg}$ is generated by the first graded component.",
"(The ring ${\\bf A}^{*}_{K+A,comb}$ is also generated by the first graded component; this is clear by construction.)",
"The restriction of $\\alpha $ to the first graded components is a group isomorphism.",
"Therefore, $\\alpha $ is surjective.",
"The map $\\alpha $ is injective.",
"Lemma 27 The ring ${\\bf A}^{*}_{K+A, alg}$ is generated by the group ${\\bf A}^{1}_{K+A, alg}$ .",
"By Theorem REF ${\\bf A}^{*}_{K} \\cong \\frac{ {\\bf A}^{*}_{K+A, alg}[T]}{\\big ( f_{A}(T), T\\cdot \\ker (i_{A}^{*})\\big )}.$ Observe that: The zero graded components of ${\\bf A}^{*}_{K+A, alg}, {\\bf A}^{*}_{K+A, comb}$ equals $\\mathbb {Z}$ .",
"The map $\\pi _{A}^*:\\;{\\bf A}^{*}_{K+A, alg} \\rightarrow {\\bf A}^{*}_{K}$ is a homomorphism of graded rings.",
"Moreover, the variable $T$ stands for the additive inverse of the class of the exceptional divisor $.",
"And so, $ T$ a degree one homogeneous element.$ Since $i_{A}^*$ is the multiplication by the cycle $(A)$ , the kernel $\\ker (i_{A}^{*})$ equals the annihilator $\\operatorname{Ann}(A)_{alg}$ in the ring ${\\bf A}^{*}_{K+A, alg}$ .",
"Since the space $\\overline{\\mathcal {M}}_{0, K+A}$ is an HI-scheme, the degree of the ideal $\\operatorname{Ann}(A)_{alg}$ is strictly positive.",
"The polynomial $f_{A}(T)$ is a homogeneous element whose degree equals the degree $\\deg _{T}(f_{A}(T))$ .",
"Besides, its coefficients are generated by elements from the first graded component since they all belong to the ring $\\alpha ({\\bf A}^{*}_{K+A, comb})$ .",
"Denote by $\\langle {\\bf A}^1_{K+A, alg}\\rangle $ the subalgebra of ${\\bf A}_{K+A, alg}$ generated by the first graded component.",
"First observe that the restriction of the map ${\\bf A}^{*}_{K+A, alg}[T] \\rightarrow {\\bf A}^{*}_{K}$ to the first graded components is injective.",
"Assuming that the lemma is not true, consider a homogeneous element $r$ of the ring ${\\bf A}^{*}_{K+A, alg}$ with minimal degree among all not belonging to $\\langle {\\bf A}^{1}_{K+A, alg}\\rangle $ .",
"There exist elements $b_{i}\\in \\langle {\\bf A}^{1}_{K+A, alg}\\rangle $ such that $b_{p}\\cdot T^{p} + \\dots + b_{1}\\cdot T + b_{0} = r$ in the ring ${\\bf A}^{*}_{K+A, alg}[T]$ .",
"The elements $b_{i}$ are necessarily homogeneous.",
"Equivalently, $b_{p}\\cdot T^{p} + \\dots + b_{1}\\cdot T + b_{0} - r$ belongs to the ideal $\\big (f_A(T), T\\cdot \\operatorname{Ann}(A_{alg})\\big )$ .",
"Therefore $b_{p}\\cdot T^{p} + \\dots + b_{1}\\cdot T + b_{0} - r=x\\cdot f_A(T) + y \\cdot T \\cdot i$ with some $x, y \\in R[T]$ and $i \\in \\operatorname{Ann}(A_{alg})$ .",
"Setting $T=0$ , we get $b_{0} - r = x_{0}\\cdot f_{0}$ .",
"If the element $x_{0}$ belongs to $\\langle {\\bf A}^{1}_{K+A, alg}\\rangle $ , then we are done.",
"Assume the contrary.",
"Then from the minimality assumption we get the following inequalities: $\\deg (b_0 - r) = \\deg (x_{0}\\cdot f_{0}) > \\deg (x_{0}) \\ge \\deg (r)$ .",
"A contradiction.",
"Lemma 28 For any ASD complex $L$ the groups ${\\bf A}^1_{L, comb}$ and ${\\bf A}^1_{L, alg}$ are isomorphic.",
"The isomorphism is induced by the homomorphism $\\alpha _{L}$ .",
"The proof analyses how do these groups change under flips.",
"We know that the claim is true for threshold complexes.",
"Due to Lemma REF we may consider flips only with $n-2>|A|>2$ .",
"Again, we suppose that the claim is true for the complex $K+B$ and will prove for the complex $K+A$ with $A \\sqcup B = [n]$ .",
"Under such flips ${\\bf A}^{1}_{comb}$ does not change.",
"The group ${\\bf A}^{1}$ does not change neither.",
"This becomes clear with the following two short exact sequences (see Theorem REF ,e): $0 &\\rightarrow {\\bf A}_{n-4}\\big (\\overline{\\mathcal {M}}_{0, \\mathcal {P}_{|A|+1}}\\big ) \\rightarrow {\\bf A}_{n-4}\\big ( \\overline{\\mathcal {M}}_{0, \\mathcal {P}_{|A|+1} } \\times \\overline{\\mathcal {M}}_{0, \\mathcal {P}_{|B|+1} } \\big ) \\oplus {\\bf A}_{n-4}\\big ( \\overline{\\mathcal {M}}_{0, K + B} \\big ) \\rightarrow {\\bf A}_{n-4}\\big (\\overline{\\mathcal {M}}_{0, K}\\big ) \\rightarrow 0,\\\\0 &\\rightarrow {\\bf A}_{n-4}\\big (\\overline{\\mathcal {M}}_{0, \\mathcal {P}_{|B|+1}}\\big ) \\rightarrow {\\bf A}_{n-4}\\big ( \\overline{\\mathcal {M}}_{0, \\mathcal {P}_{|A|+1} } \\times \\overline{\\mathcal {M}}_{0, \\mathcal {P}_{|B|+1} } \\big ) \\oplus {\\bf A}_{n-4}\\big ( \\overline{\\mathcal {M}}_{0, K + A} \\big ) \\rightarrow {\\bf A}_{n-4}\\big (\\overline{\\mathcal {M}}_{0, K}\\big ) \\rightarrow 0.$ $\\Box $ Now we know that $\\alpha :\\;{\\bf A}^{*}_{K+A, comb} \\rightarrow {\\bf A}^{*}_{K+A, alg}$ is surjective.",
"Proposition 29 Let $\\Gamma $ be a graph $Vert(\\Gamma )=[n]$ which equals a tree with one extra edge.",
"Assume that the unique cycle in $\\Gamma $ has the odd length.",
"Then the set of perfect cycles $\\lbrace (ij)\\rbrace $ corresponding to the edges of $\\Gamma $ is a basis of the (free abelian) group ${\\bf A}^{1}_{univ}$ .",
"Any element of the group ${\\bf A}^{1}_{univ}$ by definition has a form $\\sum _{ij} a_{ij} \\cdot (ij)$ with the sum ranges over all edges of the complete graph on the set $[n]$ .",
"The four-term relation can be viewed as an alternating relation for a four-edge cycle.",
"One concludes that analogous alternating relation holds for each cycle of even length.",
"Example: $(ij)-(jk)+(kl)-(lm)+(mp)-(pi)=0$ .",
"Such a cycle may have repeating vertices.",
"Therefore, if a graph has an even cycle, the perfect cycles associated to its edges are dependant.",
"It remains to observe that the graph $\\Gamma $ is a maximal graph without even cycles.",
"By Theorem REF , the Chow rings of the compactifications corresponding to complexes $K$ , $K+A$ , and $K+B$ are related in the following way: ${\\bf A}^{*}_{K} \\cong \\frac{ {\\bf A}^{*}_{K+A, alg}[T]}{\\big ( f_{A}(T), T\\cdot \\ker (i_{A}^{*})\\big )} \\cong \\frac{ {\\bf A}^{*}_{K+B}[S]}{\\big ( f_{B}(S), S\\cdot \\ker (i_{B}^{*})\\big )}.$ Now we need an explicit description of the polynomials $f_{A}$ and $f_{B}$ .",
"Assuming that $A=\\lbrace x, x_{2}, \\dots , x_{a}\\rbrace $ $B=\\lbrace y, y_{2}, \\dots , y_{b}\\rbrace $ , where $|A|= a$ and $|B| = b$ , take the generators $&\\big \\lbrace (xy); (xy_{i}), i \\in \\lbrace 2, \\dots , b\\rbrace ; (x_{j}y), j \\in \\lbrace 2, \\dots , a\\rbrace ; (yy_{2}) \\big \\rbrace \\text{ for } {\\bf A}^{*}_{K+ B} \\text{ , and }\\\\&\\big \\lbrace (xy); (xy_{i}), i \\in \\lbrace 2, \\dots , b\\rbrace ; (x_{j}y), j \\in \\lbrace 2, \\dots , a\\rbrace ; (xx_{2}) \\big \\rbrace \\text{ for } {\\bf A}^{*}_{K+ A, comb}.$ Figure: 111Denote by ${\\mathcal {A}}$ the subring of the Chow rings ${\\bf A}^{*}_{K+ A, comb}$ and ${\\bf A}^{*}_{K+ B}$ generated by the elements $\\lbrace (xy); (xy_{i}), i \\in \\lbrace 2, \\dots , b-1\\rbrace ; (x_{j}y), j \\in \\lbrace 2, \\dots , a-1\\rbrace \\rbrace $ .",
"Then ${\\bf A}^{*}_{K+ A, comb}$ is isomorphic to ${\\mathcal {A}}[I]/F_{B}(I)$ where $I:=(xx_{2})$ and $F_{B}(I)$ is an incarnation of the expression $(B) = (yy_{2})\\cdot \\dots \\cdot (yy_{b})=0$ via the generators.",
"Analogously, ${\\bf A}^{*}_{K+ B, comb} \\cong {\\mathcal {A}}[J]/F_{A}(J)$ with $V:= (yy_{2})$ .",
"The cycles $(A)$ and $(B)$ equal to the complete intersection of divisors $(xx_{2}), (xx_{3}), \\dots , (xx_{a-1})$ and $(yy_{2}), (yy_{3}), \\dots , (yy_{b-1})$ respectively.",
"So the Chern polynomials are: $f_{A}(T) = \\big (T + (xx_{2})\\big )\\cdot \\dots \\cdot \\big (T + (xx_{a-1})\\big ) \\text{ and }f_{B}(S) = \\big (S + (yy_{2})\\big )\\cdot \\dots \\cdot \\big (S + (yy_{b-1})\\big ).$ Moreover, the new variables $T$ and $S$ correspond to one and the same exceptional divisor ${A}={B}$ .",
"Relation between polynomials $f_{\\bullet }$ and $F_{\\bullet }$ are clarified in the following lemma.",
"Lemma 30 The Chow class of the image of a divisor $(ab)_{K+A}$ , $a,b \\in [n]$ under the morphism $\\pi _{A}^{*}$ equals $(ab)_{K}$ for $a \\in A, b\\in B$ , or vice versa; ${\\bf bl}_{(ab)(A)}\\big ((ab)_{K+A}\\big )$ for $\\lbrace a,b\\rbrace \\subset B$ ; ${\\bf bl}_{(A)}\\big ((ab)_{K+A}\\big ) + {A}$ for $\\lbrace a,b\\rbrace \\subset A$ .",
"In case (1), the cycle $(ab)_{K+A}$ does not intersect $(A)_{K+A}$ .",
"It is by definition $\\textbf {bl}_{(ab)\\cap (A)}\\big ((ab)_{K+A}\\big )$ .",
"Then (1) and (2) follow directly from Theorem REF ,(2) by dimension counts.",
"The claim (3) also follows from the blow-up formula Theorem REF : $\\pi _{A}^{*} (ab) = \\textbf {bl}_{(ab)\\cap (A)}\\big ((ab)_{K+A}\\big ) + j_{A, *}\\big ( g^{*}_{A} [(ab)\\cap (A)] \\cdot s(N_{{A}}\\overline{\\mathcal {M}}_{0, K}) \\big )_{n-4},$ where $N_{{A}}\\overline{\\mathcal {M}}_{0, K}$ is a normal bundle and $s(\\;)$ is a total Segre class.",
"This follows from the equalities by the functoriality of the total Chern and Segre classes and the equality $s(U)\\cdot c(U) = 1$ .",
"Namely, we have $s((ab)\\cap (A)) &= [(ab)\\cap (A)] \\cdot s(N_{(ab)\\cap (A)}(ab)_{K+A}) = [(ab)\\cap (A)]\\cdot s\\left(N_{(A)}\\overline{\\mathcal {M}}_{0, K+A}\\right), \\\\c\\left( \\frac{g^{*}_{A}N_{(A)}\\overline{\\mathcal {M}}_{0,K+A}}{N_{{A}}\\overline{\\mathcal {M}}_{0,K}} \\right) &\\cdot g^{*}_{A} [(ab)\\cap (A)]\\cdot g^{*}_{A}s\\left(N_{(A)}\\overline{\\mathcal {M}}_{0, K+A}\\right) = g^{*}_{A} [(ab)\\cap (A)] \\cdot s(N_{{A}}\\overline{\\mathcal {M}}_{0, K}).$ Finally, we note that $g^{*}_{A} [(ab)\\cap (A)] = {A}$ .",
"From Lemma REF , we have the equality $f_{A}(T) = \\pi _{A}^{*}\\big ( (xx_{2})\\cdot \\dots \\cdot (xx_{a-1}) \\big ) = \\pi _{A}^{*}(A)\\text{ and }f_{B}(S) = \\pi _{B}^{*}(B).$ Lemma 31 $\\;$ The ideal $\\operatorname{Ann}(A)_{comb}$ is generated by its first graded component.",
"More precisely, the generators of the ideal $\\operatorname{Ann}(A)_{comb}$ are the elements of type $(ab)$ with $a\\in A, b\\in B$ , and the elements of type $(a_1a_2)-(b_1b_2),$ where $a_1,a_2 \\in A$ ; $b_1,b_2 \\in B=A^c$ .",
"The annihilators $\\operatorname{Ann}(A)_{comb}$ and $\\operatorname{Ann}(B)$ are canonically isomorphic.",
"First observe that the kernel $\\ker (i_{A}^*)$ equal the annihilator of the cycle $(A)$ .",
"Set $\\kappa $ be the ideal generated by $\\ker (i_{A}^{*})\\cap A^1$ .",
"Without loss of generality, we may assume that $A=\\lbrace 1,2,...,m\\rbrace $ .",
"Observe that: If $I\\subset [n]$ has a nonempty intersection with both $A$ and $A^c$ , then $(A)(I)=0$ .",
"In this case, $(I)$ can be expressed as $(ab)(I^{\\prime })$ , where $a\\in A,\\ b \\in A^c$ .",
"If $I\\subset A$ , then $(A)\\smile (I)= (A)(m+1...m+|I|).$ In this case, the element $(I)-(m+1,...,m+|I|)$ is in $\\kappa $ .",
"Let us demonstrate this by giving an example with $A=\\lbrace 1,2,3,4\\rbrace $ , $(I)=(12)$ : $(1234)\\smile (12)=(A)\\smile ((15)+(16)-(56))= 0+0-(1234)(56)$ .",
"We conclude that $(12)-(56)\\in \\kappa $ .",
"Let us show that $(123)-(567)\\in \\kappa $ .",
"Indeed, $(123)-(567)=(12) \\smile (23)-(567)\\in \\kappa \\Leftrightarrow (56)\\smile (23)-(567) \\in \\kappa \\Leftrightarrow (56)\\smile ((23)-(67))\\in \\kappa $ .",
"Since $(23)-(67)\\in \\kappa $ , the claim is proven.",
"If $I\\subset A^c$ , then $(A)\\smile (I)= (A)(m+1,...,m+|I|).$ The element $(I)-(m+1,...,m+|I|)$ is in $\\kappa $ .",
"This follows from (1) and (2).",
"Now let us prove the lemma.",
"Assume $x\\smile (A)=0$ .",
"Let $x=\\sum _i a_i (I_1^i)...(I^i_{k_i}).$ We may assume that $x$ is a homogeneous element.",
"Modulo $\\kappa $ , each summand $(I_1)...(I_{k_i})$ can be reduced to some $ (m+1...m+r_1)(m+r_1+1,...,m+r_2)...(m+r_{k_i}+1...m^{\\prime })$ .",
"Modulo $\\kappa $ , $ (m+1...m+r_1)(m+r_1+1...m+r_2)...(m+r_{k_i}+1,...,m^{\\prime })$ can be reduced to a one-bracket element $ (m+1...m+r_1m+r_1+1...m+r_2...m+r_{k_i}+1,...,m^{\\prime \\prime })$ .",
"Indeed, for two brackets we have: $ (m+1...m+r_1)(m+r_1+1,...,m+r_2) \\equiv (m+1...m+r_1)(m+r_1,...,m+r_2-1) \\equiv (m+1...m+r_2-1)\\;\\;(\\mathrm {mod} \\; \\kappa ).$ For a bigger number of brackets, the statement follows by induction.",
"We conclude that a homogeneous $x\\in \\ker (i_{A}^*)$ modulo $\\kappa $ reduces to some $a(m+1...m+m^{\\prime })$ , where $a \\in \\mathbb {Z}$ .",
"Then $a=0$ .",
"Indeed, $(A)(m+1...m+m^{\\prime })\\ne 0$ since by Lemma REF $(A)(m+1...m+m^{\\prime })(m+m^{\\prime }...n)\\ne 0$ .",
"Remark 32 Via the four-term relation any element from b) can be expressed as a linear combination of elements from a).",
"So only a)–elements are sufficient to generate the annihilators.",
"Actually, ${\\mathcal {A}}\\cong \\mathbb {C} \\oplus \\operatorname{Ann}(A)_{comb}.$ We arrive at the following commutative diagram of graded rings: [column sep=small] A*KA*K/Ann(B) A*K/Ann(A)comb A*K:= A*K+A, comb [T]/f(A)(T) A*K+B [S]/f(B)(S) [u] A*K+B A[J]/FA(J)[uur, \"fB = B* FB\"] A*K+A, combA[I]/FB(I) [uul, \"fA = A* FA\"] A[ul, \"FA\"] [ur, \"FB\"] Therefore, the following diagram commutes: [column sep=small] 0[r] Ann(A)comb [r] [d] A*K+A, comb [f(A)] [r][d, two heads, \"\"] A*K+A, comb[f(A)]Ann(A)comb [r] [d, \"\"] 0 0 [r] Ann(A)alg [r] A*K+A, alg [f(A)][r] A*K+A, alg [f(A)]Ann(A)alg [r] 0 Here $R[g]$ denotes the extension of a ring $R$ by a polynomial $g(t)$ .",
"All three vertical maps are induced by the map $\\alpha $ ; the last vertical map is an isomorphism since both rings are isomorphic to ${\\bf A}^{*}_{K}$ .",
"The ideals $\\operatorname{Ann}(A)_{comb}$ and $\\operatorname{Ann}(A)_{alg}$ coincide, so the homomorphism $\\alpha :\\;{\\bf A}^{*}_{K+A, comb} [f_{(A)}] \\rightarrow {\\bf A}^{*}_{K+A, alg} [f_{(A)}]$ is injective, and the theorem is proven.",
"$\\Box $"
],
[
"Poincaré polynomials of ASD compactifications",
"Theorem 33 Poincaré polynomial ${\\mathcal {P}}_{q}\\big (\\overline{\\mathcal {M}}_{0, L}\\big )$ for an ASD complex $L$ equals ${\\mathcal {P}}_{q}\\big (\\overline{\\mathcal {M}}_{0, L}\\big ) = \\frac{1}{q(q-1)} \\left( (1+q)^{n-1} - \\sum \\limits _{I \\in L} q^{|I|} \\right).$ This theorem is proven by Klyachko [10] for polygon spaces, that is, for compactifications coming from a threshold ASD complex.",
"Assume that $K + A$ be a threshold ASD complex.",
"For the blow up of the space $\\overline{\\mathcal {M}}_{0, K+B}$ along the subvariety $(B)$ we have an exact sequence of Chow groups $0 \\rightarrow {\\bf A}_{p}\\big (\\overline{\\mathcal {M}}_{0, \\mathcal {P}_{|A|+1}}\\big ) \\rightarrow {\\bf A}_{p}\\big ( \\overline{\\mathcal {M}}_{0, \\mathcal {P}_{|A|+1} } \\times \\overline{\\mathcal {M}}_{0, \\mathcal {P}_{|B|+1} } \\big ) \\oplus {\\bf A}_{p}\\big ( \\overline{\\mathcal {M}}_{0, K + B} \\big ) \\rightarrow {\\bf A}_{p}\\big (\\overline{\\mathcal {M}}_{0, K}\\big ) \\rightarrow 0,$ where, as before, $A \\sqcup B = [n]$ and $p$ is a natural number.",
"We get the equality ${\\mathcal {P}}_{q}(K) = {\\mathcal {P}}_{q}(K+B) + {\\mathcal {P}}_{q}(\\mathcal {P}_{|A|+1})\\cdot {\\mathcal {P}}_{q}(\\mathcal {P}_{|B|+1})- {\\mathcal {P}}_{q}(\\mathcal {P}_{|A|+1}).$ Also, we have the recurrent relations for the Poincaré polynomials: ${\\mathcal {P}}_{q}(K + B) &= {\\mathcal {P}}_{q}(K + A) + {\\mathcal {P}}_{q}(\\mathcal {P}_{|A|+1}) - {\\mathcal {P}}_{q}(\\mathcal {P}_{|B|+1}) =\\\\&=\\frac{1}{q(q-1)}\\left((1+q)^{n-1} - \\sum \\limits _{I\\in K + A} q^{|I|} + q^{|A|} - q - q^{|B|} +q \\right) = \\\\&=\\frac{1}{q(q-1)}\\left((1+q)^{n-1} - \\sum \\limits _{I\\in K} q^{|I|} -q^{|A|} + q^{|A|} - q - q^{|B|} +q \\right) = \\\\&= \\frac{1}{q(q-1)} \\left( (1+q)^{n-1} - \\sum \\limits _{I \\in K + B} q^{|I|} \\right).$ We have used the following: if $U$ is a facet of an ASD complex, then there is an isomorphism $\\overline{\\mathcal {M}}_{0, \\mathcal {P}_{|U|+1}} \\cong \\mathbb {P}^{|U| - 2}$ (see Example REF (2) ); the Poincaré polynomial of the projective space $\\mathbb {P}^{|U| - 2}$ equals $\\frac{q^{|U|} - q}{q(q-1)}$ ."
],
[
"The tautological line bundles over $\\overline{\\mathcal {M}}_{0,K}$ and the {{formula:20554466-30da-43b7-a0b4-4747bad0c0fd}} –classes",
"The tautological line bundles $L_i, \\ i=1,...,n$ were introduced by M. Kontsevich [4] for the Deligne-Mumford compactification.",
"The first Chern classes of $L_i$ are called the $\\psi $ -classes.",
"We now mimic the Kontsevich's original definition for ASD compactifications.",
"Let us fix an ASD complex $K$ and the corresponding compactification $\\overline{\\mathcal {M}}_{0,K}$ .",
"Definition 34 The line bundle $E_i=E_{i}(L)$ is the complex line bundle over the space $\\overline{\\mathcal {M}}_{0,K}$ whose fiber over a point $(u_1,...,u_n)\\in (\\mathbb {P}^1)^n$ is the tangent lineIn the original Kontsevich's definition, the fiber over a point is the cotangent line, whereas we have the tangent line.",
"This replacement does not create much difference.",
"to the projective line $\\mathbb {P}^1$ at the point $u_i$ .",
"The first Chern class of $E_{i}$ is called the $\\psi $ -class and is denoted by $\\psi _i$ .",
"Proposition 35 For any $i \\ne j \\ne k\\in [n]$ we have $\\psi _i= (ij) +(ik)-(jk).$ The four-term relation holds true: $(ij)+(kl)=(ik)+(jl)$ for any distinct $i,j,k,l \\in [n].$ (1) Take a stable configuration $(x_1,...,x_n)\\in \\overline{\\mathcal {M}}_{0,K}$ .",
"Take the circle passing through $x_i,x_j$ , and $x_k$ .",
"It is oriented by the order $ijk$ .",
"Take the vector lying in the tangent complex line to $x_i$ which is tangent to the circle and points in the direction of $x_j$ .",
"It gives rise to a section of $E_{i}$ which is defined correctly whenever the points $x_i,x_j$ , and $x_k$ are distinct.",
"Therefore, $\\psi _i = A(ij) +B(ik)+C(jk)$ for some integer $A,B,C$ .",
"Detailed analysis specifies their values.",
"Now (2) follows since the Chern class $\\psi _i $ does not depend on the choice of $j$ and $k$ .",
"Let us denote by $|d_{1}, \\dots , d_{n}|_K$ the intersection number $\\langle \\psi _1^{ d_1} ... \\psi _k^{ d_k}\\rangle _K= \\psi _1^{\\smile d_1} \\smile ... \\smile \\psi _k^{\\smile d_k}$ related to the ASD complex $K$ .",
"Theorem 36 Let $\\overline{\\mathcal {M}}_{0,K}$ be an ASD compactification.",
"A recursion for the intersection numbers is $|d_{1}, \\dots , d_{n}|_{K} = &|d_{1}, \\dots , d_{i}+d_{j}-1, \\dots , \\hat{d_{j}}, \\dots , d_{n}|_{K_{(ij)}} + |d_{1}, \\dots , d_{i}+d_{k}-1, \\dots , \\hat{d_{k}}, \\dots , d_{n}|_{K_{(ik)}} \\\\&-|d_{1}, \\dots , d_{i}-1 ,\\dots , d_{j}+d_{k}, \\dots , \\hat{d_{j}}, \\hat{d_{k}},\\dots , d_{n}|_{K_{(jk)}},$ where $i, j, k \\in [n]$ are distinct.",
"Remind that $K_{(ij)}$ denotes the complex $K$ with $i$ and $j$ frozen together.",
"Might happen that $K_{(ij)}$ is ill-defined, that is, $(ij)\\notin K$ .",
"Then we set the corresponding summand to be zero.",
"By Proposition REF , $\\langle \\psi _1^{d_1} \\dots \\psi _{n}^{d_n} \\rangle _K = \\langle \\psi _1^{d_1-1} \\dots \\psi _{n}^{d_n} \\rangle _K \\smile \\big ((1i)+(1j)- (ij)\\big ).$ It remains to observe that $\\langle \\psi _1^{d_1-1} \\dots \\psi _{n}^{d_n} \\rangle _K\\smile (ab)$ equals the $\\langle \\psi _1^{d_1-1} \\dots \\psi _{n}^{d_n} \\rangle _{K_{(a,b)}}$ .",
"Theorem 37 Let $\\overline{\\mathcal {M}}_{0,K}$ be an ASD compactification.",
"Any top monomial in $\\psi $ -classes modulo renumbering has a form $\\psi _1^{d_1}\\smile ...\\smile \\psi _m^{d_m}$ with $\\sum _{q=1}^m d_q=n-3$ and $d_q \\ne 0$ for $q=1,...,m$ .",
"Its value equals the signed number of partitions $[n-2]=I\\cup J$ with $m+1 \\in I$ and $I,J \\subset K$ .",
"Each partition is counted with the sign $(-1)^N \\cdot \\varepsilon ,$ where $N= |J|+\\sum _{q \\in J , q\\le m} d_q, \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\varepsilon =\\left\\lbrace \\begin{array}{lll}1, & \\hbox{if \\ \\ } J\\cup \\lbrace n\\rbrace \\in K, \\hbox{and\\ \\ }J\\cup \\lbrace n-1\\rbrace \\in K;\\\\-1, & \\hbox{if \\ \\ } I\\cup \\lbrace n\\rbrace \\in K, \\hbox{and\\ \\ }I\\cup \\lbrace n-1\\rbrace \\in K;\\\\0, & \\hbox{otherwise.",
"}\\end{array}\\right.$ Proof goes by induction.",
"Although the base is trivial, let us look at it.",
"The smallest $n$ which makes sense is $n=4$ .",
"There exist two ASD complexes with four vertices, both are threshold.",
"So there exist two types of fine moduli compactifications, both correspond to the configuration spaces of some flexible four-gon.",
"The top monomials are the first powers of the $\\psi $ -classes.",
"For $l_1=1;\\ l_2=1;\\ l_3=1;\\ l_4=0,1$ we have $\\psi _1=\\psi _2=\\psi _3=0$ , and $\\psi _4=2$ .",
"Let us prove that the theorem holds for the monomial $\\psi _1$ .",
"There are two partitions of $[n-2]~=~[2]$ : $J=\\lbrace 1\\rbrace ,\\ I=\\lbrace 2\\rbrace $ .",
"Here $\\varepsilon =0$ , so this partition contributes 0.",
"$J=\\emptyset ,\\ I=\\lbrace 1,2\\rbrace $ .",
"Here $I\\notin K $ , so this partition also contributes 0.",
"For $l_1=2,9;\\ l_2=1;\\ l_3=1;\\ l_4=1$ , we have $\\psi _2=\\psi _3=\\psi _4=1$ , and $\\psi _1=-1$ .",
"Let us check that the theorem holds for the monomial $\\psi _1$ .",
"(The other monomials are checked in a similar way.)",
"There partitions of $[2]$ are the same: $J=\\lbrace 1\\rbrace ,\\ I=\\lbrace 2\\rbrace $ .",
"Here $\\varepsilon =-1, \\ N=1+1$ , so this partition contributes $-1$ .",
"$J=\\emptyset ,\\ I=\\lbrace 1,2\\rbrace $ .",
"Here $I\\notin K $ , so it contributes 0.",
"For the induction step, let us use the recursion.",
"We shall show that for any partition $[n-2]=I\\cup J$ , its contribution to the left hand side and the right hand side of the recursion are equal.",
"This is done through a case analysis.",
"We present here three cases; the rest are analogous.",
"Assume that $i,j,k \\in I$ , and $(I,J)$ contributes 1 to the left hand side count.",
"Then $\\triangleright $ $(d_{1}, \\dots , d_{i}+d_{j}-1, \\dots , \\hat{d_{j}}, \\dots , d_{n})_{K_{(ij)}}$ contributes 1 to the right hand side.",
"Indeed, neither $N$ , nor $\\varepsilon $ changes when we pass from $K$ to $K_{(ij)}$ .",
"$\\triangleright $ $(d_{1}, \\dots , d_{i}+d_{k}-1, \\dots , \\hat{d_{k}}, \\dots , d_{n})_{K_{(ik)}} $ contributes 1, and $\\triangleright $ $-(d_{1}, \\dots , d_{i}-1 ,\\dots , d_{j}+d_{k}, \\dots , \\hat{d_{j}}, \\hat{d_{k}},\\dots , d_{n})_{K_{(jk)}}$ contributes $-1$ .",
"Assume that $i\\in I,\\ j,k \\in J$ , and $(I,J)$ contributes 1 to the left hand side count.",
"Then $\\triangleright $ $(d_{1}, \\dots , d_{i}+d_{j}-1, \\dots , \\hat{d_{j}}, \\dots , d_{n})_{K_{(ij)}}$ contributes 0 to the right hand side.",
"$\\triangleright $ $(d_{1}, \\dots , d_{i}+d_{k}-1, \\dots , \\hat{d_{k}}, \\dots , d_{n})_{K_{(ik)}} $ contributes 0, and $\\triangleright $ $-(d_{1}, \\dots , d_{i}-1 ,\\dots , d_{j}+d_{k}, \\dots , \\hat{d_{j}}, \\hat{d_{k}},\\dots , d_{n})_{K_{(jk)}}$ contributes 1.",
"Indeed, $N$ turns to $N-1$ , whereas $\\varepsilon $ stays the same.",
"Assume that $i\\in J,\\ j,k \\in I$ , and $(I,J)$ contributes 1 to the left hand side count.",
"Then $\\triangleright $ $(d_{1}, \\dots , d_{i}+d_{j}-1, \\dots , \\hat{d_{j}}, \\dots , d_{n})_{K_{(ij)}}$ contributes 0.",
"$\\triangleright $ $(d_{1}, \\dots , d_{i}+d_{k}-1, \\dots , \\hat{d_{k}}, \\dots , d_{n})_{K_{(ik)}} $ contributes 0, and $\\triangleright $ $-(d_{1}, \\dots , d_{i}-1 ,\\dots , d_{j}+d_{k}, \\dots , \\hat{d_{j}}, \\hat{d_{k}},\\dots , d_{n})_{K_{(jk)}}$ contributes 1, since $N$ turns to $N-1$ , and $\\varepsilon $ stays the same.$\\Box $ This theorem was proven for polygon spaces (that is, for threshold ASD complexes) in [14]."
],
[
"Appendix. Chow rings and blow ups",
"Assume we have a diagram of a blow up $\\widetilde{Y}:={\\bf bl}_{X}(Y)$ .",
"Here $X$ and $Y$ are smooth varieties, $\\iota : X \\hookrightarrow Y$ is a regular embedding, and $\\widetilde{X}$ is the exceptional divisor.",
"In this case, $\\iota ^{*}: A^*(Y) \\rightarrow A^*(X)$ is surjective.",
"X d [r, hookrightarrow, \"\"] Y d X [r, hookrightarrow, \"\"] Y Denote by $E$ the relative normal bundle $E:= \\tau ^* N_X Y/ N_{\\widetilde{X}} \\widetilde{Y}.$ Theorem 38 [7] The Chow ring $A^{*}(\\widetilde{Y})$ is isomorphic to $\\frac{A^{*}(Y)[T]}{(P(T), T\\cdot \\ker i^*)},$ where $P(T) \\in A^*(Y)[T]$ is the pullback from $A^*(X)[T]$ of Chern polynomial of the normal bundle $N_{X}Y$ .",
"This isomorphism is induced by $\\pi ^* : A^*(Y)[T] \\rightarrow A^* (\\widetilde{Y})$ which sends $-T$ to the class of the exceptional divisor $\\widetilde{X}$ .",
"Theorem 39 [15] Let $k \\in \\mathbb {N}$ .",
"a) (Key formula) For all $x \\in A_{k}(X)$ $\\pi ^* \\iota _{*} (x) = \\theta _* (c_{d-1}(E)\\cap \\tau ^* x)$ e) There are split exact sequences $0 \\rightarrow A_{k} X \\xrightarrow{} A_{k} \\widetilde{X} \\oplus A_{k} Y \\xrightarrow{} A_{k} \\widetilde{Y} \\rightarrow 0$ with $\\upsilon (x) = \\big ( c_{d-1}(E)\\cap \\tau ^{*}x, -\\iota _{*}x \\big )$ , and $\\eta (\\tilde{x}, y) = \\theta _{*}\\tilde{x}+ \\pi ^* y$ .",
"A left inverse for $\\upsilon $ is given by $(\\tilde{x}, y) \\mapsto \\tau _{*}(\\tilde{x})$ .",
"Theorem 40 [15] $\\;$ (Blow-up Formula) Let $V$ be a $k$ -dimensional subvariety of $Y$ , and let $\\tilde{V} \\subset \\tilde{Y}$ be the proper transform of $V$ , i.e., the blow-up of $V$ along $V\\cap X$ .",
"Then $\\pi ^{*}[V] = [\\tilde{V}] + j_{*}\\lbrace c(E)\\cap \\tau ^{*} s(V\\cap X, V)\\rbrace _{k} \\text{ in }A_{k}\\tilde{Y}.$ If $\\dim V\\cap X \\le k-d$ , then $\\pi ^* [V] = [\\tilde{V}]$ .",
"An algebraic variety $Z$ is a HI–scheme if the canonical map $\\mathrm {cl}: A^{*}(Z) \\rightarrow H^{*}(Z, \\mathbb {Z})$ is an isomorphism.",
"Theorem 41 [7] If $X, \\widetilde{X}$ , and $Y$ are HI, then so is $\\widetilde{Y}$ .",
"If $X, \\widetilde{X}$ , and $\\widetilde{Y}$ are HI, then so is $Y$ .",
"Acknowledgement.",
"This research is supported by the Russian Science Foundation under grant 16-11-10039."
]
] | 1808.08600 |
[
[
"Backward Stochastic Riccati Equation with Jumps associated with\n Stochastic Linear Quadratic Optimal Control with Jumps and Random\n Coefficients"
],
[
"Abstract In this paper, we investigate the solvability of matrix valued Backward stochastic Riccati equations with jumps (BSREJ), which is associated with a stochastic linear quadratic (SLQ) optimal control problem with random coefficients and driven by both Brownian motion and Poisson jumps.",
"By dynamic programming principle, Doob-Meyer decomposition and inverse flow technique, the existence and uniqueness of the solution for the BSREJ is established.",
"The difficulties addressed to this issue not only are brought from the high nonlinearity of the generator of the BSREJ like the case driven only by Brownian motion, but also from that i) the inverse flow of the controlled linear stochastic differential equation driven by Poisson jumps may not exist without additional technical condition, and ii) how to show the inverse matrix term involving jump process in the generator is well-defined.",
"Utilizing the structure of the optimal problem, we overcome these difficulties and establish the existence of the solution.",
"In additional, a verification theorem for BSREJ is given which implies the uniqueness of the solution."
],
[
"Framework and Preliminary",
"We start with a stochastic basis $(\\Omega ,\\mathcal {F},{F},\\mathbb {P})$ with a finite time horizon $T<\\infty $ and a filtration ${F}:=\\lbrace {{F}}_{t}|t\\in [0,T]\\rbrace $ satisfying the usual conditions of right continuity and completeness, such that we can and do take all semimartingales to have right continuous paths with left limits.",
"For simplicity, we assume that ${F}_{0}$ is trivial and $\\mathcal {F}={F}_{T}.$ Denote by ${\\mathbb {E}}[\\cdot ]$ the expectation under $\\mathbb {P}$ .",
"Conditional expectations with respect to a sub-$\\sigma $ algebra ${G}$ of $\\mathcal {F}$ are denoted by $\\mathbb {E}^{{G}}[\\cdot ].$ Let ${B}(\\Lambda )$ denote the Borel $\\sigma $ -algebra of the topological space $\\Lambda .$ Let $W=\\lbrace W(t)=(W^{1}(t),W^{2}(t),\\cdots ,W^{d}(t))^{\\top }|t\\in [0,T]\\rbrace $ be a $d$ -dimensional standard Brownian motion with respect to its natural filtration under $\\mathbb {P}$ .",
"Let $(\\Lambda ,{B}(\\Lambda ))$ be a measurable space and $\\nu $ a finite measure defined on it.",
"Denote by $\\mu $ an integer-valued random measure $\\mu (de,dt)=\\lbrace \\mu (\\omega ,de,dt)|\\omega \\in \\Omega \\rbrace $ on $([0,T]\\times \\Lambda ,{B}([0,T])\\otimes {B}(\\Lambda ))$ induced by a stationary ${F}$ -Poisson point process $(p_{t})_{t\\ge 0}$ on $\\Lambda $ with the Lévy measure $\\nu .$ Let $\\tilde{\\mu }(de,dt):={\\mu }(de,dt)-\\nu (de)dt$ be the compensated Poisson random measure.",
"Suppose that the Brownian motion $W$ and the random measure $\\tilde{\\mu }(de,dt)$ are stochastically independent under $\\mathbb {P}$ .",
"Without loss of general assumptions, we assume that the filtration ${F}$ is the $\\mathbb {P}$ -augmentation of the natural filtration generated by the Brownian motion and the Poisson random measure.",
"Let ${P}$ be the ${F}$ -predictable $\\sigma $ -field on $\\Omega \\times [0,T]$ and denote ${\\tilde{P}}:={P}\\otimes {B}(\\Lambda ).$ For a ${\\tilde{P}}$ -measurable function $U$ on $\\tilde{\\Omega }$$,$ define its integration with respect to $\\mu $ (analogously for $\\nu \\otimes {\\rm Leb}$ ) by $\\int _{0}^{T}\\int _{\\Lambda }U(s,e)\\mu (de,ds)(\\omega )=\\left\\lbrace \\begin{array}{rl}{\\displaystyle \\int _{0}^{T}\\int _{\\Lambda }U(\\omega ,s,e)\\mu (\\omega ,ds,de),\\quad } & \\mbox{if finitely defined},\\\\+\\infty ,\\qquad \\qquad \\qquad \\quad & \\mbox{otherwise}.\\end{array}\\right.$ The random measure and stochastic integrals can be referred to [9], [23] for details."
],
[
"Introduction on BSREJ",
"Denote by $\\mathbb {S}^{n}$ the space of all $n\\times n$ symmetric matrices and by $\\mathbb {S}_{+}^{n}$ the space of all $n\\times n$ nonnegative matrices.",
"Throughout this paper, the following standard assumptions holds.",
"Suppose that $A,B,C,D,E,F,Q,N$ and $M$ are given random mappings such that $A:[0,T]\\times \\Omega \\rightarrow \\mathbb {R}^{n\\times n},B:[0,T]\\times \\Omega \\rightarrow \\mathbb {R}^{n\\times m};C^{i}:[0,T]\\times \\Omega \\rightarrow \\mathbb {R}^{n\\times n},D^{i}:[0,T]\\times \\Omega \\rightarrow \\mathbb {R}^{n\\times m},i=1,2,\\cdots ,d;E:[0,T]\\times \\Omega \\times \\Lambda \\rightarrow \\mathbb {R}^{n\\times n};F:[0,T]\\times \\Omega \\times \\Lambda \\rightarrow \\mathbb {R}^{n\\times m};Q:[0,T]\\times \\Omega \\rightarrow \\mathbb {R}^{n\\times n},N:[0,T]\\times \\Omega \\rightarrow \\mathbb {R}^{m\\times m};M:\\Omega \\rightarrow \\mathbb {R}^{n\\times n}$ satisfies : Assumption 1.1 $A,B,C,D,N$ and $Q$ are uniformly bounded ${F}$ -predictable stochastic processes.",
"$E$ and $F$ are uniformly bounded ${\\tilde{P}}$ -measurable stochastic processes.",
"$M$ is a uniformly bounded ${{F}}_{T}$ -measurable random variable.",
"Moreover, for a.s. a.e.",
"$(t,\\omega )\\in [0,T]\\times \\Omega $ , $Q\\in \\mathbb {S}_{+}^{n}$ and $N\\in \\mathbb {S}_{+}^{m}$ .",
"$M\\in \\mathbb {S}_{+}^{n}$ for a.e.",
"$\\omega \\in \\Omega $ .",
"And $N$ is uniformly positive, i.e.",
"for a.s. a.e.",
"$(t,\\omega )\\in [0,T]\\times \\Omega ,$ $N(t)\\ge \\delta I$ for some positive constant $\\delta $ .",
"For any $(t,K,L,R(\\cdot ))\\in [0,T]\\times \\mathbb {S}^{n}\\times (\\mathbb {S}^{n})^{d}\\times {\\mathcal {M}}^{\\nu ,2}(\\mathbb {S}^{n})$ (see the meaning of the notations in subsection REF ), define ${N}(t,K,R(\\cdot )) & := & N(t)+{\\displaystyle \\sum _{i=1}^{d}(D^{i})^{*}(t)KD^{i}(t)+\\int _{\\Lambda }F^{*}(t,e)(K+R(e))F(t,e)\\nu (de)},\\nonumber \\\\{M}(t,K,L,R(\\cdot )) & := & KB(t)+\\sum _{i=1}^{d}L^{i}D^{i}(t)+\\sum _{i=1}^{d}(C^{i})^{*}(t)KD^{i}(t)\\nonumber \\\\& & +{\\displaystyle \\int _{\\Lambda }\\Big [E^{*}(t,e)KF(t,e)+(I+E^{*}(t,e))R(e)F(t,e)\\Big ]\\nu (de),}\\\\G(t,K,L,R(\\cdot )) & := & A^{*}(t)K+KA(t)+\\sum _{i=1}^{d}L^{i}C^{i}(t)+\\sum _{i=1}^{d}(C^{i})^{*}(t)L^{i}+\\sum _{i=1}^{d}(C^{i})^{*}(t)KC^{i}(t)\\nonumber \\\\& & +{\\displaystyle \\int _{\\Lambda }R(e)E(t,e)\\nu (de)+{\\displaystyle \\int _{\\Lambda }E^{*}(t,e)R(e)\\nu (de)}}\\nonumber \\\\& & +{\\displaystyle \\int _{\\Lambda }E^{*}(t,e)(K+R(e))E(t,e)\\nu (de)}\\nonumber \\\\& & +Q(t)-{M}(t,K,L,R(\\cdot )){N}^{-1}(t,K,R(\\cdot )){M}^{*}(t,K,L,R(\\cdot )),\\nonumber $ where $I$ is $n$ -th order identity matrix and $*$ denotes the transpose of a matrix.",
"With the notations defined above, we introduce the following backward stochastic integral-differential equation driven by Brownian motion $W$ and Poisson random measure $\\tilde{\\mu }$ : $\\left\\lbrace \\begin{array}{lll}dK(t) & = & -G(t,K(t-),L(t),R(t,\\cdot ))dt+{\\displaystyle \\sum _{i=1}^{d}L^{i}(t)dW_{t}^{i}+{\\displaystyle \\int _{\\Lambda }R(t,e)\\tilde{\\mu }(de,dt),}}\\\\K(T) & = & M,\\quad L(t):=(L^{1}(t),\\cdots ,L^{d}(t)).\\end{array}\\right.$ with the unknown triple of stochastic processes $(K,L,R).$ Now we give the definition of the solution to BSREJ (REF ) as follows.",
"Definition 1.1 A triplet of stochastic processes $(K,L,R)$ valued in $\\mathbb {S}^{n}\\times (\\mathbb {S}^{n})^{d}\\times {\\cal M}^{\\nu ,2}(\\mathbb {S}^{n})$ with $K$ being ${F}$ -progressive measurable, $L$ ${F}$ -predictable and $R$ $\\tilde{{P}}$ -measurable is called a solution of BSREJ (REF ) if (i)$\\int _{0}^{T}|G(t,K(t-),L(t),R(t))|dt+\\int _{0}^{T}\\int _{\\Lambda }|R(t,e)|^{2}\\nu (de)dt+\\int _{0}^{T}|L(t)|^{2}dt<\\infty ,a.s.;$ (ii) ${N}(t,K(t-),R(t))$ is positive definite a.s.",
"a.e.",
"; (iii) for all $t\\in [0,T],$ it a.e.",
"holds that $K(t)=M+\\int _{t}^{T}G(s,K(s-),L(s),R(s))ds-\\int _{t}^{T}\\sum _{i=1}^{d}L^{i}(s)dW_{s}^{i}-\\int _{t}^{T}\\int _{\\Lambda }R(s,e)\\tilde{\\mu }(de,ds).$ This is the so-called BSREJ associated with a linear quadratic optimal control problem with jumps formulated in Section 2 (See Problem REF ).",
"When the coefficients $A,B,C,D,E,F,Q,N$ are all deterministic, then $L^{1}=\\cdots =L^{d}=R=0$ , and the BSREJ (REF ) degenerates to a deterministic Riccati integral-differential equation (see [29] for the case without jumps).",
"If $D=0$ and $F=0$ , i.e.",
"the corresponding controlled differential system does not contain control in martingale integration terms, and the second and third unknown variables $(L,R)$ only have a linear structure in the generator $G$ .",
"And in this case the solvability of BSREJ could be covered by the result of Meng [19].",
"Due to that the martingale integration parts of corresponding controlled system (REF ) contains control variable, and the system has non-Markovian structure, the associated BSREJ (REF ) is highly nonlinear with respect to the unknown triple of $(K,L,R)$ ."
],
[
"Developments of BSRE and Contributions of this Paper",
"The study of BSREs had quite a long history.",
"In the case of BSREs driven by only Brownian motion $W,$ (REF ) will reduce to the following form: $\\left\\lbrace \\begin{array}{ll}dK(t)= & -\\bigg [A^{*}(t)K(t)+K(t)A(t)+\\sum _{i=1}^{d}L^{i}(t)C^{i}(t)+\\sum _{i=1}^{d}(C^{i})^{*}(t)L^{i}(t)+\\sum _{i=1}^{d}(C^{i})^{*}(t)K(t)C^{i}(t)+Q\\\\& -[K(t)B(t)+\\sum _{i=1}^{d}L^{i}(t)D^{i}(t)+\\sum _{i=1}^{d}(C^{i})^{*}(t)K(t)D^{i}(t)][N(t)+\\sum _{i=1}^{d}(D^{i})^{*}(t)K(t)D^{i}(t)]^{-1}\\\\& \\quad \\cdot [K(t)B(t)+\\sum _{i=1}^{d}L^{i}(t)D^{i}(t)+\\sum _{i=1}^{d}(C^{i})^{*}(t)K(t)D^{i}(t)]^{*}\\bigg ]dt+{\\displaystyle \\sum _{i=1}^{d}L^{i}(t)dW_{t}^{i},}\\\\K(T)= & M,\\quad L(t):=(L^{1}(t),\\cdots ,L^{d}(t)).\\end{array}\\right.$ Historically speaking, the French mathematician Bimut [1] firstly proposed the definition of the adapted solution to (REF ) , and due to the difficulty of its solvability, it is listed as an open problem by Peng [20].",
"Until 2013, Tang [25] generally solved this open problem applying the stochastic maximum principle and using the technique of stochastic flow for the associated stochastic Hamiltonian system.",
"In 2015, Tang [26] gives the second but more comprehensive (seeming much simpler, by Doob-Meyer decomposition theorem and Dynamic programming principle) method to solve the general BSREs.",
"For earlier history on BSRE, we refer to Peng [22], Tang and Kohlmann [12], [13], Tang [25] and the plenary lecture reported by Peng [21] at the ICM in 2010.",
"For the indefinite BSRE, the reader can be referred to [2], [30], [14], [15], [24], [4].",
"Equation (REF ) is very different from equation (REF ).",
"From a direct viewpoint, Equation (REF ) is driven by both a Brownian motion $W$ and an additional compensated Poisson measure $\\tilde{\\mu }$ .",
"From an essential viewpoint, not only the first unknown element $K$ and but also the third unknown element $R$ are included in the nonlinear term ${N}(t,K(t-),R(t,\\cdot ))^{-1}$ in BSREJ (REF ).",
"For the BSRE driven only by a Brownian motion, the nonlinear term ${N}(t,K(t-),R(t,\\cdot ))^{-1}$ degenerates into $\\big [N(t)+D^{i*}(t)K(t)D^{i}(t)\\big ]^{-1}$ which is well defined since in that case we can show that $K$ is continuous and nonnegative.",
"But for the BSREJ (REF ), one only expects to prove the square integrability of the third unknown element $R$ , but this regularity is difficult to derive the non-negativity of matrix ${N}(t,K(t-),R(t,\\cdot ))$ .",
"How to show ${N}(t,K(t-),R(t,\\cdot ))$ keeping to be positive is key to give the solvability of BSREJ (REF ).",
"As far as we know, there is very few literature related to BSREJ.",
"In 2008, under partial information framework, Hu and Ãksendal [8] studied the one-dimensional SLQ problem with random coefficients and Poisson jumps, where they presented the state feedback representation of the optimal control by an one-dimensional BSREJ, but the authors did not discuss the wellposeness of the solution to BSREJ.",
"[19] is the first work addressed to the study of high dimensional SLQ with random coefficients, the author formally derived BSREJ (REF ) and utilized Bellman's principle of quasi-linearization to solve a special form of BSREJ (REF ) in which the generator $G$ only linearly depends on $L$ and $R$ .",
"Li et al [18] used so-called relax compensator to describe indefinite BSREJ and investigated the solvability BSREJ in some special cases.",
"The contributions of our paper is to establish the solvability of the general BSREJ (REF ).",
"Adapting the method proposed by Tang [26], with the help of control problem and dynamic programming principle, we use the value function and Doob-Meyer decomposition to construct the triple process $(K(t),L(t),R(t,\\cdot ))$ and later show it is nothing but the solution of BSREJ (REF ).",
"Conversely, we also could utilize the solution of BSREJ (REF ) to depict the optimal control in a feedback form.",
"One advantage of above method is to avoid the proof of the positive definiteness of the matrix process ${N}$ at the beginning.",
"In our approach, we show not only the positive definiteness of of ${N}$ , but also that of $\\int _{\\Lambda }F^{*}(t,e)(K(t-)+R(t,e))F(t,e)\\nu (de)$ .",
"The proof is based on an observation that: $\\int _{\\Lambda }R(t,e)\\mu (de,\\lbrace t\\rbrace )$ is nothing but the jump measure of $K(t)$ .",
"Hence the value $\\int _{\\Lambda }F^{*}(t,e)(K(t-)+R(t,e))F\\mu (de,\\lbrace t\\rbrace )$ vanishes except at the jump time, then it coincides with $\\int _{\\Lambda }F^{*}(t,e)K(t)F(t,e)\\mu (de,\\lbrace t\\rbrace )$ since the jump $\\Delta K_{t}=K(t)-K(t-)=R(t,\\Delta p_{t})$ , where $\\Delta p_{t}$ is the jump of underlying Poisson process.",
"Obviously (REF ) is positive once the positive definiteness of ${N}$ obtained.",
"The inverse flow of the controlled stochastic differential equation on interval $[0,T]$ is a key technique in Tang's method in [26] to give the representation of the BSREJ.",
"In some literature about stochastic differential with jumps [7], [16], [27], [3], the authors give a technical condition to guarantee its inverse flow exists on $[0,T]$ (using the notation of SDE (REF )) $I+E(t,e)\\ge \\delta I,\\quad {\\rm a.e.a.s.",
"},\\:{\\rm for}\\:{\\rm some}\\:\\delta >0.$ But this condition is not necessary for the LQ control problem.",
"In our approach, to overcome the difficulty brought from the absence of condition (REF ), we deal with SDE (REF ) in every stochastic sub-interval between every two adjacent jumping time $(\\!",
"(\\tau _{i},\\tau _{i+1})\\!",
")$ , on which SDE (REF ) has continuous trajectory solution and subsequently inverse flow without the help of condition REF .",
"Then we use the semi-martingale property of $K$ to integrate all the sub-intervals to obtain the representation of BSREJ on the whole interval $[0,T]$ .",
"The rest of this article is organized as follows.",
"In Section , we introduce some useful notations, preliminary results and the SLQ problem with jumps.",
"In Section , we list the preliminary results and the controlled SLQ problem.",
"Section gives some basic properties of the value function $V$ , and also the semimartingale property of $V$ by dynamic programming principle.",
"In Section , with the help of results in Section we show the existence of BSREJ (REF ).",
"In Section , we show the verification theorem which gives the uniqueness of the solution for BSREJ, and use the solution of BSREJ to describe the optimal control and valuation of the SLQ problem."
],
[
"Notations",
"Let $H$ be a Hilbert space.",
"The inner product in $H$ is denoted by $\\langle \\cdot ,\\cdot \\rangle ,$ and the norm in $H$ is denoted by $|\\cdot |_{H}$ or $|\\cdot |$ if there is no danger of confusion.",
"Let $p\\ge 1.$ Let ${T}$ denote the totality of all ${F}$ -stopping times taking values in $[0,T].$ Define ${T}_{\\tau }:=\\lbrace \\gamma \\in {T}:\\gamma \\ge \\tau ,\\,\\mathbb {P}{\\rm -a.s.}\\rbrace $ for $\\tau \\in {T}.$ Given $\\tau \\in {T}$ and $\\gamma \\in {F}_{\\tau }$ , the following spaces will be frequently used in this paper: ${\\mathcal {S}}_{{F}}^{p}(\\tau ,\\gamma ;H)$ : the set of all $H$ -valued ${F}$ -adapted right continuous left limit (RCLL) processes $f\\triangleq \\lbrace f(t,\\omega ),t\\in [\\!",
"[\\tau ,\\gamma ]\\!",
"]\\rbrace $ such that $\\Vert f\\Vert _{{S}_{{F}}^{p}(\\tau ,\\gamma ;H)}:=\\bigg \\lbrace {\\mathbb {E}}\\bigg [\\sup _{\\tau \\le t\\le \\gamma }|f(t)|_{H}^{p}\\bigg ]\\bigg \\rbrace ^{\\frac{1}{p}}<\\infty $ ; ${\\mathcal {M}}_{{F}}^{p}(\\tau ,\\gamma ;H)$ : the set of all $H$ -valued ${F}$ -progressively measurable processes $f\\triangleq \\lbrace f(t,\\omega ),t\\in [\\!",
"[\\tau ,\\gamma ]\\!",
"]\\rbrace $ such that $\\Vert f\\Vert _{{\\mathcal {M}}_{{F}}^{p}(\\tau ,\\gamma ;H)}:=\\bigg \\lbrace \\mathbb {E}{\\displaystyle \\bigg [\\int _{\\tau }^{\\gamma }|f(t)|_{H}^{p}dt\\bigg ]\\bigg \\rbrace ^{\\frac{1}{p}}<\\infty }$ ; ${\\cal M}_{{F}}^{2,p}(\\tau ,\\gamma ;H)$ : the set of all $H$ -valued ${F}$ -progressively measurable processes $f\\triangleq \\lbrace f(t,\\omega ),t\\in [\\!",
"[\\tau ,\\gamma ]\\!",
"]\\rbrace $ such that $\\Vert f\\Vert _{{\\cal M}_{{F}}^{2,p}(\\tau ,\\gamma ;H)}:=\\bigg \\lbrace {\\mathbb {E}}\\bigg [{\\displaystyle \\int _{\\tau }^{\\gamma }|f(t)|_{H}^{2}dt\\bigg ]^{\\frac{p}{2}}\\bigg \\rbrace ^{\\frac{1}{p}}<\\infty }$ ; ${\\cal M}^{\\nu ,2}(H):$ the set of all H-valued measurable functions $r\\triangleq \\lbrace r(e),e\\in \\Lambda \\rbrace $ defined on the measure space $({\\Lambda },{B}(\\Lambda ),\\nu )$ such that $\\Vert r\\Vert _{{\\cal M}^{\\nu ,2}(H)}:=\\sqrt{{\\displaystyle \\int _{\\Lambda }|r(e)|_{H}^{2}\\nu (de)}}<~\\infty $ ; ${\\cal M}_{{F}}^{\\nu ,2,p}(\\tau ,\\gamma ;H):$ the set of all $H$ -valued ${\\tilde{P}}$ -measurable processes $r\\triangleq \\lbrace r(t,\\omega ,e),(t,e)\\in [\\!",
"[\\tau ,\\gamma ]\\!",
"]\\times \\Lambda \\rbrace $ such that $\\Vert r\\Vert _{\\mathcal {M}_{{F}}^{\\nu ,2,p}(\\tau ,\\gamma ;H)}:=\\bigg \\lbrace \\mathbb {E}\\bigg [{\\displaystyle \\int _{\\tau }^{\\gamma }\\int _{\\Lambda }}|r(t,e)|_{H}^{2}\\nu (de)dt\\bigg ]^{\\frac{p}{2}}\\bigg \\rbrace ^{\\frac{1}{p}}<~\\infty $ ; ${\\cal M}_{{F}}^{\\nu ,p}(\\tau ,\\gamma ;H):$ the set of all $H$ -valued ${\\tilde{P}}$ -measurable processes $r\\triangleq \\lbrace r(t,\\omega ,e),(t,e)\\in [\\!",
"[\\tau ,\\gamma ]\\!",
"]\\times \\Lambda \\rbrace $ such that $\\Vert r\\Vert _{\\mathcal {M}_{{F}}^{\\nu ,p}(\\tau ,\\gamma ;H)}:=\\bigg \\lbrace \\mathbb {E}{\\displaystyle \\bigg [\\int _{\\tau }^{\\gamma }\\int _{\\Lambda }|r(t,e)|_{H}^{p}\\nu (de)dt\\bigg ]\\bigg \\rbrace ^{\\frac{1}{p}}<~\\infty }$ ; $L^{p}(\\Omega ,{{G}},\\mathbb {P};H):$ the set of all $H$ -valued ${{G}}$ -measurable random variable $\\xi $ defined on $(\\Omega ,\\mathcal {F},P)$ such that $\\Vert \\xi \\Vert _{L^{p}(\\Omega ,{{G}},P;H)}:=\\lbrace \\mathbb {E}[|\\xi |_{H}^{p}]\\rbrace ^{\\frac{1}{p}}$ where ${G}$ is a subalgebra of $\\mathcal {F}$ .",
"In the following we recall a classical theorem for the essential infimum of a family of nonnegative random variables in a probability space (see, e.g.",
"Karatzas and Shreve [11]).",
"Lemma 2.1 Let ${X}$ be a family of nonnegative integrable random variables defined on a probability space $(\\Omega ,\\mathcal {F},\\mathbb {P}).$ Then there exists an $\\mathcal {F}$ -measurable random variable $X^{*}$ such that 1. for all $X\\in {X},$ $X\\ge X^{*}$ a.s.; 2. if $Y$ is a random variable satisfying $X\\ge Y$ a.s. for all $X\\in {X},$ then $X^{*}\\ge Y$ a.s.",
"This random variable, which is unique a.s., is called the essential infimum of ${X},$ and is denoted by $\\operatornamewithlimits{ess\\,inf}{X}$ or $\\operatornamewithlimits{ess\\,inf}_{X\\in {X}}X$ .",
"Furthermore, if ${X}$ is closed under pairwise minimum (i.e.",
"$X,Y\\in {X}$ implies $X\\wedge Y\\in {X}$ ), then there exists a nondecreasing sequence $\\lbrace Z_{n}\\rbrace _{n\\in \\mathbb {N}}$ of random variables in ${X}$ such that $X^{*}=\\lim _{n\\rightarrow \\infty }Z_{n}$ a.s.",
"Moreover, for any sub-algebra ${G}$ of $\\mathcal {F},$ the ${G}$ -conditional expectation is interchangeable with the essential infimum: $\\mathbb {E}[\\operatornamewithlimits{ess\\,inf}_{X\\in {X}}X|{G}]=\\operatornamewithlimits{ess\\,inf}_{X\\in {X}}\\mathbb {E}[X|{G}].$"
],
[
"Some Basic Definition and Results on ${T}$ -System",
"For any $\\tau _{1},\\tau _{2}\\in {T},$ with $\\tau _{1}\\le \\tau _{2}$ almost surely and $\\mathbb {P}(\\tau _{1}<\\tau _{2})>0,$ let ${T}[\\tau _{1},\\tau _{2}]:=\\lbrace \\tau \\in {T}|\\tau _{1}\\le \\tau \\le \\tau _{2}~\\mathbb {P}{\\rm -a.s.}\\rbrace .$ The following classical result of aggregation of supmartingale system could be found in [5].",
"Definition 2.1 A family of random variables ${K}:=\\lbrace {K}(\\tau ),\\tau \\in {T}\\rbrace $ indexed by ${T}$ is said to be ${T}$ -system if it satisfies 1. for all $\\tau \\in {T}$ , ${K}(\\tau )$ is ${F}_{\\tau }$ -measurable random variable; 2. for all $\\tau _{1},\\tau _{2}\\in {T},$ ${K}(\\tau _{1})={K}(\\tau _{2})$ a.s. on $\\lbrace \\tau _{1}=\\tau _{2}\\rbrace ~{\\rm for}~\\tau _{1},\\tau _{2}\\in {T}.$ Definition 2.2 We call a ${T}$ -system $\\lbrace {K}(\\tau ),\\tau \\in {T}\\rbrace $ a submartingale system if the following two properties hold: (i) ${K}(\\tau )$ is integrable for any $\\tau \\in {T};$ (ii) $\\mathbb {E}^{{F}_{\\tau _{1}}}[{K}(\\tau _{2})]\\ge {K}(\\tau _{1}),$ $\\mathbb {P}$ -a.s., for all $\\tau _{1}\\in {T},\\tau _{2}\\in {T}_{\\tau _{1}}.$ We call ${T}$ -system ${K}:=\\lbrace {K}(\\tau ),\\tau \\in {T}\\rbrace $ is said to be a supermartingale system if $-{K}$ is a submartingale system, and call it a martingale if it is both a ${T}$ -supermartingale and a ${T}$ -submartingale system.",
"Definition 2.3 A ${T}$ -system $\\lbrace {K}(\\tau ),\\tau \\in {T}\\rbrace $ is called right-(resp., left-) continuous along times in expectation (RCE (resp., LCE)) if for any sequences of stopping times $(\\tau _{n})_{n\\in \\mathbb {N}}$ such that $\\tau _{n}\\searrow \\tau $ a.s.(resp., $\\tau _{n}\\nearrow \\tau $ ), one has $\\mathbb {E}[{K}(\\tau )]=\\lim _{n\\longrightarrow \\infty }\\mathbb {E}[{K}(\\tau _{n})].$ Definition 2.4 We call that an process $X=\\lbrace X(t),t\\in [0,T]\\rbrace $ aggregates the ${T}$ -system $\\lbrace {K}(\\tau ),\\tau \\in {T}\\rbrace ,$ if for any $\\tau \\in {T},$ it holds $X(\\tau )={K}(\\tau ),$ $\\mathbb {P}$ -a.s.",
"The following result could be found in [5], or adapted from [10].",
"Proposition 2.2 Let a ${T}$ -system $\\lbrace {K}(\\tau ),\\tau \\in {T}\\rbrace $ be a supermartingale system which is RCE and such that ${K}(0)<\\infty $ .",
"There then exists a RCLL adapted process denoted by $\\lbrace K(t)\\rbrace _{t\\in [0,T]}$ which aggregates ${T}$ -system $\\lbrace {K}(\\tau ),\\tau \\in {T}\\rbrace .$ Consider a supermartingale process $({K}(t))_{0\\le t\\le T}$ , by Theorem 3.13 in [10], it has a RCLL modification $K(t):=\\lim _{s\\searrow t,s\\in \\mathbb {Q}}{K}(s)$ .",
"For any stopping time $\\tau $ , define $\\tau _{n}(\\omega ):=\\frac{i}{2^{n}}$ , if $\\tau (\\omega )\\in (\\frac{i-1}{2^{n}},\\frac{i}{2^{n}}]$ for some integer $i>0$ .",
"It is easy to see that ${K}(\\tau _{n})=K(\\tau _{n})$ .",
"Then by REC of ${K}$ and uniform convergence of $\\lbrace K(\\tau _{n})\\rbrace $ (see Remark 3.12 in [10]), passing $n$ to infinity, we have ${K}(\\tau )=K(\\tau )$ a.e.",
"Thus $\\lbrace K(t)\\rbrace _{t\\in [0,T]}$ aggregates ${T}$ -system $\\lbrace {K}(\\tau ),\\tau \\in {T}\\rbrace $ .",
"For future purposes, we shall consider the conditional extension of ${T}$ -system.",
"More precisely, for a family of random variables ${{K}}:=\\lbrace {K}(\\sigma ),\\sigma \\in {T}_{\\tau }\\rbrace $ indexed by ${T}_{\\tau }$ , it is called a ${T}_{\\tau }$ -system if it satisfies 1. for all $\\sigma \\in {T}_{\\tau }$ , ${K}(\\sigma )$ is ${F}_{\\sigma }$ -measurable random variable.",
"2. for all $\\sigma _{1},\\sigma _{2}\\in {T}_{\\tau },$ ${K}(\\sigma _{1})={K}(\\sigma _{2})$ a.s. on $\\lbrace \\sigma _{1}=\\sigma _{2}\\rbrace ~{\\rm for}~\\sigma _{1},\\sigma _{2}\\in {T}_{\\tau }.$ Naturally, Definitions REF and REF can be adapted for the ${T}_{\\tau }$ -system.",
"Given a ${T}_{\\tau }$ -system ${K}$ , one can extend it to be a ${T}$ -system, still denoted by ${{K}}$ , in the following way: ${K}(\\sigma ):={K}(\\sigma )\\chi _{\\lbrace \\sigma \\ge \\tau \\rbrace }+\\mathbb {E}[{K}(\\tau )\\chi _{\\lbrace \\sigma <\\tau \\rbrace }|{F}_{\\sigma }]\\chi _{\\lbrace \\sigma <\\tau \\rbrace }.$ If the original ${T}_{\\tau }$ -system ${{K}}$ is a submartingale (resp.",
"supermartingale) system, then the extension is also a submartingale (resp.",
"supermartingale) system.",
"Moreover, the RCE (or LCE) property holds for the extension.",
"Hence, according to Proposition REF , if ${{K}}$ is a supermartingale ${T}_{\\tau }$ -system which is RCE and $E[{K}(\\tau )]<+\\infty $ , then there exists a RCLL adapted process $K$ defined on the random interval $[\\!",
"[\\tau ,T]\\!",
"]$ which aggregates ${K}$ , i.e., for any $\\sigma \\in {T}_{\\tau }$ , $K(\\sigma )={K}(\\sigma ),\\mathbb {P}-a.s..$"
],
[
"Preliminary Results for Liner SDE with Jumps",
"Let $p\\ge 2.$ For any $(\\tau ,\\xi )\\in {T}\\times L^{p}(\\Omega ,{{F}_{\\tau }},\\mathbb {P};\\mathbb {R}^{n}),$ consider the following linear SDE with jumps $\\left\\lbrace \\begin{array}{lll}dX(t) & = & [A(t)X(t-)+f(t)]dt+{\\displaystyle \\sum _{i=1}^{d}[C^{i}(t)X(t-)+g^{i}(t)]dW^{i}(t)}\\\\& & +{\\displaystyle \\int _{\\Lambda }[E(t,e)X({t-})+h(t,e)]\\tilde{\\mu }(de,dt),\\tau \\le t\\le T,}\\\\x(\\tau ) & = & \\xi ,\\end{array}\\right.$ where the coefficients satisfy the following basic assumption: Assumption 2.1 The matrix-valued processes $A:[0,T]\\times \\Omega \\rightarrow \\mathbb {R}^{n\\times n},B:[0,T]\\times \\Omega \\rightarrow \\mathbb {R}^{n\\times m};C^{i}:[0,T]\\times \\Omega \\rightarrow \\mathbb {R}^{n\\times n},i=1,2,\\cdots ,d$ are uniformly bounded and ${F}$ -predictable.",
"The matrix process $E:[0,T]\\times \\Omega \\times \\Lambda \\rightarrow \\mathbb {R}^{n\\times n}$ is uniformly bounded and ${\\tilde{P}}$ -measurable.",
"The stochastic processes $f(\\cdot ),g^{i}(\\cdot )$ belong to ${\\cal M}_{{F}}^{2,p}(0,T;\\mathbb {R}^{n})$ and $h(\\cdot ,\\cdot )$ belongs to ${\\cal M}_{{F}}^{\\nu ,p}(0,T;\\mathbb {R}^{n}).$ The following classical estimate could be found in lots of literature (see [23], [17]), the proof based on the Itô formula, Gronwall's inequality and BDG inequality is standard.",
"Lemma 2.3 Let Assumptions REF be satisfied.",
"Then the SDE (REF ) has a unique strong solution $X(\\cdot )\\in {\\cal S}_{{\\cal F}}^{p}(\\tau ,T;\\mathbb {R}^{n})$ and there is a constant $C_{p}>0$ such that for any stopping time $\\tau <T$ , $\\begin{split}\\mathbb {E}^{{F}_{\\tau }}\\bigg [\\sup _{\\tau \\le t\\le T}|X(t)|^{p}\\bigg ]\\le C_{p}\\mathbb {E}^{{F}_{\\tau }}\\bigg [|\\xi |^{p}+\\bigg (\\int _{\\tau }^{T}|f(t)|^{2}dt\\bigg )^{\\frac{p}{2}}+\\bigg (\\int _{\\tau }^{T}\\sum _{i=1}^{d}|g^{i}(t)|^{2}dt\\bigg )^{\\frac{p}{2}}+\\int _{\\tau }^{T}\\int _{\\Lambda }|h(t,e)|^{p}\\nu (de)dt\\bigg ].\\end{split}$"
],
[
"Formulation on SLQ Problem",
"In this section, we formulate the SLQ problem with jumps.",
"We first give the following definition of admissible control.",
"Definition 2.5 Let $\\tau \\in {T}.$ An ${F}$ -predictable process $u(\\cdot )$ is said to be an admissible control on the random interval $[\\!",
"[\\tau ,T]\\!",
"],$ if $u(\\cdot )\\in {\\cal M}_{{F}}^{2}(\\tau ,T;\\mathbb {R}^{m}).$ The set of all admissible control is denoted by ${U}_{\\tau }$ For any given admissible control $u(\\cdot )\\in {U}_{0}$ , consider the following controlled linear SDE with jumps: $\\left\\lbrace \\begin{array}{lll}dX(t) & = & [A(t)X(t-)+B(t)u(t)]dt+{\\displaystyle \\sum _{i=1}^{d}[C^{i}(t)X(t-)+D^{i}(t)u(t)]dW^{i}(t)}\\\\& & +{\\displaystyle \\int _{\\Lambda }[E(t,e)X({t-})+F(t,e)u(t)]\\tilde{\\mu }(de,dt),}\\\\X(0) & = & x\\end{array}\\right.$ with the cost functional $\\begin{split}J(u(\\cdot );0,x):=\\mathbb {E}\\bigg [\\langle MX(T),X(T)\\rangle +\\int _{0}^{T}\\big (\\langle Q(t)X(t),X(t)\\rangle +\\langle N(t)u(t),u(t)\\rangle \\big )dt\\bigg ].\\end{split}$ Here $A,B,C,D,E,F,Q,N$ and $M$ are given random mappings such that $A:[0,T]\\times \\Omega \\rightarrow \\mathbb {R}^{n\\times n};B:[0,T]\\times \\Omega \\rightarrow \\mathbb {R}^{n\\times m};C^{i}:[0,T]\\times \\Omega \\rightarrow \\mathbb {R}^{n\\times n},D^{i}:[0,T]\\times \\Omega \\rightarrow \\mathbb {R}^{n\\times m},i=1,2,\\cdots ,d;E:[0,T]\\times \\Omega \\times \\Lambda \\rightarrow \\mathbb {R}^{n\\times n};F:[0,T]\\times \\Omega \\times \\Lambda \\rightarrow \\mathbb {R}^{n\\times m};Q:[0,T]\\times \\Omega \\rightarrow \\mathbb {R}^{n\\times n},N:[0,T]\\times \\Omega \\rightarrow \\mathbb {R}^{m\\times m};M:\\Omega \\rightarrow \\mathbb {R}^{n\\times n}$ satisfying Assumption REF .",
"By Lemma REF , for any $u(\\cdot )\\in {{U}}_{0},$ it follows that the SDE (REF ) admits a unique strong solution in the space ${\\cal S}_{{F}}^{2}(0,T;\\mathbb {R}^{n})$ , denoted by $X^{0,x;u(\\cdot )}(\\cdot )$ .",
"We call $X(\\cdot )\\triangleq X^{0,x;u(\\cdot )}(\\cdot )$ the state process corresponding to the control process $u(\\cdot )$ and call $(u(\\cdot );X(\\cdot ))$ the admissible pair.",
"Furthermore, Assumption REF and the a priori estimate (REF ) imply that $|J(u(\\cdot );0,x)|<\\infty .$ Then our SLQ problem can be stated as follows.",
"Problem 2.4 Find an admissible control process ${\\bar{u}}(\\cdot )\\in {{U}}_{0}$ such that $J({\\bar{u}}(\\cdot );0,x)=\\inf _{u(\\cdot )\\in {{U}}_{0}}J(u(\\cdot );0,x).$ The admissible control ${\\bar{u}}(\\cdot )$ satisfying (REF ) is called an optimal control process of Problem REF .",
"Correspondingly, the state process ${\\bar{X}}(\\cdot )$ associated with ${\\bar{u}}(\\cdot )$ is called an optimal state process and $({\\bar{u}}(\\cdot );{\\bar{X}}(\\cdot ))$ is called an optimal pair of Problem REF ."
],
[
"Initial-Data-Parameterized SLQ Problem",
"This subsection is devoted to introducing the initial-data-parameterized SLQ Problem.",
"For simplicity, we define the random function $f(t,x,u):=\\langle Q(t)x,x\\rangle +\\langle N(t)u,u\\rangle ,\\quad \\forall (t,x,u)\\in [0,T]\\times \\mathbb {R}^{n}\\times \\mathbb {R}^{m}.$ Fixed initial data $(\\tau ,\\xi )\\in {T}\\times L^{2}(\\Omega ,{{F}_{\\tau }},\\mathbb {P};\\mathbb {R}^{n}),$ for any given admissible control $u(\\cdot )\\in {U}_{\\tau },$ denote by $X^{\\tau ,\\xi ;u}$ the solution of following state equation $\\left\\lbrace \\begin{array}{lll}dX(t) & = & [A(t)X(t-)+B(t)u(t)]dt+{\\displaystyle \\sum _{i=1}^{d}[C^{i}(t)X(t-)+D^{i}(t)u(t)]dW^{i}(t)}\\\\& & +{\\displaystyle \\int _{\\Lambda }[E(t,e)X({t-})+F(t,e)u(t)]\\tilde{\\mu }(de,dt),}\\\\X(\\tau ) & = & \\xi .\\end{array}\\right.$ The cost functional is defined as the following conditional expectation: $J(u(\\cdot );\\tau ,\\xi ):=\\mathbb {E}^{{F}_{\\tau }}{\\displaystyle \\bigg [{\\displaystyle \\int _{\\tau }^{T}f(s,X^{\\tau ,\\xi ;u(\\cdot )}(s),u(s))ds+\\langle MX^{\\tau ,\\xi ;u(\\cdot )}(T),X^{\\tau ,\\xi ;u(\\cdot )}(T)\\rangle \\bigg ].",
"}}$ Then the corresponding initial-data-parameterized SLQ Problem is stated as follows : Problem 3.1 Find an admissible control process ${\\bar{u}}(\\cdot )\\in {U}_{\\tau }$ such that $J({\\bar{u}}(\\cdot );\\tau ,\\xi )=\\operatornamewithlimits{ess\\,inf}_{u(\\cdot )\\in {U}_{\\tau }}J(u(\\cdot );\\tau ,\\xi ).$ We also denote the above optimal control problem by Problem ${P}_{\\tau ,\\xi }$ to stress the dependence on the parameter $(\\tau ,\\xi ).$ Clearly, for any initial data $(\\tau ,\\xi )\\in {T}\\times L^{2}(\\Omega ,{{F}_{\\tau }},\\mathbb {P};\\mathbb {R}^{n})$ and admissible control $u(\\cdot )\\in {U}_{\\tau },$ the state equation (REF ) has a unique strong solution $X(\\cdot )\\equiv X^{\\tau ,\\xi ;u(\\cdot )}$ and (REF ) is well-defined.",
"Furthermore, we can define the following conditional minimal value system ${V}(\\tau ,\\xi ):=\\operatornamewithlimits{ess\\,inf}_{u(\\cdot )\\in {U}_{\\tau }}J(u(\\cdot );\\tau ,\\xi ).$ It is obvious that ${V}(\\tau ,\\xi )$ is ${F}_{\\tau }$ -measurable random variable for any $(\\tau ,\\xi )\\in {T}\\times L^{2}(\\Omega ,{{F}_{\\tau }},\\mathbb {P};\\mathbb {R}^{n})$ .",
"The random variable ${V}(\\tau ,\\xi )$ will play an important role in the dynamic programming principle method to obtain the existence of the solution of the BSREJ (REF ).",
"The following two results Proposition REF and Theorem REF are needed in our approach.",
"The description and their proofs are more or less standard in the context of SLQ problem.",
"We just give a sketch of the proof in the case of jumps since it is similar to that in the case of Brownian motion.",
"We suggest the reader to visit Sections 2 and 3 in [26] for full details.",
"Proposition 3.2 Let Assumption REF hold.",
"(i) There is a positive constant $\\lambda $ such that for any $(\\tau ,\\xi )\\in {T}\\times L^{2}(\\Omega ,{{F}_{\\tau }},\\mathbb {P};\\mathbb {R}^{n})$ , it has $0\\le {V}(\\tau ,\\xi )\\le J(0;\\tau ,\\xi )\\le \\lambda |\\xi |^{2}.$ (ii) For any given initial data $(\\tau ,\\xi )\\in {T}\\times L^{2}(\\Omega ,{{F}_{\\tau }},\\mathbb {P};\\mathbb {R}^{n}),$ Problem ${P}_{\\tau ,\\xi }$ has a unique optimal control $\\bar{u}(\\cdot )\\in {U}_{\\tau }$ , i.e.",
"${V}(\\tau ,\\xi )=J(\\bar{u}(\\cdot );\\tau ,\\xi ),\\,\\mathbb {P}\\text{-a.s.}$ (iii) The value functional ${V}(\\tau ,\\xi )$ is quadratic with respect to $\\xi .$ Moreover, there is an $\\mathbb {S}_{+}^{n}$ -valued family ${K}:=\\lbrace {K}(\\tau ),\\tau \\in {{T}}\\rbrace $ such that ${K}(\\tau )$ is essentially bounded for any $\\tau \\in {T}$ and $\\xi \\in L^{2}(\\Omega ,{F}_{\\tau },\\mathbb {P};\\mathbb {R}^{n})$ ${\\text{$\\mathbb {V}$}}(\\tau ,\\xi )=\\langle {K}(\\tau )\\xi ,\\xi \\rangle .$ (iv) For each $x\\in \\mathbb {R}^{n},$ define the family ${V}_{x}:=\\lbrace {V}(\\tau ,x),\\tau \\in {T}\\rbrace .$ Then it is a ${T}$ -system.",
"Moreover, the family ${K}=\\lbrace {K}(\\tau ),\\tau \\in {{T}}\\rbrace $ is also a ${T}$ -system.",
"(i) Noting Assumption REF and (REF ), it is sufficient to show $J(0;\\tau ,\\xi )\\le \\lambda |\\xi |^{2}.$ In fact, from the a priori estimate (REF ), we get that $\\begin{split}J(0;\\tau ,\\xi ) & \\le C\\mathbb {E}^{{F}_{\\tau }}{\\displaystyle \\bigg [{\\displaystyle \\int _{\\tau }^{T}|X^{\\tau ,\\xi ;0}(t)|^{2}dt+|X^{\\tau ,\\xi ;0}(T)|^{2}\\bigg ]}}\\\\& \\le C\\mathbb {E}^{{F}_{\\tau }}\\bigg [\\sup _{\\tau \\le t\\le T}|X^{\\tau ,\\xi ;0}(t)|^{2}\\bigg ]\\\\& \\le C|\\xi |^{2}.\\end{split}$ (ii) Let $(\\tau ,\\xi )\\in {T}\\times L^{2}(\\Omega ,{{F}_{\\tau }},\\mathbb {P};\\mathbb {R}^{n})$ .",
"For any $u_{1}(\\cdot ),u_{2}(\\cdot )\\in {U}_{\\tau }$ , define $\\hat{u}(\\cdot ):=u_{1}(\\cdot )\\chi _{\\lbrace J(u_{1}(\\cdot );\\tau ,\\xi )\\le J(u_{2}(\\cdot );\\tau ,\\xi )\\rbrace }+u_{2}(\\cdot )\\chi _{\\lbrace J(u_{1}(\\cdot );\\tau ,\\xi )>J(u_{2}(\\cdot );\\tau ,\\xi )\\rbrace }.$ Then $X^{\\tau ,\\xi ;\\hat{u}(\\cdot )}=X^{\\tau ,\\xi ;u_{1}(\\cdot )}\\chi _{\\lbrace J(u_{1}(\\cdot );\\tau ,\\xi )\\le J(u_{2}(\\cdot );\\tau ,\\xi )\\rbrace }+X^{\\tau ,\\xi ;u_{2}(\\cdot )}\\chi _{\\lbrace J(u_{1}(\\cdot );\\tau ,\\xi )>J(u_{2}(\\cdot );\\tau ,\\xi )\\rbrace }.$ Hence $J(\\hat{u}(\\cdot );\\tau ,\\xi )= & J(u_{1}(\\cdot );\\tau ,\\xi )\\chi _{\\lbrace J(u_{1}(\\cdot );\\tau ,\\xi )\\le J(u_{2}(\\cdot );\\tau ,\\xi )\\rbrace }+J(u_{2}(\\cdot );\\tau ,\\xi )\\chi _{\\lbrace J(u_{1}(\\cdot );\\tau ,\\xi )>J(u_{2}(\\cdot );\\tau ,\\xi )\\rbrace }\\\\= & \\min \\lbrace J(u_{1}(\\cdot );\\tau ,\\xi ),J(u_{2}(\\cdot );\\tau ,\\xi )\\rbrace .$ That is $\\lbrace J(u(\\cdot );\\tau ,\\xi ):u(\\cdot )\\in {U}_{\\tau }\\rbrace $ is closed under pairwise minimum.",
"By Lemma REF , there is a sequence $\\lbrace u_{k}(\\cdot )\\rbrace _{k=1}^{\\infty }\\subset {U}_{\\tau }$ , such that $J(u_{k}(\\cdot );\\tau ,\\xi )\\searrow {V}(\\tau ,\\xi ),\\quad {\\rm as}\\,k\\rightarrow \\infty .$ By the parallelogram equality, $2J\\big (\\frac{1}{2}(u_{k}(\\cdot )-u_{l}(\\cdot ));\\tau ,\\xi \\big )+2{V}(\\tau ,\\xi )\\le & 2J\\big (\\frac{1}{2}(u_{k}(\\cdot )-u_{l}(\\cdot ));\\tau ,\\xi \\big )+2J\\big (\\frac{1}{2}(u_{k}(\\cdot )+u_{l}(\\cdot ));\\tau ,\\xi \\big )\\\\= & J(u_{k}(\\cdot );\\tau ,\\xi )+J(u_{l}(\\cdot );\\tau ,\\xi ).$ Let $k,l\\rightarrow \\infty $ in the following inequality, $0\\le 2J\\big (\\frac{1}{2}(u_{k}(\\cdot )-u_{l}(\\cdot ));\\tau ,\\xi \\big )\\le J(u_{k}(\\cdot ),\\tau ,\\xi )+J(u_{l}(\\cdot ),\\tau ,\\xi )-2{V}(\\tau ,\\xi )\\rightarrow 0,$ which means $\\lbrace u_{k}(\\cdot )\\rbrace _{k=1}^{\\infty }$ is Cauchy sequence in ${\\cal M}_{{F}}^{2}(\\tau ,T;\\mathbb {R}^{m})$ .",
"And it is easy to check that $\\bar{u}(\\cdot ):=\\lim _{k\\rightarrow \\infty }u_{k}(\\cdot )$ is the unique optimal control for problem ${P}_{\\tau ,\\xi }$ .",
"(iii) One can show that (see [6] or [26]), for any real number $\\eta >0$ , $x,y\\in \\mathbb {R}^{n}$ , ${V}(\\tau ,\\eta x) & =\\eta ^{2}{V}(\\tau ,x),\\\\{V}(\\tau ,x+y)+{V}(\\tau ,x-y) & =2{V}(\\tau ,x)+2{V}(\\tau ,y).$ So ${V}(\\tau ,x)$ is a quadratic form.",
"Let $\\begin{split}{K}(\\tau )=\\frac{1}{4}({V}(\\tau ,e_{i}+e_{j})-{V}(\\tau ,e_{i}-e_{j}))_{{i,j=1}}^{n},\\end{split}$ then we have (REF ).",
"(iv) Verifying Definition REF directly, we shall prove that ${V}_{x}$ is ${T}$ -system and and consequently so does ${K}$ ."
],
[
"Dynamical Programming Principle and the Semimartingale Property",
"The following result is the dynamical programming principle for Problem ${P}_{\\tau ,\\xi }.$ Theorem 3.3 Let Assumption REF hold.",
"(i) For $\\tau \\in {T},\\sigma \\in {T}_{\\tau },$ and $\\xi \\in L^{2}(\\Omega ,{F}_{\\tau },\\mathbb {P};\\mathbb {R}^{n}),$ $\\begin{split}{V}(\\tau ,\\xi )=\\operatornamewithlimits{ess\\,inf}_{u(\\cdot )\\in {{U}}_{\\tau }}\\mathbb {E}^{{F}_{\\tau }}\\bigg [\\int _{\\tau }^{\\sigma }f(s,X^{\\tau ,\\xi ;u(\\cdot )}(s),u(s))ds+{V}(\\sigma ,X^{\\tau ,\\xi ;u(\\cdot )}(\\sigma ))\\bigg ].\\end{split}$ And it holds that $\\begin{split}{V}(\\tau ,\\xi )=\\mathbb {E}^{{F}_{\\tau }}\\bigg [\\int _{\\tau }^{\\sigma }f(s,X^{\\tau ,\\xi ;\\bar{u}(\\cdot )}(s),\\bar{u}(s))ds+{V}(\\sigma ,X^{\\tau ,\\xi ;\\bar{u}(\\cdot )}(\\sigma ))\\bigg ]\\end{split}$ for the optimal control $\\bar{u}(\\cdot )\\in {U}_{\\tau }$ of Problem ${P}_{\\tau ,\\xi }$ .",
"(ii) For any $\\tau \\in {T}$ and $(x,u(\\cdot ))\\in \\mathbb {R}^{n}\\times {U}_{\\tau },$ the family ${J}^{\\tau ,x,u(\\cdot )}:=\\lbrace {J}^{\\tau ,x,u(\\cdot )}(\\sigma ),\\sigma \\in {T}_{\\tau }\\rbrace $ is a ${T}$ -submartingale, where ${{J}}^{\\tau ,x,u(\\cdot )}(\\sigma ):={V}(\\sigma ,X^{\\tau ,x;u(\\cdot )}(\\sigma ))+\\int _{\\tau }^{\\sigma }f(r,X^{\\tau ,x;u(\\cdot )}(r),u(r))dr,\\quad \\sigma \\in {T}_{\\tau };$ And the family ${J}^{\\tau ,x,\\bar{u}(\\cdot )}$ is a ${T}$ -martingale for the optimal control $\\bar{u}(\\cdot )\\in {U}_{\\tau }$ of problem ${P}_{\\tau ,x}$ .",
"Besides, ${{J}}^{\\tau ,x,u(\\cdot )}(\\sigma )=\\operatornamewithlimits{ess\\,inf}_{v(\\cdot )\\in {U}_{\\sigma }^{u(\\cdot )}}\\mathbb {E}^{{F}_{\\sigma }}\\bigg [\\int _{\\tau }^{T}f(r,X^{\\tau ,x;v(\\cdot )}(r),v(r))+\\langle MX^{\\tau ,x;v(\\cdot )}(T),X^{\\tau ,x;v(\\cdot )}(T)\\rangle \\bigg ],\\:u\\in {U}_{\\tau },$ where ${U}_{\\sigma }^{u(\\cdot )}:=\\big \\lbrace v(\\cdot )\\in {U}_{\\tau }|v(\\cdot )=u(\\cdot )~{\\rm on}~[\\!",
"[\\tau ,\\sigma ]\\!",
"]\\big \\rbrace .$ (iii) If $\\bar{u}(\\cdot )\\in {{U}}_{\\tau }$ such that ${J}^{\\tau ,x,\\bar{u}(\\cdot )}$ is a ${T}$ -martingale, then $\\bar{u}(\\cdot )$ is optimal for Problem ${P}_{\\tau ,x}.$ (i) Similar as (REF ), there is a minimizing sequence $\\lbrace v_{m}(\\cdot )\\rbrace \\subset {U}_{\\sigma }$ of Problem ${P}_{\\sigma ,X^{\\tau ,\\xi ;u(\\cdot )}(\\sigma )}$ such that, then we have for any $v(\\cdot )\\in {U}_{\\sigma }$ , $\\mathbb {E}^{{F}_{\\tau }}\\Big [J(v(\\cdot );\\sigma ,X^{\\tau ,\\xi ;u(\\cdot )}(\\sigma ))\\Big ]\\ge & \\mathbb {E}^{{F}_{\\tau }}\\Big [{V}(\\sigma ,X^{\\tau ,\\xi ;u(\\cdot )}(\\sigma ))\\Big ]\\\\= & \\mathbb {E}^{{F}_{\\tau }}\\Big [\\inf _{m}J(v_{m}(\\cdot );\\sigma ,X^{\\tau ,\\xi ;u(\\cdot )}(\\sigma ))\\Big ]\\\\= & \\operatornamewithlimits{ess\\,inf}_{m}\\mathbb {E}^{{F}_{\\tau }}\\Big [J(v_{m}(\\cdot );\\sigma ,X^{\\tau ,\\xi ;u(\\cdot )}(\\sigma ))\\Big ]\\\\\\ge & \\operatornamewithlimits{ess\\,inf}_{v(\\cdot )\\in {U}_{\\sigma }}\\mathbb {E}^{{F}_{\\tau }}\\Big [J(v(\\cdot );\\sigma ,X^{\\tau ,\\xi ;u(\\cdot )}(\\sigma ))\\Big ].$ Taking $\\operatornamewithlimits{ess\\,inf}_{v(\\cdot )\\in {U}_{\\sigma }}$ on the left hand side of above inequality, then the inequalities turn to equalities.",
"We have $\\operatornamewithlimits{ess\\,inf}_{v(\\cdot )\\in {U}_{\\sigma }}\\mathbb {E}^{{F}_{\\tau }}\\Big [J(v(\\cdot );\\sigma ,X^{\\tau ,\\xi ;u(\\cdot )}(\\sigma ))\\Big ]=\\mathbb {E}^{{F}_{\\tau }}\\Big [{V}(\\sigma ,X^{\\tau ,\\xi ;u(\\cdot )}(\\sigma ))\\Big ].$ Furthermore for any $u(\\cdot )\\in {U}_{\\tau }$ , $& \\mathbb {E}^{{F}_{\\tau }}\\Big [\\int _{\\tau }^{\\sigma }f(s,X^{\\tau ,\\xi ;u(\\cdot )}(s),u(s))ds+{V}(\\sigma ,X^{\\tau ,\\xi ;u(\\cdot )}(\\sigma ))\\Big ]\\\\= & \\operatornamewithlimits{ess\\,inf}_{v(\\cdot )\\in {U}_{\\sigma }}\\mathbb {E}^{{F}_{\\tau }}\\Big [\\int _{\\tau }^{\\sigma }f(s,X^{\\tau ,\\xi ;u(\\cdot )}(s),u(s))ds+J(v(\\cdot );\\sigma ,X^{\\tau ,\\xi ;u(\\cdot )}(\\sigma ))\\Big ]\\\\= & \\operatornamewithlimits{ess\\,inf}_{v(\\cdot )\\in {U}_{\\sigma }}\\mathbb {E}^{{F}_{\\tau }}\\Big [J(u(\\cdot )\\otimes v(\\cdot );\\tau ,\\xi )\\Big ],$ where $u(\\cdot )\\otimes v(\\cdot )=u(\\cdot )$ on $[\\!",
"[\\tau ,\\sigma ]\\!",
"]$ , and $u(\\cdot )\\otimes v(\\cdot )=v(\\cdot )$ on $[\\!",
"[\\sigma ,T]\\!",
"]$ .",
"(REF ) is the result of taking $\\operatornamewithlimits{ess\\,inf}_{u(\\cdot )\\in {U}_{\\tau }}$ on both sides of above equality.",
"If $\\bar{u}(\\cdot )\\in {U}_{\\tau }$ is the optimal control for ${P}_{\\tau ,\\xi }$ , then its restriction $\\bar{u}\\big |_{[\\!",
"[\\sigma ,T]\\!",
"]}(\\cdot )$ is the optimal control for ${P}_{\\tau ,X^{\\tau ,\\xi ;\\bar{u}(\\cdot )}(\\sigma )}$ .",
"Then (REF ) follows.",
"Then assertion (i) holds.",
"In view of (i), it is easy to check that (ii) and (iii) hold.",
"Lemma 3.4 Let Assumptions REF be satisfied.",
"Then for each $x\\in \\mathbb {R}^{n},$ the ${T}$ -systems ${V}_{x}$ and ${K}=\\lbrace {K}(\\tau ),\\tau \\in {{T}}\\rbrace $ are RCE.",
"For any $\\tau \\in {T}_{0}$ , $\\tau _{m}\\in {T}_{\\tau }$ satisfying that $\\tau _{m}\\searrow \\tau $ a.s. as $m\\rightarrow \\infty $ .",
"By (i) of Theorem REF , for the optimal control $\\bar{u}(\\cdot )$ of Problem ${P}_{\\tau ,x}$ , ${V}(\\tau ,x)=\\mathbb {E}^{{F}_{\\tau }}\\bigg [\\int _{\\tau }^{\\tau _{m}}f(s,X^{\\tau ,x;\\bar{u}(\\cdot )}(s),\\bar{u}(s))ds+{V}(\\tau _{m},X^{\\tau ,x;\\bar{u}(\\cdot )}(\\tau _{m}))\\bigg ].$ Since ${K}$ is uniformly bounded, $& \\mathbb {E}^{{F}_{\\tau }}\\text{$\\bigg [$}\\Big |{V}(\\tau _{m},X^{\\tau ,x;\\bar{u}(\\cdot )}(\\tau _{m}))-{V}(\\tau _{m},x)\\Big |\\bigg ]\\\\= & \\mathbb {E}^{{F}_{\\tau }}\\bigg [\\Big |\\big \\langle {K}(\\tau _{m})X^{\\tau ,x;\\bar{u}(\\cdot )}(\\tau _{m}),X^{\\tau ,x;\\bar{u}(\\cdot )}(\\tau _{m})\\big \\rangle -\\big \\langle {K}(\\tau _{m})x,x\\big \\rangle \\Big |\\bigg ]\\\\\\le & \\lambda \\bigg (\\mathbb {E}^{{F}_{\\tau }}\\Big [|x|+|X^{\\tau ,x;\\bar{u}(\\cdot )}(\\tau _{m})|\\Big ]^{2}\\bigg )^{\\frac{1}{2}}\\bigg (\\mathbb {E}^{{F}_{\\tau }}\\big [X^{\\tau ,x;\\bar{u}(\\cdot )}(\\tau _{m})-x\\big ]^{2}\\bigg )^{\\frac{1}{2}},$ and $\\mathbb {E}^{{F}_{\\tau }}\\bigg [\\int _{\\tau }^{\\tau _{m}}f(s,X^{\\tau ,x;\\bar{u}(\\cdot )}(s),\\bar{u}(s))ds\\bigg ]\\le C\\mathbb {E}^{{F}_{\\tau }}\\bigg [\\int _{\\tau }^{\\tau _{m}}\\big (\\big |X^{\\tau ,x;\\bar{u}(\\cdot )}(s)\\big |^{2}+|\\bar{u}(s)|^{2}\\big )ds\\bigg ],$ Then by (REF ), the estimate (REF ) and the dominate control theorem, we get $& \\mathbb {E}^{{F}_{\\tau }}\\bigg [\\big |{V}(\\tau ,x)-{V}(\\tau _{m},x)\\big |\\bigg ]\\\\\\le & \\mathbb {E}^{{F}_{\\tau }}\\Big [\\int _{\\tau }^{\\tau _{m}}|f(s,X^{\\tau ,x;\\bar{u}}(s),\\bar{u}(s))ds|+\\Big |{V}(\\tau _{m},X^{\\tau ,x;\\bar{u}}(\\tau _{m}))-{V}(\\tau _{m},x)\\Big |\\Big ]\\\\\\rightarrow & 0,\\qquad {\\rm as}\\:m\\rightarrow \\infty .$ This is the RCE of ${V}_{x}$ .",
"The RCE of ${K}$ is a direct inference of that of ${V}_{x}$ and (REF ).",
"Theorem 3.5 Let Assumptions REF be satisfied.",
"(i) For any $\\tau \\in {T}$ and $(x,u(\\cdot ))\\in \\mathbb {R}^{n}\\times {U}_{\\tau },$ the ${T}_{\\tau }$ -system ${J}^{\\tau ,x,u(\\cdot )}$ is RCE and aggregated by a RCLL ${F}$ -submartingale denoted by $\\lbrace \\mathbb {J}^{\\tau ,x,u(\\cdot )}(t),t\\in [\\!",
"[\\tau ,T]\\!",
"]\\rbrace $ .",
"For the optimal control $\\bar{u}(\\cdot )\\in {U}_{\\tau }$ of Problem ${P}_{\\tau ,x},$ the corresponding ${T}_{\\tau }$ -system ${J}^{\\tau ,x,\\bar{u}(\\cdot )}$ is aggregated by a RCLL ${F}$ -martingale denoted by $\\lbrace \\mathbb {J}^{\\tau ,x,\\bar{u}(\\cdot )}(t),t\\in [\\!",
"[\\tau ,T]\\!",
"]\\rbrace $ .",
"(ii) The ${T}$ -system $\\lbrace {K}(\\tau ),\\tau \\in {T}\\rbrace $ is RCE and aggregated by a RCLL process denoted by $\\lbrace K(t),t\\in [0,T]\\rbrace .$ $K$ is essentially bounded and $\\mathbb {S}_{+}^{n}$ -valued.",
"We have for any $t\\in [0,T]$ $\\begin{split}K(t)=K(0)-\\int _{0}^{t}dk(s)+\\sum _{i=1}^{d}\\int _{0}^{t}L^{i}(s)dW_{s}^{i}+\\int _{0}^{t}\\int _{\\Lambda }R(s,e)d\\tilde{\\mu }(de,ds),\\quad K(T)=M,\\end{split}$ where $k$ is an $\\mathbb {S}^{n}$ -valued predictable process of bounded variation, $L^{i}$ an $\\mathbb {S}^{n}$ -valued predictable process and $R$ a $\\tilde{{P}}$ -measurable process.",
"(iii) The condition minimal value system ${V}_{x}$ for $x\\in \\mathbb {R}^{n}$ is aggregated by the following RCLL semimartingale $V(t,x):=\\langle K(t)x,x\\rangle ,\\quad t\\in [0,T].$ In view of (REF ), the REC of family ${J}^{\\tau ,x,u(\\cdot )}$ comes from that of the ${T}$ -system ${V}_{x}$ and the a.s. right continuity of maps $t\\mapsto X_{t}^{\\tau ,x,u(\\cdot )}$ and $t\\mapsto \\int _{\\tau }^{t}f(s,X(s),u(s))ds$ .",
"Using Proposition REF , we prove the first part of assertion (i).",
"From the second part of Theorem REF , we see that ${J}^{\\tau ,x,\\bar{u}(\\cdot )}$ is a ${F}$ -martingale.",
"Now we begin to show the assertion (ii).",
"Denote by $\\tau _{k}$ the $n$ -th jump time of the Poisson point process.",
"Recall that $e_{i}$ is the unit column vector whose $i$ -th component is the number 1 for $i=1,\\cdots ,n.$ We see that for $x=e_{i},e_{i}+e_{j},e_{i}-e_{j}$ with $i,j=1,\\cdots ,n$ , the process $\\mathbb {J}^{k,x}(t):={{J}}^{\\tau _{k}\\wedge T,x,0}(t),t\\in [\\!",
"[\\tau _{k}\\wedge T,T]\\!",
"]$ is a right-continuous submartingale and $\\mathbb {J}^{k,x}(\\cdot )-\\mathbb {J}^{k,x}(\\tau _{k}\\wedge T)$ is of class D. Hence by Doob-Meyer decomposition (see [23]), it could be decomposed to an increasing, predictable process and a uniformly integrable martingale.",
"Consider an $\\mathbb {S}^{n}$ -valued ${T}_{\\tau _{k}\\wedge T}$ -system $\\Gamma _{k}:=\\lbrace \\Gamma _{k}(\\tau ),\\tau \\in {T}_{\\tau _{k}\\wedge T}\\rbrace $ defined as follows: $\\Gamma _{k}(\\tau ):=\\frac{1}{4}\\Big ({J}^{\\tau _{k}\\wedge T,e_{i}+e_{j},0}(\\tau )-{J}^{\\tau _{k}\\wedge T,e_{i}-e_{j},0}(\\tau )\\Big )_{1\\le i,j\\le n},\\quad \\tau \\in {T}_{\\tau _{k}\\wedge T}.$ In view of $X^{\\tau _{k}\\wedge T,e_{i}\\pm e_{j},0}(\\tau )=X^{\\tau _{k}\\wedge T,e_{i},0}(\\tau )\\pm X^{\\tau _{k}\\wedge T,e_{j},0}(\\tau )$ , (REF ) and the proof of (REF ), we have $& \\big ({V}(\\tau ,X^{\\tau _{k}\\wedge \\tau ,e_{i}+e_{j},0}(\\tau ))-{V}(\\tau ,X^{\\tau _{k}\\wedge \\tau ,e_{i}-e_{j},0}(\\tau ))\\big )_{1\\le i,j\\le n}\\\\= & \\big (X^{\\tau _{k}\\wedge \\tau ,e_{1},0}(\\tau ),\\ldots ,X^{\\tau _{k}\\wedge \\tau ,e_{n},0}(\\tau )\\big )^{*}{K}(\\tau )\\big (X^{\\tau _{k}\\wedge \\tau ,e_{1},0}(\\tau ),\\ldots ,X^{\\tau _{k}\\wedge \\tau ,e_{n},0}(\\tau )\\big ).$ This together with (REF ) and (REF ) yields $\\Gamma _{k}(\\tau )=\\Phi _{k}^{*}(\\tau ){K}(\\tau )\\Phi _{k}(\\tau )+\\int _{\\tau _{k}\\wedge T}^{\\tau }\\Phi _{k}^{*}(r)Q(r)\\Phi _{k}(r)dr,$ where $\\Phi _{k}(t)$ is the solution of the following linear SDE: $\\left\\lbrace \\begin{array}{l}d\\Phi (t)=A(t)\\Phi (t-)dt+\\sum _{i=1}^{d}C^{i}(t)\\Phi (t-)dW^{i}(t)+\\int _{\\Lambda }E(t,e)\\Phi ({t-})\\tilde{\\mu }(de,dt),\\\\\\Phi (\\tau _{k}\\wedge T)=I,\\qquad \\qquad t\\in (\\hspace{0.0pt}(\\tau _{k}\\wedge T,\\tau _{k+1}\\wedge T)\\hspace{0.0pt}).\\end{array}\\right.$ The ${T}_{\\tau _{k}\\wedge T}-$ system $\\Gamma _{k}$ is aggregated by the following process still denoted by $\\lbrace \\Gamma _{k}(t),t\\in [\\!",
"[\\tau _{k}\\wedge T,T)\\hspace{0.0pt})\\rbrace :$ $\\Gamma _{k}(t)=:\\frac{1}{4}(\\mathbb {J}^{k,e_{i}+e_{j}}(t)-\\mathbb {J}^{k,e_{i}-e_{j}}(t))_{1\\le i,j\\le n},t\\in [\\!",
"[\\tau _{k}\\wedge T,T)\\hspace{0.0pt}),$ which is a right-continuous semimartingale with predictable of bounded variational part.",
"We see that $\\Phi _{k}(t)$ is reversible for $t\\in (\\hspace{0.0pt}(\\tau _{k}\\wedge T,\\tau _{k+1}\\wedge T)\\hspace{0.0pt})$ brown and its inverse $\\Psi _{k}(t):=\\Phi _{k}^{-1}(t)$ satisfying $\\left\\lbrace \\begin{array}{l}d\\Psi _{k}(t)=\\Psi _{k}(t-)\\bigg [-A(t)+C^{2}(t)+\\int _{\\Lambda }E(t,e)\\nu (de)\\bigg ]dt-\\sum _{i=1}^{d}\\Psi _{k}(t-)C^{i}(t)dW^{i}(t),\\\\\\Psi _{k}(\\tau _{k}\\wedge T)=I,\\qquad \\qquad t\\in (\\hspace{0.0pt}(\\tau _{k}\\wedge T,\\tau _{k+1}\\wedge T)\\hspace{0.0pt}).\\end{array}\\right.$ It is obvious that $\\Psi _{k}(t)$ is continuous at $[\\!",
"[\\tau _{k}\\wedge T,\\tau _{k+1}\\wedge T)\\hspace{0.0pt})$ and has left-limit at $\\tau _{k+1}\\wedge T$ .",
"Define $K_{k}(t):=\\Psi _{k}^{*}(t)\\Gamma _{k}(t)\\Psi _{k}(t)-\\Psi _{k}^{*}(t)\\int _{\\tau _{k}\\wedge T}^{t}\\Phi _{k}^{*}(s)Q(s)\\Phi _{k}(s)ds\\Psi _{K}(t),\\,t\\in [\\!",
"[\\tau _{k}\\wedge T,\\tau _{k+1}\\wedge T)\\hspace{0.0pt}).$ It is continuous on $[\\!",
"[\\tau _{k}\\wedge T,\\tau _{k+1}\\wedge T)\\hspace{0.0pt})$ and has left-limit at $\\tau _{k+1}\\wedge T$ .",
"By Itô formula, $K_{k}$ is a semimartingale, i.e.",
"$K_{k}(t)=K_{k}(\\tau _{k}\\wedge T)+\\tilde{M}_{k}(t)+\\tilde{A}_{k}(t),\\quad t\\in [\\!",
"[\\tau _{k}\\wedge T,\\tau _{k+1}\\wedge T)\\hspace{0.0pt})$ where $\\tilde{M}_{k}$ with $\\tilde{M}_{k}(\\tau _{k}\\wedge T)=0$ is a local martingale and $\\tilde{A}$ with $\\tilde{A}(\\tau _{k}\\wedge T)=0$ a predictable process with finite variation.",
"We see that ${K}(\\tau )=K_{k}(\\tau )$ for $\\tau _{k}\\wedge T\\le \\tau <\\tau _{k+1}\\wedge T$ .",
"Thus ${K}$ is aggregated by the process $K(t):= & \\sum _{k=0}^{\\infty }K_{k}(t)\\chi _{\\lbrace \\tau _{k}\\wedge T\\le t<\\tau _{k+1}\\wedge T\\rbrace }\\\\= & \\Big (\\sum _{\\tau _{k+1}\\le t}\\tilde{M}_{k}\\big ((\\tau _{k+1}\\wedge T)-\\big )+\\tilde{M}_{i}(t)\\Big )+\\Big (\\sum _{\\tau _{k+1}\\le t}\\tilde{A}_{k}\\big ((\\tau _{k+1}\\wedge T)-\\big )+\\tilde{A}_{i}(t)\\Big )\\\\& \\quad +\\sum _{\\tau _{k}\\le t,k>1}\\big (K_{k}(\\tau _{k}\\wedge T)-K_{k-1}((\\tau _{k}\\wedge T)-)\\big ),$ where $i$ is the maximal integer with $\\tau _{i}\\le t$ .",
"It is easy to observe that the first term of the right hand of above equality is a continuous martingale, the second term is continuous bounded variational process, and the third term is a pure jump process.",
"By localizing method, it is easy to know the first part of last term is a local martingale, second part a finite variational predictable process.",
"According to $K_{k}$ is uniformly bounded, Theorem 35 in [23] yields the pure jump process $\\sum _{\\tau _{k}\\le t,k>1}\\big (K_{k}(\\tau _{k}\\wedge T)-K_{k-1}((\\tau _{k}\\wedge T)-)\\big )$ is a special semimartingale.",
"Thus $K$ could be canonically decomposed into the sum of an ${F}$ -predictable process $k_{t}$ with finite variation and an ${F}$ -martingale process on the whole time interval $[0,T]$ .",
"By martingale representation theorem (see [23] or [28] for a easier version), we know that $K$ can be written as (REF ).",
"At last, the assertion (iii) is just a result of (REF ).",
"Thus we finish the proof.",
"Remark 3.1 In [25], [26], the inverse flow of the controlled SDE is the key technique to show $K_{t}$ to be the fist part of the triple processes solution of BSRE.",
"And in [25], the author pays lots of calculus to prove that the inverse flow of the solution$X$ for SDE associated with the corresponding optimal control exists on the whole time interval.",
"For the SDE with jump, its inverse flow may not exists on whole time $[0,T]$ without additional condition, e.g., $I+E\\ge \\delta I,\\quad {\\rm a.e.,a.s.",
"}$ However, condition (REF ) is not necessary for the original control problem.",
"So we insist on not introducing the condition (REF ) in the formulation of our BSREJ.",
"We observe that in the form of optimal feedback (see (REF )), $K$ is independent of the state of the controlled equation, which hint us to represent $K$ by different state process in different time interval.",
"Hence to overcome the difficulty of absence of (REF ), we can piece-wisely represent $K$ by the inverse flow on sub-interval between two adjacent jump time, on which the SDE (REF ) has continuous trajectories hence an inverse flow.",
"After that we integrated the representation of $K$ from piece-wise to whole process on $[0,T]$ by the semimartingale property."
],
[
"Existence of Solutions to BSREJ",
"This section is devoted to showing that $(K,L,R)$ given by Theorem REF is nothing other than the solution of BSREJ (REF ), and to giving their estimates.",
"Thus we establish the existence of solution for BSREJ (REF ).",
"Theorem 4.1 Let Assumptions REF be satisfied.",
"Then $(K,L,R)$ given by Theorem REF satisfies BSREJ (REF ).",
"And there is a deterministic constant $C$ such that the following estimate holds: $\\mathbb {E}\\bigg (\\int _{0}^{T}\\sum _{i=1}^{d}\\big |L^{i}(t)\\big |^{2}ds\\bigg )+\\mathbb {E}\\bigg (\\int _{0}^{T}\\int _{\\Lambda }\\big |R(t,e)\\big |^{2}\\nu (de)dt\\bigg )\\le C.$ Hence ${\\displaystyle \\int _{0}}^{\\cdot }L^{i}(s)dW_{s}^{i}+{\\displaystyle \\int }_{0}^{\\cdot }\\int _{E}R(e,s)\\tilde{\\mu }(de,ds)$ is a BMO martingale.",
"Moreover ${\\displaystyle \\int }_{\\Lambda }F^{*}(t,e)(K(t-)+R(t,e))F(t,e)\\nu (de)$ is nonnegative for almost all $t$ , $P$ -a.s.. Firstly, we show that $(K,L,R)$ satisfies satisfies (REF ) a.e.a.s.",
"Define the functional $& \\mathbb {F}(t,x,u,K(t),L(t),R(t,\\cdot ))\\\\:= & 2\\langle K(t)x,A(t)x+B(t)u\\rangle +2\\sum _{i=1}^{d}\\langle L^{i}(t)x,C^{i}(t)x+D^{i}(t)u\\rangle +\\sum _{i=1}^{d}\\langle K(t)(C^{i}(t)x+D^{i}(t)u),C^{i}(t)x+D^{i}(t)u\\rangle \\\\& +2{\\displaystyle \\int _{\\Lambda }\\Big \\langle R(t,e)x,E(t,e)x+F(t,e)u\\Big \\rangle \\nu (de)}+{\\displaystyle \\int _{\\Lambda }\\Big \\langle \\big (K(t)+R(t,e)\\big )(E(t,e)x+F(t,e)u),E(t,e)x+F(t,e)u\\Big \\rangle \\nu (de).",
"}$ For $\\tau \\in {T}$ , $\\sigma \\in {T}_{\\tau }$ and $u(\\cdot )\\in {U}_{\\tau },$ applying ItÃŽ formula to $V(t,X^{\\tau ,x;u(\\cdot )}(t))=\\langle K(t)X^{\\tau ,x;u(\\cdot )}(t),X^{\\tau ,x;u(\\cdot )}(t)\\rangle $ , we get $\\begin{split} & V(\\sigma ,X^{\\tau ,x;u(\\cdot )}(\\sigma )))\\\\= & V(\\tau ,x)-\\int _{\\tau }^{\\sigma }\\langle dk(t)X,X\\rangle +\\int _{\\tau }^{\\sigma }\\mathbb {F}(t,X,u(t),K(t-),L(t),R(t,\\cdot ))dt\\\\& +\\sum _{i=1}^{d}\\int _{\\tau }^{\\sigma }\\bigg [\\langle L^{i}(t)X,X\\rangle +2\\langle K(t-)X,C^{i}(t)X+D^{i}(t)u(t)\\rangle \\bigg ]dW^{i}(t)\\\\& +\\int _{\\tau }^{\\sigma }\\int _{\\Lambda }\\langle R(t,e)(X+E(t,e)X+F(t,e)u(t)),X+E(t,e)X+F(t,e)u(t)\\rangle \\tilde{\\mu }(dt,de)\\\\& +\\int _{\\tau }^{\\sigma }\\int _{\\Lambda }\\langle K(t-)(E(t,e)X+F(t,e)u(t)),E(t,e)X+F(t,e)u(t)\\rangle \\tilde{\\mu }(dt,de)\\\\& +2\\int _{\\tau }^{\\sigma }\\int _{\\Lambda }\\langle K(t-)X,E(t,e)X+F(t,e)u(t)\\rangle \\tilde{\\mu }(dt,de),\\end{split}$ where $X$ is short for $X^{\\tau ,x;u(\\cdot )}(t-)$ .",
"Taking conditional expectation with ${F}_{\\tau }$ on both sides of the above relation and noting the fact that the conditional expectation of the stochastic integrals w.r.t.",
"the Brownian motion $W$ and the Poisson random measure $\\tilde{\\mu }$ vanishes by the localization with the stopping time, we obtain $\\begin{split} & \\mathbb {E}^{{F}_{\\tau }}[V(\\sigma ,X^{\\tau ,x;u(\\cdot )}(\\sigma ))]\\\\= & V(\\tau ,x)+\\mathbb {E}^{{F}_{\\tau }}\\bigg [\\int _{\\tau }^{\\sigma }\\Pi (t,X^{\\tau ,x;u(\\cdot )}(t-))dt\\bigg ]-\\mathbb {E}^{{F}_{\\tau }}\\bigg [\\int _{\\tau }^{\\sigma }f(t,X^{\\tau ,x;u(\\cdot )}(t-),u(t))dt\\bigg ],\\end{split}$ where $\\begin{split} & \\Pi (dt;\\tau ,x,u(\\cdot ))\\\\:= & -\\langle dk(t)X^{\\tau ,x;u(\\cdot )}(t-),X^{\\tau ,x;u(\\cdot )}(t-)\\rangle +\\mathbb {F}(t,X^{\\tau ,x;u(\\cdot )}(t-),u(t),K(t-),L(t),R(t,\\cdot ))dt\\\\& +f(t,X^{\\tau ,x;u(\\cdot )}(t-),u(t))dt.\\end{split}$ This implies that $\\begin{split} & \\mathbb {E}^{{F}_{\\tau }}\\bigg [\\int _{\\tau }^{\\sigma }\\Pi (dt;\\tau ,x,u(\\cdot ))dt\\bigg ]\\\\= & \\mathbb {E}^{{F}_{\\tau }}[V(\\sigma ,X^{\\tau ,x;u(\\cdot )}(\\sigma ))]+\\mathbb {E}^{{F}_{\\tau }}\\bigg [\\int _{\\tau }^{\\sigma }l(t,X^{\\tau ,x;u(\\cdot )}(t-),u(t))dt\\bigg ]-V(\\tau ,x).\\end{split}$ From the dynamic programming principle, we have $\\begin{split} & \\mathop {\\text{ess.",
"}\\inf }_{u(\\cdot )\\in {U}_{\\tau }}\\mathbb {E}^{{F}_{\\tau }}\\bigg [\\int _{\\tau }^{\\sigma }\\Pi (dt;\\tau ,x,u(\\cdot ))dt\\bigg ]\\\\= & \\mathop {\\text{ess.",
"}\\inf }_{u(\\cdot )\\in {U}_{\\tau }}\\bigg \\lbrace \\mathbb {E}^{{F}_{\\tau }}[V(\\sigma ,X^{\\tau ,x;u(\\cdot )}(\\sigma ))]+\\mathbb {E}^{{F}_{\\tau }}\\bigg [\\int _{\\tau }^{\\sigma }f(t,X^{\\tau ,x;u(\\cdot )}(t-),u(t))dt\\bigg ]\\bigg \\rbrace -V(\\tau ,x)\\\\= & V(\\tau ,x)-V(\\tau ,x)\\\\= & 0.\\end{split}$ Choose $\\tau _{k}$ as the $k$ -th jump time of the Poisson point process.",
"This implies that the measure $\\Pi (ds;\\tau _{k}\\wedge T,x,u(\\cdot ))dxdP$ is nonnegative on $\\lbrace (t,x,\\omega ):t\\in (\\tau _{k}(\\omega )\\wedge T,T],x\\in \\mathbb {R}^{n},\\omega \\in \\Omega \\rbrace $ for any $u(\\cdot )\\in {U}_{\\tau }.$ Therefore, for any essentially bounded nonnegative predictable field $\\eta $ defined on $[0,T]\\times \\mathbb {R}^{n}\\times \\Omega $ , we have $\\begin{split}\\mathbb {E}\\int _{\\tau _{k}\\wedge T}^{\\tau _{k+1}\\wedge T}\\int _{\\mathbb {R}^{n}}\\eta (s,X^{\\tau _{k}\\wedge T,x;u(\\cdot )}(s))\\det (\\Phi _{k}(s))\\Pi (ds;\\tau _{k}\\wedge T,x,u(\\cdot ))\\ge 0,\\quad \\forall u(\\cdot )\\in {U}_{0}\\end{split}$ with $\\Phi _{k}(s)$ being the Jacobian matrix of flow transformation $x\\longrightarrow X^{\\tau _{k}\\wedge T,x;u(\\cdot )}(s)$ for any $u(\\cdot )\\in U_{\\tau _{k}\\wedge T}$ .",
"Note that before the next jump time $\\tau _{k+1}$ , $\\Phi (s)$ is inversible, i.e., $\\det (\\Phi (s))>0$ $\\mathbb {P}$ -a.s.",
"Via a transformation of state variable $x$ , we have $\\begin{split}\\mathbb {E}\\int _{\\tau _{k}\\wedge T}^{\\tau _{k+1}\\wedge T}\\int _{\\mathbb {R}^{n}}\\eta (s,x)\\Pi (ds;\\tau _{k}\\wedge T,Y^{\\tau _{k}\\wedge T,x;u(\\cdot )}(s),u(\\cdot ))\\ge 0,\\quad \\forall u(\\cdot )\\in {U}_{0},\\end{split}$ where $Y^{\\tau _{k}\\wedge T,x;u(\\cdot )}(s)$ is the inverse of the flow $x\\longrightarrow X^{\\tau _{k}\\wedge T,x;u(\\cdot )}(s)$ for $\\tau _{k}\\wedge T\\le s<\\tau _{k+1}\\wedge T$ .",
"Incorporating $\\Pi (ds;0,\\cdot ,u(\\cdot ))\\ge 0$ with the inverse flow $Y^{\\tau _{k}\\wedge T,x;u(\\cdot )}(s),x\\in \\mathbb {R}^{n}$ we have $\\begin{split}0\\le & \\Pi (dt;\\tau _{k}\\wedge T,Y^{\\tau _{k}\\wedge T,x;u(\\cdot )}(t);u(\\cdot ))\\\\= & -\\langle dk(t)x,x\\rangle +\\mathbb {F}(t,x,u(t),K(t-),L(t),R(t,\\cdot ))dt\\\\& \\qquad +f(t,x,u(t))dt\\end{split}$ on $\\lbrace (t,\\omega ):t\\in (\\tau _{k}(\\omega )\\wedge T,\\tau _{k+1}(\\omega )\\wedge T),\\omega \\in \\Omega \\rbrace $ .",
"In a similar way, we have for a.e.",
"a.s. $(t,\\omega )\\in [0,T]\\times \\Omega ,$ $\\begin{split}0= & \\Pi (dt;\\tau _{k}\\wedge T,Y^{\\tau _{k}\\wedge T,x;\\bar{u}(\\cdot )}(t),\\bar{u}(\\tau _{k}\\wedge T,Y^{\\tau _{k}\\wedge T,x;\\bar{u}(\\cdot )}(t)))\\\\= & -\\langle dk(t)x,x\\rangle +\\mathbb {F}(t,x,\\bar{u}(\\tau _{k}\\wedge T,Y^{\\tau _{k}\\wedge T,x;\\bar{u}(\\cdot )}(t)),K(t-),L(t),R(t,\\cdot ))dt\\\\& +f(t,x,\\bar{u}(\\tau _{k}\\wedge T,Y^{\\tau _{k}\\wedge T,x;\\bar{u}(\\cdot )}(t)))dt.\\end{split}$ Therefore, we have $\\begin{split}\\langle dk(t)x,x\\rangle =\\min _{v\\in \\mathbb {R}^{n}}\\big [\\mathbb {F}(t,x,v,K(t-),L(t),R(t,\\cdot ))+f(t,x,v)\\big ]dt,\\quad t\\in (\\hspace{0.0pt}(\\tau _{k}\\wedge T,\\tau _{k+1}\\wedge T)\\hspace{0.0pt}).\\end{split}$ Since $k$ is a predictable process, it does not have a jump at the inaccessible time $\\tau _{k}$ .",
"Thus $dk$ does not contain singular measure, in other word, any $t\\in [0,T]$ , $\\langle dk(t)x,x\\rangle =\\min _{v\\in \\mathbb {R}^{n}}\\big [\\mathbb {F}(t,x,v,K(t-),L(t),R(t,\\cdot ))+f(t,x,v)\\big ]dt.$ In view of assertion (ii) in Proposition REF , the right hand side of (REF ) has a unique minimal point $\\bar{u}(t)$ , hence the minmium value is nothing but $G(t,K(t-),L(t),R(t,\\cdot ))$ and ${N}(t,K(t),R(t,\\cdot ))$ is invertible, which together with (REF ) implies that $(K,L,R)$ satisfies (REF ) a.s. Next we prove the BMO martingale property and (REF ).",
"Using (REF ) for $u(\\cdot )=0$ and $X=X^{\\tau ,x;0}$ , we have $\\left\\lbrace \\begin{array}{ll}dV(t,X(t)))= & -\\langle dk(t)X(t),X(t)\\rangle +\\mathbb {F}(t,X(t-),0,K(t-),L(t),R(t,\\cdot ))dt\\\\& +{\\displaystyle \\sum _{i=1}^{d}\\bigg [\\langle L^{i}(t)X(t-),X(t-)\\rangle +2\\langle K(t-)X(t-),C^{i}(t)X(t-)\\rangle \\bigg ]dW^{i}(s)}\\\\& +{\\displaystyle \\int _{\\Lambda }\\langle R(t,e)(X(t-)+E(t,e)X(t-)),X+E(t,e)X(t-)\\rangle \\tilde{\\mu }(de,dt)}\\\\& +{\\displaystyle \\int _{\\Lambda }\\langle K(t-)E(t,e)X(t-),E(t,e)X(t-)\\rangle \\tilde{\\mu }(de,dt)}\\\\& +2{\\displaystyle \\int _{\\Lambda }\\langle K(t-)X(t-),E(t,e)X(t-))\\rangle \\tilde{\\mu }(de,dt)},\\\\V(T,X(T))= & \\langle MX(T),X(T)\\rangle \\text{.",
"}\\end{array}\\right.$ Applying ItÃŽ formula to $|V(t,X(t))|^{2}$ , we have $& \\int _{\\tau }^{T}\\int _{\\Lambda }\\bigg |\\langle R(I+E)X,(I+E)X\\rangle +\\langle K(2I+E)X,EX\\rangle \\bigg |^{2}\\mu (de,dt)\\nonumber \\\\& \\quad +\\int _{\\tau }^{T}\\sum _{i=1}^{d}\\bigg |\\langle L^{i}X,X\\rangle +2\\langle KX,C^{i}X\\rangle \\bigg |^{2}dt\\nonumber \\\\= & |\\langle MX(T),X(T)\\rangle |^{2}-|\\langle K(\\tau )x,x\\rangle |^{2}+2\\int _{\\tau }^{T}\\langle KX,X\\rangle \\Big [\\langle dkX,X\\rangle -\\mathbb {F}(t,X,0,K,L,R)dt\\Big ]\\nonumber \\\\& \\quad -2\\int _{\\tau }^{T}\\int _{\\Lambda }\\langle KX,X\\rangle \\Big [\\langle R(I+E)X,(I+E)X\\rangle +\\langle K(2I+E)X,EX\\rangle \\bigg ]\\tilde{\\mu }(de,dt)\\\\& \\quad -2\\int _{\\tau }^{T}\\langle KX,X\\rangle {\\displaystyle \\sum _{i=1}^{d}\\bigg [\\langle L^{i}X,X\\rangle +2\\langle KX,C^{i}X\\rangle \\bigg ]dW^{i}(t)},\\nonumber $ where $X$ means $X^{\\tau ,x;0}(t-)$ , $K$ means $K(t-)$ .",
"In the following estimates the constant $C$ may change line by line.",
"Since $V(t,X(t))>0$ and the measure $\\Pi (dt;\\tau \\wedge T,x,u)dxd\\mathbb {P}$ (see (REF )) is nonnegative, we have a.e.",
"$& \\int _{\\tau }^{T}\\int _{\\Lambda }\\bigg |\\langle R(I+E)X,(I+E)X\\rangle +\\langle K(2I+E)X,EX\\rangle \\bigg |^{2}\\mu (de,dt)\\\\& \\quad +\\int _{\\tau }^{T}\\sum _{i=1}^{d}\\bigg |\\langle L^{i}X,X\\rangle +2\\langle KX,C^{i}X\\rangle \\bigg |^{2}dt\\\\\\le & |M|^{2}|X(T)|^{4}+2\\int _{\\tau }^{T}\\langle KX,X\\rangle f(t,X,0)dt\\\\& \\quad -2\\int _{\\tau }^{T}\\int _{\\Lambda }\\langle KX,X\\rangle \\Big [\\langle R(I+E)X,(I+E)X\\rangle +\\langle K(2I+E)X,EX\\rangle \\bigg ]\\tilde{\\mu }(de,dt)\\\\& \\quad -2\\int _{\\tau }^{T}\\langle KX,X\\rangle {\\displaystyle \\sum _{i=1}^{d}\\bigg [\\langle L^{i}X,X\\rangle +2\\langle KX,C^{i}X\\rangle \\bigg ]dW^{i}(t)}.$ Thanks to inequality $\\frac{1}{2}a^{2}-b^{2}\\le (a+b)^{2}$ , and the boundness of $K$ , we have $& \\int _{\\tau }^{T}\\int _{\\Lambda }\\bigg |\\langle R(I+E)X,(I+E)X\\rangle \\bigg |^{2}\\mu (de,dt)+\\int _{\\tau }^{T}\\sum _{i=1}^{d}\\bigg |\\langle L^{i}X,X\\rangle \\bigg |^{2}dt\\nonumber \\\\\\le & 2\\int _{\\tau }^{T}\\int _{\\Lambda }\\bigg |\\langle K(2I+E)X,EX\\rangle \\bigg |^{2}\\mu (de,dt)+2\\int _{\\tau }^{T}\\sum _{i=1}^{d}\\bigg |2\\langle KX,C^{i}X\\rangle \\bigg |^{2}dt\\nonumber \\\\& +2|M|^{2}|X|^{4}+4\\int _{\\tau }^{T}\\langle KX,X\\rangle f(t,X,0)dt\\nonumber \\\\& -4\\int _{\\tau }^{T}\\langle KX,X\\rangle \\sum _{i=1}^{d}\\bigg [\\langle L^{i}X,X\\rangle +2\\langle KX,C^{i}X\\rangle \\bigg ]dW^{i}(t)\\nonumber \\\\& -4\\int _{\\tau }^{T}\\int _{\\Lambda }\\langle KX,X\\rangle \\Big [\\langle R(I+E)X,(I+E)X\\rangle +\\langle K(2I+EX,EX\\rangle \\bigg ]\\tilde{\\mu }(de,dt)\\\\\\le & C\\sup _{\\tau \\le t\\le T}|X|^{4}+\\Big |4\\int _{\\tau }^{T}\\langle KX,X\\rangle \\sum _{i=1}^{d}\\langle L^{i}X,X\\rangle d^{i}W(t)\\Big |\\nonumber \\\\& +8\\Big |\\int _{\\tau }^{T}\\langle KX,X\\rangle \\sum _{i=1}^{d}\\langle KX,C^{i}X\\rangle dW^{i}(t)\\Big |\\nonumber \\\\& +4\\Big |\\int _{\\tau }^{T}\\int _{\\Lambda }\\langle KX,X\\rangle \\langle R(I+E)X,(I+E)X\\rangle \\tilde{\\mu }(de,dt)\\Big |\\nonumber \\\\& +4\\Big |\\int _{\\tau }^{T}\\int _{\\Lambda }\\langle KX,X\\rangle \\langle K(2I+E)X,E)X)\\rangle \\tilde{\\mu }(de,dt)\\Big |.\\nonumber $ This means that $& \\mathbb {E}^{{F}_{\\tau }}\\bigg [\\int _{\\tau }^{T}\\int _{\\Lambda }\\bigg |\\langle R(I+E)X,(I+E)X\\rangle \\bigg |^{2}\\mu (de,dt)\\bigg ]+\\mathbb {E}^{{F}_{\\tau }}\\bigg [\\int _{\\tau }^{T}\\sum _{i=1}^{d}\\bigg |\\langle L^{i}X,X\\rangle \\bigg |^{2}dt\\bigg ]\\nonumber \\\\\\le & C\\mathbb {E}^{{F}_{\\tau }}\\bigg [\\sup _{\\tau \\le t\\le T}|X(t)|^{4}\\bigg ]+C_{p}\\mathbb {E}^{{F}_{\\tau }}\\bigg [\\bigg |\\int _{\\tau }^{T}\\langle KX,X\\rangle {\\displaystyle \\sum _{i=1}^{d}\\langle L^{i}X,X\\rangle dW^{i}(t)\\bigg |\\bigg ]}\\\\& +C\\mathbb {E}^{{F}_{\\tau }}\\bigg [\\bigg |\\int _{\\tau }^{T}\\langle KX,X\\rangle {\\displaystyle \\sum _{i=1}^{d}\\langle KX,C^{i}X\\rangle dW^{i}(t)\\bigg |\\bigg ]}\\nonumber \\\\& +C\\mathbb {E}^{{F}_{\\tau }}\\bigg [\\bigg |\\int _{\\tau }^{\\tau }\\int _{\\Lambda }\\langle KX,X\\rangle \\langle R(I+E)X,(I+E)X\\rangle \\tilde{\\mu }(de,dt)\\bigg |\\bigg ]\\nonumber \\\\& +C\\mathbb {E}^{{F}_{\\tau }}\\bigg [\\bigg |\\int _{\\tau }^{T}\\int _{\\Lambda }\\langle KX,X\\rangle \\langle K(I+E)X,EX\\rangle \\tilde{\\mu }(de,dt)\\bigg |\\bigg ].\\nonumber $ Using BDG inequality, Hölder inequality, boundness of $K$ and the estimation Lemma REF , we have the following estimation about the every terms in right hand side of (REF ), $\\begin{split} & \\mathbb {E}^{{F}_{\\tau }}\\bigg [\\bigg |\\int _{\\tau }^{T}\\langle KX,X\\rangle \\sum _{i=1}^{d}\\langle L^{i}X,X\\rangle dW(t)\\bigg |\\bigg ]\\\\\\le & \\mathbb {E}^{{F}_{\\tau }}\\bigg [\\bigg |\\int _{\\tau }^{T}\\bigg |\\langle KX,X\\rangle {\\displaystyle \\sum _{i=1}^{d}\\langle L^{i}X,X\\rangle \\bigg |^{2}dt\\bigg |^{\\frac{1}{2}}\\bigg ]}\\\\\\le C & \\mathbb {E}^{{F}_{\\tau }}\\bigg [\\sup _{\\tau \\le t\\le T}|X|^{2}\\bigg (\\int _{\\tau }^{T}{\\displaystyle \\sum _{i=1}^{d}\\bigg |\\langle L^{i}X,X\\rangle \\bigg |^{2}dt\\bigg )^{\\frac{1}{2}}\\bigg ]}\\\\\\le \\frac{C}{\\varepsilon } & \\mathbb {E}^{{F}_{\\tau }}\\bigg [\\sup _{\\tau \\le T\\le T}|X|^{4}\\bigg ]+\\varepsilon \\mathbb {E}^{{F}_{\\tau }}\\bigg [\\bigg (\\int _{\\tau }^{T}\\sum _{i=1}^{d}\\Big |\\langle L^{i}X,X\\rangle \\Big |^{2}dt\\bigg )\\bigg ]\\\\\\le & \\frac{C}{\\varepsilon }|x|^{4}+\\varepsilon \\mathbb {E}^{{F}_{\\tau }}\\bigg [\\int _{\\tau }^{T}{\\displaystyle \\sum _{i=1}^{d}\\Big |\\langle L^{i}X,X\\rangle \\Big |^{2}dt\\bigg ]},\\end{split}$ $\\begin{split} & \\mathbb {E}^{{F}_{\\tau }}\\bigg [\\bigg |\\int _{\\tau }^{T}\\langle KX,X\\rangle {\\displaystyle \\sum _{i=1}^{d}\\langle KX,C^{i}X\\rangle dW^{i}(t)\\bigg |\\bigg ]}\\\\\\le & \\mathbb {E}^{{F}_{\\tau }}\\bigg [\\bigg (\\int _{\\tau }^{T}\\bigg |\\langle KX,X\\rangle {\\displaystyle \\sum _{i=1}^{d}\\langle KX,C^{i}X\\rangle \\bigg |^{2}dt\\bigg )^{\\frac{1}{2}}\\bigg ]}\\\\\\le & C_{p}E^{{F}_{\\tau }}\\bigg [\\sup _{\\tau \\le t\\le T}|X|^{4}\\bigg ]\\\\\\le & C_{p}|x|^{4},\\end{split}$ $\\begin{split} & \\mathbb {E}^{{F}_{\\tau }}\\bigg [\\bigg |\\int _{\\tau }^{T}\\int _{\\Lambda }\\langle KX,X\\rangle \\langle K(2I+EX,EX)\\rangle \\tilde{\\mu }(de,dt)\\bigg |\\bigg ]\\\\\\le & \\mathbb {E}^{{F}_{\\tau }}\\bigg [\\Big (\\int _{\\tau }^{T}\\int _{\\Lambda }\\bigg |\\langle KX,X\\rangle \\langle K(2I+E)X,EX\\rangle \\bigg |^{2}\\mu (de,dt)\\Big )^{\\frac{1}{2}}\\bigg ]\\\\\\le & C\\mathbb {E}^{{F}_{\\tau }}\\bigg [\\sup _{\\tau \\le t\\le T}|X|^{4}\\bigg ]\\\\\\le & C|x|^{4},\\end{split}$ $\\begin{split} & \\mathbb {E}^{{F}_{\\tau }}\\bigg [\\bigg |\\int _{\\tau }^{T}\\int _{\\Lambda }\\langle KX,X\\rangle \\langle R(I+E)X,(I+E)X\\rangle \\tilde{\\mu }(de,dt)\\bigg |\\bigg ]\\\\\\le & C\\mathbb {E}^{{F}_{\\tau }}\\bigg [\\bigg \\lbrace \\int _{\\tau }^{T}\\int _{\\Lambda }\\bigg |\\langle KX,X\\rangle \\langle R(I+E)X,(I+E)X\\rangle \\bigg |^{2}\\mu (de,dt)\\bigg \\rbrace ^{\\frac{1}{2}}\\bigg ]\\\\\\le & C\\mathbb {E}^{{F}_{\\tau }}\\bigg [\\sup _{\\tau \\le t\\le T}|X|^{2}\\bigg \\lbrace \\int _{\\tau }^{T}\\int _{\\Lambda }\\bigg |\\langle R(I+E)X,(I+E)X\\rangle \\bigg |^{2}\\mu (de,dt)\\bigg \\rbrace ^{\\frac{1}{2}}\\bigg ]\\\\\\le & \\frac{C}{\\varepsilon }\\mathbb {E}^{{F}_{\\tau }}\\bigg [\\sup _{\\tau \\le t\\le T}|X|^{4}\\bigg ]+\\varepsilon \\mathbb {E}^{{F}_{\\tau }}\\bigg [\\int _{\\tau }^{T}\\int _{\\Lambda }\\bigg |\\langle R(I+E)X,(I+E)X\\rangle \\bigg |^{2}\\mu (de,dt)\\bigg ]\\\\\\le & \\frac{C}{\\varepsilon }|x|^{4}+\\varepsilon \\mathbb {E}^{{F}_{\\tau }}\\bigg [\\int _{\\tau }^{T}\\int _{\\Lambda }\\bigg |\\langle R(I+E)X,(I+E)X\\rangle \\bigg |^{2}\\mu (de,dt)\\bigg ].\\end{split}$ Taking conditional expectation on both sides of (REF ), putting (REF )-(REF ) into it, and then letting $\\varepsilon =1/4$ , we get $\\mathbb {E}^{{F}_{\\tau }}\\bigg [\\int _{\\tau }^{T}\\int _{\\Lambda }\\bigg |\\langle R(I+E)X,(I+E)X\\rangle \\bigg |^{2}\\mu (de,dt)\\bigg ]+\\mathbb {E}^{{F}_{\\tau }}\\bigg [\\int _{\\tau }^{T}\\sum _{i=1}^{d}\\bigg |\\langle L^{i}X,X\\rangle \\bigg |^{2}dt\\bigg ]\\le C|x|^{4},$ the constant $C$ is independent of $\\tau $ and $x$ .",
"Then we have $\\mathbb {E}^{{F}_{\\tau }}\\bigg [\\int _{\\tau }^{T}\\int _{E}\\bigg |\\Phi ^{*}(I+E)^{*}R(I+E)\\Phi \\bigg |^{2}\\mu (de,dt)\\bigg ]+\\mathbb {E}^{{F}_{\\tau }}\\bigg [\\int _{\\tau }^{T}\\sum _{i=1}^{d}\\bigg |\\Phi L^{i}\\Phi \\bigg |^{2}dt\\bigg ]\\le C,$ where $\\Phi $ is the solution of matrix equation (REF ) on $[\\!",
"[\\tau \\wedge T,T]\\!",
"]$ with initial data $\\Phi (\\tau \\wedge T)=I$ .",
"Recall that $\\tau _{k}$ as the $k$ -th jump time of the Poisson point process.",
"For any stopping time $\\gamma \\le T$ , denote by $\\hat{\\tau }_{k}:=\\gamma \\vee \\tau _{k}$ the $n$ -th jump time after the stopping time $\\gamma $ .",
"Applying (REF ) for $\\tau =\\hat{\\tau }_{k}\\wedge T$ and noting that $\\Phi $ is inversible on time $[\\!",
"[\\hat{\\tau }_{k}\\wedge T,\\hat{\\tau }_{k+1}\\wedge T)\\hspace{0.0pt})$ and $\\mathbb {E}^{{F}_{\\hat{\\tau }_{k}\\wedge T}}\\bigg [\\sup _{t\\in [\\!",
"[\\hat{\\tau }_{k}\\wedge T,\\hat{\\tau }_{k+1}\\wedge T)\\hspace{0.0pt})}\\Phi ^{-4}(t)\\bigg ]$ is bounded by a constant only depending on the bound of the coefficients and $T$ (see (REF ) for details), we see that for any $k\\ge 1$ , $& \\mathbb {E}^{{F}_{\\hat{\\tau }_{k}\\wedge T}}\\bigg [\\int _{\\hat{\\tau }_{k}\\wedge T}^{\\hat{\\tau }_{k+1}\\wedge T}\\sum _{i}|L^{i}|^{2}dt\\bigg ]\\nonumber \\\\\\le & \\mathbb {E}^{{F}_{\\hat{\\tau }_{k}\\wedge T}}\\bigg [\\int _{\\hat{\\tau }_{k}\\wedge T}^{\\hat{\\tau }_{k+1}\\wedge T}\\sum _{i}|(\\Phi ^{*})^{-1}\\Phi ^{*}L^{i}\\Phi \\Phi ^{-1}|^{2}dt\\bigg ]\\nonumber \\\\\\le & \\mathbb {E}^{{F}_{\\hat{\\tau }_{k}\\wedge T}}\\bigg [\\sup _{t\\in [\\!",
"[\\hat{\\tau }_{k}\\wedge T,\\hat{\\tau }_{k+1}\\wedge T)\\hspace{0.0pt})}|\\Phi ^{-1}(t)|^{2}\\int _{\\hat{\\tau }_{k}\\wedge T}^{\\hat{\\tau }_{k+1}\\wedge T}\\sum _{i}|\\Phi L^{i}\\Phi |^{2}dt\\bigg ]\\\\\\le & \\bigg \\lbrace \\mathbb {E}^{{F}_{\\hat{\\tau }_{k}\\wedge T}}\\sup _{t\\in [\\!",
"[\\hat{\\tau }_{k}\\wedge T,\\hat{\\tau }_{k+1}\\wedge T)\\hspace{0.0pt})}|\\Phi ^{-1}(t)|^{4}\\bigg \\rbrace ^{\\frac{1}{2}}\\bigg \\lbrace \\mathbb {E}^{{F}_{\\hat{\\tau }_{k}\\wedge T}}\\bigg [\\int _{\\hat{\\tau }_{k}\\wedge T}^{\\hat{\\tau }_{k+1}\\wedge T}\\sum _{i}|\\Phi ^{*}L^{i}\\Phi |^{2}dt\\bigg ]\\bigg \\rbrace ^{\\frac{1}{2}}\\nonumber \\\\\\le & C.\\nonumber $ Similarly, we have $\\mathbb {E}^{{F}_{\\gamma }}\\bigg [\\int _{\\gamma \\wedge T}^{\\hat{\\tau }_{1}\\wedge T}\\sum _{i}|L^{i}|^{2}dt\\bigg ]\\le C.$ Then, using estimates (REF ) and (REF ), we have $\\begin{split} & \\mathbb {E}^{{F}_{\\gamma }}\\bigg [\\int _{\\gamma }^{T}\\sum _{i}|L^{i}|^{2}dt\\bigg ]\\\\= & \\mathbb {E}^{{F}_{\\gamma }}\\bigg [(\\int _{\\gamma }^{\\hat{\\tau }_{1}\\wedge T}+\\sum _{n=1}^{\\infty }\\int _{\\hat{\\tau }_{n}\\wedge T}^{\\hat{\\tau }_{n+1}\\wedge T})\\sum _{i}|L^{i}|^{2}dt\\bigg ]\\\\= & \\mathbb {E}^{{F}_{\\gamma }}\\Big [\\int _{\\gamma }^{\\hat{\\tau }_{1}\\wedge T}\\sum _{i}|L^{i}|^{2}dt\\Big ]+\\sum _{n=1}^{\\infty }\\mathbb {E}^{{F}_{\\gamma }}\\Big [\\int _{\\hat{\\tau }_{n}\\wedge T}^{\\hat{\\tau }_{n+1}\\wedge T}\\sum _{i}|L^{i}|^{2}dt\\Big ]\\\\= & \\mathbb {E}^{{F}_{\\gamma }}\\bigg [\\chi _{\\lbrace \\gamma <T\\rbrace }\\mathbb {E}^{{F}_{\\gamma }}\\Big [\\int _{\\gamma }^{\\hat{\\tau }_{1}\\wedge T}\\sum _{i}|L^{i}|^{2}dt\\Big ]+\\sum _{n=1}^{\\infty }\\chi _{\\lbrace \\hat{\\tau }_{n}<T\\rbrace }\\mathbb {E}^{{F}_{\\hat{\\tau }_{n}\\wedge T}}\\Big [\\int _{\\hat{\\tau }_{n}\\wedge T}^{\\hat{\\tau }_{n+1}\\wedge T}\\sum _{i}|L^{i}|^{2}dt\\Big ]\\bigg ]\\\\\\le & \\mathbb {E}^{{F}_{\\gamma }}\\bigg [\\Big (\\chi _{\\lbrace \\gamma <T\\rbrace }+\\sum _{n=1}^{\\infty }\\chi _{\\lbrace \\hat{\\tau }_{n}<T\\rbrace }\\Big )C\\bigg ]\\\\= & C\\mathbb {E}^{{F}_{\\gamma }}\\Big [\\mu \\big ([\\!",
"[\\gamma ,T]\\!",
"]\\times \\Lambda \\big )\\Big ].\\end{split}$ In view of the independent increment property of the Poisson point process $\\lbrace p_{t}\\rbrace _{t\\ge 0}$ , $\\mu \\big ([\\!",
"[\\gamma ,T]\\!",
"]\\times \\Lambda \\big )$ is independent of ${F}_{\\gamma }$ .",
"So we have $\\mathbb {E}^{{F}_{\\gamma }}\\Big [\\mu \\big ([\\!",
"[\\gamma ,T]\\!",
"]\\times \\Lambda )\\big )\\Big ]=\\mathbb {E}\\Big [\\mu \\big ([\\!",
"[\\gamma ,T]\\!",
"]\\times \\Lambda \\big )\\Big ]\\le \\mathbb {E}\\Big [\\mu \\big ([0,T]\\times \\Lambda \\big )\\Big ]=T\\nu (\\Lambda )$ .",
"Hence we obtain that for any stopping time $\\gamma $ valued in $[0,T]$ , $\\mathbb {E}^{{F}_{\\gamma }}\\Big [\\Big |\\sum _{i}\\int _{\\gamma }^{T}L^{i}dW^{i}(t)\\Big |^{2}\\Big ]\\le \\mathbb {E}^{{F}_{\\gamma }}\\bigg [\\int _{\\gamma }^{T}\\sum _{i}|L^{i}|^{2}dt\\bigg ]\\le C,$ which means $\\int _{0}^{\\cdot }L^{i}(s)dW_{s}^{i}$ is a BMO martingale, $i=1,\\ldots ,d$ .",
"For $\\eta _{t}:=\\int _{0}^{t}\\int _{\\Lambda }R(e,t)\\tilde{\\mu }(de,dt)$ , we see that it is a purely continuous martingale whose jumps coincide with those of $K$ .",
"Since $K$ is uniformly bounded by some constant $\\lambda $ , jumps of $\\eta $ is also uniformly bounded by $2\\lambda $ .",
"Hence we have $[\\eta ]_{T}-[\\eta ]_{\\gamma -}= & \\sum _{\\gamma \\le s\\le T}|\\Delta \\eta _{s}|^{2}=\\sum _{\\gamma \\le \\tau _{i}\\le T}|\\Delta \\eta _{s}|^{2}\\\\\\le & 4\\lambda ^{2}\\mu ([\\!",
"[\\gamma ,T]\\!",
"]\\times \\Lambda ).$ Thus $\\mathbb {E}^{{F}_{\\gamma }}\\bigg [[\\eta ]_{T}-[\\eta ]_{\\gamma -}\\bigg ]\\le 4\\lambda ^{2}\\mathbb {E}^{{F}_{\\gamma }}\\bigg [\\mu ([\\!",
"[\\gamma ,T]\\!",
"]\\times \\Lambda )\\bigg ]\\le CT\\nu (\\Lambda )<\\infty ,$ which means that $J$ is also a BMO martingale.",
"Let $\\gamma =0$ in (REF ) and (REF ), we have $\\mathbb {E}\\bigg [\\int _{0}^{T}\\sum _{i}|L^{i}|^{2}dt\\bigg ]\\le C,$ and $\\mathbb {E}\\bigg [\\int _{0}^{T}\\int _{\\Lambda }R^{2}\\nu (de)dt\\bigg ]=\\mathbb {E}[\\eta ]_{T}\\le C,$ we have estimate (REF ).",
"Last we show the nonnegativity of $\\int _{\\Lambda }F^{*}(t,e)(K(t-)+R(t,e))F(t,e)\\nu (de)$.",
"First note that the pure jump process $\\zeta _{t}:=\\int _{0}^{t}\\int _{\\Lambda }F^{*}(s,e)(K(t-)+R(s,e))F(s,e)\\mu (de,ds)$ only changes its value at the jumping time of Poisson process and $\\Delta \\zeta _{t}=\\int _{\\Lambda }F^{*}(s,e)(K(s-)+R(s,e))F(s,e)\\mu (de,\\lbrace t\\rbrace )$ .",
"Since at the jumping moment $R(s,p_{s})$ is equivalent to $K(s)-K(s-)$ , it is easy to know that $K(s-)+R(s,e)=K(s)$ is nonnegative definite (here $e$ is the jumpping amplitude at the moment), therefore for any $y\\in {\\mathcal {M}}_{{\\cal {F}}}^{\\infty }(0,T;\\mathbb {R}^{m})$ , we have $\\int _{0}^{T}\\int _{\\Lambda }y^{*}(s)F^{*}(s,e)(K(s-)+R(s,e))F(s,e)y(s)\\mu (de,ds)\\ge 0,\\quad P{\\rm -a.e.",
"}$ In view of the martingale property, $\\mathbb {E}\\big [\\int _{0}^{T}\\int _{\\Lambda }y^{*}(s)F^{*}(s,e)(K(s-)+R(s,e))F(s,e)y(s)\\tilde{\\mu }(ds,de)\\big ]=0.$ Hence $\\begin{split} & \\mathbb {E}\\big [\\int _{0}^{T}\\int _{\\Lambda }y^{*}(s)F^{*}(s,e)(K(s-)+R(s,e))F(s,e)y(s)\\nu (de)ds\\big ]\\\\= & \\mathbb {E}\\Big [\\int _{0}^{T}\\int _{\\Lambda }y^{*}(s)F^{*}(s,e)(K(s-)+R(s,e))F(s,e)y(s)[\\mu (de,ds)-\\tilde{\\mu }(de,ds)]\\Big ]\\ge 0.\\end{split}$ By the arbitrariness of $y$ , we have $\\int _{\\Lambda }F^{*}(t,e)(K(t-)+R(t,e))F(t,e)\\nu (de)$ is nonnegative for almost all $t$ , $\\mathbb {P}$ -a.s. $\\omega $ .",
"Thus, the proof is complete.",
"Remark 4.1 If we have the condition (REF ) in hand, (REF ) could be obtained from (REF ) directly like the way of (REF )-(REF ).",
"In our case, observing the structure of BSREJ and utilizing the relationship between the jump of $K$ and $R$ , we can prove (REF ) by the estimate of $K$ , and this way seemed to be easier."
],
[
"Verification theorem",
"In section , we exploit Problem REF and the dynamic programming principle to show the existence of solution for BSREJ (REF ).",
"In this section we will deal with the problem from an inverse aspect – if the BSREJ (REF ) has a solution, how to describe the corresponding optimal control problem?",
"The following Theorem REF tells us that the existence of solution for BSREJ (REF ) means the existence of the optimal control for problem (REF ).",
"Besides, the optimal control could be depicted as a linear feedback by the solution of BSREJ (REF ).",
"Theorem 5.1 Let Assumptions REF be satisfied.",
"And assume BSREJ (REF ) has a solution $(K,L,R)$ in the meaning of Definition REF .",
"Then the linear SDE $\\left\\lbrace \\begin{array}{l}d\\bar{X}^{t,x}(s)=[A(s)-B(s){N}^{-1}(s,K(s-),R(s,\\cdot )){M}^{*}(s,K(s-),L(s),R(s,\\cdot ))]\\bar{X}^{t,x}(s-)ds\\\\\\quad +\\sum _{i=1}^{d}[C^{i}(s)-D^{i}(s){N}^{-1}(s,K(s-),R(s,\\cdot )){M}^{*}(s,K(s-),L(s),R(s,\\cdot ))]\\bar{X}^{t,x}(s-)dW^{i}(s)\\\\\\quad +\\int _{\\Lambda }[E(s,e)-F(s,e){N}^{-1}(s,K(s-),R(s,\\cdot )){M}^{*}(s,K(s-),L(s),R(s,\\cdot ))]\\bar{X}^{t,x}(s-)\\tilde{\\mu }(ds,de),\\\\\\bar{X}(t)=x,\\qquad s\\in [t,T]\\end{array}\\right.$ has a unique solution $\\bar{X}^{t,x}(\\cdot )$ such that $\\mathbb {E}^{{F}_{t}}\\bigg [\\sup _{s\\in [t,T]}|\\bar{X}^{t,x}(s)|^{2}\\bigg ]<C_{x},$ where the constant $C_{x}$ is independent of initial time $t$ .",
"(ii) The given process $\\bar{u}^{t,x}(s):=-{N}^{-1}(s,K(s-),R(s,\\cdot )){M}^{*}(s,K(s-),L(s),R(s,\\cdot ))\\bar{X}(s-),\\quad s\\in [t,T]$ belongs to ${\\cal M}_{{F}}^{2}(t,T;\\mathbb {R}^{m})$ , and is the optimal control for the problem (REF ) for the initial data $(\\tau ,\\xi )=(t,x)$ .",
"(iii) The value field $V$ is given by $V(t,x)=\\langle K(t)x,x\\rangle ,(t,x)\\in [0,T]\\times \\mathbb {R}^{n}.$ Since the coefficients of the optimal SDE (REF ) are square integrable w.r.t.",
"$t$ a.s., it admits a unique strong solution $\\bar{X}(\\cdot )$ .",
"For a sufficiently large integer $j$ , define the stopping time $\\gamma _{j}$ as follows: $\\gamma _{j}^{t,x}:=T\\wedge \\inf \\lbrace s\\ge t||\\bar{X}^{t,x}(s)|\\ge j\\rbrace $ with the convention that $\\inf \\emptyset =\\infty $ .",
"It is obvious that $\\gamma _{j}^{t,x}\\uparrow T$ almost surely as $j\\uparrow \\infty $ .",
"Then by Itô formula we have $\\langle K(t)x,x\\rangle =\\mathbb {E}^{{F}_{t}}\\bigg [\\langle K(\\gamma _{j}^{t,x})\\bar{X}^{t,x}(\\gamma _{j}^{t,x}),\\bar{X}^{t,x}(\\gamma _{j}^{t,x})\\rangle +\\int _{t}^{\\gamma _{j}^{t,x}}f(s,\\bar{X}^{t,x}(s),\\bar{u}^{t,x}(s))dt\\bigg ].$ Noting that $K$ is positive and bounded by $\\lambda $ , and $N>\\delta I$ for some constant $\\delta $ (see Assumption REF ), (REF ) implies $\\mathbb {E}^{{F}_{t}}\\bigg [\\int _{t}^{\\gamma _{j}^{t,x}}\\big (\\bar{u}^{t,x}\\big )^{2}(s)ds\\bigg ]\\le \\frac{1}{\\delta }\\mathbb {E}^{{F}_{t}}\\bigg [\\int _{t}^{\\gamma _{j}^{t,x}}f(s,\\bar{X}^{t,x}(s),\\bar{u}^{t,x}(s))ds\\bigg ]\\le \\frac{1}{\\delta }\\langle K(t)x,x\\rangle \\le \\frac{\\lambda }{\\delta }|x|^{2}.$ Using Fatou's lemma, we have $\\bar{u}^{t,x}(\\cdot )\\in {\\cal M}_{{F}}^{2}(0,T;\\mathbb {R}^{m})$ .",
"Then we have the estimation (REF ) from Lemma REF .",
"Thus, Assertion (i) and the first part of the assertion (ii) have been proved.",
"Now we prove the optimality of $\\bar{u}^{t,x}(\\cdot )$ and the assertion (iii).",
"By (REF ), we know for any stopping time $\\tau $ valued in $[t,T]$ , $\\mathbb {E}^{{F}_{t}}\\Big [\\big |\\bar{X}^{t,x}(\\tau )\\big |^{2}\\Big ]\\le \\mathbb {E}^{{F}_{t}}\\bigg [\\sup _{s\\in [t,T]}|\\bar{X}^{t,x}(s)|^{2}\\bigg ]<C_{x},$ hence $|\\bar{X}^{t,x}|^{2}$ is uniformly integrable.",
"Besides (REF ) together with Chebyshev inequality shows that for any positive integer $j$ , $\\mathbb {P}\\bigg (\\sup _{s\\in [t,T]}|\\bar{X}^{t,x}(s)|\\ge j\\bigg )\\le \\frac{\\mathbb {E}^{{F}_{t}}\\bigg [\\sup _{s\\in [t,T]}|\\bar{X}^{t,x}(s)|^{2}\\bigg ]}{j^{2}}\\rightarrow 0,\\qquad {\\rm as}\\,j\\rightarrow \\infty .$ It follows that $\\,\\mathbb {P}\\lbrace \\gamma _{j}^{t,x}=T\\rbrace \\nearrow 1$ .",
"Combining the dominate convergence theorem and the boundness of $K$ , we have the first term in right hand of (REF ) $\\mathbb {E}^{{F}_{t}}\\bigg [\\langle K(\\gamma ^{t,x})\\bar{X}(\\gamma ^{t,x}),\\bar{X}(\\gamma ^{t,x})\\rangle \\bigg ]\\rightarrow \\mathbb {E}^{{F}_{t}}\\bigg [\\langle K(T)\\bar{X}^{t,x}(T),\\bar{X}^{t,x}(T)\\rangle \\bigg ]$ as $j\\rightarrow \\infty $ .",
"The $L^{2}$ -boundness of $\\bar{X}^{t,x}(\\cdot )$ and $\\bar{u}^{t,x}(\\cdot )$ yields the second term in right hand of (REF ) $\\mathbb {E}^{{F}_{t}}\\bigg [\\int _{t}^{\\gamma _{j}^{t,x}}f(s,\\bar{X}^{t,x}(s),\\bar{u}^{t,x}(s))ds\\bigg ]\\rightarrow \\mathbb {E}^{{F}_{t}}\\bigg [\\int _{t}^{T}f(s,\\bar{X}^{t,x}(s),\\bar{u}^{t,x}(s))ds\\bigg ]$ as $j\\rightarrow \\infty $ .",
"Hence (REF ) yields $& \\langle K(t)x,x\\rangle \\nonumber \\\\= & \\lim _{j\\rightarrow \\infty }\\mathbb {E}^{{F}_{t}}\\bigg [\\langle K(\\gamma _{j}^{t,x})\\bar{X}^{t,x}(\\gamma _{j}^{t,x}),\\bar{X}^{t,x}(\\gamma _{j}^{t,x})\\rangle +\\int _{t}^{\\gamma _{j}^{t,x}}f(s,\\bar{X}^{t,x}(s),\\bar{u}^{t,x}(s))ds\\bigg ]\\\\= & \\mathbb {E}^{{F}_{t}}\\bigg [\\langle K(T)\\bar{X}^{t,x}(T),\\bar{X}^{t,x}(T)\\rangle +\\int _{t}^{T}f(s,\\bar{X}^{t,x}(s),\\bar{u}^{t,x}(s))ds\\bigg ]=J(\\bar{u}^{t,x}(\\cdot );0,x).\\nonumber $ To obtain the optimality of $\\bar{u}^{t,x}(\\cdot )$ , it remain to show $J(u(\\cdot );t,x)\\ge \\langle K(t)x,x\\rangle ,\\quad \\forall u(\\cdot )\\in {\\cal M}_{{F}}^{2}(t,T;\\mathbb {R}^{m}).$ To do this, for any $u(\\cdot )\\in {\\cal M}_{{F}}^{2}(t,T;\\mathbb {R}^{m})$ , define the stopping times $\\gamma _{j}^{t,x;u(\\cdot )}=T\\wedge \\inf \\lbrace s\\ge t||X^{t,x;u(\\cdot )}(s)|\\ge j\\rbrace ,\\quad j\\in \\mathbb {Z}_{+}.$ Same as $\\gamma _{j}^{t,x}$ , $\\gamma _{j}^{t,x;u(\\cdot )}\\nearrow T$ and $\\mathbb {P}\\lbrace \\gamma _{j}^{t,x;u(\\cdot )}=T\\rbrace \\nearrow 1$ as $j\\rightarrow \\infty $ .",
"Define $\\tilde{u}(s):=-{N}^{-1}(s,K(s-),R(s,\\cdot )){M}(s,K(s-),L(s),R(s,\\cdot ))X^{0,x,u(\\cdot )}(s-),\\quad s\\in [t,T].$ Obviously, $\\mathbb {E}\\Big [\\int _{t}^{\\gamma _{j}^{t,x;u(\\cdot )}}\\big |\\tilde{u}(t)\\big |^{2}dt\\Big ]<\\infty $ .",
"Then applying Itô formula to $\\langle K(t)X^{t,x;u(\\cdot )}(t),X^{t,x;u(\\cdot )}(t)\\rangle $ and by straightforward computing, we get that $& \\mathbb {E}^{{F}_{t}}\\bigg [\\langle K(\\gamma _{j}^{tx;u(\\cdot )})X^{t,x;u(\\cdot )}(\\gamma _{j}^{t,x;u(\\cdot )}),X^{t,x;u(\\cdot )}(\\gamma _{j}^{t,x;u(\\cdot )})\\rangle +\\int _{t}^{\\gamma _{j}^{t,x;u(\\cdot )}}f(s,X^{t,x;u(\\cdot )}(s),u(s))ds\\bigg ]\\\\= & \\langle K(t)x,x\\rangle +\\mathbb {E}^{{F}_{t}}\\Big [\\int _{t}^{\\gamma _{j}^{t,x;u(\\cdot )}}\\big \\langle {N}^{-1}(s,K(s),R(s,\\cdot ))\\big (u(s)-\\tilde{u}(s)\\big ),u(s)-\\tilde{u}(s)\\big \\rangle \\Big ]\\nonumber \\\\\\ge & \\langle K(t)x,x\\rangle .\\nonumber $ Since $u(\\cdot )\\in {\\cal M}_{{F}}^{2}(t,T;\\mathbb {R}^{m})$ , according to the estimate (REF ), similar to the limitation in (REF ), we take limit in (REF ) $& J(u(\\cdot );t,x)\\\\= & \\mathbb {E}^{{F}_{t}}\\bigg [\\langle K(T)X^{t,x;u}(T),X^{t,x;u}(T)\\rangle +\\int _{t}^{T}f(s,X^{t,x;u}(s),u(s))ds\\bigg ]\\\\\\ge & \\langle K(t)x,x\\rangle .$ According to the above verification theorem, we immediately have the following uniqueness of the solution for BSREJ (REF ).",
"Theorem 5.2 Let Assumptions REF be satisfied.",
"Let $(\\tilde{K},\\tilde{L},\\tilde{R})$ be another solution of BSREJ (REF ) in the meaning of Definition REF .",
"Then $(\\tilde{K},\\tilde{L},\\tilde{R})=(K,L,R)$ .",
"In view of (REF ), the uniqueness of value function $V$ leads to that of first unknown variable $K$ of solution for BSREJ (REF ), hence $\\tilde{K}=K$ .",
"By the expression of BSREJ (REF ), the integration w.r.t.",
"$\\tilde{\\mu }$ is just pure jump martingale, hence $\\sum _{s\\le t}\\Delta K_{s} & =\\int _{0}^{t}\\int _{\\Lambda }R(t,e)\\mu (de,ds),\\\\\\sum _{s\\le t}\\Delta \\tilde{K}_{s} & =\\int _{0}^{t}\\int _{\\Lambda }\\tilde{R}(t,e)\\mu (de,ds).$ Comparing the above two equality, taking the quadratic variation (the bracket) , and then taking expectation on both sides, we have $0=\\mathbb {E}\\Big [\\sum _{s\\le t}\\Delta \\big (K_{s}-\\tilde{K}_{s}\\big ),\\sum _{s\\le t}\\Delta \\big (K_{s}-\\tilde{K}_{s}\\big )\\Big ]=\\mathbb {E}\\Big [\\int _{0}^{t}\\int _{\\Lambda }\\big (R-\\tilde{R}\\big )^{2}\\nu (de)ds\\Big ].$ This means $\\tilde{R}=R$ .",
"With the uniqueness of the first and third unknown variables $(K,R)$ in hand, the uniqueness of the optimal control and its feedback form (REF ) yields the uniqueness of the second unknown variable $L$ ."
]
] | 1808.08336 |
[
[
"A Framework on Hybrid MIMO Transceiver Design based on Matrix-Monotonic\n Optimization"
],
[
"Abstract Hybrid transceiver can strike a balance between complexity and performance of multiple-input multiple-output (MIMO) systems.",
"In this paper, we develop a unified framework on hybrid MIMO transceiver design using matrix-monotonic optimization.",
"The proposed framework addresses general hybrid transceiver design, rather than just limiting to certain high frequency bands, such as millimeter wave (mmWave) or terahertz bands or relying on the sparsity of some specific wireless channels.",
"In the proposed framework, analog and digital parts of a transceiver, either linear or nonlinear, are jointly optimized.",
"Based on matrix-monotonic optimization, we demonstrate that the combination of the optimal analog precoders and processors are equivalent to eigenchannel selection for various optimal hybrid MIMO transceivers.",
"From the optimal structure, several effective algorithms are derived to compute the analog transceivers under unit modulus constraints.",
"Furthermore, in order to reduce computation complexity, a simple random algorithm is introduced for analog transceiver optimization.",
"Once the analog part of a transceiver is determined, the closed-form digital part can be obtained.",
"Numerical results verify the advantages of the proposed design."
],
[
"Introductions",
"The great success of multiple-input multiple-output (MIMO) technology makes it widely accepted for current and future high data-rate communication systems[1].",
"Acting as a pillar to satisfy data hungry applications, a natural question is how to reduce the cost of MIMO technology, especially that of large scale antenna arrays.",
"The traditional setting of one radio-frequency (RF) chain per antenna element is too expensive for large-scale MIMO systems, especially at high frequencies, such as millimeter wave bands or Terahertz bands.",
"Hybrid analog-digital architecture is promising to alleviate the straits and strike a balance between the cost and the performance of practical MIMO systems.",
"A typical hybrid analog-digital MIMO transceiver consists of four components, i.e., digital precoder, analog precoder, analog processor, and digital processor [2].",
"In the early transceiver design, hybrid MIMO technology is often referred to antenna selection [3], [4] to reap spatial diversity.",
"In these works, analog switches are used in the radio-frequency domain.",
"Phase-shifter based soft antenna selection [5], [6], [7] has been proposed to improve performance for correlated MIMO channels.",
"Nowadays, the phase-shifter based hybrid structure has been widely used.",
"For a phase shifter, only signal phase, instead of both magnitude and phase, can be adjusted.",
"Thus, the optimization of a MIMO transceiver with phase shifters becomes complicated due to the constant-modulus constraints on analog precoder and analog processor.",
"It has been shown in [8] that the performance of a full-digital system can be achieved when the number of shifters is doubled in a phase-shifter based hybrid structure.",
"However, this can hardly be practical due to the requirement on a large number of phase shifters, especially in large-scale MIMO systems.",
"As a matter of fact, the phase shifters in large-scale MIMO systems have been considered to be a burden sometimes.",
"Thus, sub-connected hybrid structure has emerged as an alternative option [9], [10] and it has received much attention recently [9], [11], [12], [10], [13], [14].",
"Unit modulus and discrete phase make the optimization of analog transceivers nonconvex and thus difficult to address [3], [4].",
"There have been some works on hybrid transceiver optimization considering different design limitations and requirements.",
"Their motivation is to exploit the underlying structures of the hybrid transceiver to achieve high performance but with low complexity.",
"Early hybrid transceiver design is based on approximating digital transceivers in terms of the norm difference between all-digital design and the hybrid counterpart.",
"For the millimeter wave (mmWave) band channels, which are usually with sparsity, an orthogonal matching pursuit (OMP) algorithm has been used in signal recovery for the hybrid transceiver [15].",
"In order to overcome the non-convexity in hybrid transceiver optimization, some distinct characteristics of mmWave channels must be exploited [15].",
"This methodology is a compromise on the constant-modulus constraint, which has been validated in different environments, including multiuser and relay scenarios [16], [17].",
"However, it has been found later on that the OMP algorithm cannot achieve the optimal solution sometimes.",
"A singular-value-decomposition (SVD) based descent algorithm [18] has been proposed, which is nearly optimal.",
"An alternative fast constant-modulus algorithm [19] has also been developed to reduce the gap between the analog and digital precoders.",
"The above methods are hard for complex scenarios due to high computation complexity [20], [21], [22].",
"Therefore, based on the idea of unitary matrix rotation, several algorithms [23], [24] have been proposed to improve the approximation performance while maintaining a relative low complexity at the same time.",
"On the other hand, some works for hybrid precoding design are based on codebooks, which relax the problem into a convex optimization problem [25].",
"However, the codebook-based algorithm suffers performance loss if channel state information (CSI) is inaccurate [26].",
"In order to reduce the complexity of codebook design and the impact of partial CSI, special structures of massive MIMO channels [27], [28], can be exploited.",
"Recently, an angle-domain based method has been proposed from the viewpoint of array signal processing [29], [30], which provides a useful insight on hybrid analog and digital signal processing.",
"Based on the concept of the angle-domain design, some mathematical approaches, such as matrix decomposition algorithm, have been developed [31], [32].",
"Energy efficient hybrid transceiver design for Rayleigh fading channels has been investigated in [33].",
"Hybrid transceiver optimization with partial CSI and with discrete phases has been discussed in [34] and [35], respectively.",
"Hybrid MIMO transceivers are not only limited to mmWave frequency bands or terahertz frequency bands but also potentially work in other frequency bands.",
"The transceiver itself could either be linear or nonlinear.",
"Moreover, the performance metrics for MIMO transceiver could be different, including capacity, mean-squared error (MSE), bit-error rate (BER), etc.",
"A unified framework on hybrid MIMO transceiver optimization will be of great interest.",
"In this paper, we will develop a unified framework for hybrid linear and nonlinear MIMO transceiver optimization.",
"Our main contributions are summarized as follows.",
"Both linear and nonlinear transceivers with Tomlinson-Harashima precoding (THP) or deci-sion-feedback detection (DFD) are taken into account in the proposed framework for hybrid MIMO transceiver optimization.",
"Different from the existing works in which a single performance metric is considered for hybrid MIMO transceiver designs, more general performance metrics are considered.",
"Based on matrix-monotonic optimization framework, the optimal structures of both digital and analog transceivers with respect to different performance metrics have been analytically derived.",
"From the optimal structures, the optimal analog precoder and processor correspond to selecting eigenchannels, which facilitates the analog transceiver design.",
"Furthermore, several effective analog design algorithms have been proposed.",
"The rest of this paper is organized as follows.",
"In Section II, a general hybrid system model and the MSE matrices corresponding to different transceivers are introduced.",
"In Section III, a unified hybrid transceiver is discussed in detail and the related transceiver optimization is present.",
"In Section IV, the optimal structure of digital transceivers is derived based on matrix-monotonic optimization.",
"In Section V, basic properties of the optimal analog precoder and processor are investigated, based on which effective algorithms to compute the analog transceiver are proposed.",
"Next, in Section VI, simulation results are provided to demonstrate the performance advantages of the proposed algorithms.",
"Finally, Figure: General hybrid MIMO transceiver.conclusions are drawn in Section VII.",
"Notations: In this paper, scalars, vectors, and matrices are denoted by non-bold, bold lower-case, and bold upper-case letters, respectively.",
"The notations ${\\bf X}^{\\rm {H}}$ and ${\\rm {Tr}}({\\bf X})$ denote the Hermitian and the trace of a complex matrix ${\\bf X}$ , respectively.",
"Matrix ${\\bf X}^{\\frac{1}{2}}$ is the Hermitian square root of a positive semi-definite matrix ${\\bf X}$ .",
"The expression $ \\mathrm {diag} \\lbrace \\mathbf {X} \\rbrace $ denotes a square diagonal matrix with the same diagonal elements as matrix $ \\mathbf {X} $ .",
"The $ i $ th row and the $ j $ th column of a matrix are denoted as $ [\\cdot ]_{i,:} $ and $ [\\cdot ]_{:,j} $ , respectively, and the element in the $ k $ th row and the $ \\ell $ th column of a matrix is denoted as $ [\\cdot ]_{k,\\ell } $ .",
"In the following derivations, ${\\Lambda }$ always denotes a diagonal matrix (square or rectangular diagonal matrix) with diagonal elements arranged in a nonincreasing order.",
"Representation $ \\mathbf {A} \\preceq \\mathbf {B} $ means that the matrix $ \\mathbf {B} - \\mathbf {A} $ is positive semidefinite.",
"The real and imaginary parts of a complex variable are represented by $ \\Re \\lbrace \\cdot \\rbrace $ and $ \\Im \\lbrace \\cdot \\rbrace $ , respectively, and statistical expectation is denoted by $\\mathbb {E}\\lbrace \\cdot \\rbrace $ ."
],
[
"General Structure of Hybrid MIMO Transceiver",
"In this section, we will first introduce the system model of MIMO hybrid transceiver designs.",
"Then a general signal model is introduced, which includes nonlinear transceiver with THP or DFD and linear transceiver as its special cases.",
"Based on the general signal model, the general linear minimum mean-squared error (LMMSE) processor and data estimation mean-squared error (MSE) matrix are derived, which are the basis for the subsequent hybrid MIMO transceiver design."
],
[
"System Model",
"As shown in Fig.",
"REF , we consider a point-to-point hybrid MIMO system where the source and the destination are equipped with $ N $ and $ M $ antennas, respectively.",
"Without loss of generality, it is assumed that both the source and the destination have $L$ RF chains.",
"A transmit data vector ${\\bf {a}}$ $\\in \\mathbb {C}^{D\\times 1}$ is first processed by a unit with feedback operation and then goes through a digital precoder $ {\\bf F}_{\\rm D} \\in \\mathbb {C}^{L \\times D} $ and an analog precoder $ {\\bf F}_{\\rm A} \\in \\mathbb {C}^{N \\times L} $ .",
"This is a more general model as it includes both linear precoder and nonlinear precoder as its special cases.",
"For the nonlinear transceiver with THP at source, the feedback matrix $ {\\bf B}^{\\rm Tx} $ is strictly lower triangular.",
"The key idea behind THP is to exploit feedback operations to pre-eliminate mutual interference between different data streams.",
"In order to control transmit signals in a predefined region, a modulo operation is introduced for the feedback operation [36].",
"Based on lattice theory, it can be proved that the modulo operation is equivalent to adding an auxiliary complex vector ${\\bf {d}}$ whose element is with integer imaginary and real parts [36], [37].",
"The vector ${\\bf {d}}$ makes sure ${\\bf {x}}={\\bf a} + {\\bf d}$ in a predefined region [36], [37].",
"Based on this fact, the output vector ${\\bf {b}}$ of the feedback unit satisfies the following equation ${\\bf b} = ({\\bf a} + {\\bf d})-{\\bf {B}}^{\\rm {Tx}}{\\bf {b}},$ that is ${\\bf {b}}=({\\bf {I}}+{\\bf {B}}^{\\rm {Tx}})^{-1}\\underbrace{({\\bf a} + {\\bf d})}_{\\triangleq {\\bf {x}}}.$ It is worth noting that ${\\bf {d}}$ can be perfectly removed by a modulo operation [36], [37] and thus recovering ${\\bf {x}}$ is equivalent to recovering ${\\bf {a}}$ .",
"On the other hand, for linear precoder, there is no feedback operation, i.e., ${\\bf {B}}^{\\rm {Tx}}={\\bf {0}}$ and ${\\bf {d}}={\\bf {0}}$ [38].",
"Moreover, based on (REF ) we have ${\\bf {b}}={\\bf {a}}$ .",
"Then, the received signal ${\\bf {y}}$ at the destination is ${\\bf {y}}={\\bf {H}}{\\bf {F}}_{\\rm {A}}{\\bf {F}}_{\\rm {D}}({\\bf I}+{\\bf B}^{\\rm Tx})^{-1}{\\bf {x}}+{\\bf {n}},$ where $ {\\bf n} $ is an $ M \\times 1 $ additive Gaussian noise vector with zero mean and covariance $ {\\bf R}_{\\rm n} $ , $ {\\bf H} $ is an $ M \\times N $ channel matrix, and $ {\\bf B}^{\\rm Tx} $ is a general feedback matrix at source, which is determined by the types of precoders.",
"It is worth noting that $ {\\bf B}^{\\rm Tx} = {\\bf 0} $ corresponds to linear precoder without feedback operation.",
"As shown in Fig.",
"REF , after analog and digital processing at the destination, the recovered signal is given by ${\\bf {\\hat{x}}}^{\\rm {General}}={\\bf {G}}_{\\rm {D}}{\\bf {G}}_{\\rm {A}}{\\bf {y}} - {\\bf B}^{\\rm Rx}{\\bf x},$ where $ {\\bf G}_{\\rm A} \\in \\mathbb {C}^{L \\times M} $ is an analog processor, $ {\\bf G}_{\\rm D} \\in \\mathbb {C}^{D \\times L} $ is a digital processor, and $ {\\bf B}^{\\rm Rx} $ is a general feedback matrix at the destination.",
"Note that since the analog precoder $ {\\bf F}_{\\rm A} $ and analog processor $ {\\bf G}_{\\rm A} $ are implemented through phase shifters, they are restricted to constant-modulus matrices with constant magnitude elements.",
"For DFD at the receiver, the decision feedback matrix $ {\\bf B}^{\\rm Rx} $ in (REF ) is a strictly lower-triangular matrix.",
"For linear detection, the feedback matrix in (REF ) is an all-zero matrix, i.e., $ {\\bf B}^{\\rm Rx} = {\\bf 0} $ .",
"Based on (REF ) and (REF ), the recovered signal vector can be rewritten as ${\\bf {\\hat{x}}}= {\\bf {G}}_{\\rm {D}}{\\bf {G}}_{\\rm {A}}{\\bf {H}}{\\bf {F}}_{\\rm {A}} {\\bf {F}}_{\\rm {D}}( {\\bf I} + {\\bf B}^{\\rm Tx} )^{-1}{\\bf {x}} -{\\bf B}^{\\rm Rx}{\\bf {x}} + {\\bf {G}}_{\\rm {D}}{\\bf {G}}_{\\rm {A}}{\\bf {n}}.$ This is a general signal model and includes nonlinear hybrid transceivers with THP or DFD and linear hybrid transceiver as its special cases.",
"More specifically, for a linear hybrid transceiver, there is no feedback, either at the source or at the destination, i.e., $ {\\bf B}^{\\rm Tx} = {\\bf B}^{\\rm Rx} = {\\bf 0} $ .",
"Therefore, the recovered signal in (REF ) becomes ${\\bf {\\hat{x}}}^{\\rm Linear} = {\\bf {G}}_{\\rm {D}}{\\bf {G}}_{\\rm {A}}{\\bf {H}}{\\bf {F}}_{\\rm {A}}{\\bf {F}}_{\\rm {D}}{\\bf {x}}+{\\bf {G}}_{\\rm {D}}{\\bf {G}}_{\\rm {A}}{\\bf {n}}.$ For the nonlinear transceiver with THP at the source and linear decision at the destination, i.e., $ {\\bf B}^{\\rm Rx} = {\\bf 0} $ [36], [37], the detected signal vector in (REF ) becomes ${\\bf {\\hat{x}}}^{\\rm THP} = {\\bf {G}}_{\\rm {D}}{\\bf {G}}_{\\rm {A}}{\\bf {H}}{\\bf {F}}_{\\rm {A}}{\\bf {F}}_{\\rm {D}}({\\bf {I}}+{\\bf {B}}^{\\rm Tx})^{-1}{\\bf {x}}+ {\\bf {G}}_{\\rm {D}} {\\bf {G}}_{\\rm {A}} {\\bf {n}}.$ For the nonlinear transceiver with DFD at the destination and a linear precoder at the source, i.e., $ {\\bf B}^{\\rm Tx} = {\\bf 0} $ , the detected signal vector in (REF ) becomes ${\\bf {\\hat{x}}}^{\\rm DFD} = ({\\bf {G}}_{\\rm {D}}{\\bf {G}}_{\\rm {A}}{\\bf {H}}{\\bf {F}}_{\\rm {A}}{\\bf {F}}_{\\rm {D}}-{\\bf {B}}^{\\rm Rx}){\\bf {x}}+{\\bf {G}}_{\\rm {D}}{\\bf {G}}_{\\rm {A}}{\\bf {n}}.$"
],
[
"Unified MSE Matrix for Different Precoders and Processors",
"Based on the general signal model in (REF ), the general MSE matrix of the recovered signal at the destination equals $& {\\Phi }_{\\rm {MSE}}({\\bf {G}}_{\\rm {D}},{\\bf {G}}_{\\rm {A}},{\\bf {F}}_{\\rm {A}},{\\bf {F}}_{\\rm {D}},{\\bf B}^{\\rm Tx},{\\bf B}^{\\rm Rx}) \\nonumber \\\\= \\, & \\mathbb {E}\\lbrace ({\\bf {\\hat{x}}}-{\\bf {x}})({\\bf {\\hat{x}}}-{\\bf {x}})^{\\rm {H}}\\rbrace \\nonumber \\\\=\\, & \\mathbb {E}\\lbrace \\left( {\\bf {G}}_{\\rm {D}}{\\bf {G}}_{\\rm {A}}{\\bf {H}}{\\bf {F}}_{\\rm {A}} {\\bf {F}}_{\\rm {D}}({\\bf B}^{\\rm Rx}+{\\bf {I}})^{-1}{\\bf {x}}-{\\bf {x}}-{\\bf {G}}_{\\rm {D}}{\\bf {G}}_{\\rm {A}}{\\bf {n}} \\right)\\nonumber \\\\&\\times \\left( {\\bf {G}}_{\\rm {D}}{\\bf {G}}_{\\rm {A}}{\\bf {H}}{\\bf {F}}_{\\rm {A}} {\\bf {F}}_{\\rm {D}}({\\bf B}^{\\rm Rx}+{\\bf {I}})^{-1}{\\bf {x}}-{\\bf {x}}-{\\bf {G}}_{\\rm {D}}{\\bf {G}}_{\\rm {A}}{\\bf {n}} \\right)^{\\rm {H}}\\rbrace \\nonumber \\\\= \\, & \\mathbb {E} \\lbrace \\big ( {\\bf {G}}_{\\rm {D}}{\\bf {G}}_{\\rm {A}}{\\bf {H}}{\\bf {F}}_{\\rm {A}} {\\bf {F}}_{\\rm {D}} - ({\\bf B}^{\\rm Rx}+{\\bf {I}})( {\\bf I} + {\\bf B}^{\\rm Tx} ) \\big ) {\\bf b} {\\bf b}^{\\rm H}\\nonumber \\\\& \\times \\big ( {\\bf {G}}_{\\rm {D}}{\\bf {G}}_{\\rm {A}}{\\bf {H}}{\\bf {F}}_{\\rm {A}} {\\bf {F}}_{\\rm {D}} - ({\\bf B}^{\\rm Rx}+{\\bf {I}})( {\\bf I} + {\\bf B}^{\\rm Tx} ) \\big )^{\\rm H} \\rbrace \\nonumber \\\\& \\, + {\\bf {G}}_{\\rm {D}}{\\bf {G}}_{\\rm {A}}{\\bf {R}}_{\\rm {n}}{\\bf {G}}_{\\rm {A}}^{\\rm {H}}{\\bf {G}}_{\\rm {D}}^{\\rm {H}},$ where the third equality is based on ${\\bf {b}}=({\\bf B}^{\\rm Rx}+{\\bf {I}})^{-1}{\\bf {x}}$ given in (REF ).",
"Based on lattice theory, each element of ${\\bf b}$ is identical and independent distributed, i.e., $\\mathbb {E}\\lbrace {\\bf b} {\\bf b}^{\\rm H}\\rbrace \\propto {\\bf {I}}$ [37].",
"Thus, for notational simplicity, we can assume $\\mathbb {E}\\lbrace {\\bf b} {\\bf b}^{\\rm H}\\rbrace ={\\bf {I}}$ in the following derivations.",
"Denote ${\\bf {B}}={\\bf B}^{\\rm Rx}+{\\bf B}^{\\rm Tx}+{\\bf B}^{\\rm Rx}{\\bf B}^{\\rm Tx}$ , then $({\\bf B}^{\\rm Rx}+{\\bf {I}})( {\\bf I} + {\\bf B}^{\\rm Tx} )={\\bf {I}}+{\\bf {B}}.$ It is obvious that ${\\bf {B}}$ is a strictly lower-triangular matrix based on the definitions of ${\\bf B}^{\\rm Tx}$ and ${\\bf B}^{\\rm Rx}$ , which implies that using nonlinear precoding at transmitter and nonlinear detection at the receiver at the same time is equivalent to just one of two.",
"Therefore, nonlinear precoding at the transmitter and nonlinear detection at the receiver are equivalent and only one is enough.",
"Direct matrix derivation [38] yields that the optimal $ {\\bf G}_{\\rm D} $ will be ${\\bf {G}}_{\\rm D}^{\\rm opt} & = ({\\bf {I}}+{\\bf B}) ({\\bf {G}}_{\\rm {A}}{\\bf {H}}{\\bf {F}}_{\\rm {A}}{\\bf {F}}_{\\rm {D}})^{\\rm {H}}\\nonumber \\\\& \\ \\ \\ \\times [({\\bf {G}}_{\\rm {A}}{\\bf {H}}{\\bf {F}}_{\\rm {A}}{\\bf {F}}_{\\rm {D}})({\\bf {G}}_{\\rm {A}}{\\bf {H}}{\\bf {F}}_{\\rm {A}}{\\bf {F}}_{\\rm {D}})^{\\rm {H}}+{\\bf {G}}_{\\rm {A}} {\\bf {R}}_{\\rm {n}}{\\bf {G}}_{\\rm {A}}^{\\rm {H}}]^{-1}.$ That is, the general MSE matrix can be further simplified into $& {\\Phi }_{\\rm {MSE}}({\\bf {G}}_{\\rm {A}},{\\bf {F}}_{\\rm {A}},{\\bf {F}}_{\\rm {D}},{\\bf B}) \\nonumber \\\\= \\, & {\\Phi }_{\\rm {MSE}}({\\bf {G}}_{\\rm {D}}^{\\rm {opt}},{\\bf {G}}_{\\rm {A}},{\\bf {F}}_{\\rm {A}},{\\bf {F}}_{\\rm {D}},{\\bf B}^{\\rm Tx},{\\bf B}^{\\rm Rx}) \\nonumber \\\\= \\, & ( {\\bf I} + {\\bf B}) ({\\bf I} + {\\bf F}_{\\rm D}^{\\rm H} {\\bf F}_{\\rm A}^{\\rm H} {\\bf H}^{\\rm H} {\\bf G}_{\\rm A}^{\\rm H}({\\bf G}_{\\rm A} {\\bf R}_{\\rm n} {\\bf G}_{\\rm A}^{\\rm H})^{-1} {\\bf G}_{\\rm A} {\\bf H} {\\bf F}_{\\rm A}{\\bf F}_{\\rm D})^{-1}\\nonumber \\\\&\\times ({\\bf I} + {\\bf B} )^{\\rm {H}} \\nonumber \\\\\\preceq \\, & {\\Phi }_{\\rm {MSE}}({\\bf {G}}_{\\rm {D}},{\\bf {G}}_{\\rm {A}},{\\bf {F}}_{\\rm {A}},{\\bf {F}}_{\\rm {D}},{\\bf B}^{\\rm Tx},{\\bf B}^{\\rm Rx}),$ for any $ {\\bf G}_{\\rm D} $ .",
"If $ {\\bf B}= {\\bf 0} $ in (REF ) and (REF ), the results are reduced to linear transceiver.",
"Specifically, the corresponding digital LMMSE processor for linear transceiver is given as follows ${\\bf {G}}_{\\rm D,L}^{\\rm {opt}} = & ({\\bf {G}}_{\\rm {A}}{\\bf {H}}{\\bf {F}}_{\\rm {A}}{\\bf {F}}_{\\rm {D}})^{\\rm {H}}[({\\bf {G}}_{\\rm {A}}{\\bf {H}}{\\bf {F}}_{\\rm {A}}{\\bf {F}}_{\\rm {D}})({\\bf {G}}_{\\rm {A}}{\\bf {H}}{\\bf {F}}_{\\rm {A}}{\\bf {F}}_{\\rm {D}})^{\\rm {H}}\\nonumber \\\\&+{\\bf {G}}_{\\rm {A}}{\\bf {R}}_{\\rm {n}}{\\bf {G}}_{\\rm {A}}^{\\rm {H}}]^{-1},$ and the MSE matrix for linear transceiver is $& {\\Phi }_{\\rm {MSE}}^{\\rm {L}}({\\bf {G}}_{\\rm {A}},{\\bf {F}}_{\\rm {A}},{\\bf {F}}_{\\rm {D}})\\nonumber \\\\=&{\\Phi }_{\\rm {MSE}}({\\bf {G}}_{\\rm {A}},{\\bf {F}}_{\\rm {A}},{\\bf {F}}_{\\rm {D}},{\\bf 0})\\nonumber \\\\= &\\left[{\\bf {I}} + {\\bf {F}}_{\\rm {D}}^{\\rm {H}} {\\bf {F}}_{\\rm {A}}^{\\rm {H}} {\\bf {H}}^{\\rm {H}} {\\bf {G}}_{\\rm {A}}^{\\rm {H}}({\\bf {G}}_{\\rm {A}}{\\bf {R}}_{\\rm {n}}{\\bf {G}}_{\\rm {A}}^{\\rm {H}})^{-1} {\\bf {G}}_{\\rm {A}}{\\bf {H}}{\\bf {F}}_{\\rm {A}}{\\bf {F}}_{\\rm {D}}\\right]^{-1}\\nonumber \\\\\\triangleq &\\left[{\\bf {I}}+{\\Gamma }({\\bf {G}}_{\\rm {A}}, {\\bf {F}}_{\\rm {D}}, {\\bf {F}}_{\\rm {A}})\\right]^{-1},$ where ${\\Gamma }({\\bf {G}}_{\\rm {A}}, {\\bf {F}}_{\\rm {D}}, {\\bf {F}}_{\\rm {A}})={\\bf {F}}_{\\rm {D}}^{\\rm {H}} {\\bf {F}}_{\\rm {A}}^{\\rm {H}} {\\bf {H}}^{\\rm {H}} {\\bf {G}}_{\\rm {A}}^{\\rm {H}}({\\bf {G}}_{\\rm {A}}{\\bf {R}}_{\\rm {n}}{\\bf {G}}_{\\rm {A}}^{\\rm {H}})^{-1} {\\bf {G}}_{\\rm {A}}{\\bf {H}}{\\bf {F}}_{\\rm {A}}{\\bf {F}}_{\\rm {D}}$ , which is signal-to-noise ratio for single antenna case.",
"For the nolinear transceivers, ${\\bf {B}}={\\bf B}^{\\rm Tx}$ for THP or ${\\bf {B}}={\\bf B}^{\\rm Rx}$ for DFD in (REF )-(REF ).",
"Based on (REF ) and (REF ), the general MSE matrix for nonlinear transceivers can also be written in the following unified formula ${\\Phi }_{\\rm {MSE}}({\\bf {G}}_{\\rm {A}}, {\\bf {F}}_{\\rm {A}}, {\\bf {F}}_{\\rm {D}}, {\\bf {B}})&= ({\\bf {I}} + {\\bf {B}}) {\\Phi }_{\\rm {MSE}}^{\\rm {L}}({\\bf {G}}_{\\rm {A}}, {\\bf {F}}_{\\rm {A}}, {\\bf {F}}_{\\rm {D}}) \\nonumber \\\\&\\times ({\\bf {I}} + {\\bf {B}})^{\\rm H},$ which turns into the MSE matrix in (REF ) when ${\\bf {B}}={\\bf {0}}$ .",
"In the following, we will investigate unified hybrid MIMO transceiver optimization, which is applicable to various objective functions based on on the general MSE matrix (REF )."
],
[
"The Unified Hybrid MIMO Transceiver Optimization",
"Because of the multi-objective optimization nature for MIMO systems with multiple data streams, there are different kinds of objectives that reflect different design preferences [39].",
"All can be regarded as a matrix monotonic function of the data estimation MSE matrix in (REF ) [40].",
"A function $f(\\cdot )$ is a matrix monotone increasing function if $ f({\\bf X}) \\ge f({\\bf Y}) $ for $ {\\bf X} \\succeq {\\bf Y} \\succeq {\\bf 0}$ [40].",
"To avoid case-by-case discussion, we will investigate in depth hybrid MIMO transceiver optimization with different performance metrics from a unified viewpoint, in this section.",
"Based on the MSE matrix in (REF ), the unified hybrid MIMO transceiver design can be formulated in the following form $\\min _{{\\bf {G}}_{\\rm {A}},{\\bf {F}}_{\\rm {A}},{\\bf {F}}_{\\rm {D}},{\\bf {B}}} \\ & f \\left({\\Phi }_{\\rm {MSE}}({\\bf {G}}_{\\rm {A}},{\\bf {F}}_{\\rm {A}},{\\bf {F}}_{\\rm {D}},{\\bf {B}})\\right) \\nonumber \\\\{\\rm {s.t.}}",
"\\qquad & {\\rm {Tr}}({\\bf {F}}_{\\rm {A}}{\\bf {F}}_{\\rm {D}}{\\bf {F}}_{\\rm {D}}^{\\rm {H}}{\\bf {F}}_{\\rm {A}}^{\\rm {H}})\\le P \\nonumber \\\\& {\\bf {F}}_{\\rm {A}} \\in \\mathcal {F}, \\ \\ {\\bf {G}}_{\\rm {A}} \\in \\mathcal {G},$ where $f(\\cdot )$ is a matrix monotone increasing function [40].",
"The sets $\\mathcal {F}$ and $\\mathcal {G}$ are the feasible analog precoder set and analog processor set satisfying constant-modulus constraint, and $ P $ denotes the maximum transmit power at the source."
],
[
"Specific Objective Functions",
"There are many ways to choose the matrix monotone increasing function.",
"In this subsection, we will investigate the properties of different objective functions in (REF ).",
"One group of matrix monotone increasing functions can be expressed as $&f_1\\left({\\Phi }_{\\rm {MSE}}({\\bf {G}}_{\\rm {A}},{\\bf {F}}_{\\rm {A}},{\\bf {F}}_{\\rm {D}},{\\bf {B}})\\right)\\nonumber \\\\&=f_{\\rm {Schur}} \\left({\\bf {d}}({\\Phi }_{\\rm {MSE}}({\\bf {G}}_{\\rm {A}},{\\bf {F}}_{\\rm {A}},{\\bf {F}}_{\\rm {D}},{\\bf {B}}))\\right),$ where ${\\bf {d}}({\\Phi }_{\\rm {MSE}}({\\bf {G}}_{\\rm {A}},{\\bf {F}}_{\\rm {A}},{\\bf {F}}_{\\rm {D}},{\\bf {B}}))$ is a vector consisting of the diagonal elements of the matrix ${\\Phi }_{\\rm {MSE}}({\\bf {G}}_{\\rm {A}},{\\bf {F}}_{\\rm {A}},{\\bf {F}}_{\\rm {D}},{\\bf {B}})$ and $f_{\\rm {Schur}}(\\cdot )$ is a function of a vector satisfying one the following four properties discussed in Appendix : Multiplicatively Schur-convex Multiplicatively Schur-concave Additively Schur-convex Additively Schur-concave.",
"Many widely used metrics can be regarded as a special case of this group of functions [39], [37], [36].",
"Conclusion 1: For linear transceiver, the feedback matrix ${\\bf {B}}$ in (REF ) is an all-zero matrix, i.e., $ {\\bf B}^{\\rm {opt}} = {\\bf 0} $ .",
"For nonlinear transceiver, from Appendix the optimal feedback matrix ${\\bf {B}}$ for $f_1(\\cdot )$ is ${\\bf {B}}^{\\rm {opt}}={\\rm {diag}}\\lbrace [[{\\bf {L}}]_{1,1},\\cdots ,[{\\bf {L}}]_{L,L}]^{\\rm {T}}\\rbrace {\\bf {L}}^{-1}-{\\bf {I}},$ where ${\\bf {L}}$ is a lower triangular matrix of the following Cholesky decomposition ${\\Phi }_{\\rm {MSE}}^{\\rm {L}}({\\bf {G}}_{\\rm {A}},{\\bf {F}}_{\\rm {A}},{\\bf {F}}_{\\rm {D}})={\\bf {L}}{\\bf {L}}^{\\rm {H}}.$ It has been proved in [40] and [38] that for nonlinear transceiver design each data stream will have the same performance if $f_{\\rm {Schur}}(\\cdot )$ in (REF ) is multiplicatively Schur-convex.",
"On the other hand, if $f_{\\rm {Schur}}(\\cdot )$ in (REF ) is multiplicatively Schur-concave, for nonlinear transceiver design the objective function includes geometrically weighted signal-to-noise-plus-interference-ratio (SINR) maximization as its special case.",
"If $f_{\\rm {Schur}}(\\cdot )$ in (REF ) is additively Schur-convex, the objective function includes the the maximum MSE minimization and the minimum BER with the same constellation on each data stream as special cases.",
"If $f_{\\rm {Schur}}(\\cdot )$ in (REF ) is additively Schur-concave, the objective function includes weighted MSE minimization as its special case.",
"Additive Schur functions are usually used for linear transceivers (${\\bf {B}}={\\bf {0}}$ in (REF )) since closed-form solutions can be obtained in this case.",
"Besides the above group of matrix monotone increasing functions, we can choose one to reflect capacity and MSE for linear transceivers.",
"Capacity is one of the most popular performance metrics in MIMO transceiver optimization.",
"It can be expressed as the form of MSE matrix considering the well-known relationship between the MSE matrix and capacity [40], i.e., $ C = -{\\rm log}| {\\bf \\Phi }_{\\rm MSE} | $ .",
"Then, the objective can be given as $& f_{\\rm 2} (\\cdot )= {\\rm {log}}|{\\Phi }_{\\rm {MSE}}^{\\rm {L}}({\\bf {G}}_{\\rm {A}},{\\bf {F}}_{\\rm {A}},{\\bf {F}}_{\\rm {D}})|.$ MSE is another widely used performance metric that demonstrates how accurately a signal can be recovered.",
"The corresponding weighted MSE minimization objective is $& f_{\\rm 3} (\\cdot )= {\\rm {Tr}}\\left[{\\bf {A}}^{\\rm {H}}{\\Phi }_{\\rm {MSE}}^{\\rm {L}}({\\bf {G}}_{\\rm {A}},{\\bf {F}}_{\\rm {A}},{\\bf {F}}_{\\rm {D}}){\\bf {A}}\\right],$ where $ {\\bf A} $ is a general, not necessarily diagonal, weight matrix, even if it is often diagonal in many applications."
],
[
"Hybrid MIMO Transceiver Optimization",
"Denote ${\\Pi }_{\\rm {L}} & =( {\\bf {G}}_{\\rm {A}} {\\bf {R}}_{\\rm {n}} {\\bf {G}}_{\\rm {A}}^{\\rm {H}} )^{-1/2}{\\bf {G}}_{\\rm {A}}{\\bf {R}}_{\\rm {n}}^{1/2}, \\nonumber \\\\{\\Pi }_{\\rm {R}} & = {\\bf {F}}_{\\rm {A}} ( {\\bf {F}}_{\\rm {A}}^{\\rm {H}} {\\bf {F}}_{\\rm {A}} )^{-\\frac{1}{2}}, \\nonumber $ and ${\\bf {\\tilde{F}}}_{\\rm {D}} & = ( {\\bf {F}}_{\\rm {A}}^{\\rm {H}} {\\bf {F}}_{\\rm {A}} )^{\\frac{1}{2}}{\\bf { F}}_{\\rm {D}}{\\bf {Q}}^{\\rm {H}},$ where ${\\bf {Q}}$ is a unitary matrix to be determined by digital transceiver optimization in the next section.",
"Then (REF ) can be rewritten as $ {\\Gamma }({\\bf {G}}_{\\rm {A}},{\\bf {F}}_{\\rm {D}},{\\bf {F}}_{\\rm {A}})={\\bf {Q}}^{\\rm {H}}{\\tilde{\\Gamma }}({\\bf {G}}_{\\rm {A}},{\\bf {\\tilde{F}}}_{\\rm {D}},{\\bf {F}}_{\\rm {A}}){\\bf {Q}},$ where $& {\\tilde{\\Gamma }}({\\bf {G}}_{\\rm {A}},{\\bf {\\tilde{F}}}_{\\rm {D}},{\\bf {F}}_{\\rm {A}})= {\\bf {\\tilde{F}}}_{\\rm {D}}^{\\rm {H}}{\\Pi }_{\\rm {R}}^{\\rm {H}}{\\bf {H}}^{\\rm {H}}{\\bf {R}}_{\\rm {n}}^{-1/2}{\\Pi }_{\\rm {L}}^{\\rm {H}}{\\Pi }_{\\rm {L}} {\\bf {R}}_{\\rm {n}}^{-1/2}{\\bf {H}}{\\Pi }_{\\rm {R}}{\\bf {\\tilde{F}}}_{\\rm {D}}.$ The optimal ${\\bf {B}}$ is usually a function of ${\\Gamma }({\\bf {G}}_{\\rm {A}},{\\bf {F}}_{\\rm {D}},{\\bf {F}}_{\\rm {A}})$ , for all objective functions as demonstrated by (REF ) for $f_{\\rm {Schur}}(\\cdot )$ in (REF ).",
"From (REF ), we can conclude that the optimal ${\\bf {B}}$ is a function of ${\\bf {Q}}^{\\rm {H}}{\\tilde{\\Gamma }}({\\bf {G}}_{\\rm {A}},{\\bf {\\tilde{F}}}_{\\rm {D}},{\\bf {F}}_{\\rm {A}}){\\bf {Q}}$ .",
"Therefore, using (REF ) and (REF ), the objective function of (REF ) can be expressed in terms of ${\\tilde{\\Gamma }}({\\bf {G}}_{\\rm {A}},{\\bf {\\tilde{F}}}_{\\rm {D}},{\\bf {F}}_{\\rm {A}})$ as $& f \\left( {\\Phi }_{\\rm {MSE}}({\\bf {G}}_{\\rm {A}},{\\bf {F}}_{\\rm {A}},{\\bf {F}}_{\\rm {D}},{\\bf {B}}^{\\rm {opt}}) \\right) \\nonumber \\\\= & f \\big ( ({\\bf {I}}+{\\bf {B}}^{\\rm opt}) ({\\bf I} + {\\bf {Q}}^{\\rm {H}}{\\tilde{\\Gamma }} ({\\bf {G}}_{\\rm {A}},{\\bf {\\tilde{F}}}_{\\rm {D}},{\\bf {F}}_{\\rm {A}}) {\\bf {Q}} )^{-1} ({\\bf {I}}+{\\bf {B}}^{\\rm opt})^{\\rm {H}} \\big ) \\nonumber \\\\\\triangleq & f_{{\\rm S}} \\left({\\bf {Q}}^{\\rm {H}}{\\tilde{\\Gamma }}({\\bf {G}}_{\\rm {A}},{\\bf {\\tilde{F}}}_{\\rm {D}},{\\bf {F}}_{\\rm {A}}){\\bf {Q}}\\right).$ After introducing ${\\tilde{\\Gamma }}({\\bf {G}}_{\\rm {A}},{\\bf {\\tilde{F}}}_{\\rm {D}},{\\bf {F}}_{\\rm {A}})$ and a new auxiliary matrix $ {\\bf Q} $ , the objective function is transferred into $ f_{{\\rm S}}(\\cdot )$ rather than $ f(\\cdot ) $ .",
"Note that this new function notation, $ f_{{\\rm S}}(\\cdot )$ , is defined only for notational simplicity and it explicitly expresses the objective as a function of matrix variables ${\\bf Q}$ and ${\\tilde{\\Gamma }}$ .",
"Therefore, the optimization problem in (REF ) is further rewritten into the following one $\\min _{{\\bf {Q}},{\\bf {G}}_{\\rm {A}},{\\bf {\\tilde{F}}}_{\\rm {D}},{\\bf {F}}_{\\rm {A}}} \\ & f_{{\\rm S}} \\left({\\bf {Q}}^{\\rm {H}}{\\tilde{\\Gamma }}({\\bf {G}}_{\\rm {A}},{\\bf {\\tilde{F}}}_{\\rm {D}},{\\bf {F}}_{\\rm {A}}){\\bf {Q}}\\right) \\nonumber \\\\{\\rm {s.t.}}",
"\\ \\ \\ \\ \\ \\ & {\\rm {Tr}}({\\bf {\\tilde{F}}}_{\\rm {D}}{\\bf {\\tilde{F}}}_{\\rm {D}}^{\\rm {H}})\\le P \\nonumber \\\\& {\\bf {F}}_{\\rm {A}} \\in \\mathcal {F}, \\ \\ {\\bf {G}}_{\\rm {A}} \\in \\mathcal {G}.$ We will discuss in detail how to solve the optimization problem (REF ) with respect to ${\\bf {Q}}$ , ${\\bf {G}}_{\\rm {A}}$ ,${\\bf {\\tilde{F}}}_{\\rm {D}}$ , and ${\\bf {F}}_{\\rm {A}}$ subsequently.",
"In (REF ), ${\\bf {B}}$ has been formulated as a function of ${\\bf {Q}}$ , ${\\bf {G}}_{\\rm {A}}$ ,${\\bf {\\tilde{F}}}_{\\rm {D}}$ , and ${\\bf {F}}_{\\rm {A}}$ .",
"When ${\\bf {Q}}$ , ${\\bf {G}}_{\\rm {A}}$ ,${\\bf {\\tilde{F}}}_{\\rm {D}}$ , and ${\\bf {F}}_{\\rm {A}}$ are calculated, the optimal ${\\bf {B}}$ can be directly derived based on (REF )."
],
[
"Digital Transceiver Optimization",
"In the following, we focus on the digital transceiver optimization for the optimization problem (REF ).",
"More specifically, we first derive the optimal unitary matrix ${\\bf {Q}}$ and then find the optimal ${\\bf {\\tilde{F}}}_{\\rm {D}}$ ."
],
[
"Optimal $ {\\bf {Q}} $",
"At the beginning of this section, two fundamental definitions are given based on the following eigenvalue decomposition (EVD) and SVD $&{\\tilde{\\Gamma }}({\\bf {G}}_{\\rm {A}},{\\bf {\\tilde{F}}}_{\\rm {D}},{\\bf {F}}_{\\rm {A}})={\\bf {U}}_{{\\tilde{\\Gamma }}}{\\Lambda }_{{\\tilde{\\Gamma }}}{\\bf {U}}_{{\\tilde{\\Gamma }}}^{\\rm {H}} \\nonumber \\\\&{\\bf {A}}={\\bf {U}}_{{\\bf {A}}}{\\Lambda }_{{\\bf {A}}}{\\bf {V}}_{{\\bf {A}}}^{\\rm {H}},$ where $ {\\Lambda }_{\\rm \\Phi } $ and $ {\\Lambda }_{\\rm A} $ denote a diagonal matrix with the diagonal elements in nondecreasing order.",
"Denote ${\\bf {U}}_{\\rm {GMD}}$ as the unitary matrix that makes the lower triangular matrix ${\\bf {L}}$ in (REF ) has the same diagonal elements.",
"It has been shown in [39], [40], [38] that the optimal ${\\bf {Q}}$ for the first group of matrix-monotonic functions can be expressed as ${\\bf {Q}}^{\\rm {opt}} = \\left\\lbrace {\\begin{array}{l}{{\\bf {U}}_{{\\tilde{\\Gamma }}}{\\bf {U}}_{\\text{GMD}}^{\\rm {H}}}\\ \\ \\text{if} \\ f(\\cdot ) \\ \\text{is multiplicatively Schur-convex} \\\\{{\\bf {U}}_{{\\tilde{\\Gamma }}}} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\text{if} \\ f(\\cdot ) \\ \\text{is multiplicatively Schur-concave} \\\\{{\\bf {U}}_{{\\tilde{\\Gamma }}}{\\bf {U}}_{\\text{DFT}}^{\\rm {H}}}\\ \\ \\ \\text{if} \\ f(\\cdot ) \\ \\text{is additively Schur-convex}\\\\{{\\bf {U}}_{{\\tilde{\\Gamma }}}} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\text{if} \\ f(\\cdot ) \\ \\text{is additively Schur-concave}.\\end{array}} \\right.$ The above results are obtained by directly manipulating with the objective function $f(\\cdot )$ in (REF ), and thus the optimal ${\\bf {Q}}$ varies with the matrix-monotone increasing function in (REF ).",
"For the capacity maximization in (REF ), the objective function of (REF ) can be written as $& f_{\\rm S, 2}(\\cdot ) = -{\\rm {log}} |{\\bf {Q}}^{\\rm {H}} {\\tilde{\\Gamma }} ({\\bf {G}}_{\\rm {A}},{\\bf {\\tilde{F}}}_{\\rm {D}},{\\bf {F}}_{\\rm {A}}) {\\bf {Q}} + {\\bf {I}}|.$ Since the function in (REF ) is independent of $ {\\bf Q} $ as long as it is a unitary matrix, the optimal $ {\\bf Q} $ , namely $ {\\bf Q}^{\\rm opt} $ , can be any unitary matrix with proper dimension.",
"For the weighted MSE minimization given by (REF ), the objective function of (REF ) can be rewritten as $f_{\\rm S, 3} (\\cdot ) & ={\\rm {Tr}} [{\\bf {A}}^{\\rm {H}} ({\\bf {Q}}^{\\rm {H}} {\\tilde{\\Gamma }} ({\\bf {G}}_{\\rm {A}},{\\bf {\\tilde{F}}}_{\\rm {D}},{\\bf {F}}_{\\rm {A}}) {\\bf {Q}}+ {\\bf {I}})^{-1} {\\bf {A}}].$ Based on the EVD and SVD defined in (REF ) and the matrix inequality in Appendix , the optimal ${\\bf {Q}}$ is ${\\bf {Q}}^{\\rm opt} = {\\bf {U}}_{{\\tilde{\\Gamma }}}{\\bf { U}}_{{\\bf {A}}}^{\\rm {H}}.$ We have to stress that it is still hard to find the closed-form expression for the optimal ${\\bf {Q}}$ for an arbitrary function $f(\\cdot )$ .",
"However, most of the meaningful and popular metric functions have been shown included in one of the above function families, and are with the closed-form expression for optimal ${\\bf {Q}}$ ."
],
[
"Optimal ${\\bf {\\tilde{F}}}_{\\rm {D}}$",
"After substituting the optimal ${\\bf {Q}}$ into the objective function of (REF ), the objective function becomes a function of the eigenvalues of ${\\tilde{\\Gamma }}({\\bf {G}}_{\\rm {A}}, {\\bf {\\tilde{F}}}_{\\rm {D}}, {\\bf {F}}_{\\rm {A}})$ , i.e., $& f_{\\rm {S}}\\left({\\bf {Q}}^{\\rm {H}}{\\tilde{\\Gamma }}({\\bf {G}}_{\\rm {A}},{\\bf {\\tilde{F}}}_{\\rm {D}},{\\bf {F}}_{\\rm {A}}){\\bf {Q}}\\right)\\triangleq f_{\\rm {E}} \\left({\\lambda } \\Big ( {\\tilde{\\Gamma }}({\\bf {G}}_{\\rm {A}}, {\\bf {\\tilde{F}}}_{\\rm {D}}, {\\bf {F}}_{\\rm {A}}) \\Big ) \\right),$ where ${\\lambda }({X})=[\\lambda _1({X}), \\cdots , \\lambda _L({X})]^{\\rm {T}}$ and $\\lambda _i({X})$ is the $ i $ th largest eigenvalue of ${X}$ .",
"It is worth highlighting that for $f_{{\\rm {S}},1}(\\cdot )$ and $f_{{\\rm {S}},3}(\\cdot )$ based on (REF ) and (REF ) we can directly have (REF ).",
"For $f_{{\\rm {S}},2}(\\cdot )$ the optimal ${\\bf {Q}}$ can be an arbitrary unitary matrix, minimizing $f_{{\\rm {S}},2}(\\cdot )$ mathematically equals to minimizing $-\\sum _{l=1}^L{\\rm {log}}\\left(1+{\\lambda }_l ( {\\tilde{\\Gamma }}({\\bf {G}}_{\\rm {A}}, {\\bf {\\tilde{F}}}_{\\rm {D}}, {\\bf {F}}_{\\rm {A}}))\\right)$ for any ${\\bf {Q}}$ .",
"In other words, (REF ) always holds for these kinds of functions discussed above.",
"Note that the definition in (REF ) follows from the facts that the unitary matrix in ${\\tilde{\\Gamma }}({\\bf {G}}_{\\rm {A}}, {\\bf {\\tilde{F}}}_{\\rm {D}}, {\\bf {F}}_{\\rm {A}})$ has been removed by the optimal ${\\bf {Q}}$ and only its eigenvalues remain to be optimized.",
"Therefore, the unified hybrid MIMO transceiver optimization in (REF ) is simplified to $\\min _{{\\bf {G}}_{\\rm {A}},{\\bf {\\tilde{F}}}_{\\rm {D}},{\\bf {F}}_{\\rm {A}}} \\ \\ & f_{\\rm {E}}\\left({\\lambda }\\left({\\tilde{\\Gamma }}({\\bf {G}}_{\\rm {A}},{\\bf {\\tilde{F}}}_{\\rm {D}},{\\bf {F}}_{\\rm {A}})\\right)\\right) \\nonumber \\\\{\\rm {s.t.}}",
"\\ \\ \\ & {\\rm {Tr}}({\\bf {\\tilde{F}}}_{\\rm {D}}{\\bf {\\tilde{F}}}_{\\rm {D}}^{\\rm {H}})\\le P \\nonumber \\\\& {\\bf {F}}_{\\rm {A}} \\in \\mathcal {F}, \\ \\ {\\bf {G}}_{\\rm {A}} \\in \\mathcal {G}.$ By applying the obtained results of ${\\bf Q}^{\\rm opt}$ and the fact that $ f(\\cdot ) $ is a matrix-monotone increasing function, it can be concluded from the discussion in [39], [38] that $ f_{{\\rm E}}(\\cdot ) $ is a vector-decreasing function for $f_{{\\rm {S}},1}(\\cdot )$ .",
"Moreover, substituting the optimal ${\\bf {Q}}$ into the objective function of (REF ), for $f_{{\\rm {S}},2}(\\cdot )$ and $f_{{\\rm {S}},3}(\\cdot )$ we have $f_{{\\rm {E}}}(\\cdot )&=-\\sum _{l=1}^L{\\rm {log}}\\left(1+{\\lambda }_l ( {\\tilde{\\Gamma }}({\\bf {G}}_{\\rm {A}}, {\\bf {\\tilde{F}}}_{\\rm {D}},{\\bf {F}}_{\\rm {A}}))\\right), \\\\f_{{\\rm {E}}}(\\cdot )&=\\sum _{l=1}^L\\frac{\\lambda _l({\\bf {A}})}{1+{\\lambda }_l ( {\\tilde{\\Gamma }}({\\bf {G}}_{\\rm {A}}, {\\bf {\\tilde{F}}}_{\\rm {D}}, {\\bf {F}}_{\\rm {A}}))},$ respectively, which implies that $ f_{{\\rm E}}(\\cdot ) $ is also vector-decreasing.",
"In a nutshell, based on ${\\bf Q}^{\\rm opt}$ we can conclude that $ f_{{\\rm E}}(\\cdot ) $ in (REF ) is a vector-decreasing function.",
"Thus, from (REF ), the optimization becomes maximizing the eigenvalues of ${\\tilde{\\Gamma }}({\\bf {G}}_{\\rm {A}},{\\bf {\\tilde{F}}}_{\\rm {D}},{\\bf {F}}_{\\rm {A}})$ .",
"Each eigenvalue of ${\\tilde{\\Gamma }}({\\bf {G}}_{\\rm {A}},{\\bf {\\tilde{F}}}_{\\rm {D}},{\\bf {F}}_{\\rm {A}})$ corresponds to SNR of an eigenchannel.",
"In problem (REF ), the variables are still matrix variables.",
"To simplify the optimization, we will first derive the diagonalizable structure of the optimal matrix variables.",
"Based on the derived optimal structure, the dimensionality of the optimization problem are reduced significantly.",
"In order to derive the optimal structure and to avoid tedious case-by-case discussion, we consider a multi-objective optimization problem in the following.",
"Its Pareto optimal solution set contains all the optimal solutions of different types of transceiver optimizations.",
"In particular, as discussed in [40], the optimal solution of problem (REF ) with a specific objective function, i.e., $f_1(\\cdot )$ , $f_{2}(\\cdot )$ , or $f_3(\\cdot )$ , must be in the Pareto optimal solution set of the following vector optimization (multi-objective) problem $& \\ \\ \\max _{{\\bf {G}}_{\\rm {A}},{\\bf {\\tilde{F}}}_{\\rm {D}},{\\bf {F}}_{\\rm {A}}} \\ \\ {\\lambda }\\left({\\tilde{\\Gamma }}({\\bf {G}}_{\\rm {A}},{\\bf {\\tilde{F}}}_{\\rm {D}},{\\bf {F}}_{\\rm {A}})\\right) \\nonumber \\\\& \\ \\ \\ \\ \\ \\ \\ {\\rm {s.t.}}",
"\\ \\ \\ \\ \\ \\ {\\rm {Tr}}({\\bf {\\tilde{F}}}_{\\rm {D}}{\\bf {\\tilde{F}}}_{\\rm {D}}^{\\rm {H}})\\le P \\nonumber \\\\& \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ {\\bf {F}}_{\\rm {A}} \\in \\mathcal {F}, \\ \\ {\\bf {G}}_{\\rm {A}} \\in \\mathcal {G}.$ Equivalently, the vector optimization problem in (REF ) can be rewritten as the following matrix-monotonic optimization problem $& \\ \\ \\max _{{\\bf {G}}_{\\rm {A}},{\\bf {\\tilde{F}}}_{\\rm {D}},{\\bf {F}}_{\\rm {A}}} \\ \\ {\\tilde{\\Gamma }}({\\bf {G}}_{\\rm {A}},{\\bf {\\tilde{F}}}_{\\rm {D}},{\\bf {F}}_{\\rm {A}}) \\nonumber \\\\& \\ \\ \\ \\ \\ \\ \\ {\\rm {s.t.}}",
"\\ \\ \\ \\ \\ {\\rm {Tr}}({\\bf {\\tilde{F}}}_{\\rm {D}}{\\bf {\\tilde{F}}}_{\\rm {D}}^{\\rm {H}})\\le P \\nonumber \\\\& \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ {\\bf {F}}_{\\rm {A}} \\in \\mathcal {F}, \\ \\ {\\bf {G}}_{\\rm {A}} \\in \\mathcal {G}.$ It is worth noting that optimization (REF ) aims at maximizing a positive semi-definite matrix.",
"Generally speaking, maximizing a positive semi-definite matrix includes two tasks, i.e., maximizing its eigenvalues and choosing a proper EVD unitary matrix.",
"Note that in (REF ) there is no need to optimize the EVD unitary matrix, because the constraints can remain satisfied if only EVD unitary matrix changes.",
"Using the definitions in (REF ) and given analog precoder ${\\bf {F}}_{\\rm {A}}$ and analog processor ${\\bf {G}}_{\\rm {A}}$ , problem (REF ) is a standard matrix-monotonic optimization with respect to ${\\bf {\\tilde{F}}}_{\\rm {D}}$ .",
"It follows $& \\ \\ \\max _{{\\bf {\\tilde{F}}}_{\\rm {D}}} \\ \\ {\\bf {\\tilde{F}}}_{\\rm {D}}^{\\rm {H}}{\\Pi }_{\\rm {R}}^{\\rm {H}}{\\bf {H}}^{\\rm {H}}{\\bf {R}}_{\\rm {n}}^{-1/2}{\\Pi }_{\\rm {L}}^{\\rm {H}}{\\Pi }_{\\rm {L}} {\\bf {R}}_{\\rm {n}}^{-1/2}{\\bf {H}}{\\Pi }_{\\rm {R}}{\\bf {\\tilde{F}}}_{\\rm {D}} \\nonumber \\\\& \\ \\ \\ {\\rm {s.t.}}",
"\\ \\ \\ \\ {\\rm {Tr}}({\\bf {\\tilde{F}}}_{\\rm {D}}{\\bf {\\tilde{F}}}_{\\rm {D}}^{\\rm {H}})\\le P.$ Based on the matrix-monotonic optimization theory developed in [40], the optimal solution of (REF ) satisfies the following diagonalizable structure.",
"Conclusion 2: Defining the following SVD, $&{\\Pi }_{\\rm {L}}{\\bf {R}}_{\\rm {n}}^{-1/2}{\\bf {H}}{\\Pi }_{\\rm {R}}={\\bf {U}}_{{\\mathbb {H}}}{\\Lambda }_{\\mathbb {H}}{\\bf {V}}_{\\mathbb {H}}^{\\rm {H}},$ with the diagonal elements of ${\\Lambda }_{\\mathbb {H}}$ in decreasing order, the optimal ${\\bf {\\tilde{F}}}_{\\rm {D}}$ satisfies ${\\bf {\\tilde{F}}}_{\\rm {D}}^{\\rm opt} = {\\bf {V}}_{\\mathbb {H}} {\\Lambda }_{\\bf {F}} {\\bf {U}}_{\\rm {Arb}}^{\\rm {H}},$ where ${\\Lambda }_{\\bf {F}}$ is a diagonal matrix determined by the specific objective functions, e.g., sum MSE, capacity maximization, etc., as discussed in the previous section.",
"The unitary matrix ${\\bf {U}}_{\\rm {Arb}}$ can be an arbitrary unitary matrix.",
"Thus far by using Conclusion 2, the optimal ${\\bf {\\tilde{F}}}_{\\rm {D}}$ can be obtained by conducting basic manipulations as in [40] on optimizing ${\\Lambda }_{\\bf {F}}$ given a specific objective function.",
"As a result, the remaining key task is to optimize the analog precoder and processor, which is the focus of the following section."
],
[
"Analog Transceiver Optimization",
"Based on the optimal solution of digital precoder given in the previous section, we optimize the analog precoder and processor under constant-modulus constraints.",
"In the following, the optimal structure of the analog transceiver is first derived.",
"Different from existing works, we show that the analog precoder and processor design can be decoupled by using the optimal transceiver structure.",
"This optimal structure greatly simplifies the involved analog transceiver design.",
"For the analog transceiver optimization in (REF ) and using (REF ), we have the following matrix-monotonic optimization problem $& \\ \\ \\max _{{\\bf {F}}_{\\rm {A}},{\\bf {G}}_{\\rm {A}}} \\ \\ {\\bf {\\tilde{F}}}_{\\rm {D}}^{\\rm {H}}{\\Pi }_{\\rm {R}}^{\\rm {H}}{\\bf {H}}^{\\rm {H}}{\\bf {R}}_{\\rm {n}}^{-1/2}{\\Pi }_{\\rm {L}}^{\\rm {H}}{\\Pi }_{\\rm {L}} {\\bf {R}}_{\\rm {n}}^{-1/2}{\\bf {H}}{\\Pi }_{\\rm {R}}{\\bf {\\tilde{F}}}_{\\rm {D}} \\nonumber \\\\& \\ \\ \\ {\\rm {s.t.}}",
"\\ \\ \\ \\ \\ \\ {\\bf {F}}_{\\rm {A}} \\in \\mathcal {F}, \\ \\ {\\bf {G}}_{\\rm {A}} \\in \\mathcal {G}.$ Denote the SVDs $ {\\bf {R}}_{\\rm {n}}^{-1/2}{\\bf {H}}& \\triangleq {\\bf {U}}_{{\\mathcal {H}}}{\\Lambda }_{\\mathcal {H}}{\\bf {V}}_{\\mathcal {H}}^{\\rm {H}},\\\\{\\bf {R}}_{\\rm {n}}^{1/2} {\\bf {G}}_{\\rm {A}}^{\\rm {H}}& \\triangleq {\\bf {U}}_{{\\bf {R}}{\\bf {G}}} {\\Lambda }_{{\\bf {R}}{\\bf {G}}}{\\bf {V}}_{{\\bf {R}}{\\bf {G}}}^{\\rm {H}}.$ In Appendix , we prove the following conclusion on the optimal structure of ${\\bf {F}}_{\\rm {A}}$ and ${\\bf {G}}_{\\rm {A}}$ .",
"Conclusion 3: Let the SVD of ${\\bf {F}}_{\\rm {A}}$ be ${\\bf {F}}_{\\rm {A}} \\triangleq {\\bf {U}}_{{\\bf {F}}_{\\rm {A}}}{\\Lambda }_{{\\bf {F}}_{\\rm {A}}}{\\bf {V}}_{{\\bf {F}}_{\\rm {A}}}^{\\rm {H}}.$ The singular values in ${\\Lambda }_{{\\bf {F}}_{\\rm {A}}}$ do not affect the objective function in (REF ), and the unitary matrix ${\\bf {U}}_{{\\bf {F}}_{\\rm {A}}}$ for the optimal ${\\bf {F}}_{\\rm {A}}$ satisfies $ [{\\bf {U}}_{{\\bf {F}}_{\\rm {A}}}]_{:,1:L}^{\\rm {opt}}={\\rm {arg\\,max}}\\lbrace \\Vert [{\\bf {V}}_{\\mathcal {H}}]_{:,1:L} [{\\bf {U}}_{{\\bf {F}}_{\\rm {A}}}]_{:,1:L}^{\\rm {H}}\\Vert _{\\rm {F}}^2\\rbrace .$ On the other hand, denote the SVD of $ {\\bf {R}}_{\\rm {n}}^{1/2} {\\bf {G}}_{\\rm {A}}^{\\rm {H}} $ as ${\\bf {R}}_{\\rm {n}}^{1/2} {\\bf {G}}_{\\rm {A}}^{\\rm {H}} \\triangleq {\\bf {U}}_{{\\bf {R}}{\\bf {G}}} {\\Lambda }_{{\\bf {R}}{\\bf {G}}}{\\bf {V}}_{{\\bf {R}}{\\bf {G}}}^{\\rm {H}}.$ The singular values in ${\\Lambda }_{{\\bf {R}}{\\bf {G}}}$ do not affect the objective in (REF ), and the unitary matrix ${\\bf {U}}_{{\\bf {R}}{\\bf {G}}}$ for the optimal ${\\bf {G}}_{\\rm {A}}$ satisfies $[{\\bf {U}}_{{\\bf {R}}{\\bf {G}}}]_{:,1:L}^{\\rm {opt}}={\\rm {arg\\,max}}\\lbrace \\Vert [{\\bf {U}}_{\\mathcal {H}}]_{:,1:L} [{\\bf {U}}_{{\\bf {R}}{\\bf {G}}}]_{:,1:L}^{\\rm {H}}\\Vert _{\\rm {F}}^2\\rbrace .$ Based on the optimal structure given in Conclusion 3, in the following two kinds of algorithms are proposed to compute the analog precoder and processor.",
"The first one is based on phase projection, which provides better performance while the second one based on a heuristic random selection, is with low complexity."
],
[
"Phase Projection Based Algorithm",
"Analog Precoder Design From Conclusion 3, the optimal analog precoder should select the first $L$ -best eigenchannels.",
"It is challenging to directly optmize ${\\bf {F}}_{\\rm {A}}$ based on (REF ) because of the SVD of a constant-modulus matrix.",
"Alternatively, we resort to finding a matrix in the constant-modulus space with the minimum distance to the space spanned by $[{\\bf {V}}_{\\mathcal {H}}]_{:,1:L} $ .",
"Then, the corresponding optimization problem of analog precoder design can be formulated as $ &\\min _{{\\bf {F}}_{\\rm {A}},{\\mathrm {\\mathbf {\\Lambda }_A}},{\\bf {Q}}_{\\rm {A}}} \\ \\ \\Vert [{\\bf {V}}_{\\mathcal {H}}]_{:,1:L} {\\mathrm {\\mathbf {\\Lambda }_A}}{\\bf {Q}}_{\\rm {A}}-{\\bf {F}}_{\\rm {A}}\\Vert _{\\rm {F}}^2\\nonumber \\\\& \\ \\ \\ \\ {\\rm {s.t.}}",
"\\ \\ \\ \\ \\ \\ {\\bf {Q}}_{\\rm {A}}{\\bf {Q}}_{\\rm {A}}^{\\rm {H}}={\\bf {I}}\\nonumber \\\\& \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ {\\bf {F}}_{\\rm {A}} \\in \\mathcal {F}.$ Different from the existing work [24], the diagonal matrix $ \\mathrm {\\mathbf {\\Lambda }_A} $ and the unitary matrix $ \\mathrm {\\mathbf {Q}_A} $ in our work are jointly optimized to make $ {\\bf F}_{\\rm A} $ as close as possible in the space spanned by $ [{\\bf {V}}_{\\mathcal {H}}]_{:,1:L} $ in terms of Frobenius norm.",
"As there is no constraint on the diagonal matrix $ \\mathrm {\\mathbf {\\Lambda }_A} $ , given matrices $ {\\bf Q}_{\\rm A} $ and $ {\\bf F}_{\\rm A} $ , the optimal $ \\mathbf {\\Lambda }_{\\rm A} $ is $\\mathbf {\\Lambda }_{\\rm {A}}^{\\rm {opt}}= \\mathrm {diag} \\Big \\lbrace \\Re \\big ( {\\mathrm {\\mathbf {Q}_A}} {\\mathrm {\\mathbf {F}_A^H}} [{\\bf {V}}_{\\mathcal {H}}]_{:,1:L} \\big ) \\Big \\rbrace .$ Then we rewrite the objective function in (REF ) as $&\\Vert [{\\bf {V}}_{\\mathcal {H}}]_{:,1:L} \\mathbf {\\Lambda }_{\\rm {A}}{\\bf {Q}}_{\\rm {A}}-{\\bf {F}}_{\\rm {A}}\\Vert _{\\rm {F}}^2 \\nonumber \\\\=&{\\rm {Tr}}([{\\bf {V}}_{\\mathcal {H}}]_{:,1:L}\\mathbf {\\Lambda }_{\\rm {A}}^{\\rm {opt}}(\\mathbf {\\Lambda }_{\\rm {A}}^{\\rm {opt}})^{\\rm {H}}[{\\bf {V}}_{\\mathcal {H}}]_{:,1:L}^{\\rm {H}})\\nonumber \\\\&+{\\rm {Tr}}({\\bf {F}}_{\\rm {A}}{\\bf {F}}_{\\rm {A}}^{\\rm {H}})-2\\Re \\lbrace {\\rm {Tr}}({\\bf {F}}_{\\rm {A}}^{\\rm {H}}[{\\bf {V}}_{\\mathcal {H}}]_{:,1:L} \\mathbf {\\Lambda }_{\\rm {A}}^{\\rm {opt}}{\\bf {Q}}_{\\rm {A}})\\rbrace .$ To minimize (REF ) given ${\\Lambda }_{\\rm {A}}$ and ${\\bf {F}}_{\\rm {A}}$ , the term $\\Re \\lbrace {\\rm {Tr}}({\\bf {F}}_{\\rm {A}}^{\\rm {H}}[{\\bf {V}}_{\\mathcal {H}}]_{:,1:L} {\\mathrm {\\mathbf {\\Lambda }_A}^{\\rm {opt}}}{\\bf {Q}}_{\\rm {A}})\\rbrace $ should be maximized.",
"By applying the matrix inequality [41], the optimal $ {\\bf Q}_{\\rm A} $ is $\\mathbf {Q}_{\\mathrm {A}}^{\\mathrm {opt}} = \\mathbf {V}_{\\mathrm {Q}} \\mathbf {U}_{\\mathrm {Q}}^\\mathrm {H},$ where $\\mathbf {V}_{\\mathrm {Q}}$ and $\\mathbf {U}_{\\mathrm {Q}}$ are defined based on the following SVD ${\\bf {F}}_{\\rm {A}}^\\mathrm {H} \\, [{\\bf {V}}_{\\mathcal {H}}]_{:,1:L} {\\mathbf {\\Lambda }_\\mathrm {A}} = \\mathbf {U}_{\\mathrm {Q}} \\mathbf {\\Sigma }_{\\mathrm {Q}} \\mathbf {V}_{\\mathrm {Q}}^\\mathrm {H}.$ Now that for given $ {\\bf Q}_{\\rm A} $ and $\\mathbf {\\Lambda }_\\mathrm {A} $ , the optimal analog precoder $ \\mathrm {\\mathbf {F}_A} $ is [20] ${\\bf F}_{\\rm A}^{\\rm opt} = {\\bf P}_{\\mathcal {F}} \\left( [ {\\bf {V}}_{\\mathcal {H}}]_{:,1:L} {\\mathrm {\\mathbf {\\Lambda }_A}} {\\bf {Q}}_{\\rm {A}} \\right),$ where the phase projection ${\\bf P}_{\\mathcal {F}}( \\mathrm {\\mathbf {A}} )$ is defined as $\\left[ {\\bf P}_{\\mathcal {F}}( \\mathrm {\\mathbf {A}} ) \\right]_{i,j} ={\\left\\lbrace \\begin{array}{ll}\\left[ \\mathrm {\\mathbf {A}} \\right]_{i,j} / | \\left[ \\mathrm {\\mathbf {A}} \\right]_{i,j} |, & \\text{if} \\ \\left[ \\mathrm {\\mathbf {A}} \\right]_{i,j} \\ne 0 \\\\1, & \\text{otherwise}.\\end{array}\\right.",
"}$ Using (REF ), (REF ) and (REF ), the phased projection based analog precoder optimization is proposed in Algorithm REF .",
"[t] Analog Precoder Design [1] Left singular matrix of equivalent channel $ [{\\bf {V}}_{\\mathcal {H}}]_{:,1:L} $ , algorithm threshold $ \\zeta $ .",
"Initialize ${\\bf {F}}_{\\rm {A}}$ with $\\mathbf {P}_{\\mathcal {F}} \\lbrace [{\\bf {V}}_{\\mathcal {H}}]_{:,1:L} \\rbrace $ .",
"the decrement of the objective function in (REF ) is larger than $ \\zeta $ Calculate ${\\mathbf {\\Lambda }_{\\rm A}}$ based on (REF ).",
"Calculate $\\mathbf {Q}_{\\mathrm {A}} $ based on (REF ).",
"Calculate $\\mathbf {F}_{\\mathrm {A}}$ based on (REF ).",
"Update the decrement value of the objective function in (REF ).",
"$ \\mathbf {F}_{\\mathrm {A}} $ .",
"Analog Processor Design Based on Conclusion 3, the optimal structure of analog processor is similar to the analog precoder, but a bit complicated in that the noise variance is tangled in the analog processor formulation.",
"In this case, the left singular matrix of $ {\\bf {R}}_{\\rm {n}}^{1/2} {\\bf {G}}_{\\rm {A}}^{\\rm {H}} $ is required to match the first $ L $ column of left singular matrix of effective channel, i.e., $ [{\\bf {U}}_{\\mathcal {H}}]_{:,1:L} $ .",
"[ht] Iterative Analog Processor Design [1] The matrix $ [{\\bf {U}}_{\\mathcal {H}}]_{:,1:L} $ , the unitary matrix $ \\mathbf {Q}_{\\mathrm {G}} $ , the diagonal matrix $ \\mathbf {\\Lambda }_{\\mathrm {G}} $ , controlling factor $ \\eta $ , and convergent threshold $ \\upsilon $ .",
"Compute $ \\mathbf {W} $ in (REF ).",
"Initialize constant-modulus processor as ${\\mathbf {r}}_{(0)} = \\frac{\\sqrt{2}}{2}\\mathbf {1} $ .",
"The decrement of the objective function in (REF ) is larger than $ \\upsilon $ Calculate $ \\mathbf {P} $ using (REF ) based on $ \\mathbf {G}_{\\mathrm {A}} $ computed in the previous iteration.",
"Find out the optimal solution of (REF ) based on (REF ).",
"Update the decrement of the objective function in (REF ).",
"Construct $ \\mathbf {G}_{\\mathrm {A}} $ based on the optimal solution in Step 5.",
"$ {\\mathbf {G}}_{\\mathrm {A}} $ .",
"Thus, analogous to the analog precoder design in (REF ), we have the following optimization problem $\\min _{{\\bf {G}}_{\\rm {A}},{\\mathrm {\\mathbf {\\Lambda }_G}},{\\bf {Q}}_{\\rm {G}}} \\;\\; & \\big \\Vert [{\\bf {U}}_{\\mathcal {H}}]_{:,1:L}{\\mathbf {\\Lambda }_\\mathrm {G}} {\\bf {Q}}_{\\rm {G}} - {\\bf {R}}_{\\rm {n}}^{1/2} {\\bf {G}}_{\\rm {A}}^\\mathrm {H} \\big \\Vert _{\\rm {F}}^2\\nonumber \\\\{\\rm {s.t.}}",
"\\quad \\;\\; & {\\bf {Q}}_{\\rm {G}}{\\bf {Q}}_{\\rm {G}}^{\\rm {H}} = {\\bf {I}}\\nonumber \\\\& {\\bf {G}}_{\\rm {A}} \\in \\mathcal {G}.$ The optimization of unitary matrix $ {\\bf {Q}}_{\\rm {G}} $ and diagonal matrix $ {\\mathbf {\\Lambda }_\\mathrm {G}} $ in (REF ) is exactly the same as that for the analog precoder optimization.",
"However, the optimization of the analog processor, ${\\bf {G}}_{\\rm {A}}$ , in (REF ) is different.",
"When noises from different antennas are correlated the analog processor design is more challenging than the analog precoder design.",
"In order to overcome this challenge, problem (REF ) is relaxed to minimize an upper bound of the original objective function.",
"Applying $& \\big \\Vert [{\\bf {U}}_{\\mathcal {H}}]_{:,1:L}{\\mathbf {\\Lambda }_\\mathrm {G}} {\\bf {Q}}_{\\rm {G}} - {\\bf {R}}_{\\rm {n}}^{1/2} {\\bf {G}}_{\\rm {A}}^\\mathrm {H} \\big \\Vert _{\\mathrm {F}}^2 \\\\\\le \\;\\; & \\lambda _{\\mathrm {max}}( \\mathbf {R}_{\\mathrm {n}} )\\big \\Vert {\\bf {R}}_{\\rm {n}}^{-1/2} [{\\bf {U}}_{\\mathcal {H}}]_{:,1:L}{\\mathbf {\\Lambda }_\\mathrm {G}} {\\bf {Q}}_{\\rm {G}} - {\\bf {G}}_{\\rm {A}}^\\mathrm {H} \\big \\Vert _{\\mathrm {F}}^2,$ the objective function of (REF ) is relaxed with $ \\big \\Vert {\\bf {R}}_{\\rm {n}}^{-1/2} [{\\bf {U}}_{\\mathcal {H}}]_{:,1:L} {\\mathbf {\\Lambda }_\\mathrm {G}} {\\bf {Q}}_{\\rm {G}} - {\\bf {G}}_{\\rm {A}}^\\mathrm {H} \\big \\Vert _{\\mathrm {F}}^2$ .",
"Note that solving (REF ) is the same as that for the analog precoder design.",
"It is obvious that this relaxation is tight when $ {\\bf R}_{\\rm n} = \\sigma _n^2 {\\bf I} $ .",
"This relaxation may result in some performance loss.",
"Inspired by the work in [42], an iterative algorithm is also proposed to compute ${\\bf {G}}_{\\rm {A}}$ .",
"The constant modulus constraints is asymptotically satisfied via iteratively updating an additional constraint.",
"This iterative algorithm is given in Algorithm REF , and detailed derivation is given in Appendix ."
],
[
"Random Algorithm",
"The proposed phase projection based analog transceiver design suffers from high computation complexity.",
"This may prohibit the proposed analog transceiver design from practical implementation.",
"In order to reduce complexity, we can randomly generate analog precoder and processor matrices to avoid the heavy computations involved in the phase projection based algorithms.",
"In this random algorithm, we randomly select multiple matrices in the column or row space of ${\\bf H}^{\\rm {H}}$ and use their phase projections as the candidates for the analog transceiver design.",
"Then the best candidate matrix is chosen according to some criterion.",
"Specifically, the random algorithm consists of three steps.",
"First, a series of parameter matrices, denoted by $ \\lbrace \\mathbf {R}_k\\rbrace $ and $\\lbrace {\\mathbf {T}}_k\\rbrace $ , are generated, whose elements are randomly generated following a specific distribution e.g., uniform distribution or Gaussian distribution.",
"Secondly, a series of candidate analog precoder and processor matrices are computed based on the parameter matrices.",
"Specifically, based on the parameter matrices and after computing ${\\bf {H}}^{\\rm {H}}{\\bf {R}}_k$ and ${\\mathbf {T}}_k{\\bf {H}}^{\\rm {H}}$ , the constant-modulus candidate matrices are obtained using their phase projections.",
"Finally, the analog precoder and processor are chosen from these candidates according to the determinant of a certain matrix version SNR matrix.",
"The procedure is detailed in Algorithm REF .",
"[t] Random Algorithm for Analog Transceiver Design [1] The number of transmitter antennas $ N $ , number of RF-Chain $ L $ , selection number $ K $ , probability density function $ f_\\mathrm {trans} (x) $ , and ${\\bf {H}}$ Generate $ K $ parameter matrices, $ {\\bf R}_{1},\\ldots ,{\\bf R}_{K} \\in \\mathbb {C}^{ M \\times L} $ , whose elements are randomly generated based on $ f_\\mathrm {trans} (x) $ .",
"Rotate the channel as $ \\mathrm {\\mathbf {H}^H} {\\bf R}_{k} $ .",
"Calculate ${\\bf F}_{k} = \\mathrm {\\mathbf {P}}_{\\mathcal {F}} \\left( \\mathrm {\\mathbf {H}^H} {\\bf R}_{k} \\right) $ .",
"${\\bf {F}}_{\\rm {max}}={\\arg \\max }_{{\\bf {F}}_i} \\left|{ \\bf F}_{i}^{\\mathrm {H}} {\\bf H}^{\\mathrm {H}} \\mathrm { \\mathbf {R}_n^{-1} } {\\bf H}{\\bf F}_{i} \\right|$ .",
"Generate $ K $ parameter matrices $ {\\bf T}_{1},\\ldots ,{\\bf T}_{K} \\in \\mathbb {C}^{ L \\times N } $ randomly based on $ f_\\mathrm {trans} (x) $ .",
"Rotate the channel as $ {\\bf T}_{k} \\mathrm {\\mathbf {H}^H} $ .",
"Calculate $ {\\bf G}_{k} = \\mathrm {\\mathbf {P}}_{\\mathcal {F}} \\left( {\\bf T}_{k} \\mathrm {\\mathbf {H}^H} \\right) $ .",
"$ {\\bf {G}}_{\\max }={\\arg \\max }_{{\\bf {G}}_i}\\bigl \\vert {\\bf G}_{i} \\mathrm { \\mathbf {R}_n^{-1/2} } {\\bf H} {\\bf H}^{\\mathrm {H}} \\mathrm { \\mathbf {R}_n^{-1/2} } {\\bf G}_{i}^{\\mathrm {H}} \\bigr \\vert $ .",
"$ {\\bf F}_\\mathrm {A} = {\\bf F}_\\mathrm {max} $ , $ {\\bf G}_\\mathrm {A} = {\\bf G}_\\mathrm {max} $"
],
[
"Simulation Results",
"In this part, some numerical results are provided to assess the performance of the proposed hybrid transceiver design.",
"As our algorithms are applicable to any frequency band, both microwave frequency band and mmWave frequency band are simulated.",
"In addition, quantization of phase shifters is also taken into account.",
"Figure: Spectral efficiency comparison of 5 different hybrid transceiver design methods.",
"Here, 32×16 32 \\times 16 mmWave channel model is adopted in the simulation.",
"Both the transmitter and receiver are equipped with L=4 L = 4 RF-chains and the system is conveying D=4 D = 4 data streams.More specifically, both mmWave channel model $ {\\mathrm {\\mathbf {H}_{m}}} $ and classic Rayleigh channel model $ {\\mathrm {\\mathbf {H}_{r}}} $ are tested.",
"For mmWave channel, $ {\\mathrm {\\mathbf {H}_{m}}} $ , the uniformed linear arrays (ULA) is adopted.",
"Unless otherwise specified, it is assumed that 1) the mmWave channel has $ N_{\\rm cl} = 2 $ clusters with each of them containing $ N_\\mathrm {path} = 5 $ paths; 2) the azimuth angle spread of transmitter is restricted to $ 7.5^{\\circ } $ at the mean of azimuth angle $ \\hat{\\theta } = 45^{\\circ } $ , and the receiver is omni-directional; 3) the path loss factors obey the standard Gaussian distribution; 4) the inter-antenna spacing $ d $ equals to half-wavelength.",
"The channel is normalized to meet $ \\mathbb {E}{ \\left\\lbrace \\Vert {\\bf H}_{\\rm m} \\Vert _{\\rm {F}}^2 \\right\\rbrace } = NM $ .",
"For the random phase algorithm, we set $ K = 10 $ , which means that the best analog precoder and processor are selected from 10 candidates and uniform distribution is utilized, i.e., $ f_{\\mathrm { trans }}(x) = 1$ for $0 \\le x \\le 1 $ .",
"We average the result over 2,000 independent trials.",
"The transmitting power is denoted as $ P_{\\mathrm {Tx}} $ .",
"OMP and MaGiQ algorithms refers to the corresponding algorithms in [15] and [24], respectively.",
"The analog precoder and processor for the direct phase algorithm are obtained by phase projection.",
"Figure: Spectral efficiency comparison of 5 different hybrid transceiver design methods.",
"The 32×16 32 \\times 16 mmWave channel model, which involves N cl =3 N_{\\rm cl} = 3 clusters with N path =5 N_{\\rm path} = 5 multipath at each cluster, is adopted in the simulation.",
"Both the transmitter and receiver are equipped with L=6 L = 6 RF-chains and the system is conveying D=4 D = 4 data streams.Figure: Spectral efficiency comparison of 5 different hybrid transceiver design methods.",
"The 32×16 32 \\times 16 Rayleigh channel model is adopted in the simulation.",
"Both the transmitter and receiver are equipped with L=6 L = 6 RF-chains and the system is conveying D=4 D = 4 data streams.Fig.",
"REF demonstrates spectral efficiency versus the transmit power for different algorithms, where the hybrid transceiver is with $ N = 32 $ transmit antennas, $ M = 16 $ receive antennas, and 4 data streams.",
"Both the transmitter and receiver are equipped with $ L = 4 $ RF-chains.",
"From Fig.",
"REF , the proposed phased projection based hybrid transceiver design outperforms the other hybrid transceiver design algorithms.",
"The performance of the proposed algorithm is very close to the full digital one.",
"Fig.",
"REF shows the performance of the hybrid transceiver design with 6 RF-chains for channel with $ N_{\\rm cl} = 3 $ clusters, each with $ N_{\\rm path} = 5 $ paths.",
"From this figure, the proposed phase projection algorithm works well for different numbers of RF-chains and performs very close to the full-digital one and it is better than that of other hybrid transceiver designs.",
"It is worth noting that the direct phase projection method performs Figure: Spectral efficiency comparison of 5 different hybrid transceiver design methods concerning 2 bits quantization of phase shifters.",
"The 32×16 32 \\times 16 mmWave channel model is adopted in the simulation.",
"Both the transmitter and receiver are equipped with L=4 L = 4 RF-chains and the system is conveying D=4 D = 4 data streams.even better than OMP and MaGiQ.",
"This is because the error bound of the method decreases when the number of RF chains increases [20].",
"However, as the limitation that the number of data streams should be equal to that of RF-chains [24] is not satisfied in this case, MaGiQ algorithm is the worst at high SNR.",
"The following simulations focus on Rayleigh channels at micro-wave bands.",
"Under this circumstance, the $ 32 \\times 16 $ system is adopted with $ L = 6 $ RF-chains are in use transferring $ D = 4 $ data streams.",
"After performing extensive simulation compared with randomly generated codebooks or DFT codebook, we found that the codebook constructed by the phase projection, i.e., $ \\mathcal {C} = {\\bf P}_{\\mathcal {F}}( \\mathrm {\\mathbf {H}}) $ , has much better performance.",
"This codebook is used for performance comparison in the following simulation.",
"Fig.",
"REF compares the performance for the different algorithms under Rayleigh channels.",
"Figure: Spectral efficiency of random algorithm under mmWave channel.",
"L=6 L = 6 RF-chains are assumed to be equipped both transmitter and receiver.",
"Both 32×16 32 \\times 16 and 64×16 64 \\times 16 system are involved during the simulation.",
"The number of data streams is set to be D=4 D = 4 .From the figure, the proposed algorithm obtains nearly the optimal performance as the full-digital one.",
"The proposed algorithm performs better than MaGiQ algorithm in [24].",
"Moreover, it is worth noting that even with the carefully chosen codebook, the OMP algorithm exhibits a large performance gap compared with the full-digital one, which indicates that the OMP algorithm is not suitable for micro-wave frequency bands.",
"As the practical analog phase shifters are often implemented by digital controller with finite resolution, Fig.",
"REF compares the performance of different hybrid transceiver designs for $ 32 \\times 16 $ mmWave channel when phase quantization is taken into account.",
"Each hybrid transceiver design only uses the phase shifter with 2-bit resolution and $ L = 4 $ .",
"From the figure the performance of the proposed hybrid transceiver design still outperforms other hybrid transceiver designs with finite resolution phase shifters.",
"In Fig.",
"REF , both $ 32 \\times 16 $ and $ 64 \\times 16 $ mmWave channels are used to assess the performance.",
"In this case, the number of RF-chains is 6.",
"From Fig.",
"REF , with the same number of transmit antennas, the random algorithm is worse than that of the phase projection based algorithm.",
"Although the random algorithm suffers nearly $ 5\\,{\\rm {dB}} $ performance loss comparing with the full-digital one, by involving more antennas at base station, e.g., $ N = 64 $ , the performance of random algorithm will be comparable to the performance corresponding to the full-digital transmitter with 32 antennas.",
"This implies that we can obtain appropriate performance using the low complexity random algorithm by simply increasing the number of transmit antennas.",
"Because of its low complexity, the random algorithm will be a friendly algorithm for hardware realization.",
"Fig.",
"REF shows the BER performances of different kinds of hybrid MIMO transceiver designs for $ 32 \\times 16 $ Raleigh channel with 4 RF chains.",
"In this case, there are 4 data streams and 16-QAM is used.",
"From this figure, at high SNR, the BER performance of the hybrid nonlinear transceiver design is much better than that of the hybrid linear transceiver design.",
"Furthermore, the hybrid nonlinear transceivers with THP and DFD have almost the same BER performance because of the duality between precoder design and processor design.",
"Figure: BERs of the linear hybrid transceiver for capacity maximization, nonlinear transceiver with DFD and nonlinear transceiver THP.",
"The 32×16 32 \\times 16 Rayleigh channel model is involved in the simulation.",
"Both transmitter and receiver are equipped with L=4 L = 4 RF-chains transferring D=4 D = 4 data streams simultaneously."
],
[
"Conclusions",
"In this paper, we have investigated the hybrid digital and analog transceiver design for MIMO system based on matrix-monotonic optimization theory.",
"We have proposed a unified framework for both linear and nonlinear transceivers.",
"Based on the matrix-monotonic optimization theory, the optimal transceiver structure for various MIMO transceivers has been derived, from which the function of analog transceiver part can be regarded as eigenchannel selection.",
"Using the derived optimal structure, effective algorithms have been proposed considering the constant-modulus constraint.",
"Finally, it is shown that the proposed algorithms outperform existing hybrid transceiver designs."
],
[
"Preliminary Definition of Majorization Theory",
"In this appendix, some fundamental functions in majorization theory are defined for the convenience of unified framework analysis.",
"These definitions are also given in [38] and in order to make the paper self-contained, they are also given here.",
"Definition 1 [41]: For a $K\\times 1$ vector $ {\\bf x} \\in \\mathbb {R}^{K} $ , the $ \\ell $ th largest element of $ {\\bf x} $ is denoted as $ {x}_{[\\ell ]} $ , and in other words, we have $ {x}_{[1]} \\ge {x}_{[2]} \\ge \\cdots \\ge {x}_{[K]} $ .",
"Based on this definition, for two vectors $ {\\bf x}, {\\bf y} \\in \\mathbb {R}^{K} $ , it state that $ {\\bf y} $ majorizes $ {\\bf x} $ additively, denoted by $ {\\bf x} \\prec _{+} {\\bf y} $ , if and only if the following properties are satisfied $\\sum _{k = 1}^{p} {x}_{[k]} \\le \\sum _{k = 1}^{p} {y}_{[k]}, \\; p = 1,2,\\ldots , K-1 \\; \\text{and} \\; \\sum _{k = 1}^{K} {x}_{[k]} = \\sum _{k = 1}^{K} {y}_{[k]}.$ Definition 2 [41]: A function $ f(\\cdot ) $ is Schur-convex if and only if it satisfies the following property ${\\bf x} \\prec _{+} {\\bf y} \\, \\Longrightarrow \\, f( {\\bf x} ) \\le f( {\\bf y} ).$ On the other hand, a function $ f(\\cdot ) $ is additively Schur-concave if $ -f(\\cdot ) $ is additively Schur-convex.",
"Definition 3 [38]: For two $K\\times 1$ vectors $ {\\bf x}, {\\bf y} \\in \\mathbb {R}^{K} $ with nonnegative elements, it states that the vector $ {\\bf y} $ majorizes vector $ {\\bf x} $ multiplicatively, i.e., $ {\\bf x} \\prec _{\\times } {\\bf y} $ , if and only if the following properties are satisfied $\\prod _{k = 1}^{p} {x}_{[k]} \\le \\prod _{k = 1}^{p} {y}_{[k]}, \\; p = 1,2,\\ldots , K-1 \\; \\text{and} \\; \\prod _{k = 1}^{K} {x}_{[k]} = \\prod _{k = 1}^{K} {y}_{[k]}.$ Definition 4 [38]: A function $ f(\\cdot ) $ is multiplicatively Schur-convex if and only if it satisfies the following property ${\\bf x} \\prec _{+} {\\bf y} \\, \\Longrightarrow \\, f( {\\bf x} ) \\le f( {\\bf y} ).$ On the other hand, a function $ f(\\cdot ) $ is multiplicatively Schur-concave if $ -f(\\cdot ) $ is multiplicatively Schur-convex."
],
[
"The optimal ${\\bf {B}}$",
"Note that this optimal ${\\bf {B}}$ for nonlinear transceiver was previously obtained in [36] when function belongs to the family of multiplicatively Schur-concave/convex functions defined in Appendix A.",
"The following presents a slightly different proof of the optimal B, which generalizes the result to the case with an arbitrary monotone increasing function $f(\\cdot )$ .",
"Here, the function f operates only on the diagonal elements of ${\\Phi }_{\\rm {MSE}}({\\bf {G}}_{\\rm {A}}, {\\bf {F}}_{\\rm {A}}, {\\bf {F}}_{\\rm {D}},{\\bf {B}})$ and ${\\bf {B}}$ is restricted as a strictly lower triangular matrix which specifies the use of nonlinear transceiver.",
"Based on the Cholesky decomposition ${\\Phi }_{\\rm {MSE}}^{\\rm {L}}({\\bf {G}}_{\\rm {A}}, {\\bf {F}}_{\\rm {A}}, {\\bf {F}}_{\\rm {D}}) ={\\bf {L}}{\\bf {L}}^{\\rm {H}},$ we have $&{\\Phi }_{\\rm {MSE}}({\\bf {G}}_{\\rm {A}}, {\\bf {F}}_{\\rm {A}}, {\\bf {F}}_{\\rm {D}}, {\\bf {B}})\\nonumber \\\\&= ({\\bf {I}} + {\\bf {B}}) {\\Phi }_{\\rm {MSE}}^{\\rm {L}}({\\bf {G}}_{\\rm {A}}, {\\bf {F}}_{\\rm {A}}, {\\bf {F}}_{\\rm {D}}) ({\\bf {I}} + {\\bf {B}})^{\\rm H}\\nonumber \\\\&=({\\bf {I}} + {\\bf {B}}){\\bf {L}}{\\bf {L}}^{\\rm {H}}({\\bf {I}} + {\\bf {B}})^{\\rm H},$ based on which the $n$ th diagonal element of ${\\Phi }_{\\rm {MSE}}({\\bf {G}}_{\\rm {A}}, {\\bf {F}}_{\\rm {A}}, {\\bf {F}}_{\\rm {D}}, {\\bf {B}})$ equals $[{\\Phi }_{\\rm {MSE}}({\\bf {G}}_{\\rm {A}}, {\\bf {F}}_{\\rm {A}}, {\\bf {F}}_{\\rm {D}}, {\\bf {B}})]_{n,n}&=[({\\bf {I}} + {\\bf {B}}){\\bf {L}}]_{n,:}[({\\bf {I}} + {\\bf {B}}){\\bf {L}}]_{n,:}^{\\rm {H}}\\nonumber \\\\&=\\Vert [({\\bf {I}} + {\\bf {B}}){\\bf {L}}]_{n,:}\\Vert ^2.$ In addition, as ${\\bf {B}}$ is strictly lower triangular it can be calculated that the last element of the vector $[({\\bf {I}} + {\\bf {B}}){\\bf {L}}]_{n,:}$ equals $[{\\bf {L}}]_{n,n}$ , i.e., $[({\\bf {I}} + {\\bf {B}}){\\bf {L}}]_{n,:}=[\\cdots ,[{\\bf {L}}]_{n,n}].$ Therefore, from (REF ) to (REF ) the following relationship holds $[{\\Phi }_{\\rm {MSE}}({\\bf {G}}_{\\rm {A}}, {\\bf {F}}_{\\rm {A}}, {\\bf {F}}_{\\rm {D}}, {\\bf {B}})]_{n,n}&=\\Vert [({\\bf {I}} + {\\bf {B}}){\\bf {L}}]_{n,:}\\Vert ^2\\ge [{\\bf {L}}]_{n,n}^2.$ It is obvious that the above inequality can be achieved with equality as $[{\\Phi }_{\\rm {MSE}}({\\bf {G}}_{\\rm {A}}, {\\bf {F}}_{\\rm {A}}, {\\bf {F}}_{\\rm {D}}, {\\bf {B}})]_{n,n}= [{\\bf {L}}]_{n,n}^2$ when the following equality holds for different $n$ $({\\bf {I}}+{\\bf {B}}){\\bf {L}} ={\\rm {diag}}\\lbrace [[{\\bf {L}}]_{1,1},\\cdots ,[{\\bf {L}}]_{L,L}]^{\\rm {T}}\\rbrace ,$ based on which the optimal ${\\bf {B}}$ equals ${\\bf {B}}^{\\rm {opt}} ={\\rm {diag}}\\lbrace [[{\\bf {L}}]_{1,1},\\cdots ,[{\\bf {L}}]_{L,L}]^{\\rm {T}}\\rbrace {\\bf {L}}^{-1}-{\\bf {I}}.$"
],
[
"Fundamental Matrix Inequalities",
"In this appendix, two fundamental matrix inequalities are given.",
"For two positive semi-definite matrices ${X}$ and ${Y}$ , there are following EVDs defined ${X}&={\\bf {U}}_{X} {\\Lambda }_{X} {\\bf {U}}^{\\rm {H}}_{X} \\ \\ \\text{with} \\ \\ {\\Lambda }_{X}\\searrow \\nonumber \\\\{Y}&={\\bf {U}}_{Y} {\\Lambda }_{Y} {\\bf {U}}^{\\rm {H}}_{Y} \\ \\ \\text{with} \\ \\ {\\Lambda }_{Y}\\searrow \\nonumber \\\\{Y}&={\\bf {\\bar{U}}}_{Y}{\\bar{\\Lambda }}_{Y} {\\bf {\\bar{U}}}^{\\rm {H}}_{Y} \\ \\ \\text{with} \\ \\ {\\bar{\\Lambda }}_{Y} \\nearrow .$ For the trace of the two matrices, we have the following fundamental matrix inequalities [40] $&\\sum _{i=1}^{N}\\lambda _{i-1+N}({X}) \\lambda _i({Y})\\le {\\rm {Tr}}({X}{Y}) \\le \\sum _{i=1}^{N}\\lambda _i({X}) \\lambda _i({Y}),$ where $ \\lambda _i( { \\mathbf {X} } ) $ is the $ i $ th ordered eigenvalue of $ { \\mathbf {X} } $ , and the left equality holds when ${\\bf {U}}_{X}={\\bf {\\bar{U}}}_{Y}$ .",
"On the other hand, the right equality holds when ${\\bf {U}}_{X}={\\bf { U}}_{Y}$ ."
],
[
"Optimal Structure of Analog Transceiver",
"It is worth noting that the nonzero singular values of the matrix, $ {\\Pi }_{\\rm {R}} = {\\bf {F}}_{\\rm {A}} ({\\bf {F}}_{\\rm {A}}^{\\rm {H}}{\\bf {F}}_{\\rm {A}})^{-\\frac{1}{2}} $ , are all ones.",
"Similarly for ${\\Pi }_{\\rm {L}}=({\\bf {G}}_{\\rm {A}}{\\bf {R}}_{\\rm {n}}{\\bf {G}}_{\\rm {A}}^{\\rm {H}})^{-1/2}{\\bf {G}}_{\\rm {A}}{\\bf {R}}_{\\rm {n}}^{1/2}$ , the nonzero singular values of ${\\Pi }_{\\rm {L}}$ are all ones.",
"It implies that the singular values of ${\\bf {F}}_{\\rm {A}}$ and ${\\bf {G}}_{\\rm {A}}{\\bf {R}}_{\\rm {n}}^{1/2}$ do not affect the optimization problem.",
"Based on the SVDs ${\\bf {R}}_{\\rm {n}}^{-1/2}{\\bf {H}}={\\bf {U}}_{{\\mathcal {H}}}{\\Lambda }_{\\mathcal {H}}{\\bf {V}}_{\\mathcal {H}}^{\\rm {H}}$ , ${\\bf {R}}_{\\rm {n}}^{1/2} {\\bf {G}}_{\\rm {A}}^{\\rm {H}} = {\\bf {U}}_{{\\bf {R}}{\\bf {G}}} {\\Lambda }_{{\\bf {R}}{\\bf {G}}}{\\bf {V}}_{{\\bf {R}}{\\bf {G}}}^{\\rm {H}}$ , and ${\\bf {F}}_{\\rm {A}}={\\bf {U}}_{{\\bf {F}}_{\\rm {A}}} {\\Lambda }_{{\\bf {F}}_{\\rm {A}}}{\\bf {V}}_{{\\bf {F}}_{\\rm {A}}}^{\\rm {H}}$ with the singular values in decreasing order, the objective function in (REF ) becomes ${\\bf {\\tilde{F}}}_{\\rm {D}}^{\\rm {H}}{\\bf {V}}_{{\\bf {F}}_{\\rm {A}}}{\\Lambda }_{\\rm {R}}^{\\rm {T}}{\\bf {U}}_{{\\bf {F}}_{\\rm {A}}}^{\\rm {H}}{\\bf {H}}^{\\rm {H}}{\\bf {U}}_{\\rm {RG}}{\\Lambda }_{\\rm {L}}^{\\rm {T}}{\\Lambda }_{\\rm {L}}{\\bf {U}}_{\\rm {RG}}^{\\rm {H}}{\\bf {H}}{\\bf {U}}_{{\\bf {F}}_{\\rm {A}}}{\\Lambda }_{\\rm {R}}{\\bf {V}}_{{\\bf {F}}_{\\rm {A}}}^{\\rm {H}}{\\bf {\\tilde{F}}}_{\\rm {D}}$ where the diagonal elements of the diagonal matrices ${\\Lambda }_{\\rm {R}}$ and ${\\Lambda }_{\\rm {L}}$ satisfies $&[{\\Lambda }_{\\rm {R}}]_{i,i}=1, \\ \\ i\\le L \\nonumber \\\\& [{\\Lambda }_{\\rm {R}}]_{i,i}=0, \\ \\ i> L \\nonumber \\\\&[{\\Lambda }_{\\rm {L}}]_{i,i}=1, \\ \\ i\\le L \\nonumber \\\\& [{\\Lambda }_{\\rm {R}}]_{i,i}=0, \\ \\ i> L.$ Therefore, ${\\bf {F}}_{\\rm {A}}$ and ${\\bf {G}}_{\\rm {A}}$ do not affect the optimal solution.",
"Moreover, the unitary matrices ${\\bf {V}}_{{\\bf {F}}_{\\rm {A}}}$ and ${\\bf {V}}_{\\rm {RG}}$ do not affect the optimal solution as ${\\bf {\\tilde{F}}}_{\\rm {D}}$ in the constraint is unitary invariant.",
"Based on the above the discussion and (REF ), the remaining task to maximize the singular values of matrix $[{\\bf {U}}_{\\rm {RG}}^{\\rm {H}}{\\bf {H}}{\\bf {U}}_{{\\bf {F}}_{\\rm {A}}}]_{1:L,1:L}$ .",
"Note that ${\\bf {U}}_{\\rm {RG}}$ and ${\\bf {U}}_{{\\bf {F}}_{\\rm {A}}}$ are unitary matrices, for the optimal solution, the left eigenvectors of its first $ L $ largest singular values of $ {\\bf F}_{\\rm A} $ should have the maximum inner product with $[{\\bf {V}}_{\\mathcal {H}}]_{:,1:L}$ i.e., $ [{\\bf {U}}_{{\\bf {F}}_{\\rm {A}}}]_{:,1:L}^{\\rm {opt}}={\\rm {arg\\,max}}\\lbrace \\Vert [{\\bf {V}}_{\\mathcal {H}}]_{:,1:L} [{\\bf {U}}_{{\\bf {F}}_{\\rm {A}}}]_{:,1:L}^{\\rm {H}}\\Vert _{\\rm {F}}^2\\rbrace .$ Similarly for the optimal solution, the left eigenvectors of its first $ L $ largest singular values of $ {\\bf {R}}_{\\rm {n}}^{1/2} {\\bf {G}}_{\\rm {A}}^{\\mathrm {H}} $ should have the maximum inner product with $[{\\bf {U}}_{\\mathcal {H}}]_{:,1:L}$ , i.e., $ [{\\bf {U}}_{{\\bf {R}}{\\bf {G}}}]_{:,1:L}^{\\rm {opt}}={\\rm {arg\\,max}}\\lbrace \\Vert [{\\bf {U}}_{\\mathcal {H}}]_{:,1:L} [{\\bf {U}}_{{\\bf {R}}{\\bf {G}}}]_{:,1:L}^{\\rm {H}}\\Vert _{\\rm {F}}^2\\rbrace .$"
],
[
"Analog Transceiver Design",
"For fixed ${\\mathbf {\\Lambda }_\\mathrm {G}}$ and ${\\bf {Q}}_{\\rm {G}}$ , the optimization problem (REF ) can be transferred into the following vector variable optimization problem $\\min _{\\mathbf {r}} \\;\\; & {\\mathbf {r}}^{W} {\\mathbf {r}} - \\mathbf {p}^ - {\\mathbf {r}}^{p} + q \\\\\\text{s.t.}",
"\\;\\;\\; & \\mathbf {r}^{K}_{i} \\mathbf {r} = a^2, \\quad i = 1,2,\\ldots , NL.$ The vector $ \\mathbf {r} $ is constructed via vectorizing $ \\mathbf {G}_{\\mathrm {A}} $ , i.e., $\\mathbf {r} = \\big [ \\Re \\lbrace \\mathrm {vec}{( \\mathbf {G}_{\\mathrm {A}} )} \\rbrace ^ \\, \\Im \\lbrace \\mathrm {vec}{( \\mathbf {G}_{\\mathrm {A}} )} \\rbrace ^]^$ and the matrices $ \\mathbf {W}, \\, \\mathbf {K}_{i} $ and vector $ \\mathbf {p} $ are defined as follows: $\\mathbf {W} & =\\begin{bmatrix}\\; \\Re \\lbrace \\mathbf {I} \\otimes { \\mathbf {R}_{\\mathrm {n}} } \\rbrace & - \\Im \\lbrace \\mathbf {I} \\otimes { \\mathbf {R}_{\\mathrm {n}} } \\rbrace \\;\\; \\\\\\; \\Im \\lbrace \\mathbf {I} \\otimes { \\mathbf {R}_{\\mathrm {n}} } \\rbrace & \\Re \\lbrace \\mathbf {I} \\otimes { \\mathbf {R}_{\\mathrm {n}} } \\rbrace \\;\\;\\end{bmatrix}, \\\\\\mathbf {K}_{i} & = \\mathrm {diag} \\Big \\lbrace \\bigl [ \\mathbf {0}_{(i-1) \\times 1}^ 1 ,\\mathbf {0}_{(NL - 1) \\times 1}^ 1 ,\\mathbf {0}_{(NL - i) \\times 1}^] \\Big \\rbrace ,$ and $\\mathbf {p} =\\begin{bmatrix}\\Re \\lbrace \\big ( \\mathbf {I} \\otimes { \\mathbf {R}_{\\mathrm {n}}^{ {1}/{2}} } \\big )^{\\mathrm {H}} \\mathrm {vec}{ \\big ( [{\\bf {U}}_{\\mathcal {H}}]_{:,1:L} {\\mathbf {\\Lambda }_\\mathrm {G}} {\\bf {Q}}_{\\rm {G}} \\big ) } \\rbrace \\\\\\Im \\lbrace \\big ( \\mathbf {I} \\otimes { \\mathbf {R}_{\\mathrm {n}}^{ {1}/{2}} } \\big )^{\\mathrm {H}} \\mathrm {vec}{ \\big ( [{\\bf {U}}_{\\mathcal {H}}]_{:,1:L} {\\mathbf {\\Lambda }_\\mathrm {G}} {\\bf {Q}}_{\\rm {G}} \\big ) } \\rbrace \\end{bmatrix}.$ The constant scalar, $q$ , in (REF ) equals $q = || \\mathrm {vec}{ \\big ( [{\\bf {U}}_{\\mathcal {H}}]_{:,1:L} {\\mathbf {\\Lambda }_\\mathrm {G}} {\\bf {Q}}_{\\rm {G}} \\big ) } ||_2^2$ .",
"Note that because of the constant modulus constraints, the term ${\\bf {r}}^{\\rm {T}}{\\bf {r}}$ is a constant.",
"As a result, for a constant real scalar, $\\eta $ , the objective function in (REF ) is equivalent to ${\\mathbf {r}}^\\mathbf {W}+\\eta {\\bf {I}}) {\\mathbf {r}} - \\mathbf {p}^ - {\\mathbf {r}}^{p} + q$ .",
"As the constant modulus constraints in (REF ) are all quadratic equalities, the optimization problem (REF ) is nonconvex.",
"Following the idea of [42], an iterative algorithm is proposed via iteratively updating constraints to guarantee the constant modulus constraints.",
"Specifically, at the $n$ th iteration each constraint $\\mathbf {r}^{K}_{i} \\mathbf {r} = a^2$ is replaced by $\\mathbf {\\tilde{r}}_{(n-1)}^{K}_{i} \\mathbf {r}_{(n)} = a^2$ where $\\mathbf {\\tilde{r}}_{(n-1)}$ is a vector computed based on ${\\bf {r}}$ computed in the $(n-1)$ th iteration.",
"After stacking $\\mathbf {\\tilde{r}}_{(n-1)}^{K}_{i}$ for $i = 1,2,\\ldots NL$ in ${\\bf {P}}_{(n-1)}$ , optimization problem (REF ) is transferred to $\\min _{\\mathbf {r}_{(n)}} \\;\\; & {\\mathbf {r}}_{(n)}^\\mathbf {W}+\\eta {\\bf {I}}) {\\mathbf {r}}_{(n)} - \\mathbf {p}^_{(n)} - {\\mathbf {r}}_{(n)}^{p} + q \\\\\\text{s.t.}",
"\\;\\;\\; & {\\bf {P}}_{(n-1)}\\mathbf {r}_{(n)} = a^2{\\bf {1}},$ where the matrix $ \\mathbf {P}_{(n-1)} $ is defined as $&[ \\mathbf {P}_{(n-1)} ]_{\\ell ,j} \\nonumber \\\\=&{\\left\\lbrace \\begin{array}{ll}\\cos \\big ( \\angle [{\\rm {vec}}({\\bf {G}}_{{\\rm {A}},(n-1)})]_{\\ell } \\big ) & \\text{if } \\ell = j, \\; \\ell \\le NL \\\\\\sin \\big ( \\angle [ {\\rm {vec}}({\\bf {G}}_{{\\rm {A}},(n-1)}) ]_{\\ell } \\big ) & \\text{if } j = \\ell +NL, \\; \\ell \\le NL \\\\0 & \\text{otherwise.}\\end{array}\\right.",
"}$ The vector $ \\mathbf {1} $ is a column vector with all elements equal to 1.",
"As proved in [42], when $\\eta \\ge \\sigma _{\\max } NL / 8 + ||\\mathbf {p}||_2^2$ , where $ \\sigma _{\\max } $ is the largest eigenvalue of $ { \\mathbf {R}_{\\mathrm {n}} }$ , the optimal solution of the iterative optimization (REF ) minimizes the objective function and satisfies the constant modulus constraints asymptotically.",
"As (REF ) is convex at each iteration, based on its KKT conditions, at the $n$ th iteration the optimal solution of (REF ) is ${\\mathbf {r}}_{(n)} =( {\\mathbf {W}} + \\eta \\mathbf {I} )^{-1}\\left({\\bf {q}}+\\frac{\\lambda }{2}{\\bf {P}}_{(n-1)}^{\\rm {T}}\\right)$ with $\\frac{\\lambda }{2}&=\\left( {\\bf {P}}_{(n-1)}( {\\mathbf {W}} + \\eta \\mathbf {I} )^{-1}{\\bf {P}}_{(n-1)}^{\\rm {T}}\\right)^{-1}\\nonumber \\\\& \\ \\ \\ \\ \\ \\times \\left(a^2{\\bf {1}}-{\\bf {P}}_{(n-1)}( {\\mathbf {W}} + \\eta \\mathbf {I} )^{-1}{\\bf {q}}\\right).$ In a nutshell, the iterative algorithm is given in Algorithm REF .",
"Using the iterative algorithm, the numerical result of analog processor can be found."
]
] | 1808.08526 |
[
[
"Data-dependent Learning of Symmetric/Antisymmetric Relations for\n Knowledge Base Completion"
],
[
"Abstract Embedding-based methods for knowledge base completion (KBC) learn representations of entities and relations in a vector space, along with the scoring function to estimate the likelihood of relations between entities.",
"The learnable class of scoring functions is designed to be expressive enough to cover a variety of real-world relations, but this expressive comes at the cost of an increased number of parameters.",
"In particular, parameters in these methods are superfluous for relations that are either symmetric or antisymmetric.",
"To mitigate this problem, we propose a new L1 regularizer for Complex Embeddings, which is one of the state-of-the-art embedding-based methods for KBC.",
"This regularizer promotes symmetry or antisymmetry of the scoring function on a relation-by-relation basis, in accordance with the observed data.",
"Our empirical evaluation shows that the proposed method outperforms the original Complex Embeddings and other baseline methods on the FB15k dataset."
],
[
"Introduction",
"Large-scale knowledge bases, such as YAGO [16], Freebase [1], and WordNet [5] are utilized in knowledge-oriented applications such as question answering and dialog systems.",
"Facts are stored in these knowledge bases as triplets of form $(\\text{\\textit {subject entity}},\\text{\\textit {relation}},\\text{\\textit {object entity}})$ .",
"Although a knowledge base may contain more than a million facts, many facts are still missing [12].",
"Knowledge base completion (KBC) aims to find such missing facts automatically.",
"In recent years, vector embedding of knowledge bases has been actively pursued as a promising approach to KBC.",
"In this approach, entities and relations are embedded into a vector space as their representations, in most cases as vectors and sometimes as matrices.",
"A variety of methods have been proposed, each of which computes the likeliness score of given triplets using different vector/matrix operations over the representations of entities and relations involved.",
"In most of the previous methods, the scoring function is designed to cover general non-symmetric relations, i.e., relations $r$ such that for some entities $e_1$ and $e_2$ , triplet $(e_1,r,e_2)$ holds but not $(e_2, r, e_1)$ .",
"This reflects the fact that the subject and object in a relation are not interchangeable in general (e.g., parent_of).",
"However, the degree of symmetry differs from relation to relation.",
"In particular, a non-negligible number of symmetric relations exist in knowledge bases (e.g., sibling_of).",
"Moreover, many non-symmetric relations in knowledge base are actually antisymmetric, in the sense that for every distinct pair $e_1, e_2$ of entities, if $(e_1,r,e_2)$ holds, then $(e_2, r, e_1)$ never holds.Note that even if a relation is antisymmetric in the above sense, its truth-value matrix may not be antisymmetric.",
"For relations that show certain regularities such as above, the expressiveness of models to capture general relations might be superfluous, and a model with less parameters might be preferable.",
"It thus seems desirable to encourage the scoring function to produce sparser models, if the observed data suggests a relation being symmetric or antisymmetric.",
"As we do not assume any background knowledge about individual relations, the choice between these contrasting properties must be made solely from the observed data.",
"Further, we do not want the model to sacrifice the expressiveness to cope with relations that are neither purely symmetric or antisymmetric.",
"Complex Embeddings (ComplEx) [21] are one of the state-of-the-art methods for KBC.",
"ComplEx represents entities and relations as complex vectors, and it can model general non-symmetric relations thanks to the scoring function defined by the Hermitian inner product of these vectors.",
"However, for symmetric relations, the imaginary parts in relation vectors are redundant parameters since they only contribute to non-symmetry of the scoring function.",
"ComplEx is thus not exempt from the issue we mentioned above: the lack of symmetry/antisymmetry consideration for individual relations.",
"Our experimental results show that this issue indeed impairs the performance of ComplEx.",
"In this paper, we propose a technique for training ComplEx relation vectors adaptively to the degree of symmetry/antisymmetry observed in the data.",
"Our method is based on L1 regularization, but not in the standard way; the goal here is not to make a sparse, succinct model but to adjust the degree of symmetry/antisymmetry on the relation-by-relation basis, in a data-driven fashion.",
"In our model, L1 regularization is imposed on the products of coupled parameters, with each parameter contributing to either the symmetry or antisymmetry of the learned scoring function.",
"Experiments with synthetic data show that our method works as expected: Compared with the standard L1 regularization, the learned functions is more symmetric for symmetric relations and more antisymmetric for antisymmetric relations.",
"Moreover, in KBC tasks on real datasets, our method outperforms the original ComplEx with standard L1 and L2 regularization, as well as other baseline methods."
],
[
"Background",
"Let $\\mathbb {R}$ be the set of reals, and $\\mathbb {C}$ be the set of complex numbers.",
"Let $i \\in \\mathbb {C}$ denote the imaginary unit.",
"$[\\mathbf {v}]_j$ denotes the $j$ th component of vector $\\mathbf {v}$ , and $[\\mathbf {M}]_{jk}$ denotes the $(j,k)$ -element of matrix $\\mathbf {M}$ .",
"A superscript T (e.g., $\\mathbf {v}^{\\mathrm {T}}$ ) represents vector/matrix transpose.",
"For a complex scalar, vector, or matrix $\\mathbf {Z}$ , $\\overline{\\mathbf {Z}}$ represent its complex conjugate, with $\\mathop {\\text{Re}}(\\mathbf {Z})$ and $\\mathop {\\text{Im}}(\\mathbf {Z})$ denoting its real and imaginary parts, respectively."
],
[
"Knowledge Base Completion",
"Let $\\mathcal {E}$ and $\\mathcal {R}$ respectively be the sets of entities and the (names of) binary relations over entities in an incomplete knowledge base.",
"Suppose a relational triplet $(s,r,o)$ is not in the knowledge base for some $s, o \\in \\mathcal {E}$ and $r \\in \\mathcal {R}$ .",
"The task of KBC is to determine the truth value of such an unknown triplet; i.e., whether relation $r$ holds between subject entity $s$ and object entity $o$ .",
"A typical approach to KBC is to learn a scoring function $\\phi (s, r, o)$ to estimate the likeliness of an unknown triplet $(s, r, o)$ , using as training data the existing triplets in the knowledge base and their truth values.",
"A higher score indicates that the triplet is more likely to hold.",
"The scoring function $\\phi $ is usually parameterized, and the task of learning $\\phi $ is recast as that of tuning the model parameters.",
"To indicate this explicitly, model parameters $\\mathbf {\\Theta }$ are sometimes included in the arguments of the scoring function, as in $\\phi (s, r, o; \\mathbf {\\Theta })$ ."
],
[
"Complex Embeddings (ComplEx)",
"The embedding-based approach to KBC defines the scoring function in terms of the vector representation (or, embeddings) of entities and relations.",
"In this approach, model parameters $\\mathbf {\\Theta }$ consist of these representation vectors.",
"ComplEx [21] is one of the latest embedding-based methods for KBC.",
"It represents entities and relations as complex vectors.",
"Let $\\mathbf {e}_j, \\mathbf {w}_r \\in \\mathbb {C}^d$ respectively denote the $d$ -dimensional complex vector representations of entity $j \\in \\mathcal {E}$ and relation $r \\in \\mathcal {R}$ .",
"The scoring function of ComplEx is defined by $\\phi (s, r, o; \\mathbf {\\Theta }) & = \\mathop {\\text{Re}}\\left( \\mathbf {e}_s^{\\mathrm {T}}\\mathop {\\text{diag}}(\\mathbf {w}_r) \\overline{\\mathbf {e}_o} \\right) \\\\& = \\mathop {\\text{Re}}\\left( \\langle \\mathbf {w}_r, \\mathbf {e}_s, \\overline{\\mathbf {e}_o} \\rangle \\right) \\nonumber \\\\& = \\langle \\mathop {\\text{Re}}(\\mathbf {w}_r), \\mathop {\\text{Re}}(\\mathbf {e}_s), \\mathop {\\text{Re}}(\\mathbf {e}_o) \\rangle \\nonumber \\\\& \\qquad + \\langle \\mathop {\\text{Re}}(\\mathbf {w}_r), \\mathop {\\text{Im}}(\\mathbf {e}_s), \\mathop {\\text{Im}}(\\mathbf {e}_o) \\rangle \\nonumber \\\\& \\qquad + \\langle \\mathop {\\text{Im}}(\\mathbf {w}_r), \\mathop {\\text{Re}}(\\mathbf {e}_s), \\mathop {\\text{Im}}(\\mathbf {e}_o) \\rangle \\nonumber \\\\& \\qquad - \\langle \\mathop {\\text{Im}}(\\mathbf {w}_r), \\mathop {\\text{Im}}(\\mathbf {e}_s), \\mathop {\\text{Re}}(\\mathbf {e}_o) \\rangle , $ where $\\mathop {\\text{diag}}(\\mathbf {v})$ denotes a diagonal matrix with the diagonal given by vector $\\mathbf {v}$ , and $\\langle \\mathbf {u}, \\mathbf {v}, \\mathbf {w}\\rangle = \\left( \\sum _{k=1}^d [\\mathbf {u}]_k [\\mathbf {v}]_k [\\mathbf {w}]_k \\right) $ , with $\\mathbf {\\Theta } = \\lbrace \\mathbf {e}_j \\in \\mathbb {C}^d \\mid j \\in \\mathcal {E} \\rbrace \\cup \\lbrace \\mathbf {w}_r \\in \\mathbb {C}^d \\mid r \\in \\mathcal {R} \\rbrace $ .",
"The use of complex vectors and Hermitian inner product makes ComplEx both expressive and computationally efficient."
],
[
"Roles of Real/Imaginary Parts in Relation Vectors",
"Many relations in knowledge bases are either symmetric or antisymmetric.",
"For example, all 18 relations in WordNet are either symmetric (4 relations) or antisymmetric (14 relations).",
"Also, relations that take different “types” of entities as the subject and object are necessarily antisymmetric; take relation born_in for example, which is defined for a person and a location.",
"Clearly, if $(\\textit {Barack\\_Obama}, \\textit {born\\_in}, \\textit {Hawaii})$ holds, then $( \\textit {Hawaii}, \\textit {born\\_in}, \\textit {Barack\\_Obama})$ does not.",
"Now, let us look closely at the scoring function of ComplEx given by Eq.",
"(REF ).",
"We observe the following: If the relation vector $\\mathbf {w}_r$ is a real vector, then $\\phi (s,r,o) = \\phi (o,r,s)$ for any $s, o\\in \\mathcal {E}$ ; i.e., the scoring function $\\phi (s,r,o)$ is symmetric with respect to $s$ and $o$ .",
"This can be seen by substituting $\\mathop {\\text{Im}}(\\mathbf {w}_r) = \\mathbf {0}$ in Eq.",
"(), in which case the last two terms vanish.",
"If, to the contrary, $\\mathbf {w}_r$ is purely imaginary, $\\phi (s,r,o)$ is antisymmetric in $s$ and $o$ , in the sense that $\\phi (s,r,o) = -\\phi (o,r,s)$ .",
"Again, this can be verified with Eq.",
"(), but this time by substituting $\\mathop {\\text{Re}}(\\mathbf {w}_r) = \\mathbf {0}$ .",
"As we see from these two cases, the real parts in the components of $\\mathbf {w}_r$ are responsible for making the scoring function $\\phi $ symmetric, whereas the imaginary parts in $\\mathbf {w}_r$ are responsible for making it antisymmetric.",
"Each relation has a different degree of symmetry/antisymmetry, but the original ComplEx, which is usually trained with L2 regularization, does not take this difference into account.",
"Specifically, L2 regularization is equivalent to making a prior assumption that all parameters, including the real and imaginary parts of relation vectors, are independent.",
"As we have discussed above, this independence assumption is unsuited for symmetric and antisymmetric relations.",
"For instance, we expect the vector for symmetric relations to have a large number of nonzero real parts and zero imaginary parts."
],
[
"Multiplicative L1 Regularization for Coupled Parameters",
"On the basis of the observation above, we introduce a new regularization term for training ComplEx vectors.",
"This term encourages individual relation vectors to be more symmetric or antisymmetric in accordance with the observed data.",
"The resulting objective function is $\\underset{\\mathbf {\\Theta }}{\\mathrm {min}} \\sum _{(s, r, o) \\in \\Omega } \\log (1 + \\exp (-y_{rso} \\phi (s, r, o; \\mathbf {\\Theta }))) \\nonumber \\\\+ \\lambda ( \\alpha R_1(\\mathbf {\\Theta }) + (1 - \\alpha ) R_2(\\mathbf {\\Theta }))$ where $\\Omega $ is the training samples of triplets; $y_{rso} \\in \\lbrace +1, -1\\rbrace $ gives the truth value of the triplet $(s, r, o)$ ; hyperparameter $\\lambda \\ge 0$ determines the overall weight on the regularization terms; and $\\alpha \\in [0, 1]$ (also a hyperparameter) controls the balance between two regularization terms $R_1$ and $R_2$ .",
"These terms are defined as follows: $R_1(\\mathbf {\\Theta }) & = \\sum _{r \\in \\mathcal {R}} \\sum _{k=1}^d \\left| \\mathop {\\text{Re}}([\\mathbf {w}_r]_k) \\cdot \\mathop {\\text{Im}}([\\mathbf {w}_r]_k) \\right|, \\\\R_2(\\mathbf {\\Theta }) & = \\left\\Vert \\mathbf {\\Theta } \\right\\Vert ^2_2.",
"$ In Eq.",
"(), $\\mathbf {\\Theta }$ is treated as a vector, with all the parameters it contains as the vector components.",
"Eq.",
"(REF ) differs from the original ComplEx objective in that it introduces the proposed regularizer $R_1$ , which is a form of L1-norm penalty (see Equation (REF )).",
"In general, L1-norm penalty terms promote producing sparse solutions for the model parameters and are used for feature selection or the improve the interpretability of the model.",
"Note, however, that our L1 penalty encourages sparsity of pairwise products.",
"This means that only one of the coupled parameters needs to be propelled towards zero to minimize the summarized in Eq.",
"(REF ).",
"To distinguish from the standard L1 regularization, we call the regularization term in $R_1$ multiplicative L1 regularizer, since it is based on the L1 norm of the vector of the product of the real and imaginary parts of each component.",
"As we explain in the next subsection, standard L1 regularization implies the independence of parameters as a prior.",
"By contrast, our regularization term $R_1$ dictates the interaction between the real and imaginary parts of a component in a relation vector; if, as a result of L1 regularization, one of these parts falls to zero, the other can freely move to minimize the objective (REF ).",
"This encourages selecting either of the coupled parameters to be zero, but not necessarily both.",
"Unlike the standard L1 regularization, the proposed regularization term is non-convex, and makes the optimization harderNotice that the objective function in ComplEx is already non-convex without a regularization term..",
"However, in our experiments reported below, multiplicative L1 regularization outperforms the standard one in KBC, and is robust against random initialization.",
"Since the real and imaginary parts of a relation vector govern the symmetry/antisymmetry of the scoring function for the relation, this L1 penalty term is expected to help guide learning a vector for relation $r$ in accordance with whether $r$ is symmetric, antisymmetric, or neither of them, as observed in the training data.",
"For example, if the data suggests $r$ is likely to be symmetric, our L1 regularizer should encourage the imaginary parts to be zero while allowing the real parts to take on arbitrary values.",
"Because parameters are coupled componentwise, the proposed model can also cope with non-symmetric, non-antisymmetric relations with different degree of symmetry/antisymmetry."
],
[
"MAP Interpretation",
"MAP estimation finds the best model parameters $\\hat{\\mathbf {\\Theta }}$ by maximizing a posterior distribution: $\\hat{\\mathbf {\\Theta }} & = \\underset{\\mathbf {\\Theta }}{\\rm argmax} \\log p(\\mathbf {\\Theta } | \\mathcal {D}) \\nonumber \\\\& = \\underset{\\mathbf {\\Theta }}{\\rm argmax} \\log p(\\mathcal {D} | \\mathbf {\\Theta }) + \\log p(\\mathbf {\\Theta }), $ where $\\mathcal {D}$ is the observed data.",
"The first term represents the likelihood function, and the second term represents the prior distribution of parameters.",
"Our objective function Eq.",
"(REF ) can also be viewed as MAP estimation in the form of Eq.",
"(REF ); the first term in our objective corresponds to the likelihood function, and the regularizer terms define the prior.",
"Let us discuss the prior distribution implicitly assumed by using the proposed multiplicative L1 regularizer $R_1$ .For brevity, we neglect the regularizer $R_2$ and focus on $R_1$ in this discussion.",
"Let $C, C^{\\prime }, C^{\\prime \\prime }, \\ldots $ denote constants.",
"Our multiplicative L1 regularization is equivalent to assuming the prior $p(\\mathbf {\\Theta }) & = \\prod _{r\\in \\mathcal {R}} p(\\mathbf {w}_r) \\\\ \\multicolumn{2}{l}{\\text{with}}\\\\p(\\mathbf {w}_r) & = \\prod _{k=1}^d C \\exp \\left( - \\frac{ \\left|\\mathop {\\text{Re}}([\\mathbf {w}_r]_k) \\cdot \\mathop {\\text{Im}}([\\mathbf {w}_r]_k) \\right| }{C^{\\prime }} \\right).$ In other words, a 0-mean Laplacian prior is assumed on the distribution of $\\mathop {\\text{Re}}([\\mathbf {w}_r]_k) \\cdot \\mathop {\\text{Im}}([\\mathbf {w}_r]_k)$ .",
"The equivalence can be seen by $\\log p(\\mathbf {\\Theta }) & = \\log \\prod _{r\\in \\mathcal {R}} p( \\mathbf {w}_r) \\nonumber \\\\& = \\log \\!",
"\\prod _{r\\in \\mathcal {R}} \\prod _{k=1}^d \\!",
"C \\exp \\!",
"\\left( \\!\\!",
"- \\frac{ \\left|\\mathop {\\text{Re}}([\\mathbf {w}_r]_k) \\cdot \\mathop {\\text{Im}}([\\mathbf {w}_r]_k) \\right| }{C^{\\prime }} \\right) \\nonumber \\\\& = - C^{\\prime \\prime } \\!",
"\\sum _{r\\in \\mathcal {R}} \\sum _{k=1}^d \\left|\\mathop {\\text{Re}}([\\mathbf {w}_r]_k) \\cdot \\mathop {\\text{Im}}([\\mathbf {w}_r]_k) \\right| + C^{\\prime \\prime \\prime }.$ Neglecting the scaling factor $C^{\\prime \\prime }$ and the constant term $C^{\\prime \\prime \\prime }$ , we see that this is equal to the regularizer $R_1$ in Eq.",
"(REF ).",
"Now, suppose the standard L1 regularizer $R_{\\text{std L1}}(\\mathbf {\\Theta }) = \\sum _{r\\in \\mathcal {R}} \\sum _{k=1}^d ( \\left| \\mathop {\\text{Re}}([\\mathbf {w}_r]_k) \\right| + \\left| \\mathop {\\text{Im}}([\\mathbf {w}_r]_k) \\right|)$ is instead of the proposed $R_1(\\Theta )$ .",
"In terms of MAP estimation, in this case, its use is equivalent to assuming a prior distribution $p(\\mathbf {w}_r) = \\prod _{k=1}^d p(\\mathop {\\text{Re}}([\\mathbf {w}_r]_k)) \\, p(\\mathop {\\text{Im}}([\\mathbf {w}_r]_k))$ where both $p(\\mathop {\\text{Re}}([\\mathbf {w}_r]_k))$ and $p(\\mathop {\\text{Im}}([\\mathbf {w}_r]_k))$ obey a 0-mean Laplacian distribution.",
"Notice that the distribution (REF ) assumes the independence of the real and imaginary parts of components.",
"This assumption can be harmful if parameters are (positively or negatively) correlated with each other, which is indeed the case with symmetric or antisymmetric relations."
],
[
"Training Procedures",
"Optimizing our objective function (Eq.",
"(REF )) is difficult with standard online optimization methods, such as stochastic gradient descent.",
"In this paper, we extend the Regularized Dual Averaging (RDA) algorithm [22], which can produce sparse solutions effectively and is used in learning sparse word representations [4], [17].",
"Let us indicate the parameter value at time $t$ by superscript $(t)$ .",
"RDA keeps track of the online average subgradients at time $t$ : $\\sbox {\\mathbf {g}}\\widetilde{\\usebox {}}^{(t)} = (1/t) \\sum _{\\tau =1}^t \\mathbf {g}^{(\\tau )}$ , where $\\mathbf {g}^{(t)}$ is the subgradient at time $t$ .",
"In this paper, we calculate the subgradients $\\mathbf {g}^{(t)}$ in terms of only the loss function and L2 norm penalty term; i.e., they are the derivatives of the objective without the L1 penalty term.",
"The update formulas for multiplicative L1 regularizer are given as follows: $\\mathop {\\text{Re}}([\\mathbf {w}_r]_k^{(t+1)}) & = { {\\left\\lbrace \\begin{array}{ll}0, & \\text{if } \\left| [ \\sbox {\\mathbf {g}_r}\\widetilde{\\usebox {}}]_k^{(t)} \\right| \\le \\beta \\left| \\mathop {\\text{Im}}([\\mathbf {w}_r]_k^{(t)}) \\right|, \\\\\\gamma , & \\text{otherwise,}\\end{array}\\right.}}",
"\\\\\\mathop {\\text{Im}}([\\mathbf {w}_r]_k^{(t+1)}) & = { {\\left\\lbrace \\begin{array}{ll}0, & \\text{if } \\left| [ \\sbox {\\mathbf {g^{\\prime }}_r}\\widetilde{\\usebox {}} ]_k^{(t)} \\right| \\le \\beta \\left| \\mathop {\\text{Re}}([\\mathbf {w}_r]_k^{(t)}) \\right|, \\\\\\gamma ^{\\prime } & \\text{otherwise,}\\end{array}\\right.",
"}}$ where $\\beta = \\lambda \\alpha $ is a constant, $\\mathbf {g}_r, \\mathbf {g}^{\\prime }_r\\in \\mathbb {R}^d$ are the real and imaginary parts of the subgradients with respect to relation $r$ , and $\\gamma & = - \\eta t \\left( [\\sbox {\\mathbf {g}_r }\\widetilde{\\usebox {}} ]_k^{(t)} - \\beta \\left| \\mathop {\\text{Im}}([\\mathbf {w}_r]_k^{(t)}) \\right| \\mathop {\\text{sign}}([\\sbox {\\mathbf {g} _r}\\widetilde{\\usebox {}} ]_k^{(t)} ) \\right), \\\\\\gamma ^{\\prime } & = - \\eta t \\left( [\\sbox {\\mathbf {g}^{\\prime }_r}\\widetilde{\\usebox {}}]_k^{(t)} - \\beta \\left| \\mathop {\\text{Re}}([\\mathbf {w}_r]_k^{(t)}) \\right| \\mathop {\\text{sign}}([\\sbox {\\mathbf {g}^{\\prime }_r}\\widetilde{\\usebox {}} ]_k^{(t)} )\\right).$ From these formulas, we notice the interaction of the real and imaginary parts of a component; the imaginary part appears in the update formula for the real part, and vice versa.",
"The term $\\beta |\\mathop {\\text{Im}}([\\mathbf {w}_r^{(t)}]_k)|$ can be regarded as the strength of L1 regularizer specialized for $\\mathop {\\text{Re}}([\\mathbf {w}_r]_k)$ at time $t$ ; if $\\mathop {\\text{Im}}([\\mathbf {w}_r]_k)=0$ , then $\\mathop {\\text{Re}}([\\mathbf {w}_r]_k)$ keeps a nonzero value $\\gamma $ .",
"Likewise, if $\\mathop {\\text{Re}}([\\mathbf {w}_r]_k)=0$ , $\\mathop {\\text{Im}}([\\mathbf {w}_r]_k)$ is free to take on a nonzero value.",
"Notice that the above update formulas are applied only to relation vectors.",
"Entity vectors are learned with a standard optimization method."
],
[
"Knowledge Base Embedding",
"RESCAL [14] is an embedding-based KBC method whose scoring function is formulated as $\\mathbf {e}_{s}^{\\rm T}\\mathbf {W}_{r}\\mathbf {e}_{o}$ , where $\\mathbf {e}_s, \\mathbf {e}_o \\in \\mathbb {R}^d$ are the vector representations of entities $s$ and $o$ , respectively, and (possibly non-symmetric) matrix $\\mathbf {W}_r \\in \\mathbb {R}^{d\\times d}$ represents a relation $r$ .",
"Although RESCAL is able to output non-symmetric scoring functions, each relation vector $\\mathbf {W}_r$ holds $d^2$ parameters.",
"This can be problematic both in terms of overfitting and computational cost.",
"To avoid this problem, several methods have been proposed recently.",
"DistMult [23] restricts the relation matrix to be diagonal, $\\mathbf {W}_r = \\mathop {\\text{diag}}(\\mathbf {w}_r)$ , and it can compute the likelihood score in time $O(d)$ by way of $\\phi (s,r,o) = \\mathbf {e}_s^{\\mathrm {T}}\\mathop {\\text{diag}}(\\mathbf {w}_r) \\mathbf {e}_o $ .",
"However, this form of function is necessarily symmetric in $s$ and $o$ ; i.e., $\\phi (s,r,o)=\\phi (o,r,s)$ .",
"To reconcile efficiency and expressiveness, [21] complex proposed ComplEx, using the complex-valued representations and Hermitian inner product to define the scoring function (Eq.",
"(REF )).",
"ComplEx is founded on the unitary diagonalization of normal matrices [20].",
"Unlike DistMult, the scoring function can be nonsymmetric in $s$ and $o$ .",
"[6] eq found that ComplEx is equivalent to another state-of-the-art KBC method, Holographic Embeddings (HolE) [13].",
"ANALOGY [9] also assumes that $\\mathbf {W}_r$ is real normal.",
"They showed that any real normal matrix can be block-diagonalized, where each diagonal block is either a real scalar or a $2\\times 2$ real matrix of form $\\begin{pmatrix} a & b \\\\ -b & a \\end{pmatrix}$ .",
"Notice that this $2\\times 2$ matrix is exactly the real-valued encoding of a complex value $a+i b$ .",
"In this sense, ANALOGY can be regarded as a hybrid of ComplEx (corresponding to the $2\\times 2$ diagonal blocks) and DistMult (real scalar diagonal elements).",
"It can also be viewed as an attempt to reduce of the number of parameters in ComplEx, by constraining some of the imaginary parts of the vector components to be zero.",
"Although the idea resembles our proposed method, in ANALOGY, the reduced parameters (the number of scalar diagonal components) is a hyperparameter.",
"By contrast, our approach lets the data adjust the number of parameters for individual relations, by means of the multiplicative L1 regularizer.",
"Figure: Visualization of the vector representations trained on synthetic data.Each column represents a complex-valued vector component,with the upper and lower cells representing the real and imaginary parts, respectively.The black cells represent non-zero values."
],
[
"Sparse Modeling",
"Sparse modeling is often used to improve model interpretability or to enhance performance when the size of training data is insufficient relative to the number of model parameters.",
"Lasso [19] is a sparse modeling technique for linear regression.",
"Lasso uses L1 norm penalty to effectively find which parameters can be dispensed with.",
"Following Lasso, several variants for inducing structural sparsity have been proposed, e.g., [18], [7].",
"One of them, the exclusive group Lasso [7] uses an $l_{1, 2}$ -norm penalty, to enforce sparsity at an intra-group level.",
"An interesting connection exists between our regularization terms in Eq.",
"(REF ) and the exclusive group Lasso.",
"Let $R_j(\\mathbf {w}_r)$ , $j=1,2$ represents the terms in $R_j(\\mathbf {\\Theta })$ that are concerned with relation $r$ .",
"When $\\alpha = 2/3$ , we can show that the regularizer terms $\\alpha R_1(\\mathbf {w}_r) + (1 - \\alpha ) R_2 (\\mathbf {w}_r) = (1/3) \\sum _{k=1}^d ( \\mathop {\\text{Re}}([\\mathbf {w}_r]_k) + \\mathop {\\text{Im}}([\\mathbf {w}_r]_k) )^2$ .",
"The right-hand side can be seen as an instance of the exclusive group Lasso.",
"Word representation learning [10], [15] has proven useful for a variety of natural language processing tasks.",
"Sparse modeling has been applied to improve the interpretability of the learned word vectors while maintaining the expressive power of the model [4], [17].",
"Unlike our proposed method, however, the improvement of the model performance was not the main focus."
],
[
"Demonstrations on Synthetic Data",
"We conducted experiments with synthetic data to verify that our proposed method can learn symmetric and antisymmetric relations, given such data.",
"We randomly created a knowledge base of 3 relations and 50 entities.",
"We generated a total of 6,712 triplets, and sampled 5369 triplets as training set, then one-half of the remaining triplet as validation set and the other as test set.",
"The truth values of triplets were determined such that the first relation was symmetric, the second was antisymmetric, and the last was neither symmetric or antisymmetric.",
"We compared the standard and multiplicative L1 regularizers on this dataset.",
"For the standard L1 regularization, the regularizer $R_{\\text{std}}$ (Eq.",
"(REF )) was used in place of $R_1$ (Eq.",
"(REF )) in the objective function (REF ).",
"The dimension of the embedding space was set to $d=50$ .",
"For the multiplicative L1 regularizer, hyperparameters were set as follows: $\\alpha = 1.0, \\lambda = 0.05, \\eta = 0.1$ .",
"Figure REF displays which real and imaginary parts of the learned relation vectors have non-zero values.",
"The multiplicative L1 regularizer produced the expected results: most of the imaginary parts were zero in the symmetric relation and the real parts were zero in the antisymmetric relation.",
"Table REF shows the triplet classification accuracy on the test set.",
"Triplet classification is the task of predicting the truth value of given triplets in the test set.",
"For a given triplet, the prediction of the systems was determined by the sign of the output score ($+$ = true, $-$ = false).",
"The multiplicative L1 regularizer (`ComplEx w/ mul L1') outperformed the standard L1 regularizer (`ComplEx w/ std L1') considerably for both symmetric and antisymmetric relations."
],
[
"Real Datasets: WN18 and FB15k",
"Following previous work, we used the WordNet (WN18) and Freebase (FB15k) datasets to verify the benefits of our proposed method.",
"The dataset statistics are shown in Table REF .",
"Because the datasets contain only positive triplets, (pseudo-)negative samples must be generated in this experiment.",
"In this experiment, negative samples were generated by replacing the subject $s$ and object $o$ in a positive triplet $(s,r,o)$ with a randomly sampled entity from $\\mathcal {E}$ .",
"Table: Dataset statistics for FB15k and WN18.Table: Results on the WN18 and FB15k datasets: (Filtered and raw) MRR and filtered Hits@{1,3,10}\\lbrace 1,3,10\\rbrace (%).",
"* and ** denote the results reported in and , respectively.For evaluation, we performed the entity prediction task.",
"In this task, an incomplete triplet is given, which is generated by hiding one of the entities, either $s$ or $o$ , from a positive triplet.",
"The system must output the rankings of entities in $\\mathcal {E}$ for the missing $s$ or $o$ in the triplet, with the goal of placing (unknown) true $s$ or $o$ higher in the rankings.",
"Systems that learn a scoring function $\\phi (s,r,o)$ can use the score for computing the rankings.",
"The quality of the output rankings is measured by two standard evaluation measures for the KBC task: Mean Reciprocal Rank (MRR) and Hits@1, 3 and 10.",
"We here report results in both the filtered and raw settings [2] for MRR, but only filtered values for Hits@$n$ ."
],
[
"Experimental Setup",
"We selected the hyperparameters $\\lambda $ , $\\alpha $ , and $\\eta $ via grid search such that they maximize the filtered MRR on the validation set.",
"The ranges for the grid search were as follows: $\\lambda \\in \\lbrace 0.01, 0.001, 0.0001, 0 \\rbrace $ , $\\alpha \\in \\lbrace 0, 0.3, 0.5, 0.7, 1.0 \\rbrace $ , $\\eta \\in \\lbrace 0.1, 0.05 \\rbrace $ .",
"During the training, learning rate $\\eta $ was tuned with AdaGrad [3], both for entity and relation vectors.",
"The maximum number of training epochs was set to 500 and the dimension of the vector space was $d=200$ .",
"The number of negative triplets generated per positive training triplet was 10 for FB15k and 5 for WN18.",
"Table: Results on WN18 when the size of training data is reduced to a halfFigure: The box plots for the variance of filtered MRR on the FB15k dataset.Figure: The scatter plots showing the degree of sparsity against the symmetry score for each relation."
],
[
"Results",
"We compared our proposed model (`ComplEx w/ mul L1') with ComplEx and ComplEx with standard L1 regularization (`ComplEx w/ std L1').",
"Note that our model reduces to ComplEx when $\\alpha =0$ .",
"Table REF shows the results.",
"The results for TransE, DistMult, HolE, and ANALOGY are transcribed from the literature [21], [9].",
"All the compared models, except for DistMult, can produce a non-symmetric scoring function.",
"For most of the evaluation metrics, the multiplicative L1 regularizer (`ComplEx w/ mul L1') outperformed or was competitive to the best baseline.",
"Of particular interest, its performance on FB15k was much better than that of WN18.",
"Indeed, the accuracy of the original ComplEx is already quite high for WN18.",
"This difference between two datasets comes from the percentage of infrequent relations in two datasets.",
"Most of the 1,345 relations in FB15k are infrequent, i.e., each of them has only several dozen triplets in training data.",
"By contrast, of the 18 relations in WN18, most have more than one thousand training triplets.",
"Because sparse modeling is, in general, much effective when training data is scarce, this difference in the proportion of infrequent relations should have contributed to the different performance improvements on the two datasets.",
"To support this hypothesis, we ran additional experiments with WN18, by reducing the number of training samples to one half.",
"The test data remained the same as in the previous experiment.",
"The results are shown in Table REF .",
"The standard and multiplicative L1 regularizers work better than the original ComplEx and the improvement in each evaluation metric is greater than that observed with all training data.",
"The proposed method also outperformed the standard L1 regularization consistently.",
"The reliability of the result in Table REF was verified by computing the variance of the filtered MRR scores over 8 trials on FB15k.",
"For each of the trials, different random choices were used to generate the initial values of the representation vectors and the order of samples to process.",
"The result, shown in Figure REF , confirms that random initial values have little effect on the result; there is no overlapping MRR range among the compared methods, and the proposed method (`ComplEx w/ mul L1') consistently outperformed the standard L1 (`ComplEx w/ std L1') and vanilla ComplEx with only L2 regularization."
],
[
"Analysis",
"We analyzed how the two L1 regularizers, standard and multiplicative, work for symmetric and antisymmetric relations on FB15k.",
"Because of the large number of relations FB15k contains, manually extracting symmetric/antisymmetric relations is difficult.",
"To quantify the degree of symmetry of each relation $r$ , we define its “symmetry score” by $ \\mathop {\\text{sym}}(r) = |\\mathcal {T}^{\\text{sym}}_r| / |\\mathcal {T}_r| $ where $\\mathcal {T}_r & = \\lbrace (s, r, o) \\mid (s, r, o) \\in \\Omega , y_{rso} = +1 \\rbrace , \\\\\\mathcal {T}^{\\text{sym}}_r & = \\lbrace (s, r, o) \\mid (s, r, o) \\in \\Omega , y_{rso} = y_{ros} = +1 \\rbrace ,$ and compute this score for all relations in the FB15k training set.",
"By definition, symmetric relation $r$ should give $\\text{sym}(r)=1.0$ and antisymmetric relation $r$ should give $\\text{sym}(r)=0.0$ .",
"Thus, this score should indicate the degree of symmetry/antisymmetry of relations.",
"In Figure REF , we plot the percentage of non-zero elements in the real and imaginary parts of its representation vector $\\mathbf {w}_r$ against $\\text{sym}(r)$ for each relation $r$ .",
"Panels (a) and (b) are for $\\mathop {\\text{Re}}(\\mathbf {w}_r)$ and $\\mathop {\\text{Im}}(\\mathbf {w}_r)$ , respectively.",
"It is expected that the higher (resp.",
"lower) the symmetry is, the more imaginary (resp.",
"real) parts become zero.",
"With the multiplicative L1 regularizer (denoted by `mul L1' and green crosses in the figure), the density of real parts correlates with symmetry, and the density of imaginary parts inversely correlates with symmetry.",
"By contrast, correlation is weak or not observable for the standard L1 regularizer (`std L1'; blue circles).",
"As well as demonstrated on synthetic data, we conclude that the gains in Table REF come mainly from the more desirable representations for symmetric/antisymmetric relations learned with the multiplicative L1 regularization.",
"However, compared to the results on synthetic data shown in Figure REF , many non-zero values exist in $\\mathop {\\text{Re}}(\\mathbf {w}_r)$ and $\\mathop {\\text{Im}}(\\mathbf {w}_r)$ even for completely symmetric/antisymmetric relations.",
"We suspect that this is related to the fact that an antisymmetric relations does not imply an antisymmetric truth-value matrix; even if a relation is antisymmetric, there are many entity pairs $(e_1, e_2)$ such that neither of $(e_1, r, e_2)$ or $(e_2, r, e_1)$ holds."
],
[
"Conclusion",
"In this paper, we have presented a new regularizer for ComplEx to encourage vector representations of relations to be more symmetric or antisymmetric in accordance with data.",
"In the experiments, the proposed regularizer improved over the original ComplEx (with only L2 regularization) on FB15k, as well as WN18 with limited training data.",
"Recently, researchers have been making attempt to leverage background knowledge to improve KBC, such as by incorporating information of entity types, hierarchy, and relation attributes into the models [8], [11].",
"Our method does not assume background knowledge but adapt to symmetry/asymmetry of relations in a data-driven manner.",
"However, learning more diverse knowledge about relations from data should be an interesting future research topic.",
"We would also like to consider different type of regularization for entity vectors.",
"As another future research direction, we would like to develop a better strategy for sampling negative triplets that are suitable for our method."
],
[
"Acknowledgments",
"We are grateful to the anonymous reviewers for useful comments.",
"MS thanks partial support by JSPS Kakenhi Grant 15H02749."
]
] | 1808.08361 |
[
[
"Intrinsic magnetic topological insulators in van der Waals layered\n MnBi$_2$Te$_4$-family materials"
],
[
"Abstract The interplay of magnetism and topology is a key research subject in condensed matter physics and material science, which offers great opportunities to explore emerging new physics, like the quantum anomalous Hall (QAH) effect, axion electrodynamics and Majorana fermions.",
"However, these exotic physical effects have rarely been realized in experiment, due to the lacking of suitable working materials.",
"Here we predict that van der Waals layered MnBi$_2$Te$_4$-family materials show two-dimensional (2D) ferromagnetism in the single layer and three-dimensional (3D) $A$-type antiferromagnetism in the bulk, which could serve as a next-generation material platform for the state-of-art research.",
"Remarkably, we predict extremely rich topological quantum effects with outstanding features in an experimentally available material MnBi$_2$Te$_4$, including a 3D antiferromagnetic topological insulator with the long-sought topological axion states, the type-II magnetic Weyl semimetal (WSM) with simply one pair of Weyl points, and the high-temperature intrinsic QAH effect.",
"These striking predictions, if proved experimentally, could profoundly transform future research and technology of topological quantum physics."
],
[
"Intrinsic magnetic topological insulators in van der Waals layered MnBi$_2$ Te$_4$ -family materials Jiaheng Li$^{1,2}$ Yang Li$^{1,2}$ Shiqiao Du$^{1,2}$ Zun Wang$^{1,2}$ Bing-Lin Gu$^{1,2,3}$ Shou-Cheng Zhang$^{4}$ Ke He$^{1,2}$ kehe@tsinghua.edu.cn Wenhui Duan$^{1,2,3}$ dwh@phys.tsinghua.edu.cn Yong Xu$^{1,2,5}$ yongxu@mail.tsinghua.edu.cn $^1$ State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, People's Republic of China $^2$ Collaborative Innovation Center of Quantum Matter, Beijing 100084, People's Republic of China $^3$ Institute for Advanced Study, Tsinghua University, Beijing 100084, People's Republic of China $^4$ Department of Physics, McCullough Building, Stanford University, Stanford, California 94305-4045, USA $^5$ RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan The interplay of magnetism and topology is a key research subject in condensed matter physics and material science, which offers great opportunities to explore emerging new physics, like the quantum anomalous Hall (QAH) effect, axion electrodynamics and Majorana fermions.",
"However, these exotic physical effects have rarely been realized in experiment, due to the lacking of suitable working materials.",
"Here we predict that van der Waals layered MnBi$_2$ Te$_4$ -family materials show two-dimensional (2D) ferromagnetism in the single layer and three-dimensional (3D) $A$ -type antiferromagnetism in the bulk, which could serve as a next-generation material platform for the state-of-art research.",
"Remarkably, we predict extremely rich topological quantum effects with outstanding features in an experimentally available material MnBi$_2$ Te$_4$ , including a 3D antiferromagnetic topological insulator with the long-sought topological axion states, the type-II magnetic Weyl semimetal (WSM) with simply one pair of Weyl points, and the high-temperature intrinsic QAH effect.",
"These striking predictions, if proved experimentally, could profoundly transform future research and technology of topological quantum physics.",
"The correlation between topology and symmetry as a central fundamental problem of modern physics has attracted intensive research interests in condensed matter physics and material science since the discovery of topological insulators (TIs) [1], [2].",
"One essential symmetry is the time reversal symmetry (TRS), which is crucial to many kinds of topological quantum states of matter like TIs and is broken in the presence of magnetism.",
"Intriguingly, the interplay between magnetism and topology in materials could generate a variety of exotic topological quantum states [1], [2], [3], including the quantum anomalous Hall (QAH) effect showing dissipationless chiral edge states [4], [5], [6], the topological axion states displaying quantized magnetoelectric effects [7], [8], [9], and Majorana fermions obeying non-abelian statistics [2], [10], [11].",
"In this context, great research effort has been devoted to explore the novel topological quantum physics, which is of profound importance to fundamental science and future technologies, like dissipationless topological electronics and topological quantum computation [1], [2].",
"One key subject that is of crucial significance to the whole research community is to develop topological quantum materials (TQMs) showing the coexistence of topology and other quantum phases (e.g.",
"magnetism, ferroelectricity, charge density wave, and superconductivity), coined “composite TQMs” (CTQMs), which include magnetic TQMs (MTQMs) as an important subset.",
"Currently, very limited number of MTQMs are experimentally available, including magnetically doped TIs and magnetic topological heterostructures [6], [12], [13], [14], whose material fabrication, experimental measurement and property optimization are quite challenging.",
"Since the major topological properties of these existing MTQMs depend sensitively on delicate magnetic doping or proximity effects, only little preliminary experimental progress has been achieved until now, leaving many important physical effects not ready for practical use or still unproved.",
"For instance, the QAH effect was only observed in magnetically doped (Bi$_{x}$ Sb$_{1-x}$ )$_2$ Te$_3$ thin films at very low working temperatures by fine tuning chemical compositions [6], [12].",
"While a previous theory predicted the existence of topological axion states at the surface of a three-dimensional (3D) antiferromagnetic (AFM) TI [15], no 3D AFM TI has been realized experimentally, as far as we know.",
"For future research and applications, people should go beyond the existing strategy for building MTQMs and try to design intrinsic MTQMs in no need of introducing alloy/doping or heterostructures.",
"Noticeably, van der Waals (vdW) layered materials represent a large family of materials with greatly tunable properties by quantum size effects or vdW heterojunctions [16], in which a variety of quantum phases in different spatial dimensions have been discovered by the state-of-the-art research [17], [18], [19], [20], [21], [22].",
"For instance, both two-dimensional (2D) and 3D TI states were previously found in the tetradymite-type Bi$_2$ Te$_3$ -class materials [17], [18], and 2D intrinsic magnetism was recently found in ultrathin films of CrI$_3$ [23] and Cr$_2$ Ge$_2$ Te$_6$ [24] .",
"However, these topological materials are not magnetic; those magnetic material are not topological.",
"It is highly desirable to incorporate the magnetic and topological states together into the same vdW material, so as to obtain layered intrinsic MTQMs, which are able to inherit advantages of vdW-family materials.",
"In this work, based on first-principles (FP) calculations, we find a series of layered intrinsic MTQMs from the tetradymite-type MnBi$_2$ Te$_4$ -related ternary chalcogenides (MB$_2$ T$_4$ : M = transition-metal or rare-earth element, B = Bi or Sb, T = Te, Se or S), in which the intralayer exchange coupling is ferromagnetic (FM), giving 2D ferromagnetism in their septuple layer (SL); while the interlayer exchange coupling is antiferromagnetic, forming 3D $A$ -type antiferromagnetism in their vdW layered bulk.",
"Strikingly, we predict that plenty of novel 2D and 3D intrinsic magnetic topological states with outstanding features could be naturally realized in an experimentally available material MnBi$_2$ Te$_4$ , including the 3D AFM TI in the bulk together with the long-sought topological axion states on the (001) surface, large-gap intrinsic QAH insulators ($E_{\\textit {g}} \\sim $ 38 meV) with dissipaltionless and patternable chiral edge states in thin films, the type-II magnetic WSM with a single pair of Weyl points in the FM bulk, and possibly Majorana fermions if interacting with superconductivity.",
"The tetradymite-type MnBi$_2$ Te$_4$ crystalizes in a rhombohedral layered structure with the space group $R\\overline{3}m$ , and each layer has a triangular lattice with atoms ABC-stacked along the out-of-plane direction, the same as Bi$_2$ Te$_3$ .",
"Slightly differently, one layer MnBi$_2$ Te$_4$ includes seven atoms in a unit cell, forming a Te-Bi-Te-Mn-Te-Bi-Te SL, which can be viewed as intercalating a Mn-Te bilayer into the center of a Bi$_2$ Te$_3$ quintuple layer (Fig.",
"1a).",
"Noticeably, the mixing between Mn and Bi in this compound would create unstable valence states Mn$^{+3}$ and Bi$^{+2}$ , which is energetically unfavorable.",
"Thus the formation of alloys could be avoided, leading to stoichiometric compounds.",
"This physical mechanism also explains the stability of compounds like SnBi$_2$ Te$_4$ and PbBi$_2$ Te$_4$ and is applicable to many other MB$_2$ T$_4$ -family materials.",
"Importantly, our FP calculations predicted that a series of MB$_2$ T$_4$ compounds are energetically and dynamically stable, including M = Ti, V, Mn, Ni, Eu, etc., as confirmed by FP total-energy and phonon calculations [25], which could be fabricated by experiment.",
"These stable compounds are characterized by an insulating band gap in their single layers, offering a collection of layered intrinsic magnetic (topological) insulators.",
"In fact, MnBi$_2$ Te$_4$ is now experimentally available [26], which is selected to introduce properties of the material family.",
"Mn has a valence charge of $+$ 2 by losing its two 4$s$ electrons.",
"The remaining five 3$d$ electrons fill up the spin-up Mn-$d$ levels according to the Hund's rule, introducing 5 $\\mu _{B}$ magnetic moment (mostly from Mn) per unit cell.",
"By comparing different magnetic structures, we found that the magnetic ground state is a 2D ferromagnetism with an out-of-plane easy axis in the single layer (depicted Fig.",
"1a), in agreement with the previous work [27].",
"Each Mn atom is bonded with six neighboring Te atoms, which form a slightly distorted edge-sharing octahedron.",
"According to the Goodenough-Kanamori rule, the superexchange interactions between Mn-Te-Mn with a bonding angle of $\\sim 94^{\\circ }$ are ferromagnetic, similar as in CrI$_3$ and Cr$_2$ Ge$_2$ Te$_6$ [23], [24].",
"Intralayer ferromagnetic exchange couplings was also found in other stable MB$_2$ T$_4$ -family members, but the easy axis can be varied to in-plane (e.g.",
"for M = V), displaying rich 2D magnetic features.",
"Intriguingly, magnetic and topological states are well incorporated together into MnBi$_2$ Te$_4$ , where Mn introduces magnetism and the Bi-Te layers could generate topological states similar as Bi$_2$ Te$_3$ [17], as schematically depicted in Fig.",
"1b.",
"The exchange splitting between spin-up and spin-down Mn $d$ bands are extremely large ($>$ 7 eV) caused by the large magnetic moment of Mn.",
"Thus Mn $d$ -bands are far away from the band gap, and only Bi/Te $p$ -bands are close to the Fermi level.",
"The single layer has a 0.73 eV direct (1.36 eV indirect) band gap when including (excluding) spin-orbit coupling (SOC) (Figs.",
"1c and 1d).",
"While the single layer is topologically trivial ferromagnetic insulator, extremely interesting topological quantum physics emerges in the bulk and thin films, as we will demonstrate.",
"For the layered bulk, we found that the interlayer magnetic coupling is AFM, giving a $A$ -type AFM ground state (depicted in Fig.",
"2a) [25], which is similar as in Cr$_2$ Ge$_2$ Te$_6$ [24] and explained by the interlayer super-superexchange coupling.",
"The spatial inversion symmetry $P$ (i.e.",
"$P_1$ centered at O$_1$ ) is preserved, but the TRS $\\Theta $ gets broken.",
"There exist two new symmetries: $P_2 \\Theta $ and $S=\\Theta T_{1/2}$ , where $P_2$ is an inversion operation centered at O$_2$ and $T_{1/2}$ is a lattice translation (depicted in Fig.",
"2a).",
"The band structures with and without SOC are presented in Figs.",
"2b and 2c.",
"Here every band is (at least) doubly degenerate, which is ensured by the $P_2 \\Theta $ symmetry [28].",
"In the presence of $S$ , a $\\mathcal {Z}_2$ classification becomes feasible [15].",
"However, in contrast to the time-reversal invariant case, there is a $\\mathcal {Z}_2$ invariant for the $k_z = 0$ plane but not for the $k_z = \\pi $ plane.",
"$\\mathcal {Z}_2 = 1$ corresponds to a novel 3D AFM TI phase [15], which has not been experimentally confirmed.",
"Here the parity criteria can be applied to determine $\\mathcal {Z}_2$ due to the $P_1$ symmetry [29].",
"Interestingly, the parities of the valence band maximum (VBM) and conduction band minimum (CBM) at $\\Gamma $ are opposite, and both change signs by the SOC effects (Figs.",
"2b and 2c), implying a band inversion.",
"By varying the SOC strength, the band gap first closes and then reopens at $\\Gamma $ (Fig.",
"2d), showing simply one band inversion and thus implying a topological phase transition.",
"The SOC-induced band reversal is happened between Bi $p_z^+$ and Te $p_z^-$ , essentially the same as for Bi$_2$ Te$_3$ [17].",
"Our Wannier charge center (WCC) calculations for the $k_z = 0$ plane revealed $\\mathcal {Z}_2 = 1$ ($\\mathcal {Z}_2 = 0$ ) for with (without) SOC, confirming that MnBi$_2$ Te$_4$ is a 3D AFM TI.",
"The global band gap is $\\sim $ 0.16 eV direct at $Z$ , and the direct gap at $\\Gamma $ is $\\sim $ 0.18 eV.",
"One prominent feature of the AFM TI is the existence of 2D gapless surface states protected by $S$ , which is confirmed by the surface-state calculations (Figs.",
"2f and 2g).",
"The surface states are indeed gapless on the (100) termination.",
"However, they become gapped on the (001) termination due to the $S$ symmetry breaking.",
"The intrinsically gapped (001) surfaces are promising for probing the long-sought topological axion states, which give the topological quantized magnetoelectric effect related to an axion field with $\\theta = \\pi $ [7], [8], [9], [15].",
"Previous work proposed to probe the novel states by adding opposite out-of-plane ferromagnetism onto the bottom and top surfaces of 3D TIs [7], [14].",
"Such kind of TRS-broken surface states are naturally provided by even-layer MnBi$_2$ Te$_4$ films (when with negligible hybridizations between top and bottom surface states), benefitting from their $A$ -type AFM structure.",
"An additional requirement is to open a band gap at side surfaces, which is realized in relatively thin films or by breaking the $S$ symmetry on the side surfaces (e.g.",
"controlling surface morphology).",
"The simplified proposal together with the suitable material candidate MnBi$_2$ Te$_4$ could greatly facilitate the research of axion electrodynamics.",
"The AFM ground state of MnBi$_2$ Te$_4$ could be tuned, for instance, by applying an external magnetic field, to other magnetic structures.",
"Then the material symmetry changes, leading to distinct topological phases.",
"This concept is demonstrated by studying the simple FM ordering along the out-of-plane direction (Fig.",
"3a).",
"The FM structure has $P_1$ but neither $\\Theta $ nor $P_2 \\Theta $ , leading to spin-split bands, nonzero Berry curvatures and possibly nonzero topological Chern numbers that correspond to novel topological phases, like 3D QAH insulators [30] and WSMs [31], [32], [33], [22], [3].",
"The band structure of FM MnBi$_2$ Te$_4$ (Fig.",
"3b) displays a pair of band crossings at W/W$^{\\prime }$ along the $\\overline{\\textit {Z}}$ -$\\Gamma $ -Z line.",
"The band crossings are induced by interlayer orbital hybridizations and protected by the $C_3$ rotational symmetry.",
"Our WCC calculations found that W is a momentum-space monopole with a topological charge of +1 (i.e.",
"an Berry phase of $2\\pi $ ) (Fig.",
"3e), and its time-reversal partner W$^{\\prime }$ has an opposite topological charge of -1, indicating that the system is a topological WSM.",
"While in most of the momentum space the electron pocket is located above the hole pocket (e.g.",
"along the F-W-L line), the Weyl cones get tilted along the $\\overline{\\textit {Z}}$ -$\\Gamma $ -Z line (Fig.",
"3d), leaving some part of the electron pocket below the hole pocket, which is the characteristic feature of the type-II WSM [22].",
"In contrast to the time-reversal-invariant WSMs that must have even pairs of Weyl points, this ferromagnetic WSM represents the simplest one, hosting only a pair of Weyl points.",
"Moreover, our surface-state calculations clearly demonstrated the existence of Fermi arcs on the (100) and ($\\bar{1}$ 00) terminations (Figs.",
"3f and 3g), which is the fingerprint of the WSM.",
"Importantly, the Weyl points are well separated in the momentum space and very close to the Fermi level, advantageous for experimental observations.",
"The vdW layered materials are featured by tunable quantum size effects.",
"For AFM MnBi$_2$ Te$_4$ films, even layers do not possess $P$ and $\\Theta $ symmetries, but have $P_2 \\Theta $ , ensuring double degeneracy in every band.",
"Differently, odd layers have $P$ but neither $\\Theta $ nor $P_2 \\Theta $ , leading to spin-split bands.",
"The different symmetries lead to distinct topological properties in even and odd layers.",
"Specifically, the topological Chern number $\\mathcal {C} = 0$ is required by $P_2 \\Theta $ in even layers, $\\mathcal {C} \\ne 0$ is allowed in odd layers.",
"In contrast to the single and 3-layer films, where a trivial insulating gap is opened by quantum confinement effects, we found that the 5-layer film is an intrinsic QAH insulator with $\\mathcal {C} = 1 $ , as confirmed by the appearance of an quantized Hall conductance and chiral edge states within the bulk gap (Figs.",
"4a-c).",
"We also calculated a 7-layer film, which is also a QAH insulator with $\\mathcal {C} = 1 $ .",
"The exchange splitting introduced by the magnetic Mn-layers together with the strong SOC effects in the Bi-Te bands cooperatively induce the QAH effect, similar as in the magnetically doped Bi$_2$ Te$_3$ -class TIs [5].",
"However, the present material has an intrinsic magnetism and does not need uncommon mechanisms (e.g.",
"the Van-Vleck mechanism [5]) to form ferromagnetism in insulating states.",
"Moreover, since the magnetic doping is not needed, disorder-induced magnetic domains and potential fluctuations, that deteriorate the QAH effect, are avoided.",
"Furthermore, the QAH gap of the 5-layer film is 38 meV, greater than the room-temperature thermal energy of 26 meV, enabling a high working temperature.",
"The thickness dependence behaviors can be understood as follows.",
"There are intrinsically gapped surface states on both sides of thick films, as obtained for the (001) semi-infinite surface.",
"These surface gaps are opened by the TRS-breaking field near the surface, which have half quantized Hall conductances $\\sigma _{xy} = e^2/2h$ or $-e^2/2h$ when the magnetism in the surface layer is up- or down-oriented, respectively.",
"Thus, the Hall conductances of bottom and top surfaces is cancelled in even layers, giving axion insulators with $\\mathcal {C} = 0$ or topological axion states mentioned above; while they get added in odd layers, giving QAH insulators with $\\mathcal {C} = 1$ .",
"This physical picture, consistent with our calculation results, suggests an oscillation of $\\mathcal {C}$ in even and odd layers.",
"Thus chiral edge states always appear near step edges (Fig.",
"4d), which can be used for dissipationless conduction.",
"Based on this unique feature, the chiral edge states could be selectively patterned by controlling the film thickness or step edges, advantageous for building dissipationless circuits.",
"Looking back to the history of the TI research, the first- and second-generation TIs are the HgTe/CdTe quantum wells [34] and Bi-Sb alloys, respectively, which are very complex and difficult to study theoretically and experimentally.",
"Research interests have been increased exponentially since the discovery of the third-generation TIs in the intrinsic Bi$_2$ Te$_3$ -class materials [17], [18].",
"A very similar situation is faced by the research of magnetic topological physics.",
"Currently, experimental works are majorally based on magnetically doped TIs and magnetic topological heterostructures, which are quite challenging and have led to little preliminary progress.",
"Looking forwards, the research progress is expected to be greatly prompted by discovering intrinsic MTQMs that are simple and easy to control.",
"The vdW layered MnBi$_2$ Te$_4$ -family materials satisfy all these material traits.",
"More importantly, this material family could host extremely rich topological quantum states in different spatial dimensions (like 3D AFM TIs, time-reversal invariant TIs and topological semimetals, 2D QAH and QSH insulators, etc.)",
"and are promising for investigating other exotic emerging physics (like Majorana fermions), which are thus perfect next-generation MTQMs for future research.",
"Figure: Monolayer MnBi 2 _2Te 4 _4 (MBT).",
"(a) The side view of monolayer MBT.",
"(b) A schematic diagram of band structure in MBT including thin films and 3D bulk.",
"(c,d) Band structures of monolayer MBT calculated by HSE06 without/with spin-orbital coupling (SOC).",
"The blue (green) curve in the (1c) represents the spin-up (spin-down) states.",
"(e) A schematic diagram of XBT materials family.",
"The red arrows represent the magnetic moment, whose length and direction represent the magnitude and easy magnetic axis of magnetic atoms.Figure: AFM MBT.",
"(a) The side view of bulk AFM MBT crystal structure.The black dot O 1 _1 represents inversion center located at the magnetic Mn atoms.",
"The black dot O 2 _2 represents symmetry operation which combines inversion operator and the lattice translation T 1/2 T_{1/2} along c axis.",
"The schematic diagram shows MBT is stacked in the way combining the AFM order and ABC-stacking sequence.",
"Brillouin zone used in the calculation are showed below.",
"(b), (c) Band structures of 3D bulk AFM MBT without/with SOC.",
"The parity of wavefunction at high-symmetry points Γ\\Gamma is showed by the minus and plus signs.",
"Minus (plus) sign means odd (even) parity.",
"(d) Band gap at Γ\\Gamma calculated by artificially changing the strength of SOC.",
"Band gap closing happens at artificial SOC strength around 81%.",
"(e) Wannier charge centers along k 2 k_2 on the k 3 k_3=0 plane.",
"(f), (g) The calculated surface states of the AFM TI MBT on the (001) and (100) faces, respectively.",
"The inset of (f) shows gapped surface states on the (001) face clearly, however,the inset of (g) shows no apparent gap on the (100) face.Figure: FM MBT.",
"(a) Atomic structure of FM MBT.",
"Dashed rhombus represents the unit cell.",
"(b) Band structure of FM MBT.",
"(c) Brillouin zone of FM MBT.",
"The red and blue points represent a pair of type-II Weyl points W and W ' ^{\\prime }.",
"(d) Zoom-in band structure around Weyl point W from different directions.",
"(e) Motion of the sum of WCCs on the sphere around Type-II Weyl point W. (f, g) Fermi arcs on the (100) and (1 ¯00\\bar{1}00) terminations.",
"Fermi energy is fixed at the energy level of Weyl point W.Figure: MBT thin films.",
"(a) Band structure, (b) Hall conductance σ xy \\sigma _{xy} as a function of energy and (c) edge states of the 5-layer MBT.",
"(d) A schematic diagram showing dissipationless edge channels on step edges of MBT."
]
] | 1808.08608 |
[
[
"Spontaneous and Gravitational Baryogenesis"
],
[
"Abstract Some problems of spontaneous and gravitational baryogenesis are discussed.",
"Gravity modification due to the curvature dependent term in gravitational baryogensis scenario is considered.",
"It is shown that the interaction of baryonic fields with the curvature scalar leads to strong instability of the gravitational equations of motion and as a result to noticeable distortion of the standard cosmology."
],
[
"Introduction",
"Observations show that at least the region of the Universe around us is matter-dominated.",
"Though we understand how the matter-antimatter asymmetry may be created, the concrete mechanism is yet unknown.",
"The amount of antimatter is very small and it can be explained as the result of high energy collisions in space.",
"The existence of large regions of antimatter in our neighbourhood would produce high energy radiation as a consequence of matter-antimatter annihilation, which is not observed.",
"Any initial asymmetry at inflation could not solve the problem of observed excess of matter over antimatter, because the energy density associated with baryonic number would not allow for sufficiently long inflation.",
"On the other hand, matter and antimatter seem to have similar properties and therefore we could expect a matter-antimatter symmetric Universe.",
"A satisfactory model of our Universe should be able to explain the origin of the local observed matter-antimatter asymmetry.",
"The term baryogenesis means the generation of the asymmetry between baryons (basically protons and neutrons) and antibaryons (antiprotons and antineutrons).",
"In 1967 Andrey Sakharov pointed out 3 ingredients, today known as Sakharov principles, to produce a matter-antimatter asymmetry from an initially symmetric Universe.",
"These conditions include: 1) non-conservation of baryonic number; 2) breaking of symmetry between particles and antiparticles; 3) deviation from thermal equilibrium.",
"However, not all of three Sakharov principles are strictly necessary.",
"In what follows we briefly discuss some features of spontaneous baryogenesis (SBG) and concentrate in more detail on gravitational baryogenesis (GBG).",
"Both these mechanisms do not demand an explicit C and CP violation and can proceed in thermal equilibrium.",
"Moreover, they are usually most efficient in thermal equilibrium.",
"The statement that the cosmological baryon asymmetry can be created by spontaneous baryogenesis in thermal equilibrium was mentioned in the original paper by Cohen and Kaplan[1] and developed in subsequent papers[2], [3], for review see[4], [5].",
"The term \"spontaneous\" is related to spontaneous breaking of a global $U(1)$ -symmetry, which ensures the conservation of the total baryonic number in the unbroken phase.",
"This symmetry is supposed to be spontaneously broken and in the broken phase the Lagrangian density acquires the additional term ${\\cal L}_{SB} = (\\partial _{\\mu } \\theta ) J^{\\mu }_B\\, ,$ where $\\theta $ is the Goldstone field and $J^{\\mu }_B$ is the baryonic current of matter fields, which becomes non-conserved.",
"For a spatially homogeneous field, $\\theta = \\theta (t)$ , the Lagrangian is reduced to the simple form ${\\cal L}_{SB} = \\dot{\\theta }\\, n_B\\,, \\ \\ \\ \\ n_B\\equiv J^0_B,$ where time component of a current is the baryonic number density of matter, so it is tempting to identify ${\\dot{\\theta }}$ with the chemical potential, $ \\mu _B$ , of the corresponding system.",
"However, such identification is questionable and depends upon the representation chosen for the fermionic fields[6], [7].",
"It is heavily based on the assumption ${\\dot{\\theta }\\approx const}$ , which is relaxed in the work[8].",
"But still the scenario is operative and presents a beautiful possibility to create an excess of particles over antiparticles in the Universe.",
"Subsequently the idea of gravitational baryogenesis (GBG) was put forward [9], where the scenario of SBG was modified by the introduction of the coupling of the baryonic current to the derivative of the curvature scalar $R$ : ${\\cal L}_{GBG} = \\frac{1}{M^2} (\\partial _\\mu R ) J^\\mu _B\\, ,$ where $M$ is a constant parameter with the dimension of mass.",
"In the presented talk we demonstrate that the addition of the curvature dependent term (REF ) to the Hilbert-Einstein Lagrangian of General Relativity (GR) leads to higher order gravitational equations of motion, which are strongly unstable with respect to small perturbations.",
"The effects of this instability may drastically distort not only the usual cosmological history, but also the standard Newtonian gravitational dynamics.",
"We discovered such instability for scalar baryons [10] and found similar effect for the more usual spin one-half baryons (quarks) [11]."
],
[
"Gravitational baryogenesis with scalar baryons ",
"Let us start from the model where baryonic number is carried by scalar field $\\phi $ with potential $U(\\phi , \\phi ^* )$ .",
"An example with baryonic current of fermions will be considered in the next section.",
"The action of the scalar model has the form: $A = \\int d^4 x\\, \\sqrt{-g} \\left[ \\frac{m_{Pl}^2}{16\\pi } R + \\frac{1}{M^2} (\\partial _{\\mu } R) J^{\\mu } -g^{\\mu \\nu } \\partial _{\\mu }\\phi \\, \\partial _{\\nu }\\phi ^* + U(\\phi , \\phi ^*)\\right] - A_m\\, ,$ where $m_{Pl}=1.22\\cdot 10^{19}$ GeV is the Planck mass, $A_m$ is the matter action, $J^\\mu = g^{\\mu \\nu }J_\\nu $ , and $g^{\\mu \\nu }$ is the metric tensor of the background space-time.",
"We assume that initially the metric has the usual GR form and study the emergence of the corrections due to the instability described below.",
"In contrast to scalar electrodynamics, the baryonic current of scalars is not uniquely defined.",
"In electrodynamics the form of the electric current is dictated by the conditions of gauge invariance and current conservation, which demand the addition to the current of the so called sea-gull term proportional to $e^2 A_\\mu |\\phi |^2$ , where $A_\\mu $ is the electromagnetic potential.",
"On the other hand, a local $U(1)$ -symmetry is not imposed on the theory determined by action (REF ).",
"It is invariant only with respect to a $U(1)$ transformations with a constant phase.",
"As a result, the baryonic current of scalars is considerably less restricted.",
"In particular, we can add to the current an analogue of the sea-gull term, $ \\sim (\\partial _\\mu R)\\,|\\phi |^2 $ , with an arbitrary coefficient.",
"In our paper [10] we study the following two extreme possibilities, when the sea-gull term is absent and the current is not conserved, or the sea-gull term is included with the coefficient ensuring current conservation.",
"In both cases no baryon asymmetry can be generated without additional interactions.",
"It is trivially true in the second case, when the current is conserved, but it is also true in the first case despite the current non-conservation, simply because the non-zero divergence $D_\\mu J^\\mu $ does not change the baryonic number of $\\phi $ but only leads to redistribution of particles $\\phi $ in the phase space.",
"So to create any non-zero baryon asymmetry we have to introduce an interaction of $\\phi $ with other particles which breaks conservation of $B$ by making the potential $U$ non-invariant with respect to the phase rotations of $\\phi $ , as it is described below.",
"If the potential $U(\\phi )$ is not invariant with respect to the $U(1)$ -rotation, $\\phi \\rightarrow \\exp {(i \\beta )} \\phi $ , the baryonic current defined in the usual way $J_{ \\mu } = i q (\\phi ^* \\partial _{\\mu }\\phi - \\phi \\partial _{\\mu }\\phi ^*)$ is not conserved.",
"Here $q$ is the baryonic number of $\\phi $ and we omitted index $B$ in current $J_{\\mu }$ .",
"With this current and Lagrangian (REF ) the equation for the curvature scalar, $R$ , takes the form: $\\frac{m_{Pl}^2}{16\\pi }\\, R + \\frac{1}{M^2}\\left[ (R + 3 D^2) D_{\\alpha } J^{ \\alpha } + J^{\\alpha } \\,D_{\\alpha }R \\right] -D_{\\alpha } \\phi \\, D^{\\alpha } \\phi ^* + 2 U(\\phi ) = - \\frac{1}{2} \\, T_{\\mu }^{\\mu }\\, ,$ where $D_\\mu $ is the covariant derivative in metric $g_{\\mu \\nu }$ (of course, for scalars $D_\\mu = \\partial _\\mu $ ) and $T_{\\mu \\nu }$ is the energy-momentum tensor of matter obtained from action $A_m$ .",
"According to definition (REF ), the current divergence is: $D_{\\mu } J^{\\mu } = \\frac{2q^2}{M^2} \\left[ D_{\\mu } R\\, (\\phi ^* D^{\\mu }\\phi + \\phi D^{\\mu }\\phi ^*) + |\\phi |^2 D^2 R \\right]+ i q \\left(\\phi \\frac{\\partial U}{\\partial \\phi } - \\phi ^* \\frac{\\partial U}{\\partial \\phi ^*} \\right)\\,.$ If the potential of $U$ is invariant with respect to the phase rotation of $\\phi $ , i.e.",
"$U = U(|\\phi |)$ , the last term in this expression disappears.",
"Still the current remains non-conserved, but this non-conservation does not lead to any cosmological baryon asymmetry.",
"Indeed, the current non-conservation is proportional to the product $\\phi ^* \\phi $ , so it can produce or annihilate an equal number of baryons and antibaryons.",
"To create cosmological baryon asymmetry we need to introduce new types of interactions, for example, the term in the potential of the form: $ U_4 = \\lambda _4 \\phi ^4 + \\lambda _4^* \\phi ^{*4} $ .",
"This potential is surely non invariant w.r.t.",
"the phase rotation of $\\phi $ and can induce the B-non-conserving process of transition of two scalar baryons into two antibaryons, $2 \\phi \\rightarrow 2 \\bar{\\phi }$ .",
"Let us consider solution of the above equation of motion in cosmology.",
"The metric of the spatially flat cosmological FRW background can be taken as: $ds^2=dt^2 - a^2(t) d{\\bf r}^2\\, .$ In the homogeneous case the equation for the curvature scalar (REF ) takes the form: $\\frac{m_{Pl}^2}{16\\pi }\\, R + \\frac{1}{M^2}\\left[ (R + 3 \\partial _t^2 + 9 H \\partial _t) D_{\\alpha } J^{\\alpha } +\\dot{R} \\, J^0 \\right]= - \\frac{ T^{(tot)}}{2} \\,,$ where $J^0$ is the baryonic number density of the $\\phi $ -field, $H = \\dot{a}/a$ is the Hubble parameter, and $T^{(tot)}$ is the trace of the energy-momentum tensor of matter including contribution from the $\\phi $ -field.",
"In the homogeneous and isotropic cosmological plasma $T^{(tot)} = \\rho - 3 P\\, ,$ where $\\rho $ and $P$ are respectively the energy density and the pressure of the plasma.",
"For relativistic plasma $\\rho = \\pi ^2 g_* T^4/30$ with $T$ and $g_*$ being the plasma temperature and the number of particle species in the plasma.",
"The Hubble parameter is expressed through $\\rho $ as $H^2 = 8\\pi \\rho /(3m_{Pl}^2) \\sim T^4/m_{Pl}^2$ .",
"The covariant divergence of the current is given by the expression (REF ).",
"In the homogeneous case we are considering it takes the form: $D_{\\alpha } J^{\\alpha } = \\frac{2q^2}{M^2} \\left[ \\dot{R}\\, (\\phi ^* \\dot{\\phi }+ \\phi \\dot{\\phi }^*) +(\\ddot{R} + 3H \\dot{R})\\, \\phi ^*\\phi \\right]+ i q \\left(\\phi \\frac{\\partial U}{\\partial \\phi } - \\phi ^* \\frac{\\partial U}{\\partial \\phi ^*} \\right)\\,.$ To derive the equation of motion for the classical field $R$ in the cosmological plasma we have to take the expectation values of the products of the quantum operators $\\phi $ , $\\phi ^*$ , and their derivatives.",
"Performing the thermal averaging, we find $\\langle \\phi ^* \\phi \\rangle = \\frac{T^2}{12}\\, , \\ \\ \\ \\langle \\phi ^* \\dot{\\phi }+ \\dot{\\phi }^* \\phi \\rangle = 0\\, .$ Substituting these average values into Eq.",
"(REF ) and neglecting the last term in Eq.",
"(REF ) we obtain the fourth order differential equation: $\\frac{m_{Pl}^2}{16\\pi }\\, R +\\frac{q^2}{6 M^4} \\left(R + 3 \\partial _t^2 + 9 H \\partial _t \\right)\\left[\\left(\\ddot{R} + 3H \\dot{R}\\right) T^2 \\right] +\\frac{1}{M^2} \\dot{R} \\, \\langle J^0 \\rangle = - \\frac{T^{(tot)}}{2} \\, .$ Here $\\langle J^0 \\rangle $ is the thermal average value of the baryonic number density of $\\phi $ .",
"It is assumed to be zero initially and generated as a result of GBG.",
"We neglect this term, since it is surely small initially and probably subdominant later.",
"Anyhow it does not noticeably change the exponential rise of $R$ at the onset of the instability.",
"Eq.",
"(REF ) can be further simplified if the variation of $R(t)$ is much faster than the universe expansion rate or in other words $\\ddot{R} / \\dot{R} \\gg H$ .",
"Correspondingly the temperature may be considered adiabatically constant.",
"The validity of these assumption is justified a posteriori after we find the solution for $R(t)$ .",
"Keeping only the linear in $R$ terms and neglecting higher powers of $R$ , such as $R^2$ or $H R$ , we obtain the linear differential equation of the fourth order: $\\frac{d^4 R}{dt^4} + \\mu ^4 R = - \\frac{1}{2} \\, T^{(tot)}\\,,\\ {\\rm where }\\ \\ \\mu ^4 = \\frac{m_{Pl}^2 M^4}{8 \\pi q^2 T^2}\\,.$ The homogeneous part of this equation has exponential solutions $R \\sim \\exp (\\lambda t)$ with $\\lambda = | \\mu | \\exp \\left( i\\pi /4 + i \\pi n /2 \\right),$ where $n = 0,1,2,3$ .",
"There are two solutions with positive real parts of $\\lambda $ .",
"This indicates that the curvature scalar is exponentially unstable with respect to small perturbations, so $R$ should rise exponentially fast with time and quickly oscillate around this rising function.",
"Now we need to check if the characteristic rate of the perturbation explosion is indeed much larger than the rate of the universe expansion, that is: $(Re\\, \\lambda )^4 > H^4 = \\left( \\frac{ 8\\pi \\rho }{ 3 m^2_{Pl}}\\right)^2 = \\frac{16 \\pi ^6 g_*^2}{2025}\\,\\frac{ T^8}{m_{Pl}^4},$ where $\\rho = \\pi ^2 g_* T^4 /30$ is the energy density of the primeval plasma at temperature $T$ and $g_* \\sim 10 - 100$ is the number of relativistic degrees of freedom in the plasma.",
"This condition is fulfilled if $\\frac{2025}{2^9 \\pi ^7 q^2 g_*^2}\\frac{m_{Pl}^6 M^4}{T^{10}} > 1\\,,$ or, roughly speaking, if $T \\le m_{Pl}^{3/5} M^{2/5} $ .",
"Let us stress that at these temperatures the instability is quickly developed and the standard cosmology would be destroyed.",
"If we want to preserve the successful big bang nucleosynthesis (BBN) results and impose the condition that the development of the instability was longer than the Hubble time at the BBN epoch at $T \\sim 1 $ MeV, then $M$ should be extremely small, $M < 10^{-32}$ MeV.",
"The desire to keep the standard cosmology at smaller $T$ would demand even tinier $M$ .",
"A tiny $M$ leads to a huge strength of coupling (REF ).",
"It surely would lead to pronounced effects in stellar physics."
],
[
"Gravitational baryogenesis with fermions ",
"Let us now generalize results, obtained for scalar baryons, to realistic fermions.",
"We start from the action in the form $A= \\int d^4x \\sqrt{-g} \\left[\\frac{m_{Pl}^2}{16 \\pi } \\,R - {\\cal L}_{m}\\right]\\,$ with $ \\nonumber {\\cal L}_{m} &= &\\frac{i}{2} (\\bar{Q} \\gamma ^\\mu \\nabla _\\mu Q - \\nabla _\\mu \\bar{Q}\\, \\gamma ^\\mu Q) - m_Q\\bar{Q}\\,Q\\\\&+&\\frac{i}{2} (\\bar{L} \\gamma ^\\mu \\nabla _\\mu L - \\nabla _\\mu \\bar{L} \\gamma ^\\mu L)- m_L\\bar{L}\\,L \\\\ \\nonumber &+& \\frac{g}{m_X^2}\\left[(\\bar{Q}\\,Q^c)(\\bar{Q} L) + (\\bar{Q}^cQ)(\\bar{L} Q) \\right]+ \\frac{f}{m_0^2} (\\partial _{\\mu } R) J^{\\mu } + {\\cal L}_{other}\\,,$ where $Q$ is the quark (or quark-like) field with non-zero baryonic number, $L$ is another fermionic field (lepton), $\\nabla _\\mu $ is the covariant derivative of Dirac fermion in tetrad formalism.",
"$m_0$ is a constant parameter with dimension of mass and $f$ is dimensionless coupling constant which is introduced to allow for an arbitrary sign of the curvature dependent term in the above expression.",
"$J^{\\mu } = \\bar{Q} \\gamma ^{\\mu } Q$ is the quark current with $\\gamma ^{\\mu }$ being the curved space gamma-matrices, ${\\cal L}_{other}$ describes all other forms of matter.",
"The four-fermion interaction between quarks and leptons is introduced to ensure the necessary non-conservation of the baryon number with $m_X$ being a constant parameter with dimension of mass and $g$ being a dimensionless coupling constant.",
"In grand unified theories $m_X$ may be of the order of $10^{14}-10^{15}$ GeV.",
"Varying the action (REF ) over metric, $g^{\\mu \\nu }$ , and taking trace with respect to $\\mu $ and $\\nu $ , we obtain the following equation of motion for the curvature scalar: $ \\nonumber - \\frac{m_{Pl}^2}{8\\pi } R &=& m_{Q} \\bar{Q} Q + m_L \\bar{L} L+\\frac{2g}{m_X^2}\\left[(\\bar{Q}\\,Q^c)(\\bar{Q} L) + (\\bar{Q}^cQ)(\\bar{L} Q) \\right] \\\\&-&\\frac{2f}{m_0^2} (R + 3D^2) D_{\\alpha } J^{\\alpha } + T_{other}\\,,$ where $T_{other} $ is the trace of the energy momentum tensor of all other fields.",
"At relativistic stage, when masses are negligible, we can take $T_{matter} = 0$ .",
"The average expectation value of the interaction term proportional to $g$ is also small, so the contribution of all matter fields may be neglected.",
"As we see in what follows, kinetic equation leads to an explicit dependence on $R$ of the current divergence, $D_\\alpha J^\\alpha $ , if the current is not conserved.",
"As a result we obtain 4th order equation for $R$ .",
"As previously, we study solutions of Eq.",
"(REF ) in cosmology in homogeneous and isotropic FRW background with the metric $ds^2=dt^2 - a^2(t) d{\\bf r}^2 $ .",
"The curvature is a function only of time and the covariant derivative acting on a vector $V^\\alpha $ , which depends only on time and has only time component, has the form: $D_\\alpha V^\\alpha = (\\partial _t + 3 H) V^t ,$ where $H = \\dot{a}/a$ is the Hubble parameter.",
"As an example let us consider the reaction $q_1 + q_2 \\leftrightarrow \\bar{q}_3 + l_4$ , where $q_1$ and $q_2$ are quarks with momenta $q_1$ and $q_2$ , while $\\bar{q}_3$ and $l_4$ are antiquark and lepton with momenta $q_3$ and $l_4$ .",
"We use the same notations for the particle symbol and for the particle momentum.",
"The kinetic equation for the variation of the baryonic number density $n_B \\equiv J^t$ through this reaction in the FRW background has the form: $(\\partial _t +3 H) n_B = I_B^{coll},$ where the collision integral for space and time independent interaction is equal to: $&&I^{coll}_B =- 3 B_q (2\\pi )^4 \\int \\,d\\nu _{q_1,q_2} \\,d\\nu _{\\bar{q}_3, l_4}\\delta ^4 (q_1 +q_2 -q_3 - l_4)\\nonumber \\\\&& \\left[ |A( q_1+q_2\\rightarrow \\bar{q}_3 +l_4)|^2f_{q_1} f_{q_2} -|A( \\bar{q}_3 +l_4 \\rightarrow q_1+q_2 ) |^2f_{\\bar{q}_3} f_{l_4}\\right],$ where $ A( a \\rightarrow b)$ is the amplitude of the transition from state $a$ to state $b$ , $B_q$ is the baryonic number of quark, $f_a$ is the phase space distribution (the occupation number), and $d\\nu _{q_1,q_2} =\\frac{d^3 q_1}{2E_{q_1} (2\\pi )^3 }\\, \\frac{d^3 q_2}{2E_{q_2} (2\\pi )^3 } ,$ where $E_q = \\sqrt{ q^2 + m^2}$ is the energy of particle with three-momentum $q$ and mass $m$ .",
"The element of phase space of final particles, $d\\nu _{\\bar{q}_3, l_4} $ , is defined analogously.",
"We neglect the Fermi suppression factors and the effects of gravity in the collision integral.",
"This is generally a good approximation.",
"The calculations are strongly simplified if quarks and leptons are in equilibrium with respect to elastic scattering and annihilation.",
"In this case their distribution functions take the form $f = \\frac{1}{e^{(E/T - \\xi }) + 1} \\approx e^{-E/T + \\xi },$ with $\\xi = \\mu /T$ being dimensionless chemical potential, different for quarks, $\\xi _q$ , and leptons, $\\xi _l$ .",
"The assumption of kinetic equilibrium is well justified since it is usually enforced by very efficient elastic scattering.",
"Equilibrium with respect to annihilation, say, into two channels: $2\\gamma $ and $3\\gamma $ , implies the usual relation between chemical potentials of particles and antiparticles, $\\bar{\\mu }= -\\mu $ .",
"The baryonic number density is given by the expression: $ \\nonumber n_B &=& \\int \\frac{d^3 q}{2 E_q\\, (2\\pi )^3} (f_q - f_{\\bar{q}}) \\\\&=& \\frac{g_S B_q}{6} \\left(\\mu T^2 + \\frac{\\mu ^3}{ \\pi ^2}\\right) =\\frac{g_S B_q T^3}{6}\\,\\left(\\xi + \\frac{\\xi ^3}{\\pi ^2}\\right) \\,,$ where $T$ is the cosmological plasma temperature, $g_S$ and $B_q$ are respectively the number of the spin states and the baryonic number of quarks.",
"We can use another representation of the quark field: $Q_2 = \\exp ( i f R /m_0^2 )\\, Q$ analogously to what is done in our paper [8].",
"Written in terms of $Q_2$ Lagrangian (REF ) would not contain terms proportional to $f /m_0^2$ , but dependence on such terms would reappear in the interaction term as: $\\frac{2g}{m_X^2}\\left[ e^{-3ifR/m_0^2}\\, (\\bar{Q}_2\\,Q_2^c)(\\bar{Q}_2 L) +e^{3ifR/m_0^2}\\,(\\bar{Q}_2^cQ_2)(\\bar{L} Q_2) \\right] .$ Nevertheless we obtain the same fourth order equation for the evolution of curvature, as for non-rotated field $Q$ .",
"Since the transition amplitudes, which enter the collision integral, are obtained by integration over time of the Lagrangian operator (REF ), taken between the initial and final states, the energy conservation delta-function in Eq.",
"(REF ) would be modified due to time dependent factors $\\exp [\\pm 3 ifR(t)/m_0^2]$ .",
"In the simplest case, which is usually considered in gravitational (and spontaneous) baryogenesis, a slowly changing $\\dot{R}$ is taken, so we can approximate $R(t) \\approx \\dot{R}(t)\\,t$ .",
"In this case the energy is not conserved but the energy conservation condition is trivially modified, as $ \\nonumber &&\\delta [ E(q_1) + E(q_2) - E(q_3) - E(l_4) ] \\rightarrow \\\\&&\\rightarrow \\delta [ E(q_1) + E(q_2) - E(q_3) - E(l_4) - 3f\\dot{R}(t) /m_0^2\\,].$ Thus the energy is non-conserved due to the action of the external field $R(t)$ .",
"Delta-function (REF ) is not precise, but the result is pretty close to it, if $\\dot{R}(t)$ changes very little during the effective time of the relevant reactions.",
"If the dimensionless chemical potentials $\\xi _q$ and $\\xi _l$ , as well as $ f\\dot{R}(t) /m_0^2 /T $ , are small, the collision integral can be written as: $I^{coll}_B \\approx \\frac{C_I g^2 T^8}{m_X^4}\\,\\left[ \\frac{3 f\\dot{R}(t)}{ m_0^2\\,T} - 3\\xi _q + \\xi _l \\right] ,$ where $C_I$ is a positive dimensionless constant.",
"The factor $T^8$ appears for reactions with massless particles and the power eight is found from dimensional consideration.",
"Because of conservation of the sum of baryonic and leptonic numbers $\\xi _l = -\\xi _q/3 $ .",
"The case of an essential variation of $\\dot{R} (t)$ is analogous to fast variation of $\\dot{\\theta }(t)$ studied in our paper [8].",
"Clearly, it is much more complicated technically.",
"Here we consider only the simple situation with quasi-stationary background and postpone more realistic time dependence of $R(t)$ for the future work.",
"For small chemical potential the baryonic number density (REF ) is equal to $n_B \\approx \\frac{g_s B_q}{6}\\, \\xi _q T^3\\, ,$ and if the temperature adiabatically decreases in the course of the cosmological expansion, according to $\\dot{T} = - H T$ , equation (REF ) turns into $\\dot{\\xi }_q = \\Gamma \\left[ \\frac{9 f\\dot{R}(t)}{10m_0^2\\, T} - \\xi _q \\right],$ where $\\Gamma \\sim g^2 T^5/m_X^4 $ is the rate of B-nonconserving reactions.",
"If $\\Gamma $ is in a certain sense large, this equation can be solved in stationary point approximation as $\\xi _q=\\xi _q^{eq} - \\dot{\\xi }_q^{eq}/\\Gamma \\, , \\ \\ {\\rm where} \\ \\ \\xi _q^{eq} = \\frac{9}{10} \\frac{f \\dot{R}}{m_0^2 T}\\, .$ If we substitute $\\xi _q^{eq} $ into Eq.",
"(REF ) we arrive to the fourth order equation for $R$ .",
"According to the comment below Eq.",
"(REF ), the contribution of thermal matter into this equation can be neglected, and we arrive to the very simple fourth order differential equation: $\\frac{d^4 R}{dt^4} = \\lambda ^4 R,$ where $\\lambda ^4 = C_\\lambda m_{Pl}^2 m_0^4 /T^2 $ with $ C_\\lambda = 5/(36\\pi f^2 g_s B_q )$ .",
"Deriving this equation we neglected the Hubble parameter factor in comparison with time derivatives of $R$ .",
"It is justified a posteriori because the calculated $\\lambda $ is much larger than $ H $ .",
"Evidently equation (REF ) has extremely unstable solution with instability time by far shorter than the cosmological time.",
"This instability would lead to an explosive rise of $R$ , which may possibly be terminated by the nonlinear terms proportional to the product of $H$ to lower derivatives of $R$ .",
"Correspondingly one may expect stabilization when $HR \\sim \\dot{R}$ , i.e.",
"$H\\sim \\lambda $ .",
"Since $\\dot{H} + 2 H^2 = - R/6,$ $H$ would also exponentially rise together with $R$ , $ H \\sim \\exp (\\lambda t )$ and $\\lambda H \\sim R$ .",
"Thus stabilization may take place at $R \\sim \\lambda ^2 \\sim m_{Pl} m_0^2 / T$ .",
"This result should be compared with the normal General Relativity value $ R_{GR} \\sim T_{matter} /m_{Pl}^2 $ , where $T_{matter}$ is the trace of the energy-momentum tensor of matter."
],
[
"Discussion and conclusion ",
"For more accurate analysis numerical solution will be helpful, which we will perform in another work.",
"The problem is complicated because the assumption of slow variation of $\\dot{R}$ quickly becomes broken and the collision integral in time dependent background is not so simply tractable as the usual stationary one.",
"The technique for treating kinetic equation in non-stationary background is presented in Ref. [8].",
"For evaluation of $R(t)$ in this case numerical calculations are necessary, which will be presented elsewhere.",
"Here we describe only the basic features of the new effect of instability in gravitational baryogenesis.",
"To conclude we have shown that gravitational baryogenesis in the simplest versions discussed in the literature is not realistic because the instability of the emerging gravitational equations destroys the standard cosmology.",
"Some stabilization mechanism is strongly desirable.",
"Probably stabilization may be achieved in a version of $F(R)$ -theory.",
"Acknowledgement This work was supported by the RSF Grant N 16-12-10037.",
"The author expresses sincere gratitude to Harald Fritzsch for his invitation and for the opportunity to present the talk at the Conference on Particles and Cosmology.",
"She would like to thank Kok Khoo Phua for his kind hospitality at NTU, Singapore."
]
] | 1808.08577 |
[
[
"Densest vs. jammed packings of 2D bent-core trimers"
],
[
"Abstract We identify the maximally dense lattice packings of tangent-disk trimers with fixed bond angles ($\\theta = \\theta_0$) and contrast them to both their nonmaximally-dense-but-strictly-jammed lattice packings as well as the disordered jammed states they form for a range of compression protocols.",
"While only $\\theta_0 = 0,\\ 60^\\circ,\\ \\rm{and}\\ 120^\\circ$ trimers can form the triangular lattice, maximally-dense maximally-symmetric packings for all $\\theta_0$ fall into just two categories distinguished by their bond topologies: half-elongated-triangular for $0 < \\theta_0 < 60^\\circ$ and elongated-snub-square for $60^\\circ < \\theta_0 < 120^\\circ$.",
"The presence of degenerate, lower-symmetry versions of these densest packings combined with several families of less-dense-but-strictly-jammed lattice packings act in concert to promote jamming."
],
[
"Introduction",
"Jamming of anisotropic constituents has attracted great interest [1], [2], [3], [4], [5] for two reasons.",
"The first is that understanding how anistropy affects jamming is critical because most real granular materials are composed of anisotropic grains.",
"The second is that constituent-particle anistropy affects systems' jamming phenomenology and their thermal-solidification phenomenology in similar ways, and hence studying the jamming of grains of a given shape can provide insight into the thermal solidification of similarly shaped molecules and/or colloids [6], [7], [8], [9].",
"Such studies are maximally effective when they are complemented by identifying the particles' densest possible packings since the differences between densest and jammed packings are often analogous to the differences between crystals and glasses formed via thermal solidification [10], [11], [9].",
"Bent-core trimers are a simple model for multiple liquid-crystal-forming [12] and glass-forming [13], [14], [15], [16] molecules.",
"As illustrated in Figure REF , their shape can be characterized using three parameters: the bond angle $\\theta _0$ , the ratio $r$ of end-monomer radius to center-monomer radius, and the ratio $R$ of intermonomer bond length to center-monomer diameter.",
"For example, para-, meso-, and ortho-terphenyl correspond to the molecule shown in Fig.",
"REF (with $\\theta _0 = 0^\\circ ,\\ 60^\\circ ,\\ \\rm {and}\\ 120^\\circ $ , respectively), and the popular Lewis-Wahnstrom model [17] for OTP implements this molecular geometry with $R = 2^{-1/6},\\ r = 1,\\ \\theta _0 = 105^\\circ $ .",
"It is well known that the properties of systems composed of such molecules depend strongly on all three of these parameters; for example, the three terphenyl isomers form very differently structured bulk solids under the same preparation protocol [13], as do xylenes [14], diphenylcycloalkenes [15] and homologous series of cyclic stilbenes [16].",
"The structural isomers and near-isomers of more complicated small molecules also often exhibit very different solidification behavior, e.g.",
"the trisnapthylbenzenes which have attracted great interest in recent years because they have been shown to form quasi-ordered glasses when vapor-deposited [18], [19], [20], [21], [22].",
"Figure: Rigid bent-core disk-trimers with bond angle θ 0 \\theta _0.",
"Panel (a) shows the general geometry with unspecified (r,Rr,\\ R).",
"Here we study the r=R=1r = R = 1 case shown in panel (b).Our understanding of such phenomena and hence our ability to engineer crystallizability/glass-formability at the molecular level remains very limited.",
"One of the reasons why this is so is that only a few theoretical studies have isolated the role played by molecular shape using simple models.",
"Molecules like those studied in Refs.",
"[18], [19], [20], [21], [22] tend to form liquid-crystalline phases with columnar order [21].",
"Studying packing of 2D models for these molecules corresponds to studying the in-plane ordering of such anisotropic phases.",
"Optimal packing of molecules with the geometry shown in Figure REF has been investigated only minimally; Ref.",
"[23] reported the densest packings of 2D $R = 1/2$ trimers as a function of $r$ and $\\theta _0$ .",
"The tangent-disk ($r = R = 1$ ) case shown in Fig.",
"REF (b) is of considerable interest because it allows straightforward connection to results obtained for monomers – and hence isolation of the role played by the bond and angular constraints – while remaining a reasonable minimal model for terphenyl-shaped molecules.",
"In this paper, we identify and characterize the densest lattice packings of 2D bent-core tangent-disk trimers as a function of their bond angle $\\theta _0$ , and contrast them to both their nonmaximally-dense-but-strictly-jammed lattice packings and the disordered jammed packings they form under dynamic compression.",
"Figure: Structure of the putatively densest trimer packings.",
"Panel (a): lattice configuration for 0≤θ 0 ≤60 ∘ 0 \\le \\theta _0 \\le 60^\\circ .",
"Panel (b): lattice configuration for 60 ∘ ≤θ 0 ≤120 ∘ 60^\\circ \\le \\theta _0 \\le 120^\\circ .",
"Panel (c): Postulated maximal density φ max (θ 0 )\\phi _{max}(\\theta _0) [Eqs.",
", , ]."
],
[
"Densest Packings",
"The only $\\theta _0$ allowing formation of the triangular lattice [which is the densest possible 2D disk packing, with $\\phi = \\phi _{tri} = \\pi /(2\\sqrt{3}) \\simeq .9069$ ] are $0^\\circ ,\\ 60^\\circ ,\\ \\rm {and}\\ 120^\\circ $ .",
"In this section, we identify the densest packings for all $\\theta _0$ .",
"A potential geometry of the densest packings for $0 \\le \\theta _0 \\le 60^\\circ $ is shown in Fig.",
"REF (a).",
"Black circles indicate the monomer positions for a reference trimer centered at the origin.",
"For this range of $\\theta _0$ , another similarly oriented trimer can be centered at $\\vec{c}_2 = [1/2,\\sqrt{3}/2]$ , with its leftmost and rightmost monomers respectively at $\\vec{l}_2 = [1/2-\\cos (\\theta _0),\\sqrt{3}/2+\\sin (\\theta _0)]$ and $\\vec{r}_2 = [3/2,\\sqrt{3}/2]$ .",
"Then the center monomer of a third trimer with this orientation can be placed at $\\vec{c}_3 = [2+\\cos (\\theta ),-\\sin (\\theta )]$ .",
"A horizontally oriented unit cell with lattice vectors $\\vec{b}_1, \\vec{b}_2$ is obtained by rotating $\\vec{c}_2,\\ \\vec{c}_3$ through the angle $\\delta (\\theta _0) = \\tan ^{-1}(\\sin (\\theta _0)/[2+\\cos (\\theta _0)])$ .",
"The area of this unit cell is $A_1(\\theta _0) = \\det \\left( \\left[ \\begin{array}{c} \\vec{b}_1 \\\\ \\vec{b}_2 \\end{array} \\right] \\right) = \\det \\left( \\left[ \\begin{array}{c} \\vec{c}_2 \\\\ \\vec{c}_3 \\end{array} \\right] \\right),$ which yields the packing fraction of lattices with this geometry: $\\phi _1(\\theta _0) = \\displaystyle \\frac{3\\pi }{4A_1(\\theta _0)} = \\left( \\displaystyle \\frac{3}{2 + \\cos (\\theta _0) + \\sin (\\theta _0)/\\sqrt{3}} \\right) \\phi _{tri}.$ $\\phi _1(\\theta _0)$ is maximal [$\\phi _1 = \\phi _{tri}$ ] at $\\theta _0 = 0\\ \\rm {and}\\ 60^\\circ $ , and minimal [$\\phi _1 = 3\\pi /[4(1+\\sqrt{3})] \\simeq .862 \\simeq .951\\phi _{tri}$ ] at $\\theta _0 = 30^\\circ $ .",
"The factors of 3 in the numerators in Eq.",
"REF reflect the fact that there are three monomers per trimer.",
"For $\\theta _0 > 60^\\circ $ , it is impossible to center a second trimer at $[1/2,\\sqrt{3}/2]$ .",
"We postulate that the densest packings correspond to the double lattices described by Kuperberg, Torquato and Jiao [24], [25].",
"Double lattice packings consist of two lattices related by a displacement plus a $180^\\circ $ rotation of all constituents about their centers of inversion symmetry.",
"They are often the densest possible packings for both convex [24] and concave [25] particles.",
"Trimers are inversion-symmetric about the centroids of their center monomers.",
"Fig.",
"REF (b) shows our postulated lattice geometry.",
"The leftmost monomer of a second trimer may be placed at $\\vec{l}_2 = [1-\\cos (\\theta _0),\\sin (\\theta _0)]$ .",
"Then its center monomer lies at $\\vec{c}_2 = [2 - \\cos (\\theta _0), \\sin (\\theta _0)]$ and its rightmost monomer at $\\vec{r}_2 = [2,0]$ .",
"This trimer is related to the reference trimer by a $180^\\circ $ rotation about its inversion center plus displacement by $\\vec{c}_2$ .",
"The lattice vectors for this geometry are $\\vec{b}_1 = [1/2 - \\cos (\\theta _0), \\sqrt{3}/2 + \\sin (\\theta _0)]$ and $\\vec{b}_2 = [3,0]$ .",
"Its unit-cell area $A_2(\\theta _0)$ is $A_2(\\theta _0) = \\det \\left( \\left[ \\begin{array}{c} \\vec{b}_1 \\\\ \\vec{b}_2 \\end{array} \\right] \\right)$ and its packing fraction is $\\phi _2(\\theta _0) = \\displaystyle \\frac{6\\pi }{4A_2(\\theta _0)} = \\left( \\displaystyle \\frac{1}{1/2 + \\sin (\\theta _0)/\\sqrt{3}} \\right) \\phi _{tri}.$ $\\phi _2(\\theta _0)$ is maximal [$\\phi _2 = \\phi _{tri}$ ] at $\\theta _0 = 60^\\circ \\ \\rm {and}\\ 120^\\circ $ and minimal [$\\phi _2 = \\pi /(2+\\sqrt{3}) \\simeq .842 \\simeq .928\\phi _{tri}$ ] at $\\theta _0 = 90^\\circ $ .",
"The factor of 6 (rather than 3) in Eq.",
"REF reflects the fact that these packings have 2 trimers per lattice cell.",
"Our postulated maximal packing density for 2D bent-core trimers is $\\phi _{max}(\\theta _0) = \\bigg { \\lbrace } \\begin{array}{ccc}\\phi _1(\\theta _0) & , & 0 \\le \\theta _0 \\le 60^\\circ \\\\\\\\\\phi _2(\\theta _0) & , & 60^\\circ \\le \\theta _0 \\le 120^\\circ \\end{array}.$ The variation of $\\phi _{max}$ with $\\theta _0$ is illustrated in Fig.",
"REF (c).",
"As discussed above, minimal $\\phi _{max}(\\theta _0)$ occur at the $\\theta _0$ that are most distant from those commensurable with the triangular lattice, i.e.",
"$30^\\circ $ and $90^\\circ $ .",
"Kuperberg [24] identified a lower bound $\\phi _K = \\sqrt{3}/2$ for the maximal packing density of identical convex particles.",
"For $71.4^\\circ \\lesssim \\theta _0 \\lesssim 108.6^\\circ $ , $\\phi _{max} \\le \\phi _K$ , indicating trimers' concavity plays a critical role in decreasing $\\phi _{max}$ for (at least) this range of $\\theta _0$ .",
"The role of interlocking phenomena specific to concave particles [2], [25] in determining $\\phi _{max}(\\theta _0)$ and the jamming density $\\phi _J(\\theta _0)$ will be discussed further below.",
"Figure: Putatively densest packings for 2D bent-core trimers.",
"The top panels show the molecular geometries for θ 0 =0,15 ∘ ,30 ∘ ,45 ∘ ,60 ∘ ,75 ∘ ,90 ∘ ,105 ∘ , and 120 ∘ \\theta _0 = 0,\\ 15^\\circ ,\\ 30^\\circ ,\\ 45^\\circ ,\\ 60^\\circ ,\\ 75^\\circ ,\\ 90^\\circ ,\\ 105^\\circ ,\\ \\rm {and}\\ 120^\\circ .",
"The bottom panels show the bond/contact topologies for the same systems, with noncovalent contacts indicated by blue lines.",
"Black parallelograms show the unit cells, which are primitive cells for θ 0 <60 ∘ \\theta _0 < 60^\\circ and contain 2 trimers for θ 0 ≥60 ∘ \\theta _0 \\ge 60^\\circ .",
"Green lines show covalent bonds.Figure REF shows the lattice packings associated with these motifs for several representative values of $\\theta _0$ .",
"For $0 < \\theta _0 < 60^\\circ $ these consist of triple layers of triangular lattice separated by lines of “gap” defects.",
"The gaps are necessary to accommodate the incommensurability of the 3-body fixed-angle ($\\theta = \\theta _0$ ) constraints with the triangular lattice.",
"The size and shape of the gaps varies with $\\theta _0$ and determines $\\phi _{max}(\\theta _0)$ , but their overall orientation does not change.",
"For $60^\\circ < \\theta _0 < 120^\\circ $ the impossibility of centering a second trimer at $[1/2,\\sqrt{3}/2]$ (Fig.",
"REF ) makes forming triple layers of triangular lattice impossible; instead, the maximally dense packings are composed of double layers of triangular lattice separated by lines of gap defects.",
"This larger concentration of gaps is responsible for the lower $\\phi _{max}$ for $\\theta _0 > 60^\\circ $ .",
"For example, $[\\phi _{tri}-\\phi _{max}(90^\\circ )]/[\\phi _{tri}-\\phi _{max}(30^\\circ )] = 2(\\sqrt{3}-1) \\simeq 1.46$ is close to the value ($3/2$ ) that might be naively expected from the $\\theta _0 > 60^\\circ $ packings' larger gap concentration, and in fact $\\displaystyle \\frac{\\phi _{tri}-\\phi _{max}(90^\\circ )}{\\phi _{tri}-\\phi _{max}(30^\\circ )} \\cdot \\displaystyle \\frac{\\phi _{max}(30^\\circ )}{\\phi _{max}(90^\\circ )} = \\displaystyle \\frac{3}{2}.$ Further insight into the structure of these lattice packings can be gained by examining the topology of their bond/contact network.",
"As shown in the bottom panels of Fig.",
"REF , the bond/contact network is composed of triangles corresponding to monomers in close-packed layers and parallelograms corresponding to monomers bordering gaps.",
"For $\\theta _0 = 0^\\circ ,\\ 60^\\circ ,\\ \\rm {and}\\ 120^\\circ $ , the opposite corners corners of the parallelograms form additional contacts as the gaps close.",
"The average monomer coordination numbers are $Z_{mon} = \\bigg { \\lbrace } \\begin{array}{lcl}6 & , & \\theta _0 = 0,\\ 60^\\circ ,\\ \\rm {or}\\ 120^\\circ \\\\16/3 & , & 0 < \\theta _0 < 60^\\circ \\\\5 & , & 60^\\circ < \\theta _0 < 120^\\circ \\end{array},$ where $Z_{mon}$ includes both covalent bonds and noncovalent contacts.",
"The lower coordination and less efficient packing for $60^\\circ < \\theta _0 < 120^\\circ $ are both consistent with the idea that concavity plays a more important role in these systems [11], [25]; both trends arise from the inability of a reference trimer to form a bond-triangle on its concave side with a monomer belonging to a second trimer when $\\theta _0 > 60^\\circ $ , i.e.",
"they arise from the difference between the arrangements depicted in Fig.",
"REF (a-b).",
"One might expect that the parallelogramic bond/contact arrangements depicted in Fig.",
"REF are associated with soft shear modes (as they often are in monomeric systems).",
"In these maximally dense lattice packings, however, the fixed-angle constraints prevent the parallelograms from being sheared without violating hard-disc overlaps [26], [27].",
"Moreover, since monomers are highly overconstrained in these packings, trimers are necessarily also highly overconstrained.",
"We will show below that the dense lattice packings identified here are in fact all strictly jammed [27].",
"Figure: Degeneracy of the densest lattice packings.",
"The top panels illustrate one of the degeneracies for θ 0 =45 ∘ \\theta _0 = 45^\\circ : (a) shows the arrangement depicted in Fig.",
"(a), while (b) shows a degenerate flipped-and-shifted version of this arrangement with the same φ=φ max (θ 0 )\\phi = \\phi _{max}(\\theta _0).",
"The bottom panels illustrate one of the degeneracies for θ 0 =90 ∘ \\theta _0 = 90^\\circ : (c) shows the arrangement depicted in Fig.",
"(b), while (d) shows a degenerate φ=φ max (θ 0 )\\phi = \\phi _{max}(\\theta _0) arrangement with different symmetry and gap topology.A key factor influencing both jamming and glass-formation in systems of constituents that are able to crystallize is competition of degenerate crystalline structures.",
"For example, monodisperse sphere packings jam far more readily than monodisperse disk packings [28] because there are two incommensurate close-packed lattices in 3D (i.e.",
"FCC and BCC) but only a single close-packed lattice in 2D (i.e.",
"the triangular lattice).",
"Finding the geometries shown in Figs.",
"REF -REF did not address the question of degeneracy.",
"It turns out that these lattices are highly degenerate.",
"Figure REF illustrates how these degeneracies arise.",
"For $0 < \\theta _0 < 60^\\circ $ , rotating the trimers in alternating layers (here depicted in blue and red) by $180^\\circ $ about their inversion-symmetry centers and then shifting them by [$1-\\cos (\\theta _0), \\sin (\\theta _0)$ ] produces a lattice with the same $\\phi = \\phi _{max}(\\theta _0)$ .",
"For $60 < \\theta _0 < 120^\\circ $ , rotating the blue trimers so that their concave sides point away from rather than towards the centroids of the red trimers they double-contact achieves the same effect.",
"Figure: Panel (a): Schematic depiction of our adaptive shrinking cells for n tri =4n_{tri} = 4.",
"Periodic images of the n tri n_{tri} trimers are present in the ASC algorithm but are not shown here.",
"Panel (b): Comparison of analytic φ max (θ 0 )\\phi _{max}(\\theta _0) [Eqs.",
", , ; solid curve] to ASC results for n tri =2n_{tri} = 2 (red *) and n tri =4n_{tri} = 4 (green x) with X final ≥7X_{final} \\ge 7 (cf.",
"Sec.",
").",
"n tri =3n_{tri} = 3 is not considered here because lattices with bases containing an odd number of trimers cannot be double lattices and hence are generally less than maximally dense for θ 0 >60 ∘ \\theta _0 > 60^\\circ .",
"The data for n tri =2 and 4n_{tri} = 2\\ \\rm {and}\\ 4 overlap; for clarity, results are presented for alternating values of θ 0 \\theta _0.",
"Panel (c): One of the degenerate maximally dense n tri =4n_{tri} = 4 lattice packings for θ 0 =90 ∘ \\theta _0 = 90^\\circ contains alternating layers of the degenerate φ=φ max (90 ∘ )\\phi = \\phi _{max}(90^\\circ ) n tri =2n_{tri} = 2 lattices shown in Fig.",
"(c-d).The degenerate lattices depicted in Fig, REF (a-b) and (c-d) differ in their symmetries and gap topologies.",
"They represent competing ordered structures that are essentially trimeric – they are not present for monomers or dimers because their existence requires the fixed-angle ($\\theta = \\theta _0$ ) constraints.",
"Other degenerate arrangements with $\\phi = \\phi _{max}(\\theta _0)$ also exist.",
"We expect that there are in fact infinitely many of them, just as there are infinitely many variants of the close-packed lattice in 3D.",
"The arguments of Ref.",
"[28] suggest that all this degeneracy should strongly promote jamming / suppress crystallization in bent-core-trimeric systems relative to their monomeric and dimeric counterparts.",
"Below, we will investigate the degree to which this is true.",
"Jennings et.",
"al.",
"[23] found that for $R = 1/2$ trimers, double-lattice packings are not optimally dense for some $\\theta _0$ and $r$ .",
"In these special cases, the densest packings are lattices with bases containing more than two trimers.",
"To see whether this is true for our $r = R = 1$ systems, we identify maximally dense lattice packings for bases of various sizes using a variant of Torquato et.",
"al.",
"'s adaptive shrinking cell (ASC) algorithm [10], [11].",
"Figure REF (a) shows our ASC geometry for periodic cells containing $n_{tri}$ trimers.",
"The algorithm we use to obtain both optimally-dense and less-dense packings is described in detail below (in Section ).",
"Figure REF (b) compares our analytic prediction for $\\phi _{max}(\\theta _0)$ to the maximal-density lattice packings found from ASC runs for $n_{tri} \\le 4$ .",
"For both $n_{tri} = 2$ and $n_{tri} = 4$ , ASC results converge (within our numerical precision) to our putatively densest configurations or their degenerate counterparts.",
"These results indicate a key difference between the densest packings of the $r = R = 1$ trimers considered here and those of the $R = 1/2$ trimers studied in Ref.",
"[23] (wherein larger bases produce denser lattice packings for some $r$ and $\\theta _0$ .)",
"The simpler behavior for $r = R = 1$ appears to result from a reduction in the number of ways that small numbers of trimers can fit together when the trimers are composed of monodisperse tangent disks as opposed to bidisperse overlapping disks.",
"Another important difference associated with the abovementioned degeneracies appears for $n_{tri} > 2$ .",
"As shown in Fig.",
"REF (c), packings with $\\phi = \\phi _{max}(\\theta _0)$ may be formed by alternating layers of the degenerate structures identified above (Fig.",
"REF ).",
"Increasingly complicated arrangements of this type become possible as $n_{tri}$ increases.",
"This effect is analogous to the increasing number of distinguishable ways to stack $N_l$ layers of triangular lattice to form 3D close-packed structures as $N_l$ increases, and should further promote jamming."
],
[
"Statics: nonoptimally dense strictly jammed lattice packings",
"Having identified the maximally dense lattice packings, we now characterize the less-dense strictly-jammed lattice packings of these systems.",
"Our ASC algorithm [illustrated in Fig.",
"REF (a)] is implemented as follows.",
"Since translational invariance implies that trimer 0 can be centered at the origin ($x_0 = y_0 = 0$ ) without loss of generality, systems have $N_{dof} = 4 + 3(n_{tri}-1)$ degrees of freedom: the cell shape parameters $\\alpha , \\beta , \\gamma $ and the trimer-arrangement variables [$\\phi $ , and $x_i, y_i, \\phi _i$ for $i = 1, 2, ..., n_{tri}-1$ ].",
"Starting values of $\\alpha $ and $\\beta $ are chosen to be sufficiently large that all $n_{tri}$ trimers are able to freely rotate within the cell while $\\gamma = 0$ .",
"Strictly jammed packings are obtained through four types of moves: (1) random incremental changes of ($\\alpha , \\beta , \\gamma $ ) accompanied by affine displacements of the trimer centers [($x_i, y_i$ ) for $i = 1,\\ 2,\\ ...,\\ n_{tri}-1$ ]; (2) single-particle moves consisting of random incremental changes of $\\phi $ or of ($x_i, y_i, \\phi _i$ ) for some $i = 1,\\ 2,\\ ...,\\ n_{tri}-1$ ; (3) collective moves consisting of rigid translations or rotations of ($j < n_{tri}$ )-trimer subsets of trimers $1,\\ 2,\\ ...,\\ n_{tri}-1$ ; (4) changes of ($\\alpha , \\beta , \\gamma $ ) that preserve volume.",
"Type (1) moves are accepted if they reduce the cell volume $A = \\alpha \\beta \\cos (\\gamma )$ without producing any particle overlaps, while moves of types (2-4) are accepted if they produce no particle overlaps.",
"The initial maximal increment sizes for these moves are fixed at $\\lbrace |\\delta \\alpha | = .05\\cdot 2^{-X},\\ |\\delta \\beta | = .05\\cdot 2^{-X},\\ |\\delta \\phi | = 2.5^\\circ \\cdot 2^{-X},\\ |\\delta x_i |= .25\\cdot 2^{-X},\\ |\\delta y_i| = .25\\cdot 2^{-X},\\ |\\delta \\phi _i| = 2.5^\\circ \\cdot 2^{-X} \\rbrace $ with $X = 0$ .",
"After the process of compressing the system using moves of types (1-3) has converged [i.e.",
"no more moves of these types are being accepted], the system is collectively jammed [27] for the given value of $X$ .",
"Then moves of type (4) are used to check for strict jamming.",
"Successful type-4 moves indicate the system is not strictly jammed; when they occur, moves of types (1-3) are begun again.",
"This process repeats itself until the system is strictly jammed with respect to moves of types (1-4) for the given value of $X$ .",
"Then $X$ is increased by 1 and the process begins again.",
"This is repeated until satisfactory convergence is achieved; the data in Figure REF (b) indicate the maximally dense packings found by our algorithm that are strictly jammed for $X_{final} \\ge 7$ .",
"Figure: Analytic and ASC results for strictly jammed lattice packings.",
"Panel (a): Colored curves indicate the analytic φ i (θ 0 )\\phi _i(\\theta _0) (Table ) while black symbols indicate n tri =2n_{tri} = 2 ASC results.",
"Panel (b): ASC results for n tri =4n_{tri} = 4.As discussed in Refs.",
"[10], [11], improved performance of the algorithm can be obtained by adjusting the $X$ -increment and/or adopting more complicated Monte Carlo schemes such as allowing occasional acceptance of moves that increase $A$ .",
"To obtain a wide range of packings including both optimally and nonoptimally dense geometries, it is necessary to employ a wide range of trimer-arrangement initial conditions for each $\\theta _0$ .",
"This requirement combined with the fact that the computational complexity of the above-discussed type-3 moves is roughly exponential in $N_{dof}$ prohibited extending our comprehensive ASC studies to $n_{tri} > 4$ .",
"However, studies of selected $\\theta _0$ for $n_{tri} = 6$ found no packings with $\\phi > \\phi _{max}(\\theta _0)$ , and as we will show below, much insight can be combined by combining $n_{tri} \\le 4$ ASC studies with large-$n_{tri}$ molecular dynamics simulations.",
"Figure: θ 0 =θ min \\theta _0 = \\theta _{min} configurations and bond topologies of the seven nonoptimally-dense n tri =2n_{tri} = 2 lattice packings (families 3-9 in Table , Fig. )",
"that remain stable for n tri =4n_{tri} = 4.",
"The maximally-dense lattices (families 1-2) were illustrated in Figs.",
"-.",
"Note that it is the bond topologies that define the tiling types listed in Table .Table: Families of strictly jammed lattice packings for n tri =2n_{tri} = 2, in order of decreasing minimal density φ i (θ min )\\phi _i(\\theta _{min}).",
"Families 1 and 2 are respectively the maximally-symmetric maximally-dense lattice packings identified above [Fig.",
"(a-b); Eqs.",
", ] for 0≤θ 0 ≤60 ∘ 0 \\le \\theta _0 \\le 60^\\circ and 60 ∘ ≤θ 0 ≤120 ∘ 60^\\circ \\le \\theta _0 \\le 120^\\circ .",
"pp and qq are respectively equal to |30 ∘ -θ 0 ||30^\\circ - \\theta _0| and 2sin -1 [(22+2cos[θ 0 ]) -1 ]2\\sin ^{-1}[(2\\sqrt{2+2\\cos [\\theta _0]})^{-1}].",
"The “tiling types” describe the lattices' bond topology for all θ 0 \\theta _0 within the range, but in general only precisely describe the lattices' geometrical structure for specific θ 0 \\theta _0; for example, type 5's bond topology is always that of the square lattice but its geometry is only that of the square lattice for θ 0 =90 ∘ \\theta _0 = 90^\\circ .",
"All families except for 10(a-b) remain strictly jammed for n tri =4n_{tri} = 4; their n tri =4n_{tri} = 4 versions are simply two adjacent copies of the n tri =2n_{tri} = 2 lattices.As shown in Figure REF (a), at least ten distinct families of strictly jammed lattice packings exist for $n_{tri} = 2$ .",
"Each family represents a continuous set of lattice packings sharing a common bond topology (Figure REF ).",
"Since the families can be distinguished by their bond topologies, it is convenient to associate them with distinct categories of planar tilings [29].",
"Moreover, for each family $i$ , these ASC results allowed us to identify exact analytic expressions for the packing fraction $\\phi _i(\\theta _0)$ .",
"Results are summarized in Table REF .",
"All families except for 7b and 10b [30] share two common features: (i) their $\\phi _i(\\theta _0)$ are maximized at their “endpoint” $\\theta _0$ (e.g.",
"$\\theta _0 = 0\\ \\rm {and}\\ 60^\\circ $ for family 1) and minimized at their respective $\\theta _{min}$ ; (ii) they reduce to the triangular lattice at at least one of the three $\\theta _0$ allowing for it ($0,\\ 60^\\circ ,\\ \\rm {and}\\ 120^\\circ $ ).",
"Because the various $\\phi _i(\\theta _0)$ vary continuously, feature (ii) means that the various families' densities converge to each other as $\\theta _0$ approaches these special values.",
"Despite this convergence, the associated lattices remain distinct and incommensurable.",
"We expect that this incommensurability strongly promotes jamming in bulk systems.",
"Some of the nonmaximally dense families correspond to familiar 2D lattice structures, e.g.",
"family 5 is the square lattice for $\\theta _0 = 90^\\circ $ and family 10a (10b) is the kagome lattice for $\\theta _0 = 60^\\circ $ ($\\theta _0 = 0$ ).",
"Others represent less-familiar forms, such as family 8 which possesses a very intriguing bond topology consisting of five-sided polygons as well as the usual triangles and parallelograms.",
"The wide variety of mechanically stable arrangements with different symmetries and bond topologies suggests that these systems' equilibrium phase diagrams may be especially rich [31], potentially including entropically-driven solid-solid transitions, various liquid-crystalline phases (for example, $\\theta _0 = 90^\\circ \\ \\rm {and}\\ 120^\\circ $ systems might respectively form thermodynamically stable bent-core tetratic and hexatic liquid crystals), and KTHNY-style continuous melting transitions [32], [33].",
"Figure REF (b) presents our ASC results for $n_{tri} = 4$ .",
"The most obvious difference from the $n_{tri} = 2$ results is the elimination of the lowest-$\\phi $ strictly-jammed packings; this occurs because the larger basis allows for shear modes that destabilize the kagome-like lattices (family 10).",
"A second obvious difference is that there are many additional families.",
"Visual inspection indicates that a large fraction of these are formed by combining two of the families discussed above.",
"There are very many such combinations, making exhaustive cataloguing of them (as we did for $n_{tri} = 2$ ) prohibitively difficult, and continuing our ASC studies to even larger $n_{tri}$ would of course exacerbate this issue.",
"Instead we will test the extent to which the ideas presented here are useful by looking for local structural motifs within large-$n_{tri}$ jammed configurations that correspond to the nonoptimally-dense families."
],
[
"Dynamics: disordered and partially-ordered jammed packings",
"None of the above discussion addresses the dynamics of the jamming process.",
"Since the dynamics of systems' jamming transitions naturally relate to the dynamics of their glass transitions [34], we now examine the compression-rate dependence of our model trimers' athermal solidification behavior using molecular dynamics (MD) simulations.",
"Each of the $n_{tri}$ simulated trimers contains three monomers of mass $m$ .",
"The trimers are rigid; bond lengths and angles are held fixed by holonomic constraints.",
"Monomers on different trimers interact via a harmonic potential $U_{H}(r) = 10\\epsilon (1 - r/\\sigma )^2 \\Theta (\\sigma -r)$ , where $\\epsilon $ is the energy scale of the pair interactions, $\\sigma $ is monomer diameter, and $\\Theta $ is the Heaviside step function.",
"Initial states are generated by placing the trimers randomly within a square cell, with periodic boundary conditions applied along both directions.",
"Then Newton's equations of motion are integrated with a timestep $\\delta t = .005\\tau $ , where the unit of time is $\\tau =\\sqrt{m\\sigma ^2/\\epsilon }$ .",
"Systems are equlibrated at $k_BT/\\epsilon = 1$ and $\\phi = \\exp (-1)\\phi _{tri}$ until intertrimer structure has converged, then cooled to $T=0$ at a rate $10^{-4}(\\epsilon /k_B)/\\tau $ .",
"After cooling, systems are hydrostatically compressed at a true strain rate $\\dot{\\epsilon }$ , i.e.",
"the cell side length $L$ is varied as $L = L_0 \\exp (-\\dot{\\epsilon }t)$ .",
"To maintain near-zero temperature during compression, we employ overdamped dynamics with the equation of motion $m\\ddot{\\vec{r_i}} = \\vec{F} - \\Gamma \\dot{\\vec{r_i}} +h(\\lbrace \\vec{r} , \\dot{\\vec{}}{r} \\rbrace )$ where $\\vec{r}_i$ is the position of monomer $i$ , $\\vec{F}$ is the force arising from the harmonic pair interactions, the damping coefficient $\\Gamma = 10^4\\dot{\\epsilon }$ , and the $h(\\lbrace \\vec{r} , \\dot{\\vec{}}{r} \\rbrace )$ term enforces trimer rigidity [35].",
"Jamming is defined to occur when the nonkinetic part of the pressure $P$ exceeds $P_{thres}= 10^{-4}\\epsilon /\\sigma ^2$ ; choosing a lower (higher) value of $P_{thres}$ lowers (raises) $\\phi _J(\\theta _0)$ , but does not qualitatively change any of the results presented herein.",
"We choose to identify jamming with the emergence of a finite bulk modulus rather than with the vanishing of soft modes because proper handling of soft modes associated with trimeric “rattlers” [36] is highly nontrivial.",
"All MD simulations are performed using LAMMPS [37].",
"Figure: Dynamical jamming.",
"Panels (a) and (b) show φ J (θ 0 )\\phi _J(\\theta _0) and the ratio φ J (θ 0 )/φ max (θ 0 )\\phi _J(\\theta _0)/\\phi _{max}(\\theta _0) for several strain rates, and panel (c) shows shows φ J (θ 0 ;ϵ ˙)\\phi _J(\\theta _0; \\dot{\\epsilon }) for several characteristic values of θ 0 \\theta _0.",
"The dotted gray line in panel (a) shows φ max (θ 0 )\\phi _{max}(\\theta _0).",
"Dotted lines in panel (c) indicate fits to Eq.",
".",
"All results are averaged over 9 independently prepared n tri =400n_{tri} = 400 systems.Figure REF shows results for the rate-dependent $\\phi _J(\\theta _0)$ .",
"As shown in panel (a), all systems jam at densities well below the monomeric value $\\phi _J^{mon} \\simeq .84$ [36]; the reduced $\\phi _J$ relative to monomers are caused by the frozen-in 2-body and 3-body constraints [8].",
"For all compression rates, values of $\\phi _J(\\theta _0)$ clearly follow trends in $\\phi _{max}(\\theta _0)$ , exhibiting maxima at $\\theta _0 = 0,\\ 60^\\circ \\ \\rm {and}\\ 120^\\circ $ and minima at $\\theta _0 \\simeq 30^\\circ \\ \\rm {and}\\ \\simeq 90^\\circ $ .",
"There are clearly two separate branches of $\\phi _J(\\theta _0)$ : one for $0 \\le \\theta _0 \\le 60^\\circ $ and one for $60^\\circ \\le \\theta _0 \\le 120^\\circ $ .",
"However, values of $\\phi _J(\\theta _0)$ do not simply track $\\phi _{max}(\\theta _0)$ .",
"Specifically, closed ($\\theta _0 = 120^\\circ $ ) trimers have a much higher $\\phi _J$ than their open ($\\theta _0 = 0\\ \\rm {or}\\ 60^\\circ $ ) counterparts even though their $\\phi _{max}$ are identical.",
"More generally, while the first branch of $\\phi _J(\\theta _0)$ is close to symmetric about $\\theta _0 = 30^\\circ $ , the second branch is clearly asymmetric; $\\phi _J(90^\\circ + \\psi ) > \\phi _J(90^\\circ - \\psi )$ , increasingly so as $\\psi $ increases from zero towards $30^\\circ $ .",
"While it is not surprising that $\\theta _0 = 120^\\circ $ trimers are the best crystal-formers (cf.",
"Fig.",
"REF ) – they are compact and threefold-symmetric whereas lower $\\phi _J$ are expected for small-$\\theta _0$ systems owing to their larger aspect ratios [1], [8], our results show that this effect propagates downward in $\\theta _0$ as far as $\\theta _0 \\simeq 90^\\circ $ .",
"A useful metric characterizing the strength of the physical processes promoting disorder in these systems' solid-state morphologies is $f(\\theta _0; \\dot{\\epsilon }) = \\phi _J(\\theta _0; \\dot{\\epsilon })/\\phi _{max}(\\theta _0)$ .",
"This quantity is unity when systems crystallize into their maximal-density lattices during compression, and smaller when systems jam at $\\phi < \\phi _{max}(\\theta _0)$ due to the presence of disorder.",
"Roughly speaking, characterizing the decrease of $f(\\theta _0; \\dot{\\epsilon })$ with increasing $\\dot{\\epsilon }$ provides insight into the kinetics of the solidification process, while characterizing its variation with $\\theta _0$ in the low-$\\dot{\\epsilon }$ limit provides insight into how the strength of frustration/degeneracy-related effects varies with molecular shape.",
"Panel (b) presents results for $f(\\theta _0; \\dot{\\epsilon })$ for all systems.",
"For all strain rates considered here, trends in $f(\\theta _0; \\dot{\\epsilon })$ are opposite those in $\\phi _J(\\theta _0; \\dot{\\epsilon })$ and $\\phi _{max}(\\theta _0)$ .",
"Minima in the former correspond to maxima in the latter, e.g.",
"maxima in $f$ occur at $\\theta _0 = 30^\\circ \\ \\rm {and}\\ 90^\\circ $ .",
"One potential reason for this is that grains of crystals with incompatible local ordering corresponding to the different families discussed above are more likely to form at densities slightly below $\\phi _J(\\theta _0)$ and then jam as systems are further compressed for the systems with lower $f(\\theta _0; \\dot{\\epsilon })$ .",
"The fact that differences between several of the $\\phi _i(\\theta _0)$ are maximal for $\\theta _0 = 30^\\circ \\ \\rm {and}\\ 90^\\circ $ and minimal for $\\theta _0 = 0,\\ 60^\\circ ,\\ \\rm {and}\\ 120^\\circ $ supports this hypothesis; competition between differently ordered grains should be greater when their densities are closer due to the abovementioned convergence of the $\\phi _i(\\theta _0)$ .",
"Another potential reason is that jamming dynamics are controlled by the trimer mobility $\\mu $ and that $\\mu (\\phi ,\\theta _0)$ depends far more strongly on $\\phi $ than on $\\theta _0$ for $\\phi < \\phi _J$ .",
"It would be interesting to test this idea in thermalized versions of these systems by constructing isomobility curves in ($\\theta _0,\\phi $ )-space.",
"Panel (c) illustrates the compression-rate dependence of jamming in more detail for the $\\theta _0$ corresponding to extrema of $\\phi _J$ : $0,\\ 30^\\circ ,\\ 60^\\circ ,\\ 90^\\circ ,\\ \\rm {and}\\ 120^\\circ $ .",
"For all systems, results are well fit by $\\phi _J(\\theta _0; \\dot{\\epsilon }) \\simeq \\phi _{J}^0(\\theta _0)\\exp \\left[ -\\left( \\displaystyle \\frac{\\dot{\\epsilon }}{\\dot{\\epsilon }_0} \\right)^\\gamma \\right],$ where $\\phi _J^0$ is the quasistatic-limiting value of $\\phi _J$ , $\\dot{\\epsilon }_0$ is a characteristic rate, and $\\gamma $ describes the strength of the rate dependence.",
"While rigorously determining the exact functional form of $\\phi _J(\\theta _0; \\dot{\\epsilon })$ would require more computational resources than we currently possess and is beyond our present scope, the fact that the best-fit values of $\\dot{\\epsilon }_0$ and $\\gamma $ remain within relatively narrow ranges for all $0 \\le \\theta _0 \\le 120^\\circ $ – respectively $3\\cdot 10^{-4} \\lesssim \\dot{\\epsilon }_0\\tau \\lesssim 7\\cdot 10^{-4}$ and $.65 \\lesssim \\gamma \\lesssim .85$ – suggests that Eq.",
"REF is a useful approximate form for all $\\theta _0$ .",
"Similar compression-rate dependencies of $\\phi _J$ are found for dynamical jamming of a wide variety of systems [38].",
"Figure: Jammed packings of n tri =400n_{tri} = 400 bent-core trimers generated by dynamic compression.",
"Rows from top to bottom show typical marginally-jammed states for θ 0 =0,30 ∘ ,60 ∘ ,90 ∘ , and 120 ∘ \\theta _0 = 0,\\ 30^\\circ ,\\ 60^\\circ ,\\ 90^\\circ ,\\ \\rm {and}\\ 120^\\circ .",
"The left and right columns respectively indicate results for ϵ ˙τ=10 -5.5 and 10 -7 \\dot{\\epsilon }\\tau = 10^{-5.5}\\ \\rm {and}\\ 10^{-7}.",
"Colors from red to blue indicate trimers 1,2,...,4001,\\ 2,\\ ...,\\ 400.Finally we turn to a qualitative characterization of how $\\theta _0$ and $\\dot{\\epsilon }$ affect jammed systems' microstructure.",
"Typical marginally-jammed packings for the five characteristic $\\theta _0$ discussed above are shown in Figure REF .",
"For $\\theta _0 = 0,\\ 60^\\circ ,\\ \\rm {and}\\ 120^\\circ $ , triangular-crystalline grains are clearly visible.",
"This is consistent with the well-known result [28] that 2D systems of monodisperse disks have a strong propensity to crystallize.",
"Contrasting the high-$\\dot{\\epsilon }$ and low-$\\dot{\\epsilon }$ snapshots for these $\\theta _0$ suggests that systems jam via a two-stage, two-length-scale process.",
"First, randomly-oriented crystalline grains form and grow to a size that depends on both $\\theta _0$ and $\\dot{\\epsilon }$ .",
"Since these grains cannot be compressed further, they effectively behave as single nearly-rigid particles as compression continues.",
"The degree to which grains grow prior to jamming is kinetically limited and is a key factor producing the rate dependence of $\\phi _J(\\theta _0)$ .",
"Higher compression rates lead to smaller grain sizes and greater intergrain misorientation, just as is the case in monomeric systems [28], [38].",
"The results for $\\theta _0 = 30^\\circ \\ \\rm {and}\\ 90^\\circ $ are less easy to interpret since only basic features such as the decrease in the typical size of interstitials with decreasing $\\dot{\\epsilon }$ [28] are immediately apparent.",
"The $\\theta _0 = 90^\\circ $ systems clearly possess some grains with square-lattice ordering, showing that the nonoptimally-dense families identified above (Table REF , Figs.",
"REF -REF ) do indeed play a role in these systems' jamming phenomenology.",
"However, the clear crystallization kinetics observed for $\\theta _0 = 0,\\ 60^\\circ ,\\ \\rm {and}\\ 120^\\circ $ are absent here, perhaps because these systems' lower absolute $\\phi _J$ and $\\phi _{max}$ values make them appear more disordered overall.",
"The effects of degeneracy may also be larger at these $\\theta _0$ .",
"These issues might be resolved by going to much lower $\\dot{\\epsilon }$ , but doing so is not yet computationally feasible."
],
[
"Discussion and Conclusions",
"In this paper, we examined the athermal solidification behavior of 2D bent-core tangent-disk trimers.",
"We found that trends in $\\phi _J(\\theta _0)$ closely follow those in $\\phi _{max}(\\theta _0)$ , but with additional effects related to symmetry and degeneracy superposed.",
"We reported two distinct regimes of packing/jamming phenomenology, identifying the source of the difference between them as the ability (inability) of a reference trimer to form a bond-triangle on its concave side with a monomer belonging to a second trimer when $\\theta _0 < 60^\\circ $ ($\\theta _0 > 60^\\circ $ ).",
"Well-packed systems with $\\theta _0 > 60^\\circ $ are generally less dense and less hyperstatic than their $\\theta _0 < 60^\\circ $ counterparts.",
"Another key insight was that deviations of $\\theta _0$ away from the values allowing formation of the triangular lattice ($0^\\circ ,\\ 60^\\circ ,\\ \\rm {and}\\ 120^\\circ $ ) do not by themselves frustrate crystalline order.",
"Instead, crystals belonging to several families distinguished by their differing bond topologies can form.",
"We believe that it is the presence of these competing families combined with the extensive degeneracy of the densest lattices that frustrates crystallization and promotes jamming in these systems, Our work complements recent studies of the thermal solidification of Lewis-Wahnstrom-like models [39], [40], which illustrated several nontrivial effects of trimeric structure (e.g.",
"that its enhancement of the interfacial energy between crystalline and liquid phases promotes glass-formation) but did not attempt to connect their findings to the models' optimal-packing or jamming-related phenomenology.",
"One of the principal goals of soft materials science is designing materials that possess tunable solid morphology.",
"Designing custom pair interactions that yield nontriangular 2D-crystalline or non-close-packed 3D-crystalline ground states has attracted significant interest in recent years [41], [42], [43].",
"Our 2D bent-core tangent-disk trimers provide a simple example of how the same goal may be achieved with hard-disk or hard-sphere pair interactions by controlling 2-body and 3-body correlations, i.e.",
"by imposing covalent bonding and controlling the bond angle $\\theta _0$ .",
"The potential relevance to real systems is that controlling the bond angle (or analogous shape parameters) of small molecules [13], [14], [16], [15], [19], [20], [21], [22] is often easier than controlling the pair interactions of their constituent atoms.",
"Another key tuning parameter for soft materials is their degree of (dis)order.",
"Recent work [16], [15] has shown that the best glass-formers (crystal-formers) in homologous series of small molecules are those with the highest (lowest) ratio $T_g/T_m$ .",
"One might naively guess that an athermal version of this principle applicable to bent-core trimer molecules is that the best glass-formers (crystal-formers) are those with the lowest (highest) ratio $\\phi _J(\\theta _0)/\\phi _{max}(\\theta _0)$ .",
"However, the dynamic-compression results we presented here suggest that this is not the case.",
"Moreover, the multiplicity of nonoptimally-dense-but-strictly-jammed lattice packings these systems can form suggests that they may possess multiple thermodynamically stable crystalline phases and/or nontrivial liquid-crystalline phases [12].",
"We conclude that further characterizing these systems' equilibrium phase behavior, e.g.",
"by accurately determining their $\\phi _{melt}(\\theta _0)$ using techniques like those of Refs.",
"[32], [33], is necessary to flesh out this issue further.",
"This material is based upon work supported by the National Science Foundation under Grant DMR-1555242."
]
] | 1808.08488 |
[
[
"How many labeled license plates are needed?"
],
[
"Abstract Training a good deep learning model often requires a lot of annotated data.",
"As a large amount of labeled data is typically difficult to collect and even more difficult to annotate, data augmentation and data generation are widely used in the process of training deep neural networks.",
"However, there is no clear common understanding on how much labeled data is needed to get satisfactory performance.",
"In this paper, we try to address such a question using vehicle license plate character recognition as an example application.",
"We apply computer graphic scripts and Generative Adversarial Networks to generate and augment a large number of annotated, synthesized license plate images with realistic colors, fonts, and character composition from a small number of real, manually labeled license plate images.",
"Generated and augmented data are mixed and used as training data for the license plate recognition network modified from DenseNet.",
"The experimental results show that the model trained from the generated mixed training data has good generalization ability, and the proposed approach achieves a new state-of-the-art accuracy on Dataset-1 and AOLP, even with a very limited number of original real license plates.",
"In addition, the accuracy improvement caused by data generation becomes more significant when the number of labeled images is reduced.",
"Data augmentation also plays a more significant role when the number of labeled images is increased."
],
[
"Introduction",
"License plate recognition is one of the most important components of modern intelligent transportation systems.",
"It has attracted the attention of many researchers.",
"However, most existing algorithms[3], [13], [14], [18] can only work normally under certain conditions.",
"For example, some recognition systems require sophisticated hardware to shoot high-quality images, while other systems require the vehicle to slowly pass through a fixed access opening or even stop.",
"Accurately detecting license plates and recognizing characters in an open environment is a challenging task.",
"The main difficulties are different license plate fonts and colors, character distortion caused by the image capture process and non-uniform illumination, and low-quality images caused by occlusion or motion blur.",
"In this paper, we propose a license plate recognition system, in which we cope with challenge such as, low light, low resolution, motion blur, and other harsh conditions.",
"Fig.",
"REF shows the license plates which can be correctly recognized by our proposed method.",
"From top to bottom are the license plate images affected by the shooting angle, uneven illumination, low resolution, detection error and motion blur.",
"Figure: The complex license plates images.In general, supervised learning requires a large amount of labeled data in order to achieve good results.",
"However, real data is not easy to obtain, the acquisition process is slow, and the data needs to be processed and annotated before it can be used for training.",
"To achieve a higher accuracy of the annotation, manual inspection is also required.",
"However, the acquisition of a large amount of real data and manual annotations is very expensive.",
"Therefore, data generation is very important for the training of license plate recognition network.",
"We believe that the information contained in a small number of real license plates is sufficient to recognize most of the existing license plate images.",
"However, there is no clear common understanding on how much labeled data is needed to get satisfactory performance.",
"In this paper, we try to address such a question in vehicle license plate character recognition.",
"The main contributions of this paper can be summarized as the following three points: 1.We propose various methods of data generation and data augmentation.",
"As long as we have a few labeled license plate images, a large amount of generated data can be created.",
"We can achieve and even exceed the recognition accuracy and results of systems trained only on real images.",
"2.We compare the performance of various data generation and data augmentation methods to find that both data generation methods and data augmentation methods can significantly improve license plate recognition accuracy.",
"Data augmentation plays a larger role in accuracy improvement when there are many labeled license plates but when the number of labelled license plates is small, data generation more significantly increases accuracy.",
"3.We apply a network that is modified from DenseNet to license plate recognition to reduce network parameters and inference time and improve accuracy.",
"The rest of paper is arranged as follows.",
"In Section 2 we review the related works briefly.",
"In Section 3 we describe the details of networks used in our approach.",
"Experimental results are provided in Section 4, and conclusions are drawn in Section 5."
],
[
"Related Work",
"The section introduces previous work on license plate recognition and GANs."
],
[
"License Plate Recognition",
"Existing license plate recognition systems are either text segmentation-based [3], [6], or non-segmentation-based [13].",
"Methods that depend on segmentation first preprocess the license plate image and then segment individual characters through image processing.",
"After this, each character is classified by a convolutional neural network.",
"This method is very dependent on the accuracy of text segmentation, and the recognition speed is slower.",
"A recognition method that does not require segmentation is proposed by Li et al.",
"[13].",
"It is composed of a deep convolutional network and a Long Short-Term Memory(LSTM), where the deep CNN is directly applied for feature extraction, and a bidirectional LSTM network is applied for sequence labeling.",
"DenseNet[9] is a highly efficient convolutional neural network.",
"Because of its low parameter number and fast inference time, DenseNet is widely used.",
"Our method is also a segmentation-free approach based on the framework proposed by [9], where DenseNet is applied for feature extraction.",
"Data generation is used in license plate recognition to improve the accuracy of recognition.",
"The labeled license plates generated by CycleGAN as a pre-training data set for the recognition network are used in [18], and the model is fine-tuned with the real license plate data set.",
"This data generation method can significantly improve the recognition accuracy.",
"License plate detection and recognition is combined in [14], and it finally improves the recognition speed and recognition accuracy of the system."
],
[
"Generative Adversarial Networks",
"Generative adversarial networks(GANs) [2], [15] train a generator and discriminator alternatively.",
"The output of the discriminator acts as a generator's loss function.",
"Zhu et al.",
"[21] propose Cycle-Consistent Adversarial Networks(CycleGAN), which learns the mapping relationship from one domain to another and is mainly used for the style conversion of pictures.",
"Wasserstein GANs(WGAN) [1] are proposed to improve the stability of GAN training.",
"Applying Wasserstein loss to CycleGAN, creating CycleWGAN, also improves its training stability in [18].",
"Gradient penalties in WGAN(WGAN-GP) [5] are proposed to solve the WGAN generator weight distribution problem."
],
[
"Data Generation For Training",
"A large number of real labeled images are often difficult to obtain, so the role of data generation is very significant [17].",
"The synthesized images are used to train scene text detection networks [7] and recognition networks [11].",
"The generated data is shown to improve the performance of person detection [20], font recognition [19], and semantic segmentation [16].",
"However, when the difference between the generated data and the real data is very large, the performance is poor when applied to a real scene.",
"Therefore, [18] applies CycleGAN to convert the style of license plate generated by the script into a real license plate, which can greatly reduce the gap between the generated image and the real image.",
"We apply data generation and data augmentation methods at the same time, and use the data generated by different methods directly as the training set for recognition network.",
"Therefore we need very little real data.",
"In this section, the pipeline of the proposed method is described.",
"We train the GAN model using synthetic images and real images simultaneously.",
"We then use the generated images to train a model modified from DenseNet."
],
[
"CycleGAN",
"CycleGAN [21] learns to translate an image from a source domain X to a target domain Y in the absence of paired examples.",
"Our goal is to train a mapping relationship G between the script license plate domain X and the real license plate domain Y. CycleGAN contains two mapping functions $G:X\\rightarrow Y$ and $Y\\rightarrow X$ , and associated adversarial discriminators D$_Y$ , D$_X$ .",
"The techniques proposed in WGAN [1] are applied in CycleGAN, and CycleWGAN is proposed in [18].",
"WGAN points out why the traditional GAN is difficult to converge and improve during training, which greatly reduces the training difficulty and speeds up the convergence.",
"There are two main improvements: the first one is to remove the log from the loss function, and the second is to perform weight clipping after each iteration to update the weight, and limit the weight to a range (eg, the limit range is [-0.1, +0.1].",
"Outside weights are trimmed to -0.1 or +0.1).",
"CycleWGAN solves the problem of training instability and collapse mode, which makes the result more diverse.",
"We apply the techniques in WGAN-GP [5] to CycleWGAN and propose the CycleWGAN-GP.",
"WGAN-GP also proposes an improvement plan based on WGAN.",
"WGAN reduces the training difficulty of GAN, but it is still difficult to converge in some conditions, and the generated pictures are worse than DCGAN.",
"WGAN-GP applies gradient penalty, and solves the above problem along with the problems of vanishing gradient and exploding gradient during training.",
"It also converges faster than CycleWGAN and produces higher quality pictures.",
"We apply a CycleGAN equipped with WGAN and WGAN-GP techniques to train the mapping relationship between the fake license plate and the real license plate.",
"First of all, we apply OpenCV scripts to generate synthetic license plates as a source domain X, and then choose real license plates without labels as a target domain Y.",
"Before the training of CycleWGAN-GP, these license plates are randomly cropped and randomly flipped horizontally or vertically."
],
[
"Recognition network design",
"DenseNet is a densely connected convolutional neural network.",
"In this network, there is a direct connection between any two layers.",
"The input of each layer of the network is the union of the output of all previous layers, and the feature map learned by this layer is also directly transmitted to all subsequent layers.",
"DenseNet allows the input of lth Layer to directly affect all subsequent layers.",
"Its output is: $x_l=H_l([x_0,x_1,...,x_{l-1}]) $ where $H_l(\\cdot )$ refers to a composite function of three consecutive operations: batch normalization (BN) [10], followed by a rectified linear unit (ReLU), and a $3\\times 3$ convolution (Conv).",
"Additionally, since each layer contains the output information of all previous layers, it only needs a few feature maps, so the number of parameter of DenseNet is greatly reduced compared to other models.",
"Table: Construction of recognition network.",
"The output size represents w×\\times h×\\times c. Note that each ¡°conv¡± layer shown in the table corresponds the sequence BN-ReLU-ConvOur network structure is shown in Table REF , which is different from the network structure of [9], because the input license plate image is smaller and is a gray scale image of 136$\\times $ 36, so the network only has 3 dense blocks.",
"The transition layers used in our network consist of a batch normalization layer and an 1$\\times $ 1 convolutional layer followed by a 2$\\times $ 2 average pooling layer.",
"A 1$\\times $ 1 convolution can be introduced as bottleneck layer before each 3$\\times $ 3 convolution to reduce the number of input feature-maps.",
"To improve model compactness, we reduce the number of feature-maps from 192 to 128 at transition layers 2.",
"The last DenseNet layer is followed by a fully-connected layer with 68 neurons for the 68 classes of label, including 31 Chinese characters, 26 letters, 10 digits and ¡°blank¡±.",
"We train the networks with stochastic gradient descent (SGD).",
"The labelling loss is derived using Connectionist Temporal Classification (CTC) [4].",
"The optimization algorithm Adam [12] is then applied, as it converges quickly and does not require a complicated learning rate schedule.",
"Another advantage of using the modified DenseNet network is that it does not require the Long Short-Term Memory(LSTM) networks.",
"The use of LSTM complicates the solution and increases computational cost."
],
[
"Experiment",
"In this section, we conduct experiments to verify the effectiveness of the proposed methods.",
"Our network is implemented capitalizing keras.",
"The experiments are trained on a NVIDIA Tesla P40 with 24GB memory and are tested on a NVIDIA GTX745 GPU with 4GB memory."
],
[
"Dataset",
"The image in the Dataset-1 [18] are captured from a wide variety of real traffic monitoring scenes under various viewpoints, blurring and illumination.",
"Dataset-1 contains a training set of 203,774 plates and a test set of 9,986 plates.",
"The first character of Chinese license plates is a Chinese character which represents the province.",
"While there are 31 abbreviations for all of the provinces, Dataset-1 contains 30 classes of them.",
"The second data set is the application-oriented license plate (AOLP) [8] benchmark database, which has 2049 images of Taiwan license plates.",
"This database is categorized into three subsets: access control (AC) with 681 samples, traffic law enforcement (LE) with 757 samples, and road patrol (RP) with 611 samples.",
"The recognition network is shown in Table REF .",
"We implement it with Keras.",
"The images are resized to 136$\\times $ 36 and converted to gray scale and then fed to the recognition network.",
"We change the last layer of fully connected layers to 68 neurons according to the 68 classes of characters-33 Chinese characters, 24 letters, 10 digits and \"blank\".",
"We train the networks with SGD and learning rate of 0.0001.",
"The labelling loss is derived using CTC.",
"We set the training batch size as 256 and predicting size as 1."
],
[
"Evaluation Criterion",
"In this work, we evaluate the model's performance in terms of recognition accuracy and character recognition accuracy, which is similar to Wang et al.[18].",
"Recognition accuracy is defined as : $RA = \\frac{Number\\ of\\ correctly\\ recognized\\ license\\ plates}{Number\\ of\\ all\\ license\\ plates}$ Character recognition accuracy is defined as: $CRA = \\frac{Number\\ of\\ correctly\\ recognized\\ characters}{Number\\ of\\ all\\ characters}$ Figure: Three data generation methods (a)Examples of license plates generated by OpenCV scripts.",
"(b)Examples of license plates generated by CycleWGAN.",
"(c)Examples of license plates generated by CycleWGAN-GP."
],
[
"GAN Training and Testing",
"Three data generation methods are shown in Fig.",
"REF .",
"To train CycleWGAN, first we use the OpenCV scripts to generate 1000 blue fake license plates as a source domain X, and then select 1000 real blue license plates from Dataset-1 as a target domain Y.",
"We train the CycleWGAN model with these fake license plates and real license plates.",
"The training real plates do not require character labels.",
"All the images are resized to 143$\\times $ 143, cropped to 128$\\times $ 128 and randomly flipped for data augmentation.",
"We use Adam with $L_{1}=0.9$ , $L_{2}=0.999$ and learning rate of 0.0001.",
"We stop training after 300,000 steps and save the model.",
"When testing, first we use the OpenCV scripts to generate 40,000 blue fake license plates, and then we apply the last checkpoint to generate 40,000 license plates.",
"The same goes for CycleWGAN-GP.",
"Finally we get 80,000 blue license plates generated by CycleWGAN and CycleWGAN-GP."
],
[
"Data Augmentation",
"The six data augmentation methods are proposed in order to increase the training data of the recognition network.",
"The data was augmented through affine transformation, motion blurring, uneven lighting, stretching, erosion and dilation, downsampling and the application of gaussian noise.",
"Examples of these transformations are shown in Fig.",
"REF .",
"A real license plate image randomly passes through the six data augmentation methods, allowing for the creation of much more training data.",
"First, we select a small number of labeled real license plates from Dataset-1, such as 300.",
"And then using data augmentation methods in Fig.",
"REF , we generate 80,000 augmented license plates with these selected real license plates."
],
[
"Mixed Training Data",
"Our mixed training data consists of four parts, including 40,000 license plates generated by OpenCV scripts, 40,000 license plates generated by CycleWGAN, 40,000 license plates generated by CycleWGAN-GP, and 80,000 license plates augmented from a small number of labeled real license plates.",
"All 200,000 training images are generated with license plate character labels.",
"The license plates that need manual labeling are only selected from Dataset-1.",
"After converting the training data to gray scale, 400,000 more training images are obtained by flipping pixels in order to simulate gray images of yellow and green license plates.",
"Then, these images are fed to the recognition network modified from DenseNet.",
"Figure: Six data augmentation methods(a)Affine transformation (b)Motion blur (c)Uneven light (d)Stretching transformation (e)Erosion and dilation (f)Down sampling and gaussian noise."
],
[
"Performance Evaluation on Dataset-1",
"With the above methods, our mixed training data is generated from 300, 700, 3,333, 4,750 and 6,000 real license plates selected from Dataset-1 training set respectively.",
"Our baseline is the [18] using the license plate images generated by the CycleWGAN pre-training recognition network, and then using 9,000, 50,000 and 200,000 real labeled license plate images in a fine-tuning model.",
"From the results in Table REF , it is concluded that when data generation, data augmentation and DenseNet are used, we only need 300 real labeled license plates to achieve the effect of 200,000 real license plates.",
"In the same way, when the number of real license plates reaches 4,750, the final recognition accuracy has reached 99.0%, an increase of 1.4%.",
"When the number of real license plate images exceeds 4,750, license plate recognition accuracy and character recognition accuracy are not improving.",
"We conjecture that 4,750 real images contain enough information to recognize most of the license plates.",
"Thus, by increasing the number of real license plates, the total amount of information after data augmentation will not change, and the recognition accuracy will not increase any further."
],
[
"Performance Evaluation on Data Generation",
"In order to evaluate the effect of the data generated by different methods, we train the models using synthetic data generated by script, CycleGAN, CycleWGAN, and CycleWGAN-GP respectively.",
"The results are shown in Table REF .",
"When we only use the data set generated by script for training, the recognition accuracy on the test set of Dataset-1 is 42.2%.",
"As shown in Fig.",
"REF , our synthetic license plates generated by script also contain noise such as low light, low resolution, motion blur.",
"The CycleGAN images achieve a recognition accuracy of 51.2%.",
"Accuracy is not much improved because of the instability and lack of diversity in CycleGAN training.",
"As shown in Fig.",
"REF and Fig.",
"REF , the CycleWGAN and CycleWGAN-GP images display more various styles and colors, and part of them can not really distinguish from real images.",
"The CycleWGAN and CycleWGAN-GP images achieve a recognition accuracy of 62.5% and 64.5% respectively.",
"We also compare the impact of data generation and data augmentation on accuracy.",
"When the number of real license plates is 3333, the recognition accuracy of the augmented data on Dataset-1 is 97.9%, far exceeding the recognition accuracy of the generated data.",
"Table: Single data augmentation recognition results with 3333 real images compare with .",
"Recognition accuracy (%), character recognition accuracy (%) are shown.",
"\"CRA-C\" is the recognition accuracy (%) of the Chinese characters of the first character, and \"CRA-NC\" is the recognition accuracy (%) of the letters and numbers of the last six characters.In order to understand how much number of real license plates improves recognition accuracy, we compare data augmentation results from 60 to 6000 real license plates.",
"The result in Table REF shows that the greater the number of real license plates, the higher the recognition accuracy obtained.",
"Up to 4750, the highest recognition accuracy of the Dataset-1 is 99.0%.",
"Even if the number of real license plates is increased from 4750, the result is no longer improved.",
"In order to understand the impact of GAN on recognition accuracy, we did some additional comparative experiments.",
"It can also be seen in the Table REF that training data composed of data augmentation and data generation can get better results than training data composed of only data augmentation.",
"The conclusion is that the recognition accuracy of augmented data can be improved with data generation.",
"In addition, the fewer real license plates, the more recognition accuracy increases contributed from generated data.",
"Table: The effect of the number of plates and data generation on the results.",
"Recognition accuracy (%), character recognition accuracy (%) are shown.",
"RA(A) indicates the recognition accuracy of the data augmentation.",
"CRA(A) indicates the character recognition accuracy of the data augmentation.",
"RA(A+G) indicates the recognition accuracy of the training data composed of data augmentation and data generation.",
"CRA(A+G) indicates the character recognition accuracy of the mixed training data composed of data augmentation and data generation."
],
[
"Performance Evaluation on AOLP",
"For the application-oriented license plate(AOLP) dataset, the experiments are carried out by using license plates from different sub-datasets for training and test.",
"This data set is divided into three sub-datasets: access control (AC), traffic law enforcement (LE), and road patrol (RP).",
"For example, in Table REF , we use the license plates from the LE and RP sub-datasets to train the DenseNet, and test its performance on the AC sub-dataset.",
"Similarly, AC and RP are used for training and LE for test, and so on.",
"Since there is no AOLP license plate font, only the data augmentation methods are used, without script and GAN generated license plates.",
"In Table REF , through data augmentation and DenseNet, our method achieves the highest recognition accuracy on the AOLP dataset.",
"Table: The accuracy of other methods compared with the proposed method on dataset AOLP.",
"Recognition accuracy (%), character recognition accuracy (%) are shown.",
"AOLP is categorized into three subsets: access control (AC) with 681 samples, traffic law enforcement (LE) with 757 samples, and road patrol (RP) with 611 samplesIn this paper, we have investigated how many real labeled license plates are needed to train the license plate recognition system.",
"We have proposed three data generation methods and six data augmentation methods in order to fully obtain all the information in a small number of images.",
"The experimental results show that the proposed method only requires 300 real labeled license plates to achieve the effect achieved by 200,000 real license plates.",
"The result shows that the greater the number of real license plates, the higher the recognition accuracy obtained.",
"Up to 4750, the highest recognition accuracy of the Dataset-1 is 99.0%.",
"Even if the number of real license plates is increased furthermore, the result is no longer improved.",
"Additionally, training data composed of both augmented and generated data can achieve better results than training data composed of only augmented data.",
"Furthermore, the fewer real license plates, the more recognition accuracy increases contributed from generated data."
]
] | 1808.08410 |
[
[
"Single Image Dehazing Based on Generic Regularity"
],
[
"Abstract This paper proposes a novel technique for single image dehazing.",
"Most of the state-of-the-art methods for single image dehazing relies either on Dark Channel Prior (DCP) or on Color line.",
"The proposed method combines the two different approaches.",
"We initially compute the dark channel prior and then apply a Nearest-Neighbor (NN) based regularization technique to obtain a smooth transmission map of the hazy image.",
"We consider the effect of airlight on the image by using the color line model to assess the commitment of airlight in each patch of the image and interpolate at the local neighborhood where the estimate is unreliable.",
"The NN based regularization of the DCP can remove the haze, whereas, the color line based interpolation of airlight effect makes the proposed system robust against the variation of haze within an image due to multiple light sources.",
"The proposed method is tested on benchmark datasets and shows promising results compared to the state-of-the-art, including the deep learning based methods, which require a huge computational setup.",
"Moreover, the proposed method outperforms the recent deep learning based methods when applied on images with sky regions."
],
[
"Introduction",
"Images captured by the camera, are often affected by haze due to several atmospheric conditions such as fog, smoke, multiple light sources, the scattering of light, etc.",
"Presence of haze in an image obscure the clarity and hence needs to be restored.",
"The flux of light per unit area received by the camera from the scene is attenuated along the line of sight.",
"This redirection of light lessens the immediate scene transmission and replaces with a layer of scattered light known as airlight.",
"Such scattering of light due to rigid particles floating in the air reduces the visibility of the scene.",
"Figure: Sample result of the proposed approach.",
"(a) Input hazy image.",
"(b) Dehazed image using our approach.Image methods endeavor to recover original scene radiance by expelling the impact of haze from the picture(as shown as an example in Figure 1).",
"Since the effect of scattering of light at any pixel depends on the depth of the pixel, the degradation of the image due to haze is spatial-variant.",
"However, recovering the effect of scene radiance from a pixel on an object is perplexing as the measure of fog and haze relies upon the separation between the object and the camera.",
"In this way, global enhancement strategies do not function admirably for a scene image, because of the presence of objects at different depth and the background.",
"The conventional methods exploited multiple images to overcome the problem.",
"For example, the method of Narasimhan et.al.",
"[1] require multiple pictures of a similar scene taken under various climate conditions.",
"The method of Shwartz et.al.",
"[2] requires images with the various level of polarization.",
"Figure: NO_CAPTIONFigure: Result of dehazing an image.",
"(a) input haze image.",
"(b) refined transmission map after NN regularization (c) final haze-free image (d) depth map of the output Haze-free Image.However, images taken in various climatic conditions may be unavailable always.",
"Recently researches are going on for single image dehazing, where reference images for the given hazy images are unrequired.",
"After the introduction of the concept of Dark Channel Prior(DCP), people started dehazing an image without the help of a reference image [3].",
"Tan et.al.",
"proposed a DCP based strategy for the single image by maximizing the local image contrast adopting a Markov Random Field (MRF) framework [3].",
"Fattal [4] used DCP to estimate the medium transmittance accepting the surface shading, furthermore, medium transmission capacities are locally measurably uncorrelated.",
"He et al.",
"[5] proposed the concept of DCP for estimating transmission map of the image.",
"The DCP based approaches produce good result, but cannot handle locales where shading segment does not shift fundamentally contrasted with noise.",
"Nair et al.",
"[6] proposed a DCP based dehazing technique using a center surround filter to reduce the execution time.",
"Another line of thought was introduced by Fattal in [7], where the color line was used for dehazing.",
"The color line based methods approximate transmission map based on the shift of the direction of atmospheric light from the origin.",
"However, both the approaches fail to retain the original color of the objects in the hazy image.",
"Currently, some deep learning based methods have been proposed, producing much better results compared to the handcrafted features like DCP and color-line [8], [9], [10].",
"However, in case of images where sky background is present, the deep learning based methods fail to dehaze properly, as the learned features are not capable of fixing the haze at high depth areas of the image.",
"In this paper, the contribution over the state-of-the-art is three-folds.",
"Unlike the state-of-the-art methods, the proposed method for a single image assesses the benefit of the generic regularity of the image in which, pixels with moderate intensities of small image patches exhibit concordant distributions in RGB space.",
"Our second contribution is to substitute the commonly used soft matting technique [11] in assessing the transmission map for haze removal, by first discovering some settled points in a maximum filter and then applying Nearest-Neighbor (NN) regularization.",
"One example of using the transmission map in haze removal by the proposed method is shown in Figure 2.",
"We utilize pixels of small image patches as a rule accompanying a one-dimensional dispersion in RGB space, to approximate the intensity at each patch and interject at places wherein the estimate is not generally unreliable.",
"We do not consider the atmospheric light-weight.",
"Finally, unlike the state-of-the-art methods, we do not consider and estimate the transmission of the medium.",
"Alternatively, we approximate the additional airlight present in the image patch and eliminate that to clear the haze.",
"We interject and regularize the fractional evaluations of pixel intensity values into an entire transmission map, and outline an Adaptive Naive Bayes Classification approach to deal with pixels with a significant depth (such as sky region).",
"Contrary to the conventional field models which comprise of normal coupling between adjacent pixels, we settle the transmission in isolated locales by fluctuating the number of pixels.",
"We take the likelihood of more number of pixels to get the suitable pixel intensity value.",
"The rest of the paper is organised as follows.",
"Section 2 describes the problem and a survey on the existing literature.",
"Section 3 displays the mathematical and numerical foundation for the proposed strategy for dehazing.",
"Section 4 weighs up the proposed algorithm.",
"Experimental results and concluding comments are given in Sections 5 and 6 respectively."
],
[
"Related Work",
"Dehazing is a task of image remaking; the corruption of a hazy image is a direct result of the suspended particles in the turbid air.",
"Single image dehazing is a challenging task which draws significant attention from researchers of the fields of Computer Graphics and Image Processing.",
"The physical model often used to characterize the haze formation that causes degradation of an image, is known as Koschmieder's atmospheric scattering model [12]: $\\ I(x) = t(x)J(x) + (1 - t(x))A,$ where $I(x)$ is the observed intensity value of pixel $x$ in image $I$ ; $J(x)$ is the scene radiance of a haze-free image at $x$ and $t(x)$ is the medium transmission describing the portion of the light which reaches the camera without scattering.",
"$A$ is the airlight, a global vector quantity describing the ambient light.",
"The goal of haze removal task is to recover $J$ , $A$ and $t$ from $I$ .",
"In (REF ), the transmittance $t(x)$ does not change much for a given wavelength in a hazy image.",
"The term $J(x)t(x)$ of (REF ) is termed as direct attenuation [3], which provides the information about the quantity of radiance received by the observer.",
"The second part, called airlight, provides an approximation of the measure of the atmospheric light included due to the dissipating course of the viewer.",
"Several attempts have been made for fog and haze expulsion in outdoor scene pictures [13].",
"Notwithstanding, the considerable significant advance in the research on haze removal methods was the idea of DCP.",
"It is observed that, in the more considerable part of the local neighborhood regions which do not contain the sky, a few pixels in many instances get extremely low-intensity value at the minimum one color (RGB) channel (called \"dark pixels\").",
"In case of hazy image, the intensity of these dark pixels in the corresponding channel is contributed by the airlight.",
"Along these lines, these dark pixels can explicitly give a precise estimation of the fog's transmission.",
"In any case, the progression to reclassify the transmission approximation utilizing soft matting technique often leads to transmission underestimate.",
"When the density of haze varies smoothly in space, the effect of the homogeneous atmosphere transmission $t$ can be shown as: $\\ t(x) = e^{\\beta d(x)},$ where $d(x)$ is the depth at pixel $x$ and $\\beta $ is the scattering coefficient (in three dimensional space) of the atmosphere.",
"So, (REF ) specifies the amount of radiance received by the observer, which is attenuated exponentially with the depth.",
"Several algorithms for image dehazing follow the image formation model (REF ) to dehaze images by recouping $J$ .",
"However, to regain haze-free image, both (REF ) and (REF ) are followed.",
"First $J$ is obtained by subtracting a constant value corresponding to the dark pixels and then estimate the value of transmission $t$ precisely by assuming $A$ is known.",
"Haze lessens the contrast in the picture, and different techniques depend on this perception for rebuilding.",
"He et al.",
"[14] amplifies the contrast in each patch, while keeping up a global coherent image.",
"This method enhanced the contrast for image dehazing and as a result, distant objects to the viewer seemed to be smooth and over saturated, causing the dehazed image look artificial.",
"In [15], Park et al.",
"evaluated the amount of haze in the image, from the contrast between the RGB channels, which diminishes as haze increments.",
"This postulation of estimation of amount of haze is erroneous in the greay area.",
"In [16], Zhu et al.",
"assessed the haze in view of the perception that hazy regions are portrayed by high brightness and low saturation.",
"Some efforts have been made by utilizing a prior on the depth of the picture, for haze expulsion.",
"A smoothness prior is utilized in [17], expecting the image to be smooth except the pixels with depth discontinuities.",
"Nishino et al.",
"[18] expected the albedo and depth as factually free and together approximated them using priors on both.",
"The albedo-prior accepted the distribution of gradients in images of characteristic scenes displaying a heavy-tail dispersion, and it is estimated as a summed up ordinary appropriation and further approximated as Gaussian distribution.",
"The depth-prior is scene-dependent and is picked physically, either as piece-wise consistent for urban scenes or definitely changing for non-urban scenes.",
"The DCP based approaches [14] accept that within a small image patche there will be no less than one pixel with a dark color channel and utilize this negligible incentive as an approximate of the haze.",
"This prior works extremely well, with the exception of in bright regions of the scene where intensity values of the considerable number of channels are likewise high.",
"Hence, DCP cannot manage the sky area of the images in light of the fact that the dull channel pixels are potentially inaccessible in those splendid picture areas.",
"Also, DCP based techniques are tedious because of the soft matting procedure involved in the method.",
"Recently, several techniques have been proposed to address the shortcoming of DCP [14].",
"In [19], color ellipsoids are fitted in the RGB color space.",
"These ellipsoids are utilized to arrange a unified approach to deal with the constraints of the DCP based techniques, and another strategy was proposed to assess the transmission in every ellipsoid.",
"In [5], [20], color lines were fit in RGB space, searching for little fixes with a steady transmission.",
"The traditional color line based methods depend on the assumption that pixels in a haze-free image constitute color lines in RGB space [12].",
"These lines go through the inception and come from color varieties inside items.",
"As appeared in [5], the color lines in foggy scenes do not go through the starting point any longer, because of the added substance fog part.",
"Some significant attempts have been made using some priors other than the DCP and color line [21], [22], [23], [24].",
"In [23], globally guided regularization technique is applied for dehazing.",
"In [24], contrast of the image is enhanced using boundary conditions.",
"Tang et al.",
"proposed a learning based technique for image dehazing [22].",
"However, the DCP and colorline based methods continue to dominate the other methods due to better performances, except images with high depth area (such as sky area).",
"Riaz et al.",
"proposed a DCP based approach [25] where the inefficacy of DCP at sky area is handled by limiting the contrast at the sky region.",
"However, in most of the cases limiting the contrast causes loss of minute color information of the image.",
"The proposed method follows the concepts of DCP [14] and color line [5].",
"In addition to applying the above two priors, the proposed method applies a Nearest Neighbor (NN) regularization technique which helps reduce the execution time.",
"The domain-adaptive nature of the patches helps in maintaining the original color and contrast of the image.",
"Finally, the adaptive nature of the size of the mask help in maintaining the original color in high depth areas of the image (area which is far from the camera, such as, sky area).",
"Recently, deep learning based techniques are being successfully applied in all areas of Computer vision.",
"A few efforts have been made to apply deep learning techniques for haze removal [9], [10], [8].",
"Unlike the traditional image dehazing techniques, AOD-Net in [10] proposes an end-to-end system for image dehazing, where a simple CNN is introduced to directly estimate the haze-free image, without obtaining the transmission map of the image.",
"Dehazenet is proposed in [9], where a transformation map of the input image is obtained from a CNN, by a trainable model.",
"The CNN based methods usually give better results for image dehazing, compared to the traditional prior based models.",
"However, for scene images where objects are far from the camera, the CNN based methods do not work well.",
"Also, the deep network-based methods need a huge computational set up, which may not be affordable in many cases, e.g., hand-held devices.",
"Next we discuss the mathematical background of the proposed method."
],
[
"Background of the Proposed Approach",
"We assess the transmission by considering $A$ not constant for the scene.",
"We observe that (REF ) expect the pixel intensity $I(x)$ to be radiometrically-straight.",
"Hence, additionally to adjust methods that rely upon this course of action illustrate, our system requires the reversal of the procurement nonlinearities.",
"In this section we explain the mathematical background of the proposed approach and the significance of the approach in the image dehazing task."
],
[
"Dark Channel Prior (DCP)",
"The DCP is based on the perception on haze-free images: the most of the cases, the patches taken from a haze-free image must have no less than one color channel having a low intensity incentive at a few pixels [14].",
"According to this perception, for any image $J$ , we characterize: $J^{dark}(x) = \\min _{c \\in \\lbrace r,g,b \\rbrace }(\\min _{y \\in \\Omega (x) } J^{c}(y)),$ where $\\Omega (x)$ is the neighborhood patch centered at $x$ , $J^c$ is the color channel of the picture $J$ and $J^{dark}$ the dark channel of $J$ .",
"The intensity of $J^{dark}$ is low and tends to zero if $J$ is a haze-free image.",
"The low intensities in the haze-free locale are due to the following three elements: Shadows of objects: For example, the shadows of leaves, trees and rocks in landscape images or the shadows of buildings in cityscape images; Dark objects or surfaces: For example, dark tree trunk and stone; Colorful objects or surfaces: Color of any object (for example, green grass/tree/plant, blue water surface or red flower/leaf/wall) mostly tends to be very close to one of the color planes such as, Red, Green, Blue, Cyan, Magenta, Yellow, Black and White.",
"However, the DCP based techniques work well for images with high contrast and low depth region.",
"For images consisting of regions with high depth value, DCP based methods fail to maintain the original color of the region.",
"For example, sky region is usually bright and hence, do not have dark intensity values.",
"We can deal with the sky locales by utilizing the Haze atmospheric scattering model (REF ), which is applied in the proposed model and discussed in the next subsection.",
"Figure: (a), (c) Input hazy images.",
"(b), (d) Depth map of the Input Hazed Images"
],
[
"Nearest-Neighbor (NN) Regularization",
"In Figure 3, we can observe that the transmission map is of an even consistency but is having unanticipated depth jumps in the hazy image.",
"In the image-dehazing context, we assume that comparable pixels have an almost identical transmission value, according to [5], [26].",
"In order to proficiently quantify the similarity between two image pixels, we have to think about the spatial variation and smoothness.",
"Given an image pixel $x$ , we characterize its element vector for similitude estimation in light of NN regularization as given below: $f(x) = (R,G,B,\\lambda X,\\lambda Y)^T,$ where the intensities of $x$ are depicted as $R$ , $G$ and $B$ in the RGB color space, respectively; $X$ and $Y$ are the spatial coordinates of $x$ , and $\\lambda $ is the balancing factor.",
"Despite the fact that we utilize indistinguishable features from [27], however, is not the same as matting, so we adjust our technique to the important and broadly utilized conditions for matting utilizing (REF )."
],
[
"Color line model",
"Colors in the small patches of a haze-free image generally lie on the line going through the starting point as shown by Omer et al.",
"[12], if we represent colors as coordinates in the the RGB space.",
"But the line is shifted from the origin by $A$ due to the additive airlight component in the case of hazy images.",
"We assume that $A$ is constant.",
"As we expect piecewise smooth depth $d$ which gives us a smooth scattering coefficient $\\beta $ which verifies that $t(x)$ is piecewise smooth which is smooth at pixels that correspond to the same object.",
"And $t(x)$ is also varying smoothly and slowing in the scene known from equation (REF ) except at depth discontinuities.",
"This assumption of piecewise smooth geometry was used by Carr et al.",
"[28] and Nishino et al.",
"[18].",
"So, the equation (1) for a small patch can be rewritten as: $I(x) = J(x)t + (1 - t)A, x \\in \\Omega ,$ where $t$ is a fixed transmission value in the patch $\\Omega $ , and the airlight component is known to us.",
"If we estimate the color line directly, then the measurement of color line direction will be erroneous if some dark pixels are present.",
"In the proposed model, as the image is first subjected to DCP, will be less prone to error."
],
[
"Adaptive Naive Bayes Classification Model",
"Naive Bayes is a conditional likelihood model.",
"Given an issue occurrence to be characterized, spoken to by a vector ${\\displaystyle \\mathbf {x} =(x_{1},\\dots ,x_{n})}$ representing $n$ features (independent factors), it relegates to this case probabilities $p(C_{k} | x_{1},\\dots ,x_{n})$ for each one of the $K$ conceivable results or classes ${\\displaystyle C_{k}}$ .",
"The predicament with the above expression is that, if the quantity of highlights $n$ is substantial or if an element can go up against a considerable number of values, at that point applying such a model on likelihood tables is troublesome.",
"We, therefore, reformulate the model to make it less demanding to deal with.",
"Appropriating Bayes' theorem, the conditional likelihood can be calculated as: $p(C_{k} | \\mathbf {x}) = \\frac{p(C_{k}) \\cdot p(\\mathbf {x} | C_{k})}{p(\\mathbf {x})}.$ The denominator of (6) can be considered as a constant.",
"The numerator is proportional to the conditional likelihood $p(C_{k} | x_{1},\\dots ,x_{n})$ , which can be revised as given below, utilizing the chain lead for numerous implementations of the meaning of conditional likelihood: $p(C_{k}, x_{1},\\dots ,x_{n}) &= p(x_{1},\\dots ,x_{n}, C_{k}) \\\\&= p(x_{1}|x_{2},\\dots ,x_{n}, C_{k}) \\cdot p(x_{2},\\dots ,x_{n}, C_{k}) \\\\&= p(x_{1}|x_{2},\\dots ,x_{n}, C_{k}) \\cdot p(x_{2}|x_{3},\\dots ,x_{n},\\\\& \\quad C_{k}) \\cdot p(x_{3},\\dots ,x_{n}, C_{k})\\\\&= p(x_{1}|x_{2},\\dots ,x_{n}, C_{k}) \\cdot p(x_{2}|x_{3},\\dots ,x_{n},\\\\& \\quad C_{k}) \\cdot p(x_{3}|x_{4},\\dots ,x_{n}, C_{k})\\\\&= \\dots \\\\&= p(x_{1}|x_{2},\\dots ,x_{n}, C_{k}) \\cdot p(x_{2}|x_{3},\\dots ,x_{n},\\\\& \\quad C_{k}) \\cdots p(x_{n-1}|x_{n},C_{k}) \\cdot p(x_{n}|C_{k}) \\cdot p(C_{k})\\\\& \\quad $ The “naive\" conditional independence presumptions are connected here for the proposed dehazing approach.",
"Let us expect that each component ${\\displaystyle x_{i}}$ is restrictively independent of each different feature ${\\displaystyle x_{j}}$ for ${\\displaystyle j\\ne i}$ , given the classification ${\\displaystyle C}$ .",
"Which implies, $p(x_{i} | x_{i+1},\\dots ,x_{n}, C_{k}) = p(x_{i} | C_{k})$ .",
"Along these lines, the proposed conditional likelihood model can be communicated as $p(C_{k} | x_{1},\\dots ,x_{n}) &\\propto p(C_{k})p(x_{1}|C_{k})p(x_{2}|C_{k}) \\dots \\\\&= p(C_{k}) \\prod ^{n}_{i=1} p(x_{i}|C_{k}).$ where $\\propto $ denotes proportionality.",
"For evaluating the colorline heading, we develop a model that assigns the value in view of the component vector for similitude estimation in condition (4), where the element vectors are directly free.",
"In attendance, we shift the quantity of pixels given to the Naive Bayes Classification Model.",
"By changing the number of pixels we take the probability of more number of pixels by finding the probabilities utilizing the Naive Bayes' condition to compute the posterior probability for every pixel based on the inherent vector in (4).",
"The pixel with the most astounding posterior probability obtain the result of the proposed expectation model.",
"In this section, we explain the steps to dehaze an image utilizing the nearby fix demonstrate in (REF ) and its related transmission estimation system in (REF ).",
"We started with a brief review of the mathematical background in the previous section.",
"Now we demonstrate the way the mathematical models can be used to check the input image and consider small patches of pixels as possible fixes that satisfy (REF ).",
"As articulated in the previous section, pixels that relate to an almost planar surface lie on a color line in RGB space depicted by (REF ) and (REF ).",
"In this way, in each fix we run a Naive Bayes Classification Model unequivocally built using (REF ) that scans for a line bolstered by a significant number of pixels.",
"We at that point check whether the line found is reliable with our arrangement show by testing it against a rundown of conditions postured by the model.",
"A line that concludes everyone of these tests effectively is then utilized for assessing the transmission as per (REF ) and (REF ).",
"The subsequent values at that point are allocated to every one of the pixels that help in finding the color line.",
"We evaluate the transmission in patches where we neglect to discover a line that meets every one of the conditions by utilizing the NN Regularization and reapplying the above strides on the relating patch.",
"Next we discuss the process of estimating transmission map of an image using DCP."
],
[
"Transmission Estimation Using Dark Channel Prior",
"We believe that the transmission $t$ within a neighborhood patch $\\Omega (x)$ is steady.",
"Performing the minimum operation in the neighborhood patch of the hazy picture in (REF ), we have the following: $\\min _{y \\in \\Omega (x)}(I^c (y)) = \\min _{y \\in \\Omega (x) }(J^c (y))t + (1 - t)A^c.$ The above equation can be derived further as: $\\min _{y \\in \\Omega (x) }(\\frac{I^c (y)}{A^c}) = \\min _{y \\in \\Omega (x) }(\\frac{J^c (y)}{A^c})t + (1 - t).$ Considering the min operation again on the above condition, in order to get the minimum of the three colors in RGB channel, we have: $\\min _{c}(\\min _{y \\in \\Omega (x)} (\\frac{I^c (y)}{A^c}) = \\min _{c}(\\min _{y \\in \\Omega (x) }(\\frac{J^c (y)}{A^c})t) + (1 - t).$ As indicated by (REF ), the dark channel $J^{dark}$ of the haze-free pixel intensity $J$ should tend to zero.",
"Also, as $A^c$ is a positive constant, thus: $\\min _{c}\\Big (\\min _{y \\in \\Omega (x) }(\\frac{J^c (y)}{A^c})\\Big ) = 0.$ From (REF ) and (REF ), we get: $\\ t = 1 - \\min _{c}\\Big (\\min _{y \\in \\Omega (x) }(\\frac{I^c (y)}{A^c})\\Big ).$ The above mathematical formulation gives us an estimation of the transmission.",
"As stated earlier, the DCP is not a appropriate prior for the sky locales.",
"Since the sky is at boundless infinite depth and practically has zero transmission, the (REF ) smoothly tackles both sky and non-sky regions in the proposed approach.",
"We dehaze the image from the assessed approximation of transmission obtained from (REF ) and utilize the neighborhood local patch model.",
"Reasonably, even in clear environment, the atmosphere around us is not completely free of any molecule.",
"Along these lines, the haze still exist when we have a look at inaccessible items.",
"In addition, the presence of haze is a major prompt for a human to comprehend depth."
],
[
"Dehazing Algorithm",
"In this component, we illustrate the initiatives for dehazing an image using our method.",
"First, we dehaze the image by removing the dark pixels from the image without applying a soft matting algorithm [11] to convalesce the transmission.",
"We signify the transmission outline $t(x)$ and get the accompanying cost function: $E(t) = t^TLt + \\lambda (t - \\tilde{t})^{T+1},$ where $L$ is the Matting Laplacian matrix introduced by Levin [29], and $\\lambda $ is a regularization parameter.",
"We analyse the transmission estimations of a few points with low depth values, and propose recuperating the transmission estimations of the rest of the points by finding their closest match from the arrangement of exact focuses in light of the built k-d tree with the 5-dimensional component vectors characterized in (REF ).",
"Conforming transmission maps along with the input image is shown in Figure 3, from which we can perceive that the transmission outline is significantly smooth aside from unanticipated depth jumps.",
"As specified earlier, the dehazing issue is extremely under-constrained.",
"Hence, we have to surmise a few assumptions from (REF ), (REF ) and (REF ) on the normal transmission map.",
"The refined transmission outline exercising the above strategy accomplishes to capture the sharp edge discontinuities and outline the shape of the items.",
"With the transmission outline, we can remunerate the scene radiance as per (REF ).",
"Yet, the direct attenuation may be near to zero when the transmission is near to zero.",
"In this way, we limit the transmission $t(x)$ to a lower bound $t_0$ , so that a specific measure of haze are retained.",
"The scene radiance $J(x)$ is eventually recouped by: $J(x) = \\frac{I(x) - \\hat{A}}{max(t(x),t_{0})} + \\hat{A},$ where $\\hat{A}$ denotes the estimated airlight of the image.",
"A traditional range of $t_{0}$ falls in the interval [0.09, 0.1], which is fixed experimentally.",
"Since the scene radiance is generally unlike the atmospheric light, the image after disturbance by haze evacuation looks diminish.",
"Hence, we increse the insight into $J(x)$ for illustration.",
"We examine the information from the input image and consider limited windows of pixels as hopeful patches following (REF ).",
"Pixels relating to an almost-planar mono-chromatic surface lie on a color line in RGB space as indicated by (REF ).",
"We evaluate the color line by applying “Naive Bayes Classifier\" on the intensity values in RGB space and get two focuses $\\vec{p_{1}}$ and $\\vec{p_{2}}$ lying on the straight line and an arrangement of inliers.",
"The condition of the line will be $\\vec{P} = \\vec{P_{0}} + \\rho \\vec{D} $ , where $\\rho $ represent the parameter of the line, $\\vec{P_{0}} = \\vec{p_{1}}$ and the heading proportion $\\vec{D} = \\frac{\\vec{p_{2}} - \\vec{p_{1}}}{\\mid \\mid ( \\vec{p_{2}} - \\vec{p_{1}} )\\mid \\mid }$ .",
"We examine the assessed line with conditions in particular, noteworthy number of inliers, positive incline of $\\vec{D}$ and unimodality analogously in [5] and eliminate the questionable ones.",
"Figure 4 demonstrates the effect of applying the Naive-Bayes Classifier and the proposed NN regularization technique on a sample hazy image.",
"The second row of Figure 4 illustrates the effect of applying different patch sizes on the same image.",
"Figure: NO_CAPTIONFigure: Result of the proposed method with varying patch size.",
"(a) input haze image.",
"(b) Output Image without Nearest Neighbour Regularization (c) Output image without applying Naive Bayes' Classification Model (d) pixels are given to Naive Bayes Classification Model with patch size 7 ×\\times 7 (e) 15 ×\\times 15 and (f) 30 ×\\times 30."
],
[
"Computing $\\hat{A}$",
"We can make our estimates for $\\hat{A}$ using the Dark Channel [3] of the patch.",
"Dark Channel of the specific patch $\\Omega $ is determined as follows (following [30]: $Dark(\\Omega ) = \\min _{x \\in \\Omega (x) }\\Big (\\min _{c \\in {R,G,B}} I_c(x)\\Big ),$ where $I_c$ is the $c$ th color channel, for $c\\in \\lbrace R,G,B\\rbrace $ .",
"We have already defined the direction ratio $\\vec{D}$ in the previous section, considering the standard conditions, significant line support, positive reflect of $\\vec{D}$ , unimodality and valid transmission according to (REF ) and (REF ) as discussed in [5].",
"We check whether the visible pixels efficiently found are consistent with our formation model by running it following (REF ), (REF ) and (REF ).",
"With respect to $\\vec{D}$ , we conventionally consider the normal to the inclined plane from the precise origin as determined as $\\vec{N} = \\frac{\\vec{p_{2}} \\times \\vec{p_{1}}}{\\mid \\mid ( \\vec{p_{2}} \\times \\vec{p_{1}} )\\mid \\mid }$ .",
"We compute $\\hat{A}$ by minimizing the following error from the existing corresponding normals from the above defination: $E(\\hat{A}) = \\frac{1}{\\Omega } \\cdot \\sum _{i \\in {\\Omega }} (N_i \\cdot \\hat{A}),$ where the line parameters are denoted by $N_i \\cdot \\hat{A}$ of the patch pixels.",
"The $\\cdot $ typically denotes the dot-product in RGB space.",
"This reliable measure consists of projecting the line parameters onto an error function, $E(\\hat{A})$ .",
"Therefore, (REF ) vanishes over uniformly distributed pixels and becomes positive.",
"Therefore, we have $\\frac{\\partial E}{\\partial \\hat{A}} = \\hat{A} \\cdot \\Big (\\sum _{i} N_iN_i^T \\Big ) = 0.$ From the (REF ), we need the non-trivial solution as $\\hat{A}$ is a non-null vector which can be acquired by registering a covariance matrix from the normals and afterwards obtaining the result as a covariance matrix similar to the smallest eigenvalue."
],
[
"Estimating the Magnitude of Airlight",
"We can retrieve the magnitude $a(x)$ of the airlight component, from the evaluated $\\hat{A}$ and acquire the color line correlating with the patch by limiting the accompanying error: $\\ E_{line} (\\rho , s) = \\min _{\\rho , s} \\mid \\mid {P_{0}} + \\rho D - s \\hat{A} \\mid \\mid ^2.$ Here, airlight component is denoted by $s$ .",
"The solution of (REF ) is obtained following [7].",
"The registered airlight portion is then approved with the accompanying conditions: vast intersection and convergence angles, close intersection, substantial range and shading changeability.",
"Out of them, vast intersection angle and close convergence are done likewise approaches as detailed in [7].",
"We estimate $J$ from the refined transmission map and the output image obtained from the proposed method and a black image $E$ .",
"From the intermediate outcomes, it can be observed that the underlying $J$ has noticeable artefacts in the sky locale, which is progressively exterminating to amid the improvement.",
"The outcome is given in Figure 5, exhibiting the gradual decline in the graph alongside the images as output.",
"Figure: The convergence of the proposed approach.",
"The objective function in Eq.",
"() is monotonically declining.",
"The transitional consequences of JJ and 10×E10 \\times E at the given iteration."
],
[
"Interpolation of Estimate",
"For assessing the airlight part $a(x)$ we dispose of a couple of patches and introduce the estimation of airlight at each pixel.",
"This is done by limiting the accompanying capacity: $\\begin{aligned}\\psi (a(x)) = \\sum _{\\Omega } \\sum _{x \\in {\\Omega }} \\frac{(a(\\textbf {x}) - \\widetilde{a} (\\textbf {x})^2}{(\\sigma _a (\\Omega ))^2} + \\beta \\cdot \\sum _{x} \\frac{a(\\textbf {x})}{\\mid \\mid I(\\textbf {x})\\mid \\mid } \\\\+ \\alpha \\cdot \\sum _{\\Omega } \\sum _{x \\in {L(x)}} \\frac{(a(\\textbf {x}) - \\vec{a} (\\textbf {x})^2}{\\mid \\mid I(\\textbf {x}) - I(\\textbf {y}) \\mid \\mid ^2}.\\end{aligned}$ Here, $\\vec{a}(x)$ is the assessed magnitude of the airlight component, $L(x)$ is the local neighborhood of $\\textbf {x}$ and $a(\\textbf {x})$ represents the adequate segment to be registered and $\\sigma _a(\\Omega )$ is the error difference of the estimate of $\\vec{a}(x)$ inside the patch $\\Omega $ .",
"The initial two terms constitute the capacity utilized as a part of [5].",
"To obtain a better result, we include the last term which guarantees that the part would be a little portion of $I(\\textbf {x})$ .",
"The latter term of (9) is utilized to change the segment with the intensity of $I(\\textbf {x})$ .",
"Presently, to limit the vitality work given by condition (20), we change over this to the following structure: $\\Psi (a) = (a - \\widetilde{a})^T \\cdot \\Sigma (a - \\widetilde{a}) + \\alpha a^T La + \\beta b^T a,$ where $a$ and $\\widetilde{a} $ denote the vector forms of $a(\\textbf {x})$ and $\\widetilde{a}(\\textbf {x})$ respectively, $\\Sigma $ denote the covariance grid of the pixels where the approximation is made and $L$ is the Laplacian framework of the diagram built by considering each pixel as a vertex and interfacing the neighboring vertices.",
"The influence of the edge between the pair of vertices $x$ and $y$ is $\\frac{1}{\\mid \\mid I(\\textbf {x}) - I(\\textbf {y})\\mid \\mid ^2}$ .",
"Here $\\alpha $ and $\\beta $ are scalars controlling the significance of each term."
],
[
"Haze Removal",
"Airlight at every pixel is now obtained by figuring $a(\\textbf {x})\\hat{A}$ .",
"Consequently, the immediate transmission can be recouped from $J(\\textbf {x}) t(\\textbf {x}) = I(\\textbf {x}) - a(\\textbf {x})\\hat{A}.$ As we do not have $t(x)$ unequivocally, we elevate the contrast utilizing the airlight and attempt to recoup $J(\\textbf {x})$ .",
"For instance, if the recouped image is $R_im (\\textbf {x})$ , at pixel $x$ : $R_im (\\textbf {x}) = \\frac{J(\\textbf {x}) t(\\textbf {x})}{1 - Y(a(\\textbf {x})\\hat{A})},$ where $Y$ is given by the following equation: $Y(I(\\textbf {x})) = 0.2989 I_R (\\textbf {x}) + 0.5870 I_G (\\textbf {x}) + 0.1140 I_B (\\textbf {x}),$ where $Y(I(\\textbf {x}))$ processes the luma at the pixel $x$ .",
"The idea is to upgrade the pixel $x$ relying upon how much intensity (brightness) is expelled out from it.",
"In many cases, the image remains dull even after the above operation.",
"So we utilize gamma revision to reestablish the natural shine of the image."
],
[
"RESULTS",
"We assess the performance of the proposed method on the Berkeley Segmentation Dataset (BSDS300), containing natural images [31].",
"This is a diverse dataset of clear open air characteristic images and consequently represents the moderate scenes that may have been affected by haze.",
"We have evaluated airlight part in condition (17).",
"We utilize similar parameters for every images: in condition (10) we set $\\lambda $ = 0.1 and we scale $\\frac{1}{\\sigma ^2 (x)}$ in the interim [0,1] to maintain a strategic distance from numeric issues.",
"For experimenting on the synthetic dataset, we utilize the dataset proposed in [5].",
"The dataset contains eleven dehazed pictures, manufactured distance maps and relating reenacted fog pictures.",
"An indistinguishably appropriated zero-mean Gaussian commotion with three one of a kind clamour level: n = 0:01; 0:025; 0:05 was added to these images (with picture control scaled to [0; 1]).",
"Table: NO_CAPTION"
],
[
"Quantitative Analysis",
"Table I condenses the L1 errors on non-sky pixels of the transmission maps and the fog free images of the synthetic dataset.",
"The results of four sample images are given in the table.",
"We utilize a progression of assessment criteria as distant as to separate between each coordinate of the haze-free image along with the result.",
"Notwithstanding the extensively used mean square error (MSE) and the structural similarity (SSIM) [32] measures, we utilized some more assessment frameworks, for example, weighted peak signal-to-noise ratio (WSNR) and peak signal-to-noise ratio (PSNR) [33].",
"Our method is compared with seven state-of-the-art methods (including deep learning based methods) when applied on the hazy images of the BSDS dataset, results of which is shown in Table II.",
"Table: The average results of MSE, SSIM, PSNR and WSNR on the hazy Images.In Table II, the last three lines demonstrate the results of applying deep learning based techniques.",
"As we can observe, the effects of applying the proposed method on hazy images is much better than the best in class handmade approaches.",
"Additionally, the proposed method is comparable to the deep-learning based procedures as well.",
"Table: The average results of MSE, SSIM, PSNR and WSNR on the Images with the Sky Region.Figure: Graphs showing the performance of the proposed approach compared to the state-of-the-art, with increasing depth value.",
"(a) Images without sky area, (b) Images with sky area.In Table III, the result of implementing the proposed method on images containing sky and other high depth regions, compared to the state-of-the-art methods, are shown.",
"Figure 7 shows how the accuracy of the proposed methods vary with respect to depth.",
"We can observe that, as the depth increases, the proposed method gives much better result compared to the state-of-the-art, including the deep learning based methods.",
"Figure: Comparison on natural images: [Left] Input Hazy Images[Right] Our result.",
"Middle columns display results by several competing methods."
],
[
"Qualitative results",
"In Figure 8, we compare the results of the proposed approach with the state-of-the-art single image dehazing techniques [5], [14], [34], [35], [36].",
"As noted by [5], the picture after fog evacuation may look diminish, since the scene radiance is typically not as splendid as the airlight.",
"The techniques in [5], [14] provide good results, yet do not have some miniaturized scale differentiate when contrasted with [34] and to our our method.",
"In the consequence of [36] there are artefacts in the boundary in between portions.",
"Figure: Comparison on natural images: [Left] Input Hazy Images[Right] Our result.",
"Middle columns display results by several competing methods.Figure 9 demonstrates a comparison between the outcomes obtained by the proposed technique and the state-of-the-art (explicitly, profound deep-learning based techniques) when applied on images containing sky region.",
"The proposed approach outperforms the competing techniques on this kind of images.",
"Our assumption in regards to having a fog-free pixel in each haze line does not enclose as clear by a few cloudy pixels that set a most extreme radius, e.g.",
"the red structures.",
"In spite of that, the transmission in those territories is evaluated accurately because of the regularization that proliferates the depth data spatially from the other fog lines.",
"Dehazing the sky area in a hazy image is really challenging, in light of the fact that clouds and mist are similar normal phenomenons with a similar air diffusing illustrate.",
"This issue is facilitated, in any case proceeds in DCP [14], Non-Local Dehazing [34], DehazeNet [9] and MSCNN [8] results.",
"While MSCNN makes the opposite curio of overenhancement: see the sky region of Yosemite for an example (Figure 9).",
"AOD-Net [10] can oust the obscurity, without displaying fake shading tones or turned dissent shapes.",
"In any case, AOD-Net does not explicitly consider the treatment of white scenes(can be found in sky range of Yosemite and Building in Figure 9).",
"Our method appears to be capable of finding the sky region to keep the shading, and ensures a decent dehazing sway in different locale."
],
[
"Conclusion",
"In this paper, we proposed a method for Image De-Hazing in view of the dim channel earlier and shading lines pixel normality on the regular images.",
"Dark Channel Prior technique depends on insights of the regular outdoor pictures.",
"We got a local nearby formation model that reasons this consistency in foggy/ hazy scenes and portrayed how it is utilized for assessing the scene transmission.",
"We utilize the image forming equation to recuperate haze free picture.",
"We have accepted commitment $A$ to be consistent.",
"We computed transmission and in addition profundity discontinuities.",
"The proposed dehazing scheme delivers comparable outcomes with the recent deep-learning based strategies which require gigantic computational setup.",
"In addition, the proposed technique outperforms the state-of-the-art deep learning based approaches when applied on images with sky regions.",
"Stretching out the recommended model to apply on real-time images may be a conceivable future research heading, which may be extended to apply on hand held devices having less computing power."
]
] | 1808.08610 |
[
[
"A Dichotomous Analysis of Unemployment Welfare"
],
[
"Abstract In an economy which could not accommodate the full employment of its labor force, it employs some labor but does not employ others.",
"The bipartition of the labor force is random, and we characterize it by an axiom of equal employment opportunity.",
"We value each employed individual by his or her marginal contribution to the production function; we also value each unemployed individual by the potential marginal contribution the person would make if the market hired the individual.",
"We then use the aggregate individual value to distribute the net production to the unemployment welfare and the employment benefits.",
"Using real-time balanced-budget rule as a constraint and policy stability as an objective, we derive a scientific formula which describes a fair, debt-free, and asymptotic risk-free tax rate for any given unemployment rate and national spending level.",
"The tax rate minimizes the asymptotic mean, variance, semi-variance, and mean absolute deviation of the underlying posterior unemployment rate.",
"The allocation rule stimulates employment and boosts productivity.",
"Under some symmetry assumptions, we even find that an unemployed person should enjoy equivalent employment benefits, and the tax rate goes with this welfare equality.",
"The tool employed is the cooperative game theory in which we assume many players.",
"The players are randomly bipartitioned, and the payoff varies with the partition.",
"One could apply the fair distribution rule and valuation approach to other profit-sharing or cost-sharing situations with these characteristics.",
"This framework is open to alternative identification strategies and other forms of equal opportunity axiom."
],
[
"INTRODUCTION",
"The problem with which we are concerned relates to the following typical situation: consider an economy which could not achieve full employment of its labor force, and therefore some people are employed, and others are not.",
"As the employed receives wages and employment benefits (e.g., pension, health insurance, social security, education allowances, paid vacation), should the unemployed receive some unemployment welfare?",
"If YES, how much is fair?",
"In a specific jurisdiction system, the term “unemployment welfare\" here may also mean “unemployment benefits,\" “unemployment insurance,\" or even “unemployment compensation.\"",
"In an advanced economy, the answer to the first question is likely YES.",
"This paper answers the second question by justifying a fair share of unemployment welfare for the unemployed and deriving a fair tax rate for the employed.",
"Fair unemployment welfare and a fair tax rate are among the most fundamental topics of our society.",
"The fair-division problem arises in various real-world settings.",
"For a simple motivating example, let us consider a $k$ -out-of-$n$ redundant system in engineering which has $n$ identical components, any $k$ of which being in good condition makes the system work properly.",
"When valuing the importance of each component (either working or standby), one may intuitively claim that these components should be equally valued.",
"A very similar situation occurs in a simple majority voting where not all the voters would support the proposal to vote; thus, the proposal could be passed or failed.",
"Nevertheless, voters are supposed to have the same voting power no matter what they support and for whom they vote.",
"For another example, in the health insurance industry, not all of the policyholders are ill and use the insurance to cover their medical expenses.",
"The question is how to fairly share the total medical cost among both the ill policyholders and the non-ill ones.",
"In a labor market, we have a similar, but more complicated situation: on the one hand, the market could not hire all of its labor force even though everyone in the market would like to be employed; on the other hand, the participants in the market have heterogeneous performance in the production.",
"There are four common features in these examples: a coalition of players with cooperative nature, a random bipartition of the players, a payoff associated with the partition, and an objective to share the payoff with all the players.",
"This paper derives a solution for situations with these features.",
"In the $k$ -out-of-$n$ redundant system and the simple majority voting, we expect equality of outcome.",
"We face a few challenges to deal with when fairly distributing the welfare and benefits, both generated by the employed.",
"First of all, fairness may be an abstract but vague concept.",
"We believe that “fairness\" is bound with the equality of employment opportunity, not with the equality of outcome, nor with the equality of productivity.",
"Furthermore, we also believe that everyone in the labor market could contribute in some way; but the opportunities are limited.",
"Thus, unemployment is not a fault of the unemployed, nor a flaw of the labor market, but a self-adjustment mechanism toward the efficiency of the market.",
"Secondly, we attempt to apply a taxation policy to the labor market which operates in an ever-changing economy and with ever-changing productivity.",
"For it to be useful, the tax and division rule should be not only fair to all people but also be able to cope with the uncertainty and sustainability in generating and distributing the net production.",
"Ideally, it should balance the account of value generated and that of value paid in each employment contingency.",
"Thirdly, in a perfect world, a fair tax rate should depend only on observed data, to avoid any excessively political bargaining and costly strategic voting.",
"One major issue, however, is the non-observability of the heterogeneous-agent production function in all employment scenarios at all the time.",
"Another data issue is the desynchronization between the unemployment rate and the tax rate; the former is high-frequency data while the latter has a lower frequency.",
"Often in a yearly time-frame, policymakers determine the tax rate after observing most monthly unemployment rates.",
"Vast literature (e.g., Kornhauser 1995; Fleurbaey and Maniquet 2006) from various aspects has studied the fairness in taxation and unemployment payments.",
"In particular, Shapley (1953) proposes an influential axiom of fairness to develop a fair-division method, called the Shapley value, which is widely used in distributing employment compensation and welfare (see, for example, Moulin 2004; Devicienti 2010; Giorgi and Guandalini 2018; Krawczyk and Platkowski 2018).",
"Beneath the pillars of the Shapley value and the Shapley axiom, however, are two underlying assumptions: players' unanimous participation in the production, and distributor's complete information about the production function.",
"Recently, Hu (2002, 2006, 2018) relax the unanimity assumption and generalize the Shapley value, using some non-informative probability distributions for the dichotomy or bipartition of the players.",
"In particular, Hu (2018) proposes using a Beta-Binomial distribution to address the equality of opportunity.",
"This current research also capitalizes on the Beta-Binomial distributions.",
"Furthermore, we do not assume the complete information about the production function.",
"We only need its value at one observation which does occur.",
"The advantage of our approach is twofold.",
"On the one hand, we provide a game-theoretical micro-foundation for a fair-division solution to distribute the unemployment welfare and employment benefits.",
"One can apply the solution concept to many similar situations without substantive alternations, and one may also extend the framework using other identification schemes, rather than the tax policy stability or unemployment rate minimization, detailed in Section .",
"On the other hand, the fair tax rate we provide is simple enough to be used in practice.",
"It relies only on the unemployment rate and a reserved portion of production, which is not for personal use.",
"The total unemployment welfare depends only on the tax rate and the observed production.",
"We attempt to immunize our solution from any unnecessary randomness, hypotheticals, ambiguity, and latency.",
"These include, but are not limited to, the competitive and cooperative features of the labor market, endogenous employment search behavior, non-linear schedule of tax rates, the exact sizes of the labor market and time-varying unemployment population.",
"With this simplicity in hand, a certain level of abstraction is necessary and any application of the fair solution should accommodate to the concrete reality.",
"We organize the remainder of the paper as follows.",
"Section applies the framework of dichotomous valuation (or simply, “D-value\") in Hu (2002, 2006, 2018) to value each person in the labor market, assuming equal employment opportunity.",
"The two sides of D-value depend on two unknown parameters and are aggregated separately for the unemployed and the employed labor.",
"Next, Section formulates a set of fair divisions of the net production, using the aggregate components of D-value.",
"In this section, we base the set of fair tax rates on two accounting identities for a balanced budget.",
"In Section , by maximizing the stability of the tax rate or minimizing the expected posterior unemployment rate, we identify a specific relation between the fair tax rate and the unemployment rate.",
"The particular solution is robust under a few other criteria.",
"Section lists three other applications (namely, voting power, health insurance, and highway toll) of the distribution rule, and Section suggests several ways to extend this framework.",
"The account is self-contained, and the proofs are in the Appendix."
],
[
"DICHOTOMOUS VALUATION",
"Before our formal discussions, let us introduce a few notations.",
"For a general economy, we assume that its labor force consists of the employed labor and the unemployed labor, ignoring any part-time labor.",
"Let $\\mathbb {N} = \\lbrace 1, 2,\\cdots ,n\\rbrace $ denote the set of people in the labor force, indexed as $1,2,..., n$ , and let $\\mathbf {S} \\subseteq \\mathbb {N}$ denote the random subset of the employed labor in $\\mathbb {N}$ .",
"For any subset $T$ of $\\mathbb {N}$ , let $|T|$ denote its cardinality; for notational simplicity, we often use $n$ for $|\\mathbb {N}|$ , $t$ for $|T|$ , and $s$ for $|\\mathbf {S}|$ .",
"Let us also write the employment rate as $\\omega =\\frac{s}{n}$ , which is one minus the unemployment rate.",
"Besides, we employ the vinculum (overbar) in naming a singleton set; for example, “$\\overline{i}$ \" is for the singleton set $\\lbrace i \\rbrace $ .",
"Also, we use “$\\setminus $ \" for set subtraction, “$\\cup $ \" for set union, and $\\beta (\\cdot , \\cdot )$ for a two-parameter Beta function.",
"The Appendix defines $\\Delta $ , $\\Delta _1, \\cdots , \\Delta _9$ as shorthand notations."
],
[
"Equality of Employment Opportunity",
"Equal employment opportunity is widely acknowledged and is the starting point or axiom for us to study fairness and welfare.",
"In the United States, for example, equal employment opportunity has been enacted to prohibit federal contractors from discriminating against employees by race, sex, creed, religion, color, or national origin since President Lyndon Johnson signed Executive Order 11246 in 1965.",
"In the literatureBesides the justification from equality of opportunity, unemployment welfare could also come from other considerations (e.g., Sandmo 1998, Tzannatos and Roddis 1998, Vodopivec 2004).",
"These include, but are not limited to, social protection, insurance of income flow, poverty prevention, the efficiency of the labor market, and political considerations.",
"In this paper, we focus solely on equal employment opportunity., there are abundant qualitative descriptions, informal or formal, about the equal opportunity (e.g., Friedman and Friedman 1990; Roemer 1998; Rawls 1999).",
"In this section, we introduce a quantitative and probabilistic version of equal opportunity whereby the employment opportunity is assumed equitable for all persons in the labor force.",
"We assume three layers of uncertainty for the random subset $\\mathbf {S}$ while maintaining the equality of employment opportunity.",
"In the first layer, the employment size $|\\mathbf {S}|$ follows a binomial distribution with parameters $(n, p)$ .",
"When independence is assumed, $p$ is the probability of any given person being employed.",
"In the second layer, the unknown parameter $p$ has a prior Beta distribution with hyperparameters $(\\theta , \\rho )$ , where $\\theta >0$ and $\\rho >0$ are to be determined by estimation or specification.",
"Thus, the joint probability density of $p\\in (0,1)$ and $|\\mathbf {S}|=s$ is $ \\frac{p^{\\theta -1}(1-p)^{\\rho -1}}{\\beta (\\theta , \\rho )}\\left(\\begin{array}{c}n \\\\s\\end{array}\\right)p^s (1-p)^{n-s}=\\frac{n!}{s!(n-s)!",
"}\\frac{p^{\\theta +s-1}(1-p)^{\\rho +n-s-1}}{\\beta (\\theta , \\rho )}.$ Eq.",
"(REF ) implies the following marginal probability density for $|\\mathbf {S}|=s$ : $ \\begin{array}{rcl}\\mathbb {P}(|\\mathbf {S}| = s)& = &\\mathop {{\\int _0^1}}\\frac{n!}{s!(n-s)!",
"}\\frac{p^{\\theta +s-1}(1-p)^{\\rho +n-s-1}}{\\beta (\\theta , \\rho )} \\mathrm {d} p \\\\&=&\\frac{n!}{s!(n-s)!",
"}\\frac{\\beta (\\theta +s, \\rho +n-s)}{\\beta (\\theta , \\rho )}, \\hspace{28.45274pt} s=0,1,\\cdots ,n.\\end{array}$ In the third layer, for any given employment size $s$ , all subsets of size $s$ have the same probability of being $\\mathbf {S}$ .",
"As there are $\\frac{n!}{s!(n-s)!",
"}$ subsets of size $s$ in $\\mathbb {N}$ , the probability for the employment scenario $\\mathbf {S}=T$ is $ \\mathbb {P}(\\mathbf {S}=T)=\\left\\lbrace \\begin{array}{ll}\\frac{\\mathbb {P}(|\\mathbf {S}| = s)}{\\frac{n!}{s!(n-s)!}}",
"= \\frac{\\beta (\\theta +s,\\rho +n-s)}{\\beta (\\theta ,\\rho )},\\quad &\\mathrm {if} \\ t=s;\\\\\\end{array}0, & \\mathrm {otherwise}.\\right.",
"\\\\$ Clearly, the equality of employment opportunity is implied in the assumed triple-layered uncertainty.",
"Furthermore, equal opportunity is also assumed for all coalitions of the same size.",
"By Eq.",
"(REF ) and Eq.",
"(REF ), the posterior density function of $p$ given $|\\mathbf {S}|=s$ is $\\frac{\\frac{n!}{s!(n-s)!",
"}\\frac{p^{\\theta +s-1}(1-p)^{\\rho +n-s-1}}{\\beta (\\theta , \\rho )}}{\\frac{n!}{s!(n-s)!",
"}\\frac{\\beta (\\theta +s, \\rho +n-s)}{\\beta (\\theta , \\rho )}}=\\frac{p^{\\theta +s-1}(1-p)^{\\rho +n-s-1}}{\\beta (\\theta +s, \\rho +n-s)}.$ Thus, the posterior employment rate follows a Beta distribution with parameters $(\\theta +s, \\rho +n-s)$ .",
"In the following, let us use $\\tilde{p}_{n,\\omega }$ to denote the posterior employment rate given the observance of $|\\mathbf {S}|=n \\omega $ .",
"In contrast, $p$ is the unobservable prior employment rate, and $\\omega $ is the observable employment rate."
],
[
"Value of the Employed and the Unemployed",
"For any $T\\subseteq \\mathbb {N}$ , we use a heterogeneous-agent production function $v(T)$ to measure the aggregate productivity when $\\mathbf {S}=T$ .",
"We assume the net-profit production function $v(T)$ excludes the labor cost which compensates the time and efforts devoted by the employed labor in producing $v(T)$ .",
"To isolate the added value by the labor force alone, we also assume that $v(T)$ excludes the cost of consumed physical and financial resources.",
"Both the labor cost and the resource cost are exempt from taxation in a firm.",
"Thus, without loss of generality, we may assume that $v(\\emptyset )=0$ for the empty set $\\emptyset $ .",
"However, $v(T)$ does not necessarily increase with $T$ nor with its size $|T|$ .",
"To retain its labor or to minimize its labor turnover, a firm would share part of its net profit with its employees.",
"To keep things simple, we use the term “employment benefits\" to denote the employees' profit-sharing part in $v(T)$ , in contrast to the term “unemployment welfare.\"",
"Let us formally introduce two components of the D-value.",
"For any $i \\in \\mathbb {N}$ , to analyze its marginal effect on the value generating process, we consider two jointly exhaustive and mutually exclusive events: Event 1: $i \\in {\\mathbf {S}}$ .",
"Then, $i$ 's marginal effect is $v({\\mathbf {S}}) - v( {\\mathbf {S}} \\setminus \\overline{i})$ , called marginal gain, in that he or she contributes $v({\\mathbf {S}}) - v( {\\mathbf {S}} \\setminus \\overline{i})$ to the production, due to his or her existence in ${\\mathbf {S}}$ .",
"The expected marginal gain is $ \\gamma _i[v] \\stackrel{\\mbox{def}}{=} \\mathbb {E} \\left[v(\\mathbf {S}) - v(\\mathbf {S} \\setminus \\overline{i}) \\right].$ In the above, “def\" is for definition and “$\\mathbb {E}$ \" for expectation under the probability distribution specified by Eq.",
"(REF ).",
"Event 2: $i \\notin {\\mathbf {S}}$ .",
"Then, $i$ 's marginal effect is $v({\\mathbf {S}}\\cup \\overline{i}) - v({\\mathbf {S}})$ in that ${\\mathbf {S}}$ faces a marginal loss $v({\\mathbf {S}}\\cup \\overline{i}) - v({\\mathbf {S}})$ , due to $i$ 's absence from the employed labor force ${\\mathbf {S}}$ .",
"In other words, the person would increase the production by $v({\\mathbf {S}}\\cup \\overline{i} ) - v({\\mathbf {S}})$ if the market included him or her in $\\mathbf {S}$ .",
"The expected marginal loss is then $\\lambda _i[v] \\stackrel{\\mbox{def}}{=}\\mathbb {E} \\left[v({\\mathbf {S}}\\cup \\overline{i}) - v({\\mathbf {S}}) \\right].$ We let $\\gamma _i[v]$ be the employment benefits $i$ receives when he or she is employed, and let $\\lambda _i[v]$ be the unemployment welfare $i$ receives when he or she is unemployed.",
"Note that, even if $i$ always remains employed, both $\\mathbf {S}$ and $\\mathbf {S}\\setminus \\overline{i}$ change daily, if not hourly; thus $i$ 's marginal gain is not a constant.",
"Similarly, even if $i$ remains unemployed for a while, $\\mathbf {S}$ , $\\mathbf {S}\\cup \\overline{i}$ , and $i$ 's marginal loss are not constant.",
"To account for this uncertainty, we have already taken expectations in Eq.",
"(REF ) and Eq.",
"(REF ) when defining $\\gamma _i[v]$ and $\\lambda _i[v]$ .",
"A few key points worth mentioning to help us understand the profit-sharing strategy.",
"First, in addition to receiving employment benefits, the employed labor also receives the reimbursement for labor cost, which compensates its human capital usage in generating $v(\\mathbf {S})$ .",
"Human capital also accumulates in prior employment and pre-employment education.",
"Labor, physical, and financial costs are not part of the generated value $v(\\mathbf {S})$ .",
"The unemployed labor, however, only receives unemployment welfare.",
"Secondly, if $i \\in \\mathbf {S}$ , then $v(\\mathbf {S} \\setminus \\overline{i})$ is not observable while we observe $v( \\mathbf {S})$ .",
"Similarly, when $j \\notin \\mathbf {S}$ , we cannot observe both $v(\\mathbf {S} \\cup \\overline{j})$ and $v( \\mathbf {S})$ simultaneously.",
"Thus, we need to transform the aggregate marginals into observable forms, such as those in Theorem REF .",
"Thirdly, the aggregate employment benefits $\\sum \\limits _{i\\in \\mathbf {S}} \\left[v(\\mathbf {S})-v(\\mathbf {S}\\setminus \\overline{i}) \\right]$ is not necessarily equal to $v(\\mathbf {S})$ , the value collectively generated by $\\mathbf {S}$ .",
"Thus, we distribute some of the surpluses $v(\\mathbf {S}) - \\sum \\limits _{i\\in \\mathbf {S}}\\left[v(\\mathbf {S})-v(\\mathbf {S}\\setminus \\overline{i}) \\right]$ to the unemployed labor $\\mathbb {N}\\setminus \\mathbf {S}$ .",
"The distribution is not through personal giving but government taxation and unemployment payment systems.",
"This distribution channel also appeals to us for the aggregate benefits and aggregate welfare at the national level, as stated in Theorem REF .",
"The aggregate components of the D-value are $\\begin{array}{rcl}\\end{array}\\sum \\limits _{i\\in \\mathbb {N}} \\gamma _i [v]&=&\\mathbb {E} \\left[ \\sum \\limits _{i\\in \\mathbf {S}} \\left( v(\\mathbf {S}) - v(\\mathbf {S}\\setminus \\overline{i}) \\right) \\right] \\\\$ = n(+n,)(, )v(N) + TN: T= N t(+-1)-n+n-t-1 (+t, +n-t)(,)v(T), iN i [v] = E [ i NS (v(Si) - v(S) ) ] = TN:T= t(+-1)-n(-1)+t-1 (+t, +n-t)(,) v(T) - n(,+n)(,)v()."
],
[
"ACCOUNTING IDENTITIES FOR A BALANCED BUDGET",
"By Eq.",
"(REF ), the expected production is $ \\mathbb {E} \\left[ v(\\mathbf {S}) \\right]= \\sum \\limits _{T\\subseteq \\mathbb {N}} \\frac{\\beta (\\theta +t, \\rho +n-t)}{\\beta (\\theta ,\\rho )} v(T).$ Out of the $2^n$ employment scenarios for $\\mathbf {S}$ , we observe only one.",
"Let us consider this particular scenario at $\\mathbf {S}=T$ , which occurs with probability $\\frac{\\beta (\\theta +t, \\rho +n-t)}{\\beta (\\theta ,\\rho )}$ and which generates the value $v(T)$ .",
"After the production, we face a challenge of fairly dividing $v(T)$ between $T$ and $\\mathbb {N}\\setminus T$ , whom both have an entitlement to $v(T)$ .",
"Our division rule should fully respect the entitlement claim: each employed person receives his or her expected marginal gain, and each unemployed person receives his or her expected marginal loss.",
"As a contrast, the Shapley value distributes $v(\\mathbb {N})$ to all players in $\\mathbb {N}$ ; Hu(2018) formulates solutions to divide $\\mathbb {E} \\left[ v(\\mathbf {S}) \\right]$ and $v(\\mathbb {N})-\\mathbb {E} \\left[ v(\\mathbf {S}) \\right]$ .",
"Besides $\\gamma _i [v]$ and $\\lambda _i [v]$ , we should reserve a portion of $v(T)$ for the common good of the economy and society."
],
[
"A Real-Time Balanced Budget Rule",
"As noted above, we purposely divide the net production $v(T)$ into three components.",
"The first one is for employment benefits.",
"We compare the coefficients of $v(T)$ in Eq.",
"(REF ) and that of $\\mathbb {E} \\left[ \\sum \\limits _{i\\in \\mathbf {S}} \\left(v(\\mathbf {S}) - v(\\mathbf {S}\\setminus \\overline{i}) \\right) \\right]$ in Eq.",
"(REF ), ignoring the probability density for the scenario; the employed labor $T$ should retain $\\frac{t (\\theta +\\rho -1)-n\\theta }{\\rho +n-t-1} v(T)$ as their employment benefits.",
"The rest, $\\left[1 - \\frac{t(\\theta +\\rho -1)-n\\theta }{\\rho +n-t-1} \\right] v(T)$ , pays to the government as “tax\"In practice, not all unemployment welfare comes from the taxation system.",
"In the United States, for example, it is funded by a compulsory governmental insurance system, which manages the collection and payment accounts.",
"However, the contribution to and distribution from the insurance system are de facto of a type of payroll tax.",
"In Australia, unemployment welfare, as part of social security benefits, is funded through the taxation system..",
"Besides, we assume both $\\lambda _i[v]$ and $\\gamma _i[v]$ are tax-exempt in order to avoid any double taxation.",
"Thus, we define the tax rate $\\tau $ as $\\tau \\stackrel{\\mathrm {def}}{=} 1 - \\frac{s(\\theta +\\rho -1)-n\\theta }{\\rho +n-s-1},\\qquad s=1,\\cdots , n-1.$ For the time being, the rate relies on the labor market size $n$ and the employment rate $\\omega = \\frac{|\\mathbf {S}|}{n}$ , but not on the production $v(\\mathbf {S})$ .",
"In this definition, we exclude two extreme but unlikely cases at $\\omega =0$ and $\\omega =1$ .In reality, the two cases have zero probability to occur, even though the probability model (REF ) gives a very slight chance.",
"Secondly, we compare the coefficients of $v(T)$ in Eq.",
"(REF ) and that of $\\mathbb {E} \\left[ \\sum \\limits _{i \\in \\mathbb {N}\\setminus \\mathbf {S}} \\left(v(\\mathbf {S}\\cup \\overline{i}) - v(\\mathbf {S}) \\right) \\right]$ in Eq.",
"(REF ); the unemployed labor $\\mathbb {N}\\setminus T$ should claim $\\frac{t(\\theta +\\rho -1) - n(\\theta -1)}{\\theta +t-1} v(T)$ from the tax revenue $\\tau v(T)$ as its unemployment welfare.",
"Thirdly, we assume that a reserved proportion, $\\delta v(T)$ , is not individually and not directly distributed to the labor force $\\mathbb {N}$ .",
"As a consequence, the tax rate $\\tau $ includes both the reserve ratio $\\delta $ and the proportion to the unemployment welfare, i.e., $ \\tau \\equiv \\delta + \\frac{s(\\theta +\\rho -1)-n(\\theta -1)}{\\theta +s-1}.$ The reserve $\\delta v(T)$ is meant to serve the general public's interests and to have broad appeal, rather than the individual needs.",
"More specifically, $\\delta v(T)$ includes, but is not limited to, the payments to the population who are not in the labor force, to the corporate equity earnings not used for employment benefits, to the public administration and national defense, to the public welfare, to the tax deficit from the past, to the future development, and so on.",
"By admitting the corporate equity earnings to the reserve $\\delta v(T)$ , we have deliberately ignored firm-specific re-distribution processes of the net corporate earnings, and have purposely avoided the associated corporate income taxation.",
"In practice, government expenditureincluding that in all levels of government but excluding all unemployment welfare payments.",
"and corporate earnings are two major components of the reserve $\\delta v(T)$ .",
"When the government has other sources of tax revenue, a pro rata share of its expenditure will come from $\\delta v(T)$ .",
"Besides, the government could implement a countercyclical fiscal policy by adjusting its spending level in $\\delta v(T)$ such that it counteracts the ratio of the corporate earnings to the production.",
"Indeed, by Eq.",
"(REF ), $\\tau $ automatically reacts positively to the change of the corporate earning ratio, other things remaining the same.",
"Put in another way, the real-time balanced budget rule implied in Eq.",
"(REF ) and Eq.",
"(REF ) forbids any borrowing between different labor market scenarios or forbids any inter-temporal borrowing.",
"Thus, this sustainable taxation policy meets the needs of the present market scenario without compromising the ability of future market scenarios to meet their own needs.",
"In practice, however, it is challenging, if not impossible, to enforce or enact the balanced budget rule at the labor market scenario level or on a real-time basis.",
"In the United States, for example, the employment rate $\\omega $ changes daily and is recorded monthly by the Bureau of Labor Statistics; as a policy variable, the tax rate $\\tau $ changes yearly.",
"Striving for the balanced budget rule to the highest degree, one could minimize the variance of the employment rate $\\omega $ within a yearly time frame.",
"In a perfect situation, the employment rate closely follows a degenerate probability distribution and remains almost constant within the year.",
"We discuss the variance minimization in Section .",
"In contrast to the exhaustive distribution of the net production in the national accounts, a household could still maximize its utility through inter-temporal borrowing, saving, lending, and consuming."
],
[
"Sets of Feasible Solutions",
"Although the terms “tax rate\" and “tax rule\" have been used loosely and interchangeably about $\\tau $ , we must distinguish between them to have an accurate discussion.",
"As a tax rule or tax policy, $\\tau $ is a function of $(n, \\omega , \\delta )$ , subject to the equality of employment opportunity and the balance of budget as specified in Eq.",
"(REF ) and Eq.",
"(REF ).",
"In contrast, as a tax rate, $\\tau $ is merely the value of the function at a specific $(n,\\omega , \\delta )$ .",
"For a given triple of $(n, \\omega , \\delta )$ , there are three indeterminates $(\\theta , \\rho , \\tau )$ in the system of two equations Eq.",
"(REF ) and Eq.",
"(REF ).",
"Let $\\Omega _{n,\\omega ,\\delta }$ denote the set of all feasible combinations of $(\\theta , \\rho , \\tau )$ which satisfy both Eq.",
"(REF ) and Eq.",
"(REF ): $\\Omega _{n,\\omega ,\\delta }\\stackrel{\\mathrm {def}}{=} \\ \\left\\lbrace (\\theta ,\\rho , \\tau ) \\left|\\begin{array}{l}\\tau = 1 - \\frac{n \\omega (\\theta +\\rho -1)-n\\theta }{\\rho +n- n \\omega -1}, \\\\\\tau = \\delta + \\frac{n \\omega (\\theta +\\rho -1)-n(\\theta -1)}{\\theta + n \\omega -1}, \\\\0 \\le \\tau \\le 1,\\ \\theta >0,\\ \\rho >0.\\end{array}\\right.\\right\\rbrace .$ From now on, a “fair tax rate\" could mean a solution of $\\tau $ in $\\Omega _{n,\\omega ,\\delta }$ ; or it could be a limit of tax rates which satisfy Eq.",
"(REF ) and Eq.",
"(REF ).",
"In general, for a reasonable $\\delta $ , a finite $n$ , and a $\\omega \\in (0,1)$ , there are infinitely many solutions in $\\Omega _{n,\\omega ,\\delta }$ .",
"In this case, we need one more restriction to solve $(\\theta ,\\rho , \\tau )$ uniquely.",
"To do so, one could capitalize on the statistical relation between $\\omega $ and $(\\theta , \\rho )$ : the prior mean and mode of $\\omega $ are $\\frac{\\theta }{\\theta +\\rho }$ and $\\frac{\\theta -1}{\\theta +\\rho -2}$ , respectively (e.g., Johnson et al.",
"1995, Chapter 21).",
"Alongside this direction, for example, we could set $\\frac{\\theta }{\\theta +\\rho }$ (or $\\frac{\\theta -1}{\\theta +\\rho -2}$ ) to be the historical average (or mode, respectively) of $\\omega $ in the previous year.",
"Alternatively, we could set $\\frac{\\theta }{\\theta +\\rho }$ or $\\frac{\\theta -1}{\\theta +\\rho -2}$ to be a target employment rate or the natural employment rate.",
"However, this type of identification schemes requires additional input (e.g., historical average, target employment rate, or natural employment rate).",
"Furthermore, to derive a unique solution from $\\Omega _{n,\\omega ,\\delta }$ or its boundary, we should base our deviation on observed data only.",
"One concern in the input $(n, \\omega , \\delta )$ is the size of the labor market $n$ .",
"Even though the size $n$ is not random in our model of equal employment opportunity, it is likely a time-varying latent variable in that there is no clear cut between entry to and exit from the labor market; and many depressed unemployed persons may not actively seek new positions.",
"In practice, $n$ changes daily whereas $\\tau $ most likely changes yearly.",
"No matter how it changes and how much latent it is, however, we are confident that $n$ is a large number in the general economy.",
"Thus, in contrast to Eq.",
"(REF ) and Eq.",
"(REF ), we seek a fair tax rule which is valid for all large $n$ 's but is not specific to a particular $n$ .",
"As a result, the tax rule we are targeting should not involve $n$ , and we may write it as $\\tau (\\omega , \\delta ): (0,1) \\times (0,1) \\rightarrow (0,1)$ .",
"To visualize the solution set $\\Omega _{n,\\omega ,\\delta }$ , let us write $(\\theta ,\\rho )$ in terms of $(n, \\delta , \\tau , \\omega )$ using Lemma REF in the Appendix.",
"Figure REF (a-b) plots the feasible solution sets for $n=10000$ , $\\delta =.1$ , and any $\\omega \\in (0,1)$ .",
"Note that there is a straight line which dramatically raises both $\\theta $ and $\\rho $ .",
"From Figure REF (c-d), we observe that both $\\theta $ and $\\rho $ drop sharply on the other side of the straight line.",
"As a matter of fact, any point on the other side of the straight line does not represent a fair solution, owing to the positivity requirement of $\\theta $ and $\\rho $ .",
"Actually, the straight line has a tax rule $\\tau (\\omega , \\delta ) = 1-\\omega + \\delta \\omega $ .",
"This linear relation between $\\tau $ and $\\omega $ can be seen from Lemma REF : on the one hand, when $\\Delta \\equiv \\omega +\\tau -\\delta \\omega -1 > 0$ , both $\\theta =\\frac{n^2 \\omega \\Delta _1 + n\\Delta _2 +\\Delta _3 }{n\\Delta + \\Delta _3}$ and $\\rho = \\frac{n^2 (1-\\omega ) \\Delta _1 + n\\Delta _4 + \\Delta _3}{n\\Delta + \\Delta _3}$ increase to $+\\infty $ as $n\\rightarrow \\infty $ , due to the positivity of $\\Delta _1 \\equiv \\delta \\omega -\\omega -\\tau +2> 0$ ; on the other hand, when $\\Delta <0$ , both $\\theta $ and $\\rho $ decrease to $-\\infty $ as $n\\rightarrow \\infty $ .",
"Thus, the singular line is expressed as $\\Delta =0$ , or $\\tau (\\omega , \\delta ) = 1-\\omega +\\delta \\omega $ .",
"Furthermore, for any fair tax rule on the side of $\\Delta >0$ , the posterior employment rate $\\tilde{p}_{n,\\omega }$ has a degenerated limit distribution as $n\\rightarrow \\infty $ .",
"This is claimed in Theorem REF .",
"The next section answers a related question: which tax rule makes the distribution convergence fastest.",
"The answer happens to be the solution on the singular line.",
"Figure: Solutions of Eq.",
"() and Eq.",
"() for n=10000,δ=.1,ω∈(0,1)n=10000,\\ \\delta =.1,\\ \\omega \\in (0,1).For any fair tax rule $\\tau (\\omega , \\delta ) \\in (1-\\omega +\\delta \\omega ,1)$ , as $n\\rightarrow \\infty $ , $\\tilde{p}_{n,\\omega }$ converges in distribution to the degenerate probability distribution with mass at $\\omega $ ."
],
[
"AN ASYMPTOTIC RISK-FREE TAX RATE",
"In this section, we derive the limit fair tax rule $\\tau (\\omega , \\delta ) =1-\\omega +\\delta \\omega $ from a few different angles.",
"At a minimum, a good tax rule should not discourage employment incentives and productivity, such as detailed in Theorem REF and REF .",
"On the other end, we expect a good rule to be robust and optimal under multiple criteria.",
"We study five criteria in Theorem REF , REF , REF , REF , and REF , any of which uniquely identifies the solution.",
"They either minimize the employment market risk or maximize the employment expectation, under a given market capacity and a budget balance constraint.",
"First, we should heavily exploit the observed market behavior.",
"While an efficient labor market stimulates productivity $v(\\mathbf {S})$ , a higher employment rate does not imply higher productivity and vice versa — the production function $v(\\mathbf {S})$ does not necessarily increase with the employment size $|\\mathbf {S}|$ .",
"Acemoglu and Shimer (2000) find that a moderate level of unemployment could boost productivity by improving the quality of jobs.",
"Indeed, it fosters peer pressure in producing $v(\\mathbf {S})$ , allows workers to move on from declining firms, and enables rising companies and the economy as a whole to optimally respond to external shocks.",
"Therefore, our tax rule would not merely target a higher expectation of employment rate, but grounds its assumption in the observed market behavior, and lets the market itself respond with a higher employment rate, to the maximum extent permitted by the budget rule and the market capacity.",
"Secondly, from a statistical viewpoint, our tax rule relies on a realization of $p$ , not on the uncertainty of the realized value.",
"A natural way to factor out this uncertainty is to study the posterior rate $\\tilde{p}_{n,\\omega }$ , where both $s$ and $\\omega $ are no longer random.",
"As Theorem REF indicates, $\\omega $ is informative and indicative in revealing the central tendency of the posterior labor market, delineated by $\\tilde{p}_{n,\\omega }$ .",
"Asymptotic dispersion of $\\tilde{p}_{n,\\omega }$ and the market's response to the tax rule $\\tau (\\omega , \\delta )$ are among other essential ingredients in the complete profile of $\\tilde{p}_{n,\\omega }$ .",
"Thus, we set optimal criteria to minimize dispersion measures or maximize the expected market response."
],
[
"Zero Asymptotic Posterior Variance",
"Tax rate stability creates the right environment for a balance of payments, reduces the uncertainty of the labor market, and creates confidence in technology and human capital investment.",
"As a function of $\\omega $ , $\\tau $ migrates the risk from the unemployment rate to the tax rate.",
"With the absence of other exogenous shocks, indeed, stability in the tax rate is equivalent to that in the unemployment rate.",
"The same argument is also valid when the exogenous shocks are orthogonal to that in $\\omega $ .",
"When we use the variance of $\\tilde{p}_{n,\\omega }$ to measure its instability, Theorem REF states that the tax rule $\\tau (\\omega , \\delta ) = 1 - \\omega + \\delta \\omega $ minimizes the asymptotic variance of $\\tilde{p}_{n,\\omega }$ .",
"Furthermore, it is also the limit of variance-minimizing tax rules for finite labor markets.",
"We add the restriction “$\\theta , \\rho \\ge \\frac{1}{n}$ \" in the finite labor markets to ensure the positivity of the hyperparameters $(\\theta ,\\rho )$ .",
"As $n\\rightarrow \\infty $ , the tax rule $\\tau (\\omega , \\delta ) = 1 - \\omega + \\delta \\omega $ minimizes the limit variance of $\\sqrt{n} \\tilde{p}_{n,\\omega }$ , i.e., $\\mathop {\\mathrm {argmin}}\\limits _{\\tau } \\lim \\limits _{n\\rightarrow \\infty } \\mathrm {VAR} \\left(\\sqrt{n} \\tilde{p}_{n,\\omega } \\right)= 1 -\\omega +\\delta \\omega .$ Besides, $\\lim \\limits _{n\\rightarrow \\infty } \\mathop {\\mathrm {argmin}}\\limits _{\\tau } \\mathrm {VAR} \\left(\\sqrt{n} \\tilde{p}_{n,\\omega } \\bigm \\vert \\theta , \\rho \\ge \\frac{1}{n} \\right)= 1 -\\omega +\\delta \\omega .$ Furthermore, both the minimum limit variance and the limit minimum variance are zero.",
"We provide a few comments to help clarify any potential misunderstandings about the theorem.",
"First, a stable $\\sqrt{n} \\tilde{p}_{n,\\omega }$ is achieved at $n=\\infty $ where the limit variance is zero.",
"However, $\\tilde{p}_{n,\\omega }$ is still exposed to exogenous shocks, such as those studied in Pissarides (1992) and Blanchard (2000).",
"Secondly, it is worth emphasizing that $1-\\omega +\\delta \\omega $ is the limit tax rule as $n\\rightarrow \\infty $ .",
"For a large but finite $n$ , a small positive number could be added to $1-\\omega +\\delta \\omega $ to ensure the positivitySee details in the proof of Theorem REF .",
"of $\\theta $ and $\\rho $ .",
"That small positive number, however, is negligible; thus, we can practically use the rule $\\tau (\\omega , \\delta ) = 1-\\omega +\\delta \\omega $ without any addition.",
"Moreover, a higher-order approximation $\\tau =1-\\omega +\\delta \\omega +\\frac{\\omega (1-\\omega )(1-\\delta )^2}{n}$ could be an excellent alternativeSee the proof of Theorem REF for details.",
"to $\\tau =1-\\omega +\\delta \\omega $ .",
"Thirdly, with a zero or near zero variance in the unemployment rate, labor mobility means one layoff and one new hire should almost coincide to ensure the total employment size $|\\mathbf {S}|$ remains nearly constant.",
"It also means that the sizes of employment and labor market change proportionally so that their ratio remains unchanged.",
"Lastly, though the posterior distribution is skewed, the tax rule minimizes both the overall risk and the one-sided risk of $\\sqrt{n} \\tilde{p}_{n,\\omega }$ , as stated in Theorem REF .",
"In particular, a policymaker's concern is on the downside risk only.",
"As $n\\rightarrow \\infty $ , the tax rule $\\tau (\\omega , \\delta ) = 1 - \\omega + \\delta \\omega $ minimizes both the limit lower semivariance and the limit upper semivariance of $\\sqrt{n} \\tilde{p}_{n,\\omega }$ ."
],
[
"Consistency and Robustness",
"The above fair tax rule also captures several striking features of the labor market.",
"In the first place, it is the policymaker's best response to the market to stimulate employment within the framework of Eq.",
"(REF ) and Eq.",
"(REF ).",
"In the second place, we can also derive it by minimizing statistical dispersion measures other than the posterior variance or semivariance.",
"Meanwhile, it helps mitigate income inequality.",
"The rule $\\tau (\\omega , \\delta ) = 1 - \\omega + \\delta \\omega $ is an effective taxation strategy to maximally boost the employment size without breaking the opportunity equality and the budget balance.",
"For an economic policymaker, one primary concern is the forward-looking employment profile $\\tilde{p}_{n,\\omega }$ .",
"By Theorem REF , the mean of $\\tilde{p}_{n,\\omega }$ converges to $\\omega $ as $n\\rightarrow \\infty $ for any fair rule $\\tau (\\omega , \\delta ) \\in (1-\\omega +\\delta \\omega ,1)$ .",
"When $n$ is finitely large, they respond adversely to an increasing $\\tau $ (cf Theorem REF ).",
"Consequently, to maximize the posterior mean, we should minimize the tax rate $\\tau $ while still maintaining the conditions $\\tau (\\omega , \\delta ) \\in (1-\\omega +\\delta \\omega ,1)$ , $\\theta >0$ and $\\rho >0$ .",
"Thus, the limit of fair tax rates which maximize the mean of $\\tilde{p}_{n,\\omega }$ would be $1-\\omega +\\delta \\omega $ .",
"As a remark, the condition $\\omega >.5$ in Theorem REF is satisfied in the general economy.",
"For any $\\omega \\in (.5,1)$ and a finitely large $n$ , the mean of $\\tilde{p}_{n,\\omega }$ reacts negatively to an increasing tax rate $\\tau \\in (1-\\omega +\\delta \\omega ,1)$ .",
"As a consequence, $\\lim \\limits _{n\\rightarrow \\infty } \\mathop {\\mathrm {argmax}}\\limits _{\\tau } \\mathrm {MEAN}\\left(\\tilde{p}_{n,\\omega } \\bigm \\vert \\theta , \\rho \\ge \\frac{1}{n} \\right)=1 -\\omega +\\delta \\omega .$ Furthermore, the tax rule also minimizes the mean absolute deviation (thereafter, MAD) from the mean as $n \\rightarrow \\infty $ .",
"For a Beta distribution, especially with large parameters, MAD is a more robust measure of statistical dispersion than the variance.",
"The MAD around the mean for the posterior $\\tilde{p}_{n,\\omega }$ is (e.g., Gupta and Nadarajah 2004, page 37): $\\mathbb {E} \\left[ \\left|\\tilde{p}_{n,\\omega } - \\mathbb {E} (\\tilde{p}_{n,\\omega }) \\right| \\right]=\\frac{2(\\theta +s)^{\\theta +s} (\\rho +n-s)^{\\rho +n-s}}{\\beta (\\theta +s, \\rho +n-s)(\\theta +\\rho +n)^{\\theta +\\rho +n}}.$ In the next theorem, we identify the same tax rule by minimizing the asymptotic MAD.",
"As $n\\rightarrow \\infty $ , the tax rule $\\tau (\\omega , \\delta ) = 1 - \\omega + \\delta \\omega $ minimizes the MAD of $n \\tilde{p}_{n,\\omega }$ around the mean, i.e.",
"$\\lim \\limits _{n\\rightarrow \\infty } \\mathop {\\mathrm {argmin}}\\limits _{\\tau }\\mathbb {E} \\left[n \\left|\\tilde{p}_{n,\\omega } - \\mathbb {E} (\\tilde{p}_{n,\\omega }) \\right| \\biggm \\vert \\theta , \\rho \\ge \\frac{1}{n} \\right]= 1 -\\omega +\\delta \\omega .$ Besides, $\\mathop {\\mathrm {argmin}}\\limits _{\\tau } \\lim \\limits _{n\\rightarrow \\infty }\\mathbb {E} \\left[n \\left|\\tilde{p}_{n,\\omega } - \\mathbb {E} (\\tilde{p}_{n,\\omega }) \\right| \\biggm \\vert \\theta , \\rho \\ge \\frac{1}{n} \\right]= 1 -\\omega +\\delta \\omega .$"
],
[
"Equality of Outcome with Symmetric Production",
"To see the relation between productivity and the tax rate, we introduce a partial ordering in the labor market.",
"For any $i, j\\in \\mathbb {N}$ with $i\\ne j$ , we say $i$ uniformly outperforms $j$ in $v$ if $v(T\\cup \\overline{i}) - v(T) \\ge v(T\\cup \\overline{j}) - v(T)$ for any $T\\subseteq \\mathbb {N} \\setminus \\overline{i} \\setminus \\overline{j}$ ; and $v(T)-v(T\\setminus \\overline{i}) \\ge v(T)-v(T \\setminus \\overline{j})$ for any $T\\subseteq \\mathbb {N}$ with $i,j \\in T$ .",
"With these two inequality conditions, $i$ has higher marginal productivity than $j$ in all comparable employment contingencies $-$ either both employed or both unemployed.",
"As productivity is highly valued in Eq.",
"(REF ) and Eq.",
"(REF ), $j$ should receive fewer employment benefits and less unemployment welfare than $i$ does.",
"This is formally claimed in Theorem REF .",
"Besides, the theorem does not require the Beta-Binomial distribution, as long as $i$ and $j$ alone have the same chance of being employed; other players in the labor market may have unequal employment opportunities.",
"The theorem is valid for all fair tax rules, including the special one $\\tau (\\omega , \\delta )=1-\\omega +\\delta \\omega $ .",
"If $i\\in \\mathbb {N}$ uniformly outperforms $j\\in \\mathbb {N}$ in $v$ and they have equal employment opportunity, then $\\gamma _i[v] \\ge \\gamma _j[v]$ and $\\lambda _i[v] \\ge \\lambda _j[v]$ .",
"We say $i, j\\in \\mathbb {N}$ are symmetric in the production function $v$ if they uniformly outperform each other.",
"By Theorem REF , $\\lambda _i[v] =\\lambda _j[v]$ and $\\gamma _i[v] = \\gamma _j[v]$ if $i$ and $j$ are symmetric and they have equal employment opportunity.",
"In other words, they should receive the same amount of unemployment welfare if both are unemployed; they should also receive the same amount of employment benefits if both are employed.",
"Without any further analysis of or any prior knowledge about the production function, symmetry among the unemployed (or the employed) could be a reasonable a priori assumption to distribute the unemployment welfare (or employment benefits, respectively).",
"For example, the Cobb-Douglas production function is symmetric among all employed persons in the labor market.",
"Besides, if we assume symmetry among the employed labor and also assume symmetry among the unemployed labor, then the tax rule $\\tau (\\omega , \\delta ) = 1 - \\omega + \\delta \\omega $ eliminates the income inequality, when the income is either employment benefits or unemployment welfare.",
"Theorem REF affirms this equality of outcome, but an employed individual and an unemployed one may not be symmetric in $v$ .",
"Furthermore, the theorem does not restrict the size of $n$ and the specific probability distribution for the equal employment opportunity.",
"In the $k$ -out-of-$n$ redundant system mentioned in Section , accordingly, the $n$ components (either working or standby) are equally important if they have equal quality.",
"Section offers a few alternative probability distributions for the equal employment opportunity.",
"Assume equal employment opportunity in $\\mathbb {N}$ .",
"If all employed individuals are symmetric in $v$ and all unemployed individuals are also symmetric in $v$ , then $\\tau (\\omega , \\delta ) = 1 - \\omega + \\delta \\omega $ if and only if an unemployed person's unemployment welfare equals an employee's employment benefits."
],
[
"Labor Cost",
"We could use the symmetry to calibrate the labor cost, which is excluded from $v({\\mathbf {S}})$ .",
"For example, let the labor cost for $i \\in {\\mathbf {S}}$ be the minimum wage requirement from all $j \\in \\mathbb {N}\\setminus {\\mathbf {S}}$ , who either are symmetric with $i$ or uniformly outperforms $i$ in $v$ .",
"That is, some $j$ from $\\mathbb {N}\\setminus {\\mathbf {S}}$ can do $i$ 's work without compromising the production $v$ .",
"The minimum wage is called reservation wage, below which $j$ is unwilling to work.",
"At this minimum market replacement cost, ${\\mathbf {S}}$ can switch $i$ with someone else from $\\mathbb {N} \\setminus {\\mathbf {S}}$ without sacrificing its net profit.",
"In fact, $j$ does not need to be symmetric with or outperform $i$ as long as the probability of $v({\\mathbf {S}}\\cup \\overline{j}\\setminus \\overline{i}) \\ge v({\\mathbf {S}}\\cup \\overline{i}\\setminus \\overline{j})$ is significantly large and the person has a willingness to accept (e.g., Horowitz and McConnell 2003).",
"The probability of such $\\mathbf {S}$ could be difficult for job interviewers to approximate and estimate.",
"At the other end, in order to avoid creating an undue disincentive to work, $j$ 's unemployment welfare must be less than the reservation wage plus employment benefit.",
"Moreover, to ensure every one of $\\mathbb {N}$ is in the labor market, the welfare is necessary to bind to the incentive-compatibility constraint.",
"This incentive requirement places a lower bound for the labor cost."
],
[
"OTHER APPLICATIONS",
"In an abstract sense, the above account is a fair-division solution for the following game-theoretic setting: there are a large number of players; the players are randomly divided into two groups; the payoff comes from one group.",
"A wide range of applications falls into this type of games.",
"In this section, we analyze three applications other than labor markets to show how to use the formula derived in the last three sections."
],
[
"Voting Power",
"In a voting game (e.g., Shapley 1962), $v: 2^\\mathbb {N} \\rightarrow \\lbrace 0, 1\\rbrace $ is a monotonically increasing set function.",
"Let $\\mathbf {S}$ denote the random subset of voters who vote for the proposal.",
"The proposal passes when $v(\\mathbf {S})=1$ ; otherwise, it blocks when $v(\\mathbf {S})=0$ .",
"However, $v$ should not mean “production\" or alike.",
"No matter the outcome, Hu (2006) describes $\\gamma _i [v]$ as $i$ 's probability of turning a blocked result to a passed one, and $\\lambda _i [v]$ the probability of turning a passed result to being blocked.",
"Thus, the sum of $\\lambda _i [v]$ and $\\gamma _i [v]$ quantifies $i$ 's power in the game.",
"The ratio $\\delta $ plays a role in some circumstances.",
"Let us consider, for example, $10\\%$ of the voters approve just a proposal before a referendum voting on the proposal, and assume the number of other voters' support ballots follows a Beta-Binomial distribution.",
"Many voting games are symmetric.",
"In these cases, the equality of outcome becomes an egalitarian allocation of power."
],
[
"Health Insurance",
"Health insurance has two types of policyholders: some are ill and use the insurance to cover their medical expenses; others are healthy and do not use the insurance.",
"Let $\\mathbf {S}$ denote the random set of ill policyholders, $v(\\mathbf {S})$ be the total medical expenses with copays deducted, and $\\tilde{\\delta }v(\\mathbf {S})$ be the surcharge paid to the insurance company.",
"Let $\\delta = - \\tilde{\\delta }$ .",
"Then the total expenses except for the copays, $(1 - \\delta ) v(\\mathbf {S})$ , are billed to all insurance policyholders.",
"If $\\tau =1-\\omega +\\delta \\omega $ and $v$ is symmetric among the two types of policyholders, respectively, then by the equality of outcome, the cost to buy the insurance policy would be $\\frac{(1-\\delta ) \\mathbb {E} \\left[ v(\\mathbf {S}) \\right]}{n}$ per policyholder.",
"We take expectation on $\\frac{(1-\\delta ) v(\\mathbf {S})}{n}$ because the policyholders pay it upfront.",
"On the contrary, the unemployment welfare and employment benefits payments come after the production.",
"In this example, patients pay the predetermined copays.",
"In the labor market studied before, the labor costs are exempt from the distribution of the net production; they, however, have an indirect effect on $\\tau $ through the corporate earning ratio and $\\delta $ .",
"In the next example, we use the equality of outcome to derive the same type of payments as copays and labor cost."
],
[
"Highway Toll",
"The I-66 highway inside the Capital Beltway of the Washington metropolitan area has enforced a dynamic toll rule during rush hours: a carpool driver pays no toll, but a solo driver pays a dynamic toll fee, say $\\xi (n, \\omega )$ .",
"Here $n$ is the number of cars in the segment of the highway, and $\\omega $ is the percentage of solo drivers in the traffic.",
"Let $g(n)$ be the average carpool driver's cost in the traffic when the traffic volume is $n$ cars.",
"It is likely a nonlinear increasing function of $n$ .",
"An excellent choice of $g(n)$ is the expected driving time in hours multiplied by the average hourly pay rate, plus expenses on gas and vehicle depreciation.",
"Also, let $\\mathbf {S}$ denote the random set of solo drivers.",
"Then, $v(\\mathbf {S}) = ng(n) - n(1-\\omega ) g\\left(n(1-\\omega ) \\right) - n\\omega \\xi (n, \\omega ) $ is the total cost of over-traffic generated by the solo drivers, with toll fees deducted.",
"The production function $v$ is symmetric among all solo drivers and also symmetric among all carpool drivers.",
"By the equality of outcome, each driver shares the same cost $\\frac{v(\\mathbf {S})}{n}$ .",
"As a carpool driver pays no toll, his or her shared cost should exactly offset the extra cost caused by the solo drivers, which is $g(n) - g \\left(n(1-\\omega ) \\right)$ .",
"Finally, equation $\\frac{v(\\mathbf {S})}{n}=g(n) - g\\left(n(1-\\omega ) \\right)$ implies $\\xi (n, \\omega ) = g\\left(n(1-\\omega ) \\right).$ In this example, an administration surcharge $\\delta $ may apply; the carpooled passengers are free-riders, but their costs are exempt from $v(\\mathbf {S})$ ."
],
[
"CONCLUSIONS",
"In this paper, a fair-division solution is proposed to allocate the unemployment welfare in an economy, where the heterogeneous-agent production function is almost unknown.",
"We interpret “fairness\" as equal employment opportunity and model it by a Beta-Binomial probability distribution.",
"Our “sustainability\" is meant to be free of debt and free of surplus in the taxation budget.",
"To justify the value of the unemployed labor, we capitalize on the D-value concept in Hu (2018).",
"The D-value specifies how much of the net production to be retained with the employed labor, and what portion to be distributed to the unemployed.",
"Finally, we postulate that the labor market is static and identify a sustainable tax policy by minimizing the asymptotic variance of the posterior employment rate.",
"The policy can also be uniquely determined by minimizing the asymptotic posterior mean of the unemployment rate, or minimizing the downside risk of the posterior employment rate, or minimizing the posterior mean absolute deviation.",
"Surprisingly, the tax rule is not only simple enough for practical use but also motivates the unemployed to seek employment and the employed to improve productivity.",
"One could extend this framework in several ways.",
"One way is to re-specify the probability distribution of equal employment opportunity, for example, by any of the following re-specifications.",
"First, we can replace the two-parameter Beta distribution with a four-parameter one or a Beta rectangular distribution.",
"Secondly, we can let $\\theta $ and $\\rho $ be some functions of other unknown parameters.",
"Thirdly, we could substitute the Beta-Binomial distribution with a Dirichlet-Multinomial distribution or a Beta-Geometric distribution.",
"Fourthly, we could randomize $p$ without involving $(\\theta , \\rho )$ , by generating two independent three-parameter Gamma random variables $X$ and $Y$ .",
"Then, the ratio $X/(X+Y)$ is a Beta random variable.",
"In any of the four cases, however, we need additional identification restrictions to figure out a $\\tau (\\omega , \\delta )$ precisely.",
"From other angles, we could apply other identification schemes or other objective functions to find a unique fair tax rule.",
"From a statistical viewpoint, one could try the maximum likelihood estimation using the twelve months' data prior to determining the policy tax rate, or minimize the ex-ante risk of $\\omega $ , or apply the statistical methods mentioned in Section REF .",
"From an economic viewpoint, one could minimize the Gini coefficient of the Beta distribution of $\\tilde{p}_{n,\\omega }$ .",
"From a strategic game-theoretical viewpoint, one could seek a bargaining solution from the feasible solution set $\\Omega _{n,\\omega ,\\delta }$ , which may be particularly useful when $n$ is small.",
"Finally, a policymaker could treat the reserve ratio $\\delta $ as endogenous, for example, letting it be an increasing function of $\\omega $ .",
"He or she could also place a heavier weight on the marginal gain than on the marginal loss, in order to stimulate employment.",
"The simple static model, however, ignores several important aspects of a real labor market.",
"First, it does not capture the dynamic features of the income inequality, nor its rational response to the tax rule.",
"Secondly, while preventing the fungibility of borrowing funds from the future reduces the risk that a government administration piles up its national debt, it impairs that administration's ability (especially, monetary policy) to intervene in the economy.",
"The government, however, can still moderately stimulate the economy during a recession by adjusting the reserve ratio $\\delta $ .",
"Thirdly, the postulation of labor market efficiency does neglect the recent development of the incomplete-market theory (e.g., Magill and Quinzii 1996).",
"Fourthly, a multi-criteria objective function may be a viable alternative to the minimum-variance one, especially when there is a high unemployment rate $1-\\omega $ or a large $\\delta $ .",
"Lastly, a single tax rule $\\tau (\\omega , \\delta )$ could have overly simplified the complexity of the taxation system, which is also affected by other determinants.",
"These are just a few challenges our framework introduces, which require further development.",
"In summary, the fair and sustainable tax policy studied here has a solid theoretical underpinning, together with simplicity in practical use, consistency with productivity and employment incentives, and robustness to similar objectives.",
"When applying this framework to a real fair-division problem, one should also consider the benefits of alternative probability distributions for equal opportunity, alternative objective functions, alternative restrictions, and dynamic thinking."
],
[
"APPENDIX",
"Our focus in this paper is to study the relationship between the fair tax rate $\\tau $ and the employment rate $\\omega $ when the labor market size $n$ is large.",
"To analyze the limit behavior of $\\tau $ and $\\omega $ for a large $n$ , we only need the relevant asymptotic approximations.",
"We say two functions $f(n)=O \\left(g(n) \\right)$ if $\\limsup \\limits _{n\\rightarrow \\infty } \\left|\\frac{f(n)}{g(n)} \\right| < \\infty $ ; and we say $f(n) \\approx g(n)$ if $\\lim \\limits _{n\\rightarrow \\infty } \\frac{f(n)}{g(n)} =1$ .",
"For simplicity, let us denote the following shorthands: $\\begin{array}{lcl}\\Delta &\\equiv &\\omega + \\tau - \\delta \\omega - 1, \\\\\\end{array}\\Delta _1&\\equiv &\\delta \\omega - \\omega -\\tau + 2\\equiv 1 - \\Delta , \\\\$ 2- 2 - 2 + 2 + -1, 3(1- )(-) = --+ 2, 4- + + - 2+ 2 - 2 + - 2 + 3-1, 5- + - 2 + - 2 + 4-2 2+4, 6-+ +-1 2 + 3, 7-- + 2+ - 1 4 + (1-) 3, 8-- + + 3-2 3+5, 9 -2-+2+ 4-3 + 8.",
"$$ The following lemma, to be used in other proofs, re-writes Eq.",
"(REF ) and Eq.",
"(REF ).",
"In terms of $(n,\\delta ,\\tau , \\omega )$ , we can solve $(\\theta ,\\rho )$ from Eq.",
"(REF ) and Eq.",
"(REF ) as $ \\left\\lbrace \\begin{array}{l}\\theta =\\frac{n^2 \\omega \\Delta _1 + n\\Delta _2 +\\Delta _3 }{n\\Delta + \\Delta _3}, \\\\\\end{array}\\rho =\\frac{n^2 (1-\\omega ) \\Delta _1 + n\\Delta _4 + \\Delta _3}{n\\Delta + \\Delta _3}.\\right.\\qquad \\mathrm {(A.1)}$ In this proof, we use the following relation about Beta functions: $\\left\\lbrace \\begin{array}{lcll}\\beta (x-1, y+1) &=& \\frac{y}{x-1} \\beta (x, y), &x>1,\\ y>0; \\\\\\beta (x+1, y-1) &=& \\frac{x}{y-1} \\beta (x,y), &x>0,\\ y>1.",
"\\\\\\end{array}\\right.$ First, the expected aggregate marginal gain and loss are: $\\begin{array}{rcl}\\mathbb {E}\\left[ \\sum \\limits _{i\\in \\mathbf {S}} \\left( v(\\mathbf {S})-v(\\mathbf {S}\\setminus \\overline{i}) \\right) \\right]\\end{array}&=&\\sum \\limits _{T\\subseteq \\mathbb {N}} \\mathbb {P}(\\mathbf {S}=T) \\sum \\limits _{i\\in T} \\left[ v(T)-v(T\\setminus \\overline{i}) \\right] \\\\$ = i N TN: i T P(S=T) [ v(T)-v(Ti) ] = i N i[v], E[ i NS ( v(Si)-v(S) ) ] = TN P(S=T) i NT [ v(Ti)-v(T) ] = i N T N: i T P(S=T) [ v(Ti)-v(T) ] = i N i[v].",
"$$ Next, by Eq.",
"(REF )$-$ (REF ), we re-write the expected marginal gain and loss as: $ \\begin{array}{rcl}\\end{array}\\gamma _i [v]&=& \\sum \\limits _{T\\subseteq \\mathbb {N}: T\\ni i} \\frac{\\beta (\\theta +t,\\rho +n-t)}{\\beta (\\theta ,\\rho )}[v(T)-v(T\\setminus \\overline{i})] \\\\$ Z=Ti= TN: Ti (+t,+n-t)(,) v(T) - ZN i (+|Z|+1, +n-|Z|-1)(,) v(Z), i [v] =ZNi (+|Z|,+n-|Z|)(,)[v(Zi) -v(Z)] T=Zi=TN: Ti (+t-1,+n-t+1)(,) v(T) - ZNi (+|Z|,+n-|Z|)(,) v(Z).",
"By Eq.",
"(REF ), the aggregate value of the employed labor $\\sum \\limits _{i \\in \\mathbb {N}} \\gamma _i [v]$ is $\\begin{array}{rcl}&&\\sum \\limits _{i\\in \\mathbb {N}} \\sum \\limits _{T\\subseteq \\mathbb {N}: T\\ni i} \\frac{\\beta (\\theta +t, \\rho +n-t)}{\\beta (\\theta ,\\rho )}v(T)- \\sum \\limits _{i\\in \\mathbb {N}} \\sum \\limits _{Z\\subseteq \\mathbb {N} \\setminus \\overline{i}} \\frac{\\beta (\\theta +|Z|+1, \\rho +n-|Z|-1)}{\\beta (\\theta ,\\rho )} v(Z) \\\\\\end{array}&\\stackrel{T=Z}{=}& \\sum \\limits _{T\\subseteq \\mathbb {N}: T\\ne \\emptyset }\\sum \\limits _{i\\in T} \\frac{\\beta (\\theta +t, \\rho +n-t)}{\\beta (\\theta ,\\rho )}v(T)- \\sum \\limits _{T\\subseteq \\mathbb {N}: T\\ne \\mathbb {N}} \\sum \\limits _{i \\in \\mathbb {N} \\setminus T} \\frac{\\beta (\\theta +t+1, \\rho +n-t-1)}{\\beta (\\theta ,\\rho )} v(T) \\\\$ = TN: T= t (+t, +n-t)(,)v(T) - TN: T= N (n-t) (+t+1, +n-t-1)(,) v(T) = n(+n,)(, )v(N) - n(+1,+n-1)(, )v() + TN: T= , T= N t (+t, +n-t) - (n-t) (+t+1, +n-t-1)(,)v(T) = n(+n,)(,)v(N) - n+n-1(,+n)(, )v() + TN: T= N, T= [ t - (n-t) (+t)+n-t-1 ] (+t, +n-t)(,)v(T) = n(+n,)(, )v(N) + TN: T= N t(+-1)-n+n-t-1 (+t, +n-t)(,)v(T).",
"$$ Also by Eq.",
"(REF ), the aggregate value of the unemployed labor $\\sum \\limits _{i\\in \\mathbb {N}} \\lambda _i [v]$ is $\\begin{array}{rcl}&&\\sum \\limits _{i\\in \\mathbb {N}} \\sum \\limits _{T\\subseteq \\mathbb {N}: T\\ni i} \\frac{\\beta (\\theta +t-1, \\rho +n-t+1)}{\\beta (\\theta ,\\rho )}v(T)- \\sum \\limits _{i\\in \\mathbb {N}} \\sum \\limits _{Z \\subseteq \\mathbb {N} \\setminus \\overline{i}} \\frac{\\beta (\\theta +|Z|, \\rho +n-|Z|)}{\\beta (\\theta ,\\rho )} v(Z) \\\\\\end{array}&\\stackrel{T=Z}{=}& \\sum \\limits _{T\\subseteq \\mathbb {N}: T\\ne \\emptyset }\\sum \\limits _{i\\in T} \\frac{\\beta (\\theta +t-1, \\rho +n-t+1)}{\\beta (\\theta ,\\rho )}v(T)- \\sum \\limits _{T\\subseteq \\mathbb {N}: T \\ne \\mathbb {N}}\\sum \\limits _{i\\in \\mathbb {N} \\setminus T} \\frac{\\beta (\\theta +t, \\rho +n-t)}{\\beta (\\theta ,\\rho )} v(T) \\\\$ = TN: T= t(+t-1, +n-t+1)(,)v(T) - TN: T =N(n-t)(+t, +n-t)(,) v(T) = n(+n-1, +1)(,)v(N) - n(, +n)(,) v() +TN: T= , T =N t(+t-1, +n-t+1)-(n-t)(+t, +n-t)(,)v(T) = n+n-1(+n, )(,)v(N) - n(, +n)(,) v() + TN: T= , T= N [ t(+n-t)+t-1 - (n-t) ] (+t, +n-t)(,) v(T) = TN: T= t(+-1)-n(-1)+t-1 (+t, +n-t)(,) v(T) - n(, +n)(,) v().",
"$$ We can re-write Eq.",
"(REF ) and Eq.",
"(REF ) as a linear system of unknowns $(\\theta , \\rho )$ : $\\left\\lbrace \\begin{array}{rcl}(1-\\tau )(\\rho +n-s-1)&=&s(\\theta +\\rho -1)-n\\theta , \\\\\\end{array}(\\tau -\\delta )(\\theta +s-1)&=&s(\\theta +\\rho -1)-n(\\theta -1).\\right.$ As a consequence, the symbolic solution of $(\\theta , \\rho )$ is unique.",
"Let us assume Eq.",
"(REF ) and verify that it satisfies both Eq.",
"(REF ) and Eq.",
"(REF ) by the following identities, some of which are used in other proofs: $\\begin{array}{rcl}\\end{array}\\theta +s&=&\\frac{n^2 \\omega \\Delta _1 + n\\Delta _2 +\\Delta _3 }{n\\Delta + \\Delta _3}+ \\frac{n\\omega (n\\Delta + \\Delta _3)}{n\\Delta + \\Delta _3}=\\frac{n^2 \\omega + n\\Delta _6 + \\Delta _3}{n\\Delta + \\Delta _3}, \\\\$ +s-1 = n2 + n6 + 3n+ 3 - n+ 3n+ 3 = n2 + n (6-) n+ 3 = n2 + n (- 1) n+ 3, += n2 1 + n2 +3 n+ 3 + n2 (1-) 1 + n4 + 3n+ 3 = n2 1 + n 5 + 23n+ 3, +- 1 = n2 1 + n 5 + 23n+ 3 - n+ 3n+ 3 = n2 1 + n ( - 2 - 2 + 3- 1) + 3 n+ 3, ++ n = n2 1 + n 5 + 23n+ 3 + n(n+ 3)n+ 3 = n2 + n 8 + 23n+ 3, ++ n + 1 = n2 + n 8 + 23n+ 3 +n+ 3n+ 3 = n2 + n 9 + 33 n+ 3, ++ n - 1 = n2 + n 8 + 23n+ 3 - n+ 3n+ 3 = n2 + n (8 - ) + 3 n+ 3, ++ n - 2 = n2 + n 8 + 23n+ 3 - 2(n+ 3)n+ 3 = n2 + n (8-2) n+ 3, +n-s = n2 (1-) 1 + n4 + 3n+ 3 + n(1-)(n+ 3)n+ 3 = n2 (1-) + n 7 + 3n+ 3, +n-s - 1 = n2 (1-) + n 7 + 3n+ 3 -n+ 3n+ 3 = n2 (1-)+ n (1-)(-) n+ 3.",
"$Thus,$ rcl s(+- 1)-n= n[n2 1 + n ( - 2 - 2 + 3- 1) + 3] n+ 3 - n(n2 1 + n2 +3)n+ 3 = n2[(- 2 - 2 + 3- 1) -2] + n(-1)3 n+ 3 = n2(1-)(1-) + n(-1)3 n+ 3, s(+- 1)-n(-1) = [s(+- 1)-n] + n = n2(1-)(1-) + n(-1)3 n+ 3 + n(n+ 3)n+ 3 = n2 (-) + n 3 n+ 3.",
"$Therefore,$ rclcl s(+- 1)-n+n-s - 1 = n2(1-)(1-)+ n(-1)3n2 (1-)+ n (1-)(-) = 1-, s(+- 1)-n(-1)+s-1 = n2 (-) + n 3n2 + n (- 1) = - , $which are equivalent to Eq.",
"(\\ref {eq:tax_rate_def}) and Eq.",
"(\\ref {eq:tax_rate_c}), respectively.$ For any integer $z \\ge 0$ , by the proof of Lemma REF , $\\begin{array}{rcl}\\frac{\\theta +s+z}{\\theta +\\rho +n+z}\\end{array}&=&\\frac{\\frac{n^2\\omega +n\\Delta _6+\\Delta _3}{n\\Delta +\\Delta _3}+\\frac{z(n\\Delta +\\Delta _3)}{n\\Delta +\\Delta _3}}{\\frac{n^2+n\\Delta _8+2\\Delta _3}{n\\Delta +\\Delta _3}+\\frac{z(n\\Delta +\\Delta _3)}{n\\Delta +\\Delta _3}} \\\\$ = n2+n(6+z)+(1+z)3 n2+n(8+z)+(2+z)3 , as n. $$ As a function of $\\eta $ , the characteristic function of $\\tilde{p}_{n,\\omega }$ (e.g., Johnson et al.",
"1995, Chapter 21) is $\\mathbb {E} \\left[e^{i \\eta \\tilde{p}_{n,\\omega }} \\right]= 1 + \\sum \\limits _{k=1}^\\infty \\frac{(i \\eta )^k}{k!",
"}\\prod \\limits _{z=0}^{k-1} \\frac{\\theta +s+z}{\\theta +\\rho +n+z}$ where $i$ is the unit imaginary number, i.e., $i^2=-1$ .",
"We let $n\\rightarrow \\infty $ , $\\begin{array}{rcl}\\lim \\limits _{n\\rightarrow \\infty } \\mathbb {E}[e^{i \\eta \\tilde{p}_{n,\\omega }}]&=&1 + \\sum \\limits _{k=1}^\\infty \\frac{(i \\eta )^k}{k!}",
"\\lim \\limits _{n\\rightarrow \\infty }\\prod \\limits _{z=0}^{k-1}\\frac{\\theta +s+z}{\\theta +\\rho +n+z}\\\\\\end{array}&=&1 + \\sum \\limits _{k=1}^\\infty \\frac{(i \\eta \\omega )^k}{k!}",
"\\\\$ = (i ).",
"$Therefore, as $ $, $ pn,$ converges in distribution to the degenerate distribution with mass at $$,which has the characteristic function $ (i )$.$$$ As $\\tilde{p}_{n,\\omega }$ has a Beta distribution with parameters $(\\theta +s, \\rho +n-s)$ (see Section REF ), its variance is $\\frac{(\\theta +s)(\\rho +n-s)}{(\\theta +\\rho +n)^2(\\theta +\\rho +n+1)}$ (e.g., Gupta and Nadarajah 2004, page 35).",
"By the proof of Lemma REF , the variance of $\\sqrt{n} \\tilde{p}_{n,\\omega }$ is $\\begin{array}{rcl}&&\\frac{n \\frac{n^2\\omega +n\\Delta _6+\\Delta _3}{n\\Delta +\\Delta _3}\\frac{n^2(1-\\omega )+n\\Delta _7+\\Delta _3}{n\\Delta +\\Delta _3}}{ \\left(\\frac{n^2+n\\Delta _8+2\\Delta _3}{n\\Delta +\\Delta _3}\\right)^2\\frac{n^2+n\\Delta _9+3\\Delta _3}{n\\Delta +\\Delta _3}} \\\\\\end{array}&=&\\frac{n(n\\Delta +\\Delta _3)(n^2\\omega +n\\Delta _6+\\Delta _3)\\left[n^2(1-\\omega )+n\\Delta _7+\\Delta _3 \\right]}{\\left(n^2+n\\Delta _8+2\\Delta _3 \\right)^2\\left(n^2+n\\Delta _9+3\\Delta _3 \\right)} \\\\$ = (+3n) [ (1+6n+3n2 ) ] [ (1-) (1+7n(1-)+3n2(1-) ) ] (1+ 8n+23n2)2 (1+ 9n+ 33n2 ) = (1-) (+3n) [ 1+ 6n + 7n(1-) - 28n-9n + O(1n2 ) ] = (1-)+ (1-)n [ 3 + ( 6 + 71- - 28 - 9) ] + O(1n2 ) (1-) (+--1), as n .",
"To minimize $\\lim \\limits _{n\\rightarrow \\infty }\\mathrm {VAR}(\\sqrt{n} \\tilde{p}_{n,\\omega })= \\omega (1-\\omega )\\Delta $ while $\\lim \\limits _{n\\rightarrow \\infty } \\mathrm {VAR}(\\sqrt{n} \\tilde{p}_{n,\\omega } )\\ge 0$ , we have to set $\\Delta = 0$ .",
"Thus, $\\mathop {\\mathrm {argmin}}\\limits _{\\tau } \\lim \\limits _{n\\rightarrow \\infty }\\mathrm {VAR} \\left(\\sqrt{n} \\tilde{p}_{n,\\omega } \\right) = 1-\\omega +\\delta \\omega .$ However, when applying $\\tau = 1-\\omega + \\delta \\omega $ to a labor market with finite $n$ , we have $\\Delta = 0$ , $\\Delta _1 = 1$ and $\\Delta _3 = -\\omega (1-\\omega ) (1-\\delta )^2$ .",
"And Lemma REF reduces to $\\left\\lbrace \\begin{array}{rcl}\\theta &=&\\frac{n^2 \\omega + n \\Delta _2 - \\omega (1-\\omega ) (1-\\delta )^2}{-\\omega (1-\\omega ) (1-\\delta )^2}, \\\\\\end{array}\\rho &=&\\frac{n^2 (1-\\omega )+n\\Delta _4-\\omega (1-\\omega ) (1-\\delta )^2}{-\\omega (1-\\omega ) (1-\\delta )^2},\\right.$ which converge to $-\\infty $ as $n\\rightarrow \\infty $ .",
"In theory, therefore, for a large but finite $n$ , we need to choose $\\tau $ to be $1-\\omega +\\delta \\omega $ plus a small positive number to ensure that $\\theta >0$ and $\\rho >0$ .",
"To estimate the small positive number, let us first try the higher-order approximation of $\\tau = 1 - \\omega + \\delta \\omega + \\frac{c}{n},$ for some constant $c > 0$ .",
"Then $ \\begin{array}{lcl}\\Delta &=&\\frac{c}{n}, \\\\\\end{array}\\Delta _3&=&\\left[\\omega (1-\\delta ) - \\frac{c}{n} \\right]\\left[-(1-\\omega ) (1-\\delta ) - \\frac{c}{n} \\right] \\\\$ = -(1-)(1-)2 + O(1n ).",
"By the next to the last step in (REF ), $\\mathrm {VAR} \\left(\\sqrt{n} \\tilde{p}_{n,\\omega } \\Bigm \\vert \\tau = 1 - \\omega + \\delta \\omega + \\frac{c}{n} \\right)=\\frac{\\omega (1-\\omega ) c - \\omega ^2 (1-\\omega )^2 (1-\\delta )^2}{n} + O\\left(\\frac{1}{n^2} \\right).$ To minimize the above variance, we let $c = \\omega (1-\\omega )(1-\\delta )^2$ .",
"Thus, $\\tau = 1 - \\omega +\\delta \\omega + \\frac{\\omega (1-\\omega )(1-\\delta )^2}{n}$ is a higher-order approximation for $1-\\omega +\\delta \\omega $ .",
"Further high-order approximations, if necessary, could be found similarly.",
"When $\\tau = 1 - \\omega +\\delta \\omega + \\frac{c}{n}$ , let us use Eq.",
"(REF ) to calculate $ \\begin{array}{rcl}n\\Delta +\\Delta _3&=&c + \\left[\\omega (1-\\delta ) - \\frac{c}{n} \\right] \\left[-(1-\\omega ) (1-\\delta ) - \\frac{c}{n} \\right] \\\\\\end{array}&=&\\frac{(1-2\\omega ) (1-\\delta ) c}{n} + \\frac{c^2}{n^2}$ which is negative when $\\omega >.5$ and $n\\ge \\left| \\frac{\\omega (1-\\omega )(1-\\delta )}{1-2\\omega }\\right|$ .",
"As the numerators of $\\theta $ and $\\rho $ in Lemma REF are both positive for a large $n$ , $\\theta $ and $\\rho $ are negative when $n$ is large and $\\omega >.5$ .",
"Let us make another try at $\\tau = 1-\\omega +\\delta \\omega +\\frac{2c}{n}$ .",
"Similar to Eq.",
"(REF ), $\\begin{array}{rcl}n\\Delta +\\Delta _3&=&2c + \\left[\\omega (1-\\delta ) - \\frac{2c}{n} \\right] \\left[-(1-\\omega ) (1-\\delta ) - \\frac{2c}{n} \\right] \\\\\\end{array}&=&c + \\frac{2(1-2\\omega ) (1-\\delta ) c}{n} + \\frac{4c^2}{n^2}.$ $In this case, both $ $ and $$ are larger than $ 1n$ when $ n$ is large.Note from the last step of Eq.",
"(\\ref {eq:asym_var}) that $ VAR (n pn, )$ is an increasing function of $$ when $ n$ is large; thus,the small positive number to be added to $ 1-+$ is between $ cn$and $ 2cn$.",
"In practice, however,as the number is too small for a large $ n$,there is no necessity to exactly calculate it and add it to $ 1-+$.$ Let $\\tilde{\\tau }_n = \\mathop {\\mathrm {argmin}}\\limits _{\\tau } \\mathrm {VAR} \\left(\\sqrt{n} \\tilde{p}_{n,\\omega } \\bigm \\vert \\theta , \\rho \\ge \\frac{1}{n} \\right)$ .",
"We add the restriction $\\theta , \\rho \\ge \\frac{1}{n}$ to ensure the existence of $\\tilde{\\tau }_n$ .",
"As $\\mathrm {VAR} \\left(\\sqrt{n} \\tilde{p}_{n,\\omega } \\right)$ is an increasing function of $\\tau $ when $n$ is large, $\\tilde{\\tau }_n$ is unique and less than $1 - \\omega + \\delta \\omega + \\frac{2c}{n}$ when $n$ is large.",
"By Eq.",
"(REF ), $\\mathrm {VAR}(\\sqrt{n} \\tilde{p}_{n,\\omega } \\Bigm \\vert \\tau = \\tilde{\\tau }_n) = \\omega (1-\\omega ) (\\omega + \\tilde{\\tau }_n-\\delta \\omega -1) + O \\left(\\frac{1}{n} \\right).$ Finally, we use the relation $0\\le \\mathrm {VAR}\\left(\\sqrt{n} \\tilde{p}_{n,\\omega } \\Bigm \\vert \\tau = \\tilde{\\tau }_n \\right)\\le \\mathrm {VAR} \\left(\\sqrt{n} \\tilde{p}_{n,\\omega } \\Bigm \\vert \\tau = 1 - \\omega + \\delta \\omega + \\frac{2c}{n} \\right)$ to get $0\\le \\omega (1-\\omega ) (\\omega + \\tilde{\\tau }_n-\\delta \\omega -1) + O \\left(\\frac{1}{n} \\right)\\le O \\left(\\frac{1}{n} \\right).$ Letting $n \\rightarrow \\infty $ in the above inequality, we get $\\omega (1-\\omega ) (\\omega + \\tilde{\\tau }_n-\\delta \\omega -1) = O(\\frac{1}{n})$ , i.e., $\\lim \\limits _{n\\rightarrow \\infty } \\mathop {\\mathrm {argmin}}\\limits _{\\tau } \\mathrm {VAR} \\left(\\sqrt{n} \\tilde{p}_{n,\\omega } \\bigm \\vert \\theta , \\rho \\ge \\frac{1}{n} \\right) = 1 - \\omega + \\delta \\omega $ .",
"$$ In this proof, we constantly apply the identities in the proof of Lemma REF .",
"Let $\\mu _n = \\frac{\\theta +s}{\\theta +\\rho +n}$ be the mean of $\\tilde{p}_{n,\\omega }$ , and let $\\sigma _n^2$ be the variance of $\\tilde{p}_{n,\\omega }$ .",
"The lower semivariance of $\\tilde{p}_{n,\\omega }$ is calculated as $\\sigma ^2_{n-} = \\int _0^{\\mu _n} (x - \\mu _n)^2 \\frac{x^{\\theta +s-1}(1-x)^{\\rho +n-s-1}}{\\beta ( \\theta +s, \\rho +n-s)} \\mathrm {d} x.$ We apply Chebychev's inequality in terms of the lower semivariance (e.g., Berck and Hihn 1982) to get $\\mathbb {P} \\left(\\tilde{p}_{n,\\omega } \\le \\mu _n - a_n \\sigma _{n-} \\right)\\le \\frac{1}{a_n^2}, \\quad \\forall \\ a_n>0.$ By the proof of Theorem REF , $\\tilde{p}_{n,\\omega }$ has the variance $\\sigma _n^2 = \\frac{\\omega (1-\\omega )\\Delta }{n}+O(\\frac{1}{n^2})$ .",
"We let $a_n = \\frac{\\sqrt{\\frac{\\omega (1-\\omega )\\Delta }{n}}}{\\sigma _{n-}}$ , then $\\mathbb {P} \\left(\\tilde{p}_{n,\\omega } \\le \\mu _n - \\sqrt{\\frac{\\omega (1-\\omega )\\Delta }{n}} \\right)\\le \\frac{n\\sigma ^2_{n-}}{\\omega (1-\\omega )\\Delta }.\\qquad \\mathrm {(A.6)}$ Let $\\kappa _n = \\frac{\\theta +s-1}{\\theta +\\rho +n-2}$ and $\\varepsilon _n$ be the mode and median of $\\tilde{p}_{n,\\omega }$ , respectively.",
"The mode $\\kappa _n$ maximizes the density function $\\frac{x^{\\theta +s-1}(1-x)^{\\rho +n-s-1}}{\\beta (\\theta +s, \\rho +n-s)}, 0 < x < 1$ .",
"As the median lies between the mean and mode, we have $\\begin{array}{rcl}|\\mu _n - \\varepsilon _n|&\\le &|\\mu _n - \\kappa _n |=\\left| \\frac{\\theta +s}{\\theta +\\rho +n} - \\frac{\\theta +s-1}{\\theta +\\rho +n-2} \\right| \\\\\\end{array}&=&\\left| \\frac{n+\\rho -\\theta -2s}{(\\theta +\\rho +n)(\\theta +\\rho +n-2)} \\right| \\\\$ | +s(++n)(++n-2) | + | n+-s(++n)(++n-2) | = O(1n).",
"$In the following lower-bound estimation of Eq.",
"(\\ref {eq:lower1se}), we useGamma function, denoted by $ ()$,and its Stirling^{\\prime }s approximation $ (z+1)2z (z(1) )z$.$$\\begin{array}{rcl}&&\\mathbb {P} \\left(\\tilde{p}_{n,\\omega } \\le \\mu _n - \\sqrt{\\frac{\\omega (1-\\omega )\\Delta }{n}} \\right) \\\\\\end{array}&=&\\mathbb {P} \\left(\\tilde{p}_{n,\\omega } \\le \\varepsilon _n \\right)- \\int \\limits _{\\mu _n - \\sqrt{\\frac{\\omega (1-\\omega )\\Delta }{n}}}^{\\varepsilon _n}\\frac{x^{\\theta +s-1}(1-x)^{\\rho +n-s-1}}{\\beta (\\theta +s, \\rho +n-s)} \\mathrm {d} x \\\\$ 12 - | n-(n- (1-)n ) | n+s-1(1-n)+n-s-1(+s,+n-s) = 12 - | (1-)n + O(1n) | (+s-1++n-2)+s-1 (1-+s-1++n-2)+n-s-1(+s) (+n-s)(++n) = 12 - | (1-)n + O(1n) | (+s-1)+s-1(+s) (+n-s-1)+n-s-1(+n-s)(++n-2)++n-2(++n-1) (++n-1) 12 - (1-)n (+s-1)2(+s-1) (+n-s-1)2(+n-s-1) (++n-2)(++n-1) 2(++n-2) = 12 - (1-)n (++n-1) ++n-22(+s-1)(+n-s-1) 12 - (1-)n n n 2n n(1-) = 12 - 12.",
"$$ Finally, we re-write Eq.",
"(REF ) as $\\left(\\frac{1}{2}-\\frac{1}{\\sqrt{2\\pi }} \\right)\\omega (1-\\omega )\\Delta + O \\left(\\frac{1}{\\sqrt{n}} \\right)\\le n \\sigma ^2_{n-}\\le n \\sigma _n^2=\\omega (1-\\omega )\\Delta + O \\left(\\frac{1}{n} \\right).$ Letting $n\\rightarrow \\infty $ , we get $\\left(\\frac{1}{2}-\\frac{1}{\\sqrt{2\\pi }} \\right)\\omega (1-\\omega )\\Delta \\le \\liminf \\limits _{n\\rightarrow \\infty } n\\sigma _{n-}^2\\le \\limsup \\limits _{n\\rightarrow \\infty } n\\sigma _{n-}^2\\le \\omega (1-\\omega )\\Delta .$ Therefore, $\\Delta =0$ minimizes the limit of lower semivariance of $\\sqrt{n} \\tilde{p}_{n,\\omega }$ .",
"We can apply similar arguments to the upper semivariance of $\\sqrt{n} \\tilde{p}_{n,\\omega }$ .",
"$$ Note that $\\Delta _6-\\omega \\Delta _8 = (1-\\tau )(2\\omega -1)+\\omega ^2 (\\delta -1).$ By the proof of Lemma REF , $\\begin{array}{rcl}\\mathbb {E} [\\tilde{p}_{n,\\omega }]&=&\\frac{\\theta +s}{\\theta +\\rho +n}=\\frac{n^2\\omega + n \\Delta _6 + \\Delta _3}{n^2 + n\\Delta _8 + 2\\Delta _3}\\\\\\end{array}&=&\\omega + \\frac{n (\\Delta _6-\\omega \\Delta _8) + (1-2\\omega ) \\Delta _3}{n^2 + n\\Delta _8 + 2\\Delta _3} \\\\$ = + (1-)(2-1)+2 (-1)n + O(1n2).",
"$$ In the above approximation, the mean reacts negatively with an increasing $\\tau $ , when $n$ is finitely large and $\\omega >.5$ .",
"To maximize the mean, we thus minimize $\\tau \\in (1-\\omega +\\delta \\omega )$ such that $\\theta >0$ and $\\rho >0$ .",
"Particularly, using the proof of Theorem REF , we can make it smaller than $1-\\omega +\\delta \\omega + \\frac{2c}{n}$ when $n$ is large enough, i.e.",
"$1-\\omega +\\delta \\omega <\\mathop {\\mathrm {argmax}}\\limits _{\\tau } \\mathrm {MEAN}\\left(\\tilde{p}_{n,\\omega } \\bigm \\vert \\theta ,\\rho \\ge \\frac{1}{n} \\right)\\le 1-\\omega +\\delta \\omega + \\frac{2c}{n}.$ Finally, let $n\\rightarrow \\infty $ in the above inequalities to get $\\lim \\limits _{n\\rightarrow \\infty } \\mathop {\\mathrm {argmax}}\\limits _{\\tau } \\mathrm {MEAN}\\left(\\tilde{p}_{n,\\omega } \\bigm \\vert \\theta ,\\rho \\ge \\frac{1}{n} \\right)=1 -\\omega +\\delta \\omega .$ for any $\\omega \\in (.5,1)$ .",
"$$ When $n$ is large, $\\theta , \\rho \\ge \\frac{1}{n}$ implies $\\tau (\\omega , \\delta ) \\in (1-\\omega +\\delta \\omega , 1)$ .",
"By the proof of Lemma REF , both $\\theta +s\\rightarrow \\infty $ and $\\rho +n-s\\rightarrow \\infty $ as $n\\rightarrow \\infty $ .",
"Applying Stirling's formula, Johnson et al.",
"(1995, page 219) derive the following approximation for the ratio of the variance and the squared MAD around the mean: $\\lim \\limits _{\\theta +s\\rightarrow \\infty , \\rho +n-s \\rightarrow \\infty }\\frac{\\left(\\mathbb {E} \\left[ \\left| \\tilde{p}_{n,\\omega } - \\mathbb {E} (\\tilde{p}_{n,\\omega }) \\right| \\right] \\right)^2}{ \\mathrm {VAR} (\\tilde{p}_{n,\\omega })}= \\frac{2}{\\pi }.$ Thus, minimizing the MAD around the mean is equivalent to minimizing the variance of $\\tilde{p}_{n,\\omega }$ when $n$ is large.",
"By Theorem REF , we have proved Theorem REF .",
"$$ For any $i, j\\in \\mathbb {N}$ and $i\\ne j$ , if $i$ uniformly outperforms $j$ and they have equal employment opportunity, then $\\begin{array}{rcl}\\gamma _j[v]&=&\\sum \\limits _{T\\subseteq \\mathbb {N}: j\\in T}\\mathbb {P} (\\mathbf {S}=T) [v(T) - v(T\\setminus \\overline{j})] \\\\\\end{array}&=&\\sum \\limits _{T\\subseteq \\mathbb {N}: j\\in T, i\\in T}\\mathbb {P} (\\mathbf {S}=T) [v(T) - v(T\\setminus \\overline{j})] \\\\&&+\\sum \\limits _{T\\subseteq \\mathbb {N}: j\\in T, i\\notin T}\\mathbb {P} (\\mathbf {S}=T) [v(T) - v(T\\setminus \\overline{j})] \\\\$ TN: jT, iT P (S=T) [v(T) - v(Ti)] + TN: jT, iT P (S=T) [v(T) - v(Tj)] Z=Tj= TN: jT, iT P (S=T) [v(T) - v(Ti)] + ZN: jZ, iZ P (S=Zj) [v(Z j) - v(Z)] TN: jT, iT P (S=T) [v(T) - v(Ti)] + ZN: jZ, iZ P (S=Zi) [v(Z i) - v(Z)] T =Zi = TN: jT, iT P (S=T) [v(T) - v(Ti)] + TN: jT, iT P (S=T) [v(T) - v(Ti)] = TN: iT P (S=T) [v(T) - v(Ti)] = i[v].",
"$$ Neither the Beta-Binomial distribution nor Eq.",
"(REF ) is required in the proof.",
"Also, the equality of employment opportunity is not required for other players in $\\mathbb {N}$ , except that $\\mathbb {P} (\\mathbf {S}=Z\\cup \\overline{i})=\\mathbb {P} (\\mathbf {S}=Z\\cup \\overline{j})$ for any $Z\\subseteq \\mathbb {N} \\setminus \\overline{i} \\setminus \\overline{j}$ .",
"This identity implies that $i$ and $j$ have equal chance to be hired by $Z$ , when both are unemployed.",
"Similar arguments can be used to prove $\\lambda _j[v] \\le \\lambda _i[v]$ .",
"In this case, we use the identity $\\mathbb {P} (\\mathbf {S}=Z\\setminus \\overline{i})=\\mathbb {P} (\\mathbf {S}=Z\\setminus \\overline{j})$ for any $Z\\subseteq \\mathbb {N}$ such that $i,j \\in Z$ .",
"This identity implies both $i$ and $j$ have equal opportunity to be laid off from $Z$ , when both are employed in $Z$ .",
"$$ We distribute $v(\\mathbf {S})$ to $\\mathbb {N}$ : $(1-\\tau ) v(\\mathbf {S})$ to $\\mathbf {S}$ and $(\\tau -\\delta )v(\\mathbf {S})$ to $\\mathbb {N}\\setminus \\mathbf {S}$ .",
"When the employed individuals are symmetric in $v$ , Theorem REF claims that each employed person receives $\\frac{(1-\\tau ) v(\\mathbf {S})}{|\\mathbf {S}| }$ as his or her employment benefits.",
"Similarly, any unemployed person receives $\\frac{(\\tau -\\delta ) v(\\mathbf {S})}{n - |\\mathbf {S}| }$ as his or her unemployment welfare.",
"Finally, $\\frac{(1-\\tau ) v(\\mathbf {S})}{|\\mathbf {S}| }= \\frac{(\\tau -\\delta ) v(\\mathbf {S})}{n - |\\mathbf {S}| }$ is equivalent to $\\frac{(1-\\tau ) }{\\omega }= \\frac{(\\tau -\\delta )}{1 - \\omega },$ which itself is equivalent to $\\tau = 1 - \\omega +\\delta \\omega .$ $$"
]
] | 1808.08563 |
[
[
"Saliency Detection via Bidirectional Absorbing Markov Chain"
],
[
"Abstract Traditional saliency detection via Markov chain only considers boundaries nodes.",
"However, in addition to boundaries cues, background prior and foreground prior cues play a complementary role to enhance saliency detection.",
"In this paper, we propose an absorbing Markov chain based saliency detection method considering both boundary information and foreground prior cues.",
"The proposed approach combines both boundaries and foreground prior cues through bidirectional Markov chain.",
"Specifically, the image is first segmented into superpixels and four boundaries nodes (duplicated as virtual nodes) are selected.",
"Subsequently, the absorption time upon transition node's random walk to the absorbing state is calculated to obtain foreground possibility.",
"Simultaneously, foreground prior as the virtual absorbing nodes is used to calculate the absorption time and obtain the background possibility.",
"Finally, two obtained results are fused to obtain the combined saliency map using cost function for further optimization at multi-scale.",
"Experimental results demonstrate the outperformance of our proposed model on 4 benchmark datasets as compared to 17 state-of-the-art methods."
],
[
"Introduction",
"Saliency detection aims to effectively highlight the most important pixels in an image.",
"It helps to reduce computing costs and has widely been used in various computer vision applications, such as image segmentation [1], image retrieval [2], object detection [3], [4], object recognition [5], image adaptation [6], and video segmentation [7], [8].",
"Saliency detection could be summarized in three methods: bottom-up methods [9], [10], [11], top-down methods [12], [13] and mixed methods [14], [15], [16].",
"The top-down methods are driven by tasks and could be used in object detection tasks.",
"The authors in [17] proposed a top-down method that jointly learns a conditional random field and a discriminative dictionary.",
"Top-down methods could be applied to address complex and special tasks but they lack versatility.",
"The bottom-up methods are driven by data, such as color, light, texture and other basic features.",
"Itti et al [18] proposed a saliency method by using these basic features.",
"It could be effectively used for real-time systems.",
"The mixed methods are considered both bottom-up and top-down methods.",
"In this paper, we focus on the bottom-up methods, the proposed method is based on the properties of Markov model, there are many works based on Markov model, such as [19], [20].",
"Traditional saliency detection via Markov chain [21] is based on Marov model as well, but it only consider boundaries nodes.",
"However, in addition to boundaries cues, background prior and foreground prior cues play a complementary role to enhance saliency detection.",
"We consider four boundaries information and the foreground prior saliency object, using absorbing Markov chain, namely, both boundary absorbing and foreground prior are considered to get background and foreground possibility.",
"In addition, we further optimize our model by fusing these two possibilities, and exploite multi-scale processing.",
"Fig.1 demonstrates and compares the results of our proposed method with the traditional saliency detection absorbing Markov chain (MC) method [21], where the outperformance of our method is evident.",
"Figure: Comparison of the proposed method with the ground truth and MC method."
],
[
"Related works",
"There are existing many studies on saliency detection in the past decades according to the resent surveys [22], [23], [24].",
"And the proposed model ia based on absorbing Markov chain belonging to bottom-up method, therefore, in this section, we mainly focus on the traditional models and other newly models.",
"The traditional models which belong to bottom-up methods, have the features of unconscious, fast, data driven, low-level feature driven, which means without any prior knowledge, bottom-up saliency detection can achieve the goal of finding the important regions in a image.",
"The earliest researchers Itti and Koch model [18], [25], which compute saliency maps though low-level features, such as texture, orientation, intensity, and color contrast.",
"From then on, multitudinous traditional saliency models have been appeared and acquired outperformances.",
"Some methods based on pixel [26], [27], [28], Some methods based on superpixel [29], [30], and others based on multi-scale [31], [32], [33].",
"Jiang et al [21] propose saliency detection model by random walk via absorbing Markov Chain where absorbing nodes are duplicated from the four boundaries, and compute absorbing time from the transient nodes to absorbing nodes, obtain the final saliency maps.",
"Considering the importance of the transition probability matrix, Zhang et al [34] based on their aforementioned work propose a learnt transition probability matrix to improve the perfermance.",
"There are some other work based on Markov chain.",
"Zhang et al [35] propose an approach to detection salient objects by exporing both patch-level and object-level cues via absorbing Markov chain.",
"Zhang et al [36] present a data-driven salient region detection model based on absorbing Markov chain via multi-feature.",
"Zhu et al [37] integrate boundary connectivity based on geodesic distances into a cost function to get the final optimized saliency map.",
"Li et al [38] propose a saliency optimization scheme by considering both the foreground appearance and background prior.",
"Resent, due to the great developing of deep learning, there are numerous saliency detection models [11], [15], [39], [40] based on deep learning, which obtain outperformance than the traditional methods.",
"However, deep learning based methods need much dates to train, costing much time on computation, which makes the proceeding of saliency detection much complex than before.",
"In this work, based on the traditional method, we introduce bidirectional absorbing Markov chain to this kinds of work to get excellent performance in saliency detection."
],
[
"Fundamentals of absorbing Markov chain",
"In absorbing Markov chain, the transition matrix $P$ is primitive [41], by definition, state $i$ is absorbing when $P(i,i)=1$ , and $P(i,j)=0$ for all $i \\ne j$ .",
"If the Markov chain satisfies the following two conditions, it means there is at least one or more absorbing states in the Markov chain.",
"In every state, it is possible to go to an absorbing state in a finite number of steps(not necessarily in one step), then we call it absorbing Markov chain.",
"In an absorbing Markov chain, if a state is not a absorbing state, it is called transient state.",
"An absorbing chain has $m$ absorbing states and $n$ transient states, the transfer matrix $P$ can be written as: $P\\rightarrow \\left(\\begin{array}{cc}Q&R\\\\0&I\\\\\\end{array}\\right),$ where $Q$ is a n-by-n matrix, giving transient probabilities between any transient states, $R$ is a nonzero n-by-m matrix giving these probabilities from transient state to any absorbing state, 0 is a m-by-n zero matrix and $I$ is the m-by-m identity matrix.",
"For an absorbing chain $P$ , all the transient states can achieve absorbing states in one or more steps, we can write the expected number of times $N(i,j)$ (which means the transient state moves from $i$ state to the $j$ state), its standard form is written as: $N=(I-Q)^{-1},$ namely, the matrix $N$ with invertible matrix, where $n_{ij}$ denotes the average transfer times between transient state $i$ to transient state $j$ .",
"Supposing $c=[1,1,\\cdot \\cdot \\cdot ,1]_{1\\times n}^{N}$ , the absorbed time for each transient state can be expressed as: $z=N\\times c.$"
],
[
"The proposed approach",
"To obtain more robust and accurate saliency maps, we propose a method via bidirectional absorbing Markov chain.",
"This section explains the procedure to find the saliency area in an image in two orientations.",
"Simple linear iterative clustering(SLIC) algorithm [42] has been used to get the superpixels.",
"The pipeline is explained below: Figure: The processing of our proposed method"
],
[
"Graph construction",
"The SLIC algorithm is used to split the image into different pitches of superpixels.",
"Afterwards, two kinds of graphs $G^1(V^1,E^1)$ and $G^2(V^2,E^2)$ are constructed, see Figure REF for detail, Figure: The graph construction.",
"(a) The superpixels in the yellow area are the duplicated absorbing nodes.",
"(b) The foreground cues are obtained and regarded as absorbing nodes which are duplicated with red area."
],
[
"Graph construction",
"where $G^1$ represents the graph of boundary absorbing process and $G^2$ represents the graph of foreground prior absorbing process.",
"In each of the graphs, $V^1, V^2$ represent the graph nodes and $E^1,E^2$ represent the edges between any nodes in the graphs.",
"For the process of boundary absorbing, superpixels around the four boundaries as the virtual nodes are duplicated.",
"For the process of foreground prior absorbing, superpixels from the regions (calculated by the foreground prior) are duplicated.",
"There are two kinds of nodes in both graphs, transient nodes (superpixels) and absorbing nodes (duplicated nodes).",
"The nodes in these two graphs constitute following three properties: (1) The nodes (including transient or absorbing) are associated with each other when superpixels in the image are adjacent nodes or have the same neighbors.",
"And also boundary nodes (superpixels on the boundary of image) are fully connected with each other to reduce the geodesic distance between similar superpixels.",
"(2) Any pair of absorbing nodes (which are duplicated from the boundaries or foreground nodes) are not connected (3) The nodes, which are duplicated from the four boundaries or foreground prior nodes, are also connected with original duplicated nodes.",
"In this paper, the weight $w_{ij}$ of the edges is defined as $w_{ij}=e^{-\\frac{\\left\\Vert x_{i}-x_{j} \\right\\Vert }{\\sigma ^{^{2}}}}, i,j \\in V^1 \\ \\text{or} \\ i,j \\in V^2$ where $\\sigma $ is the constant parameter to adjust the strength of the weights in CIELAB color space.",
"Then we can get the affinity matrix $A$ $a_{ij}={\\left\\lbrace \\begin{array}{ll}w_{ij}, &\\text{ if $j\\in M(i)$ \\quad $1 \\le i \\le j$} \\\\1, &\\text{ if $i=j$ } \\\\0, &\\text{ otherwise},\\end{array}\\right.",
"}$ where $M(i)$ is a nodes set, in which the nodes are all connected to nodes $i$ .",
"The diagonal matrix is given as: $D = diag(\\sum _{j}a_{ij})$ , and the obtained transient matrix is calculated as: $P = D^{-1} \\times A.$"
],
[
"Saliency detection model",
"Following the aforementioned procedures, the initial image is transformed into superpixels, now two kinds of absorbing nodes for saliency detection are required.",
"Firstly, we choose boundary nodes and foreground prior nodes to duplicate as absorbing nodes and obtain the absorbed times of transient nodes as foreground possibility and background possibility.",
"Secondly, we use a cost function to optimize two possibility results together and obtain saliency results of all transient nodes."
],
[
"Absorb Markov chain via boundary nodes",
"In normal conditions, four boundaries of an image rarely have salient objects.",
"Therefore, boundary nodes are assumed as background, and four boundaries nodes set $H^1$ are duplicated as absorbing nodes set $D^1$ , $H^1,D^1 \\subset V^1$ .",
"The graph $G^1$ is constructed and absorbed time $z$ is calculated via Eq.REF .",
"Finally, foreground possibility of transient nodes $z^f = \\bar{z}(i) \\quad i=1,2,\\cdot \\cdot \\cdot ,n,$ is obtained, and $\\bar{z}$ denotes the normalizing the absorbed time vector."
],
[
"Absorb Markov chain via foreground prior nodes",
"We use boundary connectivity to get the foreground prior $\\textbf {f}_i$ without using the down-top method [37].",
"$f_i = \\sum ^N_{j=1} (1 - \\mathrm {exp}\\big (-\\frac{BC_j^2}{2\\sigma _b^2}\\big ))d_a(i,j)\\mathrm {exp}\\big (-\\frac{d_s^2(i,j)}{2\\sigma _s^2}\\big )$ where $d_a(i,j)$ and $d_s(i,j)$ denote the CIELAB color feature distance and spatial distance respectively between superpixel $i$ and $j$ , the boundary connectivity (BC) of superpixel $i$ is defined as $BC_i = \\frac{\\sum _{j\\in \\mathcal {H}}w_{ij}}{\\sqrt{\\sum ^N_{j=1}w_{ij}}}$ in Fig.",
"REF , $\\sigma _b = 1 $ , $\\sigma _s = 0.25 $ .",
"$\\mathcal {H}$ denotes the boundary area of image and $w_{ij}$ is the similarity between nodes $i$ and $j$ .",
"$N$ is the number of superpixels.",
"Afterwards, nodes ($\\lbrace i|f_i > avg(f)\\rbrace $ ) with high level values are selected to get a set $H^2$ , which are duplicated as absorbing nodes set $D^2$ , $H^2,D^2 \\subset V^2$ .",
"The graph $G^2$ is constructed and absorbed time $z$ is calculated using Eq.REF .",
"Finally, the background possibility of transient nodes $z^b = \\bar{z}(i) \\quad i=1,2,\\cdot \\cdot \\cdot ,n,$ is obtained, where $\\bar{z}$ denotes the absorbed time vector normalization.",
"Figure: An illustrative example of boundary connectivity.",
"(a) input image (b) the superpixels of input image (c) the superpixels of similarity in each pitches (d) an illustrative example of boundary connectivity."
],
[
"Saliency Optimization",
"In order to combine different cues, this paper has used the optimization model presented in [37], which fused background possibility and foreground possibility for final saliency map.",
"It is defined as $\\sum _{i=1}^{N}z^b_{i}s_{i}^{2}+\\sum _{i=1}^{N}z^f_{i}(s_{i}-1)^{2}+\\sum _{i,j}w_{ij}(s_{i}-s_{j})^{2}$ where the first term defines superpixel $i$ with large background probability $z^b$ to obtain a small value $s_i$ (close to 0).",
"The second term encourages a superpixel $i$ with large foreground probability $z^f$ to obtain a large value $s_i$ (close to 1).",
"The third term defines the smoothness to acquire continuous saliency values.",
"In this work, the used super-pixel numbers $N$ are 200, 250, 300 in the superpixel element, and the final saliency map is given as: $\\textbf {S} = \\sum _h{S^h}$ at each scale, where $h = 1, 2, 3$ .",
"The algorithm of our proposed method is summarized in Algorithm REF .",
"[htb] Saliency detection by bidirectional Markov chain.",
"[1] An image and required parameters as the input image.",
"Using SLIC method to segment the input image into superpixels, construct two graphs $G^1(V^1,E^1)$ and $G^2(V^2,E^2)$ ; Duplicate the four boundaries nodes and foreground prior nodes(which is computed by Eq.",
"REF ) as the absorbing nodes on graphs $G^1(V^1,E^1)$ and $G^2(V^2,E^2)$ , respectively; Compute the transient matrix $P$ with Eg.",
"REF on both graphs; Obtain the absorbed time $z^f$ and $z^b$ with Eq.",
"REF , respectively; Optimize the model with the Eq.",
"REF ; Compute final saliency values with $\\textbf {S} = \\sum _{h=1}^{3}\\textbf {S}^h$ ; Output a saliency map $\\textbf {S}$ with the same size as the input image."
],
[
"Experiments",
"The proposed method is evaluated on four benchmark datasets ASD [43], CSSD [31], ECSSD [31] and SED [44].",
"ASD dataset is a subset of the MSRA dataset, which contains 1000 images with accurate human-labeled ground truth.",
"CSSD dataset, namely complex scene saliency detection contains 200 complex images.",
"ECSSD dataset, an extension of CSSD dataset contains 1000 images and has accurate human-labeled ground truth.",
"SED dataset has two parts, SED1 and SED2, images in SED1 contains one object, and images in SED2 contains two objects, in total they contain 200 images.",
"We compare our model with 17 different state-of-the-art saliency detection algorithms: CA [45], FT [43], SEG [46], BM [47], SWD [48], SF [49], GCHC [50], LMLC [51], HS [31], PCA [30], DSR [52], MC [21], MR [10], MS [53], RBD [37], RR [54], MST [55].",
"The tuning parameters in the proposed algorithm is the edge weight $\\sigma ^2=0.1$ that controls the strength of weight between a pair of nodes.",
"In the following results of the experiments, we show the evaluation of our proposed saliency models based on the aforementioned datasets comparing with the best works.",
"In addition, we also give some limitation about our model and analysis the reason."
],
[
"Evaluation of the proposed model",
"The precision-recall curves and F-measure are used as performance metrics.",
"The precision is defined as the ratio of salient pixels correctly assigned to all the pixels of extracted regions.",
"The recall is defined as the ratio of detected salient pixels to the ground-truth number.",
"Which can be fomulated as, $Precision = \\frac{TP}{TP+FP},Recall = \\frac{TP}{TP+FN}£¬$ where $TP$ , $FP$ and $FN$ represent the true positive, false positive and false negative, respectively.",
"A PR curve is obtained by the threshold sliding from 0 to 255 to get the difference between the saliency map (which is calculated) and ground truth(which is labeled manually).",
"F-measure is calculated by the weighted average between the precision values and recall values, which can be regarded as overall performance measurement, given as: $F_{\\beta } = \\frac{(1+\\beta ^{2})Precision \\times Recall}{\\beta ^{2}Precision + Recall},$ we set $\\beta ^{2} = 0.3$ to stress precision more than recall.",
"PR-curves and the F-measure curves are shown in Figure REF - REF , where the outperformance of our proposed method as compared to 17 state-of-the-art methods is evident.",
"Fig.REF presets visual comparisons selected from four datasets.",
"It can be seen that the proposed method achieved best saliency results as compared to the state-of-the-art methods.",
"Figure: PR-curves and F-measure curves comparing with different methods on ASD dataset.Figure: PR-curves and F-value curves comparing with different methods on ECSSD dataset.Figure: PR-curves and F-measure curves comparing with different methods on CSSD dataset.Figure: PR-curves and F-measure curves comparing with different methods on SED dataset.Figure: Examples of output saliency maps results using different algorithms on the ASD, CSSD, ECSSD and SED datasets"
],
[
"Failure cases analysis",
"In this work, the idea of bidirectional absorbing Markov chain is first proposed.",
"Although the proposed method is effective for most images on the four datasets.",
"However, if the appearances of four boundaries and the foreground prior are similar to each other, the performance is not obviously, which is shown in Figure REF .",
"Figure: Examples of our failure examples.",
"(a) Input images.",
"(b) Ground truth.",
"(c) Saliency maps."
],
[
"Conclusion",
"In this paper, a bidirectional absorbing Markov chain based saliency detection method is proposed considering both boundary information and foreground prior cues.",
"A novel optimization model is developed to combine both background and foreground possibilities, acquired through bidirectional absorbing Markov chain.",
"The proposed approach outperformed 17 different state-of-the-art methods over four benchmark datasets, which demonstrate the superiority of our proposed approach.",
"In future, we intend to apply our proposed saliency detection algorithm to problems such as multi-pose lipreading and audio-visual speech recognition."
],
[
"Acknowledgments",
"This work was supported by China Scholarship Council, the National Natural Science Foundation of China (No.913203002), the Pilot Project of Chinese Academy of Sciences (No.XDA08040109).",
"Prof. Amir Hussain and Dr. Ahsan Adeel were supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant No.EP/M026981/1."
]
] | 1808.08393 |
[
[
"Causes of Effects via a Bayesian Model Selection Procedure"
],
[
"Abstract In causal inference, and specifically in the \\textit{Causes of Effects} problem, one is interested in how to use statistical evidence to understand causation in an individual case, and so how to assess the so-called {\\em probability of causation} (PC).",
"The answer relies on the potential responses, which can incorporate information about what would have happened to the outcome as we had observed a different value of the exposure.",
"However, even given the best possible statistical evidence for the association between exposure and outcome, we can typically only provide bounds for the PC.",
"Dawid et al.",
"(2016) highlighted some fundamental conditions, namely, exogeneity, comparability, and sufficiency, required to obtain such bounds, based on experimental data.",
"The aim of the present paper is to provide methods to find, in specific cases, the best subsample of the reference dataset to satisfy such requirements.",
"To this end, we introduce a new variable, expressing the desire to be exposed or not, and we set the question up as a model selection problem.",
"The best model will be selected using the marginal probability of the responses and a suitable prior proposal over the model space.",
"An application in the educational field is presented."
],
[
"Introduction",
"The Causes of Effects (CoE) problem concerns the study of individual causation and explicitly refers to something happened to a well identified individual.",
"This nuance of causation has received less attention than the study of the Effects of Causes (EoC), also called the general causation problem.",
"Actually, in EoC, the aim is the prediction of an outcome after the realization of an alleged cause; in CoE, instead, we want to evaluate the Probability of Causation (PC), i.e.",
"the probability the observed outcome would not have been realized if the alleged cause had not been made effective, despite the fact that the cause and the outcome were already observed.",
"Since we always refer causation to a specific individual, for the sake of simplicity, we call her Ann.",
"A simplified CoE question is the following.",
"“Ann had a headache and decided to take aspirin.",
"Her headache went away.",
"Was that caused by the aspirin?” In CoE, to evaluate the probability of causation, formally defined in Section , the relevant questions are: How might one use experimental and/or observational data, gathered on a reference group to which Ann belongs?",
"How would one find the characteristics shared with Ann by the group of individuals from which the data came?",
"These problems, also called the “Group to individual” (G2i) issue in forensic science, have generated a large debate in legal circles (see Faigman et al.",
"(2014) [10]).",
"The issue was extensively studied in the statistical literature from a technical and a philosophical point of view, reaching different and somewhat related results.",
"Essential references are Dawid (2000; 2016), [6], [3] and Pearl (2009; 2015) [14], [15].",
"Even if their approaches follow different routes, they agree about the use of counterfactuals and potential outcomes originating from Neyman ([12]) and re-introduced in the modern literature by Rubin (1974) ([17])Rubin's approach has been mainly used to solve EoC-type problems: for this reason it will be not considered in this paper..",
"The need for counterfactuals in CoE is well illustrated by the question: “What would be the probability of the response if no treatment had been provided to the individual in whom we are interested?” Since only the treatment can be assigned to the individual, the matter concerns a counterfactual, i.e.",
"an event which assumes something different from what happened.",
"Depending on the assumptions, different results emerge in the forms of a precise probability or bounds (see Section ).",
"Although not as specific as one would ideally like, such bounds can be of use.",
"In particular, if the lower bound exceeds one-half, then, in civil cases, we can infer causality “on the balance of probabilities”.",
"To evaluate the PC, we follow the approach of Dawid et al.",
"(2016) [6], that identified and detailed three fundamental conditions used to estimate from the data upper and lower bounds for the probability of causation.",
"A related problem is the choice of the reference population.",
"More specifically, how to find that group, from among those obtainable from partitioning the randomized experiment sample according to Ann's characteristics, which best fulfills the fundamental conditions.",
"How much information to take into account and how to select the best comparison group is a tricky issue, also because the choice of the reference class can affect significantly the conclusions of the inference.",
"The main aim of the present paper is to make the fundamental conditions operative by means of empirical testing of such underlying assumptions, in a way that comes up with the “best” comparison group.",
"We set the question up as a Bayesian model selection problem, where each model specifies a particular choice (more or less detailed) for the characteristics to be included, shared by Ann and the other individuals participating in the study.",
"The best model will be selected considering the marginal probabilities of the response and a satisfactory prior proposal in the model space.",
"To this end, we introduce a new variable that expresses, for each individual in the study, the desire to receive the treatment or not.",
"This variable allows introducing the fundamental conditions in the model selection procedure.",
"Such a method can have various applications in sociology and education, as well as in medicine.",
"We present an example in the field of education, where we investigate the relation between success in a test and whether or not a hint was received (taking into account the student's preference for receiving or not receiving the hint).",
"The structure of this paper is as follows.",
"We first introduce the notation and we define formally the Probability of Causation (PC).",
"After the review of some results in the CoE literature, see Section , we provide the assumptions we require, Section .",
"Then, in Section , we detail how to find the reference sample suitable for evaluating CoE by a model selection procedure.",
"After presenting the application in Section , we draw some final conclusions."
],
[
"Notation",
"We first specify the notation we need to introduce our approach.",
"Given data obtained from an ideal large randomized study concerning $n$ individuals, drawn from a population to which Ann belongs, assume that the study records the outcome $R=\\lbrace 0,1\\rbrace $ of a treatment $T=\\lbrace 0,1\\rbrace $ , which is supposed large enough that the sampling variability of the estimates is negligible.",
"Let $H$ be a large set of variables $H=\\lbrace H_1,\\dots ,H_k\\rbrace $ , characteristic of both Ann and the individuals participating in the study, where we assume that each $H_j$ , $j\\in 1,\\dots ,k$ is discrete (or even dichotomous).",
"We denote by $H^{A}$ the value that the set of variables $H$ has for Ann.",
"Since Ann not only takes the aspirin but also expresses her desire to take it, we introduce a variable $E=\\lbrace 0,1\\rbrace $ (for Ann $E=1$ ) expressing such a desire for each individual in the study.",
"This variable, first proposed as an unobservable by Dawid (2011), is here considered observable.",
"We also introduce the potential variables $R_T=\\lbrace 0,1\\rbrace $ , so that, if the triple $(T,R_0,R_1)$ were observed contemporaneously, it would be easy to solve the CoE problem by stating the probability of causation (PC), defined as $PC_A& \\stackrel{\\triangle }{=}& \\Pr (R_0=0|H^A,T=1,R_1=1) \\nonumber \\\\&=& \\dfrac{\\Pr (R_0=0,R_1=1|H^A, T=1)}{\\Pr (R_1=1|H^A,T=1)}.$ Unfortunately, $R_0,R_1$ are not jointly estimable from the data, and consequently $PC_A$ cannot be evaluated without some further assumptions."
],
[
"Some results from the literature",
"In epidemiology, $PC_A$ has often been expressed by the quantity referred to as the excess risk ratio (see for instance [16]) $ERR &=&\\dfrac{\\Pr (R_1=1|H^A)-\\Pr (R_0=1|H^A)}{\\Pr (R_1=1|H^A)}$ and sometime evaluated in terms of the observational risk ratio, $ORR&=& 1- 1/\\dfrac{\\Pr (R=1|H^A, T=1)}{\\Pr (R=1|H^A, T=0)}.$ The quantity (REF ), which plays an important role in the developments we consider, can be evaluated by using the data for the treated and untreated individuals coming from a randomized study or from observational data.",
"The choice between these two sources of data depends on the assumptions.",
"We review the following three contributions.",
"a) Pearl, 2000 (Theorem 9.2.14) [13], showed that, under Exogeneity ($R_0,R_1\\hspace{4.2679pt}\\perp \\hspace{-11.09654pt}\\perp \\hspace{4.2679pt}T |H^A$ , i.e.",
"the potential outcomes $(R_0, R_1)$ have the same joint distribution, among both treated and untreated study subjects sharing the same background information $H^A$ as Ann) and Monotonicity ($R_0=1 \\rightarrow R_1=1$ , i.e.",
"if Ann were to recover if untreated, she would certainly recover if treated), $PC_A$ is identified and equal to ORR, evaluated on observational data.",
"This result is remarkable since it ends up with a precise probability.",
"At the same time, these assumptions are not easily defensible: Exogeneity, also called Strong Ignorability, is reasonable for data coming from a randomized study but it is considered weak for observational data.",
"Monotonicity, also called No Prevention, is apparently reasonable (the treatment cannot be obtain worse results than the placebo) but for some individuals, for example those allergic to a medical treatment, it may not hold.",
"In any case, this latter assumption can not usually be verified.",
"b) Tian and Pearl, 2000 [18] demonstrated that, relaxing Exogeneity but retaining Monotonicity, the evaluation of the probability of causation can be obtained in a more refined form as $PC_A &=&\\dfrac{\\Pr (R^{obs}=1|T=1,H^A)-\\Pr (R^{obs}=0|T=1, H^A)}{\\Pr (R^{obs}=1|T=1, H^A)} + \\nonumber \\\\&+& \\dfrac{\\Pr (R^{obs}1|T=0, H^A)-\\Pr (R^{exp}=1|T=0,H^A)}{\\Pr (R^{obs}=1,T=1|H^A)} $ where $obs$ and $exp$ specify the source of the data (observational or experimental).",
"This result points out that CoE has both an experimental and an observational nature.",
"Data from a randomized experiment amount for no confounding, i.e.",
"the desirable Exogeneity property can be assumed quite safely.",
"At the same time, Ann made a choice to receive the treatment, i.e.",
"she was not forced to receive it.",
"Hence, a difference between $\\Pr (R^{obs}=1| T=0,H^A)$ and $\\Pr (R^{exp}=1|T=0,H^A)$ is plausible and must be taken into account.",
"For instance, if Ann's disease is at an advanced stage, she has little will to be treated because she perceives that her survival is almost independent of the treatment.",
"Since expression (REF ) points out the double nature of CoE, its computation requires having data from two different surveys, and this can be problematic.",
"c) Dawid et al.",
"(2016) consider the possibility of evaluating CoE by using only data coming from a randomized experiment.",
"The authors proceed in two steps.",
"First they derive bounds for the probability of causation based on the potential outcomes and the constraints implied in their joint distribution.",
"Interestingly, the relevant $PC_A$ lower bound is equal to ERR (see (REF )).",
"$ PC_A > \\max \\lbrace 0,1-1/RR_A\\rbrace $ where $RR_A$ , the risk ratio, is $RR_A=\\frac{\\Pr (R_1=1|H^A,T=1)}{\\Pr (R_0=1|H^A,T=1)}.$ Specifically, a large lower bound is the minimum probability of observing the response opposite to that actually observed if the treatment were not provided, so demonstrating that a different story would have been possible if the treatment had not been applied.",
"The question is: “Even accepting working with bounds, what cautions must be taken to estimate $\\Pr (R_1=1|H^A,T=1)$ and $\\Pr (R_0=1|H^A,T=1)$ from experimental data?” To answer the question, the authors detailed three conditions, called the fundamental conditions, to be assumed so as to estimate upper and lower bounds for $PC_A$ based on the marginal probabilities of the response.",
"The three conditions are the following: Exogeneity: Already defined in Section .",
"Comparability: Ann's potential response, $R^A_1$ , is comparable with those of the treated subjects having the same background characteristics $H^A$ as Ann.",
"Sufficiency: Ann's potential response $R^A_0$ and those of the untreated subjects, all having the same background characteristics $H^A$ as Ann, are comparable.",
"While Exogeneity follows directly from randomization, the other two conditions deserve careful reasoning, so the authors restrict their approach to when we can make good arguments for the acceptability of these fundamental conditions."
],
[
"Validating the fundamental conditions",
"Our proposal to evaluate the CoE consists in finding, among all the possible groups of individuals differing from the specification of $H$ , that one that “best fits” the conditions of Comparability and Sufficiency.",
"The problem of validating the assumptions is turned into the search for the most suitable group of experimental data supporting the fundamental conditions.",
"To take into account the experimental and observational nature of the CoE, we consider as observed the variable $E$ , (see Section ), the decision to receive the treatment.",
"For Ann, this variable provides some indirect information about her state of health, in the light of which it would no longer be appropriate to consider her similar to individuals in a pure experimental study for which this information was not available.",
"For this reason we extend our experimental data to include the desire of the individuals in the sample to be treated or not.",
"We believe that it is possible to get this information from people who have accepted a randomized treatment and we also believe that this practice is much less troublesome than to have a double survey of the same population, as in Tian and Pearl (2000) ([18]) (see Section ).",
"In this extended scenario, Comparability means that, conditional on my knowledge of the pre-treatment characteristics of Ann and the trial subjects, I regard Ann's potential response as comparable with those of the sub-group identified by ($T=1,E=1$ ) having characteristics $H^A$ .",
"In the same framework, the Sufficiency condition refers to the counterfactual scenario in which Ann was not treated.",
"In this case we do not have information about Ann's response nor the information concerning her will to receive the treatment or not.",
"Apparently, for $T=0$ , we could imagine that Ann did not desire receive the treatment (so $E=0$ ) but it might also be possible that she did not have the drug available, but her wish was to receive it (so $E=1$ ).",
"Our concern is to find the specification of $H$ that makes irrelevant the influence of $E$ on the responses in the untreated group.",
"If we can obtain reasonable support for the condition $R_0 \\hspace{4.2679pt}\\perp \\hspace{-11.09654pt}\\perp \\hspace{4.2679pt}E | H^A$ , i.e.",
"if $\\Pr (R_0=1|H^A, E=1)= \\Pr (R_0=1|H^A, E=0),$ it would be possible to estimate (REF ) by $\\Pr (R=1|H^A, T=0)$ using the data of the untreated."
],
[
"Model selection",
"To perform a selection from models characterized by different $H$ , we need to compute the marginal probabilities of the observed responses of Ann and the individuals participating in the study.",
"We assume a relevant effect of the treatment that can be modelled using partial exchangeability among treated and untreated individuals.",
"Furthermore the Comparability condition establishes that we are not able to distinguish between Ann and the group of treated individuals who desire to receive the treatment (identified by $(T=1, E=1)$ ), sharing with Ann the same characteristics $H^A$ , as it concerns the uncertainty of their responses to the treatment.",
"Since Comparability focusses on the group with the same characteristics as Ann, we don't need to detail the responses of the remaining individuals in the treated group, and we model them as exchangeable.",
"The Sufficiency condition requires that in the untreated group, individuals sharing Ann's characteristics (despite the fact that for some of them $E=0$ and for others $E=1$ ) are considered exchangeable.",
"Also here we are not interested in distinguishing between the remaining individuals in the untreated group (those not sharing Ann's characteristics), so that the untreated group is modeled as partially exchangeable.",
"Of course according to what characteristics are included in $H$ , different individuals in the randomized sample will be compared with Ann.",
"We have $2^k$ ways of selecting a set of characteristics from $H$ .",
"Let $J$ be one of these choices, identified as a subset of $\\lbrace 1,\\ldots ,K\\rbrace $ .",
"Each choice of $J$ induces a partition of the sample (treated and untreated), which defines a model $M_j$ .",
"Then, by assuming partial exchangeability and by using de Finetti's representation theorem inside every specified exchangeable group, we can evaluate the probability of observing Ann and the group of responses, induced by different subsets of $H$ .",
"In this way we turn the issue of finding the group most supporting the fundamental conditions into a model selection problem, solved, as usual, by computing the marginal probability of the data conditionally on different instantiations of $H$ .",
"Restricted to the treated group, we denote by $A_{1,1}$ the set of individuals who desire to take the treatment and share the same characteristics as Ann, with $\\bar{A}_{1,1}$ its complement, and with $\\mathbf {r}_{A_{1,1}}$ , $\\mathbf {r}_{\\bar{A}_{1,1}}$ the corresponding vector of responses, while $r_A$ denotes Ann's response.",
"In the untreated group, let $A_{0,e}$ be the sets of individuals considered, $e=\\lbrace 0,1\\rbrace $ , $A_0=A_{0,0}\\cup A_{0,1}$ and $\\bar{A}_{0}$ its complement.",
"We extend these notations in the obvious way to the vectors of responses and to the mixing parameters.",
"We have: $& \\Pr (r_A,\\mathbf {r}_{A_{1,1}}, \\mathbf {r}_{\\bar{A}_{1,1}}, \\mathbf {r}_{A_{0,1}}, \\mathbf {r}_{A_{0,0}} , \\mathbf {r}_{\\bar{A}_{0}}) |M_J)= \\\\& \\int _{\\Theta }\\Pr (r_A,\\mathbf {r}_{A_{1,1}}, \\mathbf {r}_{\\bar{A}_{1,1}}, \\mathbf {r}_{A_{0,1}}, \\mathbf {r}_{A_{0,0}} , \\mathbf {r}_{\\bar{A}_{0}} \\mid \\theta _{A_{1,1}},\\theta _{\\bar{A}_{1,1}}, \\theta _{A_{0}},\\theta _{\\bar{A}_{0}}, M_j)\\cdot \\\\& \\cdot \\pi (\\theta _{A_{1,1}},\\theta _{\\bar{A}_{1,1}}, \\theta _{A_{0}},\\theta _{\\bar{A}_{0}})d\\theta _{A_{1,1}}d\\theta _{\\bar{A}_{1,1}}d\\theta _{A_{0}}d\\theta _{\\bar{A}_{0}}= \\\\&= \\int _{\\Theta }\\Pr (r_A,\\mathbf {r}_{A_{1,1}}, \\mathbf {r}_{\\bar{A}_{1,1}}, \\mathbf {r}_{A_{0,1}}, \\mathbf {r}_{A_{0,0}} , \\mathbf {r}_{\\bar{A}_{0}} \\mid \\theta _{A_{1,1}},\\theta _{\\bar{A}_{1,1}}, \\theta _{A_{0}},\\theta _{\\bar{A}_{0}}, M_j)\\cdot \\\\& \\cdot \\pi (\\theta _{A_{1,1}})\\pi (\\theta _{\\bar{A}_{1,1}})\\pi (\\theta _{A_{0}})\\pi (\\theta _{\\bar{A}_{0}})d\\theta _{A_{1,1}}d\\theta _{\\bar{A}_{1,1}}d\\theta _{A_{0}}d\\theta _{\\bar{A}_{0}}= \\\\& \\int _{\\Theta _{A_{1,1}}} \\Pr (r_A,\\mathbf {r}_{A_{1,1}}\\mid \\theta _{A_{1,1}},M_J)\\pi (\\theta _{A_{1,1}})d\\theta _{A_{1,1}} \\cdot \\int _{\\Theta _{\\bar{A}_{1,1}}} \\Pr (\\mathbf {r}_{\\bar{A}_{1,1}}\\mid \\theta _{\\bar{A}_{1,1}},M_J)\\pi (\\theta _{\\bar{A}_{1,1}})d\\theta _{\\bar{A}_{1,1}}\\cdot \\\\ \\nonumber &\\cdot \\int _{\\Theta _{A_{0}}} \\Pr (\\mathbf {r}_{A_{0,1}}, \\mathbf {r}_{A_{0,0}}\\mid \\theta _{A_{0}},M_J)\\pi (\\theta _{A_{0}})d\\theta _{A_{0}}\\cdot \\int _{\\Theta _{\\bar{A}_{0}}} \\Pr (\\mathbf {r}_{\\bar{A}_{0}}\\mid \\theta _{\\bar{A}_{0}},M_J)\\pi (\\theta _{\\bar{A}_{0}})d\\theta _{\\bar{A}_{0}}$ By the de Finetti's Representation Theorem, the conditional probabilities of the responses are a mixture of a binomial model and a mixture distribution over the corresponding parameters $\\theta $ s that are assumed independent of each other.",
"As a consequence the overall integral easily factorizes.",
"Now we provide expressions for the above integrals.",
"All the details are in Appendix ().",
"We have: $\\int _{\\Theta _{A_{1,1}}} \\Pr (r_A,\\mathbf {r}_{A_{1,1}}\\mid \\theta _{A_{1,1}},M_J)\\pi (\\theta _{A_{1,1}})d\\theta _{A_{1,1}}=\\frac{x_{A_{1,1}}+1}{n_{A_{1,1}}+2} \\cdot \\frac{1}{n_{A_{1,1}}+1}$ where $x_{A_{1,1}}$ is the number of successes in the group $A_{1,1}$ and $n_{A_{1,1}}=|A_{1,1}|$ .",
"The notation is extended in the obvious way in the other groups.",
"$\\int _{\\Theta _{\\bar{A}_{1,1}}} \\Pr (\\mathbf {r}_{\\bar{A}_{1,1}}\\mid \\theta _{\\bar{A}_{1,1}},M_J)\\pi (\\theta _{\\bar{A}_{1,1}})d\\theta _{\\bar{A}_{1,1}}=\\frac{1}{n_{\\bar{A}_{1,1}}+1}$ $\\int _{\\Theta _{A_{0}}} \\Pr (\\mathbf {r}_{A_{0,1}}, \\mathbf {r}_{A_{0,0}}\\mid \\theta _{A_{0}},M_J)\\pi (\\theta _{A_{0}})d\\theta _{A_{0}} &=\\dbinom{n_{A_{0,0}}}{x_{A_{0,0}}}\\dbinom{n_{A_{0,1}}}{x_{A_{0,1}}}\\dbinom{n_{A_{0,0}}+n_{A_{0,1}}}{x_{A_{0,0}}+x_{A_{0,1}}}^{-1}\\cdot \\\\& \\cdot \\frac{1}{n_{A_{0,0}}+n_{A_{0,1}}+1}$ $\\int _{\\Theta _{\\bar{A}_{0}}} \\Pr (\\mathbf {r}_{\\bar{A}_{0}}\\mid \\theta _{\\bar{A}_{0}},M_J)\\pi (\\theta _{\\bar{A}_{0}})d\\theta _{\\bar{A}_{0}}= \\frac{1}{n_{\\bar{A}_{0}}+1}$ Readers might recognize the conditional Irving–Fisher exact test as part of (REF ).",
"The result is not surprising since we are looking for the set of $H$ making $E$ irrelevant, and thus supporting the hypothesis of no-difference between the success ratio in the two groups.",
"An illustration of the behavior of the hypergeometric for $n_{A_{0,0}}=n_{A_{0,1}}=10$ is given in Figure REF .",
"High support to the model is achieved when the number of successes $x_{A_{0,0}}$ and $x_{A_{0,1}}$ is almost the same in the two groups.",
"Figure: Hypergeometric behaviour for n A 0,0 =n A 0,1 =10n_{A_{0,0}}=n_{A_{0,1}}=10 and different values for x A 0,0 x_{A_{0,0}} and x A 0,1 x_{A_{0,1}}"
],
[
"Prior and posterior in the model space",
"The goal is to evaluate the posterior probability of $M_J$ given the responses observed for Ann and the sample.",
"The overall marginal likelihood of $M_j$ is the product of (REF ), (REF ), (REF ) and (REF ) $& \\Pr (r_A,\\mathbf {r}_{A_{1,1}}, \\mathbf {r}_{\\bar{A}_{1,1}}, \\mathbf {r}_{A_{0,1}}, \\mathbf {r}_{A_{0,0}} , \\mathbf {r}_{\\bar{A}_{0}}) |M_J)=\\frac{x_{A_{1,1}}+1}{n_{A_{1,1}}+2} \\cdot \\frac{1}{n_{A_{1,1}}+1}\\cdot \\frac{1}{n_{\\bar{A}_{1,1}}+1}\\\\ \\nonumber \\cdot & \\dbinom{n_{A_{0,0}}}{x_{A_{0,0}}}\\dbinom{n_{A_{0,1}}}{x_{A_{0,1}}}\\dbinom{n_{A_{0,0}}+n_{A_{0,1}}}{x_{A_{0,0}}+x_{A_{0,1}}}^{-1}\\frac{1}{n_{A_{0,0}}+n_{A_{0,1}}+1} \\cdot \\frac{1}{n_{\\bar{A}_{0}}+1}$ Concerning the prior over the space of models, the simplest choice is to consider a uniform distribution $ \\Pr (M_J)=\\frac{1}{2^k}.$ Another choice is the one proposed by Chen and Chen, (2008) [2].",
"They give the same prior probability (equal to $\\frac{1}{k+1}$ ) for all models sharing the same number of characteristics $k$ .",
"In this way, for the generic model $M_J$ , we have $\\Pr (M_J)= \\frac{1}{k+1}\\binom{k}{|M_J|}^{-1}\\cdot I(|M_J|\\le k/2)$ where the search spans all models including at most $k/2$ characteristics.",
"This last choice favors model selection according to Occam's razor principle: the fewer characteristics employed, the more probable is the model.",
"This rationale is reasonably objective.",
"Combining (REF ) or (REF ) with (REF ), we get the required posterior."
],
[
"Computational issues",
"If the model size becomes huge, we may not be in a position to evaluate the normalizing constant of the posterior distribution, but we can establish an MCMC to make an inference about the variable $M_J$ .",
"Essentially a Metropolis–Hastings would suffice, the acceptance ratio being given by (REF ), evaluated for two different elements of $M_J$ , taking into account the probability of proposing a new model."
],
[
"Application",
"We carried out an experiment at the University of Florence, School of Engineering, Fall 2017.",
"We asked 161 students to solve a simple probabilistic question and we provided randomly a hint (the treatment $T$ ).",
"Table (REF ) presents the students' background information included in the analysis (the characteristics $H$ ).",
"Before the test, we asked the students if they wished to be helped or not (the desire variable $E$ ).",
"Table: List of student characteristics included in the experimentWe wish to investigate whether there is a causal relation between the hint and the ability to solve the question, for those students who desired to receive a hint.",
"We had 8 of these cases.",
"The corresponding risk ratio (obtained considering the model best fitting the fundamental conditions) lies in the interval $[1.73, 2.43]$ and in one case it exceeds 2 ($RR=2.43$ ), which is a clue for there being a causal relation.",
"In all the other cases, the causal relation is not strongly supported, since $RR$ is close but does not reach 2 (see, in Figure (REF )), the left side of each sub-picture).",
"In the right side of each sub-picture in Figure(REF ) we illustrate how the model selection procedure proposes models respecting the fundamental conditions.",
"For the model with the highest probability, the red dot indicates, in the $x-$ axis, the success ratio for treated and, in the $y-$ axis, the ratio between the success ratio for untreated with $E=1$ and with $E=0$ , respectively.",
"These are the main forces driving the marginal probability for the responses, as shown in (REF ).",
"Ideally, comparability and sufficiency are mostly supported by the highest possible success ratio among the treated and by a ratio between untreated with E=0 and E=1 approximately equal 1.",
"As is apparent, the selected model achieves a good compromise between these requirements.",
"The set of variables selected in the 8 cases are shown in Table (REF ).",
"Note that, overall, these models include only 5 characteristics, and all of them include the educational status of the family.",
"In 4 out of the 8 cases, the same model, including family education, $1^\\text{first}$ University registration (a proxy for understanding whether the student failed earlier in their educational career) and previous exposure to statistical training was selected.",
"Figure: Educational causation.Risk Ratio and posterior probability for each explored model (left side).Success ratio for treated vs relative success ratio for untreated with E=1E=1 and E=0E=0 (right side).In the picture there is an example concerning four students who succeed.Figure: Forensic causation.Risk Ratio and posterior probability for each explored model (left side).Success ratio for treated vs relative success ratio for untreated with E=1E=1 and E=0E=0 (right side).In the picture there is an example concerning four students who DID NOT succeed.Table: Educational causation.",
"For the 8 students who desired the hint, got it, and succeed, there are reported the characteristics HH selected by the model that best supported the fundamental conditions and the Risk RatioAs a result of our experiment we also have that among the students who asked for and received the hint, 24 did not succeed.",
"We can suppose that some of them claimed that it was the hint which caused their failure.",
"In this case, the $RR$ lies in the interval $[0.86,1.60]$ , an example for four students is in Figure (REF ).",
"It is not conclusive that there is a causal relation between the hint and the failure since all the models for all the considered students provided values of RR much smaller than 2.",
"In a civil trial this would not suggest to a judge that compensation be awarded."
],
[
"Conclusions",
"We introduced a typical Causes of Effects problem by means of an archetypical example considering Ann and the effect of an aspirin on her headache.",
"We have proposed a possible solution to make operational the choice of variables to include, so as to validate the fundamental assumptions underlying the assessment of Ann's probability of causation.",
"We assume it is possible to take a randomized sample from Ann's population where, as usual, $T$ is assigned following a randomized protocol and $E$ (this is a novelty) is a question asking the members of the sample about their preference to be treated or not.",
"In the evaluation of $RR_A$ (see REF ), an extreme position would be to include all the subjects participating in the experiment so that simply belonging to the reference population would make the individuals in the sample similar to Ann.",
"On the other hand, the choice could be to find the persons most similar to Ann, i.e.",
"those matching all the available characteristics.",
"Clearly neither of these positions is safe: the former ignores some characteristics of Ann which could be very influential on her reaction to the headache after taking aspirin.",
"The latter greatly reduces the number of individuals to be employed in the estimation, so producing a very unstable inference.",
"Our approach takes a sensible middle course and provide a sensible different causal inference for individuals experiencing the same treatment.",
"Interestingly, this is exactly the aim of Precision Medicine (see Mesko (2017), [11]) which looks for different medical interventions for a group of individuals sharing some relevant (for the reaction between treatment and outcome) characteristics.",
"The next step will be to extend the method to observational studies, to make possible in a wider range of cases the evaluation of the $PC_A$ for Causes of Effects problems."
],
[
"Appendix",
"Now we detail the computation of the integrals (REF ), (REF ), (REF ) and (REF ).",
"We always assume a no informative prior for all $\\pi (\\cdot )=Beta(1,1)$ .",
"We start with (REF ).",
"$& \\int _{\\Theta _{A_{1,1}}} \\Pr (r_A,\\mathbf {r}_{A_{1,1}}\\mid \\theta _{A_{1,1}},M_J)\\pi (\\theta _{A_{1,1}})d\\theta _{A_{1,1}}= \\nonumber \\\\&= \\int _{\\Theta _{A_{1,1}}} \\Pr (r_A | \\mathbf {r}_{A_{1,1}},\\theta _{A_{1,1}},M_J)\\underbrace{\\Pr (\\mathbf {r}_{A_{1,1}}|\\theta _{A_{1,1}},M_J)\\pi (\\theta _{A_{1,1}}|M_J)}_{\\text{numerator of the }\\theta _{A_{1,1}} \\text{ updating}}d \\theta _{A_{1,1}} \\nonumber \\\\&= \\underbrace{\\int _{\\Theta _{A_{1,1}}} \\Pr (r_A|\\theta _{A_{1,1}},M_J)\\pi (\\theta _{A_{1,1}}|\\mathbf {r}_{A_{1,1}},M_J)d\\theta _{A_{1,1}}}_{\\dfrac{x_{A_{1,1}}+1}{n_{A_{1,1}}+2}}\\cdot \\nonumber \\\\& \\cdot \\underbrace{\\int _{\\Theta _{A_{1,1}}}\\Pr (\\mathbf {r}_{A_{1,1}}|\\theta _{A_{1,1}},M_J)\\Pr (\\theta _{A_{1,1}}|M_J)d\\theta _{A_{1,1}} }_{\\dbinom{n_{A_{1,1}}}{x_{A_{1,1}}}\\dfrac{\\Gamma (1+1)}{\\Gamma (1)\\Gamma (1)} \\dfrac{\\Gamma (1+x_{A_{1,1}}) \\Gamma (n_{A_{1,1}}+ 1 -x_{A_{1,1}})}{\\Gamma (n_{A_{1,1}}+ 1 +1)}} \\nonumber \\\\ & =\\frac{x_{A_{1,1}}+1}{n_{A_{1,1}}+2} \\cdot \\frac{1}{n_{A_{1,1}}+1}.",
"\\nonumber $ For (REF ) we have $&\\int _{\\Theta _{\\bar{A}_{1,1}}} \\Pr (\\mathbf {r}_{\\bar{A}_{1,1}}\\mid \\theta _{\\bar{A}_{1,1}},M_J)\\pi (\\theta _{\\bar{A}_{1,1}})d\\theta _{\\bar{A}_{1,1}}= \\\\& \\underbrace{\\int _{\\Theta _{\\bar{A}_{1,1}}}\\Pr (\\mathbf {r}_{\\bar{A}_{1,1}}|\\theta _{\\bar{A}_{1,1}},M_J)\\Pr (\\theta _{\\bar{A}_{1,1}}|M_J)d\\theta _{\\bar{A}_{1,1}} }_{\\dbinom{n_{\\bar{A}_{1,1}}}{x_{\\bar{A}_{1,1}}}\\dfrac{\\Gamma (1+1)}{\\Gamma (1)\\Gamma (1)} \\dfrac{\\Gamma (1+x_{\\bar{A}_{1,1}}) \\Gamma (n_{\\bar{A}_{1,1}}+ 1 -x_{\\bar{A}_{1,1}})}{\\Gamma (n_{\\bar{A}_{1,1}}+ 1 +1)}} \\nonumber \\\\& =\\frac{1}{n_{\\bar{A}_{1,1}}+1}$ We now compute (REF ) $&\\int _{\\Theta _{A_{0}}} \\Pr (\\mathbf {r}_{A_{0,1}}, \\mathbf {r}_{A_{0,0}}\\mid \\theta _{A_{0}},M_J)\\pi (\\theta _{A_{0}},M_J)d\\theta _{A_{0}}= \\\\&= \\int _{\\Theta _{A_{0}}} \\Pr (\\mathbf {r}_{A_{0,0}},|\\theta _{A_{0}},\\mathbf {r}_{A_{0,1}},M_J)\\underbrace{\\Pr (\\mathbf {r}_{A_{0,1}}|\\theta _{A_{0}}, M_J)\\pi (\\theta _{A_{0}}|M_J)}_{\\text{numerator of the }\\theta _{A_{0}} \\text{ updating}}d\\theta _{A_{0}} \\nonumber \\\\&= \\int _{\\Theta _{A_{0}}} \\Pr (\\mathbf {r}_{A_{0,0}}|\\theta _{A_{0}},M_J)\\pi (\\theta _{A_{0}}| \\mathbf {r}_{A_{0,1}},M_J)d\\theta _{A_{0}} \\int _{\\Theta _{A_{0}}}\\Pr (\\mathbf {r}_{A_{0,1}}|\\theta _{A_{0}},M_J)\\pi (\\theta _{A_{0}}| M_J)d\\theta _{A_{0}} \\nonumber \\\\&= \\dbinom{n_{A_{0,0}}}{x_{A_{0,0}}} \\dfrac{\\Gamma (n_{A_{0,1}}+2)}{\\Gamma (x_{A_{0,1}}+1)\\Gamma (n_{A_{0,1}}- x_{A_{0,1}}+1)} \\dfrac{\\Gamma (x_{A_{0,1} } +1 + x_{A_{0,0}})\\Gamma (n_{A_{0,0}}+ n_{A_{0,1}}- x_{A_{0,1} }+1- x_{A_{0,0}}) }{\\Gamma (n_{A_{0,0}} + n_{A_{0,1}}+2)} \\nonumber \\\\&\\cdot \\dbinom{n_{A_{0,1}}}{x_{A_{0,1}}}\\dfrac{\\Gamma (1+1)}{\\Gamma (1)\\Gamma (1)} \\dfrac{\\Gamma (1+x_{A_{0,1}}) \\Gamma (n_{A_{0,1}}+ 1 -x_{A_{0,1} })}{\\Gamma (n_{A_{0,1}} + 1 +1)} \\nonumber \\\\& = \\dbinom{n_{A_{0,0}}}{x_{A_{0,0}}}\\dbinom{n_{A_{0,1}}}{x_{A_{0,1}}}\\dbinom{n_{A_{0,0}}+n_{A_{0,1}}}{x_{A_{0,0}}+x_{A_{0,1}}}^{-1}\\frac{1}{n_{A_{0,0}}+n_{A_{0,1}}+1.",
"}$ Concerning the last term, (REF ), with a similar computation as for (REF ), it is straightforward to see that $\\int _{\\Theta _{\\bar{A}_{0}}} \\Pr (\\mathbf {r}_{\\bar{A}_{0}}\\mid \\theta _{\\bar{A}_{0}},M_J)\\pi (\\theta _{\\bar{A}_{0}})d\\theta _{\\bar{A}_{0}}= \\frac{1}{n_{\\bar{A}_{0}}+1}.$ Acknowledgement:The second author was supported by the project GESTA of the Fondazione di Sardegna and Regione Autonoma di Sardegna"
]
] | 1808.08440 |
[
[
"The Eulerian distribution on involutions is indeed $\\gamma$-positive"
],
[
"Abstract Let $\\mathcal I_n$ and $\\mathcal J_n$ denote the set of involutions and fixed-point free involutions of $\\{1, \\dots, n\\}$, respectively, and let $\\text{des}(\\pi)$ denote the number of descents of the permutation $\\pi$.",
"We prove a conjecture of Guo and Zeng which states that $I_n(t) := \\sum_{\\pi \\in \\mathcal I_n} t^{\\text{des}(\\pi)}$ is $\\gamma$-positive for $n \\ge 1$ and $J_{2n}(t) := \\sum_{\\pi \\in \\mathcal J_{2n}} t^{\\text{des}(\\pi)}$ is $\\gamma$-positive for $n \\ge 9$.",
"We also prove that the number of $(3412, 3421)$-avoiding permutations with $m$ double descents and $k$ descents is equal to the number of separable permutations with $m$ double descents and $k$ descents."
],
[
"Introduction",
"A polynomial $p(t) = a_rt^r + a_{r+1}t^{r+1}+ \\cdots + a_st^s$ is called palindromic of center $\\frac{n}{2}$ if $n = r + s$ and $a_{r+i} = a_{s-i}$ for $0 \\le i \\le \\frac{n}{2} - r$ .",
"A palindromic polynomial can be written uniquely [2] as $p(t) = \\sum _{k=r}^{\\lfloor \\frac{n}{2}\\rfloor }\\gamma _k t^k(1 + t)^{n - 2k},$ and it is called $\\gamma $ -positive if $\\gamma _k \\ge 0$ for each $k$ .",
"The $\\gamma $ -positivity of a palindromic polynomial implies unimodality of its coefficients (i.e., the coefficients $a_i$ satisfy $a_r \\le a_{r+1} \\le \\cdots \\le a_{\\lfloor n/2\\rfloor } \\ge a_{\\lfloor n/2\\rfloor + 1} \\ge \\cdots \\ge a_s$ ).",
"Let $\\mathfrak {S}_n$ be the set of all permutations of $[n] = \\lbrace 1, 2, \\dots , n\\rbrace $ .",
"For $\\pi \\in \\mathfrak {S}_n$ , the descent set of $\\pi $ is $\\operatorname{Des}(\\pi ) = \\lbrace i \\in [n-1] : \\pi (i) > \\pi (i+1)\\rbrace ,$ and the descent number is $\\operatorname{des}(\\pi ) = \\#\\operatorname{Des}(\\pi )$ .",
"The double descent set is $\\operatorname{DD}(\\pi ) = \\lbrace i \\in [n] : \\pi (i-1) > \\pi (i) > \\pi (i+1) \\rbrace $ where $\\pi (0) = \\pi (n+1) = \\infty $ , and we define $\\operatorname{dd}(\\pi ) = \\#\\operatorname{DD}(\\pi )$ .",
"Finally, a permutation $\\pi $ is said to avoid a permutation $\\sigma $ (henceforth called a pattern) if $\\pi $ does not contain a subsequence (not necessarily consecutive) with the same relative order as $\\sigma $ .",
"We let $\\mathfrak {S}_n(\\sigma _1, \\dots , \\sigma _r)$ denote the set of permutations in $\\mathfrak {S}_n$ avoiding the patterns $\\sigma _1,\\dots , \\sigma _r$ .",
"The descent polynomial $A_n(t) = \\sum _{\\pi \\in \\mathfrak {S}_n} t^{\\operatorname{des}(\\pi )}$ is called the Eulerian polynomial, and we have the following remarkable fact, which implies that $A_n(t)$ is $\\gamma $ -positive.",
"Theorem 1.1 (Foata–Schützenberger [4]) For $n \\ge 1$ , $A_n(t) = \\sum _{k = 0}^{\\lfloor \\frac{n-1}{2} \\rfloor }\\gamma _{n,k}t^k(1+t)^{n-1-2k},$ where $\\gamma _{n,k} = \\# \\lbrace \\pi \\in \\mathfrak {S}_n :\\operatorname{dd}(\\pi ) = 0, \\operatorname{des}(\\pi ) = k\\rbrace $ .",
"Similarly, let $I_n$ be the set of all involutions in $\\mathfrak {S}_n$ , and let $J_n$ be the set of all fixed-point free involutions in $\\mathfrak {S}_n$ .",
"Define $I_n(t) = \\sum _{\\pi \\in I_n} t^{\\operatorname{des}(\\pi )}, \\quad J_n(t) = \\sum _{\\pi \\in J_n} t^{\\operatorname{des}(\\pi )}.$ Note that $J_n(t) = 0$ for $n$ odd.",
"Strehl [15] first showed that the polynomials $I_n(t)$ and $J_{2n}(t)$ are palindromic.",
"Guo and Zeng [6] proved that $I_n(t)$ and $J_{2n}(t)$ are unimodal and conjectured that they are in fact $\\gamma $ -positive.",
"Our first two theorems, which we prove in Sections and , confirm their conjectures.",
"thmmIn For $n \\ge 1$ , the polynomial $I_n(t)$ is $\\gamma $ -positive.",
"thmmJn For $n \\ge 9$ , the polynomial $J_{2n}(t)$ is $\\gamma $ -positive.",
"Theorem REF and many variations of it have been proved by many methods, see for example [8], [13].",
"One of these methods uses the Modified Foata–Strehl (MFS) action on $\\mathfrak {S}_n$ [1], [11], [12].",
"In fact it follows from [1] that the same property holds for all subsets of $\\mathfrak {S}_n$ which are invariant under the MFS action.",
"The $(2413, 3142)$ -avoiding permutations are the separable permutations, which are permutations that can be built from the trivial permutation through direct sums and skew sums [7].",
"These are not invariant under the MFS action.",
"However, in 2017, Fu, Lin, and Zeng [5], using a bijection with di-sk trees, and Lin [9], using an algebraic approach, proved that the separable permutations also satisfy the following theorem.",
"Theorem 1.2 ([5]) For $n \\ge 1$ , $\\sum _{\\pi \\in \\mathfrak {S}_n(2413, 3142)} t^{\\operatorname{des}(\\pi )} = \\sum _{k = 0}^{\\lfloor \\frac{n-1}{2}\\rfloor } \\gamma _{n,k}^S t^k(1+t)^{n-1-2k},$ where $\\gamma _{n,k}^S = \\#\\lbrace \\pi \\in \\mathfrak {S}_n(2413, 3142) : \\operatorname{dd}(\\pi ) = 0, \\operatorname{des}(\\pi ) = k\\rbrace $ .",
"Two sets of patterns $\\Pi _1$ and $\\Pi _2$ are $\\operatorname{des}$ -Wilf equivalent if $\\sum _{\\pi \\in \\mathfrak {S}_n(\\Pi _1)} t^{\\operatorname{des}(\\pi )} =\\sum _{\\pi \\in \\mathfrak {S}_n(\\Pi _2)} t^{\\operatorname{des}(\\pi )},$ and are $\\operatorname{Des}$ -Wilf equivalent if $\\sum _{\\pi \\in \\mathfrak {S}_n(\\Pi _1)} \\prod _{i \\in \\operatorname{Des}(\\pi )}t_i= \\sum _{\\pi \\in \\mathfrak {S}_n(\\Pi _2)} \\prod _{i \\in \\operatorname{Des}(\\pi )}t_i.$ We also say that the permutation classes $\\mathfrak {S}_n(\\Pi _1)$ and $\\mathfrak {S}_n(\\Pi _2)$ are $\\operatorname{des}$ -Wilf or $\\operatorname{Des}$ -Wilf equivalent.",
"In 2018, Lin and Kim [10] determined all permutation classes avoiding two patterns of length 4 which are $\\operatorname{des}$ -Wilf equivalent to the separable permutations, all of which are $\\operatorname{Des}$ -Wilf equivalent to each other but not to the separable permutations.",
"One such class is the $(3412, 3421)$ -avoiding permutations, which is invariant under the MFS action.",
"A byproduct of this is that the number of $(3412, 3421)$ -avoiding permutations with no double descents and $k$ descents is also equal to $\\gamma _{n,k}^S$ .",
"In Section , we prove the following more general fact.",
"thmmdddes For $n \\ge 1$ , $\\sum _{\\pi \\in \\mathfrak {S}_n(3412, 3421)} x^{\\operatorname{des}(\\pi )}y^{\\operatorname{dd}(\\pi )} = \\sum _{\\pi \\in \\mathfrak {S}_n(2413, 3142)} x^{\\operatorname{des}(\\pi )}y^{\\operatorname{dd}(\\pi )}.$"
],
[
"Proof of the $\\gamma $ -positivity of {{formula:e9bf8e0e-d54f-4ead-ae10-133b4eddd1e0}}",
"In this section we prove Theorem , restated below for clarity.",
"Let the $\\gamma $ -expansion of $I_n(t)$ be $I_n(t) = \\sum _{k=0}^{\\lfloor \\frac{n-1}{2}\\rfloor }a_{n,k} t^k(1+t)^{n-2k-1}.$ We have the following recurrence relation for the coefficients $a_{n,k}$ .",
"Theorem 2.1 ([6]) For $n \\ge 3$ and $k \\ge 0$ , $na_{n,k} = &(k+1)a_{n-1,k} + (2n-4k)a_{n-1,k-1}+ [k(k+2) + n-1]a_{n-2,k} \\\\&+ [(k-1)(4n-8k-14) + 2n-8]a_{n-2,k-1} \\\\&+ 4(n-2k)(n-2k+1)a_{n-2,k-2},$ where $a_{n,k} = 0$ if $k < 0$ or $k > (n-1)/2$ .",
"* We will prove by induction on $n$ the slightly stronger claim that $a_{n,k} \\ge 0$ for $n \\ge 1$ , $k \\ge 0$ , and $a_{n,k} \\ge \\frac{2}{n} a_{n-1,k-1}$ if $n = 2k + 1$ and $k \\ge 4$ .",
"Assume the claim is true whenever the first index is less than $m$ .",
"We want to prove the claim for all $a_{m,k}$ .",
"If $m \\le 2000$ , we can check the claim directly (this has been done by the author using Sage).",
"Thus, we may assume that $m > 2000$ .",
"If $m \\ge 2k + 3$ , then all the coefficients in the recursion are nonnegative, so we are done by induction.",
"Thus, assume that $(m, k) = (2n+1, n)$ or $(2n+2, n)$ with $n \\ge 1000$ .",
"Case 1: $(m,k) = (2n+2, n)$ .",
"We wish to show that $a_{2n+2,n} \\ge 0$ .",
"We apply the recurrence in Theorem REF , noting that $a_{2n, n} = 0$ since $n > (2n-1)/2$ , to get $(2n+2)a_{2n+2, n} &= (n+1)a_{2n+1,n} + 4a_{2n+1,n-1}+ 24a_{2n,n-2} - (2n-2)a_{2n,n-1} \\\\&\\ge 4a_{2n+1,n-1} + 24a_{2n,n-2} - 2na_{2n,n-1} \\\\&\\ge 4a_{2n+1, n-1} + 24a_{2n,n-2} - na_{2n-1, n-1}- 4a_{2n-1, n-2} \\\\& \\quad - 24a_{2n-2, n-3}, $ where the last inequality comes from applying the recurrence once again to obtain $2n a_{2n,n-1} \\le na_{2n-1, n-1} + 4a_{2n-1, n-2}+ 24a_{2n-2, n-3}.$ Note that since $2n+1 \\ge 2(n-1) + 3$ , when we apply the recurrence relation for $a_{2n+1, n-1}$ all terms in the sum are positive.",
"We drop all terms but the $a_{2n-1, n-1}$ term and multiply by $4/(2n+1)$ to get $4a_{2n+1,n-1} &\\ge \\frac{4}{2n+1} \\left[ (n-1)(n+1) + 2n \\right] a_{2n-1,n-1} \\ge na_{2n-1, n-1}.$ Similarly, since $2n \\ge 2(n-2) + 3$ , when we use the recursion to calculate $a_{2n, n-2}$ , we can drop all terms but the $a_{2n-1, n-2}$ term, which after multiplying by $12/(2n)$ gives $12 a_{2n, n-2} \\ge \\frac{12}{2n}[(n-1) a_{2n-1, n-2}]\\ge 4 a_{2n-1, n-2}.$ Alternatively, we could have dropped all but the $a_{2n-2, n-3}$ term to get $12a_{2n, n-2} &\\ge \\frac{12}{2n}[ (n-3)\\cdot 2 + 4n - 8]a_{2n-2, n-3} \\ge 24a_{n-2, n-3}.$ Plugging the previous three inequalities into $(\\dagger )$ proves that $a_{2n+2, n} \\ge 0$ , as desired.",
"Case 2: $(m,k) = (2n+1, n)$ .",
"We want to show $(2n+1)a_{2n+1,n} \\ge 2a_{2n, n-1}$ .",
"By the recurrence relation, we have $(2n+1)a_{2n+1,n} = 2a_{2n, n-1} + 8a_{2n-1, n-2}- (6n-4)a_{2n-1, n-1}.$ Thus, it suffices to show that $8a_{2n-1,n-2} - (6n-3)a_{2n-1,n-1} \\ge 0$ .",
"Note that $8a_{2n-1,n-2} - &(6n-3)a_{2n-1,n-1} \\\\ &\\ge 8a_{2n-1,n-2} - 6a_{2n-2, n-2} - 24a_{2n-3, n-3}$ because, by applying the same recurrence relation for $a_{2n-1, n-1}$ and dropping the $-(6n-10)a_{2n-3, n-2}$ term, which is negative, we see that $(6n-3)a_{2n-1,n-1} &= 3(2n-1)a_{2n-1, n-1}\\\\&\\le 3(2a_{2n-2, n-2} + 8a_{2n-3, n-3})\\\\&= 6a_{2n-2, n-2} + 24a_{2n-3, n-3}.$ Multiplying the right hand side of $(\\ast )$ by $2n-1$ and applying the recurrence relation for $a_{2n-1, n-2}$ we get $(2n-1) \\cdot (\\ast ) &= 8(2n-1)a_{2n-1, n-2} - (12n-6)a_{2n-2, n-2}- (48n - 24)a_{2n-3, n-3} \\\\&= 8[(n-1)a_{2n-2, n-2} + 6a_{2n-2, n-3} + (n^2 - 2)a_{2n-3, n-2}\\\\&\\quad + (2n-4) a_{2n-3, n-3} + 48a_{2n-3, n-4}] \\\\&\\quad - (12n - 6)a_{2n-2, n-2} - (48n - 24)a_{2n-3, n-3} \\\\&= 48a_{2n-2,n-3} + (8n^2-16)a_{2n-3,n-2}+ 384a_{2n-3,n-4}\\\\&\\quad - (4n+2)a_{2n-2,n-2} - (32n+8)a_{2n-3,n-3}.",
"$ Now, since $2n-2\\ge 2(n-3) + 3$ , we use the recursion to calculate $a_{2n-2, n-3}$ , drop some terms, and multiply by $48/(2n-2)$ to get $48 a_{2n-2,n-3} \\ge \\frac{48}{2n-2}[(n-4)\\cdot 2 + 4n-12] a_{2n-4,n-4}\\ge 120 a_{2n-4, n-4}.$ Now we apply the recursion for $a_{2n-2, n-2}$ and multiply by $(4n+2)/(2n-2)$ , which is less than 5, to obtain $(4n + 2)a_{2n-2, n-2}& = \\frac{4n+2}{2n-2}[(n-1)a_{2n-3,n-2} + 4a_{2n-3,n-3} \\\\ &\\quad + 24a_{2n-4, n-4} - (2n-6)a_{2n-4, n-3}] \\\\&< (5n-5)a_{2n-3,n-2} + 20a_{2n-3, n-3} + 120a_{2n-4, n-4} \\\\&\\quad - (10n-30)a_{2n-4,n-3}.$ We substitute the above two bounds on $48a_{2n-2, n-3}$ and $(4n+2)a_{2n-2, n-2}$ for the corresponding terms in $(\\ast \\ast )$ to get $(\\ast \\ast ) &\\ge 120 a_{2n-4, n-4} + (8n^2-16)a_{2n-3,n-2}+ 384a_{2n-3,n-4}\\\\&\\quad - (5n-5)a_{2n-3,n-2} - 20a_{2n-3, n-3} - 120a_{2n-4, n-4} \\\\&\\quad + (10n-30)a_{2n-4,n-3} - (32n + 8)a_{2n-3, n-3} \\\\&= (8n^2 - 5n -11)a_{2n-3, n-2} + 384 a_{2n-3, n-4}+ (10n- 30)a_{2n-4, n-3} \\\\&\\quad - (32n + 28)a_{2n-3, n-3}.",
"$ Since $2n-3 = 2(n-2) + 1$ , by the $a_{2k+1, k} \\ge \\frac{2}{2k+1}a_{2k, k-1}$ part of the induction hypothesis, we have $(8n^2-5n-11)a_{2n-3,n-2} \\ge \\frac{2(8n^2-5n-11)}{2n-3}a_{2n-4, n-3}\\ge (8n + 6) a_{2n-4, n-3}.$ Plugging the previous inequality into $(\\ast \\ast \\ast )$ gives $(\\ast \\ast \\ast ) &\\ge (18n - 24)a_{2n-4, n-3}+ 384a_{2n-3, n-4} - (32n + 28)a_{2n-3,n-3}.$ Since $2n-3 \\ge 2(n-4) \\ge 3$ , when apply the recursion for $a_{2n-3, n-4}$ , we can drop the $a_{2n-5, n-6}$ term which is positive, to get (after multiplying by $384/(2n-3)$ ), $384a_{2n-3,n-4} &\\ge \\frac{384}{2n-3} [(n-3)a_{2n-4,n-4} + 10a_{2n-4,n-5} \\\\&\\quad + (n^2-4n + 4)a_{2n-5, n-4} + (10n-44)a_{2n-5,n-5}].$ Apply the recursion for $a_{2n-3, n-3}$ directly and multiply by $32n+28/(2n-3)$ to get $(32n + 28)a_{2n-3,n-3}&= \\frac{32n+28}{2n-3}[(n-2)a_{2n-4,n-3} + 6a_{2n-4,n-4} \\\\&\\quad + (n^2-2n-1) a_{2n-5,n-3} + (2n-6)a_{2n-5,n-4}\\\\&\\quad + 48a_{2n-5, n-5} ].$ Now, we will check that each of the terms in the expansion for $(32n + 28)a_{2n-3, n-3}$ is less than one of the terms in the expansion of $(18n-24)a_{2n-4,n-3}+ 384a_{2n-3,n-4}$ .",
"We have $(32n+28)/(2n-3) \\le 17$ , and we see that $17\\cdot (n-2)a_{2n-4,n-3} &\\le (18n-24)a_{2n-4,n-3}, \\\\17\\cdot 6a_{2n-4,n-4} &\\le \\frac{384(n-3)}{2n-3}a_{2n-4,n-4},\\\\17\\cdot (2n-6)a_{2n-5,n-4} &\\le \\frac{68(n^2-4n+4)}{2n-3}a_{2n-5,n-4}, \\\\17\\cdot 48a_{2n-5,n-5} &\\le \\frac{384(10n-44)}{2n-3}a_{2n-5,n-5}.$ Now, it suffices to show $17\\cdot (n^2 - 2n - 1)a_{2n-5,n-3}\\le \\frac{316(n^2-4n+4)}{2n-3}a_{2n-5,n-4}.$ It suffices to show that $9a_{2n-5,n-4} \\ge na_{2n-5,n-3}.$ By the recurrence relation we have $na_{2n-5,n-3} &\\le \\frac{n}{(2n-5)}(2a_{2n-6,n-4} + 8a_{2n-7,n-5}) \\\\4.5 a_{2n-5,n-4} &\\ge \\frac{4.5(n-3)}{(2n-5)}a_{2n-6,n-4} \\\\4.5 a_{2n-5,n-4} &\\ge \\frac{4.5(2n-8)}{(2n-5)} a_{2n-7,n-5}.$ Combining these gives the desired inequality."
],
[
"Proof of the $\\gamma $ -positivity of {{formula:5ef23ba2-5ba5-4aff-8556-9c14ddca32a6}}",
"In this section we prove Theorem , restated below.",
"Let the $\\gamma $ -expansion of $J_{2n}(t)$ be $J_{2n}(t) = \\sum _{k=1}^n b_{2n,k} t^k(1+t)^{2n-2k}.$ We have the following recurrence relation for the coefficients $b_{2n,k}$ .",
"Theorem 3.1 ([6]) For $n\\ge 2$ and $k\\ge 1$ , we have $2nb_{2n,k} &= [k(k+1) + 2n-2]b_{2n-2,k} +[2+2(k-1)(4n-4k-3)]b_{2n-2,k-1} \\\\&\\quad + 8(n-k+1)(2n-2k+1)b_{2n-2,k-2},$ where $b_{2n,k} = 0$ if $k < 1$ or $k > n$ .",
"* We will prove by induction on $n$ the slightly stronger claim that for $b_{2n,k} \\ge 0$ for $n \\ge 9, k \\ge 1$ , and $b_{2n, n} \\ge b_{2n-2,n-1}$ for $n \\ge 11$ .",
"Assume the claim is true whenever the first index is less than $m$ .",
"We want to prove the claim for all $b_{m, k}$ .",
"If $m \\le 2000$ , we can check the claim directly (this has been checked using Sage).",
"Thus, we may assume $m > 2000$ .",
"If $m > 2k$ , then all of the coefficients in the recursion are nonnegative, so we are done by induction.",
"Thus, we can assume that $(m,k) = (2n, n)$ with $n > 1000$ .",
"By the recurrence relation, we have $2nb_{2n,n} = 8b_{2n-2,n-2} - (6n-8)b_{2n-2,n-1}.$ We want to show that $8b_{2n-2,n-2} - (8n-8)b_{2n-2,n-1} \\ge 0$ .",
"We have $8b_{2n-2,n-2} - &(8n-8)b_{2n-2,n-1} \\\\&= 8b_{2n-2,n-2} - 32b_{2n-4, n-3}+ 4(6n-14)b_{2n-4, n-2}.$ Multiplying by $(2n-2)/8$ , it suffices to show $(2n-2) b_{2n-2,n-2} - (8n-8) b_{2n-4,n-3} + (6n^2 - 20n + 14)b_{2n-4,n-2} \\ge 0.$ By expanding $(2n-2)b_{2n-2, n-2}$ using the recursion, we find that the above is equivalent to $&(7n^2 - 21n + 12)b_{2n-4, n-2}+ 48b_{2n-4, n-4} - (6n-4)b_{2n-4,n-3}\\ge 0.",
"$ By the induction hypothesis, $(7n^2-21n+12)b_{2n-4,n-2} \\ge (7n^2-21n+12)b_{2n-6,n-3}.$ By the recurrence in Theorem REF , $(2n-4)_{2n-4,n-4} &\\ge (n^2 - 5n + 6)b_{2n-6, n-4}+ (10n - 48)b_{2n-6,n-5}.$ Multiplying by $48/(2n-4)$ yields $48b_{2n-4,n-4} &\\ge \\frac{48}{2n-4}[ (n^2 - 5n + 6)b_{2n-6, n-4}+ (10n - 48)b_{2n-6,n-5} ].$ Also, multiplying the recurrence for $b_{2n-4, n-3}$ by $(6n-4)/(2n-4)$ yields $(6n-4)b_{2n-4,n-3} &= \\frac{6n-4}{2n-4}[(n^2 -3n)b_{2n-6,n-3}+ (2n - 6)b_{2n-6,n-4} \\\\&\\qquad \\qquad \\qquad + 48b_{2n-6,n-5} ].$ We check that each term in this sum is less that one of the terms in the expansion of $(7n^2-21n+12)b_{2n-6,n-3} + 48b_{2n-4,n-4}$ .",
"We have $(6n-4)/(2n-4) \\le 4$ and $4(n^2-3n)b_{2n-6,n-3} &\\le (7n^2-21n+12)b_{2n-6,n-3} \\\\4(2n-6)b_{2n-6,n-4} &\\le \\frac{48(n^2-5n+6)}{2n-4}b_{2n-6,n-4} \\\\4\\cdot 48 b_{2n-6,n-5} &\\le \\frac{48(10n-48)}{2n-4}b_{2n-6,n-5}.$ Thus $(\\ddagger )$ is true, as desired."
],
[
"$(3412, 3421)$ -avoiding permutations and\nseparable permutations",
"In this section we prove Theorem , restated below.",
"* For convenience we define the following variants of the double descent set.",
"Let $\\operatorname{DD}_0(\\pi ) = \\lbrace i \\in [n] : \\pi (i-1) > \\pi (i) > \\pi (i+1)\\rbrace $ where $\\pi (0) = 0$ , $\\pi (n+1) = \\infty $ , and $\\operatorname{DD}_\\infty (\\pi ) = \\lbrace i \\in [n] : \\pi (i-1) > \\pi (i) > \\pi (i+1)\\rbrace $ where $\\pi (0) = \\infty $ , $\\pi (n+1) = 0$ .",
"Similarly define $\\operatorname{dd}_0(\\pi )$ and $\\operatorname{dd}_\\infty (\\pi )$ .",
"Finally, let $\\operatorname{des}^{\\prime }(\\pi ) = \\#(\\operatorname{Des}(\\pi ) \\setminus \\lbrace n-1\\rbrace ),\\quad \\operatorname{dd}^{\\prime }(\\pi ) = \\#(\\operatorname{DD}(\\pi ) \\setminus \\lbrace n - 1\\rbrace ).$ Let $\\mathfrak {S}_n^1 = \\mathfrak {S}_n(2413, 3142)$ and $\\mathfrak {S}_n^2 = \\mathfrak {S}_n(3412, 3421)$ .",
"For $i = 1, 2$ , define $S_i(x,y,z) &= \\sum _{n \\ge 1} \\sum _{\\pi \\in \\mathfrak {S}_n^i}x^{\\operatorname{des}(\\pi )}y^{\\operatorname{dd}(\\pi )}z^n.$ Moreover, define $F_1(x,y,z) &= \\sum _{n \\ge 1} \\sum _{\\pi \\in \\mathfrak {S}_n^1}x^{\\operatorname{des}(\\pi )}y^{\\operatorname{dd}_0(\\pi )}z^n \\\\R_1(x,y,z) &= \\sum _{n \\ge 1} \\sum _{\\pi \\in \\mathfrak {S}_n^1}x^{\\operatorname{des}(\\pi )}y^{\\operatorname{dd}_\\infty (\\pi )}z^n \\\\T_2(x,y,z) &= \\sum _{n \\ge 1} \\sum _{\\pi \\in \\mathfrak {S}_n^2}x^{\\operatorname{des}^{\\prime }(\\pi )} y^{\\operatorname{dd}^{\\prime }(\\pi )}.$ We will also use $S_i$ , $F_1$ , etc.",
"to denote $S_i(x, y,z)$ , $F_1(x, y, z)$ , etc.",
"The proof of the following lemma is very similar to the proof of [9], so it is omitted.",
"The essence of the proof is Stankova's block decomposition [14].",
"Lemma 4.1 We have the system of equations $S_1 &= z + (z + xyz)S_1 + \\frac{2xzS_1^2}{1 - xR_1F_1} +\\frac{xzS_1^2(F_1 + xR_1)}{1 - xR_1F_1}, \\\\F_1 &= z + (xzS_1 + zF_1) + \\frac{2xzF_1S_1}{1 - xR_1F_1} +\\frac{xzF_1S_1(F_1 + xR_1)}{1-xR_1F_1}, \\\\R_1 &= yz + zS_1 + xyzR_1 + \\frac{2xzR_1S_1}{1-xR_1F_1}+ \\frac{xzR_1S_1(F_1 + xR_1)}{1 - xR_1F_1}.$ Combining the first equation multiplied by $F_1$ and the second equation multiplied by $S_1$ , and combining the first equation multiplied by $R_1$ and the third equation multiplied by $S_1$ , respectively, gives us $ F_1 = \\frac{S_1+xS_1^2}{1+xyS_1}, \\quad R_1 = \\frac{yS_1 + S_1^2}{1 + S_1}.",
"$ Plugging these values into the first equation and expanding yields the following.",
"Corollary 4.2 We have $S_1(x,y,z) = xS_1^3(x,y,z) + xzS_1^2(x,y,z) + (z + xyz)S_1(x,y,z) + z.$ We will show that $S_2$ satisfies the same equation.",
"Lemma 4.3 We have the system of equations $S_2 &= z + zS_2 + (xy - x)zS_2 + xT_2S_2, \\\\T_2 &= z + (x - xy)z^2 + zS_2 + (xyz - 2xz + z)T_2 + xT_2^2.$ By considering the position of $n$ , we see that every permutation $\\pi \\in \\mathfrak {S}_n^2$ can be uniquely written as either $\\pi _1 n$ where $\\pi _1 \\in \\mathfrak {S}_{n-1}^2$ or $\\pi _1 \\ast \\pi _2$ where $\\pi _1\\in \\mathfrak {S}_{k}^2$ , $\\pi _2 \\in \\mathfrak {S}_{n-k}^2$ , $1 \\le k \\le n - 1$ , and $\\pi _1 \\ast \\pi _2 = AnB$ where $A&= \\pi _1(1) \\cdots \\pi _1(k-1) \\\\B&= (\\pi _2(1) + \\ell )\\cdots (\\pi _2(j-1) + \\ell ) \\pi _1(k) (\\pi _2(j+1) + \\ell )\\cdots (\\pi _2(n-k) + \\ell ),$ where $\\pi _2(j) = 1$ and $\\ell = k - 1$ .",
"Furthermore, $\\begin{aligned}[c]\\operatorname{des}(\\pi _1 n) &= \\operatorname{des}(\\pi _1) \\\\\\operatorname{dd}(\\pi _1 n) &= \\operatorname{dd}(\\pi _1) \\\\\\operatorname{des}^{\\prime }(\\pi _1 n) &= \\operatorname{des}(\\pi _1)\\\\\\operatorname{dd}^{\\prime }(\\pi _1 n) &= \\operatorname{dd}(\\pi _1)\\end{aligned}\\hspace{28.45274pt}\\begin{aligned}\\operatorname{des}(\\pi _1 \\ast \\pi _2) &= \\operatorname{des}^{\\prime }(\\pi _1) + \\operatorname{des}(\\pi _2) + 1 \\\\\\operatorname{dd}(\\pi _1 \\ast \\pi _2) &= \\operatorname{dd}^{\\prime }(\\pi _1) + \\operatorname{dd}(\\pi _2) \\\\\\operatorname{des}^{\\prime }(\\pi _1 \\ast \\pi _2) &= \\operatorname{des}^{\\prime }(\\pi _1) + \\operatorname{des}^{\\prime }(\\pi _2) + 1\\\\\\operatorname{dd}^{\\prime }(\\pi _1 \\ast \\pi _2) &= \\operatorname{dd}^{\\prime }(\\pi _1) + \\operatorname{dd}^{\\prime }(\\pi _2),\\end{aligned}$ with the exceptions $\\operatorname{dd}(1 \\ast \\pi _2)= \\operatorname{dd}(\\pi _2) + 1$ , $\\operatorname{des}^{\\prime }(\\pi _1 \\ast 1) = \\operatorname{des}^{\\prime }(\\pi _1)$ , $\\operatorname{dd}^{\\prime }(1 \\ast \\pi _2) = \\operatorname{dd}^{\\prime }(\\pi _2) + 1$ , and $\\operatorname{des}^{\\prime }(1 \\ast \\pi _2) = \\operatorname{des}^{\\prime }(\\pi _2)$ if $n \\le 2$ .",
"With the initial conditions $S_2(x, y, z) = z + \\cdots ,\\quad T_2(x, y, z) = z + 2z^2 + \\cdots ,$ the above implies the stated equations.",
"Solving the equations in Lemma REF shows that $S_2$ satisfies the same equation as $S_1$ ."
],
[
"Concluding remarks and open problems",
"Our proofs of the $\\gamma $ -positivity of $I_n(t)$ and $J_{2n}(t)$ are purely computational.",
"Guo and Zeng first suggested the following question.",
"Problem 5.1 (Guo–Zeng [6]) Give a combinatorial interpretation of the coefficients $a_{n,k}$ .",
"Dilks [3] conjectured the following $q$ -analog of the $\\gamma $ -positivity of $I_n(t)$ .",
"Here $\\operatorname{maj}(\\pi )$ denotes the major index of $\\pi $ , which is the sum of the descents of $\\pi $ .",
"Conjecture 5.2 (Dilks [3]) For $n \\ge 1$ , $\\sum _{\\pi \\in I_n} t^{\\operatorname{des}(\\pi )}q^{\\operatorname{maj}(\\pi )} = \\sum _{k = 0}^{\\lfloor \\frac{n-1}{2}\\rfloor } \\gamma _{n,k}^{(I)} t^k q^{\\binom{k+1}{2}} \\prod _{i = k + 1}^{n - 1 - k} (1 + tq^i),$ where $\\gamma _{n,k}^{(I)}(q) \\in \\mathbb {N}[q]$ .",
"Since the $(3412, 3421)$ -avoiding permutations are invariant under the MFS action, it would be interesting to find a combinatorial proof of Theorem , since this would lead to a group action on $\\mathfrak {S}_n(2413, 3142)$ such that each orbit contains exactly one element of $\\lbrace \\pi \\in \\mathfrak {S}_2(2413, 3142) : \\operatorname{dd}(\\pi ) = 0, \\operatorname{des}(\\pi ) = k \\rbrace $ (cf.",
"[5]).",
"Problem 5.3 Give a bijection between $(3412, 3421)$ -avoiding permutations with $m$ double descents and $k$ descents and separable permutations with $m$ double descents and $k$ descents.",
"Note that there does not exist a bijection preserving descent sets because the separable permutations are not $\\operatorname{Des}$ -Wilf equivalent to any permutation classes avoiding two patterns.",
"Finally, Lin [9] proved that the only permutations $\\sigma $ of length 4 which satisfy $\\sum _{\\pi \\in \\mathfrak {S}_n(\\sigma , \\sigma ^r)} t^{\\operatorname{des}(\\pi )}= \\sum _{k = 0}^{\\lfloor \\frac{n-1}{2}\\rfloor }\\gamma _{n,k}t^k(1 + t)^{n-1-2k}$ where $\\gamma _{n,k} = \\#\\lbrace \\pi \\in \\mathfrak {S}_n(\\sigma , \\sigma ^r): \\operatorname{dd}(\\pi ) = 0, \\operatorname{des}(\\pi ) = k\\rbrace $ are the permutations $\\sigma = 2413$ , 3142, 1342, 2431.",
"Here $\\sigma ^r$ denotes the reverse of $\\sigma $ .",
"We can similarly ask the following.",
"Problem 5.4 Which permutations $\\sigma $ of length $\\ell \\ge 6$ satisfy the above property?",
"Remark 5.5 For $\\ell = 5$ , the answer to Problem REF is $\\sigma = 13254$ , 15243, 15342, 23154, 25143 and their reverses.",
"We have verified using Sage that these are the only permutations which satisfy the property for $n = 5, 6, 7$ , and these permutation classes are all invariant under the MFS action because in these patterns, every index $i \\in [5]$ is either a valley or a peak."
],
[
"Acknowledgements",
"This research was conducted at the University of Minnesota Duluth REU and was supported by NSF / DMS grant 1650947 and NSA grant H98230-18-1-0010.",
"I would like to thank Joe Gallian for suggesting the problem, and Brice Huang for many careful comments on the paper."
]
] | 1808.08481 |
[
[
"Dialkali-Metal Monochalcogenide Semiconductors with High Mobility and\n Tunable Magnetism"
],
[
"Abstract The discovery of archetypal two-dimensional (2D) materials provides enormous opportunities in both fundamental breakthroughs and device applications, as evident by the research booming in graphene, atomically thin transition-metal chalcogenides, and few-layer black phosphorous in the past decade.",
"Here, we report a new, large family of semiconducting dialkali-metal monochalcogenides (DMMCs) with an inherent A$_{2}$X monolayer structure, in which two alkali sub-monolayers form hexagonal close packing and sandwich the triangular chalcogen atomic plane.",
"Such unique lattice structure leads to extraordinary physical properties, such as good dynamical and thermal stability, visible to near-infrared light energy gap, high electron mobility (e.g.",
"$1.87\\times10^{4}$ cm$^{2}$V$^{-1}$S$^{-1}$ in K$_{2}$O).",
"Most strikingly, DMMC monolayers (MLs) host extended van Hove singularities near the valence band (VB) edge, which can be readily accessed by moderate hole doping of $\\sim1.0\\times10^{13}$ cm$^{-2}$.",
"Once the critical points are reached, DMMC MLs undergo spontaneous ferromagnetic transition when the top VBs become fully spin-polarized by strong exchange interactions.",
"Such gate tunable magnetism in DMMC MLs are promising for exploring novel device concepts in spintronics, electronics and optoelectronics."
],
[
"Introduction",
"Although the critical importance of dimensionality in determining the extraordinary physical properties of low-dimension systems has long been recognized since Richard Feyman, the groundbreaking experiments on graphene [1], [2], [3] provide a fascinating platform for exploring exotic phenomena in a rather simple hexagonal lattice.",
"After the gold rush of graphene, the search of new two-dimensional (2D) systems have received unparalleled attention in the hope of novel physical properties and prototype functionalities.",
"However, despite the flourishing of 2D materials [4], [5], [6], [7], [8], [9], [10], the majority of 2D researches remain heavily focusing on the archetypal systems of graphene [11], transition-metal dichalcogenides (TMDCs) [12], [13], [14], [15], and few-layer black phosphorus (BP) [16], [17].",
"These 2D paradigms share the common features of weak interlayer coupling by van der Waals interactions, good thermal stability at room temperature, and high charge carrier mobility.",
"Nevertheless, very recently, there are emergent 2D materials with tempting physical properties beyond the aforementioned systems, such as the InSe family [18], [19], 2D ferromagnetic van der Waals crystals [20], [21], binary maingroup compounds with the BP-type puckering lattice [22], [23], [24], and the huge family of carbide and nitride based transition metal MXenes, in which M represents transition metal cation and X is C or N anion, respectively [8], [25].",
"Here, by using ab initio density functional calculations, we discover a large family of dialkali-metal monochalcogenides (DMMCs) with excellent dynamical and thermal stability, visible to near-infrared light energy gap, and very high electron mobility of exceeding $1.0\\times 10^{3}\\,\\textrm {cm}^{2}\\textrm {V}^{-1}\\textrm {s}^{-1}$ .",
"With the same lattice structure of 1T-TMDCs, DMMCs have the inherent layer-by-layer structure with very weak interlayer coupling, due to the intrinsic +1 oxidation state of alkali cations.",
"All these features make DMMCs a unique choice beyond TMDCs for fundamental studies of 2D systems and for developing potential electronic and optoelectronic devices.",
"Most fascinatingly, all DMMC monolayers (MLs) host extended singularity points in the density of states near the Mexican-hat shaped valence band maximum (VBM).",
"Using hole doping in the range of $\\sim 1.0\\times 10^{13}- 2.5\\times 10^{13}$ cm$^{-2}$ via electrostatic or liquid ion gating, spontaneous ferromagnetic transitions, which lead to half-metallic character with full spin polarized top valence bands, can be triggered by strong exchange interactions in these 2D systems.",
"Such gate tunable magnetism and half metallicity in DMMC MLs may pave new routes in novel device concepts for spintronics.",
"Dialkali-metal monochalcogenides form a large family with sixteen compounds in the general formula of A$_{2}$ X, where A represents an alkali atom (Na, K, Rb or Cs) and X is a chalcogen anion of O, S, Se or Te.",
"The synthesis of bulk dicesium monoxide (Cs$_{2}$ O) was first reported in 1955 [26] with the space group $R\\bar{3}m$ (No.",
"166), which has the same lattice structures as the well-known 1T-TMDCs [27] as displayed in Fig.",
"REF a and REF a.",
"By taking van der Waals (vdW) corrections into account, the optimized lattice constants of bulk Cs$_{2}$ O are $a=4.23$ Å, $b=4.23$ Å, and $c=19.88$ Å, respectively, which are in excellent agreement with the experimental values [26].",
"In the 2D limit, the monolayer of DMMCs has the space group $P\\bar{3}m1$ (No.",
"164).",
"The primitive unit cell consists of two alkali cations and one group-VIa anions, making A$_{2}$ X stoichiometry.",
"The metal-chalcogen ratio in DMMC MLs is distinct from the well-known TMDC MLs, in which the unit cell includes two chalcogenide anions and one metal cation.",
"For instance, in Cs$_{2}$ O ML, the centering O atom is surrounded by six Cs cations, forming a distinctive O-Cs octahedron, as shown in Fig.",
"REF a.",
"Such lattice structure is rooted in strong intralayer O-Cs ionic bonding, as revealed by the analysis of electron localization functions (ELFs).",
"As shown in Fig.",
"REF b, the electron densities are highly localized around Cs and O atoms with negligible inter-atom distribution, reflecting ionic bonding and electron donation from Cs to O atoms.",
"The calculated oxidation state of O anions, as represented by the Hirshfeld charge, is nearly identical to that of BaO, also confirming the dianionic character of oxygen in Cs$_{2}$ O.",
"No accumulation of electron density is observed between Cs atoms, suggesting no chemical bond between them.",
"The A-X octahedral geometry centring the chalcogen dianion is general observed for the whole family, as presented in Fig.",
"REF b for Na$_{2}$ O ML and in Supplementary Information (SI) Figure S1-S4 for the other 14 DMMC MLs.",
"Due to the intrinsic full oxidation state (+1) of Cs, the interlayer coupling between Cs sub-MLs is very weak, which only introduces marginal changes to the in-plane lattice constants.",
"The optimized $a$ and $b$ of Cs$_{2}$ O ML (4.26 Å) are almost the same as the experimental bulk value.",
"Comparing with the paradigmatic 2D systems of MoS$_{2}$ and graphite, we find that there are even less interlayer electron density localization in Cs$_{2}$ O as a result of dominant in-plane ionic bonding, which also means micromechanical exfoliation of Cs$_{2}$ O crystals can be readily achieved (Fig.",
"REF a).",
"Indeed, we calculate the exfoliation energy of Cs$_{2}$ O ML to be $\\sim 0.19$ J/m$^{2}$ (Fig.",
"REF b), which is surprisingly lower than the value of 0.33 J/m$^{2}$ for graphene [28].",
"It has to been emphasized here that the lower exfoliation energy of bulk Cs$_{2}$ O, compared with MoS$_{2}$ ($\\sim $ 0.42 J/m$^{2}$ ) and graphite, is consistently obtained by different vdW correction methods, which is summarized in SI Table S1.",
"To assess the dynamical stability, which is also crucial for the micro-exfoliation of Cs$_{2}$ O ML, we have computed the phonon dispersion with the finite displacement method.",
"As shown in Fig.",
"REF c, the good stability of ML Cs$_{2}$ O is evident by the positive values of all phonon modes.",
"We further check the room-temperature (RT) thermal stability of Cs$_{2}$ O ML by first-principle molecular dynamics simulations.",
"The fluctuation of energy and temperature as a function of time are plotted in Fig.",
"REF d. After running 4000 steps (10 ps), the 3$\\times $ 3$\\times $ 1 trigonal lattice is well sustained, and the free energy of the supercell converges.",
"Excellent dynamical and thermal stability have also been validated for the other DMMC MLs (See SI Figure S5 and S6).",
"Thus, it is feasible to experimentally exfoliate DMMC MLs for device fabrications and applications."
],
[
"Electronic Properties",
"We now elucidate the electronic structures and fundamental physical properties of DMMC MLs using two representative examples of Cs$_{2}$ O and Na$_{2}$ O MLs.",
"As shown in Fig.",
"REF a, Na$_{2}$ O ML is a 2D semiconductor with a direct bandgap of $\\sim $ 1.99 eV at the $\\Gamma $ point.",
"Noticeably, the VBM of Na$_{2}$ O ML is mainly contributed by O-2$p$ orbitals (its projection weight is not shown in the energy dispersion for clarity) hybridized with very low weight of Na-2$p$ (weighted by green triangles), while the CBM is dominated by Na-3$s$ (red squares).",
"The electronic band structure undergoes a pronounced change when Na is replaced by Cs, whose strongly delocalized 5$p$ orbitals induce significant density of states (DOS) away from the $\\Gamma $ point, which is still dominated by O-2$p$ .",
"As the energy of Cs-5$p$ orbitals is $\\sim 0.4$ eV higher than the 2$p$ orbitals of O, the inter $p$ -orbital hybridization reverses the energy levels of VB1 and VB2 along the $\\Gamma -M$ direction.",
"The resulting Cs$_{2}$ O ML is an indirect semiconductor with multiple VBMs centering the $M$ points (Fig.",
"REF b).",
"In general, for heavier cations of K, Rb and Cs, the inter $p$ -orbital hybridization with chalcogen anions is prevailing, shifting VBMs from $\\Gamma $ to the $M$ point.",
"As a result, except for Na$_{2}$ X (X=O, S, Se and Te), all DMMC MLs are indirect semiconductors.",
"The inter $p$ -orbital hybridization induced direct to indirect bandgap transition from Na$_{2}$ X to other A$_{2}$ X MLs has also been confirmed by the PBE method, as shown in SI Table S2 and SI Figure S7.",
"As summarized in Fig.",
"REF c, the energy gaps of DMMC MLs range from 1 eV to 3 eV, covering the near-infrared (K$_{2}$ O, Rb$_{2}$ O and Cs$_{2}$ O MLs) and visible light regions, which are highly desirable for optoelectronic device applications.",
"To evaluate the optical performance, the absorption coefficients of Cs$_{2}$ O and Na$_{2}$ O MLs are directly compared with bulk silicon [29].",
"As shown in Fig.",
"REF d, Na$_{2}$ O ML shows absorption attenuation above $\\sim 625$ nm, in consistent with the direct bandgap of 1.99 eV.",
"Strikingly, indirect-bandgap Cs$_{2}$ O ML exhibits unrivaled cyan-to-red light absorption efficiency, which is nearly constant over the whole visible light wavelengths.",
"Such unusual findings not only predict Cs$_{2}$ O MLs to be extraordinary optoelectronic materials, but also are rooted in the enormous, non-differentiable DOS in the vicinity of VBM, i.e.",
"van Hove singularity.",
"As shown in Fig.",
"REF a and REF b, the top VBs of DMMC MLs, dominated by the localized $p$ orbitals of chalcogen anions, are distinctive by very flat dispersion in the momentum space.",
"The inter $p$ -orbital hybridization in Cs$_{2}$ O ML drastically reduces the VB energy dispersion, which effectively enhances DOS near the VBM and creates multiple saddle points in the vicinity of $\\Gamma $ , $K$ and $M$ .",
"This leads to a prominent extended van Hove singularity in the density of states [30].",
"It is also noteworthy that the inter $p$ -orbital hybridization also greatly enhances DOS near the CBM, which contributes significantly to the extraordinary light absorption characteristics of Cs$_{2}$ O ML.",
"Charge carrier mobility is another critical parameter for the device performance of 2D material-based devices.",
"As summarized in Table REF as well as in SI Table S3 and S4, the mobility of DMMC MLs are largely asymmetric between electrons and holes due to the drastic difference in the energy dispersion between the CBs and VBs, producing highly asymmetric effective mass.",
"Noticeably, K$_{2}$ O ML has the highest electron mobility of 1.87$\\times 10^{4}$ cm$^{2}$ V$^{-1}$ S$^{-1}$ along the $y$ direction.",
"Such electron mobility is comparable to the hole mobility of few-layer BP [31], [17] and order of magnitude larger than that of MoS${_2}$ atomic layers [13].",
"Large electron mobility in DMMC MLs may hold a great promise for applications in high-performance electronics.",
"We have noticed that with the same trigonal structure ($R\\bar{3}m$ , No.166), monolayer Tl$_{2}$ O is predicted to be a semiconductor with a direct bandgap and highly anisotropic charge carrier mobilities up to $4.3\\times 10^{3}\\,\\textrm {cm}^{2}\\textrm {V}^{-1}\\textrm {s}^{-1}$ [32].",
"This is not surprising since Tl is one of the pseudo-alkali metals, in close resemblance to K both in the oxidation state and ionic radius."
],
[
"Tunable Magnetism by Hole doping.",
"The aforementioned van Hove singularities in the top VB of DMMC MLs are associated with inherent electronic instability, when hole doping pushes the Fermi level approaching the divergent point.",
"Taking Cs$_{2}$ O as an example, we find that the system undergoes a spontaneous ferromagnetic phase transition with a moderate hole doping of $n\\sim 1.0\\times 10^{13}$ cm$^{-2}$ .",
"In Fig.",
"REF a, we plot the magnetic moment and the spin polarization energy per doped hole carrier as a function of $n$ .",
"The latter represents the energy difference between the non-spin-polarized phase and the ferromagnetic phase.",
"Clearly, the top VB becomes fully spin polarized by the critical hole doping, as manifested by the constant magnetic moment of 1 $\\mu _{B}$ per carrier above $1.0\\times 10^{13}$ cm$^{-2}$ .",
"The spin polarization energy, which becomes -3.11 meV per carrier for a minimal hoel doping, monotonically decreases as a function of $n$ , approving that the doping-induced ferromagnetic ordering is energy favourable.",
"Such spontaneous ferromagnetic ordering can be well understood by the Stoner model, which has been adopted to explain the so-called “$d^{0}$ ferromagnetism” in hole doped nitrides and oxides [33].",
"In this picture, ferromagnetism can spontaneously appear when the Stoner criterion $U*g(E_{f})>$ 1 is satisfied, where $U$ is the exchange interaction strength and $g(E_{f})$ is the DOS at the Fermi energy of non-magnetic state.",
"In such condition, the energy gain by exchange interactions exceeds the loss in kinetic energy, and hence the system would favor a ferromagnetic ground state.",
"We have confirmed that the $U*g(E_{f})$ of Cs$_{2}$ O is always larger than one above the critical doping level, indicating that the huge DOS associated with the van Hove singularity plays a key role in the occurrence of ferromagnetism.",
"In Fig.",
"REF b and REF c, we plot the spin-polarized DOS of Cs$_{2}$ O with a doping level of $n=5.1\\times 10^{14}$ cm$^{-2}$ .",
"Compared with the undoped case (Fig.",
"REF b), the corresponding spin splitting process causes a significant energy shit of 0.1 eV between the spin-up and spin-down bands, leaving the Fermi level cutting through only one spin channel.",
"Such half-metal state allows fully spin-polarized transport, which is crucial for spintronics applications.",
"It is fascinating that this half-metallic ferromagnetism can be reversibly switched on and off in a $\\sim $ 1 nm thick atomic layer without extrinsic dopant and defects.",
"We have found that except for Na$_{2}$ Te, the ferromagnetic transition can be triggered in all DMMC MLs by hole doping, ranging from $\\sim 1.0\\times 10^{13}$ cm$^{-2}$ for Cs$_{2}$ O to $8.0\\times 10^{15}$ cm$^{-2}$ for Na$_{2}$ Se (see SI Figure S8-S11).",
"The special case of Na$_{2}$ Te can be understood by insufficient DOS near the VBM, which is dominated by the 5$p$ orbitals of Te.",
"Due to much lower electron negativity, the 5$p$ orbitals of Te are more delocalized than the other chalcogen elements.",
"Consequently, the high dispersion of Te-5$p$ reduce DOS near the VBM of Na$_{2}$ Te (SI Fig.",
"S8), invalidating the Stoner criterion.",
"We further evaluated the doping dependent Curie temperature ($T_{c}$ ) of DMMC MLs using the Heisenberg model.",
"As shown in Fig.",
"REF d, using $n=5.0\\times 10^{14}$ cm$^{-2}$ as a reference, which can be readily induced by ionic liquid gating or lithium glass gating [16], [34], RT ferromagnetism is achievable in K$_{2}$ O and Na$_{2}$ O MLs (SI Figure S12).",
"For Cs$_{2}$ O, the corresponding $T_{c}$ is $\\sim 200$ K, which is nearly five times higher than the recently reported ferromagnetic CrI$_{3}$ and Cr$_{2}$ Ge$_{2}$ Te$_{6}$ [20], [21].",
"In summary, we have reported a new family of 1T-TMDC structured dialkali-metal monochalcogenides (DMMCs), which show fascinating physical properties and great device application promises in the 2D limit.",
"Distinctively, DMMC MLs are characterized by large electron mobility, e.g.",
"$1.87\\times 10^{4}$ cm$^{2}$ V$^{-1}$ S$^{-1}$ for K$_{2}$ O and $5.42\\times 10^{3}$ cm$^{2}$ V$^{-1}$ S$^{-1}$ for Na$_{2}$ O, and by very flat VBs with rather weak energy dispersion.",
"The latter are responsible for the formation of extended van Hove singularities with extremely high DOS just below the VB edge of DMMC MLs.",
"By introducing moderate hole doping, these DMMC MLs are subjected to strong electronic instabilities induced by electron-electron interaction, which ultimately leads to spontaneous ferromagnetic phase transitions where the top VBs becomes fully spin-polarized.",
"Using heavy doping methods of ion liquid and solid gating, we can readily increase the $T_{c}$ and switch on and off the magnetism of DMMC MLs even at room temperature.",
"Although alkali compounds are not very stable in air by reacting with ambient moisture, the excellent dynamical and thermal stability allows DMMC MLs to be exfoliated and encapsulated in glove box environment, which is now routine research facilities for studying air-sensitive 2D materials [16], [17], [19], [20], [21].",
"It is also feasible to grow DMMC MLs by molecular beam epitaxy or by chemical vapor deposition, with abundant options of single crystal substrates of 1T- or 2H-TMDCs.",
"By forming heterostructure with high electron affinity main-group MDCs, such as SnSe$_{2}$ , DMMC MLs may become effectively hole doped by interfacial charge transferring [24], which would provide the opportunity for in-situ studying the gate-tunable ferromagnetism in DMMC MLs by scanning probe microscope.",
"Figure: (a) Structure of DMMC MLs.",
"The upper inset is the first Brillouin zone.",
"Each chalcogen anion is surrounded by six alkali cations, forming XA 6 _{6} octahedron.",
"(b-c) Normalized electron localization function (ELF) of Cs 2 _{2}O and Na 2 _{2}O, respectively.",
"ELF = 1 (red) and 0 (blue) indicate accumulated and vanishing electron densities, respectively.",
"The 2D ELF plane is defined by the green line in Fig.",
"1a.Figure: Stability and Micromechanical Exfoliation.",
"(a) Interlayer differential charge density of Cs 2 _{2}O bulk, graphite and MoS 2 _{2} bulk.",
"(b) Calculated exfoliation energy of Cs 2 _{2}O ML, in comparison with graphene (blue triangles).",
"(c) The phonon spectrum of Cs 2 _{2}O ML.",
"(d) First-principle molecular dynamics simulations of Cs 2 _{2}O ML.",
"The thermal stability of the supercell is evident by the convergence in the free energy.Figure: Electronic Properties.",
"Band structures and the corresponding DOS of Na 2 _{2}O ML (a) and Cs 2 _{2}O ML (b).",
"The contributions of alkali ss-orbital and pp-orbital are indicated by red and green circles, respectively.",
"The insets are the spatial distribution of the wave-functions for the VBM and CBM.",
"(c) Energy gaps of all DMMC MLs.",
"Strong inter pp-orbital hybridization leads to the transition from direct bandgaps in Na 2 _{2}X to indirect gaps in other DMMC MLs.",
"The gap values of DMMC MLs are mainly determined by the electron affinity of the anions.",
"The results have been confirmed by both the HSE06 and PBE methods.",
"(d) Extraordinary light absorbance of Cs 2 _{2}O and Na 2 _{2}O MLs, in comparison with silicon.Figure: Tunable Magnetism by Hole Doping.",
"(a) Magnetic moment and spin polarization energy per carrier of Cs 2 _{2}O ML.",
"Remarkably, spontaneous ferromagnetic phase transition is trigger by a moderate hole doping of 1.28×10 13 1.28\\times 10^{13} cm -2 ^{-2}.",
"(b) Spin-resolved DOS of Cs 2 _{2}O ML at n=5×10 14 n=5\\times 10^{14} cm -2 ^{-2}.",
"The inset shows the spin density distribution, apparently locating at O atoms.",
"(c) Spin-splitted top valence bands in the vicinity of MM points in Cs 2 _{2}O ML (n=5×10 14 n=5\\times 10^{14} cm -2 ^{-2}).",
"(d) Curie temperature verusverus hole doping density, showing the feasibility of RT ferromagnetism by heavy hole doping.",
"Note that antiferromagnetic ordering in DMMC MLs is energetically higher in free energy than the ferromagnetic state.Table: Mobility and effective mass along the xx and yy transport directions in four Cs 2 _{2}X MLs and K 2 _{2}O.",
"We calculated the carrier mobility using a phonon-limited scattering model, which includes deformation potential (E 1 _{1}) and elastic modulus (C 2D _{2D}) in the propagation direction of the longitudinal acoustic wave.",
"Noticeably, K 2 _{2}O has the highest electron mobility.This work is supported by the National Key R&D Program of the MOST of China (Grant Nos.",
"2016YFA0300204 and 2017YFA0303002), and the National Science Foundation of China (Grant Nos.",
"11574264 and 61574123).",
"Y.Z.",
"acknowledges the funding support from the Fundamental Research Funds for the Central Universities and the Thousand Talents Plan."
],
[
"Competing financial interests:",
"The authors declare no competing financial interests."
]
] | 1808.08461 |
[
[
"Context-Free Session Types for Applied Pi-Calculus"
],
[
"Abstract We present a binary session type system using context-free session types to a version of the applied pi-calculus of Abadi et.",
"al.",
"where only base terms, constants and channels can be sent.",
"Session types resemble process terms from BPA and we use a version of bisimulation equivalence to characterize type equivalence.",
"We present a quotiented type system defined on type equivalence classes for which type equivalence is built into the type system.",
"Both type systems satisfy general soundness properties; this is established by an appeal to a generic session type system for psi-calculi."
],
[
"Introduction",
"Binary session types [6] describe the protocol followed by the two ends of a communication medium, in which messages are passed.",
"A sound type system of this kind guarantees that a well-typed process does not exhibit communication errors at runtime.",
"Session types have traditionally been used to describe linear interaction between partners [10], but later type systems are able to distinguish between linear and unlimited channel usages.",
"In particular Vasconcelos has proposed session types with $\\textsf {lin}/\\textsf {un}$ qualifiers that describe linear interaction as well as shared resources [10].",
"Context-free session types introduced by [9] are more descriptive than the regular types described in previous type systems [6], [10] in that they allow full sequential composition of types.",
"Because of this, session types can now describe protocols that cannot be captured in the regular session types, such as transmitting complex composite data structures.",
"Many binary session type systems are concerned with languages that use the selection and branching constructs introduced in [6] in addition to the normal input and output constructs.",
"These constructs are synchronisation operations, where two processes synchronise on a channel, and are similar to method invocations found in object-oriented programming.",
"A branching process $l \\rhd \\lbrace l_1 :P_1, \\ldots , l_n : P_n \\rbrace $ continues as process $P_k$ together with $P$ if label $l_k$ is selected on channel $c$ using the selection $c \\lhd l_k.P$ .",
"In this article we consider a session type system for a version of the applied pi-calculus, due to Abadi et.",
"al.",
"[1].",
"This is an extension of the pi-calculus [8] with terms and extended processes and in our case extended further with selection and branching.",
"Our version is a “low-level” version in that allows one to build composite terms but only allows for the communication of names and nullary function symbols.",
"In this way, the resulting version is close to the versions of the pi-calculus used for encoding composite terms [8].",
"Our session type system combine ideas from the type systems from [9] and [10] into a type system in which session types are context-free and use $\\textsf {lin}/\\textsf {un}$ qualifiers in order to distinguish between linear and unbounded resources.",
"The resulting type system uses types that are essentially process terms from a variant of the BPA process calculus [4].",
"In Section we present our version of the applied pi-calculus and in Section we define the syntax and semantics of our session type system.",
"We then prove the soundess of our type system by using the general results about psi-calculi from [7].",
"This is done by showing that the applied pi-calculus is a psi-calculus, and that our type system is a instance of the generic type system presented by Hüttel in [7].",
"Lastly we present type equivalence between types by introducing a notion of type bisimulation for endpoint types.",
"By considering equivalence classes under type bisimilarity we get a new quotiented type system whose types are equivalence classess.",
"It then follows from the theorems in [7] that the general results for our first type system also hold for the quotiented type system."
],
[
"Applied pi-Calculus",
"We consider a “low-level” version of the applied pi-calculus [1] in which composite terms are allowed but only the transmission of simple data is possible: Only names $n$ , constants represented by functions $f_0$ with arity 0 and functions that evaluate to values of base types can be transmitted.",
"We use the notation $\\widetilde{M}$ to represent the sequence $M_1,\\ldots ,M_i$ and $\\widetilde{x}$ to represent the sequence of variables $x_1,\\ldots ,x_i$ .",
"We always assume that our processes are specified relative to a family of parameterized agent definitions that are on the form $N(\\widetilde{x})\\overset{\\underset{\\mathrm {def}}{}}{=}A$ and that every agent variable $N$ occurring in a process has a corresponding definition.",
"The formation rules for processes $P$ and extended processes $A$ can be seen below in ().",
"Note that we distinguish between variables ranged over by $x, y \\ldots $ and names ranged over by $m,n, \\ldots $ .",
"We let $a$ range over the union of these sets.",
"We extend the syntax of processes with branching $c \\rhd \\lbrace l_1:P_1,\\ldots ,l_k:P_k \\rbrace $ and selection $c \\lhd l.P$ where $l_1, \\ldots l_k$ are taken from a set of labels.",
"Extended processes extend processes with the ability to use active substitutions of the form $\\lbrace {M}{x}\\rbrace $ that instantiate variables.",
"P ::= 0P1 P2 !P ( n)P ( n) P if M1=M2 then P1 else P2 n(x).P nu .P c l.P c { l1:P1,...,lk:Pk } N(M) u ::= n x f0 M ::= n x f(M) A ::= P A1 A2 (n)A ( x)A {Mx} The notion of structural congruence extends that of the usual pi-calculus [8].",
"The following two further axioms that are particular to the applied pi-calculus are of particular importance, as they show the role played by active substitutions.",
"$ P \\mid \\lbrace {M}{x}\\rbrace \\equiv P\\lbrace {M}{x}\\rbrace \\mid \\lbrace {M}{x}\\rbrace & & & \\mathbf {0}\\equiv (\\nu \\, x) \\lbrace {M}{x}\\rbrace $ Together with the axioms of [8] they allow us to factor out composite terms such that they only occur in active substitutions.",
"For instance, we have that $\\texttt {if }M_1 = M_2\\texttt { then }P\\texttt { else }Q\\equiv \\nu {x}\\nu {y}(\\texttt {if }x=y\\texttt { then }P\\texttt { else }Q \\mid \\lbrace {M_1}{x}\\rbrace \\mid \\lbrace {M_2}{y}\\rbrace )$ .",
"We can therefore use a term by only mentioning the variable associated with it.",
"In our version of the applied pi-calculus we very directly make use of this.",
"Our semantics consists of reduction semantics and a labelled operational semantics that extend those of [1] with rules for branching and selection.",
"Reductions are of the form $P \\rightarrow P^{\\prime }$ , while transitions are of the form $P \\xrightarrow{} P^{\\prime }$ where the label $\\alpha $ is given by $ \\alpha ::= c \\lhd l \\mid c \\rhd l \\mid a(x) \\mid \\overline{a}x \\mid \\tau $ and $c \\lhd l$ is the co-label of $c \\rhd l$ .",
"The rules defining reductions and labelled transitions are found in Tables REF and REF , respectively.",
"Table: Reduction rulesTable: Labelled transition rules"
],
[
"A context-free session type system",
"We now present the syntax and semantics of our type system."
],
[
"Session types",
"Session types describe the communication protocol followed by a channel.",
"Consider writing a process that transmits a binary tree where each internal node containsan integer.",
"We want to transmit this tree by sending only base types (i.e.",
"the integers) on a channel.",
"The tree data type can be described with the grammar below.",
"$\\textsf {Tree} ::= (\\textsf {Int}, \\textsf {Tree}, \\textsf {Tree})\\ \\text{ ǀ\\ }\\ \\textsf {Leaf}$ Using the regular types introduced in [10], the type for a channel transmitting such a data structure could be the recursive type $\\mu z.\\oplus \\lbrace \\textsf {Leaf}: \\textsf {lin }\\textsf {skip}\\quad \\textsf {Node}:\\textsf {lin }!\\textsf {Int}.z\\rbrace $ .",
"In this type $z$ is a type variable that is recursively defined.",
"The type describes how a single node is transmitted, if it is a leaf node we do not do anything, if it is an internal node then the integer value is transmitted with the output type $!\\textsf {Int}$ and then the sub-trees of the node are transmitted with a recursive call.",
"However, if we use this regular session type, we are not able to guarantee that the tree structure is preserved.",
"The reason is that the session type describes that a list of nodes are being sent, but not the position in the tree of each node.",
"On the other hand, if we use the context-free session type disciple introduced by Thiemann and Vasconcelos [9], we can specify the preservation of tree structures by using types such as $\\mu z.\\oplus \\lbrace \\textsf {Leaf}: \\textsf {skip}\\quad \\textsf {Node}:!\\textsf {Int};z;z\\rbrace $ .",
"Using sequential composition with the $\\_;\\_$ operator, we can specify a protocol that will guarantee that the tree structure by first sending the left sub-tree and then the right sub-tree.",
"Introducing a sequential operator can introduce challenges for typing a calculus, as the following example shows.",
"If we were to reuse a channel by sending an integer after transmitting a tree, the type would be $\\mu z.\\oplus \\lbrace \\textsf {Leaf}: \\textsf {skip}\\quad \\textsf {Node}:!\\textsf {Int};z;z\\rbrace ;!\\textsf {Int}$ .",
"${\\Gamma (c)=\\oplus \\lbrace l_i: T_{E_i}\\rbrace _{i\\in I}}{\\Gamma \\vdash \\text{select } l_i \\text{ on } c}$ When typing rules are created for a calculus, it is often defined on the structure of types and terms.",
"An example of such a typing rule for a select statement is shown in (REF ), which says that a select statement is well typed if the select operation is performed on a channel with a select type.",
"If however the channel has a type as shown before, the select rule cannot be used, as the channel has the type of a sequential composition, and inside the sequential composition we have a recursive type.",
"We require an equirecursive treatment of types, which allows us to expand the type to $\\oplus \\lbrace \\textsf {Leaf}: \\textsf {skip}\\quad \\textsf {Node}: !\\textsf {Int};\\mu z.\\oplus \\lbrace \\textsf {Leaf}:\\textsf {skip}\\quad \\textsf {Node}: !\\textsf {Int};z;z\\rbrace ;\\mu z.\\oplus \\lbrace \\textsf {Leaf}: \\textsf {skip}\\quad \\textsf {Node}: !\\textsf {Int};z;z\\rbrace \\rbrace ;!\\textsf {Int}$ .",
"This is achieved by unfolding a recursive type $\\mu z.T$ to $T$ where all occurrences of $z$ in $T$ are replaced with the original $\\mu z.T$ .",
"So now we are left with a sequential composition with a select type and output type.",
"In order to transform this into a select type, we need a distributive law that allows us to move the sequential composition inside the select type, in order to obtain the type $\\oplus \\lbrace \\textsf {Leaf}:\\textsf {skip};!\\textsf {Int}\\quad \\textsf {Node}: !\\textsf {Int};\\mu z.\\oplus \\lbrace \\textsf {Leaf}: \\textsf {skip}\\quad \\textsf {Node}: !\\textsf {Int};z;z\\rbrace ;\\mu z.\\oplus \\lbrace \\textsf {Leaf}:\\textsf {skip}\\quad \\textsf {Node}: !\\textsf {Int};z;z\\rbrace ;!\\textsf {Int}\\rbrace $ .",
"Such a rule does exist, we will introduce a method to prove that these types exhibit the same behaviour in Section when we introduce type equivalence."
],
[
"The language of types",
"We denote the set of all types by $\\mathcal {T}$ ; the types $T\\in \\mathcal {T}$ are described by the formation rules in (REF ).",
"These session types are a modified version of the context-free session types presented by [9].",
"The modifications made to the types are that we allow input and output session types to transmit other session types to allow sending channels in other channels, and finally that we introduce the lin and un qualifiers.",
"We let $B$ range over a set of base types.",
"p ::= skip ?T !T &{li:TEi} {li:TEi} q ::= lin un TE ::= q p z z.TE TE1; TE2 T ::= S B TE S ::= (TE1, TE2) From the formation rules we can see that a session type $S$ is a pair of endpoint types $T_{E_1}$ and $T_{E_2}$ .",
"Endpoint types describe one end of channel, and are the types that evolve when a channel is used.",
"The qualifiers lin and un from [10] are used to describe a linear interaction between two partners and a unrestricted shared resources respectively.",
"An example of an un type could be a server, that a lot of processes have access to.",
"To ensure that no communication errors occur if multiple processes can read from a channel concurrently, we require that the type must never change behaviour.",
"This means that an un type must be the same before and after a transition."
],
[
"Transitions for types",
"We now present the transition rules for endpoint types, which describe how the types can evolve when an action is performed.",
"We use an annotated reduction semantics to describe the behaviour of our types.",
"Our labels are generated by the grammar in (REF ) where we have select, branch, input and output actions.",
"The transitions are of the form $T_E\\xrightarrow{}T_E^{\\prime }$ and are shown in Table REF .",
"We let $T_E\\lbrace {y}{x}\\rbrace $ be the endpoint type $T_E$ where all free occurrences of $x$ has been replaced with $y$ .",
"$\\lambda ::= !T_1 \\mid ?T_1 \\mid \\rhd \\, l \\mid \\lhd \\, l$ In the transition rules, we use the function $Q$ defined in (REF ) to find the qualifier of compound types such as recursive types or sequential composition types.",
"$\\begin{split}{Q}(q\\ p) &\\overset{\\underset{\\mathrm {def}}{}}{=}q \\\\{Q}(T_{E_1};T_{E_2}) &\\overset{\\underset{\\mathrm {def}}{}}{=}{Q}(T_{E_1}) \\\\{Q}(\\mu z.T_E) &\\overset{\\underset{\\mathrm {def}}{}}{=}{Q}(T_E)\\end{split}$ A relation $\\sqsubseteq =\\lbrace (\\textsf {lin},\\textsf {un}),(\\textsf {un},\\textsf {un}),(\\textsf {lin},\\textsf {lin})\\rbrace $ is defined for qualifiers in [10].",
"We also follow the definition of $q(T)$ and $q(\\Gamma )$ from that of [10].",
"In short $\\textsf {lin}(\\Gamma )$ is always satisfied, and $\\textsf {un}(\\Gamma )$ is satisfied iff all elements in $\\Gamma $ are unrestricted.",
"The type system contains the sequential operator $\\_;\\_$ as well as choice operators; select $\\&\\lbrace \\ldots \\rbrace $ and branch $\\oplus \\lbrace \\ldots \\rbrace $ .",
"This is very similar to Basic Process Algebra (BPA)[4] that contains the sequential operator $\\cdot $ and nondeterministic choice operator $+$ .",
"We also have recursive types, which corresponds to variables with recursive definitions in BPA.",
"A BPA expression is guarded if all recursive variables on the right hand side is preceded by an action $\\lambda $ [4].",
"Similarly we say that a type is guarded if every recursion variable is preceded by an input or output.",
"These similarities with BPA will become very important, as we can describe types as BPA expressions and by showing results about these expression, we can in turn show results about our types.",
"Table: Annotated reduction semantics for types"
],
[
"Typing rules",
"We now present a type system for our version of the applied pi-calculus in Table REF .",
"In this type system, the type judgements for processes are on the form $\\Gamma \\vdash P$ meaning that the process $P$ is well typed in context $\\Gamma $ .",
"The judgments for terms are on the form $\\Gamma \\vdash M:T$ meaning that the term $M$ is well typed with type $T$ in context $\\Gamma $ .",
"Lastly the judgments for extended processes are on the form $\\Gamma \\vdash _A A$ meaning that the process $A$ is well typed in context $\\Gamma $ .",
"We type processes in a type context $\\Gamma $ which contains types for the variables, names and functions symbols of a process.",
"We follow the definition of a type context from [10]: $\\emptyset $ is the empty context, $\\Gamma , x: T$ is the context equal to $\\Gamma $ except that $x$ has the type $T$ in the new context.",
"This operation is only defined when $x\\notin \\text{dom}(\\Gamma )$ .",
"Table REF shows the typing rules for processes.",
"We use the context split $\\circ $ and context update $+$ operations from [10].",
"The context split operation is used to split a context into two constituents.",
"A maximum of two processes must have access to a given linear session type; a context either contains the entire session type $S=(T_{E_1},T_{E_2})$ , or a single endpoint type $T_{E}$ .",
"When splitting a context into two, we can pass $S$ to either context, or one endpoint to each context.",
"If the context only has an endpoint type, the endpoint type can only be passed on to one of the two contexts.",
"This way we ensure that each lin endpoint of a channel is known in exactly one context.",
"Names of unrestricted type can be shared among all contexts.",
"The context update operation updates the type of a channel.",
"The $\\Gamma , x:T$ operation is only defined when $x\\notin \\text{dom}(\\Gamma )$ .",
"The $+$ operation $\\Gamma =\\Gamma _1+\\Gamma _2$ uses the type of $x$ in $\\Gamma _2$ to update the type in $\\Gamma _1$ .",
"So if $\\Gamma _1(x)=T_1$ and $\\Gamma _2(x)=T_2$ then $\\Gamma (x)=T_2$ .",
"The two operations are used in (Input) and (Output) where we must split our context into two, to type each endpoint of a linear channel in its own context.",
"Table: Typing rules for Extended ProcessesTable: Context split for Applied π\\pi -calculus, based on Table: Context update for Applied π\\pi -calculus from Table: Typing rules for termsTable: Typing rules for processes"
],
[
"Duality of types",
"The notion of type duality is central to session type systems; it expresses that the protocols followed by the two endpoints of a name must be opposites: If one end transmits a value, the other end must receive it.",
"We denote the dual of an endpoint type $\\overline{T_E}$ and define duality below, following [9].",
"$\\overline{q\\ \\textsf {skip}}&\\overset{\\underset{\\mathrm {def}}{}}{=}q\\ \\textsf {skip} && \\overline{q\\ \\&\\lbrace l_1: T_{E_1}, \\ldots , l_k:T_{E_k}\\rbrace }\\overset{\\underset{\\mathrm {def}}{}}{=}q\\ \\oplus \\lbrace l_1: \\overline{T_{E_1}}, \\ldots , l_k:\\overline{T_{E_k}}\\rbrace \\\\\\overline{q\\ ?T}&\\overset{\\underset{\\mathrm {def}}{}}{=}q\\ !T && \\overline{q\\ \\oplus \\lbrace l_1: T_{E_1}, \\ldots , l_k:T_{E_k}\\rbrace }\\overset{\\underset{\\mathrm {def}}{}}{=}q\\ \\&\\lbrace l_1: \\overline{T_{E_1}}, \\ldots , l_k:\\overline{T_{E_k}}\\rbrace \\\\\\overline{q\\ !T}&\\overset{\\underset{\\mathrm {def}}{}}{=}q\\ ?T && \\overline{\\mu z.T_E} \\overset{\\underset{\\mathrm {def}}{}}{=}\\mu z.\\overline{T_E} \\\\\\overline{T_{E_1};T_{E_2}}&\\overset{\\underset{\\mathrm {def}}{}}{=}\\overline{T_{E_1}};\\overline{T_{E_2}} && \\overline{z} \\overset{\\underset{\\mathrm {def}}{}}{=}z$ A session type $S = (T_{E_1},T_{E_2})$ is balanced iff its endpoint types are dual, that is, if $\\overline{T_{E_1}} = T_{E_2}$ .",
"A type context $\\Gamma $ is balanced iff every session type in the range of $\\Gamma $ is balanced.",
"We use the notation $\\Gamma \\vdash _\\text{bal}P$ to describe that a process $P$ is well typed in a balanced context $\\Gamma $ ."
],
[
"Applied pi-calculus as a psi-calculus instance",
"Psi-calculus is a general framework for process calculi.",
"In this section we show that the applied pi-calculus is an instance of it, and this will then be used in Section to obtain results about our type system.",
"An instance of the psi-calculus framework contains the seven elements [3] given in Table REF .",
"Table: Elements of psi-calculiThe syntax of an instance of the psi-calculus is described by the formation rules in (REF ) that generalize those of the pi-calculus.",
"The input and output constructions allows to use arbitrary terms as channels, and the input construction allows for matching on a pattern $(\\lambda \\tilde{x})N$ ; the pattern variables in $\\tilde{x}$ are bound to the subterms that match.",
"The other syntactic constructs are similar to those of the pi-calculus.",
"The case construct is a generic case of the if-construct; we generalize match conditions $M_1=M_2$ to allow any $\\phi \\in \\textbf {C}$ as a condition.",
"Lastly in psi-calculi we have a concept of assertions $\\in \\textbf {A}$ .",
"These generalize the notion of active substition found in the applied pi calculus.",
"$P :=\\ &\\mathbf {0}\\mid \\overline{M}N.P \\mid \\underline{M}(\\lambda \\widetilde{x})N.P \\mid \\textbf {case}\\ \\phi _1:P_1[]\\ldots [] \\phi _n: P_n \\\\\\text{ ǀ\\ }\\ &(\\nu a)P \\mid P\\text{ǀ}Q \\mid !P \\mid \\llparenthesis \\rrparenthesis \\mid M l.P \\mid M \\triangleright \\lbrace l_1:P_1,\\ldots ,l_k:P_k \\rbrace $ The structural operational semantics of psi-calculi has transitions of the form $\\Psi \\blacktriangleright P\\xrightarrow{}P^{\\prime }$ where the labels $\\alpha $ are given by the formation rules below [7].",
"::= M l | M l | M(va)N | KN | (va)@ (vb)(MNK) | (va)@ (MlN) The first four actions correspond to the actions we know from the applied pi-calculus: selection, branch, output and input.",
"The last two action are internal $\\tau $ -actions that correspond to internal actions that are either an input/output-exchange or a branch/select-exchange.",
"It follows from[3] that the standard pi-calculus is an instance of the psi-calculus.",
"Below we do the same for the instance APi that shows that the applied pi-calculus is also an instance of the psi-calculus.",
"In the definition $\\mathcal {N}$ is the set of all names and $\\mathcal {F}$ is the set of all function names.",
"$\\textbf {T} &\\overset{\\underset{\\mathrm {def}}{}}{=}\\mathcal {N} \\cup \\lbrace f(M_1, \\ldots M_k) \\text{ ǀ\\ } f \\in \\mathcal {F}, M_i \\in \\textbf {T}\\rbrace & &\\textbf {C} && \\overset{\\underset{\\mathrm {def}}{}}{=}\\lbrace M=N \\text{ ǀ\\ } M,N\\in \\textbf {T}\\rbrace \\\\\\textbf {A} &\\overset{\\underset{\\mathrm {def}}{}}{=}\\lbrace 1\\rbrace & &{\\hspace{-1.66656pt}\\cdot }&&\\overset{\\underset{\\mathrm {def}}{}}{=}\\lbrace ((n, n), n=n) \\text{ ǀ\\ } n \\in \\mathcal {N} \\rbrace \\\\\\otimes &\\overset{\\underset{\\mathrm {def}}{}}{=}\\lbrace ((_1, _2), 1) \\text{ ǀ\\ } \\in \\textbf {A} \\rbrace & &\\textbf {1} & &\\overset{\\underset{\\mathrm {def}}{}}{=}1 \\\\\\vdash &\\overset{\\underset{\\mathrm {def}}{}}{=}\\lbrace (1, M=M) \\text{ ǀ\\ } M\\in \\textbf {T}\\rbrace $"
],
[
"Properties of our type system",
"A generic binary session type system for psi-calculi was presented in [7].",
"The intention is that any existing binary session type systems for process calculi can be captured as special instances of the generic system, as long it satisfies four specific requirements.",
"We already know that our applied pi-calculus is a simple psi-calculus and now establish that our type system fulfils the requirements of [7].",
"This will then allow us to obtain the usual results for binary type systems as a simple corollary of the theorems for the generic system."
],
[
"Transition structure",
"Type transitions in our type system are an instance of the generic type transitions in [7].",
"Both the generic type transitions and our type transition consist of send, receive, branch and select.",
"The syntax of type transitions is uniform across the two articles as they are both generated by the grammar in (REF ).",
"This illustrates that there is a one-to-one correspondence between the type transitions in the two type systems.",
"$\\lambda ::= l \\text{ ǀ\\ } \\triangleright l \\text{ ǀ\\ } !T \\text{ ǀ\\ } ?T$"
],
[
"Revisiting duality",
"Duality of types in our pi-calculus is defined on the structure of types, as seen in Section REF .",
"In [7] duality is defined on type transitions, where (REF ) holds for dual types (we use $\\underline{\\,\\cdot \\,}$ to denote duality defined on type transitions).",
"$T_E \\xrightarrow{} T_E^{\\prime } \\Leftrightarrow \\underline{T_E} \\xrightarrow{} \\underline{T_E^{\\prime }}$ In (REF ) and () we see the duality of type transitions, as presented by [7].",
"$\\overline{!T_1} = ?T_2 && \\overline{?T_1} = !T_2 \\\\\\overline{l} = \\triangleright l && \\overline{\\triangleright l} = l$ We now show that the duality defined in Section REF upholds the property in (REF ).",
"Lemma 1 $\\underline{T_E} =\\overline{T_E}$ By induction in the structure of types."
],
[
"Checking requirements",
"In the type system presented in [7], type judgements are relative to a type context and an assignment and therefore of the form $\\Gamma ,\\vdash {J}$ , where the judgment body is either ${J}$ is either a term typing $M : T$ or $P$ , the statement that process $P$ is well-typed.",
"We write $\\Gamma ,\\vdash _\\text{min} {J}$ , if $\\Gamma ^{\\prime },^{\\prime }\\lnot \\vdash _\\text{min} {J}$ for every smaller $\\Gamma ^{\\prime }$ and $^{\\prime }$ .",
"For each requirement presented in [7] we show that it is satisfied in our type system."
],
[
"Requirement 1:",
"If $\\Gamma _1, _1 \\vdash _\\text{min}{J} $ and $\\Gamma _2, _2 \\vdash _\\text{min} {J}$ then $\\Gamma _1 = \\Gamma _2 $ and $_1 \\simeq _2$ .",
"As we only have the assertion 1 in type judgements, $_1\\simeq _2$ is trivially fulfilled as both are 1.",
"Let $\\Gamma _1,\\Gamma _2$ be type contexts such that $\\Gamma _1, \\textbf {1} \\vdash _{\\text{min}}{J}$ and $\\Gamma _2, \\textbf {1}\\vdash _{\\text{min}}{J}$ .",
"Assume that $\\Gamma _1 \\ne \\Gamma _2$ then without loss of generality there exists an $x$ such that $x\\in \\text{dom}(\\Gamma _1)$ and $x\\notin \\text{dom}(\\Gamma _2)$ .",
"Because judgments must be well-formed we know that $fn({J})\\subseteq \\text{dom}(\\Gamma _1)$ and $fn({J})\\subseteq \\text{dom}(\\Gamma _2)$ , hence $x\\notin fn({J})$ .",
"Let $\\Gamma _1^{\\prime }$ be defined as $\\Gamma _1$ except $x\\notin \\text{dom}(\\Gamma _1^{\\prime })$ , then $\\Gamma _1^{\\prime },\\textbf {1}\\vdash {J}$ and $\\Gamma _1^{\\prime }<\\Gamma _1$ thus $\\Gamma _1,\\textbf {1}\\nvdash _{\\text{min}}{J}$ .",
"This is a contradiction, hence our assumption is wrong and $\\Gamma _1=\\Gamma _2$ , which means that the requirement is fulfilled."
],
[
"Requirement 2:",
"If $\\Gamma , \\vdash M:T@c$ then $\\Gamma (c)=T_E$ for some endpoint type $T_E$ .",
"In our calculus, the only terms that can be used as channels are names.",
"The $(\\nu n)P$ construct uses a channel constructor $n$ to declare a channel with a session type $\\Gamma , \\vdash n: S@n$ .",
"$S$ is a session type $S=(T_{E_1}, T_{E_2})$ for some endpoint types.",
"When type checking $n$ , only one endpoint is present in the context $\\Gamma $ , in which case $\\Gamma (n)=T_{E_1}$ or $\\Gamma (n)=T_{E_2}$ , meaning that the requirement is satisfied."
],
[
"Requirement 3:",
"Suppose $\\widetilde{N}\\in \\textsc {match}(M, \\widetilde{x}, X)$ , $\\Gamma , \\vdash M$ and $\\Gamma _1+\\widetilde{x}:\\widetilde{T},_1\\vdash _{\\text{min}}X:\\widetilde{T}\\rightarrow U$ .",
"Then there exist $\\Gamma _{2i},_{2i}$ such that $\\Gamma _{2i}, _{2i}\\vdash _{\\text{min}}N_i:T_i$ for all $1\\le i \\le \\vert \\widetilde{x}\\vert =n$ .",
"In our calculus, the only possible match is $M \\in MATCH(M, x, x)$ .",
"As $\\vert \\widetilde{x}\\vert =1$ the requirement becomes $\\Gamma _{2},_{2}\\vdash _{\\text{min}}M : T$ .",
"From the requirement that $M$ is well typed, this is trivially fulfilled."
],
[
"Requirement 4:",
"If ${\\hspace{2.77771pt}\\text{ǀ}\\hspace{-5.0pt}=}M {\\hspace{-1.66656pt}\\cdot }K$ and $\\Gamma , \\vdash M:S$ then $\\Gamma , \\vdash K:S$ .",
"If ${\\hspace{2.77771pt}\\text{ǀ}\\hspace{-5.0pt}=}M {\\hspace{-1.66656pt}\\cdot }K$ and $\\Gamma , \\vdash M:T$ then $\\Gamma , \\vdash K:\\overline{T}$ .",
"In Section () we defined ${\\hspace{-1.66656pt}\\cdot }$ as $=$ , meaning that two channels are equal if it is the same name.",
"The first part of the requirement becomes: If ${\\hspace{2.77771pt}\\text{ǀ}\\hspace{-5.0pt}=}n = n$ and $\\Gamma ,\\vdash n:S$ then $\\Gamma , \\vdash n:S$ which is trivially fulfilled.",
"The second part happens when only one end of a channel is in the context, with an endpoint type, then the other end of the channel must have a dual endpoint type.",
"If ${\\hspace{2.77771pt}\\text{ǀ}\\hspace{-5.0pt}=}n = c$ and $\\Gamma , \\vdash n:T_E$ then $\\Gamma , \\vdash c:\\overline{T_E}$ .",
"A channel has a balanced session type $S=(T_E, \\overline{T_E})$ for some endpoint type.",
"If $n$ and $c$ have endpoint types and are the same channel, then they must also have types dual of each other.",
"If the types are not unrestricted then they can only evolve into other types dual of each other, due to the requirement that only two processes have access to the channels and that if $T_E\\xrightarrow{}T_E^{\\prime }\\Longleftrightarrow \\underline{T_E}\\xrightarrow{}\\underline{T_E^{\\prime }}$ we know that the types will stay dual of each other.",
"If the types are unrestricted then the requirement that whenever $T_E\\xrightarrow{}T_E^{\\prime }$ for some $\\lambda $ then $T_E=T_E^{\\prime }$ , ensures that the types will stay dual of each other.",
"In this section we discuss the results that we obtain by showing that our type system is an instance of the generic type system.",
"Hüttel presents and proves two main theorems for the generic type system, that we will use to show results about our type system as well.",
"The first theorem, which is about well typed $\\tau $ -actions from [7] follows below: Theorem 1 (Well-typed $\\tau $ -actions, Theorem 9 of [7]) Suppose we have $_0 \\blacktriangleright P\\xrightarrow{}P^{\\prime }$ , where $\\alpha $ is a $\\tau $ -action and that $\\Gamma ,\\vdash _{\\normalfont \\textsf {bal}}P$ and $\\le _0$ then for some $^{\\prime }\\le $ and $\\Gamma ^{\\prime }\\le \\Gamma $ we have $\\Gamma ^{\\prime },^{\\prime }\\vdash _{{\\normalfont \\text{min}}}\\alpha :(T@c,U)$ .",
"This theorem says that if a process $P$ can make a $\\tau $ -action and become $P^{\\prime }$ , and that $P$ is well typed in an environment where channels have pairs of dual endpoint types, then the action is also well-typed.",
"So internal synchronisation in a well-typed process is well typed as well.",
"The second theorem is about fidelity and follows below: Theorem 2 (Fidelity, Theorem 10 of [7]) Suppose we have $_0\\blacktriangleright P \\xrightarrow{}P^{\\prime }$ , where $\\alpha $ is a $\\tau $ -action and that $\\Gamma ,\\vdash _{\\normalfont \\textsf {bal}}P$ .",
"Then for some $\\Gamma ^{\\prime }\\le \\Gamma $ and for some $^{\\prime }\\le $ we have $\\Gamma ^{\\prime },^{\\prime }\\vdash _{\\normalfont \\textsf {min}}\\alpha :(T@c,U)$ and $\\Gamma \\pm (\\alpha ,(T@c,U)),^{\\prime }\\vdash _{\\normalfont \\textsf {bal}}\\circeq P^{\\prime }$ .",
"This theorem states that when an action performed in a well typed process and the action is well typed, which is guaranteed by the previous theorem, then the resulting process after the $\\tau $ -action is also well typed in an updated type environment.This result gives us the property that if a process is well typed, then it will never experience communication errors.",
"The theorem also tells us that processes evolve according to the types prescription."
],
[
"Type equivalence",
"In this section we discuss type equivalence in our type system and show how this leads to a new session type system."
],
[
"Why type equivalence matters",
"First we motivate type equivalence by expressing an example from [9], in our applied pi-calculus and our type system.",
"Recall the example from Section REF where the goal was to transmit a binary tree while preserving the tree structure.",
"Example 1 Consider the parameterised agent $S$ below.",
"The parameter $t$ is the tree to be transmitted, $c$ is the channel the tree is sent on, and $w$ is a channel used for initiating the transition of the right sub-tree after the transmission of the left sub-tree has finished.",
"The projection functions fst, snd and thrd are used to access the elements of a tuple.",
"$S(t, c, w) \\eqdef$ if $t=\\textsf{Leaf}$ then $c\\< \\textsf{Leaf}.\\o{w}\\l c \\r.\\nil$ else $c\\<\\textsf{Node}.$ $\\o{c}\\l\\textsf{fst(t)}\\r.$ $(\\nu n)(S(\\textsf{snd}(t), c, n)|n(x).S(\\textsf{thrd}(t), c, w))$ If we analyse the $S$ process, the endpoint type of $c$ is $T_C$ , which is the same type as was described in Section REF .",
"The type describes how a single node is transmitted, if it is a leaf node we do not do anything, if it is internal node then the value is transmitted and then the sub-trees of the node.",
"$T_C = \\mu z.\\oplus\\{$ $\\textsf{Leaf}: \\lin \\textsf{skip} $ $\\textsf{Node}: \\lin \\textsf{!Int};z;z$ $\\}$ The process below receives the transmitted tree preserving the original structure.",
"Through similar analysis as on the sending end, we can confirm that the endpoint type of $c$ in this process is $\\overline{T_C}$ .",
"$R(c, w) \\eqdef c\\>\\{$ $\\textsf{Leaf}: \\o{w}\\l c\\r.\\nil$ $\\textsf{Node}: (\\nu n)(c(x).R(c, n) | n(x).R(c, w))$ $\\}$ We can now create a process $P$ that transmits a tree on the $c$ channel.",
"$P=S(\\textsf {Tree}(1, \\textsf {Leaf}, \\textsf {Leaf}), c, w) \\text{ ǀ\\ } R(c, w)$ With an addition to $P$ , we can create a process $P^{\\prime }$ that reuses $c$ for transmitting an integer after transmitting the tree.",
"$P^{\\prime }=\\left[S(\\textsf {Tree}(1, \\textsf {Leaf}, \\textsf {Leaf}), c, w) \\text{ ǀ\\ } R(c, v)\\right]\\text{ ǀ\\ }\\left[v(c^{\\prime }).c^{\\prime }(x).\\mathbf {0}\\text{ ǀ\\ }w(c^{\\prime \\prime }).\\overline{c^{\\prime \\prime }}\\langle 1\\rangle .\\mathbf {0}\\right]$ The type of $c$ after expanding the recursive type is now $(T_C^{\\prime }, \\overline{T_C^{\\prime }})$ where $T_C' = \\lin \\oplus\\{$ $\\textsf{Leaf}: \\lin \\textsf{skip} $ $\\textsf{Node}: \\lin \\textsf{!Int};$ $\\mu z.\\oplus\\{\\textsf{Leaf}: \\lin \\textsf{skip}\\quad\\textsf{Node}:\\lin \\textsf{!Int};z; z\\};$ $\\mu z.\\oplus\\{\\textsf{Leaf}:\\lin \\textsf{skip}\\quad\\textsf{Node}:\\lin \\textsf{!Int};z;z\\}$ $\\};\\lin !\\textsf{Int}$ The last step is what motivates type equivalence; we would like to have a distributive law that allows the output to be moved into the select type, as illustrated below.",
"$T_C'' = \\lin \\oplus\\{$ $\\textsf{Leaf}: \\lin \\textsf{skip};\\lin !\\textsf{Int} $ $\\textsf{Node}: \\lin \\textsf{!Int};$ $\\mu z.\\oplus\\{\\textsf{Leaf}: \\lin \\textsf{skip}\\quad\\textsf{Node}:\\lin \\textsf{!Int};z;z\\};$ $\\mu z.\\oplus\\{\\textsf{Leaf}: \\lin \\textsf{skip}\\quad\\textsf{Node}:\\lin \\textsf{!Int};z;z\\};$ $\\lin !\\textsf{Int}$ $\\}$ As the typing rules in our applied pi-calculus are defined on the type transitions instead of the structure of a type, the distributive law is not essential in our calculus, since we require that the type exhibit specific behaviour instead of having a specific structure, but the example shows how type equivalence can be used to tell if two types can be used interchangeably.",
"In the next section we will introduce a method for checking if two types are equivalent, and we will return to Example REF , to check that it is indeed the case that $T_C^{\\prime \\prime }$ is equivalent to $T_C^{\\prime }$ ."
],
[
"Type bisimilarity",
"As described in Section REF , our types are highly reminiscent of BPA.",
"For BPA expressions, bisimulation is used to prove that two processes exhibit the same behaviour.",
"We now extend bisimulation to work on types as well.",
"The definition follows from the definition of bisimulation in [2].",
"Definition 1 (Type bisimulation) A binary relation $\\mathcal {R}$ between endpoint types is a type bisimulation iff whenever $T_{E_1} \\mathcal {R}\\ T_{E_2}$ : $Q(T_{E_1})=Q(T_{E_2})$ $\\forall \\lambda $ if $T_{E_1} \\xrightarrow{} T_{E_1}^{\\prime } $ then $\\exists T_{E_2}^{\\prime }$ such that $ T_{E_2} \\xrightarrow{} T_{E_2}^{\\prime }$ and $T_{E_1}^{\\prime } \\mathcal {R}\\ T_{E_2}^{\\prime }$ $\\forall \\lambda $ if $T_{E_2} \\xrightarrow{} T_{E_2}^{\\prime } $ then $\\exists T_{E_1}^{\\prime }$ such that $ T_{E_1} \\xrightarrow{} T_{E_1}^{\\prime }$ and $T_{E_1}^{\\prime } \\mathcal {R}\\ T_{E_2}^{\\prime }$ We write that $T_{E_1}\\sim T_{E_2}$ if $T_{E_1} \\mathcal {R} T_{E_2}$ for some type bisimulation and then say that $T_{E_1}$ and $T_{E_2}$ are type bisimilar.",
"Type bisimilar endpoint types will exhibit the same behaviour.",
"In other words, for a type $T_{E}$ , any type $T_E^{\\prime }$ where $T_E\\sim T_E^{\\prime }$ , $T_E^{\\prime }$ can be used instead of $T_E$ , without introducing communication errors.",
"Example 2 (A distributive law) Consider the types $T_C^{\\prime }$ and $T_C^{\\prime \\prime }$ from Example REF .",
"We can use type bisimulation to show that these two types describe the same communication behaviour on a channel.",
"To do so, we must provide a bisimulation that shows that $T_C^{\\prime }$ and $T_C^{\\prime \\prime }$ are type bisimilar.",
"Let $R$ be a relation over endpoint types.",
"Let $R$ be the symmetric closure of $\\lbrace (T_C^{\\prime },T_C^{\\prime \\prime }), ((\\textsf {lin }!\\textsf {int};r;r);\\textsf {lin }!\\textsf {int},\\linebreak [0]\\textsf {lin }!\\textsf {int};r;r;\\textsf {lin }!\\textsf {int})\\rbrace \\cup \\lbrace (T_E,T_E) \\text{ ǀ\\ } \\forall T_E\\in \\mathcal {T}\\rbrace $ where $r$ is the term $\\mu z.\\oplus \\lbrace \\textsf {Leaf}: \\textsf {lin }\\textsf {skip}\\quad \\textsf {Node}:\\textsf {lin }\\textsf {!Int};z;z\\rbrace $ .",
"In fact, we can generalise this result and prove the distributive law for both select and branch.",
"Lemma 2 Let $\\star $ be $\\oplus $ or $\\&$ .",
"Then $q\\ \\star \\lbrace l_i: T_{E_i}\\rbrace _{i\\in I};T_E\\sim q\\ \\star \\lbrace l_i: T_{E_i};T_E\\rbrace _{i\\in I}$ Let $R$ be a relation between types.",
"We must show that $R$ is a bisimulation of $q\\ \\star \\lbrace l_i: T_{E_i}\\rbrace _{i\\in I};T_E$ and $q\\ \\star \\lbrace l_i: T_{E_i};T_E\\rbrace _{i\\in I}$ .",
"Define $R$ as the symmetric closure of $\\lbrace (q\\ \\star \\lbrace l_i: T_{E_i}\\rbrace _{i\\in I};T_E,q\\ \\star \\lbrace l_i: T_{E_i};T_E\\rbrace _{i\\in I})\\rbrace \\cup \\lbrace (T_E,T_E) \\text{ ǀ\\ } \\forall T_E\\in \\mathcal {T}\\rbrace $ From the first requirement for a type bisimulation we have that $Q(q\\ \\star \\lbrace l_i: T_{E_i}\\rbrace _{i\\in I};T_E) = Q(q\\ \\star \\lbrace l_i: T_{E_i};T_E\\rbrace _{i\\in I}))$ , this is trivially fulfilled as $q=q$ .",
"By the (Seq) rule we have that the transitions of a sequential compositions are those of the left-hand side of the operator.",
"So the transitions of $q\\ \\star \\lbrace l_i: T_{E_i}\\rbrace _{i\\in I};T_E$ are $q\\ \\star \\lbrace l_i: T_{E_i}\\rbrace _{i\\in I};T_E\\xrightarrow{} T_{E_i};T_E$ , where $\\diamond \\in \\lbrace \\lhd ,\\rhd \\rbrace $ .",
"The available transitions for $q\\ \\star \\lbrace l_i: T_{E_i};T_E\\rbrace _{i\\in I}$ are $q\\ \\star \\lbrace l_i: T_{E_i};T_E\\rbrace _{i\\in I}\\xrightarrow{} T_{E_i};T_E$ .",
"Since $R$ is reflexive, we have that $(T_{E_i};T_E, T_{E_i};T_E)\\in R$ .",
"So any transition taken by one of the types can be matched by the other to end up with syntactically equivalent types.",
"Since the types are syntactically equivalent, all further transitions can be matched by any of the two types, and all further type pairs will be in $R$ .",
"This means that $R$ is a type bisimulation, and that $q\\ \\star \\lbrace l_i: T_{E_i}\\rbrace _{i\\in I};T_E \\sim q\\ \\star \\lbrace l_i: T_{E_i};T_E\\rbrace _{i\\in I}$ ."
],
[
"A quotiented session type system",
"We now define a quotiented session type system whose types are equivalence classes of session types from the the already existing type system.",
"We define transitions between equivalence classes instead of endpoint types as follows.",
"Definition 2 (Equivalence Classes) Let $\\vert T_E \\vert $ be the equivalence class of $T_E$ given by: $\\vert T_E \\vert = \\lbrace T_E^{\\prime }\\in \\mathcal {T} \\text{ ǀ\\ } T_E^{\\prime } \\sim T_E\\rbrace $ We denote a transition between equivalence classes with action $\\lambda $ as $\\vert T_{E_1}\\vert \\xrightarrow{} \\vert T_{E_2}\\vert $ .",
"Lemma 3 $\\vert T_E \\vert \\xrightarrow{} \\vert T_E^{\\prime } \\vert $ iff $\\forall T_{E_1} \\in \\vert T_E \\vert \\ \\exists T_{E_1}^{\\prime } \\in \\vert T_E^{\\prime } \\vert $ such that $T_{E_1} \\xrightarrow{} T_{E_1}^{\\prime }$ By the properties of type bisimilarity that states that two bisimilar types will evolve to bisimilar types for all possible transitions.",
"We use Lemma REF to define a new type system that is defined on equivalence classes of types from the previous type system.",
"In Section REF the transitions of endpoint types were of the form $T_E\\xrightarrow{}T_E^{\\prime }$ .",
"In the new type system the transitions are of the form $\\vert T_E \\vert \\xrightarrow{}\\vert T_E^{\\prime }\\vert $ .",
"The transition rules from Table REF still apply to the new type system, but where all endpoint types $T_E$ have been replaced with $\\vert T_E \\vert $ .",
"For example, the (Select) rule from Table REF would in the new type system be (REF ).",
"We also expand the $Q$ function on equivalence classes to be the result of applying $Q$ to any witness of the equivalence class.",
"$\\textsc {(Select)}\\qquad {q \\sqsubseteq Q(\\vert T_{E_k} \\vert )}{\\vert q\\ \\oplus \\lbrace l_i : \\vert T_{E_i} \\vert \\rbrace _{i\\in I} \\vert \\xrightarrow{} \\vert T_{E_k}\\vert }$ The typing rules in the new type system would also be the rules from Tables REF , REF and REF , where endpoint types $T_E$ have been replaced with equivalence classes $\\vert T_E \\vert $ .",
"In REF the (Input) rule from Table REF has been changed to fit the new type system.",
"$\\textsc {(Input)} \\qquad {\\Gamma _1 \\vdash n:\\vert T_E \\vert & \\vert T_E \\vert \\xrightarrow{} \\vert T_E^{\\prime } \\vert & \\Gamma _2, x:T + n: \\vert T_E^{\\prime } \\vert \\vdash P}{\\Gamma _1 \\circ \\Gamma _2 \\vdash n(x).P }$ The generic type system from [7] depends on the transitions available to types.",
"We have already shown that the previous type system is an instance of generic type systems.",
"From Lemma REF , we can see that equivalence classes have the exact same transitions as the old endpoint types had.",
"From these two results we can see that the type system defined on equivalence classes is an instance of the generic type system as well, allowing us to retain previously obtained results of fidelity and well typed internal actions from Section .",
"In conclusion, this gives us a type system where type equivalence is a trivial property, since two endpoint types of the old type system would be the same type in the new type system."
],
[
"Conclusion",
"In this paper we have considered a “low-level” applied pi-calculus that allows composite terms to be built but only allows for passing names and nullary function symbols.",
"In this setting, we introduced a type system for the applied pi-calculus based on the work on context-free session types by Thiemann and Vasconcelos in [9] and on the work on qualified session types by Vasconcelos in [10].",
"The type system is a context-free session type system with qualifiers.",
"The type system is an instance of the psi-calculus type system introduced in [7], and this allows us to establish a fidelity result about the type system that ensures that a well typed process continues to be well typed, and without communication errors, until termination.",
"The type system has a notion of type equivalence defined by introducing type bisimilarity.",
"Using it, we then get a type system of equivalence classes, for which type equivalence is a natural part of the type system itself.",
"The current focus is to deal with the decidability of type equivalence.",
"In [9] Thiemann and Vasconcelos show the decidability of type equivalence using a transformation from their types to guarded BPA expressions, for which bisimilarity is decidable.",
"As already discussed in this article, our types are very close to BPA, in which case we should be able to achieve the same results about decidability more directly.",
"In [5] an algorithm is presented for deciding bisimilarity for normed context free processes in polynomial time, and we conjecture that this algorithm can be adapted to our setting for checking type bisimilarity.",
"Here, it would be important to find a characterization of the class of applied pi processes that can be typed using normed session types only."
]
] | 1808.08648 |
[
[
"IIIDYT at IEST 2018: Implicit Emotion Classification With Deep\n Contextualized Word Representations"
],
[
"Abstract In this paper we describe our system designed for the WASSA 2018 Implicit Emotion Shared Task (IEST), which obtained 2$^{\\text{nd}}$ place out of 26 teams with a test macro F1 score of $0.710$.",
"The system is composed of a single pre-trained ELMo layer for encoding words, a Bidirectional Long-Short Memory Network BiLSTM for enriching word representations with context, a max-pooling operation for creating sentence representations from said word vectors, and a Dense Layer for projecting the sentence representations into label space.",
"Our official submission was obtained by ensembling 6 of these models initialized with different random seeds.",
"The code for replicating this paper is available at https://github.com/jabalazs/implicit_emotion."
],
[
"Introduction",
"Although the definition of emotion is still debated among the scientific community, the automatic identification and understanding of human emotions by machines has long been of interest in computer science.",
"It has usually been assumed that emotions are triggered by the interpretation of a stimulus event according to its meaning.",
"As language usually reflects the emotional state of an individual, it is natural to study human emotions by understanding how they are reflected in text.",
"We see that many words indeed have affect as a core part of their meaning, for example, dejected and wistful denote some amount of sadness, and are thus associated with sadness.",
"On the other hand, some words are associated with affect even though they do not denote affect.",
"For example, failure and death describe concepts that are usually accompanied by sadness and thus they denote some amount of sadness.",
"In this context, the task of automatically recognizing emotions from text has recently attracted the attention of researchers in Natural Language Processing.",
"This task is usually formalized as the classification of words, phrases, or documents into predefined discrete emotion categories or dimensions.",
"Some approaches have aimed at also predicting the degree to which an emotion is expressed in text [11].",
"In light of this, the WASSA 2018 Implicit Emotion Shared Task (IEST) [10] was proposed to help find ways to automatically learn the link between situations and the emotion they trigger.",
"The task consisted in predicting the emotion of a word excluded from a tweet.",
"Removed words, or trigger-words, included the terms “sad”, “happy”, “disgusted”, “surprised”, “angry”, “afraid” and their synonyms, and the task was to predict the emotion they conveyed, specifically sadness, joy, disgust, surprise, anger and fear.",
"From a machine learning perspective, this problem can be seen as sentence classification, in which the goal is to classify a sentence, or in particular a tweet, into one of several categories.",
"In the case of IEST, the problem is specially challenging since tweets contain informal language, the heavy usage of emoji, hashtags and username mentions.",
"In this paper we describe our system designed for IEST, which obtained the second place out of 30 teams.",
"Our system did not require manual feature engineering and only minimal use of external data.",
"Concretely, our approach is composed of a single pre-trained ELMo layer for encoding words [13], a Bidirectional Long-Short Memory Network (BiLSTM) [7], [6], for enriching word representations with context, a max-pooling operation for creating sentence representations from said word vectors, and finally a Dense Layer for projecting the sentence representations into label space.",
"To the best of our knowledge, our system, which we plan to release, is the first to utilize ELMo for emotion recognition.",
"Figure: Proposed architecture."
],
[
"Preprocessing",
"As our model is purely character-based, we performed little data preprocessing.",
"Table REF shows the special tokens found in the datasets, and how we substituted them.",
"Table: Preprocessing substitutions.Furthermore, we tokenized the text using a variation of the twokenize.pyhttps://github.com/myleott/ark-twokenize-py script, a Python port of the original Twokenize.java [5].",
"Concretely, we created an emoji-aware version of it by incorporating knowledge from an emoji database,https://github.com/carpedm20/emoji/blob/e7bff32/emoji/unicode_codes.py which we slightly modified for avoiding conflict with emoji sharing unicode codes with common glyphs used in Twitter,For example, the hashtag emoji is composed by the unicode code points U+23 U+FE0F U+20E3, which include U+23, the same code point for the # glyph.",
"and for making it compatible with Python 3."
],
[
"Architecture",
"Figure REF summarizes our proposed architecture.",
"Our input is based on Embeddings from Language Models (ELMo) by [13].",
"These are character-based word representations allowing the model to avoid the “unknown token” problem.",
"ELMo uses a set of convolutional neural networks to extract features from character embeddings, and builds word vectors from them.",
"These are then fed to a multi-layer Bidirectional Language Model (BiLM) which returns context-sensitive vectors for each input word.",
"We used a single-layer BiLSTM as context fine-tuner [7], [6], on top of the ELMo embeddings, and then aggregated the hidden states it returned by using max-pooling, which has been shown to perform well on sentence classification tasks [3].",
"Finally, we used a single-layer fully-connected network for projecting the pooled BiLSTM output into a vector corresponding to the label logits for each predicted class."
],
[
"Implementation Details and Hyperparameters",
" ELMo Layer: We used the official AllenNLP implementation of the ELMo modelhttps://allenai.github.io/allennlp-docs/api/allennlp.modules.elmo.html, with the official weights pre-trained on the 1 Billion Word Language Model Benchmark, which contains about 800M tokens of news crawl data from WMT 2011 [2].",
"Dimensionalities: By default the ELMo layer outputs a 1024-dimensional vector, which we then feed to a BiLSTM with output size 2048, resulting in a 4096-dimensional vector when concatenating forward and backward directions for each word of the sequenceA BiLSTM is composed of two separate LSTMs that read the input in opposite directions and whose outputs are concatenated at the hidden dimension.",
"This results in a vector double the dimension of the input for each time step.. After max-pooling the BiLSTM output over the sequence dimension, we obtain a single 4096-dimensional vector corresponding to the tweet representation.",
"This representation is finally fed to a single-layer fully-connected network with input size 4096, 512 hidden units, output size 6, and a ReLU nonlinearity after the hidden layer.",
"The output of the dense layer is a 6-dimensional logit vector for each input example.",
"Loss Function: As this corresponds to a multiclass classification problem (predicting a single class for each example, with more than 2 classes to choose from), we used the Cross-Entropy Loss as implemented in PyTorch [12].",
"Optimization: We optimized the model with Adam [9], using default hyperparameters ($\\beta _1=0.9$ , $\\beta _2=0.999$ , $\\epsilon =10^{-8}$ ), following a slanted triangular learning rate schedule [8], also with default hyperparameters ($cut\\_frac=0.1$ , $ratio=32$ ), and a maximum learning rate $\\eta _{max}=0.001$ , over $T=23,970$ iterationsThis number is obtained by multiplying the number of epochs (10), times the total number of batches, which for the training dataset corresponds to 2396 batches of 64 elements, and 1 batch of 39 elements, hence $2397\\times 10=23,970$ .. Regularization: we used a dropout layer [14], with probability of 0.5 after both the ELMo and the hidden fully-connected layer, and another one with probability of 0.1 after the max-pooling aggregation layer.",
"We also reshuffled the training examples between epochs, resulting in a different batch for each iteration.",
"Model Selection: To choose the best hyperparameter configuration we measured the classification accuracy on the validation (trial) set."
],
[
"Ensembles",
"Once we found the best-performing configuration we trained 10 models using different random seeds, and tried averaging the output class probabilities of all their possible $\\sum _{k=1}^{9}{\\binom{9}{k}}=511$ combinations.",
"As Figure REF shows, we empirically found that a specific combination of 6 models yielded the best results ($70.52\\%$ ), providing evidence for the fact that using a number of independent classifiers equal to the number of class labels provides the best results when doing average ensembling [1].",
"Figure: Effect of the number of ensembled models on validation performance."
],
[
"Experiments and Analyses",
"We performed several experiments to gain insights on how the proposed model's performance interacts with the shared task's data.",
"We performed an ablation study to see how some of the main hyperparameters affect performance, and an analysis of tweets containing hashtags and emoji to understand how these two types of tokens help the model predict the trigger-word's emotion.",
"We also observed the effects of varying the amount of data used for training the model to evaluate whether it would be worthwhile to gather more training data."
],
[
"Ablation Study",
"We performed an ablation study on a single model having obtained 69.23% accuracy on the validation set.",
"Results are summarized in Table REF .",
"Table: *We can observe that the architectural choice that had the greatest impact on our model was the ELMo layer, providing a $3.71\\%$ boost in performance as compared to using GloVe pre-trained word embeddings.",
"We can further see that emoji also contributed significantly to the model's performance.",
"In Section REF we give some pointers to understanding why this is so.",
"Additionally, we tried using the concatenation of the max-pooled, average-pooled and last hidden states of the BiLSTM as the sentence representation, following [8], but found out that this impacted performance negatively.",
"We hypothesize this is due to tweets being too short for needing such a rich representation.",
"Also, the size of the concatenated vector was $4096\\times 3=12,288$ , which probably could not be properly exploited by the 512-dimensional fully-connected layer.",
"Using a greater BiLSTM hidden size did not help the model, probably because of the reason mentioned earlier; the fully-connected layer was not big or deep enough to exploit the additional information.",
"Similarly, using a smaller hidden size neither helped.",
"We found that using 50-dimensional part-of-speech embeddings slightly improved results, which implies that better fine-tuning this hyperparameter, or using a better POS tagger could yield an even better performance.",
"Regarding optimization strategies, we also tried using SGD with different learning rates and a step-wise learning rate schedule as described by [4], but we found that doing this did not improve performance.",
"Finally, Figure REF shows the effect of using different dropout probabilities.",
"We can see that having higher dropout after the word-representation layer and the fully-connected network's hidden layer, while having a low dropout after the sentence encoding layer yielded better results overall.",
"Figure: *"
],
[
"Error Analysis",
"Figure REF shows the confusion matrix of a single model evaluated on the test set, and Table REF the corresponding classification report.",
"In general, we confirm what klinger2018iest report: anger was the most difficult class to predict, followed by surprise, whereas joy, fear, and disgust are the better performing ones.",
"To observe whether any particular pattern arose from the sentence representations encoded by our model, we projected them into 3d space through Principal Component Analysis (PCA), and were surprised to find that 2 clearly defined clusters emerged (see Figure REF ), one containing the majority of datapoints, and another containing joy tweets exclusively.",
"Upon further exploration we also found that the smaller cluster was composed only by tweets containing the pattern un __TRIGGERWORD__, and further, that all of them were correctly classified.",
"It is also worth mentioning that there are 5827 tweets in the training set with this pattern.",
"Of these, 5822 (99.9%) correspond to the label joy.",
"We observe a similar trend on the test set; 1115 of the 1116 tweets having the un __TRIGGERWORD__ pattern correspond to joy tweets.",
"We hypothesize this is the reason why the model learned this pattern as a strong discriminating feature.",
"Finally, the only tweet in the test set that contained this pattern and did not belong to the joy class, originally had unsurprised as its triggerwordWe manually searched for the original tweet., and unsurprisingly, was misclassified.",
"Figure: Confusion Matrix (Test Set).Table: Classification Report (Test Set)."
],
[
"Effect of the Amount of Training Data",
"As Figure REF shows, increasing the amount of data with which our model was trained consistently increased validation accuracy and validation macro F1 score.",
"The trend suggests that the proposed model is expressive enough to learn from more data, and is not overfitting the training set.",
"Figure: Effect of the amount of training data on classification performance."
],
[
"Effect of Emoji and Hashtags",
"Table REF shows the overall effect of hashtags and emoji on classification performance.",
"Tweets containing emoji seem to be easier for the model to classify than those without.",
"Hashtags also have a positive effect on classification performance, however it is less significant.",
"This implies that emoji, and hashtags in a smaller degree, provide tweets with a context richer in sentiment information, allowing the model to better guess the emotion of the trigger-word.",
"Table: *Table REF shows the effect specific emoji have on classification performance.",
"It is clear some emoji strongly contribute to improving prediction quality.",
"The most interesting ones are mask, rage, and cry, which significantly increase accuracy.",
"Further, contrary to intuition, the sob emoji contributes less than cry, despite representing a stronger emotion.",
"This is probably due to sob being used for depicting a wider spectrum of emotions.",
"Finally, not all emoji are beneficial for this task.",
"When removing sweat_smile and confused accuracy increased, probably because they represent emotions other than the ones being predicted.",
"Figure: 3d Projection of the Test Sentence Representations."
],
[
"Conclusions and Future Work",
"We described the model that got second place in the WASSA 2018 Implicit Emotion Shared Task.",
"Despite its simplicity, and low amount of dependencies on libraries and external features, it performed almost as well as the system that obtained the first place.",
"Our ablation study revealed that our hyperparameters were indeed quite well-tuned for the task, which agrees with the good results obtained in the official submission.",
"However, the ablation study also showed that increased performance can be obtained by incorporating POS embeddings as additional inputs.",
"Further experiments are required to accurately measure the impact that this additional input may have on the results.",
"We also think the performance can be boosted by making the architecture more complex, concretely, by using a BiLSTM with multiple layers and skip connections in a way akin to [13], or by making the fully-connected network bigger and deeper.",
"We also showed that, what was probably an annotation artifact, the un __TRIGGERWORD__ pattern, resulted in increased performance for the joy label.",
"This pattern was probably originated by a heuristic naïvely replacing the ocurrence of happy by the trigger-word indicator.",
"We think the dataset could be improved by replacing the word unhappy, in the original examples, by __TRIGGERWORD__ instead of un __TRIGGERWORD__, and labeling it as sad, or angry, instead of joy.",
"Finally, our studies regarding the importance of hashtags and emoji in the classification showed that both of them seem to contribute significantly to the performance, although in different measures."
],
[
"Acknowledgements",
"We thank the anonymous reviewers for their reviews and suggestions.",
"The first author is partially supported by the Japanese Government MEXT Scholarship."
]
] | 1808.08672 |
[
[
"On free subgroups in maximal subgroups of skew linear groups"
],
[
"Abstract The study of the existence of free groups in skew linear groups have been begun since the last decades of the 20-th century.",
"The starting point is the theorem of Tits (1972), now often is referred as Tits' Alternative, stating that every finitely generated subgroup of the general linear group $\\GL_n(F)$ over a field $F$ either contains a non-cyclic free subgroup or it is solvable-by-finite.",
"In this paper, we study the existence of non-cyclic free subgroups in maximal subgroups of an almost subnormal subgroup of the general skew linear group over a locally finite division ring."
],
[
"all"
]
] | 1808.08453 |
[
[
"Black holes and higher depth mock modular forms"
],
[
"Abstract By enforcing invariance under S-duality in type IIB string theory compactified on a Calabi-Yau threefold, we derive modular properties of the generating function of BPS degeneracies of D4-D2-D0 black holes in type IIA string theory compactified on the same space.",
"Mathematically, these BPS degeneracies are the generalized Donaldson-Thomas invariants counting coherent sheaves with support on a divisor $\\cal D$, at the large volume attractor point.",
"For $\\cal D$ irreducible, this function is closely related to the elliptic genus of the superconformal field theory obtained by wrapping M5-brane on $\\cal D$ and is therefore known to be modular.",
"Instead, when $\\cal D$ is the sum of $n$ irreducible divisors ${\\cal D}_i$, we show that the generating function acquires a modular anomaly.",
"We characterize this anomaly for arbitrary $n$ by providing an explicit expression for a non-holomorphic modular completion in terms of generalized error functions.",
"As a result, the generating function turns out to be a (mixed) mock modular form of depth $n-1$."
],
[
"Introduction and summary",
"The degeneracies of BPS black holes in string vacua with extended supersymmetry possess remarkable modular properties, which have been instrumental in recent progress on explaining the statistical origin of the Bekenstein-Hawking entropy in [1] and many subsequent works.",
"Namely, the indices $\\Omega (\\gamma )$ counting — with sign — microstates of BPS black holes with electromagnetic charge $\\gamma $ may often be collected into a suitable generating function which exhibits modular invariance, providing powerful constraints on its Fourier coefficients and enabling direct access to their asymptotic growth.",
"When the black holes can be realized as black strings wrapped on a circle, a natural candidate for such a generating function is the elliptic genus of the superconformal field theory supported by the black string, which is modular invariant by construction [2], [3], [4].",
"Equivalently, one may consider the partition function of the effective three-dimensional gravity living on the near-horizon geometry of the black string [5].",
"In most cases however, the BPS indices depend not only on the charge $\\gamma $ but also on the moduli $z^a$ at spatial infinity, due to the wall-crossing phenomenon: some of the BPS bound states with total charge $\\gamma $ only exist in a certain chamber in moduli space, and decay as the moduli are varied across `walls of marginal stability' which delimit this chamber.",
"At strong coupling where the black hole description is accurate, this phenomenon has a transparent interpretation in terms of the (dis)appearance of multi-centered black hole configurations, which can be used to derive a universal wall-crossing formula [6], [7], [8], [9].",
"In the case of four-dimensional string vacua with $\\mathcal {N}=4$ supersymmetry, where the BPS index is sensitive only to single-centered 1/4-BPS black holes and to bound states of two 1/2-BPS black holes, the resulting moduli dependence is reflected in poles in the generating function, requiring a suitable choice of contour for extracting the Fourier coefficients in a given chamber [10], [11], [12].",
"Upon subtracting contributions of two-centered bound states, the generating function of single-centered indices is no longer modular in the usual sense but it transforms as a mock Jacobi form with specific `shadow' — a property which is almost as constraining as standard modular invariance [13].",
"In four-dimensional string vacua with $\\mathcal {N}=2$ supersymmetry, the situation is much more complicated, firstly due to the fact that the moduli space of scalars receives quantum corrections, and secondly due to BPS bound states potentially involving an arbitrary number of constituents, resulting in an extremely intricate pattern of walls of marginal stability.",
"Thus, it does not seem plausible that a single generating function may capture the BPS indices $\\Omega (\\gamma ,z^a)$ — which are known in this context as generalized Donaldson-Thomas (DT) invariants — in all chambers.",
"Nevertheless, modular invariance is still expected to constrain them.",
"In particular, D4-D2-D0 black holes in type IIA string theory compactified on a generic compact Calabi-Yau (CY) threefold $\\mathfrak {Y}$ can be lifted to an M5-brane wrapped on a divisor $\\mathcal {D}\\subset \\mathfrak {Y}$ [2].",
"If the divisor $\\mathcal {D}$ labelled by the D4-brane charge $p^a$ is irreducible, the indices $\\Omega (\\gamma ,z^a)$ are independent of the moduli of $\\mathfrak {Y}$ , at least in the limit where the volume of $\\mathfrak {Y}$ is scaled to infinity, and their generating function is known to be a holomorphic (vector valued) modular form of weight $-{1\\over 2}\\, b_2(\\mathfrak {Y})-1$ [3], [4], [6], [14].",
"This includes the case of vertical, rank 1 D4-D2-D0 branes in K3-fibered Calabi-Yau threefolds [15], [16], [17].",
"But if the divisor $\\mathcal {D}$ is a sum of $n$ effective divisors $\\mathcal {D}_i$ , the indices $\\Omega (\\gamma ,z^a)$ do depend on the Kähler moduli $z^a$ , even in the large volume limit.",
"In general however, the black string SCFT is supposed to capture the states associated to a single $AdS_3$ throat, while for generic values of the moduli multiple $AdS_3$ throats can contribute [18], [19].",
"It is thus natural to consider the modular properties of the DT invariants $\\Omega (\\gamma ,z^a)$ at the large volume attractor pointHere $p^a$ and $q_a$ are D4 and D2-brane charges, respectively, and the index of $q_a$ is raised with help of the inverse of the metric $\\kappa _{ab}=\\kappa _{abc}p^a$ with $\\kappa _{abc}$ being the triple intersection numbers on $H_4(\\mathfrak {Y},{Z})$ .",
"$z^a_\\infty (\\gamma )= \\lim _{\\lambda \\rightarrow +\\infty }\\left(-q^a+I\\lambda p^a\\right),$ where only a single $AdS_3$ throat is allowed [20].",
"Following [21] we denote these invariants by $\\Omega ^{\\rm MSW}(\\gamma )=\\Omega (\\gamma ,z^a_\\infty (\\gamma ))$ and call them Maldacena-Strominger-Witten (MSW) invariants.",
"As we discuss in Section , the DT invariants $\\Omega (\\gamma ,z^a)$ can be recovered from the MSW invariants $\\Omega ^{\\rm MSW}(\\gamma )$ by using a version of the split attractor flow conjecture [22], [6] developed in [23], which we call the flow tree formula.",
"The case where $\\mathcal {D}$ is the sum of two irreducible divisors was first considered in [20], [24], [25], and studied more recently in [14], [26].",
"In that case, the generating function of MSW invariants turns out to be a mock modular form, with a specific non-holomorphic completion obtained by smoothing out the sign functions entering in the bound state contributions, recovering the prescription of [27], [20].",
"The goal of this paper is to extend this result to the general case where $\\mathcal {D}$ is the sum of $n$ irreducible divisors $\\mathcal {D}_i$ , where $n$ can be arbitrarily large.",
"In such generic situation, we characterize the modular properties of the generating function $h_{p,\\mu }$ of MSW invariants and find an explicit expression for its non-holomorphic completion $\\widehat{h}_{p,\\mu }$ in terms of the generating functions $h_{p_i,\\mu _i}$ associated to the $n$ constituents, multiplied by certain iterated integrals introduced in [28], [29], which generalize the usual error function appearing when $n=2$ .",
"This result implies that in this case $h_{p,\\mu }$ is a (mixed, vector valued) mock modular form of depth $n-1$ , in the sense that the antiholomorphic derivative of its modular completion is itself a linear combination of modular completions of mock modular forms of lower depth, times antiholomorphic modular forms (with the depth 1 case reducing to the standard mock modular forms introduced in [27], [30], and the depth 0 case to usual weakly holomorphic modular forms; see [31] for a more precise definition).",
"In order to establish this result, we follow the same strategy as in our earlier works [14], [26] and analyze D3-D1-D(-1) instanton corrections to the metric on the hypermultiplet moduli space $\\mathcal {M}_H$ in type IIB string theory compactified on $\\mathfrak {Y}$ , at arbitrary order in the instanton expansion.",
"After reducing on a circle and T-dualizing, this moduli space is identical to the vector multiplet moduli space in type IIA string theory compactified on $\\mathfrak {Y}\\times S_1$ , where it receives instanton corrections from D4-D2-D0 black holes winding around the circle.",
"In either case, each instanton contribution is weighted by the same generalized DT invariant $\\Omega (\\gamma )$ which counts the number of BPS black hole microstates with electromagnetic charge $\\gamma $ .",
"The modular properties of the generalized DT invariants are fixed by requiring that the quaternion-Kähler (QK) metric on $\\mathcal {M}_H$ admits an isometric action of $SL(2,{Z})$ , which comes from S-duality in type IIB, or equivalently from large diffeomorphisms of the torus appearing when viewing type IIA/$\\mathfrak {Y}\\times S_1$ as M-theory on $\\mathfrak {Y}\\times T^2$ [32].",
"This QK metric is most efficiently encoded in the complex contact structure on the associated twistor space, a ${C}P^1$ -bundle over $\\mathcal {M}_H$ [33], [34].",
"The latter is specified by a set of gluing conditions determined by the DT invariants [32], [35], which can in turn be expressed in terms of the MSW invariants using the tree flow formula.",
"An important quantity appearing in this twistorial formulationA familiarity with the twistorial formulation is not required for this work.",
"Here we use only two equations relevant for the twistorial description of D-instantons: the integral equation (REF ) for certain Darboux coordinates, which appears also in the study of four-dimensional $N=2$ gauge theory on a circle [36], and the expression for the contact potential (REF ) in terms of these Darboux coordinates.",
"These two equations lead to (REF ) and (REF ), respectively, which can be taken as the starting point of our analysis.",
"Note that in the context of gauge theories the contact potential can be interpreted as a supersymmetric index [37].",
"is the so called contact potential $e^\\phi $ , a real function on $\\mathcal {M}_H$ related to the Kähler potential on the twistor space, and afforded by the existence of a continuous isometry unbroken by D-instantons.",
"On the type IIB side, $e^\\phi $ can be identified with the four-dimensional dilaton $1/g_4^2$ .",
"When expressed in terms of the ten-dimensional axio-dilaton $\\tau =c_0+I/g_s$ (or in terms of the modulus of the 2-torus on the M-theory side), it becomes a complicated function having a classical contribution, a one-loop correction, and a series of instanton corrections expressed as contour integrals on the ${C}P^1$ fiber.",
"The importance of the contact potential stems from the fact that it must be a non-holomorphic modular form of weight $(-\\frac{1}{2},-\\frac{1}{2})$ in the variable $\\tau $ in order for $\\mathcal {M}_H$ to admit an isometric action of $SL(2,{Z})$ [32].",
"This requirement imposes very non-trivial constraints on the instanton contributions to $e^\\phi $ , which can be used to deduce the modular properties of generating functions of DT invariants, at each order in the instanton expansion.",
"This strategy was used in [14] at two-instanton order to characterize the modular behavior of the generating function of MSW invariants in the case of a divisor equal to the sum of $n=2$ irreducible divisors.",
"In this paper we generalize this result to all $n$ , by analyzing the instanton expansion to all orders.",
"Below we summarize the main steps in our analysis and our main results.",
"First, we show that the contact potential $e^\\phi $ in the large volume limit, where the Kähler parameters of the CY are sent to infinity, can be expressed (see (REF )) through another (complex valued) function $\\mathcal {G}$ (REF ) on the moduli space, which we call the instanton generating function.",
"The expansion of $\\mathcal {G}$ in powers of DT invariants is governed by a sum over unrooted trees decorated by charges $\\gamma _i$ (see (REF ) and (REF )).",
"The modularity of the contact potential requires $\\mathcal {G}$ to transform as a modular form of weight $(-\\tfrac{3}{2},\\tfrac{1}{2})$ .",
"After expressing the DT invariants $\\Omega (\\gamma ,z^a)$ through the moduli independent MSW invariants $\\Omega ^{\\rm MSW}(\\gamma )$ using the tree flow formula of [23], and expanding $\\mathcal {G}$ in powers of $\\Omega ^{\\rm MSW}(\\gamma )$ , each order in this expansion can be decomposed into a sum of products of certain indefinite theta series and of holomorphic generating functions $h_{p_i,\\mu _i}$ of the invariants $\\Omega ^{\\rm MSW}(\\gamma )$ (see (REF )), similarly to the usual decomposition of standard Jacobi forms.",
"Thus, the modular properties of the indefinite theta series $\\vartheta _{{p},{\\mu }}\\bigl (\\Phi ^{{\\rm tot}}_{n},n-2\\bigr )$ are tied with the modular properties of the generating functions $h_{p_i,\\mu _i}$ , in order for $\\mathcal {G}$ to be modular.",
"In order for an indefinite theta series $\\vartheta _{{p},{\\mu }}\\bigl (\\Phi ,\\lambda )$ to be modular, its kernel $\\Phi $ must satisfy a certain differential equation (REF ), which we call Vignéras' equation [38].",
"By construction, the kernels $\\Phi ^{\\rm tot}_n$ appearing in our problem are given by iterated contour integrals along the ${C}P^1$ fiber of the twistor space, multiplied by so-called `tree indices' $g_{{\\rm tr},n}$ coming from the expression of $\\Omega (\\gamma ,z^a)$ in terms of $\\Omega ^{\\rm MSW}(\\gamma )$ .",
"We evaluate the twistorial integrals in terms of the generalized error functions introduced in [28], [29], and show that the resulting kernels satisfy Vignéras' equation away from certain loci where they have discontinuities.",
"Furthermore, we prove that the discontinuities corresponding to walls of marginal stability cancel between the integrals and the tree indices.",
"But there are additional discontinuities coming from certain moduli independent contributions to the tree index.",
"They spoil Vignéras' equation at the multi-instanton level so that the theta series $\\vartheta _{{p},{\\mu }}\\bigl (\\Phi ^{{\\rm tot}}_{n},n-2\\bigr )$ are not modular.",
"In turn, this implies that the holomorphic generating functions $h_{p_i,\\mu _i}$ are not modular either.",
"However, we show that one can complete $h_{p,\\mu }$ into a non-holomorphic modular form $\\widehat{h}_{p,\\mu }(\\tau )$ , by adding to it a series of corrections proportional to products of $h_{p_i,\\mu _i}$ with the same total D4-brane charge $p=\\sum _i p_i$ (see (REF )).",
"The non-holomorphic functions $R_n$ entering the completion are determined by the condition that the expansion of $\\mathcal {G}$ rewritten in terms of $\\widehat{h}_{p,\\mu }$ gives rise to a non-anomalous, modular theta series.",
"Equivalently, one can work with functions $\\widehat{g}_n$ appearing in the expansion (REF ) of the holomorphic generating function of DT invariants $h^{\\rm DT}_{p,q}$ in powers of the non-holomorphic functions $\\widehat{h}_{p_i,\\mu _i}$ .",
"Imposing the conditions for modularity, we find that $\\widehat{g}_n$ can be represented in an iterative form (REF ), or more explicitly as a sum (REF ) over rooted trees with valency $\\ge 3$ at each vertex (known as Schröder trees), where $g^{(0)}_n$ are certain locally polynomial functions defined in (REF ), while $\\mathcal {E}_n$ are smooth solutions of Vignéras' equation constructed in terms of generalized error functions.",
"The functions $R_n$ are similarly given by a sum over Schröder trees (REF ) in terms of exponentially decreasing and non-decreasing parts of $\\mathcal {E}_n$ .",
"These equations represent the main technical result of this work.",
"For $n\\le 5$ our general formulas can be drastically simplified.",
"In particular, the main building blocks, the functions $g^{(0)}_n$ and $\\mathcal {E}_n$ , can be written as a sum over a suitable subset of flow trees, as in (REF ).",
"In addition, we show that $g^{(0)}_n$ has a natural extension including the refinement parameter conjugate to the spin $J_3$ of the black hole.",
"Figure: Various types of trees arising in this work.T n,m {T}_{n,m} denotes the set of unrooted trees with nn vertices and mm marks distributed between the vertices.T n ≡T n,0 {T}_n\\!\\equiv {T}_{n,0} comprises the trees without marks.T n r {T}_n^{\\rm r} is the set of rooted trees with nn vertices.T n af {T}_n^{\\rm af} is the set of attractor flow trees with nn leaves (of which we only draw the different topologies).T n S {T}_n^{\\rm S} denotes the set of Schröder trees with nn leaves.In addition, an important rôle is played by the sets T n,m ℓ {T}_{n,m}^\\ell and T n ℓ =T n,0 ℓ {T}_n^\\ell ={T}_{n,0}^\\ell of unrooted, labelled, marked trees which are obtained from T n,m {T}_{n,m} and T n {T}_{n} by assigning different labels and markings to the verticesin such way that a vertex with m 𝔳 m_\\mathfrak {v} marks is decorated by 2m 𝔳 +12m_\\mathfrak {v}+1 labels.At the end of this lengthy analysis, we thus find a modular completion of the generating functions $h_{p,\\mu }$ of MSW invariants for an arbitrary divisor, i.e.",
"decomposable into a sum of any number of irreducible divisors.",
"The result is expressed through sums of products of generalized error functions labelled by trees of various types.",
"For the reader's convenience, in Fig.",
"REF we display the various trees which appear in our construction, up to $n=4$ .",
"Since generalized error functions are known to be related to iterated Eichler integrals [14], [39], which occur in the non-holomorphic completion of mock modular forms, we loosely refer to the generating functions of MSW invariants as higher depth mock modular forms, although we do not spell out the precise meaning of this notion.",
"An unexpected byproduct of our analysis is an interesting combinatorial identity, relating rooted trees to the binomial coefficients, which plays a rôle in our derivation of the modular completion.",
"Since we are not aware of such an identity in the mathematical literature,We were informed by Karen Yeats that the special case $m=n-1$ appears in [40].",
"The denominator in (REF ) is sometimes known as the tree factorial $T^{\\prime }!$ , see (REF ).",
"we state it here as a theorem whose proof can be found in appendix .",
"Theorem 1 Let $V_T$ be the set of vertices of a rooted ordered tree $T$ and $n_v(T)$ is the number of descendants of vertex $v$ inside $T$ plus 1 (alternatively, the number of vertices of the subtree in $T$ with the root being the vertex $v$ and the leaves being the leaves of $T$ ).",
"Then for a rooted tree $T$ with $n$ vertices and $m<n$ one has $\\sum _{T^{\\prime }\\subset T}\\prod _{v\\in V_{T^{\\prime }}}\\frac{n_v(T)}{n_v(T^{\\prime })}=\\frac{n!}{m!(n-m)!",
"}\\, ,$ where the sum goes over all subtrees with $m$ vertices, having the same root as $T$ (see Fig.",
"REF ).",
"The organization of the paper is as follows.",
"In section we review known results about DT invariants, their expression in terms of MSW invariants, and specialize them to the case of D4-D2-D0 black holes in type IIA string theory on a Calabi-Yau threefold.",
"In section we present the twistorial description of the D-instanton corrected hypermultiplet moduli space in the dual type IIB string theory, evaluate the contact potential in the large volume approximation by expressing it through a function $\\mathcal {G}$ , and obtain the instanton expansion for this function via unrooted labelled trees.",
"In section we obtain a theta series decomposition for each order of the expansion of $\\mathcal {G}$ in MSW invariants and analyze the modular anomaly of the resulting theta series, implying a modular anomaly for the generating functions $h_{p,\\mu }$ .",
"In section we construct the non-holomorphic completion $\\widehat{h}_{p,\\mu }$ , for which the anomaly is cancelled, and determine its explicit form.",
"Section is devoted to discussion of the obtained results.",
"Finally, several appendices contain details of our calculations and proofs of various propositions.",
"In addition, in appendix we present explicit results up to order $n=4$ , and in appendix , as an aid to the reader, we provide an index of notations.",
"Figure: An example illustrating the statement of the Theorem for a tree with n=7n=7 vertices and subtrees with m=3m=3 vertices.The subtrees are distinguished by red color.Near each vertex of the subtrees we indicated the pair of numbers (n v (T),n v (T ' ))(n_v(T),n_v(T^{\\prime }))."
],
[
"BPS indices and wall-crossing",
"In this section, we review some aspects of BPS indices in theories with $\\mathcal {N}=2$ supersymmetry, including the tree flow formula relating the moduli-dependent index $\\Omega (\\gamma ,z^a)$ to the attractor index $\\Omega _\\star (\\gamma )$ .",
"We then apply this formalism in the context of Calabi-Yau string vacua, and express the generalized DT invariants $\\Omega (\\gamma ,z^a)$ in terms of their counterparts evaluated at the large volume attractor point (REF ), known as MSW invariants."
],
[
"Wall-crossing and attractor flows",
"The BPS index $\\Omega (\\gamma ,z^a)$ counts (with sign) micro-states of BPS black holes with total electro-magnetic charge $\\gamma =(p^\\Lambda ,q_\\Lambda )$ , for a given value $z^a$ of the moduli at spatial infinity.",
"While $\\Omega (\\gamma ,z^a)$ is a locally constant function over the moduli space, it can jump across real codimension one loci where certain bound states, represented by multi-centered black hole solutions of $N=2$ supergravity, become unstable.",
"The positions of these loci, known as walls of marginal stability, are determined by the central charge $Z_\\gamma (z^a)$ , a complex-valued linear function of $\\gamma $ whose modulus gives the mass of a BPS state of charge $\\gamma $ , while the phase determines the supersymmetry subalgebra preserved by the state.",
"Since a bound state can only decay when its mass becomes equal to the sum of masses of its constituents, it is apparent that the walls correspond to hypersurfaces where the phases of two central charges, say $Z_{\\gamma _L}(z^a)$ and $Z_{\\gamma _R}(z^a)$ , become aligned.",
"The bound states which may become unstable are then those whose constituents have charges in the positive cone spanned by $\\gamma _L$ and $\\gamma _R$ .",
"We shall assume that the charges $\\gamma _L,\\gamma _R$ have non-zero Dirac-Schwinger-Zwanziger (DSZ) pairing $\\langle \\gamma _L,\\gamma _R\\rangle \\ne 0$ , since otherwise marginal bound states may form, whose stability is hard to control.",
"The general relation between the values of $\\Omega (\\gamma ,z^a)$ on the two sides of a wall has been found in the mathematics literature by Kontsevich–Soibelman [41] and Joyce–Song [42], [43], and justified physically in a series of works [6], [7], [8], [9].",
"However, in this work we require a somewhat different result: an expression of $\\Omega (\\gamma ,z^a)$ in terms of moduli-independent indices.",
"One such representation, known as the Coulomb branch formula, was developed in a series of papers [44], [45], [46] (see [47] for a review) where the moduli-independent index is the so-called `single-centered invariant' counting single-centered, spherically symmetric BPS black holes.",
"Unfortunately, this representation (and its inverse) is quite involved, as it requires disentangling genuine single-centered solutions from so-called scaling solutions, i.e.",
"multi-centered solutions with $n\\ge 3$ constituents which can become arbitrarily close to each other [6], [48].",
"A simpler alternative is to consider the attractor index, i.e.",
"the value of the BPS index in the attractor chamber $\\Omega _*(\\gamma )\\equiv \\Omega (\\gamma ,z^a_\\star (\\gamma ))$ , where $z^a_\\star (\\gamma )$ is fixed in terms of the charge $\\gamma $ via the attractor mechanism [49] (recall that for a spherically symmetric BPS black hole with charge $\\gamma $ , the scalars in the vector multiplets have fixed value $z^a_\\star (\\gamma )$ at the horizon independently of their value $z^a$ at spatial infinity.).",
"By definition, the attractor indices are of course moduli independent.",
"The problem of expressing $\\Omega (\\gamma ,z^a)$ in terms of attractor indices was addressed recently in [23], extending earlier work in [20], [50].",
"Relying on the split attractor flow conjecture [22], [6], it was argued that the rational BPS index $\\bar{\\Omega }(\\gamma ,z^a) = \\sum _{d|\\gamma } \\frac{1}{d^2}\\,\\Omega (\\gamma /d,z^a)$ can be expanded in powers of $\\bar{\\Omega }_*(\\gamma _i)$ , $\\bar{\\Omega }(\\gamma ,z^a) =\\sum _{\\sum _{i=1}^n \\gamma _i=\\gamma }g_{{\\rm tr},n}(\\lbrace \\gamma _i\\rbrace ,z^a)\\,\\prod _{i=1}^n \\bar{\\Omega }_*(\\gamma _i),$ where the sum runs over orderedIn [23] a similar formula was written as a sum over unordered decompositions, weighted by the symmetry factor $1/|{\\rm Aut}\\lbrace \\gamma _i\\rbrace |$ .",
"Since $g_{{\\rm tr},n}(\\lbrace \\gamma _i\\rbrace ,z^a)$ is symmetric under permutations of $\\lbrace \\gamma _i\\rbrace $ , we can sum over all ordered decompositions with unit weight, at the expense of inserting the factor $1/n!$ in its definition (REF ).",
"In the sequel, all similar sums are always assumed to run over ordered decompositions.",
"decompositions of $\\gamma $ into sums of vectors $\\gamma _i\\in \\Gamma _+$ , with $\\Gamma _+$ being the set of all vectors $\\gamma $ whose central charge $Z_\\gamma (z^a_\\infty )$ lies in a fixed half-space defining the splitting between BPS particles ($\\gamma \\in \\Gamma _+$ ) and anti-BPS particles ($\\gamma \\in - \\Gamma _+$ ).",
"Such decompositions correspond to contributions of multi-centered black hole solutions with constituents carrying charges $\\gamma _i$ .",
"The coefficient $g_{{\\rm tr},n}$ , called the tree index, is defined as $g_{{\\rm tr},n}(\\lbrace \\gamma _i\\rbrace ,z^a)=\\frac{1}{n!",
"}\\sum _{T\\in {T}_n^{\\rm af}}\\Delta (T)\\, \\kappa (T) ,$ where the sum goes over the set ${T}_n^{\\rm af}$ of attractor flow trees with $n$ leaves.",
"These are unordered rooted binary treesThe number of such trees is $|{T}_n^{\\rm af}|=(2n-3)!!=(2n-3)!/[2^{n-2}(n-2)!",
"]=\\lbrace 1,1,3,15,105,945,...\\rbrace $ for $n\\ge 1$ .",
"$T$ with vertices decorated by electromagnetic charges $\\gamma _v$ , such that the leaves of the tree carry the constituent charges $\\gamma _i$ , and the charges propagate along the tree according to $\\gamma _v= \\gamma _{L(v)}+\\gamma _{R(v)}$ at each vertex, where $L(v)$ , $R(v)$ are the two childrenThis assignment requires an ordering of the children at each vertex, which can be chosen arbitrarily for each tree.",
"With such an ordering, and assuming that all the charges $\\gamma _i$ are distinct, the flow tree can be labelled by a 2-bracketing of a permutation of the set $\\lbrace 1,\\dots , n\\rbrace $ , as shown in Fig.",
"REF .",
"of the vertex $v$ (see Fig.",
"REF ).",
"The charge carried by the root of the tree is then the total charge $\\gamma =\\sum \\gamma _i$ .",
"The idea of the split attractor flow conjecture is that each tree represents a nested sequence of two-centered bound states describing a multi-centered solution built out of constituents with charges $\\gamma _i$ .",
"With this interpretation, the edges of the graph represent the evolution of the moduli under attractor flow, so that one starts from the moduli at spatial infinity $z^a_\\infty \\equiv z^a$ and assigns to the root $v_0$ the point in the moduli space $z^a_{v_0}$ where the attractor flow with charge $\\gamma $ crosses the wall of marginal stability where $\\,{\\rm Im}\\,\\bigl [Z_{\\gamma _{L(v_0)}} \\bar{Z}_{\\gamma _{R(v_0)}}(z^a_{v_0})\\bigr ]=0$ .",
"Then one repeats this procedure for every edge, obtaining a set of charges and moduli $(\\gamma _v,z^a_v)$ assigned to each vertex, with the bound state constituents and their attractor moduli $(\\gamma _i,z^a_{\\gamma _i})$ assigned to the leaves.",
"Figure: An example of attractor flow tree corresponding to the bracketing ((13)(2(45)))((13)(2(45))).Given these data, the factor $\\Delta (T)$ in (REF ) is given by $\\Delta (T)=\\prod _{v\\in V_T}\\Delta _{\\gamma _{L(v)}\\gamma _{R(v)}}^{z_{p(v)}},\\qquad \\Delta _{\\gamma _L\\gamma _R}^z={1\\over 2}\\,\\Bigl [{\\rm sgn}\\,\\,{\\rm Im}\\,\\bigl [ Z_{\\gamma _L}\\bar{Z}_{\\gamma _R}(z^a)\\bigr ]+ {\\rm sgn}(\\gamma _{LR}) \\Bigr ] ,$ where $V_T$ denotes the set of vertices of $T$ excluding the leaves, $p(v)$ is the parent of vertex $v$ , and $\\gamma _{LR}=\\langle \\gamma _L,\\gamma _R \\rangle $ .",
"This factor vanishes unless the stability conditionIn fact, the admissibility also requires $\\,{\\rm Re}\\,\\bigl [ Z_{\\gamma _{L(v)}}\\bar{Z}_{\\gamma _{R(v)}}(z^a_{v})\\bigr ]>0$ at each vertex.",
"This condition will hold automatically for the case of our interest, namely, D4-D2-D0 black holes in the large volume limit, so we do not impose it explicitly.",
"$\\gamma _{L(v)R(v)}\\, \\,{\\rm Im}\\,\\bigl [ Z_{\\gamma _{L(v)}}\\bar{Z}_{\\gamma _{R(v)}}(z^a_{p(v)})\\bigr ]>0$ is satisfied for all $v\\in V_T$ , which ensures admissibility of the flow tree $T$ , i.e.",
"the existence of the corresponding nested bound state.",
"Importantly, the sign of $\\,{\\rm Im}\\,\\bigl [ Z_{\\gamma _{L(v)}}\\bar{Z}_{\\gamma _{R(v)}}(z^a_{p(v)})\\bigr ]$ entering (REF ) can be computed recursively in terms of asymptotic data, without evaluating the attractor flow along the edges [23].",
"More precisely, the signs depend only on the stability parameters (also known as Fayet-Iliopoulos parameters) $c_i(z^a) = \\,{\\rm Im}\\,\\bigl [ Z_{\\gamma _i}\\bar{Z}_\\gamma (z^a)\\bigr ]\\, .$ Note that due to $\\gamma =\\sum _{i=1}^n \\gamma _i$ , these parameters satisfy $\\sum _{i=1}^n c_i=0$ .",
"Accordingly, we shall often denote $g_{{\\rm tr},n}(\\lbrace \\gamma _i\\rbrace ,z^a)$ by $g_{{\\rm tr},n}(\\lbrace \\gamma _i,c_i\\rbrace )$ , and always assume that $\\lbrace c_i\\rbrace $ are generic real parameters subject to the condition $\\sum _{i=1}^n c_i=0$ , such that no proper subset of them sums up to zero.",
"The second factor in (REF ) is independent of the moduli, and given by $\\kappa (T) \\equiv (-1)^{n-1} \\prod _{v\\in V_T} \\kappa ( \\gamma _{L(v)R(v)}),\\qquad \\kappa (x)=(-1)^x\\, x.$ This is simply the product of the BPS indices of the nested two-centered solutions associated with the tree $T$ .",
"Note that the signs of $\\Delta (T)$ and $\\kappa (T)$ separately depend on the choice of ordering $(\\gamma _{L(v)},\\gamma _{R(v)})$ at each vertex, but their product is independent of that choice.",
"Sometimes, it is useful to consider the refined BPS index $\\Omega (\\gamma ,z^a,y)$ which carries additional dependence on the fugacity $y$ conjugate to the spin $J_3$ .",
"All the above equations remain valid in this case as well (except for the definition of the rational invariant (REF ), which must be slightly modified, see e.g.",
"[8]), but now the function $\\kappa (x)$ appearing in (REF ) becomes a symmetric Laurent polynomial in $y$ $\\kappa (x)=(-1)^x \\, \\frac{ y^x-y^{-x}}{y-y^{-1}}\\, ,$ reducing to $(-1)^x x$ in the unrefined limit $y\\rightarrow 1$ .",
"It is easy to see that the `flow tree formula' (REF ) is consistent with the primitive wall-crossing formula [6], [7] $\\Delta \\bar{\\Omega }(\\gamma _L+\\gamma _R) = -{\\rm sgn}(\\gamma _{LR})\\, \\kappa (\\gamma _{LR})\\, \\bar{\\Omega }(\\gamma _L,z^a)\\, \\bar{\\Omega }(\\gamma _R,z^a) ,$ which gives the jump of the BPS index due to the decay of bound states after crossing the wall defined by a pair of primitiveHere, by primitive we mean that all charges with non-zero index in the two-dimensional lattice spanned by $\\gamma _L$ and $\\gamma _R$ are linear combinations $N_L \\gamma _L + N_R\\gamma _R$ with coefficients $N_L, N_R$ of the same sign.",
"charges $\\gamma _L$ and $\\gamma _R$ .",
"To this end, it suffices to consider all flow trees which start with the splitting $\\gamma \\rightarrow \\gamma _L+\\gamma _R$ at the root of the tree.",
"It is also consistent with the general wall-crossing formula of [41], provided the tree index is computed for a small generic perturbation of the DSZ matrix $\\gamma _{ij}$ [23].",
"Finally, it is useful to note that, assuming that all charges $\\gamma _i$ are distinct, the sum over splittings and flow trees in (REF ) can be generated by iterating the quadratic equation [23] $\\bar{\\Omega }(\\gamma ,z^a) &= &\\bar{\\Omega }_*(\\gamma )-{1\\over 2}\\sum _{\\begin{array}{c}\\gamma =\\gamma _L+\\gamma _R\\\\\\langle \\gamma _L,\\gamma _R \\rangle \\ne 0\\end{array}}\\Delta _{\\gamma _L\\gamma _R}^z \\, \\kappa (\\gamma _{LR})\\,\\bar{\\Omega }(\\gamma _L,z^a_{LR})\\, \\bar{\\Omega }(\\gamma _R,z^a_{LR}),$ where $z^a_{LR}$ is the point where the attractor flow of charge $\\gamma $ crosses the wall of marginal stability $\\,{\\rm Im}\\,\\bigl [Z_{\\gamma _L}\\bar{Z}_{\\gamma _R}(z^a_{LR})\\bigr ]=0$ ."
],
[
"Partial tree index",
"While the representation of the BPS index based on attractor flows is useful for many purposes, it produces a sum of products of sign functions depending on non-linear combinations of DSZ products $\\gamma _{ij}$ , which are very difficult to work with.",
"A solution to overcome this problem was found in [23].",
"The key idea is to introduce a refined index with a fugacity $y$ conjugate to angular momentum, and represent it as $g_{{\\rm tr},n}(\\lbrace \\gamma _i,c_i\\rbrace ,y)= \\frac{(-1)^{n-1+\\sum _{i<j} \\gamma _{ij} }}{(y-y^{-1})^{n-1}}\\,\\,{\\rm Sym}\\, \\Bigl \\lbrace F_{{\\rm tr},n}(\\lbrace \\gamma _i,c_i\\rbrace )\\,y^{\\sum _{i<j} \\gamma _{ij}}\\Bigr \\rbrace ,$ where $\\,{\\rm Sym}\\, $ denotes symmetrization (with weight $1/n!$ ) with respect to the charges $\\gamma _i$ , and $F_{{\\rm tr},n}$ is the `partial tree index' defined byUnlike the tree index $g_{{\\rm tr},n}$ , the partial tree index $F_{{\\rm tr},n}$ is not a symmetric function of charges $\\gamma _i$ and stability parameters $c_i$ , however we abuse notation and still denote it by $F_{{\\rm tr},n} (\\lbrace \\gamma _{i},c_i\\rbrace )$ .",
"$F_{{\\rm tr},n} (\\lbrace \\gamma _{i},c_i\\rbrace ) =\\sum _{T\\in {T}_n^{\\mbox{\\tiny af-pl}}} \\Delta (T) .$ Here the sum runs over the set ${T}_n^{\\mbox{\\scriptsize af-pl}}$ of planar flow trees with $n$ leavesThe number of such trees is the $n-1$ -th Catalan number $|{T}_n^{\\mbox{\\scriptsize af-pl}}|=\\frac{(2n-2)!}{n[(n-1)!",
"]^2}=\\lbrace 1,1,2,5,14,42,132,\\dots \\rbrace $ for $n\\ge 1$ .",
"carrying ordered charges $\\gamma _1,\\dots , \\gamma _n$ .",
"Although this is not manifest, the refined tree index (REF ) is regular at $y=1$ , and its value (computed e.g.",
"using l'Hôpital rule) reduces to the tree index (REF ).",
"The advantage of the representation (REF ) is that the partial tree index $F_{{\\rm tr},n}$ does not involve the $\\kappa $ -factors (REF ) and is independent of the refinement parameter.",
"The partial index $F_{{\\rm tr},n}$ satisfies two important recursive relations.",
"To formulate them, let us introduce some convenient notations: $S_k=\\sum _{i=1}^k c_i,\\quad \\ \\beta _{k\\ell }=\\sum _{i=1}^k \\gamma _{i\\ell },\\quad \\ \\Gamma _{k\\ell }=\\sum _{i=1}^k\\sum _{j=1}^\\ell \\gamma _{ij}.$ In terms of these notations, the partial index satisfies the iterative equation [23], $F_{{\\rm tr},n}(\\lbrace \\gamma _i,c_i\\rbrace )={1\\over 2}\\sum _{\\ell =1}^{n-1} \\bigl ( {\\rm sgn}(S_\\ell )-{\\rm sgn}(\\Gamma _{n\\ell })\\bigr )\\,F_{{\\rm tr},\\ell }(\\lbrace \\gamma _i,c_i^{(\\ell )}\\rbrace _{i=1}^\\ell )\\,F_{{\\rm tr},n-\\ell }(\\lbrace \\gamma _i,c_i^{(\\ell )}\\rbrace _{i=\\ell +1}^n),$ where $c_i^{(\\ell )}$ is the value of the stability parameters at the point where the attractor flow crosses the wall for the decay $\\gamma \\rightarrow (\\gamma _1+\\cdots +\\gamma _\\ell ,\\gamma _{\\ell +1}+\\cdots +\\gamma _n)$ , given by $c_i^{(\\ell )}=c_i -\\frac{\\beta _{ni}}{\\Gamma _{n\\ell }}\\, S_{\\ell }\\, .$ Importantly, $c_i^{(\\ell )}$ satisfies $\\sum _{i=1}^\\ell c_i^{(\\ell )}= \\sum _{i=\\ell +1}^n c_i^{(\\ell )}=0$ so that the two factors on the r.h.s.",
"of (REF ) are well-defined.",
"Note that the iterative equation (REF ) is in the spirit of the quadratic equation (REF ).",
"According to [23], the partial tree index satisfies another recursion $\\begin{split}F_{{\\rm tr},n}(\\lbrace \\gamma _i,c_i\\rbrace )=&\\,F^{(0)}_n(\\lbrace c_i\\rbrace )- \\sum _{n_1+\\cdots +n_m= n\\atop n_k\\ge 1, \\ m<n}F_{{\\rm tr},m}(\\lbrace \\gamma ^{\\prime }_k,c^{\\prime }_k\\rbrace )\\prod _{k=1}^m F^{(0)}_{n_k}(\\beta _{n,j_{k-1}+1},\\dots ,\\beta _{nj_{k}}),\\end{split}$ where the sum runs over ordered partitions of $n$ , $m$ is the number of parts, and for $k=1,\\dots ,m$ we defined $\\begin{split}&\\qquad \\qquad j_0=0,\\qquad j_k=n_1+\\cdots + n_k,\\\\&\\gamma ^{\\prime }_k=\\gamma _{j_{k-1}+1}+\\cdots +\\gamma _{j_{k}},\\qquad c^{\\prime }_k=c_{j_{k-1}+1}+\\cdots +c_{j_{k}}.\\end{split}$ The function appearing in (REF ) is simply a product of signs, $F^{(0)}_n(\\lbrace c_i\\rbrace )=\\frac{1}{2^{n-1}}\\prod _{i=1}^{n-1}\\mbox{sgn}(S_i) .$ This new recursive relation allows to express the partial index in a way which does not involve sign functions depending on non-linear combinations of parameters, in contrast to the previous relation (REF ) where such sign functions arise due to the discrete attractor flow relation (REF )."
],
[
"D4-D2-D0 black holes and BPS indices",
"The results presented above are applicable in any theory with $\\mathcal {N}=2$ supersymmetry.",
"Let us now specialize to the BPS black holes obtained as bound states of D4-D2-D0-branes in type IIA string theory compactified on a CY threefold $\\mathfrak {Y}$ .",
"In this case the moduli $z^a=b^a+It^a$ ($a=1,\\dots , b_2(\\mathfrak {Y})$ ) are the complexified Kähler moduli with respect to a basis of $H^2(\\mathfrak {Y},{Z})$ , parametrizing the Kähler moduli space $\\mathcal {M}_\\mathcal {K}(\\mathfrak {Y})$ .",
"The charge vectors $\\gamma \\in H_{\\rm even}(\\mathfrak {Y},{Q})$ have the form $\\gamma =(0,p^a,q_a,q_0)$ where the first entry corresponds to the D6-brane charge, which is taken to vanish, whereas the other components, corresponding to the D4, D2 and D0-brane charges, satisfy the following quantization conditions [51]: $p^a\\in {Z},\\qquad q_a \\in {Z}+ \\frac{1}{2} \\,(p^2)_a ,\\qquad q_0\\in {Z}-\\frac{1}{24}\\, p^a c_{2,a},$ where $c_{2,a}$ are components of the second Chern class of $\\mathfrak {Y}$ .",
"In the second relation we used the notations $(kp)_a=\\kappa _{abc}k^b p^c$ and $(lkp)=\\kappa _{abc}l^a k^b p^c$ (recall that $\\kappa _{abc}$ are the intersection numbers on $H_4(\\mathfrak {Y},{Z})$ ) which will be extensively used below.",
"The lattice of charges $\\gamma $ satisfying (REF ) will be denoted by $\\Gamma $ .",
"The cone $\\Gamma _+\\subset \\Gamma $ is obtained by imposing the further restriction that the D4-brane charge $p^a$ corresponds to an effective divisor in $\\mathfrak {Y}$ and belongs to the Kähler cone, i.e.",
"$p^3> 0,\\qquad (r p^2)> 0,\\qquad k_a p^a > 0,$ for all effective divisors $r^a \\gamma _a \\in H_4^+(\\mathfrak {Y},{Z})$ and effective curves $k_a \\gamma ^a \\in H_2^+(\\mathfrak {Y},{Z})$ , where $\\gamma _a$ denotes irreducible divisors giving an integer basis of $ \\Lambda =H_4(\\mathfrak {Y},{Z})$ , dual to the basis $\\gamma ^a$ of $\\Lambda ^*=H_2(\\mathfrak {Y},{Z})$ .",
"The charge $p^a$ induces a quadratic form $\\kappa _{ab}=\\kappa _{abc} p^c$ on $\\Lambda \\otimes {R}\\simeq {R}^{b_2}$ of signature $(1,b_2-1)$ .",
"This quadratic form allows to embed $\\Lambda $ into $\\Lambda ^*$ , but the map $\\epsilon ^a \\mapsto \\kappa _{ab} \\epsilon ^b$ is in general not surjective, the quotient $\\Lambda ^*/\\Lambda $ being a finite group of order $|\\,{\\rm det}\\, \\kappa _{ab}|$ .",
"The holomorphic central charge, governing the mass of BPS states, is given by $Z_\\gamma (z^a)=q_\\Lambda X^\\Lambda (z^a) -p^\\Lambda F_\\Lambda (z^a),$ where $X^\\Lambda (z^a)=(1,z^a)$ are the special coordinates and $F_\\Lambda =\\partial _{X^\\Lambda } F(X)$ is the derivative of the holomorphic prepotential $F(X)$ on $\\mathcal {M}_\\mathcal {K}$ .",
"In the large volume limit $t^a\\rightarrow \\infty $ , the prepotential reduces to the classical cubic contribution $F(X)\\approx F^{\\rm cl}(X)=-\\kappa _{abc}\\, \\frac{ X^a X^b X^c}{6X^0}\\, ,$ and the central charge can be approximated as $Z_\\gamma \\approx -{1\\over 2}\\, (pt^2)+I\\left( q_a t^a+(pbt)\\right)+q_0+q_a b^a+{1\\over 2}\\,(pb^2).$ Note, in particular, that it always has a large negative real part.",
"Another useful observation is that both quantities appearing in the definition of $\\Delta _{\\gamma _L\\gamma _R}^z$ (REF ) are independent of the last component $q_0$ of the charge vector.",
"Indeed, $\\begin{split}\\langle \\gamma ,\\gamma ^{\\prime }\\rangle =&\\, q_a p^{\\prime a}-q^{\\prime }_ap^a,\\\\\\,{\\rm Im}\\,\\bigl [ Z_{\\gamma }\\bar{Z}_{\\gamma ^{\\prime }}\\bigr ]=&\\, -{1\\over 2}\\,\\Bigl ((p^{\\prime } t^2)(q_{a} +(pb)_a)t^a - (p t^2) (q^{\\prime }_{a}+(p^{\\prime }b)_a) t^a\\Bigr ).\\end{split}$ The BPS index $\\bar{\\Omega }(\\gamma ,z^a)$ counting D4-D2-D0 black holes is given mathematically by the generalized Donaldson-Thomas invariant, which countsMore precisely, the generalized DT invariant computes the weighted Euler characteristic of the moduli space of semi-stable coherent sheaves [42]; in this context, the DSZ product $\\langle \\gamma ,\\gamma ^{\\prime }\\rangle $ coincides with the antisymmetrized Euler form.",
"semi-stable coherent sheaves supported on a divisor $\\mathcal {D}$ in the homology class $p^a\\gamma _a$ , with first and second Chern numbers determined by $(q_a,q_0)$ .",
"An important property of these invariants is that they are unchanged under a combined integer shift of the Kalb-Ramond field, $b^a\\mapsto b^a +\\epsilon ^a$ , and a spectral flow transformation acting on the D2 and D0 charges $q_a \\mapsto q_a - \\kappa _{abc}p^b\\epsilon ^c,\\qquad q_0 \\mapsto q_0 - \\epsilon ^a q_a + \\frac{1}{2}\\, (p\\epsilon \\epsilon ).$ The shift of $b^a$ is important since the DT invariants are only piecewise constant as functions of the complexified Kähler moduli $z^a=b^a+It^a$ due to wall-crossing.",
"In contrast, the MSW invariants ${\\bar{\\Omega }}^{\\rm MSW}(\\gamma )$ , defined as the generalized DT invariants $\\bar{\\Omega }(\\gamma ,z^a)$ evaluated at their respective large volume attractor point (REF ), are by construction independent of the moduli, and therefore invariant under the spectral flow (REF ).",
"As a result, they only depend on $p^a, \\mu _a$ and $\\hat{q}_{0}$ , where we traded the electric charges $(q_a,q_0)$ for $(\\epsilon ^a, \\mu _a,\\hat{q}_0)$ .",
"The latter comprise the spectral flow parameter $\\epsilon ^a$ , the residue class $\\mu _a\\in \\Lambda ^*/\\Lambda $ defined by the decomposition $q_a = \\mu _a + \\frac{1}{2}\\, \\kappa _{abc} p^b p^c + \\kappa _{abc} p^b \\epsilon ^c,\\qquad \\epsilon ^a\\in \\Lambda \\, ,$ and the invariant charge ($\\kappa ^{ab}$ is the inverse of $\\kappa _{ab}$ ) $\\hat{q}_0 \\equiv q_0 -\\frac{1}{2}\\, \\kappa ^{ab} q_a q_b\\, ,$ which is invariant under (REF ).",
"This allows to write ${\\bar{\\Omega }}^{\\rm MSW}(\\gamma )=\\bar{\\Omega }_{p,\\mu }( \\hat{q}_0)$ .",
"An important fact is that the invariant charge $\\hat{q}_0$ is bounded from above by $\\hat{q}_0^{\\rm max}=\\tfrac{1}{24}((p^3)+c_{2,a}p^a)$ .",
"This allows to define two generating functions $h^{\\rm DT}_{p,q}(\\tau ,z^a) &=& \\sum _{\\hat{q}_0 \\le \\hat{q}_0^{\\rm max}}\\bar{\\Omega }(\\gamma ,z^a)\\,{\\bf e}\\!\\left( -\\hat{q}_0 \\tau \\right),\\\\h_{p,\\mu }(\\tau ) &=& \\sum _{\\hat{q}_0 \\le \\hat{q}_0^{\\rm max}}\\bar{\\Omega }_{p,\\mu }(\\hat{q}_0)\\,{\\bf e}\\!\\left( -\\hat{q}_0 \\tau \\right),$ where we used notation ${\\bf e}\\!\\left( x\\right)=e^{2\\pi Ix}$ .",
"Whereas the generating function of DT invariants $h^{\\rm DT}_{p,q}$ depends on the full electric charge $q_a$ and depends on the moduli $z^a$ in a piecewise constant fashion, the generating function of MSW invariants $h_{p,\\mu }(\\tau )$ , due to the spectral flow symmetry, depends only on the residue class $\\mu _a$ .",
"This generating function will be the central object of interest in this paper, and our main goal will be to understand its behavior under modular transformations of $\\tau $ .",
"In general, the MSW invariants ${\\bar{\\Omega }}^{\\rm MSW}(\\gamma )$ are distinct from the attractor moduli $\\bar{\\Omega }_\\star (\\gamma )$ , since the latter coincide with the generalized DT invariants $\\bar{\\Omega }(\\gamma ,z^a)$ evaluated at the true attractor point $z^a_*(\\gamma )$ for the charge $\\gamma $ , while the former are the generalized DT invariants evaluated at the large volume attractor point $z^a_\\infty (\\gamma )$ defined in (REF ).",
"Nevertheless, we claim that in the large volume limit $t^a\\rightarrow \\infty $ , the tree flow formula reviewed in the previous subsections still allows to express $\\bar{\\Omega }(\\gamma ,z^a)$ in terms of the MSW invariants, namely $\\bar{\\Omega }(\\gamma ,z^a) =\\sum _{\\sum _{i=1}^n \\gamma _i=\\gamma }g_{{\\rm tr},n}(\\lbrace \\gamma _i\\rbrace , z^a)\\,\\prod _{i=1}^n {\\bar{\\Omega }}^{\\rm MSW}(\\gamma _i) \\qquad (\\,{\\rm Im}\\,z^a\\rightarrow \\infty )$ The point is that the only walls of marginal stability which extend to infinite volume are those where the constituents carry no D6-brane charge, and that non-trivial bound states involving constituents with D4-brane charge are ruled out at the large volume attractor point, similarly to the usual attractor chamber.",
"Since the r.h.s.",
"of (REF ) is consistent with wall-crossing in the infinite volume limit and agrees with the left-hand side at $z^a=z^a_\\infty (\\gamma )$ , it must therefore hold everywhere at large volume.",
"Of course, some of the states contributing to ${\\bar{\\Omega }}^{\\rm MSW}(\\gamma _i)$ may have some substructure, e.g.",
"be realized as D6-$\\overline{\\rm D6}$ bound states, but this structure cannot be probed in the large volume limit.",
"Importantly, since the quantities (REF ) entering in the definition of the tree index are independent of the D0-brane charge $q_0$ , the flow tree formula (REF ) may be rewritten as a relation between the generating functions, $h^{\\rm DT}_{p,q}(\\tau ,z^a) =\\sum _{\\sum _{i=1}^n \\check{\\gamma }_i=\\check{\\gamma }}g_{{\\rm tr},n}(\\lbrace \\check{\\gamma }_i\\rbrace ,z^a)\\,e^{\\pi I\\tau Q_n(\\lbrace \\check{\\gamma }_i\\rbrace )}\\prod _{i=1}^n h_{p_i,\\mu _i}(\\tau ),$ where $\\check{\\gamma }=(p^a,q_a)$ denotes the projection of the charge vector $\\gamma $ on $H_4\\oplus H_2$ , and the phase proportional to $Q_n(\\lbrace \\check{\\gamma }_i\\rbrace )= \\kappa ^{ab}q_a q_b-\\sum _{i=1}^n\\kappa _i^{ab}q_{i,a} q_{i,b}$ appears due to the quadratic term in the definition (REF ) of the invariant charge $\\hat{q}_0$ .",
"In this section, we switch to the dual setupOur preference for the type IIB set-up is merely for consistency with our earlier works on hypermultiplet moduli spaces in $d=4$ Calabi-Yau vacua.",
"The same considerations apply verbatim, with minor changes of wording, to the vector multiplet moduli space in type IIA string theory compactified on $\\mathfrak {Y}\\times S^1$ , which is more directly related to the counting of D4-D2-D0 black holes in four dimensions.",
"of type IIB string theory compactified on the same CY manifold $\\mathfrak {Y}$ .",
"The DT invariants, describing the BPS degeneracies of D4-D2-D0 black holes in type IIA, now appear as coefficients in front of the D3-D1-D(-1) instanton effects affecting the metric on the hypermultiplet moduli space $\\mathcal {M}_H$ .",
"The main idea of our approach is that these instanton effects are strongly constrained by demanding that $\\mathcal {M}_H$ admits an isometric action of the type IIB S-duality $SL(2,{Z})$ .",
"This constraint uniquely fixes the modular behavior of the generating functions introduced in the previous section.",
"Here we recall the twistorial construction of D-instanton corrections to the hypermultiplet metric, describe the action of S-duality, and analyze the instanton expansion of a particular function on $\\mathcal {M}_H$ known as contact potential."
],
[
"$\\mathcal {M}_H$ and twistorial description of instantons ",
"The moduli space of four-dimensional $N=2$ supergravity is a direct product of vector and hypermultiplet moduli spaces, $\\mathcal {M}_V\\times \\mathcal {M}_H$ .",
"The former is a (projective) special Kähler manifold, whereas the latter is a quaternion-Kähler (QK) manifold.",
"In type IIB string theory compactified on a CY threefold $\\mathfrak {Y}$ , $\\mathcal {M}_H$ is a space of real dimension $4b_2(\\mathfrak {Y})+4$ , which is fibered over the complexified Kähler moduli space $\\mathcal {M}_\\mathcal {K}(\\mathfrak {Y})$ of dimension $2b_2(\\mathfrak {Y})$ .",
"In addition to the Kähler moduli $z^a=b^a+It^a$ , it describes the dynamics of the ten-dimensional axio-dilaton $\\tau =c^0+I/g_s$ , the Ramond-Ramond (RR) scalars $c^a,\\tilde{c}_a,\\tilde{c}_0$ , corresponding to periods of the RR 2-form, 4-form and 6-form on a basis of $H^{\\rm even}(\\mathfrak {Y},{Z})$ , and finally, the NS-axion $\\psi $ , dual to the Kalb-Ramond two-form $B$ in four dimensions.",
"At tree-level, the QK metric on $\\mathcal {M}_H$ is obtained from the Kähler moduli space $\\mathcal {M}_{\\mathcal {K}}$ via the $c$ -map construction [52], [53] and thus is completely determined by the holomorphic prepotential $F(X)$ .",
"But this metric receives $g_s$ -corrections, both perturbative and non-perturbative.",
"The latter can be of two types: either from Euclidean D-branes wrapping even dimensional cycles on $\\mathfrak {Y}$ , or from NS5-branes wrapped around the whole $\\mathfrak {Y}$ .",
"In this paper we shall be interested only in the effects of D3-D1-D(-1) instantons, and ignore the effects of NS5 and D5-instantons, which are subleading in the large volume limit.",
"Since NS5-instantons only mix with D5-instantons under S-duality, this truncation does not spoil modular invariance [21].",
"The most concise way to describe the D-instanton corrections is to consider type IIA string theory compactified on the mirror CY threefold ${\\widehat{\\mathfrak {Y}}}$ and use the twistor formalism for quaternionic geometries [33], [34].",
"In this approach the metric is encoded in the complex contact structure on the twistor space, a ${C}P^1$ -bundle over $\\mathcal {M}_H$ .",
"The D-instanton corrected contact structure has been constructed to all orders in the instanton expansion in [32], [35], and an explicit expression for the metric has been derived recently in [54], [55].",
"Here we will present only those elements of the construction which are relevant for the subsequent analysis, and refer to reviews [56], [57] for more details.",
"The crucial point is that, locally, the contact structure is determined by a set of holomorphic Darboux coordinates $(\\xi ^\\Lambda ,\\tilde{\\xi }_\\Lambda ,\\alpha )$ on the twistor space, considered as functions of coordinates on $\\mathcal {M}_H$ and of the stereographic coordinate $t$ on the ${C}P^1$ fiber, so that the contact one-form takes the canonical form $\\mathrm {d}\\alpha +\\tilde{\\xi }_\\Lambda \\mathrm {d}\\xi ^\\Lambda $ .",
"Although all Darboux coordinates are important for recovering the metric, for the purposes of this paper the coordinate $\\alpha $ is irrelevant.",
"Therefore, we consider only $\\xi ^\\Lambda $ and $\\tilde{\\xi }_\\Lambda $ which can be conveniently packaged into holomorphic Fourier modes $\\mathcal {X}_\\gamma = {\\bf e}\\!\\left( p^\\Lambda \\tilde{\\xi }_\\Lambda -q_\\Lambda \\xi ^\\Lambda \\right)$ labelled by a charge vector $\\gamma =(p^\\Lambda ,q_\\Lambda )$ .",
"At tree level, the Darboux coordinates (multiplied by $t$ ) are known to be simple quadratic polynomials in $t$ so that $\\mathcal {X}_\\gamma $ take the formThe superscript `sf' stands for `semi-flat', which refers to the flatness of the classical geometry in the directions along the torus fibers parametrized by $\\zeta ^\\Lambda ,\\tilde{\\zeta }_\\Lambda $ .",
"$\\mathcal {X}^{\\rm sf}_\\gamma (t)={\\bf e}\\!\\left( \\frac{\\tau _2}{2}\\left(\\bar{Z}_\\gamma (\\bar{u}^a)\\,t-\\frac{Z_\\gamma (u^a)}{t}\\right)+p^\\Lambda \\tilde{\\zeta }_\\Lambda - q_\\Lambda \\zeta ^\\Lambda \\right),$ where $Z_\\gamma (u^a)$ is the central charge (REF ), now expressed in terms of the complex structure moduli $u^a$ of the CY threefold ${\\widehat{\\mathfrak {Y}}}$ mirror to $\\mathfrak {Y}$ , $\\zeta ^\\Lambda $ and $\\tilde{\\zeta }_\\Lambda $ are periods of the RR 3-form in the type IIA formulation, and $\\tau _2=g_s^{-1}$ is the inverse ten-dimensional string coupling.",
"At the non-perturbative level, this expression gets modified and the Darboux coordinates are determined by the integral equation $\\mathcal {X}_\\gamma (t) = \\mathcal {X}^{\\rm sf}_\\gamma (t)\\, {\\bf e}\\!\\left( \\frac{1}{8\\pi ^2}\\sum _{\\gamma ^{\\prime }} \\sigma _{\\gamma ^{\\prime }} \\, \\bar{\\Omega }(\\gamma ^{\\prime })\\, \\langle \\gamma ,\\gamma ^{\\prime }\\rangle \\int _{\\ell _{\\gamma ^{\\prime }} }\\frac{\\mathrm {d}t^{\\prime }}{t^{\\prime }}\\, \\frac{t+t^{\\prime }}{t-t^{\\prime }}\\,\\mathcal {X}_{\\gamma ^{\\prime }}(t^{\\prime })\\right),$ where the sum goes over all charges labelling cycles wrapped by D-branes, $\\bar{\\Omega }(\\gamma ^{\\prime })=\\bar{\\Omega }(\\gamma ^{\\prime },z^a)$ is the corresponding rational Donaldson-Thomas invariant, $\\ell _\\gamma = \\lbrace t\\in {C}P^1\\ :\\ \\ Z_\\gamma /t\\in I{R}^-\\rbrace $ is the so called BPS ray, a contour on ${C}P^1$ extending from $t=0$ to $t=\\infty $ along the direction fixed by the central charge, and $\\sigma _\\gamma $ is a quadratic refinement of the DSZ product on the charge lattice $\\Gamma $ , i.e.",
"a sign factor satisfying the defining relation $\\sigma _{\\gamma _1}\\sigma _{\\gamma _2}=(-1)^{\\langle \\gamma _1,\\gamma _2\\rangle }\\sigma _{\\gamma _1+\\gamma _2},\\qquad \\forall \\gamma _1,\\gamma _2\\in \\Gamma .$ The system of integral equations (REF ) can be solved iteratively by first substituting $\\mathcal {X}_{\\gamma ^{\\prime }}(t^{\\prime })$ on the r.h.s.",
"with its zero-th order value $\\mathcal {X}^{\\rm sf}_{\\gamma ^{\\prime }}(t^{\\prime })$ in the weak coupling limit $\\tau _2\\rightarrow \\infty $ , computing the leading correction from the integral and iterating this process.",
"This produces an asymptotic series at weak coupling, in powers of the DT invariants $\\bar{\\Omega }(\\gamma )$ .",
"Using the saddle point method, it is easy to check that the coefficient of each monomial $\\prod _i \\bar{\\Omega }(\\gamma _i)$ is suppressed by a factor $e^{-\\pi \\tau _2 \\sum _i |Z_{\\gamma _i}|}$ , corresponding to an $n$ -instanton effect [36], [58].",
"Note that multi-instanton effects become of the same order as one-instanton effects on walls of marginal stability where the phases of $Z_{\\gamma _i}$ become aligned, and that the wall-crossing formula ensures that the QK metric on $\\mathcal {M}_H$ is smooth across the walls.",
"[36], [35]."
],
[
"D3-instantons in the large volume limit",
"The above construction of D-instantons is adapted to the type IIA formulation because the equation (REF ) defines the Darboux coordinates in terms of the type IIA fields appearing explicitly in the tree level expression (REF ).",
"To pass to the mirror dual type IIB formulation, one should apply the mirror map, a coordinate transformation from the type IIA to the type IIB physical fields.",
"This transformation was determined in the classical limit in [59], but it also receives instanton corrections.",
"In order to fix the form of these corrections, we require that the metric on $\\mathcal {M}_H$ carries an isometric action of S-duality group $SL(2,{Z})$ of type IIB string theory, which acts on the type IIB fields by an element $g={\\scriptsize \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}}$ in the following way $\\begin{split}&\\tau \\mapsto \\frac{a \\tau +b}{c \\tau + d} \\, ,\\qquad t^a \\mapsto |c\\tau +d| \\,t^a,\\qquad \\begin{pmatrix} c^a \\\\ b^a \\end{pmatrix} \\mapsto \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}\\begin{pmatrix} c^a \\\\ b^a \\end{pmatrix} ,\\\\&\\qquad \\tilde{c}_a\\mapsto \\tilde{c}_a - c_{2,a} \\varepsilon (g) ,\\qquad \\begin{pmatrix} \\tilde{c}_0 \\\\ \\psi \\end{pmatrix} \\mapsto \\begin{pmatrix} d & -c \\\\ -b & a \\end{pmatrix}\\begin{pmatrix} \\tilde{c}_0 \\\\ \\psi \\end{pmatrix},\\end{split}$ where $\\varepsilon (g)$ is the logarithm of the multiplier system of the Dedekind eta function [51].",
"For this purpose, one uses the fact that any isometric action on a quaternion-Kähler manifold (preserving the quaternionic structure) can be lifted to a holomorphic contact transformation on twistor space.",
"In the present case, $SL(2,{Z})$ acts on the fiber coordinate $t$ by a fractional-linear transformation with $\\tau $ -dependent coefficients.",
"This transformation takes a much simpler form when formulated in terms of another coordinate $z$ on ${C}P^1$ (not to be confused with the Kähler moduli $z^a$ ), which is related to $t$ by a Cayley transformation, $z =\\frac{t+I}{t-I}\\, .$ Then the action of $SL(2,{Z})$ on the fiber is given by a simple phase rotationActually, this is true only when five-brane instanton corrections are ignored.",
"Otherwise, the lift also gets a non-trivial deformation [60].",
"$z\\mapsto \\frac{c\\bar{\\tau }+d}{|c\\tau +d|}\\, z\\, .$ Using the holomorphy constraint for the $SL(2,{Z})$ action on the twistor space, quantum corrections to the classical mirror map were computed in [61], [62], [21], [26], in the large volume limit where the Kähler moduli are taken to be large, $t^a\\rightarrow \\infty $ .",
"In this limit, one finds $\\begin{split}u^a=&\\,b^a+It^a-\\frac{I}{2\\tau _2}{\\sum _{\\gamma \\in \\Gamma _+}}p^a\\left[\\int _{\\ell _{\\gamma }}\\mathrm {d}z\\,(1-z)\\,H_{\\gamma }+\\int _{\\ell _{-\\gamma }}\\frac{\\mathrm {d}z}{z^3} (1-z)\\,H_{-\\gamma }\\right]\\\\\\zeta ^0=&\\,\\tau _1,\\qquad \\zeta ^a=-(c^a-\\tau _1 b^a) -3\\sum _{\\gamma \\in \\Gamma _+} p^a\\,{\\rm Re}\\,\\int _{\\ell _{\\gamma }} \\mathrm {d}z\\, z\\,H_{\\gamma },\\\\\\tilde{\\zeta }_a =&\\,\\tilde{c}_a+ \\frac{1}{2}\\, \\kappa _{abc} \\,b^b (c^c - \\tau _1 b^c)+\\kappa _{abc}t^b \\sum _{\\gamma \\in \\Gamma _+} p^c\\,{\\rm Im}\\,\\int _{\\ell _{\\gamma }} \\mathrm {d}z\\,H_{\\gamma },\\end{split}$ where we introduced the convenient notationThe functions $H_\\gamma $ have a simple geometric meaning [32], [63]: they generate contact transformations (i.e.",
"preserving the contact structure) relating the Darboux coordinates living on patches separated by BPS rays.",
"In fact, these functions together with the contours $\\ell _\\gamma $ are the fundamental data fixing the contact structure on the twistor space.",
"$H_\\gamma (t)= \\frac{\\bar{\\Omega }(\\gamma )}{(2\\pi )^2}\\,\\sigma _\\gamma \\mathcal {X}_\\gamma (t)\\, .$ Similar results are known for $\\tilde{\\zeta }_0$ and the NS-axion dual to the $B$ -field, but will not be needed in this paper.",
"Note that the integral contributions to the mirror map are written in terms of the coordinate $z$ (REF ).",
"The reason for using this variable is that, in the large volume limit, the integrals along BPS rays $\\ell _\\gamma $ in (REF ) are dominated by the saddle point [21] $z^{\\prime }_\\gamma \\approx -I\\,\\frac{(q_a+(pb)_a)\\,t^a}{(pt^2)}\\, ,$ for $(pt^2)>0$ , and $z^{\\prime }_{-\\gamma }=1/z^{\\prime }_\\gamma $ in the opposite case.",
"This shows that all integrands can be expanded in Fourier series either around $z=0$ or $z=\\infty $ , keeping constant $t^a z$ or $t^a/z$ , respectively.",
"This allows to extract the leading order in the large volume limit in a simple way.",
"Let us therefore evaluate the combined limit $t^a\\rightarrow \\infty $ , $z\\rightarrow 0$ of the system of integral equations (REF ), assuming that only D3-D1-D(-1) instantons contribute.",
"As a first step, we rewrite the tree level expression (REF ) in terms of the type IIB fields.",
"To this end, we restrict the charge $\\gamma $ to lie in the cone $\\Gamma _+$ , take the central charge as in (REF ) with the cubicThe other contributions to the prepotential, representing perturbative $\\alpha ^{\\prime }$ -corrections and worldsheet instantons, combine with D(-1) and D1-instantons, but are irrelevant for our discussion of D3-instantons in the large volume limit.",
"prepotential (REF ), and substitute the mirror map (REF ).",
"Furthermore, we change the coordinate $t$ to $z$ and take the combined limit.",
"In this way one finds $\\mathcal {X}^{\\rm sf}_\\gamma (z)=\\mathcal {X}^{\\rm cl}_\\gamma (z)\\exp \\left[2\\pi \\sum _{\\gamma ^{\\prime }\\in \\Gamma _+} (tpp^{\\prime })\\int _{\\ell _{\\gamma ^{\\prime }}} \\mathrm {d}z^{\\prime }\\,H_{\\gamma ^{\\prime }}\\right],$ where $\\mathcal {X}^{\\rm cl}_\\gamma (z) ={\\bf e}\\!\\left( - \\hat{q}_0\\tau \\right)\\mathcal {X}^{(\\theta )}_{p,q}(z)$ is the classical part of the Darboux coordinates which we represented as a product of two factors: exponential of the invariant charge (REF ) and the remaining $q_0$ -independent exponential $\\mathcal {X}^{(\\theta )}_{p,q}(z) = e^{-S^{\\rm cl}_p}\\,{\\bf e}\\!\\left( - \\frac{\\tau }{2}\\,(q+b)^2+c^a (q_a +\\textstyle {1\\over 2}(pb)_a)+I\\tau _2 (pt^2)(z^2-2zz_\\gamma )\\right),$ with $S^{\\rm cl}_p$ being the leading part of the Euclidean D3-brane action in the large volume limit given by $S^{\\rm cl}_p= \\pi \\tau _2(pt^2) - 2\\pi Ip^a \\tilde{c}_a$ .",
"Next, we can approximate $\\frac{\\mathrm {d}t^{\\prime }}{t^{\\prime }}\\, \\frac{t+t^{\\prime }}{t-t^{\\prime }}=\\frac{2\\mathrm {d}z^{\\prime }}{1-z^{\\prime 2}}\\, \\frac{1-zz^{\\prime }}{z-z^{\\prime }}\\approx \\left\\lbrace \\begin{array}{cc}\\frac{2\\mathrm {d}z^{\\prime }}{z-z^{\\prime }}\\, , \\qquad & \\gamma ^{\\prime }\\in \\Gamma _+,\\\\\\frac{2(1-zz^{\\prime })\\mathrm {d}z^{\\prime }}{z^{\\prime 3}}\\, , \\qquad & \\gamma ^{\\prime }\\in -\\Gamma _+.\\end{array} \\right.$ This shows that the contribution of $\\gamma ^{\\prime }\\in -\\Gamma _+$ is suppressed comparing to $\\gamma ^{\\prime }\\in \\Gamma _+$ and therefore can be neglected.",
"As a result, the system of integral equations (REF ) in the large volume limit where only D3-D1-D(-1) instantons contribute reduces to the following system of integral equations for $H_\\gamma $ , $H_\\gamma (z)=H^{\\rm cl}_\\gamma (z) \\, \\exp \\left[\\sum _{\\gamma ^{\\prime }\\in \\Gamma _+}\\int _{\\ell _{\\gamma ^{\\prime }}}\\mathrm {d}z^{\\prime }\\, K_{\\gamma \\gamma ^{\\prime }}(z,z^{\\prime })\\,H_{\\gamma ^{\\prime }}(z^{\\prime })\\right].$ Here $H^{\\rm cl}_\\gamma $ is the classical limit of $H_\\gamma $ , i.e.",
"the function (REF ) with $\\mathcal {X}_\\gamma $ replaced by $\\mathcal {X}^{\\rm cl}_\\gamma $ , the integration kernel is now $K_{\\gamma _1\\gamma _2}(z_1,z_2)=2\\pi \\left((tp_1p_2)+\\frac{I\\langle \\gamma _1,\\gamma _2\\rangle }{z_1-z_2}\\right),$ and the BPS ray $\\ell _{\\gamma }$ effectively extends from $z^{\\prime }=-\\infty $ to $z^{\\prime }=+\\infty $ , going through the saddle point (REF ) [21].",
"Below we shall need a perturbative solution of the integral equation (REF ).",
"Applying the iterative procedure outlined below (REF ), or equivalently using the Lagrange inversion theorem, such solution can be written as a sum over rooted trees [36], $H_{\\gamma _1}(z_1) = H^{\\rm cl}_{\\gamma _1}(z_1) \\, \\sum _{n=1}^{\\infty } \\left(\\prod _{i=2}^n \\sum _{\\gamma _i\\in \\Gamma _+}\\,\\int _{\\ell _{\\gamma _i}} \\mathrm {d}z_i\\, H^{\\rm cl}_{\\gamma _i}(z_i) \\right) \\sum _{\\mathcal {T}\\in {T}_n^{\\rm r}}\\,\\frac{\\mathcal {A}(\\mathcal {T})}{|{\\rm Aut}(\\mathcal {T})|}\\, ,$ where ${T}_n^{\\rm r}$ is the set of rooted trees with $n$ vertices and $\\mathcal {A}(\\mathcal {T}) = \\prod _{e\\in E_{\\mathcal {T}}}K_{\\gamma _{s(e)},\\gamma _{t(e)}}(z_{s(e)},z_{t(e)}).$ A rooted treeWe will use calligraphic letters $\\mathcal {T}$ for trees where charges $\\gamma _i$ are assigned to vertices to distinguish them from rooted trees $T$ where the charges are assigned to leaves (hence $T$ has always more than $n$ vertices).",
"Similarly, we will use notations $\\mathfrak {v}$ and $v$ for vertices of these two types of trees, respectively.",
"Note also that whereas $V_\\mathcal {T}$ denotes the set of all vertices, $V_T$ does not includes the leaves.",
"An example of trees of the latter type are attractor flow trees.",
"$\\mathcal {T}$ consists of $n$ vertices joined by directed edges so that the root vertex has only outgoing edges, whereas all other vertices have one incoming edge and an arbitrary number of outgoing ones.",
"We label the vertices of $\\mathcal {T}$ by $\\mathfrak {v}=1,\\dots , n$ in an arbitrary fashion, except for the root which is labelled by $\\mathfrak {v}=1$ .",
"The symmetry factor $|{\\rm Aut}(\\mathcal {T})|$ is the order of the symmetry group which permutes the labels $2,\\dots , n$ without changing the topology of the tree.",
"Each vertex is decorated by a charge vector $\\gamma _\\mathfrak {v}$ and a complex variable $z_\\mathfrak {v}\\in {C}P^1$ .",
"We denote the set of edges by $E_{\\mathcal {T}}$ , the set of vertices by $V_{\\mathcal {T}}$ , and the source and target vertex of an edge $e$ by $s(e)$ and $t(e)$ , respectively.",
"Unpacking these notations, we get, at the few leading orders, $H_{\\gamma _1} &= & H^{\\rm cl}_{\\gamma _1} + \\sum _{\\gamma _2} K_{12} H^{\\rm cl}_{\\gamma _1} \\,H^{\\rm cl}_{\\gamma _2}+ \\sum _{\\gamma _2,\\gamma _3} \\left( \\tfrac{1}{2}\\, K_{12} K_{13} + K_{12} K_{23} \\right)H^{\\rm cl}_{\\gamma _1}\\, H^{\\rm cl}_{\\gamma _2}\\, H^{\\rm cl}_{\\gamma _3}\\\\& + & \\sum _{\\gamma _2,\\gamma _3,\\gamma _4} \\!\\!\\!\\left( \\tfrac{1}{6}\\, K_{12} K_{13} K_{14}+ \\tfrac{1}{2}\\, K_{12} K_{23} K_{24} + K_{12} K_{13} K_{24} + K_{12} K_{23} K_{34} \\right)H^{\\rm cl}_{\\gamma _1} \\, H^{\\rm cl}_{\\gamma _2}\\, H^{\\rm cl}_{\\gamma _3}\\, H^{\\rm cl}_{\\gamma _4}+ \\dots \\nonumber $ where we omitted the integrals and denoted $K_{ij}=K_{\\gamma _i,\\gamma _j}(z_i,z_j)$ .",
"The expansion (REF ) is effectively a multi-instanton expansion in powers of the DT invariants $\\bar{\\Omega }(\\gamma _i)$ , which is asymptotic to the exact solution to (REF ) in the weak coupling limit $\\tau _2\\rightarrow \\infty $ ."
],
[
"From the contact potential to the instanton generating function",
"Recall that our goal is to derive constraints imposed by S-duality on the DT invariants $\\bar{\\Omega }(\\gamma )$ appearing as coefficients in the multi-instanton expansion.",
"To achieve this goal, rather than studying the full metric on $\\mathcal {M}_H$ , it suffices to consider a suitable function on this moduli space which has a non-trivial dependence on $\\bar{\\Omega }(\\gamma )$ and specified transformations under S-duality.",
"There is a natural candidate with the above properties: the so-called contact potential $e^{\\phi }$ , a real function which is well-defined on any quaternion-Kähler manifold with a continuous isometry [34].",
"Furthermore, there is a general expression for the contact potential in terms of Penrose-type integrals on the ${C}P^1$ fiber.",
"In the present case, the required isometry is the shift of the NS-axion, which survives all quantum corrections as long as NS5-instantons are switched off.",
"The contact potential is then given by the exact formula [32] $e^{\\phi } = \\frac{I\\tau _2^2}{16}\\left(\\bar{u}^\\Lambda F_\\Lambda - u^\\Lambda \\bar{F}_\\Lambda \\right)-\\frac{\\chi _{{\\widehat{\\mathfrak {Y}}}}}{192\\pi }+\\frac{I\\tau _2}{16}\\,\\sum _\\gamma \\int _{\\ell _\\gamma } \\frac{\\text{d}t}{t} \\left( t^{-1} Z_\\gamma (u^a)-t\\bar{Z}_\\gamma (\\bar{u}^a)\\right) H_\\gamma ,$ where $\\chi _{{\\widehat{\\mathfrak {Y}}}}$ is the Euler characteristic of ${\\widehat{\\mathfrak {Y}}}$ .",
"This formula indeed captures contribution from D-instantons due to the last term proportional to $H_\\gamma $ .",
"On the other hand, in the classical, large volume limit one finds $e^\\phi =\\frac{\\tau _2^2}{12}(t^3)$ , which shows that the contact potential can be identified with the four-dimensional dilaton and in this approximation behaves as a modular form of weight $(-\\tfrac{1}{2}, -\\tfrac{1}{2})$ under S-duality transformations (REF ).",
"In fact, one can show [32] that $SL(2,{Z})$ preserves the contact structure, i.e.",
"it is an isometry of $\\mathcal {M}_H$ , only if the full non-perturbative contact potential transforms in this way, $e^\\phi \\mapsto \\frac{e^\\phi }{|c\\tau +d|}\\, .$ Furthermore, since S-duality acts by rescaling the Kähler moduli $t^a$ and by a phase rotation of the fiber coordinate $z$ (see (REF )), it preserves each order in the expansion around the large volume limit.",
"This implies that the large volume limit of the D3-instanton contribution to $e^\\phi $ , which we denote by $(e^\\phi )_{\\rm D3}$ , must itself transform as (REF ).",
"It is this condition that we shall exploit to derive modularity constraints on the DT invariants.",
"To make this condition more explicit, let us extract the D3-instanton contribution to the function (REF ).",
"The procedure is the same as the one used to get (REF ), and we relegate the details of the calculation to appendix .",
"The result can be written in a concise way using the complex function defined by $\\mathcal {G}&=& \\sum _{\\gamma \\in \\Gamma _+}\\int _{\\ell _{\\gamma }} \\mathrm {d}z\\, H_{\\gamma }(z)-{1\\over 2}\\,\\sum _{\\gamma _1,\\gamma _2\\in \\Gamma _+}\\int _{\\ell _{\\gamma _1}}\\mathrm {d}z_1\\,\\int _{\\ell _{\\gamma _2}} \\mathrm {d}z_2\\, K_{\\gamma _1\\gamma _2}(z_1,z_2)\\,H_{\\gamma _1}(z_1)H_{\\gamma _2}(z_2)$ and the Maass raising operator $\\mathcal {D}_{\\mathfrak {h}} = \\frac{1}{2\\pi I}\\left(\\partial _\\tau +\\frac{\\mathfrak {h}}{2I\\tau _2}+ \\frac{It^a}{4\\tau _2}\\, \\partial _{t^a}\\right),$ which maps modular functions of weight $(\\mathfrak {h},\\bar{\\mathfrak {h}})$ to modular functions of weight $(\\mathfrak {h}+2,\\bar{\\mathfrak {h}})$ .",
"Then one has (generalizing [14] to all orders in the instanton expansion) $(e^\\phi )_{\\rm D3}=\\frac{\\tau _2}{2}\\,{\\rm Re}\\,\\left(\\mathcal {D}_{-\\frac{3}{2}}\\mathcal {G}\\right)+\\frac{1}{32\\pi ^2}\\,\\kappa _{abc}t^c\\partial _{\\tilde{c}_a}\\mathcal {G}\\partial _{\\tilde{c}_b}\\overline{\\mathcal {G}}.$ It is immediate to see that $(e^\\phi )_{\\rm D3}$ transforms under S-duality as (REF ) provided the function $\\mathcal {G}$ transforms as a modular form of weight $(-\\tfrac{3}{2},\\tfrac{1}{2})$ .",
"In order to derive the implications of this fact, we need to express $\\mathcal {G}$ in terms of the generalized DT invariants.",
"For this purpose, we substitute the multi-instanton expansion (REF ) into (REF ).",
"We claim that the result takes the simple form $\\mathcal {G}=\\sum _{n=1}^\\infty \\left[\\prod _{i=1}^{n} \\sum _{\\gamma _i\\in \\Gamma _+}\\int _{\\ell _{\\gamma _i}}\\mathrm {d}z_i\\, H^{\\rm cl}_{\\gamma _i}(z_i) \\right]\\mathcal {G}_n(\\lbrace \\gamma _i,z_i\\rbrace ),$ where $\\mathcal {G}_n(\\lbrace \\gamma _i,z_i\\rbrace )$ is now a sum over unrooted trees with $n$ vertices, $\\mathcal {G}_n(\\lbrace \\gamma _i,z_i\\rbrace )=\\sum _{\\mathcal {T}\\in \\, {T}_n} \\frac{\\mathcal {A}(\\mathcal {T})}{|{\\rm Aut}(\\mathcal {T})|}=\\frac{1}{n!",
"}\\sum _{\\mathcal {T}\\in \\, {T}_n^\\ell } \\mathcal {A}(\\mathcal {T}),$ and in the second equality we rewrote the result as a sum over unrooted labelled trees.The number of such trees is $|{T}_n^\\ell |=n^{n-2}=\\lbrace 1,1,3,16,125,1296,\\dots \\rbrace $ for $n\\ge 1$ .",
"Such trees also appear in the Joyce-Song wall-crossing formula [42], [43] and are conveniently labelled by their Prüfer code.",
"To see why this is the case, observe that under the action of the Euler operator $\\hat{D}=H^{\\rm cl}\\partial _{H^{\\rm cl}}$ rescaling all functions $H^{\\rm cl}_\\gamma $ , the function $\\mathcal {G}$ maps to the first term in (REF ), which we denote by $\\mathcal {F}$ .",
"Namely, $\\hat{D}\\cdot \\mathcal {G}=\\mathcal {F},\\qquad \\mathcal {F}\\equiv \\sum _{\\gamma \\in \\Gamma _+}\\int _{\\ell _{\\gamma }} \\mathrm {d}z\\, H_{\\gamma },$ as can be verified with the help of the integral equation (REF ).",
"The multi-instanton expansion of $\\mathcal {F}$ follows immediately from (REF ), $\\mathcal {F}= \\sum _{n=1}^{\\infty } \\left(\\prod _{i=1}^n \\sum _{\\gamma _i\\in \\Gamma _+}\\,\\int _{\\ell _{\\gamma _i}} \\mathrm {d}z_i\\, H^{\\rm cl}_{\\gamma _i}(z_i) \\right) \\sum _{\\mathcal {T}\\in {T}_n^{\\rm r}}\\,\\frac{\\mathcal {A}(\\mathcal {T})}{|{\\rm Aut}(\\mathcal {T})|}\\, .$ Integrating back the action of the derivative operator $\\hat{D}$ , we see that the sum over rooted trees in (REF ) turns into the sum over unrooted trees in (REF ).",
"At the first few orders we get, using the same shorthand notation as in (REF ), $\\begin{split}\\mathcal {G}= &\\sum _{\\gamma } H^{\\rm cl}_\\gamma + \\frac{1}{2} \\sum _{\\gamma _1,\\gamma _2}K_{12}H^{\\rm cl}_{\\gamma _1} \\,H^{\\rm cl}_{\\gamma _2}+ \\frac{1}{2} \\sum _{\\gamma _1,\\gamma _2,\\gamma _3}K_{12}K_{23}H^{\\rm cl}_{\\gamma _1}\\, H^{\\rm cl}_{\\gamma _2}\\, H^{\\rm cl}_{\\gamma _3}\\\\&+\\sum _{\\gamma _1,\\gamma _2,\\gamma _3,\\gamma _4}\\left(\\frac{1}{6} \\, K_{12} K_{13} K_{14} + \\frac{1}{2}\\,K_{12} K_{23} K_{34} \\right)H^{\\rm cl}_{\\gamma _1}\\, H^{\\rm cl}_{\\gamma _2}\\, H^{\\rm cl}_{\\gamma _3}\\, H^{\\rm cl}_{\\gamma _4}+\\dots \\end{split}$ The simplicity of the expansion (REF ), and the relation (REF ) to the contact potential, show that the function $\\mathcal {G}$ is very natural and, in some sense, more fundamentalIn [14], it was noticed that the function $\\mathcal {G}$ , denoted by $\\tilde{\\mathcal {F}}$ in that reference and computed at second order in the multi-instanton expansion, could be obtained from the seemingly simpler function $\\mathcal {F}$ by halving the coefficient of its second order contribution.",
"Now we see that this ad hoc prescription is the consequence of going from rooted to unrooted trees, as a result of adding the second term in (REF ).",
"than the naive instanton sum $\\mathcal {F}$ .",
"We shall henceforth refer to $\\mathcal {G}$ as the `instanton generating function'.",
"In the following we shall postulate that $\\mathcal {G}$ transforms as a modular form of weight $(-\\frac{3}{2},\\frac{1}{2})$ , and analyze the consequences of this requirement for the DT invariants."
],
[
"Theta series decomposition and modularity",
"In this section, we use the spectral flow symmetry to decompose the instanton generating function $\\mathcal {G}$ into a sum of indefinite theta series multiplied by holomorphic generating functions of MSW invariants.",
"We then study the modular properties of these indefinite theta series, and identify the origin of the modular anomaly."
],
[
"Factorisation",
"To derive modularity constraints on the DT invariants, we need to perform a theta series decomposition of the generating function $\\mathcal {G}$ defined in (REF ).",
"To this end, let us make use of the fact noticed in (REF ) that the DSZ products $\\langle \\gamma ,\\gamma ^{\\prime }\\rangle $ and hence the kernels (REF ) do not depend on the $q_0$ charge.",
"Choosing the quadratic refinement $\\sigma _\\gamma $ as in (REF ), which is also $q_0$ -independent, and using the factorization (REF ) of $\\mathcal {X}^{\\rm cl}_\\gamma $ , one can rewrite the expansion (REF ) as follows $\\mathcal {G}=\\sum _{n=1}^\\infty \\frac{1}{(2\\pi )^{2n}}\\left[\\prod _{i=1}^{n}\\sum _{p_i,q_i}\\sigma _{p_i,q_i}h^{\\rm DT}_{p_i,q_i}\\int _{\\ell _{\\gamma _i}}\\mathrm {d}z_i\\, \\mathcal {X}^{(\\theta )}_{p_i,q_i}(z_i) \\right]\\mathcal {G}_n(\\lbrace \\gamma _i,z_i\\rbrace ),$ where the sum over the invariant charges $\\hat{q}_{i,0}$ gave rise to the generating functions of DT invariants defined in (REF ).",
"This is not yet the desired form because these generating functions depend non-trivially on the remaining electric charges $q_{i,a}$ .",
"If it were not for this dependence, the sum over $q_{i,a}$ would produce certain non-Gaussian theta series, and at each order we would have a product of this theta series and $n$ generating functions.",
"Then the modular properties of the theta series would dictate the modular properties of the generating function.",
"Such a theta series decomposition can be achieved by expressing the DT invariants in terms of the MSW invariants, for which the dependence on electric charges $q_{i,a}$ reduces to the dependence on the residue classes $\\mu _{i,a}$ due to the spectral flow symmetry.",
"Substituting the expansion (REF ) of $h^{\\rm DT}_{p,q}$ in terms of $h_{p,\\mu }$ , the expansion (REF ) of the function $\\mathcal {G}$ can be brought to the following factorized form $\\mathcal {G}=\\sum _{n=1}^\\infty \\frac{2^{-\\frac{n}{2}}}{\\pi \\sqrt{2\\tau _2}}\\left[\\prod _{i=1}^{n}\\sum _{p_i,\\mu _i}\\sigma _{p_i}h_{p_i,\\mu _i}\\right]e^{-S^{\\rm cl}_p}\\vartheta _{{p},{\\mu }}\\bigl (\\Phi ^{{\\rm tot}}_{n},n-2\\bigr ),$ where $\\vartheta _{{p},{\\mu }}$ is a theta series (REF ) with parameter $\\lambda =n-2$ , whose kernel has the following structure $\\Phi ^{{\\rm tot}}_n({x})=\\,{\\rm Sym}\\, \\left\\lbrace \\sum _{n_1+\\cdots n_m=n \\atop n_k\\ge 1} \\Phi ^{\\scriptscriptstyle \\,\\int }_m({x}^{\\prime })\\prod _{k=1}^m \\Phi ^{\\,g}_{n_k}(x_{j_{k-1}+1},\\dots ,x_{j_k})\\right\\rbrace .$ Here the sum runs over ordered partitions of $n$ , whose number of parts is denoted by $m$ , and we adopted notations from (REF ) for indices $j_k$ .",
"The argument ${x}$ of the kernel encodes the electric components of the charges (shifted by the B-field and rescaled by $\\sqrt{2\\tau _2}$ ) and lives in a vector space $\\left(\\oplus _{i=1}^n \\Lambda _i\\right) \\otimes {R}$ of dimension $d=nb_2$ , given by $b_2$ copies of the lattice $\\Lambda $ , where the $i$ -th copy $\\Lambda _i$ carries the bilinear form $\\kappa _{i,ab}=\\kappa _{abc}p_i^c$ of signature $(1,b_2-1)$ .",
"Therefore, $\\vartheta _{{p},{\\mu }}$ is an indefinite theta series associated to the bilinear form given explicitly in (REF ), which has signature $(n,n(b_2-1))$ .",
"Finally, the kernel (REF ) is constructed from two other functions.",
"The first, $\\Phi ^{\\scriptscriptstyle \\,\\int }_n$ , is the iterated integral of the coefficient $\\mathcal {G}_n$ in the expansion (REF ), $\\Phi ^{\\scriptscriptstyle \\,\\int }_n({x}) =\\left(\\frac{\\sqrt{2\\tau _2}}{2\\pi }\\right)^{n-1}\\left[\\prod _{i=1}^n\\int _{\\ell _{\\gamma _i}}\\frac{\\mathrm {d}z_i}{2\\pi } \\, W_{p_i}(x_i,z_i)\\right]\\mathcal {G}_n(\\lbrace \\gamma _i,z_i\\rbrace ),$ weighted with the Gaussian measure factor $W_{p}(x,z)=e^{-2\\pi \\tau _2 z^2(pt^2)-2\\pi I\\sqrt{2\\tau _2}\\, z\\, (pxt)}$ coming from the $z$ -dependent part of (REF ).",
"Although this function is written in terms of $\\mathcal {G}_n$ depending on full electromagnetic charge vectors $\\gamma _i$ , it is actually independent of their $q_0$ components.",
"Indeed, using the result (REF ), it can be rewritten as $\\Phi ^{\\scriptscriptstyle \\,\\int }_n({x}) = \\frac{1}{n!",
"}\\left[\\prod _{i=1}^n\\int _{\\ell _{\\gamma _i}}\\frac{\\mathrm {d}z_i}{2\\pi } \\, W_{p_i}(x_i,z_i)\\right]\\sum _{\\mathcal {T}\\in \\, {T}_n^\\ell }\\prod _{e\\in E_\\mathcal {T}} \\hat{K}_{s(e) t(e)},$ where we introduced a rescaled version of the kernel (REF ) $\\hat{K}_{ij}(z_i,z_j)=\\left(\\left(\\sqrt{2\\tau _2}\\,t+I\\,\\frac{x_i-x_j}{z_i-z_j}\\right)p_ip_j\\right).$ Note that $\\tau _2$ and $t^a$ appear only in the modular invariant combination $\\sqrt{2\\tau _2}\\, t^a$ .",
"In (REF ) this function appears with the argument ${x}^{\\prime }$ and carries a dependence on ${p}^{\\prime }$ (not indicated explicitly) which are both $mb_2$ -dimensional vectors with components (cf.",
"(REF )) $p^{\\prime a}_k=\\sum _{i=j_{k-1}+1}^{j_k}p^a_i,\\qquad x^{\\prime a}_k=\\kappa ^{\\prime ab}_k\\sum _{i=j_{k-1}+1}^{j_k} \\kappa _{i,bc} x^c_i,$ where $\\kappa ^{\\prime }_{k,ab}=\\kappa _{abc}p^{\\prime c}_k$ .",
"The second function, $\\Phi ^{\\,g}_{n}$ , appears due to the expansion of DT invariants in terms of the MSW invariants and is given by a suitably rescaled tree index $\\Phi ^{\\,g}_{n}({x})=\\frac{\\sigma _{\\gamma }(\\sqrt{2\\tau _2})^{n-1}}{\\prod _{i=1}^n\\sigma _{\\gamma _i}}\\,g_{{\\rm tr},n}(\\lbrace \\gamma _i,c_i\\rbrace ).$ It is also written in terms of functions depending on the full electromagnetic charge vectors $\\gamma _i$ (with $\\gamma =\\gamma _1+\\cdots \\gamma _n$ ).",
"However, using (REF ), all quadratic refinements can be expressed through $(-1)^{\\langle \\gamma _i,\\gamma _j\\rangle }$ which cancel the corresponding sign factors in the tree index (see (REF )).",
"Furthermore, as was noticed in the end of section REF , the tree index is independent of the $q_0$ components of the charge vectors.",
"Therefore, it can be written as a function of $p_i^a$ , $\\mu _{i,a}$ and $x_i^a=\\sqrt{2\\tau _2}(\\kappa _i^{ab} q_{i,b}+b^a)$ .",
"Then, since after cancelling the sign factors, $g_{{\\rm tr},n}$ becomes a homogeneous function of degree $n-1$ in the D2-brane charge $q_{i,a}$ , all factors of $\\sqrt{2\\tau _2}$ in (REF ) cancel as well."
],
[
"Modularity and Vignéras' equation",
"As explained in appendix REF , the theta series $\\vartheta _{{p},{\\mu }}\\bigl (\\Phi ,\\lambda \\bigr )$ is a vector-valued modular form of weight $(\\lambda +d/2,0)$ provided the kernel $\\Phi $ satisfies Vignéras' equation (REF ) — along with certain growth conditions which we expect to be automatically satisfied for the kernels of interest in this work.",
"In our case $\\lambda =n-2$ and the dimension of the lattice is $d=nb_2(\\mathfrak {Y})$ so that the expected weight of the theta series is $(2(n-1)+nb_2/2,0)$ .",
"It is consistent with weight $(-\\tfrac{3}{2},\\tfrac{1}{2})$ of $\\mathcal {G}$ given in (REF ) only if $h_{p,\\mu }$ is a vector-valued holomorphic modular form of weight $(-b_2/2-1,0)$ .",
"However for this to be true, the kernel $\\Phi ^{{\\rm tot}}_n$ ought to satisfy Vignéras' equation.",
"Let us examine whether or not this is the case.",
"To this end, we first consider the kernel $\\Phi ^{\\scriptscriptstyle \\,\\int }_n$ (REF ).",
"In appendix we evaluate explicitly the iterated integrals defining this kernel.",
"To present the final result, let us introduce the following $d$ -dimensional vectors ${v}_{ij}, {u}_{ij}$ : $\\begin{split}({v}_{ij})_k^a=&\\, \\delta _{ki} p_j^a-\\delta _{kj} p_i^a \\qquad \\qquad \\quad \\ \\mbox{such that} \\quad {v}_{ij}\\cdot {x}=(p_ip_j(x_i-x_j)),\\\\({u}_{ij})_k^a=&\\, \\delta _{ki}(p_jt^2)t^a-\\delta _{kj} (p_it^2)t^a \\quad \\mbox{such that} \\quad {u}_{ij}\\cdot {x}=(p_jt^2)(p_ix_it)-(p_it^2)(p_jx_jt),\\end{split}$ where $k$ labels the copy in $\\oplus _{k=1}^n \\Lambda _k$ , $a=1,\\dots , b_2$ , and the bilinear form is given in (REF ).",
"The first scalar product ${v}_{ij}\\cdot {x}$ corresponds to the DSZ product $\\langle \\gamma _i,\\gamma _j\\rangle $ , whereas the second product ${u}_{ij}\\cdot {x}$ corresponds to $-2\\,{\\rm Im}\\,[Z_{\\gamma _i}\\bar{Z}_{\\gamma _j}]$ (REF ), both rescaled by $\\sqrt{2\\tau _2}$ and expressed in terms of $x_i^a$ .",
"From these vectors we can construct two sets of vectors which are assigned to the edges of an unrooted labelled tree $\\mathcal {T}$ , such as the trees appearing in (REF ) and (REF ).",
"Namely, ${v}_e=\\sum _{i\\in V_{\\mathcal {T}_e^s}}\\sum _{j\\in V_{\\mathcal {T}_e^t}}{v}_{ij},\\qquad {u}_e=\\sum _{i\\in V_{\\mathcal {T}_e^s}}\\sum _{j\\in V_{\\mathcal {T}_e^t}}{u}_{ij},$ where $\\mathcal {T}_e^s$ , $\\mathcal {T}_e^t$ are the two disconnected trees obtained from the tree $\\mathcal {T}$ by removing the edge $e$ .",
"Then the kernel $\\Phi ^{\\scriptscriptstyle \\,\\int }_n$ can be expressed as followsBoth vectors ${u}_e$ and ${v}_{s(e) t(e)}$ depend on the choice of orientation of the edge $e$ , but this ambiguity is cancelled in the function $\\widetilde{\\Phi }^M_{n-1}$ .",
"$\\Phi ^{\\scriptscriptstyle \\,\\int }_n({x})=\\frac{\\Phi ^{\\scriptscriptstyle \\,\\int }_1(x)}{2^{n-1} n!",
"}\\sum _{\\mathcal {T}\\in \\, {T}_n^\\ell }\\widetilde{\\Phi }^M_{n-1}(\\lbrace {u}_e\\rbrace , \\lbrace {v}_{s(e) t(e)}\\rbrace ;{x}).$ Here the first factor is simply a Gaussian $\\Phi ^{\\scriptscriptstyle \\,\\int }_1(x)=\\frac{e^{-\\frac{\\pi (pxt)^2}{(pt^2)}}}{2\\pi \\sqrt{2\\tau _2(pt^2)}}$ which ensures the suppression along the direction of the total charge in the charge lattice.",
"In the second factor one sums over unrooted labelled trees $\\mathcal {T}$ with $n$ vertices, with summand given by a function $\\widetilde{\\Phi }^M_n$ defined as in (REF ), upon replacing $\\Phi _n^E$ by $\\Phi _n^M$ in that expression and setting $m=n$ .",
"Both $\\Phi _n^E$ and $\\Phi _n^M$ are the so-called generalized (complementary) error functions introduced in [28] and further studied in [29], whose definitions are recalled in (REF ), () and (REF ).",
"The functions $\\widetilde{\\Phi }^M_{n-1}$ in (REF ) depend on two sets of $n-1$ $d$ -dimensional vectors: the vectors in the first set are given by ${u}_e$ defined above, whereas the vectors ${v}_{s(e) t(e)}$ in the second set coincide with ${v}_{ij}$ for $i$ and $j$ corresponding to the source and target vertices of edge $e$ of the labelled tree.",
"The remarkable property of the generalized error functions $\\Phi ^M_{n-1}$ and their uplifted versions $\\widetilde{\\Phi }^M_{n-1}$ is that, away from certain loci where these functions are discontinuous, they satisfy Vignéras' equation for $\\lambda =0$ and $n-1$ , respectively.",
"Given that $\\Phi ^{\\scriptscriptstyle \\,\\int }_1$ is also a solution for $\\lambda =-1$ , and the vector ${t}=(t^a,\\dots ,t^a)$ (such that ${t}\\cdot {x}=(pxt)$ ) is orthogonal to all vectors ${u}_e$ and ${v}_{s(e) t(e)}$ , the kernel (REF ) satisfies this equation for $\\lambda =n-2$ .",
"However, as mentioned above, it fails to do so on the loci where it is discontinuous.",
"These discontinuities arise due to dependence of the integration contours $\\ell _\\gamma $ on moduli and electric charges.",
"Of course, since the integrands are meromorphic functions, the integrals do not depend on deformations of the contours provided they do not cross the poles.",
"But this is exactly what happens when two BPS rays, say $\\ell _\\gamma $ and $\\ell _{\\gamma ^{\\prime }}$ , exchange their positions, which in turn takes place when the phases of the corresponding central charges $Z_\\gamma $ and $Z_{\\gamma ^{\\prime }}$ align, as follows from (REF ).",
"The loci where such alignment takes place are nothing else but the walls of marginal stability.",
"This point will play an important rôle in the next subsection since it makes it possible to recombine the discontinuities of the generalized error functions with discontinuities of the tree indices.",
"We now turn to the action of Vignéras' operator on $\\Phi ^{\\,g}_{n}$ .",
"To this end, it is convenient to use the representation of the tree index as a sum over attractor flow trees (REF ).",
"Let us assign a $nb_2$ -dimensional vector $\\tilde{{v}}_{v}$ to each vertex $v$ of a flow tree.",
"Denoting by $\\mathcal {I}_v$ the set of leaves which are descendants of vertex $v$ , we set $\\tilde{{v}}_v=\\sum _{i\\in \\mathcal {I}_{L(v)}}\\sum _{j\\in \\mathcal {I}_{R(v)}}{v}_{ij}.$ With these definitions the kernel (REF ) can be written as $\\Phi ^{\\,g}_n({x}) &=&(-1)^{n-1}\\sum _{T\\in {T}_n^{\\rm af}}\\prod _{v\\in V_T}(\\tilde{{v}}_v,{x})\\,\\Delta _{\\gamma _{L(v)}\\gamma _{R(v)}}^{z_{p(v)}}.$ The factors $\\Delta _{\\gamma _L\\gamma _R}^z$ are locally constant and therefore, away from the loci where they are discontinuous, the action of Vignéras operator reduces to its action on the scalar products $(\\tilde{{v}}_v,{x})$ .",
"For a single such factor one finds $V_\\lambda (\\tilde{{v}}_v,{x})=(\\tilde{{v}}_v,{x})V_{\\lambda -1}+2\\tilde{{v}}_v\\cdot \\partial _{x}.$ The crucial observation is that all vectors $\\tilde{{v}}_v$ appearing in the product (REF ) for a single tree are mutually orthogonal $(\\tilde{{v}}_v,\\tilde{{v}}_{v^{\\prime }})=0$ , which is clear because $\\langle \\gamma _{L(v)},\\gamma _{R(v)}\\rangle $ is antisymmetric in charges $\\gamma _{L(v)},\\gamma _{R(v)}$ , whereas the factors associated with vertices which are not descendants of $v$ either depend on their sum or do not depend on them at all.",
"Therefore, one obtains $V_\\lambda \\Phi ^{\\,g}_{n_k}({x})=\\Phi ^{\\,g}_{n_k}({x})V_{\\lambda -n_k+1}+2(-1)^{n_k-1}\\sum _{T\\in {T}_{n_k}^{\\rm af}}\\Delta (T)\\sum _{v\\in V_T}\\left[\\prod _{v^{\\prime }\\in V_T\\setminus \\lbrace v\\rbrace }(\\tilde{{v}}_{v^{\\prime }},{x})\\right]\\tilde{{v}}_v\\cdot \\partial _{x}.$ Let us now evaluate the action of $V_{n-2}$ on the full kernel $\\Phi ^{{\\rm tot}}_n$ .",
"Applying the result (REF ), we observe that the second term vanishes on the other factors in (REF ) due to the same reason that they either do not depend (in the case of $\\Phi ^{\\,g}_{n_{k^{\\prime }}}$ , $k^{\\prime }\\ne k$ ) or depend (in the case of $\\Phi ^{\\scriptscriptstyle \\,\\int }_{m}$ ) only on the sum of charges entering $\\Phi ^{\\,g}_{n_k}$ .",
"Therefore, one finds that, away from discontinuities of generalized error functions and $\\Delta (T)$, one has $V_{n-2}\\cdot \\Phi ^{{\\rm tot}}_n=0$ ."
],
[
"Discontinuities and the anomaly",
"Let us now turn to the discontinuities of $\\Phi ^{{\\rm tot}}_n$ which we ignored so far and which spoil Vignéras' equation and hence modularity of the theta series.",
"There are three potential sources of such discontinuities: walls of marginal stability — at these loci $\\Phi ^{\\scriptscriptstyle \\,\\int }_{m}$ are discontinuous due to exchange of integration contours and $\\Phi ^{\\,g}_{n_k}$ jump due to factors $\\Delta _{\\gamma _{L(v)}\\gamma _{R(v)}}^{z_{p(v)}}$ assigned to the root vertices of attractor flow trees; `fake walls' — these are loci in the moduli space where $\\,{\\rm Im}\\,\\bigl [ Z_{\\gamma _{L(v)}}\\bar{Z}_{\\gamma _{R(v)}}(z^a_{p(v)})\\bigr ]=0$ , and hence the corresponding $\\Delta $ -factor jumps, where $v$ is not a root vertex — they correspond to walls of marginal stability for the intermediate bound states appearing in the attractor flow; moduli independent loci where $(\\tilde{{v}}_v,{x})=0$ — at these loci the factors $\\Delta _{\\gamma _{L(v)}\\gamma _{R(v)}}^{z_{p(v)}}$ and hence $\\Phi ^{\\,g}_{n_k}$ are discontinuous due to the second term in (REF ).",
"Remarkably, the two effects due to the non-trivial charge and moduli dependence of the DT invariants and the exchange of contours cancel each other and the function $\\mathcal {G}$ turns out to be smooth at loci of the first type.",
"This is expected because the whole construction of D-instantons has been designed to make the resulting metric on the moduli space smooth across these loci, which required the cancellation of these discontinuities [36], [35].",
"Moreover, in [37] it was proven that the contact potential is also smooth, which indicates that the function $\\mathcal {G}$ must be smooth as well.",
"In appendix we present an explicit proof of this fact based on the representation in terms of trees.",
"Furthermore, in [23] it was shown that the discontinuities across `fake walls' cancel in the sum over flow trees as well.",
"In fact, this cancellation is explicit in the representation of the partial tree index given by the recursive formula (REF ) where the signs responsible for such `fake discontinuities' do not arise at all.",
"As a result, it remains to consider only the discontinuities of the third type corresponding to the moduli independent loci.",
"It is straightforward to check that already for $n=2$ these discontinuities are indeed present and do spoil modularity of the theta series.",
"For small $n$ one can explicitly evaluate the anomaly in Vignéras' equation.",
"It is given by a series of terms proportional to $\\delta (\\tilde{{v}}_v,{x})$ .",
"Note that no derivatives of delta functions or products of two delta functions arise despite the presence of the second derivative in Vignéras' operator.",
"This is because each ${\\rm sgn}(\\tilde{{v}}_v,{x})$ from $\\Delta (T)$ is multiplied by $(\\tilde{{v}}_v,{x})$ from $\\kappa (T)$ in (REF ) and one gets a non-vanishing result only if the second order derivative operator acts on both factors.",
"In particular, this implies that the anomaly is completely characterized by the action of $V_\\lambda $ on $\\Phi ^{\\,g}_{n}$ ."
],
[
"Modular completion",
"Since the theta series $\\vartheta _{{p},{\\mu }}(\\Phi ^{{\\rm tot}}_{n})$ are not modular for $n\\ge 2$ , the analysis of the previous section implies that the generating function $h_{p,\\mu }$ of the MSW invariants is not modular either whenever the divisor $\\mathcal {D}=p^a\\gamma _a$ is the sum of $n\\ge 2$ irreducible divisors.",
"However, its modular anomaly has a definite structure.",
"In particular, in [14] it was shown that for $n=2$ , $h_{p,\\mu }$ must be a vector-valued mixed mock modular form, i.e.",
"it has a non-holomorphic completion $\\widehat{h}_{p,\\mu }$ constructed in a specific way from a set of holomorphic modular forms and their Eichler integrals [30], [13].",
"In this section we generalize this result for arbitrary $n$ , i.e.",
"for any degree of reducibility of the divisor."
],
[
"Completion of the generating function",
"Let us recall the notations $\\check{\\gamma }=(p^a,q_a)$ and $Q_n$ from (REF ), and decompose the electric component $q_a$ using spectral flow as in (REF ).",
"Then we define $\\widehat{h}_{p,\\mu }(\\tau )= h_{p,\\mu }(\\tau )+\\sum _{n=2}^\\infty \\sum _{\\sum _{i=1}^n \\check{\\gamma }_i=\\check{\\gamma }}R_n(\\lbrace \\check{\\gamma }_i\\rbrace ,\\tau _2)\\, e^{\\pi I\\tau Q_n(\\lbrace \\check{\\gamma }_i\\rbrace )}\\prod _{i=1}^n h_{p_i,\\mu _i}(\\tau ).$ We are looking for non-holomorphic functions $R_n$ , exponentially suppressed for large $\\tau _2$ , such that $\\widehat{h}_{p,\\mu }$ transforms as a modular form.",
"The condition for this to be true can be found along the same lines as before: one needs to rewrite the expansion of the function $\\mathcal {G}$ as a series in $\\widehat{h}_{p,\\mu }$ and require that at each order the coefficient is given by a modular covariant theta series.",
"For such a theta series decomposition to be possible however, it is important that $\\widehat{h}_{p,\\mu }$ be invariant under the spectral flow, which implies that the functions $R_n$ be independent of the spectral flow parameter $\\epsilon ^a$ in the decomposition (REF ) of the total charge $\\check{\\gamma }$ .As a result, this parameter can be fixed to zero so that the sum over the D2-brane charges $q_{i,a}$ is restricted to those which satisfy the constraint $\\sum _{i=1}^n q_{i,a}=\\mu _a+\\frac{1}{2} \\kappa _{abc}p^b p^c$ .",
"This condition will be an important consistency requirement on our construction.",
"Rather than inverting (REF ) and substituting the result into (REF ), we can consider the generating function of DT invariants $h^{\\rm DT}_{p,\\mu }$ and, as a first step, rewrite it as a series in $\\widehat{h}_{p,\\mu }$ .",
"Denoting the coefficient of such expansion by $\\widehat{g}_n(\\lbrace \\check{\\gamma }_i\\rbrace ,z^a)$ (with $\\widehat{g}_1\\equiv 1$ ), we get $\\begin{split}h^{\\rm DT}_{p,q}(\\tau ,z^a)=&\\,\\sum _{\\sum _{i=1}^n \\check{\\gamma }_i=\\check{\\gamma }}\\widehat{g}_n(\\lbrace \\check{\\gamma }_i\\rbrace ;z^a, \\tau _2) \\,e^{\\pi I\\tau Q_n(\\lbrace \\check{\\gamma }_i\\rbrace )}\\prod _{i=1}^n \\widehat{h}_{p_i,\\mu _i}(\\tau ).\\end{split}$ Comparing with (REF ), we see that the coefficients $\\widehat{g}_n$ are a direct analogue of the tree index $g_{{\\rm tr},n}$ .In fact, they also depend on $z^a$ only through the stability parameters (REF ), so we shall often denote them by $\\widehat{g}_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace ;\\tau _2)$ .",
"The expansion of $\\mathcal {G}$ in terms of $\\widehat{h}_{p,\\mu }$ is then simply obtained by replacing $g_{{\\rm tr},n}$ by $\\widehat{g}_n$ in (REF ), which affects the kernel of the corresponding theta series.",
"Our first problem is to express these coefficients in terms of the functions $R_n$ .",
"The result can be nicely formulated using so-called Schröder trees, which are rooted ordered trees such that all vertices (except the leaves) have at least 2 children (see Fig.",
"REF ).",
"Their vertices are decorated by charges in such way that the leaves have charges $\\gamma _i$ , whereas the charges at other vertices are defined by the inductive rule $\\gamma _v=\\sum _{v^{\\prime }\\in {\\rm Ch}(v)}\\gamma _{v^{\\prime }}$ where ${\\rm Ch}(v)$ is the set of children of vertex $v$ .",
"Note that these trees should not be confused with flow trees, since they are not restricted to be binary nor do they carry moduli at the vertices.",
"We denote the set of Schröder treesThe number of Schröder trees with $n$ leaves is the $n-1$ -th super-Catalan number, $|{T}_n^{\\rm S}|=\\lbrace 1,1, 3,11,45,197,903,\\dots \\rbrace $ for $n\\ge 1$ (sequence A001003 on the Online Encyclopedia of Integer Sequences).",
"with $n$ leaves by ${T}_n^{\\rm S}$ .",
"Figure: An example of Schröder tree contributing to W 8 W_8.",
"Near each vertex we showed the corresponding factorusing the shorthand notation γ i+j =γ i +γ j \\gamma _{i+j}=\\gamma _i+\\gamma _j.Let us also introduce a convenient notation: for any set of functions $f_n(\\lbrace \\check{\\gamma }_i\\rbrace )$ depending on $n$ charges and a given Schröder tree $T$ , we set $f_{v}\\equiv f_{k_v}(\\lbrace \\check{\\gamma }_{v^{\\prime }}\\rbrace )$ where $v^{\\prime }\\in {\\rm Ch}(v)$ and $k_v$ is their number.",
"Using these notations, the expression of $\\widehat{g}_n$ in terms of $R_n$ can be encoded into a recursive equation provided by the following proposition, whose proof we relegate to appendix : Proposition 1 The coefficients $\\widehat{g}_n$ are determined recursively by the following equation $\\widehat{g}_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace )=-{1\\over 2}\\,{\\rm Sym}\\, \\left\\lbrace \\sum _{\\ell =1}^{n-1}\\Delta _{\\gamma _L^\\ell \\gamma _R^\\ell }^z\\,\\kappa (\\gamma _{LR}^\\ell )\\,\\widehat{g}_\\ell (\\lbrace \\check{\\gamma }_i,c_i^{(\\ell )}\\rbrace _{i=1}^\\ell )\\,\\widehat{g}_{n-\\ell }(\\lbrace \\check{\\gamma }_i,c_i^{(\\ell )}\\rbrace _{i=\\ell +1}^n)\\right\\rbrace +W_n(\\lbrace \\check{\\gamma }_i\\rbrace ),$ where $\\gamma _L^\\ell =\\sum _{i=1}^\\ell \\gamma _i$ , $\\gamma _R^\\ell =\\sum _{i=\\ell +1}^n\\gamma _i$ , $\\gamma _{LR}^\\ell =\\langle \\gamma _L^\\ell ,\\gamma _R^\\ell \\rangle $ , $c_i^{(\\ell )}$ are the stability parameters at the point where the attractor flow crosses the wall for the decay $\\gamma \\rightarrow \\gamma _L^\\ell +\\gamma _R^\\ell $ (cf.",
"(REF )), and $W_n$ are functions given by the sum over Schröder trees $W_n(\\lbrace \\check{\\gamma }_i\\rbrace )= \\,{\\rm Sym}\\, \\left\\lbrace \\sum _{T\\in {T}_n^{\\rm S}}(-1)^{n_T}\\prod _{v\\in V_T} R_{v}\\right\\rbrace ,$ with $n_T$ being the number of vertices of the tree $T$ (excluding the leaves).",
"The same functions $W_n$ also provide the inverse formula to (REF ), namely $h_{p,\\mu }(\\tau )= \\widehat{h}_{p,\\mu }(\\tau )+\\sum _{n=2}^\\infty \\sum _{\\sum _{i=1}^n \\check{\\gamma }_i=\\check{\\gamma }}W_n(\\lbrace \\check{\\gamma }_i\\rbrace )\\, e^{\\pi I\\tau Q_n(\\lbrace \\check{\\gamma }_i\\rbrace )}\\prod _{i=1}^n \\widehat{h}_{p_i,\\mu _i}(\\tau ).$ What are the conditions on $\\widehat{g}_n$ for the corresponding theta series to be modular?",
"Let us denote by $\\Phi ^{\\,\\widehat{g}}_{n}$ the kernel defined by $\\widehat{g}_n$ analogously to (REF ), namely $\\Phi ^{\\,\\widehat{g}}_{n}({x})=\\frac{\\sigma _{\\gamma }(\\sqrt{2\\tau _2})^{n-1}}{\\prod _{i=1}^n\\sigma _{\\gamma _i}}\\,\\widehat{g}_n(\\lbrace \\gamma _i,c_i\\rbrace ).$ Then the above analysis implies that the modularity requires $\\Phi ^{\\,\\widehat{g}}_{n}$ to satisfy Vignéras equation away from walls of marginal stability, whereas at these walls its discontinuities should follow the same pattern as before in order to cancel the discontinuities from the contour exchange in $\\Phi ^{\\scriptscriptstyle \\,\\int }_m$ .",
"Thus, the completion should smoothen out the discontinuities from the moduli independent signs ${\\rm sgn}(\\tilde{{v}}_v,{x})$ , but otherwise leave the action of Vignéras' operator unchanged.",
"Formally, this means that $\\Phi ^{\\,\\widehat{g}}_{n}$ must satisfy the following equation $V_{n-1} \\cdot \\Phi ^{\\,\\widehat{g}}_{n}=\\,{\\rm Sym}\\, \\sum _{\\ell =1}^{n-1}\\Bigl [{u}_\\ell ^2\\,\\Delta _{n,\\ell }^{\\widehat{g}} \\,\\delta ^{\\prime }({u}_\\ell \\cdot {x})+ 2{u}_\\ell \\cdot \\partial _{x}\\Delta _{n,\\ell }^{\\widehat{g}} \\,\\delta ({u}_\\ell \\cdot {x})\\Bigr ],\\qquad \\Delta _{n,\\ell }^{\\widehat{g}}=\\frac{1}{2}\\,({v}_\\ell ,{x})\\,\\Phi ^{\\,\\widehat{g}}_\\ell \\,\\Phi ^{\\,\\widehat{g}}_{n-\\ell },$ where we introduced two sets of vectors constructed from the vectors (REF ), ${v}_\\ell =\\sum _{i=1}^\\ell \\sum _{j=\\ell +1}^n{v}_{ij}\\,,\\qquad {u}_\\ell =\\sum _{i=1}^\\ell \\sum _{j=\\ell +1}^n{u}_{ij}\\,.$ Note that $({v}_\\ell ,{x})$ and $({u}_\\ell ,{x})$ correspond to the quantities $-\\Gamma _{n\\ell }$ and $-S_\\ell $ (REF ), respectively.",
"To solve the above constraints, let us consider the following iterative ansatz $\\widehat{g}_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace )=g^{(0)}_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace )-\\,{\\rm Sym}\\, \\left\\lbrace \\sum _{n_1+\\cdots +n_m= n\\atop n_k\\ge 1, \\ m<n}\\widehat{g}_m(\\lbrace \\check{\\gamma }^{\\prime }_k,c^{\\prime }_k\\rbrace )\\prod _{k=1}^m \\mathcal {E}_{n_k}(\\check{\\gamma }_{j_{k-1}+1},\\dots ,\\check{\\gamma }_{j_{k}})\\right\\rbrace ,$ where the notations for indices and primed variables are the same as in (REF ).",
"This ansatz is motivated by analogy with the iterative equation for the (partial) tree index (REF ).",
"It involves two functions to be determined below, $g^{(0)}_n$ and $\\mathcal {E}_{n}$ .",
"The former depend on the moduli through the variables $c_i$ (REF ), whereas the latter are moduli independent.",
"We set $g^{(0)}_1=\\mathcal {E}_1=1$ and also assume that $g^{(0)}_n$ have discontinuities only at walls of marginal stability, i.e.",
"at $\\sum _{i\\in \\mathcal {I}} c_i=0$ for various subsets $\\mathcal {I}$ of indices.",
"The unknown functions $g^{(0)}_n$ and $\\mathcal {E}_{n}$ together with the functions $R_n$ defining the completion, or their combinations (REF ), are fixed by the conditions (REF ) and (REF ).",
"In appendix we prove the following result: Proposition 2 Let us split $\\mathcal {E}_n=\\mathcal {E}^{(0)}_n+\\mathcal {E}^{(+)}_n$ into $\\mathcal {E}^{(+)}_n$ , which is the part exponentially decreasing for large $\\tau _2$ , and the non-decreasing remainder $\\mathcal {E}^{(0)}_n$ .",
"Then the ansatz (REF ) satisfies the recursive equation (REF ) provided the functions $g^{(0)}_n$ are subject to a similar recursive relation $\\begin{split}&\\,\\frac{1}{4}\\,{\\rm Sym}\\, \\left\\lbrace \\sum _{\\ell =1}^{n-1} \\bigl ( {\\rm sgn}(S_\\ell )-{\\rm sgn}(\\Gamma _{n\\ell })\\bigr )\\,\\kappa (\\Gamma _{n\\ell })\\,g^{(0)}_{\\ell }(\\lbrace \\check{\\gamma }_i,c^{(\\ell )}_i\\rbrace _{i=1}^\\ell )\\,g^{(0)}_{n-\\ell }(\\lbrace \\check{\\gamma }_i,c^{(\\ell )}_i\\rbrace _{i=\\ell +1}^n)\\right\\rbrace \\\\&\\,\\qquad \\qquad \\qquad =g^{(0)}_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace )-g^{(0)}_n(\\lbrace \\check{\\gamma }_i,\\beta _{ni}\\rbrace ),\\end{split}$ where $S_k$ , $\\beta _{k\\ell }$ , $\\Gamma _{k\\ell }$ and $c^{(\\ell )}_i$ were defined in (REF ), (REF ); the non-decreasing part of $\\mathcal {E}_n$ is fixed in terms of $g^{(0)}_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace )$ as $\\mathcal {E}^{(0)}_n(\\lbrace \\check{\\gamma }_i\\rbrace )=g^{(0)}_n(\\lbrace \\check{\\gamma }_i,\\beta _{ni}\\rbrace );$ its decreasing part is given by $\\mathcal {E}^{(+)}_n(\\lbrace \\check{\\gamma }_i\\rbrace )=-\\,{\\rm Sym}\\, \\left\\lbrace \\sum _{n_1+\\cdots +n_m= n\\atop n_k\\ge 1, \\ m>1}W_m(\\lbrace \\check{\\gamma }^{\\prime }_k\\rbrace )\\prod _{k=1}^m \\mathcal {E}_{n_k}(\\check{\\gamma }_{j_{k-1}+1},\\dots ,\\check{\\gamma }_{j_{k}})\\right\\rbrace .$ Furthermore, provided the functions $\\mathcal {E}_n$ depend on electric charges $q_{i,a}$ only through the DSZ products $\\gamma _{ij}$ and their kernels $\\Phi ^{\\,\\mathcal {E}}_n({x})$ defined as in (REF ) are smooth solutions of Vignéras' equation, $V_{n-1}\\cdot \\Phi ^{\\,\\mathcal {E}}_n=0,$ then the ansatz (REF ) also satisfies the modularity constraint (REF ).",
"This proposition allows in principle to fix all unknown functions.",
"Indeed, the recursive relation (REF ) determines all $g^{(0)}_n$ .",
"Then equations (REF ) and (REF ) give the two parts of $\\mathcal {E}_n=\\mathcal {E}^{(0)}_n+\\mathcal {E}^{(+)}_n$ in terms of $g^{(0)}_n$ and $W_n$ .",
"At this point the latter are still undetermined and are defined in terms of the unknown functions $R_n$ .",
"The additional constraint that $\\mathcal {E}_n$ should satisfy Vignéras' equation links together $\\mathcal {E}^{(0)}_n$ and $\\mathcal {E}^{(+)}_n$ and thus establishes a relation between $W_n$ and $g^{(0)}_n$ .",
"Lastly, inverting (REF ) generates a solution for $R_n$ .",
"We end this discussion by making an observation which will become relevant in the next subsection: by comparing (REF ) and (REF ), it is clear that $\\widehat{g}_n$ must agree with the tree index $g_{{\\rm tr},n}$ in the limit $\\tau _2\\rightarrow \\infty $ .",
"In particular, $g_{{\\rm tr},n}$ satisfies a similar ansatz as (REF ), upon replacing $\\mathcal {E}_n$ by its non-decaying part: $g_{{\\rm tr},n}(\\lbrace \\check{\\gamma }_i,c_i\\rbrace )=g^{(0)}_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace )-\\,{\\rm Sym}\\, \\left\\lbrace \\sum _{n_1+\\cdots +n_m= n\\atop n_k\\ge 1, \\ m<n}g_{{\\rm tr},m}(\\lbrace \\check{\\gamma }^{\\prime }_k,c^{\\prime }_k\\rbrace )\\prod _{k=1}^m \\mathcal {E}^{(0)}_{n_k}\\right\\rbrace .$ It follows that the function $g^{(0)}_n$ should be independent of $\\tau _2$ , or at least that any such dependence should cancel in the recursion (REF )."
],
[
"Generalized error functions and the completion",
"From the previous discussion, the first step in the construction of the modular completion is to provide an explicit expression for $g^{(0)}_n$ .",
"Once such an expression is known, all other functions can be determined algebraically.",
"The problem, however, is that the solution of the recursive relation (REF ) is not unique.",
"On the other hand, $g^{(0)}_n$ determines the non-decaying part $\\mathcal {E}^{(0)}_n$ of $\\mathcal {E}_n$ , which is strongly constrained by the fact that $\\mathcal {E}_n$ must satisfy Vignéras' equation.",
"This restriction turns out to be strong enough to remove the redundancy in the solution of (REF ), but it shows that we have to solve simultaneously two problems: satisfy the recursive relation (REF ) and ensure that it can be promoted to a solution of Vignéras' equation.",
"We will do this by first constructing a solution of Vignéras' equation with the asymptotic part possessing the properties expected from $g^{(0)}_n$ , and then proving that it does satisfy the recursive relation.",
"A hint towards a solution of these two problems can be found by examining the form of the kernel $\\Phi ^{\\scriptscriptstyle \\,\\int }_n$ (REF ).",
"The point is that both $\\Phi ^{\\scriptscriptstyle \\,\\int }_n$ and $g^{(0)}_n$ have discontinuities at walls of marginal stability which, as we know, must cancel each other.",
"Furthermore, they should recombine into a smooth solution of Vignéras' equation.",
"Thus, $\\Phi ^{\\scriptscriptstyle \\,\\int }_n$ is expected to encode at least part of the completion of $g^{(0)}_n$ .",
"In addition, as explained in appendix REF , the function $\\widetilde{\\Phi }^M_{n-1}$ , from which $\\Phi ^{\\scriptscriptstyle \\,\\int }_n$ is constructed, appears as the term with fastest decay in the expansion similar to (REF ) of the function $\\widetilde{\\Phi }^E_{n-1}$ defined in (REF ) with the arguments $\\mathcal {V}=\\lbrace {u}_e\\rbrace $ , $\\tilde{\\mathcal {V}}=\\lbrace {v}_{s(e) t(e)}\\rbrace $ , which is a smooth solution of Vignéras' equation.",
"This motivates us to consider the following functionNote that the sign factor $(-1)^{\\sum _{i<j} \\gamma _{ij} }$ is equal to the ratio of quadratic refinements appearing in (REF ).",
"$G_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace ;\\tau _2)=\\frac{(-1)^{\\sum _{i<j} \\gamma _{ij} }}{(\\sqrt{2\\tau _2})^{n-1}}\\, \\widetilde{\\Phi }^{(0)}_n({x}),$ where $\\widetilde{\\Phi }^{(0)}_n({x})$ denotes the large ${x}$ limit of the function $\\widetilde{\\Phi }_n({x})=\\frac{1}{2^{n-1} n!",
"}\\sum _{\\mathcal {T}\\in \\, {T}_n^\\ell }\\widetilde{\\Phi }^E_{n-1}(\\lbrace {u}_e\\rbrace , \\lbrace {v}_{s(e) t(e)}\\rbrace ;{x}).$ Our first goal will be to evaluate this limit explicitly.",
"Figure: An example of marked unrooted tree belonging to T 7,4 ℓ {T}_{7,4}^\\ell where near each vertex we showed in redan integer counting marks.To express the result, we need to introduce two new types of trees, beyond those already encountered.",
"First, we denote by ${T}_{n,m}^\\ell $ the set of marked unrooted labelled trees with $n$ vertices and $m$ marks assigned to vertices (see Fig.",
"REF ).",
"Let $m_\\mathfrak {v}\\in \\lbrace 0,\\dots m\\rbrace $ be the number of marks carried by the vertex $\\mathfrak {v}$ , so that $\\sum _\\mathfrak {v}m_\\mathfrak {v}=m$ .",
"Furthermore, the vertices are decorated by charges from the set $\\lbrace \\gamma _1,\\dots ,\\gamma _{n+2m}\\rbrace $ such that a vertex $\\mathfrak {v}$ with $m_\\mathfrak {v}$ marks carries $1+2m_\\mathfrak {v}$ charges $\\gamma _{\\mathfrak {v},s}$ , $s=1,\\dots ,1+2m_\\mathfrak {v}$ and we set $\\gamma _\\mathfrak {v}=\\sum _{s=1}^{1+2m_\\mathfrak {v}}\\gamma _{\\mathfrak {v},s}$ .",
"Second, we define ${T}_{n}^{(3)}$ to be the set of (unordered, full) rooted ternary trees with $n$ leaves decorated by charges $\\gamma _i$ and other vertices carrying charges defined by the inductive rule $\\gamma _v=\\gamma _{d_1(v)}+\\gamma _{d_3(v)}+\\gamma _{d_3(v)}$ where $d_r(v)$ are the three children of vertex $v$ (see Fig.",
"REF ).",
"As usual, $V_T$ denotes the set of vertices, with cardinality $|V_T|={1\\over 2}(n-1)$ (not counting the leaves).",
"Figure: An example of rooted ternary tree belonging to T 9 (3) {T}_{9}^{(3)}.In terms of these notations, we then have the following result proven, in appendix : Proposition 3 The function (REF ) is given by $G_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace ;\\tau _2)=\\scalebox {.95}{\\frac{(-1)^{n-1+\\sum _{i<j} \\gamma _{ij} }}{2^{n-1} n!",
"}}\\sum _{m=0}^{[(n-1)/2]}\\frac{(-1)^m}{(4\\pi \\tau _2)^m}\\!\\!\\!\\!\\sum _{\\mathcal {T}\\in \\, {T}_{n-2m,m}^\\ell }\\prod _{\\mathfrak {v}\\in V_\\mathcal {T}}\\mathcal {P}_{m_\\mathfrak {v}}(\\lbrace p_{\\mathfrak {v},s}\\rbrace )\\prod _{e\\in E_{\\mathcal {T}}}\\gamma _{s(e) t(e)}\\,{\\rm sgn}(S_e),$ where, for each edge $e$ joining the subtrees $\\mathcal {T}_e^s$ and $\\mathcal {T}_e^t$ , $S_e=\\sum _{i\\in V_{\\mathcal {T}_e^s}}c_i=\\sum _{i\\in V_{\\mathcal {T}_e^s}}\\sum _{j\\in V_{\\mathcal {T}_e^t}}\\,{\\rm Im}\\,\\bigl [ Z_{\\gamma _i}\\bar{Z}_{\\gamma _j}\\bigr ],$ and $\\mathcal {P}_m$ is the weight corresponding to the marks, given by $\\mathcal {P}_{m}(\\lbrace p_s\\rbrace )=\\sum _{T\\in \\, {T}_{2m+1}^{(3)}}\\frac{1}{T!",
"}\\prod _{v\\in V_T}(p_{d_1(v)}p_{d_2(v)}p_{d_3(v)}).$ Here $T!$ is the tree factorial $T!= \\prod _{v\\in V_T}n_v(T)\\, ,$ where $n_v(T)$ , as in Theorem REF , denotes the number of vertices (excluding the leaves) of the subtree of $T$ rooted at $v$ .",
"To demystify the origin of these structures, note that the sum over $m$ in (REF ) arises because of the contributions coming from the mutual action of covariant derivative operators $\\mathcal {D}({v}_{s(e) t(e)})$ (REF ) in the definition of the function $\\widetilde{\\Phi }^E_{n-1}$ (REF ).",
"Such action is non-vanishing for any pair of intersecting edges $(e_1,e_2)$ and is proportional to $(p_{\\mathfrak {v}_1}p_{\\mathfrak {v}_2}p_{\\mathfrak {v}_3})$ where $\\mathfrak {v}_1,\\mathfrak {v}_2,\\mathfrak {v}_3$ are the three vertices belonging to the edges.",
"While in appendix it is shown that such contributions cancel in the sum over trees defining $\\Phi ^{\\scriptscriptstyle \\,\\int }_n$ (REF ), this is not so for $\\widetilde{\\Phi }^{(0)}_n({x})$ : instead of the identity (REF ), one has to apply the sign identity (REF ) which produces a constant term.",
"As a result, instead of the standard sign factors assigned to $(e_1,e_2)$ , one finds the weight factor $(p_{\\mathfrak {v}_1}p_{\\mathfrak {v}_2}p_{\\mathfrak {v}_3})$ .",
"Furthermore, the sum over trees ensures that all other factors except for this weight depend on the charges $\\gamma _{\\mathfrak {v}_1},\\gamma _{\\mathfrak {v}_2},\\gamma _{\\mathfrak {v}_3}$ only through their sum.",
"One can keep track of such contributions by collapsing the corresponding pairs of edges on the tree and marking the remaining vertices with weights $m_\\mathfrak {v}$ .",
"Essentially, the only non-trivial point is to understand the form of the weight factor $\\mathcal {P}_m$ for $m>1$ .",
"In this case more than one pair of joint edges collapse.",
"The representation (REF ) in terms of rooted ternary trees is obtained by collapsing $m$ pairs of edges successively, where the coefficient $1/T!$ takes into account that such procedure leads to an overcounting of different assignments of labels.",
"Figure: Three unrooted trees constructed from the same three subtrees.Unfortunately, the function (REF ) cannot yet be taken as an ansatz for $g^{(0)}_n$ because it depends non-trivially on the modulus $\\tau _2$ , which is in tension with the observation made at the end of the previous subsection.",
"Therefore, we shall modify the function (REF ) into a function which is still a smooth solution of Vignéras' equation, but whose large ${x}$ limit is independent of $\\tau _2$ .",
"Taking cue from the structure of (REF ), we defineThe term $m=0$ in (REF ) reduces to the original function (REF ), while the terms with $m>0$ are the afore-mentioned modification.",
"$\\widetilde{\\Phi }^{\\,\\mathcal {E}}_n({x})=\\frac{1}{2^{n-1} n!",
"}\\sum _{m=0}^{[(n-1)/2]}\\sum _{\\mathcal {T}\\in \\, {T}_{n-2m,m}^\\ell }\\left[\\prod _{\\mathfrak {v}\\in V_\\mathcal {T}}\\mathcal {D}_{m_\\mathfrak {v}}(\\lbrace \\check{\\gamma }_{\\mathfrak {v},s}\\rbrace )\\right]\\widetilde{\\Phi }^E_{n-2m-1}(\\lbrace {u}_e\\rbrace , \\lbrace {v}_{s(e) t(e)}\\rbrace ;{x}),$ where $\\mathcal {D}_{m}(\\lbrace \\check{\\gamma }_s\\rbrace )=\\sum _{\\mathcal {T}^{\\prime }\\in \\, {T}_{2m+1}^\\ell } a_{\\mathcal {T}^{\\prime }}\\prod _{e\\in E_{\\mathcal {T}^{\\prime }}}\\mathcal {D}({v}_{s(e)t(v)}).$ The coefficients $a_\\mathcal {T}$ are rational numbers which depend (up to a sign determined by the orientation of edges) only on the topology of the tree.",
"We fix them by requiring that they satisfy the following system of equations $a_{\\hat{\\mathcal {T}}_1}+a_{\\hat{\\mathcal {T}}_2}-a_{\\hat{\\mathcal {T}}_3}=a_{\\mathcal {T}_1}a_{\\mathcal {T}_2}a_{\\mathcal {T}_3},$ where $\\mathcal {T}_r$ ($r=1,2,3$ ) are arbitrary unrooted trees with marked vertex $\\mathfrak {v}_r$ and $\\hat{\\mathcal {T}}_r$ are the three trees constructed from $\\mathcal {T}_r$ as shown in Fig.",
"REF .For these equations, it is important to take into account the orientation of the edges: a change of orientation of an edge flips the sign of $a_\\mathcal {T}$ .",
"The equations (REF ) are written assuming the orientation shown in Fig.",
"REF , namely $e_1=(\\mathfrak {v}_2,\\mathfrak {v}_3)$ , $e_2=(\\mathfrak {v}_1,\\mathfrak {v}_3)$ , $e_3=(\\mathfrak {v}_1,\\mathfrak {v}_2)$ .",
"This system of equations is imposed in order to ensure certain properties of the operators (REF ) which are crucial for the cancellation of $\\tau _2$ -dependent terms in the asymptotics of $\\widetilde{\\Phi }^{\\,\\mathcal {E}}_n$ (see (REF )).",
"It is readily seen that the system (REF ) fixes all coefficients $a_\\mathcal {T}$ recursively starting from $a_\\bullet =1$ , $a_{\\bullet \\!\\mbox{-}\\!\\bullet }=0$ and going to trees with $n\\ge 3$ vertices.",
"(Starting from $n=7$ , the system (REF ) is overdetermined, but it can be checked that it does have a unique solution, with $a_\\mathcal {T}=0$ for trees with even number of vertices.)",
"For a generic tree, in appendix we prove the following Proposition 4 For a tree $\\mathcal {T}$ with $n$ vertices the coefficient $a_\\mathcal {T}$ is given recursively by $a_\\mathcal {T}=\\frac{1}{n}\\sum _{\\mathfrak {v}\\in V_\\mathcal {T}} \\epsilon _\\mathfrak {v}\\prod _{s=1}^{n_\\mathfrak {v}} a_{\\mathcal {T}_s(\\mathfrak {v})},$ where $n_\\mathfrak {v}$ is the valency of the vertex $\\mathfrak {v}$ , $\\mathcal {T}_s(\\mathfrak {v})$ are the trees obtained from $\\mathcal {T}$ by removing this vertex, and $\\epsilon _\\mathfrak {v}$ is the sign determined by the choice of orientation of edges, $\\epsilon _\\mathfrak {v}=(-1)^{n_\\mathfrak {v}^+}$ with $n_\\mathfrak {v}^+$ being the number of incoming edges at the vertex.",
"(See Fig.",
"REF for an example.)",
"Figure: An example of calculation of the coefficient a 𝒯 a_\\mathcal {T} for a tree with 7 vertices.We indicated in red the only vertices which produce non-vanishing contributions to the sum over vertices.For other vertices at least one of the trees 𝒯 s (𝔳)\\mathcal {T}_s(\\mathfrak {v}) has even number of vertices and hence vanishing coefficient.Let us now return to the function (REF ), which is now fully specified given the prescription (REF ) for the coefficients $a_\\mathcal {T}$ .",
"Similarly to (REF ) (which coincides with the $m=0$ contribution in (REF )), it is a solution of Vignéras' equation for $\\lambda =n-1$ .",
"We claim that in the large ${x}$ asymptotics of this function, all $\\tau _2$ -dependent contributions, coming from the mutual action of derivative operators $\\mathcal {D}$ , cancel.",
"More precisely, the asymptotics is given by the following Proposition 5 $\\lim _{{x}\\rightarrow \\infty }\\widetilde{\\Phi }^{\\,\\mathcal {E}}_n({x})=\\frac{1}{2^{n-1} n!",
"}\\sum _{m=0}^{[(n-1)/2]}\\sum _{\\mathcal {T}\\in \\, {T}_{n-2m,m}^\\ell }\\prod _{\\mathfrak {v}\\in V_\\mathcal {T}}\\mathcal {V}_{m_\\mathfrak {v}}(\\lbrace \\check{\\gamma }_{\\mathfrak {v},s}\\rbrace )\\prod _{e\\in E_{\\mathcal {T}}}({v}_{s(e) t(e)},{x})\\,{\\rm sgn}({u}_e,{x}),$ where $\\mathcal {V}_{m}(\\lbrace \\check{\\gamma }_{s}\\rbrace )=\\sum _{\\mathcal {T}\\in \\, {T}_{2m+1}^\\ell } a_\\mathcal {T}\\prod _{e\\in E_\\mathcal {T}}({v}_{s(e)t(v)},{x}).$ Importantly the function (REF ) is locally a homogeneous polynomial of degree $n-1$ .",
"Therefore, when written in terms of charges, the $\\tau _2$ -dependence factorizes and is cancelled after the same rescaling as in (REF ).",
"This naturally leads us to the following ansatz for the function $g^{(0)}_n$ , which we prove in appendix : Proposition 6 The function $g^{(0)}_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace )=\\frac{(-1)^{n-1+\\sum _{i<j} \\gamma _{ij} }}{2^{n-1} n!",
"}\\sum _{m=0}^{[(n-1)/2]}\\sum _{\\mathcal {T}\\in \\, {T}_{n-2m,m}^\\ell }\\prod _{\\mathfrak {v}\\in V_\\mathcal {T}}\\tilde{\\mathcal {V}}_{m_\\mathfrak {v}}(\\lbrace \\check{\\gamma }_{\\mathfrak {v},s}\\rbrace )\\prod _{e\\in E_{\\mathcal {T}}}\\gamma _{s(e) t(e)}\\,{\\rm sgn}(S_e),$ where $\\tilde{\\mathcal {V}}_{m}(\\lbrace \\check{\\gamma }_{s}\\rbrace )=\\sum _{\\mathcal {T}^{\\prime }\\in \\, {T}_{2m+1}^\\ell } a_{\\mathcal {T}^{\\prime }}\\prod _{e\\in E_{\\mathcal {T}^{\\prime }}}\\gamma _{s(e)t(e)},$ satisfies the recursive equation (REF ).",
"Note that the same recursive equation (REF ) is also obeyed by the contribution of unmarked trees (i.e.",
"$m=0$ ) to $g^{(0)}_n$ , which we denote by $g^{\\star }_n$ (see (REF )).",
"However, the latter cannot be obtained as the large ${x}$ limit of a solution of Vignéras' equation, which is why we have to consider the more complicated function (REF ).",
"Given the result for $g^{(0)}_n$ and the relation (REF ), one obtains an explicit expression for the non-decreasing part of $\\mathcal {E}_n$ , $\\mathcal {E}^{(0)}_n(\\lbrace \\check{\\gamma }_i\\rbrace )=\\frac{(-1)^{\\sum _{i<j} \\gamma _{ij} }}{2^{n-1} n!",
"}\\sum _{m=0}^{[(n-1)/2]}\\sum _{\\mathcal {T}\\in \\, {T}_{n-2m,m}^\\ell }\\prod _{\\mathfrak {v}\\in V_\\mathcal {T}}\\tilde{\\mathcal {V}}_{m_\\mathfrak {v}}(\\lbrace \\check{\\gamma }_{\\mathfrak {v},s}\\rbrace )\\prod _{e\\in E_{\\mathcal {T}}}\\gamma _{s(e) t(e)}\\,{\\rm sgn}(\\Gamma _e),$ where $\\Gamma _e=\\sum _{i\\in V_{\\mathcal {T}_e^s}}\\sum _{j\\in V_{\\mathcal {T}_e^t}}\\gamma _{ij}.$ From Proposition REF we know that the kernel $\\Phi ^{\\,\\mathcal {E}}_n$ corresponding to the full function $\\mathcal {E}_n$ must satisfy Vignéras' equation and have the asymptotics captured by (REF ).",
"But we already know a solution of Vignéras' equation with a very similar asymptotics, namely the function (REF ), whose asymptotics differs only in the fact that the vectors ${v}_e$ are replaced by ${u}_e$ .",
"Since this replacement does not affect the proof of Vignéras' equation, we immediately arrive at the following result: $\\mathcal {E}_n(\\lbrace \\check{\\gamma }_i\\rbrace ;\\tau _2)=\\frac{(-1)^{\\sum _{i<j} \\gamma _{ij} }}{(\\sqrt{2\\tau _2})^{n-1}}\\, \\Phi ^{\\,\\mathcal {E}}_n({x}),$ where $\\Phi ^{\\,\\mathcal {E}}_n({x})=\\frac{1}{2^{n-1} n!",
"}\\sum _{m=0}^{[(n-1)/2]}\\sum _{\\mathcal {T}\\in \\, {T}_{n-2m,m}^\\ell }\\left[\\prod _{\\mathfrak {v}\\in V_\\mathcal {T}}\\mathcal {D}_{m_\\mathfrak {v}}(\\lbrace \\check{\\gamma }_{\\mathfrak {v},s}\\rbrace )\\right]\\widetilde{\\Phi }^E_{n-2m-1}(\\lbrace {v}_e\\rbrace , \\lbrace {v}_{s(e) t(e)}\\rbrace ;{x}).$ In words, the function $\\mathcal {E}_n$ is a sum over marked unrooted labelled trees of solutions of Vignéras equation with $\\lambda =n-1$ , obtained from the standard generalized error functions $\\Phi ^E_{n-2m-1}$ by acting with $n-1$ raising operators."
],
[
"The structure of the completion",
"After substituting into the iterative ansatz (REF ), the two results (REF ) and (REF ) completely specify the coefficients of the expansion (REF ) of the generating function of DT invariants in terms of $\\widehat{h}_{p,\\mu }$ .",
"The result of the iteration can in fact be written explicitly as a sum over Schröder trees.",
"Adopting the same notations as in Proposition REF , one has Proposition 7 The function $\\widehat{g}_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace )$ is given by the sum over Schröder trees with $n$ leaves, $\\widehat{g}_n= \\,{\\rm Sym}\\, \\left\\lbrace \\sum _{T\\in {T}_n^{\\rm S}}(-1)^{n_T-1} \\left(g^{(0)}_{v_0}-\\mathcal {E}_{v_0}\\right)\\prod _{v\\in V_T\\setminus {\\lbrace v_0\\rbrace }}\\mathcal {E}_{v}\\right\\rbrace ,$ where $v_0$ is the root of the tree.",
"We conclude that the properties of $g^{(0)}_n$ and $\\mathcal {E}_n$ ensure that the theta series appearing in the corresponding decomposition of $\\mathcal {G}$ is modular so that the modularity of $\\mathcal {G}$ requires the function $\\widehat{h}_{p,\\mu }$ to be a vector valued (non-holomorphic) modular form of weight $(-{1\\over 2}b_2-1,0)$ .",
"The functions $R_n$ entering the non-holomorphic completion $\\widehat{h}_{p,\\mu }$ , are then given by the following proposition, whose proof can again be found in appendix : Proposition 8 Inverting the relations (REF ) and (REF ), one obtainsIn the proof of this proposition we obtain a similar formula (REF ) for the coefficients $W_n$ appearing in the inverse relation (REF ).",
"$R_n= \\,{\\rm Sym}\\, \\left\\lbrace \\sum _{T\\in {T}_n^{\\rm S}}(-1)^{n_T-1} \\mathcal {E}^{(+)}_{v_0}\\prod _{v\\in V_T\\setminus {\\lbrace v_0\\rbrace }}\\mathcal {E}^{(0)}_{v}\\right\\rbrace .$ It is important to check that the resulting functions $R_n$ are invariant under the spectral flow of the total charge.",
"This is in fact a simple consequence of the fact that the functions $\\mathcal {E}_n$ entering their definition depend on the electric charges only through the DSZ products $\\gamma _{ij}$ .",
"As follows from (REF ), all such products are invariant under the spectral flow of the total charge, hence $R_n$ are invariant as well.",
"Since $\\mathcal {E}^{(0)}_n$ are $\\tau _2$ -independent, all non-holomorphic dependence of $\\widehat{h}_{p,\\mu }$ comes from the factor $\\mathcal {E}^{(+)}_{v_0}$ in the expression (REF ) for $R_n$ .",
"Expressing the holomorphic anomaly in terms of the modular functions $\\widehat{h}_{p_i,\\mu _i}$ , one obtains the following Proposition 9 The holomorphic anomaly of the completion is given by $\\partial _{\\bar{\\tau }}\\widehat{h}_{p,\\mu }(\\tau )= \\sum _{n=2}^\\infty \\sum _{\\sum _{i=1}^n \\check{\\gamma }_i=\\check{\\gamma }}\\mathcal {J}_n(\\lbrace \\check{\\gamma }_i\\rbrace ,\\tau _2)\\, e^{\\pi I\\tau Q_n(\\lbrace \\check{\\gamma }_i\\rbrace )}\\prod _{i=1}^n \\widehat{h}_{p_i,\\mu _i}(\\tau ),$ where $\\mathcal {J}_n= \\frac{I}{2}\\,{\\rm Sym}\\, \\left\\lbrace \\sum _{T\\in {T}_n^{\\rm S}}(-1)^{n_T-1} \\partial _{\\tau _2}\\mathcal {E}_{v_0}\\prod _{v\\in V_T\\setminus {\\lbrace v_0\\rbrace }}\\mathcal {E}_{v}\\right\\rbrace .$ Note that this result is consistent with the fact that $\\tau _2^2\\partial _{\\bar{\\tau }}\\widehat{h}_{p,\\mu }$ is a modular function of weight $(-{1\\over 2}b_2-3,0)$ .",
"Indeed, the sum over D2-brane charges forms a theta series for a lattice of dimension $(n-1)b_2$ (see footnote REF ) and quadratic form $-Q_n$ of signature $(n-1,(n-1)(b_2-1))$ .",
"Furthermore, since $2\\tau _2\\partial _{\\tau _2}\\mathcal {E}_n=\\frac{(-1)^{\\sum _{i<j} \\gamma _{ij} }}{(\\sqrt{2\\tau _2})^{n-1}}\\left({x}\\cdot \\partial _{x}-(n-1)\\right) \\Phi ^{\\,\\mathcal {E}}_n({x})$ and $V_{\\lambda -2}\\left({x}\\cdot \\partial _{x}-\\lambda \\right)= \\left({x}\\cdot \\partial _{x}-(\\lambda -2)\\right)V_\\lambda $ , the function $\\tau _2\\mathcal {J}_n$ is a solution of Vignéras' equation with $\\lambda =n-3$ .",
"Thus, after multiplying Eq.",
"(REF ) by $\\tau _2^2$ , the theta series generated by the sum over D2-brane charges is modular of weight $(\\frac{n-1}{2}\\, b_2+n-3,0)$ .",
"Combining it with $n$ factors of $\\widehat{h}_{p_i,\\mu _i}$ , one recovers the correct weight for $\\tau _2^2\\partial _{\\bar{\\tau }}\\widehat{h}_{p,\\mu }$ .",
"In appendix we present explicit expressions for various elements of our construction up to order $n=4$ .",
"Based on these results, we conjecture a general formula for the kernel $\\widehat{\\Phi }^{{\\rm tot}}_n$ of the theta series appearing in the expansion of $\\mathcal {G}$ in terms of $\\widehat{h}_{p,\\mu }$ : Conjecture 1 The instanton generating function has the theta series decomposition $\\mathcal {G}=\\sum _{n=1}^\\infty \\frac{2^{-\\frac{n}{2}}}{\\pi \\sqrt{2\\tau _2}}\\left[\\prod _{i=1}^{n}\\sum _{p_i,\\mu _i}\\sigma _{p_i}\\widehat{h}_{p_i,\\mu _i}\\right]e^{-S^{\\rm cl}_p}\\vartheta _{{p},{\\mu }}\\bigl (\\widehat{\\Phi }^{{\\rm tot}}_{n},n-2\\bigr ),$ where the kernels of the theta series are given by $\\widehat{\\Phi }^{{\\rm tot}}_n= \\Phi ^{\\scriptscriptstyle \\,\\int }_1\\,{\\rm Sym}\\, \\left\\lbrace \\sum _{T\\in {T}_n^{\\rm S}}(-1)^{n_T-1}\\left(\\widetilde{\\Phi }^{\\,\\mathcal {E}}_{v_0}-\\Phi ^{\\,\\mathcal {E}}_{v_0}\\right)\\prod _{v\\in V_T\\setminus {\\lbrace v_0\\rbrace }}\\Phi ^{\\,\\mathcal {E}}_{v}\\right\\rbrace .$ Note that this result automatically implies that the theta series $\\vartheta _{{p},{\\mu }}\\bigl (\\widehat{\\Phi }^{{\\rm tot}}_{n},n-2\\bigr )$ is modular, since its kernel is a solution of Vignéras' equation.",
"We end the discussion of these results by observing that the formula (REF ) allows to obtain a new representation of the tree index $g_{{\\rm tr},n}$ .",
"Indeed, as observed at the end of Subsection REF , $g_{{\\rm tr},n}$ agrees with $\\widehat{g}_n$ in the limit $\\tau _2\\rightarrow \\infty $ , which amounts to replacing $\\mathcal {E}_{v}$ by its non-decaying part $\\mathcal {E}^{(0)}_{v}$ in (REF ).",
"In fact, as shown in appendix , all contributions due to trees with non-zero number of marks in (REF ) and (REF ) cancel in the resulting expression, leaving only the contributions coming from the terms with $m=0$ .",
"Thus, one arrives at the following representation: Proposition 10 The tree index $g_{{\\rm tr},n}$ defined in (REF ) can be expressed as $\\begin{split}g_{{\\rm tr},n}=&\\, \\,{\\rm Sym}\\, \\left\\lbrace \\sum _{T\\in {T}_n^{\\rm S}}(-1)^{n_T-1} \\left(g^{\\star }_{v_0}-\\mathcal {E}^{\\star }_{v_0}\\right)\\prod _{v\\in V_T\\setminus {\\lbrace v_0\\rbrace }}\\mathcal {E}^{\\star }_{v}\\right\\rbrace ,\\end{split}$ where $g^{\\star }_n$ is defined in (REF ) and $\\mathcal {E}^{\\star }_n=g^{\\star }_n(\\lbrace \\check{\\gamma }_i,\\beta _{ni}\\rbrace )$ .",
"Note that this representation is more explicit than the one given in section REF since it does not require taking the limit $y\\rightarrow 1$ .",
"In fact, the mechanism by which contributions of marked trees cancel in the sum over Schröder trees, explained in the proof of Proposition REF , is very general and applies just as well to all the above equations (REF )-(REF ).",
"As a result, all of them remain valid if we replace $\\widetilde{\\Phi }^{\\,\\mathcal {E}}_n$ and $\\Phi ^{\\,\\mathcal {E}}_n$ , respectively, by $\\widetilde{\\Phi }_n$ (REF ) and by its cousin with the vectors $\\lbrace {u}_e\\rbrace $ replaced by $\\lbrace {v}_e\\rbrace $ , $\\Phi _n({x})=\\frac{1}{2^{n-1} n!",
"}\\sum _{\\mathcal {T}\\in \\, {T}_n^\\ell }\\widetilde{\\Phi }^E_{n-1}(\\lbrace {v}_e\\rbrace , \\lbrace {v}_{s(e) t(e)}\\rbrace ;{x}).$ Correspondingly, $g^{(0)}_n$ and $\\mathcal {E}^{(0)}_n$ should then be replaced by their asymptotics at large $\\tau _2$ , given by $G_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace )$ and $G_n(\\lbrace \\check{\\gamma }_i,\\beta _{ni}\\rbrace )$ .",
"The proof that such replacement is possible is completely analogous to the proof of Proposition REF , and we refrain from presenting it.",
"This shows that from the very beginning we could take the function $G_n$ as the ansatz for $g^{(0)}_n$ since its $\\tau _2$ -dependence cancels in the formula for the tree index.",
"We could then avoid most of complications of section REF .",
"However, we cannot avoid the introduction of marked trees since they appear in the expression for $G_n$ (REF ) anyway.",
"Moreover, we prefer to stick to the definition (REF ) because of two reasons.",
"Firstly, due to its $\\tau _2$ -independence, $g^{(0)}_n$ might itself be interpretable as an index.",
"Secondly, as we discuss in the next subsection, it is related to other potentially interesting representations, which might be at the basis of such interpretation."
],
[
"Alternative representations",
"The construction presented above provides an explicit expression for the completion $\\widehat{h}_{p,\\mu }$ and all other relevant quantities.",
"Roughly, it can be split into three levels: sum over Schröder trees (REF ); sum over marked unrooted labelled trees (REF ); sum over unrooted labelled trees defining the weights associated to marks (REF ).",
"The complication due to the appearance of marks and hence the last level arises due to the non-orthogonality of the vectors ${v}_{s(e) t(e)}$ appearing as arguments of the generalized error functions uplifted to solutions of Vignéras' equation for $\\lambda =n-1$ .",
"However, the two sums over unrooted trees are organized is such a way that all additional contributions due to this non-orthogonality cancel.",
"This suggests that there should exist a simpler representation where the vectors appearing in the second argument of functions $\\widetilde{\\Phi }^E_{n-1}(\\mathcal {V}, \\tilde{\\mathcal {V}};{x})$ are mutually orthogonal from the very beginning.",
"Below we present some results showing that such a representation does exist at least for low values of $n$ .",
"We then move on to present yet another representation of the functions $g^{(0)}_n$ which is significantly simpler than (REF ), although its equivalence with the latter remains conjectural."
],
[
"Simplified representation via flow trees",
"We have shown, analytically for $n=2,3,4$ (see appendix ) and numerically for $n=5$ , that the function (REF ) can be rewritten as $g^{(0)}_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace )= \\,{\\rm Sym}\\, \\left\\lbrace \\frac{d_n}{2^{n-1}}\\sum _{T\\in {P}_n}\\kappa (T) \\prod _{k=1}^{n-1}\\mbox{sgn}(S_k)\\right\\rbrace \\qquad (n\\le 5),$ where $\\kappa (T)$ is the factor defined in (REF ), whereas $d_n$ and ${P}_n\\subset {T}_n^{\\rm af}$ are numerical coefficients and subsets of flow trees with $n$ leaves, respectively, which can be chosen as follows: $\\begin{split}d_1=&1,\\qquad d_2={1\\over 2}\\, ,\\qquad d_3=\\frac{1}{6}\\, ,\\qquad d_4=\\frac{1}{12}\\, ,\\qquad d_5=\\frac{1}{30}\\, ,\\\\\\quad {P}_1=&\\lbrace (1)\\rbrace ,\\qquad {P}_2=\\lbrace (12)\\rbrace ,\\qquad {P}_3=\\lbrace ((12)3),(1(23))\\rbrace ,\\\\{P}_4=&\\lbrace ((1(23))4),((12)(34)),(1((23)4))\\rbrace ,\\\\{P}_5=& \\lbrace ((((31)4)2)5),(((12)(34))5),((1((23)4))5),(((12)3)(45)),\\\\&\\qquad \\ ((12)(3(45))),(1((2(34))5)),(1((23)(45))),(1(4(2(53))))\\rbrace ,\\end{split}$ where we labelled ordered flow trees using 2-bracketings as explained in footnote REF .",
"Unfortunately, the simple ansatz (REF ) fails beyond the fifth order.",
"For instance, numerical experiments indicate that for $n=6$ , one should include an additional term given by a product of $n-3=3$ sign functions.",
"This suggests the following more general ansatz: Conjecture 2 The function (REF ) can be rewritten as $g^{(0)}_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace )= \\,{\\rm Sym}\\, \\left\\lbrace \\sum _{n_1+\\cdots +n_m= n\\atop n_k\\ge 1,\\ n_k{\\rm -odd}}P_{n,m}\\prod _{k=1}^{m-1}\\mbox{sgn}(S_{j_k})\\right\\rbrace ,$ where as in (REF ) $j_k=n_1+\\cdots + n_k$ and $P_{n,m}$ are polynomials in $\\kappa (\\gamma _{ij})$ homogeneous of degree $n-1$ which can be represented as sums over subsets ${P}_{n,m}$ of flow trees with $n$ leaves $P_{n,m}=\\frac{1}{2^{n-1}}\\sum _{T\\in {P}_{n,m}}d_T\\,\\kappa (T)$ with some numerical coefficients $d_T$ .",
"In fact, such a representation (if it exists) is highly ambiguous since there are many linear relations between polynomials $\\kappa (T)$ induced by the identity $\\gamma _{12}\\gamma _{1+2,3}+\\gamma _{23}\\gamma _{2+3,1}+\\gamma _{31}\\gamma _{1+3,2}=0,$ and even more relations after multiplication by sign functions and symmetrization.",
"Eq.",
"(REF ) suggests that there should exist a choice of subsets ${P}_{n,m}$ which leads to simple values of the coefficients $d_T$ .",
"However, we do not know of a procedure which would allow to determine it systematically beyond $n=5$ .",
"The main advantage of the representation (REF ) is that the factors $\\kappa (T)$ multiplying the products of sign functions are proportional to $\\prod _{v\\in V_T}(\\tilde{{v}}_v,{x})$ where the vectors $\\tilde{{v}}_v$ defined in (REF ) are all mutually orthogonal.",
"This property allows to immediately promote the kernel corresponding to the function (REF ), or to its value at the attractor point $\\mathcal {E}^{(0)}_n(\\lbrace \\check{\\gamma }_i\\rbrace )=g^{(0)}_n(\\lbrace \\check{\\gamma }_i,\\beta _{ni}\\rbrace )=\\,{\\rm Sym}\\, \\left\\lbrace \\sum _{n_1+\\cdots +n_m= n \\atop n_k\\ge 1,\\ n_k{\\rm -odd}}P_{n,m}\\prod _{k=1}^{m-1}\\mbox{sgn}(\\Gamma _{nj_k})\\right\\rbrace ,$ to a solution of Vignéras' equation, without going through the complicated construction of subsection REF .",
"Indeed, both such kernels are linear combinations of terms, labelled by flow trees, of the type (REF ) where the vectors $\\tilde{{v}}_i$ coincide with the vectors $\\tilde{{v}}_v$ for a given flow tree.",
"Due to their mutual orthogonality, the solutions of Vignéras' equation with $\\lambda =n-1$ with such asymptotics are given by $\\Phi ^{\\,\\mathcal {E}}_n({x})=\\frac{1}{2^{n-1}} \\,{\\rm Sym}\\, \\left\\lbrace \\sum _{n_1+\\cdots +n_m= n \\atop n_k\\ge 1,\\ n_k{\\rm -odd}}\\sum _{T\\in {P}_{n,m}}d_T\\, \\widetilde{\\Phi }_{m-1,n-1}^E(\\lbrace {v}_{j_k}\\rbrace , \\lbrace \\tilde{{v}}_v\\rbrace ;{x})\\right\\rbrace ,$ and similarly for $\\widetilde{\\Phi }^{\\,\\mathcal {E}}_n$ with exchange of ${v}_{j_k}$ by ${u}_{j_k}$ .",
"The vectors ${v}_\\ell $ , ${u}_\\ell $ defined in (REF ) can be seen as vectors ${v}_e$ , ${u}_e$ for the unrooted tree of trivial topology $\\bullet \\!\\mbox{---}\\!\\bullet \\!\\mbox{--}\\cdots \\mbox{--}\\!\\bullet \\!\\mbox{---}\\!\\bullet \\,$ .",
"Thus, instead of a sum over marked unrooted trees with factors themselves given by another sum over unrooted trees as in (REF ), we arrive at a representation involving only a sum over a suitable subset of flow trees and a single unrooted tree.Of course, the function (REF ) must be equal to (REF ), which is guaranteed by the fact that they are solutions of Vignéras' equation with the same asymptotics.",
"It becomes particularly simple in the case $n\\le 5$ where, as follows from (REF ), one can drop the sum over partitions, take $m=n$ and equate all coefficients $d_T$ to $d_n$ ."
],
[
"Refinement",
"Finally, there is yet an alternative way of obtaining the functions $g^{(0)}_n$ , which also sheds light on the origin of the representation (REF ).",
"This representation is inspired by the solution for the tree index presented in section REF .",
"As in that case, the idea is to introduce a refinement parameter $y$ , perform manipulations keeping $y\\ne 1$ and take the limit $y\\rightarrow 1$ in the end.",
"Let us therefore introduce a `refined' analogue of $g^{(0)}_n$ , which we call $g^{\\rm (ref)}_n$ and which satisfies the refined version of the recursive relation (REF ) where the $\\kappa $ -factor is replaced by its $y$ -dependent version (REF ).",
"This refined equation can be solved by the following ansatz (cf.",
"(REF )) $g^{\\rm (ref)}_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace ,y)= \\frac{(-1)^{n-1+\\sum _{i<j} \\gamma _{ij} }}{(y-y^{-1})^{n-1}}\\,\\,{\\rm Sym}\\, \\Bigl \\lbrace F^{\\rm (ref)}_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace )\\,y^{\\sum _{i<j} \\gamma _{ij}}\\Bigr \\rbrace .$ Indeed, it is easy to see that $g^{\\rm (ref)}_n$ satisfies (REF ) with the $y$ -dependent $\\kappa $ -factor provided $F^{\\rm (ref)}_n$ satisfies the recursive relation $\\begin{split}&{1\\over 2}\\sum _{\\ell =1}^{n-1}\\bigl (\\mbox{sgn}(S_\\ell )-\\mbox{sgn}(\\Gamma _{n\\ell })\\bigr )\\,F^{\\rm (ref)}_\\ell (\\lbrace \\check{\\gamma }_i,c^{(\\ell )}_i\\rbrace _{i=1}^\\ell )\\,F^{\\rm (ref)}_{n-\\ell }(\\lbrace \\check{\\gamma }_i,c^{(\\ell )}_i\\rbrace _{i=\\ell +1}^n)\\\\&\\qquad \\qquad =F^{\\rm (ref)}_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace )-F^{\\rm (ref)}_n(\\lbrace \\check{\\gamma }_i,\\beta _{ni}\\rbrace ).\\end{split}$ This equation is simpler than (REF ) in that it is $y$ -independent and does not involve any $\\kappa $ -factors.",
"Furthermore, we already know one solution — the functions $F^{(0)}_n$ (REF ) which can be shown to satisfy (REF ) with help of the sign identity (REF ).",
"However, this solution is not yet suitable because, starting from $n=3$ , its substitution into the r.h.s.",
"of (REF ) does not produce a Laurent polynomial in $y$ , but rather a rational function with a pole at $y\\rightarrow 1$ .",
"The solution can be promoted to a Laurent polynomial by noting that the recursive equation (REF ) determines $F^{\\rm (ref)}_n$ in terms of $F^{\\rm (ref)}_k$ , $k<n$ only up to an additive constant $b_n$ .",
"Thus, at each order one can adjust this constant so that to ensure that $g^{\\rm (ref)}_n$ given by (REF ) is regular at $y=1$ .",
"Then we arrive at the following solution $F^{\\rm (ref)}_n(\\lbrace c_i\\rbrace )=\\frac{1}{2^{n-1}}\\sum _{n_1+\\cdots +n_m= n\\atop n_k\\ge 1}\\prod _{k=1}^m b_{n_k}\\prod _{k=1}^{m-1}\\mbox{sgn}(S_{j_k}),$ where as usual $j_k=n_1+\\cdots + n_k$ .",
"Remarkably, this solution depends only on the stability parameters $c_i$ , despite the fact that the recursion (REF ) also involves the DSZ products $\\gamma _{ij}$ .",
"It satisfies this recursion for arbitrary coefficients $b_n$ , as can be easily checked by extracting contributions with the same sets of $\\lbrace n_k\\rbrace $ and applying the sign identity (REF ).",
"Moreover, for arbitrary values of these coefficients, the numerator of (REF ) evaluated at $y=1$ turns outAfter this work was first released, Don Zagier communicated to us a proof of this assertion, and of the conjecture (REF ) below, based on the more elementary observation that $\\,{\\rm Sym}\\, F^{(0)}_n(\\lbrace c_i\\rbrace )=2^{1-n}/n $ for $n$ even, or 0 for $n$ odd, irrespective of the value of the $c_i$ 's, where $F^{(0)}_n(\\lbrace c_i\\rbrace )$ is defined in (REF ).",
"to be a constant, independent of the $c_i$ 's.",
"The coefficients $b_n$ are then fixed uniquely by requiring that this constant vanishes, namely $\\,{\\rm Sym}\\, F^{\\rm (ref)}_n(\\lbrace c_i\\rbrace )=0\\, .$ Choosing one particular configuration of the moduli, say $c_i>0$ for $i=1,\\dots ,n-1$ and $c_n=-\\sum _{i=1}^{n-1} c_i<0$ , one finds numerically that all coefficients $b_n$ with $n$ even vanish,This follows from the fact that under the permutation $c_i\\mapsto c_{n-i+1}$ one has $S_i\\mapsto -S_{n-i}$ and therefore $F^{\\rm (ref)}_n$ flips sign.",
"The deeper reason for this is that the terms which are products of even and odd number of signs cannot mix.",
"In contrast, sign identities such as (REF ) can decrease the number of signs in a product by even number.",
"Thus, a constant can appear in $F^{\\rm (ref)}_n$ only for $n$ odd.",
"The same fact ensures the vanishing of the coefficients $a_\\mathcal {T}$ in (REF ) for trees with even number of vertices.",
"while those with $n$ odd are given by $b_1=1,\\quad b_3=-\\frac{1}{3},\\quad b_5=\\frac{2}{15},\\quad b_7=-\\frac{17}{315},\\quad b_9=\\frac{62}{2835},\\quad b_{11}=-\\frac{1382}{155925},\\quad \\dots $ We observe that these coefficients coincide with the first coefficients in the Taylor series of $\\tanh x = x - \\frac{1}{3} x^3+ \\frac{2}{15} x^5 +\\dots $ .",
"We therefore conjecture that this identification continues to hold in general, so that $b_n$ is expressed in terms of the Bernoulli number $B_{n+1}$ through $b_{n-1}=\\frac{2^{n} (2^{n} - 1)}{n!",
"}\\, B_{n} .$ Using the iterative equation (REF ), the constraint (REF ) and proceeding by induction, it is easy to see that $g^{\\rm (ref)}_n$ defined by (REF ) and its first $n-2$ derivatives with respect to $y$ vanish at $y=1$ .",
"This ensures that $g^{\\rm (ref)}_n$ is smooth and its limit $y\\rightarrow 1$ is well-defined.",
"Conjecture 3 The function $g^{\\rm (ref)}_n$ , defined by (REF ), (REF ) and (REF ), reproduces the function $g^{(0)}_n$ (REF ) in the unrefined limit, $\\lim _{y\\rightarrow 1}g^{\\rm (ref)}_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace ,y)=g^{(0)}_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace ).$ We have checked this conjecture numerically up to $n=6$ evaluating the limit $y\\rightarrow 1$ using l'Hôpital's rule.",
"Unfortunately, the new representation of $g^{(0)}_n$ obtained in this way is not helpful for the purposes of this work because it does not lead to any new representation for the completion.",
"Nevertheless, the comparison of (REF ) and (REF ) shows that the sum over partitions and the simple form of products of sign functions characterizing the representation discussed in the previous subsection find their origin in the formula for $F^{\\rm (ref)}_n$ .",
"Furthermore, this construction strongly suggests that the refinement may be compatible with S-duality, and that a suitably defined generating function of refined DT invariants may also possess interesting modular properties, which can even be simpler than those of the usual DT invariants."
],
[
"Discussion",
"In this paper we studied the modular properties of the generating function $h_{p,\\mu }(\\tau )$ of MSW invariants encoding BPS degeneracies of D4-D2-D0 black holes in Type IIA string theory on a Calabi-Yau threefold, with fixed D4-charge $p^a$ , D2-brane charge (up to spectral flow) $\\mu _a$ and invariant D0-brane charge $\\hat{q}_0$ (defined in (REF )) conjugate to the modular parameter $\\tau $ .",
"These properties follow from demanding that the vector multiplet moduli space in $D=3$ (or the hypermultiplet moduli space in the dual type IIB picture) admits an isometric action of $SL(2,{Z})$ .",
"Our main result is an explicit formula for the non-holomorphic modular completion $\\widehat{h}_{p,\\mu }$ (REF ), where the functions $R_n(\\lbrace \\gamma _i\\rbrace ,\\tau _2)$ are given by Eq.",
"(REF ), with $\\mathcal {E}_v=\\mathcal {E}_v^{(0)}+\\mathcal {E}_v^{(+)}$ specified by (REF ) and (REF ).",
"This result applies for D4-branes wrapping a general effective divisor which may be the sum of an arbitrary number $n$ of irreducible divisors.",
"The existence of such completion was not guaranteed.",
"We arrived at this result by cancelling the modular anomaly of an indefinite theta series of signature $(n, n b_2(\\mathfrak {Y})-n)$ with a complicated kernel (REF ).",
"In particular, Vignéras' equation encoding this anomaly involves Kähler moduli in a non-trivial way through the walls of marginal stability, whereas the functions $R_n$ in our ansatz (REF ) can only depend (non-holomorphically) on the modular parameter $\\tau $ and on the charges $\\gamma _i$ .",
"Note also that the transformation (REF ) of the contact potential, which was the starting point of our analysis, is only a necessary condition for the existence of an isometric action of $SL(2,{Z})$ on $\\mathcal {M}_H$ .",
"To demonstrate that this condition is sufficient, one would have to construct suitable complex Darboux coordinates on the twistor space such that, after expressing them in terms of the completed generating function $\\widehat{h}_{p,\\mu }$ found in this paper, $SL(2,{Z})$ acts on these coordinates by a complex contact transformation.",
"Such coordinates were constructed at the two-instanton level in [26], and it is an interesting challenge to extend this construction to arbitrary $n$ .",
"The structure of the completion suggests that, for a divisor decomposable into a sum of $n$ effective divisors, the holomorphic generating function $h_{p,\\mu }$ is a mixed vector valued mock modular form of higher depth, equal to $n-1$ .",
"Such objects have recently appeared in various mathematical and physical contexts [39], [64], [65], [66], [31] and correspond to holomorphic functions whose completion is constructed from period (or Eichler) integrals of mock modular forms of smaller depth (with depth 0 mock modular forms being synonymous with ordinary modular forms).",
"For instance, in the case of standard mock modular forms of depth one, the completion is given by an Eichler integral of a modular function [27].",
"In our case this structure follows from the fact that the completion $\\widehat{h}_{p,\\mu }$ is built from the generalized error functions which have a representation in terms of iterated period integrals [28], [39].",
"In order to make manifest the mixed mock modular nature of $h_{p,\\mu }$ , one should represent the antiholomorphic derivative of its completion $\\widehat{h}_{p,\\mu }$ , computed in (REF ), as a sum of products of completions of mixed mock modular forms of lower depth and anti-holomorphic modular forms.",
"This is a non-trivial task which involves the technique of lattice factorization for indefinite theta series, which we leave for future research.",
"An important caveat in our construction is that we did not establish the convergence of the various generating functions and indefinite theta series.",
"Technically, we constructed a theta series $\\vartheta _{{p},{\\mu }}\\bigl (\\widehat{\\Phi }^{{\\rm tot}}_{n},n-2\\bigr )$ whose kernel satisfies Vignéras' equation (REF ), but we did not demonstrate that it decays as required by Vignéras' criterium.",
"(For $n=2$ this is easy to show [14] since the kernel is a product of an exponentially suppressed factor and a difference of two error functions defined by two positive vectors.)",
"In fact, convergence issues already arise when expressing the generating function of DT invariants in terms of MSW invariants in (REF ), and are related to the problem of convergence of the BPS black hole partition function.",
"The latter was proven to converge in the large volume limit for the case involving up to $n\\le 3$ centers in [50].",
"We hope that the results for the tree index obtained in [23] will allow to extend this result to arbitrary $n$ .",
"We note that many of the complications in our construction originate from the occurrence of polynomial factors in the kernel of theta series, which in turn can be traced to the factors of DSZ products $\\gamma _{ij}$ in the wall-crossing formula.",
"Presumably, most of these complications would disappear if one could find a generalization of the contact potential involving the refined DT invariants, so that all these prefactors can be traded for explicit powers of the fugacity $y$ , as in section REF .",
"It is possible that twistorial topological strings [67] may provide the appropriate framework for finding such a refinement, at least in the context of string vacua on non-compact Calabi-Yau threefolds.",
"On the physics side, we expect that our results will have important implications for the physics of BPS black holes.",
"Indeed, the mock modular property of the generating function $h_{p,\\mu }$ will affect the growth of its Fourier coefficients [68], and should be taken into account when performing precision tests of the microscopic origin of the Bekenstein-Hawking entropy.",
"A natural question is to understand the origin of the non-holomorphic correction terms $R_n$ , in terms of the quantum mechanics of $n$ -centered black holes [37], [69].",
"Another outstanding question is to understand the physical significance of the instanton generating function $\\mathcal {G}$ defined in (REF ).",
"Its modular properties are exactly those expected from the elliptic genus of a superconformal field theory, except for the fact that it is not holomorphic in $\\tau $ .",
"It is natural to expect that this non-holomorphy can be traced to the existence of a continuum of states with a non-trivial spectral asymmetry between bosons and fermions [70].",
"It would be very interesting to understand the origin of this continuum in terms of the worldvolume theory of an M5-brane wrapped on a reducible divisor.",
"Moreover, for special $\\mathcal {N}=2$ theories in $D=4$ obtained by circle compactification of an $\\mathcal {N}=1$ theory in $D=5$ , $\\mathcal {G}$ is closely related to the index considered in [37].",
"It would be interesting to understand the physical origin of its modular invariance in this context.",
"Finally, we expect that the structure found in the context of generalized DT-invariants on Calabi-Yau threefolds will also arise in the study of other types of BPS invariants where higher depth mock modular forms are expected to occur, such as Vafa-Witten invariants and Donaldson invariants of four-folds with $b_2^+=1$ [28], [64].",
"In particular, it would be interesting to determine the modular completion of the generating function of Vafa–Witten invariants on the complex projective plane computed for all ranks in [71], generalizing the rank 3 case studied in [64], and compare with the completion found in the present paper.",
"The authors are grateful to Sibasish Banerjee and Jan Manschot for useful discussions and collaboration on [14], [28], [26] which paved the way for the present work, to Karen Yeats for useful communication about the combinatorics of rooted trees, and to Don Zagier for providing a proof of the conjecture (REF ).",
"The hospitality and financial support of the Theoretical Physics Department of CERN, where this work was initiated, is also gratefully acknowledged.",
"The research of BP is supported in part by French state funds managed by the Agence Nationale de la Recherche (ANR) in the context of the LABEX ILP (ANR-11-IDEX-0004-02, ANR-10- LABX-63).",
"toc"
],
[
"Proof of the Theorem",
"In this appendix we prove Theorem REF presented in the Introduction.",
"The crucial ingredient is provided by the following LemmaWe were informed that this lemma appears as an exercise in [72].",
"Lemma 1 The number of ways of labelling the vertices of a rooted ordered tree $T$ with increasing labels is given by $N_T=n_T!",
"\\prod _{v\\in V_T}\\frac{1}{n_v(T)}\\, ,$ where $n_T$ is the number of vertices of the tree $T$ .",
"First, let us find a recursive relation between the numbers $N_T$ .",
"Namely, consider a vertex $v$ and the set of its children ${\\rm Ch}(v)$ .",
"Denote by $T_v$ the tree rooted at $v$ and with leaves coinciding with leaves of $T$ , for which one has $n_{T_v}=n_v$ .",
"Then it is clear that $N_{T_v}=\\frac{\\left(\\sum _{v^{\\prime }\\in {\\rm Ch}(v)}n_{v^{\\prime }}\\right)!",
"}{\\prod _{v^{\\prime }\\in {\\rm Ch}(v)}n_{v^{\\prime }}!",
"}\\prod _{v^{\\prime }\\in {\\rm Ch}(v)}N_{T_{v^{\\prime }}}\\, .$ From this relation, one easily shows by recursion that (REF ) follows.",
"Indeed, assuming that it holds for the subtrees $T_{v^{\\prime }}$ and taking into account that $\\sum _{v^{\\prime }\\in {\\rm Ch}(v)}n_{v^{\\prime }}=n_v-1$ , the r.h.s.",
"can be rewritten as $(n_v-1)!\\prod _{v^{\\prime }\\in {\\rm Ch}(v)}\\prod _{v^{\\prime \\prime }\\in V_{T_{v^{\\prime }}}}\\frac{1}{n_{v^{\\prime \\prime }}}=n_v!\\prod _{v^{\\prime }\\in V_{T_v}}\\frac{1}{n_{v^{\\prime }}},$ which coincides with (REF ) when $v$ is the root of $T$ .",
"Given this Lemma, we can now rewrite the l.h.s.",
"of (REF ) as $\\sum _{T^{\\prime }\\subset T}\\frac{N_{T^{\\prime }}}{m!",
"}\\prod _{v\\in V_{T^{\\prime }}}n_v(T)=\\frac{n!",
"}{N_{T}}\\sum _{T^{\\prime }\\subset T}\\frac{N_{T^{\\prime }}}{m!",
"}\\prod _{T_r\\subset T\\setminus T^{\\prime }}\\frac{N_{T_r}}{n_{T_r}!",
"},$ where the last product goes over the subtrees which complement $T^{\\prime }$ to the full tree $T$ .",
"On the other hand, it is easy to check that the relation (REF ) is the special case $m=1$ of a more general relation $N_{T}=\\frac{\\left(\\sum _{T_r\\subset T\\setminus T^{\\prime }}n_{T_r}\\right)!",
"}{\\prod _{T_r\\subset T\\setminus T^{\\prime }}n_{T_r}!",
"}\\sum _{T^{\\prime }\\subset T}N_{T^{\\prime }}\\prod _{T_r\\subset T\\setminus T^{\\prime }}N_{T_r}.$ where the sum runs over subtrees $T^{\\prime }$ with $m$ vertices.",
"Taking into account that $\\sum _{T_r\\subset T\\setminus T^{\\prime }}n_{T_r}=n-m$ and substituting the resulting expression into (REF ), one recovers the binomial coefficient as stated in the Theorem.",
"toc"
],
[
"D3-instanton contribution to the contact potential",
"To evaluate $(e^\\phi )_{\\rm D3}$ , we first replace in (REF ) the full prepotential $F$ by its classical, cubic part $F^{\\rm cl}$ (REF ) and take into account that the sum over $\\gamma \\in -\\Gamma _+$ is complex conjugate to the sum over $\\gamma \\in \\Gamma _+$ .",
"This gives $e^{\\phi } \\approx \\frac{\\tau _2^2}{12}\\,((\\,{\\rm Im}\\,u)^3)+\\frac{I\\tau _2}{8}\\,\\sum _{\\gamma \\in \\Gamma _+}\\,{\\rm Re}\\,\\int _{\\ell _\\gamma } \\frac{\\text{d}t}{t} \\left( t^{-1} Z_\\gamma (u^a)-t\\bar{Z}_\\gamma (\\bar{u}^a)\\right) H_\\gamma .$ Next, we substitute the quantum corrected mirror map (REF ), change the integration variable $t$ to $z$ , and take the combined limit $t^a\\rightarrow \\infty $ , $z\\rightarrow 0$ .",
"Keeping only the leading contributions, one obtains $(e^\\phi )_{\\rm D3} &=&-\\frac{\\tau _2}{2}\\sum _{\\gamma \\in \\Gamma _+}\\,{\\rm Re}\\,\\int _{\\ell _{\\gamma }}\\mathrm {d}z\\left[\\hat{q}_0+{1\\over 2}\\, (q+b)^2+2(pt^2)zz_\\gamma -\\frac{3}{2}\\, z^2(pt^2)\\right]H_{\\gamma }\\nonumber \\\\&&-\\frac{1}{4}\\sum _{\\gamma _1,\\gamma _2\\in \\Gamma _+}(tp_1p_2)\\left(\\,{\\rm Re}\\,\\int _{\\ell _{\\gamma _1}}\\!\\!\\mathrm {d}z_1\\, H_{\\gamma _1}\\right)\\left( \\,{\\rm Re}\\,\\int _{\\ell _{\\gamma _2}}\\!\\!\\mathrm {d}z_2\\, H_{\\gamma _2}\\right).$ To further simplify this expression, we note that $\\begin{split}0=&\\, \\frac{1}{4\\pi }\\sum _{\\gamma \\in \\Gamma _+}\\int _{\\ell _{\\gamma }}\\mathrm {d}z\\,\\partial _z\\left( z\\,H_\\gamma \\right)\\\\=&\\,\\sum _{\\gamma \\in \\Gamma _+}\\int _{\\ell _{\\gamma }}\\mathrm {d}z \\left[\\frac{1}{4\\pi }+\\tau _2(pt^2) (zz_\\gamma - z^2)-\\frac{I}{4}\\sum _{\\gamma ^{\\prime }\\in \\Gamma _+}\\langle \\gamma ,\\gamma ^{\\prime }\\rangle \\int _{\\ell _{\\gamma ^{\\prime }}} \\frac{\\mathrm {d}z^{\\prime }}{z-z^{\\prime }}\\, H_{\\gamma ^{\\prime }}\\right]H_\\gamma ,\\end{split}$ where we used the integral equation (REF ).",
"Multiplying this identity by 3/4 and adding its real part to (REF ), one finds $(e^\\phi )_{\\rm D3} &=&\\frac{\\tau _2}{2}\\sum _{\\gamma \\in \\Gamma _+}\\,{\\rm Re}\\,\\int _{\\ell _{\\gamma }}\\mathrm {d}z\\, a_{\\gamma ,-\\frac{3}{2}}(z)\\,H_{\\gamma }\\nonumber \\\\&&-\\frac{1}{4}\\sum _{\\gamma _1,\\gamma _2\\in \\Gamma _+}\\left[(tp_1p_2)\\left(\\,{\\rm Re}\\,\\int _{\\ell _{\\gamma _1}}\\!\\!\\mathrm {d}z_1\\, H_{\\gamma _1}\\right)\\left( \\,{\\rm Re}\\,\\int _{\\ell _{\\gamma _2}}\\!\\!\\mathrm {d}z_2\\, H_{\\gamma _2}\\right)\\right.\\\\&&\\left.\\qquad +\\frac{3}{4}\\,\\,{\\rm Re}\\,\\left(\\int _{\\ell _{\\gamma _1}}\\!\\!\\mathrm {d}z_1\\, H_{\\gamma _1}\\int _{\\ell _{\\gamma _2}}\\!\\!\\mathrm {d}z_2\\, H_{\\gamma _2} \\,\\frac{I\\langle \\gamma ,\\gamma ^{\\prime }\\rangle }{z-z^{\\prime }}\\right)\\right],\\nonumber $ where we introduced $a_{\\gamma ,\\mathfrak {h}}(z)=-\\left(\\hat{q}_0+{1\\over 2}\\,(q+b)^2+\\frac{1}{2}\\, (pt^2)zz_\\gamma +\\frac{\\mathfrak {h}}{4\\pi \\tau _2}\\right).$ The meaning of this function is actually very simple: it gives the action of the modular covariant derivative operator $\\mathcal {D}_{\\mathfrak {h}}$ (REF ) on the classical part of the Darboux coordinate (REF ), $\\mathcal {D}_{\\mathfrak {h}}\\mathcal {X}^{\\rm cl}_{\\gamma } =a_{\\gamma ,\\mathfrak {h}}(z)\\mathcal {X}^{\\rm cl}_{\\gamma }.$ Combining this fact with equation (REF ), it is easy to check that the function (REF ) satisfies, for any value of the weight $\\mathfrak {h}$ , $\\mathcal {D}_{\\mathfrak {h}}\\mathcal {G}&=& \\sum _{\\gamma \\in \\Gamma _+}\\int _{\\ell _{\\gamma }} \\mathrm {d}z\\,a_{\\gamma ,\\mathfrak {h}}(z)\\, H_{\\gamma }\\\\&&+\\frac{1}{8\\tau _2}\\sum _{\\gamma _1,\\gamma _2\\in \\Gamma _+}\\int _{\\ell _{\\gamma _1}}\\mathrm {d}z_1\\,\\int _{\\ell _{\\gamma _2}} \\mathrm {d}z_2\\left[(tp_1p_2)+2\\mathfrak {h}\\left((tp_1p_2)+\\frac{I\\langle \\gamma _1,\\gamma _2\\rangle }{z_1-z_2} \\right)\\right]H_{\\gamma _1}(z_1)H_{\\gamma _2}(z_2),\\nonumber \\\\\\partial _{\\tilde{c}_a}\\mathcal {G}&=& 2\\pi I\\sum _{\\gamma \\in \\Gamma _+}p^a \\int _{\\ell _{\\gamma }} \\mathrm {d}z\\, H_{\\gamma },$ which allows to rewrite (REF ) exactly as in (REF ).",
"toc"
],
[
"Smoothness of the instanton generating function",
"In this appendix we prove the smoothness of the function $\\mathcal {G}$ across walls of marginal stability.",
"The starting point is the representation (REF ) where the potential discontinuities are hidden in the kernel of the theta series (REF ).",
"Given that $\\Phi ^{\\scriptscriptstyle \\,\\int }_{m}$ is represented as a sum over unrooted labelled trees (REF ), whereas $\\Phi ^{\\,g}_{n_k}$ appear as sums over flow trees (REF ), the kernel $\\Phi ^{{\\rm tot}}_{n}$ can be viewed as a sum over `blooming trees' which are unrooted trees with a flow tree (the `flower') growing from each vertex.",
"Then the idea is that the discontinuities due to flow trees (i.e.",
"due to DT invariants) of a blooming tree with $m$ vertices are cancelled by the discontinuities due to exchange of integration contours in $\\Phi ^{\\scriptscriptstyle \\,\\int }_{m+1}$ corresponding to blooming trees with $m+1$ vertices.",
"Figure: Combination of trees showing cancellation of discontinuities across the wall of marginal stabilitycorresponding to the decay γ 𝔳 →γ L +γ R \\gamma _\\mathfrak {v}\\rightarrow \\gamma _L+\\gamma _R.",
"The parts corresponding to attractor flow trees are drown in blue.Following this idea, let us consider a tree which has an attractor flow tree $T_\\mathfrak {v}$ growing from a vertex $\\mathfrak {v}$ (see Fig.",
"REF ).",
"We denote by $\\mathcal {T}_i$ , $i=1,\\dots ,n_\\mathfrak {v}$ , the blooming subtrees connected to this vertex and by $T_{L}$ and $T_{R}$ the two parts (which may be trivial) of $T_\\mathfrak {v}$ with the total charges $\\gamma _L$ and $\\gamma _R$ so that $\\gamma _\\mathfrak {v}=\\gamma _L+\\gamma _R$ .",
"Together with the contribution of this tree, we consider the contributions of the trees obtained by splitting the vertex $\\mathfrak {v}$ into two vertices $\\mathfrak {v}_L$ and $\\mathfrak {v}_R$ connected by an edge, carrying charges $\\gamma _L$ and $\\gamma _R$ and all possible allocations of the subtrees $\\mathcal {T}_i$ to these two vertices.",
"Different allocations are accounted for by the sum over permutations, whilethe weight $\\frac{1}{\\ell !",
"(n_\\mathfrak {v}-\\ell )!",
"}$ takes into account the fact that permutations between subtrees connected to one vertex are redundant.",
"The attractor flow trees $T_{L}$ and $T_{R}$ are then connected to $\\mathfrak {v}_L$ and $\\mathfrak {v}_R$ , respectively, as shown in Fig.",
"REF .",
"The contribution corresponding to the first blooming tree has a discontinuity at the wall of marginal stability for the bound state $\\gamma _L+\\gamma _R$ and originating from the factor $\\Delta ^{z}_{\\gamma _L\\gamma _R}$ assigned to the root vertex of $T_\\mathfrak {v}$ .",
"The other contributions have discontinuities at the same wall due to the exchange of the contours $\\ell _{\\gamma _L}$ and $\\ell _{\\gamma _R}$ for the integrals assigned to $\\mathfrak {v}_L$ and $\\mathfrak {v}_R$ , respectively.",
"They are given by the residues at the pole of the integration kernel $K_{\\gamma _L\\gamma _R}$ .",
"It is clear that the structure of all jumps is very similar since different subtrees produce essentially the same weights.",
"Let us analyze what differences may arise.",
"First, the contributions of the flow trees $T_{L}$ and $T_{R}$ could differ in the two cases because they have different starting points for the attractor flows: for the trees on the right side of Fig.",
"REF this is $z^a\\in \\mathcal {M}_K$ , whereas for the tree on the left this is the point on the wall for $\\gamma _L+\\gamma _R$ reached by the flow from $z^a$ .",
"But we are evaluating the discontinuity exactly on the wall where the two points coincide.",
"Thus, the contributions are the same.",
"Although the subtrees $\\mathcal {T}_i$ give rise to the same contributions for all blooming trees shown in Fig.",
"REF , the contributions of the edges connecting them to either $\\mathfrak {v}$ or $\\mathfrak {v}_L$ , $\\mathfrak {v}_R$ are not exactly the same.",
"Each of them contributes the factor given by the kernel (REF ).",
"After taking the residue, the $z$ -dependence of these kernels for the trees shown on the left and the right sides of the picture is identical.",
"However, their charge dependence is different: for the tree on the left they depend on $\\gamma _\\mathfrak {v}$ , whereas for the trees on the right they depend on $\\gamma _L$ or $\\gamma _R$ , depending on which vertex they are connected to.",
"But it is easy to see that the sums over $\\ell $ and permutations produce the standard binomial expansion of a single product of kernels which all depend on $\\gamma _L+\\gamma _R=\\gamma _\\mathfrak {v}$ and thus coinciding with the contribution of the tree on the left.",
"Finally, one should take into account that the discontinuity of $\\Delta ^{z}_{\\gamma _L\\gamma _R}$ in the first contribution gives the factor $\\langle \\gamma _L,\\gamma _R\\rangle $ .",
"But exactly the same factor arises as the residue of $K_{\\gamma _L\\gamma _R}$ corresponding to the additional edge.",
"Thus, it remains only to check that all numerical factors work out correctly.",
"Leaving aside the factors which are common to both contributions, we have $-\\frac{(-1)^{\\langle \\gamma _L,\\gamma _R\\rangle }}{2}\\,\\frac{\\sigma _{\\gamma _\\mathfrak {v}}}{(2\\pi )^2}\\, \\frac{2m}{ m!",
"}+(2\\pi I)(-2\\pi I)\\,\\frac{\\sigma _{\\gamma _L}\\sigma _{\\gamma _R}}{(2\\pi )^4}\\,\\frac{m(m+1)}{(m+1)!",
"}.$ Here $-\\frac{(-1)^{\\langle \\gamma _L,\\gamma _R\\rangle }}{2}$ comes from the factor $-\\Delta ^{z}_{\\gamma _L\\gamma _R}$ in $T_\\mathfrak {v}$ , the factors with quadratic refinement are due to functions $H_\\gamma $ (REF ) assigned to $\\mathfrak {v}$ or $\\mathfrak {v}_L$ , $\\mathfrak {v}_R$ , $(2\\pi I)$ is the standard weight of the residue, $(-2\\pi I)$ is the residue of $K_{\\gamma _L\\gamma _R}$ (REF ), and factorials are the weights of the trees in the expansion (REF ).",
"Finally, the factors $2m$ and $m(m+1)$ arise due to the freedom to relabel charges assigned to the marked vertices: on the left these are vertex $\\mathfrak {v}$ and the two children of the root in $T_\\mathfrak {v}$ , whereas on the right these are $\\mathfrak {v}_L$ and $\\mathfrak {v}_R$ .",
"It is immediate to check that all these numerical weights cancel, which ensures that the function $\\mathcal {G}$ is continuous across walls of marginal stability.",
"Moreover, in this cancellation the condition that we sit on the wall was used only in locally constant factors.",
"Therefore, this reasoning proves not only that $\\mathcal {G}$ is continuous, but that it is actually smooth around these loci.",
"toc"
],
[
"Vignéras' theorem",
"Let $\\mathbf {\\Lambda }$ be a $d$ -dimensional lattice equipped with a bilinear form $({x},{y})\\equiv {x}\\cdot {y}$ , where ${x},{y}\\in \\mathbf {\\Lambda }\\otimes {R}$ , such that its associated quadratic form has signature $(n,d-n)$ and is integer valued, i.e.",
"${k}^2\\equiv {k}\\cdot {k}\\in {Z}$ for ${k}\\in \\mathbf {\\Lambda }$ .",
"Furthermore, let ${p}\\in \\mathbf {\\Lambda }$ be a characteristic vector (such that ${k}\\cdot ({k}+ {p})\\in 2{Z}$ , $\\forall \\,{k}\\in \\mathbf {\\Lambda }$ ), ${\\mu }\\in \\mathbf {\\Lambda }^*/\\mathbf {\\Lambda }$ a glue vector, and $\\lambda $ an arbitrary integer.",
"We consider the following family of theta series $\\vartheta _{{p},{\\mu }}(\\Phi ,\\lambda ;\\tau , {b}, {c})=\\tau _2^{-\\lambda /2}\\!\\!\\!\\!\\sum _{{{k}}\\in \\mathbf {\\Lambda }+{\\mu }+{1\\over 2}{p}}\\!\\!",
"(-1)^{{k}\\cdot {p}}\\,\\Phi (\\sqrt{2\\tau _2}({k}+{b}))\\, {\\bf e}\\!\\left( - \\tfrac{\\tau }{2}\\,({k}+{b})^2+{c}\\cdot ({k}+\\textstyle {1\\over 2}{b})\\right)$ defined by a kernel $\\Phi ({x})$ such that the function $f({x})\\equiv \\Phi ({x})\\, e^{\\tfrac{\\pi }{2}\\,{x}^2}\\in L^1(\\mathbf {\\Lambda }\\otimes {R})$ so that the sum is absolutely convergent.",
"Irrespective of the choice of this kernel and of the parameter $\\lambda $ , any such theta series satisfies the following elliptic properties $\\begin{split}\\vartheta _{{p},{\\mu }}\\left({\\Phi } ,\\lambda ; \\tau , {b}+{k},{c}\\right) =&(-1)^{{k}\\cdot {p}}\\,{\\bf e}\\!\\left( -\\textstyle {1\\over 2}\\, {c}\\cdot {k}\\right) \\vartheta _{{p},{\\mu }}\\left({\\Phi } ,\\lambda ; \\tau , {b},{c}\\right),\\\\\\vphantom{A^A \\over A_A}\\vartheta _{{p},{\\mu }}\\left({\\Phi }, \\lambda ; \\tau , {b},{c}+{k}\\right)=&(-1)^{{k}\\cdot {p}}\\,{\\bf e}\\!\\left( \\textstyle {1\\over 2}\\, {b}\\cdot {k}\\right) \\vartheta _{{p},{\\mu }}\\left({\\Phi } ,\\lambda ; \\tau , {b},{c}\\right).\\end{split}$ Now let us require that in addition the kernel satisfies the following two conditions: Let $D({x})$ be any differential operator of order $\\le 2$ , and $R({x})$ any polynomial of degree $\\le 2$ .",
"Then $f({x})$ defined above must be such that $f({x})$ , $D({x})f({x})$ and $R({x})f({x})\\in L^2(\\mathbf {\\Lambda }\\otimes {R})\\bigcap L^1(\\mathbf {\\Lambda }\\otimes {R})$ .",
"$\\Phi ({x})$ must satisfy $V_\\lambda \\cdot \\Phi ({x})=0,\\qquad V_\\lambda = \\partial _{{x}}^2 + 2\\pi \\left( {x}\\cdot \\partial _{{x}} - \\lambda \\right) .$ Then in [38] it was proven that the theta series (REF ) transforms as a vector-valued modular form of weight $(\\lambda +d/2,0)$ (see Theorem 2.1 in [28] for the detailed transformation under $\\tau \\rightarrow -1/\\tau $ ).",
"We refer to $V_\\lambda $ as Vignéras' operator.",
"The simplest example is the Siegel theta series for which the kernel is $\\Phi ({x})=e^{-\\pi {x}_+^2}$ where ${x}_+$ is the projection of ${x}$ on a fixed positive plane of dimension $n$ .",
"This kernel is annihilated by $V_{-n}$ .",
"In this paper we apply the Vignéras' theorem to the case of $\\mathbf {\\Lambda }=\\oplus _{i=1}^n \\Lambda _i$ .",
"Thus, the charges appearing in the description of the theta series (REF ) are of the type ${k}=(k_1^a,\\dots ,k_n^a)$ , whereas the vectors ${b}$ and ${c}$ are taken with $i$ -independent components, namely, ${b}_i^a=b^a$ , ${c}_i^a=c^a$ for $i=1,\\dots , n$ .",
"The lattices $\\Lambda _i$ carry the bilinear forms $\\kappa _{i,ab}=\\kappa _{abc}p_i^c$ which are all of signature $(1,b_2-1)$ .",
"This induces a natural bilinear form on $\\mathbf {\\Lambda }$ : ${x}\\cdot {y}=\\sum _{i=1}^n (p_ix_iy_i).$ Note also that the sign factor $(-1)^{{k}\\cdot {p}}$ in (REF ) can be identified with the quadratic refinement provided we choose the latter as $\\sigma _\\gamma =\\sigma _{p,q}\\equiv {\\bf e}\\!\\left( {1\\over 2}\\, p^a q_a\\right)\\sigma _p,\\qquad \\sigma _p={\\bf e}\\!\\left( {1\\over 2}\\, A_{ab}p^ap^b\\right).$ The matrix $A_{ab}$ , satisfying $A_{ab} p^p - \\frac{1}{2}\\, \\kappa _{abc} p^b p^c\\in {Z}\\quad \\text{for}\\ \\forall p^a\\in {Z}\\, ,$ appears due to the non-trivial quantization of charges on the type IIB side (REF ) and can be used to perform a symplectic rotation to identify them with mirror dual integer charges on the type IIA side [51].",
"It is easy to check that the quadratic refinement (REF ) satisfies (REF )."
],
[
"Generalized error functions",
"An important class of solutions of Vignéras' equation is given by the error function and its generalizations constructed in [28] and further elaborated in [29].",
"Let us take $M_1(u)&=&-\\mbox{sgn}(u)\\, \\text{Erfc}(|u|\\sqrt{\\pi }) = \\frac{I}{\\pi } \\int _{\\ell }\\frac{\\mathrm {d}z}{z}\\,e^{-\\pi z^2 -2\\pi Iz u},\\\\E_1(u)&=&\\mbox{sgn}(u)+M_1(u)\\nonumber \\\\&=& \\operatorname{Erf}(u\\sqrt{\\pi })= \\int _{{R}} \\mathrm {d}u^{\\prime } \\, e^{-\\pi (u-u^{\\prime })^2} {\\rm sgn}(u^{\\prime }),$ where the contour $\\ell ={R}-Iu$ runs parallel to the real axis through the saddle point at $z=-Iu$ .",
"Then, given a vector with a positive norm ${v}^2>0$ so that $|{v}|=\\sqrt{{v}^2}$ , we define $\\Phi _1^E({v};{x})=E_1\\left(\\frac{{v}\\cdot {x}}{|{v}|}\\right),\\qquad \\Phi _1^M({v};{x})=M_1\\left(\\frac{{v}\\cdot {x}}{|{v}|}\\right).$ It is easy to check that the first function is a smooth solution of (REF ) with $\\lambda =0$ , whereas the second is exponentially suppressed at large ${x}$ and also solves the same equation, but only away from the locus ${v}\\cdot {x}=0$ where it has a discontinuity.",
"Generalizing the integral representations (REF ) and (), we define the generalized (complementary) error functions $M_n(\\mathcal {M};\\mathbb {u})&=&\\left(\\frac{I}{\\pi }\\right)^n |\\,{\\rm det}\\, \\mathcal {M}|^{-1} \\int _{{R}^n-I\\mathbb {u}}\\mathrm {d}^n z\\,\\frac{e^{-\\pi \\mathbb {z}^{\\rm tr} \\mathbb {z}-2\\pi I\\mathbb {z}^{\\rm tr} \\mathbb {u}}}{\\prod (\\mathcal {M}^{-1}\\mathbb {z})},\\\\E_n(\\mathcal {M};\\mathbb {u})&=& \\int _{{R}^n} \\mathrm {d}\\mathbb {u}^{\\prime } \\, e^{-\\pi (\\mathbb {u}-\\mathbb {u}^{\\prime })^{\\rm tr}(\\mathbb {u}-\\mathbb {u}^{\\prime })} {\\rm sgn}(\\mathcal {M}^{\\rm tr} \\mathbb {u}^{\\prime }),$ where $\\mathbb {z}=(z_1,\\dots ,z_n)$ and $\\mathbb {u}=(u_1,\\dots ,u_n)$ are $n$ -dimensional vectors, $\\mathcal {M}$ is $n\\times n$ matrix of parameters, and we used the shorthand notations $\\prod \\mathbb {z}=\\prod _{i=1}^n z_i$ and ${\\rm sgn}(\\mathbb {u})=\\prod _{i=1}^n {\\rm sgn}(u_i)$ .",
"The detailed properties of these functions can be found in [29].",
"Here we mention only a few: $M_n$ are exponentially suppressed for large $\\mathbb {u}$ as $M_n\\sim \\frac{(-1)^n}{\\pi ^n}|\\,{\\rm det}\\, \\mathcal {M}|^{-1}\\,\\frac{e^{-\\pi \\mathbb {u}^{\\rm tr} \\mathbb {u}}}{\\prod (\\mathcal {M}^{-1}\\mathbb {u})}$ , whereas $E_n$ are locally constant for large $\\mathbb {u}$ as $E_n\\sim {\\rm sgn}(\\mathcal {M}^{\\rm tr} \\mathbb {u})$ .",
"More generally, $E_n$ can be expressed as a linear combination of $M_k$ , $k=0,\\dots ,n$ , multiplied by $n-k$ sign functions, generalizing the first relation in () (see Eq.",
"(REF ) below for a precise statement).",
"From () it follows that every identity between products of sign functions implies an identity between generalized error functions $E_n$ .",
"Moreover, expanding the $E_n$ functions in terms of $M_k$ 's and sign functions, one obtains similar identities for functions $M_n$ .",
"For instance, the identity $(\\mbox{sgn}(x_1)+\\mbox{sgn}(x_2))\\,\\mbox{sgn}(x_1+x_2)=1+\\mbox{sgn}(x_1)\\,\\mbox{sgn}(x_2)$ implies $\\begin{split}E_2((\\mathbb {v}_1,\\mathbb {v}_1+\\mathbb {v}_2);\\mathbb {u})+E_2((\\mathbb {v}_2,\\mathbb {v}_1+\\mathbb {v}_2);\\mathbb {u})=&\\, 1+E_2((\\mathbb {v}_1,\\mathbb {v}_2);\\mathbb {u}),\\\\M_2((\\mathbb {v}_1,\\mathbb {v}_1+\\mathbb {v}_2);\\mathbb {u})+M_2((\\mathbb {v}_2,\\mathbb {v}_1+\\mathbb {v}_2);\\mathbb {u})=&\\, M_2((\\mathbb {v}_1,\\mathbb {v}_2);\\mathbb {u}),\\end{split}$ where $\\mathbb {v}_1,\\mathbb {v}_2$ are two-dimensional vectors used to encode the $2\\times 2$ matrix of parameters.",
"The main reason to introduce these functions is that, similarly to the usual error and complementary error functions, they can be used to produce solutions of Vignéras' equation on ${R}^{n,d-n}$ .",
"To write them down, let us consider $d\\times n$ matrix $\\mathcal {V}$ which can be viewed as a collection of $n$ vectors, $\\mathcal {V}=({v}_1,\\dots ,{v}_n)$ .",
"We assume that these vectors span a positive definite subspace, i.e.",
"$\\mathcal {V}^{\\rm tr}\\cdot \\mathcal {V}$ is a positive definite matrix.",
"Let $\\mathcal {B}$ be $n\\times d$ matrix whose rows define an orthonormal basis for this subspace.",
"Then we define the boosted generalized error functions $\\Phi _n^M(\\mathcal {V};{x})=M_n(\\mathcal {B}\\cdot \\mathcal {V};\\mathcal {B}\\cdot {x}),\\qquad \\Phi _n^E(\\mathcal {V};{x})=E_n(\\mathcal {B}\\cdot \\mathcal {V};\\mathcal {B}\\cdot {x}).$ It can be shown that both these functions satisfy Vignéras' equation (for $\\Phi _n^M$ one should stay away from its discontinuities, i.e.",
"loci where ${\\rm sgn}((\\mathcal {B}\\cdot \\mathcal {V})^{-1}\\mathcal {B}\\cdot {x})=0$ ).",
"Moreover, they are symmetric under permutation of the vectors ${v}_i$ .",
"Since at large ${x}$ one has $\\Phi _n^E\\sim {\\rm sgn}(\\mathcal {V}^{\\rm tr}\\cdot {x})=\\prod _{i=1}^n {\\rm sgn}({v}_i\\cdot {x})$ , one can think about this function as providing the modular completion for (indefinite) theta series with kernel given by a product of signs.",
"The relation between functions $E_n$ and $M_n$ mentioned above implies a similar relation between the functions (REF ).",
"For generic $n$ , it takes the following formThis relation reduces to Eq.",
"(3.53) and (6.18) in [28] for $n=2,3$ , and follows from Eq.",
"(63) in [29] for any $n$ .",
"$\\Phi _n^E(\\mathcal {V};{x})=\\sum _{\\mathcal {I}\\subseteq {Z}_n}\\Phi _{|\\mathcal {I}|}^M(\\lbrace {v}_i\\rbrace _{i\\in \\mathcal {I}};{x})\\prod _{j\\in {Z}_n\\setminus \\mathcal {I}}{\\rm sgn}({v}_{j\\perp \\mathcal {I}},{x}),$ where the sum goes over all possible subsets (including the empty set) of the set ${Z}_{n}=\\lbrace 1,\\dots ,n\\rbrace $ , $|\\mathcal {I}|$ is the cardinality of $\\mathcal {I}$ , and ${v}_{j\\perp \\mathcal {I}}$ denotes the projection of ${v}_j$ orthogonal to the subspace spanned by $\\lbrace {v}_i\\rbrace _{i\\in \\mathcal {I}}$ .",
"The cardinality $|\\mathcal {I}|$ can also be interpreted as the number of directions in ${R}^d$ along which the corresponding contribution has an exponential fall off.",
"Finally, note that a solution of Vignéras' equation can be uplifted to a solution of the same equation with $\\lambda $ shifted to $\\lambda +1$ by acting with the differential operator $\\mathcal {D}({v})={v}\\cdot \\left({x}+\\frac{1}{2\\pi }\\,\\partial _{x}\\right),$ which realizes the action of the covariant derivative raising the holomorphic weight by 1.",
"In particular, we can construct solutions with $\\lambda =m$ which behave for large ${x}$ as products of $n$ sign functions.",
"To this end, it is enough to act on $\\Phi _n^E$ by this operator $m$ times.",
"Thus, we define the uplifted boosted error function $\\widetilde{\\Phi }_{n,m}^E(\\mathcal {V},\\tilde{\\mathcal {V}};{x})=\\left[\\prod _{i=1}^m \\mathcal {D}(\\tilde{{v}}_i)\\right]\\Phi _n^E(\\mathcal {V};{x}),$ where $\\tilde{\\mathcal {V}}=(\\tilde{{v}}_1,\\dots ,\\tilde{{v}}_m)$ encodes the vectors contracted with the covariant derivatives.",
"Since the operators $\\mathcal {D}(\\tilde{{v}}_i)$ commute, (REF ) is invariant under independent permutations of the vectors ${v}_i$ and $\\tilde{{v}}_i$ .",
"In the case where all $\\tilde{{v}}_i$ are mutually orthogonal, one finds the following asymptotics at large ${x}$ $\\lim _{{x}\\rightarrow \\infty }\\widetilde{\\Phi }_{n,m}^E(\\mathcal {V},\\tilde{\\mathcal {V}};{x})= \\prod _{i=1}^m (\\tilde{{v}}_i,{x})\\prod _{j=1}^n {\\rm sgn}({v}_j,{x}).$ Note that the derivative $\\partial _{x}$ in (REF ) does not act on sign functions since $\\Phi _n^E$ is smooth and all discontinuities due to signs are guaranteed to cancel.",
"Similarly to (REF ), we can also define $\\widetilde{\\Phi }_{n,m}^M(\\mathcal {V},\\tilde{\\mathcal {V}};{x})$ where the action of derivatives on the discontinuities of $\\Phi _n^M$ is ignored as well.",
"In the particular case $m=n$ , we will omit the second label and simply write $\\widetilde{\\Phi }_{n}^E$ or $\\widetilde{\\Phi }_{n}^M$ .",
"toc"
],
[
"Twistorial integrals and generalized error functions",
"In this appendix we evaluate the kernels $\\Phi ^{\\scriptscriptstyle \\,\\int }_n$ (REF ) and show that they can be expressed through the generalized error functions introduced in appendix REF .",
"To this end, let us note the following identity $I\\,\\mathcal {D}({v}_{ij})\\,\\frac{W_{p_i}(x_i,z_i)\\,W_{p_j}(x_j,z_j)}{z_i-z_j}= \\hat{K}_{ij}\\, W_{p_i}(x_i,z_i)\\,W_{p_j}(x_j,z_j).$ By virtue of this relation, the kernel can be represented as $\\Phi ^{\\scriptscriptstyle \\,\\int }_n({x})=\\frac{1}{n!",
"}\\sum _{\\mathcal {T}\\in \\, {T}_n^\\ell }\\left[\\prod _{e\\in E_\\mathcal {T}}\\mathcal {D}({v}_{s(e) t(e)}) \\right]\\Phi _\\mathcal {T}({x}),$ where $\\Phi _\\mathcal {T}({x}) =\\frac{I^{n-1}}{(2\\pi )^n}\\left[\\prod _{i=1}^n\\int _{\\ell _{\\gamma _i}}\\mathrm {d}z_i \\, W_{p_i}(x_i,z_i)\\right]\\frac{1}{\\prod _{e\\in E_\\mathcal {T}}\\left(z_{s(e)}-z_{t(e)}\\right)}\\, .$ One may think that the representation (REF ) misses contributions from the covariant derivatives acting on each other.",
"However, such contributions are proportional to the scalar products of two vectors ${v}_{s(e) t(e)}$ and are non-vanishing provided the two edges have a common vertex.",
"If this is the case, the two edges generate the following factor $\\frac{(p_{\\mathfrak {v}_1}p_{\\mathfrak {v}_2}p_{\\mathfrak {v}_{3}})}{(z_{\\mathfrak {v}_1}-z_{\\mathfrak {v}_3})(z_{\\mathfrak {v}_2}-z_{\\mathfrak {v}_{3}})}\\, ,$ where $\\mathfrak {v}_1,\\mathfrak {v}_2,\\mathfrak {v}_3$ are the tree vertices joint by the edges and $\\mathfrak {v}_{3}=e_1\\cap e_2$ .",
"The crucial observation is that if we pick up 3 subtrees, each with a marked vertex, there are exactly 3 ways to form a labelled tree out of them by joining the marked vertices as shown in Fig.",
"REF on page REF .",
"Each subtree contributes the same factor in all 3 cases, whereas the joining edges and the sum over trees give rise to the vanishing factor $\\frac{(p_1p_2p_3)}{(z_1-z_3)(z_2-z_3)}+\\frac{(p_1p_2p_3)}{(z_1-z_2)(z_1-z_3)}-\\frac{(p_1p_2p_3)}{(z_2-z_3)(z_1-z_2)}=0.$ This ensures that no additional contributions arise and thereby proves (REF ).",
"Note that for this proof it was crucial that inequivalent labelled trees enter the sum with the same weight.",
"Then let us do the change of variables $z_i=z^{\\prime }-\\frac{I(pxt)}{\\sqrt{2\\tau _2}(pt^2)}+\\sum _{\\alpha =1}^{n-1} e_i^\\alpha z^{\\prime }_\\alpha ,$ where $x^a=\\kappa ^{ab}\\sum \\kappa _{i,bc} x^c_i$ (cf.",
"(REF )), and $e_i^\\alpha $ are such that $\\sum _{i=1}^n \\mathfrak {p}_i e_i^\\alpha =0,\\qquad \\sum _{i=1}^n \\mathfrak {p}_i e_i^\\alpha e_i^\\beta =0, \\quad \\alpha \\ne \\beta $ and we introduced the convenient notation $\\mathfrak {p}_i=(p_it^2)$ .",
"Labeling the $n-1$ edges of the tree by the same index $\\alpha $ , one can rewrite the function (REF ) in the new variables as $\\Phi _\\mathcal {T}({x}) =\\frac{I^{n-1} \\sqrt{\\Delta }}{(2\\pi )^n}\\,e^{-\\frac{\\pi (pxt)^2}{(pt^2)}}\\int \\mathrm {d}z^{\\prime } \\, e^{-2\\pi \\tau _2(pt^2)z^{\\prime 2}}\\prod _{\\alpha =1}^{n-1}\\int \\frac{\\mathrm {d}z^{\\prime }_\\alpha \\,e^{-2\\pi \\tau _2\\, \\Delta _\\alpha (z^{\\prime }_\\alpha )^2 -2\\pi I\\sqrt{2\\tau _2}\\, w^\\alpha z^{\\prime }_\\alpha }}{\\sum _{\\beta =1}^{n-1}\\left(e^\\beta _{s(\\alpha )}-e^\\beta _{t(\\alpha )}\\right)z^{\\prime }_\\beta }\\, ,$ where $w^\\alpha =\\sum _{i=1}^n (p_ix_i t) e_i^\\alpha ,\\qquad \\Delta _\\alpha =\\sum _{i=1}^n \\mathfrak {p}_i (e_i^\\alpha )^2,\\qquad \\Delta =\\frac{\\mathfrak {p}\\prod _{\\alpha =1}^{n-1} \\Delta _\\alpha }{\\prod _{i=1}^{n}\\mathfrak {p}_i}\\, .$ The integral over $z^{\\prime }$ is Gaussian and is easily evaluated.",
"In the remaining integrals, rescaling the integration variables by $\\sqrt{2\\tau _2\\Delta _\\alpha }$ , one recognizes the generalized error functions (REF ).",
"Thus, one obtains $\\Phi _\\mathcal {T}({x}) =\\frac{1}{2^{n-1}}\\,\\frac{\\sqrt{\\Delta }|\\,{\\rm det}\\, \\mathcal {M}|}{\\prod _{\\alpha =1}^{n-1} \\Delta _\\alpha }\\,\\Phi ^{\\scriptscriptstyle \\,\\int }_1(x)\\, M_{n-1}\\left(\\mathcal {M};\\left\\lbrace \\frac{w^\\alpha }{\\sqrt{\\Delta _\\alpha }}\\right\\rbrace \\right).$ where we used the function $\\Phi ^{\\scriptscriptstyle \\,\\int }_1$ (REF ) and introduced the matrix $\\mathcal {M}$ such that $\\mathcal {M}^{-1}_{\\alpha \\beta }=(\\Delta _\\alpha \\Delta _\\beta )^{-1/2}\\left(e^\\beta _{s(\\alpha )}-e^\\beta _{t(\\alpha )}\\right).$ A simple solution to the conditions (REF ) may be constructed as follows.",
"Let $T$ be a rooted ordered binary tree with $n$ leaves decorated by $\\gamma _i$ , $i=1\\dots n$ .",
"As usual for such trees, other vertices $v$ carry charges given by the sum of charges of their children, i.e.",
"$\\gamma _v=\\sum _{i\\in \\mathcal {I}_v}\\gamma _i$ where $\\mathcal {I}_v$ is the set of leaves which are descendants of $v$ .",
"There are $n-1$ such vertices which we label by index $\\alpha $ .",
"Then we can choose $e^\\alpha _i=\\sum _{j\\in \\mathcal {I}_{L(v_\\alpha )}}\\sum _{k\\in \\mathcal {I}_{R(v_\\alpha )}} \\left(\\delta _{ij}\\,\\mathfrak {p}_k-\\delta _{ik}\\,\\mathfrak {p}_j\\right),$ which satisfy (REF ) as can be easily checked.",
"For this choice (cf.",
"(REF )) $\\begin{split}w^\\alpha =&\\, (\\tilde{{u}}_{\\alpha },{x}),\\quad \\mbox{where}\\quad \\tilde{{u}}_\\alpha =\\sum _{i\\in \\mathcal {I}_{L(v_\\alpha )}}\\sum _{j\\in \\mathcal {I}_{R(v_\\alpha )}}{u}_{ij},\\\\\\Delta _\\alpha =&\\,\\tilde{{u}}_\\alpha ^2=\\mathfrak {p}_{v_\\alpha } \\mathfrak {p}_{L(v_\\alpha )}\\mathfrak {p}_{R(v_\\alpha )}.\\end{split}$ Note that the vectors $\\tilde{{u}}_\\alpha $ are mutually orthogonal.",
"In principle, any rooted binary tree $T$ is suitable for the above construction.",
"However, given the unrooted tree $\\mathcal {T}$ , there is a simple (but non-unique) choice of $T$ which simplifies the resulting matrix $\\mathcal {M}$ .",
"To define it, let us construct a partially increasing family of subtrees of $\\mathcal {T}$ , such that two members in this family are either disjoint, or contained in one another.",
"Moreover, we require that the largest subtree is $\\mathcal {T}$ itself, while each subtree containing more than one vertex is obtained by joining two smaller subtrees along an edge of $\\mathcal {T}$ .",
"Any such family contains $2n-1$ subtrees $\\mathcal {T}_{\\hat{\\alpha }}$ labelled by $\\hat{\\alpha }=1,\\dots ,2n-1$ .",
"Among them, $n-1$ subtrees, which we label by $\\alpha =\\hat{\\alpha }=1,\\dots ,n-1$ , contain several vertices, while the remaining $n$ subtrees with label $\\hat{\\alpha }=n,\\dots ,2n-1$ have only one vertex.",
"For each subtree $T_\\alpha $ , we denote by $e_\\alpha $ the edge of $\\mathcal {T}$ which is used to reconstruct $T_\\alpha $ from two smaller subtrees $\\mathcal {T}_{\\alpha _L}, \\mathcal {T}_{\\alpha _R}$ .",
"From this data, we construct a rooted binary tree $T$ with $n-1$ vertices in one-to-one correspondence with the subtrees $\\mathcal {T}_\\alpha $ and $n$ leaves in one-to-one correspondence with the one-vertex subtrees $\\mathcal {T}_{\\hat{\\alpha }}$ with $\\hat{\\alpha }\\ge n$ .",
"In this correspondence, the two children of a vertex associated to $\\mathcal {T}_{\\alpha }$ are the vertices associated to the two subtrees $\\mathcal {T}_{\\alpha _L}, \\mathcal {T}_{\\alpha _R}$ .",
"The ordering at each vertex is defined to be such that the subtree containing the source/target vertex of the corresponding edge $e_\\alpha $ is on the left/right.The orientation of the edges of $\\mathcal {T}$ is fixed already in (REF ), but the full kernel $\\Phi ^{\\scriptscriptstyle \\,\\int }_n$ does not depend on its choice.",
"Of course, this construction is not unique since there are many ways to decompose $\\mathcal {T}$ into such a set of subtrees (see Fig.",
"REF ).",
"Figure: An example of an unrooted labelled tree with 4 vertices and two choices of decompositions into subtrees withthe corresponding rooted binary trees.",
"The edges are labelled e 1 ,e 2 ,e 3 e_1,e_2,e_3 from left to rightalong 𝒯\\mathcal {T}.Applying the above construction to this particular choice of rooted tree, one finds $\\sqrt{\\Delta _\\alpha \\Delta _\\beta }\\,\\mathcal {M}^{-1}_{\\alpha \\beta }=\\left\\lbrace \\begin{array}{ll}\\mathfrak {p}_{v_\\alpha }, \\quad & \\alpha =\\beta ,\\\\\\epsilon _{\\alpha \\beta } \\mathfrak {p}_{L(v_\\beta )}, \\quad & e_\\alpha \\cap \\mathcal {T}_{\\beta _R} \\ne \\emptyset ,\\ e_\\alpha \\nsubseteq \\mathcal {T}_\\beta ,\\\\\\epsilon _{\\alpha \\beta } \\mathfrak {p}_{R(v_\\beta )}, \\quad & e_\\alpha \\cap \\mathcal {T}_{\\beta _L} \\ne \\emptyset ,\\ e_\\alpha \\nsubseteq \\mathcal {T}_\\beta ,\\\\0, & e_\\alpha \\cap \\mathcal {T}_\\beta = \\emptyset \\ \\mbox{ or }\\ e_\\alpha \\subset \\mathcal {T}_\\beta ,\\end{array}\\right.$ where $\\epsilon _{\\alpha \\beta }=-1$ if the orientations of $e_\\alpha $ and $e_\\beta $ on the path joining them are the same and $+1$ otherwise.",
"This result shows that the matrix $\\mathcal {M}^{-1}$ turns out to be triangular which makes it much simpler to find its inverse.",
"On the basis of (REF ), below we will prove the following Lemma 2 One has $\\mathcal {B}\\cdot \\mathcal {V}=\\mathcal {M}$ and $\\mathcal {B}\\cdot {x}=\\left\\lbrace \\frac{w^\\alpha }{\\sqrt{\\Delta _\\alpha }}\\right\\rbrace $ provided $\\begin{split}\\mathcal {B}=&\\, \\left(\\frac{\\tilde{{u}}_1}{\\sqrt{\\Delta _1}},\\dots , \\frac{\\tilde{{u}}_{n-1}}{\\sqrt{\\Delta _{n-1}}}\\right)^{\\rm tr},\\\\\\mathcal {V}=&\\, \\mathfrak {p}^{-1}\\Bigl (\\sqrt{\\Delta _1}{u}_1,\\dots , \\sqrt{\\Delta _{n-1}}{u}_{n-1}\\Bigr ),\\end{split}$ where the vectors ${u}_\\alpha $ are defined in (REF ).",
"Moreover, the vectors $\\tilde{{u}}_\\alpha $ form an orthogonal basis in the subspace spanned by ${u}_\\alpha $ .",
"This lemma allows to reexpress the kernel $\\Phi _\\mathcal {T}$ (REF ) in terms of the boosted generalized error function $\\Phi ^M_{n-1}$ (REF ).",
"It is important that its argument $\\mathcal {V}$ does not depend on the choice of the binary rooted tree $T$ , but only on the unrooted tree $\\mathcal {T}$ .",
"In addition, the function actually does not depend on the normalization of the vectors composing $\\mathcal {V}$ .",
"Given also that the determinant of $\\mathcal {M}$ is found to be $|\\,{\\rm det}\\, \\mathcal {M}|=\\prod _{\\alpha =1}^{n-1}\\frac{ \\Delta _\\alpha }{\\mathfrak {p}_{v_\\alpha }}=\\prod _{i=1}^{n}\\mathfrak {p}_i\\prod _{v\\in V_{T}\\setminus {\\lbrace v_0\\rbrace }}\\mathfrak {p}_v=\\sqrt{\\mathfrak {p}^{-1}\\,\\prod _{i=1}^{n}\\mathfrak {p}_i\\,\\prod _{\\alpha =1}^{n-1} \\Delta _\\alpha },$ so that the prefactor in (REF ) cancels, one arrives at $\\Phi _\\mathcal {T}({x}) =\\frac{1}{2^{n-1}}\\,\\Phi ^{\\scriptscriptstyle \\,\\int }_1(x)\\, \\Phi ^M_{n-1}(\\lbrace {u}_e\\rbrace ;{x}).$ Finally, since the differential operator in (REF ) commutes with functions of $x$ due to the orthogonality of ${v}_{ij}$ and ${t}$ , one can write the kernel $\\Phi ^{\\scriptscriptstyle \\,\\int }_n$ as $\\Phi ^{\\scriptscriptstyle \\,\\int }_n({x})=\\frac{\\Phi ^{\\scriptscriptstyle \\,\\int }_1(x)}{2^{n-1} n!",
"}\\sum _{\\mathcal {T}\\in \\, {T}_n^\\ell }\\left[\\prod _{e\\in E_\\mathcal {T}}\\mathcal {D}( {v}_{s(e) t(e)})\\right]\\Phi ^M_{n-1}(\\lbrace {u}_\\alpha \\rbrace ;{x}).$ which is the same as (REF )."
],
[
"Proof of Lemma 2",
"We start by proving that the vectors $\\tilde{{u}}_\\alpha $ form an orthogonal basis in the subspace spanned by ${u}_\\alpha $ .",
"Since the orthogonality is ensured by construction based on a rooted binary tree, it remains to show that any vector ${u}_\\alpha $ can be decomposed as a linear combination of $\\tilde{{u}}_\\alpha $ .",
"To this end, we show that the determinant of the Gram matrix constructed from the set of vectors $\\lbrace \\tilde{{u}}_\\alpha \\rbrace _{\\alpha =1}^{n-1}\\cup \\lbrace {u}_\\beta \\rbrace $ vanishes.",
"This requires to calculate the scalar product $(\\tilde{{u}}_\\alpha , {u}_\\beta )$ which can be done using $({u}_{ij},{u}_{kl})=\\left\\lbrace \\begin{array}{ll}\\mathfrak {p}_i\\mathfrak {p}_j\\mathfrak {p}_{i+j},\\qquad & i=k,\\ j=l,\\\\\\mathfrak {p}_i\\mathfrak {p}_j\\mathfrak {p}_{l},\\qquad & i=k,\\ j\\ne l,\\\\0,\\qquad & i,j\\ne k,l.\\end{array}\\right.$ Summing $i,j,k,l$ over appropriate subsets, it is immediate to see that $(\\tilde{{u}}_\\alpha , {u}_\\beta )=0$ if $\\mathcal {T}_{\\alpha }\\subset \\mathcal {T}_\\beta ^s$ or $\\mathcal {T}_\\beta ^t$ , which are the two trees obtained by dividing the tree $\\mathcal {T}$ into two parts by cutting the edge $e_\\alpha $ .",
"In other words, it is non-vanishing only if $e_\\beta \\subseteq \\mathcal {T}_\\alpha $ .",
"Then there are two cases which give $(\\tilde{{u}}_\\alpha , {u}_\\beta )=\\left\\lbrace \\begin{array}{ll}\\mathfrak {p}_\\alpha ^L\\,\\mathfrak {p}_\\alpha ^R\\,\\mathfrak {p}, \\quad &\\alpha =\\beta ,\\\\\\mathfrak {p}_{\\alpha \\beta }^{Ls}\\,\\mathfrak {p}_\\alpha ^R\\,\\mathfrak {p}_\\beta ^t-\\mathfrak {p}_{\\alpha \\beta }^{Lt}\\,\\mathfrak {p}_\\alpha ^R\\,\\mathfrak {p}_\\beta ^s+\\mathfrak {p}_{\\alpha }^{L}\\,\\mathfrak {p}_\\alpha ^R\\,\\mathfrak {p}_\\beta ^s=\\mathfrak {p}_{\\alpha \\beta }^{Ls}\\,\\mathfrak {p}_\\alpha ^R\\,\\mathfrak {p},\\quad & e_\\beta \\subset \\mathcal {T}_\\alpha ,\\end{array}\\right.$ where we introduced $\\mathfrak {p}_\\alpha ^L=\\sum _{i\\in \\mathcal {I}_{L(v_\\alpha )}}\\mathfrak {p}_i,\\qquad \\mathfrak {p}_{\\beta }^{s}=\\sum _{i\\in V_{\\mathcal {T}_\\beta ^s}}\\mathfrak {p}_i,\\qquad \\mathfrak {p}_{\\alpha \\beta }^{Ls}=\\sum _{i\\in \\mathcal {I}_{L(v_\\alpha )}\\cap V_{\\mathcal {T}_\\beta ^s}}\\mathfrak {p}_i,\\qquad \\mathfrak {p}_{\\alpha \\beta }^{st}=\\sum _{i\\in V_{\\mathcal {T}_\\alpha ^s}\\cap V_{\\mathcal {T}_\\beta ^t}}\\mathfrak {p}_i,$ and similarly for variables with labels $R$ and $t$ .",
"In (REF ) in the second case we assumed that the orientation of edges is such that $e_\\beta \\subset \\mathcal {T}_\\alpha ^s$ and $e_\\alpha \\subset \\mathcal {T}_\\beta ^t$ .",
"If this is not the case, one should replace $s$ by $t$ , $L$ by $R$ and flip the sign for each change of orientation.",
"Below we use the same assumption, but the computation can easily be generalized to a more general situation.",
"Given the result (REF ), $(\\tilde{{u}}_\\alpha , \\tilde{{u}}_\\beta )=\\Delta _\\alpha \\delta _{\\alpha \\beta }$ and $({u}_\\beta , {u}_\\beta )=\\mathfrak {p}_\\beta ^s\\,\\mathfrak {p}_\\beta ^t\\,\\mathfrak {p}$ , the determinant of the Gram matrix is easily found to be $\\,{\\rm det}\\, {\\rm Gram}(\\tilde{{u}}_1,\\cdots , \\tilde{{u}}_{n-1},{u}_\\beta )=\\mathfrak {p}\\prod _{\\alpha =1}^{n-1}\\Delta _\\alpha \\left[\\mathfrak {p}_\\beta ^s\\,\\mathfrak {p}_\\beta ^t-\\mathfrak {p}\\sum _{T_\\alpha \\supseteq e_\\beta }\\frac{(\\mathfrak {p}_{\\alpha \\beta }^{Ls})^2\\,\\mathfrak {p}_\\alpha ^R}{\\mathfrak {p}_{v_\\alpha }\\mathfrak {p}_\\alpha ^L } \\right].$ Note that the subtrees $\\mathcal {T}_\\alpha $ containing the edge $e_\\beta $ form an ordered set so that the sum in the square brackets goes over $\\alpha _\\ell $ , $\\ell =1,\\dots , m$ , such that $\\mathcal {T}_{\\alpha _\\ell }\\subset \\mathcal {T}_{\\alpha _{\\ell +1}}$ .",
"The first element of this set $\\alpha _1=\\beta $ , whereas the last corresponds to the total tree, $\\mathcal {T}_{\\alpha _m}=\\mathcal {T}$ .",
"Due to $\\mathfrak {p}_{\\alpha _m}^L=\\mathfrak {p}_{\\alpha _m}^s$ , $\\mathfrak {p}_{\\alpha _m}^R=\\mathfrak {p}_{\\alpha _m}^t$ and $\\mathfrak {p}_{\\alpha _m\\beta }^{Ls}=\\mathfrak {p}_\\beta ^s$ , the first term in the square brackets together with the term in the sum corresponding to $\\alpha _m$ gives $\\mathfrak {p}_\\beta ^s\\left(\\mathfrak {p}_\\beta ^t -\\frac{\\mathfrak {p}_\\beta ^s\\,\\mathfrak {p}_{\\alpha _m}^t}{\\mathfrak {p}_{\\alpha _m}^s}\\right)=\\frac{\\mathfrak {p}\\,\\mathfrak {p}_\\beta ^s\\,\\mathfrak {p}_{\\alpha _m\\beta }^{st}}{\\mathfrak {p}_{\\alpha _m}^s},$ where we have used that $\\mathfrak {p}_\\beta ^t=\\mathfrak {p}_{\\alpha _m\\beta }^{st}+\\mathfrak {p}_{\\alpha _m}^t$ and $\\mathfrak {p}_{\\alpha _m}^s=\\mathfrak {p}_{\\alpha _m\\beta }^{st}+\\mathfrak {p}_\\beta ^s$ .",
"Thus, the expression in the square brackets in (REF ) becomes $\\frac{\\mathfrak {p}}{\\mathfrak {p}_{\\alpha _m}^s}\\left[\\mathfrak {p}_\\beta ^s\\,\\mathfrak {p}_{\\alpha _m\\beta }^{st}-\\mathfrak {p}_{\\alpha _m}^s\\sum _{\\ell =1}^{m-1}\\frac{(\\mathfrak {p}_{\\alpha _\\ell \\beta }^{Ls})^2\\,\\mathfrak {p}_{\\alpha _\\ell }^R}{\\mathfrak {p}_{v_{\\alpha _\\ell }}\\mathfrak {p}_{\\alpha _\\ell }^L }\\right].$ The new expression in the square brackets is exactly the same as in (REF ) where tree $\\mathcal {T}$ was replaced by subtree $\\mathcal {T}_{\\alpha _m}^s=\\mathcal {T}_{\\alpha _{m-1}}$ .",
"Thus, one can repeat the above manipulation until one exhausts all terms in the sum.",
"As a result, the determinant of the Gram matrix turns out to be proportional to $\\mathfrak {p}_{\\alpha _1\\beta }^{st}$ .",
"But since $\\alpha _1=\\beta $ , this quantity, and hence the whole determinant, trivially vanish.",
"Next, we prove that (REF ) is consistent with $\\mathcal {B}\\cdot {x}=\\left\\lbrace \\frac{w^\\alpha }{\\sqrt{\\Delta _\\alpha }}\\right\\rbrace $ and $\\mathcal {B}\\cdot \\mathcal {V}=\\mathcal {M}$ .",
"The first relation is a direct consequence of (REF ).",
"The second relation requires to show that $(\\tilde{{u}}_\\alpha , {u}_\\beta )=\\mathfrak {p}\\,\\Delta _\\alpha ^{1/2}\\Delta _\\beta ^{-1/2}\\mathcal {M}_{\\alpha \\beta }$ or equivalently $\\sum _{\\gamma }(\\tilde{{u}}_\\alpha , {u}_\\gamma ) \\sqrt{\\Delta _\\gamma \\Delta _\\beta }\\mathcal {M}_{\\gamma \\beta }^{-1}=\\mathfrak {p}\\,\\Delta _\\alpha \\delta _{\\alpha \\beta }$ .",
"Using the result (REF ) for the matrix $\\mathcal {M}_{\\gamma \\beta }^{-1}$ , this relation can be written explicitly as $\\mathfrak {p}_{v_\\beta }(\\tilde{{u}}_\\alpha , {u}_\\beta )+\\left(\\sum _{e_\\gamma \\in E_\\beta ^R}\\epsilon _{\\gamma \\beta } \\mathfrak {p}_{L(v_\\beta )}+\\sum _{e_\\gamma \\in E_\\beta ^L}\\epsilon _{\\gamma \\beta } \\mathfrak {p}_{R(v_\\beta )}\\right)(\\tilde{{u}}_\\alpha , {u}_\\gamma )=\\mathfrak {p}\\,\\Delta _\\alpha \\delta _{\\alpha \\beta },$ where $E_{\\beta }^{L}=\\lbrace e\\in E_\\mathcal {T}:\\ e\\cap \\mathcal {T}_{\\beta _L}\\ne \\emptyset , \\ e\\nsubseteq \\mathcal {T}_\\beta \\rbrace $ and similarly for $E_{\\beta }^{R}$ .",
"Consider first the case $\\alpha =\\beta $ .",
"From (REF ), it immediately follows that the first term gives $\\mathfrak {p}\\,\\Delta _\\alpha $ .",
"On the other hand, the second contribution sums over edges for which $\\mathcal {T}_\\alpha \\subseteq \\mathcal {T}_\\gamma ^s$ or $\\mathcal {T}_\\gamma ^t$ , and as noted above (REF ) this leads to vanishing of the scalar product.",
"Thus, in this case the relation (REF ) indeed holds.",
"Let us now show that it holds as well for $\\alpha \\ne \\beta $ .",
"To this end, one should consider several options.",
"If $\\mathcal {T}_\\alpha \\cap \\mathcal {T}_\\beta =\\emptyset $ , then $\\mathcal {T}_\\alpha \\subset \\mathcal {T}_\\beta ^s$ or $\\mathcal {T}_\\beta ^t$ which implies vanishing of the first term.",
"But the second term vanishes as well because the conditions $e_\\gamma \\subseteq \\mathcal {T}_\\alpha $ and $e_\\gamma \\cap \\mathcal {T}_\\beta \\ne \\emptyset $ are inconsistent with $\\mathcal {T}_\\alpha \\cap \\mathcal {T}_\\beta =\\emptyset $ .",
"Similarly, if $\\mathcal {T}_\\alpha \\subset \\mathcal {T}_\\beta $ , one has $\\mathcal {T}_\\alpha \\subset \\mathcal {T}_\\beta ^s$ or $T_\\beta ^t$ which again leads to the vanishing of the first term, whereas the vanishing of the second is a consequence of that $e_\\gamma \\subseteq \\mathcal {T}_\\alpha $ implies $e_\\gamma \\subset \\mathcal {T}_\\beta $ so that the sum over $e_\\gamma $ is empty.",
"It remains to consider the case $\\mathcal {T}_\\beta \\subset \\mathcal {T}_\\alpha $ .",
"It is clear that $\\mathcal {T}_\\beta \\subset \\mathcal {T}_\\alpha ^s$ or $\\mathcal {T}_\\alpha ^t$ .",
"Without loss of generality, let us assume that $\\mathcal {T}_\\beta \\subset \\mathcal {T}_\\alpha ^s$ and $e_\\alpha \\subset \\mathcal {T}_\\beta ^t$ .",
"Then according to (REF ), the first term gives $\\mathfrak {p}_{v_\\beta }\\,\\mathfrak {p}_{\\alpha \\beta }^{Ls}\\,\\mathfrak {p}_\\alpha ^R\\,\\mathfrak {p}$ .",
"If instead we have chosen the orientation such that $e_\\alpha \\subset \\mathcal {T}_\\beta ^s$ , then we would find $-\\mathfrak {p}_{v_\\beta }\\,\\mathfrak {p}_{\\alpha \\beta }^{Lt}\\,\\mathfrak {p}_\\alpha ^R\\,\\mathfrak {p}$ .",
"Similar results are obtained for each term in the sum of the second contribution.",
"Again without loss of generality we assume that for all relevant edges $e_\\gamma $ one has $e_\\alpha \\subset \\mathcal {T}_\\gamma ^t$ , otherwise one flips their orientation.",
"Then the l.h.s.",
"of (REF ) is proportional to $\\mathfrak {p}_{v_\\beta }\\,\\mathfrak {p}_{\\alpha \\beta }^{Ls}+\\sum _{e_\\gamma \\in E_{\\beta }^{R}}\\epsilon _{\\gamma \\beta } \\mathfrak {p}_\\beta ^L\\,\\mathfrak {p}_{\\alpha \\gamma }^{Ls}+\\sum _{e_\\gamma \\in E_{\\beta }^{L}}\\epsilon _{\\gamma \\beta } \\mathfrak {p}_\\beta ^R \\,\\mathfrak {p}_{\\alpha \\gamma }^{Ls}.$ Let $e_\\alpha ^\\star \\in E_{\\beta }^{R}$ is the edge belonging to the path from $\\mathcal {T}_\\beta $ to $e_\\alpha $ (which may coincide with $e_\\alpha $ ).",
"Then our choice of orientation implies that $\\epsilon _{\\gamma \\beta }=-1$ for $e_\\gamma \\in E_\\beta ^L\\cup \\lbrace e_\\alpha ^\\star \\rbrace $ and $\\epsilon _{\\gamma \\beta }=1$ for $e_\\gamma \\in E_\\beta ^R\\setminus \\lbrace e_\\alpha ^\\star \\rbrace $ .",
"Furthermore, one has $\\mathfrak {p}_{\\alpha \\gamma _\\alpha ^\\star }^{Ls}=\\mathfrak {p}_{v_\\beta }+\\sum _{e_\\gamma \\in E_\\beta ^L\\cup E_\\beta ^R\\setminus \\lbrace e_\\alpha ^\\star \\rbrace }\\mathfrak {p}_{\\alpha \\gamma }^{Ls},\\qquad \\mathfrak {p}_{\\alpha \\beta }^{Ls}=\\mathfrak {p}_\\beta ^L+\\sum _{e_\\gamma \\in E_\\beta ^L}\\mathfrak {p}_{\\alpha \\gamma }^{Ls}.$ As a result, one finds $(\\mathfrak {p}_\\beta ^L+\\mathfrak {p}_\\beta ^R)\\left(\\mathfrak {p}_\\beta ^L+\\sum _{e_\\gamma \\in E_\\beta ^L}\\mathfrak {p}_{\\alpha \\gamma }^{Ls}\\right)-\\mathfrak {p}_\\beta ^L \\,\\mathfrak {p}_{\\alpha \\gamma _\\alpha ^\\star }^{Ls}+\\sum _{e_\\gamma \\in E_{\\beta }^{R}\\setminus \\lbrace e_\\alpha ^\\star \\rbrace } \\mathfrak {p}_\\beta ^L\\,\\mathfrak {p}_{\\alpha \\gamma }^{Ls}-\\sum _{e_\\gamma \\in E_{\\beta }^{L}}\\mathfrak {p}_\\beta ^R \\,\\mathfrak {p}_{\\alpha \\gamma }^{Ls}=0.$ This completes the proof of the required statement (REF ).",
"toc"
],
[
"Proofs of propositions",
"In this appendix we prove several propositions which we stated in the main text."
],
[
"Proposition ",
"To prove the recursive equation (REF ), let us note that the tree index $g_{{\\rm tr},n}$ satisfies a very similar equation (cf.",
"(REF ) or (REF )) which can be seen as the origin of its representation (REF ) in terms of attractor flow trees.",
"The only difference is the absence of the last term in (REF ).",
"Therefore, it is easy to see that this equation implies a simple relation between $\\widehat{g}_n$ and $g_{{\\rm tr},n}$ $\\widehat{g}_n(\\lbrace \\check{\\gamma }_i\\rbrace ,z^a)=\\sum _{n_1+\\cdots +n_m= n\\atop n_k\\ge 1}g_{{\\rm tr},m}(\\lbrace \\check{\\gamma }^{\\prime }_k\\rbrace ,z^a)\\prod _{k=1}^m W_{n_k}(\\check{\\gamma }_{j_{k-1}+1},\\dots ,\\check{\\gamma }_{j_{k}}),$ where as usual we use notations from (REF ).",
"Next, we substitute this relation into the expansion (REF ).",
"The result can be represented in the following form $h^{\\rm DT}_{p,q}=\\sum _{\\sum _{i=1}^n \\check{\\gamma }_i=\\check{\\gamma }}g_{{\\rm tr},n}(\\lbrace \\check{\\gamma }_i\\rbrace ,z^a)\\,e^{\\pi I\\tau Q_n(\\lbrace \\check{\\gamma }_i\\rbrace )}\\prod _{i=1}^n h^R_{p_i,\\mu _i}(\\tau ),$ where we introduced $\\begin{split}h^{R}_{p,\\mu }=&\\, \\sum _{\\sum _{i=1}^n \\check{\\gamma }_i=\\check{\\gamma }}W_n(\\lbrace \\check{\\gamma }_i\\rbrace )\\,e^{\\pi I\\tau Q_n(\\lbrace \\check{\\gamma }_i\\rbrace )}\\prod _{i=1}^n \\widehat{h}_{p_i,\\mu _i}(\\tau )\\\\=&\\, \\sum _{\\sum _{i=1}^n \\check{\\gamma }_i=\\check{\\gamma }}\\left[\\sum _{T\\in {T}_n^{\\rm S}}(-1)^{n_T}\\prod _{v\\in V_T} R_{v}\\right]\\,e^{\\pi I\\tau Q_n(\\lbrace \\check{\\gamma }_i\\rbrace )}\\\\&\\, \\times \\prod _{i=1}^n \\left(h_{p_i,\\mu _i}+\\sum _{n_i=2}^\\infty \\sum _{\\sum _{j=1}^{n_i} \\check{\\gamma }^{\\prime }_j=\\check{\\gamma }}R_{n_i}(\\lbrace \\check{\\gamma }^{\\prime }_j\\rbrace )\\, e^{\\pi I\\tau Q_{n_i}(\\lbrace \\check{\\gamma }^{\\prime }_j\\rbrace )}\\prod _{j_i=1}^{n_i} h_{p^{\\prime }_{j_i},\\mu ^{\\prime }_{j_i}}\\right)\\end{split}$ and in the last relation we used the definition of $W_n$ (REF ) and the expansion of $\\widehat{h}_{p,\\mu }$ (REF ).",
"The crucial observation is that if one picks up a factor $R_{n_i}$ from the second line of (REF ), appearing due to the expansion of $\\widehat{h}_{p_i,\\mu _i}$ , and combines it with the contribution of a tree $T$ from the first line, one finds the opposite of the contribution of another tree obtained from $T$ by adding $n_i$ children to its $i$ th leaf.",
"As a result, all such contributions cancel and the function (REF ) reduces to the trivial term $h^{R}_{p,\\mu }=h_{p,\\mu }.$ Substituting this into (REF ), it gives back the original expansion (REF ) of the generating function of DT invariants, which proves the recursive equation (REF ).",
"Finally, let us evaluate (REF ) at the attractor point $z^a_\\infty (\\gamma )$ .",
"At this point the DT invariants coincide with MSW invariants so that the l.h.s.",
"becomes the generating function $h_{p,\\mu }$ .",
"Meanwhile, all factors $\\Delta _{\\gamma _L\\gamma _R}^z$ vanish at the attractor point and $\\widehat{g}_n$ reduces to $W_n$ .",
"As a result, the relation (REF ) reproduces (REF ), which completes the proof of the proposition.",
"We prove Proposition REF by induction.",
"For $n=2$ the recursive relation (REF ) reads $g^{(0)}_2(\\check{\\gamma }_1,\\check{\\gamma }_2;c_1)-g^{(0)}_2(\\check{\\gamma }_1,\\check{\\gamma }_2;\\beta _{21})=-\\frac{1}{4}\\, \\bigl (\\mbox{sgn}(c_1)-\\mbox{sgn}(\\beta _{21}) \\bigr )\\, \\kappa (\\gamma _{12}),$ where we took into account that $S_1=c_1$ and $\\Gamma _{21}=\\beta _{21}=-\\gamma _{12}$ .",
"Since $g^{(0)}_2$ is supposed to have discontinuities only at walls of marginal stability, it must not involve signs of DSZ products.",
"Therefore, we are led to take $g^{(0)}_2(\\check{\\gamma }_1,\\check{\\gamma }_2;c_1)=-\\frac{1}{4}\\, \\mbox{sgn}(c_1) \\kappa (\\gamma _{12}).$ Then (REF ) and (REF ) imply $\\mathcal {E}_2=\\frac{1}{4}\\,\\mbox{sgn}(\\gamma _{12})\\,\\kappa (\\gamma _{12})+R_2,$ so that the ansatz (REF ) reads $\\widehat{g}_2(\\check{\\gamma }_1,\\check{\\gamma }_2;c_1)=g^{(0)}_2(\\check{\\gamma }_1,\\check{\\gamma }_2;c_1)-\\mathcal {E}_2(\\check{\\gamma }_1,\\check{\\gamma }_2)=-\\frac{1}{4}\\,\\Bigl [\\mbox{sgn}(c_1)+\\mbox{sgn}(\\gamma _{12})\\Bigr ]\\,\\kappa (\\gamma _{12})-R_2(\\check{\\gamma }_1,\\check{\\gamma }_2),$ which reproduces the recursive equation (REF ).",
"Furthermore, in appendix REF it is shown that there is a smooth solution of Vignéras' equation which asymptotes the function $({v},{x}){\\rm sgn}({v},{x})$ , coinciding with the (rescaled) first term in $\\mathcal {E}_2$ for ${v}={v}_{12}$ (REF ).",
"It is given by $\\widetilde{\\Phi }_1^E({v},{v};{x})={v}\\cdot \\left({x}+\\frac{1}{2\\pi }\\,\\partial _{x}\\right)\\operatorname{Erf}\\left(\\frac{\\sqrt{\\pi }{v}\\cdot {x}}{|{v}|}\\right),$ which corresponds to the following choice of $R_2$ [14] $R_2=\\frac{(-1)^{\\gamma _{12}}}{8\\pi }\\, |\\gamma _{12}|\\,\\beta _{\\frac{3}{2}}\\!\\left({\\frac{2\\tau _2\\gamma _{12}^2 }{(pp_1p_2)}}\\right),$ where $\\beta _{\\frac{3}{2}}(x^2)=2|x|^{-1}e^{-\\pi x^2}-2\\pi \\text{Erfc}(\\sqrt{\\pi } |x|)$ .",
"Note that the resulting $\\mathcal {E}_2$ depends on the electric charges only through the DSZ product $\\gamma _{12}$ .",
"Finally, it is immediate to see that the kernel $\\Phi ^{\\,\\widehat{g}}_2$ (REF ) satisfies (REF ).",
"Now we assume that (REF ) is consistent with the ansatz (REF ) for all orders up to $n-1$ and check it at order $n$ .",
"Denoting the second term in (REF ) by $\\widehat{g}^{(+)}_n$ and substituting this ansatz into the r.h.s.",
"of (REF ), one finds $\\qquad \\qquad \\,{\\rm Sym}\\, \\left\\lbrace \\sum _{\\ell =1}^{n-1}g_2^\\ell \\,\\Bigl [g^{(0)}_\\ell \\,g^{(0)}_{n-\\ell }-\\widehat{g}^{(+)}_\\ell \\,\\widehat{g}_{n-\\ell }-\\widehat{g}_\\ell \\,\\widehat{g}^{(+)}_{n-\\ell }-\\widehat{g}^{(+)}_\\ell \\,\\widehat{g}^{(+)}_{n-\\ell }\\Bigr ]_{c_i\\rightarrow c_i^{(\\ell )}}\\right\\rbrace +W_n,$ where $g_2^\\ell =-{1\\over 2}\\,\\Delta _{\\gamma _L^\\ell \\gamma _R^\\ell }^z\\,\\kappa (\\gamma _{LR}^\\ell )=\\frac{1}{4}\\, \\bigl (\\mbox{sgn}(S_\\ell )-\\mbox{sgn}(\\Gamma _{n\\ell })\\bigr )\\,\\kappa (\\Gamma _{n\\ell }).$ In the first term proportional to $g^{(0)}_\\ell \\,g^{(0)}_{n-\\ell }$ , one can apply the relation (REF ), which together with (REF ) gives $g^{(0)}_n-\\mathcal {E}^{(0)}_n$ .",
"The other terms in the sum over $\\ell $ can be reorganized as follows $\\begin{split}&\\,-\\,{\\rm Sym}\\, \\left\\lbrace \\sum _{\\ell =1}^{n-1} g_2^\\ell \\sum _{n_1+\\cdots +n_m= n\\atop n_k\\ge 1, \\ m<n, \\ \\ell \\in \\lbrace j_k\\rbrace } \\widehat{g}_{k_0}(c_i^{(\\ell )})\\,\\widehat{g}_{m-k_0}(c_i^{(\\ell )})\\prod _{k=1}^m \\mathcal {E}_{n_k}\\right\\rbrace \\\\=&\\,-\\,{\\rm Sym}\\, \\left\\lbrace \\sum _{n_1+\\cdots +n_m= n\\atop n_k\\ge 1, \\ 1<m<n}\\left[\\,\\sum _{k_0=1}^{m-1} g_2^{j_{k_0}}\\,\\widehat{g}_{k_0}(c_i^{(\\ell )})\\, \\widehat{g}_{m-k_0}(c_i^{(\\ell )})\\,\\right]\\prod _{k=1}^m \\mathcal {E}_{n_k}\\right\\rbrace .\\end{split}$ Here we first combined three contributions into one sum over splittings by adding the condition $\\ell \\in \\lbrace j_k\\rbrace $ , with $k_0$ being the index for which $j_{k_0}=\\ell $ , and then interchanged the two sums which allows to drop the condition $\\ell \\in \\lbrace j_k\\rbrace $ , but adds the requirement $m>1$ (following from $\\ell \\in \\lbrace j_k\\rbrace $ in the previous representation).",
"In square brackets one recognizes the first contribution from the r.h.s.",
"of (REF ) with $n$ replaced by $m<n$ .",
"Hence, it is subject to the induction hypothesis which allows to replace this expression by $\\widehat{g}_{m}(\\lbrace \\gamma ^{\\prime }_k\\rbrace ,z^a)-W_m(\\lbrace \\gamma ^{\\prime }_k\\rbrace )$ .",
"Combining all contributions together, one concludes that (REF ) is equal to $g^{(0)}_n-\\mathcal {E}^{(0)}_n-\\,{\\rm Sym}\\, \\left\\lbrace \\sum _{n_1+\\cdots +n_m= n\\atop n_k\\ge 1, \\ 1<m<n}\\bigl (\\widehat{g}_{m}-W_m\\bigr )\\prod _{k=1}^m \\mathcal {E}_{n_k}\\right\\rbrace +W_n.$ The contributions involving $W$ 's can be combined into one sum by dropping the condition $m<n$ .",
"The resulting sum coincides with the r.h.s.",
"of (REF ) so that these contributions can be replaced by $-\\mathcal {E}^{(+)}_n$ .",
"Combined with $-\\mathcal {E}^{(0)}_n$ , this gives $-\\mathcal {E}_n$ and allows to drop the condition $m>1$ in the remaining sum with $\\widehat{g}_{m}$ .",
"As a result, (REF ) becomes equivalent to the r.h.s.",
"of (REF ), which proves the consistency of this ansatz with the recursive equation.",
"Finally, let us show that the ansatz satisfies the modularity constraint (REF ).",
"The crucial observation is that the vectors ${v}_{ij}$ and ${u}_{ij}$ (REF ) satisfy $({v}_{i+j,k},{v}_{ij})=({u}_{i+j,k},{v}_{ij})=0,$ where we abused notation and denoted ${v}_{i+j,k}={v}_{ik}+{v}_{jk}$ , etc.",
"These orthogonality relations, together with the assumption that $\\mathcal {E}_n$ depend on electric charges only through the DSZ products $\\gamma _{ij}\\sim ({v}_{ij},{x})$ , imply factorization of the action of Vignéras' operator on the kernel corresponding to the second term in (REF ).",
"Indeed, all contributions of $\\partial _{x}^2$ where two derivatives act on different factors vanish and the action reduces to the sum of terms where Vignéras' operator acts on one of the factors.",
"But since it is supposed to vanish on $\\Phi ^{\\,\\mathcal {E}}_n$ , one obtains the simple result $V_{n-1} \\cdot \\Phi ^{\\,\\widehat{g}}_{n}=V_{n-1} \\cdot \\Phi ^{\\,g^{(0)}}_{n}-\\,{\\rm Sym}\\, \\left\\lbrace \\sum _{n_1+\\cdots +n_m= n\\atop n_k\\ge 1, \\ m<n}(V_{m-1}\\cdot \\Phi ^{\\,\\widehat{g}}_m)\\prod _{k=1}^m \\Phi ^{\\,\\mathcal {E}}_{n_k}\\right\\rbrace .$ Since for $n=2$ the constraint was already shown to hold, one can proceed by induction.",
"Then in the second term one can substitute the r.h.s.",
"of (REF ), whereas the first term can be evaluated using the recursive relation (REF ).",
"First of all, by the same reasoning as above, away from discontinuities, the action of Vignéras' operator is factorized and actually vanishes.",
"Furthermore, since $g^{(0)}_n$ have discontinuities only at walls of marginal stability, to obtain the complete action, it is enough to consider it only on $\\mbox{sgn}(S_\\ell )$ .",
"Since at $S_\\ell =0$ one has $c^{(\\ell )}_i=c_i$ (see (REF )), one finds that $\\Phi ^{\\,g^{(0)}}_n$ satisfy exactly the same constraint as (REF ).",
"Thus, one can rewrite (REF ) as $V_{n-1} \\cdot \\Phi ^{\\,\\widehat{g}}_{n}&=&\\,{\\rm Sym}\\, \\sum _{\\ell =1}^{n-1}\\Bigl ({u}_\\ell ^2\\,\\Delta _{n,\\ell }^{g^{(0)}} \\,\\delta ^{\\prime }({u}_\\ell \\cdot {x})+ 2{u}_\\ell \\cdot \\partial _{x}\\Delta _{n,\\ell }^{g^{(0)}} \\,\\delta ({u}_\\ell \\cdot {x})\\Bigr )\\\\&&-\\,{\\rm Sym}\\, \\left\\lbrace \\sum _{n_1+\\cdots +n_m= n\\atop n_k\\ge 1, \\ 1<m<n}\\left[\\sum _{\\ell =1}^{n-1}\\Bigl ({u}_\\ell ^2\\,\\Delta _{m,\\ell }^{\\widehat{g}} \\,\\delta ^{\\prime }({u}_\\ell \\cdot {x})+2{u}_\\ell \\cdot \\partial _{x}\\Delta _{m,\\ell }^{\\widehat{g}} \\,\\delta ({u}_\\ell \\cdot {x})\\Bigr )\\right]\\prod _{k=1}^m \\Phi ^{\\,\\mathcal {E}}_{n_k}\\right\\rbrace .\\nonumber $ Note that the orthogonality relation allows to include $\\Phi ^{\\,\\mathcal {E}}_{n_k}$ under the derivative in the last term.",
"Then one can perform the same manipulations with the sum over splittings as in (REF ) but in the inverse direction, which directly leads to the constraint (REF ).",
"Our goal is to find an explicit expression for $\\widetilde{\\Phi }^{(0)}_n$ .",
"Using the fact that the limit of large ${x}$ of the generalized error function $\\Phi _{n}^E$ is the product of $n$ sign functions, one immediately obtains $\\widetilde{\\Phi }^{(0)}_n({x})=\\frac{1}{2^{n-1} n!",
"}\\sum _{\\mathcal {T}\\in \\, {T}_n^\\ell }\\left[\\prod _{e\\in E_\\mathcal {T}}\\mathcal {D}( {v}_{s(e) t(e)})\\right]\\left[\\prod _{e\\in E_{\\mathcal {T}}}{\\rm sgn}({u}_e,{x})\\right],$ where $\\mathcal {D}({v})$ are the covariant derivative operators (REF ).",
"The action of the derivatives $\\partial _{x}$ on the sign functions can be ignored (since the original function is smooth), however there are additional contributions due to the mutual action of the operators $\\mathcal {D}$ .",
"Similar contributions were discussed in a similar context in appendix , where they were shown to cancel, but here they turn out to leave a finite remainder.",
"The contribution generated by the mutual action of two operators $\\mathcal {D}( {v}_{s(e) t(e)})$ is non-vanishing only if the two edges $e_1,e_2$ have a common vertex.",
"In this case it contributes the factor $\\frac{(p_{\\mathfrak {v}_1}p_{\\mathfrak {v}_2}p_{\\mathfrak {v}_{3}})}{2\\pi }\\, {\\rm sgn}({u}_{e_1},{x})\\, {\\rm sgn}({u}_{e_2},{x}) ,$ where $\\mathfrak {v}_1,\\mathfrak {v}_2,\\mathfrak {v}_3$ are the tree vertices joint by the edges $e_1=(\\mathfrak {v}_2,\\mathfrak {v}_3)$ , $e_2=(\\mathfrak {v}_1,\\mathfrak {v}_3)$ .",
"Again, considering the three trees shown in Fig.",
"REF , one can note that the vectors ${u}_e$ defined by these trees satisfy the following relations: ${u}_{e_1}^{(1)}=-{u}_{e_3}^{(2)}$ , ${u}_{e_2}^{(1)}={u}_{e_3}^{(3)}$ , ${u}_{e_2}^{(2)}={u}_{e_1}^{(3)}$ and ${u}_{e_2}^{(2)}={u}_{e_1}^{(1)}+{u}_{e_2}^{(1)}$ , where $e_3=(\\mathfrak {v}_1,\\mathfrak {v}_2)$ and we indicated by upper index the tree with respect to which the vector is defined.",
"Therefore, the contributions generated by these trees combine into $&& \\frac{(p_{\\mathfrak {v}_1}p_{\\mathfrak {v}_2}p_{\\mathfrak {v}_{3}})}{2\\pi }\\Bigl [ {\\rm sgn}({u}_{e_1}^{(1)},{x})\\, {\\rm sgn}({u}_{e_2}^{(1)},{x})+{\\rm sgn}({u}_{e_2}^{(2)},{x})\\, {\\rm sgn}({u}_{e_3}^{(2)},{x})-{\\rm sgn}({u}_{e_1}^{(3)},{x})\\, {\\rm sgn}({u}_{e_3}^{(3)},{x})\\Bigr ]\\nonumber \\\\&=& -\\frac{(p_{\\mathfrak {v}_1}p_{\\mathfrak {v}_2}p_{\\mathfrak {v}_{3}})}{2\\pi },$ where we used the above relations between the vectors and the sign identity (REF ) for $x_s=({u}_{e_s}^{(1)},{x})$ , $s=1,2$ , Thus, unlike in (REF ), we now get a non-vanishing result, due to the mutual action of derivative operators.",
"Each pair of intersecting edges leads to a contribution which recombines the contributions of the edges from the three trees into a single factor (REF ).",
"Furthermore, the sum over trees implies that we have to sum over all possible subtrees $\\mathcal {T}_1,\\mathcal {T}_2,\\mathcal {T}_3$ , in particular, over all possible allocations of different subtrees to the vertices $\\mathfrak {v}_1,\\mathfrak {v}_2,\\mathfrak {v}_3$ .",
"The sign factors ${\\rm sgn}({u}_e,{x})$ for edges of these subtrees do not depend on this allocation, but the factors $\\mathcal {D}( {v}_{s(e) t(e)})$ do depend for the edges connecting to one of these vertices.",
"It is easy to see that the sum over allocations effectively replaces the three vertices by a single one labelled by the total charge $p_{\\mathfrak {v}_1}+p_{\\mathfrak {v}_2}+p_{\\mathfrak {v}_{3}}$ .",
"As a result, one obtains that the function (REF ) can be represented as a sum over marked trees where a mark corresponds to a collapse of a pair of intersecting edges and contributes the factor (REF ).",
"More precisely, one has $\\widetilde{\\Phi }^{(0)}_n({x})=\\frac{1}{2^{n-1} n!",
"}\\sum _{m=0}^{[(n-1)/2]}\\frac{(-1)^m}{(2\\pi )^m}\\sum _{\\mathcal {T}\\in \\, {T}_{n-2m,m}^\\ell }\\left[\\prod _{\\mathfrak {v}\\in V_\\mathcal {T}}\\mathcal {P}_{m_\\mathfrak {v}}(\\lbrace p_{\\mathfrak {v},s}\\rbrace )\\right]\\prod _{e\\in E_{\\mathcal {T}}}({v}_{s(e) t(e)},{x})\\,{\\rm sgn}({u}_e,{x}),$ where $\\mathcal {P}_{m}(\\lbrace p_{s}\\rbrace )=\\sum _{\\mathcal {I}_1\\cup \\cdots \\cup \\mathcal {I}_{m}={Z}_{2m+1}}\\, \\prod _{j=1}^{m}(p_{j_1}p_{j_2}p_{j_3}).$ This factor collects the weights (REF ) assigned to a vertex due to collapse of $m$ pairs of edges.",
"It is represented as a sum over all possible splittings of the set ${Z}_{2m+1}=\\lbrace 1,\\dots ,2m+1\\rbrace $ into union of triples $\\mathcal {I}_j=\\lbrace j_1,j_2,j_3\\rbrace $ such that the labels in one triple $\\mathcal {I}_j$ are all different, two different triples can have at most one common label, and there are no closed cycles in the sense that there are no subsets $\\lbrace {j_k}\\rbrace _{k=1}^r$ such that $\\mathcal {I}_{j_k}\\cap \\mathcal {I}_{j_{k+1}}\\ne \\emptyset $ where $j_{r+1}\\equiv j_1$ .",
"This sum simply counts all possible splittings of a collection of $2m$ joint edges into $m$ intersecting pairs, suppressing for each pair the distinction between the three configurations of Fig.",
"REF .",
"However, the representation (REF ) is not very convenient for our purposes.",
"An alternative representation can be obtained by noting that, instead of collapsing all edges at once, one can collapse first one pair, sum over all configurations (i.e.",
"allocations of subtrees), then collapse another pair, and so on.",
"In this approach at each step the sum over different configurations ensures that all factors assigned to the elements of the surviving tree depend only on the sum of collapsing charges.",
"As a result, one obtains a hierarchical structure described by a rooted ternary tree $T$ , where the leaves correspond to the vertices of the original unrooted tree which have collapsed into one vertex with $m_\\mathfrak {v}$ marks corresponding to the root of $T$ .",
"The other vertices of $T$ are then in one-to-one correspondence with marked vertices appearing at intermediate stages of the above process.",
"This procedure gives rise to the representation (REF ) of the weight factor $\\mathcal {P}_m$ in terms of a sum over rooted ternary trees.",
"A non-trivial point which must be taken into account is that the procedure leading to this representation overcounts different configurations.",
"As a result, each tree is weighted by the rational coefficient $N_{\\hat{T}}/m!$ where $\\hat{T}$ is the rooted tree obtained from $T$ by dropping all leaves and $N_{\\hat{T}}$ is the number of ways of labelling the vertices of $\\hat{T}$ with increasing labels, which already appeared in appendix .",
"Here the numerator takes into account that the tree $T$ is generated $N_{\\hat{T}}$ times in the collapsing process, whereas the denominator removes the overcounting produced by specifying the order in which the $m$ pairs of edges are collapsed.",
"Finally, we apply Lemma REF from appendix .",
"Since $n_{\\hat{T}}=m$ , the coefficients coincides with the inverse of the tree factorial (REF ).",
"Taking into account that substitution of (REF ) into (REF ) gives exactly (REF ), this completes the proof of the proposition.",
"Our aim is to show that the recursive formula (REF ) solves the equations (REF ).",
"We will proceed by induction, starting with the case $n=3$ where there is a single unrooted tree and the system (REF ) contains a single equation corresponding to the trivial trees $\\mathcal {T}_r$ consisting of one vertex.",
"Thus, it is solved by $a_{\\bullet \\!\\mbox{-}\\!\\bullet \\!\\mbox{-}\\!\\bullet }=\\frac{1}{3}$ consistently with (REF ).",
"Let us now consider trees with $n$ vertices assuming that for all trees with less number of vertices (REF ) holds.",
"We start by noting that for every vertex $\\mathfrak {v}\\in V_{\\mathcal {T}_r}$ , among the subtrees $\\mathcal {T}_{r,s}(\\mathfrak {v})\\subset \\mathcal {T}_r$ obtained by removing the vertex $\\mathfrak {v}$ , there is one which contains $\\mathfrak {v}_r$ , which we denote by $\\mathcal {T}_{r,s_0}(\\mathfrak {v})$ .",
"(If $\\mathfrak {v}=\\mathfrak {v}_r$ , we take $\\mathcal {T}_{r,s_0}(\\mathfrak {v})=\\emptyset $ .)",
"Since every tree $\\hat{\\mathcal {T}}_r$ is a union of the three trees $\\mathcal {T}_r$ , substituting (REF ) into the l.h.s.",
"of (REF ), one obtains that the sum over vertices of $\\hat{\\mathcal {T}}_r$ can be represented as $\\frac{1}{n}\\sum _{r=1}^3 \\sum _{\\mathfrak {v}\\in V_{\\mathcal {T}_r}} \\epsilon _\\mathfrak {v}\\left(a_{\\hat{\\mathcal {T}}_{1,r}(\\mathfrak {v})} +a_{\\hat{\\mathcal {T}}_{2,r}(\\mathfrak {v})} -a_{\\hat{\\mathcal {T}}_{3,r}(\\mathfrak {v})}\\right)\\prod _{s=1\\atop s\\ne s_0}^{n_\\mathfrak {v}} a_{\\mathcal {T}_{r,s}(\\mathfrak {v})},$ where $\\hat{\\mathcal {T}}_{r^{\\prime },r}(\\mathfrak {v})$ is obtained from $\\hat{\\mathcal {T}}_{r^{\\prime }}$ by replacing $\\mathcal {T}_r$ by $\\mathcal {T}_{r,s_0}(\\mathfrak {v})$ .",
"Applying the equations (REF ), this expression gives $\\frac{1}{n}\\left[ a_{\\mathcal {T}_2} a_{\\mathcal {T}_3}\\sum _{\\mathfrak {v}\\in V_{\\mathcal {T}_1}}\\epsilon _\\mathfrak {v}\\prod _{s=1}^{n_\\mathfrak {v}} a_{\\mathcal {T}_{1,s}(\\mathfrak {v})}+a_{\\mathcal {T}_1} a_{\\mathcal {T}_3}\\sum _{\\mathfrak {v}\\in V_{\\mathcal {T}_2}}\\epsilon _\\mathfrak {v}\\prod _{s=1}^{n_\\mathfrak {v}} a_{\\mathcal {T}_{2,s}(\\mathfrak {v})}+a_{\\mathcal {T}_1} a_{\\mathcal {T}_2}\\sum _{\\mathfrak {v}\\in V_{\\mathcal {T}_3}}\\epsilon _\\mathfrak {v}\\prod _{s=1}^{n_\\mathfrak {v}} a_{\\mathcal {T}_{3,s}(\\mathfrak {v})}\\right].$ Since the trees $\\mathcal {T}_{r}$ have less than $n$ vertices, they are subject to the induction hypothesis which allows to replace the sums over vertices by $n_r a_{\\mathcal {T}_r}$ where $n_r$ is the number of vertices in $\\mathcal {T}_r$ .",
"Taking into account that $n_1+n_2+n_3=n$ , the expression (REF ) reduces to $a_{\\mathcal {T}_1} a_{\\mathcal {T}_2}a_{\\mathcal {T}_3}$ , which proves that the equations (REF ) are indeed satisfied.",
"The evaluation of the large ${x}$ limit of the function $\\widetilde{\\Phi }^{\\,\\mathcal {E}}_n({x})$ is very similar to the calculation done in the proof of Proposition REF .",
"First, using the asymptotics of the generalized error functions, we can write $\\lim _{{x}\\rightarrow \\infty }\\widetilde{\\Phi }^{\\,\\mathcal {E}}_n({x})=\\frac{1}{2^{n-1} n!",
"}\\!\\!\\sum _{m=0}^{[(n-1)/2]}\\!\\!\\!\\sum _{\\mathcal {T}\\in \\, {T}_{n-2m,m}^\\ell }\\left[\\prod _{\\mathfrak {v}\\in V_\\mathcal {T}}\\mathcal {D}_{m_\\mathfrak {v}}(\\lbrace \\check{\\gamma }_{\\mathfrak {v},s}\\rbrace )\\right]\\left[\\prod _{e\\in E_\\mathcal {T}}\\mathcal {D}( {v}_{s(e) t(e)})\\right]\\left[\\prod _{e\\in E_{\\mathcal {T}}}{\\rm sgn}({u}_e,{x})\\right]\\!",
".$ We then observe that the two last factors depend only on sums of charges appearing in the operators $\\mathcal {D}_{m_\\mathfrak {v}}$ in the first factor.",
"This implies that the vectors on which these factors depend are orthogonal to the vectors determining the operators in (REF ).",
"Therefore, these operators effectively act on a constant and can be expanded as $\\mathcal {D}_{m}\\cdot 1 = \\sum _{k=0}^m\\mathcal {V}_{m,k},$ where $\\mathcal {V}_{m,k}$ are homogeneous polynomials in ${x}$ of degree $2(m-k)$ .",
"In particular, the highest degree term coincides with the function defined in (REF ), $\\mathcal {V}_{m,0}(\\lbrace \\check{\\gamma }_{s}\\rbrace )= \\mathcal {V}_{m}(\\lbrace \\check{\\gamma }_{s}\\rbrace )$ .",
"Next, the mutual action of the derivative operators from the product over edges in the second factor generates contributions described by trees with pairs of collapsed edges replaced by marks.",
"The difference here is that the original trees were also marked.",
"This fact does not change the structure of the result, which is again given by a sum over marked trees, but it affects the weight associated with marks.",
"Denoting this weight for a vertex with total $m$ marks (old and new) by $\\mathcal {V}^{\\rm tot}_m$ , we recover the equation (REF ) in the statement of the proposition, provided that $\\mathcal {V}^{\\rm tot}_m=\\mathcal {V}_m$ .",
"We now proceed to prove the latter identity.",
"The weight factor $\\mathcal {V}^{\\rm tot}_m$ coming from the above procedure is given by $\\small \\mathcal {V}^{\\rm tot}_m(\\lbrace \\check{\\gamma }_{s}\\rbrace )=\\,{\\rm Sym}\\, \\!\\!",
"\\left\\lbrace \\sum _{m_0=0}^{m}\\frac{(-1)^{m_0}}{(2\\pi )^{m_0}}\\!\\!\\!\\!\\sum _{\\sum \\limits _{r=1}^{2m_0+1}m_r=m-m_0}\\!\\!\\!\\!\\!\\!",
"\\!\\!\\!\\!\\!\\!C(\\lbrace m_r\\rbrace )\\,\\mathcal {P}_{m_0}(\\lbrace p^{\\prime }_{r}\\rbrace )\\!\\!\\prod _{r=1}^{2m_0+1}\\sum _{k_r=0}^{m_r}\\mathcal {V}_{m_r,k_r}(\\check{\\gamma }_{j_{r-1}+1},\\dots ,\\check{\\gamma }_{j_{r}})\\right\\rbrace ,$ where the second sum goes over all ordered decompositions of $m-m_0$ into non-negative integers, $C(\\lbrace m_r\\rbrace )=\\frac{(2m+1)!",
"}{\\prod _r (2m_r+1)!",
"}$ , and we used notations similar to (REF ), $j_0=0,\\qquad j_r=m_1+\\cdots + m_r,\\qquad \\gamma ^{\\prime }_r=\\gamma _{j_{r-1}+1}+\\cdots +\\gamma _{j_{r}}.$ Here the first factor $\\mathcal {P}_{m_0}$ arises due to collapse of $m_0$ pairs of edges in the marked trees one sums over in (REF ), whereas the factors given by the sums over $k_r$ are the ones corresponding to the “old\" marks assigned to that trees.",
"Now let us use the equations (REF ) determining the coefficients $a_\\mathcal {T}$ .",
"It is easy to see that they imply the following constraint on the next-to-highest degree term in the expansion (REF ) $\\mathcal {V}_{m,1}(\\lbrace \\check{\\gamma }_{s}\\rbrace )=\\frac{1}{6\\cdot 2\\pi } \\sum _{m_1+m_2+m_3=m-1\\atop m_r\\ge 0}C(\\lbrace m_r\\rbrace )\\,{\\rm Sym}\\, \\left\\lbrace (p^{\\prime }_1 p^{\\prime }_2 p^{\\prime }_3)\\prod _{r=1}^3 \\mathcal {V}_{m_r}(\\lbrace \\check{\\gamma }_i\\rbrace _{i=j_{r-1}+1}^{j_r})\\right\\rbrace ,$ where $j_r=\\sum _{s=1}^r (2m_s+1)$ and $p^{\\prime }_r=\\sum _{i=1}^{2m_r+1}p_{j_{r-1}+i}$ .",
"Applying this constraint recursively, one can express all $\\mathcal {V}_{m,k}$ for $k>0$ through $\\mathcal {V}_{m}$ .",
"The idea is to replace the factors $\\mathcal {V}_{m_r}$ by the operators $\\mathcal {D}_{m_r}$ .",
"Then one can realize that, extracting from the resulting function the terms homogeneous in ${x}$ of order $2(m-2)$ (which have two factors of $(p^3)$ ), one obtains the result for $2\\mathcal {V}_{m,2}$ .",
"Proceeding in this way, one arrives at the representation very similar to (REF ), $\\mathcal {V}_{m,k}(\\lbrace \\check{\\gamma }_{s}\\rbrace )=\\frac{1}{(2\\pi )^k}\\,{\\rm Sym}\\, \\left\\lbrace \\sum _{\\sum \\limits _{r=1}^{2k+1}m_r=m-k}\\!\\!\\!\\!",
"C(\\lbrace m_r\\rbrace )\\,\\mathcal {P}_{k}(\\lbrace p^{\\prime }_{r}\\rbrace )\\prod _{r=1}^{2k+1}\\mathcal {V}_{m_r}(\\check{\\gamma }_{j_{r-1}+1},\\dots ,\\check{\\gamma }_{j_{r}})\\right\\rbrace .$ Substituting it into (REF ) and using the expression (REF ) for the factors $\\mathcal {P}_{m}$ through the sum over rooted ternary trees, one can recombine all these sums in the following way $\\begin{split}\\mathcal {V}^{\\rm tot}_m=&\\,\\,{\\rm Sym}\\, \\Biggl \\lbrace \\sum _{k=0}^{m}\\sum _{\\sum \\limits _{r=1}^{2k+1}m_r=m-k}\\!\\!\\!\\!\\!\\!",
"C(\\lbrace m_r\\rbrace )\\sum _{T\\in \\, {T}_{2k+1}^{(3)}(\\lbrace \\check{\\gamma }^{\\prime }_r\\rbrace )}\\frac{1}{T!",
"}\\sum _{T^{\\prime }\\subseteq T} (-1)^{n_{T^{\\prime }}}\\prod _{v\\in V_{T^{\\prime }}}\\frac{n_v(T)}{n_v(T^{\\prime })}\\\\&\\,\\times \\prod _{v\\in V_{T}}\\frac{(p_{d_1(v)}p_{d_2(v)}p_{d_3(v)})}{2\\pi }\\prod _{r=1}^{2k+1}\\mathcal {V}_{m_r}(\\check{\\gamma }_{j_{r-1}+1},\\dots ,\\check{\\gamma }_{j_{r}})\\Biggr \\rbrace ,\\end{split}$ where the sum over $T^{\\prime }$ is the sum over subtrees of $T$ having the same root.",
"In terms of the variables appearing in (REF ), one can identify $k=m_0+\\sum _r k_r$ and $n_{T^{\\prime }}=m_0$ .",
"Thus, $T^{\\prime }$ is the tree appearing in the decomposition of $\\mathcal {P}_{m_0}$ , whereas $T$ is its union with $2m_0+1$ trees $T_r$ appearing in the decomposition of $\\mathcal {P}_{k_r}$ , i.e.",
"$T=T^{\\prime }\\cup \\left(\\cup _r T_r\\right)$ where $T_r$ are the trees rooted at leaves of $T^{\\prime }$ , Finally, we took into account that for such trees one has $T!=T^{\\prime }!\\,\\prod _r T_r!\\,\\prod _{v\\in V_{T^{\\prime }}}\\frac{n_v(T)}{n_v(T^{\\prime })}\\,.$ Remarkably, the sum over subtrees in (REF ) factorizes and for a fixed number of vertices $n_{T^{\\prime }}=m_0$ is subject to Theorem REF where the rôle of the trees is played by rooted ternary trees $T$ and $T^{\\prime }$ after stripping out their leaves.",
"As a result, one finds for $k>0$ $\\sum _{T^{\\prime }\\subseteq T} (-1)^{n_{T^{\\prime }}}\\prod _{v\\in V_{T^{\\prime }}}\\frac{n_v(T)}{n_v(T^{\\prime })}=\\sum _{m_0=0}^k\\frac{(-1)^{m_0}k!}{m_0!(k-m_0)!",
"}=(1-1)^k = 0.$ Thus, the only non-vanishing contribution is the one with $k=0$ which coincides with $\\mathcal {V}_m$ .",
"This is what we had to show and therefore completes the proof of the proposition.",
"For simplicity, let us first show that the recursive equation (REF ) is satisfied by the contribution to $g^{(0)}_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace )$ given by the trees without any marks, i.e.",
"by the function $g^{\\star }_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace )=\\frac{(-1)^{n-1+\\sum _{i<j} \\gamma _{ij} }}{2^{n-1} n!",
"}\\sum _{\\mathcal {T}\\in \\, {T}_{n}^\\ell }\\prod _{e\\in E_{\\mathcal {T}}}\\gamma _{s(e) t(e)}\\,{\\rm sgn}(S_e).$ Then the inclusion of marks will be straightforward because the corresponding contributions can be dealt with essentially in the same way as the contribution (REF ).",
"To start with, we substitute $g^{\\star }_n$ into the l.h.s.",
"of the recursive equation and decompose $\\Gamma _{n\\ell }=-\\sum _{i=1}^\\ell \\sum _{j=\\ell +1}^n \\gamma _{ij}$ .",
"Then the crucial observation is that this double sum, the sum over $\\ell $ and the two sums over trees (over ${T}_\\ell ^\\ell $ and ${T}_{n-\\ell }^\\ell $ ) are all equivalent to a single sum over trees with $n$ vertices, i.e.",
"over ${T}_n^\\ell $ , supplemented by the sum over edges.",
"Namely, one can do the following replacement ${1\\over 2}\\,{\\rm Sym}\\, \\sum _{\\ell =1}^{n-1}\\frac{1}{\\ell !",
"(n-\\ell )!",
"}\\sum _{\\mathcal {T}_L\\in \\, {T}_\\ell ^\\ell }\\sum _{\\mathcal {T}_R\\in \\, {T}_{n-\\ell }^\\ell }\\sum _{i=1}^\\ell \\sum _{j=\\ell +1}^n=\\frac{1}{n!",
"}\\sum _{\\mathcal {T}\\in \\, {T}_n^\\ell }\\sum _{e\\in E_\\mathcal {T}}.$ The idea is that on the l.h.s.",
"one sums over all possible splittings of unrooted labelled trees with $n$ vertices into two trees with $\\ell $ and $n-\\ell $ vertices.",
"Such splitting can be done by cutting an edge and then $i,j$ correspond to the labels of the vertices joined by the cutting edge.",
"The binomial coefficient $\\frac{n!",
"}{\\ell !",
"(n-\\ell )!",
"}$ takes into account that after splitting the vertices of the first tree can have arbitrary labels from the set $\\lbrace 1,\\dots ,n\\rbrace $ and not necessarily $\\lbrace 1,\\dots ,\\ell \\rbrace $ , whereas ${1\\over 2}$ avoids doubling due to the symmetry between $\\mathcal {T}_L$ and $\\mathcal {T}_R$ .",
"It is easy to check that all factors in (REF ) fit this interpretation and the l.h.s.",
"takes the following form $\\frac{(-1)^{n-1+\\sum _{i<j} \\gamma _{ij} }}{2^{n-1} n!",
"}\\sum _{\\mathcal {T}\\in \\, {T}_n^\\ell }\\prod _{e\\in E_{\\mathcal {T}}}\\gamma _{s(e) t(e)}\\sum _{e\\in E_\\mathcal {T}} \\bigl ( {\\rm sgn}(S_e)-{\\rm sgn}(\\Gamma _{e})\\bigr )\\prod _{e^{\\prime }\\in E_{\\mathcal {T}}\\setminus \\lbrace e\\rbrace }{\\rm sgn}\\left(S_{e^{\\prime }}-\\frac{\\Gamma _{e^{\\prime }}}{\\Gamma _{e}}\\, S_e\\right),$ where $\\Gamma _e$ was defined in (REF ).",
"Finally, we apply the following sign identity, established in [23], $\\sum _{\\beta =1}^m \\left(\\mbox{sgn}(x_\\beta )-1\\right) \\prod _{\\alpha =1\\atop \\alpha \\ne \\beta }^m \\mbox{sgn}(x_\\alpha -x_\\beta )= \\prod _{\\alpha =1}^m \\mbox{sgn}(x_\\alpha )-1,$ where one should take the label $\\alpha $ to run over $m=n-1$ edges of a tree $\\mathcal {T}$ , identify $x_\\alpha =S_\\alpha /\\Gamma _{\\alpha }$ , and multiply it by $\\prod _{\\alpha =1}^{n-1}\\mbox{sgn}(\\Gamma _\\alpha )$ .",
"Then the expression (REF ) reduces to $g^{\\star }_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace )-g^{\\star }_n(\\lbrace \\check{\\gamma }_i,\\beta _{ni}\\rbrace )$ , i.e.",
"the r.h.s.",
"of (REF ) evaluated for function (REF ).",
"This proves that this function solves the recursive equation.",
"The generalization of this proof to the full ansatz (REF ) is elementary.",
"Instead of the relation (REF ), one now has ${1\\over 2}\\,{\\rm Sym}\\, \\sum _{\\ell =1}^{n-1}\\frac{1}{\\ell !",
"(n-\\ell )!",
"}\\sum _{k_L=0}^{[(\\ell -1)/2]}\\!\\!\\sum _{\\mathcal {T}_L\\in \\, {T}_{\\ell -2k_L,k_L}^\\ell }\\!\\!\\!\\!\\!\\sum _{k_R=0}^{[(n-\\ell -1)/2]}\\!\\!\\!\\!\\sum _{\\mathcal {T}_R\\in \\, {T}_{n-\\ell -2k_R,k_R}^\\ell }\\!\\!\\sum _{i=1}^\\ell \\sum _{j=\\ell +1}^n\\!=\\frac{1}{n!",
"}\\sum _{k=0}^{[(n-1)/2]}\\!\\!\\!\\!\\sum _{\\mathcal {T}\\in \\, {T}_{n-2k,k}^\\ell }\\sum _{e\\in E_\\mathcal {T}},$ which again reflects the fact the sum over marked unrooted labelled trees can be represented as a sum over all possible splittings into two such trees by cutting them along an edge.",
"If the cutting edge joins a marked vertex, it counts $1+2m_\\mathfrak {v}$ times, producing the right factor $\\gamma _{\\mathfrak {v}_L\\mathfrak {v}_R}$ , which is reflected by the fact that the sums over $i,j$ run over $\\ell $ and $n-\\ell $ values, respectively.",
"Other numerical factors work in the same way as before.",
"Thus, the l.h.s.",
"of (REF ) can again be rewritten as in (REF ) with the only difference that $\\sum _{\\mathcal {T}\\in \\, {T}_n^\\ell }$ should now be replaced by $\\sum _{k=0}^{[(n-1)/2]}\\sum _{\\mathcal {T}\\in \\, {T}_{n-2k,k}^\\ell }\\prod _{\\mathfrak {v}\\in V_\\mathcal {T}}\\tilde{\\mathcal {V}}_\\mathfrak {v}$ .",
"Applying the same sign identity (REF ) for $m=n-1-2k$ and the same identification for $x_\\alpha $ , one recovers the r.h.s.",
"of (REF ).",
"This completes the proof of the proposition.",
"This proposition trivially follows from (REF ).",
"The easiest way to prove (REF ) is to substitute it into (REF ) and then check that the result is consistent with the constraint (REF ).",
"The substitution generates a sum over trees which resemble the blooming trees of appendix : these are trees with vertices from which other trees grows.",
"But now the two types of trees, representing the `base' and the `flowers', are actually the same — both of them are Schröder trees.",
"The only difference is that vertices of the `base' carry weights $\\mathcal {E}^{(+)}_v$ , whereas the vertices of `flowers' have weights $\\mathcal {E}^{(0)}_v$ .",
"Since the leaves of a `flower' are in one-to-one correspondence with the children of the vertex of the `base' tree from which this flower grows, such blooming trees can be equivalently represented by the usual Schröder trees obtained by replacing the vertices of the `base' by their `flowers'.",
"In this way, we obtain $W_n= \\,{\\rm Sym}\\, \\left\\lbrace \\sum _{T\\in {T}_n^{\\rm S}}(-1)^{n_T} \\sum _{\\cup \\, T^{\\prime }=T}\\prod _{T^{\\prime }}\\left[\\mathcal {E}^{(+)}_{v_0(T^{\\prime })}\\prod _{v\\in V_{T^{\\prime }}\\setminus \\lbrace v_0(T^{\\prime })\\rbrace }\\mathcal {E}^{(0)}_{v}\\right]\\right\\rbrace ,$ where the second sum goes over decompositions of $T$ into subtrees such that the root $v_0(T^{\\prime })$ of a subtree $T^{\\prime }$ is a leaf of another subtree (except, of course, the root of the total tree).",
"Let us now consider a vertex $v$ whose only children are leaves of $T$ .",
"Then all decompositions into subtrees $T=\\cup _i\\, T^{\\prime }_i$ can be split into pairs such that two decompositions differ only by whether $v$ (together with its leaves) represents a separate subtree or it is a part of a bigger subtree.",
"The contributions of two such decompositions into (REF ) differ only by the factors assigned to the vertex $v$ and therefore they combine into the factor $\\mathcal {E}_v$ assigned to this vertex.",
"As a result, $\\mathcal {E}_v$ appears as a common factor and the vertex $v$ can be excluded from the following consideration.",
"Proceeding in the same way with the tree obtained by removing this vertex, one finds that the sum over decompositions can be evaluated explicitly and gives $W_n= \\,{\\rm Sym}\\, \\left\\lbrace \\sum _{T\\in {T}_n^{\\rm S}}(-1)^{n_T} \\mathcal {E}^{(+)}_{v_0}\\prod _{v\\in V_T\\setminus {\\lbrace v_0\\rbrace }}\\mathcal {E}_{v}\\right\\rbrace .$ Given this result, the proof of the constraint (REF ) is analogous to the proof of the relation (REF ): the contribution of each tree $T$ (from the sum in (REF )) and a splitting with $n_k>1$ (from the sum in (REF )) is cancelled by the contribution of another tree obtained from $T$ by adding $n_k$ children to its $k$ th leaf and the same splitting but with $n_k$ replaced by $1+\\cdots +1$ (repeated $n_k$ times).",
"The only contribution which survives is the one generated by the tree with a single vertex and $n$ leaves and the splitting with all $n_k=1$ .",
"It is given by $\\mathcal {E}^{(+)}_n$ , which verifies the constraint and proves the proposition.",
"Our starting point to prove the proposition is the formula $\\partial _{\\bar{\\tau }}\\widehat{h}_{p,\\mu }(\\tau )=\\frac{I}{2}\\sum _{n=2}^\\infty \\sum _{\\sum _{i=1}^n \\check{\\gamma }_i=\\check{\\gamma }}\\partial _{\\tau _2}R_n(\\lbrace \\check{\\gamma }_i\\rbrace ,\\tau _2)\\, e^{\\pi I\\tau Q_n(\\lbrace \\check{\\gamma }_i\\rbrace )}\\prod _{i=1}^n h_{p_i,\\mu _i}(\\tau ).$ Substituting $\\partial _{\\tau _2}R_n$ following from (REF ) and the inverse formula (REF ) expressing $h_{p,\\mu }$ in terms of the completion, with the functions $W_n$ found in (REF ), one arrives at the result (REF ) where the functions $\\mathcal {J}_n$ are given by $\\mathcal {J}_n= \\frac{I}{2}\\,{\\rm Sym}\\, \\left\\lbrace \\sum _{T\\in {T}_n^{\\rm S}}(-1)^{n_T-1} \\partial _{\\tau _2}\\mathcal {E}_{v_0}\\sum _{T^{\\prime }\\subseteq T}\\prod _{v\\in V_{T^{\\prime }}\\setminus {\\lbrace v_0\\rbrace }}\\mathcal {E}^{(0)}_{v}\\prod _{v\\in L_{T^{\\prime }}}\\mathcal {E}^{(+)}_{v}\\prod _{v\\in V_T\\setminus (V_{T^{\\prime }}\\cup L_{T^{\\prime }})}\\mathcal {E}_{v}\\right\\rbrace ,$ where the second sum goes over all subtrees $T^{\\prime }$ of $T$ containing its root and $L_{T^{\\prime }}$ is the set of their leaves.",
"Here the subtree $T^{\\prime }$ corresponds to the tree in the formula (REF ) for $R_n$ , whereas the subtrees starting from its leaves correspond to the trees in the expression (REF ) for $W_n$ .",
"The sum over subtrees can be evaluated in the same way as the sum over decompositions into subtrees in (REF ): the two contributions differing only by whether the vertex $v$ belongs to $V_{T^{\\prime }}$ or $L_{T^{\\prime }}$ combine into the factor $\\mathcal {E}_v$ assigned to this vertex.",
"Performing this recombination for all vertices of the tree $T$ , one obtains the expression (REF ), which proves the proposition.",
"Our goal is to prove that in the expression $g_{{\\rm tr},n}= \\,{\\rm Sym}\\, \\left\\lbrace \\sum _{T\\in {T}_n^{\\rm S}}(-1)^{n_T-1} \\left(g^{(0)}_{v_0}-\\mathcal {E}^{(0)}_{v_0}\\right)\\prod _{v\\in V_T\\setminus {\\lbrace v_0\\rbrace }}\\mathcal {E}^{(0)}_{v}\\right\\rbrace ,$ following from (REF ), all contributions due to marked trees cancel leaving only contributions of trees without marks, i.e with $m=0$ .",
"There are several possible situations which we need to analyze.",
"Figure: Combination of two Schröder trees ensuring the cancellation of contributions generated by marked trees.First, let us consider the contributions generated by non-trivial marked trees, i.e.",
"trees having more than one vertex and at least one mark.",
"Let us focus on the contribution corresponding to a vertex $\\mathfrak {v}$ with $m_\\mathfrak {v}>0$ marks of a tree $\\mathcal {T}$ , which appears in the sum over marked unrooted trees living at a vertex $v$ of a Schröder tree $T$ .",
"Let $k=n_v$ be the number of children of the vertex $v$ and $\\gamma _i$ ($i=1,\\dots ,k$ ) their charges so that $\\gamma _s$ ($s=1,\\dots ,2m_\\mathfrak {v}+1$ ) are the charges labelling the marked vertex $\\mathfrak {v}$ .",
"Note that $k\\ge 2m_\\mathfrak {v}+2$ because the tree $\\mathcal {T}$ has at least one additional vertex except $\\mathfrak {v}$ .",
"Then the contribution we described is cancelled by the contribution coming from another Schröder tree, which is obtained from $T$ by adding an edge connecting the vertex $v$ to a new vertex $v^{\\prime }$ , whose children are the $2m_\\mathfrak {v}+1$ children of $v$ in $T$ carrying charges $\\gamma _s$ (see Fig.",
"REF ).The new tree is of Schröder type because its vertex $v$ has $k-2m_{\\mathfrak {v}}\\ge 2$ children and vertex $v^{\\prime }$ has $2m_\\mathfrak {v}+1\\ge 3$ children.",
"Indeed, choosing the same tree $\\mathcal {T}$ as before in the sum over marked trees at vertex $v$ , but now with $m_\\mathfrak {v}=0$ , and in the sum at vertex $v^{\\prime }$ the trivial tree having one vertex and $m_\\mathfrak {v}$ marks, one gets exactly the same contribution as before, but now with an opposite sign due to the presence of an additional vertex in the Schröder tree.",
"Thus, all contributions from non-trivial marked trees are cancelled.",
"As a result, we remain only with the contributions generated by trivial marked trees, i.e.",
"having only one vertex and $m_\\mathfrak {v}$ marks.",
"One has to distinguish two cases: either the corresponding vertex $v$ of the Schröder tree is the root or not.",
"In the former case, this contribution is trivially cancelled in the difference $g^{(0)}_{v_0}-\\mathcal {E}^{(0)}_{v_0}$ in (REF ).",
"In the latter case, this is precisely the contribution used above to cancel the contributions from non-trivial marked trees.",
"This exhausts all possibilities and we arrive at the formula (REF ).",
"toc toc"
],
[
"Explicit results up to 4th order",
"In this appendix we provide explicit expressions for various functions appearing in our construction up to the forth order.",
"To write them down, we will use the shorthand notation $\\gamma _{i+j}=\\gamma _i+\\gamma _j$ , $c_{i+j}=c_i+c_j$ , etc.",
"as well as indicate the arguments of functions through their indices, for instance, $\\mathcal {E}_{i_1\\cdots i_n}=\\mathcal {E}_n(\\check{\\gamma }_{i_1},\\dots ,\\check{\\gamma }_{i_n})$ .",
"These expressions are obtained by simple substitutions using the results found in the main text and the sets of trees shown in Fig.",
"REF .",
"For $n=2$ they all agree with the results of [14].",
"The results (REF ) and (REF ) generate the following expansions $h^{\\rm DT}_{p,q}&=&\\widehat{h}_{p,\\mu }+\\sum _{\\check{\\gamma }_1+\\check{\\gamma }_2=\\check{\\gamma }}\\left[g^{(0)}_{12}-\\mathcal {E}_{12}\\right]e^{\\pi I\\tau Q_2(\\lbrace \\check{\\gamma }_i\\rbrace )}\\widehat{h}_{p_1,\\mu _1}\\widehat{h}_{p_2,\\mu _2}\\nonumber \\\\&& +\\sum _{\\sum _{i=1}^3 \\check{\\gamma }_i=\\check{\\gamma }}\\left[g^{(0)}_{123}-\\mathcal {E}_{123}-2\\left(g^{(0)}_{1+2,3}-\\mathcal {E}_{1+2,3}\\right)\\mathcal {E}_{12}\\right] e^{\\pi I\\tau Q_3(\\lbrace \\check{\\gamma }_i\\rbrace )}\\prod _{i=1}^3 \\widehat{h}_{p_i,\\mu _i}\\\\&&+\\sum _{\\sum _{i=1}^4 \\check{\\gamma }_i=\\check{\\gamma }}\\left[ g^{(0)}_{1234}-\\mathcal {E}_{1234}-2\\left(g^{(0)}_{1+2+3,4}-\\mathcal {E}_{1+2+3,4}\\right)\\left(\\mathcal {E}_{123} -2\\mathcal {E}_{1+2,3}\\mathcal {E}_{12}\\right)\\right.\\nonumber \\\\&& \\left.\\qquad -3\\left(g^{(0)}_{1+2,34}-\\mathcal {E}_{1+2,34}\\right)\\mathcal {E}_{12}+\\left(g^{(0)}_{1+2,3+4}-\\mathcal {E}_{1+2,3+4}\\right)\\mathcal {E}_{12}\\mathcal {E}_{34}\\right]e^{\\pi I\\tau Q_4(\\lbrace \\check{\\gamma }_i\\rbrace )}\\prod _{i=1}^4 \\widehat{h}_{p_i,\\mu _i}+\\cdots ,\\nonumber $ $\\widehat{h}_{p,q}&=&h_{p,\\mu }+\\sum _{\\check{\\gamma }_1+\\check{\\gamma }_2=\\check{\\gamma }}\\mathcal {E}^{(+)}_{12}\\,e^{\\pi I\\tau Q_2(\\lbrace \\check{\\gamma }_i\\rbrace )}h_{p_1,\\mu _1}h_{p_2,\\mu _2}\\nonumber \\\\&& +\\sum _{\\sum _{i=1}^3 \\check{\\gamma }_i=\\check{\\gamma }}\\Bigl [\\mathcal {E}^{(+)}_{123}-2\\mathcal {E}^{(+)}_{1+2,3}\\mathcal {E}^{(0)}_{12}\\Bigr ] \\,e^{\\pi I\\tau Q_3(\\lbrace \\check{\\gamma }_i\\rbrace )}\\prod _{i=1}^3 h_{p_i,\\mu _i}\\\\&&+\\sum _{\\sum _{i=1}^4 \\check{\\gamma }_i=\\check{\\gamma }}\\Bigl [\\mathcal {E}^{(+)}_{1234}-2\\mathcal {E}^{(+)}_{1+2+3,4}\\left(\\mathcal {E}^{(0)}_{123} -2\\mathcal {E}^{(0)}_{1+2,3}\\mathcal {E}^{(0)}_{12}\\right)-3\\mathcal {E}^{(+)}_{1+2,34}\\mathcal {E}^{(0)}_{12}\\Bigr .\\nonumber \\\\&& \\Bigl .",
"\\qquad +\\mathcal {E}^{(+)}_{1+2,3+4}\\mathcal {E}^{(0)}_{12}\\mathcal {E}^{(0)}_{34}\\Bigr ]\\, e^{\\pi I\\tau Q_4(\\lbrace \\check{\\gamma }_i\\rbrace )}\\prod _{i=1}^4 h_{p_i,\\mu _i}+\\cdots ,\\nonumber $ where the functions $g^{(0)}_n$ and $\\mathcal {E}_n$ can be read off from (REF ), (REF ) and (REF ), $\\begin{split}g^{(0)}_2=&\\, \\frac{(-1)^{1+\\gamma _{12}}}{4}\\, \\gamma _{12}\\,\\mbox{sgn}(c_1),\\\\g^{(0)}_3=&\\, \\frac{(-1)^{1+\\gamma _{12}+\\gamma _{1+2,3}}}{8}\\,\\,{\\rm Sym}\\, \\biggl \\lbrace \\gamma _{12}\\,\\gamma _{23}\\,\\mbox{sgn}(c_1)\\, \\mbox{sgn}(c_3)+\\frac{1}{3}\\,\\gamma _{12}\\,\\gamma _{23} \\biggr \\rbrace ,\\\\g^{(0)}_4=&\\, \\frac{(-1)^{\\gamma _{12}+\\gamma _{1+2,3}+\\gamma _{1+2+3,4}}}{16}\\,\\,{\\rm Sym}\\, \\biggl \\lbrace \\gamma _{12}\\gamma _{23}\\gamma _{34}\\,\\mbox{sgn}(c_1)\\mbox{sgn}(c_{1+2})\\mbox{sgn}(c_4)\\\\&\\, \\qquad -\\frac{1}{3}\\,\\gamma _{12}\\gamma _{23}\\gamma _{24}\\,\\mbox{sgn}(c_1)\\mbox{sgn}(c_3)\\mbox{sgn}(c_4)-\\frac{1}{3}\\, \\gamma _{12}\\gamma _{23} \\gamma _{1+2+3,4}\\mbox{sgn}(c_4)\\biggr \\rbrace ,\\end{split}$ $\\mathcal {E}_{12} &=&\\frac{(-1)^{\\gamma _{12}}}{4\\sqrt{2\\tau _2}}\\,\\widetilde{\\Phi }_{1}^E\\bigl ({v}_{12},{v}_{12}\\bigr )\\nonumber \\\\&=&\\frac{(-1)^{\\gamma _{12}}}{4}\\left[\\gamma _{12}\\, E_1\\left(\\frac{\\sqrt{2\\tau _2}\\gamma _{12} }{\\sqrt{(pp_1p_2)}}\\right)+\\frac{\\sqrt{(pp_1p_2)}}{\\pi \\sqrt{2\\tau _2}}\\, e^{-\\frac{2\\pi \\tau _2\\gamma _{12}^2 }{(pp_1p_2)}} \\right] ,\\nonumber \\\\\\mathcal {E}_{123} &=&\\frac{(-1)^{\\gamma _{12}+\\gamma _{1+2,3}}}{8}\\,\\,{\\rm Sym}\\, \\left\\lbrace \\frac{1}{2\\tau _2}\\,\\widetilde{\\Phi }_{2}^E\\bigl (({v}_{1,2+3},{v}_{1+2,3}),({v}_{12},{v}_{23})\\bigr )\\right.\\nonumber \\\\&&\\left.",
"\\qquad -\\frac{1}{3}\\left(\\gamma _{12}\\gamma _{23}-\\frac{(p_1p_2p_3)}{4\\pi \\tau _2} \\right)\\right\\rbrace ,\\\\\\mathcal {E}_{1234} &=&\\frac{(-1)^{\\gamma _{12}+\\gamma _{1+2,3}+\\gamma _{1+2+3,4}}}{16\\sqrt{2\\tau _2}}\\,\\,{\\rm Sym}\\, \\biggl \\lbrace \\frac{1}{2\\tau _2}\\biggl (\\widetilde{\\Phi }_{3}^E\\bigl (({v}_{1,2+3+4},{v}_{1+2,3+4},{v}_{1+2+3,4}),({v}_{12},{v}_{23},{v}_{34})\\bigr )\\nonumber \\\\&&+\\frac{1}{3}\\,\\widetilde{\\Phi }_{3}^E\\bigl (({v}_{1,2+3+4},{v}_{1+2+4,3},{v}_{1+2+3,4}),({v}_{12},{v}_{23},{v}_{24})\\bigr )\\biggr )\\nonumber \\\\&& -\\frac{1}{3}\\left(\\gamma _{12}\\gamma _{23}-\\frac{(p_1p_2p_3)}{4\\pi \\tau _2}\\right)\\widetilde{\\Phi }_{1}^E\\bigl ({v}_{1+2+3,4},{v}_{1+2+3,4}\\bigr )\\biggr \\rbrace ,\\nonumber $ where all generalized error functions are evaluated at ${x}=\\sqrt{2\\tau _2}({q}+{b})$ .",
"The last terms appearing in the above quantities for $n=3$ and $n=4$ correspond to contributions of trees with one mark ($m=1$ ).",
"In fact, at these orders these results can be rewritten in a simpler form, which coincides with the representations (REF ) and (REF ) (in the latter formula one should drop the sum over partitions and take $d_T=d_n$ ): $g^{(0)}_3&=& \\frac{(-1)^{1+\\gamma _{12}+\\gamma _{1+2,3}}}{12}\\,\\,{\\rm Sym}\\, \\Bigl \\lbrace \\gamma _{12}\\,\\gamma _{1+2,3}\\,\\mbox{sgn}(c_1)\\, \\mbox{sgn}(c_3) \\Bigr \\rbrace ,\\\\g^{(0)}_4&=&\\textstyle {\\frac{(-1)^{\\gamma _{12}+\\gamma _{1+2,3}+\\gamma _{1+2+3,4}}}{96}}\\,{\\rm Sym}\\, \\!\\Bigl \\lbrace \\!",
"\\Bigl (2\\,\\gamma _{23}\\gamma _{1,2+3}\\gamma _{1+2+3,4}+\\gamma _{12}\\gamma _{34}\\gamma _{1+2,3+4}\\Bigr )\\mbox{sgn}(c_1)\\mbox{sgn}(c_{1+2})\\mbox{sgn}(c_4)\\Bigr \\rbrace ,\\nonumber $ $\\mathcal {E}_{123} &=&\\frac{(-1)^{\\gamma _{12}+\\gamma _{1+2,3}}}{24\\tau _2}\\,\\,{\\rm Sym}\\, \\left\\lbrace \\widetilde{\\Phi }_{2}^E\\bigl (({v}_{1,2+3},{v}_{1+2,3}),({v}_{12},{v}_{1+2,3})\\bigr )\\right\\rbrace ,\\\\\\mathcal {E}_{1234} &=&\\frac{(-1)^{\\gamma _{12}+\\gamma _{1+2,3}+\\gamma _{1+2+3,4}}}{96(2\\tau _2)^{3/2}}\\,\\,{\\rm Sym}\\, \\biggl \\lbrace 2\\,\\widetilde{\\Phi }_{3}^E\\bigl (({v}_{1,2+3+4},{v}_{1+2,3+4},{v}_{1+2+3,4}),({v}_{23},{v}_{1,2+3},{v}_{1+2+3,4})\\bigr )\\nonumber \\\\&&+\\widetilde{\\Phi }_{3}^E\\bigl (({v}_{1,2+3+4},{v}_{1+2,3+4},{v}_{1+2+3,4}),({v}_{12},{v}_{34},{v}_{1+2,3+4})\\bigr )\\biggr )\\biggr \\rbrace .\\nonumber $ The simplest way to prove the equality of the two representations of $g^{(0)}_n$ is to expand the DSZ products appearing in (REF ) into elementary $\\gamma _{ij}$ 's and then, using symmetrization, bring all their products to the form appearing in (REF ).",
"These products are multiplied by combinations of sign functions which can be recombined with the help of the identity (REF ).",
"As an example, let us perform these manipulations for $n=3$ : $\\begin{split}&\\,\\,{\\rm Sym}\\, \\Bigl \\lbrace \\gamma _{12}\\,\\gamma _{1+2,3}\\,\\mbox{sgn}(c_1)\\, \\mbox{sgn}(c_3) \\Bigr \\rbrace =\\,{\\rm Sym}\\, \\Bigl \\lbrace \\gamma _{12}\\,\\gamma _{23}\\bigl (\\mbox{sgn}(c_1)-\\mbox{sgn}(c_2) \\bigr )\\,\\mbox{sgn}(c_3)\\Bigr \\rbrace \\\\=&\\,\\,{\\rm Sym}\\, \\Bigl \\lbrace \\gamma _{12}\\,\\gamma _{23}\\Bigl (\\mbox{sgn}(c_1)\\,\\mbox{sgn}(c_3)+{1\\over 2}\\, \\mbox{sgn}(c_{1+3})\\bigl (\\mbox{sgn}(c_1)+\\mbox{sgn}(c_3) \\bigr )\\Bigr )\\Bigr \\rbrace \\\\=&\\,{1\\over 2}\\,\\,{\\rm Sym}\\, \\Bigl \\lbrace \\gamma _{12}\\,\\gamma _{23}\\Bigl (3\\, \\mbox{sgn}(c_1)\\,\\mbox{sgn}(c_3)+1\\Bigr )\\Bigr \\rbrace ,\\end{split}$ where we used that $c_2=-(c_1+c_3)$ .",
"This identity then shows the equality of the two forms of $g^{(0)}_3$ given above.",
"For $g^{(0)}_4$ the manipulations are very similar, but a bit more cumbersome.",
"The equality of the two forms of $\\mathcal {E}_n$ follows from the equality of their asymptotics $\\mathcal {E}^{(0)}_n$ , which is in turn ensured by the same identities as for $g^{(0)}_n$ .",
"Finally, we provide expressions for the kernels $\\widehat{\\Phi }^{{\\rm tot}}_n$ of the indefinite theta series appearing in the expansion (REF ) of $\\mathcal {G}$ in powers of $\\widehat{h}_{p,\\mu }$ .",
"For the first two orders, one has $\\begin{split}\\widehat{\\Phi }^{{\\rm tot}}_1=&\\,\\Phi ^{\\scriptscriptstyle \\,\\int }_1,\\\\\\widehat{\\Phi }^{{\\rm tot}}_2=&\\,\\Phi ^{\\scriptscriptstyle \\,\\int }_2+\\Phi ^{\\scriptscriptstyle \\,\\int }_1 \\Phi ^{\\,\\widehat{g}}_2=\\frac{1}{4}\\, \\Phi ^{\\scriptscriptstyle \\,\\int }_1\\left(\\widetilde{\\Phi }_1^E({u}_{12},{v}_{12})-\\widetilde{\\Phi }_1^E({v}_{12},{v}_{12})\\right),\\end{split}$ where $\\Phi ^{\\scriptscriptstyle \\,\\int }_1(x)$ is defined in (REF ).",
"At the next order, $\\widehat{\\Phi }^{{\\rm tot}}_3=\\Phi ^{\\scriptscriptstyle \\,\\int }_3+2\\,{\\rm Sym}\\, \\Bigl \\lbrace \\Phi ^{\\scriptscriptstyle \\,\\int }_2(x_{1+2},x_3)\\Phi ^{\\,\\widehat{g}}_2(x_1,x_2) \\Bigr \\rbrace +\\Phi ^{\\scriptscriptstyle \\,\\int }_1 \\Phi ^{\\,\\widehat{g}}_3.$ To get an explicit expression in terms of smooth solutions of Vignéras' equation, one should use the relation (REF ).",
"Applying it to the case $n=2$ with $\\mathcal {V}=({u}_{1,2+3},{u}_{1+2,3})$ , using the orthogonality properties ${u}_{(1,2+3)\\perp (1+2,3)}={u}_{12},\\qquad {u}_{(1+2,3)\\perp (1,2+3)}={u}_{23},$ and acting by the operator $\\mathcal {D}({v}_{12})\\mathcal {D}({v}_{23})$ , one can show that $&&\\,{\\rm Sym}\\, \\Bigl \\lbrace \\widetilde{\\Phi }^E_{2}\\bigl (({u}_{1,2+3},{u}_{1+2,3}), ({v}_{12},{v}_{23})\\bigr )\\Bigr \\rbrace =\\,{\\rm Sym}\\, \\biggl \\lbrace \\widetilde{\\Phi }^M_{2}\\bigl (({u}_{1,2+3},{u}_{1+2,3}), ({v}_{12},{v}_{23})\\bigr )\\nonumber \\\\&& \\qquad +({v}_{12},{x})\\,\\mbox{sgn}({u}_{12},{x})\\, \\widetilde{\\Phi }^M_1\\bigl ({u}_{1+2,3},{v}_{1+2,3}\\bigr )+({v}_{12},{x})\\,({v}_{23},{x})\\,\\mbox{sgn}({u}_{1,2+3},{x})\\,\\mbox{sgn}({u}_{1+2,3},{x})\\nonumber \\\\&& \\qquad -\\frac{(p_1p_2p_3)}{6\\pi }\\biggr \\rbrace .$ This result allows to obtain the following representation for the kernel $\\widehat{\\Phi }^{{\\rm tot}}_3&=&\\frac{1}{8}\\,\\Phi ^{\\scriptscriptstyle \\,\\int }_1 \\,{\\rm Sym}\\, \\biggl \\lbrace \\widetilde{\\Phi }^E_{2}\\bigl (({u}_{1,2+3},{u}_{1+2,3}), ({v}_{12},{v}_{23})\\bigr )- \\widetilde{\\Phi }^E_{2}\\bigl (({v}_{1,2+3},{v}_{1+2,3}), ({v}_{12},{v}_{23})\\bigr )\\nonumber \\\\&&-\\left(\\widetilde{\\Phi }_1^E({u}_{1+2,3},{v}_{1+2,3})-\\widetilde{\\Phi }_1^E({v}_{1+2,3},{v}_{1+2,3})\\right)\\widetilde{\\Phi }^E_1({v}_{12},{v}_{12})\\biggr \\rbrace .$ Using the identity between functions $\\widetilde{\\Phi }^E_{2}$ implied by the identity (REF ), the kernel can also be rewritten as $\\widehat{\\Phi }^{{\\rm tot}}_3&=&\\frac{1}{8}\\,\\Phi ^{\\scriptscriptstyle \\,\\int }_1 \\,{\\rm Sym}\\, \\biggl \\lbrace \\frac{2}{3}\\, \\Bigl (\\widetilde{\\Phi }^E_{2}\\bigl (({u}_{1,2+3},{u}_{1+2,3}), ({v}_{12},{v}_{1+2,3})\\bigr )- \\widetilde{\\Phi }^E_{2}\\bigl (({v}_{1,2+3},{v}_{1+2,3}), ({v}_{12},{v}_{1+2,3})\\bigr )\\Bigr )\\nonumber \\\\&&-\\left(\\widetilde{\\Phi }_1^E({u}_{1+2,3},{v}_{1+2,3})-\\widetilde{\\Phi }_1^E({v}_{1+2,3},{v}_{1+2,3})\\right)\\widetilde{\\Phi }^E_1({v}_{12},{v}_{12})\\biggr \\rbrace ,$ so that the vectors appearing in the second argument of $\\widetilde{\\Phi }_2^E$ are now mutually orthogonal.",
"For $n=4$ , the kernel is given by $\\begin{split}\\widehat{\\Phi }^{{\\rm tot}}_4=&\\, \\Phi ^{\\scriptscriptstyle \\,\\int }_4+\\,{\\rm Sym}\\, \\Bigl \\lbrace 3\\Phi ^{\\scriptscriptstyle \\,\\int }_3(x_{1+2},x_3,x_4)\\Phi ^{\\,\\widehat{g}}_2(x_1,x_2)+\\Phi ^{\\scriptscriptstyle \\,\\int }_2(x_{1+2},x_{3+4})\\Phi ^{\\,\\widehat{g}}_2(x_1,x_2)\\Phi ^{\\,\\widehat{g}}_2(x_3,x_4)\\\\&\\,+2\\Phi ^{\\scriptscriptstyle \\,\\int }_2(x_{1+2+3},x_4)\\Phi ^{\\,\\widehat{g}}_3(x_1,x_2,x_3)\\Bigr \\rbrace +\\Phi ^{\\scriptscriptstyle \\,\\int }_1 \\Phi ^{\\,\\widehat{g}}_4.\\end{split}$ Proceeding in the same way as for $n=3$ , obtaining a generalization of (REF ) to $n=4$ , one arrives at $\\widehat{\\Phi }^{{\\rm tot}}_4&=&\\frac{1}{16}\\, \\Phi ^{\\scriptscriptstyle \\,\\int }_1\\,{\\rm Sym}\\, \\biggl \\lbrace \\widetilde{\\Phi }^E_{3}\\bigl (({u}_{1,2+3+4},{u}_{1+2,3+4},{u}_{1+2+3,4}), ({v}_{12},{v}_{23},{v}_{34})\\bigr )\\nonumber \\\\&&\\qquad -\\widetilde{\\Phi }^E_{3}\\bigl (({v}_{1,2+3+4},{v}_{1+2,3+4},{v}_{1+2+3,4}), ({v}_{12},{v}_{23},{v}_{34})\\bigr )\\nonumber \\\\&&\\quad +\\frac{1}{3}\\left(\\widetilde{\\Phi }^E_{3}\\bigl (({u}_{1,2+3+4},{u}_{1+2+4,3},{u}_{1+2+3,4}), ({v}_{12},{v}_{24},{v}_{34})\\bigr )\\right.\\nonumber \\\\&&\\qquad \\left.-\\widetilde{\\Phi }^E_{3}\\bigl (({v}_{1,2+3+4},{v}_{1+2+4,3},{v}_{1+2+3,4}), ({v}_{12},{v}_{24},{v}_{34})\\bigr )\\right)\\nonumber \\\\&&-\\left(\\widetilde{\\Phi }^E_{2}\\bigl (({u}_{1+2,3+4},{u}_{1+2+3,4}), ({v}_{1+2,3},{v}_{34})\\bigr )-\\widetilde{\\Phi }^E_{2}\\bigl (({v}_{1+2,3+4},{v}_{1+2+3,4}), ({v}_{1+2,3},{v}_{34})\\bigr )\\right.\\nonumber \\\\&&\\quad +{1\\over 2}\\left(\\widetilde{\\Phi }^E_{2}\\bigl (({u}_{1+2+4,3},{u}_{1+2+3,4}), ({v}_{1+2,3},{v}_{1+2,4})\\bigr )\\right.\\\\&&\\quad \\qquad \\left.\\left.-\\widetilde{\\Phi }^E_{2}\\bigl (({v}_{1+2+4,3},{v}_{1+2+3,4}), ({v}_{1+2,3},{v}_{1+2,4})\\bigr )\\right)\\right)\\widetilde{\\Phi }^E_1({v}_{12},{v}_{12})\\nonumber \\\\&&-\\left(\\widetilde{\\Phi }_1^E({u}_{1+2+3,4},{v}_{1+2+3,4})-\\widetilde{\\Phi }_1^E({v}_{1+2+3,4},{v}_{1+2+3,4})\\right)\\widetilde{\\Phi }^E_{2}\\bigl (({v}_{1,2+3},{v}_{1+2,3}), ({v}_{12},{v}_{23})\\bigr )\\nonumber \\\\&&+\\left(\\widetilde{\\Phi }_1^E({u}_{1+2+3,4},{v}_{1+2+3,4})-\\widetilde{\\Phi }_1^E({v}_{1+2+3,4},{v}_{1+2+3,4})\\right)\\widetilde{\\Phi }^E_1({v}_{12},{v}_{12})\\widetilde{\\Phi }^E_1({v}_{1+2,3},{v}_{1+2,3})\\nonumber \\\\&&+\\frac{1}{4}\\left(\\widetilde{\\Phi }_1^E({u}_{1+2,3+4},{v}_{1+2,3+4})-\\widetilde{\\Phi }_1^E({v}_{1+2,3+4},{v}_{1+2,3+4})\\right)\\widetilde{\\Phi }^E_1({v}_{12},{v}_{12})\\widetilde{\\Phi }^E_1({v}_{34},{v}_{34})\\biggr \\rbrace .\\nonumber $ It is possible also to rewrite this expression in terms of generalized error functions $\\widetilde{\\Phi }^E_n(\\mathcal {V},\\tilde{\\mathcal {V}})$ where the vectors entering the second argument are mutually orthogonal, as in (REF ).",
"The reader can easily guess the result by comparing (REF ) and (REF ).",
"The explicit results for $\\widehat{\\Phi }^{{\\rm tot}}_n$ , $n\\le 4$ , presented above are the basis for the conjectural formula (REF ).",
"Note that all terms in these expressions have the sum of ranks of the generalized error functions equal to $n-1$ .",
"This shows that all contributions due to trees with non-zero number of marks cancel in the sum over Schröder trees.",
"toc"
],
[
"Index of notations",
"lp10cml Symbol Description Appears/defined in $\\mathcal {A}(\\mathcal {T}) $ contribution of tree $\\mathcal {T}$ to the integrand of the multi-instanton expansion of $H_\\gamma $ and $\\mathcal {G}$ (REF ) $a_\\mathcal {T}$ coefficient of unrooted labelled tree $\\mathcal {T}$ in $\\mathcal {D}_{m}(\\lbrace \\check{\\gamma }_s\\rbrace )$ (REF ), (REF ) $\\beta _{k\\ell }$ DSZ product $\\langle \\gamma _1+\\dots +\\gamma _k,\\gamma _\\ell \\rangle $ (REF ) $b_2=b_2(\\mathfrak {Y})$ second Betti number of $\\mathfrak {Y}$ p. REF $b^a=\\,{\\rm Re}\\,(z^a)$ periods of the Kalb-Ramond field p. REF $b_n$ rational coefficients in the expansion of $F^{\\rm (ref)}_n$ (REF ), (REF ) $c_i$ stability parameters (REF ) $c_i^{(\\ell )}$ stability parameters after attractor flow (REF ) $c_{2,a}$ components of the second Chern class of $\\mathfrak {Y}$ (REF ) $d=n b_2$ dimension of the lattice $\\mathbf {\\Lambda }=\\oplus _{i=1}^n \\Lambda _i$ p. REF $d_n,d_T$ rational weights in the representation of $g^{(0)}_n$ via flow tree (REF ),(REF ) $\\Delta (T)$ , $\\Delta _{\\gamma _L\\gamma _R}^z$ sign factors assigned to attractor flow tree $T$ (REF ) $\\mathcal {D}_{\\mathfrak {h}}$ Maass raising operator (REF ) $\\mathcal {D}({v})$ modular-covariant derivative contracted with vector ${v}$ (REF ) $\\mathcal {D}_{m}(\\lbrace \\check{\\gamma }_s\\rbrace ) $ derivative operator assigning weight to vertices with $m$ marks (REF ) $E_n(\\mathcal {M};\\mathbb {u})$ generalized error function on ${R}^n$ () $\\mathcal {E}_n=\\mathcal {E}^{(0)}_n+\\mathcal {E}^{(+)}_n$ function encoding the modular completion (REF ), (REF ) $\\phi $ (logarithm of) contact potential on $\\mathcal {M}_H$ (REF ) $\\Phi _n^E$ , $\\Phi _n^M$ boosted (complementary) error functions (REF ) $\\widetilde{\\Phi }_{n,m}^E$ , $\\widetilde{\\Phi }_{n,m}^M$ uplifted boosted error functions in the kernel of $\\mathcal {V}_ m$ (denoted by $\\widetilde{\\Phi }^E_n$ , $\\widetilde{\\Phi }^M_n$ when $n=m$ ) (REF ) $\\Phi _\\mathcal {T}$ contribution of tree $\\mathcal {T}$ to $\\Phi ^{\\scriptscriptstyle \\,\\int }_n$ (REF ), (REF ) $\\Phi ^{\\scriptscriptstyle \\,\\int }_n$ kernel defined by twistorial integrals (REF ), (REF ) $\\Phi ^{\\,\\mathcal {E}}_n$ kernel corresponding to function $\\mathcal {E}_n$ (REF ) $\\widetilde{\\Phi }^{\\,\\mathcal {E}}_n$ kernel promoting $g^{(0)}_n$ to a solution of Vignéras' equation (REF ) $\\Phi ^{\\,g}_{n}$ kernel corresponding to the tree index (REF ), (REF ) $\\Phi ^{\\,\\widehat{g}}_{n}$ kernel corresponding to completed tree index $\\widehat{g}_n$ (REF ) $\\Phi ^{{\\rm tot}}_n$ total kernel in the expansion of $\\mathcal {G}$ in terms of $h_{p,\\mu }$ (REF ) $\\widehat{\\Phi }^{{\\rm tot}}_n$ total kernel in the expansion of $\\mathcal {G}$ in terms of $\\widehat{h}_{p,\\mu }$ (REF ) $F(X)$ holomorphic prepotential p. REF $F_{{\\rm tr},n} (\\lbrace \\gamma _{i}\\rbrace ,z^a)$ partial tree index (REF ) $F^{\\rm (ref)}_{n} (\\lbrace c_i\\rbrace )$ partial contribution in $g^{\\rm (ref)}_n$ (REF ) $\\mathcal {F}$ image of $\\mathcal {G}$ under Euler operator (REF ) $\\Gamma $ charge lattice inside $H_{\\rm even}(\\mathfrak {Y},{Q})$ (REF ) $\\Gamma _+$ positive cone in the charge lattice (REF ) $\\Gamma _{k\\ell }, \\Gamma _e$ sums of DSZ products (REF ), (REF ) $\\gamma = (0,p^a,q_a,q_0)$ charge vector of a generic D4 (or D3) brane (REF ) $\\check{\\gamma }=(p^a,q_a)$ projection of $\\gamma $ on $H_4(\\mathfrak {Y},\\mathbb {Q})\\oplus H_2(\\mathfrak {Y},\\mathbb {Q})$ (REF ) $\\gamma _{ij}=\\langle \\gamma _i,\\gamma _j\\rangle $ Dirac-Schwinger-Zwanziger product, or Euler pairing (REF ) $g_{{\\rm tr},n}(\\lbrace \\gamma _i\\rbrace ,z^a)$ tree index, also denoted by $g_{{\\rm tr},n}(\\lbrace \\gamma _i,c_i\\rbrace )$ (REF ) $\\widehat{g}_n(\\lbrace \\check{\\gamma }_i\\rbrace ,z^a,\\tau _2)$ completed tree index, also denoted by $\\widehat{g}_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace )$ (REF ), (REF ) $g^{(0)}_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace )$ seed term in recursion for $\\widehat{g}_n$ (REF ) $g^{\\rm (ref)}_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace ,y)$ refined version of $g^{(0)}_n$ (REF ) $\\mathcal {G}$ instanton generating function (REF ) $\\mathcal {G}_n(\\lbrace \\gamma _i,z_i\\rbrace )$ integrand in the $n$ -instanton contribution to $\\mathcal {G}$ (REF ) $G_n(\\lbrace \\check{\\gamma }_i,c_i\\rbrace ;\\tau _2)$ large ${x}$ limit of $\\widetilde{\\Phi }_n({x})$ (REF ), (REF ) $h^{\\rm DT}_{p,q}(\\tau ,z^a)$ generating function of DT invariants (REF ) $h_{p,\\mu }(\\tau ) $ generating function of MSW invariants () $\\widehat{h}_{p,\\mu }(\\tau )$ modular completion of $h_{p,\\mu }(\\tau )$ (REF ) $H_\\gamma (z)$ generator of contact transformation across $\\ell _\\gamma $ (REF ) $H^{\\rm cl}_\\gamma (z)$ classical, large volume limit of $H_\\gamma (z)$ (REF ) $\\kappa (T)$ weight of attractor flow tree $T$ (REF ) $\\kappa (x)$ BPS index of two-centered solutions (REF ) $\\kappa _{abc}$ intersection numbers on a fixed basis of $H_4(\\mathfrak {Y})$ p. REF $K_{\\gamma _1\\gamma _2}(z_1,z_2)$ integration kernel in large volume limit (REF ) $\\hat{K}_{ij}(z_i,z_j)$ rescaled integration kernel in large volume limit (REF ) $\\mathcal {J}_n(\\lbrace \\check{\\gamma }_i\\rbrace ,\\tau _2)$ coefficient in the formula for the shadow of $\\widehat{h}_{p,\\mu }$ (REF ) $\\Lambda =H_4(\\mathfrak {Y},{Z})$ lattice equipped with quadratic form $\\kappa _{ab}=\\kappa _{abc} p^a$ p. REF $\\Lambda _i=H_4(\\mathfrak {Y},{Z})$ lattice equipped with quadratic form $\\kappa _{i,ab}=\\kappa _{abc} p_i^a$ p. REF $\\mathbf {\\Lambda }$ lattice $\\mathbf {\\Lambda }=\\oplus _{i=1}^n \\Lambda _i$ spanned by $n$ D1-brane charges p. REF $\\lambda $ eigenvalue under Vignéras' operator (REF ) $\\ell _\\gamma $ BPS ray on the twistor fiber, or its large volume limit (REF ), p. REF $\\mu _a$ residue class of $q_a$ modulo spectral flow (REF ) $M_n(\\mathcal {M};\\mathbb {u})$ generalized complementary error function on ${R}^n$ (REF ) $\\mathcal {M}_{\\alpha \\beta }$ matrix of parameters in the generalized error functions (REF ), () $\\mathcal {M}_{\\mathcal {K}}(\\mathfrak {Y})$ complexified Kähler moduli space of $\\mathfrak {Y}$ p. REF $\\mathcal {M}_H$ hypermultiplet moduli space in IIB/$\\mathfrak {Y}$ , or vector multiplet moduli space in IIA/$(\\mathfrak {Y}\\times S^1)=$ M/$(\\mathfrak {Y}\\times T^2)$ p. $n_T$ number of vertices of rooted tree $T$ excluding the leaves (REF ) $n_v(T)$ number of descendants of the vertex $v$ in $T$ plus one (REF ) $n_\\mathfrak {v}$ valency of vertex $\\mathfrak {v}$ of an unrooted tree (REF ) $p^a$ homology class of the divisor wrapped by the D3-brane (REF ) $\\mathcal {P}_{m}(\\lbrace p_s\\rbrace )$ weight of a vertex with $m$ marks in $G_n$ (REF ) $\\hat{q}_0$ invariant D0-brane charge (REF ) $Q_n(\\lbrace \\check{\\gamma }_i\\rbrace )$ difference of quadratic forms for constituents and the total charge (REF ) $R_n(\\lbrace \\check{\\gamma }_i\\rbrace ,\\tau _2)$ non-holomorphic correction in the completion $\\widehat{h}_{p,\\mu }(\\tau )$ (REF ), (REF ) $\\sigma _\\gamma $ quadratic refinement (REF ), (REF ) $S_k, S_e$ sums of stability parameters (REF ), (REF ) $S^{\\rm cl}_p$ classical action of a D3-instanton in large volume limit (REF ) $\\vartheta _{{p},{\\mu }}\\bigl (\\Phi ,\\lambda )$ indefinite theta series with kernel $\\Phi $ (REF ) $\\tau =\\tau _1+I\\tau _2$ 4D axio-dilaton in IIA, or torus modulus in M theory p. REF $t$ complex coordinate on the twistor fiber p. REF $t^a=\\,{\\rm Im}\\,(z^a)$ Kähler moduli on $\\mathfrak {Y}$ p. REF $T$ rooted tree with charges assigned to the leaves footnote REF $\\mathcal {T}$ tree with charges assigned to vertices footnote REF ${T}_n, {T}_n^\\ell $ set of unrooted (labelled) trees with $n$ vertices (REF ) ${T}_{n,m}, {T}_{n,m}^\\ell $ set of unrooted (labelled) trees with $n$ vertices and $m$ marks (REF ) ${T}_n^{\\rm r}$ set of rooted trees with $n$ vertices (REF ) ${T}_n^{\\rm af}$ set of attractor flow trees with $n$ leaves (REF ) ${T}_n^{\\rm S}$ set of Schröder trees with $n$ leaves (REF ) ${T}_{2m+1}^{(3)}$ set of rooted ternary trees with $n$ leaves (REF ) $u^\\Lambda =(1,u^a)$ complex structure moduli of mirror threefold ${\\widehat{\\mathfrak {Y}}}$ p. REF ${u}_{ij}$ , ${u}_e$ , ${u}_\\ell $ vectors in ${R}^d$ associated to $-2\\,{\\rm Im}\\,[Z_{\\gamma _i}\\bar{Z}_{\\gamma _j}]$ , $-S_e$ and $-S_\\ell $ (REF ), (REF ), (REF ) ${v}_{ij}$ , ${v}_e$ , ${v}_\\ell $ vectors in ${R}^d$ associated to $\\langle \\gamma _i,\\gamma _j\\rangle $ , $\\Gamma _e$ and $-\\Gamma _{n\\ell }$ (REF ), (REF ), (REF ) $V_T$ set of vertices of rooted tree $T$ excluding leaves p. REF $V_\\lambda $ Vignéras' operator (REF ) $\\mathcal {V}_m$ weight of a vertex with $m$ marks in large ${x}$ limit of $\\widetilde{\\Phi }^{\\,\\mathcal {E}}_n$ (REF ) $\\tilde{\\mathcal {V}}_m$ weight of a vertex with $m$ marks in $g^{(0)}_n$ (REF ) $W_n(\\lbrace \\check{\\gamma }_i\\rbrace ,\\tau _2)$ coefficient in the formula for $h_{p,\\mu }$ in terms of $\\widehat{h}_{p_i,\\mu _i}$ (REF ) $\\mathcal {X}_\\gamma $ holomorphic Fourier modes on the twistor space of $\\mathcal {M}_H$ (REF ) $\\mathcal {X}^{\\rm sf}_\\gamma $ semi-flat limit of $\\mathcal {X}_\\gamma $ (REF ) $\\mathcal {X}^{\\rm cl}_\\gamma $ classical limit of $\\mathcal {X}_\\gamma $ (REF ) $\\mathcal {X}^{(\\theta )}_{p,q}$ $\\hat{q}_0$ -independent part of $\\mathcal {X}^{\\rm cl}_\\gamma $ (REF ) ${x}$ $d$ -dimensional vector, argument of kernels of theta series p. REF $z$ coordinate on the twistor fiber, after Cayley transf.",
"(REF ) $z_\\gamma $ saddle point on twistor fiber (REF ) $z^a=b^a+It^a$ complexified Kähler moduli of $\\mathfrak {Y}$ p. REF $z^a_*(\\gamma )$ attractor moduli for charge $\\gamma $ p. REF $z^a_\\infty (\\gamma )$ large volume attractor point for charge $\\gamma $ (REF ) $Z_\\gamma (z^a)$ central charge (REF ) $\\Omega (\\gamma ,z^a)$ generalized Donaldson-Thomas invariant p. REF $\\bar{\\Omega }(\\gamma ,z^a)$ rational DT invariant (REF ) $\\bar{\\Omega }_*(\\gamma )$ attractor index p. REF ${\\bar{\\Omega }}^{\\rm MSW}(\\gamma )$ MSW invariant, also denoted by $\\bar{\\Omega }_{p,\\mu }( \\hat{q}_0)$ p. REF toc"
]
] | 1808.08479 |
[
[
"Massive gravity with Lorentz symmetry breaking: black holes as heat\n engines"
],
[
"Abstract In extended phase space, a static black hole in massive gravity is studied as a holographic heat engine.",
"In the massive gravity theory considered, the graviton gain a mass due to Lorentz symmetry breaking.",
"Exact efficiency formula is obtained for a rectangle engine cycle for the black hole considered.",
"The efficiency is computed by varying two parameters in the theory, the scalar charge Q and $\\lambda$.",
"The efficiency is compared with the Carnot efficiency for the heat engine.",
"It is observed that when Q and $\\lambda$ are increased that the efficiency for the rectangle cycle increases.",
"When compared to the Schwarzschild AdS black hole, the efficiency for the rectangle cycle is larger for the Massive gravity black hole."
],
[
" Introduction",
"Black holes in anti-de Sitter space as a thermodynamical system has attracted lot of attention in a variety of contexts: when the negative cosmological constant is taken as the thermodynamical pressure of the black hole with the relation $ P = - \\frac{\\Lambda }{ 8 \\pi }$ , the resulting thermodynamics lead to interesting features.",
"In this extended phase space, the first law of thermodynamics is modified by a term $V dP$ and the mass $M$ of the black hole is treated as the enthalpy rather than the internal energy $E$ of the black hole [1] [2].",
"Many black holes in the context of extended phase space has demonstrated Van der Waals type phase transitions between small and large black holes.",
"Due to the large number of work published related to this topic we will mention few here: [3] [4] [5] [6] [7][8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18][19].",
"There is a nice review on Black hole chemistry written by Kubiznak et.al.",
"which gives a comprehensive summary of the interesting thermodynamical features of black holes with a cosmological constant [20].",
"In classical thermodynamics, there are four basic thermodynamical processes.",
"They are isothermal, adiabatic, isobaric, and isochoric processes.",
"In each of these processes the thermodynamical quantities temperature, entropy, pressure, and, volume are kept constant respectively.",
"In a heat engine, a thermodynamical cycle is chosen which consist of aforementioned processes.",
"For example, the Carnot cycle, which has the highest efficiency, have two isothermal and two adiabatic processes in the cycle.",
"Brayton cycle has two isobaric and two adiabatic processes.",
"In this paper, the goal is to study a black hole in massive gravity as a heat engine.",
"The idea that black holes could be used as a working substance in a heat engine was first presented by Johnson [21].",
"In that paper, charged black hole in AdS space in $D = 4$ was presented as an example.",
"In an extension of that work, Johnson presented the Born-Infeld AdS black hole as a heat engine in [22].",
"In [23], heat engines from dilatonic Born-Infeld black holes were analyzed.",
"Black holes in conformal gravity as heat engine was presented by Xu et.al.",
"in [24].",
"Charged BTZ black hole in 2+1 dimensions as a heat engine was studied by Mo et.al.",
"in [25].",
"Effects of dark energy on the efficiency of heat engines of AdS black holes were analyzed in detail by Liu and Meng in [26].",
"Class of black holes in massive gravity as heat engines were discussed by Hendi et.al.",
"[27].",
"The first study of rotating black holes as holographic heat engines were done by Henniger et.al.",
"[28].",
"Heat engines are defined for space-times that are not black holes as well: Johnson [29] studied Taub-Bolt space-time as an example of a heat engine and compared with the analogous Schwarzschild black hole as a heat engine.",
"Massive gravity theories have become popular as a mean of explaining accelerated expansion of the universe without having to introduce a component of “dark energy.” There are many theories of massive gravity in the literature.",
"There are large volume of work related to theories in massive gravity [30] [31] [32] [33]: here we will mention two nice reviews on the subject by de Rham [34] and Hinterbichler [35].",
"In this paper we will consider a massive gravity theory where the graviton acquire a mass due to the Lorenz symmetry breaking.",
"Here, Higgs mechanism for gravity is introduced with space-time depending scalar fields that are coupled to gravity [36].",
"The resulting theory will exhibit modified gravitational interactions at large scale.",
"These models are free from ghosts and tachyonic instabilities around curved space as well as in flat space [37] [38].",
"A review of Lorentz violating massive gravity theory can be found in [39] [40].",
"The details of the theory is discussed in section 2.",
"The paper is organized as follows: in section 2, the black hole in massive gravity is introduced.",
"Thermodynamics and phase transitions of the massive gravity black hole are discussed in section 3.",
"Black hole as a heat engine with a rectangle cycle is introduced in section 4 and in section 5 the efficiencies are computed.",
"Finally the conclusion is given in section 6."
],
[
" Introduction to AdS black holes in massive gravity with Lorentz symmetry breaking",
"The action for the massive gravity theory considered in this paper given by, $S = \\int d^4 x \\sqrt{-g } \\left[ - \\frac{1}{ 16 \\pi } \\mathcal {R} + \\Omega ^4 \\mathcal {F}( X, W^{ij}) \\right]$ Here $\\mathcal {F}$ is a function of our scalar fields $\\phi ^{\\mu }$ .",
"$\\phi ^{\\mu }$ are minimally coupled to gravity by covariant derivatives.",
"$\\mathcal {F}$ depends on two combinations of the scalar fields given by $X$ and $W$ defined in terms of the scalar fields as, $X = \\frac{\\partial ^{\\mu } \\phi ^0 \\partial _{\\mu } \\phi ^0 }{ \\Omega ^4}$ $W^{i j} = \\frac{\\partial ^{\\mu } \\phi ^i \\partial _{\\mu } \\phi ^j}{ \\Omega ^4} - \\frac{\\partial ^{\\mu } \\phi ^i \\partial _{\\mu } \\phi ^0 \\partial ^{\\nu } \\phi ^j \\partial _{\\nu } \\phi ^0 }{ \\Omega ^8 X}$ The constant $\\Omega $ has dimensions of mass: it is in the order of $ \\sqrt{ m_g M_{pl}}$ where $m_g$ is the graviton mass and $M_{pl}$ the Plank mass [39] [36] [37] [38].",
"The scalar fields $\\phi ^0, \\phi ^i$ are responsible for breaking Lorentz symmetry spontaneously when they acquire a vacuum expectation value.",
"More details on the theory can be found in [39] [36] [37] [38] [40].",
"A class of black hole solutions to the above action was derived in [41] and [42].",
"$ ds^2 = - f(r) dt^2 + \\frac{ dr^2}{ f(r)} + r^2 ( d \\theta ^2 + sin^2 \\theta d \\phi ^2)$ where, $ f(r) = 1 - \\frac{ 2 M}{ r} - \\gamma \\frac{ Q^2}{r^{\\lambda }} - \\frac{\\Lambda r^2}{3}$ Here, the parameter $Q$ represents a scalar charge related to massive gravity and $\\gamma = \\pm 1$ .",
"The constant $\\lambda $ in is an integration constant and is positive.",
"$\\Lambda $ is the cosmological constant.",
"A detailed description of this black hole is given by Fernando in [43].",
"Since for $\\lambda <1$ , the ADM mass become divergent, such solutions will not be considered.",
"When $\\lambda >1$ , for $ r \\rightarrow \\infty $ the metric approaches the Schwarzschild-AdS black hole with a finite mass $M$ ; hence we will choose $\\lambda >1$ in the rest of the paper.",
"When $ \\gamma = 1$ , the space-time of the black hole is very similar to the Schwarzschild-AdS black hole and for $\\gamma = -1$ , the geometry is similar to the Reissner-Nordstrom-AdS charged black hole.",
"There are several works related to the black holes given above.",
"P-V criticality and phase transitions of masive gravity black holes with a negative cosmological constant were presented by Fernando in [43] [44].",
"Thermodynamics for $\\Lambda =0$ case were studied in [45] [46].",
"Stability and quasi normal modes were studied in [47][48] [49] [50]."
],
[
"Thermodynamics and phase transitions",
"In this section we will derive the thermodynamical quantities and present the phase transitions that could occur in massive gravity black holes."
],
[
" Defining the thermodynamical quantities",
"The temperature of the black hole could be obtained by using the definition of Hawking temperature: it is based on the surface gravity $\\kappa $ at the outer horizon, $r_h$ .",
"It is given by, $T_H = \\frac{ \\kappa }{ 2 \\pi } = \\frac{ 1}{ 4 \\pi } \\left| \\frac{ df(r)}{ dr} \\right|_ { r = r_h} = \\frac{1}{ 4 \\pi } \\left(\\frac{ 2 M}{ r_h^2} + \\frac{ \\gamma Q^2 \\lambda }{ r_h^{ \\lambda + 1}} - \\frac{ 2 \\Lambda r_h}{ 3}\\right)$ Since $ f(r_h) = 0$ , the mass of the black hole could be written as, $M = r_h \\left( \\frac{1}{2} - \\frac{ \\gamma Q^2}{ 2 r_h^{( \\lambda )} }- \\frac{ r_h^2 \\Lambda }{6} \\right)$ The value of the mass $M$ could be substituted to eq$(\\ref {tempe1})$ and rewrite temperature in terms of $r_h, Q,$ and $\\lambda $ as, $T = \\frac{ 1 }{ 4 \\pi } \\left( \\frac{1}{ r_h} - r_h \\Lambda + \\frac{ \\gamma ( \\lambda -1)Q^2}{ r_h^{ \\lambda +1} } \\right)$ The conjugate quantity for temperature, the entropy S is given by the area law, $ S = \\pi r_h^2$ for this black hole.",
"As discussed in the introduction, in the extended phase space, the pressure $P$ is given by, $P = -\\frac{ \\Lambda }{ 8 \\pi }$ The conjugate quantity for P, the volume $V$ , is given by, $\\frac{ 4 \\pi r_h^3}{3}$ .",
"The scalar potential which is conjugated to the scalar charge $Q$ is given by, $\\Phi = - \\frac{ \\gamma Q}{ r_h^{ \\lambda -1}}$ One could rewrite the temperature in terms of $S$ and $P$ as follows: $T_H = \\frac{1}{ 4 \\sqrt{ \\pi S}} \\left( 1 + 8 P S + \\gamma ( \\lambda - 1) Q^2 \\left(\\frac{\\pi }{ S}\\right)^{\\lambda /2} \\right)$ The temperature is plotted against the entropy $S$ in Fig.$(\\ref {temp1})$ and Fig.$(\\ref {temp2})$ .",
"For $\\gamma =1$ , the temperature has a minimum: hence black holes cannot exist below this minimum for a given $\\Lambda $ value.",
"For $\\gamma =-1$ , the temperature could have an inflection point.",
"Figure: The figure shows TT vs SS for γ=1 \\gamma = 1.",
"Here λ=2.305,P=0.905\\lambda = 2.305, P = 0.905, and Q=0.115 Q = 0.115.Figure: The figure shows TT vs SS for γ=-1 \\gamma =-1.",
"Here λ=2.215\\lambda = 2.215, and P=0.05 P = 0.05.",
"The top graph is at Q=0Q=0 and the rest have Q=0.128,0.222,0.5Q = 0.128, 0.222, 0.5"
],
[
" First law in the extended phase space",
"In the extended phase space, the mass of the black hole M is not considered as the internal energy as it is usually done.",
"Instead, it is considered as the enthalpy H. Hence, $ M = H = U + P V$ .",
"Hence, the first law of the given black hole is given by, $dM = T dS + \\Phi dQ + V dP$ It is possible to combine the thermodynamical quantities, $M, P, V, S, T, \\Phi $ , and $Q$ to obtain the Smarr formula as, $M = 2 T S + \\frac{\\lambda }{2} \\Phi Q - 2 P V$ which also could be obtained using the scaling argument presented by Kastor et.al [1].",
"When $\\lambda =2$ , the Smarr formula simplifies to the one for the Reissner-Nordstrom-AdS black hole obtained by Kubiznak and Mann in [3]."
],
[
" Behavior of pressure",
"In eq.$(\\ref {tempe1})$ , one could substitute $ \\Lambda = - 8 \\pi P$ and rearrange the equation to obtain $P$ as a function of $r_h$ and $T$ as, $P = - \\frac{ 1}{ 8 \\pi r_h^2} + \\frac{ T}{ 2 r_h} + \\frac{ Q^2 \\gamma ( 1 - \\lambda )}{ 8 \\pi r_h^{ 2 + \\lambda }}$ Since the black hole radius $r_h$ is given by, $r_h = \\left(\\frac{ 3 V}{4 \\pi }\\right)^{1/3}$ one could rewrite $P$ in terms of $V$ as, $P = - \\frac{ 1}{ 8 \\pi } \\left( \\frac{ 4 \\pi }{ 3 V} \\right) ^ {2/3} + \\frac{T}{2} \\left( \\frac{ 4 \\pi }{ 3 V} \\right) ^ {1/3} +\\frac{ Q^2 \\gamma ( 1 - \\lambda )}{ 8 \\pi } \\left( \\frac{ 4 \\pi }{ 3 V} \\right) ^ {(\\lambda + 2)/3}$ The pressure is plotted vs V as in Fig.$(\\ref {pvsr2})$ and Fig.$(\\ref {pvsr1})$ .",
"For $\\gamma =1$ , there is a maximum pressure given by, $P_{max} = \\frac{ 4 \\pi r_h T ( 1 + \\lambda ) - \\lambda }{ 8 \\pi r_h^2 ( 2 + \\lambda )}$ Since $r_h$ and $V$ are related, $P_{max}$ also could be rewritten in terms of $V_m$ at which the pressure is maximum as, $P_{max} = \\frac{ \\left( (4 \\pi )^{2/3} ( 3 V_m)^{1/3} T ( 1 + \\lambda ) - \\lambda \\right)}{ 2 ( 2 + \\lambda ) (4 \\pi )^{1/3} ( 3 V_m)^{2/3}}$ Hence when the $P > P_{max}$ black holes dose not exist.",
"This implies that there is a maximum value of the cosmological constant that the black holes could exist for a given horizon radius (or volume).",
"For $\\gamma =-1$ , the behavior is quite different from what is of $\\gamma =1$ .",
"There are critical points as demonstrated from Fig$(\\ref {pvsr1})$ .",
"Figure: The figure shows PP vs r h r_h for γ=1 \\gamma =1 for varying temperature.",
"Here λ=2.865\\lambda = 2.865, and Q=0.334 Q = 0.334.",
"For large temperature the peak is higher.Figure: The figure shows PP vs VV for γ=-1 \\gamma = -1 for varying temperature.",
"Here λ=1.98\\lambda = 1.98, and Q=0.094 Q = 0.094."
],
[
" Phase transitions",
"For $\\gamma = -1$ , there are phase transitions between small and large black holes.",
"These phase transitions are first order and are similar to Van der Waals phase transitions between gas and liquid under constant temperature.",
"A thorough analysis of the phase transitions were discussed in the paper by the current author in [43].",
"There is a critical temperature $T_c$ at which the temperature which $P$ vs $V$ curve has an inflection point.",
"At that point, $\\frac{ \\partial P}{ \\partial V } = \\frac{ \\partial ^2 P}{ \\partial V^2 } =0$ The inflection point occur at the volume $V_c$ given by, $V_c = \\frac{ 4 \\pi r_{hc}^3}{ 3}$ where $r_{hc} = \\frac{1}{2} \\left( Q^2 ( \\lambda ^2 - 1) ( \\lambda +2 ) 2 ^{\\lambda -1}\\right)^{1/\\lambda }$ The corresponding $T_c$ and $P_c$ are given by, $T_c = \\frac{ \\lambda }{ \\pi ( \\lambda +1)} \\left( Q^2 ( \\lambda ^2 - 1) ( \\lambda +2 ) 2 ^{\\lambda -1}\\right)^{-1/\\lambda }$ $P_c = \\frac{ \\lambda }{ 2 \\pi ( \\lambda +2)} \\left( Q^2 ( \\lambda ^2 - 1) ( \\lambda +2 ) 2 ^{\\lambda -1}\\right)^{-2/\\lambda }$"
],
[
" Specific heat capacities",
"There are two different heat capacities for a thermodynamical system: heat capacity at constant pressure, $C_P$ , and heat capacity at constant volume, $C_V$ .",
"They are given by, $C_P = T \\left.",
"\\frac{\\partial S}{ \\partial T}\\right|_P = \\frac{ 2 S \\left( 8 P S^{\\frac{2 + \\lambda }{2}} + S ^{\\frac{\\lambda }{2}} + \\pi ^{\\frac{\\lambda }{2}} Q^2 \\gamma ( -1 + \\lambda )\\right) }{ \\left( 8 P S^{\\frac{2 + \\lambda }{2}} - S ^{\\frac{\\lambda }{2}} - \\pi ^{\\frac{\\lambda }{2}} Q^2 \\gamma ( -1 + \\lambda ^2)\\right) }$ $C_V = T \\left.",
"\\frac{\\partial S}{ \\partial T}\\right|_V = 0$ $C_V$ is zero because the entropy $S$ is proportional to $V$ ."
],
[
" Black hole as a heat engine",
"Since the black hole is considered as a thermodynamical system with a $P dV$ term, one could extract mechanical work from the black hole.",
"The given black hole has an equation of state which clearly articulate the relation between $P$ and $V$ .",
"Net work can be extracted from a given cycle in state space.",
"If the net heat input is $Q_H$ , net heat out put is $Q_C$ and the net work out put from the system is $W_{net}$ , the relation between them are, $Q_H = W_{net} + Q_C$ ."
],
[
"Carnot cycle efficiency",
"If the given thermodynamical cycle is given by two isothermal and two isentropic paths, then the efficiency of the heat engine is given by, $\\eta _c = \\frac{ W_{net}}{ Q_H} = \\left( 1 - \\frac{ T_C}{T_H} \\right)$"
],
[
" Efficiency of a cycle with two isochoric and two isobaric paths",
"In this paper, we focus on a thermodynamical path consisting of two constant pressure and two constant volume paths as shown in Fig.$(\\ref {pvcycle})$ .",
"The heat supplied along the isobaric path $1 \\rightarrow 2$ which is the higher temperature path is given by, $Q_{1 \\rightarrow 2} = Q_H = \\int _1^2 d Q = \\int _1^2 T dS$ Since $S = \\pi r_h^2$ , eq.$(\\ref {heat1})$ can be rewritten as, $Q_H = \\int _1^2 T_H d ( \\pi r_h^2) = 2 \\pi \\int _{r_1}^{r_2} T_H(r_h) r_h dr_h$ Since the pressure $P$ is constant along the path $1 \\rightarrow 2$ , and $Q, \\lambda $ are also kept constant, $T_H$ becomes a function of $r_h$ along $1 \\rightarrow 2$ .",
"Hence the integral in eq.$(\\ref {integral})$ could be computed to be, $Q_H = \\left( \\frac{r_h}{2} - \\frac{ \\gamma Q^2}{ 2 r_h^{( \\lambda -1 )} }- \\frac{ 4 \\pi r_h^3 P}{3} \\right)^{r_2}_{r_1} = M(r_2) - M(r_1)$ Similarly the net heat out put from the system could be calculated along the path $4 \\rightarrow 3$ as, $Q_{ 3 \\rightarrow 4} = Q_{C} = M(r_3) - M(r_4)$ Therefore, the efficiency of the the heat engine for this particular cycle is, $\\eta = \\frac{W_{net}}{ Q_H} = 1 - \\frac{ (M(r_3) - M(r_4) )}{ (M(r_2) - M(r_1) )}$ The above formula was obtained by Johnson [22] by using enthalpy and the first law of thermodynamics.",
"It is also noted that since volume is a function of entropy in this black hole, isochoric path also represents adiabatic path.",
"Figure: The figure shows PP vs VV for γ=-1 \\gamma = -1 for varying temperature.",
"Here λ=1.98\\lambda = 1.98, and Q=0.094 Q = 0.094.",
"The maximum pressure, P 1 =P 2 =0.925P_1 = P_2 = 0.925 and minimum pressure, P 3 =P 4 =0.503P_3 = P_4 = 0.503."
],
[
"Efficiency of the heat engine for varying parameters of the black hole",
"The goal of this section is to compute the efficiency of the black hole heat engine and compare to the Carnot efficiency $\\eta _C$ .",
"We will restrict the black holes with $\\gamma = -1$ for the rest of the paper.",
"There are two parts to this computation: first we will change $Q$ for three values and perform the calculations, and second we will change $\\lambda $ and perform the computations.",
"In both cases, $V$ (or $r_h$ ) and the $P$ for the coordinates 1 and 4 are kept constant.",
"Then $V$ (or $r_h$ ) for coordinates 2 and 3 are varied to observe how the efficiency varies.",
"Before we proceed, some cautionary remarks are in order: for a given mass, the horizon radius $r_h$ is possible only if the parameters of the black hole are chosen appropriately.",
"To clarify this further we have shown an example in the Fig.$(\\ref {frvsmass})$ of the function $f(r)$ whose roots lead to the horizon radius $r_h$ .",
"In this example, when the mass is low, there are no horizons.",
"Only for a mass greater than a critical value that the horizon radius exists.",
"On the other hand, we also would like to make sure that for all the points on the chosen thermodynamical cycle that there is a real mass value.",
"In Fig.$(\\ref {mvsr})$ , $M$ is plotted vs $r_h$ for various values of $P$ .",
"The values of the pressures are chosen so that they are between the maximum and the minimum pressure of the cycle considered.",
"It is clear that the mass $M$ is positive for all values of $P$ chosen.",
"However, it is interesting to note that the mass has a minimum value for a given $P$ .",
"We did a spot check to make sure that for the range of $r_2$ chosen, that $M(r_2), M(r_4) > 0$ .",
"Figure: The figure shows f(r)f(r) vs rr for γ=-1 \\gamma = -1 for varying mass.",
"Here λ=4,Λ=-0.2\\lambda = 4, \\Lambda = -0.2, and Q=0.3 Q = 0.3.Figure: The figure shows MM vs r h r_h for γ=-1 \\gamma = -1 for varying pressure.",
"Here λ=1.98\\lambda = 1.98, and Q=0.094 Q = 0.094.",
"The values of the pressures are chosen so that they are between the maximum and the minimum pressures of the cycle, P 1 =P 2 =0.925P_1 = P_2 = 0.925 and P 3 =P 4 =0.503P_3 = P_4 = 0.503 respectively."
],
[
"Efficiency for varying scalar charge $Q$",
"In this section, the scalar charge $Q$ will be varied for three values.",
"Here, $P_1, P_4, r_4= r_1$ will be kept constant.",
"In the thermodynamical cycle in Fig.$(\\ref {pvcycle})$ , the smallest temperature is at $T_4$ .",
"The cycle is chosen such that it does not include the region where the phase transition occur.",
"Due to this reason, the critical temperature $T_c$ needs to be smaller than $T_4$ for all chosen $Q$ values.",
"$T_c$ is computed by eq.$(\\ref {tc})$ .",
"$T_4$ is computed as, $T_4 = \\frac{ 1 }{ 4 \\pi } \\left( \\frac{1}{ r_4} + 8 \\pi P_4 r_4 + \\frac{ \\gamma ( \\lambda -1)Q^2}{ r_4^{ \\lambda +1} } \\right)$ $T_c$ and $T_4$ are plotted together in the same graph in Fig.$(\\ref {tct4})$ for varying $Q$ .",
"For large $Q$ , $T_c > T_4$ .",
"Hence one need to choose the correct range of $Q$ values to make sure that $T_c < T_4$ .",
"Figure: The figure shows T c ,T 4 T_c, T_4 vs QQ for γ=-1 \\gamma = -1.",
"Here λ=1.98\\lambda = 1.98, and P 4 =0.509P_4 = 0.509.When $Q$ is fixed, $M(r_3)$ and $M(r_2)$ will depend only on $r_3 = r_2, P_1$ and $P_4$ .",
"Now, $\\eta , \\eta _C$ are calculated using eq.$(\\ref {eta})$ and plotted for 3 different values of $Q$ in Fig.$(\\ref {etavsq})$ .",
"Figure: The figure shows η,η c \\eta , \\eta _c vs r 2 r_2 for γ=-1 \\gamma = -1 for varying scalar charge QQ.",
"Here λ=1.98\\lambda = 1.98Figure: The figure shows η η c \\frac{\\eta }{\\eta _c} vs r 2 r_2 for γ=-1 \\gamma = -1 for varying scalar charge QQ.",
"Here λ=1.98\\lambda = 1.98"
],
[
"Efficiency for varying $\\lambda $",
"In this section, the value of $\\lambda $ is varied for three values.",
"Once again, $P_1, P_4, r_4= r_1$ will be kept constant.",
"It is also important to make sure that $T_c < T_4$ for chosen values of $\\lambda $ .",
"In Fig.$(\\ref {tct4lambda})$ , $T_c, T_4$ are plotted against $\\lambda $ .",
"It is clear that there is a value $\\lambda $ that $T_c > T_4$ .",
"Hence values of $\\lambda $ are chosen such that $T_c < T_4$ .",
"Figure: The figure shows T c ,T 4 T_c, T_4 vs λ\\lambda for γ=-1 \\gamma = -1.",
"Here Q=0.094Q = 0.094, and P 4 =0.509P_4 = 0.509.Figure: The figure shows η,η c \\eta , \\eta _c vs r 2 r_2 for γ=-1 \\gamma = -1 for varying λ\\lambda .",
"Here Q=0.094Q = 0.094Figure: The figure shows η η c \\frac{\\eta }{\\eta _c} vs r 2 r_2 for γ=-1 \\gamma = -1 for varying λ\\lambda .",
"Here Q=0.094Q = 0.094"
],
[
" Conclusion",
"In this paper, we have studied black holes in massive gravity with a negative cosmological constant.",
"in the extended phase space.",
"Here the black hole has a pressure given by, $P = -\\frac{\\Lambda }{ 8 \\pi }$ .",
"There are two types of black holes for the values of $\\gamma $ in the theory.",
"Thermodynamical behavior differ significantly for $\\gamma =1$ and $\\gamma =-1$ .",
"For $\\gamma =1$ , the black hole behaves similar to the Schwarzschild-anti-de Sitter black hole: the pressure has a maximum and the temperature has a minimum.",
"For $\\gamma =-1$ , the black hole exhibits phase transitions for certain range of temperatures: for higher temperatures, the black holes behave like an ideal gas.",
"Phase transitions are between large black holes and small black holes.",
"A detailed analysis of the black holes in this context is published by the current author in [43].",
"The main goal of this paper is to study the massive gravity black hole as a heat engine.",
"The thermodynamical cycle for the heat engine considered here is a rectangle in $P, V$ space with two isobaric and two isochoric processes.",
"The efficiency of the heat engine taking black hole as the working substance is computed for the rectangle cycle as well as for the Carnot cycle by varying $Q, \\lambda , r_2$ .",
"When $Q$ is increased, the efficiency for the rectangle cycle increases, but, the Carnot efficiency decreases.",
"The ratio $\\frac{\\eta }{\\eta _c}$ decreases when $Q$ is increased.",
"Hence to achieve better efficiency, a higher $Q$ is appropriate.",
"Larger the volume $V_2$ (or $r_2$ ), higher the $\\eta $ of the rectangle cycle.",
"When $Q =0$ , we get the Schwarzschild AdS black hole.",
"Hence from the graphs it can be concluded that the Schwarzschild AdS black hole has a smaller efficiency for the rectangle cycle compared with the massive gravity black hole.",
"When $\\lambda $ is increased, the efficiency of the rectangle cycle as the Carnot cycle increases.",
"Both efficiencies increase with volume $V_2$ (or $r_2$ ).",
"The ratio $\\eta /\\eta _c$ decreases with $\\lambda $ .",
"The Reissner-Nordstrom AdS (RNAdS) black hole has the metric with $f(r) = 1 - \\frac{2 M}{r} + \\frac{Q_e^2}{r^2} - \\Lambda r^2$ .",
"When $Q_e = Q$ and $\\lambda =2$ , the massive gravity black hole and the RNAdS black hole are the same.",
"Hence one can conclude that for $\\lambda >2$ with $ Q_e = Q$ , the massive gravity black hole will have higher efficiency.",
"When $ 1 \\le \\lambda < 2 $ , massive gravity black hole will have lower efficiency."
]
] | 1808.08595 |
[
[
"On Realizability Of Gauss Diagrams And Constructions Of Meanders"
],
[
"Abstract The problem of which Gauss diagram can be realized by plane curves is an old one and has been solved in several ways.",
"In this paper, we present a direct approach to this problem.",
"We show that needed conditions for realizability of a Gauss diagram can be interpreted as follows \"the number of exits = the number of entrances\" and the sufficient condition is based on Jordan curve Theorem.",
"We give a matrix approach of realization of Gauss diagrams and then we present an algorithm to construct meanders"
],
[
"Introduction",
"In the earliest time of the Knot Theory C.F.",
"Gauss defined the chord diagram (= Gauss diagram).",
"C.F.",
"Gauss [3] observed that if a chord diagram can be realized by a plane curve, then every chord is crossed only by an even number of chords, but that this condition is not sufficient.",
"The aim of this paper is to present a direct approach to the problem of which Gauss diagram can be realized by knots.",
"This problem is an old one, and has been solved in several ways.",
"In 1936, M. Dehn [2] found a sufficient algorithmic solution based on the existence of a touch Jordan curve which is the image of a transformation of the knot diagram by successive splits replacing all the crossings.",
"A long time after in 1976, L. Lovasz and M.L.",
"Marx [5] found a second necessary condition and finally during the same year, R.C.",
"Read and P. Rosenstiehl [7] found the third condition which allowed the set of these three conditions to be sufficient.",
"The last characterization is based on the tripartition of graphs into cycles, cocycles and bicycles.",
"In [8] the notation of oriented chord diagram was introduced and it was showed that these diagrams classify cellular generic curves on oriented surfaces.",
"As a corollary a simple combinatorial classification of plane generic curves was derived, and the problem of realizability of these diagrams was also solved.",
"However all these ways are indirect; they rest upon deep and nontrivial auxiliary construction.",
"There is a natural question: whether one can arrive at these conditions in a more direct and natural fashion?",
"We believe that the conditions for realizability of a Gauss diagram (by some plane curve) should be obtained in a natural manner; they should be deduced from an intrinsic structure of the curve.",
"In this paper, we suggest an approach, which satisfies the above principle.",
"We use the fact that every Gauss diagram $\\mathfrak {G}$ defines a (virtual) plane curve ${C}(\\mathfrak {G})$ (see [4]), and the following simple ideas: (1) For every chord of a Gauss diagram $\\mathfrak {G}$ , we can associate a closed path along the curve ${C}(\\mathfrak {G})$ .",
"(2) For every two non-intersecting chords of a Gauss diagram $\\mathfrak {G}$ , we can associate two closed paths along the curve ${C}(\\mathfrak {G})$ such that every chord crosses both of those chords correspondences to the point of intersection of the paths.",
"(3) If a Gauss diagram $\\mathfrak {G}$ is realizable (say by a plane curve ${C}(\\mathfrak {G})$ ), then for every closed path (say) ${P}$ along ${C}(\\mathfrak {G})$ we can associate a coloring another part of ${C}(\\mathfrak {G})$ into two colors (roughly speaking we get “inner” and “outer” sides of ${P}$ cf.",
"Jordan curve Theorem).",
"If a Gauss diagram is not realizable then ([4]) it defines a virtual plane curve ${C}(\\mathfrak {G})$ .",
"We shall show that there exists a closed path along ${C}(\\mathfrak {G})$ for which we cannot associate a well-defined coloring of ${C}(\\mathfrak {G})$ , i.e., ${C}(\\mathfrak {G})$ contains a path is colored into two colors.",
"Using these ideas we solve the problem of which Gauss diagram can be realized by knots.",
"We then give a matrix approach of realization of Gauss diagrams and then we present an algorithm to construct meanders."
],
[
"Preliminaries",
"Recall that classically, a knot is defined as an embedding of the circle $S^1$ into $\\mathbb {R}^3$ , or equivalently into the 3-sphere $S^3$ , i.e., a knot is a closed curve embedded on $\\mathbb {R}^3$ (or $S^3$ ) without intersecting itself, up to ambient isotopy.",
"The projection of a knot onto a 2-manifold is considered with all multiple points are transversal double with will be call crossing points (or shortly crossings).",
"Such a projection is called the shadow by the knots theorists [1], [9], following [8] we shall also call these projections as plane curves.",
"A knot diagram is a generic immersion of a circle $S^1$ to a plane $\\mathbb {R}^2$ enhanced by information on overpasses and underpasses at double points."
],
[
"Gauss Diagrams",
"A generic immersion of a circle to a plane is characterized by its Gauss diagram [6].",
"Figure: The plane curve and its Gauss diagram are shown.Definition 1.1 The Gauss diagram is the immersing circle with the preimages of each double point connected with a chord.",
"On the other words, this natation can be defined as follows.",
"Let us walk on a path along the plane curve until returning back to the origin and then generate a word $W$ which is the sequence of the crossings in the order we meet them on the path.",
"$W$ is a double occurrence word.",
"If we put the labels of the crossing on a circle in the order of the word $W$ and if we join by a chord all pairs of identical labels then we obtain a chord diagram (=Gauss diagram) of the plane curve (see Figure REF ).",
"A virtual knot diagram [4] is a generic immersion of the circle into the plane, with double points divided into real crossing points and virtual crossing points, with the real crossing points enhanced by information on overpasses and underpasses (as for classical knot diagrams).",
"At a virtual crossing the branches are not divided into an overpass and an underpass.",
"The Gauss diagram of a virtual knot is constructed in the same way as for a classical knot, but all virtual crossings are disregarded.",
"Theorem 1.2 [4] A Gauss diagram defines a virtual knot diagram up to virtual moves.",
"Arguing similarly as in the real knot case, one can define a shadow of the virtual knot (see Figure REF ).",
"Figure: The chord diagram and the shadow of the virtual knot are shown.",
"Here xx and yy are the virtual crossing points."
],
[
"Conway's Smoothing",
"We frequently use the following notations.",
"Let $K$ be a knot, ${C}$ its shadow and $\\mathfrak {G}$ the Gauss diagram of ${C}$ .",
"For every crossing $c$ of ${C}$ we denote by $\\mathfrak {c}$ the corresponding chord of $\\mathfrak {G}$ .",
"If a Gauss diagram $\\mathfrak {G}$ contains a chord $\\mathfrak {c}$ then we write $\\mathfrak {c}\\in \\mathfrak {G}$ .",
"We denote by $\\mathfrak {c}_0$ , $\\mathfrak {c}_1$ the endpoints of every chord $\\mathfrak {c}\\in \\mathfrak {G}$ .",
"We shall also consider every chord $\\mathfrak {c} \\in \\mathfrak {G}$ together with one of two arcs are between its endpoints, and a chosen arc is denoted by $\\mathfrak {c}_0\\mathfrak {c}_1$ .",
"Further, $\\mathfrak {c}_\\times $ denotes the set of all chords cross the chord $\\mathfrak {c}$ and $\\mathfrak {c}_\\parallel $ denotes the set of all chords do not cross the chord $\\mathfrak {c}$ .",
"We put $\\mathfrak {c} \\notin \\mathfrak {c}_\\times $ , and $\\mathfrak {c} \\in \\mathfrak {c}_\\parallel $ .",
"Throughout this paper we consider Gauss diagrams such that $\\mathfrak {c}_\\times \\ne \\varnothing $ for every $\\mathfrak {c \\in G}$ .",
"As well known, John Conway introduced a “surgical” operation on knots, called smoothing, consists in eliminating the crossing by interchanging the strands (Figure REF ).",
"Figure: The Conway smoothing the crossings are shown.We aim to specialize a Conway smoothing a crossing of a plane curve to an operation on chords of the corresponding Gauss diagram.",
"Let $K$ be a knot, ${C}$ its shadow, and $\\mathfrak {G}$ the Gauss diagram of ${C}$ .",
"Take a crossing point $c$ of ${C}$ and let $D_c$ be a small disk centered at $c$ such that $D_c\\cap {C}$ does not contain another crossings of ${C}$ .",
"Denote by $\\partial D_c$ the boundary of $D_c$ .",
"Starting from $c$ , let us walk on a path along the curve ${C}$ until returning back to $c$ .",
"Denote this path by ${L}_c$ and let $c c_a^lc_z^lc$ be the sequence of the points in the order we meet them on ${L}_c$ , where $ \\lbrace c_a^l,c_z^l\\rbrace = {L}_c \\cap \\partial D_c$ .",
"After returning back to $c$ let us keep walking along the curve ${C}$ in the same direction as before until returning back to $c$ .",
"Denote the corresponding path by ${R}_c$ and let $cc_a^rc_z^rc$ be the sequence of the points in the order we meet them on ${R}_c$ , where $\\lbrace c_a^r, c_z^r\\rbrace = \\partial D_c \\cap {R}_c$ .",
"Figure: The Conway smoothing the crossing cc and the chord 𝔠\\mathfrak {c} are shown.Next, let us delete the inner side of $D_c \\cap {C}$ and attach $c_l^a$ to $c_r^a$ , and $c^r_z$ to $c^l_z$ .",
"We thus get the new plane curve $\\widehat{{C}}_c$ (see Figure REF ).",
"It is easy to see that this curve is the shadow of the knot, which is obtained from $K$ by Conway's smoothing the crossing $c$ .",
"Let $\\widehat{\\mathfrak {G}}_\\mathfrak {c}$ be the Gauss diagram of $\\widehat{{C}}_c$ .",
"We shall say that the Gauss diagram $\\widehat{\\mathfrak {G}}_\\mathfrak {c}$ is obtained from the Gauss diagram $\\mathfrak {G}$ by Conway's smoothing the chord $\\mathfrak {c}$ .",
"As an immediate consequence of the preceding discussion, we get the following proposition.",
"Proposition 1.1 Let $\\mathfrak {G}$ be a Gauss diagram and $\\mathfrak {c}$ be its arbitrary chord.",
"Then $\\widehat{\\mathfrak {G}}_{\\mathfrak {c}}$ is obtained from $\\mathfrak {G}$ as follows: (1) delete the chord $\\mathfrak {c}$ , (2) if two chords $\\mathfrak {a}, \\mathfrak {b} \\in \\mathfrak {c}_\\times $ intersect (resp.",
"do not intersected) in $\\mathfrak {G}$ then they do not intersect in $\\mathfrak {\\widehat{G}_c}$ (resp.",
"intersected), (3) another chords keep their positions.",
"Indeed, let $W$ be the word which is the sequence of the crossings in the order we meet them on the curve ${C}$ .",
"Since $\\mathfrak {c}_\\times \\ne \\varnothing $ , $W$ can be written as follows $W = W_1 c W_2 c W_3$ , where $W_1,W_2,W_3$ are subwords of $W$ and at least one of $W_1,W_3$ is not empty.",
"Define $W_2^R$ as the reversal of the word $W_2$ .",
"Then, from the preceding discussion, the word $\\widehat{W}_c: = W_1 W_2^R W_3$ gives $\\widehat{\\mathfrak {G}}_{\\mathfrak {c}}$ (see Figure REF ) and the statement follows."
],
[
"Partitions of Gauss Diagrams",
"In this section we introduce notations, whose importance will become clear as we proceed.",
"Definition 2.1 Let $\\mathfrak {G}$ be a Gauss diagram and $\\mathfrak {a}$ a chord of $\\mathfrak {G}$ .",
"A $C$ -contour, denoted $C(\\mathfrak {a})$ , consists of the chord $\\mathfrak {a}$ , a chosen arc $\\mathfrak {a}_0\\mathfrak {a}_1$ , and all chords of $\\mathfrak {G}$ such that all their endpoints lie on the arc $\\mathfrak {a}_0\\mathfrak {a}_1$ .",
"We call a chord from the set $\\mathfrak {a}_\\times $ the door chord of the $C$ -contour $C(\\mathfrak {a})$.",
"Let us consider a plane curve ${C}:S^1 \\rightarrow \\mathbb {R}^2$ and let $\\mathfrak {G}$ be its Gauss diagram.",
"Every chord $\\mathfrak {c \\in G}$ correspondences to the crossing $c$ of ${C}$ .",
"Thus for every $C$ -contour $C(\\mathfrak {c})$ , we can associate a closed path ${C}(c)$ along the curve ${C}$ .",
"We call ${C}(c)$ the loop of the curve ${C}$ .",
"It is obviously that there is the one-to-one correspondence between self-intersection points of ${C}(c)$ and all chords from $C(\\mathfrak {c})$ .",
"Figure: Every CC-contour of the Gauss diagram correspondences to the closed path along the plane curve and vise versa.",
"We see that the chord 6 correspondences to the self-intersection point 6 of the dotted loop.Example 2.2 In Figure REF the plane curve ${C}$ and its Gauss diagram are shown.",
"Consider the (cyan) $C$ -contour $C(5)$ .",
"We see that ${C}(5)$ is the closed path along the curve.",
"It is the self-intersecting path and we see that the crossing 6 correspondences to the chord from the set $5_\\parallel $ .",
"Further, the red closed path ${C}(0)$ correspondences to the red $C$ -contour $C(0)$ .",
"Definition 2.3 Let $\\mathfrak {G}$ be a Gauss diagram, $\\mathfrak {a},\\mathfrak {b}$ its intersecting chords.",
"An $X$ -contour, denoted $X(\\mathfrak {a},\\mathfrak {b})$ , consists of two non-intersecting arcs $\\mathfrak {a}_0\\mathfrak {b}_0$ , $\\mathfrak {a}_1\\mathfrak {b}_1$ and all chords of $\\mathfrak {G}$ such that all their endpoints lie on $\\mathfrak {a}_0\\mathfrak {b}_0$ or on $\\mathfrak {a}_1\\mathfrak {b}_1$ .",
"A chord is called the door chord of the $X$ -contour $X(\\mathfrak {a},\\mathfrak {b})$ if only one of its endpoints belongs to $X(\\mathfrak {a},\\mathfrak {b})$ .",
"We say that the $X$ -contour $X(\\mathfrak {a},\\mathfrak {b})$ is non-degenerate if it has at least one door chord, and it does not contain all chords of $\\mathfrak {G}$ .",
"Example 2.4 Let us consider the Gauss diagram in Figure REF.",
"The black chords are the door chords of the black $X$ -contour $X(1,3)$ .",
"We see that the door chords correspondence to “entrances” and “exits” of the black closed path along the curve.",
"We also see that this $X$ -contour is non-degenerate.",
"The previous Example implies a partition of a Gauss diagram (resp.",
"a plane curve) into two parts.",
"Definition 2.5 (An $X$ -contour coloring) Given a Gauss diagram $\\mathfrak {G}$ and its an $X$ -contour.",
"Let us walk along the circle of $\\mathfrak {G}$ in a chosen direction and color all arcs of $\\mathfrak {G}$ until returning back to the origin as follows: (1) we don't colors the arcs of the $X$ -contour, (2) we use only two different colors, (3) we change a color whenever we meet an endpoint of a door chord.",
"Similarly, one can define a $C$ -contour coloring of a Gauss diagram $\\mathfrak {G}$ .",
"Remark 2.6 Let $\\mathfrak {G}$ be a Gauss diagram and ${C}$ the corresponding (may be virtual) plane curve, i.e., $\\mathfrak {G}$ determines the curve ${C}$ .",
"For every $X$ -contour $X(\\mathfrak {a},\\mathfrak {b})$ in $\\mathfrak {G}$ , we can associate the closed path along the curve ${C}$ .",
"We call this path the ${X}$ -contour and denote by ${X}(a,b)$ .",
"Similarly one can define door crossing for ${X}(a,b)$ .",
"Further, for the $X(\\mathfrak {a},\\mathfrak {b})$ -contour coloring of $\\mathfrak {G}$ , we can associate ${X}(a,b)$ -contour coloring of the curve ${C}$.",
"Next, let $\\mathfrak {G}$ be a realizable Gauss diagram determines the plane curve ${C}$ and let $X(\\mathfrak {a},\\mathfrak {b})$ be an $X$ -contour of $\\mathfrak {G}$ such that ${X}(a,b)$ is the non-self-intersecting path (= the Jordan curve).",
"Then the ${X}(a,b)$ -contour coloring of ${C}$ divides the curve ${C}$ into two colored parts, cf.",
"Jordan curve Theorem.",
"Figure: For the X(1,3)X(1,3)-contour coloring of the Gauss diagram, we associate the plane curve coloring.",
"We see that the X{X}-contour X(1,3){X}(1,3) (= black loop) divides the plane curve into two parts."
],
[
"The Even and The Sufficient Conditions",
"If a Gauss diagram can be realized by a plane curve we then say that this Gauss diagram is realizable, and non-realizable otherwise.",
"So, in this section, we give a criterion allowing verification and comprehension of whether a given Gauss diagram is realizable or not.",
"Moreover, we give an explanation allowing comprehension of why the needed condition is not sufficient for realizability of Gauss diagrams."
],
[
"The Even Condition",
"Proposition 3.1 Let ${C}:S^1 \\rightarrow \\mathbb {R}^2$ be a plane curve and $\\mathfrak {G}$ its Gauss diagram.",
"Then (1) $|\\mathfrak {a}_\\times \\cap \\mathfrak {b}_\\times | \\equiv 0 \\bmod 2$ for every two non-interesting chords $\\mathfrak {a,b \\in G}$ , (2) $|\\mathfrak {c}_\\times | \\equiv 0 \\bmod 2$ for every chord $\\mathfrak {c \\in G}$ .",
"Let $\\mathfrak {a},\\mathfrak {b}\\in \\mathfrak {G}$ be two non-intersecting chords of $\\mathfrak {G}$ .",
"Take two $C$ -contours $C(\\mathfrak {a})$ , $C(\\mathfrak {b})$ such that their arcs $\\mathfrak {a_0a_1}$ , $\\mathfrak {b_0b_1}$ do not intersect.",
"It is obvious that for the loops ${C}(a)$ , ${C}(b)$ , we can associate the one-to-one correspondence between the set ${C}(a) \\cap {C}(b)$ and the set $\\mathfrak {a}_\\times \\cap \\mathfrak {b}_\\times $ .",
"Because, by Proposition REF , all chord from the set $\\mathfrak {a}_\\times \\cap \\mathfrak {b}_\\times $ keep their positions in Gauss diagram $\\mathfrak {\\widehat{G}_c}$ (= Conway's smoothing the chord $\\mathfrak {c}$ ) for every $\\mathfrak {c} \\in \\mathfrak {a}_\\parallel \\cap \\mathfrak {b}_\\parallel \\setminus \\lbrace \\mathfrak {a},\\mathfrak {b}\\rbrace $ , it is sufficient to prove the statement in the case $\\mathfrak {a}_\\parallel \\cap \\mathfrak {b}_\\parallel = \\lbrace \\mathfrak {a},\\mathfrak {b}\\rbrace $ , i.e., the loops ${C}(a)$ , ${C}(b)$ are non-self-intersecting loops (= the Jordan curves).",
"From Jordan curve Theorem, it follows that the loop ${C}(a)$ divides the curve ${C}$ into two regions, say, ${I}$ and ${O}$ .",
"Assume that $b\\in {O}$ and let us walk along the loop ${C}(b)$ .",
"We say that an intersection point $c \\in {C}(a) \\cap {C}(b)$ is the entrance (resp.",
"the exit) if we shall be in the region ${I}$ (resp.",
"${O}$ ) after meeting $c$ with respect to our walk.",
"Since a number of entrances has to be equal to the number of exits, then $|\\mathfrak {a}_\\times \\cap \\mathfrak {b}_\\times | \\equiv 0\\bmod 2$ .",
"Arguing similarly, we prove $|\\mathfrak {c}_\\times | \\equiv 0 \\bmod 2$ for every chord $\\mathfrak {c}$ .",
"As an immediate consequence of Proposition REF we get the following.",
"Corollary 3.1 (The Even Condition) If a Gauss diagram is realizable then the number of all chords that cross a both of non-intersecting chords and every chord is even (including zero).",
"We conclude this subsection with an explanation why the even condition is not sufficient for realizability of Gauss diagrams.",
"Roughly speaking, from the proof of Proposition REF it follows that every plane curve can be obtained by attaching its loops to each other by given points.",
"Conversely, if a Gauss diagram satisfies the even condition then it may be non-releasible.",
"Indeed, when we attach a loop, say, ${C}(b)$ to a loop ${C}(a)$ , where $\\mathfrak {b} \\in \\mathfrak {a}_\\parallel $ , by given points (=elements of the set $\\mathfrak {a}_\\times \\cap \\mathfrak {b}_\\times $ ) then the loop ${C}(b)$ can be self-intersected curve, which means that we get new crossings (= virtual crossings), see Figure REF .",
"To be more precisely, we have the following proposition.",
"Proposition 3.2 Let $\\mathfrak {G}$ be a non-realizable Gauss diagram which satisfies the even condition.",
"Let $\\mathfrak {G}$ defines a virtual plane curve ${C}$ (up to virtual moves).",
"There exist two non-intersecting chords $\\mathfrak {a,b \\in G}$ such that there are paths $c \\rightarrow x \\rightarrow d$ , $e \\rightarrow x \\rightarrow f$ on a loop ${C}(b)$ , where $\\mathfrak {c,d,e,f \\in a_\\times \\cap b_\\times }$ are different chords and $x$ is a virtual crossing of ${C}$ .",
"Let $\\mathfrak {a,b \\in G}$ be two non-intersecting chords.",
"Take non-intersecting $C$ -contours $C(\\mathfrak {a})$ , $C(\\mathfrak {b})$ .",
"Hence we may say that the loop ${C}(b)$ attaches to the loop ${C}(a)$ by the given points $p_1,\\ldots ,p_n $ , where $\\lbrace \\mathfrak {p_1,\\ldots ,p_n}\\rbrace = \\mathfrak {a_\\times \\cap b_\\times }$ .",
"Since $\\mathfrak {G}$ is not realizable and satisfies the even condition then a virtual crossing may arise only as a self-intersecting point of, say, the loop ${C}(b)$ .",
"Indeed, when we attach ${C}(b)$ to ${C}(a)$ by $p_1,\\ldots ,p_n $ we may get self-interesting points, say, $q_1,\\ldots ,q_m$ of the loop ${C}(b)$ .",
"If $\\mathfrak {G}$ contains all chords $\\mathfrak {q_1},\\ldots ,\\mathfrak {q}_m$ for every such chords $\\mathfrak {a,b}$ , then $\\mathfrak {G}$ is realizable.",
"Thus, a virtual crossing $x$ does not belong to $\\lbrace p_1,\\ldots , p_n\\rbrace = {C}(a) \\cap {C}(b)$ for some non-intersecting chords $\\mathfrak {a,b \\in G}$ .",
"Then we get two paths $c \\rightarrow x \\rightarrow d$ , $e \\rightarrow x \\rightarrow f$ , where $c,d,e,f \\in {C}(a) \\cap {C}(b)$ are different chords, as claimed.",
"Figure: It shows that the even condition is not sufficient for realizability of the Gauss diagrams.",
"We see that the plane curve can be obtained by attaching the white-black loop to the dotted one by the points 3,4,5,63,4,5,6, and thus the dotted loop has to have “new” crossings (= self-intersections) x 1 ,x 2 ,x 3 x_1,x_2,x_3."
],
[
"The Sufficient Condition",
"Definition 3.1 Let $\\mathfrak {G}$ be a Gauss diagram (not necessarily realizable) and $X(\\mathfrak {a,b})$ its $X$ -contour.",
"Take the $X$ -contour coloring of $\\mathfrak {G}$ .",
"A chord of $\\mathfrak {G}$ is called colorful for $X(\\mathfrak {a,b})$ if its endpoints are in arcs which have different colors.",
"Similarly, one can define a colorful chord for a $C$ -contour $C(\\mathfrak {a})$ of $\\mathfrak {G}$ .",
"Example 3.2 Let us consider the Gauss diagram, which is shown in Figure REF .",
"One can easy check that this Gauss diagram is not realizable.",
"Let us consider the orange $X$ -contour $X(1,3)$ and the $X(1,3)$ -coloring of $\\mathfrak {G}$ .",
"The chord with the endpoints 5 is colorful for the $X$ -contour $X(1,3)$ .",
"It is interesting to consider the corresponding coloring of the virtual plane curve: one can think that we forget to change color when we cross the gray loop, i.e., the gray loop “does not divide” the curve into two parts.",
"We shall show that this observation is typical for every non-realizable Gauss diagram.",
"Figure: This Gauss diagram satisfies even condition but is non-realizable.",
"There are colorful chords (e.g.",
"the chord with endpoints 5).We have seen that if a Gauss diagram is realizable then there is no colorful chord, with respect to every $X$ -contour.",
"We shall show that it is sufficient condition for realizability of a Gauss diagram.",
"Proposition 3.3 Let $\\mathfrak {G}$ be a non-realizable Gauss diagram but satisfy the even condition.",
"Then there exists an $X$ -contour and a colorful chord for this $X$ -contour.",
"By Theorem REF , $\\mathfrak {G}$ defines a virtual curve ${C}$ (= the shadow of a virtual knot diagram) up to virtual moves.",
"Starting from a crossing, say, $o$ , let us walk along ${C}$ till we meet the first virtual crossing, say, $x$ .",
"Next, let us keep walking in the same direction till we meet the first real crossing, say $d$ .",
"Denote this path by ${P}$ .",
"Just for convinces, let us put the labels, say, $x_0,x_1$ of the virtual crossing $x$ on the circle of $\\mathfrak {G}$ in the order we meet them on ${P}$ .",
"We can take two real crossings, say, $a,b$ such that the path $a \\rightarrow x \\rightarrow b$ does not contain another real crossings and $\\mathfrak {a,b} \\in \\mathfrak {o}_\\times .$ Indeed, from Proposition REF it follows that for a chord $\\mathfrak {o}\\in \\mathfrak {G}$ we can find $n \\ge 1$ chords $\\mathfrak {o}^1,\\ldots ,\\mathfrak {o}^n$ such that, for every $1\\le i \\le n$ , we have: (1) the chords $\\mathfrak {o}, \\mathfrak {o}^i$ do not intersect, (2) the loop ${C}(o)$ contains the following paths $a^i \\rightarrow x \\rightarrow b^i$ , $c^i \\rightarrow x \\rightarrow d^i$ where $\\mathfrak {a}^i,\\mathfrak {b}^i,\\mathfrak {c}^i,\\mathfrak {d}^i \\in \\mathfrak {o}_\\times \\cap \\mathfrak {o}^i_\\times $ are different chords.",
"Thus, for some $1\\le i,j \\le n$ we have an arc, say, $\\mathfrak {a}^i_1\\mathfrak {b}^j_1$ contains only one of $x_0$ or $x_1$ , no endpoints of another chords, and $\\mathfrak {a}^i,\\mathfrak {b}^j \\in \\mathfrak {o}_\\times $ .",
"Denote this arc by $\\mathfrak {a_1b_1}$ and assume that $x_0$ lies on $\\mathfrak {a_1b_1}$ .",
"We next construct an $X$ -contour contains the arc $\\mathfrak {a_1b_1}$ .",
"To do so, we have to consider the following two cases: Case 1.",
"The chords $\\mathfrak {a,b}$ intersect.",
"Figure: Since the curve C{C} is determined up to virtual moves then the cyan line has to cross the olive line.Take the $X$ -contour $X(\\mathfrak {a,b})$ which contains the arc $\\mathfrak {a_1b_1}$ .",
"It is easy to see that this $X$ -contour has at least one real door chord, because its another arc $\\mathfrak {a_0b_0}$ has no virtual crossing thus it has to have at least one real crossing (see Figure REF).",
"Case 2.",
"The chords $\\mathfrak {a,b}$ do not intersect.",
"Figure: The “general” position of chords 𝔞,𝔟,𝔠,𝔡\\mathfrak {a,b,c,d} is shown.",
"Our walk along the path P{P} correspondences to 𝔬 0 →𝔬 ˜ 0 →𝔟 0 →⋯→𝔡 1 \\mathfrak {o_0 \\rightarrow \\widetilde{o}_0 \\rightarrow b_0\\rightarrow \\cdots \\rightarrow \\mathfrak {d_1}}.Let us walk along the circle of $\\mathfrak {G}$ in the direction $\\mathfrak {b_1} \\rightarrow x_0 \\rightarrow \\mathfrak {a}_1$ (Figure REF ) till we meet the first chord, say, $\\widetilde{\\mathfrak {o}}$ such that $\\mathfrak {a,b,o} \\in \\widetilde{\\mathfrak {o}}_\\times $ .",
"If we cannot find such chord we set $\\widetilde{\\mathfrak {o}}:=\\mathfrak {o}$ .",
"Take the $X$ -contour $X(\\widetilde{\\mathfrak {o}},\\mathfrak {b})$ contains the arc $\\mathfrak {a_1b_1}$ .",
"This $X$ -contour is non-degenerate.",
"Indeed, let $\\widetilde{\\mathfrak {o}}:=\\mathfrak {o}$ and endpoints of all chords, which start in the arc $\\mathfrak {b_1}x_1\\mathfrak {d}_1$ , lie on the arc $\\mathfrak {o}_0\\mathfrak {b_0}$ .",
"We then get the situation is shown in Figure REF .",
"Figure: The dotted blue line gives another path from cc to dd.It follows that we can take another path ${P}$ without the virtual crossing $x$ .",
"This contradiction implies that the $X$ -contour $X(\\mathfrak {o,b})$ is non-degenerate.",
"So, we have a non-degenerate ${X}$ -contour of the curve ${C}$ such that $x$ is its virtual door.",
"If the $X$ -contour of $\\mathfrak {G}$ contains another chords such that all their endpoints lie on an arc of this $X$ -contour, then Proposition REF implies that Conway's smoothing all such chords does not change the ${X}$ -contour coloring of ${P}$ .",
"Thus we may assume that this ${X}$ -contour is the closed curve without self-intersections.",
"Hence, by Jordan curve Theorem, it divides ${P}$ into two parts.",
"Let us color these parts into two different colors.",
"If the crossing $x$ would be a real door of this ${X}$ -contour, then by Definition REF , we can take such ${X}$ -contour coloring of ${P}$ as before.",
"But since $x$ is not real door then after meeting it at the second time we do not change the color and thus we get an intersection point of two lines which have different colors, i.e., the corresponding chord is colorful.",
"This completes the proof.",
"Lemma 3.1 Let $\\mathfrak {G}$ be a Gauss diagram.",
"Consider a $C(\\mathfrak {a})$ -contour coloring of $\\mathfrak {G}$ for some chord $\\mathfrak {a\\in G}$ .",
"If there exists a colorful chord for the $C$ -contour $C(\\mathfrak {a})$ then the diagram $\\mathfrak {G}$ does not satisfy the even condition.",
"Indeed, let $\\mathfrak {b}$ be a colorful chord for the $C$ -contour $C(\\mathfrak {a})$ .",
"First note that, the chord $\\mathfrak {b}$ cannot cross $\\mathfrak {a}$ because otherwise $\\mathfrak {b}$ should be a door chord of the $C$ -contour $C(\\mathfrak {a})$ .",
"Next, if the chord $\\mathfrak {b}$ is colorful then it crosses an odd number of door chords of $C(\\mathfrak {a})$ .",
"Hence $|\\mathfrak {a}_\\times \\cap \\mathfrak {b}_\\times | \\equiv 1 \\bmod 2$ , as claimed.",
"Figure: The Gauss diagram does not satisfy the even condition; both the chords 1, 6 are crossed by only one chord 2.Example 3.3 Let us consider the Gauss diagram which is shown in Figure REF .",
"We have the orange $C$ -contour $C(1)$ and the $C(1)$ -contour coloring of the Gauss diagram and the corresponding (virtual) curve.",
"We see that there are two chords (namely 5 and 6) which are colorful and $5_\\times \\cap 1_\\times = 6_\\times \\cap 1_\\times = \\lbrace 2\\rbrace $ .",
"Lemma 3.2 Let $\\mathfrak {G}$ be a Gauss diagram and $\\mathfrak {a}, \\mathfrak {b} \\in \\mathfrak {G}$ be its intersecting chords.",
"Suppose that there exists a colorful chord $\\mathfrak {c}$ for an $X$ -contour $X(\\mathfrak {a},\\mathfrak {b})$ .",
"Then there exists a $C$ -contour of the Gauss diagram $\\widehat{\\mathfrak {G}}_\\mathfrak {b}$ (= Conway's smoothing the chord $\\mathfrak {b}$ ) such that the chord $\\mathfrak {c}$ is colorful for this $C$ -contour in $\\widehat{\\mathfrak {G}}_\\mathfrak {b}$ .",
"Indeed, consider the Gauss diagram $\\widehat{\\mathfrak {G}}_\\mathfrak {b}$ .",
"From Proposition REF it follows that after Conway's smoothing the chord $\\mathfrak {b}$ , the chord $\\mathfrak {c}$ does not intersect $\\mathfrak {a}$ and intersects the same door chords of the $X$ -contour $X(\\mathfrak {a},\\mathfrak {b})$ as in $\\mathfrak {G}$ .",
"Further, let us consider the $C$ -contour $C(\\mathfrak {a})$ in $\\widehat{\\mathfrak {G}}_\\mathfrak {b}$ such that it does not contain the chord $\\mathfrak {c}$ .",
"By Proposition REF , the chord $\\mathfrak {a}$ crosses in $\\widehat{\\mathfrak {G}}_\\mathfrak {b}$ only the chord that are door chords of the $X$ -contour $X(\\mathfrak {a},\\mathfrak {b})$ .",
"Hence, by Definition REF , we may take the $C(\\mathfrak {a})$ -contour coloring of $\\widehat{\\mathfrak {G}}_\\mathfrak {b}$ such that $\\mathfrak {c}$ is the colorful chord for this $C$ -contour.",
"Proposition 3.4 Let a Gauss diagram $\\mathfrak {G}$ satisfy the even condition.",
"Then $\\mathfrak {G}$ is realizable if and only if $\\widehat{\\mathfrak {G}}_\\mathfrak {c}$ satisfies the even condition for every chord $\\mathfrak {c} \\in \\mathfrak {G}$ , $\\widehat{\\mathfrak {G}}_\\mathfrak {c}$ .",
"Indeed, let $\\mathfrak {G}$ be a non-realizable Gauss diagram and let $\\mathfrak {G}$ satisfy the even condition.",
"By Proposition REF , there exists a colorful chord (say $\\mathfrak {c}$ ) for a $X$ -counter $X(\\mathfrak {a},\\mathfrak {b})$ of $\\mathfrak {G}$ .",
"By Lemma REF , the chord $\\mathfrak {c}$ is the colorful chord in $\\widehat{\\mathfrak {G}}_\\mathfrak {b}$ .",
"Hence from Lemma REF it follows that $\\widehat{\\mathfrak {G}}_\\mathfrak {b}$ does not satisfy the even condition, and the statement follows.",
"We can summarize our results in the following theorem.",
"Theorem 3.4 A Gauss diagram $\\mathfrak {G}$ is realizable if and only if the following conditions hold: (1) the number of all chords that cross a both of non-intersecting chords and every chord is even (including zero), (2) for every chord $\\mathfrak {c} \\in \\mathfrak {G}$ the Gauss diagram $\\widehat{\\mathfrak {G}}_\\mathfrak {c}$ (= Conway's smoothing the chord $\\mathfrak {c}$ ) also satisfies the above condition."
],
[
"Matrixes of Gauss Diagrams",
"In this section we are going to translate into matrix language the conditions of realizability of Gauss diagrams.",
"We consider all matrixes over the field $\\mathbb {Z}_2$ .",
"Definition 4.1 Given a Gauss diagram $\\mathfrak {G}$ contains $n$ chords, say, $\\mathfrak {c}_1,\\ldots ,\\mathfrak {c}_n$ .",
"Construct a $n \\times n$ matrix $\\mathsf {M}(\\mathfrak {G}) = (\\mathsf {m}_{i,j})_{1 \\le i,j\\le n}$ as follows: (1) put $\\mathsf {m}_{ii} = 0$ , $1 \\le i \\le n$ , (2) $\\mathsf {m}_{ij} = 1$ if the chords $\\mathfrak {c}_i$ , $\\mathfrak {c}_j$ intersect and $\\mathsf {m}_{ij} = 0$ in otherwise.",
"It is obviously that $\\mathsf {M}(\\mathfrak {G})$ is a symmetric matrix with respect to the main diagonal.",
"Next, let $\\mathsf {M}(\\mathfrak {G})$ be a $n\\times n$ matrix of a Gauss diagram $\\mathfrak {G}$ .",
"Let $\\cup _{1 \\le i \\le n}\\lbrace \\mathsf {m}_i = (\\mathsf {m}_{i1}, \\ldots , \\mathsf {m}_{in})\\rbrace $ be all its strings.",
"Define a scalar product of strings as follows: $\\langle \\mathsf {m}_i, \\mathsf {m}_j \\rangle :=\\mathsf {m}_{i1}\\mathsf {m}_{j1} + \\cdots + \\mathsf {m}_{in}\\mathsf {m}_{jn}.$ It is clear that we can define “scalar product” of any number of strings as follows $\\langle \\mathsf {m}_{i_1}, \\ldots , \\mathsf {m}_{i_k} \\rangle : = \\mathsf {m}_{i_11} \\cdots \\mathsf {m}_{i_k1} + \\cdots + \\mathsf {m}_{i_1n} \\cdots \\mathsf {m}_{i_kn}.$ Lemma 4.1 Let $\\mathfrak {G}$ be a Gauss diagram contains $n$ chords $\\mathfrak {c}_1,\\ldots ,\\mathfrak {c}_n$ .",
"Consider the corresponding matrix $\\mathsf {M}(\\mathfrak {G})$ .",
"Let the chords $\\mathfrak {c}_1,\\ldots ,\\mathfrak {c}_n$ correspondence to the strings $\\mathsf {m}_1,\\ldots ,\\mathsf {m}_n$ , respectively.",
"$| {\\mathfrak {c}_{i_1}}_\\times \\cap \\cdots \\cap {\\mathfrak {c}_{i_k}}_\\times | = \\langle \\mathsf {m}_{i_1}, \\ldots , \\mathsf {m}_{i_k} \\rangle ,$ for every $\\lbrace i_1,\\ldots ,i_k\\rbrace \\subseteq \\lbrace 1,2,\\ldots ,n\\rbrace $ .",
"Indeed, let $\\mathsf {m}_{i_1t} \\cdots \\mathsf {m}_{i_kt} \\ne 0$ , for some $1 \\le t \\le n$ , then the chord $\\mathfrak {c}_t$ intersect chords $\\mathfrak {c}_{i_1}, \\ldots , \\mathfrak {c}_{i_k}$ .",
"This completes the proof.",
"Theorem 4.2 Let $\\mathfrak {G}$ be a realizable Gauss diagram contains $n$ chords.",
"Then its matrix $\\mathsf {M}(\\mathfrak {G})$ satisfies the following conditions: (1) $\\langle \\mathsf {m}_i, \\mathsf {m}_i \\rangle \\equiv 0 \\bmod (2)$ , $1 \\le i \\le n$ , (2) $\\langle \\mathsf {m}_i, \\mathsf {m}_j \\rangle \\equiv 0 \\bmod (2)$ , if the corresponding chords do not intersect, (3) $\\langle \\mathsf {m}_i, \\mathsf {m}_j \\rangle + \\langle \\mathsf {m}_i, \\mathsf {m}_k \\rangle + \\langle \\mathsf {m}_j, \\mathsf {m}_k \\rangle \\equiv 1 \\bmod (2)$ , if the corresponding chords pairwise intersect.",
"The first two conditions follow from the first condition of Theorem REF and Lemma REF .",
"Assume that $\\mathfrak {G}$ does not contain three pairwise intersecting chords.",
"Then $\\mathfrak {G}$ is realizable if and only if the first condition of Theorem REF holds and we get the first two conditions for $\\mathsf {M}(\\mathfrak {G})$ .",
"Next, let $\\mathfrak {a},\\mathfrak {b},\\mathfrak {c}$ be three pairwise intersecting chords.",
"Consider the following sets of chords (see Figure REF ): $&A: = \\mathfrak {a}_\\times \\setminus \\lbrace \\mathfrak {b}_\\times , \\mathfrak {c}_\\times \\rbrace ,\\\\& B:= \\mathfrak {b}_\\times \\cap \\mathfrak {c}_\\times \\setminus \\left\\lbrace \\lbrace \\mathfrak {a}\\rbrace , \\mathfrak {a}_\\times \\right\\rbrace ).",
"$ Figure: The sets AA, BB are roughly shown.By Proposition REF , chords $\\mathfrak {b}, \\mathfrak {c}$ does not intersect in $\\widehat{\\mathfrak {G}}_\\mathfrak {a}$ , and the set $\\mathfrak {b}_\\times \\cap \\mathfrak {c}$ of chords in $\\widehat{\\mathfrak {G}}_\\mathfrak {a}$ is equal to the set $A \\cup B$ .",
"Hence, by Theorem REF , $|A| + |B| \\equiv 0 \\bmod (2).$ Further, using Lemma REF , we obtain: $&|A| = \\langle \\mathfrak {a}, \\mathfrak {a} \\rangle - \\langle \\mathfrak {a}, \\mathfrak {b} \\rangle - \\langle \\mathfrak {a}, \\mathfrak {c} \\rangle - \\langle \\mathfrak {a}, \\mathfrak {b}, \\mathfrak {c} \\rangle ,\\\\&|B| = \\langle \\mathfrak {b}, \\mathfrak {c} \\rangle -1- \\langle \\mathfrak {a}, \\mathfrak {b}, \\mathfrak {c} \\rangle .$ It follows that $|A| + |B| & =& \\langle \\mathfrak {a}, \\mathfrak {a} \\rangle - \\langle \\mathfrak {a}, \\mathfrak {b} \\rangle - \\langle \\mathfrak {a}, \\mathfrak {c} \\rangle - \\langle \\mathfrak {a}, \\mathfrak {b}, \\mathfrak {c} \\rangle \\\\&&+ \\langle \\mathfrak {b}, \\mathfrak {c} \\rangle -1- \\langle \\mathfrak {a}, \\mathfrak {b}, \\mathfrak {c} \\rangle \\\\&\\equiv & \\langle \\mathfrak {a}, \\mathfrak {b} \\rangle + \\langle \\mathfrak {a}, \\mathfrak {c} \\rangle + \\langle \\mathfrak {b}, \\mathfrak {c} \\rangle \\\\&\\equiv & 1 \\bmod (2),$ because, as we have already discussed, $\\langle \\mathfrak {a}, \\mathfrak {a} \\rangle \\equiv 0 \\bmod (2)$ .",
"As claimed.",
"Remark 4.3 The number of matrix satisfy the above conditions is bigger than a number of realizable Gauss diagrams.",
"The reason is the following observation.",
"A matrix which satisfies the above conditions “knows” only intersections but don't know positions of chords.",
"However there is a very important sort of plane curves (=meanderes) such that there is a one-to-one correspondence between Gauss diagrams of these curves and the corresponding matrixes."
],
[
"Thurston Generators of Braid Groups",
"We will use as generators for $B_n$ the set of positive crossings, that is, the crossings between two (necessary adjacent) strands, with the front strand having a positive slope.",
"We denote these generators by $\\sigma _1, \\ldots , \\sigma _{n-1}$ .",
"These generators are subject to the following relations: ${\\left\\lbrace \\begin{array}{ll}\\sigma _i\\sigma _j = \\sigma _j\\sigma _i, \\mbox{ if } |i-j| >1,\\\\\\sigma _i\\sigma _{i+1}\\sigma _i = \\sigma _{i+1}\\sigma _i\\sigma _{i+1}.\\end{array}\\right.",
"}$ One obvious invariant of an isotopy of a braid is the permutation it induces on the order of the strands: given a braid $B$ , the strands define a map $p(B)$ from the top set of endpoints to the bottom set of endpoints, which we interpret as a permutation of $\\lbrace 1, \\ldots , n\\rbrace $ .",
"In this way we get a homomorphism $p:B_n \\rightarrow \\mathfrak {S}_n$ , where $\\mathfrak {S}_n$ is the symmetric group.",
"The generator $\\sigma _i$ is mapped to the transposition $s_i = (i,i+1)$ .",
"We denote by $S_n = \\lbrace s_1, \\ldots , s_{n-1}\\rbrace $ the set of generators for the symmetric group $\\mathfrak {S}_n$ .",
"Now we want to define an inverse map $p^{-1}:\\mathfrak {S}_n \\rightarrow B_n$ .",
"To this end, we need the following definition [10] Definition 5.1 Let $S = \\lbrace s_1, \\ldots , s_{n-1}\\rbrace $ be the set of generators for $\\mathfrak {S}_n$ .",
"Each permutation $\\pi $ gives rise to a total order relation $\\le _\\pi $ on $\\lbrace 1, \\ldots ,n\\rbrace $ with $i \\le _\\pi j$ if $\\pi (i) < \\pi (j)$ .",
"We set $R_\\pi : = \\lbrace (i,j) \\in \\lbrace 1, \\ldots , n\\rbrace \\times \\lbrace 1,\\ldots ,n\\rbrace |i<j, \\, \\pi (i) > \\pi (j)\\rbrace .$ Lemma 5.1 [10] A set $R$ of pairs $(i,j)$ , with $i <j$ , comes from some permutation if and only if the following two conditions are satisfied: i) If $(i,j) \\in R$ and $(j,k) \\in R$ , then $(i,k) \\in R$ .",
"ii) If $(i,k) \\in R$ , then $(i,j) \\in R$ or $(j,k) \\in R$ for every $j$ with $i<j<k$ .",
"Now we will define ([10]) a very important concept of non-repeating braid (=simple braid).",
"Definition 5.2 Recall that our set of generators $S_n$ includes only positive crossings; the positive braid monoid is denoted by $B^+ =B_n^+$ .",
"We call a positive braid non-repeating if any two of its strands cross at most once.",
"We define $D =D_n \\subset B^+_n$ as the set of classes of non-repeating braids.",
"The following lemma summarizes all the above mentioned concepts and notations.",
"Lemma 5.2 [10] The homomorphism $p:B_n^+ \\rightarrow \\mathfrak {S}_n$ is restricted to a bijection $D \\rightarrow \\mathfrak {S}_n$ .",
"A positive braid $B$ is non-repeating if and only if $|B| = |p(B)|$ (here $|?|$ means the length of a word).",
"If a non-repeating braid maps to a permutation $\\pi $ , two strands $i$ and $j$ cross if and only if $(i,j) \\in R_\\pi $ .",
"Example 5.3 Let us consider the following permutation $\\pi = \\begin{pmatrix}1&2&3&4&5&6 \\\\ 4&2&6&1&5&3 \\end{pmatrix} \\in \\mathfrak {S}_6.$ We obtain $& {\\left\\lbrace \\begin{array}{ll}1<2,\\\\ \\pi (1) > \\pi (2),\\end{array}\\right.}",
"\\quad {\\left\\lbrace \\begin{array}{ll}1<4, \\\\ \\pi (1) > \\pi (4),\\end{array}\\right.}",
"\\quad {\\left\\lbrace \\begin{array}{ll}1<6, \\\\ \\pi (1) > \\pi (6),\\end{array}\\right.}",
"\\quad {\\left\\lbrace \\begin{array}{ll}2<4, \\\\ \\pi (2) > \\pi (4),\\end{array}\\right.",
"}\\\\& {\\left\\lbrace \\begin{array}{ll}3<4,\\\\ \\pi (3) > \\pi (4),\\end{array}\\right.}",
"\\quad {\\left\\lbrace \\begin{array}{ll}3<5,\\\\ \\pi (3) > \\pi (5),\\end{array}\\right.}",
"\\quad {\\left\\lbrace \\begin{array}{ll}3<6,\\\\ \\pi (3) > \\pi (6), \\end{array}\\right.}",
"\\quad {\\left\\lbrace \\begin{array}{ll}5<6,\\\\ \\pi (5) > \\pi (6).\\end{array}\\right.",
"}$ hence $R(\\pi ) = \\lbrace (1,2),(1,4),(1,6),(2,4),(3,4),(3,5),(3,6),(5,6)\\rbrace ,$ and we get a non-repeating braid is shown in Figure REF .",
"Figure: The simple braid R π R_\\pi is shown."
],
[
"Meanders and its Gauss Diagrams",
"In this section we deal with closed meanders.",
"We show that any closed meander defines a special sort of Gauss diagrams (= Gauss diagrams of meanders) and then we will see that these diagrams are completely determined by its matrixes (= meander matrix) i.e., there is a one-to-one correspondence between meander matrixes satisfy the conditions of Theorem REF .",
"It allows us to describe an algorithm constructs all closed meanders.",
"Definition 6.1 A (closed) meander is a planar configuration consisting of a simple closed curve and on orientied line, that cross finitely many times and intersect only transversally.",
"Two meanders are equivalent if there exists a homeomorphism of the plane that maps one to the other.",
"Figure: An Example of a closed meander.Definition 6.2 A matrix $\\mathbf {M} = (\\mathbf {m}_{ij}) \\in \\mathsf {Mat}_{n+1}(\\mathbb {Z}_2)$ is called meander matrix if the following conditions hold: (1) its main diagonal contains only 0, (2) $\\mathbf {M}$ is symmetric, (3) it has at leas one string $\\mathbf {m}_i = (\\mathbf {m}_{i1}, \\ldots , \\mathbf {m}_{i, n+1})$ with $\\mathbf {m}_{ik} = 1$ for every $1 \\le k\\le n+1$ , $k \\ne i$ , (4) $\\mathbf {M}$ satisfies the conditions of Theorem REF , (5) if $\\mathbf {m}_{ij}=1$ and $\\mathbf {m}_{jk} = 1$ then $\\mathbf {m}_{ik}=1$ for all $1 \\le i,j,k \\le n+1$ , (6) if $\\mathbf {m}_{ik}=1$ , then $\\mathbf {m}_{ij}=1$ or $\\mathbf {m}_{jk}=1$ for every $j$ with $1 \\le i<j<k \\le n+1$ .",
"Figure: A Gauss diagram of the meander, which is shown in Figure ., is shownProposition 6.1 Every meander matrix determines a unique closed meander (up to the equivalence).",
"Let $\\mathbf {M}$ be a meander matrix.",
"Take $n+1$ lines on the plane $\\mathbb {R}^2$ and mark them by numbers $1,2,\\ldots ,n+1$ .",
"Conditions (1) and (2) imply that the matrix $\\mathbf {M}$ can be considered as a matrix of intersection of the lines: $\\mathbf {m}_{ij} = 1$ if the lines with numbers $i$ , $j$ intersect and $\\mathbf {m}_{ij} = 0$ in otherwise.",
"Further, from condition (3) it follows that there is a line is marked by, say $r$ , such that it crossed by all other lines.",
"Next, by Lemmas REF , REF , and by conditions (5), (6), all lines with exception of the line is marked by $r$ can be considered as a non-repeating braid.",
"This gives rise a Gauss diagram (see Figure REF .",
"Condition (4) implies that this Gauss diagram is releasable by a plane curve, say ${C}$ .",
"Finally, the chord, which is marked by $r$ , correspondences to a loop ${C}(r)$ of ${C}$ .",
"It is obviously that this loop contains all crossing of the curve ${C}$ .",
"We then get the meander ${C}$ up to the equivalence, as claimed."
],
[
"Construction of Meanders",
"We start with the following example.",
"Example 7.1 Let us consider the meander is shown in Figure REF , its Gauss diagram is shown in Figure REF .",
"The correspondence meander matrix has the following form ${\\begin{bordermatrix }& 0 & 1 & 2 & 3 & 4 & 5 & 6 \\\\0 & \\mathbf {0} & 1 & 1 & 1 & 1 & 1 & 1 \\\\1 & 1 &\\mathbf {0} & 0& 0 & 0 & 0 & 1 \\\\2 & 1 & 0 & \\mathbf {0} & 0 & 0 & 0 & 1 \\\\3 & 1 & 0 & 0 & \\mathbf {0} & 1 & 1 & 1 \\\\4 & 1 & 0 & 0 & 1 & \\mathbf {0} & 1 & 1 \\\\5 & 1 &0 &0 & 1 & 1 & \\mathbf {0} & 1 \\\\6 & 1 &1 &1 & 1 & 1 & 1 & \\mathbf {0} \\\\\\end{bordermatrix }}$ This matrix is constructed as follows.",
"Take a table with seven rows and seven columns.",
"Number its columns from left to right and rows from top to bottom by numbers $0,1,\\ldots , 6$ .",
"Fill the main diagonal by zeros.",
"Table: NO_CAPTIONSince we start from 0 and go then to 3, then the chord 3 intersects chords $0,3,5,6$ , and hence we get Table: NO_CAPTIONNext, we go to 4 and then the chord 4 crosses chords $0,3,5,6$ and we get Table: NO_CAPTIONand etc.",
"Note that we changed the parity of numbers of strings.",
"Lemma 7.1 To make by step-by-step a meander matrix, the following hold: (1) every string is filled as follows: every its empty cell is to the left of the main diagonal is filled by 0 and every its empty cell is to the right of the main diagonal is filled by 1, (2) to fill the next string its number has to have another parity than a number of the previous string.",
"Indeed, assume that we go from $i$ to $j$ in a meander.",
"If they have the same parity then there are odd number of points between them Figure.",
"REF.",
"This implies that the trajectory has to have self intersection points, i.e., we get a no meander.",
"Hence the numbers $i$ , $j$ have to have different parity.",
"This gives condition (2).",
"Figure: If i,ji,j have the same parity there are odd number of points between them then.Next, as we have already seen in the previous example, after choosing a string with number, say, $i$ , the corresponding chord has to intersect chords with numbers $0,i+1,\\ldots , n+1$ , here $n$ is the number of all chords.",
"This gives condition (1).",
"Definition 7.2 A string which is filled as in condition (1) of Lemma REF is called $\\Delta $ -filled.",
"Lemma 7.2 Let in a step-by-step filling of a matrix a string with number $i$ is $\\Delta $ -filled.",
"Then $\\bigl < \\mathsf {i,j} \\bigr > \\equiv 1 \\bmod (2),$ for any $j>i$ .",
"Indeed, by Lemma REF , $\\mathbf {m}_{ij}=1$ for every $j>i$ in the matrix $\\mathbf {M}$ , and then by condition (3) of Theorem REF , $\\bigl <\\mathsf {0,i} \\bigr > + \\bigl < \\mathsf {0,j} \\bigr > + \\bigl <\\mathsf {i,j} \\bigr > \\equiv 1 \\bmod (2)$ .",
"We thus get $\\bigl <\\mathsf {0,i} \\bigr > + \\bigl < \\mathsf {0,j} \\bigr > + \\bigl <\\mathsf {i,j} \\bigr >= \\bigl < \\mathsf {i,j}\\bigr >,$ because of $\\bigl <\\mathsf {0,i} \\bigr > = \\bigl <\\mathsf {i,i} \\bigr > - 1$ , $1 \\le i \\le n$ , as claimed.",
"Theorem 7.3 Let $\\mathbf {M} \\in \\mathsf {Mat}_{n}(\\mathbb {Z}_2)$ be a symmetric matrix with zero main diagonal.",
"$\\mathbf {M}$ is a meander matrix if and only if the following hold: (1) there is a string $\\mathbf {m}_i$ such that $\\mathbf {m}_{ij}=1$ if $j \\ne i$ and $\\mathbf {m}_{ii}=0$ , (2) if $\\mathbf {m}_{ij} = 1$ , $\\mathbf {m}_{jk} =1$ , then $\\mathbf {m}_{ik}=1$ , for all $i,j,k \\in \\lbrace 1,2, \\ldots , n\\rbrace $ , (3) if $\\mathbf {m}_{ik}=1$ then $\\mathbf {m}_{ij}=1$ or $\\mathbf {m}_{jk}=1$ for all $i \\le j \\le k$ , (4) for every $1 \\le i,j\\le n$ we have $\\bigl < \\mathbf {m}_i, \\mathbf {m}_j \\bigr > &\\equiv & 0 \\bmod (2), \\qquad \\mbox{if $i=j$ or $\\mathbf {m}_{ij} = 0$},\\\\\\bigl < \\mathbf {m}_i, \\mathbf {m}_j \\bigr > &\\equiv & 1 \\bmod (2), \\qquad \\mbox{if $\\mathbf {m}_{ij} = 1$}.$ By Definition REF and Proposition REF , it is sufficient to prove that condition (3) of Theorem REF is equivalent to the last condition.",
"It is clear that an equality $\\bigl < \\mathbf {m}_i, \\mathbf {m}_i \\bigr > \\equiv 0 \\bmod (2)$ implies that only an even number (including zero) of chords intersect chord correspondences to string $\\mathbf {m}_i$ .",
"Next, an equality $\\mathbf {m}_{ij}=0$ implies that chord $i$ do not intersect chord $j$ , and an equality $\\mathbf {m}_{ij}=0$ implies that only even number of chords intersect both of chords $i$ , $j$ .",
"We thus get the first two conditions of Theorem REF .",
"Further, by Lemma REF , from the third condition of Theorem REF it follows (REF ).",
"Assume now that (REF ) holds and consider three pairwise intersecting chords, say $i,j,k$ .",
"Then $\\bigl < \\mathsf {m}_i, \\mathsf {m}_j \\bigr > \\equiv 1 \\bmod (2)$ , $\\bigl < \\mathbf {m}_i, \\mathbf {m}_k \\bigr > \\equiv 1 \\bmod (2)$ , and $\\bigl < \\mathbf {m}_j, \\mathbf {m}_k \\bigr > \\equiv 1 \\bmod (2)$ and we complete the proof.",
"From this Theorem it follows the following meanders construction algorithm.",
"REQUIRE an even number $N$ , $S_1 = \\lbrace 1,3,\\ldots , N-1\\rbrace $ , $S_0 = \\lbrace 2,4, \\ldots ,N\\rbrace $ ; ENSURE $n_1,\\ldots , n_N \\in S_1 \\cup S_0$ ; 1. take a $(N+1) \\times (N+1)$ tableau with empty cells and fill the main diagonal by zeros; 2. number strings from left to right, and number columns from top to bottom by numbers $0,1, \\ldots , N$ ; 3.",
"$\\Delta $ -fill string with number 0; 4. choose a string with an odd number $n\\in S_1$ and $\\Delta $ -fill it and column with the same number; 5.",
"$i=1$ ; 6.",
"PRINT $n$ ; 6.",
"$S_i: = S_i \\setminus \\lbrace n\\rbrace $ ; 7.",
"IF $S_i = \\varnothing $ THEN GOTO 13 ELSE GOTO 8; 8 $i:=i+1 \\bmod (2)$ ; 9. choose a string (a column) with number $m \\in S_i$ ; 10.",
"IF the string (the column) can be $\\Delta $ -filled THEN GOTO 11 ELSE choose another $m^{\\prime } \\in S_i\\setminus \\lbrace m\\rbrace $ GOTO 10; 11. using (REF ), () get a system of equations for empty cells; 12.",
"IF the system can by solved THEN GOTO 6 ELSE choose a string (column) with another number $m^{\\prime } \\in S_i\\setminus \\lbrace m\\rbrace $ GOTO 10; 13.",
"END Example 7.4 Let us construct a $9\\times 9$ meander matrix.",
"(0) Take $9\\times 9$ tableau and partially fill it as follows Table: NO_CAPTION (1) Choose a string and column 5 and $\\Delta $ -fill them Table: NO_CAPTION By(), ${\\left\\lbrace \\begin{array}{ll}\\bigl < 5, 6 \\bigr > = 1 + (6,7) + (6,8) \\equiv 1 \\bmod (2),\\\\\\bigl < 5, 7 \\bigr > = 1 + (6,7) + (7,8) \\equiv 1 \\bmod (2),\\\\\\bigl < 5, 8 \\bigr > = 1 + (6,8) + (7,8) \\equiv 1 \\bmod (2),\\end{array}\\right.",
"}$ it implies that $(6,7) = (6,8) = (7,8) = {a} \\in \\lbrace 0,1\\rbrace ,$ and we thus get Table: NO_CAPTION (2) We have to choose a string (column) with an even number.",
"(i) Take a string (column) 2, we then get $(1,2) = 0, (2,3) = (2,4) =0, (2,6) = (2,7) = (2,8)=1,$ it follows that $\\bigl <\\mathsf {1,2} \\bigr > \\equiv 0 \\bmod (2)$ .",
"We have $\\bigl <\\mathsf {1,2} \\bigr > &=& 1 + (1,3) + (1,4) \\\\&&+ (1,6)+ (1,7) + (1,8) \\equiv 0 \\bmod (2),$ and we do not get any contradiction, hence this string may be chosen.",
"(ii) Similarly, by the straightforward verification, one can easy verify that strings 4, 6 can be chosen.",
"(iii) Take string 8.",
"We get $(1,8) = (2,8) = (3,8) = (4,8) = 0= {a} = 0,$ and hence $\\bigl <\\mathsf {1,8} \\bigr > \\equiv 0 \\bmod (2)$ , but the tableau implies that $\\bigl <\\mathsf {1,8} \\bigr > = 1$ .",
"Therefore this string cannot be chosen in this step.",
"(3) Chose string 2.",
"We then get Table: NO_CAPTION by (), ${\\left\\lbrace \\begin{array}{ll}\\bigl < \\mathsf {2,3} \\bigr > = 1 + (3,4) + (3,6) + (3,7) + (3,8) \\equiv 1 \\bmod (2),\\\\\\bigl < \\mathsf {2,4} \\bigr > = 1 + (3,4) + (4,6) + (4,7) + (4,8) \\equiv 1 \\bmod (2),\\\\\\bigl < \\mathsf {2,6} \\bigr > = 1 + (3,6) + (4,6) + {a} + {a} \\equiv 1 \\bmod (2),\\\\\\bigl < \\mathsf {2,7} \\bigr > = 1 + (3,7) + (4,7) + {a} + {a} \\equiv 1 \\bmod (2),\\\\\\bigl < \\mathsf {2,8} \\bigr > = 1 + (3,8) + (4,8) + {a} + {a} \\equiv 1 \\bmod (2),\\end{array}\\right.",
"}$ and we then obtain $& (3,6) = (4,6) = b \\in \\lbrace 0,1\\rbrace \\\\& (3,7) = (4,7) = c \\in \\lbrace 0,1\\rbrace \\\\& (3,8) = (4,8) = d \\in \\lbrace 0,1\\rbrace \\\\& (3,4) = b+c+d \\bmod (2).$ Next, using the condition of Theorem REF , $\\bigl < \\mathsf {3,3} \\bigr > \\equiv 0 \\bmod (2)$ , we get $(1,3) = 0$ .",
"Similarly, one can get $(1,4) = 0$ .",
"Using $\\bigl < \\mathsf {6,6} \\bigr > \\equiv \\bigl < \\mathsf {7,7} \\bigr > \\equiv \\bigl < \\mathsf {8,8} \\bigr > \\equiv 0 \\bmod (2)$ we get $(1,6) = (1,7) = (1,8) = 1$ .",
"We thus have Table: NO_CAPTION (4) We have to chose a string with an odd number $1,3$ or 7.",
"Since string 4 has not been chosen and $(1,4)=0$ then string 1 cannot be $\\Delta $ -filled.",
"Next, if we chose string 7 we then get $(6,7) = 0$ , $(7,8) = 1$ , but $(6,7) = (7,8) = a$ , we then get a contradiction.",
"Further, one can easy verify that string 3 can be chosen.",
"(5) Take string 3.",
"It follows that $b=c=d=1$ and we then get Table: NO_CAPTION We see that strings $1,4$ are automatically $\\Delta $ -filled.",
"So, we have two possibilities: 1) $a=0$ , and 2) $a = 1$ .",
"These cases correspondence to the following possibilities for our meander (see Figure REF ).",
"Figure: The meanders which are correspondence to case a=0a=0 (in the top) and a=1a=1 (in the bottom)."
]
] | 1808.08542 |
[
[
"Advances in Computational Methods for Phylogenetic Networks in the\n Presence of Hybridization"
],
[
"Abstract Phylogenetic networks extend phylogenetic trees to allow for modeling reticulate evolutionary processes such as hybridization.",
"They take the shape of a rooted, directed, acyclic graph, and when parameterized with evolutionary parameters, such as divergence times and population sizes, they form a generative process of molecular sequence evolution.",
"Early work on computational methods for phylogenetic network inference focused exclusively on reticulations and sought networks with the fewest number of reticulations to fit the data.",
"As processes such as incomplete lineage sorting (ILS) could be at play concurrently with hybridization, work in the last decade has shifted to computational approaches for phylogenetic network inference in the presence of ILS.",
"In such a short period, significant advances have been made on developing and implementing such computational approaches.",
"In particular, parsimony, likelihood, and Bayesian methods have been devised for estimating phylogenetic networks and associated parameters using estimated gene trees as data.",
"Use of those inference methods has been augmented with statistical tests for specific hypotheses of hybridization, like the D-statistic.",
"Most recently, Bayesian approaches for inferring phylogenetic networks directly from sequence data were developed and implemented.",
"In this chapter, we survey such advances and discuss model assumptions as well as methods' strengths and limitations.",
"We also discuss parallel efforts in the population genetics community aimed at inferring similar structures.",
"Finally, we highlight major directions for future research in this area."
],
[
"Introduction",
"Hybridization is often defined as reproduction between members of genetically distinct populations [4].",
"This process could occur in various spatial contexts, and could have impacts on speciation and differentiation [2], [3], [74], [75], [99], [1], [76].",
"Furthermore, increasing evidence as to the adaptive role of hybridization has been documented, for example, in humans [94], macaques [111], [6], [88], mice [107], [66], butterflies [114], [135], and mosquitoes [32], [119].",
"Hybridization is “generically\" used to contain two different processes: hybrid speciation and introgression [31].",
"In the case of hybrid speciation, a new population made of the hybrid individuals forms as a separate and distinct lineage from either of its two parental populations.In this chapter, we do not make a distinction between species, population, or sub-population.",
"The modeling assumptions and algorithmic techniques underlying all the methods we describe here neither require nor make use of such a distinction.",
"Introgression, or introgressive hybridization, on the other hand, describes the incorporation of genetic material into the genome of a population via interbreeding and backcrossing, yet without creating a new population [45].",
"As Harrison and Larson noted [45], introgression is a relative term: alleles at some loci introgress with respect to alleles at other loci within the same genomes.",
"From a genomic perspective, and as the basis for detection of hybridization, the general view is that in the case of hybrid speciation, regions derived from either of the parental ancestries of a hybrid species would be common across the genomes, whereas in the case of introgression, regions derived from introgression would be rare across the genomes [31].",
"Fig.",
"REF illustrates both hybridization scenarios.",
"Figure: Hybrid speciation and introgression.",
"(a) A phylogenetic tree on three taxa, A, B, and C, and a gene tree within itsbranches.",
"Genetic material is inherited from ancestors to descendants and it is expected that loci across the genome would have the shown gene tree.",
"(b) A hybrid speciation scenario depicted by a phylogenetic network,where B is a hybrid population that is distinct from its parental species.",
"Shown within the branches of the network are two gene trees, both of which are assumed to be very commonacross the genome.",
"(c) An introgression scenario.",
"Through hybridization and backcrossing, genetic material from (an ancestor of) C is incorporated into the genomes of individuals in (an ancestor of) B.",
"The introgressedgenetic material would have the gene tree shown in red, and the majority of loci in B's genomes would have the gene tree shown in blue.",
"Incomplete lineage sorting would complicate all threescenarios by giving rise to loci with other possible gene trees and by changing the distribution of the various gene trees.A major caveat to the aforementioned general view is that, along with hybridization, other evolutionary processes could also be at play, which significantly complicates the identification of hybrid species and their parental ancestries.",
"Chief among those processes are incomplete lineage sorting (ILS) and gene duplication and loss.",
"Indeed, various studies have highlighted the importance of accounting for ILS when attempting to detect hybridization based on patterns of gene tree incongruence [114], [77], [127], [32], [16], [94], [119], [135], [90].",
"Furthermore, gene duplication and loss are very common across all branches of the Tree of Life.",
"While the main focus of this chapter is on modeling and inferring hybridization, a discussion of how ILS is accounted for is also provided since recent developments have made great strides in modeling hybridization and ILS simultaneously.",
"While signatures of gene duplication and loss are ubiquitous in genomic data sets, we do not include a discussion of these two processes in this chapter since methods that account for them in the context of phylogenetic networks are currently lacking.",
"When hybridization occurs, the evolutionary history of the set of species is best modeled by a phylogenetic network, which extends the phylogenetic tree model by allowing for “horizontal\" edges to denote hybridization and to facilitate modeling bi-parental inheritance of genetic material.",
"Fig.",
"REF shows two phylogenetic networks that model hybrid speciation and introgression.",
"It is very important to note, though, that from the perspective of existing models, both phylogenetic networks are topologically identical.",
"This issue highlights two important issues that must be thought about carefully when interpreting a phylogenetic network.",
"First, neither the phylogenetic network nor the method underlying its inference distinguish between hybrid speciation and introgression.",
"This distinction is a matter of interpretation by the user.",
"For example, the phylogenetic network in Fig.",
"REF (c) could be redrawn, without changing the model or any of its properties, so that the introgression is from (an ancestor of) A to (an ancestor of) B, in which case the “red\" gene tree would be expected to appear with much higher frequency than the “blue\" gene tree.",
"In other words, the way a phylogenetic network is drawn could convey different messages about the evolutionary history that is not inherent in the model or the inference methods.",
"This issue was importantly highlighted with respect to data analysis in [119] (Figure 7 therein).",
"Second, the phylogenetic network does not by itself encode any specific backbone species tree that introgressions could be interpreted with respect to.",
"This, too, is a matter of interpretation by the user.",
"This is why, for example, Clark and Messer [16] recently argued that “perhaps we should dispense with the tree and acknowledge that these genomes are best described by a network.\"",
"Furthermore, recent studies demonstrated the limitations of inferring a species tree “despite hybridization\" [106], [138].",
"With the availability of data from multiple genomic regions, and increasingly often from whole genomes, a wide array of methods for inferring species trees, mainly based on the multispecies coalescent (MSC) model [20], have been developed [67], [81], [68].",
"Building on these methods, and often extending them in novel ways, the development of computational methods for inferring phylogenetic networks from genome-wide data has made great strides in recent years.",
"Fig.",
"REF summarizes the general approaches that most phylogenetic network inference methods have followed in terms of the data they utilize, the model they employ, and the inferences they make.",
"Figure: Phylogenetic network inference process and approaches.",
"The process of phylogenetic network inference starts with collecting the genomicdata and identifying the orthology groups of unlinked loci.",
"Multiple sequence alignments or single bi-allelic markers corresponding to the unlinked loci arethen obtained; phylogenetic network inference methods use one of these two types of data.",
"In two-step inference methods, gene trees are first estimated for theindividual loci from the sequence alignment data, and these gene tree estimates are used as the input data for network inference.",
"If incomplete lineage sorting (ILS)is not accounted for, a smallest displaying network of the gene tree estimates is sought.",
"If ILS is accounted for, parsimony inference based on the MDC (minimizingdeep coalescences) criterion, a maximum likelihood estimate (MLE), a maximum a posteriori (MAP) estimate, or samples of the posterior distribution can beobtained.",
"In the direct inference approach, whether based on sequence alignment or bi-allelic marker data, a MAP estimate or samples of the posterior distributioncan be obtained directly from the data.",
"The two parsimony methods consider only the topologies of the gene tree estimates as input (i.e., they ignore gene tree branch lengths) and return as output phylogenetic network topologies.",
"The likelihood and Bayesian methods that take gene tree estimates as input can operate on gene tree topologies alone or gene trees with branch lengths as well.",
"Both methods estimate phylogenetic network topologies along with branch lengths (in coalescent units) and inheritance probabilities.",
"The direct inference methods estimate the phylogenetic network along with its associated parameters.The overarching goal of this chapter is to review the existing methods for inferring phylogenetic networks in the presence of hybridization,We emphasize hybridization (in eukaryotic species) here since processes such as horizontal gene transfer in microbial organisms result in reticulate evolutionary histories, but the applicability of methods we describe in this chapter has not been investigated or explored in such a domain.",
"describe their strengths and limitations, and highlight major directions for future research in this area.",
"All the methods discussed hereafter make use of multi-locus data, where a locus in this context refers to a segment of genome present across the individuals and species sampled for a given study and related through common descent.",
"A locus can be of varying length, coding or non-coding, and can be either functional or non-functional.",
"Therefore, the use of the term “gene trees\" is only historical; we use it to mean the evolutionary history of an individual locus, regardless of whether the locus overlaps with a coding region or not.",
"Care must be taken with increasingly long loci spanning hundreds or thousands of contiguous basepairs, however, as many methods assume a locus has not been affected by recombination.",
"Multi-locus methods are fairly popular because the model fits several types of reduced representation genomic data sets commonly generated to study biological systems.",
"Reduced representation refers to capturing many segments scattered throughout a genome, but only covering a fraction of the total genome sequence [37].",
"Reduced representation data sets which have been used with multi-locus methods include RAD-seq and genotyping by sequencing (GBS), which capture loci of roughly 100 bp associated with palindromic restriction enzyme recognition sites [30].",
"Another family of techniques often applied to studies of deeper time scales, sequence capture, extracts conserved sequences using probes complementary to targetted exons or ultraconserved elements [41].",
"Sequence capture can also be performed in silico when whole genomes are available [52].",
"The rest of the chapter is organized as follows.",
"We begin in Section 2 by defining terminology for the non-biologist, and give a very brief review of phylogenetic trees and their likelihood.",
"In Section 3, we describe the earliest, and simplest from a modeling perspective, approaches to inferring parsimonious phylogenetic networks from gene tree topologies by utilizing their incongruence as the signal for hybridization.",
"To account for ILS, we describe in Section 4 the multispecies network coalescent, or MSNC, which is the core model for developing statistical approaches to phylogenetic network inference while accounting for ILS simultaneously with hybridization.",
"In Sections 5 and 6 we describe the maximum likelihood and Bayesian methods for inferring phylogenetic networks from multi-locus data.",
"In Section 7, we briefly discuss an approach aimed at detecting hybridization by using phylogenetic invariants.",
"This approach does not explicitly build a phylogenetic network.",
"In Section 8 we briefly discuss the efforts for developing methods for phylogenetic network inference that took place in parallel in the population genetics community (they are often referred to as “admixture graphs\" in the population genetics literature).",
"In Section 9, we summarize the available software for phylogenetic network inference, discuss the data that these methods use, and then list some of the limitations of these methods in practice.",
"We conclude with final remarks and directions for future research in Section 10."
],
[
"Background for Non-biologists",
"In this section we define the biological terminology used throughout the chapter so that it is accessible for non-biologists.",
"We also provide a brief review of phylogenetic trees and their likelihood, which is the basis for maximum likelihood and Bayesian inference of phylogenetic trees from molecular sequence data.",
"Excellent books that cover mathematical and computational aspects of phylogenetic inference include [29], [102], [34], [117], [110]."
],
[
"Terminology",
"As we mentioned above, hybridization is reproduction between two members of genetically distinct populations, or species (Fig.",
"REF ).",
"Diploid species (e.g., humans) have two copies of each genome.",
"Aside from a few unusual organisms such as parthenogenic species, one copy will be maternal in origin and the other paternal.",
"When the hybrid individual (or $F_1$ ) is also diploid this process is called homoploid hybridization.",
"Figure: Hybridization, recombination, and the generation of a mosaic genome.",
"Diploid individual a from species A and individual b from species B mate, resulting ina diploid hybrid individual with one copy of its genome inherited from parent a and the other copy inherited from parent b.",
"A recombination event results in the “swapping\" of entire regionsbetween two copies of the genome.",
"After multiple generations in which more recombination happens, the genome becomes a mosaic.",
"Walking across the genomefrom left to right, the color switches back and forth between red and blue, where switches happen at recombination breakpoints.",
"Shown are four different loci.",
"Loci 1 and 3 are not appropriatefor tree inference since they span recombination breakpoints and, thus, include segments that have different evolutionary histories.",
"Loci 2 and 4 are the “ideal\" loci for analyses by methodsdescribed in this chapter.While each of the two copies of the genome in the hybrid individual traces its evolution back to precisely one of the two parents, this picture becomes much more complex after several rounds of recombination.",
"Recombination is the swapping of a stretch of DNA between the two copies of the genome.",
"Mathematically, if the two copies of the genome are given by strings $u$ and $v$ (for DNA, the alphabet for the strings is $\\lbrace A,C,T,G\\rbrace $ ), then recombination results in two strings $u^{\\prime }=u_1u_2u_3$ and $v^{\\prime }=v_1v_2v_3$ , where $u_1$ , $u_2$ , $u_3$ , $v_1$ , $v_2$ , and $v_3$ are all strings over the same alphabet, and $u_1v_2u_3=u$ and $v_1u_2v_3=v$ ; that is, substrings $u_2$ and $v_2$ were swapped.",
"Observe that when this happens, $u_1$ and $u_3$ in the copy $u^{\\prime }$ are inherited from one parent, and $v_2$ , also in the copy $u^{\\prime }$ , is inherited from a different parent.",
"A similar scenario happens in copy $v^{\\prime }$ of the diploid genome.",
"This picture gets further complicated due to backcrossing, which is the mating between the hybrid individual, or one of its descendants, with an individual in one of the parental species.",
"For example, consider a scenario in which descendants of the hybrid individual in Fig.",
"REF repeatedly mate with individuals from species A.",
"After several generations, it is expected that the genomes of the hybrid individuals become more similar to the genomes of individuals in species A, and less similar to the genomes of individuals in species B (using the illustration of Fig.",
"REF , the two copies of the genome would have much more red in them than blue).",
"Most models and methods for phylogenetic inference assume the two copies of a diploid genome are known separately and often only one of them is used to represent the corresponding individual.",
"However, it is important to note that knowing the two copies separately is not a trivial task.",
"Sequencing technologies produce data on both copies simultaneously, and separating them into their constituent copies is a well-studied computational problem known as genome phasing.",
"Biologists often focus on certain regions within the genomes for phylogenetic inference.",
"If we consider the genome to be represented as a string $w$ over the alphabet $\\lbrace A,C,T,G\\rbrace $ , then a locus is simply a substring of $w$ given by the start and end positions of the substring in $w$ .",
"The size of a locus can range anywhere from a single position in the genome to a (contiguous) stretch of 1 million or more positions in the genome.",
"As we discussed above, when recombination happens, an individual copy of the genome would have segments with different ancestries (the blue and red regions in Fig.",
"REF ).",
"A major assumption underlying phylogenetic tree inference is that the sequence data of a locus used for inference has evolved down a single tree.",
"Therefore, the more recombination which has occured within a locus over its evolutionary history (limited to the history connecting the species being studied), the less suitable it will be for phylogenetic inference.",
"Conversely, loci with low recombination rates may be more suitable in terms of avoiding intra-locus recombination, although such loci are more susceptible to linked selection [44]."
],
[
"Phylogenetic Trees and Their Likelihood",
"An unrooted binary phylogenetic tree $T$ on set ${{X}}$ of taxa (e.g., ${{X}}=\\lbrace humans,chimp,gorilla\\rbrace $ ) is a binary tree whose leaves are bijectively labeled by the elements of ${{X}}$ .",
"That is, if $|{{X}}|=n$ , then $T$ has $n$ leaf nodes and $n-2$ non-leaf (internal) nodes (each leaf node has degree 1 and each internal node has degree 3).",
"A rooted binary phylogenetic tree is a directed binary tree with a single node designated as the root and all edges are directed away from the root.",
"For $n$ taxa, a rooted binary tree has $n$ leaves and $n-1$ internal nodes (each leaf node has in-degree 1 and out-degree 0; each internal node except for the root has in-degree 1 and out-degree 2; the root has in-degree 0 and out-degree 2).",
"Modern methods for phylogenetic tree inference make use of molecular sequence data, such as DNA sequences, obtained from individuals within the species of interest.",
"The sequences are assumed to have evolved from a common ancestral sequence (we say the sequences are homologous) according to a model of evolution that specifies the rates at which the various mutational events could occur (Fig.",
"REF ).",
"Figure: Sequence evolution on a tree.",
"At the top is the ancestral sequence for a certain locus in the genome of an individual.",
"Through cell division and DNA replication, thissequence is inherited from parent to children.",
"However, mutations could alter the inherited sequences.",
"Boxes indicate letters that were deleted due to mutation.",
"Letters in blueindicate substitutions (a mutation that alters the state of the nucleotide).",
"The letter in green has mutated more than once during its evolutionary history.",
"With respect to sequenceU 1 U_1, sequence U 2 U_2 has two deletions at the 3rd and 8th positions, and a substitution (C to G) at the 5th position.",
"With respect to sequence U 1 U_1, sequenceS 3 S_3 has 7 deletions at positions 3–9, and substitutions at the first two positions (A to T, and A to C).",
"Sequences S 1 S_1, S 2 S_2, S 3 S_3, and S 4 S_4, with the boxes and colorsunknown, are often the data forphylogenetic inference.",
"That is, the four sequences used as data here are: AAATGTTAA, AAAGGTTAA, TCAA, and TCCACGAA.For example, to infer a phylogenetic tree $T$ on set ${{X}}=\\lbrace X_1,\\ldots ,X_n\\rbrace $ of taxa, the sequence $S_1$ of a certain locus is obtained from the genome of an individual in species $X_1$ , the sequence $S_2$ of a certain locus is obtained from the genome of an individual in species $X_2$ , and so on until $n$ sequences $S_1,\\ldots ,S_n$ are obtained.",
"To perform phylogenetic tree inference, the $n$ sequences must satisfy two important conditions (see Fig.",
"REF ): The sequences are homologous: The obtained sequences must have evolved down a single tree from a single sequence in an individual in an ancestral species.",
"Two sequences are homologous if they evolved from a common ancestor, including in the presence of events such as duplication.",
"Two homologous sequences are orthologs if they evolved from a common ancestor solely by means of DNA replication and speciation events.",
"Two homologous sequences are paralogs if their common ancestor had duplicated to give rise to the two sequences.",
"The sequences are aligned: While the obtained homologous “raw\" sequences might be of different lengths due to events such as insertions and deletions, the sequences must be made to be the same length before phylogenetic inference is conducted so that positional homology is established.",
"Intuitively, positional homology is the (evolutionary) correspondence among sites across the $n$ sequences.",
"That is, the sequences must be made of the same length so that the $i$ -th site in all of them had evolved from a single site in the sequence that is ancestral to all of them.",
"Figure: From homologous sequences to a phylogenetic tree.",
"Identifying homologous sequences across genomes is the first step towards a phylogenetic analysis.",
"The homologoussequences, once identified, are not necessarily of the same length, due to insertions and deletions.",
"Multiple sequence alignment is performed on the homologous sequences and the resultis sequences of the same length where boxes indicate deleted nucleotides.",
"Finally, a phylogenetic tree is constructed on the aligned sequences.Identifying homologous sequences across genomes is not an easy task; see, for example, [84] for a recent review of methods for homology detection.",
"Multiple sequence alignment is also a hard computational problem, with a wide array of heuristics and computer programs currently available for it; see, for example, [13] for a recent review.",
"We are now in position to define a basic version of the Phylogeny Inference Problem: Input: Set $S=\\lbrace S_1,\\ldots ,S_n\\rbrace $ of homologous sequences, where sequence $S_i$ is obtained from taxon $X_i$ , and the $n$ sequences are aligned.",
"Output: A phylogenetic tree $T$ on set ${{X}}$ of taxa such that $T$ is optimal, given the sequences, with respect to some criterion $\\Phi $ .",
"The books we cited above give a great survey of the various criterion that $\\Phi $ could take, as well as algorithms and heuristics for inferring optimal trees under the different criteria.",
"Here, we focus on the main criterion in statistical phylogenetic inference, namely likelihood.",
"We will make two assumptions when defining the likelihood that are (1) sites are identically and independently distributed, and (2) following a DNA replication event, the two resulting sequences continue to evolve independently of each other.",
"To define the likelihood of a tree $T$ , we first assign lengths $\\lambda :E(T) \\rightarrow {\\mathbb {R}}^+$ to its branches, so that $\\lambda (b)$ is the length of branch $b$ in units of expected number of mutations per site per generation.",
"Furthermore, we need a model of sequence evolution ${\\cal M}$ .",
"Most models of sequence evolution are Markov processes where the probability of observing a sequence $S$ at node $u$ depends only on the sequence at $u$ 's parent, the length of the branch that links $u$ to its parent, and the parameters of the model of sequence evolution.",
"If we denote by $p^{(i)}_{uv}(t)$ the probability that the $i$ -th nucleotide in the sequence at node $u$ evolves into the $i$ -th nucleotide in the sequence at node $v$ over time $t$ (measured in units of expected number of mutations as well), then the likelihood of a tree $T$ and its branch lengths $\\lambda $ is $L(T,\\lambda |S) = P(S|T,\\lambda ) = \\prod _{i} \\left(\\sum _R \\left(p(root^{(i)}) \\cdot \\prod _{b=(u,v)\\in E(T)} p^{(i)}_{uv}(\\lambda _b) \\right) \\right).$ Here, the outer product is taken over all sites $i$ in the sequences; i.e., if each of the $n$ sequences is of length $m$ , then $1 \\le i \\le m$ .",
"The summation is taken over $R$ , which is the set of all possible labelings of the internal nodes of $T$ with sequences of length $m$ .",
"Inside the summation, $p(root^{(i)})$ gives the stationary distribution of the nucleotides at position $i$ .",
"The likelihood as given by Eq.",
"(REF ) is computed in polynomial time in $m$ and $n$ using Felsenstein's “pruning\" algorithm [27].",
"Finally, the maximum likelihood estimate for solving the Phylogeny Inference Problem is given by $(T^*,\\lambda ^*) \\leftarrow {\\rm argmax}_{(T,\\lambda )} L(T,\\lambda |S).$ Computing the maximum likelihood estimate from a set $S$ of sequences is NP-hard [15], [100].",
"However, much progress has been made in terms of developing heuristics that scale up to thousands of taxa while achieving high accuracy, e.g., [109]."
],
[
"From Humble Beginnings: Smallest Displaying Networks",
"Early work and, still, much effort in the community has focused on inferring the topology of a phylogenetic network from a set of gene tree topologies estimated for the individual loci in a data set.",
"In this section, we discuss parsimony approaches to inferring phylogenetic network topologies from sets of gene trees."
],
[
"The Topology of a Phylogenetic Network",
"As discussed above, a reticulate, i.e., non-treelike, evolutionary history that arises in the presence of processes such as hybridization and horizontal gene transfer is best represented by a phylogenetic network.",
"A phylogenetic ${{X}}$ -network (Fig.",
"REF ), or ${{X}}$ -network for short, $\\Psi $ is a rooted, directed, acyclic graph (rDAG) with set of nodes $V(\\Psi )=\\lbrace r\\rbrace \\cup V_L \\cup V_T \\cup V_N $ , where $indeg(r)=0$ ($r$ is the root of $\\Psi $ ); $\\forall {v \\in V_L}$ , $indeg(v)=1$ and $outdeg(v)=0$ ($V_L$ are the external tree nodes, or leaves, of $\\Psi $ ); $\\forall {v \\in V_T}$ , $indeg(v)=1$ and $outdeg(v) \\ge 2$ ($V_T$ are the internal tree nodes of $\\Psi $ ); and, $\\forall {v \\in V_N}$ , $indeg(v) = 2$ and $outdeg(v) = 1$ ($V_N$ are the reticulation nodes of $\\Psi $ ).",
"For binary phylogenetic networks, the out-degree of the root and every internal tree node is 2.",
"The network's set of edges, denoted by $E(\\Psi ) \\subseteq V \\times V$ is bipartitioned into reticulation edges, whose heads are reticulation nodes, and tree edges, whose heads are tree nodes (internal or external).",
"Finally, the leaves of $\\Psi $ are bijectively labeled by the leaf-labeling function $\\ell :V_L \\rightarrow {{X}}$ .",
"Figure: An example of a phylogenetic network Ψ\\Psi with a single reticulation event.",
"This network is made up of leaf nodes V L ={v 1 ,v 2 ,v 3 ,v 4 }V_L = \\lbrace v_1,v_2,v_3,v_4\\rbrace , internal tree nodes V T ={u 1 ,u 2 ,u 3 }V_T=\\lbrace u_1,u_2,u_3\\rbrace , reticulation nodes V N ={h}V_N=\\lbrace h\\rbrace , and the root rr.",
"The nodes are connected by branches belonging to the set of phylogenetic network edges E(Ψ)E(\\Psi ).",
"The branches are:(r,u 1 )(r,u_1), (r,u 2 )(r,u_2), (u 1 ,h)(u_1,h), (u 2 ,h)(u_2,h), (h,u 3 )(h,u_3), (u 1 ,v 1 )(u_1,v_1), (u 2 ,v 4 )(u_2,v_4), (u 3 ,v 2 )(u_3,v_2), and (u 3 ,v 3 )(u_3,v_3).",
"The leaves are labeled by set X={A,B,C,D}{{X}}=\\lbrace A,B,C,D\\rbrace of taxa: ℓ(v 1 )=A\\ell (v_1)=A, ℓ(v 2 )=B\\ell (v_2)=B, ℓ(v 3 )=C\\ell (v_3)=C, and ℓ(v 4 )=D\\ell (v_4)=D."
],
[
"Inferring Smallest Displaying Networks",
"Early work on phylogenetic networks focused on the problem of identifying a network with the fewest number of reticulation nodes that summarizes all gene trees in the input.",
"More formally, let $\\Psi $ be a phylogenetic network.",
"We say that $\\Psi $ displays phylogenetic tree $t$ if $t$ can be obtained from $\\Psi $ by repeatedly applying the following operations until they are not applicable: For a reticulation node $h$ with two incoming edges $e_1=(u_1,h)$ and $e_2=(u_2,h)$ , remove one of the two edges.",
"For a node $u$ with a single parent $v$ and a single child $w$ , remove the two edges $(v,u)$ and $(u,w)$ , and add edge $(v,w)$ .",
"The set of all trees displayed by the phylogenetic network is ${\\cal T}(\\Psi )=\\lbrace t: \\; \\Psi \\; {\\rm displays} \\; t\\rbrace .$ For example, for the phylogenetic network $\\Psi $ of Fig.",
"REF , we have ${\\cal T}(\\Psi )=\\lbrace T_1,T_2\\rbrace $ , where $T_1=((A,(B,C)),D)$ and $T_2=(A,((B,C),D))$ .",
"Using this definition, the earliest phylogenetic network inference problem was defined as follows: Input: A set ${\\cal G}=\\lbrace g_1,g_2,\\ldots ,g_m\\rbrace $ of gene trees, where $g_i$ is a gene tree for locus $i$ .",
"Output: A phylogenetic network $\\Psi $ with the smallest number of reticulation nodes such that ${\\cal G} \\subseteq {\\cal T}(\\Psi )$ .",
"This problem is NP-hard [116] and methods were developed for solving it and variations thereof, some of which are heuristics [115], [123], [89], [124].",
"Furthermore, the view of a phylogenetic network in terms of the set of trees it displays was used for pursuing other questions in this domain.",
"For example, the topological difference between two networks could be quantified in terms of the topological differences among their displayed trees [83].",
"The parsimony and likelihood criteria were extended to the case of phylogenetic networks based on the assumption that each site (or, locus) has evolved down one of the trees displayed by the network [80], [54], [53], [56], [55], [57].",
"The concepts of character compatibility and perfect phylogeny were also extended to phylogenetic networks based on the notion of displayed trees [116], [82], [60].",
"Furthermore, questions related to distinguishability of phylogenetic networks based on their displayed trees have been pursued [59] and relationships between networks and trees have been established in terms of this definition [33], [134].",
"However, the computational complexity of this problem notwithstanding, the problem formulation could be deficient with respect to practical applications.",
"For one thing, solving the aforementioned problem only yields the topology of a phylogenetic network, but no other parameters.",
"In practice, biologists would be interested in divergence times, population parameters, and some quantification of the amount of introgression in the genomes.",
"These quantities are not recoverable under the given formulation.",
"Moreover, for the biologist seeking to analyze her data with respect to hypotheses of reticulate evolutionary events, solving the aforementioned problem could result in misleading evolutionary scenarios for at least three reasons.",
"First, the smallest number of reticulations required in a phylogenetic network to display all trees in the input could be arbitrarily far from the true (unknown) number of reticulations.",
"One reason for this phenomenon is the occurrence of reticulations between sister taxa, which would not be detectable from gene tree topologies alone.",
"Second, a smallest set of reticulations could be very different from the actual reticulation events that took place.",
"Third, and probably most importantly, some or even all of gene tree incongruence in an empirical data set could have nothing to do with reticulation.",
"For example, hidden paralogy and/or incomplete lineage sorting could also give rise to incongruence in gene trees.",
"When such phenomena are at play, seeking a smallest phylogenetic network that displays all the trees in the input is the wrong approach and might result in an overly complex network that is very far from the true evolutionary history.",
"To address all these issues, the community has shifted its attention in the last decade toward statistical approaches that view phylogenetic networks in terms of a probability distribution on gene trees that could encompass a variety of evolutionary processes, including incomplete lineage sorting."
],
[
"Phylogenetic Networks as Summaries of Trees",
"Before we turn our attention to these statistical approaches, it is worth contrasting smallest phylogenetic networks that display all trees in the input to the concept of consensus trees.",
"In the domain of phylogenetic trees, consensus trees have played an important role in compactly summarizing sets of trees.",
"For example, the strict consensus tree contains only the clusters that are present in the input set of trees, and nothing else.",
"The majority-rule consensus tree contains only the clusters that appear in at least 50% of the input trees.",
"When there is incongruence in the set of trees, these consensus trees are most often non-binary trees (contain “soft polytomies\") such that each of the input trees can be obtained as a binary resolution of the consensus tree.",
"Notice that while the consensus tree could be resolved to yield each tree in the input, there is no guarantee in most cases that it cannot also be resolved to generate trees that are not in the input.",
"Smallest phylogenetic networks that display all trees in the input could also be viewed as summaries of the trees, but instead of removing clusters that are not present in some trees in the input, they display all clusters that are present in all trees in the input.",
"Similarly to consensus trees, a smallest phylogenetic network could also display trees not in the input (which is the reason why we use $\\subseteq $ , rather than $=$ , in the problem formulation above).",
"These issues are illustrated in Fig.",
"REF .",
"Figure: Consensus trees and phylogenetic networks as two contrasting summary methods.",
"(a)-(c) Three (input) gene trees whose summary is sought.",
"(d) The strict consensus of the input trees.",
"(e) The 70% majority-rule consensus of the input trees.",
"(f) A smallest phylogenetic network that displays all three trees iin the input.",
"The strict consensus could be resolved to yield 15 different binary trees, only three of which are in the input.",
"The majority-rule consensus tree could beresolved to yield three possible trees, two of which are the trees in (a) and (d), but the third, which is (((a,b),d),c), is not in the input.",
"Furthermore, the tree in panel (b)is not included in the summary provided by the majority-rule consensus.",
"The phylogenetic network displays four trees, three of which are the input trees, and thefourth is ((a,(b,c)),d), which is not in the input.As discussed above, ILS is another process that could cause gene trees to be incongruent with each other and complicates the inference of phylogenetic networks since incongruence due to ILS should not induce additional reticulation nodes.",
"Before we move on to discuss statistical approaches that account for ILS in a principled probabilistic manner under the coalescent, we describe an extension of the minimizing deep coalescences, or, MDC, criterion [73], [72], [112], to phylogenetic networks, which was devised in [125]."
],
[
"A Step Towards More Complexity: Minimizing Deep Coalescences",
"Let $\\Psi $ be a phylogenetic network and consider node $u \\in V(\\Psi )$ .",
"We denote by $B_u \\subseteq V(\\Psi )$ the set of nodes in $\\Psi $ that are below node $u$ (that is, the set of nodes that are reachable from the root of $\\Psi $ via at least one path that goes through node $u$ ).",
"A coalescent history of a gene tree $g$ and a species (phylogenetic) network $\\Psi $ as a function $h:V(g) \\rightarrow V(\\Psi )$ such that the following conditions hold: if $w$ is a leaf in $g$ , then $h(w)$ is the leaf in $\\Psi $ with the same label (in the case of multiple alleles, $h(w)$ is the leaf in $\\Psi $ with the label of the species from which the allele labeling leaf $w$ in $g$ is sampled); and, if $w$ is a node in $g_v$ , then $h(w)$ is a node in $B_{h(v)}$ .",
"Given a phylogenetic network $\\Psi $ and a gene tree $g$ , we denote by $H_{\\Psi }(g)$ the set of all coalescent histories of gene tree $g$ within the branches of phylogenetic network $\\Psi $ .",
"Given a coalescent history $h$ , the number of extra lineages arising from $h$ on a branch $b=(u,v)$ in phylogenetic network $\\Psi $ is the number of gene tree lineages exiting branch $b$ from below node $u$ toward the root, minus one.",
"Finally, $XL(\\Psi ,h)$ is defined as the sum of the numbers of extra lineages arising from $h$ on all branches $b \\in E(\\Psi )$ .",
"Using coalescent histories, the minimum number of extra lineages required to reconcile gene tree $g$ within the branches of $\\Psi $ , denoted by $XL(\\Psi ,g)$ is given by $XL(\\Psi ,g)=\\min _{h \\in H_{\\Psi }(g)}XL(\\Psi ,h).$ Under the MDC (minimizing deep coalescence) criterion, the optimal coalescent history refers to the one that results in the fewest number of extra lineages [73], [112], and thus, $XL(\\Psi , g) = \\sum _{e \\in E(\\Psi )}[k_e(g)-1]$ where $k_e(g)$ is the number of extra lineages on edge $e$ of $\\Psi $ in the optimal coalescent history of gene tree $g$ .",
"Figure: The MDC criterion on phylogenetic networks.",
"A phylogenetic network and coalescent histories within its branches of the three gene trees in Fig.",
"(a)-(c).",
"The highlighted branch that separates the hybridization event from the MRCA of B and C has two extra lineages arising from the threeshown coalescent histories.",
"All other branches have 0 extra lineages.",
"Therefore, the total number of extra lineages in this case is 2.A connection between extra lineages and the displayed trees of a phylogenetic network is given by the following observation.",
"Observation 1 If gene tree $g$ is displayed by phylogenetic network $\\Psi $ , then $XL(\\Psi ,g)=0$ .",
"The implication of this observation is that if one seeks the phylogenetic network that minimizes the number of extra lineages, the problem can be trivially solvable by finding an overly complex network that displays every tree in the input.",
"Therefore, inferring a phylogenetic network $\\Psi $ from a collection of gene tree topologies ${{G}}$ based on the MDC criterion is more appropriately defined by $\\hat{\\Psi }(m) = \\operatornamewithlimits{argmin}_{\\Psi (m)} \\left( \\sum _{g \\in {{G}}} XL(\\Psi (m),g) \\right),$ where we write $\\Psi (m)$ to denote a phylogenetic network with $m$ reticulation nodes.",
"While the number of reticulations $m$ is unknown and is often a quantity of interest, there is a trade-off between the number of reticulation nodes and number of extra lineages in a network: Reticulation edges can be added to reduce the number of extra lineages.",
"Observing this reduction in the number of extra lineages could provide a mechanism to determine when to stop adding reticulations to the network [125].",
"In the previous section, we focused on two parsimony formulations for inferring a phylogenetic network from a collection of input gene tree topologies: The first seeks a network with the fewest number of reticulations that displays each of the input gene trees, and the second seeks a network that does not have to display every gene tree in the input, but must minimize the number of “extra lineages\" that could arise within a given number of reticulations.",
"Both formulations result in phylogenetic network topologies alone and make use of only the gene tree topologies.",
"In this section, we introduce the multispecies network coalescent, or MSNC [120], as a generative process that extends the popular multispecies coalescent, or MSC [20], that is the basis for most multi-locus species tree inference methods.",
"The MSNC allows for the coalescent to operate within the branches of a phylogenetic network by viewing a set of populations—extant and ancestral—glued together by a rooted, directed, acyclic graph structure."
],
[
"Parameterizing the Network's Topology",
"In addition to the topology of a phylogenetic network $\\Psi $ , as given by Definition REF above, the nodes and edges are parameterized as follows.",
"Associated with the nodes are divergence/reticulation times, $\\tau : V(\\Psi ) \\rightarrow {\\mathbb {R}}^+$ , where $\\tau (u)$ is the divergence time associated with tree node $u$ and $\\tau (v)$ is the reticulation time associated with reticulation node $v$ .",
"All leaf nodes $u$ in the network have $\\tau (u)=0$ .",
"Furthermore, if $u$ is on a path from the root of the network to a node $v$ , then $\\tau (u) \\ge \\tau (v)$ .",
"Associated with the edges are population mutation rate parameters, $\\theta : E(\\Psi ) \\rightarrow {\\mathbb {R}}^+$ , where $\\theta _b=4N_b \\mu $ is the population mutation rate associated with edge $b$ , $N_b$ is the effective population size associated with edge $b$ , and $\\mu $ is the mutation rate per site per generation.",
"Divergence times associated with nodes in the phylogenetic network could be measured in units of years, generations, or coalescent units.",
"Branch lengths in gene trees are often given in units of expected number of mutations per site.",
"The following rules are used to convert back and forth between these units: Given divergence time $\\tau $ in units of expected number of mutations per site, mutation rate per site per generation $\\mu $ and the number of generations per year $g$ , $\\tau / (\\mu g)$ represents divergence times in units of years.",
"Given population size parameter $\\theta $ in units of population mutation rate per site, $2\\tau / \\theta $ represents divergence times in coalescent units.",
"In addition to the divergence times and population size parameters, the reticulation edges of the network are associated with inheritance probabilities.",
"For every reticulation node $u \\in V_N$ , let $left(u)$ and $right(u)$ be the “left\" and “right\" edges incoming into node $u$ , respectively (which of the two edges is labeled left and which is labeled right is arbitrary).",
"Let $E_R \\subseteq E(\\Psi )$ be the set of reticulation edges in the network.",
"The inheritance probabilities are a function $\\gamma : E_R \\rightarrow [0,1]$ such that for every reticulation node $u \\in V_N$ , $\\gamma (left(u))+\\gamma (right(u)) = 1$ .",
"In the literature, $\\gamma $ is sometimes described as a vector $\\Gamma $ ."
],
[
"The Multispecies Network Coalescent and Gene Tree Distributions",
"As an orthologous, non-recombining genomic region from a set ${{X}}$ of species evolves within the branches of the species phylogeny of ${{X}}$ , the genealogy of this region, also called the gene tree, can be viewed as a discrete random variable whose values are all possible gene tree topologies on the set of genomic regions.",
"When the gene tree branch lengths are also taken into account, the random variable becomes continuous.",
"Yu et al.",
"[126] gave the probability mass function (pmf) for this discrete random variable given the phylogenetic network $\\Psi $ and an additional parameter $\\Gamma $ that contains the inheritance probabilities associated with reticulation nodes, which we now describe briefly.",
"The parameters $\\Psi $ and $\\Gamma $ specify the multispecies network coalescent, or MSNC (Fig.",
"REF ), and allow for a full derivation of the mass and density functions of gene trees when the evolutionary history of species involves both ILS and reticulation [126], [127].",
"This is a generalization of the multispecies coalescent, which describes the embedding and distribution of gene trees within a species tree without any reticulate nodes [20].",
"Figure: Layers of the multispecies network coalescent model.",
"A phylogenetic network describes the relationship between species (top).",
"The MSNC describes the distribution of gene trees within the network, in which alleles from the same species can have different topologies and inheritance histories due to reticulation and/or ILS (middle).",
"Some kind of mutation process occurs along the gene trees, resulting in observed differences between alleles in the present, which vary between genes based on their individual trees.It is important to note that a single reticulation edge between two nodes does not mean a single hybridization event.",
"Rather, a reticulation edge abstracts a continuous epoch of repeated gene flow between the two species, as illustrated in Fig.",
"REF .",
"Figure: Reticulation edges as abstractions of gene flow epochs.",
"(Left) An epoch of gene flow from one population to another with migration rate α\\alpha per generation.",
"(Right) A phylogenetic networkwith a single reticulation edge that abstracts the gene flow epoch, with inheritance probability γ\\gamma .The two models shown in Fig.",
"REF were referred to as the “gene flow\" model (left) and “intermixture\" model (right) of hybridization in [70].",
"While the “gene flow” model is used by the IM family of methods [47] to incorporate admixture, the MSNC adopts the “intermixture” model.",
"In this model, the $\\gamma $ inheritance probabilities indicate the ratio of genetic materials of a hybrid coming from its two parents.",
"This means that unlinked loci from a hybrid species will have independent evolutionary histories, and will have evolved through the “left” or “right” parent with some probability $\\gamma $ and $1 - \\gamma $ respectively.",
"The performance of phylogenetic network inference on data simulated under the gene flow model was demonstrated in [121].",
"Wen and Nakhleh [121] derived the density function of the probability of gene trees given a phylogenetic network, with its topology, divergence/migration times, population mutation rates and inheritance probabilities.",
"The divergence/migration times are in units of expected number of mutations per site, and population mutation rates are in units of population mutation rate per site.",
"Based on the MSNC, and by integrating out all possible gene trees, Zhu et al.",
"[137] developed an algorithm to compute the probability of a bi-allelic genetic marker given a phylogenetic network."
],
[
"Maximum Likelihood Inference of Phylogenetic Networks",
"Phylogenetic networks are more complicated than a tree with some reticulation edges.",
"The gene tree topology with highest mass probability may not be one of the backbone trees of the network with 4 or more taxa [138].",
"Also, not all networks can be obtained by simply adding edges between the original edges of a tree [33].",
"Therefore inferring phylogenetic networks is not a trivial extension of methods to infer species trees.",
"Most phylogenetic network methods [125], [127], [120], [121], [133], [137], [136], [128] sample from whole-network space rather than simply adding reticulations to a backbone tree.",
"As such methods walk the space from one phylogenetic network to another, the point estimate or posterior distribution of networks is not tree-based, does not return or imply a backbone tree, and can include networks which cannot be described by merely adding reticulate branches between tree branches."
],
[
"Inference",
"Sequential inference was initially developed to estimate species trees under the multispecies coalescent [78], [69], and in recent years has been extended to species networks [129], [127], [128], [104], [120].",
"These methods follow a two-step approach, where the first step is to estimate gene trees from multiple sequence alignments.",
"The second step is to estimate a species tree or network from the distribution of estimated gene tree topologies.",
"As described above and illustrated in Fig.",
"REF , two key requirements have facilitated the development of several methods for phylogenetic inference from multi-locus data, including those that follow the two-step approach.",
"One key requirement for current methods is that the segments are the result of speciation and not gene duplication, that is, sequences from different individuals and species are orthologs and not paralogs.",
"Meeting this requirement ensures that the nodes in each gene tree represent coalescent events and can be fit to a coalescent model within each species network branch.",
"A model which accounts for gene duplication and loss in addition to coalescent events has been developed to reconcile gene family trees with a fixed species tree [98], [132].",
"The most recent implementation of this model can also use it to estimate the species tree [21], but this model has yet to be extended to work with species networks.",
"A second key requirement is that the evolutionary history of the locus can be accurately modeled using a single tree.",
"Recombination or gene fusion should not have occurred within a locus, otherwise a gene network would be required to model that locus, breaking the MSNC model of gene trees within a species network [130].",
"Because of this requirement, multi-locus methods should be used with short contiguous sequences.",
"The results of a previous study on mammal phylogenetics suggest that individual exons are an appropriate target sequence [101].",
"Under these two key requirements, each gene tree is considered to be a valid and independent sample from an underlying distribution of gene trees conditional on some unobserved species network.",
"Of course this assumption in sequential inference is violated as the gene trees are only estimates.",
"Particular methods may be more or less sensitive to gene tree estimation errors.",
"For species trees, methods which infer unrooted species trees (e.g.",
"ASTRAL [78]) appear to be more robust relative to methods which infer rooted species trees (e.g.",
"MP-EST [69]).",
"This is because unrooted methods take unrooted gene trees as input and do not rely on correct rooting of the gene trees [103].",
"An estimate $\\hat{g}$ of the true gene tree is typically made using phylogenetic likelihood (see Section 2).",
"The phylogenetic likelihood of the sequence alignments can be combined sequentially or simultaneously with the MSNC probability densities of the gene trees to estimate a species network from sequence data.",
"Given gene trees where each node represents a coalescent event, the probability densities and masses of those gene trees given a species network can be calculated [127].",
"This can be based on the topologies and node heights of the gene trees, or based on the topologies alone (see Section REF ).",
"Maximum likelihood (ML) methods seek a phylogenetic network (along with its parameters) that maximizes some likelihood function.",
"In a coalescent context, these methods search for the species network which maximizes the likelihood of observing a sample of gene trees given the proposed species network.",
"The sample of gene trees can include branch lengths, in which case the likelihood is derived from the time intervals between successive coalescent events [127].",
"In the absence of branch lengths, the likelihood is derived from the probability mass of each gene tree topology [126], [127].",
"This probability is marginalized over every coalescent history $h$ , which is all the ways for a gene tree to follow the reticulate branching of the network: $P(g|\\Psi ,\\Gamma ,\\theta ) = \\sum _{h \\in H_\\Psi (g)}P(h|\\Psi ,\\Gamma ,\\theta )$ and the ML species network is therefore: $\\hat{\\Psi } = \\operatornamewithlimits{argmax}_\\Psi {\\prod _{g \\in {{G}}}{P(g|\\Psi ,\\Gamma ,\\theta )}}.$ ML inference of species networks has been implemented as the InferNetwork_ML command in PhyloNet [127], which identifies the ML species network up to a maximum number of reticulations.",
"Similar to our discussion of the MDC criterion above, absent any explicit stopping criterion or a penalty term in the likelihood function, obtaining an ML estimate according to Eq.",
"(REF ) can result in overly complex phylogenetic networks since adding more reticulations often improves the likelihood of the resulting network.",
"Therefore, it is important to parameterize the search by the number of reticulations sought, $m$ , and solve $\\hat{\\Psi }(m) = \\operatornamewithlimits{argmax}_{\\Psi (m)}{\\prod _{g \\in {{G}}}{P(g|\\Psi (m),\\Gamma ,\\theta )}},$ where the value $m$ is experimented with by observing the improvement in the likelihood for varying values of $m$ (for example, maximum likelihood inference of phylogenetic networks in PhyloNet implements information criteria, such as AIC and BIC, for this purpose [127]).",
"The computational cost of the likelihood calculation increases with larger species networks and gene trees.",
"Not only does this increase the number of branches and coalescent times, but as more reticulations are added many more possible coalescent histories exist to be summed over.",
"Even with one reticulation edge attached onto a tree, the difficulty of the problem is exponential to tree cases.",
"The computational complexity of the likelihood calculation is highly related to the size of the set of all coalescent histories of a gene tree conciliated in a network.",
"Zhu et al.",
"[136] proposed an algorithm to compute the number of coalescent histories of a gene tree for a network, and demonstrated that the number can grow exponentially after adding merely one reticulation edge to a species tree.",
"Figure: Running time of computing the likelihood of a phylogenetic network given gene tree topologies.",
"150 1-reticulation phylogenetic networks with 5 species and 4 individuals per species was used, and the data consisted of 10,000 bi-allelic markers.",
"The networks varied in terms of the diameter of a reticulation node (the number of edges on the cycle in the underlyingundirected graph) and the number of taxa (leaves) under the reticulation nodes.To show how running time of likelihood computation varies in a network with a single reticulation, we generated 150 random 1-reticulation networks with 5 taxa, then simulated 10,000 bi-allelic markers with 4 individuals per species.",
"When a reticulation node exists in a phylogenetic network, this will induce a cycle in the unrooted equivalent of the acyclic rooted network.",
"The “diameter” is the length of that cycle, and we ran the likelihood computation in [137] and summarized the maximum running time according to values of diameter and number of taxa under reticulation.",
"Fig.",
"(REF ) shows that the complexity of the likelihood computation is highly related to the structure of the network.",
"The running time in the worst case is hundreds of times slower than that of the best case.",
"A faster way to estimate a species network is to calculate a pseudolikelihood instead of a full likelihood.",
"The InferNetwork_MPL command in PhyloNet implements a maximum pseudo-likelihood (MPL) method for species networks.",
"This method is based on rooted triples, which is akin to the MP-EST method for species tree inference [128], [69].",
"Unlike phylogenetic trees, a given phylogenetic network is not necessarily uniquely distinguished by its induced set of rooted triples.",
"Therefore this method cannot distinguish the correct network when other networks induce the same sets of rooted triples [128].",
"However, it is much more scalable than ML methods in terms of the number of taxa [46].",
"Another MPL method, SNaQ, is available as part of the PhyloNetworks software [104], [105].",
"SNaQ is based on unrooted quartets, akin to the ASTRAL method for species tree inference [78].",
"It is even more scalable than InferNetwork_MPL [46], but can only infer level-1 networks (Fig.",
"REF ).",
"Figure: Level-1 network definition.",
"Reticulation nodes induce cycles in the (undirected graphs underlying the) phylogenetic networks.",
"The edges of the cycles are highlighted with red lines.",
"(a) A level-1 network is one where no edge of the network is shared by two or more cycles.",
"(b) A non-level-1 network is one where at least one edge is shared by at least two cycles (the shared edge in this case is the one inside the blue circle)."
],
[
"Bayesian Inference of Phylogenetic Networks",
"Maximum likelihood estimation of phylogenetic networks, as described in the previous section, has three main limitations: As discussed, without penalizing the complexity of the phylogenetic network, the ML estimate could be an overly complex network with many false reticulations.",
"The inference results in a single point estimate that does not allow for assessing confidence in the inferred network.",
"The formulation does not allow for making use of the sequence data directly, but is based on gene tree estimates.",
"One way of addressing these limitations is to adopt Bayesian inference where an estimate of the posterior distribution on networks is sought directly from the sequence data of the individual loci, and where the prior distribution on phylogenetic networks accounts for model complexity in a principled manner.",
"Before we describe the work on Bayesian inference, it is important to note that while maximum likelihood estimation is not satisfactory, we cannot say that Bayesian estimation is without challenges.",
"Such methods like [121], [133], [137] are based on reversible-jump MCMC [38] with varying numbers of parameters.",
"Mixing problems arise when they involve dimension changing moves: adding a reticulation and removing a reticulation.",
"This is because while walking over the space of phylogenetic networks, these methods jump between probability spaces of different models.",
"Therefore moves should be carefully designed to account for mixing issues."
],
[
"Probability Distributions Over Species Networks",
"It is useful to define probability distributions over species trees or networks without reference to sequence data or gene trees.",
"Among other uses, these probability distributions can be applied as prior distributions in Bayesian inference.",
"The two most common types of prior distributions used for species trees are birth-death tree priors and compound priors.",
"Both types have been extended to create probability distributions over species networks.",
"As their name implies, birth-death tree priors combine a rate of birth with a rate of death.",
"These are the rates at which one lineage splits into two, and one lineage ceases to exist respectively [36].",
"In the context of species trees these rates are more informatively called speciation and extinction.",
"When the extinction rate is set to zero, this is known as a Yule prior [131].",
"Birth-death tree priors have been extended to support incomplete sampling in the present, and sampling-through-time [108].",
"A birth-death prior for species networks has been developed, called the birth-hybridization prior.",
"This prior combines a rate of birth (or speciation) with a rate of hybridization, which is the rate at which two lineages merge into one.",
"This model does not include a rate of extinction [133].",
"All birth-death tree priors induce a uniform probability distribution over ranked tree topologies, regardless of the rates of speciation and extinction.",
"This means that birth-death priors favor symmetric over asymmetric trees, as symmetric trees have more possible ranked histories.",
"Empirical trees generally have more asymmetric shapes than predicted by birth-death models [43].",
"For the birth-hybridization prior the probability distribution over network topologies is not invariant to the hybridization rate, which when set to zero reduces to the Yule model, and any topology containing a reticulation will have zero probability.",
"All birth-death priors are generative, as is the birth-hybridization prior.",
"This means that not only can these distributions be used as priors for Bayesian inference, but they can be used to simulate trees and networks.",
"These simulated distributions can then be used for ABC inference, which is used for models which are difficult to implement using MCMC.",
"They can also be used for posterior predictive checks, which is an absolute measure of model goodness of fit [35].",
"While birth-death priors induce a probability distribution over topologies and branch lengths, compound tree and network priors are constructed from separate distributions on both.",
"Typically compound tree priors combine a uniform distribution over unranked tree topologies, favoring more asymmetric trees.",
"Empirical trees generally have more symmetric shapes than predicted by this distribution [43].",
"Then a continuous distribution such as gamma can be applied to branch lengths or node heights.",
"Compound priors are used for network inference by adding a third distribution describing the number of reticulations [121].",
"A Poisson distribution is a natural fit for this parameter as it describes a probability on non-negative integers.",
"The probability distribution for each network topology can still be uniform for all networks given $k$ reticulations.",
"Unlike birth-death priors, compound priors are not generative, so it is not straightforward to simulate trees or networks from those distributions.",
"The most obvious way to simulate such trees and networks would be running an existing Markov Chain Monte Carlo sampler without any data, and subsampling states from the chain at a low enough frequency to ensure independence between samples."
],
[
"Sampling the Posterior Distribution",
"The ML species network with $k + 1$ reticulations will always have a higher likelihood than the ML network with $k$ reticulations.",
"For this reason, some threshold of significance must be applied to estimate the number of reticulations.",
"This threshold may be arbitrary or it may be theoretically based, for example the Akaike information criterion (AIC) and Bayesian information criterion (BIC) measures of relative fit [10].",
"In contrast, Bayesian methods of species network inference are able to naturally model the probability distribution over species networks including the number of reticulations by using a prior (see Section REF ).",
"In a Bayesian model, the posterior probability of a species network $P(\\Psi )$ is proportional to the likelihood of the gene trees $P(G|\\Psi ,\\Gamma ,\\theta )$ , multiplied by the prior on the network and other parameters of the model $P(\\Psi ,\\Gamma ,\\theta )$ , and marginalized over all possible values of $\\Gamma $ and $\\theta $ : $P(\\Psi ) \\propto \\iint P(G|\\Psi ,\\theta ) \\cdot P(\\Psi ,\\Gamma ,\\theta ) \\mathop {}\\!\\mathrm {d}\\Gamma \\mathop {}\\!\\mathrm {d}\\theta .$ When a decaying prior is used on the number of reticulations or on the rate of hybridization, the prior probability of species networks with large numbers of reticulations will be very low, and so will the posterior probability (Fig.",
"REF ).",
"Figure: Bayesian inference of the number of reticulations.",
"In this example the topology of the network Ψ\\Psi is fixed, the true number of reticulations is 1, and the likelihood is calculated for a topology with no reticulations, the true reticulation, and additional reticulations, with maximum likelihood branch lengths.",
"The posterior probability was normalized to sum to 1, although as this is not integrated over branch lengths, the typical Bayesian posterior probability might be a bit different.Bayesian methods for phylogenetic inference typically use the Metropolis-Hastings Markov Chain Monte Carlo (MCMC) algorithm to estimate the posterior distribution of trees or networks.",
"MCMC is a random walk where each step depends on the previous state, and it is flexible enough to be used for implementing extremely complex models such as species network inference with relative ease.",
"Bayesian estimation of species networks from gene trees is implemented in the PhyloNet command MCMC_GT.",
"The posterior probability of a species network is equal to the integral in equation REF multiplied by a normalizing constant $Z$ known as the marginal likelihood.",
"In the case of sequential multilocus inference, this constant is equal to $Z = P(G)^{-1}$ The marginal likelihood is usually intractable to calculate, but MCMC sidesteps this calculation by sampling topologies and other parameters with frequencies proportional to their probability mass or density.",
"The posterior probability of a species network $\\Psi $ can therefore be approximated as the proportion of steps in the MCMC chain where the network topology $\\Psi _i$ at the end of the step $i$ is equal to $\\Psi $ .",
"The value of any particular parameter, for example an inheritance probability $\\gamma $ for a given reticulation node $v$ , can be estimated by averaging its value over the set of $X$ steps where the state includes that parameter.",
"In this case it is averaged over the states where the species network includes that node, i.e.",
"the set $X = \\lbrace i: v \\in \\Psi _i\\rbrace $ : $E(\\gamma ) = \\frac{1}{|X|} \\sum ^i_{X} \\gamma _i$ Bayesian inference has also enabled the inference of species trees and networks directly from multilocus sequence data.",
"Instead of first estimating individual gene trees from multiple sequence alignments, these methods jointly infer the gene trees and species network using an application of Bayes' rule: $P(G,\\Psi ) \\propto \\iint P(D|G) \\cdot P(G|\\Psi ,\\Gamma ,\\theta ) \\cdot P(\\Psi ,\\Gamma ,\\theta ) \\mathop {}\\!\\mathrm {d}\\Gamma \\mathop {}\\!\\mathrm {d}\\theta .$ Here $P(D|G)$ is the likelihood of the data over all gene trees.",
"In practice this is the sum of phylogenetic likelihoods $\\sum _i P(d_i|g_i)$ for every sequence alignment $d$ and associated gene tree $g$ .",
"As with sequential Bayesian inference of species trees and networks, the use of MCMC avoids the calculation of the marginal likelihood, which for joint inference can be expressed as $Z = P(D)^{-1}$ .",
"Joint Bayesian inference was first developed for species trees, and now has several popular implementations including StarBEAST2 [86] and BPP [96].",
"Joint Bayesian inference of species networks has been implemented independently as the PhyloNet command MCMC_SEQ, and as the BEAST2 package “SpeciesNetwork” [121], [133].",
"These two methods are broadly similar in their model and implementation, with a few notable differences.",
"MCMC_SEQ uses a compound prior on the species network, whereas SpeciesNetwork has a birth-hybridization prior (see Section REF ).",
"SpeciesNetwork is able to use any of the protein and nucleotide substitution models available in BEAST2.",
"MCMC_SEQ can be used with any nested GTR model but with fixed rates and base frequencies.",
"So the rates (e.g.",
"the transition/transversion ratio for HKY) must be estimated before running the analysis, or Jukes-Cantor is used where all rates and base frequencies are equal."
],
[
"Inference Under MSC vs. MSNC When Hybridization Is Present",
"We simulated 128 loci on the phylogenetic network of Fig.",
"REF (a).",
"The program ms [48] was used to simulate 128 gene trees on the network, and each gene tree was used to simulate a sequence alignment of 500 sites using the program Seq-gen [95] under the GTR model and $\\theta = 0.036$ for the population mutation rate.",
"The exact command used was: seq-gen -mgtr -s$0.018$ -f$0.2112,0.2888,0.2896,0.2104$ -r$0.2173,\\\\0.9798,0.2575,0.1038,1,0.2070$ -l500 We then ran both StarBEAST and MCMC_SEQ, as inference methods under the MSC and MSNC models, respectively, for $6\\times 10^7$ iterations each.",
"The results are shown in Fig.",
"REF .",
"Figure: Inference under the MSC and MSNC when the evolutionary history involves hybridization.",
"(a) The true phylogenetic network with the shown inheritance probabilities and branch lengths (in coalescent units).",
"(b) The MPP (maximum a posteriori probability) species tree estimated under the MSC by StarBEAST (frequency of 94% in the 95%95\\% credible set) with the average divergence times.",
"(c) The MPP phylogenetic network along with the inheritance probabilities estimated under the MSNC by MCMC_SEQ (the only network topology in the 95%95\\% credible set).",
"The scale bar of divergence times represents 1 coalescent unit for (a-c).",
"(d) The coalescent times of the MRCAs of (C,G), (A,Q), (A,C), (Q,R) from co-estimated gene trees inferred by StarBEAST (green) and MCMC_SEQ (blue).A few observations are in order.",
"First, while StarBEAST is not designed to deal with hybridization, it inferred the tree topology (Fig.",
"REF (b)) that is obtainable by removing the two hybridization events (the two arrows) from the true phylogenetic network (the backbone tree).",
"Second, MCMC_SEQ identified the true phylogenetic network as the one with the highest posterior (Fig.",
"REF (c)).",
"Furthermore, the estimated inheritance probabilities are very close to the true ones.",
"Third, and most interestingly, since StarBEAST does not account for hybridization, it accounts for all heterogeneity across loci as being caused by incomplete lineage sorting (ILS) by underestimating all branch lengths (that is, “squashing\" the divergence times so as to explain the heterogeneity by ILS).",
"Indeed, Fig.",
"REF (d) shows that the minimum coalescent times of the co-estimated gene trees by StarBEAST force the divergence times in the inferred species tree to be very low.",
"MCMC_SEQ, on the other hand, accurately estimates the branch lengths of the inferred phylogenetic network since networks differentiate between divergence and hybridization times.",
"For example, Fig.",
"REF (d) shows that the coalescent times of clade (C,G) across all co-estimated gene trees is a continuum with a minimum value around 2, which defines the divergence time of these two taxa in the phylogenetic network.",
"MCMC_SEQ clearly identifies two groups of coalescent times for each of the two clades (A,C) and (Q,R): The lower group of coalescent times correspond to hybridization, while the upper group of coalescent times correspond to the coalescences above the respective MRCAs of the clades.",
"We also note that the minimum value of coalescent times corresponding to (Q,R) is larger than that corresponding to (A,Q), which correctly reflects the fact that hybridization from R to Q happened before hybridization from C to A, as indicated in the true phylogenetic network.",
"Finally, for clade (A,Q), three groups of coalescence times are identified by MCMC_SEQ, which makes sense since there are three common ancestors of A and Q in the network: at the MRCA of (A,Q) in the case of no hybridization involving either of the two taxa, at the MRCA of (A,Q,L,R) in the case of the hybridization involving Q, and at the root of the network in the case of the hybridization involving A.",
"More thorough analysis and comparison of inferences under the MSC and MSNC can be found in [121].",
"These results illustrate the power of using a phylogenetic network inference method when hybridization is involved.",
"In particular, if hybridization had occurred, and the practitioner did not suspect it and ran StarBEAST instead, they would get wrong inferences.",
"In this case, the errors all have to do with the divergence time estimates.",
"However, the topology of the inferred tree could be wrong as well, depending on the hybridization scenarios."
],
[
"Phylogenetic Invariants Methods",
"The focus of this chapter up to this point has largely been on the MSC and MSNC models.",
"A parallel effort has been led to detect reticulate evolution by using the notion of phylogenetic invariants [12], [64].",
"Phylogenetic invariants are polynomial relationships satisfied by frequencies of site patterns at the taxa labeling the leaves of a phylogenetic tree (and given a model of sequence evolution).",
"Invariants that are predictive of particular tree topologies could then be used for inferring the tree topology by focusing on the space of site patterns rather than the space of tree topologies [28].",
"As Felsenstein wrote in his book, “invariants are worth attention, not for what they do for us now, but what they might lead to in the future.\"",
"With the availability of whole-genome data and, consequently, the ability to obtain better estimates of site frequencies, the future is here.",
"Indeed, methods like SVDQuartets [14] use phylogenetic invariants to estimate species trees under the MSC model.",
"A detailed discussion of phylogenetic invariants in general is beyond the scope of this manuscript.",
"Interested readers should consult the excellent exposition on the subject in Felsenstein's seminal book (Chapter 22 in [28]).",
"In this section, we briefly review phylogenetic invariants-based methods for detecting reticulation, starting with the most commonly used one, known as the $D$ -statistic or the “ABBA-BABA\" test.",
"The $D$ -Statistic [39] is a widely known and frequently applied statistical test for inferring reticulate evolution events.",
"The power of the test to infer reticulate evolution derives from the likelihood calculations of the MSNC.",
"Despite this, the test itself is simple to calculate and formalize.",
"The $D$ -Statistic is given by $\\frac{N_{ABBA} - N_{BABA}}{N_{ABBA} + N_{BABA}}$ To calculate these quantities, we are given as input the four taxon tree including outgroup of Fig.",
"REF and a sequence alignment of the genomes of P1, P2, P3, and O.",
"Given this alignment, $N_{ABBA}$ is calculated as the number of occurrences of single sites in the alignment where P1 and O have the same letter and P2 and P3 have the same letter, but these two letters are not the same i.e.",
"CTTC or GCCG.",
"Similarly, $N_{BABA}$ can be calculated as the number of occurrences in the alignment where the letters of $P1 = P3$ and $P2 = O$ with no other equalities between letters.",
"Upon calculating the $D$ -Statistic, a significant deviation away from a value of 0 gives evidence for reticulate evolution.",
"As shown in Fig.",
"REF , a strong positive value implies introgression between P2 and P3 while a strong negative value implies introgression between P1 and P3.",
"No such conclusions can be made from a D value very close to 0.",
"The crux of the theory behind the $D$ -Statistic lies in the expectation of the probabilities of discordant gene trees given the overall phylogeny of Fig.",
"REF .",
"If we remove the two reticulation events in Fig.",
"REF we end up with a species tree $\\Psi $ .",
"Figure: The four-taxon tree topology used for the DD-Statistic.",
"Significant deviations away from a value of 0 of the DD-statistic (Eq.",
"()) support introgression between P3 and either P1 or P2.",
"As shown, a significant positive value supports introgression between P2 and P3.",
"A significant negative value supports introgression between P1 and P3.Given $\\Psi $ , the two gene trees whose topologies disagree with that of the species tree are equally probable under the MSC.",
"Figure: The three scenarios of probabilities of the two gene trees that are discordant with the species tree in the case of a single reticulation event between P3 and one of the othertwo (in-group) species.",
"If the evolutionary history of the species is a tree (Ψ)(\\Psi ), the two discordant gene trees are equally probable.",
"However, if the evolutionary history of the speciesis non-treelike, as given by phylogenetic networks Ψ 1 \\Psi _1 and Ψ 2 \\Psi _2, then the probabilities of the two discordant gene trees are unequal in different ways.On the other hand, when a reticulation between P1 and P3 occurs, this results in an increase in the probability of the discordant gene tree that groups P1 and P3 as sister taxa, as compared to the other discordant gene tree.",
"Similarly, when a reticulation between P2 and P3 occurs, this results in an increase in the probability of the discordant gene tree that groups P2 and P3 as sister taxa.",
"These three scenarios are illustrated in Fig.",
"REF .",
"Assuming an infinite sites model of sequence evolution, the frequencies of gene trees ((P1,(P2,P3)),O) and (((P1,P3),P2),O) directly correlate with the values $N_{ABBA}$ and $N_{BABA}$ , respectively, explaining the rationale behind Eq.",
"(REF ).",
"To apply the $D$ -statistic, frequencies of the $ABBA$ and $BABA$ site patterns are counted across an alignment of four genomes, the value of Eq.",
"(REF ) is calculated, and deviation from 0 is assessed for statistical significance.",
"A significant deviation is taken as evidence of introgression.",
"Since the introduction of the $D$ -Statistic, work has been done to extend this framework.",
"Recently, the software package HyDe [5] was introduced with several extensions including handling multiple individuals from four populations as well as identifying individual hybrids in a population based on the method of [62].",
"In HyDe, higher numbers of individuals are handled through calculating statistics on all permutations of quartets of the individuals.",
"Another recent extension to move the $D$ -Statistic beyond four taxa is the DFOIL framework introduced by [91].",
"In it, we see the same derivation used in the $D$ -Statistic on a particular five-taxon tree.",
"This derivation includes isolating gene trees whose probabilities go from equal to unequal when going from the tree case to the network case as well as converting these gene trees to corresponding site patterns to count in an alignment.",
"Finally, Elworth et al.",
"recently devised a heuristic, $D_{GEN}$ , for automatically deriving phylogenetic invariants for detecting hybridization in more general cases than the $D$ -Statistic and DFOIL can handle [25].",
"The rationale behind the approach of Elworth et al.",
"is that invariants could be derived by computing the probabilities of gene trees under a given species tree (e.g., using the method of [19]) and then computing the probabilities of the same trees under the same species tree with any reticulation scenarios added to it (using the method of [126]), and contrasting the two to identify sets of gene trees whose equal probabilities under the tree model get violated under the network model.",
"The $D$ -Statistic is very simple to implement and understand, and it can be calculated on 4-genome alignments very efficiently, making it an appealing choice of a test for detecting introgression.",
"Indeed, applications of the $D$ -Statistic are widespread in the literature, reporting on introgression in ancient hominids [39], [22], butterflies [114], bears [63], and sparrows [23], just to name a few.",
"However, it is important to note here that the derivation of the $D$ -Statistic (and its extensions) relies on many assumptions that can easily be violated in practice.",
"One major cause of such a violation is that the mathematics behind the $D$ -Statistic relies on the coalescent which makes many simplifying assumptions about the evolutionary model and processes taking place.",
"Of course, this shortcoming applies as well to all phylogenetic inference methods that employ the MSC or MSNC models.",
"A second cause of such a violation is that in practice, more than a single reticulation could have taken place and ignoring those could result in erroneous inferences [25].",
"A third violation stems from the way the $D$ -Statistic is applied.We especially thank David Morrison for requesting that we highlight this issue.",
"In propositional logic, the statement “If $p$ , then $q$ \" and its converse “If $q$ , then $p$ \" are logically not equivalent.",
"That is, if one is true, it is not necessarily the case the other is.",
"Looking back at Fig.",
"REF , the statement illustrated by the figure is: If there is no reticulation (i.e., the species phylogeny is a tree), then the probabilities of the two discordant trees are equal.",
"The converse (if the probabilities of the two discordant trees are equal, then the species phylogeny is a tree) does not follow logically.",
"However, this is how the test is used in practice.",
"In all fairness, though, this logical fallacy is commonplace in inferences in biology, including when inferring species trees and networks under the MSC and MSNC, respectively.",
"The fallacy is always dealt with by resorting to the “simplest possible explanation\" argument.",
"For example, why various scenarios could have given rise to equal frequencies of the frequencies of the $ABBA$ and $BABA$ site patterns, the species tree scenario is considered the simplest such possible explanation and is invoked as such.",
"Last but not least, Peter [92] recently provided a review and elegant connections between the $D$ -Statistic and a family of statistics known as the $F$ -statistics."
],
[
"Phylogenetic Networks in the Population Genetics Community",
"The population genetics community has long adopted rooted, directed acyclic graphs as a model of evolutionary histories, typically of individuals within a single species.",
"Ancestral recombination graphs, or ARGs, were introduced [49], [40] to model the evolutionary history of a set of genomic sequences in terms of the coalescence and recombination events that occurred since their most recent common ancestor.",
"Statistical methods for inference of ARGs from genome-wide data have also been developed [97].",
"Gusfield's recent book [42] discusses algorithmic and combinatorial aspects of ARGs.",
"However, while ARGs take the shape of a phylogenetic network as defined above, they are aimed at modeling recombination and methods for their inference are generally not applicable to hybridization detection.",
"Efforts in the population genetics community that are aimed at modeling admixture and gene flow are more relevant to hybridization detection.",
"Here we discuss one of the most popular methods in this domain, namely TreeMix [93].",
"In population genetics, the counterparts to species trees and phylogenetic networks are population trees and admixture graphs, respectively.",
"The difference between these models boils down to what labels their leaves: If the leaves are labeled by different species, then the models are called species trees/networks, and if the leaves are labeled by different sub-populations of the same species, then the models are called population trees and admixture graphs.",
"Of course it is not always easy to identify whether a species or sub-populations have been delimited, and hence what particular tree/network should be called in that case, as species and populations may exist on a continuum [18]."
],
[
"TreeMix",
"TreeMix [93] models the evolution of a set of SNPs where the input data consists of allele frequencies of these SNPs in a set of populations whose evolutionary history is given by a population tree (in the case of no migration) or an admixture graph (in the case where migration is included).",
"The basis of the model used in TreeMix is in the notion of modeling drift over time as a diffusion process, where an original allele frequency of $x_1$ of a given SNP undergoes drift by an amount $c$ to give rise to a new allele frequency $x_2$ [11], [85], [17], as given by $x_2 = x_1 + N(0,c \\cdot x_1[1-x_1]).$ It is worth noting here that, as pointed out in [93], $c ~= t/2N_e$ for drift over small time scales where the time scale is on the same order of the effective population size [85].",
"When there are multiple populations under the effects of drift that evolved down a tree, the drift processes become linked and can no longer be described with independent Gaussian additions.",
"This process is modeled with a covariance matrix derived from the amounts of drift occurring along the branches of the evolutionary tree.",
"Finally, to incorporate reticulate evolution into the model one only needs to alter this covariance matrix based on the rate of gene flow along reticulate edges in the admixture graph.",
"In its current implementation, the authors of TreeMix assume the evolutionary history of the sampled, extant populations is very close to a tree-like structure.",
"Based on this assumption, the search for a maximum likelihood admixture graph proceeds by first estimating a rooted tree, and then adding migration events one at a time until they are no longer statistically significant (however, as the authors point out, they “prefer to stop adding migration events well before this point so that the result graph remains interpretable.\").",
"Clearly, adding additional edges connecting the edges of a tree in this way will infer a tree-based network, which is a more limited class of networks compared with phylogenetic networks [33]."
],
[
"Data, Methods, and Software",
"Given the interest in reticulate evolution from both theoreticians and empirical researchers, it is perhaps unsurprising that software to infer hybridization has proliferated in recent years.",
"Such methods have been developed for a variety of data types, including multilocus data, SNP matrices, and whole genomes (Table ).",
"Some of these methods are able to infer a phylogenetic network, whereas others infer introgression between species tree lineages.",
"With the exception of TreeMix and MixMapper, methods which infer networks are not constructed around a backbone tree, and so do not assume that tree-like evolution is the dominant process.",
"Regardless of the input and output, most of these methods allow for ILS in addition to hybridization (Table ), which is necessary to infer phylogenetic networks representing reticulate evolution in biological systems where ILS is a possibility.",
"Likelihood (including Bayesian) methods incorporate the possibility or effect of ILS into the likelihood function.",
"Maximum parsimony methods that minimize deep coalescences, for example InferNetwork_MP, essentially attempt to infer the tree or network that minimizes the quantity of ILS, but do not necessarily eliminate all genetic discordance.",
"Methods which do not allow for ILS will instead infer phylogenetic networks representing conflicting signals [9].",
"Reticulation is one such conflicting signal, but so is ILS, so reticulate branches in these networks should not be blindly interpreted as necessarily representing introgression or hybridization.",
"Common methods to infer hybridization Table: NO_CAPTION"
],
[
"Limitations",
"The biggest limitation of methods to infer introgression and hybridization, including species network methods, is scalability.",
"Methods which infer a species network directly from multilocus sequences have only been used with a handful of taxa, and less than 200 loci.",
"A systematic study of the species tree method StarBEAST found that the number of loci used has a power law relationship with a large exponent with the required computational time, making inference using thousands of loci intractable [87].",
"Although no systematic study of computational performance has been conducted for equivalent species network methods such as MCMC_SEQ, anecdotally they suffer from similar scaling issues.",
"Methods which scale better than direct multilocus inference have been developed, but they are no silver bullet.",
"Species networks can be estimated directly from unlinked biallelic markers by integrating over all possible gene trees for each marker, which avoids having to sequentially or jointly estimate gene trees.",
"Biallelic methods make the inference of species trees and networks from thousands of markers possible, at the cost of using less informative markers.",
"Pseudolikelihood inference has been developed for both biallelic and multilocus methods [136], [128].",
"This reduces the computational cost of computing the likelihood of a species network as the number of taxa increases, and enabled the reanalysis of an empirical data set with 1070 genes from 23 species [128].",
"The ABBA-BABA test and similar phylogenetic invariant methods are capable of analyzing an enormous depth of data (whole genomes), but can be limited in taxonomic breadth based on hard limits of four or five taxa for the D-Statistic and $D_{FOIL}$ , respectively, or by computational requirements for the case of $D_{GEN}$ (Table ).",
"In addition, the D-Statistic and $D_{FOIL}$ are limited to testing a specific hypothesis for introgression given a fixed species tree topology of a specific shape.",
"This can be understood as a trade off, where the flexibility of species network methods is sacrificed for the ability to use more data.",
"Beyond scalability, another present limitation is visualizing or summarizing posterior or bootstrap distributions of networks.",
"Methods have been developed to visualize whole distributions of trees, or summarize a distribution as a single tree.",
"Equivalent tools for networks are underdeveloped, leaving researchers to report the topology or set of topologies with the highest posterior or bootstrap support."
],
[
"Conclusions and Future Directions",
"Great strides have been made over the past decade in the inference of evolutionary histories in the presence of hybridization and other processes, most notably incomplete lineage sorting.",
"Species networks can now be inferred directly from species-level data which do not assume any kind of backbone tree, and instead put reticulate evolution on an equal basis with speciation.",
"To some extent the development of species network methods have recapitulated the development of species tree methods, starting with maximum parsimony and transitioning to likelihood methods, both maximum likelihood and Bayesian.",
"To improve computational performance and enable the analysis of large data sets, pseudo-likelihood species network methods have been developed, inspired by similar species tree methods.",
"Phylogenetic invariant methods such as the ABBA-BABA test are able to test for reticulate evolution across whole genomes, uncovering chromosomal inversions and other features associated with hybridization and introgression.",
"Last but not least, the population genetics community has long been interested in and developing methods for phylogenetic networks mainly to model the evolution of sub-populations in the presence of admixture and gene flow.",
"In this chapter, we surveyed the recent computational developments in the field and listed computer software programs that enable reticulate evolutionary analyses for the study of hybridization and introgression, and generally to infer more accurate evolutionary histories of genes and species.",
"Empirical biologists feel constrained by the computational performance of existing species network methods.",
"For species trees, phylogenetic invariant methods can be combined with quartet reconciliation to infer large species trees from genomic data, as in SVDquartets [14].",
"For networks, phylogenetic invariant methods to identify the true network with a limited number of edges need to be developed, as do methods to reconcile the resulting subnets.",
"Even for species trees, Bayesian methods have practical limitations in terms of the amount of data they can be used with.",
"Bayesian methods for trees and networks, with few exceptions, have been built on Markov chain Monte Carlo (MCMC).",
"This technique is inherently serial and hence unsuited to modern workstations, which contain many CPU and GPU cores working in parallel.",
"It is important to continue to explore other Bayesian algorithms which work in parallel such as sequential Monte Carlo [7], or algorithms which are orders of magnitude faster than MCMC such as variational Bayes [118].",
"Phylogenetic methods for species tree inference have a huge head start on methods for species network inference.",
"Not only is the problem of species network inference much more complicated, but species tree methods have been in development for much longer.",
"For example, MDC for species trees was first described in 1997, and extended to phylogenetic networks 14 years later [71], [130].",
"In this light the progress made is remarkable.",
"However, as evolutionary biology is moving towards data sets containing whole genomes for hundreds or even thousands of taxa, methods developers must focus on improving the scalability of their methods without sacrificing accuracy so that the full potential of this data may be realized.",
"While preliminary studies exist of the performance of the different methods for phylogenetic network inference [58], more thorough studies are needed to assess the accuracy as well as computational requirements of the different methods.",
"Last but not least, it is important to highlight that all the development described above excludes processes such as gene duplication and loss, and so may be susceptible to errors and artifacts which can be present in data such as hidden paralogy.",
"Furthermore, the multispecies network coalescent already has its own population-genetic assumptions, almost all of which are not necessarily realistic for analyses in practice.",
"Accounting for these is a major next step (though it is important to point out that these have not been fully explored in the context of species tree inference either), but the mathematical complexity will most likely add, extensively, to the computational complexity of the inference step.",
"Luay Nakhleh started working on phylogenetic networks in 2002 in a close collaboration with Tandy Warnow, Bernard M.E.",
"Moret, and C. Randal Linder.",
"At the time, their focus was on phylogenetic networks in terms of displaying trees.",
"This focus led to work on inference of smallest phylogenetic networks that display a given set of trees, as well as on comparing networks in terms of their displayed trees.",
"While the approaches pursued at the time were basic, they were foundational in terms of pursuing more sophisticated models and approaches by Nakhleh and his group.",
"Therefore, we would like to acknowledge the role that Bernard played in the early days of (explicit) phylogenetic networks.",
"The authors would also like to acknowledge James Mallet, Craig Moritz, David Morrison, and Mike Steel for extensive discussions and detailed comments that helped us significantly improve this chapter.",
"The authors thank Matthew Hahn and Kelley Harris for their discussion of the definition of phylogenetic invariant methods.",
"This work was partially supported by NSF grants DBI-1355998, CCF-1302179, CCF-1514177, CCF-1800723, and DMS-1547433."
]
] | 1808.08662 |
[
[
"Parametrized Measuring and Club Guessing"
],
[
"Abstract We introduce Strong Measuring, a maximal strengthening of J. T. Moore's Measuring principle, which asserts that every collection of fewer than continuum many closed bounded subsets of $\\omega_1$ is measured by some club subset of $\\omega_1$.",
"The consistency of Strong Measuring with the negation of CH is shown, solving an open problem from about parametrized measuring principles.",
"Specifically, we prove that Strong Measuring follows from MRP together with Martin's Axiom for $\\sigma$-centered forcings, as well as from BPFA.",
"We also consider strong versions of Measuring in the absence of the Axiom of Choice."
],
[
"Background",
"We review some background material and notation which is needed for understanding the paper.",
"Let $\\mathfrak {c}$ denote the cardinality of the continuum $2^\\omega $ .",
"A set $S \\subseteq [\\omega ]^\\omega $ is a splitting family if for any infinite set $x \\subseteq \\omega $ , there exists $A \\in S$ such that $A$ splits $x$ in the sense that both $x \\cap A$ and $x \\setminus A$ are infinite.",
"The splitting number $\\mathfrak {s}$ is the least cardinality of some splitting family.",
"Given functions $f, g : \\omega \\rightarrow \\omega $ , we say that $g$ dominates $f$ if for all $n < \\omega $ , $f(n) < g(n)$ .",
"We say that $g$ eventually dominates $f$ if there is some $m<\\omega $ such that $f(n)<g(n)$ for all $n>m$ .",
"A family $B \\subseteq \\omega ^\\omega $ is bounded if there exists a function $g \\in \\omega ^\\omega $ which eventually dominates every member of $B$ , and otherwise it is unbounded.",
"The bounding number $\\mathfrak {b}$ is the least cardinality of some unbounded family.",
"Both cardinal characteristics $\\mathfrak {s}$ and $\\mathfrak {b}$ are uncountable.",
"Let $\\mathbb {P}$ be a forcing poset.",
"A set $X \\subseteq \\mathbb {P}$ is centered if every finite subset of $X$ has a lower bound.",
"We say that $\\mathbb {P}$ is $\\sigma $ -centered if it is a union of countably many centered sets.",
"Martin's Axiom for $\\sigma $ -centered forcings ($\\textsf {MA}$ ($\\sigma $ -centered)) is the statement that for any $\\sigma $ -centered forcing $\\mathbb {P}$ and any collection of fewer than $\\mathfrak {c}$ many dense subsets of $\\mathbb {P}$ , there exists a filter on $\\mathbb {P}$ which meets each dense set in the collection.",
"More generally, let $\\mathfrak {m}$ ($\\sigma $ -centered) be the least cardinality of a collection of dense subsets of some $\\sigma $ -centered forcing poset for which there does not exist a filter which meets each dense set in the collection.",
"Note that $\\textsf {MA}$ ($\\sigma $ -centered) is equivalent to the statement that $\\mathfrak {m}$ ($\\sigma $ -centered) equals $\\mathfrak {c}$ .",
"The Bounded Proper Forcing Axiom (BPFA) is the statement that whenever $\\mathbb {P}$ is a proper forcing and $\\langle A_i : i < \\omega _1 \\rangle $ is a sequence of maximal antichains of $\\mathbb {P}$ each of size at most $\\omega _1$ , then there exists a filter on $\\mathbb {P}$ which meets each $A_i$ ([9]).",
"We note that BPFA implies $\\mathfrak {c} = \\omega _2$ ([12]).",
"It easily follows that BPFA implies Martin's Axiom, and in particular, implies $\\textsf {MA}$ ($\\sigma $ -centered).",
"The forcing axiom BPFA is equivalent to the statement that for any proper forcing poset $\\mathbb {P}$ and any $\\Sigma _1$ statement $\\Phi $ with a parameter from $H(\\omega _2)$ , if $\\Phi $ holds in a generic extension by $\\mathbb {P}$ , then $\\Phi $ holds in the ground model ([7]).",
"An open stationary set mapping for an uncountable set $X$ and regular cardinal $\\theta > \\omega _1$ is a function $\\Sigma $ whose domain is the collection of all countable elementary substructures $M$ of $H(\\theta )$ with $X \\in M$ , such that for all such $M$ , $\\Sigma (M)$ is an open, $M$ -stationary subset of $[X]^\\omega $ .",
"By open we mean in the Ellentuck topology on $[X]^\\omega $ , and $M$ -stationary means meeting every club subset of $[X]^\\omega $ which is a member of $M$ (see [12] for the complete details).",
"In this article, we are only concerned with these ideas in the simplest case that $X = \\omega _1$ and for each $M \\in \\mathrm {dom}(\\Sigma )$ , $\\Sigma (M) \\subseteq \\omega _1$ .",
"In this case, being open is equivalent to being open in the topology on $\\omega _1$ with basis the collection of all open intervals of ordinals, and being $M$ -stationary is equivalent to meeting every club subset of $\\omega _1$ in $M$ .",
"For an open stationary set mapping $\\Sigma $ for $X$ and $\\theta $ , a $\\Sigma $ -reflecting sequence is an $\\in $ -increasing and continuous sequence $\\langle M_i : i < \\omega _1 \\rangle $ of countable elementary substructures of $H(\\theta )$ containing $X$ as a member satisfying that for all limit ordinals $\\delta < \\omega _1$ , there exists $\\beta < \\delta $ so that for all $\\beta \\le \\xi < \\delta $ , $M_\\xi \\cap X \\in \\Sigma (M_\\delta )$ .",
"The Mapping Reflection Principle (MRP) is the statement that for any open stationary set mapping $\\Sigma $ , there exists a $\\Sigma $ -reflecting sequence.",
"We will use the fact that for any open stationary set mapping $\\Sigma $ , there exists a proper forcing which adds a $\\Sigma $ -reflecting sequence ([12]).",
"Consequently, MRP follows from PFA."
],
[
"Parametrized Measuring and Club Guessing",
"Let $X$ and $Y$ be countable subsets of $\\omega _1$ with the same supremum $\\delta $ .",
"We say that $X$ measures $Y$ if there exists $\\beta < \\delta $ such that $X \\setminus \\beta $ is either contained in, or disjoint from, $Y$ .",
"Measuring is the statement that for any sequence $\\langle c_\\alpha : \\alpha \\in \\omega _1 \\cap \\mathrm {Lim}\\rangle $ , where each $c_\\alpha $ is a closed and cofinal subset of $\\alpha $ , there exists a club $D \\subseteq \\omega _1$ such that for all limit points $\\alpha $ of $D$ , $D \\cap \\alpha $ measures $c_\\alpha $ .",
"The next two results are due to J. T. Moore ([8]).",
"Theorem 2.1 MRP implies Measuring.",
"Theorem 2.2 BPFA implies Measuring.",
"We now describe parametrized forms of measuring which were introduced in [2].",
"Let $\\vec{\\mathcal {C}} = \\langle \\mathcal {C}_\\alpha : \\alpha \\in \\omega _1 \\cap \\textrm {Lim} \\rangle $ be a sequence such that each $\\mathcal {C}_\\alpha $ is a collection of closed and cofinal subsets of $\\alpha $ .",
"A club $D \\subseteq \\omega _1$ is said to measure $\\vec{\\mathcal {C}}$ if for all $\\alpha \\in \\lim (D)$ and all $c \\in \\mathcal {C}_\\alpha $ , $D \\cap \\alpha $ measures $c$ .",
"Definition 2.3 For a cardinal $\\kappa $ , let Measuring$_{< \\kappa }$ denote the statement that whenever $\\vec{\\mathcal {C}} = \\langle \\mathcal {C}_\\alpha : \\alpha \\in \\omega _1 \\cap \\textrm {Lim} \\rangle $ is a sequence such that each $\\mathcal {C}_\\alpha $ is a collection of fewer than $\\kappa $ many closed and cofinal subsets of $\\alpha $ , then there exists a club $D \\subseteq \\omega _1$ which measures $\\vec{\\mathcal {C}}$ .",
"For a cardinal $\\lambda $ , let $\\textsf {Measuring}_\\lambda $ denote $\\textsf {Measuring}_{< \\lambda ^+}$ .",
"Observe that the principle Measuring is the same as $\\textsf {Measuring}_1$ .",
"If $\\kappa < \\lambda $ , then clearly Measuring$_{< \\lambda }$ implies Measuring$_{<\\kappa }$ .",
"It is easy to see that Measuring$_{\\mathfrak {c}}$ is false.",
"Definition 2.4 Strong Measuring is the statement that Measuring$_{<\\mathfrak {c}}$ holds.",
"Since the intersection of countably many clubs in $\\omega _1$ is club, Measuring easily implies Measuring$_\\omega $ .",
"In particular, Measuring together with CH implies Strong Measuring.",
"We will prove in Section 3 the consistency of Strong Measuring together with $\\lnot \\textsf {CH}$ .",
"We also observe at the end of that section that Measuring does not imply Measuring$_{\\omega _1}$ .",
"Proposition 2.5 ([2]) Measuring$_{\\mathfrak {s}}$ is false.",
"Fix a splitting family $S$ of cardinality $\\mathfrak {s}$ .",
"For each limit ordinal $\\alpha < \\omega _1$ , fix a function $f_\\alpha : \\omega \\rightarrow \\alpha $ which is increasing and cofinal in $\\alpha $ .",
"For each $A \\in S$ , let $c_{\\alpha ,A} = \\bigcup \\lbrace (f_\\alpha (n),f_\\alpha (n+1)] : n \\in A \\rbrace $ , which is clearly closed and cofinal in $\\alpha $ .",
"Let $\\mathcal {C}_\\alpha := \\lbrace c_{\\alpha ,A} : A \\in S \\rbrace $ .",
"Then $\\vec{\\mathcal {C}} := \\langle \\mathcal {C}_\\alpha : \\alpha \\in \\omega _1 \\cap \\mathrm {Lim}\\rangle $ is a sequence such that for each $\\alpha $ , $\\mathcal {C}_\\alpha $ is a collection of at most $\\mathfrak {s}$ many closed and cofinal subsets of $\\alpha $ .",
"Let $D \\subseteq \\omega _1$ be a club.",
"Fix $\\alpha \\in \\lim (D)$ .",
"We will show that there exists a member of $\\mathcal {C}_\\alpha $ which $D \\cap \\alpha $ does not measure.",
"Define $x := \\lbrace n < \\omega : D \\cap (f_\\alpha (n),f_{\\alpha }(n+1)] \\ne \\emptyset \\rbrace $ .",
"Since $\\alpha \\in \\lim (D)$ , $x$ is infinite.",
"As $S$ is a splitting family, we can fix $A \\in S$ which splits $x$ .",
"So both $x \\cap A$ and $x \\setminus A$ are infinite.",
"We claim that $D \\cap \\alpha $ does not measure $c_{\\alpha ,A}$ .",
"Suppose for a contradiction that for some $\\beta < \\alpha $ , $(D \\cap \\alpha ) \\setminus \\beta $ is either a subset of, or disjoint from, $c_{\\alpha ,A}$ .",
"Since $A \\cap x$ is infinite, we can fix $n \\in A \\cap x$ such that $f_\\alpha (n) > \\beta $ .",
"Then $n \\in x$ implies that $D \\cap (f_\\alpha (n),f_{\\alpha }(n+1)] \\ne \\emptyset $ , and $n \\in A$ implies that $(f_{\\alpha }(n),f_{\\alpha }(n+1)] \\subseteq c_{\\alpha ,A}$ .",
"It follows that $(D \\cap \\alpha ) \\setminus \\beta $ meets $c_{\\alpha ,A}$ .",
"By the choice of $\\beta $ , this implies that $(D \\cap \\alpha ) \\setminus \\beta $ is a subset of $c_{\\alpha ,A}$ .",
"But $x \\setminus A$ is also infinite, so we can fix $m \\in x \\setminus A$ such that $f_{\\alpha }(m) > \\beta $ .",
"Then $m \\in x$ implies that $D \\cap (f_\\alpha (m),f_\\alpha (m+1)] \\ne \\emptyset $ , and $m \\notin A$ implies that $(f_{\\alpha }(m),f_{\\alpha }(m+1)]$ is disjoint from $c_{\\alpha ,A}$ .",
"Thus, there is a member of $(D \\cap \\alpha ) \\setminus \\beta $ which is not in $c_{\\alpha ,A}$ , which is a contradiction.",
"We will prove later in this section that Measuring$_{\\mathfrak {b}}$ is also false.",
"We now turn to parametrized club guessing.",
"We recall some standard definitions.",
"Consider a sequence $\\vec{L} = \\langle L_\\alpha : \\alpha \\in \\omega _1 \\cap \\mathrm {Lim}\\rangle $ , where each $L_\\alpha $ is a cofinal subset of $\\alpha $ with order type $\\omega $ (that is, a ladder system).",
"We say that $\\vec{L}$ is a club guessing sequence, weak club guessing sequence, or very weak club guessing sequence, respectively, if for every club $D \\subseteq \\omega _1$ , there exists a limit ordinal $\\alpha < \\omega _1$ such that: $L_\\alpha \\subseteq D$ , $L_\\alpha \\setminus D$ is finite, or $L_\\alpha \\cap D$ is infinite, respectively.",
"We say that Club Guessing, Weak Club Guessing, or Very Weak Club Guessing holds, respectively, if there exists a club guessing sequence, a weak club guessing sequence, or a very weak club guessing sequence, respectively.",
"It is well known that Measuring implies the failure of Very Weak Club Guessing (see Proposition 2.8 below).",
"Definition 2.6 Let $\\vec{\\mathcal {L}} = \\langle \\mathcal {L}_\\alpha : \\alpha \\in \\omega _1 \\cap \\mathrm {Lim}\\rangle $ be a sequence where each $\\mathcal {L}_\\alpha $ is a non–empty collection of cofinal subsets of $\\alpha $ with order type $\\omega $ .",
"The sequence $\\vec{\\mathcal {L}}$ is said to be a club guessing sequence, weak club guessing sequence, or very weak club guessing sequence, respectively, if for every club $D \\subseteq \\omega _1$ , there exists a limit ordinal $\\alpha < \\omega _1$ and some $L \\in \\mathcal {L}_\\alpha $ such that: $L \\subseteq D$ , $L \\setminus D$ is finite, or $L \\cap D$ is infinite, respectively.",
"Definition 2.7 For a cardinal $\\kappa $ , let CG$_{< \\kappa }$ , WCG$_{<\\kappa }$ , and VWCG$_{<\\kappa }$ , respectively, be the statements that there exists a club guessing sequence, weak club guessing sequence, or very weak club guessing sequence $\\langle \\mathcal {L}_\\alpha : \\alpha \\in \\omega _1 \\cap \\mathrm {Lim}\\rangle $ , respectively, such that for each $\\alpha $ , $|\\mathcal {L}_\\alpha | < \\kappa $ .",
"Let CG$_{\\kappa }$ , WCG$_{\\kappa }$ , and VWCG$_{\\kappa }$ denote the statements CG$_{< \\kappa ^+}$ , WCG$_{< \\kappa ^+}$ , and VWCG$_{< \\kappa ^+}$ , respectively.",
"Clearly, if $\\kappa < \\lambda $ , then CG$_{< \\kappa }$ implies CG$_{<\\lambda }$ , and similarly with WCG and VWCG.",
"Observe that Club Guessing, Weak Club Guessing, and Very Weak Club Guessing are equivalent to CG$_{1}$ , WCG$_{1}$ , and VWCG$_{1}$ , respectively.",
"Obviously, CG$_{\\mathfrak {c}}$ is true.",
"The weakest forms of club guessing principles which are not provable in ZFC are when the index is $< \\mathfrak {c}$ .",
"Proposition 2.8 For any cardinal $\\kappa \\ge 2$ , Measuring$_{< \\kappa }$ implies the failure of VWCG$_{< \\kappa }$ .",
"Suppose for a contradiction that Measuring$_{< \\kappa }$ and VWCG$_{<\\kappa }$ both hold.",
"Fix a very weak club guessing sequence $\\vec{\\mathcal {L}} = \\langle \\mathcal {L}_\\alpha : \\alpha \\in \\omega _1 \\cap \\mathrm {Lim}\\rangle $ such that each $\\mathcal {L}_\\alpha $ has cardinality less than $\\kappa $ .",
"Observe that for each $\\alpha $ , every member of $\\mathcal {L}_\\alpha $ is vacuously a closed subset of $\\alpha $ since it has order type $\\omega $ .",
"By Measuring$_{< \\kappa }$ , there exists a club $D \\subseteq \\omega _1$ which measures $\\vec{\\mathcal {L}}$ .",
"Let $E$ be the club set of indecomposable limit ordinals $\\alpha >\\omega $ in $\\lim (D)$ such that $\\mathrm {ot}(D \\cap \\alpha ) = \\alpha $ .",
"Since $\\vec{\\mathcal {L}}$ is a very weak club guessing sequence, there exists a limit ordinal $\\alpha $ and $L \\in \\mathcal {L}_\\alpha $ such that $L \\cap E$ is infinite.",
"In particular, $\\alpha $ is a limit point of $E$ , and hence of $D$ .",
"Since $D$ measures $\\vec{\\mathcal {L}}$ and $L \\in \\mathcal {L}_\\alpha $ , $D \\cap \\alpha $ measures $L$ .",
"So we can fix $\\beta < \\alpha $ such that $(D \\cap \\alpha ) \\setminus \\beta $ is either a subset of, or disjoint from, $L$ .",
"Now $L \\cap E$ , and hence $L \\cap D$ , is infinite.",
"As $L$ has order type $\\omega $ , this implies that $L \\cap D$ is cofinal in $\\alpha $ .",
"By the choice of $\\beta $ , $(D \\cap \\alpha ) \\setminus \\beta $ must be a subset of $L$ .",
"But since $\\alpha \\in E$ , $\\mathrm {ot}(D \\cap \\alpha ) = \\alpha $ and $\\alpha $ is indecomposable, which implies that $\\mathrm {ot}((D \\cap \\alpha ) \\setminus \\beta ) = \\alpha $ .",
"As $\\alpha > \\omega $ , this is impossible since $(D \\cap \\alpha ) \\setminus \\beta $ is a subset of $L$ and $L$ has order type $\\omega $ .",
"In particular, since Strong Measuring is consistent, so is the failure of VWCG$_{< \\mathfrak {c}}$ .",
"(The consistency of $\\lnot \\textsf {VWCG}_{< \\mathfrak {c}}$ together with $\\mathfrak {c}$ arbitrarily large was previously shown in [4].)",
"Proposition 2.9 (Hrušák [5]) VWCG$_{\\mathfrak {b}}$ is true.",
"Fix an unbounded family $\\lbrace r_\\alpha : \\alpha < \\mathfrak {b} \\rbrace $ in $\\omega ^\\omega $ .",
"For each limit ordinal $\\delta < \\omega _1$ , fix a cofinal subset $C_\\delta $ of $\\delta $ with order type $\\omega $ and a bijection $h_\\delta : \\omega \\rightarrow \\delta $ .",
"Let $C_\\delta (n)$ denote the $n$ -th member of $C_\\delta $ for all $n < \\omega $ .",
"For all limit ordinals $\\delta < \\omega _1$ and $\\alpha < \\mathfrak {b}$ , define $A_\\delta ^\\alpha := C_\\delta \\cup \\bigcup \\lbrace h_\\delta [r_\\alpha (n)] \\setminus C_\\delta (n) : n < \\omega \\rbrace .$ It is easy to check that for all $\\delta $ and $\\alpha $ , $A_\\delta ^\\alpha $ has order type $\\omega $ and $\\sup (A_\\delta ^\\alpha ) = \\delta $ .",
"Given a club $C \\subseteq \\omega _1$ , let $\\delta $ be a limit point of $C$ and let $g_{C, \\delta } : \\omega \\rightarrow \\omega $ be the function given by $g_{C, \\delta }(n) =\\min \\lbrace m < \\omega :h_\\delta (m) \\in C \\setminus C_\\delta (n) \\rbrace .$ Now let $\\alpha < \\mathfrak {b}$ be such that $r_\\alpha (n) > g_{C, \\delta }(n)$ for infinitely many $n$ .",
"It then follows that $|A_\\delta ^\\alpha \\cap C| = \\omega $ .",
"By Propositions 2.8 and 2.9, the following is immediate.",
"Corollary 2.10 Measuring$_{\\mathfrak {b}}$ is false.",
"An obvious question is whether the parametrized versions of club guessing are actually the same as the usual ones.",
"We conclude this section by showing that they are not.",
"Recall that a forcing poset $\\mathbb {P}$ is $\\omega ^\\omega $ -bounding if every function in $\\omega ^\\omega \\cap V^\\mathbb {P}$ is dominated by a function in $\\omega ^\\omega \\cap V$ .",
"Lemma 2.11 (Hrušák) Assume that VWCG fails.",
"Let $\\mathbb {P}$ be any $\\omega _1$ -c.c., $\\omega ^\\omega $ -bounding forcing.",
"Then $\\mathbb {P}$ forces that VWCG fails.",
"Since $\\mathbb {P}$ is $\\omega _1$ -c.c.",
"and $\\omega ^\\omega $ -bounding, a standard argument shows that whenever $p \\in \\mathbb {P}$ and $p$ forces that $\\dot{b} \\in \\omega ^\\omega $ , then there exists a function $b^* \\in \\omega ^\\omega $ such that $p$ forces that $b^*$ dominates $\\dot{b}$ .",
"Let us show that whenever $p \\in \\mathbb {P}$ , $\\delta < \\omega _1$ , and $p$ forces that $\\dot{X}$ is a cofinal subset of $\\delta $ of order type $\\omega $ , then there exists a set $Y$ with order type $\\omega $ such that $p$ forces that $\\dot{X} \\subseteq Y$ .",
"To see this, fix a bijection $f : \\omega \\rightarrow \\delta $ and a strictly increasing sequence $\\langle \\alpha _n : n < \\omega \\rangle $ cofinal in $\\alpha $ with $\\alpha _0 = 0$ .",
"We claim that there exists a $\\mathbb {P}$ -name $\\dot{b}$ for a function from $\\omega $ to $\\omega $ such that $p$ forces that for all $n < \\omega $ , $\\dot{b}(n)$ is the least $m < \\omega $ such that $\\dot{X} \\cap [\\alpha _n,\\alpha _{n+1}) \\subseteq f[m]$ .",
"This is true since $p$ forces that $\\dot{X}$ has order type $\\omega $ and hence that $\\dot{X} \\cap [\\alpha _n,\\alpha _{n+1})$ is finite for all $n < \\omega $ .",
"Fix a function $b^* : \\omega \\rightarrow \\omega $ such that $p$ forces that $b^*$ dominates $\\dot{b}$ .",
"Now let $Y := \\bigcup \\lbrace f[b^*(n)] \\cap [\\alpha _n,\\alpha _{n+1}) : n < \\omega \\rbrace .$ It is easy to check that $Y$ has order type $\\omega $ and $p$ forces that $\\dot{X} \\subseteq Y$ .",
"Now we are ready to prove the proposition.",
"So suppose that $p \\in \\mathbb {P}$ forces that $\\langle \\dot{X}_\\alpha : \\alpha \\in \\omega _1 \\cap \\mathrm {Lim}\\rangle $ is a very weak club guessing sequence.",
"By the previous paragraph, for each limit ordinal $\\alpha < \\omega _1$ we can fix a cofinal subset $Y_\\alpha $ of $\\alpha $ with order type $\\omega $ such that $p$ forces that $\\dot{X}_\\alpha \\subseteq Y_\\alpha $ .",
"We claim that $\\langle Y_\\alpha : \\alpha \\in \\omega _1 \\cap \\mathrm {Lim}\\rangle $ is a very weak club guessing sequence in the ground model, which completes the proof.",
"So consider a club $C \\subseteq \\omega _1$ .",
"Then $C$ is still a club in $V^\\mathbb {P}$ .",
"Fix $q \\le p$ and a limit ordinal $\\alpha < \\omega _1$ such that $q$ forces that $\\dot{X}_\\alpha \\cap C$ is infinite.",
"Then clearly $q$ forces that $Y_\\alpha \\cap C$ is infinite, so in fact, $Y_\\alpha \\cap C$ is infinite.",
"Proposition 2.12 It is consistent that $\\lnot \\textsf {VWCG}$ and $\\textsf {CG}_{\\omega _1}$ both hold.",
"Let $V$ be a model in which CH holds and VWCG fails.",
"Such a model was shown to exist by Shelah [13].",
"Let $\\mathbb {P}$ be an $\\omega _1$ -c.c., $\\omega ^\\omega $ -bounding forcing poset which adds at least $\\omega _2$ many reals; for example, random real forcing with product measure is such a forcing.",
"We claim that in $V^\\mathbb {P}$ , CG$_{\\omega _1}$ holds but VWCG fails.",
"By Lemma 2.11, VWCG is false in $V^\\mathbb {P}$ .",
"In $V$ , define $\\vec{\\mathcal {L}} = \\langle \\mathcal {L}_\\alpha : \\alpha \\in \\omega _1 \\cap \\mathrm {Lim}\\rangle $ by letting $\\mathcal {L}_\\alpha $ be the collection of all cofinal subsets of $\\alpha $ with order type $\\omega $ .",
"Since CH holds, the cardinality of each $\\mathcal {L}_\\alpha $ is $\\omega _1$ .",
"If $C$ is a club subset of $\\omega _1$ in $V^\\mathbb {P}$ , then since $\\mathbb {P}$ is $\\omega _1$ -c.c., there is a club $D \\subseteq \\omega _1$ in $V$ such that $D \\subseteq C$ .",
"In $V$ , fix $d \\subseteq D$ with order type $\\omega $ , and let $\\alpha := \\sup (d)$ .",
"Then $d \\in \\mathcal {L}_\\alpha $ and $d \\subseteq C$ .",
"Thus, $\\vec{\\mathcal {L}}$ witnesses that CG$_{\\omega _1}$ holds in $V^\\mathbb {P}$ ."
],
[
"The Consistency of Strong Measuring and $\\lnot \\textsf {CH}$",
"As we previously mentioned, Measuring is equivalent to Measuring$_\\omega $ , and therefore under CH, Measuring is equivalent to Strong Measuring.",
"In this section we establish the consistency of Strong Measuring with the negation of CH.",
"More precisely, we will prove that MRP together with MA($\\sigma $ -centered) implies Strong Measuring, and BPFA implies Strong Measuring.",
"Recall that both MRP and BPFA imply that $\\mathfrak {c} = \\omega _2$ ([12]).",
"A set $M$ is suitable if for some regular cardinal $\\theta > \\omega _1$ , $M$ is a countable elementary substructure of $H(\\theta )$ .",
"We will follow the conventions introduced in Section 1 that the properties “open” and “$M$ -stationary” refer to open and $M$ -stationary subsets of $\\omega _1$ (where $\\omega _1$ is considered as a subspace of $[\\omega _1]^\\omega $ ).",
"Proposition 3.1 Assume that $M$ is suitable.",
"Let $\\delta := M \\cap \\omega _1$ .",
"Suppose that $\\mathcal {Y}$ is a collection of open subsets of $\\delta $ such that for any finite set $a \\subseteq \\mathcal {Y}$ , $\\bigcap a$ is $M$ -stationary.",
"Then there exists a $\\sigma $ -centered forcing $\\mathbb {P}$ and a collection $\\mathcal {D}$ of dense subsets of $\\mathbb {P}$ of size at most $|\\mathcal {Y}| + \\omega $ such that whenever $G$ is a filter on $\\mathbb {P}$ in some outer model $W$ of $V$ with $\\omega _1^V = \\omega _1^W$ which meets each member of $\\mathcal {D}$ , then there exists a set $z \\subseteq \\delta $ in $W$ which is open, $M$ -stationary, and satisfies that for all $X \\in \\mathcal {Y}$ , $z \\setminus X$ is bounded in $\\delta $ .",
"Define a forcing poset $\\mathbb {P}$ to consist of conditions which are pairs $(x,a)$ , where $x$ is an open and bounded subset of $\\delta $ in $M$ and $a$ is a finite subset of $\\mathcal {Y}$ .",
"Let $(y,b) \\le (x,a)$ if $y$ is an end-extension of $x$ , $a \\subseteq b$ , and $y \\setminus x \\subseteq \\bigcap a$ .",
"Since $M$ is countable, there are only countably many possibilities for the first component of a condition.",
"If $(x,a_0), \\ldots , (x,a_n)$ are finitely many conditions with the same first component, then easily $(x,a_0 \\cup \\ldots \\cup a_n)$ is a condition in $\\mathbb {P}$ which is below each of the conditions $(x,a_0), \\ldots , (x,a_n)$ .",
"It follows that $\\mathbb {P}$ is $\\sigma $ -centered.",
"For each $X \\in \\mathcal {Y}$ , let $D_X$ denote the set of conditions $(x,a)$ such that $X \\in a$ .",
"Observe that $D_X$ is dense.",
"For every club $C$ of $\\omega _1$ which is a member of $M$ , let $E_C$ denote the set of conditions $(x,a)$ such that $x \\cap C$ is non–empty.",
"We claim that $E_C$ is dense.",
"Let $(x,a)$ be a condition.",
"Since $\\bigcap a$ is $M$ -stationary and $\\lim (C) \\setminus (\\sup (x)+1)$ is a club subset of $\\omega _1$ in $M$ , we can find a limit ordinal $\\alpha $ in $C \\cap (\\bigcap a)$ which is in the interval $(\\sup (x),\\delta )$ .",
"Since $\\alpha \\in \\bigcap a$ and $\\bigcap a$ is open, we can find $\\beta < \\gamma < \\delta $ such that $\\alpha \\in (\\beta ,\\gamma ) \\subseteq \\bigcap a$ .",
"As $\\sup (x) + 1 < \\alpha $ , without loss of generality $\\sup (x) < \\beta $ .",
"By elementarity, the interval $b := (\\beta ,\\gamma )$ is in $M$ .",
"It follows that $(x \\cup b,a)$ is a condition, $x \\cup b$ end-extends $x$ , and $(x \\cup b) \\setminus x = b \\subseteq \\bigcap a$ .",
"Thus, $(x \\cup b,a) \\le (x,a)$ , and since $\\alpha \\in C$ , $(x \\cup b,a) \\in E_C$ .",
"Let $\\mathcal {D}$ denote the collection of all dense sets of the form $D_X$ where $X \\in \\mathcal {Y}$ , or $E_C$ where $C$ is a club subset of $\\omega _1$ belonging to $M$ .",
"Then $|\\mathcal {D}| \\le |\\mathcal {Y}| + \\omega $ .",
"Let $G$ be a filter on $\\mathbb {P}$ in some outer model $W$ with $\\omega _1^V = \\omega _1^W$ which meets each dense set in $\\mathcal {D}$ .",
"Define $z := \\bigcup \\lbrace x : \\exists a \\ (x,a) \\in G \\rbrace $ .",
"Note that since $z$ is a union of open sets, it is open (using the fact that being open is absolute between $V$ and $W$ ).",
"For each club $C \\subseteq \\omega _1$ which lies in $M$ , there exists a condition $(x,a)$ which belongs to $G \\cap E_C$ , and thus $x \\cap C \\ne \\emptyset $ .",
"Therefore, $z \\cap C \\ne \\emptyset $ .",
"Hence, $z$ is $M$ -stationary.",
"It remains to show that for all $X \\in \\mathcal {Y}$ , $z \\setminus X$ is bounded in $\\delta $ .",
"Consider $X \\in \\mathcal {Y}$ .",
"Then we can fix $(x,a) \\in G \\cap D_X$ , which means that $X \\in a$ .",
"Now the definition of the ordering on $\\mathbb {P}$ together with the fact that $G$ is a filter easily implies that $z \\setminus x \\subseteq X$ .",
"Therefore, $z \\setminus X \\subseteq x$ , and hence $z \\setminus X$ is bounded in $\\delta $ .",
"Corollary 3.2 Assume that $M$ is suitable.",
"Let $\\delta := M \\cap \\omega _1$ .",
"Suppose that $\\mathcal {Y}$ is a collection of less than $\\mathfrak {m}$ ($\\sigma $ -centered) many open subsets of $\\delta $ such that for any finite set $a \\subseteq \\mathcal {Y}$ , $\\bigcap a$ is $M$ -stationary.",
"Then there exists a set $z \\subseteq \\delta $ which is open, $M$ -stationary, and satisfies that for all $X \\in \\mathcal {Y}$ , $z \\setminus X$ is bounded in $\\delta $ .",
"Fix a $\\sigma $ -centered forcing $\\mathbb {P}$ and a collection $\\mathcal {D}$ of dense subsets of $\\mathbb {P}$ of size at most $|\\mathcal {Y}| + \\omega $ as described in Proposition 3.1.",
"Since $\\mathfrak {m}$ ($\\sigma $ -centered) is uncountable, $| \\mathcal {D} | < \\mathfrak {m}(\\textrm {$$-centered})$ .",
"Hence, there exists a filter $G$ on $\\mathbb {P}$ which meets each dense set in $\\mathcal {D}$ .",
"By Proposition 3.1, there exists a set $z \\subseteq \\delta $ which is open, $M$ -stationary, and satisfies that for all $X \\in \\mathcal {Y}$ , $z \\setminus X$ is bounded in $\\delta $ .",
"Proposition 3.3 Let $\\vec{\\mathcal {C}} = \\langle \\mathcal {C}_\\alpha : \\alpha \\in \\omega _1 \\cap \\textrm {Lim} \\rangle $ be a sequence such that each $\\mathcal {C}_\\alpha $ is a collection of less than $\\mathfrak {m}$ ($\\sigma $ -centered) many closed and cofinal subsets of $\\alpha $ .",
"Then there exists an open stationary set mapping $\\Sigma $ such that, if $W$ is any outer model with the same $\\omega _1$ in which there exists a $\\Sigma $ -reflecting sequence, then there exists in $W$ a club subset of $\\omega _1$ which measures $\\vec{\\mathcal {C}}$ .",
"For each limit ordinal $\\alpha < \\omega _1$ , let $\\mathcal {D}_\\alpha := \\lbrace \\alpha \\setminus c : c \\in \\mathcal {C}_\\alpha \\rbrace $ .",
"Observe that each $\\mathcal {D}_\\alpha $ is a collection of fewer than $\\mathfrak {m}$ ($\\sigma $ -centered) many open subsets of $\\alpha $ .",
"We will define $\\Sigma $ to have domain the collection of all countable elementary substructures $M$ of $H(\\omega _2)$ .",
"Consider such an $M$ and we define $\\Sigma (M)$ .",
"Note that $M$ is suitable.",
"Let $\\delta := M \\cap \\omega _1$ .",
"We consider two cases.",
"In the first case, there does not exist a member of $\\mathcal {D}_\\delta $ which is $M$ -stationary.",
"Define $\\Sigma (M) = \\delta $ , which is clearly open and $M$ -stationary.",
"In the second case, there exists some member of $\\mathcal {D}_\\delta $ which is $M$ -stationary.",
"A straightforward application of Zorn's lemma implies that there exists a non–empty set ${\\mathcal {Y}}_M \\subseteq \\mathcal {D}_\\delta $ such that for any $a \\in [{\\mathcal {Y}}_M]^{<\\omega }$ , $\\bigcap a$ is $M$ -stationary, and moreover, $\\mathcal {Y}_M$ is a maximal subset of $\\mathcal {D}_\\delta $ with this property.",
"Since $\\mathcal {Y}_M \\subseteq \\mathcal {D}_\\delta $ , $|\\mathcal {Y}_M| < \\mathfrak {m}(\\textrm {$$-centered})$ .",
"So the collection $\\mathcal {Y}_M$ satisfies the assumptions of Corollary 3.2.",
"It follows that there exists a set $z_M \\subseteq \\delta $ which is open, $M$ -stationary, and satisfies that for all $X \\in \\mathcal {Y}_M$ , $z_M \\setminus X$ is bounded in $\\delta $ .",
"Now define $\\Sigma (M) := z_M$ .",
"This completes the definition of $\\Sigma $ .",
"Consider an outer model $W$ of $V$ with the same $\\omega _1$ , and assume that in $W$ there exists a $\\Sigma $ -reflecting sequence $\\langle M_\\delta : \\delta < \\omega _1 \\rangle $ .",
"Let $\\alpha _\\delta := M_\\delta \\cap \\omega _1$ for all $\\delta < \\omega _1$ .",
"Let $D$ be the club set of $\\delta < \\omega _1$ such that $\\alpha _\\delta = \\delta $ .",
"We claim that $D$ measures $\\vec{\\mathcal {C}}$ .",
"Consider $\\delta \\in \\lim (D)$ .",
"Then $\\delta = \\alpha _\\delta = M_\\delta \\cap \\omega _1$ .",
"Let $M := M_\\delta $ .",
"We first claim that if $c \\in \\mathcal {C}_\\delta $ and $\\delta \\setminus c$ is not $M$ -stationary, then for some $\\beta < \\delta $ , $(D \\cap \\delta ) \\setminus \\beta \\subseteq c$ .",
"Fix a club subset $E$ of $\\omega _1$ in $M$ which is disjoint from $\\delta \\setminus c$ .",
"By the continuity of the $\\Sigma $ -reflecting sequence, there exists $\\beta < \\delta $ such that $E \\in M_\\beta $ .",
"We claim that $(D \\cap \\delta ) \\setminus \\beta \\subseteq c$ .",
"Let $\\xi \\in (D \\cap \\delta ) \\setminus \\beta $ .",
"Then $E \\in M_\\xi $ , and hence by elementarity, $\\xi = M_\\xi \\cap \\omega _1 \\in E$ .",
"Since $E$ is disjoint from $\\delta \\setminus c$ , $\\xi \\in c$ .",
"We split the argument according to the two cases in the definition of $\\Sigma (M)$ .",
"In the first case, there does not exist a member of $\\mathcal {D}_\\delta $ which is $M$ -stationary.",
"Consider $c \\in \\mathcal {C}_\\delta $ .",
"Then $\\delta \\setminus c$ is not $M$ -stationary.",
"By the previous paragraph, there exists $\\beta < \\delta $ such that $(D \\cap \\delta ) \\setminus \\beta \\subseteq c$ .",
"In the second case, there exists a member of $\\mathcal {D}_\\delta $ which is $M$ -stationary.",
"Consider $c \\in \\mathcal {C}_\\delta $ .",
"Then $X := \\delta \\setminus c \\in \\mathcal {D}_\\delta $ .",
"We consider two possibilities.",
"First, assume that $X$ is in $\\mathcal {Y}_M$ .",
"By the choice of $\\mathcal {Y}_M$ and $z_M$ , we know that $z_M \\setminus X$ is bounded in $\\delta $ .",
"So fix $\\beta _0 < \\delta $ so that $z_M \\setminus \\beta _0 \\subseteq X$ .",
"By the definition of being a $\\Sigma $ -reflecting sequence, there exists $\\beta _1 < \\delta $ so that for all $\\beta _1 \\le \\xi < \\delta $ , $M_\\xi \\cap \\omega _1 \\in \\Sigma (M) = z_M$ .",
"Let $\\beta := \\max \\lbrace \\beta _1, \\beta _2 \\rbrace $ .",
"Consider $\\xi \\in (D \\cap \\delta ) \\setminus \\beta $ .",
"Then $\\xi \\ge \\beta _1$ implies that $\\xi = M_\\xi \\cap \\omega _1 \\in z_M$ .",
"So $\\xi \\in z_M \\setminus \\beta _0 \\subseteq X = \\delta \\setminus c$ .",
"Secondly, assume that $X$ is not in $\\mathcal {Y}_M$ .",
"By the maximality of $\\mathcal {Y}_M$ , there exists a set $a \\in [\\mathcal {Y}_M]^{<\\omega }$ such that $X \\cap \\bigcap a$ is not $M$ -stationary.",
"Fix a club $E$ in $M$ which is disjoint from $X \\cap \\bigcap a$ .",
"By the continuity of the $\\Sigma $ -reflecting sequence, there exists $\\beta < \\delta $ such that $E \\in M_\\beta $ .",
"Consider $\\xi \\in (D \\cap \\delta ) \\setminus \\beta $ .",
"Then $E \\in M_\\xi $ , which implies that $\\xi = M_\\xi \\cap \\omega _1 \\in E$ .",
"Thus, $\\xi $ is not in $X \\cap \\bigcap a$ .",
"On the other hand, letting $a = \\lbrace X_0, \\ldots , X_n \\rbrace $ , for each $i \\le n$ the previous paragraph implies that there exists $\\beta _i < \\delta $ such that $(D \\cap \\delta ) \\setminus \\beta _i \\subseteq X_i$ .",
"Let $\\beta ^*$ be an ordinal in $\\delta $ which is larger than $\\beta $ and $\\beta _i$ for all $i \\le n$ .",
"Consider $\\xi \\in (D \\cap \\delta ) \\setminus \\beta ^*$ .",
"Then by the choice of $\\beta $ , $\\xi \\notin X \\cap \\bigcap a$ .",
"By the choice of the $\\beta _i$ 's, $\\xi \\in \\bigcap a$ .",
"Therefore, $\\xi \\notin X = \\delta \\setminus c$ , which means that $\\xi \\in c$ .",
"Thus, $(D \\cap \\delta ) \\setminus \\beta ^* \\subseteq c$ .",
"Corollary 3.4 Assume MRP and MA($\\sigma $ -centered).",
"Then Strong Measuring holds.",
"Let $\\vec{\\mathcal {C}} = \\langle \\mathcal {C}_\\alpha :\\alpha \\in \\omega _1 \\cap \\textrm {Lim} \\rangle $ be a sequence such that each $\\mathcal {C}_\\alpha $ is a collection of fewer than $\\mathfrak {c}$ many closed and cofinal subsets of $\\alpha $ .",
"We claim that there exists a club subset of $\\omega _1$ which measures $\\vec{\\mathcal {C}}$ .",
"By MA($\\sigma $ -centered), $\\mathfrak {m}$ ($\\sigma $ -centered) equals $\\mathfrak {c}$ .",
"So each $\\mathcal {C}_\\alpha $ has size less than $\\mathfrak {m}$ ($\\sigma $ -centered).",
"By Proposition 3.3, there exists an open stationary set mapping $\\Sigma $ such that, if $W$ is any outer model with the same $\\omega _1$ in which there exists a $\\Sigma $ -reflecting sequence, then there exists in $W$ a club subset of $\\omega _1$ which measures $\\vec{\\mathcal {C}}$ .",
"Applying MRP, there exists a $\\Sigma $ -reflecting sequence in $V$ .",
"Thus, in $V$ there exists a club subset of $\\omega _1$ which measures $\\vec{\\mathcal {C}}$ .",
"Corollary 3.5 Assume BPFA.",
"Then Strong Measuring holds.",
"Let $\\vec{\\mathcal {C}} = \\langle \\mathcal {C}_\\alpha :\\alpha \\in \\omega _1 \\cap \\textrm {Lim} \\rangle $ be a sequence such that each $\\mathcal {C}_\\alpha $ is a collection of fewer than $\\mathfrak {c} = \\omega _2$ many closed and cofinal subsets of $\\alpha $ .",
"We claim that there exists a club subset of $\\omega _1$ which measures $\\vec{\\mathcal {C}}$ .",
"Since $\\mathfrak {c} = \\omega _2$ , $\\vec{\\mathcal {C}}$ is a member of $H(\\omega _2)$ .",
"Thus, the existence of a club subset of $\\omega _2$ which measures $\\vec{\\mathcal {C}}$ is expressible as a $\\Sigma _1$ statement involving a parameter in $H(\\omega _2)$ .",
"By BPFA, it suffices to show that there exists a proper forcing which forces that such a club exists.",
"Now BPFA implies Martin's Axiom, and in particular, that $\\mathfrak {m}$ ($\\sigma $ -centered) is equal to $\\mathfrak {c}$ .",
"So each $\\mathcal {C}_\\alpha $ has size less than $\\mathfrak {m}$ ($\\sigma $ -centered).",
"By Proposition 3.3, there exists an open stationary set mapping $\\Sigma $ such that, if $W$ is any outer model with the same $\\omega _1$ in which there exists a $\\Sigma $ -reflecting sequence, then there exists in $W$ a club subset of $\\omega _1$ which measures $\\vec{\\mathcal {C}}$ .",
"By [12], there exists a proper forcing $\\mathbb {P}$ which adds a $\\Sigma $ -reflecting sequence, so in $V^\\mathbb {P}$ there is a club subset of $\\omega _1$ which measures $\\vec{\\mathcal {C}}$ .",
"We now sketch a proof that MRP alone does not imply Strong Measuring.",
"In particular, Measuring does not imply Strong Measuring.",
"Start with a model of CH in which there exists a supercompact cardinal $\\kappa $ .",
"Construct a forcing iteration $\\mathbb {P}$ in the standard way to obtain a model of MRP.",
"To do this, fix a Laver function $f : \\kappa \\rightarrow V_\\kappa $ .",
"Then define a countable support forcing iteration $\\langle \\mathbb {P}_\\alpha , \\dot{\\mathbb {Q}}_\\beta : \\alpha \\le \\kappa , \\beta < \\kappa \\rangle $ as follows.",
"Given $\\mathbb {P}_\\alpha $ , consider $f(\\alpha )$ .",
"If $f(\\alpha )$ happens to be a $\\mathbb {P}_\\alpha $ -name for some open stationary set mapping, then let $\\dot{\\mathbb {Q}}_\\alpha $ be a $\\mathbb {P}_\\alpha $ -name for a proper forcing which adds an $f(\\alpha )$ -reflecting sequence.",
"Otherwise let $\\dot{\\mathbb {Q}}_\\alpha $ be a $\\mathbb {P}_\\alpha $ -name for $Col(\\omega _1,\\omega _2)$ .",
"Now define $\\mathbb {P}:= \\mathbb {P}_\\kappa $ .",
"Arguments similar to those in the standard construction of a model of PFA can be used to show that $\\mathbb {P}$ forces MRP.",
"The forcing for adding a $\\Sigma $ -reflecting sequence for a given open stationary set mapping does not add reals ([12]).",
"In particular, it is vacuously $\\omega ^\\omega $ -bounding.",
"The property of being proper and $\\omega ^\\omega $ -bounding is preserved under countable support forcing iterations ([1]), so $\\mathbb {P}$ is also $\\omega ^\\omega $ -bounding.",
"In particular, $V \\cap \\omega ^\\omega $ is an unbounded family in $V^\\mathbb {P}$ , and it has size $\\omega _1$ since CH holds in $V$ .",
"It follows that the bounding number $\\mathfrak {b}$ is equal to $\\omega _1$ .",
"But by Corollary 2.9, Measuring$_{\\mathfrak {b}}$ is false.",
"So $\\mathbb {P}$ forces that $\\textsf {Measuring}_{\\omega _1}$ is false.",
"As $\\mathfrak {c} = \\omega _2$ in $V^\\mathbb {P}$ , Strong Measuring fails in $V^\\mathbb {P}$ .",
"We also note that Strong Measuring plus $\\mathfrak {c} = \\omega _2$ is consistent with the existence of an $\\omega _1$ -Suslin tree.",
"Namely, both the forcing for adding a $\\Sigma $ -reflecting sequence for a given open stationary set mapping $\\Sigma $ , as well as any $\\sigma $ -centered forcing, preserve Suslin trees ([11]).",
"And the property of being proper and preserving a given Suslin tree is preserved under countable support forcings iterations ([10]).",
"So starting with a model in which there exists an $\\omega _1$ -Suslin tree $S$ and a supercompact cardinal $\\kappa $ , we can iterate forcing similar to the argument in the preceding paragraphs to produce a model of MA($\\sigma $ -centered) plus MRP in which $S$ is an $\\omega _1$ -Suslin tree.",
"By Corollary 3.4, Strong Measuring holds in that model."
],
[
"Measuring Without the Axiom of Choice",
"Another natural way to strengthen $\\operatorname{\\textsf {Measuring}}$ is to allow, in the sequence to be measured, not just closed sets, but also sets of higher Borel complexity.",
"This line of strengthenings of $\\operatorname{\\textsf {Measuring}}$ was also considered in [2].",
"For completeness, we are including here the corresponding observations.",
"The version of $\\operatorname{\\textsf {Measuring}}$ where one considers sequences $\\vec{X}=\\langle X_\\alpha \\,:\\,\\alpha \\in \\omega _1\\cap \\mathrm {Lim}\\rangle $ , with each $X_\\alpha $ an open subset of $\\alpha $ in the order topology, is of course equivalent to $\\operatorname{\\textsf {Measuring}}$ .",
"A natural next step would therefore be to consider sequences in which each $X_\\alpha $ is a countable union of closed sets.",
"This is obviously the same as allowing each $X_\\alpha $ to be an arbitrary subset of $\\alpha $ .",
"Let us call the corresponding statement $\\operatorname{\\textsf {Measuring}}^\\ast $ : Definition 4.1 $\\operatorname{\\textsf {Measuring}}^\\ast $ holds if and only if for every sequence $\\vec{X}=\\langle X_\\alpha \\,:\\,\\alpha \\in \\omega _1\\cap \\mathrm {Lim}\\rangle $ , if $X_\\alpha \\subseteq \\alpha $ for each $\\alpha $ , then there is some club $D\\subseteq \\omega _1$ such that for every limit point $\\delta \\in D$ of $D$ , $D\\cap \\delta $ measures $X_\\delta $ .",
"It is easy to see that $\\operatorname{\\textsf {Measuring}}^\\ast $ is false in $\\textsf {ZFC}$ .",
"In fact, given a stationary and co-stationary $S\\subseteq \\omega _1$ , there is no club of $\\omega _1$ measuring $\\vec{X}=\\langle S\\cap \\alpha \\,:\\,\\alpha \\in \\omega _1\\cap \\mathrm {Lim}\\rangle $ .",
"The reason is that if $D$ is any club of $\\omega _1$ , then both $D\\cap S\\cap \\delta $ and $(D\\cap \\delta )\\setminus S$ are cofinal subsets of $\\delta $ for each $\\delta $ in the club of limit points in $\\omega _1$ of both $D\\cap S$ and $D\\setminus S$ .",
"The status of $\\operatorname{\\textsf {Measuring}}^\\ast $ is more interesting in the absence of the Axiom of Choice.",
"Let $\\mathcal {C}_{\\omega _1}=\\lbrace X\\subseteq \\omega _1\\,:\\,C \\subseteq X\\mbox{ for some club $C$ of $\\omega _1$}\\rbrace $ .",
"Observation 4.2 ($\\textsf {ZF}$ + $\\mathcal {C}_{\\omega _1}$ is a normal filter on $\\omega _1$ ) Suppose $\\vec{X}=\\langle X_\\delta \\,:\\,\\delta \\in \\omega _1\\cap \\mathrm {Lim}\\rangle $ is such that $X_\\delta \\subseteq \\delta $ for each $\\delta $ .",
"For each club $C\\subseteq \\omega _1$ , there is some $\\delta \\in C$ such that $C\\cap X_\\delta \\ne \\emptyset $ , and there is some $\\delta \\in C$ such that $(C\\cap \\delta )\\setminus X_\\delta \\ne \\emptyset $ .",
"Then there is a stationary and co-stationary subset of $\\omega _1$ definable from $\\vec{X}$ .",
"We have two possible cases.",
"The first case is that in which for all $\\alpha <\\omega _1$ , either $W_\\alpha ^0=\\lbrace \\delta <\\omega _1\\,:\\,\\alpha \\notin X_\\delta \\rbrace $ is in $\\mathcal {C}_{\\omega _1}$ , or $W_\\alpha ^1=\\lbrace \\delta <\\omega _1\\,:\\,\\alpha \\in X_\\delta \\rbrace $ is in $\\mathcal {C}_{\\omega _1}$ .",
"For each $\\alpha <\\omega _1$ , let $W_\\alpha $ be $W_\\alpha ^\\epsilon $ for the unique $\\epsilon \\in \\lbrace 0, 1\\rbrace $ such that $W_\\alpha ^\\epsilon \\in \\mathcal {C}_{\\omega _1}$ , and let $W^\\ast =\\Delta _{\\alpha <\\omega _1} W_\\alpha \\in \\mathcal {C}_{\\omega _1}$ .",
"Then $X_{\\delta _0}=X_{\\delta _1}\\cap \\delta _0$ for all $\\delta _0<\\delta _1$ in $W^\\ast $ .",
"It then follows, by (2), that $S=\\bigcup _{\\delta \\in W^\\ast }X_\\delta $ , which of course is definable from $\\vec{C}$ , is a stationary and co-stationary subset of $\\omega _1$ .",
"Indeed, suppose $C\\subseteq \\omega _1$ is a club, and let us fix a club $D\\subseteq W^\\ast $ .",
"There is then some $\\delta \\in C\\cap D$ and some $\\alpha \\in C\\cap D\\cap X_\\delta $ .",
"But then $\\alpha \\in S$ since $\\delta \\in W^\\ast $ and $\\alpha \\in W^\\ast \\cap X_\\delta $ .",
"There is also some $\\delta \\in C\\cap D$ and some $\\alpha \\in C\\cap D$ such that $\\alpha \\notin X_\\delta $ , which implies that $\\alpha \\notin S$ by a symmetrical argument, using the fact that $X_{\\delta _0}=X_{\\delta _1}\\cap \\delta _0$ for all $\\delta _0<\\delta _1$ in $W^\\ast $ .",
"The second possible case is that there is some $\\alpha <\\omega _1$ with the property that both $W^0_\\alpha $ and $W^1_\\alpha $ are stationary subsets of $\\omega _1$ .",
"But now we can let $S$ be $W^0_\\alpha $ , where $\\alpha $ is first such that $W^0_\\alpha $ is stationary and co-stationary.",
"It is worth comparing the above observation with Solovay's classic result that an $\\omega _1$ –sequence of pairwise disjoint stationary subsets of $\\omega _1$ is definable from any given ladder system on $\\omega _1$ (working in the same theory).",
"Corollary 4.3 ($\\textsf {ZF}$ + $\\mathcal {C}_{\\omega _1}$ is a normal filter on $\\omega _1$ ) The following are equivalent.",
"$\\mathcal {C}_{\\omega _1}$ is an ultrafilter on $\\omega _1$ ; $\\operatorname{\\textsf {Measuring}}^\\ast $ ; For every sequence $\\langle X_\\alpha \\,:\\,\\alpha \\in \\omega _1\\cap \\mathrm {Lim}\\rangle $ , if $X_\\alpha \\subseteq \\alpha $ for each $\\alpha $ , then there is a club $C\\subseteq \\omega _1$ such that either $C\\cap \\delta \\subseteq X_\\delta $ for every $\\delta \\in C$ , or $C\\cap X_\\delta =\\emptyset $ for every $\\delta \\in C$ .",
"(3) trivially implies (2), and by the observation (1) implies (3).",
"Finally, to see that (2) implies (1), note that the argument right after the definition of $\\operatorname{\\textsf {Measuring}}^\\ast $ uses only $\\textsf {ZF}$ together with the regularity of $\\omega _1$ and the negation of (1).",
"In particular, the strong form of $\\operatorname{\\textsf {Measuring}}^\\ast $ given by (3) in the above observation follows from $\\textsf {ZF}$ together with the Axiom of Determinacy.",
"We finish this digression into set theory without the Axiom of Choice by observing that any attempt to parametrize $\\operatorname{\\textsf {Measuring}}^\\ast $ , in the same vein as we did with Measuring, gives rise to principles vacuously equivalent to $\\operatorname{\\textsf {Measuring}}^\\ast $ itself, at least when the parametrization is done with the alephs.This was pointed out by Asaf Karagila.",
"Specifically, given an aleph $\\kappa $ , let us define $\\operatorname{\\textsf {Measuring}}^*_\\kappa $ as the statement that for every sequence $\\langle \\mathcal {X}_\\alpha \\,:\\,\\alpha \\in \\omega _1\\cap \\mathrm {Lim}\\rangle $ , if each $\\mathcal {X}_\\alpha $ is a set of cardinality at most $\\kappa $ consisting of subsets of $\\alpha $ , then there is a club $D\\subseteq \\omega _1$ such that for every limit point $\\delta \\in D$ of $D$ , $D\\cap \\delta $ measures $X$ for all $X\\in \\mathcal {X}_\\delta $ .",
"Then $\\operatorname{\\textsf {Measuring}}^\\ast _{\\omega }$ is clearly equivalent to $\\operatorname{\\textsf {Measuring}}^\\ast $ under $\\textsf {ZF}$ together with the normality of $\\mathcal {C}_{\\omega _1}$ and the Axiom of Choice for countable families of subsets of $\\omega _1$ (which of course follows from the Axiom of Choice for countable families of sets of reals, and therefore also from $\\textsf {ZF} + \\textsf {AD}$ ).",
"On the other hand, working in $\\textsf {ZF}$ + $\\mathcal {C}_{\\omega _1}$ is a normal filter on $\\omega _1$ , we have that $\\operatorname{\\textsf {Measuring}}^\\ast _{\\omega _1}$ follows vacuously from $\\operatorname{\\textsf {Measuring}}^\\ast $ simply because under $\\operatorname{\\textsf {Measuring}}^\\ast $ there is no sequence $\\langle \\mathcal {X}_\\alpha \\,:\\,\\alpha \\in \\omega _1\\cap \\mathrm {Lim}\\rangle $ as in the definition of $\\operatorname{\\textsf {Measuring}}^\\ast _{\\omega _1}$ and such that $\\vert \\mathcal {X}_\\alpha \\vert =\\omega _1$ for some $\\alpha $ ; indeed, $\\operatorname{\\textsf {Measuring}}^\\ast $ implies, over this base theory, that $\\mathcal {C}_{\\omega _1}$ is an ultrafilter (Corollary REF ), and if $\\mathcal {C}_{\\omega _1}$ is an ultrafilter then there is no $\\omega _1$ -sequence of distinct reals, whereas the existence of a family of size $\\omega _1$ consisting of subsets of some fixed countable ordinal clearly implies that there is such a sequence.",
"We conclude the article with two natural questions.",
"Question 4.4 Is Measuring$_{\\mathfrak {p}}$ false?",
"Question 4.5 Are Measuring and Strong Measuring equivalent statements assuming Martin's Axiom?"
]
] | 1808.08596 |
[
[
"Interferometric Observations of Cyanopolyynes toward the G28.28-0.36\n High-Mass Star-Forming Region"
],
[
"Abstract We have carried out interferometric observations of cyanopolyynes, HC$_{3}$N, HC$_{5}$N, and HC$_{7}$N, in the 36 GHz band toward the G28.28$-$0.36 high-mass star-forming region using the Karl G. Jansky Very Large Array (VLA) Ka-band receiver.",
"The spatial distributions of HC$_{3}$N and HC$_{5}$N are obtained.",
"HC$_{5}$N emission is coincident with a 450 $\\mu$m dust continuum emission and this clump with a diameter of $\\sim 0.2$ pc is located at the east position from the 6.7 GHz methanol maser by $\\sim 0.15$ pc.",
"HC$_{7}$N is tentatively detected toward the clump.",
"The HC$_{3}$N : HC$_{5}$N : HC$_{7}$N column density ratios are estimated at 1.0 : $\\sim 0.3$ : $\\sim 0.2$ at an HC$_{7}$N peak position.",
"We discuss possible natures of the 450 $\\mu$m continuum clump associated with the cyanopolyynes.",
"The 450 $\\mu$m continuum clump seems to contain deeply embedded low- or intermediate-mass protostellar cores, and the most possible formation mechanism of the cyanopolyynes is the warm carbon chain chemistry (WCCC) mechanism.",
"In addition, HC$_{3}$N and compact HC$_{5}$N emission is detected at the edge of the 4.5 $\\mu$m emission, which possibly implies that such emission is the shock origin."
],
[
"Introduction",
"Cyanopolyynes (HC$_{2n+1}$ N, $n=1,2,3,...$ ) are one of the representative carbon-chain species.",
"In low-mass star-forming regions, carbon-chain molecules are known as early-type species; they are abundant in young starless cores and deficient in star-forming cores [31], [15].",
"In contrast to the general picture, cyanoacetylene (HC$_{3}$ N), the shortest member of cyanopolyynes, is detected from various regions such as infrared dark clouds [28], molecular outflows [1], protoplanetary disks [23], [4], and comets [22] and it is interesting to trace cyanopolyyne chemistry for better understanding of the molecular evolution during star/planet formation process.",
"Cyanopolyynes attract astrobiological as well as astrochemical interests.",
"Since they contain the nitrile bond (–C$\\equiv $ N), cyanopolyynes have been suggested as possible intermediates in the synthesis of simple amino acids [11], [5].",
"Saturated complex organic molecules (COMs), consisting of more than six atoms with rich hydrogen atoms, are abundant around protostars.",
"Such chemistry is known as hot core in high-mass star-forming regions and hot corino in low-mass star-forming regions.",
"In addition to hot corino, around a few low-mass protostars, carbon-chain molecules are formed from CH$_{4}$ evaporated from dust grains, which is known as warm carbon chain chemistry [27].",
"Progress in observational studies of carbon-chain molecules in high-mass star-forming regions has been slower, compared to low-mass star-forming regions.",
"Regarding hot cores, HC$_{5}$ N has been detected in chemically rich sources, Orion KL [9] and Sgr B2 [2], while only a tentative detection of HC$_{7}$ N in Orion KL was reported [10].",
"[6] performed a chemical network simulation and suggested that cyanopolyynes could be formed in a hot core from C$_{2}$ H$_{2}$ evaporated from grain mantles.",
"Motivated by the chemical network simulation, [13] carried out survey observations of HC$_{5}$ N toward 79 hot cores associated with the 6.7 GHz methanol masers and reported its detection in 35 sources.",
"However, the association with the maser is questionable, because they used a large beam (0.95) and a low-excitation energy line ($J=12-11$ ; $E_{\\rm u}/k = 10.0$ K), which can be excited even in dark clouds.",
"[33] carried out observations of long cyanopolyynes (HC$_{5}$ N and HC$_{7}$ N) toward four massive young stellar objects, where [13] had reported the HC$_{5}$ N detection, using the Green Bank 100-m and the Nobeyama 45-m radio telescopes, and detected high-excitation energy lines ($E_{\\rm u}/k \\approx 100$ K) of HC$_{5}$ N. The detection of such lines means that HC$_{5}$ N exists at least in the warm gas, not in cold molecular clouds ($T_{\\rm {kin}} \\simeq 10$ K).",
"[34] found that the G28.28$-$ 0.36 high-mass star-forming region is a particular cyanopolyyne-rich source with less COMs compared with other sources.",
"Hence, G28.28$-$ 0.36 is considered to be a good target region to study the cyanopolyyne chemistry around massive young stellar objects (MYSOs).",
"Using the Nobeyama 45-m radio telescope, [35] investigated the main formation mechanism of HC$_{3}$ N in G28.28$-$ 0.36 from its $^{13}$ C isotopic fractionation.",
"The reaction of “C$_{2}$ H$_{2}$ + CN\" was proposed as the main formation pathway of HC$_{3}$ N, which is consistent with the chemical network simulation conducted by [6] and the WCCC model [14].",
"Figure: Spitzer's IRAC 3.6 μ\\mu m image toward G28.28--0.36.",
"The open circle and cross indicate the 6.7 GHz methanol maser and ultracompact H2 (UCH2) region , respectively.In this paper, we carried out interferometric observations of cyanopolyynes (HC$_{3}$ N, HC$_{5}$ N, and HC$_{7}$ N) toward the G28.28$-$ 0.36 high-mass star-forming region ($d = 3$ kpc) with the Karl G. Jansky Very Large Array (VLA).",
"Figure REF shows the Spitzer IRAC 3.6 $\\mu $ m imagehttp://sha.ipac.caltech.edu/applications/Spitzer/SHA/ toward the region.",
"G28.28$-$ 0.36 is classified as an Extended Green Object (EGO) source [8] from the Spitzer Galactic Legacy Infrared Mid-Plane Survey Extraordinaire [3].",
"In Figure REF , the open circle and cross indicate the 6.7 GHz methanol maser [7] and ultracompact H2 (UCH2) region [36], respectively.",
"The 6.7 GHz maser is considered to give us the exact position of MYSOs [37].",
"A UCH2 region seems to heat the environment.",
"As shown in Figure REF , the ring structure around the UCH2 region is suggestive of expanding motion and on-going massive star formation.",
"We describe the observational details and data analyses in Section .",
"The resultant images and spectra of cyanopolyynes are presented in Section .",
"We compare the spatial distributions of cyanopolyynes with the infrared images and discuss possible formation mechanisms in Section ."
],
[
"Observations",
"The observations of G28.28$-$ 0.36 using the VLA Ka-band receiver were carried out in the C configuration with the 27 $\\times $ 25-m antennas on March 20th, 2016 (Proposal ID = 16A-084, PI; Kotomi Taniguchi).",
"The field of view (FoV) is $\\sim $ 604.",
"Four spectral windows of the correlator were set at our target lines summarized in Table .",
"All of these target lines were simultaneously observed.",
"The channel separation of the correlator is 0.5 km s$^{-1}$ .",
"The angular resolutions and Position Angles (PA) for each line are summarized in Table .",
"The phase reference center was set at ($\\alpha _{2000}$ , $\\delta _{2000}$ ) = (18$^{\\rm h}$ 44$^{\\rm m}$ 133, -0418030), the 6.7 GHz methanol maser position.",
"The pointing source is J1832$-$ 1035 at ($\\alpha _{2000}$ , $\\delta _{2000}$ ) = (18$^{\\rm h}$ 32$^{\\rm m}$ 20836, $-$ 1035112).",
"The absolute flux density calibration and the bandpass calibration were conducted by observing 3C286 at ($\\alpha _{2000}$ , $\\delta _{2000}$ ) = (13$^{\\rm h}$ 31$^{\\rm m}$ 0828798, +3030329589).",
"The gain/phase calibration was conducted by observing J1851+0035 at ($\\alpha _{2000}$ , $\\delta _{2000}$ ) = (18$^{\\rm h}$ 51$^{\\rm m}$ 467217, +003532414).",
"cccccc Summary of target lines 0pt Species Transition Rest Frequency $E_{\\rm {u}}/k$ Angular PA (GHz) (K) Resolution (deg) HC$_{3}$ N $J=4-3$ 36.39232 4.4 084 $\\times $ 063 -9.92 HC$_{5}$ N $J=14-13$ 37.276994 13.4 081 $\\times $ 063 -11.04 HC$_{7}$ N $J=33-32$ 37.22349 30.4 082 $\\times $ 063 -10.20 CH$_{3}$ CN $J_{\\rm K} =2_{0} - 1_{0}$ 36.7954747 2.6 083 $\\times $ 064 -11.22 Rest frequencies are taken from the Cologne Database for Molecular Spectroscopy [21] and the Jet Propulsion Laboratory catalog [24].",
"We conducted data reduction using the Common Astronomy Software Application [20].",
"We used the VLA calibration pipelinehttps://science.nrao.edu/facilities/vla/data-processing/pipeline provided by the National Radio Astronomical Observatory (NRAO) to perform basic flagging and calibration.",
"The data cubes were imaged using the CLEAN task.",
"Natural weighting was applied.",
"The pixel size and image size are $0.2$ and $1000 \\times 1000$ pixels.",
"After the CLEAN, we smoothed the cube using the “imsmooth\" command, applying $1\\times 1$ and the position angle of 0 with the gaussian kernel.",
"The spatial resolution of 1 of the resultant images corresponds to $\\sim 0.015$ pc.",
"The $1 \\sigma $ values are approximately 0.6, 0.7, 0.7, and 0.6 mK for HC$_{3}$ N, HC$_{5}$ N, HC$_{7}$ N, and CH$_{3}$ CN, respectively.",
"We made the moment zero images of HC$_{3}$ N and HC$_{5}$ N using the “immoments\" task in CASA."
],
[
"Results",
"Figure REF shows the moment zero images of (a) HC$_{3}$ N and (b) HC$_{5}$ N in G28.28$-$ 0.36.",
"The velocity components in the range $V_{\\rm {LSR}} = 47.5 - 51.5$ km s$^{-1}$ were integrated in these moment zero images.",
"The spatial distribution of HC$_{3}$ N is more extended than that of HC$_{5}$ N, because of their excitation energies of the observed lines (Table ).",
"The observed HC$_{3}$ N line has lower excitation energy ($E_{\\rm {u}}/k = 4.4$ K) than that of HC$_{5}$ N ($E_{\\rm {u}}/k = 13.4$ K), and colder envelopes could be traced by HC$_{3}$ N. Figure: Moment zero images of (a) HC 3 _{3}N and (b) HC 5 _{5}N obtained with the VLA, including the data above 2σ2 \\sigma .",
"The contour levels are 0.2, 0.4, 0.6, and 0.8 of their peak levels, where the peak intensities are 27.9 and 13.5 (mK ·\\cdot km s -1 ^{-1}) for (a) HC 3 _{3}N and (b) HC 5 _{5}N, respectively.",
"The rms noise levels are 1.8 and 1.3 (mK ·\\cdot km s -1 ^{-1}) in the images of HC 3 _{3}N and HC 5 _{5}N, respectively.",
"The orange open circle and magenta cross indicate the 6.7 GHz methanol maser and UCH2 region , respectively.",
"The filled circles at the bottom left corner represent the angular resolution of these images (1).Regarding HC$_{7}$ N, the signal-to-noise ratio is low, which precludes determination of its spatial distribution.",
"The bottom panel of Figure REF shows HC$_{7}$ N spectra, as well as HC$_{3}$ N and HC$_{5}$ N spectra, observed toward its four peak positions A$-$ D indicated in the panel (b) of Figure REF .",
"In order to improve the signal-to-noise ratio, we applied the 1 uvtaper for HC$_{7}$ N data.",
"The intensities of these spectra are estimated within $1.5 $ regions, which corresponds to the spatial resolution of HC$_{7}$ N with the uvtaper.",
"HC$_{7}$ N is detected around the regions where HC$_{5}$ N is detected as shown in the panel (b) of Figure REF .",
"CH$_{3}$ CN emission is undetected at the rms noise level of 0.6 mK.",
"Figure: HC 3 _{3}N, HC 5 _{5}N, and HC 7 _{7}N spectra at four HC 7 _{7}N peak positions labeled as A--D, denoting in the panel (b) of Figure .",
"The gray vertical lines show the systemic velocity of the G28.28-0.36 MYSO (49 km s -1 ^{-1}).",
"The rms noise levels are 0.16, 0.20, and 0.12 K for HC 3 _{3}N, HC 5 _{5}N, and HC 7 _{7}N spectra, respectively."
],
[
"Comparisons of Cyanopolyyne Ratios",
"We derived the column densities of HC$_{3}$ N, HC$_{5}$ N, and HC$_{7}$ N at Position A (Figure REF ) assuming the local thermodynamic equilibrium (LTE).",
"We use the following formulae [12]: $ \\tau = - {\\mathrm {ln}} \\left[1- \\frac{T_{\\rm b} }{\\left\\lbrace J(T_{\\rm {ex}}) - J(T_{\\rm {bg}}) \\right\\rbrace } \\right],$ where $ J(T) = \\frac{h\\nu }{k}\\Bigl \\lbrace \\exp \\Bigl (\\frac{h\\nu }{kT}\\Bigr ) -1\\Bigr \\rbrace ^{-1},$ and $ N = \\tau \\frac{3h\\Delta v}{8\\pi ^3}\\sqrt{\\frac{\\pi }{4\\mathrm {ln}2}}Q\\frac{1}{\\mu ^2}\\frac{1}{J_{\\rm {lower}}+1}\\exp \\Bigl (\\frac{E_{\\rm {lower}}}{kT_{\\rm {ex}}}\\Bigr )\\Bigl \\lbrace 1-\\exp \\Bigl (-\\frac{h\\nu }{kT_{\\rm {ex}}}\\Bigr )\\Bigr \\rbrace ^{-1}.$ In Equation (REF ), $\\tau $ and $T_{\\rm b}$ denote the optical depth and brightness temperature, respectively.",
"$T_{\\rm {ex}}$ and $T_{\\rm {bg}}$ are the excitation temperature and the cosmic microwave background temperature ($\\simeq 2.73$ K), respectively.",
"$J$ ($T$ ) in Equation (REF ) is the effective temperature equivalent to that in the Rayleigh-Jeans law.",
"In Equation (REF ), N, $\\Delta v$ , $Q$ , $\\mu $ , and $E_{\\rm {lower}}$ denote the column density, line width (FWHM), partition function, permanent electric dipole moment, and energy of the lower rotational energy level, respectively.",
"The brightness temperatures and line widths are obtained by the gaussian fitting of spectra.",
"Figure REF in Appendix shows the fitting results for each spectra, and the obtained spectral line parameters and permanent electric dipole moments of each species are summarized in Table .",
"We derived the column densities assuming the excitation temperatures of 15, 20, 30, and 50 K, respectively.",
"The column densities of cyanopolyynes and the HC$_{3}$ N : HC$_{5}$ N : HC$_{7}$ N ratios at each excitation temperature are summarized in Table REF .",
"The uncertainties in the excitation temperatures do not significantly affect the derived column densities of HC$_{5}$ N and HC$_{7}$ N, and their derived column densities agree with each other within their the standard deviation errors.",
"On the other hand, the uncertainties in the excitation temperatures bring larger differences in the HC$_{3}$ N column density.",
"ccccc Column densities of cyanopolyynes at Position A 0pt $T_{\\rm {ex}}$ $N$ (HC$_{3}$ N) $N$ (HC$_{5}$ N) $N$ (HC$_{7}$ N) HC$_{3}$ N : HC$_{5}$ N : HC$_{7}$ N (K) ($\\times 10^{14}$ cm$^{-2}$ ) ($\\times 10^{13}$ cm$^{-2}$ ) ($\\times 10^{13}$ cm$^{-2}$ ) 15 $1.36 \\pm 0.14$ $4.4 \\pm 0.9$ $4.0 \\pm 1.2$ 1.00 : 0.32 : 0.29 20 $1.47 \\pm 0.15$ $4.3 \\pm 0.9$ $3.1 \\pm 0.9$ 1.00 : 0.29 : 0.21 30 $1.84 \\pm 0.19$ $4.9 \\pm 1.1$ $2.6 \\pm 0.8$ 1.00 : 0.26 : 0.14 50 $2.7 \\pm 0.3$ $6.5 \\pm 1.4$ $2.8 \\pm 0.8$ 1.00 : 0.24 : 0.11 The errors represent the standard deviation.",
"The errors of column densities are derived from uncertainties of the gaussian fitting (see Table in Appendix).",
"The HC$_{3}$ N : HC$_{5}$ N : HC$_{7}$ N ratios at Position A are derived to be 1.0 : $\\sim 0.3$ : $\\sim 0.2$ .",
"The ratios in L1527, which is one of the WCCC sources, are 1.0 : $0.3-0.6$ : $\\sim 0.1$ [25], [26].",
"The tendency of the ratios at Position A seems to be similar to that in L1527."
],
[
"Comparison of Spatial Distributions between Cyanopolyynes and 450 $\\mu $ m Dust Continuum",
"Figure REF shows 450 $\\mu $ m dust continuum images overlaid by the black contours of moment zero images of (a) HC$_{3}$ N and (b) HC$_{5}$ N, respectively.",
"The 450 $\\mu $ m data, which are available from the James Clerk Maxwell Telescope (JCMT) Science Archivehttp://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/en/jcmt/index.html, were obtained with the SCUBA installed on the JCMT.",
"The main beam size of the SCUBA is $7.9$ at 450 $\\mu $ m, corresponding to $\\sim 0.11$ pc.",
"Three strong 450 $\\mu $ m continuum emission peaks can be recognized; the UCH2 position, the north-west position from the UCH2, and the east position from the 6.7 GHz methanol maser.",
"The spatial distribution of HC$_{5}$ N is consistent with the 450 $\\mu $ m continuum peak of the east position from the 6.7 GHz methanol maser (hereafter Cyanopolyyne-rich clump, panel (b) of Figure REF ).",
"The spatial distribution of HC$_{3}$ N seems to surround the Cyanopolyyne-rich clump (panel (a) of Figure REF ), not only at the Cyanopolyyne-rich clump.",
"HC$_{3}$ N emission is also located at the west position of the 6.7 GHz methanol maser position, or the edge of the 450 $\\mu $ m continuum.",
"Small HC$_{5}$ N emission region is seen at the same edge of the 450 $\\mu $ m continuum.",
"We will briefly discuss a possible origin of this emission region in Section REF .",
"Figure: 450 μ\\mu m continuum image obtained with the SCUBA installed on the JCMT overlaid by black contours of (a) HC 3 _{3}N and (b) HC 5 _{5}N moment zero images, the same ones as in Figure .",
"The contour levels are 0.2, 0.4, 0.6, and 0.8 of their peak intensities 27.9 and 13.5 (mK ·\\cdot km s -1 ^{-1}) for (a) HC 3 _{3}N and (b) HC 5 _{5}N, respectively.",
"The blue filled circle and blue cross indicate the 6.7 GHz methanol maser and UCH2 region, respectively.",
"The A–D positions are indicated as the green filled circles.",
"The filed white circles at the bottom left corner represent the angular resolution of the images of HC 3 _{3}N and HC 5 _{5}N (1)."
],
[
"Possible Nature of the 450 $\\mu $ m Continuum Peak Position associated with Cyanopolyynes",
"We examine four possible types of objects of the Cyanopolyyne-rich clump (Section REF ); a hot core, a starless clump, low- or intermediate-mass protostellar core(s), and a photo-dominated region (PDR) driven by the associated UCH2 region.",
"Figure REF shows the Spitzer 3-color (3.6 $\\mu $ m, 4.5 $\\mu $ m, 8.0 $\\mu $ m) imageshttp://atlasgal.mpifr-bonn.mpg.de/cgi-bin/ATLASGAL$_{-}$ DATABASE.cgi, overlaid by yellow contours showing (b) 450 $\\mu $ m continuum, (c) HC$_{3}$ N moment zero image, and (d) HC$_{5}$ N moment zero image, respectively.",
"In panel (b), the green contours show the 8.3 mm continuum emission obtained simultaneously with cyanopolyynes by the VLA.",
"The 8.3 mm continuum peak is compact and well consistent with the UCH2 region.",
"At the Cyanopolyyne-rich clump, no point source can be recognized from the Spitzer image, which suggests that no massive young protostar is currently present at the clump position.",
"Therefore, the possibility of hot core is unrealistic.",
"We derived the average column density of H$_{2}$ , $N$ (H$_{2}$ ), of the Cyanopolyyne-rich clump from the 450 $\\mu $ m continuum data using the following formula [30]: $ N({\\rm {H}}_2) = 2.02 \\times 10^{20} {\\rm {cm}}^{-2} \\Bigl (e^{1.439(\\lambda /{\\rm {mm}})^{-1}(T/10 {\\rm {K}})^{-1}}-1\\Bigr )\\Bigl (\\frac{\\kappa _{\\nu }}{0.01\\: {\\rm {cm}}^{2}\\: {\\rm {g}}^{-1}}\\Bigr )^{-1}\\Bigl (\\frac{S_{\\nu }^{\\rm {beam}}}{{\\rm {mJy}}\\: {\\rm {beam}}^{-1}}\\Bigr )\\Bigl (\\frac{\\theta _{\\rm {HPBW}}}{10\\: {\\rm {arcsec}}}\\Bigr )^{-2}\\Bigl (\\frac{\\lambda }{{\\rm {mm}}}\\Bigr )^{3}.$ We estimated the continuum flux ($S_{\\nu }^{\\rm {beam}}$ ) toward the Cyanopolyyne-rich clump within a size of 14, and the flux intensity is $9.7 \\times 10^{3}$ mJy beam$^{-1}$ .",
"We assumed that $\\kappa _{\\nu }$ is 0.0619 cm$^{2}$ g$^{-1}$ at $\\lambda = 450$ $\\mu $ m. $T$ is the dust temperature, and we assumed it to be $30 \\pm 15$ K. The derived $N$ (H$_{2}$ ) value is ($2.8^{+8.1}_{-1.3}$ )$\\times 10^{22}$ cm$^{-2}$ , which is similar to the typical value of massive clumps forming young star clusters [29].",
"The value is an average for the clump and then it can be considered as the lower limits for cores.",
"The derived $N$ (H$_{2}$ ) value for the Cyanopolyyne-rich clump is higher than the threshold value for star formation cores, $N$ (H$_{2}$ ) $\\approx 9 \\times 10^{21}$ cm$^{-2}$ [32].",
"Hence, the Cyanopolyyne-rich clump is considered to contain deeply embedded low- or intermediate-mass protostellar core(s), and is not a starless clump.",
"In case that a low- or intermediate-mass protostar has been born, the WCCC mechanism is likely to work; cyanopolyynes are formed from CH$_{4}$ evaporated from grain mantles in the lukewarm gas ($T = 20-30$ K).",
"This may be supported by similar HC$_{3}$ N : HC$_{5}$ N : HC$_{7}$ N ratios between this clump and L1527 (Section REF ).",
"In order to investigate the possibility of the PDR, we compare the fractional abundances of cyanopolyynes at the Cyanopolyyne-rich clump with those derived from a PDR chemical network simulation [19].",
"We derived the fractional abundances, $X$ ($a$ )$= N$ ($a$ )$/N$ (H$_{2}$ ), of cyanopolyynes at Position A as summarized in Table REF .",
"Recent chemical network simulation about the Horsehead nebula, which is one of the most studied PDRs, estimated the HC$_{3}$ N and HC$_{5}$ N abundances [19].",
"The PDR and Core positions are positions with a visual extinctions ($A_{v}$ ) of $\\sim 2$ mag and $\\sim 10-20$ mag, respectively.",
"We summarize the fractional abundances of these two positions in Table REF .",
"The $X$ (HC$_{3}$ N) and $X$ (HC$_{5}$ N) values at the Cyanopolyyne-rich clump are significantly higher than those in PDR position by three order of magnitudes and more than four order of magnitudes, respectively.",
"Moreover, the HC$_{3}$ N and HC$_{5}$ N fractional abundances at the Cyanopolyyne-rich clump are still higher than those in Core position in the PDR model by a factor of $\\sim 55$ and $\\sim 80$ , respectively.",
"Therefore, cyanopolyynes at the Cyanopolyyne-rich clump in the G28.28$-$ 0.36 high-mass star-forming region are not formed by the PDR chemistry.",
"If we take large uncertainties in deriving $N$ (H$_{2}$ ) values into consideration, the differences between the Cyanopolyyne-rich clump and the PDR models are plausible.",
"However, we cannot completely exclude any effects from the UCHII region such as UV radiation.",
"lccccc Comparison of fractional abundances of cyanopolyynes 0pt Source Method $X$ (HC$_{3}$ N) $X$ (HC$_{5}$ N) $X$ (HC$_{7}$ N) Referencea ($\\times 10^{-9}$ ) ($\\times 10^{-9}$ ) ($\\times 10^{-9}$ ) Cyanopolyyne-rich clump Observation $6.6^{+5.5}_{-4.9}$ $1.7^{+1.5}_{-1.3}$ $0.9^{+0.8}_{-0.7}$ 1 Horsehead nebula (PDR position) Simulation $3.1 \\times 10^{-3}$ $4.3 \\times 10^{-5}$ ... 2 Horsehead nebula (Core position) Simulation 0.12 0.022 ... 2 L1527 (WCCC) Observation 0.43 – 0.96 0.05 – 0.24 $\\sim 0.05$ 3,4,5 The errors represent the standard deviation.",
"a(1) This work; (2) [19]; (3) [17]; (4) [25]; (5) [26] We also compare the fractional abundances of HC$_{3}$ N, HC$_{5}$ N, and HC$_{7}$ N at the Cyanopolyyne-rich clump to those in L1527.",
"These fractional abundances at the Cyanopolyyne-rich clump are higher than those in L1527 by a factor of $7-15$ , $7-37$ , and $\\sim 17$ , respectively.",
"Hence, the cyanopolyynes at the Cyanopolyyne-rich clump seem to be more abundant compared to L1527, even if we take uncertainties in fractional abundances into account.",
"It cannot be excluded that several low-mass protostars are concentrated within a small region (e.g., $\\sim 0.02$ pc corresponding to $1.5$ at the distance of 3 kpc).",
"As another interpretation, these results may imply that there is an intermediate-mass protostar because an intermediate-mass protostar heats its wider surroundings compared to a low-mass protostar; the size of lukewarm envelopes will be larger and the column densities of carbon-chain species in such lukewarm envelopes will increase.",
"This could explain the results that the cyanopolyynes at the Cyanopolyyne-rich clump are more abundant than those in L1527.",
"In Figure REF (d), compact HC$_{5}$ N emission region is located at the west position of the 6.7 GHz methanol maser position.",
"This position corresponds to the edge of the 4.5 $\\mu $ m emission, which seems to trace shock regions [8].",
"Figure REF shows spectra of HC$_{3}$ N and HC$_{5}$ N at the shock region.",
"The line widths (FWHM) derived from the Gaussian fit are $2.7 \\pm 0.2$ and $2.6 \\pm 0.2$ km s$^{-1}$ for HC$_{3}$ N and HC$_{5}$ N, respectively.",
"These line widths at the shock region are wider those at the Cyanopolyyne-rich clump, $2.05 \\pm 0.16$ and $1.4 \\pm 0.2$ km s$^{-1}$ for HC$_{3}$ N and HC$_{5}$ N, respectively.",
"These results probably support that the cyanopolyynes are the shock origin.",
"Molecules evaporated from grain mantles such as C$_{2}$ H$_{2}$ may be parent species of the cyanopolyynes.",
"It is still unclear whether HC$_{5}$ N is formed and can survive in shock regions, but this may be the first observational result showing that HC$_{5}$ N can be enhanced in shock regions.",
"Figure: Spectra of HC 3 _{3}N and HC 5 _{5}N at the shock region.",
"The red lines show the Gaussian fitting results."
],
[
"Comparison with Previous Single-Dish Telescope Observations",
"[33] reported the detection of high-excitation-energy ($E_{\\rm {u}}/k \\simeq 100$ K) lines of HC$_{5}$ N with the Nobeyama 45-m radio telescope (HPBW = 18) toward the methanol maser position, which is considered to be a MYSO position [37].",
"The significant HC$_{5}$ N peak is not seen at the methanol maser position in the VLA map.",
"This is probably caused by the different excitation energies of the observed lines.",
"The lines observed with the VLA have low excitation energies (Table ), and preferably trace lower-temperature regions such as low- or intermediate-mass protostellar cores or envelopes.",
"On the other hand, HC$_{5}$ N emission observed with the Nobeyama 45-m telescope seems to come from higher-temperature regions closer to the methanol maser or the MYSO.",
"In these higher-temperature regions, HC$_{5}$ N should be highly excited and low excitation energy lines observed with the VLA should be weak.",
"In contrast, the hotter components detected with the Nobeyama 45-m telescope were not detected with the VLA, because the low-excitation-energy lines are not suitable tracers of the hot components.",
"Combining with the VLA and Nobeyama 45-m telescope results, both the MYSO associated with the methanol maser and low- or intermediate-mass protostellar core(s) at the Cyanopolyyne-rich clump appear to be rich in cyanopolyynes.",
"It is still unclear why the G28.28$-$ 0.36 high-mass star-forming region is a cyanopolyyne-rich/COMs-poor source [34].",
"Studies about chemical diversity among high-mass star-forming regions will become a key to our understanding of massive star formation processes.",
"We need the high-spatial-resolution and higher-frequency band observations in order to investigate the spatial resolution of higher temperature components of cyanopolyynes to confirm that cyanopolyynes exist at the hot core position."
],
[
"Conclusions",
"We have carried out interferometric observations of cyanopolyynes (HC$_{3}$ N, HC$_{5}$ N, and HC$_{7}$ N) toward the G28.28$-$ 0.36 high-mass star-forming region with the VLA Ka-band.",
"We obtained the moment zero images of HC$_{3}$ N and HC$_{5}$ N and tentatively detected HC$_{7}$ N. The spatial distributions of HC$_{3}$ N and HC$_{5}$ N are consistent with the 450 $\\mu $ m dust continuum clump, i.e., the Cyanopolyyne-rich clump.",
"The HC$_{3}$ N : HC$_{5}$ N : HC$_{7}$ N column density ratios are estimated at 1.0 : $\\sim 0.3$ : $\\sim 0.2$ at Position A.",
"The Cyanopolyyne-rich clump seems to contain deeply embedded low- or intermediate-mass protostellar core(s).",
"The most probable formation mechanism of the cyanopolyynes at the Cyanopolyyne-rich clump is the WCCC mechanism.",
"We possibly found the HC$_{3}$ N and HC$_{5}$ N emission in the shock region.",
"We express our sincere thanks and appreciate for the staff of the National Radio Astronomy Observatory.",
"The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. KT appreciates support from a Granting-Aid for Science Research of Japan (17J03516).",
"Karl G. Jansky Very Large Array (VLA) Common Astronomy Software Applications (CASA)"
],
[
"Gaussian Fitting Results and Spectral Line Parameters",
"We show the gaussian fitting results for the spectra of HC$_{3}$ N, HC$_{5}$ N, and HC$_{7}$ N at Position A in Figure REF .",
"The obtained spectral line parameters are summarized in Table .",
"Figure: Spectra of HC 3 _{3}N, HC 5 _{5}N, and HC 7 _{7}N at Position A obtained with the VLA.",
"The best gaussian fits are shown overlaid in red.ccccc Spectral line parameters at Position A 0pt Species $\\mu $ $T_{\\rm {b}}$ FWHM $\\int T_{\\mathrm {b}}dv$ (Debye) (K) (km s$^{-1}$ ) (K km s$^{-1}$ ) HC$_{3}$ N 3.73 $5.4 \\pm 0.4$ $2.05 \\pm 0.16$ $11.7 \\pm 1.2$ HC$_{5}$ N 4.33 $2.3 \\pm 0.3$ $1.4 \\pm 0.2$ $3.5 \\pm 0.8$ HC$_{7}$ N 4.82 $1.0 \\pm 0.2$ $1.3 \\pm 0.3$ $1.3 \\pm 0.4$ The errors represent the standard deviation.",
"Values of their permanent electric dipole moments are taken from the Cologne Database for Molecular Spectroscopy [21]."
]
] | 1808.08435 |
[
[
"Predicting a solar cycle before its onset using a flux transport dynamo\n model"
],
[
"Abstract We begin with a review of the predictions for cycle~24 before its onset.",
"After summarizing the basics of the flux transport dynamo model, we discuss how this model had been used to make a successful prediction of cycle~24, on the assumption that the irregularities of the solar cycle arise due to the fluctuations in the Babcock--Leighton mechanism.",
"We point out that fluctuations in the meridional circulation can be another cause of irregularities in the cycle."
],
[
"Introduction",
"Let us begin with a disclaimer.",
"This review will focus on the physics of predicting solar cycles from dynamo models and will refrain from presenting any detailed prediction for the upcoming cycle 25, which is nowadays becoming a hot topic of research.",
"Figure: Different predictions of the strength of cycle 24, adopted from Pesnell (2008).The two predictions based on theoretical dynamo models (Dikpati et al.",
"2006; Choudhuriet al.",
"2007) are indicated by arrows.",
"The horizontalline added by us indicates the actual peak strength of cycle 24 reached around April 2014.Now that we know what the present cycle 24 has been like, let us take a look at the many predictions of cycle 24 before its onset.",
"Pesnell (2008) produced a plot combining all the different predictions of the peak sunspot number of cycle 24.",
"Figure 1 is adopted from this plot, indicating the two predictions based on theoretical dynamo models.",
"The first theoretical prediction by Dikpati et al.",
"(2006) was that cycle 24 would be a very strong cycle, whereas the other prediction by Choudhuri et al.",
"(2007) was that it would be a fairly weak cycle.",
"All the other predictions shown in Figure 1 were based on various precursors and empirical projections.",
"We can see that the predictions covered almost the entire range of possible values of the peak sunspot number from $\\approx $ 40 to $\\approx $ 190.",
"The horizontal line indicates the actual peak sunspot number of cycle 24 and was added by us while preparing this presentation.",
"It is clear that Choudhuri et al.",
"(2007) predicted the cycle 24 peak almost correctly.",
"If several people make several predictions covering the entire possible range, then somebody's prediction has got to come out right!",
"Were Choudhuri et al.",
"(2007) simply the lucky persons whose prediction accidentally turned out to be correct?",
"Or did they get it correct because they figured out the correct physics for making such predictions?",
"We would like to argue that they figured out the correct physics partially, but not fully.",
"Their success in predicting cycle 24 was due to a combination of intuition and luck.",
"In a classic work, Parker (1955) envisaged that the solar cycle is produced by an oscillation between the toroidal and poloidal magnetic fields of the Sun.",
"Sunspots form out of the toroidal field due to magnetic buoyancy and provide an indication of the strength of the toroidal field.",
"On the other hand, the magnetic fields in the polar regions of the Sun are a manifestation of the poloidal field.",
"We now know that there is truly an oscillation between these two field components.",
"The polar (i.e.",
"poloidal) field becomes strongest around the time when the sunspot number (i.e.",
"toroidal field) has its lowest value and vice versa.",
"Svalgaard et al.",
"(2005) and Schatten (2005), whose predictions for cycle 24 are included in Figure 1, suggested that the polar field at the beginning of a cycle is a good precursor for the strength of the cycle and used the weak polar field at the beginning of cycle 24 to predict essentially the same low value of the cycle peak that was predicted from the theoretical dynamo calculations of Choudhuri et al.",
"(2007).",
"While discussing the physics of cycle prediction, we need to address the question of why the polar field at the beginning of a cycle acts as such a good precursor of the cycle strength."
],
[
"Basics of flux transport dynamo model",
"The flux transport dynamo model, which started being developed in the 1990s (Wang et al.",
"1991; Choudhuri et al.",
"1995; Durney 1995; Dikpati & Charbonneau 1999; Nandy & Choudhuri 2002) and has been recently reviewed by several authors (Choudhuri 2011, 2014; Charbonneau 2014; Karak et al.",
"2014a), has emerged as an attractive theoretical model of the solar cycle.",
"We describe the basics of this model in this Section before getting into the question of cycle prediction in the next Section.",
"In order to have an oscillation between the toroidal and poloidal magnetic components, we need to have mechanisms to generate each component from the other.",
"The generation of the toroidal component from the poloidal component due to stretching by differential rotation is a straightforward process.",
"To complete the loop, we need a mechanism to generate the poloidal component back from the toroidal component.",
"The flux transport dynamo model assumes that this is achieved by the Babcock–Leighton (BL) mechanism (Babcock 1961; Leighton 1964), in which the toroidal field first gives rise to bipolar sunspots which emerge with a tilt due to the action of the Coriolis force (D'Silva & Choudhuri 1993) and then the poloidal field arises from the decay of these tilted bipolar sunspots (Hazra et al.",
"2017).",
"A dynamo model based just on the idea of toroidal field generation by differential rotation and poloidal field generation by the BL mechanism gives a poleward propagating dynamo wave—in accordance with what is called the Parker–Yoshimura sign rule (Parker 1955; Yoshimura 1975).",
"In order to explain the observation that sunspots appear increasingly at lower latitudes with the progress of the solar cycle, we need an extra mechanism to reverse the direction of the dynamo wave and make it propagate equatorward.",
"Choudhuri et al.",
"(1995) realized that this extra mechanism is provided by the meridional circulation (MC) of the Sun, which is observed to be poleward near the surface of the Sun.",
"In order to avoid mass piling up in the polar region, there has to be a subsurface reverse flow towards the equator.",
"The majority of flux transport dynamo models assume an one-cell MC with the equatorward flow at the bottom of the convection zone.",
"However, Hazra et al.",
"(2014) have shown that the flux transport model can work even with a multi-cell MC structure as long as there is an equatorward flow at the bottom of the convection zone.",
"The toroidal field produced there due to the strong differential rotation discovered by helioseismology is advected by this flow equatorward, making sure that sunspots appear at lower latitudes with the progress of the cycle.",
"The poloidal field generated by the BL mechanism near the surface is advected poleward by the poleward MC there—in accordance with observational data.",
"Perhaps, at this stage, it is a good idea to state clearly what we mean by flux transport dynamo.",
"We would refer to a solar dynamo model as a flux transport dynamo model if it satisfies the following characteristics: (i) the Babcock–Leighton (BL) mechanism for generating the poloidal field at the solar surface is included in the model; and (ii) the meridional circulation (MC) plays an important role by transporting the poloidal field poleward near the surface and the toroidal field equatorward at the bottom of the convection zone.",
"While different authors in the past might have meant slightly different things by flux transport dynamo, we believe that the definition we give here has come to be regarded as the universally accepted definition of flux transport dynamo at the present time.",
"Detailed theoretical calculations based on the flux transport dynamo model are broadly in agreement with observational data pertaining to the solar cycle.",
"A flux transport dynamo code based on mean field equations tends to give a periodic solution, unless something special is done to make the solution irregular.",
"After our brief summary of the flux transport dynamo model, we come to the question as to what causes irregularities of the cycle and whether an understanding of that will help us in predicting future cycles."
],
[
"Fluctuations in the Babcock–Leighton (BL) mechanism",
"It has been known for some time that fluctuations in the poloidal field generation mechanism can cause irregularities in the cycle (Choudhuri 1992).",
"Choudhuri et al.",
"(2007) suggested that the fluctuations in the BL mechanism would be the main source of irregularities in the dynamo process.",
"It is not difficult to understand how the fluctuations in the BL mechanism may arise.",
"This mechanism depends on the tilts of bipolar sunspot pairs—a pair with a larger tilt making a more significant contribution to the poloidal field.",
"The average tilt of sunspot pairs is given by Joy's law.",
"However, one finds a distribution of tilt angles around this average (Stenflo & Kosovichev 2012)—presumably caused by the effect of the turbulence in the solar convection zone through which flux tubes rise to form bipolar sunspot pairs (Longcope & Choudhuri 2002).",
"Figure: A sketch explaining how the correlation between the polar field at theend of a cycle and the strength of the next cycle arises.",
"From Jiang et al.",
"(2007).Let us first come to the question how the observed correlation between the polar field at the beginning of a solar cycle and the strength of the cycle arises.",
"Refer to Fig.",
"2 taken from Jiang et al.",
"(2007), who provided an explanation of this.",
"The BL mechanism produces poloidal field at C in mid-latitudes near the surface.",
"This poloidal field will simultaneously be advected poleward by the meridional circulation (MC) to produce the polar field at P at the beginning of the next cycle and will also diffuse to T at the bottom of the convection zone, where it will act as the seed of the next cycle.",
"If the fluctuations in the BL mechanism make the poloidal field produced at C in a cycle stronger than the usual, then both the polar field at P at the beginning of the next cycle and the seed of the next cycle at T will be stronger than usual.",
"On the other hand, if the poloidal field produced at C is weaker than the usual, then the opposite of this will happen.",
"We believe that this is how the correlation between the polar field at the beginning of a cycle and the strength of the cycle arises.",
"It may be pointed out that the turbulent diffusion has to be sufficiently high to ensure that the poloidal field diffuses from C to T in time of the order of 5–10 yr, in order to produce the required correlation.",
"Choudhuri et al.",
"(2007) used a value of turbulent diffusivity on the basis of mixing length arguments, which made the correlation come out beautifully.",
"However, Dikpati et al.",
"(2006) used a rather unrealistically low value of turbulent diffusion and the diffusion time across the convection zone in their model is of the order of 200 yr. Their model would not give a correlation between the polar field at the beginning of a cycle and the strength of the cycle.",
"It may be mentioned that solar dynamo models with high and low diffusivities have very different characteristics (Jiang et al.",
"2007; Yeates et al.",
"2008).",
"There is enough evidence now that models with higher diffusivity are closer to reality.",
"Higher diffusivity helps in explaining such observational features as the dipolar parity (Chatterjee et al.",
"2004; Hotta & Yokoyama 2010) and the lack of hemispheric asymmetry (Chatterjee & Choudhuri 2006; Goel & Choudhuri 2009).",
"In order to model actual cycles, one needs to incorporate the actual fluctuations in the BL mechanism into the code.",
"Choudhuri et al.",
"(2007) devised a scheme of figuring out actual fluctuations of the BL mechanism from the observational data of the poloidal fields and then incorporating these in the dynamo code.",
"Since such data are available only from the 1970s, actual cycles could be modelled only from that time.",
"Choudhuri et al.",
"(2007) succeeded in modelling cycles 21–23 reasonably well and cycle 24 was predicted to be a weak cycle.",
"Their prediction of cycle 24 was a robust prediction, since they had incorporated the weakness of the polar field at the beginning of cycle 24 in their model and the high diffusivity of their model would make this correlated with the strength of cycle 24.",
"As we have already pointed out, this prediction has been borne out triumphantly—making this the first successful prediction of a solar cycle based on a theoretical dynamo model in the history of this subject."
],
[
"Fluctuations in the meridional circulation (MC)",
"When Choudhuri et al.",
"(2007) made their prediction, it was not realized that the meridional circulation (MC) probably has occasional large fluctuations and these also may produce irregularities in the solar cycle.",
"These fluctuations are different from the periodic variation of MC with the solar cycle, presumably due to the Lorentz force of the dynamo-generated magnetic field (Hazra & Choudhuri 2017).",
"It has been known for some time that the strength of MC determines the period of the dynamo—faster MC making cycles shorter and vice versa (Dikpati & Charbonneau 1999).",
"Although we have MC data only for about two decades, we have data pertaining to durations of cycles for about two centuries and these data indicate that there have fluctuations in MC with correlation time of the order of a few decades (Karak & Choudhuri 2011).",
"What kinds of irregularities will the fluctuating MC introduce?",
"Suppose MC has slowed down, making cycles longer.",
"Then diffusion will have more time to act, making cycles weaker.",
"We may thus expect longer cycles to be weaker and shorter cycles stronger.",
"Since a shorter cycle would imply a faster rise time of the cycle, we may expect a correlation between the rise time and the strength of a cycle.",
"Such a correlation is found in the observational data and is known as the Waldmeier effect.",
"Karak & Choudhuri (2011) gave an explanation of this effect by introducing fluctuations in MC in their model.",
"On the basis of such studies, we conclude that there are two important sources of irregularities in the solar cycle—fluctuations in the BL mechanism and fluctuations in MC.",
"Choudhuri et al.",
"(2007) made their prediction of cycle 24 on the basis of the assumption that the irregularities are produced by fluctuations in the BL mechanism alone.",
"Presumably their prediction was so successful because there had not been large fluctuations in MC in the last few years.",
"Now that we know the fluctuations of MC to be another factor introducing irregularities in solar cycles, we need to develop cycle prediction methods taking this into account.",
"We are presently working on this problem.",
"It may be mentioned that one big challenge in this field is to develop a theory of grand minima like the Maunder minimum in the 17th century.",
"While it has been shown that grand minima can be induced by fluctuations in the BL mechanism alone (Choudhuri & Karak 2009) or by fluctuations in MC alone (Karak 2010), we need both of these to develop a comprehensive theory of grand minima (Choudhuri & Karak 2012; Karak & Choudhuri 2013)."
],
[
"Conclusion",
"Within the last few years, the flux transport dynamo model has emerged as an attractive model for explaining the solar cycle and there is increasing evidence that other solar-like stars also may have similar dynamos working inside them (Karak et al.",
"2014b; Choudhuri 2017).",
"It is important that we understand how the irregularities in the cycle arise, since such an understanding may enable us to predict a future cycle before its onset.",
"It appears that fluctuations in the BL mechanism and fluctuations in MC are the two main sources of irregularities in the solar cycle.",
"Before the beginning of cycle 24, the role of MC fluctuations was not generally appreciated.",
"The successful theoretical prediction of Choudhuri et al.",
"(2007) was based on the assumption that irregularities in the solar cycle are caused only by fluctuations in the BL mechanism.",
"With the realization that MC fluctuations also can introduce additional irregularities, it is necessary to develop prediction methods taking this into account.",
"Acknowledgments.",
"I thank Gopal Hazra for help in preparing the manuscript.",
"My research is supported by DST through a J.C. Bose Fellowship."
]
] | 1808.08543 |
[
[
"Scale Drift Correction of Camera Geo-Localization using Geo-Tagged\n Images"
],
[
"Abstract Camera geo-localization from a monocular video is a fundamental task for video analysis and autonomous navigation.",
"Although 3D reconstruction is a key technique to obtain camera poses, monocular 3D reconstruction in a large environment tends to result in the accumulation of errors in rotation, translation, and especially in scale: a problem known as scale drift.",
"To overcome these errors, we propose a novel framework that integrates incremental structure from motion (SfM) and a scale drift correction method utilizing geo-tagged images, such as those provided by Google Street View.",
"Our correction method begins by obtaining sparse 6-DoF correspondences between the reconstructed 3D map coordinate system and the world coordinate system, by using geo-tagged images.",
"Then, it corrects scale drift by applying pose graph optimization over Sim(3) constraints and bundle adjustment.",
"Experimental evaluations on large-scale datasets show that the proposed framework not only sufficiently corrects scale drift, but also achieves accurate geo-localization in a kilometer-scale environment."
],
[
"Introduction",
"Camera geo-localization from a monocular video in a kilometer-scale environment is a essential technology for AR, video analysis, and autonomous navigation.",
"To achieve accurate geo-localization, 3D reconstruction from a video is a key technique.",
"Incremental structure from motion (SfM) and visual simultaneous localization and mapping (visual SLAM) achieve large-scale 3D reconstructions by simultaneously localizing camera poses with six degrees-of-freedom (6-DoF) and reconstructing a 3D environment map [1], [2].",
"Unlike for a stereo camera, an absolute scale of the real world cannot be derived using a single observation from a monocular camera.",
"Although it is possible to estimate an environment's relative scale from a series of monocular observations, errors in the relative scale estimation accumulate over time, and this is referred to as scale drift [3], [4].",
"For an accurate geo-localization not affected by scale drift ,prior information in a geographic information system (GIS) has been utilized in previous studies.",
"For example, point clouds, 3D models, building footprints, and road maps have been proven to be efficient for correcting reconstructed 3D maps [5], [6], [7], [8], [9].",
"However, these priors are only available in limited situations, e.g., in an area that is observed in advance, or in an environment consisting of simply-shaped buildings.",
"Therefore, there is a good chance that other GIS information can help to extend the area in which a 3D map can be corrected.",
"Hence, in this paper, motivated by the recent availability of massive public repositories of geo-tagged images taken all over the world, we propose a novel framework for correcting the scale drift of monocular 3D reconstruction by utilizing geo-tagged images, such as those in Google Street View [10], and achieve accurate camera geo-localization.",
"Owing to the high coverage of Google Street View, our proposal is more scalable than those in previous studies.",
"The proposed framework integrates incremental SfM and a scale drift correction method utilizing geo-tagged images.",
"Our correction method begins by computing 6-DoF correspondences between the reconstructed 3D map coordinate system and the world coordinate system, by using geo-tagged images.",
"Owing to significant differences in illumination, viewpoint, and the environment resulting from differences in time, it tends to be difficult to acquire correspondences between video frames and geo-tagged images (Fig.",
"REF ).",
"Therefore, a new correction method that can deal with the large scale drift of a 3D map using a limited number of correspondences is required.",
"Bundle adjustment with constraints of global position information, which represents one of the most important correction methods, cannot be applied directly.",
"This is because bundle adjustment tends to get stuck in a local minimum when starting from a 3D map including large errors [4].",
"Hence, the proposed correction method consists of two coarse-to-fine steps: pose graph optimization over Sim(3) constraints, and bundle adjustment.",
"In these steps, our key idea is to extend the pose graph optimization method proposed for the loop closure technique of monocular SLAM [4], such that it incorporates the correspondences between the 3D map coordinate system and the world coordinate system.",
"This step corrects the large errors, and enables bundle adjustment to obtain precise results.",
"After implementing this framework, we conducted experiments to evaluate the proposal.",
"The contributions of this work are as follows.",
"First, we propose a novel framework for camera geo-localization that can correct scale drift by utilizing geo-tagged images.",
"Second, we extend the pose graph optimization approach to dealing with scale drift using a limited number of correspondences to geo-tags.",
"Finally, we validate the effectiveness of the proposal through experimental evaluations on kilometer-scale datasets.",
"Incremental SfM and visual SLAM are important approaches to reconstructing 3D maps from monocular videos.",
"Klein et al.",
"proposed PTAM for small AR workspaces [11].",
"Mur-Artal et al.",
"developed ORB-SLAM, which can reconstruct large-scale outdoor environments [2].",
"For accurate 3D reconstruction, the loop closure technique has commonly been employed in recent SLAM approaches [4], [2].",
"Loop closure deals with errors that accumulate between two camera poses that occur at the same location, i.e., when the camera trajectory forms a loop.",
"Lu and Milios [12] formulated this technique as a pose graph optimization problem, and Strasdat et al.",
"[4] extended pose graph optimization to deal with scale drift for monocular visual SLAM.",
"It is certain that loop closure can significantly improve 3D maps, but this is only effective if a loop exists in the video."
],
[
"Geo-registration of Reconstructions",
"Correcting reconstructed 3D maps by using geo-referenced information has been regarded as a geo-registration problem.",
"Kaminsky et al.",
"proposed a method that aligns 3D reconstructions to 2D aerial images [13].",
"Wendel et al.",
"used an overhead digital surface model (DSM) for the geo-registration of 3D maps [14].",
"Similar to our work, Wang et al.",
"used Google Street View geo-tagged images and a Google Earth 3D model for the geo-registration of reconstructed 3D maps [15].",
"However, because all these methods focus on estimating a best-fitting similarity transformation to geo-referenced information, they only correct the global scale in terms of 3D map correction.",
"Methods for geo-registration using non-linear transformations have also been proposed.",
"To integrate GPS information, Lhuillier et al.",
"proposed incremental SfM using bundle adjustment with constraints from GPS [16], and Rehder et al.",
"formulated a global pose estimation problem using stereo visual odometry, inertial measurements, and infrequent GPS information as a 6-DoF pose graph optimization problem [17].",
"In terms of correcting camera poses using sparse global information, Rehder's method is similar to our pose graph optimization approach.",
"However, our 7-DoF pose graph optimization differs in focusing on scale drift resulting from monocular 3D reconstruction, and in utilizing geo-tagged images.",
"In addition to GPS information, various kinds of reference data have been used for the non-linear geo-registration or geo-localization of a video, such as point clouds [5], [6], 3D models [7], building footprints [8], and road maps [9].",
"In this paper, we address a method that introduces geo-tagged images to the non-linear geo-registration of 3D maps."
],
[
"Proposed Method",
"Fig.",
"REF provides a flowchart of the proposed framework, which is roughly divided into three parts.",
"The first part is incremental SfM, and is described in Sec.",
"REF .",
"The second part computes 6-DoF correspondences between the 3D map coordinate system and the world coordinate system (as defined below), by making use of geo-tagged images (Sec.",
"REF ).",
"The third part then uses the correspondences to correct the scale drift of the 3D map, by applying pose graph optimization over Sim(3) constraints (Sec.",
"REF ) and bundle adjustment (Sec.",
"REF ) incrementally.",
"The initialization of the scale drift correction method is described in Sec.",
"REF ."
],
[
"World Coordinate System",
"In this paper, the world coordinates are represented by 3D coordinates $(x, y, z)$ , where the $xz$ -plane corresponds to the Universal Transverse Mercator (UTM) coordinate system, which is an orthogonal coordinate system using meters, and $y$ corresponds to the height from the ground in meters.",
"The UTM coordinates can be converted into latitude and longitude if necessary."
],
[
"Incremental SfM ",
"As large-scale incremental SfM, we use ORB-SLAM [2] (with no real-time constraints).",
"This is one of the best-performing monocular SLAM systems.",
"Frames that are important for 3D reconstruction are selected as keyframes by ORB-SLAM.",
"Every time a new keyframe is selected, our correction method is performed, and the 3D map reconstructed up to that point is corrected.",
"In the 3D reconstruction, we identify 3D map points and their corresponding 2D keypoints in the keyframes (collectively denoted by $C_{\\scalebox {0.7}{map-kf}}$ ).",
"Our proposed framework does not depend on a certain 3D reconstruction method, and can be applied to the other monocular 3D reconstruction methods, such as incremental SfM and feature-based visual SLAM."
],
[
"Obtaining Correspondences between 3D Map and World Coordinates ",
"Here, we describe the second part of the proposed method, which uses geo-tagged images to compute a 6-DoF correspondence, $C_{\\scalebox {0.7}{map-world}}$ , between the 3D map and world coordinate system.",
"For this purpose, we modify Agarwal's method [18] to integrate it into ORB-SLAM.",
"This part consists of the following four steps: geo-tagged image collection, similar geo-tagged image retrieval, keypoint matching, and geo-tagged image localization.",
"Geo-tagged Image Collection.",
"Google Street View [10] is a browsable street-level GIS, which is one of the largest repositories of global geo-tagged images (i.e., images and their associated geo-tags).",
"All images are high-resolution RGB panorama images, containing highly accurate world positions [19].",
"We make use of this data by converting each panorama image into eight rectilinear images with the same field-of-view as our input video, with eight horizontal directions.",
"Note that because each geo-tag has a position and rotation in the world coordinates, we can obtain the 6-DoF correspondences between the 3D map coordinate system and world coordinate system if geo-tagged images are localized in the 3D map coordinate system.",
"Similar Geo-tagged Image Retrieval.",
"When a new keyframe is selected, we retrieve the top-$k$ similar geo-tagged images.",
"The retrieval system employs a bag-of-words approach based on SIFT descriptors [18].",
"Keypoint Matching.",
"Given the pairs of keyframes and retrieved geo-tagged images, we detect ORB keypoints [20] from the pairs and perform keypoint matching.",
"Because the matching between video frames and Google Street View images tends to include many outliers [21], we use a virtual line descriptor (kVLD) [22], which can reject outliers by using a graph matching method even when inlier rate is around 10 Geo-tagged Image Localization.",
"To compute $C_{\\scalebox {0.7}{map-world}}$ , we first compute 3D-to-2D correspondences $C_{\\scalebox {0.7}{map-geo}}$ between 3D map points and their corresponding 2D keypoints in geo-tagged images.",
"In particular, we obtain $C_{\\scalebox {0.7}{map-geo}}$ by combining the 2D keypoint matches (computed in the previous step) with the correspondences $C_{\\scalebox {0.7}{map-kf}}$ between 3D map points and their corresponding 2D keypoints in keyframes (computed in 3D reconstruction).",
"Then, we obtain the 6-DoF camera poses of geo-tagged images in the 3D map coordinate system by minimizing the re-projection errors of $C_{\\scalebox {0.7}{map-geo}}$ , using the LM algorithm.",
"Finally, we obtain $C_{\\scalebox {0.7}{map-world}}$ by combining the camera poses of geo-tagged images and 6-DoF camera poses of the associated geo-tags.",
"Figure: An example of the proposed pose graph optimization.",
"This optimization maintains overall relative poses, except for gradual scale changes, and keeps camera poses of geo-tagged images close to the positions of the corresponding geo-tags."
],
[
"Initialization (INIT) ",
"As the initialization, two kinds of linear transformations are performed on the 3D map, because the positions and scales of the 3D map coordinates and world coordinates are significantly different.",
"Initialization is applied once, when the $i$ -th geo-tagged image is localized.",
"We set $i = 4$ .",
"Given the first to $i$ -th $C_{\\scalebox {0.7}{map-world}}$ , the first transformation assumes that all camera poses are approximately located in one plane, and rotates the 3D map to align that plane to the world $xz$ -plane.",
"The best-fitting plane can be estimated by a principal component analysis.",
"Next, we estimate the best-fitting transformation matrix given by Eq.",
"REF , which transforms a point in the 3D map coordinate system $\\mathbf {p}_{\\scalebox {0.6}{SLAM},k}$ to be closer to a corresponding point in the world coordinate system $\\mathbf {p}_{world,k}$ ($\\mathbf {p}_{\\scalebox {0.6}{SLAM},k}$ and $\\mathbf {p}_{world,k}$ are denoted using a homogeneous representation): $\\mathbf {A} =\\begin{bmatrix}s * \\cos (\\theta ) & 0 & -s * \\sin (\\theta ) & a\\\\0 & s & 0 & 1 \\\\s * \\sin (\\theta ) & 0 & s * \\cos (\\theta ) & b\\\\0 & 0 & 0 & 1\\end{bmatrix}$ Using the first to $i$ -th $C_{\\scalebox {0.7}{map-world}}$ , we estimate the four matrix parameters $[a, b, s, \\theta ]$ by minimizing the following cost using RANSAC [23] and the Levenberg-Marquart (LM) algorithm: $E =\\sum _{k \\in 1,2...i} \\Vert \\mathbf {p}_{world,k} - \\mathbf {A} \\mathbf {p}_{\\scalebox {0.6}{SLAM},k} \\Vert ^{2}$ The camera poses of the geo-tagged images in $C_{\\scalebox {0.7}{map-world}}$ , keyframes, and 3D map point can then be transformed using the resulting matrix."
],
[
"Pose Graph Optimization over Sim(3) Constraints (PGO)",
"We correct the 3D map focusing on scale drift by using the newest three of $C_{\\scalebox {0.7}{map-world}}$ .",
"This correction is performed every time a new $C_{\\scalebox {0.7}{map-world}}$ is found after initialization.",
"Then, we propose a graph-based non-linear optimization method (pose graph optimization) on Lie manifolds, which simultaneously corrects the scale drift and aligns the 3D map with the world coordinates.",
"Notation.",
"A 3D rigid body transformation $\\mathbf {G} \\in \\mathrm {SE}(3)$ and a 3D similarity transformation $\\mathbf {S} \\in \\mathrm {Sim}(3)$ are defined by Eq.",
"REF , where $\\mathbf {R} \\in \\mathrm {SO}(3)$ , $\\mathbf {t} \\in \\mathbb {R}^3$ , and $s \\in \\mathbb {R}^{+}$ .",
"Here, $\\mathrm {SO}(3)$ , $\\mathrm {SE}(3)$ , and $\\mathrm {Sim}(3)$ are Lie groups, and $\\mathfrak {so}(3)$ , $\\mathfrak {se}(3)$ , and $\\mathfrak {sim}(3)$ are their corresponding Lie algebras.",
"A Lie group can be transformed into a Lie algebra using its exponential map, and the inverse transformation is defined by the inverse logarithm map.",
"Each Lie algebra is represented by a vector of its coefficients.",
"For example, $\\mathfrak {sim}(3)$ is represented as the seven-vector $\\xi = (\\omega _{1},\\omega _{2},\\omega _{3}, \\sigma , \\nu _{1}, \\nu _{2}, \\nu _{3})^{\\mathrm {T}} = (\\omega , \\sigma , \\nu )^{\\mathrm {T}}$ , and the exponential map $\\exp _{\\mathrm {Sim}(3)}$ and logarithm map $\\log _{\\mathrm {Sim}(3)}$ are defined as in Eq.",
"REF and Eq.",
"REF , respectively, where $\\mathbf {W}$ is a term similar to Rodriguez's formula.",
"Further details of $\\mathrm {Sim}(3)$ are given in [4].",
"$\\mathbf {G}=\\begin{bmatrix}\\mathbf {R} & \\mathbf {t}\\\\\\mathbf {0} & 1\\end{bmatrix}&\\qquad \\mathbf {S}=\\begin{bmatrix}s \\mathbf {R} & \\mathbf {t}\\\\\\mathbf {0} & 1\\end{bmatrix}$ $\\begin{split}\\exp _{\\mathrm {Sim}(3)}(\\xi )&=\\begin{bmatrix}e^{\\sigma } \\exp _{\\mathrm {SO}(3)}(\\omega ) & \\mathbf {W} \\nu \\\\\\mathbf {0} & 1\\end{bmatrix} = \\mathbf {S}\\end{split}$ $\\log _{\\mathrm {Sim}(3)}(\\mathbf {S}) = {\\exp _{\\mathrm {Sim}(3)}}^{-1}(\\mathbf {S}) = \\xi $ Proposed pose graph optimization.",
"In a general pose graph optimization approach [12], [17], camera poses and relative transformations between two camera poses are represented as elements of $\\mathrm {SE}(3)$ .",
"However, in our approach, 6-DoF camera poses and relative transformations are converted into 7-DoF camera poses, represented by elements of $\\mathrm {Sim}(3)$ .",
"This is achieved by leaving the rotation $R$ and translation $\\mathbf {t}$ of a camera pose unchanged, and setting the scale $s$ to 1.",
"The idea that camera poses and relative pose constraints can be handled in $\\mathrm {Sim}(3)$ was proposed by Strasdat et al.",
"[4], for dealing with the scale drift problem in monocular SLAM.",
"In this paper, we introduce 7-DoF pose graph optimization, which has previously only been used in the context of loop closure, to correct 3D reconstruction by utilizing sparse correspondences between two coordinate systems.",
"Our pose graph contains two kinds of nodes and three kinds of edges, as follows (see Fig.",
"REF ): Node $\\mathbf {S}_{n} \\in \\mathrm {Sim}(3)$ , where $n \\in C_{1} $ : the camera pose of the $n^{th}$ keyframe.",
"Node $\\mathbf {S}_{m} \\in \\mathrm {Sim}(3)$ , where $m \\in C_{2}$ : the camera pose of the $m^{th}$ geo-tagged image.",
"Edge $\\mathbf {e}_{1_{i, j}}$ , where $(i, j) \\in C_{3}$ : the relative pose constraint between the $i^{th}$ and $j^{th}$ keyframes.",
"(Eq.",
"REF ) Edge $\\textbf {e}_{2_{k,l}}$ , where $(k, l) \\in C_{4}$ : the relative pose constraint between the $k^{th}$ keyframe and the $l^{th}$ geo-tagged image.",
"(Eq. )",
"Edge $\\textbf {e}_{3_{m}}$ , where $m \\in C_{2}$ : the distance error between the position of the $m^{th}$ geo-tagged image and the world position $\\mathbf {y}_{m}$ of the corresponding geo-tag.",
"(Eq. )",
"$\\textbf {e}_{1_{i,j}} = \\log _{\\mathrm {Sim}(3)}(\\Delta \\mathbf {S}_{i,j} \\cdot \\mathbf {S}_{i} \\cdot \\mathbf {S}_{j}^{-1}) \\in \\mathbb {R}^7 \\\\\\textbf {e}_{2_{k,l}} = \\log _{\\mathrm {Sim}(3)}(\\Delta \\mathbf {S}_{k,l} \\cdot \\mathbf {S}_{k} \\cdot \\mathbf {S}_{l}^{-1}) \\in \\mathbb {R}^7\\\\\\textbf {e}_{3_{m}} = \\mathrm {trans}(\\mathbf {S}_{m}) - \\mathbf {y}_{m} \\in \\mathbb {R}^3$ where $\\mathrm {trans}(\\mathbf {S}) \\equiv (\\mathbf {S}_{1,4}, \\mathbf {S}_{2,4}, \\mathbf {S}_{3,4})^{\\mathrm {T}}$ .",
"Here, $N$ is the total number of keyframes, and $M$ is the total number of geo-tagged images that have correspondences to keyframes.",
"The set $C_{1}$ contains all the keyframes positioned between the two that have the newest and the third newest $C_{\\scalebox {0.7}{map-world}}$ .",
"The set $C_{2}$ contains the newest three of $C_{\\scalebox {0.7}{map-world}}$ .",
"The set $C_{3}$ contains the pairs of keyframes that observe the same 3D map point in 3D reconstruction, and $C_{4}$ contains pairs of keyframes and their corresponding geo-tagged images.",
"Finally, $\\Delta \\mathbf {S}_{i,j}$ is the converted $\\mathrm {Sim}(3)$ relative transformation between $\\mathbf {S}_{i}$ and $\\mathbf {S}_{j}$ , which is calculated before the optimization and remains fixed during the optimization.",
"Note that we newly introduced the nodes $\\mathbf {S}_{m}$ , edges $\\textbf {e}_{2_{k,l}}$ , and edges $\\textbf {e}_{3_{m}}$ to Strasdat's pose graph optimization.",
"Minimizing $\\textbf {e}_{1_{i, j}}$ and $\\textbf {e}_{2_{k,l}}$ suppresses changes in the relative transformations between camera poses, with the exception of gradual scale changes.",
"Minimizing $\\textbf {e}_{3_{m}}$ keeps the positions of the geo-tagged images close to the positions obtained from the associated geo-tags.",
"Our overall cost function $E_{PGO}$ is defined as follows: $\\begin{split}E_{PGO}(\\bigl \\lbrace \\mathbf {S}_{i} \\bigl \\rbrace _{i \\in C_{1} \\cup C_{2}})&= \\lambda _{1}\\sum _{(i,j) \\in C_{3}} \\textbf {e}_{1_{i,j}}^{\\mathrm {T}} \\textbf {e}_{1_{i,j}} \\\\&+\\lambda _{2}\\sum _{(k,l) \\in C_{4}} \\textbf {e}_{2_{k,l}}^{\\mathrm {T}} \\textbf {e}_{2_{k,l}} +\\lambda _{3}\\sum _{m \\in C_{2}} \\textbf {e}_{3_{m}}^{\\mathrm {T}} \\textbf {e}_{3_{m}}\\end{split}$ The corrected camera poses of keyframes $\\mathbf {S}_{n}$ and geo-tagged images $\\mathbf {S}_{m}$ are obtained by minimizing the cost function $E_{PGO}$ on Lie manifolds using the LM algorithm.",
"Following this optimization, we also reflect this correction in the 3D map points, as in [4]."
],
[
"Bundle Adjustment (BA) ",
"Following the pose graph optimization, we refine the 3D reconstruction by applying bundle adjustment with the constraints of the geo-tagged images.",
"Bundle adjustment is a classic method that jointly refines the 3D structure and camera poses (and camera intrinsic parameters) by minimizing the total re-projection errors.",
"Each re-projection error $\\mathbf {r}_{i,j}$ between the $i^{th}$ 3D point and $j^{th}$ camera is defined as: $\\mathbf {r}_{i,j} = \\mathbf {x_{i}} - \\pi (\\mathbf {R}_{j} \\mathbf {X}_{i} + \\mathbf {t}_{j})$ $\\pi (\\mathbf {p}) = [f_{x} \\dfrac{\\mathbf {p}_{x}}{\\mathbf {p}_{z}} + c_{x},~f_{y} \\dfrac{\\mathbf {p}_{y}}{\\mathbf {p}_{z}} + c_{y}]^{\\mathrm {T}}$ where $\\mathbf {X}_{i}$ is a 3D point and $\\mathbf {x}_{i}$ is the 2D observation of that 3D point; $\\mathbf {R}_{j}$ and $\\mathbf {t}_{j}$ are the rotation and translation of the $j^{th}$ camera pose, respectively; $\\mathbf {p} = [\\mathbf {p}_{x}, \\mathbf {p}_{y}, \\mathbf {p}_{z}]^{\\mathrm {T}}$ is a 3D point; $\\pi (\\cdot ): \\mathbb {R}^3 \\mapsto \\mathbb {R}^2$ is the projection function; $(f_{x}, f_{y})$ is the focal length; and $(c_{x}, c_{y})$ is the center of projection.",
"To incorporate global position information of geo-tagged images with bundle adjustment, we add a penalty term corresponding to the constraint for a geo-tagged image [16].",
"The total cost function with this constraint is given by: $\\begin{split}E_{BA}(\\bigl \\lbrace \\mathbf {X}_{i} \\bigl \\rbrace _{i \\in C_{5}}, \\bigl \\lbrace \\mathbf {T}_{j} \\bigl \\rbrace _{j \\in C_{1}}) = \\sum _{(i,j) \\in C_{\\scalebox {0.7}{map-kf}}} \\rho (\\mathbf {r}_{i,j}^{\\mathrm {T}} \\mathbf {r}_{i,j}) + \\lambda \\sum _{m \\in C_{3}} \\Vert \\mathbf {t}_{m} - \\mathbf {y}_{m} \\Vert ^{2}\\end{split}$ where $\\mathbf {T}$ is a camera pose of a keyframe represented as an element of $\\mathrm {SE}(3)$ , $\\rho $ is the Huber robust cost function, $C_{5}$ consists of map points observed by keyframes in $C_{1}$ , and $C_{1}$ and $C_{3}$ are defined in Sec.",
"REF .",
"Both the positions of 3D points and the camera poses of keyframes are optimized by minimizing the cost function on Lie manifolds using the LM algorithm.",
"This step can potentially correct the 3D map more precisely when it starts from a reasonably good 3D map."
],
[
"Experiments",
"In this section, we evaluate the proposed method on the Málaga dataset [24], using geo-tagged images obtained from Google Street View.",
"We also investigate the performance of pose graph optimization and bundle adjustment using the KITTI Dataset [25]."
],
[
"Implementation",
"We obtained geo-tagged images from Google Street View at intervals of 5 m within the area where the video was captured.",
"We set the cost function weights to $\\lambda _{1} = \\lambda _{2} = 1.0 \\times {10}^{5}$ and $\\lambda _{3} = 1.0$ , and we employed the g2o library [26] for the implementation of the pose graph optimization and bundle adjustment."
],
[
"Performance of the Proposed Method ",
"To verify the practical effectiveness of the proposed method, we evaluate it on the Málaga dataset using geo-tagged images obtained from Google Street View.",
"The Málaga Stereo and Laser Urban Data Set (the Málaga dataset) [24]—a large-scale video dataset that captures Street-View-usable areas—is employed in this experiment.",
"The Málaga dataset contains a driving video captured at a resolution of 1024 $\\times $ 768 at 20 fps in a Spanish urban area.",
"We extracted two video clips (video 1 and video 2) from the video, and used these for the evaluation.",
"The two video clips contain no loops, and their trajectories are over 1 km long.",
"All frames in the videos contain inaccurate GPS positions, which are sometimes confirmed to contain errors of more than 10 m. Because of the inaccuracies, we manually assigned the ground truth positions to some selected keyframes by referring to the videos, inaccurate GPS positions, and Google Street View 3D Map.",
"Fig.",
"REF presents an example of inaccurate GPS data and our assigned ground truth.",
"Because the ground truth positions are assigned by taking into account the lane from which the video was taken, the errors in the ground truth are considered to be within 2 m, and these errors are sufficiently small for this experiment.",
"Figure: The left figure shows an example of inaccurate GPS data (brown dots) and manually assigned ground truth positions (back crosses) on Google Maps.",
"Although we use Google Maps to visualize the results clearly, the shapes of roads are not sufficiently accurate.",
"Our ground truth positions are always assigned in the appropriate lane of the road, as seen in the satellite image (white crosses in the center figure).",
"The right figure shows an example of a video frame captured at the left of the two ground truth positions in the left figure.We evaluated the proposed method on the two videos by comparing the proposal and a baseline method that uses a similarity transformation (like a part of [15]).",
"For the baseline method, we apply the initialization (INIT: described in Sec.",
"REF ) without applying pose graph optimization and bundle adjustment.",
"We did not employ a global similarity transformation as a baseline because it cannot be applied until the end of the whole 3D reconstruction.",
"To evaluate the proposed method quantitatively, we considered the average (Ave) and standard deviation (SD) of 2D distances between the ground truth positions and corresponding keyframe positions in the UTM coordinate system (in meters).",
"Table REF presents the quantitative results, and Fig.",
"REF visualizes the results on Google Maps.",
"As is clearly shown in these results, the baseline results accumulate scale errors, resulting in large errors of over 50 m. This is because the trajectories of these videos are long (greater than 1 km) and contain no loops.",
"The proposed method sufficiently corrects scale drift, and significantly improves the 3D map by using geo-tagged images.",
"In (b) and (e) of the visualized results, the 3D map points corrected using the proposed method are projected onto Google Maps, and it is shown that the 3D map points are correctly aligned to the map.",
"To visualize all the correspondences between the 3D map coordinate system and the world coordinate system used in the proposal, we present the correspondences between the positions of geo-tagged images transformed by initialization and the positions of the corresponding geo-tags.",
"These correspondences are employed incrementally for the correction.",
"Table: Results of our proposed method on the Málaga dataset using Google Street View.Figure: Results of our proposed method visualized on Google Maps.",
"Top: results on video 1.",
"Bottom: results on video 2.",
"In (a) and (d), red and blue dots—which appear like lines—indicate the positions of keyframes corrected using a global similarity transformation (INIT) and our proposed method (Ours), respectively.",
"In (b) and (e), 3D map points corrected by our method are depicted by green dots.",
"(c) and (f) show all of the employed correspondences between the positions of geo-tagged images transformed using a global similarity transformation (green crosses) and the positions of the corresponding geo-tags (red pin icons).",
"The correspondences are applied incrementally for scale drift correction in our proposed method."
],
[
"Performance of PGO and BA",
"To investigate the performance of the pose graph optimization and the bundle adjustment in our proposed method, we evaluated the performance using different combinations of these when varying the interval of $C_{\\scalebox {0.7}{map-world}}$ .",
"Through the previous experiment, we found that the geo-tag location information of Google Street View and the manually assigned ground truths of the Málaga dataset occasionally had errors of several meters.",
"In this experiment, we control the interval of $C_{\\scalebox {0.7}{map-world}}$ , and use high-accuracy ground truths and geo-tags by using the KITTI dataset.",
"The odometry benchmark of KITTI dataset [25] contains 11 sequences of stereo videos and precise location information obtained from RTK-GPS/IMU, and unfortunately Google Street View is not available in Germany where this dataset was captured.",
"The experiment was conducted on two sequences, which include the largest and second-largest errors when applying ORB-SLAM: sequences 02 and 08 (containing 4660 and 4047 frames, respectively).",
"The left images of the stereo videos are used as input, and pairs of a right image and location information are identified as geo-tagged images.",
"All the location information associated with keyframes is used as the ground truth.",
"In this experiment with KITTI dataset, we can compare the performances of correction methods accurately for the following reasons: geo-tag information and ground truths are sufficiently precise (open sky localization errors of RTK-GPS/IMU $<$ 5 cm); and errors in geo-tagged image localization are sufficiently small, because keypoint matching between corresponding left and right images performs very well.",
"For the comparison, we present the results of the methods employing the initialization + the pose graph optimization (INIT+PGO), and initialization + the bundle adjustment (INIT+BA).",
"The correction method of INIT + BA is the same as [16], which is often used with a GPS location information.",
"Ours includes the initialization, the pose graph optimization and the bundle adjustment.",
"We changed the interval of geo-tagged images from 100 frames to 500 frames.",
"For an equal initialization, we set geo-tagged images in the interval of 50 frames from the first to the 200th frame.",
"Table: Results of the experiments on the KITTI dataset: sequences 02 and 08.",
"Values denote average 2D errors between ground truth positions and the corresponding keyframe positions [m].",
"Ours consists of INIT, PGO, and BA.Figure: Results of the experiment on the KITTI dataset when the interval of geo-tagged images is 300 frames.",
"Keyframe trajectories estimated by INIT+BA, INIT+PGO, and Ours are visualized.Fig.",
"REF visualizes the ground truth and keyframe trajectories estimated by INIT+BA, INIT+PGO, and Ours when the interval of geo-tagged images is 300 frames.",
"Table REF presents the quantitative results of the experiment, where the values represent the average 2D errors between ground truth positions and the corresponding keyframe positions in the UTM coordinate system (in meters).",
"Moreover, we report the errors of the global linear transformation on the sequence 02 and 08 by aligning the keyframe trajectory obtained by ORB-SLAM with ground truths through a similarity transformation: $20.15$ and $25.12$ , respectively.",
"The results show that bundle adjustment with geo-tag constraints, which is typically employed in the fusion of 3D reconstruction and GPS information [16], is not suitable when the interval of $C_{\\scalebox {0.7}{map-world}}$ is large.",
"It can also be seen that Ours (the combination of initialization, pose graph optimization, and bundle adjustment) often estimates the keyframe positions more accurately than any other method."
],
[
"Scale Drift Correction",
"To confirm that scale drift is corrected incrementally, we visualize the change in scale factor of the proposed method on the KITTI dataset sequences 02 and 08.",
"Fig.",
"REF shows that ORB-SLAM with the initialization accumulates scale errors, and our method can keep the scale factor around 1.",
"Figure: Change in scale factor of the proposed method on the KITTI dataset sequences 02 and 08."
],
[
"Conclusion",
"In this paper, we propose a novel framework for camera geo-localization that can correct scale drift by utilizing massive public repositories of geo-tagged images, such as those provided by Google Street View.",
"By virtue of the expansion of such repositories, this framework can be applied in many countries around the world, without requiring the user to observe an environment.",
"The framework integrates incremental SfM and a scale drift correction method utilizing geo-tagged images.",
"In the correction method, we first acquire sparse 6-DoF correspondences between the 3D map coordinate system and the world coordinate system by using geo-tagged images.",
"Then, we apply pose graph optimization over $\\mathrm {Sim}(3)$ constraints and bundle adjustment.",
"Our experiments on large-scale datasets show that the proposed framework sufficiently improves the 3D map by using geo-tagged images.",
"Note that our framework not only corrects the scale drift of 3D reconstruction, but also accurately geo-localizes a video.",
"Our results are no less accurate than those of mobile devices (between 5 and 8.5 m) that use a cellular network and low-cost GPS [27], and those using monocular video and road network maps [9] (8.1 m in the KITTI sequence 02 and 45 m in sequence 08).",
"This implies that geo-localization using geo-tagged images is sufficiently useful compared with methods using other GIS information."
],
[
"Acknowledgment",
"This work was partially supported by VTEC laboratories Inc."
]
] | 1808.08544 |